Handbook of Nuclear Physics 9811963444, 9789811963445

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Isao Tanihata Hiroshi Toki Toshitaka Kajino Editors

Handbook of Nuclear Physics

Handbook of Nuclear Physics

Isao Tanihata • Hiroshi Toki • Toshitaka Kajino Editors

Handbook of Nuclear Physics With 1499 Figures and 153 Tables

Editors Isao Tanihata Research Center for Nuclear Physics Osaka University Ibaraki, Osaka, Japan

Hiroshi Toki Research Center for Nuclear Physics Osaka University Ibaraki, Osaka, Japan

Beihang University Haidian, Beijing, China Toshitaka Kajino Beihang University Haidian, Beijing, China

ISBN 978-981-19-6344-5 ISBN 978-981-19-6345-2 (eBook) https://doi.org/10.1007/978-981-19-6345-2 © Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

We would like to dedicate this book to the late Professor Kenzo Sugimoto, who supervised one of the Chief Editors and conceived the idea of utilizing secondary beams of unstable nuclei. Additionally, we wish to honor Professor Kiyomi Ikeda and the late Akito Arima for their contributions in elucidating the unique properties of unstable nuclei, which have greatly advanced the field of nuclear physics.

Preface

This handbook presents a comprehensive and systematic knowledge of modern nuclear physics. Emphasis is put on the development of experimental and theoretical development on the nuclei off the stability line and their exotic structures discovered together with the development of radioactive beam in 1980s. An accelerated beam of unstable nuclei, also called radioactive ion (RI) beam, developed in the late 1980s, provided a revolutionary impact in nuclear physics. Since the discovery of nuclei by Ernest Rutherford, the properties of nuclei have been studied mainly by the scattering experiment except for hyperfine spectra of atoms and nuclear decay observations. Scattering of accelerated particles provided a means to determine the sizes of nuclei, to excite nuclei, to produce short-lived nuclei, to observe dynamic properties of nuclei, and to study many other things. Before RI beam, however, an available beam and target species were restricted to nuclei that exist on the earth. Those are stable nuclei except for a few longlived nuclei. Stable nuclei locate along the so-called valley of the stability forming a line in the nuclear chart (The Segrè Chart). Naturally, most of the knowledge obtained for nuclei had been restricted to the valley of the stability except the information obtained through the decay studies of nuclei far from the valley. The naturally available nuclides on the earth are about 280. Studies of structures of such nuclei were developed to a pretty mature stage and considered that nuclei were well understood. However, at that stage, not much information was accumulated for nuclei far from the stability line. As an important example, even nuclear sizes had not been studied except for some nuclei of special elements. Short-lived nuclei near the limit of stability had not been studied even if some of them had been produced. Before 1980, such limit of nuclear existence of neutron-rich nuclei, called neutron dripline, has reached only up to Be isotopes, just only for four elements among more than 100 known elements. The discovery of projectile fragmentation of high-energy heavy ions at Berkely Bevatron opened up a powerful method (that will be presented in the first chapter of the book) to produce nuclei far from the stability line and more than 4000 nuclides have been discovered by now. Recently, other methods of producing nuclei even farther are being developed. The first RI beam has been produced using projectile fragmentation. It then opened up the possibilities to make scattering of short-lived nuclei. It became possible to study many properties of short-lived nuclei which could not be studied before. vii

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Indeed, it was called Renaissance in the field, and opened up modern nuclear physics. The important difference in studies of short-lived nuclei provided an essential difference in nuclear physics. (1) Nuclei along the valley of stability are bound with almost the same energy (separation energies of nucleon are 6 to 8 MeV); however, short-lived nuclei present largely varying separation energy in both larger and smaller (30 to 0 MeV); (2) numbers of proton and neutrons are balanced but spread widely in short-lived nuclei providing studies in wide range of isospin; (3) Fermi energies of protons and neutrons are same but could be different by tenth of MeV for short-lived nuclei. The effect of such variations could not be studied before RI beams. Indeed, RI beam resulted in discoveries of novel exotic structures of nuclei, e.g., halo nucleus, neutron skin, and new magic numbers and soft modes of excitation in far from the stability region. They drove the drastic development of nuclear structure theories in the 1990s in order to understand such new properties of nuclei. In particular, the effect of weak binding and connection to an unbound system. Also, the change of magic numbers raised questions about the robustness of the shell structure of nuclei. At present, such advanced contents are found only in some of the textbooks published recently, but the comprehensively summarized literature is not found yet. This handbook aims to present a comprehensive, systematic source of modern nuclear physics. To provide complete coverage, the book will not only introduce advanced topics on unstable nuclei, but also include conventional understandings of stable nuclei. Reflecting recent synergistic progress involving both experiment and theory, the sections of the book are not completely separated for theoretical and experimental studies. In cosmology and astrophysics, nuclei play an important role in the evolution of the universe and stars in energy generation and element synthesis and more. Not only stable nuclei but also unstable nuclei including short-lived nuclei contribute strongly to those processes. Many processes in the universe, in particular hot environments, require the knowledge of unstable nuclei and their reactions. Masses, decay properties, and reactions of unstable nuclei now can be studied with usage of unstable nuclei. Therefore, this book includes a section regarding so-called cosmonuclear physics. Cosmo-nuclear physics is the result of the RI beam technology developments, in which cosmological phenomena of stellar burning processes and nucleosynthesis are discussed in the context of nuclear physics. This handbook is organized into three parts edited by the three editors. The first part edited by I. Tanihata forms Vols. 1 and 2 of the handbook with six sections. The second part edited by H. Toki forms Vols. 3 and 4 of the handbook with five sections. The third part edited by T. Kajino forms Vol. 5 with five sections. Each section includes chapters that describe detailed subjects of nuclear physics. Each section of the handbook is assembled by the Section Editor selecting the subjects and the authors of the chapters. The handbook is the fruit of Section Editors and their great efforts. Last but not least, we send our deep gratitude to the authors of the chapters. We hope the handbook will be read for years attracting young scientists and promoting nuclear physics to a higher stage.

Preface

ix

Volumes 1 and 2 present properties of nuclei for the most part from experimental viewpoints. The chapters in Section I, Exotic Nuclei: Production, Separation, Masses and Lifetimes, present the new production methods of nuclei far from the stability and the RI beams that made the revolution in nuclear science followed by the description of the bulk properties of nuclei with some emphasis on exotic nuclei. Section II, Static Properties and Decays, describes static properties and decays of nuclei. In Section III, Dynamical Properties, a new mode of oscillation, pigmy resonance (or soft resonance), is presented in several chapters. Also, the behavior of iso-scalar and iso-vector resonances is of interest. Section IV, Fission of Nuclei, presents a strong development of this subject. Section V, Halo and Unstable Nuclei, characterizes the recent development of nuclear physics in the region far from the stability line presenting new structure and phenomena. Nuclear reactions are a central tool for studying the structure of nuclei and they always become progressively important. Section VI, Nuclear Reaction for Structure Studies, thus presents nuclear reactions. Not only for nuclear structure studies, reaction studies become important for reactions at astrophysical sites with extremely low energy and thus difficult to study directly. The chapter "Indirect Methods in Nuclear Astrophysics with Transfer Reactions" presents such studies. In contrast, high-energy reactions become common for studies of nuclear structure, and it is presented in the chapter “Influence of Nuclear Structure in Relativistic Heavy-Ion Collisions.” Another new development is the ab initio nuclear reaction theory, and it is presented in Chapter “Ab Initio Nuclear Reaction Theory with Applications to Astrophysics.” Volumes 3 and 4 delves into theoretical nuclear physics, specifically the study of nuclear structure from a theoretical standpoint. This part comprises five sections, with the first Section VII, Nucleon-Nucleon Interactions, centering on nucleonnucleon interactions, which are obtained by fitting to the scattering data. All theoretical models are based on the special features of the NN interactions. Section VIII, Models of Nuclear Structure, explores various theoretical models for describing nuclear structure, including that of unstable nuclei. These theoretical models have been developed along with various findings in experiments. Section IX, Tensor Interaction in Nuclei, is dedicated to the tensor interaction produced by one pion exchange. Although its importance in providing saturation of nuclear matter was known, its role in nuclear structure was previously overlooked. We believe it is important to highlight the development of the role of the tensor interaction, which induces high momentum components and further contributes to the shell structure. Section X, Mesonic- and Hypernuclei, covers exotic nuclei, including hypernuclei and exotic atoms. These hadrons and mesons in nuclei add richness to nuclear physics and clarify many interesting aspects of strongly interacting systems. Section XI, Quark Nuclear Physics, discusses Quark Nuclear Physics, focusing on the role of quarks and gluons in the dynamics of nucleons and nuclei. These basic ingredients provide the special properties of the nucleons and their interactions. Understanding the physics of phenomena and unifying various aspects of strong interactions among nucleons require a theoretical description of experimental results. The application of nuclear physics to cosmo-physics also heavily relies on

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theoretical models, which act as essential ingredients for cosmo activities. Quark Nuclear Physics plays a fundamental role in this area, as it provides the basis for nuclear physics. Confinement and chiral symmetry breaking at the QCD level are crucial in producing the massive compact objects known as nucleons and the Goldstone bosons, such as pions, that play an essential role in the NN interaction. Volume 5 consists of five sections of interdisciplinary sciences of Cosmic and Galactic Chemical Evolution, Nuclear and Particle Processes in Big Bang Cosmology, Evolution of Stars and Nucleosynthesis, Neutrino Processes in Nuclear Astrophysics, and Supernovae and Neutron Star Mergers. We present in Vol. 5 how the hundreds of atomic nuclei were created over the entire history of cosmic evolution from the early hot Big Bang universe 13.8 billion years ago, through the galactic and stellar evolution until the epoch of the solar-system formation 4.56 billion years ago. Modern cosmo-nuclear physics stands today at the dawn of multi-messenger astronomy and nuclear astrophysics. Gravitational waves, photons, neutrinos, and atomic nuclei are the messengers of four fundamental forces in nature, i.e., gravity, electromagnetism, weak force, and strong nuclear force. Detection of these multi-messengers provides precise knowledge of what sorts of physical and chemical processes operate in the evolution of the universe, galaxies, and stars. On the extreme celestial conditions of high temperatures or high densities such as those in supernova explosions, stellar burnings, and neutron stars, many short-lived radioactive nuclei are produced and play key roles in energy generation to promote cosmic and galactic evolution and the nucleosynthesis of the stable nuclei on the earth. In Section XII, Cosmic and Galactic Chemical Evolution, the equation of state in neutron stars and supernovae, astronomical observation and theoretical progress in galactic chemo-dynamical evolution, and cosmic radioactivity are presented. In Section XIII, Nuclear and Particle Processes in Big Bang Cosmology, the inflationary scenario and perturbation for cosmic structure formation in the early universe as well as the Big Bang nucleosynthesis are presented. In Section XIV, Evolution of Stars and Nucleosynthesis, both theoretical and experimental aspects of stellar evolution and stellar nuclear reactions are presented. The roles of weak interactions in stellar nucleosynthesis and neutron stars also are presented. In Section XV, Neutrino Processes in Nuclear Astrophysics, various aspects of the cosmic and stellar neutrino interactions with nuclei and the neutrino oscillations induced by the quantum many-body effects in the medium are presented. The experimental detection of diffuse relic supernova neutrinos also is presented. In the last Section XVI, Supernovae and Neutron Star Mergers, all possible astrophysical sites for explosive synthesis of heavy nuclei including rapid-neutron capture elements are presented exhaustedly both theoretically and observationally. Ibaraki, Japan Ibaraki, Japan Haidian, China August 2023

Isao Tanihata Hiroshi Toki Toshitaka Kajino

Contents

Volume 1 Section I Exotic Nuclei: Production, Separation, Masses, and Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hans Geissel 1

2

1

Exotic Nuclei and Their Separation, Electromagnetic Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hans Geissel and D. J. Morrissey

3

Exotic Nuclei and Their Separation, Using Atomic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hans Geissel and D. J. Morrissey

63

3

Reactions for Production of Exotic Nuclei . . . . . . . . . . . . . . . . . . . . . N. Antonenko, J. Benlliure, A. Karpov, and D. J. Morrissey

91

4

Masses of Exotic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Klaus Blaum, Sergey Eliseev, and Stephane Goriely

141

5

Lifetimes of Exotic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Mukha, H. Koura, and T. Tachibana

179

Section II Static Properties and Decays . . . . . . . . . . . . . . . . . . . . . . . . . . Juha Äystö

215

6

Matter Radii and Density Distributions . . . . . . . . . . . . . . . . . . . . . . . Akira Ozawa

217

7

Nuclear Charge Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Nörtershäuser and I. D. Moore

243

8

Spins and Electromagnetic Moments of Nuclei . . . . . . . . . . . . . . . . . Ruben P. de Groote and Gerda Neyens

313

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9

Beta Decay: Probe for Nuclear Structure and the Weak Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Rubio, W. Gelletly and O. Naviliat-Cuncic

349

10

Charged-Particle Radioactive Decays . . . . . . . . . . . . . . . . . . . . . . . . B. Blank and R. D. Page

399

11

Heaviest Elements: Decay and Laser Spectroscopy . . . . . . . . . . . . . Michael Block, Sebastian Raeder, and Rolf-Dietmar Herzberg

443

12

Nuclear Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Philip M. Walker and Zsolt Podolyák

487

13

In-Beam Spectroscopy of Nuclear Electromagnetic Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Görgen and W. Korten

Section III Dynamical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Muhsin N. Harakeh 14

Theoretical Methods for Giant Resonances . . . . . . . . . . . . . . . . . . . . Gianluca Colò

15

Experimental Techniques in Study of Giant Resonances: Magnetic Spectrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mamoru Fujiwara

525

575

577

607

16

Isoscalar Giant Resonances: Experimental Studies . . . . . . . . . . . . . Umesh Garg

631

17

Theoretical Description of Pygmy (Dipole) Resonances . . . . . . . . . Edoardo G. Lanza and Andrea Vitturi

675

18

Pygmy Dipole Resonance: Experimental Studies by Different Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andreas Zilges and Deniz Savran

711

Excitation of Isovector Giant Resonances Through Charge-Exchange Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remco G. T. Zegers

739

19

Volume 2 Section IV Fission of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter G. Thirolf 20

The Multi-humped Fission Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . Andreas Oberstedt and Stephan Oberstedt

775

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21

Microscopic Theory of Nuclear Fission . . . . . . . . . . . . . . . . . . . . . . . Nicolas Schunck

829

22

Photofission Studies: Past and Future . . . . . . . . . . . . . . . . . . . . . . . . Lorant Csige and Dan Mihai Filipescu

867

23

Multinucleon-Transfer-Induced Fission . . . . . . . . . . . . . . . . . . . . . . . Katsuhisa Nishio

901

24

The Fission Barrier of Heaviest Nuclei from a Macroscopic-Microscopic Perspective . . . . . . . . . . . . . . . . . . . . . . . . Michał Kowal and Janusz Skalski

Section V Halo and Unstable Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isao Tanihata

945

983

25

Halo Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isao Tanihata and Björn Jonson

985

26

Theory of Halo Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027 H.-W. Hammer

27

Beta Decay of Halo Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Karsten Riisager

28

Radii and Momentum Distribution of Unstable Nuclei . . . . . . . . . . 1081 Rituparna Kanungo

29

Low-Energy Reactions with Halo Nuclei . . . . . . . . . . . . . . . . . . . . . . 1125 C. Signorini, M. Mazzocco, and D. Pierroutsakou

30

Coulomb Breakup and Soft E1 Excitation of Neutron Halo Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 Takashi Nakamura

31

Unbound Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243 Haik Simon

32

Magic Numbers Off the Stability Line . . . . . . . . . . . . . . . . . . . . . . . . 1267 Tohru Motobayashi

33

Nuclei Near and at the Proton Dripline . . . . . . . . . . . . . . . . . . . . . . . 1295 Marek Pfützner and Chiara Mazzocchi

Section VI Nuclear Reaction for Structure Studies . . . . . . . . . . . . . . . . 1337 Carlos A. Bertulani 34

Indirect Methods in Nuclear Astrophysics with Transfer Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1339 Aurora Tumino and Stefan Typel

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35

Probing Nuclear Structure with Photon Beams . . . . . . . . . . . . . . . . 1371 Johann Isaak and Norbert Pietralla

36

Direct Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415 Carlos A. Bertulani and Angela Bonaccorso

37

Theoretical Studies of Low-Energy Nuclear Reactions . . . . . . . . . . 1451 Pierre Descouvemont

38

Influence of Nuclear Structure in Relativistic Heavy-Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485 Yu-Gang Ma and Song Zhang

39

Modern Approaches to Optical Potentials . . . . . . . . . . . . . . . . . . . . . 1515 Jeremy W. Holt and Taylor R. Whitehead

40

Ab Initio Nuclear Reaction Theory with Applications to Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545 Petr Navrátil and Sofia Quaglioni

41

Electron Scattering Off Stable and Unstable Nuclei . . . . . . . . . . . . 1591 Toshimi Suda

42

Sub-barrier Fusion Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615 K. Hagino

Volume 3 Section VII Nucleon-Nucleon Interactions . . . . . . . . . . . . . . . . . . . . . . . 1641 Ruprecht Machleidt 43

NN Experiments and NN Phase-Shift Analysis . . . . . . . . . . . . . . . . . 1643 Enrique Ruiz Arriola and Rodrigo Navarro Pérez

44

Phenomenology and Meson Theory of Nuclear Forces . . . . . . . . . . 1707 Ruprecht Machleidt

45

Quark Models for Baryon-Baryon Interactions . . . . . . . . . . . . . . . . 1761 P. G. Ortega, D. R. Entem, and F. Fernández

46

Lattice QCD and Baryon-Baryon Interactions . . . . . . . . . . . . . . . . . 1787 Sinya Aoki and Takumi Doi

47

Local Two- and Three-Nucleon Interactions Within Chiral Effective Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1819 Maria Piarulli, Jason Bub, and Ingo Tews

48

Semi-local Nuclear Forces from Chiral EFT: State-of-the-Art and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1853 Evgeny Epelbaum, Hermann Krebs, and Patrick Reinert

Contents

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49

Nonlocal Chiral Nuclear Forces up to N5 LO . . . . . . . . . . . . . . . . . . . 1879 D. R. Entem, Ruprecht Machleidt, and Y. Nosyk

50

Nucleon-Antinucleon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1913 Jean-Marc Richard

Section VIII Jie Meng

Models of Nuclear Structure . . . . . . . . . . . . . . . . . . . . . . . . 1935

51

Model for Independent Particle Motion . . . . . . . . . . . . . . . . . . . . . . . 1937 A. V. Afanasjev

52

Model for Collective Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1977 Z. P. Li and D. Vretenar

53

Models for Pairing Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2011 Xiang-Xiang Sun and Shan-Gui Zhou

54

Algebraic Models of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2045 P. Van Isacker

55

Nuclear Density Functional Theory (DFT) . . . . . . . . . . . . . . . . . . . . 2081 Gianluca Colò

56

Relativistic Density Functional Theories . . . . . . . . . . . . . . . . . . . . . . 2111 Jie Meng and Pengwei Zhao

57

Model for Collective Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2143 Haozhao Liang and Elena Litvinova

58

Configuration Interaction Approach to Atomic Nuclei: The Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2179 Yusuke Tsunoda and Takaharu Otsuka

59

Symmetry Restoration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2229 Jiangming M. Yao

60

Quantum Microscopic Dynamical Approaches . . . . . . . . . . . . . . . . 2265 Cédric Simenel

Section IX Tensor Interaction in Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . 2301 Hiroshi Toki 61

Hadrons from Quarks and Chiral Symmetry . . . . . . . . . . . . . . . . . . 2303 Atsushi Hosaka

62

Pion Exchange Interaction in Bonn Potential and Relativistic and Non-relativistic Framework in Nuclear Matter . . . . . . . . . . . . 2335 Jinniu Hu and Chencan Wang

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63

Extended Hartree-Fock Theory with Strong Tensor Correlation and the Tensor-Optimized Shell Model . . . . . . . . . . . . 2367 Hiroshi Toki

64

Measurements of NN Correlations in Nuclei . . . . . . . . . . . . . . . . . . 2385 E. I. Piasetzky and L. B. Weinstein

65

Many-Body Correlations in Light Nuclei with the Tensor-Optimized Antisymmetrized Molecular Dynamics . . . . . . . 2407 Takayuki Myo

66

Effects of Tensor Interactions in Nuclei . . . . . . . . . . . . . . . . . . . . . . . 2437 Isao Tanihata

Volume 4 Section X Mesonic- and Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2463 Emiko Hiyama 67

What Is Hypernuclear Physics and Why Studying Hypernuclear Physics Is Important . . . . . . . . . . . . . . . . . . . . . . . . . . 2465 Emiko Hiyama and Benjamin F. Gibson

68

High-Precision γ -Ray Spectroscopy of  Hypernuclei . . . . . . . . . . 2483 Hirokazu Tamura

69

 Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2519 Tomofumi Nagae

70

Experimental Aspect of S = −2 Hypernuclei . . . . . . . . . . . . . . . . . . 2535 Kazuma Nakazawa

71

Theoretical Studies in S = −1 and S = −2 Hypernuclei . . . . . . . . 2595 Emiko Hiyama and Benjamin F. Gibson

72

Theoretical Study of Deeply Bound Pionic Atoms with an Introduction to Mesonic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2625 Satoru Hirenzaki and Natsumi Ikeno

73

Pionic Atoms in Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2657 Kenta Itahashi

74

Kaonic Nuclei from the Experimental Viewpoint . . . . . . . . . . . . . . . 2699 Masahiko Iwasaki

75

Theory of Kaon-Nuclear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2749 Tetsuo Hyodo and Wolfram Weise

76

The η- and η -Nucleus Interactions and the Search for η, η - Mesic States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2783 Steven D. Bass, Volker Metag, and Pawel Moskal

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Section XI Quark Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811 Hideo Suganuma 77

Quantum Chromodynamics, Quark Confinement, and Chiral Symmetry Breaking: A Bridge Between Elementary Particle Physics and Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2813 Hideo Suganuma

78

Spontaneous Breaking of Chiral Symmetry in QCD . . . . . . . . . . . . 2861 Yoshimasa Hidaka

79

QCD Vacuum as Dual Superconductor: Quark Confinement and Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2897 Maxim Chernodub

80

Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2939 Hiroshi Toki

81

Quark Nuclear Physics with Heavy Quarks . . . . . . . . . . . . . . . . . . . 2963 Nora Brambilla

82

Generalization of Global Symmetry and Its Applications to QCD-Related Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3007 Yuya Tanizaki

83

Chiral Magnetic Effect: A Brief Introduction . . . . . . . . . . . . . . . . . . 3027 Dmitri E. Kharzeev

84

Crossover Between Quark Nuclear Matter and Condensed-Matter Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3041 Tomáš Brauner and Naoki Yamamoto

85

Hadrons, Quark-Gluon Plasma, and Neutron Stars . . . . . . . . . . . . 3067 Akira Ohnishi

Volume 5 Section XII Cosmic and Galactic Chemical Evolution . . . . . . . . . . . . . . 3125 Toshitaka Kajino 86

Equation of State in Neutron Stars and Supernovae . . . . . . . . . . . . 3127 Kohsuke Sumiyoshi, Toru Kojo, and Shun Furusawa

87

Galactic Chemical Evolution, Astronomical Observation from Metal-Poor Stars to the Solar System . . . . . . . . . . . . . . . . . . . . 3179 Wako Aoki and Miho N. Ishigaki

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Contents

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Chemo-dynamical Evolution of Galaxies . . . . . . . . . . . . . . . . . . . . . . 3211 Chiaki Kobayashi and Philip Taylor

89

Cosmic Radioactivity and Galactic Chemical Evolution . . . . . . . . . 3261 Roland Diehl and Nikos Prantzos

Section XIII Nuclear and Particle Processes in Big Bang Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3345 Grant J. Methews 90

Overview of Big Bang Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3347 Grant J. Mathews and Guobao Tang

91

Big Bang Thermodynamics and Cosmic Relics . . . . . . . . . . . . . . . . 3359 Grant J. Mathews and Guobao Tang

92

Big Bang Nucleosynthesis: Nuclear Physics in the Early Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3379 Brian D. Fields

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Inflation, Perturbations, and Structure Formation . . . . . . . . . . . . . 3407 Grant J. Mathews and Guobao Tang

Section XIV Evolution of Stars and Nucleosynthesis . . . . . . . . . . . . . . 3433 Ken’ichi Nomoto 94

Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3435 Friedrich-Karl Thielemann and Thomas Rauscher

95

Experimental Nuclear Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . 3491 Michael Wiescher, Richard James deBoer, and René Reifarth

96

Slow Neutron-Capture Process in Evolved Stars . . . . . . . . . . . . . . . 3537 Raphael Hirschi

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Weak Interactions in Evolving Stars . . . . . . . . . . . . . . . . . . . . . . . . . 3573 Toshio Suzuki

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Stellar Evolution and Nuclear Reaction Rate Uncertainties . . . . . . 3595 Ken’ichi Nomoto and Wenyu Xin

99

Thermal Evolution of Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . 3643 Sachiko Tsuruta and Ken’ichi Nomoto

Section XV Neutrino Processes in Astrophysics . . . . . . . . . . . . . . . . . . 3667 Akif Baha Balantekin 100

Neutrino Charged and Neutral Current Opacities in the Decoupling Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3669 Ermal Rrapaj and Sanjay Reddy

Contents

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101

Nuclear Physics Constraints on Neutrino Astrophysics . . . . . . . . . 3677 Myung-Ki Cheoun, Kyungsik Kim, Eunja Ha, Heamin Ko, and Dukjae Jang

102

Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713 Evan Grohs and George M. Fuller

103

Neutrinos and Heavy Element Nucleosynthesis . . . . . . . . . . . . . . . . 3735 Xilu Wang and Rebecca Surman

104

Many-Body Collective Neutrino Oscillations: Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3755 Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, and Akif Baha Balantekin

105

Fast Flavor Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3771 Sherwood Richers and Manibrata Sen

106

Diffuse Supernova Neutrino Background . . . . . . . . . . . . . . . . . . . . . 3789 Anna M. Suliga

Section XVI Supernovae and Neutron Star Mergers . . . . . . . . . . . . . . . 3807 Hans-Thomas Janka 107

Nucleosynthesis and Tracer Methods in Type Ia Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3809 Ivo Rolf Seitenzahl and Rüdiger Pakmor

108

Nucleosynthesis in Neutrino-Heated Ejecta and Neutrino-Driven Winds of Core-Collapse Supernovae: Neutrino-Induced Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3843 Shinya Wanajo

109

Nucleosynthesis in Jet-Driven and Jet-Associated Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3877 Martin Obergaulinger and Moritz Reichert

110

R-Process Nucleosynthesis in Neutron Star Merger Ejecta and Nuclear Dependences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915 Stephane Goriely and Ina Kullmann

111

Observations of R-Process Stars in the Milky Way and Dwarf Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3941 Anna Frebel and Alexander P. Ji

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Contents

112

Dynamics and Equation of State Dependencies of Relevance for Nucleosynthesis in Supernovae and Neutron Star Mergers . . . 4005 Hans-Thomas Janka and Andreas Bauswein

113

Measurements of Radioactive 60 Fe and 244 Pu Deposits on Earth and Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4103 Anton Wallner

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4151

About the Editors

Professor Isao Tanihata is a specially appointed professor at the Research Center of Nuclear Physics (RCNP) of Osaka University (Japan) and a professor at the School of Physics at Beihang University (China). He received his Doctor of Science degree from Osaka University in 1975. After his doctoral program, he moved to the Lawrence Berkeley National Laboratory (LBL) as a fellow. At that time, the first high-energy heavy ion was accelerated at the Bevatron and Bevalac, and he joined an experimental group led by Shoji Nagamiya, Owen Chamberlain, and Emilio Segrè and worked on heavyion collisions to look for extreme states of nuclear matter. Together with K. Sugimoto, his doctoral supervisor, he discovered the possibility of using radioactive nuclear beams, working at the University of California, Berkeley to develop the method. The first experiment that led to the discovery of neutron halos was started in 1983 under the collaboration between the LBL and the Institute of Nuclear Study, The University of Tokyo. The first several papers he published in 1985 and the following years opened a new era in nuclear physics. After serving as an assistant professor and associate professor at The University of Tokyo, he moved to RIKEN as a chief scientist. His main research subject was nuclear physics using the Ring Cyclotron in RIKEN and the Heavy-Ion Synchrotron SIS-18 facility at the GSI Helmholtz Centre for Heavy Ion Research in Germany. Observing a steady development of radioactive ion (RI) beam science, together with Masayasu Ishihara and Yasushige Yano he proposed

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About the Editors

the RI Beam Factory project (currently called RIBF), which is currently a leading facility in the field of nuclear physics. After leaving RIKEN in 2004, he worked at the Argonne National Laboratory in the United States, TRIUMF in Canada, and GANIL in France and was appointed to his current position at RCNP in 2012. He was also appointed as a Thousand Talent Program professor in 2010 and has served in another current position at Beihang University since 2016. After an accident at the Fukushima nuclear power plant following the 2011 Japan earthquake and tsunami, he organized a national project of for measurement of radioactivity in soil in the greater Fukushima region within 150 km from the plant, and the environmental radioactivity research continues as of 2023. He also develops the education system for environmental radiation for university students in all fields of science, including natural and social sciences. He has been honored with various awards: The Millar Award in 1979 from the Miller Institute for Basic Research in Science, the University of California; the Nishina Prize in 1989; was named an APS fellow in 1994; was named Professor honoris causa by Bucharest University in Romania in 1997; became a Thousand Talents Plan professor in 2010; and received the Humboldt Scientific Award in 2011. He is also a member of the Academy of Europe. Dr. Hiroshi Toki is a distinguished professor at RCNP, Osaka University, where he retired in 2011 after a remarkable career spanning several decades. He served as the director of RCNP in 2001–2007, and his exceptional leadership skills and expertise made a significant impact on the center. Despite his retirement in 2011, he continues to contribute to the academic community as a specially appointed professor in the Co-creation Center, Graduate School of Engineering Science, and Health and Counseling Center at Osaka University. His numerous outstanding contributions to the field of nuclear physics, as well as his expertise in various other fields, continue to inspire students and researchers worldwide. With over 400 peer-reviewed

About the Editors

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journal articles to his name, numerous edited books, conference proceedings, and several authored books in both English and Japanese, he has an impressive track record as an author, editor, and conference organizer. His areas of expertise include theoretical nuclear physics and noise theory in electric circuits, and he has played a leading role in nuclear physics conferences since 1995, including co-chairing the XVI International Conference on Particles and Nuclei in Osaka in 2002. Prof. Toki is a prominent researcher in nuclear physics with diverse interests spanning several fields. He has worked on nuclear deformation, structural changes in high spin states, pion physics in nuclei and deeply bound pionic atoms, and the role of chiral symmetry breaking in quark nuclear physics. In recent years, he has focused on understanding the role of the tensor interaction caused by pion exchange among nucleons in nuclear structure. His work sheds light on the fundamental nature of nuclear interactions, facilitating better understanding of the behavior of nuclear matter under extreme conditions. In addition to nuclear physics, he has expanded his research interests to other fields including the noise theory of electric circuits, mutation and evolution in biological systems, and the application of machine learning technology in health science. He obtained his Ph.D. from Osaka University in 1974. After his Ph.D. program, he worked as a researcher at Kernforschungsanlage Jülich in 1974– 1977 and as a guest associate professor at Regensburg University in West Germany in 1977–1980. He moved to the United States and worked as an associate professor at Michigan State University in 1980–1983. He then returned to Japan and continued his research and education as an associate professor at Tokyo Metropolitan University in1983–1994. Later, he served as a professor at Osaka University until his retirement. He received the Humboldt Research Award in 2001. He was also awarded the Japan Physical Society Paper Prize in 1993 and 2011.

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About the Editors

Dr. Toshitaka Kajino received his Ph.D. from The University of Tokyo in 1984. After his Ph.D. program, he joined Tokyo Metropolitan University as an assistant professor, moved to the National Astronomical Observatory of Japan, and joined the faculty of the Department of Astronomy, the Graduate School of Science, The University of Tokyo in 1993. In 2004 he was elected a fellow of the American Physical Society for significant contributions to nuclear astrophysics and theoretical nuclear physics and for the promotion of scientific exchange between Japan and the international science community. In 2016, he was honored with a Thousand Talents Plan foreign expert in China and is currently a university distinguished professor at the School of Physics, Beihang University. He is also the first director of the International Research Center for Big-Bang Cosmology and Element Genesis and a professor of the Peng Huanwu Collaborative Center for Research and Education at Beihang University. His research areas of expertise include nucleosynthesis in the big-bang, stars, supernovae and neutron star mergers, nuclear structure and reactions of astrophysical interest, neutrino oscillation, galactic chemical evolution, particle cosmology, and astrobiology. He has published more than 250 peer-reviewed journal articles and has presented over 300 international conference talks. He serves on various domestic and international review committees and editorial boards of research journals and has been the chairman of the International Symposium “Nuclei in the Cosmos (NIC)” and many other international conferences. He is a proponent of the bi-annual international conference series “Origin of Matter and Evolution of Galaxies (OMEG),” which is the oldest conference in the interdisciplinary field of nuclear physics, astrophysics, cosmology, astronomy, and space science, inaugurated in Tokyo in 1988.

Section Editors

Juha Äystö Department of Physics University of Jyväskylä Jyväskylä, Finland

Akif Baha Balantekin Department of Physics University of Wisconsin Madison, WI, USA

Carlos A. Bertulani Department of Physics and Astronomy Texas A&M University-Commerce Commerce, TX, USA

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Section Editors

Hans Geissel Justus-Liebig University Giessen Giessen, Germany GSI Helmholtzzentrum für Schwerionenforschung GmbH Darmstadt, Germany

Muhsin N. Harakeh University of Groningen Groningen, The Netherlands

Emiko Hiyama Department of physics Tohoku University Sendai, Japan

Hans-Thomas Janka Max Planck Institute for Astrophysics Garching, Germany

Section Editors

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Ruprecht Machleidt Department of Physics University of Idaho Moscow, ID, USA

Grant J. Mathews Department of Physics and Astronomy University of Notre Dame Notre Dame, IN, USA

Jie Meng School of Physics, Peking University Beijing, China

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Section Editors

Ken’ichi Nomoto Kavli Institute for the Physics and Mathematics of the Universe The University of Tokyo Kashiwa, Chiba, Japan

Hideo Suganuma Division of Physics and Astronomy Graduate School of Science, Kyoto University Kyoto, Japan

Peter G. Thirolf Faculty of Physics Ludwig-Maximilians-Universität München Garching, Germany

Contributors

A. V. Afanasjev Department of Physics and Astronomy, Mississippi State University, Mississippi State, MS, USA N. Antonenko Joint Institute for Nuclear Research, Dubna, Russia Sinya Aoki Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan Wako Aoki National Astronomical Observatory, Mitaka, Japan Enrique Ruiz Arriola Departamento de Física Atómica, Molecular y Nuclear, Instituto Carlos I de Física Teórica y Computacional, Facultad de Ciencias, Avda. Fuentenueva s/n Universidad de Granada, Granada, Spain Akif Baha Balantekin University of Wisconsin, Madison, WI, USA Steven D. Bass Kitzbühel Centre for Physics, Kitzbühel, Austria Marian Smoluchowski Institute of Physics and Institute of Theoretical Physics, Jagiellonian University, Kraków, Poland Andreas Bauswein GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany J. Benlliure Universidade de Santiago de Compostela, Santiago de Compostela, Spain Carlos A. Bertulani Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX, USA B. Blank Laboratoire de Physique des Deux Infinis de Bordeaux, Gradignan, France Klaus Blaum MPI for Nuclear Physics, Heidelberg, Germany Michael Block GSI Helmholtzzentrum fur Schwerionenforschung, Darmstadt, Germany Helmholtz Institute Mainz (HIM), University of Mainz, Mainz, Germany Angela Bonaccorso Istituto Nazionale di Fisica Nucleare, Pisa, Italy xxix

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Contributors

Nora Brambilla Physik-Department, Technische Universität München, Garching, Germany Institute for Advanced Study, Technische Universität München, Garching, Germany Munich Data Science Institute, Technische Universität München, Garching, Germany Tomáš Brauner Department of Mathematics and Physics, University of Stavanger, Stavanger, Norway Jason Bub Department of Physics, Washington University in Saint Louis, Saint Louis, MO, USA Michael J. Cervia George Washington University, Washington, DC, USA University of Maryland, College Park, MD, USA Myung-Ki Cheoun Soongsil University and OMEG Institute, Seoul, South Korea Maxim Chernodub Institut Denis Poisson, Université de Tours, Tours, France Gianluca Colò Dipartimento di Fisica, Università degli Studi di Milano, Milano, Italy INFN, Sezione di Milano, Milano, Italy Lorant Csige Laboratory of Nuclear Physics, Institute for Nuclear Research (ATOMKI), Debrecen, Hungary Richard James deBoer Department of Physics and Astronomy and the Joint Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, IN, USA Ruben P. de Groote Instituut voor Kern- en Stralingsfysica, KU Leuven, Leuven, Belgium Pierre Descouvemont Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles (ULB), CP229, Brussels, Belgium Roland Diehl Max Planck Institut für extraterrestrische Physik, Garching, Germany Takumi Doi Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Japan Sergey Eliseev MPI for Nuclear Physics, Heidelberg, Germany D. R. Entem Grupo de Física Nuclear, IUFFyM, Universidad de Salamanca, Salamanca, Spain Evgeny Epelbaum Institut für Theoretische Physik II, Ruhr-Universität Bochum, Bochum, Germany

Contributors

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F. Fernández Grupo de Física Nuclear, IUFFyM, Universidad de Salamanca, Salamanca, Spain Brian D. Fields Departments of Astronomy and Physics, University of Illinois, Urbana, IL, USA Illinois Center for the Advanced Study of the Universe, Urbana, IL, USA Dan Mihai Filipescu Horia Hulubei National Institute of Physics and Nuclear Engineering – IFIN-HH, Bucharest, Romania Anna Frebel Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA, USA Joint Institute for Nuclear Astrophysics–Center for Evolution of the Elements (JINA), East Lansing, MI, USA Mamoru Fujiwara Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka, Japan George M. Fuller Department of Physics, University of California San Diego, La Jolla, CA, USA Shun Furusawa College of Science and Engineering, Kanto Gakuin University, Kanazawa-ku, Yokohama, Kanagawa, Japan Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama, Japan Umesh Garg Physics Department, University of Notre Dame, Notre Dame, IN, USA Hans Geissel GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany W. Gelletly Physics Department, University of Surrey, Guildford, UK Benjamin F. Gibson Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA Los Alamos National Laboratory, Los Alamos, NM, USA A. Görgen Department of Physics, University of Oslo, Oslo, Norway Stephane Goriely Institut d’Astronomie et d’Astrophysique, CP-226, Université Libre de Bruxelles, Brussels, Belgium Evan Grohs Department of Physics, North Carolina State University, Raleigh, NC, USA Eunja Ha Hanyang University, Seoul, South Korea K. Hagino Department of Physics, Kyoto University, Kyoto, Japan

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Contributors

H.-W. Hammer Department of Physics, Technische Universität Darmstadt, Darmstadt, Germany ExtreMe Matter Institute EMMI and Helmholtz Forschungsakademie Hessen für FAIR (HFHF), GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany Rolf-Dietmar Herzberg University of Liverpool, Liverpool, UK Yoshimasa Hidaka KEK Theory Center, Tsukuba, Japan Graduate University for Advanced Studies (Sokendai), Tsukuba, Japan RIKEN iTHEMS, RIKEN, Wako, Japan Department of Physics, The University of Tokyo, Tokyo, Japan Satoru Hirenzaki Department of Physics, Nara Women’s University, Nara, Japan Raphael Hirschi Astrophysics Group, Keele University, Keele, UK Institute for Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Japan Emiko Hiyama Department of Physics, Tohoku University, Sendai, Japan Jeremy W. Holt Department of Physics and Astronomy and Cyclotron Institute, Texas A&M University, College Station, TX, USA Atsushi Hosaka Research Center for Nuclear Physics, Osaka University, Mihogaoka, Ibaraki, Japan Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki, Japan Jinniu Hu School of Physics, Nankai University, Tianjin, China Shenzhen Research Institute of Nankai University, Shenzhen, China Tetsuo Hyodo Tokyo Metropolitan University, Hachioji, Japan Natsumi Ikeno Department of Agricultural, Life and Environmental Sciences, Tottori University, Tottori, Japan Johann Isaak Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt, Germany P. Van Isacker GANIL, CEA/DRF-CNRS/IN2P3, Boulevard Henri Becquerel, Caen, France Miho N. Ishigaki National Astronomical Observatory, Mitaka, Japan Kenta Itahashi RIKEN Nishina Center for Accelerator-Based Science, RIKEN, Saitama, Japan RIKEN Cluster for Pioneering Research, RIKEN, Saitama, Japan

Contributors

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Masahiko Iwasaki RIKEN Nishina Center for Accelerator-Based Science and RIKEN Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama, Japan Dukjae Jang Center for Relativistic Laser Science, IBS, Gwangju, South Korea Hans-Thomas Janka Max-Planck-Institute for Astrophysics, Garching, Germany Alexander P. Ji Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA Joint Institute for Nuclear Astrophysics–Center for Evolution of the Elements (JINA), East Lansing, MI, USA Björn Jonson Department of Physics, Chalmers University of Technology, Göteborg, Sweden Rituparna Kanungo Saint Mary’s University, Halifax, NS, Canada TRIUMF, Vancouver, BC, Canada A. Karpov Joint Institute for Nuclear Research, Dubna, Russia Dmitri E. Kharzeev Center for Nuclear Theory, Stony Brook University, Stony Brook, NY, USA Brookhaven National Laboratory, Upton, NY, USA Kyungsik Kim Korea Aerospace University, Koyang, South Korea Heamin Ko Soongsil University and OMEG Institute, Seoul, South Korea Chiaki Kobayashi Department of Physics, Astronomy and Mathematics, Centre for Astrophysics Research, University of Hertfordshire, Hatfield, UK Toru Kojo Department of Physics, Tohoku University, Sendai, Japan W. Korten IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France H. Koura Advanced Science Research Center, Japan Atomic Energy Agency, Ibaraki, Japan Michał Kowal National Centre for Nuclear Research, Warsaw, Poland Hermann Krebs Institut für Theoretische Physik II, Ruhr-Universität Bochum, Bochum, Germany Ina Kullmann Institut d’Astronomie et d’Astrophysique, CP-226, Université Libre de Bruxelles, Brussels, Belgium Edoardo G. Lanza INFN, Sezione di Catania, Catania, Italy Z. P. Li School of Physical Science and Technology, Southwest University, Chongqing, China

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Contributors

Haozhao Liang Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo, Japan Elena Litvinova Department of Physics, Western Michigan University, Kalamazoo, MI, USA National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, USA Yu-Gang Ma Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai, China Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai, China Ruprecht Machleidt Department of Physics, University of Idaho, Moscow, ID, USA Grant J. Mathews Department of Physics and Astronomy, Center for Astrophysics, University of Notre Dame, Notre Dame, IN, USA Chiara Mazzocchi Faculty of Physics, University of Warsaw, Warszawa, Poland M. Mazzocco Dipartimento di Fisica e Astronomia, Università di Padova and INFN, Sezione di Padova, Padova, Italy Jie Meng School of Physics, Peking University, Beijing, China Volker Metag II. Physikalisches Institut, University of Giessen, Giessen, Germany I. D. Moore Department of Physics, University of Jyväskylä, Jyväskylä, Finland D. J. Morrissey National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, USA Pawel Moskal Marian Smoluchowski Institute of Physics and Center for Theranostics, Jagiellonian University, Kraków, Poland Tohru Motobayashi RIKEN Nishina Center, Wako, Saitama, Japan I. Mukha GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany Takayuki Myo General Education, Faculty of Engineering, Osaka Institute of Technology, Osaka, Japan Tomofumi Nagae Department of Physics, Kyoto University, Kyoto, Japan

Kitashirakawa,

Takashi Nakamura Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo, Japan Kazuma Nakazawa Faculty of Education, Graduate School of Engineering, Gifu University,Gifu, Japan

Contributors

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O. Naviliat-Cuncic Laboratoire de Physique Corpusculaire de Caen, IN2P3/CNRS, ENSICAEN, Université de Caen Normandie, Caen, France Facility for Rare Isotope Beams & Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA Petr Navrátil TRIUMF, Vancouver, BC, Canada Gerda Neyens Instituut voor Kern- en Stralingsfysica, KU Leuven, Leuven, Belgium Katsuhisa Nishio Japan Atomic Energy Agency, Tokai, Japan Ken’ichi Nomoto Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, Kashiwa, Japan W. Nörtershäuser Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt, Germany Y. Nosyk Department of Physics, University of Idaho, Moscow, ID, USA Martin Obergaulinger Departament d’Astonomia i Astrofísica, Universitat de València, Edifici d’Investigació Jeroni Munyoz, Burjassot (València), Spain Andreas Oberstedt Extreme Light Infrastructure – Nuclear Physics, BucharestMagurele, Romania Stephan Oberstedt European Commission, Joint Research Centre (JRC), Geel, Belgium Akira Ohnishi Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan P. G. Ortega Dpto. Física Fundamental, Universidad de Salamanca, Salamanca, Spain Takaharu Otsuka RIKEN Nishina Center, Wako, Hirosawa, Japan Department of Physics, University of Tokyo, Tokyo, Japan Akira Ozawa University of Tsukuba, Ibaraki 305-8571, Japan R. D. Page Department of Physics, Oliver Lodge Laboratory, University of Liverpool, Liverpool, UK Rüdiger Pakmor Max Planck Institute for Astrophysics, Garching, Germany Amol V. Patwardhan SLAC National Accelerator Laboratory, Menlo Park, CA, USA Rodrigo Navarro Pérez Department of Physics, San Diego State University, San Diego, CA, USA Marek Pfützner Faculty of Physics, University of Warsaw, Warszawa, Poland

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Contributors

Maria Piarulli Department of Physics, Washington University in Saint Louis, Saint Louis, MO, USA McDonnell Center for the Space Sciences, Washington University, St. Louis, MO, USA E. I. Piasetzky Tel Aviv University, Tel Aviv, Israel D. Pierroutsakou INFN, Sezione di Napoli, Napoli, Italy Norbert Pietralla Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt, Germany Zsolt Podolyák Department of Physics, University of Surrey, Guildford, UK Nikos Prantzos Institut d’Astrophysique, Paris, France Sofia Quaglioni Lawrence Livermore National Laboratory, Livermore, CA, USA Sebastian Raeder GSI, HIM, Mainz, Germany Thomas Rauscher Department of Physics, University of Basel, Basel, Switzerland Centre for Astrophysics Research, University of Hertfordshire, Hatfield, UK Sanjay Reddy Department of Physics, University of Washington, Seattle, WA, USA Institute for Nuclear Theory, Seattle, WA, USA Moritz Reichert Departament d’Astonomia i Astrofísica, Universitat de València, Edifici d’Investigació Jeroni Munyoz, Burjassot (València), Spain René Reifarth Goethe University, Frankfurt, Germany Patrick Reinert Institut für Theoretische Physik II, Ruhr-Universität Bochum, Bochum, Germany Jean-Marc Richard Institut de Physique des 2 Infinis de Lyon, IN2P3 & Université de Lyon, Villeurbanne, France Sherwood Richers Department of Physics and Astronomy, University of Tennessee, Knoxville, TN, USA Karsten Riisager Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark Ermal Rrapaj Department of Physics, University of California, Berkeley, CA, USA RIKEN iTHEMS, Wako, Saitama, Japan University of California, Berkeley, CA, USA B. Rubio IFIC, CSIC-Universidad de Valencia, Paterna, Spain

Contributors

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Deniz Savran GSI, Darmstadt, Germany Nicolas Schunck Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, CA, USA Ivo Rolf Seitenzahl School of Science, University of New South Wales Canberra, The Australian Defence Force Academy, Canberra, ACT, Australia Manibrata Sen Max-Planck-Institut für Kernphysik, Heidelberg, Germany C. Signorini Dipartimento di Fisica e Astronomia, Università di Padova and INFN, Sezione di Padova, Padova, Italy Cédric Simenel Department of Fundamental and Theoretical Physics, Research School of Physics, The Australian National University, Canberra, ACT, Australia Haik Simon Helmholtzzentrum füer Schwerionenforschung GmbH (GSI), Darmstadt, Germany Pooja Siwach University of Wisconsin, Madison, WI, USA Janusz Skalski National Centre for Nuclear Research, Warsaw, Poland Toshimi Suda Research Center for Electron-Photon Science, Tohoku University, Sendai, Japan Hideo Suganuma Department of Physics, Graduate School of Science, Kyoto University, Kyoto, Japan Anna M. Suliga Department of Physics, University of California Berkeley, Berkeley, CA, USA Department of Physics, University of Wisconsin–Madison, Madison, WI, USA Kohsuke Sumiyoshi National Institute of Technology, Numazu College, Numazu, Shizuoka, Japan Xiang-Xiang Sun School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China Rebecca Surman Department of Physics, University of Notre Dame, Notre Dame, IN, USA Toshio Suzuki Department of Physics, College of Humanities and Sciences, Nihon University, Tokyo, Japan T. Tachibana Waseda University, Tokyo, Japan Hirokazu Tamura Department of Physics, Tohoku University, Sendai, Japan Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan

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Contributors

Guobao Tang Department of Physics and Astronomy, Center for Astrophysics, University of Notre Dame, Notre Dame, IN, USA Isao Tanihata School of Physics, Beihang University, Beijing, China Research Center of Nuclear Physics, Osaka University, Osaka, Japan Yuya Tanizaki Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan Philip Taylor Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT, Australia Ingo Tews Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA Friedrich-Karl Thielemann Department of Physics, University of Basel, Basel, Switzerland GSI Helmholtz Center for Heavy Ion Research, Darmstadt, Germany Hiroshi Toki Research Center for Nuclear Physics (RCNP), Osaka University, Toyonaka, Osaka, Japan Yusuke Tsunoda Center for Nuclear Study, University of Tokyo, Tokyo, Japan Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki, Japan Sachiko Tsuruta Department of Physics, Montana State University, Bozeman, MT, USA Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, Kashiwa, Japan Aurora Tumino Facoltà di Ingegneria e Architettura, Università degli Studi di Enna “Kore”, Enna, Italy INFN-Laboratori Nazionali del Sud, Catania, Italy Stefan Typel Fachbereich Physik, Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt, Germany Theorie, GSI Helmholtzentrum für Schwerionnenforschung GmbH, Darmstadt, Germany Andrea Vitturi Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Padova, Italy INFN, Sezione di Padova, Padova, Italy D. Vretenar Faculty of Science, Department of Physics, University of Zagreb, Zagreb, Croatia State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing, China

Contributors

xxxix

Philip M. Walker Department of Physics, University of Surrey, Guildford, UK Anton Wallner Accelerator Mass Spectrometry and Isotope Research, HelmholtzZentrum Dresden-Rossendorf (HZDR), Institute of Ion Beam Physics and Materials Research, Dresden, Germany Institute of Nuclear and Particle Physics, Technische Universität Dresden, Dresden, Germany Shinya Wanajo Max-Planck-Institut für Gravitationsphysik (Albert-EinsteinInstitut), Potsdam-Golm, Germany Chencan Wang School of Physics, Nankai University, Tianjin, China Xilu Wang Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China L. B. Weinstein Old Dominion University, Norfolk, VA, USA Wolfram Weise Physics Department, Technical University of Munich, Garching, Germany Taylor R. Whitehead Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI, USA Michael Wiescher Department of Physics and Astronomy and the Joint Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, IN, USA Wenyu Xin Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing, China Naoki Yamamoto Department of Physics, Keio University, Yokohama, Japan Jiangming M. Yao School of Physics and Astronomy, Sun Yat-sen University, Zhuhai, China Remco G. T. Zegers Facility for Rare Isotope Beams and the Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA Song Zhang Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai, China Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai, China Pengwei Zhao School of Physics, Peking University, Beijing, China Shan-Gui Zhou CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, China Andreas Zilges University of Cologne, Cologne, Germany

Section I Exotic Nuclei: Production, Separation, Masses, and Lifetimes Hans Geissel

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Exotic Nuclei and Their Separation, Electromagnetic Devices Hans Geissel and D. J. Morrissey

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Reactions to Create Exotic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Is Separation Required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key Features of Production Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Targetry, Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermalization in Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of Production Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separation in Pure Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ion-Optical Definitions and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laterally Dispersive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Longitudinally Dispersive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 6 7 9 15 19 24 25 26 32 42 53 53

Abstract

The discovery of short-lived, exotic nuclei has opened new horizons in subatomic physics and allowed many novel applications in other fields, e.g., the understanding of the evolution of matter and production of the atomic elements in the universe. The effective and highly selective separation of the exotic nuclei produced in these reactions from other more abundant products and the primary beam is

H. Geissel () GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany e-mail: [email protected] D. J. Morrissey National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_100

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H. Geissel and D. J. Morrissey

critical for the study of exotic species. The high sensitivity of separators can reach single ions produced with sub-picobarn cross sections. Depending on the type of production reaction and the kinematics of the products, different accelerator facilities and separation methods have been constructed and applied. A reactionproduct separator is the heart of each such facility. The presentation of fragment separators and their application has been divided here in two parts. Here in this chapter, a brief overview of the relevant production reactions for modern-day separation techniques will be presented along with the ion-optical features of fragment separator design and operation. These devices can consist of pure dispersive electromagnetic elements or can incorporate specific atomic interactions of the ions in additional matter along their path through the separator. Extension to systems that incorporate atomic interactions and combinations of these devices with additional high-resolution ion-optical systems to provide unique opportunities to study exotic nuclei immediately after they pass through the system are presented in the next chapter. Nuclear physics laboratories around the world have built reaction product separators, have plans for new separation facilities, and are constructing new machines with enhanced capabilities that rely on these devices. The whole presentation concludes in the next chapter with a discussion of key goals, achievements, and limitations in the separation and science of exotic nuclei.

Introduction Since the earliest studies of atomic nuclei and the subsequent identification of the nature of isotopes of chemical elements, there has been a relentless search to find the physical limits to nuclear stability. The length of the Periodic Table of elements has been established from hydrogen (Z=1) up to oganesson (Z=118), but the breadth in neutron number for each of these elements remains an important area of research. At present the number of identified nuclear isotopes is greater than 3000; see Fig. 1 and the review with more recent updates in Thoennessen and Sherrill (2011) and Thoennessen (2018). This two-dimensional spectrum (Z,N) of nuclei has been shown to decay by beta emission, proton emission, alpha emission, and spontaneous fission. While this number is large, there is an even larger area of potentially bound nuclei that lie outside the range of present-day experimental techniques. In order to be established as having a “bound nucleus,” the isotope must be produced, separated from other copious nuclear products, and both the atomic number and neutron number must be established. This chapter will provide an overview of the experimental approaches to this task. The detailed features of the nuclear reactions used at present to produce the most exotic nuclei are presented in a subsequent chapter; here a brief overview is given to provide insight into the requirements and difficulties of each approach. Note that since all exotic nuclei have to be produced in nuclear reactions, the availability of intense primary beams is the starting point for their production. Also it should be clear that the most exotic, presently unknown nuclei will be produced with extremely low probabilities relative to the other products. Thus, highly selective

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Fig. 1 Chart of Nuclides (Koura et al. 2019) made up of the known isotopes with an indication of the primary decay branch of each following the color scheme in the legend. Most of the stable isotopes were discovered by Aston (1920) with mass measurements using an electromagnetic separator. The gray area represents the area of potentially bound nuclei that might be possible to produce, but they remain beyond our present experimental capabilities. (Some of the history of nuclide discovery is illustrated in the lower insert adapted from the reviews Thoennessen 2018)

physical separation of individual exotic nuclei from the primary beam and from other less interesting or previously studied nuclei has been at the heart of progress in nuclear physics for many years. Here, an overview of the principles and concepts that underlie various modern separators that rely on technical advances and clever ideas to attain an extreme level of sensitivity is presented. For example, modern machines provide single ions produced at femtobarn cross sections, that is, separation factors >1016 against other products. The design and effectiveness of the device depend on the nuclear reaction mechanism, the kinematics of the reaction products, and to some extent the properties of the primary beam. A number of very large facilities around the world have been constructed, and others continue to be built or improved for the production and study of exotic nuclei. In each case a reaction-product separator of some design is the heart of each of these facilities. The separators that consist of purely dispersive electromagnetic components are presented here in this chapter: the separators that incorporate specific atomic interactions as the ions pass through additional matter are presented in the next chapter. The reaction product separators can be combined with additional high-resolution ion-optical systems that open up unique opportunities to study

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exotic nuclei as quickly as they can pass through the system. An overview of specific designs of separators at various facilities will also be presented in the next chapter that concludes with a brief discussion of key goals, achievements, and present limitations in the production and separation of the most exotic nuclei.

Nuclear Reactions to Create Exotic Nuclei Many different reaction mechanisms have been used over the years to produce exotic nuclei as discussed in  Chap. 2, “Exotic Nuclei and Their Separation, Using Atomic Interactions”; however, two processes have become mainstays of modern production for practical reasons, and recently a third possibility is being explored. A schematic view of the regions in the Chart of Nuclides in which each reaction is predominant in producing exotic nuclei is shown in Fig. 2. The important reactions that will be considered here are (1) fusion of a light nucleus with another nucleus that occurs at energies near the reaction threshold, (2) fragmentation of a heavy nucleus by a very light nucleus at relatively high kinetic energies, and (3) multinucleon transfer between two large nuclei that can occur at the Coulomb barrier. Historically the fragmentation of a heavy target nucleus by a very light ion has been called spallation. In addition, very heavy nuclei produced in any reaction can decay by fission that will produce medium-mass neutron-rich products. These reactions each

Fig. 2 Chart of Nuclides with schematic representation of the present extent of known nuclei, the driplines where Sn,p = 0, and the regions in which the various nuclear reactions are predominantly used to produce exotic nuclei, or are predicted to do so

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have specific characteristics that must be considered in order to readily separate any exotic nuclear products, and very different approaches to using these reactions have been developed. This chapter provides a brief overview of the important features of nuclear reactions used extensively to produce exotic nuclei and some kinematic properties of those nuclei relevant to separation.

Why Is Separation Required? There are two general approaches to produce nuclei for study in the laboratory: (a) individual nuclear reactions can be used to produce exotic nuclei that remain ionized and are passed directly to the separator, or (b) very intense low-mass beams can be reacted with bulk materials followed by effusion, diffusion, and ionization before being passed to the separator. Note that the total production rate, dn/dt, depends on the product of beam flux, Φ, in particles per second combined with an effective number density of target atoms, N0 , the areal number density in atoms/cm2 , and the production cross section, σ . The cross section for producing an exotic nucleus is by its nature a very small number and is a fundamental property of the reaction beyond experimental control. In addition, the rate of separated exotic nuclei available for study must include the total efficiency of the separation technique and any corrections due to decay before delivery, discussed in the next chapter. First one should consider the simple fraction of nuclear reactions expected to occur in a target and the fraction that are expected to produce exotic nuclei. Then one should consider some details of the different production reactions that impact separation techniques. The production techniques for exotic nuclei to date have relied on a beam striking a fixed target in the laboratory system; see Fig. 3 for an overview. The crosssectional area of the beam is almost always smaller than that of the target, and the target should be uniform throughout, so that one can combine the beam intensity and target thickness into a luminosity, L = ΦN0 in atoms/cm2 /s. Both Φ and N0 will have physical constraints from the reaction mechanism and from the accelerator

Fig. 3 Principal components of a rare-isotope facility. The accelerator provides the primary beam, generally at high energy, that is transported to the production target area. The “target system” can range from quite complex to simple, e.g., production of low-energy target fragments, and a combination of the actual production target with an ion source is usually realized. On the other hand, projectile fragment and fusion residue ions are produced in transmission targets at various distances from the entrance to the separator. The emergent ions are purified in the separator. The exotic isotope of interest is then prepared for delivery to the “experimental setup” of the end user which could be a simple set of detectors, a collector ring, or even a reaccelerator system

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system, but L can be used to compare production reactions. The beam flux, Φ is clearly dependent on the capabilities of the accelerator system but is often limited by the power deposited in the target by the energy lost by the beam as it penetrates the target. Beam intensities on the order of 1 particle-nA (10−9 A/e = 6.24×109 particles/s) can generally be used with standard metal targets, and beam intensities on the order of 1 particle-μA (6.24×1012 particles/s) are presently possible with specialized high-power targets. The number of target atoms per square centimeter N0 = ρNA /A where ρ is the target mass density in grams/cm3 , NA is Avogadro’s Number, A is the atomic mass in grams/mol, and  is the target thickness in cm. Target materials are generally metals to dissipate heat, and the mass densities of all metals range from ≈1 to 20 g/cm3 , while the atomic masses range from ≈6 to 240 grams/mol. Since these values are divided by one another and somewhat correlated, the range of ρNA /A is remarkably small, for example, beryllium metal has 1.23×1023 atoms/cm3 , while uranium metal has 4.82 × 10 atoms/cm3 . The target thickness on the other hand, , has to be thin enough to allow the fusion or fragmentation products to escape with sufficient kinetic energy for the separation process although it can be quite large for target fragmentation reactions (spallation) but generally not longer than the range of the beam in the target. For convenience of measurement, thin metal targets are characterized by weighing a known area to give the areal mass density, ρ ×  = ρA in milligrams/cm2 . A typical thin target for a fusion reaction might be 1 milligram/cm2 = 6×1020 /A atoms/cm2 , whereas a typical thick target for a highenergy reaction might be 1 gram/cm2 = 6 × 1023 /A atoms/cm2 . Thus, the variation of usable target thicknesses is as large as that of the beam intensities. Estimates of the typical luminosities obtained to date are given in Table 1. Returning to the cross sections, the total reaction cross section for the interaction of a heavy ion with a metal target is on the order of 2 barns (1 barn = 100 fm2 = 1 × 10−24 cm2 ). The total reaction rate is ΦN0 σ in /s but concentrating on the fraction of beam particles that react one has N0 σ . Thus, starting with low values, the fraction that reacts in a 1 mg/cm2 uranium target is flow = 6 × 1020 /238/cm2 ×

Table 1 Typical beam intensities, energies, targets, and luminosities for different accelerator systems used to produce exotic nuclei in solid targets. (Data for the ISOL targets is from Stora 2013) Accelerator Heavy-ion LINAC Cyclotrons LINAC Synchrotron Proton Cyclotron Synchrotron

Target A Z, g cm−2

Luminosity cm−2 s−1

HI fusion Proj. frag. Proj. frag. Proj. frag., fission

249 Cf,

0.01

1 1 9 Be, 8

2 × 1033 4 × 1035 3 × 1037 3 × 1035

Spallation, fission Spallation, fission

238 UCx, 25 UCx, 45

4 × 1036 1 × 1036

Intensity s−1

Energy MeV/u

Reaction

6 × 1013 6 × 1012 5 × 1014 5 × 1011

10 350 200 1500

6 × 1013 1 × 1013

500 1400

9 Be, 12 C,

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2 × 10−24 cm2 ≈ 5 × 10−6 , whereas for the high values, the fraction that reacts in a 1 g/cm2 beryllium target is fhigh = 6 × 1023 /9 /cm2 × 2 × 10−24 cm2 ≈ 0.13. In all cases the large majority (or the vast majority) of beam particles pass through the target without reacting and must be fully separated from the reaction products. In most cases the unreacted beam also retains a high thermal power and must be carefully captured. Going further, the cross sections for the production of exotic nuclei can be extremely small. For example, the production cross sections of the heaviest elements in fusion reactions are presently picobarns (10−12 b) or femtobarns (10−15 b). The fractions of exotic fused nuclei are 5×10−18 per pb and 5×10−21 per fb, and similarly, the fractions for exotic projectile fragments with the same cross sections are 5×10−13 per pb and 5×10−16 per fb. These extraordinary large decontamination factors must be applied to select the nuclei of interest, and the separation process must be carefully controlled to ensure that the unreacted beam is captured while the desired product passes through the system. The propagation of a beam through an electromagnetic system depends on the atomic mass-to-charge ratio, m/q, of the ion. The atomic charge state depends primarily on the ion velocity and the target material and only indirectly on the nuclear reaction. Since the nuclear reaction yield will be distributed over the atomic charge state distribution and the m/q acceptance of realistic separators is limited, it is critically important to minimize the width of the charge state distribution. Ideally one would concentrate the yield in a single charge state. At low energies separators have been developed to maintain and transport ions with the average charge state in a gas. On the other hand, using the highest possible beam energies will concentrate the yield in full-stripped ions. These techniques are applied to different production mechanism as discussed below in Section “Overview of Production Concepts.”

Key Features of Production Reactions With the general features of the nuclear reactions that are used to produce exotic nuclei and with estimates of the level of separation required to obtain exotic nuclei, it is also important to consider the general kinematic properties of the products of these nuclear reactions. The kinematical properties of each reaction will further define the opportunities and the requirements of a workable reaction product separator. The key features of each reaction mechanism will be discussed here including some of the advantages and challenges for each. Historically, the fusion process is the oldest and best known nuclear reaction to produce exotic nuclei. It relies on the formation of a compound nucleus, generally with some excitation energy, followed by evaporation or fission. The fusion products continue forward in the laboratory with a relatively small dispersion in proportion to the extent of the evaporation de-excitation process. Very asymmetric fusion reactions in normal kinematics (mBeam < mtarget ) have a small forward momentum, and evaporation can cause significant angular broadening, whereas fusion products in reverse kinematics (mBeam > mtarget ) retain a large forward momentum, and the effects of evaporation are minimized (this is a general feature of reverse kinematics

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called kinematic focusing). Fusion has been used to produce a large range of neutron deficient nuclei across the Periodic Table due to the preferential emission of neutrons by highly excited nuclei. In fact, the proton drip line was defined to a large extent by fusion reactions. The target thickness in fusion reactions is severely limited by the energy loss by the beam and the heavy recoiling fusion products. The collection of products is also hampered by broad charge state distributions due to the low kinetic energies and high atomic numbers. So-called vacuum separators have been developed to accept a small number of charge states. In addition, lowpressure gas-filled separators have been developed to maintain and transmit the ions in an average charge state. At present, fusion is the predominant process to form the heaviest nuclei due to it being the only process shown to substantially add nucleons to a heavy nucleus (Oganessian and Utyonkov 2015). One should note that there are theoretical predictions that multinucleon transfer reactions (MNT), a near-barrier subset of deep-inelastic collisions, may provide a new route to some neutron-rich heavy nuclei discussed in in the  Chap. 3, “Reactions for Production of Exotic Nuclei”. The MNT reactions produce two large nuclei at relatively low kinetic energies that are concentrated at the grazing angle, Θg , relative to the beam direction but distributed uniformly in azimuthal angle. Thus, MNT reactions have been pursued so far with a hybrid production, capture, and ionization technique followed by electromagnetic separation at very low energies; see below. The process of high-energy spallation of a heavy target by a very intense light ion beam (also called target fragmentation) or the complementary process of projectile fragmentation of a high-energy heavy ion beam by a light target in reverse kinematics has been used to produce an extremely wide range of nuclei, albeit the products are essentially all lighter than the nucleus being fragmented in both cases. Note that the exotic nuclei produced by target fragmentation in normal kinematics (spallation) are nearly at rest in the laboratory frame so that techniques are needed to rapidly extract the products from the bulk target material followed by ionization and then injection into an electromagnetic separator. Advantages of this technique lie in the availability of very high-power light ion beams that have long ranges in target materials leading to a high luminosity combined with the ability to produce ions in a single charge state. Difficulties involve the chemical selectivity as the products move through and out of the matrix material while maintaining the overall integrity of the target material. The delay due to chemical process can lead to significant decay losses. Earlier systems used plasma ionization of effused reaction products, but recent systems use laser ionization to select an individual chemical element. On the other hand, projectile fragmentation can produce exotic nuclei similar to those from target fragmentation, but in this case, the products have large kinetic energies and retain a large forward momentum and relatively high charge states that can be rapidly separated from the unreacted beam and other uninteresting products. The available beam intensities and target thicknesses are significantly smaller than those for target fragmentation, but there is no chemical selectivity and no decay losses due to beta decay. In addition, the acceptance of modern separators can approach 100% of the typical momentum distribution.

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Finally, note that very heavy reaction products can de-excite by fission into two neutron-rich products and fission at the lowest excitation energies will lead to the most neutron-rich products. Sequential fission does not significantly change the collection process for target fragmentation due to retention in the thick targets. Sequential fission can affect the collection of MNT products, but the amount depends on the details of the actual collection system. On the other hand, sequential fission of fusion products has a dramatic negative effect on the collection efficiency since the isotropic momentum distributions of the fission fragments in the moving frame are much larger than that due to sequential evaporation. Sequential fission of projectile fragments has been studied for the heaviest ions at the highest beam energies in order to reduce the momentum broadening.

Production Cross Sections, σ (Z, A) It is useful to have a list of the production cross sections for individual, representative exotic nuclei so that one can estimate the yield at the end of the electromagnetic separator. Nuclei with closed shells of nucleons, particularly doubly closed shells, provide important tests of nuclear structure models. Table 2 contains some information on a range of important nuclei that have been produced and separated for study. The reported cross sections for the production of the nuclei are given in the table. In general, the fusion cross sections are largest near the

Table 2 Nuclear reactions and measured cross sections to produce some of the highly sought after exotic nuclei Beam energy Cross section MeV/u cm2

Isotope Production reaction 58

48 Ni

P. Frag: N i( N i, x)

78 Ni

P. Fission: Be(

78 Ni

P. Frag: Be( Kr, x)

100 Sn

P. Frag: Be(

102 Sn

Fusion:

132 Sn

P. Fission: d(

132 Sn

T. Frag:

202 Os

P. Frag: Be(

292 Og

Fusion:

9

238

9

86

9

124

50

U, f )

Xe, x)

58

Cr( N i, αpn)

238

9

249

238

U, f )

U Cx(p, x)

238

132

U, x)

48

Cf ( Ca, 3n)

Reference

74.5

0.05 ± 0.02 pb

Blank et al. (2000)

750

0.3 ± 0.2 nb

Engelmann et al. (1995)

140

0.02 ± 0.01 pb

Hosmer et al. (2005)

1000

5.8 ± 2.1 pb

Hinke et al. (2012)

6

0.6 μb

Karny et al. (2006)

1000

0.4 μb

Pereira et al. (2007)

2 mb

ISOLDE and Collaboration (2021)

1000

4.4 ± 2.0 pb

Kurcewicz et al. (2012)

5.2

0.5 + 1.5 − 0.3 pb Oganessian et al. (2006)

Sn 1400

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Coulomb barrier, and, more so, the cross section for a particular product will then be maximum in a narrow range of energies due to sequential evaporation of nucleons (usually neutrons). Thus, there is a limited range of primary beam energy and target thickness that will be able to produce a given exotic nucleus in a fusion reaction. This is particularly true for the heaviest elements where the energy losses per unit target thickness are the greatest. Although there are not extensive data available, the production cross sections of various exotic nuclei in target and projectile fragmentation reactions do not have large variations when the kinetic energy is five or more times the Coulomb energy since the nuclei approach each other along straight-line trajectories and the cross section approaches the geometric limit. Thus, the yield of a given projectile fragment can be maximized by using a bombarding energy and target thickness that will leave the products in a full-stripped charge state. An overview of some of the production cross sections with different reactions for a wide range of the tin isotopes is shown in Fig. 4. One can see that the projectile (or target) fission of uranium is strongly favored for producing the most neutron-rich tin isotopes, while the fragmentation of nearby xenon isotopes or low-energy fusion reactions is favored for the most neutron-deficient nuclei.

Fig. 4 Measured and calculated production cross sections for tin isotopes. The open and filled circles represent measured production cross sections for projectile fragmentation of 124 Sn and 129 Sn, respectively. The filled triangles represent measured values for uranium fission fragments, and the stars represent calculated fusion reaction values. The lines represent the results of theoretical model calculations

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Kinematical Considerations The kinematics of the nuclear reaction that produces the exotic nuclei play no significant role in Isotope Separator On-Line (ISOL) systems that rely on target spallation (target fragmentation) or target fission since these targets are very thick compared to the range of the products. The products stop in the material and then diffuse out. The physiochemical issues related to the migration have been extensively studied, and various overviews are in the literature (Duppen 1998; Köster 2002; Stora 2013). The properties of the ISOL beam relevant for the electromagnetic separator are defined by the ion source, and the transverse emittance of the beam is generally very good (i.e., small). Note also that ISOL ion sources produce ions in the 1+ charge state and the electromagnetic separator only has to accept a single charge state. The ions are generally produced on a high-voltage platform (of order 50 kV) and accelerated into the separator that is on ground potential. Thus the ions have relatively low velocities, and the separator generally uses room temperature (resistive) magnets and electrostatic focusing elements, for example, the CARIBU separator is a compact, high-resolution device (Davids and Peterson 2008) for online delivery of short-lived exotic nuclei from spontaneous fission that have been thermalized in gas, whereas OFFLINE2 at ISOLDE is a recent, moderate resolution device to separate long-lived isotopes (Warren et al. 2020). On the other hand, systems that incorporate in-flight production of exotic nuclei depend critically on the kinematic properties of the products and on atomic interactions in the target and are subject to having a distribution of charge states. The in-flight separator, nonetheless, has to maximize collection of the exotic nuclei and the rejection of unreacted primary beam and other unwanted, much more intense products. The kinematic issues are very different for the low-energy fusion, fission, and MNT reactions compared to the high-energy projectile fragmentation reactions and have led to different approaches to separator design. Although the projectile fragmentation separators were developed later, the kinematics and atomic charge state distributions are simpler, leading to somewhat simpler separator design. The key feature of projectile fragmentation reactions is that the nuclei move on nearly straight-line trajectories in peripheral collisions and the products retain very nearly the same energy per nucleon as the incident beam. The cross sections for these reactions, although not studied in detail, do not vary much with beam energies. Using the highest beam energy has the advantages of thick production targets (perhaps 10% to 30% of the ion range), and the atomic charge state distribution can be collapsed into only fully stripped ions. The nuclear collision and atomic energy-loss processes lead to a dispersion of the fragment momenta around the beam momentum that depends on the amount of nuclear matter lost but is nonetheless rather narrow and provides significant kinematic focusing in angle and in momentum; see Fig. 5. Similarly, Coulomb fission of a high-energy projectile also provides significant kinematic focusing, but only a fraction of the products will be within the separator’s acceptance. From a practical standpoint, the

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Fig. 5 Right side: (top) comparison of the calculated phase space for 100

132

Sn fission fragments

Δp p

and (bottom) Sn projectile fragments at 400 MeV/u. and σα represent the momentum and angular spread, respectively. Left side: the variation of the phase-space parameters for the two reaction types as a function of incident beam energy

cost of constructing a high-power, high-energy heavy ion accelerator has limited the beam energies to a few hundred up to a thousand MeV/u. In fact, if the exotic nuclei are to be slowed sown and collected for study (see below), then production at the lower end of the energy range is favored. One finds that the range straggling of a heavy ion in solid matter is proportional to the total range of that ion. Thus, the highest energies produce very large range straggling making the collection of the products in relatively thin high-resolution detectors (e.g., 500 μm silicon strip detectors) difficult. The fusion and MNT reaction cross sections are peaked near the Coulomb barrier, and the production reaction usually requires a low excitation energy to evaporate a small and specific number of light particles. Thus, the useful bombarding energy is usually limited to a narrow range near the Coulomb barrier, and the production target will have an optimum thickness that will cover this narrow bombarding energy range. Note that the kinetic energy per nucleon of all fusion products will be lower than that of the beam and the mass (and charge) will be higher so that the energy loss as the product exits the target will be higher than that of the beam. The fusion products will have a moderately narrow momentum distribution around the beam

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

15

axis, but they will have a significant distribution of atomic charge states. Conversely, MNT reaction products will be emitted near the grazing angle (usually tens of degrees in the laboratory frame relative to the beam axis), and they will be emitted with a random azimuth angle, cylindrically symmetric around the beam axis. The direct collection of MNT productions in electromagnetic separators is thus limited to a small subset within the angular acceptance of the device. A different approach has been developed to thermalize the products in a buffer gas and then extract them; see below.

Atomic Charge State Considerations The problem of collecting recoiling exotic nuclei that have a distribution of atomic charge states is particularly severe for low-energy reactions, but the charge state distribution of products always has to be evaluated. Two very different approaches to obtain the highest yields have been developed: (1) the so-called low-energy vacuum separators are used primarily for fusion reactions and generally have high resolution in m/q to separate and identify multiple charge states of the evaporation residues and (2) gas-filled magnetic separators that rely on maintaining and transporting the average ionic charge state. The highest energy fragmentation reactions (E/A ≈ 1000 MeV/u) can produce fully stripped (q = Z) ions, and intermediate energy reactions (E/A ≈150 MeV/u) can produce fully stripped ions across the lower half of the Periodic Table. Thus, vacuum separators are exclusively used to separate projectile fragmentation products. The accelerator infrastructure to provide the high-energy heavy ion beams is generally more substantial than that of the separators, and several generations of separators have been constructed and operated at major laboratories. The gas-filled separators have been developed to collect very large fractions of the fusion reaction product velocity and charge state distributions and provide reasonable mass separation. Gas-filled separators rely on two fundamental properties: low-energy heavy ions (Bρ ≈ 1 T m) will rapidly reach an equilibrium charge state when they travel through a low-pressure gas (typically He at 5 mbar), and this charge state is approximately proportional to the ion’s velocity (Fulmer and Cohen 1958; Armbruster et al. 1971; Paul et al. 1989; Leino 1997; Khuyagbaatar et al. 2013). Gas-filled separators are particularly well suited to collect exotic ions from reactions at very low yields and are relatively small devices.

Targetry, Practical Considerations It is clear that the production in the target of exotic nuclei will scale with the primary beam intensity and should be optimized. The maximum primary beam intensity is clearly established by the accelerator system and is taken as given here. However, the maximum primary beam power can be higher than can be sustained by the target. An overview of the beam intensities and beam power plus an estimate of the beam power absorbed in the target are contained in Table 3. The beam power limitations for ISOL targets can be minimized by using refractory elements

Accelerator Heavy-ion LINAC Cyclotrons LINAC Synchrotron Proton Cyclotron Synchrotron

Beam A 48

238 238 238 1

1

Z 20

92 92 92 1

1

Energy MeV/u 10 350 200 1500 500 1400

Intensity s−1 6 × 1013

6 × 1012 5 × 1014 5 × 1011 6 × 1013

1 × 1013 Spallation, fission

Proj. frag. Proj. frag. Proj. frag., fission Spallation, fission

Reaction HI Fusion 1 0.3 9 Be, 8 238 UCx, 18 UCx, 45

12 C,

9 Be,

Target A Z, g cm−2 249 Cf, 0.01

0.05

0.4 0.30 0.39 0.08

Range fraction 0.15

0.1

32 130 11 1

Target kW 0.7

2.7

51 299 18 4

Dump kW 4.1

Table 3 Typical beam intensities, energies, targets, fraction of ion range in the target, and rough estimate of beam power in target material and of the remainder delivered to the beam dump. The canonical systems are those in Table 1

16 H. Geissel and D. J. Morrissey

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

17

(e.g., Group 5 such as Nb and Ta) with the concomitant issues of limited atomic mobility and release from the materials; see the review in Stora (2013). For inflight targets, the constraints are more severe, the beam spot should also be as small as possible to maximize the acceptance of the separator (see below), and the energy loss of the primary beam and exotic nuclear products must be managed to allow subsequent separation. Placing the target material on a rapidly rotating wheel distributes the beam power over a larger area while allowing cooling processes to take place before the beam spot returns to the irradiation point. See the review for a discussion of the extensive use in heavy-ion fusion reactions by Türler and Gregorich (2014) and an example for projectile fragmentation reactions (Yoshida et al. 2004). Another important aspect of the interaction of heavy ion beams with solid target materials is radiation damage of the material, particularly atomic displacements that cause the material to swell and become brittle (Lang et al. 2001). The number of “displacements per target atom” can be quite large and can change the effective thickness of the target and change the momenta on which the separation of the emerging primary beam and reaction products relies (see below). The rotating target wheels that distribute beam power in in-flight targets can also mitigate the effects of radiation damage by distribution over a large radial volume, but radiation damage will ultimately limit the useful lifetime of the target. Note also that the remainder of the primary beam power that is not absorbed in the production target must be separated from the emerging exotic nuclei of interest and contained in a beam dump. The design of a dedicated beam dump can be relatively simple when the degraded primary beam can be easily separated from the products of interest as in ISOL targets since the thermalized atoms emerge from the surface of the target material. On the other hand, the unreacted primary beam and exotic nuclei emerge in the same direction from an in-flight target, and separation will rely on exploiting differences in magnetic rigidity. The beam dump for the inflight separators for heavy-ion fusion reactions can rely on the fact that the magnetic rigidity of the fusion products is always higher than that of the beam. In addition, the kinematical constraints and in some cases scarcity of actinide target material limit the overall beam power. Thus, a relatively simple beam dump can be placed on one side of the dispersive plane of the separator; see the recent example of GARIS-II (Kaji et al. 2013). However, the intense residual primary beam power and the fact that projectile fragments can be either more or less rigid than the emerging primary beam seriously complicate the requirements on the beam dump. Issues with the beam dump for Big-RIPS have been present by Yoshida et al. (2013) and for the FRIB/ARIS separator under construction by Avilov et al. (2016); see also the series of workshops on High Power Targetry (Okuno and Makimura 2022). The high-intensity primary beams that react in the substantial targets with the remainder collected in beam dumps create substantial prompt radiation emerging from the target and beam dumps as well as long-lived activation products. In general, the charged particles including the primary beam can be stopped in a relatively small amount of material although, as indicated above, substantial thermal cooling may be necessary. On the other hand, the prompt neutrons created from nuclear

18

H. Geissel and D. J. Morrissey

reactions can be very penetrating. The energetic neutrons will have to be thermalized in thick, low-Z shielding and eventually absorbed by the shielding to mitigate any external radiation. The shielding necessary to thermalize and absorb the neutrons is generally suitable to mitigate any gamma radiation. Order of magnitude estimates of the neutron radiation produced in the canonical reaction targets are given in Table 4. Note that the estimates use simple one-significant-figure estimates of the neutron multiplicities from the reactions and a canonical two barn cross section. Nonetheless, the neutron production rates are remarkably high and can be on the order of the neutron flux in a nuclear power reactor but with significantly higher kinetic energies. The activation of metal components in the target area of the separator and beam dump can also be significant. Thus, the shielding requirements for the target and beam dump areas for the routine production of exotic nuclei are substantial and have to be fully integrated into the design of the facility and be accommodated by the separator layout. The target areas can be expected to have a high radiation level and thus need to be heavily shielded and separated from the experimental hall. The high radiation level exists not only during the period when the intense beam interacts with the target material but also after target irradiation due to the activation products. For example, besides the local concrete and iron shielding, an additional 8 m of earth is used to cover the target area at ISOLDE. In addition, ventilation systems with filters take care of the volatile radioactive species. Target changing, storage, and maintenance are performed at ISOLDE with mobile industrial robots. The robots and any controls use radiation-hard materials, and the area is operated as an enclosed hot cell, similar to those necessary in nuclear reactors. Quite similar technical efforts with respect to the handling of the radiation and radioactivity have to be made in the target and

Table 4 Qualitative prompt neutron radiation produced in the targets for the facilities and luminosities from Table 1 Accelerator & reaction Heavy Ion LINAC: Fusion Cyclotrons: Proj. frag. LINAC: Proj. frag. Synchrotron: Proj. frag. Proton Cyclotron: Spallation, fission Synchrotron: Spallation, fission a

Prompt neutron Luminosity cm−2 s−1 Multiplicity Ratea s−1 33 2 × 10 ≈2 3 × 109

Energy MeV few MeV

Angular distribution Slightly forward

4 × 1035

≈5

2 × 1012

300 MeV

Strongly forward

3 × 1037 3 × 1035

≈5 ≈5

1 × 1014 1 × 1012

4 × 1036

≈10

6 × 1014

200 MeV Strongly forward 1,000 MeV Very strongly forward few MeV Isotropic

1 × 1036

≈10

1 × 1014

few MeV

Isotropic

Canonical reaction cross section of 2 b was folded with the order-of-magnitude neutron multiplicity

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

19

Fig. 6 All separator types are well shielded in their first section. As an example, schematic diagram of the target area of the ARIS separator (Cooper et al. 2014). The whole area is deep below ground and heavily shielded. The primary beam enters from the left (red arrow), and the unreacted beam is directed downward towards the beam dump after the first dipole

pre-separator areas of the next generation of in-flight facilities; see, e.g., the ARIS separator at FRIB (Hausmann et al. 2013). A schematic diagram of the target area and first section of the pre-separator (see discussion below) are shown in Fig. 6. Besides being deep underground and heavily shielded, the unreacted primary beam is directed downward by the first dipole towards a high-power beam dump.

Thermalization in Gas The diffusion of exotic nuclei through a solid target and the effusion from the surface into an ion source clearly limit the range of chemical elements that can be studied and can introduce large losses due to radioactive decay. The early studies of exotic nuclei (at the time) used solid foils to collect the low-energy recoils from thin targets. The range of the recoils was very short in solids (10 to 100 μm), and it was recognized that the recoils could be stopped and very rapidly thermalized in a buffer gas. Thus, the effects of radioactive decay could be reduced. Moreover, the effects of chemical reactions would be greatly reduced, for example, refractory elements that are not accessible with ISOL techniques could be collected. If the recoiling products are thermalized in a pure noble gas (helium or argon), then they will remain ionized in the 1+ or 2+ state depending on the ionization potential of the chemical element. The products can then be rapidly swept from the gas volume through an orifice and enter an electromagnetic separator system. An important fact in all of these devices is that the stopping and thermalization of the exotic ions lead to significant ionization of the buffer gas. Helium is the best buffer gas since it has the highest first ionization potential of any element; argon

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H. Geissel and D. J. Morrissey

has also been used. The buffer gas must be extremely pure, better than parts per billion (ppb) of all impurities, so that the ionization remains on the buffer gas and the exotic ions, a fact that was not recognized in the early systems. However, in practical devices, many impurities, generally small molecules, are present which can cause multiple problems. For example, if the residence time of the ions in the gas is comparable or longer than the mean time to collision with the impurities, then in most cases exotic ions will be efficiently neutralized by an impurity molecule. In a related effect, helium ions from the buffer gas will charge exchange with ALL molecules and create a mixture of uninteresting positive ions that will be extracted. Various approaches have been developed to reduce the contaminants to sub ppb levels including passing the buffer gas through active and passive getters, through cryogenic traps, and even cooling the entire gas cell to cryogenic temperatures (≤50 K is desirable) (Ranjan et al. 2015; Reiter et al. 2016; Lund et al. 2020).

Low-Energy Gas Cells Low-energy gas cells typically use gas pressures on the order of 1/3 bar and have thicknesses of less 10 cm that are optimal to efficiently stop a variety of recoil ions due to their short ranges. Solid degrader foils can be placed after the target or source to decrease the range of the products and the amount of ionization of the buffer gas. The residence time in the gas can be quite short, ≈1 ms, for suitably small volumes and high flow rates. The high velocity gas minimizes charge exchange problems. The fastest and smallest devices for collection of reaction products do not use internal electric fields to manipulate the ions; larger devices include static and/or dynamic electric fields to move ions towards the orifice (see below). The exotic nuclei exit the gas cell through an orifice and into a differentially pumped system to remove the buffer gas and deliver the ions to the acceleration region for transport into the electromagnetic separator. The size and shape of the orifice depend to a large extent on the capabilities of the differential pumping system. A multipole ion guide is used to capture the ions exiting the orifice to increase the acceptance of the acceleration system. A number of facilities have been developed to thermalize low-energy recoils from nuclear reactions and sources in gas beginning with the long-running IGISOL facility in Finland (Äystö 2001) and includes the ShipTrap system at GSI in Germany (Dilling et al. 2000), the ANL-CPT for low-energy reactions. Savard et al. 252 (2003), and the large CaRIBU system for Cf fission fragments at ANL in the US (Savard et al. 2016). The collection efficiency of devices for near-Coulomb barrier energies can be large, but the fraction that remains ionized is subject to intensity and gaseous impurities. An important extension of this technique is to reionize the thermalized neutrals in the gas volume or after they exit the gas volume with lasers; the details are part of the recent presentation of the large impact of LASER spectroscopy in nuclear science (Campbell et al. 2016). Example facilities include the LISOL system in Belgium (Van Duppen et al. 2000), the IGISOL-IV system

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

21

in Finland (Dicker et al. 2014), the buffer gas catchers for the heaviest elements at TRIGA-SPEC in Germany (Backe et al. 2015), and the KISS device in Japan (Hirayama et al. 2017). Thermalization in gas may provide the most effective means to collect the exotic MNT products that have a huge angular spread and energy distribution by using shaped collection volumes that contain the ionization caused by the primary beam and physically collect the MNT products at large angles to the beam and accept 2π in azimuthal angle (Peräjärvi et al. 2005; Choi et al. 2020; Hirayama et al. 2020).

High-Energy Gas Cells The production of exotic beams by projectile fragmentation provides beams on very short time scales without regard to chemical properties. The conversion of highenergy fragments into low-energy exotic beams including refractory elements and the most exotic very short-lived nuclides is important because it provides access to a variety of low-energy experimental techniques that are not possible with fast beams. Nonetheless, capturing and thermalizing high-energy fragment beams within a limited size helium-filled cylindrical chamber are challenging tasks due not only to the long ranges of the ions but also the momentum distribution of the fast beam and range straggling. The total momentum spread of projectile fragments exiting high-energy electromagnetic separators is on the order of a few percent due to (1) the nuclear reaction mechanism itself, (2) the differential energy loss and energy straggling in the production target, and (3) the straggling in the achromatic wedge used for beam purification. In order to be stopped in the gas cell, the separated beam has to be slowed down with additional solid degraders that will further increase the momentum spread. Ion-optical range compression of the secondary-beam momentum distribution has been developed that employs an additional magnetic system with momentum dispersion of the beam on a wedge-shaped solid degrader to provide a reasonably mono-energetic beam (Geissel et al. 1989; Weick et al. 2000). The transformation of the beam phase space presents a compromise to keep the length of the gas cell size reasonable while increasing its diameter. A dedicated beam line was constructed at the National Superconducting Cyclotron Laboratory (NSCL) to “compress” the momentum of fragments coming from the A1900 fragment separator into a 20 cm wide, approximately parallel beam. However, only a momentum spread of Δp/p ≤ 2% compared to Δp/p ≤ 5% available from the separator could be accommodated (Cooper et al. 2014; Sumithrarachchi et al. 2020). The beam then encounters precise and uniform metal degrader plates with a monochromatic metal wedge before entering the gas cell; see Fig. 7. A similar beam line is under construction for the FAIR facility. A summary of the first generation devices for thermalization of fast exotic ions can be found in the review by Wada et al. (2003), and an overview of the genealogy along with a detailed review of the underlying concepts and gas cell designs is also available (Wada 2013). The gas cells are quite large, linear gas-filled chambers of order 0.5 m long into which the beam enters along the central axis and exits

22

H. Geissel and D. J. Morrissey Degrader

Extraction System

Gas Cell

1

2

3

4

5

6 1

extracted beam

2Tesla Magnet

injected beam

Pumping

2 3

4

6 7 8

RFQ Ion Guide

RF Carpet

Charge Collection Electrodes

10 mbar He

Entrance window/degrader

5

Test (Fission) Source

Fig. 7 (Top) Schematic diagram of a large linear gas cell for fast projectile fragments showing the components necessary to thermalize the exotic ions and extract them (Cooper et al. 2014). The sections inside the gas cell represent all-metal RF walls and RF funnel leading to the nozzle (Savard et al. 2003). (Bottom) Schematic concept of a gas-filled reverse cyclotron that can provide sufficient path length in gas to thermalize energetic exotic ion beams (Bollen et al. 2005)

through an orifice at the downstream end of the cell (Sikler et al. 2003; Savard et al. 2003; Wada et al. 2003; Weissman et al. 2005). An alternative geometry with a perpendicular extraction was used in Japan (Wada et al. 2003; Sonoda et al. 2013), see principle in Fig. 8, and has been proposed for a new device at the FAIR facility (Dickel et al. 2016). The ejection orifice is usually a de Laval nozzle with a throat diameter of approximately 1 mm. The nozzle directs the gas flow into a multiple ion guide that captures the ions. The volumes of the gas cells are so large (≈0.25 m3 ) that the gas velocity is minimal except very close to the exit, and the macroscopic gas flow does not play a significant role in moving the ions into the exit region. The large gas cells for projectile fragments use internal static and dynamic electric fields to collect and transport the (positive) ions to the orifice. A static gradient on the order of 10 V/cm that is applied along the beam axis rapidly separates the more mobile electrons from the ionized buffer gas atoms and the incident ions. This creates a

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

23

Fig. 8 Schematic diagram of the PALIS gas cell that collects rejected exotic beams in the BigRiPS separator with an orthogonal extraction system of neutral atoms and laser reionization. (Figure from Sonoda et al. 2013)

weak, positive plasma or positive space charge along the beam axis that tends to hinder ion migration to the exit and drives the ions towards the walls (Huyse et al. 2002; Wada et al. 2003; Takamine et al. 2005). These devices now all include an inhomogeneous alternating electric field (RF frequency of order 1 MHz) on closely spaced electrode structures centered on the orifice to repel the ions and thus maintain them as gas-phase ions. These devices have become known as “RF carpets.” The repelling force was originally provided by a standing wave RF field applied to concentric electrodes on a flat surface and required a static drift field to migrate the ions. Subsequently, a variety of three-dimensional electrode structures have been developed, for example, an all-metal funnel systems developed at ANL (Savard et al. 2003, 2016; Sumithrarachchi et al. 2020). The combination of a standing wave RF potential and drift field can be replaced by a traveling wave which simplifies the construction and can attain high transport velocities (Bollen 2011; Brodeur et al. 2013; Lund et al. 2020). While it might seem that the buffer gas pressure should be as high as possible to maximize the stopping power, the effective repelling potential close to the surface of the RF carpet falls with the square of the pressure: Fmax ∝ m(VRF /p)2 /r 3 where m is the ion mass, VRF is the applied voltage, p is the buffer gas pressure, and r is a measure of the electrode spacing (Takamine et al. 2005). Thus, a balance has to be struck between increasing the stopping power versus losing the repelling force

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H. Geissel and D. J. Morrissey

and practical systems use pressures on the order of 100 mbar of helium. It has been suggested that the limited stopping power due to low pressure and relatively short path length can be overcome by confining the fast ions in a strong magnetic field while they slow down (Katayama et al. 1998; Bollen et al. 2005, 2008). The fast ions are injected into a cyclotron magnet at the outer radius and spiral towards the center as they lose kinetic energy. The principle of such a cyclotron gas-filled catcher is depicted in Fig. 7 where RF carpets are used to migrate the thermalized ions to the center for axial extraction. Such devices have been called a reverse cyclotron or a cyclotron stopper. One large-scale device has been constructed with a dedicated high-resolution beam line at the NSCL that is ready for commissioning (Schwarz et al. 2020). All of these buffer gas thermalization systems provide a relatively low emittance beam of exotic nuclei in one charge state at very low kinetic energies for separation or direct study.

Overview of Production Concepts The various reactions to produce exotic nuclei have various requirements for the accelerators and constraints on the targets that have led to different production facilities. There are two major classes of facilities. In-flight systems produce the exotic nuclei in targets that are thin enough that they directly emerge from the target and enter the electromagnetic separator. The velocity of the escaping nuclei is significant, and they remain ionized and travel through the system without betadecay losses. These separators have been used for both fusion reactions in vacuum (Münzenberg et al. 1979) and with gas filling (Oganessian 2007; Morita et al. 1992; Semchenkov et al. 2008) and high-energy projectile fragmentation reactions (Dufour et al. 1986; Geissel et al. 1992a; Kubo et al. 1992; Morrissey and Sherrill 1998). In an alternative approach, the exotic nuclei are produced in thick targets (or thin targets with catcher devices) that thermalize the products before transporting them to the separator. The products that are generally neutralized in the collection process have to be reionized to the 1+ state for electromagnetic separation at generally very low kinetic energies. Recent advances in laser ionization now allow elemental selection of the thermalized products before electromagnetic separation. These systems have been called Isotope Separator On-Line (ISOL) (Jonson and Riisager 2010). Recent extensions to the ISOL concept that use thin targets for faster extraction include the Ion-Guide Isotope Separator On-Line (IGISOL) in which the exotic nuclei are thermalized in helium or argon gas, remain ionized, and are passed to a low velocity separator (Äystö et al. 2014) and the KEK Isotope Separation System (KISS) system to collect neutralized and thermalized MNT products in argon gas which are reionized by lasers before electromagnetic separation (Hirayama et al. 2020). A schematic overview of the relationships among the systems to provide exotic nuclei for subsequent study is shown in Fig. 9. The major differences among the various systems are primarily at the top of the figure, whereas the techniques to study the exotic nuclei after separation are often the same.

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

25

Fig. 9 Schematic representations of the major production efforts: (left) the in-flight production and separation scheme (fusion and projectile fragmentation) and (right) the ISOL production and separation scheme (target fragmentation) and (left to right) the hybrid reaction product thermalization schemes (IGISOL and MNT). The electromagnetic separator stages are indicated in the middle of the figure and provide the exotic nuclei for study. See the text for details

Separation in Pure Electromagnetic Fields In the following sections of this contribution, the basic isotope separation tools will be presented which have been successfully used and also can be expected to be implemented in planned modern exotic nuclear beam facilities in the future. The properties of the different nuclear reactions and thus the production cross section are fixed and inherent physical quantities. Therefore, the major experimental innovations have focused on increasing the intensity of the primary beam, the effective target thickness, and the overall separation efficiency and sensitivity. Moreover, the separators have been combined with high-resolution ion-optical systems and efficient detector arrays. The atomic interaction combined with ionoptical systems has a key role in modern facilities at all energies. The international conference series on Electromagnetic Isotope Separators and Related Topics (EMIS) provides a forum for presentation of progress in this field, and their published proceedings contain an important record and detailed descriptions of the performance of existing and planned separator facilities over

26

H. Geissel and D. J. Morrissey

many decades. Equipped with the knowledge of fundamental properties outlined in this contribution, a student or newcomer in the field will be better equipped to understand the information in the EMIS conference publications. In the following sections, the main techniques to separate exotic nuclei from the primary beam and other reaction products are presented (Münzenberg 1992; Münzenberg 2013; Geissel et al. 1995) that are applied in Isotope Separator OnLine (ISOL) and in-flight separators. The principles used in each category provide selected typical examples. It is not our intention to describe and compare all of the existing and planned facilities but rather to demonstrate the key properties of the separators needed to attain the best yields of the exotic nuclei.

Ion-Optical Definitions and Tools As shown in Section “Thermalization in Gas,” the kinematic properties of rare isotopes emerging from the production target crucially determine the applicable method of separation and thus determine the physical properties and layout of the appropriate separator. The energy and angular distributions as well as the atomic charge-state populations are characteristics of the type of nuclear reactions used to produce exotic nuclei. In this section, the basic separation principles and tools which can be applied are presented. In charged particle separators, electromagnetic fields are employed to separate the desired nuclear species characterized by their mass-to-charge ratio m q and the different kinematics determined by the energy-angular distribution. The motion and transport of ions with a velocity v in static electromagnetic fields are described by the Lorentz force F: F=

d(mv) dp = = q(E + v × B), dt dt

(1)

where p and q are the momentum and charge of the ion and E and B are the electric and magnetic field strengths governing the ion motion (Carey 1986; Wollnik 2021; Yavor 2009). According to relativistic theory, the mass of a moving particle is dependent on its velocity: m= 

m0 1−

v2 c2

,

(2)

where m0 is the rest mass and c the light velocity in vacuum. In ion separators sometimes electrostatic but mostly magnetostatic fields are applied in which the energy and the phase space of the transported ions are conserved during motion through a system with such fields. Note that the situation is quite different if the ions interact with matter placed along the path in such a device. Also note that electrostatic fields are not generally applied in separators for high-energy ions since providing equivalent magnetic and electric forces would imply that 1 T ≡ 300 MV/m for the same charge state q. Even at ultrahigh vacuum

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

27

conditions and with specially prepared and highly polished electrodes, the attainable electric fields are on the order of ≈10 MV/m. Therefore, pure magnetostatic fields are used throughout separators for the highest velocity ions. From the Lorentz force, it follows that the kinetic energy and the absolute value of the velocity are conserved for the transported ions. An ion will move along a circular path with a radius (ρ) in a homogeneous magnetic dipole field perpendicular to the direction of motion; see Eq. 1: mv 2 mv p = qvB, or rearranging: Bρ = = , ρ q q

(3)

which provides a definition of the magnetic rigidity Bρ. When an ion moves in a pure electrostatic field E perpendicular to the field lines, in analogy to motion in the magnetostatic field, the direction of the ion’s velocity can be similarly changed but not its magnitude. This condition is fulfilled by cylindrical and spherical electric fields which are often utilized in low-energy separators. The corresponding electric rigidity, Eρ, also follows from the Lorentz force: mv 2 pv mv 2 = qE, again rearranging: Eρ = = ρ q q

(4)

The different dependencies of the magnetic and electric rigidities on velocity indicate that a pure magnetic or electric field cannot spatially focus ions with different velocities. However, the combination of magnetic and electric sector fields in a device can provide a velocity selection when the bending in the magnetic dipole field is exactly counterbalanced by that from an appropriate electric field. This combination leads to a pure velocity separation given by the ratio: Δv Δ(Eρ) Δ(Bρ) = − v Eρ Bρ

(5)

Ion separators based on this principle with perpendicular field arrangements are called velocity filters or Wien filters (Münzenberg et al. 1979). They transmit ions without deflection that have a single velocity, v, which is given by the ratio of the electric and magnetic field strengths v = E/B. Wien filters have a high separation power for nuclear reaction products if the mean velocity of the ion of interest is quite different from that of the residual primary beam and of the abundant contaminants. Furthermore, the velocity spread of the ions of interest needs to be relatively small. In the coming sections, this category of separators will be reviewed in more detail as they have found important applications in recent experiments. Any significant velocity spread of the desired reaction products presents a large challenge to separators with Wien filters and thus strongly affects their performance. For example, the velocity spread caused by the nuclear reaction itself and energy straggling due to atomic interactions in the target material used to produce the exotic nuclei is inevitable. Therefore, it is desirable to eliminate or at least reduce the influence of any velocity spread from the final mass-over-charge resolution. This

28

H. Geissel and D. J. Morrissey

can be achieved with so-called double focusing spectrometers with magnetic and electric sector fields. In such devices, a twofold Bρ resolution is counterbalanced by the Eρ resolution such that the velocity spread is canceled. This condition can also be directly shown in the relations of magnetic and electric rigidity: Δm/q 2Δ(Bρ) Δ(Eρ) = − m/q Bρ Eρ

(6)

Velocity filters and double-focusing separators have successfully contributed to the research and application of exotic nuclei at low energies, that is, close to the Coulomb barrier. Other technical and experimental methods must be implemented for the separation of exotic nuclei produced at higher energies due to the technical limitations on applicable electric fields. Before considering these requirements and design solutions, one should introduce the description of charged-particle transport through electromagnetic fields in more detail to understand the required ion-optical focal plane conditions and the crucial aspect of transmission probability through the separator. An ensemble of ions moving in electromagnetic fields can be described by Hamiltonian methods. This method provides an overall framework for the ion motion within the allowed phase space using generalized coordinates. The Hamiltonian function, including the Lorentz force and its potentials, can be written as: 1

H = qΦ + c{m20 c2 + (p − qA)2 } 2 ,

(7)

with B = ∇ × A,

−∇Φ = E +

∂A , ∂t

(8)

where A and Φ are the vector and scalar potentials of the electromagnetic field. The motion of ions in the phase-space volume can be obtained from the Hamiltonian function and the corresponding canonical equations. It follows directly that the particle density in the phase space under the action of the Lorentz force must be invariant in time. This property is called the Liouville’s theorem and has many practical consequences for the description of ion motion in electromagnetic fields. To a good approximation, the particle densities in sub-phase-space volumes are also preserved. However, this Liouvillian phase-space conservation is only fulfilled when conservative forces are present (Geissel et al. 1989), i.e., no atomic or nuclear interactions take place within the ion-optical system. A so-called vacuum separator, i.e., with no material in the path of the beam, would fulfill this requirement. However, gas-filled separators and high-energy separators with internal degraders would not. An important goal for the application of ion-optical calculations to separators is to find the transfer function that would create an image of the initial phase space at any desired position along the electromagnetic system. So-called ray-tracing

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(following a detailed description of the ions path) and matrix (creating a map of the coordinate transformation) calculations represent two important methods to determine the required transfer function. In the ray-tracing method, the equations of motion are numerically integrated for particle-by-particle many times along the field configuration to obtain a set of particle positions and angles at any given plane. The most important plane is certainly the final focal plane or exit from the system. On the other hand, the matrix method calculates a transfer function for each individual ion-optical element in the system with the goal to combine these individual functions (matrix multiplication) to provide the set of final coordinates of the particles transported through the ion-optical system from the set of initial coordinates. Thus, the transfer maps include all information about the particle motion and can be carried out more efficiently than the ray-tracing. In recent years, the design of and ultimate description of separator ion-optical systems have been primarily made by first applying the matrix method (also used here) and then subsequently verified with a ray-trace analysis. The transfer function can be expanded in a Taylor series of the coefficients as elements in the matrix calculation. In this transformation, it is convenient to use a coordinate system moving with a “reference charged particle” that moves along the central trajectory, the optical axis. The coordinates of such a particle trajectory form a curvilinear coordinate system given by a six dimensional set of points (x, x  , y, y  , s, δ) (Carey 1986). In this set, x is the direction of any dispersion in momentum, and y is perpendicular to both x and the path length s along the central trajectory. Both x  and y  are the derivatives of the x and y coordinates with respect to s and thus represent the corresponding angles. Finally, δ = Δp/p0 represents the relative momentum deviation. In the vacuum-separator and vacuum-spectrometer devices, the velocity of a particle is preserved in pure static electromagnetic fields; therefore, the path length, s, is a convenient and important coordinate in ion-optical calculations and can be easily interpreted as the time coordinate for motion through the system. Another important convention that is applied to the 6th coordinate is that the deviation in Bρ from the central value is used, or it is even split into kinetic energy and mass coordinates. The use of such coordinates are advantageous for separators with a combination of electric and magnetic sector fields (Wollnik 2021). Note that the convention is to write x, x  , y, y  , s, and δ in that specific order. Unfortunately, ion-optical computer codes sometimes use simpler designations or names for these coordinates while maintaining the order, e.g., (a) the six coordinates are simply assigned the numbers from 1 to 6, e.g., x=:1, x =: 2, etc., or (b) the angles in x- and y-direction are labeled a and b, or θ and φ, respectively. The equations of motion in static electromagnetic fields can be calculated by linear second-order differential equations using these curvilinear coordinates. For a dispersive system with static fields, the equations of motion are reduced to:

x  + kx (s)x =

1 Δp ρ p

and

y  + ky (s)y = 0

(9)

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H. Geissel and D. J. Morrissey

with x  =

∂ 2x ∂s 2

and

y  =

∂ 2y . ∂s 2

(10)

The solution of these trajectory equations, Eqs. 9 and 10, can be evaluated in the matrix method by using a Taylor series expansion in terms of the derivatives with respect to s. The Taylor coefficients then represent matrix elements which can be obtained in the different orders of the expansion. For example, the firstorder matrix elements represent the image magnification and the dispersion of an object at the entrance of the ion-optical system transformed to the plane of interest. The dispersion coefficient describes the contribution of the momentum or energy deviation to the image size. Now a discussion of a simple dispersive magnetic system that is often used as a building block (i.e., an elementary component) in various modern separators will be presented. The transport of an ion with a coordinate, Xi , at the starting (target) position T A to a final position F 1 (focus 1) by an ion-optical system can be written in terms of the matrix elements up to the second order as: Xi (F 1) =

6 

Rij Xj (T A) +

j =1

6 6  

Rij k Xj (T A)Xk (T A),

(11)

j =1 k=1

where Rij is the element in the i-th row and j-th column of the first-order matrix and similarly for the Rij k for the second-order matrix. Many systems are symmetric with respect to the mid-plane in y-direction which simplifies the motion in x-direction and so that the first-order approximation to the transformation becomes: xF 1 = R11 xT A + R12 xT A + R16 δ

(12)

Importantly, recall that the matrix elements used in different ion-optical computer codes can use different nomenclatures. The convention from the TRANSPORT (Brown et al. 1980) and MIRKO (Franczak 1984; Franczak et al. 2022) codes is used here. This choice was made because the results of most of the ion-optical calculations and the figures that appear in this chapter were obtained with the MIRKO program. However, it is important to recognize that many separator systems have been designed and their properties calculated with the GICOSYReference and COSY (Makino and Berz 2006) codes, so that one would write this expression in corresponding nomenclature as: R11 := (x|x),

R12 := (x|a),

R16 := (x|δ)

(13)

and analogously for the other coordinates and higher-order matrix elements. For example, major higher-order chromatic and geometric aberration terms that are important to consider are R126 := (x|aδ) and R122 := (x|aa).

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The initial coordinate vector X(i) and the corresponding final coordinate vector X(f ) are related by the multiplication with the overall transfer matrix R which is the matrix product of all sub-matrices: X(f ) = RX(i) R = Rf · · ·

(14) R2 R1

(15)

Thus, the imaging condition of an ion-optical system can be represented by a 6 × 6 matrix in first order, and a system consisting of several stages can be simply described by multiplication of the individual matrices as in Eq. 14. In addition, the coordinates can be obtained from the individual matrices at any position along the path. The phase-space volume of an ensemble of ions also consists of three position and momentum coordinates that, to a good approximation, form an ellipsoid. For a Liouvillian system, as indicated above, the phase-space volume is conserved throughout the ion-optical system, and the matrices will have unit determinants, a condition that also holds for all of the corresponding sub-systems. Similar to the coordinate transformation of a single particle with the matrix formalism, the transport of an ensemble of ions (i.e., a beam) within a phase-space ellipsoid can be described during its passage through a separator system. The phasespace projections in the x −x  and y −y  planes are called the transverse emittances. These areas represent the maximum extents in x and x  directions. The beam at the entrance of an optical system σ (T A) can be transported to any subsequent focal plane F 1 using the transfer matrix R(F1): σ (F 1) = R(F 1)σ (T A)R(F 1)T ,

(16)

where R(F 1)T is the transpose of the transport matrix. The maximum extents in x and x  directions of the beam in a non-dispersive system are: √ σ11 √ = σ22

xmax =

(17)

 xmax

(18)

The diagonal elements of the σ -matrix represent the correlation between x and x  and thus give the orientation (slope) of the ellipse. For example, an upright ellipse (called a waist) means there is no correlation between x and x  , a feature that has very important properties in beam transport. A waist can be used to minimize the phase-space growth due to atomic or nuclear interaction due to any matter that is inserted in the beam path (e.g., a detector). Therefore, a goal of separator design is often to require both an image and a waist at the focal planes where an increase of the phase space is inevitable. Good designs will have the image and waist separated by only a few centimeters or less from each other. Often it is not sufficient to

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Fig. 10 Schematic illustration of the correlations in the three-dimensional sub-phase space (x, x  , δ) demonstrating the growth of the resulting total ellipsoid due to the dispersive coordinate

consider the beam matrix only in the position-angular sub-space, but one has to include the momentum deviation δ. In this case, one has three correlations and a large increase of the total phase-space ellipse; see Fig. 10. For example, the maximal beam envelope in x-direction becomes: xmax =

 2 σ σ11 + R16 66

(19)

Laterally Dispersive Systems With the previously described tools and definitions, one can now calculate the ionoptical properties of a simple dispersive stage, which is, in principle, a building block in almost every separator. Figure 11 gives a schematic representation of the magnetic elements and the calculated envelopes. This representative magnetic system consists of three quadrupole lenses (called a triplet) to prepare the beam for bending in a dipole magnet and three quadrupole lenses at the exit to refocus the emerging beam as indicated in the three-dimensional graphic in the upper part of Fig. 11. The transport matrix of a system with a mid-plane y-symmetry is defined by: ⎤ ⎡ (x|x) (x|a) 0 0 R11 R12 0 0 0 R16 ⎥ ⎢ (a|x) (a|a) ⎢R R 0 0 0 0 0 R 26 ⎥ ⎢ ⎢ 21 22 ⎥ ⎢ ⎢ 0 0 (y|y) (y|a) 0 0 R R 0 0 ⎥ ⎢ ⎢ 33 34 R=⎢ ⎥≡⎢ ⎢ 0 0 R43 R44 0 0 ⎥ ⎢ 0 0 (b|y) (b|b) ⎥ ⎢ ⎢ ⎣ R51 R52 0 0 1 R56 ⎦ ⎣ (l|x) (l|a) 0 0 0 0 0 0 0 00 1 ⎡

⎤ 0 (x|δ) 0 (x|δ) ⎥ ⎥ ⎥ 0 0⎥ ⎥ 0 0⎥ ⎥ 1 (l|δ) ⎦ 0 1

,

(20)

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Fig. 11 Example of a one-stage dispersive magnet system. A dipole magnet combined with two multipole focusing systems at the entrance and exit forms a very simple dispersive separator stage. The spatial separation according to the magnetic rigidity is shown for ions for three momentum deviations ( Δp p = 0, ±1%) in the dispersive plane represented by the coordinate axis x. The quadrupole magnets colored in red are focusing in x-direction, the blue ones in y-direction. The size of the boxes represents the apertures of the elements

where the two equivalent representations of the first-order matrix elements are given. Note that a large number of matrix elements are zero due to the symmetry of this example and the lack of coupling of the motion between the horizontal and vertical planes in this first-order approximation. In most separators, the interesting exotic nuclei are produced in a target or emerge from a coupled ion source; both are positioned at the entrance of the separator. In the lower panel of Fig. 11, the calculated envelopes are shown in a plan view of the dispersive x-plane for three different momenta (δ = 0, ±1%). The emittance of the initial beam was chosen to 20 π mm mrad with an object size x0 = 1 mm. The emittance characterizes the sub-phase-space volumes of an ion ensemble, in this case those entering the separator. The transverse (x,y) and the longitudinal emittance (δ) are considered separately. The transverse emittance represents the volume formed by the angular and position distributions, while the longitudinal emittance includes the momentum (energy) spread. The entrance quadrupole triplet illuminates the magnetic volume of the dipole magnet to achieve the desired resolving power and, in addition, can be tuned to reach the required transmission. Furthermore, the entrance triplet can provide a narrow focused beam in y-direction inside the dipole magnet; see Fig. 12. This focusing allows minimization of the dipole gap size reducing the amount of iron, which is an important cost factor for dipole magnets. The exit quadrupole triplet is used to produce the required focal

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Fig. 12 Results from calculations of the example dispersive stage with the magnetic elements shown in Fig. 11. The envelopes were calculated in x- and y-directions. Note that a small y-envelope is required inside the dipole magnet to maximize transmission and minimize magnet cost. The illumination of the aperture in the dispersive x plane results in a momentum resolving power of 1350 for the entering emittance of 20 π mm mrad. The dispersion line for a momentum deviation of +1 % is also presented in this compact ion-optical plot by the red line. In the lower part of the figure, the entrance and exit sub-phase-space ellipses in x − x  plane are presented with the calculated main matrix elements associated with this particular ion transport

plane conditions. In principle, the three lenses can be used to simultaneously set three independent conditions at the focal plane. Figure 12 shows a commonly used representation of the calculated beam envelopes simultaneously in x- and y-direction as well as the dispersion line for +1% momentum deviation as a function of distance, s, along the central axis. This presentation has four important features that are each shown relative to the centerline of the separator. The representation of the apertures of the magnets in the x- and y-directions is shown on either side of the centerline along with the beam envelope in each dimension. The dispersion is also shown along the centerline. Note that in this case the vertical scales for the x and y coordinates are the same with the symmetric quadrupole apertures, but the dipole magnet is asymmetric being larger in the dispersive, x, direction. Note also that the dispersion line shares its axis with the x-coordinate. The first-order ion-optical properties of this example dispersive stage are summarized in the first column of Table 5. The required conditions at the focal plane,

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Table 5 Calculated 1st -order matrix elements from target position to the final focal planes F1 for different experimental conditions. The resolving power is given for a x0 = 1 mm wide beam spot at the entrance of the system. The linear dimensions (x, y components) are given in meters, the angular ones (x  , y  components) in radians, and the momentum deviation δ in parts of the nominal value. In column (a) the priority has the parallel dispersion line and in (b) the smaller spot size. The latter is achieved by a significantly smaller dispersion coefficient (R16 ) and magnification in x-direction. The path length in the spectrometer is given at the end of the table Matrix elements R11 R12 R16 R22 R26 R33 R34 R44 Resolving power Path length [m]

(a) –1.95 0.00 5.27 –0.51 0.00 0.74 0.00 1.35 1350 21.98

(b) –0.74 0.00 2.00 –1.34 0.00 1.11 0.00 0.90 1350 21.98

F 1, were to have images in both transverse directions and a parallel dispersion line. This combination of conditions, the so-called image conditions, creates an image size in x- and y-directions that is independent of the incident angular distributions. This requires that the magnetic fields in the quadrupole magnets have to be set such that the two matrix elements from Ta to F1, i.e., R12 and R34 , must be zero. Using a quadrupole triplet allows the creation of an additional focal plane condition. In the example (a), a parallel dispersion line (R26 = 0 ) is required in addition to the positional image conditions since this constraint can be very advantageous when an extended detector setup resides at F 1. Of course, one could also use the triplet to create smaller dispersion and magnification values instead. This would be necessary if a very compact detector is placed at F 1. The corresponding changes of the matrix elements for the condition with R16 = 2.00 m are given in the column (b) of the Table 5. The actual performance of a dispersive separator stage to provide the exotic isotope of interest depends strongly on the achieved resolving power. In first-order the resolving power for Bρ is determined by the accepted emittance (object size) of the system and the field area enclosed by the x-envelope of the beam in the dipole magnet: RP1st =

1 1  2x0 · 2x0 Bρ0

B df.,

(21)

where x0 and x0 refer to the size and angle at the object position and Bρ0 to the central magnetic rigidity. The entrance quadrupole triplet uses two waist conditions in x- (σ12 = 0) and y- direction (σ34 = 0) and an x-illumination of σ11 = 79 mm at the middle of the dipole magnet. A waist is ion-optically defined by an upright

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H. Geissel and D. J. Morrissey

ellipsoid formed by the position and corresponding angular axis. The waist in y-direction has the relatively small size of 2.4 mm at the center of the dipole magnet, D/2. In practice, the resolving power of a dispersive system can be increased by increasing the bending angle of the dipole magnet. However, often it is more advantageous to use two dipole magnets whose resolving power is added instead of one large magnet. Such a two-stage system with intermediate focal planes has a smaller global dispersion and can include several slit systems along the path to remove extraneous particles. In addition, the apertures of the multipole lenses can be smaller reducing the overall impact of image aberration. The directions of the two bending angles determine the number of required intermediate foci to obtain an overall dispersive system (Winfield et al. 2021). The resulting resolving power of separators and spectrometers is certainly important, but the overall transmission of the rare nuclei is similarly important. Both aspects were included in formulating Equation 21. For more complex systems consisting of many separator stages, a socalled quality factor has been developed to characterize well angle-focused, laterally dispersive separators (Wollnik 2021). As indicated above, the sub-phase space is also conserved in such pure electromagnetic systems, i.e., the area of the phase ellipsoid x-x’ is the same at each position along the ion-optical axis. This property is illustrated by the x − x  ellipses at the entrance and exit of the example dispersive stage for δp0 in the bottom of Fig. 11 that form the three-dimensional ellipsoid, such as shown schematically in Fig. 10.

ISOL Separators The separation performance of a single-stage dispersive device can be very powerful when the reaction products are transported in the same atomic charge state and with a relatively small velocity spread. Such beams can be obtained with singly charged reaction products which are extracted at low kinetic energies from the production target with a small emittance. These requirements are characteristic of the ion sources of an ISOL facility so that a relatively simple dispersive vacuum separator can be linked directly to the production target-ion -source. Recall that an ISOL facility generally produces exotic nuclei with high-intensity light primary beams which can be neutrons, electrons, protons, deuterons, or other light ions (3 He to 12 C) that directly interact with a thick target or to induce target fragmentation (spallation) and fission reactions. The highest production rates occur when the projectiles penetrate very thick targets, for example, when light ions at relativistic kinetic energies that have huge atomic ranges in matter are used. For reference, a 1.4 GeV/u proton beam has a range of 587 g/cm2 in carbon. The effective nuclear interaction range can be even larger when fast neutrons from the breakup of highenergy deuterons are used. A quite different and specific two-stage production scheme for ISOL facilities uses intense (50–100) MeV electron beams that create Bremsstrahlung radiation during slowing down that then produces photofission in thick uranium or other actinide targets. The ISOL facilities at CERN (Jonson and Richter 2000) and at TRIUMF (Bricault et al. 1997) use 2 μA 1.4 GeV/u and 100 μA

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0.5 GeV/u proton primary beams, respectively. Both ISOL facilities produce exotic nuclei in thick (100–200) g/cm2 solid or molten targets (Köster 2001) which are more than an order of magnitude thicker than targets that can be used at in-flight facilities operating with heavy-ion projectiles at the similar velocities. The ISOL ion sources are at an electric potential on the order of 60 kV which determines the kinetic energy of the emerging fragment beams, and the ions emerge predominantly in the 1+ ionic charge state. In an ISOL system, isobars of the desired exotic isotope are the main contaminants which can hinder the study of very exotic drip-line nuclei. These isobars are generally produced at much higher cross sections in the nuclear reactions and can overwhelm the detector systems. Achieving the required mass-resolving power is a great challenge for purely electromagnetic separators that are generally implemented in ISOL systems. The required mass resolving power to separate isobars with a mass of 100 amu is illustrated in Fig. 13 that shows the mass difference in keV as a function of atomic number (Audi et al. 2014). This figure demonstrates the general feature of isobaric chains that for nuclides far from the valley of stability, the required mass resolving power is modest being between (1–2) ·104 , whereas it is largest nearest stability. Recently, additional elementally selective experimental techniques such as laser ionization are combined with the separation power of a conventional mass separator to provide the required purity for the spectroscopy of very rare isotopes. This additional element selectivity before separation is based on the different electron binding energies in a specific atom and thus represents a unique characteristic (finger print) for each element and, in the highest resolution, each isotope. A Resonance Ionization Laser Ion Source (RILIS) (Mishin et al. 1993; Marsh et al. 2013) is based on a stepwise

Fig. 13 The relative mass differences and the half-lives of the A=100 isobars based on the data in Audi et al. (2014). The required mass resolving power (m/Δm) for separation of an isobar separator is shown for the most neutron-rich, low-Z, nuclei

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H. Geissel and D. J. Morrissey

excitation and ionization of the outer atomic electron of neutral radioactive atoms via intense laser beams provided from the combination of several tunable dye lasers. The resonant ionization process consists of typically two or three excitation steps and is very efficient and element selective. In this way, purified beams of one isotope species and even pure nuclear isomers can be separated for experimental studies at the final focal planes of the ISOL separators. Thus, resonant photo-ionization by intense laser light is in this sense a powerful experimental tool for combination with mass separation. The ISOL ion sources are installed at the object planes of dispersive highresolution electromagnetic mass separators. At ISOLDE either of two different mass separators can be used, the so-called General Purpose Separator (GPS) and the High Resolution Separator (HRS). The purpose of the HRS is to achieve a high resolving power to remove the major isobaric backgrounds from the fragment beam of interest. The mass resolving power of the HRS is about a factor five higher than the GPS and consists of two stages with a 90◦ and a 60◦ dipole magnet equipped with higher-order hexapole and octupole correction coils. The resolving power of the two dipole stages is added which requires one intermediate focus in the dispersive x-plane. The mass resolving power can be adjusted with the illumination lenses as desired. A singlet quadrupole after the second magnet helps to attain the required image conditions at the final focal plane. The last element of the HRS system is a third multipole corrector. In general, most of the principles and features described in this section hold with small modifications also for the other ISOL facilities presently in operation worldwide; see, for example, Table 6. The main goals for future ISOL facilities are to increase the intensity and purity of the most exotic nuclei characterized both by small production cross sections and short half-lives.

Dispersion Matching, In-flight Separators In general, the emittance of the exotic nuclei emerging from an in-flight production target is quite large; in fact, additional experimental tools have to be applied to achieve a high-purity spatial separation for the ions of interest or to perform high-resolution secondary reaction studies with exotic nuclei. One solution to this problem is to individually track each particle with transmission detectors placed in the separator sections. This data allows the magnetic rigidity to be determined on an event-by-event basis by the position and angle measurements at different

Table 6 Representative ISOL facilities and some general characteristics Facility TRIUMF CERN-ISOLDE JYFL-IGISOL GANIL-SPIRAL1

Primary beam energy p, 500 MeV p, 1400 MeV p,30; HI,k130 HI(50–95 MeV/u)

Intensity (pμA) 100 2 20 0.2

Target thickness U,Ta 300 g/cm2 U,Ta 300 g/cm2 U, 10 mg/cm2 C, 25×0.7 mm

Extracted energy 60 keV 60 keV 30 keV 37 keV

Post accelerator ISAC-II HIE-ISOLDE none CIME

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dispersive focal planes. When combined with the time-of-flight between detectors, the velocity of each particle can be determined. Such tracking measurements are often performed with exotic nuclei, particularly in secondary reaction studies with spectrometers or other tracking systems; see, for example, the SPEG (Bianchi et al. 1989) and the S800 (Bazin et al. 2003) spectrometers. However, using singleparticle detectors places a strong limit on the particle rate, particularly if a large number of “uninteresting” nuclei are mixed with the exotic nuclei. For example, a typical high detector rate of ≈106 /s of all products emerging from the production target would imply that exotic nuclei would be present in a tiny fraction of the recorded data. Thus, particle tracking is only feasible in later stages of the separator system after the abundant contaminants have been removed from the beam. A similar situation arises for another class of experiments in which the phase space of the primary beam is too large for high-resolution reaction experiments. Using tracking detectors to measure the large phase space of the primary beam is even more difficult than measuring that of the reaction products due to the even higher data rate. An elegant solution is to use an ion-optical system that provides a spatial resolving power independent of the momentum and angular spread of the incident primary beam or beam of reaction products at the final focal plane. In such highresolution reaction experiments, the relatively large phase space of a primary beam is eliminated by coupling two dispersive ion-optical systems with the reaction target placed at the intermediate focal plane. These ion-optical devices consist of an Analyzer section and a Spectrometer section and are generally called dispersionmatched systems; see Fig. 14. Usually the two dispersive sections have the same resolving power. The lateral and angular matching conditions can be written in terms of the matrix elements for the dispersion terms of the Analyzer Aik and the Spectrometer Sik as: S11 A16 + S12 A26 + S16 = 0

(22)

S21 A16 + S22 A26 + S26 = 0

(23)

Usually, image conditions are required for the Analyzer at F1 and for the Spectrometer section at F2, i.e., S12 = A12 = 0, which reduces the conditions to the well-known achromatic requirements. Of course, if the kinematic factors or the relative momentum spread changes in the interaction zone, additional conditions have to be included in the optical solution (Geissel et al. 2013). Spectrometer systems that meet all of these conditions are often called fully dispersion matched, and the properties of the matrix elements indicate that the complete system is also achromatic. Achromatism requires that the momentum and angular dispersion coefficients of the overall system are zero at the final focal plane. Such dispersion-matched systems are very versatile and are utilized at many accelerator laboratories around the world, for example, the Grand Raiden spectrometer (Wakasa et al. 2002) and others (Geissel et al. 2013; Takeda et al. 2020; Winfield et al. 2021). Often the Analyzer and Spectrometer sections consist of several dispersive systems which are added in resolving power. High-resolution

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Fig. 14 Results of ion-optical calculations of an achromatic system created by two symmetric dipole magnet stages consisting of two 40◦ dipole magnets bending in the same direction (C-shape) with focusing multipoles before and after the dipoles. Upper panel: The beam envelopes corresponding to an emittance of 20 π mm mrad in x- and y-direction with no incident momentum spread. The dispersion line for a momentum deviation of +1% is also shown in red. Lower panels: Simulated spatial resolution demonstrating the principle of a dispersion-matched system. In this case, the incident beam had a ±1% spread in magnetic rigidity, and the overall goal is to measure a change in Bρ at F1 that is one order of magnitude smaller (0.1%). The momentum and position distributions are shown at the entrance, the intermediate, and the final focal plane

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experiments can then be performed with large-emittance fragment beams or with primary beams when large energy-angular spreads, due to the accelerator performance, would limit high-resolution reaction studies. It is also important to note that dispersion-matched systems play a fundamental role in the production and delivery of exotic beams from projectile fragmentation reactions in high-energy in-flight separators. The large phase space of the reaction products emerging from the target is dispersed at a central focus for the application of a correlated momentum (or energy) shift that facilitates separation of the exotic nuclei. This application is discussed below. An achromatic system has also favorable properties for time-of-flight (TOF) detectors used to directly measure the ion velocity since the position and angular matrix elements for the path length, R51 and R52 , are tiny compared to the corresponding matrix elements in a dispersive system. The TOF detectors are best placed at the entrance and exit of the ion-optical device. Indeed, when the precise velocity measurement for particle identification is crucial, one can even apply the condition that R56 = 0, meaning that particles with different velocities have the same path length (isodrom) between the start and stop detectors. In this mode, the dispersion line is very different from that in the achromatic system, i.e., it has a zero crossing in the second dipole magnet to achieve the R56 = 0 condition, and thus only momentum compaction (R16 = 0) can be achieved in the example system, instead of achromatism at the focal plane F2. A powerful in-flight separator, the Super Separator Spectrometer (S3) (Déchery et al. 2016), is presently under construction at GANIL to handle very high-intensity primary beams from protons to heavy ions (up to 1014 ions/s). Its primary task is to separate in-flight fusion and multinucleon transfer products at the driplines and beyond the presently known isotopes and elements. S3 consists of two major separator types, a momentum achromat followed by an m/q separator. The momentum achromat is ion-optically similar to the dispersion-matched magnetic system described above, i.e., it consists of two highly symmetric dipole stages with multipole lenses. The dipole magnets bend in the same direction (C-shape configuration) with an intermediate focal plane characterized by image conditions (R12 = 0, (R34 = 0) and a parallel dispersion line (R26 = 0). The subsequent m/q separator also consists of symmetric ion-optical cells arranged in the same way, but the first dispersive element is an electrostatic dipole field matched by a subsequent magnetic dipole stage. The principle of this m/q separator can be described by Equation 6. The operating domain of S3 is given by Bρmax =1.8 Tm, Eρmax = 12 MV, with angular and Bρ acceptances of ±50 mrad and ±7%, respectively. A recent overview of the experimental techniques and existing vacuum recoil separators has been given by Ruiz et al. (2014). Some examples include the longrunning Separator of Heavy Ion reaction Products (SHIP) at GSI (Münzenberg et al. 1979) and Recoil Mass Analyzer (RMA) at Argonne National Lab (Davids et al. 1992), the related Electromagnetic Mass Analyzer (EMMA) at TRIUMF (Davids et al. 2019), and the recent SEparator for CApture Reactions (SECAR) at Michigan

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State University based on designs optimized for very low-energy nuclear reactions (Berg et al. 2018).

Longitudinally Dispersive Systems In the previous section, the spatial separation of ions with laterally dispersive electromagnetic systems was described. Another category of separation is based on longitudinally dispersive devices, where high-resolution time-of-flight (TOF) or frequency measurements in isochronous modes provide identification of the exotic isotopes. The isochronous condition requires that the transit or revolution time through the spectrometer-separator, for an ion with a given mass-to-charge ratio, is independent of the velocity spread. This condition can be achieved when ions with a higher velocity travel on longer trajectories so that the velocity difference is completely canceled in the flight time and vice versa for slower ions (Wollnik et al. 2019). One of the first isochronous separators for target fragments was operated at LAMPF (Wouters et al. 1987). This time-of-flight recoil separator, called TOFI, consisted of four symmetric dipole magnets each with an 81 degrees bending angle, for a total of 324 degrees. TOFI had a mass resolving power of 2000 and accepted (low mass) target fragments within a solid angle of 2.8 msr and a momentum-tocharge spread of ±2% . The kinetic energies of the accepted target fragments were approximately 2 MeV/u. This mass resolving power was achieved by the condition that the transit time over the full length of 14 m was independent of the entrance position, angle, and velocity with the dispersion in the m/q values of the accepted exotic nuclei. Excellent mass measurements of light exotic nuclei were performed with TOFI (Vieira et al. 1986). Present-day mass measurement devices have to have a resolving power high enough to clearly distinguish between the mass of the ground and low-lying isomeric states that might be on the order of 100 keV or less. The resolving power of TOFI does not meet this requirement and would have to be improved, especially for medium and high-mass target fragments. It is obvious that the resolving power can be increased by having a much longer isochronous flight path. Ion-optically this can be achieved with a system with a closed path (i.e., a ring) that allows multiple turn measurements. The first storage ring used for exotic beams measurements was the Experimental Storage Ring (ESR) (Franzke 1987) at GSI which started to operate a few years after the TOFI spectrometer. The ESR was designed to accumulate, store, and in particular cool all heavy ion beams up to uranium with a maximum magnetic rigidity of 10 Tm. The magnetic lattice of the ESR is shown in Fig. 15. The main beam transport components are dipole, quadrupole, and hexapole magnets arranged in a symmetric ring with a circumference of 108.4 m. The versatile ion-optical system of the ESR can be operated in an isochronous or standard (asynchronous) storage mode. The standard and isochronous modes of a ring such as the ESR differ mainly by the dispersion lines along the magnetic lattice, the operating energies, and the phase-space acceptances (Franzke et al. 2008). In the isochronous mode,

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Fig. 15 The ion-optical layout of the Experimental Storage Ring (ESR), the first storage ring used for exotic nuclei. Upper panel: Top view of the ESR with the injection and extraction channels E1 and E2. The important lattice elements for the isochronous mode are the six dipole magnets and interleaved focusing multipole magnets. Lower panel: Calculated beam envelopes for an emittance of 20 π mm mrad in x- and y-directions. The dispersion line corresponding to a momentum deviation of 0.2 % is also shown in red. In the isochronous measurements, the electron cooler (EC) is not used. The TOF detector has multiple tasks in precision isochronous mass spectrometry; it should be located in the ring where the periodic dispersion has a suitable magnitude to allow a momentum measurement in addition to the standard revolution time of each ion

the dispersion is quite large inside the dipole magnets to achieve the condition that the faster ions are sent on a longer trajectory to match the condition that for a given mass-over-charge ratio the revolution time is independent of the velocity spread. The condition of isochronicity can be explained with a few equations. The revolution time T of an ion with velocity v in a ring with circumference C is given as: T = C/v

(24)

with the differentials, one obtains the relations: ΔT 1 C = ΔC − 2 Δv T vT v T

(25)

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ΔC Δv ΔT = − . T C v

(26)

The relative change of the orbit in the lattice due to a difference in the magnetic rigidity of various ions is given by the so-called momentum compaction factor, αp : αp =

ΔC C ΔBρ Bρ

=

1 (γt )2

,

(27)

where γt is the transition point. The momentum compaction factor can also be written as:  m 

2 ΔT Δf 1 Δ q Δv γ =− = 2  + (28) − 1 2 m T f v γt γt q

Δv 1 ΔBρ = 2 v γ Bρ

(29)

where γ is the relativistic Lorentz factor corresponding to the central Bρ value. Notice that the ratio of m/q is a discreet number since the charge is an integer; therefore, any variation in the time or frequency comes from the second term in Eq. 28. An important goal for accurate mass measurements using the isochronous mode is to cancel the term with the contribution of the velocity spread. This cancellation can be achieved by tuning the ion transition point, γt , to be equal to the Lorentz factor γ of one selected m/q. During an experimental measurement, it is more practicable to set the ring to a fixed γt and change the velocity of the injected ions until the condition γ = γt is reached. Using Eq. 28, the mass resolving power R can be shown to be given by: R=

1 T m = 2 . Δm γt ΔT

(30)

The cancellation goal can thus only be reached at a fixed velocity of the fragment of interest, i.e., the velocity of the stored ions has to exactly match the ion-optical parameter γt , the transition point; see Equation 27. Generally, the value of γt is chosen such that exotic nuclei with quite unusual m/q values (most interesting to study) stay below the maximum Bρ acceptance of the isochronous ring. A good value established for the ESR is γt = 1.41. The calculated envelopes and dispersion line for this case are shown in the lower panel of Fig. 15. In the standard storage mode, the transition point is in the range of γt = 2.6 − 2.9. The ESR facility is also equipped with an electron cooler to reduce the relative momenta of the circulating ions and with RF cavities to vary the kinetic energy of the stored ions over the large range of 3 to 800 MeV/u (Franzke 1987). The time needed

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for electron cooling of the ions is presently a few seconds which is too long for many nuclei to survive radioactive decay, especially near the driplines. Isochronous spectrometry does not need cooling and is only limited by the revolution time and the required number of turns. For the presented γt , the time for a single revolution in the ESR is 509 ns. Thus, an advantage of using the isochronous mode is that very short-lived exotic nuclei can be investigated (Franzke et al. 2008) since precise mass measurements can be performed with a few hundred turns, thus allowing studies of nuclei with sub-millisecond lifetimes. However, one should note that important measurements have been made with single ions orbiting in the ESR (Litvinov et al. 2004). The isochronous condition γ = γt is manifested in an isochronous curve which has been measured for stored ions at different Bρ values. Experimentally, the verification of the isochronicity was performed with the primary beam at different velocities. The velocity change was done via the terminal voltage of the electron cooler. In principle, one could have done this procedure also with the injected beam at different incident velocities provided directly by the accelerator, but for this case, a tedious and time-consuming scaling of the complete ion-optical system between the accelerator and the ESR would have had to be done many times. The first experiments applying the isochronous mode (Hausmann et al. 2000) of the ESR achieved a mass resolving power of 150,000. This resolving power being two orders higher than that of TOFI indicates that any isomeric states can be easier identified and separated in the ESR. The plateau of the isochronicity can be improved by further tuning of the chromaticity of the ESR as shown in Fig. 16 where the relative difference in frequency is shown as a function of relative momentum. Multi-turn isochronous measurements have also been carried out at storage ring facilities in China (Wang et al. 2009) and in Japan (Ozawa et al. 2012). In general, new mass determinations are performed with relative measurements using reference ions, characterized by precisely known masses, moving simultaneously in the same electromagnetic fields. Practically this means that all ion species circulate in the storage ring with the same magnetic rigidity, i.e., the velocity is different for ions with different m/q values. Only in the special case of cooled stored ions are their mean velocities identical. Therefore, precise isochronous mass measurements for non-cooled ions with different m/q values require measurement of the revolution time and an additional, independent velocity or more generally a Bρ measurement (Geissel et al. 2006; Chen et al. 2018). These additional measurements can be carried out in several ways, for example, two time-of-flight detectors have been installed in a straight section of the storage ring in Lanzhou, and a similar set is also planned for the future FAIR Collector RING (CR) (Geissel and Litvinov 2005). Another approach to achieve the same goal is to make an in-ring position measurement relying on the lateral periodic dispersion of the closed orbits. The beam at the position of the TOF detector in the ESR has a well-suited periodic lateral dispersion; see Fig. 15. The ion-optical description of a beam in a storage ring is quite similar to the description of the transport through a separator-spectrometer system, but in a ring, a different boundary condition has to be fulfilled in order to have stable revolutions.

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Fig. 16 Calculated change of the revolution frequencies as a function of the relative momentum, dp/p, for the isochronous and standard storage modes of the ESR. The momentum of the stored ions was varied, and the corresponding deviation in the revolution frequency f was calculated with the ion-optical code MIRKO (Franczak et al. 2022). The three isochronous curves correspond to different settings of the hexapole fields to optimize the chromaticity correction. Note that the deviation of the revolution frequency is roughly a factor of 100 smaller than the change of the momentum in the isochronous mode. The corresponding dramatic increase in frequency in the standard storage mode (STD) is shown for comparison

In principle, the stored particles should be able to orbit an infinite number of turns in the periodic lattice structure. The equation of motion in a cyclic ring is similar to the differential equations, Eqs. 9 and 10, but now with the addition of periodicity: k(s) = k(s + C),

(31)

where C represents a full turn for a closed orbit. A necessary condition for stable revolutions is that the trace of the transformation matrix for a full circumference is smaller than 2 (the stability criterion). The closed periodic dispersion functions at any point s, D(s) and D’(s), have the following relation to the transport matrices that were introduced above for non-periodic optical systems: D(s) =

R12 (s)R26 (s) + (1 − R22 (s))R16 (s) (1 − R11 (s))(1 − R22 (s)) − R12 (s)R21 (s)

(32)

D  (s) =

R21 (s)R16 (s) + (1 − R11 (s))R26 (s) (1 − R11 (s))(1 − R22 (s)) − R12 (s)R21 (s)

(33)

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A special case occurs when the periodic angular dispersion D’(s) = 0, as it is the case for the position of the TOF detector inside the ESR; see Fig. 15. In that case is: D(s) =

R16 (s) (1 − R11 )

(34)

So that a position measurement with the TOF detector, installed at a position with a suitable D(s) value, can provide a Bρ measurement in addition to the revolution time measurement. The resolution of the tracking measurements, when stored ions pass through the secondary electron emitting detector foil to create the TOF signal, can be less than 1 mm. Averaging of the measured position at each turn and using the periodic betatron functions (Wiedemann 2015), one can obtain a Bρ resolving power of better than 10−4 . In summary, a secondary electron TOF detector as described in reference Kuzminchuk-Feuerstein et al. (2016) can provide three experimental observables, timing, Bρ measurement, and yield of secondary electrons for each circulating ion. The present TOF detector in the ESR is well suited to this challenge since one can optimize one branch of the output signal for timing and a second for precise position measurement; see Fig. 17. It has been experimentally proven that the yield of secondary electrons is proportional to the stopping power over many orders of the ion velocity (Rothard et al. 1990). Thus, the electron yield is characteristic of the atomic number of the circulating ion passing a thin-foil TOF detector. Another promising experimental direction to meet this challenge is the measurement of the intensity and position of a circulating ion with a resonant cavity (Sanjari et al. 2015). A relatively easy Bρ measurement in addition to the timing in an isochronous storage ring can be performed if singleparticle identification with high resolution is possible before the ions are injected. For example, this has been done with the combination of the in-flight separator BigRIPS (Kubo 2003) and the R3 ring at RIKEN-RIBF (Ozawa et al. 2012). Storage rings coupled to synchrotrons normally rely on fast bunch extraction time structures that do not allow the operation with coincident single-particle detectors due to the high instantaneous rate. Most present and planned storage rings have been located at exotic beam facilities with relatively high kinetic energies. Transverse dispersive separators generally provide spatial pre-separation of the exotic ions before they are injected into the storage ring. An overview of the characteristic properties of heavy ion storage rings (Steck and Litvinov 2020) is given in Table 7. Another longitudinally dispersive system based on isochronicity is the multiplereflection time-of-flight mass spectrometer (MR-TOF-MS). This device is designed to operate with thermalized reaction products. The exotic ions are extracted as thermalized ions from a buffer gas as described above at very low kinetic energies and then cooled, bunched, and injected into an electrostatic mirror system that is the heart of the MR-TOF-MS. The bulk of the resolving power is achieved by isochronicity in very long flight paths. Of course, the accepted emittance also contributes via aberrations to the possible resolving power. The long flight paths are achieved by having the ions traverse several hundred up to a few thousand reflections

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25 20 15

x[mm]

10 5 0 -5 -10 -15 0

50

100

150

200

250

300

Turns

Fig. 17 Simultaneous revolution time and magnetic rigidity measurements are possible at the present ESR due to the periodic dispersion (x-direction) at the position of the TOF detector. The two channel-plate branches are ideally suited to measure both observables at each turn. The simulated periodic position measurements are shown on the right-hand side for 2 particles separated by ΔBρ = 10−3 during 300 turns. The slope of the periodic structure is due to the energy loss in a 30 μg/cm2 carbon foil. The shapes of the peaks are determined by the betatron functions from which the Bρ-value can be directly deduced with an accuracy of better than 10−4

Table 7 Properties of storage rings operating with exotic nuclei Storage ring ESR CSRe R3 CRYRING

Circumference m 108.36 128.80 60.35 54.17

Bρmax Tm 10 9.4 4.05 1.44

γt (isochronous) 1.39 − 1.4 1.4 1.22 −

Acceptance % ±0.3 ±0.25 ±0.5 ±0.5

Cooling MeV/u 4−430 25−400 − 0.3 − 4

Institute GSI IMP Lanzhou RIKEN GSI-FAIR

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in a drift length of about 1 m. The kinetic energy of the stored ions traveling between the electrostatic mirrors is typically 1 keV. A schematic setup of an MR-TOF-MS is shown in the upper panel of Fig. 18. In addition, the measured and calculated mass resolving powers are given for typical numbers of reflections and the corresponding flight times (Ayet et al. 2019). The mass resolving power depends critically on the turnaround time and the time-of-flight differences in each turn due to aberrations ΔTa . The mass resolving power for a multiple-pass TOF-MS is given by: R=

T0 m Nturn + Ta = , Δm 0 2 2 ( NΔT ) + (ΔT ) a turn

(35)

where T0 is the time of flight from the start position directly to the detector without using mirrors, Ta is the time of flight for a single turn in the mirror system, and Nturn is the number of passes in the mirror system. One can also provide isotopically pure nuclear beams with longitudinally dispersive systems. For example, the time distribution of ions emerging from an MR-TOF-MS can be used to make a spatial separation by using an ion deflector (Bradbury–Nielsen gate) instead of the TOF detector (Dickel et al. 2015a). A pioneering experiment clearly demonstrated this technique when 211 Po ions were injected into the high-resolution MR-TOF-MS at the FRS (Dickel et al. 2015b) and the isomeric-to-ground state ratio of the ions was first determined from the measured mass spectrum. In a second step, the isomers were spatially separated from the ions in the ground state by an ion deflector and then implanted on a silicon detector for decay spectroscopy. This successful experiment indicates that unique isomer-resolved studies are now possible with this device. Furthermore, such an MR-TOF-MS can be used for the unambiguous particle identification with respect to the proton number (Z) and mass number (A) at extremely low kinetic energies (by exact mass determination) in cases where it was not possible before, for example, very rare isotopes produced via fusion reactions such as 100 Sn hidden under huge isobaric contamination or the heaviest elements where the alpha decay chains do not end in known long-lived isotopes but rather in fission. Note that the flight time of a very low-energy ion traveling in the MR-TOFMS can be perturbed by the positive space charge when a large number of ions is in the device. Maintaining the effectiveness for studying rare, exotic isotopes with both storage rings and MR-TOF-MS’s can be significantly enhanced when a laterally dispersive separator is used to deliver the ions (Geissel et al. 1992b; Wang et al. 2009; Ozawa et al. 2012). The first separator can remove the abundant, uninteresting, potentially perturbative, contaminant ions and thus maintain the performance characteristics of the high-resolution longitudinally dispersive device. However, the important idea here is that there often needs to be some pre-separation, and a broad-band MR-TOF-MS with a relatively low resolving power could also serve as the pre-separator stage (Jesch et al. 2017) for a subsequent high-resolution Penning trap system (Dilling et al. 2003; Reiter et al. 2020). Another example of a combined system is an ISOL production with a lateral magnetic separator

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Fig. 18 Upper panel: Schematic layout of an MR-TOF-MS. The ion bunches are injected from the left into the electrostatic mirror system and after a selected number of reflections are extracted and strike the TOF detector. Lower panel: Measured (full squares) and calculated (full, dashed lines) mass resolving power (FWHM) as a function of the number of turns or flight times (Ayet et al. 2019). The calculations are based on the dependence given in equation 35. The kinetic energy of the ions was 1.3 keV in the mirror system and the repetition rate of the injected bunches 50 Hz. Thus, this repetition frequency limits the maximum flight time to 20 ms, nonetheless, the maximum measured mass resolving power reached 620,000

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combined with an MR-TOF-MS. Such systems now exist at TRIUMF (Dickel et al. 2019) and ISOLDE (Wolf et al. 2013) that can completely purify a beam of target fragmentation products even from a strong isobaric background. The combination of conventional spectrometers and separators with MR-TOF-MS has opened up new fields, especially also for low-energy nuclear physics experiments where in the past a complete and universal particle identification was not possible.

Combination of Laterally and Longitudinally Dispersive Systems Very often the main goal of a high-resolution measurement requires studies of single or of a few exotic nuclei that require enormous decontamination factors. In order to reach such high factors, a laterally dispersive spectrometer is used as a pre-separator, and the longitudinally dispersive system performs the final highresolution measurement itself or provides a highly purified mono-isotopic beam for a subsequent measurement. Very recently, almost each low-energy nuclear accelerator laboratory has installed the versatile MR-ToF-MS as a longitudinally dispersive separator. At high incident energies, the combination of a projectile fragment separator with a storage-cooler ring has opened up a new potential for research with bare and few electron ions. This combination provides access to precision experiments that can be performed with ions that even have sub-millisecond half-lives. A pioneering example was the combination of the FRS fragment separator and the ESR storagecooler ring at GSI (Geissel et al. 1992b). A similar system was installed a few years later at the MPI Lanzhou (Ma 1998). Both combined facilities have substantially contributed to the knowledge of ground-state masses and also to the discovery and investigation of bound-state β decay (Ohtsubo et al. 2002). The ion-optical coupling of a fragment separator to a storage ring represents a strong experimental challenge because it is not sufficient just to match the geometrical acceptance of the ring but it is necessary to match the complete phase space in order to achieve stable orbits in the ring without losses. Based on the experience with the FRS-ESR combination, the new combined system of the SUPER-FRS and the Collector Ring (CR) has been designed and is under construction at FAIR. Special efforts have made to have more space for experiments and a dedicated intermediate beam line to perform the phase-space matching; see Fig. 19. Figure 19 shows the calculated results for the complete phase-space matching of the separator and the isochronous collector ring (CR). The criterion for a matched phase-space optics is that stable closed orbits are achieved in the ring for an arbitrarily high number of turns. The matching is done with three achromatic units, i.e., the super-FRS ring branch, the injection beam line, and the CR all tuned to be independent achromatic systems. This overall achromatic condition is particularly important when beam energy degraders are used at MF2. The use of internal degraders in achromatic separators is discussed in the next section. In this example, the ring parameters have been chosen to be Qh = 2.1, Qv = 4.2 and γt = 1.67.

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Fig. 19 Ion-optical example of the combination of a laterally and a longitudinally dispersive ion-optical facility that indicates the importance of an intermediate matching section. Here the optics for the coupling of the main separator of the super-FRS and the collector ring are shown (Winfield et al. 2021). The matching section has to perform the complete phase-space matching to the isochronous operating mode of CR. The dispersion line, in red, is given for a momentum deviation of +0.5%. The transverse emittance is 200 π mm mrad in the x− and the y-directions

The inevitable high-order aberrations cause losses of less than 10% which occur after a few tens of turns in the simulation. In the isochronous mode, the dispersion is much larger inside the CR lattice compared to that in the storage mode. This fact is immediately visible by the dispersion line for ions with a momentum deviation of 0.5% in Fig. 19. The super-FRS and the isochronous CR have a much larger overall acceptance than the present FRS-ESR system. Therefore, mass measurements of very shortlived exotic nuclei will be uniquely possible with the isochronous CR. The revolution time of the stored ions will be measured with two time-of-flight detectors in coincidence with an accuracy of approximately 50 to 100 keV. It should be obtained based on the experience with the FRS-ESR combination (Knöbel et al. 2016). Besides the additional velocity measurements in the new ring, a Bρ measurement is also possible, as explained in the previous section. Of course, lifetime and isomer data can be simultaneously recorded in the same experimental campaigns.

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Summary At this point one can see that a very large range of production reactions and separation techniques are available that rely solely on the propagation of the exotic nuclei through electromagnetic systems. In the next chapter of this work, the modification of the beam properties by atomic interactions is presented, and their importance for separation of fast ion beams is demonstrated. Acknowledgments The authors would like to thank T. Kubo and I. Tanihata for a thorough review of this manuscript and a fruitful long-term collaboration in this field. The authors have benefited from additional long-standing collaborations on many of the challenges and new developments for separating exotic nuclei with M. Huyse, B. Jonson, G. Münzenberg, J.A. Nolen, Jr., C. Scheidenberger, B.M. Sherrill, and P. Van Duppen and have also benefited from extensive collaborations with the exotic nuclei groups at GSI-FAIR, JINR, GANIL, IMP/HIAF, ISOLDE, MSU/NSCL, RIBF, and TRIUMF.

References P. Armbruster, J. Eidens, J. Grüter, H. Lawin, E. Roeckl, K. Sistemich, Are gas-filled magnetic separators a useful tool to investigate heavy fast recoils from nuclear reactions? Nucl. Instrum. Methods 91, 499–507 (1971) F. Aston, Isotopes and atomic weights. Nature 105(2646), 617–619 (1920) G. Audi, M. Wang, A. Wapstra, F. Kondev, M. MacCormick, X. Xu, The 2012 atomic mass evaluation and the mass tables. Nucl. Data Sheets 120, 1–5 (2014) M. Avilov, A. Aaron, A. Amroussia, W. Bergez, C. Boehlert, T. Burgess, A. Carroll, C. Colin, F. Durantel, P. Ferrante, T. Fourmeau, V. Graves, C. Grygiel, J. Kramer, W. Mittig, I. Monnet, H. Patel, F. Pellemoine, R. Ronningen, M. Schein, Thermal, mechanical and fluid flow aspects of the high power beam dump for FRIB. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 376, 24–27 (2016). Proceedings of the XVIIth International Conference on Electromagnetic Isotope Separators and Related Topics (EMIS2015), Grand Rapids, 11–15 May 2015 S.S.A. Ayet, C. Hornung, J. Ebert, W.R. Plaß, T. Dickel, H. Geissel, C. Scheidenberger, J. Bergmann, F. Greiner, E. Haettner et al., High-resolution, accurate multiple-reflection timeof-flight mass spectrometry for short-lived, exotic nuclei of a few events in their ground and low-lying isomeric states. Phys. Rev. C 99(6), 064313 (2019) J. Äystö, Development and applications of the IGISOL technique. Nucl. Phys. A 693(1), 477–494 (2001). Radioactive Nuclear Beams J. Äystö, T. Eronen, A. Jokinen, A. Kankainen, I.D. Moore, H. Penttilä (eds.), Three Decades of Research Using IGISOL Technique at the University of Jyväskylä (Springer Nature, Switzerland, 2014) H. Backe, W. Lauth, M. Block, M. Laatiaoui, Prospects for laser spectroscopy, ion chemistry and mobility measurements of superheavy elements in buffer-gas traps. Nucl. Phys. A 944, 492–517 (2015). Special Issue on Superheavy Elements D. Bazin, J. Caggiano, B. Sherrill, J. Yurkon, A. Zeller, The s800 spectrograph. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 629–633 (2003). 14th International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications G. Berg, M. Couder, M. Moran, K. Smith, M. Wiescher, H. Schatz, U. Hager, C. Wrede, F. Montes, G. Perdikakis, X. Wu, A. Zeller, M. Smith, D. Bardayan, K. Chipps, S. Pain, J. Blackmon, U. Greife, K. Rehm, R. Janssens, Design of secar a recoil mass separator for astrophysical

54

H. Geissel and D. J. Morrissey

capture reactions with radioactive beams. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 877, 87–103 (2018) L. Bianchi, B. Fernandez, J. Gastebois, A. Gillibert, W. Mittig, J. Barrette, SPEG: an energy loss spectrometer for GANIL. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 276(3), 509–520 (1989) B. Blank, M. Chartier, S. Czajkowski, J. Giovinazzo, M.S. Pravikoff, J.-C. Thomas, G. de France, F. de Oliveira Santos, M. Lewitowicz, C. Borcea, R. Grzywacz, Z. Janas, M. Pfützner, Discovery of doubly magic 48 Ni. Phys. Rev. Lett. 84, 1116–1119 (2000) G. Bollen, Ion surfing with radiofrequency carpets. Int. J. Mass Spectrom. 299(2), 131–138 (2011) G. Bollen, D. Morrissey, S. Schwarz, A study of gas-stopping of intense energetic rare isotope beams. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 550(1–2), 27–38 (2005) G. Bollen, C. Campbell, S. Chouhan, C. Guénaut, D. Lawton, F. Marti, D. Morrissey, J. Ottarson, G. Pang, S. Schwarz, A. Zeller, P. Zavodszky, Manipulation of rare isotope beams – from high to low energies. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 266(19), 4442–4448 (2008). Proceedings of the XVth International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications P. Bricault, M. Dombsky, P. Schmor, G. Stanford, Radioactive ion beams facility at triumf. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 126(1–4), 231–235 (1997) M. Brodeur, A. Gehring, G. Bollen, S. Schwarz, D. Morrissey, Experimental investigation of the ion surfing transport method. Int. J. Mass Spectrom. 336, 53–60 (2013) K.L. Brown, F. Rothacker, D.C. Carey, C. Iselin, Transport-a computer program for designing charged particle beam transport systems. Technical Report CERN–80-04, European Organization for Nuclear Research (1980) P. Campbell, I. Moore, M. Pearson, Laser spectroscopy for nuclear structure physics. Prog. Part. Nucl. Phys. 86, 127–180 (2016) D.C. Carey, The Optics of Charged Particles Beams Harwood Academic Pub., New York, NY (1986) R. Chen, X. Yan, W. Ge, Y. Yuan, M. Wang, M. Sun, Y. Xing, P. Zhang, C. Fu, P. Shuai et al., A method to measure the transition energy γ t of the isochronously tuned storage ring. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 898, 111–116 (2018) H. Choi, Y. Hirayama, S. Choi, T. Hashimoto, S.C. Jeong, H. Miyatake, J.Y. Moon, M. Mukai, T. Niwase, M. Oyaizu, M. Rosenbusch, P. Schury, A. Taniguchi, Y.X. Watanabe, M. Wada, In-gas-cell laser ionization spectroscopy of 194,196 Os isotopes by using a multireflection timeof-flight mass spectrograph. Phys. Rev. C 102, 034309 (2020) K. Cooper, C.S. Sumithrarachchi, D.J. Morrissey, A. Levand, J.A. Rodriguez, G. Savard, S. Schwarz, B. Zabransky, Extraction of thermalized projectile fragments from a large volume gas cell. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 763, 543–546 (2014) C.N. Davids, D. Peterson, A compact high-resolution isobar separator for the caribu project. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 266(19), 4449–4453 (2008). Proceedings of the XVth International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications C. Davids, B. Back, K. Bindra, D. Henderson, W. Kutschera, T. Lauritsen, Y. Nagame, P. Sugathan, A. Ramayya, W. Walters, Startup of the fragment mass analyzer at atlas. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1), 358–365 (1992) B. Davids, M. Williams, N. Esker, M. Alcorta, D. Connolly, B. Fulton, K. Hudson, N. Khan, O. Kirsebom, J. Lighthall, P. Machule, Initial operation of the recoil mass spectrometer EMMA at the ISAC-II facility of triumf. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 930, 191–195 (2019) F. Déchery, H. Savajols, M. Authier, A. Drouart, J. Nolen, D. Ackermann, A. Amthor, B. Bastin, A. Berryhill, D. Boutin et al., The super separator spectrometer s3 and the associated detection

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

55

systems: sirius & leb-reglis3. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 376, 125–130 (2016) A.R. Dicker, P. Campbell, B. Cheal, D. Gorelov, J. Hakala, D.J.S. Johnson, A. Jokinen, V. Kolhinen, J. Koponen, I.D. Moore, H. Penttilä, I. Pohjalainen, M. Reponen, S. Rinta-Antila, V. Sonnenschein, A. Voss, Laser spectroscopy at IGISOL IV. Hyperf. Interact. 227(1), 139–145 (2014) T. Dickel, W. Plaß, S. Ayet San Andres, J. Ebert, H. Geissel, E. Haettner, C. Hornung, I. Miskun, S. Pietri, S. Purushothaman, M. Reiter, A.-K. Rink, C. Scheidenberger, H. Weick, P. Dendooven, M. Diwisch, F. Greiner, F. Heiße, R. Knöbel, W. Lippert, I. Moore, I. Pohjalainen, A. Prochazka, M. Ranjan, M. Takechi, J. Winfield, X. Xu, First spatial separation of a heavy ion isomeric beam with a multiple-reflection time-of-flight mass spectrometer. Phys. Lett. B 744, 137–141 (2015a) T. Dickel, W. Plaß, A. Becker, U. Czok, H. Geissel, E. Haettner, C. Jesch, W. Kinsel, M. Petrick, C. Scheidenberger, A. Simon, M. Yavor, A high-performance multiple-reflection time-of-flight mass spectrometer and isobar separator for the research with exotic nuclei. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 777, 172–188 (2015b) T. Dickel, W. Plaß, H. Geissel, F. Heiße, I. Miskun, S. Purushothman, M. Reiter, A.-K. Rink, C. Scheidenberger, Conceptional design of a novel next-generation cryogenic stopping cell for the low-energy branch of the super-FRS. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 376, 216–220 (2016). Proceedings of the XVIIth International Conference on Electromagnetic Isotope Separators and Related Topics (EMIS2015), Grand Rapids, 11–15 May 2015 T. Dickel, S.A. San Andrés, S. Beck, J. Bergmann, J. Dilling, F. Greiner, C. Hornung, J. rew, G. Kripko-Koncz, A. Kwiatkowski et al., Recent upgrades of the multiple-reflection time-offlight mass spectrometer at TITAN, TRIUMF. Hyperf. Interact. 240(1), 1–9 (2019) J. Dilling, D. Ackermann, J. Bernard, F. Hessberger, S. Hofmann, W. Hornung, H.J. Kluge, E. Lamour, M. Maier, R. Mann, G. Marx, R. Moore, G. Münzenberg, W. Quint, D. Rodriguez, M. Schädel, J. Schönfelder, G. Sikler, C. Toader, L. Vermeeren, C. Weber, G. Bollen, O. Engels, D. Habs, P. Thirolf, H. Backe, A. Dretzke, W. Lauth, W. Ludolphs, M. Sewtz, The shiptrap project: a capture and storage facility at GSI for heavy radionuclides from ship. Hyperf. Interact. 127(1–4), 491–496 (2000) J.E.A. Dilling, P. Bricault, M. Smith, H.-J. Kluge, T. Collaboration et al., The proposed TITAN facility at ISAC for very precise mass measurements on highly charged short-lived isotopes. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 492–496 (2003) J. Dufour, R. Del Moral, H. Emmermann, F. Hubert, D. Jean, C. Poinot, M. Pravikoff, A. Fleury, H. Delagrange, K.-H. Schmidt, Projectile fragments isotopic separation: application to the lise spectrometer at GANIL. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 248(2–3), 267–281 (1986) P.V. Duppen, New results from advances in ISOL techniques. AIP Conf. Proc. 455(1), 919–926 (1998) C. Engelmann, F. Ameil, P. Armbruster, M. Bernas, S. Czajkowski, P. Dessagne, C. Donzaud, H. Geissel, A. Heinz, Z. Janas, C. Kozhuharov, C. Miehé, G. Münzenberg, M. Pfützner, C. Röhl, W. Schwab, C. Stéphan, K. Sümmerer, L. Tassan-Got, B. Voss, Production and identification of heavy ni isotopes: evidence for the doubly magic nucleus 78-ni. Z. Phys. A Hadrons Nuclei 352(4), 351–352 (1995) B. Franczak, Mirko – an interactive program for beam lines and synchrotrons. In: W. Busse, and R. Zelazny, (ed.) Lecture Notes in Physics. Springer-Verlag, Berlin Heidelberg, 215, 170–175, (1984) B. Franczak, H. Geissel, E. Haettner, Y. Tanaka, Mirko – an interactive program for heavy ion accelerator-based experiments, in Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms (2022), To be Published B. Franzke, The heavy ion storage and cooler ring project ESR at GSI. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 24–25, 18–25 (1987) B. Franzke, H. Geissel, G. Münzenberg, Mass and lifetime measurements of exotic nuclei in storage rings. Mass Spectrom. Rev. 27(5), 428–469 (2008)

56

H. Geissel and D. J. Morrissey

C.B. Fulmer, B.L. Cohen, Equilibrium charges of fission fragments in gases. Phys. Rev. 109, 94–99 (1958) H. Geissel, Y.A. Litvinov, Precision experiments with relativistic exotic nuclei at GSI. J. Phys. G: Nucl. Part. Phys. 31(10), S1779 (2005) H. Geissel, T. Schwab, P. Armbruster, J.P. Dufour, E. Hanelt, K.-H. Schmidt, B.M. Sherrill, G. Münzenberg, Ions penetrating through ion-optical systems and matter – non-liouvillian phase-space modelling. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 282(1), 247–260 (1989) H. Geissel, P. Armbruster, K.H. Behr, A. Brünle, K. Burkard, M. Chen, H. Folger, B. Franczak, H. Keller, O. Klepper, B. Langenbeck, F. Nickel, E. Pfeng, M. Pfützner, E. Roeckl, K. Rykaczewski, I. Schall, D. Schardt, C. Scheidenberger, K.-H. Schmidt, A. Schröter, T. Schwab, K. Sümmerer, M. Weber, G. Münzenberg, T. Brohm, H.-G. Clerc, M. Fauerbach, J.J. Gaimard, A. Grewe, E. Hanelt, B. Knödler, M. Steiner, B. Voss, J. Weckenmann, C. Ziegler, A. Magel, H. Wollnik, J.P. Dufour, Y. Fujita, D.J. Vieira, B.M. Sherrill, The GSI projectile fragment separator (FRS): a versatile magnetic system for relativistic heavy ions. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1), 286–297 (1992a) H. Geissel, K. Beckert, F. Bosch, H. Eickhoff, B. Franczak, B. Franzke, M. Jung, O. Klepper, R. Moshammer, G. Münzenberg, F. Nickel, F. Nolden, U. Schaaf, C. Scheidenberger, P. Spädtke, M. Steck, K. Sümmerer, A. Magel, First storage and cooling of secondary heavy-ion beams at relativistic energies. Phys. Rev. Lett. 68, 3412–3415 (1992b) H. Geissel, G. Münzenberg, K. Riisager, Secondary exotic nuclear beams. Ann. Rev. Nucl. Part. Sci. 45(1), 163–203 (1995) H. Geissel, R. Knoebel, Y. Litvnov, A new experimental approach for isochronous mass measurements of short-lived exotic nucleiwith the FRS-ESR facility. Hyperf. Interact. 173(1), 205–210 (2006) H. Geissel, J. Winfield, G. Berg, B. Franczak, N. Iwasa, G. Münzenberg, C. Nociforo, W. Plaß, C. Scheidenberger, H. Weick, M. Winkler, M. Yavor, Dispersion-matched spectrometer in the low-energy branch of the super-FRS for high-resolution measurements with large-emittance relativistic fragment beams. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 277–283 (2013). XVIth International Conference on ElectroMagnetic Isotope Separators and Techniques Related to Their Applications, 2–7 Dec 2012 at Matsue M. Hausmann, F. Attallah, K. Beckert, F. Bosch, A. Dolinskiy, H. Eickhoff, M. Falch, B. Franczak, B. Franzke, H. Geissel, T. Kerscher, O. Klepper, H.-J. Kluge, C. Kozhuharov, K. Löbner, G. Münzenberg, F. Nolden, Y. Novikov, T. Radon, H. Schatz, C. Scheidenberger, J. Stadlmann, M. Steck, T. Winkler, H. Wollnik, First isochronous mass spectrometry at the experimental storage ring ESR. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 446(3), 569–580 (2000) M. Hausmann, A.M. Aaron, A.M. Amthor, M. Avilov, L. Bandura, R. Bennett, G. Bollen, T. Borden, T. Burgess, S.S. Chouhan, V.B. Graves, W. Mittig, D.J. Morrissey, F. Pellemoine, M. Portillo, R.M. Ronningen, M. Schein, B.M. Sherrill, A.F. Zeller, Design of the advanced rare isotope separator ARIS at FRIB. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 349–353 (2013). XVIth International Conference on ElectroMagnetic Isotope Separators and Techniques Related to their Applications, 2–7 Dec 2012 at Matsue C.B. Hinke, M. Böhmer, P. Boutachkov, T. Faestermann, H. Geissel, J. Gerl, R. Gernhäuser, M. Górska, A. Gottardo, H. Grawe, J.L. Grebosz, R. Krücken, N. Kurz, Z. Liu, L. Maier, F. Nowacki, S. Pietri, Z. Podolyák, K. Sieja, K. Steiger, K. Straub, H. Weick, H.-J. Wollersheim, P.J. Woods et al., Superallowed gamow–teller decay of the doubly magic nucleus 100-sn. Nature 486(7403), 341–345 (2012) Y. Hirayama, Y.X. Watanabe, M. Mukai, M. Oyaizu, M. Ahmed, H. Ishiyama, S.C. Jeong, Y. Kakiguchi, S. Kimura, J.Y. Moon, J.H. Park, P. Schury, M. Wada, H. Miyatake, Doughnutshaped gas cell for kek isotope separation system. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 412, 11–18 (2017) Y. Hirayama, Y.X. Watanabe, M. Mukai, M. Ahmed, H. Ishiyama, S.C. Jeong, Y. Kakiguchi, S. Kimura, J.Y. Moon, M. Oyaizu, J.H. Park, P. Schury, M. Wada, H. Miyatake, Nuclear

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

57

spectroscopy of r-process nuclei using KEK isotope separation system. J. Phys.: Conf. Ser. 1643, 012138 (2020) P.T. Hosmer, H. Schatz, A. Aprahamian, O. Arndt, R.R.C. Clement, A. Estrade, K.-L. Kratz, S.N. Liddick, P.F. Mantica, W.F. Mueller, F. Montes, A.C. Morton, M. Ouellette, E. Pellegrini, B. Pfeiffer, P. Reeder, P. Santi, M. Steiner, A. Stolz, B.E. Tomlin, W.B. Walters, A. Wöhr, Halflife of the doubly magic r-process nucleus 78 Ni. Phys. Rev. Lett. 94, 112501 (2005) M. Huyse, M. Facina, Y. Kudryavtsev, P. Van Duppen, I. Collaboration et al., Intensity limitations of a gas cell for stopping, storing and guiding of radioactive ions. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 187(4), 535–547 (2002) ISOLDE and Collaboration, CERN-ISOLDE (2021) C. Jesch, T. Dickel, W.R. Plaß, D. Short, S.A. San Andres, J. Dilling, H. Geissel, F. Greiner, J. Lang, K.G. Leach et al., The MR-TOF-MS isobar separator for the TITAN facility at TRIUMF, in TCP 2014 (Springer, 2017), pp. 175–184 B. Jonson, A. Richter, More than three decades of ISOLDE physics. Hyperf. Interact. 129(1), 1–22 (2000) B. Jonson, K. Riisager, The ISOLDE facility. Scholarpedia 5(7), 9742 (2010). revision #90796 D. Kaji, K. Morimoto, N. Sato, A. Yoneda, K. Morita, Gas-filled recoil ion separator GARIS-II. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 311–314 (2013) M. Karny, L. Batist, A. Banu, F. Becker, A. Blazhev, B.A. Brown, W. Brüchle, J. Döring, T. Faestermann, M. Górska, H. Grawe, Z. Janas, A. Jungclaus, M. Kavatsyuk, O. Kavatsyuk, R. Kirchner, M. La Commara, S. Mandal, C. Mazzocchi, K. Miernik, I. Mukha, S. Muralithar, C. Plettner, A. Płochocki, E. Roeckl, M. Romoli, K. Rykaczewski, M. Schädel, K. Schmidt, ˙ R. Schwengner, J. Zylicz, Beta decay of the proton-rich nuclei 102sn and 104sn. Eur. Phys. J. A – Hadrons Nuclei 27(2), 129–136 (2006) I. Katayama, M. Wada, H. Kawakami, J. Tanaka, K. Noda, Cyclotron ion guide for energetic radioactive nuclear ions. Hyperf. Interact. 115(1), 165–170 (1998) J. Khuyagbaatar, V.P. Shevelko, A. Borschevsky, C.E. Düllmann, I.Y. Tolstikhina, A. Yakushev, Average charge states of heavy and superheavy ions passing through a rarified gas: theory and experiment. Phys. Rev. A 88, 042703 (2013) R. Knöbel, M. Diwisch, H. Geissel, Y.A. Litvinov, Z. Patyk, W. Plaß, C. Scheidenberger, B. Sun, H. Weick, F. Bosch et al., New results from isochronous mass measurements of neutron-rich uranium fission fragments with the FRS-ESR-facility at GSI. Eur. Phys. J. A 52(5), 1–11 (2016) U. Köster, ISOLDE target and ion source chemistry. Radiochim. Acta 89(11–12), 749–756 (2001) U. Köster, Intense radioactive-ion beams produced with the ISOL method. Eur. Phys. J. A 15, 255–263 (2002) H. Koura, T. Tachibana, J. Katakura, F. Minato, Jaea chart of the nuclides 2018. Japan Atomic Energy Agency (2019). https://wwwndc.jaea.go.jp/CN14/index.html T. Kubo, In-flight ri beam separator bigrips at RIKEN and elsewhere in Japan. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 97–113 (2003) T. Kubo, M. Ishihara, N. Inabe, H. Kumagai, I. Tanihata, K. Yoshida, T. Nakamura, H. Okuno, S. Shimoura, K. Asahi, The RIKEN radioactive beam facility. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1–4), 309–319 (1992) J. Kurcewicz, F. Farinon, H. Geissel, S. Pietri, C. Nociforo, A. Prochazka, H. Weick, J. Winfield, A. Estradé, P. Allegro, A. Bail, G. Bélier, J. Benlliure, G. Benzoni, M. Bunce, M. Bowry, R. Caballero-Folch, I. Dillmann, A. Evdokimov, J. Gerl, A. Gottardo, E. Gregor, R. Janik, A. Keli´c-Heil, R. Knöbel, T. Kubo, Y. Litvinov, E. Merchan, I. Mukha, F. Naqvi, M. Pfützner, M. Pomorski, Z. Podolyák, P. Regan, B. Riese, M. Ricciardi, C. Scheidenberger, B. Sitar, P. Spiller, J. Stadlmann, P. Strmen, B. Sun, I. Szarka, J. Taïeb, S. Terashima, J. Valiente-Dobón, M. Winkler, P. Woods, Discovery and cross-section measurement of neutron-rich isotopes in the element range from neodymium to platinum with the FRS. Phys. Lett. B 717(4), 371–375 (2012) N. Kuzminchuk-Feuerstein, B. Fabian, M. Diwisch, W.R. Plaß, H. Geissel, S.A. San Andrés, T. Dickel, R. Knöbel, C. Scheidenberger, B. Sun et al., Efficiency and rate capability studies of the time-of-flight detector for isochronous mass measurements of stored short-lived nuclei

58

H. Geissel and D. J. Morrissey

with the FRS-ESR facility. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 821, 160–168 (2016) M. Lang, F. Djurabekova, N. Medvedev, M. Toulemonde, C. Trautmann, Fundamental phenomena and applications of swift heavy ion irradiations (2001) M. Leino, In-flight separation with gas-filled systems. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 126(1–4), 320–328 (1997) Y.A. Litvinov, H. Geissel, Y.N. Novikov, Z. Patyk, T. Radon, C. Scheidenberger, F. Attallah, K. Beckert, F. Bosch, M. Falch et al., Precision experiments with time-resolved schottky mass spectrometry. Nucl. Phys. A 734, 473–476 (2004) K. Lund, G. Bollen, D. Lawton, D. Morrissey, J. Ottarson, R. Ringle, S. Schwarz, C. Sumithrarachchi, A. Villari, J. Yurkon, Online tests of the advanced cryogenic gas stopper at nscl. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 463, 378–381 (2020) X. Ma, Plans for atomic physics with highly-charged ions at lanzhou. Hyperf. Interact. 115(1), 107–114 (1998) K. Makino, M. Berz, Cosy infinity version 9. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 558(1), 346–350 (2006) B. Marsh, B. Andel, A. Andreyev, S. Antalic, D. Atanasov, A. Barzakh, B. Bastin, C. Borgmann, L. Capponi, T.E. Cocolios et al., New developments of the in-source spectroscopy method at RILIS/ISOLDE. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 550–556 (2013) V. Mishin, V. Fedoseyev, H.-J. Kluge, V. Letokhov, H. Ravn, F. Scheerer, Y. Shirakabe, S. Sundell, O. Tengblad, I. Collaboration et al., Chemically selective laser ion-source for the CERNISOLDE on-line mass separator facility. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 73(4), 550–560 (1993) K. Morita, A. Yoshida, T. Inamura, M. Koizumi, T. Nomura, M. Fujioka, T. Shinozuka, H. Miyatake, K. Sueki, H. Kudo et al., RIKEN isotope separator on-line GARIS/IGISOL. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1–4), 220–225 (1992) D.J. Morrissey, B.M. Sherrill, Radioactive nuclear beam facilities based on projectile fragmentation. Phil. Trans. R. Soc. A 356, 1985–2006 (1998) G. Münzenberg, The separation techniques for secondary beams. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1), 265–275 (1992) G. Münzenberg, Development of mass spectrometers from thomson and aston to present. Int. J. Mass Spectr. 349, 9–18 (2013) G. Münzenberg, W. Faust, S. Hofmann, P. Armbruster, K. Güttner, H. Ewald, The velocity filter ship, a separator of unslowed heavy ion fusion products. Nucl. Instrum. Methods 161(1), 65–82 (1979) Y. Oganessian, Heaviest nuclei from 48ca-induced reactions. J. Phys. G: Nucl. Part. Phys. 34(4), R165–R242 (2007) Y. Oganessian, V. Utyonkov, Superheavy nuclei from 48ca-induced reactions. Nucl. Phys. A 944, 62–98 (2015). Special Issue on Superheavy Elements Y.T. Oganessian, V.K. Utyonkov, Y.V. Lobanov, F.S. Abdullin, A.N. Polyakov, R.N. Sagaidak, I.V. Shirokovsky, Y.S. Tsyganov, A.A. Voinov, G.G. Gulbekian, S.L. Bogomolov, B.N. Gikal, A.N. Mezentsev, S. Iliev, V.G. Subbotin, A.M. Sukhov, K. Subotic, V.I. Zagrebaev, G.K. Vostokin, M.G. Itkis, K.J. Moody, J.B. Patin, D.A. Shaughnessy, M.A. Stoyer, N.J. Stoyer, P.A. Wilk, J.M. Kenneally, J.H. Landrum, J.F. Wild, R.W. Lougheed, Synthesis of the isotopes of elements 118 and 116 in the 249 Cf and 245 Cm +48 Ca fusion reactions. Phys. Rev. C 74, 044602 (2006) T. Ohtsubo, F. Bosch, H. Geissel, C. Scheidenberger, F. Attallah, K. Beckert, P. Beller, T. Fästermann, B. Franzke, M. Hausmann et al., Direct observation of bound state beta decay of bare 207tl at FRS-ESR. Prog. Theor. Phys. Suppl. 146, 493–497 (2002) H. Okuno (chair), S. Makimura (co-chair), High power targetry workshop-2022 (2022 [Online]) A. Ozawa, T. Uesaka, M. Wakasugi, The Rare-RI Ring Collaboration, The rare-RI ring. Prog. Theor. Exp. Phys. 2012(1), 03C009 (2012)

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

59

M. Paul, B. Glagola, W. Henning, J. Keller, W. Kutschera, Z. Liu, K. Rehm, B. Schneck, R. Siemssen, Heavy ion separation with a gas-filled magnetic spectrograph. Nucl. Instrum. Methods Phys. Res. Sect. A 277(1), 418–430 (1989) J. Pereira, J. Benlliure, E. Casarejos, P. Armbruster, M. Bernas, A. Boudard, S. Czajkowski, T. Enqvist, R. Legrain, S. Leray, B. Mustapha, M. Pravikoff, F. Rejmund, K.-H. Schmidt, C. Stéphan, J. Taïeb, L. Tassan-Got, C. Volant, W. Wlazlo, Isotopic production cross sections and recoil velocities of spallation-fission fragments in the reaction 238 U (1a GeV) + d. Phys. Rev. C 75, 014602 (2007) K. Peräjärvi, J. Cerny, J. Hakala, J. Huikari, A. Jokinen, P. Karvonen, J. Kurpeta, D. Lee, I. Moore, H. Penttilä, A. Popov, J. Äystö, New ion-guide for the production of beams of neutron-rich nuclei between z=20–28. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 546(3), 418–425 (2005) M. Ranjan, P. Dendooven, S. Purushothaman, T. Dickel, M. Reiter, S. Ayet, E. Haettner, I. Moore, N. Kalantar-Nayestanaki, H. Geissel et al, Design, construction and cooling system performance of a prototype cryogenic stopping cell for the super-FRS at fair. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 770, 87–97 (2015) M. Reiter, A.-K. Rink, T. Dickel, E. Haettner, F. Heiße, W. Plaß, S. Purushothaman, F. Amjad, S.A. San Andrés, J. Bergmann et al., Rate capability of a cryogenic stopping cell for uranium projectile fragments produced at 1000 Mev/u. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 376, 240–245 (2016) M. Reiter, F. Ames, C. Andreoiu, S.A. San Andrés, C. Babcock, B. Barquest, J. Bergmann, J. Bollig, T. Brunner, T. Dickel et al., Improved beam diagnostics and optimization at ISAC via TITAN’s MR-TOF-MS. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 463, 431–436 (2020) H. Rothard, K. Kroneberger, A. Clouvas, E. Veje, P. Lorenzen, N. Keller, J. Kemmler, W. Meckbach, K.-O. Groeneveld, Secondary-electron yields from thin foils: a possible probe for the electronic stopping power of heavy ions. Phys. Rev. A 41(5), 2521 (1990) C. Ruiz, U. Greife, U. Hager, Recoil separators for radiative capture using radioactive ion beams – recent advances and detection techniques. Eur. Phys. J. A 50(6), 99 (2014) M. Sanjari, X. Chen, P. Hülsmann, Y.A. Litvinov, F. Nolden, J. Piotrowski, M. Steck, T. Stöhlker, Conceptual design of elliptical cavities for intensity and position sensitive beam measurements in storage rings. Phys. Scr. 2015(T166), 014060 (2015) G. Savard, J. Clark, C. Boudreau, F. Buchinger, J. Crawford, H. Geissel, J. Greene, S. Gulick, A. Heinz, J. Lee et al., Development and operation of gas catchers to thermalize fusion– evaporation and fragmentation products. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 582–586 (2003) G. Savard, A. Levand, B. Zabransky, The caribu gas catcher. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 376, 246–250 (2016). Proceedings of the XVIIth International Conference on Electromagnetic Isotope Separators and Related Topics (EMIS2015), Grand Rapids, 11–15 May 2015 S. Schwarz, G. Bollen, K. Lund, C. Magsig, D. Morrissey, J. Ottarson, C. Sumithrarachchi, A. Villari, The nscl cyclotron gas stopper – preparing to go ‘online’. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 463, 293–296 (2020) A. Semchenkov, W. Brüchle, E. Jäger, E. Schimpf, M. Schädel, C. Mühle, F. Klos, A. Türler, A. Yakushev, A. Belov, T. Belyakova, M. Kaparkova, V. Kukhtin, E. Lamzin, S. Sytchevsky, The transactinide separator and chemistry apparatus (tasca) at GSI – optimization of ion-optical structures and magnet designs. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 266(19), 4153–4161 (2008). Proceedings of the XVth International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications G. Sikler, D. Ackermann, F. Attallah, D. Beck, J. Dilling, S. Elisseev, H. Geissel, D. Habs, S. Heinz, F. Herfurth et al., First on-line test of shiptrap. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 482–486 (2003) T. Sonoda, M. Wada, H. Tomita, C. Sakamoto, T. Takatsuka, T. Noto, H. Iimura, Y. Matsuo, T. Kubo, T. Shinozuka, T. Wakui, H. Mita, S. Naimi, T. Furukawa, Y. Itou, P. Schury,

60

H. Geissel and D. J. Morrissey

H. Miyatake, S. Jeong, H. Ishiyama, Y. Watanabe, Y. Hirayama, Development of a gas cellbased laser ion source for RIKEN palis. Hyperf. Interact. 216(1), 103–107 (2013) M. Steck, Y.A. Litvinov, Heavy-ion storage rings and their use in precision experiments with highly charged ions. Prog. Part. Nucl. Phys. 115, 103811 (2020) T. Stora, Recent developments of target and ion sources to produce ISOL beams. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 402–410 (2013). XVIth International Conference on ElectroMagnetic Isotope Separators and Techniques Related to Their Applications, 2–7 Dec 2012 at Matsue C. Sumithrarachchi, D. Morrissey, S. Schwarz, K. Lund, G. Bollen, R. Ringle, G. Savard, A. Villari, Beam thermalization in a large gas catcher. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 463, 305–309 (2020) A. Takamine, M. Wada, Y. Ishida, T. Nakamura, K. Okada, Y. Yamazaki, T. Kambara, Y. Kanai, T.M. Kojima, Y. Nakai, N. Oshima, A. Yoshida, T. Kubo, S. Ohtani, K. Noda, I. Katayama, P. Hostain, V. Varentsov, H. Wollnik, Space-charge effects in the catcher gas cell of a RF ion guide. Rev. Sci. Instrum. 76(10), 103503 (2005) H. Takeda, Y. Tanaka, H. Suzuki, D. Ahn, B. Franczak, N. Fukuda, H. Geissel, E. Haettner, N. Inabe, K. Itahashi et al., New ion-optical operating modes of the bigrips and zerodegree spectrometer for the production and separation of high-quality rare isotopes beams and highresolution spectrometer experiments. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 463, 515–521 (2020) M. Thoennessen, 2018 update of the discoveries of nuclides. Int. J. Mod. Phys. E 28(1–2), 19300002 (2018) M. Thoennessen, B. Sherrill, From isotopes to the stars. Nature 473, 25–26 (2011) A. Türler, K. Gregorich, Experimental techniques, in The Chemistry of Superheavy Elements, ed. by M. Schädel, D. Shaughnessy (Springer, Berlin/Heidelberg, 2014) P. Van Duppen, B. Bruyneel, M. Huyse, Y. Kudryavtsev, P. Van Den Bergh, L. Vermeeren, Beams of short lived nuclei by selective laser ionization in a gas cell. Hyperf. Interact. 127(1), 401–408 (2000) D.J. Vieira, J.M. Wouters, K. Vaziri, R.H. Kraus, H. Wollnik, G.W. Butler, F.K. Wohn, A.H. Wapstra, Direct mass measurements of neutron-rich light nuclei near n=20. Phys. Rev. Lett. 57, 3253–3256 (1986) M. Wada, Genealogy of gas cells for low-energy ri-beam production. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 450–456 (2013). XVIth International Conference on ElectroMagnetic Isotope Separators and Techniques Related to Their Applications, 2–7 Dec 2012 at Matsue M. Wada, Y. Ishida, T. Nakamura, Y. Yamazaki, T. Kambara, H. Ohyama, Y. Kanai, T.M. Kojima, Y. Nakai, N. Ohshima et al., Slow RI-beams from projectile fragment separators. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 570–581 (2003) T. Wakasa, K. Hatanaka, Y. Fujita, G. Berg, H. Fujimura, H. Fujita, M. Itoh, J. Kamiya, T. Kawabata, K. Nagayama, T. Noro, H. Sakaguchi, Y. Shimbara, H. Takeda, K. Tamura, H. Ueno, M. Uchida, M. Uraki, M. Yosoi, High resolution beam line for the grand Raiden spectrometer. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 482(1), 79–93 (2002) M. Wang, H.S. Xu, J.W. Xia, X.L. Tu, R.S. Mao, Y.J. Yuan, Z.G. Hu, Y. Liu, H.B. Zhang, Y.D. Zang, T.C. Zhao, X.Y. Zhang, F. Fu, J.C. Yang, L.J. Mao, C. Xiao, G.Q. Xiao, H.W. Zhao, W.L. Zhan, First isochronous mass measurements at csre. Int. J. Mod. Phys. E 18(02), 352–358 (2009) S. Warren, T. Giles, C. Pequeno, A. Ringvall-Moberg, Offline 2, ISOLDE’s target, laser and beams development facility. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 463, 115–118 (2020) H. Weick, H. Geissel, C. Scheidenberger, F. Attallah, T. Baumann, D. Cortina, M. Hausmann, B. Lommel, G. Münzenberg, N. Nankov et al., Slowing down of relativistic few-electron heavy ions. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 164, 168–179 (2000)

1 Exotic Nuclei and Their Separation, Electromagnetic Devices

61

L. Weissman, D. Morrissey, G. Bollen, D. Davies, E. Kwan, P. Lofy, P. Schury, S. Schwarz, C. Sumithrarachchi, T. Sun et al., Conversion of 92 MeV/u 38ca/37k projectile fragments into thermalized ion beams. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 540(2–3), 245–258 (2005) H. Wiedemann, Particle Accelerator Physics Springer Cham, (2015) J. Winfield, H. Geissel, B. Franczak, T. Dickel, E. Haettner, E. Kazantseva, T. Kubo, S. Litvinov, W. Plaß, S. Ratschow, C. Scheidenberger, Y. Tanaka, H. Weick, M. Winkler, M. Yavor, Ionoptical developments tailored for experiments with the super-FRS at fair. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 491, 38–51 (2021) R. Wolf, F. Wienholtz, D. Atanasov, D. Beck, K. Blaum, C. Borgmann, F. Herfurth, M. Kowalska, S. Kreim, Y.A. Litvinov et al., Isoltrap’s multi-reflection time-of-flight mass separator/spectrometer. Int. J. Mass Spectrom. 349, 123–133 (2013) H. Wollnik, Optics of Charged Particles, 2nd edn. Academic Press. (2021) H. Wollnik, M. Wada, P. Schury, M. Rosenbusch, Y. Ito, H. Miyatake, Time-of-flight mass spectrographs of high mass resolving power. Int. J. Mod. Phys. A 34(36), 1942001 (2019) J. Wouters, D. Vieira, H. Wollnik, G. Butler, R. Kraus, K. Vaziri, The time-of-flight isochronous (TOFI) spectrometer for direct mass measurements of exotic light nuclei. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 26(1), 286–293 (1987) M. Yavor, Optics of Charged Particle Analyzers. In: Advances in Imaging and Electron Physics. Elsevier, 157, 1–381 (2009) A. Yoshida, K. Morita, K. Morimoto, D. Kaji, T. Kubo, Y. Takahashi, A. Ozawa, I. Tanihata, Highpower rotating wheel targets at RIKEN. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 521(1), 65–71 (2004). Accelerator Target Technology for the 21st Century. Proceedings of the 21st World Conference of the International Nuclear Target Society K. Yoshida, N. Fukuda, Y. Yanagisawa, N. Inabe, Y. Mizoi, T. Kubo, High-power beam dump system for the bigrips fragment separator at RIKEN ri beam factory. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 373–380 (2013). XVIth International Conference on ElectroMagnetic Isotope Separators and Techniques Related to Their Applications, 2–7 Dec 2012 at Matsue

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Exotic Nuclei and Their Separation, Using Atomic Interactions Hans Geissel and D. J. Morrissey

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Separation Combined with Atomic Interactions . . . . . . . . . . . . . . . . . . . . . . . Energy-Bunching with a Monoenergetic Degrader in a Dispersive Separator Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial Isotopic Separation with Energy Degraders in Achromatic In-Flight Separators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge-Changing Collisions During Penetration of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of the Different Separation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary, Key Achievements, and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 64 68 70 77 82 83 85

Abstract

The discovery of short-lived, exotic nuclei has opened new horizons in subatomic physics and allowed many novel applications in other fields, e.g., the understanding of the evolution of matter and production of the atomic elements in the universe. The effective and highly selective separation of the exotic nuclei produced in these reactions from other more abundant products and the primary beam is critical for the study of exotic species. The high sensitivity of separators can reach single ions produced with sub-picobarn cross sections. Depending on the type of production reaction and the kinematics of the products, different

H. Geissel () GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany e-mail: [email protected] D. J. Morrissey National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_132

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accelerator facilities and separation methods have been constructed and applied. A reaction-product separator is the heart of each such facility. The presentation of fragment separators and their application have been divided here in two parts. A brief overview of the relevant production reactions for modern-day separation techniques and then the ion-optical features of fragment separator design and operation were presented in Part 1. These devices can consist of pure dispersive electromagnetic elements or can incorporate specific atomic interactions of the ions in additional matter along their path through the separator. Extension to systems that incorporate atomic interactions and combinations of these devices with additional high-resolution ion-optical systems to provide unique opportunities to study exotic nuclei immediately after they pass through the system is presented here in Part 2. Nuclear physics laboratories around the world have built reaction-product separators, have plans for new separation facilities, and are constructing new machines with enhanced capabilities that rely on these devices. Finally, the whole presentation concludes with a discussion of key goals, achievements, and limitations in the separation and science of exotic nuclei.

Introduction This chapter builds on the overview of nuclear reactions to produce exotic nuclei and the detailed discussion of the separation of rate exotic nuclei from the overwhelming number of less exotic nuclei. Some of the considerations for primary beams, targets, and target shielding were presented in Part 1 along with a discussion of physical separation techniques that only rely on electromagnetic systems. Here in Part 2, the discussion is extended to systems that include components that introduce atomic interactions that allow degeneracies in the momentum distributions of the product beams to be broken. Examples of large facilities that provide exotic nuclei using these techniques are presented at the end along with a short presentation of the present key goals, achievements, and their limitations.

Electromagnetic Separation Combined with Atomic Interactions The laterally and longitudinally dispersive systems and their combinations are basic building blocks used in modern heavy-ion facilities for nuclear physics research and also for medical and technical applications. The purely electromagnetic ion separators all have the fundamental property that the phase-space volume is preserved throughout since only conservative forces enter the Hamiltonian (described in detail in Part 1) describing the ion transport. Of course, this statement is only valid for the particle ensemble which is fully transmitted through the separator. Losses due to dispersion that extends outside apertures and slit systems clearly reduce the phasespace volume.

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An additional separation tool, applicable over the full energy range of ion beams, comes from the atomic interactions of ions with matter. The atomic collisions between energetic ions and the atoms in the material cause an average energy loss with a characteristic energy fluctuation, a mean angular scattering, and changes of the ionic charge state. In all experiments with exotic nuclei, an accurate knowledge of the atomic processes must be taken into account when the ions interact with any matter, e.g., the ions are created in some target material with a thickness, often pass through so-called stripper foils and gas-filled magnet sections, and are stopped in detector layers or gas-filled buffer gas cells. The production and use of projectile fragmentation products has blossomed due to the use of atomic interactions in laterally dispersive separators. In addition, a special application that has been recognized for some time is the use of highly penetrating heavy-ion beams for medical treatment originally at the BEVALAC and later at the dedicated HIMAC facility (Castro et al. 1980; Ebner and Kamada 2016). More recently, short-lived positron emitters (β + ) produced at high energies can be used for simultaneous medical diagnostics and treatment (Durante and Paganetti 2016; Durante and Parodi 2020). The important atomic collision processes have huge cross sections compared to nuclear collisions, a fact that is obvious in comparing the major forces acting on atoms and nuclei, the Coulomb and the nuclear force, respectively. The short range of the strong interaction is limited to the femtometer (10−15 m) domain which is five orders of magnitude smaller than the angstrom domain of atoms. Note that the atomic energy loss, angular scattering, and charge-changing processes in matter that is placed inside ion-optical systems cannot be described by conservative forces in the Hamiltonian and thus it is non-Liouvillian. The non-Liouvillian character of these atomic interactions and their incorporation into ion-optical systems have been described in some detail by Geissel et al. (1989). Some atomic collision processes are inevitable in accelerator-based ion separators, and others are applied deliberately. When heavy ions penetrate through matter, the important atomic interactions consist of mainly three different microscopic processes: (1) elastic collisions of the ion with the target atom, (2) inelastic collisions of the ion with the target electrons, and (3) inelastic collisions of the projectile electrons with the atom (Sigmund 2014; Geissel et al. 2002). The number of collisions is usually enormous and the net results can be treated statistically. The results of these interactions are distributions of energy loss, angular scattering, and ionic charge state of the ensemble. The widths of the energy and angular distributions are called straggling widths that result from the statistical variation of the impact parameters in the collisions and thus produce fluctuations in the kinetic energy transfer. Therefore, matter that deliberately produces atomic collisions is generally placed in a field-free drift space. This inevitably produces a phasevolume enlargement for an ensemble of (ideally) monoenergetic ions with no angular divergence. However, depending on the kinetic energy domain, the incident ion beam is often characterized by a large energy spread that can be bunched or compressed when the ion slows down in matter. This process mainly occurs at velocities where the slope of the stopping power versus the kinetic energy is

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positive ( ∂(dE/dX) > 0). This condition is fulfilled on the rising edge of the ∂E stopping power curve (Bragg peak) before the ion is stopped. In this case, the more energetic ions experience a larger stopping force, and thus the energy distribution can become narrower (see Geissel et al. (1989)). This situation is not only valid for bound electrons in atoms or compounds but holds also for free electrons, e.g., the application of electron cooling in heavy-ion storage rings; however, it only applies to slow ions in a limited energy domain. Accurate slowing-down and ionic charge-state measurements have been performed over a large energy region that provides benchmark data for the theoretical predictions of the ion-atom interactions. Important results from such atomic collision experiments with the FRS separator-spectrometer (Geissel et al. 1992) operated in a high-resolution achromatic mode have shown that the long-held Bethe description of energy-loss mechanisms (Bethe 1930) systematically fails for fully ionized heavy ions, both for stopping powers and energy straggling (Scheidenberger et al. 1994, 1996; Weick et al. 2000a; Geissel et al. 2002). The new experimental data initiated a novel theoretical effort by Lindhard and Sørensen (1996) (LS) which has resolved the observed experimental differences with the previously established models. The new LS theory for stopping powers of heavy ions was formulated as a correction term ΔL to the well-known Bethe theory:   4π Z12 e4 NZ2 dE 2mc2 β 2 2 2 = + ln γ · ln − β + ΔL , dX I mc2 β 2

(1)

where Z1 e represents the charge of the projectile moving with a velocity β = v/c, N Z2 is the density of target electrons, m is the rest mass of the electron, I is the mean ionization potential of the target atoms, and γ is the usual Lorentz factor. The correction term inside the logarithm, ΔL, includes the Bloch term, the shell, and the Barkas correction as described in Lindhard and Sørensen (1996). At relativistic energies, when the fragment velocities far exceed the electron velocity in the K-shell, the ions are predominantly fully ionized. Under this condition, the Lindhard-Sorensen theory (Lindhard and Sørensen 1996) has been found to provide reliable predictions for all ions since it takes into account the deviation from the first Born approximation in the Bethe theory and uses the solution of the Dirac equation including the relativistic Mott scattering. When lower energy projectiles start to carry electrons during their penetration of matter, partial stopping powers must be used (Weick et al. 2000b) and summed up according to their statistical weight in the charge-state population. The LS theory does not include parameters adjusted to experimental heavy-ion data; nonetheless, it can describe the results of measurements within experimental uncertainties over a huge energy range. The latter statement is important since thick solids are commonly used as energy degraders in modern high-energy heavy-ion separators for high-energy heavy-ion beams and fragments. Calculations of the magnetic field settings of a separator and the expected spatial separation of desired rare isotopes require reliable calculations of the energy-loss processes. Accuracies of the energy-loss prediction need to be

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better than 1% to make reliable magnetic field settings to attain reliable spatial isotopic separation on the order of millimeters and for the efficient complete slowing down in gas-filled stopping cells or thin silicon detectors (Plaß et al. 2013). In other words, an accurate knowledge of stopping powers allows one to obtain the mean ranges of very penetrating heavy ions. The range distribution of an ensemble of monoenergetic ions, incident with a given direction, in a thick layer of matter is defined as the probability of finding the projectiles at rest after complete loss of the initial kinetic energy. To a good approximation, the expression for the mean range of heavy ions in matter, R(E), with an incident kinetic energy of E0 in the continuous-slowing-down model (Sigmund 2014) is given by the following:  R(E0 ) =

E0

(dE/dX)−1 dE,

(2)

Emin

where Emin is the thermal energy of the stopping medium. In this model, the stopping power is the sum of the inelastic and elastic contributions. The statistical nature of energy deposition and angular scattering is manifested in the range distribution of an ensemble of ions entering the medium. Range distributions can also be calculated within the continuous-slowing-down transport theory (Sigmund 2014). The solutions for the transport equations are developed in terms of moments. Relatively simple analytical solutions have been extracted in the energy range where the elastic collisions are dominant. A good approximation for the variance of the range distribution for incident monoenergetic and collimated ion beams is the following relation:  ΩR2 (E0 ) =

E0

dΩ 2 /dX

Emin

(dE/dX)3

dE

(3)

where the differential energy straggling, dΩ 2 /dX, is the sum of variances of different energy straggling contributions. The energy and angular straggling distributions due to all of atomic collisions are in general much smaller than the corresponding contribution from the nuclear reaction that produces the exotic isotope. This observation particularly applies to nuclear reactions that create the exotic nuclei at high kinetic energies where the nuclear component becomes important for calculations of in-flight separators. It was shown in Part 1 that the heavy projectile fission process creates an even larger phase-space population than projectile fragmentation. For example, the phase space for 100 Sn (fission fragment) and 78 Ni (projectile fragment) can be compared. Here, Fig. 1 shows a comparison of the nuclear and atomic contributions to the energy and angular straggling of the uranium projectile fragment 215 Pb as a function of incident energy after they pass through a beryllium degrader adjusted at each energy to be 40% of the ion range. Such thick degraders are often used in laterally dispersive achromatic separators to separate projectile fragments as it will be discussed in the following section.

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U + Be

215

Pb

10

Degrader 5

8

8,0x10

nuclear

6 4 2

VD[mrad]

VE / E [10-3]

238

4 nuclear

3 2 1

to

0

0 0

400

800 1200 1600

0

400

800 1200 1600

E [MeV/u] Fig. 1 Calculated energy and angular straggling of 215 P b projectile fragments from a 238 U beam as a function of incident kinetic energy after passing through a degrader. Right panel: The standard deviation of the angular distribution caused by a fragmentation reaction in beryllium metal from the semiempirical model of Morrissey (1989) compared to the atomic multiple angular scattering of the ions based on Moliere’s theory. The degrader thickness was adjusted to 40% of the total range (d/R = 0.4) in each case, and the production target was negligibly thin. Left panel: Calculated energy spread for the same projectile fragmentation reaction. The calculation for the pure atomic interaction was performed assuming a monoenergetic incident 215 P b ion beam penetrating a beryllium degrader with d/R = 0.4

The only case shown in which the two contributions are nearly equal is the angular straggling of the 215 Pb product at the highest energy (1600 MeV/u).

Energy-Bunching with a Monoenergetic Degrader in a Dispersive Separator Stage In general, the range straggling of projectile fragments at high incident energies is dominated by the incident momentum spread of fragments created via nuclear reactions and by the stopping power difference between that of the projectile and of the fragment of interest in the thick production targets. The latter effect, often called differential energy loss, is simply caused by the fact that a fragment can be produced by the beam at any point along its path through the target. Thus, if the exotic isotope of interest is significantly different from the projectile, particularly in atomic number, then the ion will have moved through the target material with two significantly different energy-loss functions. Moreover, the distribution of energy lost will be uniform (flat) across the target thickness. The consequence of this differential energy loss is that the overall resulting range straggling of projectile fragments is too large for complete implantation in thin layers of matter, e.g., in silicon detectors and more so in gas-filled stopping cells.

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An ion-optical solution to the problem of a large intrinsic energy distribution is to laterally disperse the projectile beam on shaped degraders to produce a monoenergetic beam. The so-called monoenergetic degrader is placed at the focal plane of a dispersive separator stage and with a thickness that varies along the dispersive coordinate with a slope such that it ideally forces all of the isotopic fragments to emerge from the degrader with essentially the same mean velocity. Several things are clear if this approach is to work: first, the monoenergetic degrader has to correspond not only to the ion-optical dispersion, but it will have to match the energy-loss function of a particular isotope at a particular incident energy. From the ion-optical standpoint, it is clear that having a large resolving power in the preparative dispersive stage is critical along with the mechanical properties of the degrader to minimize the overall range straggling of the emerging ions. Recall from Part 1 that the resolving power of such a system includes both the dispersion and the spot size. The beam spot at the monoenergetic degrader would ideally only be determined by the dispersion (negligible object spot size), and then the corresponding thickness can compensate for the incident momentum spread. An example of the range straggling of 300 MeV/u tin fission fragments created by uranium projectile fission as a function of the ion-optical resolving power of the dispersive dipole stage followed by a monoenergetic degrader is shown in Fig. 2. In these calculations and in general, having fully ionized fragments is important, because if the position at the degrader is partially determined by the differences among ionic charge states of the same fragment, then the single mechanical slope of the thickness variation cannot match the mixed distribution of incident momenta.

Range compression with a dispersive system 132

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Pioneering work with specially shaped material layers placed within dispersive ion-optical stages was done in the 1970s at biomedical laboratories in Los Alamos (Paciotti et al. 1975). In that case, the goal was to spatially separate high-energy, which is relativistic, muons from more abundant electrons and pions emerging from a production target. These light particles have essentially the same velocity so, at the same magnetic rigidity and with quite different masses, they undergo different energy losses during penetration of a thick beryllium degrader and thus can be spatially separated in a subsequent magnetic section. Furthermore, the degrader was shaped to bunch the momentum spread as indicated above in the example of a heavy fragmentation ion. The ion-optical stage following the shaped degrader in the biomedical channel at Los Alamos was used to obtain an achromatic muon beam for biological applications. The successful application of atomic interaction within separator stages was also strongly dependent on precise knowledge of the atomic slowing-down processes of these light particles.

Spatial Isotopic Separation with Energy Degraders in Achromatic In-Flight Separators The separation of high-energy exotic nuclei created in projectile fragmentation reactions is in many aspects quite different from the separation of the fully relativistic, singly charged particles at Los Alamos. For example, the heavy-ion fragments emerging from the target have roughly the same mean velocity, although there is some variation with atomic number due to atomic interactions in the target, and are not generally fully relativistic. The width of the velocity distribution is determined by the spread in the momenta of the products from the nuclear reaction combined with the atomic interactions in the production target, as discussed above. Atomic slowing down in thick degrader materials creates significantly different Bρ values for emergent fragments that have different atomic numbers and masses. Furthermore, the atomic interactions of heavy ions with matter is more complex than that of the particles separated in LAMPF biomedical separator due to the potential for ionic charge-exchange processes and any deviations from otherwise precise theoretical perturbation treatments. Nevertheless, the exploitation of inflight isotopic separation of projectile fragments with a Bρ analysis system both in front and behind a shaped energy degrader at a dispersive focal plane has become a copious source of exotic nuclei for modern nuclear science. This technique was initially developed in the mid-1980s at several accelerator-based research centers around the world (Dufour et al. 1986; Schmidt et al. 1987) and has become a central feature of modern laboratories. The principles behind the ion-optical properties of such high-energy in-flight separators are shown by an achromatic system as calculated in Fig. 3. In this example, four stages of lateral dispersion are combined with a degrader for isotopic separation via the so-called Bρ-ΔEBρ method. The degrader is placed in the intermediate dispersive focal plane. The resolving powers of the first two sections are added at the center on the degrader as shown by the dispersion line (red) in the figure, and the magnets have relatively

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large apertures. The large dispersion is then matched in the last two sections to provide an achromatic focus at the end. The magnetic rigidities of the two sections before are set to that of the isotope of interest which will include a range of reaction products. The shape of the central degrader is matched to the dispersion and energyloss properties of the isotope of interest, and the magnetic rigidities of the last two sections provide spatial achromatic separation at the final focal plane. A slit system is used at the final focal plane to remove unwanted products, and the isotope of interest can be passed to subsequent equipment. This type of in-flight separator, characterized by one dipole-magnet stage in front and behind the energy-loss degrader, was first realized at GANIL (France) (Dufour et al. 1986; Anne et al. 1987) with LISE, originally designed for atomic physics studies, and then at the RCNP (Japan), Shimoda et al. (1992), and at the JINR (Russia) (Artukh et al. 1991). These first separators were upgraded over time to improve their performance, for example, LISE3 (Anne and Mueller 1992) and SISSI (Anne 1997), also combined with SPEG (Bianchi et al. 1989). The next step in the development of such in-flight separators was to increase the ion-optical resolving power by the addition of two dispersive dipole stages, one before and one after the energyloss degrader. These next generation in-flight separators were constructed at GSI [FRS] (Germany) (Geissel et al. 1992), at RIKEN [RIPS] (Japan) (Kubo et al. 1992), and at MSU [A1200, A1900] (USA) (Sherrill et al. 1991; Morrissey et al. 2003).

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These devices still operate or have been replaced with even more powerful separators and still form the basis of the latest generation of in-flight separators for projectile fragments, the Big-RIPS (Kubo 2003), the new ARIS separator at FRIB (Hausmann et al. 2013), and the Super-FRS under construction (Geissel et al. 2003). With increased ion-optical resolving power of the FRS separator, a feature that is decisive for isotopic separation (Geissel et al. 1989), and very high fragment velocities (i.e., β = 0.9), it became possible to spatially separate even the heaviest fragments from a uranium beam (Magel et al. 1994). As noted above, the ideal operating domain of this Bρ-ΔE-Bρ method includes the requirements that the fragments are fully ionized and that a relatively thick degrader is used but on the other hand keeps the losses due to secondary nuclear reactions at less than 50%. The condition of full ionization of the fragments emerging from the production target can be easily achieved when the velocity is high enough that even the K-shell electrons are removed in atomic collisions. The details of the charge-state distributions are discussed in the next section. The spatial separation increases with the thickness of the degrader and the energy loss. The boundaries of these two criteria in the space of beam energy and fragment atomic number are illustrated in Fig. 4. In most practical cases at the highest energies, the energy degrader is much thicker than the production target at the object point of the separator system. A typical setup to obtain spatially separated fragments at an energy of ≈1000 MeV/u includes a degrader that is approximately 50% of the atomic range. Thinner degrader thicknesses are generally needed at facilities with lower beam energies, approximately 30% of the range, due to increased phase-space growth, and even thinner degraders can be used to pass a larger set of products onto experiments when desired. Be aware that tertiary fragments from nuclear reactions in the degrader will represent a challenge for any experimental setup when a high rate of contaminant fragments impinges on thick degraders. In such cases, other techniques have to be employed to reduce the rate of contaminants. The present generation of in-flight projectile fragment separators has solved the challenge of secondary reactions by applying multiple degrader stages (Geissel et al. 2003) to the separation process. This requires, of course, a significant change in the layout of the facility due to the large scale of the required magnets and has been implemented in the RIBF facility at RIKEN (Kubo 2003). The new powerful separators ARIS (Hausmann et al. 2013) in the USA and HFRS (Sheng et al. 2020) in China have also been designed with two sequential Bρ-ΔE-Bρ separation stages. The first separator stage serves then as a pre-separator to make an initial selection in magnetic rigidity and importantly capture the residual primary beam followed by a main separator which performs the final isotopic separation and can include particle detectors for event-by-event particle identification in-flight. This identification is possible since the number of secondary fragments passing through the main separator stage is greatly reduced by the pre-separator compared to a single separator. Some of the benefits of this new separation method can be clearly seen in the discovery of numerous previously unobserved neutron-rich exotic nuclides using the BigRIPS separator (Ohnishi et al. 2010) that is now the central part of the

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Fig. 4 Operating domain of the Bρ-ΔE-Bρ separation method to achieve spatial isotopic separation of reaction products created in-flight at high energy (see the text). In this example, an aluminum degrader with a mean thickness equal to one-half of the range of the fragment defines the upper limit (red line), while the percentage of fully ionized fragments, q = Z, define the lower limits (black lines)

RIKEN high-energy Radioactive Isotope Beam Factory (RIBF) (Sakurai 2018). These results were obtained after only a short commissioning phase and used beams reaching up to the heaviest stable projectiles at 345 MeV/u in projectile fragmentation and projectile fission reactions. BigRIPS consists of two achromatic spectrometer sections; the pre-separator and main separator have a total length of 78 m, (see Fig. 5). The production target is at the entrance of the pre-separator, and the dump for the primary beam is partially the first dipole magnet with a special high-power catcher system directly behind it. The BigRIPS pre-separator has two 30◦ dipole stages bending in the same direction. Large-aperture quadrupole triplets and higher multipole magnets are placed before and after the dipoles. The first energy degrader is placed in the center of the system at a stigmatic focus which preserves the achromatism of the pre-separator. In the standard operating mode of BigRIPS, the two spatial-separation stages are subtractive in their resolving powers. However, the additive mode can also be applied (Takeda et al. 2020) in the ion-optical matching section labeled

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F 2 − F 2.5 − F 3 which can increase the spatial-separation power, and thus the isotopic background can be substantially reduced. A comparison of the features of the additive and standard modes has been illustrated in simulations of the separation power with different combinations of degrader thicknesses placed at F 1 and F 5 (see Fig. 6 and the work by Iwasa et al. (1997), Tarasov and Bazin (2008), and Franczak (1984)). In this representative example, 62 Ge isotopes were produced by the fragmentation of a 345 MeV/u 78 Kr beam in a 10 mm-thick Be target at F 0. In these simulations, Iwasa et al. (1997) and Tarasov and Bazin (2008), the positions of the 62 Ge isotopes at the final focal plane F 7 were obtained for different combinations of achromatic degraders at F 1 and F 5, as indicated in the figure. The achromatic degrader at F 1 was 4 mm thick in all cases. BigRIPS has been shown to be a very versatile magnetic spectrometer with six dispersive dipole stages and is quite similar to the Super-FRS (Geissel et al. 2003) under construction at GSI-FAIR. The Super-FRS will be operated with a significantly higher fragment energy, corresponding to a maximum magnetic rigidity of 20 Tm. The flexibility with respect to operating modes is mainly due to the large number of quadrupole lenses with independent power supplies. The ion-optical system of the Super-FRS will be even more flexible than BigRIPS because the focal-plane areas are larger in the longitudinal direction. Of course, when the focal-plane length is enlarged, the transmission is usually slightly reduced. Even bearing this in mind, the design of FRS (Geissel et al. 1992) and Super-FRS

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(Geissel et al. 2003) kept the central focal planes larger to accommodate a variety of different and changeable experimental equipment. The new Chinese in-flight separator (HFRS) (Sheng et al. 2020) presently under construction will have a quite similar ion-optical structure to BigRIPS and the Super-FRS. In addition, the HFRS, with a bending power of 22 Tm, can accept the highest magnetic rigidity fragments of this separator type. This latter feature is particularly important for studies of exotic hypernuclei. The production of hypernuclei requires higher energies than

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that to fragment stable nuclei consisting of protons and neutron since the production threshold for strange hyperons must be crossed to produce hypernuclei. The in-flight separators often pass the separated exotic nuclei on to subsequent spectrometers or storage rings to perform a large variety of nuclear structure and reaction studies (Winfield et al. 2021; Sakurai 2018). At the NSCL, the A1900 separator and now the ARIS separator are connected to the S800 spectrograph and a reaccelerator. At the RIBF, BigRIPS is coupled to the ZeroDegree (Kubo et al. 2012), SAMURI (Kobayashi et al. 2013), and SHARAQ (Uesaka et al. 2008; Kawabata et al. 2008) spectrometers and also to the Rare Radioactive Isotope Ring (R3 ). The BigRIPS and ZeroDegree spectrometer are two independent yet coupled achromatic systems that can be operated in dispersion-matched mode. The beamline from the superconducting ring cyclotron and BigRIPS can be operated as a dispersion-matched system for high-resolution spectroscopy experiments with highintensity light projectiles. These different ion-optical developments have become the basis for a new category of experiments exploring exotic nuclei and mesic atoms (Nishi et al. 2018). In summary of this section, the incorporation of an energy degrader into a laterally dispersive separator system based on magnetic dipoles has become an essential element of modern nuclear science. It is particularly effective for fully ionized heavy fragments. These systems routinely operate with ions having kinetic energies that greatly exceed the bending power of electrostatic dipole fields that are often used for isotope separation at low kinetic energies. Analytic and numerical approximations have been developed so that energy degraders can be incorporated as special ion-optical elements in routine calculations. Table 1 contains a summary of the important characteristics of the in-flight projectile fragment separators.

Table 1 General characteristics of present-day in-fight separators. All of the facilities can provide exotic nuclei across the periodic table. The values of ion-optical parameters ΔBρ, Δa, and Δb represent the maximum separator acceptances. The angular acceptance is correlated with the Bρ acceptance, and depending on the experiment, one can increase the one at the cost of the other. The thicknesses of the production targets are strongly dependent on the projectile element and its kinetic energy and, of course, on the prime goal of the specific experiment. For example, the production of new isotopes and the initial decay spectroscopic work might use a typical thickness of ≈(10–20)% of the atomic range (R ∝ ZA2 ) of the incident projectile in the target material Facility LISE RIPS ARIS Big-RIPS FRS Super-FRS HFRS

Bρmax T m 3.2 5.76 8 9.5 18 20 25

ΔBρ ± 2.5 ±3 ±5 ±3 ± 1.8 ± 2.5 ±2

Δa, Δb mrad ±15, ±15 ±40, ±40 ±40, ±40 ±40, ±50 ±13, ±10 ±40, ±20 ±30, ±15

Reference Anne et al. (1987) Kubo et al. (1992) Hausmann et al. (2013) Kubo (2003) Geissel et al. (1992) Geissel et al. (2003) Sheng et al. (2020)

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Charge-Changing Collisions During Penetration of Matter The ionic charge-state distribution (CSD) of heavy projectiles and reaction products is very important for the application of an efficient transport and separation of exotic nuclei, as well as for unambiguous particle identification of those particles in energy-deposition detectors. It is, of course, highly desirable to have all ions in a single charge state (fully ionized being the simplest case) when they emerge from the production target, during their interaction with matter, and during transport through the electromagnetic fields. These issues were already mentioned in previous sections in connection with energy-bunching and spatial separation with the Bρ-ΔE-Bρ method. However, it is not always possible to meet the optimal condition of full ionization! Therefore, a broad understanding of the expected distribution of atomic ionization is required for swift heavy ions passing through matter. The goal to maintain the highest fraction of fully ionized projectile fragments can in principle be attained by using the highest possible kinetic energy for the incident projectiles. However, experiments that require that the spatially separated reaction products be completely slowed down in matter pose restrictions for the maximum energy due to significant losses due to secondary nuclear reactions during the stopping process. Besides the impact of the kinetic energy on the fraction of fully stripped ions, the choice of the target and degrader material themselves are important. With the optimum choice of the material, one can enhance the fraction of bare ions by a factor of 2 and the hydrogen-like and He-like ions by even larger factors (see the example of uranium ions in Fig. 7). In this example, the equilibrium CSDs were calculated for 1000 MeV/u 238 U ions in different materials, and the fractions are shown as a function of atomic number of the degrader.

Fig. 7 Calculated fractions for the most strongly populated charge states of 300 MeV/u (left) and 1000 MeV/u (right) 238 U ions, Z1 = 92, after reaching equilibrium in different mono-atomic solids are shown on a logarithmic scale as a function of atomic number Z2 (Scheidenberger et al. 1998). The symbols and colors are associated with the same charge states in both panels

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In principle, beryllium metal is an ideal target for the production of projectile fragments due to its mechanical properties and low atomic number (low multiple scattering probability). However, in order to achieve predominantly fully ionized fragments at intermediate energies, the figure indicates that one should choose a medium Z material. Therefore, in practice, a thin layer, several hundred mg/cm2 , of niobium on the downstream side of the beryllium is often used to enhance the rate of bare fragments before they enter the separator fields. The production conditions and requirements for the charge-state populations of heavy-ion reaction products created exclusively at velocities (i.e., in fusion reactions) near the Coulomb barrier are completely different. Here, it is unavoidable that the ions emerge from the production target with a broad CSD and, therefore, quite different experimental techniques have to be applied to obtain efficient and selective isotope separation compared to the high-energy projectile fragments. A brief discussion of the features of atomic charge-changing collisions is needed to provide a framework for understanding the experimental developments to overcome the inherent challenges of the broad ionic charge-state distributions of low-energy heavy exotic nuclei that need to be separated in-flight. The charge state of an individual energetic ion penetrating through matter fluctuates as a result of electron loss and capture with the target atoms. After a sufficiently high number of charge-exchange collisions, the moving projectile reaches an equilibrium CSD that is independent of the incident charge state that can be characterized by a mean or average charge state. The distance in material needed to attain this average distribution is called the equilibrium path length. The equilibrium path length and the mean charge value both depend on the atomic number and the velocity of the projectile. These properties of the CSD have been studied in measurements with many different projectiles at low energies. The CSD experiments with heavy ions go back to the discovery and study of fission fragments (Lassen 1945) emitted by exited fissile heavy atoms in the neighborhood of the element uranium. Niels Bohr and his colleagues (Bohr 1948) postulated that a heavy ion, penetrating through matter with a velocity v, loses all bound electrons that have an orbital velocity smaller than that velocity, v. The fission fragments from a fissile source in rest have a mean kinetic energy on the order of (0.4–1) MeV/u with a broad velocity and a broad CSD due to the statistical nature of the nuclear creation process. This velocity cutoff is called the Bohr stripping criterion, and it directly follows in the framework of the Thomas1

Fermi atomic model that the mean charge q¯ should be proportional to Z13 and to the fission fragment velocity v: 1

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simultaneous focusing (or compression) in the velocity and in the ionic chargestate distribution can be achieved for a selected isotope. This solution is also applicable for various heavy ion reaction products, e.g., fusion products, moving in a similar velocity regime. The experimental solution is a relatively simple, gas-filled dispersive magnetic dipole stage (Cohen and Fulmer 1958; Leino 1997; Gregorich 2013). When the number of collisions in the gas-filled volume is sufficiently large, an ensemble of moving ions is deflected by the dipolar magnetic field according to the mean charge of the ensemble. The same ensemble of ions would be very widely dispersed and suffer large transmission losses in a corresponding vacuum-filled, so to speak, magnetic separator. Deflection in a gas-filled dipole stage not only focuses in the charge-state coordinate but also decreases the velocity spread and thus the image size in the final focal plane. This secondary effect can be directly seen, when q is replaced by q¯ from Eq. 4 in the definition of the bending power or magnetic rigidity: mv 2 mv p =qvB, ¯ or rearranging: Bρ= = , ρ q¯ q¯ − 13

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where A1 and Z1 are the mass and atomic number of the moving ion, respectively, and the constants have been evaluated. The filling gas in these separators are the lightest elements at the highest chemical purity, such as hydrogen or helium, to avoid significant energy-loss straggling and angular scattering. Also the gas thickness is held to the smallest value that will produce a charge-state equilibrium but yet has small loses due to energy-angular straggling. The velocity dependence in the Bohr formula for q¯ has been shown to be 2

approximately valid in the velocity region 1 < v/v0 < Z13 indicating that the gasfilled separators are well-suited for not only fission products but also evaporation residues from fusion reactions. The gas-filled separators can effectively suppress the background from primary beam and from target-like recoils and similar reactionproduct nuclei in fusion experiments and have become extremely useful in a variety of experiments to synthesize and study the properties of the heaviest elements. The traditional gas-filled separators are configured with the gas-filled dipole magnet directly after the production target to collect and analyze the reaction products which are subsequently focused with a quadruple doublet on implantation detectors for decay measurements. Systematic measurements of the mean charge for a variety of heavy ions have been performed with the Dubna gas-filled separator (Subotic et al. 2002) and have demonstrated the velocity dependence of q¯ over the velocity range relevant for the synthesis of the heaviest elements (see Fig. 8). There are a large number of examples of gas-filled separators for fusion products of the heaviest elements that include SASSY (Ghiorso et al. 1988), GARIS (Morita et al. 1992a), the Dubna GFRS (Lazarev et al. 1994), the ANL-GFS (Rehm et al. 1996), the JYFL-GFS (Enqvist et al. 1997), the GSI-GFS (Ninov et al. 1995), the BGS (Ninov et al. 1998),

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Fig. 8 Measured equilibrium mean charge state of low-energy heavy ions in low pressure hydrogen gas with the Dubna gas-filled recoil separator (DGFRS) as a calibration of CSD focusing for experiments to synthesize new elements. (Courtesy: Figure was taken from Oganessian 2007)

TASCA (Semchenkov et al. 2008), and the conversion of the modern vacuum separator VAMOS (Pullanhiotan et al. 2008) into VAMOS-GFS (Theisen 2017). As an example, the ultrahigh sensitivity and selectivity of the gas-filled separator GARIS (Morita et al. 1992b) was demonstrated by the discovery of the element nihonium (Z = 113) through the separation and identification of three individual fusion events observed during 576 days of irradiation (Morita et al. 2012). The element 113 nuclei were discovered in cold fusion reactions despite the very small production cross section of 22f b = 22 · 10−39 cm2 . The synthesis of the heaviest elements, i.e., beyond Z = 113, has relied on the so-called hot fusion reactions with 48 Ca beams (Oganessian et al. 2006) that produce evaporation residues at higher excitation energies than so-called cold fusion reactions. The background from various reaction products in hot fusion can be higher than in cold fusion, and, therefore, an upgrade of the traditional gas-filled separator design was needed. An example of an advanced gas-filled separator is GARIS-II which has more ion-optical elements with a larger acceptance for hot fusion and transfer reactions. The typical ion-optical elements and the optical properties of a modern gas-filled separator GARIS-II in Japan (Kaji et al. 2013) are presented in Fig. 9. The small dipole-magnet D2 at the end of the separator (see Fig. 9) functions to further reduce the background of scattered particles with

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its small dispersion. Such a small dipole magnet (in this case 7.5◦ bending angle) was also placed at the exit of the velocity filter separator SHIP (Hofmann and Münzenberg 2000) to provide a similar background reduction. The major scientific goal of the new generation of gas-filled and vacuum separators is to attempt to extend the superheavy elements beyond the presently known element Og (Z = 118) by studies of the products of hot fusion reactions and multinucleon transfer reactions. There are several features of the CSDs of low-energy heavy ions that should be noted. In the low-velocity regime, the mean charge states of heavy ions exiting solid materials are generally higher than those exiting or in gases at the low densities used in gas-filled separators. The reason for this is that the ions undergoing collisions in the gas are generally in their ground states due to the longer mean free path of ions in the low pressure gas. On the other hand, in solids the mean free path, so to speak, is many orders of magnitude shorter so that the time in between the collisions is much shorter and collisions with the exotic ions in excited states generally result in a higher ionization states. A related feature is that the charge-state distributions of recoiling, low-energy, exotic nuclei in vacuum can be shifted after they emerge from the target material if they are in highly excited atomic or nuclear states. For example, relatively fast decays that emit conversion electrons and/or isomeric states that decay with lifetimes less than a few nanoseconds can perturb the equilibrium charge-state distribution established in the target material. To avoid the losses due to this effective broadening of the CSD, an additional thin charge-reset foil can be mounted a short distance downstream from the target to reestablish the “normal” or expected equilibrium CSD for the mean velocity of the reaction products exiting the target. This will work even for species that emerge from the target in highly excited atomic or nuclear states which might have quite shifted CSDs and thus have

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significant transmission losses. For typical fusion products, this charge-reset foil can be on the order of 20–30 μg/cm2 thickness. Such a thin carbon charge-changing foil is usually mounted after the target and before the ion-optical entrance lenses at the velocity filter SHIP (Münzenberg et al. 1979).

Overview of the Different Separation Methods Having reviewed the basic operating principles and ion-optical properties of the various reactions to produce and the systems to separate the exotic nuclei, this is a reasonable point to summarize the overall production rates. A word of caution is necessary here; the values that were given in various tables are order-of-magnitude estimates and should be viewed as such. Here a general framework to compare systems is provided. One will see a remarkable similarity in the overall production rates. Recall that the final intensity I ( 1s ) of an exotic-nuclear product is determined by the product of the cross section σ (E) (cm2 ), the number of incident projectile nuclei Nproj ( 1s ), the effective number of target atoms Ntar ( cm1 2 ), and the overall efficiency of the separation process ε: I = σ (E)Nproj Ntar ε

(7)

While this equation may appear simple, the production cross section is usually energy dependent, and the primary beam loses energy during passing through the target so that the intensity has to be calculated by integrating over the target thickness taking into account the energy loss and at low energies ends at the Coulomb barrier. At high energies, the cross sections are less dependent on energy, but the number of beam projectiles decreases due to the induced reactions. The exotic nuclei themselves can be either lost or formed in very thick targets due to secondary reactions. In this expression, ε represents the efficiency of the entire separation process. This efficiency is defined as the ratio of the final secondary beam intensity that arrives at the experimental setup relative to the intensity of the reaction products created in the target. Therefore, ε is a product of a series of partial efficiencies each of which depends on the technique applied for the production of the radioactive ion beam. All in all, the estimated order-of-magnitude efficiencies of the separator systems are close to unity with the exception of the target fragmentation or ISOL systems due to the effect of migration through the target matrix and volatilization of the products. These processes are extremely dependent on the chemical nature of the product and also on its half-life so that only a very general estimate of epsilon was made. A detailed discussion of the measured yields from the ISOLDE facility was given by Lukic et al. (2006). With all of these caveats, one can see in Table 2 the lower limits of the production rates are similar being on the order of 10−2 /s while highest rates from the most advanced systems are three or four orders of magnitude larger.

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Table 2 Typical values of key parameters from the representative accelerator systems used to produce exotic nuclei. The values of luminosity are taken from Table 1 and the cross sections from Table 2, both in Part 1. These numbers are solely for comparison and do not necessarily represent actual results

4 × 1036

Isotope, Cross section cm2 292 Og, 0.5 pb 100 Sn, 5.8 pb 78 Ni, 0.02 pb 100 Sn, 5.8 pb 78 Ni, 0.02 pb 100 Sn, 5.8 pb 78 Ni, 0.02 pb 132 Sn, 2mb

1 × 1036

132 Sn,

Accelerator HI-LINAC Cyclotrons

Energy MeV/u 10 350

Reaction HI Fusion Proj. Frag.

Luminosity cm−2 s−1 2 × 1033 4 × 1035

LINAC

200

Proj. Frag.

3 × 1037

Proj. Frag., Fission Spallation, Fission Spallation, Fission

3 × 1035

HI 1500 Synchrotron Cyclotron 500 Proton 1400 Synchrotron

2mb

Separator Efficiency 0.9 0.9

0.1

Overall rate s−1 0.0007 2. 0.008 20. 0.08 2. 0.005 7 × 108

0.1

3 × 108

0.9 1

Summary, Key Achievements, and Future Prospects At this point, one can see that a very large range of production methods and separation techniques have been developed over many years to provide the widest possible range of exotic nuclei for study. The properties of the nuclear reaction products of interest have dictated the general features of the separators, and the details of the separator design have been adapted over time to increasing beam power and decreasing production cross sections. The general realm of the presentday exotic beam production facility is indicated in Fig. 10 that indicates the heavy-ion beam energy as a function of atomic number. The first impression from this presentation is the large extent of the coverage available, ranging from the Coulomb barrier up to relativistic energies. Another important feature to note is that beams of exotic nuclei across the full range of atomic numbers are available as fully stripped ions at the highest beam energies. While not discussed here, the low-energy and thermalized exotic nuclei are now being fed into reaccelerators to provide low-emittance beams at near-Coulomb barrier energies for nuclear reaction studies. A number of accelerator facilities around the world were constructed to use the techniques described above to produce secondary beams of exotic nuclei for nuclear science. In the first generations, the equipment was often adapted from a variety of purposes (Grunder 2002; Jonson 2002; Kubo 2003), but subsequent generations have been constructed through upgrades, improvements, or completely new construction (Henning 2003; Kubo et al. 2007; Gales 2011; Gade and Sherrill 2016). The fusion process to produce the heaviest chemical elements now rely on

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Fig. 10 Realm of typical exotic-nuclear beam separators and reaccelerator systems is shown in the space of beam energy (MeV/u) and atomic number. The Coulomb barrier for reactions with an uranium target and the energy limit for the condition that at least 90% of the ions are fully stripped are included for orientation. Complimentary facilities are applied in the different energy domains characterized by different production reactions and separation methods. The description of more developments in this field in detail can be found in the Proceedings of the EMIS conferences over several decades

very long runs using upgraded high-power accelerators and highly refined separators that can select individual nuclei. The two main production techniques for exotic heavy-ion beams (see Figure 9 in Part 1) of target spallation and fission (ISOL) and projectile fragmentation and fission are completely complimentary and, in fact, rely on very similar underlying nuclear reactions. Present-day facilities also rely on highintensity primary heavy-ion beams and large separators that provide intense beams of specific, individual exotic nuclei that can be separated from literally thousands of other reaction products. The fraction of exotic fused nuclei that have been produced, separated, and studied are 5 × 10−18 per pb and 5 × 10−21 per fb, and similarly, the fractions of exotic projectile fragments produced and studied have similar cross sections of 5 × 10−13 per pb and 5 × 10−16 per fb. These remarkably selective systems support an impressive set of capabilities to the nuclear science community. An overview of the wide-ranging nuclear science programs carried out by researchers around the globe is shown in Fig. 11. The program addresses broad challenges to our understanding of nuclear structure and reactions by employing

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Fig. 11 Illustration of some of the achievements and future prospective discoveries made possible with the exotic-nuclear beam facilities and dedicated separator systems (see the text)

sensitive experimental techniques that fully rely on the exotic nuclei produced and separated by the techniques described here. The highlighted examples show the broad range of novel nuclear phenomena that have been established and the forefront science questions such as the limits of nuclear existence in neutron number (both low and high) and in atomic number, the nature of the nuclear force and new forms of nuclear matter, novel decay modes and shapes of the most unstable nuclei, and the origin of the elements in nature and the astrophysical processes that produce them, extending the period table. In addition, these systems have become important sources of exotic nuclei for nuclear medicine and biomedical research.

References R. Anne, Sissi at ganil. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 126(1), 279–283 (1997). International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications R. Anne, D. Bazin, A. Mueller, J. Jacmart, M. Langevin, The achromatic spectrometer lise at ganil. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 257(2), 215–232 (1987) R. Anne, A. Mueller, Lise 3: a magnetic spectrometer-wien filter combination for secondary radioactive beam production. Nucl. Instrum. Methods Phys. Res. B 70(1–4), 276–285 (1992) A. Artukh, Y.L. Obukhov, V. Stchepunov, M. Nagayenko, Y.P. Severgin, V. Titov, V. Belavenko, I. Vishnievski, The magnetic channel designed for separating beams of radioactive nuclei combas. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 306(1–2), 123–127 (1991) H. Bethe, Zur theorie des durchgangs schneller korpuskularstrahlen durch materie. Ann. Phys. (Leipzig) 5, 325 (1930)

86

H. Geissel and D. J. Morrissey

L. Bianchi, B. Fernandez, J. Gastebois, A. Gillibert, W. Mittig, J. Barrette, Speg: an energy loss spectrometer for ganil. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 276(3), 509–520 (1989) N. Bohr, The Penetration of Atomic Particles Through Matter (Munksgaard, Copenhagen, 1948) J. Castro, J. Quivey, J. Lyman, G. Chen, T. Phillips, C. Tobias, Radiotherapy with heavy charged particles at lawrence berkeley laboratory. J. Can. Assoc. Radiol. 31(1), 30–34 (1980) B.L. Cohen, C.B. Fulmer, Fission-fragment mass separator and the nuclear charge distribution of fission fragments of a single mass. Nucl. Phys. 6, 547–560 (1958) J. Dufour, R. Del Moral, H. Emmermann, F. Hubert, D. Jean, C. Poinot, M. Pravikoff, A. Fleury, H. Delagrange, K.-H. Schmidt, Projectile fragments isotopic separation: application to the lise spectrometer at ganil. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 248(2–3), 267–281 (1986) M. Durante, H. Paganetti, Nuclear physics in particle therapy: a review. Rep. Prog. Phys. 79(9), 096702 (2016) M. Durante, K. Parodi, Radioactive beams in particle therapy: past, presentand future. Front. Phys. 8, 00326, (2020) https://www.frontiersin.org/articles/10.3389/fphy.2020.00326 D.K. Ebner, T. Kamada, The emerging role of carbon-ion radiotherapy. Front. Oncol. 6, 140 (2016) T. Enqvist, J.Uusitalo, S. Hurskanen, P. Kuusiniemi, M. Leino, W. Trzaska, K. Helariutta, P. Jones, R. Julin, S. Juutinen, A. Savelius, K. Eskola, R. Allatt, R. Page, V. Ninov, C. Stodel, Recent applications of the JYFL gas-filled recoil separator. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 126(1–4), 344–346 (1997) B. Franczak, Mirko—an interactive program for beam lines and synchrotrons. Computing in Accelerator Design and Operation (Springer, 1984). Springer Berlin Heidelberg, pp. 170–175 A. Gade, B. Sherrill, NSCL and FRIB at michigan state university: nuclear science at the limits of stability. Phys. Scr. 91, 053003 (2016) S. Gales, GANIL-SPIRAL2: a new era. J. Phys.: Conf. Ser. 267, 012009 (2011) H. Geissel, P. Armbruster, K.H. Behr, A. Brünle, K. Burkard, M. Chen, H. Folger, B. Franczak, H. Keller, O. Klepper, B. Langenbeck, F. Nickel, E. Pfeng, M. Pfützner, E. Roeckl, K. Rykaczewski, I. Schall, D. Schardt, C. Scheidenberger, K.H. Schmidt, A. Schröter, T. Schwab, K. Sümmerer, M. Weber, G. Münzenberg, T. Brohm, H.-G. Clerc, M. Fauerbach, J.J. Gaimard, A. Grewe, E. Hanelt, B. Knödler, M. Steiner, B. Voss, J. Weckenmann, C. Ziegler, A. Magel, H. Wollnik, J.P. Dufour, Y. Fujita, D.J. Vieira, B.M. Sherrill, The GSI projectile fragment separator (FRS): a versatile magnetic system for relativistic heavy ions. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1), 286–297 (1992) H. Geissel, T. Schwab, P. Armbruster, J.P. Dufour, E. Hanelt, K.-H. Schmidt, B.M. Sherrill, G. Münzenberg, Ions penetrating through ion-optical systems and matter – non-liouvillian phase-space modelling. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 282(1), 247–260 (1989) H. Geissel, H. Weick, C. Scheidenberger, R. Bimbot, D. Gardes, Experimental studies of heavyion slowing down in matter. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 195(1–2), 3–54 (2002) H. Geissel, H. Weick, M. Winkler, G. Münzenberg, V. Chichkine, M. Yavor, T. Aumann, K.H. Behr, M. Böhmer, A. Brünle, K. Burkard, J. Benlliure, D. Cortina-Gil, L. Chulkov, A. Dael, J.-E. Ducret, H. Emling, B. Franczak, J. Friese, B. Gastineau, J. Gerl, R. Gernhäuser, M. Hellström, B. Jonson, J. Kojouharova, R. Kulessa, B. Kindler, N. Kurz, B. Lommel, W. Mittig, G. Moritz, C. Mühle, J.A. Nolen, G. Nyman, P. Roussell-Chomaz, C. Scheidenberger, K.-H. Schmidt, G. Schrieder, B.M. Sherrill, H. Simon, K. Sümmerer, N.A. Tahir, V. Vysotsky, H. Wollnik, A.F. Zeller, The super-FRS project at GSI. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 71–85 (2003). 14th International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications A. Ghiorso, S. Yashita, M. Leino, L. Frank, J. Kalnins, P. Armbruster, J.-P. Dufour, P. Lemmertz, Sassy, a gas-filled magnetic separator for the study of fusion reaction products. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 269(1), 192–201 (1988)

2 Exotic Nuclei and Their Separation, Using Atomic Interactions

87

K. Gregorich, Simulation of recoil trajectories in gas-filled magnetic separators. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 711, 47–59 (2013) H.A. Grunder, Radioactive nuclear beam facilities in north America: status and outlook. Nucl. Phys. A 701(1), 43–48 (2002). 5th International Conference on Radioactive Nuclear Beams M. Hausmann, A.M. Aaron, A.M. Amthor, M. Avilov, L. Bandura, R. Bennett, G. Bollen, T. Borden, T. Burgess, S.S. Chouhan, V.B. Graves, W.Mittig, D.J. Morrissey, F. Pellemoine, M. Portillo, R.M. Ronningen, M. Schein, B.M. Sherrill, A.F. Zeller, Design of the advanced rare isotope separator aris at FRIB. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 349–353 (2013). XVIth International Conference on ElectroMagnetic Isotope Separators and Techniques Related to Their Applications, 2–7 Dec 2012 at Matsue W.F. Henning, Plans for the future facility at GSI. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 725–729 (2003). 14th International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications S. Hofmann, G. Münzenberg, The discovery of the heaviest elements. Rev. Mod. Phys. 72(3), 733 (2000) N. Iwasa, H. Geissel, G. Münzenberg, C. Scheidenberger, T. Schwab, H. Wollnik, Mocadi, a universal monte carlo code for the transport of heavy ions through matter within ion-optical systems. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 126(1–4), 284– 289 (1997) B. Jonson, Present and future RNB facilities in europe. Nucl. Phys. A 701(1), 35–42 (2002). 5th International Conference on Radioactive Nuclear Beams D. Kaji, K. Morimoto, N. Sato, A. Yoneda, K. Morita, Gas-filled recoil ion separator GARIS-II. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 311–314 (2013) T. Kawabata, G. Berg, T. Kubo, H. Sakai, S. Shimoura, T. Uesaka, High resolution beam line for the sharaq spectrometer. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 266(19), 4201–4204 (2008). Proceedings of the XVth International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications T. Kobayashi, N. Chiga, T. Isobe, Y. Kondo, T. Kubo, K. Kusaka, T. Motobayashi, T. Nakamura, J. Ohnishi, H. Okuno, H. Otsu, T. Sako, H. Sato, Y. Shimizu, K. Sekiguchi, K. Takahashi, R. Tanaka, K. Yoneda, Samurai spectrometer for ri beam experiments. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 294–304 (2013). XVIth International Conference on ElectroMagnetic Isotope Separators and Techniques Related to Their Applications, 2–7 Dec 2012 at Matsue T. Kubo, In-flight ri beam separator BigRIPS at riken and elsewhere in Japan. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 97–113 (2003) T. Kubo, M. Ishihara, N. Inabe, H. Kumagai, I. Tanihata, K. Yoshida, T. Nakamura, H. Okuno, S. Shimoura, K. Asahi, The riken radioactive beam facility. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1–4), 309–319 (1992) T. Kubo, K. Kusaka, K. Yoshida, A. Yoshida, T. Ohnishi, M. Ohtake, Y. Yanagisawa, N. Fukuda, T. Haseyama, Y. Yano, N. Kakutani, T. Tsuchihashi, K. Sato, Status and overview of superconducting radioactive isotope beam separator BigRIPS at riken. IEEE Trans. Appl. Supercond. 17(2), 1069–1077 (2007) T. Kubo, D. Kameda, H. Suzuki, N. Fukuda, H. Takeda, Y. Yanagisawa, M. Ohtake, K. Kusaka, K. Yoshida, N. Inabe, T. Ohnishi, A. Yoshida, K. Tanaka, Y. Mizoi, BigRIPS separator and ZeroDegree spectrometer at RIKEN RI beam factory. Prog. Theor. Exp. Phys. 2012(1), 03C003 (2012) N. Lassen, H ρ-distribution of fission fragments. Phys. Rev. 68(5–6), 142 (1945) Y.A. Lazarev, Y.V. Lobanov, Y.T. Oganessian, V.K. Utyonkov, F.S. Abdullin, G.V. Buklanov, B.N. Gikal, S. Iliev, A.N. Mezentsev, A.N. Polyakov, I.M. Sedykh, I.V. Shirokovsky, V.G. Subbotin, A.M. Sukhov, Y.S. Tsyganov, V.E. Zhuchko, R.W. Lougheed, K.J. Moody, J.F. Wild, E.K. Hulet, J.H. McQuaid, Discovery of enhanced nuclear stability near the deformed shells n = 162 and z = 108. Phys. Rev. Lett. 73, 624–627 (1994) M. Leino, In-flight separation with gas-filled systems. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 126(1–4), 320–328 (1997)

88

H. Geissel and D. J. Morrissey

J. Lindhard, A.H. Sørensen, Relativistic theory of stopping for heavy ions. Phys. Rev. A 53, 2443 (1996) S. Lukic, F. Gevaert, A. Kelic, M. Ricciardi, K.-H. Schmidt, O. Yordanov, Systematic comparison of isolde-sc yields with calculated in-target production rates. Nucl. Instrum. Methods Phys. Res. A 565, 784–800 (2006) A. Magel, H. Geissel, B. Voss, P. Armbruster, T. Aumann, M. Bernas, B. Blank, T. Brohm, H.-G. Clerc, S. Czajkowski et al., First spatial isotopic separation of relativistic uranium projectile fragments. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 94(4), 548–554 (1994) K. Morita, A. Yoshida, T. Inamura, M. Koizumi, T. Nomura, M. Fujioka, T. Shinozuka, H. Miyatake, K. Sueki, H. Kudo et al., Riken isotope separator on-line GARIS/IGISOL. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1–4), 220–225 (1992a) K. Morita, A. Yoshida, T. Inamura, M. Koizumi, T. Nomura, M. Fujioka, T. Shinozuka, H. Miyatake, K. Sueki, H. Kudo et al., Riken isotope separator on-line GARIS/IGISOL. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1–4), 220–225 (1992b) K. Morita, K. Morimoto, D. Kaji, H. Haba, K. Ozeki, Y. Kudou, T. Sumita, Y. Wakabayashi, A. Yoneda, K. Tanaka et al., New result in the production and decay of an isotope, 278113, of the 113th element. J. Phys. Soc. Jpn. 81(10), 103201 (2012) D.J. Morrissey, Systematics of momentum distributions from reactions with relativistic ions. Phys. Rev. C 39, 460–470 (1989) D.J. Morrissey, B.M. Sherrill, M. Steiner, A. Stolz, I. Wiedenhoever, Commissioning the a1900 projectile fragment separator. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 90–96 (2003). 14th International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications G. Münzenberg, W. Faust, S. Hofmann, P. Armbruster, K. Güttner, H. Ewald, The velocity filter ship, a separator of unslowed heavy ion fusion products. Nucl. Instrum. Methods 161(1), 65–82 (1979) V. Ninov, P. Armbruster, F. Heßberger, S. Hofmann, G. Münzenberg, Y. Fujita, M. Leino, A. Lüttgen, Separation of actinide-made transurania by a gas-filled magnetic separator. Nucl. Inst. Methods Phys. Res. A 357(2–3), 486–494 (1995) V. Ninov, K.E. Gregorich, C.A. McGrath, The berkeley gas-filled separator. AIP Conf. Proc. 455(1), 704–707 (1998) T. Nishi, K. Itahashi, G. Berg, H. Fujioka, N. Fukuda, N. Fukunishi, H. Geissel, R. Hayano, S. Hirenzaki, K. Ichikawa et al., Spectroscopy of pionic atoms in sn 122 (d, he 3) reaction and angular dependence of the formation cross sections. Phys. Rev. Lett. 120(15), 152505 (2018) Y. Oganessian, Heaviest nuclei from 48ca-induced reactions. J. Phys. G: Nucl. Part. Phys. 34(4), R165–R242 (2007) Y.T. Oganessian, V.K. Utyonkov, Y.V. Lobanov, F.S. Abdullin, A.N. Polyakov, R.N. Sagaidak, I.V. Shirokovsky, Y.S. Tsyganov, A.A. Voinov, G.G. Gulbekian, S.L. Bogomolov, B.N. Gikal, A.N. Mezentsev, S. Iliev, V.G. Subbotin, A.M. Sukhov, K. Subotic, V.I. Zagrebaev, G.K. Vostokin, M.G. Itkis, K.J. Moody, J.B. Patin, D.A. Shaughnessy, M.A. Stoyer, N.J. Stoyer, P.A. Wilk, J.M. Kenneally, J.H. Landrum, J.F. Wild, R.W. Lougheed, Synthesis of the isotopes of elements 118 and 116 in the 249 Cf and 245 Cm +48 Ca fusion reactions. Phys. Rev. C 74, 044602 (2006) T. Ohnishi, T. Kubo, K. Kusaka, A. Yoshida, K. Yoshida, M. Ohtake, N. Fukuda, H. Takeda, D. Kameda, K. Tanaka et al., Identification of 45 new neutron-rich isotopes produced by inflight fission of a 238u beam at 345 MeV/nucleon. J. Phys. Soc. Jpn. 79(7), 073201 (2010) M. Paciotti, J. Bradbury, J. Helland, R. Hutson, E. Knapp, O. Rivera, H. Knowles, G. Pfeufer, Tuning of the first section of the biomedical channel at LAMPF. IEEE Trans. Nucl. Sci. 22(3), 1784–1789 (1975) W. Plaß, T. Dickel, S. Purushothaman, P. Dendooven, H. Geissel, J. Ebert, E. Haettner, C. Jesch, M. Ranjan, M. Reiter et al., The FRS ion catcher–a facility for high-precision experiments with stopped projectile and fission fragments. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 317, 457–462 (2013)

2 Exotic Nuclei and Their Separation, Using Atomic Interactions

89

S. Pullanhiotan, M. Rejmund, A. Navin, W. Mittig, S. Bhattacharyya, Performance of VAMOS for reactions near the coulomb barrier. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 593(3), 343–352 (2008) E. Rehm, C. Jiang, M. Paul, D. Blumenthal, J. Gehring, D. Henderson, J. Nickles, R. Pardo, A. Roberts, J. Schiffer, R. Segel, The use of a gas-filled-magnet in experiments with radioactive ion beams. Nucl. Instrum. Methods Phys. Res. Sect. A 370(1), 438–444 (1996) H. Sakurai, Nuclear physics with ri beam factory. Front. Phys. 13(6), 1–15 (2018) C. Scheidenberger, H. Geissel, H. Mikkelsen, F. Nickel, T. Brohm, H. Folger, H. Irnich, A. Magel, M. Mohar, G. Münzenberg et al., Direct observation of systematic deviations from the bethe stopping theory for relativistic heavy ions. Phys. Rev. Lett. 73(1), 50 (1994) C. Scheidenberger, H. Geissel, H. Mikkelsen, F. Nickel, S. Czajkowski, H. Folger, H. Irnich, G. Münzenberg, W. Schwab, T. Stöhlker et al., Energy-loss-straggling experiments with relativistic heavy ions in solids. Phys. Rev. Lett. 77(19), 3987 (1996) C. Scheidenberger, T. Stöhlker, W. Meyerhof, H. Geissel, P. Mokler, B. Blank, Charge states of relativistic heavy ions in matter. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 142(4), 441–462 (1998) K.-H. Schmidt, E. Hanelt, H. Geissel, G. Münzenberg, J.P. Dufour, The momentum-loss achromat – a new method for the isotopical separation of relativistic heavy ions. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 260(2), 287–303 (1987) A. Semchenkov, W. Brüchle, E. Jäger, E. Schimpf, M. Schädel, C. Mühle, F. Klos, A. Türler, A. Yakushev, A. Belov, T. Belyakova, M. Kaparkova, V. Kukhtin, E. Lamzin, S. Sytchevsky, The transactinide separator and chemistry apparatus (tasca) at GSI – optimization of ion-optical structures and magnet designs. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 266(19), 4153–4161 (2008). Proceedings of the XVth International Conference on Electromagnetic Isotope Separators and Techniques Related to Their Applications. L. Sheng, X. Zhang, J. Zhang, J. Yang, Z. Sun, L. Mao, W. Wu, D. Yin, S. Ruan, G. Shen et al., Ion-optical design of high energy fragment separator (HFRS) at HIAF. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 469, 1–9 (2020) B.M. Sherrill, D.J. Morrissey, J.A. Nolen, J.A. Winger, The a1200 projectile fragment separator. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 56–57, 1106–1110 (1991) T. Shimoda, H. Miyatake, S. Morinobu, Design study of the secondary-beam line at RCNP. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 70(1–4), 320–330 (1992) P. Sigmund, Particle Penetration and Radiation Effects Volume 2, Springer Series in Solid State Sciences, vol. 179 (2014) K. Subotic, Y.T. Oganessian, V. Utyonkov, Y.V. Lobanov, F.S. Abdullin, A. Polyakov, Y.S. Tsyganov, O. Ivanov, Evaporation residue collection efficiencies and position spectra of the Dubna gas-filled recoil separator. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 481(1–3), 71–80 (2002) H. Takeda, Y. Tanaka, H. Suzuki, D. Ahn, B. Franczak, N. Fukuda, H. Geissel, E. Haettner, N. Inabe, K. Itahashi et al., New ion-optical operating modes of the BigRIPS and ZeroDegree spectrometer for the production and separation of high-quality rare isotopes beams and highresolution spectrometer experiments. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 463, 515–521 (2020) O. Tarasov, D. Bazin, Lise++: radioactive beam production with in-flight separators. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 266(19–20), 4657–4664 (2008) C. Theisen, Perspectives of super-heavy nuclei research with the upcoming separator-spectrometer setup s3 at ganil/spiral2 – the VAMOS gas-filled separator and agata. EPJ Web Conf. 163, 00059 (2017) T. Uesaka, S. Shimoura, H. Sakai, G. Berg, K. Nakanishi, Y. Sasamoto, A. Saito, S. Michimasa, T. Kawabata, T. Kubo, The high resolution sharaq spectrometer. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 266(19), 4218–4222 (2008). Proceedings of the XVth International Conference on Electromagnetic Isotope Separators and Techniques Related to THEIR Applications

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H. Weick, H. Geissel, C. Scheidenberger, F. Attallah, T. Baumann, D. Cortina, M. Hausmann, B. Lommel, G. Münzenberg, N. Nankov et al., Slowing down of relativistic few-electron heavy ions. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 164, 168–179 (2000a) H. Weick, H. Geissel, C. Scheidenberger, F. Attallah, D. Cortina, M. Hausmann, G. Münzenberg, T. Radon, H. Schatz, K. Schmidt et al., Drastic enhancement of energy-loss straggling of relativistic heavy ions due to charge-state fluctuations. Phys. Rev. Lett. 85(13), 2725 (2000b) J. Winfield, H. Geissel, B. Franczak, T. Dickel, E. Haettner, E. Kazantseva, T. Kubo, S. Litvinov, W. Plaß, S. Ratschow, C. Scheidenberger, Y. Tanaka, H. Weick, M. Winkler, M. Yavor, Ionoptical developments tailored for experiments with the super-FRS at fair. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 491, 38–51 (2021)

3

Reactions for Production of Exotic Nuclei N. Antonenko, J. Benlliure, A. Karpov, and D. J. Morrissey

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Energy Nuclear Reactions to Create Exotic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fusion Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of Fusion Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capture of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formation of the Compound Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decay of Compound Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peculiarities of Fusion Reactions Leading to Superheavy Nuclei . . . . . . . . . . . . . . . . . . . . Multinucleon Transfer Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-Energy Nuclear Reactions to Create Exotic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Description of Fragmentation Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual Nuclei Produced in Fragmentation Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fragmentation of Fissile Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fragmentation of Neutron-Rich Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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N. Antonenko · A. Karpov () Joint Institute for Nuclear Research, Dubna, Russia e-mail: [email protected]; [email protected] J. Benlliure () Universidade de Santiago de Compostela, Santiago de Compostela, Spain e-mail: [email protected] D. J. Morrissey National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_99

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Abstract

An overview of the nuclear reactions that are used to produce exotic nuclei along with the relevant theoretical models and predictions of their cross sections will be presented in this chapter. The chapter concludes with a discussion of key goals and present-day limitations in the production of exotic nuclei from the standpoint of nuclear reactions.

Introduction Models of nuclear structure were developed over the last century by measuring and then predicting the properties of the stable isotopes (less than 250) and other nearby radioactive ones. However, it became clear that many more unstable nuclei could be produced and the prediction of their existence and their properties would provide important tests of those models. The drive to establish the limits of nuclear stability has led to the exploration of nuclear reaction mechanisms and the development of novel separation techniques and instruments that can collect and identify single isotopes from the enormous overwhelming flood of other reaction products. The study of these short-lived and exotic nuclei has played a critical role in the recent advances in theoretical nuclear physics and more recently in nuclear astrophysics. The predictions of models of nuclear reactions and structure based on the relatively few stable nuclei have been shown to be inadequate to describe exotic nuclei that lie far in proton or neutron number from stability. The shell structure that is so robust for stable nuclei has been clearly shown to evolve, and new features of nuclear size, shape, and decay modes have been shown to exist among the most exotic nuclei. In addition, in recent years, the prediction of the properties of the most neutron-rich nuclei has been shown to be critical in predicting the production of heavy elements in stars and stellar evolution. The importance of exotic nuclei in unravelling various aspects of nuclear structure and astrophysical production of all nuclei can be seen in the schematic overview of the impact of various nuclear reaction mechanisms on the chart of nuclides shown in Fig. 1. This chapter contains an overview of the nuclear reactions with the kinematic and other properties of the exotic isotopes that are important and useful followed by detailed presentations of the most important reactions used to produce the most exotic nuclei. All exotic nuclei have to be produced in nuclear reactions, and thus the availability of intense primary beams is the starting point for their production. In addition, it should be clear that the most exotic, unknown nuclei will be produced with extremely low probabilities. Thus, highly selective physical separation of individual exotic nuclei from the primary beam and from other less interesting or previously studied nuclei has been at the heart of progress in nuclear physics for many years. A discussion of the principles and concepts that underlie the different separation methods applied to these reaction products is presented in the preceding chapter.

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Superheavy Elements 114

Bf = 4 MeV

Shapes & Collective Phenomena

Z

s-process

Sp= 0

162

New

82

126

Territory

New Decay Modes

S n= 0

rp-process 50

r-process

2p-radioactivity

Stellar Matter 82 Skin-nuclei

28

p-Halo-nuclei 20 50 8

28 20

2 2

8

n-Halo-nuclei

New Shell Structure New Forms of Matter

N

Fig. 1 An overview of various research possible with exotic nuclei shown on a chart of nuclides including the known limits of bound nuclei called the driplines where the separation energies, Sp or Sn , become zero. Previously unknown and unexpected properties of exotic nuclei such as novel decay modes or nuclear matter distributions (halos, skins, cluster formation) that have been studied with exotic nuclei lie in the yellow region, while a large area of neutron-rich nuclei is predicted to be waiting to be discovered in the dotted region stretching out to the neutron dripline in the heavy mass region. Presently available nuclei just begin to reach the nuclei thought to be critical for production in stars lie in red or narrow blue regions

The basic properties of the products of nuclear reactions are important in order to understand the opportunities and requirements for separators. Individual nuclear reactions in thin targets can be used to produce exotic nuclei in-flight for prompt separation as well as bulk reactions in thick targets for delayed release and separation. The total production rate depends on the product of beam fluence, the effective number density of target atoms, and the cross section. However, to obtain the rate of separated exotic nuclei, one must include the total efficiency of the separator and any corrections due to decay before delivery. There are certain nuclear reactions that have come to play a preeminent role in the production of exotic nuclei due to either very high production rates, e.g., target spallation by intense proton beams, or due to very favorable kinematics for efficient collection before decay, e.g., projectile fragmentation. This chapter provides a broad overview of important features of nuclear reactions used extensively to produce exotic nuclei, model predictions, and some kinematic properties of those nuclei.

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Virtually, all nuclear reactions that are used to produce exotic nuclei occur in collisions in which one of the reacting nuclei is at rest (i.e., the target nucleus) while the other (the projectile nucleus) is provided by an accelerator. In the present discussion of the production of exotic nuclei, at least one of the participants in the reaction needs to be a relatively large nucleus for reasons that will become clear below. If the projectile is lighter than the target nucleus, one has normal kinematics in which any light product is generally fast moving and the heavy product is slow moving and can have a wide angular distribution in the laboratory frame. Collection of any slow-moving products puts relatively large constraints on the separator. On the other hand, if the projectile is heavier than the target nucleus, then one has reverse kinematics in which the heavy product is fast moving and can have a narrow angular distribution of the projectile fragments. The production of exotic nuclei in reverse kinematics has had a large beneficial impact on the production and study of the most exotic nuclei. The production of exotic nuclei can occur via a variety of different reaction mechanisms, and it is probably true that all reaction mechanisms have been used at one time or another to produce what were exotic nuclei at the time. This chapter will concentrate on the most important processes used at present. Weisskopf presented a simple conceptual model for categorizing light ion-induced nuclear reactions (Weisskopf 1957) according to their absorption by a heavy nucleus. The production of exotic nuclei by protons and deuterons generally relies on target fragments, also called spallation, from central collisions and recently by direct reactions in reverse kinematics. The reactions induced by heavy ions with targets other than the very lightest nuclei have certain unique characteristics that are qualitatively different from light ion-induced reactions. First, as both nuclei are positively charged, the repulsive Coulomb force acting between the beam and target nucleus puts a significant threshold on the reactions at low energies and dominates the kinematics of the products. Second, the de Broglie wavelength of a heavy ion even at the low energy of 5 MeV/nucleon (generally above the barrier) is small compared to the dimensions of the nuclei. As a result, the interactions of these ions can be described semi-classically, and the small wavelengths allow relatively large orbital angular momentum to be present in these collisions. Hodgson et al. (1997) extended the Weisskopf framework to provide a general classification of reactions that occur between heavy ions as a function of impact parameter. The upper panel of Fig. 2 illustrates the relationship among the various nuclear reaction mechanisms operating at low bombarding energies as a function of impact parameter. The most distant collisions lead to elastic scattering and Coulomb excitation. Coulomb excitation is the transfer of energy between the two nuclei via the long-range Coulomb interaction that excites levels above the ground state in the target or the projectile nucleus (or both). More central, grazing collisions lead to inelastic processes and the onset of nucleon exchange through the shortranged nuclear force. Head-on or near head-on collisions at low kinetic energy can lead to fusion of the reacting nuclei and to the formation of a compound nucleus (CN), particularly if the reaction partners have very different masses. For impact

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Fig. 2 A schematic view of nuclear reactions as a function of impact parameter and separated into two classes by the center-of-mass kinetic energy relative to the Coulomb barrier, VCoul . The kinematics of low-energy reactions are dominated by Coulomb repulsion with complex mechanisms, while the kinematics and mechanisms at high energies are much simpler

parameters between the grazing and head-on collisions, one observes an unusual type of nuclear reaction mechanism called deep inelastic scattering (Schroder and Huizenga 1977) or more recently multinucleon transfer. In deep inelastic scattering, the colliding nuclei touch, partially amalgamate, exchange substantial amounts of energy and mass, continue to rotate due to the large angular momenta present as a dinuclear complex, and then reseparate primarily due to their mutual Coulomb repulsion. This process is also called “quasi-fission” as there is substantial mass and energy exchange between the projectile and target nuclei without the equilibration characteristic of isotropic fragment emission expected following compound nucleus formation in fusion (Hinde et al. 2021) followed by fission. This process was initially explored many years ago and has recently been considered as a production method for exotic nuclei near in mass to the projectile or target (Hirayama et al. 2020). One should note that the products of distant collisions at low energies are concentrated at the so-called grazing angle, determined by the point when the ions just come into contact. On the other hand, central collisions can lead to compound

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nucleus formation and the identity of the projectile is essentially lost. The longtime evolution of fusion reactions and the amnesia of the compound nucleus as to its mode of formation requires that the angular distribution of the products must be symmetric about 90 degrees in the frame of the moving compound nucleus. Fusion continues to play an important role in the production of the heaviest exotic nuclei, but fission of the product nuclei is the most common de-excitation mode which destroys the heavy nucleus (Itkis et al. 2015). The two fission fragments are emitted isotropically in the slow moving frame that gives a wide angular distribution in normal kinematics which becomes narrower in reverse kinematics. The fission process is an important source of neutron-rich medium-mass nuclei, and low-energy fission produces the most neutron-rich products at present. Therefore, Coulomb fission, neutron-induced fission, and spontaneous fission are the most effective at producing the most exotic nuclei, but difficulties remain in collecting and separating short-lived products in normal kinematics. The lower panel of Fig. 2 indicates the so-called kinematic focusing that occurs in heavy-ion reactions when the incident kinetic energy is significantly higher than the Coulomb barrier. If the collision occurs at relatively large impact parameters, then remnants of the projectile and the target can survive and de-excite sometimes forming very unusual nuclear products. The projectile fragment will retain essentially the same velocity as the incident projectile and can be rapidly separated by physical means without regard to their chemical nature (Geissel et al. 1992; Morrissey and Sherrill 1998; Morrissey et al. 2003). On the other hand, target fragments can be produced at high rates by intense proton bombardment and remain nearly at rest; thus, the separation is sensitive to the detailed chemistry of each element (Jonson and Riisager 2010; ISOLDE Collaboration 2021). Central collisions at high energy generally lead to disintegration of the system into many, small nuclear fragments in a process called multifragmentation (Borderie and Rivet 2008). In summary, many different reaction mechanisms have been used over the years to produce exotic nuclei. However, three processes have become mainstays of production for practical reasons, and an overview of the regions on the chart of nuclides prominently produced by these processes is shown in Fig. 3. The fusion process is the predominant process for forming the heaviest nuclei due to it being the only process shown to substantially add nucleons to a heavy nucleus and is discussed in the following section. The fusion process also forms some of the most neutron-deficient nuclei by extensive neutron evaporation. There are indications that deep-inelastic collisions or multi-nucleon transfer (MNT) reaction may provide a new route to heavy neutron-rich nuclei and are discussed next. On the other hand, the reverse-kinematic process of projectile fragmentation has been used to produce an extremely wide range of nuclei lighter than the projectile that are easily and very rapidly separated. Whereas target fragmentation or spallation by very intense light ion beams can produce a similarly large range of products, various techniques have had to be developed to rapidly extract them from the bulk target material for online separation. High-energy projectile fragmentation and fission reactions are then discussed in the next section. Finally, this chapter concludes with a discussion of key goals and limitations in the present-day production of exotic nuclei.

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Fig. 3 Illustrations of the major nuclear reactions used to create exotic nuclei by location in the chart of nuclides. Fission and fragmentation have been used to reach the most neutron-rich below uranium and all of the lightest exotic isotopes. The heaviest elements have been synthesized with cold and hot fusion reactions. Fusion reactions are also well suited to produce the most n-deficient nuclei reaching out to the proton dripline (Sp = 0) above calcium. Multi-nucleon transfer (MNT) reactions may be able to create neutron-rich heavy nucleons up to the heaviest known elements

Fig. 4 Schematic representation of the contributions of different partial waves in terms of partial cross sections to various reaction channels in the collision of heavy ions: fusion, deep-inelastic processes (DIC), and quasi-elastic and elastic scattering. (The figure is taken from the book by Zagrebaev 2019)

Low-Energy Nuclear Reactions to Create Exotic Nuclei Currently, four important reactions provide pathways to the most exotic nuclei: fusion, fragmentation, fission, and multi-nucleon transfer (MNT) (Figs. 3 and 4).

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Fragmentation of heavy nuclei at relativistic energies is very efficient to produce isotopes across the nuclidic chart, from neutron-deficient to neutron-rich; however, it is limited to elements lighter than uranium (Z = 92), since uranium is the heaviest available high-energy beam or bulk target material for irradiation. Fission reactions lead typically to neutron-rich intermediate mass nuclides. Fusion so far offers the only possible way to synthesize transuranium and superheavy nuclei, but it results in rather neutron-deficient isotopes, i.e., those on the left side of the stability line. This is due to the fact that the beta-stability line moves to larger neutron-to-proton ratios with increasing atomic numbers. The prospective use of radioactive ion beams may solve this difficulty, but these beams are characterized by rather low intensities limiting possibility of their use at present. Thus, the presently known limits of the nuclide chart are determined by the capabilities of the available reactions. The smallest cross sections which are currently accessible in these reactions are on the level of picobarns (10−12 barns), which results in average yields of one nucleus per day in fragmentation or fusion reactions of the most exotic nuclei. The predicted cross section limits for MNT reactions are rather unclear at present, but they are definitely not lower than 1 pb. These limits originate from available beam intensities, separator efficiencies, and practical target thicknesses for the respective reactions coupled with the stability of targets under intense irradiation by a high-intensity heavy-ion beam.

Fusion Reactions One of the open reaction channels between two heavy nuclei is the fusion one. Fusion occurs in more or less central collisions of the reaction partners. They completely lose their individual identities to form an excited compound nucleus with ZCN = Zt + Zp and ACN = At + Ap , where Z and A are atomic and mass numbers. This concept of compound nucleus is the central one in the theory of nuclear fusion introduced by Niels Bohr (1936) and includes postulating statistical independence of the formation and decay of this state of a nuclear system, which in turn is based on assumption of different time-scales of fusion (a relatively fast process) and decay of a compound system (a relatively slow process). The total cross section of all reaction channels characterized by a large energy loss and/or a large number of transferred nucleons is conventionally divided into the cross section for deep inelastic reactions and the capture cross section (σcap ) as indicated in Fig. 5. This figure indicates the dominate reaction channel as a function of the entrance channel angular momentum or partial wave. The capture cross section also called the contact cross section can be written as: σcap (E) =

∞ π h¯ 2  (2l + 1)Tl (Ec.m. ), 2μEc.m.

(1)

l=0

where μ is reduced mass, Ec.m. is the center-of-mass bombarding energy, l is the angular momentum of two colliding nuclei, and Tl is the penetrability of the fusion

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Fig. 5 Pictorial representation of the primary stages of low-energy reactions between heavy ions. The entrance channel is shown on the left, and the four reaction exit channels discussed in the text are shown as they progress to the right. The products of intermediate channels are shown in gray, and the final goal of fusion is the evaporation residue (EvR) on the right

barrier at that value of l. After contact, the nuclei may form a compound nucleus (i.e., fuse) with the probability PCN . This leads to the following expression for the fusion probability or fusion cross section (σfus ): σfus (E) =

∞ π h¯ 2  (2l + 1)Tl (Ec.m. ) PCN (l, Ec.m. ), 2μEc.m.

(2)

l=0

Another process that competes with fusion is quasi-fission in which large nuclei reseparate without fully equilibrating and the capture cross section is the sum of fusion and quasi-fission cross sections: σcap = σfus + σqf . Obviously, the fusion and capture cross sections are equal when the compound nucleus is formed from the configuration of two touching nuclei with unit probability (PCN = 1), i.e., when the quasi-fission cross section is negligibly small. This assumption is justified in the case of the fusion of relatively light nuclei, while the reactions leading to the formation of the heaviest nuclei are characterized by the dominant contribution of quasi-fission to the capture cross section. The compound nucleus formed as a result of the fusion of two heavy ions has an excitation energy defined as: E ∗ = Ec.m. + Qfus ,

(3)

where Qfus is the fusion energy (Q-value): Qfus = M(At , Zt ) + M(Ap , Zp ) − M(ACN , ZCN ),

(4)

where M is the ground-state mass of each nucleus. When light nuclei fuse, the reaction Q-value is generally greater than zero, since the binding energy per nucleon

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generally increases up to iron. However, Qfus < 0 when medium and heavy nuclei fuse. In this case fusion requires that Ec.m. > |Qfus |. This restriction is generally fulfilled at any energy domain where fusion has been studied experimentally since |Qfus | is much lower than the height of the Coulomb barrier, since the fusion cross section becomes negligibly small because of the low penetrability below the barrier. The decay of an excited heavy nucleus proceeds by the emission of light particles (neutrons, protons, α-particles), emission of γ -quanta and/or fission. Thus, the fusion cross section is the sum of the cross section for the formation of so-called evaporation residues (EvR) that result from the survival of the CN relative to fission. Thus, the fusion-fission cross section is σfus = σEvR + σff . The competition between survival and fission processes is determined primarily by the difference between the fission barrier and the neutron binding energy of the evaporating nuclei.

Measurement of Fusion Cross Sections The compound nuclei formed in the fusion reaction are recoiling forward (in the laboratory system) with well-determined momenta due to conservation of energy and momentum. In order to measure the fusion cross section, one can detect the final products formed after the de-excitation. It is not possible to directly observe the yield of compound nuclei, since they are formed in excited states and decay on their way to the detector. Thus, in general case, one needs to measure all possible final products, i.e., the fusion-fission σff and fusion-survival σEvR cross sections and distinguish them from the products of other reactions. Technically, such complete experiments are very complicated, since they require different devices to observe the various final products. It can be that σff  σEvR (for superheavy nuclei, e.g.,) or vice versa for lighter systems σff  σEvR . In such cases much less effort is required to determine the fusion cross section by concentrating on detecting the most significant exit channel. To measure the yield of evaporation residues at forward angles, we must separate them first of all from the unreacted beam particles and from the products of other nuclear reactions. For this purpose, various sorts of electromagnetic separators can be used as described in the previous chapter. In addition, radiochemical methods have been used to separate relatively long-lived EvRs. To measure the fission cross section, one must determine the total yield of fission fragments emitted at an angle of 180◦ in the center-of-mass system and having a sum of masses equal to ACN while taking into account the possible evaporation of several neutrons (mostly), and sometimes protons and/or αs both from an excited compound nucleus (so-called prescission particles) as well as from excited fission fragments themselves (postscission particles). An additional challenging task is to distinguish fusion-fission products from others that originate from other types of nuclear reactions which can contribute to the same mass domain. Importantly, all products from quasi-fission can even dominate the reaction cross section with heavy targets and projectiles. This, for example, complication is present in fusion reactions for synthesis of superheavy nuclei. An even more complicated case is the reaction

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between nuclei that are close in mass when all of the main reaction types (elastic, inelastic, and deep inelastic scattering, quasi-fission, and fusion-fission) contribute in the same mass domain (Figs. 4 and 5).

Capture of Nuclei Note that in many cases of reactions between not very heavy nuclei there is a general absence of quasi-fission, that is, PCN = 1), so the capture and fusion cross sections are nearly equal. In this subsection, however, the general term “capture cross section” will be used. According to Eq. 1 the capture cross section is determined by the penetrability of the Coulomb barrier for each partial wave, Tl . At energies well above the Coulomb barrier nuclei, the surfaces of the nuclei can come into contact for all partial waves from 0 up to a certain maximal value, lfus , with a probability close to unity. Thus, Tl  1 for l = 0 . . . lfus and then drops rapidly to zero for l > lfus (Fig. 6). In such cases one can obtain a simple relation for the capture cross section by assuming that the colliding nuclei are spherical and that the position of the fusion barrier, RB , does not depend on angular momentum l. In this case, the fusion barrier for an arbitrary value of l can be calculated as the sum of the fusion barrier for central collisions, VB , and the centrifugal energy taken at the barrier position for each partial wave: h¯ 2 l(l + 1)/2μRB2 . The value of lfus can be estimated from the equality:

300

contact distance

250

potential energy, MeV

Fig. 6 Potential energy for the system 40 Ca+90 Zr for different values of angular momentum l. The lowest partial wave, l = 0, shows the deepest minimum (black dashed line). The minimum decreases with increasing l and disappears at lfus (red solid line)

lfus

200

l>lfus Ec.m.

l0μ m 0. This typically happens when εn1 0 for scattering states). The reduced mass μ is defined by μ=

m1 m 2 . m1 + m2

(2)

In nuclear physics, the potential V (R) contains a short-range nuclear term VN (R) and a long-range Coulomb term VC (R). A nuclear interaction is typically described by the Woods-Saxon potential VN (R) =

V0 , 1 + exp[(R − R0 )/a]

(3)

where V0 , R0 and a are parameters which are in general fitted to experiment. The Coulomb contribution is often taken as the potential of a uniformly charged sphere of radius RC , VC (R) = =

Z1 Z2 e2 [3 − (R/RC )2 ] 2RC

for R ≤ RC ,

Z1 Z2 e2 R

for R > RC .

(4)

In a first step, we neglect the Coulomb potential. This approximation makes calculations and notations simpler. The generalization to the Coulomb interaction will be presented in the next subsection. The scattering cross sections are based on the long-range behavior of the wave function Ψ (E, R). At large distances, the potential V (R) is assumed to decrease faster than 1/R. The second-order differential equation (1) provides two independent solutions, and Ψ (E, R) tends to   exp(ikR) , Ψ (E, R) −→ A exp(ik · R) + f (θ ) R→∞ R

(5)

where the wave number k is defined by  k=

2μE h¯ 2

.

(6)

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P. Descouvemont

In Eq. (5), the first term corresponds to an incoming plane wave and the second term to an outgoing spherical wave. It is easy to show that both contributions satisfy Eq. (1) at large R. Function f (θ ) is called the scattering amplitude and provides the elastic cross section from dσ = |f (θ )|2 . dΩ

(7)

Various techniques permit the calculation of the scattering amplitude (Joachain 1983; Canto and Hussein 2013). We will discuss in detail the phase-shift method. As long as only elastic scattering is concerned, the amplitude A in Eq. (5) does not play any role. If the wave function is used in further calculations, such as in radiative capture or in transfer reactions, this amplitude should be defined consistently. A frequent choice is √ A = 1/ v,

(8)

where v = hk/μ is the relative velocity. This amplitude corresponds to a unit flux ¯ (see Joachain 1983 for details).

Phase Shifts The phase-shift method (Joachain 1983; Canto and Hussein 2013; Thompson and Nunes 2009) is based on the expansion of the scattering wave function as Ψ (E, R) =

L ∞ 1   CLM (E) uL (E, R) YLM (θ, φ), R

(9)

L=0 M=−L

where we assume that the z-axis is along the beam direction k. In Eq. (9), CLM (E) are coefficients to be determined, and YLM (θ, φ) are spherical harmonics. For a given partial wave L, the radial function uL (E, R) is obtained from   TL + V (R) uL (E, R) = EuL (E, R).

(10)

  h¯ 2 d 2 L(L + 1) . TL = − − 2μ dR 2 R2

(11)

with

The factor 1/R in Eq. (9) cancels out the first-order derivative in the kinetic energy. Let us consider large R values, where the potential V (R) tends to zero. In this range, the solutions of (10) are expressed from linear combinations of the spherical Bessel function jL and of the spherical Neumann function nL (Abramowitz and Stegun 1972). More precisely, they are written as

37 Theoretical Studies of Low-Energy Nuclear Reactions

 uL (E, R) → AL R jL (kR) − tan δL (E) nL (kR) ,

1455



(12)

where δL (E) is defined as the phase shift and AL a constant which does not affect the phase shift. Some properties of the Bessel and Neumann functions are jL (x) −→ x→0

xL , (2L + 1)!!

nL (x) −→ − x→0

(2L − 1)!! , x L+1

(13)

and π 1  sin x − L , x→∞ x 2  π 1 nL (x) −→ − cos x − L . x→∞ x 2

jL (x) −→

(14)

When the potential V (R) = 0, the radial function only contains the Bessel function, since nL (kR) diverges for small R. In that case, the solution of Eq. (10) is uL (E, R) = AL RjL (kR).

(15)

Comparing (12) and (15), we see that the phase shift δL modifies the asymptotic form (12) according to the potential. It is related to the scattering matrix UL (also called “collision matrix”) as UL = exp(2iδL ),

(16)

which, strictly speaking, is a 1 × 1 matrix in single-channel calculations. It can be, however, generalized to multichannel problems, where UL is a true matrix. Important properties of the phase shift are: • It is a continuous function of E, as well as its derivatives. • As shown by Eq. (16), it is determined within nπ , where n is an integer. The Levinson theorem (see Joachain 1983), however, says that δL (E = 0) − δL (E = ∞) = NL π,

(17)

where NL is the number of bound states (E < 0) in the partial wave L. In general, the phase shift must be computed numerically. A simple analytical problem, however, is illustrated by the hard-sphere potential

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P. Descouvemont

V (R) = ∞

for R ≤ a,

V (R) = 0

for R > a,

(18)

where a is the sphere radius. The phase shift is then obtained from the boundary condition uL (E, a) = 0,

(19)

which provides tan δL =

jL (ka) . nL (ka)

(20)

The hard-sphere phase shift is displayed by the solid lines in Fig. 1, where it is plotted as a function of ka. A strong difference between L = 0 and L = 0 shows up at low energies (or ka). For L = 0, we have δ0 = −ka.

(21)

This behavior is typical of neutral systems. There is no centrifugal barrier for L = 0 and the low-energy phase shifts are quite different for L = 0. This property is not true for charged systems, where a Coulomb barrier is present for all partial waves. The calculation of the scattering amplitude f (θ ) is performed by comparing (5) and (9) with the long-range limit (12). This provides the well-known expansion

Fig. 1 Hard-sphere phase shifts (20) for neutral systems (solid lines) and (35) for charged systems with η = 1 (dashed lines)

37 Theoretical Studies of Low-Energy Nuclear Reactions

f (θ ) =

1457

∞   1  (2L + 1) exp(2iδL ) − 1 PL (cos θ ), 2ik

(22)

L=0

where PL are the Legendre polynomials. In practice, a limited number of partial waves is sufficient to guarantee the convergence. This is particularly true at low energies, where δL tends rapidly to zero when L increases. At high energies, however, the number of partial waves increases, and other methods, such as the eikonal approximation (Glauber 1959; Joachain 1983), are used. The reader is referred to Joachain (1983), Satchler (1983), Suzuki et al. (2003), Bertulani and Danielewicz (2004), and Canto and Hussein (2013) for details.

Generalizations Coulomb Scattering In the presence of the Coulomb interaction, the asymptotic behavior of the radial function uL is obtained from the Coulomb equation   Z1 Z2 e2 as TL + uL = Euas L, R

(23)

whose solutions are the regular and irregular Coulomb functions FL (η, kR) and GL (η, kR) (Abramowitz and Stegun 1972; Thompson 2010). The Coulomb functions depend on the (dimensionless) Sommerfeld parameter η=

Z1 Z2 e2 . hv ¯

(24)

At low energies, η is large and Coulomb effects are important. The low-energy regime is typical of nuclear astrophysics (Iliadis 2007), where the relevant cross sections are extremely low, due to tunneling effects through the Coulomb barrier. At high energies, η is small and the Coulomb interaction plays a minor role. At short distances, the Coulomb functions tend to FL (η, x) −→ aL x L+1 , x→0

GL (η, x) −→ x→0

1 x −L , (2L + 1)aL

with  a0 =

2π η exp(2π η) − 1

1/2 ,

(25)

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P. Descouvemont

 L 

η2 1/2 a0 1+ 2 aL = . (2L + 1)!! n

(26)

n=1

The limits at large distances are given by FL (η, x) −→ sin(x − Lπ/2 − η log(2x) + σL ), x→∞

GL (η, x) −→ cos(x − Lπ/2 − η log(2x) + σL ), x→∞

(27)

where σL is the Coulomb phase shift σL = arg[Γ (1 + L + iη)] = σ0 + ωL , ωL =

L 

arctan

n=1

η n

.

(28)

The Coulomb interaction therefore modifies the asymptotic form of the Bessel functions jL (x) and nL (x) (see Eqs. (14)) by an additional factor depending on η. In practice, we use the incoming and outgoing Coulomb functions (Lane and Thomas 1958) IL = GL − iFL , OL = GL + iFL ,

(29)

which present the asymptotic limits   IL (η, x) −→ exp −i(x − Lπ/2 − η log(2x) + σL ) , x→∞   OL (η, x) −→ exp i(x − Lπ/2 − η log(2x) + σL ) . x→∞

(30)

With these definitions, Eq. (12) is generalized to   uL (E, R) → CL IL (η, kR) − UL OL (η, kR) ,

(31)

which can be written as   uL (E, R) → −2iCL exp(iδL ) FL (η, kR) cos δL + GL (η, kR) sin δL ,

(32)

where UL = exp(2iδL ) is the scattering matrix associated with the nuclear potential.

(33)

37 Theoretical Studies of Low-Energy Nuclear Reactions

1459

Again, a simple example is the hard-sphere potential between charged particles V (R) = ∞ V (R) =

for R ≤ a,

Z1 Z2 e2 R

for R > a,

(34)

which provides tan δL = −

FL (η, ka) GL (η, ka)

(35)

The hard-sphere phase shift (35) is presented by the dashed lines in Fig. 1 for η = 1. At high energies (large ka values), the role of the Coulomb potential decreases. At low energies, however, the Coulomb barrier makes the phase shifts much smaller than in neutral systems. This effect is particularly strong for L = 0. Although potential (34) is too simple to be realistic, these effects remain qualitatively valid for any nuclear potential. The calculation of the scattering amplitude f (θ ) could be done as in (22) by including the Coulomb phase shifts. In practice, however, this approach is not suitable since the Coulomb phase shifts ωL tend to zero very slowly (see, e.g., Fig. 3.2 of Canto and Hussein 2013). Equation (22) is therefore rewritten as f (θ ) = f N (θ ) + f C (θ ),

(36)

where the nuclear and Coulomb amplitudes are defined by f N (θ ) =

1  (2L + 1) exp(2iωL )(UL − 1)PL (cos θ ), 2ik L

1  (2L + 1)(exp(2iωL ) − 1)PL (cos θ ) 2ik L   η 2 (θ/2) . exp −iη log sin =− 2k sin2 (θ/2)

f C (θ ) =

(37)

The advantage of decomposition (36) is that the nuclear amplitude f N (θ ) converges rapidly and that the Coulomb amplitude f C (θ ) is known analytically. The Rutherford cross section is obtained from dσR η2 = |f C (θ )|2 = = dΩ 4k 2 sin4 (θ/2)



Z1 Z2 e2 4E

2

1 4

sin (θ/2)

,

(38)

and diverges for θ → 0. It is minimal at θ = 180◦ . It is customary to present low-energy cross sections as the ratio to the Rutherford cross section

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P. Descouvemont

  N  f (θ ) + f C (θ ) 2 dσ  , =   dσR f C (θ )

(39)

which is close to unity at small angles. This property is used in elastic scattering experiments to normalize the cross sections.

Complex Potentials Up to now, the potential V (R) was supposed to be real, which means that δL is real and that |UL | = 1. This approximation is well adapted to structureless particles and to low-energy scattering. The presence of composite particles and of open channels (inelastic, transfer, etc.), however, introduces absorption effects which are not accounted for by a real potential. These effects can be simulated by the introduction of an imaginary part, which leads to the notion of optical potentials V (R) = U (R) + iW (R),

(40)

where W (R) is negative and simulates absorption effects. The formal scattering theory (Feshbach 1958; Hodgson 1971) shows that the imaginary term is nonlocal and energy dependent. In practice, however, it is often replaced by a local phenomenological term, fitted to the experimental data. A recent review of the developments of the optical model can be found in Dickhoff and Charity (2019). Owing to the absorption, the scattering matrix UL (16), (33) is now written as UL = ηL exp(2iδL ),

(41)

with the modulus ηL ≤ 1. The scattering amplitude is still determined with (37). The imaginary part of the optical potential provides the reaction cross section defined as π  (2L + 1)τL , k2

(42)

2 τL = 1 − ηL .

(43)

σR =

L

where

The reaction cross section (42) corresponds to all absorption channels or, in other words, to the total flux removed from the elastic channel. It does not specify how this flux is distributed over the different open channels. It can be shown Canto and Hussein (2013) that (43) is equivalent to the integral definition 8 τL = − h¯



∞

W (R)|uL (E, R)|2 dR,

(44)

0

with the normalization (8) of the scattering function. This definition is useful when the imaginary potential W (R) is split in different contributions, associated with the

37 Theoretical Studies of Low-Energy Nuclear Reactions

1461

open channels (Assunção and Descouvemont 2013). In addition, it is numerically more accurate than Eq. (43) at low energies, where ηL ∼ 1. Optical potentials for nucleon + nucleus systems have been obtained in various compilations (Becchetti and Greenlees 1969; Varner et al. 1991; Koning and Delaroche 2003) where the authors fit the available elastic cross sections. Parametrizations for reactions induced by deuterons (Han et al. 2006), 3 H or 3 He (Pang et al. 2009), and α particles (Avrigeanu et al. 2009) also exist in the literature.

Folding Potentials Many works have been devoted to the determination of optical potentials from a nucleon-nucleon interaction (see Dickhoff and Charity 2019 for a recent review). A frequent approach is known as the folding method (Satchler 1983; Khoa 1988), which is based on a nucleon-nucleon interaction vNN and on nuclear densities ρ(s), defined by ρ J0 M0 (s) = Ψ J0 M0 |



δ(r i − s)|Ψ J0 M0 ,

(45)

i

where Ψ J0 M0 is the A-body wave function of the nucleus with angular momentum J0 M0 and r i are the nucleon coordinates. The proton (or “charge”) and neutron densities can be obtained by introducing an isospin factor in the summation. The matter density is normalized as ρ J0 M0 (s)ds = A,

(46)

where A is the nucleon number. It can be expanded over multipoles as ρ J0 M0 (s) =

 J0 M0 λ0|J0 M0 ρλ (s) Yλ0 (Ωs ).

(47)

λ

For nuclei with spin J0 = 0 or J0 = 1/2, the density (47) only contains a monopole term (λ = 0). Let us consider two nuclei with densities ρ1 (s 1 ) and ρ2 (s 2 ). The folding potential is defined as VF (R) =

ρ1 (s 1 )ρ2 (s 2 )vNN (|R + s 2 − s 1 |)ds 1 ds 2 .

(48)

This multi-dimensional integral can be performed by using Fourier transforms with V˜F (q) =

exp(iq · R)VF (R)dR = ρ˜1 (q)ρ˜2 (q)v˜NN (q),

(49)

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P. Descouvemont

where ρ˜i (q) and v˜NN (q) are the Fourier transforms of the densities and of the nucleon-nucleon interaction, respectively. In the frequent case of radial densities ρi (s), the folding potential (48) only contains a monopole term VF (R) and is obtained from 1 VF (R) = 2π 2

j0 (qR)ρ˜1 (q)ρ˜2 (q)v˜NN (q)q 2 dq,

(50)

where ρ˜i (q) = 4π

j0 (qR)ρi (R)R 2 dR,

v˜NN (q) = 4π

j0 (qR)vNN (R)R 2 dR,

(51)

are the Fourier transforms and where j0 (x) = sin(x)/x is the spherical Bessel function of order 0. For some specific functions (Gaussians, zero-range interactions), the integrals (51) can be performed analytically. These equations are simplified for nucleon + nucleus systems, where a single folding is used. Various nucleon-nucleon interactions are used in the literature. The M3Y (Bertsch et al. 1977) and its density-dependent extension DDM3Y (Kobos et al. 1982) are based on Yukawa forms; several improvements have been proposed in Khoa et al. (1997). The São Paulo potential (Chamon et al. 2002, 2021) is defined from a zero-range function. It was shown in Goldfarb (1978) that the folding potential (50) is mainly sensitive to small q values, which means that the r.m.s. radius of the density and the volume integral of the potential are the essential inputs. Let us notice that the folding potential (48) ignores exchange effects, due to the antisymmetrization between the nucleons. These effects can be approximated in various ways (see Khoa 1988; Khoa and Satchler 2000 and references therein) but introduce non-local terms in the potential. The folding method can be extended to multichannel problems, provided that densities of the excited states are known (see Assunção and Descouvemont (2013) for an application to the 12 C+12 C reaction with 12 C excited states). Frequently, the nucleon-nucleon interaction vNN (r) is real. This means that the folding potential (48) is also real and that a phenomenological imaginary term must be introduced. Another frequent approach is the assumption that the real and imaginary parts have the same shape which defines the optical potential by V (R) = (NR + iNI )VF (R),

(52)

where NR and NI are two adjustable parameters (see, e.g., Alvarez et al. 2003). Parameter NR may be slightly different from unity to account for higher-order effects.

37 Theoretical Studies of Low-Energy Nuclear Reactions

1463

Multichannel Systems In the previous sections, we have considered systems composed of two spinless particles. This assumption greatly simplifies the notations and the theoretical developments. In practice, however, the spins of the colliding nuclei must often be taken into account. Let us denote as I1 and I2 these spins, associated with internal wave functions φ I1 M1 and φ I2 M2 (the corresponding parities π1 and π2 are understood). The relative orbital momentum is denoted as L, and the good quantum numbers are the total angular momentum J and parity π = π1 π2 (−1)L . The channel spin I is defined as the coupling of the nuclear spins I = I 1 ⊕ I 2,

(53)

J = I ⊕ L.

(54)

which provides

After introduction of the spins, the spinless definition of a partial wave 1 uL (E, R)YLM (Ω) R

(55)

1  Jπ J Mπ uLI (E, R) ϕLI (Ω), R

(56)

Ψ LM (E, R) = is generalized to Ψ J Mπ (E, R) =

LI

where the channel function is defined by  J M J Mπ ϕLI (Ω) = [φ I1 ⊗ φ I2 ]I ⊗ YL (Ω) .

(57)

For a given angular momentum J and parity π , the (L, I ) values are defined from the angular momentum couplings (53) and (54) and from the parity selection rule. Some examples are given in Table 1. A further extension is given by multichannel models. In that case, several states of the colliding nuclei are considered and the expansion (56) is generalized to Ψ J Mπ (E, R) =

1  Jπ J Mπ uαLI (E, R) ϕαLI (Ω), R

(58)

αLI

where index α denotes the channel. The radial functions are obtained from a set of differential equations as 

 Jπ Jπ TL + Ec − E uJc π (R) + Vcc

(R)uc (R) = 0 c

(59)

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P. Descouvemont

Table 1 Values of I and L for different systems and different J π values System I1 = 1/2+ , I2 = 0 I1 = 3/2+ , I2 = 0

J = 1/2+ (1/2, 0)

J = 1/2− (1/2, 1)

J = 3/2+ (1/2, 2)

J = 3/2− (1/2, 1)

J = 5/2+ (1/2, 2)

J = 5/2− (1/2, 3)

(3/2, 2)

(3/2, 1)

(3/2, 0)

(3/2, 1)

(3/2, 2)

(3/2, 1)

I1 = 2+ , I2 = 1/2+

(3/2, 2)

(3/2, 1)

(3/2, 2) (3/2, 0)

(3/2, 3) (3/2, 1)

(3/2, 4) (3/2, 2)

(3/2, 3) (3/2, 1)

(5/2, 2)

(5/2, 3)

(3/2, 2) (5/2, 2) (5/2, 4)

(3/2, 3) (5/2, 1) (5/2, 3)

(3/2, 4) (5/2, 0) (5/2, 2) (5/2, 4)

(3/2, 3) (5/2, 1) (5/2, 3) (5/2, 5)

where index c stands for c = (α, L, I ) and where Ec are the threshold energies (for the sake of simplicity, the energy dependence is omitted in the radial functions J π (R) depend on the model, but the general uJc π (R)). The coupling potentials Vcc

equation (59) is common to multichannel problems. Typical examples are models involving deformed nuclei (Raynal 1981; Hagino et al. 1999) or CDCC approaches (Yahiro et al. 2012). Notice that the wording “multichannel” may be ambiguous. It is in general used for systems involving several α values, i.e., several physical channels. However, when the spins I1 and I2 are different from zero, the wave function (56) contains several (L, I ) values. The radial wave functions are obtained from system (59) although, strictly speaking, the system involves a single physical channel. The scattering matrices UL , introduced previously, are now true matrices whose dimension is given by the number of (αLI ) values. In other words, this dimension generally depends on J and π (see Table 1). The scattering matrix U J π is associated with the long-range part of the radial functions uJc π (R) as  1 Jπ uJc π (R) −→ √ IL (kc R)δcω − Ucω OL (kc R) , vc

(60)

where ω is the entrance channel. The normalization factor (vc is the relative velocity in channel c) ensures that the scattering matrix is symmetric and that the flux is unity for real potentials. Notice that it is assumed in Eq. (60) that all channels are open. The treatment of closed channels is simple, and we refer to Descouvemont and Baye (2010) for further details. The calculation of the elastic and inelastic cross sections from the scattering matrices can be found in many textbooks (Satchler 1983; Canto and Hussein 2013).

Non-local Potentials Non-locality naturally appears in the formulation of the optical potential (Dickhoff and Charity 2019). For example, exchange effects introduce non-local terms (Perey

37 Theoretical Studies of Low-Energy Nuclear Reactions

1465

and Buck 1962; Horiuchi 1977; Timofeyuk and Johnson 2013; Dohet-Eraly and Descouvemont 2021). A non-local potential is an integral operator U , acting on a wave function as U Ψ (R) =

U (R, R )Ψ (R )dR .

(61)

The kernel U (R, R ) is symmetric and can be expanded in multipoles as U (R, R ) =

1  μ μ vλ (R, R )Yλ (ΩR )Yλ (ΩR ). R R

(62)

λμ

The radial equation (10) is extended to 

 TL + V (R) uL (R) +



vL (R, R )uL (R )dR = EuL (R).

(63)

This equation can be easily solved with the R-matrix method (Hesse et al. 2002; Descouvemont and Baye 2010). Some techniques, such as the “local energy approximation” (Perey and Buck 1962), permit to find approximated solutions of the non-local equation (63). Notice that the multichannel equation (59) may also involve non-local terms.

Two-Potential Formulas We present hereuseful formulas which make a link between the scattering matrices associated with two different potentials V0 (R) and V1 (R). For a given partial wave L, the corresponding Schrödinger equations are L (TL + V0 )uL 0 = Eu0 L (TL + V1 )uL 1 = Eu1 ,

(64)

with the asymptotic conditions  1 L uL 0 (R) → √ IL (kR) − U0 OL (kR) v  1 L uL 1 (R) → √ IL (kR) − U1 OL (kR) v

(65)

L Multiplying the first equation (64) by uL 1 and the second by u0 , we get, after subtraction and integration over R

1466

P. Descouvemont

1 i h¯

U1L = U0L +



 L uL 0 (R) V1 (R) − V0 (R) u1 (R)dR,

(66)

where we have used the Wronskian relation W[OL (x), IL (x)] = 2i.

(67)

Equation (66) provides an exact relation between the scattering matrices associated with two different potentials. It is at the basis of various approximations. Let us first suppose that V0 = 0 and that V1 = VN + VC . Equation (66) provides the scattering matrix (33) 1 U = √ i h¯ v

jL (kR)[VN (R) + VC (R)]uL 1 (R)dR.

L

(68)

This definition can be used if approximate solutions uL 1 (R) are known. In general, such solutions are available over a limited range. A more accurate approximation is V0 = VC which provides 1 √ i h¯ v

U L = UCL +

FL (kR)VN (R)uL 1 (R)dR,

(69)

where UCL = exp(2iσL ) and where the integral can be performed over short distances only, owing to the short range of the nuclear potential VN (R). This equation can be also used to test the consistency of the calculation. Equation (68) can be further simplified by assuming that the potential V1 (R) is small compared to the energy E. In these conditions, the approximated scattering matrix U1L ≈

1 i hv ¯

[jL (kR)]2 V1 (R)dR

(70)

corresponds to the well-known Born approximation (Joachain 1983), which permits a direct calculation of the scattering matrix from the potential. The single-channel equation (66) can be extended to multichannel problems. 0 (R) and V 1 (R), we have, for the U J π With the coupled-channel potentials Vαβ αβ 0 and U J1 π scattering matrices Jπ U1,ij

=

Jπ U0,ij

1  + i h¯



αβ  π αβ π (R) V1 (R) − V0 (R) uJ1,jβ (R)dR. uJ0,iα

(71)

α,β

This integral definition has been used, for example, in Descouvemont et al. (2017) to separate the nuclear and Coulomb contributions in breakup reactions.

37 Theoretical Studies of Low-Energy Nuclear Reactions

1467

These developments can be also generalized to inhomogeneous equations. Let us consider the two equations (TL + V − E)uL 0 (R) = 0,

(72)

(TL + V − E)uL (R) = −ρω (R),

(73)

where ρω (r) is a source term arising from a channel ω. The asymptotic limits are  1 L uL 0 (R) → √ IL (kR) − U0 OL (kR) , v

(74)

1 uL (R) → − √ UωL OL (kR). v

(75)

Again, after multiplication of (72) and (73) by uL and by uL 0 , respectively, we obtain UωL =

1 i h¯

uL 0 (R)ρω (R)dR.

(76)

This definition is used in various applications, such as in transfer reactions.

Solving the Scattering Equation The single-channel (10) or the multichannel (59) equations must be solved to determine the scattering matrices and the associated wave functions (see Thompson and Nunes 2009 for an overview). Various types of methods exist in the literature: finite-difference techniques (Raynal 1972; Zhao and Corless 2006), models based on a set of basis functions (Descouvemont and Baye 2010; Burke 2011; Zhang et al. 1988), or integral definitions (Viviani et al. 2017). We discuss in more detail the Numerov method (Raynal 1972) and the R-matrix theory (Lane and Thomas 1958; Descouvemont and Baye 2010; Burke 2011).

The Numerov Method Let us first discuss the single-channel equation (10). The Numerov method consists in discretizing the R coordinate in N intervals of length h. If the product Nh is larger than the range of the nuclear potential, the numerical wave function can be matched to the asymptotic limit (12) (for neutral systems) or (31), (32) (for charged systems). The radial Schrödinger equation can be written in simplified notations as f

(R) = V (R)f (R).

(77)

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P. Descouvemont

In the Numerov method (Raynal 1972), one uses an auxiliary function   h2 ξ(R) = 1 − V (R) f (R) 12

(78)

which provides the recurrence ξ(R + h) = 2ξ(R) − ξ(R − h) + w(R).

(79)

Function w(R) is defined by  w(R) =

h2 V (R) 1−

h2 12 V (R)

 ξ(R),

(80)

with a truncation error proportional to h6 (the cumulative error, however, is proportional to h4 (Sloan 1968)). The wave function f (R) is obtained from f (R) = ξ(R) +

w(R) . 12

(81)

The algorithm (79) starts with ξ(0) = 0 ξ(h) = ,

(82)

where  is an arbitrary number, which only affects the overall amplitude of the wave function. The numerical values are matched to the asymptotic limit (31) or (32) for R-values large enough to make the nuclear interaction negligible. A strong test of the calculation is the stability with respect to N and to h. Typical values are h ≈ 0.1 fm and N ≈ 100 − 200, depending on the range of the potential. At high energies, the mesh size should be reduced since it must be smaller than the wavelength. An enhanced version of the Numerov algorithm is presented in Thorlacius and Cooper (1987). These algorithms can be extended to multichannel systems (Raynal 1972; Thompson 1988) and to non-local potentials (Michel 2009; Arellano and Blanchon 2019).

The R-Matrix Method The R-matrix method is an efficient tool to study scattering states. It is widely used in atomic (Burke 2011) and in nuclear (Lane and Thomas 1958; Descouvemont and Baye 2010) physics. The main idea is to divide the space into two regions, separated by the channel radius a. In the internal region, the scattering wave function

37 Theoretical Studies of Low-Energy Nuclear Reactions

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is expanded over a set of N basis functions ϕi (R). For R ≤ a, the radial wave function is therefore written as uint L (R) =

N 

fiL ϕi (R),

(83)

i=1

where fiL are the coefficients. Typical examples of basis functions are Gaussian functions (either with different centers or with different widths) or Lagrange functions (Baye 2015). In the external region, and by definition of the channel radius, only the Coulomb interaction contributes and the external wave function reads uext L (R) = IL (kR) − UL OL (kR),

(84)

where we assume a single-channel problem for the sake of simplicity. The extension to multichannel systems is presented, for example, in Descouvemont and Baye (2010). The quantities to be determined are the scattering matrix UL and the coefficients fiL . The wave function uL (R) should not sensitively depend on a, neither on N, provided that these parameters are chosen properly. The choice is essentially guided by two requirements: (i) the channel radius must be large enough to guaranty that the nuclear interaction is negligible; and (ii) the number of basis functions N must be large enough to describe accurately the wave function (83). This stability is a strong test of R-matrix calculations. Over a finite interval [0, a], the kinetic energy is not an Hermitian operator. The hermiticity is restored with the surface Bloch operator (Bloch 1957) defined as   h¯ 2 d B L(B) = δ(R − a) − , 2μ dR R

(85)

where B is an arbitrary constant. The Bloch operator makes TL + L(B) Hermitian over the range [0, a]. The Schrödinger equation (10) is replaced by the BlochSchrödinger equation

int ext TL + V (R) + L(B) − E)uint L = L(B)uL = L(B)uL

(86)

The second equality in (86) stems from the continuity of the wave function at the channel radius. Notice that the Bloch operator guaranties the continuity of the derivative of uL (R)



ext uint L (a) = uL (a)

(87)

for any choice of B. In earlier developments of the R-matrix theory (Lane and Thomas 1958), the Bloch operator was not introduced, and condition (87) was approximately satisfied by an appropriate choice of the basis functions. The

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introduction of the Bloch operator makes that requirement (87) is satisfied for any choice of the basis functions provided that this basis is large enough to reproduce the exact wave function. In addition to the stability mentioned before, condition (87) is a further test of the calculation. A poor matching at R = a indicates that the calculation is inaccurate. Typical examples are presented in Descouvemont and Baye (2010). The calculation of the scattering matrix is performed with Eq. (86) and definitions (83) and (84). After projection of the Bloch-Schrödinger equation on ϕi (R), we get 

CijL fjL = ϕi (R)|L(B)|uext L ,

(88)

j

with CijL = ϕi |TL + V (R) + L(B) − E|ϕj int ,

(89)

and where subscript “int” means that the matrix elements are calculated over the internal region. A simple calculation provides UL =

IL (ka) 1 − (L∗L − B)RL , OL (ka) 1 − (LL − B)RL

(90)

where the R-matrix is defined by RL =

h¯ 2  ϕi (a)(C L )−1 ij ϕj (a). 2μa

(91)

ij

In Eq. (90), the first factor corresponds to the hard-sphere phase shift (35), and constant LL is associated with the Coulomb functions as LL = ka

OL (ka) = SL + iPL , OL (ka)

(92)

where SL and PL are the shift and penetration factors, respectively. At low energies, the penetration factor varies rapidly, whereas the shift factor is a smooth function of the energy (see examples in Descouvemont and Baye 2010). For real potentials, the R-matrix is real, and |UL | = 1. It can be shown that the scattering matrix is independent of the boundary parameter B (Descouvemont and Baye 2010). In most applications, B = 0 is adopted. The R-matrix method can be extended to multichannel calculations (Hesse et al. 1998; Descouvemont and Baye 2010). It is particularly efficient for systems involving closed channels, since it does not present numerical instabilities, as in finite-difference methods. The R-matrix theory can be also extended to inhomogeneous Schrödinger equations

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(Lei and Descouvemont 2020), to non-local potentials (Hesse et al. 2002), and to the calculation of bound states and of resonances (Descouvemont and Vincke 1990). Let us briefly discuss the choice of the R-matrix parameters a and N (these quantities, however, are not strictly speaking “parameters” since the scattering wave function should not depend on their value). The channel radius a must be sufficiently large so that the external wave function has reached its asymptotic form (12). On the other hand, increasing a means that the number of basis functions N should be also increased since the internal wave function must be described over a wider range. The choice of a and N therefore results from a compromise between these requirements. Single-channel calculations are extremely fast on modern computers. Recent multichannel calculations, however, are very time-consuming (see, e.g., Descouvemont 2018, 2020) and an optimal choice is a critical issue. A frequent problem is the long range of non-monopole terms in the Coulomb interaction, which requires large channel radii, and therefore a large number of basis functions. In multichannel calculations, the coupling potentials between two channels c and c arising from the Coulomb potential are proportional to Vcc (R) ∼

1 R λ+1

(93)

at large distances. For dipole (λ = 1) or quadrupole (λ = 2) terms, this potential may extend to very large distances before being negligible (compared to the monopole term Z1 Z2 e2 /R). This problem is well known and has received several solutions (Rhoades-Brown et al. 1980; Raynal 1981; Christley and Thompson 1994) based on distorted Coulomb functions. In the R-matrix formalism, this issue can be addressed by using large channel radii (see, e.g., Druet and Descouvemont 2012). Propagation techniques (Baluja et al. 1982) permit to speed up the numerical calculations by splitting the interval [0, a] in several sub-intervals (see Descouvemont 2016 for more detail). The presentation is general for any choice of the basis functions. However, it has been shown in many references (see, e.g., Hesse et al. 1998; Descouvemont and Baye 2010; Druet et al. 2010) that Lagrange functions (Baye 2015) are very well adapted. The reason is that the matrix elements (89), which are integrals over the relative coordinate, can be easily computed and do not require any numerical quadrature if they are performed at the Gauss approximation. The calculation of matrix C (see Eq. (89)) is therefore extremely fast for simple potentials. The main part of the numerical calculation is the matrix inversion in Eq. (91). A Fortran subroutine for R-matrix calculations is available in Descouvemont (2016). Finally, let us mention that the R-matrix theory can also be adapted to microscopic cluster calculations (Baye and Heenen 1974; Baye et al. 1977; Baye and Descouvemont 1983; Descouvemont and Dufour 2012) or to ab initio calculations (Quaglioni and Navrátil 2009). The same formalism is valid, the only difference is in the calculation of the matrix elements in Eq. (89). Another application of the R-matrix theory is the determination of three-body phase shifts (Descouvemont et al. 2006). The very long range of the potentials (up to ≈1000 fm), however, makes these calculations rather difficult.

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The Continuum Discretized Coupled Channel (CDCC) Method In its original version, the purpose of the CDCC method was to determine the scattering and dissociation cross sections of a nucleus which can be broken up in the nuclear or Coulomb field of a target (for simplicity, the target is assumed to remain in its ground state). The final states may thus involve three (or more) particles: the target and the fragments of the projectile. The relative motion of these fragments is described by approximate continuum wave functions at discrete energies. The CDCC method was suggested by Rawitscher (1974) and first applied to deuteron + nucleus elastic scattering and breakup reactions. It was then extensively developed and used by several groups (Austern et al. 1987; Nunes and Thompson 1999). Its interest has been still revived by the availability of radioactive beams of weakly bound nuclei dissociating into three fragments, such as 6 He whose first dissociation channel is α+n+n (Matsumoto 2004; Rodríguez-Gallardo et al. 2008). Variants of the CDCC method also exist in atomic physics (Bartschat et al. 1996). To simplify the presentation, we assume a spin zero for the constituents of the projectile and for the target t. The internal structure of the particles is neglected. Let us consider a system constituted by a two-body projectile (A1 and A2 being the masses of the fragments and A12 = A1 + A2 ) and a structureless target. The internal coordinate in the projectile is denoted as r, and the relative coordinate between the projectile and the target as R. The three-body Hamiltonian is given by

A2  A1  H = H0 + TR + Vt1 R + r + Vt2 R − r , A12 A12

(94)

where H0 is the two-body Hamiltonian of the projectile H0 = Tr + V12 (r).

(95)

In general, potential V12 (r) associated with the projectile is real, whereas the interactions Vti between the fragments i and the target t are derived from the optical model, and thus complex. In a schematic notation, the wave function associated with (94) can be expanded from the eigenfunctions of Hamiltonian (95) as Ψ (R, r) =



φB (r)uB (R) +

φk (r)uk (R)dk,

(96)

B

where B denotes the bound states of the projectile, and φk (r) are two-body scattering wave functions with wave number k. The first term represents the elastic and inelastic channels, and the second term is associated with the breakup contribution. In practice, two methods are available to perform the continuum discretization, i.e., to discretize the integral over k. In the “pseudo-state” approach, it is replaced by a sum over square-integrable positive-energy eigenstates of Hamiltonian (95). The

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projectile Hamiltonian H0 is diagonalized over a finite basis, yielding the squareintegrable radial functions φi (r) at energies Ei , H0 φi (r)Ym (Ωr ) = Ei φi (r)Ym (Ωr ).

(97)

These functions are associated with bound states (i = B, Ei < 0) or represent square-integrable approximations of continuum wave functions (Ei > 0). The alternative is to replace the integral over k by averages of exact scattering states over a range of energies (“bin” method) (Austern et al. 1987). This approach also provides square-integrable basis functions. The total wave function (96) is then rewritten, for an angular momentum J and parity π = (−1)l+L , as Ψ J Mπ (R, r) =

1  Jπ u (R)ϕc (r, ΩR ), R c c

(98)

where L is the projectile-target angular momentum and where index c stands for c = (Li). The channel functions are defined by ϕc (r, ΩR ) = φi (r) [YL (ΩR ) ⊗ Y (Ωr )]J M .

(99)

The relative wave functions uJc π (R) are obtained from a set of two-body coupled equations  

Jπ Jπ Vcc TL + Ec − E uJc π (R) +

(R)uc (R) = 0.

(100)

c

Of course, the sum over  in (98) and (100) must be truncated at some value max . The sum over the pseudo-states i is limited by the number of basis states and can be reduced further by eliminating states above a maximum energy Emax . The CDCC problem is therefore equivalent to a system of coupled equations where the J π (R) are given by potentials Vcc

A2  A1  Jπ Vcc r + Vt2 R − r |ϕc (r, ΩR ) .

(R) = ϕc (r, ΩR )|Vt1 R + A12 A12

(101)

This matrix element represents a five-dimensional integral over (ΩR , Ωr , r). The optical potentials are expanded into multipoles as

A2  A1   r + Vt2 R − r = Vλ (R, r)Pλ (cos θ ), Vt1 R + A12 A12

(102)

λ

where θ is the angle between R and r, and Pλ (x) a Legendre polynomial. In practice, the number of λ values is limited by angular-momentum couplings. The

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four angular integrals in (101) are performed analytically, whereas the integration over r may require a numerical approach. The system (100) can be solved by various methods, as discussed in the previous section. Notice that the fragmenttarget potentials Vt1 and Vt2 must be central. The extension of this standard CDCC method can be done in two directions: (1) three-body projectiles (Matsumoto 2004; Rodríguez-Gallardo et al. 2008) and (2) calculations involving the breakup of both nuclei. Typical examples are 11 Be + d (Descouvemont 2018) and 11 Li + d (Descouvemont 2020), where 11 Be and 11 Li are described by 10 Be + n and 9 Li + n + n configurations. These systems involve a large number of channels, given by the product of the numbers of pseudo-states in both nuclei. For “2+2” systems, the Hamiltonian is generalized to

H = H1 (r 1 ) + H2 (r 2 ) + TR +

2 2  

Vij (R, r 1 , r 2 ),

(103)

i=1 j =1

where r 1 and r 2 are the internal coordinates in the projectile and in the target, and Vij are cluster-cluster optical potentials. The channel functions are now defined as

ϕcJ Mπ (ΩR , r 1 , r 2 ) =

 J M  I1 I φk1 (r 1 ) ⊗ φkI22 (r 2 ) ⊗ YL (ΩR ) ,

(104)

where φkI11 (r 1 ) and φkI22 (r 2 ) are the pseudo-state wave functions and where index c stands for c = (I1 , k1 , I2 , k2 , I, L). The calculations are much more time-consuming since the channel wave functions depend on two radial coordinates. Consequently, even though the formulation is similar, the calculation of the coupling potentials (101) raises important numerical difficulties. The application of the CDCC method to “3+2” systems (such as 11 Li+d) is even more challenging, but possible with modern computers and efficient optimization of the codes. These traditional CDCC calculations, however, present shortcomings. The Hamiltonian associated with the system requires optical potentials between the target and the projectile constituents. If optical potentials are in general available for nucleons and α particles, they are often unknown for heavier nuclei, owing to the lack of data on elastic scattering cross sections. Therefore, approximations must be used, either by scaling optical potentials from neighboring nuclei or by evaluating folding potentials. Another limitation comes from the potential model description of the projectile. If this approximation is, in most cases, reasonable, it may introduce inaccuracies in the cross section. In the microscopic CDCC approach (Descouvemont and Hussein 2013), the projectile (with Ap nucleons) is described by a many-body Hamiltonian

37 Theoretical Studies of Low-Energy Nuclear Reactions

H0 =

Ap 

ti +

i=1

Ap 

vij ,

1475

(105)

i 40, e < 50 mev. Phys. Rev. 182, 1190–1209 (1969) G. Bertsch, J. Borysowicz, H. Mcmanus, W.G. Love, Nucl. Phys. A 284, 399 (1977) C.A. Bertulani, P. Danielewicz, Introduction to Nuclear Reactions (Institute of Physics Publishing, London, 2004) C. Bloch, Nucl. Phys. 4, 503 (1957)

1482

P. Descouvemont

P.G. Burke, R-Matrix Theory of Atomic Collisions. Application to Atomic, Molecular and Optical Processes. Springer Series on Atomic, Optical, and Plasma Physics, vol. 61 (Springer, Berlin/Heidelberg, 2011) L.F. Canto, M.S. Hussein, Scattering Theory of Molecules, Atoms and Nuclei (World Scientific Publishing, Singapore, 2013) L.F. Canto, P.R.S. Gomes, R. Donangelo, J. Lubian, M.S. Hussein, Recent developments in fusion and direct reactions with weakly bound nuclei. Phys. Rep. 596, 1 (2015) L.C. Chamon, B.V. Carlson, L.R. Gasques, D. Pereira, C. De Conti, M.A.G. Alvarez, M.S. Hussein, M.A. Cândido Ribeiro, E.S. Rossi, C.P. Silva, Toward a global description of the nucleusnucleus interaction. Phys. Rev. C 66, 014610 (2002) L.C. Chamon, B.V. Carlson, L.R. Gasques, São paulo potential version 2 (spp2) and brazilian nuclear potential (BNP). Comput. Phys. Commun. 267, 108061 (2021) J.A. Christley, I.J. Thompson, Crcwfn: coupled real coulomb wavefunctions. Comput. Phys. Commun. 79, 143 (1994) D.D. Clayton, Principles of Stellar Evolution and Nucleosynthesis (The University of Chicago Press, Chicago, 1983) S.R. Cotanch, C.M. Vincent, Channel coupling and nonorthogonality in elastic and transfer processes. Phys. Rev. C 14, 1739 (1976) P. Descouvemont, An r-matrix package for coupled-channel problems in nuclear physics. Comput. Phys. Commun. 200, 199 (2016) P. Descouvemont, Low-energy 6 He scattering in a microscopic model. Phys. Rev. C 93, 034616 (2016) P. Descouvemont, Four-body extension of the continuum-discretized coupled-channels method. Phys. Rev. C 97, 064607 (2018) P. Descouvemont, Low-energy 11 Li + p and 11 Li + d scattering in a multicluster model. Phys. Rev. C 101, 064611 (2020) P. Descouvemont, D. Baye, The r-matrix theory. Rep. Prog. Phys. 73, 036301 (2010) P. Descouvemont, M. Dufour, Clusters in Nuclei, vol. 2. (Springer, 2012) P. Descouvemont, M.S. Hussein, Towards a microscopic description of reactions involving exotic nuclei. Phys. Rev. Lett. 111, 082701 (2013) P. Descouvemont, M. Vincke, Phys. Rev. A 42, 3835 (1990) P. Descouvemont, E.M. Tursunov, D. Baye, Nucl. Phys. A 765, 370 (2006) P. Descouvemont, L.F. Canto, M.S. Hussein, Coulomb and nuclear effects in breakup and reaction cross sections. Phys. Rev. C 95, 014604 (2017) W.H. Dickhoff, R.J. Charity, Recent developments for the optical model of nuclei. Prog. Part. Nucl. Phys. 105, 252–299 (2019) J. Dohet-Eraly, P. Descouvemont, Exchange effects in nucleus-nucleus reactions. Phys. Rev. C 103, 034619 (2021) T. Druet, P. Descouvemont, Continuum effects in the scattering of exotic nuclei. Eur. Phys. J. A 48, 147 (2012) T. Druet, D. Baye, P. Descouvemont, J.-M. Sparenberg, Cdcc calculations with the lagrange-mesh technique. Nucl. Phys. A 845, 88 (2010) H Feshbach, The optical model and its justification. Ann. Rev. Nucl. Sci. 8(1), 49–104 (1958) R.J. Glauber, High Energy Collision Theory. Lectures in Theoretical Physics, vol. 1 (Interscience, New York, 1959) N.K. Glendenning, Direct Nuclear Reactions (World Scientific, Singapore, 2004) L.J.B. Goldfarb, Inferences concerning the real part of the heavy-ion optical potential through folding techniques. Nucl. Phys. A 301(3), 497–510 (1978) L.J.B. Goldfarb, K. Takeuchi, Effects of non-orthogonality and virtual excitations in direct reactions (I). Nucl. Phys. A 181(2), 609–638 (1972) L.J.B. Goldfarb, K. Takeuchi, Effects of non-orthogonality and virtual excitations in direct reactions (II). Nucl. Phys. A 218(2), 396–404 (1974) K. Hagino, N. Rowley, A.T. Kruppa, A program for coupled-channel calculations with all order couplings for heavy-ion fusion reactions. Comput. Phys. Commun. 123(1), 143–152 (1999)

37 Theoretical Studies of Low-Energy Nuclear Reactions

1483

Y. Han, Y. Shi, Q. Shen, Deuteron global optical model potential for energies up to 200 mev. Phys. Rev. C 74, 044615 (2006) M. Hesse, J.M. Sparenberg, F. Van Raemdonck, D. Baye, Nucl. Phys. A 640, 37 (1998) M. Hesse, J. Roland, D. Baye, Nucl. Phys. A 709, 184 (2002) P.E. Hodgson, The nuclear optical model. Rep. Prog. Phys. 34(2), 765–819 (1971) H. Horiuchi, Kernels of GCM, RGM and OCM and their calculation methods. Prog. Theor. Phys. Suppl. 62, 90 (1977) C. Iliadis, Nuclear Physics of Stars (Wiley-VCH Verlag GmbH, Weinheim, 2007) C. Joachain, Quantum Collision Theory (Elsevier Science, Amsterdam, 1983) J. Gómez Camacho, A.M. Moro, A Pedestrian approach to the theory of transfer reactions: application to weakly-bound and unbound exotic nuclei (Springer, Berlin/Heidelberg, 2014), pp. 39–66 R.C. Johnson, Theory of the a(d,p)b reaction as a tool for nuclear structure studies. J. Phys. G 41(9), 094005 (2014) R.C. Johnson, P.C. Tandy, An approximate three-body theory of deuteron stripping. Nucl. Phys. A 235(1), 56–74 (1974) D.T. Khoa, Exchange effects in nuclear rainbow scattering. Nucl. Phys. A 484(2), 376 (1988) D.T. Khoa, G.R. Satchler, Generalized folding model for elastic and inelastic nucleus-nucleus scattering using realistic density dependent nucleon-nucleon interaction. Nucl. Phys. A 668, 3 (2000) D.T. Khoa, G.R. Satchler, W. von Oertzen, Nuclear incompressibility and density dependent NN interactions in the folding model for nucleus-nucleus potentials. Phys. Rev. C 56, 954–969 (1997) A.M. Kobos, B.A. Brown, P.E. Hodgson, G.R. Satchler, A. Budzanowski, Folding model analysis of alpha-particle elastic scattering with a semirealistic density-dependent effective interaction. Nucl. Phys. A 384, 65 (1982) A.J. Koning, J.P. Delaroche, Local and global nucleon optical models from 1 kev to 200 mev. Nucl. Phys. A 713, 231 (2003) A.M. Lane, R.G. Thomas, Rev. Mod. Phys. 30, 257 (1958) J. Lei, P. Descouvemont, Lagrange-mesh r-matrix method for inhomogeneous equations. Phys. Rev. C 102, 014608 (2020) T. Matsumoto, E. Hiyama, K. Ogata, Y. Iseri, M. Kamimura, S. Chiba, M. Yahiro, Continuumdiscretized coupled-channels method for four-body nuclear breakup in he6 + c12 scattering. Phys. Rev. C 70, 061601 (2004) N. Michel, A simple and efficient numerical scheme to integrate non-local potentials. Eur. Phys. J. A 42(3), 523–527 (2009) A.M. Moro, F.M. Nunes, R.C. Johnson, Theory of (d, p) and (p, d) reactions including breakup: comparison of methods. Phys. Rev. C 80, 064606 (2009) F.M. Nunes, I.J. Thompson, Multistep effects in sub-coulomb breakup. Phys. Rev. C 59, 2652 (1999) T. Ohmura, B. Imanishi, M. Ichimura, M. Kawai, Study of deuteron stripping reaction by coupled channel theory. I: variational formulation and discussion on basic equations. Prog. Theor. Phys. 41(2), 391–418 (1969) N. Oulebsir, F. Hammache, P. Roussel, M.G. Pellegriti, L. Audouin, D. Beaumel, A. Bouda, P. Descouvemont, S. Fortier, L. Gaudefroy, J. Kiener, A. Lefebvre-Schuhl, V. Tatischeff, Indirect study of the 12c(α,γ )16o reaction via the 12c(7li, t)16o transfer reaction. Phys. Rev. C 85, 035804 (2012) D.Y. Pang, P. Roussel-Chomaz, H. Savajols, R.L. Varner, R. Wolski, Global optical model potential for a = 3 projectiles. Phys. Rev. C 79, 024615 (2009) F. Perey, B. Buck, A non-local potential model for the scattering of neutrons by nuclei. Nucl. Phys. 32, 353–380 (1962) E.C. Pinilla, P. Descouvemont, Microscopic description of 7 Li in 7 Li + 12 C and 7 Li + 28 Si elastic scattering at high energies. Phys. Rev. C 89, 054615 (2014)

1484

P. Descouvemont

S. Quaglioni, P. Navrátil, Ab initio many-body calculations of nucleon-nucleus scattering. Phys. Rev. C 79, 044606 (2009) G.H. Rawitscher, Effect of deuteron breakup on elastic deuteron – nucleus scattering. Phys. Rev. C 9, 2210 (1974) J. Raynal, in Computing as a Language of Physics, Trieste, 1971 (IAEA, Vienna, 1972), p. 281 J. Raynal, Recurrence relations for distorted-wave born approximation coulomb excitation integrals and their use in coupled channel calculations. Phys. Rev. C 23, 2571 (1981) M. Rhoades-Brown, M.H. Macfarlane, S.C. Pieper, Techniques for heavy-ion coupled-channels calculations. I. Long-range coulomb coupling. Phys. Rev. C 21, 2417 (1980) M. Rodríguez-Gallardo, J.M. Arias, J. Gómez-Camacho, R.C. Johnson, A.M. Moro, I.J. Thompson, J.A. Tostevin, Four-body continuum-discretized coupled-channels calculations using a transformed harmonic oscillator basis. Phys. Rev. C 77, 064609 (2008) G.R. Satchler, Direct Nuclear Reactions (Oxford University Press, New York, 1983) B.P. Schoenborn, A.C. Nunes, Neutron scattering. Annu. Rev. Biophys. Bioeng. 1(1), 529–552 (1972) Shubhchintak, P. Descouvemont, Transfer reactions with the Lagrange-mesh method. Phys. Rev. C 100(3), 034611 (2019) I.H. Sloan, Note errors in the numerov and runge-kutta methods. J. Comput. Phys. 2(4), 414–416 (1968) Y. Suzuki, R.G. Lovas, K. Yabana, K. Varga, Structure and Reactions of Light Exotic Nuclei (Taylor & Francis, London, 2003) T. Tamura, Phys. Rep. 14C, 59 (1974) I. Tanihata, H. Savajols, R. Kanungo, Recent experimental progress in nuclear halo structure studies. Prog. Part. Nucl. Phys. 68, 215 (2013) I.J. Thompson, Comput. Phys. Rep. 7, 167 (1988) I.J. Thompson, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge/New York, 2010) I.J. Thompson, F.M. Nunes, Nuclear Reactions for Astrophysics: Principles, Calculation and Applications of Low-Energy Reactions (Cambridge University Press, Cambridge, 2009) A.E. Thorlacius, E.D. Cooper, An algorithm for integrating the schrödinger equation. J. Comput. Phys. 72(1), 70–77 (1987) N.K. Timofeyuk, R.C. Johnson, Nonlocality in deuteron stripping reactions. Phys. Rev. Lett. 110, 112501 (2013) R.L. Varner, W.J. Thompson, T.L. McAbee, E.J. Ludwig, T.B. Clegg, A global nucleon optical model potential. Phys. Rep. 201(2), 57 (1991) M. Viviani, A. Deltuva, R. Lazauskas, A.C. Fonseca, A. Kievsky, L.E. Marcucci, Benchmark calculation of p-3 H and n-3 He scattering. Phys. Rev. C 95, 034003 (2017) K. Wimmer, Nucleon transfer reactions with radioactive beams. J. Phys. G 45(3), 033002 (2018) M. Yahiro, T. Matsumoto, K. Minomo, T. Sumi, S. Watanabe, Recent development of cdcc. Prog. Theor. Phys. Suppl. 196, 87 (2012) J.Z.H. Zhang, S.I. Chu, W.H. Miller, Quantum scattering via the s-matrix version of the kohn variational principle. J. Chem. Phys. 88(10), 6233–6239 (1988) J. Zhao, R.M. Corless, Compact finite difference method for integro-differential equations. Appl. Math. Comput. 177(1), 271–288 (2006)

Influence of Nuclear Structure in Relativistic Heavy-Ion Collisions

38

Yu-Gang Ma and Song Zhang

Contents A Brief Introduction to the Relativistic Heavy-Ion Collisions and the Initial State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Brief Introduction to the Nuclear Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of α-Clustering Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of Neutron Skin Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of the Deformation in Isobaric Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Many probes are proposed to determine the quark-gluon plasma and explore its properties in ultra-relativistic heavy-ion collisions. Some of them are related to initial states of the collisions, such as collective flow, Hanbury-Brown-Twiss (HBT) correlation, chiral magnetic effects, and so on. The initial states can come from geometry overlap of the colliding nuclei, fluctuations, or nuclear structure with the intrinsic geometry asymmetry. The initial geometry asymmetry can transfer to the final momentum distribution in the aspect of hydrodynamics during the evolution of the fireball. Different from traditional methods for nuclear structure study, the ultra-relativistic heavy-ion collisions could provide a potential platform to investigate nuclear structures with the help of the

Y.-G. Ma () · S. Zhang Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai, China Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai, China e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_5

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final-state observables after the fireball expansion. This chapter first presents a brief introduction of the initial states in relativistic heavy-ion collisions and then delivers a mini-review for the nuclear structure effects on experimental observables in the relativistic energy domain.

A Brief Introduction to the Relativistic Heavy-Ion Collisions and the Initial State Relativistic heavy-ion collisions aim at investigating a new state of matter, quark-gluon plasma (QGP), which was predicted by quantum chromodynamics (QCD) (Karsch 2002) and is considered to be produced at the early stage of central nucleus-nucleus collisions in experiments (Adams et al. 2005; Müller 2012). The collective motions in partonic level were reported by the RHIC-STAR collaboration (Adams et al. 2004). Recently, a mount of experimental results are obtained to investigate the relationship between the initial states of the collision system and the properties of QGP, such as collective flow (Adamczyk et al. 2013; Adare et al. 2018), Hanbury-Brown-Twiss (HBT) correlation (Adamczyk et al. 2015a), fluctuation (Adamczyk et al. 2015b), and so on. In theoretical side, the initial geometry fluctuations are studied by various models, such as a multiphase transport (AMPT) model (Ma et al. 2016), hydrodynamics (Sievert and NoronhaHostler 2019), etc. Some methods are proposed to perform collective flow and geometry eccentricity analysis related to initial fluctuations (Alver and Roland 2010). The probes, such as collective flow, HBT correlation, or fluctuations, have been extensively investigated, and a new state of matter, so-called QGP, was declared to be created in ultra-relativistic heavy-ion collisions, such as in Au + Au or Pb + Pb collisions, as introduced above. Some similar phenomena are also observed in small systems (p + Al, p + Au, d + Au, 3 He + Au, p + Pb, and p + p) with high multiplicity events (Adare et al. 2018; Abelev et al. 2013) as in large systems (Au + Au and Pb + Pb). It is an open question how to understand transformation coefficient from initial geometry distribution or fluctuation to momentum distribution at final stage in hydrodynamic mechanism (Gardim et al. 2012; Sievert and Noronha-Hostler 2019) and if the matter created in different-size systems undergoes the similar dynamical process and has similar viscosity (Shuryak 2017). There are already a lot of theoretical works contributing to physics explanation and analysis method in this subject (Sievert and Noronha-Hostler 2019; Zhang et al. 2020). Some detailed introduction and discussion can be found in some recent review articles (Nagle and Zajc 2018). Suggestions to perform small collision system experiments could provide great opportunities to uncover the initial geometry effect in different-size collision systems. The collective flow and the related observables play the key roles in experimental and theoretical investigations, and great progress has been made at relativistic energies (Herrmann et al. 1999). Systematic measurements of collective flow at

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center of mass energy from GeV to TeV in nucleus-nucleus collisions have been conducted at RHIC and LHC energies (Adams et al. 2005; Müller 2012) and the number of constituent quark (NCQ) scaling law of elliptic flow (Adams et al. 2004) becomes one of the significant probes to demonstrate the partonic level collectivity of the super hot-dense matter. It arises more efforts to understand the transformation of asymmetry from initial geometry space into final momentum space. Specifically, the system size dependence of collective flow has been analyzed in experiments (Adare et al. 2018; Abelev et al. 2013), and the initial state fluctuations and shear viscosity effect are taken into account in theoretical works (Shuryak 2017). The viscous relativistic hydrodynamics models (Shuryak 2017) suggest a relationship between the collective flow vn and the initial geometry eccentricity εn , ln(vn /εn ) ≈ −n2  ηs (T )Nch −1/3 , where ηs (T ) represents the shear viscosity over entropy density and Nch is the number of charge particles. The intuitive picture of the initial state fluctuations is shown in Fig. 1 (Alver and Roland 2010), where the participants contribute to triangularity or higher-order geometrical eccentricities, and the triangular flow and higher-order flow harmonics are discussed and measured in literature such as Alver and Roland (2010) and Aamodt et al. (2011). Similarly, the number of nucleon (NN) scaling of elliptic flow was first proposed in Yan et al. (2006) by using the quantum molecular dynamics model simulation for lower energy heavy-ion collisions. Even though this conclusion was drawn from low-energy heavy-ion collisions, but it also works at relativistic heavy-ion collisions (Yin et al. 2017). Actually, this NN scaling of elliptic flow was confirmed √ by the STAR Collaboration for light nuclei d, t, 3 H e (for sN N = 200, 62.4, √ 39, 27, 19.6, 11.5, and 7.7 GeV) and anti-deuteron ( sN N = 200, 62.4, 39, 27, √ and 19.6 GeV) and anti-helium 3 ( sN N = 200 GeV) (Collaboraton et al. 2016). The harmonic coefficients of collective flow can be calculated from the Fourier series of particle’s azimuthal distribution (Poskanzer and Voloshin 1998),

10

PHOBOS Glauber MC 5

y(fm)

Fig. 1 Nucleon distribution on the transverse plane in Au √ + Au collisions at sN N = 200 GeV simulated by the Glauber Monte Carlo. The nucleons in the two nuclei are shown in gray and black. Wounded nucleons (participants) are indicated as solid circles, while spectators are dotted circles (Alver and Roland 2010)

0

-5

NPart = 91, H3 = 0.53 -10

-10

0

x(fm)

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d 2N 1 d 3N E 3 = 2π pT dpT dy d p

 1+

N 

 2vn cos[n(φ − ΨRP )] ,

(1)

n=1

where E is the energy, pT is transverse momentum, y is rapidity, and φ is azimuthal angle of the particle. ΨRP denotes the reaction plane angle. The Fourier coefficients vn (n = 1, 2, 3, . . .) are collective flows to characterize different orders of azimuthal anisotropies with the form, vn = cos(n[φ − ΨRP ]),

(2)

where the bracket  means statistical averaging over particles and events. The true reaction plane angle ΨRP is always estimated by event plane angle (Poskanzer and Voloshin 1998) or by participant plane angle (Alver and Roland 2010). The harmonic flow can be calculated with respect to participant plane angle or event plane angle, called as participant plane (PP-) method and event plane (EP-) method, respectively. Some methods avoiding to reconstruct the reaction plane are developed, such as Q-cumulant (QC-) method (Bilandzic et al. 2011; Ma et al. 2016) and two-particle correlation (2PC-) method with rapidity gap (Chatrchyan et al. 2013). In fact, two-particle correlation method has been already developed earlier in low-intermediate energy heavy-ion collision for flow analysis, e.g., see Ma and Shen (1995). The collective flow is driven from the initial anisotropy in geometry space. To investigate transformation from initial geometry to final momentum space, the initial geometry eccentricity coefficients εn are calculated from the participants via (Ma et al. 2016; Alver and Roland 2010; Gardim et al. 2012), En ≡ εn einΦn ≡ −

n einφpart  rpart n  rpart

,

(3)

 2 + y 2 and φ where, rpart = xpart Part are coordinate position and azimuthal angle part of initial participants in the collision zone in the recentered coordinates system (xpart  = ypart  = 0). Φn is the initial participant plane and εn = |En |2 1/2 . The bracket  means the average over the transverse position of all participants event 2 by event. Note that for the definition of eccentricity coefficients εn , rpart weight is alternative, and it was discussed in Gardim et al. (2012) and Alver and Roland (2010). From the above introduction, it can be seen that the collective flow and the related observables are sensitive to the initial state in collisions, which may come from the overlap of the collision zone between the collided nuclei, the initial fluctuation, and the intrinsic geometry of the nuclei. Next the chapter mainly focuses on how the intrinsic geometry (nuclear structure) affects the observables at the final state in the relativistic heavy-ion collisions.

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A Brief Introduction to the Nuclear Structure There are a lot of theoretical works (or models) to describe nuclear structure, such as Woods-Saxon model, Shell-model, effective field theory, or Ab initio calculations. Here the Woods-Saxon model is only introduced, and a short review is made for the exotic nuclear structures, specifically for α-clustering nuclei as well as neutron (proton) skin nuclei in theories and experiments. This is due to that the main purpose of this chapter is to investigate how the exotic nuclear structures affect the final observables in relativistic heavy-ion collisions or on the other hand how the final observables can be used to distinguish the exotic nuclear structures. The Woods-Saxon potential was introduced by R. D. Woods and D. S. Saxon in 1954 to describe nucleon-nucleus scattering, and written as V (r) =

V0 , 1 + e(r−R)/a

(4)

here r is radial distance of nucleon and V0 , R, a are different parameters. The ground state of nuclei can be obtained by solving the Schrödinger equation by using this potential (Fl¨gge 1999). In relativistic heavy-ion collisions, the nucleon density distribution in the initial collided nuclei can be usually parameterized by a three-parameter Fermi (3pF) distribution (Miller et al. 2007), namely, ρ(r) = ρ0

1 + ω(r/R)2 ,  1 + exp r−R a

(5)

where ρ0 is the nucleon density in the center of the nucleus and the three parameters R, a, and ω are related to the nuclear radius, the (surface) skin depth, and deviations from a spherical shape. If ω sets to zero, it becomes the so-called two-parameter Fermi (2pF) distribution (Ilkka et al. 2017; Wei et al. 2014; Tarbert et al. 2014): ρ(r) =

ρ0 . 1 + e(r−R)/a

(6)

To describe nonspherical nuclei, the extended formula can be found in Deng et al. (2016) and Li et al. (2018) by introducing spherical harmonics, and the spatial distribution of deformed nuclei in the rest frame could be written in the WoodsSaxon form: ρ(r, θ ) =

ρ0 1 + exp{[r − R0 − β0 R0 Y20 (θ )]/a}

,

(7)

where ρ0 is the normal nuclear density, R0 and a represent the “radius” of the nucleus and the surface diffuseness parameter, respectively, and β2 was the deformation of the nucleus.

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The Woods-Saxon distribution of nucleons, introduced above, always gives the same radius of proton and neutron distribution in the nucleus, denoted as Rp and Rn , respectively. However, if the density profiles are the same for protons and neutrons in neutron-rich nucleus, the radius of neutron distribution should be larger than that of proton’s, i.e., Rn > Rp , which is called the neutron skin. From the measurements at lower energies, the neutron skin thickness, which is defined as the difference between the root mean square radii for the neutron and proton distributions, is presented, and the mechanism is investigated in Tarbert et al. (2014). To configure a nucleus with neutron skin in a model, one can refer to literature Wei et al. (2014), and here a method based on the two-parameter Fermi (2pF) distribution in equation (6) is introduced. For a nucleus with the neutron skin, the nuclear density is written by ρ A (r) = ρ p,A (r) + ρ n,A (r), and different choice of parameters for protons and neutrons results in the neutron skin effect. In nature, cluster structure is an ordinary phenomenon that can be observed everywhere, such as cloud in the sky, water clusters into ice in the river, and atom forms crystal. In general, it can be considered the constituents to form a stable structure through interaction with each other even though some of the potential among constituents has not been found very well. Keep in mind that the nuclear structure effects are taken into account in heavy-ion collisions, or on the other hand, it will be discussed for the investigation of the exotic nuclear structures from the final observables, and then a brief introduction to α-clustered nuclei will be given for theoretical works and experimental progress. The history of clustered nuclei hypothesis which was first proposed by Gamow (1931) can be traced back decades ago, and the light 4n nuclei is considered to be composed of α particles, which is the so-called α cluster model. And then in the last decades, there are amount of works contributing to this exciting subject of exotic nuclear structure (von Oertzen et al. 2006). Some recent reviews can be found in von Oertzen et al. (2006). Fifty years ago, the famous threshold diagram was put forward, as shown in Fig. 2, which pointed out that cluster structures were mainly found close to cluster decay thresholds (von Oertzen et al. 2006). In this prediction, for example, the ground state of 8 Be would be considered as a bound state of 2αs. Later on, this diagram was extended to N = Z nuclei which can be composed of αs with valence neutrons in von Oertzen et al. (2006), as shown in Fig. 3. From the above introduction, it can be found that there is a chain of clustering nuclei from α, 8 Be, 12 C, 16 O . . . , and another chain for Nα+mn (N , m the number of α, and neutron) nuclei. Actually the investigation on α-conjugate nuclei, such as 8 Be, 12 C, and 16 O, is still an open question for exotic nuclear structures (Schuck et al. 2017). Effort on investigation of 12 C is very impressive, and the obtained precise structural data of 12 C provides a benchmark for verifying first-principles calculations. Fermionic molecular dynamics (FMD), anti-symmetrized molecular dynamics (AMD), and covariant density functional theory (Liu and Zhao 2012) support that ground state of 12 C is in a triangle-like structure with three αs. The Brink type THSR-wave function was employed to demonstrate that there is strong two-α

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Fig. 2 Ikeda threshold diagram for α-clustered nuclei. (The figure is taken from von Oertzen et al. 2006)

correlation in ground state of 12 C (Zhou et al. 2014). The experimental evidence was reported for triangular symmetry in 12 C at the ground state (Marín-Lámbarri et al. 2014). The spectra of giant dipole resonance (GDR) was proposed as fingerprint for α-clustering light nuclei by using an extended quantum molecular dynamics (EQMD) (He et al. 2014). 16 O is another interesting nucleus with an additional α with respect to 12 C. The four αs in 16 O are considered to be arranged in regular tetrahedral distribution, which corresponds to the ground state in the Ikeda diagram as in Fig. 2. Chiral nuclear effective field theory (Epelbaum et al. 2014) and covariant density functional theory (Liu and Zhao 2012) support this regular tetrahedral structure in α-clustered 16 O. The algebraic cluster model (ACM) (Bijker and Iachello 2002; Epelbaum et al. 2014) was used to produce the rotation-vibration spectrum of 16 O which is comparable to that in experiment and provides a strong evidence for tetrahedral symmetry in 16 O. The EQMD calculations presented that 16 O ground state shows the tetrahedral structure, and the corresponding characteristic spectra of GDR are comparable to the experimental result (He et al. 2014).

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Fig. 3 Extended threshold diagram. (The figure is taken from von Oertzen et al. 2006)

Influence of α-Clustering Effects It is an open question how the initial geometrical asymmetry transfers to final momentum space and how the intrinsic deformation of nuclei plays an important role for the collective motion properties in relativistic heavy-ion collisions. The α-clustered light nuclei are arranged in some special geometry structures with the constituent α clusters, such as 12 C in triangle with 3-αs (Liu and Zhao 2012; Zhou et al. 2014; Marín-Lámbarri et al. 2014; He et al. 2014). Broniowski and Ruiz Arriola (2014) and Bo˙zek et al. (2014) proposed an α-clustered carbon colliding against a heavy ion, 12 C + Au, to investigate the collective flow in the evolution of the fireball. The large deformation in the initial intrinsic nucleus was transformed into the anisotropy of the final momentum space in the fireball. The configuration of the α-clustered carbon is described as follows: the centers of three clusters were placed in an equilateral triangle of

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edge length l, and the rms radius of the α-cluster is rα . The model parameters l and rα are optimized by fitting the radial density distribution from the so-called BEC model (Funaki et al. 2006) and the VMC calculations (Pieper et al. 2002) as provided at http://www.phy.anl.gov/theory/research/density. Within the Glauber framework (Rybczy´nski et al. 2014), the so-called mixed model (Back et al. 2002) was used to simulate the initial state of the collisions, and the event-by-event (3+1)-dimensional viscous hydrodynamics (Bo˙zek 2012) was employed to simulate the evolution of the created fireball. In those works, the initial fluctuations and the intrinsic geometry from α-clusters were simulated and discussed. Figure 4 from Bo˙zek et al. (2014) simulating with clustered and uniform 12 C showed that the geometry enhances the triangularity at high values of the number of wounded nucleons, Nw (corresponding to central collisions), and raises ellipticity at lower values of Nw (corresponding to peripheral collisions). With the help of hydrodynamics (Gardim et al. 2012), the ratio of vn {4}/vn {2} was proposed as a probe to distinguish the initial α cluster structure of 12 C for the relationship to the initial geometry deformations, εn {4} vn {4}  , εn {2} vn {2}

(8)

Fig. 4 The eccentricity simulated by the GLISSANDO model for the Bose-Einstein condensation (BEC) case at RHIC as a function of the wounded nucleons Nw which is corresponding to centralities (Bo˙zek et al. 2014)

n{2}

where vn {k} or εn {k} means nth-order harmonic flow coefficients or eccentricity coefficients via k-particle cumulant moments as defined in Bo˙zek et al. (2014). Figure 5 showed the simulation results of vn {4}/vn {2}, and it indicated that the geometric triangularity increased for collisions with a larger number participants, corresponding to central collisions, which was straightforward to measure in ultrarelativistic heavy-ion collisions with standard techniques devoted to analysis of harmonic flow (Bo˙zek et al. 2014). Recently the simulations involving α-clustered 7,9 Be, 12 C, and 16 O were presented in ultra-relativistic heavy-ion collisions (Rybczy´nski et al. 2018). The

0.5

2{2} 3{2} 2{2} 3{2}

0.45 0.4

(BEC) (BEC) (uniform) (uniform) 10%

0.35

1%

0.1%

0.3 0.25 30

40

50

60

70 Nw

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vn{4}/vn{2}

Fig. 5 Ratios of four-particle to two-particle cumulants plotted as a function of the total number of wounded nucleons for the Bose-Einstein condensation (BEC) case and uniform case (Bo˙zek et al. 2014)

1 10%

0.1%

1%

0.9 0.8 0.7

n=2 (BEC) n=3 (BEC) n=2 (uniform) n=3 (uniform)

0.6 0.5

30

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Nw

5

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RDS 20 25

15

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vn{4}/vn{2}

vn{4}/vn{2}

0

0.7 0.6

10%

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n=2, uniform n=3, uniform

n=2 n=3

9Be + 208Pb

0.5

1% 0.1%

30

40 NW

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10 15

RDS 20 25 30 35 40 45 50 55 n=2, uniform n=3, uniform

n=2 n=3

0.7 0.6

16O + 208Pb

0.5 0

10

20

30

10%

40

50

60

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1% 0.1%

80

90

NW

Fig. 6 Ratios of four-particle to two-particle cumulants for 9 Be + Pb (left) and 16 O + Pb (right) collisions, plotted as a function of the total number of wounded nucleons (Rybczy´nski et al. 2018)

influence of fluctuation and the intrinsic geometry effect were investigated sys√ tematically at the SPS collision energy of sN N = 17 GeV. In Rybczy´nski et al. (2018), the ratios of v2 {4}/v2 {2} presented increasing (decreasing) trend with the increasing of the wounded nucleons Nw in the collisions with 7,9 Be beams for α-clustered (uniform) configuration, and the ratios of v3 {4}/v3 {2} gave the similar trend in 16 O + Pb collisions as shown in Fig. 6. As proposed by Citron et al. (2019) and STAR Collaboration (2019), the collisions with 16 O beam were studied in details (Rybczy´nski et al. 2018; Li et al. 2020a) and were considered as a platform to investigate the effect from the intrinsic geometry of nuclei at RHIC or LHC. The effect from the intrinsic geometry was also studied in a multiphase transport (AMPT) model (Lin et al. 2005; Lin and Zheng 2021). AMPT model describes the whole heavy-ion collision processes dynamically as follows: the binary collisions among the initial nucleons result in the exited strings and mini-jet partons which all will fragment into partons, and the partons undergo a cascade process till reaching a so-called parton freeze-out status, the freeze-out partons coalesce into hadrons via a simple coalescence model, and then the hadrons participate in the

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final rescattering process. AMPT was extensively used in the community and can successfully describe various physics results in relativistic heavy-ion collisions at the RHIC and LHC energies (Lin et al. 2005; Lin and Zheng 2021). During the initialization in the original version of AMPT, the initial nucleon coordinate space in nuclei is described by the HIJING model (Gyulassy and Wang 1994) with the Woods-Saxon distribution (Lin et al. 2005). However, the α-clustered nuclear structure in the initial condition embodies via reconfiguring in HIJING as following Li et al. (2020a), He et al. (2020), and Zhang et al. (2017). Three αs in equilateral triangle structure are placed at each vertice with side length l3α and four αs in tetrahedral structure are placed at each vertice with side length l4α , for 12 C and 16 O, respectively. The parameters of l 3α = 1.8 f m and l4α = 3.42 f m were inherited from the EQMD calculation (He et al. 2014), and nucleons in the α follow the Woods-Saxon distribution. And the α-clustered 12 C and 16 O give the rms radius, 2.47 f m and 2.699 f m, respectively, and the rms radii of 12 C and 16 O configured by the Woods-Saxon distribution were 2.46 f m and 2.726 f m, which were all consistent with the experimental data (Angeli and Marinova 2013), 2.47 f m for 12 C and 2.6991 f m for 16 O. Though the 12 C with 3 αs in chain structure was unlikely in the ground state, it was also simulated to investigate the initial intrinsic geometry effect, where the 3 αs were placed in a line with equivalent distance 2.19 f m giving a rms radius of 12 C 2.47 f m. Once it is possible to provide the reasonable initial stable nuclei with different intrinsic geometry structure, they were introduced to the initial state in HIJING process of AMPT model to investigate the α cluster influence in relativistic heavyion collisions. The initial geometry in 12 C + 197 Au collisions was characterized by the eccentricity coefficients εn calculated using Equation (3). Since the initial fluctuation should be also taken into account, the initial coordinates of participants were used to calculate εn and to reconstruct the so-called participant plane (Ma et al. 2016; Alver and Roland 2010):        atan2 r 2 sin nϕpart , r 2 cos nϕpart + π , Ψn {PP } = n

(9)

where, Ψn {PP } is the nth-order participant plane angle, r and ϕpart are coordinate position and azimuthal angle of participants in the collision zone at initial state, and the average · · ·  denotes density weighting. Then the harmonic flow coefficients with respect to participant plane can be defined as vn ≡ cos(n[φ − Ψn {PP }]).

(10)

Figure 7 from Zhang et al. (2017) showed the calculated second- and third-order √ eccentricity coefficients, ε2 {PP } and ε3 {PP } in 12 C + 197 Au collisions at sN N = 200 and 10 GeV for different configurations of 12 C, i.e., the α-clustered chain and triangle structure as well as nucleon distribution in the Woods-Saxon distribution. √ At the top RHIC energy ( sN N = 200 GeV), the chain (triangle) structure enhanced the seconded (third)-order geometry coefficient, and ε2 {PP } (ε3 {PP })

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3{PP}

0.35

2

{PP}

0.6

(a) 12 C+ Au@200 GeV 12 C, Chain 12 C, Woods-Saxon 12 C,Triangle

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Au@10 GeV

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{PP}

3{PP}

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Ntrack

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(d)

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Ntrack

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Ntrack

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√ Fig. 7 In 12 C + 197 Au collisions at sN N = 10 and 200 GeV, second and third order participant eccentricity coefficients, ε2 {PP } and ε3 {PP }, as a function of Ntrack (Zhang et al. 2017)

increased with the increasing of the number of particles, Ntrack , in the collisions. In the Woods-Saxon case, the geometry coefficients (ε2 {PP } and ε3 {PP }) slightly decreased with the increasing of Ntrack , which indicated the fluctuation contribution gradually weakened from peripheral collisions to central collisions. These initial geometry properties were consistent with those in Broniowski and Ruiz Arriola (2014) and Bo˙zek et al. (2014) and comparable with the presented results in Fig. 4. With the help of AMPT model, the evolution of collisions was simulated dynamically from the initial nucleon-nucleon collisions, partonic interaction to the hadronic rescattering, and the initial geometry asymmetry is transferred to the final momentum space. Figure 8 presented the ratios of ε3 {PP }/ε2 {PP } and v3 {PP }/v2 {PP } which reflected the initial geometry properties: v3 {PP }/v2 {PP } from triangle structure configuration of 12 C displayed an increasing trend with the increasing of Ntrack , that from the Woods-Saxon configuration kept a flat pattern in a low pT region, and that from the chain structure configuration of 12 C also kept flat pattern and lower than that from the Woods-Saxon configuration. It indicated that the flow coefficient ratio of v3 {PP }/v2 {PP } was sensitive to the initial geometry properties. Via comparing the results from the Woods-Saxon case and the α-clustered triangle case, the ratio of v3 {PP }/v2 {PP } could be taken as a probe to distinguish the exotic nuclear structure in heavy-ion collisions; note that the chain structure of 12 C was always considered in excited state.

38 Influence of Nuclear Structure in Relativistic Heavy-Ion Collisions

1.5

197

(a) 12 C+ Au@200 GeV 12 C, Chain 12 C, Woods-Saxon 12 C,Triangle

1

1

T

Solid:1

. As noted, Kurasawa and Suzuki examined the fourth moment, < rc4 >, for the first time and discovered that it includes a contribution of the neutron-distribution radius (Kurasawa and Suzuki 2019), which can be described as follows: 10 2 < Rp(point) >< rp2 > 3 10 N 2 + < Rn(point) + rel.corr., >< rn2 > 3 Z

4 >+ < rc4 > =< Rp(point)

(29)

where the neutron-distribution radius, < Rn(point) >, appears explicitly. 2 Knowing < Rp(point) > (Campbell 2016) and < rp2 >, < rn2 > (Workman et al. 2022) determined by independent measurements, one can extract < Rn(point) > from < rc4 > measured by electron scattering. The experimental determination of the < rc4 > of the charge-density distribution using electron scattering is twofold. They are (i) using Eq. (18) with ρc (r) as determined via elastic electron scattering measured under a wide range of q and (ii) using the Taylor expansion of the form factor, Eq. (17), at the low q region. The first method (i) involves the use of the charge-density distributions determined according to the inverse Fourier transformation of the measured charge form factor (Eq. (14)) and is applied for the charge-density distributions of the doubly magic stable nuclei, 48 Ca and 208 Pb, precisely known through elastic electron scattering. The neutron-distribution radii extracted from the , as calculated based on Eq. (18), have been reported to be 5.736 ± 0.013 fm and 3.596 ± 0.009 fm (Kurasawa et al. 2021), respectively, which appears to be consistent with the neutron radius of 208 Pb recently determined via parity-violating electron scattering (Adhikari et al. 2021) (5.795 ± 0.075 fm). However, it is difficult to apply this approach for unstable nuclei, since the precise charge-density distribution is needed, which requires a high luminosity for measuring the elastic cross section, including the high q region. For unstable nuclei with an extremely low luminosity, the second method using Taylor’s expansion could present a way to study the neutron distribution. Due to the 1/q 4 dependence of the Mott cross section, the elastic cross section becomes very large at a low q, thus compensating for the low luminosity of the electron

1612

T. Suda

scattering for unstable nuclei. Measuring the elastic cross section at a low q allows for extracting the < rc4 > from the form factor at the low q region using Eq. (17). 2 On obtaining the < Rp(point) > for exotic nuclei via isotope-shift measurements, the neutron radius described in Eq. (29) can be determined from < rc4 >.

Photonuclear Response Finally, one additional research possibility at a low-luminosity electron scattering facility for exotic nuclei is discussed: possible measurement of the photonuclear response of exotic nuclei covering the giant dipole resonance (GDR) region. The photonuclear response at the GDR is known to reflect the ground-state nuclear shapes (Berman and Fultz 1975). Due to the fact that the atomic nucleus is a quantum system with a rather sharp surface, the nuclear shape is well defined. Generally speaking, however, it has proven to be difficult to study the ground-state deformation of atomic nuclei via electron scattering (Hersman et al. 1986). Here, it should be noted that elastic electron scattering for 0+ nuclei is only sensitive to the spherically symmetric part of the charge distributions, while the nucleus is deformed. Thus far, the Coulomb excitation in a heavy-ion collision has presented the only way to explore the photonuclear response of exotic nuclei, assuming that they are excited only by the Coulomb interaction. In addition to the fact that the interpretation of experimental data is model-dependent, it is difficult to access the whole GDR region since the photon flux in even high-energy heavy-ion collision is strongly hindered at high photon energies, such as Eγ ∼ 30 MeV. Inelastic electron scattering at the ultra-forward angle, θ ∼ 0◦ , could potentially eliminate all these problems. The virtual photon theory (Bertulani 2007) allows for relating the inelastic electron scattering cross section to the photonuclear cross section. Numerical simulations concluded (Suda and Simon 2017) that the ultraforward electron inelastic scattering under the luminosity of ∼1027 /cm2 /s allows for easily determining the photonuclear responses covering the whole GDR region. Since such inelastic electron scattering would appear to be virtually the only way to reliably determine the photonuclear response around the GDR in structure studies of exotic nuclei, including their shapes, it is highly recommended that electron scattering facilities are designed in such a way that also allows for measuring the ultra-forward inelastic electron scattering. The readers are encouraged to refer to a  Chap. 35, “Probing Nuclear Structure with Photon Beams” by J. Isaak and N. Pietralla in this Handbook about the photonuclear reaction study by real photons.

References D. Adhikari et al., Phys. Rev. Lett. 126, 172502 (2021) I. Angeli, K.P. Marinova, At. Data Nucl. Data Tab. 99, 69–95(2013)

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H. Baghaei et al., Phys. Rev. C42, 2358 (1990) B.L. Berman, S.C. Fultz, Rev. Mod. Phys. 47, 713 (1975) W. Bertozzi et al., Phys. Lett. 41B, 408 (1972) C.A. Bertulani, Phys. Rev. C75, 024606 (2007) P. Campbell, I.D. Moore, M.R. Pearson, Prog. Part. Nucl. Phys. 86, 127–180 (2016) J.M. Cavedon et al., Phys. Rev. Lett. 49, 978 (1982) J.M. Cavedon et al., Phys. Rev. Lett. 58, 195–198 (1987) A. D’Alessio et al., Phys. Rev. C102, 011302(R) (2020) T. Deforest, J.D. Walecka, Adv. Phys. 15, 1 (1966) H. de Vries, C.W. de Jager, C. de Vries, At. Data Nucl. Data Tab. 36, 495 (1987) T.W. Donnelly, J.D. Walecka, Ann. Rev. Nucl. Part. Sci. 25, 329(1975) T.W. Donnely, I. Sick, Rev. Mod. Phys. 56, 461 (1984) B. Dreher et al., Nucl. Phys. A235, 219–248 (1974) H. Euteneuer, J. Friedrich, N. Voegler, Nucl. Phys. A298, 452 (1978) J.R. Ficenec et al., Phys. Lett. 32B, 460 (1970) J.R. Ficenec et al., Phys. Lett. 42B, 213 (1972) G. Fricke et al., At. Data Nucl. Data Tab. 60, 177–285 (1995) J. Friedrich, N. Voegler, Nucl. Phys. A373, 192 (1982) R.F. Frosch et al., Phys. Rev. 174, 1380 (1968) W.J. Gerace, G.C. Hamilton, Phys. Lett. 39B, 481 (1972) J. Heisenberg et al., Phys. Rev. Lett. 23, 1402–1405 (1969) J. Heisenberg, H.P. Blok, Ann. Rev. Nucl. Part. Sci. 33, 569 (1983) F.W. Hersman et al., Phys. Rev. C33, 1905–1916 (1986) R. Hofstadter, Nobel Lecture Archive. (1961) https://www.nobelprize.org/prizes/physics/1961/ hofstadter/lecture/ R. Hofstadter, Ann, Rev. Nucl. Sci. 7, 231(1957) R. Hofstadter, H.R. Fechter, J.A. McIntyre, Phys. Rev. 92, 978 (1953) H.V. Hynes et al., Phys. Rev. Lett. 42, 1444 (1979) F.P. Juster et al., Phys. Rev. Lett. 55, 2261 (1985) N. Kalanter-Nayestanaki et al., Phys. Rev. Lett. 60, 1707 (1988) T. Katayama et al., Nucl. Phys. A626, 545c–560c (1997) V. Khvastun et al., Nucl. Phys. A146, 15 (1970) F.J. Kline et al., Nucl. Phys. A209, 381 (1973) H. Koura, T. Tachibana, M. Uno, M. Yamamada, Prog. Theor. Phys. 113, 305 (2005) C. Kremer et al., Phys. Rev. Lett. 117, 172503 (2016) H. Kurasawa, T. Suzuki, Prog. Theor. Exp. Phys. 2019, 113D01 (2019) H. Kurasawa, T. Suzuki, Prog. Theor. Exp. Phys. 2022, ptac008 (2022) H. Kurasawa, T. Suda, T. Suzuki, Prog. Theor. Exp. Phys. 2021, 013D02 (2021) L. Lapikas, Nucl. Phys. A553, C297 (1993) G. Muenzenberg et al., Nucl. Phys. A626, 249c–252c (1997) T. Ohnishi et al., Nucl. Instrum. Method B317, 357–360 (2013) I. Sick, Phys. Lett. 44B, 62 (1973) I. Sick, H. Flocard, M. Veneroni Phys. Lett. 39B, 443 (1972) R.P. Singhal et al., Nucl. Phys. A216, 29 (1973) B.B.P. Sinha et al., Phys. Lett. 35B, 217 (1971) T. Suda, J. Phys. 1643, 012159 (2020) T. Suda, H. Simon, Prog. Part. Nucl. Phys. 96, 1–13 (2017) T. Suda et al., Phys. Rev. Lett. 102, 102501 (2009) I. Tanihata, H. Savajols, R. Kanungo, Prog. Part. Nucl. Phys. 68, 215–313, (2013) K. Tsukada et al., Phys. Rev. Lett. 118, 262501 (2017) H. Uberall, Electron Scattering from Complex Nuclei (Academic, New York, 1971)

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M. Wakasugi, T. Suda, Y. Yano, Nucl. Instrum. Method 532, 3–14 (2004) M. Wakasugi et al., Phys. Rev. Lett. 100, 16 (2008) M. Wakasugi et al., Nucl. Instrum. Method B317, 668–673 (2013) M. Wakasugi et al., Rev. Sci. Instrum. 89, 095107 (2018) J.D. Walecka, Electron Scattering for Nuclear and Nucleon Structure. Cambridge Monographs on Particle Physics (Cambridge University Press, Cambridge, 2001) R.L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2022)

Sub-barrier Fusion Reactions

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A General Introduction to Heavy-Ion Fusion Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . Earlier Review Articles and Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential Model and the Wong Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparisons with Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fusion of Deformed Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupled-Channels Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fusion Barrier Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deep Subbarrier Fusion Hindrance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fusion of Neutron-Rich Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fusion Reactions for Superheavy Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superheavy Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heavy-Ion Fusion Reactions for Superheavy Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Modelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hot Versus Cold Fusion Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Role of Deformation in Hot Fusion Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1616 1616 1618 1619 1619 1621 1621 1624 1628 1630 1631 1632 1632 1633 1634 1635 1636 1638

Abstract

The concept of compound nucleus was proposed by Niels Bohr in 1936 to explain narrow resonances observed in scattering of a slow neutron off atomic nuclei. A compound nucleus is a metastable state with a long lifetime, in which all the degrees of freedom are in a sort of thermal equilibrium. Fusion reactions are defined as reactions to form such compound nucleus by merging two

K. Hagino () Department of Physics, Kyoto University, Kyoto, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_9

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atomic nuclei. Here a short description of heavy-ion fusion reactions at energies close to the Coulomb barrier is presented. This includes (i) an overview of a fusion process, (ii) a strong interplay between nuclear structure and fusion, (iii) fusion and multidimensional/multiparticle quantum tunneling, and (iv) fusion for superheavy elements.

Introduction A General Introduction to Heavy-Ion Fusion Reactions A fusion reaction is defined as a reaction to form a compound nucleus, the concept of which was originally proposed by Niels Bohr in 1936. In the year earlier, Enrico Fermi performed experiments with slow neutrons and observed many narrow resonances in scattering cross sections. The width of those resonances was typically order of a few eV (see, e.g., Asghar et al. 1966), which is much smaller than a typical nuclear scale of order of MeV. This implies that the resonances formed by reactions of slow neutrons are very long lived. Bohr considered that the energy of the incident neutron was distributed among the other nucleons in a nucleus after many collisions, and a kind of thermal equilibrium state was formed. This is the concept of compound nucleus which Bohr proposed. Since a nucleus is a finite system, the energy may be concentrated once again to one of the neutrons in a nucleus, and that neutron is scattered off from a nucleus. This happens only at a long time after the compound nucleus is formed, leading to narrow resonance widths. Similar compound nuclei are formed in heavy-ion fusion reactions by bombarding two heavy nuclei. Such fusion reaction plays an important role in several phenomena in nuclear physics, such as the energy production in stars, nucleosyntheses, and formations of superheavy elements. Theoretically, fusion and fission are large-amplitude motions of quantum many-body systems with a strong interaction, and their microscopic understanding is one of the ultimate goals of nuclear physics (Bender et al. 2002). Figure 1 shows a schematic illustration of a fusion process. At first a projectile nucleus (“P”) collides with a target nucleus (“T”) and forms a unified nucleus (“P+T”), i.e., a compound nucleus. Since the projectile nucleus brings the energy and the angular momentum into a system, the compound nucleus is at high excitation energies with large angular momenta. For light compound nucleus with a mass number less than about 170, the compound nucleus decays mainly by emitting neutrons, protons, alpha particles, and gamma rays. Such process is called an evaporation process. For heavy compound nucleus with a mass number larger than about 220, fission dominates the decay process of the compound nucleus. This is particularly the case for fusion reactions relevant to superheavy nuclei. For compound nuclei with intermediate mass numbers, the evaporation and the fission processes compete with each other. In any case, fusion cross sections are measured by detecting residues of the evaporation process (called “evaporation residues”) and/or fission fragments.

42 Sub-barrier Fusion Reactions

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Fig. 1 A schematic illustration of heavy-ion fusion reactions

Potential (MeV)

16

100 80 60 V 40 b 20 0 -20 -40 -60 -80 5

O+

144

Sm

Coulomb interaction

R touch R b

nuclear interaction

10

15

20

r (fm) Fig. 2 An internucleus potential between two nuclei as a function of the distance between them (the solid line). The 16 O+144 Sm reaction is considered as a typical example. The Coulomb and the nuclear contributions are denoted by the dotted and the dashed lines, respectively. Rb and Vb denote the position and the height of the Coulomb barrier, respectively. Rtouch is the distance at which two nuclei touch with each other

Figure 2 shows a typical potential between two nuclei as a function of the distance r between them. Two different interactions are involved here. Firstly, a nucleus has a positive charge, and the Coulomb interaction acts between two nuclei. This is a long-range and repulsive interaction. When the distance between the two nuclei gets smaller, an attractive short-range nuclear interaction (i.e., the strong interaction) becomes active. Because of the cancellation of these two, a potential barrier, referred to as the Coulomb barrier, is formed at some distance Rb , which is usually larger than the touching radius Rtouch at which the two nuclei touch with

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each other. The height of the Coulomb barrier, Vb , specifies the energy scale of the reaction system. In this chapter, we overview the fusion dynamics at energies around the Coulomb barrier, that is, subbarrier fusion reactions. There are two obvious reasons why the subbarrier region is important. One is related to fusion reactions to form superheavy elements. Usually such experiments are carried out at energies slightly above the Coulomb barrier. For instance, in the 209 Bi(70 Zn,n)278 Nh reaction to form the element 113 (nihonium), the experiments were performed at Ec.m. = 262 MeV in the center of mass frame (Morita et al. 2004, 2007, 2012), while a barrier height for this reaction is around 260 MeV if the Bass potential (Bass 1980) is employed. Fusion reactions for superheavy elements will be further discussed in a later section in this chapter. The second obvious reason to discuss the subbarrier region is in connection to nuclear astrophysical reactions. Nuclear fusion reactions in stars, such as the 12 C+12 C reaction, take place at extremely low energies, for which direct measurements of fusion cross sections are difficult. One thus has to extrapolate measured fusion cross sections at higher energies down to the energy region relevant to nuclear astrophysics. In order to do reliable extrapolations, deep understandings of the fusion dynamics in the subbarrier region are crucially important. Besides these two obvious reasons, the reaction dynamics of subbarrier fusion is intriguing in its own. Firstly, it is known that nuclear structure affects significantly nuclear fusion, and thus there is a strong interplay between nuclear structure and nuclear reaction there. This is in contrast to high-energy nuclear reactions, at which the reaction dynamics is much simpler. Secondly, subbarrier fusion reactions can be regarded as a typical example of many-particle tunneling phenomena. In order for fusion to take place, two nuclei have to get close at least to the touching radius, and thus fusion occurs only by quantum tunneling effect when the incident energy is below the Coulomb barrier (see Fig. 2). An interesting fact in atomic nuclei is that there are many types of intrinsic degrees of freedom which can affect quantum tunneling, that is, there are several types of collective vibrations as well as nuclear deformation with several multipolarities. Moreover, the energy dependence of the tunneling rate can be studied in fusion reactions by varying the incident energy, in a marked contrast to other tunneling phenomena in nuclear physics, such as alpha decays, for which the energy is basically fixed by the decay Q-value. Heavy-ion fusion reactions can be thus considered as an ideal playground to study manyparticle quantum tunneling with many degrees of freedom.

Earlier Review Articles and Textbooks A few review articles have been published on the subject of subbarrier fusion reactions. While Balantekin and Takigawa (1998) and Hagino and Takigawa (2012) discuss theoretical aspects of subbarrier fusion reactions, Dasgupta et al. (1998) summarizes experimental observations in subbarrier fusion reactions from a viewpoint of the so-called fusion barrier distributions. Back et al. (2014), Montagnoli and Stefanini (2017), and Jiang et al. (2021) discuss heavy-ion fusion

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reactions at deep subbarrier energies, at which fusion cross sections appear to be hindered as compared to simple extrapolations of fusion cross sections at subbarrier energies. Subbarrier fusion reactions are discussed also in many textbooks of nuclear reactions; see, e.g., Fröbrich and Lipperheide (1996), Bertulani and Danielewicz (2004), Thompson and Nunes (2009), and Canto and Hussein (2013).

Potential Model Potential Model and the Wong Formula The simplest approach to fusion reactions is to employ the potential model, in which one considers inert projectile and target nuclei and assumes some potential between them. For fusion reactions of medium-heavy nuclei, it is considered to be a good approximation to assume that a compound nucleus is formed automatically once the touching position is achieved. Fusion cross sections σfus (E) are then given by σfus (E) =

π  (2l + 1)Pl (E), k2

(1)

l

 where E is the bombarding energy in the center of mass frame and k = 2μE/h¯ 2 is the wave number for the relative motion between the two nuclei with the reduced mass μ. l is the orbital angular momentum for the relative motion, and Pl (E) is the probability to reach the touching configuration. The factor 2l + 1 is simply a statistical weight coming from the fact that the probability Pl does not depend on the z-component of l. Notice that Pl (E) is nothing but the penetrability of the Coulomb barrier. This can be evaluated, e.g., by adding a short-range absorbing potential to an internucleus potential. Such absorbing potential in general describes any process besides elastic scattering, but to a good approximation it simulates the compound nucleus formation as long as it is well confined inside the Coulomb barrier. Based on this approach, Wong has derived a simple compact formula for fusion cross sections (Wong 1973) (see also Rowley and Hagino 2015). To this end, he first approximated the Coulomb barrier by an inverted parabola, 1 V (r) ∼ Vb − μΩ 2 (r − Rb )2 , 2

(2)

for which the penetrability can be given analytically as P0 (E) =

1 + exp



1 2π h¯ Ω (Vb

− E)

.

(3)

For nonzero partial waves, he considered l-independent barrier position and curvature and replaced Pl (E) with

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 Pl (E) ∼ P0

l(l + 1)h¯ 2 E− 2μRb2

 (4)

.

Finally, Wong replaced the sum in Eq. (1) by an integral σfus (E) =

 π  π (2l + 1)P (E) → dl (2l + 1)P (l, E) l k2 k2

(5)

l

to obtain the so-called Wong formula given by

 hΩ 2π ¯ 2 R ln 1 + exp σfus (E) = (E − Vb ) . 2E b hΩ ¯

(6)

Notice that the first energy derivative of Eσfus from this formula is proportional to the s-wave penetrability of the Coulomb barrier, d [Eσfus (E)] = π Rb2 P0 (E), dE

(7)

where P0 (E) is given by Eq. (3). Figure 3 shows calculated fusion cross sections for the 16 O+144 Sm system. The dashed lines are obtained with the Wong formula, while the solid lines are obtained by numerically solving the Schrödinger equation for each partial wave to obtain Pl (E). The left and the right panels show the results in the linear and the logarithmic scales, respectively. One can see that the Wong formula works well, except for the region well below the Coulomb barrier (the height of the Coulomb barrier is about Vb = 61.25 MeV for this system; see Fig. 2), at which the Wong formula overestimates the fusion cross sections. The overestimate of fusion cross sections at 500

σfus (mb)

400 300 200

16

O+

144

Sm

Exact Wong

100 0 55

10 10

2

10 10 10

60 65 Ec.m. (MeV)

3

1

0

-1

-2

70 10 55

60 65 Ec.m. (MeV)

70

Fig. 3 A comparison of the fusion cross sections obtained with the Wong formula (the dashed lines) with the exact fusion cross sections (the solid lines) for the 16 O+144 Sm system. The left and the right panels show the results with the linear and the logarithmic scales, respectively

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energies well below the barrier is because the parabolic approximation (2) used in the Wong formula underestimates the width of the potential barrier, which results in the overestimate of the penetrabilities.

Comparisons with Experimental Data Figure 4 compares fusion cross sections obtained with the potential model to the experimental data for the 14 N+12 C (the left panel) and the 16 O+154 Sm (the right panel) systems. The height of the Coulomb barrier is around Vb ∼ 6.9 MeV for the 14 N+12 C system and V ∼ 59 MeV for the 16 O+154 Sm system. For the 14 N+12 C b system, one can see that the potential model works well. On the other hand, for the 16 O+154 Sm system, the potential model largely underestimates the fusion cross sections at energies below the Coulomb barrier, even though it works well at energies above the barrier. This phenomenon is referred to as the subbarrier enhancement of fusion cross sections and has been systematically observed in a number of systems (Beckerman 1985, 1988).

Fusion of Deformed Nuclei The subbarrier enhancement of fusion cross sections for the 16 O+154 Sm system shown in Fig. 4 can be easily explained if one notices that the 154 Sm nucleus is a typical deformed nucleus. This nucleus exhibits characteristic rotational excitations, for which the energy of a state with the angular momentum I is proportional to I (I + 1) (see Fig. 5). This is interpreted as that 154 Sm is statically deformed in the

10

σfus (mb)

10

3

2

10 10 10 10

1

0

14

-1

16

O+

154

Sm

potential model

potential model

-2

10

12

N+ C

-3

-4

10 3

4

5

7 6 Ec.m. (MeV)

8

9 50

55

60 65 Ec.m. (MeV)

70

75

Fig. 4 Fusion cross sections for the 14 N+12 C system (the left panel) and for the 16 O+154 Sm system (the right panel) obtained with the potential model. The arrows indicate the height of the Coulomb barrier for each system. (The experimental data are taken from Switkowski et al. 1977 and Leigh et al. 1995)

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8+

0.544

6+

0.267

4+

0.082 0

2++ 0

154Sm

154

1 EI (MeV)

(MeV) 0.903

Sm

0.8

0.6

0.4 0.2 0 0

20

40 I(I+1)

60

80

Fig. 5 The spectrum of the 154 Sm nucleus. Each level is specified by its angular momentum I and parity π as I π Fig. 6 A schematic view of nuclear deformation; see also Naito et al. (2021)

z

Oblate

Spherical

Prolate

z Rz

R0 Rx

x

ground state (see Fig. 6). For axially symmetric shape, the nuclear deformation is often characterized by the deformation parameters {βλ } defined as  R(θ ) = R0 1 +



 βλ Yλ0 (θ ) ,

(8)

λ

where R(θ ) is the angle-dependent radius of a nucleus, R0 is the radius in the spherical limit, θ is the angle measured from the symmetric axis, and Yλ0 (θ ) is the spherical harmonics (see the lower figure of Fig. 6). For the 154 Sm nucleus, the

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q 154Sm

16O

Fig. 7 The angle-dependent internucleus potential (the left panel) for the 16 O+154 Sm system, for which the orientation angle of the deformed 154 Sm is denoted by θ. The corresponding penetrabilities for s-wave scattering are shown in the right panel. The dashed lines show the potential (the left panel) and the penetrability (the right panel) in the spherical limit

deformation parameters are β2 ∼ 0.30 for λ = 2 (the quadrupole deformation) and β4 ∼ 0.05 for λ = 4 (the hexadecapole deformation) (Leigh et al. 1993, 1995). When a target nucleus is deformed, the internucleus potential depends on the orientation angle of the deformed nucleus. When the projectile nucleus approaches the target nucleus from the direction of the longer axis of the target, the Coulomb barrier is lowered as compared to the potential in the spherical limit. This is because the nuclear attraction acts from longer distances. The lowering of a barrier results in an enhancement of the penetrability. For a prolately deformed nucleus with β2 > 0, this corresponds to the angle θ = 0 (see Fig. 7). The opposite happens when the projectile approaches from the direction of the shorter axis of the target. For a prolately deformed nucleus with β2 > 0, this corresponds to the angle θ = π/2. The total penetrability is computed by averaging the angle-dependent penetrability as  1 P (E) = d(cos θ ) P0 (E; θ ), (9) 0

where P0 (E; θ ) is the penetrability for the orientation angle θ . The thick solid line in the left panel of Fig. 8 is obtained in this way. Since the tunneling probability has an exponential dependence on the energy, the enhancement of the penetrability due to θ ∼ 0 leads to the main contribution to the total penetrability at energies below the barrier. The total penetrability is thus enhanced at these energies as compared to the penetrabilities in the spherical limit. This is the main mechanism for the subbarrier enhancement of fusion cross sections shown in Fig. 4. The formula (9) can be actually extended to fusion cross sections σfus as

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K. Hagino 3

10

1

16

O+

2

Spherical θ = 0 deg. θ = 90 deg. average

0.5

0 50

55

60

Ec.m.

65 (MeV)

σfus (mb)

P (l = 0)

10

10

Sm

1

0

10

Expt. Pot. Model Deformation

-1

10

-2

10 -10

70

154

-5

0 5 E-Vb (MeV)

10

Fig. 8 (The left panel) The same as the right panel of Fig. 7, but with the penetrability obtained by averaging all the orientation angles (the thick solid line). (The right panel) The same as the right panel of Fig. 4, but with the fusion cross sections obtained by averaging all the orientation angles (the solid line). The fusion cross sections are plotted as a function of energy relative to the height of the Coulomb barrier



1

σfus (E) = 0

(0)

d(cos θ ) σfus (E; θ ).

(10)

The solid line in the right panel of Fig. 8 is obtained in this way. One can see that the subbarrier enhancement of fusion cross sections for this system is well accounted for by taking into account the deformation of the 154 Sm nucleus. This clearly demonstrates that there is a strong interplay between nuclear structure and heavy-ion fusion reactions at subbarrier energies.

Coupled-Channels Approach The subbarrier fusion enhancement discussed in the previous section has been observed also in systems with non-deformed target nuclei. It has been understood by now that the subbarrier fusion enhancement is caused by couplings of the relative motion between colliding nuclei to several low-lying collective excitations in the nuclei as well as particle transfer processes (Balantekin and Takigawa 1998; Hagino and Takigawa 2012; Dasgupta et al. 1998). The deformation effect discussed in the previous section is a special case, in which the rotational excitations due to a nuclear deformation can be taken into account in terms of orientation-dependent internucleus potential (Hagino and Takigawa 2012). In order to take into account such coupling effects, the coupled-channels approach has been developed (Hagino and Takigawa 2012; Hagino et al. 2022; Tamura 1965; Satchler 1983; Broglia and Winther 2004). This is a quantal reaction theory schematically illustrated in Fig. 9. In this figure, couplings of the relative motion to a state in the target nucleus with the angular momentum I = 2 and positive parity π = + are considered. At the initial stage of the reaction, the target nucleus is in the ground state, I π = 0+ . The wave function for the relative motion for this configuration is denoted by ψ0 (r). During

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Fig. 9 A schematic illustration of the coupled-channels approach. Here, excitations to a state with I π = 2+ in a target nucleus are considered. ψ(r) is the wave function for the relative motion, which depends on the target states

the reactions, due to the interaction between the projectile and the target nuclei, the target nucleus may be excited to the I π = 2+ state, and at the same time, the relative wave function is changed to ψ1 (r). The coupling is taken into account by the off-diagonal components of the potential V (r). The I π = 2+ state may be deexcited to the 0+ state, and thus the two wave functions ψ0 (r) and ψ1 (r) are coupled to each other. One then solves in a non-perturbative manner coupled Schrödinger equations, which are referred to as coupled-channels equations, to determine the S-matrix, from which several reaction observables can be constructed. A few computer codes are available for coupled-channels calculations, such as ECIS (Raynal, Saclay Report No. DPh-T 69/42, unpublished; Lépine-Szily and Lichtenthäler 2021), FRESCO (Thompson and Nunes 2009; Thompson 1988), and CCFULL (Hagino et al. 1999). As an example, Fig. 10 shows fusion cross sections for the 58 Ni+58 Ni system calculated with the code CCFULL. Here, the quadrupole excitations up to the double-phonon states are taken into account in each of the 58 Ni nucleus. By taking into account the excitations of the 58 Ni nuclei, the subbarrier enhancement of fusion cross sections is well reproduced. One may regard this as a clear example of coupling-assisted tunneling phenomenon. It is instructive to discuss how the subbarrier enhancement of fusion cross sections is realized using a schematic model. To this end, let us consider a twochannel problem for scattering in one dimension (Dasso et al. 1983a, b) and solve the coupled-channels equations in the form of

h¯ 2 d 2 − + 2m dx 2



V (x) F (x) F (x) V (x) + ε



 −E

u0 (x) u1 (x)

 = 0.

(11)

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K. Hagino

σfus(E) (mb)

10 10

3

10 10 10

58

2

58

Ni + Ni

1

0

Pot. model coupled-ch.

-1

-2

10 90

95

100 105 Ec.m. (MeV)

110

Fig. 10 Fusion cross sections for the 58 Ni+58 Ni system. The dashed line shows the results of the potential model. The solid line shows the results of the coupled-channels calculations, in which the quadrupole excitations up to the double-phonon states in each 58 Ni are taken into account. (The experimental data are taken from Beckerman et al. 1981)

Here, V (x) describes a potential barrier, and F (x) denotes the coupling potential between the two channels. ε is the energy of the excited state relative to the ground state. Assuming that the particle is incident from the right-hand side of the potential barrier, these equations are solved with the boundary conditions of u0 (x) → e−ik0 x − R0 eik0 x → T0 e

−ik0 x

(x → ∞),

(12)

(x → −∞),

(13)

and  u1 (x) → −  →

k0 R1 eik1 x k1 k0 T1 e−ik1 x k1

(x → ∞),

(14)

(x → −∞),

(15)

  with k0 = 2mE/h¯ 2 and k1 = 2m(E − ε)/h¯ 2 . The penetrability of the barrier is then given by P (E) = |T0 |2 + |T1 |2 .

(16)

The upper panel of Fig. 11 shows the penetrability P (E) so obtained with a 2 2 Gaussian barrier given by V (x) = V0 e−x /2σ . The coupling potential is also 2 2 assumed to have a Gaussian form, F (x) = F0 e−x /2σ . The parameters are set to be V0 = 100 MeV, σ = 3 fm, and F0 = 3 MeV, together with ε = 1 MeV and m = 29 × 938 MeV/c2 . The dashed line shows the result without the coupling

42 Sub-barrier Fusion Reactions

1 0.8 P

0.6 0.4 No coupling C.C.

0.2 0 0.4 -1

dP/dE (MeV )

Fig. 11 (The upper panel) The penetrability of a one-dimensional Gaussian barrier in the presence of the channel coupling effects. The height of the barrier is set to be 100 MeV. The dotted line denotes the result without the channel coupling, while the solid line shows the result of the coupled-channels calculations. (The lower panel) The energy derivative of the penetrability shown in the upper panel

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0.3 0.2 0.1 0 90

95

100 105 E (MeV)

110

(i.e., the case with F0 = 0), while the solid line is obtained by solving the coupledchannels equations. As in the subbarrier fusion enhancement, one can see that the penetrability is enhanced at energies below the barrier. When the excitation energy ε is zero, the coupled-channels equations (11) can be transformed to two decoupled equations,  h¯ 2 d 2 + V (x) ± F (x) − E u± (x) = 0, − 2m dx 2

(17)

√ with u± (x) = (u0 (x) ± u1 (x))/ 2. That is, the wave functions u± (x) are governed by the potentials V (x) ± F (x), one of which lowers the barrier and the other raises the barrier. The penetrability is then given by P (E) =

1 [P0 (E; V (x) + F (x)) + P0 (E; V (x) − F (x))] , 2

(18)

where P0 (E; V (x)) is the penetrability for a potential barrier V (x). Similar to fusion of deformed nuclei, the total penetrability is enhanced as compared to the penetrability for the no-coupling case, because the contribution of the lowered barrier is much more significant than the contribution of the higher barrier. This remains the same even with a finite excitation energy, ε (Hagino et al. 1997).

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Fusion Barrier Distributions Equation (18) can be generalized to cases with more than two barriers as P (E) =



wα P0 (E; Vα (x)),

(19)

α

where α denotes “eigenchannels” with the potential Vα (x) and wα is the weight factor for each eigenchannel. That is, the penetrability is given as a weighted sum of the penetrability for each eigenchannel α. In this case, a single barrier is replaced by a set of distributed barrier. The corresponding formula for fusion cross sections reads σfus (E) =



(0)

wα σfus (E; Vα (r)).

(20)

α

In the case of fusion of deformed nuclei, Eq. (10), the eigenchannel α corresponds to the orientation angle θ with the weight factor wθ = 2π sin θ . Since the penetrability P varies from zero to one at energies around the barrier height, its energy derivative shows a Gaussian-like peak centered at the barrier height energy (see the dashed line in the lower panel of Fig. 11) (For a classical penetrability, P (E) = θ (E − Vb ), the energy derivative is given by a delta function, dP /dE = δ(E − Vb ).). This implies that the energy derivative of Eq. (19) shows many peaks centered at the barrier height for each eigenchannel and that the height of each peak is proportional to the corresponding weight factor, wα . This is demonstrated in the lower panel of Fig. 11 for a two-channel case. Noticing the relation given by Eq. (7), one finds that the corresponding quantity for fusion cross sections is the second energy derivative of Eσfus (E) given by Dfus (E) =

d 2 (Eσfus (E)) . dE 2

(21)

This quantity is referred to as the fusion barrier distribution (Dasgupta et al. 1998; Rowley et al. 1991) and has been experimentally extracted for several systems (Dasgupta et al. 1998; Leigh et al. 1995). Similar barrier distributions have been extracted using also quasi-elastic scattering (i.e., a sum of elastic, inelastic, and transfer processes) at backward angles (Timmers et al. 1995; Hagino and Rowley 2004). The fusion barrier distribution converts the exponential behavior of fusion excitation functions to the linear scale and is suitable to visualize details of the underlying dynamics of subbarrier fusion reactions. Figure 12 shows the barrier distribution for the 16 O+154 Sm system as an example. The solid line shows the barrier distribution obtained with Eq. (10), while the dashed lines show the contribution of different orientation angles. The fusion barrier distribution is structured because of the distribution of many barriers. It has been shown that the shape of the fusion

42 Sub-barrier Fusion Reactions

800 16

Dfus (mb / MeV)

Fig. 12 The fusion barrier distribution for the 16 O+154 Sm system. The solid line is obtained by averaging all the orientation angles of the deformed 154 Sm by Eq. (10). The dashed lines show the contributions of different orientation angles. (The experimental data are taken from Leigh et al. 1995)

1629

600

O+

154

Sm

Eq. (10)

400 200 0 -200 50

55

60 Ec.m. (MeV)

65

70

Fig. 13 The barrier distribution for the 24 Mg+90 Zr system extracted from quasi-elastic scattering at backward angles. The right panel shows the posterior probability distribution for the Bayesian analysis for the quadrupole and the hexadecapole deformation parameters of 24 Mg. (Taken from Gupta et al. 2020)

barrier distribution is sensitive to the deformation parameters used in the calculation (Dasgupta et al. 1998; Leigh et al. 1995). Using such sensitivity of the barrier distribution to the deformation parameters, the quadrupole and the hexadecapole deformation parameters of the 24 Mg nucleus have been extracted recently (Gupta et al. 2020). To this end, the barrier distribution extracted from the quasi-elastic scattering for the 24 Mg+90 Zr system was analyzed with the Bayesian statistics. From this analysis, the hexadecapole deformation parameter of 24 Mg, β4 = −0.11 ± 0.02, has been determined precisely for the first time (see Fig. 13).

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Deep Subbarrier Fusion Hindrance At energies well below the Coulomb barrier, that is, at E  Vb , the Wong formula (6) is reduced to σfus (E) ∼

 hΩ 2π ¯ (E − Vb ) . Rb2 exp − hΩ 2E ¯

(22)

That is, fusion cross sections fall off exponentially. This has been generally observed experimentally. However, as the energy decreases further down, it has been systematically observed that fusion cross sections fall off much steeper (Jiang et al. 2002, 2021; Back et al. 2014). This phenomenon has been referred to as deep subbarrier fusion hindrance. It has been considered that the hindrance is attributed to the dynamics after two colliding nuclei touch with each other (Ichikawa et al. 2007; Jiang et al. 2021). As an example, Fig. 14 shows the fusion cross sections for the 64 Ni+64 Ni system. The results of the standard coupled-channels calculations are denoted by the dashed and the dot-dashed lines. These calculations well reproduce the experimental data at energies larger than about 89 MeV (see the arrow). However, at lower energies the experimental data show hindrance as compared to the standard coupled-channels calculation and fall off much steeper. The solid line models the deep subbarrier hindrance by quenching the coupling strengths after two nuclei

10

3

Cross Section (mb)

102

64

Ni +

64

Ni

101 10

0

10

–1

10

–2

10

–3

10

–4

Exp. Woods-Saxon YPE(NC) YPE + damping

10–5 10–6

85

90

95 100 Ec.m. (MeV)

105

110

Fig. 14 The fusion cross sections for the 64 Ni+64 Ni system. The dashed and the dot-dashed lines show the results of the standard coupled-channels calculations with two different potentials. The solid line is obtained with the adiabatic model for deep subbarrier fusion hindrance, which introduces a quenching of coupling strengths after the touching. The arrow indicates the threshold energy for the deep subbarrier fusion hindrance. (Taken from Ichikawa 2015)

42 Sub-barrier Fusion Reactions

1631

touch each other (Ichikawa 2015) (see Mi¸sicu and Esbensen (2006) for another modeling of deep subbarrier fusion hindrance, which introduces a repulsive core to an internucleus potential). This calculation well accounts for the data, clearly indicating an importance of the dynamics of a transition from a two-body system with two separate nuclei to a one-body mono-nuclear system after the touching. See Jiang et al. (2021) for a recent review article on this topic.

Fusion of Neutron-Rich Nuclei One of the main research fields in modern nuclear physics is physics of unstable nuclei, especially neutron-rich nuclei far from the stability line. Those nuclei are weakly bound and are characterized by a spatially extended density distribution. It is likely that excited states of those nuclei are in the continuum spectrum and thus the breakup process plays an important role when such nuclei are used either as a projectile or as a target in nuclear reactions. In fusion of weakly bound nuclei, several effects may interplay with each other. Those are: 1. A lowering of the Coulomb barrier due to the extended density distribution (Takigawa and Sagawa 1991). 2. The breakup process, which may hinder fusion cross sections since the lowering of the Coulomb barrier disappears. At the same time, it may also enhance fusion cross sections if couplings to a breakup channel dynamically lower the Coulomb barrier (Dasso and Vitturi 1994; Hagino et al. 2000; Diaz-Torres and Thompson 2002). 3. The transfer processes. Since the Q-value is positive for neutron-rich nuclei, it may significantly affect the fusion process (Choi et al. 2018; Moschini et al. 2021). Furthermore, the breakup process significantly complicates the reaction dynamics of complete fusion and incomplete fusion in a nontrivial way (see Fig. 15). Here, the complete fusion is the process in which all the breakup fragments are absorbed by a target nucleus, while the incomplete fusion refers to the process in which only a part of the breakup fragments is absorbed. A theoretical model which coherently incorporates all of these effects has still yet to be developed, even though the continuum discretized coupled-channels (CDCC) method has been developed for the breakup process (Yahiro et al. 2012) (notice also that there have been recent developments in theoretical descriptions of inclusive breakup processes (Lei and Moro 2019a, b; Rangel et al. 2020; Cortes et al. 2020)). See Hagino et al. (2022) and Canto et al. (2006, 2015) for review articles on fusion of weakly bound nuclei. It is worth noticing that fusion of neutron-rich nuclei is important for nuclear astrophysics (Chamel and Haensel 2008; Steiner 2012) as well as for superheavy elements (Loveland 2007).

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K. Hagino

1 2

DCF

T

DCF 1 2

CN

ACN = Ap +AT

CN

ACN = Ap +AT

CF

T 2

B-Up 1 2

1 T

T

SCF 1 T 2

1 T

2

ICF 1

CN

2

ICF

ACN < Ap +AT

2

T

1

ICF 2

Elastic Breakup

CN

T

1

1 2

Fig. 15 A schematic illustration of fusion dynamics in the presence of breakup of the projectile. CF and ICF refer to the complete fusion and the incomplete fusion, respectively. The complete fusion is further subdivided into the direct complete fusion (DCF) and the sequential complete fusion (SCF) processes. (Taken from Canto et al. 2006)

Fusion Reactions for Superheavy Nuclei Superheavy Nuclei The elements heavier than plutonium (the atomic number Z = 94) are all unstable and do not exist in nature. Yet, one can artificially synthesize them using nuclear reactions. There have been continuous efforts since the 1950s, and the elements up to Z = 118 have been synthesized by now. The transactinide elements, that is, the elements with Z ≥ 104, are referred to as superheavy elements and have attracted lots of attention in recent years (Hofmann and Münzenberg 2000; Hamilton et al. 2013; Düllmann et al. 2015; Hinde et al. 2021; Nazarewicz 2018; Giuliani et al. 2019; Hagino 2019). One of the main motivations to study superheavy elements, in addition to synthesizing new elements, is to explore the island of stability, which was theoretically predicted some 50 years ago (Myers and Swiatecki 1966; Sobiczewski et al. 1966). While heavy nuclei in the transactinide region are unstable against alpha decay and spontaneous fission, the shell effect due to magic numbers can stabilize a certain number of nuclei in that region. The predicted proton and neutron magic

42 Sub-barrier Fusion Reactions

1633

numbers are Z = 114 and N = 184, respectively (Myers and Swiatecki 1966; Sobiczewski et al. 1966). The region around these magic numbers is referred to as the island of stability, where nuclei may have a life time as long as 103 years (Koura et al. 2018). More modern Hartree-Fock calculations have been also predicted (Z, N ) = (114, 184), (120, 172), and (126,184) for candidates for the next double magic nucleus beyond 208 Pb (Bender et al. 1999). The island of stability has not yet been reached experimentally. In fact, the heaviest Fl element (Z = 114) synthesized so far is 289 175 Fl (Pakou et al. 2004), which is 9 neutrons less from the the predicted magic number, N = 184. This implies that neutron-rich beams are indispensable in order to reach the island of stability. An experimental technique has yet to be developed to deal with the low intensity of such beams.

Heavy-Ion Fusion Reactions for Superheavy Nuclei Heavy-ion fusion reactions at energies around the Coulomb barrier have been used as a standard tool to synthesize those superheavy elements (Hofmann and Münzenberg 2000; Hamilton et al. 2013). Figure 16 schematically illustrates fusion reactions to form superheavy nuclei (see also Fig. 1). In the first phase of reaction, two nuclei approach each other to reach the touching configuration after the Coulomb barrier is overcome. A compound nucleus is formed almost automatically for medium-heavy systems once the touching configuration is achieved. In contrast, in the superheavy region, there is a huge probability for the touching configuration to reseparate due to a strong Coulomb repulsion between the two nuclei. This process is referred to as quasi-fission. Furthermore, even if a compound nucleus is formed with a small probability, it decays most likely by fission, again due to the strong Coulomb interaction. For heavy systems, quasi-fission characteristics significantly overlap with fission of the compound nucleus, and a detection of fission events itself does not guarantee a formation of the compound nucleus. Therefore, a formation of

Fig. 16 A schematic illustration of heavy-ion fusion reactions to synthesize superheavy nuclei

capture

compound nucleus (CN)

re-separation (quasi-fission)

fission

evaporation residue (ER) experimentally indistinguishable

1634

10 10

3

10 10 10

U

286-x

Cn + xn

capture

0

-1

-3

-5

-6

10 10 10 10

1

238

-4

10 10

Ca +

-2

10 10

48

2

10

cross section (mb)

Fig. 17 The experimental evaporation residue cross sections for the 48 Ca+238 U reaction leading to the formation of Cn (Z = 112) element. The filled circles denote the capture cross sections (Kozulin et al. 2014) to form the touching configuration. The filled squares and triangles show the evaporation residue cross sections (Oganessian et al. 2000, 2004), for which the former and the latter correspond to the 3n (emission of 3 neutrons) and the 4n (emission of 4 neutrons) channels, respectively

K. Hagino

-7 -8 -9

evaporation residues 3n

-10

10

-11

170

180 190 200 Ec.m. (MeV)

4n 210

superheavy elements has been identified by measuring evaporation residues. These are extremely rare events, in which a compound nucleus is survived against fission. As an example, Fig. 17 shows the measured cross sections for the 48 Ca+238 U reaction forming the Cn (Z = 112) element. The filled circles show the capture cross sections (Kozulin et al. 2014) to form the touching configuration shown in Fig. 16. On the other hand, the filled squares and triangles denote the evaporation residue cross sections for emissions of three and four neutrons, respectively (Oganessian et al. 2000, 2004). One can observe that the evaporation residue cross sections are indeed much smaller than the capture cross sections, by about 11 orders of magnitude.

Theoretical Modelings Based on the time-scale of each process, the formation process of evaporation residues can be conceptually divided into a sequence of the following three processes (see Fig. 16). The first phase is a process in which two separate nuclei form the touching configuration after overcoming the Coulomb barrier. After two nuclei touch with each other, a huge number of nuclear intrinsic motions are activated, and the energy for the relative motion of the colliding nuclei is quickly dissipated to internal energies. Because of the strong Coulomb interaction, the touching configuration appears outside a fission barrier, which has to be thermally

42 Sub-barrier Fusion Reactions

1635

activated to form a a compound nucleus against a severe competition to the quasifission process. The Langevin approach has often been used to describe this process ´ (Swiatecki et al. 2003, 2005; Abe et al. 2000; Shen et al. 2002; Aritomo and Ohta 2004; Zagrebaev and Greiner 2015). The third process is a statistical decay of the compound nucleus (Lü et al. 2016a), with strong competitions between fission and particle emissions (i.e., evaporations). Here, the fission barrier height is one of the most important parameters which significantly affect evaporation residue cross sections (Lü et al. 2016b). For a given partial wave l, the probability for a formation of an evaporation residue is given as a product of the probability for each of the three processes, Pl , PCN , and Wsur , that is, PER (E, l) = Pl (E)PCN (E, l)Wsur (E ∗ , l),

(23)

where E and E ∗ are the bombarding energy in the center of mass frame and the excitation energy of the compound nucleus, respectively. Cross sections for a formation of evaporation residues are then given by σER (E) =

π  (2l + 1)Pl (E)PCN (E, l)Wsuv (E ∗ , l). k2

(24)

l

For medium-heavy systems, the probability for the second phase, PCN , is almost unity, and Eq. (24) is reduced to Eq. (1) when fission cross sections are added to it. In contrast, for superheavy nuclei, PCN is significantly smaller than unity. As has been mentioned, this quantity cannot be determined experimentally, which causes large ambiguities in theoretical calculations.

Hot Versus Cold Fusion Reactions Since a formation probability of evaporation residues is extremely small, it is important to choose appropriate combinations of the projectile and the target nuclei in order to efficiently synthesize superheavy elements. For this purpose, two different experimental strategies have been employed. In the so-called cold fusion reactions, 208 Pb and 209 Bi are used for the target nuclei so that the compound nucleus is formed with relatively low excitation energies. The competition between neutron emissions and fission can be minimized, which in turn maximizes Wsur in Eq. (24) (Hofmann and Münzenberg 2000; Hamilton et al. 2013). An advantage of this strategy is that alpha decays of the evaporation residues end up in the known region of nuclear chart, and thus superheavy elements can be identified unambiguously. On the other hand, in the so called hot fusion reactions, the neutronrich double magic nucleus 48 Ca has been used as a projectile to optimize the formation probability of the compound nucleus, PCN (Hofmann and Münzenberg 2000; Hamilton et al. 2013). This strategy has been successfully employed by the

1636

K. Hagino

σER (pb)

7

10 6 10 5 10 4 10 3 10 2 10 1 10 0 10 -1 10 -2 10 -3 10 100

Hot Fusion Reactions Cold Fusion Reactions

105

110 Z

115

120

Fig. 18 The measured evaporation residue cross sections as a function of the atomic number Z of a compound nucleus. The filled circles denote the results of the hot fusion reactions with 48 Ca projectile. The open circles show the fusion cross sections obtained with the cold fusion reactions, in which 208 Pb or 209 Bi nuclei are used as a target. The maximum of a sum of the 3n and 4n cross sections and that of the 1n cross sections are shown for each Z for the hot and the cold fusion reactions, respectively. (The experimental data are taken from Zagrebaev et al. 1999, Oganessian et al. 2013, and Morita et al. 2004, 2007, 2012)

experimental group at Dubna, led by Oganessian, to synthesize superheavy elements up to Z = 118. Figure 18 shows the measured evaporation residue cross sections for the hot fusion reactions (the filled circles) and for the cold fusion reactions (the open circles). For the cold fusion reactions, the cross sections drop rapidly as a function of Z of the compound nucleus. It would therefore be difficult to go beyond nihonium using this strategy. In contrast, for the hot fusion reactions, the cross sections remain relatively large between Z = 113 and 118. This may be due to the fact that the survival probability, Wsur , is increased because the compound nuclei formed are in the proximity of the predicted island of stability (Myers and Swiatecki 1966; Sobiczewski et al. 1966). An increase of nuclear dissipation at high temperatures may also play a role (Yanez et al. 2014).

Role of Deformation in Hot Fusion Reactions The incident energy for fusion reactions to synthesize superheavy nuclei is usually taken at energies slightly above the Coulomb barrier. This is because the compound nucleus formed has to be as cold as possible, but yet the capture probability, Pl (E), has to be large enough. In the hot fusion reactions, with the 48 Ca projectile, the corresponding target nuclei are in the actinide region, in which the nuclei are well deformed in the ground state. It has been argued that the collision with θ = π/2 (i.e., the “side collision”; see Fig. 7) plays an important role in the hot fusion, since the touching configuration is more compact than that formed with the “tip collision” with θ = 0, and thus the

42 Sub-barrier Fusion Reactions

0.1 -1

Dqel (MeV )

0.08

1637

48

Ca +

248

Cm

296-x

Lv + xn

0.06 0.04 0.02 0 -0.02

σER (pb)

10

1

4n 10

0

3n

-1

10 160 170 180 190 200 210 220 230 Ec.m. (MeV) Fig. 19 (Upper panel) The barrier distribution for the capture process for the 48 Ca+248 Cm system extracted from quasi-elastic scattering at backward angles. The solid line is obtained with the coupled-channels calculation which takes into account the deformation of the 248 Cm target. The octupole phonon excitation of 48 Ca and a one-neutron transfer process are also taken into account. The dashed line shows the contribution of the side collision with θ = π/2. (The experimental data are taken from Tanaka et al. 2018). (Lower panel) The evaporation residue cross sections for the same system. (The experimental data are taken from Oganessian et al. 2004 and Hofmann et al. 2012)

effective barrier height for the diffusion process is low (Hinde et al. 1995; Hagino 2018). Recently, barrier distributions for the capture process have been extracted for several hot fusion systems, and the notion of compactness has been confirmed experimentally for the first time (Tanaka et al. 2018, 2020). As an example, the top panel of Fig. 19 shows the experimental barrier distribution for the 48 Ca+248 Cm system (Tanaka et al. 2018) and its comparison to the coupled-channels calculations which take into account the deformation of the 248 Cm nucleus. The coupledchannels calculation provides information on the energy region corresponding to the side collision, which is indicated by the dashed line in the figure. The measured evaporation residue cross sections σER for this system (Oganessian et al. 2004; Hofmann et al. 2012) are shown in the lower panel of the figure. One can clearly see that the peak of evaporation residue cross sections, σER , appears in the energy region corresponding to the side collision, which is compatible with the notion of compactness.

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K. Hagino

Acknowledgments I thank L. Felipe Canto for his careful reading of the manuscript and useful suggestions. I also thank Tomoya Naito for his help in preparing Fig. 6. This work was supported in part by JSPS KAKENHI Grant Numbers JP19K03861 and JP21H00120.

References Y. Abe, D. Boilley, B.G. Giraud, T. Wada, Phys. Rev. E 61, 1125 (2000) Y. Aritomo, M. Ohta, Nucl. Phys. A 744, 3 (2004) M. Asghar et al., Nucl. Phys. 85, 305 (1966) B.B. Back, H. Esbensen, C.L. Jiang, K.E. Rehm, Rev. Mod. Phys. 86, 317 (2014) A.B. Balantekin, N. Takigawa, Rev. Mod. Phys. 70, 77 (1998) R. Bass, Nuclear Reactions with Heavy-Ions (Springer, Berlin, 1980) M. Beckerman, Phys. Rep. 129, 145 (1985) M. Beckerman, Rep. Prog. Phys. 51, 1047 (1988) M. Beckerman, J. Ball, H. Enge, M. Salomaa, A. Sperduto, S. Gazes, A. DiRienzo, J.D. Molitoris, Phys. Rev. C 23, 1581 (1981) M. Bender, K. Rutz, P.-G. Reinhard, J.A. Maruhn, W. Greiner, Phys. Rev. C 60, 034304 (1999) M. Bender et al., J. Phys. G 47, 113002 (2002) C.A. Bertulani, P. Danielewicz, Introduction to Nuclear Reactions (IOP Publishing, Bristol, 2004) N. Bohr, Nature 137, 351 (1936) R.A. Broglia, A. Winther, Heavy-ion Reactions (Westview Press, Cambridge, MA, 2004) L.F. Canto, M.S. Hussein, Scattering Theory of Molecules, Atoms and Nuclei (World Scientific Publishing Co. Pte. Ltd., Singapore, 2013) L.F. Canto, P.R.S. Gomes, R. Donangelo, M.S. Hussein, Phys. Rep. 424, 1 (2006) L.F. Canto, P.R.S. Gomes, R. Donangelo, J. Lubian, M.S. Hussein, Phys. Rep. 596, 1 (2015) N. Chamel, P. Haensel, Living Rev. Rel. 11, 10 (2008) K.S. Choi, M.K. Cheoun, W. So, K. Hagino, K. Kim, Phys. Lett. B 780, 455 (2018) M.R. Cortes, J. Rangel, J.L. Ferreira, J. Lubian, L.F. Canto, Phys. Rev. C 102, 064628 (2020) M. Dasgupta, D.J. Hinde, N. Rowley, A.M. Stefanini, Annu. Rev. Nucl. Part. Sci. 48, 401 (1998) C.H. Dasso, A. Vitturi, Phys. Rev. C 50, R12 (1994) C.H. Dasso, S. Landowne, A. Winther, Nucl. Phys. A 405, 381 (1983a) C.H. Dasso, S. Landowne, A. Winther, Nucl. Phys. A 407, 221 (1983b) A. Diaz-Torres, I.J. Thompson, Phys. Rev. C 65, 024606 (2002) C.E. Düllmann, R.-D. Herzberg, W. Nazarewicz, Y. Oganessian, Nucl. Phys. A 944, 1 (2015). Special Issue on Superheavy Elements P. Fröbrich, R. Lipperheide, Theory of Nuclear Reactions (Clarendn Press, Oxford, 1996) S.A. Giuliani, Z. Matheson, W. Nazarewicz, E. Olsen, P.-G. Reinhard, J. Sadhukhan, B. Schuetrumpf, N. Schunck, P. Schwerdtfeger, Rev. Mod. Phys. 91, 011001 (2019) Y.K. Gupta, B.K. Nayak, U. Garg, K. Hagino, K.B. Howard, N. Sensharma, M. Senyi˘ ¸ git, W.P. Tan, P.D. O’Malley, M. Smith, R. Gandhi, T. Anderson, R.J. deBoer, B. Frentz, A. Gyurjinyan, O. Hall, M.R. Hall, J. Hu, E. Lamere, Q. Liu, A. Long, W. Lu, S. Lyons, K. Ostdiek, C. Seymour, M. Skulski, B. Vande Kolk, Phys. Lett. B 806, 135473 (2020) K. Hagino, Phys. Rev. C 98, 014607 (2018) K. Hagino, AAPPS Bull. 29, 31 (2019) K. Hagino, N. Rowley, Phys. Rev. C 69, 054610 (2004) K. Hagino, N. Takigawa, Prog. Theor. Phys. 128, 1061 (2012) K. Hagino, N. Takigawa, A.B. Balantekin, Phys. Rev. C 56, 2104 (1997) K. Hagino, N. Rowley, A.T. Kruppa, Comput. Phys. Commun. 123, 143 (1999) K. Hagino, A. Vitturi, C.H. Dasso, S.M. Lenzi, Phys. Rev. C 61, 037602 (2000) K. Hagino, K. Ogata, A.M. Moro, Prog. Part. Nucl. Phys. 125, 103951 (2022) J.H. Hamilton, S. Hofmann, Y.T. Oganessian, Ann. Rev. Nucl. Part. Sci. 63, 383 (2013)

42 Sub-barrier Fusion Reactions

1639

D.J. Hinde, M. Dasgupta, J.R. Leigh, J.P. Lestone, J.C. Mein, C.R. Morton, J.O. Newton, H. Timmers, Phys. Rev. Lett. 74, 1295 (1995) D.J. Hinde, M. Dasgupta, E.C. Simpson, Prog. Part. Nucl. Phys. 118, 103856 (2021) S. Hofmann, G. Münzenberg, Rev. Mod. Phys. 72, 733 (2000) S. Hofmann et al., Eur. Phys. J. A 48, 62 (2012) T. Ichikawa, Phys. Rev. C 92, 064604 (2015) T. Ichikawa, K. Hagino, A. Iwamoto, Phys. Rev. C 75, 064612 (2007) C.L. Jiang, H. Esbensen, K.E. Rehm, B.B. Back, R.V.F. Janssens, J.A. Caggiano, P. Collon, J. Greene, A.M. Heinz, D.J. Henderson, I. Nishinaka, T.O. Pennington, D. Seweryniak, Phys. Rev. Lett. 89, 052701 (2002) C.L. Jiang, B.B. Back, K.E. Rehm, K. Hagino, G. Montagnoli, A.M. Stefanini, Eur. Phys. J. A 57, 235 (2021) H. Koura, J. Katakura, T. Tachibana, F. Minato, Chart of the Nuclides. Japan Atomic Energy Agency (2018) E.M. Kozulin, G.N. Knyazheva, I.M. Itkis, M.G. Itkis, A.A. Bogachev, E.V. Chernysheva, L. Krupa, F. Hanappe, O. Dorvaux, L. Stuttgé, W.H. Trzaska, C. Schmitt, G. Chubarian, Phys. Rev. C 90, 054608 (2014) J. Lei, A.M. Moro, Phys. Rev. Lett. 122, 042503 (2019a) J. Lei, A.M. Moro, Phys. Rev. Lett. 123, 232501 (2019b) J.R. Leigh, N. Rowley, R.C. Lemmon, D.J. Hinde, J.O. Newton, J.X. Wei, J.C. Mein, C.R. Morton, S. Kuyucak, A.T. Kruppa, Phys. Rev. C 47, R437 (1993) J.R. Leigh, M. Dasgupta, D.J. Hinde, J.C. Mein, C.R. Morton, R.C. Lemmon, J.P. Lestone, J.O. Newton, H. Timmers, J.X. Wei, N. Rowley, Phys. Rev. C 52, 3151 (1995) A. Lépine-Szily, R. Lichtenthäler, Euro. Phys. J. A 57, 99 (2021) W. Loveland, Phys. Rev. C 76, 014612 (2007) H. Lü, A. Marchix, Y. Abe, D. Boilley, Comput. Phys. Commun. 200, 381 (2016a) H. Lü, D. Boilley, Y. Abe, C. Shen, Phys. Rev. C 94, 034616 (2016b) S. ¸ Mi¸sicu, H. Esbensen, Phys. Rev. Lett. 96, 112701 (2006) G. Montagnoli, A.M. Stefanini, Eur. Phys. J. A 53, 169 (2017) K. Morita et al., J. Phys. Soc. Jpn. 73, 2593 (2004) K. Morita et al., J. Phys. Soc. Jpn. 76, 5001 (2007) K. Morita et al., J. Phys. Soc. Jpn. 81, 3201 (2012) L. Moschini, A.M. Moro, A. Vitturi, Phys. Rev. C 103, 014604 (2021) W.D. Myers, W.J. Swiatecki, Nucl. Phys. 81, 1 (1966) T. Naito, S. Endo, K. Hagino, Y. Tanimura, J. Phys. B 54, 165201 (2021) W. Nazarewicz, Nat. Phys. 14, 537 (2018) Y.T. Oganessian, V.K. Utyonkov, Y.V. Lobanov, F.S. Abdullin, A.N. Polyakov, I.V. Shirokovsky, Y.S. Tsyganov, G.G. Gulbekian, S.L. Bogomolov, B.N. Gikal, A.N. Mezentsev, S. Iliev, V.G. Subbotin, A.M. Sukhov, O.V. Ivanov, G.V. Buklanov, K. Subotic, M.G. Itkis, K.J. Moody, J.F. Wild, N.J. Stoyer, M.A. Stoyer, R.W. Lougheed, C.A. Laue, Y.A. Karelin, A.N. Tatarinov, Phys. Rev. C 63, 011301 (2000) Y.T. Oganessian, V.K. Utyonkov, Y.V. Lobanov, F.S. Abdullin, A.N. Polyakov, I.V. Shirokovsky, Y.S. Tsyganov, G.G. Gulbekian, S.L. Bogomolov, B.N. Gikal, A.N. Mezentsev, S. Iliev, V.G. Subbotin, A.M. Sukhov, A.A. Voinov, G.V. Buklanov, K. Subotic, V.I. Zagrebaev, M.G. Itkis, J.B. Patin, K.J. Moody, J.F. Wild, M.A. Stoyer, N.J. Stoyer, D.A. Shaughnessy, J.M. Kenneally, P.A. Wilk, R.W. Lougheed, R.I. Il’kaev, S.P. Vesnovskii, Phys. Rev. C 70, 064609 (2004) Y.T. Oganessian, V.K. Utyonkov, F.S. Abdullin, S.N. Dmitriev, R. Graeger, R.A. Henderson, M.G. Itkis, Y.V. Lobanov, A.N. Mezentsev, K.J. Moody, S.L. Nelson, A.N. Polyakov, M.A. Ryabinin, R.N. Sagaidak, D.A. Shaughnessy, I.V. Shirokovsky, M.A. Stoyer, N.J. Stoyer, V.G. Subbotin, K. Subotic, A.M. Sukhov, Y.S. Tsyganov, A. Türler, A.A. Voinov, G.K. Vostokin, P.A. Wilk, A. Yakushev, Phys. Rev. C 87, 034605 (2013) A. Pakou, N. Alamanos, G. Doukelis, A. Gillibert, G. Kalyva, M. Kokkoris, S. Kossionides, A. Lagoyannis, A. Musumarra, C. Papachristodoulou, N. Patronis, G. Perdikakis, D. Pierroutsakou, E.C. Pollacco, K. Rusek, Phys. Rev. C 69, 054602 (2004)

1640

K. Hagino

J. Rangel, M.R. Cortes, J. Lubian, L.F. Canto, Phys. Lett. B 803, 135337 (2020) N. Rowley, K. Hagino, Phys. Rev. C 91, 044617 (2015) N. Rowley, G.R. Satchler, P.H. Stelson, Phys. Lett. B 254, 25 (1991) G.R. Satchler, Direct Nuclear Reactions (Clarendn Press, Oxford, 1983) C.W. Shen, G. Kosenko, Y. Abe, Phys. Rev. C 66, 061602 (2002) A. Sobiczewski, F.A. Gareev, B.N. Kalinkin, Phys. Lett. 22, 500 (1966) A.W. Steiner, Phys. Rev. C 85, 055804 (2012) ´ W.J. Swiatecki, K. Siwek-Wilczy´nska, J. Wilczy´nski, Acta Phys. Pol. B 34, 2049 (2003) ´ W.J. Swiatecki, K. Siwek-Wilczy´nska, J. Wilczy´nski, Phys. Rev. C 71, 014602 (2005) Z.E. Switkowski, R.G. Stokstad, R.M. Wieland, Nucl. Phys. A 279, 502 (1977) N. Takigawa, H. Sagawa, Phys. Lett. B 265, 23 (1991) T. Tamura, Rev. Mod. Phys. 37, 679 (1965) T. Tanaka, Y. Narikiyo, K. Morita, K. Fujita, D. Kaji, K. Morimoto, S. Yamaki, Y. Wakabayashi, K. Tanaka, M. Takeyama, A. Yoneda, H. Haba, Y. Komori, S. Yanou, B.J.-P. Gall, Z. Asfari, H. Faure, H. Hasebe, M. Huang, J. Kanaya, M. Murakami, A. Yoshida, T. Yamaguchi, F. Tokanai, T. Yoshida, S. Yamamoto, Y. Yamano, K. Watanabe, S. Ishizawa, M. Asai, R. Aono, S. Goto, K. Katori, K. Hagino, J. Phys. Soc. Jpn. 87, 014201 (2018) T. Tanaka, K. Morita, K. Morimoto, D. Kaji, H. Haba, R.A. Boll, N.T. Brewer, S. Van Cleve, D.J. Dean, S. Ishizawa, Y. Ito, Y. Komori, K. Nishio, T. Niwase, B.C. Rasco, J.B. Roberto, K.P. Rykaczewski, H. Sakai, D.W. Stracener, K. Hagino, Phys. Rev. Lett. 124, 052502 (2020) I.J. Thompson, Comput. Phys. Rep. 7, 167 (1988) I.J. Thompson, F.M. Nunes, Nuclear Reactions for Astrophysics (Cambridge University Press, Cambridge, 2009) H. Timmers, J.R. Leigh, M. Dasgupta, D.J. Hinde, R.C. Lemmon, J.C. Mein, C.R. Morton, J.O. Newton, N. Rowley, Nucl. Phys. A 584, 190 (1995) C.Y. Wong, Phys. Rev. Lett. 31, 766 (1973) M. Yahiro, K. Ogata, T. Matsumoto, K. Minomo, Prog. Theor. Exp. Phys. 2012, 01A206 (2012) R. Yanez, W. Loveland, L. Yao, J.S. Barrett, S. Zhu, B.B. Back, T.L. Khoo, M. Alcorta, M. Albers, Phys. Rev. Lett. 112, 152702 (2014) V.I. Zagrebaev, A.S. Denikin, A.V. Karpov, A.P. Alekseev, M.A. Naumenko, V.A. Rachkov, V.V. Samarin, V.V. Saiko, NRV web knowledge base on lowenergy nuclear physics (1999). http:// nrv.jinr.ru/ V.I. Zagrebaev, W. Greiner, Nucl. Phys. A 944, 257 (2015)

Section VII Nucleon-Nucleon Interactions Ruprecht Machleidt

NN Experiments and NN Phase-Shift Analysis

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Enrique Ruiz Arriola and Rodrigo Navarro Pérez

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Setup in the Spinless Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic Unitarity and Complete Set of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotational Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting Scattering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Confidence Limits and Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NN Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering with Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wolfenstein Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Partial-Wave Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixed States and Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Work supported by project PID2020-114767GB-I00 funded by MCIN/AEI/10.13039/ 501100011033 as well as Junta de Andalucía (grant FQM-225) E. R. Arriola () Departamento de Física Atómica, Molecular y Nuclear, Instituto Carlos I de Física Teórica y Computacional, Facultad de Ciencias, Avda. Fuentenueva s/n Universidad de Granada, Granada, Spain e-mail: [email protected] R. N. Pérez Department of Physics, San Diego State University, San Diego, CA, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_47

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NN Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NN Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possible NN Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real NN Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NN Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phenomenological Analysis of NN Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The NN Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1678 1681 1681 1682 1684 1686 1686 1695 1703 1703

Abstract

A comprehensive partial-wave analysis of proton-proton and neutron-proton elastic scattering phenomenology from threshold to pion production threshold is presented as a guideline starting from elementary scattering theory and nuclear physics as well as statistical concepts to describe up-to-date calculations. An attempt is made to highlight and justify the essential theoretical and phenomenological features which enables a statistically satisfactory determination of NN scattering amplitudes using the large body of pp and np data including differential and polarization observables collected over 70 years. This includes proper consideration of analyticity in terms of a NN potential whose longrange contribution incorporates charge-dependent one pion exchange, Coulomb effects, vacuum polarization, magnetic moment interactions and relativistic corrections on the theoretical side, and a statistically motivated selection of mutually consistent data on the experimental side. The approach is based on a partial-wave analysis and a coarse graining of the short-range component of the nuclear interaction and enables not only a sound determination of statistical uncertainties and their subsequent propagation to ab initio nuclear structure and nuclear reactions calculations but also a precise and accurate evidence of the one-pion exchange mechanism above a relative distance of 3 fm.

Introduction The main goal of theoretical nuclear physics is to understand and describe in a quantitative manner the properties of atomic nuclei and their interactions throughout the periodic table in terms of their basic hadronic constituents, the proton and the neutron, and their interactions. There are multiple direct and indirect experimental sources providing hints on the main features of the basic NN interaction, but one should note from the start that the study of the nuclear force cannot be completely separated from nuclear physics as a whole. However, the role played by the underlying composition of nucleons in terms of quarks and gluons while being more fundamental is less effective. The present contribution will substantiate this statement.

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Before embarking on the modern determination based on phase shifts and a large body of experimental np+pp scattering data, it is worth establishing these qualitative characteristics which already implement simplifying assumptions for a more quantitative analysis and which are ultimately validated using probabilistic and statistical means. The most characteristic feature of the nuclear force is the rather small range of the interaction, about 1 fm, which together with quantum mechanics fixes the relevant scales for typical wavelike and interference phenomena. The very existence of finite nuclei is an indication of the strong nuclear attraction which overcomes the long- range Coulomb repulsion. Actually, the finite size of atomic nuclei, the N = Z stability for just about Z ≤ 100, and the saturation property of an average binding energy per particle suggest, B/A ∼ 8 MeV that the interaction holding nuclei together is short range. Further quantum mechanical evidence on the deuteron properties suggest a more quantitative range of about 1.4–2 fm. The fact that atomic nuclei are finite size systems with a rough surface also indicates that the NN interaction is short range. Electron scattering experiments on atomic nuclei reveal an exponential falloff with a diffuseness of 0.7 fm. A second important characteristic of the nuclear force is its non- central anisotropic character due to the spin of the particles and which major evidence is provided by the nonvanishing quadrupole moment of the deuteron, the bound state made of a proton and a neutron. This is the analogous magnetic dipole interaction for the nuclear force. This so-called tensor force opens the possibility to a rich variety of experiments where beams and targets can be initially polarized and further transformed through the nuclear interaction. A third important characteristic is the exchange character of nuclear forces, a phenomenon that contradicts classical intuition. In a neutron-proton collision where the neutron is the beam and the proton is the target, the recoiling target particle becomes the neutron or the proton with similar probability very much as if they were identical particles. All these features are seen in NN scattering experiments. The constructive and historic way of fixing the interaction out of experimental data with a minimum amount of theoretical bias has been reported in the older textbooks and subsequent reviews which still keep their insightful value. Particularly recommendable references are Blatt and Weisskopf (1952), Breit and Gluckstern (1953), Breit (1962), Breit and Haracz (1967), Phillips (1959), Mac Gregor et al. (1960), Signell (1969), de Swart and Nagels (1978), and Machleidt (1989) for the period between 1950 and 1990. This period can be characterized by the single statement: no theory describes all data. This statement requires qualification and it will be clarified along the presentation. The great achievement of the Nijmegen group 30 years ago was to provide for the first time a statistically satisfactory description of a large amount of np+pp scattering data (Stoks et al. 1993, 1994). This was possible because of two good reasons. First, charge dependence (CD) of the pion exchange interaction and tiny electromagnetic effects such as vacuum polarization, magnetic moments interac-

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tions (which requires summing up over thousand partial waves), and relativistic corrections to the Coulomb scattering were incorporated. Second, a suitable selection of all the data was implemented. This success was followed by other authors pursuing alternative and equally satisfactory fits along with a steady upgrade of the NN database (Wiringa et al. 1995; Machleidt 2001; Gross and Stadler 2008). The Granada-2013 database (Navarro Pérez et al. 2013c, d, 2014c, d, e) relies on a similar approach, but with three significant improvements: the number of data is almost twice, the selection process has been made self-consistent as suggested by Gross and Stadler (2008), and the interaction has been coarse-grained to reduce the number of fitting parameters with negligible statistical correlations. For the most recent state of affairs, see, e.g., Navarro Pérez and Ruiz Arriola (2020) and Ruiz Arriola et al. (2020) and references therein. The present contribution makes an attempt to justify why still today the partialwave analysis in conjunction with the large body of scattering data, proper statistical treatment including data selection, and a minimum of theoretical apparatus provides a satisfactory and bench-marking description of NN scattering amplitudes. In so doing, some important aspects such as a comprehensive discussion of deuteron properties (Gilman and Gross 2002), the role of dispersion relations, and the application of the purely field theoretical chiral approach to nuclear interactions will be left out for later chapters in this handbook. This contribution is addressed to graduate and postgraduate students with working knowledge on quantum mechanics including elementary scattering theory (the Born approximation and phase shifts), some familiarity with nuclear physics including isospin formalism (Bethe and Morrison 1956; Segre et al. 2013; Preston and Bhaduri 1975), and the standard error analysis which is applied in the basic physics lab (Taylor 1997). Many of the aspects covered here are basic, and an attempt is made to stress some of the issues which are usually not contained in textbooks. Lack of space prevents a very detailed derivation of some of the relevant equations (Coulomb scattering or numerical determination of phase shifts) for which there exists abundant literature. Some of the relevant properties of protons, neutrons, and pions are listed in Table 1. At all times natural units will be used where h¯ = c = 1, so that mass Table 1 Values of fundamental constants used

Constant h¯ c mπ 0 mπ ± Mp Mn me α −1 f2 μp μn

Value 197.327053 134.9739 139.5675 938.27231 939.56563 0.510999 137.035989 0.075 2.7928474 −1.9130427

Units MeV fm MeV/c2 MeV/c2 MeV/c2 MeV/c2 MeV/c2

μ0 μ0

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and energy are inverse to each other with the conversion factor 197.327053 fm MeV = 1.

General Setup in the Spinless Case It is instructive to review firstly the spinless case ignoring the spin complications in the interaction, which happens to be a good approximation in np and pp scattering below 50 MeV LAB energy. Most of the material presented here is well known and belongs to quantum scattering theory textbooks for which good references are Landau (1990), Newton (1966), and Goldberger and Watson (1964). Actually, much of the progress on scattering theory itself has been motivated by the study of the nuclear force which takes the form of an inverse problem; to what extent NN scattering experiments set constraints on the NN interaction. The simplified presentation is meant as a reminder which should be read in conjunction with Landau (1990), Newton (1966), and Goldberger and Watson (1964) and will be illustrated with actual NN results whose ultimate justification is postponed for later sections.

Statement of the Problem Scattering experiments are designed after the original Rutherford experiment which lead to the discovery of the atomic nucleus in 1908 (for a historical overview, see, e.g., Fernandez and Ripka 2013). Nin particles emitted from a source of surface S are collimated forming a beam which is scattered at a given solid angle  ≡ (θ, φ), and Nout particles are counted on a detector at a far distance R. The differential cross section is defined as the ratio: σexp (θ, φ) =

Nout (θ, φ)/Δ , Nin /S

(1)

over a given time interval and detector of surface SD corresponding to an angular resolution Δ = sin θ Δθ Δφ = SD /R 2 . In general, absolute determinations of differential cross sections are difficult, and there exists a normalization constant, which can be determined by comparing with  a theoretically known cross section or by checking the total cross section σT ≡ dσ () with a forward transmission experiment where the mean free path is determined l = 1/nσT and the density of scatterers per unit volume, n, is known. In nuclear physics total cross sections are measured in barns, 1barn = 10−24 cm2 = 100 fm2 . Compared to the classical geometrical value of a disk of radius a, one has σ = π a 2 , whereas two billiard balls of radii a and b have σ = π(a + b)2 , which means that, for instance, σ = 10 m barn corresponds to a + b ∼ 0.56 fm. Experiments are often carried out in the LAB system where the targets at rest are bombarded by the beam containing projectiles. For the case of NN scattering

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E. R. Arriola and R. N. Pérez

masses of target particles and beam particles are very similar, so the CM system is particularly adequate to describe the collision theoretically. Protons and neutrons are nearly degenerate in mass, so one could use the rest nucleon mass as the average MN = (Mp + Mn )/2 or the inertial nucleon mass as 2/MN = 1/Mp + 1/Mn since the CM energy for small CM momentum is given by

ECM =



2 pCM

+ Mn2 +



2 pCM

+ Mp2

p2 = Mp +Mn + CM 2



 1 1 + . . . (2) + Mp Mn

Actually, at sufficiently high energy, one may produce additional particles, the pion this happens for CM energy  being the lightest. Using relativistic kinematics √ 2 2 2 pCM + MN = 2MN + mπ , or pCM ∼ MN mπ since mπ  MN . Below this pion production threshold scattering is elastic, NN → NN and σel = σN N , and above it one has NN → NN + NNπ , with an inelastic component and so that σinel = σN N π and the total cross section is σT = σel + σinel . Elastic and total cross sections can experimentally be discriminated by comparing the corresponding mean free paths lel = 1/nσel and lT = 1/nσT in transmission experiments checking or not for energy conservation in the final nucleons, respectively. Figure 1 illustrates the case of proton-proton (pp) scattering above pCM = 150 MeV, indicating that max = 400 MeV. scattering is purely elastic up to pCM max 2 2 Since (pCM ) /MN ∼ 0.16, one can consider the role of relativity to be marginal for purely elastic scattering.

Fig. 1 The pp total (red) and total elastic (blue) cross section as a function of the CM momentum. Single π -production and the single and double Δ-production thresholds are marked. (Data from PDG (Particle Data Group et al. 2022))

43 NN Experiments and NN Phase-Shift Analysis

1649

Figure 1 also displays the quantum phenomenon of shadow scattering σel ∼ σinel which is observed above the  double Δ(1232)production process NN → ΔΔ → 2 + M 2 = 2 p 2 + M 2 or p N Nππ corresponding to 2 pCM CM ∼ 800 MeV. Δ N CM

Counting Fluctuations All scattering experiments are based on detecting particles in a given final state and ultimately on counting particles hitting on a detector. In general, the number of counts Nout is a random variable subjected to fluctuations. If for a fixed solid angle (θ, φ) the scattering probability is denoted by p, the statistics of counting k events out of N is given by the binomial distribution:  PN,k =

N k



p k (1 − p)N −k ⇒

N 

PN,k = 1 ,

(3)

k=0

which is normalized to unity. The mean k¯ and variance (Δk)2 are given by k¯ = Np ,

2

(Δk)2 = (k − k)2 = k 2 − k = Np(1 − p) .

(4)

In practice, p  1  k  N one has the sequence of binomial → Poisson → Gauss distributions: PN,k  → e

−Np (Np)k

k!

p1

→ e 

−(k−Np)2 /2

√

2πΔk

,

(5)

k 1

where in Fig. 2 the situation for the case p = 0.1 and N = 50 is illustrated. Thus, one may consider that Nout (and hence σ (θ, E)) is Gauss distributed. If ξ is a normally distributed variable, i.e., ξ ∈ N(0, 1) the probability density is √ 2 P (ξ ) = e−ξ /2 / 2π with ξ¯ = 0 and Δξ = 1, and hence one may write Nout = N¯ out + ξ ΔNout . Thus, for a 68% confidence level, one writes as usual Nout = N¯ out ± ΔNout ,

 ΔNout =

N¯ out .

(6)

The same applies to Nin = N¯ in ± ΔNin which can be determined by removing the target, but here √ fluctuations are almost negligible since Nin Nout and so ΔNin /Nin = 1/ Nin  1. All this is summarized by stating a statistical error estimate with a 68% confidence level for the differential cross section, σ (θ, E) = σ¯ (θ, E) ± Δσ (θ, E).

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E. R. Arriola and R. N. Pérez

Fig. 2 Binomial (histogram), Poisson (points), and Gauss (line) distributions for p = 0.1 and N = 50

Classical Scattering The consideration of classical scattering, although well known (Newton 1966), is pertinent as it provides some insight into the analysis of NN scattering experiments. For simplicity nonrelativistic elastic scattering for two spinless particles with masses M1 and M2 and interacting by a central potential V (r) (characterized by the spherical symmetry) is assumed. As usual, the CM system, characterized by an effective reduced mass μ = M1 M2 /(M1 + M2 ), is taken. The relation between the kinetic LAB energy and the CM momentum is given by ELAB = 2ECM = 2 /μ ∼ 2p 2 /M . pCM N CM At the classical level, one solves Newton’s equation for the relative coordinate: μx (t) = −∇V (x) = −ˆxV (r) ,

(7)

subjected to the asymptotic scattering conditions corresponding to free particles x(t)  → b + vt , t→−∞

x(t)  → b + v t .

(8)

t→+∞

where b and b are the impact parameters which corresponds to the minimal distance to the target when coming from the remote past or the remote future t → ∓∞, respectively. They are perpendicular to the incoming and outgoing velocities respectively so that b · v = b · v = 0, vˆ · vˆ = bˆ · bˆ = cos θ with θ the scattering angle. Because the force is time-independent and central, energy and angular momentum are conserved:

43 NN Experiments and NN Phase-Shift Analysis

E=

dE p2 + V (x) ⇒ = 0, 2μ dt

L = x ∧ p ⇒

dL = 0, dt

1651

(9) (10)

where p(t) = μv(t). For t → ∓∞ energy conservation and V (∞) = 0 imply E = μv2 /2 = μv 2 /2 so that v = Rv with R a rotation matrix. Likewise, for t → ∓∞ angular momentum conservation implies |L(t)| = |x(t) ∧ p(t)| = bp = bμv = b μv so that b = b . Finally the time reversal invariance requires x(−t) = Rx(t) so that b = Rb. For a beam of N particles in a given time, the classical differential cross section is defined as the ratio between the outgoing N/Δ distribution and the incoming flux (in the z-direction) N/Δbx Δby : σcl (θ, φ) =

b

db

d 2b =

. d sin θ dθ

(11)

An important consequence of all this is that for a finite range interaction, i.e., V (r) = 0 for r > a, so that σcl (θ, φ) = 0 for b > a. This sets a limit on the maximal angular momentum Lmax = pa compatible with the occurrence of scattering. In particular, if one has a constant repulsive finite range potential V (r) = V0 > E scattering only for b < a, b = a cos(θ/2), so that dσ/d = a 2 /4 and  occurrs 2 hence σT = d b = π a 2 which correspond to a geometrical cross section. For pp Coulomb scattering V (r) = e2 /r ≡ α/r, the equation for the trajectory, b = α cot(θ/2)/4E (E is the proton energy in CM system), implies the timehonored Rutherford’s formula: α2 dσC = . d 16E 2 sin4 (θ/2)

(12)

Due to the  long range ofthe interaction, the total classical cross section diverges, σT ≡ dσcl () = d 2 b = ∞ due to the large divergent contribution of small angle scattering. In the nuclear case, this divergence is purely mathematical for electromagnetic interactions since physically one has charge neutrality and screening due to electrons. For the case of proton-proton scattering, the beam is an accelerated proton, and the target is usually molecular hydrogen, H2 , which is electrically neutral, in which case scattering is zero for impact parameters larger than the molecule size ∼1 Å = 105 fm which is a large but not infinite range. The meson exchange picture yields Yukawa-like forces among hadrons V (r) ∼ e−(r/a) /r with the longest range corresponding to a ∼ h¯ /mπ c ∼ 1.4 fm (Yukawa 1935). It is worth reminding that the same divergent result, σT = d 2 b = ∞, also holds for Yukawa interactions. In turn, this means that the mean free path vanishes, l = 1/nσT = 0, in flagrant contradiction with transmission experiments. The best known way to cure this problem is by posing the scattering problem quantum mechanically.

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Quantum Scattering The proper and time-dependent description of quantum scattering in terms of wave packets can be found in standard textbooks (Landau 1990; Newton 1966; Goldberger and Watson 1964) which for pure energy states allow the more elementary and conventional stationary treatment. Quantum mechanically one solves the stationary Schrödinger equation in the CM system: −

h¯ 2 2 ∇ ψ(x) + V (r)ψ(x) = Eψ(x) , 2μ

(13)

for positive energies E = h¯ 2 k 2 /2μ > 0 with the asymptotic boundary condition ψk (x) → eik·x + f (θ, φ)

eikr , r

(14)

made of a superposition of an incoming plane wave and an outgoing spherical wave, whose modulating complex factor, f (θ, φ) is the scattering amplitude. It can also ˆ and a final state |kˆ  pointing be written as a transition between an initial state |k ˆ = kˆ |f |k, ˆ and kˆ |k ˆ = along directions kˆ and kˆ , respectively, f (θ, φ) = f (kˆ , k) (2) δ (kˆ − kˆ ). One consequence of the stationary equation is the conservation of the transition current involving states with the same energy but different directions kˆ and kˆ : J=

 1  ∗ ψk ∇ψk − ∇ψk∗ ψk ⇒ ∇ · J = 0 . 2μi

(15)

In the spinless case, the only observable is the differential cross section, which is obtained as the ratio of the outcoming flux over the incoming flux:   dσ (θ, φ) 2 rˆ · Jout ˆ kˆ )|2 , = |f (θ, φ)|2 = |f (k, = lim r r→∞ d kˆ · Jin

(16)

which for a fixed energy E warrants access to the modulus of the scattering amplitude |f (θ, φ)| but not its phase. The finiteness of the total cross section ( d 2 kˆ is solid angle  = (θ, φ) along kˆ ):  σT =

ˆ kˆ )|2 , d 2 kˆ |f (k,

(17)

requires the scattering amplitude to be square integrable. Besides, time reversal ˆ = f (−k, ˆ −kˆ ), whereas parity f (kˆ , k) ˆ = f (−kˆ , −k) ˆ so that implies f (kˆ , k) ˆ = f (k, ˆ kˆ ). The finiteness of the total cross section together with the PTf (kˆ , k) ˆ be square integrable both in kˆ and kˆ in the unit symmetry requires that f (kˆ , k) ˆ (|m| ≤ l = 0, 12, . . . ), as a sphere, thus using the spherical harmonics, Ylm (k) complete and orthonormal system:

43 NN Experiments and NN Phase-Shift Analysis





∗ ˆ Ylm (kˆ )Ylm (k) = δ (2) (kˆ − kˆ ) ,

1653

ˆ lm (k) ˆ = δll δmm , d 2 kYl∗ m (k)Y

(18)

lm

one has the double expansion: ˆ = f (kˆ , k)



ˆ ∗, fl m ;lm Yl m (kˆ )Ylm (k)

(19)

l m ;lm

with fl m ;lm = flm;l m .

Elastic Unitarity and Complete Set of Experiments From ∇·J = 0 global flux  conservation implies, upon use of the divergence theorem, limr→∞ r 2 dJr = d 3 x∇·J = 0. For elastic scattering, and taking the transition ∗ (x) and ψ (x) implies currents involving outgoing directions kˆ and −kˆ with ψ−k −k the off-forward optical theorem (Glauber and Schomaker 1953): ˆ − f (k, ˆ kˆ )∗ = 2ik f (kˆ , k)



ˆ , d 2 kˆ f (kˆ , kˆ )∗ f (kˆ , k)

(20)

whence the usual (forward) optical theorem follows ˆ k) ˆ = k Imf (k, 4π



ˆ kˆ )|2 = k σT d 2 kˆ |f (k, 4π

(21)

Taken as an integral operator on the unit sphere, one can rewrite the off-forward optical theorem in operator form f − f † = 2ikf † f . One defines the S-matrix: ˆ ≡ δ (2) (kˆ − kˆ ) + 2kif (kˆ , k) ˆ , S(k , k)

(22)

which operationally reads S = 1 + 2ikf . The off-forward optical theorem implies ˆ ≡ kˆ |S|k. ˆ Note that equivalently, the unitarity of the S-matrix, SS † = 1 if S(k , k) Sl m ;lm = δl l δm m + 2ikflm;l m . As already mentioned, experimentally one has direct access to the modulus of the scattering amplitude only leaving the phase undetermined in principle. However, ˆ ˆ = |f (kˆ , k)|e ˆ iα(kˆ ,k) writing f (kˆ , k) the off-forward optical theorem relation becomes:  ˆ ˆ ˆ ˆ ˆ sin α(k , k)|f (k , k)| = k d 2 kˆ |f (kˆ , kˆ )||f (kˆ , k)| (23) ˆ , cos[α(kˆ , kˆ ) − α(kˆ , k)]

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E. R. Arriola and R. N. Pérez

ˆ = α(). This allows to reconstruct which is an integral relation in the phase α(kˆ , k) the phase so that complete knowledge of the differential cross section, σ () at a given fixed energy E for all angles, allows for a complete determination of the scattering amplitude at that energy (Puzikov et al. 1957). There remains an ambiguity α() → π − α() or equivalently f () → −f ()∗ which leaves both Eq. (24) and σ () invariant. This solves the quantum scattering problem at the experimental level, provided one has a sufficiently large number of measurements at different scattering angles σ (1 ), . . . σ (N ) obtained from Eq. (1) yielding f (1 ), . . . , f (N ) after solving Eq. (24) on a finite grid. Already at this level, it should be noted that besides the experimental uncertainties associated with Eq. (1) due to fluctuations in the counting rate or to systematic effects, one has also the uncertainties deduced from the use of a finite grid. This can be avoided using a partial-wave analysis (PWA) to be discussed below. Discrete ambiguities will be shown below to disappear in an energy-dependent context invoking analyticity of the scattering amplitude in the complex energy plane.

Rotational Invariance Phase Shifts ˆ = f (R kˆ , R k) ˆ (R is a In the spinless case, rotational invariance requires f (kˆ , k) rotation matrix) so that the scattering amplitude must be a function of kˆ · kˆ = cos θ only. Therefore, f (θ, φ) = f (θ ) and fl ,m ;l,m = δm,m δl,l fl with Sl = 1 + 2ikfl . The off-forward unitarity relation implies fl − fl∗ = 2ikfl∗ fl so that fl−1 = gl−1 − ik, with gl = gl∗ but also Sl Sl∗ = 1. Thus, defining Sl = e2iδl with δl the phase shifts, one has gl = k cot δl . Using the addition theorem of spherical harmonics 4π



ˆ ∗ (kˆ ) = (2l + 1)Pl (cos θ ) , Ylm (k)Y lm

(24)

m

with Pl (z) the Legendre polynomials (P0 = 1, P1 = x, P2 = (3x 2 − 1)/2, . . . ), for a spherically symmetric potential V (r), the scattering amplitude depends just on the scattering angle θ and admits a partial-wave expansion (PWE) of the form: ∞  e2iδl (p) − 1 Pl (cos θ ) , f (θ ) = (2l + 1) 2ip l=0

E=

p2 . 2μ

(25)

The total cross section becomes: σT =

∞ 4π  (2l + 1) sin2 δl (p) . p2 l=0

(26)

43 NN Experiments and NN Phase-Shift Analysis

1655

The phase shifts δl (p) are computed by solving the reduced Schrödinger equation for the reduced radial wave function ul (r) (Ψ (x) = (ul (r)/r)Yl,m (θ, φ)): − u l (r) +



 l(l + 1) + 2μV (r) ul (r) = p2 ul (r) , r2

(27)

with the asymptotic conditions (assuming non-singular potentials r 2 V (r) → 0) → r l+1 ul (r)  r→0

,

  lπ + δl . ul (r)  → sin pr − 2

(28)

r→∞

Taking l 1 so that L2 = l(l + 1)h¯ 2 ∼ (l +  1/2)2 h¯ 2 and using the classical relation L = pb, the sum in l becomes an integral, l → dl, which in terms of b becomes dl = db/p. Assuming a large phase shift, one may replace sin2 δl → 1/2 (its mean value), and the classical expression σT → d 2 b is recovered. The fact that |δl | 1 in the classical limit can be seen from direct application of the WKB method (Galindo and Pascual 1991). For a finite range potential, V (r) ∼ 0 for r  a. The no-scattering condition corresponds to δl (p) ∼ 0 for b  a or equivalently lmax + 12 ∼ pa ∼ p/mπ . In this case the total cross section is now convergent for Yukawa forces. The (truncated) partial-wave analysis (PWA) describes scattering data in terms of phase shifts and known angular dependence of Pl (cos θ ), so that the differential cross section becomes a finite polynomial in cos θ : lmax max

2 2l dσ 1



= |f (θ )|2 = 2 (2l + 1)eiδl sin δl Pl (cos θ ) = an (cos θ )n , d p l=0

(29)

n=0

where an suitable coefficients which depends on the phase shifts and ultimately on the potential V (r). Alternatively, these coefficients can directly be determined from experiment at a fixed energy provided one has a least 2lmax ∼ 2pa angle measurements θ1 , . . . , θ2lmax . However, in this case the determination of phase shifts is ambiguous, since δl → δl − π or δl → −δl does not change an .

Coulomb Scattering For charged particles one has the Coulomb potential VC (r) = e2 /r which is long range and produces a divergent total cross section. The exact quantum mechanical treatment is mathematically involved and can be found in Landau (1990), Newton (1966), and Goldberger and Watson (1964), where the asymptotic behavior of the radial wave functions is   lπ ul (r)  → sin pr − + δlC − ln 2kr/kR , 2 r→∞

(30)

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where R = 1/Mp e2 = 28.8199 fm is the pp Bohr radius and the Coulomb phase shifts are δlC = Im log Γ (l + 1 + i/kR), so that the scattering amplitude reads fC (θ ) =

∞ 

(2l + 1)

l=0

C (p)

e2iδl

2ip

−1

Pl (cos θ ) ,

(31)

which correspond to Eq. (25) replacing δl → δlC . The sum can be computed analytically (Landau 1990; Newton 1966; Goldberger and Watson 1964) with the result that the pure Coulomb amplitude yields the classical Rutherford’s formula: fC (θ ) =

eiαC (θ) 2

4Rk 2 sin (θ/2)

⇒

α2 dσC = |fC (θ )|2 = , d 16E 2 sin4 (θ/2)

(32)

Note that the Coulomb phase αC (θ ) = − log(1 − cos θ )/kR + π + δ0C generates an expansion in powers of e2 in the scattering amplitude to all orders, so that dropping the phase of fC (θ ) yields accidentally the same differential cross section. When a short-range nuclear potential V (r) is added to the Coulomb potential, the long-distance behavior is modified to be   lπ ul (r)  → sin pr − + δl + δlC − ln 2kr/kR , 2

(33)

r→∞

where δl are the modified nuclear phase shifts. Note that in this definition, they are not the phase shifts corresponding to the pure short-range nuclear potential V (r) unless one takes the limit e2 → 0. This is so because the sum of potentials V (r) + e2 /r generally does not correspond to the sum of phase shifts. The total scattering amplitude is written by separating explicitly the Coulomb contribution as follows: ∞  C e2iδl (p) − 1 Pl (cos θ ) . (2l + 1)e2iδl (p) f (θ ) = fC (θ ) + 2ip

(34)

l=0

The convergence of the partial-wave expansion in the second term is similar to the case without Coulomb force and is determined from the short-range component of the potential, i.e., lmax + 12 ∼ pa.

Identical Particles For identical particles interacting by a potential V (r), the scattering amplitude has to be replaced by f (θ ) → f± (θ ) = f (θ ) ± f (π − θ ) depending on the symmetry or antisymmetry of the orbital wave function, respectively; taking into account their possible exchange in the final state which in the CM system translates into θ → π − θ . Since this corresponds to Pl (cos θ ) → Pl (− cos θ ) = (−)l Pl (cos θ ), these combinations f± (θ ) cancel the odd or even powers in the partial-wave expansion, respectively.

43 NN Experiments and NN Phase-Shift Analysis

1657

For particles with spin 1/2 such as nucleons, the total antisymmetry depends also on the spin state; the spin singlet is antisymmetric in spin, and hence the orbital part must be symmetric, and the spin triplet is symmetric in spin, and hence the orbital must be antisymmetric. If the interaction V (r) does not depend on spin and f (θ ) is the corresponding scattering amplitude, one has differential cross sections: dσ

= |f (θ ) + f (π − θ )|2 ,

d singlet

dσ

= |f (θ ) − f (π − θ )|2 . (35)

d triplet

For particles with no spin orientation in initial or final state, the average cross section corresponds to statistically weighted average: 1 dσ

d σ¯ 3 dσ

= + ,

d 4 d singlet 4 d triplet

(36)

which is an even function of θ . However, the total cross section corresponds to half the integrated differential cross section to avoid double counting between the  incoming and outgoing particles, σ¯ T = d(d σ¯ /d)/2. For the proton-proton case, one has to add the Coulomb component which dominates at sufficiently small angles as illustrated in Fig. 3. One sees a flat region at larger energies which could be identified by Jastrow in 1950 as a repulsive core at short distances ac ∼ 0.6 fm. The case of neutron-proton scattering is also shown; despite not being identical particles, they do resemble approximately the pattern of being an even function, a feature noted by Serber in 1950 and corresponding to an exchange symmetry between protons and neutrons. The amazing fact about this is that classically such an exchange pattern is not possible. One may imagine two billiard balls of different color colliding in a head-to-head event in the LAB system. Clearly the incoming projectile will always be behind the recoiling target along the scattering direction as they cannot pass into each other. Exchange symmetry clearly corresponds to assume that this is in fact possible and almost with equal probability. This can be stated symbolically as np → np + pn. In the particle

Fig. 3 Differential cross section for pp (left panel) and np (right panel) for several LAB energy values 1 (solid blue), 5, 10, 25, 50 (solid read), 100, 150, 200, 250, 300 (solid black), 350 MeV

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E. R. Arriola and R. N. Pérez

exchange picture, this phenomenon is interpreted as the exchange of a charged pion, np → (pπ − )p → p(π − p) → pn or np → n(π + n) → (nπ + )n → pn.

Analytical Properties The scattering problem has so far been discussed for a fixed given energy E, but it is possible to deduce useful and compelling conditions on the energy dependence. In the meson exchange picture, one can derive NN potentials from a perturbative matching between quantum field theory and the quantum mechanical problem, which turn out to be a superposition of Yukawa-like potentials: 1 V (r) = − 4π



∞

dμρ(μ) μ0

e−μr , r

(37)

with ρ(μ) a suitable spectral function which is nonvanishing above a given threshold value μ0 , typically μ0 , mπ , 2mπ , etc. From this representation one can deduce ˆ in the Mandelstam important constraints for the scattering amplitude f (kˆ , k), 2 2 2 variables t = −(kf − ki ) and s = 4(k + MN ) is not fully arbitrary; it fulfills a Mandelstam double spectral representation (Mandelstam 1959) in the complex plane. Equivalently, the partial wave amplitudes fl (p) as a function of the CM energy E = p2 /2μ = p2 /MN (Blankenbecler et al. 1960), namely, • fl (E) is an analytical function in the complex plane with the exception of some subsets on the real axis. • fl (E) is real for −μ20 /4MN < E < 0. • fl (E)∗ = fl (E ∗ ), i.e., Schwarz’s reflection principle is satisfied. • fl (E) has branch point at E = 0 and a branch cut for E ∈ (0, ∞) separating the first and second Riemann sheets. The discontinuity is given by the unitarity relation: Imfl (E) = k|fl (E)|2 ,

E = k 2 /M .

• For E > 0 one has fl (k) = 1/(gl (k) − ik), so that the function gl (k) = gl (−k) (even). • fl (E) has simple poles on the first (second) Riemann sheet for negative energy states E = −|En | located at the bound (virtual) states energies. • fl (E) has a cut in E ∈ (−∞, −μ20 /4M), where the discontinuity is proportional to the spectral function ρ • In the limit E → 0 one has fl (E) ∼ E l . These features are illustrated in Fig. 4 for the case of NN scattering in the LAB energy variable. The analytical properties listed above only depend on the spectral representation, Eq. (37), for the long-distance component of the potential; if one separates the potential as

43 NN Experiments and NN Phase-Shift Analysis

1659

Fig. 4 The LAB energy complex plane, showing the partial waves left-cut structure due to multiple pion (and σ, ρω) exchange along with the right-cut structure due to pion production. left TLAB = (. . . , −375.3, −260.6, −166.8, −93.8, −41.7, −10.4) MeV. The outer/inner circles correspond to LAB energies of 350/125 MeV, respectively

V (r) = VS (r)θ (rc − r) + VL (r)θ (r − rc ) ,

(38)

it suffices that VL (r) satisfies Eq. (37) for any arbitrary VS (r) and any separation distance (Ruiz de Elvira and Ruiz Arriola 2018). The bottom line of all these properties is that one generally expects a continuous and smooth behavior of the partial-wave amplitudes along the scattering line E > 0, and hence interpolation between different measured energies makes sense. A further consequence of analyticity is the removal of scattering ambiguities.

The Low Energy Limit In the long wavelength limit, ka  1, due to the analytical property fl (k) ∼ k 2l only l = 0 survives and scattering becomes isotropic (see Fig. 3), i.e.,

f (θ ) =

eiδ0 (k) sin δ0 (k) sin2 δ0 (k) dσ + O(k 2 a 2 ) ⇒ = . k d k2

(39)

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E. R. Arriola and R. N. Pérez

Thus, the total cross section becomes: σT =

k 2 cot δ

4π . 2 2 0 (k) + k

(40)

In this limit for r → ∞, one has u0 (r) → A + Br, and matching to u0 (r) → sin (kr + δ0 ), one gets in the limit p → 0 that δ0 (p) → −α0 p. Thus, f (θ ) → −α0 and σT → 4π α02 which has a classical geometrical interpretation of a disk surface of a radius α0 . The number α0 is the scattering length which has the unrestricted range −∞ < α0 < ∞ depending on the particular interaction. If α0 ∼ a is considered to be naturally large and unnaturally large for |α0 | a or small |α0 |  a. In any case one has the following effective range expansion: k cot δ0 = −

1 1 + r0 k 2 + v2 k 4 + . . . α0 2

(41)

where r0 is the effective range and v2 a curvature parameter. The case of Coulomb scattering modifies the formula to be 2π/R 1 1 cot δ0 + hC (1/kR)/R = − + r0 k 2 + v2 k 4 + . . . α0 2 −1

e2π/kR

(42)

 2 2 with the function hC (x) = −2 log x + 2x 2 ∞ n=1 1/n(x + n ). The limit R → ∞ reduces to the previous case. Using these expansions the parameters α0 and r0 can be determined by extrapolating low energy measurements. Taking into account the spin dependence one has spin singlet S-wave for pp and nn and both spin singlet (1 S0 ) and triplet (3 S1 ) for np, one has the empirical or recommended values in Table 2. The large values of the scattering length compared to the range of the interaction |α0 | a make the low energy cross section much larger than the classical geometrical cross section.

Virtual and Bound States One of the consequences of analytical properties is the extrapolation to the complex plane and the possibility of predicting bound states from scattering data. If one uses the low energy limit of the previous section, one has the S-wave scattering amplitude: f0 (k) =

1 1 = . 1 1 2 k cot δ0 − ik − α0 + 2 r0 k + v2 k 4 + · · · − ik

Table 2 Scattering length and effective range for NN α0 (fm) r0 (fm)

pp −7.8196(26) 2.790(14)

np –1 S0 −23.740(20) 2.77(5)

np –3 S1 5.419(7) 1.753(8)

nn −18.95(40) 2.86(10)

(43)

43 NN Experiments and NN Phase-Shift Analysis

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If one keeps only the term in α0 , the poles are k = iγ with γ = 1/α0 , i.e., purely imaginary. For this case E = k 2 /2μ = −1/2μα02 < 0. The sign of γ determines whether one has the pole on the first or second Riemann sheet in the complex energy plane in which case one has a bound or virtual state respectively. For the values of Table 2, one obtains k = iγ with γ > 0 only for the np triplet state 3 S1 . This bound state corresponds to the deuteron whose binding energy would be ED = −2.2 MeV. All other singlet interactions for pp, np, or nn correspond to virtual states. Going further in the effective range expansion, the poles are determined by −

1 1 + r0 k 2 − ik = 0 . α0 2

(44)

This quadratic equation contains two solutions but only one which goes smoothly to k = i/α0 . The discriminant −1 + 2r0 /α0 is negative for the values of Table 2 so that again the solution is purely imaginary, so that for k = iγ , one has γ =

  1  1 r0 1 − 1 − 2r0 /α0 ∼ + 2 + ... r0 α0 2α0

(45)

The effective range corrections refine the value provided one deals with a weakly bound state. In the general case, of course, one uses the full partial-wave amplitude and predicts a value which may be compared with the experimental value Bd = 2.224575(9) MeV which can be determined very well by the photodisintegration process γ + d → np.

Square Wells All these features at low energies are well illustrated with an attractive square well potential V (r) = −V0 θ (a−r), where matching the wave function inside and outside the well by the logarithmic derivative condition at r = a, u 0 (a)/u0 (a), yields k cot δ0 (k) =

k(K cot ka cot Ka + k) k cot ka − K cot Ka

(46)

 where K = k 2 + 2μV0 . The effective range expansion yields the scattering length and effective range: α0 = a − 

√ tan 2μV0 √ 2μV0

1 a2 r0 = a 1 − − 2 2α0 aμV0 3α0

(47)  .

(48)

The first equation shows that due to the appearence of the tan function, the scattering length α0 may take any possible value, −∞ < α0 < ∞ and only for very special cases is α0 ∼ a.

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E. R. Arriola and R. N. Pérez

For this attractive potential, it is straightforward to check that the poles of the scattering amplitude, i.e., the zeros of k cot δ0 − ik = 0, correspond to the condition of a bound state of negative energy whose wave function outside the potential r > a falls off exponentially as ∼ e−γ r , i.e., u0 (r) = AS e−γ r

r ≥ a,

(49)

where AS can be determined from the normalization condition using that inside the r > a one has u0 (r) = B sin Kr. The only difference with the virtual state is that outside the potential, one has an exponential growth as ∼ e+γ r . However, for small values of γ , the difference of the wave  √ function inside the potential is small due to the fact that K = 2μV0 + γ 2 ∼ 2μV0 . From Table 2 one may deduce the values of V0 and a and predict the virtual np and nn states or the deuteron np binding energy to good accuracy with the experimental value as well as AS or r 2  which can be measured by electron scattering on the deuteron. The square well with finite range a provides some insight into the convergence of the partial-wave amplitude which as it has already been mentioned has a maximal angular momentum roughly corresponding to an impact parameter of bmax = a. This argument ignores interferences between the different partial waves which tend to cancel. Numerical calculations show that this is equivalent to an effective smaller √ impact parameter beff ∼ b/ 2 (Simo et al. 2018).

Inverse Scattering Ambiguities The square well example also shows the use of analyticity providing this connection between scattering and bound states, which experimentally require quite different setups. Unfortunately, it also displays a recurrent issue in nuclear physics which has to do with the inverse scattering problem dealt with here. α0 and r0 , are V0 √ Given √ and a unique?. Clearly for large α0 one may change 2μV0 → 2μV0 + nπ and in this case r0 = a. More generally, one may change the functional dependence V (r) without changing the phase shifts δ0 but introducing additional bound states. Thus, for instance, the deuteron state discussed here would be the least bound of all these states or the most excited bound state. According to the oscillation theorem, the bound state wave function of the n-th excited state has n-nodes, and for instance the r 2  value depends on this fact, but experiment (for instance electron scattering) will discard this scattering ambiguity declaring the deeply bound states as spurious. The inclusion of these extra states has a consequence also at the scattering level, namely, by the application of Levinson’s theorem which reads δl (0) − δl (∞) = nπ ,

(50)

with n the number of bound states. Taking into account that one never √ goes to infinite energy and elastic scattering is indeed restricted to 0 ≤ k ≤ MN mπ , this contingency may actually happen in the analysis. Thus, when fixing a potential

43 NN Experiments and NN Phase-Shift Analysis

1663

with scattering data, one must pay attention to the possible appearance of spurious bound states.

Volume Integrals and Symmetries Another method to detect spurious bound states is by considering even volume integrals of the potential and their moments: C2l =

(−)l (2l + 1)!!

 d 3 x r 2l V (r) .

(51)

The normalization is chosen to correspond to the power series expansion of the Fourier transformation of the potential. These quantities change dramatically under these scattering ambiguities and turn out to be relatively independent on the particular potential shape (Navarro Pérez et al. 2016a). This model independence can be promoted to become a way of specifying the potential itself and characterize symmetry patterns of the potential which are not completely obvious from visual analysis. For instance, in the square potential well case, the lowest volume integrals are  C0 =

4π d x V (r) = − V0 a 3 , 3 3

1 C2 = − 6

 d 3 x V (r) =

4π V0 a 5 , 30

(52)

so that the values corresponding to fix α0 and r0 in the np 1 S0 and 3 S1 are (C0 (1 S0 ), C0 (3 S1 )) = (1030, 1288) MeV fm3 ,

(53)

(C2 ( S0 ), C2 ( S1 )) = (715, 554) MeV fm ,

(54)

1

3

5

which turn out to be very similar despite their corresponding α0 and r0 being very different. In the isospin formalism, the generalized Pauli principle states that (−)L+S+T = −1, so that for L = 0 the spin singlet corresponds to an isospin triplet and the spin triplet corresponds to an isospin singlet. The numerical similarity of these volume integrals suggests that p ↑, p ↓, n ↑, n ↓ behave as a degenerate quadruplet. This fact was discovered by Wigner in 1937 and is the basis of a supermultiplet SU(4) symmetry pattern for light nuclei (Parikh 1978).

Statistical Analysis Going beyond a rough description of experimental data requires a massive use of statistical methods involving the usual fitting parameter procedure based on least squares minimization, confidence limits, error analysis, and uncertainty propagation. While this material is mostly discussed in standard textbooks (see, e.g., Eadie et al. 1971; Evans and Rosenthal 2004), it is worth to review it here focusing on the most relevant aspects within our scattering context (see also Arndt and Macgregor

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E. R. Arriola and R. N. Pérez

(1966a) for an early account and Dobaczewski et al. (2014) for a more general nuclear physics point of view). The ultimate goal is a serious verification or validation of the theory describing the data. This implies in particular that some data could be discarded as outliers. This is a subtle issue, since not always do nice and beautiful theories describe data reasonably well. Within this context the key question is this: Is the theory or the data wrong or both? Statistical methods allow to make a decision within an expected probabilistic confidence level, so that a theory may be discarded or falsified because it appears too improbable according to reasonable expectations. The validation of the theory actually never happens because there will always remain unmeasured data which might falsify the theory. Nonetheless, one abusively speaks of validation meaning in fact that falsification has not occurred yet.

Fitting Scattering Data The general fitting problem corresponds to a situation where on the one hand N data with given uncertainties Oi ± ΔOi have been measured and the other hand a theory depending on M-parameters p = (p1 , . . . , pM ) predicting Oi (p) is proposed. The question “Does theory explain data?” which admits only yes/no answer needs some qualification and can indeed be answered probabilistically as follows. If all uncertainties follow an independent Gaussian distribution for a choice of parameters p, one writes exp

Oi

= Oith + ξi ΔOi ,

ξi ∈ N(0, 1) ,

(55)

and defines the minimized least squares sum

2 χmin

≡ minp χ (p) = χ (p0 ) , 2

2

  exp 2 N  Oi (p) − Oi χ (p) = . ΔOi 2

(56)

i=1

This condition effectively eliminates M-independent variables, so that for the remaining degrees of freedom ν = N − M, one has the following χ 2 probability density distribution evaluating all possible values of ξ1 , . . . , ξν with a fixed χ 2 Pν (χ ) = 2

 ν  n=1

e−ξi /2 dξi √ 2π −∞ ∞

2

 δ(χ 2 −

ν  n=1

e−χ χ ν−2  , 2ν/2 Γ ν2 2

ξn2 ) =

(57)

which is plotted in Fig. 5 as a function of χ 2 /ν, and shows the drastic narrowness for a large number of data. The mean and variance χ 2  =√ν, (χ 2 − χ 2 )2  = 2ν 2 . For √ are given by 2 2 ν 1 the χ ∈ N(ν, 2ν) whence χ = ν ± 2ν at the 68% confidence level.

43 NN Experiments and NN Phase-Shift Analysis

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2 Fig. ν = 10 (blue),100 (pink),1000 (brown),10000 (green). Left panel: √ 5 The χ distribution for ∞ 2νPν (χ) as a function of χ 2 /ν. Right panel: p-value defined as p = 100 × χ 2 Pν (χ)dχ min

Table 3 Values of acceptable fits for some sample sizes at the 1σ -level ν χ 2 /ν (68%)

10 1 ± 0.447

100 1 ± 0.141

1000 1 ± 0.044

10,000 1 ± 0.014.

Thus, the assumption that data differ from theory by fluctuations, Eq. (55), holds at Nσ -standard deviations level if 2 χmin = 1 ± Nσ ν



2 , ν

ν =N −M

, d.o.f (degrees of freedom) .

(58)

2 /ν outside the confidence interval Table 3 provides the case Nσ = 1. Thus, χmin is unlikely (for Nσ = 1, 2, 3 is less than 32, 5, 1% respectively) and implies either a bad model or bad data or both. On the contrary, an acceptable χ 2 /ν suggests consistency between model and data, and, more importantly, errors on the parameters reflect statistical uncertainties of the input data p = p0 + Δp which can be propagated to functions of the parameters F (p) not involved in the fitting procedure. More sophisticated methods of testing a posteriori the validity of Eq. (55) based on residuals analysis are thoroughly discussed in Navarro Pérez et al. (2014a, 2015, 2016a). Of course, once estimates for the most likely values of the theory parameters have been obtained, one needs to check a posteriori that they naturally fall within some given expectations if they exist. If the reasonable expectations are not fulfilled, the fit should be discarded as unnatural. A way of incorporating these expectations a priori is by using an augmented χ 2 function where one adds to the usual one, which 2 , an extra theoretical term: one may denote as χexp

χth2

=

 2 P  pi − pith i=1

Δpith

,

(59)

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E. R. Arriola and R. N. Pérez

to favor values of pi within a theoretical estimate pith ± Δpith , where Δpith → ∞ corresponds to complete ignorance and Δpith → 0 to complete certainty on the 2 + χ 2 , the first term will typically dominate over theory side. In the total χ 2 = χexp th the second for N P , and an adequate weighted balance depends on our degree of belief, which may be a controversial and clearly subjective concept since there is a potential risk of overweighting the theory. This is the essence of the Bayesian approach (D’Agostini 2003) (for a recent presentation within NN scattering, see, e.g., Melendez et al. 2017).

Fitting Strategies Single-Energy Fits The simplest situation corresponds to have complete data in a given energy E (or momentum p), namely, (σ (θ1 , E), . . . , σ (θN , E)). In this case one can determine the lmax ∼ pa phase shifts directly from the data as fitting parameters (δ0 (E), . . . , δlmax (E)) by minimizing χ (δ1 (E), . . . , δlmax (E), Z) = 2

N  exp  σ (θi , E) − Zσ th (θi , δ1 (E), . . . , δl



(E))

2

Δσ (θi , E)

i=1

+

max

1−Z ΔZ

2 .

(60)

which has at least one solution for N ≥ lmax ∼ ka corresponding to the minimal number of necessary measurements at the fixed momentum k. Here the normalization Z with estimated uncertainty ΔZ (provided by experimentalists) is common for one energy. Phase shifts become “experimental” and exp exp model-independent observables, δl (E) ± Δδl (E) for l = 0, . . . , lmax . The main drawback of the method is that a sufficient number of measurements must be available at a fixed energy, a condition which is not always fulfilled particularly when spin and polarization are included.

Multiple-Energy Fits If one has incomplete data in energies and angles given by the set of N data (σ (θ1 , E1 ) ± Δσ (θ1 , E1 ), . . . , σ (θN , EN ) ± Δσ (θN , EN )) ,

(61)

one cannot generally determine phase shifts δl (Ei ) at those energies. Instead, a model-dependent interpolation with fitting parameters p in the energy is needed. A typical example  is to take a rational representation of the K−matrix, of the L 2n / form k 2l+1 cot δl = K a k b k 2n where the corresponding coefficients n n n=0 n=0 build the vector of fitting parameters p = (a0 , . . . , aK ; b0 , . . . , bL ) with M = K + L + 2. This parameterization incorporates some analytical properties: the

43 NN Experiments and NN Phase-Shift Analysis

1667

threshold behavior and the fact k 2l+1 cot δl (k) is an even function of k. Thus, in this case one minimizes χ (p, Z) = 2

2 N   σ (θi , Ei )exp − Zσ th (θi , Ei , p) i=1

Δσ (θi , Ei )

 +

1−Z ΔZ

2 .

(62)

Different experiments or different LABs have different normalizations so that generally χ 2 (p, Z1 , . . . ZE ) =

E 

χi2 (p, Zi ) .

(63)

i=1

The advantage of this method is that phases are assumed to be continuous functions (a consequence of the analyticity properties mentioned earlier) which can be accessed at any arbitrary energy; this way data at different energies are intertwined. This raises the issue on the possible exclusion of outliers and consequently on the selection of mutually consistent data very much dependent on the model. Therefore, the best model would hopefully congregate the maximal number of mutually consistent data.

Potential Models Analyticity of the scattering amplitude corresponding to potentials obeying the spectral representation in terms of Yukawa’s of Eq. (37) generates a continuous energy dependence which can be extended to the complex energy plane. Such a potential can be deduced from quantum field theory (QFT) by matching the perturbative calculation of relevant Feynman diagrams implementing the meson exchange picture with quantum mechanics order by order, yielding VL (r) = VQFT (r). Longrange electromagnetic effects implement QED perturbative corrections and are thus also included in VQFT (r). This perturbative matching becomes a valid expansion at long distances (where the potential is small) leaving the shortest distance potential piece, VS (r), undetermined. If rc is a given separation distance, one writes: V (r) = VS (r)θ (rc − r) + VQFT (r)θ (r − rc ) .

(64)

Within this potential model approach to the problem, there also arises the question on the adequate number of fitting parameters p. On physical grounds and for a model potential VS (r) with range rc , one can argue that for a maximum CM momentum pmax , the best possible resolution corresponds to Δr ∼ h¯ /pmax which roughly corresponds to sample the potential at equidistant points rn = nΔr. This coarse graining reduces a continuous function to a finite number of sampled points V (r) → V (rn ) which act themselves as fitting parameters (Ruiz Arriola et al. 2016a). For a central potential, one has to add the centrifugal barrier l(l + 1)/2μr 2 , and the points rn below the barrier correspond to the classically forbidden region, so

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E. R. Arriola and R. N. Pérez

they will not contribute. Taking into account that lmax + 1/2 ∼ pmax a, the number of points is (Fernandez-Soler and Ruiz Arriola 2017) N=

   2 θ pmax − l(l + 1)/rn2 ∼ (pmax rc )2 /2 ,

(65)

nl 2 where for  l 1 one  approximates l(l + 1) ∼ (l + 1/2) and replaces sums by integrals, l → dl. Thus, in the nonrelativistic approximation, the number of fitting parameters grows linearly with the maximum energy.

Confidence Limits and Error Propagation Once one enters the least squares minimization in analysis of the scattering problem, the standard methodology follows, but the goal is quite simple: How much can one change the parameters of the fit so that one can still claim the theory is not falsified by the data? This implies providing error bars either directly on the phase shifts (energy-dependent analysis) or on the potential model parameters first and on the phase shifts second (multienergy analysis). One should stress that only after the goodness of the fit has been established, it becomes possible to propagate the (direct or indirect) uncertainties on the phase shifts to the evaluation of the uncertainties, say on the scattering amplitude, f (θ ), which is not directly accessible but needs to be reconstructed by using the integral equation Eq. (24). The standard uncertainty propagation assumes a Gaussian distribution of parameters with small errors which may be correlated through the covariance matrix and is well documented (Arndt and Macgregor 1966a; Eadie et al. 1971; Evans and Rosenthal 2004). Another more elaborated technique, the bootstrap method, is based on generating a set of possible new experiments (resampling), α = 1, . . . , K based on the Gaussian distribution Oi ± ΔOi yielding new synthetic data Oiα and fit the parameters p = (p1 , . . . , pM ) for any single new experiment producing a distribution of α ) which can be used to estimate their mean and parameters pα = (p1α , . . . , pM standard deviation (see, e.g., Navarro Pérez et al. (2014b) and references therein).

Systematic Uncertainties The strategies based on fitting directly phase shifts at different fixed energies δl (En ) or the parameters of a given potential VS (r) at a set of fitting energies have pros and cons. On the one hand, the direct δ minimization requires a complete set of experiments for different angles, θ1 , . . . , θm , at any given energy En and is free from theoretical systematic uncertainties but presents the problem of interpolation between different energies and hence on the uniqueness of the solution. On the other hand, the indirect δ minimization through a given potential VS (r) with some potential parameters can be undertaken without complete sets of measurements and

43 NN Experiments and NN Phase-Shift Analysis

1669

provides an interpolation in the energy, but the specific form of the potential is not determined. Moreover, while both methods should be virtually identical, it is in practice observed that any different potential parameterization of VS (r) leads to similar statistical uncertainties and often much larger discrepancies! In essence the reason is that the minimizations implicit in the multienergy fit via potential model parameters may be slightly worse than the direct minimization in terms of phase shifts (Escalante et al. 2021). Denoting the phase shifts δα where α = (En , l) runs over all the available energies and relevant angular momenta, written in compact form, one has min χ 2 (δ) ≡ χ 2 (δ ∗ ), δ

(66)

where δα∗ are the minimizing phase shifts whose statistical uncertainties, Δδα , and correlation matrix can be obtained from the standard covariance matrix. In the energy-dependent strategy, a potential with parameters is used, which are denoted as Vi for short, one actually has that effectively the phase shifts become functions of the potential parameters δα (V ) ≡ δα (V1 , . . . , VM ) so that χ 2 (V ) = χ 2 (δ(V )).

(67)

Thus, from a variational point of view, the minimization with respect to the potential parameters operates in general as a restriction and hence minimization means min χ 2 (δ(V )) ≡ χ 2 (δ(V ∗ )) ≥ min χ 2 (δ) ≡ χ 2 (δ ∗ ). V

δ

(68)

Now, in the limit of small deviations, one sets δα (V ∗ ) = δα∗ + εα , and because of the stationary condition, one gets that the accuracy in the minimum is quadratic in the small deviation: χ 2 (δ(V ∗ )) − χ 2 (δ ∗ ) = χ 2 (δ ∗ + ε) − χ 2 (δ ∗ ) = O(ε2 ).

(69)

Thus, by construction one expects the energy-dependent strategy to provide only O(ε) accurate phase shifts for O(ε2 ) accurate values of χ 2 . In other words, using a potential one can get rather good fits with not so precise phase shifts. Any potential taken as a basis for the PWA will induce a bias, and one can only hope that by using different potentials with different biases, the global bias will be removed in average. In the above notation if one takes different potentials, say V (1) , V (2) , . . . , in average εα  = 0 but their mean squared standard deviation Std(εα ) corresponds to an average systematic uncertainty to be compared to the statistical uncertainty, Δδα . In Fig. 6 the situation is illustrated for 13 PWA statistically satisfactory fits to np data (see below also). Clearly Std(εα ) Δδα . Of course, one can improve on this by introducing arbitrarily many parameters in the potential, a procedure that may produce over-fitting and hence strong correlations among the fitting parameters.

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E. R. Arriola and R. N. Pérez

63

11

45 27 9

4 1

-3

S0

-10

-9 -3.5 -10.5 -17.5 -24.5

1

P1

144 112

4.2

-9

-17.5

80

3

-15

48

1.8

-21

16

0.6

3

P1

-31.5

10.8

27 1

6

δ (deg.)

-3.5 -10.5 -24.5

-31.5 8.4

D2

3

21

3.6

10

9

6

1.2

3

2

-0.4

5.4

-1.2

4.2

-2

3

-4.9

F3

-2.8

-6.3 2.25 1.75

3

-3.6

F3

9 1

1.25

7

G4

5.4

S1

3

3

D1

3

F2

3

G3

3

H4

3

I5

1.5

2

1.1

-1.75

0.7

-2.45

0.3

-3.15 7.2 3

D3

5.6

-0.1 -0.6

3

-1.8

4

-3

1.8

2.4

-4.2

0.6

0.8

-5.4

2.8

G4

-27

-1.05

P2

3.6 3

5

1

2

-0.2 3

-0.6

F4

-1

0.9 0.7

4

0.5

0.75

3

1.2

-1.4

0.3

0.25

1

0.4

-1.8

0.1

-0.25

-0.14

-0.75 -1.25

P0

-3

-0.35

14

-2.1 1

3

18

D2

15

-0.7 -3.5

3

-17

1

-1.75

H5

3.15

0

2.45

-0.7

-0.2

1.75

-0.75

1.05

-1.05

0.35

-1.35

-0.98

-2.25

3

H5

-0.4

-1.26 50

150

250

350

-0.15

0.2

-0.42

3

G5

-0.6 50

150

250

350

50

150

250

TLAB (MeV)

350

5

50

-0.45

150

250

350

50

150

250

350

Fig. 6 np phase shifts in degrees for all partial waves with J ≤ 5 in 13 PWA. The dark blue band represents the mean and standard deviation of 13 different determinations of the NN interaction to their contemporary database (Stoks et al. 1993, 1994; Wiringa et al. 1995; Machleidt 2001; Gross and Stadler 2008; Navarro Pérez et al. 2013c, d, 2014c, d, e). The red, green, olive green, light blue, light red, and light green bands represent the statistical uncertainty of the DS-OPE (Navarro Pérez et al. 2013c, d), DS-χTPE (Navarro Pérez et al. 2014c, d), Gauss-OPE (Navarro Pérez et al. 2014e), Gauss-χTPE, DS-ΔBO, and Gauss-ΔBO potentials, respectively

To conclude this section, it should be noted that here is also another source of systematic uncertainties. The discussion above assumes from the start that all measurements are independent. When there exist correlations among several data, the direct least squares minimization in the presence of known correlations suffers from a deficiency, dubbed as the d’Agostini bias (D’Agostini 1994), which needs to be properly handled (see, e.g., Alekhin 2000; Ball et al. 2010).

NN Scattering Theory Having exposed the method of determining the scattering amplitude from data in the simplest spinless case, one may pass to the more complicated case of two particles with spin 1/2. The NN scattering process will be reviewed from a theoretical perspective along the new pertinent observables. This framework sets theoretical constraints on the definition and analysis of NN experimental data in terms of differential (angle dependent) observables. Given the fact that nucleons are spin 1/2 particles and that the interaction is anisotropic, complications arise, and polarization phenomena become key aspects of both theory and experiment. For a

43 NN Experiments and NN Phase-Shift Analysis

1671

readable account on the early polarization experiments, see, e.g., Segrè (1956) and Chamberlain (1985).

Scattering with Spin In the NN case including spin, the relative NN wave function in the CM system fulfills the generalized scattering condition for pure uncorrelated initial spin states: ψk,m1 ,m2 (x) → eik·x χm1 ,m2 +

eikr  (θ, φ)χm ,m , M 1 2 r m1 ,m2 ;m1 ,m2

(70)

m1 ,m2

where χm1 ,m2 = χm1 χm2 and χm are Pauli spinors with components m = ± 12 corresponding to spin up and spin down states. Here, the scattering amplitude becomes a 4 × 4 matrix in the tensor product space of two Pauli spinors due to the possibility of spin flips. For the particular process (kˆ i ; m1 , m2 ) → (kˆ f ; m 1 , m 2 ), the differential cross section in the CM system is given by 

dσ d

 m1 ;m2 →m 1 ,m 2

= |Mm 1 ,m 2 ;m1 ,m2 (θ, φ)|2 ,

(71)

where Mm 1 ,m 2 ;m1 ,m2 (kˆ f , kˆ i ) = χm† χm† M(kˆ f , kˆ i )χm1 χm2 . 1

(72)

2

This corresponds to a situation where particles both in the initial and final states have a well-defined spin, i.e., fully polarized beam and target before and after the collision. For elastic scattering the off-forward optical theorem, Eq. (20), is also fulfilled in matrix sense by replacing f and f ∗ of the spinless case with M and M † , respectively. This allows to reconstruct (Puzikov et al. 1957) likewise the phases of the amplitude from the differential cross sections up to discrete ambiguities (Schumacher and Bethe 1961). In principle, one has a total of 4 × 4 = 16 possible scattering processes, but due to parity P and time-reversal T symmetries, they are not all independent, and only five independent amplitudes survive (Hoshizaki 1969; Bystricky et al. 1978; LaFrance and Winternitz 1983) (see below) which determine the modulus of all the amplitudes. The use of off-forward optical theorem fixes then the corresponding phases by solving a coupled system of nonlinear integral equations up to discrete ambiguities (Schumacher and Bethe 1961). While this program for experimentally determining the scattering amplitude seems rather straightforward, the main obstacle hindering its practical implementation is often a lack of fully polarized beams and targets, i.e., states with a well-defined spin orientation such as χm1 or χm2 , so that in practice one may have

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beams and targets which are an ensemble of different spin directions, i.e., with probabilities Am1 ,m2 and Bm 1 ,m 2 in the initial and final state, respectively: 

dσ

 =4 d A→B

 m1 ,m2 ;m 1 m 2

Am1 ,m2 Bm 1 ,m 2 |Mm 1 ,m 2 ;m1 ,m2 |2

(73)

  with 0 ≤ Am1 ,m2 , Bm 1 ,m 2 ≤ 1 and m1 ,m2 Am1 ,m2 = m 1 ,m 2 Bm 1 ,m 2 = 1. Defining the preparation A and the measurement B in terms of the density matrices ρA and ρB respectively 

ρA =

χm1 ,m2 Am1 ,m2 χm† 1 ,m2 = ρA†

(74)

χm 1 ,m 2 Bm 1 ,m 2 χm† ,m = ρB†

(75)

m1 ,m2



ρB =

m 1 ,m 2

1

2

one has that the cross section for measuring the average A from an initial average B can be written as a trace in four dimensions: 

  dσ

 = 4Tr ρB MρA M † d A→B

(76)

The factor 4 corresponds to detect all particles with final spins m 1 , m 2 and efficiency Bm 1 ,m 2 . The simplest unpolarized state case corresponds to take equal probabilities Am1 ,m2 = Bm 1 ,m 2 = 1/4 so that ρA = ρB = 1/4 ≡ ρ0 and 

 dσ

1   = Tr MM † d 0→0 4

(77)

If one has an initial mixed state ρi and measure final pure states χm 1 and χm 2 , one has 

dσ

 = χm† ,m Mρi M † χm 1 ,m 2 . 1 2 d i→m 1 m 2

(78)

Given a spin observable Om1 ,m2 , one can define its expectation value in the final state as  Of =

m 1 ,m 2

dσ   Om 1 ,m 2  d  Tr OMρi M † i→m 1 m 2

  = dσ Tr MM †   d 0→0

(79)

43 NN Experiments and NN Phase-Shift Analysis

1673

Wolfenstein Parameters As it was mentioned earlier, the general scattering amplitude has 16 complex components which transforms as a scalar under rotations. In the CM system, one has the final and initial relative nucleon momenta respectively , kf = k kˆ f , ki = k kˆ i as vectors and the single nucleon Pauli matrices σ 1 and σ 2 as axial vectors. Because of the property σi σj = δij + iεij k σk , any function of spin becomes a linear funcion. This means that one may have only 16 independent matrices 1, σ 1 , σ 2 , σ 1 × σ 2 . Defining l, m, n as the three unitary orthogonal vectors along the directions of kˆ f + kˆ i , kˆ f − kˆ i and kˆ f ∧ kˆ i and using on-shell elastic condition kf = ki ≡ k, one has the identity σ 1 · σ 2 = (σ 1 · l)(σ 2 · l) + (σ 1 · m)(σ 2 · m) + (σ 1 · n)(σ 2 · n)

(80)

which reflects the completeness of the basis l, m, n. Finally imposing P (parity) and T (time reversal) which imply the relations P :

σ i → +σ i

kˆ f , kˆ i → −kˆ f , −kˆ i ,

(81)

T :

σ i → −σ i

kˆ f , kˆ i → −kˆ i , −kˆ f ,

(82)

respectively, the complete on-shell NN scattering amplitude contains five independent complex quantities (Okubo and Marshak 1958), which are chosen for definiteness as the Wolfenstein parameters (Glöckle 1983): M(kˆ f , kˆ i ) = a + m(σ 1 · n)(σ 2 · n) + (g − h)(σ 1 · m)(σ 2 , m) +(g + h)(σ 1 · l)(σ 2 · l) + c(σ 1 + σ 2 ) · n ,

(83)

where a, m, g, h, c depend on energy E and the scattering angle θ only. For illustration, in Fig. 7 real and imaginary parts of a, m, g, h, c for np scattering at several energies from several analyses (Stoks et al. 1993, 1994; Wiringa et al. 1995; Machleidt 2001; Gross and Stadler 2008; Navarro Pérez et al. 2013c, d, 2014c, d, e) are shown. The relation to the canonical basis {ˆex , eˆ y , eˆ z } for the geometry kˆ i = kˆez and kˆ f = {sin θ cos φ, sin θ sin φ, cos θ } can be deduced from the relations: ˆl = sin θ cos φ eˆ x + sin θ sin φ eˆ y + cos θ eˆ z , 2 2 2 θ θ θ ˆ = cos cos φ eˆ x + cos sin φ eˆ y − sin eˆ z , m 2 2 2 nˆ = sin φ eˆ x − cos φ eˆ y .

(84) (85) (86)

1674

E. R. Arriola and R. N. Pérez

Re(g) [fm]

Re(m) [fm]

Re(c) [fm]

Re(a) [fm]

TLAB = 50MeV

0.88

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60

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90 120 150 180

θc.m. [deg] TLAB = 50MeV

Im(g) [fm]

TLAB = 350MeV

0.9

0

Im(h) [fm]

TLAB = 200MeV

1.06

-0.2 Re(h) [fm]

TLAB = 100MeV

0.9

30

60

90 120 150 180

-0.3 0

θc.m. [deg] TLAB = 100MeV

30

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θc.m. [deg] TLAB = 200MeV

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60

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θc.m. [deg]

0 0

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90 120 150 180

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θc.m. [deg] TLAB = 350MeV

1.02

-0.023

30

-0.006 0

30

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90 120 150 180

θc.m. [deg]

0

30

60

90 120 150 180

θc.m. [deg]

Fig. 7 Same as Fig. 6 for the real (top panel) and imaginary (bottom panel) parts of the Wolfenstein parametrization of the np scattering amplitude in fm as a function of center of mass scattering angle at TLAB = 50, 100, 200, 350 MeV

43 NN Experiments and NN Phase-Shift Analysis

1675

This amplitude commutes with S2 , with S = (σ 1 + σ 2 )/2 the total spin, so one can rewrite the scattering amplitude in the χsm basis: 

χsm =

m1 ,m2

1 1 C( , , s|m1 , m2 , m)χm1 χm2 , 2 2

(87)

so that Mm1 ,m 2 ,m1 ,m2 =

 m,m ,s,s

1 1 1 1 s C( , , s|m1 , m2 , m)C( , , s|m 1 , m 2 , m )Mm,m , (88) 2 2 2 2

where C( 12 , 12 , s|m1 , m2 , m) are the Clebsch-Gordan coefficients, with |m| ≤ s, and s = 0, 1 corresponding to spin singlet and triplet, respectively. In matrix form one has ⎛ M=

M M √10 √10 2 2 ⎜ M ⎜ √012 M002+M0 M002−M0 ⎜ M00 −M0 M00 +M0 ⎜ M 01 ⎝ √2 2 2 M−10 √ M−11 M√1−0 2 2

M11

M1,−1 M0,−1 √ 2 M0,−1 √ 2

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

(89)

M−1,−1

The unitary S-matrix in this case is given by the expression: S(kˆ f , kˆ i ) = δ (2) (kˆ − kˆ )1 + 2kiM(kˆ f , kˆ i ) ,

(90)

with 1 the 4-dimensional unit matrix.

The Partial-Wave Expansion Besides other symmetries the NN scattering amplitude conserves the total angular momentum J = L + S, the spin (S)2 , and hence a complete set of commuting observables is given by {J 2 , Jz , S 2 } where a suitable basis is provided by the vector ˆ given by the decomposition: spherical harmonics YJ LSM (k), ˆ = YJ LSM (k)



ˆ S,Ms . C(LSJ |ML MS M)YL,ML (k)χ

(91)

ML ,MS

This reduces the action of the integral operator such as the S−matrix to finite matrix multiplication: ˆ = S YJ LSM (k)

 L

J,S ˆ SL,L YJ L SM (k) .

(92)

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E. R. Arriola and R. N. Pérez

The angular momentum composition implies that for S = 0, one has L = J , and ˆ = for S = 1 one may have L, L = J ± 1, J . Due to parity invariance P YJ LSM (k) L ˆ ˆ ˆ YJ LSM (−k) and since YL,ML (−k) = (−) YL,ML (k) one has that for S = 1 either L = J ± 1 for unnatural parity states or J = L for natural parity states. One may denote these states in the standard spectroscopic notation as 2S+1 LπJ with the letters S, P , D, F, G, H, I, J, K for the values L = 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. Coupled states are denoted as 2S+1 (J − 1)πJ −2S+1 (J + 1)πJ . Finally, assuming isospin invariance the generalized Pauli principle requires that only states with (−)T +S+L = −1 are allowed. For instance, for J = 0 one has the states 1 S0 and 3 P0 , for J = 1 one has the states 3 S1 −3 D1 ,1 P1 ,3 P1 , etc. The analysis of NN scattering has been traditionally carried out by a decomposition of the scattering amplitude in partial waves. For this amplitude the partial-wave expansion in this case reads: s Mm ,m (θ ) =

1   4π(2L + 1)YmL −m (θ, 0) 2ik J,L ,L



J,S × C(L J S|m − m , m , m)i L−L (SL,L − δL ,L )

(93)

C(LJ S|0, m, m) ,

where S is the unitary coupled channel S-matrix and the C s are Clebsch-Gordan J,S coefficients. Denoting the phase shifts as δL,L , for the singlet (S = 0, L = L = J ) and triplet uncoupled (S = 1, L = L = J ) channels, the S matrix is simply e2iδL , in the triplet-coupled channel (S = 1, L = J ±1, l = J ±1) it reads in the so-called nuclear bar convention: JS

 SJ =

J,1

e2iδJ −1 cos 2εJ J,1

J,1

iei(δJ −1 +δJ +1 ) sin 2εJ

J,1

J,1

iei(δJ −1 +δJ +1 ) sin 2εJ J,1

e2iδJ +1 cos 2εJ

 ,

(94)

with εJ the mixing angle. Because of unitarity, in the reduced subspace, one has that SJ S = (MJ S − i1)(MJ S + i1)−1 with (MJ S )† = MJ S a Hermitian coupled channel matrix (also known as the K-matrix). As already mentioned, the main advantage is that for finite range interactions of range a, one expects the partial wave sum to be truncated at about Lmax + 1/2 ∼ ka. At low energies for |p| ≤ mπ /2, the scattering amplitude is analytic and one has the Taylor expansion (Pavon Valderrama and Ruiz Arriola 2005; Navarro Pérez et al. 2016b): ∞

 1 JS −1 J S )l,l + (r0 )Jl,lS p2 + (v2n )Jl,lS p2n . pl+l +1 Ml,l (p) = −(α 2

(95)

n=2

These parameters have been evaluated from different data analyses and show a large degree of universality (Navarro Pérez et al. 2016b).

43 NN Experiments and NN Phase-Shift Analysis

1677

Mixed States and Density Matrix For simplicity, in the previous description fixed, incoming energy and outgoing angle scattering have tacitly been assumed. Neither of those properties are strictly fulfilled in experiment, and in general one deals with mixed states which are best described by the density matrix. A proper treatment in the spinless case can be first found in a paper by Newton (1979) where it is shown that in a multiple scattering event, the final outgoing state can be taken as the initial incoming state of the next collision. Assuming this issue is properly handled here, pure energy and angle states but mixed spin states will be regarded, and the main features will be briefly reviewed (a comprehensive account can be found, e.g., in Blum 2012 and Glöckle 1983.) The most general initial pure energy and spin mixed states of two spin 1/2 particles is described by a 4-dimensional self-adjoint density matrix operator, ρ12 = † ρ12 , and which can be expanded in the complete basis of 16 self-adjoint operators 1, σ 1 , σ 2 , σ 1 σ 2 given by 1 ρ12 = ρ1 ρ2 + σ 1 Cσ 2 , 4

(96)

where ρi = (1 + σ i · Pi )/2 is the reduced single particle two-dimensional density matrix, and Pi its polarization (tensor product is overunderstood, ρ1 ρ2 ≡ ρ1 ⊗ ρ2 ). The correlation component is described by C = C ∗ is a three-dimensional real matrix, however, Cij = Cj i , with coefficients indexed after an orthonormal system which is chosen for every particle independently. For instance for the same choice (x, y, z) one has Cxx , Cxy , Cxz etc. From the properties of the Pauli matrices, one has the normalization condition Tr(ρ12 ) = 1 with Tr1 = 4 the full spin trace, and also P1 = σ 1  = Tr(σ 1 ρ12 ) ,

(97)

P2 = σ 1  = Tr(σ 2 ρ12 ) ,

(98)

Cij = σ 1i σ2j  = Tr(σi1 σj 2 ρ12 ) − Pi1 Pi2 .

(99)

Thus, the general NN mixed state requires 16 independent numbers, P1 , P1 , and C. With these definitions independent single particle mixed states correspond to ρ12 = ρ1 ρ2 , i.e., Cij = 0. The measurement of polarization is a single particle operation which requires counting individual states corresponding to particle 1 or 2 with two opposite spin orientations. For the case of, e.g., the z-direction, one has exp

Pz exp

=

N↑z − N↓z , N↑z + N↓z

(100)

≤ 1. The inequalities saturate their extreme values Pz = clearly −1 ≤ Pz ±1 when states are fully polarized, i.e., | ↑z  or | ↓z  respectively. In general,

1678

E. R. Arriola and R. N. Pérez

however, the measurement in the x and y directions becomes necessary so that in practice for a stationary beam, they can be measured sequentially one after another. Experimental uncertainties include besides systematic estimates the conventional statistical binomial fluctuations of different counting rates. The determination of spin correlations in a given two-particle state requires a coincidence measurement where both spin orientations of the two particles are detected, so that exp

Cij =

N↑↑ + N↓↓ − N↑↓ − N↓↑ , N↑↑ + N↓↓ + N↑↓ + N↓↑

(101)

where ↑ and ↓ are the spin directions in the chosen quantization axis for any of the scattered particles (Kanellopoulos and Brown 1957; Brown and Kanellopoulos 1957). Clearly, both the polarizations and the correlations depend on the angle and the energy.

NN Observables For unpolarized initial and final states, the differential cross section, denoted as I0 , becomes: 1   dσ = |Mm 1 ,m 2 ;m1 ,m2 (kˆ f , kˆ i )|2 d 4 m ,m 1

2

m1 ,m2

 1  = I0 = tr MM † = |a|2 + |m|2 + 2|c|2 + 2|g|2 + 2|h|2 . 4

(102)

Using the general decomposition of Eq. (96) for the density matrix 1 (1 + σ 1 · P1i + σ 2 · P2i + σ 1 Ci σ 2 ) 4  I0  1 + σ 1 · P1f + σ 2 · P2f + σ 1 Cf σ 2 → ρf = 4  = a + mσ 1 · nσ 2 · n + (g − h)σ 1 · mσ 2 , m + (g + h)σ 1 · lσ 2 · l  +c(σ 1 + σ 2 ) · n ρi =

1 (1 + σ 1 · P1 + σ 2 · P2 + σ 1 Ci σ 2 ) 4 × [a + mσ 1 · nσ 2 · n + (g − h)σ 1 · mσ 2 , m + (g + h)σ 1 · lσ 2 · l ×

+ c(σ 1 + σ 2 ) · n] .

(103)

43 NN Experiments and NN Phase-Shift Analysis

1679

To get a general idea of the calculation, it is best to proceed in a more compact notation by using the self-adjoint basis of SU(4) generators Sμ with Sμ† = Sμ given   by S0 = 1, S1 = σ1x , . . . , S15 = σ1z σ2z one has Tr Sμ Sν = 4δμν and one can write the density matrix as ρ = ρμ S μ /4(repeated indices Einstein’s summation convention) so that ρμ = Sμ  = Tr Sμ ρ . The amplitude and its adjoint can then be written as M = Mμ Sμ and M † = Mμ∗ Sμ , respectively, so that the final density matrix reads ρf = ρμf S μ /4 = I0 Mν S ν ραi S α ρβif S β /4 ,

(104)

and hence the final observables are     Of = Mν∗ ραi Mβ Tr OS ν S α S β = Sα i Mν∗ Mβ Tr OS ν S α S β ,

(105)

where, for instance, O = O μ Sμ . Thus, the outcome is that one gets a linear mapping depending on the polarization of either beam or target and their correlation before and after the collision: (P1i , P2i , Ci ) → (P1f , P2f , Cf ) .

(106)

Rotational as well as parity and time-reversal invariances imply that this mapping requires adding also ki , kf with real coefficients and preserving their vector or axial nature. The general explicit expressions and comprehensive discussions can be found in Hoshizaki (1969) and Bystricky et al. (1978). Besides the differential cross section, I0 , for unpolarized beam and target in both initial and final states, the singly induced polarization in this case is given by  ∗  ˆ P0 = σ  = n2Re c (a + m) /I0 ,

(107)

which points in a perpendicular direction of the scattering plane. For illustration, in Fig. 8, P0 is shown from the 2013-Granada pp+np analysis (to be discussed later) at several energies. One can see that P0 becomes significant above 25 MeV. When particle 1 (beam) is initially polarized with P1i , one has the relations:   dσ

1 = Tr M [1 + σ 1 · P1i ]M †

d i0→00 4

P1f

= I0 (1 + P0 · P1i ) ,   1 † = σ 1 f = Tr σ 1 M [1 + σ 1 · P1i ]M 4     = nˆ P0 + DP · nˆ + lˆ Dln P1i · lˆ + Dlm P1i · m ˆ   ˆ , +m ˆ Dml P1i · lˆ + Dmm P1i · m

(108)

(109)

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E. R. Arriola and R. N. Pérez

Fig. 8 Polarization P0 and depolarization D and Dt Observables for pp (left panel) and np (right panel) for several LAB energy values 1 (solid blue), 5, 10, 25, 50 (solid read), 100, 150, 200, 250, 300 (solid black), 350 MeV

where the depolarization tensor is defined as I0 Dαβ =

1 TrMσ1α M † σ1β . 4

(110)

The polarization of particle 2 in this case is given by   1 P2f = σ 2 f = Tr σ 2 M [1 + σ 1 · P1i ]M † 2     = nˆ P0 + Knn P1i · nˆ + lˆ Kln P1i · lˆ + Klm P1i · m ˆ   ˆ , +m ˆ Kml P1i · lˆ + Kmm P1i · m

(111)

43 NN Experiments and NN Phase-Shift Analysis

1681

with the polarization transfer tensor defined as I0 Kαβ =

1 TrMσ1α M † σ2β 4

(112)

Another interesting observable is spin correlation tensor for initial unpolarized particles I0 Cαβ =

1 TrMM † σ1α σ2β . 4

(113)

NN Experiments Possible NN Experiments For spinless particles the measurement of the differential cross section in the elastic regime as a function of the scattering angle and for a given fixed energy allows to reconstruct the full scattering amplitude. When spin is added to either beam or target, it becomes necessary to determine the corresponding cross sections for the given spin orientations. Finally, when both beam and target have spin, the number of fully polarized projectiles and targets is thus needed in principle. This makes a total of 16 preparations of the initial state (beam and/or target) which together with the 16 possible measurements of the final state (beam and/or target) gives a total number of 256 possibilities, although symmetry principles allow to reduce the number of independent parameters (Wolfenstein 1956; Faissner 1959; Bilenky et al. 1964; Ohlsen 1972). As already mentioned, expectation values depend generally on the averages Sμ Sν Sα Sβ  with S0 = 1, S1 = σ1x , . . . S15 = σ1z σ2z . Experiments are carried out in the LAB system, so observables are naturally defined in that system where the orthogonal basis is given by the tree vectors kˆ ≡ L ˆ ≡ p ˆL ˆL ˆL ˆL ˆ ∧ pˆ whose relation with the CM is pL i /|pi |, n f ∧p i /|p f ∧p i | and sˆ ≡ n given by ( nˆ is invariant) θ θ ˆ , kˆ = cos ˆl − sin m 2 2 θ θ ˆ . sˆ = sin ˆl + cos m 2 2

(114)

Therefore one may rewrite the SU(4) basis Sμ also in the CM or LAB orthogonal ˆ etc. system, say σ1n ≡ σ 1 · n, Depending on the n, s, k polarized or not 0 case for beam target before and after collision, one introduces the notation for polarization P00n0 , P0n00 , analyzing power A00n0 , the spin correlation parameters A00nn ,A00ss , A00sk , A00kk , the spin transfer parameters K0nn0 , K0ss0 , K0sk0 , the depolarization parameters D0n0n ,D0s0s ,D0s0k , and the three-spin parameters N0nkk ,N0skn ,N0ssn , N0sns .

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E. R. Arriola and R. N. Pérez

In particular, one has the quantities for initially polarized beam D ≡ Dnn , R ≡ Dls =, A ≡ Dlp , R ≡ Dms , A ≡ Dmp which describe the change in polarization in the normal direction to scattering plane (D), and the possible spin nonflip (R, R )and spin flip (A, A ) in the scattering plane: I0 D = |a|2 + |m|2 + 2|c|2 − 2|g|2 − 2|h|2 (115)   θ  θ 2 I0 R = sin |a| − |m|2 − |g − h|2 + |g + h|2 + cos Im 2c(a ∗ − m∗ ) 2 2 (116)   θ  θ 2 I0 R = cos |a| − |m|2 − |g − h|2 + |g + h|2 − sin Im 2c(a ∗ − m∗ ) 2 2 (117)    θ  θ I0 A = sin |a|2 − |m|2 − |g − h|2 + |g + h|2 + cos Im 2c(a ∗ − m∗ ) 2 2 (118)    θ  θ |a|2 − |m|2 − |g − h|2 + |g + h|2 + cos Im 2c(a ∗ − m∗ ) I0 A = sin 2 2 (119) These parameters are illustrated in Figs. 8 and 9 for several LAB energies. Similar expressions follow for the quantities Dt , At , A t , Rt , Rt when the target is polarized. ˆ n} ˆ , one has the correlation coefficients Finally, the orthogonal CM basis {ˆl, m, Cll = (g + h)a ∗ + (a − m)g ∗ + (a + m)h∗ + (h − g)m∗   Clm = Cml = 2i hc∗ − ch∗ Cll = (g − h)a ∗ + (a − m)g ∗ − (a + m)h∗ − (g + h)m∗ Cnn = ma ∗ + 2cc∗ − 2gg ∗ + 2hh∗ + am∗ Cln = Cnl = Cmn = Cnm = 0

(120)

Real NN Experiments Experiments dealing with polarized nucleons are far more involved than differential or total cross sections, particularly because at low energies the generated or induced degree of polarization becomes small. Wolfenstein (1956) proposed triple NN scattering experiments in order to study polarization. Indeed, if the interaction depends on spin, any unpolarized beam and target results after the collision in partially polarized scattered final states. This can be done, for instance, with carbon targets which act as polarimeters achieving a high degree of polarization and corresponding to the preparation of the initial state. A further and secondary

43 NN Experiments and NN Phase-Shift Analysis

1683

Fig. 9 Differential beam polarization observables for pp (left panel) and np (right panel) for several LAB energy values 1, 5, 10, 25, 50, 100, 150, 200, 250, 300, 350 MeV

scattering event tests the polarization properties of the nucleon-nucleon interaction. A third scattering event allows to analyze polarization by measuring the relevant polarization observables in the final state. According to this method, NN scattering

1684

E. R. Arriola and R. N. Pérez

observables can be divided into single, double, and triple scattering experimental parameters. Obviously, if polarized beams and targets are available, single scattering suffices. The cross section for a unpolarized beam is a single scattering parameter, whereas the polarization of the beam after having scattered with the target is a double scattering parameter. The depolarization D and the R, A, R , A rotation parameters are triple scattering parameters. The analysis of the recoiling target allows to determine similarly Dt , Rt , At , Rt , A t . Coincidence measurements of the scattered beam state and the recoiling target allow to determine correlations from initially unpolarized beam and target.

NN Database The earliest NN scattering experiments date back almost a century ago. The compilation by Hess by 1958 already covers a great deal of different copious experiments which can be classified in (1) total cross sections, (2) differential cross sections, and (3) polarization experiments. The first complete set of measurements (Chamberlain et al. 1957) was subsequently analyzed by means of a partial-wave analysis (Stapp et al. 1957) at fixed 310 MeV LAB energy and eight different solutions were found displaying the expected ambiguites of such an analysis, although five were discarded by invoking other features. A compilation of pp and np data up to 2013 has been made in the Granada analysis (Navarro Pérez et al. 2013d) and is illustrated by the abundance plots in Fig. 10 (see top and bottom left panels) where every point represents a measured observable (cross section, polarization, etc.) of a total of about 8000 pp+np data. The disparity in the data calls for a multiple energy analysis. Furthermore, within such a large number of data, it is almost unavoidable to encounter inconsistency between different experiments, and hence a selection strategy becomes crucial in

Fig. 10 Abundance plots for pp (top panels) and np (bottom panels) scattering data. Full data base (left panel). Standard 3σ criterion (middle panels). Self-consistent 3σ criterion (right panels). Accepted data (blue), rejected data (red), and recovered data (green) are shown

43 NN Experiments and NN Phase-Shift Analysis

1685

order to obtain a statistically acceptable fit to a set of mutually compatible data. Under such circumstances one may wonder which experiment or datum including exp its error estimate, Oi ± ΔOi , is correct. For observables, say the differential cross section at a given energy and angle, O = σ (θ, E), which are measured by different experiments or in different laboratories, two data are roughly inconsistent if their values do not overlapp within uncertainties. To elaborate on this further, one may assume two experiments which yield the measurements Oexp1 ± ΔOexp1 and Oexp2 ± ΔOexp2 . If the theoretical estimate is Oth , one has 

Oexp1 − Oth χ = ΔOexp1 2

2



Oexp2 − Oth + ΔOexp2

2 (121)

Minimizing respect to Oth , one gets: 2 = χmin

(Oexp1 − Oexp2 )2 2 2 ΔOexp1 + ΔOexp2

(122)

 2 2 , in which which becomes larger than 1 for |Oexp1 − Oexp2 | ≥ ΔOexp1 + ΔOexp2 case one has two inconsistent measurements. The important question is whether both measurements are wrong or just only one. The term wrong here does not necessarily mean an incorrect measurement; it suffices if one or both errors ΔOexp1 and ΔOexp2 are unrealistically small. In case of a discrepancy, one may reanalyze the experiment or simply ask the experts, an unfeasible strategy for the experiments performed in the time span of 80 years comprising the analysis. The advantage of the statistical method is that, for a large number of experiments, the systematic errors are also randomized and one may rule out some experiments in a kind of majority vote argument. However, if the energy or the angle are close but experimentally distinguishable, it is not obvious how their possible inconsistency may be declared. Establishing data inconsistencies may be eased with the help of a model where extrapolation from one measure to another becomes possible. To be more precise, if LAB-1 measures σ1 (θ1 , E1 ) ± Δσ1 (θ1 , E1 ) and LAB-2 measures σ2 (θ2 , E2 ) ± Δσ2 (θ2 , E2 ) are these measurements incompatible? A given model will produce a function σ (θ, E) so that both measurements become intertwined through the model parameters. Actually, by a simultaneous fit of the given model to all data, one can establish if the full database is built from mutually consistent data. Within this scenario it should be reminded that the main purpose of a fit is to determine the true values of certain parameters with a given and admissible confidence level. Over the years the selection of NN data has been routinely applied in different forms (Arndt and MacGregor 1966b; Mac Gregor et al. 1968; Arndt and Roper 1972). In the Nijmegen 3σ criterion (Stoks et al. 1993), one searches for a maximization of experimental consensus by excluding data sets inconsistent with the rest of the database within the fitting model. A self-consistency criterion for data selection was proposed by Gross

1686

E. R. Arriola and R. N. Pérez

and Stadler (2008) and implemented by the Granada group (Navarro Pérez et al. 2013c). It consists of the following steps: 1. 2. 3. 4. 5.

Fit to all data. If χ 2 /ν < 1 you can stop. If not proceed further. Remove data sets with improbably high or low χ 2 (3σ criterion) Refit parameters for the remaining data. Reapply 3σ criterion to all data. Repeat until no more data are excluded or recovered.

As already mentioned, a particular model becomes indispensable to implement such a data selection procedure, but this raises the problem on the possible model bias discussed in the spinless case. Hence, the goal is to look for the least biased model which might naturally gather the maximal number of available data. In the next section, the models used for such an analysis will be motivated and described.

Phenomenological Analysis of NN Experiments The NN Potential One of the good reasons to discuss t he NN scattering problem is to design suitable NN interactions which can be used in few and many body calculations and, in particular, address the problem of binding in atomic nuclei. In the most popular Hamiltonian approach, the interaction is characterized by a potential. Moreover, provided the potential fulfills a spectral representation as a superposition of Yukawa one-pion exchange (OPE) form, the scattering amplitude can be shown to possess analytical properties in both the CM energy and the scattering angle (or equivalently Mandelstam s, t variables). This also guarantees the smoothness in a given energy interpolation, a feature which proves useful in fitting data. For LAB energies below 350 MeV relativity plays a small but significant role (relativistic Coulomb corrections will be needed). The proper incorporation of relativity requires also to take into account retardation effects and in practice many schemes are possible. The standard Bethe-Salpeter equation (Salpeter and Bethe 1951) is an exact four-dimensional integral equation which besides facing technical difficulties poses theoretical issues in practice since any finite truncation of irreducible diagrams generates spurious effects and inconsistencies in the amplitude or generate fake results in the heavy-light limiting case. One may consider instead three-dimensional reductions of the Blankenblecker-Sugar (1966), Gross (1969), or Kadyshevsky form (1968), among the many possible schemes, fulfilling or focusing on some special properties. This relativistic ambiguity would come in addition to several ones discussed below. The concept of a potential is essentially nonrelativistic and the procedure to obtain it is to match perturbatively the nonrelativistic quantum mechanical potential to the same scattering amplitude obtained in quantum field theory, namely,

43 NN Experiments and NN Phase-Shift Analysis

Born fQFT (θ, E) = −

2μ 4π



1687

d 3 xeik ·x V (x, p)e−ik·x

(123)

where the on-shell condition is understood k = k = p. This already incorporates an ambiguity, since one may add terms which vanish on the mass shell. In the static limit of heavy nucleons, the ambiguity disappears and the potential deduced from field theory becomes local, which is of great practical advantage since one obtains a Schrödinger differential equation, for which very efficient methods exist. This efficiency is ultimately very convenient since the fitting least squares minimization process to be described below requires multiple evaluations of the χ 2 function.

NN Potential Components Assuming isospin invariance, the most general form of the NN interaction can be written in momentum space as (Okubo and Marshak 1958) V (p , p) = VC + τ 1 · τ 2 WC + [ VS + τ 1 · τ 2 WS ] σ 1 · σ 2 + [ VLS + τ 1 · τ 2 WLS ] (−iS · (q × P) ) + [ VT + τ 1 · τ 2 WT ] σ 1 · q σ 2 · q   + VQ + τ 1 · τ 2 WQ σ 1 · (q × P ) σ 2 · (q × P ) + [ VP + τ 1 · τ 2 WP ] σ 1 · P σ 2 · P ,

(124)

where p and p denote the final and initial nucleon momenta in the CMS, respectively. Moreover, q = p − p is the momentum transfer, P = (p + p)/2 the average momentum and S = (σ 1 + σ 2 )/2 the total spin, with σ 1,2 and τ 1,2 the spin and isospin operators, of nucleon 1 and 2, respectively. For the on-shell situation, Vi and Wi (where i is equal to C, S, LS, T , Q or P ) can be expressed as functions of q = |q | and p = |p | = |p |, only. Furthermore, as pointed out in Okubo and Marshak (1958), the terms corresponding to VP and WP can be rewritten in terms of other operators provided particles are on-shell, i.e., P · q = p 2 − p2 = 0. However, this is in general not the case. For instance, the exchange of an A1 meson does produce such a contribution to order 1/MN2 .

Locality and Semilocality This previous form of the potential is too general since the functional dependence of any of the scalar functions Vi and Wi involves three variables p2 , p 2 , p·p . However, in the static limit, i.e., for infinitely heavy nucleons, MN → ∞, on physical grounds one expects that the functional dependence reduces to just one single variable q2 = (p − p )2 where q is the momentum transfer. Thus, mathematically one has lim V (p , p) →

MN →∞





d 3 xei(p−p )·x V (x) ,

(125)

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E. R. Arriola and R. N. Pérez

which corresponds to a local potential. In this limit the potential components can be written as Fourier transformations of radial functions, say Vi (r) and Wi (r), which are consistent with solving Schrödinger equation in configuration space. The identification of local potentials with static energies, VN N (r) = EN N (r) − 2MN + O(MN−1 ), is particularly appealing since some of the essential long-range effects (see below) can only be handled efficiently in configuration space. Of course, nucleons are not infinitely heavy, and there will be some hopefully small deviations from this locality which is usually termed semilocal potentials. In the so-called Argonne basis, the potential containing at most two powers of momentum can be written as V =

21 

On Vn (r) ,

(126)

n=1

where the operators On contain the linear momentum operator only through the angular momentum operator L = r ∧ p and are matrix-multiplicative when acting on the partial wave basis, i.e., On YJ LSM (ˆr ) =

 S (On )JL,L r ). YJ L SM (ˆ

(127)

L

This can be generalized to higher orders in p or equivalently in L. Here, the first 14 operators are the same charge-independent ones used in the Argonne v14 potential and are given by O n=1,14 = 1, τ1 ·τ2 , σ1 ·σ2 , (σ1 ·σ2 )(τ1 ·τ2 ), S12 , S12 (τ1 ·τ2 ), L·S, L·S(τ1 ·τ2 ), L2 , L2 (τ1 ·τ2 ), L2 (σ1 ·σ2 ), L2 (σ1 ·σ2 )(τ1 ·τ2 ), (L·S)2 , (L·S)2 (τ1 ·τ2 ) .

(128)

These 14 components are denoted by the abbreviations c, τ , σ , σ τ , t, tτ , ls, lsτ , l2, l2τ , l2σ , l2σ τ , ls2, and ls2τ . The remaining terms are O n=15,21 = T12 , (σ1 ·σ2 ), T12 S12 T12 , (τz1 + τz2 ) , (σ1 ·σ2 )(τz1 + τz2 ) , L2 T12 , L2 (σ1 ·σ2 )T12 .

(129)

These terms are charge dependent and are labeled as T , σ T ,tT , σ τ z, l2T , and l2σ T . The matrix elements of these potentials in the partial wave basis can be found, e.g., in Navarro Pérez et al. (2013a). The advantage of the Argonne basis is that the problem may directly be handled in configuration space yielding to matrix-like Schrödinger equation with one or two components. The resulting coupled channel differential equations in the partial wave basis and the relation of their asymptotic form to the S-matrix of Eq. (94) are listed,

43 NN Experiments and NN Phase-Shift Analysis

1689

e.g., in Ruiz Arriola et al. (2020) (see, e.g., Goldberger and Watson (1964) for a derivation including also Coulomb interactions).

Long-Range Effects The hadronic QFT calculable contribution is separated into two pieces, the strong (pion exchange) piece and the purely EM piece: VQFT = Vπ (r) + VEM (r) .

(130)

The charge-dependent OPE potential in the long-range part of the interaction can be obtained from field theory by evaluating the Feynman diagrams included in Fig. 11 and is the same as the one used by the Nijmegen group on their 1993 PWA (Stoks et al. 1993) and subsequent analyses. One defines the combinations: fp2 = fπ 0 pp fπ 0 pp ,

(131)

f02 = −fπ 0 nn fπ 0 pp ,

(132)

2fc2 = fπ − pn fπ + np .

(133)

The relevant relationships between the pseudo-scalar pion coupling constants, gπ N N , and the pseudo-vector one, fπ N N , are given by gπ2 a N N = 4π



M N + MN mπ +

2 fπ2a N N ,

(134)

where N, N = n, p and π a = π 0 , π ± (the factor mπ ± is conventional). Thus, one may define g02 , gc2 , gp2 , and gn2 . Usually, the charge symmetry breaking is restricted

Fig. 11 Feynman diagrams contributing to the charge-dependent one-pion √ exchange interaction. The couplings are assumed to be in the isospin limit g = gπ 0 pp = gπ ± np / 2 = −gπ 0 nn

1690

E. R. Arriola and R. N. Pérez

to mass differences by setting fp = −fn = fc = f ; see Table 1. Then, one has  Vm,OPE (r) = f

2

m mπ ±

2

 1  m Ym (r)σ 1 · σ 2 + Tm (r)S1,2 , 3

(135)

being f the pion coupling constant, σ 1 and σ 2 the single nucleon Pauli matrices, S1,2 the tensor operator, and Ym (r) and Tm (r) the usual Yukawa and tensor functions: e−mr , mr  −mr  e 3 3 + . Tm (r) = 1 + mr mr (mr)2

Ym (r) =

(136)

Charge dependence is introduced by the difference between the charged mπ ± and neutral mπ 0 pion mass by setting VOPE,pp (r) = Vmπ 0 ,OPE (r), VOPE,np (r) = −Vmπ 0 ,OPE (r) + (−)(T +1) 2Vmπ ± ,OPE (r).

(137)

The neutron-proton electromagnetic potential includes only a magnetic moment interaction: αμn VEM,np (r) = VMM,np (r) = − 2Mn r 3



 μp S1,2 L·S , + 2Mp μnp

(138)

where μn and μp are the neutron and proton magnetic moments, Mn is the neutron mass, Mp the proton one and L·S is the spin orbit operator. The EM terms in the proton-proton channel include one- and two-photon exchange, vacuum polarization, and magnetic moment: VEM,pp (r) = VC1 (r) + VC2 (r) + VVP (r) + VMM,pp (r)

(139)

where α , r αα VC2 (r) = − , Mp r 2  √ 2  1 x −1 2αα ∞ −2me rx 1+ 2 VVP (r) = e dx , 3π r 1 2x x2

VC1 (r) =

(140) (141) (142)

43 NN Experiments and NN Phase-Shift Analysis

VMM,pp (r) = −

 α  2 μ S + 2(4μ − 1)L·S . 1,2 p 4Mp2 r 3 p

1691

(143)

As it will be discussed below, these potentials are only used above a certain elementarity radius rc = 3fm, and thus form factors accounting for the finite size of the nucleon can be set to one. Energy dependence is present through the parameter 1 + 2k 2 /Mp2 α = α  , 1 + k 2 /Mp2

(144)

where k is the center of mass momentum and α the fine structure constant.

NN Anatomy and Short Distance Potential The idea of coarse graining the short range potential VS (r) already discussed before (see Eq. (64)) relies on the identification of a resolution Δr and of a short distance cutoff rc above which the meson exchange picture is assumed to hold and which becomes the actual operating range of VS (r). This amounts to sample the potential at points rn = nΔr ≤ rc and taking V (rn ) as fitting parameters. From a general nuclear physics point of view, the relevant length scales are (a) the mean interparticle separation distance d = 1.8fm as obtained from nuclear matter saturation density ρ0 = 1/d 3 = 0.17 fm−3 , (b) the Fermi momentum 1 kF = (3/2) 3 /d ∼ 250 MeV which gives a wavelength of about h¯ /kF = 0.8 fm, (c) minimal relative CM √ de Broglie wavelength corresponding to the pion production threshold λ = h¯ / MN mπ ∼ 0.5 fm, and (d) the pion Compton wavelength 1/mπ = 1.4 fm. The situation is presented pictorially in Fig. 12 suggesting that for the description of the ground state in light nuclei, both the short distance core and the role of explicit pions become marginal although, similarly to the EM piece, they turn out to be crucial for a proper extraction of the short distance piece. Scale Resolution Most high-quality fits (Stoks et al. 1993, 1994; Wiringa et al. 1995; Machleidt 2001; Gross and Stadler 2008) obtained in the past which have historically been capable of fitting their contemporary NN scattering data with χ 2 /d.o.f  1 at their time required about 40 parameters for the unknown part of the interaction. The reason for this can be understood by dissecting the anatomy of the NN interaction below pion production threshold as sketched in Fig. √ 14. The first inelastic process NN → NNπ occurs at CM momentum pCM ∼ mπ MN . This corresponds to a reduced de Broglie wavelength Δr ∼ h¯ /pmax ∼ 0.6fm, which is identifed with the shortest resolution scale. For comparison a free spherical √ wave, sin(pr), with p = 2kF mπ MN is also depicted, a case relevant for describing a back-to-back scattering of nucleons at the Fermi surface in nuclear matter. This √ is about the largest relevant p which accidentally but fortunately fulfills 2kF ∼ mπ MN . The numerical coincidence in these two scales suggests that inelastic effects NN are not expected to become key ingredients in the description of nuclear matter and finite nuclei.

1692

E. R. Arriola and R. N. Pérez

Fig. 12 Cartoon of a nucleus, displaying the size of the nucleons as compared to the typical distance to nearest neighbors and the shortest wavelength wave functions

Fig. 13 Point-like (solid-blue) and extended (dashed-red) proton-proton Coulomb interaction as a function of distance

Effective elementarity NN potential models regard nucleons as elementary and point-like particles. However, nucleons are composite and extended particles made mainly of three quarks, p = uud and n = udd; thus, one must distinguish between overlapping and nonoverlapping regions as measured by the interaction. For instance, the classical electrostatic interaction the pp potential at a distance r would be

43 NN Experiments and NN Phase-Shift Analysis

1693

Fig. 14 Anatomy of the NN interaction showing the different regions as a function of the distance (in fm) for a resolution Δr = 0.6 fm (see main text)

 Vpp,EM (r) =

d 3 r1 d 3 r2 

=

ρp (r1 )ρp (r2 ) |r1 − r2 − r|

d 3 q 4π e2 e2 2 iq·r |G (q)| e ∼ E,p r (2π )3 q2

r ≥ rc = 1.8 fm

(145)

where GE,p (q) is the proton electric form factor (which is taken as a dipole; see Fig. 13). Thus, regarding EM interaction the proton behaves as a point-like particle for r ≥ rc = 1.8 fm. This figure is, again, fortunately but accidentally comparable with the average distance separation of nucleons in nuclear matter, 1 d = (3/2) 2 /kF ∼ 1.8 fm. Number of fitting parameters For the decomposition of Eq. (64), the maximal angular momentum needed for convergence of the partial wave expansion is lmax = pmax rc = rc /Δr = N. On the other hand, the minimal distance where 2 the centrifugal barrier dominates corresponds to l(l + 1)/rmin ≤ p2 , which is rrmin = 0.7, 1.2, 1.7, 2.2, 2.7 fm for l = 1, 2, 3, 4, 5, respectively. Thus, for a given

1694

E. R. Arriola and R. N. Pérez

l ≤ lmax , one can count √ the number of points between rmin and rc sampled at a resolution Δr = 1/ MN mπ ; see Fig. 14, which means lmax = N = 3, 4, 5 for rc = 1.8, 2.4, 3. One counts partial waves according to their threshold behavior in coupled channels (Pavon Valderrama and Ruiz Arriola 2005), namely, 2S+1 LJ = O(p2L ), EJ = O(p2J ) with EJ being the tensor mixing waves. The number of parameters for an unknown interaction below rc > 2 fm and momentum p ≤ pmax = 2 fm−1 becomes: Npar (rc ) ∼ 60, 38, 21

rc = 3, 2.4, 1.8 fm

(146)

A complementary view of the problem is provided in Fig. 15 where an effective impact parameter scaling analysis explicitly shows for the NN case which partial waves become active at a given energy. The number of volume integrals As it was already mentioned in the spinless case, volume integrals are largely model independent, so that one may use them directly as independent fitting parameters. For the general case of a nonlocal potential, it can be shown that these volume integrals correspond to a polynomial expansion of the partial-wave matrix elements of the potential V (p , p), namely,

√ Fig. 15 Effective impact parameter beff = b/ 2 as a function of the LAB energy for different partial waves according to their angular momentum L and the formula bp = (L+1/2) with TLAB = 2p2 /MN . This probes the region of a short distance potential Vshort (r) which vanishes above a certain distance (see main text). The horizontal dotted lines represent the distances above which the NN potential is described by pion exchanges in the Granada analyses (Navarro Pérez et al. 2013c, d, 2014c, d, e). Here VN N (r) = V1π (r) for r > rc = 3 fm and VN N (r) = V1π (r) + V2π (r) for r > rc = 1.8 fm

43 NN Experiments and NN Phase-Shift Analysis l l VlS,J ,l (p , p) = p p

 ν

1695 S,J,(N,k)

cl ,l

(p )ν pν−2k

(147)

k

. Thus, the counting to order ν in momentum gives with real coefficients clS,J,(N,k) ,l the total number N(O(pν )) = 2, 7, 19, 41 of clS,J,(N,k) parameters for ν = ,l 0, 2, 4, 6, respectively. The expansion has a convergence radius of |p |, |p| < mπ /2, which is extended to nmπ /2 after additive inclusion of nπ exchange. Thus, for p  3mπ /2 one needs 2π exchange and just 9 coefficients (Ekstrm et al. 2013). This corresponds to take Δr ∼ 1 fm or ELAB ∼ 90 MeV.

Data Analysis The most recent statistical analysis of the full database was done by the Granada group along the lines reviewed here (about 8000) available data pp+np scattering from 1950 till 2013 below 350 MeV LAB energy. The particular form of the short range potential corresponded to a sum of delta-shells (Aviles 1972; Entem et al. 2008; Navarro Pérez et al. 2012): V (r) =



Oi Vi (r) ,

Vi (r) =



Vi (rn )Δrδ(r − rn ) ,

(148)

n

i

where rn = nΔr and Δr = h/p ¯ max = 0.6 fm. and rc = 3fm and Vi (rn ) were the fitting parameters. Using the χ 2 with normalization constants, the goal was to obtain 2 χmin = min χ 2 . Vi (rn )

(149)

A selection of data became necessary to exclude outliers according to the selfconsistent 3σ exclusion criterion. The effect of the selection criterion was to go from χ 2 /ν|all = 1.41 to χ 2 /ν|selected = 1.05 with a reduction in the number of data from NData = 8173 to NData = 6713. While this seems a drastic rejection, it is the self-consistent fit to the largest date base below 350 MeV. For this number of data, this is not a minor improvement; it makes the difference between having an estimated p-value of p = 10−20 or p = 0.68. A representative comparison of data and fits can be seen in Figs. 16 and 17 for np and pp scattering, respectively. The set of 32 scattering observables involved in the fit to the selected database are listed in Tables 4 and 5 for pp and np decomposing the contributions to the total χ 2 both in terms of the fitted observables and in different energy bins. As can be seen, the sizes of the contributions χ 2 /N are at similar levels for most observables with the qualification that observables with a considerable larger or smaller χ 2 /N are also observables with a small number of data and therefore larger statistical fluctuations are expected. This means that essentially the description of the different observables is rather homogeneous.

1696

E. R. Arriola and R. N. Pérez I0 25.8 MeV

34.1

I0 50.0 MeV

19

0.255

32.3

17

30.5

15

0.075

28.7

13

-0.015

0.165

-0.105

11

26.9 1.48

10.8

D 212.0 MeV

0.96

8.4

0.44

6

0.05

3.6

-0.09

-0.08

0.19

0.7

-0.02

0.5

-0.3

0.3

0.15

-0.58

0.1

-0.03

Rt 325.0 MeV 0

30

60

At 325.0 MeV

120

150

180

Experimental δ-shell AV18 Nijm PWA

0.51 0.33

-0.1 90

Dt 325.0 MeV

-0.23

1.2 0.26

-0.86

P 50.0 MeV

0.33

I0 324.1 MeV

-0.21 0

30

60

90

120

150

P 325.0 MeV 0

180

30

60

90

120

150

180

θc.m. [deg]

Fig. 16 Comparing potentials and some experimental np data

0.84

R 141.0 MeV

0.52

0.245

0.86

0.135

0.58

0.2

0.025

0.3

-0.12

-0.085

0.02 P 142.0 MeV

-0.195

-0.44

-0.26

0.305

0.86

0.72

0.215

0.58

0.4

0.125

0.3

0.08

0.035

0.02

Cnn 143.0 MeV

1.04

-0.055

-0.24 0

30

60

90

P 210.0 MeV 0

30

D 142.0 MeV

Experimental δ-shell AV18 Nijm PWA

R 209.1 MeV

-0.26 60

90

0

30

60

90

θc.m. [deg]

Fig. 17 Comparing potentials and some experimental pp data

Likewise, one can also break up the contributions in order to see the significance of different energy intervals; see Table 6. It is found that, in agreement with the Nijmegen analysis (see Stoks and de Swart (1993, 1995) for comparisons with previous potentials), there is a relatively large degree of uniformity in describing data at different energy bins. It is also noteworthy that the fit in the low-energy region below 2 MeV gives the largest values for χ 2 /N.

Scattering Amplitudes and Phase Shifts From an analysis of the data, the resulting pp and np phase shifts are presented in Fig. 18 for the lowest partial waves including their statistical uncertainties. The scattering amplitude components are given in terms of the real and imaginary parts of Wolfenstein parameters a, m, c, g, h in Fig. 7.

43 NN Experiments and NN Phase-Shift Analysis Table 4 Contributions to the total χ 2 for different pp observables (Navarro Pérez et al. 2017a; Ruiz Arriola et al. 2016b). The notation of Hoshizaki (1969) and Bystricky et al. (1978) is used

Table 5 Contributions to the total χ 2 for different np observables (Navarro Pérez et al. 2017a; Ruiz Arriola et al. 2016b). The notation is that of Hoshizaki (1969) and Bystricky et al. (1978)

1697

Observable

Code

Npp

2 χpp

2 /N χpp pp

dσ/d Ayy D P Azz R A Axx Ckp R Ms 0sn Ns 0kn Azx A

DSG AYY D P AZZ R A AXX CKP RP MSSN MSKN AZX AP

935 312 104 807 51 110 79 271 2 29 18 18 264 6

903.5 339.0 135.1 832.4 47.4 112.8 70.5 250.7 3.1 11.9 13.1 8.5 250.6 0.8

0.97 1.09 1.30 1.03 0.93 1.03 0.89 0.92 1.57 0.41 0.73 0.47 0.95 0.14

Observable

Code

Nnp

2 χnp

2 /N χnp np

dσ/d Dt Ayy D P Azz R Rt Rt At D0s 0k N0s kn N0s sn N0nkk A σ ΔσT ΔσL

DSG DT AYY D P AZZ R RT RPT AT D0SK NSKN NSSN NNKK A SGT SGTT SGTL

1712 88 119 29 977 89 5 76 4 75 29 29 30 18 6 411 20 16

1803.4 83.7 96.0 37.1 941.7 108.1 4.5 72.2 1.4 77.0 44.0 25.5 20.3 13.5 2.9 500.2 26.3 18.4

1.05 0.95 0.81 1.28 0.96 1.21 0.91 0.95 0.35 1.03 1.52 0.88 0.68 0.75 0.49 1.22 1.31 1.15

Nuclear Potentials At this point it is worth recalling that there are a set of ingredients which have proven to become absolutely essential for a good fit to the NN scattering data in the elastic region. Those are the long-range calculable effects such as CD-OPE, vacuum polarization, magnetic moments interaction, and relativistic Coulomb effects. This part of the interaction can in principle be fixed independently of NN scattering data just using Table 1.

Npp

103 82 92 124 111 261 152 301 882 898

Bin (MeV)

0.0−0.5 0.5−2 2−8 8−17 17−35 35−75 75−125 125−183 183−290 290−350

107.2 58.8 80.1 100.3 85.5 231.2 154.8 300.5 905.0 956.1

2 χpp

1.04 0.72 0.87 0.81 0.77 0.89 1.02 1.00 1.03 1.06

2 /N χpp pp

46 50 122 229 346 513 399 372 858 798

Nnp 88.2 92.8 151.0 183.9 324.2 559.7 445.2 381.7 841.4 808.1

2 χnp

1.92 1.86 1.24 0.80 0.94 1.09 1.12 1.03 0.98 1.01

2 /N χnp np

149 132 214 353 457 774 551 673 1740 1696

N 195.4 151.5 231.0 284.1 409.7 790.9 600.0 682.2 1746.4 1764.1

χ2

1.31 1.15 1.08 0.80 0.90 1.02 1.09 1.01 1.00 1.04

χ 2 /N |fit

1± 0.11 1± 0.12 1± 0.10 1± 0.08 1± 0.07 1± 0.05 1± 0.06 1± 0.05 1± 0.03 1± 0.03

χ 2 /N |th

Table 6 The χ 2 results of the main combined pp and np partial-wave analysis (Navarro Pérez et al. 2017a; Ruiz Arriola √ et al. 2016b) for the ten single-energy bins in the range 0 < TLAB < 350MeV. The fit χ 2 /N |fit is compared with the theoretical expectation χ 2 /N |th = 1 ± 2/N

1698 E. R. Arriola and R. N. Pérez

43 NN Experiments and NN Phase-Shift Analysis

pp

1699

np

np (b)

(a)

63

144

27 9

1

1

S0

-9

48

(e)

4 -3

16

3

3

P0

-17

-17.5

P0

-24.5

(g)

(h)

3

3

P1

-15

P1

-21 -27 27

14

21

3

10

3

P2

6

P2

(j)

2

3

D1

3

D2

15 9

(k)

(l)

3

10.8

5.4 1

6

1

D2

3.6

D2

4.2

1.53 3

(n) (q) 3

F2

-2.1

F2

-3.5

0.17

-6.3

2

(s)

2

(t)

3 1.8

-3.15

0.6 250

350 50

150

250

F3

4.2

-2.45 150

1

5.4

-1.75

0 50

(o) (r)

0.6 -0.7

-4.9

-1.05

D3

1.8

0.51 -0.35

3

3

(m) (p)

1.2

0.85

(i)

-9

18

1.19

P1

-3

-31.5

8.4

1

-31.5

-10.5 -24.5

(f)

-3.5 -10.5

-3.5 -17.5

(c) S1

80

S0

(d)

11

-10

3

112

45

350

1 0 50

(u) 150

250

350

TLAB [MeV] Fig. 18 (Color online) Phase shifts obtained from a partial-wave analysis to pp and np data and statistical uncertainties. Blue band from Navarro Pérez et al. (2013d) from a fit with fixed f 2 and red band (Navarro Pérez et al. 2017a) from a fit with charge symmetry breaking on the 3 P0 , 3 P1 , and 3 P2 partial waves and in the coupling constants f02 , fp2 , and fc2

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The short-range part is unknown and can only be obtained from fits to the NN scattering data in a more or less efficient way. Along the fitting process, it helps that the short-range parameterization provides small statistical correlations, which is an indication of a good choice of parameters. This has turned out to be the case in the coarse graining approach; the about 50 parameters Vi (rn ) turned out to be highly decorrelated in the partial-wave basis (Navarro Pérez et al. 2014a). Lack of correlations in the fitting parameters has the side benefit of speeding up the least squares minimization process. In Fig. 19 the 12 components in momentum space for the AV18 potential (Wiringa et al. 1995) in the local approximation compared to the several Granada potentials. Despite that these NN potentials produce high-quality fits to the NN database and have the same long-range contribution, their behavior is rather different. This illustrates a basic and old problem in nuclear physics which is related to the fact that in quantum mechanics, the potential is not an observable and hence is ambiguous even though the S-matrix can essentially be determined from a complete set of experimental data. The physical reason is that phase shifts only provide the NN relative wave function in the region outside the interaction range, i.e., for r a but not inside the potential. This ambiguity in the potential is a serious issue, and it is transferred dramatically to the nuclear few- and many-body problem. The current interpretation is that, in order to describe the binding of finite nuclei and nuclear matter, information on three-nucleon and perhaps on four-nucleon forces is needed.

Fig. 19 Momentum space local components of potentials as a function of the momentum transfer for several potentials: AV18 (Wiringa et al. 1995) and the Granada DS-OPE (Navarro Pérez et al. 2013c, d), DS-χTPE (Navarro Pérez et al. 2014c, d) and Gauss-OPE (Navarro Pérez et al. 2014e)

43 NN Experiments and NN Phase-Shift Analysis

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Multinucleon forces have been a worry since the early days of applications of quantum field theory in nuclear physics (Drell and Huang 1953; Fujita and Miyazawa 1957), and it is likely that the problem will be solved with the introduction of chiral symmetry ideas in nuclear physics proposed by Weinberg in the form of an effective field theory (EFT) (Weinberg 1990) (see, e.g., Bedaque and van Kolck (2002), Epelbaum et al. (2009), Machleidt and Entem (2011), and Hammer et al. (2020) for reviews) only after 30 years of theoretical discussions and massive calculations addressing this issue. The most complete analyses to date of chiral potentials are by the Idaho-Salamanca group (Entem et al. 2017) and the Bochum group (Reinert et al. 2018) which are validated by basically using the Granada 2013 database and nicely outperform the traditional potentials such as those of the Nijmegen group and AV18 with much less parameters. It is expected that in the near future, the data selection carried out so far ignoring these important theoretical developments will ultimately be undertaken incorporating the EFT-chiral approach. Many of these aspects will be discussed in later chapters in the present handbook. In this regard it is worth noting that the volume integrals discussed above and which seem to be rather universal (Navarro Pérez et al. 2016a) are used as primary fitting parameters in the EFT-chiral approach and termed counterterms there. This terminology originates from the concept of renormalization in QFT, which generally imply a dependence on a finite renormalization scale, Λ. In this particular case of NN interaction, the renormalization scale corresponds to the largest momentum used in the fit, Λ = pmax which was taken to be 400MeV. Obviously, the fitting parameters change with pmax , and consequently the corresponding volume integrals depend on Λ (Navarro Pérez et al. 2013b). The complementarity of this renormalization and the idea of coarse graining implemented in the partialwave analysis discussed here has also been discussed and the quantitative relation Λ = π/2rc with rc the short distance cutoff derived (Entem et al. 2008) (see also Ruiz Arriola (2016) for a study in the context of symmetries in the Λ  mπ limit).

The Pion Exchange Potentials The original Yukawa potential (Yukawa 1935) identified the pion as the exchanged particle which becomes responsible for the short- range interaction of nuclear forces but did not predict the coupling f 2 which appears in Table 1. The Feynman diagram depicted in Fig. 11 suggests that the vertices appearing can also be combined to describe the π N → π N process. During many years it was believed that this was the most natural and possibly precise way to determine f 2 , perhaps because in π N scattering the pion becomes a real particle. Experimentally, however, one can only consider scattering of charged pions π + so that not all the vertices in Fig. 11 are probed. Long after a pioneering determination (Signell 1960), the Nijmegen group analysis suggested that NN scattering might be an advantageous way to determine the coupling even though the neutral and charged exchanged pions are virtual (de Swart et al. 1997) (see, e.g., Matsinos (2019) for a historic account) yielding a recommended value of f 2 = 0.0750(9). The 2% accuracy depends on the number of data which was about 4000 np+pp. For a large number of data, one

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√ roughly expects a relative accuracy of Δg/g ∼ 1/ N . This point has been analyzed by the Granada group with about 7000 np+pp data (Navarro Pérez et al. 2017b) yielding f 2 = 0.0763(1) which becomes 0.2% in relative terms with χ 2 /ν = 1.04. In Fig. 20 a chronological recreation of fp2 and f 2 determinations of the Granada group is shown (Ruiz Arriola et al. 2016a). At any rate the NN data of the complete 3σ -consistent database measured up to a given year was considered, and it expectedly resembles the historic plot (Sainio 1999). A discussion on discriminating the differences of np and pp couplings on coupling constants for neutral and charged pions (see Fig. 11) was promoted by the Nijmegen group (de Swart et al. 1997), answered in the affirmative by the Granada group (Navarro Pérez et al. 2017b), and contested by the Bochum group more recently (Reinert et al. 2021). While the global fit is an indirect verification of the CD-OPE potential, it is instructive to provide a more direct evidence by explicitly varying the pion masses away from their input physical values, as illustrated in Fig. 21. There is a sharp

0.084

fp2 from pp data f from N N data Nijmegen determinations

0.082

2

0.08 0.078 0.076 0.074 0.072 0.07

1960

1970

1980

1990 Year

2000

2010

Fig. 20 (Color online) Chronological recreation of pion-nucleon coupling constants determinations from 3σ selected NN data compared to several Nijmegen determinations

Fig. 21 Total χ 2 /N as a function of the neutral and charged pion masses taken as variables

43 NN Experiments and NN Phase-Shift Analysis

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minimum of the χ 2 observed for the physical values of the masses; the neutral pion mass is better determined due to the high accuracy pp scattering data, whereas the charged pion mass is just a charged exchange contribution to the pn scattering (see Fig. 11). Therefore, the direct evidence for (virtual) neutral and charged pion exchange in NN scattering is very robust. This feature calls for extending this inevitability to the shorter-range two-pion exchange, and here the evidence is theoretically compelling and welcome but not completely indispensable.

Conclusions NN experiments have been studied by a phase-shift analysis since the earliest complete sets of observables were determined by the mid- 1950s. Although one may have phase shifts without potentials, the large body of accumulated np+pp data over the years requires using in fact a NN potential to do the analysis. Benchmarking fits are possible allowing for determinations of the coupling of nucleons to pions to a high level of precision imposing strong constraints on the interaction, but for the purposes of nuclear physics and its predictive power, this is not the whole story. As it was stated in the introduction, the study of the nuclear force cannot be completely separated from the study of finite nuclei which besides the deuteron is notoriously more involved. Nonetheless triton and 3 He and 4 He are numerically solvable including three and four body forces. Current ab initio calculations have gone up to 40 Ca with a huge computing effort and correspondingly less precision in an attempt to have some predictive power between the input involving A = 2, 3, 4 and the output A ≥ 5. Uncertainties become crucial in order to validate or falsify a theoretical model of the NN interaction on the basis of the current scattering data, and in this regard one wonders which should be a balanced precision between input and output in order to validate or not the predictive power in nuclear physics. Clearly, the tight constraints imposed by the existing abundant NN data are very valuable on their own, but are perhaps too precise to resolve this particular issue. It is about time for new insights.

References S.I. Alekhin, (2000). hep-ex/0005042 R. Arndt, M. Macgregor, Methods Comput. Phys. 6 253 (1966a) R.A. Arndt, M.H. MacGregor, Phys. Rev. 141, 873 (1966b) R. Arndt, L. Roper, Nucl. Phys. B50, 285 (1972) J.B. Aviles, Phys. Rev. C 6, 1467 (1972) NNPDF, R.D. Ball et al., JHEP 05, 075 (2010). 0912.2276 P.F. Bedaque, U. van Kolck, Ann. Rev. Nucl. Part. Sci. 52, 339 (2002). nucl-th/0203055 H.A. Bethe, P. Morrison, Elementary Nuclear Theory (John Wiley and Sons, Inc., New York, 1956) S.M. Bilenky, L.I. Lapidus, R.M. Ryndin, Usp. Fiz. Nauk 84, 243 (1964) R. Blankenbecler, R. Sugar, Phys. Rev. 142, 1051 (1966) R. Blankenbecler et al., Ann. Phys. 10, 62 (1960)

1704

E. R. Arriola and R. N. Pérez

J.M. Blatt, V.F. Weisskopf, Theoretical Nuclear Physics (Springer, New York, 1952) K. Blum, Density matrix theory and applications, in Springer Series on Atomic, Optical and Plasma Physics, vol. 64 (Springer, Berlin/Heidelberg, 2012) G. Breit, Rev. Mod. Phys. 34, 766 (1962) G. Breit, R.L. Gluckstern, Ann. Rev. Nucl. Part. Sci. 2, 365 (1953) G. Breit, R.D. Haracz, High Energy Physics. EHS Burlop, Academic Press Inc., New York, vol. 1, p. 21 (1967) G. Brown, T. Kanellopoulos, Proc. Phys. Soc. Sect. A 70, 703 (1957) J. Bystricky, F. Lehar, P. Winternitz, J. Phys.(France) 39, 1 (1978) O. Chamberlain, A personal history of nucleon polarization experiments. Le Journal de Physique Colloques, 46(C2), C2-743 (1985) O. Chamberlain et al., Phys. Rev. 105, 288 (1957) G. D’Agostini, On the use of the covariance matrix to fit correlated data. Nucl. Inst. Methods Phys. Res. A: Accelerators, Spectrometers, Detectors and Associated Equipment, 346(1-2), 306–311 (1994) G. D’Agostini, Bayesian Reasoning in Data Analysis: A Critical Introduction (World Scientific, Singapore, 2003) J.J. de Swart, M.M. Nagels, Fortsch. Phys. 26, 215 (1978) J. de Swart, M. Rentmeester, R. Timmermans, PiN News Lett. 13, 96 (1997). nucl-th/9802084 J. Dobaczewski, W. Nazarewicz, P.G. Reinhard, Error estimates of theoretical models: a guide. Journal of Physics G: Nuclear and Particle Physics, 41(7), 074001 (2014). 1402.4657 S. Drell, K. Huang, Phys. Rev. 91, 1527 (1953) W.T. Eadie, D. Drijard, F.E. James, Statistical methods in experimental physics. (Amsterdam, North-Holland, 1971) A. Ekstrm et al., Phys. Rev. Lett. 110, 192502 (2013). 1303.4674 D. Entem et al., Phys. Rev. C77, 044006 (2008). 0709.2770 D.R. Entem, R. Machleidt, Y. Nosyk, Phys. Rev. C96, 024004 (2017). 1703.05454 E. Epelbaum, H.W. Hammer, U.G. Meissner, Rev. Mod. Phys. 81, 1773 (2009). 0811.1338 A.G. Escalante, R. Navarro Pérez, E. Ruiz Arriola, Phys. Rev. C 104, 054002 (2021). 2010.03885 M.J. Evans, J.S. Rosenthal, Probability and Statistics: The Science of Uncertainty (Macmillan, Toronto, 2004) H. Faissner, Polarisierte nucleonen. In: Ergebnisse der exakten Naturwissenschaften. Ergebnisse der Exakten Naturwissenschaften, vol 32. (Springer, Berlin, Heidelberg, 1959), pp. 180–346 B. Fernandez, G. Ripka, Unravelling the Mystery of the Atomic Nucleus: A Sixty Year Journey 1896-1956 (Springer, 2013) P. Fernandez-Soler, E. Ruiz Arriola, Phys. Rev. C96, 014004 (2017). 1705.06093 J.i. Fujita, H. Miyazawa, Prog. Theor. Phys. 17, 360 (1957) A. Galindo, P. Pascual, The WBK Method, in Quantum Mechanics II (Springer, 1991), pp. 88–121 R.A. Gilman, F. Gross, J. Phys. G 28 R37 (2002). nucl-th/0111015 R. Glauber, V. Schomaker, Phys. Rev. 89, 667 (1953) W. Glöckle, The Quantum Mechanical Few-Body Problem (Springer, Berlin, 1983) M.L. Goldberger, K.M. Watson, Collision Theory (John Wiley and Sons. INC. New York-LondonSydney, 1964) F. Gross, Phys. Rev. 186, 1148 (1969) F. Gross, A. Stadler, Phys.Rev. C78, 014005 (2008). 0802.1552 H.W. Hammer, S. König, U. van Kolck, Rev. Mod. Phys. 92, 025004 (2020). 1906.12122 W.N. Hess, Rev. Mod. Phys. 30, 368 (1958) N. Hoshizaki, Prog. Theor. Phys. Suppl. 42, 107 (1969) V.G. Kadyshevsky, Nucl. Phys. B6, 125 (1968). T. Kanellopoulos, G. Brown, Proc. Phys. Soc. Sect. A 70, 690 (1957) P. LaFrance, P. Winternitz, Phys. Rev. D27, 112 (1983) R.H. Landau, Quantum Mechanics, A Second Course in Quantum Theory, vol. 2 (1990) M.H. Mac Gregor, M.J. Moravcsik, H.P. Stapp, Ann. Rev. Nucl. Part. Sci. 10, 291 (1960) M.H. Mac Gregor, R.A. Arndt, R.M. Wright, Phys. Rev. 169, 1128 (1968)

43 NN Experiments and NN Phase-Shift Analysis

1705

R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989) R. Machleidt, Phys. Rev. C63, 024001 (2001). nucl-th/0006014 R. Machleidt, D. Entem, Phys. Rep. 503, 1 (2011). 1105.2919 S. Mandelstam, Phys. Rev. 115, 1741 (1959) E. Matsinos, (2019). 1901.01204 J.A. Melendez, S. Wesolowski, R.J. Furnstahl, Phys. Rev. C 96, 024003 (2017). 1704.03308 R. Navarro Pérez, E. Ruiz Arriola, Eur. Phys. J. A 56, 99 (2020) R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Prog. Part. Nucl. Phys. 67, 359 (2012). 1111.4328 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C 88, 064002 (2013a). 1310.2536; [Erratum: Phys. Rev. C 91, 029901 (2015)] R. Navarro Perez, J.E. Amaro, E. Ruiz Arriola, Few Body Syst. 54, 1487 (2013b). 1209.6269 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C88, 024002 (2013c). 1304.0895 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C88, 064002 (2013d). 1310.2536 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C 89, 064006 (2014a). 1404.0314 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Lett. B 738, 155 (2014b). 1407.3937 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C89,024004 (2014c). 1310.6972 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Error analysis of nuclear matrix elements. FewBody Syst. 1 55(8-10), 977–981 (2014d). ISO 690, 1310.8167 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C89, 064006 (2014e). 1404.0314 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, J. Phys. G 42, 034013 (2015). 1406.0625 R. Navarro Perez, J.E. Amaro, E. Ruiz Arriola, Int. J. Mod. Phys. E 25, 1641009 (2016a). 1601.08220 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, J. Phys. G 43, 114001 (2016b). 1410.8097 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C95 064001 (2017a). 1606.00592 R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C 95, 064001 (2017b). 1606.00592 R.G. Newton, Found. Phys. 9, 929 (1979) R.G. Newton, Scattering Theory of Waves and Particles, McGraw-Hill, New York, (1966) G.G. Ohlsen, Rept. Prog. Phys. 35, 717 (1972) S. Okubo, R.E. Marshak, Ann. Phys. 4, 166 (1958) J.C. Parikh, Group Symmetries in Nuclear Structure (Springer Science & Business Media, 1978) Particle Data Group, R.L. Workman, Others, PTEP 2022, 083C01 (2022) M. Pavon Valderrama, E. Ruiz Arriola, Phys. Rev. C72, 044007 (2005) R. Phillips, Rep. Prog. Phys. 22, 562 (1959) M.A. Preston, R.K. Bhaduri, Structure of the Nucleus by MA Preston and RK Bhaduri (CRC Press, Addison-Wesley, 1975) L. Puzikov, R. Ryndin, Y.A. Smorodinsky, Sov. Phys. JETP 5, 459 (1957) P. Reinert, H. Krebs, E. Epelbaum, Eur. Phys. J. A54, 86 (2018). 1711.08821 P. Reinert, H. Krebs, E. Epelbaum, Phys. Rev. Lett. 126, 092501 (2021). 2006.15360 E. Ruiz Arriola, Symmetry 8, 42 (2016) E. Ruiz Arriola, J.E. Amaro, R. Navarro Pérez, Mod. Phys. Lett. A 31, 1630027 (2016a). 1606.02171 E. Ruiz Arriola, J.E. Amaro, R. Navarro Pérez, Mod. Phys. Lett. A31, 1630027 (2016b) E. Ruiz Arriola, J.E. Amaro, R. Navarro Pérez, Front. Phys. 8, 1 (2020). 1911.09637 J. Ruiz de Elvira, E. Ruiz Arriola, Eur. Phys. J. C78, 878 (2018). 1807.10837 M. Sainio, PiN News Lett. 15, 156 (1999). hep-ph/9912337 E.E. Salpeter, H.A. Bethe, Phys. Rev. 84, 1232 (1951) C.R. Schumacher, H.A. Bethe, Phys. Rev. 121, 1534 (1961) E. Segrè, Physica 22, 1079 (1956) E. Segre et al., Nuclei and Particles: An Introduction to Nuclear and Subnuclear Physics. Dover Books on Physics (Dover Publication, 2013) P.S. Signell, Phys. Rev. Lett. 5, 474 (1960) P. Signell, The nuclear potential, In: Baranger, M., Vogt, E. (eds) Advances in Nuclear Physics (Springer, New York, NY, 1969), pp. 223–294 I.R. Simo et al., J. Phys. G 45, 035107 (2018). 1705.06522

1706

E. R. Arriola and R. N. Pérez

H.P. Stapp, T.J. Ypsilantis, N. Metropolis, Phys. Rev. 105, 302 (1957) V. Stoks, J.J. de Swart, Phys. Rev. C47, 761 (1993) V.G.J. Stoks, J.J. de Swart, Phys. Rev. C52, 1698 (1995). nucl-th/9411002 V. Stoks et al., Phys. Rev. C48, 792 (1993) V. Stoks et al., Phys. Rev. C49, 2950 (1994). nucl-th/9406039 J.R. Taylor, Introduction to Error Analysis, the Study of Uncertainties in Physical Measurements, 2nd Edition, University Science Books, 648 Broadway, Suite 902, New York, NY 10012 (University Science Books, Sausalito, 1997) S. Weinberg, Phys. Lett. B251, 288 (1990) R.B. Wiringa, V. Stoks, R. Schiavilla, Phys. Rev. C51, 38 (1995). nucl-th/9408016 L. Wolfenstein, Ann. Rev. Nucl. Part. Sci. 6, 43 (1956) H. Yukawa, Proc. Phys. Math. Soc. Jpn. 17, 48 (1935)

Phenomenology and Meson Theory of Nuclear Forces

44

Ruprecht Machleidt

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First Attempts and the Early Pion Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The One-Boson-Exchange Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding Meson-Exchange the Easy Way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sketch of the Field Theoretic Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantitative OBE Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beyond the OBE Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge Independence Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The High-Accuracy Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models for Nuclear Many-Body Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diverse Nuclear Many-Body Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic Meson-Theoretic Approaches to Nuclear Structure . . . . . . . . . . . . . . . . . . . . Nucleon-Nucleon Scattering Above the Inelastic Threshold . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1708 1708 1710 1713 1714 1716 1722 1723 1725 1728 1730 1734 1736 1736 1740 1743 1746 1756

Abstract

Historically, the first fundamental idea for a theory of nuclear forces was advanced by the Japanese physicist Hideki Yukawa in 1935. He proposed that the exchange of subnuclear particles (eventually called mesons) between nucleons would create the force. The resulting meson theory was the most popular approach to nuclear forces for more than half a century. However, with the

R. Machleidt () Department of Physics, University of Idaho, Moscow, ID, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_48

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advancement of quantum chromodynamics (QCD) to the fundamental theory of strong interactions, meson theory had to be demoted to the level of a model. Yet, among all approaches to the nuclear force, the meson model remains the most insightful as well as the most quantitative one. Nucleon-nucleon potentials based upon meson-exchange are still today in frequent use. Because of its historical and conceptual value as well as its quantitative strength, the meson model is discussed in detail in this chapter. Since meson theory and phenomenology have been entwined throughout history, we also review the phenomenological approach to nuclear forces.

Introduction The development of a proper theory of nuclear forces has occupied the minds of some of the brightest physicists for almost nine decades and has been one of the main topics of physics research in the twentieth century. The early idea was that the force is generated by the exchange of lighter particles (than nucleons) known as mesons, and this idea gave rise to the birth of a new subfield of modern physics, namely, (elementary) particle physics. The modern perception of the nuclear force is that it is a residual interaction (similar to the van der Waals force between neutral atoms) of the even stronger force between quarks, which is mediated by the exchange of gluons. Still, meson models provide the most plausible and pedagogical explanation of the force between two nucleons. Moreover, meson-exchange potentials easily generate a very quantitative reproduction of nucleon-nucleon scattering data and the properties of the deuteron, which is why, even today, they are applied regularly. Thus, meson models continue to be of significance. We, therefore, devote the present chapter to this model and the related phenomenology.

First Attempts and the Early Pion Theories After the discovery of the neutron by Chadwick in 1932, it was clear that the atomic nucleus is built up from protons and neutrons. In such a system, electromagnetic forces cannot be the reason why the constituents of the nucleus are sticking together. Therefore, the concept of strong nuclear interactions was introduced (A detailed historical account of this early phase is presented in: BrownRechenberg 1996). In 1935, a theory for this new force was started by the Japanese physicist Hideki Yukawa (1935), who suggested that the nucleons would exchange particles between each other and this mechanism would create the force. Yukawa constructed his theory in analogy to the theory of the electromagnetic interaction where the exchange of a (massless) photon is the cause of the force. In electrodynamics, the Coulomb potential φ(r) =

q 1 4π r

(1)

44 Phenomenology and Meson Theory of Nuclear Forces

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is the solution of Poisson’s equation ∇ 2 φ(r) = −q δ (3) (r) .

(2)

However, in the case of the nuclear force, Yukawa assumed that the “force-makers” (which were eventually called “mesons”) carry a mass a fraction of the nucleon mass. This would limit the effect of the force to a finite range—as necessary for the interaction among nucleons. When adding a mass term to the l.h.s. of Eq. (2) (and flipping the sign on the r.h.s. to adjust for scalar coupling), one obtains (Throughout this chapter, units such that h¯ = c = 1 are used.) (∇ 2 − m2 )ϕ(r) = g δ (3) (r) ,

(3)

the solution of which becomes ϕ(r) = −

g e−mr , 4π r

(4)

which is the scalar field generated by one nucleon. A second nucleon, with also coupling g, at a distance r from the first one will be exposed to the interaction energy V (r) = −

g 2 e−mr , 4π r

(5)

which is known as the Yukawa potential. The exponential in this expression, that is due to the mass m of the meson, restricts the potential to a finite range, which was the desired goal. For m → 0, one gets back to the form of the Coulomb potential. Similar to other theories that were floating around in the 1930s (like the Fermi-field theory (Fermi 1934)), Yukawa’s meson theory was originally meant to represent a unified field theory for all interactions in the atomic nucleus (weak and strong, but not electromagnetic). But after about 1940, it was generally agreed that strong and weak nuclear forces should be treated separately. Yukawa’s proposal did not receive much attention until the discovery of the muon in cosmic ray (Neddermeyer and Anderson 1937) in 1937 after which, however, the interest in meson theory exploded. In his first paper of 1935, Yukawa had envisioned a scalar field theory, but when the spin of the deuteron ruled that out, he considered vector fields (Yukawa and Sakata 1937). Kemmer considered the whole variety of nonderivative couplings for spin-0 and spin-1 fields (scalar, pseudoscalar, vector, axial-vector, and tensor) (Kemmer 1938). By the early 1940s, the pseudoscalar theory was gaining in popularity, since it provided a more suitable force for light nuclei. In 1947, a strongly interacting meson was found in cosmic ray (Lattes et al. 1947) and, in 1948, in the laboratory (Gardner and Lattes 1948): the isovector pseudoscalar pion with mass around 138 MeV. It appeared that, finally, the right quantum of strong interactions had been identified.

1710

R. Machleidt

Originally, the meson theory of nuclear forces was perceived as a fundamental relativistic quantum field theory (QFT), similar to quantum electrodynamics (QED), the exemplary QFT that was so successful. In this spirit, a lot of effort was devoted to pion field theories in the early 1950s (Taketani et al. 1952; Brueckner and Watson 1953; Gartenhaus 1955; Schweber et al. 1955; Moravcsik 1963). Ultimately, all of these meson QFTs failed. In retrospect, they would have been replaced anyhow, because mesons and nucleons are not elementary particles and QCD is the correct QFT of strong interactions. However, the meson field concept failed long before QCD was invented since, even when considering mesons are elementary, the theory was beset with problems that could not be resolved. Assuming the renormalizable pseudoscalar (γ5 ) coupling between pions and nucleons, gigantic virtual pair terms emerged that were not seen experimentally in pion-nucleon (π N) or nucleon-nucleon (NN) scattering. Using the pseudovector or derivative coupling (γ5 γ μ ∂μ ), these pair terms were suppressed, but this type of coupling was not renormalizable (Schweber et al. 1955). Moreover, the large coupling constant (gπ2 /4π ≈ 14) made perturbation theory questionable. Last not least, the pionexchange potential contained unmanageable singularities at short distances.

Phenomenology One way out of the above dilemma was to simply forget about theory and resort to pure phenomenology, i. e., to just describe the properties of nuclear interactions as evidenced from observation. Some such properties can be deduced from the characteristics of nuclei. Nuclei exhibit a phenomenon known as saturation: the volume of nuclei increases proportionally to the number of nucleons, once the number of nucleons is above 4. This property suggests that the nuclear (central) force is of short range (a few fm) and strongly attractive within that range (Wigner 1933). Besides this, the nuclear force has also a very complex spin-dependence. Evidence for this property first came from the observation that the deuteron (the proton-neutron bound state, the smallest atomic nucleus) deviates slightly from a spherical shape: it has a nonvanishing quadrupole moment (Kellog et al. 1939). This suggests a force that depends on the orientation of the spins of the nucleons with regard to the vector joining the two nucleons (a “tensor force”). In heavier nuclei, a shell structure has been observed which, according to a proposal by M. G. Mayer and J. H. D. Jensen (1949; 1949), can be explained by a strong force between the spin of the nucleon and its orbital motion (the “spin-orbit force”). More clear-cut evidence for the spin-dependence is extracted from scattering experiments where one nucleon is scattered off another nucleon, with distinct spin orientations. In such experiments, the existence of the nuclear spin-orbit and tensor forces has clearly been established (Chamberlain et al. 1957; Gammel and Thaler 1957). Scattering experiments at higher energies (more than 200 MeV) provide evidence that the

44 Phenomenology and Meson Theory of Nuclear Forces

1711

nucleon-nucleon interaction turns repulsive at short internucleon distances (smaller than 0.5 fm, the ‘hard core’) (Jastrow 1951). Besides the force between two nucleons (2NF), there are also three-nucleon forces (3NFs), four-nucleon forces (4NFs), and so on. However, the 2NF is much stronger than the 3NF, which in turn is much stronger than the 4NF, etc. In exact calculations of the properties of light nuclei based upon the “elementary” nuclear forces, it has been shown that 3NFs are important. Their contribution is small, but crucial. The need for 4NFs for explaining nuclear properties has not (yet) been established. Nuclear forces are approximately “charge-independent” meaning that the force between two protons, two neutrons, and a proton and a neutron is nearly the same (in the same quantum mechanical state) when electromagnetic forces are ignored. The next step then is to find a mathematical expression for the nucleon-nucleon (N N ) potential that reproduces the above two-nucleon properties and can also be used as input to quantum many-body calculations to explain the properties of atomic nuclei and their reactions. To obtain the structure for a suitable expression to represent the NN potential, Okubo and Marshak (1958) imposed the following symmetries: • • • • • • • •

Translational invariance, Galilean invariance, Rotational invariance, Space reflection invariance, Time reversal invariance, Invariance under the interchange of particle 1 and 2, Isospin symmetry, Hermiticity,

to obtain a general expression for the NN potential (in position space) that consists of six terms: V = VC + τ 1 · τ 2 WC + [ VS + τ 1 · τ 2 WS ] σ 1 · σ 2 + [ VLS + τ 1 · τ 2 WLS ] L · S + [ VT + τ 1 · τ 2 WT ] S12 (ˆr ) + [ Vσ L + τ 1 · τ 2 Wσ L ] Q12   + Vσp + τ 1 · τ 2 Wσp σ 1 · p σ 2 · p

central spin − spin spin − orbit tensor σ −L σ −p

(6)

with S12 (ˆr ) ≡ 3σ 1 · rˆ σ 2 · rˆ − σ 1 · σ 2 , Q12 ≡

1 [σ 1 · L σ 2 · L + σ 2 · L σ 1 · L] , 2

(7) (8)

1712

R. Machleidt

and ≡ r1 − r2 ≡ r/r ≡ 12 (p1 − p2 ) ≡ L1 + L2 = r × p = −ir × ∇

relative coordinate, unit vector for relative coordinate, relative momentum, total orbital angular momentum in position, space, total spin, S ≡ 12 (σ 1 + σ 2 ) (9) where r1,2 , p1,2 , L1,2 , σ 1,2 , and τ 1,2 denote position, momentum, angular momentum, spin, and isospin, respectively, of nucleon 1 and 2. The Vi and Wi , with i = C, S, LS, T , σ L, σp, are functions of r 2 , p2 , and L2 only, i.e., r rˆ p L

Vi = Vi (r 2 , p2 , L2 ) ,

(10)

Wi = Wi (r 2 , p2 , L2 ) .

(11)

Charge independence or isospin invariance requires that the potential is a scalar in the isospin space of the two nucleons. The only such scalars are 1 and τ 1 · τ 2 , which explains the isospin structures in Eq. (6). If energy is conserved in the scattering process (“on-shell”), then there are only five independent terms, and the σ − p term can be expressed as a combination of the other five terms. Note, however, that when a potential is applied in a scattering equation (Schroedinger or Lippmann-Schwinger equation), the potential goes off shell. Potentials which are based upon the operator structure Eq. (6) with functions Vi and Wi chosen such as to fit the NN data or phase shifts are called phenomenological potentials. To keep things simple, most phenomenological potentials do not include all six terms. A minimal set for a realistic potential is the central, spin-spin, spin-orbit, and tensor term. It should be noted that, because the reality of the one-pion-exchange (1PE) was recognized as early as 1956 (1956), all phenomenological NN potentials constructed after 1960 include the 1PE to describe the long-range part of the nuclear force and the lion’s share of the tensor force (needed for the deuteron). In fact, when discussing potential models, it is insightful to divide the range of the nuclear interaction into three regions, as suggested first by Taketani, Nakamura, and Sasaki in 1951 (Taketani et al. 1951): namely, a long-range region (r > ∼ 2 fm) ruled by 1PE. The short range (r < ∼ 1 fm) is typically treated phenomenologically by introducing a hard or soft repulsive core and/or applying form factors to make < the potential regular at the origin. For the intermediate region (1 fm < ∼r ∼2 fm), two-pion-exchange (2PE) appears to be the natural choice. The best known 2PE potentials from the early 1950s are the ones by Taketani, Machida, and Onuma (1952), Brueckner and Watson (1953), and Gartenhaus (1955) which, however, did not describe the NN scattering data well, mainly due to a lack of spin-orbit force.

44 Phenomenology and Meson Theory of Nuclear Forces

1713

A substantial advance occurred in 1957, when Gammel and Thaler (1957) presented the first quantitative phenomenological NN potential, which crucially included a spin-orbit term (besides central and tensor). This discovery triggered the construction of the semi-phenomenological Signell-Marshak potential (Signell and Marshak 1958), which consisted of the Gartenhaus 2PE potential (Gartenhaus 1955) plus a phenomenological spin-orbit force. In the early 1960s, these models were further improved by the Hamada-Johnston (1962) and Yale (1962) hard-core potentials. Hard- and soft-core models were presented by Reid in 1968 (Reid 1968). The Hamada-Johnston potential was used frequently in the 1960s, and the Reid soft-core model became the most popular potential of the 1970s. “Supersoft” potentials were constructed by Sprung and coworkers in the 1970s (De Tourreil et al. 1975). The construction of purely phenomenological NN potentials continued through the 1980s and 1990s with the Urbana v14 (UV14) (Lagaris and Pandharipande 1981), the Argonne v14 (AV14) (Wiringa et al. 1984), and the Argonne v18 (AV18) (Wiringa et al. 1995) potentials. The UV14, AV14, and AV18 potentials are represented in terms of the first eight operators of Eq. (6) plus the following six operators: [ 1 , τ 1 · τ 2 , σ 1 · σ 2 , (σ 1 · σ 2 ) (τ 1 · τ 2 ) ] ⊗ L2 ,

(12)

[ 1 , τ 1 · τ 2 ] ⊗ (L · S)2 .

(13)

In addition, the AV18 potential includes four operators that break charge independence.

The One-Boson-Exchange Model Another way out of the dilemma of the failure of the pion theories in the late 1950s was to introduce more mesons into the meson theory of nuclear forces than just the pion. The phenomenological evidence for a repulsive core and a strong spinorbit force lead Sakurai (1960) and Breit (1960) to postulate the existence of a neutral vector meson (ω meson), which would create both these features. Moreover, Nambu (1957) and Frazer and Fulco (1959) showed that a ω meson and a 2π P -wave resonance (ρ meson) would explain the electromagnetic structure of the nucleons. Soon after these predictions, heavier (non-strange) mesons were found in experiment, notably the vector (spin-1) mesons ρ(770) and ω(782) (Maglic et al. 1961; Erwin et al. 1961). It became now fashionable, to add these newly discovered mesons to the meson theory of the nucleon-nucleon interaction. However, to avoid the problems with multi-meson-exchanges and higher order corrections, the various mesons were now exchanged just singly (i. e., in lowest order). In addition, one would multiply the meson-nucleon vertices with form factors (“cutoffs”) to remove the singularities at short distances. Clearly, this is not QFT anymore. It is a model motivated by the meson-exchange idea. These models became known as one-bosonexchange (OBE) models, which were started in the early 1960s (Bryan and Scott

1714

R. Machleidt

1964) and turned out to be very successful in terms of phenomenology. Thus, the OBE model broke the deadlock encountered at the end of the 1950s. The popularity of OBE potentials extended all the way into the 1990s (Machleidt 1989; Stoks et al. 1994; Machleidt 2001).

Understanding Meson-Exchange the Easy Way Let’s first address the question of which mesons to consider. When deriving the nuclear force, one has generally more confidence in the predictions for the longer ranged parts (Taketani et al. 1951). Since the range, Rα , of the force created by the exchange of a meson is inversely proportional to the meson mass, mα , i.e., Rα ∼

1 , mα

(14)

one starts with the lightest mesons and moves, step by step, up to mesons with masses of about the nucleon mass. This includes essentially six mesons (Tanabashi et al. 2018), namely, π(138), η(548), σ (500), ρ(770), ω(782), and a0 (980) (cf. Fig. 1), where the numbers in parentheses denote the masses in MeV. As it turns out, η and a0 are not very important and, so, the focus here will be on • • • •

the pseudoscalar isovector pion (0− , 1), the scalar isoscalar sigma (0+ , 0), the vector isoscalar omega (1− , 0), the vector isovector rho (1− , 1),

where the parenthetical information, (J P , I ), summarizes spin J , parity P , and isospin I for each particle. The contributions to the nuclear force from these four mesons can be understood in simple terms as follows (cf. also Table 1 and Fig. 2):

Fig. 1 The one-bosonexchange model for the N N interaction. Solid lines denote nucleons and the dashed line represents mesons. The δ signifies the a0 meson

44 Phenomenology and Meson Theory of Nuclear Forces

1715

Table 1 The four most important mesons and their contributions to the major components of the two-nucleon force Meson π(138) σ (500)

ω(782)

ρ(770)

Central — strong, attractive, intermediateranged strong, repulsive, short-ranged —

Spin-Spin weak, long-ranged —

Tensor strong, long-ranged —

Spin-Orbit — moderate,

—

—

intermediate-ranged

—

—

strong , short-ranged,

— weak, short-ranged,

coherent with σ —

—

coherent with π

— moderate, short-ranged, opposite to π

—

Fig. 2 Cartoon of the nuclear force in the meson picture highlighting the mesons that contribute to the major components of the N N potential, as denoted. The central force is shown explicitly as a function of r with the vertical lines separating the three ranges into which the nuclear force is commonly subdivided (Taketani et al. 1951)

• Pion: Since the pion is a pseudoscalar particle, it has to couple to a pseudoscalar vertex to form a scalar (parity is conserved in strong interactions). A (nonrelativistic) pseudoscalar is (σ · q) with q the momentum of the pion. Since this happens for each of the two nucleons, there is (σ 1 · q) (σ 2 · q), which can be rewritten as (σ 1 · q) (σ 2 · q) = [ S12 (q) + σ 1 · σ 2 q2 ]/3 ,

(15)

where the momentum-space tensor operator has been introduced, which is defined by S12 (q) ≡ 3(σ 1 · q)(σ 2 · q) − q2 σ 1 · σ 2 .

(16)

Thus, one understands how the pseudoscalar nature of the pion leads straightforwardly to a tensor force.

1716

R. Machleidt

• Sigma: Scalar particle exchange creates attraction as first pointed out by Yukawa in 1935 (Yukawa 1935) and are derived in Eqs. (3)–(5) in terms of classical field theory. • Omega: The ω is a vector boson like the photon (except that the ω has a mass). The force mediated by a photon between like charges is repulsive and so is the force between nucleons (baryons of positive baryon number) as mediated by the ω. In tune with this, the force between nucleon and antinucleon (positive versus negative baryon number) is attractive, as is the Coulomb force between positive and negative electric charges. • Spin-orbit forces emerge from the lower components of the Dirac spinors representing nucleons (see derivations, below). Thus, spin-orbit forces are relativistic effects. These relativistic effects can be understood in analogy to the spin-orbit splitting in the hydrogen atom. But, in the nuclear case, the sign of the spin-orbit force is opposite to the atomic case, because the Coulomb force in hydrogen is attractive, while the “Coulomb force” between two nucleons is repulsive. Thus, the nuclear spin-orbit potential is negative. [Vector particle exchanges (ω, γ ) create central and spin-orbit forces of opposite sign, since spin-orbit forces are d typically ∼ 1r dr VC (r)L · S with VC the Coulomb(-like) central force and L and S the angular momentum and spin of the particle(s) involved.] • Rho: The nonrelativistic version of the (tensor) coupling of the ρ meson to the nucleon is (σ × q). Thus, for two nucleons, one has (σ 1 × q) · (σ 2 × q) = σ 1 · σ 2 q2 − (σ 1 · q) (σ 2 · q) = [ −S12 (q) + 2 σ 1 · σ 2 q ]/3 2

(17) (18)

Hence, the ρ creates a tensor force opposite to the one from the pion, while the spin-spin forces from π and ρ add up coherently. The potential created by the ρ is analogous to the interaction energy between two magnetic dipoles m1 and m2 : (m1 × ∇) · (m2 × ∇)

1 1 = −[ 3 (m1 · rˆ ) (m2 · rˆ ) − m1 · m2 ] 4π r 4π r 3 2 − m1 · m2 δ (3) (r) . (19) 3

A pedagogical introduction in the meson model can be found in Machleidt (1989).

Sketch of the Field Theoretic Derivations Yukawa’s original considerations used classical field theory. A more proper derivation should be based upon quantum field theory, as will be done now. Single meson-exchanges are represented by the Feynman diagram Fig. 3, where a meson α is exchanged between two nucleons. Calculations are performed in momentum space and in the center-of-mass (CMS) system of the two interacting nucleons,

44 Phenomenology and Meson Theory of Nuclear Forces

1717

Fig. 3 Feynman diagram describing the exchange of a boson α between two nucleons

where p1 = p = (E, p) and p2 = (E, −p) in the initial states and p1 =  p = (E  , p ) and p2 = (E  , −p ) in the final states with E = p2 + M 2 and M the nucleon mass. Spin and isospin of nucleon 1 and 2 will be denoted by σ 1,2 , and τ 1,2 , respectively. The momentum transfer between the two nucleons is q = p1 − p1 = p − p = (E  − E, p − p) = (q0 , q), which is the momentum of the exchanged meson. The Feynman diagram gives rise to the amplitude Fα (p , p) =

u¯ 1 Γ1 u1 Pα u¯ 2 Γ2 u2 , q 2 − m2α

(20)

where Pα /(q 2 − m2α ) is the propagator with mα the meson mass. The vertices, Γi (with i = 1, 2), are extracted from the meson-nucleon Lagrangians (see examples below). In practice, it is customary to multiply the vertices with form factors (“cutoffs”), which suppress high-momentum components to ensure the convergence of the scattering equation. A simple form for these cutoffs is 

Λ2α − m2α Λ2α + q2

nα ,

(21)

where the cutoff mass Λα is typically chosen in the range 1.3–2.0 GeV. The multiplication by these form factors is not explicitly shown in the formulas given below. Nucleons are represented by Dirac spinors; e.g., for incoming nucleon 1:  E+M 1 u1 (p1 , s1 ) = χs1 σ 1 ·p 2M E+M   1 ≈ σ 1 ·p χs1 

 ≈

2M

χs1 0

(22)

(23)

 ,

(24)

where the “1” inside the Dirac spinor stands for a 2 × 2 unit matrix and χs1 is a twocomponent Pauli spinor describing the spin of the nucleon (spinors characterizing the isospin are suppressed). Equation (23) is the static approximation (E ≈ M), and

1718

R. Machleidt

Eq. (24) is the nonrelativistic approximation. The generic formula for the adjoint spinor is u¯ = u† γ0 , and the normalization of our Dirac spinors is uu ¯ = 1.

One-Pion-Exchange Potential Starting point is a Lagrangian that describes the interaction of pions with nucleons (Throughout this article, notation and conventions as in Schwartz (2014) are used, except for Dirac spinors, cf. Eq. (22).) Lπ N N = −

fπ N N ψ¯ γ μ γ5 τ ψ · ∂μ φ (π ) , m ˆπ

(25)

where ψ represents the nucleon field and φ (π ) the (isovector) pion fields. m ˆπ is a scaling mass to make fπ N N [the (pseudovector) π -N coupling constant] dimensionless. It is customary to choose m ˆ π = mπ ± . However, the main point to notice here is that the scaling mass m ˆ π is independent of the mass of the pion being exchanged, which is denoted by mπ . The constructions of Lagrangians are essentially guided by the symmetries of the fields and the fact that the Lagrangian has to come out to be a Lorentz scalar. Note that the pion field is a pseudoscalar. The vertex implied by this Lagrangian is i times the Lagrangian stripped off the fields, which is for an incoming pion Γπ N N = −i

fπ N N μ fπ N N γ γ5 τ (−iqμ ) ≈ (σ · q) τ . m ˆπ m ˆπ

(26)

In the approximate expression, the lower components are left out. The second vertex in the Feynman diagram Fig. 3 has the opposite sign because the pion is outgoing (q is replaced by −q). Thus, assembling the Feynman amplitude according to Eq. (20) and multiplying with i to obtain the “potential,” one gets Vπ = iFπ = −

fπ2N N (σ 1 · q)(σ 2 · q) τ1 · τ2 , m ˆ 2π q2 + m2π

(27)

where Pπ = i is applied for the propagator and approximated the pion fourmomentum squared by q 2 = −q2 . The necessary distinction between the scaling mass m ˆ π and the mass of the exchanged meson, mπ , is now clearly demonstrated. As customary, the potential is written in terms of an operator in spin and isospin space, which is why the two-component Pauli spinors are omitted. The one-pionexchange potential can then be rewritten in terms of the tensor operator f2 Vπ = π N2N 3m ˆπ

−

q2 S12 (q) σ1 · σ2 − 2 2 2 q + mπ q + m2π

 τ1 · τ2

(28)

44 Phenomenology and Meson Theory of Nuclear Forces

=

fπ2N N 3m ˆ 2π



1719

  mπ 2 S12 (q) − 1 σ · σ − τ1 · τ2 . 1 2 q2 + m2π q2 + m2π

(29)

Thus, the potential is broken up into a spin-spin Yukawa and delta-function term and a tensor force; see Eq. (48) for the equivalent position-space expressions.

One-Sigma-Exchange Potential The simplest way to describe the interaction of a scalar-isoscalar particle (frequently called a σ or ε boson or a f0 meson (Tanabashi et al. 2018)) with a nucleon is given by the Lagrangian Lσ N N = −gσ ψ¯ ψ φ (σ ) ,

(30)

with φ (σ ) representing the field of the σ boson. The vertex is just (−igσ ). Approximating the Dirac spinors by Eq. (23), the left side of the diagram becomes u¯ 1 Γ1 u1

≈ −igσ

(σ 1 · p )(σ 1 · p) 1− 2M · 2M

 (31)

 p · p + iσ 1 · (p × p) = −igσ 1 − 4M 2

k2 − 14 q2 − σ 1 · L = −igσ 1 − , 4M 2

(32) (33)

where k = 12 (p + p) denotes the average momentum and L = −i(p × p) is the total orbital angular momentum. Performing a similar calculation for the right side of the diagram and putting everything together with Pσ = i, one obtains for the one-sigma-exchange potential g2 Vσ = iFσ = 2 σ 2 q + mσ



q2 L·S k2 − − −1 + 2M 2 8M 2 2M 2

,

(34)

where only terms up to momentum squared are retained. [S = 12 (σ 1 + σ 2 ) denotes the total spin of the two-nucleon system.] Thus, σ -exchange provides mainly an attractive central potential and and a spin-orbit force that emerges from the (relativistic) lower components of the Dirac spinors.

One-Omega-Exchange Potential Since the ω is likea heavy photon, its Lagrangian is modeled after the QED Lagrangian and, thus, reads LωN N = −gω ψ¯ γ μ ψ φμ(ω) ,

(35)

1720

R. Machleidt

with φ (ω) the field of the ω meson. The vertex is (−igω γ μ ). Keeping only the μ = 0 term of the vertex and approximating the Dirac spinors by Eq. (23), the left side of the diagram becomes

 (σ 1 · p )(σ 1 · p) − igω u¯ 1 γ 0 u1 ≈ −igω 1 + 2M · 2M

 σ1 · L , = −igω 1 − 4M 2

(36) (37)

where the momentum-dependent terms are left out. For massive vector bosons, the numerator of the propagator is Pω = −i gμν + i

qμ qν m2ω

(38)

with gμν the metric tensor. Since the nucleon current is conserved, the second term can be dropped. Thus, putting everything together, one obtains Vω = iFω =

gω2 q2 + m2ω

  L·S 1− . 2M 2

(39)

Including also the γ i (i = 1, 2, 3) terms in the vertices enhances the spin-orbit term by a factor of 3, and, thus, the final result is Vω =

gω2 2 q + m2ω

  L·S 1−3 . 2M 2

(40)

The main effect of ω-exchange is a repulsive central potential and a strong spin-orbit force.

One-Rho-Exchange Potential For the tensor coupling of the ρ-meson, the Lagrangian reads (tensor) =− LρN N

  fρ (ρ) ψ¯ σ μν τ ψ · ∂μ φ (ρ) , − ∂ φ ν ν μ 4Mˆ

(41)

where φ (ρ) represents the (isovector) ρ fields and Mˆ is a scaling mass to make fρ dimensionless. It is customary to choose Mˆ = Mp with Mp the proton mass, but the main point here to notice is that Mˆ is not necessarily the mass of the nucleon(s) involved. For an incoming meson, the vertex implied by this Lagrangian is given by (tensor)

ΓρN N

=−

fρ μν fρ μν fρ σ (qμ − qν )τ = − σ qμ τ ≈ − (σ × q) τ 4Mˆ 2Mˆ 2Mˆ

(42)

44 Phenomenology and Meson Theory of Nuclear Forces

1721

In the approximate expression, the lower components are left out. The second vertex in the Feynman diagram Fig. 3 has the opposite sign because the meson is outgoing (q is replaced by −q). Applying the vector-meson propagator Eq. (38) and keeping only the space components of the vertices, one obtains Vρ(tensor) = iFρ(tensor) = −

=−

fρ2 (σ 1 × q) · (σ 2 × q) τ1 · τ2 q2 + m2ρ 4Mˆ 2

fρ2 σ 1 · σ 2 q2 − (σ 1 · q)(σ 2 · q) τ1 · τ2 q2 + m2ρ 4Mˆ 2

fρ2 = 12Mˆ 2 fρ2 = 12Mˆ 2



S12 (q) 2 q2 σ1 · σ2 + 2 − 2 q + m2ρ q + m2ρ

(43)

(44)

τ1 · τ2

(45)



mρ 2 S12 (q) − 1 σ1 · σ2 + 2 2 τ 1 · τ 2 . (46) q2 + m2ρ q + m2ρ



Compared to the 1PE potential, the spin-spin force carries the same sign, while the tensor force is of opposite sign. The momentum-space expressions can be Fourier transformed, 1 Vα (r) = (2π )3

 d 3 q eiq·r Vα (q) ,

(47)

to yield the equivalent position-space potentials: Vπ (r) =

1 fπ2N N 3 mπ 12π m ˆ 2π  + 1+

Vσ (r) = −

gσ2 mσ 4π

+



1  mσ  2 2 M

Vω (r) =

1− 



Y (mπ r) −

3 3 + mπ r (mπ r)2



 4π (3) δ (r) σ1 · σ2 m3π

 Y (mπ r)S12 (ˆr ) τ 1 · τ 2 ,

(48)

  1  mσ 2 1  2 Y (mσ r) + ∇ Y (mσ r) + Y (mσ r)∇ 2 2 4 M 4M

1 1 + mσ r (mσ r)2



 Y (mσ r)L · S ,

(49)

   1 1 gω2 3  mω 2 Y (m mω Y (mω r) − + r)L · S , ω 4π 2 M mω r (mω r)2 (50)

1722

R. Machleidt

4π (3) 2 Y (mρ r) − 3 δ (r) σ 1 · σ 2 mρ



1 fρ2 3 Vρ (r) = m 48π Mˆ 2 ρ

   3 3 − 1+ r)S (ˆ r ) τ1 · τ2 , Y (m + ρ 12 mρ r (mρ r)2

(51)

with the “Yukawa function,” Y (x) = e−x /x ,

(52)

and the position-space tensor operator, S12 (ˆr ), given in Eq. (7). Note that in the derivations Eqs. (26)-(46) and in the expressions Eqs. (48)-(51), many approximations have been applied—for pedagogical reasons—to highlight the dominant features of each meson-exchange. However, in quantitative meson models, these approximations should not be made and the more complete expressions given in Appendix B should be used. Moreover, both vector mesons, ω and ρ, couple via a vector and a tensor coupling. However, the tensor coupling of the ω is negligible, and the vector coupling of the ρ is very weak, such that they were neglected them in the above (qualitative) considerations.

Quantitative OBE Potentials Classic examples for OBE potentials (OBEPs) are the Bryan-Scott potentials started in the early 1960s (Bryan and Scott 1964), but soon many other researchers got involved, too (Nagels et al. 1978). Since it is suggestive to think of a potential as a function of r (where r denotes the distance between the centers of the two interacting nucleons), the OBEPs of the 1960s were represented as local r-space potentials. Some groups continued to hold on to this tradition, and, thus, the construction of improved r-space OBEPs continued well into the 1990s (Stoks et al. 1994). An important advance during the 1970s was the development of the relativistic OBEP (Gersten 1971; Erkelenz 1974; Holinde and Machleidt 1975) (cf. Appendix A). In this model, the full, relativistic Feynman amplitudes for the various oneboson-exchanges are used to define the potential. These nonlocal expressions do not pose any numerical problems when used in momentum space and allow for a more quantitative description of NN scattering. The high-accuracy CD-Bonn (Machleidt 2001) as well as the Stadler-Gross (Gross and Stadler 2008) potentials are of this nature.

44 Phenomenology and Meson Theory of Nuclear Forces

1723

Beyond the OBE Approximation Historically, one must understand that, after the failure of the pion theories in the 1950s, the OBE model was considered a great success in the 1960s. On the other hand, one has to concede that the OBE model is a great simplification of the complicated scenario of a full meson theory for the NN interaction. Therefore, in spite of the quantitative success of the OBEPs, one should be concerned about the approximations involved in the model. Major critical points include: • The scalar isoscalar σ “meson” of about 500 MeV. • The neglect of all non-iterative diagrams. • The role of meson-nucleon resonances. Two pions, when “in the air,” can interact strongly. When in a relative P -wave (L = 1), they form a proper resonance, the ρ meson. They can also interact in a relative S-wave (L = 0), which gives rise to the σ boson. Whether the σ is a proper resonance is controversial, even though the Particle Data Group lists an f0 (500) or σ (500) meson, but with a width of 400–700 MeV (Tanabashi et al. 2018). What is for sure is that two pions have correlations, and if one doesn’t believe in the σ as a two-pion resonance, then one has to take these correlations into account. There are essentially two approaches that have been used to calculate these two-pionexchange contributions to the NN interaction (which generates the intermediate range attraction): dispersion theory and field theory. In the 1960s, dispersion theory was developed out of frustration with the failure of a QFT for strong interactions in the 1950s (Moravcsik 1963). In the dispersiontheoretic approach, the NN amplitude is connected to the (empirical) π N amplitude by causality (analyticity), unitarity, and crossing symmetry. Schematically this is shown in Fig. 4. The total diagram (a) is analyzed in terms of two “halves” (b). The hatched ovals stand for all possible processes which a pion and a nucleon can undergo. This is made more explicit in (d) and (e). The hatched boxes represent baryon intermediate states including the nucleon. (Note that there are also crossed pion-exchanges which are not shown.) The shaded circle stands for π π scattering. Quantitatively, these processes are taken into account by using empirical information from π N and π π scattering (e.g., phase shifts) which represents the input for such a calculation. Dispersion relations then provide an on-shell NN amplitude, which—with some kind of plausible prescription—is represented as a potential. The Stony Brook (Chemtob et al. 1972) and Paris (Cottingham 1973) groups have pursued this approach. They could show that the intermediate-range part of the nuclear force is, indeed, described about right by the 2π -exchange as obtained from dispersion integrals. To construct a complete potential, the 2π exchange contribution is complemented by one-pion and ω exchange. In addition to

1724

R. Machleidt

Fig. 4 The 2π -exchange contribution to the N N interaction as viewed by dispersion theory. Solid lines represent nucleons and dashed lines pions. Further explanations are given in the text

Fig. 5 (a) Field-theoretic model for the 2π -exchange. Notation as in Fig. 4. Double lines represent isobars. The hatched circles are π π correlations. (b) πρ contributions to the N N interaction. Further explanations are given in the text

this, the Paris potential (Lacombe 1980) contains a phenomenological short-range part for r < 1.5 fm to improve the fit to the NN data. For further details, we refer the interested reader to a pedagogical article by Vinh Mau (1979). A first field-theoretic attempt toward the 2π -exchange was undertaken by Lomon and Partovi (1970). Later, the more elaborated model shown in Fig. 5a was developed by the Bonn group (Machleidt et al. 1987). The model includes

44 Phenomenology and Meson Theory of Nuclear Forces

1725

contributions from isobars as well as from π π correlations. This can be understood in analogy to the dispersion relations picture. In general, only the lowest π N resonance, the so-called Δ isobar (spin 3/2, isospin 3/2, mass 1232 MeV), is taken into account. The contributions from other resonances have proven to be small for the low-energy N N processes under consideration. A field-theoretic model treats the Δ isobar as an elementary (Rarita-Schwinger) particle. The six upper diagrams of Fig. 5a represent uncorrelated 2π exchange. The crossed (non-iterative) twoparticle exchanges (second diagram in each row) are important. They guarantee the proper (very weak) isospin-dependence due to characteristic cancelations in the isospin-dependent parts of box and crossed box diagrams. Furthermore, their contribution is about as large as the one from the corresponding box diagrams (iterative diagrams); therefore, they are not negligible. In addition to the processes discussed, also correlated 2π exchange has to be included [lower two rows of Fig. 5a]. Quantitatively, these contributions are about as sizable as those from the uncorrelated processes. Graphs with virtual pairs are left out, because the pseudovector (gradient) coupling is used for the pion, in which case pair terms are small. Besides the contributions from two pions, there are also contributions from the combination of other mesons. The combination of π and ρ is particularly significant, Fig. 5b. This contribution is repulsive and important to suppress the 2π exchange contribution at high momenta (or small distances), which is too strong by itself. The Bonn full model (Machleidt et al. 1987) includes all the diagrams displayed in Fig. 5 plus single π and ω exchange. Besides this, the model also includes some 3π -exchanges that can be approximated in terms of π σ diagrams and 4π -exchanges of π ω type. The sum of the latter two groups is small, indicating convergence of the diagrammatic expansion. Having highly sophisticated models at hand, like the Paris and the Bonn potentials, allows to check the approximations made in the simple OBE model. As it turns out, the complicated 2π exchange contributions to the NN interaction tamed by the πρ diagrams can well be simulated by the single scalar isoscalar boson, the σ , with a mass around 550 MeV. In retrospect, this fact provides justification for the simple OBE model. To illustrate this point, in Fig. 6, phase shift predictions are shown from the Bonn (Machleidt et al. 1987)) and Paris (Lacombe 1980) potentials as well as a relativistic OBEP (Machleidt 1989). A concise comparison between some classic meson-theoretic NN potentials is provided in Table 2.

Charge Dependence By definition, charge independence is invariance under any rotation in isospin space. A violation of this symmetry is referred to as charge dependence or charge independence breaking (CIB). Charge symmetry is invariance under a rotation by 1800 about the y-axis in isospin space if the positive z-direction is associated with the positive charge. The violation of this symmetry is known as charge symmetry breaking (CSB). Obviously, CSB is a special case of charge dependence.

1726

R. Machleidt

Fig. 6 Phase parameters for np scattering from some “classic” meson-exchange models for the N N interaction. Predictions are shown for the Bonn full model (Machleidt et al. 1987) (solid line), the Paris potential (Lacombe 1980) (dashed), and a (relativistic) OBEP (Machleidt 1989) (dotted). Phase parameters with total angular momentum J ≤ 2 are displayed for laboratory energies up to 325 MeV. Symbols represent results from N N phase shift analyses

CIB of the strong NN interaction means that, in the isospin T = 1 state, the proton-proton (Tz = +1), neutron-proton (Tz = 0), or neutron-neutron (Tz = −1) interactions are (slightly) different, after electromagnetic effects have been removed. CSB of the NN interaction refers to a difference between proton-proton (pp) and neutron-neutron (nn) interactions, only. For reviews, see (Machleidt 1989; Miller et al. 1990; Li and Machleidt 1998a, b; Machleidt and Slaus 2001). The charge dependence of the NN interaction is subtle, but in the 1 S0 state it is well established. The observation of small charge-dependent effects in this state is possible because the scattering length of an almost bound state acts like a powerful magnifying glass on the interaction. The latest empirical values for the singlet (1 S0 ) scattering length a and effective range r are: N app = −17.30 ± 0.40 fm (Miller et al. 1990), N rpp = 2.85 ± 0.04 fm (Miller et al. 1990);

(53)

44 Phenomenology and Meson Theory of Nuclear Forces

1727

Table 2 Comparison of some “classic” (pre-1990) meson-theoretic nucleon-nucleon potentials Nijmegen (Nagels et al. 1978) 15

# of free parameters Theory includes OBE terms Yes 2π exchange No πρ diagrams No Relativity No χ 2 /datum for fit of world NN dataa All pp data 2.06 All np data 6.53 (np without σtot ) (3.83) All pp and np 5.12 a b

Paris (Lacombe 1980) ≈ 60

Bonn full model (Machleidt et al. 1987) 12

Yes Yes No No

Yes Yes Yes Yes

2.31 4.35 (1.98) 3.71

1.94b 1.88 (1.89) 1.90

From Machleidt and Li (1994) The pp version of the Bonn full model was applied, which is presented in the Appendix of Machleidt and Li (1994)

N ann = −18.95 ± 0.40 fm (González Trotter et al. 2006; Chen et al. 2008), N rnn = 2.75 ± 0.11 fm Miller et al. (1990);

(54)

anp = −23.740 ± 0.020 fm (Machleidt 2001), rnp = 2.77 ± 0.05 fm (Machleidt 2001).

(55)

The values given for pp and nn scattering refer to the nuclear part of the interaction as indicated by the superscript N; i. e., electromagnetic effects have been removed from the experimental values. The above values imply that charge symmetry is broken by N N ΔaCSB ≡ app − ann = 1.65 ± 0.60 fm,

(56)

N N ΔrCSB ≡ rpp − rnn = 0.10 ± 0.12 fm;

(57)

and the following CIB is observed: 1 N N (a + ann ) − anp = 5.62 ± 0.60 fm, 2 pp 1 N N + rnn ) − rnp = 0.03 ± 0.13 fm. ≡ (rpp 2

ΔaCI B ≡

(58)

ΔrCI B

(59)

In summary, the NN singlet scattering lengths show a small amount of CSB and a clear signature of CIB.

1728

R. Machleidt

The current understanding is that—on a fundamental level—the charge dependence of nuclear forces is due to a difference between the up and down quark masses and electromagnetic interactions among the quarks. A consequence of this are mass differences between hadrons of the same isospin multiplet and meson mixing. Therefore, if CIB is calculated based upon hadronic models, the mass differences between hadrons of the same isospin multiplet, meson mixing, and irreducible meson-photon exchanges are considered as major causes.

Charge Symmetry Breaking The difference between the masses of neutron and proton represents the most basic cause for CSB of the nuclear force. Therefore, it is important to have a very thorough accounting of this effect. The most trivial consequence of nucleon mass splitting is a difference in the kinetic energies: for the heavier neutrons, the kinetic energy is smaller than for protons. This raises the magnitude of the nn scattering length by 0.25 fm as compared to pp. Besides the above, nucleon mass splitting has an impact on all meson-exchange diagrams that contribute to the nuclear force. Based upon the Bonn full model for the N N interaction (Machleidt et al. 1987), a thorough calculation of these CSB effects has been conducted (Li and Machleidt 1998a). The results can be summarized as follows: For this, the total number of meson-exchange diagrams that is involved in the nuclear force is divided into several classes, and the result for each class is reported. 1. One-boson-exchange. Contributions mediated by π 0 (135), ρ 0 (770), ω(782), a0 /δ(980), and σ  (550), where the σ  (550) stands for the correlated 2π exchange in π π -S-wave [cf. Fig. 5a]. Charge symmetry is broken by the fact that for pp scattering, the proton mass is used in the Dirac spinors representing the four external legs, while for nn scattering the neutron mass is applied. The CSB effect from the OBE diagrams is very small (cf. Table 3). 2. 2π -exchange diagrams. This class consists of three groups, namely, the diagrams with NN, NΔ, and ΔΔ intermediate states [cf. Fig. 5a]. The most important group is the one with NΔ intermediate states which is shown in Fig. 7 explicitly, including all details relevant for CSB. Part (a) of Fig. 7 applies to pp scattering, while part (b) refers to nn scattering. When charged-pion-

Table 3 CSB differences of the 1 S0 effective range parameters caused by nucleon mass splitting. 2π denotes the sum of all 2π -contributions and πρ the sum of all πρ-contributions. TBE (noniterative two-boson-exchange) is the sum of 2π , πρ, and (π σ + π ω) ΔaCSB (fm) ΔrCSB (fm)

kin. en. 0.246 0.004

OBE 0.013 0.001

2π 2.888 0.055

πρ −1.537 −0.031

πσ + πω −0.034 −0.001

TBE 1.316 0.023

Total 1.575 0.028

Empirical 1.65 ± 0.60 fm 0.10 ± 0.12 fm

44 Phenomenology and Meson Theory of Nuclear Forces

1729

exchange is involved, the intermediate-state nucleon (and its mass) differs from the one of the external legs. This is one of the sources for CSB from this group of diagrams. In principle, also the mass differences between the different charge-states of the Δ-isobar would contribute to CSB. However, since the mass splittings of the Δ are not well determined (cf. Tanabashi et al. (2018)), this source of CSB is not taken into account. The 2π class of diagrams causes the largest CSB effect (cf. Table 3). 3. πρ-exchanges. Graphically, the πρ diagrams can be obtained by replacing in each 2π diagram (e. g., in Fig. 7) one pion by a ρ-meson of the same chargestate. The effect is typically opposite to the one from 2π exchange. 4. Further 3π and 4π contributions (π σ +π ω). The Bonn potential also includes some 3π -exchanges that can be approximated in terms of π σ diagrams and 4π -exchanges of π ω type. The sum of the two groups is small, indicating convergence of the diagrammatic expansion. The CSB effect from this class is essentially negligible. The total CSB difference of the singlet scattering length caused by nucleon mass splitting amounts to 1.58 fm (cf. Table 3) which agrees well with the empirical value of 1.65 ± 0.60 fm. Thus, nucleon mass splitting alone can explain the entire empirical CSB of the singlet scattering length. This is a remarkable result. The focus here has been on the S-wave; however, moderate CSB is also predicted for higher partial waves (Li and Machleidt 1998a), which is important for the explanation of the Nolen-Schiffer anomaly (Nolen and Schiffer 1969) as demonstrated in Machleidt and Muether (2001).

Fig. 7 Two-pion-exchange contributions with N Δ intermediate states to (a) pp and (b) nn scattering

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R. Machleidt

Fig. 8 ρ 0 -ω exchange contributions to the nuclear force that violate charge symmetry. The solid lines are either all protons or all neutrons leading to contributions of opposite sign. Hm represents the meson mixing interaction

Before finishing this subsection, a word is in place concerning other mechanisms that cause CSB of the nuclear force. Traditionally, it was believed that ρ 0 -ω mixing (Fig. 8) explains essentially all CSB in the nuclear force (Miller et al. 1990; Machleidt and Muether 2001). However, doubt has been cast on this paradigm. The issue is unresolved. A good summary of the controversial points of view can be found in Machleidt and Muether (2001). Finally, for reasons of completeness, it is mentioned that irreducible diagrams of π and γ exchange between two nucleons create a charge-dependent nuclear force. These contributions have been calculated to leading order in chiral perturbation theory (van Kolck et al. 1998). It turns out that to this order, the π γ force is chargesymmetric (but does break charge independence; see below).

Charge Independence Breaking The major cause of CIB in the NN interaction is pion mass splitting. Based upon the Bonn full model for the NN interaction (Machleidt et al. 1987), the CIB due to pion mass splitting has been calculated carefully and systematically in Li and Machleidt (1998b). The various classes of diagrams and their contributions to CIB will be discussed now: 1. One-pion-exchange. The CIB effect is created by replacing the diagram Fig. 9a by the two diagrams Fig. 9b. The effect caused by this replacement can be understood as follows: In nonrelativistic approximation, (For pedagogical reasons, we use simple, approximate expressions to discuss the effects from pion-exchange. Note, however, that in the calculations of Li and Machleidt (1998b), relativistic time-ordered perturbation theory is applied in its full complexity and without approximations.) 1PE is given by V1π (gπ , mπ ) = −

gπ2 (σ 1 · q) (σ 2 · q) τ 1 · τ 2 Fπ2N N (Λπ N N , |q|) 4M 2 m2π + q2

(60)

44 Phenomenology and Meson Theory of Nuclear Forces

1731

Fig. 9 One-pion-exchange (1PE) contributions to (a) pp and (b) np scattering

with M the average nucleon mass, mπ the pion mass, and q the momentum transfer. The relationship between the various types of π N N coupling constants is gπ fπ N N , = 2M mπ ±

(61)

where the scaling mass mπ ± is used for π ± as well as π 0 coupling to the nucleon to avoid the introduction of unmotivated CSB. The above expression includes a π N N vertex form factor, Fπ N N , which depends on the cutoff mass Λπ N N and the magnitude of the momentum transfer |q|. For S = 0 and T = 1, where S and T denote the total spin and isospin of the two-nucleon system, respectively, one has 01

V1π (gπ , mπ ) =

q2 gπ2 F 2 (Λπ N N , |q|), 2 + q 4M 2 π N N

m2π

(62)

where the superscripts 01 refer to ST . In the 1 S0 state, this potential expression is repulsive. The charge-dependent 1PE is then, 01

pp

V1π =

01

V1π (gπ 0 , mπ 0 )

(63)

for pp scattering and 01

np

V1π = 2 01 V1π (gπ ± , mπ ± ) −

01

V1π (gπ 0 , mπ 0 )

(64)

for np scattering. Assuming charge independence of gπ (i. e., gπ 0 = gπ ± ), all CIB comes from the charge splitting of the pion mass, which is Tanabashi et al. (2018) mπ 0 = 134.9766 MeV,

(65)

mπ ± = 139.5702 MeV.

(66)

1732

R. Machleidt

Table 4 CIB contributions to the 1 S0 scattering length, ΔaCI B , and effective range, ΔrCI B , from various components of the N N interaction. 1PE refers to one-pion-exchange, while 2π denotes the sum of all 2π -contributions, and πρ the sum of all πρ-contributions. TBE (non-iterative twoboson-exchange) is the sum of 2π , πρ, and (π σ + π ω) ΔaCI B (fm) ΔrCI B (fm)

1PE 3.243 0.099

2π 0.360 0.002

πρ −0.383 −0.006

πσ + πω 1.426 0.020

TBE 1.403 0.016

πγ −0.405 −0.004

Total 4.241 0.111

Empirical 5.62 ± 0.60 0.03 ± 0.13

Since the pion mass appears in the denominator of 1PE, the smaller π 0 -mass exchanged in pp scattering generates a larger (repulsive) potential in the 1 S0 state as compared to np where also the heavier π ± -mass is involved. Moreover, the π 0 -exchange in np scattering carries a negative sign, which further weakens the np 1PE potential. The bottom line is that the pp potential is more repulsive than the np potential. The quantitative effect on ΔaCI B is such that it explains about 60% of the empirical value (cf. Table 4). This has been known for a long time. Due to the small mass of the pion, 1PE is a sizable contribution in all partial waves including the higher ones; and due to the pion’s relatively large mass splitting (3.4%), 1PE creates relatively large charge-dependent effects in all partial waves. 2. 2π -exchange diagrams. Next is the CIB created by the 2π exchange contribution to the N N interaction. There are many diagrams that contribute (see Li and Machleidt (1998b) for a complete overview). For the qualitative discussion here, the largest of all 2π diagrams is considered, namely, the box diagrams with NΔ intermediate states, Fig. 10. Disregarding isospin factors and using some drastic approximation, the amplitude for such a diagram is V2π (gπ , mπ ) = −

gπ4 72 16M 4 25



[σ · q S · q]2 d 3 p 3 (2π ) (m2π + q2 )2 (Ep + EpΔ − 2Ep )

×Fπ2N N (Λπ N N , |q|) Fπ2N Δ (Λπ N Δ , |q|),

(67)

where q = p − p with p the relative momentum in the initialand final state (for simplicity, a diagonal matrix element is considered); Ep = M 2 + p2 and 

2 + p2 with M EpΔ = MΔ Δ = 1232 MeV the Δ-isobar mass; S is the spin transition operator between nucleon and Δ. For the π N Δ coupling constant, 2 fπ N Δ , the quark-model relationship fπ2N Δ = 72 25 fπ N N is used (Machleidt et al. 1987). For small momentum transfers q, this attractive contribution is roughly proportional to m−4 π . Thus for the 2π exchange, the heavier pions will provide less attraction than the lighter ones. Charged and neutral pion-exchanges occur for pp as well as for np, and it is important to take the isospin factors carried by the various diagrams into account. They are given in Fig. 10 below each diagram.

44 Phenomenology and Meson Theory of Nuclear Forces

1733

Fig. 10 2π -exchange box diagrams with N Δ intermediate states that contribute to (a) pp and (b) np scattering. The numbers below the diagrams are the isospin factors

For pp scattering, the diagram with double π ± exchange carries the largest factor, while double π ± exchange carries only a small relative weight in np scattering. Consequently, pp scattering is less attractive than np scattering which leads to an increase of ΔaCI B by 0.79 fm due to the diagrams of Fig. 10. The crossed diagrams of this type reduce this result, and including all 2π exchange diagrams, one finds a total effect of 0.36 fm (Li and Machleidt 1998b). 3. πρ-exchanges. This group is, in principle, as comprehensive as the 2π exchanges discussed above. Graphically, the πρ diagrams can be obtained by replacing in each 2π -diagram one of the two pions by a ρ-meson of the same charge-state. This contribution to CIB is typically opposite to the one from 2π (cf. Table 4). 4. Further 3π and 4π contributions (π σ + π ω). As discussed, the Bonn potential also includes some 3π -exchanges that can be approximated in terms of π σ diagrams and 4π -exchanges of π ω type. These diagrams carry the same isospin factors as 1PE. The CIB effect from this class of diagrams is sizeable in 1 S0 and has the same sign as the effect from 1PE. The reason for the 1PE character of this

1734

R. Machleidt

contribution is that π σ prevails over π ω and, thus, determines the character of this contribution. Since sigma-exchange is negative and since, furthermore, the propagator in between the π and the σ exchange is also negative, the overall sign of the π σ exchange is the same as 1PE. Thus, it acts like a short-ranged 1PE contribution. The CIB contributions discussed explain about 80% of ΔaCI B (cf. Table 4). Ericson and Miller (1983) arrived at a very similar result using the meson-exchange model of Partovi and Lomon (1970). Another CIB contribution to the nuclear force is irreducible pion-photon (π γ ) exchange. Traditionally, it was believed that this contribution would take care of the remaining 20% of ΔaCI B (Ericson and Miller 1983). However, a recently derived π γ potential based upon chiral perturbation theory (van Kolck et al. 1998) decreases ΔaCI B by about 0.5 fm, making the discrepancy even larger. Thus, it is a matter of fact that about 25% of the charge dependence of the singlet scattering length is not explained in terms of hadronic physics.

The High-Accuracy Potentials Initiated by the Nijmegen group, one focus in the 1990s has been on the quantitative aspects of NN potentials. Even the best NN models of the 1980s (Lacombe 1980; Machleidt et al. 1987) fit the NN data typically with a χ 2 /datum ≈ 2 or more (cf. Table 2). This is still substantially above the perfect χ 2 /datum ≈ 1. To put microscopic nuclear structure theory to a reliable test, one needs a perfect NN potential such that discrepancies in the predictions cannot be blamed on a bad fit of the N N data. To prepare for the construction of potentials that could have a chance to produce a χ 2 /datum ≈ 1, the Nijmegen group created a base of NN scattering data below 350 MeV laboratory energy that allowed for a phase shift analysis with a χ 2 /datum ≈ 1 (Stoks et al. 1993). Based upon this analysis and database, new charge-dependent N N potentials were constructed in the 1990s. The groups involved and the names of these new “high-accuracy potentials” are—in chronological order: • Nijmegen group (Stoks et al. 1994): Nijm-I, Nijm-II, and Reid93 potentials. • Argonne group (Wiringa et al. 1995): AV18 potential. • Bonn group (Machleidt 2001): CD-Bonn potential. All these potentials have in common that they take charge dependence into account and use about 45 parameters to reproduce the 1992 Nijmegen data base (Stoks et al. 1993) with a χ 2 /datum ≈ 1. However, since 1993, the pp database, in particular, has substantially expanded, and for a more current database (Machleidt 2001), the χ 2 /datum produced by some of these potentials is not so perfect anymore (cf. Table 5).

44 Phenomenology and Meson Theory of Nuclear Forces

1735

Table 5 χ 2 /datum for the reproduction of various N N data bases (below 350 MeV) by the CDBonn potential (Machleidt 2001), the Nijmegen phase shift analysis (PSA) (Stoks et al. 1993), and the Argonne v18 potential (AV18) (Wiringa et al. 1995). The 1992 database is defined in Stoks et al. (1993) and after-1992 data are given in Machleidt (2001)

Proton-proton data 1992 pp database (1787 data) After-1992 pp data (1145 data) 1999 pp database (2932 data) Neutron-proton data 1992 np database (2514 data) After-1992 np data (544 data) 1999 np database (3058 data) pp and np data 1992 N N database (4301 data) 1999 N N database (5990 data)

CD-Bonn potential

Nijmegen phase shift analysis

AV18 potential

1.00 1.03 1.01

1.00 1.24 1.09

1.10 1.74 1.35

1.03 0.99 1.02

0.99 0.99 0.99

1.08 1.02 1.07

1.02 1.02

0.99 1.04

1.09 1.21

Concerning the theoretical basis of these potential, one could say that they are all—more or less—constructed “in the spirit of meson theory” (e.g., all potentials include the charge-dependent one-pion-exchange contribution). However, there are considerable differences in the details leading to considerable off-shell differences among the potentials. The CD-Bonn potential uses the full, original, nonlocal Feynman amplitude for 1PE, while all other potentials apply local approximations. As a consequence of this, the CD-Bonn potential has a weaker tensor force as compared to all other potentials. This is reflected in the predicted D-state probabilities of the deuteron, PD , which is a measure of the strength of the nuclear tensor force. While CD-Bonn predicts PD = 4.85%, the other potentials yield PD = 5.7(1)% (cf. Table 6). These differences in the strength of the tensor force lead to considerable differences in nuclear structure predictions. An indication of this is given in Table 6. The CD-Bonn potentials predicts 8.00 MeV for the triton binding energy, while the local potentials predict only 7.62 MeV. The relationship between the nonlocality of the NN potential and the predicted triton binding energy has been investigated systematically by Doleschall and coworkers within a phenomenological framework (Doleschall et al. 2003). The 1PE contribution to the nuclear force essentially takes care of the longrange interaction and the tensor force. In addition to this, all models must describe the intermediate and short-range interactions, for which very different approaches are taken. The CD-Bonn includes (besides the pion) the vector mesons ρ(769) and ω(783) and two scalar-isoscalar bosons, σ , using the full, nonlocal Feynman amplitudes for their exchanges. Thus, all components of the CD-Bonn are nonlocal, and the off-shell behavior is the original one as determined from relativistic field theory.

1736

R. Machleidt

Table 6 Modern high-accuracy N N potentials and their predictions for the two- and threenucleon bound states (The predictions for the deuteron quadrupole moment do not include corrections from meson-exchange currents and relativity) CD-Bonn (Machleidt 2001) Character Nonlocal Deuteron properties Quadr. 0.270 moment (fm2 ) Asymptotic 0.0256 D/S state D-state 4.85 probab. (%) Triton binding 8.00 (MeV) a Central

Nijm-I (Stoks et al. 1994) Mixeda

Nijm-II (Stoks et al. 1994) Local

Reid93 (Stoks et al. 1994) Local

AV18 (Wiringa et al. 1995) Local

NATURE

Nonlocal

0.272

0.271

0.270

0.270

0.2859(3)

0.0253

0.0252

0.0251

0.0250

0.0256(4)

5.66

5.64

5.70

5.76

–

7.72

7.62

7.63

7.62

8.48

force nonlocal, tensor force local

The models Nijm-I and Nijm-II are based upon the Nijmegen78 potential (Nagels et al. 1978) which is constructed from approximate one-boson-exchange (OBE) amplitudes. Whereas Nijm-II uses the totally local approximations for all OBE contributions, Nijm-I keeps some nonlocal terms in the central force component (but the Nijm-I tensor force is totally local). However, nonlocalities in the central force have only a very moderate impact on nuclear structure. Thus, it would be more important to keep the tensor force nonlocalities. The Reid93 (Stoks et al. 1994) and Argonne V18 (Wiringa et al. 1995) potentials do not use meson-exchange for intermediate and short range; instead, a phenomenological parametrization is chosen. The Argonne V18 uses local functions of Woods-Saxon type, while Reid93 applies local Yukawa functions of multiples of the pion mass, similar to the original Reid potential of 1968 (Reid 1968). At very short distances, the potentials are regularized either by exponential (V18 , Nijm-I, Nijm-II) or by dipole (Reid93) form factors, which are all local functions. The high-accuracy potentials discussed in this section have been applied intensely in ab initio calculations of nuclear structure and reactions from around 1995 to about 2010 and are still in frequent use today.

Models for Nuclear Many-Body Interactions Diverse Nuclear Many-Body Interactions Originally, it was hoped that the structure of finite nuclei could be understood in terms of just the two-nucleon force (2NF) (Negele 1970)—if one would only find the “right” 2NF. However, in the course of the 1970s, when more reliable microscopic

44 Phenomenology and Meson Theory of Nuclear Forces

1737

calculations became available, growing evidence accumulated that showed that it was impossible to saturate nuclear matter at the right energy and density when applying only 2NFs (Day 1983; Machleidt 1989). Another problem was the triton binding energy, which was considerably underpredicted with the 2NFs available at the time (Brandenburg 1974; Brandenburg et al. 1977). These failures were interpreted as an indication for the need of nuclear many-body forces. Strictly speaking, many-nucleon forces are an artifact of theory. They are created by freezing out non-nucleonic degrees of freedom contained in the full-fledged problem. Examples are given in Fig. 11. In part (a) of the figure, the frozen degree of freedom is a nucleon resonance (here: the Δ(1232) isobar with spin and isospin 3 2 ), in (b) an antinucleon, and in (c) meson resonances (here: the σ and ρ bosons). Fig. 11 Diverse three-nucleon force diagrams. Solid lines (nucleons) are upward directed unless noted otherwise. Dashed lines represent various mesons, as appropriate. Diagram (d) is the 2π -exchange three-nucleon force of the Tucson-Melbourne type (Coon et al. 1979), where the shaded oval represents the (off-shell) π N amplitude with the forward propagating Born term subtracted

1738

R. Machleidt

The oldest examples of three-body forces are those derived by Primakoff and Holstein (1939) in 1939. They arise from particle-antiparticle pairs, Fig. 11b, which, in a nonrelativistic model, are represented by three-body potential terms. While these forces turned out to be negligible for atomic systems, they were found to be sizable for nuclear systems (Primakoff and Holstein 1939). This fact is demonstrated in the so-called Dirac-Brueckner approach to nuclear matter (Brockmann and Machleidt 1984, 1990; Ter Haar and Malfliet 1987), where diagram Fig. 11b plays a crucial role (see below). Diagram (a) of Fig. 11 was first considered by Fujita and Miyazawa (FM) (Fujita and Miyazawa 1957) in 1957, and both diagrams (a) and (b) were taken into account by the Catania group (Li et al. 2008). Diagram (c), with all mesons involved of the scalar type, was evaluated by Barshay and Brown (1975) and found to be fairly large. For a reliable approach to three-nucleon forces (3NFs), two aspects need to be considered: First, as far as possible, one should take into account all processes that may create a 3NF, and, second, the strength of the contributions should be consistent with their size in other hadronic reactions. In this context, it was noticed early on that the internal parts of Fig. 11a–c are major contributions to pion-nucleon scattering (Brown 1972). This suggests the idea to use the empirical π N amplitude as starting point, where however the pions are on their mass shell. On the other hand, within a 3NF diagram, the pions are virtual and space-like, and, therefore, the amplitude must be extrapolated off mass shell. In the work of Coon et al. (1979), this is accomplished by applying constraints based upon current algebra and partial conservation of axial current (PCAC). This approach has become known as the Tucson-Melbourne (TM) 3NF, shown symbolically in Fig. 11d. The shaded area in that figure contains everything that contributes to π N scattering—except a positiveenergy single-nucleon intermediate state to avoid double counting, since the latter is automatically generated by the iteration of the 1PE two-nucleon force. It is instructive to note that, for the 2PE 3NF, the same dichotomy exists as for the 2PE contribution to the 2NF. The TM 3NF is based upon dispersion theory, while the alternative, a field-theoretic approach based upon Lagrangians, was pursued by Robilotta and coworker (1983), known as the Brazilian 3NF. Moreover, in analogy to the πρ contributions to the 2NF (Fig. 5b), later work on the 3NF also included ρ-exchange in the diagram of Fig. 11d (Robilotta and Isidro Filho 1984; Coon and Pena 1993). The attraction provided by the FM or TM 2PE 3NFs has proven to be useful in explaining the binding energies of light nuclei (particularly, 3 H and 4 He) which are, in general, underbound when only 2NFs are applied. However, this added attraction leads to overbinding and too high a saturation density in nuclear matter (cf. Fig. 12, curve labeled UV14+TM). Therefore, some groups added a repulsive short-range 3NF which ameliorates the problem, but does not solve it (Day 1983; Carlson et al. 1983) (Fig. 12, curve UV14+UV). In the work by the Urbana group, many versions of such 3NF were developed with Urbana IX (UIX) being the most popular one—applied in light nuclei, nuclear matter (Fig. 12, curve AV18+UIX), and neutron matter (Akmal et al. 1998). In later work (Pieper et al. 2001), the Urbana group extended their model for the 3NF by including the 2π -exchange S-wave

44 Phenomenology and Meson Theory of Nuclear Forces

1739

-8 AV18+UIX UV14+UV

E/A (MeV)

-12

-16

AV18 2NF

UV14 2NF -20 UV14+TM -24

0

0.1

0.2

0.3

0.4

ρ (fm-3)

0.5

0.6

Fig. 12 Energy per nucleon, E/A, as a function of density, ρ, of symmetric nuclear matter. Black dashed curves represent predictions applying phenomenological 2NFs with UV14 referring to the Urbana v14 2NF (Lagaris and Pandharipande 1981) and AV18 signifying the Argonne v18 2NF (Wiringa et al. 1995). Solid black curves include phenomenological 3NFs with TM denoting the Tucson-Melbourne (Coon et al. 1979), UV the Urbana V (Carlson et al. 1983), and UIX the Urbana IX 3NF models. The shaded box marks the area in which empirical nuclear matter saturation is presumed to occur

Fig. 13 Diagrams (a)–(d) show the three-body force Feynman diagrams that define the Illinois 3NF (Pieper et al. 2001), with (a) the Fujita-Miyazawa, (b) the two-pion S-wave, and (c) and (d) three-pion-exchange ring diagrams with one Δ isobar in intermediate states. Diagram (e) is a two-meson-exchange 2NF diagram involving one Δ isobar. The double slash on the intermediate nucleon line indicates the change of the propagator in the nuclear medium ((a)–(d) reproduced from Pieper et al. (2001) with permission)

contribution (which according to Friar et al. (1988) can be sizable) plus three-pionexchange ring diagrams with one Δ excitation (Fig. 13a–d). The peculiar spin and isospin dependencies of Δ-ring diagrams were found to be helpful in the explanation of spectra of light nuclei. This has become known as the Illinois 3NFs (Pieper et al. 2001), which so far have evolved up to Illinois-7 (IL7) (Pieper 2008).

1740

R. Machleidt

The 3NFs of the Urbana type, adjusted to the ground state and the spectra of light nuclei, do not saturate nuclear matter properly (Carlson et al. 1983; Akmal et al. 1998) (Fig. 12) and severely underbind intermediate-mass nuclei (Lonardoni et al. 2017). The AV18 2NF plus IL7 3NF yields a pathological equation of state of pure neutron matter (Maris et al. 2013). In addition, the so-called Ay puzzle of nucleon-deuteron scattering (Entem et al. 2002) is not resolved by any of the phenomenological 3NFs (Kievsky et al. 2010). Last but not least, it is mentioned that models do exist where the degree of freedom responsible for the generation of 3NF-like contributions is not frozen out. Triton calculations in which the nucleon and the Δ isobar are regarded on an equal footing have been performed by the Hannover group (1983) and Picklesimer and coworkers (1995). In such coupled systems, diagrams of the type displayed in Fig. 13a, c, and d (and many more which also include two and three Δ’s) are generated automatically. A consistent treatment of the Δ isobar also affects the two-nucleon force when inserted into the nuclear medium, as indicated in Fig. 13e. This effect is repulsive (Holinde and Machleidt 1977; Machleidt and Holinde 1980) and essentially cancels the attraction produced by the Δ-induced 3NF contributions, leaving basically no net effect (Machleidt 1989; Picklesimer et al. 1995). The bottom line is that while phenomenological and meson-theoretic 2NFs are able to provide an excellent description of NN scattering, phenomenological 3NFs are doing rather poorly and are certainly not able to explain nuclear matter saturation properly (cf. Fig. 12). In fact, before the year of 2010 (when the application of chiral 3NFs in nuclear matter started full-scale (Holt et al. 2010; Hebeler et al. 2011; Sammarruca et al. 2012; Coraggio et al. 2014; Sammarruca et al. 2015)), the only microscopic approach that was able to explain nuclear matter saturation was the relativistic Dirac-Brueckner-Hartree-Fock (DBHF) method (Brockmann and Machleidt 1984, 1990; Ter Haar and Malfliet 1987), which will be discussed in the next subsection.

Relativistic Meson-Theoretic Approaches to Nuclear Structure In the 1970s, a relativistic approach to nuclear structure was developed by Miller and Green (1972). They studied a Dirac-Hartree model for the groundstate of nuclei which was able to reproduce the binding energies, the root-mean-square radii, and the single-particle levels, particularly the spin-orbit splittings. Their potential consisted of a strong (attractive) scalar and (repulsive) vector component. The Dirac-Hartree(-Fock) model was further developed by Brockmann (1978) and by Horowitz and Serot (1981; 1986). At about that same time, Clark and coworkers applied a Dirac equation containing a scalar and vector field to protonnucleus scattering (Arnold et al. 1979). The most significant result of this Dirac phenomenology is the quantitative fit of spin observables which are only poorly described by the Schrödinger equation (Wallace 1987). This success and Walecka’s

44 Phenomenology and Meson Theory of Nuclear Forces

1741

theory on highly condensed matter (Walecka 1974) made relativistic approaches very popular in the 1980s and 1990s. Inspired by these achievements, a relativistic extension of Brueckner theory has been suggested by Shakin and collaborators (1983), frequently called the DiracBrueckner approach. The advantage of a Brueckner theory is that the free NN interaction is used; thus, there are no parameters in the force which are adjusted in the many-body problem. The essential point of the Dirac-Brueckner approach is to apply the Dirac equation for the single-particle motion in nuclear matter. In the work done by the Brooklyn group, the relativistic effect is calculated in first order perturbation theory. This approximation is avoided, and a full self-consistency of the relativistic single-particle energies and wave functions is performed in the subsequent work by Brockmann and Machleidt (1984; 1990) and by ter Haar and Malfliet (1987). Formal aspects involved in the derivation of the relativistic Gmatrix have been discussed in detail by Horowitz and Serot (1984; 1987). As mentioned, the essential point of the Dirac-Brueckner approach is to use the Dirac equation for the single-particle motion in nuclear matter: ( p − M − U ) u(p, ˜ s) = 0

(68)

(α · p + βM + βU ) u(p, ˜ s) = εp u(p, ˜ s) ,

(69)

U = US + γ 0 UV ,

(70)

or, in Hamiltonian form,

with

where US is an attractive scalar and UV (the time component of) a repulsive vector field. M denotes the mass of the free nucleon. The fields, US and UV , are in the order of several hundred MeV and strongly density-dependent. In nuclear matter, they can be determined self-consistently. The resulting fields are in close agreement with those obtained in the Dirac phenomenology of scattering. The solution of Eq. (68) is  u(p, ˜ s) =

E˜ p + M˜ 2M˜



1

σ ·p E˜ p +M˜

χs ,

(71)

with M˜ = M + US , E˜ p =

 M˜ 2 + p2 ,

(72) (73)

1742

R. Machleidt

and χs a Pauli spinor. The covariant normalization is u(p, ˜¯ s) u(p, ˜ s) = 1. Notice that the Dirac spinor Eq. (71) is obtained from the free Dirac spinor by simply replacing ˜ M by M. As in conventional Brueckner theory, the basic quantity in the Dirac-Brueckner approach is a G-matrix, which satisfies an integral equation. In this relativistic approach, a relativistic three-dimensional equation is chosen, e. g., the Thompson equation (Thompson 1970), which is a relativistic three-dimensional reduction of the Bethe-Salpeter equation (Salpeter and Bethe 1951). In operator form, the equation reads: ˜ z) = V˜ + V˜ P (˜z)Q G(˜ ˜ z) , G(˜

(74)

with P (˜z) the relativistic two-nucleon propagator in the medium and Q the Pauli operator, which projects onto unoccupied states. The parameter z˜ is defined below. For more details, see Brockmann and Machleidt (1990). The energy in nuclear matter is then obtained, in lowest order, from E 1  M˜ = m|γ · pm + M|m A A E˜m m≤k F

+

1  M˜ 2 ˜ z)|mn − nm − M . mn|G(˜ ˜ m E˜ n 2A E m,n≤k

(75)

F

The single-particle potential U (m) =

M˜ M˜ M˜ m|U |m = m|US + γ 0 UV |m = U S + UV E˜m E˜m E˜m

(76)

is the self-energy of the nucleon in nuclear matter, which is defined in terms of the G-matrix in formally the same way as in the nonrelativistic version of Brueckner theory: U (m) = Re

 n≤kF

M˜ 2 ˜ z)|mn − nm , mn|G(˜ E˜n E˜m

(77)

where m denotes a state below or above the Fermi surface (continuous choice) and z˜ = εm + εn ,

(78)

with the single-particle energy, εm , given by εm =

M˜ m|γ · pm + M|m + U (m) E˜m

(79)

44 Phenomenology and Meson Theory of Nuclear Forces

=

M M˜ + p2m + U (m) E˜m

= E˜m + UV .

1743

(80) (81)

In the above equations, the states |m and |n are represented by Dirac spinors of the kind Eq. (71) and an appropriate isospin wavefunction, m| and n| are the adjoint Dirac spinors u¯˜ = u˜ † γ 0 . The states of the nucleons in nuclear  matter, w, are † ˜ E˜ × u˜ which to be normalized by w w = 1. This is achieved by defining w ≡ M/ ˜ E˜ in the above equations. explains the frequent factors of M/ The expression for the energy, Eq. (75), is known as the Dirac-BruecknerHartree-Fock (DBHF) approach. If M˜ is replaced by M, essentially the conventional Brueckner-Hartree-Fock (BHF) approximation is obtained. The essential difference to standard Brueckner theory is the use of the potential V˜ in Eq. (74). Indicated by the tilde, this meson-theoretic potential is evaluated by using the spinors of Eq. (71) instead of the free spinors applied in scattering as well ˜ are strongly as in conventional (nonrelativistic) Brueckner theory. Since US (and M) 2 ˜ ˜ ˜ ˜ density-dependent, so is the effective potential (M/E) V , since M decreases with density. The essential effect in nuclear matter is a suppression of the (attractive) ˜ E) ˜ 2 . This suppression increases with σ -exchange by the scalar density factor (M/ density, providing the additional saturation. It turns out that this effect is so strongly density-dependent that the empirical nuclear matter saturation can be reproduced; see Fig. 14.

Nucleon-Nucleon Scattering Above the Inelastic Threshold One-boson-exchange potentials are real and, therefore, only able to describe NN scattering below the inelastic threshold. Above Tlab ≈ 290 MeV, pions can be produced in NN collisions. A model that is expected to have validity at intermediate energies needs to take the inelasticity due to pion production into account. Below about 1.5 GeV, pion production proceeds mainly through the formation of the Δ(1232) isobar which is a pion-nucleon resonance with spin and isospin 3/2. The next higher resonance is the N ∗ (1440), also known as Roper resonance, with spin and isospin 1/2 (Tanabashi et al. 2018). This resonance was included in a meson model for NN scattering up to 2 GeV constructed by Lee (1984) and found to contribute less than 1 mb to the inelastic cross section even at 2 GeV. An exclusive measurement of two-pion production in pp scattering at 775 MeV finds cross sections that can be attributed to the Roper resonance of less than 0.1 mb (Pätzold et al. 2003). Thus below 2 GeV, the N ∗ (1440) is much less important than the Δ(1232). Therefore, in this energy regime, it is sufficient to introduce only the Δ as an additional degree of freedom. Thus, besides the NN channel, two more twobaryon channels show up, namely, NΔ and ΔΔ.

1744

R. Machleidt

Fig. 14 Energy per nucleon, E /A, as a function of the Fermi momentum, kF , for symmetric nuclear matter. Solid lines represent predictions from the DBHF approach, while dashed lines are based upon the conventional BHF approximation. Symbols in the background are the saturation points from nonrelativistic BHF calculations applying a large number of two-nucleon potentials. These background symbols define what is known as the Coester band. The shaded box marks the area in which empirical nuclear matter saturation is presumed to occur

Since all channels have baryon number two, transitions between these channels are allowed, i. e., the channels “couple.” Mathematically this produces a system of coupled equations for the scattering amplitudes. In operator notation, one can write: Tij = Vij +



Vik gk Tkj ,

(82)

k

where each subscript i, j, and k denotes a two-baryon channel (NN , NΔ, or ΔΔ), and gk is the appropriate two-baryon propagator. In principle, there are nine transition potentials, Vij , which reduce to six due to time reversal. Three of them, namely, VN Δ,N Δ , VN Δ,ΔΔ , VΔΔ,ΔΔ , involve ΔΔα vertices, where α is a nonstrange meson. Exploiting the usual symmetries, ΔΔα vertices can be constructed; however, there is no way to test empirically if the assumptions about these vertices and their coupling strengths are realistic. Therefore, such constructs are beset with large uncertainties, which is why it may be a reasonable assumption to omit them. A consequence of their omission is that the system of coupled equations, Eq. (82), decouples and the T -matrix of NN scattering, T ≡ TN N,N N , is the solution of just one integral equation: T = Veff + Veff gN N T ,

(83)

44 Phenomenology and Meson Theory of Nuclear Forces

1745

Fig. 15 Two-mesonexchange box-diagram contributions to N N scattering above the inelastic threshold as generated in a coupled-channel approach. Notation as in Fig. 5

with Veff = VN N,N N + VN N,N Δ gN Δ VN Δ,N N + VN N,ΔΔ gΔΔ VΔΔ,N N ,

(84)

where VN N,N N is an OBEP (Fig. 1) and the last two terms on the r.h.s. of the above equation are depicted in Fig. 15. Because of isospin conservation, the transition potentials containing NΔα vertices can only involve isovector mesons. Thus, VN N,N Δ =



VNα N,N Δ ,

(85)

VNα N,ΔΔ .

(86)

α=π ,ρ

VN N,ΔΔ =



α=π ,ρ

The amplitudes, VNα N,N Δ and VNα N,ΔΔ , with α = π , ρ, are derived from the interaction Lagrangians: LN Δπ = − LN Δρ = i

fN Δπ μ ¯ ψTψ ∂μ ϕ (π ) + H.c., mπ

fN Δρ (ρ) ¯ 5 γ μ Tψ ν (∂μ ϕ (ρ) ψγ ν − ∂ν ϕ μ ) + H.c., mρ

(87) (88)

where ψμ is a Rarita-Schwinger field (Rarita and Schwinger 1941) describing the (spin 32 ) Δ-isobar and T denotes an isospin transition operator that acts between isospin- 12 and isospin- 32 states. H.c. stands for Hermitian conjugate. The transition potentials VNπN,N Δ and VNπN,ΔΔ can be found in Holinde and Machleidt (1977) and ρ ρ VN N,N Δ , and VN N,ΔΔ are given in Holinde et al. (1978). The two-baryon propagators involved in Eqs. (83) and (84) are, 1 gN N = √ , s − 2Ek + iε

(89)

1 gN Δ = √ √ , s − Ek − E˜ kΔ ( s)

(90)

1746

R. Machleidt

1 gΔΔ = √ √ , s − 2E˜ kΔ ( s)

(91)

 √ √ √ 2 (√s) with M ˜ Δ ( s) = MΔ − iΓ ( s)/2 a complex where E˜ kΔ ( s) = k 2 + M˜ Δ Δ-mass. The real part of the Δ mass is the well-known physical mass, MΔ = 1232 MeV. The imaginary part, which is associated with the decay-width of the Δ-isobar, creates the inelasticity in the model and simulates pion production. It is calculated from the self-energy of the Δ-isobar that is obtained from a solution of the Dyson equation in which the Δ is coupled virtually to the π N decay channel √ (van Faassen and Tjon 1984; Ter Haar and Malfliet 1987; Elster et al. 1988). Γ ( s), is energy√ dependent and the CMS √ threshold is s = 2M + mπ for diagrams with one intermediate Δ√state and s = 2M + 2mπ for two intermediate Δ. Below these thresholds, Γ ( s) vanishes. Note that, due to isospin conservation, NΔ diagrams contribute only in isospin T = 1 NN-states, while ΔΔ diagrams contribute to all states. Consequently, in T = 0, only double-Δ diagrams contribute (besides the usual OBE contributions). This explains the thresholds for the inelasticity parameters seen in Figs. 16 and 17. From the NN T -matrix, Eq. (83), phase shifts and inelasticity parameters can be calculated, which are displayed in Figs. 16, 17, and 18 (from Eyser et al. (2004)). A model of this kind is obviously able to describe NN scattering up to about 1 GeV in semiquantitative terms (cf. also van Faassen and Tjon (1984); Ter Haar and Malfliet (1987); Elster et al. (1988) where similar results are obtained). It is seen that several phase shifts are predicted quantitatively, notably the S waves; others are semiquantitative, like the P waves which, typically, show too much attraction at intermediate energies. The cusps that are known to be the signature of the Δ threshold (Verwest 1982) also show up clearly: the shape of the 1 D2 phase shift is well reproduced while, in 3 F3 and 3 P2 , only the trends are right. Inelasticities are by-and-large described well, but in the crucial cases, namely, 1 D2 and 3 F3 , the inelasticity is predicted too small. This is a well-known problem (van Faassen and Tjon 1984; Ter Haar and Malfliet 1987; Elster et al. 1988). However, overall one may conclude that there is at least a qualitative understanding of NN scattering up to about 1 GeV in terms of a relativistic meson model extended by the Δ(1232) resonance. Much more problematic is the energy regime above 1 GeV, for which we refer the interested reader to Eyser et al. (2004).

Conclusions This chapter has thoroughly reviewed phenomenological and meson-theoretic approaches to nuclear forces, providing pedagogical insights as well as demonstrating quantitative strengths (particularly, for the two-nucleon interaction). After all this, it is mandatory to mention that, in recent years, an alternative approach to nuclear forces has become very popular, namely, chiral effective field

44 Phenomenology and Meson Theory of Nuclear Forces

Phase Shift (deg)

Inelasticity P. (deg)

1S 0

100 50 0 -50

1S 0

50

25

0 0

200 400 600 800 Lab. Energy (MeV)

0

1000

200 400 600 800 Lab. Energy (MeV)

1000

60

3P 0

Inelasticity P. (deg)

Phase Shift (deg)

50

0

-50

3P 0

40

20

0 0

200 400 600 800 Lab. Energy (MeV)

1000

1P 1

0

Inelasticity P. (deg)

Phase Shift (deg)

25

0

-25

200 400 600 800 Lab. Energy (MeV)

1000

1P 1

50

25

0

-50 0

200 400 600 800 Lab. Energy (MeV)

0

1000

200 400 600 800 Lab. Energy (MeV)

1000

75

3P 1

Inelasticity P. (deg)

Phase Shift (deg)

40

0

-40

3P

1

50

25

0 200 400 600 800 Lab. Energy (MeV)

50

3S 1

200

0

1000

Inelasticity P. (deg)

Phase Shift (deg)

0

100

0

200 400 600 800 Lab. Energy (MeV)

3S

40

1000

1

30 20 10 0

0

200 400 600 800 Lab. Energy (MeV)

1000

0

200 400 600 800 Lab. Energy (MeV)

1000

20

0

1

Inelasticity P. (deg)

3D

10 Phase Shift (deg)

Fig. 16 Phase shifts and inelasticity parameters of N N scattering up to 1 GeV laboratory energy for states with J = 0, 1. The solid curve represents the predictions by the coupled channel model described in the text. The solid dots and open circles represent results from N N phase shift analysis

1747

-10 -20 -30

3D

1

0

-20

-40 0

200 400 600 800 Lab. Energy (MeV)

1000

0

200 400 600 800 Lab. Energy (MeV)

1000

1748

1

20

D2

Inelasticity P. (deg)

Phase Shift (deg)

30

10 0

1

40

D2

20

0

-10 0

200 400 600 800 Lab. Energy (MeV)

0

1000

200 400 600 800 Lab. Energy (MeV)

1000

3D

Inelasticity P. (deg)

Phase Shift (deg)

40

2

20

0

3D

8

2

4

0 0

200 400 600 800 Lab. Energy (MeV)

3P 2

30

0

1000

Inelasticity P. (deg)

Phase Shift (deg)

40

20 10

0

200 400 600 800 Lab. Energy (MeV)

3P

50

1000

2

25

1000

1F 3

0

Inelasticity P. (deg)

Phase Shift (deg)

10

0

-10

200 400 600 800 Lab. Energy (MeV)

1000

1F 3

8

4

0

-20

10

200 400 600 800 Lab. Energy (MeV)

3

5

1000

F3

0

Inelasticity P. (deg)

0

Phase Shift (deg)

200 400 600 800 Lab. Energy (MeV)

0

0

0 -5 -10

200 400 600 800 Lab. Energy (MeV)

3

40

1000

F3

20

-15 0

10

200 400 600 800 Lab. Energy (MeV)

3

0

1000

D3

Inelasticity P. (deg)

0

Phase Shift (deg)

Fig. 17 Same as Fig. 16, but for J = 2, 3

R. Machleidt

5

0

200 400 600 800 Lab. Energy (MeV)

3

30

1000

D3

20 10 0

-5 0

200 400 600 800 Lab. Energy (MeV)

1000

0

200 400 600 800 Lab. Energy (MeV)

1000

12 8 4 0 0

Mixing Parameter (deg)

Mixing Parameter (deg)

ε1

16

200 400 600 800 Lab. Energy (MeV)

10

5

0 0

200 400 600 800 Lab. Energy (MeV)

6 4 2 0 -2 0

2

ε3

15

1000

1749

ε2

8

1000

Mixing Parameter (deg)

Mixing Parameter (deg)

44 Phenomenology and Meson Theory of Nuclear Forces

200 400 600 800 Lab. Energy (MeV)

1000

ε4

0

-2

0

200 400 600 800 Lab. Energy (MeV)

1000

Fig. 18 Mixing parameters for J ≤ 4 and laboratory energies up to 1 GeV. Notation as in Fig. 16

theory (EFT) (Machleidt and Entem 2011). Arguments in favor of chiral EFT—that are typically advanced—are: Chiral EFT • is closely related to QCD; • comes with an organizational scheme (power counting) that allows to estimate the accuracy of the predictions (at a given order); • generates two- and many-body forces on an equal footing. That’s all fine. However, there is also a caveat in place. Since chiral EFT is a low-momentum expansion, we have to expect limitations concerning its applicability. This is demonstrated in Fig. 19, where phase shift predictions by various NN potentials up to 1000 MeV (lab. energy for the incident nucleon) are shown and compared to the results from phase shift analyses. The figure includes one representative from meson theory, namely, the CD-Bonn potential (Machleidt 2001) (solid line), which obviously predicts the phase shifts correctly up to the highest energies displayed, even though in its construction it was adjusted only up to 350 MeV. On the other hand, the chiral NN potentials at order N3 LO (dashed line (Entem and

80

1S 0

40 0 -40 -80 0

Phase Shift (deg)

R. Machleidt

Phase Shift (deg)

1750

250 500 750 1000 Lab. Energy (MeV)

3P 0

80 40 0 -40 0

250 500 750 1000 Lab. Energy (MeV)

Fig. 19 np phase shifts of the 1 S0 and 3 P0 partial waves for lab. energies up to 1000 MeV. The solid curve shows the phase shifts predicted by the CD-Bonn potential, which is based upon meson theory. Note that this curve is hardly visible because it agrees with the data up to 1000 MeV and, thus, is buried under the symbols representing the data. The dashed and the dotted lines are the predictions by the N3 LO chiral potentials constructed by the Idaho (Entem and Machleidt 2003) and the Bochum (Epelbaum et al. 2005) groups, respectively. The solid dots and open circles represent results from N N phase shift analysis

Machleidt 2003) and dotted line (Epelbaum et al. 2005)) do not make any reasonable predictions beyond about 300 MeV lab. energy. This is, of course, not unexpected since chiral EFT applies only for momenta below the chiral symmetry breaking scale Λχ ≈ 1 GeV which, in general, is enforced by a regulator function which typically starts affecting the potential for momenta around 0.4 GeV/c. Thus, chiral potentials are reliable only for CMS momenta p, p  2 fm−1 . That may be o.k. for microscopic predictions of ground states or low-energy excitation spectra of finite nuclei. However, it may be questionable in other nuclear systems. For example, a Fermi momentum kF ≈ 2 fm−1 is equivalent to a neutron matter density ρ ≈ 1.7ρ0 , where ρ0 denotes normal nuclear matter density. Thus, neutron matter calculations in which chiral nuclear potentials are applied may be trusted only for those low densities, which are totally insufficient for reliable predictions of the properties of neutron stars. In contrast, relativistic meson theory can be trusted up to considerably higher momenta (cf. the CD-Bonn curve in Fig. 19) and densities equivalent to those high momenta, which easily cover the densities in the interior of a neutron star. Acknowledgments This work was supported in part by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-FG02-03ER41270.

Appendix A: The Relativistic One-Boson-Exchange Potential Popular Lagrangians for meson-nucleon coupling are ¯ 5 ψϕ (ps) Lps = −gps ψiγ Lpv = −

fps ¯ μ γ 5 ψ∂μ ϕ (ps) ψγ m ˆ ps

(92) (93)

44 Phenomenology and Meson Theory of Nuclear Forces (s) ¯ Ls = −gs ψψϕ

1751

(94)

¯ μ ψϕμ(v) − Lv = −gv ψγ

fv ¯ μν ψ(∂μ ϕν(v) − ∂ν ϕμ(v) ) ψσ 4Mˆ

(95)

(α)

with ψ the nucleon and ϕ(μ) the meson fields (notation and conventions as in Schwartz (2014)). m ˆ ps and Mˆ are scaling masses to keep the associated coupling constants dimensionless. For isospin-1 mesons, ϕ (α) is to be replaced by τ ·ϕ (α) with τ l (l = 1, 2, 3) the usual Pauli matrices for isospin- 12 . The sub- and superscripts ps, pv, s, and v denote pseudoscalar, pseudovector, scalar, and vector couplings/fields, respectively. The one-boson-exchange potential (OBEP) is defined as a sum of one-particleexchange amplitudes of certain bosons with given mass and coupling. Using the six non-strange bosons with masses below 1 GeV/c2 , one has

VOBEP =



VαOBE

(96)

α=π,η,ρ,ω,δ,σ

with π and η pseudoscalar, σ and δ/a0 scalar, and ρ and ω vector particles. The contributions from the isovector bosons π, δ/a0 and ρ include a factor τ1 · τ2 . The above Lagrangians imply the following OBE contributions: (Note that the potential is defined as i times the Feynman amplitude; furthermore, there is a factor of i for each vertex and propagator; since i 4 = 1, the factors of i are not explicitly shown.)

OBE p λ1 λ2 |Vps |pλ1 λ2

=−

2 gps

(2π )3

  u(p ¯  , λ1 )iγ 5 u(p, λ1 )u(−p ¯ , λ2 )iγ 5 u(−p, λ2 )/

[(p − p)2 + m2ps ]; OBE |pλ1 λ2 p λ1 λ2 |Vpv

=

2 1 fps   u(p ¯  , λ1 )γ 5 γ μ i(p − p)μ u(p, λ1 )u(−p ¯ , λ2 )γ 5 γ μ i (2π )3 m ˆ 2ps

(p − p)μ u(−p, λ2 )/ [(p − p)2 + m2ps ]

(97)

1752

=

R. Machleidt 2 fps 4M 2   {u(p ¯  , λ1 )γ 5 u(p, λ1 )u(−p ¯ , λ2 )γ 5 u(−p, λ2 ) 3 (2π ) m ˆ 2ps   +[(E  − E)/(2M)]2 u(p ¯  , λ1 )γ 5 γ 0 u(p, λ1 )u(−p ¯ , λ2 )γ 5 γ 0 u(−p, λ2 )   +[(E  − E)/(2M)][u(p ¯  , λ1 )γ 5 u(p, λ1 )u(−p ¯ , λ2 )γ 5 γ 0 u(−p, λ2 )   +u(p ¯  , λ1 )γ 5 γ 0 u(p, λ1 )u(−p ¯ , λ2 )γ 5 u(−p, λ2 )]}/

[(p − p)2 + m2ps ];

(98)

p λ1 λ2 |VsOBE |pλ1 λ2 =−

gs2   u(p ¯  , λ1 )u(p, λ1 )u(−p ¯ , λ2 )u(−p, λ2 )/ (2π )3

[(p − p)2 + m2s ];

(99)

p λ1 λ2 |VvOBE |pλ1 λ2 =

1 fv {gv u(p ¯  , λ1 )γμ u(p, λ1 ) + u(p ¯  , λ1 )σμν i(p − p)ν u(p, λ1 )} 3 (2π ) 2Mˆ fv     ×{gv u(−p u(−p ¯ ¯ , λ2 )γ μ u(−p, λ2 ) − , λ2 )σ μν i(p − p)ν u(−p, λ2 )}/ ˆ 2M

[(p − p)2 + m2v ]   1 M gv + fv = u(p ¯  , λ1 )γμ u(p, λ1 ) (2π )3 Mˆ fv u(p ¯  , λ1 )[(p + p)μ + (E  − E)(gμ0 − γμ γ 0 )]u(p, λ1 ) 2Mˆ   M   u(−p ¯ , λ2 )γ μ u(−p, λ2 ) × gv + fv ˆ M



−

 fv     μ0 μ 0 u(−p ¯ − , λ2 )[(p + p)μ + (E − E)(g − γ γ )]u(−p, λ2 ) / 2Mˆ

[(p − p)2 + m2v ].

(100)

Working in the two-nucleon CMS, the momenta of the two incoming (outgoing)      2 2 2 nucleons are p and −p (p and −p ). E ≡ M + p and E ≡ M + p 2 . Using the BbS (Blankenbecler and Sugar 1966) or Thompson (Thompson 1970) equations, the four-momentum transfer between the two nucleons is (p −p) = (0, p −p). The Dirac equation is applied repeatedly in the evaluations of the pv-coupling, and the

44 Phenomenology and Meson Theory of Nuclear Forces

1753

Gordon identity (Schwartz 2014) is used in the case of the v-coupling. (Note that in Ep. (100), second line from the bottom, the term (p + p)μ carries μ as a subscript to ensure the correct sign of the space component of that term.) The propagator for vector bosons is i

−gμν + (p − p)μ (p − p)ν /m2v −(p − p)2 − m2v

(101)

where the (p − p)μ (p − p)ν -term is dropped which vanishes on-shell, anyhow, since the nucleon current is conserved. The off-shell effect of this term was examined in Holinde and Machleidt (1975) and was found to be unimportant. The Dirac spinors in helicity representation are given by  u(p, λ1 ) =  u(−p, λ2 ) =

E+M 2M E+M 2M

 

1

2λ1 |p| E+M

1

2λ2 |p| E+M

|λ1

(102)

|λ2 ,

(103)



where the helicity λi of particle i (with i = 1 or 2) is the eigenvalue of the helicity operator 12 σ i · pi /|pi | which is ± 12 . They are normalized covariantly, that is u(p, ¯ λ)u(p, λ) = 1.

(104)

with u¯ = u† γ 0 . To ensure convergence of the scattering equation, it is customary to multiply each meson-nucleon vertex with a form factor, for which the following form may be chosen, Fα [(p − p)2 ] =



Λ2α − m2α 2 Λα + (p − p)2

 nα (105)

with mα the mass of the meson involved, Λα the so-called cutoff mass, and nα an exponent. Thus, the OBE amplitudes Eqs. (97)–(100) are multiplied by Fα2 . The coupling constants for the two different couplings for ps particles are related by fps gps = . 2M m ˆ ps

(106)

Further developments, like partial-wave decomposition and the application of the OBEP in appropriate relativistic three-dimensional scattering equations, are given in Machleidt (1993).

1754

R. Machleidt

Appendix B: Nonrelativistic Approximations and Position-Space Potentials The momentum-space expressions for the OBE amplitudes given in Appendix A depend on two momentum variables, namely, the incoming and outgoing relative momenta p and p , respectively. A Fourier transformation of these expressions into position space would yield functions of r and r , the relative distance between the two in- and outgoing nucleons, i. e. a non-local potential. Because of the complexity of the expressions, this Fourier transformation cannot be done analytically. Both features mentioned are rather inconvenient for position-space (r-space) calculations. Therefore, it is customary to simplify the momentum-space expressions such that an analytic Fourier transformation becomes possible. This is achieved by using Dirac spinors in the representation  u(p, s) =

E+M 2M



 1

σ ·p E+M

χs

(107)

(with χs a Pauli spinor) for the evaluation of the OBE amplitudes of Appendix A and defining the momentum variables q = p − p k=

1  (p + p) . 2

(108) (109)

By dropping χs , the resulting potential is an operator in spin space, as customary. The relativistic energies are expanded in powers of q2 and k2 keeping the lowest order. In this way, one obtains the following simple momentum-space expressions: 2 gps (σ 1 · q)(σ 2 · q) 2 4M q2 + m2ps

gs2 q2 i k2 Vs (q, k) = − 2 + − S · (q × k) 1− q + m2s 2M 2 8M 2 2M 2  1 q2 3i 3k2 2 Vv (q, k) = 2 − + S · (q × k) g 1 + v 2 2 2 q + mv 2M 8M 2M 2

Vps (q) = −

(110)

(111)

 q2 1 σ · σ + (σ · q)(σ · q) 1 2 1 2 4M 2 4M 2

2  q 4i q2 1 gv fv − + S · (q × k) − σ 1 · σ 2 + (σ 1 · q)(σ 2 · q) + M M M M 2Mˆ  f2  + v −q2 σ 1 · σ 2 + (σ 1 · q)(σ 2 · q) (112) 4Mˆ 2 −

44 Phenomenology and Meson Theory of Nuclear Forces

1755

with S = 12 σ 1 + σ 2 the total spin of the two-nucleon system. These expressions contain nonlocalities due to k2 and (q × k) terms. The latter leads to the orbital angular momentum operator L = −ir × ∇ in r-space, whereas the former provides ∇ 2 -terms. A quadratic spin-orbit term, 12 (σ 1 ·L σ 2 ·L+σ 2 ·L σ 1 ·L), is obtained when terms up to the fourth power in the momentum are retained, leading to substantially more comprehensive potential expressions. However, within a consistent meson model, these quadratic spin-orbit terms as well as the other terms of higher momenta do not improve the fit to the NN data but cause serious mathematical problems. If substantial improvements over the above expressions are desired, the use of the full unabridged momentum-space expressions of Appendix A is recommended. The role of the quadratic spin-orbit term is different if it is used as a phenomenological term to be fitted to the data. Then, particularly, an improvement of the 1D2 and 3D2 phase shifts can be achieved (Hamada and Johnston 1962).  The Fourier transform, V (r) = (2π )−3 d 3 keiq·r V (q), which can now be performed analytically, yields:

 2  m 2 1 gps 4π (3) ps mps Vps (mps , r) = Y (mps r) − 3 δ (r) σ 1 · σ 2 12 4π M mps  +Z(mps r)S12 (ˆr )

(113)



  1  ms 2 1  2 2 1− Y (ms r) + ∇ Y (m r) + Y (msr)∇ s 4 M 4M 2  1 + Z1 (ms r)L · S (114) 2

g2 Vs (ms , r) = − s ms 4π

Vv (mv , r) =



  1  mv 2 3  2 2 1+ Y (mv r) − ∇ Y (m r) + Y (m r)∇ v v 2 M 4M 2  1  mv 2 3 1 + Y (mv r)σ 1 · σ 2 − Z1 (mv r)L · S − Z(mv r)S12 (ˆr ) 6 M 2 12    mv 2 2  mv 2 1 gv fv M mv Y (mv r) + Y (mv r)σ 1 · σ 2 + 2 4π M 3 M Mˆ  1 −4Z1 (mv r)L · S − Z(mv r)S12 (ˆr ) 3 gv2 mv 4π

f2 + v 4π



    M 2 1 mv 2 1 mv Y (mv r)σ 1 · σ 2 − Z(mv r)S12 (ˆr ) 6 M 12 Mˆ

(115)

with Y (x) = e−x /x

(116)

1756

R. Machleidt

  m 2  3 3 α Z(x) = 1 + + 2 Y (x) M x x  m 2 1 d α Y (x) Z1 (x) = − M x dx   m 2  1 1 α = + 2 Y (x) . M x x

(117)

(118)

Similar to the σ 1 · σ 2 part of the ps potential, there are δ (3) (r) function terms in the central force and spin-spin part of the vector potential which are left out (since they drop out anyhow, see below). A form factor, Eq. (105), with nα = 1 can be taken into account by using for each meson potential Vα (r) = Vα (mα , r) −

Λ2α,2 − m2α Λ2α,2 − Λ2α,1

Vα (Λα,1 , r) +

Λ2α,1 − m2α Λ2α,2 − Λ2α,1

Vα (Λα,2 , r)

(119) where Λα,1 = Λα + ε and Λα,2 = Λα − ε with ε/Λα  1. The easy way to understand the effect of a cutoff is to consider the case of nα = 1/2 in Eq. (105), i.e., a factor Fα2 = (Λ2α − m2α )/(Λ2α + q2 ) is applied to an OBE contribution. The effect of such a cutoff is that from an OBE potential, Eqs. (113)– (115), the same expression is subtracted with the meson mass replaced by the cutoff mass (and using the same coupling constant), i. e. Vα (r) = Vα (mα , r) − Vα (Λα , r). This removes the δ (3) (r)-function terms as well as the r −3 singularities, and, in terms of momentum-space language, it damps the potential at high momenta.

References A detailed historical account of this early phase is presented in: L. M. Brown and H. Rechenberg, The Origin of the Concept of Nuclear Forces (IOP Publishing, Bristol and Philadelphia, 1996) A. Akmal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C 58, 1804 (1998) M.R. Anastasio, L.S. Celenza, W.S, Pong, C.M. Shakin, Phys. Reports 100, 327 (1983) L.G. Arnold, B.C. Clark, R.L. Mercer, Phys. Rev. C 19, 917 (1979) S. Barshay, G.E. Brown, Phys. Rev. Lett. 34, 1106 (1975) R. Blankenbecler, R. Sugar, Phys. Rev. 142, 1051 (1966) R.A. Brandenburg, Y.E. Kim, A. Tubis, Phys. Lett. 49B, 205 (1974) R.A. Brandenburg, P.U. Sauer, R. Machleidt, Z. Physik A280, 93 (1977) G. Breit, Phys. Rev. 120, 287 (1960) R. Brockmann, R. Machleidt, Phys. Lett. 149B, 283 (1984) R. Brockmann, R. Machleidt, Phys. Rev. C 42, 1965 (1990) R. Brockmann, Phys. Rev. C 18, 1510 (1978); R. Brockmann, W. Weise, Nucl. Phys. A355, 365 (1981) G.E. Brown, Comments Nucl. Part. Phys. 5, 6 (1972) K.A. Brueckner, K.M. Watson, Phys. Rev. 90, 699; 92, 1023 (1953) R.A. Bryan, B.L. Scott, Phys. Rev. 135, B434 (1964); 164, 1215 (1967); 177, 1435, (1969); and references therein

44 Phenomenology and Meson Theory of Nuclear Forces

1757

J. Carlson, V.R. Pandharipande, R.B. Wiringa, Nucl. Phys. A 401, 59 (1983) O. Chamberlain, E. Segre, R.D. Tripp, C. Wiegand, T. Ypsilantis, Phys. Rev. 105, 288 (1957) M. Chemtob, J.W. Durso, D.O. Riska, Nucl. Phys. B38, 141 (1972); A.D. Jackson, D.O. Riska, B. Verwest, Nucl. Phys. A249, 397 (1975) Q. Chen et al., Phys. Rev. C 77, 054002 (2008) H.T. Coelho, T.K. Das, M.R. Robilotta, Phys. Rev. C 28, 1812 (1983) S.A. Coon, M.T. Pena, Phys. Rev. C 48, 2559 (1993) S.A. Coon, M.D. Scadron, P.C. McNamee, B.R. Barrett, D.W.E. Blatt, B.H.J. McKellar, Nucl. Phys. A 317, 242 (1979) L. Coraggio, J.W. Holt, N. Itaco, R. Machleidt, L.E. Marcucci, F. Sammarruca, Phys. Rev. C 89, 044321 (2014) W.N. Cottingham, M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, Phys. Rev. D 8, 800 (1973); R. Vinh Mau, J.M. Richard, B. Loiseau, M. Lacombe, W.N. Cottingham, Phys. Lett. B 44, 1 (1973) B.D. Day, Comments Nucl. Part. Phys. 11, 115 (1983) R. De Tourreil, B. Rouben, D.W.L. Sprung, Nucl. Phys. A 242, 445 (1975) P. Doleschall, I. Borbely, Z. Papp, W. Plessas, Phys. Rev. C 67, 064005 (2003) C. Elster, K. Holinde, D. Schütte, R. Machleidt, Phys. Rev. C 38, 1828 (1988) D.R. Entem, R. Machleidt, Phys. Rev. C 68, 041001 (2003) D.R. Entem, R. Machleidt, H. Witala, Phys. Rev. C 65, 064005 (2002) E. Epelbaum, W. Glöckle, U.-G. Meißner, Nucl. Phys. A747, 362 (2005) T.E.O. Ericson, G.A. Miller, Phys. Lett. 132B, 32 (1983) K. Erkelenz, Phys. Reports 13C, 191 (1974) A.R. Erwin et al., Phys. Rev. Lett. 6, 628 (1961) K.O. Eyser, R. Machleidt, W. Scobel, Eur. Phys. J. A 22, 105 (2004) E. Fermi, Z. Phys. 88, 161 (1934) W.R. Frazer, J.R. Fulco, Phys. Rev. Lett. 2, 365 (1959); Phys. Rev. 117, 1609 (1960) J.L. Friar, B.F. Gibson, G.L. Payne, S.A. Coon, Few-Body Systems 5, 13 (1988) J.-I. Fujita, H. Miyazawa, Prog. Theor. Phys. 17, 360 (1957) J.L. Gammel, R.M. Thaler, Phys. Rev. 107, 291, 1337 (1957) E. Gardner, C.M.G. Lattes, Science 107, 270 (1948) S. Gartenhaus, Phys. Rev. 100, 900 (1955) A. Gersten, R. Thompson, A.E.S. Green, Phys. Rev. D 3, 2076 (1971); G. Schierholz, Nucl. Phys. B40, 335 (1972); J. Fleischer, J.A. Tjon, Nucl. Phys. B84, 375 (1975); Phys. Rev. D 24, 87 (1980) D.E. González Trotter et al., Phys. Rev. C 73, 034001 (2006) F. Gross, A. Stadler, Phys. Rev. C 78, 014005 (2008) C. Hajduk, P.U. Sauer, W. Struve, Nucl. Phys. A 405, 581 (1983) T. Hamada, I.D. Johnston, Nucl. Phys. 34, 382 (1962) O. Haxel, J.H.D. Jensen, H.E. Suess, Phys. Rev. 75, 1766 (1949) K. Hebeler, S.K. Bogner, R.J. Furnstahl, A. Nogga, A. Schwenk, Phys. Rev. C 83, 031301(R) (2011) K. Holinde, R. Machleidt, Nucl. Phys. A 280, 429 (1977) K. Holinde, R. Machleidt, Nucl. Phys. A247, 495 (1975) K. Holinde, R. Machleidt, M.R. Anastasio, A. Faessler, H. Müther, Phys Rev C 18, 870 (1978) J.W. Holt, N. Kaiser, W. Weise, Phys. Rev. C 81, 024002 (2010) C.J. Horowitz, B.D. Serot, Nucl. Phys. A368, 503 (1981) C.J. Horowitz, B.D. Serot, Phys. Lett. 137B, 287 (1984) C.J. Horowitz, B.D. Serot, Nucl. Phys. A464, 613 (1987) R. Jastrow, Phys. Rev. 81, 165 (1951) J. Kellog, I. Rabi, N.F. Ramsey, J. Zacharias, Phys. Rev. 55, 318 (1939); ibid. 56, 728 (1939); ibid. 57, 677 (1940)

1758

R. Machleidt

N. Kemmer, Proc. Roy. Soc. (London) A166, 127 (1938) A. Kievsky, M. Viviani, L. Girlanda, L.E. Marcucci, Phys. Rev. C 81, 044003 (2010) M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. Côté, P. Pirès, R. de Tourreil, Phys. Rev. C 21, 861 (1980) I.E. Lagaris, V.R. Pandharipande, Nucl. Phys. A 359, 331 (1981) K.E. Lassila, M.H. Hull, H.M. Ruppel, F.A. McDonald, G. Breit, Phys. Rev. 126, 881 (1962) C.M.G. Lattes, G.P.S. Occhialini, C.F. Powell, Nature 160, 453, 486 (1947) T.-S.H. Lee, Phys. Rev. C 29, 195 (1984) G.Q. Li, R. Machleidt, Phys. Rev. C 58, 1393 (1998a) G.Q. Li, R. Machleidt, Phys. Rev. C 58, 3153 (1998b) Z.H. Li, U. Lombardo, H.-J. Schulze, W. Zuo, Phys. Rev. C 77, 034316 (2008) D. Lonardoni, A. Lovato, S.C. Pieper, R.B. Wiringa, Phys. Rev. C 96(2), 024326 (2017) R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989) R. Machleidt, in Computational Nuclear Physics 2—Nuclear Reactions, ed. by K. Langanke, J.A. Maruhn, S.E. Koonin, (Springer, New York, 1993), Chapter 1, pp. 1–29 R. Machleidt, Phys. Rev. C 63, 024001 (2001) R. Machleidt, D.R. Entem, Phys Reports 503, 1 (2011) R. Machleidt, K. Holinde, Nucl. Phys. A 350, 396 (1980) R. Machleidt, G.Q. Li, Phys. Reports 242, 5 (1994) R. Machleidt, H. Muether, Phys. Rec. C 63, 034005 (2001) R. Machleidt, I. Slaus, J. Phys. G: Nucl. Part. Phys. 27, R69 (2001) R. Machleidt, K. Holinde, Ch. Elster, Phys. Reports 149, 1 (1987) B.C. Maglic et al., Phys. Rev. Lett. 7, 178 (1961) P. Maris, J.P. Vary, S. Gandolfi, J. Carlson, S.C. Pieper, Phys. Rev. C 87(5), 054318 (2013) R.V. Mau, in Mesons in Nuclei, Vol. I, ed. by M. Rho, D.H. Wilkinson (North-Holland, Amsterdam, 1979), pp. 151–196 M.G. Mayer, Phys. Rev. 75, 1969 (1949) L.D. Miller, A.E.S. Green, Phys. Rev. C 5, 241 (1972) G.A. Miller, M.K. Nefkens, I. Slaus, Phys. Rep. 194, 1 (1990) M. J. Moravcsik, The Two-Nucleon Interaction (Clarendon Press, Oxford, 1963) M.M. Nagels, T.A. Rijken, J.J. de Swart, Phys. Rev. D 17, 768 (1978) Y. Nambu, Phys. Rev. 106, 1366 (1957) S.H. Neddermeyer, C.D. Anderson, Phys. Rev. 51, 884 (1937) J.W. Negele, Phys. Rec. C 1, 1260 (1970) J.A. Nolen, J.P. Schiffer, Annu. Rev. Nucl. Sci. 19, 471 (1969) S. Okubo, R.E. Marshak, Ann. Phys. (N. Y.) 4, 166 (1958) J. Pätzold et al., Phys. Rev. C 67, 052202 (2003) M.H. Partovi, E.L. Lomon, Phys. Rev. D 2, 1999 (1970) A. Picklesimer, R.A. Rice, R. Brandenburg, Few Body Syst. 19, 47 (1995) S.C. Pieper, AIP Conf. Proc. 1011, 143 (2008) S.C. Pieper, V.R. Pandharipande, R.B. Wiringa, J. Carlson, Phys. Rev. C 64, 014001 (2001) H. Primakoff, T. Holstein, Phys. Rev. 55, 1218 (1939) W. Rarita, J. Schwinger, Phys. Rev. 60, 61 (1941) R.V. Reid, Jr., Annals Phys. 50, 411 (1968) M.R. Robilotta, M.P. Isidro Filho, Nucl. Phys. A 414, 394 (1984) J.J. Sakurai, Phys. Rev. 119, 1784 (1960) E.E. Salpeter, H.A. Bethe, Phys. Rev. 84, 1232 (1951) F. Sammarruca, B. Chen, L. Coraggio, N. Itaco, R. Machleidt, Phys. Rev. C 86, 054317 (2012) F. Sammarruca, L. Coraggio, J.W. Holt, N. Itaco, R. Machleidt, L.E. Marcucci, Phys. Rev. C 91, 054311 (2015) M.D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge University Press, New York, 2014) S.S. Schweber, H.A. Bethe, F. de Hoffmann, Mesons and Fields, Volume I Fields (Row, Peterson and Company, Evanston, IL, 1955)

44 Phenomenology and Meson Theory of Nuclear Forces

1759

B.D. Serot, J.D. Walecka, Adv. Nucl. Phys. 16, 1 (1986) P.S. Signell, R.E. Marshak, Phys. Rev. 109, 1229 (1958) V.G.J. Stoks, R.A.M. Klomp, M.C.M. Rentmeester, J.J. de Swart, Phys. Rev. C 48, 792 (1993) V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen, J.J. de Swart, Phys. Rev. C 49, 2950 (1994) M. Taketani, S. Nakamura, M. Sasaki, Proc. Theor. Phys. (Kyoto) 6, 581 (1951) M. Taketani, S. Machida, S. Onuma, Prog. Theor. Phys. (Kyoto) 7, 45 (1952) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, 030001 (2018) B. Ter Haar, R. Malfliet, Phys. Rept. 149, 207 (1987) R.H. Thompson, Phys. Rev. D 1, 110 (1970) E.E. van Faassen, J.A. Tjon, Phys. Rev. C 30, 285 (1984); ibid. 33, 2105 (1986) U. van Kolck, M.C.M. Rentmeester, J.L. Friar, T. Goldman, J.J. de Swart, Phys. Rev. Lett. 80, 4386 (1998) B.J. Verwest, Phys. Rev. C 25, 482 (1982) J.D. Walecka, Ann. Phys. (N. Y.) 83, 491 (1974) S.J. Wallace, Ann. Rev. Nucl. Part. Sci. 37, 267 (1987) E. Wigner, Phys. Rev. 43, 252 (1933) R.B. Wiringa, R.A. Smith, T.L. Ainsworth, Phys. Rev. C 29, 1207 (1984) R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, Phys. Rev. C 51, 38 (1995) H. Yukawa, Proc. Phys. Math. Soc. Jpn. 17, 48 (1935) H. Yukawa, S. Sakata, Proc. Phys. Math. Soc. Jpn. 19, 1084 (1937) M. Taketani, Proc. Theor. Phys. (Kyoto), Suppl. 3, 1–174 (1956)

Quark Models for Baryon-Baryon Interactions

45

P. G. Ortega, D. R. Entem, and F. Fernández

Contents Symmetries and Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Symmetry and the Constituent Quark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nucleon-Nucleon Interaction in Constituent Quark Models . . . . . . . . . . . . . . . . . . . . . . The Resonating Group Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Shifts in the Constituent Quark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bound States and Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperon-Hyperon and Nucleon-Antinucleon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Since the discovery of quarks, it has been a challenge to describe the nuclear force in terms of quark degrees of freedom. This is not an easy task because, at the scale of nuclear phenomenology, QCD is non-perturbative, and one has to resource to models that catch the most important properties of QCD. In this article, the constituent quark model for baryon-baryon interactions will be reviewed. The emphasis is on the foundations of the approach and the main features of the model. The origins of the short-range repulsion in the nuclear force and other baryonic interactions are discussed.

P. G. Ortega Dpto. Física Fundamental, Universidad de Salamanca, Salamanca, Spain e-mail: [email protected] D. R. Entem () · F. Fernández Grupo de Física Nuclear, IUFFyM, Universidad de Salamanca, Salamanca, Spain e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_49

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Symmetries and Quarks Symmetries have been a useful tool for the development of physics, in particular for the classification and understanding of the phenomenology of the elementary particles. In the late 1960s, hundreds of strong interacting particles (hadrons) were known. They were classified according to their quantum numbers which include, in addition to spin and parity, some related to internal symmetries like the charge number, the baryon number (B), the isospin (I ), or strangeness (S). More quantum numbers like charm, bottom, and top refer to heavy quarks, which are outside of the scope of this article. A combination of the baryon number and strangeness (excluding charm and bottom), the hypercharge Y = B + S, is sometimes used. Gell-Mann was the first to recognize that if the particle properties were plotted in a certain way (GellMann 1964), taking as axes the third component of the isospin and the hypercharge, and selecting those particles with the same baryon number, spin, and parity, there were only three forms of graph. These graphs were one octet of J P = 1− mesons (B = 0), a second octet of J P = 1/2+ baryons (B = 1), and a decuplet of baryons with J P = 3/2+ (see Fig. 1). It is important to notice that the last particle belonging to the decuplet, the so-called Ω − , was not known at that time and was discovered at Brookhaven in 1964 (Barnes et al. 1964), 2 years after the prediction by Gell-Mann. Gell-Mann realized that these patterns correspond to the weight diagrams of the representations of the SU (3) group. This scheme was dubbed the “eightfold way” (Gell-Mann 1961; Ne’eman 1961). The name remembers the Buddhist eightfold path that was meant as a guideline to promote learning as a process of self-discovery. Underlying the success of the “eightfold way” and the symmetry group SU (3) is the existence of a substructure of matter, with constituents called quarks by Gell-Mann and aces by Zweig (1964). These objects belong to the two fundamental representations of the SU (3) group which correspond to three quarks and three antiquarks. They belong to a family of three fractional charged members distinguished by a new degree of freedom called flavor. These are the u (up), d (down), and s (strange) quarks. The quantum numbers for these quarks are summarized in Table 1. For a discussion of the SU (3) group and its application to the hadron description, see, for example, Lipkin (1973), Lichtenberg (1978), or the original papers of Gell-Mann and Ne’eman (Gell-Mann 1961; Ne’eman 1961). The fact that particles with fractional charge have not been seen in nature requires that quarks combine with other quarks or antiquarks producing only integer charged particles. The baryons should be made of three quarks and mesons of a quark and an antiquark. It is true that, as stated in the original paper of Gell-Mann, they are not the only combination to produces integral charge particles. For example, two quarks and two antiquarks may form also an integer charged particle. The treatment of these quark molecules are also outside the scope of this article. The SU (3) group contains two subgroups: the SU (2) which corresponds to the isospin degree of freedom and the U (1) corresponding to strangeness. Two of the quark flavors were chosen to be the two degrees of freedom associated with the

45 Quark Models for Baryon-Baryon Interactions

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Fig. 1 Classification of the lightest mesons (upper frame) and baryons (lower two frames) in two octets and a decuplet

Table 1 Quantum numbers of the three lightest quarks

u d s

J

B

1 2 1 2 1 2

1 3 1 3 1 3

S 0

Y

I

I3

Q

1 2 1 2

1 2

2 3

0

1 3 1 3

− 21

− 13

−1

− 23

0

0

− 13

isospin I = 1/2 (up and down quarks), whereas the third quark flavor takes into account the strangeness quantum number (strange quark). Soon after the development of the “eightfold way” scheme, experiments of high energy electron beams on proton targets (Bloom et al. 1969) suggested that the proton charge was localized on discrete scattering centers (Bjorken 1967, 1969)

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without internal structures. These components of the protons with spin 1/2 are called partons and were identified with the quarks or aces of Gell-Mann and Zweig. In addition to flavor, quarks have another internal degree of freedom called color. As quarks are particles with spin 1/2, the wave function of a collection of quarks should be antisymmetric under the interchange of any two of them. However this requirement is not fulfilled with the degrees of freedom introduced until now. The paradigmatic case is the particle Δ++ with a quark spin contents u ↑ u ↑ u ↑ and therefore with a totally symmetric wave function which violates the Pauli principle. To circumvent this difficulty, a new quark degree of freedom called color was proposed in such a way that the hadron wave function is antisymmetric in the color variable. As a baryon contains three quarks, it is natural to assume that the color degree of freedom takes three values. This new degree of freedom became more relevant than a simple artifact to solve the antisymmetrization problem. Just as the photon carries the electromagnetic interaction between charged particles, so should exist field quanta which carry the interaction between quarks. Following this idea, one can develop a theory similar to quantum electrodynamics (QED) based on the carrier of the interaction between colored quarks, that from now on, they will be called gluons. QED is a gauge theory based on the U (1) symmetry. Taking into account that color symmetry is an exact SU (3) symmetry, the same procedure of QED to construct a gauge theory based in the SU (3) color symmetry can be followed. The quark Lagrangian is   L = q¯j iγ μ ∂μ − m qj

(1)

to be invariant under a local SU (3) transformation q(x) → e−˙ı αa (x)Ta q(x)

(2)

with Ta the SU (3) generators and αa (x) the local phases. The locality of the transformation induces extra terms that have to be absorbed defining a covariant derivative Dμ = ∂μ − ı˙gTa Gaμ

(3)

where Gaμ is the gluon field and transforms under infinitesimal transformations as Gaμ → Gaμ −

1 ∂μ αa + fabc αb Gcμ . g

(4)

Finally the Lagrangian is given by   1 LQCD = q¯ iγ μ Dμ − m q − Gaμν Gμν a 4

(5)

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or    μ  1 LQCD = q¯ iγ μ ∂μ − m q + g qγ ¯ Ta q Gaμν − Gaμν Gμν a 4

(6)

This theory called quantum chromodynamics (QCD) (Fritzsch et al. 1973; Gross and Wilczek 1973; Politzer 1973; Weinberg 1973) is considered the true theory of the strong interaction. An important difference between QED and QCD is that SU (3) is not an abelian group, which is reflected in the fact that the structure constants fabc are different from zero, and therefore, the gluon fields do not commute with each other, producing two peculiar properties of the QCD Lagrangian. The first one is called asymptotic freedom and consists of the fact that the effective interaction between quarks decreases as the momentum transfer increases. This property allows to make perturbative calculation at this range of momenta. On the other hand, the QCD coupling constant increases as the transferred momentum decreases making QCD highly non-perturbative at the hadron scale. It has been speculated that this behavior leads to the confinement of colored states. Although it has not been yet proved how QCD really produces confinement, it has been demonstrated in lattice calculations that the long range part of the quarkquark interaction behaves as a linear rising potential which would be screened by meson pair creation (Bali et al. 2005). The difficulties to solve QCD at low momentum transfer makes necessary to resort to models that mimic the most important properties of QCD. These models allow to describe in terms of quark degrees of freedom the hadron masses and other hadron properties. Much research has been devoted to study the hadron phenomenology often with considerable success. Among them, the pioneering work of De Rújula et al. (1975) is of special relevance. These authors argue that the effective quark-quark interaction at short range arises from the one gluon exchange between quarks. This interaction resembles the Fermi-Breit interaction between electrons generalized to fermions with color and arbitrary masses. The model is able to explain successfully hadron mass relationships and develop a quantitative understanding of many features of the hadron mass spectrum and electromagnetic properties like the hadron magnetic moments. The price to pay for this description is to accept an effective quark mass of the order of 300 MeV/c2 .

Chiral Symmetry and the Constituent Quark Model Chiral symmetry has been shown to be a crucial symmetry in hadron and nuclear physics. The basic idea is that for zero quark masses, which is usually refereed as the chiral limit, the quarks have well-defined right-handed or left-handed chirality. The right-handed and left-handed quark fields are defined as

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qR = PR q

qL = PL q

(7)

with γ5 = iγ 0 γ 1 γ 2 γ 3 . PR =

1 1 + γ5 2

PL =

1 1 − γ5 2

(8)

The QCD Lagrangian can be rewritten as 1 LQCD = q¯R iγ μ Dμ qR + q¯L iγ μ Dμ qL − q¯L mqR − q¯R mqL − Gaμν Gμν a 4

(9)

so in the chiral limit, the non-diagonal terms on chirality cancel, and chirality is conserved. However if chiral symmetry was conserved, this symmetry should be seen in the hadron spectra. For example, chiral symmetry would imply the existence of a partner of opposite parity for each meson, which is clearly not the case in the experimental meson spectra. Chiral symmetry is explicitly broken since the current quark masses are nonzero; however, in the light sector, they have very small values, and one would not expect to describe the large deviations observed with such small values. For this reason it is assumed that chiral symmetry is spontaneously broken by the QCD vacuum. The Goldstone theorem tells us that the breaking of a continuous symmetry implies the existence of Goldstone bosons, that in the chiral limit would be massless, but with a small explicit breaking, they would be expected to be light bosons. The pions are perfect candidates. The spontaneous chiral symmetry breaking effect has a long history. A model inspired in the instanton liquid model (Dyakonov and Petrov 1986) will be used here. It provides a very natural non-perturbative explanation of this effect. The model provides an effective Lagrangian for quarks and Goldstone bosons given by ¯ / − M(q 2 )U γ5 )ψ L = ψ(i∂ i λa φ γ

(10)

with ∂/ ≡ γ μ ∂μ , ψ is the quark field, U γ5 = e fπ a 5 is a unitary transformation, and φa is the Goldstone boson field matrix. M(q 2 ) is the dynamical (constituent) quark mass that vanishes at large momenta, and it is frozen at low momenta to a value around 300 MeV/c2 . This dynamical mass justifies the effective quark mass needed to explain the baryon magnetic moments in the naive quark model. It is important to notice that even in the chiral limit, the dynamical quark masses do not vanish due to the spontaneous chiral symmetry breaking effect. This effect has been seen on the lattice (see Fig. 1 of Bowman et al. 2005) and has been compared with the instanton liquid model prediction (see Fig. 1 of Musakhanov 2012) or the predictions in the Dyson-Schwinger approach (Brodsky et al. 2012) (see Fig. 1). The other important consequence is the appearance of the interaction between quarks and pions that would generate the interaction between hadrons and pions. For

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example, the pion-exchange interaction of the NN force, which is the best known part of the nuclear interaction. Based on these ideas, a constituent quark model was developed by the groups of Tuebingen and Salamanca (Fernandez et al. 1993c; Entem et al. 2000). Since some of the results presented in this chapter have been obtained within this model, its most important ingredients will be described here. Although the function M(q 2 ) can be obtained from the instanton liquid model, it will be parametrized as  M(q 2 ) = mq

Λ2χ

1/2

Λ2χ + q 2

(11)

where mq ∼ 300 MeV/c2 is the constituent quark mass and Λχ fixes the chiral symmetry breaking scale. Expanding the matrix U γ5 on the Goldstone boson fields U γ5 = 1 +

i 1 γ5 λ a φa − φa φa + . . . fπ 2fπ2

(12)

The first term in the expansion generates the dynamical quark mass term. The second term generates the coupling of dynamical quarks and Goldstone bosons, and the first consequence is the appearance of the one-boson-exchange interactions between quarks. The third term generates the coupling between the dynamical quarks and two Goldstone bosons. This term will contribute to one-loop diagrams for the interaction between quarks, i.e., two-pion exchange terms in the light sector. The main contribution of this kind of diagrams is given by the scalar part, and this contribution will be mimicked by the exchange of a σ scalar boson. The expression of the one-boson-exchange interactions generated by this effect is in momentum space VijP S ( q) = − VijS ( q) = −

2 Λ2χ gch ( σi · q)( σj · q) 1 ( τi · τj ) 3 2 2 2 2 (2π ) 4mq Λχ + q mP S + q 2

(13)

2 Λ2χ gch 1 3 2 2 2 (2π ) Λχ + q mS + q 2

(14)

where q = p − p is the momentum transfer, the σ (τ ) are the spin (isospin) Pauli matrices, mq is the constituent quark mass, and gch = mq /fπ . The pseudo-scalar P S boson is the pion (mP S = mπ ), and the scalar S is identified with the π π resonance f0 (500). Here, momentum states normalized to 1 are used. However, as already mentioned, the spontaneous chiral symmetry breaking is not the only non-perturbative effect from QCD. One important aspect is that free quarks have not been observed in experiments. They have only been observed confined in hadrons, so in colorless objects. In hadron collisions, if a quark is ejected

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from a hadron, the necessary energy to separate it becomes so big that it is more efficient to create quark-antiquark pairs and create two separate colorless objects. The potential grows linearly at small distances, but at a certain value the string breaks and generates q q¯ pairs, with an interaction energy much smaller between the new hadrons. To describe such a behavior, the model uses an screened linear confinement potential of the form i · λ j ) VijCON (r) = [−ac (1 − e−μc r ) + Δ](λ

(15)

i · λ  j ), The potential at small r  1/μc is linear with a string strength σ = −ac μc (λ while for large r 1/μc tends to a constant. Finally at short distances one would expect that perturbation theory will work. The model also incorporates perturbative effects in this range that at first order are given by the exchange of gluons. The expression of the potential is 1 1 i · λ  j ) 4π αs (λ VijOGE ( q) = (2π )3 4



1 1 − q2 4m2q 

2

2 1 1 + 2 2 q ⊗ q · σi ⊗ σj 4mq q



 2 1 + ( σi · σj ) + 3 (16)

where αs = g/4π . It is important to notice that confinement and the one-gluon-exchange potential  i ·λ  j , and so the interaction between colorless objects cancel. As are proportional to λ previously seen, all hadrons are colorless, and so this part of the interaction does not give a contribution to the interaction between hadrons.(As it will be shown, when there is also an exchange of quarks between hadrons, there is a contribution which is a short-range contribution.) However, the Goldstone boson exchange does not have this factor, and it is of most importance for the study of the interaction between hadrons.

The Nucleon-Nucleon Interaction in Constituent Quark Models As explained in the previous section, at low energies constituent quark degrees of freedom can be considered as nonrelativistic particles (or with relativistic kinematics) that interact among them. In the quark model picture the nucleonnucleon (or in general the hadron-hadron) interaction is a residual force from the underlying quark-quark potential, in a similar way the atom-atom interaction is given in terms of its constituents. However there is a very important difference; most of the quark-quark interactions are color interactions as seen previously, and for this reason the contribution to the interaction between colorless objects cancel (It will be seen that there is a short-range contribution due to the Pauli principle.). This implies that at large distances the dominant terms are given by contributions

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without color. In the model previously covered, these are given by the exchange of Goldstone bosons due to the spontaneous chiral symmetry breaking. In this way the well-known one-pion-exchange contribution to the nucleon-nucleon interaction is included, which is the longest and best known component of the NN interaction. The quark degrees of freedom are more relevant at short distances, where the Pauli principle allows to exchange quarks between nucleons, and so the color interactions do not cancel anymore. In fact this mechanism was proposed in 1975 to explain the short-range repulsion or hard core of the NN interaction (Neudatchin et al. 1975, 1977). First attempts to describe the NN interaction in terms of quark degrees of freedom were performed in potential models (Liberman 1977), in bag models (DeTar 1978) or in the Born-Oppenheimer approach (Born and Oppenheimer 1927). In 1980, the resonating group method (RGM) (Wheeler 1937), which was used to study composite systems as nuclear systems (Wildermuth and Tang 1977), was first used to study the N N interaction in terms of its quark constituents (Oka and Yazaki 1980; Ribeiro 1980; Toki 1980). The question of the origin of the hard core was revisited by Harvey (1981), who obtained, instead of a hard core, some weak attraction that was attributed to the inclusion of hidden-color components. However, the situation was restored, even with hidden-color components, when a more consistent model was used (Faessler et al. 1982). The first quantitative models to describe the NN interaction were done in two ways. The first one introduced effective meson exchange potentials in the RGM equations (Oka and Yazaki 1983; Maltman and Isgur 1984; Fujiwara et al. 2001). The second one coupled meson degrees of freedom directly to quark degrees of freedom (Faessler and Fernandez 1983; Shimizu 1984; Oka et al. 2000). The main difference between the two ways is on the short range part, which can be relevant in certain cases (Fernández 1987). As mentioned previously, a natural way to couple mesons and quarks is given by models in which meson degrees of freedom appear as the Goldstone bosons of the spontaneously broken chiral symmetry. Additionally, chiral symmetry imposes stronger constraints in the couplings between quarks and mesons.

The Resonating Group Method To obtain the interaction between nucleons from the quark interactions, mainly two approaches have been used: the Born-Oppenheimer approximation (Born and Oppenheimer 1927) and the resonating group method (RGM) (Wheeler 1937). Here, an overview of the second one will be given. The main idea of the RGM is that the degrees of freedom of the particles inside a cluster are frozen, fixing the wave function of the internal degrees of freedom, and the interactions only contribute to the dynamics of relative degrees of freedom between clusters.

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Traditionally the RGM has been developed in coordinate space. However antisymmetry generates non-localities in the potentials between clusters, so the final RGM equation is an integro-differential equation which makes more it complicated to solve it. However, it can also be formulated in momentum space, and in this way the treatment of local or non-local interactions is completely equivalent, getting always an integral equation. It is also important to notice that in momentum space the coupling between different channels is very easily implemented, while in coordinate space it is much more complicated. The baryon total wave function is written as ψB = φB (pξ1 , pξ2 )χB ξc 13

(17)

where B can be a nucleon or its isospin excitation the Δ, φB is the spatial wave function depending on the internal Jacobi momenta pξ1 and pξ2 of the quarks inside the baryon 1 (p1 − p2 ) 2 2 1 = p3 − (p1 + p2 ) 3 3

pξ1 = pξ2

(18)



χB is the spin-isospin wave function of the baryon, and ξc 13 is the color singlet wave function. The Jacobi coordinates used are for quarks with equal masses. Ideally one would use for φB the wave function obtained from the solution of the three body problem; however to simplify the calculations, one usually uses gaussian wave functions of the form 

2b2 φB (pξB ) ≡ φB (pξ1 , pξ2 ) = π

 34 e

−b2 pξ2

1



3b2 2π

 34

2

e

− 3b4 pξ2

2

(19)

which is symmetric to the exchange of quarks. When b is conveniently chosen, the potential obtained with the solution of the Schrödinger equation or only these two gaussians is very similar (Valcarce et al. 1996) and strongly simplifies the calculation. Now, the two-baryon wave function can be written as

ψB1 B2 = A χ (P )ψBST1 B2 = A φB1 (pξB1 )φB2 (pξB2 )χ (P )χBST1 B2 ξc 23 (20) where χBST1 B2 is the wave function of the two-baryon system coupled to total spin S

and total isospin T , ξc 23 is the product of the two color singlet wave functions, A the antisymmetrizer operator, and χ (P ) is the relative wave function between clusters.

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In the case of the B1 B2 interaction for N and Δ baryons, the three quarks in each nucleon are identical, and the antisymmetrizer can be written as A =

1 (1 − P )(1 − 9P36 ) 2

(21)

with P = P14 P25 P36 . The term 12 (1 − P ) fixes the symmetry of the wave function at the baryon level. It was shown in Fernández et al. (1993a) that this operator is properly taken into account writing the wave function as μJ LST C

ψB1 B2

=

1 2(1 + δB1 B2 )

C C (ψBJ 1LST + (−1)μ+S1 +S2 −S+I1 +I2 −I ψBJ 2LST B2 B1 )

(22)

where μ fixes the symmetry of the wave function at the baryon level and L + μ = odd. For N N states, the well-known relation L + S + I = odd is obtained. Thus the antisymmetrizer reduces to A = (1 − 9P36 ). The dynamics of the system is given by the Hamiltonian H =

N   pi2 + Vij − TCM 2mq

(23)

i 0. With the S matrix the observables for scattering processes can be obtained.

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Phase Shifts in the Constituent Quark Model Now, Some results for NN phase shifts in the constituent quark model will be given. It is important to noticed that the pseudo-scalar boson exchange coming from the spontaneously symmetry breaking will be identified with the one pion exchange interaction (OPE). The coupling constant gch is then chosen to obtain the long-range Yukawa potential for OPE. The Fourier transform of the direct kernel corresponding to the pion exchange interaction between quarks is given by VcP S =

2 −mπ r m2π 1 gch 2e ρ(im ˜ π) 2 3 4π 4mq r

 2 5 ( σiN · σjN )( τiN · τjN ) 3

(34)

where ρ(q) ˜ is the Fourier transform of the quark density normalized to ρ(q ˜ = 0) = 1. The usual OPE contribution is written as VcOP E = and using ρ(q) ˜ = e−

b2 q 2 6

1 fπ2N N e−mπ r N ( σi · σjN )( τiN · τjN ) 3 4π r

obtained from RGM, one gets the relation 2 gch = 4π

f2

(35)

 2 2 fπ N N 4m2q − b2 m2π 5 e 3 3 4π m2π

(36)

Taking the value πNN 4π = 0.0749 (Bergervoet et al. 1990), the value given in Table 2 is obtained. When two nucleons interact, the interaction between quarks can produce excitations in the nucleon, producing, for example, a Δ resonance. For this reason, when calculating NN scattering, ΔΔ intermediate states in the I = 0 channel, and ΔΔ and N Δ states in the I = 1 channel are also included. The most important channels are, of course, NΔ which are important to describe with the same model the two isospin channels. However, it is important to notice that here there is no new parameter including these channels, since everything is fixed from the underlying quark substructure. The most relevant partial waves for the NN interaction are the S waves. The NN state can be in 1 S0N N and 3 S1N N partial waves, where the spectroscopic notation is used. The second one is the deuteron channel, while in the first one there is a virtual N N state. For the 1 S0N N channels, there is no coupling with other NN partial waves. However it can couple to the 5 D0N Δ channel and 1 S0ΔΔ and 5 D0ΔΔ channels. The results are shown in Fig. 3a. The dashed line includes only NN states, the dotted line includes also NΔ states, and the solid line also includes ΔΔ states. A more refined calculation, including also isospin breaking terms, can be found in Entem et al. (1999).

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Fig. 3 N N S wave phase shifts for I = 1 (a) and I = 0 (b). Experimental points with and without error bars correspond to the Energy-independent and energy-dependent solutions of Arndt et al. (1997), respectively. The phase shifts shown and the analysis correspond to neutron-proton. (a) Dashed line represents the calculation including N N channels only, dotted line includes also N Δ components, and solid line is the full calculation with N N , N Δ and ΔΔ channels. (b) Dashed line is the calculation with N N only, and solid line is the full calculation including N N and ΔΔ channels. (Figure reprinted from Entem et al. (2000) with permission. Copyright(2000) by the American Physical Society)

Table 3 Low-energy scattering parameters from Entem et al. (2000). The result of the OBEP and Paris potential are from Machleidt (1989) and Lacombe et al. (1980), respectively. Experimental data are from Houk (1971), Dilg (1975), Klarsfeld et al. (1984) anp (fm) rnp (fm) at (fm) rt (fm)

Quark -27.010 2.64 5.437 1.779

OBEP -23.750 2.71 5.424 1.761

Paris -17.612 2.88 5.427 1.766

Exp. -23.748(10) 2.75(5) 5.419(7) 1.754(8)

In Fig. 3b the results for the 3 S1N N partial wave are shown. In this case the 3 D1N N partial wave is coupled to the S wave, and 3 S1ΔΔ , 3 D1ΔΔ , 7 D1ΔΔ , and 7 GΔΔ partial 1 waves are also coupled. It can be seen that the full result is in good agreement with empirical phase shifts. The low-energy parameters for S waves are given in Table 3. One can see that the low-energy parameters are well reproduced. It is also interesting to notice that the short-range repulsion of the nuclear force is well described in the quark model, showing a repulsive character at high energies. Another typical example of the repulsive core is given by the 1 P1N N interaction. In Fig. 4 the result of the for this partial wave is shown in two cases, when the exchange terms due to antisymmetry are not included (dash-dotted line) and the full result (solid line). The first one shows an attractive character, while the second one is repulsive due to the short-range repulsion, in agreement with experimental phase shits.

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Fig. 4 N N 1 P1 phase-shift. Dashed and solid lines have the same meanings as in Fig. 3b. Dashed-dotted line shows the effect of antisymmetry, corresponding to the result when all the exchange kernels are removed. (Figure reprinted from Entem et al. (2000) with permission. Copyright(2000) by the American Physical Society)

The phase shifts for L > 1 are in reasonable agreement with experiment, as seen in Figs. 4–9. However, as explained in Entem et al. (2000), the interaction shows a lack of spin-orbit. The spin-orbit force has been always a problem in constituent quark models. Usually these models produce the spin-orbit force through the one gluon exchange interaction, which is not enough to explain the data. Other sources of spin-orbit interaction were studied in Valcarce et al. (1995). Here the spin-orbit coming from the one sigma exchange was studied. It was found that a combination of the scalarmeson-exchange and the OGE leads to a satisfactory description of the P -wave N N phase shifts. A similar conclusion can be found in Takeuchi (1994). Another possible source is the one arising as a relativistic effect from the confinement potential (the-so called Thomas term). In Koike (1986), this interaction is studied using a particular model of confinement (flip-flop model). The spin-orbit force generated by one-gluon exchange and by a flip-flop model for confinement gives results which are qualitatively similar to those reported by Valcarce et al. (1995). Recent attempts to overcome this problem have been done by Chen et al. using an extension of the quark delocalization color screening model (QDCSM) (Wang et al. 1992) which include a one-pion exchange with a short-range cutoff in the QDCSM Hamiltonian (Lu et al. 2003). The quark delocalization is achieved by writing wave function of each nucleon as a linear combination of left and right Gaussians in a two center cluster model approximation where the mixing parameter  is determined by the six quark dynamics. They obtain similar results as the Salamanca version of the constituent quark model (Entem et al. 2000) but replacing the σ -meson exchange by the quark delocalization and color screening mechanism (Chen et al. 2007). However this new mechanism does not contribute to solve the spin-orbit problem (Huang et al. 2008). Then, one has to conclude that the situation of the spin-orbit force in quark potential models is still quite controversial. To remove the remaining uncertainties,

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Fig. 5 N N 3 PJ phase shifts. (a) Curves have the same meaning as in Fig. 3a, b, and c. Dashed line corresponds to the result with N N channels only, and the solid line includes N N and N Δ channels. (Figure reprinted from Entem et al. (2000) with permission. Copyright(2000) by the American Physical Society)

a better understanding of the quark confinement is clearly needed (see also the discussion of this issue in Myhrer and Wroldsen 1988).

Bound States and Resonances The N N system has only one bound state, the deuteron. The quantum numbers of the deuteron are J P = 1+ with I = 0, which are given by 3 S1N N and 3 D1N N partial waves. In the previous section, the phase shifts in these two partial waves were shown to be in good agreement with experiment.

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Fig. 6 N N 1 D2 phase-shift. Solid, dashed, and dotted lines have the same meanings as in Fig. 3a. Dashed-dotted line represents the result without exchange kernels. (Reprinted figure with permission from Entem et al. (2000) Copyright (2000) by the American Physical Society)

The binding energy of the deuteron is known to high precision, and usually every model take it as an observable to fit the parameters. In the model described previously (Entem et al. 2000), this was done including NN and ΔΔ channels. The properties of the bound state obtained are given in Table 4, which are in agreement with experimental data. The results labelled as Quarkb are including only NN channels and slightly changing the coupling constant for the scalar exchange to get the correct deuteron binding energy. This shows that the effect of ΔΔ components is small and is only of relevance if one wants to describe the I = 0 and I = 1 S waves with the same set of parameters. The effect of NN ∗ states has been analyzed in Juliá-Díaz et al. (2002), showing that this component gives also an small effect. However, there are other possible resonance that can have effects in NN scattering. One example is NΔ resonances. In Entem et al. (2003) the scattering above the pion production threshold was studied within the same model. There, the phase shifts for the 1 D2N N and 3 F3N N partial waves show a resonance behavior, shown in the Argand plots (Fig. 12 of Entem et al. 2003). The reason is that in these cases the NN −N Δ coupling is very important. The resonance behavior of these two partial waves has been extensively discussed in the literature. The first interpretation in terms of dibaryon resonances was given by Hoshizaki (1978). Parametrizing the resonance part as a Breit-Wigner, he obtained 2.17 and 2.22 GeV for the masses of the dibaryons in these two partial waves. This behavior can also be explained due to the opening of the NΔ threshold; however the quark model also predicts a NΔ bound state in the 5 S2N Δ partial wave at an energy of 0.141 MeV below the NΔ threshold, which corresponds to a mass of 2.17 GeV (Mota et al. 2002). This is not the only dibaryon resonance predicted by the quark model. Also a ΔΔ bound state in the J P = 3+ I = 0 channel was predicted in the same reference (see Qing et al. (2000) and Ping et al. (2001) for other theoretical calculations). This is a candidate to explain the ABC effect measured by the WASA/CELSIUS Collaboration (Adlarson et al. 2011), although other explanations are possible (Ikeno et al. 2021).

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Fig. 7 N N 3 DJ phase shifts. (a) Lines are labeled as in Fig. 3b. (b) and (c) Solid line corresponds to the result with N N channels only. (Figure reprinted from Entem et al. (2000) with permission. Copyright(2000) by the American Physical Society)

Hyperon-Hyperon and Nucleon-Antinucleon Interactions The great advantage of formulating the nucleon-nucleon interaction in terms of quark degrees of freedom is that it can be migrated to other baryon systems with little or no modification. In this section three examples will be shown: the nucleonantinucleon interaction, the nucleon-hyperon interaction, and the pp¯ → ΛΛ¯ system. The nucleon-antinucleon potential is related with the quark model based nucleonnucleon one by the G-parity transformation defined by G = C eiπ I2

(37)

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Fig. 8 N N F phase shifts. In the I = 0 sector, the solid line includes N N channels only. In the I = 1 sector, the dashed line corresponds to including N N channels only, and the solid line considers also N Δ channels. (Figure reprinted from Entem et al. (2000) with permission. Copyright(2000) by the American Physical Society)

where C is the C-parity operator and I2 is the operator associated with the second component of the isospin. Then, starting from the quark-quark interaction described in the previous section, one can obtain the nucleon-antinucleon interaction in the following way. The N N¯ potential will have two pieces derived from the one pion and one sigma exchange by the G-parity transformation. They would not have a direct one gluon exchange because they involve quark exchange diagrams that in this case are forbidden. However, they will have annihilation contribution trough gluon and pion exchanges. All these contributions are parameter-free because they are fixed in the nucleon-nucleon sector.

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Fig. 9 N N G phase shifts. Lines have the same meanings as in Fig. 8. (Figure reprinted from Entem et al. (2000) with permission. Copyright(2000) by the American Physical Society) Table 4 Deuteron properties obtained from a quark model with ΔΔ channels (a) and without (b). The result of the OBEP and Paris potential are from Machleidt (1989) and Lacombe et al. (1980), respectively. Experimental data are from Martorell et al. (1995), Rodning and Knutson (1986), Ericson and Rosa-Clot (1983), Van Der Leun and Alderliesten (1982) ED (MeV) rm (fm) AS (fm−1/2 ) η

Quarka 2.2246 1.985 0.8941 0.0250

Quarkb 2.2246 1.976 0.8895 0.0251

OBEP 2.2246 1.9688 0.8860 0.0264

Paris 2.2249 1.9717 0.8869 0.0261

Exp. 2.224575(9) 1.971(6) 0.8846(8) 0.0256(4)

On top of that a realistic model should also include annihilation into mesons. Certainly these annihilation processes would be described at quark level, but the microscopic models are complicated, and most of the time they do not include all the possible contributions.

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Usually one chooses to describe such processes as an optical potential to simplify the calculations

Vopt ( q ) = iW e

−q 2 b2 3

(38)

where W gives the strength an b the range. The values of these parameters are fixed by fitting the total annihilation cross section for the pp¯ system. Following this scheme, in Entem and Fernández (2006, 2007), possible baryonium bound or resonant states were investigated in a quark model-based N N¯ interaction supplemented by a spin- and isospin-independent optical potential of the form given by Eq. (38). The major finding of these calculations is that the model is able to reproduce the isospin dependence of the N N¯ cross section using only the dependence coming from the real part of the interaction which precisely is the one constrained by the NN sector. The model is able to reproduce the energy level shift in protonium using a generalized Trueman formula showing that the large 3 P0 protonium energy shift can be justified as a combined effect of the pp¯ and the nn¯ coupled channel. Finally the model does not show any N N¯ bound state or quasibound state despite the fact that the near threshold enhancements in the pp¯ invariant mass spectrum reported by several collaborations in the B → Xp p¯ and J /ψ → Xpp¯ decays have been interpreted as a narrow baryonium state X(1835) (Loiseau and Wycech 2005). However in Entem and Fernández (2007) it was shown that the near threshold enhancement can be understood in terms of final state interaction in the outgoing pp¯ system, which in some sense explains the universality of the decay energy dependence. These results have been corroborated in a different model in Kang et al. (2015). The second example of a possible extension of the NN quark-model-based interaction to another sectors is the nucleon-hyperon interaction. Originally the model from Eq. (10) was introduced for SU (2), so λa where the Pauli matrices and φa were the pion fields. However, it can be obviously extended to SU (3) being now λa the eight Gell-Mann matrices and φa the octet meson fields from Fig. 1. This generalization was done by Zhang et al. (1994). The parameters of the model are fixed in the same way as in Fernández et al. (1993b), but now the oscillator parameter for the √ strange quark bs is obtained by scaling the parameter of the light quark bu as bs = mu /ms bu where mu and ms are the constituent mass of the light and the strange quark, respectively. Using this model, Zhang et al. (1994) studied the binding energy of the deuteron, the N N scattering phase Shifts, and the hyperon-nucleon cross section, obtaining results reasonably consistent with the experiment. More examples of the description of the baryon-baryon interactions in the SU (6) quark model can be found in Fujiwara et al. (2007). Until now the baryon-baryon interaction made in terms of quark degrees of freedom describes the same physics than other models, despite the fact that one

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can discuss if it is more fundamental or it is less accurate due to the lack of enough freedom in the parameters. Then it is interesting to look to those processes or systems in which the quark model description involves completely different physics from the other models. One of these processes is the pp¯ → ΛΛ¯ hyperon production process. From one experimental point of view, this process has two characteristic features. The first one is that the description of the energy dependence of the cross section after the threshold should involve higher partial waves than the S-wave. The second feature is that the ΛΛ¯ production occurs predominantly in a spin triplet state, being the singlet contribution almost completely suppressed. Two different models are used to describe the pp¯ → ΛΛ¯ process. At the quark level the ΛΛ¯ is produced via annihilation of a uu¯ quark pair in the pp¯ state following by a creation of a s s¯ pair by gluon and kaon s-channel exchanges. The conventional description in terms of meson degrees of freedom relies on the t-channel meson exchange. Examples of this two models can be found in Fujiwara et al. (2007) and Ortega et al. (2011). The two types of models involve different physics. The spin triplet state dominance can be understood in quark models because the s s¯ pair is produced by vector exchanges (gluons) which give rise to a spin 1 for the ΛΛ¯ states. In meson models, to produce the triplet spin dominance, one must add together the tensor pieces of the K and K ∗ exchanges, which may introduce a model dependence. The real part of the pp¯ and the ΛΛ¯ interaction is derived in both models using a G-parity transformation. In the case of the quark model, the interaction is the one described in the former paragraphs. In both cases the interactions are supplemented by an optical potential. Although both models reproduce the total and differential cross section with reasonable accuracy, the description of the polarization observables shows completely different patterns. The results of the quark model calculations provide a better description of the spin observables. However, the scarce and inaccurate existing data prevents for any definitive conclusion.

Conclusions In this article, a description of the nucleon-nucleon interaction in terms of quark degrees of freedom was presented. As QCD is non-perturbative at the scale of the nuclear phenomena, its most important characteristics using the constituent quark model were modeled. Leaving aside that this model is not derived from the fundamental theory, the use of quark and gluon degrees of freedom allows us to better understand the physics underlying some phenomena, like the hard core of the nuclear force or the role played by the quark antisymmetry, which in other models can be hidden in the employed parameters. The model allows for a reasonable description of the nucleon-nucleon phenomenology despite the fact that a quark-based model is less flexible to reproduce the experimental data. However, its main advantage is that it can describe a huge

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variety of phenomena, baryon and meson spectrum, baryon-baryon interactions, and few nucleon systems (deuteron, triton), with a unified (and sometimes reduced) set of parameters with a reasonable quality compared with other models. The main drawback of this kind of models is the way the values of its parameters are set. This makes it difficult to determine errors of the calculated observables and impossible to improve the model order by order. Acknowledgments This work has been partially funded by Ministerio de Ciencia, Innovación y Universidades under Contract No. PID2019-105439GB-C22/AEI/10.13039/501100011033 and by the EU Horizon 2020 research and innovation program, STRONG-2020 project, under grant agreement No. 824093.

References P. Adlarson et al., Phys. Rev. Lett. 106, 242302 (2011) R.A. Arndt, C. Heon Oh, I.I. Strakovsky, R.L. Workman, F. Dohrmann, Phys. Rev. C 56, 3005 (1997) G.S. Bali, H. Neff, T. Düssel, T. Lippert, K. Schilling (SESAM Collaboration), Phys. Rev. D 71, 114513 (2005) V.E. Barnes, P.L. Connolly, D.J. Crennell, B.B. Culwick, W.C. Delaney, W.B. Fowler, P.E. Hagerty, E.L. Hart, N. Horwitz, P.V.C. Hough, J.E. Jensen, J.K. Kopp, K.W. Lai, J. Leitner, J.L. Lloyd, G.W. London, T.W. Morris, Y. Oren, R.B. Palmer, A.G. Prodell, D. Radojiˇci´c, D.C. Rahm, C.R. Richardson, N.P. Samios, J.R. Sanford, R.P. Shutt, J.R. Smith, D.L. Stonehill, R.C. Strand, A.M. Thorndike, M.S. Webster, W.J. Willis, S.S. Yamamoto, Phys. Rev. Lett. 12, 204 (1964) J.R. Bergervoet, P.C. van Campen, R.A.M. Klomp, J.-L. de Kok, T.A. Rijken, V.G.J. Stoks, J.J. de Swart, Phys. Rev. C 41, 1435 (1990) J. Bjorken, Conf. Proc. C 670828, 490 (1967) J.D. Bjorken, Phys. Rev. 179, 1547 (1969) E.D. Bloom, D.H. Coward, H. DeStaebler, J. Drees, G. Miller, L.W. Mo, R.E. Taylor, M. Breidenbach, J.I. Friedman, G.C. Hartmann, H.W. Kendall, Phys. Rev. Lett. 23, 930 (1969) M. Born, R. Oppenheimer, Ann. Phys. 389, 457 (1927) P.O. Bowman, U.M. Heller, D.B. Leinweber, M.B. Parappilly, A.G. Williams, J. Zhang, Phys. Rev. D 71, 054507 (2005) S.J. Brodsky, C.D. Roberts, R. Shrock, P.C. Tandy, Phys. Rev. C 85, 065202 (2012) L. Chen, H. Pang, H. Huang, J. Ping, F. Wang, Phys. Rev. C76, 014001 (2007). http://arxiv.org/ abs/nucl-th/0703103, arXiv:nucl-th/0703103 [nucl-th] A. De Rújula, H. Georgi, S.L. Glashow, Phys. Rev. D 12, 147 (1975) C. DeTar, Phys. Rev. D 17, 323 (1978) W. Dilg, Phys. Rev. C11, 103 (1975) D. Dyakonov, V. Petrov, Nucl. Phys. B 272, 457 (1986) D.R. Entem, F. Fernández, Phys. Rev. C 73, 045214 (2006) D.R. Entem, F. Fernández, Phys. Rev. D 75, 014004 (2007) D. Entem, F. Fernández, A. Valcarce, Phys. Lett. B 463, 153 (1999) D.R. Entem, F. Fernández, A. Valcarce, Phys. Rev. C 62 (2000). https://doi.org/10.1103/physrevc. 62.034002 D.R. Entem, F. Fernández, A. Valcarce, Phys. Rev. C 67, 014001 (2003) T. Ericson, M. Rosa-Clot, Nucl. Phys. A 405, 497 (1983) A. Faessler, F. Fernandez, Phys. Lett. B 124, 145 (1983) A. Faessler, F. Fernandez, G. Lübeck, K. Shimizu, Phys. Lett. B 112, 201 (1982) F. Fernández, Phys. Lett. B 188, 314 (1987)

45 Quark Models for Baryon-Baryon Interactions

1785

F. Fernández, A. Valcarce, P. González, V. Vento, Phys. Rev. C 47, 1807 (1993a) F. Fernández, A. Valcarce, U. Straub, A. Faessler, J. Phys. G: Nucl. Part. Phys. 19, 2013 (1993b) F. Fernandez, A. Valcarce, U. Straub, A. Faessler, J. Phys. G: Nucl. Part. Phys. 19, 2013 (1993c) H. Fritzsch, M. Gell-Mann, H. Leutwyler, Phys. Lett. B 47, 365 (1973) Y. Fujiwara, M. Kohno, C. Nakamoto, Y. Suzuki, Phys. Rev. C 64, 054001 (2001) Y. Fujiwara, Y. Suzuki, C. Nakamoto, Prog. Part. Nucl. Phys. 58, 439 (2007) M. Gell-Mann, The Eightfold Way: A Theory of Strong Interaction Symmetry (1961). https://doi. org/10.2172/4008239 M. Gell-Mann, Phys. Lett. 8, 214 (1964) D.J. Gross, F. Wilczek, Phys. Rev. D 8, 3633 (1973) M. Harvey, Nucl. Phys. A 352, 326 (1981) N. Hoshizaki, Prog. Theor. Phys. 60, 1796 (1978). https://academic.oup.com/ptp/article-pdf/60/6/ 1796/5223022/60-6-1796.pdf T.L. Houk, Phys. Rev. C3, 1886 (1971) H.-X. Huang, J.-L. Ping, L.-Z. Chen, F. Wang, H.-R. Pang, Chin. Phys. Lett. 25, 1617 (2008) N. Ikeno, R. Molina, E. Oset, Phys. Rev. C 104, 014614 (2021) B. Juliá-Díaz, D.R. Entem, A. Valcarce, F. Fernández, Phys. Rev. C 66, 047002 (2002) X.-W. Kang, J. Haidenbauer, U.-G. Meißner, Phys. Rev. D 91, 074003 (2015) S. Klarsfeld, J. Martorell, D.W.L. Sprung, J. Phys. G10, 165 (1984) Y. Koike, Nucl. Phys. A 454, 509 (1986) M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. Cote, P. Pires, R. De Tourreil, Phys. Rev. C21, 861 (1980) D.A. Liberman, Phys. Rev. D 16, 1542 (1977) D.B. Lichtenberg, Unitary Symmetry and Elementary Particles, Academic Press, New York (1978) H.J. Lipkin, Phys. Rep. 8, 173 (1973) B. Loiseau, S. Wycech, Phys. Rev. C 72, 011001 (2005) X.-F. Lu, J.-L. Ping, F. Wang, Chin. Phys. Lett. 20, 42 (2003). [NUCL-TH] R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989) K. Maltman, N. Isgur, Phys. Rev. D 29, 952 (1984) J. Martorell, D.W.L. Sprung, D.C. Zheng, Phys. Rev. C 51, 1127 (1995) R.D. Mota, A. Valcarce, F. Fernández, D.R. Entem, H. Garcilazo, Phys. Rev. C 65, 034006 (2002) M. Musakhanov, PoS, Baldin-ISHEPP-XXI, 008 (2012). http://arxiv.org/abs/1212.0947, arXiv:1212.0947 [hep-ph] F. Myhrer, J. Wroldsen, Rev. Mod. Phys. 60, 629 (1988) Y. Ne’eman, Nucl. Phys. 26, 222 (1961) V.G. Neudatchin, I.T. Obukhovsky, V.I. Kukulin, N.F. Golovanova, Phys. Rev. C 11, 128 (1975) V.G. Neudatchin, Y.F. Smirnov, R. Tamagaki, Prog. Theor. Phys. 58, 1072 (1977) M. Oka, K. Yazaki, Phys. Lett. B 90, 41 (1980) M. Oka, K. Yazaki, Nucl. Phys. A 402, 477 (1983) M. Oka, K. Shimizu, K. Yazaki, Prog. Theor. Phys. Suppl. 137, 1 (2000) P.G. Ortega, D.R. Entem, F. Fernandez, Phys. Lett. B696, 352 (2011) J. Ping, F. Wang, T. Goldman, Nucl. Phys. A 688, 871 (2001) H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973) D. Qing, H.-M. Sun, F. Wang, Phys. Rev. C 62, 018201 (2000) J.E.F.T. Ribeiro, Z. Phys. C: Part. Fields 5, 27 (1980) N.L. Rodning, L.D. Knutson, Phys. Rev. Lett. 57, 2248 (1986) K. Shimizu, Phys. Lett. B 148, 418 (1984) S. Takeuchi, Phys. Rev. Lett. 73, 2173 (1994) H. Toki, Z. Phys. A: At. Nucl. 294, 173 (1980) A. Valcarce, A. Buchmann, F. Fernández, A. Faessler, Phys. Rev. C 51, 1480 (1995) A. Valcarce, P. González, F. Fernández, V. Vento, Phys. Lett. B 367, 35 (1996) C. Van Der Leun, C. Alderliesten, Nucl. Phys. A 380, 261 (1982) F. Wang, G.-H. Wu, L.-J. Teng, J.T. Goldman, Phys. Rev. Lett. 69, 2901 (1992). http://arxiv.org/ abs/nucl-th/9210002, arXiv:nucl-th/9210002 [nucl-th]

1786

P. G. Ortega et al.

S. Weinberg, Phys. Rev. Lett. 31, 494 (1973) J.A. Wheeler, Phys. Rev. 52, 1083 (1937) K. Wildermuth, Y.C. Tang, A Unified Theory of the Nucleus (Vieweg and Teubner Verlag, Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1977) Z.-Y. Zhang, A. Faessler, U. Straub, L. Glozman, Nucl. Phys. A 578, 573 (1994) G. Zweig, An SU(3) model for strong interaction symmetry and its breaking. CERN Yellow Report CERN-TH-401 (1964)

46

Lattice QCD and Baryon-Baryon Interactions Sinya Aoki and Takumi Doi

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HAL QCD Potential Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Extension: Coupled Channel Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NN Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Central and Tensor Interactions in Parity-Even Channels . . . . . . . . . . . . . . . . . . . . . . . . . . Central, Tensor, and Spin-Orbit Interactions in Parity-Odd Channels . . . . . . . . . . . . . . . . Three-Nucleon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon Interactions in the Flavor SU(3) Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H Dibaryon in the Flavor SU(3) Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ΛΛ−N Ξ Interactions at the Almost Physical Point and the Fate of the H-Dibaryon . . . N Ξ Interactions at the Almost Physical Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dibaryons at the Almost Physical Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Most Strange Dibaryon Ωsss Ωsss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Most Charming Dibaryon Ωccc Ωccc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparisons of Two Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N Ωsss Dibaryon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S. Aoki () Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan e-mail: [email protected] T. Doi Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_50

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Abstract

In this chapter, the current status on baryon-baryon interactions such as nuclear forces in lattice quantum chromodynamics (QCD) is reviewed. In studies of baryon-baryon interactions in lattice QCD, the most reliable method so far is the potential method, proposed by the Hadrons to Atomic nuclei from Lattice QCD (HAL QCD) collaboration, whose formulation, properties, and extensions are explained in detail. Using the HAL QCD potential method, potentials between nucleons (proton and neutron, denoted by N) in the derivative expansion have been extracted in various cases. The lattice QCD results shown in this chapter include a leading order (LO) central potential in the parity-even NN(1 S0 ) channel, LO central and tensor potentials in the parity-even NN(3 S1 -3 D1 ) channel, and a next-to-leading order (NLO) spin-orbit potential as well as LO potentials in the parity-odd channels. Preliminary results at the almost physical pion and kaon masses, in addition to exploratory studies on three-nucleon potentials, are presented. Interactions between generic baryons including hyperons, made of one or more strange quarks as well as up and down quarks, have also been investigated. Universal properties of potentials between baryons become manifest in the flavor SU(3) symmetric limit, where masses of three quarks, up, down, and strange, are all equal. In particular, it is observed that one bound state, traditionally called the H -dibaryon, appears in the flavor singlet representation of SU(3). A fate of the H dibaryon is also discussed with flavor SU(3) breaking taken into account at the almost physical point. Finally, various kinds of dibaryons, bound or resonate states of two baryons, including charmed dibaryons, have been predicted by lattice QCD simulations at the almost physical point.

Introduction Properties of nuclei and hypernuclei are ultimately controlled by quantum chromodynamics (QCD), which governs the dynamics of quarks and gluons. It is, however, too challenging to investigate nuclei directly from QCD, and thus the most important theoretical task is to determine hadron interactions such as nuclear forces and hyperon forces without introducing any model parameters other than the gauge coupling constant of QCD and quark masses. Currently, the only method which can systematically incorporate non-perturbative nature of QCD dynamics is the first-principles lattice QCD numerical simulations. Indeed, not only hadron masses themselves (Durr et al. 2008) but also hadron mass splittings such as a proton-neutron mass difference can be precisely determined in latest lattice QCD calculations (Borsanyi et al. 2015). In lattice QCD, hadron-hadron interactions have been mainly investigated by two methods, the finite volume method (Luscher 1991) and the HAL QCD potential method (Ishii et al. 2007; Aoki et al. 2010). In the finite volume method, twohadron spectra in finite volume are converted to scattering phase shifts through

46 Lattice QCD and Baryon-Baryon Interactions

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the Lüscher’s finite volume formula, while Nambu-Bethe-Salpeter (NBS) wave functions for two hadrons are converted to an interaction kernel or a non-local potential between two hadrons in the HAL QCD method, with which scattering phase shifts are extracted through the Schrödinger equation. Both methods are theoretically equivalent, but they have their own pros and cons in practice. The finite volume method has been mainly used for meson-meson systems in both center of mass (CM) and moving frames (Rummukainen and Gottlieb 1995) (see Briceno et al. (2018) for a review). On the other hand, the HAL QCD potential method has been applied mainly to baryon-baryon systems in the CM frame, and an extension to the moving frame has just begun (Aoki and Akahoshi 2022; Akahoshi and Aoki 2023). Some reviews are given in Aoki et al. (2008, 2012), Aoki (2010, 2011, 2013), and Aoki and Doi (2020). In the past, there were some controversies over the consistency between the finite volume method and the HAL QCD potential method. In the case of nucleon-nucleon (NN) interactions at heavier pion masses, while finite volume spectra predict that both deuteron and dineutron are bound (Yamazaki et al. 2011, 2012, 2015; Beane et al. 2012, 2013; Berkowitz et al. 2017; Orginos et al. 2015; Wagman et al. 2017), N N potential in the HAL QCD method indicates that two nucleons are unbound in both channels (Ishii et al. 2007, 2012; Aoki et al. 2010; Inoue et al. 2012). Then it has been pointed out that incorrect extractions of NN finite volume spectra are responsible for the NN controversy (Iritani et al. 2016, 2017, 2019a, b; Aoki et al. 2016, 2018) (see also Aoki and Doi 2020 for a summary), and this claim is being confirmed by recent studies on NN finite volume spectra (Francis et al. 2019; Hörz et al. 2021; Hörz 2022; Green et al. 2022; Nicholson et al. 2022; Wagman 2022) using more sophisticated methods such as the variational method (Luscher and Wolff 1990). Now it is the lattice QCD community’s consensus that NN systems are unbound at heavier pion masses in both channels. With the reliability of the HAL QCD method well established, results on baryon-baryon (BB) interactions in the HAL QCD method will be exclusively reported in this chapter.

HAL QCD Potential Method In this section, the HAL QCD potential method (Ishii et al. 2007; Aoki et al. 2010, 2012) is briefly reviewed.

Basic Formulation A key quantity in the HAL QCD method is the equal time NBS wave function, defined by  1 B1 B2 ψW (r)e−W t ≡  Ω|B1 (x + r, t)B2 (x, t)|B1 B2 ; W , ZB1 ZB2 x

(1)

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where Ω| is the QCD vacuum (bra-)state and |B1 B2 ; W  is a QCD eigenstate for  two baryons with equal masses mB1 = mB2 = mB , the CM energy W =

2 p2W + m2B , and a relative momentum pW . For simplicity, other quantum numbers such as spin and helicity are suppressed. A baryon operator B1,2 (x, t) is made of quarks, which is given by, e.g., for a local nucleon operator,   B1,2 (x) = Nq (x) = εabc uaT (x)Cγ5 d b (x) q c (x),

x = (x, t),

where C = γ2 γ4 is a charge conjugation matrix and q = u(d) for a proton (neutron). Extensions of formula in this section to unequal masses mB1 = mB2 are straightforward, but some formulae become more complicated. If the total energy lies below the inelastic threshold of one meson production as W < Wth := 2mB + mM , where mM is a mass of the lightest meson M coupled to this channel, the above NBS wave function satisfies the Helmholtz equation at large r = |r| > R as 

 B1 B2 2 pW + ∇ 2 ψW (r)  0,

pW = |pW |,

(2)

where R is a scale of the two-baryon interaction, which is short-ranged in QCD. More precisely, the NBS wave function for given orbital angular momentum  and total spin s behaves asymptotically as B1 B2 ψW (r; s) 

sin(pW r − π /2 + δs (pW )) iδs (pW ) e pW r

(3)

for large r > R, where δs (pW ) is a phase of the QCD S-matrix for B1 B2 scattering below the inelastic threshold, which is real thanks to the unitarity of the S-matrix (Lin et al. 2001; Aoki et al. 2005). The HAL QCD method introduces a non-local but energy-independent potential U (r, r ) from the NBS wave function as B1 B2 (EW −H0 )ψW (r)

p2 ∇2 B1 B2  = d 3 r  U (r, r )ψW (r ), EW = W , H0 = − , (4) 2μ 2μ

where μ = mB /2 is a reduced mass and W < Wth is always required. It is important to note that non-relativistic approximation is not required to define U (r, r ) even though Eq. (4) has a form of the Schrödinger equation. NBS wave functions for ∀ W < Wth are necessary to determine the non-local potential U (r, r ), which correctly reproduces scattering phase shifts δs (pW ) for all , s, and pW with W < Wth . In lattice QCD simulations, performed in finite boxes, however, only a limited number of NBS wave functions are available, so that a direct determination of the non-local potential is impractical. Instead, one needs to expand the non-local potential in terms of the velocity (derivative) with local coefficient functions as U (r, r ) = V (r, ∇)δ (3) (r − r ), where

46 Lattice QCD and Baryon-Baryon Interactions

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V (r, ∇) = V0 (r) + Vσ (r)σ 1 · σ 2 + VT (r)S12 + VLS (r)L · S +O(∇ 2 )



 LO

(5)

NLO

at lowest few orders for the NN case with a given isospin. Here V0 (r) is a central potential, Vσ is a spin dependent central potential with a Pauli matrix σi acting on a spinor index of an i-th baryon, VT (r) is a tensor potential with a tensor operator S12 = 3(r · σ 1 )(r · σ 2 )/r 2 − σ 1 · σ 2 , and VLS (r) is a spin-orbit potential with an angular momentum L = r × p and a total spin S = (σ 1 + σ 2 )/2. Each coefficient function is further decomposed into its flavor components. For example, in the case of NN interactions, the decomposition reads VX (r) = VX0 (r) + VXτ (r)τ 1 · τ 2 ,

X = 0, σ , T, LS, · · · ,

(6)

where τ i is the Pauli matrix acting on the SU(2) flavor index of the i-th baryon. The above form of the velocity expansion has already been found in Okubo and Marshak (1958) using a similar argument based on symmetries, though an expansion of the non-local potential is not unique. The leading order potential is simply obtained from one NBS wave function as V LO (r) =

B1 B2 (EW − H0 )ψW (r) B1 B2 ψW (r)

= V0 (r) + Vσ (r)σ 1 · σ 2 + VT (r)S12 .

(7)

Solving the Schrödinger equation with V LO (r) in an infinite volume, one obtains LO (q), which satisfy δ LO (p ) = δ (p ) since ψ B1 B2 (r) scattering phase shifts δs W s W W s LO (q) = is a solution to the Schrödinger equation with V LO (r) at q = pW , but δs δs (q) for general q = pW . Hereafter, a momentum pW , which gives a correct phase shift δs (pW ), is called an “anchor.” If NBS wave functions at two different energies W1 and W2 are obtained, the non-local potential can be determined at NLO (q) not the NLO in the velocity expansion. The NLO scattering phase shifts δs NLO NLO only satisfy δs (pW1 ) = δs (pW1 ) and δs (pW2 ) = δs (pW2 ) but also give good approximations of δs (q) between two anchors, pW1 and pW2 . In this way, predictions of scattering phase shifts in the HAL QCD method can be improved by increasing order of the velocity expansion of the non-local potential. See some explicit demonstrations in Aoki (2019) and Aoki and Yazaki (2022). Since NBS wave functions cannot be obtained directly in lattice QCD, a correlation function for two baryons is considered instead, FJB1 B2 (r, t) =

 Ω|B1 (x + r, t + t0 )B2 (x, t + t0 )JB†1 B2 (t0 )|Ω,

(8)

x

where JB†1 B2 (t0 ) is a source operator which creates two baryon states at time t0 . If a time separation t is large enough to suppress inelastic contributions to the correlation function, it is shown that

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FJB1 B2 (r, t) =



−Wn t 1 B2 B1 B2 AB + ··· , J,n ψWn (r)e

(9)

n

 † 1 B2 where AB J,n = ZB1 ZB2 B1 B2 ; Wn |JB1 B2 (0)|Ω and ellipses represent contributions from inelastic states such as B1 B2 M, which are suppressed at least as e−Wth t . In the limit of t → ∞, the correlation function reduces to the NBS wave function with the lowest energy as   −W0 t 1 B2 B1 B2 lim FJB1 B2 (r, t) = AB + O e−Wn=0 t , J,0 ψW0 (r)e

(10)

t→∞

where W0 is the lowest energy of B1 B2 states. Wn=0 − W0 = O(L−2 ) in lattice QCD on a finite box with a volume L3 , where we neglect the effect of a binding energy (if there exists a bound state) for simplicity. (See Gongyo and Aoki (2018) if a bound state appears.) The leading order potential is obtained as V LO (r) = lim V LO (r, t), t→∞

V LO (r, t) :=

(EW − H0 )FJB1 B2 (r, t) FJB1 B2 (r, t)

,

(11)

which is independent on the source operator JB†1 B2 , since a source-dependent

1 B2 constant AB J,0 is cancelled in the above ratio as t → ∞. For the above procedure

to work, the ground-state saturation in FJB1 B2 , Eq. (10), must be satisfied by taking a large t. In practice, however, FJB1 B2 becomes very noisy at large enough t, in particular for two baryons. To overcome this difficulty, an alternative extraction of potentials has been introduced. A ratio of correlation functions is defined by FJB1 B2 (r, t) , GB1 (t)GB2 (t)  ¯ GB (t) := Ω|B(x, t)B(0, 0)|Ω  ZB e−mB t + · · · ,

RJB1 B2 (r, t) :=

(12)

x

which behaves RJB1 B2 (r, t) 

1 B2  AB J,n

n

ZB1 ZB2

B1 B2 ψW (r)e−ΔWn t , n

ΔWn := Wn − 2mB ,

(13)

for t 1/Wth , where inelastic contributions can be neglected. Using ΔW =

2 pW (ΔW )2 − , mB 4mB

B1 B2 B1 B2 (EW − H0 )ψW (r) = V (r, ∇)ψW (r),

(14)

46 Lattice QCD and Baryon-Baryon Interactions

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the ratio is shown to satisfy   ∂ 1 ∂2 RJB1 B2 (r, t) = V (r, ∇)RJB1 B2 (r, t), −H0 − + ∂t 4mB ∂t 2

(15)

where V (r, ∇) can be extracted in terms of the velocity expansion as mentioned before. The required condition, t 1/Wth (saturation of elastic states), is much easier to be achieved than the condition in the usual finite volume method, t 1/(W1 − W0 )  O(L2 ) (saturation of the ground state). Equation (15) is called the time-dependent HAL QCD method (Ishii et al. 2012).

An Extension: Coupled Channel Potentials The HAL QCD potential method can be extended to certain types of coupled channel systems (Aoki et al. 2011; Sasaki 2010). For simplicity, a case that X1 +X2 → Y1 +Y2 scattering occurs in addition to an elastic scattering X1 +X2 → X1 + X2 is considered, where X1,2 , Y1,2 are one-particle states of hadrons, and mX1 + mX2 < mY1 + mY2 < W < Wth with W and Wth being the total energy and the inelastic threshold of the coupled channel system, respectively. For simplicity, mX1 = mX2 = mX and mY1 = mY2 = mY are assumed in the following. In this case, NBS wave functions are generalized as ΨWX (r)e−W t =  ΨWY (r)e−W t = 

 Ω|X1 (x + r, t)X2 (x, t)|X + Y ; W ,

1 ZX1 ZX2 1 ZY1 ZY2

(16)

x

 Ω|Y1 (x + r, t)Y2 (x, t)|X + Y ; W ,

(17)

x

where |X + Y ; W  is a QCD eigenstate in the coupled channel system, which may be expressed as |X + Y ; W  = cX |X1 X2 ; W  + cY |Y1 Y2 ; W  + · · · ,  |H1 , pin ⊗ |H2 , −pin |H1 H2 ; W  =

(18) (19)

H |p|=pW

for H = X, Y . Here |H1,2 , pin is an in state for a hadron H1,2 with a momentum H is a magnitude of a relative momentum in the channel H , given by W = p, and pW  H )2 + m2 for H = X, Y . 2 (pW H A coupled channel non-local potential is defined by 

H )2 (pW ∇2 + 2μH 2μH

 ΨWH (r)

=

 H  =X,Y



d 3 r  U H H  (r, r )ΨWH (r ),

(20)

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where μH = mH /2 is a reduced mass in a channel H . At the leading order of the velocity expansion, two pairs of NBS wave functions at two different energies W1 and W2 satisfy  K

H

Wi (r)

:=

H )2 (pW i

∇2 + 2μH



ΨWHi (r) 2μH   = V H H  (r)ΨWHi (r), i = 1, 2,

(21)

H  =X,Y

where V H H  (r) is a local potential matrix at the leading order, which is solved as 

V X X (r) V X Y (r) V Y X (r) V Y Y (r)



 =

K X W1 (r) K X W2 (r) K Y W1 (r) K Y W2 (r)



ΨWX1 (r) ΨWX2 (r) ΨWY 1 (r) ΨWY 2 (r)

−1 . (22)

Once the coupled channel potential matrix V H H  (r) is obtained, physical observables such as scattering phase shifts and inelasticities can be extracted as a function of W by solving a coupled channel Schrödinger equation in an infinite volume with appropriate boundary conditions. As in the case of the single channel potential, NBS wave functions are extracted in lattice QCD from correlation functions as FJH (r, t) :=

 Ω|H1 (x + r, t)H2 (x, t)JH† (0)|Ω x

H −W0 t H −W1 t + AH + O(e−W2 t )  AH J,0 ΨW0 (r)e J,1 ΨW1 (r)e

(23)

for large t, where W0 < W1 < W2 are three lowest eigen-energies. Using two † † different source operators, J1,H and J2,H , we may extract ΨWH0 (r) and ΨWH1 (r) for H = X, Y , but this may be a harder task than an extraction of ΨWX0 (r) in the single channel. Therefore the time-dependent method is also employed in the coupled channel case. Using

RJHi (r, t) :=

FJHi (r, t) GH1 (t)GH2 (t)



 AH Ji ,n n

ZH1 ZH2

ΨWHn (r)e−ΔWn t , H

(24)

for H = X, Y and i = 1, 2, where Ji represent two different source operators, ΔWnH = Wn − mH1 − mH2 , and t is taken to be large enough to ignore inelastic contributions, it is easy to show

46 Lattice QCD and Baryon-Baryon Interactions



∇2 ∂ 1 ∂2 − + 2μH ∂t 8μH ∂t 2

1795

 RJHi (r, t)





Δ

H

d 3r 

H  (t)

H 

U H H  (r, r )RJHi (r , t),

(25)

where  Δ

H

H  (t)

=

ZH1 ZH2 e−(mH1 +mH2 )t , ZH1 ZH2 e−(mH1 +mH2 )t

(26)

which is necessary to correct differences in masses and Z factors between two channels. Denoting the left-hand side of Eq. (25) as K H Ji (r, t), the LO potential, U H H  (r, r ) = V H H  (r)δ (3) (r − r ), is extracted as 

ΔX Y (t)V X Y (r) V X X (r) Y Y V Y Y (r) Δ X (t)V X (r)





=

 K X J1 (r, t) K X J2 (r, t) K Y J1 (r, t) K Y J2 (r, t) −1  RJX1 (r, t) RJX2 (r, t) . RJY1 (r, t) RJY2 (r, t)

(27)

NN Interactions In this section, results on nucleon-nucleon (NN) interactions in the HAL QCD potential method are summarized.

Central and Tensor Interactions in Parity-Even Channels In the parity-even channels between two nucleons, 1 S0 (the isospin triplet and the spin singlet S wave) and 3 S1 − 3 D1 (the isospin singlet and the spin triplet S and D waves) have been investigated by the LO analysis. Since the statistical fluctuations for two nucleons increase as a pion mass decreases, the studies have been performed mainly at heavy pion masses. The LO potential for 1 S0 is simply obtained from Eq. (15), while the leading order potential in the 3 S1 − 3 D1 channel has two terms, the central potential and the tensor potential as 

1 ∂2 ∂ −H0 + + ∂t 4mN ∂t 2



  RJN N (r, t)  VCI (r) + VTI (r)S12 RJN N (r, t), (28)

where I = 0 represents a total isospin of two nucleons for the 3 S1 − 3 D1 channel. To disentangle a central potential VCI (r) and a tensor potential VTI (r), RJN N (r, t)

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+

is decomposed into two contributions, P A1 RJN N (r, t) and (1 − P A1 )RJN N (r, t), + where P A1 is a projection operator to the A+ 1 representation of the cubic group + A+ while 1 − P 1 is a projection orthogonal to P A1 . Since the A+ 1 representation of + the cubic group on a cubic box contains  = 0, 4, 6 partial waves, P A1 RJN N (r, t) + and (1 − P A1 )RJN N (r, t) are expected to be dominated by the  = 0 component (the S wave) and the  = 2 component (the D wave) at low energies, respectively. These expectations are indeed confirmed numerically (Aoki et al. 2010). Using these two components, VCI (r) and VTI (r) are extracted by solving linear equations, as in the couple channel case. For a more sophisticated partial wave decomposition, see Miyamoto et al. (2020). Figure 1 (left) shows the central potential in the 1 S0 channel, obtained in (2+1)-flavor lattice QCD (Ishii 2013) using ensembles generated by the PACS-CS Collaboration at the lattice spacing a  0.091 fm (a −1 = 2.176(31) GeV) on 323 × 64 (Aoki et al. 2009). Quark masses of these ensembles correspond to (mπ , mN )  (701, 1584) MeV (blue), (570, 1412) MeV (green), and (411, 1215) MeV (red). The central potential at each pion mass reproduces qualitative features of the phenomenological NN potential, namely, a repulsive core at short distance surrounded by an attractive well at medium and long distances. A range of a tail structure at long distance in the central potential becomes wider as pion mass decreases. This behavior may be understood from a viewpoint of the one-pion exchange between nucleons. At short distance, on the other hand, a height of the repulsive core increases as pion mass decreases. This pion mass dependence of the short range repulsion could be explained by the one-gluon exchange in the quark model, whose strength is proportional to an inverse of the constituent quark mass. Figure 1 (right) shows the scattering phase shifts, extracted through the Schrödinger equation in the infinite volume with the corresponding central potential VC1 (r), which indicates that there is no bound state in the channel at this range of pion masses. While behaviors of scattering phase shifts are qualitatively similar to experimental ones, the strength

4500 15

4000 80

mπ =411 MeV mπ =570 MeV mπ =700 MeV

60 40

2500

10 δ(1S0) [degree]

VC (r; 1 S0 ) [MeV]

3500 3000

20

2000

0 1500

-20

1000

-40

500

mpion =700 MeV mpion =570 MeV mpion =411 MeV

5 0 -5

0

0.5

1

1.5

2

2.5

0 -10

-500 0

0.5

1.5

1 r [fm]

2

2.5

0

50

100

150 E [MeV]

200

250

300

Fig. 1 (Left) The N N central potential VC1 (r) in the 1 S0 channel, obtained from (2+1)-flavor lattice QCD at mπ  411 (red), 570 (green), 701 (blue) MeV. (Right) Corresponding scattering phase shifts as a function of energy. (Figures are taken from Ishii 2013)

46 Lattice QCD and Baryon-Baryon Interactions 20

3500

0

80

2500

mπ =411 MeV mπ =570 MeV mπ =700 MeV

60 40

2000

VT (r; 3 S1 -3 D1 ) [MeV]

3000 VC(r; 3S1-3D1) [MeV]

1797

20

1500

0 1000

-20

500

-40 0

0

0.5

1

1.5

2

2.5

-500

-20 -40 -60 -80 -100 mπ =411 MeV mπ =570 MeV mπ =700 MeV

-120 -140

0

0.5

1.5

1 r [fm]

2

2.5

0

0.5

1.5

1

2

2.5

r [fm]

Fig. 2 The N N potential in the 3 S1 − 3 D1 channel, obtained from (2+1)-flavor lattice QCD at mπ  411 (red), 570 (green), 701 (blue) MeV. (Left) The central potential VC0 (r). (Right) The tensor potential VT0 (r). (Figures are taken from Ishii 2013)

of the attraction is weaker due to heavier pion masses. Therefore, it is expected that experimental data are reproduced at the physical pion mass up to systematic errors such as the lattice artifact. Central and tensor potentials, VC0 (r) (left) and VT0 (r) (right), in the 3 S1 − 3 D1 channel from same ensembles are presented in Fig. 2. While the central potential has similar shape and quark mass dependence to those in the 1 S0 channel, the tensor potential is strong with a negative sign at all distances and becomes stronger at lighter pion masses. As mentioned before, it is necessary to extract NN potentials at the physical pion mass, in order to confirm whether NN interactions in the HAL QCD method correctly reproduce experimental results including the binding energy of deuteron, up to systematic errors such as lattice artifacts. As the pion mass decreases, however, statistical fluctuations become larger and larger, so that reliable extractions of NN potentials become difficult. For example, Fig. 3 shows LO NN potentials in the 3 S − 3 D channel, V 0 (r) (left) and V 0 (r) (right), obtained at m  146 MeV 1 1 π C T (almost physical pion mass) (Doi et al. 2018), which are much noisier than those in Fig. 2 at heavier pion masses. Note that large fluctuations of these potentials are mainly caused by contaminations from higher partial waves such as  = 4, as evident form comb-like structures, which could be reduced by the partial wave decomposition method (Miyamoto et al. 2020).

Central, Tensor, and Spin-Orbit Interactions in Parity-Odd Channels In the next-to-leading order (NLO) of the derivative expansion, there appears spinorbital forces (LS) in NN potentials, which are known to play important roles in the LS splittings of nuclear spectra and nuclear magic numbers. It is also argued that the LS force in the 3 P2 −3 F2 channel is responsible for the P -wave superfluidity in neutron stars, which affects a cooling process of neutron stars.

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Fig. 3 The N N potential in the 3 S1 − 3 D1 channel, obtained at t = 8 − 11, using the (2+1)-flavor lattice QCD ensemble with mπ  146 MeV at a  0.0846 fm on a 964 lattice. (Left) The central potential VC0 (r). (Right) The tensor potential VT0 (r). (Figures are taken from Doi et al. 2018) 450

400

V(r;1P1) V(r;33P0) V(r, P1) V(r;3P2)

400 350 300 V(r) [MeV]

V(r) [MeV]

200

0 VI=0 C;S=0 VI=1 C;S=1

-200

1 r [fm]

150 50 0

VI=1 LS 0

200 100

VI=1 T -400

250

-50 2

0

0.5

1 r [fm]

1.5

2

Fig. 4 (Left) Central (S = 0 and 1), tensor, and spin-orbit potentials in parity-odd channels. (Right) Potentials for the 1 P1 , 3 P0 , 3 P1 , and 3 P2 channels. (Figures are taken from Murano et al. 2014)

LO and NLO potentials in parity-odd channels have been calculated in twoflavor lattice QCD with a very heavy quark mass corresponding to (mπ , mN )  (1133, 2158) MeV, at a  0.156 fm on a 163 × 32 lattice (Murano et al. 2014). To access parity-odd channels in lattice QCD, source operators should have nonzero momenta, which require more computational costs and make data noisier. To overcome these difficulties, heavy pion mass was used in this study. Figure 4 (left) gives LO potentials, VCI =0,S=0 (r) (green), VCI =1,S=1 (r) (red), I =1 (r) (blue). It is and VTI =1 (r) (black), as well as the LS potential at NLO, VLS observed that both central potentials VCI =0,S=0 (r) and VCI =1,S=1 (r) are repulsive at all distances and the tensor potential VTI =1 (r) is positive but very weak compared I =1 (r) is strong and negative at all to central potentials, while the LS potential VLS distances. These features are qualitatively consistent with those of phenomenological potentials. Potentials in 1 P1 , 3 P0 , 3 P1 , and 3 P2 channels are reconstructed from these four potentials as V (r; 1 P1 ) = VCI =0,S=0 (r), V (r; 3 P0 ) = VCI =1,S=1 (r) −

15 10 5 0 -5 -10 -15 -20 -25 -30 -35

(a) phase shift (degree)

phase shift (degree)

46 Lattice QCD and Baryon-Baryon Interactions

1P1 3P0 3P1 1P1 (EXP) 3P0 (EXP) 3P1 (EXP) 0

50

250 200 150 100 2 Elab = 2 k /mN [MeV]

300

350

1799 18 16 14 12 10 8 6 4 2 0 -2 -4

(b)

0

50

3

P 3 2 F2 mixing parameter 3 P2 (EXP) 3 F2 (EXP) mixing parameter (EXP)

250 200 150 100 2 Elab = 2 k /mN [MeV]

300

350

Fig. 5 (Left) Scattering phase shifts in the 1 P1 , 3 P0 , and 3 P1 channels as a function of energy, together with experimental data for comparisons. (Right) Scattering phase shifts and mixing parameters (with Stapp’s convention) in the 3 P2 –3 F2 channel as a function of energy, together with experimental data for comparisons. (Figures are taken from Murano et al. 2014)

I =1 (r), V (r; 3 P ) = V I =1,S=1 (r) + 2V I =1 (r) − V I =1 (r), and 4VTI =1 (r) − 2VLS 1 C T LS I =1,S=1 I =1 (r). Figure 4 (right) shows V (r; 1 P ) 3 V (r; P2 ) = VC (r)− 25 VTI =1 (r)+VLS 1 (magenta), V (r; 3 P0 ) (blue), V (r; 3 P1 ) (green), and V (r; 3 P2 ) (red). Scattering observables (scattering phase shifts and mixing angles) extracted from these potentials are plotted in Fig. 5. Compared with experimental phase shifts, qualitative features of scattering phase shifts are roughly reproduced, while their magnitudes are much smaller probably due to the much heavier pion mass in this study. For the scattering phase shift in the 3 P0 channel (red data in the left figure), an attractive behavior at low energies is missing compared with the experimental one (red line), which is likely due to a weak tensor force VTI =1 (r) (black in the left of Fig. 4) caused by the much heavier pion mass. The most interesting feature is an attractive behavior of the phase shift in the 3 P2 channel (red in the right of Fig. 5), I =1 (r) (blue in the which originates from the strong attraction of the LS potential VLS left of Fig. 4). As mentioned, this LS interaction is relevant to the pairing correlation of neutrons and possible P -wave superfluidity in neutron stars. The next step in future studies will be calculations of parity-odd potentials at lighter pion masses, in order to see the tensor potential and the LS potential become larger in magnitudes so as to reproduce attractive behaviors of scattering phase shifts in 3 P0 channel and 3 P2 channel, respectively.

Three-Nucleon Interactions Not only two-body NN interactions but also three-body NNN interactions are necessary for nuclear physics. Three-nucleon interactions play important roles in nuclear spectra and structures such as binding energies of light nuclei and properties of neutron-rich nuclei. In addition, they are important pieces for properties of nuclear matters such as an equation of state (EoS) at high density, relevant to

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structures of neutron stars and nucleosynthesis at binary neutron star mergers. Although there have been many investigations for constructions of three-nucleon interactions by phenomenological (Coon and Han 2001; Pieper et al. 2001) or chiral EFT (Weinberg 1992; Epelbaum et al. 2009; Machleidt and Entem 2011; Hammer et al. 2020) approaches, it would be most desirable to determine three-nucleon interactions directly from the first principles lattice QCD. Unlike N N potentials, which can be derived with relativistic kinematics in the HAL QCD method, derivations of three-nucleon potentials are currently restricted to non-relativistic kinematics in the HAL QCD method. See Aoki et al. (2013a, b) for definitions of n-body potentials in the HAL QCD method. A correlation function for three nucleons is defined by FJ3N (r, ρ, t) = 3

 Ω|N(x1 , t + t0 )N (x2 , t + t0 )N (x3 , t + t0 )J3† (t0 )|Ω, (29) R

where R := (x1 + x2 + x3 )/3, r := x1 − x2 , ρ := x3 − (x1 + x2 )/2 are Jacobi coordinates and J3† (t0 ) is a creation operator for three nucleons at time t0 . The corresponding R-correlator is given by RJ3N (r, ρ, t) = 3

(r, ρ, t) FJ3N 3 (GN (t))3

.

(30)

A genuine three-nucleon potential V3N F (r, ρ) at the LO analysis is extracted from the time-dependent Schrödinger equation as ⎡ ⎣−

∇r2 2μr

−

∇ρ2 2μρ

+

 i 0 and q denoting the four-momentum of the virtual photon, depend on the isoscalar charge density operator that has been worked out in chiral EFT to a high accuracy. In Fig. 5, the results for the charge and quadrupole FF of the deuteron are shown at N4 LO (Filin et al. 2020, 2021). Here, the empirical results were used for the nucleon form factors to parametrize

48 Semi-local Nuclear Forces from Chiral EFT: State-of-the-Art and Challenges 1

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100

|GC(Q)|

GQ(Q)

0.1

10

0.01

1

0.001 0

N4LO truncation error (68% DoB)

1

2

3 Q [fm-1 ]

0.1 4

5

6

0

1

2

3 Q [fm-1 ]

4

5

6

Fig. 5 The deuteron charge (left panel) and quadrupole (right panel) form factors calculated at N4 LO for the cutoff Λ = 500 MeV. Bands between dashed red lines correspond to a 1σ error in the determination of the 2N short-range contributions to the charge density operator. For references to the experimental data and their parametrization shown by the black solid circles see Filin et al. (2021)

the single-nucleon contributions to the charge density operator without relying on the chiral expansion. The relativistic corrections and the contributions of the 2N charge density operators were also taken into account. The latter depend on two LECs that have been fixed from the experimental data for GC (Q) and GQ (Q); see Fig. 5. With all LECs being determined as described above, a prediction for the deuteron structure radius rstr and the quadrupole moment Qd = GQ (0) was made. The structure radius denotes the contribution to the deuteron charge radius, which 2 C (Q ) is related to the FF GC via r 2 = −6 dGdQ 2 , that arises from the nuclear 2 Q =0

binding mechanism. Up to the so-called Darwin term, rstr can be interpreted as the charge radius of the deuteron made out of structureless nucleons. See Filin et al. (2021) for more details and Epelbaum et al. (2022) for the interpretation of the FF in terms of the charge density distribution. The final predictions for the deuteron structure radius and quadrupole moment in Filin et al. (2021) read rstr = 1.9729+0.0015 −0.0012 fm,

2 Qd = 0.2854+0.0038 −0.0017 fm ,

(6)

where the quoted errors include the statistical uncertainties of various LECs, the uncertainty in the parametrizations of the nucleon FFs, and the estimated N4 LO truncation uncertainty. These values are to be compared with the determinations from laser spectroscopy experiments (Jentschura et al. 2011; Puchalski et al. 2020): exp

rstr = 1.97507(78) fm,

exp

Qd

exp

= 0.285699(23) fm2 .

(7)

Here, the quoted value for rstr was obtained using the mean square neutron exp radius of rn2 = −0.114(3) fm2 . Alternatively, the prediction for rstr was used in combination with the very precise experimental data on the deuteron-proton chargeradius difference from Jentschura et al. (2011) to update the value of the mean 2 square neutron radius rn2 = −0.105+0.005 −0.006 fm .

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Beyond the NN System SMS Three-Nucleon Force at N2 LO The leading 3NF at N2 LO is generated by the tree-level topologies visualized in Fig. 6a; see also Table 1. It is straightforward to regularize the corresponding expressions using the semi-local cutoff in momentum space and employing the same convention for the subtraction terms as in the SMS 2NF of Reinert et al. (2018):

VΛ3N =

2 gA

8Fπ4



  σ1 · q1 σ3 · q3 T13 2c3 q1 · q3 − 4c1 Mπ2 + c4 T132 q1 × q3 · σ2 2 2 2 2 (q1 + Mπ ) (q3 + Mπ ) + C

+ C

σ1 · q1  q12 + Mπ2 σ3 · q3  q32 + Mπ2

2c3 T13 σ3 · q1 + c4 T132 q1 × σ3 · σ2

2c3 T13 σ1 · q3 + c4 T132 σ1 × q3 · σ2







  − q12 +Mπ2 − q32 +Mπ2 Λ2 Λ2 e + C 2 2c3 T13 σ1 · σ3 + c4 T132 σ1 × σ3 · σ2 e

2  q 2 +Mπ p2 +p 2 − 12 2 12 − 3 2 Λ Λ σ1 · q3 + C σ1 · σ3 e e

σ3 · q3

−

gA D T13 8Fπ2

+

p +p 1 − 12 2 12 − Λ E T12 e e 2

q32 + Mπ2 2

2

3Q23 +3Q32 4Λ2

+ 5 permutations,

(8)

where qi = p1 − pi is the momentum transfer of nucleon i with pi and p1 being the corresponding initial and final momenta, respectively. Further, p12 = (p1 − p2 )/2  3 = 2(p3 − (p1 + p2 )/2)/3 are the Jacobi momenta in the initial state and and Q  = (p   = 2(p  − (p  + p  )/2)/3 in the final state. The isospin p12 1 − p2 )/2 and Q 3 3 1 2 operators Tij and Tij k are defined as Tij ≡ τ i · τ j and Tij k ≡ τ i × τ j · τ k . The subtraction constant C is specified in Eq. (3).

a)

b) ci

D

E

limq

0

q

=

0

Fig. 6 (a) Feynman diagrams contributing to the 3NF at N2 LO. Solid and dashed lines refer to nucleons and pions, respectively. (b) Chiral symmetry constraint on the π NN interaction

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Nucleon-Deuteron Scattering Three-nucleon scattering and bound state observables are calculated by solving the Faddeev equations in momentum space in the partial wave basis; see Maris et al. (2021) for details. The partial wave decomposition of a general 3NF is carried out numerically as detailed in Hebeler et al. (2015). The leading 3NF depends on the LECs D and E, whose values are extracted from low-energy 3N observables. Specifically, following Epelbaum et al. (2019) and Maris et al. (2021), the authors first require that the 3 H binding energy is correctly reproduced. This constraint fixes the value of the LEC E for a given value of D. To reliably determine this remaining LEC, it is essential to employ observables that are not strongly correlated with the 3 H binding energy (Wesolowski et al. 2021). The authors of Epelbaum et al. (2019) have studied the constraints imposed on the value of the LEC D by the experimental data for the Nd doublet scattering length, the total Nd scattering cross section, and the differential cross section minimum in elastic Nd scattering at several energies. It was found that the high-precision experimental data of Sekiguchi et al. (2002) for the differential cross section at the proton energy of Ep = 70 MeV, see Fig. 7, provide a particularly strong constraint on D. This finding is in line with the known sensitivity of the cross section minimum to 3NF effects; see Glöckle et al. (1996) and references therein. The LECs D and E determined from the 3 H binding energy and the Nd cross section minimum at 70 MeV were shown to result in a consistent description of all observables mentioned above; see Fig. 2 of Epelbaum et al. (2019) and Fig. 2 of Maris et al. (2021). Having determined the LECs D and E as described above, it is interesting to consider selected predictions for Nd scattering observables up to N2 LO. In Fig. 7, it

Fig. 7 Results for selected observables in elastic Nd scattering at laboratory energy of EN = 70 MeV at NLO (yellow bands) and N2 LO (green bands) for Λ = 500 MeV. The light- (dark-) shaded bands indicate the estimated 95% (68%) DoB intervals. Open circles are proton-deuteron data from Sekiguchi et al. (2002)

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is demonstrated that the NLO and N2 LO results for the differential cross section and selected polarization observables in elastic Nd scattering at 70 MeV are in agreement with the experimental data. For the analyzing powers Ayy and Axz , the discrepancies between the calculations and the experimental data are comparable with the truncation errors at N2 LO. These discrepancies are not resolved by higherorder corrections to the 2NF (Epelbaum et al. 2020) and thus indicate the important role played by subleading 3NF contributions. The residual cutoff dependence of the calculated cross sections is, in general, compatible with the estimated truncation errors. In particular, the results for Λ = 500 MeV shown in Fig. 7 are very similar to those using Λ = 450 MeV and shown in Fig. 3 of Maris et al. (2021). The dependence of the obtained predictions on the different choices of semilocal regulators (i.e., coordinate-space versus momentum-space) and the subtraction conventions for the 3NF is also fully consistent with the truncation uncertainty; see Maris et al. (2021) and Epelbaum et al. (2019) for details. The estimated truncation errors for Nd scattering observables were further validated by analyzing the contributions of selected higher-order short-range 3NF terms in Epelbaum et al. (2020). Finally, in Fig. 8, the predictions for the total cross section at EN = 70 and 135 MeV are shown for all four cutoff values. These calculations are based on the SMS 2NF at LO, NLO, N2 LO, N3 LO, and N4 LO+ from Reinert et al. (2018). Starting from N2 LO, the leading 3NF contributions specified in Eq. (8) are also taken into account. For each combination of the 2NF and 3NF and for each cutoff value, the LECs D and E are fixed to reproduce the 3 H binding energy and the Nd cross section minimum at 70 MeV as described above. Since the 3NF is included only at N2 LO, one can regard the predictions based on the 2NF at N3 LO and N4 LO+ as alternative N2 LO calculations when estimating the truncation error. The total cross section is underestimated by ∼3.5% (∼7%) at EN = 70 MeV (EN = 135 MeV) when using the high-precision 2NF at N4 LO+

Fig. 8 Predictions for the Nd total cross section at 70 MeV (left panel) and 135 MeV (right panel) based on the SMS chiral interactions at different orders (shown by solid symbols with error bars). 3NF is included at N2 LO only. Error bars show the EFT truncation uncertainty (68% DoB intervals). For the incomplete calculations at N3 LO and N4 LO+ , the quoted errors are the N2 LO truncation uncertainties. Gray open symbols without error bars show the results based on the 2NF only. Horizontal bands are experimental data from Abfalterer et al. (1998)

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alone. Similar discrepancies that tend to increase with the energy were also observed in calculations based on high-precision phenomenological 2N potentials (Abfalterer et al. 1998). Adding the leading 3NF is essential to bring the chiral EFT predictions at N2 LO in agreement with the data. Moreover, the 3NF contributions to the total cross section appear to be comparable to the NLO truncation errors in agreement with the Weinberg power counting (Weinberg 1991). Also the differences between the N2 LO and N3 LO predictions that originate from the N3 LO contributions to the 2NF are comparable with the estimated N2 LO truncation errors.

Heavier Systems The SMS chiral interactions have also been applied by the LENPIC collaboration to predict the ground and excited state energies and radii of light- and mediummass nuclei. In Fig. 9, the results for the ground state energies of selected nuclei up to A = 12 are shown using the NLO 2NF (left symbols), the N2 LO 2NF (middle symbols), and the N2 LO 2NF in combination with the N2 LO 3NF (right symbols) for the cutoff Λ = 450 MeV as a representative example. Notice that the calculated energies are pure predictions since the LECs in the nuclear Hamiltonian are determined from the 2N and 3N systems only. For nuclei with A ≤ 10, the leading 3NF significantly improves the agreement with the data by increasing the binding energy. For heavier nuclei, the N2 LO Hamiltonian is systematically too attractive (but the ground state energies of both 12 B and 12 C are still in agreement with the data within 1.5σ ). This tendency continues and increases with the mass number (Maris et al. 2021). The origin of the systematic overbinding in heavier nuclei is currently under investigation by the LENPIC collaboration. More results

(0+, 0)

(0+, 1)

(0+, 2)

(1+, 0) (3/2-, 1/2)

NLO, 68% DoB N2LO, 95% DoB

(2+, 1)

N2LO, 68% DoB

(0+, 1)

(3+, 0)

(JP, T) (1+, 1)

NLO N2LO without 3N forces

(0+, 0)

N2LO including 3N forces Experimental values 4He

6He

6Li

7Li

8He

8Li

10Be

10B

12B

12C

Fig. 9 Predictions for ground state energies of selected nuclei with A = 4 –12 at NLO and N2 LO for Λ = 450 MeV using the ab-initio No-Core Configuration Interaction method (NCCI). Black error bars indicate the NCCI uncertainties, while shaded bars refer to the EFT truncation errors (not shown for incomplete N2 LO calculations based on 2NF only). (See Maris et al. (2021) for details)

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for p-shell nuclei based on semi-local chiral EFT interactions can be found in Binder et al. (2016, 2018), Epelbaum et al. (2019), and Maris et al. (2021).

Towards Consistent Regularization Beyond the 2N System The missing three- and four-nucleon forces and exchange current operators at N3 LO and beyond are becoming more and more of a bottleneck for ab initio low-energy nuclear theory. Although the N3 LO and even some of the N4 LO contributions to the many-body forces and currents have been worked out using dimensional regularization (DR), see Table 1, the existing expressions can not be directly employed in few-nucleon calculations due to the inconsistencies caused by combining the dimensional and cutoff regularizations (Epelbaum et al. 2020). Below, an explicit example will be given to demonstrate such an inconsistency for the 3NF regularized in a naive way using both (semi-) local and nonlocal cutoffs.

Statement of the Problem Both the 2NF and 3NF need to be regularized in order to obtain a well- defined solution of the Faddeev equations. High-momentum components in the integrals appearing in the iterations of the Faddeev equation generate contributions involving positive powers and logarithms of the cutoff which diverge in the Λ → ∞ limit and are supposed to get absorbed by the available short-range interactions. The momentum dependence of such contact interactions beyond the 2N sector is, however, severely constrained by the spontaneously broken chiral symmetry of QCD. In particular, in the limit of exact chiral symmetry (i.e., for Mπ → 0), only derivative pion couplings are allowed in the effective Lagrangian according to the Goldstone theorem. In the 2N sector, the tree-level short-range interactions do not involve any pion couplings, and their momentum dependence is therefore not restricted by the chiral symmetry. This is in contrast to the D-like 3NF interactions, which are constrained by the chiral symmetry as visualized in Fig. 6b. These constraints lead to inconsistencies that plague calculations involving the 3NF derived using DR and regularized additionally by multiplying with a local or nonlocal cutoff. As will be exemplified below, the resulting mismatch between the two regularization schemes can not be compensated by shifting the values of the available LECs. To be specific, the authors focus here on the example discussed in Epelbaum et al. (2020) and consider the relativistic correction to 2π -exchange 3NF proportional to 2 (Bernard et al. 2011): gA 2π, 1/m

V3N

=i

2 gA σ1 · q1 σ3 · q3 τ 1 · (τ 2 × τ 3 )(2k1 · q3 + 4k3 · q3 32mFπ4 (q12 + Mπ2 )(q32 + Mπ2 )

+ i [ q1 × q3 ] · σ2 ) + 5 permutations,

(9)

48 Semi-local Nuclear Forces from Chiral EFT: State-of-the-Art and Challenges

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  with ki = pi + pi /2. Consider now the first iteration of these N3 LO contributions with the LO 1π -exchange 2N potential 

1π V2N

gA =− 2Fπ

2 τ1 · τ2

σ1 · q σ2 · q . q 2 + Mπ2

(10)

Regularization of these 2NF and 3NF is achieved by multiplying them with a local Gaussian cutoff 2π,1/m

V3N,

2π,1/m

= V3N

e

−

2 q12 +Mπ Λ2

e

−

2 q32 +Mπ Λ2

,

1π 1π V2N,

= V2N e

−

2 q 2 +Mπ Λ2

.

(11)

Performing a large-Λ expansion leads to 2π,1/m

V3N,

2π, 1/m

1π 1π G0 V2N,

+ V2N,

G0 V3N,

=Λ

g4 q2 · σ2 q3 · σ3 (τ 2 · τ 3 − τ 1 · τ 3 ) 2 √ A 3/2 6 q3 + Mπ2 128 2π Fπ

−Λ

g4 q3 · σ3 q3 · σ1 τ1 · τ3 2 + ... , √ A 3/2 6 q3 + Mπ2 96 2π Fπ

(12)

where G0 is the free resolvent operator and the ellipses refer to all permutations of the nucleon labels and to terms that are finite in the Λ → ∞-limit. The last term on the right-hand side (rhs) of Eq. (12) has the form of the D-term of the N2 LO 3NF, and it therefore can be absorbed into a redefinition of the LEC D. In contrast, the first term on the rhs of Eq. (12) cannot be absorbed into a redefinition of the LECs entering the N2 LO 3NF since the corresponding structure is not allowed by the chiral symmetry. One therefore expects this problematic term to cancel against some other contribution in the 3N amplitude. The only other term with the desired 4 , whose combination of the LECs is the 1π -2π -exchange 3NF at N3 LO ∝ gA expression was derived in Bernard et al. (2008) using DR. Had one used the same cutoff regularization also in the calculation of this 3NF, one would obtain a linearly divergent term 2π −1π V3N,

= −Λ

−Λ

g4 q2 · σ2 q3 · σ3 (τ 2 · τ 3 − τ 1 · τ 3 ) 2 √ A q3 + Mπ2 128 2π 3/2 Fπ6 g4 q3 · σ3 q3 · σ1 τ1 · τ3 2 + ... , √ A 3/2 6 q3 + Mπ2 32 2π Fπ

(13)

where the ellipses refer to terms which are finite in the Λ → ∞-limit that coincides with the DR result of Bernard et al. (2008). As expected, the problematic term in Eq. (12) cancels exactly by the first term on the rhs of Eq. (13), while the second

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term can, again, be absorbed into a redefinition of the LEC D. Clearly, the cancellation is only operative if one uses the same cutoff regularization in the derivation of the N3 LO 3NF and in the Faddeev equation. A naive approach by multiplying the N3 LO 3NF expressions, derived using DR, with some cutoff regulators obviously fails to ensure the cancellation of the chiral-symmetry-violating UV divergences and results in an uncontrolled approximation for the amplitude, which cannot be renormalized. It is important to emphasize that the above inconsistency is by no means restricted to the usage of local regulators for long-range interactions. Indeed, repeating the same exercise using a nonlocal regulator of a Gaussian type, 2π,1/m

V3N,

2π,1/m

= V3N

e

−

2 +P  2 P12 12 Λ2

e

−3

Q23 +Q32 4Λ2

,

1π 1π V2N,

= V2N e

−

2 +P  2 P12 12 Λ2

,

(14) one obtains for the first iteration of the Faddeev equation with the 1π -exchange 2NF being antisymmetrized in the 12-subsystem: 2π,1/m

V3N,

2π, 1/m

1π 1π G0 V2N,

+ V2N, G0 V3N,

=Λ

4 gA (7 k1 − 3 k2 ) · σ1 q3 · σ3 i τ · (τ × τ ) 1 2 3 1536 (2π )3/2 Fπ6 q32 + Mπ2

−Λ

4 gA q3 · σ3 q3 · σ1 τ1 · τ3 2 + ... . 3/2 6 384 (2π ) Fπ q3 + Mπ2

(15)

Again, the first term on the rhs of Eq. (15) violates the chiral symmetry and cannot be absorbed into a redefinition of the LEC D. The nonlocality of the regulator thus does not cure the problem. It actually introduces additional complications by affecting the analytic structure of the long-range potentials and making the derivation of consistent cutoff-regularized 3NF technically more demanding. The issue with inconsistent regularization affects not only three- and morenucleon forces, but it is also relevant for exchange currents at and beyond N3 LO. In particular, analogous considerations for the axial vector current at N3 LO demonstrate the appearance of chiral-symmetry-violating UV divergences when mixing the DR and cutoff regularization (Krebs 2020).

Possible Solutions The above inconsistencies can be rectified by using the same regulator in the derivation of the nuclear forces and currents and iterations of the dynamical equation. Such a regulator has to respect all the relevant symmetries. One option is to implement the regulator at the level of the effective Lagrangian. One can, in particular, require for q 2 +M 2

the regularized pion propagator to take the form exp(− Λ2 π )/(q 2 +Mπ2 ). This can be achieved by adding specific higher-derivative terms to the effective Lagrangian,

48 Semi-local Nuclear Forces from Chiral EFT: State-of-the-Art and Challenges

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which disappear in the limit Λ → ∞. Such a higher-derivative regularization was introduced by Slavnov a long time ago to regularize the nonlinear sigma model (Slavnov 1971). This idea can be used to construct a Λ-dependent effective Lagrangian for pions that is manifestly invariant under global chiral transformations and yields long-range nuclear interactions regularized in a local way. On top of it, one can employ a nonlocal higher-derivative regularization for contact interactions. The implementation of these ideas in the 2N and 3N sectors is in progress. Another possibility to implement a symmetry-preserving regulator is given by the so-called gradient flow regularization approach proposed originally by Lüscher (2010); see also Grabowska and Kaplan (2016) and Kaplan and Grabowska (2016). The idea behind this method is similar to that of the stochastic quantization by introducing a fifth dimension. The original pion field that depends on space-time coordinates is to be replaced by a field that depends, in addition, on a fictitious time t. This field satisfies a gradient flow equation and reduces to the original pion field in the limit t → 0. Chiral perturbation theory with this kind of regulator was discussed in Bär and Golterman (2014); see also a related work in Aoki et al. (2015). Last but not least, one can also employ a lattice regularization in chiral EFT. The regularized version of the effective pion Lagrangian on the lattice can be found in Borasoy et al. (2004). Nuclear forces and currents regularized in this way are guaranteed to respect the underlying symmetries and may be particularly useful for ongoing efforts to extend nuclear lattice EFT simulations to higher orders (Lähde and Meißner 2019; Lee 2020).

Summary and Outlook To summarize, this chapter focused on the new generation of nuclear interactions derived from chiral EFT using semi-local regulators (Epelbaum et al. 2015a, b; Reinert et al. 2018, 2021). At N4 LO+ , the highest order available, the resulting SMS potentials were used to perform, for the first time, a full-fledged partial wave analysis of proton-proton and neutron-proton scattering data in the framework of chiral EFT (Reinert et al. 2021; Reinert 2022). The resulting near-perfect description of NN data up to the pion-production threshold leaves little room for improvement and suggests no need to extend the EFT expansion beyond N4 LO+ given the available NN data. The recent high-accuracy calculation of the deuteron charge and quadrupole form factors (Filin et al. 2020, 2021) were also briefly reviewed. The novel SMS interactions have been used to analyze 3N scattering observables and selected properties of light- and medium-mass nuclei (Epelbaum et al. 2019; Epelbaum et al. 2020; Binder et al. 2016, 2018; Maris et al. 2021). The accuracy of these studies is limited by that of the 3NF, which is only available at N2 LO. At this chiral order, the predicted Nd scattering observables and ground state energies of nuclei with A ≤ 12 agree with the data within truncation error. N3 LO and some of the N4 LO contributions to the 3NF and 4NF have already been derived using dimensional regularization (Epelbaum 2006, 2007; Bernard et al. 2008, 2011; Girlanda et al. 2011; Krebs et al. 2012, 2013); see Table 1.

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Unfortunately, these expressions cannot be employed to calculate observables. This is because mixing the dimensional and cutoff regularizations when calculating scattering amplitudes violates the chiral symmetry and results in uncontrolled approximations beyond the 2N system. This issue is not restricted to a particular type of cutoff regulator and applies to local, semi-local, and nonlocal cutoffs. It also plagues calculations involving exchange currents at N3 LO and beyond. A solution of this challenge requires a complete re-derivation of many-body forces and exchange currents using a cutoff regulator that respects the underlying symmetries such as, e.g., the higher-derivative regularization (Slavnov 1971). Work along this line is in progress and will open an avenue for performing high-accuracy chiral EFT calculations beyond the 2N system. Acknowledgments The authors are grateful to Vadim Baru, Arseniy Filin, Ashot Gasparyan, Jambul Gegelia, Christopher Körber, Ulf Meißner, and to all members of the LENPIC Collaboration for sharing their insights into the topics addressed in this chapter. This work was supported by BMBF (contract No. 05P18PCFP1), by DFG and NSFC through funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 11621131001, DFG Project-ID 196253076 – TRR 110), by ERC AdG NuclearTheory (grant No. 885150), and by the EU Horizon 2020 research and innovation program (STRONG-2020, grant agreement No. 824093).

References W.P. Abfalterer et al., Phys. Rev. Lett. 81, 57–60 (1998) S. Aoki, K. Kikuchi, T. Onogi, JHEP 04, 156 (2015) O. Bär, M. Golterman, Phys. Rev. D 89(3), 034505 (2014) [erratum: Phys. Rev. D 89(9), 099905 (2014)] V. Bernard, E. Epelbaum, H. Krebs, U.-G. Meißner, Phys. Rev. C 77, 064004 (2008) V. Bernard, E. Epelbaum, H. Krebs, U.-G. Meißner, Phys. Rev. C 84, 054001 (2011) S. Binder et al. [LENPIC], Phys. Rev. C 93(4), 044002 (2016) S. Binder et al. [LENPIC], Phys. Rev. C 98(1), 014002 (2018) L. Bird, P. Christmas, A. Taylor, E. Wood, Nucl. Phys. 27, 586 (1961) B. Borasoy, R. Lewis, P.P.A. Ouimet, Nucl. Phys. B Proc. Suppl. 128, 141–147 (2004) G.F. Cox et al., Nucl. Phys. B 4, 353–373 (1968) J.J. de Swart, C.P.F. Terheggen, V.G.J. Stoks, 3rd International Symposium on Dubna Deuteron 95 (1995) A. Ekström et al., Phys. Rev. C 91(5), 051301 (2015) D.R. Entem, R. Machleidt, Y. Nosyk, Phys. Rev. C 96(2), 024004 (2017) E. Epelbaum, Phys. Lett. B 639, 456–461 (2006) E. Epelbaum, Eur. Phys. J. A 34, 197–214 (2007) E. Epelbaum, PoS CD2018, 006 (2019) E. Epelbaum, J. Gegelia, Eur. Phys. J. A 41, 341–354 (2009) E. Epelbaum et al., Phys. Rev. Lett. 129 (1), 012001. (2022) E. Epelbaum, H. Krebs, U.-G. Meißner, Eur. Phys. J. A 51(5), 53 (2015a) E. Epelbaum, H. Krebs, U.-G. Meißner, Phys. Rev. Lett. 115(12), 122301 (2015b) E. Epelbaum, A.M. Gasparyan, J. Gegelia, U.-G. Meißner, Eur. Phys. J. A 54, 186 (2018) E. Epelbaum et al. [LENPIC], Phys. Rev. C 99(2), 024313 (2019) E. Epelbaum, H. Krebs, P. Reinert, Front. Phys. 8, 98 (2020) E. Epelbaum et al., Eur. Phys. J. A 56(3), 92 (2020)

48 Semi-local Nuclear Forces from Chiral EFT: State-of-the-Art and Challenges

1877

A.A. Filin et al., Phys. Rev. Lett. 124(8), 082501 (2020) A.A. Filin et al., Phys. Rev. C 103(2), 024313 (2021) R.J. Furnstahl et al., Phys. Rev. C 92(2), 024005 (2015) A. M. Gasparyan, E. Epelbaum, Phys. Rev. C 105(2), 024001 (2022) A. Gezerlis et al., Phys. Rev. Lett. 111(3), 032501 (2013) L. Girlanda, A. Kievsky, M. Viviani, Phys. Rev. C 84(1), 014001 (2011) [erratum: Phys. Rev. C 102(1), 019903 (2020)] L. Girlanda, A. Kievsky, L.E. Marcucci, M. Viviani, Phys. Rev. C 102, 064003 (2020) W. Glöckle, H. Witała, D. Hüber, H. Kamada, J. Golak, Phys. Rep. 274, 107–285 (1996) D.M. Grabowska, D.B. Kaplan, Phys. Rev. Lett. 116(21), 211602 (2016) F. Gross, A. Stadler, Phys. Rev. C 78, 014005 (2008) K. Hebeler et al., Phys. Rev. C 91(4), 044001 (2015) M. Hoferichter et al., Phys. Rev. Lett. 115(19), 192301 (2015) O.N. Jarvis, C. Whitehead, M. Shah, Phys. Lett. B 36(4), 409–411 (1971) U.D. Jentschura et al., Phys. Rev. A 83, 042505 (2011) N. Kaiser, R. Brockmann, W. Weise, Nucl. Phys. A 625, 758–788 (1997) D.B. Kaplan, D.M. Grabowska, PoS LATTICE2016, 018 (2016) E.G. Kessler Jr., Phys. Lett. A 255, 221 (1999) H. Krebs, Eur. Phys. J. A 56(9), 234 (2020) H. Krebs, A. Gasparyan, E. Epelbaum, Phys. Rev. C 85, 054006 (2012) H. Krebs, A. Gasparyan, E. Epelbaum, Phys. Rev. C 87(5), 054007 (2013) T.A. Lähde, U.-G. Meißner, Lect. Notes Phys. 957, 1–396 (2019) D. Lee, Front. Phys. 8, 174 (2020) P.W. Lisowski et al., Phys. Rev. Lett. 49, 255–259 (1982) M. Lüscher, JHEP 08, 071 (2010) [erratum: JHEP 03, 092 (2014)] R. Machleidt, Phys. Rev. C 63, 024001 (2001) P. Maris et al., Phys. Rev. C 103(5), 054001 (2021) R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C 95(6), 064001 (2017) R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C 88(6), 064002 (2013) [erratum: Phys. Rev. C 91(2), 029901 (2015)] R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C 88, 024002 (2013) [erratum: Phys. Rev. C 88(6), 069902 (2013)] M. Puchalski, J. Komasa, K. Pachucki, Phys. Rev. Lett. 125(25), 253001 (2020) P. Reinert, H. Krebs, E. Epelbaum, Eur. Phys. J. A 54(5), 86 (2018) P. Reinert, H. Krebs, E. Epelbaum, Phys. Rev. Lett. 126(9), 092501 (2021) P. Reinert, PhD thesis, Ruhr-Universität Bochum, to be published (2022) N.L. Rodning, L.D. Knutson, Phys. Rev. C 41, 898–909 (1990) K. Schoen et al., Phys. Rev. C 67, 044005 (2003) K. Sekiguchi et al., Phys. Rev. C 65, 034003 (2002) A.A. Slavnov, Nucl. Phys. B 31, 301–315 (1971) V.G.J. Stoks et al., Phys. Rev. C 48, 792–815 (1993) V.G.J. Stoks et al., Phys. Rev. C 49, 2950–2962 (1994) A.E. Taylor, E. Wood, L. Bird, Nucl. Phys. 16(2), 320–330 (1960) C. Van Der Leun, C. Alderliesten, Nucl. Phys. A 380, 261–269 (1982) S. Weinberg, Nucl. Phys. B 363, 3–18 (1991) S. Wesolowski et al., Phys. Rev. C 104(6), 064001 (2021)

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective Field Theory for Low-Energy QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetries of Low-Energy QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Effective Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Forces from EFT: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Perturbation Theory and Power Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ranking of Nuclear Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantitative Chiral NN Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NN Contact Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of NN Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regularization and Nonperturbative Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . NN Potentials Order by Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Many-Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Nucleon Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D. R. Entem Grupo de Física Nuclear, IUFFyM, Universidad de Salamanca, Salamanca, Spain e-mail: [email protected] R. Machleidt () · Y. Nosyk Department of Physics, University of Idaho, Moscow, ID, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_55

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Abstract

During the past two decades, chiral effective field theory (EFT) has evolved into a powerful tool to derive nuclear forces from first principles. This review is started with a pedagogical introduction, and then it is shown in detail how, governed by chiral symmetry, the long- and intermediate-range parts of the NN potential unfold order by order. The expansion proceeds up to sixth order in small momenta, where convergence is achieved. A remarkable point of chiral EFT is that it produces not only two-nucleon forces but also many-body forces on an equal footing with the two-body forces. Therefore, this review finishes with an overview of the systematic development of three- and four-nucleon forces, which play a subtle, but crucial, role in microscopic nuclear structure calculations.

Introduction The theory of nuclear forces has a long history. The first serious attempt toward a theory was launched in 1935, when the Japanese physicist Yukawa (1935) suggested that nucleons would exchange quanta between each other to create the force. Yukawa constructed his theory in analogy to the theory of the electromagnetic interaction where the exchange of a (massless) photon is the cause of the force. However, in the case of the nuclear force, Yukawa assumed that the “force-makers” carry a mass that was in-between the masses of the electron and the proton (which is why these particles were eventually called “mesons”). The mass of the mesons limits the effect of the force to a finite range, since the uncertainty principal allows massive virtual particles to travel only a finite distance. The meson predicted by Yukawa was finally found in 1947 in cosmic ray and in 1948 in the laboratory and called the pion. Yukawa was awarded the Nobel Prize in 1949. In the 1950s and 1960s, more mesons were found in accelerator experiments, and the meson theory of nuclear forces was extended to include many mesons. These models became known as oneboson-exchange models, which is a reference to the fact that the different mesons are exchanged singly in this model. The one-boson-exchange model is very successful in explaining essentially all properties of the nucleon-nucleon interaction at low energies (Bryan and Scott 1969; Erkelenz 1974; Holinde et al. 1976; Machleidt 1989, 2001). In the 1970s and 1980s, also meson models were developed that went beyond the simple single-particle exchange mechanism. These models included, in particular, the explicit exchange of two pions with all its complications. Well-known representatives of the latter kind are the Paris (Lacombe et al. 1980) and the Bonn potentials (Machleidt et al. 1987). Since these meson models were quantitatively very successful, it appeared that they were the solution of the nuclear force problem. However, with the discovery (in the 1970s) that the fundamental theory of strong interactions is quantum chromodynamics (QCD) and not meson theory, all “meson theories” had to be viewed as models, and the attempts to derive the nuclear force from first principals had to start all over again.

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The problem with a derivation of nuclear forces from QCD is twofold. First, each nucleon consists of three valence quarks, quark-antiquark pairs, and gluons such that the system of two nucleons is a complicated many-body problem. Second, the force between quarks, which is created by the exchange of gluons, has the feature of being very strong at the low-energy scale that is characteristic of nuclear physics. This extraordinary strength makes it difficult to find converging expansions. Therefore, during the first round of new attempts, QCD-inspired quark models became popular. The positive aspect of these models is that they try to explain nucleon structure (made up from three constituent quarks) and nucleon-nucleon interactions (six quarks) on an equal footing. Some of the gross features of the two-nucleon force, like the “hard core,” are explained successfully in such models. However, from a critical point of view, it must be noted that these quark-based approaches are yet another set of models and not a theory. Alternatively, one may try to solve the sixquark problem with brute computing power, by putting the six-quark system on a four-dimensional lattice of discrete points which represents the three dimensions of space and one dimension of time. This method has become known as lattice QCD and is making progress. However, such calculations are computationally very expensive and cannot be used as a standard nuclear physics tool. Around 1980/1990, a major breakthrough occurred when the Nobel Laureate Steven Weinberg applied the concept of an effective field theory (EFT) to lowenergy QCD (Weinberg 1979, 1990, 1991). He simply wrote down the most general theory that is consistent with all the properties of low-energy QCD, since that would make this theory equivalent to low-energy QCD. A particularly important property is the so-called chiral symmetry, which is “spontaneously” broken. Massless spin1 2 fermions possess chirality, which means that their spin and momentum are either parallel (“right-handed”) or antiparallel (“left-handed”) and remain so forever. Since the quarks, which nucleons are made of (“up” and “down” quarks), are almost massless, approximate chiral symmetry is a given. Naively, this symmetry should have the consequence that one finds in nature hadrons of the same mass, but with opposite parities (“parity doublets”). However, this is not the case and such failure is termed a “spontaneous” breaking of the symmetry. According to a theorem first proven by Goldstone, the spontaneous breaking of a symmetry creates a particle, here, the pion. Thus, the pion becomes the main player in the production of the nuclear force. The interaction of pions with nucleons is weak as compared to the interaction of gluons with quarks. Therefore, pionnucleon processes can be calculated without problem. Moreover, this effective field theory can be expanded in powers of momentum over “scale,” where scale denotes the “chiral-symmetry breaking scale” which is about 1 GeV. This scheme is also known as chiral perturbation theory (ChPT) (Gasser and Leutwyler 1984; Gasser et al. 1986) and allows to calculate the various terms that make up the nuclear potential systematically power by power or order by order (Ordóñez et al. 1996). Another advantage of the chiral EFT approach is its ability to generate not only the force between two nucleons but also many-nucleon forces, on the same footing (Weinberg 1992; van Kolck 1994). In modern theoretical nuclear physics,

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the chiral EFT approach has gained great popularity and is applied with outstanding success (Machleidt and Entem 2011; Epelbaum et al. 1773; Hammer et al. 2020). The purpose of this article is twofold: namely, firstly, to provide a pedagogical introduction into chiral EFT and, secondly, to review the developments of chiral nuclear interactions up to the highest order reached so far (which is the sixth power).

Effective Field Theory for Low-Energy QCD The first step toward the development of an EFT is the identification of appropriate scales. The large difference between the masses of the pions and the masses of the vector mesons, like ρ(770) and ω(782), provides a clue. From that observation, one is prompted to take the pion mass as the identifier of the soft scale, Q ∼ mπ , while the rho mass sets the hard scale, Λχ ∼ mρ , often referred to as the chiral-symmetry breaking scale. It is then natural to consider an expansion in terms of Q/Λχ . Concerning the choice of the degrees of freedom, it is noted that, for conventional nuclear physics, quarks and gluons are ineffective—making nucleons and pions the appropriate degrees of freedom. Of course, one does not wish to construct yet another phenomenological model, and, therefore, our EFT must be firmly linked with QCD. This strong link is present if the EFT is required to be consistent with the symmetries of QCD. The meaning and relevance of such statement is expressed in the so-called folk theorem by Weinberg (1979): If one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition, and the assumed symmetry principles.

To summarize, the development of a proper EFT must proceed as follows: 1. Identify the low- and high-energy scales and the degrees of freedom suitable for (low-energy) nuclear physics. 2. Recognize the symmetries of low-energy QCD and explore the mechanisms responsible of their breakings. 3. Build the most general Lagrangian which respects those (broken) symmetries. 4. Formulate a scheme to organize contributions in order of their importance. Clearly, this amounts to performing an expansion in terms of (low) momenta. 5. Using the expansion mentioned above, evaluate Feynman diagrams to any desired accuracy. In what follows, each of the above steps will be discussed. Since the first one has already been dealt with, the second one will now be addressed.

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Symmetries of Low-Energy QCD This is just a short introduction into (low-energy) QCD, with particular attention to the symmetries and their breakings. For a more detailed presentation, the interested reader is referred to Machleidt and Entem (2011), Scherer (2003).

Chiral Symmetry Starting point is the QCD Lagrangian, 1 LQCD = q(iγ ¯ μ Dμ − M )q − Gμν,a Gaμν 4

(1)

with the gauge-covariant derivative Dμ = ∂μ − ig

λa Aμ,a 2

(2)

and the gluon field strength tensor (For SU (N) group indices, Latin letters are used, . . . , a, b, c, . . . , i, j, k, . . . , and, in general, no distinction is made between subscripts and superscripts.) Gμν,a = ∂μ Aν,a − ∂ν Aμ,a + gfabc Aμ,b Aν,c .

(3)

In the above, q denotes the quark fields and M the quark mass matrix. Further, g is the strong coupling constant and Aμ,a are the gluon fields. Moreover, λa are the Gell-Mann matrices and fabc the structure constants of the SU (3)color Lie algebra (a, b, c = 1, . . . , 8); summation over repeated indices is always implied. The gluongluon term in the last equation arises from the non-Abelian nature of the gauge theory and is the reason for the peculiar features of the color force. The current masses of the up (u), down (d), and strange (s) quarks are in a MS scheme at a scale of μ ≈ 2 GeV (Zyla et al. 2020): mu = 2.2 ± 0.5 MeV,

(4)

md = 4.7 ± 0.5 MeV,

(5)

ms = 93 ± 11 MeV.

(6)

These masses are small as compared to a typical hadronic scale such as the mass of a light hadron other than a Goldstone bosons, e.g., mρ = 0.78 GeV ≈ 1 GeV. Thus it is relevant to discuss the QCD Lagrangian in the case when the quark masses vanish:

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1 0 LQCD = qiγ ¯ μ Dμ q − Gμν,a Gaμν . 4

(7)

Right- and left-handed quark fields are defined as qR = PR q ,

qL = PL q ,

(8)

with PR =

1 1 (1 + γ5 ) , PL = (1 − γ5 ) . 2 2

(9)

Then the Lagrangian can be rewritten as 1 0 LQCD = q¯R iγ μ Dμ qR + q¯L iγ μ Dμ qL − Gμν,a Gaμν . 4

(10)

This equation revels that the right- and left-handed components of massless quarks do not mix in the QCD Lagrangian. For the two-flavor case, this is SU (2)R ×SU (2)L symmetry, also known as chiral symmetry. However, this symmetry is broken in two ways: explicitly and spontaneously.

Explicit Symmetry Breaking The mass term −qM ¯ q in the QCD Lagrangian Eq. (1) breaks chiral symmetry explicitly. To better see this, let’s rewrite M for the two-flavor case:  M =

mu 0 0 md



=

    1 1 10 1 0 (mu + md ) + (mu − md ) 01 0 −1 2 2

=

1 1 (mu + md ) I + (mu − md ) τ3 . 2 2

(11)

The first term in the last equation in invariant under SU (2)V (isospin symmetry) and the second term vanish for mu = md . Therefore, isospin is an exact symmetry if mu = md . However, both terms in Eq. (11) break chiral symmetry. Since the up and down quark masses [Eqs. (4) and (5)] are small as compared to the typical hadronic mass scale of ∼ 1 GeV, the explicit chiral-symmetry breaking due to nonvanishing quark masses is very small.

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Spontaneous Symmetry Breaking A (continuous) symmetry is said to be spontaneously broken if a symmetry of the Lagrangian is not realized in the ground state of the system. There is evidence that the (approximate) chiral symmetry of the QCD Lagrangian is spontaneously broken—for dynamical reasons of nonperturbative origin which are not fully understood at this time. The most plausible evidence comes from the hadron spectrum. From chiral symmetry, one naively expects the existence of degenerate hadron multiplets of opposite parity, i.e., for any hadron of positive parity, one would expect a degenerate hadron state of negative parity and vice versa. However, these “parity doublets” are not observed in nature. For example, take the ρ-meson which is a vector meson of negative parity (J P = 1− ) and mass 776 MeV. There does exist a 1+ meson, the a1 , but it has a mass of 1230 MeV and, therefore, cannot be perceived as degenerate with the ρ. On the other hand, the ρ meson comes in three charge states (equivalent to three isospin states), the ρ ± and the ρ 0 , with masses that differ by at most a few MeV. Thus, in the hadron spectrum, SU (2)V (isospin) symmetry is well observed, while axial symmetry is broken: SU (2)R × SU (2)L is broken down to SU (2)V . A spontaneously broken global symmetry implies the existence of (massless) Goldstone bosons. The Goldstone bosons are identified with the isospin triplet of the (pseudoscalar) pions, which explains why pions are so light. The pion masses are not exactly zero because the up and down quark masses are not exactly zero either (explicit symmetry breaking). Thus, pions are a truly remarkable species: they reflect spontaneous as well as explicit symmetry breaking. Goldstone bosons interact weakly at low energy. They are degenerate with the vacuum, and, therefore, interactions between them must vanish at zero momentum and in the chiral limit (mπ → 0).

Chiral Effective Lagrangians The next step in our EFT program is to build the most general Lagrangian consistent with the (broken) symmetries discussed above. An elegant formalism for the construction of such Lagrangians was developed by Callan, Coleman, Wess, and Zumino (CCWZ) (Coleman et al. 1969) who developed the foundations of nonlinear realizations of chiral symmetry from the point of view of group theory. (An accessible introduction into the rather involved CCWZ formalism can be found in Scherer (2003).) The Lagrangians to be applied are built upon the CCWZ formalism. As addressed before, the appropriate degrees of freedom are pions (Goldstone bosons) and nucleons. Because pion interactions must vanish at zero momentum transfer and in the limit of mπ → 0, namely, the chiral limit, the Lagrangian is expanded in powers of derivatives and pion masses. More precisely, the Lagrangian is expanded in powers of Q/Λχ where Q stands for a (small) momentum or pion

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mass and Λχ ≈ 1 GeV is identified with the hard scale. These are the basic steps behind the chiral perturbative expansion. Schematically, the effective Lagrangian is written as L = Lπ π + Lπ N + LN N + . . . ,

(12)

where Lπ π deals with the dynamics among pions, Lπ N describes the interaction between pions and a nucleon, and LN N contains two-nucleon contact interactions which consist of four nucleon fields (four nucleon legs) and no meson fields. The ellipsis stands for terms that involve two nucleons plus pions and three or more nucleons with or without pions, relevant for nuclear many-body forces. The individual Lagrangians are organized order by order: (4) Lπ π = Lπ(2) π + Lπ π + . . . ,

(13)

(2) (3) (4) (5) Lπ N = Lπ(1) N + Lπ N + Lπ N + Lπ N + Lπ N + . . . ,

(14)

and (0)

(2)

(4)

(6)

LN N = LN N + LN N + LN N + LN N + . . . ,

(15)

where the superscript refers to the number of derivatives or pion mass insertions (chiral dimension) and the ellipsis stands for terms of higher dimensions. The heavybaryon formulation of the Lagrangians is used, the explicit expressions of which can be found in Machleidt and Entem (2011), Krebs et al. (2012).

Nuclear Forces from EFT: Overview Further steps are taken toward a derivation of nuclear forces from EFT. In this section, the expansion is discussed in more detail as well as the various Feynman diagrams as they emerge order by order.

Chiral Perturbation Theory and Power Counting An infinite number of Feynman diagrams can be evaluated from the effective Lagrangians, and so one needs to be able to organize these diagrams in order of their importance. Chiral perturbation theory (ChPT) provides such organizational scheme. As discussed, in ChPT graphs are analyzed in terms of powers of small external momenta over the large scale: (Q/Λχ )ν , where Q is generic for a momentum (nucleon three-momentum or pion four-momentum) or a pion mass and Λχ ∼ 1

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GeV is the chiral-symmetry breaking scale (hadronic scale, hard scale). Determining the power ν has become known as power counting. For the moment, only so-called irreducible graphs are considered. By definition, an irreducible graph is a diagram that cannot be separated into two by cutting only nucleon lines. Following the Feynman rules of covariant perturbation theory, a nucleon propagator carries the dimension Q−1 , a pion propagator Q−2 , each derivative in any interaction is Q, and each four-momentum integration Q4 . This is also known as naive dimensional analysis. Applying then some topological identities, one obtains for the power of an irreducible diagram involving A nucleons (Machleidt and Entem 2011) ν = −2 + 2A − 2C + 2L +



Δi ,

(16)

i

with the “index of the interaction” Δi ≡ di +

ni − 2. 2

(17)

In the above equations, C represents the number of individually connected parts of the diagram, while L is the number of loops; moreover, for each vertex i, di indicates how many derivatives or pion masses are present, and ni is the number of nucleon fields. The summation extends over all vertices present in that particular diagram. Notice also that chiral symmetry implies Δi ≥ 0. Interactions among pions have at least two derivatives (di ≥ 2, ni = 0), while interactions between pions and a nucleon have one or more derivatives (di ≥ 1, ni = 2). Finally, pure contact interactions among nucleons (ni = 4) have di ≥ 0. In this way, a low-momentum expansion based on chiral symmetry can be constructed. Naturally, the powers must be bounded from below for the expansion to converge. This is in fact the case, with ν ≥ 0. To further illustrate the power formula Eq. (16), it is applied to diagrams shown in Fig. 1. From the LO row, consider the one-pion exchange diagram (second diagram). Each vertex in this diagram is a small dot which consists of one derivative (di = 1) and two nucleon legs/fields (ni = 2); thus, for the small-dot vertices, one has Δi = 1 +

2 − 2 = 0. 2

(18)

Moreover, A = 2, C = 1, and L = 0, and so Eq. (16) results into ν = −2 + 2 × 2 − 2 × 1 + 2 × 0 +



0 = 0,

(19)

i

as it should for LO. Moving on to NLO, let us pick one triangular diagram from the second row of Fig. 1. All three small-dot vertices in this diagram have Δi = 0. Furthermore, A = 2, C = 1, and L = 1 (one loop). Hence,

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ν = −2 + 2 × 2 − 2 × 1 + 2 × 2 +



0 = 2,

(20)

i

as required for NLO. Alternatively, one can also calculate the power of a diagram “from scratch,” i.e., without the help of Eq. (16). For the NLO triangular diagram just discussed, this goes like this: Each vertex contains one derivative, each meson propagator is (−2), the nucleon propagator is (−1), and the loop integration is 4; thus, ν = 1 + 1 + 1 − 2 − 2 − 1 + 4 = 2,

(21)

in agreement with what is obtained from Eq. (16). By the way, the power formula Eq. (16) also allows to predict the leading orders of connected multi-nucleon forces. Consider a m-nucleon irreducibly connected diagram (m-nucleon force) in an A-nucleon system (m ≤ A). The number of separately connected  pieces is C = A − m + 1. Inserting this into Eq. (16) together with L = 0 and i Δi = 0 yields ν = 2m − 4. Thus, two-nucleon forces (m = 2) appear at ν = 0, three-nucleon forces (m = 3) at ν = 2 (but they happen to cancel at that order), and four-nucleon forces at ν = 4 (they don’t cancel). More about this in the next subsection. For later purposes, it is noted that for an irreducible NN diagram (A = 2, C = 1), the power formula collapses to the very simple expression ν = 2L +



Δi .

(22)

i

To summarize, at each order ν, there is only a well-defined number of diagrams, which renders the theory feasible from a practical standpoint. The magnitude of what has been left out at order ν can be estimated (in a very simple way) from (Q/Λχ )ν+1 . The ability to calculate observables (in principle) to any degree of accuracy gives the theory its predictive power.

The Ranking of Nuclear Forces As shown in Fig. 1, nuclear forces appear in ranked orders in accordance with the power counting scheme. The lowest power is ν = 0, also known as the leading order (LO). At LO, there are only two contact contributions with no momentum dependence (∼ Q0 ). They are signified by the four-nucleon-leg diagram with a small-dot vertex shown in the first row of Fig. 1. Besides this, there is the static one-pion exchange (1PE), also shown in the first row of Fig. 1. In spite of its simplicity, this rough description contains some of the main attributes of the NN force. First, through the 1PE it generates the tensor component of the force known to be crucial for the two-nucleon bound state. Second, it predicts correctly NN phase parameters for high partial waves. At LO, the two terms which

49 Nonlocal Chiral Nuclear Forces up to N5 LO

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Fig. 1 Hierarchy of nuclear forces in ChPT. Solid lines represent nucleons and dashed lines pions. Small dots, large solid dots, solid squares, triangles, diamonds, and stars denote vertices of index Δi = 0, 1, 2, 3, 4, and 6, respectively. Further explanations are given in the text

result from a partial-wave expansion of the contact term impact states of zero orbital angular momentum and produce attraction at short and intermediate range. Notice that there are no terms with power ν = 1, as they would violate parity conservation and time-reversal invariance. The next order is then ν = 2, next-to-leading order, or NLO. Note that the two-pion exchange (2PE) makes its first appearance at this order, and thus it is referred to as the “leading 2PE.” As is well known from decades of nuclear physics, this contribution is essential for a realistic account of the intermediate-range attraction. However, the leading 2PE has insufficient strength, for the following reason: the loops present in the diagrams which involve pions carry the power ν = 2 [cf. Eq. (22)], and so only π N N and π π N N vertices with Δi = 0

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are allowed at this order. These vertices are known to be weak. Moreover, seven new contacts appear at this order which impact L = 0 and L = 1 states. (As always, two-nucleon contact terms are indicated by four-nucleon-leg diagrams and a vertex of appropriate shape, in this case a solid square.) At this power, the appropriate operators include central, spin-spin, spin-orbit, and tensor terms, namely, all the spin operator structures needed for a realistic description of the 2NF, although the medium-range attraction still lacks sufficient strength. At the next order, ν = 3 or next-to-next-to-leading order (NNLO), the 2PE contains the subleading π π N N seagull vertices with two derivatives (Kaiser et al. 1997). These vertices, denoted by a large solid dot in Fig. 1, simulate correlated 2PE and intermediate Δ(1232)-isobar contributions. Consistent with what the meson theory of the nuclear forces (Lacombe et al. 1980; Machleidt et al. 1987) has shown since a long time concerning the importance of these effects, at this order the 2PE finally provides medium-range attraction of realistic strength, bringing the description of the NN force to an almost quantitative level. No new contacts become available at NNLO. An important advantage of ChPT is that it generates two- and many-nucleon forces on an equal footing. Thus, three-nucleon forces appear for the first time at NLO, but their net contribution vanishes at this order (Weinberg 1992). The first nonzero 3NF contribution is found at NNLO (van Kolck 1994; Epelbaum et al. 2002). It is therefore easy to understand why 3NF are very weak as compared to the 2NF which contributes already at (Q/Λχ )0 . For ν = 4, or next-to-next-to-next-to-leading order (N3 LO), some representative diagrams are displayed in Fig. 1. There is a large attractive one-loop 2PE contribution (the bubble diagram with two large solid dots), which slightly overestimates the 2NF attraction at medium range. Two-pion exchange graphs with two loops are seen at this order, together with three-pion exchange (3PE), which was determined to be very weak at N3 LO (Kaiser 2000a, b). The most important feature at this order is the presence of 15 additional contacts ∼ Q4 , signified by the four-nucleon-leg diagram in the figure with the diamond-shaped vertex. These contacts impact states with orbital angular momentum up to L = 2 and are the reason for the quantitative description of the two-nucleon force (up to approximately 300 MeV in terms of laboratory energy) at this order (Machleidt and Entem 2011; Entem and Machleidt 2003). More 3NF diagrams show up at N3 LO, as well as the first contributions to four-nucleon forces (4NF). It is then seen that forces involving more and more nucleons appear for the first time at higher and higher orders, which gives theoretical support to the fact that 2NF  3NF  4NF. . . . Further 2PE and 3PE occur at N4 LO (fifth order). The contribution to the 2NF at this order has been first calculated by Entem et al. (2015a). It turns out to be moderately repulsive, thus compensating for the attractive surplus generated at N3 LO by the bubble diagram with two solid dots. The long- and intermediate-range 3NF contributions at this order have been evaluated (Krebs et al. 2012, 2013), but not yet applied in nuclear structure calculations. They are expected to be sizeable. Moreover, a new set of 3NF contact terms appears (Girlanda et al. 2011) that had a successful application in the context of the so-called Ay puzzle of nucleon-deuteron

49 Nonlocal Chiral Nuclear Forces up to N5 LO

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scattering (Girlanda et al. 2019). The N4 LO 4NF has not been derived yet. Due to the subleading π π N N seagull vertex (large solid dot), this 4NF could be sizeable. Finally turning to N5 LO (sixth order), the dominant 2PE and 3PE contributions to the 2NF have been derived by Entem et al. in Entem et al. (2015b), which represents the most advanced investigation conducted in chiral EFT for the NN system. The effects are small indicating the desired trend toward convergence of the chiral expansion for the 2NF. Moreover, a new set of 26 NN contact terms ∼ Q6 occurs that contributes up to F waves (represented by the NN diagram with a star in Fig. 1) bringing the total number of NN contacts to 50 (Entem and Machleidt 1993). The three-, four-, and five-nucleon forces of this order have not yet been evaluated. To summarize, Fig. 2 displays the contributions to the phase shifts of peripheral N N scattering through all orders from LO to N5 LO as obtained from a perturbative calculation. Note that the difference between the LO prediction (one-pion exchange,

Fig. 2 Phase shifts of neutron-proton scattering in G waves at all orders of ChPT from LO to N5 LO. The filled and open circles represent the results from the Nijmegen multi-energy np phaseshift analysis (Rentmeester 1993) and the GWU single-energy np analysis SP07 (Briscoe 2007), respectively

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dotted line) and the data (filled and open circles) is to be provided by two- and three-pion exchanges, i.e., the intermediate-range part of the nuclear force. How well that is accomplished is a crucial test for any theory of nuclear forces. NLO produces only a small contribution, but N2 LO creates substantial intermediaterange attraction (most clearly seen in 1 G4 , 3 G5 ). In fact, N2 LO is the largest contribution among all orders. This is due to the one-loop 2π -exchange triangle diagram which involves one π π N N contact vertex (large solid dot). As discussed, the one-loop 2π -exchange at N2 LO is attractive and describes the intermediaterange attraction of the nuclear force about right. At N3 LO, more one-loop 2PE is added by the bubble diagram with two large solid dots, a contribution that seemingly is overestimating the attraction. This attractive surplus is then compensated by the prevailingly repulsive two-loop 2π - and 3π -exchanges that occur at N4 LO and N5 LO. In this context, it is worth noting that also in conventional meson theory (Machleidt et al. 1987), the one-loop models for the 2PE contribution always show some excess of attraction (cf. Fig. 2 of Entem and Machleidt (2002) and Fig. 10 of Machleidt and Entem (2011)). The same is true for the dispersion theoretic approach pursued by the Paris group (see, e.g., the predictions for 1 D2 , 3 D2 , and 3 D in Fig. 8 of Vinh Mau (1979) which are all too attractive). In conventional 3 meson theory, this attraction is reduced by heavy-meson exchanges (ρ-, ω-, and πρ-exchange) which, however, has no place in chiral effective field theory (as a finite-range contribution). Instead, in the latter approach, two-loop 2π - and 3π -exchanges provide the corrective action.

Quantitative Chiral NN Potentials In the previous section, the pion exchange contributions to the NN interaction were mainly discussed. They describe the long- and intermediate-range parts of the nuclear force, which are governed by chiral symmetry and rule the peripheral partial waves (cf. Fig. 2). However, for a “complete” nuclear force, all partial waves have to be described correctly, including the lower ones. In fact, in calculations of N N observables at low energies (cross sections, analyzing powers, etc.), the partial waves with L ≤ 2 are the most important ones, generating the largest contributions. The same is true for microscopic nuclear structure calculations. The lower partial waves are dominated by the dynamics at short distances. Therefore, a close look at the short-range part of the NN potential has to be taken.

NN Contact Terms In conventional meson theory (Machleidt 1989; Machleidt et al. 1987), the shortrange nuclear force is described by the exchange of heavy mesons, notably the ω(782). Qualitatively, the short-distance behavior of the NN potential is obtained by Fourier transform of the propagator of a heavy meson:

49 Nonlocal Chiral Nuclear Forces up to N5 LO

 d 3q

1893

eiq·r e−mω r ∼ . r m2ω + q2

(23)

ChPT is an expansion in small momenta Q, too small to resolve structures like a ρ(770) or ω(782) meson, because Q Λχ ≈ mρ,ω . But the latter relation allows us to expand the propagator of a heavy meson into a power series: 1 1 ≈ 2 m2ω + Q2 mω

 1−

 Q2 Q4 + − + . . . , m2ω m4ω

(24)

where the ω is representative for any heavy meson of interest. The above expansion suggests that it should be possible to describe the short-distance part of the nuclear force simply in terms of powers of Q/mω , which fits in well with our overall power expansion since Q/mω ≈ Q/Λχ . In Fig. 1, such terms are denoted by four-nucleonleg diagrams, like the first diagram in the first and the second row of the figure. Since in these diagrams, the nucleons come infinitely close to each other (no meson exchanges in-between them that would keep them apart), these diagrams are dubbed contact terms. Contact terms play an important role in renormalization. Regularization of the loop integrals that occur in multi-pion exchange diagrams typically generates polynomial terms with coefficients that are, in part, infinite or scale dependent (cf. Appendix B of Machleidt and Entem (2011)). Contact terms absorb infinities and remove scale dependences. Due to parity, only even powers of Q are allowed. Thus, the expansion of the contact potential is formally given by (0)

(2)

(4)

(6)

Vct = Vct + Vct + Vct + Vct + . . . ,

(25)

where the superscript denotes the power or order. The contact terms of the various orders are given below. Zeroth-Order (LO) NN Contact Potential

Vct(0) (p , p) = CS + CT σ 1 · σ 2 ,

(26)

where p and p denote the final and initial nucleon momenta in the center-ofmass system, respectively, and σ 1 and σ 2 represent the spin operators for nucleons 1 and 2. In terms of partial waves, there is 1 S = 4π (CS − 3 CT ) Vct(0) (1 S0 ) = C 0 3 S = 4π (CS + CT ) . Vct(0) (3 S1 ) = C 1

(27)

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Second-Order (NLO) NN Contact Potential Vct (p , p) = C1 q 2 + C2 k 2  + C 3 q 2 + C4 k 2 σ 1 · σ 2 (2)

+ C5 (−iS · (q × k)) + C6 (σ 1 · q) (σ 2 · q) + C7 (σ 1 · k) (σ 2 · k) ,

(28)

with q = p − p, k = (p + p)/2 and S = (σ 1 + σ 2 )/2 the total spin. Partial-wave decomposition yields Vct (1 S0 ) = C1 S0 (p2 + p ) 2

(2)

Vct (3 P0 ) = C3 P0 pp

(2)

Vct (1 P1 ) = C1 P1 pp

(2)

Vct(2) (3 P1 ) = C3 P1 pp

Vct (3 S1 ) = C3 S1 (p2 + p ) 2

(2)

(2)

Vct (3 S1 −3 D1 ) = C3 S1 −3 D1 p2 Vct(2) (3 D1 −3 S1 ) = C3 S1 −3 D1 p

2

Vct (3 P2 ) = C3 P2 pp , (2)

(29)

which obviously contributes up to P waves. Fourth-Order (N3 LO) NN Contact Potential Vct(4) (p , p) = D1 q 4 + D2 k 4 + D3 q 2 k 2 + D4 (q × k)2  + D5 q 4 + D6 k 4 + D7 q 2 k 2 + D8 (q × k)2 σ 1 · σ 2  + D9 q 2 + D10 k 2 (−iS · (q × k))  + D11 q 2 + D12 k 2 (σ 1 · q) (σ 2 · q)  + D13 q 2 + D14 k 2 (σ 1 · k) (σ 2 · k) + D15 (σ 1 · (q × k) σ 2 · (q × k)) .

(30)

49 Nonlocal Chiral Nuclear Forces up to N5 LO

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The rather lengthy partial-wave expressions at this order are given in Appendix E of Machleidt and Entem (2011). These contacts affect partial waves up to D waves. Sixth Order (N5 LO) At sixth order, 26 new contact terms appear, bringing the total number to 50. These terms as well as their partial-wave decomposition have been worked out in Entem and Machleidt (1993). They contribute up to F waves. Except for the F -wave terms, these contacts have not been used in the construction of NN potentials.

Definition of NN Potential Now everything has been rounded up that is needed for a realistic nuclear force— long-, intermediate-, and short-ranged components—and so one can finally proceed to the lower partial waves. However, here one encounters another problem. The twonucleon system at low angular momentum, particularly in S waves, is characterized by the presence of a shallow bound state (the deuteron) and large scattering lengths. Thus, perturbation theory does not apply. In contrast to π -π and π -N , the interaction between nucleons is not suppressed in the chiral limit (Q → 0). Weinberg (1991) showed that the strong enhancement of the scattering amplitude arises from purely nucleonic intermediate states (“infrared enhancement”). He therefore suggested to use perturbation theory to calculate the NN potential (i.e., the irreducible graphs) and to apply this potential in a scattering equation to obtain the NN amplitude. This prescription will be followed here. The potential V as discussed in previous sections is, in principle, an invariant amplitude and, thus, satisfies a relativistic scattering equation, for which the BbS equation (Blankenbecler and Sugar 1966) is chosen, which reads explicitly

T (p , p) = V (p , p) +



2 1 d 3 p



MN T (p

, p) V (p , p ) 3 2 Ep

p − p

2 + iε (2π )

(31)

with Ep

≡ MN2 + p

2 and MN the nuclear mass. The advantage of using a relativistic scattering equation is that it automatically includes relativistic corrections to all orders. Thus, in the scattering equation, no propagator modifications are necessary when raising the order to which the calculation is conducted. Defining

(p , p) ≡ 1 V (2π )3



MN V (p , p) Ep



MN Ep

(32)

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and 1 T(p , p) ≡ (2π )3



MN T (p , p) Ep



MN , Ep

(33)

where the factor 1/(2π )3 is added for convenience, the BbS equation collapses into the usual, nonrelativistic Lippmann-Schwinger (LS) equation: (p , p) + T(p , p) = V



(p , p

) d 3 p

V

MN p2

− p

2

+ iε

T(p

, p) .

(34)

 satisfies Eq. (34), it can be used like a nonrelativistic potential, and T may Since V be perceived as the conventional nonrelativistic T -matrix.

Regularization and Nonperturbative Renormalization  in the LS equation, Eq. (34), requires cutting V  off for high momenta Iteration of V to avoid infinities. This is consistent with the fact that ChPT is a low-momentum expansion which is valid only for momenta Q Λχ ≈ 1 GeV. Therefore, the  is multiplied with the regulator function f (p , p): potential V (p , p) −→ V (p , p) f (p , p) V

(35)



f (p , p) = exp −(p /Λ)2n − (p/Λ)2n ,

(36)

with

such that  (p , p) 1 − (p , p) f (p , p) ≈ V V







p

Λ

2n +

 p 2n Λ



 + ...

.

(37)

Typical choices for the cutoff parameter Λ that appears in the regulator are Λ ≈ 0.5 GeV < Λχ ≈ 1 GeV. Equation (37) provides an indication of the fact that the exponential cutoff does not necessarily affect the given order at which the calculation is conducted. For sufficiently large n, the regulator introduces contributions that are beyond the given order. Assuming a good rate of convergence of the chiral expansion, such orders are small as compared to the given order and, thus, do not affect the accuracy at the given order. (In calculations, one uses, of course, the exponential form, Eq. (36), and not the expansion Eq. (37).) It is pretty obvious that results for the T -matrix may depend sensitively on the regulator and its cutoff parameter. This is acceptable if one wishes to build models. For example, the meson models of the past (Machleidt 1989; Machleidt et al. 1987)

49 Nonlocal Chiral Nuclear Forces up to N5 LO

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always depended sensitively on the choices for the cutoff parameters, and they were welcome as additional fit parameters to further improve the reproduction of the NN data. However, the EFT approach wishes to be more fundamental in nature and not just another model. In field theories, divergent integrals are not uncommon and methods have been devised for how to deal with them. One regulates the integrals and then removes the dependence on the regularization parameters (scales, cutoffs) by renormalization. In the end, the theory and its predictions do not depend on cutoffs or renormalization scales. Renormalizable quantum field theories, like QED, have essentially one set of prescriptions that takes care of renormalization through all orders. In contrast, EFTs are renormalized order by order, i.e., each order comes with the contact terms needed to renormalize that order. Note that this applies only to perturbative calculations. The NN potential is calculated perturbatively and hence properly renormalized. However, the story is different for the NN amplitude (T -matrix) that results from a solution of the LS equation, Eq. (34), which is a nonperturbative resummation of the potential. This resummation is necessary in nuclear EFT because nuclear physics is characterized by bound states which are nonperturbative in nature. EFT power counting may be different for nonperturbative processes as compared to perturbative ones. Such difference may be caused by the infrared enhancement of the reducible diagrams generated in the LS equation. Weinberg’s discussion in Weinberg (1990, 1991) may suggest that the contact terms introduced to renormalize the perturbatively calculated potential, based upon naive dimensional analysis (“Weinberg counting”), may also be sufficient to renormalize the nonperturbative resummation of the potential in the LS equation. Weinberg’s alleged assumption may not be correct as first pointed out by Kaplan, Savage, and Wise (KSW) (1996) who, therefore, suggested to treat 1PE perturbatively—a prescription which, however, has convergence problems (Fleming et al. 2000). The KSW critique resulted in a flurry of publications on the renormalization of the NN amplitude, and the interested reader is referred to Section 4.5 of Machleidt and Entem (2011) for an account of the first phase of discussion. However, even today, no generally accepted solution to this problem has emerged, and some more recent proposals can be found in (Hammer et al. 2020; Nogga et al. 2005; Birse 2006; Long and Yang 2012; Long 2016; Valderrama 2011; Pavon Valderrama 2011; Valderrama 2016; Pavon Valderrama et al. 2017; Epelbaum et al. 2017; K˝onig et al. 2017; Epelbaum et al. 2018; van Kolck 2020; Valderrama 1902; Entem and Oller 2021). Concerning the construction of quantitative NN potential (by which NN potentials suitable for use in contemporary many-body nuclear methods are meant), only Weinberg counting has been used with success during the past 25 years (Ordóñez et al. 1996; Entem and Machleidt 2003; Epelbaum et al. 2000, 2005, 2015a, b; Ekstr˝om et al. 2013; Gezerlis et al. 2014; Piarulli et al. 2015, 2016; Navarro Pérez et al. 2015; Carlsson et al. 2016; Reinert et al. 2018; Ekstr˝om et al. 2015; Ekstr˝om et al. 2018; Entem et al. 2017).

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In spite of the criticism, Weinberg counting may be perceived as not unreasonable by the following argument. For a successful EFT (in its domain of validity), one must be able to claim independence of the predictions on the regulator within the theoretical error. Also, truncation errors must decrease as one goes to higher and higher orders. These are precisely the goals of renormalization. Lepage (1977) has stressed that the cutoff independence should be examined for cutoffs below the hard scale and not beyond. Ranges of cutoff independence within the theoretical error are to be identified using Lepage plots (Lepage 1977). A systematic investigation of this kind has been conducted in Marji et al. (2013). In that work, the error of the predictions was quantified by calculating the χ 2 /datum for the reproduction of the np elastic scattering data as a function of the cutoff parameter Λ of the regulator function Eq. (36). Predictions by chiral np potentials at order NLO and NNLO were investigated applying Weinberg counting for the NN contact terms. It is found that the reproduction of the np data at lab. energies below 200 MeV is generally poor at NLO, while at NNLO the χ 2 /datum assumes acceptable values (a clear demonstration of order-by-order improvement). Furthermore, at NNLO, a “plateau” of constant low χ 2 for cutoff parameters ranging from about 450 to 850 MeV can be identified. This may be perceived as cutoff independence (and, thus, successful renormalization) for the relevant range of cutoff parameters. Alternatively, one may go for a compromise between Weinberg’s prescription of full resummation of the potential and Kaplan, Savage, and Wise’s (1996) suggestion of perturbative pions—as discussed in van Kolck (2020): 1PE is resummed only in lower partial waves and all corrections are included in distorted-wave perturbation theory. However, since current ab initio calculations are tailored such that they need a potential as input, the question is if there is a way to reconcile those (low-cutoff) potentials with the approach of partially perturbative pions. A first attempt to address this issue has recently been undertaken by Valderrama (1902).

NN Potentials Order by Order N N potentials depend on two different sets of parameters, the π N and the NN LECs. The π N LECs are the coefficients that appear in the π N Lagrangians. They are determined in π N analysis (Hoferichter et al. 2015). The NN LECs are the coefficients of the NN contact terms. They are fixed by an optimal fit to the NN data below pion-production threshold; see Entem et al. (2017) for details. N N potentials are then constructed order by order and the accuracy improves as the order increases. How well the chiral expansion converges in important lower partial waves is demonstrated in Figs. 3 and 4, where phase parameters for potentials developed through all orders from LO to N4 LO (Entem et al. 2017) are shown. (See also the chiral NN potentials of (Epelbaum et al. 2005, 2015a, b; Ekstr˝om et al. 2013; Gezerlis et al. 2014; Piarulli et al. 2015, 2016; Navarro Pérez et al. 2015; Carlsson et al. 2016; Reinert et al. 2018; Ekstr˝om et al. 2015; Ekstr˝om et al. 2018).) These figures clearly reveal substantial improvements in the reproduction of the empirical phase shifts with increasing order.

49 Nonlocal Chiral Nuclear Forces up to N5 LO

1899 60

1 80

S0

40

LO NNLO N4LO N3LO NLO

0 -40 0

Phase Shift (deg)

Phase Shift (deg)

120

0

3P

-20 0

N4LO N3LO NNLO NLO

0

100 200 300 400 Lab. Energy (MeV)

0

P1 LO

-10 -20

N3LO N4LO NLO NNLO

-30 0

LO N3LO N4LO NNLO NLO

-100

P1

-10

NLO LO NNLO N3LO N4LO

-20 -30 0

10

S1

100 0

3 0

100 200 300 400 Lab. Energy (MeV)

3

200

Phase Shift (deg)

10

1

Phase Shift (deg)

Phase Shift (deg)

LO 20

100 200 300 400 Lab. Energy (MeV)

10

Phase Shift (deg)

40

100 200 300 400 Lab. Energy (MeV)

3

0

D1

-10 NLO N3LO N4LO NNLO LO

-20 -30 -40

0

100 200 300 400 Lab. Energy (MeV)

0

100 200 300 400 Lab. Energy (MeV)

Fig. 3 Chiral expansion of neutron-proton scattering as represented by the phase shifts for J ≤ 1. Five orders ranging from LO to N4 LO are shown as denoted. Filled and open circles as in Fig. 2 (From Entem et al. (2017))

The χ 2 /datum for the reproduction of the NN data at various orders of chiral EFT is shown in Table 1 for different energy intervals below 290 MeV laboratory energy (Tlab ). The bottom line of Table 1 summarizes the essential results. For the close to 5000 pp plus np data below 290 MeV (pion-production threshold), the χ 2 /datum is 51.4 at NLO and 6.3 at NNLO. Note that the number of NN contact terms is the same for both orders. The improvement is entirely due to an improved description of the 2PE contribution, which is responsible for the crucial intermediate-range attraction of the nuclear force. At NLO, only the uncorrelated 2PE is taken into account which is insufficient. From the classic meson theory of

ε1

10

LO NNLO N3LO N4LO

5 0

NLO

Phase Shift (deg)

D. R. Entem et al. Mixing Parameter (deg)

1900

N3LO N4LO NNLO

5 NLO LO 0

100 200 300 400 Lab. Energy (MeV)

0

40 30

30

D2

NNLO N4LO N3LO LO NLO

20 10 0 0

N3LO NLO N4LO

-2

LO NNLO -4

100 200 300 400 Lab. Energy (MeV)

3

P2

20

NLO N3LO N4LO

10

NNLO LO

0

100 200 300 400 Lab. Energy (MeV)

ε2

0

Phase Shift (deg)

3

0

Phase Shift (deg)

Phase Shift (deg)

D2

-5 0

Mixing Parameter (deg)

1

10

8

100 200 300 400 Lab. Energy (MeV)

3

D3

4

N3LO N4LO

0 NNLO LO NLO

-4 -8

0

100 200 300 400 Lab. Energy (MeV)

0

100 200 300 400 Lab. Energy (MeV)

Fig. 4 Same as Fig. 3, but for 3 P2 , D waves, and mixing parameters ε1 and ε2

nuclear forces (Machleidt et al. 1987), it is well known that π -π correlations and nucleon resonances need to be taken into account for a realistic model of 2PE. As discussed, in the chiral theory, these contributions are encoded in the subleading π N vertexes. These enter at NNLO and are the reason for the substantial improvements encountered at that order. To continue on the bottom line of Table 1, after NNLO, the χ 2 /datum then further improves to 1.63 at N3 LO and, finally, reaches the almost perfect value of 1.15 at N4 LO—a fantastic convergence. The evolution of the deuteron properties from LO to N4 LO of chiral EFT is shown in Table 2. In all cases, the deuteron binding energy is fit to its empirical

49 Nonlocal Chiral Nuclear Forces up to N5 LO

1901

Table 1 χ 2 /datum for the fit of the 2016 N N database by N N potentials at various orders of chiral EFT (From Entem et al. (2017)) Tlab bin (MeV) Proton-proton 0–100 0–190 0–290 Neutron-proton 0–100 0–190 0–290 pp plus np 0–100 0–190 0–290

No. of data

LO

NLO

NNLO

N3 LO

N4 LO

795 1206 2132

520 430 360

18.9 43.6 70.8

2.28 4.64 7.60

1.18 1.69 2.09

1.09 1.12 1.21

1180 1697 2721

114 96 94

7.2 23.1 36.7

1.38 2.29 5.28

0.93 1.10 1.27

0.94 1.06 1.10

1975 2903 4853

283 235 206

11.9 31.6 51.5

1.74 3.27 6.30

1.03 1.35 1.63

1.00 1.08 1.15

Table 2 Two- and three-nucleon bound-state properties as predicted by N N potentials at various orders of chiral EFT (Λ = 500 MeV in all cases). (Deuteron: Binding energy Bd , asymptotic S state AS , asymptotic D/S state η, structure radius rstr , quadrupole moment Q, D-state probability PD ; the predicted rstr and Q are without meson-exchange current contributions and relativistic corrections. Triton: Binding energy Bt .) Bd is fitted; all other quantities are predictions (From Entem et al. (2017)) Deuteron Bd (MeV) AS (fm−1/2 ) η rstr (fm) Q (fm2 ) PD (%) Triton Bt (MeV)

LO

NLO

NNLO

N3 LO

N4 LO

Empiricala

2.224575 0.8526 0.0302 1.911 0.310 7.29

2.224575 0.8828 0.0262 1.971 0.273 3.40

2.224575 0.8844 0.0257 1.968 0.273 4.49

2.224575 0.8853 0.0257 1.970 0.271 4.15

2.224575 0.8852 0.0258 1.973 0.273 4.10

2.224575(9) 0.8846(9) 0.0256(4) 1.97507(78) 0.2859(3) –

11.09

8.31

8.21

8.09

8.08

8.48

a See

Table XVIII of Machleidt (2001) for references; the empirical value for rstr is from Jentschura et al. (2011)

value of 2.224575 MeV using the nonderivative 3 S1 contact. All other deuteron properties are predictions. Already at NNLO, the deuteron has converged to its empirical properties and stays there through the higher orders. At the bottom of Table 2, the predictions for the triton binding as obtained in 34-channel charge-dependent Faddeev calculations using only 2NFs are shown. The results show smooth and steady convergence, order by order, toward a value around 8.1 MeV that is reached at the highest orders shown. This contribution from the 2NF

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will require only a moderate 3NF. The relatively low deuteron D-state probabilities (≈ 4.1% at N3 LO and N4 LO) and the concomitant generous triton binding energy predictions are a reflection of the fact that the NN potentials are soft (which is, at least in part, due to their nonlocal character).

Nuclear Many-Body Forces Two-nucleon forces derived from chiral EFT have been applied, often successfully, in the many-body system. On the other hand, over the past several years, one has learned that, for some few-nucleon reactions and nuclear structure issues, 3NFs cannot be neglected. The most well-known cases are the so-called Ay puzzle of N -d scattering (Entem et al. 2002), the ground state of 10 B (Caurier et al. 2002), and the saturation of nuclear matter (Sammarruca et al. 2012; Coraggio et al. 2014; Sammarruca et al. 2015; Machleidt and Sammarruca 2016; Sammarruca et al. 1807). As observed previously, the EFT approach generates consistent two- and manynucleon forces in a natural way (cf. the overview given in Fig. 1). The focus is now shifted to chiral three- and four-nucleon forces.

Three-Nucleon Forces Weinberg (1992) was the first to discuss nuclear three-body forces in the context of ChPT. Not long after that, the first 3NF at NNLO was derived by van Kolck (1994). For a 3NF, A = 3 and C = 1 and, thus, Eq. (16) implies ν = 2 + 2L +



Δi .

(38)

i

This equation will be used to analyze 3NF contributions order by order. Next-to-Leading Order The lowest possible power is obviously  ν = 2 (NLO), which is obtained for no loops (L = 0) and only leading vertices ( i Δi = 0). As discussed by Weinberg (1992) and van Kolck (1994), the contributions from these diagrams vanish at NLO. So, the bottom line is that there is no genuine 3NF contribution at NLO. The first nonvanishing 3NF appears at NNLO. Next-to-Next-to-Leading Order The  power ν = 3 (NNLO) is obtained when there are no loops (L = 0) and i Δi = 1, i.e., Δi = 1 for one vertex, while Δi = 0 for all other vertices. There are three topologies which fulfill this condition, known as the 2PE, 1PE, and contact graphs (van Kolck 1994; Epelbaum et al. 2002) (Fig. 5).

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Fig. 5 The three-nucleon force at NNLO with (a) 2PE, (b) 1PE, and (c) contact diagrams. Notation as in Fig. 1

(a)

(b)

(c)

The 2PE 3N potential is derived to be  3NF V2PE

=

gA 2fπ

2

(σ i · qi )(σ j · qj ) 1  F ab τ a τ b 2 + m2 )(q 2 + m2 ) ij k i j 2 (q π π i j i =j =k

(39)

with qi ≡ p i − pi , where pi and p i are the initial and final momenta of nucleon i, respectively, and   4c1 m2π c4  abc c 2c3 + + q · q ε τk σ k · [qi × qj ] . Fijabk = δ ab − i j fπ2 fπ2 fπ2 c

(40)

It is interesting to observe that there are clear analogies between this force and earlier 2PE 3NFs already proposed decades ago, particularly the FujitaMiyazawa (1957) and the Tucson-Melbourne (TM) (Coon et al. 1979) forces. In fact, based upon the chiral 3NF at NNLO, the TM force was corrected (Friar et al. 1999) leading to what became known as the TM’ or TM99 force (Coon and Han 2001). The 2PE 3NF does not introduce additional fitting constants, since the π N LECs are already present in the 2PE 2NF. The other two 3NF contributions shown in Fig. 5 are the 1PE contribution 3NF V1PE = −D

gA  σ j · qj (τ i · τ j )(σ i · qj ) 8fπ2 q 2 + m2π i =j =k j

(41)

and the 3N contact potential Vct3NF = E

1  τi · τj . 2

(42)

i =j =k

These 3NF potentials introduce two additional constants, D and E, which can be constrained in more than one way. One may use the triton binding energy and the nd doublet scattering length 2 and (Epelbaum et al. 2002) or an optimal global fit of the properties of light nuclei (Navratil 2007). Alternative choices include the binding energies of 3 H and 4 He (Nogga et al. 2006) or the binding energy of 3 H and the point charge radius of 4 He (Hebeler et al. 2011). Another method makes

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use of the triton binding energy and the Gamow-Teller matrix element of tritium β-decay (Marcucci et al. 2012). When the values of D and E are determined, the results for other observables involving three or more nucleons are true theoretical predictions. Applications of the leading 3NF include few-nucleon reactions (Epelbaum et al. 2002; Navratil et al. 2010; Viviani et al 2013), structure of light- and medium-mass nuclei (Navratil et al. 2007; Roth et al. 2011, 2012; Hagen et al. 2012a, b; Barrett et al. 2013; Hergert et al. 2013; Hagen et al. 2014a; Binder et al. 2014; Hagen et al. 2016; Lapoux et al. 2016; Simonis et al. 2016, 2017; Morris et al. 2018; Somà et al. 2020), and infinite matter (Sammarruca et al. 2012; Coraggio et al. 2014; Sammarruca et al. 2015; Machleidt and Sammarruca 2016; Sammarruca et al. 1807; Hebeler et al. 2011; Hebeler and Schwenk 2010; Hagen et al. 2014b; Coraggio et al. 2013), often with satisfactory results. Some problems, though, remain unresolved, such as the well-known “Ay puzzle” in nucleon-deuteron scattering (Entem et al. 2002; Epelbaum et al. 2002). Predictions which employ only 2NFs underestimate the analyzing power in p-3 He scattering to a larger degree than in p-d. Although the p-3 He Ay improves considerably (more than in the p-d case) when the leading 3NF is included (Viviani et al 2013), the disagreement with the data is not fully removed. Also, predictions for light nuclei are not quite satisfactory. In summary, the leading 3NF of ChPT is a remarkable contribution. It gives validation to, and provides a better framework for, 3NFs which were proposed already five decades ago; it alleviates existing problems in few-nucleon reactions and the spectra of light nuclei. Nevertheless, there are still several challenges. With regard to the 2NF, as discussed earlier, it is necessary to go to order four or even five for convergence and high-precision predictions. Thus, the 3NF at N3 LO must be considered simply as a matter of consistency with the 2NF sector. At the same time, one hopes that its inclusion may result in further improvements with the aforementioned unresolved problems. Next-to-Next-to-Next-to-Leading Order At N3 LO, there are loop and tree diagrams. For the loops (Fig. 6), L = 1 and, therefore, all Δi have to be zero to ensure ν = 4. Thus, these one-loop 3NF diagrams can include only leading order vertices, the parameters of which are fixed from π N and NN analysis. The diagrams have been evaluated by the Bochum-Bonn

(a)

(b)

(c)

(d)

(e)

Fig. 6 Leading one-loop 3NF diagrams at N3 LO. One representative example for each of five topologies is shown, which are (a) 2PE, (b) 1PE-2PE, (c) ring, (d) contact-1PE, and (e) contact2PE. Notation as in Fig. 1

49 Nonlocal Chiral Nuclear Forces up to N5 LO

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group (Bernard et al. 2008, 2011). The long-range part of the chiral N3 LO 3NF has been tested in the triton and in three-nucleon scattering (Golak et al. 2014) leaving the N − d Ay puzzle unresolved. The long- and short-range parts of this force have been applied in nuclear and neutron matter calculations (Kr˝uger et al. 2013; Drischler et al. 2016; Hebeler et al. 2015; Drischler et al. 2019; Sammarruca and Millerson 2020) as well as in the structure of medium-mass nuclei (Hoppe et al. 2019; Hüther et al. 2020) with, partially, great success. The 3NF At N4 LO In regard to some unresolved issues, one may go ahead and look at the next order of 3NFs, which is N4 LO or ν = 5. The loop contributions that occur at this order are obtained by replacing in the N3 LO loops one vertex by a Δi = 1 vertex (with LEC ci ), Fig. 7, which is why these loops may be more sizable than the N3 LO loops. The 2PE, 1PE-2PE, and ring topologies have been evaluated (Krebs et al. 2012, 2013) so far. In addition, there are three “tree” topologies (Fig. 8), which include a new set of 3N contact interactions that has been derived by the Pisa group (Girlanda et al. 2011). The N 4 LO 3NF contacts have been applied with success in calculations of few-body reactions at low energy solving the p-d Ay puzzle (Girlanda et al. 2019).

Four-Nucleon Forces For connected (C = 1) A = 4 diagrams, Eq. (16) yields ν = 4 + 2L +



(43)

Δi .

i

(a)

(b)

(c)

(d)

(e)

Fig. 7 Subleading one-loop 3NF diagrams which appear at N4 LO with topologies similar to Fig. 6. Notation as in Fig. 1 Fig. 8 3NF tree graphs at N4 LO (ν = 5) denoted by (a) 2PE, (b) 1PE-contact, and (c) contact. Notation as in Fig. 1

(a)

(b)

(c)

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Fig. 9 Leading four-nucleon force at N3 LO

It is seen that the first (connected) nonvanishing 4NF is generated at ν = 4 (N3 LO), with all vertices of leading type, Fig. 9. This 4NF has no loops and introduces no novel parameters (Epelbaum 2007). For a reasonably convergent series, terms of order (Q/Λχ )4 should be small, and, therefore, chiral 4NF contributions are expected to be very weak. This has been confirmed in calculations of the energy of 4 He (Rozpedzik et al. 2006) as well as neutron matter and symmetric nuclear matter (Kr˝uger et al. 2013). The effects of the leading chiral 4NF in symmetric nuclear matter and pure neutron matter have been worked out by Kaiser et al. (2012; 2015; 2016).

Uncertainty Quantification When applying chiral two- and many-body forces in ab initio calculations producing predictions for observables of nuclear structure and reactions, major sources of uncertainties are (Furnstahl et al. 2015): 1. Experimental errors of the input NN data that the 2NFs are based upon and the input few-nucleon data to which the 3NFs are adjusted. 2. Uncertainties in the Hamiltonian due to a. Uncertainties in the determination of the NN and 3N contact LECs, b. Uncertainties in the π N LECs, c. Regulator dependence, d. EFT truncation error. 3. Uncertainties associated with the few- and many-body methods applied. For a thorough discussion of all aspects, see Entem et al. (2017), where it was concluded that regulator dependence and EFT truncation error are the major source of uncertainty.

49 Nonlocal Chiral Nuclear Forces up to N5 LO

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The choice of the regulator function and its cutoff parameter creates uncertainty. Originally, cutoff variations were perceived as a demonstration of the uncertainty at a given order (equivalent to the truncation error). However, in various investigations (Sammarruca et al. 2015; Epelbaum et al. 2015a), it has been shown that this is not correct and that cutoff variations, in general, underestimate this uncertainty. Therefore, the truncation error is better determined by sticking literally to what “truncation error” means, namely, the error due to ignoring contributions from orders beyond the given order ν. The largest such contribution is the one of order (ν + 1), which one may, therefore, consider as representative for the magnitude of what is left out. This suggests that the truncation error at order ν can reasonably be defined as ΔXν = |Xν − Xν+1 | ,

(44)

where Xν denotes the prediction for observable X at order ν. If Xν+1 is not available, then one may use ΔXν = |Xν−1 − Xν |Q/Λ ,

(45)

choosing a typical value for the momentum Q, or Q = mπ . Alternatively, one may also apply more elaborate definitions, like the one given in Epelbaum et al. (2015a). Note that one should not add up (in quadrature) the uncertainties due to regulator dependence and the truncation error, because they are not independent. In fact, it is appropriate to leave out the uncertainty due to regulator dependence entirely and just focus on the truncation error (Epelbaum et al. 2015a). The latter should be estimated using the same cutoff (e.g., Λ = 500 MeV) in all orders considered. The bottom line is that the most substantial uncertainty is the truncation error. This is the dominant source of (systematic) error that can be reliably estimated in the EFT approach.

Conclusions One of the most fundamental aims in theoretical nuclear physics is to understand nuclear structure and reactions in terms of the basic forces between nucleons. Research pursuing this goal has two major ingredients: nuclear forces and quantum many-body theory. Concerning the second ingredient, during the past two decades, there has been great progress in the development and refinement of diverse many-body methods. Among them are the no-core shell model (Barrett et al. 2013), coupled cluster theory (Hagen et al. 2014a), self-consistent Green’s functions (Somà et al. 2014), quantum Monte Carlo (Carlson et al. 2015), and the in-medium similarity renormalization group method (Hergert et al. 2016). Benchmark calculations have shown that these many-body methods are very well under control.

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Thus, whenever there are problems with predictions, then it is most likely due to the forces applied. Therefore, the focus of this paper has been on those forces. Specifically, it can be argued that the chiral EFT approach to nuclear forces has considerable advantages, because chiral EFT: • Is rooted in low-energy QCD; • Is, in principle, model-independent; • Occurs with an organizational scheme (“power counting”) that allows to quantify the uncertainty of predictions; • Generates two- and many-body forces on an equal footing . But the strength of chiral EFT extends beyond a purely formal level. Ab initio calculations applying chiral nuclear forces have been very successful. One example is the explanation of nuclear matter saturation. Though it was speculated early on that the addition of 3NFs might solve the problem, nonrelativistic calculations with phenomenological 3NFs failed to do so (Day et al. 1983; Carlson et al. 1983; Akmal et al. 1998). In contrast, the chiral 3NF at only leading order (NNLO) has the ability to solve that problem (Sammarruca et al. 2012; Coraggio et al. 2014; Sammarruca et al. 2015; Machleidt and Sammarruca 2016; Sammarruca et al. 1807; Hebeler et al. 2011). Similar observations can be made about intermediate-mass nuclei: While the traditional approach fails badly in the intermediate-mass region (Lonardoni et al. 2017), chiral EFT-based two- and three-body forces generate excellent predictions (Ekstr˝om et al. 2015; Hagen et al. 2016; Simonis et al. 2016, 2017; Morris et al. 2018; Somà et al. 2020; Hüther et al. 2020). The N -d Ay puzzle, which could never be resolved with phenomenological 3NFs (Kievsky et al. 2010), is another example for the success of EFT-based 3NFs (Girlanda et al. 2019). Thus, it is fair to say that chiral EFT has brought about substantial progress and improvement. But since EFT is a field theory, the standards to which it must measure up are higher than for a model. A sound EFT must be renormalizable and allow for a proper power counting (order-by-order arrangement). The presently used chiral nuclear potentials are based on naive dimensional analysis (“Weinberg counting”) and apply a cutoff regularization scheme. In that scheme, one wishes to see cutoff independence of the results. Such independence is seen to a good degree below the breakdown scale (Marji et al. 2013; Coraggio et al. 2014, 2013), but to which extent that is satisfactory is controversial. The problem is due to the nonperturbative resummation necessary for typical nuclear physics problems (bound states). However, there is hope that from the current discussion (Hammer et al. 2020; Nogga et al. 2005; Birse 2006; Long and Yang 2012; Long 2016; Valderrama 2011; Pavon Valderrama 2011; Valderrama 2016; Pavon Valderrama et al. 2017; Epelbaum et al. 2017; K˝onig et al. 2017; Epelbaum et al. 2018), constructive solutions may emerge (van Kolck 2020; Valderrama 1902; Entem and Oller 2021). In conclusion, considering both formal aspects and evidence of successful applications, one may say that chiral EFT has substantially advanced the field of theoretical nuclear physics (Machleidt and Sammarruca 2020). It is unclear, though,

49 Nonlocal Chiral Nuclear Forces up to N5 LO

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whether the remaining unresolved issues will be settled in a satisfactory manner in the near future. Acknowledgments This work has been supported in part by the Ministerio de Economía, Industria y Competitividad under Contract No. FPA2016-77177-C2-2-P; by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DEFG0203ER41270; and by the EU STRONG-2020 project under the program H2020-INFRAIA-2018-1, grant agreement no. 824093.

References A. Akmal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C 58, 1804 (1998) B.R. Barrett, P. Navratil, J.P. Vary, Prog. Part. Nucl. Phys. 69, 131 (2013) V. Bernard, E. Epelbaum, H. Krebs, Ulf-G. Meißner, Phys. Rev. C 77, 064004 (2008) V. Bernard, E. Epelbaum, H. Krebs, Ulf-G. Meißner, Phys. Rev. C 84, 054001 (2011) S. Binder, J. Langhammer, A. Calci, R. Roth, Phys. Lett. B 736, 119 (2014) M.C. Birse, Phys. Rev. C 74, 014003 (2006); ibid. 76, 034002 (2007); ibid. 77, 047001 (2008); Phil. Trans. Roy. Soc. Lond. A 369, 2662 (2011) R. Blankenbecler, R. Sugar, Phys. Rev. 142, 1051 (1966) W.J. Briscoe, I.I. Strakovsky, R.L. Workman, SAID Partial-Wave Analysis Facility, Data Analysis Center, The George Washington University, Solution SP07 (Spring, 2007) R.A. Bryan, B.L. Scott, Phys. Rev. 1964, 135, B434-B450; ibid. 1969, 177, 1435–1442 J. Carlson, V.R. Pandharipande, R.B. Wiringa, Nucl. Phys. A 401, 59 (1983) J. Carlson, S. Gandolfi, F. Pederiva, S.C. Pieper, R. Schiavilla, K.E. Schmidt, R.B. Wiringa, Rev. Mod. Phys. 87, 1067 (2015) B.D. Carlsson et al., Phys. Rev. X 6, 011019 (2016) E. Caurier, P. Navratil, W.E. Ormand, J.P. Vary, Phys. Rev. C 66, 024314 (2002) S. Coleman, J. Wess, B. Zumino, Phys. Rev. 177, 2239 (1969); C.G. Callan, S. Coleman, J. Wess, B. Zumino, ibid. 177, 2247 (1969). 288, 249 (2001) S.A. Coon, H.K. Han, Few Body Syst. 30, 131 (2001) S.A. Coon, M.D. Scadron, P.C. McNamee, B.R. Barrett, D.W.E. Blatt, B.H.J. McKeller, Nucl. Phys. A317, 242 (1979) L. Coraggio, J.W. Holt, N. Itaco, R. Machleidt, F. Sammarruca, Phys. Rev. C 87, 014322 (2013) L. Coraggio, J.W. Holt, N. Itaco, R. Machleidt, L.E. Marcucci, F. Sammarruca, Phys. Rev. C 89, 044321 (2014) B.D. Day, Comments Nucl. Part. Phys. 11, 115 (1983) C. Drischler, A. Carbone, K. Hebeler, A. Schwenk, Phys. Rev. C 94, 054307 (2016) C. Drischler, K. Hebeler, A. Schwenk, Phys. Rev. Lett. 122, 042501 (2019) A. Ekstr˝om et al., Phys. Rev. Lett. 110, 192502 (2013) A. Ekstr˝om et al., Phys. Rev. C 91, 051301 (2015) A. Ekstr˝om, G. Hagen, T.D. Morris, T. Papenbrock, P.D. Schwartz, Phys. Rev. C 97, 024332 (2018) D.R. Entem, R. Machleidt, unpublished (1993) D.R. Entem, R. Machleidt, Phys. Lett. B 524, 93 (2002) D.R. Entem, R. Machleidt, Phys. Rev. C 68, 041001 (2003) D.R. Entem, J.A. Oller, Non-perturbative methods for NN singular interactions. [arXiv:2103.02069 [nucl-th]] D.R. Entem, R. Machleidt, H. Witala, Phys. Rev. C 65, 064005 (2002) D.R. Entem, N. Kaiser, R. Machleidt, Y. Nosyk, Phys. Rev. C 91, 014002 (2015) D.R. Entem, N. Kaiser, R. Machleidt, Y. Nosyk, Phys. Rev. C 92, 064001 (2015) D.R. Entem, R. Machleidt, Y. Nosyk, Phys. Rev. C 96, 024004 (2017) E. Epelbaum, Eur. Phys. J. A34, 197 (2007)

1910

D. R. Entem et al.

E. Epelbaum, H.-W. Hammer, U.-G. Meißner, Rev. Mod. Phys. 2009, 81 (1773) E. Epelbaum, W. Glöckle, U.-G. Meißner, Nucl. Phys. A671, 295 (2000) E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. Meißner, H. Witala, Phys. Rev. C 66, 064001 (2002) E. Epelbaum, W. Glöckle, U.-G. Meißner, Nucl. Phys. A747, 362 (2005) E. Epelbaum, H. Krebs, U.-G. Meißner, Eur. Phys. J. A51, 53 (2015a) E. Epelbaum, H. Krebs, U.-G. Meißner, Phys. Rev. Lett. 115, 122301 (2015b) E. Epelbaum, J. Gegelia, U. G. Meißner, Nucl. Phys. B 925, 161 (2017) E. Epelbaum, A.M. Gasparyan, J. Gegelia, U.G. Meißner, Eur. Phys. J. A 54, 186 (2018) K. Erkelenz, Phys. Rep. 13C, 191–258 (1974) S. Fleming, T. Mehen, I.W. Stewart, Nucl. Phys. A 677, 313 (2000) J.L. Friar, D. Huber, U. van Kolck, Phys. Rev. C 59, 53 (1999) J.-I. Fujita, H. Miyazawa, Prog. Theor. Phys. 17, 360 (1957) R.J. Furnstahl, D.R. Phillips, S. Wesolowski, J. Phys. G 42, 034028 (2015) J. Gasser, H. Leutwyler, Ann. Phys. 158, 142 (1984); Nucl. Phys. B250, 465 (1985) J. Gasser, M.E. Sainio, A. Švarc, Nucl. Phys. B307, 779 (1986) A. Gezerlis, I. Tews, E. Epelbaum, M. Freunek, S. Gandolfi, K. Hebeler, A. Nogga, A. Schwenk, Phys. Rev. C 90, 054323 (2014) L. Girlanda, A. Kievsky, M. Viviani, Phys. Rev. C 84, 014001 (2011) [erratum: Phys. Rev. C 102(1), 019903 (2020)] L. Girlanda, A. Kievsky, M. Viviani, L.E. Marcucci, Phys. Rev. C 99, 054003 (2019) J. Golak et al., Eur. Phys. J. A 50, 177 (2014) H. Hagen, M. Hjorth-Jensen, G.R. Jansen, R. Machleidt, T. Papenbrock, Phys. Rev. Lett. 108, 242501 (2012a) H. Hagen, M. Hjorth-Jensen, G.R. Jansen, R. Machleidt, T. Papenbrock, Phys. Rev. Lett. 109, 032502 (2012b) G. Hagen, T. Papenbrock, M. Hjorth-Jensen, D. J. Dean, Rept. Prog. Phys. 77, 096302 (2014a) G. Hagen, T. Papenbrock, A. Ekstr˝om, K.A. Wendt, G. Baardsen, S. Gandolfi, M. Hjorth-Jensen, C.J. Horowitz, Phys. Rev. C 89, 014319 (2014b) G. Hagen, G.R. Jansen, T. Papenbrock, Phys. Rev. Lett. 117, 172501 (2016) H.W. Hammer, S. König, U. van Kolck, Rev. Mod. Phys. 92, 025004 (2020) K. Hebeler, A. Schwenk, Phys. Rev. C 82, 014314 (2010) K. Hebeler, S.K. Bogner, R.J. Furnstahl, A. Nogga, A. Schwenk, Phys. Rev. C 83, 031301(R) (2011) H. Hergert, S.K. Bogner, S. Binder, A. Calci, J. Langhammer, R. Roth, A. Schwenk, Phys. Rev. C 87, 034307 (2013) K. Hebeler, H. Krebs, E. Epelbaum, J. Golak, R. Skibinski, Phys. Rev. C 91, 044001 (2015) H. Hergert, S.K. Bogner, T.D. Morris, A. Schwenk, K. Tsukiyama, Phys. Rep. 621, 165 (2016) M. Hoferichter, J. Ruiz de Elvira, B. Kubis, U.-G. Meißner, Phys. Rev. Lett. 115, 192301 (2015); Phys. Rep. 625, 1 (2016) K. Holinde, R. Machleidt, Nucl. Phys. A247, 495–520 (1975); ibid. A256, 479–496 (1976) J. Hoppe, C. Drischler, K. Hebeler, A. Schwenk, J. Simonis, Phys. Rev. C 100, 024318 (2019) T. Hüther, K. Vobig, K. Hebeler, R. Machleidt, R. Roth, Phys. Lett. B 808, 135651 (2020) U.D. Jentschura, A. Matveev, C.G. Parthey, J. Alnis, R. Pohl, Th. Udem, N. Kolachevsky, T.W. H˝ansch, Phys. Rev. A 83, 042505 (2011) S. K˝onig, H.W. Grießhammer, H.-W. Hammer, U. van Kolck, Phys. Rev. Lett. 118, 202501 (2017) N. Kaiser, Phys. Rev. C 61, 014003 (2000a) N. Kaiser, Phys. Rev. C 62, 024001 (2000b) N. Kaiser, Eur. Phys. J. A 48, 135 (2012) N. Kaiser, R. Milkus, Phys. Rev. C 92, 064001 (2015) N. Kaiser, R. Milkus, Eur. Phys. J. A 52, 4 (2016) N. Kaiser, R. Brockmann, W. Weise, Nucl. Phys. A625, 758 (1997) D.B. Kaplan, M. Savage, M.B. Wise, Nucl. Phys. B478, 629 (1996); Phys. Lett. B424, 390 (1998); Nucl. Phys. B534, 329 (1998)

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A. Kievsky, M. Viviani, L. Girlanda, L.E. Marcucci, Phys. Rev. C 81, 044003 (2010) T. Kr˝uger, I. Tews, K Hebeler, A. Schwenk, Phys. Rev. C 88, 025802 (2013) H. Krebs, A. Gasparyan, E. Epelbaum, Phys. Rev. C 85, 054006 (2012) H. Krebs, A. Gasparyan, E. Epelbaum, Phys. Rev. C 87, 054007 (2013) M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. Côté, P. Pires, R. de Tourreil, Phys. Rev. C 21, 861–873 (1980) V. Lapoux, V. Somà, C. Barbieri, H. Hergert, J.D. Holt, S.R. Stroberg, Phys. Rev. Lett. 117, 052501 (2016) G.P. Lepage, How to Renormalize the Schrödinger Equation (1977). arXiv:nucl-th/9706029 D. Lonardoni, A. Lovato, S.C. Pieper, R.B. Wiringa, Phys. Rev. C 96(2), 024326 (2017) B. Long, Int. J. Mod. Phys. E 25, 1641006 (2016) B. Long, C. Yang, Phys. Rev. C 86, 024001 (2012) R. Machleidt, K. Holinde, Ch. Elster, Phys. Rep. 149, 1–89 (1987) R. Machleidt, Adv. Nucl. Phys. 19, 189–376 (1989) R. Machleidt, Phys. Rev. C 63, 024001 (2001) R. Machleidt, D.R. Entem, Phys Reports 503, 1 (2011) R. Machleidt, F. Sammarruca, Phys. Scr. 91, 083007 (2016) R. Machleidt, F. Sammarruca, Eur. Phys. J. A 56, 95 (2020) L.E. Marcucci, A. Kievsky, S. Rosati, R. Schiavilla, M. Viviani, Phys. Rev. Lett. 108, 052502 (2012) E. Marji et al., Phys. Rev. C 88, 054002 (2013) T.D. Morris, J. Simonis, S.R. Stroberg, C. Stumpf, G. Hagen, J.D. Holt, G.R. Jansen, T. Papenbrock, R. Roth, A. Schwenk, Phys. Rev. Lett. 120, 152503 (2018) R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C 91, 054002 (2015), and references to the comprehensive work by the Granada group therein P. Navratil, Few Body Syst. 41, 117 (2007) P. Navratil, V.G. Gueorguiev, J.P. Vary, W.E. Ormand, A. Nogga, Phys. Rev. Lett. 99, 042501 (2007) P. Navratil, R. Roth, S. Quaglioni, Phys. Rev. C 82, 034609 (2010) A. Nogga, R.G.E. Timmermans, U. van Kolck, Phys. Rev. C 72, 054006 (2005) A. Nogga, P. Navratil, B.R. Barrett, J.P. Vary, Phys. Rev. C 73, 064002 (2006) C. Ordóñez, L. Ray, U. van Kolck, Phys. Rev. C 53, 2086 (1996) M. Pavon Valderrama, Phys. Rev. C 84, 064002 (2011) M. Pavon Valderrama, M. Sanchez Sanchez, C.J. Yang, B. Long, J. Carbonell, U. van Kolck, Phys. Rev. C 95, 054001 (2017) M. Piarulli, L. Girlanda, R. Schiavilla, R. Navarro Pérez, J.E. Amaro, E. Ruiz Arriola, Phys. Rev. C 91, 024003 (2015) M. Piarulli, L. Girlanda, R. Schiavilla, A. Kievsky, A. Lovato, L.E. Marcucci, Steven C. Pieper, M. Viviani, R.B. Wiringa, Phys. Rev. C 94, 054007 (2016) P. Reinert, H. Krebs, E. Epelbaum, Eur. Phys. J. A 54, 86 (2018) V.G.J. Stoks, R.A.M. Klomp, M.C.M. Rentmeester, J.J. de Swart, Phys. Rev. C 48, 792 (1993) R. Roth, J. Langhammer, A. Calci, S. Binder, P. Navratil, Phys. Rev. Lett. 107, 072501 (2011) R. Roth, S. Binder, K. Vobig, A. Calci, J. Langhammer, P. Navratil, Phys. Rev. Lett. 109, 052501 (2012) D. Rozpedzik, J. Golak, R. Skibinski, H. Witala, W. Glöckle, E. Epelbaum, A. Nogga, H. Kamada, Acta Phys. Polon. B37, 2889 (2006); arXiv:nucl-th/0606017 F. Sammarruca, R. Millerson, Phys. Rev. C 102, 034313 (2020) F. Sammarruca, L.E. Marcucci, L. Coraggio, J.W. Holt, N. Itaco, R. Machleidt, arXiv:1807.06640 [nucl-th] F. Sammarruca, B. Chen, L. Coraggio, N. Itaco, R. Machleidt, Phys. Rev. C 86, 054317 (2012) F. Sammarruca, L. Coraggio, J.W. Holt, N. Itaco, R. Machleidt, L.E. Marcucci, Phys. Rev. C 91, 054311 (2015) S. Scherer, Adv. Nucl. Phys. 27, 277 (2003) J. Simonis, K. Hebeler, J.D. Holt, J. Menendez, A. Schwenk, Phys. Rev. C 93, 011302 (2016)

1912

D. R. Entem et al.

J. Simonis, S.R. Stroberg, K. Hebeler, J.D. Holt, A. Schwenk, Phys. Rev. C 96, 014303 (2017) V. Somà, A. Cipollone, C. Barbieri, P. Navratil, T. Duget, Phys. Rev. C 89, 061301(R) (2014) V. Somà, P. Navrátil, F. Raimondi, C. Barbieri, T. Duguet, Phys. Rev. C 101, 014318 (2020) M.P. Valderrama, Scattering amplitudes versus potentials in nuclear effective field theory: is there a potential compromise? arXiv:1902.08172 [nucl-th] M.P. Valderrama, Phys. Rev. C 83, 024003 (2011) M.P. Valderrama, Int. J. Mod. Phys. E 25, 1641007 (2016) U. van Kolck, Phys. Rev. C 49, 2932 (1994) U. van Kolck, Front. Phys. 8, 79 (2020) R. Vinh Mau, in Mesons in Nuclei, Vol. I, ed. by M. Rho, D.H. Wilkinson (North-Holland, Amsterdam, 1979), pp. 151–196 M. Viviani, L. Girlanda, A. Kievsky, L.E. Marcucci, Phys. Rev. Lett. 111, 172302 (2013) S. Weinberg, Physica 96A, 327 (1979) S. Weinberg, Phys. Lett. B251, 288 (1990) S. Weinberg, Nucl. Phys. B363, 3 (1991) S. Weinberg, Phys. Lett. B 295, 114 (1992) H. Yukawa, Proc. Phys. Math. Soc. Jpn. 17, 48–57 (1935) P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020)

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Nucleon-Antinucleon Interaction Jean-Marc Richard

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Nucleon-Nucleon to Antinucleon-Nucleon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case of Strong Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consequences of the G-Parity Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NN Interaction in Effective Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annihilation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annihilation Viewed in Terms of Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phenomenology of Annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NN Interaction and Hadron Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antinucleon-Nucleus Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron-Antineutron in Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antiprotonic Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exotic Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Level Rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Protonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antiproton-Nucleus Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Invited contribution to “Handbook of Nuclear Physics”, Springer, 2022, I. Tanihata, H. Toki and T. Kajino, Eds., Section “Nuclear Interactions” coordinated by R. Machleidt J.-M. Richard () Institut de Physique des 2 Infinis de Lyon, IN2P3 & Université de Lyon, Villeurbanne, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_53

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Antiprotonic Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Day-Snow-Sucher Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antiprotons in the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

A review is presented of the antinucleon-nucleon interaction and some related issues such as fundamental symmetries, annihilation mechanisms, antinucleonnucleus scattering, antiprotonic atoms, and neutron-antineutron oscillations. The overall perspective is historical, but the modern approaches are also presented.

Introduction The positron was discovered in 1932 by Anderson (1932), and further studies confirmed that it has the expected properties, in particular, the same mass as the electron and the opposite charge. It was thus reasonably anticipated that the proton should also possess an associated antiparticle, the antiproton, though the proton is not exactly a Dirac particle since its magnetic moment substantially deviates from the Bohr magneton, as shown by Stern in 1933 (see, e.g., Estermann and Foner 1975). It was also expected that the antiproton would hardly be detected with cosmic ray experiments, and a dedicated accelerator was built at Berkeley, the Bevatron (at that time, 109 eV was denoted 1 BeV), to produce and study antiprotons. This was a remarkable technical achievement, both for the accelerator and the detector. The antiproton was discovered in 1955 by a team led by Chamberlain and Segrè. One year later, the charge-exchange cross-section pp ¯ → nn ¯ gave access to the antineutron n, ¯ partner of the neutron n. See, e.g., the Nobel lectures (Nobel Foundation 1998). In successive experiments, the antiproton-proton symmetry has been checked with a greater and greater accuracy. The charge-to-mass ratios |q|/m turn out identical to about 10−10 and the magnetic moments |μ| to about 10−9 (Zyla et al. 2020). Before the Berkeley measurements, one anticipated an analogy of the NN interaction with the e+ e− one in QED, where the elastic interaction is mediated by photon-exchange and annihilation by fusion into two or three photons, as shown in Fig. 1.

Fig. 1 Contributions to the elastic and annihilation e+ e− cross-sections

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Fig. 2 Simple mechanisms contributing to the N N interaction: one-pion exchange and annihilation into ρπ by baryon exchange

If one naively believes that the antiproton-proton interaction is dominated by the mechanisms shown in Fig. 2, with one-pion exchange and annihilation mediated by a very short-range baryon exchange, one predicts the following hierarchy for the annihilation (pp ¯ → mesons), elastic (pp ¯ → pp), ¯ and charge-exchange (pp ¯ → nn) ¯ cross-sections σann < σel < σce ,

(1)

the second inequality resulting straightforwardly from a ratio of isospin ClebschGordan coefficients. Actually, the inverse ordering was observed, with the annihilation cross section σann nearly twice larger than the elastic one, σel , and the charge-exchange one, σce , being rather suppressed. This pattern of the antiproton cross sections motivated a flurry of phenomenological studies which eventually forced to a drastic revision of our view of the annihilation mechanisms. In this review, we follow the traditional path which describes the antinucleonnucleon interaction (NN) as a long-range part due to meson exchanges “supplemented” by annihilation. In fact, the main pattern of the NN interaction is that of a very strong annihilation with a tail of Yukawa-type contribution. Filters are required to see the latter on the top a very strong absorption. We then resume the discussion by an outline of the more modern approach, based on effective theories. For an introduction to this physics, see, e.g., Amsler and Myhrer (1991), Dover et al. (1992), Klempt et al. (2002, 2005), and Haidenbauer (2019). A word about the experimental framework. An intense program of studies of antiproton-induced has been performed, in particular, at CERN and Brookhaven, using secondary beams, i.e., antiprotons just produced on a production target. Such beams had a large momentum spread and were much contaminated by other negatively charged particles such as kaons (K − ) and pions (π − ). For a summary of early results, see, e.g., Flaminio et al. (1970). The method of stochastic cooling has been developed at CERN (Van Der Meer 1985). The main purpose was to build a high-energy collider to discover the W ± and Z 0 , the bosons mediating the week interactions. A low-energy branch, LEAR (Low Energy Antiproton Ring), was also set-up (Koziol and Mohl 2004) and used to study the interaction of slow antiprotons. Stochastic cooling has also be implemented at Fermilab with highenergy pp ¯ collision leading, in particular, to the discovery of the top quark and also medium-energy antiproton hitting a hydrogen gas target to form charmonium resonances (Garzoglio et al. 2004), as pioneered in the R704 experiment at

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CERN (Baglin et al. 1986). At CERN, the LEAR facility has been dismantled, but the ELENA decelerator gives access to very-low-energy experiments (Oelert 2017). For years, it has been proposed to resume experiments with antiprotons in the GeV energy range, as in the “SuperLEAR” project (Dalpiaz et al. 1987; Hertzog 1993). This will eventually be the case with the program around PANDA (Belias 2020) at Darmstadt.

From Nucleon-Nucleon to Antinucleon-Nucleon The Case of QED In principle, the same amplitude M (s, t) describes both e− e− and e+ e− scattering, the former for s > 4 m2e and t < 0 and the latter for s < 0 and t > 4 m2e . Here, me is the electron mass, and s and t are two of the Mandelstam variables describing the 1 + 2 → 3 + 4 reaction with quadri-momenta p˜ i , s = (p˜ 1 + p˜ 2 )2 ,

t = (p˜ 1 − p˜ 3 )2 ,

u = (p˜ 1 − p˜ 4 )2 ,

(2)

which fulfill s + t + u = m21 + · · · + m24 . It is simpler, however, to compare the amplitudes Ma (s, t) for e− e− → e− e− and Mb (s, t) for e+ e− → e+ e− for the same values of s and t. The result, sometimes referred to as the C-conjugation rule, is the following. If

(3) −Ma1

Mb2

+ + · · · , that is to say, the contributions with the exchange then Mb = of an odd number of photons flip sign.

The Case of Strong Interactions Nothing prevents from using the C-conjugation rule for hadrons. In particular, if M (pp → pp) = Mπ + Mρ + Mω + · · · according to the mesons that are exchanged, then M (pp ¯ → pp) ¯ = Mπ − Mρ − Mω + · · · The rule holds also for a continuum of exchanges, e.g., for two-pion exchange, π 0 π 0 and the C-even part ¯ while the C-odd of π + π − exchanges are unchanged when going from pp to pp, part of π + π − flips sign. However, it is convenient to relate amplitudes with the same isospin, for instance, pp → pp for NN and np ¯ → np ¯ for NN. This is achieved with the G-parity rule of Fermi and Yang (1949): if in isospin I

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M I (NN) = MπI + MπI π + MωI + · · · ,

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(4)

according to the mesons that are exchanged (ρ is included here in MπI π ), then M I (NN) = −MπI + MπI π − MωI + · · ·

(5)

This was the basis of nearly 50 years of phenomenological studies.

Consequences of the G-Parity Rule First a remark on the isospin structure. The charge-exchange cross section being governed by Mce ∝ M 0 (NN) − M 1 (NN),

(6)

its suppression results from a cancellation between the I = 0 and I = 1 amplitudes. This turns out to be a very severe constraint. The second observation deals with the average properties of the long-range interaction. The NN potential is weakly attractive, as seen from the shallow binding of the deuteron and the absence of dineutron. In the conventional picture of meson exchanges, this is understood by some cancellation between the attractive contribution of f0 exchange (Better known in the nuclear physics community as  or σ exchange, or as the scalar-isoscalar part of two-pion exchange), and repulsive contributions such as ω exchange. Once the G-parity rule is applied, one gets a coherent attraction in the NN case. Of course, the risk is that models with large coupling constants, but properly tuned by cancellations to reproduce the NN data, would give an unrealistic NN attraction. Already in Fermi and Yang (1949), and later in the “bootstrap” era (Chew 1966), this attraction suggested a composite picture of mesons as NN bound states. Among the many difficulties in this approach, one can mention the breaking of “exchange degeneracy”: for instance, the ω meson with isospin I = 0 and the ρ meson with I = 1, both with spin-parity J P = 1− , have nearly the same mass, while the meson-exchange part of the NN interaction is more attractive for I = 0 than for I = 1 (Ball et al. 1966). Two decades after Fermi and Yang (1949), Shapiro and his collaborators (Shapiro 1978), and others (Buck et al. 1979), came back on this idea and proposed that the N N bound states and resonances are associated with new kind of mesons, preferentially coupled to NN , named “quasi-nuclear” states or “baryonia.” We return to this question in the section devoted to hadron spectroscopy. Another consequence of the G-parity rule is a change of the spin dependence of the interaction. In the NN case, the most salient feature is the presence of a strong spin-orbit component, at work in nucleon scattering and in the spectroscopy of nuclear levels. For NN, the spin-orbit component is moderate, but a very strong coherence is observed in the tensor component, especially for I = 0. In nuclear

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physics, it is well known that the tensor potential is crucial to achieve the binding of the deuteron, but the fraction of 3 D1 admixture into the dominant 3 S1 remains small, of the order of 5% (In the limit of very strong tensor forces, this percentage can reach 2/3). In the case of NN, there are cancellations among the various meson exchanges contributions to the spin-orbit potential, but coherences for the tensor component, especially for I = 0 (Dover and Richard 1978).

Optical Models Once the long-range NN interaction is derived from the NN one by the G-parity rule, it has to be supplemented empirically by short-range terms. The elastic (pp ¯ → pp), ¯ charge-exchange (pp ¯ → nn), ¯ and annihilation (pp ¯ → mesons) integrated cross-sections have been analyzed by Ball and Chew (1958), Lévy (1960), . . . , who concluded that one needs a strong absorption even in the partial waves with angular momentum  > 0. This was confirmed in explicit fits with optical potentials (Gourdin et al. 1958; Bryan and Phillips 1968). One may wonder: why an optical potential? Because this is a valuable tool to study antiprotonic atoms, antiproton-nucleus, and antinucleus-nucleus interaction. Some of the applications will be outlined in the next sections. If, for instance, one studies the strangeness-exchange reaction pp ¯ → ΛΛ, one should in principle setup a cumbersome system of coupled channels pp ¯ ↔ mesons ↔ ΛΛ, in which the mechanism of K, K ∗ exchange, or internal conversion π π → KK could be somewhat lost. Instead, an optical model provides the adequate distorted waves for the initial and final states. Anyhow, further optical potentials were elaborated in the same spirit as (Bryan and Phillips 1968), namely: start from a meson-exchange model of the long-range nucleon-nucleon interaction, apply the G-parity rule, and replace its short-range part by an empirical complex core whose parameters are adjusted to fit the NN data. One may cite the Dover-Richard (1980) or Kohno-Weise (1986) potentials, with always the same conclusion: one needs a very strong absorption up to at least 0.8 fm. In a baryon-exchange picture, this implies very wide form factors; this means it becomes more appropriate to speak of the size rather than of the range of annihilation. See, also, Mull and Holinde (1995).

Spin Observables There has been early measurements of the polarization (or analyzing power) in the elastic reaction. In the LEAR era, it could have been envisaged a thorough investigation of the spin observables, but this was not approved by the CERN committee in charge, in view of the very packed program of experiments. The aim was twofold: probe our understanding of the NN interaction and, possibly, build a set-up for polarizing antiprotons, by filtering or transfer. For a summary of the available data, see, e.g., Klempt et al. (2002) and Haidenbauer (2019).

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Fig. 3 Analyzing power of pp ¯ → pp ¯ at plab = 679 MeV/c (Kunne et al. 1988)

Figure 3 shows the analyzing power of elastic pp ¯ scattering at momentum plab = 679 MeV/c (Kunne et al. 1988). The moderate values can be understood to be due to a moderate spin-orbit component, or to a strong tensor component acting beyond first order, or to a combination of both. As for the phenomenology of spin observables, it has been noticed that they are sensitive to the high partial waves and thus to the meson-exchange tail of the interaction. Simulations have been attempted with a spin- and isospin-independent complex core and a long-range part given by the G-parity rule (Joseph and Soffer 1981; Dover and Richard 1982). The polarization (or analyzing power) is rather moderate, but some rank 2 observables sensitive to the tensor interaction are rather pronounced. This implies that experiments should be performed with two simultaneous spin measurements and beam or target polarized longitudinally rather than transversally. A striking prediction is that a charge-exchange reaction on a longitudinally polarized proton target will produce polarized antineutrons. Interesting spin effects are not restricted to NN → NN. They are also at work in other N -induced reactions such as pp ¯ → π + π − , K + K − or ΛΛ. The latest measurements of annihilation into two pseudoscalars, π π or KK, have been done by the collaboration PS172 with a polarized target (Hasan et al. 1992). The analyzing power (or asymmetry, somewhat improperly called polarization) turns out extremal (|An | ∼ 1) in some wide ranges of energy and angles, as seen in Fig. 4. This means that one of the transversity amplitudes dominates. The pattern observed in Fig. 4 has been explained from different viewpoints, such as initial (Elchikh and Richard 1993) or final state (Takeuchi et al. 1993) interaction. The pp ¯ → ΛΛ (and also some other hyperon-antihyperon final states) reaction has been measured at LEAR by the PS185 collaboration. Even without polarized target, interesting results can be obtained (Barnes et al. 1996), as the weak decay of Λ or Λ informs about its spin. The most striking result is the suppression of the spin-singlet fraction. A complete reconstruction was possible with the data taken using a polarized proton target (Paschke et al. 2006). The detailed measurement of the pp ¯ → ΛΛ has reactivated the studies about the constraints among the various spin observables. Very often, when two observables X and Y which belong to the interval [−1, +1], the set {X, Y } is restricted to a subset of the square [−1, +1]2 , such as the disk X2 + Y 2 ≤ 1 or the triangle Y − 2 |X| + 1 ≤ 0. For triples of observables, a variety of subdomains of the cube [−1, +1]3 are obtained (Artru et al. 2009). Checking such constraints is a prerequisite for any amplitude analysis.

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NN Interaction in Effective Theories Chiral effective theory has been proposed by Weinberg for the study of the nucleonnucleon interaction and other hadronic systems at low energy. It has become very fashionable. The NN potentials based on this approach have now become the ultimate standard in the field. For an introduction, see, e.g., van Kolck (2021), and ¯ for the application to NN, Dai et al. (2017) and references there. In this approach, the N N interaction consists of one-pion exchange and a series of contact terms with an increasing power of the incoming and outgoing momenta. A sample of diagrams picturing the first contributions is shown in Fig. 5. For more details, see Dai et al. (2017). The aim is to provide a more consistent and systematic approach. However, ¯ phase-shifts that are not free the low-energy constants are tuned to fit a set of NN from ambiguities. This strategy could perhaps be improved. Remember that early N N potentials were fitted to reproduce some phase-shifts (see, e.g., De Tourreil and Sprung 1973), while more recent models are tuned directly to the NN observables (see, e.g., Lacombe et al. 1980).

Fig. 4 Asymmetry in the reactions pp ¯ → π π (left) and pp ¯ → KK (right), as measured by the PS172 collaboration (Hasan et al. 1992)

Fig. 5 Some contributions to the N N interaction in chiral effective field theory. The disk and square contact terms differ by the number of powers of the momenta

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Annihilation Mechanisms General Considerations Several important questions are related to annihilation: 1. Can one account for the overall strength of the absorptive component of the NN interaction? 2. Can one understand the main patterns of the observed branching ratios pp ¯ → ¯ π π π , . . . at rest and in flight? What is the role of final-state interaction? π π , KK, 3. Is annihilation an adequate doorway to study the spectroscopy of the light mesons and identify some light exotics? 4. Is the optical model well suited to account for annihilation? 5. Can one understand annihilation in terms of the quark content of the nucleon and antinucleon? Anyhow, annihilation is a fascinating and long-debated issue, and there is no consensus at present whether annihilation shall be described at the hadronic level or at the quark level.

Baryon Exchange As reminded in the introduction, the analogy with the e+ e− annihilation in QED suggests a mechanism of baryon exchange, with some examples in Fig. 6. Also shown is an example of iteration that contributes to the NN amplitude. The large mass of baryons implies a range of about 0.1 fm, but some form factor corrections are in order, in which case it is more appropriate to talk about the size rather than the range of the annihilation. Note that if the nucleon or nucleon resonance is replaced by a hyperon, then some strange mesons can be produced. The appealing aspect is that one can build NN potential in which both the elastic and absorptive parts are described in terms of hadrons, with the same coupling constants; see, e.g. Hippchen et al. (1991). However, all the diagrams with baryons in the mass range 1–3 GeV have about the same range, and it is not clear how the series of diagrams converges.

¯ N

π

¯ N

N

π

N

m

m

π π

¯ N

π

¯ N

¯ N

π

¯ N

π π

N

π

N

N

π

N

Fig. 6 Some annihilation diagrams driven by baryon exchange: production of two pions, production of several pions though meson resonances, iteration of two-pion production without and with form factors

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Annihilation Viewed in Terms of Quarks Already at the beginning of the quark model, or, say, of the SU(3) flavor symmetry, a systematics of the branching ratios was attempted (without an estimate of the overall annihilation cross-section) (Rubinstein and Stern 1966). The mechanism corresponds to the rearrangement, as schematically shown in Fig. 7. It was revisited in the 1980s, in particular by Green and Niskanen (1984) and by Ihle et al. (1988), who demonstrated that it gives the right order of magnitude for the strength of annihilation. The validity of this approach was much debated. First it was argued that the annihilation has to be short-ranged, on the basis of very general properties of scattering amplitudes (Martin 1961). However, again, this is not a question of range, but a problem of size, as strictly speaking, rearrangement is not “annihilation.” Once the diagram of Fig. 7 is estimated with harmonic oscillator wave functions for mesons and baryons, or, more precisely, its iteration NN → mesons → NN can be estimated, the result is a separable interaction (Green and Niskanen 1984; Ihle et al. 1988) with the form factors directly related to the size of the hadrons. Another concern is related to the belief that planar diagrams should dominate. In the terminology spelled out in Fig. 8, the planar diagrams are A2 or A3. The dominance of planar diagrams has been revisited by Pirner (1988), who concluded that the rearrangement diagrams are not suppressed. The respective role of the A2, A3, R2, and R3 diagrams was also discussed phenomenologically; see, e.g., Dover et al. (1986, 1992) and Maruyama et al. (1987). Clearly, R3 alone cannot produce ¯ is introduced. On the other kaons, unless some rescattering such as π π → KK hand, A2 and A3 tend to produce too many kaons and thus require the introduction of an empirical “strangeness suppression factor.”

Fig. 7 Rearrangement of the three quarks of a nucleon and the three antiquarks of an antinucleon into three mesons

R3

R2

A3

Fig. 8 Some diagrams contributing to the N N annihilation

A2

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Phenomenology of Annihilation Many data have been accumulated over the years, in particular, at Brookhaven and CERN. The measurements at rest are listed together with the pressure in the target, from which one can infer the percentage of S-wave annihilation, as seen in the section on antiprotonic atoms. One should stress that probing an annihilation mechanism is not an easy task. Each branching ratio is typically of the form B(NN → m1 m2 . . .) = P(NN) × PS × |M (NN → m1 m2 . . .)|2 ,

(7)

involving the probability P to find NN in the appropriate partial wave, the phasespace factor PS, and the square of the amplitude. In protonium (for annihilation at rest), and to a lesser extent in flight, there are dramatic differences among the various S-wave or P-wave probabilities P corresponding to different spin or isospin. Predicting the P factors would require a reliable model of the NN interaction probed with spin observables that have never been measured. For instance, an observation that B(ρφ)  B(ωφ) would indicate either that the short-range protonium wave function has more I = 0 than I = 1 or that the quark diagram interferes constructively for ωφ and destructively for ρφ.

NN Interaction and Hadron Spectroscopy The physics of antiprotons has always been closely related to the hadron spectroscopy. Even before the era of stochastic cooling, a lot of data have been collected on annihilation at rest, and several light meson resonances have been identified thanks to NN. This search has been resumed at LEAR, in particular, with the Asterix, Obelix, and Cristal-Barrel experiments with the aim to detect new kinds of light mesonic resonances, for instance, q qg ¯ hybrids. For a review of light-meson production at LEAR, see, e.g., Klempt et al. (2005) and Amsler (2019). Historically, the physics of low-energy antiprotons has been developed in the 1980s to study baryonium. The name “baryonium” denotes mesons that are preferentially coupled to baryon-antibaryon channels. The baryonium was predicted by Rosner (1968), on the basis of duality arguments (for a review, see, e.g., Phillips and Roy 1974): schematically, a reaction a + b → c + d can be described either as a sum of s-channel resonances or t-channel exchanges; then, a coherent dynamics in the t-channel implies the existence of s-channel resonances; for NN , the meson-exchanges in the t-channel are dual of s-channel NN resonances. Some indications were found in the 1970s, as reviewed by Montanet in (1980), and these discoveries have motivated the construction of the low-energy facility LEAR at CERN. Unfortunately, none of the peaks discovered in antiproton-induced reactions were confirmed at LEAR. On the other hand, baryon-antibaryon pairs are sometimes

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Fig. 9 String picture of mesons, baryons, and tetraquarks (baryonia in the light sector)

observed in the decay of heavy quarkonia of flavored mesons. See, e.g., the review on non-q q¯ mesons by Amsler and Hanhart in (2020). On the theory side, there has been several approaches to baryonium, anticipating the complementarity and emulation between the tetraquark models and the molecular picture of exotics. The baryonium has been modeled as a color-3¯ diquark and a color-3 antidiquark (without any strict derivation of such clustering). Even more speculative is the so-called mock-baryonium with a color-6 diquark and a color-6¯ antidiquark. In the string dynamics, one can view mesons, baryons, and baryonia as successive stages of the construction, as pictured in Fig. 9. For its link to QCD, see e.g., Montanet et al. (1980) and Rossi and Veneziano (2016). Another approach is based on the N N interaction. Unlike some speculations in the 1960s, tentatively associating NN states with ordinary mesons, Shapiro and his followers associated such NN states with new kind of mesons, the baryonia (Buck et al. 1979; Shapiro 1978). This was named “quasi-nuclear” picture, but it became “molecular.” In the above references, the NN spectrum was first calculated using the real part of the interaction, and then the effect of annihilation was discussed in a rather empirical (and optimistic) manner. More serious calculations, using the whole optical potential, have shown that most of the states are washed out by annihilation (Myhrer and Gersten 1977; Dalkarov and Myhrer 1977), while a few states might survive (Wycech et al. 2015).

Antinucleon-Nucleus Interaction Elastic Scattering At the start of the LEAR facility, the angular distribution of p¯ 12 C, p¯ 40 Ca and p¯ 208 Pb elastic scattering have been measured (Garreta et al. 1984). Some of the results are shown in Fig. 10. Data have been collected later at other energies and with other targets. The results have been interpreted in terms of an optical potential, either empirical ¯ amplitudes with the nuclear density; see, or derived by folding the elementary NN e.g., Adachi and Von Geramb (1987), Suzuki (1985), and Heiselberg et al. (1986). A comparison of 16 O and 18 O isotopes, see Fig. 11, does not indicate any striking isospin dependence of the pN ¯ interaction, when averaged on spins. However, at very low energy, some isospin dependence is suggested by the data and analyses by the PS179 and OBELIX (PS201) (Balestra et al. 1989; Botta 2001).

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Fig. 10 Angular distribution for p¯ scattering on the nuclei 12 C, 40 Ca, and 208 Pb at kinetic energy Tp¯ = 180 MeV (Garreta et al. 1984). The electronic retrieving of the data is due to Matteo Vorabbi

Fig. 11 Left: angular distribution for p¯ scattering on 18 O and 16 O at 178.7 MeV (Bruge et al. ∗ 1986). Right: angular distribution of the 12 C(p, ¯ p) ¯ 12 C reaction for the 3− excited state at 9.6 MeV. The incident p¯ has an energy of 179.7 MeV

Inelastic Scattering In between elastic scattering and annihilation, there is an interesting window of inelastic scattering, pA ¯ → pA ¯ ∗ , where A denotes an excited state of the nucleus ¯ amplitude, A. It gives access to the spin-isospin dependence of the elementary NN as stressed in Dover et al. (1983). A generalization is the charge-exchange reaction pA ¯ → nB ¯ (∗) .

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Some results have been obtained by the PS184 collaboration on 12 C and (Janouin et al. 1986) and analyzed in Dover et al. (1983, 1984). See Fig. 11.

18 O

Neutron-Antineutron in Nuclei The antinucleon-nucleon interaction plays a crucial role when estimating the lifetime of nuclei from a given rate of neutron-to-antineutron transitions, as predicted in some theories of grand unification. The instability of nuclei is an efficient alternative to free-neutron experiments, such as the one performed at Grenoble (Baldo-Ceolin et al. 1994), with a limit of about τnn¯  10−8 s for the oscillation period. A crucial feature is that the neutron-to-antineutron transition and the subsequent annihilation occur at the nuclear surface. This alleviates the fear (Kabir 1983) that the phenomenon could be obscured in nuclei by uncontrolled medium corrections. Another consequence is that the schematic modelization of nucleons and antineutrons evolving in a box, feeling an average potential Vn or Vn¯ , does not work too well. It is important to account for the tail of the neutron distribution, where n and n¯ are almost free. In practice, there are several variants; see, e.g., Alberico (1998). The simplest is based on the Sternheimer equation, which gives the first-order correction to the wave function without summing over unperturbed states. In a shell model with realistic neutron (reduced) radial wave functions unJ (r) with shell energy EnJ , the induced n¯ component is given by −

 (r) wnJ ( + 1)   + + Vn¯ (r) wnJ (r) − EnJ wnJ (r) = γ unJ (r). μ μ r2

(8)

Here μ is the reduced mass of the n-(A ¯ − 1) system, Vn¯ the complex (optical) n-(A ¯ − 1) potential, and γ = 1/τnn¯ the strength of the transition. Once wnJ is calculated, one can estimate the second-order correction to the energy, and in particular, the width ΓnJ of this shell is given by 

∞

ΓnJ = −2 0

 ImVn¯ |wnJ (r)|2 dr = −2 γ

∞

unJ (r) ImwnJ (r) dr,

(9)

0

and is readily seen to scale as ΓnJ ∝ γ 2 .

(10)

An averaging over the shells gives a width per neutron Γ associated with a lifetime T T = Tr τn2n¯ ,

(11)

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where Tr is named either the “reduced lifetime” (in s−1 ) or the “nuclear suppression factor.” The spatial distribution of the wnJ and the integrands in (9) and the relative contribution to Γ clearly indicate the peripheral character of the process. See, e.g., Barrow et al. (2019, 2021) for an application to a simulation in the forthcoming DUNE experiment and to Super-Kamiokande and refs. there to earlier estimates. For the deuteron, an early calculation by Dover et al. (1983) gave Tr 2.5 × 1022 s−1 . Oosterhof et al. (2019), in an approach based on effective chiral theory, found a value significantly smaller, Tr 1.1 × 1022 s−1 . However, their calculation has been revisited by Haidenbauer and Meißner (2019), who got almost perfect agreement with Dover et al. For 40 Ar relevant to the DUNE experiment, the result of Barrow et al. (2019) is Tr 5.6 × 1022 s−1 .

Antiprotonic Atoms Exotic Atoms Exotic atoms are systems in which an electron is replaced by a heavier negatively charged particle. Among them, the muonic atoms μ− p and μ− A, where A denotes a nucleus, are useful tools to probe the structure of the positive kernel. However, it should be noted that the stability does not always survive the substitution e− → μ− . For instance, pμ− e− is unstable, unlike H− (pe− e− ). Among exotic atoms, hadronic atoms h− p or h− A, where h− denotes π − , K − , or p, ¯ are of special interest, as they probe the interplay between the long-range Coulomb interaction and the short-range strong interaction. The simplest model consists of the Schrödinger equation [−Δ/(2 μ) + Vc + Vs − E] Ψ = 0,

(12)

where μ is the reduced mass, Vc is the Coulomb term, Vs the h− -nucleus stronginteraction potential, and E the energy, whose difference δE = E − Ec with respect to the pure Coulomb case Ec is referred to as the level shift. In most cases, ordinary perturbation theory is not suited to estimate δE and the deformation of the wave function Ψ . For instance, a hard core of small radius gives a tiny δE, while the firstorder correction is infinite! The expansion scheme is actually the ratio a/R of the scattering length in as to the Bohr radius. For S waves, the leading term is δE −Ec

4a , nR

(13)

where n is the principal quantum number. This formula, referred to as the DeserTrueman formula (Deser et al. 1954; Trueman 1961), works quite well, as long as |a|  R. It can be extended to non-Coulomb interaction in the long range; see, e.g., Combescure et al. (2011). More recently, the problem has been formulated in the framework of effective theories (Holstein 1999).

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Level Rearrangement The changes undergone by (13) at large |a| are better seen the way initiated by Zel’dovich (1960) and Shapiro (1978), who rewrote (12) with a potential Vc + λ Vs and studied how the spectrum evolves as λ is varied. A typical example is shown in Fig. 7. If the range of Vs is made shorter, then the behavior is sharper, almost like a staircase function. See, e.g., Deloff (2003) and Combescure et al. (2007). As a consequence, an anomalously large energy shift would indicate a bound or virtual state near the threshold (Fig. 12).

Protonium The analysis of protonium is more delicate than the above one-channel modeling. At long distance, the wave function is pure pp. ¯ At shorter distances, there is an admixture of nn, ¯ and one of the isospin components, I = 0 or I = 1, dominates, depending on the partial wave. It is important to understand the branching ratios of protonium annihilation. For angular momentum  > 1, the energy shifts are negligible. For  = 1, they are dominated by the one-pion exchange tail of the interaction, as shown in Kaufmann and Pilkuhn (1978). For l = 0, there is a net effect of the absorptive part of the interaction, with δE complex, and a positive real part, i.e., a repulsive effect. For the experimental results, see, e.g., Augsburger et al. (1999) and refs. therein. The average shift is δE = (712.5 ± 20.3) − i (527 ± 33) eV. An attempt to separate 1 S and 3 S gives a smaller shift and a larger width for the former, in agreement 0 1 with the potential-model calculations (Richard and Sainio 1982; Carbonell et al. 1992; Gutsche et al. 1999).

Antiproton-Nucleus Atoms The latest calculation of the antiprotonic-deuterium atom has been performed in Lazauskas and Carbonell (2021), with an optical potential input into the Faddeev Fig. 12 The first three levels of the S-wave solution of (12) with V (r) = −1/r + λ a 2 exp(−a r). The pattern of rearrangement is shown here for 2 μ = 1 in (12) and a = 100 in the potential V (r)

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equations. References can be found there to earlier estimates based on mere folding of the NN amplitude with the deuterium wave function. For heavier nuclei, there are fits based on an empirical Wood-Saxon potential or on a potential proportional to the nuclear density (r) (Batty et al. 1997)

U (r) = −

2π  μ 1+ a¯ (r), μ m

(14)

where m is the antiproton mass, μ the reduced mass, and a¯ an effective (complex) scattering length. A good fit is obtained with Rea¯ ∼ Ima¯ ∼ 1 fm. More ambitious are the attempts to derive the potential from the “elementary” NN interaction (Dumbrajs et al. 1986; Adachi and Von Geramb 1987; Suzuki and Narumi 1983).

Antiprotonic Helium In 1964, Condo proposed that metastable states could be formed in exotic helium atoms He h− e− (Condo 1964; Russell 1969). Such states have been observed at CERN in antiprotonic helium (Yamazaki et al. 1993). At first, such states look just as a curiosity of the three-body problem. Actually, antiprotonic atoms have emerged as a remarkable precision laboratory to measure the antiproton properties (mass, charge, . . . ) and even the fine-structure constant (Yamazaki et al. 2002).

Day-Snow-Sucher Effect The process of formation of antiprotonic atoms has been much studied (Borie and Leon 1980). A low-energy antiproton is slowed down by electromagnetic interactions and captured in some high orbit. The electrons of the initial atoms are expelled during this capture and the series of transitions of the antiproton towards lower states, preferentially through circular orbits with  = n − 1. In a vacuum, protonium annihilation is negligible in high orbits with l > 1, at the level of about 1% in 2P, and 100% in 1S (Kaufmann and Pilkuhn 1978). However, in a dense target, it often happens that the protonium state goes inside the ordinary atoms, where it experiences a Stark mixing of levels with the same principal number n but orbital momentum lower than  = n − 1, eventually down to  = 0, where annihilation might occur. This is the Day-Snow-Sucher effect (Day et al. 1960). It plays a crucial role when analyzing the results of annihilation at rest: the changes of the branching ratios as a function of the density of the target allow one to determine the contributions of S-waves vs. the ones of P-waves.

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Antiprotons in the Universe A major issue in cosmology is why matter is seemingly dominant over antimatter and whether some pieces of antimatter remain. For instance, the AMS experiment, installed on the International Space Station, has measured a p/p ¯ ratio of about 0.2× 10−3 Aguilar et al. (2016). AMS has also detected antinuclei in cosmic rays, and one should discuss whether they are primary or secondary objects. The question of antimatter in the Universe is recurrently addressed. Till the 1970s, some models were elaborated with a symmetric (matter vs. antimatter) Universe, with the key issue of how matter domains have been separated from the antimatter ones. In this context, an amusing correlation was made between a positive energy shift in protonium and a repulsive interaction (due to annihilation) between matter and antimatter (Caser and Omnes 1972). The more recent scenarios assume that thanks to CP violation (charge conjugation times parity) (Sakharov 1967), the Universe is dominated by matter from its very beginning. Now a very small amount of CP violation has been detected in neutral kaons (Christenson et al. 1964) and later in mesons carrying heavy flavor (Bigi and Sanda 2009), but this is just of beginning.

Outlook The physics of low-energy antiprotons is extremely rich at the interface of nuclear physics and quark physics and also atomic physics and physics beyond the standard model. The experimental activity has been dramatically vitalized in the 1980s with the advent of stochastic cooling. Today, a very ambitious program of highprecision measurements is carried out with very-low-energy antiprotons. In the near future, the physics of charm and strangeness should get interesting information from medium-energy antiprotons. Acknowledgments The author thanks M. Asghar for useful comments.

References S. Adachi, H.V. Von Geramb, Microscopic analysis of antiproton – nucleus elastic scattering. Nucl. Phys. A 470, 461–476 (1987) M. Aguilar et al., Antiproton flux, antiproton-to-proton flux ratio, and properties of elementary particle fluxes in primary cosmic rays measured with the alpha magnetic spectrometer on the international space station. Phys. Rev. Lett. 117(9), 091103 (2016) W.M. Alberico, Neutron antineutron oscillations: theory, in Lepton and Baryon Number Violation in Particle Physics, Astrophysics and Cosmology. Proceedings, 1st International Symposium, Lepton-Baryon’98, Trento, April 20–25, pp. 94–108 (1998) C. Amsler, Nucleon-antinucleon annihilation at LEAR. 2019, in Talk at the ECT∗ Workshop on Antiproton Physics, Trento, 17–21 June (2019) C. Amsler, F. Myhrer, Low-energy antiproton physics. Ann. Rev. Nucl. Part. Sci. 41, 219–267 (1991)

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C.D. Anderson, The apparent existence of easily deflectable positives. Science 76(1967), 238–239 (1932) X. Artru, M. Elchikh, J.-M. Richard, J. Soffer, O.V. Teryaev, Spin observables and spin structure functions: inequalities and dynamics. Phys. Rep. 470(1–2), 1–92 (2009) M. Augsburger, D.F. Anagnostopoulos, G. Borchert, D. Chatellard, J.-P. Egger, P. El-Khoury, H. Gorke, D. Gotta, P. Hauser, P. Indelicato, K. Kirch, S. Lenz, K. Rashid, T. Siems, L.M. Simons, Measurement of the strong interaction parameters in antiprotonic hydrogen and probable evidence for an interference with inner bremsstrahlung. Nucl. Phys. A 658(2), 149– 162 (1999) C. Baglin et al., Search for the p wave singlet charmonium state in pp ¯ annihilations at the CERN intersecting storage rings. Phys. Lett. B 171, 135–141 (1986) M. Baldo-Ceolin et al., A new experimental limit on neutron – antineutron oscillations. Z. Phys. C63, 409–416 (1994) F. Balestra et al., Evidence of isospin effects in antiproton – nucleus annihilation. Nucl. Phys. A491, 572–586 (1989) J.S. Ball, G.F. Chew, Nucleon-antinucleon interaction at intermediate energies. Phys. Rev. 109, 1385–1391 (1958) J.S. Ball, A. Scotti, D.Y. Wong, One boson exchange model of N N and N N¯ interaction. Phys. Rev. 142, 1000–1012 (1966) P.D. Barnes et al., Observables in high statistics measurements of the reaction pp ¯ → ΛΛ. Phys. Rev. C 54, 1877–1886 (1996) J.L. Barrow, E.S. Golubeva, E. Paryev, J.-M. Richard, Progress and simulations for intranuclear neutron–antineutron transformations in 40 18 Ar (2019) arXiv:1906.02833 J.L. Barrow, A.S. Botvina, E.S. Golubeva, J.-M. Richard, A new model of intranuclear neutronantineutron transformations in 16 8 O (2021) C.J. Batty, E. Friedman, A. Gal, Strong interaction physics from hadronic atoms. Phys. Rep. 287, 385–445 (1997). See, also, E. Friedman and A. Gal. In-medium nuclear interactions of lowenergy hadrons. Phys. Rept., 452:89–153, 2007. A. Belias, FAIR status and the PANDA experiment. JINST 15(10), C10001 (2020) I.I. Bigi, A.I. Sanda, CP Violation. Cambridge Monographs on Particle Physics, Nuclear Physics, and Cosmology (Cambridge University Press, Cambridge, 2009) E. Borie, M. Leon, X-ray yields in protonium and mesic hydrogen. Phys. Rev. A 21(5), 1460 (1980) E. Botta, Antineutron nucleus annihilation. Nucl. Phys. A692, 39–46 (2001) G. Bruge et al., Comparative study of the elastic scattering of 178.4-MeV antiprotons by the 16 O and 18 O isotopes. Phys. Lett. 169B, 14–16 (1986) R.A. Bryan, R.J.N. Phillips, Nucleon antinucleon potential from single meson exchanges. Nucl. Phys. B5, 201–219 (1968) W.W. Buck, C.B. Dover, J.-M. Richard, The interaction of nucleons with antinucleons. 1. General features of the N¯ N spectrum in potential models. Ann. Phys. 121, 47 (1979) J. Carbonell, J.-M. Richard, S. Wycech, On the relation between protonium level shifts and nucleon – antinucleon scattering amplitudes. Z. Phys. A343, 325–329 (1992) S. Caser, R. Omnes, A possible test of matter-antimatter separation. Phys. Lett. B 39, 369–370 (1972) G.F. Chew, The Analytic S-Matrix: A Basis for Nuclear Democracy (Benjamin, New York, 1966) J.H. Christenson, J.W. Cronin, V.L. Fitch, R. Turlay, Evidence for the 2π decay of the K20 meson. Phys. Rev. Lett. 13, 138–140 (1964) M. Combescure, A. Khare, A. Raina, J.-M. Richard, C. Weydert, Level rearrangement in exotic atoms and quantum dots. Int. J. Mod. Phys. B21, 3765–3781 (2007). See, also, A. Gal, E. Friedman, and C. J. Batty. On the interplay between Coulomb and nuclear states in exotic atoms, Nucl. Phys. A, 606:283–291, 1996. M. Combescure, C. Fayard, A. Khare, J.M. Richard, Exotic atoms in two dimensions. J. Phys. A Math. Gen. 44(27), 275302 (2011) G.T. Condo, On the absorption of negative pions by liquid helium. Phys. Lett. 9(1), 65–66 (1964)

1932

J.-M. Richard

L.-Y. Dai, J. Haidenbauer, U.-G. Meißner, Antinucleon-nucleon interaction at next-to-next-to-nextto-leading order in chiral effective field theory. JHEP 07, 078 (2017) O.D. Dalkarov, F. Myhrer, A simple model for proton-antiproton scattering at low-energies. Nuovo Cim. A 40, 152 (1977) P. Dalpiaz, R. Klapisch, P. Lefevre, M. Macri, L. Montanet, D. Mohl, A. Martin, J.M. Richard, H.J. Pirner, L. Tecchio, Physics at superlear, in Les Houches Workshop on Theoretical Physics: The Elementary Structure of Matter, vol. 2 (1987) T.B. Day, G.A. Snow, J. Sucher, High-orbital S-state capture of π -mesons by protons. Phys. Rev. 118, 864–866 (1960) A. Deloff, Fundamentals in hadronic atom theory (World Scientific, River Edge, 2003), 352p S. Deser, M.L. Goldberger, K. Baumann, W.E. Thirring, Energy level displacements in pi mesonic atoms. Phys. Rev. 96, 774–776 (1954) R. De Tourreil, D.W.L. Sprung, Construction of a nucleon-nucleon soft-core potential (in French). Nucl. Phys. A 201, 193–214 (1973) C.B. Dover, J.-M. Richard, Strong tensor forces and the possibility of high spin states in the nucleon-antinucleon system. Phys. Rev. D17, 1770–1777 (1978) C.B. Dover, J.-M. Richard, Elastic, charge exchange, and inelastic pp cross sections in the optical model. Phys. Rev. C 21, 1466–1471 (1980) C.B. Dover, J.-M. Richard, Spin observables in low-energy nucleon antinucleon scattering. Phys. Rev. C25, 1952–1958 (1982) C.B. Dover, M.E. Sainio, G.E. Walker, Antinucleon as a probe of nuclear spin and isospin excitations. Phys. Rev. C28, 2368–2373 (1983) C.B. Dover, A. Gal, J.-M. Richard, Neutron antineutron oscillations in nuclei. Phys. Rev. D27, 1090–1100 (1983) C.B. Dover, M.A. Franey, W.G. Love, M.E. Sainio, G.E. Walker, Antinucleon nucleus inelastic scattering and the spin dependence of the N¯ N annihilation potential. Phys. Lett. 143B, 45–49 (1984) C.B. Dover, P.M. Fishbane, S. Furui, Dynamical selection rules in N N¯ annihilation. Phys. Rev. Lett. 57, 1538–1541 (1986) C.B. Dover, T. Gutsche, M. Maruyama, A. Faessler. The physics of nucleon – antinucleon annihilation. Prog. Part. Nucl. Phys. 29, 87–174 (1992) O. Dumbrajs, H. Heiselberg, A.S. Jensen, A. Miranda, G.C. Oades, J.M. Richard, Microscopic calculation of antiproton atomic like bound states in light nuclei. Nucl. Phys. A 457, 491–517 (1986) M. Elchikh, J.-M. Richard, Meson exchange and nonlocality effects in proton antiproton annihilation into two pseudoscalar mesons. Phys. Rev. C48, R17–R20 (1993) I. Estermann, S.N. Foner, History of molecular beam research: personal reminiscences of the important evolutionary period 1919–1933. Citation: Am. J. Phys. 43, 661 (1975) E. Fermi, C.-N. Yang, Are mesons elementary particles? Phys. Rev. 76, 1739–1743 (1949) V. Flaminio, J.D. Hansen, D.R. Ogston Morrison, N. Tovey, Compilation of Cross-Sections (CERN, Geneva, 1970). Superseded by CERN HERA 73-01 D. Garreta et al., Elastic scattering of antiprotons from carbon, calcium and lead at 180-MeV. Phys. Lett. 149B, 64–68 (1984) G. Garzoglio et al., Experiment E835 at Fermilab. Nucl. Instrum. Methods A 519, 558–609 (2004) M. Gourdin, B. Jancovici, L. Verlet, On the nucleon-antinucleon interactions. Il Nuovo Cimento 8(3), 485–496 (1958) A.M. Green, J.A. Niskanen, From phenomenological to microscopic descriptions of NN annihilation. Int. Rev. Nucl. Phys. 1, 569–700 (1984) T. Gutsche, R. Vinh Mau, M. Strohmeier-Presicek, A. Faessler, Radiative proton – antiproton annihilation and isospin mixing in protonium. Phys. Rev. C 59, 630–641 (1999) J. Haidenbauer, Talk at the ECT∗ Workshop on Antiproton Physics, Trento, 17–21 June (2019). https://indico.ectstar.eu/event/41 J. Haidenbauer, U.-G. Meißner, Neutron-antineutron oscillations in the deuteron studied with N N and N¯ N interactions based on chiral effective field theory (2019). arXiv:1910.14423 [hep-ph]

50 Nucleon-Antinucleon Interaction

1933

A. Hasan et al., Differential cross-sections and analyzing powers for pp ¯ → π − π + and K − K + from 360-MeV/c to 1550-MeV/c. Nucl. Phys. B 378, 3–51 (1992) H. Heiselberg, A.S. Jensen, A. Miranda, G.C. Oades, J.-M. Richard, Microscopic calculation of antiproton nucleus elastic scattering. Nucl. Phys. A451, 562–580 (1986) D.W. Hertzog, The physics of superLEAR. Nucl. Phys. A 558, 499C–518C (1993) T. Hippchen, J. Haidenbauer, K. Holinde, V. Mull, Meson – baryon dynamics in the nucleon – antinucleon system. 1. The Nucleon – antinucleon interaction. Phys. Rev. C 44, 1323–1336 (1991) B.R. Holstein, Hadronic atoms and effective interactions. Phys. Rev. D 60, 114030 (1999) G. Ihle, H.J. Pirner, J.-M. Richard, s Wave nucleon – antinucleon interaction in the constituent quark model. Nucl. Phys. A485, 481–508 (1988) S. Janouin et al., Optical model analysis of antiproton nucleus elastic scattering at 50-MeV and 180-MeV. Nucl. Phys. A451, 541–561 (1986) C. Joseph, J. Soffer (eds.), Proceedings, International Symposium on High-Energy Physics with Polarized Beams and Polarized Targets, Basel, vol. 38 (Birkhaeuser, 1981) P.K. Kabir, Limits on nn¯ oscillations. Phys. Rev. Lett. 51, 231 (1983) W.B. Kaufmann, H. Pilkuhn, Black sphere model for the line widths of p state protonium. Phys. Rev. C17, 215–218 (1978) E. Klempt, F. Bradamante, A. Martin, J.-M. Richard, Antinucleon nucleon interaction at low energy: scattering and protonium. Phys. Rep. 368, 119–316 (2002) E. Klempt, C. Batty, J.-M. Richard, The Antinucleon-nucleon interaction at low energy: annihilation dynamics. Phys. Rep. 413, 197–317 (2005) M. Kohno, W. Weise, Proton – antiproton scattering and annihilation into two mesons. Nucl. Phys. A454, 429–452 (1986) H. Koziol, D. Mohl, The CERN low-energy antiproton programme: the synchrotrons. Phys. Rep. 403–404, 271–280 (2004) R.A. Kunne et al., Asymmetry in pp ¯ elastic scattering. Phys. Lett. B 206, 557–560 (1988) M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. Cote, P. Pires, R. De Tourreil, Parametrization of the Paris N N potential. Phys. Rev. C 21, 861–873 (1980) R. Lazauskas, J. Carbonell, Antiproton-deuteron hydrogenic states in optical models. Phys. Lett. B 820, 136573 (2021) M. Lévy, Range of proton-antiproton annihilation. Phys. Rev. Lett. 5, 380–381 (1960) A. Martin, Range of the nucleon-antinucleon annihilation potential. Phys. Rev. 124, 614–615 (1961) M. Maruyama, S. Furui, A. Faessler, p p¯ annihilation and scattering in a quark model. Nucl. Phys. A 472, 643–700 (1987) L. Montanet, G.C. Rossi, G. Veneziano, Baryonium physics. Phys. Rep. 63, 149–222 (1980) V. Mull, K. Holinde, Combined description of N¯ N scattering and annihilation with a hadronic model. Phys. Rev. C51, 2360–2371 (1995) F. Myhrer, A. Gersten, One boson exchange potentials and the nucleon-antinucleon scattering. Nuovo Cim. A 37, 21 (1977) Nobel Foundation, Nobel Lectures in Physics. vol. 1942–1962 (World Scientific, 1998). p. 489 (Chamberlain), p. 508 (Segrè) W. Oelert, The ELENA project at CERN. Acta Phys. Polon. B 48(10), 1895–1902 (2017) F. Oosterhof, B. Long, J. de Vries, R.G.E. Timmermans, U. van Kolck, Baryon-number violation by two units and the deuteron lifetime. Phys. Rev. Lett. 122(17), 172501 (2019) ¯ at K.D. Paschke et al., Experimental determination of the complete spin structure for pp ¯ → ΛΛ 1.637 GeV/c. Phys. Rev. C74, 015206 (2006) R.J.N. Phillips, D.P. Roy, Duality. Rep. Prog. Phys. 37, 1035–1097 (1974) H.J. Pirner, N N¯ annihilation in the large Nc limit. Phys. Lett. B209, 154–158 (1988) J.-M. Richard, M.E. Sainio, Nuclear effects in protonium. Phys. Lett. 110B, 349–352 (1982) J.L. Rosner, Possibility of baryon – antibaryon enhancements with unusual quantum numbers. Phys. Rev. Lett. 21, 950–952 (1968)

1934

J.-M. Richard

G. Rossi, G. Veneziano, The string-junction picture of multiquark states: an update. JHEP 06, 041 (2016) H.R. Rubinstein, H. Stern, Nucleon – antinucleon annihilation in the quark model. Phys. Lett. 21, 447–449 (1966) J.E. Russell, Metastable states of απ − e− , αK − e− , and α pe ¯ − atoms. Phys. Rev. Lett. 23, 63–64 (1969) A.D. Sakharov, Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe. Pisma Zh. Eksp. Teor. Fiz. 5, 32–35 (1967) I.S. Shapiro, The physics of nucleon – antinucleon systems. Phys. Rep. 35, 129–185 (1978) T. Suzuki, Analysis of antiproton nucleus scattering with the medium corrected optical potential. Nucl. Phys. A444, 659–677 (1985) T. Suzuki, H. Narumi, Anti-proton nucleus interaction at low-energy. Phys. Lett. B 125, 251–254 (1983) S. Takeuchi, F. Myhrer, K. Kubodera, Maximum asymmetry phenomena in pp ¯ → π − π + and − + pp ¯ → K K reactions. Nucl. Phys. A 556, 601–620 (1993) T.L. Trueman, Energy level shifts in atomic states of strongly-interacting particles. Nucl. Phys. 26, 57–67 (1961) S. Van Der Meer, Stochastic cooling and the accumulation of antiprotons. Rev. Mod. Phys. 57, 689–697 (1985) U. van Kolck, Nuclear effective field theories: reverberations of the early days. Few Body Syst. 62, 85 (2021) S. Wycech, J.P. Dedonder, B. Loiseau, Baryonium, a common ground for atomic and high energy physics. Hyperfine Interact. 234(1–3), 141–148 (2015) T. Yamazaki et al., Formation of long-lived gas-phase antiprotonic helium atoms and quenching by H2. Nature 361(6409), 238–240 (1993) T. Yamazaki, N. Morita, R.S. Hayano, E. Widmann, J. Eades, Antiprotonic helium. Phys. Rep. 366, 183–329 (2002) Y.B Zel’dovich, Energy levels in a distorted coulomb field. Sov. Phys. Solid State 1, 1497 (1960) P.A. Zyla et al., Review of particle physics. PTEP 2020(8), 083C01 (2020)

Section VIII Models of Nuclear Structure Jie Meng

Model for Independent Particle Motion

51

A. V. Afanasjev

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Independent Particle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformed Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cranked Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial Densities of the Single-Particle States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microscopic+Macroscopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Consistent Approaches: Covariant Density Functional Theory . . . . . . . . . . . . . . . . . . . Manifestation of Independent Particle Motion in Non-rotating and Rotating Nuclei . . . . . . Global Shell Structure at Spin Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superheavy Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superdeformation at High Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Phenomenon of Band Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Particle States in Deformed Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Particle and Polarization Effects Due to the Occupation of Single-Particle Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1938 1938 1940 1943 1946 1949 1953 1954 1957 1957 1959 1961 1962 1964 1968 1971 1972

Abstract

Independent particle model in nuclear physics assumes that the nucleon in the nucleus moves in the average (mean field) potential generated by all other nucleons. This chapter gives a short overview of basic features of the independent particle motion in atomic nuclei and its theoretical realization in the framework of shell models for spherical, deformed, and rotating nuclei as well as in more

A. V. Afanasjev () Department of Physics and Astronomy, Mississippi State University, Mississippi State, MS, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_10

1937

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A. V. Afanasjev

sophisticated approaches such as microscopic+macroscopic model and density functional theories. Independent particle motion of nucleons leads to global and single-particle consequences. The global ones manifest themselves in the shell structure, and its consequences for the global structure of nuclear landscape, the existence of superheavy nuclei, and the superdeformation at high spin are briefly reviewed. The latter shows itself in the single-particle properties such as energies, alignments, and densities; their manifestations are illustrated on specific examples.

Introduction The basic idea of the independent particle model is that nucleon (proton or neutron) moves in an average (mean field) nuclear potential and this motion is independent of motion of other nucleons. This allows to replace the complicated solution of the problem of A interacting particles (A is the total number of nucleons in the nucleus) with much simpler problem of A noninteracting particles in the mean field potential. This mean field potential can be either designed in a phenomenological way or calculated fully self-consistently from effective interaction. As discussed in this chapter, many properties of the nuclei have been described with high accuracy within such frameworks, and many new phenomena (e.g., superheavy nuclei and superdeformation at high spin) have been predicted theoretically and later observed experimentally. Independent particle model in its different realizations provides also the basis [mean field] for the models which take into account residual interactions such as pairing, vibrations, particle-vibration coupling, etc. These interactions are typically included either by adding respective terms to the version of independent particle model or by the modification of the formalism. The basic features of independent particle model and its realization in the shell model variants, microscopic+macroscopic model, and covariant density functional theory will be discussed in this chapter. In addition, some manifestations of the independent particle motion emerging from the shell structure and singleparticle properties will be considered. In no way, this chapter should be considered as “all-inclusive”: it only scratches the surface of huge body of experimental and theoretical results. More comprehensive and detailed reviews on specific phenomena/theoretical approaches are quoted in respective sections. In addition, some aspects of the independent particle motion are discussed in the books (Bohr and Mottelson (1969), Ring and Schuck (1980), Casten (1990), and Nilsson and Ragnarsson (1995)).

Independent Particle Model The solution of the many-body nuclear problem for a nucleus consisting of A nucleons (Z protons and N neutrons) requires the solution of the eigenvalue problem (Bohr and Mottelson 1969; Ring and Schuck 1980)

51 Model for Independent Particle Motion

1939

H Ψα (r , σ, τ ) = Eα Ψα (r , σ, τ ).

(1)

The Hamiltonian of the system could be in either non-relativistic (Schrödinger equation) or relativistic (Dirac equation) forms. It contains the kinetic and potential energy contributions from each nucleon H =

A  h¯ 2 2  ∇ + Vij , 2mi i

(2)

i=j

i=1

and includes two-body interaction between i- and j -th nucleons. The total wave function Ψα (r , σ, τ ) of the nucleus in the state with quantum numbers α has to be expressed in terms of the ones of the individual nucleons ψi (ri , σi , τi ). Here, r, σ , and τ are position, spin, and isospin variables, respectively. For simplicity, in further discussion, ri will represent all independent variables of the i-th nucleon. Total wave function of the system of the A particles has to be antisymmetric with respect of the exchange of the coordinates of two particles. Thus, a many-body state is written in the form of the Slater determinant (Ring and Schuck 1980)    ψ1 (r1 ) ψ1 (r2 ) . . . ψ1 (rA )     ψ2 (r1 ) ψ2 (r2 ) . . . ψ2 (rA )  1 .  Ψα (r1 , r2 , . . . , rA ) = √ det  . . . . . . . . .  A!  ...  ψ (r ) ψ (r ) . . . ψ (r )  A 1 A 2 A A

(3)

The factor √1 is due to normalization. The choice of the single-particle wave funcA! tions defines the type of the many-body state. It is important to remember that the single-particle spectrum is an infinite one. Thus, in practical applications, respective Hilbert space is truncated most frequently based on energy considerations. In nuclear systems, it is possible to recast Eq. (1) in the form H =

A 

A 

h(ri ) +

V˜ (ri , rj ),

(4)

i=j =1

i=1

where V˜ (ri , rj ) is the residual two-body interaction and h(ri ) is the single-particle Hamiltonian. This recast essentially means that the part of original two-body interaction Vij is moved to the single-particle Hamiltonian h(ri ). The latter can be chosen in such a way that the contribution of residual interaction becomes rather small or even negligible. Independent particle model corresponds to the situation when the second term in Eq. (4) is neglected. Thus, the nuclear Hamiltonian is a sum of single-particle terms H =

A  i=1

h(ri ),

(5)

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A. V. Afanasjev

and eigenvalue problem is reduced to h(ri )ψk (ri ) = ek ψk (ri ),

(6)

where ek stands for the single-particle energy. Independent particle model allows to reduce the nuclear many-body problem from complicated two-body treatment to much simpler one-body one. It also relies on the use of the single-particle potentials. Independent particle model corresponds to the description of a nucleus in terms of noninteracting particles in the orbitals of these single-particle potentials which itself are produced by all the nucleons. Thus, nucleons move essentially free in these potentials. In this model, the ground state is formed by filling all the single-particle states located below the Fermi level. The excitation of the particle from the occupied state below this level to an empty one above it leads to an excited state. This process is called as a particle-hole excitation. The justifications for an independent particle model have been discussed in a number of publications (Bohr and Mottelson 1969; Casten 1990). The mean free path between collisions of the constituent nucleons is large compared to the average distance between them, and in some cases, it could be larger than the dimensions of the nucleus. As a consequence, the interactions between nucleons contribute mostly to the smoothly varying average potential in which the particles moves independently. One can arrive to the same conclusion by taking into account the fact that the nucleons occupy only approximately 1% of the volume of the nucleus. This estimate follows from the consideration of nucleons as hard spheres of radius c ≈ 0.5 fm. This value of c corresponds to the half of the closest distance which two nucleons can approach each other due to infinite repulsion (see discussion in Sec. 2–5b of Bohr and Mottelson 1969). For such value of c, nuclear matter behaves close to the free gas of nucleons with very rare head-on collisions of nucleons. The Pauli principle and the weakness of strong nuclear interaction when compared with characteristic kinetic energies of nucleons inside the nucleus are other contributing factors for the justification of independent particle model (Bohr and Mottelson 1969; Casten 1990).

Spherical Shell Model The discussion of the previous section clearly illustrates the need for the introduction of the single-particle potential. This one-body potential can be introduced either in phenomenological or in a fully self-consistent way. In the former case, one deals with phenomenological Nilsson, Woods-Saxon, or folded Yukawa potentials. Self-consistent single-particle potentials are formed as a result of the solution of the many-body nuclear problem within non-relativistic and relativistic density functional theories (DFTs); note that these potentials are not treated separately from the rest of the nuclear many-body problem. To illustrate the major physics aspects behind these potentials and for pedagogical reasons, the harmonic oscillator (HO) potential and its modifications relevant for

51 Model for Independent Particle Motion

1941

nuclear physics problems will be considered here. The single-particle Hamiltonian of the nucleon with mass m entering into Eq. (5) is given by h(r ) = −

h¯ 2 2 ∇ + V (r ), 2m i

(7)

1 mω 0 r 2 2

(8)

where the central potential V (r ) =

is pure HO potential representing one-body (mean) potential (field). This Hamiltonian in spherical coordinates is given by h(r ) = −

h¯ 2 1 ∂ 2 l 2 (θ, φ) r + + V (r), 2m r ∂r 2 2mr 2

(9)

and the wave function ψ representing the solution of the eigenvalue problem is separable in angular (θ , φ) and radial (r) coordinates: ψ = R(r)Ylm (θ, φ). Here, R(r) and Ylm (θ, φ) are radial and angular [given by spherical harmonics] wave functions, respectively. Note that the latter is the eigenfunction of the angular momentum operator l 2 : l 2 Ylm (θ, φ) = h¯ 2 l(l + 1)Ylm (θ, φ).

(10)

The solutions of the Hamiltonian with only central potential included are shown in the left column of Fig. 1. The spectrum of single-particle states, characterized by principal quantum number N, is equidistant in energy with high degree of the degeneracy of the single-particle states. The realistic nuclear potential is located somewhat between pure HO and square well potentials. To correct for that, a centrifugal potential has to be added to V (r); it is usually parametrized as (Nilsson and Ragnarsson 1995; Nilsson 1955; Ragnarsson et al. 1978):     2 2 Vcorr = −μ hω . ¯ 0 l − l N

(11)

Within the N-shell, this term leads to a lowering in energy of high-l states relative to low-l ones and to a removal of the degeneracy between the l states (see middle panel of Fig. 1). In addition, there is a coupling between spin (s = 1/2) and orbital motion of the single particle which, in general, is given by the following interaction term: VLS = λ

1 ∂VSO (r)  l · s. r ∂r

(12)

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A. V. Afanasjev

2h 1k μ′= 0.021

N=7 κ = 0.06 μ′ = 0.024

N=6 κ = 0.06 μ′= 0.024

2h11/2

4s 3d 1j 2g

184

1i 3p

126

N=5

1h

3p3/2

1h9/2 2f7/2

N=4

1g

2f5/2 1i13/2

82

3s 2d

κ = 0.06 μ′ = 0.024

1j3/2

4s 3d3/2 1/2 2g 3d5/2 7/2 1j15/2 1i11/2 2g9/2 3p1/2

2f

κ = 0.06 μ′ = 0.024

1k17/2

50

1h11/2 3s1/2 2d3/2 1g 2d5/2 7/2 1g9/2

N=3 κ = 0.075 μ′ = 0.0263

2p

2p1/2

1f

2p3/2

1f5/2

28 1f7/2

20 μ′ = 0 N = 2 κ = 0.08

Hosc

1d3/2

2s+1d _

2

-μ′h ω0 [l -N(N+3)/2]

_

-2κh ω0 l·s

2s1/2 1d5/2

Fig. 1 The sequential build-up of realistic nucleonic potential. The left column shows the singleparticle states of the pure harmonic oscillator. The employed parameters κ and μ of the MO potential are displayed: note that they are different for the different N -shells. The modifications introduced by Eq. (11) are shown in the middle column. Finally, the right column shows the impact of spin-orbit interaction on the energies of single-particle states and their ordering. Particle numbers corresponding to spherical shell closures are encircled. Black and red colors are used for positive and negative parity states, respectively. The figure is based on the results presented in Fig. 6.3 of Nilsson and Ragnarsson (1995)

Here, VSO indicates the spin-orbit potential which may be different from the central one, and λ is the coupling constant of spin-orbit interaction. In the case of harmonic oscillator potential, this term can be further simplified to VLS = −2κ hω ¯ 0 l · s. The presence of spin-orbit potential leads to the lowering and rising in energy of the j = l + 1/2 and j = l − 1/2 states emerging from the state with a given orbital angular

51 Model for Independent Particle Motion

1943

momentum l (see right column in Fig. 1). Only with this interaction included, it is possible to reproduce experimentally observed shell closures at particle numbers 2, 8, 20, 28, 50, 82, and 126 (Bohr and Mottelson 1969; Casten 1990; Ragnarsson et al. 1978). Note that the single-particle state is completely defined by a set of quantum numbers [N lj ]. The combined potential VMO = V (r) + Vcorr + VLS

(13)

is usually called as modified oscillator (MO) or Nilsson potential. This potential is defined by three parameters ω0i , κi , and μi for each kind of nucleons (i = π or ν). ω0i defines the radius of respective matter distribution, μi simulates the surface diffuseness depth, and κi is the strength of spin-orbit interaction. The Coulomb potential is not directly included into the MO potential, but it is effectively accounted by the differences of the abovementioned parameters in the proton and neutron subsystems. Although the MO potential is still extensively used in nuclear structure studies, more realistic treatment of the single-particle degrees of freedom is achieved by means of the Woods-Saxon (WS) (Gareev et al. 1973; Cwiok et al. 1987; Bengtsson et al. 1989) and folded Yukawa (FY) (Möller and Nix 1981; Dobrowolski et al. 2016) potentials. This is because they have more realistic shape of the potential (thus eliminating the need for Vcorr ) and explicitly include the Coulomb interaction. Their structure is illustrated below for the WS potential VW S = VW S (r) + VLS + VC ,

(14)

where central potential is given by VW S (r) =

V0 1 + exp[(r − R)/a]

(15)

with V0 being the potential depth, a the surface thickness, and R = r0 A1/3 the nuclear radius (with typical value of r0 ≈ 1.2 fm). Here, VC represents the Coulomb potential. However, numerical realization of these potentials is more complicated as compared with the MO potential. The detailed comparison of the WS and MO potentials is presented in Bengtsson et al. (1989).

Deformed Shell Model In deformed nuclei, the shape of the nuclear surface is generally parametrized by means of a multipole expansion of the radius in terms of the shape parameters (Ring and Schuck 1980); a typical parametrization is

1944

A. V. Afanasjev

⎡ R(θ, φ) = R0 ⎣1 +



⎤ αλμ Yλμ (θ, φ)⎦ ,

(16)

λμ

where αλμ are the deformation parameters and R0 is the radius of the sphere with the same volume. For axially symmetric nuclear shapes μ = 0 and quadrupole deformation β2 = α20 is dominant (Ring and Schuck 1980; Chasman et al. 1977). For simplicity, higher multipolarity deformations such as octupole β3 , hexadecapole β4 , etc. are not taken into account in the present discussion. Note that in the literature, different parametrizations of the deformations of the single-particle potential exist (see discussion in Nilsson and Ragnarsson 1995; Chasman et al. 1977). There are two facts which affect the consideration of single-particle potentials in deformed nuclei (Nilsson 1955). First, this potential should follow the nuclear density distribution. Second, in deformed nuclei, the oscillator frequencies ω i (i = x, y and z) are different along different principal axis of nuclear ellipsoid. As a result, the central potential of Eq. (8) is modified in the following way: V (r ) =

 m 2 2 ω x x + ω2y y 2 + ω2z z 2 . 2

(17)

Since nuclear matter is highly incompressible, the change of the shape of the nucleus from spherical to ellipsoidal should not modify the volume of the nucleus. This is accounted by the volume-conservation condition ω30 = ω x ω y ω z .

(18)

Assuming axial symmetry around the z-axis, i.e., ω x = ω y , the central potential can be rewritten as V (r ) =

1 mω20 r 2 − β2 mω20 r 2 Y20 (θ, φ), 2

(19)

where β2 stands for the quadrupole deformation of the potential (the measure of the deviation from spherical shape). Thus, the Hamiltonian of the Nilsson model becomes h(r ) = −

h¯ 2 2 1 ∇ + mω20 r 2 − β2 mω20 r 2 Y20 (θ, φ) 2m i 2     2 2 − 2κ h¯ ω 0 l · s. − μ hω ¯ 0 l − l N

(20)

Note that for simplicity, only general outline of the Nilsson model is provided here (Nilsson 1955). Technical details of the solution of the Nilsson potential are discussed in Nilsson and Ragnarsson (1995), Nilsson (1955), and Bengtsson et al. (1989). Note also that there are generalizations of the Woods-Saxon and folded

[64

2]

114

1/2

[52

106

1] 3/2 [52 1]

1i13/2

-5 -6

104 102

7/2

9/2 [

-4

9/2[624]

50 5]

-3

2f7/2

92 254

No

[63

3]

4] [51 7/25 96 /2[ 64 1/2 2] 0] [53 [40 0] 1/2

-5

3d3/2 3d5/2

-9

0.1

0.2

β2 deformation

0.3

0

20]

5] [61 152 9/2[7 34]

9/2

138

NL3*

1/2[6

3/2[6

22]

2g9/2 1j15/2

1i11/2 0

]

[725

/2 13] 11

7/2[6

-7 -8

4]

60

2[ 9/

2g7/2 164

-6

3]

1 5/2[6

[7] 1 /2

-2

-4

11 /2[ 60 6]

3/2

2f5/2

51] 1/2[6

Single-proton energies ei [MeV]

-1

1945

Single-neutron energies ei [MeV]

51 Model for Independent Particle Motion

5/2 [75 2]

5/2 1/ [622] 2[ 63 1] 148 7/2 1] [74 [50 3] 1/2

4]

7/2[62

0.1

0.2

β2 deformation

0.3

Fig. 2 Single-particle energies, i.e., the diagonal elements of the single-particle Hamiltonian h in the canonical basis (Ring and Schuck 1980), for the lowest in total energy solution in the nucleus 254 No calculated as a function of the quadrupole equilibrium deformation β for covariant energy 2 density functional NL3*. Solid and dashed lines are used for positive- and negative-parity states, respectively. Relevant spherical and deformed gaps are indicated. (Figure taken from Dobaczewski et al. 2015)

Yukawa potentials to deformed shapes (Gareev et al. 1973; Cwiok et al. 1987; Möller and Nix 1981; Dobrowolski et al. 2016; Chasman et al. 1977). As a result, the focus here is on the consequences of the breaking of spherical symmetry on the single-particle states. They are usually illustrated by means of the Nilsson diagrams which show the evolution of the energies of deformed singleparticle states as a function of deformation (see, e.g., Figs. 3 and 5 in Nilsson (1955), Figs. 8.3 and 8.5 in Nilsson and Ragnarsson (1995), and Fig. 3 in Afanasjev et al. (1999) for such diagrams obtained with the Nilsson potential). The general structures (and frequently fine details) of the Nilsson diagrams obtained in phenomenological single-particle potentials and in self-consistent models are very similar. This clearly indicates that the former have deep microscopic roots. Thus, the results obtained in covariant density functional theory presented in Fig. 2 are used here to illustrate the impact of deformation on the single-particle states. Several major features emerge on the transition from spherical to deformed shapes. First, it removes the 2j + 1 degeneracy of the spherical subshells, and deformed single-particle states are only twofold degenerate. Second, the deformed single-particle states are defined by approximate Nilsson quantum numbers Ω[N nz Λ] (note they are frequently shown in inverted order of [Nnz Λ]Ω). Here, Ω (Λ) stands for the projection of the total (orbital) single-particle angular momentum on the axis of symmetry, N is the principal quantum number, and nz is the number of nodes of the wave function along the symmetry axis. Note that the parity π of the state is defined by (−1)N . For prolate (β2 > 0) shapes, the deformed orbitals emerging from a given spherical j -subshell are split in such a way that the state with Ω = 1/2 is always lowest in energy, while that with Ω = j is the highest one. The states with intermediate Ω  values are arranged in such a way that the energies E

1946

A. V. Afanasjev

of two neighboring deformed states satisfy the condition E(Ω  + 1) > E(Ω  ) (see Fig. 2).The order of the states is inverted for the oblate (β2 < 0) shapes (Nilsson and Ragnarsson 1995; Nilsson 1955). Third, as a consequence of these modifications of the single-particle states with deformation, spherical shell gaps disappear, and new deformed gaps, which are comparable in size with minor shell gaps at spherical shape, appear. These gaps are encircled in Fig. 2.

Cranked Shell Model Rotation is the phenomenon which appears in all branches of physics, from galaxies down to atomic nuclei. In the latter, it is a collective phenomenon in which many nucleons define a nuclear deformation and contribute to rotational motion. Note that contrary to classical mechanics, a collective rotation along the symmetry axis of nuclear density distribution is forbidden in quantum mechanics. The simplest way to describe the properties of rotating nuclei in a microscopic way is to use the cranking-model approximation suggested by Inglis (1954, 1956). Over the years, this model has been successfully applied to the description of different phenomena in rotating nuclei (Nilsson and Ragnarsson 1995; Afanasjev et al. 1999; de Voigt et al. 1983; Szyma´nski 1983; Meng et al. 2013). The consideration here is restricted to one-dimensional cranking in which the nuclear field is rotated externally with a constant angular velocity ω around a principal axis usually defined as the x-axis. This is done in order to outline the basic features of this model. The basic idea of the cranking model is that a nucleus with angular momentum I = 0 can be described in terms of an intrinsic state Ψ ω at rest in a rotating frame. In the one-dimensional cranking approximation for collective rotation, the total cranking Hamiltonian (or Routhian) for a system of independent particles is given by H ω = H − ωIx =



hωi ,

(21)

i occ

where H is the total Hamiltonian in the laboratory system, Ix is the x-component of the total angular momentum, and hω is the single-particle Hamiltonian in the rotating system hω = h − ωjx

(22)

with h being the single-particle Hamiltonian in the laboratory system and jx the xcomponent of the single-particle angular momentum. In Eq. (21), the sum extends over the occupied proton and neutron orbitals. The term −ωIx in Eq. (21) is analogous to the Coriolis and centrifugal forces in classical mechanics. Then the total energy Etot in the laboratory system is given as

51 Model for Independent Particle Motion

1947

Fig. 3 Single-neutron energy levels obtained with the Woods-Saxon potential as a function of rotational frequency for deformation parameters β2 = 0.61 and β4 = 0.11 typical for the yrast superdeformed band in 152 Dy and its closest neighbors. The orbitals are defined by the parity π and signature quantum number α. Solid, dotted, short-dashed, and long-dashed lines are used for the orbitals with (π = −, α = −1/2)(π = −, α = +1/2), (π = +, α = +1/2), and (π = +, α = −1/2), respectively. The dominant Nilsson components of the wave functions of the levels are shown at the lowest and highest calculated frequencies. (Figure is reproduced from Robin et al. 2008)

Etot =



ψiω |h|ψiω =

i occ

 i occ

eiω + ω



ψiω |jx |ψiω ,

(23)

i occ

and the total spin I by I ≈ Ix =



ψiω |jx |ψiω ,

(24)

i occ

where ψiω are the single-particle eigenfunctions in the rotating system and eiω = ψiω |hω |ψiω the corresponding eigenvalues (single-particle Routhians). The approximation used in Eq. (24) is valid only in the limit of high spin (I 1). The evolution of the single-particle states in rotating potential is shown in Fig. 3. There are several important features. First, the time-reversal symmetry is broken in rotating potential, and twofold degeneracy of deformed single-particle states, which exist in non-rotating nuclei, is removed. As a consequence, each single-particle orbital has additionally to be characterized by signature quantum number (either ri or αi ) (Nilsson and Ragnarsson 1995; Afanasjev et al. 1999; de Voigt et al. 1983).

1948

A. V. Afanasjev

This quantum number is a consequence of the fact that full cranking Hamiltonian is invariant with respect to a rotation through an angle π around the cranking axis (x-axis) Rx = exp(−iπjx ),

Rx ψi = exp(−iπjx )ψi .

(25)

The eigenvalues of Rx are exp(−iπ α), where α is the signature exponent quantum number. Alternatively, one can define signature quantum number as r = exp(−iπ α). Signature quantum number αi (ri ) of a single-particle orbital could take the αi = + 12 (ri = −i) or αi = − 12 (ri = +i) values. Similar to parity, this classification of the single-particle states is an important tool for identifying the nucleon orbitals in the rotating nuclear potential. Second, the coupling between the different single-particle orbitals increases with increasing rotational frequency. As illustrated in Fig. 3, this leads to the change of the dominant Nilsson components of the wave function. Third, the slope of the orbitals in Fig. 3 corresponds to the single-particle alignment jx i (Nilsson and Ragnarsson 1995; de Voigt et al. 1983) jx i = −

∂eiω ∂ω

(26)

for the case of cranking at fixed deformation. Note that single-particle Routhians are always plotted at fixed deformation in phenomenological potentials. In contrast, they are given along the equilibrium deformation path in the DFT calculations. Thus, the condition (26) is only approximately satisfied in the latter case provided that deformation changes with increasing rotational frequency are modest. Note that in general jx i depends on signature and this dependence is especially pronounced for the single-particle orbitals with low value of Ω. The knowledge of single-particle alignments is extremely useful for an understanding of physical situation and interpretation of experimental data. For example, high-j or high-N intruder orbitals (such as those with Nilsson labels 1/2[770] and 1/2[880]) have large values of jx i , and, as a consequence, they are strongly downsloping with increasing rotational frequency (see Fig. 3). Large energy splitting between two signature partner orbitals of a given single-particle state appear for the states with Ω = 1/2, but no such splitting exists for the states with large value of Ω. For example, this is a case for the 1/2[770] state for which a large signature splitting between the α = ±1/2 branches leads to a formation of large N = 85 gap at ω ≈ 0.3 MeV which is absent at ω = 0 MeV (see Fig. 3). The advantage of the cranking model is that it provides a microscopic description of rotating nuclei, where the total angular momentum is described as a sum of single-particle angular momenta, and thus collective and “non-collective” rotations can be treated on the same footing. On the other hand, there are limitations. The cranking-model approximation is semiclassical, because the rotation is imposed externally. The model also breaks the rotational invariance, since a fixed rotation axis is used. These latter limitations are not very important for very fast rotation

51 Model for Independent Particle Motion

1949

(I 1). Another shortcoming of the cranking model is that the wave functions are not eigenstates of the angular momentum operator, which can lead to difficulties; for example, a proper calculation of electromagnetic transition probabilities requires the use of projection techniques (Ring and Schuck 1980). Furthermore, the cranking model is a poor approximation in the crossing region of two weakly interacting bands, but this difficulty can be overcome by the removal of such crossings (Afanasjev et al. 1999).

Spatial Densities of the Single-Particle States The total density of the nucleus is defined as the sum of the single-particle densities of occupied states ρtot (r ) =



ρk (r ),

(27)

ρk (r ) = ψk∗ (r )ψk (r )

(28)

k

which are given by

for the k−th single-particle state. The single-particle wave function ψk (r ) can be expanded into basis states |μ > ψk (r ) =



ckμ |μ >,

(29)

μ

where μ represents the set of quantum numbers defining the basis state and ckμ the expansion coefficients. Of special interest are the cases when this expansion is dominated by a single basis state n (ckn ≈ 1) since then the nodal structure of the wave function of the state ψk (r ) and consequently its single-particle density will be predominantly defined by this basis state. This takes place either at spherical shape or at extremely elongated nuclear shapes typical for rod shape structures or megadeformation (MD) (Afanasjev and Abusara 2018). In the latter case, the wave function defined by asymptotic Nilsson quantum number is expanded into the basis states characterized by μ = [Nnz Λ]Ω (Nilsson and Ragnarsson 1995; Afanasjev and Abusara 2018). For extremely elongated shapes in light nuclei, this expansion is dominated by a single basis state (Afanasjev and Abusara 2018). This is illustrated in Fig. 4 which shows the single-particle densities of indicated states at the deformation β2 ≈ 1.6 typical for the MD shapes. They are characterized by axially or nearly axially symmetric spheroidal/ellipsoidal like density clusters formed for the single-particle states with the [NN 0]1/2 Nilsson quantum numbers, doughnut density distributions for the [N 01]Ω states and multiply (two for nz = 1 and three for nz = 2) ring shapes for the [N, N − 1, 1]Ω Nilsson states with N = 2 and 3. Figure 4 clearly indicates the importance of the

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A. V. Afanasjev

g9/2

0

Neutron single-particle energies ei [MeV]

f5/2 -10

-20

f7/2

[321]1/2 /2 2]5 [440]1 [20

/2

d3/2 s1/2

[321]3/2 /2 [211]1 [330] 1/2 [211]3/2 /2 1]1 3/2 [10 [101]

d5/2 -30 p1/2 -40

[220]1/2

p3/2 [110]1/2

40 -50

Ca

[000]1/2

s1/2 0

0.5 

1

1.5

- deformation

Fig. 4 Left panel: The Nilsson diagram for the neutron single-particle states in 40 Ca. It is based on the results obtained in axial relativistic Hartree-Bogoliubov calculations with the NL3* functional. Right panel: Single-neutron density distributions due to the occupation of indicated Nilsson states. The box in the right bottom corner exemplifies physical dimensions of the nucleus as well as the colormap used for single-particle densities. Other density plots are reduced down to the shape and size of the nucleus which is indicated by black solid line corresponding to total neutron density line of ρ = 0.001 fm−3 . The colormap shows the densities as multiplies of 0.001 fm−3 ; the plotting of the densities starts with yellow color at ρ = 0.001 fm−3 . (Figure taken from Afanasjev (2018), and it is based on the results presented in Afanasjev and Abusara 2018)

deformation which has two critical effects. First, it leads to the formation of density clusters with specific nodal structure and to the separation of the clusters in space. Second, it lowers the energies of the Nilsson states of the [NN0]1/2 type which favors the α-clusterization and leads to the configurations in which all occupied single-particle states have this type of structure. It is impossible to experimentally measure the single-particle density distributions in rotating nuclei. That is a reason why contrary to the single-particle energies and alignments, they have not been used as fingerprints of independent particle motion. However, they are important for an understanding of the αclusterization and the evolution of extremely elongated shapes in light nuclei such as ellipsoidal and rod shapes and nuclear molecules (Afanasjev and Abusara 2018; Afanasjev 2018). The examples of total neutron densities of the latter two types of nuclear shapes are provided in Fig. 5. A rod-shaped nuclear configuration in 12 C is built from a linear chain of three α-clusters and could be well understood

51 Model for Independent Particle Motion

1951

0.1 rod−shape, I = 8.8

0.06 0 0.04 −5

12 C

(a) −5 0 5 Symmetry axis Z (fm)

0.1

0.02 0

[31,31], I = 21

5

0.08 Y (fm)

Y (fm)

5

0.08 0.06

0 0.04 −5

36 Ar

(d)

−5 0 5 Symmetry axis Z (fm)

0.02 0

Fig. 5 Total neutron densities [in fm3 ] of indicated configurations in the 12 C and 36 Ar nuclei at specified spin values. The plotting of the densities starts with yellow color at 0.001 fm3 . The results are based on the cranked relativistic mean field calculations with the NL3* functional. (Figure taken from Afanasjev and Abusara 2018)

from the summation of the single-particle densities of occupied single-particle states (Afanasjev and Abusara 2018) (see also Ichikawa et al. (2011), Yao et al. (2014), and Zhao et al. (2015) for additional discussion of rod-shaped nuclei built on linear chains of the α particles). Rod-shaped configurations appear also in heavier nuclei such as 42 Ca and 44 Ti (Afanasjev and Abusara 2018). However, their densities are more uniform since the α-clusterization is suppressed by the occupation of the single-particle orbitals which have either doughnut- or ring-type single-particle density distributions (see Fig. 4 and Afanasjev and Abusara 2018; Ebran et al. 2018). Nuclear molecules are characterized by the formation of the neck between two nuclear fragments: the configuration of 36 Ar is an example of such structure (see Fig. 5). A knowledge of the nodal structure of the single-particle densities of the states allows to understand the microscopic mechanism of the transition between different nuclear shapes. For example, in order to build nuclear molecules from typical ellipsoidal density distributions, one has to move matter from the neck (equatorial region) into the polar one. This can be achieved by means of specific particle-hole excitations which move the particles from (preferentially) doughnut-type orbitals or from the orbitals which have a density ring in an equatorial plane into the orbitals (preferentially of the [NN0]1/2 type) building the density mostly in the polar regions of the nucleus (Afanasjev and Abusara 2018; Afanasjev 2018). Another example of important role of the single-particle density of the states is seen in so-called “bubble” structures of the ground states of spherical nuclei. This is due to a specific radial nodal structure of the wave functions of the singleparticle states (see Fig. 6.2 in Nilsson and Ragnarsson 1995): only s (l = 0) states build the density in the center of the nucleus. On the contrary, the l > 0 states do not participate in the building of the densities in the center of the nucleus due to the presence of the centrifugal barrier (see also Fig. 2 in Karakatsanis et al. 2017). The impact of this feature is especially pronounced in proton subsystem of 34 Si (Karakatsanis et al. 2017; Grasso et al. 2009). Left panel of Fig. 6 shows the evolution of the proton densities in the N = 20 isotones. The 40 Ca, 38 Ar, and 36 S

1952

A. V. Afanasjev

40

0.1

Ca Ar S 34 Si 36

-3

proton density (fm )

38

0.05

0

2

4

r (fm)

Fig. 6 Left panel: Proton densities of the N = 20 isotones obtained in the spherical relativistic Hartree-Bogoliubov calculations with the DD-ME2 functional. Right panels: Neutron and proton densities of the spherical 292 120172 nucleus obtained in non-relativistic Skyrme and Gogny as well as covariant DFT calculations. Employed functionals are indicated. (Figures are taken from Karakatsanis et al. 2017; Afanasjev and Frauendorf 2005a)

nuclei in which the proton 2s1/2 orbital is occupied show similar proton density patterns with the density peak at the center. In contrast, the emptying of the proton 2s1/2 orbital in 34 Si leads to a considerable depletion of the proton density in the central region of nucleus. This “bubble” phenomenon has been indirectly confirmed in experiment (Mutschler et al. 2017). Similar depletion of the density in the central region of the nucleus exist also in superheavy nuclei (see Afanasjev and Frauendorf (2005a), Bender et al. (1999), Schuetrumpf et al. (2017) and right panel of Fig. 6). However, the mechanism of its creation is different as compared with the one active in 34 Si since the emptying of the s states is not possible in such nuclei. It relies on the fact that the filling of low-j /high-j orbitals builds the density in the central/surface region of the nucleus (Afanasjev and Frauendorf 2005a; Bender et al. 1999). Thus, starting from the flat density distribution in 208 Pb (which is experimentally verified; see Fig. 2.4 in Nilsson and Ragnarsson 1995), one can build “bubble”-type structure in the 292 120 172 superheavy nucleus by occupying predominantly high-j orbitals outside the 208 Pb core (Afanasjev and Frauendorf 2005a). It was suggested in Schuetrumpf et al. (2017) that the depletion of density in central region is mostly due to Coulomb

51 Model for Independent Particle Motion

1953

interaction. This, however, contradicts to the observation that spherical superheavy nuclei with Z = 126 have significantly smaller depletion of the density in the central region as compared with the 292 120172 one (Afanasjev and Frauendorf 2005a).

Microscopic+Macroscopic Models The “shell model” concepts discussed above assume some fixed mean field properties which do not depend on nucleonic configuration. In addition, they require the use of deformation parameters in the case of deformed and cranked shell models which have to be defined either from experimental data or from higher-level model. To overcome these deficiencies, the microscopic+macroscopic (mic+mac) model has been suggested in the 1960s (Strutinsky 1967; Nilsson et al. 1969; Brack et al. 1972) which can be considered as an approximation to the Hartree-Fock approach (Brack et al. 1972). In this model, the total energy of the nucleus Etot is separated into two parts, a macroscopic part Emacro and a microscopic part Emicro , Etot = Emacro + Emicro .

(30)

The macroscopic energy Emacro is defined by some version of the liquid-drop model, while the microscopic energy Emicro is obtained from quantal shell corrections Esh calculated from a phenomenological potential using the Strutinsky prescription (Strutinsky 1967). The shape of the nuclear surface is parametrized by means of a multipole expansion of the radius in terms of the shape parameters (see Eq. (16)). Then the equilibrium deformation in a specific nucleonic configuration is determined by a minimization of the total energy Etot with respect to the shape parameters. For simplicity, only the basic features of the mic-mac method for the rotating nuclei are outlined here since the case of no rotation is easy to obtain by dropping respective terms. The total nuclear energy Etot at a specific deformation β¯ = β2 , γ , β4 , . . . and spin I0 is given as a sum of the rotating liquid drop energy and the shell energy



¯ I0 = ELD β, ¯ I =0 + Etot β,

1 ¯ 2Jrig (β)



¯ I0 . I02 + Esh β,

(31)

The shell energy is defined as the difference between the discrete and smoothed (indicated by  ) single-particle energy sums, Esh (I0 ) =



 ¯  ei (ω, β)

I =I0

  ¯  ei ( − ω, β) 

I=I0

(32)

with both terms evaluated at the same spin value I0 . The smoothed sum is calculated using the Strutinsky procedure (Strutinsky 1967). Then the total nuclear energy

1954

A. V. Afanasjev



¯ I0 and equilibrium deformation β¯eq of specific nucleonic configuration can Etot β, be calculated as a function of spin I (defined via Eq. (24)). By considering numerous configurations, defined by the occupation of single-particle orbitals with the specific sets of quantum numbers such as parity, signature, etc., one can build comprehensive spectra of multiply rotational bands and compare them with experimental data. Note that in the mic+mac approach, the moment of inertia, defined from smoothed single-particle quantities, has to be renormalized to a rotating liquid behavior. The renormalization of the moment of inertia is especially important in the Nilsson potential because of the l 2 -term, but it is also required in the WoodsSaxon potential because the potential radius is often different from the nuclear matter radius (see Afanasjev et al. (1999) and references therein). The consideration above is restricted to the situation of no pairing correlations. The experience and detailed studies show that this is quite accurate approximation at high spins (Afanasjev et al. 1999; Vretenar et al. 2005). However, the pairing correlations play an important role at low and medium spins. In the mic+mac approaches, they are usually taken into account at the BCS level by adding pairing energy term Epairing (Pomorski and Dudek 2003; Carlsson et al. 2008; Möller et al. 2016) pair

Etot = Etot + Epairing .

(33)

The mic+mac model is simpler than self-consistent approaches, and it is also substantially cheaper in numerical calculations. Despite approximations and simplifications employed by this method, it provides an accurate description of the ground state energies and deformations as well as fission barrier heights (Pomorski and Dudek 2003; Möller et al. 2016) and rotational properties of multiply bands in different nuclei (Afanasjev et al. 1999; Carlsson et al. 2008).

Self-Consistent Approaches: Covariant Density Functional Theory There are several types of self-consistent density functional theories (DFTs) based either on the non-relativistic Shrödinger equation with finite range Gogny or zero range Skyrme forces or on the relativistic Dirac equation. The latter is called as covariant density functional theory (CDFT) (Vretenar et al. 2005), and its brief outline will be presented here. It was very successful in the description of many physical phenomena (see reviews in Vretenar et al. 2005; Meng et al. 2006; Nikši´c et al. 2011; Meng 2016). The detailed reviews of the Skyrme and Gogny DFTs are presented in Bender et al. (2003) and Peru and Martini (2014). There are several classes of the CDFT models, and, for simplicity, only mesonexchange (ME) models will be considered here. In these models, the nucleus is described as a system of Dirac nucleons interacting via the exchange of mesons with finite masses leading to finite-range interactions. The starting point of the ME

51 Model for Independent Particle Motion

1955

models is a standard Lagrangian density (Gambhir et al. 1990)   1−τ3 1 1 A) − m − gσ σ ψ + (∂σ )2 − m2σ σ 2 L = ψ¯ γ · (i∂−gω ω−gρ ρ τ − e 2 2 2 1 1 1 1 1 − Ωμν Ω μν + m2ω ω2 − Rμν R μν + m2ρ ρ 2 − Fμν F μν , 4 2 4 2 4

(34)

which contains nucleons described by the Dirac spinors ψ with the mass m and several effective mesons characterized by the quantum numbers of spin, parity, and isospin. The Lagrangian (34) contains as parameters the meson masses mσ , mω , and mρ and the coupling constants gσ , gω , and gρ . e is the charge of the protons, and it vanishes for neutrons. The coupling constants are density dependent in the densitydependent meson exchange (DDME) class of covariant energy density functionals (CEDFs) (Typel and Wolter 1999; Lalazissis et al. 2005). In contrast, they are constant in the so-called non-linear (NL) CEDFs in which the density dependence is introduced via the powers of the σ -meson (Boguta and Bodmer 1977): 1 1 LN L = L − g2 σ 3 − g3 σ 4 . 3 4

(35)

The solution of these Lagrangians leads to the relativistic Hartree-Bogoliubov (RHB) equations (Vretenar et al. 2005). They are illustrated below on the example of the cranked RHB (CRHB) equations for the fermions in the rotating frame (in one-dimensional cranking approximation) (Afanasjev et al. 2000) 

Δˆ hˆ D − λτ − Ωx Jˆx ∗ ∗ ∗ ˆ ˆ −Δ −hD + λτ + Ωx Jˆx



   Uk (r) Uk (r) = Ek , Vk (r) Vk (r)

(36)

where λτ (τ = p, n) are chemical potentials defined from the average particle number constraints for protons and neutrons; Uk (r) and Vk (r) are quasiparticle Dirac spinors; Ek denotes the quasiparticle energies; and Jˆx is the angular momentum component entering into the Coriolis term −Ωx Jx . Here, hˆ D is the Dirac Hamiltonian for the nucleon with mass m hˆ D = α(−i∇ − V (r)) + V0 (r) + β(m + S(r)),

(37)

which contains an attractive scalar potential S(r) S(r) = gσ σ (r),

(38)

a repulsive vector potential V0 (r) V0 (r) = gω ω0 (r) + gρ τ3 ρ0 (r) + e

1 − τ3 A0 (r), 2

(39)

1956

A. V. Afanasjev

and a magnetic potential V (r) V (r) = gω ω(r) + gρ τ3 ρ(r) + e

1 − τ3 A(r). 2

(40)

These equations are solved numerically in triaxial harmonic oscillator basis, and signature basis is used for the single-particle states. The CRHB framework is applicable to the description of both rotating nuclei in the paired regime and one-/two-quasiparticle configurations in rotating and non-rotating nuclei. An approximate particle number projection by means of the Lipkin-Nogami (LN) method is also used in it (Afanasjev et al. 2000), but its discussion is omitted for simplicity. The CRHB+LN calculations have been successful in the description of rotating nuclei in paired regime (Vretenar et al. 2005; Afanasjev et al. 2000; Afanasjev and Abdurazakov 2013) and one-quasiparticle configurations in odd-A nuclei (see discussion of Fig. 12). Moreover, the CRHB framework can be either reduced to unpaired regime, leading to so-called cranked relativistic mean field (CRMF) approach (Koepf and Ring 1989; Afanasjev et al. 1998), or upgraded to the two- and three-dimensional cranking approximation (Meng et al. 2013). By putting Ωx = 0, it can also be applied to the description of non-rotating nuclei, but more specialized versions of the RHB computer codes (without cranking and signature basis) designed for triaxial, axially symmetric, and spherical shapes also exist (Nikši´c et al. 2014). It turns out that without any assumptions on the form of the single-particle potential, self-consistent calculations generate single-particle properties (energies, alignments) which in general are similar to those obtained in phenomenological potentials. This is illustrated in Fig. 9 which compares the single-particle spectra of the superheavy 292 120172 nucleus obtained with phenomenological folded Yukawa (FY) potential; non-relativistic Skyrme functionals SkP, SkM*, SLy6, SLy7, SkI1, SkI4, and Sk3; and CEDFs NL3, NL-Z, NL-Z2, and NL-VT1. The Nilsson diagrams obtained in the RHB calculations with the CEDF NL3* (see Fig. 2) show a lot of similarities with those obtained in the Woods-Saxon potential (see, e.g., Figs. 3 and 4 in Chasman et al. 1977). The behavior of the single-particle states in the rotating potential obtained in the Woods-Saxon potential (see Fig. 3) is very similar to that obtained in the CRMF calculations (see Fig. 4 in Afanasjev et al. 1998). However, there is a principal difference in the physical mechanisms related to the singleparticle degrees of freedom between two types of approaches, and it is related to time-odd mean fields (Dobaczewski and Dudek 1995; Afanasjev and Ring 2000a). They are absent in phenomenological potentials but are present in self-consistent models. For example, in the CRHB framework, they are related to the terms which break time-reversal symmetry, namely, a magnetic potential V (r) (see Eq. (40)) and the Coriolis term −Ωx Jx . In the DFT approaches, time-odd mean fields modify single-particle energies (Afanasjev and Abusara 2010; Schunck et al. 2010) and single-particle alignments (Afanasjev and Ring 2000a) and have large impact on rotational properties of nuclei (Dobaczewski and Dudek 1995; Afanasjev and Ring 2000a).

51 Model for Independent Particle Motion

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Manifestation of Independent Particle Motion in Non-rotating and Rotating Nuclei When considering the manifestations of independent particle motion, one should separate the effects which emerge from the shell structure (as a coherent effect of motion of many particles) and those from individual motion of the particles. The configuration-mixing interactions (which are accumulated in residual interaction term of the Hamiltonian given by Eq. (4)) such as pairing and the coupling to the low-lying collective vibrational degrees of freedom act destructively on the individual properties of the single-particle orbitals. In contrast, the global effects emerging from the underlying shell structure are not that much affected by these residual interactions. Thus, a global shell structure at spin zero, the superdeformation at high spin, and the existence of superheavy nuclei are considered as the examples of such effects. Then the physical properties which sensitively depend on the single-particle features of individual orbitals are discussed. First, the energies of experimental and calculated one-quasiparticle states in non-rotating deformed nuclei are compared. Then, the analysis is extended to rotating nuclei at low and medium spins. The rotation acts as a tool to significantly reduce the role of pairing interaction (Shimizu et al. 1989; Nazarewicz et al. 1989; Afanasjev and Frauendorf 2005b). Thus, nuclear systems at very high spins in which the impact of pairing is negligible are considered, and the single-particle and polarization effects due to the occupation of specific orbitals are analyzed. In addition, the phenomenon of band termination is discussed as an example of the competition of the collective and single-particle degrees of freedom.

Global Shell Structure at Spin Zero The state-of-the-art view on the nuclear landscape is shown in Fig. 7. It is based on the results of the calculations presented in Agbemava and Afanasjev (2021) and Agbemava et al. (2014, 2019). The bands (shown by gray color) of spherical nuclei along proton and neutron numbers 8, 20, 28, 50, and 82 (as well as for neutron number N = 126) are seen in this figure. They are due to large spherical shell gaps at these particle numbers which provide an extra stability of the nuclei. Note that these bands are more pronounced in neutron subsystem. Outside of these bands of spherical nuclei, the ground states of the Z ≤ 120 nuclei are either oblate or prolate being in typical range of quadrupole deformations |β2 | < 0.4. Similar structures and features are also obtained in non-relativistic calculations of Möller et al. (2016), Delaroche et al. (2010), and Erler et al. (2012), but they are restricted to the nuclei below Z ≈ 120. In general, existing experimental data confirms these model predictions (Sorlin and Porquet 2008), but there may be some differences between specific model and experiment. With increasing proton number, these classical features disappear, and only toroidal shapes are calculated as the lowest in energy. This region (shown in white

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Proton number Z

200

DD-PC1

160 120 N=258

Z=82

80

N=184

Z=50

40

N=126 50

0 0

40

N=82

80

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8

120 160 200 240 280 320 360 400 440

Neutron number N Fig. 7 Nuclear landscape at spin zero as obtained in the RHB calculations with the DD-PC1 functional. For proton numbers Z ≤ 130, the ground states have ellipsoidal shapes. These nuclei are shown by the squares, the color of which indicates the equilibrium quadrupole deformation β2 (see colormap). Toroidal shapes dominate nuclear landscape for higher proton numbers (white region located between two-proton and two-neutron drip lines for toroidal nuclei shown by solid black lines). (Figure taken from Agbemava and Afanasjev 2021)

color between two black lines in Fig. 7) is penetrated only by three islands (shown in gray color) of potentially stable spherical hyperheavy nuclei. Their existence is due to substantial proton Z = 154 and 186 and neutron N = 228, 308, and 406 spherical shell gaps (Agbemava and Afanasjev 2021) and substantial fission barriers around spherical minima (Agbemava and Afanasjev 2021; Agbemava et al. 2019). However, these states are highly excited with respect to minima corresponding to toroidal shapes. They could become the ground states if relevant toroidal minima are unstable with respect to so-called sausage deformations (Agbemava and Afanasjev 2021; Wong 1973). Thus, the richness of nuclear structure seen in experimentally known part of nuclear landscape is replaced by a more uniform structure of the nuclear landscape in the region of hyperheavy (Z ≥ 126) nuclei dominated by toroidal nuclei (Agbemava and Afanasjev 2021; Agbemava et al. 2019). This transition from compact ellipsoidal-like shapes to non-compact toroidal shapes is driven by the enhancement of the role of Coulomb interaction with increasing Z (Agbemava et al. 2019). Thus, the former shapes become either unstable against fission or energetically unfavored in hyperheavy nuclei. Single-particle degrees of freedom play also an important role in the definition of the boundaries of nuclear landscape. For example, the position of two-neutron drip line of ellipsoidal nuclei depends sensitively on the single-particle energies of high-j states located in the vicinity of neutron continuum threshold (Afanasjev et al. 2015). The transition from ellipsoidal to toroidal shapes drastically modifies the underlying single-particle structure and as a result lowers the energy of the Fermi level for protons (Agbemava and Afanasjev 2021). As a consequence, a substantial

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shift of the two-proton drip line from its expected position for ellipsoidal shapes (shown by dashed orange line in Fig. 7) toward more proton-rich nuclei for toroidal shapes (shown by solid line in Fig. 7) takes place (Agbemava and Afanasjev 2021).

Superheavy Nuclei With increasing proton number beyond Z = 100, the fission barriers provided by the liquid drop become rather small, and then for higher Z, they disappear (see Fig. 10.6 in Nilsson and Ragnarsson 1995). It turns out that the existence of the heaviest nuclei with Z > 104 is primarily determined by shell effects due to the quantummechanical motion of protons and neutrons inside the nucleus (Sobiczewski et al. 1966; Meldner 1967; Oganessian and Utyonkov 2015; Giuliani et al. 2019; Adamian et al. 2021). In these nuclei, the heights of fission barriers are entirely determined by the shell corrections, and they would not exist without shell effects. These effects also play a central role for the production, stability, and spectroscopy of superheavy nuclei. Although state-of-the-art theoretical models provide a reasonable description of many aspects of the physics of superheavy nuclei, they face substantial challenges in the prediction of the location of next spherical shell closures (Bender et al. 1999; Sobiczewski and Pomorski 2007; Oganessian and Utyonkov 2015; Giuliani et al. 2019; Adamian et al. 2021; Agbemava et al. 2015). These challenges are illustrated in Fig. 8 which shows the map of calculated ground state quadrupole deformations obtained with two covariant energy density functionals. The predictions of these two functionals are drastically different in the vicinity of the Z = 120 and N = 184 lines. The PC-PK1 functional predicts wide bands of spherical nuclei in the (Z, N ) chart along Z = 120 and N = 184. In contrast, in the calculations with DD-PC1, the band along Z = 120 does not exist, and narrower band along N = 184 is seen only for Z ≤ 114. These discrepancies are due to modest differences in the singleparticle properties of these functionals and related differences in the competition of shell effects at spherical and deformed shapes (Agbemava et al. 2015). Note that the

Fig. 8 Charge quadrupole deformations β2 of even-even superheavy nuclei obtained in the RHB calculations with indicated functionals. Experimentally known nuclei are indicated by white circles. (Figure taken from Agbemava et al. 2015)

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Fig. 9 Single-particle energies of proton (top) and neutron (bottom) states in the 292 120172 nucleus obtained at spherical shape with indicated non-relativistic and relativistic functionals. (Figure taken from Bender et al. 1999)

width of the band of spherical nuclei in the (Z, N ) chart along a specific particle number corresponding to a shell closure indicates the impact of this shell closure on the structure of neighboring nuclei. There is a wide variety of the predictions for proton and neutron spherical shell closures in superheavy nuclei obtained in different models. These are proton numbers at Z = 114, 120, and 126 and neutron numbers at N = 172 and 184 (see Fig. 9 and Bender et al. 1999, Sobiczewski and Pomorski 2007, Giuliani et al. 2019, and Agbemava et al. 2015). However, in the CDFT, the N = 172 could be eliminated as a potential candidate when the deformation effects are taken into account (Agbemava et al. 2015), and the Z = 120 nuclei become deformed in

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a number of functionals when both deformation and correlations beyond mean field effects are considered (Shi et al. 2019). The challenges the models face in the prediction of spherical shell closures are due to two factors. At present, selfconsistent models do not provide a spectroscopic quality of the description of the single-particle spectra (Dobaczewski et al. 2015). In contrast, the phenomenological potentials better describe single-particle spectra in known nuclei (see Sobiczewski and Pomorski (2007) and references therein), but they fail to incorporate selfconsistency effects (such as a depletion of the density in central region of nucleus) which are important in superheavy nuclei (see discussion of Fig. 6). Figure 8 illustrates that only for a few experimentally known Z = 116 and 118 nuclei, the CDFT predictions differ. In general, with possible exception of these high-Z isotopic chains, existing experimental data on superheavy nuclei are described with comparable level of accuracy by all existing models (Giuliani et al. 2019; Adamian et al. 2021; Agbemava et al. 2015). This is undeniable success of independent particle model, the global consequences of which (shell structure) led to the predictions of superheavy nuclei in the 1960s within relatively simple models (Sobiczewski et al. 1966; Meldner 1967). Experimental data collected over these years fully confirmed these predictions, and some differences in the predictions of next spherical shell closures are secondary to this success. This field of research (both experimental and theoretical) remains extremely active which is illustrated by recent reviews (Oganessian and Utyonkov 2015; Giuliani et al. 2019; Adamian et al. 2021). However, at present, available experimental data does not allow to give a preference to the predictions of one or another model and to decide where next (if any) proton and neutron spherical shell closures are located or whether there is an island of spherical superheavy nuclei.

Superdeformation at High Spin Figure 7 shows that the ground states of the nuclei in experimentally known part of the nuclear chart are characterized by quadrupole deformation β2 which is located typically in the range −0.35 ≤ β2 ≤ +0.35. Strutinsky has predicted excited minimum with β2 ∼ 0.6 in the second (superdeformed, SD) potential well of potential energy curves of deformed nuclei in Strutinsky (1967). At low spin, such minima are located at high excitation energies, but they can be brought down to the yrast line by fast rotation since it favors extremely deformed shapes. This was confirmed later in the mic+mac calculations of Andersson et al. (1976) which predicted the doubly magic SD band in 152 Dy with 2:1 semi-axis ratio. The existence of the SD bands is due to the shell effects associated with large proton and neutron SD shell gaps. For example, in the A ∼ 150 region of superdeformation, these shell effects are produced by large proton Z = 66 and neutron N = 86 SD shell gaps which exist both in phenomenological potentials (see Fig. 3 in this paper and Nazarewicz et al. 1989; Ragnarsson 1993) and in the DFT calculations (Afanasjev et al. 1998).

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The phenomenon of superdeformation at high spin has been confirmed experimentally by the observation of the first SD band in 152 Dy in 1986 (Twin et al. 1986). From that time, a significant amount of experimental data on the SD bands in different mass regions has been collected (Singh et al. 2002). The mic+mac and DFT-based cranked approaches rather well describe experimental observables such as dynamic J (2) and kinematic J (1) moments of inertia and transitional quadrupole moments Qt of these bands (see Afanasjev et al. (2000), Afanasjev et al. (1998), Nazarewicz et al. (1989), Ragnarsson (1993) and references quoted therein). The calculations also reveal a substantial dependence of these physical observables on the occupation of high-N intruder orbitals; here, N stands for the principal quantum number of dominant component of the wave function. For example, they strongly depend on the number of occupied N = 6 protons and N = 7 neutrons in the A ∼ 150 region of superdeformation. This region is also of special interest since pairing is negligible in the majority of the SD bands. As a consequence, it provides one of the best examples of independent particle motion in nuclear physics (see detailed discussion in the last section).

The Phenomenon of Band Termination One of the clear manifestations of the independent particle motion is the phenomenon of band termination (Afanasjev et al. 1999; Häusser et al. 1972; Watt et al. 1980). It definitely reveals the fact that each single-particle orbital occupied in nucleonic configuration possesses a limited and state-dependent angular momentum. Of special interest are so-called smooth terminating bands which show a continuous transition from high collectivity at low and medium spin values to a pure particle-hole (terminating) state at the maximum spin which can be built within the configuration (Afanasjev et al. 1999; Ragnarsson et al. 1995). Note that this feature of finite multi-fermion systems has so far only been observed and studied in atomic nuclei. In the terminating state, the symmetry axis coincides with the rotation axis. Since collective rotation along the symmetry axis is forbidden in quantum mechanics, no further angular momentum can be brought into the system with the same occupation of single-particle orbitals, and thus this state represents the termination of the rotational band. The phenomenon of band termination has been observed in several regions of nuclear chart (see review in Afanasjev et al. 1999). However, the best examples of smooth terminating bands are seen in the A ≈ 110 region in which the nuclei have several valence particles and holes outside the 100 Sn core. A classical example of terminating bands in 109 Sb (Afanasjev et al. 1999; Ragnarsson et al. 1995) is considered here in order to illustrate the major features of smooth band termination. A critical feature of these bands is the fact that their dynamic moments of inertia J (2) gradually decrease with increasing rotational frequency to unusually low values (near 1/3 of rigid-body value) at the highest observed frequencies. This is definite indication of significant suppression of pairing correlations (Afanasjev et al. 1999)

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0

2

L 1

3

E− 0.013 I(I+1) [MeV]

−1

109

−2

Sb expt.

0

[21,3]

+/−

87/2 83/2 [21,2]

−1

− +

89/2

109

−2 10

+

−

Sb theory 15

20

25

30

35

40

45

50

−

I [h ]

Fig. 10 The comparison of experimental and calculated E − ERLD curves for the bands 1 − 3 in 109 Sb. The energies have been normalized in such a way that calculated and experimental curves for the band 1 have the same value at the minimum of the E − ERLD curve. The energies of the unlinked experimental bands 2 and 3 are chosen so that their E − ERLD minima have the same energies as theoretical counterparts. Terminating states are indicated by large open circles, and their spins are displayed. Solid and dashed lines are used for the configurations with π = + and π = −, respectively. The states with signature αtot = +1/2 and αtot = −1/2 are shown by solid an open symbols, respectively. (Figure taken from Afanasjev et al. 1999)

making this type of rotational structures as one of the best examples of independent particle motion. Figure 10 compares the experimental and calculated energies E of rotational bands with respect to rigid rotor reference ERLD = AI (I + 1), where A is the moment of inertia parameter. The calculations are performed in configuration-dependent cranked Nilsson-Strutinsky (CNS) approach (Afanasjev et al. 1999; Bengtsson and Ragnarsson 1985). The configurations relative to the 100 Sn core are labeled using the shorthand notation [p1 p2 , n]αtot ≡ [π(g9/2 )−p1 (h11/2 )p2 (g7/2 d5/2 )Z−50+p1 −p2 ⊗ ν(h11/2 )n (g7/2 d5/2 )N −50−n ]αtot , where αtot is the total signature of the configuration (only sign is shown) and p1 , p2 , and n are the numbers of proton holes in the g9/2 orbitals, of protons in the h11/2 orbitals, and of neutrons in the h11/2 orbitals, respectively. One can see that the CNS calculations without pairing very well reproduce experimental data. Some discrepancies seen at low I ≤ 20h¯ spin are due to the neglect of pairing.

1964

A. V. Afanasjev 4.0 109

Sb − band 1

3.5

109

60°

50°

40°

−

[21,2]

3.0

Sb

Qt (eb)

0.2

30°

2.5 2.0

[21,3]− 39.5+

0.1

37.5− [21,2]−

20°

40.5+ [21,3]+

1.5

γ

1.0

16

20

24

28 − I (h )

32

36

40

10° 21.5+ 19.5− 22.5+

0.0

0.1

ε2

0.2

0°

0.3

Fig. 11 (left panel) Calculated deformation paths in the (ε2 , γ ) plane of the configurations assigned to smooth terminating bands 1–3 in 109 Sb. The spins of some states are shown. (right panel) The comparison of experimental and calculated transition quadrupole moments Qt of the band 1 and assigned theoretical configuration. (Figure taken from Afanasjev et al. 1999)

The CNS calculations suggest that rotational bands of interest are near prolate at low spin and thus they involve collective rotation about an axis perpendicular to the symmetry axis. With increasing spin, the valence nucleons gradually align their spin vectors with the axis of rotation via the Coriolis interaction. This causes the nuclear shape to gradually trace a path through the triaxial (γ ) plane, toward the non-collective oblate shape at γ = +60◦ (see left panel of Fig. 11). After the available spin is exhausted, consistent with the Pauli principle, the band terminates. This gradual change from collective near-prolate (γ ∼ 0◦ ) to non-collective oblate (γ = +60◦ ) shape leads to a gradual decrease of transition quadrupole moment Qt which agrees with available experimental data (see right panel in Fig. 11). The termination spins, which depend on the configuration, are well-defined property for terminating bands, and they confirm the termination process. For example, the − state in 109 Sb detailed structure of the [21, 2]− (band 1) terminating I = 83 2 6 2 1 2 is π(g9/2 )−2 8 (g7/2 d5/2 )6 (h11/2 )5.5 ⊗ ν(g7/2 d5/2 )12 (h11/2 )10 . Note that fully selfconsistent cranked relativistic mean field calculations confirm this physical picture (Vretenar et al. 2005). However, due to technical reasons, it is difficult to follow smooth terminating bands up to their terminating states in such calculations.

Single-Particle States in Deformed Nuclei Non-rotating Nuclei The detailed comparison of experimental and theoretical information on the properties of specific individual states can shed additional light on the validity in independent particle motion in atomic nuclei. In non-rotating nuclei, such infor-

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NL1

NL3*

D1M

D1S

UNEDF2

Cf

1

11/2[606]

9/2[604]

7/2[514] 1/2[521]

9/2[615] 1/2[7] 11/2[725] 3/2[622] 7/2[613] 1/2[620]

7/2[633] 3/2[521] 1/2[400] 5/2[642] 1/2[550]

9/2[734] 5/2[622]

3/2[651]

-1

7/2[624]

5/2[523]

7/2[743]

7/2[743]

11/2[505] 1/2[6]

1/2[6] 1/2[550]

-2

5/2[613] 9/2[604]

251

Bk

One-quasiparticle energies [MeV]

SLy4

5/2[613]

1/2[651]

249

0

exp.

NL3*

NL1

D1S

D1M

UNEDF2

2

SLy4

exp.

mation is provided by the energies of the single-particle states, by their densities, and by the transitions between the single-particle states. However, such densities are not accessible experimentally, and calculated transition probabilities are prone to substantial theoretical errors. One could imagine that spherical nuclei would provide the cleanest and simplest set of data on the single-particle properties. It turns out that this is not a case because of substantial residual interaction due to (quasi)particlevibration (quasiparticle-phonon) coupling. As a consequence, the wave functions of the states in the spherical nuclei are not of pure single-particle nature since they are substantially polluted by the vibrational admixtures (Mahaux et al. 1985; Afanasjev and Litvinova 2015; Cao et al. 2014). The impact of the quasiparticle-vibration coupling is reduced in deformed nuclei since the part of such correlations is accounted at the level of deformed mean field. Indeed, the admixtures of the phonons to the structure of the ground and low-lying states in deformed rare-earth and actinide odd-mass nuclei are relatively small (especially, when compared with spherical open shell nuclei (Afanasjev and Litvinova 2015)) according to the quasiparticle-phonon model (Alikov et al. 1988; Shirikova et al. 2015). Thus, deformed nuclei can provide a better and cleaner example of independent particle motion. Figure 12 compares experimental and calculated energies of one-quasiparticle qp states Ei in odd-A 249 Bk and 251 Cf nuclei. The results of the calculations with

1/2[631]

1/2[631]

Fig. 12 Experimental and calculated quasiparticle spectra in 249 Bk and 251 Cf. Solid and dashed lines are used for positive- and negative-parity states, respectively. The states are labeled by the Nilsson label of the dominant component of the wave function when the squared amplitude of this component exceeds 50%. Otherwise, they are labeled by Ω[N ] where N is the principal quantum number of the components of wave function whose cumulative contribution into the wave function is dominant. (Figure taken from Dobaczewski et al. 2015)

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non-relativistic Skyrme (SLy4, UNEDF2), Gogny (D1S, D1M), and covariant (NL1, NL3*) energy density functionals are shown in this figure. These are fully selfconsistent calculations which means that the total energies Ei of the nucleonic configurations with blocked i-th single-particle state of interest are obtained within Hartree-Fock-Bogoliubov or RHB frameworks (Ring and Schuck 1980). Then the ground state is associated with nucleonic configuration which has the lowest energy qp qp Elowest , and the energies of the excited states Ei are defined as Ei = Ei −Elowest . qp qp Note that in Fig. 12, the particle and hole states are plotted at Ei and −Ei , respectively. This is done in order to facilitate the comparison of these selfconsistent results with the Nilsson diagrams. One can see that the energies of some deformed one-quasiparticle states are rather well described in specific functionals, but others deviate appreciably from the experiment. The detailed comparison of experimental data with calculations is presented in Dobaczewski et al. (2015). Considering an average level of the accuracy of the description of experimental data, it is difficult to give a clear preference to one or another functional. At present, a systematic analysis of the accuracy of the reproduction of the single-particle spectra in deformed nuclei is available only in the RHB framework (Afanasjev and Shawaqfeh 2011), and smaller in scope (only for the Rb, Yb, and Nb isotopic chains) analysis is carried out in the HFB calculations with Gogny forces (see Rodriguez-Guzman et al. (2011) and references therein). These investigations reveal two sources of inaccuracies in the description of the energies of the single-particle states, namely, low effective mass leading to a stretching of the energy scale of the calculated results as compared with experimental ones and incorrect relative positions of some single-particle states (Dobaczewski et al. 2015; Afanasjev and Shawaqfeh 2011). On the absolute scale, these deficiencies are not large especially considering the fact that no (or very limited) information on single-particle degrees of freedom has been taken into account in the fitting protocols of covariant (non-relativistic) energy density functionals. The accounting of quasiparticle-vibration coupling (QVC) improves the agreement with experiment by both improving the description of the energies of individual states and increasing the density of the single-particle states in the vicinity of the Fermi level. This was illustrated for the 251 Cf nucleus in the RHB+QVC framework in Zhang et al. (2022). In general, phenomenological potentials provide better description of the energies of the single-particle states in deformed odd-A nuclei because they are fitted to this type of experimental data (Cwiok et al. 1987; Bengtsson et al. 1989; Bengtsson and Ragnarsson 1985). However, such accuracy is obtained at the cost of neglect of self-consistency effects (Afanasjev and Shawaqfeh 2011) and effective incorporation of the effects of particle-vibration coupling (Zhang et al. 2022).

Rotating Nuclei in the Pairing Regime As discussed earlier, the rotation of the nuclei can generate substantial modifications of the single-particle energies and provides single-particle alignments jx i as a new measure of the single-particle properties. This leads to a new and very robust probe of the single-particle structure since some pairs of the orbitals emerging

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Fig. 13 Experimental (left panel) and calculated (right panel) Routhians relative to the g-band reference Routhian Eref as a function of rotational frequency h¯ ω1 . Note that the data are restricted to the simplest configurations since two-quasiproton and several three-quasineutron configurations are left away. The convention for (π, α) is the following: open and solid symbols are used for π = + and π = −, respectively, and the α = +1/2 and α = −1/2 states are shown by triangles up and triangles down, respectively. (Figures taken from Frauendorf 2018)

from a given non-rotating state show significant signature splitting (see, e.g., the [770]1/2(α = ±1/2) and [761]3/2(α = ±1/2) pairs of the orbitals in Fig. 3), while other pairs (such as [532]5/2(α = ±1/2) and [633]7/2(α = ±1/2) in Fig. 3) show no signature splitting. These features can be used for a reliable interpretation of experimental data. This is demonstrated in Fig. 13 which compares experimental Routhians with calculated ones for the selected set of rotational bands in odd-neutron 163 Er nucleus. The excitation energies are taken relative to a reference configuration which is the ground state (g-) configuration in even-even nucleus. Experimental Routhian is well approximated by the Harris expression Eref =

ω2 ω4 h¯ 2 Θ0 + Θ1 + 2 4 8Θ0

(41)

with Θ0 = 32h¯ 2 MeV−1 and Θ1 = 32h¯ 4 MeV−3 extracted from the ground state band in 164 Er. The calculations are performed in the cranked shell model (CSM) (Bengtsson and Frauendorf 1979) which employs fixed mean field parameters for the Nilsson potential and pairing. The parameters of its Eref reference are adjusted to the calculated Routhian of the g-configuration. The quasiparticle orbitals, on which rotational bands are build, are labeled by the letters of the alphabet A, B, C , and D (for high-j intruder states) and E, F, G, etc. (for normal parity states) (see Frauendorf (2018) for details of this labeling convention). One can see that this relatively simple model can describe rather well experimentally observed features such as (i) large signature splitting in the pair of bands A/B; (ii) the lack or small amount of signature splitting in the pairs of bands E/F, X/Y, and G/H; (iii) the presence of paired band crossing at h¯ ω1 ≈ 0.25 MeV in the E/F, G/H, and C bands which reflects itself in the change of the slope of the Routhians as a function of rotational frequency; and

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(iv) the absence of paired band crossing in the A/B and X/Y pairs. The latter is due to the fact that the occupation of either of these orbitals blocks paired band crossing in neutron subsystem. Over the years, huge amount of experimental data on rotating nuclei has been accumulated, and more sophisticated theoretical tools have been developed and successfully applied for the description of rotating nuclei in both paired and unpaired regimes. They go beyond the basic assumption of the CSM on the constancy of the mean and pairing fields and take into account their configuration dependence either on the level of mic+mac model or on a fully self-consistent level. These include cranked Nilsson-Strutinsky approach (Afanasjev et al. 1999; Carlsson et al. 2008), cranked approaches based on the Skyrme and Gogny DFTs (Molique et al. 2000; Egido and Robledo 1994; Duguet et al. 2001) and CDFT (Afanasjev and Abdurazakov 2013; Koepf and Ring 1989; Zhang et al. 2020), and others.

Single-Particle and Polarization Effects Due to the Occupation of Single-Particle Orbitals The addition or removal of particle(s) to the nucleonic configuration modifies the total physical observables. But it also creates the polarization effects on the physical properties (in both time-even and time-odd channels) of initial configuration. The comparison of relative properties of two configurations can shed important light both on the impact of the added/removed particle(s) in specific orbital(s) on physical observable of interest and on the related polarization effects. In addition, the comparison with experimental data on such properties can provide a measure of the accuracy of the description of single-particle properties in model calculations. Let us consider an example of rotational bands in the regime of weak pairing. The relative properties of different physical observables such as relative charge quadrupole moments ΔQ(ω) = QB (ω) − QA (ω),

(42)

and relative effective alignments (Ragnarsson 1993) B,A (ω) = IB (ω) − IA (ω), ieff

(43)

are defined as the differences between either charge quadrupole moments Q or the spins I of the bands A and B at the same rotational frequency ω with the band A being a reference band. They provide sensitive probes of the single-particle motion and polarization effects induced by the occupation of specific single-particle orbitals. In addition, the experimental values of relative charge quadrupole moments are to a large degree free from the uncertainties due to stopping powers which impact the absolute values of these moments. The ΔQ(ω) probes the differences in time-even mean fields generated by the addition or removal of particle(s) from the configuration of the reference band (Afanasjev et al. 1998; Satuła et al. 1996).

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Table 1 Experimental and calculated relative charge quadrupole moments ΔQ = Q(Band) − Q(152 Dy(1)) of the 149 Gd(1), 151 Tb(1), and 151 Dy(1) superdeformed bands. The detailed structure 152 of the configurations of these bands relative  to the doubly magic Dy superdeformed core is given in column 2. Column 5 shows the sum i ΔQi of the “independent” contributions ΔQi of i-th particle to the charge quadrupole moment calculated at rotational frequency ω = 0.50 MeV. Note that the values in columns 3 and 4 are averaged over the observed spin range. Based on the results of the CRMF calculations with the NL1 functionals and non-relativistic Skyrme DFT calculations with SkP and SkM* functionals presented in Afanasjev et al. (1998) and Satuła et al. (1996)  i Band Configuration ΔQexp (eb) ΔQth (eb) i ΔQ 1 2 3 4 5 149 Gd(1) ν[770]1/2(r = −i)−1 (π [651]3/2)−2 −2.5(0.3) −2.41 [NL1] −2.44 [NL1] −2.42 [SkP] −2.32 [SkM*] 151 Tb(1) π [651]3/2(r = +i)−1 −0.7(0.7) −1.01 [NL1] −0.99 [NL1] −0.96 [SkP] −0.96 [SkM*] 151 Dy(1) ν[770]1/2(r = −i)−1 −0.6(0.4) −0.53 [NL1] −0.55 [NL1] −0.57 [SkP] −0.48 [SkM*]

The effective alignment ieff depends both on the alignment properties of singleparticle orbital(s) by which the two bands differ and on the polarization effects (in both time-even and time-odd channels of the DFTs) induced by the particles in these orbitals (Afanasjev et al. 1998). It can also be used to investigate experimentally the structure of the underlying single-particle orbitals in the configurations of interest (Afanasjev et al. 1998; Ragnarsson 1993). Note that additivity principle for the single-particle observables has been tested for the first time on the example of effective alignments of superdeformed bands in the A ∼ 150 mass region (Ragnarsson 1993). The comparison of experimental and calculated ΔQ(ω) values for a selected set of superdeformed bands in the nuclei near 152 Dy is presented in Table 1 (for a more extensive set of data, see Table 2 in Satuła et al. 1996). One can see that these quantities (column 3) are well described in model calculations (columns 4 or 5). In addition, the additivity rule of quadrupole moments (Satuła et al. 1996) which states that relative quadrupole moments of two configurations ΔQ(ω) can be well approximated by the sum of individual contributions ΔQi of i-th particles to the charge quadrupole moment defined with respect to core SD configuration in 152 Dy ΔQ(ω) ≈



ΔQi

(44)

i

is rather well fulfilled in both relativistic and non-relativistic DFT calculations (compare columns 4 and 5 in Table 1 and see Satuła et al. 1996). This means that the polarization effects due to individual particles or holes are largely independent.

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CRMF - solid lines CNS - dashed lines

Exp. data - unlinked circles

3.0

(a)

1.0 149

2.0

Gd(1)/

150

(b)

Tb(1)

π[651]3/2(r = + ι)

0.0 151

1.0

Tb(1)/152Dy(1)

-1.0 0.0 π[651]3/2(r = – ι)

-1.0

Effective alignment ιeff

3.0

-2.0 2.0

150

Tb(1)/

151

152

Tb(1) 1.0

Dy(1)/153Dy(1)

(d)

2.0

(c)

ν[761]3/2(r = + ι)

0.0

1.0 0.0 ν[770]1/2(r = – ι)

-1.0

0.4

π[301]1/2

0.2

r= – ι

(e)

0.2

ν[402]5/2(r = ± ι)

(f)

r=+ι

-0.4 -0.6

0.0 -0.2

-0.2

Tb(4)/152Dy(1) r= – ι

0.0 0.4

151

152

Dy(1)/153Dy(2,3)

-0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Rotational frequency Ω (MeV)

-0.8 -1.0

151

Tb(2)/152Dy(1)

r=+ι

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Rotational frequency Ω (MeV)

Fig. 14 Effective alignments, ieff (in units h), ¯ extracted from experimental data are compared with those obtained in the CRMF and CNS calculations of Afanasjev et al. (1998) and Ragnarsson (1993). The experimental effective alignment between bands A and B is indicated as “A/B.” The band A in the lighter nucleus is taken as a reference, so the effective alignment measures the effect of the additional particle. The calculated configurations differ in the occupation of the orbitals indicated by the quantum numbers in the panels. (Figure taken from Afanasjev and Ring 2000b)

Experimental and calculated ieff values for single-particle orbitals active in the vicinity of the SD shell closures in the A ≈ 150 region of superdeformation are compared in Fig. 14. The calculations are carried out within the cranked relativistic mean field (Afanasjev et al. 1998) and the cranked Nilsson-Strutinsky (Ragnarsson 1993) approaches. The CRMF calculations reproduce in average the experimental ieff values better than the CNS approach. This indicates that alignment properties

51 Model for Independent Particle Motion

1971

of single-particle orbitals and their polarization effects are correctly accounted for in the CRMF approach. The discrepancy between the CRMF calculations and experiment seen in Fig. 14a at Ω ≤ 0.5 MeV is most likely due to the fact that pairing correlations play some role at low rotational frequency in the 149 Gd(1) band. It is necessary to point on principal differences in the description of these relative observables in the DFT- and mic+mac-based approaches. The relative quadrupole moments are self-consistently described in the DFT-based approaches (Afanasjev et al. 1998; Satuła et al. 1996; Matev et al. 2007). In contrast, the effective single-particle quadrupole moments in the mic+mac method are not uniquely defined because of the lack of self-consistency between the microscopic and macroscopic contributions (Karlsson et al. 1998). In the mic+mac models (e.g., CNS), polarization effects caused by time-odd fields are neglected, and therefore ieff is predominantly defined by the alignment properties of the active single-particle orbitals (Ragnarsson 1993). On the contrary, they play an important role in the DFT models (Dobaczewski and Dudek 1995; Afanasjev and Ring 2000a). Similar studies of relative properties of rotational bands have been performed in different regions of nuclear chart (Afanasjev and Frauendorf 2005b; Matev et al. 2007; Karlsson et al. 1998)). For example, a statistical analysis of significant number of rotational configurations in the A ∼ 130 region of superdeformation confirms the additivity principle for quadrupole moments and effective alignments (Matev et al. 2007). This justifies the use of an extreme single-particle model in an unpaired regime typical of high angular momentum. Note that the basic idea behind the additivity principle for one-body operators is rooted in the independent particle model.

Conclusions The concept of independent particle motion is the foundation of the absolute majority of the nuclear structure models. It leads to verifiable consequences such as shell structure and individual single-particle properties of nucleons. Both mic+mac and DFT models are used nowadays for the description of numerous aspects of low energy nuclear phenomena. The former allows significant flexibility (such as the calculation of large number of the nucleonic configurations in a single nucleus as it is done in the CNS approach (Afanasjev et al. 1999)) but lacks selfconsistency. This neglect of self-consistency limits the applicability of the mic+mac models to ellipsoidal nuclei with similar density distributions in proton and neutron subsystems. In contrast, the DFT models are fully self-consistent which makes them applicable to very exotic nuclear shapes including clustered, “bubble,” and toroidal ones. However, the calculation of many nucleonic configurations in a given nucleus involving one or several blocked single-particle states still remains a challenge especially in the formalism which includes pairing. At present, the use of these two theoretical frameworks should be considered as complimentary. The validity of independent particle model has been confirmed by experimental observation of superheavy nuclei and the phenomenon of superdeformation at high

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spin; these observations were motivated by model predictions. They together with global structure of the nuclear landscape are the consequences of underlying shell structure, which emerges from independent particle motion of the nucleons in the nucleus. The energies and alignments of the single-particle states serve as a clear fingerprint of individual properties of the single-particle orbitals. Signature splitting properties are of particular value since either their large values or no splittings help in the identification of the single-particle states involved in the structure of observed experimental bands. The experimental features seen at extremely high spins where the pairing correlations are negligible provide the best [at the level of single-particle states] examples of independent particle motion; these are found in smooth terminating bands of the A ∼ 110 mass region and in superdeformed bands of the A ∼ 150 mass region. Acknowledgments This material is based upon work supported by the US Department of Energy, Office of Science, Office of Nuclear Physics, under Award No. DE-SC0013037. I would like to thank I. Debes and J. Dudek for the creation of improved quality Fig. 3. Useful discussions with I. Ragnarsson are greatly appreciated.

References G.G. Adamian, N.V. Antonenko, H. Lenske, L.A. Malov, S.-G. Zhou, Self-consistent methods for structure and production of heavy and superheavy nuclei. Eur. Phys. J A57, 89 (2021) A.V. Afanasjev, Cluster structures, ellipsoidal shapes and nuclear molecules in light A = 12 − 50 nuclei. EPJ Web Conf. 194, 06001 (2018) A.V. Afanasjev, O. Abdurazakov, Pairing and rotational properties of actinides and superheavy nuclei in covariant density functional theory. Phys. Rev. C 88, 014320 (2013) A.V. Afanasjev, H. Abusara, Time-odd mean fields in covariant density functional theory: nonrotating systems. Phys. Rev. C 81, 014309 (2010) A.V. Afanasjev, H. Abusara, From cluster structures to nuclear molecules: the role of nodal structure of the single-particle wave functions. Phys. Rev. C 97, 024329 (2018) A.V. Afanasjev, S. Frauendorf, Central depression in nuclear density and its consequences for the shell structure of superheavy nuclei. Phys. Rev. C 71, 024308 (2005a) A.V. Afanasjev, S. Frauendorf, Description of rotating N = Z nuclei in terms of isovector pairing. Phys. Rev. C 71, 064318 (2005b) A.V. Afanasjev, E. Litvinova, Impact of collective vibrations on quasiparticle states of open-shell odd-mass nuclei and possible interference with the tensor force. Phys. Rev. C 92, 044317 (2015) A.V. Afanasjev, P. Ring, Time-odd mean fields in the rotating frame: microscopic nature of nuclear magnetism. Phys. Rev. C 62, 031302(R) (2000a) A.V. Afanasjev, P. Ring, Superdeformations in relativistic and non-relativistic mean field theories. Phys. Scripta. T 88, 10 (2000b) A.V. Afanasjev, S. Shawaqfeh, Deformed one-quasiparticle states in covariant density functional theory. Phys. Lett. B 706, 177 (2011) A.V. Afanasjev, G. Lalazissis, P. Ring, Relativistic mean field theory in rotating frame: singleparticle properties at superdeformation. Nucl. Phys. A 634, 395 (1998) A.V. Afanasjev, D.B. Fossan, G.J. Lane, I. Ragnarsson, Termination of rotational bands: disappearance of quantum many-body collectivity. Phys. Rep. 322, 1 (1999) A.V. Afanasjev, P. Ring, J. König, Cranked relativistic Hartree-Bogoliubov theory: formalism and application to the superdeformed bands in the A ∼ 190 region. Nucl. Phys. A676, 196 (2000) A.V. Afanasjev, S.E. Agbemava, D. Ray, P. Ring, Neutron drip line: single-particle degrees of freedom and pairing properties as sources of theoretical uncertainties. Phys. Rev. C 91, 014324 (2015)

51 Model for Independent Particle Motion

1973

S.E. Agbemava, A.V. Afanasjev, Hyperheavy spherical and toroidal nuclei: the role of shell structure. Phys. Rev. C 103, 034323 (2021) S.E. Agbemava, A.V. Afanasjev, D. Ray, P. Ring, Global performance of covariant energy density functionals: ground state observables of even-even nuclei and the estimate of theoretical uncertainties. Phys. Rev. C 89, 054320 (2014) S.E. Agbemava, A.V. Afanasjev, T. Nakatsukasa, P. Ring, Covariant density functional theory: reexamining the structure of superheavy nuclei. Phys. Rev. C 92, 054310 (2015) S.E. Agbemava, A.V. Afanasjev, A. Taninah, A. Gyawali, Extension of the nuclear landscape to hyperheavy nuclei. Phys. Rev. C 99, 034316 (2019) B.A. Alikov, K.N. Badalov, V.O. Nesterenko, A.V. Sushkov, J. Wawryszczuk, On the role of the Coriolis and quasiparticle-phonon interactions in describing E1 transition probabilitues in odd Eu and Tb isotopes. Z. Phys. A 331, 265 (1988) G. Andersson, S.E. Larsson, G. Leander, P. Möller, S.G. Nilsson, I. Ragnarsson, S. Åberg, R. Bengtsson, J. Dudek, B. Nerlo-Pomorska, K. Pomorski, Z. Szyma´nski, Nuclear shell structure at very high angular momentum. Nucl. Phys. A 268, 205 (1976) M. Bender, K. Rutz, P.-G. Reinhard, J.A. Maruhn, W. Greiner, Shell structure of superheavy nuclei in self-consistent mean-field models. Phys. Rev. C 60, 034304 (1999) M. Bender, P.-H. Heenen, P.-G. Reinhard, Self-consistent mean-field models for nuclear structure. Rev. Mod. Phys. 75, 121 (2003) R. Bengtsson, S. Frauendorf, Quasiparticle spectra near the yrast line. Nucl. Phys. A 327, 139 (1979) T. Bengtsson, I. Ragnarsson, Rotational bands and particle-hole excitations at very high spin. Nucl. Phys. A 436, 14 (1985) R. Bengtsson, J. Dudek, W. Nazarewicz, P.Olanders, A systematic comparison between the Nilsson and Woods-Saxon deformed shell model potentials. Phys. Scripta 39, 196 (1989) J. Boguta, R. Bodmer, Relativistic calculation of nuclear matter and the nuclear surface. Nucl. Phys. A 292, 413 (1977) A. Bohr, B.R. Mottelson, Nuclear Structure, vol. 1 (W. A. Benjamin, Inc., New York, 1969) M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky, C.Y. Wong, Funny Hills: the shell-correction approach to nuclear shell effects and its application to the fission process. Rev. Mod. Phys. 44, 320 (1972) L.-G. Cao, G. Colò, H. Sagawa, P.F. Bortignon, Properties of single-particle states in a fully selfconsistent particle-vibration coupling approach. Phys. Rev. C 89, 044314 (2014) B.G. Carlsson, I. Ragnarsson, R. Bengtsson, E.O. Lieder, R.M. Lieder, A.A. Pasternak, Triaxial shape with rotation around the longest principal axis in 142 Gd. Phys. Rev. C 78, 034316 (2008) R.F. Casten, Nuclear Structure from a Simple Perspective (Oxford University Press, New York, 1990) R.R. Chasman, I. Ahmad, A.M. Friedman, J.R. Erskine, Survey of single-particle states in the mass region A > 228. Rev. Mod. Phys. 49, 833 (1977) S. Cwiok, J. Dudek, W. Nazarewicz, J. Skalski, T. Werner, Single-particle energies, wave functions, quadrupole moments and g-factors in an axially deformed Woods-Saxon potential with applications to the two-centre-type nuclear problems. Comput. Phys. Commun. 46, 379 (1987) J.-P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire, S. Peru, N. Pillet, G.F. Bertsch, Structure of even-even nuclei using a mapped collective Hamiltonian and the D1S Gogny interaction. Phys. Rev. C 81, 014303 (2010) M.J.A. de Voigt, J. Dudek, Z. Szyma´nski, High-spin phenomena in atomic nuclei. Rev. Mod. Phys. 55, 949 (1983) J. Dobaczewski, J. Dudek, Time-odd components in the mean field of rotating superdeformed nuclei. Phys. Rev. C 52, 1827 (1995) J. Dobaczewski, A.V. Afanasjev, M. Bender, L.M. Robledo, Y. Shi, Properties of nuclei in the nobelium region studied within the covariant, Skyrme, and Gogny energy density functionals. Nucl. Phys. A 944, 388 (2015) A. Dobrowolski, K. Pomorski, J. Bartel, Solving the eigenvalue problem of the nuclear Yukawafolded mean-field Hamiltonian. Comput. Phys. Commun. 199, 118 (2016)

1974

A. V. Afanasjev

T. Duguet, P. Bonche, P.-H. Heenen, Rotational properties of 252,253,254 No: influence of pairing correlations. Nucl. Phys. A 679, 427 (2001) J.-P. Ebran, E. Khan, R.-D. Lasseri, D. Vretenar, Single-particle spatial dispersion and clusters in nuclei. Phys. Rev. C 97, 061301(R) (2018) J.L. Egido, L.M. Robledo, High spin properties of the Er and Yb isotopes with the Gogny force. Nucl. Phys. A 570, 69 (1994) J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A.M. Perhac, M. Stoitsov, The limits of the nuclear landscape. Nature 486, 509 (2012) S. Frauendorf, Beyond the unified model. Phys. Scr. 93, 043003 (2018) Y.K. Gambhir, P. Ring, A. Thimet, Relativistic mean field theory for finite nuclei. Ann. Phys. (NY) 198, 132 (1990) F.A. Gareev, S.P. Ivanova, V.G. Soloviev, S.I. Fedotov, Single-particle energies and wave functions of the Saxon-Woods potentials and nonrotational states of odd-mass nuclei in the region 150 < A < 190. Phys. Elem. Part. Atom. Nuclei 4, part 2, pp. 357–455 (1973) S.A. Giuliani, Z. Matheson, W. Nazarewicz, E. Olsen, P.-G. Reinhard, J. Sadhukhan, B. Schuetrumpf, N. Schunck, P. Schwerdtfeger, Colloquium: superheavy elements: oganesson and beyond. Rev. Mod. Phys. 91, 011001 (2019) M. Grasso, L. Gaudefroy, E. Khan, T. Nikši´c, J. Piekarewicz, O. Sorlin, N. Van Giai, D. Vretenar, Nuclear “bubble” structure in 34 Si. Phys. Rev. C 79, 034318 (2009) O. Häusser, A.J. Ferguson, A.B. Mcdonald, I.M. Szöghy, T.K. Alexander, D.L. Disdier, High spin states in 20 Ne from 16 O(α, α) scattering. Nucl. Phys. A 179, 465 (1972) T. Ichikawa, J.A. Maruhn, N. Itagaki, S.Ohkubo, Linear chain structure of four-α clusters in 16 O. Phys. Rev. Lett. 107, 112501 (2011) D.R. Inglis, Particle derivation of nuclear rotation properties associated with a surface wave. Phys. Rev. 96, 1059 (1954) D.R. Inglis, Nuclear moments of inertia due to nucleon motion in a rotating well. Phys. Rev. 103, 1786 (1956) K. Karakatsanis, G.A. Lalazissis, P. Ring, E. Litvinova, Spin-orbit splittings of neutron states in N = 20 isotones from covariant density functionals and their extensions. Phys. Rev. C 95, 034318 (2017) L.B. Karlsson, I. Ragnarsson, S. Åberg, Polarization effects in superdeformed nuclei. Nucl. Phys. 639, 654 (1998) W. Koepf, P. Ring, A relativistic description of rotating nuclei: the yrast line of 20 Ne. Nucl. Phys. A 493, 61 (1989) G.A. Lalazissis, T. Nikši´c, D. Vretenar, P. Ring, New relativistic mean-field interaction with density-dependent meson-nucleon couplings. Phys. Rev. C 71, 024312 (2005) C. Mahaux, P.F. Bortignon, R.A. Broglia, C.H. Dasso, Dynamics of the shell model. Phys. Rep. 120, 1 (1985) M. Matev, A.V. Afanasjev, J. Dobaczewski, G.A. Lalazissis, W. Nazarewicz, Additivity of effective quadrupole moments and angular momentum alignments in A ∼ 130 nuclei. Phys. Rev. C 76, 034304 (2007) H. Meldner, Predictions of new magic regions and masses for super-heavy nuclei from calculations with realistic shell model single particle Hamiltonians. Ark. Fys. 36, 593 (1967) J. Meng (ed.), Relativistic Density Functional for Nuclear Structure. International Review of Nuclear Physics, vol. 10 (World Scientific Publishing Co, New Jersey, 2016) J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long, L.S. Geng, Relativistic continuum HartreeBogoliubov theory for ground state properties of exotic nuclei. Prog. Part. Nucl. Phys. 57, 470 (2006) J. Meng, J. Peng, S.-Q. Zhang, P.-W. Zhao, Progress on tilted axis cranking covariant density functional theory for nuclear magnetic and antimagnetic rotation. Front. Phys. 8, 55 (2013) H. Molique, J. Dobaczewski, J. Dudek, Superdeformed bands in 32 S and neighboring nuclei predicted within the Hartree-Fock method. Phys. Rev. C 61, 044304 (2000) P. Möller, J.R. Nix, Nuclear mass formula with a Yukawa-plus-exponential macroscopic model and a folded-Yukawa single-particle potential. Nucl. Phys. A 361, 117 (1981)

51 Model for Independent Particle Motion

1975

P. Möller, A.J. Sierk, T. Ichikawa, H. Sagawa, Nuclear ground-state masses and deformations: FRDM(2012). Atomic Data and Nucl. Data Tables 109–110, 1 (2016) A. Mutschler, A. Lemasson, O. Sorlin, D. Bazin, C. Borcea, R. Borcea, Z. Dombrádi, J.-P. Ebran, A. Gade, H. Iwasaki, E. Khan, A. Lepailleur, F. Recchia, T. Roger, F. Rotaru, D. Sohler, M. Stanoiu, S.R. Stroberg, J.A. Tostevin, M. Vandebrouck, D. Weisshaar, K. Wimmer, A proton density bubble in the doubly magic 34 Si nucleus. Nat. Phys. 13, 152 (2017) W. Nazarewicz, R. Wyss, A. Johnson, Structure of superdeformed bands in the A ≈ 150 mass region. Nucl. Phys. A 503, 285 (1989) T. Nikši´c, D. Vretenar, P. Ring, Relativistic nuclear energy density functionals: mean-field and beyond. Prog. Part. Nucl. Phys. 66, 519 (2011) T. Nikši´c, N. Paar, D. Vretenar, P. Ring, DIRHB – a relativistic self-consistent mean-field framework for atomic nuclei. Comput. Phys. Commun. 185, 1808 (2014) S.G. Nilsson, Binding states of individual nucleons in strongly deformed nuclei. Mat.-Fys. Medd.K.Dan. Vidensk. Selsk. 29(16), 69 (1955) S.G. Nilsson, I. Ragnarsson, Shapes and Shells in Nuclear Structure (Cambridge University Press, Cambridge, 1995) S.G. Nilsson, C.F. Tsang, A. Sobiczewski, Z. Szyma´nski, S. Wycech, C. Gustafsson, I.-L. Lamm, P. Möller, B. Nilsson, On the nuclear structure and stability of heavy and superheavy elements. Nucl. Phys. A 131, 1 (1969) Y.T. Oganessian, V.K. Utyonkov, Super-heavy element research. Rep. Prog. Phys. 78, 036301 (2015) S. Peru, M. Martini, Mean field based calculations with the Gogny force: some theoretical tools to explore the nuclear structure. Eur. Phys. J A50, 88 (2014) K. Pomorski, J. Dudek, Nuclear liquid-drop model and surface-curvature effects. Phys. Rev. C 67, 443161 (2003) I. Ragnarsson, Orbital and spin assignment of SD bands in the Dy/Gd region – identical bands. Nucl. Phys. A 557, 167c (1993) I. Ragnarsson, S. G. Nilsson, R. K. Sheline, Shell structure in nuclei. Phys. Rep. 45, 1 (1978) I. Ragnarsson, V.P. Janzen, D.B. Fossan, N.C. Schmeing, R. Wadsworth, Smooth termination of collective rotational bands. Phys. Rev. Lett. 74, 3935 (1995) P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin, 1980) J. Robin, T. Byrski, G. Duchêne, F.A. Beck, D. Curien, N. Dubray, J. Dudek, A. Gó´zd´z, A. Odahara, N. Schunck, N. Adimi, D.E. Appelbe, P. Bednarczyk, A. Bracco, B. Cederwall, S. Courtin, D.M. Cullen, O. Dorvaux, S. Ertück, G. de France, B. Gall, P. Joshi, S.L. King, A. Korichi, K. Lagergren, G. Lo Bianco, S. Leoni, A. Lopez-Martens, S. Lunardi, B. Million, A. Nourredine, E. Pachoud, E.S. Paul, C. Petrache, I. Piqueras, N. Redon, A. Saltarelli, J. Simpson, O. Stezowski, R. Venturelli, J.P. Vivien, K. Zuber, Extended investigation of superdeformed bands in 151,152 Tb nuclei. Phys. Rev. C 77, 014308 (2008) R. Rodriguez-Guzman, P. Sarriguren, L.M. Robledo, Shape evolution in yttrium and niobium neutron-rich isotopes. Phys. Rev. C 83, 044307 (2011) W. Satuła, J. Dobaczewski, J. Dudek, W. Nazarewicz, Additivity of quadrupole moments in superdeformed bands: single-particle motion at extreme conditions. Phys. Rev. Lett. 77, 5182 (1996) B. Schuetrumpf, W. Nazarewicz, P.-G. Reinhard, Central depression in nucleonic densities: trend analysis in the nuclear density functional theory approach. Phys. Rev. C 96, 024306 (2017) N. Schunck, J. Dobaczewski, J. McDonnell, J. Moré, W. Nazarewicz, J. Sarich, M.V. Stoitsov, Onequasiparticle states in the nuclear energy density functional theory. Phys. Rev. C 81, 024316 (2010) Z. Shi, A.V. Afanasjev, Z.P. Li, J. Meng, Superheavy nuclei in a microscopic collective Hamiltonian approach: the impact of beyond-mean-field correlations on ground state and fission properties. Phys. Rev. C 99, 064316 (2019) Y.R. Shimizu, J.D. Garrett, R.A. Broglia, M. Gallardo, E. Vigezzi, Pairing fluctuations in rapidly rotating nuclei. Rev. Mod. Phys. 61, 131 (1989) N.Y. Shirikova, A.V. Sushkov, L.A. Malov, R.V. Jolos, Structure of the low-lying states of the odd-neutron nuclei with Z ≈ 100. Eur. Phys. J. A 51, 21 (2015)

1976

A. V. Afanasjev

B. Singh, R. Zywina, R.B. Firestone, Table of superdeformed nuclear bands and fission isomers. Nucl. Data Sheets 97, 241 (2002) A. Sobiczewski, K. Pomorski, Description of structure and properties of superheavy nuclei. Prog. Part. Nucl. Phys. 58, 292 (2007) A. Sobiczewski, F.A. Gareev, B.N. Kalinkin, Closed shells for Z ≥ 82 and N ≥ 126 in a diffuse potential well. Phys. Lett. 22, 500 (1966) O. Sorlin, M.-G. Porquet, Nuclear magic numbers: new features far from stability. Prog. Part. Nucl. Phys. 61, 602 (2008) V.M. Strutinsky, Shell effects in nuclear masses and deformation energies. Nucl. Phys. A 95, 420 (1967) Z. Szyma´nski, Fast Nuclear Rotation (Claredon Press, Oxford, 1983) P.J. Twin, B.M. Nyakó, A.H. Nelson, J. Simpson, M.A. Bentley, H.W. Cranmer-Gordon, P.D. Forsyth, D. Howe, A.R. Mokhtar, J.D. Morrison, J.F. Sharpey-Schafer, G. Sletten, Observation of a discrete-line superdeformed band up to 60h¯ in 152 Dy. Phys. Rev. Lett. 57, 811 (1986) S. Typel, H.H. Wolter, Relativistic mean field calculations with density dependent meson-nucleon coupling. Nucl. Phys. A 656, 331 (1999) D. Vretenar, A.V. Afanasjev, G.A. Lalazissis, P. Ring, Relativistic Hartree-Bogoliubov theory: static and dynamic aspects of exotic nuclear structure. Phys. Rep. 409, 101 (2005) A. Watt, D. Kelvin, R.R. Whitehead, Shell-model calculations in the sd shell: X. Termination of rotational bands. J. Phys. G 6, 35 (1980) C.Y. Wong, Toroidal and spherical bubble nuclei. Ann. Phys. 77, 279 (1973) J.M. Yao, N. Itagaki, J. Meng, Searching for a 4α linear-chain structure in excited states of 16 O with covariant density functional theory. Phys. Rev. C 90, 054307 (2014) Y. Zhang, A. Bjel˘ci´c, T. Nik˘si´c, E. Litvinova, P. Ring, P. Schuck, A many-body approach to superfluid nuclei in axial geometry. Phys. Rev. C 105, 044326 (2022) Z.-H. Zhang, M. Huang, A.V. Afanasjev, Rotational excitations in rare-earth nuclei: a comparative study within three cranking models with different mean fields and treatments of pairing correlations. Phys. Rev. C 101, 054303 (2020) P.W. Zhao, N. Itagaki, J. Meng, Rod-shaped nuclei at extreme spin and isospin. Phys. Rev. Lett. 115, 022501 (2015)

Model for Collective Motion

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Shape Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Surface Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Rotation-Vibration Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microscopic Derivation of the Collective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Concepts of the Generator Coordinate Method . . . . . . . . . . . . . . . . . . . . . . . . . . The Gaussian Overlap Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microscopic Collective Hamiltonian Based on Density Functional Theory . . . . . . . . . . . . . The Five-Dimensional Collective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shape Coexistence in 76 Kr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrupole-Octupole Collective Hamiltonian for Pear-Shaped Nuclei . . . . . . . . . . . . . . Time-Dependent GCM+GOA for Nuclear Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1978 1979 1981 1985 1989 1989 1992 1996 1996 1999 2001 2003 2007 2008

Abstract

Collective motion is a manifestation of emergent phenomena in medium-heavy and heavy nuclei. A relatively large number of constituent nucleons contribute coherently to nuclear excitations (vibrations, rotations) that are characterized by large electromagnetic moments and transition rates. Basic features of collective Z. P. Li () School of Physical Science and Technology, Southwest University, Chongqing, China e-mail: [email protected] D. Vretenar Faculty of Science, Department of Physics, University of Zagreb, Zagreb, Croatia State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_11

1977

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excitations are reviewed, and a simple model is introduced that describes largeamplitude quadrupole and octupole shape dynamics, as well as the dynamics of induced fission. Modern implementations of the collective Hamiltonian model are based on the microscopic framework of energy density functionals that provide an accurate global description of nuclear ground states and collective excitations. Results of illustrative calculations are discussed in comparison with available data.

Introduction A medium-heavy or heavy atomic nucleus presents a typical example of a complex quantum system, in which different interactions between a relatively large number of constituent nucleons give rise to physical phenomena that are qualitatively different from those exhibited by few-nucleon systems. There are a number of features that characterize complex systems, but for the topic of the present chapter, of particular interest is the emergence of collective structures and dynamics that do not occur in light nuclei composed of only a small number of nucleons. Collective motion is the simplest manifestation of emergent phenomena in atomic nuclei. It can be interpreted as a kind of motion in which a large number of nucleons contribute coherently to produce a large-amplitude oscillation of one or more electromagnetic multipole moments. Collective motion gives rise to excited states that are characterized by large electromagnetic transition rates to the ground state, that is, rates that correspond to many single-particle transitions (Rowe 1970). This chapter will mainly focus on low-energy large-amplitude collective motion (LACM), such as surface vibrations, rotations, and fission. Theoretical studies of LACM started as early as in the 1930s. Based on the liquid drop model of the atomic nucleus (Weizsäcker 1935), Flügge applied Rayleigh’s normal modes (Rayleigh 1879) to a classical description of low-energy excitations of spherical nuclei (Flügge 1941). The model of Flügge was quantized by Bohr (1952), who formulated a quantum model of surface oscillations of spherical nuclei and also introduced the concept of intrinsic frame of reference for a quadrupole deformed nuclear surface characterized by the Euler angles and the shape parameters β and γ (nowadays often called the Bohr deformation parameters). Subsequently, Bohr and Mottelson (1953) generalized the model to vibrations and rotations of deformed nuclei. A generalization of the Bohr Hamiltonian to describe large-amplitude collective quadrupole excitations of even-even nuclei of arbitrary shape was introduced by Belyaev (1965) and Kumar and Baranger (1967). Several specific forms of the collective Hamiltonian, designed to describe collective excitations in nuclei of particular shape, were also considered (Wilets and Jean 1956; Davydov and Filippov 1958; Davydov and Chaban 1960; Faessler and Greiner 1962a, b; Faessler et al. 1965). In the past several decades, enormous progress has been made in developing microscopic many-body theories of nuclear systems. However, a description of collective phenomena starting from single-nucleon degrees of freedom still presents

52 Model for Collective Motion

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a considerable challenge. One of the methods that has been used to obtain such a description is the adiabatic time-dependent HFB theory (ATDHFB) (Belyaev 1965; Baranger and Kumar 1968; Baranger and Vénéroni 1978) which, in the case of quadrupole collective coordinates, leads to the Bohr Hamiltonian. Another approach to collective phenomena that is based on microscopic degrees of freedom is the generator coordinate method (GCM) (Hill and Wheeler 1953). In the Gaussian overlap approximation (GOA) (Brink and Weiguny 1968; Onishi and Une 1975), the GCM also leads to the Bohr collective Hamiltonian (Une et al. 1976; Gó´zd´z et al. 1985).

Nuclear Shape Parameters Excitation spectra of even-even nuclei in the energy range of up to ≈3 MeV exhibit characteristic band structures that are interpreted as vibrations and rotations of the nuclear surface in the geometric collective model, first introduced by Bohr (1952) and Bohr and Mottelson (1953) and further elaborated by Faessler and Greiner (1962a, b). For low-energy excitations, the compression mode is not relevant because of the high incompressibility of nuclear matter, and the diffuseness of the nuclear surface layer can also be neglected to a good approximation. One therefore starts with the model of a nuclear liquid drop of constant density and sharp surface. With these assumptions, the time-dependent nuclear surface can, quite generally, be described by an expansion in spherical harmonics with shape parameters as coefficients: ⎛ R(θ, φ; t) = R0 ⎝1 +

λ ∞  

⎞ αλμ (t)∗ Yλμ (θ, φ)⎠

(1)

λ=0 μ=−λ

where R(θ, φ; t) denotes the nuclear radius in spherical coordinates (θ, φ) and R0 is the radius of a sphere with the same volume. The shape parameters αλμ (t) play the role of collective dynamical variables, and their physical meaning will be discussed for increasing values of λ (Fig. 1). To the lowest order, the dipole mode λ = 1 corresponds to a translation of the nucleus as a whole and, therefore, is not considered for low-energy excitations.

Fig. 1 Nuclear shapes with dipole (λ = 1), quadrupole (λ = 2), octupole (λ = 3), and hexadecupole (λ = 4) deformations

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Dynamical quadrupole deformations, that is, the mode with λ = 2, turn out to be the most relevant low-lying collective excitations. Most of the following discussion of collective models will focus on this case, so a more detailed description is included below. Octupole dynamical deformations, λ = 3, are the principal asymmetric modes of a nucleus associated with negative-parity bands. While there is no evidence for pure hexadecupole excitations in low-energy spectra, this mode plays an important role as admixture to quadrupole excitations and for fission dynamics. Shape oscillations of higher multipoles are not relevant for low-energy excitations. For the case of pure quadrupole deformation, the nuclear surface is parameterized: ⎛ R(θ, φ) = R0 ⎝1 +

2 

⎞ αμ∗ Y2μ (θ, φ)⎠

(2)

μ=−2

where the time dependence is implicit for dynamical variables. If the shape of the nucleus is an ellipsoid, its three principal axes (x, y, z) are linked with the (X, Y , Z) axes of a Cartesian coordinate system in the laboratory frame. From the symmetry of the ellipsoid, it follows that a1 = a−1 = 0, a2 = a−2 , where aμ are the shape parameters in the principal-axis system. Obviously, the five coefficients αμ in the laboratory frame reduce to two real independent variables a0 and a2 in the principal-axis system, which, together with the three Euler angles, provide a complete parameterization of the nuclear surface. The details of the transformation between αμ and aμ are included below. In the principal-axis system, the nuclear radius is given by   R(θ  , φ  ) = R0 1 + a0 Y20 (θ  , φ  ) + a2 Y22 (θ  , φ  ) + a2 Y2−2 (θ  , φ  ) .

(3)

Two parameters (a0 , a2 ) are generally used to describe quadrupole deformations, but, instead of a0 and a2 , the polar coordinates β and γ are usually employed. They are defined as follows: a0 = β cos γ , (4)

1 a2 = √ β sin γ . 2

Using Eqs. (3) and (4), we can express the increments of the three semi-axes in the principal-axis system: δRκ = R0

 5 2π β cos γ − κ , 4π 3

κ = 1, 2, 3

(5)

52 Model for Collective Motion

1981

Fig. 2 Nuclear shapes in the (β, γ ) plane. Panel (a) is from Fortunato (2005)

where κ = 1, 2, 3 correspond to x, y, z, respectively. The parameters β and γ only describe exactly ellipsoidal shapes in the limit of small β-values. Figure 2 displays nuclear shapes in the (β, γ ) plane. At γ = 0 the nucleus is elongated along the z axis, with the x and y semi-axes being equal. This axially symmetric shape is somewhat reminiscent of a cigar and is called prolate. As γ increases, the x semi-axis lengthens with respect to the y and z semi-axes through a region of triaxial shapes with three unequal semi-axes, until axial symmetry is again reached at γ = 60◦ , but now with the z and x semi-axes equal in length. These two are longer than the y semi-axis. The flat, pancake-like shape is called oblate. This pattern is repeated: every 60◦ axial symmetry recurs, and prolate and oblate shapes alternate but with the semi-axes permuted in their relative length.

Nuclear Surface Oscillations A simple framework that accurately describes the most prominent features of lowenergy collective excitations is provided by the model of surface oscillations. Let the surface of the nucleus be parametrized by Eq. (1), and if the coefficients αλμ are small, the deformation potential energy and the associated kinetic energy take the forms V =

2 1  Cλ αλμ , 2 λμ

T =

2 1  Bλ α˙ λμ , 2

(6)

λμ

respectively. The dot denotes a derivative with respect to time, and the quantities Bλ and Cλ depend on the assumed properties of nucleonic matter. For an incompressible nucleus of constant density ρ0 , one finds Bλ = λ−1 ρ0 R05 , assuming nuclear matter to exhibit irrotational flow. If, moreover, the charge of the nucleus Ze is uniformly distributed over its volume, one obtains

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Cλ = (λ − 1)(λ + 2)R02 S −

3 λ − 1 Z 2 e2 2π 2λ + 1 R0

(7)

where S is the surface tension. It is convenient to make a change of coordinates in such a way that the kinetic energy separates into a vibrational and a rotational part. This coordinate transformation will be considered in the following paragraphs for deformations of order λ = 2. For convenience, B2 and C2 are simplified to B and C. Consider a coordinate system K  whose axes coincide with the principal axes of the ellipsoid. The coordinate transformation from the five αμ to the new coordinates a0 , a2 and θi must be given by αμ =



2∗ aν Dμν (θi ) ,

(8)

ν 2∗ (θ ) is the Wigner function and θ ≡ (θ , θ , θ ) are the three Euler where Dμν i i 1 2 3 angles which describe the orientation of the body-fixed axes in space. Using Eq. (4) 2 , one obtains and the unitary character of Dμν

 2  αμ = aν2 = a02 + 2a22 = β 2 . μ

(9)

ν

From equations (6) and (9), it follows V = 12 Cβ 2 for the potential energy of quadrupole deformation. To express the kinetic energy of the oscillating nucleus, from Eq. (8) one derives α˙ μ =



2∗ a˙ ν Dμν (θi ) +

ν



aν θ˙j

νj

∂ 2∗ D (θi ) . ∂θj μν

(10)

If this expression is used in Eq. (6), the kinetic energy can be written as a sum of three terms. The first term is quadratic in a˙ ν and represents shape vibrations of the ellipsoid that retains its orientation. The second term, quadratic in θ˙i , represents a rotation of the ellipsoid without change of its shape. The third term, which contains mixed time derivatives a˙ ν θ˙i , vanishes as can be shown from simple properties of the 2 and their derivatives. Wigner coefficients Dμν Therefore, the kinetic energy reads T = Tvib + Trot .

(11)

For the vibrational energy, one obtains Tvib =

 1  1 |a˙ ν |2 = B β˙ 2 + β 2 γ˙ 2 B 2 2 ν

(12)

52 Model for Collective Motion

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by means of Eq. (4). To derive a convenient form for Trot , the most natural description is achieved if the principal axes of the nucleus determine the body-fixed system, because then the inertia tensor is diagonal. The classical rotational energy reads 1 Ji ωi2 , 2 3

Trot =

(13)

i=1

Ji = 4Bβ 2 sin2 (γ − i

2π ), 3

i = 1, 2, 3

where Ji is the moment of inertia about the i-th principal axis of the nucleus and ωi is the corresponding angular velocity of rotation. The next step is the quantization of the classical Hamiltonian. As it is well known, there is no unique prescription for such a quantization in the general case. Usually one adopts the Pauli prescription, for which the Laplace operator is expressed in curvilinear coordinates. The final form of the Hamiltonian when expressed in β and γ , as was done originally by Bohr (1952), reads 



3  Iˆi2 1 + Cβ 2 2Ji 2 i=1 (14) where Iˆi are operators of the projections of the nuclear angular momenta on the principal axes. However, the operators Iˆi cannot be identified with the standard angular-momentum operators in the laboratory frame. Their commutation relations look similar to those of the laboratory-fixed operators, but with a crucial change in sign: 2

h¯ Hˆ = − 2B

1 ∂ 4 ∂ ∂ 1 1 ∂ β + 2 sin(3γ ) ∂γ β 4 ∂β ∂β β sin(3γ ) ∂γ

  Iˆ1 , Iˆ2 = −ih¯ Iˆ3 ,

  Iˆ2 , Iˆ3 = −ih¯ Iˆ1 ,

+

  Iˆ3 , Iˆ1 = −ih¯ Iˆ2 .

(15)

Detailed expression for these operators can be found in Davydov (1965). The stationary wave functions separate in the following way: Ψ (β, γ , θi ) = f (β)Φ (γ , θi ) .

(16)

The solution proceeds through the familiar procedure of separation of variables, and, thus, the normalized solution for the β part of the wave function reads ⎡ fnl (β) = (−1)n ⎣

⎤1/2

2(n!)3

Γ n+l+

5 2

⎦

β l e−β

2 /2

l+3/2

Ln

 β2 ,

(17)

where n is the principal quantum number, while l is the quantum number that corresponds to the solution of the eigenvalue equation of the Casimir operator of

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the SO(5) group (also called SO(5) seniority). The total energy E of the fivedimensional quadrupole oscillator reads 5 5 E = (2n + l + )hω ¯ = (N + )hω, ¯ 2 2

N = 0, 1, · · ·

(18)

The wave function Φ can be written in the form I ΦMlν (γ , θi ) =

I 

I I∗ gKlν (γ )DMK (θi )

(19)

K=−I

where I , M, and K are the quantum numbers for the total angular momentum, its projections on Z axis in the laboratory frame, and on the third axis in the intrinsic frame, respectively. ν is a label that distinguishes between multiple sets of the same quantum numbers I , M in a given SO(5) irreducible representation. The functions g are given explicitly in Corrigan et al. (1976). In addition, the above expression must be symmetrized because of the ambiguity in the choice of intrinsic axes. The total number of possible choices is 24, and they can be related by 3 basic transformations Rˆ 1 , Rˆ 2 , Rˆ 3 , shown in Fig. 3. Under these transformations the collective wave functions must satisfy Rˆ 1 : Ψ (β, γ , θ1 , θ2 , θ3 ) = Ψ (β, γ , θ1 + π, π − θ2 , −θ3 ) ,

π , Rˆ 2 : Ψ (β, γ , θ1 , θ2 , θ3 ) = Ψ β, −γ , θ1 , θ2 , θ3 + 2 

π π 2π , θ1 , θ2 + , θ3 + . Rˆ 3 : Ψ (β, γ , θ1 , θ2 , θ3 ) = Ψ β, γ + 3 2 2

(20)

By applying Rˆ 22 to the wave function (19), one obtains that K must be even. I ∗ (θ ) into (−1)I D I ∗ I Rˆ 1 will transform DMK M−K (θ ) and require that g−Klν (γ ) = I I (−1) gKlν (γ ). Finally, the wave function is written to explicitly display the symmetry property which restricts the sum to K ≥ 0:

Fig. 3 Three basic transformations Rˆ 1 , Rˆ 2 , and Rˆ 3 that define the symmetry of the collective wave function. (a) Rˆ 1 . (b) Rˆ 2 . (c) Rˆ 3

52 Model for Collective Motion I I ΨMnlν fnl (β) (β, γ , θi ) = Nnlν

1985



I I gKlν (γ )φMK (θi ) ,

K≥0 even

(21)

where  I φMK (θi ) =

  2I + 1 I∗ I I∗ D (θ ) + (−1) D (θ ) i i MK M,−K 16(1 + δK0 )π 2

(22)

I normalare the properly symmetrized wave functions for the symmetric rotor. Nnlν izes the γ part of Eq. (21). Detailed expression for the wave function can be found in Corrigan et al. (1976).

The Rotation-Vibration Model The most important special case of collective surface motion is that of welldeformed nuclei, whose potential energy surfaces exhibit a deep axially deformed minimum like in the simple rotor model but with the additional feature of small oscillations around that minimum in both β and γ degrees of freedom (see the illustration in Fig. 4). As the harmonic vibrations are easier to deal with in Cartesian coordinates, the potential is expressed in the coordinates (a0 , a2 ) instead of (β, γ ). Assuming the potential minimum is located at a0 = β0 and a2 = 0, the small displacement from the equilibrium deformation can be written in the form ξ = a0 − β0 ,

η = a2 = a−2 ,

(23)

and it is easy to show that the Hamiltonian can be expressed in the form  1 1 2 1 B ξ˙ + 2η˙ 2 + Ji ωi2 + C0 ξ 2 + C2 η2 . 2 2 2 3

H = Tvib + Trot + V =

(24)

i=1

Note the additional factor of 2 in the η-dependent part of the potential, which takes into account that η represents the two coordinates a2 and a−2 . In a quantum mechanical description, a nucleus cannot rotate around the symmetry axis. However, here the vibrations make such rotations possible dynamically, and they are coupled to the dynamic fluctuations from axial symmetry described by η. The moments of inertia are given by the lowest-order expressions J ≡ J1 = J2 = 3Bβ02 ,

J3 = 8Bη2 ,

and the simplest expression for the kinetic energy reads

(25)

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Z. P. Li and D. Vretenar

Fig. 4 Schematic plot of the collective potential V (β, γ ) for 0 ≤ γ ≤ π/3, shape of global minimum, vibrations in β and γ directions, and rotation

T =

 1  1 2 B ξ˙ + 2η˙ 2 + J ω12 + ω22 + 4Bη2 ω32 . 2 2

(26)

Similarly, the quantized Hamiltonian can be written in the form −h¯ 2 Hˆ (ξ, η, θ ) = 2B



 ˆ2 ˆ2 ˆ2 I1 + I2 I3 − h¯ 2 1 ∂2 1 ∂2 + + + + C0 ξ 2 +C2 η2 . 2 ∂η2 2J ∂ξ 2 16Bη2 2

(27)

The term with η−2 is similar to the centrifugal potential and can be treated accordingly. The eigenfunctions of the rotational operator in the Hamiltonian correspond to those of the rigid rotor, while the ξ -dependent and η-dependent parts can obviously be separated. Thus, a trial wave function takes the form I∗ ψ(ξ, η, θ ) = g(ξ )χ (η)DMK (θi ).

(28)

The next step is to complete the separation of the ξ and η dependence, and finally they correspond to typical eigenvalue problems with one-dimensional and threedimensional harmonic oscillator potentials, respectively. The total energy reads



   h¯ 2  1 1 n 2n + h ω |K| + 1 + I (I + 1) − K 2 , Enβ nγ I K = hω + + ¯ γ ¯ β β γ 2 2 2J (29) with C0 C2 ωβ = , ωγ = , nβ or nγ = 0, 1, · · · (30) B B

52 Model for Collective Motion

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For the eigenfunctions that depend on the ξ (β) coordinate, the one-dimensional harmonic oscillator eigenfunctions can be used directly, and they are simply written as ξ |nβ . For the γ coordinate, the solution for the three-dimensional harmonic oscillator potential is used: χKnγ (η) = NKnγ



|η|η

K/2 −λη2 /2

e

 3 2 , 1 F1 −nγ , lK + , λη 2

(31)

with λ = 2Bωγ /h, ¯ lK = (−1 ± K)/2, and NKnγ the normalization factor. After symmetrization, the wave functions read   I I ψMKn (ξ, η, θ ) = ξ | nβ χKnγ (η)φMK (θi ) , β nγ

(32)

I and φMK (θi ) are the symmetrized eigenfunctions for the symmetric rotor of Eq. (22). The allowed values of the quantum numbers are

K = 0, 2, 4, . . .  K, K + 1, K + 2, . . . for K = 0 I= 0, 2, 4, . . . for K = 0

(33)

M = −I, −I + 1, . . . , +I. The structure of a typical spectrum is shown in Fig. 5. The bands are characterized by a given set of K, nβ , nγ , and the excitation energies follow the I (I + 1) rule of a rotor. The principal bands are • the ground-state band (g. s.) includes the states |I M000 with I even. The excitation energies are given by h¯ 2 I (I + 1)/2J . • the β band includes the states |I M010 with one quantum of vibration in the β direction. It starts at h¯ ωβ above the ground state and also contains only even angular momenta. • the γ band corresponds to K = 2. It is characterized by coupling between rotation (around the third principal axis) and γ -vibration, induced by the term (Iˆ32 − h¯ 2 )/16Bη2 in the Hamiltonian. In the spectra it is easy to distinguish it from the β band, since it starts with 2+ and contains both even and odd angular momenta. • the next higher bands are the γ band with nγ = 1 and the one with K = 4, the second β band with nβ = 2. The simplest observables that can be calculated and compared to data are the electric quadrupole moments and transition probabilities. The collective quadrupole operator in the intrinsic system (in lowest order of deformation parameters) is given by

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Fig. 5 Structure of the excitation spectrum of the rotation-vibration model. The probability density distributions in the (β, γ ) plane for the band-heads are shown above each band. The notation and the corresponding quantum numbers are indicated below the band-heads



  2∗ 2∗ 2∗ ˆ 2μ = 3Ze R02 Dμ0 (θi )(β0 + ξ ) + Dμ2 (θi ) + Dμ−2 (θi ) η . Q 4π

(34)

The collective quadrupole moment is defined by QI Knβ nγ =

 16π  ˆ I M = I Knβ nγ Q 20 I M = I Knβ nγ . 5

(35)

It does not depend on nβ and nγ , and its value reads QI K = Q0

3K 2 − I (I + 1) , (I + 1)(2I + 3)

(36)

where the intrinsic quadrupole moment is defined 3ZeR 2 Q0 = √ 0 β0 . 5π

(37)

The reduced electric quadrupole transition probabilities are calculated using the expression

52 Model for Collective Motion

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! B E2; Ii → If =

  2 1 ˆ i . If ||Q||I 2Ii + 1

(38)

The so-called stretched B(E2)-values in a rotational band read 5 I +2 2 I 2 C , 16π K 0 K

(39)

5 3 (I + 1)(I + 2) . 16π 2 (2I + 3)(2I + 5)

(40)

B(E2; I + 2 → I ) = Q20 and, for K = 0 bands, one obtains B(E2, I + 2 → I ) = Q20

The transition probabilities allow the determination of the intrinsic quadrupole moment and thus the deformation itself, whereas the energy spacings in the excitation spectra determine the moment of inertia.

Microscopic Derivation of the Collective Hamiltonian The general form of the collective Bohr Hamiltonian has been derived starting from a microscopic Hamiltonian or from an energy density functional in two rather different ways: (i) using the generator coordinate method (GCM) (Haff and Wilets 1973; Banerjee and Brink 1973; Giraud and Grammaticos 1974; Onishi and Une 1975; Ring and Schuck 1980) and (ii) applying the time-dependent Hartree-Fock (TDHF) theory (Baranger and Kumar 1968; Baranger and Vénéroni 1978; Goeke and Reinhard 1978). Here, the former one is presented, taking into account that the derivation based on the GCM is fully quantum mechanical. It relies on the validity of the Gaussian overlap approximation (GOA) for the overlaps between configurations with different deformations and on the assumption that the collective velocities are small, i.e., that the expansion in the collective momenta up to second order is adequate. The validity of these approximations is demonstrated in a comparison with a full GCM calculation for a nucleus characterized by shape coexistence: 76 Kr (Yao et al. 2014).

General Concepts of the Generator Coordinate Method In the generator coordinate method, the nuclear wave function |Φ can be expressed as a continuous superposition of generating functions |ϕ(β) that are labeled by an arbitrary number of real or complex parameters {β} = β1 , β2 , . . . , βi , the so-called generator coordinates: " |Φ =

dβf (β)|ϕ(β) .

(41)

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Z. P. Li and D. Vretenar

The weight f (β) is assumed to be a well-behaved function of the variables β, and the multidimensional integral includes all real and imaginary parameters β. The set of generating wave functions ϕ(β) is often determined by a specific problem or geometry. In many cases, the intrinsic states of a nucleus are chosen as the generating states and the corresponding shape deformation parameters as the generator coordinates. See the schematic illustration in Fig. 6. The variational principle implies δE(Φ) = δ

 1 Φ|Hˆ |Φ = Φ|Hˆ − E|δΦ + δΦ|Hˆ − E|Φ = 0. Φ|Φ Φ|Φ (42)

By inserting Eq. (41) into Eq. (42), and performing the variation with respect to f (β), an integral equation is obtained: "

ϕ(β)|Hˆ |ϕ(β  ) f (β  )dβ  = E

"

ϕ(β)|ϕ(β  ) f (β  )dβ  .

(43)

This is the Hill-Wheeler equation, and it can formally be written in the form Hf = ENf,

(44)

with the overlap functions H (β, β  ) = ϕ(β)|Hˆ |ϕ(β  ) 



N (β, β ) = ϕ(β)|ϕ(β )

The energy kernel

(45)

The overlap kernel

as integral kernels.

E

_)²

E Fig. 6 The nuclear wave function |Φ as a continuous superposition of generating functions |ϕ(β) , labeled by the parameters β

52 Model for Collective Motion

1991

To solve this equation, a transformation is defined from the space of the generating coordinates to the coordinate space in which the wave functions specified by the true coordinates η are orthogonal. The transformation β → η is defined in such a way that N(β, β  ) → δ(η − η ). In the first step, the eigenvalue equation for the overlap kernel is solved: "

N(β, β  )χ (s, β  )dβ  = ν(s)χ (s, β).

(46)

The eigenfunctions specified by a new variable s form a complete orthonormalized set: " " ∗   χ (s, β)χ (s , β)dβ = δ(s − s ); χ ∗ (s, β)χ (s, β  )ds = δ(β − β  ). (47) Next, a transformation kernel is introduced: "

1 2

N (β, η) =

1

χ (s, β)ν 2 (s)χ ∗ (s, η)ds,

(48)

which represents the transition from δ(η − η ) to N(β, β  ) through "

1

1

N 2 ∗ (β, η)δ(η − η )N 2 (η , β  )dηdη =

" "

=

χ ∗ (s, β  )ν(s)χ (s, β)ds N(β, β  )δ(β  − β  )dβ 

= N(β, β  ).

(49)

1

Obviously, the inverse N − 2 has to be determined to transform N(β, β  ) to δ(η − η ). Accordingly, the energy kernel should be transformed to the coordinate representation by using the narrowing kernel L(η, η ) =

"

1

1

N − 2 ∗ (β, η)H (β, β  )N − 2 (β  , η )dβdβ  ,

(50)

and the new wave function g(η) in coordinate representation is related to the generating function f (β) " g(η) =

1

N 2 (β, η)f (β)dβ.

(51)

By replacing each factor in Eq. (43) with the corresponding one after transformation, the orthogonal Hill-Wheeler equation is obtained:

1992

Z. P. Li and D. Vretenar

"

L(η, η )g(η )dη = Eg(η),

(52)

or in matrix form Lg = Eg.

(53)

Equation (52) can be considered as a stationary Schrödinger equation, where L is the collective Hamiltonian and g the corresponding eigenfunction in the new representation. To derive the collective Hamiltonian, one needs an explicit 1 expression for the transformation kernel N 2 . This is achieved by introducing the Gaussian overlap approximation.

The Gaussian Overlap Approximation A simple and pedagogical derivation of the collective Hamiltonian can be presented by using a single generator coordinate. A set of time-reversal invariant generating functions |β = |ϕ(β) , which depend on a single collective parameter β, can be obtained by constrained self-consistent mean-field calculations. To derive the collective Hamiltonian, the Gaussian overlap approximation (GOA) is based on two assumptions: (1) generally, the overlap function N = β|β  rapidly becomes smaller with the increase of the distance |β − β  |; and (2) both the energy kernel H and overlap kernel N are well-behaved functions. Therefore, a Gaussian function is assumed for the overlap kernel N: 1

 2

N(β, β  ) = e− 2 γ0 (β−β ) .

(54)

Note that Eq. (54) is written in a homogeneous form, namely, γ0 is constant. The overlap kernel is a function that only depends on the difference (β − β  ). In this case, the solutions of equation (46) are  ν(s) =

2π − 2γs 2 e 0; γ0

1 χ (s, β) = √ eisβ , 2π

(55) 1

for the eigenvalue and eigenfunction, respectively. Hence, the kernel N 2 can be expressed as 1

N 2 (β, η) =

"

1

χ ∗ (s, β)ν 2 (s)χ (s, η)ds = (

2γ0 1 −γ0 (β−η)2 )4 e . π

(56) 1

Starting from Eq. (50), and using the expressions above for χ (s, β), ν(s), and N − 2 , Onishi and Une derived the collective Hamiltonian in 1975. Here, the approach

52 Model for Collective Motion

1993

proposed by Ring and Schuck in their textbook The Nuclear Many-Body Problem (Ring and Schuck 1980) will be described. One starts from the expectation of the Hamiltonian E = Φ|Hˆ |Φ =

"

f ∗ (β)H (β, β  )f (β  )dβdβ  ,

(57)

and, to transform f (β) and H (β, β  ) to g(η) and L(η, η ) as in Eq. (52), the reduced energy kernel h(β, β  ) = H (β, β  )/N(β, β  ) is introduced. The diagonal element of the reduced energy kernel is just the expectation value of the Hamiltonian in the state |β . Then Eq. (57) takes the form " E=

f ∗ (β)N (β, β  )h(β, β  )f (β  )dβdβ  .

(58)

Inserting Eq. (49) into Eq. (58), the following expression is obtained: " E=

1

1

f ∗ (β)N 2 ∗ (β, η)δ(η − η )h(β, β  )N 2 (η , β  )f (β  )dβdβ  dηdη .

(59)

Next, h(β, β  ) is expanded to second order around the point β = η, β  = η : h(β, β  ) = h + hβ (β − η) + hβ  (β  − η )  1 hββ (β − η)2 + 2hββ  (β − η)(β  − η ) + hβ  β  (β  − η )2 . + 2 (60) To calculate the derivatives h, the Hamiltonian kernel and overlap kernel are written in the form   ˆ ˆ H (β, β  ) = η|e−i(β−η)P /h¯ Hˆ ei(β −η )P /h¯ |η

ˆ





ˆ

N(β, β  ) = η|e−i(β−η)P /h¯ ei(β −η )P /h¯ |η

(61)

∂ is the momentum operator. Then the derivative h can be where Pˆ = −i h¯ ∂β expressed as

h

β

∂ = hβ = ∂β



 H (β, β  ) i  ˆ ˆ η| P H |η = −  = 0, h¯ N(β, β  ) β=β  =η=η η=η

(62)

where η = η because of δ(η − η ) in Eq. (59). Note that the generating functions |η are time-reversal invariant states, and Pˆ is a time-odd operator. Therefore, the first derivatives vanish, while the second derivatives read

1994

Z. P. Li and D. Vretenar

hβ  β  hββ 

 2 ˆ   2  ˆ ˆ ˆ = hββ = − 2 η|P H |η − η|H |η η|P |η h¯ η=η

 1 = 2 η|Pˆ Hˆ Pˆ |η − η|Hˆ |η η|Pˆ 2 |η . h¯ η=η 1

In addition, the term

∂ ∂η hβ

(63)

= hββ + hββ  will be used when performing partial 1

1

integration. Setting N1 = N 2 (β, η) and N2 = N 2 (β  , η ) and inserting the expansion of h into Eq. (59), one obtains " E=

 dβdβ  dηdη f ∗ (β)f (β  )N1 N2 h + hβ (β − η) + hβ  (β  − η )

 1 + hββ (β − η)2 + 2hββ  (β − η)(β  − η ) + hβ  β  (β  − η )2 . 2

(64)

From Eq. (56), (β − η)N1 and (β − η)2 N1 can be expressed in the form (β − η)N1 =

1 ∂ N1 2γ0 ∂η

1 ∂2 1 (β − η) N1 = ( 2 2 + )N1 . 2γ0 4γ0 ∂η

(65)

2

Obviously, (β  − η )N2 and (β  − η )2 N2 have the same forms as in Eq. (65), except that the derivatives are with respect to η . Then Eq. (64) becomes  hββ + hβ  β  1 dβdβ  dηdη δ(η − η )f ∗ (β)f (β  ) (h + )N1 N2 + hβ N1 N2 4γ0 2γ0 # 1 1 1 1 + hβ  N1 N2 + hββ N1 N2 + hββ  N1 N2 + hβ  β  N1 N2 . 2γ0 8γ02 4γ02 8γ02 (66)

" E=

After partial integration, and inserting Eq. (51), the final result is obtained neglecting higher-order derivatives: " E=

$

# hββ  ∂ 1 ∂ g δ(η − η ) h − + (hββ − hββ  ) gdηdη . 2γ0 ∂η 4γ02 ∂η ∗



Comparing to Eq. (52), the collective Hamiltonian L(η, η ) finally reads

(67)

52 Model for Collective Motion

1995

$

hββ  ∂ 1 ∂ L(η, η ) = δ(η − η ) h − + (hββ − hββ  ) 2γ0 ∂η 4γ02 ∂η   1 ∂ ∂  + V (η) , = δ(η − η ) − ∂η 2M(η) ∂η 

#



(68)

with the collective mass and potential 1 −1 (hββ − hββ  ), = M(η) 2γ02 hββ  , V (η) = h − 2γ0

(69)

respectively. In a more general case, namely, for inhomogeneous GOA, γ0 in Eq. (54) depends on the collective coordinates. However, the inhomogeneous GOA can always be brought into a homogeneous form by performing a transformation from %β  γ (β  )/γ0 dβ  (Ring and Schuck 1980). Finally, to β to a new coordinate α = obtain the general form of the collective Hamiltonian, √ a transformation back to the original coordinate is carried out by inserting dη = γ /γ0 dξ into Eq. (67) and using a new wave function

g˜ =

−1/4 γ0 g

"  14 2 2 = e−γ (β−ξ ) f (β)dβ. π

(70)

Then " E=

√

  hββ  1 ∂ √ 1 ∂ +√ gdξ ˜ dξ  . γ g˜ ∗ δ(ξ − ξ  ) h − γ 2 (hββ − hββ  ) 2γ γ ∂ξ ∂ξ 4γ (71)

Note that hββ and hββ  will involve γ0 /γ when transforming back to the original coordinate. The general form of the collective Hamiltonian reads   ∂ 1 ∂ √ 1 + V (ξ ) , L(ξ, ξ  ) = δ(ξ − ξ  ) − √ γ γ ∂ξ 2M(ξ ) ∂ξ

(72)

where the collective mass and collective potential have the same forms as in Eq. (69), with γ0 replaced by γ .

1996

Z. P. Li and D. Vretenar

Microscopic Collective Hamiltonian Based on Density Functional Theory In the previous section, the collective Bohr Hamiltonian was derived starting from a microscopic GCM framework. To illustrate possible applications of the collective Hamiltonian to various low-energy structure and dynamical phenomena, selected examples that use different collective coordinates will be presented. The Hamiltonian that will be employed in the following examples is constructed using covariant density functional theory (CDFT, details can be found in  Chap. 56, “Relativistic Density-Functional Theories”), but, of course, it can equally well be built from any of a number of successful non-relativistic density functionals. In the first representative case, the quadrupole five-dimensional collective Hamiltonian (5DCH) is applied to the phenomenon of shape coexistence in the neutron-deficient nucleus 76 Kr. The spectroscopy of pear-shaped nuclei can be described using a quadrupole-octupole collective Hamiltonian. In the final example, nuclear fission dynamics is modeled by the time evolution of a collective wave packet in the space of quadrupole and octupole axially symmetric deformations.

The Five-Dimensional Collective Hamiltonian To describe complex phenomena that originate in collective quadrupole excitations of nuclei, e.g., shape phase transitions, shape coexistence, and super-deformed bands, the simple Bohr Hamiltonian of Eq. (14) has to be extended to a general five-dimensional collective Hamiltonian (5DCH): Hˆ (β, γ , θ ) = Tˆvib + Tˆrot + Vcoll  &  ∂ ∂ 1 r 4 ∂ r 3 ∂ h¯ 2 β Bγ γ − β Bβγ =− √ ∂β ∂β w ∂γ 2 wr β 4 ∂β w  ' ∂ 1 ∂ 1 r ∂ r ∂ − sin 3γ Bβγ + sin 3γ Bββ + β sin 3γ ∂γ w ∂β β ∂γ w ∂γ +

3 1  Jˆk2 + Vcoll . 2 Ik

(73)

k=1

where Jˆk denotes the components of the angular momentum in the body-fixed frame of a nucleus and both the mass parameters Bββ , Bβγ , and Bγ γ and the moments of inertia Ik depend on the quadrupole deformation variables β and γ . Vcoll is the collective potential. Two additional quantities that appear in Tˆvib , namely, r = 2 , B1 B2 B3 (Bk are related to Ik (Nikši´c et al. 2009)) and w = Bββ Bγ γ − Bβγ determine the volume element in the collective space. Details about the DFT-based 5DCH can be found in Nikši´c et al. (2009), Libert et al. (1999), and Prochniak et al. (2004).

52 Model for Collective Motion

1997

The 5DCH describes quadrupole vibrations, rotations, and the coupling of these collective modes. The corresponding eigenvalue equation is solved by expanding the eigenfunctions on a complete set of basis functions that depend on the deformation variables β and γ and the Euler angles (Prochniak et al. 2004). The results are the excitation energy spectrum EαI and the collective wave functions ΨαI M (β, γ , θ ) =



I I ψαK (β, γ )φMK (θ ),

(74)

K∈ΔI

where M and K are the projections of angular momentum I on the third axis in the laboratory and intrinsic frames, respectively, and α denotes additional I (θ ) is defined by Eq. (22). Using the quantum numbers. The rotational function φMK collective wave functions, various observables, such as E2 transition probabilities, can be calculated: B(E2; αI → α  I  ) =

1 ˆ |α  I  ||M(E2)||αI |2 , 2I + 1

(75)

ˆ where M(E2) is the electric quadrupole operator. The next step is to calculate the collective inertia parameters B and Ik and the collective potential Vcoll , using solutions of microscopic self-consistent deformation-constrained triaxial CDFT calculations (c.f.  Chap. 56, “Relativistic Density-Functional Theories”). In principle, one can use results obtained with the GCM+GOA framework. However, in practice, the resulting inertia parameters are generally too small (Baranger and Vénéroni 1978; Goeke and Reinhard 1980). The adiabatic time-dependent Hartree-Fock (ATDHF) theory (Baranger and Vénéroni 1978) provides an alternative way to derive a classical collective Hamiltonian, and, after requantization, a Bohr Hamiltonian of the same structure is obtained but with different microscopic expressions for the inertia parameters (Villars 1977). This method has the advantage that the time-odd components of the microscopic wave functions are also included and, in this sense, the full dynamics of a nuclear system. In many applications a further simplification is thus introduced in terms of cranking formulas for the inertia parameters and zero-point energy corrections (Girod and Grammaticos 1979). The entire map of the energy surface as a function of quadrupole deformations is obtained by imposing constraints on the axial and triaxial mass quadrupole moments:  ˆ 2μ − q2μ )2 , H + C2μ (Q (76) μ=0,2

where q2μ is the constrained value of the multipole moment and C2μ is the ˆ 2μ denotes the corresponding stiffness constant (Ring and Schuck 1980). Q expectation value of the mass quadrupole operator: ˆ 22 = x 2 − y 2 . ˆ 20 = 2z2 − x 2 − y 2 and Q Q

(77)

1998

Z. P. Li and D. Vretenar

The single-nucleon wave functions, energies, and occupation factors, generated from constrained triaxial CDFT calculations, provide the microscopic input for the parameters of the collective Hamiltonian. The moments of inertia are calculated according to the Inglis-Belyaev formula (Inglis 1956; Belyaev 1961):  ui vj − vi uj Ik = Ei + Ej

!2   ˆ 2 i Jk j ,

k = 1, 2, 3,

(78)

i,j

where k denotes the axis of rotation, and the summation runs over the proton and neutron quasiparticle states. Ei , vi , and |i are the quasiparticle energies, occupation probabilities, and single-nucleon wave functions, respectively. The massparameters ˆ 20 and q2 = associated with the two quadrupole collective coordinates q0 = Q   ˆ 22 are also calculated in the cranking approximation (Girod and Grammaticos Q 1979):   −1 −1 Bμν (q0 , q2 ) = h¯ 2 M(1) M(3) M(1)

μν

,

(79)

with    ˆ ˆ  i Q 2μ j j Q 2v i !2 M(n),μν (q0 , q2 ) = ui vj + vi uj . !n Ei + Ej i,j

(80)

The collective energy surface includes the energy of zero-point motion, which has to be subtracted. The collective zero-point energy (ZPE) corresponds to a superposition of zero-point motion of individual nucleons in the single-nucleon potential. In practice, ZPE corrections originating from the vibrational and rotational kinetic energy are considered. These are given by the following expressions (Girod and Grammaticos 1979): ΔVvib (q0 , q2 ) =

 1  −1 Tr M(3) M(2) , 4

(81)

and ΔVrot (q0 , q2 ) = ΔV−2−2 (q0 , q2 ) + ΔV−1−1 (q0 , q2 ) + ΔV11 (q0 , q2 ) ,

(82)

with ΔVμν (q0 , q2 ) =

1 M(2),μν (q0 , q2 ) . 4 M(3),μν (q0 , q2 )

(83)

52 Model for Collective Motion

1999

Finally, the collective potential Vcoll in Eq. (73) is obtained by subtracting the ZPE corrections from the total mean-field energy: Vcoll (q0 , q2 ) = Etot (q0 , q2 ) − ΔVvib (q0 , q2 ) − ΔVrot (q0 , q2 ) .

(84)

Shape Coexistence in 76 Kr The observation that an atomic nucleus with a particular combination of neutron N and proton Z numbers can exhibit states characterized by different shapes, i.e., shape coexistence, appears to be a unique feature of finite many-body quantum systems (Heyde and Wood 2011). The low-lying states of neutron-deficient eveneven krypton isotopes are of particular interest due to their rapid structural change with neutron number and the presence of multiple shape coexistence, i.e., several 0+ states with different intrinsic shapes coexist at low excitation energy. The irregularities observed in the ground-state bands at low spin in 74,76 Kr were explained by shape coexistence in Piercey et al. (1981). As an example of the application of the quadrupole collective Hamiltonian, here the CDFT-based 5DCH model is used to analyze shape coexistence in 76 Kr. A systematic study of shape evolution and shape coexistence in Kr isotopes has been reported in Fu et al. (2013). Figure 7 displays the collective potential energy surface in the β-γ plane for the even-even nucleus 76 Kr, with the ZPEs of rotational and vibrational motion subtracted from the total deformation energy (cf. Eq. (84)). As shown in the panel on the right, after subtraction of the ZPEs the prolate deformed minimum becomes deeper, and the energy with respect to the spherical shape is reduced by ≈0.9 MeV. This leads to a coexistence picture of competing spherical and prolate minima on the

Fig. 7 (a) Collective potential energy surface of 76 Kr (Fu et al. 2013) in the β-γ plane calculated using the constrained triaxial CDFT with the PC-PK1 functional (Zhao et al. 2010). (b) Comparison of the total axial deformation energy and the collective potential of76 Kr. All energies are normalized to that of the spherical shape

2000

Z. P. Li and D. Vretenar

energy surface. When correlations related to restoration of broken symmetries are taken into account in the 5DCH calculation, the ground state of 76 Kr is dominated by the prolate deformed configurations. The 5DCH low-spin excitation spectrum of the nucleus 76 Kr is compared with the available data in Fig. 8. The main features are reproduced very well by the model calculation, in particular for the ground-state band and the low-lying 0+ 2 + state. The observed large E2 transition strength from the 0+ 2 state to the 21 state is underestimated by about a factor of four. This may imply that the mixing between + the two calculated bands is too weak. In contrast, the large B(E2; 0+ 2 → 21 ) can be reproduced by a similar 5DCH calculation that uses the Gogny D1S functional, which predicts a rather large mixing between the two 0+ bands, characterized by a + very large electric monopole transition rate ρ(E0; 0+ 2 → 01 ) (Clement et al. 2007). The density distribution of the collective states, which takes the form ρI α (β, γ ) =



I |ψαK (β, γ )|2 β 3 ,

(85)

K∈ΔI

with the normalization "

"

∞

2π

βdβ 0

ρI α (β, γ )| sin(3γ )|dγ = 1,

(86)

0

provides further insight into shape coexistence. Figure 9 displays ρI α in the β-γ 76 plane for the first two 0+ states and the 2+ 1 state in Kr. Obviously, the dominant configurations of the ground-state band correspond to the large prolate deformed shape at (β ∼ 0.50). The distribution of probability density ρI α (β, γ ) indicates a prolate-oblate mixed configuration for the 0+ 2 state. From the potential energy 76 surface shown in Fig. 4, one notices that Kr is rather soft with respect to γ deformation, and this is reflected in the structure of the 0+ 2 state.

Fig. 8 (Color online) Low-spin excitation spectrum of 76 Kr (Fu et al. 2013), in comparison with available data (Clement et al. 2007; National Nuclear Data Center). The E2 transition strengths are in units e2 fm4

52 Model for Collective Motion

2001

Fig. 9 Distribution of the probability density ρI α (β, γ ) for the first two 0+ states and the 2+ 1 state in 76 Kr (Fu et al. 2013)

Quadrupole-Octupole Collective Hamiltonian for Pear-Shaped Nuclei Even though most deformed medium-heavy and heavy nuclei exhibit quadrupole, reflection-symmetric equilibrium shapes, there are regions in the mass table where octupole deformations (pear shapes) occur, in particular nuclei with neutron (proton) number N(Z) ≈ 34, 56, 88 and 134. Pear shapes are characterized by the occurrence of low-lying negative-parity bands (c.f. Fig. 10), as well as pronounced electric octupole transitions (Gaffney et al. 2013). The physics of octupole correlations was extensively explored several decades ago (see the review of Butler and Nazarewicz 1996), but there has also been a strong revival of interest in pear shapes more recently, as shown by a series of experimental studies (Butler et al. 2019, 2020; Chishti et al. 2020). The quadrupole-octupole collective Hamiltonian (QOCH), which can simultaneously treat the axially symmetric quadrupole and octupole vibrational and rotational excitations, is expressed in terms of two deformation parameters β2 and β3 and the Euler angles θ that define the orientation of the intrinsic principal axes in the laboratory frame: Hˆ coll

$ ∂ I ∂ I ∂ h¯ 2 ∂ B33 B23 =− √ − ∂β2 ∂β2 w ∂β3 2 wI ∂β2 w # I ∂ I ∂ ∂ ∂ B23 B22 − + ∂β3 w ∂β2 ∂β3 w ∂β3 +

(87)

Jˆ2 + Vcoll (β2 , β3 ). 2I

Jˆ denotes the component of angular momentum perpendicular to the symmetry axis in the body-fixed frame of a nucleus. The mass parameters B22 , B23 , and B33 , the moments of inertia I , and collective potential Vcoll depend on the quadrupole

2002

Z. P. Li and D. Vretenar

Fig. 10 The low-lying rotational spectrum of 224 Ra, compared with that of the H35 Cl molecule (Butler and Nazarewicz 1996). Schematic shapes for the two systems are also shown

and octupole deformation variables β2 and(β3 . The additional quantity that appears 2 , determines the volume in the vibrational kinetic energy, w = B22 B33 − B23 element in the collective space. Just as in the case of the 5DCH model, all the collective parameters are calculated from the self-consistent solutions of constrained reflection-asymmetric CDFT, using cranking formulas (Xia et al. 2017). As an illustrative example, the CDFT-based QOCH is used to calculate the lowenergy excitation spectra and electromagnetic transitions of Ra isotopes (Sun et al. 2019). In Fig. 11 it is shown that already at the mean-field level, the calculation predicts a very interesting structural evolution in Ra isotopic chain. Quadrupole deformation increases starting from 222 Ra, and one also notices the emergence of octupole deformation with β3 ∼ 0.11. For 224,226 Ra the occurrence of a rather strongly marked octupole minimum is predicted. The deepest octupole minimum is calculated in 226 Ra, and the octupole deformation energy is ∼0.94 MeV. In 228 Ra the deformation energy surface exhibits a softer minimum in the β3 direction, and the octupole deformation starts to decrease. In Fig. 12 the low-energy excitation spectra of positive- and negative-parity states, the corresponding B(E2) values for intraband transitions, and the interband B(E3) values, calculated with the QOCH based on the PC-PK1 functional, are compared with data for the nuclei 222−228 Ra (Gaffney et al. 2013). The level

52 Model for Collective Motion

2003

Fig. 11 Deformation energy surfaces of the nuclei 222−228 Ra (Sun et al. 2019) in the β2 -β3 plane calculated with the CDFT, using the PC-PK1 functional (Zhao et al. 2010). For each nucleus the energies are normalized with respect to the binding energy of the global minimum. Contour lines are separated by 0.5 MeV (between neighboring solid curves) and 0.25 MeV (between neighboring dashed and solid curves), respectively. The density distributions of the global equlibrium minima of 224,226 Ra are also shown in panels on the right

schemes show that the lowest negative-parity bands are located close in energy to the corresponding ground-state positive-parity bands. In fact, one notices that the lowest positive- and negative-parity bands form a single, alternating-parity band, starting with angular momentum J ≈ 5. Except for the negative-parity bandheads in 222 Ra and 228 Ra that are calculated somewhat higher and lower than their corresponding experimental counterparts, respectively, the overall structure agrees very well with the available data. The calculated E2 and E3 transition rates are also in reasonable agreement with the experimental values. The shape transition is also confirmed by other characteristic collective observables, such as + + + + + the B(E3; 3− 1 → 01 ) and B(E2; 21 → 01 ). The B(E2; 21 → 01 ) values 222 228 increase from Ra to Ra, indicating an enhancement of quadrupole collectivity. + The theoretical B(E3; 3− 1 → 01 ) values can be used as a measure of octupole collectivity, and exhibit a maximum in 226 Ra, together with E3 rates for higher spin states.

Time-Dependent GCM+GOA for Nuclear Fission As a final example, the time-dependent generator coordinate method plus Gaussian overlap approximation (TDGCM+GOA) is applied to nuclear fission dynamics (Tao et al. 2017). In the exothermic process of fission decay, an atomic nucleus splits into two or more independent fragments. A quasi-stationary initial state evolves

2004

Z. P. Li and D. Vretenar

Fig. 12 (Color online) The excitation spectrum, intraband B(E2) (in W.u.), and interband B(E3) (in W.u.) values of 222−228 Ra, calculated with the QOCH based on the functional PC-PK1 and compared to experimental results (National Nuclear Data Center; Gaffney et al. 2013; Butler et al. 2020)

in time through a sequence of increasingly deformed shapes. The system reaches the outer saddle point on the deformation energy surface and continues to deform, while a neck appears that eventually becomes so thin that scission occurs. Fission can be described as a slow adiabatic process determined by only a few collective degrees of freedom, for example, the quadrupole β2 and octupole β3 deformation parameters. In the TDGCM+GOA fission dynamics is modeled by a local, timedependent Schrödinger-like equation in the space of collective coordinates i h¯

∂g(β2 , β3 , t) = Hˆ coll (β2 , β3 )g(β2 , β3 , t), ∂t

(88)

with the collective Hamiltonian 2

h¯ Hˆ coll (β2 , β3 ) = − 2

 ∂ ∂ [B −1 (β2 , β3 )]kl + V (β2 , β3 ). ∂βk ∂βl kl

(89)

52 Model for Collective Motion

2005

Fig. 13 Potential energy surface of 228 Th in the (β2 , β3 ) plane, calculated using the reflectionasymmetric CDFT with the PC-PK1 functional. The nucleon density distributions at (β2 , β3 ) = (3.6, 2.56) and (5.36, 0) are shown in the panels on the right

g(β2 , β3 , t) is a time-dependent complex wave function in the (β2 , β3 ) plane. Equation (88) describes how an atomic nucleus characterized by the collective mass Bkl evolves in time in the collective potential V . As an illustration, the left panel of Fig. 13 displays the potential energy surface of 228 Th, calculated using CDFT with the PC-PK1 functional (Zhao et al. 2010). From (β2 , β3 ) ≈ (0.9, 0.0) to ≈(3.6, 2.56), an asymmetric fission valley extends on the energy surface, with two saddle points located at (β2 , β3 ) ≈ (1.2, 0.35) and ≈(2.1, 1.0). For elongations β2 > 2.0, a symmetric valley extends up to the scission point at β2 ≈ 5.36. The symmetric and asymmetric fission valleys are separated by a ridge from (β2 , β3 ) ≈ (1.6, 0.0) to ≈(3.8, 1.2). In the panels on the right of Fig. 13, the nucleon density distribution is plotted at (β2 , β3 ) = (3.6, 2.56) and (5.36, 0), two points on the potential energy surface that are close to scission. The nascent fission fragments can be clearly identified, and, therefore, the number of particles in the connecting neck can be defined: "

"

2π

Q=

dϕ 0

"

∞

rdr 0

+∞ −∞

  ρ(r, ϕ, z) exp −(z − zn )2 dz

(90)

where ρ is the nucleon density distribution in cylindrical coordinates and zn is the point of minimum density on the z-axis. When the number of nucleons in the neck region falls below a certain value, typically few nucleons, scission occurs. Considering the entire PES, the scission points define the scission hypersurface. Given an appropriate initial state, Eq. (88) evolves the collective wave function from the inner region with a single nuclear density distribution to the external region that contains the two fission fragments (see the illustration in Fig. 14). The flux of the probability current through the scission hypersurface provides a measure of the probability of observing a given pair of fragments at time t.

2006

Z. P. Li and D. Vretenar

Fig. 14 Time evolution of the probability density |g|2 in the (β2 , β3 ) plane. The solid contour corresponds to the scission hypersurface

The expression for the probability current J in the collective space reads Jk (β2 , β3 , t) =

3 h¯  Bkl (β2 , β3 )[g ∗ (β2 , β3 , t) 2i

(91)

l=2

∂g(β2 , β3 , t) ∂g ∗ (β2 , β3 , t) − g(β2 , β3 , t) ]. ∂βl ∂βl The integrated flux for a given scission surface element associated with a specific pair of fragments is computed from "

∞

Y =

 dt J · ds.

(92)

0

By considering all points on the scission contour, the final charge or mass distribution of fission fragments is obtained. Figure 15 displays the charge yields (normalized to 200%) for induced fission of 228 Th, calculated using the TDGCM+GOA, in comparison with data of photo-induced fission at an excitation energy ≈11 MeV (Schmidt et al. 2001). The calculation reproduces the trend of the data except that, without particle number projection, the model obviously cannot describe the oddeven staggering of the experimental charge yields.

52 Model for Collective Motion

2007

Fig. 15 The charge yields (normalized to 200%) for induced fission of 228 Th, calculated using the TDGCM+GOA, in comparison with data of photo-induced fission at an excitation energy ≈11 MeV (Schmidt et al. 2001)

Further Reading This chapter has focused on the basic features of large-amplitude collective motion in atomic nuclei. Various additional topics related to nuclear collective motion are thoroughly covered in the following textbooks: • • • •

P. Ring and P. Schuck, The nuclear many-body problem, Springer-Verlag, 2004. W. Greiner and J. A. Maruhn, Nuclear models, Springer-Verlag, 1996. D. J. Rowe, Nuclear collective motion, World Scientific, 2010. D. J. Rowe and J. L. Wood, Fundamentals of nuclear models, World Scientific, 2010. • W. Younes and W. D. Loveland, An introduction to nuclear fission, SpringerVerlag, 2021. Different aspects of quadrupole collective dynamics, and more detailed derivations of specific implementations of the collective model, are discussed in review articles: • L. Próchniak, and S. G. Rohozi´nski, Quadrupole collective states within the Bohr collective Hamiltonian, J. Phys. G: Nucl. Part. Phys. 36, 123101 (2009). • T. Nikši´c, D. Vretenar, and P. Ring, Relativistic nuclear energy density functionals: Mean-field and beyond, Prog. Part. Nucl. Phys. 66, 519 (2011).

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• S. G. Rohozi´nski, Gaussian overlap approximation for the quadrupole collective states, J. Phys. G: Nucl. Part. Phys 39, 095104 (2012). • S. G. Rohozi´nski, On the Gaussian overlap approximation for the collective excitations of odd nuclei, J. Phys. G: Nucl. Part. Phys. 42, 025109 (2015). • K. Matsuyanagi, M. Matsuo, T. Nakatsukasa, K. Yoshida, N. Hinohara, and K. Sato, Microscopic derivation of the quadrupole collective Hamiltonian for shape coexistence/mixing dynamics, J. Phys. G: Nucl. Part. Phys 43, 024006 (2016). • J. L. Egido, State-of-the-art of beyond mean field theories with nuclear density functionals, Phys. Scr. 91, 073003 (2016).

References B. Banerjee, D.M. Brink, Z. Phys. A 258, 46 (1973) M. Baranger, K. Kumar, Nucl. Phys. A 122, 241 (1968) M. Baranger, M. Vénéroni, Ann. Phys. (NY) 114, 123 (1978) S.T. Belyaev, Nucl. Phys. 24, 322 (1961) S.T. Belyaev, Nucl. Phys. 64, 17 (1965) A. Bohr, Dan. Mat. Fys. Medd. 26, 14 (1952) A. Bohr, B.R. Mottelson, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 27, 16 (1953) D.M. Brink, A. Weiguny, Nucl. Phys. A 120, 59 (1968) P.A. Butler, W. Nazarewicz, Rev. Mod. Phys. 68, 349 (1996) P.A. Butler et al., Nat. commun. 10, 2473 (2019) P.A. Butler et al., Phys. Rev. Lett. 124, 042503 (2020) M.M.R. Chishti et al., Nat. Phys. 16, 853–856 (2020) E. Clement et al., Phys. Rev. C 75, 054313 (2007) T.M. Corrigan, F.J. Margetan, S.A. Williams, Phys. Rev. C 14, 2279 (1976) A.S. Davydov, Quantum Mechanics, 1st edn. (Pergamon Press Ltd., 1965), pp. 157–161 A.S. Davydov, A.A. Chaban, Nucl. Phys. 20, 499 (1960) A.S. Davydov, G.F. Filippov, Nucl. Phys. 8, 237 (1958) A. Faessler, W. Greiner, Z. Phys. 168, 425 (1962a) A. Faessler, W. Greiner, Z. Phys. 170, 105 (1962b) A. Faessler, W. Greiner, R.K. Sheline, Nucl. Phys. 80, 417 (1965) S. Flügge, Ann. Phys. Lpz. 431, 373 (1941) L. Fortunato, Eur. Phys. J. A 26, 1 (2005) Y. Fu, H. Mei, J. Xiang, Z.P. Li, J.M. Yao, J. Meng, Phys. Rev. C 87, 054305 (2013) L.P. Gaffney, P.A. Butler, M. Sheck et al., Nature 497, 199 (2013) B. Giraud, B. Grammaticos, Nucl. Phys. A 233, 373 (1974) M. Girod, B. Grammaticos, Nucl. Phys. A 330, 40 (1979) K. Goeke, P.-G. Reinhard, Ann. Phys. (NY) 112, 328 (1978) K. Goeke, P.-G. Reinhard, Ann. Phys. (NY) 124, 249 (1980) A. Gó´zd´z, K. Pomorski, M. Brack, E. Werner, Nucl. Phys. A 442, 26 (1985) P.K. Haff, L. Wilets, Phys. Rev. C 7, 951 (1973) K. Heyde, J.L. Wood, Rev. Mod. Phys. 83, 1467 (2011) D.L. Hill, J.A. Wheeler, Phys. Rev. 89, 1102 (1953) D.R. Inglis, Phys. Rev. 103, 1786 (1956) K. Kumar, M. Baranger, Nucl. Phys. A 92, 608 (1967) J. Libert, M. Girod, J.-P. Delaroche, Phys. Rev. C 60, 054301 (1999) National Nuclear Data Center, Brookhaven National Laboratory, http://www.nndc.bnl.gov/ T. Nikši´c, Z.P. Li, D. Vretenar, L. Prochniak, J. Meng, P. Ring, Phys. Rev. C 79, 034303 (2009)

52 Model for Collective Motion

2009

N. Onishi, T. Une, Local Gaussian approximation in the generator coordinate method. Prog. Theor. Phys. 53, 504 (1975) R.B. Piercey et al., Phys. Rev. Lett. 47, 1514 (1981) L. Prochniak, P. Quentin, D. Samsoen, J. Libert, Nucl. Phys. A 730, 59 (2004) L. Rayleigh, Proc. R. Soc. 29, 71 (1879) P. Ring, P. Schuck, The Nuclear Many-Body Problem, 1st edn. (Springer, Berlin/Heidelberg, 1980), pp. 424–428 D.J. Rowe, Nuclear Collective Motion, 1st edn. (World Scientific Publishing Co. Pte. Ltd., 1970), Singapore, pp. 4–17 K.-H. Schmidt, J. Benlliure, A.R. Junghans, Nucl. Phys. A 693, 169 (2001) W. Sun, S. Quan, Z.P. Li, J. Zhao, T. Nikši´c, D. Vretenar, Phys. Rev. C 100, 044319 (2019) H. Tao, J. Zhao, Z.P. Li, T. Nikši´c, D. Vretenar, Phys. Rev. C 96, 024319 (2017) T. Une, A. Ikeda, N. Onishi, Prog. Theor. Phys. 55, 498 (1976) F. Villars, Nucl. Phys. A 285, 269 (1977) C.F.V. Weizsäcker, Z. Physik. 96, 431 (1935) L. Wilets, M. Jean, Phys. Rev. 102, 788 (1956) S.Y. Xia, H. Tao, Y. Lu et al., Phys. Rev. C 96, 054303 (2017) J.M. Yao, K. Hagino, Z.P. Li, J. Meng, P. Ring, Phys. Rev. C 89, 054306 (2014) P.W. Zhao, Z.P. Li, J.M. Yao, J. Meng, Phys. Rev. C 82, 054319 (2010)

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Xiang-Xiang Sun and Shan-Gui Zhou

Contents Effects of Nucleon Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Pairing Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Pairing Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Seniority Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The BCS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The BCS Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The BCS Approximation with Resonant States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Generalized Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hartree-Fock-Bogoliubov Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selected Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blocking Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Issues with Particle Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exact Solutions for Pairing Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle Number Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2012 2015 2015 2017 2018 2020 2020 2024 2026 2027 2029 2034 2035 2035 2037 2039 2040

X.-X. Sun School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China e-mail: [email protected] S.-G. Zhou () CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_12

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Abstract

Pairing effects manifest themselves in many aspects in nuclear systems ranging from finite nuclei to nuclear matter and compact stars. Although with some specific features for nuclear systems, the mechanism of pairing between nucleons in these systems resembles that of electrons in superconductors. The BardeenCooper-Schrieffer (BCS) theory, the first successful and microscopic theory for superconductivity, and the Bogoliubov transformation, the generalization of the BCS theory, have been widely used to describe pairing correlations in nuclear systems. To deal with the problem of particle number nonconservation in the BCS method and generalized Bogoliubov transformation, particle number projection techniques as well as several approaches which keep the particle number conserved have been proposed. In the study of exotic nuclei, which are quantum open systems, the continuum contributions have to be taken into account. In this chapter, a thorough but brief discussion of pairing effects in nuclear systems will be introduced. Then nuclear models dealing with pairing correlations in nuclear structure properties will be presented to different extent of details. Although formulas are given, the emphasis is mainly put on the basic ideas concerning these models.

Effects of Nucleon Pairing Since the early stage of nuclear physics, it has been known that an atomic nucleus is more stable if it consists of even number(s) of protons and/or neutrons, as is clearly seen both from the fact that there are much more even-even stable isotopes than oddmass and odd-odd ones and from the odd-even effects in nuclear masses or binding energies. Such observations indicate that protons (neutrons) like to be coupled into pairs in atomic nuclei. Indeed, pairing effects have been observed in many essential nuclear phenomena (Bohr and Mottelson 1998a, b; Ring and Schuck 1980; Dean and Hjorth-Jensen 2003; Brink and Broglia 2005; Broglia and Zelevinsky 2013): The odd-even effect in binding energy The binding energy of an odd-even nucleus is found to be smaller than the arithmetic mean of binding energies of its two neighboring even-even nuclei. One can verify this fact by calculating, e.g., the neutron gaps for even-even nuclei 1 Δ(N, Z) = − [EB (Z, N − 1) + EB (Z, N + 1) − 2EB (Z, N )], 2

(1)

with the latest nuclear masses given in AME2020 (Kondev et al. 2021; Huang et al. 2021; Wang et al. 2021) and can find that they are almost larger than 1 MeV systematically. N and Z are the number of neutrons and charge number, respectively. Similar conclusions can also be drawn for the proton gaps. Note that, besides pairing effects, the odd-even mass staggering is also related to other

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mechanisms (Satula et al. 1998; Xu et al. 1999; Dobaczewski et al. 2001), such as the shell structure, single-particle properties, and deformations. Energy spectra The ground states of all even-even nuclei have the spin-parity I π = 0+ , which can be understood as that two nucleons occupying time-reversal states couple with each other via the pairing interaction. For an even-even nucleus, the excitation energy of the first non-collective excited state can be well estimated as the energy corresponding to a broken pair (Ring and Schuck 1980). The situation in an odd-even nucleus is different: The ground state of an oddeven nucleus is almost completely determined by the last unpaired nucleon, and it can have many excited states in the same energy interval. The moment of inertia The moment of inertia of deformed nuclei can be extracted from the rotational spectra. The systematics of excitation spectra show that the moment of inertia from the pure single-particle scheme deviates by a factor of 2 from the experimental values (Ring and Schuck 1980). But when the pairing is included, theoretical calculations are in good agreement with experimental data. This is due to the fact that the paired nucleons with the spin zero contribute little to the rotational angular momentum (von Oertzen and Vitturi 2001). The level density In the case of a few nucleons occupying a single j -shell, the energy spectra can be constructed and are all energetically degenerate corresponding to the various possibilities of angular momentum coupling. The number of states per energy unit can easily be estimated, and it is found that the energy spectrum is directly related to the pairing strength. Deformations In the well-accepted mean-field picture, if proton or neutron orbitals below a major shell are all occupied, the nucleus generally has a spherical shape in its ground state. However, for open-shell nuclei, single-particle levels can be partially filled due to pairing correlations. The quantum correlations originating from the mix of different single-particle orbitals drive these nuclei to be deformed. This is reflected by the fact that there is a shape transition from spherical shapes for closed-shell nuclei to well-deformed shapes for nuclei with half-filled shells (Ring and Schuck 1980). The nuclear fission The spontaneous fission is a decay mode in which a nucleus splits into two or more lighter nuclei. This process requires the system to tunnel under the fission barrier. The character of the system, whether it is superfluid or not, affects much the fission dynamics (Bertsch 2012). The occurrence of a systematic difference between even- and odd-mass nuclei in the fission of, e.g., U isotopes is associated with pairing effects (Bohr and Mottelson 1998b). Halos Most of halo nuclei are close to or located at drip lines (Tanihata et al. 2013). For nuclei close to drip lines, the valence nucleons are weakly bound, and pairing correlations provide the possibility of scattering valence nucleons back and forth in the continuum. Once the valence nucleons occupy orbitals with low orbital angular momenta, say s- or p-wave, a halo can be formed. Therefore pairing correlations play a crucial role in the formation of nuclear halos. The occupation of s- or p-wave orbitals by weakly bound nucleon(s) leads to that halo nuclei have a pure neutron (proton) matter with a very low density surrounding

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a dense core (Meng and Ring 1996; Meng et al. 2006; Meng and Zhou 2015), where pair condensate or strong di-neutron correlations may occur (Hagino et al. 2007; Sagawa and Hagino 2015). Thus, halo nuclei may be an ideal prototype to search for evidence of induced pairing through surface vibrational coupling (Kanungo and Tanihata 2012). The pygmy dipole resonances (Nakamura et al. 2006; Kanungo et al. 2015) as a result of the oscillation between the core and the low-density neutron matter are also connected with pairing correlations. Furthermore, halo configurations are related to other aspects of nuclear structures, including deformations and the shell evolution, and the interplay among them results in more fascinating and complicated phenomena (Nakamura et al. 2014; Zhou 2017; Sun et al. 2018; Sun and Zhou 2021). Two-nucleon decays Nuclei are unbound with respect to nucleon(s) emission beyond the drip lines. This feature implies some new radioactivities of which mostly discussed are one- or two-nucleon decays. A handful of ground state twoproton emitters have been observed due to the Coulomb interaction which leads to a Coulomb barrier hindering the escape of protons from the parent nucleus (Blank and Płoszajczak 2008; Pfützner et al. 2012). There may be also twoneutron radioactivity and the only candidate identified so far is 26 O (Lunderberg et al. 2012). The information of two-nucleon decay is particularly important for revealing the angular and energy correlations between emitted nucleons (Michel et al. 2010), which are highly correlated to the pairing at the initial stage of the process. The backbending In a well-deformed superfluid nucleus, the rotational spectrum follows the I (I + 1) law, and the moment of inertia is almost a constant below a certain critical value of rotational frequency. In some deformed nuclei, the observed spectra indicate that there is a very steep increase of moment of inertia at high angular momenta. This is because when a deformed nucleus starts to rotate, the Coriolis force acts in opposite directions on the two nucleons of each time-reversal pair. As a consequence, the rotation tends to align the spins of the nucleons by successive breaking of pairs with high angular momentum. The neutron-proton pairing For nuclei with N ≈ Z, the protons and neutrons near the Fermi levels occupy identical orbitals, which allows for the appearance of pairs consisting of a neutron (n) and a proton (p). The binding energies show the characteristic T (T + 1) isorotational dependence on isospin T , which means the presence of an isovector pair condensate that rotates in isospace. Such rotation implies that the existence of the np condensate is on an equal footing with the nn and pp condensates (Frauendorf and Macchiavelli 2014). Pairing rotations The energies of I π = 0+ states relative to that of a reference nucleus are quadratic as a function of the number of additional pairs. A typical example is the energies of 0+ states of Sn isotopes with the reference nucleus 116 Sn (Broglia et al. 2000; Potel et al. 2011). This parabolic behavior is similar to the rotational band in deformed nuclei. The pairing rotation in atomic nuclei is one kind of Nambu-Goldstone modes as a sequence of the spontaneous symmetry breaking in the U (1) gauge space, namely, the condensation of Cooper pairs (Broglia et al. 2000; Broglia 1973; Brink and Broglia 2005; Potel et al. 2013).

53 Models for Pairing Phenomena Table 1 Possible types of nucleon-nucleon pairs with isospin (T and Tz ) and total spin (S)

Type pp pn nn pn

2015 T 1 1 1 0

Tz −1 0 1 0

S 0 0 0 1

In one word, pairing manifests itself in many aspects of nuclear physics. Besides the abovementioned nuclear structure features, pairing is also related to nuclear astrophysics, such as the thermal evolution of neutron stars and glitches in pulsar stars (Dean and Hjorth-Jensen 2003). In nuclear reactions, a typical example is that the enhanced two-nucleon transfer cross section is understood as arising from the collective pairing states (von Oertzen and Vitturi 2001). This chapter mainly focuses on pairing effects in the study of nuclear structure.

The Pairing Mechanism The Pairing Forces Atomic nuclei are quantum many-body systems composed of protons and neutrons. Apart from relatively weak electric forces, the interaction between two protons is very similar to that between two neutrons. The isospin degree of freedom is used to distinguish proton (τ = 1/2, τz = −1/2) and neutron (τ = 1/2, τz = +1/2), and a nuclear state can be labeled with the isospin quantum number T , with the third component Tz = (N − Z)/2 for a nucleus with N neutrons and Z protons. When the relative orbital angular momentum L is even, two nucleons can couple to T = 0 (isoscalar) and T = 1 (isovector) pairs with the spin S of 1 or 0, respectively. Thus two neutrons can form a pair with T = 1, Tz = 1, and S = 0 and two protons with T = 1, Tz = −1, and S = 0. A neutron and a proton can couple to S = 0 and isospin T = 1 with Tz = 0 (isovector) or T = 0 (isoscalar) and S = 1 in order to ensure the antisymmetry of the total wave function of the two nucleons. The possible types of nucleon-nucleon pairs are listed in Table 1. Pairing correlations and the phenomena associated with superfluidity in nuclear physics directly rely on the underlying interaction, i.e., the nucleon-nucleon (NN) interaction. Nowadays, the bare NN interaction, the interaction of two nucleons in the free space, can be constructed by using some methods to reproduce nucleonnucleon scattering phase shifts in different partial waves labeled by 2S+1 LJ , where S, L, and J represent the total spin, orbital, and total angular momenta, respectively. The bare NN interaction is characterized by a strong repulsive core at short distances. The pairing gap is mainly determined by the attractive part of N N interactions. In the 1 S0 channel, the NN interaction is attractive when the internucleon distance is larger than ∼0.6 fm. In pure neutron matter, neutrons can form Cooper pairs in the weak-coupling limit with this attractive interaction. One can also expect that the system undergoes a Bose-Einstein condensation into a single

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quantum state in the strong-coupling regime (Dean and Hjorth-Jensen 2003). The pairing interactions in other channels are also very important, especially for the study of dynamical and thermal evolution of neutron stars. For finite nuclei, although various approaches starting from realistic nuclear forces have been developed and produced encouraging results, there are still some issues to be explored further, such as the understanding of pairing correlations from bare NN interactions, the influence of the medium polarization on pairing in finite nuclei, and impacts from Coulomb and three-nucleon forces (Duguet 2012). Thus it is more straightforward to understand the pairing force in a phenomenological way. The idea of a pairing interaction was already proposed in the early developments of the traditional shell model (Mayer and Jensen 1955), in which single particles move under a central potential with a strong spin-orbit interaction. One of the most important effects of the pairing interaction is that it can couple two identical nucleons to spin zero. This can be understood in the case of two nucleons interacting with a short-range attractive effective interaction in a single j -shell. The simplest example is the δ-interaction with the strength V0 , V (r12 ) = −4π V0 δ(r 1 − r 2 ).

(2)

Two identical nucleons in a shell model orbital with angular momentum j coupling to a total angular momentum J have a wave function |jj J M, and the interaction energy is  2   1  1  2j + 1  V0 I (j )  Jj 0 j  , EJ = jj J M|V (r12 )|jj J M = − 2 2 2

(3)

where  I (j ) =

2 dr12 , Rj2 (r12 )r12

(4)

  depending on the radial wave function Rj (r12 ) of the level j , and is an integral  Jj 0 12 j 12 is the Clebsch-Gordan coefficient. The energy of the J = 0 state can be simplified as E0 = −

2j + 1 V0 I (j ), 2

(5)

and the energies of other states can be obtained, E2 ∼

1 E0 , 4

E4 ∼

9 E0 , 64

E6 ∼

25 E0 , · · · . 256

(6)

This result shows that the interaction energy of the J = 0 state is much smaller than other states and thus the J = 0 pair is energetically favored.

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Besides this δ interaction, many phenomenological pairing forces have been proposed, such as a constant pairing force (Ring and Schuck 1980), zero-range pairing force with or without density dependence (Dobaczewski et al. 1996; Meng 1998), Gogny force (Dechargé and Gogny 1980), and separable pairing force of finite range (Tian et al. 2009). They have been widely applied in modern nuclear density functional theories and can describe nuclear superfluidity successfully (Bender et al. 2003; Meng et al. 2006; Vretenar et al. 2005; Robledo et al. 2019). Moreover, to simultaneously describe the density dependence of the neutron pairing gap for both symmetric and neutron matter, an isospin dependence in the effective pairing interaction has also been developed (Margueron et al. 2007; Zhang et al. 2010) and applied to study the properties of finite nuclei (Margueron et al. 2008; Yamagami et al. 2009; Chen et al. 2015). As mentioned above, the long-range attractive part of the bare NN force can lead to the nuclear superfluidity. Thus many efforts have been made to incorporate realistic forces, such as low-momentum NN interactions and Argonne v14 and v18 , to investigate nuclear structure by using mean-field methods (Barranco et al. 2004; Duguet and Lesinski 2008; Hebeler et al. 2009; Pankratov et al. 2011). It has been shown that energy gaps for semi-magic nuclei can be reproduced in such kinds of calculations. It should be noted that the implementation of realistic forces is technically much more complicated than phenomenological pairing forces due to the complexities of bare NN forces. The microscopic understanding of pairing correlations starting from the underlying forces is still an open question up to now.

Pairing Models Various approaches have been developed to understand the pairing phenomena in nuclear systems. The seniority model (Racah 1942; Mayer 1950) solves the problem of N particles occupying the single j -shell and can be used to understand why the ground states of even-even nuclei have spin zero and the spin of an oddmass nucleus is the same as the spin of the last unpaired nucleon. In the 1950s the superconductivity in metals was understood by that two electrons of opposite momenta attract each other to form a bound state with zero momentum, which is known as the Cooper pair and behaves like a boson (Cooper 1956; Bardeen et al. 1957). This is the basic idea of the BCS theory of superconductivity (Bardeen et al. 1957). The characteristic of a metallic superconductor is primarily the large energy gap in the spectrum, corresponding to the energy required to break a Cooper pair. Following the suggestions of Bohr et al. (1958), the BCS theory was used to study atomic nuclei by Belyaev (1959). The BCS wave function can be generalized by using the Bogoliubov-Valatin transformation (Bogoljubov 1958; Valatin 1961). Both the BCS approximation and generalized Bogoliubov transformation have been widely applied in various mean-field models nowadays to incorporate pairing correlations. Since the BCS-type wave function does not keep the conservation of particle number, to deal with this problem many methods have been proposed, such as the Lipkin-Nogami method (Lipkin 1960; Nogami

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1964), exact solutions (Richardson and Sherman 1964a, b) and particle number conserved methods (Zeng and Cheng 1983) for the pairing Hamiltonian, and the particle number projection technique in mean-field models (Sheikh and Ring 2000; Anguiano et al. 2001, 2002). In the following sections, the basic formulas of these methods and several relevant topics are introduced.

The Seniority Model The ground state of an even-even nucleus has the spin-parity I π = 0+ , and the spin of an odd-even nucleus is the same as that of the last unpaired nucleon. This fact can be understood by using the seniority model, in which N nucleons interact through a constant pairing force in a single j -shell with the degeneracy of 2j + 1 (Racah 1942). Setting single-particle energies of the orbitals in the j -shell to be zero, then the pairing Hamiltonian is H = −G

 m, m >0

† † ˆ ˆ am  a−m a−m am = −GS+ S− ,

(7)

where m is the projection of j on the z-axis and the pair creation and annihilation operators are defined as Sˆ+ ≡

 m

(m)

s+ =

Ω 

† † am a−m ,

 † Sˆ− = Sˆ+ ,

(8)

m=1

with the number of paired states Ω = (2j + 1)/2. The quasi-spin operator proposed in Kerman (1961) is one of the methods to solve this problem and will be introduced here. For each substate m, let (m) † † s+ = am a−m , (m)

s− = a−m am ,  1 † (m) † am am + a−m s0 = a−m − 1 . 2

(9)

One can get the commutation relations as

(m) (m) (m) s+ , s− = 2s0 ,

(m) (m) (m) s0 , s + = s + ,

(m) (m) (m) s0 , s− = −s− .

(10)

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It is found that such commutation properties are the same as those of angular momentum operators. s0(m) has two eigenvalues of −1/2 and 1/2 corresponding to whether the pair (m, −m) is full or empty. Thus the operator s (m) is similar to the spin operator and is called “quasi-spin.” The total spin vector is S=



s (m) ,

(11)

m>0

and its third component is S0 =



(m)

s0

m>0

=

  1 1  † † Nˆ − Ω , am am + a−m a−m − 1 = 2 2

(12)

m>0

ˆ with the particle number operator N. The pairing Hamiltonian can be rewritten as   H = −G S 2 − S02 + S0 ,

(13)

and the pairing energy reads  1 1 E(N, S) = −G S(S + 1) − (N − Ω)2 + (N − Ω) . 4 2

(14)

When the number of particles is even, S = Ω/2, Ω/2 − 1, · · · , and |N − Ω|/2. When the particle number is odd, S = (Ω − 1)/2, (Ω − 3)/2, · · · , |N − Ω|/2. If one defines S = (Ω − σ )/2 with σ being the seniority quantum number, the energy is E(N, σ ) = −

G σ (σ + 1) − 2σ (Ω + 1) + 2N(Ω + 1) − N 2 . 4

(15)

The seniority quantum number represents the number of unpaired particles in j shell. The corresponding pairing gaps are G(2Ω + 1)/2 for even particle number and GΩ for odd particle number. This simple model shows that • The ground state for an even system has the minimal seniority σ = 0 and σ = 1 for an odd system; • For even N, the first excited state is given by σ = 2 and the excitation energy is GΩ, which is independent on N; • When the particle number is much smaller than the degeneracy of single j -shell (N  Ω), the ground state energy increases with GΩ multiplied by the number of pairs. This is related to the pair vibrational spectrum, and a typical example is neutron pair vibration based on 208 Pb (Mottelson 1976).

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X.-X. Sun and S.-G. Zhou

The nucleus 210 Po can be taken an example. Its low-lying excited states with = 2+ , 4+ , and 6+ are nearly degenerate, and the excitation energy of the 2+ state is 1.18 MeV. This can be well explained by the seniority scheme as two protons occupying the 1h9/2 orbital, which gives the excitation energy of the first excited state about 1 MeV with the pairing gap estimated by the empirical formula (13.43 ± 1.38)A−0.48±0.03 MeV (Changizi et al. 2015). In addition, the seniority model is useful for nuclei with the number of protons (neutrons) close to magic numbers for which the configurations can be well described by single j components. There are many states which are the mixture of several j configurations even in semi-magic nuclei (Gottardo et al. 2012; Simpson 2014). To study these nuclei, the seniority model has been generalized to the case of multi-j -shell (Arima and Ichimura 1966; Talmi 1971; Shlomo and Talmi 1972). Recently, the generalized seniority model has been successfully applied to study various properties of Sn isotopes (Morales et al. 2011; Jia and Qi 2016; Maheshwari and Jain 2016). Iπ

The BCS Model Although the seniority model is successful for the description of the energy gaps related with the nuclear superfluidity, it is limited to the study of nuclei close to magic ones. The BCS model was firstly proposed to study superfluidity in mentals (Bardeen et al. 1957) and later applied to nuclear systems. This method can be combined easily with the mean-field models as shown in Reinhard et al. (1986) and Ring (1996) and has been widely applied to study nuclei close to the β-stability line. In this section, the BCS theory will be presented.

The BCS Approximation In fermion systems like atomic nuclei, the Kramers degeneracy ensures the existence ¯ Nucleons of pairs of degenerate and mutually time-reversal conjugate states (k, k). occupying such states could be coupled strongly by a short-range force. Under the independent particle approximation, the ground state properties are described by filling the single-particle levels from the bottom up to the Fermi level. As will be shown below, in this case the occupation probability of each single-particle level is either zero or one. Due to the pairing interaction, pairs of nucleons can be scattered from the levels below the Fermi level to those above. One then has to deal with the occupation probabilities ranging from zero to one. Mathematically, this could be solved by introducing the concept of quasiparticles. For stable nuclei, the BCS approximation has turned out to be very useful. The model Hamiltonian with a single-particle term and a residual two-body interaction is

53 Models for Pairing Phenomena

H =



2021

εn1 n2 an†1 an2 +

n1 n2

1  v¯n n n n a † a † an an , 4 n n n n 1 2 3 4 n1 n2 4 3

(16)

1 2 3 4

where v¯n1 n2 n3 n4 = n1 n2 |v|n3 n4  − n1 n2 |v|n4 n3 . This problem can be solved by using the BCS approximation. The wave function for an even-even nucleus can be approximated by the BCS wave function as |BCS =

(uk + vk ak† ak†¯ )|0,

(17)

k>0

where uk and vk represent variational parameters. The product runs over half of the configuration space (k > 0). |0 represents the vacuum state of single particles. vk2 ¯ is occupied and and u2k represent the probability that a certain pair of states (k, k) empty, respectively. The norm of the state requires u2k + vk2 = 1.

(18)

It should be mentioned that generally uk and vk are complex and it is reasonable to choose real uk and vk to satisfy the variation principle (Ring and Schuck 1980). For k > 0, that means k¯ < 0, one has uk¯ := uk ,

vk¯ := −vk .

(19)

uk and vk can be determined in such a way that the total energy of the system is minimal. As is seen in Eq. (17), the BCS wave function does not conserve the particle number. Under the condition that the expectation value of the particle number operator has the desired value N ˆ BCS|N|BCS =2



vk2 = N,

(20)

k>0

the variational Hamiltonian reads H  = H − λNˆ .

(21)

λ is the chemical potential or Fermi energy, which represents the increase of the energy with respect to a change in the particle number, λ=

dE . dN

(22)

The particle number uncertainty reads (ΔN )2 := BCS|Nˆ 2 |BCS − N 2 = 4

 k>0

u2k vk2 .

(23)

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X.-X. Sun and S.-G. Zhou

The expectation value of H  with respect to the BCS state is   1 2 2 + v¯kk  kk  vk vk  + v¯k kk BCS|H |BCS = ¯  k¯ uk vk uk  vk  . 2  k k kk  (24) Under the condition of vk2 + u2k = 1, the BCS wave function can be determined after obtaining vk . The variation condition 





(εkk − λ)vk2

δBCS|H  |BCS = 0,

(25)

yields 

∂ vk ∂ BCS|H  |BCS = 0. − ∂vk uk ∂uk

(26)

After differentiating, one can get the BCS equation 2˜εk uk vk + Δk (vk2 − u2k ) = 0,

k > 0,

(27)

with ε˜ k = εkk +

1 2 (v¯kk  kk  + v¯kk ¯  kkk ¯  )vk  − λ, 2 

(28)

k

and Δk = −

 k  >0

v¯k kk ¯  k¯ uk  vk  .

(29)

There is a trivial solution for the BCS equation (cf. Eq. (27)), i.e., the pairing gap Δ = 0, vk2 = 1 and u2k = 0 for all single-particle states below the Fermi level (˜εk ≤ λ). The nontrivial solution of the BCS equation is ⎤ ⎡ 1 ε ˜ k ⎦, vk2 = ⎣1 −  2 2 2 ε˜ k + Δk

⎤ ⎡ 1 ε ˜ k ⎦. u2k = ⎣1 +  2 2 2 ε˜ k + Δk

(30)

Then the gap equation can be obtained by inserting Eq. (30) into Eq. (29) Δk = −

Δk  1 . v¯k k¯  k k¯  2  ε˜ k  + Δk 

(31)

k >0

In general, these equations are nonlinear and can be solved iteratively. One can get the occupation probability of each single-particle level determined by the mean-field

53 Models for Pairing Phenomena

2023

Fig. 1 Occupation number as a function of single-particle energy for pairing gap Δ = 0 (left panel) and Δ = 0 (right panel)

potential, shown in Fig. 1. For the trivial solution, vk2 is a step function of the singleparticle energy. In the case of finite pairing gap, the single-particle levels around the Fermi energy λ are partially occupied, and particles can be scattered from below to above the Fermi level due to pairing correlations. The abovementioned variation procedure usually leads to the ground state of the system, which can be compared with the results given by the seniority model to address the limitation of the BCS method when applying to nuclear systems. In the case of N particles occupying single j -shell, for the ground state with the seniority zero, one can find  vk =

N , 2Ω

 uk =

1−

N . 2Ω

(32)

The pairing energy with the BCS model is (N ) EBCS

  N N 1 , + = − GNΩ 1 − 2 2Ω 2Ω 2

(33)

and this is close to the result obtained from Eq. (15) when Ω  N. The uncertainty in the particle number is given by ΔN = N



N 2 − . N Ω

(34)

Thus the BCS approximation is suitable to study the ground state of a system with a large particle number. For those systems with small particle number, the uncertainty from the BCS method is relatively large. A way to suppress the error due to particle number fluctuation is the LipkinNogami approximation (Lipkin 1960; Nogami 1964), in which the variation Hamiltonian reads

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X.-X. Sun and S.-G. Zhou

ˆ − λ2 Nˆ 2 , H  = H − λ1 N

(35)

where λ1 represents the Lagrange multiplier used to constrain the average particle number and the parameter λ2 is determined by   Hˆ ΔNˆ 22 . λ2 =  Nˆ 2 ΔNˆ 22

(36)

Nˆ 2 is the term of the particle number operator which projects onto two-quasiparticle states and ΔNˆ 22 = Nˆ 22 − Nˆ 22  represents its variance. In this case, the Fermi energy is given by λ = λ1 + 4λ2 (N + 1). The Lipkin-Nogami method is an approximation to the particle number projected BCS theory and has also been applied in modern density functional calculations (Valor et al. 1996; Reinhard et al. 1996; Bender et al. 2000; Nikši´c et al. 2006) to consider the energy corrections caused by the particle number fluctuation. The wave function given in Eq. (17) corresponds to the case where all particles are paired. But for odd-mass or odd-odd nuclei, the valence particle is unpaired. Therefore, Eq. (17) cannot be directly used. One needs to block the valence orbital kb with vk2b = 1 and its time-reversal state is empty. The corresponding wave function reads (Ring and Schuck 1980; Zhou et al. 2001) ak†b

k>0, k =kb

(uk + vk ak† ak†¯ )|0.

(37)

Applying this wave function to solve the variation problem, one can find that the occupation probabilities for other states are the same as Eq. (30). The unpaired particle does  not contribute to the pairing gap and the total particle number is 1 + 2 k =kb vk2 . The changes on pairing gaps and occupation probabilities caused by blocking unpaired particle are called blocking effects. It should be mentioned that this blocking procedure breaks the time-reversal symmetry and leads to the appearance of currents, which can be avoided by using the equal filling approximation (Perez-Martin and Robledo 2008; Schunck et al. 2010). This method will be introduced in section “Blocking Effects”.

The BCS Approximation with Resonant States Nowadays, with the worldwide development of radioactivity-ion-beam facilities, more and more exotic nuclear phenomena have been observed (Tanihata et al. 2013; Nakamura et al. 2017; Zhou 2017; Yamaguchi et al. 2021). For a suitable description of exotic nuclei, the contribution from the continuum should be included (Dobaczewski et al. 2007; Michel et al. 2010; Johnson et al. 2020). However, the conventional BCS method is only valid for bound states and not justified for

53 Models for Pairing Phenomena

2025

Fig. 2 The schematic picture for the difference in pairing correlations of stable nuclei (left panel) and drip line nuclei (right panel). Normally the last occupied nucleon in stable nuclei has about 8 MeV binding, while it is near the threshold of particle emission in drip line nuclei. Pairing correlations in exotic nuclei provide the possibility of scattering valence nucleons back and forth in the continuum. (Taken from Meng et al. 2006)

exotic nuclei because it cannot include properly the contribution of continuum states (Bulgac 1999; Dobaczewski et al. 1984). For the extremely neutron-rich (or protonrich) nuclei near drip lines, the contribution from the continuum plays an important role as schematically shown in Fig. 2. The conventional BCS approach involves unphysical states, and the density contributed from continuum is nonlocal. For these nuclei, one must either investigate the detailed properties of continuum states and include the coupling between the bound state and the continuum by extending the BCS method to resonant BCS (rBCS) theory or treat pairing correlations by using the generalized Bogoliubov transformation. In the following, the way to include the contribution of resonant states within the rBCS method will be given. Resonant states are important for determining the pairing properties of the ground state of nuclei far from the β-stability line (Meng et al. 2006). Considering the contribution of continuum with the BCS approximation can make this method suitable for the study of exotic nuclei. This idea was first realized in the HartreeFock framework (Sandulescu et al. 2000). The main point is that the continuum can be discretized and regarded as a set of discrete states with the corresponding level density; more details can be found in Sandulescu et al. (2000, 2003). In the rBCS method, pairing gaps for bound states are Δi =



Viij j uj vj +

j



 Vii,νεν νεν

gν (ε)uν (ε)vν (ε)dε,

(38)

Iν

ν

and averaged pairing gaps for resonances are Δν =

 j

Vνεν νεν ,j j uj vj +

 ν

 Vνεν νεν ,ν  ε

ν ν

ε  ν

gν  (ε )uν  (ε )vν  (ε )dε , Iν 

(39) where V is the interaction matrix element and gν (ε) is the total level density, which takes into account the variation of the localization of scattering states in the energy

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X.-X. Sun and S.-G. Zhou

Fig. 3 The rms neutron radii (left panel) and the two-neutron separation energies (right panel) of even Zr isotopes as functions of mass number A. (Taken from Sandulescu et al. 2003)

region of a resonance (i.e., the width effect) and goes to a delta function in the limit of a very small width. Both bound and resonant states contribute to the total particle number, which reads   2 N= vi + gν (ε)vν2 (ε)dε. (40) i

ν

Iν

The implementation of the rBCS method in mean-field models can be achieved straightforwardly by substituting the gap equations in the conventional BCS method by Eqs. (38), (39), and (40). The wave functions, energies, and widths of resonant states can be obtained by using the scattering-type boundary conditions. This approximation scheme has been realized in the relativistic mean-field (RMF) model (Sandulescu et al. 2003), called the RMF+rBCS method, which has been applied to study the neutron-rich Zr isotopes. It has been shown in Sandulescu et al. (2003) that the sudden increase of the neutron radius close to the neutron drip line depends on a few resonances embedded in the continuum. As shown in Fig. 3, the neutron radii and neutron separation energies from RMF+rBCS calculations are consistent with the results from the relativistic continuum HartreeBogoliubov (RCHB) calculations, in which pairing correlations are treated by using the generalized Bogoliubov transformation and RCHB equations are solved in coordinate space (Meng 1998).

The Generalized Bogoliubov Transformation Many properties of nuclei can be described in terms of independent particle moving in a mean-field potential (Ring and Schuck 1980). In modern nuclear density functional theories, this mean-field potential can be obtained by using the Hartree or Hartree-Fock (HF) method. By combining mean-field models with the BCS method, the pairing interaction is treated as a residual interaction, and the occupation

53 Models for Pairing Phenomena

2027

probabilities are determined by solving the BCS equations after obtaining singleparticle levels. In other words, the BCS method does not treat the mean-field and pairing correlations on the same footing. As a generalization of the HF+BCS method, the Hartree-Fock-Bogoliubov (HFB) theory can treat the mean-field and pairing correlations self-consistently. It has been shown that by formulating the generalized Bogoliubov transformation in the coordinate-space representation, the continuum effects can be properly taken into account (Bulgac 1999; Dobaczewski et al. 1984, 1996; Dobaczewski and Nazarewicz 2012). In this section, the main formulas of the HFB theory will be shown and some related topics will be given. For odd-mass or odd-odd nuclei, the blocking effects must be considered. Therefore the blocking method in the HFB theory is also presented.

The Hartree-Fock-Bogoliubov Theory The HFB wave function |Φ is represented as the vacuum with respect to the quasiparticles βk |Φ = 0,

k = 1, . . . , M,

(41)

where βk is the quasiparticle annihilation operator (Ring and Schuck 1980) and M is the number of quasiparticle states. The HFB wave function can be constructed as |Φ =

βk |−,

(42)

k

with the bare vacuum |−. The most general linear transformation from the particle operators {al† , al } to the quasiparticle operators {βk† , βk } has the form 

β β†





U† V † = V T UT

     a † a ≡W . a† a†

(43)

Since the quasiparticle operators {βk† , βk } should obey the same fermion commutation relations as the particles, the matrix W is unitary, i.e., W W † = 1. The corresponding density matrices are defined as ρll  = Φ|al† al |Φ =



Vl  k Vlk∗ ,

k

κll  = Φ|al  al |Φ =



Ul  k Vlk∗ .

(44)

k

ρ and κ are called the normal and abnormal densities (or density matrix and pairing tensor), respectively.

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X.-X. Sun and S.-G. Zhou

A generalized density matrix can be constructed from ρ and κ 

   Φ|al† al |Φ Φ|al  al |Φ ρ κ . R= = −κ ∗ 1 − ρ ∗ Φ|al† al† |Φ Φ|al  al† |Φ

(45)

As for the eigenvalue of R, one has 

   Φ|βl† βl |Φ Φ|βl  βl |Φ 00 W RW = = , 01 Φ|βl† βl† |Φ Φ|βl  βl† |Φ †

(46)

and R 2 = R,which  the eigenvalues of R are 0 and 1 with corresponding  means  ∗that U V . eigenvectors and U∗ V The expectation value of H  (cf. Eq. (21)) with respect to the quasiparticle vacuum reads E [R] = Φ|H  |Φ   1  εl1 l2 − λδl1 l2 ρl2 l1 + = v¯l1 l2 l3 l4 ρl3 l1 ρl4 l2 2 l2 l2

l1 l2 l3 l4

(47)

1  v¯l1 l2 l3 l4 κl∗1 l2 κl3 l4 , 4

+

l1 l2 l3 l4

which is a functional of the general density matrix R. The variation of the energy functional is δE = E [R + δR] − E [R] =



Hkk  δRkk  ,

(48)

kk 

where the Hamiltonian matrix H is defined as Hkk  =

∂E [R] . ∂Rk  k

(49)

Since  ∂E = εkk  − λδkk  + v¯kp k  p ρpp ≡ εkk  − λδkk  + Γkk  , ∂ρk  k  pp

1

∂E − ∗ = ∂κk  k 2

pp

(50) v¯

kk  pp

κ

pp

≡Δ

kk 

,

53 Models for Pairing Phenomena

2029

H can be expressed explicitly as  h Δ , −Δ∗ −h∗

 H =

(51)

where h = ε−λ+Γ is the single-particle Hamiltonian and Δ is the pairing potential. One gets the HFB equation 

h Δ −Δ∗ −h∗

    Uk Uk = Ek , Vk Vk

(52)

where Ek is the quasiparticle energy and (Uk , Vk )T are quasiparticle wave functions. It should be noted that for the study of ground state of an even-even nucleus, the generalized Bogoliubov transformation can be reduced to the BCS transformation by transforming the quasiparticle basis to the canonical basis (Ring and Schuck 1980). The HFB theory is the basis of modern nuclear density functional theory, which provides an amazingly successful description of the complicated many-body system in nuclei all over the chart of nuclides (Negele 1982; Ring 1996; Bender et al. 2003; Vretenar et al. 2005; Meng et al. 2006; Jones 2015). The HFB theory provides a unified description of particle-hole (ph) and particle-particle (pp) correlations (Ring and Schuck 1980) on a mean-field level by using two average potentials: the self-consistent HF field Γ which is attributed to long-range ph-correlations and a pairing potential Δ which corresponds to the pp-correlations. In nuclear density functional theory, the effective nucleon-nucleon interactions are constructed from basic symmetries of the nuclear force, and the involved parameters are determined by fitting to characteristic experimental data of finite nuclei and nuclear matter. These effective interactions are usually adopted in ph-channel in nonrelativistic density functional or covariant density functional theories, while phenomenological interactions are mostly used for pp-channel. It should be noted that in HFB calculation with the Gogny force, the interaction used in ph-channel is a finiterange central potential in the pp-channel (Péru and Martini 2014; Berger et al. 2017; Robledo et al. 2019).

Selected Topics As mentioned before, the generalized Bogoliubov transformation has been widely used in the study of nuclear structure because it can provide a self-consistent treatment of mean-field and pairing correlations, which is particularly important for describing the properties of exotic nuclei. Thus in this part, several selected topics related to exotic nuclei will be introduced, including the continuum effects, the influence of pairing on nuclear size, and various pairing forces adopted in density functional theory calculations.

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X.-X. Sun and S.-G. Zhou

The Generalized Bogoliubov Transformation and Continuum Spectra The advantage of the HFB theory is that, based on the quasiparticle transformation, it unifies the self-consistent description of single-particle orbitals and the BCS pairing theory into a single variation theory. By solving the HFB equation, one can get the quasiparticle energy spectrum, which contains discrete bound states, resonances, and nonresonant continuum states (Dobaczewski and Nazarewicz 2012). For a self-bound system, i.e., λ < 0, the matter density calculated from quasiparticle wave functions Vk with their quasiparticle energies being positive is always local (Dobaczewski et al. 1984). Therefore the contribution of continuum effects can be self-consistently taken into account in the HFB theory. The solution of the HFB equations is not so straightforward because the continuum states must satisfy the scattering boundary conditions. It has been shown that using the Green’s function method in coordinate space or the Berggren basis, the ground state from the HFB theory can be obtained and the bound, resonant, and scattering quasiparticles are well defined (Belyaev et al. 1987; Michel et al. 2008; Zhang et al. 2011; Sun et al. 2019). Alternatively, solving the HFB equation in coordinate space with the box boundary condition, the quasiparticle spectrum consists of bound and discretized continuum states, and the corresponding ground state can be approximately obtained. It has been shown that by choosing an appropriate box size, the ground state can be accurately calculated (Meng 1998; Grasso et al. 2001; Pei et al. 2011; Typel 2018). Besides, several basis expansion methods have also been proven to be valid for weakly bound nuclei (Pannert et al. 1987; Price and Walker 1987; Stoitsov et al. 1998; Zhou et al. 2003; Nakada and Takayama 2018). With the box boundary condition in HFB calculations, the resonant states can be well located by using the stabilization method (Pei et al. 2011). A schematic picture of the HFB quasiparticle spectrum is shown in Fig. 4. The bound states locate in the region 0 < Ek < −λ. When solving the HFB equation with the box boundary condition, the continuum is discretized and can be replaced by a set of discrete nonresonant states with −λ < Ek < Ecut , where Ecut is the cutoff energy in the quasiparticle space. With the increase of the box size, the level density in continuum becomes more dense. It should be noted that when studying exotic nuclei by solving the wave function in a box, a proper description should be independent on Ecut and the box size RBox . For the HFB theory in coordinate representation, the contribution from the nonresonance continuum state plays an important role and can be included by calculating the densities with the lower component of quasiparticle wave function, which is always local (Dobaczewski et al. 1984, 1996). By using the canonical transformation (Ring and Schuck 1980), one can get the single-particle levels in the canonical basis. It is found that some bound single-particle states might correspond to quasiparticle states with Ek > −λ such that the couplings between single-particle states below the Fermi energy and those in the continuum can be properly treated. Additionally, the treatment and influence of the quasiparticle state with energy larger than Ecut , including resonant and nonresonant ones, can be found in Pei et al. (2011) and are not discussed here.

53 Models for Pairing Phenomena

2031

Fig. 4 Schematic picture of the HFB quasiparticle spectrum. When solving the HFB equation with the box boundary condition, the quasiparticle spectrum consists of bound states with quasiparticle energy smaller than −λ and discretized continuum states with −λ < Ek < Ecut . For each quasiparticle state, the quasiparticle energy is plotted as a function of the box size. The blue lines are used to indicate how to find resonant states by using the stabilization method. After getting all resonances, the HFB quasiparticle spectrum is composed of bound states, continuum resonant states, and nonresonant continuum states

Pairing correlations can significantly affect properties of the nuclei close to drip lines due to the presence of the vast continuum space available for pair scattering (Dobaczewski et al. 1996). A typical example is the effect of pairing on nuclear halos, in which the continuum induced by pairing correlations changes the asymptotic behavior of particle density and the occupation probabilities of single-particle states near the Fermi energy in even-even weakly bound system, thus influencing its spatial extension (Bennaceur et al. 2000; Zhou et al. 2010; Hagino and Sagawa 2011; Pei et al. 2013, 2014; Chen et al. 2014; Meng and Zhou 2015; Sun et al. 2020). In addition, the pairing coupling to positive-energy single-particle states also influences the nuclear binding (Dobaczewski et al. 1996). Particularly, the strong coupling to the continuum lowers the Fermi energy, thus influencing the range of bound nuclei and impacting the limit of the nuclear landscape (Goriely et al. 2009; Xia et al. 2018; Zhang et al. 2022).

Phenomenological Pairing Force: Finite Range vs. Zero Range There are two kinds of pairing interactions commonly used in modern nuclear density functional calculations: zero-range forces with or without a density dependence (Dobaczewski et al. 1996; Meng 1998) and finite-range forces. The latter include Gogny (Dechargé and Gogny 1980; Berger et al. 1984; Gonzalez-Llarena et al. 1996) and separable pairing forces (Tian et al. 2009). The zero-range force is relatively simple for numerical calculations, but it allows a coupling to the very

2032

X.-X. Sun and S.-G. Zhou 5

S2n (MeV)

Neutron Pairing Energies [MeV]

90

30

Exp. RCHBGogny HFB (SkP) RCHB δ HFB(SIII)

20

10 Ni

RCHBδ( Ni)

Ni

RCHBGogny

0

52

RCHBδ( Ni)

−5

−10

0

−15 20

30

40

50

N

60

70

20

30

40

50

60

70

N

Fig. 5 Left panel: Two-neutron separation energies S2n of even Ni isotopes as a function of N , including the experimental data (solid points) and calculation results from the RCHB with δ-force (open circles), RCHB with Gogny force (triangles), HFB with SkP interaction (stars), and HFB with SIII interaction (pluses). Right panel: The neutron pairing energies for RCHB with Gogny force (triangles) and δ-force (open circles). The open circles connected by dashed line or solid line are, respectively, the pairing energies for RCHB with δ-force by fitting that of RCHB with Gogny force at 52 Ni or 90 Ni. (Taken from Meng 1998)

highly excited states. Therefore an energy cutoff has to be introduced, and the interaction strength has to be properly renormalized with respect to, e.g., pairing gaps (Bender et al. 2000). The Gogny force has a better treatment for the coupling to the highly excited states, but it involves more sophisticated numerical techniques. The separable pairing force is numerically simpler than the Gogny force and has also been widely used nowadays. In Meng (1998), the proper form of the pairing interaction is discussed in the framework of the RCHB theory. The even-even Ni isotopes ranging from the proton drip line to the neutron drip line are taken as examples. The pairing correlations are described by using a density-dependent force with zero-range and the finite-range Gogny force, and the results from these two pairing forces were compared. Through the comparisons of the two-neutron separation energies S2n , the neutron, proton, and matter rms radii, good agreements have been found between the calculations with both interactions and the empirical values. In Fig. 5, for example, the two-neutron separation energies of Ni isotopes are shown as a function of the neutron number N , including the experimental data (solid points) and results from the RCHB with δ-force (open circles), RCHB with Gogny force (triangles), and HFB with SkP interaction (Dobaczewski et al. 1996) (stars) and SIII interaction (Beiner et al. 1975) (pluses). The RCHB results with δ-force and Gogny force are almost identical. There is a strong kink at N = 28 and a weaker one at N = 50. The neutron drip line position is predicted at 100 Ni in both calculations. The empirical data are known only up to N = 50. Comparing with the available empirical data, the general trend and gradual decline of S2n have been well reproduced. It is shown clearly in Meng (1998) that after proper renormalization (e.g., fixing the corresponding pairing energies), observables including two-neutron separation energies and rms

53 Models for Pairing Phenomena

2033

radii remain the same, even when the interaction strength is changed within a reasonable region. A further study in Tian et al. (2009) has shown that the same average gaps can be given by the Gogny force D1S and δ force if the size of the strength is adjusted properly, but the individual matrix elements of the forces and the matrix elements of the pairing field are very different from each other. Additionally, the separable approximation is very similar to the full Gogny force. In conclusion, for the study of ground state properties by using mean-field approaches, the results from the abovementioned pairing forces are almost the same.

Pairing Correlations and Nuclear Size The radius of a nucleus is determined by the spatial distribution of the matter density. The asymptotic behavior of neutron ground state density from HF calculations can be approximated as (Bennaceur et al. 2000) ! ρ(r) ∝ exp (−2μr)/r 2 ,

(53)

√ where μ = −2mεk /h¯ and εk is the single-particle energy of the least bound orbital with l = 0. The corresponding mean square radius deduced from the asymptotic solution is r 2 HF ∝

h¯ 2 , 2m|εk |

(54)

which diverges in the limit εk → 0. This mechanism, i.e., valence nucleon(s) occupying weakly bound and low-l orbital, has been used in early interpretations of the nuclear halo phenomenon (Bertsch et al. 1989; Sagawa 1992; Zhu et al. 1994). When pairing correlations are included, the mean square radius deduced from the asymptotic HFB density is r 2 HFB ∝

h¯ 2 2m(Ek − λ)

(55)

 with the lowest discrete quasiparticle energy Ek = (εk − λ)2 + Δ2k . When the pairing gap Δk is finite, the radius does not diverge in the limit of small separation energy εk  λ → 0. The asymptotic HF and HFB densities characterized by l = 0 orbitals were compared in Bennaceur et al. (2000), and it has been emphasized that pairing correlations reduce the nuclear size and an extreme halo with infinite radius cannot be formed in superfluid nuclear systems. In this sense, pairing correlations act against the formation of an infinite matter radius. This is the so-called pairing antihalo effect (Bennaceur et al. 2000). In fact, the limiting condition εk → 0 and the radius deduced from this l = 0 orbital alone correspond to an extremely ideal situation, which is difficult to be found in real nuclei. In Chen et al. (2014), the influences of pairing correlations on

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X.-X. Sun and S.-G. Zhou

the nuclear size and on the formation of a nuclear halo were studied in details by using the self-consistent RCHB theory (Meng 1998). It has been shown that pairing correlations not only influence the radius of the orbital but also affect the occupation probabilities of the orbitals close to the Fermi level. As a result, the nuclear radius is dominated by their competition.

Blocking Effects To describe odd-mass or odd-odd nuclei, the blocking effect has to be taken into account. In this part, the method describing the blocking effect in the HFB approach is introduced. To treat pairing correlations with the generalized Bogoliubov transformation, the quasiparticle concept is adopted, and the ground state of an even-even nucleus |Φ (cf. Eq. (42)) is represented as a vacuum with respect to quasiparticles (Ring and Schuck 1980). For odd-mass nuclei, in practice, the ground state can be constructed as one quasiparticle state |Φ1  = β1† |Φ0  = β1†

βk |0,

(56)

k

where β1† corresponds to the quasiparticle level to be blocked. This one quasiparticle state |Φ1  can be regarded as the vacuum with respect to the set of  ) with quasiparticle operators (β1 , . . . , βM β1 = β1† ,

β2 = β2 ,

...,

 βM = βM ,

(57)

and the exchange of the operators β1† ↔ β1 forms a new set of quasiparticle  , β † , . . . , β † ), which corresponds to the exchange of operators (β1 , . . . , βM M 1 ∗ ) in the matrix W (cf. Eq. (43)). Therefore, the columns (Ul1 , Vl1 ) ←→ (Vl1∗ , Ul1 the blocking effect in the odd system can be realized by exchanging the creator β1† with the corresponding annihilator β1 in the quasiparticle space. Next the procedure will be presented of implementing the blocking in axially deformed nuclei (Li et al. 2012). For a fully paired and axially symmetric deformed system with the time-reversal symmetry, the projection of the total angular momentum on the symmetry axis Ω is a good quantum number, and each single-particle state has a degeneracy of two. The HFB equation can be reduced to half dimension M/2 and decomposed into degenerate blocks with quantum numbers +Ω or −Ω. The dimension of the corresponding density and abnormal density matrices is M. For an odd system with the kb -th level blocked in the +Ω subspace, the timereversal symmetry is violated, and currents appear in the system. These currents are axially symmetric, i.e., Ω remains a good quantum number, but the quasiparticle energies are no longer degenerate for the two subspaces, because the subspace with +Ω contains the odd particle and the subspace with −Ω contains an empty level. Therefore, in principle, one has to diagonalize the HFB equation in the whole

53 Models for Pairing Phenomena

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quasiparticle space composed of +Ω and −Ω subspaces. With the equal filling approximation (Perez-Martin and Robledo 2008; Schunck et al. 2010), which has been shown to be valid in dealing with blocking effects (Bertsch et al. 2009), one can average over the two configurations of a particle in the +Ω space and in the −Ω space. The corresponding currents in two subspaces cancel each other and can be neglected. In this way one can diagonalize the HFB equation in the +Ω subspace or the −Ω subspace, and the resulted fields are time-reversal symmetric. In practice the density matrix ρ and the abnormal density κ in two subspaces are   ρ = V ∗V T

M×M

  κ = V ∗U T

M×M

 1 ∗ T Ukb Uk∗T − V V kb kb , b 2  1 ∗ T Ukb Vk∗T − + V U kb kb , b 2

+

(58) (59)

where Vkb and Ukb are column vectors in the matrices V and U corresponding to the blocked level.

Issues with Particle Number Both the BCS method and the HFB theory have been widely applied to describe nuclear superfluidity, but the BCS-type and HFB wave functions do not keep the particle number conserved, which is connected with the Nambu-Goldstone mode of a broken U (1) phase symmetry (Broglia et al. 2000). In addition, the spontaneous breaking of U (1) symmetry leads to a sharp phase transition, i.e., the pairing energy turns from zero to a finite value at a critical pairing strength (Ring and Schuck 1980). Many efforts have been made to remedy the problem of particle number violation in the mean-field model, configuration space, and ab initio calculations (see Sheikh et al. 2021 and references therein). The simplest way to treat the uncertainty in the particle number is the Lipkin-Nogami method mentioned before, but which only considers the corrections on the energy. Many exact solutions of the pairing Hamiltonian have also been proposed (Dukelsky et al. 2004). In mean-field models, the restoration of particle number of BCS-type or HFB wave function can be achieved by using the particle number projection (PNP) method. In this section, PNP in mean-field models and a particle number conservation method in the manybody configuration space are introduced.

Exact Solutions for Pairing Hamiltonian The exact numerical solution of the pairing model has been proposed in the 1960s by Richardson and Sherman (1964a, b). This method has been extended to a family of exactly solvable models, called the Richardson-Gaudin (RG) models, and widely applied in various areas of quantum many-body systems, such as mesoscopic systems, condensed matter, quantum optics, cold atomic gases, and atomic nuclei

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X.-X. Sun and S.-G. Zhou

(Dukelsky and Pittel 2012; Dukelsky et al. 2004; Qi and Chen 2015). In this method, one does not need to diagonalize the pairing Hamiltonian, but instead to solve a set of nonlinear equations, called Richardson’s equation, for parameters in the pairing wave functions. More details of this method can be found in Brink and Broglia (2005), Dukelsky and Pittel (2012), and Dukelsky et al. (2004). In this part, a shell-model-like approach (SLAP) for the pairing, dubbed the particle number conservation (PNC) method developed in the 1980s (Zeng and Cheng 1983), will be introduced. In the PNC method, the pairing Hamiltonian is directly diagonalized in the many-body configuration space, and it has been shown to be more accurate than the BCS calculation as compared with the exact solution (Molique and Dudek 1997). Furthermore, blocking effects are taken into account automatically, and both odd-mass and even-even nuclei can be treated on the same footing. It has been demonstrated that the number of configurations with significant contributions to the low-lying excited states of a nucleus is quite limited (Wu and Zeng 1989). Consequently the concept of many-body configuration truncation is introduced instead of the single-particle state truncation used in the BCS or generalized Bogoliubov method. Extensive studies and discussions on the validity of the truncated many-body configuration spaces as well as the application of the PNC method can be found in Wu and Zeng (1991a, b) and Zeng et al. (1994, 2001, 2002). In particular the presence of the low-lying seniority σ = 0 solutions, which are usually poorly described by using the standard BCS approximation or HFB theory, has been found to play a role in the interpretation of the spectra of rotating nuclei. In the SLAP for pairing, the model Hamiltonian reads H = Hs.p. + Hpair ,

(60)

 μ =ν where Hs.p. = ν εν aν† aν and Hpair = −G μ,ν>0 aμ† aμ†¯ aν¯ aν with G the average strength, εν the single-particle energy, and ν the notation of the each level. In the case of axially deformed nuclei, ν ≡ (Ω, π ). ν¯ represents the time-reversal state of ν. For an even-even nucleus with the total particle number N = 2n, the multiparticle configurations (MPCs) used to diagonalize the Hamiltonian are constructed as the following: (a) The fully paired configurations with the seniority σ = 0: |ρ1 ρ¯1 · · · ρn ρ¯n  = aρ†1 aρ†¯1 · · · aρ†n aρ†¯n |0 ,

(61)

(b) The configurations with two unpaired particles, i.e., seniority σ = 2: |μνρ1 ρ¯1 · · · ρn−1 ρ¯n−1  = aμ† aν† aρ†1 aρ†¯1 · · · aρ†n−1 aρ†¯n−1 |0 ,

(62)

where μ and ν denote two unpaired levels. The MPCs with larger σ can also be constructed in this way (Zeng and Cheng 1983).

53 Models for Pairing Phenomena

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In realistic calculations, the MPC space has to be truncated. Only configurations with excitation energies smaller than Ec are used to diagonalize the Hamiltonian (60), where Ec is the cutoff energy. The corresponding nuclear wave function can be expanded as 

ψβ =

ρ1 ,··· ,ρn

+



Vρβ1 ,··· ,ρn |ρ1 ρ¯1 · · · ρn ρ¯n  

Vρβ(μν) |μ¯ν ρ1 ρ¯1 · · · ρn−1 ρ¯n−1  + · · · , 1 ,··· ,ρn−1

(63)

μ,ν ρ1 ,··· ,ρn−1

where β = 0 (ground state), 1, 2, 3, · · · (excited states). The occupation probability of the ith level for the state β is β

ni =

 ρ1 ,··· ,ρn−1

 2   β  Vρ1 ,··· ,ρn−1 ,i  +

 2  β(μν)  Vρ1 ,··· ,ρn−2 ,i  +· · · , i = 1, 2, 3, · · ·.



μ,ν ρ1 ,··· ,ρn−2

(64) This SLAP for pairing has been implemented in the relativistic mean-field model (Meng et al. 2006; Shi et al. 2018; Xiong 2020) and Skyrme Hartree-Fock model (Liang et al. 2015; Dai et al. 2019), and it turns out that this hybrid model is valid for both ground state properties and low-lying spectra. In addition, it has also been applied in several cranking models to study the rotational properties of ground state bands and low-lying high-K multi-quasiparticle bands (Fu et al. 2013; Liang et al. 2015; Shi et al. 2018; Xiong 2020; Zhang et al. 2020).

Particle Number Projection In the HFB theory, the wave functions are the vacua of the corresponding quasiparticle operators, which do not represent states with good particle number. The nonconservation of particle number can be restored by projecting an HFB state |Ψ  onto a state with good particle number (Sheikh and Ring 2000; Anguiano et al. 2001, 2002; Bender et al. 2009; Sheikh et al. 2002) 1 2π

|Φ N  = Pˆ N |Ψ  =



2π

1 eiϕN

0

ˆ

ei N ϕ |Ψ ,

(65)

with the particle number projection operator 1 Pˆ N = 2π Then the projected energy is given by

 0

2π

ˆ

eiϕ(N −N ) dϕ.

(66)

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X.-X. Sun and S.-G. Zhou

EN =

Φ N |Hˆ |Φ N  = Φ N |Φ N 



2π

dϕE[ϕ]N

N

(ϕ),

(67)

0

with E[ϕ] =

Φ0 |Hˆ |Φϕ  , Φ0 |Φϕ 

N

N

(ϕ) =

e−iN ϕ Φ0 |Φϕ  , 2π Φ N |Φ N 

(68)

ˆ

where |Φϕ  = eiϕ N |Ψ . It has been shown in Sheikh and Ring (2000) and Anguiano et al. (2001) that the energy kernel E[ϕ] is similar to Eq. (47) with modified expressions for the pairing field and the HF potential and can be calculated by the generalized Wick’s theorem E[ϕ] ≡

Φ0 |Hˆ |Φϕ   1  ρρ 1  κκ ϕ0∗ 0ϕ 0ϕ 0ϕ 0ϕ tμμ ρμμ + v¯μνμν ρμμ ρνν + v¯μμν = ¯ ν¯ κμμ¯ κν ν¯ , Φ0 |Φϕ  2 4 μ μν μν (69)

where v¯ ρρ and v¯ κκ denote the effective vertices in the ph- and pp-channels. The normal and anomalous transition density matrices are 0ϕ ρμν =

Φ0 |aν† aμ |Φϕ  , Φ0 |Φϕ 

0ϕ κμν =

Φ0 |aν aμ |Φϕ  , Φ0 |Φϕ 

ϕ0∗ κμν =

Φ0 |aμ† aν† |Φϕ  Φ0 |Φϕ 

(70)

.

The projected HFB equation reads 

ε N + Γ N + λN ΔN N ∗ N −Δ −[ε + Γ N + λN ]∗ N

N

  N  N U U = E . VN VN

(71)

N = − ∂E . All the quantities keep the good quantum where Γ N = ∂E ∂ρ and Δ ∂κ ∗ number N. Detailed formulas can be found in Sheikh et al. (2002). The implementation of PNP in the BCS model is relatively simple, which was applied using different methods shown in Bayman (1960), Dietrich et al. (1964), Fomenko (1970), Janssen and Schuck (1981), and Ring and Schuck (1980). The PNP technique has been applied in mean-field models based on the HFB or HF+BCS equations. One can make the projection before or after variation, and the latter is technically much more easier and has been widely used in beyond-meanfield calculations nowadays. The mean field calculations with and without the PNP method have shown that the uncertainties caused by the particle number fluctuation

53 Models for Pairing Phenomena

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might be important for determining the positions of drip lines and the stability of rare isotopes (Schunck and Egido 2008; Schunck and Robledo 2016; An et al. 2020; Verriere et al. 2021). More details and related topics on PNP can be found in a recent review (Sheikh et al. 2021).

Further Reading Pairing effects are directly related to many aspects of nuclear physics. In this chapter, only the basic picture and methodology of the pairing effects in atomic nuclei are introduced, especially focusing on the nuclear structure study. Various additional topics related to pairing are thoroughly covered in the following textbooks: • P. Ring and P. Schuck, The nuclear many-body problem, Springer-Verlag, 1980. • W. Greiner and J.A. Maruhn, Nuclear models, Springer-Verlag, 1996. • D.M. Brink and R.A. Broglia, Nuclear Superfluidity: Pairing in Finite Systems, Cambridge University Press, 2005. • D.J. Rowe and J.L. Wood, Fundamentals of nuclear models, World Scientific, 2010. • R.A. Broglia and V. Zelevinsky (eds.), Fifty Years of Nuclear BCS: Pairing in Finite Systems, World Scientific, Singapore, 2013 More aspects of pairing effects, underlying mechanism, and more detailed derivations of specific implementations of pairing models are discussed in the following review articles: • R.A. Broglia, J. Terasaki, and N. Giovanardi, The Anderson–Goldstone–Nambu mode in finite and in infinite systems, Phys. Rep. 335, 1 (2000) • W. von Oertzen and A. Vitturi, Pairing correlations of nucleons and multinucleon transfer between heavy nuclei, Rep. Prog. Phys. 64, 1247 (2001) • D.J. Dean and M. Hjorth-Jensen, Pairing in nuclear systems: From neutron stars to finite nuclei, Rev. Mod. Phys. 75, 607 (2003) • J. Dukelsky, S. Pittel, and G. Sierra, Colloquium: Exactly solvable RichardsonGaudin models for many-body quantum systems, Rev. Mod. Phys. 76, 643 (2004) • J. Meng, H. Toki, S.-G. Zhou, S.Q. Zhang, W.H. Long, and L.S. Geng, Relativistic continuum Hartree-Bogoliubov theory for ground-state properties of exotic nuclei, Prog. Part. Nucl. Phys. 57, 470 (2006) • G.C. Strinati, P. Pieri, G. Röpke, P. Schuck, and M. Urban, The BCS–BEC crossover: From ultra-cold Fermi gases to nuclear systems, Phys. Rep. 738, 1 (2008) • S. Frauendorf and A. Macchiavelli, Overview of neutron–proton pairing, Prog. Part. Nucl. Phys. 78, 24 (2014) • N.Q. Hung, N.D. Dang, and L.G. Moretto, Pairing in excited nuclei: A review, Rep. Prog. Phys. 82, 056301 (2019)

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Acknowledgments The authors would like to thank Wen-Hui Long, Peng-Wei Zhao, Kai-Yuan Zhang, Shuang-Quan Zhang, and Zhen-Hua Zhang for their comments and suggestions. The authors have been partly supported by the National Key R&D Program of China (Grant No. 2018YFA0404402), the National Natural Science Foundation of China (Grants No. 11525524, No. 12070131001, No. 12047503, No. 11961141004, No. 11975237, No. 11575189, and No. 11790325), the Key Research Program of Frontier Sciences of Chinese Academy of Sciences (Grant No. QYZDB-SSWSYS013), the Strategic Priority Research Program of Chinese Academy of Sciences (Grants No. XDB34010000 and No. XDPB15), the Inter-Governmental S&T Cooperation Project between China and Croatia, and the IAEA Coordinated Research Project “F41033.”

References R. An, G.F. Shen, S.S. Zhang, L.S. Geng, Chin. Phys. C 44(7), 074101 (2020) M. Anguiano, J. Egido, L. Robledo, Nucl. Phys. A 696(3), 467 (2001) M. Anguiano, J. Egido, L. Robledo, Phys. Lett. B 545(1), 62 (2002) A. Arima, M. Ichimura, Prog. Theor. Phys. 36(2), 296 (1966) J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957) F. Barranco, R.A. Broglia, G. Coló, G. Gori, E. Vigezzi, P.F. Bortignon, Eur. Phys. J. A 21(1), 57 (2004) B. Bayman, Nucl. Phys. 15, 33 (1960) M. Beiner, H. Flocard, N. Van Giai, P. Quentin, Nucl. Phys. A 238(1), 29 (1975) S.T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31(11), 1 (1959) S.T. Belyaev, A.V. Smirnov, S.V. Tolokonnikov, S.A. Fayans, Sov. J. Nucl. Phys. 45, 783 (1987) M. Bender, K. Rutz, P.G. Reinhard, J.A. Maruhn, Eur. Phys. J. A 8(1), 59 (2000) M. Bender, P.H. Heenen, P.G. Reinhard, Rev. Mod. Phys. 75, 121 (2003) M. Bender, T. Duguet, D. Lacroix, Phys. Rev. C 79, 044319 (2009) K. Bennaceur, J. Dobaczewski, M. Płoszajczak, Phys. Lett. B 496(3–4), 154 (2000) J. Berger, M. Girod, D. Gogny, Nucl. Phys. A 428, 23 (1984) J.F. Berger, J.P. Blaizot, D. Bouche, P. Chaix, J.P. Delaroche, M. Dupuis, M. Girod, J. Gogny, B. Grammaticos, D. Iracane, J. Lachkar, F. Mariotte, N. Pillet, N. Van Giai, Eur. Phys. J. A 53(10), 214 (2017) G.F. Bertsch, in Fifty Years of Nuclear BCS: Pairing in Finite Systems, ed. by R.A. Broglia, V. Zelevinsky (World Scientific, Singapore, 2012), pp. 26–39 G.F. Bertsch, B.A. Brown, H. Sagawa, Phys. Rev. C 39, 1154 (1989) G.F. Bertsch, C.A. Bertulani, W. Nazarewicz, N. Schunck, M.V. Stoitsov, Phys. Rev. C 79, 034306 (2009) B. Blank, M. Płoszajczak, Rep. Prog. Phys. 71(4), 046301 (2008) N.N. Bogoljubov, Il Nuovo Cimento 7(6), 794 (1958) A. Bohr, B.R. Mottelson, Nuclear Structure, Volume 1: Single Particle Motion (World Scientific, Singapore, 1998a) A. Bohr, B.R. Mottelson, Nuclear Structure, Volume 2: Nuclear Deformations (World Scientific, Singapore, 1998b) A. Bohr, B.R. Mottelson, D. Pines, Phys. Rev. 110, 936 (1958) D.M. Brink, R.A. Broglia, Nuclear Superfluidity: Pairing in Finite Systems (Cambridge University Press, New York, 2005) R.A. Broglia, Ann. Phys. 80(1), 60 (1973) R.A. Broglia, V. Zelevinsky (eds.), Fifty Years of Nuclear BCS: Pairing in Finite Systems (World Scientific, Singapore, 2013) R. Broglia, J. Terasaki, N. Giovanardi, Phys. Rep. 335(1), 1 (2000) A. Bulgac, arXiv:nucl-th/9907088 (1999) S. Changizi, C. Qi, R. Wyss, Nucl. Phys. A 940, 210 (2015) Y. Chen, P. Ring, J. Meng, Phys. Rev. C 89, 014312 (2014)

53 Models for Pairing Phenomena

2041

W.J. Chen, C.A. Bertulani, F.R. Xu, Y.N. Zhang, Phys. Rev. C 91, 047303 (2015) L.N. Cooper, Phys. Rev. 104, 1189 (1956) A.C. Dai, F.R. Xu, W.Y. Liang, Chin. Phys. C 43(8), 084101 (2019) D.J. Dean, M. Hjorth-Jensen, Rev. Mod. Phys. 75, 607 (2003) J. Dechargé, D. Gogny, Phys. Rev. C 21, 1568 (1980) K. Dietrich, H.J. Mang, J.H. Pradal, Phys. Rev. 135, B22 (1964) J. Dobaczewski, W. Nazarewicz, in Fifty Years of Nuclear BCS: Pairing in Finite Systems, ed. by R.A. Broglia, V. Zelevinsky (World Scientific, Singapore, 2012), pp. 40–60 J. Dobaczewski, H. Flocard, J. Treiner, Nucl. Phys. A 422(1), 103 (1984) J. Dobaczewski, W. Nazarewicz, T.R. Werner, J.F. Berger, C.R. Chinn, J. Dechargé, Phys. Rev. C 53, 2809 (1996) J. Dobaczewski, P. Magierski, W. Nazarewicz, W. Satuła, Z. Szyma´nski, Phys. Rev. C 63, 024308 (2001) J. Dobaczewski, N. Michel, W. Nazarewicz, M. Płoszajczak, J. Rotureau, Prog. Part. Nucl. Phys. 59(1), 432 (2007) T. Duguet, in Fifty Years of Nuclear BCS: Pairing in Finite Systems (World Scientific, Singapore, 2012), pp. 229–242 T. Duguet, T. Lesinski, Eur. Phys. J. Spec. Top. 156(1), 207 (2008) J. Dukelsky, S. Pittel, in Fifty Years of Nuclear BCS: Pairing in Finite Systems, ed. by R.A. Broglia, V. Zelevinsky (World Scientific, Singapore, 2012), pp. 200–211 J. Dukelsky, S. Pittel, G. Sierra, Rev. Mod. Phys. 76, 643 (2004) V.N. Fomenko, J. Phys. A: Gen. Phys. 3(1), 8 (1970) S. Frauendorf, A. Macchiavelli, Prog. Part. Nucl. Phys. 78, 24 (2014) X.M. Fu, F.R. Xu, J.C. Pei, C.F. Jiao, Y. Shi, Z.H. Zhang, Y.A. Lei, Phys. Rev. C 87, 044319 (2013) T. Gonzalez-Llarena, J.L. Egido, G.A. Lalazissis, P. Ring, Phys. Lett. B 379(1), 13 (1996) S. Goriely, N. Chamel, J.M. Pearson, Phys. Rev. Lett. 102, 152503 (2009) A. Gottardo, J.J. Valiente-Dobón, G. Benzoni, R. Nicolini, A. Gadea, S. Lunardi, P. Boutachkov, A.M. Bruce, M. Górska, J. Grebosz, S. Pietri, Z. Podolyák, M. Pfützner, P.H. Regan, H. Weick, J. Alcántara Núñez, A. Algora, N. Al-Dahan, G. de Angelis, Y. Ayyad, N. Alkhomashi, P.R.P. Allegro, D. Bazzacco, J. Benlliure, M. Bowry, A. Bracco, M. Bunce, F. Camera, E. Casarejos, M.L. Cortes, F.C.L. Crespi, A. Corsi, A.M. Denis Bacelar, A.Y. Deo, C. Domingo-Pardo, M. Doncel, Z. Dombradi, T. Engert, K. Eppinger, G.F. Farrelly, F. Farinon, E. Farnea, H. Geissel, J. Gerl, N. Goel, E. Gregor, T. Habermann, R. Hoischen, R. Janik, S. Klupp, I. Kojouharov, N. Kurz, S.M. Lenzi, S. Leoni, S. Mandal, R. Menegazzo, D. Mengoni, B. Million, A.I. Morales, D.R. Napoli, F. Naqvi, C. Nociforo, A. Prochazka, W. Prokopowicz, F. Recchia, R.V. Ribas, M.W. Reed, D. Rudolph, E. Sahin, H. Schaffner, A. Sharma, B. Sitar, D. Siwal, K. Steiger, P. Strmen, T.P.D. Swan, I. Szarka, C.A. Ur, P.M. Walker, O. Wieland, H.J. Wollersheim, F. Nowacki, E. Maglione, A.P. Zuker, Phys. Rev. Lett. 109, 162502 (2012) M. Grasso, N. Sandulescu, N. Van Giai, R.J. Liotta, Phys. Rev. C 64, 064321 (2001) K. Hagino, H. Sagawa, Phys. Rev. C 84, 011303(R) (2011) K. Hagino, H. Sagawa, J. Carbonell, P. Schuck, Phys. Rev. Lett. 99, 022506 (2007) K. Hebeler, T. Duguet, T. Lesinski, A. Schwenk, Phys. Rev. C 80, 044321 (2009) W. Huang, M. Wang, F. Kondev, G. Audi, S. Naimi, Chin. Phys. C 45(3), 030002 (2021) D. Janssen, P. Schuck, Z. Physik A 301(3), 255 (1981) L.Y. Jia, C. Qi, Phys. Rev. C 94, 044312 (2016) C.W. Johnson, K.D. Launey, N. Auerbach, S. Bacca, B.R. Barrett, C.R. Brune, M.A. Caprio, P. Descouvemont, W.H. Dickhoff, C. Elster, P.J. Fasano, K. Fossez, H. Hergert, M. HjorthJensen, L. Hlophe, B. Hu, R.M.I. Betan, A. Idini, S. König, K. Kravvaris, D. Lee, J. Lei, A. Mercenne, R.N. Perez, W. Nazarewicz, F.M. Nunes, M. Płoszajczak, J. Rotureau, G. Rupak, A.M. Shirokov, I. Thompson, J.P. Vary, A. Volya, F. Xu, R.G.T. Zegers, V. Zelevinsky, X. Zhang, J. Phys. G: Nucl. Part. Phys. 47(12), 123001 (2020) R.O. Jones, Rev. Mod. Phys. 87, 897 (2015) R. Kanungo, I. Tanihata, in Fifty Years of Nuclear BCS: Pairing in Finite Systems, ed. by R.A. Broglia, V. Zelevinsky (World Scientific, Singapore, 2012), pp. 379–392

2042

X.-X. Sun and S.-G. Zhou

R. Kanungo, A. Sanetullaev, J. Tanaka, S. Ishimoto, G. Hagen, T. Myo, T. Suzuki, C. Andreoiu, P. Bender, A.A. Chen, B. Davids, J. Fallis, J.P. Fortin, N. Galinski, A.T. Gallant, P.E. Garrett, G. Hackman, B. Hadinia, G. Jansen, M. Keefe, R. Krücken, J. Lighthall, E. McNeice, D. Miller, T. Otsuka, J. Purcell, J.S. Randhawa, T. Roger, A. Rojas, H. Savajols, A. Shotter, I. Tanihata, I.J. Thompson, C. Unsworth, P. Voss, Z. Wang, Phys. Rev. Lett. 114, 192502 (2015) A.K. Kerman, Ann. Phys. 12(2), 300 (1961) F. Kondev, M. Wang, W. Huang, S. Naimi, G. Audi, Chin. Phys. C 45(3), 030001 (2021) W.Y. Liang, C.F. Jiao, Q. Wu, X.M. Fu, F.R. Xu, Phys. Rev. C 92, 064325 (2015) L.L. Li, J. Meng, P. Ring, E.G. Zhao, S.G. Zhou, Chin. Phys. Lett. 29(4), 042101 (2012) H.J. Lipkin, Ann. Phys. 9(2), 272 (1960) E. Lunderberg, P.A. DeYoung, Z. Kohley, H. Attanayake, T. Baumann, D. Bazin, G. Christian, D. Divaratne, S.M. Grimes, A. Haagsma, J.E. Finck, N. Frank, B. Luther, S. Mosby, T. Nagi, G.F. Peaslee, A. Schiller, J. Snyder, A. Spyrou, M.J. Strongman, M. Thoennessen, Phys. Rev. Lett. 108, 142503 (2012) B. Maheshwari, A.K. Jain, Phys. Lett. B 753, 122 (2016) J. Margueron, H. Sagawa, K. Hagino, Phys. Rev. C 76, 064316 (2007) J. Margueron, H. Sagawa, K. Hagino, Phys. Rev. C 77, 054309 (2008) M.G. Mayer, Phys. Rev. 78, 22 (1950) M.G. Mayer, J.H.D. Jensen, Elementary Theory of Nuclear Shell Structure (Wiley, New York, 1955) J. Meng, Nucl. Phys. A 635(1–2), 3 (1998) J. Meng, Phys. Rev. C 57, 1229 (1998) J. Meng, P. Ring, Phys. Rev. Lett. 77, 3963 (1996) J. Meng, S.G. Zhou, J. Phys. G: Nucl. Part. Phys. 42(9), 093101 (2015) J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long, L.S. Geng, Prog. Part. Nucl. Phys. 57(2), 470 (2006) J. Meng, J.Y. Guo, L. Liu, S.Q. Zhang, Front. Phys. China 1(1), 38 (2006) N. Michel, K. Matsuyanagi, M. Stoitsov, Phys. Rev. C 78, 044319 (2008) N. Michel, W. Nazarewicz, J. Okołowicz, M. Płoszajczak, J. Phys. G: Nucl. Part. Phys. 37(6), 064042 (2010) H. Molique, J. Dudek, Phys. Rev. C 56, 1795 (1997) I.O. Morales, P. Van Isacker, I. Talmi, Phys. Lett. B 703(5), 606 (2011) B. Mottelson, Rev. Mod. Phys. 48, 375 (1976) H. Nakada, K. Takayama, Phys. Rev. C 98, 011301(R) (2018) T. Nakamura, A.M. Vinodkumar, T. Sugimoto, N. Aoi, H. Baba, D. Bazin, N. Fukuda, T. Gomi, H. Hasegawa, N. Imai, M. Ishihara, T. Kobayashi, Y. Kondo, T. Kubo, M. Miura, T. Motobayashi, H. Otsu, A. Saito, H. Sakurai, S. Shimoura, K. Watanabe, Y.X. Watanabe, T. Yakushiji, Y. Yanagisawa, K. Yoneda, Phys. Rev. Lett. 96, 252502 (2006) T. Nakamura, N. Kobayashi, Y. Kondo, Y. Satou, J.A. Tostevin, Y. Utsuno, N. Aoi, H. Baba, N. Fukuda, J. Gibelin, N. Inabe, M. Ishihara, D. Kameda, T. Kubo, T. Motobayashi, T. Ohnishi, N.A. Orr, H. Otsu, T. Otsuka, H. Sakurai, T. Sumikama, H. Takeda, E. Takeshita, M. Takechi, S. Takeuchi, Y. Togano, K. Yoneda, Phys. Rev. Lett. 112, 142501 (2014) T. Nakamura, H. Sakurai, H. Watanabe, Prog. Part. Nucl. Phys. 97, 53 (2017) J.W. Negele, Rev. Mod. Phys. 54, 913 (1982) T. Nikši´c, D. Vretenar, P. Ring, Phys. Rev. C 74, 064309 (2006) Y. Nogami, Phys. Rev. 134, B313 (1964) S.S. Pankratov, M.V. Zverev, M. Baldo, U. Lombardo, E.E. Saperstein, Phys. Rev. C 84, 014321 (2011) W. Pannert, P. Ring, J. Boguta, Phys. Rev. Lett. 59, 2420 (1987) J.C. Pei, A.T. Kruppa, W. Nazarewicz, Phys. Rev. C 84, 024311 (2011) J.C. Pei, Y.N. Zhang, F.R. Xu, Phys. Rev. C 87, 051302(R) (2013) J.C. Pei, G.I. Fann, R.J. Harrison, W. Nazarewicz, Y. Shi, S. Thornton, Phys. Rev. C 90, 024317 (2014) S. Perez-Martin, L.M. Robledo, Phys. Rev. C 78, 014304 (2008)

53 Models for Pairing Phenomena

2043

S. Péru, M. Martini, Eur. Phys. J. A 50(5), 88 (2014) M. Pfützner, M. Karny, L.V. Grigorenko, K. Riisager, Rev. Mod. Phys. 84, 567 (2012) G. Potel, F. Barranco, F. Marini, A. Idini, E. Vigezzi, R.A. Broglia, Phys. Rev. Lett. 107, 092501 (2011) G. Potel, A. Idini, F. Barranco, E. Vigezzi, R.A. Broglia, Rep. Prog. Phys. 76, 106301 (2013) C.E. Price, G.E. Walker, Phys. Rev. C 36, 354 (1987) C. Qi, T. Chen, Phys. Rev. C 92, 051304(R) (2015) G. Racah, Phys. Rev. 62, 438 (1942) P.G. Reinhard, M. Rufa, J. Maruhn, W. Greiner, J. Friedrich, Z. Phys. A 323, 13 (1986) P.G. Reinhard, W. Nazarewicz, M. Bender, J.A. Maruhn, Phys. Rev. C 53, 2776 (1996) R. Richardson, N. Sherman, Nucl. Phys. 52, 221 (1964a) R. Richardson, N. Sherman, Nucl. Phys. 52, 253 (1964b) P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996) P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin/Heidelberg, 1980) L.M. Robledo, T.R. Rodríguez, R.R. Rodríguez-Guzmán, J. Phys. G: Nucl. Part. Phys. 46(1), 013001 (2019) H. Sagawa, Phys. Lett. B 286(1), 7 (1992) H. Sagawa, K. Hagino, Eur. Phys. J. A 51(8), 102 (2015) N. Sandulescu, N. Van Giai, R.J. Liotta, Phys. Rev. C 61, 061301(R) (2000) N. Sandulescu, L.S. Geng, H. Toki, G.C. Hillhouse, Phys. Rev. C 68, 054323 (2003) W. Satula, J. Dobaczewski, W. Nazarewicz, Phys. Rev. Lett. 81, 3599 (1998) N. Schunck, J.L. Egido, Phys. Rev. C 78, 064305 (2008) N. Schunck, L.M. Robledo, Rep. Prog. Phys. 79(11), 116301 (2016) N. Schunck, J. Dobaczewski, J. McDonnell, J. Moré, W. Nazarewicz, J. Sarich, M.V. Stoitsov, Phys. Rev. C 81, 024316 (2010) J.A. Sheikh, P. Ring, Nucl. Phys. A 665(1), 71 (2000) J.A. Sheikh, P. Ring, E. Lopes, R. Rossignoli, Phys. Rev. C 66, 044318 (2002) J.A. Sheikh, J. Dobaczewski, P. Ring, L.M. Robledo, C. Yannouleas, J. Phys. G: Nucl. Part. Phys. 48, 123001 (2021) Z. Shi, Z.H. Zhang, Q.B. Chen, S.Q. Zhang, J. Meng, Phys. Rev. C 97, 034317 (2018) S. Shlomo, I. Talmi, Nucl. Phys. A 198(1), 81 (1972) G.S. Simpson, G. Gey, A. Jungclaus, J. Taprogge, S. Nishimura, K. Sieja, P. Doornenbal, G. Lorusso, P.A. Söderström, T. Sumikama, Z.Y. Xu, H. Baba, F. Browne, N. Fukuda, N. Inabe, T. Isobe, H.S. Jung, D. Kameda, G.D. Kim, Y.K. Kim, I. Kojouharov, T. Kubo, N. Kurz, Y.K. Kwon, Z. Li, H. Sakurai, H. Schaffner, Y. Shimizu, H. Suzuki, H. Takeda, Z. Vajta, H. Watanabe, J. Wu, A. Yagi, K. Yoshinaga, S. Bönig, J.M. Daugas, F. Drouet, R. Gernhäuser, S. Ilieva, T. Kröll, A. Montaner-Pizá, K. Moschner, D. Mücher, H. Naïdja, H. Nishibata, F. Nowacki, A. Odahara, R. Orlandi, K. Steiger, A. Wendt, Phys. Rev. Lett. 113, 132502 (2014) M. Stoitsov, P. Ring, D. Vretenar, G.A. Lalazissis, Phys. Rev. C 58, 2086 (1998) X.X. Sun, S.G. Zhou, Sci. Bull. 66(20), 2072 (2021) X.X. Sun, J. Zhao, S.G. Zhou, Phys. Lett. B 785, 530 (2018) T.T. Sun, Z.X. Liu, L. Qian, B. Wang, W. Zhang, Phys. Rev. C 99, 054316 (2019) X.X. Sun, J. Zhao, S.G. Zhou, Nucl. Phys. A 1003, 122011 (2020) I. Talmi, Nucl. Phys. A 172(1), 1 (1971) I. Tanihata, H. Savajols, R. Kanungo, Prog. Part. Nucl. Phys. 68, 215 (2013) Y. Tian, Z.Y. Ma, P. Ring, Phys. Lett. B 676(1), 44 (2009) S. Typel, Front. Phys. 6, 73 (2018) J.G. Valatin, Phys. Rev. 122, 1012 (1961) A. Valor, J.L. Egido, L.M. Robledo, Phys. Rev. C 53, 172 (1996) M. Verriere, N. Schunck, D. Regnier, Phys. Rev. C 103, 054602 (2021) W. von Oertzen, A. Vitturi, Rep. Prog. Phys. 64(10), 1247 (2001) D. Vretenar, A.V. Afanasjev, G.A. Lalazissis, P. Ring, Phys. Rep. 409(3–4), 101 (2005) M. Wang, W. Huang, F. Kondev, G. Audi, S. Naimi, Chin. Phys. C 45(3), 030003 (2021) C.S. Wu, J.Y. Zeng, Phys. Rev. C 39, 666 (1989)

2044

X.-X. Sun and S.-G. Zhou

C.S. Wu, J.Y. Zeng, Phys. Rev. Lett. 66, 1022 (1991a) C.S. Wu, J.Y. Zeng, Phys. Rev. C 44, 2566 (1991b) X.W. Xia, Y. Lim, P.W. Zhao, H.Z. Liang, X.Y. Qu, Y. Chen, H. Liu, L.F. Zhang, S.Q. Zhang, Y. Kim, J. Meng, At. Data Nucl. Data Tables 121–122, 1 (2018) B.W. Xiong, Phys. Rev. C 101, 054305 (2020) F.R. Xu, R. Wyss, P.M. Walker, Phys. Rev. C 60, 051301(R) (1999) M. Yamagami, Y.R. Shimizu, T. Nakatsukasa, Phys. Rev. C 80, 064301 (2009) T. Yamaguchi, H. Koura, Y. Litvinov, M. Wang, Prog. Part. Nucl. Phys. 120, 103882 (2021) J.Y. Zeng, T.S. Cheng, Nucl. Phys. A 405(1), 1 (1983) J.Y. Zeng, Y.A. Lei, T.H. Jin, Z.J. Zhao, Phys. Rev. C 50, 746 (1994) J.Y. Zeng, S.X. Liu, Y.A. Lei, L. Yu, Phys. Rev. C 63, 024305 (2001) J.Y. Zeng, S.X. Liu, L.X. Gong, H.B. Zhu, Phys. Rev. C 65, 044307 (2002) S.S. Zhang, L.G. Cao, U. Lombardo, E.G. Zhao, S.G. Zhou, Phys. Rev. C 81, 044313 (2010) Y. Zhang, M. Matsuo, J. Meng, Phys. Rev. C 83, 054301 (2011) Z.H. Zhang, M. Huang, A.V. Afanasjev, Phys. Rev. C 101, 054303 (2020) K. Zhang, M.K. Cheoun, Y.B. Choi, P.S. Chong, J. Dong, Z. Dong, X. Du, L. Geng, E. Ha, X.T. He, C. Heo, M.C. Ho, E.J. In, S. Kim, Y. Kim, C.H. Lee, J. Lee, H. Li, Z. Li, T. Luo, J. Meng, M.H. Mun, Z. Niu, C. Pan, P. Papakonstantinou, X. Shang, C. Shen, G. Shen, W. Sun, X.X. Sun, C.K. Tam, Thaivayongnou, C. Wang, X. Wang, S.H. Wong, J. Wu, X. Wu, X. Xia, Y. Yan, R.W.Y. Yeung, T.C. Yiu, S. Zhang, W. Zhang, X. Zhang, Q. Zhao, S.G. Zhou, At. Data Nucl. Data Tables 144, 101488 (2022) S.G. Zhou, PoS INPC2016, 373 (2017) S.G. Zhou, C. Zheng, J. Hu, Phys. Rev. C 63, 047305 (2001) S.G. Zhou, J. Meng, P. Ring, Phys. Rev. C 68, 034323 (2003) S.G. Zhou, J. Meng, P. Ring, E.G. Zhao, Phys. Rev. C 82, 011301(R) (2010) Z.Y. Zhu, W.Q. Shen, Y.H. Cai, Y.G. Ma, Phys. Lett. B 328(1–2), 1 (1994)

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry Methods in Quantum Many-Body Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-Particle States in Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle-Number Non-conserving Dynamical Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Dynamical Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry in the Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nuclear Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairing and Quasi-spin SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation and SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairing with Neutrons and Protons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry in the Interacting Boson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Interacting Boson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bosons with F Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bosons with Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bosons with Intrinsic Spin and Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2046 2047 2047 2048 2049 2050 2051 2052 2052 2054 2057 2062 2065 2065 2066 2068 2070 2073 2075 2077 2077

Abstract

This chapter discusses symmetry methods as used in models of atomic nuclei. After an introduction of the notions of dynamical algebra, symmetry algebra,

P. Van Isacker () GANIL, CEA/DRF-CNRS/IN2P3, Boulevard Henri Becquerel, Caen, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_13

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and dynamical symmetry, relevant to any quantum many-body system, the focus is on a few historical models of seminal importance to our understanding of the structure of nuclei. They include the quasi-spin SU(2) model of pairing [and its neutron-proton SO(8) generalization], the SU(4) supermultiplet model, the SU(3) model of deformed nuclei, and the U(6) interacting boson model of collective nuclear excitations. The guiding thread of this review is that all these models can be understood from a common perspective based on the analysis of their symmetry character, which unveils the solvability of classes of model Hamiltonians and enables to establish connections between the single-particle and collective descriptions of nuclei.

Introduction Algebraic models of quantum-mechanical systems rely on the application of group theory to obtain insight in and to seek solutions of models in quantum mechanics. Since group theory is concerned with the mathematics of symmetry, algebraic models therefore propose a symmetry perspective of quantum mechanics. They stress the existence of conserved quantities, leading to a group-theoretical labeling of eigenstates, analytical expressions for eigenvalues and eigenfunctions, and selection rules for transition processes. With reference to the solution of the hydrogen atom based on SO(4) symmetry (Pauli 1926), it can be claimed that the algebraic approach did have its practitioners from the inception of quantum mechanics. Symmetries also played a vital role at the beginning of nuclear physics: first with the proposal of the isospin SU(2) algebra (Heisenberg 1932) and a few years later with the extension to the spin-isospin SU(4) algebra (Wigner 1937). In the subsequent decades nuclear physics saw the emergence of two contrasting views of the nucleus. The nuclear shell model (Mayer 1949; Jensen et al. 1949), on the one hand, assumes as a starting point that the nucleus is a collection of independent nucleons in a spherical mean field with a strong spin-orbit coupling. The geometric collective model (Rainwater 1950; Bohr 1952; Bohr and Mottelson 1953), on the other hand, starts from the premise that the nucleus can be treated as a dense, liquid drop that vibrates and, if deformed, also rotates. Significant progress in the understanding of the structure of both models has been made by applying the algebraic approach. Two achievements stand out. The first is the seniority model, which, although first developed to classify electron states in an atom (Racah 1943), has later found more fruitful use to understand pairing correlations in nuclei. The second achievement is the SU(3) model (Elliott 1958), which showed how deformation may arise in the spherical shell model, hence providing a link between the two abovementioned contrasting views. Collective models of the nucleus also lend themselves well to an algebraic treatment and none more so than the interacting boson model (IBM) of Arima and Iachello (1975), which provides a simple description of many nuclear properties based on a U(6) algebra.

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To review exhaustively a topic that has evolved over a period of nearly a century is an impossible task. Choices have to be made. This chapter, after a section on general features of symmetry methods, focuses on the few seminal algebraic models in nuclear physics, as sketched above, giving sufficient details to acquire some understanding of them at a formal level. Each discussion typically ends with a more recent example or an outlook on possible applications. Needless to say, many important aspects and studies are left out. One glaring omission is the algebraic treatment of the Bohr Hamiltonian, extensively discussed by Rowe and Wood (2010). It is hoped that the section on the IBM can serve as a proxy, as grouptheoretical aspects of the geometric collective model and the IBM are similar. Notions of group theory are used without much explanation. The interested reader may consult one of the many books that have covered the topic (Hamermesh 1962; M. Moshinsky 1968; Wybourne 1970; Elliott and Dawber 1979; Frank and Van Isacker 1994; Iachello 2006). All along this chapter, group-theoretical results are taken from the book by Iachello (2006).

Symmetry Methods in Quantum Many-Body Systems Since the notion of a dynamical symmetry is most conveniently discussed in second quantization, a brief reminder of some essentials of this formalism is given.

Many-Particle States in Second Quantization Particle creation and annihilation operators are denoted as cα† and cα , respectively. The index α stands for the set of labels of a single-particle state, which may include intrinsic quantum numbers such as spin, isospin, color, etc. The particles are either fermions or bosons, for which the notations c ≡ a and c ≡ b are used, respectively. They obey Fermi or Bose statistics, which in second quantization is imposed through the (anti-)commutation properties of creation and annihilation operators: {aα , aβ† } = δαβ ,

{aα† , aβ† } = {aα , aβ } = 0,

(1)

[bα , bβ† ] = δαβ ,

[bα† , bβ† ] = [bα , bβ ] = 0.

(2)

and

A many-body state is written as follows in second quantization: |n ¯ ≡

 (c† )nα √α |o, nα ! α

(3)

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where |o is the vacuum state, satisfying cα |o = 0 for all α. The many-body state (3) is determined by the number of particles nα in each single-particle state α, and the set of numbers nα is collectively denoted as n¯ ≡ {n1 , n2 , . . .}. For fermions only nα = 0 and nα = 1 are allowed due to the Pauli principle, but for bosons no restriction on nα exists.

Dynamical Algebras The properties of a quantal system of n interacting particles are determined by the secular equation associated with the Hamiltonian Hˆ =



εα cα† cα +

α

1 4



υαβγ δ cα† cβ† cδ cγ + · · · ,

(4)

αβγ δ

containing one-body terms εα , two-body interactions υαβγ δ , and possibly higherorder interactions. By construction, the Hamiltonian (4) conserves the number of particles, which covers the cases of interest in the applications discussed in this chapter. Since cβ† cδ = φ(cδ cβ† −δβδ ), where φ = +1 for bosons and φ = −1 for fermions, the Hamiltonian (4) can be written in a different form as Hˆ =

 αγ

εα δαγ − 14 φ

 β

  υαβγβ uˆ αγ + 14 φ υαβγ δ uˆ αδ uˆ βγ + · · · ,

(5)

αβγ δ

where the notation uˆ αγ ≡ cα† cγ is introduced. With use of the relations (1) or (2) the commutator of the uˆ αβ operators can be worked out, [uˆ αβ , uˆ γ δ ] = uˆ αδ δβγ − uˆ γβ δαδ ,

(6)

which is valid for bosons as well as for fermions. The uˆ αβ operators therefore close under commutation and generate the unitary (Lie) algebra U(Ω), where Ω is the dimension of the single-particle space. The conversion of the Hamiltonian from its normal ordered to its multipole form is always possible for any order of the interaction. In other words, the Hamiltonian appropriate for the description of n interacting particles can always be expressed in terms of the generators of U(Ω), which for this reason is called the spectrum-generating or dynamical algebra. The (irreducible) representations of the dynamical algebra acquire a particular relevance since they encode the action of the uˆ αβ operators on many-body states. The solution of eigenvalue problem for n interacting particles requires the diagonalization of the Hamiltonian in the symmetric representation [n] of U(Ω) in case of bosons or in its antisymmetric representation [1n ] ≡ [1, 1, . . . ] in case of fermions. While this, in general, is a numerical problem that quickly becomes intractable with an increasing number n of

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particles and/or with increasing dimension Ω of the single-particle space, particular classes of Hamiltonians can be solved by symmetry methods.

Dynamical Symmetries The preceding subsection defined the notion of a dynamical algebra, which henceforth is denoted as Gdyn . This is not the symmetry algebra of the problem since, in general, not all generators of Gdyn commute with Hˆ . Assume that the Hamiltonian does have a symmetry algebra Gsym , which implies that [Hˆ , g] ˆ = 0,

∀gˆ ∈ Gsym .

(7)

The symmetry algebra necessarily must be a subalgebra of the dynamical algebra, Gdyn ⊃ Gsym . It is therefore possible to look for chains of nested algebras of the type Gdyn ≡ G1 ⊃ G2 ⊃ · · · ⊃ Gs ≡ Gsym ,

(8)

where for subsequent convenience the algebras are enumerated, the dynamical (symmetry) algebra being G1 (Gs ). While the dynamical algebra may be unitary, Gdyn = U(Ω), the subalgebras in Eq. (8) can be unitary, orthogonal, or symplectic and, in some cases, even exceptional. To appreciate the relevance of the chain (8) of nested algebras, one first needs to introduce the notion of a Casimir operator of an algebra G. It is an invariant operator, denoted in the following as Cˆ m [G], which commutes with all generators of G: [Cˆ m [G], g] ˆ = 0,

∀gˆ ∈ G,

(9)

where m is the order of Cˆ m [G] in the generators g. ˆ For any (semi-simple) Lie algebra there exists a procedure, based on the commutation properties of the generators, to construct the Casimir operator of order m. The expressions for the Casimir operators of all Lie algebras are well known and can be found in Chapter 5 of the book by Iachello (2006). Associated with the chain (8) of nested algebras is the Hamiltonian Hˆ =

s  

i ˆ κm Cm [Gi ],

(10)

i=1 m i are arbitrary coefficients. The sum in i runs over all algebras in Eq. (8), where κm while the sum in m goes up to the maximum order of the interaction between the particles. Since for any pair i ≤ i all elements of Gi also belong to Gi , the operators in the expansion (10) satisfy

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[Cˆ m [Gi ], Cˆ m [Gi ]] = 0,

∀i, i ,

∀m, m .

(11)

This shows that the Hamiltonian (10) is written as a sum of commuting operators. Its eigenstates are therefore characterized by the quantum numbers associated with these operators, that is, by the labels of the representations of the different algebras Gi , which again are well known for all Lie algebras; see Chapter 4 of the book by Iachello (2006). The Hamiltonian (10), obtained from the general expression (4) for specific combinations of the coefficients εα , υαβγ δ . . . , can therefore be solved analytically. The most important property of the Hamiltonian (10) is that its eigenstates have a i . The labeling of its eigenstates fixed structure, independent of the coefficients κm can be summarized as follows: Gdyn ⊃ G2 ⊃ · · · ⊃ Gsym , ↓ ↓ ↓ η12 Σ2 ηs−1,s Σs Σ1

(12)

where underneath the algebra Gi is given the associated label or set of labels Σi . There appears an additional label ηi−1,i (also known as a missing label), which is needed if Σi occurs more than once in the representation Σi−1 . The eigenvalues of the Hamiltonian (10) are given in closed form as Hˆ |Σ1 η12 Σ2 . . . ηs−1,s Σs  =

s  

i κm Em (Σi )|Σ1 η12 Σ2 . . . ηs−1,s Σs ,

(13)

i=1 m

where Em (Σi ) is the eigenvalue of the Casimir operator Cˆ m [Gi ] in the representation Σi , known from the standard theory of Lie groups; see Chapter 5 of the book by Iachello (2006). This establishes a general procedure for finding Hamiltonians (4) that can be solved analytically. It amounts to finding chains of nested algebras of the type (8), which is a mathematical problem. Many-body Hamiltonians associated with a chain of nested algebras are said to have a dynamical symmetry. The dynamical algebra Gdyn is not a symmetry of the Hamiltonian but is broken dynamically until the only remaining symmetry is Gsym . Examples of dynamical symmetries exist in many different fields of physics. Two cases of relevance to nuclear physics are discussed in this chapter: SU(3) in the shell model and the dynamical symmetries of the interacting boson model.

Particle-Number Non-conserving Dynamical Algebras The dynamical algebra Gdyn constructed from the unitary operators uˆ αβ can become very large, resulting in a difficult group-theoretical analysis of the chain (8) of nested algebras. In some cases a more economical method is available based on

54 Algebraic Models of Nuclei

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the introduction of generators that do not conserve particle number in addition to † the unitary operators. Such generators are denoted as sˆαβ ≡ cα cβ and sˆαβ ≡ cα† cβ† , † which implies that sˆαβ = (ˆsβα )† . To arrive at a set of operators that close under commutation, the particle-number conserving generators are modified to uˆ αβ ≡ uˆ αβ + 12 φδαβ . This modified expression does not alter the commutation relations between them,

[uˆ αβ , uˆ γ δ ] = uˆ αδ δβγ − uˆ γβ δαδ ,

(14)

since the operators uˆ αβ and uˆ αβ differ by a constant only. With use of the relations (1) and (2), the commutator of uˆ αβ with sˆγ δ and sˆγ† δ can be worked out: [uˆ αβ , sˆγ δ ] = −ˆsγβ δαδ − sˆβδ δαγ ,

† [uˆ αβ , sˆγ† δ ] = sˆαδ δβγ + sˆγ† α δβδ .

(15)

Finally, the commutator of sˆαβ with sˆγ† δ is [ˆsαβ , sˆγ† δ ] = uˆ δβ δαγ + uˆ γ α δβδ + φ(uˆ δα δβγ + uˆ γβ δαδ ),

(16)

which shows that the modification uˆ αβ → uˆ αβ is necessary to ensure closure. The

† number of independent generators in the set {uˆ αβ , sˆαβ , sˆαβ } is n(2n+1) or n(2n−1) for bosons or fermions, forming the algebras Sp(2n) or SO(2n), respectively. † The addition of the pair creation and annihilation operators sˆαβ and sˆαβ does not lead to a simplification of the algebraic structure of the problem since it enlarges the dimension of the dynamical algebra. The simplification is obtained by considering † specific linear combinations of uˆ αβ , sˆαβ , and sˆαβ such that these form a subalgebra of either Sp(2n) or SO(2n). Two cases of relevance to nuclear physics are discussed in this chapter: the quasispin SU(2) model of pairing and the SO(8) model of T = 0 and T = 1 pairing.

Partial Dynamical Symmetries The classification (12) can be rewritten as ⊃ G2 ⊃ G3 ⊃ · · · ⊃ Gsym Gdyn , ↓ ↓ ↓ ↓ Σs ≡ Λ Σ1 ≡ [hn ] Σ2 Σ3

(17)

where for the present discussion missing labels can be omitted. It is assumed in this section that particle number is conserved, in which case the representation [hn ] is either symmetric [n] (bosons) or antisymmetric [1n ] (fermions). In the case of a dynamical symmetry associated with Eq. (17), all levels are labeled by |[hn ]Σ2 Σ3 . . . Λ, the whole spectrum is known analytically, and all eigenstates

2052

P. Van Isacker

have a fixed structure, independent of the parameters in the Hamiltonian. These conditions can be relaxed so that they are only partially valid, leading to partial dynamical symmetries, which can be of three different types: 1. All labels Σ2 , Σ3 , . . . are conserved for part of the eigenstates. 2. Part of the labels Σ2 , Σ3 , . . . are conserved for all eigenstates. 3. Part of the labels Σ2 , Σ3 , . . . are conserved for part of the eigenstates. Partial dynamical symmetries, originally suggested by Leviatan and collaborators (Leviatan et al. 1986; Alhassid and Leviatan 1992; Leviatan 1996), have been applied in diverse areas of physics; see the review by Leviatan (2011). For a partial dynamical symmetry of the first kind, the following general construction procedure is available. Just as eigenstates can be labeled as |[hn ]Σ2 . . . Λ, operators can be classified according to their tensor character under (17) as m] Tˆσ[h2 ...λ . Of particular relevance to a partial dynamical symmetry are the m-particle annihilation operators with the property (0) m] Tˆσ[h2 ...λ |[hn ]Σ2 . . . Λ = 0,

(18)

(0)

for all states contained in a given representation Σ2 of G2 . Any interaction written in terms of these annihilation operators and their Hermitian conjugates can be added to the Hamiltonian (10), without mixing the labels of states with Σ2 = Σ2(0) . The annihilation condition (18) is satisfied if none of the representations Σ2 of G2 contained in the representation [hn−m ] of Gdyn belongs to the Kronecker product (0) Σ2 × σ2 . So the problem of finding partial dynamical symmetries of the first kind is reduced to a Kronecker product (Leviatan and Van Isacker 2002).

Symmetry in the Shell Model The Nuclear Shell Model For an introduction to the nuclear shell model, the reader is referred to the many comprehensive discussions in several books that appeared over the years (de-Shalit and Talmi 1963; Bohr and Mottelson 1969; Brussaard and Glaudemans 1977; Lawson 1980; Heyde 1990; Talmi 1993). In this section a review is given of its properties from an algebraic perspective, indicating what are the appropriate dynamical and symmetry algebras. It is assumed that the nucleus can be described as a system of interacting (possibly only valence) neutrons and protons, occupying a number of shells of a harmonic oscillator. Different situations may arise: 1. The valence shells are occupied by either neutrons or protons. The dynamical algebra is U(Ω), where Ω is the dimension of the single-particle space, Ω =  (2j + 1). The generators of U(Ω) are aj†mj aj m j or j

54 Algebraic Models of Nuclei

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(J ) (J ) Aˆ MJ (j, j ) ≡ (aj† × a˜ j )MJ =

 mj m j

(j mj j m j |J MJ )aj†mj a˜ j m j ,

(19)

where the symbol in round brackets is a Clebsch-Gordan coefficient (Talmi 1993) and a˜ j mj ≡ (−)j +mj aj −mj is a modified annihilation operator. Since the nuclear shell model Hamiltonian is invariant under rotations, the total angular momentum J of the nucleons is conserved, and the symmetry algebra is SUJ (2). 2. The valence shells are occupied by neutrons and protons. The dynamical algebra is U(Ω), where Ω is the dimension of the single-particle space, which is assumed to be the same for neutrons and protons, Ω = 2 j (2j + 1). As in most shell model calculations, the total isospin T of the nucleons is assumed to be a conserved quantum number. The generators of U(Ω) can be written in isospin formalism as (J T ) (J T ) Aˆ MJ MT (j, j ) ≡ (aj†1/2 × a˜ j 1/2 )MJ MT ,

(20)

where a˜ j mj 1/2mt ≡ (−)j +mj +1/2+mt aj −mj 1/2−mt . The symmetry algebra is SUJ (2) ⊗ SUT (2). 3. The valence shells are occupied by neutrons and protons, and the Hamiltonian is independent of the spin and isospin of the nucleons. Thedynamical algebra is U(4Ωorb ), where Ωorb is the orbital degeneracy, Ωorb = l (2l + 1), and the factor 4 accounts for the spin-isospin degeneracy. The generators of U(4Ωorb ) can be written as (LST ) (LST ) Aˆ ML MS MT (l, l ) ≡ (al†1/21/2 × a˜ l 1/21/2 )ML Ms MT ,

(21)

where a˜ lml 1/2ms 1/2mt ≡ (−)l+ml +1/2+ms +1/2+mt aj −mj 1/2−ms 1/2−mt . The total angular momentum J and the total isospin T of the nucleons, as well as their total orbital angular momentum L and total spin S, are conserved quantum numbers, and the symmetry algebra is SOL (3) ⊗ SUS (2) ⊗ SUT (2). Whether the j t or the lst representation of nucleon operators is more convenient depends on the application. Results should be independent of the particular representation used because the operators are connected by the unitary transformation: aj†mj 1/2mt =



† (lml 1/2ms |j mj )alm . l 1/2ms 1/2mt

(22)

ml ms

In general, energy spectra and eigenfunctions in the shell model are obtained by diagonalizing a nucleon-number conserving Hamiltonian in the antisymmetric representation [1n ] of U(Ω), which has the dimension 

Ω n

 =

Ω! . n!(Ω − n)!

(23)

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P. Van Isacker

This dimension can be reduced by exploiting symmetries of the Hamiltonian (conservation of separate neutron and proton numbers, of angular momentum J , of isospin T ,. . . ), but it is clear that the numerical diagonalization of the Hamiltonian matrix rapidly becomes a formidable problem with increasing Ω and/or n. Hence the interest of establishing dynamical symmetries and analytical solutions of the shell model.

Pairing and Quasi-spin SU(2) Consider a single-j shell occupied by a single type of nucleon, leading to the dynamical algebra U(2j + 1) with the generators (19) of case 1. Furthermore, it is assumed that the nucleons interact through a pairing force, j j Vˆpairing = −g Sˆ+ Sˆ− ,

(24)

with j Sˆ+ =

1 2

 2j + 1(aj† × aj† )(0) 0 ,

 j † j Sˆ− = Sˆ+ .

(25)

This interaction can be diagonalized analytically for an n-body system by considerj ing, in addition to Sˆ± , also the operator j Sˆz = 14 (2nˆ j − Ω),

where nˆ j = relations



† m aj m aj m

(26)

is the nucleon-number operator. The commutation

j j j [Sˆz , Sˆ± ] = ±Sˆ± ,

j j j [Sˆ+ , Sˆ− ] = 2Sˆz ,

(27)

j j show that the three operators {Sˆz , Sˆ± } form an SU(2) algebra, which is referred to as the quasi-spin algebra (Kerman 1961; Helmers 1961). Since

 2  j 2 j j j Sˆ+ Sˆ− = Sˆ j − Sˆz + Sˆz ,

(28)

it follows that the pairing interaction can be written as a combination of Casimir operators belonging to the chain of nested algebras: j SU(2) ⊃ SO(2) ≡ {Sˆz } . ↓ ↓ S MS

(29)

54 Algebraic Models of Nuclei

2055

The eigenvalues of the pairing Hamiltonian (24) follow immediately: E(S, MS ) = −g[S(S + 1) − MS (MS − 1)].

(30)

This derivation shows that an eigenstate of a system of n nucleons interacting via a pairing force is characterized by two quantum numbers S and MS , in addition to the usual J and MJ that follow from the rotational invariance of the Hamiltonian. The quantum numbers S and MS can be replaced by the more usual ones: S = 14 (Ω − 2υ),

MS = 14 (2n − Ω),

(31)

where the second relation follows immediately from Eq. (26) and shows that MS and the nucleon number n are equivalent. A state |SMS J MJ  can therefore also be denoted as |j n υJ MJ  such that the eigenvalue expression (30) becomes E(n, υ) = − 14 g(n − υ)(Ω − n − υ + 2).

(32)

j Since E(n = υ, υ) = 0, it follows that Sˆ− |j υ υJ MJ  = 0, that is, the state υ |j υJ MJ  does not contain a nucleon pair coupled to J = 0. The repeated action j of Sˆ+ on the state |j υ υJ MJ  produces a chain of states with υ + 2, υ + 4, . . . nucleons, which all have the same quantum number υ. The state |j υ υJ MJ  therefore acts as a parent state for a whole class of states n |j υJ MJ , and for this reason υ is called seniority. The seniority quantum number was introduced in 1943 by Racah (1943) for the classification of atomic spectra. Despite pairing being a poor approximation to the Coulomb interaction, Racah considered it to distinguish states of n electrons in an l shell with a given orbital angular momentum L and spin S. The seniority scheme was later adapted (Racah 1952; Flowers 1952) to jj coupling appropriate for nuclei. In subsequent studies (Racah and Talmi 1952; Schwartz and de Shalit 1954), summarized in the book by Talmi (1993), it was shown that in a single-j shell seniority is conserved for a much wider class of interactions than just pairing and that the interaction between nucleons of the same kind conserves seniority to a good approximation. More recently, interest in seniority was rekindled (Rowe and Rosensteel 2001; Escuderos and Zamick 2006; Van Isacker and Heinze 2008; Qi 2011) because in a single-j shell examples occur of the conservation of seniority for some but not all states, that is, examples of a partial dynamical symmetry. An obvious limitation of the cases discussed so far is that they are confined to a single-j shell, which seldom is a reasonable approximation to a realistic valence space. The quasi-spin algebra can be generalized in a straightforward way to the  j j case of several  degenerate j shells by making the substitutions S+ → S+ ≡ j S+ and Ω → j (2j + 1). All of the characteristic predictions of the SU(2) quasispin solution of the pairing Hamiltonian (i.e., an n-independent excitation energy of the 2+ 1 level in even-even nuclei, the linear variation of two-nucleon separation

2056

P. Van Isacker

energies as a function of n, the odd-even staggering in nuclear binding energies, the enhancement of two-nucleon transfer) remain valid for identical nucleons interacting through a pairing force and distributed over several degenerate j shells. If the nucleons interact through a pairing force and are distributed over several nondegenerate j shells, the eigenvalue problem can still be solved by a technique proposed by Richardson (1963). This method and its generalization to other classes of integrable pairing models are reviewed by Dukelsky et al. (2004). The nondegeneracy of the j shells implies that the structure of the eigenstates is no longer independent of the parameters in the pairing Hamiltonian. Since this property is one of the defining features of a dynamical symmetry, the Richardson solution is not discussed further here. An even more generally valid approach is obtained if one imposes the following condition on a shell-model Hamiltonian:  α 2 α α [[Hˆ GS , Sˆ+ ], Sˆ+ ] = Δ Sˆ+ , (33) α =  α S j creates the lowest two-particle eigenstate where Δ is a constant and Sˆ+ j j + of Hˆ GS with energy E0 . Hamiltonians that obey the condition (33) are said to have generalized seniority (Talmi 1971), which is a much weaker assumption than that of a pairing interaction. Generalized seniority Hamiltonians do not generally have a dynamical symmetry but satisfy a number of simple properties such as an even-even α )n/2 |o with energy nE + 1 n(n − 1)Δ, which reduces ground state of the form (Sˆ+ 0 2 to Racah’s seniority formula (32) for E0 = − 14 g(Ω + 1) and Δ = 12 g. Another property of generalized seniority Hamiltonians is the constancy of the excitation energy of the 2+ 1 level. The conservation of seniority also affects electromagnetic transitions and leads to selection rules and to simple predictions for the dependence of the transition probabilities on the number of valence nucleons. This is well known for even-tensor electric (e.g., E2) transitions in a single-j shell (Talmi 1993). It can be extended to even- and odd-tensor transitions between eigenstates of a generalized seniority Hamiltonian, as shown with an application to high-spin isomers in the tin isotopes (Maheshwari and Jain 2016). Not all simple predictions of single-j seniority remain valid for a generalized seniority Hamiltonian. This is illustrated in Fig. 1, which summarizes the situation for the tin isotopes as regards the excitation energy of the 2+ 1 level and its reduced E2 transition probability to the ground state. While the 2+ 1 level is calculated at a constant excitation energy, its E2 decay to the ground state does not follow the simple parabolic behavior expected in a single-j or degenerate multi-j shell. It should be stressed, however, that the theoretical result shown in Fig. 1 is not a + prediction but rather a fit to the data. Specifically, the B(E2; 2+ 1 → 01 ) values are 13 17 obtained assuming two nondegenerate shells with j = /2 and /2 with a fractional filling that depends on the neutron number N . Details of this approach are given by Morales et al. (2011) and adapted here to account for the revised E2 data that are reported by Allmond et al. (2015).

54 Algebraic Models of Nuclei

2057

Fig. 1 The excitation energy of the 2+ 1 level and its reduced E2 transition probability to the ground + + state in the tin isotopes. The Ex (2+ 1 ) energies are taken from ENSDF and the B(E2; 21 → 01 ) values from Allmond et al. (2015). The data (black circles) are compared with a generalized seniority calculation assuming a single-j or degenerate multi-j shell (blue triangles) or assuming two nondegenerate shells with j = 13/2 and 17/2 (red squares)

Deformation and SU(3) Consider an entire major shell of the harmonic oscillator, with major quantum number N, occupied by neutrons and protons. In addition, it is assumed, following Wigner (1937), that the shell model Hamiltonian Hˆ is invariant under rotations in spin and isospin space. The dynamical algebra is therefore U(4Ωorb ) where Ωorb = (N + 1)(N + 2)/2 with generators (21) of case 3. They can be divided into those that are scalar in orbital angular momentum, (ST ) Aˆ MS MT ≡

√

(0ST )

2l + 1(al†1/21/2 × a˜ l 1/21/2 )0MS MT ,

(34)

l

and those that are scalar in spin-isospin, † Aˆ (L) ˜ l 1/21/2 )(L00) ML (l, l ) ≡ (al 1/21/2 × a ML 00 .

The two sets have one common element, namely, the number operator:

(35)

2058

P. Van Isacker (00) nˆ = 2Aˆ 00 = 2

√

(0) 2l + 1 Aˆ 0 (l, l).

(36)

l

The spin, isospin and spin-isospin operators can be written as Sˆμ =

√

(10) 2Aˆ μ0 ,

Tˆν =

√

(01) 2Aˆ 0ν ,

Yˆμν = Aˆ (11) μν ,

(37)

and, together with the number operator, they generate the algebra UST (4). If the nuclear Hamiltonian satisfies the commutation relations [Hˆ , n] ˆ = [Hˆ , Sˆμ ] = [Hˆ , Tˆμ ] = [Hˆ , Yˆμν ] = 0,

(38)

as was assumed by Wigner (1937), the symmetry algebra of Hˆ is UST (4). The generators of UL (Ωorb ) and those of UST (4) close under commutation. In particular, the following commutator property among the generators of UL (Ωorb ) is valid:

ˆ (λ ) [Aˆ (λ) μ (l1 , l2 ), Aμ (l3 , l4 )] =

1 2



(2λ + 1)(2λ + 1)



(λμ λ μ |λ μ )

λ μ



(λ ) l1 +l4 +λ λ λ λ δl2 l3 Aˆ μ (l1 , l4 ) × (−) l4 l1 l2 

λ λ λ (λ ) δl1 l4 Aˆ μ (l3 , l2 ) , −(−)l2 +l3 +λ+λ l3 l2 l1

(39)

where the symbol in curly brackets is a 6j symbol (Talmi 1993). Furthermore, the fact that the generators of UL (Ωorb ) and those of UST (4) commute which each other implies the classification U(4Ωorb ) ⊃ UL (Ωorb ) ⊗ UST (4) ↓ ↓ ↓ . ¯ [h] [h¯ ] [1n ]

(40)

¯ ≡ [h1 , . . . , hΩorb ] in orbital Many-body states are characterized by the labels [h] space and the corresponding spin-isospin labels [h¯ ] ≡ [h 1 , h 2 , h 3 , h 4 ], where h1 +· · ·+hΩorb = h 1 +h 2 +h 3 +h 4 = n. The symmetry under the partial exchange of coordinates (e.g., of only the spatial or only the spin-isospin coordinates) can be specified by a Young diagram. Since the overall wave function is antisymmetric, the ¯ and [h¯ ] must be obtained from each other by interchanging two Young diagrams [h] rows and columns, that is, they must be conjugate. Overall anti-symmetry of the many-body wave function therefore imposes certain restrictions on the allowed orbital Young diagrams: they cannot have rows with more than four boxes, 4 ≥ h1 ≥ h2 ≥ · · · ≥ hΩorb ≥ 0. This condition expresses the fact that one orbital

54 Algebraic Models of Nuclei

2059

single-particle state cannot be occupied by more than four particles, namely, the four nucleonic intrinsic states with spin and isospin up or down. Since the number operator nˆ belongs to UL (Ωorb ) as well as UST (4), a proper direct product of algebras is UL (Ωorb ) ⊗ SUST (4), where SUST (4) no longer contains n. ˆ The representations of SUST (4) are characterized by the three labels: λ = h 1 − h 2 ,

μ = h 2 − h 3 ,

ν = h 3 − h 4 ,

(41)

and the classification (40) can be replaced by an equivalent one: U(4Ωorb ) ⊃ UL (Ωorb ) ⊗ SUST (4) . ↓ ↓ ↓ ¯ [h] (λ , μ , ν ) [1n ]

(42)

The central idea of Wigner’s supermultiplet model is that the attractive shortrange character of the nucleon-nucleon interaction favors many-body states with maximal spatial or minimal spin-isospin symmetry. Such states occur at low energy in the spectrum and are said to belong to the “favored supermultiplet.” The separation of the supermultiplets [i.e., of the representations of SUST (4)] can be achieved with the operator Cˆ 2 [UL (Ωorb )] = 4

 ll λ

√  (0) (−)l+l +λ 2λ + 1 Aˆ (λ) (l, l ) × Aˆ (λ) (l , l) 0 ,

(43)

which is the quadratic Casimir operator of UL (Ωorb ) since it commutes with all its generators, ˆ [Aˆ (λ) μ (l, l ), C2 [UL (Ωorb )]] = 0.

(44)

Casimir operators are determined up to an overall factor only; the coefficient 4 in Eq. (43) ensures that the expectation value of Cˆ 2 [UL (Ωorb )] yields the eigenvalue of UL (Ωorb ) known from classical group theory (Iachello 2006): ¯ = EU(Ω) (h1 , h2 , . . . , hΩ ) ≡ EU(Ω) (h)

Ω 

hi (hi + Ω + 1 − 2i).

(45)

i=1

The operator Cˆ 2 [UL (Ωorb )] can be considered as a Majorana operator since it splits states with different orbital symmetry. For a given nucleus with N neutrons and Z protons, the possible supermultiplet labels are restricted because the nucleus’ isospin projection Tz = 12 (N − Z) requires that (λ , μ , ν ) must contain at least one isospin with T ≥ |Tz |. In a given supermultiplet the allowed values of S and T are found from the branching rule for

2060

P. Van Isacker

SUST (4) ⊃ SUS (2) ⊗ SUT (2) ↓ ↓ ↓ , S T (λ , μ , ν )

(46)

which can be derived in general. The SU(3) model of Elliott (1958) can now be introduced. It is a further elaboration upon Wigner’s supermultiplet model since it proposes the following chain of nested algebras starting from the orbital dynamical algebra UL (Ωorb ): UL (Ωorb ) ⊃ UL (3) ⊃ SUL (3) ⊃ SOL (3) ↓ ↓ ↓ ↓ . ¯ (λ, μ) KL [h] [h 1 , h 2 , h 3 ]

(47)

The algebra UL (3) contains the number operator nˆ [see Eq. (36)] and the components of the angular-momentum and quadrupole operators, Lˆ μ =





l

 ˆμ = Q

4l(l + 1)(2l + 1) ˆ (1) Aμ (l, l), 3

64π  lr 2 Y2 l  ˆ (2) Aμ (l, l ), √ 5 5 ll

(48)

ˆ μ only. In Q ˆ μ appear the reduced matrix while SUL (3) consists of Lˆ μ and Q 2 elements of r Y2 , which are well known for a harmonic oscillator (Talmi 1993):  lr Y2 l = −(2N + 3) 2

 lr Y2 l + 2 = − 2

5 l(l + 1)(2l + 1) , 16π (2l − 1)(2l + 3)

5 6(l + 1)(l + 2)(N − l)(N + l + 3) . 16π 2l + 3

(49)

ˆ μ } close under commutation, The eight operators {Lˆ μ , Q √ [Lˆ μ , Lˆ ν ] = − 2 (1μ 1ν|1μ + ν)Lˆ μ+ν , √ ˆ ν ] = − 6 (1μ 2ν|2μ + ν)Qˆ μ+ν , [Lˆ μ , Q √ ˆ μ, Q ˆ ν ] = 3 10 (2μ 2ν|1μ + ν)Lˆ μ+ν , [Q

(50)

and form a subalgebra of UL (Ωorb ). This proves that the chain of nested algebras (47) can be constructed for any major oscillator shell N . ¯ follow from Wigner’s supermultiplet model; see The UL (Ωorb ) labels [h] Eq. (42). The representations of UL (3) are labeled by a three-rowed Young diagram and those of SUL (3) by the differences

54 Algebraic Models of Nuclei

2061

λ = h 1 − h 2 ,

μ = h 2 − h 3 .

(51)

The quantum number L in Eq. (47) is the total orbital angular momentum of the nucleons. The allowed UL (3) labels [h¯ ] or, equivalently, the allowed SUL (3) ¯ of UL (Ωorb ), are determined from labels (λ, μ) for a given Young diagram [h] the branching rule for UL (Ωorb ) ⊃ UL (3). In general, no closed expression is available for this rule, but it can be derived by a procedure known as the plethysm of S functions (Littlewood 1940; Wybourne 1970). Likewise, the branching rule for SUL (3) ⊃ SOL (3) can be obtained from S-function theory for orthogonal algebras. Since some L values may occur more than once in a given representation (λ, μ), a multiplicity label K is needed. From these branching rules the following picture emerges. Each representation (λ, μ) contains the orbital angular momenta L of a rotational band up to some upper value. Also, the label K can be interpreted as the projection of L on the axis of symmetry of the rotating deformed nucleus (Elliott 1958). ˆ ·Q ˆ commutes with From the relations (50) it can be shown that 12 Lˆ · Lˆ + 16 Q ˆ ˆ Lμ and Qμ , which therefore can be identified with the quadratic Casimir operator of SUL (3), with eigenvalues known from classical group theory (Iachello 2006): ESU(Ω) (h1 , h2 , . . . , hΩ−1 , 0) =

  Ω   h h hi − + 2Ω − 2i . hi − Ω Ω

(52)

i=1

For Ω = 3 this eigenvalue expression can be rewritten in terms of the SU(3) labels ˆ ·Q ˆ has the (λ, μ) with the result that the rescaled Casimir operator 3Lˆ · Lˆ + Q eigenvalues 4[λ(λ + 3) + μ(μ + 3) + λμ]. In summary, within one major shell of the harmonic oscillator, the Hamiltonian ˆ ·Q ˆ + κ Lˆ · L, ˆ Hˆ 1SU(3) = κM Cˆ 2 [UL (Ωorb )] + κ Q

(53)

is a combination of Casimir operators belonging to the chain (47) of nested algebras. As a consequence, it has a dynamical symmetry with the n-nucleon eigenstates: ¯ μ)KL × [h¯ ]ST , |[1n ]; [h](λ,

(54)

with eigenvalues ¯ + 4κ[λ(λ + 3) + μ(μ + 3) + λμ] + (κ − 3κ)L(L + 1). κM EU(Ωorb ) (h)

(55)

The SU(3) model implies that a quadrupole interaction, assuming it is scalar in spin and isospin and acts in a complete, degenerate shell of the harmonic oscillator, leads to a rotational classification of states as a result of the mixing of spherical configurations. Elliott’s work demonstrated that deformed nuclear shapes, as they arise in the geometric collective model, can be described with the spherical shell model.

2062

P. Van Isacker

The SU(3) model can be extended to cover two consecutive major harmonic oscillator shells described by the Hamiltonian (Van Isacker and Pittel 2016): ˆ − + κ+ Q ˆ + + κQ ˆ ·Q ˆ + κ Lˆ · L, ˆ− ·Q ˆ+ ·Q ˆ Hˆ 2SU(3) = ε− nˆ − + ε+ nˆ + + κ− Q

(56)

ˆ ±,μ are the number, angular-momentum, and quadrupole where nˆ ± , Lˆ ±,μ and Q ˆ μ are summed operators for the lower and upper shells, respectively, and Lˆ μ and Q ˆμ ≡ Q ˆ −,μ + Q ˆ +,μ . An application of the operators, Lˆ μ ≡ Lˆ −,μ + Lˆ +,μ and Q two-shell SU(3) Hamiltonian (56) to 20 Ne is shown in Fig. 2. The SU(3) model provides an explanation of deformed nuclear states, but it is based on Wigner’s SU(4) symmetry, which is broken in most nuclei mainly due to the strong spin-orbit coupling for nucleons. Since Elliott’s work, many studies have been devoted to arrive at an SU(3)-type understanding of nuclear deformation starting from a jj rather than an LS classification. Three such extensions are called pseudo-, quasi-, and proxy-SU(3), respectively. A pseudo-LS classification is obtained for a specific ratio of the spin-orbit and orbit-orbit strengths. It is the consequence of a pseudo-spin symmetry of the Hamiltonian, which results in degenerate pseudo-spin doublets in the nuclear mean field (Hecht and Adler 1969; Arima et al. 1969) and which is a property of the Dirac equation if the scalar and vector potentials are equal in size but opposite in sign (Ginocchio 1997). QuasiSU(3) is an approximate symmetry based on the similarities of matrix elements of the quadrupole operator in the jj - and LS-coupling schemes (Zuker et al. 1995). Proxy-SU(3) is an approximate, analytic treatment of the Nilsson model (1955), which applies to heavy, well-deformed nuclei (Bonatsos et al. 2017). These three SU(3) approximations are schematically illustrated in Fig. 3 for the sdg shell of the harmonic oscillator.

Pairing with Neutrons and Protons The pairing interaction considered in the SU(2) quasi-spin model acts between identical nucleons in a single-j shell coupled to angular momentum J = 0. If neutrons and protons are considered, the isospin of a pair of nucleons is not necessarily T = 1 but can also be T = 0, and the pairing interaction can be generalized. The most convenient way to do so is to consider the lst representation of nucleon operators with a pairing interaction between pairs of nucleons coupled to total orbital angular momentum L = 0. The antisymmetric nature of the pairs implies that for L = 0 one has S = 0, T = 1 or S = 1, T = 0. The pairing interaction is therefore either of spin-scalar, isovector or of spin-vector, isoscalar character with two independent strength parameters. This leads to a generalized pairing interaction of the form 01 ˆ 01 10 ˆ 10 Vˆpairing = −g01 Sˆ+ · S− − g10 Sˆ+ · S− ,

(57)

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Fig. 2 The observed energy spectrum of 20 Ne taken from ENSDF compared with the eigenspectrum of the two-shell SU(3) Hamiltonian (56) for 16 nucleons in the p-sd shells with parameters (in MeV) Δε ≡ ε+ − ε− = 7.60, κ− = κ+ = −0.0125, κ = −0.025 and κ = 0.16. The rotational bands are labeled by K π in (a) and by the SU(3) quantum numbers (λ, μ) in (b)

where the dot indicates a scalar product in spin or in isospin and with 01 Sˆ+,μ =

√1 2

√

(001) 2l + 1(al†1/21/2 × al†1/21/2 )00μ ,

 01 † 01 Sˆ−,μ = Sˆ+,μ ,

(010) 2l + 1(al†1/21/2 × al†1/21/2 )0μ0 ,

 10 † 10 Sˆ−,μ = Sˆ+,μ .

l

10 = Sˆ+,μ

√1 2

√ l

(58)

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Fig. 3 Schematic illustration of three SU(3) approximations for the sdg shell of the harmonic oscillator. The exact SU(3) dynamical symmetry requires a vanishing spin-orbit coupling as illustrated with the single-particle energies shown in (a). The strong nuclear spin-orbit coupling splits the d and g doublets and introduces the 0h11/2 orbital. While the 0h11/2 orbital (red) is not included in quasi- or pseudo-SU(3), both approaches propose an SU(3)-type treatment of other orbitals. (b) Quasi-SU(3) is based on the observation that the spaces constructed from 0g9/2 , 1d5/2 , 2s1/2 (blue) and from 0g7/2 , 1d3/2 (purple) approximately decouple and that each space has approximate SU(3) properties. (c) Pseudo-SU(3) arises if the 0g9/2 orbital (red) can be omitted from the valence space and if 0g7/2 , 1d5/2 and 1d3/2 , 2s1/2 are doublets, which then enables their treatment in terms of a pseudo-pf shell (blue). (d) In proxy-SU(3) the deformed 0g9/2 levels (red) are not included, but they are replaced by all but one of the deformed 0h11/2 levels, of which it can be shown that they behave approximately as the former

To recognize the algebraic structure of this model, one considers, in addition to the pair creation and annihilation operators, nucleon-number conserving operators (Flowers and Szpikowski 1964; Pang 1969; Evans et al. 1981). This is similar to the construction of the SU(2) quasi-spin algebra, where the pair creation and annihilation operators Sˆ± are supplemented with Sˆz to obtain a closed algebraic structure. To ensure closure in the neutron-proton case, one is compelled to add several more 10 , Sˆ 01 }, which operators to arrive at the set of 28 operators {n, ˆ Sˆμ , Tˆν , Yˆμν , Sˆ±,μ ±,μ form SO(8), the dynamical algebra of the model. Besides the pair creation and annihilation operators, it contains the number operator n, ˆ the spin and isospin operators Sˆμ and Tˆν , and the spin-isospin operators Yˆμν , defined in the supermultiplet model. Physically relevant classifications in the SO(8) model conserve spin S and isospin T , and hence one is interested in algebraic reductions of SO(8) that contain SUS (2) ⊗ SUT (2). Three such limits exist, And they are specified by the lattice of algebras: ⎧ ⎫ ⎨ SOT (5) ⊗ SUS (2) ⎬ SO(8) ⊃ ⊃ SUS (2) ⊗ SUT (2), SU(4) ⎩ ⎭ SOS (5) ⊗ SUT (2)

(59)

54 Algebraic Models of Nuclei

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01 }, SO (5) ≡ {n, 10 } and SU(4) ≡ where SOT (5) ≡ {n, ˆ Tˆμ , Sˆ±,μ ˆ Sˆμ , Sˆ±,μ S {Sˆμ , Tˆμ , Yˆμν }. The relevance of this lattice with respect to the pairing interaction (57) is found by noting the following relations: 01 ˆ 01 2Sˆ+ · S− = Cˆ 2 [SOT (5)] − Cˆ 2 [SUT (2)] − 14 (2Ω − n)(2Ω ˆ − nˆ + 6), 01 ˆ 01 10 ˆ 10 2S+ · S− + 2S+ · S− = Cˆ 2 [SO(8)] − Cˆ 2 [SO(6)] − 14 (2Ω − n)(2Ω ˆ − nˆ + 12), 10 ˆ 10 2Sˆ+ · S− = Cˆ 2 [SOS (5)] − Cˆ 2 [SUS (2)] − 14 (2Ω − n)(2Ω ˆ − nˆ + 6).

(60) In three cases the pairing Hamiltonian (57) can therefore be written as a combination of Casimir operators belonging to a chain of nested algebras: g10 = 0, g01 = g10 and g01 = 0. They are the dynamical symmetries of the SO(8) model. The upper and lower rows in the lattice (59) are obtained in the case of pure isovector and isoscalar pairing, respectively. Both cases are equivalent mathematically under the exchange of S and T and have been analyzed by Hecht (1965, 1967, 1989). The middle row in the lattice (59) is obtained with equal pairing strengths, in which case the pairing Hamiltonian can be written in terms of the generators of Wigner’s SU(4) algebra. Because of these properties SO(8) is a schematic model that can be used to study the competition between T = 0 and T = 1 pairing. While the consequences of isovector pairing are by now well established, those of its isoscalar sibling are still poorly understood, especially in N ∼ Z nuclei; see the review by Frauendorf and Macchiavelli (2014). The SO(8) model can, however, only have limited applicability since it assumes an LS classification and no spin-orbit coupling. Once a spin-orbit term is added, the subspace constructed out of L = 0 pairs no longer is decoupled, and one is forced to solve the eigenproblem in the full shell-model space.

Symmetry in the Interacting Boson Model The Interacting Boson Model In the interacting boson model (IBM), it is assumed that low-lying collective states of an even-even nucleus can be described in terms of N s and d bosons (Arima and Iachello 1975), where N is half the number of valence nucleons. The dynamical algebra of the IBM is therefore U(Ω) with Ω = 1 + 5 = 6. The separate boson numbers ns and nd are not necessarily conserved, but their sum ns + nd = N is. Eigenenergies and eigenfunctions are obtained by diagonalizing a Hamiltonian in the symmetric representation [N] of U(6), which has the dimension 

Ω +N −1 N

 =

(Ω + N − 1)! , N !(Ω − 1)!

(61)

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with Ω = 6. The IBM Hamiltonian is invariant under rotations, leading to the conservation of the total angular momentum J of the bosons and the existence of an SO(3) symmetry. As the spin carried by the bosons is an integer, the total angular momentum must also be an integer, and for this reason it usually has been denoted as L (Iachello and Arima 1987). Following the notation introduced for the shell model, the symbol J is used to stress that it corresponds to the total angular momentum of the nucleons and not to their total orbital angular momentum. The Hamiltonian of the IBM is of the general form: Hˆ = E0 + Hˆ 1 + Hˆ 2 + Hˆ 3 + · · · ,

(62)

where the subscript refers to the order of the interaction between the bosons. The first term is a constant, which accounts for the binding energy of the core, and the second is the one-body term Hˆ 1 = εs nˆ s + εd nˆ d ,

(63)

where εs and εd are single-boson energies. The two-body interaction can be written as Hˆ 2 =

 1 ≤2 , 1 ≤ 2 ,J

υJ  1

2 1 2

 † (0) (b1 × b†2 )(J ) × (b˜ 2 × b˜ 1 )(J ) 0 ,

(64)

where b˜m ≡ (−)+m b−m . The υ coefficients are related to the interaction matrix elements,  υJ    1 2 1 2

=

2J + 1 1 2 ; J |Hˆ 2 | 1  2 ; J , (1 + δ1 2 )(1 + δ 1  2 )

(65)

which are seven in number. This analysis can be extended to higher orders, and sometimes three-body interactions 1 2 3 ; J |Hˆ 3 | 1  2  3 ; J  are considered. It should be emphasized that a numerical diagonalization of the general Hamiltonian is always possible since for all cases of interest (N  25) the dimension (61) with Ω = 6 remains manageable.

Dynamical Symmetries † The generators of the dynamical algebra U(6) are bm b or   m (J ) (J ) Bˆ MJ (,  ) ≡ (b† × b˜ )MJ =

 m m 

† ˜ (m  m  |J MJ )bm b ,   m

(66)

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which satisfy the commutation relations (λ )

[Bˆ μ(λ) (1 , 2 ), Bˆ μ (3 , 4 )] =

λ μ

1 +4 +λ λ λ λ δ2 3 Bˆ μ(λ ) (1 , 4 ) × (−) 4 1 2 

(λ ) 2 +3 +λ+λ λ λ λ ˆ δ1 4 Bμ (3 , 2 ) . −(−) 3 2 1



  (2λ + 1)(2λ + 1) (λμ λ μ |λ μ )

(67)

The symmetry algebra SO(3) consists of the three components of the angular √ ˜ (1) momentum operator Jˆμ = 10(d † × d) μ . The dynamical symmetries of the IBM are found by enumerating all chains of nested algebras between U(6) and SO(3). They have been established by Arima and Iachello (1976, 1978, 1979) and can be summarized with the following lattice of algebras: ⎧ ⎫ ⎨ U(5) ⊃ SO(5) ⎬ U(6) ⊃ ⊃ SO(3). SU(3) ⎩ ⎭ SO(6) ⊃ SO(5)

(68)

The realization of the different subalgebras of U(6) in terms of boson creation and annihilation operators can be found, for example, in the book by Iachello and Arima (1987). Associated with the algebras appearing in the lattice (68) are two linear and six quadratic Casimir operators, leading to the following Hamiltonian up to second order: Hˆ 1+2 = ε0 Cˆ 1 [U(6)] + κ0 Cˆ 2 [U(6)] + κ0 Cˆ 1 [U(6)]Cˆ 1 [U(5)] + ε1 Cˆ 1 [U(5)] + κ1 Cˆ 2 [U(5)] + κ2 Cˆ 2 [SU(3)] + κ3 Cˆ 2 [SO(6)] + κ4 Cˆ 2 [SO(5)] + κ5 Cˆ 2 [SO(3)].

(69)

Since the nine coefficients match the number of one- and two-body interactions quoted in the previous section, one concludes that the most general IBM Hamiltonian with up to two-body interactions can also be written in terms of Casimir operators. In the following the Casimir operators associated with U(6) are omitted. This can be done if one is only interested in the spectrum of a single nucleus, in which case Cˆ m [U(6)] reduces to a constant. The Hamiltonian (69) has a dynamical symmetry if some of the coefficients εi and κi vanish such that Hˆ 1+2 contains Casimir operators of subalgebras belonging to a single nested chain of subalgebras in Eq. (68). Three cases exist, the vibrational U(5), the rotational SU(3), and the γ -unstable SO(6) limit, with the respective Hamiltonians: Hˆ U(5) = ε1 Cˆ 1 [U(5)] + κ1 Cˆ 2 [U(5)] + κ4 Cˆ 2 [SO(5)] + κ5 Cˆ 2 [SO(3)],

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Hˆ SU(3) = κ2 Cˆ 2 [SU(3)] + κ5 Cˆ 2 [SO(3)], Hˆ SO(6) = κ3 Cˆ 2 [SO(6)] + κ4 Cˆ 2 [SO(5)] + κ5 Cˆ 2 [SO(3)].

(70)

Since in each case the Hamiltonian is written as a sum of commuting operators, the quantum numbers associated with the different algebras are conserved. They can be summarized as follows: U(6) ⊃ U(5) ⊃ SO(5) ⊃ SO(3) ↓ ↓ ↓ ↓ , υ νΔ J [N ] nd U(6) ⊃ SU(3) ⊃ SO(3) ↓ ↓ ↓ , [N ] (λ, μ) KJ U(6) ⊃ SO(6) ⊃ SO(5) ⊃ SO(3) ↓ ↓ ↓ ↓ , [N] σ υ νΔ J

(71)

with the corresponding analytic eigenvalues EU(5) (nd , υ, J ) = ε1 nd + κ1 nd (nd + 4) + κ4 υ(υ + 3) + κ5 J (J + 1), ESU(3) (λ, μ, J ) = κ2 [λ(λ + 3) + μ(μ + 3) + λμ] + κ5 J (J + 1), ESO(6) (σ, υ, J ) = κ3 σ (σ + 4) + κ4 υ(υ + 3) + κ5 J (J + 1).

(72)

If combinations of certain coefficients εi and κi vanish, the Hamiltonian (69) has a dynamical symmetry. The converse is not necessarily true: in some cases the Hamiltonian Hˆ 1+2 has a dynamical symmetry even if all parameters are nonzero. This is a consequence of the existence of transformations in the parameter space {εi , κi } that preserve the eigenspectrum of the Hamiltonian Hˆ 1+2 (Shirokov et al. 1998). Finally, the U(5) and SO(6) limits are connected by an integrable path in terms of the product algebra SUs (1, 1) ⊗ SUd (1, 1) (Pan and Draayer 1998).

The Classical Limit A geometric interpretation of the IBM with s and d bosons can be given by means of the coherent state (Ginocchio and Kirson 1980; Dieperink et al. 1980; Bohr and Mottelson 1980): N   † † |N ; α2μ  ∝ s + α2μ dμ |o, μ

(73)

54 Algebraic Models of Nuclei

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where |o is the boson vacuum. The α2μ are five complex variables, which can be identified with the quadrupole variables and their conjugate momenta. For static properties the α2μ can be taken real. Furthermore, they can be converted to two intrinsic shape variables a0 and a−2 = a+2 ≡ a2 (with a−1 = a+1 = 0) and three Euler angles {θ1 , θ2 , θ3 } that determine the orientation of the body fixed to the laboratory frame (Bohr and Mottelson 1975). Following the standard notation a0 = β cos γ and a2 = √1 β sin γ , the intrinsic coherent state can be rewritten as 2

  |N ; βγ  ∝ s † + β cos γ d0† +

√1 2

  † † N sin γ d−2 + d+2 |o.

(74)

For any IBM Hamiltonian Hˆ IBM , which depends on a set of parameters {εi , κi , . . . }, an energy surface can be defined as E(N, εi , κi , . . . ; β, γ ) =

N ; βγ |Hˆ IBM |N; βγ  . N; βγ |N ; βγ 

(75)

This energy surface depends on the boson number N and on the parameters in the Hamiltonian, which can be considered as control parameters, and on the shape variables β and γ , which play the role of order parameters. One can now perform a catastrophe analysis (Gilmore 1981) of the energy surface, that is, determine the nature of the quadrupole shapes that can be obtained as a function of {N, εi , κi , . . . }. This analysis has been carried out for the general IBM Hamiltonian with up to twobody interactions (López-Moreno and Castaños 1996) and is illustrated here for the simplified IBM Hamiltonian (Jolie et al. 2001): ˆ · Q. ˆ Hˆ ECQF = ε nˆ d + κ Q

(76)

This is the Hamiltonian of the extended consistent-Q formalism (ECQF) (Warner and Casten 1983; Lipas et al. 1985), which captures some basic features of nuclear ˆ ·Q ˆ structural evolution with a vibrational term nˆ d and a quadrupole interaction Q with † ˜ (2) ˆ μ = (s † × d˜ + d † × s)(2) Q μ + χ (d × d)μ .

(77)

The spectrum of the ECQF Hamiltonian is, up to a scale factor, determined by the ratio ε/κ and χ . Furthermore,√the three limits of the IBM are obtained with κ = 0 for U(5), (ε, χ ) = (0, ± 12 7) for SU± (3), and (ε, χ ) = (0, 0) for SO(6). The parameter space of the ECQF Hamiltonian can therefore be represented on a triangle. Each point of this so-called “Casten” triangle corresponds to a specific couple (ε/κ, χ ) And its vertices to the values appropriate for the√three limits U(5), SU(3), or SO(6). With two possible choices for SU(3), χ = ± 12 7, the triangle is extended to cover negative as well as positive values of χ (Jolie et al. 2001). The energy surface (75) associated with the ECQF Hamiltonian is

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Fig. 4 Phase diagram of the ECQF Hamiltonian. The black dots correspond to the dynamical symmetries of the ECQF Hamiltonian. The triangle is divided into three regions where the energy surface has an absolute minimum that is (a) spherical, (b) prolate deformed, or (c) oblate deformed. These regions are separated by Maxwell sets (dashed lines), which meet in a triple point (grey dot). The shaded area corresponds to a region where the energy surface has two minima, a spherical and a deformed one (Iachello et al. 1998)

Nεβ 2 N(5 + (1 + χ 2 )β 2 ) EECQF (N, ε, κ, χ ; β, γ ) = + κ (78) 1 + β2 1 + β2    N(N − 1) 2 2 4 2 3 2 , χ β −4 χβ cos 3γ + 4β + 7 (1 + β 2 )2 7 and its catastrophe analysis is summarized in the phase diagram of Fig. 4.

Bosons with F Spin Since the bosons of the IBM are interpreted as pairs of nucleons, it is natural to introduce bosons for neutrons and for protons (Arima et al. 1977), with the † † associated creation operators bm and bm , respectively. This is appropriate  ,ν  ,π if the neutrons and protons occupy different valence shells, and there is no need for neutron-proton pairs, discussed in the subsequent sections. Since the number of neutron bosons Nν and proton bosons Nπ is fixed for a given nucleus, the dynamical algebra of the neutron-proton IBM, known as IBM-2, is Uν (6) ⊗ Uπ (6). A possible classification is therefore Uν (6) ⊗ Uπ (6) ⊃ Uν+π (6) , ↓ ↓ ↓ [Nπ ] [N − f, f ] [Nν ]

(79)

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where N = Nν + Nπ . The generators of Uν+π (6) are sums of corresponding generators of Uν (6) and Uπ (6): † † (b,ν × b˜ ,ν )MJ + (b,π × b˜ ,π )MJ . (J )

(J )

(80)

Dynamical symmetries of the IBM-2 are determined by the study of the subalgebras of the dynamical algebra Uν (6) ⊗ Uπ (6). An important class of IBM-2 dynamical symmetries starts from Eq. (79), leading to the three limits known from the standard IBM (Van Isacker et al. 1986). What is the meaning of the label f in the classification (79)? If the bosons are all neutron or all proton bosons, the representation of Uν+π (6) must be symmetric, either [Nν ] or [Nπ ]. For a system of neutron and proton bosons, this is no longer necessary, and states can be of mixed symmetry, as can be expressed with a Young diagram (Iachello 2006). Since there are two species (ν and π ), the Young diagram can have at most two rows, with lengths N − f and f , respectively. It is customary to introduce the F -spin quantum number (Otsuka et al. 1978), which is half the difference in length between the two rows, F = 12 N − f = 12 N, 12 N − 1, . . . . There exists an alternative way to introduce F spin. Just as a nucleon carries an isospin 1/2 and its projection differs for a neutron or a proton, one assigns an F spin quantum number 1/2 to an IBM-2 boson such that its projection distinguishes between a neutron and a proton boson. The operators (J F ) (J F ) Bˆ MJ MF (,  ) ≡ (b†1/2 × b˜ 1/2 )MJ MF ,

(81)

with b˜m 1/2mf ≡ (−)+m +1/2+mf b−m 1/2−mf , form the algebra U(12), which contains Uν (6) ⊗ Uπ (6) as a subalgebra. The generators (81) can be divided into those that are scalar in angular momentum, (F ) Bˆ M ≡ F

√

) 2 + 1(b†1/2 × b˜1/2 )(0F 0MF ,

(82)



and those that are scalar in F spin, (J ) (J 0) Bˆ MJ (,  ) ≡ (b†1/2 × b˜ 1/2 )MJ 0 .

(83)

The explicit expressions for the former are Bˆ 0(0) =

√1 2

 † sν s˜ν + sπ† s˜π + dν† · d˜ν + dπ† · d˜π =

ˆ √1 N, 2

(1) Bˆ −1 = sν† s˜π + dν† · d˜π ≡ Fˆ− ,  (1) Bˆ 0 = √1 −sν† s˜ν + sπ† s˜π − dν† · d˜ν + dπ† · d˜π = 2

(1) Bˆ +1

=

sπ† s˜ν

+ dπ† · d˜ν ≡ Fˆ+ .

√1 2



√ −Nˆ ν + Nˆ π ≡ 2Fˆz , (84)

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They consist of the boson number operator Nˆ and the three operators {Fˆ± , Fˆz }, of which it can be shown that they satisfy the commutation relations [Fˆz , Fˆ± ] = ±Fˆ± ,

[Fˆ+ , Fˆ− ] = 2Fˆz ,

(85)

forming the F -spin algebra SUF (2). The four operators Bˆ MF close under commuta(J ) tion and so do the 36 operators Bˆ MJ (,  ). Since in addition both sets of operators commute, the following chain of algebras is valid: (F )

U(12) ⊃ UJ (6) ⊗ UF (2) . ↓ ↓ ↓ [N ] [N − f, f ] [N − f, f ]

(86)

(J ) (,  ) of UJ (6) are, up to a constant, identical to the summed The generators Bˆ M J operators (80) and therefore UJ (6) and Uν+π (6) are one and the same algebra. Because the representation [N] of U(12) is symmetric, it follows that the representations of UJ (6) and UF (2) must be the same. Since UF (2) can have at most two rows, so can UJ (6). If f = 0 or F = 12 N , states belong to the symmetric representation [N, 0] ≡ [N ] of UJ (6) and are the analogues of the states in IBM-1. A new class of states (Iachello 1984) is obtained for f = 1 or F = 12 N − 1, corresponding to the mixed-symmetric representation [N − 1, 1]. They occur in the three F -spin symmetric limits of the IBM-2 with typical patterns as illustrated in Fig. 5. Of particular relevance are 1+ states because they do not occur in IBM-1 but are allowed in IBM-2; they are thus

Fig. 5 Schematic illustration of part of the energy spectra in the three dynamical symmetries of the IBM-2 that conserve F -spin symmetry. Levels are labeled by the angular momentum and parity J π and by the UJ (6) labels [N − f, f ]

54 Algebraic Models of Nuclei

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necessarily of mixed-symmetric character. The M1 excitation of 1+ states in IBM-2 satisfies the sum rule (Ginocchio 1991): 

+ B(M1; 0+ 1 → 1i ) =

i

6N 3 (gν − gπ )2 Nν Nπ 0+ ˆ d |0+ 1 |n 1 , 4π N −1

(87)

where gν and gπ are the boson g factors and the matrix element is the expectation value of the d-boson number in the ground state. While it vanishes in the U(5) limit, in the SU(3) and SO(6) limits of the IBM-2, all M1 strength is concentrated in the first-excited 1+ state, and a simple expression is available for the corresponding strength (Van Isacker et al. 1986). Since the first experimental identification of an M1-excited 1+ state in 156 Gd (Bohle et al. 1984), extensive studies have shown the occurrence of mixed-symmetry states in many nuclei (Heyde et al. 2010). The raising and lowering operators Fˆ± change a neutron into a proton boson (or vice versa) and connect states in different nuclei but with Nν + Nπ constant. Therefore, a single representation [N ] of U(12) contains several nuclei with Nν + Nπ = N. This is the definition of an F -spin multiplet (von Brentano et al. 1985). If the IBM-2 Hamiltonian has SUF (2) symmetry, that is, if it satisfies [Hˆ , Fˆ± ] = [Hˆ , Fˆz ] = 0, then the spectra before and after the application of Fˆ± are identical, and the binding energies and energy spectra of all nuclei belonging to an F -spin multiplet [N] are the same. This condition can be relaxed to [Hˆ , Fˆ 2 ] = 0, which still implies identical spectra but ground-state binding energies that satisfy Egs (MF ) = κ0 + κ1 MF + κ2 MF2 ,

(88)

with MF = 12 (−Nν + Nπ ) the eigenvalue of Fˆz . In analogy with the isobaric multiplet mass equation (IMME), this can be called an F -spin multiplet mass equation (FMME) (Frank et al. 2019).

Bosons with Isospin If neutrons and protons occupy the same valence shell, the inclusion of only neutron and proton bosons is not well founded. They correspond to a nucleon pair consisting of two neutrons or of two protons, but there is no reason why not to consider a neutron-proton pair as well. In fact, isospin invariance requires that J = 0 and J = 2 states occur at the same energy for neutron-neutron, neutron-proton, and proton-proton configurations. In the isospin version of the IBM, called IBM-3, one introduces, besides the ν and π bosons, also a neutron-proton δ boson (Elliott and White 1980). The bosons are members of two T = 1 triplets, one for J = 0 and another for J = 2. Since entire isospin multiplets are considered, states with good isospin can be constructed in the IBM-3. If neutrons and protons occupy the same valence shell, this is not possible in IBM-2 (Elliott and Evans 1987).

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To determine the dynamical algebra of IBM-3, one notes that the total number of bosons Nν + Nδ + Nπ = N is constant (half the number of valence nucleons) but that, unlike in IBM-2, the separate boson numbers are not conserved. For example, a system of two neutrons and two protons corresponds to Nν = Nπ = 1 and Nδ = 0 ) † but also to Nν = Nπ = 0 and Nδ = 2. Therefore the operators (b,ρ, × b˜ ,ρ )(J MJ should be included with ,  = 0, 2 and ρ, ρ = ν, δ, π, of which there are 324, generating the algebra U(18). The index ρ can be considered as an intrinsic quantum † number, giving rise to boson creation operators bm that are members of an  1mt isospin triplet with mt = −1, 0, and +1 for the ν, δ, and π bosons, respectively. The generators of U(18) can then be written alternatively as (J T ) (J T ) † Bˆ MJ MT (,  ) ≡ (b1 × b˜ 1 )MJ MT ,

(89)

where b˜m 1mt ≡ (−)+m +1+mt b−m 1−mt . Dynamical symmetries of the IBM-3 are found by constructing chains of nested algebras between the dynamical algebra U(18) and the symmetry algebra SOJ (3) ⊗ SOT (3), associated with angular momentum and isospin. A possible classification starts with U(18) ⊃ UJ (6) ⊗ UT (3) , ↓ ↓ ↓ [N] [N1 , N2 , N3 ] [N1 , N2 , N3 ]

(90)

where UT (3) is generated by (T ) Bˆ MT ≡

√

(0T ) † 2 + 1(b1 × b˜1 )0MT ,

(91)



while UJ (6) consists of the isoscalar operators (J ) (J 0) † Bˆ MJ (,  ) ≡ (b1 × b˜ 1 )MJ 0 .

(92)

Many-boson states belong to the symmetric representation [N ] of U(18), and as a result the representations of UJ (6) and UT (3) must be identical. Since the bosons come in three different species, ν, δ, and π , the Young diagrams can have at most three rows or three labels Ni . The orbital classification now proceeds as in the standard IBM, leading to three different limits; see Eq. (68). The three components (1) of the isospin operator, Tˆν ∝ Bˆ ν , belong to UT (3), and the relevant reduction for the isospin part is therefore UT (3) ⊃ SUT (3) ⊃ SOT (3) ↓ ↓ ↓ . (λ, μ) T [N1 , N2 , N3 ]

(93)

54 Algebraic Models of Nuclei

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Dynamical symmetries of the IBM-3 with SUT (3) charge symmetry have been studied in some detail (García-Ramos and Van Isacker 1999). Dynamical symmetries that break SUT (3) charge symmetry have also been proposed (Ginocchio 1996; Kota 1998) and are based on the reduction U(18) ⊃ SO(18) ⊃ · · · ⊃ SOJ (3) ⊗ SOT (3).

Bosons with Intrinsic Spin and Isospin Pairs of nucleons can be written in terms of the lst representation of nucleon operators and mapped onto bosons with corresponding quantum numbers, ) † (al†1/21/2 × al†1/21/2 )(LST ML MS MT → bLML SMS T MT ,

(94)

where L + S + T must be odd because of anti-symmetry. The bosons are assumed to obey exact commutation relations, whereas they are only approximately satisfied by the fermion pairs. The quantum numbers of these ideal bosons are henceforth denoted with small letters and their shell model origin implies that , s, and t are nonnegative integers (with s, t ≤ 1) and that  + s + t must be odd. This defines therefore an st representation of bosons, that is, bosons with an intrinsic spin s in addition to the isospin quantum number t of IBM-3. This extension of the IBM is often referred to as IBM-4 (Elliott and Evans 1981). The dynamical algebra of the IBM-4 consists of the operators (LST ) (LST ) † Bˆ ML MS MT (st,  s t ) ≡ (bst × b˜ s t )ML MS MT ,

(95)

 which generate U(Ω) with Ω = st (2 + 1)(2s + 1)(2t + 1). Furthermore, one introduces operators that are scalar in L, (ST ) Bˆ MS MT (st, s t ) ≡

√ (0ST ) † × b˜s t )0MS MT , 2 + 1(bst

(96)



and operators that are scalar in spin-isospin, (L) (,  ) ≡ Bˆ M L



† (2s + 1)(2t + 1)(bst × b˜ st )(L00) ML 00 .

(97)

st

In this section the labels st are left unspecified, besides the constraint that  + s + t must be odd, leading to a generic formulation of the boson model (Van Isacker 2018). Of particular relevance is the question whether an SU(4) algebra can be constructed in terms of boson operators (95) with the same interpretation as in the supermultiplet model. Two cases must be distinguished:  is even or  is odd. Bosons with even . If  is even, it follows that (st) = (01) or (10) and that the number of spin-isospin states is six. Dynamical symmetries with labels that also appear in the shell model are obtained from the following chain of nested algebras:

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 U(6Ωorb ) ⊃ UL (Ωorb ) ⊃ · · · ⊃ SOL (3)  ⊗ UST (6) ⊃ SUST (4) ⊃ SOS (3) ⊗ SOT (3) , where the orbital degeneracy is denoted as Ωorb = SUST (4) are Sˆμe =

√

2Bˆ μ0 (10, 10), (10)

Tˆνe =



√

 (2

(98)

+ 1). The generators of

2Bˆ 0ν (01, 01), (01)

e (11) (11) = Bˆ μν (10, 01) ± Bˆ μν (01, 10). Yˆμν

(99)

These operators satisfy the same SU(4) commutation relations as the fermion operators (37). Bosons with odd . If  is odd, it follows that (st) = (00) or (11) and that the number of spin-isospin states is ten. The relevant dynamical symmetries are obtained from  U(10Ωorb ) ⊃ UL (Ωorb ) ⊃ · · · ⊃ SOL (3)  ⊗ UST (10) ⊃ SUST (4) ⊃ SOS (3) ⊗ SOT (3) .

(100)

In this case SUST (4) is generated by Sˆμo =

√

6Bˆ μ0 (11, 11), (10)

Tˆνo =

√ (01) 6Bˆ 0ν (11, 11),

o (11) (11) (11) = Bˆ μν (00, 11) + Bˆ μν (11, 00) ± 2Bˆ μν (11, 11). Yˆμν

(101)

Again, these operators satisfy the same SU(4) commutation relations as the fermion operators (37). As a result a generic boson model can be constructed with bosons (01) and (10) for even  as well as bosons (00) and (11) for odd . The dynamical algebra is U(6Ω bosons, Ωe =  e + 10Ωo ), where Ωe is the orbital degeneracy of the even-   even (2 + 1), and Ωo is that of the odd- bosons, Ωo =  odd (2 + 1). This dynamical algebra allows the following classification: U(6Ωe + 10Ωo ) ⊃ U(6Ωe ) ⊗ U(10Ωo )  ⊃ UL (Ωe ) ⊗ UL (Ωo ) ⊃ · · · ⊃ SOL (3)  ⊗ UeST (6) ⊗ UoST (10) ⊃ SUeST (4) ⊗ SUoST (4) ⊃ SUST (4) .

(102)

The orbital classification, indicated by the dots, depends on the dimensions Ωe and Ωo , and ends in the angular momentum algebra SOL (3). Examples of particular models of the type (102) exist. The most elementary version adopts bosons with  = 0, implying Ωe = 1 and Ωo = 0 and resulting in the dynamical algebra U(6) (Van Isacker et al. 1998). The IBM-4 of Elliott and Evans (1981) considers bosons with  = 0 and  = 2, implying Ωe = 6 and

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Ωo = 0 and resulting in the dynamical algebra U(36). Classifications with odd- bosons have rarely been considered (Van Isacker 2018). The classification (102) defines the most general boson model, in which the full panoply of shell model labels L, S, T , and (λμν) can be represented.

Concluding Remarks The central result of this chapter is Eq. (12), which shows a chain of nested Lie algebras stretching from the dynamical to the symmetry algebra of a model Hamiltonian. This single equation is at the basis of all results presented in this chapter, be they for the shell model or for the interacting boson model. Many-body Hamiltonians associated with a chain of nested algebras have eigenstates with a fixed structure, independent of the parameters in the Hamiltonian, and characterized by a set of labels proper to the algebras in the chain. This procedure provides an analytic solution of particular classes of model Hamiltonians based on symmetry principles. It has been found that in many cases these classes of analytically solvable Hamiltonians can be used as benchmarks in the comparison with data. The main theme of this chapter is that symmetries are crucial for improving our understanding of the structure of models. This, one may hope, is by now a widely accepted opinion. What is perhaps less well acknowledged is the fact that symmetries also play an important role for establishing connections between models. This is illustrated by the development of the various versions of the IBM and their relation to the shell model. While the s and d bosons have been introduced originally mainly on empirical grounds, they acquired a more microscopic status with each refinement of the model, because in later versions of the IBM the role of shell model labels such as isospin, spin, and orbital angular momentum could be clarified. The author closes this chapter with the conjecture that symmetry arguments will remain of pivotal importance for our understanding of nuclei.

References Y. Alhassid, A. Leviatan, J. Phys. A: Math. Gen. 25, L1265 (1992) J.M. Allmond, A.E. Stuchbery, A. Galindo-Uribarri, E. Padilla-Rodal, D.C. Radford, J.C. Batchelder, C.R. Bingham, M.E. Howard, J.F. Liang, B. Manning, S.D. Pain, N.J. Stone, R.L. Varner, C.-H. Yu, Phys. Rev. C 92, 041303 (2015) A. Arima, F. Iachello, Phys. Rev. Lett. 35, 1069 (1975) A. Arima, F. Iachello, Ann. Phys. (NY) 99, 253 (1976) A. Arima, F. Iachello, Ann. Phys. (NY) 111, 201 (1978) A. Arima, F. Iachello, Ann. Phys. (NY) 123, 468 (1979) A. Arima, M. Harvey, K. Shimizu, Phys. Lett. B 30, 517 (1969) A. Arima ,T. Ohtsuka, F. Iachello, I. Talmi, Phys. Lett. B 66, 205 (1977) D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N. Lo Iudice, F. Palumbo, O. Scholten, Phys. Lett. B 137, 27 (1984) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26(14), 1 (1952)

2078

P. Van Isacker

A. Bohr, B.R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27(160), 1 (1953) A. Bohr, B.R. Mottelson, Nuclear Structure. I Single-Particle Motion (Benjamin, New York, 1969) A. Bohr, B.R. Mottelson, Nuclear Structure. II Nuclear Deformations (Benjamin, New York, 1975) A. Bohr, B.R. Mottelson, Phys. Scripta 22, 468 (1980) D. Bonatsos, I.E. Assimakis, N. Minkov, A. Martinou, R.B. Cakirli, R.F. Casten, K. Blaum, Phys. Rev. C 95, 064325 (2017) P.J. Brussaard, P.W.M. Glaudemans, Shell-Model Applications in Nuclear Spectroscopy (NorthHolland, Amsterdam, 1977) A. de-Shalit, I. Talmi, Nuclear Shell Theory (Academic, New York, 1963) A.E.L. Dieperink, O. Scholten, F. Iachello, Phys. Rev. Lett. 44, 1747 (1980) J. Dukelsky, S. Pittel, G. Sierra, Rev. Mod. Phys. 76, 643 (2004) J.P. Elliott, Proc. R. Soc. (London) A 245, 128; 562 (1958) J.P. Elliott, P.G. Dawber, Symmetry in Physics, vols. 1 & 2 (Oxford University Press, Oxford, 1979) J.P. Elliott, J.A. Evans, Phys. Lett. B 101, 216 (1981) J.P. Elliott, J.A. Evans, Phys. Lett. B 195, 1 (1987) J.P. Elliott, A.P. White, Phys. Lett. B 97, 169 (1980) A. Escuderos, L. Zamick, Phys. Rev. C 73, 044302 (2006) J.A. Evans, G.G. Dussel, E.E. Maqueda, R.P.J. Perazzo, Nucl. Phys. A 367, 77 (1981) B.H. Flowers, Proc. R. Soc. (London) A 212, 248 (1952) B.H. Flowers, S. Szpikowski, Proc. Phys. Soc. 84, 673 (1964) A. Frank, P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics (WileyInterscience, New York, 1994) A. Frank, J. Jolie, P. Van Isacker, Symmetries in Atomic Nuclei. From Isospin to Supersymmetry (Springer, Berlin, 2019) S. Frauendorf, A.O. Macchiavelli, Prog. Part. Nucl. Phys. 78, 24 (2014) J.E. García-Ramos, P. Van Isacker, Ann. Phys. (NY) 274, 45 (1999) R. Gilmore, Catastrophe Theory for Scientists and Engineers (Wiley, New York, 1981) J.N. Ginocchio, Phys. Lett. B 265, 6 (1991) J.N. Ginocchio, Phys. Rev. Lett. 77, 28 (1996) J.N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997) J.N. Ginocchio, M.W. Kirson, Phys. Rev. Lett. 44, 1744 (1980) M. Hamermesh, Group Theory and Its Application to Physical Problems (Addison-Wesley, Reading, 1962) K.T. Hecht, Phys. Rev. 139, B794 (1965) K.T. Hecht, Nucl. Phys. A 102, 11 (1967) K.T. Hecht, Nucl. Phys. A 493, 29 (1989) K.T. Hecht, A. Adler, Nucl. Phys. A 137, 129 (1969) W. Heisenberg, Z. Phys. 77, 1 (1932) K. Helmers, Nucl. Phys. 23, 594 (1961) K.L.G. Heyde, The Nuclear Shell Model (Springer, Berlin, 1990) K. Heyde, P. von Neumann-Cosel, A. Richter, Rev. Mod. Phys. 82, 2365 (2010) F. Iachello, Phys. Rev. Lett. 53, 1427 (1984) F. Iachello, Lie Algebras and Applications (Springer, Berlin, 2006) F. Iachello, A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987) F. Iachello, N.V. Zamfir, R.F. Casten, Phys. Rev. Lett. 81, 1191 (1998) J.H.D. Jensen, H. Suess, O. Haxel, Die Naturwissenschaften 36, 155 (1949) J. Jolie, R.F. Casten, P. von Brentano, V. Werner, Phys. Rev. Lett. 87, 162501 (2001) A.K. Kerman, Ann. Phys. (NY) 12, 300 (1961) V.K.B. Kota, Ann. Phys. (NY) 265, 101 (1998) R.D. Lawson, Theory of the Nuclear Shell Model (Clarendon Press, Oxford, 1980) A. Leviatan, Phys. Rev. Lett. 77, 818 (1996) A. Leviatan, Prog. Part. Nucl. Phys. 66, 93 (2011) A. Leviatan, P. Van Isacker, Phys. Rev. Lett. 89, 222501 (2002)

54 Algebraic Models of Nuclei

2079

A. Leviatan, A. Novoselsky, I. Talmi, Phys. Lett. B 172, 144 (1986) P.O. Lipas, P. Toivonen, D.D. Warner, Phys. Lett. B 155, 295 (1985) D.E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups (Clarendon, Oxford, 1940) E. López-Moreno, O. Castaños, Phys. Rev. C 54, 2374 (1996) B. Maheshwari, A.K. Jain, Phys. Lett. B 753, 122 (2016) M.G. Mayer, Phys. Rev. 75, 1969 (1949) M. Moshinsky, Group Theory and the Many-Body Problem (Gordon & Breach, New York, 1968) I.O. Morales, P. Van Isacker, I. Talmi, Phys. Lett. B 703, 606 (2011) S.G. Nilsson, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 29, 16 (1955) T. Otsuka, A. Arima, F. Iachello, I. Talmi, Phys. Lett. B 76, 139 (1978) F. Pan, J.P. Draayer, Nucl. Phys. A 636, 156 (1998) S.C. Pang, Nucl. Phys. A 128, 497 (1969) W. Pauli, Z. Phys. 36, 336 (1926) C. Qi, Phys. Rev. C 83, 014307 (2011) G. Racah, Phys. Rev. 63, 367 (1943) G. Racah, L. Farkas Memorial Volume (Research council of Israel, Jerusalem, 1952), p. 294 G. Racah, I. Talmi, Physica 18,1097 (1952) J. Rainwater, Phys. Rev. 79, 432 (1950) R.W. Richardson, Phys. Lett. 3, 277 (1963) D.J. Rowe, G. Rosensteel, Phys. Rev. Lett. 87, 172501 (2001) D.J. Rowe, J.L. Wood, Fundamentals of Nuclear Models. Foundational Models (World Scientific, Singapore, 2010) C. Schwartz, A. de Shalit, Phys. Rev. 94, 1257 (1954) A.M. Shirokov, N.A. Smirnova, Y.F. Smirnov, Phys. Lett. B 434, 237 (1998) I. Talmi, Nucl. Phys. A 172, 1 (1971) I. Talmi, Simple Models of Complex Nuclei. The Shell Model and Interacting Boson Model (Harwood Academic, Chur, 1993) P. Van Isacker, Jap. Phys. Soc. Conf. Proc. 23, 012027 (2018) P. Van Isacker, S. Heinze, Phys. Rev. Lett. 100, 052501 (2008) P. Van Isacker, S. Pittel, Phys. Scr. 91, 023009 (2016) P. Van Isacker, K. Heyde, J. Jolie, A. Sevrin, Ann. Phys. (NY) 171, 253 (1986) P. Van Isacker, J. Dukelsky, S. Pittel, O. Juillet, J. Phys. G: Nucl. Part. Phys. 24, 1261 (1998) P. von Brentano, A. Gelberg, H. Harter, P. Sala, J. Phys. G: Nucl. Part. Phys. 11, L85 (1985) D.D. Warner, R.F. Casten, Phys. Rev. C 28, 1798 (1983) E.P. Wigner, Phys. Rev. 51, 106 (1937) B.G. Wybourne, Symmetry Principles and Atomic Spectroscopy (Wiley-Interscience, New York, 1970) A.P. Zuker, J. Retamosa, A. Poves, E. Caurier, Phys. Rev. C 52, R1741 (1995)

Nuclear Density Functional Theory (DFT)

55

Gianluca Colò

Contents Introduction (with a Few Historical Remarks) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hartree-Fock with Density-Dependent Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From HF to DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Motivation and DFT for Coulomb Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nuclear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Local Densities and Generalized EDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry Breaking and Pairing Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Functional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2082 2084 2086 2088 2092 2096 2098 2104 2106 2108

Abstract

The present chapter is devoted to explaining the basics of nuclear Density Functional Theory (DFT). At the start, a reminder is provided of some of the many empirical pieces of evidence that point to the fact that a description of the atomic nucleus in terms of independent particles is a reasonable approximation. Accordingly, Hartree-Fock (HF) has been one of the most widely used methods throughout the second half of the last century. However, HF has been successful (mainly) in connection with density-dependent Hamiltonians; then, at a given point, it has been concluded that this is a mere realization of DFT in nuclear physics, as explained in the text. The chapter also focuses on recent developments that include avoiding underlying Hamiltonians and building functionals with diverse densities, implementing new symmetry-breaking formulations, or

G. Colò () Dipartimento di Fisica, Università degli Studi di Milano, Milano, Italy INFN, Sezione di Milano, Milano, Italy e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_14

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improving the pairing sector. DFT for Coulomb systems will be used as a paradigm: similarities and differences will be pointed out. The chapter deals only with nonrelativistic DFT and single-reference implementations, as other topics will be covered in other chapters.

Introduction (with a Few Historical Remarks) Since the early days of nuclear structure physics, much evidence has been collected pointing to the fact that nucleons inside the nucleus move almost independently, to a first approximation: “the structure of nuclear matter is much closer to that of the free gas than to the crystalline state” is a statement that one can find in Bohr and Mottelson (1969). The validity of a mean-field approximation, in which each nucleon moves in an average potential that is generated by all the others, seems very surprising at first glance, knowing that the nucleon-nucleon (NN) interaction is “strong”. However, one should consider that at the largest nuclear density which is reached in nuclei, ρ0 ≈ 0.16 fm−3 , the distance between nucleons is of the order of 2r0 (where r0 is defined by ρ10 = 43 π r03 ) and is somewhat larger than the range of the nuclear force. This fact, together with the effect of the Pauli exclusion principle, explains why the mean free path of nucleons in the nucleus is larger than the nuclear radius (cf. e.g. Fig. 7 of Lopez et al. 2014). According to this picture, the so-called mean-field models have been intensively developed with various degrees of success over several decades. After the discovery of the nuclear shell structure in the 1940s (that has led, later, to the Nobel prize awarded to Maria Goeppert-Mayer and J. Hans D. Jensen), it has been customary to describe nuclear structure by assuming that nucleons occupy orbitals in some empirical potential, like the Woods-Saxon (WS) potential. Microscopically, an average potential can be generated starting from a Hamiltonian H that includes the kinetic energy of the nucleons plus an interparticle interaction, if one treats this Hamiltonian within the Hartree-Fock approximation. Nevertheless, it took quite some effort to elaborate on the phenomenological evidence, and on the microscopic HF theory (Negele 1982), and eventually develop successful models like Skyrme-HF (Vautherin and Brink 1972) or GognyHF (Dechargé and Gogny 1980) that are still in use today. The reader must pay attention to the fact that in these models one introduces effective interactions that are supposed to be employed only within the HF framework and that contain a number of free parameters (typically of the order of ≈10). These parameters are fitted so that many nuclear properties like masses or charge radii are well reproduced at the mean-field level. HF has been extended to Hartree-Fock-Bogoliubov (HFB) to treat open-shell nuclei. The status of mean-field HF and HFB models with effective interactions until the turn of the last century is very nicely reviewed in Bender et al. (2003). Another key point that one should keep in mind from the very beginning is that effective interactions that have been widely employed within HF(B) are all densitydependent. Density-dependent interactions are not elementary two-body forces and

55 Nuclear Density Functional Theory (DFT)

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can create some conceptual problems. In what follows, according to the current view, it will be advocated that density-dependent interactions are merely generators of Energy Density Functionals. This paradigm shift from HF(B) to DFT will be discussed below. Energy Density Functionals (EDFs) are the basic objects to describe fermion systems within DFT. Theory ensures that an exact EDF exists (Hohenberg and Kohn 1964) for any Fermi system. The EDFs proposed in the literature are approximations of the exact, unknown EDF for the given Fermi system. However, the development of DFT for electronic systems and nuclear systems has followed somehow different paths. The success of electronic DFT stems from the fact that the Coulomb force is known, and the dominant contribution to the total energy is the simple Hartree term (Parr and Yang 1994). Electronic DFT, whose main features will be mentioned below (avoiding a lengthy treatment but referring to external pedagogical sources), is an extension of mean-field theory (i.e., HF): in HF, one includes the direct and exchange terms of the Coulomb force, while in DFT, one aims to include (all possible) correlations beyond the mean-field level. One could have followed the same path in nuclear physics, had not the problem of the bare NN interaction itself prevented from doing so. The situation is quite different from what has just been mentioned for the Coulomb case. The nuclear interaction has a complicated operator structure and is not uniquely determined. There are attempts to derive it rigorously from Quantum Chromo Dynamics (QCD) by using the ideas by S. Weinberg on low-energy effective QCD Lagrangians based on chiral symmetry. These so-called chiral NN forces (Machleidt and Entem 2011; Epelbaum and Meißner 2012) are experiencing tremendous progress while this chapter is written, but all the attempts have not yet come to a fully satisfactory conclusion (Machleidt and Sammarruca 2020). Three-body forces, or NNN interactions, are known to play an important role and cannot be neglected at all (Hebeler 2021). Because of these reasons, extracting EDFs from ab initio is still in its infancy, and, so far, the EDFs in use are generalizations of EDFs produced by effective interactions or other kinds of empirical EDFs. To end this Introduction, it must be stressed once again that, even if calculations performed with effective interactions of generalizations thereof are still named as HF(B) calculations, the presence of the so-called rearrangement terms makes them analogous to the Kohn-Sham (KS) equations of electronic DFT. The basics of all this will be discussed, as well as the various types of nonrelativistic nuclear EDFs; “generalized” densities, recent developments, and peculiarities associated with the pairing sector will be discussed. There will be little, or no mention, of multi-reference implementations and relativistic (or covariant) EDFs because these are subjects of other chapters. The literature on nuclear DFT is huge. Recently, a dedicated book has been published (Schunck 2019). A short introduction is also available in Colò (2020). In this chapter, ri labels the space coordinate of the i-th particle and σi labels its spin coordinate. xi ≡ (ri , σi ), and the symbol dx means integration over space and sum over spin. Latin letters are also used to label the single-particle states, so that a single-particle wave function will be denoted by φi . Greek indices will

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instead refer to the spinor structure of this wave function: φiα (x) ≡ fi (r)χiα (σ ), with α = 1, 2. The same notation will be used for operators and corresponding eigenvalues unless it creates confusion. In this latter case, a hat will be put over the operator and will distinguish in this way the number of particles N from the number ˆ operator N.

Hartree-Fock with Density-Dependent Forces The starting point is a Hamiltonian H = T + V [ρ],

(1)

where T =



−

i

h¯ 2 2 ∇ 2m i

(2)

is the kinetic energy of the nucleons (the small difference between neutron and proton masses is neglected), and V [ρ] =

1 v(xi , xj ; ρ) 2

(3)

i=j

is a two-body interaction between nucleons. The assumption of the HF theory is that the A-particle wave function is a (totally anti-symmetric) product of single-particle wave functions, that is, a Slater determinant:    φ (x ) . . . φA (x1 )   1  1 1 Ψ (x1 . . . xA ) = √  . . . . . . . . .  . A!  φ (x ) . . . φ (x )  1 A A A

(4)

Consequently, the Hartree-Fock theory amounts to the minimization of the total energy in the Hilbert space of Slater determinants. One usually sets the constraint that the single-particle wave functions form an orthonormal set, so that the minimization turns into a constrained one: ⎛ δ ⎝E −



 εij

⎞ dx φi∗ (x)φj (x)⎠ = 0.

(5)

ij

When one uses a Slater determinant of the type (4), the expectation values of 1-body and 2-body operators can be easily calculated. For the kinetic energy, one obtains

55 Nuclear Density Functional Theory (DFT)

Ψ |T |Ψ =

A  i=1

h¯ 2 − 2m

2085



dx φi∗ (x)∇ 2 φi (x),

(6)

while for the two-body potential, the result is 1 Ψ |V |Ψ = 2



dx dx φi∗ (x)φj∗ (x )v(x, x ; ρ)φi (x)φj (x ) +

ij

−

 1 2

dx dx φi∗ (x)φj∗ (x )v(x, x ; ρ)φi (x )φj (x).

(7)

ij

In the case of HF, the variational quantities are the single-particle orbitals. If the interaction is density-dependent, the rules of functional derivation (see the Appendix) can be used to write δ ∗ δφi (y)

 . . . v[ρ] . . . =

∂v ∂v ∂v ∂ρ(y) ≈ = φi (y) . ∗ ∗ ∂φi (y) ∂ρ(y) ∂φi (y) ∂ρ(y)

(8)

Then, by performing the variation of Eq. (5) with the use of φi∗ , one arrives at the following set of equations:  h¯ 2 2 ∇i φi (x) + 2m A

−



j =1



∂v φj (x )φi (x) dx φj∗ (x ) v(x, x ; ρ) + ρ(x) ∂ρ(x)

 εij φj (x). −φj (x)φi (x ) =

(9)

ij

These are a set of single-particle Schrödinger equations in a non-canonical form. The matrix εij can be diagonalized, and this leads to the HF set of equations:  h¯ 2 2 ∇i φi (x) + 2m A

−



j =1



∂v φj (x )φi (x) dx φj∗ (x ) v(x, x ; ρ) + ρ(x) ∂ρ(x)

−φj (x)φi (x ) = εi φi (x).

(10)

In HF, the s.p. potential depends on the wave functions themselves, so that the solution must be found in an iterative manner. One can write the set (10) in the form h¯ 2 2 − ∇ φi (x)+ 2m i





dx v(x, x ; ρ)+vrearr (x, x ; ρ) ρ(x ) ×φi (x)

− Exchange term = εi φi (x).

(11)

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On the one hand, the so-called Hartree term has been separated from the exchange, or Fock one. It becomes clear that the Hartree term represents the interaction of a nucleon with the environment in which other nucleons are distributed with the density ρ: it is, in other terms, the classical mean-field term. The Fock term is due, instead, to the fermionic nature of the nucleons. On the other hand, it has been emphasized in (11) that the two-body interaction that appears is not simply v but includes the so-called rearrangement term. Once the convergence to the HF solution has been achieved, the total energy of the system is the sum of (6) and (7), calculated with the wave functions that minimize these expressions. Using the notation h¯ 2 i|t|j = − 2m



dx φi∗ (x)∇ 2 φj (x),

(12)

and  ij |v[ρ]|kl AS =



dx dx φi∗ (x)φj∗ (x )v(x, x ; ρ) φk (x)φl (x ) − φk (x )φl (x) , (13)

the total energy becomes, in the HF basis, E=

  1 1 i|t|i + ij |v[ρ]|ij AS ≡ i|t|i + Ui , 2 2 i

ij

i

(14)

i

where in the last step a mean field Ui is defined for the state i. One could write the same quantity, expressed in a different single-particle basis: it is enough, in fact, to define the density matrix associated with that basis, that is, ρij = Ψ |aj† ai |Ψ

(15)

(where a and a † are the usual fermion annihilation and creation operators). Then, the total energy becomes (Ring and Schuck 1980) E=

 1 i|t|j ρj i + ρki ij |v[ρ]|kl AS ρlj . 2 ij

(16)

ij kl

From HF to DFT Equation (16) shows that, without loss of generality, the HF energy is a functional of the density ρ. As it was said, most of the widely adopted interactions are density-dependent: their parameters and the kind of density dependence (which is a power law in the Skyrme case) are chosen so that the energy reproduces the experimental findings for known nuclei. In this respect, E[ρ] is the key quantity,

55 Nuclear Density Functional Theory (DFT)

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and the Hamiltonian (1) can be merely seen as the generator of the energy functional E[ρ]. It must be added that, with an effective Hamiltonian, one can generate a large variety of functionals – but not necessarily any generic functional E[ρ] can be derived from an effective Hamiltonian. This is why practitioners, at present, write directly the energy functional, seeking to be more general. Some physical understanding related to nuclear phenomenology will be further provided, with the goal of justifying why an energy functional is needed to describe nuclei. In fact, the argument is borrowed from Weisskopf (1957) (note that similar arguments have been already introduced in Nakatsukasa et al. 2016 and Davesne et al. 2018). It is easier to refer to uniform nuclear matter, where the momentum k is a good quantum number. The relationship  kF =

3π 2 ρ 2

1/3 (17)

,

for symmetric matter, holds also in the mean-field approximation. The mean field U may or may not depend on the specific state i. The total energy (14), E=

  1 1 i|t|i + ij |v[ρ]|ij AS = i|t|i + Ui , 2 2 i

ij

i

i

can be also written as A E = AT¯ + U¯ , 2

(18)

where the bar indicates an average value. The separation energy is 1 S ≡ E(A) − E(A + 1) = −T¯ − U¯ . 2

(19)

However, due to the Koopmans’ theorem, one can also write the separation energy as the negative of the Fermi energy: S = −TF − UF .

(20)

By comparing the last two expressions, one can deduce that S = 2S − S = −2T¯ − U¯ + TF + UF .

(21)

Since T¯ = 35 TF , one arrives at 1 S = − TF − U¯ + UF . 5

(22)

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G. Colò

The latter equation shows that the mean field must be state-dependent: if U¯ were equal to UF , the separation energy would be negative. The mean field could be made state-dependent in some way (e.g., with a momentum-dependent potential or effective mass). Of course, many-body frameworks exist that go beyond the mean-field level and are, in principle, exact. However, it is worth stressing that DFT is also, in principle, exact and that its Kohn-Sham formulation retains the simplicity of the mean-field formalism nonetheless. DFT circumvents the problem that has just been described in the same way as HF with density-dependent forces does. As it has been shown, in the mean field that enters the HF equations (11) and in the total energy (14), different two-body interactions v enter (with and without rearrangement terms). This breaks the relationship between U and E of the previous equations and solves the contradiction shown in Eq. (22). Some general motivation for DFT will now be provided, its features in the case of Coulomb systems will be briefly explained, and the transferability to the nuclear many-body problem will be discussed.

General Motivation and DFT for Coulomb Systems One might believe that knowing the total wave function is mandatory when dealing with N-body systems, but this is not the case. The total wave function Ψ is a very complicated antisymmetric function of the N coordinates, and it is not measurable and interesting per se. One is interested in calculating measurable quantities and comparing them with the experimental findings. A measurable quantity can be the expectation value of a one-body operator, namely Oˆ =



O(xi ).

(23)

i

Then,  O = N

dx1 . . . dxN Ψ † (x1 . . . xN )O(x1 )Ψ (x1 . . . xN )

 =

 dx1 O(x1 )N

dx2 . . . dxN Ψ † (x1 . . . xN )Ψ (x1 . . . xN ).

(24)

This latter equation shows that it is enough to know the one-body density, defined by  ρ(x1 ) = N

dx2 . . . dxN Ψ † (x1 . . . xN )Ψ (x1 . . . xN ),

and calculate the expectation value (24) as

(25)

55 Nuclear Density Functional Theory (DFT)

2089

 O =

dx1 O(x1 )ρ(x1 ).

(26)

All this provides a strong motivation to consider density as a basic variable. Of course, one is often interested in expectation values of two-body operators (or even three-body, etc.). In that case, a calculation similar to the one that has just been sketched shows that knowing the two-body density,  ρ(x1 , x2 ) = N(N − 1)

dx3 . . . dxN Ψ † (x1 . . . xN )Ψ (x1 . . . xN ),

(27)

would be in order. Assuming that the two-body density factorizes into a product of one-body densities amounts to assuming the overall validity of the HF approximation. It makes life simple, but it is a strong assumption. In this context, DFT stands out as a real breakthrough. DFT shows that it is indeed possible, at least in principle, to express all quantities as a functional of the one-body density. Therefore, it is not surprising that DFT is one of the most popular approaches in physics and chemistry. As the subject is also included in standard textbooks (cf., e.g., Parr and Yang 1994; Martin 2004; Giuliani and Vignale 2005; Martin et al. 2016), only the main facts will be sketched here. A Hamiltonian of the type H = T + V + vext ,

(28)

where the kinetic energy T is the same as in (2), the second term is a two-body interaction, and the last term is a possible external potential, is assumed to govern the systems under study. In the case of a Coulomb system with N electrons and M fixed ions (labeled by α, and having charge Zα as well as coordinate Rα ), the Hamiltonian (28) for the electrons reads H =

 i

−

  Zα e2 e2 h¯ 2 2 1  ∇i + + . 2m 2 |ri − rj | |ri − Rα | α i=j

(29)

i

Among the countless strategies to solve the many-electron problem, DFT stands out for its conceptual elegance and simplicity. Its foundation lies in the theorems that bear the name of Hohenberg and Kohn (HK) (Hohenberg and Kohn 1964). According to these theorems, the total energy of a system of fermions associated with the Hamiltonian (28) can be written as a functional, in terms only of the fermion density ρ:  Evext [ρ] = Ψ |T + V + vext |Ψ = F [ρ] +

d 3 r vext (r)ρ(r).

(30)

The first equality is just the definition of total energy. In contrast, the second equality defines the functional: the contribution of the external potential is singled out, and

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G. Colò

the universal functional F is introduced. The functional E, Eq. (30), has a minimum at the exact ground-state density where it becomes equal to the exact energy. It is hard to underestimate the value of Eq. (30) because it tells us that the whole information about the ground-state of the system is contained in the one-body density ρ. In principle, even the expectation values of two-body, three-body, etc. operators would be functionals of the one-body density. This does not happen, in practice, because our functionals are not accurate enough. The notation in the last paragraph is that of the original paper, in which the ground state is not degenerate, and spin polarization is ignored (this explains the small discrepancy between the one-body density ρ(x) defined by (25) and ρ(r) which is summed over possible spin orientations). Extensions of the theorems to the case of degenerate ground states, spin-polarized systems, finite temperature, and relativistic case can be found in the literature (Dreizler and Gross 1990). The real drawback of the HK theorems is that a mere proof of the existence of the universal functional F is provided. There is no constructive proof, and the strategy to build the functional is left as an open problem. However, a step forward is represented by the Kohn-Sham (KS) scheme (Kohn and Sham 1965), in which it is assumed that the density ρ is represented in terms of auxiliary single-particle orbitals φi (r), that is, ρ(r) =



|φi (r)|2 .

(31)

i

Within the KS framework, the total kinetic energy can be written in the standard form of independent particles, like in Eq. (12): T =



−

i

h¯ 2 2m



d 3 r φi∗ (r)∇ 2 φi (r).

(32)

In the case of electrons, the direct Coulomb energy (Hartree energy) can also be singled out: EHartree =

e2 2



d 3r d 3r

ρ(r)ρ(r ) . |r − r |

(33)

In this way, the energy functional E, Eq. (30), can be written as  EKS = T + EHartree + Exc +

d 3 r vext (r)ρ(r),

(34)

where now the unknown part is the so-called “exchange-correlation” term, or Exc . As it has been alluded to, in the Coulomb case the kinetic and Hartree contributions are dominant, and one is left with a somehow small unknown part to be fixed. The search for the minimum of the functional (34) with respect to the density must be carried out with the constraint that the orbitals are orthonormal, exactly as

55 Nuclear Density Functional Theory (DFT)

2091

in Eq. (5). The whole procedure follows closely the one that has been discussed in the HF case and leads to the Kohn-Sham equations that are written here directly in their canonical form (and still ignoring spin polarization): 

h¯ 2 2 e2 ∇ + − 2m i 2



 δExc ρ(r ) + + vext (r) φi (r) = εi φi (r). d r |r − r | δρ(r) 3

(35)

Although the exact Exc is unknown, one can resort to a series of approximations, and at the same time, the Eqs (35) keep their simple structure. In the words of W. Kohn, the Kohn-Sham method is “an exactification of Hartree-Fock.” The simplest approximation for Exc is the Local Density Approximation (LDA), in which one calculates it in uniform matter and exploits the result also for finite systems. Formally, this means that one writes  Exc =

d 3 r εxc [ρ(r)],

(36)

and uses the same exchange-correlation energy density εxc in the finite systems, by assuming that the energy density at each point is the same as in a uniform system with the same density. In passing, it can be noticed that if one extended the same treatment to the kinetic energy, Eq. (32) could be easily calculated using plane waves and would give h¯ 2 5/3 1/3 T = 3 π 5m

 d 3 r ρ 5/3 (r).

(37)

Using this kinetic energy, together with the Hartree energy and possibly the external potential, amounts to adopting the so-called Thomas-Fermi (TF) approximation. The TF approach can be seen as a simplified DFT: it is not realistic, but it is straightforward to build and can be seen as a pedagogical tool. Coming back to LDA, it usually produces an overbinding of atoms and molecules by about a few percent, while lattice constants of solids are underestimated by approximately the same amount. These shortcomings can be corrected by introducing a dependence of εxc on the gradient of the density, ∇ρ (Generalized Gradient Approximation, or GGA). At present, functionals that depend also on higher derivatives of the density, like meta-GGA, have also been proposed. This means that  Exc = d 3 r εxc [ρ(r), ∇ρ(r), ∇ 2 ρ(r) . . .], (38) and the corresponding xc-potential in the KS equations can be obtained from Eq. (75), as one has cast the xc-functional in the form of (74). These various degrees of sophistication are pictorially referred to as a “Jacob’s ladder” (see Fig. 1 and Perdew and Schmidt 2001; Perdew et al. 2005).

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G. Colò

Fig. 1 The Jacob’s ladder of approximations to Exc or to εxc . (The figure is taken from Perdew et al. 2005). Note that the density is here denoted by n instead of ρ and that the starting point is not LDA but the local “spin” density approximation or LSD

These functionals that are constructed considering increasingly sophisticated densities can be directly compared with the nuclear case, as discussed at length in the following sections. However, a somehow different strategy is followed by the so-called “hybrid” functionals in which a mixture of Exc and of the exact (Fock) exchange energy is considered, with optimized weights. The reader can browse papers that briefly review the state-of-the-art Coulomb DFT to get an idea of the performance of existing EDFs and of the open questions (Burke 2012; Becke 2014; Yu et al. 2016).

The Nuclear Case Different density-dependent forces have been proposed and used in the nuclear case with some degree of success. On the one hand, Skyrme and Gogny forces have been employed by a more significant number of groups; still, one should not forget other types of forces that have appeared in the literature. The Skyrme ansatz takes its name from the ideas put forward by T.H.R. Skyrme already in the 1950s (Skyrme 1956, 1958; Bell and Skyrme 1956). In fact, the Skyrme force has been for the first time employed in HF calculations for spherical nuclei in Vautherin and Brink (1972). In its standard form (Chabanat et al. 1997, 1998), it reads  1 VSkyrme (r1 , r2 ) = t0 (1 + x0 Pσ ) δ(r1 − r2 ) + t1 (1 + x1 Pσ ) k†2 δ(r1 − r2 ) 2  +δ(r1 − r2 )k2 + t2 (1 + x2 Pσ ) k† · δ(r1 − r2 )k

55 Nuclear Density Functional Theory (DFT)

2093

1 + t3 (1 + x3 Pσ ) δ(r1 − r2 )ρ α 6



r1 + r2 2



+ iW0 (σ1 + σ2 ) · k† × δ(r1 − r2 )k,

(39)

where Pσ = 12 (1 + σ 1 · σ 2 ) is the spin-exchange operator, k = − 2i (∇1 − ∇2 ) is the relative momentum operator acting at right, and k† is its adjoint operator acting at left. There are 10 free parameters to be adjusted that are the ti , xi as well as the power α. The terms in t0 and t3 are, respectively, attractive and repulsive. When the density increases, the contribution to the energy given by the repulsive term increases, due to the dependence on ρ α , until a balance is reached, that corresponds to nuclear saturation. In fact, saturation in symmetric nuclear matter could be realized through an effective force that includes these two terms only (cf. Eq. (41) below). The momentum-dependent terms in t1 and t2 are needed to reproduce experimental data in a realistic way, and the same holds true for the spin-orbit term in W0 . The t1 and t2 terms mimic the finite range (cf. the short discussion below on the Density Matrix Expansion). As it has been already emphasized, a force like (39) and like those described below in this section must be seen as a tool to generate an energy functional through Eq. (14) that can be written as the sum of (6) and (7): E = Ψ |T + V |Ψ ,

(40)

where Ψ is the most general Slater determinant (consistent with the symmetries of the system). It is a useful warm-up exercise to check that a simplified Skyrme force without momentum and spin dependence, 1 V (r1 , r2 ) = t0 δ(r1 − r2 ) + t3 δ(r1 − r2 )ρ α 6



r1 + r2 2

 ,

(41)

generates, in the case of even-even systems, the functional  E[ρn , ρp ] =

d 3 r ε[ρn , ρp ],



   h¯ 2 1 1 1 2 2 2 α+2 1 α 2 2 τ + t0 ρ − ρ + ρp + t3 ρ − ρ ρn +ρp . ε[ρn , ρp ] = 2m 2 2 n 12 2 (42) The kinetic term of the Hamiltonian has led to the appearance of the so-called kinetic density, τ (r) =

 i

|∇φi (r)|2 .

(43)

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G. Colò

By plugging the whole force (39) into Eq. (40), a more general functional is obtained, in which ρ, its gradient, and τ appear. In fact, the spin-orbit term which is the last term in (39), together with exchange terms in Ψ |V |Ψ produced by the central terms of the force, lead to a dependence of the functional on the so-called spin-orbit density J(r) =



∗ φiα (r)∇φiβ (r) × α|σ |β .

(44)

i,αβ

It has also been suggested that tensor terms can be added to the standard Skyrme force (39). This leads to a functional with a richer dependence on the spin-orbit density and helps explain a number of experimental data. The review paper Sagawa and Colò (2014) includes an exhaustive discussion on the tensor terms of the Skyrme force. In summary, an EDF derived from a standard Skyrme force will be a function of all these densities, 

ESkyrme = d 3 r ε ρq , ∇ρq , τq , Jq , (45) at least for even-even nuclei. q labels protons and neutrons. The detailed expression of ε can be found in Chabanat et al. (1997, 1998). In this case, the coefficients will bear some definite relationships with the aforementioned parameters ti , xi and α. A more general Skyrme functional will have the same type of form but, in principle, arbitrary values for the coefficients. The dependences that are shown in (45) are the same as in Fig. 1 (except for the spin-orbit density, but it is known that spin orbit is much weaker in electronic systems than in nuclei). A functional of the type (42) can be said to be local, because the energy density at a point r depends on the density at the very same point. A Skyrme functional like (45) is said to be quasi-local because the energy density can be shown to depend on quantities evaluated at the very same point, but these quantities include derivatives that are sensitive to variations around the point. In this respect, quasilocal functionals are intermediate between fully local and fully non-local ones that will be introduced now. Non-local functionals can be generated via Eq. (40) if one adopts a finite-range interaction V . The most important and most widely used finite-range interaction in the nuclear context is the Gogny force (Dechargé and Gogny 1980; Berger et al. 1991) that reads VGogny (r1 , r2 ) =

2 

e

|r1 −r2 |2 μ2j

  Wj + Bj Pσ − Hj Pτ − Mj Pσ Pτ

j =1

+ t3 (1 + x0 Pσ ) δ(r1 − r2 )ρ α



r1 + r2 2

+ iWls (σ 1 + σ 2 ) · k† × δ(r1 − r2 )k,



(46)

55 Nuclear Density Functional Theory (DFT)

2095

where part of the notation has been already introduced when defining the Skyrme force, Pτ = 12 (1 + τ 1 · τ 2 ) is the isospin-exchange operator (analogous to Pσ ), and there are 14 parameters to be adjusted. This interaction is the sum of two Gaussians with exchange operators, a density-dependent term and a spin-orbit term. The density-dependent term has zero range, in order to avoid ambiguities on the point where the density must be evaluated, while the spin-orbit term has zero range for simplicity. No significant qualitative differences appear in self-consistent calculations when systematically comparing Skyrme and Gogny results. As already mentioned, the finite-range character of the Gogny force is effectively mocked up by the momentum-dependent terms of Skyrme. The so-called Density Matrix Expansion (DME) is a technique that allows mapping non-local EDFs into quasi-local ones, and it has been applied in Carlsson and Dobaczewski (2010), demonstrating our previous statement quantitatively. The actual difference between Skyrme and Gogny forces is instead in the pairing channel, as discussed below. Other effective forces have been considered, throughout the years, as generators of EDFs. Semi-realistic interactions of Yukawa-type with density-dependent terms have been proposed by Nakada (2008). Indeed, in the spirit of effective field theory and along the line of the above discussion, there should be no fundamental difference between generating an EDF through δ-functions, Gaussians, or Yukawas. Another kind of EDF has been proposed by S.A. Fayans and collaborators (Fayans et al. 1994; Fayans 1998). In the original works, one can find a detailed discussion of the features of the Fayans-type EDFs; here, it is useful to stress that the energy density ε has a dependence on gradients ∇ρ both in the numerator and in the denominator of a fraction. The underlying idea is that a Padé approximation should be superior to a Taylor expansion, at variance with the Skyrme ansatz. Before reporting about recent steps forward in generalizing the nuclear EDFs, it is better to stop and stress some critical points often not clear enough in the literature when comparing the Coulomb case and the nuclear case. 1. In the case of Coulomb EDFs, the fundamental interaction is known, and the Hartree term is well-defined so that the “functional” is defined by the choice for Exc . Hartree is the dominant part, while exchange and correlation partly cancel each other: the success of Coulomb DFT relies on specific features of the Coulomb force. HF and KS-DFT are clearly distinguished for Coulomb systems. 2. In the nuclear case, so far, all EDFs are based on an ansatz for the form of ε and a parameter fit. There is no underlying “fundamental force,” so that Hartree, exchange, and correlations are mixed up in the terms of the EDF. HF with effective forces and KS-DFT are connected. Usually, one speaks of the latter when a Hamiltonian is assumed (like in this section), and one speaks of the former when the EDF is written directly without any underlying Hamiltonian (as discussed below). There is no considerable difference between HF and KS-DFT, though. 3. In the case of electronic systems, there is an external potential that also constrains the origin of the electron coordinates. Nuclei are self-bound objects. This has

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G. Colò

some consequences. First, the original proof of the HK theorems should be better replaced by the Levy and Lieb approach as explained, e.g., in Parr and Yang (1994). Secondly, the density defined in the laboratory frame becomes meaningless because the nucleus can translate. The density in the laboratory can be spread everywhere in space and negligible at each specific point. This has given rise to ample discussions (Engel 2007; Barnea 2007; Messud et al. 2009). The conclusion is that one should ground the construction of DFT, starting from the fundamental theorems, on intrinsic densities. It has been proven (Valiev and Fernando 1997) that, at least in principle, given an arbitrary Hermitian operator ˆ one can build an EDF depending on Q(r) ≡ Q(r) ˆ Q, that is universal in the HK sense and has its minimum at the correct value of Q with the correct energy. One can replace the laboratory density with the intrinsic density in this respect.

Generalized Local Densities and Generalized EDFs As already said, the very idea that a Hamiltonian including a density-dependent force is the tool to generate an EDF has been largely abandoned. A functional can be directly parameterized in terms of local densities. A pioneering step in this direction was taken by Reinhard and Flocard (1995), already more than 20 years ago. They wrote the spin-orbit part of their EDFs without reference to a specific form of the spin-orbit force. Later, this has become the customary procedure. As a paradigmatic example, the functionals of the UNEDF family (Kortelainen et al. 2014) are discussed here. They have the form, for the even-even systems,  E=

  d 3 r εkin + εSkyrme + εCoulomb + εpairing .

(47)

In this equation, εkin is the usual kinetic energy density for independent nucleons like the first term at r.h.s. of (42). The pairing energy density will be discussed below, while the Coulomb part does not deserve special comments here. As for the remaining term, 2

2

εSkyrme = C ρρ [ρ]ρ 2 + C ρτ ρτ + C (∇ρ) (∇ρ)2 + C J J2 + C ρ∇·J ρ∇ · J.

(48)

Note that, for the sake of simplicity, the label q, for either neutrons  ρρand protons, is omitted: in reality, the first term in the l.h.s. should read q Cq ρq2 , and so on. Alternatively, one can express everything in terms of isoscalar and isovector densities. In principle, all the coupling constants C could be density-dependent; in practice, in most of the cases so far, only the C ρρ coupling constant is taken as density-dependent. It has the form C ρρ = A + Bρ α ,

(49)

55 Nuclear Density Functional Theory (DFT)

2097

as in a functional of the type (42). All the possible local densities that can appear in an EDF have been classified in Engel et al. (1975), Dobaczewski and Dudek (1995, 1996), and Bender et al. (2003) (cf. also Dobaczewski and Dudek 1997), and a thorough discussion can be also found in Sec. 7.3.4 of Duguet (2004). Once densities are defined, the nuclear EDF must be invariant with respect to parity, time-reversal, rotational, translational, and isospin transformations. If the single-particle wave functions are written as φiα (x) ≡ fi (r)χiα (σ ), the density is a 2×2 matrix ραβ (r, r ). Thus, it can be expanded as ραβ (r, r ) = ρ(r, r )1 + s(r, r ) · σ .

(50)

This writing defines the density ρ and the spin density s. Local densities are defined by setting r = r:  ρ(r) = ρ(r, r )r =r ,  s(r) = s(r, r )r =r .

(51) (52)

Given these starting densities, one has to apply derivative operators in a systematic manner and up to a given order. The procedure is outlined here up to the second order, employing the usual center-of-mass and relative coordinates, 

r+r

2 , r − r .

R= r=

(53)

Since the EDF must be invariant under space translations, only derivatives with respect to the relative coordinate, viz. ∇ −∇ , must be taken into account. Moreover, due to the identity  2  2 ∇ − ∇ = ∇ + ∇ − 4∇ · ∇ ,

(54)

the action of the second derivative with respect to the relative coordinate can be turned into the action of ∇ · ∇ . Eventually, starting from ρ one obtains the current j(r, r ) =

   1  ∇ − ∇ ρ(r, r ) 2i r =r

(55)

and the kinetic density  τ (r, r ) = ∇ · ∇ ρ(r, r )r =r . In a fully analogous manner, starting from the spin density, one obtains the spin current

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G. Colò

   1 

 ∇ − ∇ ⊗ s(r, r ) J= 2i r =r

(56)

and the spin kinetic density  T = ∇ · ∇ s(r, r )r =r .

(57)

A few comments are in order here. Time-reversal plays a unique role. While the density, the kinetic density, and the spin current are even under time-reversal (time-even), the current, the spin density, and the spin kinetic density are odd under time-reversal (time-odd). The time-odd densities vanish in even-even systems. Thus, any EDF can be built with terms that are bilinear in either type of densities, but the terms that are bilinear in time-odd densities are present only in the systems with an odd number of particles. The pseudo-tensor quantity (56) can be decomposed into a pseudo-scalar, vector, and pseudo-tensor term as Jμν =

1 1 (2) εμνκ Jκ(1) + Jμν . δμν J (0) + 3 2 κ

(58)

The vector term is precisely the spin-orbit density that has been introduced in the previous section. A currently open question for practitioners is to which extent higher gradients of the density are needed for an EDF to satisfactorily account for nuclear properties (Raimondi et al. 2011; Becker et al. 2015; Davesne et al. 2015). Even if one limits oneself to quadratic terms in the densities, the number of such terms can grow dramatically if one allows higher-order derivatives (cf. Raimondi et al. (2011), and in particular Table XXVII therein). If the EDF is too involved, fitting all its parameters may become prohibitive. Gauge invariance may be imposed to reduce the number of terms, but this provides only a partial solution. Related to this question is the analysis of correlations and redundancies. Correlation analysis has become a tool that EDF practitioners use more and more (Ireland and Nazarewicz 2015). This analysis is meant to determine if new terms in an EDF bring further information, if a new term correlates with previous ones, or if the nuclear data employed do not constrain that new term.

Symmetry Breaking and Pairing Correlations Generally speaking, if the Hamiltonian of a system commutes with a given symmetry operator S, [H, S] = 0,

(59)

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one expects to find eigenstates of H , which are also eigenstates of S. Often, problems with this principle statement come when one treats strongly correlated systems and is obliged to resort to severe approximations. In that case, a symmetry-conserving solution might be unpractical or too hard to find, within the approximated framework, than a symmetry-breaking one. To substantiate this statement, one can start with a simple and yet famous example, i.e., that of translational symmetry within the independent-particle picture, or HF and KS schemes. In these cases, one assumes that the total wave function is a Slater determinant built up with localized orbitals like in Eq. (4). This wave function is not an eigenstate of the total momentum operator P (Lipkin 1960), although clearly [H, P] = 0. In other words, the Slater determinant is not translationally invariant. One cannot reconcile the independent-particle character of the total wave function with one of the symmetries of the system; and in keeping with all the enormous advantages associated with the HF and KS schemes that have been discussed in this chapter, one better gives up, at this point, the requirement of translational invariance. Indeed, all the densities that have been introduced are localized densities, i.e., they are defined with respect to a fixed origin. Nuclear DFT is grounded on this choice of localized, translational symmetry-breaking densities. Other types of symmetry breaking, i.e., those associated with shape and pairing correlations, have been identified as the source of the most important correlations in finite nuclei for several decades. In addition to translational symmetry, rotational symmetry also holds for the nuclear Hamiltonian. Exactly in the same manner as P is the generator of the center-of-mass translations, the total angular momentum J is the generator of the rotations R(Ω), where Ω is a set of Euler angles. The motivation to introduce densities that break the rotational invariance (i.e., that are non-spherical) comes from the nuclear phenomenology. In particular, nuclei with strong intrinsic quadrupole deformation can be identified through their rotational bands, that is, by the fact that their spectra correspond to that of a rotor (Bohr and Mottelson 1975). One is led, then, to considering localized deformed densities. This example is relevant also for the introduction of symmetry restoration. The rotational symmetry must be restored in the laboratory system. This means, according to our previous discussion, that one has to recover states that have the total angular momentum J as a good quantum number. Intuitively, by resorting to the uncertainty principle, one can think that a deformed density with a given orientation provides the largest uncertainty on the angular momentum and the maximal symmetry breaking. Furthermore, by superimposing the different orientations of the nucleus that are produced by rotating one of them through R(Ω), one can fix the angular momentum and restore the symmetry (cf. the schematic picture of Fig. 2). These interesting aspects are the subject of another chapter on multi-reference DFT (MR-DFT). For the mere sake of illustration, the formal equation that does the job will be introduced, in the case of deformed densities that keep an intrinsic symmetry axis z and have a good projection K of the angular momentum along that axis, like in Fig. 2. The projection operator on good angular momentum J M in the laboratory system reads

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Fig. 2 Schematic view (in two dimensions) of the deformed nucleus in the intrinsic frame that rotates in the laboratory frame. See the text for a discussion. (Figure taken and adapted from Nakatsukasa et al. 2016)

 PJ M = N

 ∗ J dΩ DKM (Ω) R(Ω),

(60)

where N is a normalization factor and D is a Wigner function. Using this background on the breaking and restoration of rotational symmetry, one can introduce the more abstract reasoning associated with the breaking of number symmetry and with pairing correlations (Brink and Broglia 2005). The number operator N is another operator which commutes with H . The densities that have been discussed so far correspond to a fixed particle number. The motivation to introduce number symmetry-breaking densities comes from the empirical evidence that open-shell nuclei are characterized by a fraction of nucleons that display superfluidity: the odd-even staggering in the mass formula, the gap between the ground and first excited state in the even-even nuclei, the moment of inertia of deformed nuclei are all pieces of evidence that point in that direction. Pairing is the subject of another chapter. For more than half a century, starting from the proposal by J. Bardeen, L. Cooper, and R. Schrieffer of the theory that bears their names (i.e., the BCS theory), it has been evident that describing superfluidity and superconductivity becomes more effective in a symmetry-breaking framework. The basic idea behind BCS is that fermions can form the so-called Cooper pairs and that superfluidity and superconductivity are associated with the coherent behavior of these pairs that act as quasi-bosons. Accordingly, the BCS ansatz for the many-body wave function |Ψ reads   |Ψ = Πj uj + vj aj† a †˜ |− , j

(61)

where j labels a set of single-particle orbitals like the orbitals φj that have been already discussed throughout this chapter; aj and aj† are, respectively, the annihilation and creation operators that destroy or add a particle in these orbitals;

55 Nuclear Density Functional Theory (DFT)

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and the symbol j˜ indicates the time-reversal operation on j . |− is the particle vacuum. The wave function (61) can be seen as a generalization of the Slater determinant (4). vj and uj can be seen as the probability amplitudes that the orbital j is occupied or empty, respectively. The Slater determinant (4) can be reproduced if the set of orbitals are assumed to be fully occupied from the one that has the lowest energy up to the Fermi energy. Moreover, (61) displays a superposition of components having a different particle number. In analogy with the previous cases, the BCS wave function keeps the simplicity and transparency associated with a description in terms of single-particle orbitals and emphasizes the presence of possible pairs of fermions in time-reversal states; these advantages outmatch the shortcoming of the number-symmetry breaking. One could, also in this case, project on good particle number by using the operator  PN = N

ˆ

dφ eiN φ e−i N φ ,

(62)

where one has to distinguish the number operator Nˆ from the given particle number ˆ and where e−i N φ can be seen as a rotation by the angle φ generated by the number operator. φ is called the gauge angle. A more general theory than BCS is HFB, which has been already mentioned above without explanations. Within HFB, the wave function is more general than (61), and it includes Cooper pairs that are not limited to particles in time-reversal states. The Cooper pairs are assumed to be the most general pairs that are compatible with the symmetries of the system; in even-even nuclei, pairs are coupled to J π = 0+ , but, formally, it is possible to write down other versions of the theory. HFB has many properties that make it the counterpart of HF with density-dependent forces. In particular, by introducing quasi-particles (that are mixtures of particles and holes), the HFB wave function can be written as a Slater determinant. The reader can grasp all the main facts from the short but very effective résumé in Bulgac (1999) and find more comprehensive treatments in Ring and Schuck (1980) and Dobaczewski et al. (1996). In this Handbook, some notions about HFB are included in the  Chap. 14, “Theoretical Methods for Giant Resonances,” as well as in a few others. HFB becomes superior to BCS in nuclear physics when dealing with weakly-bound systems. From the DFT viewpoint, the breaking of the particle-number symmetry implies that EDFs can also depend on a generalized density that is called abnormal density (or pairing tensor) and reads κ(r, r ) = a(r)a(r ) (where a(r) is the annihilation operator of a particle at point r and a † (r) is the corresponding creation operator). The corresponding local density reads  κ(r) = κ(r, r )r =r .

(63)

It is useful to remind, at this point, that the normal density can be written as ρ(r) = a † (r)a(r) in second quantization. The aforementioned HFB with density-

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dependent forces can be seen as a realization of KS-DFT when ρ is complemented with κ. The discussion of the previous section can be repeated for the abnormal sector. In principle, one could follow all the steps and derive in the same systematic manner as above the abnormal spin density and all the generalized densities that are produced by the action of derivatives. In practice, however, there is no empirical evidence that a sophisticated functional that depends on the abnormal counterparts of ∇ρ, τ, J, as in Eq. (45), is needed. Rather, Skyrme EDFs are usually complemented, in the pairing sector, by something that resembles more the simple functional of Eq. (42). This seems to be enough to obtain a satisfactory agreement with data. It means that, usually,  Epairing =

 d 3 r εpairing =

d 3r

1 C(ρ)κ 2 (r), 2

(64)

with

   ρ(r) γ . C(ρ) = V0 1 − x ρ0

(65)

The parameters V0 , x, ρ0 and γ are usually fit to observables like the pairing gap Δ in one or several open-shell nuclei. x = 0 corresponds to the so-called volume pairing. x = 1 (surface pairing) introduces a dependence of the pairing strength on the density, following the suggestion made in Bertsch and Esbensen (1991). It has to be stressed that the Gogny interaction (46) can also be used in the pairing sector to generate a pairing functional. When it is inserted in Eq. (40), and the Slater determinant Ψ is replaced with a Slater determinant of quasi-particles, the energy density ε does acquire a pairing term. The details can be found in Dechargé and Gogny (1980) and Ring and Schuck (1980). A key point to be kept in mind is that the normal sector and the pairing sector of the EDF must remain decoupled: they correspond to different physics and are sensitive to different observables. The clever idea by D. Gogny is that, if the parameter x0 is fixed at +1, the t3 term cannot contribute to the pairing channel because this requires T = 1, S = 0 (in the case of either neutron-neutron or proton-proton pairs coupled to J π = 0+ ). In electronic DFT, the extension to superconductors has been proposed in Oliveira et al. (1988). Formally, the ideas are the same as described in this section: the authors introduce the abnormal density or pairing tensor in a Kohn-Sham scheme and acknowledge that these quantities have been earlier introduced in nuclear HFB. However, one should remember that, at least in the case of “conventional” electronic superconductivity, it is clear that the pairing phenomenon is originated by the coupling of electrons to lattice vibrations (electron-phonon interaction). The origin of the strong nuclear pairing force is much less clear. The discussion so far has concerned the pairing functionals generated by the pairing interaction between equal particles (with either neutron-neutron or protonproton pairs). One speaks, in this case, of T = 1 pairing. While there are

55 Nuclear Density Functional Theory (DFT) Table 1 The different types of symmetry breaking that are discussed in this section. The right column shows the corresponding generalization in the densities. It is meant here that each successive row introduces a new symmetry breaking on top of those defined in the previous rows

2103 Symmetry breaking Translation Rotation Particle number Charge symmetry ...

Density Localised ρ(r) Non-spherical ρ(r) κ(r) added ρ1,2 added in Eq. (66) ...

several pieces of evidence that T = 0 pairing, with an associated neutronproton component, should also exist, there are also many open questions. Currently employed EDFs do not incorporate, as a rule, the T = 0 pairing. Nevertheless, there is active research in this field that has been summarized in review papers Frauendorf and Macchiavelli (2014) and Sagawa et al. (2016). Proton-neutron pairing causes charge-mixing if treated within BCS or HFB. In simple terms, a wave function of the type (61) breaks one more symmetry if the pair is a proton-neutron pair. For the sake of completeness, one can investigate DFT with a full proton-neutron mixing in all sectors (both normal and abnormal); that is, one can at this stage avoid the asymmetry between a normal density ρ without charge mixing and an abnormal density κ with charge mixing. To clarify the meaning of charge mixing, by recalling Eq. (50), one can write a similar one with the isospin instead of the spin Pauli matrices: ραβ (r, r ) = ρ(r, r )1 +



ρi (r, r )τ i .

(66)

i=1,2,3

Whereas, so far, only ρ and the third component ρ3 appearing in the last term have been used, charge-mixing amounts to introducing explicitly the components ρ1,2 or ρ± in Eq. (66). Developing EDFs that include these charge-breaking densities in a local form (r = r) has been the goal of a series of works. The proper definition of all densities that involve an arbitrary proton-neutron mixing together with the formal writing of the most general EDF that is quadratic in those densities can be found in Perli´nska et al. (2004). Applications of these EDF can be found in Sato et al. (2013) and Sheikh et al. (2014). Table 1 summarizes, in the left column, all the types of symmetry breaking that have been introduced in this section. In the right column, the changes in the densities that have to be inserted in an appropriate EDF are displayed. All the generalized densities that have been introduced above should be considered, although they are not shown in the table for the sake of simplicity. Our treatment is simplified and suitable for a general reader, and it avoids the discussion of mathematical aspects that are treated in the pedagogical reference (Duguet 2004). This includes a full-fledged introduction to symmetry breaking in DFT, connected with group

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Fig. 3 Difference between theoretical and experimental total energies obtained using two of the local EDFs of the last decade: the left panel is taken from Kortelainen et al. (2012) and refers to the UNEDF1 functional, whereas the right panel is taken from Kortelainen et al. (2014) and corresponds to the UNEDF2 functional

theory. It is also helpful to remind that a textbook treatment of symmetry breaking is available in Blaizot and Ripka (1986). Active research concerns deepening and applying the symmetry-breaking formalisms that are briefly resumed by Table 1. A further aspect is the implementation of projection methods that can restore symmetries at the level of all the rows in the table. Problems associated with the number projection within DFT are discussed in Dobaczewski et al. (2007) (cf. also the references therein, some of which deal with problems associated with projected DFT in the case of other types of symmetry breaking). Isospin-invariant DFT seems to better allow symmetry restoration through projection (cf. Satuła et al. (2010) and references therein). The reader can find a general review of projection methods in the recent paper Sheikh et al. (2021).

Examples of Calculations In the nuclear case, KS-DFT can be applied and has been applied to study a vast number of nuclear properties. Many of these are the subject of other chapters in the Handbook, and consequently, this section includes only very few examples. The nuclear masses are the main ground-state property. In Fig. 3, one can see predictions for a large number of nuclei, obtained with two recent functionals of the UNEDF family. The figure displays the difference between theory and experiment (as for this difference, speaking either of masses or binding energies is equivalent). The main message of the figure is that current EDFs reproduce masses at the level of ≈ MeV. The reader can also compare with similar figures that are found in the chapter of this Handbook concerning covariant EDFs, or with Fig. 14 of Roca-

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Fig. 4 The nuclear landscape, defined by the neutron and proton drip lines as predicted by a number of models. See the discussion in the main text. (Figure taken from Afanasjev et al. 2013)

Maza and Paar (2018). Although the figure suggests that the accuracy of the models does not vary too much for neutron numbers between ≈20 and 120, one should note the “arches” displayed by the trends that span the ranges between consecutive magic numbers. These indicate that current EDFs do not have sufficient quality to reproduce at the same level closed-shell and open-shell nuclei. Nuclear binding energies or masses determine the limit of nuclear existence. In particular, the neutron and proton drip lines can be defined as the loci where the two-neutron or two-proton separation energies change their sign and become negative. In Fig. 4, we show predictions for the drip lines obtained with Skyrme EDFs (SEDF, labelled as SDFT in the figure) and covariant EDFs (CEDF, labeled as CDFT in the figure). The figure is taken from Afanasjev et al. (2013), and the SEDF results are taken from Erler et al. (2012). Errors complement the SEDF and CEDF results, and remarkably, these results are compatible to a large extent. Results from Gogny (GEDF, labeled as GDFT in the figure) and microscopic-macroscopic models (mic-mac) that are not discussed in this chapter are also displayed. The spread between predictions increases with the neutron and proton numbers and may become significant for the heavy systems. Reducing this spread is one of the main goals of the research in nuclear DFT. One should add that the proton and neutron drip lines are not the only limits that define the nuclear landscape. Superheavy elements are another frontier, and an up-to-date discussion of all these limits can be found in Nazarewicz (2018) and references therein. Masses and separation energies are not the only outcome of DFT calculations. A (not fully exhaustive) list of problems that have been tackled includes charge radii, neutron radii, and nuclear shapes (quadrupole, octupole, and more exotic deformations), low-lying states or nuclear spectroscopy (mainly, but not only, in the framework of MR-DFT methods), high-lying nuclear states like Giant Resonances, direct reactions like transfer, fission and other examples of large-amplitude motion, extrapolations to neutron matter, and neutron stars. There are sections and chapters

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in this Handbook for many of these topics, whereas in this chapter only the general formulation of the nonrelativistic, single-reference version has been laid out.

Appendix: Functional Derivatives One starts from the definition of a mapping between infinitely differentiable functions in one variable and real numbers: F : C ∞ → R, f (x) → F [f (x)].

(67)

F is called a functional. The definition can be extended to functions of several variables, e.g., in R3 . This latter case is the subject of this chapter; however, starting from the simple situation of functions of a single variable is quite illustrative (cf. Fig. 5). One can perform a small variation of the argument f (x), that becomes f (x) + εη(x), with ε small and η(x) an arbitrary function that keeps the incremented function within C ∞ . If the resulting value of F [f (x) + εη(x)] can be expanded in a Taylor series in ε,   dF [f (x) + εη(x)] ε + ..., F [f (x) + εη(x)] = F [f (x)] + dε ε=0

(68)

then the equation

Fig. 5 (Upper part) A schematic view of a functional F as a map between differentiable functions and real numbers. (Lower part) Representation of a small variation of a function f (x), that allows defining the functional derivative as described in the text

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   dF δF  [f (x) + εη(x)] η(x) = dx dε δf (x) ε=0

(69)

becomes a definition of the functional derivative δfδF(x) , which is another function of x. The meaning of this definition can be understood in other ways. The previous two equations are consistent with the fact that, at lowest order in ε,    dF δF [f (x) + εη(x)] εη(x). δF = ε = dx dε δf (x) ε=0

(70)

If discretized, the latter equation becomes δF =

 δF i

δfi

δfi ,

(71)

which is what any reader is familiar with, if written in terms of partial derivatives (i.e., with δ → ∂). The functional derivative can also be introduced through the formula F [f (x) + εδ(x − y)] − F [f (x)] δF = lim , δf (y) ε→0 ε

(72)

which is also the analogous of a formula that everyone knows from the previous knowledge of partial derivatives, namely F [fi + εδij ] − F [fi ] δF . = lim ε→0 δfi ε

(73)

In practice, in this chapter only functionals of the form  F =

f (ρ, ∇ρ, ∇ 2 ρ . . .),

(74)

have been shown or implicitly introduced. By using the above rules and integration by parts, it is easy to show that ∂f ∂f ∂f δF = −∇ · + ∇2 2 + . . . δρ ∂ρ ∂∇ρ ∂∇ ρ

(75)

This latter formula is applied in a few points of the chapter and is meant to be used by any reader if she or he wishes to make detailed calculations.

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References A.V. Afanasjev, S.E. Agbemava, D. Ray, P. Ring, Nuclear landscape in covariant density functional theory. Phys. Lett. B 726(4), 680–684 (2013) N. Barnea, Density functional theory for self-bound systems. Phys. Rev. C 76, 067302 (2007) A.D. Becke, Perspective: fifty years of density-functional theory in chemical physics. J. Chem. Phys. 140(18), 18–301 (2014) P. Becker, D. Davesne, J. Meyer, A. Pastore, J. Navarro, Tools for incorporating a d-wave contribution in Skyrme energy density functionals. J. Phys. G: Nucl. Part. Phys. 42(3), 034001 (2015) J.S. Bell, T.H.R. Skyrme, CVIII. The nuclear spin-orbit coupling. Philos. Mag.: J. Theor. Exp. Appl. Phys. 1(11), 1055–1068 (1956) M. Bender, P.-H. Heenen, P.-G. Reinhard, Self-consistent mean-field models for nuclear structure. Rev. Mod. Phys. 75, 121–180 (2003) J.F. Berger, M. Girod, D. Gogny, Time-dependent quantum collective dynamics applied to nuclear fission. Comput. Phys. Commun. 63(1), 365 (1991) G.F. Bertsch, H. Esbensen, Pair correlations near the neutron drip line. Ann. Phys. 209(2), 327–363 (1991) J.P. Blaizot, G. Ripka, Quantum Theory of Finite Systems (The MIT Press, Cambdridge/London, 1986) A. Bohr, B.R. Mottelson, Nuclear Structure, vol. I (Benjamin, New York/Amsterdam, 1969) A. Bohr, B.R. Mottelson, Nuclear Structure, vol. II (Benjamin, New York/Amesterdam, 1975) D.M. Brink, R.A. Broglia, Nuclear Superfluidity: Pairing in Finite Systems (Cambridge University Press, Cambridge, 2005) A. Bulgac, Hartree-Fock-Bogoliubov approximation for finite systems. arXiv:nucl-th/9907088 (1999) K. Burke, Perspective on density functional theory. J. Chem. Phys. 136(15), 150901 (2012) B.G. Carlsson, J. Dobaczewski, Convergence of density-matrix expansions for nuclear interactions. Phys. Rev. Lett. 105, 122501 (2010) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, A Skyrme parametrization from subnuclear to neutron star densities. Nucl. Phys. A 627(4), 710 (1997) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, A Skyrme parametrization from subnuclear to neutron star densities. Part II: Nuclei far from stabilities. Nucl. Phys. A 635, 231 (1998) G. Colò, Nuclear density functional theory. Adv. Phys.: X 5(1), 1740061 (2020) D. Davesne, J. Navarro, P. Becker, R. Jodon, J. Meyer, A. Pastore, Extended Skyrme pseudopotential deduced from infinite nuclear matter properties. Phys. Rev. C 91, 064303 (2015) D. Davesne, J. Navarro, J. Meyer, K. Bennaceur, A. Pastore, Two-body contributions to the effective mass in nuclear effective interactions. Phys. Rev. C 97, 044304 (2018) J. Dechargé, D. Gogny, Hartree-Fock-Bogolyubov calculations with the D1 effective interaction on spherical nuclei. Phys. Rev. C 21, 1568–1593 (1980) J. Dobaczewski, J. Dudek, Time-odd components in the mean field of rotating superdeformed nuclei. Phys. Rev. C 52, 1827–1839 (1995) J. Dobaczewski, J. Dudek, Time-odd components in the rotating mean field and identical bands. Acta Phys. Pol. B 27, 95 (1996) J. Dobaczewski, J. Dudek, Erratum: time-odd components in the mean field of rotating superdeformed nuclei [Pys. Rev. C 52, 1827 (1995)]. Phys. Rev. C 55, 3177–3177 (1997) J. Dobaczewski, W. Nazarewicz, T.R. Werner, J.F. Berger, C.R. Chinn, J. Dechargé, Mean-field description of ground-state properties of drip-line nuclei: pairing and continuum effects. Phys. Rev. C 53, 2809–2840 (1996) J. Dobaczewski, M.V. Stoitsov, W. Nazarewicz, P.-G. Reinhard, Particle-number projection and the density functional theory. Phys. Rev. C 76, 054315 (2007)

55 Nuclear Density Functional Theory (DFT)

2109

R.M. Dreizler, E.K.U. Gross, Density Functional Theory (Springer, Berlin/Heidelberg/New York, 1990) T. Duguet, Lecture Notes in Physics, vol. 879 (Springer, Berlin/Heidelberg/New York, 2004), p. 293 J. Engel, Intrinsic-density functionals. Phys. Rev. C 75, 014306 (2007) Y.M. Engel, D.M. Brink, K. Goeke, S.J. Krieger, D. Vautherin, Time-dependent Hartree-Fock theory with Skyrme’s interaction. Nucl. Phys. A 249(2), 215 (1975) E. Epelbaum, U.-G. Meißner, Chiral dynamics of few- and many-nucleon systems. Annu. Rev. Nucl. Part. Sci. 62(1), 159–185 (2012) J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A.M. Perhac, M. Stoitsov, The limits of the nuclear landscape. Nature 486, 509 (2012) S.A. Fayans, Towards a universal nuclear density functional. J. Exp. Theor. Phys. Lett. 68(3), 169– 174 (1998) S.A. Fayans, E.L. Trykov, D. Zawischa, Influence of effective spin-orbit interaction on the collective states of nuclei. Nucl. Phys. A 568(3), 523 (1994) S. Frauendorf, A.O. Macchiavelli, Overview of neutron proton pairing. Prog. Part. Nucl. Phys. 78, 24–90 (2014) G. Giuliani, G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005) K. Hebeler, Three-nucleon forces: implementation and applications to atomic nuclei and dense matter. Phys. Rep. 890, 1–116 (2021) P. Hohenberg, W. Kohn, Inhomogeneous electron gas. Phys. Rev. 136, 864–871 (1964) D.G. Ireland, W. Nazarewicz, Enhancing the interaction between nuclear experiment and theory through information and statistics. J. Phys. G: Nucl. Part. Phys. 42(3), 030301 (2015) W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, 1133–1138 (1965) M. Kortelainen, J. McDonnell, W. Nazarewicz, P.-G. Reinhard, J. Sarich, N. Schunck, M.V. Stoitsov, S.M. Wild, Nuclear energy density optimization: large deformations. Phys. Rev. C 85, 024304 (2012) M. Kortelainen, J. McDonnell, W. Nazarewicz, E. Olsen, P.-G. Reinhard, J. Sarich, N. Schunck, S.M. Wild, D. Davesne, J. Erler, A. Pastore, Nuclear energy density optimization: shell structure. Phys. Rev. C 89, 054314 (2014) H.J. Lipkin, Collective motion in many-particle systems: Part 1. The violation of conservation laws. Ann. Phys. 9(2), 272–291 (1960) O. Lopez, D. Durand, G. Lehaut, B. Borderie, J.D. Frankland, M.F. Rivet, R. Bougault, A. Chbihi, E. Galichet, D. Guinet, M. La Commara, N. Le Neindre, I. Lombardo, L. Manduci, P. Marini, P. Napolitani, M. Pârlog, E. Rosato, G. Spadaccini, E. Vient, M. Vigilante, In-medium effects for nuclear matter in the Fermi-energy domain. Phys. Rev. C 90, 064602 (2014) R. Machleidt, D.R. Entem, Chiral effective field theory and nuclear forces. Phys. Rep. 503(1), 1–75 (2011) R. Machleidt, F. Sammarruca, Can chiral EFT give us satisfaction? Eur. Phys. J. A 56(3), 56–95 (2020) R.M. Martin, Electronic Structure: Basic Theory and Practical Methods (Cambridge University Press, Cambridge, 2004) R.M. Martin, L. Reining, D.M. Ceperley, Interacting Electrons: Theory and Computational Approaches (Cambridge University Press, Cambridge, 2016) J. Messud, M. Bender, E. Suraud, Density functional theory and Kohn-Sham scheme for self-bound systems. Phys. Rev. C 80, 054314 (2009) H. Nakada, Mean-field approach to nuclear structure with semi-realistic nucleon-nucleon interactions. Phys. Rev. C 78, 054301 (2008) T. Nakatsukasa, K. Matsuyanagi, M. Matsuo, K. Yabana, Time-dependent density-functional description of nuclear dynamics. Rev. Mod. Phys. 88, 045004 (2016) W. Nazarewicz, The limits of nuclear mass and charge. Nat. Phys. 14(6), 537–541 (2018)

2110

G. Colò

J.W. Negele, The mean-field theory of nuclear structure and dynamics. Rev. Mod. Phys. 54, 913– 1015 (1982) L.N. Oliveira, E.K.U. Gross, W. Kohn, Density-functional theory for superconductors. Phys. Rev. Lett. 60, 2430–2433 (1988) R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, Oxford, 1994) J.P. Perdew, K. Schmidt, Jacob’s ladder of density functional approximations for the exchangecorrelation energy. AIP Conf. Proc. 577(1), 1–20 (2001) J.P. Perdew, A. Ruzsinszky, J. Tao, V.N. Staroverov, G.E. Scuseria, G.I. Csonka, Prescription for the design and selection of density functional approximations: more constraint satisfaction with fewer fits. J. Chem. Phys. 123(6), 062201 (2005) E. Perli´nska, S.G. Rohozi´nski, J. Dobaczewski, W. Nazarewicz, Local density approximation for proton-neutron pairing correlations: formalism. Phys. Rev. C 69, 014316 (2004) F. Raimondi, B.G. Carlsson, J. Dobaczewski, Effective pseudopotential for energy density functionals with higher-order derivatives. Phys. Rev. C 83, 054311 (2011) P.-G. Reinhard, H. Flocard, Nuclear effective forces and isotope shifts. Nucl. Phys. A 584(3), 467 (1995) P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin/Heidelberg/New York, 1980) X. Roca-Maza, N. Paar, Nuclear equation of state from ground and collective excited state properties of nuclei. Prog. Part. Nucl. Phys. 101, 96–176 (2018) H. Sagawa, G. Colò, Tensor interaction in mean-field and density functional theory approaches to nuclear structure. Prog. Part. Nucl. Phys. 76, 76–115 (2014) H. Sagawa, C.L. Bai, G. Colò, Isovector spin-singlet (T = 1, S = 0) and isoscalar spin-triplet (T = 0, S = 1) pairing interactions and spin-isospin response. Phys. Scr. 91(8), 083011 (2016) K. Sato, J. Dobaczewski, T. Nakatsukasa, W. Satuła, Energy-density-functional calculations including proton-neutron mixing. Phys. Rev. C 88, 061301 (2013) W. Satuła, J. Dobaczewski, W. Nazarewicz, M. Rafalski, Isospin-symmetry restoration within the nuclear density functional theory: formalism and applications. Phys. Rev. C 81, 054310 (2010) N. Schunck (ed.), Energy Density Functional Methods for Atomic Nuclei (IoP Publishing, Bristol, 2019) J.A. Sheikh, N. Hinohara, J. Dobaczewski, T. Nakatsukasa, W. Nazarewicz, K. Sato, Isospininvariant Skyrme energy-density-functional approach with axial symmetry. Phys. Rev. C 89, 054317 (2014) J.A. Sheikh, J. Dobaczewski, P. Ring, L.M. Robledo, C. Yannouleas, Symmetry restoration in mean-field approaches. J. Phys. G: Nucl. Part. Phys. 48(12), 123001 (2021) T.H.R. Skyrme, CVII. The nuclear surface. Philos. Mag.: J. Theor. Exp. Appl. Phys. 1(11), 1043– 1054 (1956) T.H.R. Skyrme, The effective nuclear potential. Nucl. Phys. 9(4), 615–634 (1958) M. Valiev, G.W. Fernando, Generalized Kohn-Sham Density-Functional Theory via Effective Action Formalism. arXiv:cond-mat/9702247 (1997) D. Vautherin, D.M. Brink, Hartree-Fock calculations with Skyrme’s interaction. I. Spherical nuclei. Phys. Rev. C 5, 626–647 (1972) V.F. Weisskopf, The problem of an effective mass in nuclear matter. Nucl. Phys. 3(3), 423–432 (1957) H.S. Yu, S.L. Li, D.G. Truhlar, Perspective: Kohn-sham density functional theory descending a staircase. J. Chem. Phys. 145(13), 130901 (2016)

Relativistic Density-Functional Theories

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advantages of Relativistic DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Achievements of Relativistic DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic DFT in This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirac Equation with Scalar and Vector Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meson-Exchange Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Matter with the σ -ω Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear and Density-Dependent Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Point-Coupling Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter introduces the relativistic density-functional theory in nuclear physics. After a brief introduction on the history, advantages, and achievements of the relativistic density-functional theory, the solution of a Dirac equation with large scalar and vector potentials, which is the most essential ingredient of relativistic density functional theories, will be provided. The relativistic density functionals based on both meson-exchange and point-coupling interactions are formulated, and the corresponding relativistic Kohn-Sham equations are discussed. The applications of the relativistic density functional theory are preluded by the σ -ω model for infinite nuclear matter, which emphasizes the basic ideas

J. Meng () · P. Zhao School of Physics, Peking University, Beijing, China e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_15

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for the relativistic description of nuclear systems. The applications for finite nuclei are complemented with the nuclear landscape description by the modern relativistic density functional theory. Other interesting applications and related topics, e.g., pairing density functionals, exchange (Fock) correlations, nuclear excitations, and relativistic energy density functionals from first principle, are briefly discussed as well.

Introduction A Brief History Relativistic density functional theory takes into account Lorentz invariance, which is one of the basic symmetries of quantum chromo-dynamics (QCD), the basic theory of the strong interaction. It starts from a field theory Lagrangian. In the most popular version of relativistic density functional theory, the nucleus is described as a system of Dirac nucleons that interact with each other via the exchange of mesons with various relativistic spin and isospin quantum numbers. The scalar-isoscalar, vectorisoscalar, scalar-isovector, and vector-isovector mesons build the minimal set of fields that, together with the electromagnetic field, are necessary for a description of the bulk and single-particle properties of nuclei. Solving nuclear many-body systems based on relativistic field theories can be traced back to at least the early 1950s (Schiff 1951; Johnson and Teller 1955). In 1974, a relativistic, many-body, quantum field theory was developed by Walecka to discuss the properties of cold, condensed stellar objects, such as neutron stars. The original model contained neutrons, protons, and isoscalar, Lorentz scalar and vector mesons, which are coupled to the scalar density and the conserved nucleon currents, respectively. In the nonrelativistic limit, this gives rise to the Yukawa potential, which generates the main features of the nucleon-nucleon interaction, i.e., a strong, short-range repulsion and a medium-range attraction. The fluctuations in the meson fields were ignored, and thus, the meson fields could be replaced by their classical expectation values or mean fields. The Walecka model was not only used to study the equation of state (EOS) for nuclear matter but also applied to the bulk and single-particle properties of finite nuclei (Serot and Walecka 1986). The scalar and vector mean fields were found to be roughly several-hundred MeV at the saturation density for infinite nuclear matter. This provides automatically, for finite nuclear systems, large single-particle spin-orbit potentials, which are roughly the same size as the observed ones. In other words, the relativistic mean-field (RMF) model predicts the existence of the nuclear single-particle shell model, without any ad hoc adjustments to the spin-orbit force. Due to this success, there have been tremendous efforts to improve the original Walecka model for a better description of finite nuclei and infinite nuclear matter. The isovector-vector ρ meson and photon were included to distinguish protons and neutrons. The proper treatment of nuclear matter and surface properties requires a density dependence of the effective meson-nucleon vertices. It can be introduced

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in the Lagrangian by nonlinear meson self-interactions or by an explicit density dependence for the couplings parameters. The nonlinear self-couplings of the scalar σ field (Boguta and Bodmer 1977), vector ω field (Sugahara and Toki 1994), and isovector ρ field (Long et al. 2004) were incorporated in the RMF models, which reproduced bulk and single-particle nuclear observables, and are better than any other concurrent models. Because the exchange of heavy mesons is associated with short-distance dynamics that cannot be resolved at low energies, relativistic density functionals with point-coupling interactions have been proposed as an alternative. Here, in each channel (scalar-isoscalar, vector-isoscalar, scalar-isovector, and vector-isovector), the meson exchange is replaced by a corresponding local four-point interaction between the nucleons. Medium effects can be taken into account by including higher-order couplings or by assuming a density dependence of the strength parameters. Numerous efforts have been devoted to developing relativistic DFT based on the finite-range meson-exchange and the zero-range point-coupling models. The successes of the mean-field approximation in the RMF models could be understood in the framework of density functional theory (DFT), which is based on the original theorem of Hohenberg and Kohn (1964). It proves that for a manyparticle system, the ground-state density constitutes a “basic variable,” i.e., all properties of the system, in particular, its ground-state energy can be written as unique functionals of the ground-state density. Moreover, the ground-state density can be determined from the ground-state energy functional via the variational principle with respect only to the density. In a covariant generalization of DFT applied to nuclei, the ground-state energy of a nucleus is written as functionals of the scalar density and the baryon currents for nucleons. RMF models are analogs of the Kohn-Sham formalism of DFT (Kohn and Sham 1965) with the local scalar and vector fields appearing in the role of relativistic Kohn-Sham potentials (Serot and Walecka 1997). The DFT provides, in principle, a framework including all correlation effects if the exact density functional is identified, while the mean-field models approximate the exact functional using powers and gradients of auxiliary meson fields or nucleon densities.

Advantages of Relativistic DFT The relativistic DFT provides an efficient description of nuclei with the underlying large scalar and vector fields of the order of a few hundred MeV, which are hidden in the non-relativistic DFTs. As a result, the relativistic framework can provide a good description of nuclear matter and finite nuclei using the Brueckner techniques with only the bare nucleon-nucleon interactions (Shen et al. 2019). In the non-relativistic framework, however, three-nucleon interactions, which are still very uncertain, have to be introduced for a reasonable description of nuclear matter and/or finite nuclei. The nucleons are described by Dirac spinors in relativistic DFT, which automatically provides the spin degree of freedom and the large spin-orbit potentials in nuclear systems. In non-relativistic density functionals, one needs additional terms

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and parameters accounting for the spin-orbit interactions. Moreover, the hidden pseudo-spin symmetry, i.e., the quasi-degeneracy between single-particle orbitals with the spherical quantum numbers (n, l, j = l+1/2) and (n−1, l+2, j = l+3/2), has been recognized as a relativistic symmetry since the 1990s (Ginocchio 1997; Meng et al. 1998). Later on, the spin symmetry for anti-nucleon has also been proposed (Zhou et al. 2003), and many other special features and concepts have been introduced, see, e.g., a recent review article Liang et al. (2015). In relativistic DFT, the nucleon currents are connected with the time-odd parts of the Kohn-Sham potentials, i.e., the spatial-like components of the vector fields, and due to the Lorentz symmetry, they share the same coupling constant as the time-like components. Therefore, no new parameters are required for the time-odd fields in the relativistic DFT formulations, despite the fact that only time-even fields (timelike components of the vector fields) are finely constrained by fitting the nuclear density functionals to experimental ground-state properties. In the non-relativistic DFT, however, the time-odd fields show some ambiguity. The consistent treatment of time-odd fields in the relativistic DFT is especially important for describing nuclear excitations, such as nuclear rotations (Meng et al. 2013; Vretenar et al. 2005). The relativistic DFT could serve as the basis to study dense and hot nuclear matter, where relativity is certainly important. Moreover, for the structure of normal nuclei, it should be noted that the relativistic DFT is successful not because of the relativistic kinematics in nuclei but because of the strong relativistic fields in the nucleus describing the velocity and spin dependence in an appropriate way. This leads to a number of important effects, which are automatically reproduced in relativistic calculations (Ring 2012). One can certainly take into account these effects in non-relativistic calculations but might be at the expense of additional elusive parameters. Recently, a successful non-relativistic reduction of relativistic DFT was achieved via the similarity renormalization method (Ren and Zhao 2020), and this provides a possible means to bridge the relativistic and nonrelativistic DFTs in the future.

Achievements of Relativistic DFT Due to these advantages, relativistic DFT has received wide attention over the past decades and achieved a successful description of a large number of nuclear phenomena (Meng 2015; Lalazissis et al. 2004; Ring 1996; Meng et al. 2006; Vretenar et al. 2005; Nikši´c et al. 2011). The relativistic DFT has been successfully applied to describe a large variety of nuclear phenomena from nuclear matter to finite nuclei, from stable nuclei to extremely unstable ones, from spherical nuclei to abnormally deformed ones, from nuclear ground states to excited ones, from nuclear structure to reactions, and from normal nuclear matter to hyper nuclear matter with strangeness degrees of freedom. Moreover, the relativistic DFT has been widely used to investigate the equation of state for compact stars and provides important observables for nucleosynthesis in

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astrophysics. Due to the length limit, we cannot cover all these topics in this chapter, but the reader is referred to the literature including other chapters of this book for more discussions. Relativistic DFT can provide a good description for the ground-state properties of almost all nuclei in the whole nuclear chart. Most many-body correlations are taken into account via the mechanism of spontaneous symmetry breaking (Ring and Schuck 1980). Particular attentions have been focused on the neutron-rich exotic nuclei, where various novel phenomena including halo (Meng and Ring 1996, 1998; Zhou et al. 2010) and shell structure evolution (Long et al. 2009; Liu et al. 2020) have been studied. Constrained relativistic DFT incorporating various nuclear deformations has been developed, and it provides a powerful tool to investigate exotic deformation, shape isomers, shape coexistence, fission landscape, etc. In order to microscopically predict the chiral nuclei (Frauendorf and Meng 1997), which host a pair of nearly degenerate ΔI = 1 bands with the same parity, the adiabatic and configuration-fixed constrained triaxial relativistic DFT was developed to search for the existences of the triaxial shape as well as the high-j proton and neutron configurations in nuclei. A new phenomenon, multiple chiral doublets (Mχ D), i.e., more than one pair of chiral doublet bands in one single nucleus, was suggested for 106 Rh based on the triaxial deformations and their corresponding proton and neutron configurations. After the Mχ D prediction, lots of efforts were made to search for its existence, and it was verified in the nucleus 133 Ce (Ayangeakaa et al. 2013). Moreover, the multi-dimensionally constrained relativistic DFT implemented in the harmonic oscillator space allows for a description of nuclear states with both reflection symmetry and axial symmetry breaking (Lu et al. 2012, 2014; Zhou 2016). Recently, significant progresses for a direct solution of the relativistic DFT equations in threedimensional coordinate space, which can describe any nuclear shape, have also been achieved (Tanimura et al. 2015; Ren et al. 2017, 2019; Li et al. 2020). The relativistic DFT can be transformed to a rotating frame (Koepf and Ring 1989; Madokoro and Matsuzaki 1997) to describe the nuclear rotational excitation with a static mean-field (König and Ring 1993; Afanasjev et al. 1999; Meng et al. 2013; P. Zhao 2015). In particular, the novel rotational modes including magnetic and antimagnetic rotations (Frauendorf 2001) and chiral rotations (Frauendorf and Meng 1997) have been successfully described (Peng et al. 2008; Zhao et al. 2011a, b; Zhao 2017). Beyond mean-field correlations arisen from symmetry restoration and fluctuations around the mean-field equilibrium solutions have to be taken into account for describing nuclear low-lying spectra associated with, for instance, shape coexistence and shape phase transitions (Nikši´c et al. 2011). This can be achieved by expressing the total energy as a functional of all transition density matrices between various Slater determinants of different deformations and orientations (Nikši´c et al. 2006; Yao et al. 2010). Such calculations are numerically expensive for heavy nuclei, so a considerable simplification is achieved by using the constrained relativistic meanfield calculations for the derivation of a collective Hamiltonian (Nikši´c et al. 2009; Li et al. 2009). A recent new method to go beyond the mean field is the configuration

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interaction projected density functional theory, where one keeps all the advantages of the mean-field theory and goes beyond mean field through mixing of states with good symmetries in a shell-model way (Zhao et al. 2016). The relativistic random-phase approximation (RPA) has been formulated in the small-amplitude limit of the time-dependent relativistic DFT (Vretenar et al. 2005). It was found that a fully consistent inclusion of the Dirac sea of negative-energy states is necessary for a quantitative description of the giant resonances (Ring et al. 2001). Moreover, the exchange (Fock) terms also play an essential role for the self-consistency of the framework (Liang et al. 2008). The relativistic RPA has been successfully applied to study nuclear compression modes, multipole giant resonances, β decays, electron captures, etc. In addition, the relativistic particlevibrational coupling model has been developed for the decay width of the excitations as well as the fragmentation of single-particle states (Litvinova and Ring 2006). With the great successes of the relativistic DFTs over the years, there has been growing interest to examine their applicability in understanding nucleosynthesis (Sun et al. 2008), the structure of neutron stars (Horowitz and Piekarewicz 2001; Fattoyev et al. 2010, 2018; Tong et al. 2020), the neutrinoless double beta decays (Song et al. 2014), etc.

Relativistic DFT in This Chapter In the spirit of this book, this chapter will be focused on the ideas and concepts in relativistic DFT and will discuss its simplest application. We will start with the solution of a Dirac equation with scalar and vector potentials, which is the most essential ingredient of relativistic DFT. Then, the framework of the relativistic DFT is introduced in terms of two typical functionals, meson-exchange and pointcoupling ones. The application of the σ -ω model for infinite nuclear matter is represented to give the basic ideas of a relativistic description. Among numerous successful applications of the relativistic DFT for finite nuclei, only the description of nuclear landscape is illustrated in this chapter. More interesting topics and applications are provided in the part of further reading with related references.

Dirac Equation with Scalar and Vector Potentials In relativistic density functional theory, the most essential ingredient is the nucleon wave function, which is the solution of a Dirac equation with large scalar and vector potentials. Before introducing the relativistic density functional theory, the solution including spectra and wave functions for a Dirac particle will be discussed in this section. The Dirac equation reads, [α · p + V (r) + β (M + S(r))] ψi (r) = εi ψi (r),

(1)

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where V (r) is the time component of the vector potential, S(r) is the scalar potential, and M is the nucleon mass. The natural units h¯ = c = 1 are adopted. The matrices α and β can be written as  α=

0σ σ 0



 ,

β=

I 0 0 −I

 ,

(2)

where σ is the Pauli matrix, and I denotes a unit matrix. For nuclei with spherical shape, which are true for most nuclei with the proton and/or neutron numbers being the magic numbers, i.e., 8, 20, 28, 50, 82, 126, . . ., the potentials V (r) and S(r) are spherical. It is convenient to work with the spherical polar coordinates. The kinetic energy operator α · p can be written as 1 ∂ − α · (er × l) ∂r r αr ∂ = −iαr + i (σ · l), ∂r r

α · p = −iα · er

(3)

where the identity, (σ · A)(σ · B) = A · B + iσ · (A × B), is used. The single-particle wave function ψ(r) can be written as the following ansatz by separating the radial and angular parts:  ψ(r) =

i Gnκr (r) Fnκ (r) r σr

 Yjl m (θ, ϕ),

(4)

with σr = σ · er . Here, Gnκ (r) and Fnκ (r) are respectively the large and small components of the radial wave functions with κ = (−1)j +l+1/2 (j + 1/2), while Yjl m (θ, ϕ) is the spinor spherical harmonics (Varshalovich et al. 1988). Because 

σr Yjl m (θ, ϕ) = −Yjl m (θ, ϕ)

(5)

with l  = 2j − l, the single-particle wave functions in Eq.(4) can also be expressed as 1 ψ(r) = r



iGnκ (r)Yjl m (θ, ϕ)  −Fnκ (r)Yjl m (θ, ϕ)

 .

(6)

From the equations (3) and (6), the radial Dirac equation can be derived from Eq. (1) as

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⎞ κ d     − + Gnκ (r) ⎟ Gnκ (r) ⎜ V (r) + S(r) + M dr r = εnκ , ⎠ ⎝ κ d Fnκ (r) Fnκ (r) + V (r) − S(r) − M dr r ⎛

(7)

where (σ · l)Yjl m = −(κ + 1)Yjl m is used. In Eq. (7), if the radial potentials V (r) and S(r) are known, e.g., a WoodsSaxon potential, the radial Dirac equation can be easily solved for given κ, and the eigenvalue εnκ and the corresponding radial wave functions Gnκ (r) and Fnκ (r) can be obtained. The quantum number n labels the energy and relates to the number of nodes in the radial wave function Gnκ (r). In Fig. 1, the single-particle potentials for neutrons in 208 Pb are shown, which are calculated by the relativistic density functional PC-PK1 (Zhao et al. 2010). There are two distinct potentials in the positive and negative energy regimes. The nucleons are in the positive potential V + S + M of several-dozen MeV, while the anti-nucleons are in the negative potential V − S − M of a few-hundred MeV. The large potential in the negative-energy regime is related to the appearance of the large spin-orbit potential in nuclei. The solution of the radial Dirac equation (7) with given radial potentials V (r) and S(r) can be carried out by the shooting method, which is one of the most efficient methods for solving the coupled differential equations. The r space is discretized in a box with the size R. For a single-particle state with nκ, the procedures to solve Eq. (7) are as follows (Meng 1998): 0 , which may start at the depth of the 1. Choosing an initial single-particle energy εnκ potential V + S + M; 2. From the proper boundary conditions, G(r = 0) and F (r = 0), integrating Eq. (7) outward from r = 0 to a matching point r = rmatch numerically, for example, by the Runge-Kutta algorithms. 3. From the proper boundary conditions, G(r = R) and F (r = R), integrating Eq. (7) inward from r = R to the matching point r = rmatch numerically. 4. The two sets of solution for G(r = rmatch ) and F (r = rmatch ) from the last two steps are used to construct the following matching matrix:



G(i) G(o) F (i) F (o)

 ,

(8)

where G(i) and F (i) represent the solutions obtained from the inward integration, while G(o) and F (o) represent those from the outward integration. 5. The single-particle energy εnκ is then determined by requiring the determinant of the matching matrix vanishes. The quantum number n − 1 is the number of nodes for the corresponding radial wave function Gnκ (r). From the single-particle potentials in Fig. 1 for neutrons in 208 Pb, all singleparticle energies and their corresponding radial wave functions can be obtained.

56 Relativistic Density Functional Theories

Fig. 1 Single-particle potentials for neutrons in functional PC-PK1 (Zhao et al. 2010).

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208 Pb

calculated with the relativistic density

There are 126 neutrons in 208 Pb that occupy the single-particle states by the order of energy. In Fig. 2, the single-particle states below the threshold for neutron in 208 Pb by the relativistic density functional PC-PK1 (Zhao et al. 2010) are shown, where the neutron mass M has been taken as reference. The states with a degeneracy of 2j + 1 are labeled by the quantum number n, orbital angular momentum l, and total angular momentum j of the large components of the Dirac spinors. The large spin-orbit splittings in the spectrum are given automatically from the solutions of the Dirac equation. Due to the large splitting between the spin-orbit doublets 1i13/2 and 1i11/2 , only the lower state 1i13/2 is occupied, and this forms the neutron magic number N = 126.

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V(r)+S(r) [MeV]

0

-20

-40

4s1/2

3s1/2

2s1/2

2g7/2

3d3/2

1i11/2 1j15/2

3p1/2 3d5/2 2f 5/2 2g 9/2 1h9/2 3p3/2 1i13/2 2d3/2 2f7/2 1g7/2 1h11/2 2p1/2 2d5/2 1f5/2 1g 9/2

2p3/2

1d3/2 1f7/2 1p1/2 1d5/2

-60

-80 0

1s1/2 1p 3/2

2

208Pb

4

6

8

Neutron PC-PK1 10

12

r [fm] Fig. 2 Single-particle states below threshold for neutrons in 208 Pb calculated by the relativistic density functional PC-PK1 (Zhao et al. 2010). The potential V (r) + S(r) is also shown as the solid line.

As examples, the single-particle wave functions of the 2s1/2 and 1d3/2 states, including both the upper G(r) and lower F (r) components, are illustrated in Fig. 3. It is seen that the upper components are considerably larger than the lower ones. The upper component of the single-particle state 2s1/2 has one node, and that of 1d3/2 has no node, so there is a distinct radial shape. However, their lower components are almost identical except at the surface region. This is actually associated with the origin and explanation of the pseudospin symmetry in the single-particle spectrum in nuclei (Ginocchio 1999, 2005; Liang et al. 2015). In the language of pseudospin, the states 2s1/2 and 1d3/2 are actually a pair of pseudospin doublets with the corresponding pseudo-orbital angular momentum l  = 2j − l = 1 as shown in Eq. (6), which is nothing but the orbital angular momentum of the lower component of the Dirac spinor. By removing the upper component of the Dirac spinor in Eq. (6), the deducing equation in the lower component of the Dirac spinor will explain the origin of the pseudospin symmetry in the single-particle spectrum and the spin symmetry in the single antiparticle spectrum (Meng et al. 1998; Zhou et al. 2003).

Meson-Exchange Functionals The relativistic density functional of nuclei is usually derived from a field Lagrangian obeying necessary symmetries, such as Lorentz invariance, locality, and causality. The Walecka model (Walecka 1974) provides an ideal starting point for this purpose, where the interactions between nucleons are provided by the exchange of mesons with proper relativistic quantum numbers. The obtained relativistic Kohn-Sham equation is the Dirac equation with the Hamiltonian derived

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Fig. 3 Neutron single-particle wave functions of the 2s1/2 and 1d3/2 states in 208 Pb calculated by the relativistic density functional PC-PK1 (Zhao et al. 2010).

from the energy functional. In this derivation, the meson fields are not quantized but treated in a classical manner. As a result, the obtained Dirac equation contains only Hartree terms, and the Fock terms are neglected. The nucleon fields are quantized, while the effects of the Dirac sea are neglected, i.e., the vacuum polarization is not taken into account. In principle, both the effects of Fock terms and Dirac sea should be considered in the calculations. However, because the energy density functional here is anyhow phenomenological, it is reasonable to assume that these effects can be absorbed in a phenomenological way in the coupling constants of the model justified by experimentally observables. Nevertheless, there are also numerous efforts to study the role of the Fock terms (Long et al. 2007; Wang et al. 2020; Liu et al. 2020) and Dirac sea (Chin 1977; Zhu et al. 1994). It turns out that the Walecka model is very useful for the implementation of relativistic DFT for nuclei. However, it was soon recognized that the model with only linear meson-nucleon couplings does not provide the proper density dependence of nuclear matter. The incompressibility is much too large, and the nuclear surface properties are not described properly. Therefore, an additional density dependence has been introduced either by nonlinear couplings in the meson sector (Boguta and Bodmer 1977; Sugahara and Toki 1994; Long et al. 2004) or by an explicit density dependence of the meson-nucleon coupling vertices (Lenske and

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Fuchs 1995; Typel and Wolter 1999; Long et al. 2004; Lalazissis et al. 2005). In this section, we will first introduce the simplest case of the linear σ -ω model with the application to infinite nuclear matter, and then the nonlinear couplings as well as the density-dependent couplings.

Nuclear Matter with the σ -ω Model Nuclear matter is defined as a static, homogeneous infinite system of nucleons, and the Coulomb interaction is neglected. It is a very useful ideal system for exploring both many-body effects and nuclear interactions for nuclear systems. The starting point of the σ -ω model for nuclear matter is a Lagrangian density of the form

L = ψ¯ iγ μ ∂μ − M − gσ σ − gω γ μ ωμ ψ 1 1 1 1 + ∂ μ σ ∂μ σ − m2σ σ 2 − Ω μν Ωμν + m2ω ωμ ωμ , 2 2 4 2

(9)

where M, mσ , and mω are the masses of nucleons, σ meson, and ω meson, respectively, while gσ and gω are the coupling constants of the nucleons and the mesons. Here, Ω μν = ∂ μ ων − ∂ ν ωμ ,

(10)

is the field tensor for the ω meson. In the following, Greek indices μ and ν run over the Minkowski indices 0, 1, 2, 3 or t, x, y, z, while Roman indices i, j , etc. denote the spatial components. In this Lagrangian density, the isoscalar-scalar σ meson provides the mid-range and long-range attractive parts of the nuclear interactions, whereas the short-range repulsive part is provided by the isoscalar-vector ω meson. For practical applications, the field operators of the σ and ω mesons are replaced with their expectation values, i.e., σ = σ , ω = ω, which is the so-called mean-field approximation. As a result, the mesons serve only to introduce the mean potentials in the nucleus, and the nucleons move independently in these potentials. Therefore, the nucleon field operator can be expanded in terms of the single-particle states with the index k as ψ(x) =



ψk (x)ck

k

and

ψ † (x) =



ψk† (x)ck† ,

(11)

k

where ck is the annihilation operator for a nucleon in the state k, and ψk is the corresponding single-particle wave function. The operators ck and its conjugate ck† satisfy the anticommutation rules, {ck , ck† } = δkk 

and

{ck , ck  } = {ck† , ck† } = 0.

(12)

56 Relativistic Density Functional Theories

2123

The ground state can be constructed as |Φ =



ci† |0

with Φ|Φ = 1.

(13)

i0

ν ρ ν∗ ν∗ ρ ν  ρ12 ρ21 2 1 1 2 − , ω − ων + iδ ω + ων − iδ

(6)

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being the spectral expansion over the exact excited states |ν of the many-body system with the energies ων = Eν − E0 measured from the ground-state energy. The residues of this expansion are the products of the transition densities ν = 0|ψ2† ψ1 |ν, ρ12

(7)

which represent the weights of the pure particle-hole configurations in the singleparticle basis {1} on top of the ground state |0 in the exact excited states |ν. The EOM for the response function can be generated by the differentiation of Eq. (1) with respect to the time arguments. Differentiation with respect to t leads to (i∂t +ε12 )R12,1 2 (t −t  ) = δ(t −t  )N121 2 +iT [V , ψ1† ψ2 ](t)(ψ2† ψ1 )(t  ),

(8)

where the norm kernel is introduced as N121 2 = [ψ1† ψ2 , ψ2† ψ1 ] = δ22 ψ1† ψ1  − δ11 ψ2† ψ2 .

(9)

With the diagonal one-body density matrix, the norm simplifies to the form of N121 2 = δ11 δ22 (n1 − n2 ) ≡ δ11 δ22 N12 , where n1 = ψ1† ψ1  is identified with the occupancy of the fermionic state 1. In Eq. (8) and in the following, ε12 = ε1 −ε2 with ε1 and ε2 being the eigenvalues of the one-body part of the Hamiltonian (4). The differentiation of the last term on the right-hand side of Eq. (8) with respect to t  generates the second EOM, ← − iT [V , ψ1† ψ2 ](t)(ψ2† ψ1 )(t  )(−i ∂t  − ε2 1 ) = − δ(t − t  )[[V , ψ1† ψ2 ], ψ2† ψ1 ] + iT [V , ψ1† ψ2 ](t)[V , ψ2† ψ1 ](t  ).

(10)

Combining Eqs. (10) and (8), after the Fourier transformation to the energy domain, one obtains 

(0)

R12,1 2 (ω) = R12,1 2 (ω) +

343 4

(0)

(0)

R12,34 (ω)T34,3 4 (ω)R3 4 ,1 2 (ω),

(11)

with the uncorrelated particle-hole response R (0) (ω), (0) R12,1  2 (ω) =

n1 − n2 N121 2 = δ11 δ22 , ω − ε21 ω − ε21

(12)

and the T -matrix T (ω), which is the Fourier image of  T12,1 2 (t − t  ) = N12−1 − δ(t − t  )[[V , ψ1† ψ2 ], ψ2† ψ1 ]  + iT [V , ψ1† ψ2 ](t)[V , ψ2† ψ1 ](t  ) N1−1  2 .

(13)

57 Model for Collective Vibration

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Thereby, the T -matrix is decomposed naturally into the instantaneous (static) T (0) and time-dependent (dynamical) T (r) terms,   (0) (r) −1   T12,1 2 (t − t  ) = N˜121  2 T12,1 2 δ(t − t ) + T12,1 2 (t − t ) ,

(14)

with N˜121 2 = N12 N1 2 and (0) † † T12,1  2 = −[[V , ψ1 ψ2 ], ψ2 ψ1 ],

T12,1 2 (t − t  ) = iT [V , ψ1† ψ2 ](t)[V , ψ2† ψ1 ](t  ), (r)

(15)

where the superscript “(r)” indicates the retarded character of the dynamical term. It is convenient to further transform Eq. (11) into a formally closed equation for R(ω), similar to the Dyson equation for the one-fermion propagators, by introducing the kernel K(ω) irreducible with respect to the uncorrelated particle-hole response R (0) , i.e., R(ω) = R (0) (ω) + R (0) (ω)K(ω)R(ω),

(16)

T (ω) = K(ω) + K(ω)R (0) (ω)T (ω).

(17)

where

In other words, K(ω) = T irr (ω), and it can also be decomposed into the instantaneous and time-dependent parts as   K(t − t  ) = N˜ −1 K (0) δ(t − t  ) + K (r) (t − t  ) ,

(18)

with K (0) = T (0) ,

K (r) (t − t  ) = T (r)irr (t − t  ).

(19)

In a complete analogy to the case of one-fermion EOM, the decomposition of the interaction kernel in Eqs. (14) and (18) into the static and time-dependent, or dynamical, parts is a generic feature of the in-medium interaction in the particle-hole channel and the direct consequence of the time independence of the bare interaction V of Eq. (5). After evaluating the commutators in Eq. (15) and introducing the two-fermion density (2)

ρij,kl = ψk† ψl† ψj ψi  = ρik ρj l − ρil ρj k + σij,kl ,

(20)

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where σij,kl represents its fully correlated part and the Roman indices have the same meaning as the number indices, the static kernel takes the form of (0)

K12,1 2 = N12 v¯21 12 N1 2 +



(2)

v¯2j 2 k σ1 k,1j +



jk

1 − δ11 2 −

1 2

 j kl

(2)

v¯1 k1j σ2j,2 k

jk

1 (2) δ22 v¯2j kl σkl,2 j − 2 (2)

v¯ij 2 1 σ1 2,ij −

ij

1 2



v¯j i1k σ1(2)  k,j i

ij k (2)

v¯21 kl σkl,12 .

(21)

kl

In this form, the first term isolates the contribution from the bare interaction, where the norm factors are compensated by their inverses in Eq. (14). The remaining terms of T 0 with the single-particle mean field are absorbed in the single-particle energies  (0) (0) † by the substitution ε1 → ε˜ 1 = ε1 + Σ11 , where Σ11 = ij v¯ 1i1j ψi ψj , in the uncorrelated response of Eq. (12). Thereby, Eq. (21) gives the exact form of the static kernel which, in the absence of correlations contained in the quantities σ (2) and T (r) , reduces the EOM (16) to the well-known RPA. The static part of the in-medium two-fermion interaction kernel in the particle-hole channel is shown diagrammatically in Fig. 1. The dynamical part T (r) of the kernel T , after evaluating the commutators of Eqs. (15), is decomposed as

2 2

T 1

(0)

2’

2

2’

v−

2’

2

− 1_

+

= 1’

− v 1

1

1’

− v −

1

− v

2’

2’

1

1’

− v 1

1’

2

2

1_ 2

2

2

1’

1’

−

− 1_

v−

2

2’

1_ 2

1’ 2’

1

(0)

Fig. 1 Diagrammatic representation of the static part of the kernel T12,1 2 defined by Eqs. (15) and (21). The lines with arrows denote fermionic propagators, while the rectangular blocks stand for the antisymmetrized nucleon-nucleon interaction v¯ and the two-body density ρ of Eq. (20). (The figure is adopted from Litvinova and Schuck 2019)

57 Model for Collective Vibration

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T12,1 2 (t − t  ) = T12,1 2 (t − t  ) + T12,1 2 (t − t  ) + T12,1 2 (t − t  ) + T12,1 2 (t − t  ), (22) where (r)

(r;11)

T12,1 2 (t − t  ) = − (r;11)

(r;12)

(r;21)

(r;22)

 i  † † v¯j 2kl T (ψ1† ψj† ψl ψk )(t) (ψm ψn ψp ψ1 )(t  )v¯nm2 p , (23) 4 mnp j kl

T12,1 2 (t − t  ) = (r;12)

 † i  v¯j 2kl T (ψ1† ψj† ψl ψk )(t) (ψ2 ψn† ψq ψp )(t  )v¯n1 pq , 4 npq

(24)

 i  † † v¯j i1k T (ψi† ψj† ψk ψ2 )(t) (ψm ψn ψp ψ1 )(t  )v¯nm2 p , 4 mnp

(25)

j kl

T12,1 2 (t − t  ) = (r;21)

ij k

T12,1 2 (t − t  ) = − (r;22)

 † i  v¯j i1k T (ψi† ψj† ψk ψ2 )(t) (ψ2 ψn† ψq ψp )(t  )v¯n1 pq , 4 npq

(26)

ij k

and shown diagrammatically in Fig. 2. The operator products in Eqs. (23), (24), (25), and (26) define the correlated 2p2h propagators G(543 1 , 5 4 31) = T (ψ1† ψ3† ψ5 ψ4 )(t)(ψ4† ψ5† ψ3 ψ1 )(t  ).

(27)

Thus, the particle-hole response function (1) is the solution of the integral equation R12,1 2 (ω) = R˜ 12,1 2 (ω) +

 343 4

where as

(r) K12,1 2 (ω)

=

(r)irr T12,1 2 (ω)

  (0) (r) R˜ 12,34 (ω) K34,3 4 + K34,3 4 (ω) R3 4 ,1 2 (ω),

(28) ˜ and the mean-field response R12,1 2 (ω) is defined

n˜ 1 − n˜ 2 , R˜ 12,1 2 (ω) = δ11 δ22 ω − ε˜ 21

(29)

with the occupancies n˜ i in the basis, which diagonalizes εi . Because of the two-time nature of the response function, its EOM reduces to the equation with only one energy (frequency) variable in the energy domain. The kernel K(ω), which is split into the static and dynamical parts, contains all the in-medium nucleonic correlations, which are, in principle, completely determined by the bare interaction v. ¯ However, in practice, consistent calculations of both the static and dynamical kernels are technically very difficult, and the difficulties are of a conceptual character. An accurate computation of the static kernel requires the correlated twobody density, and the dynamical kernel is based on the correlated 2p2h propagator. While the two-body density can be extracted from the response function in the

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H. Liang and E. Litvinova

4

(r;11)

2

T121’2’ =

4’

v−

G

3

5

v−

3’ 5’

1

1’ 4

(r;12)

2

T 121’2’ =

(r;21) T 121’2’ =

v−

2’ 3

3’

v−

G

5

5’

1

4’

2

5’ 4

1

v−

v−

4’

G

2’ 4

1

2’

1’

2

(r;22)

1’

3’

3

5

T 121’2’ =

2’

v−

4’

G

3

5

3’

v−

1’

5’

(r)  Fig. 2 Diagrammatic representation of the four components of the dynamical kernel T12,1  2 (t −t ) of Eqs. (23), (24), (25), and (26). The blocks G are associated with the corresponding time-ordered operator products

static limit, the correlated 2p2h propagator, in principle, requires a solution of the respective EOM. Alternatively, the 2p2h propagator can be approximated by various cluster expansions over lower-rank propagators. The solutions neglecting completely the dynamical kernel are the simplest approaches to the nuclear response, which correspond to the RPA. Typically the RPA is formulated in terms of the transition densities (7). Indeed, substituting the spectral form of R12,1 2 (ω) in Eq. (6) into Eq. (28) and dropping the dynamical kernel K (r) , in the vicinity of the pole ω → ων , one obtains the equation ν ρ12 =

n˜ 2 − n˜ 1  (0) ν K21,43 ρ34 ω − ε˜ 12

(30)

34

for the transition density ρ ν . Identifying its matrix elements with the Xν and ν = ρ ν and Y ν = ρ ν , Eq. (30) can be brought to the Y ν amplitudes as Xph ph ph hp conventional 2 × 2 block matrix form (Ring and Schuck 1980). Retaining the

57 Model for Collective Vibration

2153

full correlated static kernel leads to the self-consistent RPA (Schuck et al. 2021), while neglecting the terms with two-body correlations σ (2) implies the ordinary RPA. In the latter case, for a reasonable description of the observed spectra, the bare interaction should be replaced by an effective interaction, either schematic or derived from the DFT. Such interactions are designed to take into account the neglected correlations in a static approximation by absorbing them in the parameters. This type of approaches comprises most of the modern applications of RPA. The quasiparticle RPA (QRPA) (Ring and Schuck 1980) can be obtained analogously in the basis of the Bogoliubov quasiparticles (Bogoljubov et al. 1958). Another type of approaches goes beyond (Q)RPA by retaining also the dynamical kernel K (r) . These approaches can be further classified by the approximations employed for the treatment of this kernel, which is irreducible in the particlehole channel. The possible cluster expansions of the generic 2p2h propagator (27) entering all the four terms (23), (24), (25), and (26), which satisfy this condition and do not involve the correlated propagators of more than two fermions, are the following: (i) The completely uncorrelated factorization reads G(0)irr (543 1 , 5 4 31) = T (ψ1† ψ3† )(t)(ψ3 ψ1 )(t  )0 ×T (ψ5 ψ4 )(t)(ψ4† ψ5† )(t  )0 ,

(31)

i.e., it is a product of two uncorrelated antisymmetrized propagators, such as T (ψ1† ψ3† )(t)(ψ3 ψ1 )(t  )0 = T ψ1† (t)ψ1 (t  )0 T ψ3† (t)ψ3 (t  )0 −T ψ1† (t)ψ3 (t  )0 T ψ3† (t)ψ1 (t  )0 ,

(32)

where the subscript “0” indicates the uncorrelated character of the quantity. This type of dynamical kernel is shown diagrammatically in Fig. 3 for the term (r;11)irr T12,1 2 (t − t  ) of Eq. (23). It illustrates explicitly the 2p2h content of this approach, which corresponds to various versions of the second RPA (Yannouleas et al. 1983; Drozdz et al. 1990; Grasso and Gambacurta 2020; Papakonstantinou and Roth 2009; Raimondi and Barbieri 2019).

4 2

− v 3 5

1

G (0)irr

3’

4’ − v

4 2

2’

=

5’ 1’

− v 5

1

4’ 3

3’

− v

2’

− AS

5’ 1’

Fig. 3 Diagrammatic representation of the uncorrelated contributions (i) to the (11)-component (r;11)irr (t − t  ) irreducible with respect to the particle-hole propagator. of the dynamical kernel T12,1  2 “AS” includes all the antisymmetrized contributions

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H. Liang and E. Litvinova

(ii) Another possibility is to retain correlations in one of the two-body propagators in the factorization of the dynamical kernel. In this case, the irreducible correlated 2p2h propagator G reads G(c)irr (543 1 , 5 4 31) = T (ψ1† ψ3† )(t)(ψ3 ψ1 )(t  )T (ψ5 ψ4 )(t)(ψ4† ψ5† )(t  )0 + T (ψ1† ψ3† )(t)(ψ3 ψ1 )(t  )0 T (ψ5 ψ4 )(t)(ψ4† ψ5† )(t  ) + T (ψ1† ψ5 )(t)(ψ5† ψ1 )(t  )T (ψ3† ψ4 )(t)(ψ4† ψ3 )(t  )0 + T (ψ1† ψ5 )(t)(ψ5† ψ1 )(t  )0 T (ψ3† ψ4 )(t)(ψ4† ψ3 )(t  ) − AS,

(33)

where the upper index “(c)” indicates the presence of one two-fermion correlation function in each term of the expansion and “AS” stands for all the antisymmetric terms corresponding to the terms shown explicitly. This approximation to the (r;11)irr dynamical kernel is illustrated diagrammatically in Fig. 4 for the term T12,1 2 (t − t  ) of Eq. (23). It can be mapped to the class of approaches, which model the dynamical kernel in terms of the particle-vibration coupling. This mapping is exact, if the pairing γ μ± and normal g μ± phonon vertices as well as the respective (±) (±) propagators Δμ and Dμ are defined as μ(+)

γ12

=



μ

μ(−)

v1234 α34 ,

γ12

=



34 ) Δ(σ μ (ω) =

μ

β34 v3412 ,

34

σ (σ σ )

ω − σ (ωμ

− iδ)

,

ν(σ ) ν ν∗ g13 = δσ,+1 g13 + δσ,−1 g31 ,

ν g13 =



ν v¯1234 ρ42 ,

24

σ , Dν(σ ) (ω) = ω − σ (ων − iδ)

(34)

with σ = ±1, via the normal and anomalous transition densities ν ρ12 = 0|ψ2† ψ1 |ν,

μ

α12 = 0|ψ2 ψ1 |μ,

μ

β12 = 0|ψ2† ψ1† |μ.

(35)

This mapping is shown diagrammatically in Fig. 5, and it expresses the mechanism of emergence of the collective degrees of freedom associated with phonons. One can see, for instance, that, if this mapping is applied to the first and the third terms on the right-hand side of Fig. 4, together with an analogous mapping of the remaining components of T (r)irr , the dynamical kernel takes the form of the conventional NFTPVC approach (Bertsch et al. 1983; Kamerdzhiev et al. 1997; Mahaux et al. 1985; Tselyaev 1989; Litvinova et al. 2007, 2008; Litvinova and Wibowo 2018; Tselyaev et al. 2016; Niu et al. 2015; Egorova and Litvinova 2016; Robin and Litvinova

57 Model for Collective Vibration 4 2

− v 3

3’

G

4

4’ − v

2’

2

1 4

+

− v 5

3

(pp)

3’

5

2’

− v

1’

3’

− v 5’

2

+

− v

1’

3

3’

R

(hh)

− v

2’

5’ 1’

1

4’ 3

3’

− v

2’

− AS

5’

5

R 1

4’

5

4 2’

4 − v

2

+

5’

4’

R (ph) 3

4’

R

1

1’

2

− v

=

5’

5

2155

(ph)

1

1’

Fig. 4 Diagrammatic representation of the singly correlated approximation (ii) to the (11)(r;11)irr component of the dynamical kernel T12,1 2 (t − t  ) irreducible with respect to the particle-hole propagator. The rectangular blocks R (ph) correspond to the particle-hole response, while those containing R (pp) and R (hh) are the analogous correlated propagators of two particles and two holes, respectively

=

v

R (ph)

v

=

v

G (pp)

v

Fig. 5 The exact mapping of the phonon vertices (empty and filled circles) and propagators (wavy lines and double lines) onto the bare interaction (squares, antisymmetrized v¯ and plain v) and two-fermion correlation functions (rectangular blocks R (ph) and G(pp) ) in a diagrammatic form. Lines with arrows stand for fermionic particles (right arrows) and holes (left arrows). Top: normal (particle-hole) phonon. Bottom: pairing (particle-particle) phonon, as introduced in Eq. (34). (The figure is adopted from Litvinova and Schuck 2019)

2018). The remaining terms, such as the second and fourth terms on the right-hand side of Fig. 4, are not explicitly associated with the phonon exchange and represent a different type of correlations. (iii) More of the essential dynamics can be included if both of the pairwise propagators in the factorization of the dynamical kernel are exact and contain all correlations. The irreducible intermediate 2p2h propagator in this case reads G(cc)irr (543 1 , 5 4 31) = T (ψ1† ψ3† )(t)(ψ3 ψ1 )(t  )T (ψ5 ψ4 )(t)(ψ4† ψ5† )(t  ) + T (ψ1† ψ5 )(t)(ψ5† ψ1 )(t  )T (ψ3† ψ4 )(t)(ψ4† ψ3 )(t  ) − AS,

(36)

and its diagrammatic image is given in Fig. 6. Remarkably, this type of dynamical kernel absorbs the maximal amount of the correlations via the two-fermion correlation functions, and it takes the simplest form among the three dynamical

2156

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4

4’

− v

v−

5 1

H. Liang and E. Litvinova

3

G

4 2

2’

v−

3’

=

5’

4’

R (ph) 3’

3

v−

4 2’

+

5’

5

2

1

4’ 3

R

(pp)

3’

− v

2’

− AS

5’

5

R (ph) 1’

− v

R (hh) 1’

1

1’

Fig. 6 Same as in Fig. 4 but for the doubly correlated approximation (iii)

kernels defined by Eqs. (31), (33), and (36). It has the minimal amount of terms and eliminates the single-particle propagators. By the mapping shown in Fig. 5, this kernel can be related with the QPM or multiphonon models (Soloviev 1992; Ponomarev et al. 1999; Lo Iudice et al. 2012; Lenske and Tsoneva 2019) and with the extensions of the PVC approaches (Litvinova and Schuck 2019; Litvinova et al. 2010; Shen et al. 2020). It should be noted that most of the implementations of the dynamical kernels (i)–(iii) listed above used the effective interactions, instead of the bare interaction v¯ and, simultaneously, the same effective interactions to approximate the entire static kernel. This means that in such approaches, the many-body correlations are distributed differently between the static and dynamical kernels. Such a deficiency is, then, compensated by a subtraction procedure proposed in Tselyaev (2013). On the other hand, the class of beyond-(Q)RPA approaches based on the bare interaction (Knapp et al. 2014; Papakonstantinou and Roth 2009; Bacca et al. 2013; Raimondi and Barbieri 2019; Bacca 2014) does not demonstrate a consistent performance of the quality comparable to that of the approaches employing the effective interactions. The listed ab initio implementations have a common feature that the dynamical kernels are not related to the static kernels as it is required by the exact EOM. Instead, various preprocessing methods, such as the in-medium similarity renormalization group (Hergert et al. 2016) or Brückner’s G-matrix (Dickhoff and Barbieri 2004), are applied to the bare interaction, with a subsequent application of one of the standard many-body methods. Another difficulty may arise from the use of the reference mean fields in the numerical implementation. Such auxiliary mean fields have to be subtracted from the interacting part of the Hamiltonian and, thus, go through all the commutators in both the static and dynamical kernels. This is done, in particular, for the static one-fermion kernel in Dickhoff and Barbieri (2004). However, new terms with few-fermion propagators appear in the dynamical kernels that produce additional nontrivial nonlinearities in the resulting EOMs. These terms and their roles in the emergent properties of the dynamical kernels have to be carefully analyzed, both analytically and quantitatively.

Nuclear Spectral Calculations The response theory and the associated diagonalization approach are widely applied to the description of nuclear excited states. The response function (1) is partic-

57 Model for Collective Vibration

2157

ularly convenient as it is directly related to the excitation energies and transition probabilities, which is obvious from its spectral expansion (6). In experiments, the transition probabilities and their derivatives with respect to the energy variable are commonly extracted from the measured cross sections. The strength function as a result of the response to a given external field associated with the operator F can be defined as   S(ω) = |ν|F † |0|2 δ(ω − ων ) − |ν|F |0|2 δ(ω + ων ) , (37) ν>0

where the summation over ν runs through all the excited states |ν. The matrix element of the transition between the ground and excited states, in the case of onebody external field operator, reads ν|F † |0 =

  ∗ † ∗ ν∗ ν|F12 ψ2 ψ1 |0 = F12 ρ21 . 12

(38)

12

In the numerical implementations, the delta-functions in Eq. (37) are approximated by the Lorentz distribution, δ(ω − ων ) =

Δ 1 lim , π Δ→0 (ω − ων )2 + Δ2

(39)

so that S(ω) =

  Δ Δ 1 2 lim |ν|F † |0|2 − |ν|F |0| π Δ→0 ν (ω − ων )2 + Δ2 (ω + ων )2 + Δ2

=−

 |ν|F † |0|2 |ν|F |0|2  1 lim − π Δ→0 ν ω − ων + iΔ ω + ων + iΔ

=−

1 lim Π (ω). π Δ→0

(40)

The quantity Π (ω) is the polarizability of the many-body system,   |ν|F † |0|2 |ν|F |0|2   B¯ ν Bν − = − ω − ων + iΔ ω + ων + iΔ ω − ων + iΔ ω + ων + iΔ ν ν (41) with the transition probabilities defined as

Π (ω) =

Bν = |ν|F † |0|2 ,

B¯ ν = |ν|F |0|2 .

(42)

Thus, the strength function associated with the given external field operator F can be computed with the aid of the response function (1):

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H. Liang and E. Litvinova

SF (ω) = −

 1 lim F12 R12,1 2 (ω + iΔ)F1∗ 2 . π Δ→0  

(43)

121 2

In principle, the strength function (43) should reproduce all the states excited by the operator F , if the response function (1) is computed exactly. The latter is, however, a difficult task, as it follows from its EOM (28) and from the explicit expressions for the respective interaction kernels K (0) and K (r) . As discussed before, in practice, various approximations are applied to these kernels. The finite imaginary part Δ of the energy variable introduces a smooth envelope of the strength function; otherwise, it would look as a series of infinitely narrow peaks of infinite height. Smoothing of such a distribution is, thus, useful for representation purposes and has no physical meaning, if the theory is exact. Since the latter is practically never the case, the smearing parameter may absorb the effects that are not taken into account explicitly in the dynamical kernel K (r) and, thus, mimic the missing fragmentation effects. As the experimental data usually have finite-energy resolution, the resulting spectral peaks also have finite widths and heights. The common agreement is, thus, that, for a fair comparison between theory and experiment, the smearing parameter Δ used in the calculations should be comparable with the experimental energy resolution. As the physical observable is the transition probability, its value should not depend on the smearing parameter. Indeed, at the peaks of the strength function, the following relationship holds: Bν = lim π ΔS(ων ), Δ→0

(44)

which points out that the choice of the smearing parameter does not affect the transition probabilities, although Eq. (44) gets, obviously, less accurate with larger Δ. In the next subsections, the authors overview some examples of realistic nuclear response calculations, on both the RPA and beyond-RPA levels. The major focus is put on the recent theoretical achievements in the description of nuclear collective modes, while comprehensive reviews including early studies can be found in Ishkhanov and Kapitonov (2021), Paar et al. (2007), Roca-Maza and Paar (2018), Bertsch et al. (1983), Kamerdzhiev et al. (1997, 2004), Drozdz et al. (1990), and Garg and Colò (2018). Electromagnetic and Isoscalar Response The electromagnetic response is the most studied type of nuclear response as it can be induced by the most accessible experimental probes with photons (Ishkhanov and Kapitonov 2021; Harakeh and can der Woude 2001; Savran et al. 2013). The corresponding excitation operators are classified by the transferred angular momentum L and parity π . The electric operators have natural parity, i.e., π = (−1)L , and are defined as

57 Model for Collective Vibration

F00 = e

Z 

ri2 ,

F1M =

i=1

FLM = e

Z 

2159 Z N eN  eZ  ri Y1M (ˆri ) − ri Y1M (ˆri ), A A i=1

L ≥ 2,

ri L YLM (ˆri ),

i=1

(45)

i=1

where e stands for the proton charge, YLM (ˆr) are the spherical harmonics, and Z and N are the numbers of protons and neutrons in a nucleus, respectively. The expression for L = 1 contains the “kinematic” charges to account for the centerof-mass motion. Otherwise, the electric excitation operators (45) imply only the interaction of the projectiles with the charged protons and no interaction with the neutrons. The corresponding isoscalar operators with zero isospin transfer contain summations over all the nucleons and no electric charge, if they are not associated with the electric probes. The isoscalar dipole operator reads (0)

F1M =

A  (ri3 − ηri )Y1M (ˆri ),

(46)

i=1

where η = 5r 2 /3, and the second term in the brackets eliminates the spurious translational mode (Garg and Colò 2018). The superscript “(0)” indicates the isoscalar character of the operator, ΔT = 0, in contrast to the operators (45), which are often classified as isovector ones with ΔT = 1. The magnetic multipole operators are of the unnatural parity π = (−1)L+1 and of a more complex nature. Magnetic resonances are associated with the spin transfer and, generally, do not exhibit pronounced collectivity (Tselyaev et al. 2020). Therefore, the authors focus on the electric multipole transitions in this section. The response of strongly correlated systems to external perturbations manifests some generic features of the excitation spectra, which can be captured by a schematic model proposed by Brown and Bolsterli (1959), see also Ring and Schuck (1980). In this model, which adopts a separable effective multipole-multipole interaction, the RPA excitation spectrum contains two highly collective states, the low-frequency and the high-frequency ones. These two states are formed by the coherent particle-hole contributions from the uncorrelated ph-excitations (with respect to the Hartree-Fock or the phenomenological mean-field vacuum), when the interaction is switched on. The remaining ph states are mostly non-collective and lie between the two collective solutions. In the RPA calculations with more realistic interactions, the resulting spectrum depends on the nature of the residual interaction and on the quality of the numerical implementation. The general gross structure of the spectrum remains as in the Brown-Bolsterli model, but the main collective solutions undergo some fragmentation, the so-called Landau damping. Typically, the low-energy solutions are not very collective in the L = 0 and L = 1 channels but acquire collectivity at larger angular momentum transfer L ≥ 2. The

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high-energy ones are associated with collective oscillations, which involve all the nucleons. Taking into account the dynamical kernel, in any of the approximations discussed above, induces further fragmentation of the ph states due to their coupling to more complex configurations. This effect is a consequence of the pole structure of the dynamical kernel. The fine details of the obtained spectra vary depending on the approximation to K (r) . Figure 7 illustrates microscopic calculations of the isoscalar monopole, isoscalar quadrupole, and isovector dipole responses in 208 Pb of Tselyaev et al. (2016). The strength distributions were obtained with various Skyrme interactions in both the RPA and beyond-RPA model that includes the particle-vibration coupling in 208

strength

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strength

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Pb L=0, T=0 fraction EWSR / MeV

0

2

4

6

8 10 12 14 16 18 E [MeV]

5

10

15 20 E [MeV]

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Fig. 7 Isoscalar monopole (left), isoscalar quadrupole (middle), and isovector dipole (right) strength distributions in 208 Pb in the units of the energy-weighted sum rules per MeV for L = 0 and L = 2 and as a photoabsorption cross section [σ (E) defined in Eq. (47)] for the dipole mode. The RPA results are shown by the blue dashed lines, and the beyond-RPA PVC extensions are given by the red solid lines. The experimental data for the GDR (Belyaev et al. 1995) and for the GMR and GQR (Youngblood et al. 2004) are shown by the brown lines with error bars. (The figure is adopted from Tselyaev et al. 2016)

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the time blocking approximation (TBA), which results in the PVC dynamical kernel of the NFT type (ii). The calculations are done with a relatively large smearing parameter of the order of 1 MeV. The case of the quadrupole response is a clear illustration of the Brown-Bolsterli picture, while in the dipole and monopole channels the low-energy peak is not distinguishable. It is possibly too weak in the electromagnetic dipole channel, while in the monopole case it can be suppressed by the monopole selection rule. The giant resonances at high frequencies are, however, well pronounced showing up as broad peaks dominating the spectra. The fragmentation due to PVC shows up as a broadening of the giant resonance also in all the three channels; however, the effect is weaker for the monopole response. The latter occurs due to the partial cancellation between the self-energy K (r;11) , K (r;22) and phonon-exchange K (r;12) , K (r;21) terms, that is typical for the PVC kernels in the L = 0 channel (Bortignon et al. 1998; Litvinova et al. 2007). The use of a large smearing parameter in the calculations for the dipole response to reproduce the data, as compared with the experimental energy resolution, indicates the deficiency of fragmentation originating from the PVC mechanism. This can be attributed to the underestimated phonon collectivity in the Skyrme-RPA calculations employed for obtaining the phonon characteristics and/or to the deficiency of the PVC model space. While the choice of the smearing parameter looks more adequate with respect to the experimental resolution in the L = 0 and L = 2 cases, the RPA-PVC calculations underestimate the peak height of the GMR and, in some cases, the centroid of the GQR. Thus, the currently available Skyrme-RPA-PVC results for 208 Pb call for further refinement of the Skyrme interactions and/or of the employed many-body calculation schemes. Figure 8 shows the cross sections of the total dipole photoabsorption in four medium-mass spherical nuclei obtained within the relativistic QRPA (Paar et al. 2003) (RQRPA, black dashed curves) and the relativistic quasiparticle TBA (Litvinova et al. 2008) (RQTBA, red solid curves), compared with the neutron data (blue error bars) from National Nuclear Data Center. This cross section is defined as σE1 (E) =

16π 3 e2 E SE1 (E), 9hc ¯

(47)

i.e., with the additional energy factor in front of the strength distribution, which slightly emphasizes the high-energy part of the response. These calculations also employ the PVC dynamical kernel in its NFT form, which has been generalized to the superfluid phase, i.e., to the coupling between the superfluid quasiparticles and phonons (Litvinova et al. 2008; Litvinova and Tselyaev 2007). The in-medium interaction is of the effective meson-exchange origin and adjusted to bulk nuclear properties in the framework of the covariant DFT (Vretenar et al. 2005; Meng et al. 2006; Meng 2016) with the NL3 parametrization (Lalazissis et al. 1997). In the fully self-consistent calculation scheme, the RQRPA generally produces the dipole strength, which is mostly concentrated in a narrow energy region. The localization of the centroid is reproduced fairly well, as compared with the data.

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Fig. 8 Total dipole photoabsorption cross section in stable medium-mass nuclei. (The figure is adopted from Broglia and Zelevinsky 2013; Meng 2016)

The total transition probability is another characteristic of the GDR, which is typically reproduced well in the (Q)RPA approaches. The most robust related quantity is the energy-weighted sum rule (EWSR), SE1 =

 ν

Eν Bν =

9h¯ 2 e2 NZ , 8mp A

(48)

which is proportional to the cross section integrated over the energy variable. The right-hand side of Eq. (48) is calculated by transforming the sum into a double commutator of the dipole excitation operator and the system Hamiltonian, under the assumption that the interaction between nucleons has no momentum dependence. In this case, the potential energy part commutes with the excitation operator and, thus, does not contribute to the sum rule. The relation (48) is, therefore, valid for any Hamiltonian without momentum dependence in the two-body sector and known as Thomas-Reiche-Kuhn sum rule. Modern energy density functionals (EDFs), such as the Skyrme, Gogny, and relativistic ones, yield the effective interactions, which depend on the nucleonic momenta, so that a 10–20% or even larger enhancement of the dipole EWSR can be obtained in the (Q)RPA calculations (Trippa et al. 2008) as well as in experiments, where the measurements span sufficiently broad energy intervals (National Nuclear Data Center). Adding the dynamical kernels, which satisfy the consistency conditions between the self-energy K (r;11) , K (r;22) and the exchange K (r;12) , K (r;21) terms, should

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not violate the EWSR (Tselyaev 2007). In particular, the (quasi)particle-vibration coupling (QPVC) kernels of the NFT form (ii) satisfy this condition, if the numerical implementation is performed properly. Thus, the EWSR conservation serves as a very good test for such implementations. Accordingly, the energy centroid remains intact. The subtraction procedure (Tselyaev 2013), which is applied to eliminate the double counting of the QPVC effects in EDFs, induces a slight violation of the EWSR, because it modifies the static part of the kernel and pushes the centroid slightly upward, so that the resulting position of the major peak is back to its (Q)RPA position. Otherwise, the dynamical kernel alone shifts the major peak to lower energy. This is a desirable feature in the ab initio implementations, such as the second RPA of Papakonstantinou and Roth (2009). However, if an effective interaction is employed for the dynamical kernel, the major peak is already well positioned in (Q)RPA, so that its downward shift by the dynamical kernel is well compensated by the subtraction. This procedure is quite simple and consists of the replacement, K˜ (0) + K˜ (r) (ω) → K˜ (0) + δ K˜ (r) (ω) = K˜ (0) + K˜ (r) (ω) − K˜ (r) (0),

(49)

i.e., the dynamical kernel in the static approximation ω = 0 is subtracted from the dynamical kernel itself. The energy-independent combination K˜ (0) − K˜ (r) (0), thus, stands for the effective interaction freed from the long-range effects taken into account by K˜ (r) (ω). The “” sign in Eq. (49) marks the kernels, where the effective interaction is employed, that is, the entire static kernels K˜ (0) and the interaction matrix elements in the topologically equivalent dynamical kernels K˜ (r) (ω) computed in various approximations. One can further see from Fig. 8 that the coupling between the superfluid quasiparticles and phonons included within the RQTBA provides a sizable fragmentation of the GDR. Due to the inclusion of a large number of the phonon modes, the final strength distribution acquires nearly a Lorentzian shape, though relatively small values of the smearing parameter, Δ = 200 keV for the Sn isotopes and Δ = 400 keV for Sr and Zr, were used in both the RQRPA and RQTBA calculations. The choice of these parameters was based on the estimate of the continuum contribution, which was not included explicitly. In principle, the particle escaping to the continuum plays a role in the formation of the width of the high-frequency resonances above the particle emission threshold. The latter is the minimal energy, at which the nucleon emission is possible, often called nucleon binding, or separation, energy, and its typical value is ∼7–10 MeV for stable medium-mass and heavy nuclei. Loosely bound exotic nuclei with strong dominance of one type of nucleons (protons or neutrons) are characterized by lower separation energies for the excess nucleons. For example, in neutron-rich nuclei, neutrons are loosely bound and have a lower separation energy than protons and vice versa. The effect of the single-particle continuum in the (Q)RPA and beyond(Q)RPA calculations can be taken into account within the method first proposed in Shlomo and Bertsch (1975) for RPA, later extended to QRPA (Tselyaev et al. 2016; Kamerdzhiev et al. 1998; Hagino and Sagawa 2001; Matsuo 2002; Khan

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et al. 2002; Daoutidis and Ring 2009) and QRPA+QPVC (Litvinova and Tselyaev 2007). The complete inclusion of the single-particle continuum in these methods is achieved by employing the coordinate-space representation for the (Q)RPA propagator and the final EOM, while the QPVC part of the propagator in Litvinova and Tselyaev (2007) is transformed to the coordinate space via the single-particle wave functions. In Tselyaev et al. (2016), a modification of this method was proposed for the numerical solution of the response EOM in the discrete basis of the single-particle states with the box boundary condition. Both the original and modified methods are based on constructing the mean-field propagator from the regular and irregular single-particle wave functions as the mean-field solutions with the Coulomb asymptotics. The single-particle continuum included in the calculations presented in Fig. 7 does not play a very important role in the description of medium-mass and heavy nuclei, producing a typical continuum width of the order of 100 keV for each single peak in the spectrum above the particle threshold, although the role of continuum increases dramatically in light nuclei, especially the loosely bound ones. Clear examples are given in Tselyaev et al. (2016). The inclusion of multiparticle continuum in (Q)RPA and its extensions was not addressed in the nuclear physics literature until now, though effects of two-nucleon evaporation should become sensible already at the GDR centroid energy and further escape of more nucleons can affect the GDR’s high-energy shoulder. Further details of the calculations presented in Figs. 7 and 8 can be found in Litvinova et al. (2008) and Tselyaev et al. (2016), respectively. The described models with various types of dynamical kernels, although quite successful, still have not reached the spectroscopic accuracy of even hundreds of keV in the description of excitation spectra and other properties of mediummass and heavy nuclei, which can be associated with these spectra. Despite the convincing progress on both the beyond-(Q)RPA methods and the EDFs, it remains unclear to what degree the lack of accuracy should be attributed to the imperfections of the EDFs, truncations in the beyond-(Q)RPA calculation schemes, unavoidable with the present computational capabilities, or principal limitations of these manybody methods. Up until now, the best-quality nuclear response calculations beyond (Q)RPA include up to the (correlated) 2p2h (Litvinova et al. 2008, 2010; Niu et al. 2015; Gambacurta et al. 2015; Robin and Litvinova 2016, 2019), in rare cases 3p3h (Lo Iudice et al. 2012; Litvinova and Schuck 2019; Lenske and Tsoneva 2019; Ponomarev 1999; Savran et al. 2011), configuration complexity with the current computational capabilities. Direct comparison between the 2p2h and 3p3h calculations within the same implementation schemes indicates that the latter higher-rank configurations (i) improve the results noticeably and (ii) the effect of the inclusion of 3p3h configurations, in addition to 2p2h ones, is weaker than the effect of the inclusion of 2p2h configurations beyond (Q)RPA. The former points to the importance of the 3p3h configurations, and the latter means that the theory exhibits saturation with respect to the configuration complexity. An example is given in Fig. 9, where 3p3h configurations were included in the “two quasiparticles coupled to two phonons” (2q ⊗ 2phonon) scheme for the electromagnetic dipole response of 42,48 Ca. This was achieved by implementing the

57 Model for Collective Vibration

2165

250

42

σ [mb]

200

R(Q)RPA: 2q R(Q)TBA: 2q+phonon

Ca

48

3

EOM/R(Q)TBA : 2q+2phonon Exp: NNDC

EME1

Ca

EME1

150

100

50

0

5

10

15

20

E [MeV]

25

5

10

15

20

25

30

E [MeV]

Fig. 9 Giant dipole resonance in 42,48 Ca calculated within the R(Q)RPA, R(Q)TBA, and EOM/R(Q)TBA3 approaches (Litvinova and Schuck 2019), in comparison with the experimental data (Erokhova et al. 2003; National Nuclear Data Center). (The figure is adopted from Litvinova and Schuck 2019)

dynamical QPVC kernel of type (iii) in an iterative cycle. Namely, after computing and selecting the most relevant RQRPA phonon modes (without dynamical kernels), the dynamical QPVC kernel (ii) was constructed, and the RQTBA response was calculated for the most relevant J π (J ≤ 6) channels of natural parity. After that, the obtained response functions were recycled in the dynamical kernel (iii) of the EOM for the dipole response. This scheme was originally proposed in Litvinova (2015) within the quasiparticle time blocking approximation and later re-derived starting from the bare Hamiltonian and implemented numerically in Litvinova and Schuck (2019). The approach was named EOM/RQTBA3 due to its construction. The total photoabsorption cross section obtained within EOM/RQTBA3 (red solid curves) is plotted in Fig. 9 together with the results of RQRPA (black dot-dashed curves), RQTBA (blue dashed curves), and experimental data (green curves and circles) of National Nuclear Data Center. The GDR in calcium isotopes was investigated within the RQTBA framework in Egorova and Litvinova (2016) with the focus on the role of the 2q ⊗ phonon configurations in the width of the GDR. It was found that these configurations result in the formation of the spreading width and improve significantly the agreement to data as compared to RQRPA. Nevertheless, although the authors used a large model space of the 2q ⊗ phonon configurations with the RQRPA phonons, the total width of the GDR was still underestimated. In addition, on the high-energy shoulder of the GDR, the cross sections were systematically underestimated. A similar situation was reported in Tselyaev et al. (2016), for QTBA calculations with various Skyrme forces. These observations pointed out that further refinement of the dynamical kernels may be necessary. Now with the EOM/RQTBA3 , taking into account more complex 2q ⊗ 2phonon configuration, we can see that these problems can be potentially resolved. Indeed, from Fig. 9 one can see that the new

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higher-rank configurations in EOM/RQTBA3 cause additional fragmentation of the GDR and, thus, intensify the spreading of the strength to both higher and lower energies. Technically, this is the consequence of the appearance of the new poles in the resulting response function. These new poles rearrange the energy balance of the strength distribution in both the low-energy and the higher-energy sectors, however, without violating the dipole EWSR (Litvinova and Schuck 2019). Response of nonspherical nuclei to external probes, in general, is more difficult to calculate microscopically. Already on the QRPA level, the 2q model space expands dramatically, as compared to the spherical case. The reason is the lifted degeneracy of j -orbitals, because the total angular momentum is not a good quantum number in nonspherical geometries. Therefore, QRPA calculations are numerically very expensive even in axially deformed nuclei (Arteaga and Ring 2008). In particular, such calculations require numerical evaluation of the enormous amount of matrix elements of the nucleon-nucleon interaction, which makes deformed QRPA prohibitively difficult even in the DFT frameworks. See also Péru and Goutte (2008) and Toivanen et al. (2010) for the studies along this direction. A very elegant numerical solution was proposed in Nakatsukasa et al. (2007), where the finiteamplitude method (FAM), avoiding direct computation of the interaction matrix elements, was developed and employed for RPA calculations of the response of deformed nuclei. Later on, the FAM-RPA was generalized to superfluid nuclei as FAM-QRPA (Oishi et al. 2016; Nikši´c et al. 2013; Kortelainen et al. 2015). An example of FAM-QRPA calculations for the GDR in axially deformed nuclei is shown in Fig. 10, in terms of the total dipole photoabsorption cross sections (Oishi et al. 2016). The calculations were performed with the Skyrme DFT. The available experimental data were also shown. In general, the double-hump shape of the GDR in axially deformed nuclei is attributed to the deformation, while the ratio of the peak energies corresponds to the ratio of the major axes of the nuclear ground-state ellipsoid. The main observation from the Skyrme FAM-QRPA calculations shown in Fig. 10 is that the typical frequencies of the GDR are fairly well reproduced. The width and the plateau top of the distribution are well understood as the total J = 1 strength is a sum of K = 0 and |K| = 1 modes excited on ground states with prolate deformations. The remaining discrepancies between the QRPA calculations and experimental data were attributed by the authors to the peculiarities of the Skyrme interaction and to the missing effects beyond QRPA. Indeed, since considerably larger smearing than the experimental energy resolution is needed in these FAM-QRPA calculations to reproduce the data, the effects beyond QRPA are necessary for further improvements. The work in this direction is under way (Zhang et al. 2022). Spin-Isospin Response The nuclear spin-isospin response, also known as the charge-exchange excitations, corresponds to the transitions from the ground state of the nucleus (N, Z) to the final states in the neighboring nuclei (N ∓ 1, Z ± 1) in the isospin lowering T− and raising T+ channels, respectively. These excitations can take place spontaneously,

57 Model for Collective Vibration

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Fig. 10 Dipole photoabsorption cross sections for Gd, Dy, and Er (left), and for Yb, Hf, and W (right) isotopes as a function of photon energy, obtained in the FAM-QRPA calculation with the Skyrme interaction and the smearing parameter Δ = 1.0 MeV. Calculations with Δ = 0.5 MeV are given for comparison (dotted lines). (The figure is adopted from Oishi et al. 2016)

e.g., in the famous β decays, or be induced by external fields, e.g., in the chargeexchange reactions, such as (p, n) or (3 He, t). Nuclear spin-isospin responses are categorized into different modes according to the nucleons with spin-up and spindown oscillating either in phase, the non-spin-flip modes with S = 0, or out of phase, the spin-flip modes with S = 1. The important modes, which have attracted an extensive attention experimentally and theoretically, include the isobaric analog state with S = 0, J π = 0+ , Gamow-Teller resonance with S = 1, J π = 1+ , and spin-dipole resonance with S = 1, J π = 0− , 1− , 2− (Osterfeld 1992; Ichimura et al. 2006; Paar et al. 2007; Roca-Maza and Paar 2018). The corresponding operators of these charge-exchange excitations read

± = FIAS

A 

τ± (i),

i=1 ± FGTR =

A  → [1 ⊗ − σ (i)]J =1 τ± (i), i=1

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± FSDR =

A  → [ri Y1 (i) ⊗ − σ (i)]J =(0,1,2) τ± (i),

(50)

i=1

where Y is the spherical harmonics and σ and τ are the Pauli matrices of spin and isospin degrees of freedom, respectively. Thecorresponding non-energy-weighted  sum rules (NEWSR), S − − S + = ν Bν− − ν Bν+ , are − + SIAS − SIAS = N − Z, − + SGTR − SGTR = 3(N − Z),  9  − + Nr 2 n − Zr 2 p , SSDR − SSDR = 4π

(51)

where the GTR one is the famous model-independent Ikeda sum rule, while the SDR one involves the root-mean-square radii of protons and neutrons and is considered to be an alternative way for measuring neutron skin thickness (Krasznahorkay et al. 1999; Yako et al. 2006). For neutron-rich nuclei, the excitations in the T+ channel are significantly suppressed by the Pauli principle, and thus S − alone approximately represents the NEWSR. The GTR, which is the most studied nuclear spin-isospin response, is related to both the spin-orbit and isospin properties of nuclear systems. Although this relationship is not direct and clouded by complex many-body correlations, experimental data on the GTR can be used to constrain the respective terms in the effective interactions and EDFs. For instance, one of the recently developed and widely used Skyrme effective interactions, SAMi (Roca-Maza et al. 2012), has acquired improved spin-isospin properties by achieving an accurate description of GTR peak energies. Meanwhile, in the relativistic framework, it is found that on the (Q)RPA level an accurate description of GTR peak energies can be achieved in a fully selfconsistent way by taking the Fock terms of the meson-exchange interactions into account (Liang et al. 2008; Niu et al. 2013, 2017). Overall, RPA and QRPA with effective interactions (Borzov 2003; Sarriguren 2013; Paar et al. 2004; Liang et al. 2008; Niu et al. 2013, 2017) produce reasonable results for the major GTR peak. However, reproducing the detailed strength distribution is impossible within these approaches neglecting the dynamical correlations. Moreover, since the total strength is constrained by the model-independent Ikeda sum rule, it is exhausted within the relatively narrow energy interval, because of the model space limitations of (Q)RPA. This causes, to a large extent, the wellknown quenching problem (Osterfeld 1992). The overall situation is similar to that with electromagnetic excitations, and the GTR is considerably affected by the effects beyond RPA. While SRPA calculations for the GTR have been reported already in 1990 (Drozdz et al. 1990), calculations with the (Q)PVC kernels based on modern density functionals, both relativistic NL3 (Marketin et al. 2012; Robin and Litvinova 2016, 2018, 2019) and nonrelativistic Skyrme (Niu et al. 2015), have become available more recently. Lately, SRPA calculations were also advanced to

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the Skyrme EDF framework (Gambacurta et al. 2020). Despite technical differences between the various implementations, all the extensions beyond (Q)RPA improve the description of the GTR considerably. In the cases of neutron-rich nuclei, where the low-energy part of the GTR spectrum is associated with spontaneous beta decay, the description of beta decay rates are improved by up to one or two orders of magnitude, compared to those obtained in (Q)RPA (Robin and Litvinova 2016; Litvinova et al. 2020). The role of QPVC effects is illustrated in Fig. 11 for the response of the neutron-rich tin isotopes 130,132,136 Sn to the GT− operator, obtained within the proton-neutron version of RQTBA (pnRQTBA) with the QPVC dynamical kernel, which was originally developed in Robin and Litvinova (2016). These calculations are compared to the proton-neutron RQRPA (pnRQRPA) without the dynamical kernel, and the presented spectra are displayed on the energy scales relative to the parent nuclei. The most general observation from these calculations is that QPVC leads to a similar degree of fragmentation as in non-charge-exchange channels, which is somewhat higher in nuclei with larger isospin asymmetry. In turn, this fragmentation redistributes the strength in the low-energy sector, in particular, in the Qβ energy window. This leads to faster beta decay in the pnRQTBA calculations, improving significantly the agreement with experimental data (National Nuclear Data Center), as compared to pnRQRPA. The corresponding half-lives are shown in the right panel of Fig. 11. More examples, details, and discussions are presented in Robin and Litvinova (2016). Furthermore, the pnRTBA has been generalized recently to finite temperature in Litvinova et al. (2020), which allowed applications to beta decay in stellar environments.

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Fig. 11 GT− strength distributions for 130,132,136 Sn nuclei at zero temperature in the pnRQTBA, compared to the pnRQRPA (a–c). Beta decay half-lives in neutron-rich tin isotopes extracted from the pnRQRPA (diamonds) and pnRQTBA (triangles) strength distributions, compared to data (circles) (National Nuclear Data Center) (d). (The figure is adopted from Litvinova et al. 2020)

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The most advanced calculations with the PVC kernel included, in addition to the standard NFT terms, also the ground-state correlations (GSC) caused by PVC (GSCP V C ). These correlations were introduced and discussed in detail, for instance, in Kamerdzhiev et al. (1997, 2004), where their role in the spin-flip magnetic dipole excitations was found significant. As the GTR also involves the spin-flip process, an important contribution from the GSC-PVC is expected. It can be especially significant in the GT+ branch in neutron-rich nuclei, where these correlations were found to be solely responsible for the unblocking mechanism (Robin and Litvinova 2019). An example is given in Fig. 12, where the GT± strength distributions in 90 Zr are shown in comparison with the data of Yako et al. (2005) and Wakasa et al. (1997). In the GT− branch of the response, the inclusion of the PVC effects within the pnRTBA leads to an overall fragmentation and broadening of the strength distribution, as compared to the pnRRPA (not shown). In the GT+ branch, in principle, the GSC of RPA (GSCRP A ) can unlock transitions from particle to hole states, but such transitions appear only above 7 MeV with very low probabilities. The inclusion of PVC in the pnRTBA with only the standard NFT forward-going diagrams in the PVC kernel induces almost no change. However, the inclusion of the GSCP V C associated with backward-going PVC processes has a very strong effect on the GT+ strength. These correlations cause fractional occupancies of the singleparticle states of the parent nucleus, which leads to new transitions from particle to

EXP GT pn-RTBA no GSCPVC GT pn-RTBA with GSCPVC GT+IVSM pn-RTBA no GSCPVC GT+IVSM pn-RTBA with GSCPVC

-1 SGT+(E) (MeV )

-1

SGT-(E) (MeV )

4

GT-

3 2 1

0 0.4

GT+

0.3 0.2 0.1 0.0

0

4

8

12

16

20

24

28

32

36

40

44

48

Excitation energy E (MeV) Fig. 12 GT strength distributions for the transitions 90 Zr → 90 Nb (top) and 90 Zr → 90 Y (bottom). The pure GT strength and the mixed GT+IVSM strength of pnRTBA without GSCP V C (dashed-dotted and dotted blue) and with GSCP V C (solid and dashed red) are displayed, in comparison with the experimental data (Yako et al. 2005; Wakasa et al. 1997). (The figure is adopted from Robin and Litvinova 2019)

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particle state and from hole to hole state. For instance, the peak around 4.5 MeV appears mainly due to the π 1g9/2 → ν1g7/2 and π 2p3/2 → ν2p1/2 transitions, with the corresponding absolute values of the transition densities (7) of 0.347 and 0.182, respectively. In the calculations shown in Fig. 12, the theoretical GT+ and GT− strength distributions were smeared with a parameter Δ = 2 and 1 MeV, respectively, to match the experimental energy resolutions. As in the case of electromagnetic excitations, the pnRRPA calculations do not provide a good agreement with data; therefore, they are not shown. In the GT− channel, the pnRTBA with GSCP V C demonstrates a good agreement with the data up to ∼25 MeV, except for a small mismatch of the position of the low-lying state. Remarkably, in the GT+ channel, the GSC induced by PVC are solely responsible for the appearance of both the lowenergy peak at 4 MeV and the higher-energy strength up to ∼50 MeV. Above the low-lying peak, even the pnRTBA GT+ strength alone largely underestimates the data. It is well known, however, that at large excitation energy contributions of the isovector spin-monopole (IVSM) mode become important. The data of Yako et al. (2005) and Wakasa et al. (1997), in particular, also contain the contribution of the IVSM excitations, which could not be disentangled from the GT transitions due to technical difficulties. The IVSM modes are generated by response to the operator  2 − → FI±V SM = i r (i) Σ (i)τ± (i), which should be mixed with the GT response, for instance, following the procedure of Terasaki (2018). It introduces the mixed  − → 2 operator Fα± = i [1 + αr (i)] Σ (i)τ± (i), where α is a parameter adjusted to reproduce the magnitude of the theoretical low-energy GT strength. In this way, the values α = 9.1 × 10−3 and α = 7.5 × 10−3 fm−2 were adopted for the GT+ and GT− branches, respectively. After that, as one can see from Fig. 12, the resulting strength above 25–30 MeV reasonably describes the data in the (p, n) branch, thus highlighting the importance of both the GSCP V C and the IVSM contribution. In the (n, p) channel, the results are also improved after adding the GSCP V C and the IVSM in pnRTBA, so that a very good agreement of the overall strength distribution is obtained also for GT− . Further details and discussions of this case can be found in Robin and Litvinova (2019). The nucleon-nucleon tensor force is another hot topic in the past two decades (Sagawa and Colò 2014; Otsuka et al. 2020). Figure 13 displays the SDR strength distributions in 208 Pb in the T− channel. Panels (a)–(c) and (e)–(f) show the J π = 0− , 1− , 2− multipoles, respectively, and panels (d) and (h) illustrate the total strength distributions. The experimental data with the multipole decomposition were obtained only recently with the polarized proton beam (Wakasa et al. 2012). It is remarkable that the centroid energies of SDR in 208 Pb are found to be E(1− ) ≈ E(2− ) < E(0− ), instead of E(2− ) < E(1− ) < E(0− ), conjectured by the most conventional shell model and RPA calculations. The study of Bai et al. (2010) analyzed the sensitivity of the SDR centroids to the properties of the tensor force. It is shown that the conventional Skyrme RPA calculations without the tensor force, in general, fail in reproducing the experimentally established relationship between the centroids of the three components of the SDR. Only when the tensor force with

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Fig. 13 Spin-dipole strength distributions in 208 Pb, calculated by Skyrme RPA without and with the tensor force. The discrete RPA results have been smoothed by Δ = 2 MeV and compared with the experimental data (Wakasa et al. 2012). (The figure is taken from Bai et al. 2010)

specific signs and strengths is included as, e.g., in the effective interactions T43 and SLy5+TW , the centroid of the 1− component is significantly pushed down, while that of the 2− component is slightly pushed up. In such a way, the experimental data on the centroids order are reproduced. In the relativistic framework, to include the tensor force, the Fock terms of the meson exchange must be taken into account. This is the relativistic HartreeFock (RHF) theory (Bouyssy et al. 1987; Long et al. 2006). It is, however, not straightforward to identify the tensor effects in the RHF theory, because the tensor

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force is mixed together with other components, such as the central and spin-orbit ones. For example, simply excluding the pion-nucleon coupling, which is known as the most important carrier of the tensor force, leads to substantial changes also in the central part of the mean field. The quantitative analysis of tensor effects in the RHF theory was achieved for the first time in Wang et al. (2018), which allows fair and direct comparisons with the corresponding results in the nonrelativistic framework. It is found that the strengths of tensor force in the existing RHF effective interactions are, in general, weaker than those in the nonrelativistic Skyrme and Gogny theories (Wang et al. 2018, 2020). So far, the SDR in 208 Pb has not been reproduced within the RHF+RPA scheme yet, with reasonable strengths of the tensor force constrained by the covariant symmetry. The study of this open question is in progress. One of the possible ways is to establish a bridge between the relativistic and nonrelativistic DFTs, by performing the nonrelativistic expansion with fast convergence (Guo and Liang 2019, 2020; Ren and Zhao 2020).

Implications for Astrophysics and Outlook In this chapter, the authors discuss the nuclear response theory – the exact equation of motion and its hierarchy of approximations to the response function of an atomic nucleus. On the one hand, it can be seen from the selected applications shown above that the recent theoretical developments beyond RPA and their numerical implementations have substantially improved the microscopic description of nuclear spectral properties, in particular, compared to the phenomenological models and the conventional RPA methods. On the other hand, it is also seen that further efforts on advancing the nuclear response theory are needed to obtain an even more accurate description of nuclear spectral properties. Besides being an interesting theoretical problem, the response theory has many applications, where accurate nuclear excitation spectra are required, especially at the extremes of energy, mass, isospin, and temperature. The most prominent example is nuclear astrophysics, in particular, the rapid neutron capture process (r-process) nucleosynthesis in kilonova, core-collapse supernovae, and neutron star mergers (Kajino et al. 2019). The nuclear response to the electric and magnetic dipole, Gamow-Teller, and spin-dipole operators are the microscopic sources of the major astrophysical reaction rates, such as the radiative neutron capture (n, γ ), electron capture, β-decay, and β-delayed neutron emission. These rates are very sensitive to the fine details of the calculated response, or strength functions, in the given channels and needed for many nuclei, including those that are not accessible in laboratory. The low-energy parts of the listed strength distributions are of particular importance. The low-lying dipole strength, which is relevant for the (n, γ ) rates, was studied very intensively during the past decades and associated with the neutron skin oscillations. In the neutron-rich nuclei, lying on the r-process path in the nuclear landscape, such oscillations form the pygmy dipole resonance, which can affect the (n, γ ) rates considerably (Litvinova et al. 2009a, b; Savran et al. 2013; Paar et al. 2007). The low-energy parts of the GTR and SDR are responsible for

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the beta decay and electron capture rates (Nikši´c et al. 2005; Niu et al. 2013; Mustonen and Engel 2016; Dzhioev et al. 2020). The recent developments have demonstrated, in particular, that the weak reaction rates are affected considerably by the nuclear correlations beyond (Q)RPA (Niu et al. 2013, 2015; Robin and Litvinova 2016, 2018; Litvinova et al. 2020; Litvinova and Robin 2021). Nevertheless, the simplistic (Q)RPA theoretical reaction rates as well as the mean-field nuclear matter equation of state are still employed in most of astrophysical simulations, while the deficiencies of these approaches are even more amplified in stellar environments (Arnould et al. 2007; Mumpower et al. 2016; Langanke et al. 2021; Cowan et al. 2021). Therefore, adopting the microscopic methods advanced beyond QRPA for astrophysical simulations can be the first step on the way to a high-quality nuclear physics input for such simulations. The inability of the theory to provide accurate nuclear spectra impedes the progress on other related disciplines, including the searches for the new physics beyond the standard model in the nuclear domain, such as the neutrinoless double β-decay and the electric dipole moment. These applications involve a delicate interplay of numerous emergent effects beyond QRPA and, thus, also require computation of consistency and accuracy, which are beyond the limits of current state-of-the-art theoretical and computational approaches to nuclear response. A major hope to resolve the issues discussed above is to reconcile consistently the static and dynamical kernels of the EOMs for the nuclear response in various channels, based on the lessons learned from the existing approaches. This has to be complemented by a strong effort on the nuclear interactions, both on the bare and the effective interactions, where the meson-exchange interactions (Machleidt 1989), the chiral effective field theory (χ EFT) (Epelbaum et al. 2020; Van Kolck 2020), and the DFT (Meng 2016; Colò 2020) are the most promising ones. Acknowledgments This work is supported in part by the US-NSF Career Grant PHY-1654379, the JSPS Grant-in-Aid for Early-Career Scientists under Grant No. 18K13549, the JSPS Grantin-Aid for Scientific Research (S) under Grant No. 20H05648, RIKEN iTHEMS program, and RIKEN Pioneering Project: Evolution of Matter in the Universe.

References M. Arnould, S. Goriely, K. Takahashi, Phys. Rep. 450, 97 (2007) D.P. Arteaga, P. Ring, Phys. Rev. C 77, 034317 (2008) S. Bacca, Phys. Rev. C 90, 064619 (2014) S. Bacca, N. Barnea, G. Hagen, G. Orlandini, T. Papenbrock, Phys. Rev. Lett. 111, 122502 (2013) C.L. Bai et al., Phys. Rev. Lett. 105, 072501 (2010) G.C. Baldwin, G.S. Klaiber, Phys. Rev. 71, 554 (1947) S.N. Belyaev et al., Phys. At. Nucl. 58, 1833 (1995) G. Bertsch, P. Bortignon, R. Broglia, Rev. Mod. Phys. 55, 287 (1983) N.N. Bogoljubov, V.V. Tolmachov, D.V. Širkov, Fort. Phys. 6, 605 (1958) D. Bohm, D. Pines, Phys. Rev. 82, 625 (1951) A. Bohr, B.R. Mottelson, Nuclear Structure, vol. 1 (Benjamin, New York, 1969) A. Bohr, B.R. Mottelson, Nuclear Structure, vol. 2 (Benjamin, New York, 1975)

57 Model for Collective Vibration

2175

P.F. Bortignon, A. Bracco, R.A. Broglia, Giant Resonances: Nuclear Structure at Finite Temperature. Contemporary Concepts in Physics (Harwood, Amsterdam, 1998) I.N. Borzov, Phys. Rev. C 67, 025802 (2003) A. Bouyssy, J.-F. Mathiot, N. Van Giai, S. Marcos, Phys. Rev. C 36, 380 (1987) R. Broglia, V. Zelevinsky (eds.), Fifty Years of Nuclear BCS: Pairing in Finite Systems (World Scientific, Singapore, 2013) G.E. Brown, M. Bolsterli, Phys. Rev. Lett. 3, 472 (1959) A. Cipollone, C. Barbieri, P. Navrátil, Phys. Rev. Lett. 111, 062501 (2013) G. Colò, Adv. Phys.: X 5, 1740061 (2020) J.J. Cowan et al., Rev. Mod. Phys. 93, 015002 (2021) J. Daoutidis, P. Ring, Phys. Rev. C 80, 024309 (2009) G. De Gregorio, J. Herko, F. Knapp, N. Lo Iudice, P. Veselý, Phys. Rev. C 95, 024306 (2017) W.H. Dickhoff, C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004) S. Drozdz, S. Nishizaki, J. Speth, J. Wambach, Phys. Rep. 197, 1 (1990) A.A. Dzhioev, K. Langanke, G. Martínez-Pinedo, A.I. Vdovin, C. Stoyanov, Phys. Rev. C 101, 025805 (2020) I.A. Egorova, E. Litvinova, Phys. Rev. C 94, 034322 (2016) J.P. Elliot, B.H. Flowers, Proc. R. Soc. Lond. 242, 57 (1957) E. Epelbaum, H. Krebs, P. Reinert, Front. Phys. 8, 98 (2020) V.A. Erokhova et al., Bull. Russ. Acad. Sci.: Phys. 67, 1636 (2003) D. Gambacurta, M. Grasso, J. Engel, Phys. Rev. C 92, 034303 (2015) D. Gambacurta, M. Grasso, J. Engel, Phys. Rev. Lett. 125, 212501 (2020) U. Garg, G. Colò, Prog. Part. Nucl. Phys. 101, 55 (2018) M. Goldhaber, E. Teller, Phys. Rev. 74, 1046 (1948) M. Grasso, D. Gambacurta, Phys. Rev. C 101, 064314 (2020) Y. Guo, H. Liang, Phys. Rev. C 99, 054324 (2019) Y. Guo, H. Liang, Phys. Rev. C 101, 024304 (2020) K. Hagino, H. Sagawa, Nucl. Phys. A 695, 82 (2001) M.N. Harakeh, A. can der Woude, Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitation (Oxford University Press, Oxford, 2001) H. Hergert, S.K. Bogner, T.D. Morris, A. Schwenk, K. Tsukiyama, Phys. Rep. 621, 165 (2016) M. Ichimura, H. Sakai, T. Wakasa, Prog. Part. Nucl. Phys. 56, 446 (2006) B.S. Ishkhanov, I.M. Kapitonov, Phys.-Uspekhi 64, 141 (2021) T. Kajino et al., Prog. Part. Nucl. Phys. 107, 109 (2019) S.P. Kamerdzhiev, G.Y. Tertychny, V.I. Tselyaev, Phys. Part. Nucl. 28, 134 (1997) S. Kamerdzhiev, R.J. Liotta, E. Litvinova, V. Tselyaev, Phys. Rev. C 58, 172 (1998) S. Kamerdzhiev, J. Speth, G. Tertychny, Phys. Rep. 393, 1 (2004) E. Khan, N. Sandulescu, M. Grasso, N. Van Giai, Phys. Rev. C 66, 024309 (2002) F. Knapp et al., Phys. Rev. C 90, 014310 (2014) M. Kortelainen, N. Hinohara, W. Nazarewicz, Phys. Rev. C 92, 051302 (2015) A. Krasznahorkay et al., Phys. Rev. Lett. 82, 3216 (1999) G.A. Lalazissis, J. König, P. Ring, Phys. Rev. C 55, 540 (1997) K. Langanke, G. Martínez-Pinedo, R. Zegers, Rept. Prog. Phys. 84, 066301 (2021) H. Lenske, N. Tsoneva, Eur. Phys. J. A 55, 238 (2019) H. Liang, N. Van Giai, J. Meng, Phys. Rev. Lett. 101, 122502 (2008) H. Liang, J. Meng, S.-G. Zhou, Phys. Rep. 570, 1 (2015) E. Litvinova, Phys. Rev. C 91, 034332 (2015) E. Litvinova, P. Ring, Phys. Rev. C 73, 044328 (2006) E. Litvinova, C. Robin, Phys. Rev. C 103, 024326 (2021) E. Litvinova, P. Schuck, Phys. Rev. C 100, 064320 (2019) E. Litvinova, V. Tselyaev, Phys. Rev. C 75, 054318 (2007) E. Litvinova, H. Wibowo, Phys. Rev. Lett. 121, 082501 (2018) E. Litvinova, P. Ring, V. Tselyaev, Phys. Rev. C 75, 064308 (2007) E. Litvinova, P. Ring, V. Tselyaev, Phys. Rev. C 78, 014312 (2008)

2176

H. Liang and E. Litvinova

E. Litvinova, P. Ring, V. Tselyaev, K. Langanke, Phys. Rev. C 79, 054312 (2009a) E. Litvinova et al., Nucl. Phys. A 823, 26 (2009b) E. Litvinova, P. Ring, V. Tselyaev, Phys. Rev. Lett. 105, 022502 (2010) E. Litvinova, C. Robin, H. Wibowo, Phys. Lett. B 800, 135134 (2020) N. Lo Iudice, V.Y. Ponomarev, C. Stoyanov, A.V. Sushkov, V.V. Voronov, J. Phys. G 39, 043101 (2012) W.-H. Long, N. Van Giai, J. Meng, Phys. Lett. B 640, 150 (2006) J.-J. Lu, Z.-H. Li, C.-Y. Chen, M. Baldo, H. Schulze, Phys. Rev. C 96, 044309 (2017) R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989) C. Mahaux, P. Bortignon, R. Broglia, C. Dasso, Phys. Rep. 120, 1 (1985) T. Marketin, E. Litvinova, D. Vretenar, P. Ring, Phys. Lett. B 706, 477 (2012) M. Matsuo, Prog. Theor. Phys. Suppl. 146, 110 (2002) M.G. Mayer, J.H. Jensen, Elementary Theory of Nuclear Shell Structure (Wiley, New York, 1955) J. Meng (ed.), Relativistic Density Functional for Nuclear Structure. International Review of Nuclear Physics, vol. 10 (World Scientific, Singapore, 2016) J. Meng et al., Prog. Part. Nucl. Phys. 57, 470 (2006) A. Migdal, Zh. Eksp. Teor. Fiz. 15, 81 (1945) A. Migdal, Theory of Finite Fermi Systems and Application to Atomic Nuclei (Wiley-Interscience Publ., New York, 1967) M. Mumpower, R. Surman, G. McLaughlin, A. Aprahamian, Prog. Part. Nucl. Phys. 86, 86 (2016) M.T. Mustonen, J. Engel, Phys. Rev. C 93, 014304 (2016) T. Nakatsukasa, T. Inakura, K. Yabana, Phys. Rev. C 76, 024318 (2007) National Nuclear Data Center, https://www.nndc.bnl.gov T. Nikši´c, T. Marketin, D. Vretenar, N. Paar, P. Ring, Phys. Rev. C 71, 014308 (2005) T. Nikši´c, N. Kralj, T. Tutiš, D. Vretenar, P. Ring, Phys. Rev. C 88, 044327 (2013) Z.M. Niu et al., Phys. Lett. B 723, 172 (2013) Y. Niu, Z. Niu, G. Colò, E. Vigezzi, Phys. Rev. Lett. 114, 142501 (2015) Z.M. Niu, Y.F. Niu, H.Z. Liang, W.H. Long, J. Meng, Phys. Rev. C 95, 044301 (2017) T. Oishi, M. Kortelainen, N. Hinohara, Phys. Rev. C 93, 034329 (2016) F. Osterfeld, Rev. Mod. Phys. 64, 491 (1992) T. Otsuka, A. Gade, O. Sorlin, T. Suzuki, Y. Utsuno, Rev. Mod. Phys. 92, 015002 (2020) N. Paar, P. Ring, T. Nikši´c, D. Vretenar, Phys. Rev. C 67, 034312 (2003) N. Paar, T. Nikši´c, D. Vretenar, P. Ring, Phys. Rev. C 69, 054303 (2004) N. Paar, D. Vretenar, E. Khan, G. Coló, Rep. Prog. Phys. 70, 691 (2007) S. Pankratov, M. Zverev, M. Baldo, U. Lombardo, E. Saperstein, Phys. Rev. C 84, 014321 (2011) P. Papakonstantinou, R. Roth, Phys. Lett. B 671, 356 (2009) S. Péru, H. Goutte, Phys. Rev. C 77, 044313 (2008) V.Y. Ponomarev, Nucl. Phys. A 649, 243 (1999) V.Y. Ponomarev et al., Phys. Rev. Lett. 83, 4029 (1999) F. Raimondi, C. Barbieri, Phys. Rev. C 99, 054327 (2019) Z.X. Ren, P.W. Zhao, Phys. Rev. C 102, 021301 (2020) P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin/Heidelberg, 1980) C. Robin, E. Litvinova, Eur. Phys. J. A 52, 205 (2016) C. Robin, E. Litvinova, Phys. Rev. C 98, 051301(R) (2018) C. Robin, E. Litvinova, Phys. Rev. Lett. 123, 202501 (2019) X. Roca-Maza, N. Paar, Prog. Part. Nucl. Phys. 101, 96 (2018) X. Roca-Maza, G. Colò, H. Sagawa, Phys. Rev. C 86, 031306(R) (2012) D.J. Rowe, Rev. Mod. Phys. 40, 153 (1968) H. Sagawa, G. Colò, Prog. Part. Nucl. Phys. 76, 76 (2014) E.E. Saperstein, M. Baldo, N.V. Gnezdilov, S.V. Tolokonnikov, Phys. Rev. C 93, 034302 (2016) P. Sarriguren, Phys. Rev. C 87, 045801 (2013) D. Savran et al., Phys. Rev. Lett. 97, 172502 (2006) D. Savran et al., Phys. Rev. C 84, 024326 (2011) D. Savran, T. Aumann, A. Zilges, Prog. Part. Nucl. Phys. 70, 210 (2013)

57 Model for Collective Vibration

2177

P. Schuck, Z. Phys. A 279, 31 (1976) P. Schuck et al., Phys. Rep. 929, 1 (2021) S. Shen et al., Phys. Rev. C 101, 044316 (2020) S. Shlomo, G. Bertsch, Nucl. Phys. A 243, 507 (1975) V. Soloviev, Theory of Atomic Nuclei: Quasiparticles and Phonons (Institute of Physics Publishing, Bristol/Philadelphia, 1992) V. Soma, C. Barbieri, T. Duguet, Phys. Rev. C 87, 011303 (2013) J. Terasaki, Phys. Rev. C 97, 034304 (2018) J. Toivanen et al., Phys. Rev. C 81, 034312 (2010) L. Trippa, G. Colo, E. Vigezzi, Phys. Rev. C 77, 061304 (2008) V. Tselyaev, Sov. J. Nucl. Phys. 50, 780 (1989) V.I. Tselyaev, Phys. Rev. C 75, 024306 (2007) V.I. Tselyaev, Phys. Rev. C 88, 054301 (2013) V. Tselyaev, N. Lyutorovich, J. Speth, S. Krewald, P.G. Reinhard, Phys. Rev. C 94, 034306 (2016) V. Tselyaev, N. Lyutorovich, J. Speth, P.G. Reinhard, Phys. Rev. C 102, 064319 (2020) N. Van Giai, P.V. Thieu, Phys. Lett. B 126, 421 (1983) U. Van Kolck, Front. Phys. 8, 79 (2020) D. Vretenar, A.V. Afanasjev, G.A. Lalazissis, P. Ring, Phys. Rep. 409, 101 (2005) T. Wakasa et al., Phys. Rev. C 55, 2909 (1997) T. Wakasa et al., Phys. Rev. C 85, 064606 (2012) J.D. Walecka, Ann. Phys. 83, 491 (1974) Z. Wang, Q. Zhao, H. Liang, W.H. Long, Phys. Rev. C 98, 034313 (2018) Z. Wang, T. Naito, H. Liang, W.H. Long, Phys. Rev. C 101, 064306 (2020) D.H. Wilkinson, Physica 22, 1039 (1956) K. Yako et al., Phys. Lett. B 615, 193 (2005) K. Yako, H. Sagawa, H. Sakai, Phys. Rev. C 74, 051303(R) (2006) C. Yannouleas, M. Dworzecka, J.J. Griffin, Nucl. Phys. A 397, 239 (1983) D.H. Youngblood et al., Phys. Rev. C 69, 034315 (2004) Y. Zhang, A. Bjelˇci´c, T. Nikši´c, E. Litvinova, P. Ring, P. Schuck, Phys. Rev. C 105, 044326 (2022)

Configuration Interaction Approach to Atomic Nuclei: The Shell Model

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Yusuke Tsunoda and Takaharu Otsuka

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell Structure and Magic Numbers: Traditional View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell-Model Calculation: CI Calculation Beyond the IPM . . . . . . . . . . . . . . . . . . . . . . . . . . . Valence Shell and Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-Body Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Particle Energies for the Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective N N Interaction for the Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of Shell-Model Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Shell Model: Computational Breakthrough and More . . . . . . . . . . . . . . . . . . . Basic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Generation of Basis Vectors by Variational Method . . . . . . . . . . . . . . . . . . . . . Extrapolation to Exact Energy Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional Remarks on the MCSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell Evolution Due to Monopole Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monopole Matrix Element and Monopole Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Central, 2-Body Spin-Orbit and Tensor Parts of the NN Interaction . . . . . . . . . . . . . . . . . Monopole Interaction of the Central Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monopole Interaction of the Tensor Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monopole Interaction Effects from the Central and Tensor Forces Combined . . . . . . . . . N=34 New Magic Number as a Consequence of the Shell Evolution . . . . . . . . . . . . . . . . Monopole Interaction of the 2-Body Spin-Orbit Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monopole Interaction from the Three-Nucleon Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . Short Summary of This Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Y. Tsunoda () Center for Nuclear Study, University of Tokyo, Tokyo, Japan Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki, Japan e-mail: [email protected] T. Otsuka () RIKEN Nishina Center, Wako, Hirosawa, Japan Department of Physics, University of Tokyo, Tokyo, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_17

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Correlations Among Valence Nucleons and Monte Carlo Shell Model . . . . . . . . . . . . . . . . . Shape Deformation, Quadrupole Interaction, and Rotational Band . . . . . . . . . . . . . . . . . . Type II Shell Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Doubly Closed Nucleus 68 Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformed Shapes and Potential Energy Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation Parameters and Comparison to Calculations with Gogny Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T-Plot Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell Evolution and Surface Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Short Summary of This Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2209 2209 2210 2210 2211 2215 2218 2221 2222 2224 2224

Abstract

The atomic nucleus comprises protons and neutrons, with complex quantum many-body structure, arising from these two kinds of constituents and also from complicated forces binding them (nuclear forces). Nevertheless, atomic nuclei exhibit simple and beautiful features, unexpected from the complexities. The gap between the complexity and the simplicity/beauty can be filled by the shell model, the nuclear physics terminology of configuration Interaction (CI) approach. This article presents basic ideas and formulations of the shell model, up to recent developments. The computational aspect is quite crucial for the shell model, because the Schrödinger equation has to be solved with the nuclear forces and the two kinds of fermions. The traditional approach based on direct diagonalization of Hamiltonian matrix has been used since the 1950s with technical improvements. Besides this approach, a different CI methodology, Monte Carlo shell model (MCSM), was proposed in the 1990s and has been developed. These methodologies are explained in a pedagogical way. Ni and Cu isotopes are discussed as examples of various appearances of low-lying deformed states coexisting with spherical ground states. The T-plot analysis is explained as a unique way to unveil nuclear shapes contained in the MCSM wave functions. The original version of the shell model was conceived by Mayer and Jensen. Recent studies show definite departures from this picture: the evolution of the shell structure, or the shell evolution, in exotic nuclei. The shell evolution is briefly sketched, with a certain emphasis on the prominent role of the tensor force. The shell evolution is extended from a single-particle-type feature to highly correlated many-body features such as the collective motion leading to surface deformation, as referred to as type II shell evolution. Thus, this article overviews the basic and contemporary facets of the nuclear shell model in simple terms.

Introduction It is of particular interest to describe the quantum many-body structure of atomic nuclei, largely because the atomic nucleus is an isolated object comprising many protons and neutrons, called nucleons collectively, interacting through highly complex and yet strongly attractive forces, called nuclear forces. Visible materials

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Fig. 1 Schematic illustration of the compound nucleus by Niels Bohr. (Taken from Fig. 1 of Bohr 1936b)

in the universe, except for hydrogen atoms, are composed of atomic nuclei packed with nucleons bound by such nuclear forces. The description of atomic nuclei is a major challenge, however. Such description was attempted with the picture of “compound nucleus” by Bohr (1936a), where the motion of nucleons is considered to be so chaotic that their individual motions cannot be traced in terms of single-particle degrees of freedom (see Fig. 1, Bohr 1936b, 1937). This picture looks like a quantum mechanical version of the liquid drop model in the classical-mechanical view. With emerging experimental evidences pointing to single-particle features of atomic nuclei, about a decade later, Mayer (1949) and Jensen (Haxel et al. 1949) proposed the shell model, where protons and neutrons occupy single-particle orbits forming shells and magic numbers. As this description succeeded in explaining key experimental features, the shell model has become the standard theoretical framework and has been developed significantly in various directions. The initial view by Mayer and Jensen can be characterized as the independent particle model (IPM) with a proper potential. The aspects of many-body physics were developed, for instance, by Talmi (1962), treating the interactions between nucleons more explicitly. Over several decades, there have been significant developments (see, for instance, a review article (Caurier et al. 2005) or introductory textbooks Heyde 1994, 2004). The shell model has become an indispensable approach to describe and analyze the atomic nuclei based on the nucleon-nucleon (NN) interactions, without resorting to models and/or approximations linked directly to the features to be described. In this short article, basic ideas and procedures of the shell model are presented. One of the advantages of the shell model is to include various correlations in many-body systems as the NN interactions produce. Representative features of them are sketched. The atomic nucleus is composed of Z protons and N neutrons. Their sum is called the mass number A = Z + N . Among atomic nuclei, stable nuclei are

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characterized by their infinite or almost infinite lifetimes and are characterized by rather balanced Z to N ratios, with N/Z ranging from about 1 up to about 1.5. There are about 300 stable nuclei. Other nuclei are called exotic (or unstable) nuclei. The total number of exotic nuclei is unknown but seems to be between 7000 and 10,000, providing a huge show case of various properties as well as the paths of stellar nucleosynthesis (see, for instance, Gade and Glasmacher 2008; Nakamura et al. 2017; Sorlin and Porquet 2008). The exotic nuclei decay, through β (i.e., weak) processes, to other nuclei where Z and N are better balanced, as the β decay changes a neutron to a proton or vice versa. This decay occurs successively, until the process ends at a stable nucleus. Thus, only stable nuclei exist on earth, while exotic nuclei do not, being exotic literally. The shell model works equally well for stable and exotic nuclei, exhibiting both usual and unusual properties of these nuclei as consequences of nuclear forces. Features of exotic nuclei are a current frontier of nuclear physics, and this article is expected to provide a useful guide to ongoing and future studies on exotic nuclei. After this Introduction in section “Introduction”, this article presents an overview of the traditional pictures of the shell structure and the magic number in section “Shell Structure and Magic Numbers: Traditional View”, along the IPM picture of Mayer and Jensen. The shell-model calculations including configuration mixings due to NN interactions are discussed in section “Shell-Model Calculation: CI Calculation Beyond the IPM”, beyond the IPM. The concepts like valence shell, model space, shell-model dimension, effective interaction, etc. will be introduced, but the methodology will be restricted to the traditional one by the matrix diagonalization. The Monte Carlo shell model (MCSM) will be introduced and explained in a rather pedagogical manner in section “Monte Carlo Shell Model: Computational Breakthrough and More”, as a breakthrough of major difficulty of the traditional method. Some of recent developments are presented there. The article then sheds light, in section “Shell Evolution Due to Monopole Interaction”, on a more conceptual development: the evolution of the shell structure with excess neutrons, abbreviated as the shell evolution, and its relation to the monopole interaction. The monopole effects from the central, tensor, and three-nucleon forces are sketched with some examples like the theoretical prediction and experimental verification of the new magic number N = 34. In section “Correlations Among Valence Nucleons and Monte Carlo Shell Model”, correlations among valence nucleons are discussed by referring to results of the MCSM calculations. Some basic concepts of nuclear ellipsoidal shapes will be presented, and MCSM wave functions are analyzed in terms of the T-plot. Type II shell evolution will be discussed for Ni isotopes, demonstrating the monopole and quadrupole interactions work together, yielding low-lying intruder deformed bands. The ground-state spin inversion in exotic Cu isotopes, which is a typical shell evolution phenomenon, will be analyzed as to how it survives after the correlations by the NN interaction are included. This article thus overviews basic formulation, current methodologies, and selected recent outcomes, as pedagogically as possible. The atomic nucleus might look like chaotic as N. Bohr conceived, once one looks into degrees of freedom involved. It is, fortunately, still possible to describe its structure starting from

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

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the interactions between its constituents, providing beautiful features with clean regularities. It is a great interest how far the shell model keeps penetrating into the incredible complexities of nuclear structure, identified as being chaotic by N. Bohr.

Shell Structure and Magic Numbers: Traditional View Figure 2 depicts the basic idea and consequences of Mayer-Jensen’s model. The nuclear matter composed of protons and neutrons is displayed in Fig. 2a: this matter shows an almost constant density of nucleons inside the surface which is a sphere as a natural assumption. Because of the short-range character of nuclear forces, this constant density results in a mean potential with a constant depth inside the surface, as shown in Fig. 2b. Let’s assume that the density distribution is isotropic, producing an isotropic mean potential. Figure 2b also suggests that the harmonic oscillator (HO) potential is a good approximation to this mean potential as long as the mean potential shows negative values (or attraction) as a function of r, the radius from the center of the nucleus. The mean potential is then switched to the HO potential, which is analytically more tractable. Thus, the HO potential can be introduced from the constant density (sometimes referred to as “density saturation”) and the short-range attraction due to nuclear forces. The eigenstates of the HO potential are single-particle states shown in the farleft column of Fig. 2c with associated magic numbers and HO quanta, N. These HO magic numbers do not change by adding the minor correction of the 2 term, the scalar product of the orbital angular momentum l (see the second column from left in Fig. 2c; for details see Bohr and Mottelson 1969). The crucial factor introduced by Mayer and Jensen was the spin-orbit (SO) term, (l · s), the effect of which is shown in the third column from left in Fig. 2c. The two orbits with the same orbital angular momentum, , and the same HO quanta are denoted as j> =  + 1/2 and j< =  − 1/2,

(1)

where 1/2 is due to the spin, s = 1/2. The notation of j> and j< will be used throughout this article. The spin-orbit term, vls = fls (l · s),

(2)

is added to the HO + 2 potential, where fls is the strength parameter. With fls < 0 as is the case for nuclear forces, the j> state is lowered in energy, whereas the j< state is raised. The value of fls is known empirically to be about −20A−2/3 MeV (see eq. (2-132) of Bohr and Mottelson 1969). The final pattern of the single-particle energies (SPE) is shown schematically in Fig. 2c. The single-particle states are labelled in the standard way up to their j values, and both HO and spin-orbit magic gaps are indicated in black and red, respectively. The magic numbers have been considered to be Z, N = 2, 8, 20, 28,

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Fig. 2 Schematic illustration of (a) density distribution of nucleons in atomic nuclei, (b) a mean potential (solid line) produced by nucleons in atomic nuclei and an approximation by a harmonic oscillator (HO) potential (dashed line). In (a) and (b), the horizontal line denotes the radius from the center of the nucleus. (c) The shell structure produced and the resulting magic numbers in circles. (left column) Only the HO potential is taken with HO quanta shown as N=0, N=1, . . . (N here does not mean the neutron number, N .) (middle column) The so-called 2 term is added to the HO potential, where the magic gaps are shown in circles. The single-particle orbits are labeled in the standard way to the left. (right column) The spin-orbit term, (l · s), is included further, and magic gaps emerging from this term are shown in red. The single-particle orbits are labeled to the right, including j = l + s. The magic gaps are classified as “HO” and “SO” for the HO potential and the spin-orbit origins, respectively. (Taken from Fig. 2 of Otsuka et al. (2020), which was based on Ragnarsson and Nilsson 1995)

50, 82, and 126, because the effect of the spin-orbit term becomes stronger as j becomes larger. In fact, the magic numbers 28, 50, 82, and 126 are all due to this effect. Instead, the HO magic numbers beyond 20 were considered to be absent or to show only minor effects. They shall be looked back on, from modern views of the nuclear structure covering stable and exotic nuclei. It is now investigated to what extent magic gaps in Fig. 2c have been observed. Figure 3 displays the observed excitation energies of the first 2+ states of eveneven nuclei as a function of N , where even-even stands for even-Z-even-N. These excitation energies tend to be high at the magic numbers, because excitations across the relevant magic gap are needed. The conventional magic numbers of Mayer and Jensen, N = 2, 8, 20, 28, . . . 126, are expected to arise, and sharp spikes are indeed seen at these magic numbers in Fig. 3a where the excitation energies are shown for stable and long-lived (i.e. meta stable) nuclei. The panel (b) includes all measured

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model 8000 6000

N=8

(a) stable and long-lived

N=20 N=28

N=126

4000

+

Ex(2 1) (keV)

Fig. 3 Systematics of the first 2+ excitation energies (Ex (2+ 1 ), for (a) stable and long-lived nuclei and (b) all nuclei measured up to 2016, as functions of the neutron number. Peaks in (a) are labelled by the neutron number (N ), while the names of the nuclei are displayed for some new points (red symbols and letters) in (b). In (b), the new data of 78 Ni (green symbol and letter) is added. (Modified from Fig. 4 of Otsuka et al. 2020)

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N=50

N=82

2000 0 0 8000

20

40

60

80

100

120

140

(b) all measured (as of 2016) 6000

24

O16 52

4000

Ca32 54

Ca34

2000 0 0

20

40

78

Ni50

60 80 100 neutron number

120

140

first 2+ excitation energies as of 2016. In addition to the spikes in panel (a), one sees some new ones. One of them is at N =40, which corresponds to 68 Ni40 , representing a HO magic gap at N=40. There are three others corresponding to the nuclei, 24 O16 , 52 Ca , and 54 Ca , as marked in red in the panel. The 2+ excitation energies of 32 34 these nuclei are about a factor of two higher than the overall trend, suggesting that N=16, 32, and 34 can be magic numbers, although none of them is present in Fig. 2c. These new possible magic numbers are consequences of what are missing in the argument for deriving magic gaps in Fig. 2c. This exciting subject will be discussed, but before that, the shell-model calculation will be formulated.

Shell-Model Calculation: CI Calculation Beyond the IPM The shell model proposed by Mayer and Jensen is considered to be the independentparticle model (IPM), where nucleons stay forever in designated single-particle orbits of the given potential well. The actual nuclear structure cannot be so simple in general: nucleons scatter one another by the NN interaction and consequently may change their orbits. This kind of processes, being beyond the IPM, yields additional piece of the binding energy, called correlation energies. One then has to evaluate and include the correlation energies into the theoretical description. In this section, a sketch how to perform such calculations is drawn from the general viewpoint, followed by two examples.

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Valence Shell and Hamiltonian The Hamiltonian is written in general as Hˆ = Hˆ 0 + Vˆ ,

(3)

where Hˆ 0 denotes the one-body term given by Hˆ 0 =



p

p

0;j nˆ j +

j



n 0;j nˆ nj ,

(4)

j

p,n

p,n

where nˆ j means the proton or neutron number operator for the orbit j and 0;j implies proton or neutron SPE of the orbit j . This SPE is composed of the kinetic energy of the orbit j and the energy shift of the orbit j produced by all nucleons in the inert core. In Eq. (3), Vˆ stands for the nucleon-nucleon (NN) interaction, written in the standard way as 1  Vˆ = vαβγ δ aα† aβ† aδ aγ , 4

(5)

α,β,γ ,δ

where α, β, γ , and δ are single-particle states of nucleons. The labels α, β, γ , and δ denote the orbital j , its magnetic substate m, and the index of proton or neutron of each state. Here, vαβγ δ denotes an antisymmetric two-body matrix element of the interaction Vˆ . The single-particle states appearing in Eqs. (4), (5) are explained now. Singleparticle orbits are assumed as shown in Fig. 2. Note that each single-particle state is a magnetic substate of one of the single-particle orbit j . The single-particle orbits of a given nucleus are generally grouped into closed shell, valence shell, and higher shells as shown in Fig. 4a. The closed shell comprises the orbits lowest in energy and are fully occupied (see the blue boxes in Fig. 4). The valence shell is partly occupied as indicated by green boxes in Fig. 4. Protons (neutrons) in the valence shell are called valence protons (neutrons), which are frequently mentioned in shellmodel studies. Likewise, single-particle orbits in the valence shell are referred to as valence orbits and so on. In shell-model calculations, the single-particle orbits above the valence shell are treated to be empty. The effective NN interaction is defined for valence nucleons, while effects of virtual excitations of nucleons to higher orbits are supposed to be implicitly included in this effective NN interactions through their renormalization. In general, there are certain variations of the effective NN interaction for a given valence shell, corresponding to its origin, derivation, tunings, etc. Common features and varying properties of such effective NN interactions shall be discussed. Figure 4a and b depicts different valence shells: panels a-1 and a-2 represent normal shell structure à la Mayer and Jensen, while panel b schematically displays

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

a-1

Energy

2p1/2 1f 5/2 2p3/2 1f 7/2

a-2

b

28 2

28 2

20

20

1d3/2 2s1/2 1d5/2 1p1/2 1p3/2

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valence shell 8

8

2

2

8

closed shell

2

1s1/2

Fig. 4 Schematic illustrations of the shells and the magic numbers of neutrons. (The same feature appears for protons.) Horizontal bars indicate the energies of single-particle orbits, labelled to the left. Magic numbers are shown in larger circles with yellow highlight. Filled and open small circles imply neutrons. The lowest three orbits (1s1/2 , 1p3/2,1/2 ) are fully occupied, forming the closed shell (blue box). On top of it, the next shell between the magic numbers 8 and 20, called the sd shell, is partially occupied. If a shell is partially occupied, it is called a valence shell (green box). (a-1, a-2) Different occupation patterns in the sd shell as the valence shell. Two neutrons shown by open circles in panel (a-1) scatter each other, and resulting in panel (a-2), causing a configuration mixing. Shells above the magic number 20 are empty. The shell consisting of the 2p1/2,3/2 , 1f5/2,7/2 is called the pf shell. (b) Two major shells, sd and pf , are merged, and neutrons are distributed over the sd-pf shell

shell structure that may appear, for instance, in some exotic nuclei, as two conventional shells are merged into a new single shell. Thus, Vˆ in Eq. (3) actually refers to effective NN interactions between valence (i.e. active) nucleons. This means that the single-particle states in Eq. (5) are among those of the valence shell.

Many-Body Schrödinger Equation The purpose of the shell-model calculation is to solve the many-body Schrödinger equation, Hˆ Ψ = E Ψ,

(6)

where Ψ is an eigenstate for an eigenvalue E. This general problem is solved within the valence shell by expanding Ψ in terms of many-body states in the valence shell for the given numbers of valence protons and valence neutrons. To

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be ideal, such many-body states are given by a complete set of the many-body states with these valence particles. A natural choice is given by all possible Slater determinants formed by valence protons and neutrons in the valence shell. Such Slater determinants are represented as φ1 , φ2 , ..., φi , ..., and φnd ,

(7)

where nd is the total number of all possible Slater determinants, and φi = aα†i aβ†i ....|0,

(8)

with |0 being the appropriate closed shell (see blue boxes in Fig. 4). The matrix elements, φ1 | H | φ1 , φ1 | H | φ2 , ..., φnd | H | φnd ,

(9)

form the Hamiltonian matrix ⎞ φ1 | H | φ1  φ1 | H | φ2  ... φ1 | H | φnd  ⎜ φ2 | H | φ1  φ2 | H | φ2  ... φ2 | H | φn  ⎟ d ⎟ H = ⎜ ⎠. ⎝ .... .... .... ... φnd | H | φ1  φnd | H | φ2  ... φnd | H | φnd  ⎛

(10)

With this matrix, the Schrödinger equation is rewritten as ⎛

⎛ ⎞ ⎞⎛ ⎞ φ1 | H | φ1  φ1 | H | φ2  ... φ1 | H | φnd  c1 c1 ⎜ φ2 | H | φ1  φ2 | H | φ2  ... φ2 | H | φn  ⎟ ⎜ c2 ⎟ ⎜ c2 ⎟ d ⎟⎜ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ... ⎠ = E ⎝ ... ⎠ . .... .... .... ... cnd cnd φnd | H | φ1  φnd | H | φ2  ... φnd | H | φnd 

(11)

where ci ’s are amplitudes, and the corresponding eigenstate Ψ is given by the superposition Ψ =



ci φi .

(12)

i

In most of practical calculations, the aμ† creates a proton or neutron (specified by μ) in the state μ which is the magnetic substate with a definite z-component jz of the angular momentum j . In this scheme, the total value of jz s, denoted as Jz , is a definite value. The Hamiltonian H conserves the total angular momentum, J , and its z-component, and its eigenvalue does not change among states differing only in Jz . This means that if the Schrödinger equation is solved for one of Jz = −J , −J +1, ... , J − 1, J , one obtains all the essential information. Thus, the above calculation

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

conventinal shell model 10

20

2189

110

Zr

Monte Carlo Shell Model

12

C

68

Ni

dimension

10 64 56

Ni

Ge

Be

128

Xe

1010

52

32

Fe 56

27 22 20 19

100

F 1960

Ne

Na

50

Al

48

Cr

Mg

Ni

Cr

28

Si

1980

2000

2020

year Fig. 5 Shell-model dimension as a function of the year. The maximum value of the shell-model dimension is indicated with the name of the nucleus for the year of the paper publication or the actual calculation. (Modified from Fig. 6 of Shimizu et al. 2017)

is carried out for one of the Jz values, and the Slater determinants in Eq. (8) are taken from those having the specified Jz value. The nd value is then called shellmodel dimension, which equals the dimension of the matrix in Eq. (10). For a fixed J , Jz = ±J is taken usually, because the shell-model dimension is smaller than the cases of |Jz | < J . It is noted that because of the time-reversal invariance, the nd value does not change if only Jz changes its sign but all the other quantum numbers remain unchanged. Obviously, the shell-model dimension, nd , indicates the size or the difficulty of the calculation from the viewpoint of the numerical computation. If there is only one valence orbit and only one valence nucleon, the shell-model dimension is one. But this dimension increases very rapidly as the number of valence orbits increases or the number of valence nucleons increases. Figure 5 exhibits the increase of the shell-model dimension as a function of the year of the publication or the actual calculation. It is noted that the dimension for the 32 Mg case is much larger than other nuclei with A ∼30, because two shells, sd and pf , are merged in this case as shown in Fig. 4b (see Tsunoda et al. 2020). There are quite a few computer codes for performing shell-model calculations. Among those being used widely, ANTOINE (Caurier and Nowacki 1999), BIGSTICK (Johnson et al. 2013), KSHELL (Shimizu et al. 2019), and NuShellX (Brown and Rae 2014) are accessible. These shell-model calculations are based on the traditional scheme besides many sophisticated technicalities. In addition to such traditional shell-model calculations, the Monte Carlo shell model has been proposed in order to overcome the difficulty of the shell-model dimension stated above.

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Single-Particle Energies for the Shell Model The single-particle energies (SPE) in Eq. (4) are one of the essential factors of the shell-model calculations. These values are either empirically obtained from measured values or calculated from fundamental principles. In this article, because of the length limitation, how to fix these values is not discussed, and in the examples discussed below, the SPEs are estimated, in some proper ways, with respect to the relevant closed shells. The most naïve way is to take their values from the experimental energy levels of the nuclei comprising a closed shell plus a nucleon. In many cases, the values of the SPEs are fitted so as to reproduce a large number of observed energy levels. This latter method has been shown to work quite well.

Effective NN Interaction for the Shell Model The interaction Vˆ in Eq. (3) (or Eq. (5)) is another crucial input to the shell-model calculation. It is often called the shell-model interaction. It is expressed in terms of the so-called two-body-matrix element (TBME) defined as j1 , j2 ; J, T | Vˆ | j3 , j4 ; J, T ,

(13)

where j1 , ... stand for the valence orbits and J and T imply, respectively, the angular momentum and the isospin formed by the two-nucleon system being considered. As discussed above, Vˆ conserves the J value, and the same J should appear in the ket and bra states. Usually the isospin is conserved by the NN interaction, keeping T as well. The rotational invariance of Vˆ makes the TBME independent of Jz if all the other quantum numbers are unchanged. Thus the TBME in Eq. (13) is independent of Jz . The TBMEs have been obtained for various valence shells. When the effective interaction Vˆ is discussed, the valence shell is often called the model space, which may be done also in this article. Because of various renormalization effects coming from single-particle orbits below and/or above the valence shell, Vˆ varies, in principle, as the model space is changed. In early studies, the TBMEs were fitted purely empirically, for instance, for nuclei with 3 ≤ Z ≤8 (Cohen and Kurath 1965). This kind of approach was limited up to Z=10 (Ne) (Arima et al. 1968). After a number of valuable attempts to obtain effective interactions, the so-called realistic shell-model interaction was obtained for the sd-shell nuclei (8 ≤ Z, N ≤20), named USD (-family) (Brown and Wildenthal 1988), with further extensions, USDA and USDB (Brown and Richter 2006; Richter et al. 2008). The effective interaction for the pf -shell nuclei (20 ≤ Z, N ≤40) was obtained as the KB3 (-family) interaction (Poves and Zuker 1981) and later as the GXPF1 (-family) interaction (Honma et al. 2002, 2004). These interactions were obtained in two steps: the starting point was given by microscopic G-matrix NN interactions proposed initially by Kuo and Brown (Hjorth-Jensen

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et al. 1995; Kuo and Brown 1966), and as the second step, certain phenomenological improvements were made by the fit to large numbers of experimental energy levels. It is mentioned that some main features, for instance, the tensor force component, remain unchanged by this fit (Otsuka et al. 2010a). Many other valuable shell-model interactions, for instance, (so-called) Kuo-Herling (Brown 2000), sn100pn (Brown et al. 2005), JUN45 (Honma et al. 2009), and LNPS (Lenzi et al. 2010) interactions, have been constructed from the G-matrix interactions sometimes with refinements like monopole adjustments. It should be noticed that these shell-model interactions are derived microscopically to a large extent and that they should be distinguished from purely phenomenological interactions in earlier times mentioned above. The M3Y interaction (Bertsch et al. 1977) is related to the G-matrix, too. The original methodological and conceptual contribution of the so-called G-matrix approach to the effective NN interaction (Hjorth-Jensen et al. 1995; Kuo and Brown 1966) is highly appreciated. The effective NN interaction was derived, more recently, from the chiral effective field theory (EFT) interaction, e.g. Machleidt and Entem (2011). It is first processed by the Vlow−k method (Bogner et al. 2002; Nogga et al. 2004), the similarity renormalization group (SRG) method (Bogner et al. 2010), or by other methods for transforming the nuclear forces in the free space into tractable forms for further treatments. The outcome is then processed by the in-medium SRG method (Hergart et al. 2016), by the extended Krenciglowa-Kuo (EKK) method (Takayanagi 2011a, b; Tsunoda et al. 2014a), or other methods for incorporating medium effects like the core polarization. The Vlow−k method has been adopted for the derivation of some modern shell-model interactions, for instance, the one by Coraggio et al., for Sn (Coraggio et al. 2009) and Cr-Fe (Amswald et al. 2017) regions as well as the EEdf1 interaction (Tsunoda et al. 2017). The EEdf1 interaction has been applied to various studies for the nuclei around N =20 up to the neutron dripline (see Tsunoda et al. 2020). The SRG and IM-SRG methods have been studied extensively also but are not discussed in this article due to the length limitation.

Example of Shell-Model Calculation The result of the calculation with the GXPF1 interaction for 57 Ni is shown in Fig. 6 as an example of the shell-model calculation (Honma et al. 2004). One sees a remarkable agreement between the calculation and the experiment. The GXPF1 interaction was not fitted particularly to the nucleus 57 Ni. There have been a large number of shell-model calculations, as good fractions of them are reviewed in Caurier et al. (2005) and Otsuka et al. (2020). It should be stressed that such shellmodel calculations have significantly contributed to deeper and wider understanding of the quantum many-body structure contained in experimental data on stable and exotic nuclei. The diversity in the shell-model applications to interdisciplinary researches may be exemplified with the nuclear matrix element (NME) of the neutrinoless double beta decay. This quantity has been calculated in a variety of theoretical approaches,

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23/2−(2) 23/2− 23/2−(1) 21/2−(3)

8

19/2−(1)

Ex (MeV)

6 13/2−(3) 13/2−(2)

4

11/2−(2) 9/2−(3)

5/2−(3)

21/2−(2)

(21/2−) (19/2−) (19/2) 21/2−

21/2−(1) 19/2−(3) 19/2−(2)

(19/2) (17/2) 19/2−

17/2−(3)

(17/2−) (15/2)

17/2−(2)

17/2−(1)

17/2− (9/2−) (13/2)

15/2−(3)

15/2−

15/2−(2) 15/2−(1)

15/2− 13/2− 15/2−

(11/2) 13/2− 13/2− 11/2−

13/2−(1) 1/2−(2) 11/2−(3)

na 9/2− 9/2−

9/2−(2) 3/2−(3) 11/2−(1) 9/2−(1) 7/2−(3) 3/2−(2) 7/2−(2)

(5/2)− 7/2− 7/2− 3/2−

na 11/2− 11/2− 3/2− 9/2− (5/2−,7/2−)

7/2−

5/2−(2) 7/2−(1)

5/2−

2 1/2−(1)

1/2−

5/2−(1)

0

5/2−

3/2−

3/2−(1)

th.

57

exp. Ni

Fig. 6 Energy levels of 57 Ni as a typical example of the shell model calculation. The results obtained from the GXPF1A interaction (th.) are compared to the experimental ones (exp.). Yrast levels (energetically lowest states for a given spin/parity) are connected by dotted lines. (Taken from Fig. 19 of Honma et al. 2004)

as shown in Fig. 7. The NME is needed to derive the neutrino mass from the lifetime of this decay, among other factors needed as well, see, for instance, a recent review (Agostini et al. 2022). Figure 7 suggests rather smaller values by the shell model than other theoretical calculations. As various correlations contribute to the NME, the shell model should play crucial roles because different correlations are treated on an equal footing.

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EDF

7

IBM QRPA

6

NSM IMSRG

0ν

M long

5

CC

4 3 2 1 0

48

Ca

76

82

Ge

Se

100

Mo

116

Cd

130

Te

136

Xe

150

Nd

Fig. 7 Nuclear matrix elements of neutrinoless double beta decay calculated by various theoretical methods. The mother nuclei are shown below the panel. The shell model results are shown as “NSM” in the inset. (Taken from Fig. 8 of Agostini et al. 2022)

Monte Carlo Shell Model: Computational Breakthrough and More The shell-model calculation is a powerful method to describe nuclear properties starting from a given NN interaction. It does not depend on a particular model assumption for the result to be obtained. However, the actual application encounters a difficulty of the huge shell-model dimension. Figure 5 shows the dimensions for various nuclei with relevant model spaces. Certainly, there are interesting and important cases beyond the limit.

Basic Formulation The Monte Carlo shell model (MCSM) has been proposed and developed in order to overcome this dimension problem (Honma et al. 1995; Otsuka et al. 1998; Otsuka et al. 2001a; Shimizu et al. 2012, 2017). As the shell-model dimension is the major obstacle for the conventional shell-model calculation, the MCSM can be regarded as a game changer, because many cases with much larger shell-model dimensions become tractable. In this section, the outline of the MCSM is presented. The k-th eigenstate of spin/parity J π is expressed in the MCSM as Ψ (J π , k) =



fi (J π , k) Pˆ (J π ) Φi (J π ),

(14)

i

where i is the index of the basis vector and Φi (J π ) and fi (J π , k) mean, respectively, the i-th basis vector, which is a Slater determinant, and its amplitude.

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It is noted that the basis vectors, Φi (J π ), i =1, 2, ..., are selected for the specific J π. This amplitude is scaled so that the state Ψ (J π , k) is normalized. It is stressed here that the sets {Φi ; i =1, 2, ...} and {fi ; i =1, 2, ...} vary as functions of J π . The z-component Jz is not shown here, because the energy eigenvalue remains unchanged if only Jz is varied. In actual calculations, one of possible Jz values is taken. The Jz dependences of other physical observables are known and are correctly treated in actual applications. In some case, an additional quantum number(s), such as the isospin, may have to be specified, but the required extension is trivial and is not discussed here. The symbol Pˆ (J π ) implies the projection onto the quantum number J π , and it must project on the Jz chosen as stated above, but this process is not explicitly argued here for brevity. The J π projection contains some technical complexities as stated elsewhere. Equation (14) resembles Eq. (12), but there are distinct differences. One of clear differences lies in the basis vectors: the basis vector φi in Eq. (12) is a Slater determinant given by the product of usual single-particle states as shown in Eq. (8). On the other hand, the basis vector Φi in Eq. (14) is a Slater determinant given by the product, † (J π ,i) † (J π ,i) b2 ....|0,

Φi (J π ) = b1

(15)

where the proton or neutron creation operator b† is defined as † (J bm

π ,i)

=



π

(J ,i) † Dn,m an ,

m = 1, 2, .... .

(16)

n

Here D denotes a matrix element, and a † already appeared in Eq. (8). Equation (16) implies that the operator b† is a proton or neutron creation operator for a state given by a superposition of all single-particle states in the model space. This superposition can make each Φi more optimum in the description of the eigenstates of the specified J π value. It is noted that each MCSM basis vector can be called a “deformed” Slater determinant where single-particle states are superpositions of the original single-particle states (e.g., those of the HO potential). In this point, Eq. (14) differs from Eq. (12). Another major difference is the projection operator Pˆ (J π ). The matrix D thus transforms the original (naïve) single-particle states to the single-particle states specific for the i-th MCSM basis vector for the specific eigenstate of the designated J π value. Thus, it has a superscript labelling (J π , i) (see Eq. (16)). The search for the optimum D is crucial for the actual MCSM calculations. The matrix D is fixed successively. One can start with an initial guess, denoted by u, of the first MCSM basis vector Φi=1 (J π ), utilizing mean-field techniques, e.g., a deformed Hartree-Fock (HF) calculation for the shell-model Hamiltonian being used. Another candidate, denoted by v, is generated by a random minor change of u, and the energy is calculated from v. If this energy of v is lower than the energy of u, u is abandoned, and v is kept for the next step. If the other

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way around, v is rejected, and u is carried over to the next step. This competition process is repeated until the energy gain is saturated, meaning that the gain cannot exceed the prefixed minimum requirement, after some trials. The first MCSM basis vector, Φi=1 (J π ), is thus determined. In short, it is a search for an optimum “deformed” Slater determinant by utilizing random variations of candidate states. As the deformed HF is obtained without the angular momentum projection but the present MCSM process is performed with good angular momentum and parity, the obtained Φi=1 (J π ) naturally differs from the deformed HF state. 64 Ge nucleus (Shimizu Figure 8 shows an example for the 0+ 1 ground state of 2013). The energy monitored is the ground-state energy in this case. The blue symbols (labelled “MC sampling”) indicate an MCSM calculation, and Φi=1 (J π ) gives the energy slightly above −303 MeV. The second basis Φi=2 (J π ) is searched similarly. It is different from the i=1 process in that the energy is calculated in a two-dimensional space spanned by the current trial state and the first basis vector Φi=1 (J π ) which has been fixed. The number of basis vectors, denoted by nb , is two now, and the eigenvalue of nb =2 is indicated around −303.5 MeV by the blue symbol in Fig. 8. Once the second basis vector Φi=2 (J π ) is fixed, the third basis vector is searched in a similar way. With this successive process, Φi=1 (J π ), Φi=2 (J π ), Φi=3 (J π ), ..., Φi=nb (J π ) are obtained. These basis vectors are collectively called MCSM basis π vectors and are actually represented by the matrix D(J ,i) , i = 1, 2, 3, ..., nb . It is mentioned that the MCSM basis vectors, Φi (J π ), i = 1, 2, ..., are not mutually orthogonal. They are, however, linearly independent, and this independence gives additional dimensions to the many-body Hilbert space of the MCSM calculation. It is noted that because each basis vector takes the form of Slater determinant, the non-orthogonality is inevitable. By adding basis vectors, the energy eigenvalue is lowered. In other words, the calculated energy eigenvalue decreases as a function of nb . The blue symbol in Fig. 8 displays such energy eigenvalues for each value of nb . The eigenvalue comes down to about −305.7 MeV at nb =100, where the eigenvalue is changing rather slowly, providing an expectation that the exact value may not be too far. Nevertheless, it is desired to have a quicker convergence to the exact eigenvalue.

Fig. 8 Convergence pattern of the ground-state energy from an example case with 64 Ge. (Modified from Fig. 3 of Shimizu 2013)

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Advanced Generation of Basis Vectors by Variational Method The basis vector generation process has substantially been improved by introducing a variational process. A basis vector Φi (J π ) is good in the level of random search, but in general, there is a certain room of the improvement because the random search is a discrete sampling and may miss possible improvements between sampled cases. Once the basis vector Φi (J π ) is fixed by the random sampling, it is improved by the variational process called the conjugate gradient (CG) method (Shimizu 2013). In π this process, the variation is made on the matrix D(J ,i) following the prescription of the CG method, monitoring the energy calculated with the angular momentum and parity projections. Thus an advanced version of the MCSM has been achieved. Red symbols in Fig. 8 display the energy eigenvalues obtained by this advanced version. Figure 8 implies that this process produces a considerable effect, more than 0.7 MeV, already for the first basis vector (nb =1). The convergence is improved also, and the derivative becomes smaller near nb =100.

Extrapolation to Exact Energy Eigenvalue Figure 8 depicts that the energy eigenvalues are on trajectories converging to the exact value. The plot in Fig. 8 may not be the best way for estimating the exact eigenvalue, as nb can be as large as 1014 in this case. Of course, the convergence occurs much earlier, but it is computationally heavy to trace the evolution of the eigenvalue as a function of nb . Instead, a very powerful method has been introduced, as briefly explained below. The temporary energy eigenvalue corresponding to the number of the adopted π basis vectors, nb , is denoted by E (J ,k) (nb ), and its eigen wavefunction is by π π Ω (J ,k) (nb ). In other words, Ω (J ,k) (nb ) is the eigen wave function, when the π Schrödinger equation is solved with these nb basis vectors. Thus, Ω (J ,k) (nb ) generally differs from the exact eigen wave function, while the former may serve as a good approximation to the latter. It is now attempted to gauge the difference between the two. First, the following quantity is introduced, E (J

π ,k)

(nb ) = Ω (J

π ,k)

(nb )|Hˆ |Ω (J

π ,k)

(nb )

(17)

where Hˆ is the Hamiltonian defined in Eq. (3). The energy variance is then defined as Δ(J

π ,k)

(nb ) = Ω (J

π ,k)

π π π (nb )|Hˆ 2 |Ω (J ,k) (nb ) − Ω (J ,k) (nb )|Hˆ |Ω (J ,k) (nb )2 . (18)

Both terms on the right-hand side are real numbers due to the hermiticity of Hˆ . The π π variance Δ(J ,k) (nb ) is positive definite or zero. It is pointed out that Δ(J ,k) (nb ) π becomes vanished if Ω (J ,k) (nb ) happens to be an eigenstate of Hˆ . Figure 9 shows

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Fig. 9 Extrapolation for the example of 64 Ge. (Modified from Fig. 4 of Shimizu et al. 2010)

+ π E (J ,k) (nb ) against Δ(J ,k) (nb ) for J π = 0+ 1 (ground state) and J = 21 , where the index k is shown as the superscripts. (The index k will be indicated in this way hereafter.) Namely, each symbol is plotted at the coordinate designated by the π π variance, Δ(J ,k) (nb ), on the horizontal axis and the expectation value, E (J ,k) (nb ), on the vertical axis. Thus, the left edge of the panel stands for the variance equal to zero, implying the exact solution. For the 0+ 1 state, the calculations with the variational improvement (red symbol) and without it (blue symbol) are shown. It is clear that the plots can be extrapolated by smooth curves. The plotted energies are fitted by π

π

E = E0 + a Δ(J

π ,k)

(nb ) + b {Δ(J

π ,k)

(nb )}2 + ...

(19)

where E is the energy and E0 is the value of E with a vanished variance. As mentioned above, E0 is the exact eigenvalue, if the fit works as expected. The energy fitted with the polynomial up to the quadratic term is shown by lines in Fig. 9. One sees that the fit indeed works well. Thus, quite reliable values for the energies can be obtained. It is further emphasized that the excitation energies converge earlier than the energies themselves, as demonstrated by Fig. 9.

Additional Remarks on the MCSM In many actual MCSM calculations, the energies of several eigenstates of a given J π are monitored, for instance, by their average, in the generation process of the MCSM basis vectors. That kind of calculations are more sophisticated than those monitoring the energy of a single eigenstate but can be done. By doing this, the results for those states are obtained at once, and the orthogonalities among the obtained eigenstates are guaranteed. It is noted, however, that the accuracy may somewhat vary among those states.

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It is mentioned that the quasi-particle vacua shell model (QVSM), which is the most advanced extension of the MCSM, has recently been proposed, where the basis vectors are not Slater determinants but particle-number-projected quasiparticle vacua (Shimizu et al. 2021). Like the Bogoliubov method, the pairing correlations over many single-particle orbits, which are treated by superposing Slater determinants in the usual MCSM, can be contained in individual basis vectors to a certain extent in the QVSM, which makes calculations for heavy nuclei more tractable. Before discussing actual applications of the MCSM, a basic feature of the nuclear shell structure is discussed in the next section, for a more transparent understanding.

Shell Evolution Due to Monopole Interaction p

n in Eq. (4), are given, in the IPM The single-particle energies (SPEs), 0;j and 0;j of Mayer and Jensen, by the HO potential, the spin-orbit splitting, and the minor correction by the so-called 2 term (see Fig. 2). The SPEs are then constants within a given nucleus and vary only gradually as Z or N changes. In actual shell-model calculations, the SPEs are adjusted, in some cases, so as to describe experimental data such as the ground-state energies and level energies of excited states. However, as long as stable nuclei are concerned, the basic trends, such as the magic gaps and the orderings of single-particle orbits, do not differ from those depicted by Fig. 2. The shell model, as the physics of many-body correlations rather than the IPM, includes an N N interaction in Eq. (5) and solves the Schrödinger equation in Eq. (6) for the Hamiltonian in Eq. (3). The SPEs are then crucial in determining the basic properties of the eigenstates, but it is also possible that such an NN interaction may change the SPEs in an effective way. This aspect has been discussed from early times, but its greater importance and wider appearance have been recognized in exotic nuclei, with emerging differences from the patterns shown in Fig. 2. The SPEs substantially vary as Z or N changes: in some cases, new magic numbers absent in Fig. 2 show up, while some conventional magic numbers are weakened or even disappear. Such phenomena are collectively called the shell evolution and occur due to particular basic features of the NN interaction. This section is devoted to the shell evolution: how it occurs and what consequences arise. The following part of this section is largely based on Otsuka (2022), as they are self-contained to a good extent.

Monopole Matrix Element and Monopole Interaction The interaction Vˆ in Eq. (5) can be decomposed, in general, into the two components: monopole and multipole interactions (Poves and Zuker 1981), irrespectively of its origin, derivation, or parameters. The monopole interaction is expressed in terms of the monopole matrix element defined for single-particle orbits j and j  as,

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

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Monopole matrix element between orbits j and j’

v

v

v v

v

number of matrix elements in the summation : magnetic substates of orbit j’

: magnetic substates of orbit j

Fig. 10 Schematic illustration of the monopole matrix element. (Taken from Fig. 7 of Otsuka et al. 2020)

V mono (j, j  ) =

j  , m |Vˆ |j, m ; j  , m  , (m,m ) 1

(m,m ) j, m ;

(20)

where m and m are magnetic substates of j and j  , respectively, and the summation over m, m is taken for all ordered pairs allowed by the Pauli principle. The monopole matrix element represents, as displayed schematically in Fig. 10, an orientation average for two nucleons in the orbits j and j  . See Otsuka et al. (2020) for more detailed and pedagogical descriptions. The monopole interaction between two neutrons is given as the averaged interaction in this manner. For example, the monopole interaction between the two orbits j = j  takes the following form, mono Vnn (j, j  )

 μ,μ

aμ† aμ†  aμ aμ ,

(21)

where μ (μ ) is a magnetic substate of the orbit j (j  ). The monopole interaction between two neutrons can be expressed as mono = Vˆnn

 j

mono Vnn (j, j )

 1 n n mono nˆ j (nˆ j − 1) + Vnn (j, j  ) nˆ nj nˆ nj , 2 

(22)

j and j  = j< ) with the identical radial wave functions (see Eq. (1)), for instance, 1f7/2 and 1f5/2 . Another example is the coupling between unique-parity orbits, such as 1g9/2 and 1h11/2 , for which the radial wave functions are similar because of no radial node. These types of strong correlations were pointed out by Federman and Pittel (1977), where the total effect of the 3 S1 channel of the NN interaction was discussed without the reference to the monopole interaction.

Monopole Interaction of the Tensor Force Another important source of the monopole interaction with strong orbital dependences is the tensor force. The tensor force produces very unique effects on the ESPE. This is displayed in Fig. 11: the intuitive argument in Otsuka et al. (2005, 2020) proves that the monopole interaction of the tensor force is attractive between a nucleon in an orbit j< and another nucleon in an orbit j> , whereas it becomes repulsive for combinations, (j> , j> ) or (j< , j orbit, the ESPE of the proton orbit j> is raised, whereas that of the proton orbit j< is lowered. This is nothing but a reduction of a proton spinorbit splitting due to a specific neutron configuration. The amount of the shift is proportional to the number of neutrons in this configuration, as shown in Eq. (32) and in Fig. 12c. Other cases follow the same rule shown in Fig. 11. These general features have been pointed out in Otsuka et al. (2005) with an analytic formula and an intuitive description of its origin.

Fig. 11 Monopole interaction of the tensor force. (Taken from Fig. 2 of Otsuka et al. 2005)

(b)

(a)

j
’

attraction spin

j>

j>’

repulsion wave function of relative motion

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Y. Tsunoda and T. Otsuka

(b)

j>’

j
’

proton

j> proton

neutron

j>’

j


j>’

j


j
’

j
 . (b) The shifts of the proton ESPEs due to two (valence) neutrons in the orbit j> . (c) The same as (b) except for four neutrons. (d, e) Type II shell evolution due to neutron particle-hole excitations. (Taken from Fig. 1 of Otsuka and Tsunoda 2016)

Monopole Interaction Effects from the Central and Tensor Forces Combined The combined effects of the central and tensor forces were discussed in Otsuka et al. (2010a) in terms of realistic shell-model interactions, USD (Brown and Wildenthal 1988) and GXPF1A (Honma et al. 2002, 2004). The VMU interaction was then introduced as a general and simple shell-model NN interaction. Its central part consists of Gaussian interactions with spin/isospin dependences, and their strength parameters are determined so as to simulate the overall features of the monopole matrix elements of the central part of USD (Brown and Wildenthal 1988) and GXPF1A (Honma et al. 2002) interactions. Its tensor part is taken from the standard π - and ρ-meson exchange potentials (Bäckman et al. 1985; Osterfeld 1992; Otsuka et al. 2005). Thus, the VMU interaction is defined as a function of the relative distance of two nucleons with spin/isospin dependences, which enables us to use it in a variety of regions of the nuclear chart, as shall be seen. A wide model space, typically a full HO shell or more, is required in order to obtain reasonable results, though. Figure 13 depicts some examples: panel (a) displays the transition from a standard (à la Mayer-Jensen) N=20 magic gap to an exotic N =16 magic gap by plotting ˆjn  within the filling scheme (see Eq. (31)), as Z decreases from 20 to 8. The tensor monopole interaction between the proton d5/2 and the neutron d3/2 orbits plays an important role. The small N =20 magic gap for Z=8–12 is consistent with the island of inversion picture (see reviews, e.g., Caurier et al. 2005; Otsuka et al. 2020). Panel (b) depicts the inversion between the proton f5/2 and p3/2 p orbits as N increases in Ni isotopes, by showing ˆj  (see Eq. (30)). The figure

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

2205

4

p 3/2

H (MeV)

0

(a) neutron ESPE of N=20 isotones

-10

p 1/2

-10

d 3/2

d 3/2

2

p 3/2

g 7/2

f 7/2

-20

h 11/2

s 1/2

f 5/2

f 7/2 s1/2

(c) neutron ESPE for N=50

(b) proton ESPE of Ni isotopes

0

d 5/2

-20 8

14 16

Z

20

40

N

50

40

Z

50

Fig. 13 Effective single-particle energies calculated by the VMU interaction. The dashed lines are obtained by the central force only, while the solid lines include both the central force and the tensor force contributions. (Taken Fig. 3 of from Otsuka et al. 2010a)

exhibits exotically ordered single-particle orbits for N > 44. The tensor monopole interactions between the proton f7/2,5/2 and the neutron g9/2 orbits produce crucial effects. Panel (c) shows significant changes of the neutron single-particle levels from 90 Zr to 100 Sn, in terms of ˆ jn . Without the tensor force, the near degeneracy of g7/2 and d5/2 orbits in 100 Sn does not show up. These changes of the shell structure as a function of Z and/or N were collectively called shell evolution in Otsuka et al. (2005). The splitting between proton g7/2 and h11/2 in Sb isotopes shows a substantial widening as N increases from 64 to 82 as pointed out by Schiffer et al. (2004), which was one of the first experimental supports to the shell evolution partly because this was not explained otherwise. Note that while the origin of the shell evolution can be any part of the NN interaction, its appearance is exemplified graphically in Fig. 12a, b, c for the tensor force. The shell-evolution trend depicted in Fig. 13 appears to be consistent with experiment (ENSDF; Ichikawa et al. 2019; Liddick et al. 2006; Otsuka et al. 2010a, 2020; Sahin et al. 2017; Seweryniak et al. 2007). The monopole properties discussed in this subsection are consistent with the results shown by Smirnova et al. (2010) obtained through the spin-tensor decomposition (see, e.g., Otsuka et al. (2020) for some account). Some of the emerging concepts were conceived in the study of exotic nuclei, particularly by looking at the shell structure and magic numbers of them. The obtained concepts were found later not to be limited to exotic nuclei but to be general. In this way, after the initial trigger by exotic nuclei, the overall picture of the shell structure has been renewed, and this section is devoted to a sketch of it with two major keywords, the monopole interaction and the shell evolution.

N=34 New Magic Number as a Consequence of the Shell Evolution Among various cases of shell evolution, a notable impact was made by predicting a new magic number N =34. Figure 14 displays the shell evolution of some neutron single-particle orbits from Ni back to Ca isotopes, as Z decreases from 28 to 20.

Y. Tsunoda and T. Otsuka

34 2p 1/2

2p 1/2

2p

2p

32 3/2

1f 7/2

1f 7/2 Ni

proton

3/2

proton

3 52

neutron

N = 30 N = 32 N = 34

Ca

54

Ca

2

1 18

Ca neutron

Level energy (MeV)

2206

22

26

30

Proton number, Z

34

Fig. 14 (left) Schematic illustration of the shell evolution from Ni back to Ca for neutron orbits. Blue circles denote protons. The wavy line is the interaction between the proton 1f7/2 orbit and the neutron 1f5/2 orbit. The numbers in circles indicate magic numbers. (Taken from Fig. 3 of Otsuka and Tsunoda 2016). (right) Observed excitation energies of the 2+ 1 states. (Taken from Fig. 2 c of Steppenbeck et al. 2013)

The 1f5/2 orbit is between the 2p3/2 orbit and the 2p1/2 orbit in Mayer-Jensen’s shell model (see Fig. 2). By losing eight protons in Ni isotopes, this canonical shell structure is destroyed as the 1f5/2 orbit moves up above the 2p1/2 orbit. This movement of 1f5/2 orbit creates the N =32 gap as a by-product (Huck et al. 1985). The energy shift of the 1f5/2 orbit is due to the central and tensor forces by almost equal amounts. It is mentioned that the emergence of the N =34 magic number in Ca isotopes and the appearance of the Mayer-Jensen scheme in Ni isotopes are consistent from the viewpoint of the basic features of the NN interaction. The appearance of the N =34 magic number was predicted as a result of a spin-isospin interaction in Otsuka et al. (2001b). However, 12 years were required (Janssens 2005) until the experimental verification became feasible (Steppenbeck et al. 2013) (see Fig. 14 (right)). The measured 2+ energy levels are included in Fig. 3b. More details are presented in Otsuka et al. (2020). Further evidences have been obtained recently by different experimental probes as reported in Michimasa et al. (2018) and Chen et al. (2019).

Monopole Interaction of the 2-Body Spin-Orbit Force It is a natural question what effect can be expected from the 2-body spin-orbit force of the N N interaction. This force can be well described by the M3Y interaction, and its monopole effects were analyzed in detail in Otsuka et al. (2020), including its supplementary document. Although the monopole effects of this force contribute to the spin-orbit splitting (Otsuka et al. 2020), apart from this bulk-type property, its effect on the shell evolution is much weaker than the tensor force in most cases. An interesting case is, however, found in the coupling between an s orbit and the p3/2,1/2 orbits. There is no monopole effect from the tensor force, if an s

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

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orbit is involved. Instead, the s-p coupling due to 2-body spin-orbit force can be exceptionally strong as stressed in Otsuka et al. (2020) from both quantitative and intuitive viewpoints.

Monopole Interaction from the Three-Nucleon Force The three-nucleon force (3NF) is currently of intense interest (see, for instance, a review Hammer et al. 2013). Among various aspects, Otsuka et al. (2010b) showed that one of the characteristic features of the monopole interaction of the effective N N interaction originates in the Fujita-Miyazawa 3NF (Fujita and Miyazawa 1957). A brief description is presented here (see Otsuka (2022) for a somewhat more detailed explanation). Figure 15a displays the effect of the Δ excitation in NN interaction. The Δ-hole excitation from the inert core changes the SPE of the orbit j as shown in panel (b) where m is one of the magnetic substates of the orbit j and m means any state. This diagram renormalizes the SPE, and observed SPE should include this contribution. If there is a valence nucleon in the state m as in panel (c), the process in panel (b) is Pauli-forbidden. However, in the shell-model and other nuclear structure calculations, the SPE containing the effect of panel (b) is used. One has to somehow incorporate the Pauli effect of panel (c), and a solution is the introduction of the process in panel (d). In this process, the state m doubly appears in the intermediate state, but one can evaluate the Pauli effect by including panels (b) and (d) consistently. So, the process of panel (d) is included as an additional term

m

m’

∆

m’

m’

m (a)

∆

m (b)

m

(i) Energies calculated

(c)

m’

Energy (MeV)

∆

0

m

from G-matrix NN + 3N (Δ) forces −20

−40

Exp. NN + 3N (Δ) NN 14 16

m’

m

−60

m’

m

8

∆

∆

m’

(d)

m

(j) Schematic picture of twovalence-neutron interaction induced from 3N force

(e)

16

O core

(f )

(g)

20

Neutron Number (N)

(h)

Fig. 15 Schematic illustration of the 3NF. (Based on Figs. 3 and 4 of Otsuka et al. 2010b)

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Y. Tsunoda and T. Otsuka

to the effective N N interaction Vˆ in Eqs. (3), (5). This is a usual mathematical trick and enables us to correctly treat the Pauli principle within the simple framework. The actual modification of the effective NN interaction can be carried out in terms of the Fujita-Miyazawa 3NF. Panel (e) displays the Fujita-Miyazawa 3NF, but the states m and m are assigned so that the situation in panel (d) is represented. Clearly the state m appears in double, as mentioned above. If the nucleon line in the middle is taken for all the states in the inert core, panel (e) becomes equivalent to panel (d). By taking any possible magnetic substates for the left and right nucleon lines of panel (d), and summing over all inert-core states for the middle nucleon line, the modification to the effective NN interaction for valence nucleons can be introduced. The Pauli-blocking effect discussed above can then be incorporated by the monopole term of the present additional NN interaction. Naturally, the shell evolution due to the Δ-hole blocking is described by this particular monopole interaction obtained from the Fujita-Miyazawa 3NF. This monopole interaction is repulsive because of its origin in the blocking effect. (As a detail, it is noted that direct terms do not contribute to this monopole term because of parity-changing vertex for the pion exchange process (see Otsuka et al. 2005, 2010b).) The 3NF may also arise from other sources, but the main contribution to the shell evolution is expected to be generated by the present mechanism. A similar treatment can be carried out in the chiral EFT framework. Panel (f) corresponds to panel (e), but the violation of Pauli principle is kind of hidden, because of a vertex in the middle (depicted by a square) instead of the Δ excitation. In this argument, the 3NF produces a repulsive monopole NN interaction in the valence space, after the summation over the hole states of the inert core (see panel (j)), which corresponds to the normal ordering in other works. Panel (i) indicates an example of the repulsive effect on the ground-state energy of oxygen isotopes, locating the oxygen dripline at the right place or solving the oxygen anomaly (Otsuka et al. 2010b). This is rather strong repulsive monopole interaction, which is a consequence of inert core. This means that the present case is irrelevant to the no-core shell model or other many-body approaches without the inert core (e.g., GFMC) (Carlson et al. 2015). This feature has caused some confusions in the past, but the difference is clear. The present repulsive monopole effect is much stronger than the other effects of the 3NF (Tsunoda et al. 2020), and the latter will be better clarified by further developments of the chiral EFT for 3NF in the future. It is noted that the present valence-space repulsive effect was noticed by Talmi in the 1960s (Talmi 1962).

Short Summary of This Section The shell evolution is found in many isotopic and isotonic chains and sometimes results in the formation of new magic gaps or the disappearance of old ones, as departures from the IPM. Figure 3b displays the emergence of such new magic numbers N=16, 32, and 34. More changes may appear in the future studies.

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

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Interestingly, these findings are neither isolated nor limited to particular singleparticle aspects but are related to other features of the nuclear structure as depicted below.

Correlations Among Valence Nucleons and Monte Carlo Shell Model The N N interaction acting on valence nucleons produces the shell evolution through its monopole component. Besides this, it produces other important phenomena as combined effects of the monopole and multipole components (see Eq. (29)). These effects are linked to the mixings of different configurations of valence nucleons like the mixing between two configurations in Fig. 4 a-1 and a-2. Such configuration mixings are found in amplitudes in the vector in Eq. (11) and are caused by correlations among valence nucleons. Here the correlations mean that different configurations are superposed so as to gain certain many-body features. Of course, these correlations are consequences of the NN interaction. The energy gain by these correlations are often called correlation energy, for which the shell model seems to be advantageous over other models, because its solution is built directly on the NN interaction. This section sheds lights on some features of such correlations.

Shape Deformation, Quadrupole Interaction, and Rotational Band The most visible outcome of the correlations is probably the surface deformation from a sphere, in particular ellipsoidal shapes or quadrupole deformations. The quadrupole deformation has been a very important subject since the 1950s, as initiated by Rainwater (1950) and by Bohr and Mottelson (Bohr 1952, 1975; Bohr and Mottelson 1953, 1969, 1975). In the shell model, the quadrupole deformation is considered to be driven by the quadrupole interaction, a part of the multipole interaction in Eq. (28). However, the quadrupole interaction is a somewhat vague idea. It can mean the rank 2 component for the proton-neutron interaction, but this is already not applicable for the neutron-neutron or the proton-proton interactions because of ambiguities in the angular momentum recoupling. (The rank 2 component of the proton-neutron interaction means a part of this interaction comprising products of the proton and neutron one-body operators, each of which carries angular momentum 2. The coefficient of individual product is given by the NN interaction.) Apart from such complexities in the interactions, there are certain features that can be simulated by the (scalar) coupling of the quadrupole-moment operators. In fact, the effects of the rank 2 part of the proton-neutron interaction may be approximated by those of such “quadrupole-moment interaction” particularly for low-lying states with strong quadrupole deformations, because fine details can be smeared out in coherent contributions from various configurations. Although the quadrupolemoment interaction with appropriately chosen strength parameters may be thus

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Y. Tsunoda and T. Otsuka

useful for intuitive interpretations or even simple calculations, the quadrupole interaction refers, in this article, to the rank 2 component of the proton-neutron interaction. The basic idea here is that the monopole (rank 0) and quadrupole (rank 2) interactions are taken from the common underlying NN interaction. The actual calculations for physical quantities are performed by including all other multipole interactions contained in realistic NN interactions, while the contributions of such other multipole interactions may be modest or negligible for low-lying ellipsoidally deformed states. It is noted that the quadrupole-moment interaction was used in the SU(3) model for the sd shell (Elliott 1958a, b), the Pairing+QQ model (Bes and Sorensen 1969; Kumar and Baranger 1968), etc. It is also pointed out that rotational bands of atomic nuclei can be described with realistic NN interactions, as shown, for instance, Caurier et al. (1994), or even with ab initio interaction (Otsuka et al. 2022b). If the quadrupole moments are larger, i.e., a stronger quadrupole deformation occurs, the nucleus gains more binding energy from the above argument on the quadrupole interaction. This is a very general phenomenon, and because of this mechanism, the ground and low-lying states of many nuclei are ellipsoidally deformed. Once the nucleus is deformed to an ellipsoid, this ellipsoid rotates and produces rotational bands, in the way analogous to the Nambu-Goldstone mechanism.

Type II Shell Evolution Among many aspects of the surface deformation, the discussions here are focused on the crossroad of the deformation and the shell evolution. The shell evolutions shown in Fig. 12b and c occur, respectively, due to the addition of two or four neutrons into the orbit j> . Instead of adding, one can put neutrons into the orbit j> by taking the neutrons out of some orbits below j> or equivalently by creating holes there, as shown in Fig. 12d. If such a lower orbit happens to be the j orbit. However, because holes are created in j (see Fig. 12d). Thus, the particle-hole (ph) excitation of the two neutrons in Fig. 12d reduces the proton j> -j< splitting even more than in Fig. 12b. This reduction becomes stronger with the ph excitations of four neutrons, as depicted in Fig. 12e. Such strong reduction of the spin-orbit splitting produces interesting consequences beyond shell structure changes. This type of the shell structure change within the same nucleus is called Type II shell evolution.

A Doubly Closed Nucleus 68 Ni The type II shell evolution was first discussed for 68 Ni in Tsunoda et al. (2014b). Figure 16 shows the theoretical and experimental energy levels of 68 Ni. The

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

5

(a) calc. 8

+

4 Ex (MeV)

+

4

+

6+

4 +

3 2

6

2+ 0+

2+ + 0

2211

(b) exp. 73654-

5

+

8 6+ + (4 ) 4+ + 2+ 0 + 2 + 0

-

7(6-) (3 ) 5

(5-) (4-)

1 0

0+ 0+ spherical oblate prolate negative positive negative shape shape shape parity parity parity

Fig. 16 Level scheme of 68 Ni. (Taken from Fig. 2 of Tsunoda et al. 2014b)

theoretical results were obtained with the A3DA-m interaction through the Monte Carlo shell model (MCSM) calculation. It is noted that the theoretical calculations presented in the following parts of this section are obtained by the A3DA-m interaction unless otherwise mentioned. Because Z=28 is an SO magic number and N =40 is an HO magic number (see Fig. 2), the ground state of 68 Ni is primarily a doubly closed shell. Indeed, in the theoretical ground state, the neutron pf -shell is completely occupied with a large probability (54%), although the neutron g9/2 orbit holds about one neutron in average (∼ 2× 0.46) as a consequence of the correlations due to the NN interaction. In contrast, the 0+ 3 state located at the excitation energy, Ex∼ 3 MeV, is the band head of a rotational band of an ellipsoidal shape. The neutron pf -shell closure is severely broken, with the occupation number of the neutron g9/2 orbit being as large as ∼4. The mechanism shown in Fig. 12e is emerging in this case, reducing the proton f5/2 -f7/2 splitting. A reduced splitting facilitates the configuration mixing between these two orbits, which can produce notable effects on the quadrupole deformation as stated below.

Deformed Shapes and Potential Energy Surface The energy of 68 Ni is illustrated in Fig. 17 (left) for various ellipsoidal shapes, spherical, prolate, oblate, and in between (called triaxial), which will be explained below. The energy is calculated by the constrained Hartree-Fock (CHF) calculation with the same shell-model Hamiltonian as in Fig. 16. The imposed constraints are given in terms of the quadrupole moments in the intrinsic (body-fixed) frame, denoted by Q0 and Q2 (Bohr and Mottelson 1975), to be discussed now. The ellipsoid “observed” in the intrinsic frame can be schematically illustrated as in Fig. 18 with the (x, y, z) axes of the intrinsic (body-fixed) frame. There is a

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Y. Tsunoda and T. Otsuka

68

ob de late fo rm at io

o e at ati ol pr form e d

n

n

spherical

Energy (MeV)

Ni

20 18 16 14 12 10 8 6 4 2 0 -400 -300 -200 -100

0

100

200

300

400

2

Quadrupole Moment, Q0 (fm )

Fig. 17 (left) Potential energy surface (PES) of 68 Ni. (Taken from Fig. 5 of Otsuka and Tsunoda 2016). (right) PES of 68 Ni for axially symmetric shapes. The solid line shows the PES of the full Hamiltonian, whereas the dashed line is the PES with practically no tensor force contribution. (Taken from Fig. 6 of Otsuka and Tsunoda 2016)

a. Legend

b. Rz > Rx > Ry

>

z

view A

view A

Rz

Rz

view B

c. γ = 60 : Rz = Rx > Ry

>

y

Ry

Rz

Rx

>

view A

Ry

x

Ry

Rx

Rx

Rx

Rx

view B

view B

Fig. 18 Schematic illustration of ellipsoids of uniform-density matter. The principal axes, Rx , Ry , and Rz of the x, y, and z axes are indicated, respectively. (a) Legend indicating that the views A and B are along the y and z axes, respectively. (b) The general case with Rz > Rx > Ry is displayed. (c) A special case with Rz = Rx > Ry is shown

freedom to choose these axes. In this article, the traditional convention is adopted: the (x, y, z) axes are assigned so that the three principal axes of the ellipsoid are placed on the (x, y, z) axes, with the relation Rz ≥ Rx ≥ Ry where (Rx , Ry , Rz ) denote, respectively, the lengths of the three principal axes (Bohr and Mottelson 1975). Figure 18b depicts the general situation of this assignment. If Rz ≥ Rx = Ry holds, the shape is called prolate. A special case of Rz = Rx > Ry is displayed in Fig. 18c, which corresponds to an oblate shape. If two of the principal axes have the same lengths, the shape is called axially symmetric. The shapes other than the prolate and oblate shapes are called triaxial, except for the trivial spherical shape.

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

2213

For a classical uniform density ellipsoid, the quadrupole moments are expressed as Q0 =

dxdydz (2z2 − x 2 − y 2 ) ρ0 ,

(34)

V

and Q2 =

dxdydz (x 2 − y 2 ) ρ0 ,

3/2

(35)

V

where ρ0 is the value of the uniform density and V to the integral symbol means the interior of the ellipsoid. With the above convention, both Q0 and Q2 are positive definite (or zero). Moving back to the quantum mechanics, for a given state η in the intrinsic frame, a matrix is calculated as ⎞ η | i (x 2 )i | η η | i (xy)i | η η | i (xz)i | η ⎝η | (yx)i | η η | i (y 2 )i | η η | i (yz)i | η⎠ , i η | i (zx)i | η η | i (zy)i | η η | i (z2 )i | η ⎛

(36)

where the index, i, runs over all valence nucleons. The (x, y, z) axes are redefined so that this hermitian matrix becomes diagonal. The principal axes are placed on the new (x, y, z) as to fulfill the relation among the eigenvalues: Rz ≥ Rx ≥ Ry  axes, so with Rz = (5/3)  i (z2 )i , etc. The two intrinsic quadrupole moments are then given, similarly to Eqs. (34) and (35), by Q0 = η|

 (2z2 − x 2 − y 2 )i |η ,

(37)

i

and Q2 =

3/2 η|

 (x 2 − y 2 )i |η ,

(38)

i

where the index, i, runs over all valence nucleons. (See, e.g., Ring and Schuck (1980) for more technical details about the intrinsic shapes.) Note that the nucleons in the closed shell do not contribute to the quadrupole moments. Because of the construction of the intrinsic frame, Q0 and Q2 are positive definite or zero. If the state η is invariant with respect to the rotation about the z axis of the intrinsic frame, the relation Q2 =0 holds. As Q0 ≥ 0, this quantum mechanical case is also called the prolate shape, consistently with the classical prolate case of Rx = Ry in Fig. 18b. Such axially symmetric prolate shapes have been believed to be dominant in heavy nuclei, but a different picture is emerging in recent years, as sketched later.

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Y. Tsunoda and T. Otsuka

Figure 18c depicts the axially symmetric oblate shapes: the wave function is invariant with respect to the rotation about the y axis. This case can be described, equivalently, by exchanging the Rz and Ry axes (see, for instance, Ring and Schuck (1980) for details). This exchange makes Q0 negative (becauseof Rz < Ry , Rx ) and Q2 = 0. In fact, by this exchange, Q0 is replaced by − Q20 + 2 Q22 . This expression of the oblate shape is sometimes used, as seen just below. It is noted that the physics does not change between the two ways of describing the ellipsoidal properties. The plot of the CHF energy against Q0 and Q2 is called the potential energy surface (PES).  Figure 17 (left) exhibits an example. The distance from the origin

stands for Q20 + 2 Q22 . The Q0 value increases from 0 along the axis toward the blue symbol, where Q2 = 0 is kept, representing prolate shapes. The Q2 value is indicated by the angle from the axis of Q0 > 0 and Q2 = 0, and the value of this angle is given by √ γ = arctan { 2 Q2 /Q0 } .

(39)

The γ value is positive or zero in the default convention, where Q0,2 ≥ 0. The range 0◦ ≤ γ ≤ 60◦ is sufficient (Ring and Schuck 1980). The axis toward the green object in Fig. 17 (left) corresponds to γ = 60◦ , which is nothing but the axially symmetric oblate case and can be expressed by negative Q0 and Q2 =0 as discussed just above. The minimum energy in Fig. 17 (left) is found at the spherical shape (red sphere), where Q0 =Q2 =0. The constraints are changed to a more prolate deformed ellipsoid (blue object in Fig. 17(left) along the upper-right axis (“prolate deformation” in the figure), where Q0 increases but Q2 =0 is kept. This means that the vertical axis (z axis) of the blue object is the stretched direction of the ellipsoid, and the cross section of the ellipsoid perpendicular to this axis is a circle. The green object in Fig. 17(left) is shrunk in the y axis, and the cross section perpendicular to it is a circle, an oblate axially symmetric deformation. The green object is placed so that its y axis is vertical. The area in between represents a wide area of triaxial deformations. In most of the discussions of this section, the axially symmetric deformation is assumed for the sake of simplicity, but the triaxiality is discussed later for other nuclei. Figure 17(left) shows that as Q0 is increased from Q0 =0 to the upper right direction, the energy relative to the minimum energy climbs up by 6 MeV. This is because protons and neutrons must be excited across the magic gaps from the doubly closed shell in order to create states producing imposed Q0 values. After reaching the local peak or the “pass,” the energy starts to come down. It is lowered by 3 MeV from this “pass” to a “basin,” or the area around the local minimum. This lowering is due to the quadrupole interaction. Beyond the basin, the effect of the quadrupole interaction is saturated, and it cannot compete the energy needed for exciting more protons and neutrons across the gaps to fulfill the constraints. This explanation is the usual one for the appearance of the deformed local minimum in

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

2215

the PES. The appearance of two (or more) different shapes with rather small energy difference is one of the phenomena frequently seen in atomic nuclei and is called the shape coexistence (see a review Heyde and Wood (2011), for instance). The shape coexistence is seen, in the energy levels and the PES, between the spherical ground state and the prolate deformed excited state now. The quadrupole interaction is undoubtedly among the essential factors of the shape coexistence. But this may not be a full story. Figure 17 (right) exhibits the same energy along the axes of the left panel. In the right half of the right panel, Q0 is varied from 0 fm2 to 400 fm2 while Q2 =0 is kept. The left half of the right panel stands for γ = 60◦ , oblate shapes. As discussed above, another convention with the y and z exchanged is taken so that the ellipsoidal deformation is described with negative Q0 values and Q2 = 0. Thus, Fig. 17 (right) shows Q0 from −400 fm2 to 400 fm2 , which actually represents the two axes in Fig. 17 (left) connecting the green to the red and to the blue objects. The red solid line shows the CHF results of the full Hamiltonian, whereas for the dashed line, the tensor monopole interactions between the neutron (g9/2 , f5/2 ) orbits and the proton (f5/2 , f7/2 ) orbits are practically removed. This removal means no effects depicted in Fig. 12d, e. The dashed line displays a less pronounced prolate local minimum at weaker deformation with much higher excitation energy. The significant difference between the solid and dashed lines suggests that the monopole effects are crucial to lower this local minimum and stabilize it. Some details of the mechanism for this difference are touched upon now. With the tensor monopole interaction, once sufficient neutrons are in g9/2 , the proton f5/2 -f7/2 splitting is reduced. As the quadrupole interaction generally tends to mix different single-particle orbits, this reduced splitting facilitates the effects driven by the quadrupole interaction, leading to stronger quadrupole deformations. Thus, type II shell evolution increases the deformation. The tensor monopole interaction involving the neutron g9/2 orbit produces extra binding energy, if more protons are in f5/2 and less are in f7/2 . This extra binding energy lowers the deformed states, otherwise they are high in energy because of the energy cost for promoting neutrons from the pf shell to g9/2 due to single-particle energies. Thus, a strong interplay emerges between the monopole interaction and the quadrupole interaction, and type II shell evolution strengthens this interplay. It enhances the deformation and lowers the energy of deformed states. Without this interplay, as indicated by blue dashed line in Fig. 17 (right), the local minimum is pushed up by 2 MeV and may be dissolved into the sea of many other states not shown in the PES. It is obvious that this interplay mechanism works selfconsistently.

Deformation Parameters and Comparison to Calculations with Gogny Interaction The deformation parameters, β2 and γ , are introduced now, as more direct measures of the quadrupole deformation carried by wave functions.

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Y. Tsunoda and T. Otsuka

The axis lengths, Rz , Ry , and Rx , are parametrized (Bohr and Mottelson 1975) by variables, R0 , β2 , and γ , Rz = {1 + 0.63 β2 cos γ } R0 ,

(40)

Rx = {1 + 0.63 β2 sin (γ − 30◦ )} R0 ,

(41)

Ry = {1 − 0.63 β2 cos (60◦ − γ )} R0 ,

(42)

where R0 is the average of Rx,y,z and β2 , called deformation parameter, represents the magnitude of the ellipsoidal deformation from the sphere (β2 =0). The angle γ is the same as the one introduced in Eq. (39), as explained below. It is known that the range γ = 0◦ − 60◦ suffices to specify the ratios among Rx , Ry , and Rz . The parameter β2 represents the magnitude of the ellipsoidal deformation from a sphere. The β2 is sometimes denoted by β for brevity. The value of β2 can be obtained, in some approximation, from Q0 and Q2 obtained in the shell model calculation. Such Q0 and Q2 values are appropriate for analyzing the properties of the shell-model states formulated in the given model space. The β2 value, however, represents the actual shape of the nucleus and is connected to its actual quadrupole moments. As the shell-model quadrupole moments are transformed to the actual values by in-medium (e.g., core-polarization) corrections, the β2 value is obtained, in a reasonable approximation, through the formula (Utsuno et al. 2015), β2 =



5/16π {(e + ep + en )/e} (4π/3R02 A5/3 ) (Q0 )2 + 2(Q2 )2 ,

(43)

where e is the unit charge, ep (en ) denotes proton (neutron) effective charge induced by in-medium effects, and R0 stands for the radius parameter of the droplet model (see Otsuka et al. (2022a) for some detailed explanation). Here, Q0 and Q2 imply the quadrupole moments of the valence-nucleon system, and in-medium correction is needed as represented by the multiplication factor (e + ep + en )/e, where the ep and en appear in symmetric way, because the matter quadrupole moments are relevant rather than the charge ones. The value of the parameter γ does not change from the one in Eq. (39), because Q0 and Q2 are rescaled by the same amount. The relations in Eqs. (43) and (39) have been shown to work very well in many studies, for instance, Leoni et al. (2017), M˘arginean et al. (2020), Marsh et al. (2018), Otsuka et al. (2019), Tsunoda and Otsuka (2021). Figure 19b shows the PES for 68−78 Ni as a function of β2 (β in the figure) assuming the axial symmetry, in comparison to the corresponding PES obtained by the calculation with the Gogny interaction (Girod et al. 1988). The convention of Q0 < 0 ( i.e., β2 < 0) and Q2 = 0 (i.e., γ = 0) is taken for axially symmetric oblate shapes, with β2 ranging from ∼ − 0.5 to ∼0.5. The energy minimum for 68 Ni is adjusted to be equal between the two calculations, as indicated by blue thin solid line. Both the present and the Gogny calculations produce similar PES curves near the minima. The local bump appears around β2 = 0.2−0.3 in the prolate side in both

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model a Gogny

2217

b A3DA-m 70 68 66 64 62 60

68

Ni

58 56 54 52 50 48

70

Ni

46 44 42 40

E (MeV)

38 36

72

Ni

34 32 30 28 26

74

Ni

24 22 20 18

76

Ni

16 14 12 10 8

78

Ni

6 4 2 0 -2 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1

0

0.1 0.2 0.3 0.4 0.5 0.6

β

Fig. 19 Potential energy surface (PES) of 68−78 Ni with axially symmetric shapes. (a) PES by the Gogny interaction. (Taken from Fig. 4 of Girod et al. 1988). (b) PES by the A3DA-m interaction. The horizontal thin blue, thick red, and pink dashed lines indicate, respectively, the lowest energy, the prolate local minimum in panel (b), and the same one in panel (a), for 68 Ni

calculations. The major difference arises for the local minimum around β2 ∼ 0.4: its excitation energy appears to be higher than 5 MeV for the Gogny calculation (pink dashed line in Fig. 19), whereas it is lower than 3 MeV in the present calculation (red solid line in Fig. 19). This notable difference is ascribed, to a large extent, to the type II shell evolution driven by the tensor interaction. The tensor interaction is included in the present calculation, whereas not (at least explicitly) in the Gogny calculation. This observation is consistent with the major conclusion obtained from Fig. 17 (right). The difference between the two calculations becomes smaller as moving from 68 Ni to 78 Ni. This trend is natural because the neutron excitations from the pf -shell are suppressed as the 1g9/2 orbits are filled by more neutrons, and consequently the type II shell evolution occurs more weakly. It is noted that the oblate local minima gain the energy through the monopole tensor force, but the effect is smaller.

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Y. Tsunoda and T. Otsuka

Fig. 20 PES and T-plot for 68−78 Ni. (Taken from Fig. 3 of Tsunoda et al. 2014b )

T-Plot Analysis The PES discussed so far is very useful but cannot pin down shapes carried by individual MCSM wave functions. This subsection is devoted to a powerful methodology for this purpose, called T-plot (Tsunoda et al. 2014b). Each basis vector Φi (J π ) in Eq. (14) has intrinsic quadrupole moments, ˆ 0 |Φi (J π ) and Φi (J π )|Q ˆ 2 |Φi (J π ), Φi (J π )|Q

(44)

ˆ 0,2 imply the operators for Q0,2 mentioned above. It is assumed that the where Q axes of the intrinsic frame are taken according to the default convention, i.e., the traditional way. The T-plot circle for Φi (J π ) is placed according to those values on the PES. The area of the circle is proportional to the overlap probability with the eigenstate of interest, Ψ , as shown in Eq. (14). Such T-plot circles are displayed in Fig. 20 for 68−78 Ni. The small yellow circles are located at the (Q0 , Q2 ) values of MCSM basis vectors, with their sizes representing their relevance to the eigenstate, as stated above. With the relations in Eqs. (43) and (39), the T-plot can utilize the (β2 , γ ) as partial but useful labeling of the basis vector in Eq. (14). The fully correlated eigenstates are described in terms of (β2 , γ ) with their mean values and fluctuations with respect to quadrupole shapes. The evolution of nuclear shapes is clearly seen in Fig. 20. Certain advantages of mean-field approaches are now nicely incorporated into the shell model. Figure 20 shows that the T-plot circles are concentrated near the spherical limit for the ground state of 68 Ni. The 68 Ni nucleus has three 0+ states of different

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

2219

Fig. 21 PES and T-plot for 66 Ni. (Taken from Fig. 1 of Leoni et al. 2017)

+ characters: spherical, oblate, and prolate. The 2+ 1 and 22 states depict, respectively, + + T-plot patterns similar to those of the 02 and 03 states, as a clear indication of the formation of collective bands. The prolate local minimum disappears in 74 Ni and beyond. The 78 Ni nucleus shows a steep minimum, but the T-plot pattern of the ground state shows notable fluctuations. The fluctuations in 78 Ni are stronger than those in 68 Ni, and this difference is understood by the jj (LS) neutron closed shell for 78 Ni (68 Ni) where the dynamical quadrupole deformation occurs (does not occur) over the magic gap at N =50 (40). This subtlety of magic SO and HO magic numbers is a fascinating feature. Lighter Ni isotopes have been studied with the same Hamiltonian as the one for Fig. 20. The obtained T-plot circles are shown, in a three-dimensional plot, in Fig. 21 for 66 Ni (Leoni et al. 2017). The same set of the MCSM basis vectors is used for the four 0+ states. The white circles represent the MCSM basis vectors mainly forming the ground state, while the red circles indicate those mainly composing the 0+ 4 state, which is strongly deformed. Although there is no local minimum for oblate shapes, the T-plot circles of the major MCSM basis vectors for the 0+ 2 state are concentrated around moderately oblate shapes (green circles). It is mentioned that the intrinsic shapes are also analyzed, in the method of Poves et al. (2020), within the conventional shell model by using Kumar invariants of quadrupole matrix elements for a set of the eigenstates belonging to the (almost) same shape. It is of interest how the prolate deformed state changes its excitation energy as the neutron number, N , is varied within the Ni isotopes. Figure 22 displays + + the calculated excitation energies of the 2+ 1 , oblate 0 , and prolate 0 states, in comparison to experimental counterparts, for example, those obtained by Flavigny et al. (2015), Leoni et al. (2017), M˘arginean et al. (2020), Morales et al. (2017), where the shape assignment may be subject to further confirmation. The 2+ 1 levels go down fast as N departs from N=40 in both sides (except for N =50) and then stay

2220

Y. Tsunoda and T. Otsuka

prolate 0+

7

no tensor

6

quadratic modeling

Ex (MeV)

5

oblate 0+ 2+

4 3 2 1 0 34

36

38

40

42

44

46

48

50

N

Fig. 22 Excitation energies for 64−78 Ni. Symbols are experimental data. Lines are theoretical calculations. The excitation energies of strongly prolate deformed states are shown by red squares and line. Those of moderately oblate deformed states are shown by blue squares and line. Black circles and line indicate the excitation energies of the 2+ 1 states. The estimated excitation energies of strongly prolate deformed states calculated without major monopole contributions of the tensor force are shown by red dashed line. Pink dotted line indicates the simple modeling of deformed intruder states (see the text), with parameters fitted to the estimated values without the tensor force effect

nearly constant, as reproduced well by the present calculation. It is noted that the levels of 78 Ni are calculated by another Hamiltonian (see Taniuchi et al. 2019). The prolate deformed 0+ state shows a pattern of particular interest. As this state gains binding energies from the deformation, it is expected to come down as N moves away from the N=40 magic number, because the deformation tends to become stronger toward the middle of the shell, which is considered to be at N=34, a half way down to N=28. In fact, this 0+ state comes down in energy as N increases from 40. On the contrary, the prolate 0+ state is shifted higher in energy as N decreases from 40, and moreover this trend is described well by the present MCSM calculation. In order to understand the meanings of this trend unexpected in a sense, the excitation energy is estimated with the Hamiltonian without major tensor force effects (Otsuka and Tsunoda 2016). Figure 17 depicts that the excitation energy of the prolate state of 68 Ni is raised by ∼4 MeV if the tensor force effect is cut off. Similarly, the excitation energies without major tensor force effects are estimated for 64,66 Ni as shown by red dashed line in Fig. 22. The trend can be compared to what is expected conventionally. In fact, the conventionally expected trend is modeled by a parabola like Ex ∼ a(N − Nc )2 + b where a and b are parameters, N denotes the neutron number, and Nc implies N for the middle of the relevant shell. In the present case, the major shell N = 28 − 40 is taken, yielding Nc =34. The pink dotted line represents how such a simple model looks like, where the present result for N=36 and the average of them for N =38 and 40 are used to fix the parameters

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

2221

a and b. As this is an extremely simple modeling, its predictions cannot be very quantitative. Nevertheless, it is of interest that for N =34, its prediction is close to the extrapolation of the red line, the actual MCSM results. It is natural that type II shell evolution is weakened as N becomes smaller from N=40, because the neutron Fermi surface becomes far below the g9/2 orbit reducing neutron excitations from the pf -shell to the g9/2 orbit. In this way, the “unexpected” trend of the prolate-state excitation energy for N < 40 can be naturally understood. Thus, the present systematic trend further supports the crossroad scenario between the monopole and quadrupole (deformation) effects. It is of interest to clarify the trends further away in both sides of Fig. 22 as well as in other regions of the nuclear chart.

Shell Evolution and Surface Deformation The shell evolution discussed earlier in this article can, in principle, be washed away by correlations produced by the shell-model NN interaction. The inversion − between the 3/2− 1 and 5/21 states in Cu isotopes is discussed from the viewpoint of the straightforward shell evolution in Fig. 13b. This inversion can be studied in the presence of such correlations. Figure 23 shows the energy of the 5/2− 1 state relative to the 3/2− 1 state as a result of the full diagonalization of the A3DA-m Hamiltonian, − as a function of the neutron number. The spacing between the 3/2− 1 and 5/21 states is reduced from the shell-evolution estimate due to the monopole interaction (see “shell evolution” in Fig. 13b). This reduction is basically a consequence of the correlations due to the multipole interaction in the shell-model calculation, which are primarily, in this case, the excitations of the corresponding Ni core. The magnitude of these correlation effects does not change much as a function of the neutron number. In fact, if the monopole interaction relevant to this inversion is removed from the NN interaction, the shell model calculation produces the excitation energy of the 5/2− 1 state equal to 1–1.5 MeV (see “core excitation” in Fig. 13b), leading to no inversion. Thus, it is clear that the shell evolution mechanism lowers the 5/2− 1 state as the neutron number increases, while the core excitation yields a kind of background shift, being constant to a good extent, of the excitation energy. More details of the core excitation can be visualized by T-plots as shown in Fig. 24. The T-plot is quite similar between a given Cu isotope and its Ni core. This similarity suggests that the odd proton does not disturb the Ni core too much, which − is consistent with the almost constant shift of the 3/2− 1 -5/21 energy difference, shown in Fig. 23. The validity of the MCSM wave functions is verified by the magnetic moments (see Fig. 25), where the experimental values agree with the MCSM value. The prolate strongly deformed states of 68,70 Ni isotopes discussed above are formed on top of massive particle-hole excitations. Figure 24 suggests that no − T-plot circles appear for large (Q0 , Q2 ) values in the 3/2− 1 or 5/21 state. Modest

2222

Excitation energy [keV]

1500

(a) 5/2 -

(1/2-, 3/2-)

1/2 -

(1/2-, 3/2-)

66.2 keV T1/2=149 ns 61.7 keV

1000 5/2-

T1/2=310 ns

75

Cu

(5/2 -)

500 (1/2 -)

(3/2 -) (5/2 -)

0

3000

Energy [keV]

Fig. 23 Shell evolution in neutron-rich Cu isotopes. (a) Experimental systematics of energy levels for odd-A Cu isotopes. Inset exhibits experimental details. (b) Calculated energy levels of the 5/2− 1 state relative to the 3/2− 1 state, in three different methods with the same Hamiltonian, A3DA-m. The red squares depict the MCSM calculation with full of correlations, whereas the blue line indicates the shell evolution assuming the simple filling configuration in the g9/2 orbit of neutrons. The green dashed line represents the MCSM calculation without major shell evolution effects. (Taken from Fig. 1 of Ichikawa et al. 2019)

Y. Tsunoda and T. Otsuka

3/2 -

3/2 -

(1/2 -)

3/2 -

(b)

5/2 -

5/2 -

full calculation shell evolution core excitation

2000

1000

0 69Cu

71Cu

73Cu

75Cu

77Cu

particle-hole excitations are seen experimentally in 77 Cu (Sahin et al. 2017) and 79 Cu (Olivier et al. 2018) isotopes with rather clear separation from the low-lying − the 3/2− 1 and 5/21 states, suggesting that the Z=28 gap still remains for such lowest states as shown in Fig. 13b.

Short Summary of This Section Type II shell evolution occurs in various cases, especially in a number of shape coexistence cases, providing deformed intruder states with stronger deformation, lower excitation energies, and more stabilities. It is an appearance of the monopolequadrupole interplay and plays crucial roles in various phenomena in other nuclei including the first-order phase transition (Zr isotopes Kremer et al. 2016; Singh et al. 2018; Togashi et al. 2016), the second-order phase transition (Sn isotopes Togashi et al. 2018), and the multiple even-odd phase transitions (Hg isotopes Marsh et al. 2018). The shell model can describe the monopole-quadrupole interplay, including type II shell evolution, as far as the model space is sufficiently wide and the Hamiltonian is realistic. This advantage becomes more important in other subjects of nuclear structure, being one of the front lines of the shell model studies. In some of such cutting-edge calculations, the merging of conventional shells occurs as shown in Fig. 4b, which may open new scopes but may bring more difficulties to actual computations.

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

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a. Schematic view : proton and core (a-1) 69Cu

p

(a-2) 75Cu

68

Ni

p

(a-3) 79Cu

74

Ni

p

78

Ni

b. 3/2− states in Cu isotopes (b-1) 69Cu

ellipsoidal shapes ellip

(b-2) 75Cu

oblate prolate te Q (fm

2

2)

Q

m (f

)

spherical 0

c. 5/2− states in Cu isotopes (c-1) 75Cu

(c-2) 79Cu

d. Ground states (0＋) in Ni isotopes (d-1) 68Ni

(d-2) 74Ni

(d-3) 78Ni

Fig. 24 T-plots of Ni and Cu isotopes. (a) Schematic illustrations of the Ni core and the odd − proton. (b, c) For 69,75,79 Cu isotopes, the 3/2− 1 and 5/21 states are shown. (d) The corresponding Ni ground states are taken. (Taken from Fig. 4 of Ichikawa et al. 2019)

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Magnetic moment [ μN ]

Fig. 25 Magnetic moments of Cu isotopes. “This work” in the figure means the measured value reported in Ichikawa et al. (2019). (Taken from Fig. 3 of Ichikawa et al. 2019)

Y. Tsunoda and T. Otsuka

μ(π p3/2) 3/2-

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μ(π f5/2) Exp. 3/2Exp. 5/2Calc. 3/2Calc. 5/2-

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0 69

Cu

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Remarks The major purpose of this article is to present a quick overview of the contemporary shell model studies, starting from a scratch. Quite often, for the shell model, computed numbers and computational feasibility are regarded as the shell model itself, but this does not fully cover what the shell model can provide. Although the computational capability of the shell model is very important and even crucial to some studies of nuclear physics, the shell model has provided and can provide us with clear, useful and sometimes novel pictures or ideas for a variety of facets of the nuclear structure. These merits have been developed partly by importing models and concepts of other theories of nuclear structure such as mean field models. In this sense, the contemporary shell model is a combined product of various models and theories of nuclear structure. Along this line, there are quite a few developments in recent years, including new pictures of collective rotational bands in heavy nuclei (Otsuka et al. 2019), new mechanism of neutron driplines (Tsunoda et al. 2020), and new view of the α-clustering in light nuclei (Otsuka et al. 2022b). These are reviewed in a compact form (Otsuka 2022) or reported very recently, and are not discussed in this article. Acknowledgments This work was supported in part by MEXT as “Program for Promoting Researches on the Super computer Fugaku” (Simulation for basic science: from fundamental laws of particles to creation of nuclei) and by JICFuS. This work was supported by JSPS KAKENHI Grant Numbers JP19H0514, JP21H00117.

References M. Agostini, G. Benato, J.A. Detwiler, J. Menéndez, F. Vissani, Toward the Discovery of Matter Creation with Neutrinoless Double-Beta Decay (2022). arXiv:2202.01787v1 [hep-ex] K. Arnswald et al., Phys. Lett. B 772, 599 (2017) A. Arima, S. Cohen, R.D. Lawson, M.H. Macfarlane, Nucl. Phys. A 108, 94 (1968) S.-O. Bäckman, G.E. Brown, J.A. Niskanen, Phys. Rep. 124, 1 (1985) R.K. Bansal, J.B. French, Phys. Lett. 11, 145 (1964)

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

2225

M. Baranger, Nucl. Phys. A 149, 225 (1970) G. Bertsch, J. Borysowicz, H. McManus, W.G. Love, Nucl. Phys. A 284, 399 (1977) D.R. Bes, R.A. Sorensen, Advances in Nuclear Physics, vol. 2, ed. by M. Baranger, E. Vogt (Plenum, New York, 1969; Springer, Berlin, 2012), pp. 129–222 S. Bogner, T.T.S. Kuo, L. Coraggio, A. Covello, N. Itaco, Phys. Rev. C 65, 051301 (2002) S.K. Bogner, R.J. Furnstahl, A. Schwenk, Prog. Part. Nucl. Phys. 65, 94 (2010) N. Bohr, Nature 137, 344 (1936a) N. Bohr, Nature 137, 351 (1936b) N. Bohr, Science 86, 161 (1937) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, 14 (1952) A. Bohr, Nobel Lectures, Physics 1971–1980, ed. by S. Lundqvist (World Scientific, Singapore, 1992), pp. 213–232 A. Bohr, B.R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, 16 (1953) A. Bohr, B.R. Mottelson, Nuclear Structure I (Benjamin, New York, 1969) A. Bohr, B.R. Mottelson, Nuclear Structure II (Benjamin, New York, 1975) B.A. Brown, Phys. Rev. Lett. 85, 5300 (2000) B.A. Brown, W.D.M. Rae, Nucl. Data Sheets 120, 115 (2014) B.A. Brown, W.A. Richter, Phys. Rev. C 74, 034315 (2006) B.A. Brown, B.H. Wildenthal, Annu. Rev. Nucl. Part. Sci. 38, 29 (1988) B.A. Brown, N.J. Stone, J.R. Stone, I.S. Towner, M. Hjorth-Jensen, Phys. Rev. C 71, 044317 (2005) J. Carlson et al., Rev. Mod. Phys. 87, 1067 (2015) E. Caurier, F. Nowacki, Acta Phys. Polon. B 30, 705 (1999) E. Caurier, A. Zuker, A. Poves, G. Martiínez-Pinedo, Phys. Rev. C 50, 225 (1994) E. Caurier, G. Martínez-Pinedo, F. Nowacki, A. Poves, A.P. Zuker, Rev. Mod. Phys. 77, 427 (2005) S. Chen et al., Phys. Rev. Lett. 123, 142501 (2019) S. Cohen, D. Kurath, Nucl. Phys. 73, 1 (1965) L. Coraggio, A. Covello, A. Gargano, N. Itaco, Phys. Rev. C 80, 021305 (2009) J.P. Elliott, Proc. R. Soc. 245, 128 (1958a) J.P. Elliott, Proc. R. Soc. 245, 562 (1958b) Evaluated nuclear structure data file, http://www.nndc.bnl.gov/ensdf/ P. Federman, S. Pittel, Phys. Lett. B 69, 385 (1977) F. Flavigny et al., Phys. Rev. C 91, 034310 (2015) J. Fujita, H. Miyazawa, Prog. Theor. Phys. 17, 360 (1957) A. Gade, T. Glasmacher, Prog. Part. Nucl. Phys. 60, 161 (2008) M. Girod, P. Dessagne, M. Langevin, F. Pougheon, P. Roussel, Phys. Rev. C 37, 2600 (1988) H. Grawe, in The Euroschool Lectures on Physics with Exotic Beams, vol. I, ed. by J. Al-Khalili, E. Roeckl (Springer, Berlin/Heidelberg, 2004), pp. 33–75 H.-W. Hammer, A. Nogga, A. Schwenk, Rev. Mod. Phys. 85, 197 (2013) O. Haxel, J.H.D. Jensen, H.E. Suess, Phys. Rev. 75, 1766 (1949) H. Hergart, S.K. Bogner, T.D. Morris, A. Schwenk, K. Tsukiyama, Phys. Rep. 621, 165 (2016) K. Heyde, The Nuclear Shell Model (Springer, Berlin, 1994) K. Heyde, Basic Ideas and Concepts in Nuclear Physics: An Introductory Approach (Institute of Physics Publishing, London, 2004) K. Heyde, J.L. Wood, Rev. Mod. Phys. 83, 1467 (2011) M. Hjorth-Jensen, T.T.S. Kuo, E. Osnes, Phys. Rep. 261, 125 (1995) M. Honma, T. Mizusaki, T. Otsuka, Phys. Rev. Lett. 75, 1284 (1995) M. Honma, T. Otsuka, B.A. Brown, T. Mizusaki, Phys. Rev. C 65, 061301(R) (2002) M. Honma, T. Otsuka, B.A. Brown, T. Mizusaki, Phys. Rev. C 69, 034335 (2004) M. Honma, T. Otsuka, T. Mizusaki, M. Hjorth-Jensen, Phys. Rev. C 80, 064323 (2009) A. Huck, G. Klotz, A. Knipper, C. Miehé, C. Richard-Serre, G. Walter, A. Poves, H.L. Ravn, G. Marguier, Phys. Rev. C 31, 2226 (1985) Y. Ichikawa et al., Nat. Phys. 15, 321 (2019) R.V.F. Janssens, Nature 435, 897 (2005) C.W. Johnson, W.E. Ormand, P.G. Krastev, Comput. Phys. Commun. 184, 2761 (2013)

2226

Y. Tsunoda and T. Otsuka

C. Kremer et al., Phys. Rev. Lett. 117, 172503 (2016) K. Kumar, M. Baranger, Nucl. Phys. A 110, 529 (1968) T.T.S. Kuo, G.E. Brown, Nucl. Phys. 85, 40 (1966) S.M. Lenzi, F. Nowacki, A. Poves, K. Sieja, Phys. Rev. C 82, 054301 (2010) S. Leoni et al., Phys. Rev. Lett. 118, 162502 (2017) S.N. Liddick et al., Phys. Rev. Lett. 97, 082501 (2006) R. Machleidt, D.R. Entem, Phys. Rep. 503, 1 (2011) B.A. Marsh et al., Nat. Phys. 14, 1163 (2018) M.G. Mayer, Phys. Rev. 75, 1969 (1949) S. Michimasa et al., Phys. Rev. Lett. 121, 022506 (2018) A.I. Morales et al., Phys. Lett. B 765, 328 (2017) S. M˘arginean et al., Phys. Rev. Lett. 125, 102502 (2020) T. Nakamura, H. Sakurai, H. Watanabe, Prog. Part. Nucl. Phys. 97, 53 (2017) A. Nogga, S.K. Bogner, A. Schwenk, Phys. Rev. C 70, 061002 (2004) L. Olivier et al., Phys. Rev. Lett. 119, 192501 (2018) F. Osterfeld, Rev. Mod. Phys. 64, 491 (1992) T. Otsuka, Physics 4, 258 (2022) T. Otsuka, Y. Tsunoda, J. Phys. G 43, 024009 (2016) T. Otsuka, T. Mizusaki, M. Honma, Phys. Rev. Lett. 81, 1588 (1998) T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu, Y. Utsuno, Prog. Part. Nucl. Phys. 47, 319–400 (2001) T. Otsuka, R. Fujimoto, Y. Utsuno, B.A. Brown, M. Honma, T. Mizusaki, Phys. Rev. Lett. 87, 082502 (2001) T. Otsuka, T. Suzuki, R. Fujimoto, H. Grawe, Y. Akaishi, Phys. Rev. Lett. 95, 232502 (2005) T. Otsuka, T. Suzuki, M. Honma, Y. Utsuno, N. Tsunoda, K. Tsukiyama, M. Hjorth-Jensen, Phys. Rev. Lett. 104, 012501 (2010) T. Otsuka, T. Suzuki, J.D. Holt, A. Schwenk, Y. Akaishi, Phys. Rev. Lett. 105, 032501 (2010) T. Otsuka, Y. Tsunoda, T. Abe, N. Shimizu, P. Van Duppen, Phys. Rev. Lett. 123, 222502 (2019) T. Otsuka, A. Gade, O. Sorlin, T. Suzuki, Y. Utsuno, Rev. Mod. Phys. 92, 015002 (2020) T. Otsuka, N. Shimizu, Y. Tsunoda, Phys. Rev. C 105, 014319 (2022) T. Otsuka, T. Abe, T. Yoshida, Y. Tsunoda, N. Shimizu, N. Itagaki, Y. Utsuno, J. Vary, P. Maris, H. Ueno, Nat. Commun. 13, 2234 (2022) A. Poves, A. Zuker, Phys. Rep. 70, 235 (1981) A. Poves, F. Nowacki, Y. Alhassid, Phys. Rev. C 101, 054307 (2020) I. Ragnarsson, S. Nilsson, Shapes and Shells in Nuclear Structure (Cambridge University Press, Cambridge, 1995) J. Rainwater, Phys. Rev. 79, 432 (1950) W.A. Richter, S. Mkhize, B.A. Brown, Phys. Rev. C 78, 064302 (2008) P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin, 1980) E. Sahin et al., Phys. Rev. Lett. 118, 242502 (2017) J.P. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004) D. Seweryniak et al., Phys. Rev. Lett. 99, 022504 (2007) N. Shimizu, Genshikaku Kenkyu (in Japanese) 58, 96 (2013) N. Shimizu, Y. Utsuno, T. Mizusaki, T. Otsuka, T. Abe, M. Honma, Phys. Rev. C 82, 061305(R) (2010) N. Shimizu, T. Abe, Y. Tsunoda, Y. Utsuno, T. Yoshida, T. Mizusaki, M. Honma, T. Otsuka, Prog. Theor. Exp. Phys. 2012, 01A205 (2012) N. Shimizu, T. Abe, M. Honma, T. Otsuka, T. Togashi, Y. Tsunoda, Y. Utsuno, T. Yoshida, Phys. Scr. 92, 063001 (2017) N. Shimizu, T. Mizusaki, T. Utsuno, Y. Tsunoda, Comput. Phys. Commun. 244, 372 (2019) N. Shimizu, Y. Tsunoda, Y. Utsuno, T. Otsuka, Phys. Rev. C 103, 014312 (2021) P. Singh et al., Phys. Rev. Lett. 121, 192501 (2018) N.A. Smirnova, B. Bally, K. Heyde, F. Nowacki, K. Sieja, Phys. Lett. B 686, 109 (2010) O. Sorlin, M.-G. Porquet, Prog. Part. Nucl. Phys. 61, 602 (2008)

58 Configuration Interaction Approach to Atomic Nuclei: The Shell Model

2227

D. Steppenbeck et al., Nature 502, 207 (2013) M. Storm, A. Watt, R. Whitehead, J. Phys. G 9, L165 (1983) K. Takayanagi, Nucl. Phys. A 852, 61 (2011a) K. Takayanagi, Nucl. Phys. A 864, 91 (2011b) I. Talmi, Rev. Mod. Phys. 34, 704 (1962) R. Taniuchi et al., Nature 569, 53 (2019) T. Togashi, Y. Tsunoda, T. Otsuka, N. Shimizu, Phys. Rev. Lett. 117, 172502 (2016) T. Togashi, Y. Tsunoda, T. Otsuka, N. Shimizu, M. Honma, Phys. Rev. Lett. 121, 062501 (2018) Y. Tsunoda, T. Otsuka, Phys. Rev. C 103, L021303 (2021) N. Tsunoda, K. Takayanagi, M. Hjorth-Jensen, T. Otsuka, Phys. Rev. C 89, 024313 (2014) Y. Tsunoda, T. Otsuka, N. Shimizu, M. Honma, Y. Utsuno, Phys. Rev. C 89, 031301(R) (2014) N. Tsunoda, T. Otsuka, N. Shimizu, M. Hjorth-Jensen, K. Takayanagi, T. Suzuki, Phys. Rev. C 95, 021304(R) (2017) N. Tsunoda, T. Otsuka, K. Takayanagi, N. Shimizu, T. Suzuki, Y. Utsuno, S. Yoshida, H. Ueno, Nature 587, 66 (2020) Y. Utsuno, N. Shimizu, T. Otsuka, T. Yoshida, Y. Tsunoda, Phys. Rev. Lett. 114, 032501 (2015)

Symmetry Restoration Methods

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Jiangming M. Yao

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry and Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry Breaking in Mean-Field Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry Restoration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generator Coordinate Method (GCM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction of Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Typical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximate Treatments with Power Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electric Multiple Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Illustrative Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact of Particle-Number Projection in Low-Lying States . . . . . . . . . . . . . . . . . . . . . . . . The Densities of Symmetry-Restored States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Correlation Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triaxiality in Atomic Nuclei with Shape Coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical and Static Octupole Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of Shell Structure in Neutron-Rich Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Symmetry techniques based on group theory play a prominent role in the analysis of nuclear phenomena and in particular in the understanding of observed regular patterns in nuclear spectra and selection rules for electromagnetic transitions. A variety of symmetry-based nuclear models have been developed in nuclear physics, providing efficient tools of choice to interpret nuclear spectroscopic

J. M. Yao () School of Physics and Astronomy, Sun Yat-sen University, Zhuhai, China e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_18

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data. This chapter provides a pedagogical introduction to the basic idea of symmetry-breaking mechanism and symmetry-restoration methods in modeling atomic nuclei.

Introduction Symmetries and conservation laws are indispensable keys to understanding the structure of atomic nuclei the dynamics of which are described fundamentally in terms of quarks and gluons interacting with the strong interaction governed by the theory of quantum chromodynamics. Due to the non-perturbative nature of the strong interaction in low-energy region, atomic nuclei are usually modeled in terms of nucleon degrees of freedom instead. Therefore, the symmetries of atomic nuclei are determined by the Hamiltonian composed of nucleon-nucleon interactions. Like those of many macroscopic systems, nuclear Hamiltonian possesses the symmetries associated with geometric transformations in space-time coordinates, including translational and rotational symmetries, and space-inversion invariance. Furthermore, atomic nucleus is a quantum many-body system that is also characterized by the symmetries defined in abstract spaces, such as the spin and isospin symmetries as proposed by Wigner (1937) and the rotation invariance in Fock space corresponding to particle-number conservation. The exploration of these symmetries and their mathematical representations is essential to modeling atomic nuclei as it provides not only constraints on the nucleon-nucleon interactions, but also guidelines on the choices of model spaces for nuclear wave functions. The latter will be the main focus of this chapter. The exact solution to a nuclear many-body problem is computational challenging. One of the important approaches to atomic nuclei is the interacting shell model which represents nuclear wave function as a superposition of all possible symmetry-conserving configurations. The shell model has achieved great success in understanding the structure of atomic nuclei, but its applicability has been strongly limited by the exponential growth of the number of configurations with nucleon numbers. To mitigate the computational challenge, symmetry techniques can be exploited to optimize model space for atomic nuclei with strong collective correlations (Launey et al. 2016). Alternatively, symmetry has also been exploited to describe atomic nuclei in terms of collective degrees of freedoms, such as the collective models (Bohr 1976; Mottelson 1976) developed by Bohr and Mottelson and the interacting boson model (Arima and Iachello 1975) developed by Arima and Iachello. In the former, the nuclear Hamiltonian is constructed in terms of collective coordinates characterizing certain shapes of the nuclear surface preserving the corresponding geometrical symmetries. In the latter, the Hamiltonian is constructed based on the algebra of symmetry group in terms of interacting bosons of different ranks which are composite operators for pairs of correlated nucleons. These two methods have turned out to be very successful in describing a wide variety of nuclear properties, especially nuclear spectroscopy of vibration and rotation excitations. The content of this chapter will be mainly about another type of nuclear models starting from the reference state of a mean-field calculation. Mean-field

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approximation has been frequently employed in modeling quantum many-body systems. An atomic nucleus is no exception. In this approximation, the atomic nucleus is described as a bound system of independent nucleons trapped in a mean-field potential determined self-consistently by themselves. The nuclear wave function becomes a single Slater determinant of single-particle or quasiparticle basis functions (Meng et al. 2006), and the many-body problem is reduced to an effective one-body problem that can be easily solved. It turns out that this simplified nuclear wave function is often incapable of simultaneously describing essential correlations of an atomic nucleus and also satisfying the conservation laws. Therefore, certain conservation law is allowed to be violated in the meanfield potential to incorporate more correlations. In this case, the state of the system is referred to as a state of broken symmetry, and it is described by the wave function that does not possess corresponding quantum numbers. The use of such a symmetry-breaking wave function for atomic nucleus suffers from some drawbacks as the interpretation of nuclear spectroscopic data becomes obscure. To solve this problem, symmetry-restoration methods are introduced to recover missing quantum numbers. This strategy is the core idea of the symmetry-projected generator coordinate method (PGCM), in which the nuclear wave function is constructed as a linear combination of nonorthogonal basis functions that are projected out from “deformed” mean-field wave functions (Bender et al. 2003). Following this idea, many advanced nuclear models have been established, including the multireference energy-density-functional (MR-EDF) method (Nikši´c et al. 2011; Egido 2016; Robledo et al. 2019), Monte-Carlo shell model (MCSM) (Otsuka et al. 2001), and the projected shell model (PSM) based on either a schematic pairing-plusquadrupole Hamiltonian (Hara and Sun 1995) or the effective interaction derived from a covariant density functional theory (Zhao et al. 2016; Wang et al. 2022). These methods have been frequently employed to interpret nuclear spectroscopic data in different mass regions. A detailed introduction to symmetry breaking in nuclear mean fields and the restoration of broken symmetries with projection techniques can be found in textbooks Ring and Schuck (1980), Blaizot and Ripka (1986) and in the recent review papers Sheikh et al. (2021), Yao et al. (2022) and the references therein. Recently, this idea has also been implemented into ab initio studies of atomic nuclei based on the many-body expansion methods, including the symmetry broken and restored coupled-cluster theory (Duguet 2015; Duguet and Signoracci 2017; Signoracci et al. 2015; Hagen et al. 2022); in-medium generator coordinate method (IM-GCM) (Yao et al. 2018, 2020) – a new variant of multireference in-medium similarity renormalization group method (Hergert et al. 2016); and many-body perturbation theory (MBPT) (Frosini et al. 2022a, b, c). This chapter provides a pedagogical introduction to the symmetries of nuclear Hamiltonian that are allowed to be broken in nuclear mean-field potentials to incorporate many-body correlations, with special emphasis on the methods that have been frequently employed to restore these broken symmetries to achieve an accurate description of nuclear structure and decay properties. Some illustrative applications to selected nuclear structure problems are also discussed. The frontier and recent exciting progress in modeling atomic nuclei based on symmetry techniques are briefly mentioned.

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Symmetry and Group Representations Let us start with the basic knowledge of symmetry and group theory. Considering a set of transformations G = {e, ˆ gˆ 1 , gˆ 2 , · · · }, the set G forms a group if the following conditions are fulfilled for all the elements gi belonging to the group (Stancu 1996; Frank et al. 2009): identity : eˆgˆ i = gˆ i eˆ = gˆ i ,

(1a)

inverse : gˆ i gi−1 = e, ˆ

(1b)

closure : gˆ i gˆ j = gˆ ∈ G,

(1c)

associativity : (gˆ i gˆ j )gˆ k = gˆ i (gˆ j gˆ k ).

(1d)

In the group G, the element eˆ is the unity or identity element, and gˆ i−1 is the inverse of the element gˆ i . The group G is called a finite group if the number of the element is finite, and this number is called the order of the finite group. There are also groups with an infinite number of elements. These groups can be either the discrete groups of infinite order or the continuous groups. For the latter, the group ˆ element denoted as R(ϕ) depends on a finite set of continuously varying parameters ϕ = {ϕ1 , ϕ2 , · · · , ϕr }. In quantum mechanics, group theory offers a systematic way of finding the properties of eigenstates under various transformation from the symmetries of Hamiltonian. The eigenstates form linear spaces providing matrix representations of the group transformations G. Specifically, for the Hamiltonian Hˆ 0 of a quantum system which is invariant under the transformations g, ˆ one has [g, ˆ Hˆ 0 ] = 0,

∀gˆ ∈ G.

(2)

Considering a linear space L composed of N orthonormal basis states {|Φ1 , |Φ2 , · · · , |ΦN }, any group element gˆ acting on the states induces a N ×N matrix Dj i (g), i.e., g|Φ ˆ i =

N 

  Dj i (g) Φj ∈ L,

for

∀|Φi  ∈ L,

and

∀gˆ ∈ G,

(3)

j =1

where the indices i, j run from 1 to N . The N × N matrices D(g) with the element determined by Dj i (g) = Φj | gˆ |Φi  preserve the multiplicative structure of the group G and form a matrix representation of the group. The group G can have a different matrix representation denoted as D  (g) if a different linear space L is employed. If the two representations D(g) and D  (g) are connected by a similarity transformation X for all the group elements, these two representations are called equivalent. By applying different similarity transformations X(i) , one can generate many equivalent representations D (i) , among which, if there is one with the following form:

59 Symmetry Restoration Methods

X

−1

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  λ D 1 (g) T 12 (g) , D(g)X = 0 D λ2 (g)

for

∀gˆ ∈ G,

(4)

then, the representation D(g) is called reducible. The block matrices D λ1 (g) and D λ2 (g) are two square matrices with dimensions dλ1 and dλ2 , respectively. It indicates that there is a subspace L1 ⊂ L, ∀|Φlλ1  ∈ L1 is transformed into another state g|Φ ˆ lλ1  which also belongs to the subspace, and this is true for all the transformations of the group, namely, g|Φ ˆ lλ1  =



λ1 Dkl (g)|Φkλ1 ,

for

∀gˆ ∈ G.

(5)

k

The subspace L1 fulfilling the above requirement is called invariant subspace. Furthermore, if the matrix T 12 (g) = 0 in (4), the representation D(g) is called fully reducible. In this case, the full L space can be written as a sum of two invariant subspaces L = L1 ⊕ L2 and the representation D(g) = D λ1 (g) ⊕ D λ2 (g).

(6)

This property has been frequently exploited to split a large Hilbert space into a set of independent subspaces with smaller dimensions. The representation D λ (g) is called irreducible if there is no similarity transformation which brings the matrices D λ (g) into block diagonal form simultaneously for ∀gˆ ∈ G. The irreducible representation is sometimes abbreviated as irrep for brevity. It is worth mentioning that an irrep of one full group may be reducible for its subgroup. For example, the spherical harmonics Ym with m = −, · · · ,  forms an invariant subspace of 2 + 1 dimension for the rotational group SO(3) with the irrep of D  (g), which is however reducible for the subgroup SO(2) related to the rotation Rˆ z (ϕ) in two-dimensional space (along z-axis): ˆ Rˆ z (ϕ) = e−iϕ Lz ,

d . Lˆ z = −i h¯ dϕ

(7)

In other words, the irrep of D  (Rz ) can be further reduced into a block diagonal form: D  (Rz ) =

  imϕ 0 e , 0 e−imϕ

∀Rˆ z (ϕ) ∈ SO(2).

(8)

The basis functions form multiplets of the orbital angular momentum, each of which constitutes a one-dimensional irreducible invariant subspace for the SO(2) group. If the group G is a finite group or a compact Lie group like the orthogonal group O(n) and the unitary group U(n), it can be proven that any matrix representation of the group G is equivalent to a unitary representation (Stancu 1996) in which the

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matrices are unitary for all group elements. Supposing Dijλ (g) and Dijλ (g) are two ˆ unitary irreps of the group element R(g), where the matrix element is determined by λ(λ )

Dkl

λ(λ )

(g) = Φk

λ(λ )

|g|Φ ˆ l

,

(9)

with the indices k, l running from 1 to dλ , one has the orthogonality relation 1  λ ⨋ dgDkl (g)[Dkλ l  (g)]∗ = 0, nG

(10)

for non-equivalent representations and 1 δλλ  λ ⨋ dgDkl (g)[Dkλ l  (g)]∗ = Xkk  Xl−1 l , nG dλ

(11)

for equivalent representations. The symbol ⨋ dg stands for the summation or integral over all the elements of the group G, and nG is the order (the number of elements or the volume) of the group, dλ is the dimension of the irrep, and X is  the similarity transformation which connects the two irreps of D λ and D λ . If the two irreps are identical, then the similarity transformation is just a unity matrix, i.e., X = 1.

Symmetry Breaking in Mean-Field Approximations Symmetries can be classified according to the order of the groups, including the symmetries associated with discrete groups (such as parity, charge conjugation, time reversal, permutation symmetry) consisting of a finite number of group elements and the symmetries associated with continuous groups (such as translations and rotations symmetries in space-time or gauge angle coordinate system) consisting of an infinite number (continuum) of elements. According to Noether’s theorem (Noether 1971), any continuous symmetry of the action (Lagrangian or Hamiltonian) of a physical system with conservative forces has a corresponding conservation law. The abovementioned symmetries imply the conservation of total momentum (energy), angular momentum, and particle numbers in atomic nuclei, respectively. On the other hand, symmetries can also be classified according to the space where the symmetries are defined, including the symmetries defined in the space-time coordinate system and the symmetries defined in abstract Hilbert spaces. The symmetry of a physical system could be broken explicitly or spontaneously. The former is due to the presence of an additional interaction term that is not invariant under the symmetry transformation, such as the isospin SU(2) symmetry in atomic nuclei which breaks into SO(2) symmetry. These symmetries are called dynamical symmetries in the sense that the nuclear Hamiltonian breaks the symme-

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tries but preserves the symmetries defined by their subgroups (Frank et al. 2009). The breaking of this type of symmetries in atomic nuclei is not necessary to be restored as it is determined by the nature of nuclear force. In the latter case, the Hamiltonian is kept to be invariant under the symmetry transformation, but the physical state chooses one of the multiple degenerate (ground) states which are not the eigenstates of (but connected by) the generator operator of the symmetry group. This spontaneous symmetry breaking phenomenon could be caused by the presence of external perturbations and the thermodynamic limit. In fact, any other choice of the solutions would have exactly the same energy, which for continuous symmetry implies the existence of a massless Nambu–Goldstone boson. The spontaneous symmetry breaking usually happens in infinite systems studied in condensed matter or high-energy physics, such as the superconductors, ferromagnets, and the Higgs mechanism that generates the masses of elementary particles (Brauner 2010). In nuclear physics, the atomic nucleus is a finite system, for which, the phenomenon of spontaneous symmetry breaking arises only as a result of approximations. That is, if the nuclear many-body problem is solved exactly, symmetry should not be broken in nuclear states. This is exactly the reason why one needs to restore the symmetries broken by the employed approximations in nuclear models. In the self-consistent mean-field approaches, such as Hartree-Fock (HF) or Hartree-Fock-Bogoliubov (HFB), a single Slater determinant is chosen as the trial wave function of an atomic nucleus which is determined with the variational principle, that is, the minimization of the expectation value of the Hamiltonian to the wave function. The variational principle which guarantees the solution corresponding to the local energy minimum within the restricted Hilbert space cannot ensure the symmetry structure of the Hamiltonian. As a result, the symmetries of a given nuclear Hamiltonian are not fully retained in the solutions. If one constrains the solution to preserve a certain symmetry throughout the variational calculation, one may end up with a solution that does not incorporate sufficient correlations of atomic nuclei and cannot achieve a satisfying description of nuclear basic quantities, such as binding energies. Therefore, the mean-field approaches which allow for the breaking of the symmetries of the Hamiltonian in the mean-field potentials (also referred to as symmetry-breaking mechanism) during the variational procedure have been widely employed in nuclear physics. One of the celebrated successes of these symmetrybreaking mean-field approaches is shown in the description of nuclear deformation and pairing correlations (Ring and Schuck 1980). In molecular physics, an interplay between degenerate electronic states (singleparticle motions) and a molecular vibration (collective motions) gives rise to a spontaneous breaking of a molecular symmetry. This is the so-called Jahn-Teller effect (Jahn and Teller 1937), which turns out to be a general feature of quantum many-body systems induced by a linear coupling between their microscopic and collective degrees of freedom (Reinhard and Otten 1984). In open-shell nuclei, the degeneracy of eigenvalues of a (spherical) nuclear single-particle Hamiltonian around the Fermi level leads to the solution unstable with respect to shape vibrations. Additional correlation energies can be gained if the degeneracy of the single-particle

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Hamiltonian is lifted by allowing the breaking of the symmetry. As a result, a “deformed” solution emerges in the variational calculation. The obtained wave function Φ(r1 , · · · , rA ; q) abbreviated as Φ(q) of a A-body nuclear system from, for instance, the HFB approach can be generally labeled with a set of collective coordinates denoted with the symbol q; see Fig. 1a, which includes for instance the pairing amplitude κ or particle-number fluctuation ΔN 2 ≡ Φ(q)| (Nˆ − N0 )2 |Φ(q) ,

Φ(q)| Nˆ |Φ(q) = N0 ,

(12)

and the dimensionless multiple deformation parameters βλμ =

 4π   Qλμ ij Φ(q)| ai† aj |Φ(q), λ 3AR

(13)

ij

† † where the particle-number operator Nˆ = i ci ci with ci , ci being the singleparticle creation and annihilation operators in a certain basis, respectively, and the multipole moment tensor operator Qλμ = r λ Yλμ . The nuclear size R = 1.2A1/3 fm, with A being the mass number of the nucleus. The value of μ = −λ, −λ+1, . . . , λ. The onset of nonzero value of pairing amplitude κ or ΔN 2 is the result of the mixture of single-particle creation and annihilation operators, leading to the violation of gauge rotation symmetry described by the U(1) group which will be introduced in detail later. Figure 1b displays some cartoon pictures for nuclear shapes characterized by the deformation parameter βλμ . • The spherical shape corresponds to βλμ = 0 for all values of λ, μ. • The positive and negative value of β20 corresponds to prolate and oblate deformed shapes, respectively. Nuclear triaxiality is defined by the nonzero value of β22 . The occurrence of a nonzero quadrupole deformation β2μ induces the

Fig. 1 (a) A schematic representation of a general nuclear wave function |Ψ  expanded in terms of nonorthogonal basis functions |Φ(q) in multi-dimensional collective space labeled with the coordinate vector q = {βλμ , κ, · · · , ωc }. (b) A schematic show of some typical nuclear shapes characterized with multipole deformation parameters βλμ , taken from Lu et al. (2012). ((b) reprinted with the courtesy of B. N. Lü)

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mixture of single-particle states with different angular momentum, leading to the violation of rotational SO(3) symmetry – a special orthogonal group in three dimensions. • If the β3μ = 0 for |Φ(q), the density distribution of the solution |Φ(q) has a reflection-asymmetric shape. The occurrence of nonzero octupole moments induces the mixture of single-particle states with positive and negative parities. The resultant wave function |Φ(q) is thus not an eigenstate of the parity operator Pˆ . Therefore, the wave function |Φ(q) from the mean-field calculations usually mixes different irreps |Φα (q) of symmetry groups that may include the gauge U(1) symmetry, the rotational SO(3) symmetry, the space-reversal Z2 symmetry, etc. Thus, the mean-field wave function can be generally decomposed as follows: |Φ(q) =



cα (q) |Φα (q).

(14)

α

where the symbol α = {J NZπ, · · · } stands for a set of quantum numbers, such as angular momentum J , particle numbers (N, Z), parity π , etc. The methods to extract the component |Φα (q) with the correct quantum numbers α will be introduced subsequently.

Symmetry Restoration Methods Despite the great success of mean-field approaches for nuclear ground state, the deficiency of using a symmetry-breaking wave function |Φ(q) in (14) shows up in the analysis of nuclear spectroscopic properties. The restoration of missing quantum numbers in nuclear wave functions due to the admixture of different irreps is essential in the interpretation of spectroscopic data to ensure the selection rules. This can be done by projecting out the irreps |Φα (q) of the symmetry groups from the “deformed” wave function in (14). This procedure is usually called symmetry restoration in nuclear physics.

Generator Coordinate Method (GCM) ˆ If the Hamiltonian Hˆ 0 is invariant under the transformation R(ϕ), or equivalently, ˆ ˆ [H0 , R(ϕ)] = 0, then all the rotated mean-field states are degenerate: ˆ Φ(q)| Rˆ † (ϕ)Hˆ 0 R(ϕ) |Φ(q) = Φ(q)| Hˆ 0 |Φ(q).

(15)

Therefore, a natural way to restore the symmetry G associated with this rotation is ˆ to use a linear superposition of a family of the rotated states for all R(ϕ) ∈ G:

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|Φα (q) =

ˆ dϕF α (ϕ)R(ϕ) |Φ(q) .

(16)

This is essentially the basic idea of the generator coordinate method (GCM) (Hill and Wheeler 1953), and the rotation parameter ϕ can be interpreted as a generator coordinate which generates the wave functions in different orientations. The use of the trial wave function in (16) with the unknown expansion coefficient F α (ϕ) as the variational parameter is equivalent to the projection of the states with good symmetry (Blaizot and Ripka 1986). It will be shown later that the F α (ϕ) can be derived analytically from the group theory which defines the so-called projection operator. This procedure can be carried out in two different schemes. In the projectionafter-variation (PAV) scheme, both the mean-field wave function |Φ(q) and the coefficient F α (ϕ) are changing simultaneously to search for the minimum of the energy expectation to the symmetry-projected wave function |Φα (q). In contrast, the wave function |Φα (q) in the variation-after-projection (VAP) scheme is extracted from the mean-field wave function |Φ(q) determined after the variational calculation. In other words, the variation of |Φ(q) and F α (ϕ) is carried out sequentially. The parameter space in the VAP is generally larger than that in the PAV; thus the VAP is expected to provide a closer-to-exact solution. In particular, the VAP scheme can prevent the occurrence of sharp phase transition in the mean-field solution of atomic nuclei due to the artificial symmetry breaking (Mang et al. 1976; Ring and Schuck 1980). However, the VAP scheme is more computation demanding and is thus frequently employed for parity and particle-number projections, scarcely for angular momentum projection. In many applications, the wave function by either the PAV or VAP scheme provides a good starting point of a more advanced nuclear model. According to GCM, nuclear wave function is constructed as a linear superposition of all the symmetry-projected wave functions |Φα (q):

|Ψα  =

dqfα (q) |Φα (q).

(17)

Here the collective coordinate q is the generator coordinate which is integrated out and does not appear in the final GCM wave function. The coefficient fα (q), also known as collective wave function or generator function, is determined by the variational principle which leads to the following integral equation, referred to as Hill-Wheeler-Griffin equation (Hill and Wheeler 1953; Griffin and Wheeler 1957):

  dq Φα (q)| Hˆ |Φα (q ) − Eα Φα (q)| Φα (q ) fα (q ) = 0.

(18)

It is worth pointing out that the projected wave functions |Φα (q) do not form an orthogonal basis and thus the weights fα are not orthogonal functions. In most cases, the function gα (q) is introduced as follows:

59 Symmetry Restoration Methods

gα (q) =

2239

 1/2 dq Φα (q)| Φα (q ) fα (q ),

dq|gα (q)|2 = 1,

(19)

which fulfills the normalization condition and reflects how configurations mix in the GCM wave function |Ψα . To distinguish the procedure of only performing a symmetry-restoration calculation with projection operators (even though this procedure also belongs to GCM) from the procedure of also mixing the projected wave functions with different collective coordinate q, the former procedure is referred to as symmetry restoration or projection, while the latter procedure is called symmetry-projected GCM (PGCM).

Construction of Projection Operators Basic Properties of a Projection Operator A straightforward way to extract the component |Φα (q) with the quantum numbers α is by means of projection operators. Supposing |Φα (q) provides a basis function of unitary irrep of the group G, a projection operator Pˆ λ is defined in such a way that Pˆ λ |Φα (q) = δλα |Φλ (q).

(20)

A repeated use of the projection operator would not change the result: (Pˆ λ )2 |Φα (q) = δλα Pˆ λ |Φλ (q) = δλα |Φλ (q).

(21)

It gives rise to the important properties of the projection operator (idempotent and Hermitian): (Pˆ λ )2 = Pˆ λ ,

Pˆ λ† = Pˆ λ .

(22)

Because the basis functions belonging to different representation spaces are orthogonal to each other, i.e., Φλ | Φα  = δλα , the projection operator Pˆ λ can be constructed formally in terms of the basis function as below Pˆ λ =



|Φλσ  Φλσ |,

(23)

σ

where σ denotes all the set of quantum numbers other than λ.

The Projection Operator from Group Theory λ is introduced based on the orthogonalIn group theory, the projection operator Pˆμμ λ ity relation (11) of the irreps Dμμ (g) of the group G (Cornwell 1997)

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dλ λ ≡ Pˆμμ nG



λ∗ ˆ (ϕ)R(ϕ), dϕDμμ

(24)

where the symbol μ is introduced to distinguish basis functions of the invariant subspace λ. The projection operator (24) has the following properties: λ† λ Pˆμμ = Pˆμμ ,

λ ˆλ λ Pˆμμ . Pμμ = Pˆμμ

(25)

λ only picks up the component One can prove that the projection operator Pˆμμ λ |Φμ (q) from the mean-field wave function (14) λ Pˆμμ |Φ(q) = cλ,μ (q) |Φμλ (q),

(26)

where the coefficient is cλ,μ (q) determined by λ |Φ(q). |cλ,μ (q)|2 = Φ(q)| Pˆμμ

(27)

It defines a normalized symmetry-projected wave function: 1 λ |Φμλ (q) = Pˆμμ |Φ(q). λ ˆ Φ(q)| Pμμ |Φ(q)

(28)

The expectation value of a general scalar operator Oˆ with respect to this wave function |Φμλ (q) is determined by Oλ,μ (q, q) =

Φμλ (q)| Oˆ |Φμλ (q) Φμλ (q)| 1 |Φμλ (q)

=

λ |Φ(q) Φ(q)| Oˆ Pˆμμ . λ |Φ(q) Φ(q)| Pˆμμ

(29)

The kernels can be rewritten explicitly in term of the integration over all the rotation angles ϕ, where the integrand is the overlap of the operator Oˆ between the original and rotated mean-field wave functions, weighted by the Wigner-D function λ Φ(q)| Oˆ Pˆμμ |Φ(q)

dλ λ∗ ˆ dϕDμμ (ϕ) Φ(q)| Oˆ |Φ(q; ϕ) · Φ(q)| R(ϕ) |Φ(q), = nG

(30)

where the rotated quasiparticle vacuum |Φ(q, g) is defined as below |Φ(q, ϕ) ≡

ˆ R(ϕ) |Φ(q) . ˆ Φ(q)| R(ϕ) |Φ(q)

(31)

59 Symmetry Restoration Methods

2241

Some Typical Examples In general, the HFB wave function |Φ(q) may break several symmetries simultaneously. To project out the component with correct quantum numbers, one needs to introduce all the corresponding projection operators. In the following, the projection operators for some typically broken symmetries in |Φ(q) will be introduced, separately.

The Space-Reversal Symmetry Z2 The space-reversal (parity) transformation Pˆ forms the abelian group Z2 with two ˆ There are two irreps Pφ(r) ˆ ˆ P}. elements {1, = φ(−r) = π φ(r), where the eigenvalue π = ± corresponds to even or odd parity, respectively. In atomic nuclei, parity is conserved in the strong interaction, and thus it serves as one of the quantum numbers used to label nuclear states. In atomic nuclei of some particular mass regions with neutron or proton numbers around 34, 56, 88, and 134, the negativeparity states of vibration or rotation excitations are observed near their ground states. This phenomenon can be naturally explained in mean-field approaches that the onset of large octupole correlations is attributed to the existence of pairs of singleparticle states with opposite parities around the Fermi surface such as g9/2 − p3/2 , h11/2 − d5/2 , and i13/2 − f7/2 , j15/2 − g9/2 which are strongly coupled by octupole moments (Butler and Nazarewicz 1996). The emergence of nonzero moments of odd multiplicity such as octupole moments induces the mixing of the states with different parity. As a result, the obtained nuclear wave function |Φ(q) does not have a definite parity. According to (28), the normalized wave function with a definite parity π can be recovered by projecting out one of the two irreps with a parityprojection operator |Φπ (q) =

1 Φ(q)| Pˆ π

|Φ(q)

Pˆ π |Φ(q),

(32)

where the parity-projection operator Pˆ π is defined as   1 1 + π Pˆ , Pˆ π ≡ 2

(Pˆ π )2 = Pˆ π .

(33)

When the space-reversal operator Pˆ acts on the product many-body wave function |Φ(q), it induces a factor of (−1) to the expansion coefficient of each singlej nucleon wave function on the spherical HO basis ϕnj mj = Rn (r)[Y ⊗ χs ]mj , where Ym and χsms are the spherical harmonic and spin wave function, respectively. Alternatively, one can define a many-body operator Pˆ = exp(iπ Nˆ − ) (Egido and ˆ where the operator Nˆ − =  c† c Robledo 1991) for the space-reversal operator P, k k k is similar to the particle-number operator, but with the summation over all the k restricted to the single-particle states with negative parity only. The expectation

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J. M. Yao

value of a general parity-conserving operator Oˆ to the state |Φπ (q) with the definite parity π is thus determined by Oπ (q, q) =

Φ(q)| Oˆ Pˆ π |Φ(q) Φπ (q)| Oˆ |Φπ (q) . = Φπ (q)| 1 |Φπ (q) Φ(q)| Pˆ π |Φ(q)

(34)

The Gauge Symmetry U(1) Nuclear state |ΦAτ  is labeled with a definite number Aτ of nucleons (neutrons or protons) which is an eigenstate of the rotation operator ˆ R(ϕ) |ΦAτ  = e−iϕAτ |ΦAτ ,

(35)

where the symbol Aτ stands for either neutron (τ = n) or proton (τ = p). This is a local transformation in the sense that the symmetry transformation is related to pairing vibration in which nucleon number Aτ serves as the differentiate coordinate for the dynamics of the pairing vibration in the nuclear system. The group element ˆ R(ϕ) can be defined in terms of the particle-number operator Aˆ τ ˆ ˆ R(ϕ) = eiϕ Aτ ,

Aˆ τ =

N (Z) 

cj† cj ,

(36)

j =1

where ϕ ∈ [0, 2π ] is the gauge angle. This transformation is called gauge rotation. The symmetry is described by the unitary group acting in one-dimensional complex space and is thus called U(1) gauge symmetry. The wave function |ΦAτ  forms an irrep of the group. Similar to electrons in a superconducting metal, nucleons inside an atomic nucleus also exhibit pairing correlations, as supported by many evidences including the odd-even staggering of binding energies and the energy gap in the low-lying energy spectra of atomic nuclei (Bohr et al. 1958). This pairing correlation can be excellently described with the BCS or HFB theory in terms of a quasiparticle vacuum state |Φ(q) which is constructed as an admixture of wave functions with different nucleon numbers. In other words, the obtained wave function |Φ(q) is a superposition of different irreps of the U(1) group usually differing from each other by two nucleons. The use of this particle-number-violating wave function for an atomic nucleus may cause serious problems as it contains the component of neighboring atomic nuclei. Some of these problems can be remedied with the help of particle-number projection (PNP) operator Pˆ Aτ which can project out the component |ΦAτ (q) with the particle number Aτ |ΦAτ (q) =

1 Φ(q)| Pˆ Aτ |Φ(q)

Pˆ Aτ |Φ(q),

where the particle-number projection operator is defined according to (24)

(37)

59 Symmetry Restoration Methods

2243

1 Pˆ Aτ = 2π



2π

ˆ dϕe−iϕAτ R(ϕ).

(38)

0

For atomic nuclei, the numbers of neutrons and protons are conserved separately. The violation of particle number in the quasiparticle vacuum |Φ(q) can be seen clearly by decomposing it into a linear combination of common eigenstates of the ˆ of neutrons (and protons) particle-number operators Nˆ (and Z) |Φ(q) =



cN0 Z0 (q) |ΦN0 Z0 (q) ,

ˆ Z) ˆ |ΦN0 Z0 (q) = N0 (Z0 ) |ΦN0 Z0 (q), N(

N 0 Z0

(39) where the expansion coefficient cN0 Z0 of the component with N0 neutrons and Z0 protons can be determined by Φ(q)| Pˆ N0 Pˆ Z0 |Φ(q) =



∗ cN ΦN Z (q)| cN0 Z0 |ΦN0 Z0 (q) = |cN0 Z0 |2 . 0 Z0

NZ

(40) The expectation value of a general particle-number-conserving operator Oˆ to the wave function |ΦN0 Z0 (q) is given by ON0 Z0 (q, q) =

ΦN0 Z0 (q)| Oˆ |ΦN0 Z0 (q) Φ(q)| Oˆ Pˆ N0 Pˆ Z0 |Φ(q) = , (41) ΦN0 Z0 (q)| 1 |ΦN0 Z0 (q) Φ(q)| Pˆ N0 Pˆ Z0 |Φ(q)

where the kernel can be rewritten explicitly in terms of an integration over all the overlaps of the operator Oˆ between the unrotated and rotated mean-field wave functions at different gauge angles ( ϕn , ϕp ) Φ(q)| Oˆ Pˆ N0 Pˆ Z0 |Φ(q)

2π −iϕp Z0

2π −iϕn N0 e e ˆ p ˆ i Nˆ ϕn ei Zϕ dϕn dϕp Φ(q)| Oe |Φ(q). = 2π 2π 0 0

(42)

If the mean-field wave function |Φ(q) has a definite number parity (−1)N0 (Z0 ) , the interval of the gauge angles can be reduced to be within [0, π ] (Bally and Bender 2021). The above integration can be carried out using the trapezoidal rule based on the Fomenko expansion method (Fomenko 1970) in which the particle-number projection operator becomes   L 1  i Aˆ τ −Aτ ϕm Aτ ˆ P = e , L m=1

ϕm =

m π, L

(43)

where the gauge angle ϕτ ∈ [0, π ] is discretized with L points. In atomic nuclei, the wave function |Φ(q) is smeared out only a few particle numbers around its mean value; only a small value of L is sufficient to achieve a rather exact projection.

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J. M. Yao

Typically, the odd value of L = 7 or 9 is employed to avoid the singularity problem of choosing the point ϕ = π/2 (Tajima et al. 1992; Donau 1998; Anguiano et al. 2001). An alternative way to evaluate the integral over the gauge angle ϕτ is to carry out the integration analytically using the residues theorem, as employed by Dietrich et al. (1964). The projection operator Pˆ Aτ in (38) can be rewritten in terms of a complex variable z = eiϕ : 1 Pˆ Aτ = 2π i



ˆ

zAτ zAτ +1

dz.

(44)

The projected wave function becomes |ΦAτ (q) = Pˆ Aτ =2p |Φ(q) =

1 2π i

 ζ

τ  dζ   uk + vk ζ ak+ ak+ |−, p+1

(45)

k=1

where the variable z has been replaced by ζ = z2 . The p = Aτ /2 is the number of pairs. The integration just picks the component with ζ −1 , that is, the component with p pairs. The advantage of using the residue theorem is that it provides an analytical way to explore the structural properties of particle-number projected wave functions. Besides, the variation after particle-number projection (PNVAP) can be carried in a similar way to the ordinary variational calculation (Dietrich et al. 1964). Compared to the Fomenko expansion, the downside of this method is that it becomes difficult to be implemented in combination with the projection operators of other symmetries, like the rotational symmetry described by the SO(3) group.

The Rotational Symmetry SO(3) The mean-field wave function |Φ(q) for an open-shell nucleus is usually an admixture of the components |ΦJ M (q)  cJ M |ΦJ M (q), (46) |Φ(q) = JM

where |ΦJ M (q) is the eigenfunction of squared-angular-momentum operator Jˆ2 and its projection along z-axis Jˆz , Jˆ2 |ΦJ M  = J (J + 1)h¯ 2 |ΦJ M ,

(47a)

Jˆz |ΦJ M  = M h¯ |ΦJ M .

(47b)

The component |ΦJ M (q) can be obtained by applying an angular-momentum J projection (AMP) operator PˆMM onto the symmetry-violating wave function. The J ˆ AMP operator PMM can be constructed as a product of two projection operators defined according to Löwdin’s method (Löwdin 1964)

59 Symmetry Restoration Methods



J = PˆMM

I =J

2245

 Jˆz − K Jˆ2 − I (I + 1) . J (J + 1) − I (I + 1) M −K

(48)

K=M

From Eqs. (46) and (48), one immediately finds J PˆMM |Φ(q) = cJ M |ΦJ M (q) .

(49)

J with the following form: Alternatively, one can introduce an operator PˆMK

2J + 1 J PˆMK ≡ 8π 2





2π

0



π

2π

sin βdβ

dα 0

0

J∗ ˆ dγ DMK (Ω)R(Ω),

(50)

ˆ where the rotational operator R(Ω) is defined as a rotation about the origin of threedimensional Euclidean space, specified by three real parameters (α, β, γ ) (Varshalovich et al. 1988): ˆ ˆ ˆ ˆ R(Ω) = e−iα Jz e−iβ Jy e−iγ Jz .

(51)

The irrep of the SO(3) group is the Wigner-D function J −iMα J ˆ DMK (Ω) = ΦJ M |R(Ω)|Φ dMK (β)e−iKγ , JK = e

(52)

ˆ

J (β) = Φ −iβ Jy |Φ where the small d-function is defined as dMK JM| e J M . With the orthogonality relation

2J + 1 8π 2





J∗ J (Ω)DK dΩDMK  M  (Ω) = δJ J  δMK  δKM  ,

(53)

one can prove that the operator (50) extracts the component |ΦJ M  from the symmetry-breaking wave function |Φ with the weight cJ K : J PˆMK |Φ(q) = cJ K |ΦJ M .

(54)

J can also be written as Thus, the operator PˆMK J PˆMK =



|ΦJσM ΦJσK |,

(55)

σ

where σ characterizes all quantum numbers besides the J, M required to completely specify the states. From the above, one arrives at the following relation: J J ˆJ PˆMK PˆM  K  = δJ J  δKM  PMK  ,

J† J PˆMK = PˆKM .

(56)

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J. M. Yao

J In other words, the operator PˆMK is a true projection operator only if J ˆ M = K. In a general case, the operator PMK with M = K is sometimes referred to as transfer operator. See such as Bally and Bender (2021) and reference therein for more discussions. The wave function of a triaxially deformed nucleus (β22 = 0) with the good angular momentum J can be constructed as (Yao et al. 2008, 2009)

|ΦJ M (q) =



J ˆJ gK PMK |Φ(q).

(57)

K

In the simple case that the wave function |Φ has axial symmetry, the K value is conserved. Otherwise, one needs to mix the components with different K values. J is determined by minimizing the energy of state which is The mixing coefficient gK given by  E (q, q) = J

ΦJ M (q)|Hˆ 0 |ΦJ M (q)



ΦJ M (q) | ΦJ M (q)

= KK



KK 

J ∗gJ H J gK K  KK  I ∗gJ N J gK K  KK 

(58)

J Hˆ Pˆ J ˆ 0 Pˆ J  is used. The Hamiltonian and norm where the relation PˆKM 0 MK  = H KK kernels are defined below:      ˆ ˆJ  J (59a) HKK  = Φ(q) H0 PKK   Φ(q)      ˆJ  J NKK (59b)  = Φ(q) PKK   Φ(q) . J is the solution to the generalized eigenvalue problem: The coefficient gK

 K

J J J HKK  gK  = E

 K

J J NKK  gK  .

(60)

This equation is equivalent to the diagonalization of the Hamiltonian in the space J |Φ(q). spanned by the nonorthogonal basis functions PˆMK Before ending this subsection, it is worth mentioning that the deformed state |Φ(q) becomes a stationary state if the atomic nucleus is very large (Blaizot and Ripka 1986). Considering the nuclear system evolves with time t from the deformed state |Φ(q), the wave function at the time t is given by ˆ

|Ψ (t) = e−i H0 t/h¯ |Φ(q) = e−iE0 t/h¯



cJ M e−iJ (J +1)t/2J |ΦJ M (q)

(61)

JM

where the |ΦJ M (q) is assumed to be the eigenstate of the Hamiltonian Hˆ 0 (independent of time) with the eigenvalue EJ taking the following value:

59 Symmetry Restoration Methods

2247

Hˆ 0 |ΦJ M (q) = EJ |ΦJ M (q) ,

EJ = E0 + J (J + 1)/2J .

(62)

In the limit that the moment of inertial J → ∞ for a very large nuclear system, the wave function in (61) becomes |Ψ (t)J →∞ = e−iE0 t/h¯



cJ M |ΦJ M (q) = e−iE0 t/h¯ |Φ(q) .

(63)

JM

The above formulas indicate that the deformed state |Φ(q) from mean-field calculations is approximately a stationary state for a very large nucleus. In other words, for an infinite large system, the energy splitting between different J states vanishes; the direct couplings between any two rotated symmetry-breaking states ˆ 1 ) |Φ(q) and |Φ(q, Ω2 ) with Ω1 = Ω2 are zero, i.e., |Φ(q, Ω1 ) ≡ R(Ω Φ(q, Ω1 )| Φ(q, Ω2 ) = δ(Ω1 − Ω2 ).

(64)

It means that the time required for the nuclear system to pass from one oriented state to another oriented state becomes infinite long. It explains why the restoration of rotational symmetry is more important in light deformed nuclear systems than in heavy ones.

Isospin Symmetry SU(2) As discussed before, the isospin symmetry is a dynamical symmetry in atomic nuclei, described by the special unitary group acting in two-dimensional complex space, namely, the SU(2) group. It plays an important role in understanding the asymmetry between the energy spectra of mirror nuclei and nuclear β decays. The isospin symmetry is largely preserved by the strong interactions with a small violation from weakly charge-dependent components and the mass difference between neutrons and protons. Besides, the main source of isospin breaking is the electromagnetic interaction. For the convenience of discussion, the total nuclear Hamiltonian Hˆ = Hˆ 0 + Hˆ 1 ,

(65)

is separated into a sum of isospin-rotation invariant term Hˆ 0 and isospin-rotation breaking term Hˆ 1 dominated by the Coulomb interaction VC [Hˆ 0 , Rˆ y (θ )] = 0,

[Hˆ 1 , Rˆ y (θ )] = 0,

(66)

ˆ where the operator Rˆ y (θ ) = e−iθ Ty is defined as a rotation about the y-axis in the isospin space with the angle θ with the Tˆy being the y-component of isospin operator. With the Hˆ 0 term only, the Hamiltonian commutes with the total isospin Tˆ 2 . Nuclear states can then be labeled with the quantum number T , and the states |T MT  form degenerate multiplets consisting of 2T + 1 components with different

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J. M. Yao

MT . With the VC , the degeneracy between the multiplet components is lifted. This phenomenon is shown to be a general feature in the spectra of isobaric isotopes. In addition to the Hˆ 1 term which breaks the isospin symmetry explicitly, in the mean-field description of atomic nuclei with N = Z, the presence of the neutron or proton excess automatically yields isovector mean fields, i.e., different mean-field potentials for protons and neutrons. As a result, the isospin invariance is artificially broken by the mean-field approximation even in the case that the isospin-invariant Hamiltonian is used (Brink and Svenne 1970; Engelbrecht and Lemmer 1970). It indicates that more efforts are needed to take care of the isospin mixing in atomic nuclei (especially the neutron-rich nuclei with large neutron excess) if one starts from the mean-field approximation. Starting from an isospin-invariant Hamiltonian Hˆ 0 , the breaking of isospin invariance in the mean-field solution is then due to the omission of neutron-proton correlations and can be recovered with the randomphase approximation (RPA) (Engelbrecht and Lemmer 1970). From this point of view, the RPA provides an excellent framework to consider isospin-breaking effect in atomic nuclei, which is essential to achieve an accurate calculation of the isospin corrections for superallowed Fermi beta decay (Liang et al. 2009). Alternatively, one can also take into account the isospin mixing effects with the isospin-symmetry projection operator (Caurier et al. 1980; Satula et al. 2009). This projection method applies not only to the ground state but also to nuclear excited states. In this framework, the nuclear state |n, MT  with a proper isospin mixing can be spanned in terms of the irreps |T MT  of the SU(2) group |n, MT  =

 T ≥|MT |

aTn MT |T MT 

(67)

T where the |T MT  is generated by acting the isospin-projection operator PˆM T MT on top of the mean-field wave function |ΦN0 Z0  (Note that a three-dimensional isospin projection operator is needed if a particle-number-violating mean-field wave function |Φ(q) is employed.) which is chosen as the eigenstate of neutron- and proton-number operators (Satula et al. 2009, 2010)

1 T |T MT  = |ΦN0 Z0  PˆM T MT T ˆ ΦN0 Z0 |PMT MT |ΦN0 Z0 

1 2T + 1 π T dθ sin θ dM (θ )Rˆ y (θ )|ΦN0 Z0 , = T MT 2 T 0 ˆ ΦN0 Z0 |PMT MT |ΦN0 Z0  (68) where dTTz Tz (β) is the Wigner d-function and MT = (N0 − Z0 )/2 is the third component of the total isospin T . The coefficient aTn MT defining the degree of isospin mixing is determined by the variational principles using the full Hamiltonian in Eq. (65). Usually, an isospin-mixing parameter αC is introduced. For the lowest 2   energy solution with n = 1, it is defined as αC = 1 − aTn=1 MT  with T = MT .

59 Symmetry Restoration Methods

2249

Approximate Treatments with Power Expansions Extracting the irreps of a symmetry group of a given nuclear Hamiltonian with the projection operator usually requires a multi-dimensional integration of many overlaps between two general Slater determinants for all the group elements. The calculation of these overlaps is generally very time-consuming. This is particularly true if the VAP scheme is employed where this kind of integration needs to be carried out at each iteration. Historically, some approximate treatments of the symmetry-restoration effect on nuclear energy have been proposed, such as the Lipkin (1960) and Kamlah (1968) power expansion methods. The basic idea of these expansion methods is to eliminate the symmetry-violation effect on the energy eigenvalue by introducing a model Hamiltonian (Lipkin 1960) ˆ H  = Hˆ 0 − f (X),

(69)

for which all the states |ΦX0  belonging to different irreps (labeled with X0 ) of the symmetry group G are degenerate. Here, Hˆ 0 is the original Hamiltonian, and the ˆ is chosen as a power series expansion of the operator Xˆ f (X) ˆ = f (X)

M 

λk (Xˆ − X0 )k ,

Xˆ |ΦX0  = X0 |ΦX0 ,

(70)

k=1

where the operator Xˆ is given by the generators of the symmetry group and can be chosen as, for instance, the angular-momentum operator Jˆ2 or particle-number operator Nˆ if the rotational SO(3) symmetry or the gauge U(1) symmetry is violated, respectively. To this end, the expansion coefficients λk are determined in such a way that the quantity ΦX0 | Hˆ  |ΦX0  is independent of the X0 . Lipkin’s idea was adopted by Nogami (1964) to suppress the impact of particlenumber fluctuation of the BCS wave function on the binding energy of open-shell atomic nuclei by truncating the expansion in (70) up to the k = 2 terms: ˆ = λ1 (Nˆ − N0 ) + λ2 (Nˆ 2 − N02 ). f (N)

(71)

The above prescription is referred to as the Lipkin-Nogami method. The Lipkin method can be alternatively understood as an expansion of the energy of the symmetry-projected wave function, which leads to Kamlah’s method (Kamlah 1968), where a general scheme to approximate the energy of momentum projection was derived, and it was shown that the cranking model is simply the variational problem in the first-order approximation to the energy of the angular-momentum projected state. Following Ring and Schuck (1980), this method is introduced for ˆ ˆ the case of a one-dimensional rotation R(ϕ) = eiϕ X , where the parameter ϕ can be either the Euler angle associated with angular momentum J (As shown in (50), a three-dimensional rotation operator should be employed in principle to define the angular-momentum projection. The Kamlah expansion for the general three-

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J. M. Yao

dimensional rotation case was presented by Beck et al. 1970.) or the gauge angle associated with particle number N . The energy of the projected state labeled with X0 is generally written as  E X0 (q, q) =

     ˆ  dϕe−iϕX0 Φ(q) Hˆ 0 eiϕ X  Φ(q) dϕe−iϕX0 h(ϕ)    =      dϕe−iϕX0 n(ϕ) dϕe−iϕX0 Φ(q) eiϕ Xˆ  Φ(q)

(72)

where the short-hand notations h(ϕ) and n(ϕ) were introduced for Hamiltonian and norm overlaps, respectively:      ˆ n(ϕ) = Φ(q) eiϕ X  Φ(q) .

    ˆ  h(ϕ) = Φ(q) Hˆ 0 eiϕ X  Φ(q) ,

(73)

In the case that the deformation or pairing correlations are strong in |Φ(q), one expects the h(ϕ) and n(ϕ) to be peaked at ϕ = 0 and to be very small elsewhere in such a way that the quotient h(ϕ)/n(ϕ) behaves smoothly. In this case, one can expand h(ϕ) in terms of n(ϕ) in the following way: h(ϕ) =

M 

hm Kˆ m n(ϕ),

m=0

1 ∂ ˆ − Φ|X|Φ, Kˆ = i ∂ϕ

(74)

where the Kamlah operator Kˆ was introduced. In this case, the energy of the projected state becomes  X0 E[M]

=

M  m  ˆm dϕe−iϕX M m=0 hm K n(ϕ) ˆ |Φ ,  = hm Φ| (ΔX) −iϕX dϕe n(ϕ)

(75)

m=0

with ΔXˆ ≡ Xˆ − Φ| Xˆ |Φ. The expansion coefficients hm are determined by ˆ . . ., Kˆ M and taking the limit ϕ → 0. One applying the Kamlah operators 1, K, has the following relation: ˆ n |Φ = Φ| Hˆ (ΔX)

M 

ˆ m+n |Φ. hm Φ| (ΔX)

(76)

m=0

Different levels of approximations can be defined according to the truncation in the expansion order M. • Up to the first order (M = 1): In this case, Eq. (76) produces two equations: ˆ |Φ Φ| Hˆ |Φ = h0 + h1 Φ| (ΔX) ˆ |Φ = h0 Φ| (ΔX) ˆ |Φ + h1 Φ| (ΔX) ˆ 2 |Φ. Φ| Hˆ (ΔX)

(77a) (77b)

59 Symmetry Restoration Methods

2251

The projected energy in (75) is composed of two terms: X0 ˆ |Φ. = h0 + h1 Φ| (ΔX) E[1]

(78)

The wave function |Φ is usually determined by the minimization of the projected X0 ˆ |Φ = 0 which in the case of Xˆ = Jˆ energy E[1] with the constraint Φ| (ΔX) leads to the equation in the cranking model (Ring and Schuck 1980)   δ ˆ ˆ Φ|H0 |Φ − h1 Φ|X|Φ = 0, δΦ

h1 =

ˆ |Φ Φ| Hˆ 0 (ΔX) , ˆ Φ| (ΔX)2 |Φ

(79)

where h1 is referred to as the cranking frequency. • Up to the second order (M = 2): In this case, the above relation leads to a set of equations corresponding to the choices of n = 0, 1, 2, respectively, ˆ |Φ + h2 Φ| (ΔX) ˆ 2 |Φ Φ| Hˆ 0 |Φ = h0 + h1 Φ| (ΔX) ˆ |Φ = Φ| Hˆ 0 (ΔX)

2 

(80a)

ˆ m+1 |Φ, hm Φ| (ΔX)

(80b)

ˆ m+2 |Φ. hm Φ| (ΔX)

(80c)

m=0

ˆ 2 |Φ = Φ| Hˆ 0 (ΔX)

2  m=0

One can determine the coefficients h0 , h1 , and h2 from the above equations with ˆ |Φ = 0 and the omission of the Φ| (ΔX) ˆ 3 |Φ term the constraint Φ| (ΔX) ˆ 2 |Φ , h0 = Φ| Hˆ 0 |Φ − h2 Φ| (ΔX) h2 =

h1 =

ˆ |Φ Φ| Hˆ 0 (ΔX) , ˆ 2 |Φ Φ| (ΔX)

ˆ 2 |Φ − Φ| Hˆ 0 |Φ Φ| (ΔX) ˆ 2 |Φ Φ| Hˆ 0 (ΔX) , ˆ 4 |Φ − Φ| (ΔX) ˆ 2 |Φ2 Φ| (ΔX)

(81a)

(81b)

from which one obtains the energy of the projected state X0 ˆ |Φ + h2 Φ| (ΔX) ˆ 2 |Φ. E[2] = h0 + h1 Φ| (ΔX)

(82)

X0 . For the The wave function |Φ is determined by minimizing the energy E[2] ˆ ˆ case of X = N, the truncation at this order is nothing but the formulas proposed by Nogami (1964). For simplicity, the h2 is often kept to be constant and is determined after the variation.

As mentioned before, the truncation of the Kamlah expansion in (74) converges rather rapidly for the atomic nuclei with strong correlations for which case

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the overlap functions h(ϕ) and n(ϕ) are strongly peaked at the origin and the ˆ 2 |Φ is very large. For example, the wave function |Φ with fluctuation Φ| (ΔX) a large fluctuation in either angular momentum Φ| (ΔJˆ)2 |Φ or particle number ˆ 2 |Φ indicating the spread Φ| Δϕ 2 |Φ in the corresponding Euler angle Φ| (ΔN) or gauge angle is small. In other words, the orientation of the nucleus is almost fixed in space. For the nuclei with weak collective correlations (small fluctuation), however, one may need to include higher-order terms in the expansion (Wang et al. 2014). Finally, it is worth noting that these approximate methods only provide a correction to energy. The resultant wave function |Φ(q) is still an admixture of different irreps of the symmetry group, and thus the quantum number X0 is not recovered. One still needs to implement the projection operator to extract the component |ΦX0 (q) with correct quantum numbers from the symmetry-breaking wave function |Φ(q) to study (electromagnetic multiple) transitions between different states.

Electric Multiple Transitions Symmetry restoration plays an essential role in the studies of transition between nuclear states with different spin-parity quantum numbers. It is also one of the main motivations to implement symmetry-restoration methods into symmetry-breaking calculations. With the symmetry-projected wave functions |ΦJi/f ,σi/f  of the initial and find states defined in (57), where σ distinguishes different states for the same angular momentum J , one can evaluate the transition strength between these two ˆ λμ using the Wigner-Eckart theorem states under a general tensor operator Q     2 e2   ˆ  ΦJf ,σf (qf ) Qλ  ΦJi ,σi (qi )  , B(Eλ; Ji σi → Jf σf ) = 2Ji + 1 

(83)

where the reduced transition matrix element is determined by 

   ˆ  ΦJf ,σf (qf ) Q λ  ΦJi ,σi (qi )

= (2Jf + 1)

 Ki Kf

J ∗

Ji gKff ,σf gK i ,σi

 μK 

      ˆ λμ Pˆ Ji  Φ qi . × Φ qf Q K Ki 

(−1)Jf −Kf +λ



Jf λ Ji −Kf μ K 



(84)

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In the above derivation, the following relation is used: J ˆ λμ Pˆ J   = J  M  λμ|J M PˆKM Q MK

 ˆ λμ Pˆ J   , J  νλμ |J KQ νK

(85)

νμ

with J  M  λμ|J M being a Clebsch-Gordan coefficient. It is noted that in Eq. (84), the mean-field wave functions of the initial and final states are characterized with different collective coordinates qi and qf . In the PGCM, these collective coordinates are also integrated out weighted by the functions fJi ,σi (qi ) and fJf ,σf (qf ), respectively, as shown in Eq. (17).

Some Illustrative Applications Impact of Particle-Number Projection in Low-Lying States J Pˆ Z0 Pˆ N0 |Φ(q) of mean-field state Figure 2 displays the norm kernel Φ(q)| Pˆ00 24 |Φ(q) of Mg with projection onto angular momentum J = 0, 2, 4, 6, neutron number N0 = 12, and proton number Z0 = 12 as a function of quadrupole deformation parameter β20 from the calculation of a covariant density functional theory (CDFT) (Yao et al. 2011) with the PC-PK1 (Zhao et al. 2010) force. The

J Pˆ N Pˆ Z |Φ(q) as a function of the quadrupole deformation Fig. 2 The norm kernel Φ(q)| Pˆ00 parameter β20 for 24 Mg, where the intrinsic wave functions |Φ(β20  are from the constrained CDFT calculations. The number L of meshes in gauge angles is set to either 1 or 7, labeled with “no PNP” and “with PNP”, respectively

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norm kernels without the particle-number projection are also shown for comparison. According to (27), the norm kernel determines the weight |cJ N0 Z0 (q)|2 of the component |ΦJ N0 Z0 (q) in the mean-field wave function |Φ(q). It is shown clearly that the spherical mean-field state with β20 = 0 contains only the J = 0 component. Besides, one notices that the particle-number projection does not change the kernels of prolate deformed states with β20 ≥ 0.4, indicating that particle numbers are not violated in these states. It is because of the collapse of pairing correlations in these states. In contrast, pairing correlation is shown to be strong in oblate deformed states with the deformation parameter β20  −0.3, leading to a strong violation of particle numbers. As a result, these oblate states contain the component with particle numbers other than N0 = Z0 = 12 and the weight |cJ N0 Z0 (q)|2 is significantly decreased when the particle-number projection is turned on. Starting from each mean-field state |Φ(q), one can evaluate the energy of the symmetry-projected state |ΦJ N0 Z0 (q) for different value of J , and it is usually called projected energy surface. The results of different spin states are shown in Fig. 3, where the violation of particle numbers is treated in three different ways. The panels (a) and (c) show the results without and with the particlenumber projection, respectively, while the panel (b) shows the approximate way to consider the symmetry-violation effect on the projected energy, in which the original Hamiltonian H0 is replaced with the following one: Hˆ  = Hˆ 0 − λp (Zˆ − Z0 ) − λn (Nˆ − N0 ),

(86)

where the Lagrange multiplier λτ is chosen to be the Fermi energy, which is determined by requiring the correct average particle number Φ(q)| Nˆ τ |Φ(q) =

Fig. 3 Total energy curves projected onto angular momentum (J = 0, 2, · · · , 6), as well as the projected GCM states in 24 Mg for the cases of (a) without PNP, (b) with an approximate particlenumber correction (86), and (c) with exact PNP. The projected energy curves are plotted as a function of the intrinsic deformation β20 of the mean-field states. The energies of projected GCM states are indicated by bullets and horizontal bars placed at the average deformation β¯J,σ defined in (87)

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N0 (Z0 ). It has been shown in Yao et al. (2011) that the deviation of the average particle number from the correct one can be as large as 0.4 particles, both for neutrons and protons. The subsidiary term could bring an evident correction to nuclear total energy. Moreover, this deviation displays a pronounced dependence on both the angular momentum and deformation. One can see from Fig. 3a that the violation of particle numbers in oblate states produces a wrong ordering of the projected energy curves for which the J = 0 (red) one is even higher than the J = 6 (pink). With the correction considered in (86), the energy ordering of the energy curves becomes normal and closer to the results of the exact particle-number projection calculation. Finally, PGCM calculation was carried out by mixing all the axially deformed states, c.f. (17), where the collective coordinate q is discretized. One obtains discrete states with their energies indicated by dots centered at their mean deformations β¯J,σ defined as   2 β¯J,σ = β(qi ) gJ,σ (qi ) , (87) qi

where gJ,σ has been defined in (19). It is shown in Fig. 3 that a rather well-defined + + + rotational band (0+ 1 , 21 , 41 , 61 ) is built upon the prolate deformed configurations around β20 = 0.5. The energy spectrum of this rotational band is slightly more stretched in the calculation with the particle-number projection.

The Densities of Symmetry-Restored States The effect of restoring angular momenta for nuclear low-lying states can be clearly visualized from their density distribution ρ J (r) ≡ ΦJ N0 Z0 (q)|



δ (r − ri ) |ΦJ N0 Z0 (q),

(88)

i

where index i runs over occupied single-particle states for neutrons or protons. The projected density ρ J (r) has the following form (Yao et al. 2015a): ρ J (r) =



J 0L0 | J 0ρLJ (r)YL0 (ˆr),

(89)

L

where rˆ stands for the angular part of the coordinate r. The radial part ρLJ (r) of the L-component of the density is defined as ρLJ (r) =

 ∗ (−1)K J KL − K | J 0 d rˆ ρJ K (r, rˆ )YLK (ˆr) K

with the quantity ρJ K (r, rˆ ) given by

(90)

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2J + 1 π J∗ ρJ K (r) ≡ sin θ dθ dK0 (θ ) Φ(q)| 2 0  ˆ δ (r − ri ) e−iθ Jy Pˆ N0 Pˆ Z0 |Φ(q).

(91)

i

Equation (89) corresponds to the multipole expansion of the density ρ J (r) of the projected state labeled with a definite angular momentum J . As a special case, the density of the ground state with J = 0 is simplified as 1 ρ J =0 (r) = √ Y00 (ˆr) 4π



d rˆ  ρ00 (r, rˆ  ).

(92)

Figure 4 illustrates how the density distribution of neutrons (upper panels) and protons (lower panels) is modified at different levels of calculation for 34 Si. The left column shows contour plots of both neutron and proton densities for the particle-number projected HFB+Lipkin-Nogami state |Φ(q) with β2 = 0.26. Because of nonzero quadrupole deformation, the rotation symmetry is violated in the densities, and a deformed semi-bubble structure is observed. After projection on total angular momentum J = 0 (middle column), the density is obtained in the laboratory frame and becomes spherical. The configuration mixing in the PGCM calculation increases the central proton density again and simultaneously reduces the value at the bulge. As a result, the depletion factor defined as Fmax ≡

Fig. 4 The contour plots of the neutron (upper panels) and proton (lower panels) densities of 34 Si in the y = 0 plane for the particle number projected HFB+Lipkin-Nogami state with β2 = 0.26 (left column), its projection on both particle numbers and total angular momentum J = 0 (middle column) and for the 0+ 1 GCM ground state (right column). The Skyrme energy density functional SLy4 (Chabanat et al. 1998) was employed in the calculations. (Taken from Yao et al. (2012). Figure reprinted with permission from the American Physical Society)

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(ρmax,p −ρcent,p )/ρmax,p , which measures the reduction of the density at the nucleus center relatively to its maximum value, is reduced by the beyond-mean-field effect.

Dynamical Correlation Energies Symmetry restoration is equivalent to the diagonalization of nuclear Hamiltonian in a set of degenerate rotated states |Φ(q, ϕ), which generally leads to an additional (dynamical) correlation energy correction to nuclear ground state. The correction energy from the restoration of rotation symmetry for the deformed state |Φ(q) can be calculated as follows: ΔEJ =0 (q)) = E(q) − EJ =0 (q) =

Φ(q)| Hˆ |Φ(q) ΦJ (q)| Hˆ |ΦJ (q) − . Φ(q)| 1ˆ |Φ(q) ΦJ (q)| 1ˆ |ΦJ (q) (93)

Since this energy correction varies with nuclear deformation, the location of the energy minimum on the energy surface might be shifted after symmetry restoration. Given this situation, it is more natural to define the rotational energy correction as the difference between the energies of the lowest mean-field state |Φ(qmf ) and the lowest angular-momentum projected state |Φ(q0 ): ΔEJ =0 = E(qmf ) − EJ =0 (q0 ).

(94)

5

5

4

4 ΔEJ=0 [MeV]

ΔEJ=0 [MeV]

The rotational energy correction ΔEJ =0 for 605 nuclei from a MR-EDF calculation based on the SLy4 force is shown in Fig. 5 as a function of the quadrupole deformation β2 at the mean-field energy minimum or as a function of the nuclear mass number A, where both particle-number and angular momentum projection are implemented for axially deformed states. For simplification, the so-called topological Gaussian overlap approximation is employed for angular

3 2 1 0 −0.4

(a) −0.2

0.0 β2 (qmf )

0.2

0.4

3 2 1 0

(b) 0

50

100 150 200 Mass number A

250

Fig. 5 Dynamical correction energy ΔEJ =0 as a function of the (a) quadrupole deformation of the mean-field energy minimum state or (b) the mass number A for the 605 nuclei, where the mean-field states are generated from the HF+BCS calculation using the Skyrme interaction SLy4 (Chabanat et al. 1998). (Results are taken from Bender et al. 2006)

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momentum projection (Bender et al. 2006). It is seen that the ΔEJ =0 is small for spherical/weakly deformed nuclei and is about 3 MeV for either oblate or prolate deformed nuclei. Besides, the ΔEJ =0 s for light nuclei spread in between 1 and 4 MeV. In contrast, the ΔEJ =0 is close to 3.0 MeV in heavy nuclei. The size of energy correlation from symmetry restoration seems to be not sensitive to the employed energy density functionals (Rodríguez et al. 2015; Zhang et al. 2022).

Triaxiality in Atomic Nuclei with Shape Coexistence The PGCM provides a powerful tool of choice to study nuclei with strong shape mixing (the wave function of one nuclear state is an admixture of prolate and oblate deformed shapes) or shape coexistence. The latter is a nuclear phenomenon that the low-lying states of an atomic nucleus consist of two or more states with similar energies which have well-defined and distinct properties and can be interpreted in terms of different intrinsic shapes. The occurrence of shape coexistence affects the evolution behavior of nuclear low-lying states with spin and isospin. Neutrondeficient krypton isotopes are typical examples. Figure 6 displays the low-lying spectra of 76 Kr from the PGCM calculation by mixing axially deformed only or also triaxially deformed configurations, in comparison with data. It is shown that a restriction to axial states fails to reproduce the low-energy structures of the spectrum (including both excitation energies and electric quadrupole transitions), demonstrating the important role of including the triaxially deformed configurations in the description of the structure of the low-lying states of 76 Kr.

Fig. 6 Low-lying energy spectra of 76 Kr from the MR-CDFT calculation with the mixing of (c) axially and (b) triaxially (β22 = 0) deformed states projected onto both particle numbers and different angular momenta, in comparison with data (a). The electric quadrupole transition strengths B(E2) (in e2 fm4 ) are indicated on arrows. (Adapted from Yao et al. (2014). Figure reprinted with permission from the American Physical Society)

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Dynamical and Static Octupole Deformation The atomic nucleus 224 Ra is usually suggested to be a stable pear-shaped nucleus based on the measured electric multiple (Eλ) transitions (Ahmad and Butler 1993; Butler and Nazarewicz 1996). According to the rotation energy spectrum of asymmetric diatomic molecular, one is expected to observe a rotational band with alternating parity for the states with even and odd angular momenta in 224 Ra. However, this feature is shown only in the high-spin states of 224 Ra, not in the low-spin states. It has been interpreted as the rotation excitations of a quadrupole deformed “rotor” on top of octupole vibration motions. Figure 7 shows the calculated low-lying energy spectra of 224 Ra. It is seen that the spectra and the E2, E3 transitions can be reproduced reasonably well using only the energy-minimum configuration. However, the energy displacement between the parity doublets in the low-spin region can only be reproduced in the PGCM calculation by mixing the configurations around the equilibrium shape. As shown in Yao et al. (2015b), the collective wave function of the ground state is broadly distributed in the deformation plane, indicating a large shape fluctuation. With the increase of angular momentum, the collective wave functions of positiveparity states become gradually concentrated around the energy minimum and close to that of negative-parity states, displaying the classical picture of the stabilization of nuclear shapes with the increase of rotation frequency.

Fig. 7 Low-lying energy spectra for 224 Ra. The results from full configuration and the single energy-minimum configuration calculations are compared to available data. The numbers on arrows are E2 (red [light gray] color) and E3 (blue [dark gray] color) transition strengths (Weisskopf units). (Taken from Yao et al. (2015b). Figure reprinted with permission from the American Physical Society)

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Fig. 8 Systematics of the excitation energy of 2+ 1 state and the electric quadrupole transition + strength of 0+ 1 → 21 in the neutron-rich magnesium isotopes from the PGCM calculations based on two different energy density functionals (Rodin et al. 2006; Yao et al. 2011) and the ab initio VS-IMSRG calculation based on the NO2B approximation (Miyagi et al. 2020), in comparison with available data (National Nuclear Data Center 2020)

Evolution of Shell Structure in Neutron-Rich Nuclei The information on the evolution of shell structure can be learned from the properties of nuclear low-lying states. Generally speaking, a nucleus with a large shell gap is dominated by spherical or weakly deformed shapes with high excitation energy of the first 2+ state and low E2 transition strength between this state and the ground state. The description of the weakening of the neutron shell gap at N = 20 is challenging for many nuclear models. For example, the spectroscopic data for the low-lying states of 32 Mg indicate that this nucleus is deformed in the ground state. However, it is predicted to be spherical in almost all the mean-field approaches. Figure 8 shows the systematic evolution of the excitation energy of 2+ 1 state + and the electric quadrupole transition strength of 0+ 1 → 21 in the neutron-rich magnesium isotopes from two multi-reference EDF calculations (Rodin et al. 2006; Yao et al. 2011) and the ab initio VS-IMSRG calculation (Miyagi et al. 2020). The data show that the N = 20 and N = 28 shell gaps are melted in magnesium isotopes. This phenomenon has been well described by the non-relativistic EDF D1S force, partially by the relativistic EDF PC-F1 force, but poorly by the current implementation of valence-space IMSRG based on the normal-ordering two-body approximation with a chiral NN+3N interaction. The description of nuclear large deformation and shape coexistence is still a challenge to nuclear ab initio methods.

Concluding Remarks Symmetry and group theory play an important role in modeling atomic nuclei. In many mean-field-based nuclear models, certain symmetries of nuclear Hamiltonian are allowed to be broken in mean-field potentials to incorporate some essential

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many-body correlations, which leads to a symmetry-breaking nuclear wave function. This symmetry breaking is introduced artificially because of the employed approximations, and the broken symmetries need to be restored. In this chapter, the basic idea of this symmetry-breaking mechanism in mean-field approaches and the methods used to restore the broken symmetries at the beyond-mean-field level have been introduced with some illustrative examples. In the meantime, the success of these methods in the description of nuclear low-lying states has been demonstrated. Symmetry restoration has been popularly implemented in energy density functional calculations. However, most of the currently employed nuclear energy functionals were parameterized at the mean-field level. The inclusion of beyondmean-field effects from symmetry restoration and also configuration mixing may worsen the description of nuclear properties. A new parametrization of energy density functionals including in the fitting procedure the beyond-mean-field effects is required to improve further the accuracy of the calculation with nuclear multireference energy density functionals. Besides, it is worth mentioning that symmetryrestoration methods may meet the singularity and self-interaction problems if one starts from a general energy density functional (Anguiano et al. 2001; Bender et al. 2009; Duguet et al. 2009). This discussion is not covered in this chapter. Lots of efforts are devoted to finding out an energy functional free of these problems. This problem does not exist in the Hamiltonian-based approaches. Therefore, the implementation of symmetry-restoration methods into ab initio methods starting from a realistic nuclear force derived from, for instance, the chiral effective field theory proposed by Weinberg (Weinberg 1991) becomes very attractive. The abovementioned problems are not shown in these ab initio frameworks. This development has stimulated great research interest in the application of symmetryrestoration methods to nuclear structure and decay properties related to fundamental interactions and symmetries, such as the clustering structure in light nuclei, isospinsymmetry breaking correction in the superallowed β-decay, Schiff moment in octupole deformed nuclei, and the nuclear matrix elements of neutrinoless doublebeta decay. Acknowledgments I would like to thank J. Meng for constructive discussion during the preparation of this manuscript and I. Ivanov for his careful reading of it. Besides, I thank all my collaborators in the development of symmetry-restoration methods starting from various nucleonnucleon interactions or energy density functionals, including B. Bally, M. Bender, J. Engel, Y. Fu, K. Hagino, P.-H. Heenen, H. Hergert, C.F. Jiao, Z. P. Li, H. Mei, P. Ring, T. R. Rodriguez, D. Vretenar, X. Y. Wu, E.F. Zhou, and many others. This work was supported in part by the National Natural Science Foundation of China (Grant No. 12141501) and the Fundamental Research Funds for the Central Universities, Sun Yat-sen University.

References I. Ahmad, P.A. Butler, Ann. Rev. Nucl. Part. Sci. 43, 71 (1993) M. Anguiano, J.L. Egido, L.M. Robledo, Nucl. Phys. A 696, 467 (2001), nucl-th/0105003 A. Arima, F. Iachello, Phys. Rev. Lett. 35, 1069 (1975). https://doi.org/10.1103/PhysRevLett.35. 1069

2262

J. M. Yao

B. Bally, M. Bender, Phys. Rev. C 103, 024315 (2021), https://doi.org/10.1103/PhysRevC.103. 024315 R. Beck, H.J. Mang, P. Ring, Z. Phys. 231, 26 (1970) M. Bender, P.-H. Heenen, P.-G. Reinhard, Rev. Mod. Phys. 75, 121 (2003), https://doi.org/10.1103/ RevModPhys.75.121 M. Bender, G.F. Bertsch, P.-H. Heenen, Phys. Rev. C 73, 034322 (2006), https://doi.org/10.1103/ PhysRevC.73.034322 M. Bender, T. Duguet, D. Lacroix, Phys. Rev. C 79, 044319 (2009), https://doi.org/10.1103/ PhysRevC.79.044319 J.P. Blaizot, G. Ripka, Quantum Theory of Finite Systems (The MIT Press, Cambridge, MA, 1986) A. Bohr, Rev. Mod. Phys. 48, 365 (1976) A. Bohr, B.R. Mottelson, D. Pines, Phys. Rev. 110, 936 (1958) T. Brauner, Symmetry 2, 609 (2010), 1001.5212 D.M. Brink, J.P. Svenne, Nucl. Phys. A 154, 449 (1970) P.A. Butler, W. Nazarewicz, Rev. Mod. Phys. 68, 349 (1996) E. Caurier, A. Poves, A. Zuker, Phys. Lett. B 96, 11 (1980), ISSN 0370-2693, https://www. sciencedirect.com/science/article/pii/0370269380902002 E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, Nucl. Phys. A 635, 231 (1998) [Erratum: Nucl. Phys. A 643, 441–441 (1998)] J.F. Cornwell, Group Theory in Physics: An Introduction. Academic Press (1997) K. Dietrich, H.J. Mang, J.H. Pradal, Phys. Rev. 135, B22 (1964) F. Donau, Phys. Rev. C 58, 872 (1998) T. Duguet, J. Phys. G 42, 025107 (2015), 1406.7183 T. Duguet, A. Signoracci, J. Phys. G 44, 015103 (2017) [Erratum: J. Phys. G 44, 049601 (2017)], 1512.02878 T. Duguet, M. Bender, K. Bennaceur, D. Lacroix, T. Lesinski, Phys. Rev. C 79, 044320 (2009), https://doi.org/10.1103/PhysRevC.79.044320 J.L. Egido, Phys. Scripta 91, 073003 (2016), https://doi.org/10.1088/0031-8949/91/7/073003 J.L. Egido, L.M. Robledo, Nucl. Phys. A 524, 65 (1991) C.A. Engelbrecht, R.H. Lemmer, Phys. Rev. Lett. 24, 607 (1970), https://doi.org/10.1103/ PhysRevLett.24.607 V.N. Fomenko, J. Phys. A: Gen. Phys. 3, 8 (1970), https://doi.org/10.1088/0305-4470/3/1/002 A. Frank, J. Jolie, P. Van Isacker, Symmetries in Atomic Nuclei: From Isospin to Supersymmetry, Springer New York, NY (2009) M. Frosini, T. Duguet, J.-P. Ebran, V. Somà, Multi-reference many-body perturbation theory for nuclei: I. Novel PGCM-PT formalism. Eur. Phys. J. A58, 62 (2022a) M. Frosini, T. Duguet, J.-P. Ebran, B. Bally, T. Mongelli, T.R. Rodríguez, R. Roth, V. Somà, Multireference many-body perturbation theory for nuclei: II. Ab initio study of neon isotopes via PGCM and IM-NCSM calculations. Eur. Phys. J. A58, 63 (2022b) M. Frosini, T. Duguet, J.-P. Ebran, B. Bally, H. Hergert, T.R. Rodríguez, R. Roth, J.M. Yao, V. Somà, Multi-reference many-body perturbation theory for nuclei: III. Ab initio calculations at second order in PGCM-PT. Eur. Phys. J. A58, 64 (2022c) J.J. Griffin, J.A. Wheeler, Phys. Rev. 108, 311 (1957), https://doi.org/10.1103/PhysRev.108.311 G. Hagen, S.J. Novario, Z.H. Sun, T. Papenbrock, G.R. Jansen, J.G. Lietz, T. Duguet, A. Tichai Angular-momentum projection in coupled-cluster theory: structure of 34Mg. Phys. Rev. C 105, 064311 (2022) K. Hara, Y. Sun, Int. J. Mod. Phys. E 4, 637 (1995) H. Hergert, S.K. Bogner, T.D. Morris, A. Schwenk, K. Tsukiyama, Phys. Rep. 621, 165 (2016), http://www.sciencedirect.com/science/article/pii/S0370157315005414 D.L. Hill, J.A. Wheeler, Phys. Rev. 89, 1102 (1953), https://doi.org/10.1103/PhysRev.89.1102 H.A. Jahn, E. Teller, Proc. R. Soc. Lond. Ser. A 161, 220 (1937) A. Kamlah, Z. Phys. A 216, 52 (1968) K.D. Launey, T. Dytrych, J.P. Draayer, Prog. Part. Nucl. Phys. 89, 101 (2016), 1612.04298 H. Liang, N. Van Giai, J. Meng, Phys. Rev. C 79, 064316 (2009), 0904.3673

59 Symmetry Restoration Methods

2263

H.J. Lipkin, Ann. Phys. 9, 272 (1960), ISSN 0003-4916, https://www.sciencedirect.com/science/ article/pii/0003491660900324 P. Löwdin, Rev. Mod. Phys. 36, 966 (1964), https://doi.org/10.1103/RevModPhys.36.966 B.-N. Lu, J. Zhao, E.-G. Zhao, S.-G. Zhou, EPJ Web Conf. 38, 05003 (2012), 1303.0621 H.J. Mang, B. Samadi, P. Ring, Z. Phys. A 279, 325 (1976) J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long, L.S. Geng, Prog. Part. Nucl. Phys. 57, 470 (2006), nucl-th/0508020 T. Miyagi, S. Stroberg, J. Holt, N. Shimizu, arXiv:2004.12969 [nucl-th] (2020), https://arxiv.org/ abs/2004.12969 B. Mottelson, Rev. Mod. Phys. 48, 375 (1976) National Nuclear Data Center, NuDat 2 Database (2020), https://www.nndc.bnl.gov/nudat2 T. Nikši´c, D. Vretenar, P. Ring, Prog. Part. Nucl. Phys. 66, 519 (2011), ISSN 0146-6410, https:// www.sciencedirect.com/science/article/pii/S0146641011000561 E. Noether, Transp. Theory Stat. Phys. 1, 186 (1971), physics/0503066 Y. Nogami, Phys. Rev. 134, B313 (1964) T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu, Y. Utsuno, Prog. Part. Nucl. Phys. 47, 319 (2001), ISSN 0146-6410, https://www.sciencedirect.com/science/article/pii/S0146641001001570 P.-G. Reinhard, E. Otten, Nucl. Phys. A 420, 173 (1984), ISSN 0375-9474, https://www. sciencedirect.com/science/article/pii/0375947484904378 P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer, New York, 1980) L.M. Robledo, T.R. Rodríguez, R.R. Rodríguez-Guzmán, J. Phys. G 46, 013001 (2019), 1807.02518 V. Rodin, A. Faessler, F. Šimkovic, P. Vogel, Nucl. Phys. A 766, 107 (2006), ISSN 0375-9474, https://www.sciencedirect.com/science/article/pii/S0375947405012455 T.R. Rodríguez, A. Arzhanov, G. Martínez-Pinedo, Phys. Rev. C 91, 044315 (2015), 1407.7699 W. Satula, J. Dobaczewski, W. Nazarewicz, M. Rafalski, Phys. Rev. Lett. 103, 012502 (2009), 0903.1182 W. Satula, J. Dobaczewski, W. Nazarewicz, M. Rafalski, Phys. Rev. C 81, 054310 (2010), 0912.4381 J.A. Sheikh, J. Dobaczewski, P. Ring, L.M. Robledo, C. Yannouleas, J. Phys. G 48, 123001 (2021), 1901.06992 A. Signoracci, T. Duguet, G. Hagen, G. Jansen, Phys. Rev. C 91, 064320 (2015), 1412.2696 F. Stancu, Group Theory in Subnuclear Physics. Oxford Studies in Nuclear Physics, vol. 19 (1996) N. Tajima, H. Flocard, P. Bonche, J. Dobaczewski, P.H. Heenen, Nucl. Phys. A 542, 355 (1992) D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum. World Scientific Publishing Co. (1988) X.B. Wang, J. Dobaczewski, M. Kortelainen, L.F. Yu, M.V. Stoitsov, Phys. Rev. C 90, 014312 (2014), 1403.6364 Y.K. Wang, P.W. Zhao, J. Meng, Phys. Rev. C 105, 054311 (2022), https://doi.org/10.1103/ PhysRevC.105.054311 S. Weinberg, Nucl. Phys. B 363, 3 (1991) E. Wigner, Phys. Rev. 51, 106 (1937), https://doi.org/10.1103/PhysRev.51.106 J.M. Yao, J. Meng, D.P. Arteaga, P. Ring, Chin. Phys. Lett. 25, 3609 (2008), 0811.3058 J.M. Yao, J. Meng, P. Ring, D.P. Arteaga, Phys. Rev. C 79, 044312 (2009), https://doi.org/10.1103/ PhysRevC.79.044312 J.M. Yao, H. Mei, H. Chen, J. Meng, P. Ring, D. Vretenar, Phys. Rev. C 83, 014308 (2011), https:// doi.org/10.1103/PhysRevC.83.014308 J.-M. Yao, S. Baroni, M. Bender, P.-H. Heenen, Phys. Rev. C 86, 014310 (2012), 1205.2262 J.M. Yao, K. Hagino, Z.P. Li, J. Meng, P. Ring, Phys. Rev. C 89, 054306 (2014), https://doi.org/10. 1103/PhysRevC.89.054306 J.M. Yao, M. Bender, P.-H. Heenen, Phys. Rev. C 91, 024301 (2015a), https://doi.org/10.1103/ PhysRevC.91.024301 J.M. Yao, E.F. Zhou, Z.P. Li, Phys. Rev. C 92, 041304 (2015b), https://doi.org/10.1103/PhysRevC. 92.041304

2264

J. M. Yao

J.M. Yao, J. Engel, L.J. Wang, C.F. Jiao, H. Hergert, Phys. Rev. C 98, 054311 (2018), https://doi. org/10.1103/PhysRevC.98.054311 J.M. Yao, B. Bally, J. Engel, R. Wirth, T.R. Rodríguez, H. Hergert, Phys. Rev. Lett. 124, 232501 (2020), https://doi.org/10.1103/PhysRevLett.124.232501 J.M. Yao, J. Meng, Y. Niu, P. Ring, Prog. Part. Nucl. Phys. 103965 (2022), ISSN 0146-6410, https://www.sciencedirect.com/science/article/pii/S0146641022000266 K. Zhang et al. (DRHBc Mass Table), Atom. Data Nucl. Data Tabl. 144, 101488 (2022), 2201.03216 P.W. Zhao, Z.P. Li, J.M. Yao, J. Meng, Phys. Rev. C 82, 054319 (2010), https://doi.org/10.1103/ PhysRevC.82.054319 P.W. Zhao, P. Ring, J. Meng, Phys. Rev. C 94, 041301 (2016), https://doi.org/10.1103/PhysRevC. 94.041301

Quantum Microscopic Dynamical Approaches

60

Cédric Simenel

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-Body States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Particle States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Distinguishable Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Indistinguishable Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-Fermion States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Can’t the Many-Body Time-Dependent Schrödinger Equation Be Solved Exactly? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational Principle with the Dirac Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Dependent Hartree-Fock (TDHF) Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TDHF Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liouville Form of the TDHF Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving TDHF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Static HF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of TDHF Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limitations of TDHF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Justification of the Mean-Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random-Phase Approximation (RPA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harmonic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strength Function from the TDHF Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reduced Electric Transition Probability and Deformation Parameters . . . . . . . . . . . . . . . Giant Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RPA Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2266 2268 2268 2268 2268 2269 2269 2270 2270 2272 2272 2273 2275 2276 2277 2278 2279 2279 2280 2280 2280 2281 2282 2283 2283 2283

C. Simenel () Department of Fundamental and Theoretical Physics, Research School of Physics, The Australian National University, Canberra, ACT, Australia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_19

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RPA Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Widths of Giant Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TDHF Versus RPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Dependent Hartree-Fock-Bogoliubov Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manifestation of Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Including Correlations Via Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairing Correlations Through Breaking of Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . Restoring a Good Particle Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Applications of TDHFB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Balian-Vénéroni Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Balian-Vénéroni Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exact Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TDHF from the BV Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Dependent Random-Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2286 2287 2288 2288 2288 2289 2289 2292 2292 2293 2293 2294 2295 2298 2300

Abstract

Nuclear physics is ideal to test and develop techniques to describe the microscopic dynamics of quantum many-body systems. At low energy, nuclear dynamics is described with non-relativistic approaches based on the meanfield approximation and its extensions. Variational principles based on the stationarity of the action are introduced to build theoretical models with different levels of approximation. In particular, the time-dependent Hartree-Fock (TDHF) equation for mean-field dynamics and its linear approximation, also known as the Random-Phase Approximation (RPA), are derived. Predictions of vibrational spectra at the RPA level are presented as an application. The inclusion of beyond TDHF correlations and fluctuations are then discussed. In particular, pairing correlations are treated at the BCS and Bogoliubov levels. The BalianVénéroni variational principle is finally introduced. In addition to providing some insight into mean-field limitations, it offers a possibility to incorporate quantum fluctuations of one-body observables with the time-dependent RPA formalism.

Introduction The goal of this chapter is to introduce tools to describe the dynamics of nuclear systems out of equilibrium through their explicit time-evolution. This allows to investigate nuclear responses to external excitations leading to collective motion such as vibration, rotation, and fission. The same tools are also used to study heavyion collisions such as fusion and (multi)nucleon transfer reactions. Our focus in on the tools and concepts rather than applications. The ambition, here, is not to provide a review with an exhaustive list of references. The methods discussed in this chapter, as well as their applications to nuclear dynamics, have been recently reviewed in Simenel (2012), Lacroix and Ayik (2014), Nakatsukasa et al. (2016), Simenel and Umar (2018), Sekizawa (2019), and Stevenson and Barton

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(2019), which the reader is invited to consult for further details. See also the Sky3D solver (Maruhn et al. 2014; Schuetrumpf et al. 2018). Our framework is limited to low-energy nuclear dynamics, with typically few MeV per nucleon. In this case, the internal structure of nucleons in terms of quark and gluon degrees of freedom can be neglected. The nucleons are approximated as point-like spin 1/2 fermions without internal excitations. They also carry an isospin 1/2 to distinguish between protons and neutrons. Another consequence of the limitation to low-energy dynamics is that relativistic effects can be neglected in the first approximation. The nuclei are then treated as non-relativistic quantum many-body systems, with up to ∼500 nucleons in the case of actinide collisions (the heaviest nuclear systems that can be studied on Earth). Ideally, one has to solve the time-dependent Schrödinger equation ⎡ ⎤ A A 2   d p(i) ˆ i h¯ |Ψ (t) = ⎣ + v(i, ˆ j )⎦ |Ψ (t), dt 2m i=1

(1)

i>j =1

where |Ψ (t) is the time-dependent many-body quantum state and v(1, ˆ 2) the interaction between 2 nucleons. (For simplicity, three-body interactions and higher are omitted.) There are two main problems here. The first one is that v(1, ˆ 2) is not well known. Unfortunately, this interaction is different from the one between two isolated nucleons (which can be measured through scattering experiments), and connecting the interaction in vacuum with the interaction in medium is not straightforward. This is in part due to the fact that the nucleons are not point-like objects and could be polarized by surrounding nucleons, thus modifying the interaction in a nonstraightforward way. Another difficulty is that relativity leaves significant traces in the nuclear interaction such as spin-orbit interaction terms. The second problem is that it is usually impossible to solve Eq. (1) exactly. As a result, approximations to the quantum many-body problem are needed. A compromise is often needed between the complexity and level of precision of the mechanism one wants to describe in one hand, and, in the other hand, the computational capabilities that are available. This is the problem that is addressed here, leaving the very interesting and highly challenging issue with the in-medium interaction to others. It is thus assumed that v(1, ˆ 2) is “known,” and the focus of this chapter is on building efficient approximations to the time-dependent quantum many-body problem to describe aspects of low-energy nuclear dynamics. As a reminder (as well as an introduction to the notation), this chapter starts with a description of many-body states. The question “Why can’t we solve the timedependent Schrödinger equation exactly for many-body systems?” is then answered. Variational principles are introduced as a tool to optimize many-body dynamics. The time-dependent Hartree-Fock (TDHF) theory and its linearization leading to the Random-Phase (RPA) approximation are described in the following two sections. The inclusion of pairing correlations is then done within the time-dependent

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Hartree-Fock-Bogoliubov (TDHFB) and BCS approaches. The Balian-Vénéroni variational principle and its application to one-body fluctuations through the timedependent RPA are discussed in the last section.

Many-Body States One-Particle States The quantum state of a single-particle is noted |ϕ. It belongs to the Hilbert space of one particle, noted H1 . In the framework of second quantization, such a state is written |ϕ = aˆ ϕ† |−, where |− is the state of the vacuum and aˆ ϕ† creates a particle in the state |ϕ.

Two Distinguishable Particles When two particles are distinguishable, such as a proton and an electron in the Hydrogen atom, or a proton and a neutron in a deuteron, the state of each particle can be labeled, e.g., with a number. Two cases are considered: one where the particles are independent, and one where they are correlated.

Independent Particles The state of two independent particles can be written as |Φ = |1 : ϕ1 , 2 : ϕ2 . In this case, the state of one particle does not depend on the state of the other particle. The notations |1 : ϕ1 , 2 : ϕ2  = |ϕ1  ⊗ |ϕ2  = |ϕ1 |ϕ2  are used equivalently. Correlated Particles The state of two  correlated particles is written as a sum of independent particle states: |Ψ  = α Cα |1 : ϕα , 2 : ϕα . The correlation reads as follows: • If particle 1 is in ϕa , then particle 2 is in ϕa • If particle 1 is in ϕb , then particle 2 is in ϕb • ··· The coefficients Cα are the amplitude of probability for each configuration. For example, in the case of a deuteron, if a proton is found in r, the neutron is at a nearby position and not elsewhere. In the case of a prolately deformed nucleus, if one nucleon is found at one tip of the nucleus, then another one has to be present at the other tip.

Two Indistinguishable Fermions Now consider the case of two identical fermions, which are indistinguishable. In this case, the states |1 : ϕ1 , 2 : ϕ2  and |1 : ϕ2 , 2 : ϕ1  are equivalent and must

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both appear with the same probability in the two-body state. For two independent fermions, √ |Φ = |ϕ1 ϕ2  = (|1 : ϕ1 , 2 : ϕ2  − |1 : ϕ2 , 2 : ϕ1 )/ 2. The minus sign accounts for the fact that the state must be antisymmetric for fermions (there would be a plus sign for bosons). The antisymmetry is obvious as |ϕ1 ϕ2  = −|ϕ2 ϕ1 . In particular, |ϕϕ = 0, which is a manifestation of the Pauli principle. The state |Φ can be written in the form of a Slater determinant   1 |1 : ϕ1  |1 : ϕ2  = A |1 : ϕ1 , 2 : ϕ2 , |Φ = √  2 |2 : ϕ1  |2 : ϕ2  where A is the antisymmetrization operator. In the second quantization, |Φ = aˆ 2† aˆ 1† |−, with the antisymmetry ensured by † † ˆ 2 } = 0. The generalization to correlated the anticommutation relationship {aˆ 1, a two-particle states becomes |Ψ  = α Cα aˆ α†2 aˆ α†1 |−. Both |Φ and |Ψ  belong to the Hilbert space of two identical fermions H2 .

Many-Fermion States The generalization to the case of N identical fermions is now straightforward. For independent particles, the many-body state is described by a Slater determinant    |1 : ϕ1  · · · |1 : ϕN     1   .. .. † † |Φ = √   = A |1 : ϕ1 , · · · , N : ϕN  = aˆ N · · · aˆ 1 |−. . .   N!  |N : ϕ1  · · · |N : ϕN  As before, a correlated state is written as a sum of independent particle states  |Ψ  = α Cα |Φα . As a result, a complete set of orthonormal Slater determinants constitute a possible basis of the Hilbert space of N particles HN .

Why Can’t the Many-Body Time-Dependent Schrödinger Equation Be Solved Exactly? Consider a Slater (uncorrelated many-body state) |Ψ (t0 ) = |Φ0  at initial time t0 . The goal is to evaluate the state at a later time t1 . This can be done using the evolution operator ˆ

|Ψ (t1 ) = e−i H (t1 −t0 )/h¯ |Φ0 ,

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with the Hamiltonian Hˆ =

A  p(i) ˆ 2 i=1

2m

+

A 1  v(i, ˆ j ). 2 i,j =1

The first term of the Hamiltonian (the kinetic energy) is a one-body operator, while the second term (the interaction) is a two-body operator. (The 12 factor is to avoid double counting interactions between pairs of particles.) If vˆ = 0 (no interaction), then the particles remain independent and evolve pˆ 2 according to their own kinetic energy operator 2m . In this case, the state remains a single Slater determinant, and the problem becomes trivial. Of course, the difficulty comes from the interaction when it is non-zero. The exponential of an operator is usually expressed as a Taylor expansion. As the latter is an infinite sum, it needs to be truncated. For a time evolution operator, this is done with a small time increment Δt. The evolution over Δt is then repeated many times to reach the desired time t1 . ˆ

|Ψ (t1 ) = · · · e−i H Δt/h¯ · · · |Φ0 . Choosing Δt small enough, the exponential can be approximated in the first order, ˆ e−i H Δt/h¯  1 − i Hˆ Δt/h. ¯ The first iteration reads |Ψ (t0 + Δt)  |Φ0  − i Hˆ Δt|Φ ¯ One then has to  0 /h. compute the action of the interaction on the initial Slater, A ˆ j )|Φ0 . i,j =1 v(i, Let us use some simple (and maybe not so rigorous) arguments to evaluate  the complexity of the problem. If there was only one sum i instead of two ( i,j ), then the problem would be mathematically similar to the vˆ = 0 case, i.e.,  the Slater would remain a Slater at all times. However, there is an additional sum A j =1 , which means that the state at t0 + Δt is a sum of A (the number of particles) Slaters. One then expects A2 Slaters at t0 + 2Δt and, more generally, An Slaters at t0 + nΔt. This exponential increase is not tractable. As a result, the many-body Schrödinger equation can usually not be solved by brute force, and an approach that allows to build various levels of approximation is required.

Variational Principles Neglecting relativistic effects, the exact evolution is supposed to be given by the time-dependent Schrödinger equation. Thus, any theory that is equivalent to Schrödinger when no approximations are made is as “good” as Schrödinger.

Variational Principle with the Dirac Action There are several variational principles that could be used. For the moment, consider a variational principle based on Dirac’s action

60 Quantum Microscopic Dynamical Approaches



T

S= 0

2271

 d ˆ − H |Ψ . dt Ψ | i h¯ dt

Requiring the stationarity of the action, δS = 0, leads to the Schrödinger equation. Remember that for a complex variable z = Re(z) + i Im(z), there are two independent variables, Re(z) and Im(z), or, equivalently, z = Re(z) + i Im(z) and z∗ = Re(z) − i Im(z). Similarly, |Ψ  and Ψ | can be treated as independent variational quantities, and the property Ψ | = (|Ψ )† can be restored at the end. The stationarity condition is then expressed as δS δS = 0 and = 0 for 0 ≤ t ≤ T . δΨ | δ|Ψ  The first condition gives 

d − Hˆ |Ψ (t  ) = 0 dt  Ψ (t  )| i h¯ dt 0

 T  d  δΨ (t )| ˆ i h¯ − H |Ψ (t  ) = 0 dt δΨ (t)| dt 0 

d ˆ − H |Ψ  = 0 i h¯ dt

δ δΨ (t)|



T



(t )|  where δΨ δΨ (t)| = δ(t − t ) was used. The last line is the Schrödinger equation. The second condition can be expressed as

δ δ|Ψ (t)

0

T



d − Hˆ |Ψ (t  ) = 0. dt  Ψ (t  )| i h¯ dt

One has to be careful with the

d dt

term. Using integration by part,

 d |Ψ (t  ) = dt 0

  T d δ    Ψ (t )| |Ψ (t ) = dt δ|Ψ (t) dt 0

 T  T d δ|Ψ (t  )  δ|Ψ (t )   Ψ (t )| = Ψ (t )| − dt δ|Ψ (t) 0 dt δ|Ψ (t) 0  T d  δ|Ψ (t ) Ψ (t )| − Ψ (t)|. δ|Ψ (t) 0 dt δ δ|Ψ (t)

T Ψ (t  )|Ψ (t  ) 0 −

(Note that

δΨ (t  )| δΨ (t)|



T

dt  Ψ (t  )|



= δ(t − t  ) in the [· · · ]T0 term was not used intentionally.)

(2)

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The term with Hˆ is trivial if the latter does not depend on |Ψ : δ δ|Ψ (t)



T

dt  Ψ (t  )|Hˆ |Ψ (t  ) = Ψ (t)|Hˆ .

(3)

0

Combining Eqs. (2) and (3),

 δ|Ψ (T ) δ|Ψ (0) d i h¯ Ψ (T )| − Ψ (0)| − Ψ (t)| − Ψ (t)|Hˆ = 0. δ|Ψ (t) δ|Ψ (t) dt

(4)

Now “restore” the property Ψ | = (|Ψ )† . As shown earlier, imposing

δS δΨ | = 0 d Ψ | = −i h¯ dt

leads to the Schrödinger equation. Taking its Hermitian conjugate, δS Ψ |Hˆ . Comparing with Eq. (4), it is seen that to have δ|Ψ  = 0, one needs to have Ψ (T )|

δ|Ψ (0) δ|Ψ (T ) − Ψ (0)| = 0. δ|Ψ (t) δ|Ψ (t)

This is obtained by forbidding variations at the initial and final times, i.e., δ|Ψ (0) = δ|Ψ (T ) = 0.   T d To summarize, defining the Dirac action S = 0 dt Ψ | i h¯ dt − Hˆ |Ψ  and δS δS requiring δΨ | = 0 lead to Schrödinger equation, while δ|Ψ  = 0 gives its Hermitian conjugate if it is required that Ψ cannot vary at t = 0 and t = T .

Variational Space To get Schrödinger from the variational principle δS = 0, the variational space for Ψ was not restricted, i.e., it spans the entire Hilbert space HN . If the variational space was restricted to a sub-space FN of HN , one would expect to get a solution to the variational principle |Ψ (t) that is not solution to the Schrödinger equation, but which is an optimized approximation of the exact evolution within FN . The role of a theorist is then to choose a sub-space FN that (i) contains enough physics to give a good approximation to the evolution of the system of interest, and (ii) is simple enough so that the resulting equations of motion can be solved analytically or numerically.

Time-Dependent Hartree-Fock (TDHF) Theory If the state of the system was to remain a Slater at all time, then the evolution would be in principle easier to evaluate. The TDHF theory, as proposed by Dirac in 1930, is built upon the approximation that the state of the many-body system remains in

60 Quantum Microscopic Dynamical Approaches

2273

the sub-space FN of Slater determinants. To get the TDHF equation, one then needs to solve the variational principle δS = 0 within this subspace.

TDHF Equation The Dirac action is St0 ,t1 [Φ] =

t1

t0



d ˆ − H |Φ(t) dt Φ(t)| i h¯ dt

where |Φ ∈ FN is a Slater determinant built from the A-occupied single-particle states |ϕi , 1 ≤ i ≤ A. The expectation value of the time derivative is computed first. From

 A  d d d |Φ = A |ϕA  · · · |ϕ1  = A |ϕj  · · · |ϕ1  |ϕA  · · · dt dt dt j =1

one gets Φ|

 A  d d |Φ = [ϕ1 | · · · ϕA |A ] A |ϕj  · · · |ϕ1  |ϕA  · · · dt dt j =1

=

 A  d ϕ1 |ϕ1  · · · ϕj | |ϕj  · · · ϕA |ϕA  dt j =1

=

A  d ϕj | |ϕj . dt j =1

Now compute the expectation value of the Hamiltonian and write it as an energy density functional (EDF) E[ρ]: Φ|Hˆ |Φ = ϕ1 · · · ϕA |Hˆ |ϕA · · · ϕ1  = E[ϕ1 · · · ϕA ] ≡ E[ρ], where ρ is the one-body density matrix. Its elements are defined as ραβ = aˆ β† aˆ α .

(5)

For a Slater, ραβ = ϕ1 · · · ϕA |aˆ β† aˆ α |ϕA · · · ϕ1  =

A  j =1

ϕj |βα|ϕj .

(6)

2274

C. Simenel

For instance, in the position-spin-isospin basis of H1 , {|r, s, q}, one has   

ρ(rsq, r s q ) =

A 

ϕj∗ (r s  q  )ϕj (rsq).

j =1

The one-body density matrix is expressed as a function of all the occupied singleparticle states. It contains the same information on the system as the Slater. This is why the functional E[ϕ1 · · · ϕA ] can be replaced by the energy density functional E[ρ]. The EDF accounts for the kinetic energy and the interaction between the particles. The most popular EDF are the Skyrme and Gogny functionals (see Bender et al. (2003) for a review). The action is then written as S=



t1

dt t0

=



i=1



t1

dt t0

N  d i h¯ ϕi | |ϕi  − E[ρ(t)] dt

i h¯

N 

dx ϕi∗ (x, t)

i=1

d ϕi (x, t) − E[ρ(t)] dt



(7)    where the notations x ≡ (rsq) and dx = qs dr are introduced. The variational principle δS = 0 is solved by considering the variation of all the independent variables ϕi and ϕj∗ , 1 ≤ j ≤ A: δS =0 δϕj (x, t)

δS = 0. δϕj∗ (x, t)

and

(8)

The variation over ϕ ∗ gives d δS = i h¯ ϕj (x, t) − δϕj∗ (x, t) dt



t1

t0

dt 

δE[ρ(t  )] . δϕj∗ (x, t)

(9)

The functional derivative of E can be re-written, thanks to a change of variable δE[ρ(t  )] = δϕj∗ (x, t)



dy dy 

δE[ρ(t  )] δρ(y, y  ; t  ) . δρ(y, y  ; t  ) δϕj∗ (x, t)

(10)

Using δρ(y, y  ; t  ) = ϕj (y, t  ) δ(y  − x) δ(t − t  ) δϕj∗ (x, t)

(11)

60 Quantum Microscopic Dynamical Approaches

2275

and noting the single-particle Hartree-Fock Hamiltonian h with matrix elements δE[ρ(t)] , δρ(y, x; t)

h(x, y; t) =

(12)

one gets the TDHF equation for the set of occupied states i h¯

d ϕj (x, t) = dt

dy h(x, y; t) ϕj (y, t) .

(13)

As in the Schrödinger case, the variation over ϕ leads to the complex conjugate of the TDHF equation. The TDHF equation can also be written in a matrix form i h¯

d ϕj (t) = h(t)ϕj (t) , 1 ≤ j ≤ A, dt

or, using Dirac notation, i h¯

d ˆ |ϕj (t) = h(t)|ϕ j (t). dt

(14)

Note that the single-particle Hamiltonian h is self-consistent, i.e., it depends on the density matrix ρ. It is then also a function of time. As a result, the TDHF equation is equivalent to a set of non-linear single-particle Schrödinger equations. The nonlinearity comes from the self-consistency of h[ρ]: The Hamiltonian is a function of the states on which it acts.

Liouville Form of the TDHF Equation The one-body density matrix can be written as an operator of H1 : ρˆ =

A 

|ϕj ϕj |.

j =1

Indeed, its matrix elements ραβ = α|ρ|β ˆ =

A  ϕj |βα|ϕj  j =1

are the same as for a Slater determinant in Eq. (6). Multiplying Eq. (14) by ϕj | on the right and summing over j ,

2276

C. Simenel A  j =1



 A  d ˆ j ϕj |. |ϕj  ϕj | = i h¯ h|ϕ dt

(15)

j =1

Similarly, multiplying the Hermitian conjugate of Eq. (14) by |ϕj  on the left and summing over j leads to A 

−i h|ϕ ¯ j

j =1

  A d ˆ ϕj | = |ϕj ϕj |h. dt

(16)

j =1

Taking the difference between Eqs. (15) and (16) gives the Liouville form of the TDHF equation A  j =1

i h¯

A     d  ˆ j ϕj | − |ϕj ϕj |hˆ |ϕj ϕj | = h|ϕ dt j =1

i h¯

  d ˆ ρˆ . ρˆ = h, dt

(17)

This can be written also in the matrix form i h¯

d ρ = [h, ρ] . dt

(18)

Solving TDHF The easiest way to solve the TDHF equation numerically is to start from the nonlinear Schrödinger equation for occupied single-particle wave functions i h¯

 d ϕj (rsq, t) = dt  



d 3 r  h(rsq, r s  q  ; t) ϕj (r s  q  , t) ,

1 ≤ j ≤ A.

sq

Because h ≡ h[ρ(t)] is time-dependent, one cannot use e−ih(t1 −t0 )/h¯ as an evolution operator, unless one considers an evolution over a time interval that is short enough so that the variation of h(t) during this time interval can be neglected, i.e., ϕj (t + Δt)  e−ihΔt/h¯ ϕj (t). To ensure that the evolution is unitary (required for energy and particle-number conservations), one needs the same operator (up to Hermitian conjugation) to go from t → t +Δt as the one to do the time-reversed operation t +Δt → t. Therefore, h has to be estimated at t + Δt 2 .

60 Quantum Microscopic Dynamical Approaches

2277

A possible algorithm is {ϕj (t)} ⇒ ⇑ Δt Δt ϕi (t + Δt) = e−i h¯ h(t+ 2 ) ϕi (t) ⇑ ⇐ ρ(t + h(t + Δt 2 )

ρ(t)

⇒

h(t) ⇓ −i Δt ϕ˜i (t + Δt) = e h¯ h(t) ϕi (t) ⇓ ρ(t+Δt)+ρ(t) ˜ Δt ) = ⇐ ρ(t ˜ +Δt) 2 2 (19)

The initial state {ϕj (t0 )} could be a Hartree-Fock (HF) ground state, which is put into motion, thanks to an external one-body time-dependent potential added to the mean field. For collisions, one usually starts with two non-overlapping HF ground states at some finite distance in a large box, and a Galilean boost eikn ·r (n = 1, 2 denotes the two nuclei) is applied to induce a momentum hk ¯ n to the nucleons at the initial time t0 . Typical TDHF solvers use Cartesian grids with mesh spacing Δx = 0.6−1.0 fm. Several types of boundary conditions can be used: • Hard (particles are reflected by an infinite potential on the edges of the box) • Periodic (particles reaching one edge reappear on the other side) • Absorbing (particles are absorbed on the edge) Absorbing boundary conditions can be obtained with an additional layer of imaginary potential outside the box, or by “twisting” the phase on the boundaries (Schuetrumpf et al. 2016).

Static HF Static HF ground states are required to build the TDHF initial condition. The static HF equation reads [h[ρ], ρ] = 0. As they commute, one can find a single-particle states basis that diagonalizes both ρ and h simultaneously. (Note that to solve TDHF, a basis needs to be chosen. TDHF equations are usually solved in the “canonical” basis in which ρ is diagonal. In this basis, h is not diagonal if h and ρ do not commute, i.e., when the state evolves in time: [h, ρ] = i h¯ ρ.) ˙ In principle the HF ground states could be determined from any HF code as long as the same energy density functional is used as in the TDHF evolution. However, to minimize numerical errors, it is best to use an HF code that uses the same numerical approximations as in TDHF, in particular in terms of mesh grid and spatial derivatives.

2278

C. Simenel

One way to do this is by solving the HF equation with the imaginary time method, i.e., replacing t → −iτ in the TDHF equation. This makes an initial state (usually built from Nilsson or harmonic oscillator wave functions) converge toward the ground state. Of course, in the Hartree-Fock theory, the Hamiltonian is self-consistent, so an iterative process with imaginary time step Δτ is needed. In addition, e−hτ/h¯ is not unitary, so an orthonormalization of the occupied single-particle wave-functions is required, e.g., with a Gram-Schmidt procedure.

Examples of TDHF Applications Here, some applications are only briefly mentioned. See Reviews Simenel (2012), Lacroix and Ayik (2014), Nakatsukasa et al. (2016), Simenel and Umar (2018), Sekizawa (2019), and Stevenson and Barton (2019) for more detailed examples. TDHF calculations are now standard to investigate various aspects of nuclear dynamics. Collective vibrations can be studied by computing the time evolution of multipole moments of a nucleus following a collective boost excitation (see next section on RPA). Although most applications have been dedicated to giant resonances, low-lying collective vibrations have been also studied. Fusion barriers can be obtained in a straightforward manner by searching for fusion thresholds in head-on collisions. Below this threshold, the two fragments re-separate after contact. Once this threshold is overcome, the fragments merge in a single one. Compared to barrier heights of the “bare” nucleus-nucleus potential, this fusion threshold is often found at a lower energy due to dynamical effects, such as vibration and transfer in the entrance channel. Applying the same method for searching for fusion thresholds at finite impact parameters allows the computation of the above barrier fusion cross-sections. Even if the nuclei do not fuse after contact, part of the wave function can be transferred from one fragment to the other in quasi-elastic collisions. The final fragments being a coherent superposition of eigenstates of the particle-number operator, one obtains a distribution of probability to find a given number of nucleons in the final fragments. Naturally, both fragments are entangled so that “measuring” the number of protons or neutrons of one fragment projects the other fragment into a state with the corresponding number of nucleons as the total number of particles is conserved in TDHF. Significant efforts have been dedicated in the past decade to study heavy-ion collision at energies well above the barrier, leading to deep inelastic collisions (DIC), as well as in heavy systems leading to quasi-fission where fission-like fragments are produced without the intermediate formation of an equilibrated compound nucleus. These reactions offer a wide variety of observables to compare with experiment, such as fragment mass and charge distributions, scattering angle, and final kinetic energy. Comparison with theory offers valuable information on expected contact times between the fragments, as well as equilibration and dissipation mechanisms.

60 Quantum Microscopic Dynamical Approaches

2279

Limitations of TDHF The TDHF theory is based on the mean-field approximation. It describes the evolution of independent particles in a mean-field potential produced by the ensemble of particles. As a result, some correlations are neglected. This is the case of pairing correlations that lead to the formation of Cooper pairs and induce a superfluid behavior to some nucleons in the nucleus. These correlations are generated by the pairing residual interaction that is neglected in the Hartree-Fock approximation. Another limitation is the lack of quantum tunneling of the many-body wave function. This is due to the restriction of the variational space to a single Slater determinant. In reality, colliding nuclei have a non-zero probability to fuse even at sub-barrier energies, thanks to tunneling. However, to describe a coherent superposition of a fused system and a system of two outgoing fragments, one would need at least two Slaters, one for the compound nucleus and for the outgoing fragments. Standard applications of TDHF are then unable to describe sub-barrier fusion or spontaneous fission. There are, however, techniques such as DensityConstrained TDHF that allow to extract a nucleus-nucleus potential from TDHF simulations of two colliding nuclei (Umar and Oberacker 2006). Such a potential can then be used to evaluate tunneling probabilities with standard techniques and then predict fusion cross-sections even below the barrier. Alternatively, tunneling can be studied at the mean-field level through a Wick rotation changing real time into imaginary time (Levit 1980). However, this approach has not reached the level of realistic applications yet. It is also well known that TDHF lacks quantum fluctuations. This is evident from the comparisons between theoretical predictions and experimental measurements of fragment particle-number distributions in deep-inelastic collisions in which TDHF underpredicts widths of these distributions. This is again a limitation due to the restriction to a single Slater determinant. Indeed, in TDHF, one Slater is used to describe all exit channels. While the mean-field dynamics of this Slater is expected to give a reasonable description of the most likely outcome, it is unlikely to be realistic for exit channels that deviate significantly from the average one.

Justification of the Mean-Field Approximation It may seem strange that an independent particle approximation works for the nucleus, which is a very dense object of nucleons very close to each other and interacting via the strong and Coulomb interactions. What makes it work is the Pauli exclusion principle. The latter prevents the collision of two nucleons to happen inside the nucleus if the final state of the collision is already occupied. Consider two nucleons with energies E1 and E2 before they interact and energies E1 and E2 after the collision. Because of energy conservation, and assuming the state of the other nucleons is not changed, one has E1 + E2 = E1 + E2 . In the ground state, or a low-excitation-energy state, the nucleons have initially an energy

2280

C. Simenel

lower than the Fermi energy EF below which single-particle states are essentially occupied and above which they are mostly empty. This means that E1 < EF and E2 < EF . Thus, even if one final state has Ei > EF , the other must have an energy lower than EF , and then one nucleon ends up in a state which is likely to be already occupied, which is forbidden by Pauli. This “Pauli blocking” increases the mean-free path of a nucleon to about the size of the nucleus, reducing the effect of collisions and thus allowing the mean-field approximation to work.

Random-Phase Approximation (RPA) (The term “random phase” refers to another way of deriving the RPA equations.)

Harmonic Approximation The RPA is used to study small amplitude oscillations of many-body systems. It is a “harmonic” approximation as in the small amplitude limit (i.e., the system is only allowed small deformations around the ground state) the potential energy varies quadratically, V ∝ (Q − Qg.s. )2 , with respect to the deformation Q (e.g., multipole ω moment). In the harmonic picture, the relationship between the frequency 2π of the 1 oscillation and the energy spectrum of the harmonic oscillator En = (n + 2 )hω ¯ can then be used.

Transition Amplitude In addition to the energy h¯ ω of the vibration, one is also interested in its collectivity ˆ between quantified by the transition amplitude qν = ν|Q|0  the ground state |0 ˆ = A and the first phonon |ν of the vibrational spectrum. Q ˆ is a one-body i=1 q(i) operator, often chosen as a multipole moment ˆ LM = Q

A 

ˆ φ). ˆ rˆ L+2δL,0 YLM (θ,

i=1

 ˆ the In the expression for the transition amplitude qν = ν| A i=1 q(i)|0, contributions of each nucleon to the many-body state |ν are summed over. Then, the larger the magnitude of qν , the larger the collectivity. Both h¯ ων and qν can be extracted from TDHF using the linear response theory.

60 Quantum Microscopic Dynamical Approaches

2281

Linear Response Theory The time evolution of an observable Q(t) is computed from the time-dependent state |Ψ (t) after an excitation induced by a small boost on the ground state |0: ˆ ˆ |Ψ (0) = e−i Q |0 = |0 − i Q|0 + O( 2 ).

Inserting 1ˆ = {Eν },

 ν

(20)

|νν| where {|ν} are the eigenstates of Hˆ with eigenenergies |Ψ (0) = |0 − i



qν |ν + O( 2 ).

ν

The time evolution of the state reads ˆ

|Ψ (t) = e−i H t/h¯ |Ψ (0)  =e

−iE0 t/h¯

|0 − i



 qν e

−iων t

|ν + O( 2 ),

ν

(21) with hω ¯ ν = Eν − E0 . The response to this excitation can be written as ˆ (t) − 0|Q|0 ˆ Q(t) = Ψ (t)|Q|Ψ       ∗ iων t −iω t 2 ν ˆ |0 − i ˆ qν e ν| Q qν e |ν −0|Q|0+O( ) = 0| + i = −2



ν

ν

|qν |2 sin ωt + O( 2 ).

(22)

ν

The energies h¯ ων and transition amplitudes qν can be extracted from the strength function −1 ∞ RQ (ω) = lim dt Q(t) sin(ωt). (23) →0 π  0 Indeed, using the expression (22) for Q(t), ∞ 2 2 |qν | dt sin(ωt) sin(ων t) RQ (ω) = π ν 0  = |qν |2 δ(ω − ων ). ν

(24) (25)

2282

C. Simenel

Strength Function from the TDHF Evolution The evolution of Q(t) can be computed directly from the TDHF codes. The boost of Eq. (20) can be applied directly to the single-particle wave functions according to ˆ

|Ψ (0) = e−i Q |0 = e−i

A

ˆ i=1 q(i)

    |ϕ1 · · · ϕA  = | e−iq ϕ1 · · · e−iq ϕA .

Strength function Fourier transform

Octupole moment

To ensure that the response is in the linear regime, various values are usually used ˆ to check that Q(t) ∝ ε. ˆ An example of octupole response Q(t) in the linear regime is shown in Fig. 1 in 40 Ca and 56 Ni. The computation of the strength function can be done with Eq. (23). However, the computational time being finite, the integral cannot be performed up to t → ∞. Performing the integration up to T without modifying Eq. (23) would induce spurious oscillations in the spectrum as can be expected from the Fourier transform of a signal convoluted with a step function. To avoid such spurious oscillations, one usually multiplies Q(t) by a damping function f (t) (e.g., a cos2 or a Gaussian function) decreasing slowly from f (0) = 1 to f (T ) = 0. The damping function induces a small width proportional to 1/T in the strength function. The strength functions associated with the octupole responses in 40 Ca and 56 Ni are shown in Fig. 1. The rapid oscillation of the octupole moment in 56 Ni is associated with a higher energy hω ¯ ∼ 9 MeV than in 40 Ca where the main lowlying octupole vibration is found at ∼4 MeV. Other states with smaller strengths are also observed at higher energy. As these nuclei have ground-state spin and parity 0+ , and as the octupole operator allows for a change of angular momentum ΔL = 3h, ¯ these vibrational states have a spin-parity 3− . The area of the peaks correspond to the transition probabilities |qν |2 .

40

Ca Ni

40 56

Ca Ni

56

0

2

4

t (zs)

6

8

0

10

20

E (MeV)

30

40

Fig. 1 (left) Time evolution of the octupole moment following an octupole boost in 40 Ca and 56 Ni. (right) Corresponding strength functions. The vertical axes are in arbitrary units

60 Quantum Microscopic Dynamical Approaches

2283

Reduced Electric Transition Probability and Deformation Parameters The transition amplitudes can be used to evaluate other useful quantities, such as the reduced electric transition probability and the deformation parameters. The former can be evaluated according to B(Eλ; 0+ gs → ν) =

Z2 2 e |qν |2 , A2

where the proton density is assumed to be proportional to the neutron density. For + instance, one can evaluate B(E2; 0+ g.s. → 2 ) from the TDHF evolution of the quadrupole moment Q20 (t) following a quadrupole boost. The deformation parameter can be computed from (ν)

βλ =

4π |qν | , 3AR0λ

where R0  r0 A1/3 is the nuclear radius with r0  1.1–1.2 fm. It is often useful to compare deformations of different nuclei, as well as to account for couplings to low-lying vibrations in coupled-channel calculations of heavy-ion collisions.

Giant Resonances Giant resonances are highly collective nuclear vibrations that can usually decay by emitting nucleons. Their first phonons are often found between 10 and 30 MeV, depending on their multipolarity as well as their scalar/vector and isoscalar/isovector characteristics. Although most TDHF applications to nuclear vibrations are dedicated to giant resonances, the fact that they decay via particle emission means that such calculations has to be treated with care to avoid spurious finite-size effects of the numerical box. To avoid using very large boxes (which often limit the calculations to spherical symmetric systems), one can use absorbing boundary conditions. In this case, the resulting strength functions correspond to continuum-RPA calculations. An example of the TDHF calculation of a monopole giant resonance in 208 Pb is shown in Fig. 2. The direct decay of the oscillation amplitude is due to the evaporation of nucleon wave functions that are leaving the nucleus. This decay is exponential and contributes to the width of the peak in the strength function.

RPA Equation TDHF can be used in the small amplitude regime to extract energies and transition amplitudes of vibrational states at the RPA level. This is not, however, how a

2284

C. Simenel

Fig. 2 (left) Time evolution of the monopole moment following a monopole boost in 208 Pb. (middle) Corresponding strength functions. (right) Amplitude of the oscillations (in logarithmic vertical scale). (From Avez 2009)

standard RPA code works. The goal is now to show the connection between linearized TDHF and RPA in more details. ˆ When a small boost e−iεQ is applied to the wave function, it induces a perturbation to the one-body density matrix ρ = ρ (0) + ερ (1) + O(ε2 )

(26)

 (H F ) (H F ) where ρ (0) = A ϕh | is the one-body density matrix of the Hartreeh=1 |ϕh Fock ground state. In TDHF, the many-bodystate remains a Slater with the A corresponding one-body density matrix ρ = i=1 |ϕi ϕi |. One can see that ρ is a projector onto the sub-space of occupied single-particle states by computing  ρ2 = A |ϕ ϕ |ϕ ϕ | = ρ as ϕ |ϕ  = δ . Using this property with the i i j j i j ij i,j =1 expression (26) gives   2 ρ 2 = ρ (0) + ε ρ (0) ρ (1) + ρ (1) ρ (0) + O(ε2 ) = ρ = ρ (0) + ερ (1) + O(ε2 ). 2

As a result, ρ (0) = ρ (0) and ρ (1) = ρ (0) ρ (1) + ρ (1) ρ (0) . In the single-particle basis that diagonalizes ρ (0) ,

ρ (0)

 ⎞ ⎛ (0) (0)

 ρhp ρhh 10 ⎝ ⎠ =  (0)   (0)  = 00 ρph ρpp

where h and p denote hole and particle states, respectively.

(27)

60 Quantum Microscopic Dynamical Approaches

2285

Let us write ρ (1) in this basis and use Eq. (27) to get ⎛

ρ (1)

 ⎞ ⎛  (1) (1) ρhp 2 ρhh ⎠, = ⎝ (1)   (1) ⎠ = ⎝  (1)  ρph ρph ρpp 0

(1)

(1) ρhh



(1) ρhp

⎞

(1)

implying that ρhh = ρpp = 0. As a result, ρ (1) has only particle-hole non-zero matrix elements. The RPA equation is an equation for ρ (1) . It is obtained from the linearization of the TDHF equation, i.e., keeping only terms linear in ε:    d  (0) d ρ = [h, ρ] ⇒ i h¯ ρ + ερ (1) = h[ρ (0) + ερ (1) ], ρ (0) + ερ (1) . dt dt (28) Expanding the Hamiltonian as i h¯

h[ρ (0) + ερ (1) ] = h[ρ (0) ] +

δh (1) ερ + O(ε2 ) δρ

leads to i h¯ Note that

δh (1) δρ ρ

 d (1)  (0) (1)  δh (1) (0) . ρ = h[ρ ], ρ ρ ,ρ + dt δρ

is a shorthand notation for  ph



   δh  δh  (1) (1) ρ + ρ . δρph ρ=ρ (0) ph δρhp ρ=ρ (0) hp

One then gets the RPA equation i h¯

d (1) ρ = M ρ (1) , dt

(29)

where    δh · , ρ (0) M · = h[ρ (0) ], · + δρ is the RPA matrix acting only on the particle-hole (ph) space, and ph, residual interaction.

δh δρ

is the RPA, or

2286

C. Simenel

RPA Modes Let us decompose the ph density matrix as ρ (1) (t) =



ρν(1) eiων t + H.c.,

ν

where H.c. stands for “Hermitian conjugate.” The RPA equation then becomes i h¯



   † † ρν(1) iων eiων t − ρν(1) iων e−iων t = M ρν(1) eiων t + ρν(1) e−iων t .

ν

ν

(30) leading to (1) (1) − hω and ¯ ν ρν = M ρν

†

†

(1) (1) hω ¯ ν ρν = M ρν .

(31)

This is an eigenvalue problem with the RPA modes ν of energy hω ¯ ν. These energies hω ¯ ν are the same as the ones obtained from the linear response of TDHF, corresponding to peaks in the strength functions such as the ones plotted in Fig. 1. The task of an RPA code is then to diagonalize the RPA matrix in order to get these energies and, from the corresponding eigenstates, the transition amplitudes. Unlike TDHF, no explicit time evolution is required. RPA goes beyond HF, thanks to the residual interaction δh δρ . In HF, [h, ρ] = 0 which, in the basis which diagonalizes both h and ρ, gives ⎛ h(0) = ⎝

(0)

hhh 0



⎞ 

0 (0)

hpp

⎠ ,

    (0) (0) with hhh and hpp diagonal matrices with single-particle energies eh and ep on their diagonal. In RPA, there is also the residual interaction δh[ρ]ph δ 2 E[ρ] = δρp h δρhp δρp h in the particle-hole channel. This residual interaction is what is responsible for the collectivity of vibrational state, allowing many 1p1h excitations to contribute coherently. This is illustrated in Fig. 3 where an RPA response (from TDHF) is compared to an “unperturbed” one obtained without the residual interaction. Large peaks in the strength function, corresponding to collective vibrations, are indeed only observed once the RPA residual interaction is taken into account. Collective excitations in RPA (and in TDHF in the linear regime) are then coherent superpositions of one-particle one-hole (1p1h) excitations. One then understands why large differences between strength functions of various nuclei are

60 Quantum Microscopic Dynamical Approaches

1

1

2

3 4 time (zs)

5

6

7

2287

10

20 30 E (MeV)

40

Fig. 3 (left) Time evolution of the octupole moment following an octupole boost in 208 Pb with (solid line) and without (dashed line) RPA residual interaction. (right) Corresponding strength functions. (From Simenel 2012)

sometimes found, as illustrated in the octupole responses of 40 Ca and 56 Ni in Fig. 1. Indeed, despite the fact that these nuclei are both N = Z doubly magic isotopes, their low-lying collective octupole vibrations have very different properties, with the 56 40 40 3− 1 energy in Ni more than twice the one in Ca. This is because in Ca, 1p1h − configurations coupling to 3 spin parity can be formed by promoting nucleons from the sd shells to the empty 1f7/2 level sitting just above the Fermi level. In 56 Ni, however, the 1f 7/2 level is fully occupied, and coupling a hole in this level to a particle in the other fp states cannot produce the desired spin parity. As a result, the first 3− configurations can only be obtained by coupling 1f7/2 holes with the particle states in the higher energy 1g9/2 level, and/or by coupling sd holes with fp (excluding 1f7/2 ) particle states, producing a collective 3− 1 states at much higher energy than in 40 Ca.

Widths of Giant Resonances The width of a giant resonance (GR) has three components: • The Landau dampings is due to the coupling of the collective (coherent) superposition of 1p1h configurations forming the GR to incoherent 1p1h excitations. • The escape width is due to the emission of particles. It requires a proper treatment of the continuum. • The spreading width comes from the coupling of 1p1h configurations to the 2p2h states. RPA and TDHF account for the Landau damping and the escape width (if the continuum is properly accounted for), but not the spreading width that requires 2p2h residual interaction. This is the purpose of the “second-RPA” and extended TDHF approaches. As a result, the RPA and TDHF are known to underestimate the width of GR.

2288

C. Simenel

TDHF Versus RPA Although TDHF, in the linear limit, is formally equivalent to RPA, there are pros and cons in their implementation, and depending on the application, one may favor the use of one solver against the other. • RPA is sometimes solved with a simplified residual interaction in the ph channel (e.g., neglecting spin-orbit or Coulomb interaction). This can lead to spurious modes that need to be removed. TDHF, on the contrary, gives a fully selfconsistent RPA response, i.e., the same interaction (or functional) is used to compute ρ (0) and ρ (1) . Naturally, fully self-consistent RPA codes that account for all terms in the residual interaction are free of such spurious modes. • Unlike RPA, TDHF is not limited to the linear response. Nonlinearities in TDHF can be used to investigate other terms of the residual interaction than 1p1h, such as 3p1h and 3h1p. • A proper treatment of the continuum is necessary for unbound states, such as giant resonances. In continuum-RPA, this is done with Coulomb (for protons) and Hankel (for neutrons) functions. In TDHF, this is often done with absorbing boundary conditions. • RPA gives the amplitudes of the single particles contributing to a collective state more directly than TDHF. • TDHF may require long computational times, in particular if one wants to achieve a high resolution in the strength functions.

Time-Dependent Hartree-Fock-Bogoliubov Theory Pairs of nucleons with a strong overlap of their wave functions (e.g., Cooper pairs of nucleons in time-reversed states |j m and |j −m) are particularly sensitive to the short-range part of the nuclear interaction, leading to a pairing energy and pairing correlations between the particles: if one particle goes from one energy level to another, then the other one is expected to follow. The resulting correlated state is a coherent superposition of 2p2h configurations.

Manifestation of Pairing In condensed matter, Cooper pairs are responsible for superconductivity and superfluidity. In nuclear physics, pairing induces odd-even mass staggering (even numbers of neutrons or protons are more bound than odd numbers), enhanced pair transfer in heavy-ion collisions (with respect to the transfer of 2 independent nucleons), backbending effect (moment of inertia suddenly increases when there is enough rotational energy to break a pair, which in turn slows down the rotation), and a first (non-collective) excitation in mid-shell even-even nuclei at ∼2 MeV (energy needed to break a pair).

60 Quantum Microscopic Dynamical Approaches

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Note that the pairing residual interaction can only move a pair by ∼2 MeV, so only nucleons near the Fermi surface can be paired. Indeed, the scattering of more bound nucleons would be blocked by the Pauli exclusion principle as the final state would be already occupied. As a result, the contribution of pairing to the total binding energy is relatively small.

Including Correlations Via Symmetry Breaking Before discussing the specific case of pairing, let us review the general technique of including correlations at the mean-field level through symmetry breaking. The idea is that by allowing the mean-field Hamiltonian to break a symmetry of the exact Hamiltonian, one can include some of the residual interaction while preserving the simplicity of a mean-field treatment.

Translational Invariance As a first example, let us discuss the case of translational invariance. The interaction between two nucleons depends on their relative distance, not the position of their center of mass, so the interaction is of the form V (|r1 − r2 |). Iftranslational A ˆ invariance is imposed to the mean-field Hamiltonian Hˆ MF = i=1 h[ρ](i), ˆ however, one can see that h[ρ], and therefore, ρ itself has to be flat (constant in space). In this case the eigenstate of hˆ are plane waves occupying the entire space: There are no spatial correlations between the nucleons and therefore no nucleus. To allow a mean-field treatment of the nucleus, translational invariance needs then to be broken with a position dependent mean-field potential trapping the nucleons. Rotational Invariance The same approach can be used by breaking rotational invariance of Hˆ MF , i.e., with a deformed mean field along a particular direction. This induces a deformation of ρ and allows for a mean-field treatment of long-range correlations responsible for static deformations in nuclei.

Pairing Correlations Through Breaking of Gauge Invariance A mean-field treatment of pairing correlations can be obtained by breaking gauge invariance responsible for particle-number conservation.

Generalized One-Body Density Matrix † Recall that the one-body density ρ associated with a Slater |Φ = aˆ A · · · aˆ 1† |− contains the same information as the Slater itself. One can then focus the reasoning on how to treat pairing with an object like ρ.

2290

C. Simenel

A Slater has a “good” (i.e., a well-defined) number of particles A. Thus, only the matrix elements ραβ = Φ|aˆ β† aˆ α |Φ are needed as Φ|aˆ β† aˆ α† |Φ = Φ|aˆ β aˆ α |Φ = 0. If one now considers a state that does not have a good number of particles, then the so-called anomalous density καβ = Φ|aˆ β aˆ α |Φ and its complex conjugate ∗ = Φ|a καβ ˆ α† aˆ β† |Φ do not vanish. In this case, a “generalized” one-body density matrix R that contains both ρ and κ is used. It is defined as  ρ κ . −κ ∗ 1 − ρ ∗

R=

Quasi-particle Vacuum † The focus is now on the state |Ψ . For a Slater, |Φ = aˆ A · · · aˆ 1† |−. In order to preserve the simplicity of mean-field equations, it is desirable to keep a similar form of a product of independent operators. However, to describe a system that does not have a good number of particles, a Slater cannot be used. The form of a state describing a coherent superposition of various numbers of pairs of particles can be guessed as something like |Ψ  ∝ |− +



(Vij aˆ i† aˆ j† )|− +

ij

 (Vij aˆ i† aˆ j† )(Vkl aˆ k† aˆ l† )|− + · · ·

(32)

ij kl

where the first term in the r.h.s. contains zero pair, the second contains one pair, the third contains two pairs, etc. This state can also be written as |Ψ  ∝ Πij (Uij + Vij aˆ i† aˆ j† )|−

(33)

or, equivalently, |Ψ  ∝ Πj

 

 Uij∗ aˆ i

+ Vij∗ aˆ i†

|−

(34)

i

Note that U and V are generic matrices. Their complex conjugation in Eq. (34) is a convention. To show that Eqs. (33) and (34) are equivalent, start from (34), develop, bring the aˆ † to the left using {aˆ i† , aˆ j† } = {aˆ i , aˆ j } = 0 and {aˆ i† , aˆ j } = δij , and use a|− ˆ = 0. This is easier to show at the BCS level where U and V couple time-reversed states i and i¯ only:    |Ψ  ∝ Πi ui aˆ i − vi aˆ i¯† ui aˆ i¯ + vi aˆ i† |−   ∝ Πi u2i aˆ i aˆ i¯ + ui vi aˆ i aˆ i† − vi ui aˆ i¯† aˆ i¯ − vi2 aˆ i¯† aˆ i† |−   ∝ Πi ui + vi aˆ i† aˆ i¯† |−.

60 Quantum Microscopic Dynamical Approaches

2291

   ∗ † ∗a U through a Introducing the quasiparticle operator βˆi = a ˆ + V ˆ i i ij ij i Bogoliubov transformation, one gets |Ψ  = Πi βˆi |−. The U and V matrices can be chosen such that the quasi-particle operators obey the Fermionic anticommutation relationship {βˆi† , βˆj† } = {βˆi , βˆj } = 0 and {βˆi† , βˆj } = δij . In this case, βˆi |Ψ  = 0, ∀i, i.e., |Ψ  is a quasiparticle vacuum. A state with, say, N quasiparticle † excitations, β1† · · · βN |Ψ , is clearly a state of independent quasiparticles. This is the generalization of a Slater that was looked for.

Non-conservation of Particle Number Before going to the TDHFB equation, let us discuss why the particle-number conservation had to be broken in the first place. The problem comes from the 2p2h component of |Ψ , which is associated with the two-body density matrix with elements aˆ p† aˆ p†  aˆ h aˆ h . In (TD)HFB, these terms are approximated by aˆ p† aˆ p†  aˆ h aˆ h  (involving κ and κ ∗ ) which, to be non-zero, require the state to be a superposition of Slaters with different particle numbers, and which is described as a quasi-particle vacuum. TDHFB Equation The TDHFB equation can be obtained from the variational principle δS = 0 with the Dirac action and the variational space defined by independent quasiparticles. Note, however, that this variational space spans subspaces of HN , HN ±2 , HN ±4 , · · · . The TDHFB equation is given here without derivation: i h¯

d R = [H , R] , dt

(35)

where H [R] is the self-consistent generalized Hamiltonian of the form

H =

 h Δ , −Δ∗ −h∗

Δij =

δE[ρ, κ, κ ∗ ] δκij

and

is the pairing field.

Static HFB Equation It can be shown that Nˆ  is in fact conserved in TDHFB, i.e., the particle number is conserved in average. However, this is not the case in the algorithms used to solve the static HFB equation [H , R] = 0.

(36)

2292

C. Simenel

It is therefore necessary to add a constraint on the particle number, e.g., replacing Hˆ by Hˆ + λ(Nˆ − N0 )2 , where λ is a Lagrange parameter adjusted to force the system to have Nˆ  = N0 particles at its minimum of energy.

Restoring a Good Particle Number Physicists are not always at ease with extracting observables (whether for structure or reaction studies) from states that have a number of particles defined only in average. Therefore, they sometimes use a particle-number projection technique to transform the HFB state into a state with well-defined particle number. The projector onto a state with N particles acts as 1 PˆN |Ψ  = 2π Indeed, decomposing |Ψ  =





2π

ˆ

dϕ eiϕ(N −N ) |Ψ .

0

n Cn |Ψn 

with Nˆ |Ψn  = n|Ψn  and n ∈ N, one gets

2π 1  PˆN |Ψ  = Cn dϕ eiϕ(n−N ) |Ψn . 2π n 0 Noting that

 2π 0

dϕ eiϕ(n−N ) = 0 for n = N, one finally obtains PˆN |Ψ  = CN |ΨN .

 2π ˆ A potential problem is that 0 dϕ eiϕ N |Ψ  is a sum over quasiparticle vacuum. This takes the state outside the variational space of independent quasiparticle states. As a result, the final projected state is not entirely consistent with the variational approach.

Some Applications of TDHFB Pairing Vibrations Pairing vibrations are associated with oscillations of κ(t). They can be studied in the same way as “normal” vibrations with ρ(t) using the small amplitude limit. The quasiparticle RPA (QRPA) can also be obtained from the linearization of the TDHFB equation. Pairing vibrations are probed in pair transfer reactions with an operator containing aˆ † aˆ † and/or aˆ aˆ terms. For instance,   Fˆ = d 3 r f (r) aˆ † (r, ↑)aˆ † (r, ↓) + a(r, ˆ ↑)a(r, ˆ ↓)

60 Quantum Microscopic Dynamical Approaches

2293

induces ΔL = 0 transitions, e.g., from 0+ ground state of a nucleus with A nucleons to 0+ states in nuclei with A ± 2 nucleons. The TDHFB linear response Fˆ (t) contains modes from both final nuclei. As in RPA, some modes are clearly collective (larger peaks in the strength function). They are called pairing vibrations and are expected to be significantly populated in pair-transfer reactions.

Fusion Barrier Recently, full TDHFB simulations of fusion reactions have shown a possible hindrance of fusion due to different gauge angles (created by the gauge symmetry breaking) between the colliding nuclei and producing a domain wall at the neck (Magierski et al. 2017; Scamps 2018).

Balian-Vénéroni Variational Principle The possibility of having different variational principles was mentioned earlier. Indeed the action is not unique, and as long as one recovers Schrödinger in the exact case, i.e., without restriction of the variational space, they could equivalently be considered in order to build approximated dynamics. It turns out that TDHF is usually recovered at the simplest level, both from variational and non-variational approaches. It is often interesting to see different approaches to derive an equation such as TDHF as they usually give us new perspectives on the range of possible applications and limitations of the theory.

Balian-Vénéroni Action The Dirac action has two variational quantities, Ψ (t) and Ψ ∗ (t), both describing the time evolution of the many-body state. They lead to the Schrödinger picture in which the time evolution is carried by the state, while the observables are timeindependent quantities. This is not the only possible picture. For instance, in the Heisenberg picture, the observables evolve in time, while the state remains static. Balian and Vénéroni (BV) proposed an action that mixes both Schrödinger and Heisenberg pictures. The BV action has two variational quantities, namely, the many-body state described by its density matrix ˆ D(t) = |Ψ (t)Ψ (t)|, ˆ (Balian and Vénéroni 1981). This way, one can recover the and the observable A(t) exact dynamics in both the Schrödinger and Heisenberg pictures. The BV action is defined as      t1 ˆ d D(t) i ˆ 1 )D(t ˆ ˆ 1) − ˆ S = Tr A(t + [Hˆ , D(t)] (37) dt Tr A(t) dt h¯ t0

2294

C. Simenel

with the boundary conditions ˆ 0 ) = Dˆ 0 , D(t

(38)

(the initial state of the system Dˆ 0 at the initial time t0 is known) and ˆ 1 ) = Aˆ 1 , A(t

(39)

(the goal is to compute the expectation value Aˆ 1  at the final time t1 > t0 ).

Exact Evolution Schrödinger Equation Let us verify that the exact evolution is recovered if no restrictions on the variational spaces are imposed. As the approach contains two variational quantities, the variational principle δS = 0 is obtained by imposing δA S = 0 and δD S = 0, ˆ where δX induces small variations with respect toX(t).  ˆ 1 )D(t ˆ 1 ) = 0. As a result, ˆ As A(t1 ) is fixed by a boundary condition, δA A(t

t1

t0





ˆ d D(t) i ˆ ˆ + [Hˆ , D(t)] dt Tr δA A(t) dt h¯

 = 0.

ˆ For this to be true for all variations of A(t), the term in brackets must be zero, leading to the Liouville form of the Schrödinger equation i h¯

  ˆ d D(t) ˆ = Hˆ , D(t) . dt

(40)

Heisenberg Equation ˆ Now consider the variations with respect to D(t). It is easier to first rewrite the integral term in the action (37) using an integration by part:

     ˆ d D(t) i ˆ ˆ 1 )D(t ˆ ˆ 1 ) − Tr A(t ˆ 0 )D(t ˆ 0) + [Hˆ , D(t)] = Tr A(t dt Tr A(t) dt h¯   t1   ˆ d A(t) i t1 ˆ ˆ ˆ − dt Tr A(t)[ Hˆ , D(t)] . (41) dt Tr D(t) + dt h¯ t0 t0

t1 t0





  ˆ 0 ) is fixed, the variation of Tr A(t ˆ 0 )D(t ˆ 0 ) vanishes. Conveniently, the As D(t   ˆ 1 )D(t ˆ 1 ) will cancel with the first term in the action (37). In addition, the Tr A(t last term can be rearranged using

60 Quantum Microscopic Dynamical Approaches

2295

ˆ Hˆ , D]) ˆ = −Tr(D[ ˆ Hˆ , A]). ˆ Tr(A[ One then obtains δD S =

t1 t0





ˆ d A(t) i ˆ ˆ + [Hˆ (t), A(t)] dt Tr δD D(t) h¯ dt

 = 0.

(42)

ˆ As this must be true for all variations of D(t), one concludes that i h¯

  ˆ d A(t) ˆ = Hˆ , A(t) , dt

(43)

ˆ in the Heisenberg picture. which gives the exact evolution of A(t)

TDHF from the BV Variational Principle ˆ Let us restrict the variational space of D(t) to independent particle states, i.e., ˆ ˆ to one-body D = |ΦΦ| where |Φ is a Slater, and the variational space of A(t) operators, i.e., in the second quantization, Aˆ =



Aαβ aˆ α† aˆ β .

αβ

Because the variations are arbitrary, one is free to choose ˆ = aˆ α† aˆ β , t0 ≤ t ≤ t1 , δA A(t) where α and β are arbitrary. δSA = 0 leads to

 d i |ΦΦ| + [Hˆ , |ΦΦ|] = 0. Tr aˆ α† aˆ β dt h¯    i  d  † Tr aˆ α aˆ β |ΦΦ| + Tr aˆ α† aˆ β Hˆ , |ΦΦ| = 0. h¯ dt ˆ = Form the definition of the trace Tr(O) HN , one has



ˆ

ν Ψν |O|Ψν ,

where {|Ψν } is a basis of

   Tr aˆ α† aˆ β |ΦΦ| = Ψν |aˆ α† aˆ β |ΦCν∗ = Φ|aˆ α† aˆ β |Φ = ρβα , ν

(44)

(45)

2296

C. Simenel

 where the state is decomposed into |Φ = ν Cν |Ψν , and the definition of the one-body density matrix in Eq. (5) was used. The second trace in Eq. (44) can be computed in the same way:        Tr aˆ α† aˆ β Hˆ , |ΦΦ| = Tr aˆ α† aˆ β Hˆ |ΦΦ| − Tr aˆ α† aˆ β |ΦΦ|Hˆ   = Φ|aˆ α† aˆ β Hˆ |Φ − Tr Hˆ aˆ α† aˆ β |ΦΦ| = Φ|aˆ α† aˆ β Hˆ |Φ − Φ|Hˆ aˆ α† aˆ β |Φ   = Φ| aˆ α† aˆ β , Hˆ |Φ.

(46)

The goal is now to show that this last term is just [h, ρ]. This is usually done by introducing an explicit expression for Hˆ . However, here, one wishes to remain general so that the approach is still valid in the EDF formalism. Introducing a basis {|ν} of eigenstates of Hˆ with eigenvalues {Eν }, Φ|



aˆ α† aˆ β , Hˆ



 |Φ = Φ| =



aˆ α† aˆ β ,



 ˆ |νν|H |Φ

ν

  Eν Φ| aˆ α† aˆ β , |νν| |Φ

ν

One can write |ν in the n−particle n−hole basis built from |Φ as |ν = C0ν |Φ +





ν † Cph aˆ p aˆ h |Φ +

ph

pp hh

ν Cpp ˆ p† aˆ p†  aˆ h aˆ h |Φ + · · ·  hh a

(47)

ν = Φ|a with Cph ˆ h† aˆ p |ν. As a result,

     ν ν ∗ . Φ| aˆ α† aˆ β , Hˆ |Φ = Eν Cβα C0ν ∗ − C0ν Cαβ

(48)

ν

Now rearrange the [h, ρ] term in order to express it as a function of the same quantities: [h, ρ]αβ

    δE[ρ] δE[ρ] hαγ ργβ − ραγ hγβ = = ργβ − ραγ δργ α δρβγ γ γ

Noting E[ρ] = Φ|Hˆ |Φ ≡ Hˆ  and ραβ = Φ|aˆ β† aˆ α |Φ ≡ aˆ β† aˆ α ,

60 Quantum Microscopic Dynamical Approaches

[h, ρ]αβ =





δHˆ  δaˆ α† aˆ γ 

γ

2297

aˆ β† aˆ γ  − aˆ γ† aˆ α 

δHˆ  δaˆ γ† aˆ β 

 .

(49)

Notice that in the canonical basis, aˆ β† aˆ γ  and aˆ γ† aˆ α  are non-zero only if γ is a  hole state. Using Hˆ  = ν Eν φ|νν|φ and the decomposition (47), one can see that Eq. (49) includes terms like δ(φ|νν|φ) δaˆ α† aˆ γ 

=



ν Cph

ph

δaˆ p† aˆ h  δaˆ α† aˆ γ 

ν|φ +



ν ∗ Cph φ|ν

ph

δaˆ h† aˆ p  δaˆ α† aˆ γ 

ν = Cαγ ν|φ + Cγν α ∗ φ|ν.

(50)

Only Cph terms are non-zero and, for a Slater, ρhh = δhh and ραp = ρpα = 0. As a result, Cγν α ργβ = Cγν α ρβγ = 0 as γ cannot be a particle and a hole at the same time. Equation (49) then becomes [h, ρ]αβ =

 ν

=



Eν

  ν ν ∗ Cαγ ν|φργβ − ραγ Cβγ φ|ν γ

  ν ν ∗ ν Eν Cαβ C0ν ∗ − Cβα C0 ,

(51)

ν

Comparing with Eq. (48), the conclusion is that   [h, ρ]αβ = φ| aˆ β† aˆ α , Hˆ |φ.

(52)

Combining Eqs. (44), (45), (46), and (52), d i ρβα + [h, ρ]βα = 0 dt h¯ which, after rearranging, leads to the TDHF equation in the Liouville form i h¯

d ρ = [h, ρ] . dt

This demonstration shows that the BV variational principle with independent particles and one-body operator variational spaces leads to the TDHF equation. Note that no explicit form for Hˆ was invoked, so this derivation still holds in the energy density functional approach. Importantly, the fact that TDHF was obtained by restricting the variational space for the observables to one-body operators tells us that TDHF is only optimized to compute expectation values of one-body observables. In particular, fluctuations of one-body observables,

2298

C. Simenel

 σF = Fˆ 2  − Fˆ 2 ,  with Fˆ = αβ fαβ aˆ β† aˆ α are outside the variational space. Indeed, Fˆ 2 contains twoˆ This explains why TDHF underestimates the widths body terms of the form aˆ † aˆ † aˆ a. of fragment mass and charge distributions in deep-inelastic collisions.

Time-Dependent Random-Phase Approximation To estimate one-body fluctuations and correlations, Balian and Vénéroni used their ˆ variational principle with a larger variational space for the operators A(t) (Balian † and Vénéroni 1984). Instead of aˆ aˆ (one-body), they used a variational space for † operators of the form eaˆ aˆ . Then, the fluctuations of an operator Fˆ can be computed from ˆ Aˆ 1 = e−εF

using  1  lnAˆ 1  = −εFˆ + ε2 Fˆ 2  − Fˆ 2 + O(ε3 ). 2 Indeed, the ε2 term is σF2 /2. How to solve the BV variational principle in this space is beyond the scope of this chapter and can be found in detail in Simenel (2012). The idea is to take small values of ε, which means that small fluctuations of the action around the stationary path are considered. By analogy with the RPA (small fluctuation around a local static state minimizing the energy), this approach is called the time-dependent RPA (TDRPA). Note that it can be also used to get correlations (in addition to fluctuations) between one-body observables. Correlations of two (commuting) one-body operators, Xˆ and Yˆ , are given by  σXY =

ˆ Yˆ . Xˆ Yˆ  − X

(53)

ˆ just use the above formula with Xˆ = Yˆ . It can be shown To get the fluctuations of X, that, in TDHF, the correlations at t1 are obtained from 2 σXY

(T DH F )

(t1 ) = Tr {Yρ(t1 )X[I − ρ(t1 )]} ,

(54)

where I is the identity, and X and Y are the matrices representing the operators Xˆ and Yˆ .

60 Quantum Microscopic Dynamical Approaches

2299

One can also show that, in the TDRPA, this expression becomes 2 σXY

(T DRP A)

Tr {[ρ(t0 ) − ρX (t0 , )] [ρ(t0 ) − ρY (t0 , )]} , →0 2 2

(t1 ) = lim

(55)

where the one-body density matrices ρX (t, ) are solutions to the TDHF equation with the boundary condition ρX (t1 , ) = eiX ρ(t1 )e−iX .

(56)

To calculate the correlations, the state at t1 is propagated backwards in time to the initial time t0 , explaining why the correlations at the time of interest, t1 , depends on the density matrices at the initial time, t0 . Equations (54) and (55) are not equivalent. In fact, fluctuations computed from TDRPA are larger (in particular in DIC) than with TDHF, in better agreement with experiment. To investigate fluctuations of particle number in outgoing fragments of deepinelastic collisions, one can use  Xˆ = Nˆ V = dr aˆ † (rsq)a(rsq) ˆ Θ(r), (57) sq

where Θ(r) = 1 in the volume V containing the fragment, and 0 elsewhere. According to Eq. (56), the single-particle states (protons and/or neutrons) are transformed at t1 as ψiX (rsq, t1 ; ) = e−iΘ(r) ϕi (rsq, t1 ).

(58)

These transformed states are then propagated backwards in time to the initial time t0 for various (small) values of ε. The succession of forward and backward propagations is illustrated in Fig. 4. To obtain the correlations (e.g., between proton and neutron numbers) and fluctuations, one then evaluates Eq. (55), which can be reduced to

Fig. 4 Schematic representation of the TDRPA computational method to determine the fluctuations of N at final time t1 . (From Simenel 2012)

2300

C. Simenel

σXY (t0 ) =

η00 (t0 ) + ηXY (t0 ) − η0X (t0 ) − η0Y (t0 ) , 2 2

(59)

where ηXX is ηXX =

A ! "   X  X 2  ψi ψj  ,

(60)

i,j =1

with X = 0 and Y = 0 denoting the use of untransformed states, ϕi . Acknowledgments This work has been supported by the Australian Research Council under Grant No. DP190100256.

References B. Avez, PhD thesis, University of Paris XI (2009) R. Balian, M. Vénéroni, Phys. Rev. Lett. 47, 1353 (1981) R. Balian, M. Vénéroni, Phys. Lett. B 136, 301 (1984) M. Bender, P.-H. Heenen, P.-G. Reinhard, Rev. Mod. Phys. 75, 121 (2003) D. Lacroix, S. Ayik, Eur. Phys. J. A 50, 95 (2014) S. Levit, Phys. Rev. C 21, 1594 (1980) P. Magierski, K. Sekizawa, G. Wlazlowski, Phys. Rev. Lett. 119, 042501 (2017) J.A. Maruhn, P.-G. Reinhard, P. D. Stevenson, A.S. Umar, Comput. Phys. Commun. 185, 2195 (2014) T. Nakatsukasa, K. Matsuyanagi, M. Matsuo, K. Yabana, Rev. Mod. Phys. 88, 045004 (2016) G. Scamps, Phys. Rev. C 97, 044611 (2018) B. Schuetrumpf, W. Nazarewicz, P.-G. Reinhard, Phys. Rev. C 93, 054304 (2016) B. Schuetrumpf, P.-G. Reinhard, P.D. Stevenson, A.S. Umar, J.A. Maruhn, Comput. Phys. Commun. 229, 211 (2018) K. Sekizawa, Front. Phys. 7, 20 (2019) C. Simenel, Eur. Phys. J. A 48, 152 (2012) C. Simenel, A.S. Umar, Prog. Part. Nucl. Phys. 103, 19 (2018) P.D. Stevenson, M.C. Barton, Prog. Part. Nucl. Phys. 104, 142 (2019) A.S. Umar, V.E. Oberacker, Phys. Rev. C 74, 021601 (2006)

Section IX Tensor Interaction in Nuclei Hiroshi Toki

Hadrons from Quarks and Chiral Symmetry

61

Atsushi Hosaka

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Symmetry and Its Spontaneous Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview with Some History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possible Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One Pion Exchange Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The NJL Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial Vector Coupling Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exotic Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X(3872) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pc Pentaquarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2304 2305 2305 2307 2307 2310 2312 2315 2318 2318 2319 2321 2325 2325 2327 2331 2332

Abstract

Chiral symmetry with its spontaneous breaking is one of the greatest achievements and has provided the basis of theoretical methods in various fields in modern physics. The idea has been developed in hadron physics even before quantum chromodynamics (QCD) was established. This chapter discusses several important hadron phenomena including the recent active discussions in A. Hosaka () Research Center for Nuclear Physics, Osaka University, Mihogaoka, Ibaraki, Japan Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_56

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exotic hadrons with the basis of quark dynamics with chiral symmetry implemented. The pion as the Nambu-Goldstone boson coupled to the light constituent u, d quarks gives a simple picture for hadron structure and interaction.

Introduction In the last two decades of the twenty-first century, hadron physics has been experiencing remarkable progress. On one hand, high energy accelerator facilities have been reporting phenomena implying new hadrons, as triggered by signals of exotic hadrons (Choi et al. 2003; Nakano et al. 2003; Hosaka et al. 2016) (References in this note are not quite complete. However, further related references are found easily at iNSPIRE: https://inspirehep.net/). On the other hand, developments in supercomputers and theories enable first principle calculations for hadrons including their interactions by lattice QCD simulations (Zyla et al. 2020; Ishii et al. 2007) (Latest progresses in lattice simulations for hadron spectroscopy are found in Zyla et al. (2020)). Yet there are many unsettled issues in dynamical processes such as resonance formations (excitation mechanism) and their interactions. As compiled in the Particle Data (Zyla et al. 2020), by now several hundreds of hadrons are identified. This is the issue of spectroscopy which is the most fundamental field of hadron physics. In principle, by knowing interactions of hadrons, it is possible to construct scattering amplitudes from which various properties of hadrons can be extracted and predicted. However, difficulties lie in that hadrons are composite objects of quarks and gluons. Due to the non-perturbative nature of the fundamental theory of the strong interaction, quantum chromodynamics (QCD), hadron structures from the “bare” quarks and gluons of QCD are highly nontrivial. For this, lattice simulations of QCD have been successful in explaining properties of stable hadrons including their interactions. Stable here means that they do not decay via the strong interaction. However, it is not easy to extract physical mechanism that explains the phenomena. Now, from the observation of systematics in hadron spectrum, for instance, baryons as shown in Fig. 1, it is natural to expect a simple mechanism that governs their properties. Here it is noted that the resonances are observed up to excitation energy of around 1 GeV. Above that energy, the observed spectrum loses prominent structure, becoming a rather smooth continuum. To describe the phenomena in the relevant energy region, effective models of hadrons are useful. The most standard one is the quark model; quarks were introduced by Gell-Mann (1964) and Zweig (1964) prior to the establishment of QCD, and later the model was elaborated by implementing important ingredients of QCD (Isgur and Karl 1978, 1979a, b). Here the quarks are not the “bare” quarks of QCD but rather “constituent” quarks that construct hadrons in a simple manner. Despite its simplicity, the model has explained many features of hadrons. Key ingredients are dynamically generated constituent quark masses and their interactions by gluon exchange and confinement. Furthermore, chiral symmetry

61 Hadrons from Quarks and Chiral Symmetry

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Fig. 1 Excitation energies of various baryon resonances of positive parity (Takayama et al. 1999)

and its spontaneous breaking naturally incorporate the pion coupling to the quarks. The so-called chiral quark model (Manohar and Georgi 1984), though not directly derived from QCD, is a good working hypothesis for hadron physics. Having these observations, this chapter overviews hadron physics in an effective theory of constituent quarks and pions which are the effective degrees of freedom (quasiparticles) of QCD. An essential ingredient here is the spontaneous breaking of chiral symmetry. Therefore, this chapter starts with the discussion of its idea, and the Nambu-Jona-Lasinio model is described in some detail (Nambu and JonaLasinio 1961a, b). The model leads naturally to the basic structure of the quark model in terms of the constituent quarks and pions. Having this framework, some recent topics of exotic hadrons will be discussed, X(3872) and Pc pentaquarks that are the well-recognized states among various candidates.

Chiral Symmetry and Its Spontaneous Breaking Overview with Some History The discovery of chiral symmetry and its spontaneous breaking is one of the great achievements in physics in the twentieth century. The idea and methods apply to various fields of quantum many-body systems, explaining how the varieties emerge from simple laws. A seed of the idea may be dated back to the theory of the strong interaction by Yukawa (1935). He introduced a hypothetical particle, the pion, to explain the binding of nucleons to form atomic nuclei. From the typical size of nuclei, the mass of the pion was predicted to be around 200 MeV. An important point of his prediction in relation with chiral symmetry is that the pion is significantly lighter than the other hadrons with typical mass around 1 GeV. Because of the light mass, the pion plays an important role as the mediator of the long range part of the nuclear

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interaction. In the present day, the pion exchange interaction is the best established part of the nuclear interaction at long distances. A new phase of the development started in 1958 when Goldberger and Treiman derived the well-known formula relating the beta decay of the neutron and the weak decay of the pion (Goldberger and Treiman 1958). It was then realized by Nambu that the relation could be derived by the partially conserved axial vector current (PCAC) (Nambu 1960). Together with the conservation of the vector current, the idea was summarized as (approximate) SU(2)L × SU(2)R chiral symmetry. A small violation of the symmetry, or the nonconservation of the current, could be attributed to the small mass of the pion. But then the question would arise; if the symmetry is realized manifestly as it relates one state to another by symmetry (algebraic) transformations, there must be degenerate pairs of particles with opposite parities, which in reality is not the case. This lead to the idea of spontaneous breaking of chiral symmetry. The crucial step was made by Nambu and Jona-Lasinio who formulated the mechanism of the spontaneous breaking, which is now well appreciated as the NJL model (Nambu and Jona-Lasinio 1961a, b; Hatsuda and Kunihiro 1994). The model explains how the SU(2)L × SU(2)R chiral symmetry breaks spontaneously with leaving the diagonal isospin vector symmetry SU(2)V . Without much input for the details of the interaction, the proposed mechanism yields rich consequences of strong interaction dynamics, answering the fundamental questions such as how nucleons obtain a mass while pions stay light (almost massless) in a manner consistent with conservation of axial current (In QCD, axial currents do not conserve due to anomaly. In this note, however, this feature is not discussed). (In particular, the (almost) massless pion has been established as the realization of the NambuGoldstone (approximate) zero modes (Goldstone 1961)). The spontaneous symmetry breaking and the appearance of the zero mode are seen in various many-body systems. A simple example in quantum mechanics (not the field theory as for the pion) is translational invariance associated to the symmetry of parallel displacement in coordinate space when there is no external positiondependent potential. If a composite system allows a bound state, the position of its center of mass can be chosen arbitrarily or spontaneously. Because of translational invariance, the bound object can be moved, in particular, in the limit of slow motion without spending energy, which is the origin of the zero mode. When quantized, the motion of the center of mass is described by a plane wave of a fixed momentum to recover the broken translational symmetry. The momentum states form a continuum spectrum starting from zero value corresponding exactly to the zero energy mode. Another example is the rotation of a deformed object which generates a series of rotational spectra. Classically the deformed object can rotate with any value of energy forming continuum spectrum, while quantum mechanically, the finite volume of the phase space of rotational motion results in discrete spectrum. The original NJL model was formulated by the nucleons. Today the idea is often applied to quarks of light up, down, and strange (u, d, s) flavors (Hatsuda and Kunihiro 1994). This could be the theoretical ground of empirically successful quark model with constituent quarks with a finite mass.

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Possible Scenario Spontaneous breaking of chiral symmetry is a phenomena of phase transition where one of order parameters is the chiral condensate qq. ¯ A nonzero value of qq ¯ is an evidence of the symmetry breaking. In QED, in the absence of source charges such as electrons and protons, the photon field obeys the linear field equation, the Maxwell equation, and hence its time-independent static solution is trivial, Aμ = 0. Contrary, gluons as color SU (3) gauge field interact among themselves. The resulting nonlinear field equation allows nontrivial solutions even without the source of color charge of quarks. The solutions are the so-called instantons that make the vacuum nontrivial (Belavin et al. 1975). They are the classical solutions in the Euclidean space-time and are classified by topological numbers. In the real time, the solutions connect degenerate vacua which are topologically distinguished by tunneling mechanism. As lattice simulations indicate, the QCD vacuum is dynamic, where instantons and anti-instantons are created and annihilated. As discussed in Diakonov (2003), the instantons couple the left and right components of quarks. Consequently, the instantons generate the Dirac zero-eigenmodes, causing spontaneous breaking of chiral symmetry with a finite quark condensate qq ¯ = 0 as obeying the Banks-Casher theorem (Banks and Casher 1980). The theorem has been also studied in the lattice simulations (Fukaya et al. 2010, 2011), where they have shown the accumulation of the density of states of the Dirac zero-eigenmode. Instantons not only drive spontaneous breaking of chiral symmetry but also can be the source of the spin-dependent interactions among quarks (Takeuchi and Oka 1991; Takeuchi 1998). The latter explains mass splittings of hadrons with different spins, which is the issue of hadron spectroscopy. It is often considered gluon exchange interaction provides spin-spin and spin-orbit interactions, yielding fine and hyperfine structures in the spectrum. However, their phenomenologically determined strengths are not consistent with what are naively expected from the quark-gluon interaction in QCD. For instance, the phenomenological spin-spin interaction seems too large, while the spin-orbit interaction too small especially for light baryons. It was discussed that these inconsistencies could be resolved by the instantons which constructively or destructively contributed to the spin-spin or spinorbit interaction, respectively (Takeuchi and Oka 1991; Takeuchi 1998).

One Pion Exchange Potential The interaction between two nucleons (NN) mediated by one pion exchange is the best established one among various hadron interactions. Originally, the interaction was introduced to explain the mechanism of the nuclear binding (Yukawa 1935). Today the presence of that interaction can be regarded as an evidence of spontaneous breaking of chiral symmetry.

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Considering the spin and parity of the nucleon (J P = 1/2+ ) and pion (J P = 0− ), there are two types of interaction Lagrangians, pseudoscalar and pseudovector (axial vector) types, ¯ 5 τN · π , igπ N N Nγ

gA ¯ Nγμ γ5 τN · ∂ μ π , 2fπ

(1)

where N is the four-component Dirac spinor of spin 1/2 and two-component isospin spinor for the nucleon field, and π the isovector and pseudoscalar pion field. Parameters gπ N N ∼ 13, gA ∼ 1.25 and fπ ∼ 93 MeV (Reinert et al. 2021) are the pion-nucleon coupling constant, axial coupling constant, and pion decay constant, respectively (Recent discussions on the determination of these parameters are found in Reinert et al. (2021)). For an on-mass-shell nucleon, the two Lagrangians are equivalent as shown by using the equation of motion of the nucleon, (i∂ /−mN )N = 0 with the mass of the nucleon mN . In the nonrelativistic approximation which is well satisfied in many nuclear physics problems, they reduce to the well-known form in the momentum representation, once again for the on-mass shell nucleon, gπ N N † χ ( σ · q) τ χ · π , 2mN

gA † χ ( σ · q) τ χ · π , 2fπ

(2)

where q is the momentum carried by the pion and χ is a two-component spinor and isospinor for the nonrelativistic nucleon field. The two interactions are equivalent when gπ N N gA = mN fπ

(3)

which is the celebrated Goldberger-Treiman relation (Goldberger and Treiman 1958). By using the actual values of coupling constants and mN ∼ 940 MeV, the relation is well satisfied with an accuracy of only a few percent. Now applying the standard Feynman’s method, it is possible to derive the one pion exchange potential (OPEP) between two nucleons as 

N gA VπN N ( q) = − 2fπ

2

( σ1 · q )( σ2 · q ) τ1 · τ2 . q 2 + m2π

(4)

Several comments are in order: • In the four momentum transfer q 2 = (q 0 )2 − q2 appearing in the Feynman propagator i/(q 2 − m2π ), the energy transfer q 0 vanishes for elastic scattering, and so q 2 → − q 2 . This is no longer the case for inelastic scatterings where masses of scattering particles change (Yamaguchi et al. 2020). • The minus sign in (4) is a general property of the second order perturbation; the energy correction to the ground state is always negative.

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• Though it looks rather trivial, the momentum dependence on q is a consequence of spontaneous breaking of chiral symmetry; pions interact with (small) momentum-dependent coupling, σ · q. At low energies, the NN interaction (4) is of order O( q 2 ). In addition to the above basic features, another important issue in practice is the structure of the nucleon with a finite spatial size. To see this point, let us first make the following rearrangement:  VπN N ( q) =

N gA

2fπ

2

1 3



m2 −1 + 2 π 2 q + mπ



− q2 ˆ 2 σ1 · σ2 + S12 (q) q + m2π

 τ1 · τ2 , (5)

where the tensor operator is defined by ˆ = 3( σ1 · q)( ˆ σ2 · q) ˆ − σ1 · σ2 . S12 (q)

(6)

The first term of (5) is the spin- and isospin-dependent central interaction, which has been further decomposed into the constant and q2 -dependent terms. The constant term takes on the form of the δ-function in the coordinate space. This singularity appears because the nucleon has been treated as a point-like particle. In reality, nucleons have finite structure the effect of which is to smear the delta function. Instead in nuclear physics, the constant term is often subtracted. One reason is that the hard core in the nuclear interaction suppresses the wave function at short distances and the δ-function term is practically ineffective. Then a form factor is introduced to incorporate the structure of the nucleon. A widely used form is the monopole type normalized at q 2 = m2π , F (q 2 ) =

Λ2 − m2π , q 2 ∼ − q2 , Λ2 − q 2

(7)

where Λ is the cutoff parameter that is often determined phenomenologically. The form factor is related to the structure of the nucleon by the Fourier transform of a relevant density distribution ρ( x ),  F (q ) = 2

d 3 ρ( x )ei q·x .

(8)

Thus, knowing the distribution ρ( x ), the cutoff Λ can be estimated. For the NN interaction, F ( q ) is multiplied at each vertex, and so (after removing the delta function term),  VπN N ( q)

→

N gA F ( q) 2fπ

2

  m2π 1 − q2 τ1 · τ2 . (9) σ  · σ  + S ( q) ˆ 1 2 12 3 q 2 + m2π q 2 + m2π

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Performing the Fourier transform, the OPEP in the coordinate space is obtained, 

VπN N (r )

d3 q V ( q )ei q·r (2π )3   N 2

gA 1 σ1 · σ2 C(r; mπ , Λ) + S12 (ˆr )T (r; mπ , Λ) τ1 · τ2 . (10) = 2fπ 3 =

where C(r; m, Λ) and T (r; m, Λ) are given by  e−mr e−Λr (Λ2 − m2 ) −Λr , (11) − − e r r 2Λ 

e−mr

−Λr 1 2 2 e T (r; m, Λ) = 3 + 3mr + m2 r 2 − 3 + 3Λr + Λ r 4π r3 r3  m2 − Λ2 e−Λr + (1 + Λr) . (12) 2 r

m2 C(r; m, Λ) = 4π



In the following sections, similar pion exchange interaction is applied to the interaction between light quarks. The quarks there are the constituent quarks that are dressed by the interaction with gluons and hence have internal structure.

Chiral Symmetry In this section, chiral symmetry and its spontaneous breaking are discussed. Much part of discussions follow (Hosaka and Toki 2001). A somewhat heuristic way to introduce chiral symmetry is to consider a massless spin 1/2 fermion whose Lagrangian is given by ¯ /ψ . Lm=0 = ψi∂

(13)

Defining the chirality left- and right-handed components by applying the projection operators PL,R , ψ = ψL + ψR ; ψL,R = PL,R ψ , PL,R =

1 ∓ γ5 , 2

(14)

the Lagrangian (13) is written as ¯ /ψ = ψ¯ L i∂ Lm=0 = ψi∂ /ψL + ψ¯ R i∂ /ψR .

(15)

Now introduce an internal symmetry of, to be specific, isospin and let ψ be the two-component isospinors, ψ = (u, d). Then consider two isospin symmetries for

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ψL = (uL , dL ) and ψR = (uR , dR ) independently. The Lagrangian (13) is then invariant under the chiral transformations, ψL → gL ψL , ψR → gR ψR ,

(16)

where gL ∈ SU (2)L and gR ∈ SU (2)R . This is the chiral symmetry SU (2)L × SU (2)R . The “chirality” is to differentiate an object from the mirror imaged (parity transformed) one. In the present context, it can be characterized by the helicity, which is the spin projection along the momentum axis, h ≡ Σ · p/|  p|.  Here, Σ is the four component spin matrix, Σ = diag( σ /2, σ /2). It is shown that left- and righthanded components of massless fermions are nothing but the helicity h = ∓1/2 states, respectively. Furthermore, ψL transforms into ψR and vice versa under the parity transformation, ψ(t, x) → γ0 ψ(t, − x ). The former property holds in the massless limit, while the latter does for an arbitrary mass value. To see explicitly the relation of the L, R components with the helicity states, let us write the spinor of helicity h = +1/2 in the Dirac representation, ⎛

ψp,h=+1/2 

⎞ 1 ∼ ⎝ σ · p ⎠ χ↑ E+m ⎞ ⎛ ⎛ p p ⎞ 1⎝ 1− E+m ⎠ 1 ⎝1 + E + m ⎠ = χ↑ + χ↑ p p 2 −1 + 2 1+ E+m E+m ≡ ψL + ψR ,

(17)

where in the second equation, L, R decomposition is shown explicitly. For a finite is decomposed into two nonvanishing components of ψL and ψR . For m, ψp,h=+1/2  massless case m = 0, ψL vanishes and the positive helicity state reduces precisely to the right chirality state. The mixing of the L and R chirality states for a massive fermion is also verified by decomposing the Lagrangian of the massive fermion,   / − m) ψ = ψ¯ L i∂ /ψL + ψ¯ R i∂ /ψR − m ψ¯ L ψR + ψ¯ R ψL , Lm = ψ¯ (i∂

(18)

where it is observed that the mass term mixes them. The left-right decomposition is also possible for the coupling term with a vector field, ¯ μ Aμ ψ = ψ¯ L γμ Aμ ψL + ψ¯ R γμ Aμ ψR . ψγ

(19)

Therefore, in QCD, chiral symmetry is well satisfied for the light u and d (and often s) quarks. The “light” here means that these quarks are sufficiently lighter than the QCD scale ΛQCD ∼ several hundred MeV. In contrast, protons and neutrons

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with a mass significantly larger than ΛQCD seem to contradict chiral symmetry. It is now believed that spontaneous breaking of chiral symmetry explains how the (almost) massless quarks generate finite masses. This explains the finite mass of the proton and neutron in a way consistent with chiral symmetry. Furthermore, it also explains the emergence of the (almost) massless pions.

The Linear Sigma Model The linear sigma model of Gell-Mann-Levi is a simple model to realize the spontaneous breaking of chiral symmetry (Gell-Mann and Levy 1960). It is a theory of massless fermion ψ with an interaction with σ and π mesons satisfying chiral symmetry. As in the previous subsection, approximately massless u, d quarks are considered, ψ = (u, d) (each u and d is a Dirac spinor) with isospin chiral symmetry, SU (2)L × SU (2)R . The model Lagrangian is given by ¯ /ψ − g ψ¯ (σ + i τ · π γ5 ) ψ − V (σ 2 + π 2 ) Lσ = ψi∂

(20)

with the potential V (σ 2 + π 2 ) =

μ2 2 λ (σ + π 2 ) + (σ 2 + π 2 )4 . 2 4

(21)

Here σ is a scalar and isoscalar meson, and π a (a = 1, 2, 3) the isovector and pseudoscalar meson, that is, the pion. With an assumption that the set of four components (σ, π ) transforms as the (1/2, 1/2) representation of SU (2)L × SU (2)R , or equivalently as a four vector under O(4) transformations, it is shown that the Lagrangian (20) is invariant under chiral transformations. For instance, the second term of (20) takes on the form of an inner product of two O(4) vectors, ¯ i ψ¯ τγ5 ψ). Several comments are in order: (σ, π ) and (ψψ, • The σ and π mesons have a common mass term of μ2 . • Parameter λ must be positive definite when requiring the theory stable. • The mass parameter μ2 can be both positive and negative. Depending on the sign of the mass parameter μ2 , the properties of the vacuum and hence the resulting particle spectrum change. • μ2 > 0 The potential shape is shown in the left panel of Fig. 2. Therefore, the vacuum, the minimum of the potential, is uniquely determined at φ 2 = 0, the origin of the  Consequently the masses of σ and π four-dimensional space of (σ, π ) = (0, 0). mesons are equal, μ, which is the result that chiral symmetry is realized manifestly.

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Fig. 2 A plot for the potential V (φ 2 ), where φ 2 = σ 2 + π 2 . The three axes for π are shown by one axis. The potential shape changes depending on the sign of the mass term, μ2 > 0 (left panel) or μ2 < 0 (right panel)

• μ2 < 0 This is the case where chiral symmetry is spontaneously broken, for which the potential shape is shown in the right panel of Fig. 2. The potential minimum can be at an arbitrary point on the circle of radius |φ| = −μ2 /λ ≡ fπ as drawn by red circle in the figure. Here fπ ∼ 93 MeV is the pion decay constant as it is shown that μ the (isospin) axial current J A

μ JA (x) = −fπ ∂ μ π (x)

(22)

determines the pion decay through,

0|JA (x)|π b (q) = iδab fπ q μ e−iqx , aμ

(23)

where a, b are isospin indices. Now one has to spontaneously choose a minimum point on the circle as the vacuum. Due to the requirement that the vacuum has positive parity, the axis connecting the origin and the minimum point is defined to be the σ -axis. This is the axis passing through the point A shown in the figure. If one choses a point B as the vacuum, then the axis can be rotated to redefine the axis passing though the point B as the σ -axis. By determining the σ -axis in this way, the vacuum point is  (fπ , 0).

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To see particle spectrum (masses), redefine the fields as, (σ, π ) → (fπ + σ, π )

(24)

Inserting this into the Lagrangian (20), the following expression is found: 1 1 L = ψ¯ (i∂/ − gfπ − g(σ + iγ5 π · τ)) ψ + (∂μ σ )2 + (∂μ π )2 2 2 1 λ − μ2 + fπ2 − λfπ2 σ 2 − λfπ σ (σ 2 + π 2 ) − (σ 2 + π 2 )2 . 4 4

(25)

From this, the following observations are made: • Apparently massless fermion in (20) acquires a finite mass m = gfπ , the dynamically generated mass. This is the essence of mass generation for quarks, the link connecting the (almost) massless bare quarks of QCD to the finite mass constituent quarks of the quark model. • The mass relation is nothing but the Goldberger-Treiman relation for the axial coupling gA = 1, gA g . = fπ m

(26)

• The degeneracy of the σ and π is resolved; the σ meson remains massive, m2σ = 2λfπ2 = −2μ2 , while the pion becomes massless. The latter is a realization of the Nambu-Goldstone theorem, guaranteeing the appearance of massless particles when symmetry is spontaneously broken. The Lagrangian (25) can be used in many applications in hadron physics when q the fermions are regarded as light u, d quarks. Reexpressing g → gπ qq , gA → gA to indicate that they are parameters for quarks, these parameters are employed: q

fπ = 93 MeV, gπ qq = 4, gA = 1 .

(27)

The value of gA = 1 is expected for a fermion of fundamental representation of chiral symmetry (1/2, 0) + (0, 1/2) (Weinberg 1969, 1990a). The GoldbergerTreiman relation (26) implies (m → mq ) mq ∼ 370 MeV .

(28)

which agrees well with the empirical value of the constituent quark mass (Manohar and Georgi 1984; Karliner and Rosner 2014). By using the quark model wave functions as briefly discussed in the next section, the above parameters explain various properties of hadrons. For instance, let us consider the pion-nucleon coupling from the pion-quark coupling in (20):

61 Hadrons from Quarks and Chiral Symmetry

Lπ qq = −igπ qq ψ¯ τ · π γ5 ψ → −

2315

gπ qq † χ σ · qτ · πχ  , 2mq

(29)

where in the second step, a nonrelativistic approximation has been used. Taking a nucleon matrix element of Lπ qq , the well-known factor 5/3 is obtained, which is N in the quark model: σ 3 τ 3  = 5/3. Thus the pionnothing but the nucleon gA nucleon coupling constant in the quark model is identified to be gπ N N =

5 mN gπ qq ∼ 17 3 mq

(30)

The numerical value computed here gπ N N ∼ 17 is overestimated. This is related N . By setting g N ∼ 1.25 the expected value to the overestimate of the nucleon gA A gπ N N ∼ 13 is found. N from 5/3 to 1.25 (about 30% reduction) can be understood The reduction of gA when considering the relativistic effect of the confined quark motion (Chodos et al. 1974; Arifi et al. 2021). As discussed in some detail in the next section, the effect explains well not only the pion-nucleon coupling constant but also the decays of hadrons containing heavy quarks such as D ∗ → Dπ and Σc → Λc π .

The NJL Model The Nambu-Jona-Lasinio (NJL) model provides an essential idea of the spontaneous breaking of chiral symmetry (Nambu and Jona-Lasinio 1961a, b; Hatsuda and Kunihiro 1994). In comparison with the Gell-Mann-Levi’s linear sigma model, the NJL model is regarded as a microscopic model as the BCS model for the LandauGinzburg model of superconductor (Miransky 1994). The spontaneous symmetry breaking in the linear sigma model is controlled by the parameter μ2 , while in the NJL model it is dictated by an interaction between the fermions. Moreover, instructive is that the physical vacuum is dynamically realized as the ground state of the interacting fermion system. The problem is essentially a (infinitely) many-body problem, where the formation of a mean field of an interacting field theory leads to the spontaneous breaking. As anticipated, quarks of isospin 1/2, ψ = (u, d) is considered. Let us start with the Lagrangian with a four point interaction: ¯ /ψ + LN J L = ψi∂

g ¯ 2 + (ψi ¯ τγ5 ψ)2 . (ψψ) 2

(31)

The bare quarks ψ are massless to start with but interact with each other in a chiral invariant manner. The invariance can be checked explicitly by performing the chiral transformations (16). Now the task is to find the ground state that minimizes the Hamiltonian:

2316

A. Hosaka



 − g (ψψ) ¯ 2 + (ψi ¯ τγ5 ψ) . H = −i ψ¯ γ · ∇ψ 2

(32)

In the mean field approach, expectation values are introduced: ¯ ψψ =−

m , ψ¯ τγ5 ψ = 0 , g

(33)

which are the matrix elements taken by the physical vacuum and is the order parameter for the phase transition of the spontaneous breaking of chiral symmetry. The reason that only the scalar in (33) takes a finite value is that the ground state should have positive parity and also zero isospin. This is similar to the choice of the σ axis as discussed in the previous subsection. The negative sign with m ≥ 0 is due to the fact that the expectation value is contributed from antiparticles or negative energy states. In the basis of massless quarks, pairs of a left and right quark condensate in the vacuum, which is an indication of chiral symmetry breaking. The physical and perturbative vacua are related by the Bogoliubov transformation (Hatsuda and Kunihiro 1994; Miransky 1994). By using (33), the ground state energy per unit volume can be computed as (note ¯ 2  → ψψ ¯ 2 , etc.) also (ψψ)



 + m ψ − g ψψ ¯ 2 + ψi ¯ ¯ τγ5 ψ2 − mψψ E(m) ≡ H  = ψ¯ γ · ∇ 2

2  + m ψ + m → ψ¯ γ · ∇ 2g  Λ 3  d p m2 2 + m2 + . (34) p  = −12 3 2g 0 (2π ) An important observation here is that the energy E is regarded as a function of the “mass” m. The expectation value in the first and the second line is taken by the physical vacuum. Hence, in the last step of (34), the (negative energy) solution to the Dirac equation for quarks of mass m has been used. The factor 12 reflects the spin (2), isospin (2), and color (3) degrees of freedom (2 × 2 × 3), and the cutoff parameter Λ is introduced to make the integral finite. Physically this reflects the spatially extended structure of the quarks due the interaction, which smears the point-like interaction in (31). Its typical value is about 1 GeV for low energy dynamics of hadrons. Now let us study the energy density E(m) for various g and Λ. Performing the integral of (34), it is found that      Λ3 m2 Λ 12  2 m4 Λ + Λ2 + m2 m2 2 + ln . (35) Λ +m E(m) = − 2 − + 4 8 8 m 2g 2π

The resulting E(m) is shown in Fig. 3. From (34), (35), and Fig. 3, important observations are made.

61 Hadrons from Quarks and Chiral Symmetry

2317

Fig. 3 Energy per unit volume in units of fm−4 as function of the mass m. The cutoff is set at Λ = 3 [fm−1 ]

• The minimum is given by the condition ∂ E(m) = 0 ∂m

 →

Λ

−12 0

d 3p m m  =0. + g (2π )3 p2 + m2

(36)

This is the gap equation to determine m. • For a fixed Λ, the minimum occurs at m = 0 for smaller g. For g larger than the critical strength gc , the minimum point occurs at a finite m. This means that spontaneous symmetry breaking is caused by sufficient attraction between a quark and an antiquark. • For a fixed g, the cutoff Λ must be sufficiently large, for spontaneous symmetry breaking. The reason is again that sufficient amount of the attraction is necessary. • The critical value is extracted by looking at the coefficient of m2 of E(m) when expanded in powers of m2 . It is the sum of attractive component from the integral term of (35) and the repulsive term of m2 /g. The condition is gc =

π2 3Λ2

(37)

giving the turning point from convex to concave shapes of E(m) as shown in Fig. 3. By setting Λ ∼ 3 fm−1 , for instance, the critical coupling constant is gc ∼ 0.37 fm2 . • The equation (36) has been interpreted as the gap equation for a quark interacting with quarks in the vacuum. It is also interpreted as the eigenvalue equation for the pion channel with the vanishing eigenenergy. This is a consequence of chiral symmetry; one equation can be interpreted in two different ways. The attraction of the pion channel precisely compensates quark masses to ensure the pion massless.

2318

A. Hosaka

Just as the Gell-Mann-Levi’s linear sigma model, an important implication of the NJL model for quarks is that it can be a basis for the “quark model” with a a constituent mass m of order 300 MeV. Having a few numbers of parameters, it is possible to discuss various hadron properties. The following section is for the discussions of several examples for them.

Hadrons Mesons Standard mesons are pairs of a quark and an antiquark, qq. ¯ In the nonrelativistic scheme of the quark model, the states are classified by spin S = 0, 1 of the two quarks, orbital angular momentum L = 0, 1, 2, · · · of the relative motion of the two quarks and the total angular momentum (often referred to simply as spin) J = L + S = 0, 1, 2, · · · . As usual, the spectroscopic notation is employed for L, and so, meson states are labeled as 1 S0 ,3 P1 , · · · , etc. In addition, parity P = −(−1)L and flavor are specified. Charge conjugation C is defined for selfcharge conjugate states. Finally, for isospin 1/2 mesons formed by u, d quarks, G-parity is also assigned. Among the above quantum numbers, orbital angular momentum L and spin S are not an observable but can be assigned (guessed) by the help of the quark model from the parity, J and (excited) mass. For more details of quantum numbers, refer to Zyla et al. (2020). In this note, mostly the spin and parity J P are considered. In Table 1, typical mesons are listed with their spin, parity, and possible quark model configurations. The quark model assumes a Hamiltonian consisting of the nonrelativistic kinetic term, two-body spin-independent interaction of the Coulomb type and linear confinement, and spin-dependent short-range interactions such as spin-spin and spin-orbit interactions (Silvestre-Brac 1996). With this Hamiltonian, various meson masses are reproduced reasonably well. The one-gluon exchange contains the Coulomb and spin-spin interactions as color-electric and color-magnetic terms, respectively. Furthermore, spin-orbit and tensor forces are also generated. This, however, does not necessarily mean that the origin of these interactions is exclusively perturbative. As anticipated in the previous section, part of them may be of non-perturbative nature such as an instanton induced one (Takeuchi and Oka 1991; Takeuchi 1998). Table 1 Examples of qq ¯ mesons and their properties (Zyla et al. 2020) Mass (MeV) Quark content Spin-parity, J P Configuration

π+ 140 ud¯

ρ+ 770 ud¯

0− 1S 0

1+ 3S 1

σ 400 – 550 uu¯ + d d¯ 0+ 3P 0

a1+ 1230 ud¯ 1+ 3P 1

K+ 494 u¯s 0− 1S 0

K ∗+ 890 u¯s 1+ 3S 1

D¯ + 1870 cd¯

D ∗+ 2010 cd¯

0− 1S 0

1− 3S 1

61 Hadrons from Quarks and Chiral Symmetry

2319

The structure of mass spectrum for excited states is determined by different energy scales of the interactions of the Hamiltonian. The spin-independent Coulomb and confinement forces are stronger than spin-dependent forces. Therefore, large energy splittings are expected between different orbital excitations, with degenerate states with various spin and orbital angular momentum states when spin-dependent force is ignored. The degenerate states are split by spin-dependent forces, resulting in fine and hyperfine structures. Therefore, the lightest hadrons have quarks in the orbitally ground S-states (L = 0). They split into two states of spin S = 0, 1. When the linear confinement dominates for binding of quark and antiquark pair, the first orbitally excited states are in p-wave, L = 1. Indeed this is the case for light quark ¯ systems (qq ¯ and qQ). ¯ In contrast, for heavy charm or bottom sectors (cc ¯ and bb), the Coulomb term dominates and the states of L = 1 and n = 1 (n: the number of nodes in radial wave function) are getting degenerated. These states are then split by the spin-dependent interactions, forming a spectrum resembling the one of hydrogen-like atoms.

Baryons Quark Model Wave Functions Baryons are made of three quarks, qqq. The constituent quarks have a finite mass of order 300 MeV as generated by spontaneous breaking of chiral symmetry (see eq. (28)). As in the case of mesons, the states are classified by the spin of the three quarks S = 1/2, 3/2, orbital angular momentum L = 0, 1, · · · , and the total spin J = 1/2, 3/2, · · · . Removing the center-of-mass motion, the orbital motion of three quarks is classified by the motions of λ and ρ degrees of freedom , the Jacobi coordinates, as defined in Fig. 4. Now how baryon wave functions are constructed is outlined (Hosaka and Toki 2001; Close 1979). For this, first consider the case of approximately equal quark masses as for u, d, s quarks. If the constituent quarks are assumed to move nonrelativistically, the spin decouples from the orbital angular momentum. Moreover the excitation energies of λ and ρ motions are degenerate. In this case they are regarded as identical particles with different spin, flavor, orbital, and color quantum numbers. For spin and flavor part, there are six elements u↑, u↓, d↑, d↓, s↑, s↓, which can be regarded as the fundamental representation of SU (6) ⊃ SU (2) × SU (3) symmetry. Having these elements, baryon wave functions are formed by the direct products of spin (S), flavor (F ), orbital (O), and color (C) wave functions of the three quarks:

Fig. 4 The internal λ and ρ degrees of freedom (Jacobi coordinates) of qqq baryons

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A. Hosaka

ΨB ∼ ψS ⊗ ψF ⊗ ψO ⊗ ψC .

(38)

Now, because quarks are fermions, baryon wave functions are totally antisymmetric under odd permutations of the three quarks due to the Pauli’s exclusion principle. Since color states are totally antisymmetric for colorless (singlet) baryons, the spin, flavor, and orbital part ψS ⊗ ψF ⊗ ψO must be totally symmetric. For example, ground state baryons have three quarks in the same lowest S-wave orbit. Thus, the spin-flavor part is also totally symmetric, which is the 56-dimensional representation in the following irreducible decomposition: 6 × 6 × 6 = 56 + 70 + 70 + 20 .

(39)

The two 70-dimensional representations are of mixed symmetry, and 20-ones of total antisymmetry. They become relevant for orbitally excited states. The 56-dimensional representations are further decomposed into spin and flavor representations as 56 → 28 +4 10 (56 = 2 × 8 + 4 × 10) ,

(40)

where the upper indices 2 and 4 show the spin degeneracy of the three quarks, 2S+1, and 8 and 10 are the dimensions of flavor SU (3) representations. The eight- and tendimensional representations correspond to the well-known octet baryons of spin 1/2 (N, Σ, Λ, Ξ ) and decuplet baryons of spin 3/2 (Δ, Σ ∗ , Ξ ∗ , Ω), respectively. The correlation between the spin and flavor representations is due to the Pauli’s exclusion principle. Further classification schemes can be applied to orbitally excited states of p-wave, d-wave, nodal excitations, and so on. In these cases, the symmetry properties of the ρ and λ motions are properly taken into account with combination with spin and flavor symmetries. More details of how to construct wave functions explicitly are found in Hosaka and Toki (2001).

Magnetic Moments An evidence for constituent quarks at work for hadron structure is in the magnetic moments of baryons. To the leading order of nonrelativistic expansion, magnetic moments are the matrix elements of the sum of quark’s magnetic moments assuming that they behave like bare Dirac particles (Weinberg 1990b), μ =

 i

ei

σi , 2mi

(41)

where ei , σi and mi are the charge, spin matrix, and constituent mass for the i-th quark. Matrix elements are computed by the SU (6) baryon wave functions. Relativistic corrections are also estimated (Manohar and Georgi 1984). The parameters are the quark masses, where u, d quark masses are set equal, while s quark mass is slightly heavier. The best fit is obtained by mu,d = 360 MeV and ms = 540 MeV.

61 Hadrons from Quarks and Chiral Symmetry Table 2 Magnetic moments of various baryons in units of the nuclear magneton, 1/2mN ; mN = 940 MeV is the mass of the nucleon

p n Σ+ Λ Σ− Ξ0 Ξ− Ω− χ2

2321 Non-rel 2.61 −1.74 2.51 −0.58 −0.97 −1.35 −0.48 −1.92 0.1

+ Rel corr 2.78 −1.90 2.35 −0.61 −1.15 −1.25 −0.68 −2.26 0.01

Exp 2.79 −1.91 2.46 −0.61 −1.16 −1.25 −0.65 −2.02 −

These values consistently agree with those obtained by phenomenological analysis of baryon masses (Karliner and Rosner 2014) and also with that in (28). The results of the magnetic moments are shown in Table 2.

Heavy Baryons Heavy baryons contain one or more heavy quarks: charm or bottom quarks, c, b. Their masses are significantly heavier than ΛQCD and are regarded approximately as a static color source with their spin-dependent properties suppressed. Here important features of singly heavy baryons Qqq are briefly described, where Q and q denote heavy (c, b) and light quarks (u, d, s), respectively. An important feature of the heavy quark is its suppressed spin-dependent interaction. This is so because the fundamental interaction of the quarks is brought by gluons that are the gauge fields of color SU (3), where the magnetic interaction appears at the order O(1/mQ ). Therefore, states of Qqq form spin multiplets of J = SQ + jqq , where SQ = 1/2 is the spin of the heavy quark Q and jqq is the net spin including spin and orbital angular momentum of the qq system. In literatures, the object qq is called brown muck (Neubert 1994). Another feature of the heavy baryons is that the λ and ρ mode excitations are no longer degenerate, and the λ mode is lowered. Here the λ mode is the relative motion between Q and qq, and the ρ mode the relative motion between q and q as shown in Fig. 4. The splitting of the two modes is purely kinematic effects where the inertia mass of Q-qq system is heavier than that of qq system, like the isotope shift (Copley et al. 1979). Therefore, heavy baryons of spin J may allow distinct states depending on orbital angular momentum of Lλ or Lρ . As anticipated, a negative parity states of the λ-mode (Lλ , Lρ ) = (1, 0) is lighter than that of ρ-mode (Lλ , Lρ ) = (0, 1). Observed data so far are consistent with the quark model calculations (Copley et al. 1979; Yoshida et al. 2015).

Axial Vector Coupling Constants As discussed in the previous sections, axial vector coupling constants gA (often abbreviated as axial coupling) are important quantities reflecting hadron structures.

2322

A. Hosaka

The nucleon’s axial coupling is often employed as a benchmark test for lattice simulations (Ishikawa et al. 2021). Through the Goldberger-Treiman relation, they also dictate one pion emission decays of hadrons, providing a testing ground for spontaneous breaking of chiral symmetry. Because of these reasons, some more details for the axial vector coupling constants are discussed.

Neutron Beta Decay The axial coupling gA is defined by the matrix element of the axial current Aμ for a hadronic weak process. A typical example is the beta decay of the neutron, n → p + e + ν¯ , where the hadronic matrix element is written as

τa p(pf )|Aaμ |n(pi ) = u¯ p (pf ) gA (q 2 )γ μ γ5 + h(q 2 )q μ γ5 un (pi ) . 2

(42)

The axial coupling is defined at q 2 = (pf − pi )2 = 0, gA = gA (q 2 = 0). In the chiral limit (the mass of the pion is zero, mπ = 0), the axial current conserves, ∂ μ Aaμ = 0, and so   0 = u¯ p (pf ) gA (q 2 )q/γ5 + h(q 2 )q 2 γ5 un (pi )   = u¯ p (pf ) 2mN gA (q 2 ) + h(q 2 )q 2 γ5 un (pi ) ,

(43)

where the equation of motion for the Dirac spinor, (p/i − mN )up (pf ) = 0 and the one for un (pi ) are used. Therefore, hA (q 2 ) = −2mN

gA . q2

(44)

This indicates that the hA term is induced by the one pion pole coupled to the nucleons (see Fig. 5). This term can be expressed by the pion-nucleon coupling constant gπ N N and the pion decay constant fπ : hA (q 2 ) = −2fπ gπ N N

1 . q2

Comparing (44) and (45), the Goldberger-Treiman relation is obtained:

Fig. 5 The contents of the neutron beta decay

(45)

61 Hadrons from Quarks and Chiral Symmetry

2323

gA gπ N N = . fπ mN

(46)

gA = 1.25, fπ = 93 MeV, gπ N N = 13, mN = 940 MeV

(47)

The actual values

satisfy the relation with about 3% accuracy.

Quark Model Estimate In the quark model, the axial coupling is computed as the matrix element of the axial current: q

Aaμ = gA



q¯n γμ γ5

n

τa qn 2

(48) q

sandwiched by hadron states. Here the sum goes over the light u, d quarks, and gA is the axial coupling of the constituent quarks which is usually taken to be unity. In principle, it can take any values as the constituent quarks have structure. However, as discussed in the previous section, it may be set to be unity, which is the case when the quarks are the fundamental representations of chiral symmetry (Weinberg 1969, 1990a, b). The current is also responsible for various transitions accompanied by one pion emission or absorption through the interaction: Lπ qq =

1 a A ∂μ π a . fπ μ

(49)

In the leading order of nonrelativistic expansion, the relevant components are the spatial ones of (48) (μ = i): Aai =

 n

σi (n)

τa (n) . 2

(50)

Quark model matrix element for the nucleon is well known and the result is N|Aai |N  =

5 τ a (N ) σi (N ) 3 2

→

N gA =

5 . 3

(51)

This overestimates the experimental value as shown in (47).

Other Transitions The overestimate of the axial coupling in the quark model (or equivalently, suppression of the experimental gA from what is expected from the quark model) has been observed in various hadron’s decays through one pion emission. In literatures, decays of K ∗ , D ∗ mesons and also those of heavy baryons such as Σc (2455) and

2324

A. Hosaka

Fig. 6 Quark description of the decay of K ∗ → Kπ

Σc∗ (2520) have been discussed (Nagahiro et al. 2017). These decays are transitions associated with spin flip of light quarks in the ground state as shown in Fig. 6. For instance, in K ∗ → Kπ , u-quark spin-up state turns into spin down state. Let us compute the decay of K ∗ which is caused by the interaction (Yamaguchi et al. 2020): Lπ KK ∗ = −igK † (∂μ τ · π )Kμ∗ + h.c.

(52)

The decay width is obtained by the formula ΓK ∗+ →K + π 0 =

q3 g2 . 24π mK ∗ (EK + Eπ )

(53)

Notations in these equations are g: the coupling constant which is related to the ¯ K where m ¯ K = (m transition gA by gA /fπ = g/m K + mK ∗ )/2, q the relative

momentum of the decaying π 0 , and K + , Eπ,K = m2π,K + q 2 . From the data ΓK ∗+ →K + π 0 = 50 × 1/3 MeV (Zyla et al. 2020), it turns out that g ∼ 6.4 and the resulting axial coupling are gA ∼ 0.67. Hence once again the suppression of q gA is observed. Similar suppression has been observed in other processes such as decays D ∗ → Dπ, Σc∗ (2520) → Λc π and Σc (2450) → Λc π (Yamaguchi et al. 2020; Nagahiro et al. 2017). Because of these observation, effectively suppressed q quark gA is employed (Yan et al. 1992). However, this does not seem to solve the problem consistently when examining the decays of Roper resonance and its siblings. Experimental data indicate that they have a large decay rate (Arifi et al. 2021). The Roper resonance is the first nucleon excited state having the same spin and parity as the ground state nucleon. In the quark model, it is a radial excitation with the same spin structure as the nucleon. As observed in the structure of (49) and (52), one pion coupling to a hadron transition takes the form in the nonrelativistic  Therefore, the matrix element of the Roper resonance decay, approximation q · S. ∗  N| q · S|N , vanishes in the limit q → 0 due to the orthogonality of the radial wave functions of the Roper resonance N ∗ and the nucleon N . The suppression of q the axial coupling gA does not help explain the large decay width. Long ago it was pointed out that higher order relativistic effects are important in the context of electromagnetic transitions (Kubota and Ohta 1976). A similar mechanism may be applied to the pion emission. Using the Foldy-Wouthuysen transformation for the pion-quark coupling, the following result is found (Arifi et al. 2021):

61 Hadrons from Quarks and Chiral Symmetry

Lπ qq = +

2325

 q  gA ω σ · q + ( σ · q − σ · pi ) 2fπ 2mq q  gA  2 mπ + σ · q + 2 σ · ( q − 2pi ) × ( q × pi ) , 2 16mq

(54)

where pi is the momentum of the i-th light quark. The second line of order 1/m2q is the key ingredient, as it reflects the relativistic corrections of the quark motion inside the baryons. It has been shown that the transition matrix element Nπ |Lπ∗qq |N ∗  has a nonvanishing contribution from the pi2 term in (54) as proportional to 1/α 2 , where α is a size parameter of a hadron. The inclusion of such terms improves the decay of the Roper resonance significantly. Such relativistic corrections have been revisited for other hadrons (Arifi et al. 2021, 2022), where it is shown that they improve the overestimated gA ’s of the nucleon, heavy baryons, and mesons. For ground state hadrons, total spin has a contribution, not only from the quark intrinsic spin but also from orbital angular momentum from the lower component of the Dirac spinor. Consequently, the expectation value of the intrinsic spin to gA is suppressed. The bag model where quarks are explicitly relativistic has already shown that gA of the nucleon is reduced ∼1.09 (Chodos et al. 1974) to be compared with the nonrelativistic value of 5/3.

Exotic Hadrons A remarkable progress has been made in the twenty-fist century, in particular in the last decade, in a series of discoveries of unexpected signals of hadrons (Hosaka et al. 2016; Zyla et al. 2020). They cannot be explained in the quark model as qq ¯ mesons or qqq baryons, hence called exotic hadrons. In this section, among many candidates, X(3872) and Pc pentaquarks are discussed in some detail.

X(3872) One of the early-day discovery is the X(3872) in e+ e− collisions (Choi et al. 2003). A very sharp peak was observed near the D D¯ ∗ threshold, with unexpected decay pattern violating isospin conservation, X(3872) → J /ψρ and J /ψω, where the former decay channel is isospin triplet while that in the latter isospin singlet. High statistics experiments have resolved even a several MeV split in different charge states, D 0 D¯ ∗0 and D + D ∗− . The fact that X(3872) is located nearly on the D 0 D¯ ∗0 threshold (In fact, D¯ 0 D ∗0 is also possible. For simplicity, the notation D 0 D¯ ∗0 is understood to include both D 0 D¯ ∗0 and D¯ 0 D ∗0 when needed. Furthermore, upper indices for charges are shown or not depending on the necessity of discussions) implies a hadronic molecule of loosely bound D 0 D¯ ∗0 in S-wave, while other options are not excluded (Chen et al. 2016). The D 0 D¯ ∗0 structure can naturally explain

2326

A. Hosaka

the violation of isospin symmetry. The latest experimental values of the mass are 3871.65 ± 0.06 MeV, to be compared to the threshold masses of D 0 D¯ ∗0 (3871.69 ± 0.05 MeV) and of D + D ∗− (3879.92 ± 0.05 MeV). The spin-parity quantum numbers are also determined, J P C = 1++ (Aaij et al. 2015a), which is naturally explained by the S-wave structure of D D¯ ∗ . The molecular structure of D 0 D¯ ∗0 can naturally explain the isospin violating decay pattern. Yet a somewhat peculiar fact is the large production rate in hard collision process (Esposito et al. 2015). If the structure of X(3872) is exclusively molecular-like, it is hard to expect that it can be produced abundantly. Prior to the experimental observation, it was proposed that the X(3872) could be an admixture of D D¯ ∗ molecule with a cc ¯ compact structure (Takizawa and Takeuchi 2013; Yamaguchi et al. 2020). It is often discussed whether exotic states are bound, resonant, or virtual states. However, an important question to be addressed is what interaction causes such a peak structure; whichever the nature of the state is, a sharp peak implies an attractive interaction between D D¯ ∗ . The one pion exchange interaction is one of possible sources of attraction. In particular the tensor force provides attraction via the second order process of the SD-wave mixing (Fig. 7a). For a D D¯ ∗ system, the pion couples to a light quark in the D (∗) meson, for example, the u-quark in D (∗)− ∼ cu ¯ (Here D (∗) denotes either ∗ D or D ). In such a construction, an assumption is that pion coupling is given as discussed in section “Hadrons”, the principle of which is chiral symmetry with its spontaneous breaking. In Yamaguchi et al. (2020), based on this idea, one pion exchange interaction was constructed. It turned out that the tensor force and the resulting SD-wave mixing provided attraction, which however was not sufficient to explain the peak structure at the observed energy. Additional attractive force was postulated as provided by the coupling of the molecular component with a compact object of cc. ¯ Assuming that its intrinsic mass is larger than the D D¯ ∗ threshold mass, such a coupling generates attraction for the D D¯ ∗ (Takizawa and Takeuchi 2013). By tuning the coupling strength, it was successful to explain the data. The contribution of the coupling to the compact object and the pion exchange was found to be roughly equal. The analysis was also made for the mixing ratio of the D D¯ ∗ and compact components; about 90% is D D¯ ∗ (dominated almost by D 0 D¯ ∗0 ) and ∼10% compact components. Interestingly this rate is consistent with what is expected from the hard production

Fig. 7 (a): Pion exchange for SD-wave mixing, (b) (c): Pion exchange between light quarks for meson-meson and meson-baryon molecules, respectively

61 Hadrons from Quarks and Chiral Symmetry

2327

Fig. 8 Schematic picture for X(3872) as an admixture of D 0 D¯ ∗0 -D + D¯ − -cc ¯ configurations

reaction (Esposito et al. 2015). In this way, a picture of X(3872) can be drawn: the admixture of a soft D D¯ ∗ molecule and a compact cc ¯ configurations as shown in Fig. 8. In particular the molecule is dominated by the neutral D 0 D¯ 0∗ whose spatial size is as large as several fm as expected from a tiny binding energy (Hosaka et al. 2016; Yamaguchi et al. 2020).

Pc Pentaquarks In 2015, LHCb observed a signal of pentaquarks in the invariant mass analysis of J /ψp decaying from Λb (Aaij et al. 2015b). Because of the observed channel, the minimum quark content is ccuud ¯ with isospin 1/2, hence a hidden charm pentaquark named Pc . Since the nominated states Pc (4450) and Pc (4380) were close to ¯ a molecular structure has been immediately the thresholds of Σc D¯ ∗ and Σc∗ D, suggested, while other options were not excluded (Chen et al. 2016). Later in 2019, LHCb further reported updated result indicating the three possible states, Pc (4312), Pc (4440), and Pc (4457) (Aaij et al. 2019). Here the earlier signal of Pc (4450) has split into the latter two, while the evidence of the lower one Pc (4380) has become less significant. The observation of the three states in 2019 has pushed forward the speculation of the meson-baryon molecular structure (Yamaguchi et al. 2020; Liu et al. 2019). Assuming that constituent meson and baryon are in the lowest S-state, the lower one Pc (4312) is likely to be a spin singlet of J P = 1/2− formed by Σc D¯ (J P = 1/2+ , 0− ), and the higher two are spin doublet of J P = 1/2− , 3/2− formed by Σc D¯ ∗ (J P = 1/2+ , 1− ). The negative parity of these states is naturally expected from those of Σc D¯ (∗) . In the context of the present note, once again the role of the pion exchange interaction is noted as acting between the light quarks in Σc and D (∗) as shown in the left panel of Fig. 9. As in the case of X(3872), the tensor force is expected to provide attraction through the second order SD-wave mixing. As discussed in the previous subsections, the interaction contains least parameters as determined by chiral symmetry and the structure of Σc and D (∗) in terms of quarks. In the following subsection, discussions made in Yamaguchi et al. (2017, 2020) are introduced, where coupled channel analysis has been performed. Predictions were made in the earlier reference Yamaguchi et al. (2017) which has been updated after the LHCb’s observation in 2019, where the role of the pion exchange interaction, in particular the tensor force, was carefully investigated not only for masses but also for decay widths and spin-parity determination (Yamaguchi et al. 2020).

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Fig. 9 Two types of (∗) interactions for Σc D (∗) : pion exchange and the coupling to a compact state

¯ Λc D¯ ∗ , Coupled channels are prepared by the nearby threshold channels, Λc D, ∗ ∗ ∗ ∗ ¯ Σc D, ¯ Σc D¯ , and Σc D¯ . Their S and D waves (and even higher partial waves Σc D, if necessary) are also included as coupled by the tensor force. The Lagrangian for the pion coupling with various heavy hadrons are constructed with the heavy quark spin symmetry under which D¯ and D¯ ∗ mesons are regarded as components of the same spin multiplet. Hence, the π D¯ (∗) D¯ (∗) coupling is given by Casalbuoni et al. (1997)

μ M Lπ H H = gA Tr Hb γμ γ5 Aba H¯ a ,

(55)

where Ha is the heavy quark spin doublet of spin zero and spin one mesons (D and ∗ γμ − D ¯ a γ5 ](1 + γμ v μ )/2 and H¯ a = γ0 Ha† γ0 . The trace Tr [· · · ] D ∗ ), Ha = [D¯ aμ is taken over the gamma matrices. The subscripts a, b are for two components of isospin, and vμ is the four-velocity of the heavy quark in the heavy meson. The axial M = 0.59 is tuned to reproduce the decay rate D ∗ → Dπ (Yamaguchi coupling gA et al. 2020) (with the understanding that relativistic effects are included and

so it †

i i μ † is smaller than unity). A = 2 ξ ∂μ ξ − ξ ∂μ ξ with ξ = exp 2fπ τ · π is the pion’s axial current, fπ ∼ 93 MeV being the pion decay constant. Similarly, for baryons, Σc and Σc∗ are regarded as heavy quark spin doublet of spin 1/2 and 3/2, respectively, and their couplings to the pion are given by Yan et al. (1992) Lπ BB =



3 g1 (ivκ )εμνλκ tr S¯μ Aν Sλ + g4 tr S¯ μ Aμ Λc + H.c., 2 (∗)

(56)

where tr [· · · ] is over the flavor space. In this equation π Λc Σc , coupling is (∗) (∗) (∗) also included as independent from the π Σc Σc couplings, as Λc is a heavy (∗) quark spin singlet. √ The spin doublet of Σc is implemented by the superfield Sμ = Σcμ − 1/ 3(γμ + vμ )γ5 Σc . The coupling constants g1 and g4 are determined √ (∗) by the decay of Σc and were determined as g1 = ( 8/3)g4 ∼ 1. Again this q strength is supposed to include the relativistic effect as the corresponding gA is smaller than unity (Maeda et al. 2018).

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The Lagrangians (55) and (56) lead to the one pion exchange interaction in momentum space 

M gB gA A V ( q) = − 4fπ2 π



(S1 · q )(S2 · q )   T1 · T2 , q 2 + m2π

(57)

B is a suitable combination of g and g depending on the initial and final where gA 1 4 baryons. In addition to the pion exchange, a short range interaction was introduced through the coupling to a compact configuration of five quarks. For X(3872), the compact component is assumed to be cc ¯ as the possible minimal quark content (see section “X(3872)”). Contrary, for Pc s, it is formed by five quarks ccuud. ¯ In principle, it would be possible to have a configuration of only uud component. However, in the relevant energy region of Pc , it is natural to include cc ¯ component; configurations of only uud would form a continuum spectrum in such a high (∗) energy region. Hence the couplings of D (∗) Σc (and also D (∗) Λc ) to a compact state ccuud ¯ are expected to be proportional to the overlap between them. Such factors are known to be as spectroscopic S-factors in nuclear physics. For the fivequark configurations, those in the quark model favored by the one-gluon exchange interaction are considered (Takeuchi and Takizawa 2017). Having the only one parameter for a common coupling strength as denoted by f , all couplings of D (∗) Σc(∗) to a compact state are fixed. Another ingredient of the interaction is the form factor, whose cutoff can be fixed at a typical hadron scale of 1 GeV. Now the coupled channel equations for meson-baryons and five-quarks are rewritten as the effective coupled channel equations by eliminating the five-quark states. It is a set of equations for multicomponent wave function for meson-baryon channels ψ (Yamaguchi et al. 2020):

 K +Vπ +v

 1 v ψ =ψ, E − E5q

(58)

where K is for kinetic energies of meson-baryon channels, v the coupling between the meson-baryon and five-quark channels as written as the S-factor and the strength f , and E5q the energies of the five-quark channels. By tuning the coupling strength f , the mass spectrum is obtained in the threshold region of D (∗) Σc(∗) as shown in Fig. 10. The states appear as quasi-bound (resonant) states due to the coupling to lower channels. The mechanism of the resonance generation is the Feshbach mechanism (Feshbach 1958, 1962). Since detailed discussions are found in Yamaguchi et al. (2020), several important points are listed here: • By setting f at an appropriate value (attraction with f/f0 ∼ 50 − 80; f0 is the reference strength as defined in Yamaguchi et al. 2020), quasi-bound (resonant) states appear at all threshold regions. The parity of all states is negative. At f/f0 ∼ 50 or less, one (or more) quasi-bound states disappear. Although the

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A. Hosaka [MeV] 4550

EXP

4527:1/2

4457 4440

unbound: 1/2

¯ 6 c D

4511:3/2 4497:5/2

4527

4468:5/2

4462:1/2

¯ 6 c D 4462

4440:3/2 4415:3/2

4400 4380

6 c D¯ 4385

4371:3/2

4350

4300

f / f0 = 80 4521:1/2

4524:3/2

4500

4450

f / f0 = 50

4350:3/2 4312

6 c D¯

4313:1/2 4299:1/2

4321

Fig. 10 Mass spectrum for various Pc pentaquarks from Yamaguchi et al. (2020). Red bars are calculated states with their middle points for their mass values and with their lengths being twice of half decay widths. The black bars on the left are for the observed states

•

• •

•

method of Yamaguchi et al. (2020) does not identify the disappeared states, they are likely to be virtual states. To develop quasi-bound states near the threshold regions, both pion exchange and coupling to compact states are needed with almost equal amount of attraction. This seems similar to the case of X(3872). All these states are dominated by the nearby threshold channels. For instance, the dominant component of Pc (4313) is Σc D¯ of J P = 1/2− . For Σc D¯ ∗ and Σc∗ D¯ ∗ , among possible spin multiplets, the higher spin states are lighter than the lower spin states, which would be of counter-intuition. As discussed in Yamaguchi et al. (2020), this is due to the tensor force of the pion exchange interaction; the higher spin states receive more attraction from the second order effect of the tensor force. Not only the ordering of masses within a spin multiplet, decay widths systematically increase as the spin J increases. Hence the determination of spin and parity of these states is very important to know more about the structure of Pc s, especially the interaction among Σc(∗) D¯ (∗) .

In the end of this section, it is emphasized that the determination of hadron interaction is an important issue, rather than talking whether the observed peaks are associated with resonances, bound states, or virtual states (cusps). As is well known, the two nucleon system of 3 S1 generates a bound state (the deuteron), while

61 Hadrons from Quarks and Chiral Symmetry

2331

that of 1 S0 a virtual state. The difference is in a rather small difference in the strength of attraction. Perhaps, more constructive and important is to clarify the origin of the interaction. It has been a fortunate situation that the low energy nuclear interaction is approximated by the t-channel process as meson exchanges and is expressed in the form of (almost) local potential to establish the realistic nuclear interaction with high precision (Machleidt and Entem 2011). In general, s or u-channel dynamics will become relevant as discussed in this section: t-channel by the pion exchange and s-channel via coupling to compact states. In the latter case, hadron structures become more relevant, and therefore, consistent understanding of both structure and interactions is further developed.

Remarks and Comments In this chapter, the structure of low-lying hadrons is discussed in the quark model whose ground may be traced in chiral symmetry and its spontaneous breaking of QCD. The light bare quarks of the fundamental theory of QCD acquire a finite mass and become effective degrees of freedom for hadrons. At the same time, light pseudoscalar mesons emerge as the Nambu-Goldstone bosons. This framework, a working hypothesis for many years, is now waiting for further test not only for the conventional hadrons but also for exotic hadrons. Among yet unanswered questions, diquarks are considered to be another important effective degree of freedom. Though there are many suggestions for diquarks at work (Anselmino et al. 1993; Wilczek 2005), decisive evidence is not yet in full consensus. The diquarks are expected to play not only in spectroscopy but also in high density quark matter, where color superconducting phase appears (Alford et al. 1998, 1999; Rapp et al. 1998). Diquarks in hadrons can be a reference when extrapolating to those in high density matter. The subject is now extensively discussed in neutron star properties where the equation of state of high density and low temperature hadronic matter is the issue to be solved (Baym et al. 2018; Fukushima and Hatsuda 2011). Here heavy quarks are expected to isolate the diquarks in a hadron and extract their properties. An experimental project at JPARC, hadron hall extension, is now extensively discussed for the exploration of strange and heavy baryon spectroscopy and hypernuclei toward high density matter (Aoki et al. 2021). To further explore, hadron interactions should be better understood. For this, the use of correlation functions, the femtoscopy, is a promising approach to extract the hadron interactions that cannot be obtained by traditional accelerator experiments (Greiner and Muller 1989; Morita et al. 2015; Acharya et al. 2022). Theoretical method of the lattice QCD is also developed extensively (Ishii et al. 2007; Sasaki et al. 2015; Ikeda et al. 2016; Gongyo et al. 2018; Iritani et al. 2019). Systematic studies of structure and interactions from both theory and experiment will open further activities in hadron physics.

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Acknowledgments The author acknowledges his colleagues for discussions, comments, and encouragements in collaborations. They are Jafar Arifi, Emiko Hiyama, Hideko Nagahiro, Hiroyuki Noumi, Makoto Oka, Elena Santopinto, Daiki Suenaga, Sachiko Takeuchi, Makoto Takizawa, Kiyoshi Tanida, Yasuhiro Yamaguchi, and Shigehiro Yasui. This work is supported in part by Grants-in Aid for Scientific Research on Innovative Areas (No. 18H05407).

References R. Aaij et al. [LHCb], Phys. Rev. D 92(1), 011102 (2015a). https://doi.org/10.1103/PhysRevD.92. 011102 [arXiv:1504.06339 [hep-ex]] R. Aaij et al. [LHCb], Phys. Rev. Lett. 115, 072001 (2015b). https://doi.org/10.1103/PhysRevLett. 115.072001 [arXiv:1507.03414 [hep-ex]] R. Aaij et al. [LHCb], Phys. Rev. Lett. 122(22), 222001 (2019). https://doi.org/10.1103/ PhysRevLett.122.222001 [arXiv:1904.03947 [hep-ex]] S. Acharya et al. [ALICE], (2022). [arXiv:2201.05352 [nucl-ex]] M.G. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. B 422, 247–256 (1998). https://doi.org/10. 1016/S0370-2693(98)00051-3 [arXiv:hep-ph/9711395 [hep-ph]] M.G. Alford, K. Rajagopal, F. Wilczek, Nucl. Phys. B 537, 443–458 (1999). https://doi.org/10. 1016/S0550-3213(98)00668-3 [arXiv:hep-ph/9804403 [hep-ph]] M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson, D.B. Lichtenberg, Rev. Mod. Phys. 65, 1199–1234 (1993). https://doi.org/10.1103/RevModPhys.65.1199 K. Aoki, H. Fujioka, T. Gogami, Y. Hidaka, E. Hiyama, R. Honda, A. Hosaka, Y. Ichikawa, M. Ieiri, M. Isaka et al., (2021). [arXiv:2110.04462 [nucl-ex]] A.J. Arifi, D. Suenaga, A. Hosaka, Phys. Rev. D 103(9), 094003 (2021). https://doi.org/10.1103/ PhysRevD.103.094003 [arXiv:2102.03754 [hep-ph]] A.J. Arifi, D. Suenaga, A. Hosaka, Y. Oh, Phys. Rev. D 105(9), 094006 (2022). https://doi.org/10. 1103/PhysRevD.105.094006 [arXiv:2201.10427 [hep-ph]] T. Banks, A. Casher, Nucl. Phys. B 169, 103–125 (1980). https://doi.org/10.1016/05503213(80)90255-2 G. Baym, T. Hatsuda, T. Kojo, P.D. Powell, Y. Song, T. Takatsuka, Rept. Prog. Phys. 81(5), 056902 (2018). https://doi.org/10.1088/1361-6633/aaae14 [arXiv:1707.04966 [astro-ph.HE]] A.A. Belavin, A.M. Polyakov, A.S. Schwartz, Y.S. Tyupkin, Phys. Lett. B 59, 85–87 (1975). https:// doi.org/10.1016/0370-2693(75)90163-X R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, G. Nardulli, Phys. Rep. 281, 145–238 (1997). https://doi.org/10.1016/S0370-1573(96)00027-0 [arXiv:hep-ph/9605342 [hep-ph]] H.X. Chen, W. Chen, X. Liu, S.L. Zhu, Phys. Rep. 639, 1–121 (2016). https://doi.org/10.1016/j. physrep.2016.05.004 [arXiv:1601.02092 [hep-ph]] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn, Phys. Rev. D 10, 2599 (1974). https://doi.org/10. 1103/PhysRevD.10.2599 S.K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003). https://doi.org/10.1103/PhysRevLett. 91.262001 [arXiv:hep-ex/0309032 [hep-ex]] F.E. Close, An Introduction to Quarks and Partons (Academic, 1979) L.A. Copley, N. Isgur, G. Karl, Phys. Rev. D 20, 768 (1979) [erratum: Phys. Rev. D 23, 817 (1981)]. https://doi.org/10.1103/PhysRevD.20.768 D. Diakonov, Prog. Part. Nucl. Phys. 51, 173–222 (2003). https://doi.org/10.1016/S01466410(03)90014-7 [arXiv:hep-ph/0212026 [hep-ph]] A. Esposito, A.L. Guerrieri, L. Maiani, F. Piccinini, A. Pilloni, A.D. Polosa, V. Riquer, Phys. Rev. D 92(3), 034028 (2015). https://doi.org/10.1103/PhysRevD.92.034028 [arXiv:1508.00295 [hep-ph]] H. Feshbach, Ann. Phys. 5, 357–390 (1958). https://doi.org/10.1016/0003-4916(58)90007-1 H. Feshbach, Ann. Phys. 19, 287–313 (1962). https://doi.org/10.1016/0003-4916(62)90221-X

61 Hadrons from Quarks and Chiral Symmetry

2333

H. Fukaya et al. [JLQCD], Phys. Rev. Lett. 104, 122002 (2010) [erratum: Phys. Rev. Lett. 105, 159901 (2010)]. https://doi.org/10.1103/PhysRevLett.104.122002 [arXiv:0911.5555 [hep-lat]] H. Fukaya et al. [JLQCD and TWQCD], Phys. Rev. D 83, 074501 (2011). https://doi.org/10.1103/ PhysRevD.83.074501 [arXiv:1012.4052 [hep-lat]] K. Fukushima, T. Hatsuda, Rep. Prog. Phys. 74, 014001 (2011). https://doi.org/10.1088/00344885/74/1/014001 [arXiv:1005.4814 [hep-ph]] M. Gell-Mann, Phys. Lett. 8, 214-215 (1964). https://doi.org/10.1016/S0031-9163(64)92001-3 M. Gell-Mann, M. Levy, Nuovo Cim. 16, 705 (1960). https://doi.org/10.1007/BF02859738 M.L. Goldberger, S.B. Treiman, Phys. Rev. 110, 1178–1184 (1958). https://doi.org/10.1103/ PhysRev.110.1178 J. Goldstone, Nuovo Cim. 19, 154–164 (1961). https://doi.org/10.1007/BF02812722 S. Gongyo, K. Sasaki, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, T. Inoue, T. Iritani, N. Ishii, T. Miyamoto et al., Phys. Rev. Lett. 120(21), 212001 (2018). https://doi.org/10.1103/ PhysRevLett.120.212001 [arXiv:1709.00654 [hep-lat]] C. Greiner, B. Muller, Phys. Lett. B 219, 199–204 (1989). https://doi.org/10.1016/03702693(89)90377-8 T. Hatsuda, T. Kunihiro, Phys. Rep. 247, 221–367 (1994). https://doi.org/10.1016/03701573(94)90022-1 [arXiv:hep-ph/9401310 [hep-ph]] A. Hosaka, H. Toki, Quarks, Baryons and Chiral Symmetry. World Scientific, Singapore (2001) A. Hosaka, T. Iijima, K. Miyabayashi, Y. Sakai, S. Yasui, PTEP 2016(6), 062C01 (2016). https:// doi.org/10.1093/ptep/ptw045 [arXiv:1603.09229 [hep-ph]] Y. Ikeda et al. [HAL QCD], Phys. Rev. Lett. 117(24), 242001 (2016). https://doi.org/10.1103/ PhysRevLett.117.242001 [arXiv:1602.03465 [hep-lat]] T. Iritani et al. [HAL QCD], Phys. Lett. B 792, 284–289 (2019). https://doi.org/10.1016/j.physletb. 2019.03.050 [arXiv:1810.03416 [hep-lat]] N. Isgur, G. Karl, Phys. Rev. D 18, 4187 (1978). https://doi.org/10.1103/PhysRevD.18.4187 N. Isgur, G. Karl, Phys. Rev. D 19, 2653 (1979a) [erratum: Phys. Rev. D 23, 817 (1981)]. https:// doi.org/10.1103/PhysRevD.19.2653 N. Isgur, G. Karl, Phys. Rev. D 20, 1191–1194 (1979b). https://doi.org/10.1103/PhysRevD.20. 1191 N. Ishii, S. Aoki, T. Hatsuda, Phys. Rev. Lett. 99, 022001 (2007). https://doi.org/10.1103/ PhysRevLett.99.022001 [arXiv:nucl-th/0611096 [nucl-th]] K.I. Ishikawa et al. [PACS], Phys. Rev. D 104(7), 074514 (2021). https://doi.org/10.1103/ PhysRevD.104.074514 [arXiv:2107.07085 [hep-lat]] M. Karliner, J.L. Rosner, Phys. Rev. D 90(9), 094007 (2014). https://doi.org/10.1103/PhysRevD. 90.094007 [arXiv:1408.5877 [hep-ph]] T. Kubota, K. Ohta, Phys. Lett. B 65, 374–376 (1976). https://doi.org/10.1016/03702693(76)90246-X M.Z. Liu, Y.W. Pan, F.Z. Peng, M. Sánchez Sánchez, L.S. Geng, A. Hosaka, M. Pavon Valderrama, Phys. Rev. Lett. 122(24), 242001 (2019). https://doi.org/10.1103/PhysRevLett.122.242001 [arXiv:1903.11560 [hep-ph]] R. Machleidt, D.R. Entem, Phys. Rept. 503, 1–75 (2011). https://doi.org/10.1016/j.physrep.2011. 02.001 [arXiv:1105.2919 [nucl-th]] S. Maeda, M. Oka, Y.R. Liu, Phys. Rev. C 98(3), 035203 (2018). https://doi.org/10.1103/ PhysRevC.98.035203 [arXiv:1803.11349 [nucl-th]] A. Manohar, H. Georgi, Nucl. Phys. B 234, 189–212 (1984). https://doi.org/10.1016/05503213(84)90231-1 V.A. Miransky, Dynamical Symmetry Breaking in Quantum Field Theories. World Scientific, Singapore (1994) K. Morita, T. Furumoto, A. Ohnishi, Phys. Rev. C 91(2), 024916 (2015). https://doi.org/10.1103/ PhysRevC.91.024916 [arXiv:1408.6682 [nucl-th]] H. Nagahiro, S. Yasui, A. Hosaka, M. Oka, H. Noumi, Phys. Rev. D 95(1), 014023 (2017). https:// doi.org/10.1103/PhysRevD.95.014023 [arXiv:1609.01085 [hep-ph]]

2334

A. Hosaka

T. Nakano et al. [LEPS], Phys. Rev. Lett. 91, 012002 (2003). https://doi.org/10.1103/PhysRevLett. 91.012002 [arXiv:hep-ex/0301020 [hep-ex]] Y. Nambu, Phys. Rev. Lett. 4, 380–382 (1960). https://doi.org/10.1103/PhysRevLett.4.380 Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345–358 (1961a). https://doi.org/10.1103/PhysRev. 122.345 Y. Nambu, G. Jona-Lasinio, Phys. Rev. 124, 246–254 (1961b). https://doi.org/10.1103/PhysRev. 124.246 M. Neubert, Phys. Rep. 245, 259–396 (1994). https://doi.org/10.1016/0370-1573(94)90091-4 [arXiv:hep-ph/9306320 [hep-ph]] R. Rapp, T. Schäfer, E.V. Shuryak, M. Velkovsky, Phys. Rev. Lett. 81, 53–56 (1998). https://doi. org/10.1103/PhysRevLett.81.53 [arXiv:hep-ph/9711396 [hep-ph]] P. Reinert, H. Krebs, E. Epelbaum, Phys. Rev. Lett. 126(9), 092501 (2021). https://doi.org/10.1103/ PhysRevLett.126.092501 [arXiv:2006.15360 [nucl-th]] K. Sasaki et al. [HAL QCD], PTEP 2015(11), 113B01 (2015). https://doi.org/10.1093/ptep/ptv144 [arXiv:1504.01717 [hep-lat]] B. Silvestre-Brac, Few Body Syst. 20, 1–25 (1996). https://doi.org/10.1007/s006010050028 M. Takayama, H. Toki, A. Hosaka, Prog. Theor. Phys. 101, 1271–1283 (1999). https://doi.org/10. 1143/PTP.101.1271 S. Takeuchi, Nucl. Phys. A 642, 543–561 (1998). https://doi.org/10.1016/S0375-9474(98)005399 [arXiv:hep-ph/9807240 [hep-ph]] S. Takeuchi, M. Oka, Phys. Rev. Lett. 66, 1271–1274 (1991). https://doi.org/10.1103/PhysRevLett. 66.1271 S. Takeuchi, M. Takizawa, Phys. Lett. B 764, 254–259 (2017). https://doi.org/10.1016/j.physletb. 2016.11.034 [arXiv:1608.05475 [hep-ph]] M. Takizawa, S. Takeuchi, PTEP 2013, 093D01 (2013). https://doi.org/10.1093/ptep/ptt063 [arXiv:1206.4877 [hep-ph]] S. Weinberg, Phys. Rev. 177, 2604–2620 (1969). https://doi.org/10.1103/PhysRev.177.2604 S. Weinberg, Phys. Rev. Lett. 65, 1177–1180 (1990). https://doi.org/10.1103/PhysRevLett.65.1177 S. Weinberg, Phys. Rev. Lett. 65, 1181–1183 (1990). https://doi.org/10.1103/PhysRevLett.65.1181 F. Wilczek, (2005) https://doi.org/10.1142/9789812775344_0007 [arXiv:hep-ph/0409168 [hepph]] Y. Yamaguchi, A. Giachino, A. Hosaka, E. Santopinto, S. Takeuchi, M. Takizawa, Phys. Rev. D 96(11), 114031 (2017). https://doi.org/10.1103/PhysRevD.96.114031 [arXiv:1709.00819 [hepph]]. Y. Yamaguchi, A. Hosaka, S. Takeuchi, M. Takizawa, J. Phys. G 47(5), 053001 (2020). https://doi. org/10.1088/1361-6471/ab72b0 [arXiv:1908.08790 [hep-ph]] Y. Yamaguchi, H. García-Tecocoatzi, A. Giachino, A. Hosaka, E. Santopinto, S. Takeuchi, M. Takizawa, Phys. Rev. D 101(9), 091502 (2020). https://doi.org/10.1103/PhysRevD.101. 091502 [arXiv:1907.04684 [hep-ph]] T.M. Yan, H.Y. Cheng, C.Y. Cheung, G.L. Lin, Y.C. Lin, H.L. Yu, Phys. Rev. D 46, 1148–1164 (1992) [erratum: Phys. Rev. D 55, 5851 (1997)]. https://doi.org/10.1103/PhysRevD.46.1148 T. Yoshida, E. Hiyama, A. Hosaka, M. Oka, K. Sadato, Phys. Rev. D 92(11), 114029 (2015). https:// doi.org/10.1103/PhysRevD.92.114029 [arXiv:1510.01067 [hep-ph]] H. Yukawa, Proc. Phys. Math. Soc. Jpn. 17, 48–57 (1935). https://doi.org/10.1143/PTPS.1.1 G. Zweig, CERN-TH-412 (1964) P.A. Zyla et al. [Particle Data Group], PTEP 2020(8), 083C01 (2020). https://doi.org/10.1093/ptep/ ptaa104

Pion Exchange Interaction in Bonn Potential and Relativistic and Non-relativistic Framework in Nuclear Matter

62

Jinniu Hu and Chencan Wang

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charged-Dependent Bonn Potential with Pseudovector Coupling . . . . . . . . . . . . . . . . . . . . . The Nuclear Matter in BHF and RBHF Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Several New Relativistic Ab Initio Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hartree-Fock with UCOM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Relativistic Hartree-Fock Model with High Momentum Components . . . . . . . . . . . . Summaries and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2336 2339 2344 2351 2351 2356 2362 2363

Abstract

As the residual interaction of quantum chromodynamics in low-energy region, the nucleon-nucleon (NN) potential can only be exactly described by the model picture now. In the Bonn potential, one of the most well-known NN interaction models, the nucleons interact with each other through exchanging the pion and several heavier mesons, where the pion plays an essential role. It provides a partial contribution of tensor force in the intermediate-range region and the main component in the long-range region in NN potential. However, it is very difficult to be treated in the nuclear many-body system due to its pseudovector or pseudoscalar property. Recently, three high-precision charge-dependent Bonn

J. Hu () School of Physics, Nankai University, Tianjin, China Shenzhen Research Institute of Nankai University, Shenzhen, China e-mail: [email protected] C. Wang School of Physics, Nankai University, Tianjin, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_57

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J. Hu and C. Wang

potentials were proposed with pseudovector coupling types and different pionnucleon coupling strengths and applied them to study the properties of nuclear matter and neutron stars in the non-relativistic and relativistic frameworks. Furthermore, to properly deal with the strong short-range repulsion and tensor force of the NN potential, some new relativistic ab initio methods have also been developed in the past decade to discuss the role of pion and relativistic effects in nuclear matter.

Introduction Most finite nuclei in the nuclide chart composed of protons and neutrons are complex quantum many-body systems. They are self-bound together with the nucleon-nucleon (NN) potential, which is considered a residual part of one of the four fundamental forces in the universe, with strong interaction at a low-energy scale. In principle, it should be solved by using quantum chromodynamics (QCD) theory. However, it is a long way to directly generate the NN potential from QCD theory due to its unperturbation of the nucleon. The first successful attempt to describe the NN potential with a serious theoretical framework was made by Yukawa. He assumed that the interaction between two nucleons is generated by a particle with a mass of around 100 MeV, i.e., a pion through analogy with the photon in electromagnetic force (Yukawa 1935). Finally, the pion was discovered in the cosmic rays (Lattes et al. 1947). With the development of accelerators, a lot of NN scattering data about protonproton and proton-neutron was generated at terrestrial laboratories. The basic characters of the NN potential were extracted from the phase-shift analysis. There are strong short-range repulsion and attraction in the intermediate range between two nucleons. When the distance between two nucleons is larger than 2 fm, their strong interaction becomes weak and is denominated by the pion. Firstly, the NN force model was generated by the free NN scattering information, such as phase shifts, polarizations, and cross-sections in the 1950s, which was called the realistic NN potential. However, in complex nuclei, it was found that the repulsion of NN interaction is much weaker than the realistic NN potential due to the nuclear manybody medium effects. As a result, many effective NN interactions have recently been constructed by reproducing the properties of infinite nuclear matter and finite nuclei (Bender and Heenen 2003; Stone et al. 2007; Ring 1996; Vretenar et al. 2005; Meng et al. 2006; Nikši´c et al. 2011). In this paper, the realistic NN potential is mainly discussed. In the NN scattering, several important symmetries, i.e., rotation invariance, translation invariance, and space reflection invariance, are kept, which should be embodied in the theoretical framework to describe the NN interaction. Furthermore, these scattering processes are strongly spin-dependent. In the 1960s, the NN forces were associated with terms including the spin, momentum, orbital-angular

62 Pion Exchange Interaction in Bonn Potential and Relativistic and. . .

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momentum, and coordinate operators, which accord with the above symmetries, like the Hamada-Johnston potential (Hamada and Johnston 1962) and Reid68 potential (Reid 1968). At the same time, several heavier mesons were found in the accelerators, which were adopted to describe the short- and intermediate-region interactions by the Bonn group with one-boson-exchange (OBE) potential (Erkelenz 1974) based on the quantum field theory. Furthermore, the other baryon degrees of freedom, such as Δ isobar and the multi-meson exchange terms, were included in the Bonn potential in the 1980s (Machleidt et al. 1987; Machleidt 1989). In the 1990s, the charge independence breaking (CIB) and charge symmetry breaking (CSB) effects were considered in NN interaction for the NN scattering phase shift analysis. Therefore, many high-precision NN forces were generated, such as Reid93, Nijmegen 93, Nijmegen I, Nijmegen II, and AV18 potentials (Stoks et al. 1994; Wiringa et al. 1995). In 2000, Machleidt proposed a charge-dependent Bonn (CD-Bonn) potential as a high-precision version of the Bonn potential (Machleidt 2001). The ω, ρ, π mesons and two scalar mesons σ1 and σ2 are exchanged by two nucleons in CDBonn potential. It has been widely applied to study the properties of nuclear systems, from light nuclei to heavy nuclei, and infinite nuclear matter. Furthermore, the chiral effective field theory was applied to describe the NN potential, which has been expanded up to the fifth-order expansion (Weinberg 1990, 1991, 1992; Ordóñez et al. 1994, 1996; Epelbaum et al. 1998, 2000, 2005, 2015a, b; Entem and Machleidt 2003; Entem et al. 2015, 2017; Reinert et al. 2018; Lu et al. 2022). Due to the strong repulsion of the NN potential at a short-range distance, which was first proposed by Jastrow from abundant experimental data of NN scattering (Jastrow 1951), the nuclear many-body system cannot be described by the NN potential with the perturbation theory. Furthermore, the tensor force mainly generated by the pion is also very difficult to include due to its non-central character. Both of them can raise the strong high momentum correlations between two nucleons in the nuclei. Therefore, the realistic NN potential should be renormalized at low momentum in the nuclear medium to generate a softer interaction so that finite nuclei and symmetric nuclear matter can form the bound states. The earliest NN renormalization scheme was given by Brueckner et al., where the strong repulsion is reduced by the Bethe-Goldstone equation (Brueckner et al. 1954; Bethe 1956). Meanwhile, Jastrow exhibited another way to treat the high momentum correlations with a variational method (Jastrow 1955). Besides these two schemes, many relevant nuclear many-body methods with realistic NN potential to calculate the finite nuclei and infinite nuclear matter have been developed, i.e., ab initio method, such as the Brueckner-Hartree-Fock (BHF) method (Li et al. 2006; Baldo and Maieron 2007; Baldo and Burgio 2016), lowest-order constrained variational method (Modarres 1993), quantum Monte Carlo methods (Akmal et al. 1998; Carlson et al. 2015), self-consistent Green’s function method (Dickhoff and Barbieri 2004), coupledcluster method (Hagen et al. 2014a, b), many-body perturbation theory (Carbone

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et al. 2013, 2014; Drischler et al. 2014), functional renormalization group (FRG) method (Drews and Weise 2015; Drews and Weise 2017), and so on. However, most of the nuclear ab initio methods were constructed in the nonrelativistic framework, which cannot completely reproduce the empirical nuclear saturation properties, E/A = −16 ± 1 MeV at n0 = 0.16 ± 0.01 fm−3 and the ground-state properties of finite nuclei only with the present two-body realistic nuclear force. The additional three-body nuclear potential must be included Li et al. (2006), Hu et al. (2017), Sammarruca et al. (2018), and Logoteta (2019). In the 1980s, Anastasio et al. proposed the relativistic version of the BHF model, i.e., the relativistic Brueckner-Hartree-Fock (RBHF) method (Ansatasio et al. 1983). It was developed later by Horowitz and Serot (1987) and Brockmann and Machleidt (1990). A repulsive contribution is provided due to the relativistic effect in the RBHF model, which can reproduce the empirical nuclear saturation properties with the Bonn potential. It was explained that the nucleon-antinucleon excitation in the relativistic effect has a similar contribution to that of the three-nucleon force (Li et al. 2008). Furthermore, the superfluity of nuclear matter, properties of the neutron star, and nuclear density functional theories were studied within the RBHF model (Alonso and Sammarruca 2003; Krastev and Sammarruca 2006; Sammarruca 2010; Dalen and Muether 2010). Recently, a fully self-consistent calculation of the RBHF model in finite nuclei system and extended this framework on the neutron drops were realized (Shen et al. 2016, 2017, 2019). However, there are still many unsolved problems in the relativistic ab initio method. The nucleon should be considered as a Dirac particle in the relativistic framework. Therefore, nucleon-nucleon potentials must be constructed in the quantum field theory. There are only Bonn and CD-Bonn potentials until 2018, which can be used in the RBHF model. However, the Bonn potential was generated at the beginning of 1990. It does not consider the CIB and CSB effects and is not a high-precision NN potential. Furthermore, the coupling between pion and nucleon in the CD-Bonn potential is the pseudoscalar type, which produces an additional attractive contribution (Fuchs et al. 1998). Hence, it is urgent to develop a highprecision NN potential that can be adopted by the RBHF model. On the other hand, the normalization schemes from the Brueckner theory cannot clearly distinguish the roles of short-range correlation and tensor correlation from the pion. It is necessary to develop the new relativistic ab initio beyond the meanfield approximation, which can treat the strong repulsion at short-range distance and attraction at intermediate range from pion with realistic NN potential. In this paper, our recent progress in the high-precision NN potential and relativistic ab initio method will be shown. In section “Charged-Dependent Bonn Potential with Pseudovector Coupling”, the revised CD-Bonn potentials with pseudovector coupling and different tensor components are given. In section “The Nuclear Matter in BHF and RBHF Models”, the properties of nuclear matter and neutron stars with the revised CD-Bonn potential will be discussed within the RBHF model. In section “Several New Relativistic Ab Initio Methods”, several new relativistic ab initio methods are shown beyond the relativistic Hartree-Fock model. In the last section, the summary and perspective will be given.

62 Pion Exchange Interaction in Bonn Potential and Relativistic and. . .

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Charged-Dependent Bonn Potential with Pseudovector Coupling The behavior of NN interaction in the long-range region is mainly denominated by the pion, which also represents the chiral symmetry of QCD theory as a Goldstone boson, but the coupling types between nucleon and pion have two schemes, i.e., the pseudoscalar (PS) and pseudovector (PV) coupling. The on-shell amplitudes of PS and PV coupling between pion and nucleon are completely the same. However, the PV coupling was preferred in the calculations of pion-nucleon scatterings, such as π N to the π π  N N¯ and so on (Lacombe et al. 1975; Jackson et al. 1975). In the chiral perturbation theory, the PV type was also taken in low-energy region to investigate the pion electroproduction and photoproduction (Drechsel and Tiator 1992; Drechsel et al. 1999). On the other hand, an additional strong attraction was obtained for its strong coupling to negative energy states, when the NN potentials including the PS coupling were applied to the relativistic framework. Instead, the PV coupling can suppress this contribution due to the vanishment of the matrix element between the antinucleon and nucleon in the on-shell scattering (Fuchs et al. 1998). The high-precision CD-Bonn potential proposed by Machleidt (2001) with the analogous framework of Bonn potential should be adopted in the RBHF model. However, its coupling scheme between pion and nucleon is the PS type. Therefore, in a relativistic framework, it will provide a very attractive contribution and the empirical saturation properties; thus it cannot be reproduced. Therefore, the PV coupling is adopted instead of the PS one in the original CD-Bonn potential. In the CD-Bonn potential, five mesons are considered to be exchanged in the NN interaction. Two scalar mesons are σ1 and σ2 . Two vector mesons are ω and ρ. The pion is recognized as the PV meson. The Lagrangian between them and nucleons can be written as fπ N N ¯ 5 γ μ ψ · ∂μ φ (π 0 ) , ψγ mπ ± √ 2fπ N N ¯ 5 γ μ τ± ψ · ∂μ φ (π ± ) , ψγ =− mπ ±

Lπ 0 N N = − Lπ ± N N

(1)

(σ ) ¯ , Lσ N N = −gσ N N ψψφ

¯ μ ψφμ(ω) , LωN N = −gωN N ψγ ¯ μ τψ φμ(ρ) − LρN N = −gρN N ψγ

fρN N ¯ μν τψ∂μ φν(ρ) , ψσ 2Mp

where ψ and φ denote the nucleon and meson fields, respectively. τ is the isospin operator. g and f are the coupling constants. mπ and Mp are the masses of pion and nucleon, respectively. The NN interaction can be obtained by the Feynman scattering amplitude between meson and nucleon

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V¯α (q  , q) = −u¯ 1 (q  )Γα(1) u1 (q)

iPα k

2

+ m2α

u¯ 2 (−q  )Γα(2) u2 (−q),

(2)

where ui is the spinor at free space  ui (q) = E=



E+M 2M



1

σ i ·q 2M

q 2 + M 2 and E  =



 ,



u¯ i (q ) =

E + M 2M

  σ i · q 1, − . 2M

(3)

 q  2 + M 2 are the input and output energies of on-shell

nucleon. Γα is the coupling vertex between meson and nucleon. q  and q are the corresponding momenta. iPα /(k 2 + m2α ) is the meson propagator without retarded effect. k = q  −q is the transferring momentum. Therefore, the NN interaction from various mesons can be summarized as    M M  Fα (k 2 )V¯α (q  , q), (4) V (q , q) =  E E 0 ± (i)

α=π ,π ,ω,ρ,σ

  M where M E E  comes from the minimal relativity. Furthermore, a form factor to denote the size of nucleon must be included in the vertex between meson and nucleon  nα Λ2α − m2α 2 Fα (k ) = (5) Λ2α + k 2 to regularize the high momentum behaviors of NN interaction. Λα is called the momentum cut-off, and in Bonn potential, nα = 2 is the dipole case. In the NN scattering and infinite nuclear matter, the partial wave presentation is convenient  sj |V (q  , q)|sj , which is transformed from the helicity presentation by λ1 λ2 |V (q  , q)|λ1 λ2  → λ1 λ2 |Vj (q  , q)|λ1 λ2  →  sj |V (q  , q)|sj ,

(6)

Three pion-nucleon coupling strengths are considered to discuss the role of pion in NN interaction as shown in Table 1 (Wang et al. 2019). Due to the largest cutoff in the C case, it has the strongest pion contribution in NN potential. For the ω and ρ mesons, their coupling constants with nucleon and cut-offs are taken as the conventional magnitudes from the hadron physics given in Table 2. Table 1 Three kind values of gπ2 /4π and cut-offs Λπ , which are named as A, B, and C

A B C

mπ 0 [MeV] 134.9766 134.9766 134.9766

mπ ± [MeV] 139.5702 139.5702 139.5702

gπ2 /4π 13.9 13.7 13.6

Λπ [GeV] 1.12 1.50 1.72

62 Pion Exchange Interaction in Bonn Potential and Relativistic and. . . Table 2 The parameters of vector mesons, including their mass, m; coupling strengths, g 2 /4π ; and cut-offs Λ

Mesons ρ0, ρ± ω

m [GeV] 0.770 0.783

g 2 /4π 0.84 20

2341 f/g 6.1 0

Λ [GeV] 1.31 1.50

Fig. 1 The tensor contributions from the pion and vector mesons in 3 S1 − 3 D1 (upper panels) and 3 P − 3 F channels (lower panels). In panel (a), only pion is included. In panel (b), pion and ρ 2 2 meson are considered. In panel (c), all mesons are taken into account. Here, the output momentum is fixed as q  = 0.265 GeV

The pion, ω, and ρ mesons all can generate the tensor force, which is denoted as the matrix elements between the  and  + 2 states, such as S − D or P − F channels. In Fig. 1, the half-on-shell matrix elements at 3 S1 − 3 D1 and3 P2 − 3 F2 channels are shown with a fixed output momentum q  = 0.265 GeV. Pion provides the most attractive contributions at S − D channels. When the ρ meson is included, the magnitudes are largely reduced. Meanwhile, the ω meson can also generate some attractive components. In the P − F channel, the matrix elements of pion are positive, while those from the ρ and ω mesons are negative, which are strongly dependent on the isospin characters of NN potential. The parameters about the scalar mesons, gσ21 /4π, mσ1 , gσ22 /4π, mσ2 , are determined by the partial wave phase shifts at each channel extracted from the NN scattering and the binding energy of deuteron. In the center of mass system (c.m.), the scattering process is described by the Lippmann-Schwinger (LS) equation: T (q  , q) = V (q  , q) +



d 3k M T (k, q). V (q  , k) 3 2 (2π ) q − k 2 + iε

(7)

In the partial wave basis |sj  with the conservations of total spin and angular momentum, this equation can be expanded as

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T sj (q  , q) = V sj (q  , q) +



k 2 dk V  sj (q  , k)



M q2

− k 2 + iε

T sj (k, q), (8)

where V sj =  j s|V |j s. The phase shift, δj , can be related to the T matrix through the S matrix. The parameters about two scalar mesons are obtained by minimizing the function with the least square method: χ2 =

N 2 1  δ − δN ij m i . N

(9)

i=1

The input energies of nucleon at laboratory frame, Elab , are less than 300 MeV, here. δN ij m is the Nijmegen partial wave analysis (PWA). First, the proton-proton (pp) potential is generated by this fitting process. Then, for the proton-neutron (np) and neutron-neutron (nn) potentials, the CSB and CIB effects are considered. Finally, three high-precision charge-dependent Bonn potentials with PV coupling are obtained, which can nicely reproduce the data from Nijmegen PWA, i.e., pvCDBonn A, B, and C potentials. In Fig. 2, the phase shifts of pp scattering from pvCDBonn A, B, and C potentials are shown and compared to those from the Nijmegen PWA, original CD-Bonn, and latest chiral NN potential at different channels. It can be found that all of them can nicely reproduce the data from the Nijmegen PWA and are consistent with the results from the other high-precision NN potential. The only difference appears in the mixing parameter ε2 at 3 P2 −3F2 channel, which is related to the strengths of tensor components in these three potentials. There is a similar situation for the np scattering at the 3 S1 −3D1 channel. In Fig. 3, the matrix elements of np components from pvCDBonn A, B, and C potentials are shown in the two-dimensional contour as functions of momenta. It is clear that most differences among them at 3 S1 and 3 S1 −3D1 channels, which

Fig. 2 The phase shifts of pp from pv CDBonn A, B, and C potentials and compared to those from the Nijmegen PWA, origin CD-Bonn potential and latest chiral potential

62 Pion Exchange Interaction in Bonn Potential and Relativistic and. . .

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Fig. 3 The matrix elements of np components from pvCDBonn potentials. They are rescaled with V˜ = Mπ V /2 whose unit is fm

are denominated by the pion. At the low momentum region, there is attractive interaction at 3 S1 channel for pvCDBonn A potential, while they become the repulsive one at pvCDBonn C potential, which has the strongest pion contribution. Meanwhile, the situation is the opposite at the 3 S1 −3D1 channel. The pvCDBonn C has the strongest attractive contribution in the low-moment region. Therefore, the role of the pion in the NN potential is mainly embodied in the coupled channels. When the relative momentum q is very small, the phase shift at the S wave can be denoted by the scattering length, a, and effective range, r q cot δS ≈ −

1 1 2 + rq + O(q 4 ), a 2

(10)

which can be measured from the experiments. Their values from the pvCDBonn potential are shown in Table 3, which are consistent with results from the original CD-Bonn potential and the empirical data from experiments. The lightest nuclear bound state, deuteron, can be solved with an on-shell LS equation Ψ (q) = −

M γd2 + q 2



+∞

q 2 dq  V (q, q  )Ψ (q  ),

(11)

0

where γd2 /M = Bd is the bound energy of deuteron and M the mean mass of proton and neutron. The wave function Ψ should have two components at S and D waves. The properties of the deuteron, the asymptotic S-state amplitude (AS ), the ratio of the D/S states (η), the root-mean-square radius of the deuteron (rd ),

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J. Hu and C. Wang

Table 3 The scattering length (a) and effective range (r) for pvCDBonn A, B, and C potential and the empirical data from experiments. Their units are fm. at (np) and rt (np) correspond the scattering length and effective range from spin triplet channel at 3 S1 state

a(nn)

A B C −18.936 −18.931 −18.917

CD-Bonn (Machleidt 2001) −18.968

r(nn) a(np)

2.778 2.770 2.765 −23.711 −23.757 −23.752

2.819 −23.738

r(np)

2.649

2.649

2.642

2.671

at (np)

5.429

5.418

5.420

5.420

rt (np)

1.773

1.757

1.761

1.751

Emp. −18.95(40) (Howell et al. 1998; González Trotter et al. 1999) 2.75(11) (Miller et al. 1990) −23.749(20) (Houk 1971; Machleidt 2001) 2.77(5) (Houk 1971; Machleidt 2001) 5.419(7) (Houk 1971; Machleidt 2001) 1.753(8) (Houk 1971; Machleidt 2001)

Table 4 The properties of deuteron from pvCDBonn potentials, original CD-Bonn potential, and experiment data

AS [fm−1 ] η

CD-Bonn (Machleidt 2001) A B C 0.8818 0.8828 0.8856 0.8846 0.0237 0.0246 0.0250 0.0256

rd [fm] Qd [fm2 ] PD [%]

1.970 0.258 4.279

1.967 0.269 5.493

1.969 0.275 6.152

1.966 0.270 4.850

Exp. 0.8846(9) (Kermode et al. 1983) 0.0256(4) (Rodning and Knutson 1990) 1.97507(78) (Martorell et al. 1995) 0.2859(3) (Bishop and Cheung 1979)

the quadrupole moment (Qd ), and the D-state probability (PD ) from pvCDBonn potentials, original CD-Bonn potential, and experiment data are given in Table 4. They are consistent with the experiment data. The D-state probability represents the strength of the tensor component in an NN potential. Therefore, there is the strongest tensor component in the pvCDBonn C potential.

The Nuclear Matter in BHF and RBHF Models The infinite nuclear matter is an idea object consisting of protons and neutrons only with the NN interaction. To avoid the divergence of energy, the Coulomb interaction between protons is neglected. For symmetric nuclear matter with the same number of protons and neutrons, its saturation properties, such as saturation density and binding energy, can be extracted from the center region of heavy nuclei. The properties of nuclear matter, which is not only dependent on the density but also on the fractions of proton and neutron, especially in the high-density region, are very important for the compact star and supernova explosion. There are several important properties to describe nuclear matter, i.e., binding energy per nucleon

62 Pion Exchange Interaction in Bonn Potential and Relativistic and. . .

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(E/A), nuclear saturation density (n0 ), the symmetry energy (Esym ), the slope of symmetry energy at n0 (L), the incompressibility (K), and the effective nucleon matter (MN∗ ). The nuclear matter can be investigated both by the density functional theories (Bender and Heenen 2003; Stone et al. 2007) and the ab initio methods (Li et al. 2006; Dickhoff and Barbieri 2004; Hagen et al. 2014b; Carbone et al. 2013; Drischler et al. 2014; Hu et al. 2017). However, there are large uncertainties in the high-density region for binding energy per nucleon from density functional theories (Li et al. 2019). As a result, it is more reasonable to treat the nuclear matter with ab initio methods. In this section, the properties of nuclear matter will be studied in BHF and RBHF models. In the non-relativistic framework, the nucleon moving in the nuclear matter is given by a plane wave function, ψ ∼ exp(−ip · x). In the relativistic framework, the nucleon is described by the Dirac equation (p / − M − Σ)u(p, λ) = 0,

(12)

Σ = Σs − γ 0 Σ0 + γ · pΣv

(13)

where

is the self-energy of the nucleon in the nuclear medium. It has three components, scalar one Σs , time component, Σ0 , and Σv , which are density-, momentum-, and isospin-dependent. The effective mass and energy are defined as M∗ =

M + Σs , 1 + Σv

E∗ =

E + Σ0 . 1 + Σv

(14)

Therefore, the Dirac equation about the nucleon in nuclear medium has a solution  E (p) = p 2 + M ∗2 , ∗

 u(p, λ) =

E∗ + M ∗ 2M ∗



1

σ ·p M ∗ +E ∗

|λ.

(15)

The normalization of the NN potential in Brueckner theory is based on the Bethe-Brueckner-Goldstone (BBG) equation, which is reduced from the BetheSalpeter equation in the nuclear medium from four dimension to three dimension with Blankenbecler-Sugar choice: Gτ τ  (q  , q|x) = Vτ τ  (q  , q) + ×



Mτ∗ Mτ∗ d 3k   (q , k) V τ τ Eτ∗ (k)Eτ∗ (k) (2π )3

Qτ τ  (k, x) 2Wk Gτ τ  (k, q|x). W0 + Wk W0 − Wk + iε 

Here, Gτ τ  is the effective NN potential in nuclear medium x = {W0 , P , u, kFτ , kFτ }. τ denotes the isospin degree of freedom of nucleon. P = pτ + p τ  is the total

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J. Hu and C. Wang

Fig. 4 The G matrix elements of pvCDBonn A, B, and C potentials at 3 S1 channel with BBG equation and compared to the realistic ones, where n = 0.16 fm−3 and the starting energy is W0 = 6.7 fm−1 . The unit of this figure is the same as Fig. 3

momentum, and u = P /(Eτ∗ + Eτ∗ ) is the relative velocity of the c.m. frame to the rest frame of nuclear matter. V is the realistic NN potential. Q is the Pauli operator, which prevents the states scattering from above Fermi momentum states to the occupied states. The matrix elements of G-matrix from pvCDBonn A, B, and C potentials at 3 S channel are plotted and compared to the realistic ones in Fig. 4. With the 1 normalization of the BBG equation, the pvCDBonn potentials are much softer. The very strong repulsion of pvCDBonn C potential is completely removed at the lowmoment region and transforms into an attractive contribution. In the high-density region, the repulsive components are also weakening. With the G-matrix, the binding energy per nucleon in nuclear matter, EB /A, can be obtained based on the Hartree-Fock approximation as EB (n, α) = Ekin + Epot , A

τ 1  kF d 3 p Mτ∗ u(p, ¯ λ)|p / + Mτ |u(p, λ) − M, Ekin = n (2π )3 Eτ∗ (p) τ,λ

Epot

1  = 2n   τ,τ λ,λ



p

even>

12 16

A

28 32

40

Fig. 2 Spin and isospin expectation values of mirror pairs. Values obtained from the pair of mirror nuclei are shown. The single particle values are shown by the horizontal lines for corresponding orbitals. Dashed lines are just for guiding eyes

66 Effects of Tensor Interactions in Nuclei

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Table 2 Comparison of the experimental data of isoscalar spin expectation value to theoretical values. The second-order configuration mixing due to the tensor interaction is made by HamadaJonson Potential and shown in the last column A 15 17 39 41

Orbital p1/2 d5/2 d3/2 f7/2

δ < Σσ > exp / < Σσ > % −49.4 −13.6 −61.9 −11.9

δ < Σσ > / < Σσ > (H.J.)% −63 −18 −55 −19

j = l + 1/2 and j = l − 1/2 orbitals. It also explains that most of the expectation values sit in between j = l + 1/2 and j = l − 1/2 single-particle values. We will not discuss further on this subject here because it is not directly related to the tensor interactions. Interested readers may refer to the section on shell models or in ref. (Arima and Horie 1955). Those expectation values should be equal to the single-particle values for the mirror pairs of “doubly closed shell ± 1” nuclei because the first-order configuration mixing does not affect the expectation values. In fact, the values are very close to the single-particle values indicating that they are good single-particle states. However, if you see the details, the experimental values for such nuclei deviate from the single-particle values visibly as shown in Table 2 for pairs of A = 15, 17, 39, and 41. It is known that the isovector spin expectation values are affected by many other effects such as meson exchange current,  existence in nuclei and more. We, therefore, restrict our discussion on the isoscalar expectation values to the followings. The important facts are: 1. If a nucleon-nucleon interaction is central, the spin is a constant of motion. 2. The expectation values of spin in nuclei and LS-closed shell±one nucleon are then the same as their single-particle values, irrespective of the degree of configuration mixing. 3. In order to modify the values, non-central interaction has to be introduced. 4. Pion exchange potential of nuclear forces includes a strong tensor force. The tensor forces contribute a second-order mixing of wave functions that is exactly the same mechanism of D-wave mixing in deuteron. Theoretical calculations including the tensor forces have been made for the second-order effects including high-excitation energy (Arima et al. 1985). The examples of the results by H. J. potential are shown in Table 1. They could reproduce the experimental values with several interactions such as Hamada-Jonson (H. J.) and Paris potentials. The convergence of the calculation is slow so that highly excited state has to be introduced. Reasonable agreements were seen in all the cases when the mixing of up to 12ω excitations was included in the calculations. The theoretical conclusions for the spin expectation values are summarized as follows:

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1. First-order configuration mixing is very important. 2. The -hole excitation does not contribute to first-order perturbation to the isoscalar magnetic moment. 3. The contribution from meson exchange currents and second-order -hole excitation is negligibly small. 4. Thus, the difference of the from the single-particle value provides the best evidence for the second-order configuration mixing mainly due to the tensor force. 5. The slow convergence of the result to 12ω indicates that high excited states or high momentum nucleons have large effects on the ground state magnetic moments.

Spin-Orbit Coupling It is an important question to ask how spin-orbit (l•s) interactions emerge in one particle mean field potential in nuclei. The Thomas term due to the relativistic effect, known well in atomic physics, is found to be small and even an opposite sign from the observed splittings (Inglis 1936). The two-body l•s forces studied by nucleonnucleon scatterings are known to be too small to explain the l•s strength in nuclei. The possibility of tensor force as an origin of l•s splitting was studied under the same consideration as magnetic moments calculations. Second-order configuration mixing effects due to the tensor interactions are studied for light nuclei (Terasawa 1960; Arima and Terasawa 1960). Basic reason for the l•s splitting is the blocking of the 2p-2 h excitation by the tensor interaction. Figure 3 shows the simplest case for the 5 He nucleus where two particles are excited from the lowest configuration. The lowest configuration is described as four nucleons in the 0s orbital and one neutron in the p shell. The spin of this configuration can be J = 1/2 or 3/2. In the figure, (a) presents excitations of two nucleons in the core to upper orbitals other than the porbital. (b) is the excitation of one nucleon from the core together with the original p-wave neutron. (c) is the excitation of two core nucleons but one excitation goes to the p-orbital where already one neutron exists. In (d) both of the core nucleons excite to p-orbital. Pauli principle is important for (c) and (d) cases. The blocking

Fig. 3 Configurations of 5 He with 2p-2 h excitations

66 Effects of Tensor Interactions in Nuclei

2445

configurations are different for J = 1/2 and 3/2. For example, an excitation of proton and neutron pair in the core with S = 1 and T = 0 can be made freely for the J = 3/2 state but some restrictions occur due to the neutron occupation for the J = 1/2 state. It is due to that fact that the occupation of neutron in p1/2 orbital blocks the excitation from s1/2 orbital to p1/2 orbital. The calculations have been done with meson exchange interactions and also for Serber forces. It is found that the l•s splitting is in the same direction as the observed splitting but about half of the experimental value. As a conclusion, the tensor interactions are definitely one of the origins of the l•s splitting in nuclei. This study points out that the Pauli blocking of the configurations excited by the tensor interactions plays an important role in binding energy. One of the clear situations is shown in the following as the parity mixing of the neutron halo wave function.

s1/2 and p1/2 Mixing in a Halo Nucleus 11 Li

Fig. 4 Transverse momentum (px ) distribution of 10 Li, reconstructed from the momenta of 9 Li and the neutron measured in coincidence. Dashed and dotted lines represent (1s1/2 )2 and (0p1/2 )2 single-particle momentum distributions calculated for one of the halo neutrons in 11 Li, including core shadowing and the experimental resolution. The solid line represents the best fit to the data obtained with a 45% (1s1/2 )2 contribution (Simon et al. 1999)

ds / dpx (mb/(MeV/c))

As neutron number 8 is a magic number, then 11 Li nuclei is expected to have (1p1/2 ) closed shell. However, the detailed studies of nuclei show an almost equal mixing of (2s1/2 )2 and (1p1/2 )2 wave in the two-neutron halo of 11 Li. The existence of an s wave enhances the effect of neutron halo and is very important for understanding the halo (see Part I section 5 “Halo and Unstable Nuclei” for nuclear halos). Why 2s1/2 wave contribute strongly to this nucleus? Such evidences have been seen in several different observations. Figure 4 presents the transverse momentum distribution of 10 Li fragments obtained by summing the final state 9 Li momentum (p9Li ) and the forward neutron momentum

1 1p1/2

0.1

-200

2s1/2

-100

0

px (MeV/c)

100

200

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(pn ) measured by Simons et al. (Simon et al. 1999). The figure is made under the assumption that the momentum distribution of the 10 Li fragments is produced by one neutron removal from 11 Li. They concluded that the neutron wave function of ground state 11 Li includes two components with different parity single-particle wave function as,  

11

      2  2  + 0+ 1p1/2 , Li =  9 Li • 0+ 2s1/2

(9)

where two neutrons in s and p orbitals are mixed with almost equal amplitudes. Although other components such as (1p3/2 )2 may be included, its contribution was suggested to be small from the analysis of the angular correlations shown below. The mixing of different parity single-particle states is also observed in the angular correlation of the fragments. Figure 5 shows the distribution of the angles (θ nf ) between the direction of the 10 Li momentum and the direction of the relative coordinate of 9 Li and the neutron. The forward-backward asymmetry, the inclusion of odd term in cos(θ nf ), clearly indicates the mixing of different parity states. The observed strong asymmetry in the angular correlation gives a direct and model independent indication of a mixing of states with different parities (1p1/2 and 2s1/2 ) in the ground state of 11 Li. The contribution of the (2s1/2 )2 state in the 11 Li groundstate wave function is determined to be (45 ± 10)%. The mixing of s- and p-waves was also observed in the two-neutron transfer reaction at low energy. The experiment was done at the ISAC-2 facility at TRIUMF that provides a high-intensity 11 Li beam up to 55 MeV (Tanihata et al. 2008). The angular distribution of 11 Li(p, t)9 Li reactions determined for 36.9 MeV incident 11 Li is shown in Fig. 6. Theoretical calculations were made including one-step and twostep sequential transfer processes via 10 Li as shown in the figure. The wave function

9

Li

1000 ds / d cos(qnf)

Fig. 5 Distribution of the decay neutrons from 10 Li formed in 11 Li neutron knockout reactions. The inset shows a schematic diagram of the reaction where θ nf is the angle between the momentum direction of 10 Li and the direction of the n +9 Li relative momentum pnf . The distribution asymmetry can be explained only if one assumes contributions from interfering s and p states in 10 Li (Simon et al. 1999)

qnf n2

q

800 n1

600 400

-0.5

cos(qnf)

0.5

66 Effects of Tensor Interactions in Nuclei

2447

2

(p1/2) P0 model P2 model P3 model

Fig. 6 Differential cross sections of 11 Li(p, t)9 Li reaction to the ground state of 9 Li. Theoretical predictions were shown by curves. See text for the difference of the wave function

(p1/2 )2 with no n-n correlation gives very small cross sections (dashed curve) that are far from the measured values. Also the P0 wave function, with n-n correlation but with a small (s1/2 )2 mixing amplitude, gives too small cross sections. The results of the P2 (31% s1/2 ) and P3 (45% s1/2 ) wave functions, that include the correlation and large mixing of (s1/2 )2 , fit the forward angle data reasonably well but the fitting near the minimum of the cross section is unsatisfactory. Although the calculations do not give perfect agreement to the entire angular distribution data, the comparison of the magnitude of the data and the theory indicates the importance of the mixing of (p1/2 )2 and (s1/2 )2 waves in addition to the strong correlation between two halo neutrons and the core. Such a mixing of s and p waves in 11 Li halo has been observed also in the betadecay measurement and gave 30–50% contribution from s1/2 , which is consistent with the reaction measurement shown above (Borge et al. 1977, 1997). Although an equal mixing of 2s1/2 and 1p1/2 waves is observed in experiments, it was not explained by the theoretical models for a while. A model that accounts for the tensor forces shows an important possibility. The Tensor-Optimized Shell Model (TOSM). (Myo et al. 2007) explicitly includes the second-order mixing due to the tensor interactions in 2p-2 h shell space. It is similar to the calculations made by Arima et al. shown above in the discussion of magnetic moments. However, the

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10

10

AV8’

5/2  7/2 

Ex [MeV]

8 6 T-1

4 2

1+ 2+

T-1

1/2 

2+ 3+ 0+

3/2 

1+

Expt. Theor.

0 5

Li

6

Li

8

1/2  3/2  5/2  5/2 

1+

6

4+ 2+ 1+ 2+ 1+ 0+ 3+

7/2 

7/2  3/2  5/2 

2

1+ 1/2  3/2 

7

Li

1/2  3/2 

2+

8

Li

4

9

0

Li

Fig. 7 Results of TOSM calculation for Li isotopes. Av8 potentials are used for this calculations

size parameter of each orbital is treated as variational parameters to obtain a fast convergence of the energy and can include high-momentum nucleons quickly. This model reproduces the binding of 4 He and also for Li isotopes very well as shown in Fig. 7. The excitation spectra of Li isotopes are well reproduced. Based on this model the mixing of s- and p-waves are successfully explained as a Pauli blocking effect by the tensor interactions and pairing interactions. Under this model, as shown in Fig. 8, the ground state of 9 Li includes configurations denoted as “Pairing” and “Tensor” in addition to 0p-0 h configurations. “Pairing” configuration has a large two-neutron amplitude in 1p1/2 orbital. “Tensor” configuration has a proton and a neutron in 1p1/2 orbital excited from 1s1/2 orbital. It is due to the selection rule of the tensor interaction,L = 2, S = 2. For forming 11 Li two possibilities can be considered, one is to put two neutrons in the 1p1/2 orbital (1p)2 and the other is to put them in the 2s1/2 orbital (2s)2 . It is not possible to put two neutrons in the 1p1/2 orbital for “Pairing” nor “Tensor” configurations because those orbitals are already occupied. It means that the energy gain from paring and tensor interactions existing in 9 Li is lost in this configuration. On the other hand, two neutrons can be in 2s1/2 orbitals without disturbing “Pairing” and “Tensor” configurations and thus gaining energy through those interactions. It should be noted that it is not necessary that the single-particle orbitals 2s1/2 and 1p1/2 are very close to each other. Although the 2s1/2 orbital is still higher than the 1p1/2 orbital, the (1p)2 and (2s)2 configurations are very close in energy and thus the mixing occurs. The result of the TOSM calculations shows a good agreement with experimental values of the mixing amplitude P(s2 ) and matter radius. Other observables such as nuclear radii and magnetic moments were also explained well.

66 Effects of Tensor Interactions in Nuclei

2449

2s1/2 9Li

(GS)

11Li

(1p)2

11Li

(2s)2

1p1/2

1p3/2 1s1/2

2s1/2

1p1/2

1p3/2 1s1/2 2s1/2

1p1/2

1p3/2 1s1/2

Fig. 8 Schematic illustration for the Pauli blocking in 11 Li

This mechanism of configuration mixing in particular the 2p-2h excitation depends on the available configurations in the region of a nucleus. Therefore, a sudden change of nuclear character may appear at certain proton or neutron numbers. Further studies are necessary to see such effects on heavier halo nuclei. The blocking of tensor configuration is sensitive to the symmetry of the occupied orbitals of proton and neutron. Therefore, it may drive exotic phenomena when nuclei move from near the stability line to the drip lines.

High-Momentum Component in Wave Function Theoretical Considerations As already mentioned above, high-momentum components of nucleons in nuclei play an important role to gain the binding energy through tensor interactions. How do the tensor interactions affect the momentum distribution of nucleons in nuclei? Figure 9 explains the relation between the momentum distribution and physics origins. The most important origin of the momentum distribution is the movement of nucleons in a nuclear mean field potential. Although the detailed momentum distributions depend on an orbital of nucleons, the main feature of the momentum

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(p) (log) r

1

2

3

4

Momentum [fm-1]

Fig. 9 Origin of the high momentum components of nucleons inside nuclei

distribution is determined by the size of the potential as typically expressed as Fermi momentum. It thus mainly has momentum below 1 fm−1 as shown by the red line in Fig. 9. The momentum distributions are also affected by the nucleonnucleon correlations in particular at short distances. One of the well-known origins is the short-range repulsion of the central forces. The characteristic momentum is determined by the range and the sharpness of the rise of the short-range repulsion. The rough sketch of the momentum distribution by this origin is shown by the green curve in the figure. The tensor forces also give a characteristic range as shown in the N-N interaction potential (top-left figure in Fig. 9) and make a large contribution at momentum around 2 fm−1 as shown in the figure by the blue curve. Predictions of the momentum distributions with tensor interactions have been made by several theoretical models, GFMC (Schiavilla et al. 2007), UCOM (Neff and Feldmeier 2003) as shown in Fig. 10. Both of them show the enhancement of the momentum density at around 2 fm−1 due to the tensor force. In GFMC calculations, the momentum density is enhanced by 2–3 orders of magnitude when the AV18 potential is used. In UCOM, the range parameter of the tensor force is not uniquely determined, and thus several range parameters were assumed as shown by α, β, and γ. The amount of enhancement strongly depends on the parameter but again the enhancement is seen at ∼2 fm−1 .

Electron Scattering Experiments High-energy electron scattering provides a mean to study quasi-free knock-out processes of proton from nuclei. With (e, e p) experiments single-particle shellmodel spectroscopic factors of a nucleon in nuclei were studied in detail and found that the spectroscopic factors are always smaller than the value expected by

66 Effects of Tensor Interactions in Nuclei

5

2451

1)

10

AV6´

10

AV4´ ρNN(q,Q=0) (fm˚)

4

He

AV18/UIX

4

3

10

AV18

4

He

4

He

2

H

2

10

1

10

0

10

-1

10

0

1

2

3

4

5

q (fm-1)

1.

AV18 16O

2) ρn(k) / A [fm3]

0.1 0.01

g b a

0.001 0.0001

radial

0.00001

VMC LDA 0

1

2

k [fm−1]

3

4

Fig. 10 Momentum distributions of nucleons inside nuclei, by (1) GFMC, (2) UCOM, and (3) three-body multi-cluster model. All of them present the enhancement of a magnitude of momentum distribution at around 2 fm−1

the shell model. It indicates that some part of nucleons is in configurations that cannot be described in the shell model (See Fig. 11.). Those could be due to the correlations in nuclei such as collective effects (long-range correlations) and shortrange correlations. The observation of protons emitted leaving the recoil nucleus to the highly excited states is observed in large cross sections. The momentum distribution of proton in light nuclei was studied also. In plane-wave impulse approximation (PWIA) the cross section of (e, e p) is written as   dσ = Kσep S Emiss , p miss , dvd e dE miss d p

(10)

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Fig. 11 The fractional spectroscopic factors (the ratio of measured cross sections to those calculated with the independent particle shell model) for valence nucleon knockout (e, e p). (Adapted from Lapikas et al. 1993)

where K = Ep pp /(2π )3 is a kinematical factor, Ep and pp are the energy and momentum of the outgoing proton, σ ep is the electron cross section for scattering by a bound proton, and S is the spectral function, the probability of finding a nucleon in the nucleus with momentum pmiss and separation energy Emiss . p miss = q − pp ,

(11)

Emiss = v − Tp − TA−1 ,

(12)

where Tp and TA − 1 are the kinetic energies of the detected proton and residual A-1 nucleus. The kinematics is shown in Fig. 12. (Hen et al. 2017) The equation presents how to obtain the momentum distribution of a proton inside a nucleus from cross section measurement. It should be noted that a distorted wave consideration is necessary for more precise information. The experimental data and theoretical results are shown in Fig. 13. It is seen that the observed momentum distributions increase from smooth drop-off at above 2 fm−1 suggesting a high momentum tail due to the short-range correlations. However, it was not possible to discuss the origin of the high momentum enhancement from what type of short-range correlations are making it.

66 Effects of Tensor Interactions in Nuclei

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Fig. 12 The (e, e p) reaction in the plane wave impulse approximation. A nucleus of fourmomentum PA emits a nucleon of four-momentum Pmiss that absorbs a virtual photon of four-momentum q to make a nucleon of four-momentum Pmiss + q with (Pmiss + q)2 = M2 , where M is the nucleon mass. The blob represents the in-medium electromagnetic form factors

Fig. 13 The nucleon momentum distributions n0 (k) (dashed lines) and n(k) (solid lines) plotted vs momentum in fm−1 for the deuteron, 4 He,12 C, and 56 Fe (Ciofi degli Atti and Simula 1996)

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Fig. 14 Ratio of the cross sections are plotted against the missing momentum. All data show that the correlated two nucleons are mostly pn pairs

Quasi-free knock-out of two nucleons was studied by (e,e pN) reaction. (Hen et al. 2017) The idea is again under an impulse assumption of the processes. Let us consider the correlated pair with small total momentum. When a proton in the correlated pair is knocked out, the remaining nucleon in the pair also has a large momentum. The advantage of this method is that pp and pn pairs can be   distinguished by the final state. The various ratios among 12 C(e, e p),12 C(e, e pn),  12 and C(e, e pp) are presented in Fig. 14. It can be seen that the observed correlated pair are mostly pn pair, and pp pair is only a several percent among the pair emissions. Assuming the isospin symmetry of proton and neutron, the contribution from T = 1 pn pair is considered to be as small as that of pp pairs. Therefore, it is a good indication of the pn pair originated by the tensor interactions. It is considered so because the correlation should be same for pp, pn, and nn pairs if the short-range correlations occur due to the repulsive core of the central interactions. Then the large amount of the pn cross section is from T = 0 pair.

With Strong Interacting Probes To observe a momentum distribution of nucleons in the nucleus, quasi-free knockout reactions (p,2p) measurement may be considered. Although it is a powerful tool to see the distribution at small momenta, they are not the best tool to see

66 Effects of Tensor Interactions in Nuclei

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A momentum distribution of ejected protons

102

6

100

6

-2

-1 -2

ρ(k) [fm3]

10

0

Li d

10-4 10-6 10-8 -10

10 10

-3

He

-12

0

2

4

6

8

10

k [fm-1]

-4

Fig. 15 The observation of momentum distribution by (p,2p), reactions. A peak of observed momentum distribution is from 0 momentum nucleons

the high-momentum component that is orders of magnitudes smaller in amplitude. The reason for this is briefly explained in Fig. 15. When knocked-out protons are observed, they form a strong peak at two-body kinematics between the incident particle and the kicked-out proton that had an internal momentum near 0, where the momentum density is maximum. The high momentum part of the interest has a density that is 4 orders of magnitudes smaller. Therefore, any complications apart from the pure quasi-free process, such as final state interactions and multiple scatterings, make it difficult to observe the original momentum amplitude if they affect more than 10−4 in the spectrum. It is extremely difficult, if not impossible, to determine such effects with accuracy of 10−4 . A possible method that is insensitive to the low momentum component is a nucleon pick-up reaction such as (p, d) or (p, dp) reactions. The reason is similar to the backward scattering of pd elastic scattering. In pd elastic scattering, a strong increase in the differential cross section is observed at backward angles (see Fig. 16). This increase is understood as an exchange reaction of neutrons or pick up of neutron from the target deuteron. A high-momentum neutron in a target deuteron is picked up by an incident proton to form a deuteron in the final state. The transferred momentum PF is exactly the internal momentum of the neutron in the initial state. The cross section σ F under the Born approximation is written as:  2  2 2 Pd  2  σF = K N (PF ) BD + (p − Pd /2)  ϕ(r), ei(p−Pd •r/2)  , p M

(13)

where K is a phase space constant, BD is the deuteron binding energy, and M is the nucleon mass. The deuteron wave function is expressed by φ(r). The last term of

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(P)

Pd – Pp = PF (Pd)

p

d

PF

d (Pdinc=0)

p -PF

(-PF)

d in forward

Fig. 16 A pd elastic scattering cross section at 135 MeV. The backward increase is due to the neutron pick up reaction (Sekiguchi et al. 2004)

the equation indicates that the cross section is proportional to the amplitude of the nucleon with an internal momentum exactly the same as the transferred momentum (Chew and Goldberger 1950). A (p, d) reaction at a forward angle (the deuteron at small scattering angles) occurs with exactly the same mechanism. The incident proton picks up a neutron in the target with internal momentum exactly the same as the transferred momentum. The cross section is proportional to the amplitude of the momentum distribution with that momentum. Therefore, it is most suitable to see the momentum distributions by changing the transferred momentum. It is desirable to do the experiment at 0 degree scattering angle so that a complication due to reaction mechanisms is minimum. Another possibility is quasi-free (p, pd) and (p, nd) reaction at forward deuteron emission. Because a high-momentum neutron is expected to be correlated with another nucleon, this nucleon may be emitted to the backward direction just like the proton emitted in the backward angle in a pd elastic scattering. This proton carries also the momentum as an internal momentum. The (p, pd) reaction picks up a neutron from a correlated pn pair and thus sensitive to the tensor interactions. On the other hand, the (p, nd) reaction picks up a neutron from a correlated pair

66 Effects of Tensor Interactions in Nuclei

2457

Fig. 17 (p,pd) scattering in quasi-elastic domain

of two neutrons and is sensitive only to the short-range correlation due to the central repulsive core. Therefore, the comparison of those reactions provides a unique information on the effect of tensor interactions. In addition, the ratio of those cross sections is rather insensitive to the final state interactions because those two reactions occur almost under identical kinematics. 16 O(p, pd) reaction studies provide the best indication of the tensor interactions at high momentum. For a correlated pair of proton and neutron (p-n pair) the isoscalar (S = 1, T = 0) or isovector (S = 0, T = 1) pair may exist in a nucleus. The tensor interactions work only on an isoscalar pair so that the amplitude of highmomentum component should be larger for isoscalar pairs if tensor interactions contribute. It can be observed if one compares the cross section between (p,pd) and (p,nd) scattering. However, (p,pd) reaction solely can also see the difference between isoscalar pairs and isovector pairs. The principle is shown in Fig. 17. Let us assume that the target nucleus has (Jπ = 0+ , T = 0) and a correlated pair is separated from other nucleons (quasi-free assumption). After picking up the highmomentum neutron in a correlated p-n pair, the proton in the pair is emitted out from the target nucleus because the momentum of the proton is large and much different from other nucleons. In quasi-free condition the remaining final state nucleus thus has the same spin and isospin as the escaped p-n pair, namely J = 1+ , T = 0 state if the pair was (S = 1, T = 0) and J = 1+ , T = 1 if the pair was (S = 0, T = 1). Ratios of cross sections to those two states, with different momentum transfer, present a change of the momentum components in isoscalar and isovector p-n pairs. Experimental data are available for a low energy beam (75 MeV) and higher energy (392 MeV) to see the ratio for different momentum transfers. Figure 18 shows those data for 16 O(p, pd)14 N reaction. A state of J = 1+ , T = 0 is seen at 3.95 MeV and J = 0+ , T = 1 is seen at 2.31 MeV. Although the ground state of 14 N has J = 1+ , T = 0, the reaction to this final state is not discussed because this transition is known to include l = 2 component and thus theoretical discussion is complicated. Transitions to 3.95 and 2.31 transition are known to be l = 0 and

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Counts (/250keV) 2000

10 3 10 2

1500 10

1

0

-4

4

8

12

16

1000

500

-4

0

4

8

12 16 14 Ex( N) (MeV)

Fig. 18 Top figure: the excitation spectrum of 16 (Op, pd)14 N reaction at 75 MeV. The peak corresponding the 2.31 excited state is clearly seen. Bottom figure: the excitation spectrum at higher energy (394 MeV). The peak for the 2.31 MeV is very weak compared with the peak for 2.95 MeV (Terashima et al. 2018)

66 Effects of Tensor Interactions in Nuclei

2459

distorted wave impulse approximation (DWIA) is expected to be appropriate. So, we compare the reactions to those two states. The transition to the J = 0+ , T = 1 state is clearly  seen in 75 MeV data and the relative strength R0+ /1+ ≡ σ J = 0+ , T = 1 /σ J = 1+ , T = 0 is about 0.36 ± 0.05. On the other hand, the J = 0+ , T = 1 state is relatively much weaker in high energy data R0+ /1+ = 0.05 ± 0.03. It shows that the (S = 1, T = 0) pairs become dominated at high energy or large momentum transfer (∼1.5fm−1 ). A DWIA calculation predicts R0+ /1+ to be 0.18 at the high energy. The calculation uses the deuteron wave function so that the effect of tensor interactions is included and expected to be compared directly with the data if (S = 1, T = 0) pairs have the same property as deuteron. Qualitatively the decrease of R0+ /1+ is reproduced by the calculation but there still remains a difference in the value of R0+ /1+ . Further studies are awaited. It should be noted that the PWIA (Plain Wave Impulse Approximation) analysis gives the same value as that from DWIA calculations. It clearly shows that R0+ /1+ is not sensitive to the details of the reaction mechanisms. It thus shows that tensor interactions generate high momentum nucleons in nuclei through (S = 1, T = 0) correlated pair. Neutron pick-up reaction (p,d) also provides similar sensitivity to highmomentum nucleons. This reaction studies the comparison of the parity of the final state in 15 O is the key. The simple reason is illustrated in Fig. 19. Oxygen-16

Fig. 19 Neutron pick up reactions from 16 O and its final states. Positive parity states is excited strongly when two-particle two hole states by tensor interactions are mixed in the ground state configurations

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is a double closed shell nucleus and has doubly closed shell configuration as the main component. The selection rules of tensor interaction for exciting two nucleons are S = 2, L = 2, and J = 0. A typical correlated pair excitation in shell model picture is shown in the left bottom panel of Fig. 19. A p-n pair is excited to the next measure shell orbitals which have different parity. Picking up a neutron from normal shell model configuration makes a hole state in the p-shell and thus produces negative parity states. In contrast a neutron picked up from a correlated pair state produces positive parity states. The difference in the behavior of the cross section for positive and negative parity final states will provide an information on the effect of tensor interaction. Such a data is shown in Fig. 20 for different incident energies that changes the transferred momentum. Let us put an attention to the thee lowest excited states. The first peak is the transition to the ground state (1/2− ) of 15 O and the third peak is that to the 6.18 MeV excited state (3/2− ). Both are transitioning to the negative parity states. The second peak in contrast is positive parity states at 5.2 MeV excitation. Two states 1/2+ and 5/2+ are known at this energy but they are not separated. But both are positive parity states. It is seen that the ratio of the two negative parity states does not change much from low energy to higher energies. In contrast positive parity state becomes stronger for higher energies. It is even stronger than the transition to the ground state at 392 MeV incident energy for that momentum transfer of the neutron is near 2 fm−1 . A more than an order of magnitude increase of the cross section suggests a good indication of the tensor correlation. The details analysis shows, although no realistic wave function of 16 O wave function including tensor correlation is available, that this increase is consistent with the effect of high-momentum nucleon due to the tensor interactions calculated for 4 He nucleus. Another important fact is that many strong peaks, comparable to the ground state, are observed at high energy data. Since tensor interactions excite correlated pairs to higher shell orbitals, remember the discussion on the magnetic moment where 12ω excitation has to be included, it is natural to observe neutron pick-up transitions from such pairs which become stronger for a larger momentum transfer. Many strong peaks at higher excitation energies suggest such highly excited correlated pairs. For these considerations one important fact has to be noted. There are no reliable reaction theories for pick-up reactions at high energy. Developments of reaction theory are necessary for quantitative discussion at energies from several hundred MeV to a few GeV incident energy reaction. However, the above discussion, as long as quantitative discussion is concerned, is valid because only the ratio of the cross sections at almost same kinematics is used.

Summary It is shown that the tensor interactions affect the structure of nuclei. One important fact is that the two-nucleon correlations produced by the tensor interaction produce high-momentum nucleons in the ground state of a nucleus. Those part of the wave function does not belong to the usual so-called shell model space that is covered by nucleons with smaller momenta less than the Fermi momentum. It is

66 Effects of Tensor Interactions in Nuclei

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Fig. 20 Typical excitation energy spectra for the 16 O(p, d)15 O reactions obtained at proton energies (a) 392 MeV and (b) 198 MeV. The deuterons were detected at 10◦ with respect to the incident beam. The level scheme for 15 O is shown at the top for reference. For clarity, only the well-established spin parities are shown. (c) A spectrum for Ep = 45.34 MeV measured at a deuteron-scattering angle of 20. 1◦ was replicated from the figure in Ref. (Ong et al. 2013), and is shown for comparison

usually considered that low-energy characters of nuclei are determined by the lowmomentum space and the high-momentum component does not affect the details of the character except for the normalization of the wave functions. However, as seen in this chapter, the low-energy characters are also strongly affected by the

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high momentum nucleons. First of all, large amounts of binding energies of nuclei are from tensor interaction through the high momentum component. Therefore, the small change in the configurations even at low excitation energies affects the mixing of configurations due to the tensor interactions and changes the binding energy or the arrangement of the single-particle orbitals. It is also found that properties of ground states such as magnetic moments are affected very much by the mixing of highly excited configurations. Systematic change of nuclear structure is expected from near the stability line to the drip lines due to the large change in the ratio of proton and neutron numbers. Tensor interaction affects the nuclear structure through J = 1+ , T = 0 pair of a proton, and a neutron and therefore it would be one of the important ingredients for understanding the systematic change of nuclear structure. It is extremely important to study the effect of tensor interactions through highmomentum nucleons for understanding nuclear structure for nuclei far from the stability line as well as nuclei near the stability line.

References A. Arima, H. Horie, Progr. Theor. Phys. 12, 127 (1955) A. Arima, T. Terasawa, Progr. Theor. Phys. 23(1), 115 (1960) A. Arima, T. Cheon, K. Shimizu, Hyperfine Interact. 21, 79 (1985) M.J. Borge et al., Phys. Rev. C 55, R8 (1977) M.J. Borge et al., Nucl. Phys. A 613, 199 (1997) G.F. Chew, M.L. Goldberger, Phys. Rev. 77, 470 (1950) C. Ciofi degli Atti, S. Simula, Phys. Rev. C 53, 1689 (1996) O. Hen, G.A. Miller, E. Piasetzky, L.B. Weinstein, Rev. Mod. Phys. 89, 045002 (2017) K. Ikeda, T. Myo, K. Kato, H. Toki, Lect. Notes Phys. 818, 165 (2010) D.R. Inglis, Phys. Rev. 50, 783 (1936)., 75 (1949), 1767 L. Lapikas et al., Nucl. Phys. A 553(1993), 297 (1993) T. Myo et al., Progr. Theor. Phys. 117, 257 (2007) T. Neff, H. Feldmeier, Nucl. Phys. A 713, 311 (2003) H.J. Ong et al., Phys. Lett. B 725, 277 (2013) C. Pieper, R.B. Wiringa, Ann. Rev. Nucl. Part. Sci. 51, 53 (2001) R. Schiavilla, R.B. Wiringa, S.C. Pieper, J. Carlson, Phys. Rev. Lett. 98, 132501 (2007) K. Sekiguchi et al., Phys. Rev. Lett. 95, 162301 (2004) H. Simon et al., Phys. Rev. Lett. 83, 83 (1999) I. Tanihata et al., Phys. Rev. Lett. 100, 192502 (2008) T. Terasawa, Progr. in Theor. Phys. 23, 87–105 (1960) S. Terashima et al., Phys. Rev. Lett. 121, 242501 (2018) R.B. Wiringa, S.C. Pieper, Phys. Rev. Lett. 89, 182501 (2002)

Section X Mesonic- and Hypernuclei Emiko Hiyama

What Is Hypernuclear Physics and Why Studying Hypernuclear Physics Is Important

67

Emiko Hiyama and Benjamin F. Gibson

Contents Hyperons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2465 2467 2471 2480

Abstract

Since the initial discovery of hypernuclei in 1952, a number of interesting hypernuclear phenomena have been observed. In this chapter several highlights of hypernuclear physics are introduced: the identification of single-particle energies in the medium mass Λ hypernucleus, 89 Λ Y, the determination that the Y N spin-orbit interaction differs significantly from that of the NN , the important role of ΛN − ΣN coupling, and the observation of double-Λ hypernuclei and Ξ hypernuclei. Finally, a brief introduction is provided to the role of hyperons in the equation of state at high density and their effect in neutron stars.

Hyperons A nucleus is composed of neutrons and protons, members of the baryon family of hadrons. Baryons are characterized by having three valence quarks. There are

E. Hiyama () Department of Physics, Tohoku University, Sendai, Japan e-mail: [email protected] B. F. Gibson Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_29

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six species of quarks: u(up), d(down), s(strange), c(charm), b(bottom), and t(top). For example, a neutron is a hadron having a (ddu) triplet of valence quarks, and a proton is a hadron having a (uud) triplet of valence quarks. If a baryon has one or more of its three valence quarks replaced by one or more strange (s) quarks, that baryon is labeled a hyperon. Because neutrons and protons contain no strange valence quark, they are called nucleons, and one says that they have the Strangeness quantum number 0. Hyperons with one s valence quark (the Λ or Σ) are said to have Strangeness −1; hyperons with two s valence quarks (the Ξ ) have Strangeness −2; finally, a hyperon with three s valence quarks (the Ω) has Strangeness −3. Hadrons with only two valence quarks are known as mesons; mesons with the lightest mass (pions) have no strange valence quarks and, therefore, have Strangeness 0. For reference we list the baryons and mesons in Table 1. We point out that the Λ, Σ, Ξ , and Ω hyperons have lifetimes much shorter than the neutron, which in free space decays into a proton, electron, and antineutrino in somewhat more than 13 min. Many scientists have expended significant effort to study ordinary (non-strange) nuclei, i.e., nuclear phenomena such as size, shape, shell evolution, cluster and halo structure, rotational and vibrational excitations, etc., as is discussed in other chapters in this handbook. Such nuclear structure properties extend to different composite baryonic systems. A hypernucleus is composed of nucleons and one or more hyperons: a Λ hypernucleus is a bound system of nucleons plus a single Λ. A double-Λ hypernucleus is a bound system of nucleons plus two Λs. A Ξ

Table 1 (a) Mesons and (b) baryons. Mass (MeV), lifetime (s), spin, Strangeness (S) value, and isospin are listed (a)Mesons Particle Pion

Kaon η-Meson (b) Baryon Particle Proton(uud) Neutron(ddu) Lambda(uds) Sigma(uus,uds,dds)

Cascade(uss,dss) Omega(sss)

Notation π+ π0 π− K± η0 Notation p n Λ Σ+ Σ0 Σ− Ξ0 Ξ− Ω−

Mass (MeV/c2 ) 140 135 140 493.68 549 Mass (MeV/c2 ) 938.3 939.6 1116 1189 1193 1197 1315 1321 1672

Lifetime (s)

Spin (S)

S

I

2.6 × 10−8 8.7 × 10−17 2.6 × 10−8 1.2 × 10−8 6.0 × 10−19

0 0 0 0 0

0 0 0 ±1 0

1 1 1 1/2 0

Lifetime (s)

Spin (S)

S

I

stable 889 2.6 × 10−10 8.0 × 10−11 6.0 × 10−20 1.5 × 10−10 2.9 × 10−10 1.6 × 10−10 8.2 × 10−11

1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 3/2

0 0 −1 −1 −1 −1 −2 −2 −3

1/2 1/2 0 1 1 1 1/2 1/2 0

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hypernucleus is a bound system of nucleons plus a Ξ . We explore in this chapter what is hypernuclear physics (strangeness physics) and why studying hypernuclear physics is important.

Experimental Overview

fΛ 1

dΛ

-14°

pΛ

sL

σ 2°

Fig. 1 Single-particle Λ energies in 89 Λ Y. (The data is from Hotchi et al. 2001)

(μb/sr/0.25 MeV)

A hypernucleus was initially discovered in 1952 by Danysz and Pniewski (1953) in a balloon-flown emulsion plate experiment. They analyzed the interaction of high-energy cosmic ray protons with nuclei in the emulsion, from which a bound hypernucleus emerged, came to rest, and subsequently decayed into three charged particles. A 3Λ H composed of a deuteron plus a Λ particle had been created. Here, the notation for a Λ hypernucleus is A Λ Z, where A is the number of baryons and Z is the isotope of the nuclear core state in which the Λ is bound. This discovery was followed by an extensive effort through the middle of the 1970s to produce hypernuclei using emulsion techniques. Hypernuclei with masses up to A = 15 (Juric et al. 1973; Davis 2005) were observed, primarily measurements of groundstate binding energies and of branching ratios for different decay channels. To investigate hypernuclei more systematically, the (K − , π − ), (K − , π 0 ), + (π , K + ), and (e, e K + ) reactions were then developed. From the mid-1970s to the mid-1990s, (K − , π − ) and (π + , K + ) reactions were employed to obtain data on excited states of hypernuclei. The (π + , K + ) reaction proved to be an especially powerful tool in the 1980s; such experiments were initiated at Brookhaven National Laboratory (BNL) (Milner et al. 1985; Pile et al. 1991) and afterward were also performed at KEK. Using this reaction, experimentalists succeeded in creating Λ hypernuclei with A = 9 to 89 (Akei et al. 1991; Hasegawa et al. 1995, 1996; Hashimoto et al. 1998; Hotchi et al. 2001). One especially noteworthy measurement using the (π + , K + ) reaction obtained a clear determination of shell-model singleparticle states, 0s to 0f in 89 Λ Y as shown in Fig. 1 (Hotchi et al. 2001).

0

−20

0

-BΛ(MeV)

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Utilizing the (π + , K + ) reaction at KEK in the middle of the 1990s, reaction 208 spectra of 139 Λ La and Λ Pb were obtained (Hasegawa et al. 1995, 1996). Finally, Λ hypernuclei with mass A = 3 to 208 were created, so that the depth of the Λ singleparticle potential UΛ could be estimated to be about −30 MeV. This is only about 3/5 the depth of the attractive nuclear single-particle potential. Contemporary facilities for performing hypernuclear experiments now exist in Germany at Mainz and at GSI, in Switzerland at CERN, in Italy at DAΦNE, in the USA at JLab, and recently the new facility in Japan at J-PARC. These facilities have produced important new data on hypernuclei and hold promise of even more important measurements in the near future. The more exotic double-Λ hypernuclei were first observed in 1963 by Danysz et al., when 10 ΛΛ Be (a 2α +2Λ system) were seen in an emulsion experiment (Danysz et al. 1963). Following that discovery, observations of 6ΛΛ He (Prowse 1966) and 13 B (Aoki et al. 1991) were reported. However, it was difficult to confirm these ΛΛ two double-Λ hypernuclei without ambiguity. In 2001 a surprising new observation of 6ΛΛ He was reported (Takahashi et al. 2001), one which indicated a very different binding energy of 6.91±0.16 MeV with respect to the α+Λ+Λ three-body breakup threshold. From this binding energy, it was understood that the strength of the ΛΛ interaction should be an order-of-magnitude smaller than that of the ΛN interaction (the details can be found in  Chap. 69, “ Hypernuclei”). Since the discovery of hypernuclei, we have been learning invaluable information about many-body systems. A primary goal in nuclear physics is to understand nuclear many-body phenomena using our knowledge of nuclear forces beginning with the NN interaction. In a similar vein, in hypernuclear physics it is important to understand baryon many-body systems in terms of Y N and Y Y interactions. Moreover, a hyperon is free from the Pauli principle that governs the interaction of multiple neutrons and protons in nuclei. Therefore, it is anticipated that interesting phenomena, quite different from those observed in ordinary, non-strange nuclei, should make an appearance. That is, a hyperon can behave as an “impurity” in a hypernucleus. To explore such phenomena, it is important to have “reliable” Y N and Y Y interactions. However, because it is extremely difficult to perform hyperon scattering experiments due to the short lifetime of hyperons, the Y N and Y Y interactions are not well determined in comparison with the NN interaction. Indeed the number of Y N scattering data is about 40, which are only differential cross sections, and there is no Y Y scattering data. (Cf. The number of NN scattering data is about 4000.) Because the scattering data are so sparse, it is difficult to specify the details of the Y N and Y Y interactions. For instance, we had no information on the 1 S0 and 3 S1 partial waves of the ΛN interaction. Historically, we obtained information on these terms by studying the spin and parity of observed few-body hypernuclei from the weak decay process in which Λ → N + π . Consider 3Λ H. As shown in Fig. 2, the binding energy of the ground state of 3Λ H extracted from emulsion experiments is 2.36±0.05 MeV with respect to the n + p + Λ three-body breakup threshold. For the ground state, we need to know the spin and parity. There exist two possible candidate J π s: (i) 1/2+ in which the

67 What Is Hypernuclear Physics and Why Studying Hypernuclear. . . Fig. 2 The observed binding energy of 3Λ H relative to the d + Λ and the n + p + Λ breakup thresholds

2469 n+p+ Λ

0 MeV

d+ Λ

−2.22 MeV

−2.36 n

p Λ

3 Λ

H

spins of the two nucleons and that of the Λ are antiparallel or (ii) 3/2+ in which the spins of the two nucleons and the Λ are aligned. Thus, if the J π = 1/2+ state is the ground state of 3Λ H, then the 1 S0 interaction should be more attractive than the 3 S interaction. If the J π = 3/2+ state is the ground state, then the 3 S interaction 1 1 should be more attractive than the 1 S0 interaction. Experimentally, the weak decay of 3Λ H was observed, namely, 3Λ H →3 He + π − . From the properties of the decay process, it was inferred that the spin and parity of the ground state of 3Λ H is 1/2+ . As a result, it is understood that 1 S0 should be more attractive than 3 S1 . In this way, from the structure of 3Λ H, one was able to extract information about the spin dependence of the s-wave ΛN interaction. Theoretically, using the Y N and NN data together with SU (3) symmetry, the first realistic Y N and Y Y interactions were created, the Nijmegen hard-core A, D, and F potentials (Nagels et al. 1973, 1975, 1977, 1979), which were based on one-boson-exchange interaction models. In these potentials short-range elements were represented phenomenologically by hard cores. After these potentials were published, several other potentials such as Juelich A (JA) and B (JB) (Reuber et al. 1992) as well as the Nijmegen soft-core potential model ‘89 (NSC89) (Maessen et al. 1989) were generated. However, all of these potentials have a large degree of ambiguity. For instance, consider the spin-spin interaction mentioned before. The 4Λ H and 4 He iso-doublet are a system well suited to explore in order to extract information Λ about the spin-spin aspects of the ΛN interaction, because these mirror hypernuclei have both a bound ground state and a bound spin-flip excited state. In Fig. 3 are illustrated the spin-doublet states, the 0+ and the 1+ states. Here, the ground states of the core nuclei, 3 H and 3 He, are 1/2+ states. When a Λ particle is coupled to a core nucleus, due to the spin-spin term of the ΛN interaction, we have resulting 0+ and 1+ states, where the 0+ state is dominated by the spin 0 interaction between the nucleons in the core nucleus and Λ, while the 1+ state is dominated by the spin 1 interaction. From studying 3Λ H, we know that the 1 S0 interaction is more attractive than the 3 S1 interaction. Therefore, the ground states of the A = 4 Λ hypernuclei are 0+ states. In Fig. 3, we show the calculated spin-doublet states, 0+ and 1+ states of 4Λ H and 4Λ He for the potential models JA, JB, ND, NF, and NSC89. Because the 3 S1 interaction in JA, JB, and ND is more attractive than is the 1 S0 interaction, J π = 1+ is the ground state in the A = 4 Λ hypernuclei, which is inconsistent with

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4H ( 4He) Λ Λ

1.0 0.0 +

−1.0

1

−2.0

0

+

−3.0 Exp.

JA

JB

ND

NF

NS

Fig. 3 The calculated spin-doublet states of the A = 4 Λ hypernuclei within the framework of a 3 H(3 He)+Λ two-body model using the JA, JB, ND, NF, and NS potential models. The experimental values are also shown as dashed lines

experiment. On the other hand, for the models in which the 1 S0 interaction is more attractive than the 3 S1 interaction, the 0+ state is the ground state, but the predicted energies of these states are not in good agreement with experiment. That is, none of these potential models reproduces the data due in part to ambiguity in the spin-spin term. Therefore, it is crucial to uncover additional information regarding the Y N and Y Y interactions. However, because it is difficult to perform Y N and Y Y scattering experiments, as an alternative tool it is important to study hypernuclear structure. Historically, there has also been a major effort to extract information about the ΣN interaction. The isospin and spin of the Σ are 1 and 1/2, respectively. Thus, we need information on (I, S) = (1/2, 0), (1/2, 1), (3/2, 0), and (3/2, 1) components of the ΣN interaction. The first report of Σ hypernuclear states with narrow widths came from CERN, where a 9 Be(in-flight K − , π − ) spectrum was observed (Bertini et al. 1980). Following this experiment, there was much discussion leading to a definitive measurement (Bart et al. 1999) showing that there existed no 9 Σ Be bound system, but only a broad quasi-free structure. In the late 1980s, a bound Σ hypernucleus, 4Σ He with I = 1/2, J π = 0+ , was first reported at BNL (Hayano et al. 1989) in a stopped K− experiment. This discovery was confirmed in the late 1990s by an in-flight (K− , π − ) and K− , π + ) measurement (Nagae et al. 1998). From the experimental data, it was understood that the I = 1/2, S = 0 interaction should be attractive, among the four ΣN interaction components. In a further investigation of the ΣN interaction, a search for medium-heavy Σ hypernuclei was performed using a 28 Si target (Noumi et al. 2002). From the resulting data, it was concluded that the Σ-nucleus potential must be repulsive. Details regarding Σ hypernuclei are covered elsewhere in  Chap. 74, “Kaonic Nuclei from the Experimental Viewpoint”. Regarding Ξ hypernuclei, the Ξ N threshold lies above the ΛΛ threshold by about 28 MeV, so that “bound” Ξ hypernuclei are questionable. So far, there has been an intensive effort to search for Ξ hypernuclei experimentally. At BNL experimentalists attempted to produce 11 B + Ξ using the (K − , K + ) reaction on

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Fig. 4 The Kiso event for the 14 N + Ξ hypernucleus

a 12 C target. However, it was difficult to see any peak clearly in the reaction cross section (Khaustov et al. 2000). Therefore, one obtained no information about the Ξ N interaction. Afterward, emulsion experiments at KEK and at J-PARC sought to produce S = −2 hypernuclei such as double-Λ hypernuclei and Ξ hypernuclei. Finally, in 2015 a bound 14 N + Ξ hypernucleus was observed as shown in Fig. 4 (Nakazawa et al. 2015). This was the first observation of a Ξ hypernucleus, the KISO event. From this observation it was inferred that the Ξ -nucleus potential should be attractive. Searches for bound Ξ hypernuclei are ongoing. Detail is provided in  Chaps. 69, “ Hypernuclei” and  70, “Experimental Aspect of S = −2 Hypernuclei” in this section.

Theoretical Overview Theoretically, in the 1970s shell model calculations within the [(0s)4N (0p)A−5 N (0s)Λ ] configuration for Λ hypernuclei (A < 16) were performed (Gal et al. 1975, 1971, 1972, 1998; Dalitz and Gal 1978). Also, cluster model calculations were initiated by Dalitz and Rajasekarn (1964), Bodmer and Ali (1964); Bodmer and Murphy (1964, 1965), and Tang and Herndon (1965, 1966). These calculations were focused entirely on the ground-state properties of Λ hypernuclei. After excited states of Λ hypernuclei were observed using (K − in-flight, π − ), (π + , K + ), and (K − stopped, π − ) reactions, extended shell model calculations were carried out for p-shell Λ hypernuclei (Majling et al. 1980; Revai and Zofka 1981; Majling et al. 1983). One important shell model calculation succeeded in interpreting the low-lying spectrum of 13 Λ C (Auerbach et al. 1981, 1983). In the 1980s, a cluster model calculation for p-shell Λ hypernuclei as an α+x+Λ (x = n, p, d, t,3 He and α) three-body model was developed by Motoba, Bando, and Ikeda (Bando et al. 1981, 1982; Motoba et al. 1983, 1984). Within the framework of this three-body cluster model, several novel phenomena in Λ hypernuclei were explored. In particular, they predicted new types of hypernuclear states, which are not observed in non-strange nuclei, by modeling the structure of 9Λ Be as an ααΛ three-body model. Their predicted energy spectrum for 9Λ Be is depicted in Fig. 5. They categorized three types of states in 9Λ Be: (i) “8 Be-analogue states,” (ii) “9 Beanalogue states,” and (iii) “genuine hypernuclear states.” The 8 Be-analogue states correspond to a Λ residing in a (0s)-orbit added to 8 Be with [s 4 p4 ](λμ) = (40) in the shell model limit. When a Λ particle in a (0p)-orbit is coupled to the core nucleus 8 Be, two types of excited bands of 9Λ Be can be produced, [s 4 p5 ](λμ] =

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Fig. 5 Calculated energy spectra of (Motoba et al. 1985)

E. Hiyama and B. F. Gibson

9 Λ Be

within the framework of the ααΛ three-body model

(50) and (31). Of these two excited bands, (31) corresponds to the 9 Be-analogue states, which means a Λ particle in a 0p-orbit is aligned parallel to the axis of the α − α core in the cluster model. On the other hand, in the band (50), a Λ in the 0porbit is aligned perpendicular to the axis of the α − α core, which is not allowed in ordinary non-strange nuclei (e.g., in 9 Be) due to the Pauli principle. Dalitz and Gal (1976, 1981) labeled states belonging to this band “super symmetric states.” Motoba and Bando called the states in this band “genuine hypernuclear states” in the cluster model (Motoba et al. 1983, 1984). Later, the (π + , K + ) reaction was utilized at KEK (Hashimoto et al. 1998) to observe such a genuine hypernuclear state (1− in 9Λ Be) that was consistent with the theoretical prediction of Motoba and Bando. Another interesting phenomenon in hypernuclear physics is the dynamical contraction of the nuclear core due to the addition of a Λ. Because a Λ is a baryon with Strangeness −1, it differs from a Strangeness 0 nucleon and suffers no Pauli principle repulsion when added to a nucleus. That is, it can occupy a (0s)-orbit along with the nucleons. Thus, 5Λ He can be bound by the attractive ΛN force, whereas 5 He is unbound because the fifth nucleon cannot occupy a (0s)-orbit, which is completely filled by the four nucleons in the 4 He nuclear core. The Λ

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causes a compression of the core nucleus in 5Λ He and 4Λ He, which increases the size of the Coulomb energy arising from the two protons in the respective nuclear core states, 4 He and 3 He (Friar and Gibson 1978). However, the Coulomb energy is not a physical observable; recall that the calculated Coulomb energy in 3 He of approximately 0.64 MeV differs from the observable binding energy difference between 3 H and 3 He of 0.76 MeV. Motoba et al. (1983, 1984) recognized that the resultant contraction of the hypernuclear core nucleus, labeled by some as the “shrinkage effect,” could be observed experimentally by measuring B(E2) values, the B(E2) operator corresponding to r 2 Y2μ (θ, φ). They calculated α − x(x = d, t, and α) radii within the framework of an α + x + Λ three-body model and predicted a reduction in the radii of some 20%. In the 1990s it was pointed out that to confirm the shrinkage effect of the core nucleus, the optimum system to see a change in the B(E2) value would be the 5/2+ → 1/2+ in 7Λ Li within the framework of a 5 He + n + p model (Hiyama et al. 1999). A contraction of 22% in the core nucleus, Λ 6 Li, due to the addition of a Λ was predicted, because 6 Li is weakly bound and easily compressed. A KEK experiment measured the B(E2) value to be 3.6 ± 0.5 e2 fm4 , confirming that the core nucleus, 6 Li, had shrunk by 19% ± 4% (Tanida et al. 2001). This was the world’s first measurement of a hypernuclear B(E2) value. (Detail is provided in  Chap. 68, “High-Precision γ -Ray Spectroscopy of  Hypernuclei”.) At the end of the 1980s, the powerful Gaussian expansion method (GEM) was introduced to perform accurate three- and four-body calculations (Kamimura 1988; Hiyama et al. 2003). This computational method has been successfully applied to various few-body problems: three- and four-nucleon systems, hypernuclei, and pentaquark systems. The energy and wave function are generated with great precision. The accuracy was confirmed by a benchmark test for the calculation of the four-nucleon bound state 4 He using realistic NN interactions by the FaddeevYakubovsky method, the Green’s function Monte Carlo method, the hyperspherical harmonics method, the no-core shell model, the stochastic variational method, and the effective interaction of hyperspherical harmonics method (Kamada et al. 2001). The results from these methods were in good agreement with one another. Because of the limited Y N scattering data, hypernuclear structure calculations and especially calculations of the binding energies of hypernuclei using, for example, the GEM have been essential in testing our interaction models. In particular, such methods as the GEM have been applied to calculate the structure of few-body hypernuclei in order to infer information regarding the Y N and Y Y interactions. Consider the role of 3Λ H, 4Λ H, and 4Λ He in the investigation of ΛN − ΣN coupling in the Y N interaction. By the late 1980s, computation methods had developed to the point that three- and four-body calculations taking into account ΛN −ΣN coupling became possible. Initial separable potential model calculations using Faddeev methods demonstrated that without ΛN − ΣN coupling 3Λ H would most likely not be bound (Afnan and Gibson 1989). Miyagawa et al. (2000) then performed three-body calculations of 3Λ H using the more realistic Nijmegen NSC97(a-f) soft core potentials (Rijken et al. 1999) as a test of how well these one-boson-exchange (OBE) potential models could reproduce the hypertriton binding energy. The results confirmed that ΛN − ΣN coupling is essential to producing a bound 3Λ H.

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Initially, to understand the important role of this mixing in the Y N interaction, a calculation of 4Λ H and 4Λ He was performed in the 1970s within the framework of a 3 He(3 H)+Λ/Σ two-body model (Gibson et al. 1972, 1973). Coupled-channel fourbody calculations of (3N +Λ)+(3N +Σ) were later performed using s-wave separable Y N potentials in the late 1970s (Gibson and Lehman 1979). Carlson performed four-body calculations of these hypernuclei with the NSC89 potential model using a variational Monte Carlo method (Carlson 1999) and obtained the binding energies with statistical errors of 100 keV. In the early 2000s, the GEM (Hiyama et al. 2002) and Yakubovsky equations (Nogga et al. 2002) were utilized to explore in detail 4Λ H and 4Λ He ground and excited states using the NSC97 soft-core potentials (Rijken et al. 1999). In particular, they explored the importance of explicit ΛN − ΣN coupling in calculating the binding energies of these hypernuclei. As shown in Fig. 6 based upon the NSC97f potential, 4Λ He is not bound by the NNNΛ channel alone, but when the N N NΣ channel is included, the A = 4 hypernucleus is bound. Thus, one finds that ΛN −ΣN coupling is crucial in producing bound A = 4 hypernuclei. Moreover, the energy of the ground state is consistent with the observed data. However, the calculated excited state is less bound than is observed experimentally. Because theorists often work with effective ΛN interactions in modeling heavier hypernuclei, it should be pointed out that ΛN − ΣN coupling plays an important role as an effective ΛNN three-body force. This is easily pictured in few-body hypernuclei. As shown in Fig. 7, one can visualize the physical effect of Σ mixing in A = 4 Λ hypernuclei: The first process (i) represents the renormalized ΛN effective two-body force, while the second process (ii) represents the effective ΛNN threebody force acting in the NNNΛ space. One sees that process (i) makes a large enough contribution to produce bound 0+ and 1+ states. Moreover, one finds that the contribution of process (ii) is significant for the 0+ state: it provides an attractive contribution of 0.62 MeV to the binding energy of the 0+ state. In

Fig. 6 The calculated energy spectra of 4Λ He together with the experimental data

67 What Is Hypernuclear Physics and Why Studying Hypernuclear. . .

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Fig. 7 The calculated energy spectra of 4Λ He in the case of (i) including only the effective two-body contribution and (ii) including the effective two-body and three-body contributions in comparison with the experimental data. (The figure is taken from Hiyama et al. 2002)

contrast it provides a small repulsive contribution of 0.09 MeV to the binding energy of the 1+ state. It should be noted that the energy of the 1+ state is not consistent with the observed value in this calculation, which uses the NSC97f model Y N interaction. Therefore, one needs to investigate the ΛN − ΣN coupling effect. Nogga et al. performed sophisticated four-body calculations using realistic N N and Y N interactions (the NSC97a-f interactions), including the Σ 0 , Σ + , Σ − mass differences, using the Faddeev-Yakubovsky approach. They discussed many interesting issues in the A = 4 system, for example, the charge-symmetry breaking effect and the role of ΛN − ΣN coupling (Nogga et al. 2002). However, no realistic Y N interactions reproduced the observed data. Recently, modern Y N interactions based on chiral effective field theory were proposed by Haidenbauer et al. (2013, 2020). Calculations of A = 3, 4 hypernuclei were performed, but still there exists ambiguity in the modeling of ΛN − ΣN coupling. Further investigation of ΛN − ΣN coupling may benefit from studying the structure of neutron-rich Λ hypernuclei. If the isospin of the core nuclei is larger, then it is possible that the contribution of ΛN − ΣN coupling will become larger. Since 2012 several exploratory experiments involving neutron-rich Λ hypernuclei have been performed. Detailed study of neutron-rich Λ hypernuclei is discussed in  Chap. 71, “Theoretical Studies in S = −1 and S = −2 Hypernuclei”. As theoretical computational methods have developed, it has become possible to test realistic Y N interactions utilizing ab initio calculations. Experimentally, high-resolution γ -ray experiments were developed in the mid-1990s, so that it was possible to measure spin-doublet energies with an accuracy of several keV. Combining theory and experiment, a new strategy to obtain information about the Y N and Y Y interactions was established, as shown in Fig. 8.

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Fig. 8 A strategy for determining the Y N and Y Y interactions

First, one models candidate Y N and Y Y interactions. Next, one measures hypernuclear spectroscopic data; however, the experimental data do not provide any direct information on these interactions. Therefore, using the candidate interactions, one performs accurate calculations of the hypernuclear structure. Then, one compares the results with the experimental data in order to infer needed improvements in the underlying interaction models. As a successful example of extracting Y N interaction information, consider the Y N spin-orbit force and its relation to the study of Λ hypernuclear structure. In nonstrange nuclei, it is well known that there is a strong NN spin-orbit force, which results in magic number nuclei. In hypernuclear physics the long-standing issue has been: How large is the Y N spin-orbit force compared with the NN spin-orbit force? Historically, it was well known that the antisymmetric spin-orbit force (ALS) based on quark models (Fujiwara et al. 1995; Morimatsu et al. 1984) provides significant strength with opposite sign to the symmetric spin-orbit force (SLS), while the ALS force of OBE models is much smaller in strength. Therefore, in order to extract information about the Y N spin-orbit force, it is essential to study Λ spin-orbit splittings in hypernuclei. For this purpose, it was useful to study 9Λ Be and 13 Λ C. The core nuclei 8 Be and 12 C are well described by 2α- and 3α-cluster models. Furthermore, the spin-spin component of the ΛN interaction vanishes, and the tensor component does not contribute to the Λα folding potential. Thus, it is useful to examine the difference of spin-orbit splitting within the framework of 2α + Λ and 3α + Λ three- and four-body models using folded Λα potentials based on the quark model and OBE model. High-resolution γ -ray experiments were performed for 9Λ Be and 13 Λ C: As shown in Fig. 9, γ -rays were measured at BNL (Akikawa et al. 2002; Tamura et al. 2005) from 3/2+ → 1/2+ and 5/2+ → 1/2+ to determine the spin-orbit splitting of 5/2+ − 3/2+ in 9Λ Be, whose dominant component is 8 Be(2+ ) ⊗ Λ(0s 1/2 ). Also, at BNL (Ajimura et al. 2001) were measured γ -rays of 3/2− → 1/2− and 1/2− → 1/2+ to determine the spin-orbit splitting of 12 3/2− − 1/2− in 13 Λ C, whose dominant component is C ⊗ Λ(0p3/2,1/2 ).

67 What Is Hypernuclear Physics and Why Studying Hypernuclear. . . Fig. 9 The energy splittings of 9Λ Be and 13 ΛC

2477 Λ(0s1/2)

α

α

＋

3/2+ 2+

γ

5/2+ γ

0+

1/2+ 9Be

8Be

Λ

1/23/2α

γ

α γ α

0+

1/2+ 12C

Λ(0s1/2)

13C

Λ

Before these measurements, the energy splittings of 9Λ Be and 13 Λ C were predicted within the framework of 2α + Λ and 3α + Λ three- and four-body cluster models, respectively, using various Nijmegen model potentials (Hiyama et al. 2000). As noted above, the core nuclei 8 Be and 12 C in these two hypernuclei are well described by the 2α- and 3α-cluster models. The calculated energy splittings are listed in Table 2. One observes that the calculated energy splittings are 0.15 to 0.2 MeV and 0.39 to 0.96 MeV in 9Λ Be and 13 Λ C, respectively. The values in parentheses are calculated splittings using only symmetric spin-orbit terms. One finds that the antisymmetric spin-orbit term reduces by about 20% to 35% the splittings obtained using only the symmetric spin-orbit term. In Kohno et al. (1999), information on a quarkmodel-based Y N spin-orbit potential was used: the strength of ALS amounts to approximately 85% of that of SLS and with opposite sign. Therefore, one can 9 estimate the energy splittings of 9Λ Be and 13 Λ C to be 0.035 to 0.040 MeV in Λ Be and 0.15 to 0.20 MeV in 13 Λ C, respectively. Following these predictions, results from the experiments for the energy splittings in 9Λ Be and 13 Λ C were reported to be 0.043± 0.005 (Akikawa et al. 2002; Tamura et al. 2005) and 0.152±0.054±0.036 (Ajimura et al. 2001), respectively. In comparison with the observed data, the calculated energy splittings for these hypernuclei using the Nijmegen models are significantly larger than experiment, while those using quark-model-based spin-orbit forces are in reasonable agreement with the data. A need to improve the Nijmegen potential

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Table 2 Energy splittings, ΔE (MeV), of (a) 9Λ Be and (b)13 Λ C for the NSC97a, NSC97f, ND, and NF potentials together with the observed data. The values in parentheses are the calculated splittings using only the symmetric spin-orbit force (a)9Λ Be ΔE

NSC97a 0.08

NSC97f 0.16

ND 0.15

NF 0.20

(0.14)

(0.23)

(0.23)

(0.25)

NSC97a 0.39

NSC97f 0.78

ND 0.75

NF 0.96

(0.67)

(1.09)

(1.09)

(1.19)

Exp. 0.043 ± 0.005 (Akikawa et al. 2002; Tamura et al. 2005) –

(b) 13 ΛC ΔE

Exp. 0.152 ± 0.054 ± 0.036 (Ajimura et al. 2001) –

model Y N spin-orbit force is suggested from the comparison with the observed data. 16 11 High-resolution γ -ray experiments for 7Λ Li, 10 Λ B, Λ B, Λ O, etc. have been performed (Tamura et al. 2000; Ukai et al. 2004; Miura et al. 2005). By combining the data with shell model calculations (Millener 2001, 2005), information on spin-dependent terms of spin-spin, spin-orbit, and tensor forces have been extracted. Once reliable Y N and Y Y interactions are established, one can explore novel hypernuclear phenomena and predict properties of the inner core of the enigmatic neutron stars. Regarding hypernuclear phenomena, we have previously provided some references earlier in this chapter, for instance, the shrinkage effect and genuine hypernuclear states. Also, it is possible to study the structure of medium mass Λ hypernuclei. Now, let us consider briefly the inner core of neutron stars. It was originally thought that the maximum mass of a neutron star was limited to 1.5 times the mass of our sun, a solar mass symbolized as M . However, in 2010 a neutron star was observed with a mass of (1.97 ± 0.04)M (Demorest et al. 2010). After this observation, other massive neutron stars, whose masses were about 2M , were reported (Antoniadis et al. 2013; Cromartie et al. 2020). Because of these observations, it is important that one understands the nuclear physics that makes possible such massive neutron stars and the structure of the inner part of neutron stars. So far, it is understood that a strongly repulsive effect in the highdensity region of the nuclear equation of state (EOS), such as a three-nucleon repulsive interaction, can produce a massive neutron star. However, once hyperons are included in the neutron-star modeling, the EOS is softened, and the maximum mass for a neutron star becomes smaller. This effect has been labeled the “hyperon puzzle” in neutron stars. To solve this “hyperon puzzle” in neutron stars, many theoretical solutions have been proposed: relativistic mean field theories (Weissenborn et al. 2012, 2014; Bednarek et al. 2012; Jiang et al. 2012; Gupta et al. 2013), Hartree-Fock extensions (Dapo et al. 2010; Massot et al. 2012), quark mean field models (Hu et al. 2014), quantum hydrodynamic models (Lopes and Menezes 2014), density functional

67 What Is Hypernuclear Physics and Why Studying Hypernuclear. . .

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theory (van Dalen et al. 2014), and Brueckner-Hartree-Fock approximations (Baldo et al. 2000; Nishizaki et al. 2002; Schulze et al. 2006; Vidana et al. 2011; Schulze and Rijken 2011; Yamamoto et al. 2014). Among the many possible solutions that have been proposed, it is considered that microscopic calculations using realistic baryon-baryon interactions are required. As an example, Akmal et al. (1998) performed a sophisticated Fermi hypernetted chain variational calculation with the Argonne V18 (AV18) NN potential (Wiringa et al. 1995) and the Urbana three-nucleon potential (Carlson et al. 1983; Pudliner et al. 1995). The resulting nuclear EOS is considered one of the “standard” EOSs. However, it should be noted that this EOS does not include any hyperons. In 2015, in reference Lonardoni et al. (2015), ab initio calculations of Λ hypernuclei were performed for a wide range of masses as well as for the neutron-star EOS employing the auxiliary field diffusion Monte Carlo method and phenomenological central ΛN and ΛNN (three-body) potentials. In another microscopic calculation, the cluster variational method was applied to calculate the neutron-star EOS using the AV18 NN and Urbana three-nucleon potential (Togashi et al. 2016). They also used a ΛN interaction which was determined from the study of the structure of p-shell Λ hypernuclei (Hiyama et al. 2006). Moreover, they introduced even- and odd-state terms in the ΛΛ interaction for this study. However, there is ambiguity in the odd-state part of ΛΛ interaction; therefore, they employed the following four types of odd-state terms: Type 1 is the most attractive, having the same attraction as the Nijmegen model D (Nagels et al. 1975, 1977); Type 2 is less attractive; Type 3 is slightly repulsive; and Type 4 is the most repulsive. As shown in Fig. 10, if there is no hyperon in the EOS, the maximum mass of the neutron star can reproduce the observed data. However, once hyperons are included, it is found the EOS is softened. Also, it is clearly seen that repulsion in the odd-state part of the ΛΛ interaction contributes about 9% to the maximum

Fig. 10 Mass-radius relations of neutron stars using four types of EOS which correspond to different odd-state terms in the ΛΛ interaction. See the text for details of the ΛΛ interaction

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mass of the neutron star. Thus, it is important to determine the odd-state component of the ΛΛ interaction. Recently, from the observation of Ξ hypernuclei, it is clear that one is required to include the Ξ N interaction in the microscopic approach to modeling the EOS in the future. Therefore, it is important to investigate in detail the Ξ N interaction via study of the structure of Ξ hypernuclei. At present, we clearly need detailed information about the Y N and Y Y interactions: the three-body Y NN and Y Y N interactions, the odd state of the ΛΛ interaction, the Ξ N interactions, etc. For the study of these interactions, we require additional experimental data. In brief summary, in this chapter a historical overview, the present status, and the future prospects of investigating hypernuclei with S = −1 and S = −2 have been introduced from the experimental point of view.

References M. Danysz, J. Pniewski, Philos. Mag. 44, 348 (1953) M. Juric` et al., Nucl. Phys. B52, 1 (1973) D.H. Davis, Nucl. Phys. A 754, 3c (2005) C. Milner et al., Phys. Rev. Lett. 54, 1237 (1985) P.H. Pile et al., Phys. Rev. Lett. 66, 2585 (1991) M. Akei et al. Nucl. Phys. A 534, 478 (1991) T. Hasegawa et al., Phys. Rev. Lett. 74, 224 (1995) T. Hasegawa et al., Phys. Rev. C 53, 1210 (1996) O. Hashimoto et al., Nucl. Phys. A 639, 93c (1998) H. Hotchi et al., Phys. Rev. C 64, 044302 (2001) M. Danysz et al., Phys. Rev. Lett. 11, 29 (1963) D.J. Prowse, Phys. Rev. Lett. 17, 782 (1966) S. Aoki et al., Prog. Theor. Phys. 85, 1287 (1991) H. Takahashi et al., Phys. Rev. Lett. 87, 212502 (2001) M.M. Nagels, T.A. Rijken, J.J. de Swart, Ann. Phys. 79, 338 (1973) M.M. Nagels, T.A. Rijken, J.J. de Swart, Phys. Rev. D 12, 744 (1975); 15, 2547 (1977) M.M. Nagels, T.A. Rijken, J.J. de Swart, Phys. Rev. D 20, 1633 (1979) A. Reuber, K. Holinde, J. Speth, Czech. J. Phys. 42, 1115 (1992) P.M.M. Maessen, T.A. Rijken, J.J. de Swart, Phys. Rev. C 40, 2226 (1989) R. Bertini et al., Phys. Lett. B90, 375 (1980) S. Bart et al., Phys. Rev. Lett. 83, 5238 (1999) R.S. Hayano et al., Nuovo Cim. A 102, 439 (1989) T. Nagae et al., Phys. Rev. Lett. 80, 1605 (1998) H. Noumi et al., Phys. Rev. Lett. 89, 072301 (2002) K. Khaustov et al., Phys. Rev. C 61, 054603 (2000) K. Nakazawa et al., Prog. Theor. Exp. Phys. 2015, 033D022 (2015) A. Gal, M. Baranger, E. Vogt (eds.), Advances in Nuclear Physics, vol. 8 (Plenum, New York, 1975), p. 1 A. Gal, J.M. Soper, R.H. Dalitz, Ann. Phys. 63, 53 (1971); 72, 445 (1972); 79, 113 (1998) R.H. Dalitz, A. Gal, Ann. Phys. (N. Y.) 116, 167 (1978); J. Phys. G 4, 889 (1978) R.H. Dalitz, G. Rajasekaran, Nucl. Phys. 50, 450 (1964) A.R. Bodmer, A. Ali, Nucl. Phys. 56, 657 (1964) A.R. Bodmer, J.W. Murphy, Nucl. Phys. 64, 593 (1965); 73, 664 (1965) Y.C. Tang, R.C. Herndon, Phys. Rev. 138, B637 (1965); Nuovo Cimento 66, 117 (1966) L. Majling, M. Sotona, J. Žofka, V.N. Fetisov, R.A. Eramzhyan, Phys. Lett. B92, 256 (1980) J. Revai, J. Žofka, Phys. Lett. B101, 228 (1981)

67 What Is Hypernuclear Physics and Why Studying Hypernuclear. . .

2481

L. Majling, M. Sotona, J. Žofka, V.N. Fetisov, R.A. Eramzhyan, Phys. Lett. B130, 235 (1983) E.H. Auerbach et al., Phys. Rev. Lett. 47, 1110 (1981); Ann. Phys. 148, 381 (1983) H. Band¯o, M. Seki, Y. Shono, Prog. Theor. Phys. 66, 2118 (1981); 68, 346 (1982) T. Motoba, H. Band¯o, K. Ikeda, Prog. Theor. Phys. 70, 189 (1983); 71, 222 (1984) T. Motoba, H. Band¯o, K. Ikeda, T. Yamada, Prog. Theor. Phys. Suppl. 81, 42 (1985) R.H. Dalitz, A. Gal, Phys. Rev. Lett. 36, 362 (1976); Ann. Phys. 131, 314 (1981) J.L.Friar, B.F. Gibson, Phys. Rev. C 18, 908 (1978) E. Hiyama, M. Kamimura, K. Miyazaki, T. Motoba, Phys. Rev. C 59, 2351 (1999) K. Tanida et al., Phys. Rev. Lett. 86, 1982 (2001) M. Kamimura, Phys. Rev. A 38, 621 (1988) E. Hiyama, Y. Kino, M. Kamimura, Prog. Part. Nucl. Phys. 51, 223 (2003) H. Kamada et al., Phys. Rev. C 64, 044001 (2001) I.R. Afnan, B.F. Gibson, Phys. Rev. C 40, R7 (1989) K. Miyagawa, H. Kamada, W. Glöckle, H. Yamamura, T. Mart, C. Bennhold, Few-Body Syst. Suppl. 12, 234 (2000) T.A. Rijken, V.G.J. Stoks, Y. Yamamoto, Phys. Rev. C 59, 21 (1999); V.G.J. Stoks, T.A. Rijken, Phys. Rev. C. 59, 3009 (1999) B.F. Gibson, A. Goldberg, M.S. Weiss, Phys. Rev. C 6, 741 (1972); Phys. Rev. C 8, 837 (1973) B.F. Gibson, D.R. Lehman, Nucl. Phys. A239, 308 (1979) J. Carlson, AIP Conf. Proc. 224, 198 (1999) E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Phys. Rev. C 66, 024007 (2002) A. Nogga, H. Kamada, W. Gloeckle, Phys. Rev. Lett. 88, 172501 (2002) J. Haidenbauer, S. Petschauer, N. Kaiser, U.-G. Meißner, A. Nogga, W. Weise, Nucl. Phys. A 915, 24 (2013) J. Haidenbauer, U.-G. Meißner, A. Nogga, Eur. Phys. J. A 56, 91 (2020) C. Nakamoto, Y. Suzuki, Y. Fujiwara, Prog. Theor. Phys. 94, 65 (1995); Y. Fujiwara, C. Nakamoto, Y. Suzuki, Prog. Theor. Phys. 94, 215 (1995) O. Morimatsu, S. Ohta, K. Shimizu, K. Yazaki, Nucl. Phys. A420, 573 (1984) H. Akikawa et al., Phys. Rev. Lett. 88, 082501 (2002) H. Tamura et al., Nucl. Phys. A754, 58c (2005) S. Ajimura et al., Phys. Rev. Lett. 86, 4255 (2001) E. Hiyama, M. Kamimura, T. Yamada, Y. Yamamoto, Phys. Rev. Lett. 85, 270 (2000) M. Kohno, Y. Fujiwara, T. Fujita, C. Nakamoto, Y. Suzuki, Proceedings of the APCTP Workshop on Strangeness Nuclear Physics, Seoul, 1999, Abstracts of Contribution Paper, p. 77 H. Tamura et al., Phys. Rev. Lett. 84, 5963 (2000) M. Ukai et al., Phys. Rev. Lett. 93, 232501 (2004) Y. Miura et al., Nucl. Phys. A 754, 75c (2005) D.J. Millener, Nucl. Phys. A 691, 93c (2001) D.J. Millener, Nucl. Phys. A 754, 48c (2005) P.B. Demorest, T. Pennuci, S.M. Ransom, M.S.E. Roberts, J.W.T. Hessels, Nature 467, 1081 (2010) J. Antoniadis et al., Science 340, 1233232 (2013) H.T. Cromartie et al., Nat. Astron. 4, 72 (2020) S. Weissenborn, D. Chatterjee, J. Schaffner-Bielich, Phys. Rev. C 85, 065802 (2012); 90, 019904 (2014) I. Bednarek, P. Haensel, J.L. Zdunik, M. Bejger, R. Manka, Astron. Astrophys. 543, A157 (2012) W.-Z. Jiang, B.-A. Li, L.-W. Chen, Astrophys. J. 756, 56 (2012) N. Gupta, P. Arumugam, Phys. Rev. C 88, 015803 (2013) H. Dapo, B.-J. Schaefer, J. Wambach, Phys. Rev. C 81, 035803 (2010) E. Massot, J. Margueron, G. Chanfray, Europhys. Lett. 97, 39002 (2012) J.N. Hu, A. Li, H. Toki, W. Zuo, Phys. Rev. C 89, 025802 (2014) L.L. Lopes, D.P. Menezes, Phys. Rev. C 89, 025805 (2014) E.N.E. van Dalen, G. Colucci, A. Sedrakian, Phys. Lett. B734, 383 (2014) M. Baldo, G.F. Burgio, H.-J. Schulze, Phys. Rev. C 61, 055801 (2000)

2482

E. Hiyama and B. F. Gibson

S. Nishizaki, Y. Yamamoto, T. Takatsuka, Prog. Theor. Phys. 108, 703 (2002) H.-J. Schulze, A. Polls, A. Ramos, I. Vidaña, Phys. Rev. C 73, 058801 (2006) I. Vidaña, D. Logoteta, C. Providência, A. Polls, I. Bombaci, Europhys. Lett. 94, 11002 (2011) H.-J. Schulze, T. Rijken, Phys. Rev. C 84, 035801 (2011) Y. Yamamoto, T. Furumoto, N. Yasutake, T.A. Rijken, Phys. Rev. C 90, 045805 (2014) A. Akmal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C 58, 1804 (1998) R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, Phys. Rev. C 51, 38 (1995) J. Carlson, V.R. Pandharipande, R.B. Wiringa, Nucl. Phys. A401, 59 (1983) B.S. Pudliner, V.R. Pandharipande, J. Carlson, R.B. Wiringa, Phys. Rev. Lett. 74, 4396 (1995) D. Lonardoni, A. Lovato, S. Gandolfi, F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015) H. Togashi, E. Hiyama, Y. Yamamoto, M. Takano, Phys. Rev. C 93, 035808 (2016) E. Hiyama, Y. Yamamoto, T.A. Rijken, T. Motoba, Phys. Rev. C 74, 054312 (2006)

High-Precision γ -Ray Spectroscopy of  Hypernuclei

68

Hirokazu Tamura

Contents Overview of Hypernuclear γ -Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of  Hypernuclei and Their γ Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physics Motivations of Hypernuclear γ -Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production of Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of γ Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study of N Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-Spin Interaction from s-shell Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p-shell Hypernuclei Revealing Spin-Dependent N Interactions . . . . . . . . . . . . . . . . . . . Consistency Test for the Spin-Dependent Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test of Theoretical N Interaction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sd-shell Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study of Charge Symmetry Breaking (CSB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N Interaction in Nuclear Matter (NN Three-Body Force) . . . . . . . . . . . . . . . . . . . . . . . Study of Impurity Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study of Baryon Properties in Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2484 2486 2486 2488 2488 2489 2490 2493 2493 2494 2505 2506 2507 2509 2511 2511 2514 2515 2515

Abstract

The γ -ray spectroscopy study of  hypernuclei is reviewed, focusing on a series of experiments for high-precision γ -ray measurement using dedicated germanium (Ge) detector arrays. γ transitions from various  hypernuclei in

H. Tamura () Department of Physics, Tohoku University, Sendai, Japan Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_30

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the mass range of A = 4 − 19 were observed, and precise data of their level energies were used to extract the strengths of the spin-spin, spin-orbit, and tensor components of the N interaction. γ -ray data for 4 He confirmed existence of a large effect of charge symmetry breaking (CSB) in  hypernuclei. A B(E2) value for a transition in 7 Li confirmed “shrinking effect” of hypernuclei. Future plans of hypernuclear γ -ray spectroscopy experiments, including a measurement of a B(M1) value for a -spin-flip transition, are discussed.

Overview of Hypernuclear γ -Ray Spectroscopy High-precision γ -ray spectroscopy has played crucial roles in the development of nuclear physics. As for ordinary nuclei composed of nucleons, numerous γ -ray data on energies, yields, and transition probabilities have been measured for thousands of nuclei, revealing their level schemes and properties. Here, use of germanium (Ge) detectors, which allows precise energy measurement of γ rays (in ∼2 keV FWHM resolution for 1 MeV γ ray), is essential. γ spectroscopy can also be applied to  hypernuclei since most of the γ transitions are faster than the weak decay lifetime of  in the nucleus (∼10−10 s). Low-lying excited states of  hypernuclei deexcite via γ transitions, until they reach the ground states of hypernuclei, which then disintegrate via weak decay of the  hyperon. However, measurement of γ rays emitted from  hypernuclei, particularly with Ge detectors, had experimental difficulties caused by huge radiation background originated from high-energy beams used to produce hypernuclei. Before 1998, only four hypernuclear γ transitions, 4 H(1+ → 0+ ) (Bedjidian et al. 1976, 1979), 7 Li(5/2+ → 1/2+ ) (May et al. 1983), 9 Be(5/2+ , 3/2+ → 1/2+ ) (May   − → 1/2+ ) (May et al. 1997), were identified, and et al. 1983), and 13  C(1/2 all of them were measured by employing NaI(Tl) scintillation counters with an energy resolution several tens of times worse than Ge detectors. High-precision measurement of hypernuclear γ rays with Ge detectors was successful for the first time in 1998 (Tamura et al. 2000), when a Ge detector array, “Hyperball,” dedicated to hypernuclear γ -ray spectroscopy was developed and applied to a γ -ray spectroscopy experiment at KEK-PS (Proton Synchrotron at the High Energy Accelerator Research Organization). Then, a series of experiments were carried out at KEK-PS, BNL-AGS (Alternating Gradient Synchrotron at Brookhaven National Laboratory), and J-PARC (Japan Proton Accelerator Research Complex), where  hypernuclei were produced by using meson (π + and K − ) beams generated from high-energy (10–30 GeV) protons. Figure 1 shows γ transitions and reconstructed level schemes of  hypernuclei observed so far. Experiments with Ge detector arrays (Hyperball, Hyperball2, and Hyperball-J) were carried out at KEK-PS for (c) 7 Li (Tamura et al. 2000; Tanida 12 et al. 2001), (f) 11  B (Miura et al. 2005; Tamura et al. 2010), and (g)  C (Hosomi et al. 2015) via (π +, K + ) reaction and at BNL-AGS for (d) 9 Be (Akikawa et al 15 16 2002; Tamura 2005), (e) 10  B (Tamura 2005), (i)  N (Ukai et al. 2008), (j)  O

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Fig. 1 All the γ -ray transitions and reconstructed level schemes of  hypernuclei observed so far. See text for details

(Ukai et al. 2004, 2007), and (c) 7 Li (Ukai et al. 2006), as well as at J-PARC for (b) 4 He (Yamamoto et al. 2015) and (k) 19 F (Yang et al. 2018), via (K − , π − ) reaction.   Employing NaI counters, (h) 13  C was studied again at BNL-AGS (Ajimura et al. 2001). The 4 H transition measured at CERN via stopped K − absorption in the 1970s (Bedjidian et al. 1976, 1979) is also shown in (a). Almost all the s- and p-shell hypernuclei that can be produced by meson beams have been investigated, and the study of heavier ones beyond p-shell hypernuclei has started.

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Structure of  Hypernuclei and Their γ Transitions Figure 2 schematically shows a typical level scheme and γ transitions of a  hypernucleus. When a  is bound in the 0s orbit of the “core nucleus,” (The ordinary nucleus part of the hypernucleus other than the hyperon.) each state of the core nucleus with a spin J (=0) splits into a doublet with spins J ± 1/2 due to the spin-spin coupling between the  and the core nucleus. The energy spacing of the splitting is caused by the spin-dependent components of the N interactions (spinspin, spin-orbit, and tensor interactions). Since these interactions are much weaker than those of the NN case, by one order of magnitude or more as described later, the doublet splitting is generally small, less than a few hundred keV except for very light hypernuclei. Therefore, a hypernucleus is expected to have a level structure similar to that of the core nucleus. In the studies of  hypernuclear structure, the assumption of “weak coupling” between the core and the  that the presence of a  does not change the structure of the core nucleus works well. Hypernuclear γ transitions are classified into three types. The γ transition from the upper to the lower member in a spin doublet is caused by flipping the  spin. This type of M1 transition, denoted as γs in Fig. 2, is called “-spin-flip transition.” The transition from a state in a certain spin doublet to another state in a different spin doublet, denoted as γc in Fig. 2, is essentially the same as the corresponding transition in the core nucleus (γcore ) and called “core transition.” The third type is an E1 transition emitted when the  changes its orbit as p → s . It is called “intershell transition” and denoted as γi in the figure.

Physics Motivations of Hypernuclear γ -Ray Spectroscopy Studies of  hypernuclear structure via γ -ray spectroscopy have been carried out under the following motivations: (1) baryon-baryon interactions, (2) impurity effects of hyperons, and (3) baryon properties in nuclear matter.

Fig. 2 Schematic level scheme and γ transitions of a  hypernucleus with a  hyperon in the 0s (and 0p) orbits. γs , γc , and γi represent three types of hypernuclear γ transitions, standing for the -spin-flip transition, the core transition, and the intershell transition, respectively

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(1) Baryon-baryon interactions Hypernuclei have played important roles in the studies of hyperon-nucleon (Y N) interactions, because of experimental difficulties in Y N scattering experiments caused by short lifetimes of hyperons. Hyperon binding energies in , , and Ξ hypernuclei provide information on the hyperonnucleus potential and consequently on the underlying Y N interaction. For  hypernuclei, observation of ’s single-particle states ((0s) , (0p) , (0d) ,. . . ) in a wide mass range of  hypernuclei provided via missing-mass spectroscopy, particularly with (π + , K + ) and (e, e K + ) reactions (Hashimoto and Tamura 2006), determined the depth of ’s nuclear potential of 30 MeV (Gal et al. 2016), giving a strength of the spin-averaged N interaction. However, spin dependence of the N interaction makes the ’s single-particle states split into doublets (see Fig. 2), which are generally difficult to separately observe in those missing-mass spectroscopy experiments with resolutions of 0.5–3 MeV (FWHM). In addition, it is to be mentioned that, since those spin-dependent components are difficult to investigate via N two-body scattering experiments. Thus, high-precision γ -ray spectroscopy plays a unique and important role to reveal the level scheme like Fig. 2 and clarify the spin-dependent components of the N interaction. The Hamiltonian of a  hypernucleus is written as H = Hcore + T +

A−1 

VN ,

i=1

where the potential of the two-body N effective interaction VN is a sum of the s 0 , and the spin-dependent part, V s , as V 0 spin-averaged part, VN N = VN +VN . N Here, the spin-dependent part may be expressed as s = Vσ (r) s  s N + V (r) l N s  + VN (r) l N s N + VT (r)S12 , VN

(1)

where S12 = 3(σ  rˆ )(σ N rˆ ) − σ  σ N . Here, Vσ , V , VN , and VT stand for radial dependence of the potentials for the spin-spin force, the -spin-dependent spin-orbit force, the nucleon-spin-dependent spin-orbit force, and the tensor force, respectively. The spin-orbit forces (the second and the third terms in the right side of Eq. 1) are also written as VSLS (r)l N (s  + s N ) + VALS (r)l N (s  − s N ), where the first and the second terms are called symmetric and antisymmetric LS forces, respectively. The structure of p-shell hypernuclei with a core nucleus of A = 5–16 is suitable for investigating these spin-dependent N interactions as described in section “Study of N Interaction.” (2) Impurity effects of hyperons Another interesting aspect of hypernuclear studies is “impurity effects” that hyperons give to the nuclear structure. Since a hyperon in a nucleus is free from Pauli principle by nucleons, it can stay in the 0s orbit and attracts the surrounding nucleons, which results in shrinkage of the nuclear size as well as change of the nuclear shape or clustering features. Due to the lack of Pauli principle, it can also create new types of states that are forbidden in ordinary nuclei. In addition, since the  binding energy is sensitive to the

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wavefunction overlap between the core nucleons and the  hyperon, it can be used as a probe to investigate the shape, deformation, and clustering of ordinary nuclei. In γ -ray spectroscopy, such impurity effects can be investigated by measuring an E2 transition probability, B(E2), and a level scheme of the hypernucleus. (3) Baryon properties in nuclear matter Structure and properties of nucleons in the nucleus may be modified from those in the free space, but a clear evidence does not exist except for EMC effect. Since a hyperon can stay in the 0s orbit and behaves as a particle distinguishable from the surrounding nucleons. Thus, through measurement of electromagnetic or weak properties of the  in a nucleus, such as the magnetic moment and the beta-decay rate of the , possible modification of baryon’s properties in a nucleus may be detected. γ -ray spectroscopy will be used to measure the B(M1) value of a -spin-flip transition to investigate possible change of the ’s magnetic moment in a nucleus. In addition, precise and detailed level scheme data of low-lying  hypernuclear states may give a hint to investigate property changes of the  embedded in the nucleus.

Experimental Methods Production of Hypernuclei In order to detect γ rays from hypernuclei buried in a huge background, production of hypernuclear states should be unambiguous identified by the missing mass (the hypernuclear mass) of the reaction or by the pion momentum from hypernuclear weak decay. Usually,  hypernuclei are abundantly produced via (K − , π − ) or (π + , K + ) reactions with meson beams around 1 GeV/c momenta at proton accelerator facilities, by converting a neutron in the target nucleus into a  hyperon. In the (K − , π − ) reaction called “strangeness exchange reaction,” the elementary reaction, K − n → π − , has a large cross section of 1–5 mb/sr and a small momentum transfer to the  less than 0.2 GeV/c for the beam momentum of pK = 0.7–1.8 GeV/c and the scattering angle of θ(K,π ) < 10 deg (Hashimoto and Tamura 2006). In this condition, hypernuclear states reachable via a small angular momentum transfer ( L ≤ 1) are abundantly produced. Therefore, this reaction has been used at BNL-AGS (D6 line) and J-PARC (K1.8 line), where intense and pure K − beams less than 2 GeV/c are available with the mass-separated beam lines equipped with double-stage electrostatic mass separators. The large cross sections of hypernuclear production and the high purity of the kaon beams result in cleaner environment with less radiation backgrounds, allowing for easier operation of Ge detectors due to smaller counting rates and less effects of radiation damage to the detectors. On the other hand, compared with the (K − , π − ) reaction, the (π + , K + ) reaction based on π + n → K + (“associated production”) has a much smaller cross section (0.2–0.5 mb/sr around 1 GeV/c) and a larger momentum transfer (∼0.35 GeV/c), but the available pion beam intensity is by two orders of

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magnitude larger than the K − beam intensity. Therefore, the (π + , K + ) reaction was also used at KEK-PS where intense kaon beams were not available.

Measurement of γ Rays In hypernuclear experiments, γ -ray detectors suffer from severe background radiation of high-energy pions, photons, and neutrons, etc., caused by high-energy (>1 GeV/c) meson beams interacting with nuclei in a thick (∼10 g/cm2 ) target. The kaon and pion beams are also accompanied with high-energy halo particles (pions and muons) and high-energy photons from in-flight decays of K − and π + beams. Not only the single counting rate but also the energy deposit rate in each Ge detector is extremely high, being of the order of 1 TeV/s in a typical condition of γ -ray measurement, due to charged particles penetrating the Ge detector and high-energy photons from π 0 decays, each of which gives a several tens- MeV energy to one Ge detector. In order to detect hypernuclear γ rays with energies less than a few MeV in such conditions, the readout electronics of the Ge detector should be carefully designed. In our Ge detector arrays (Hyperball, Hyperball2, and Hyperball-J), transistor-reset-type preamplifiers with a much lower gain than usual are introduced. To minimize baseline distortion after a rapid dump of the preamplifier output signal at the reset timing, we introduced a fast-shaping amplifier module equipped with a gated integrator (ORTEC 973U), together with a software correction of the baseline shift after the reset timing. In order to suppress background in the γ -ray spectrum, each Ge detector is surrounded by BGO (Bi4 Ge3 O12 ) scintillation counters for Hyperball and Hyperball2 (and PWO (PbWO4 ) scintillation counters for Hyperball-J), of which the signal is used to veto the Ge detector signal. This type of detectors are commonly used for suppressing Compton scattering events in Ge detector arrays for γ -ray spectroscopy of ordinary nuclei. In our case, those counters are also used to effectively reject highenergy particles and photons, which also hit the Ge detector. In our Hyperball, the continuum background in the γ -ray spectrum was reduced by a factor of 3 or 4 at 1 MeV and more than 4 above 3 MeV. Figure 3 illustrates our original Ge detector array, Hyperball (left), and the upgraded array, Hyperball2 (right). The Hyperball array consists of 14 coaxial Ge detectors with an efficiency of about 60% relative to a 3 ×3 φ NaI counter, and the total photopeak efficiency was 2.3% at 1 MeV for a point source. The Ge detectors are arranged almost spherically at a distance of ∼15 cm from the target center to the surface of each Ge detector. Later, Hyperball was upgraded to Hyperball2 by adding six Clover-type Ge detectors. Hyperball2 has ∼5% photopeak efficiency for a point source. In order to cope with a higher beam intensity at J-PARC, a new type of Ge detector array, Hyperball-J, was developed. The effect of neutron damage to Ge detectors is known to be reduced when the Ge crystal temperature is kept lower than that with liquid nitrogen (LN2 ) cooling (∼90 K). Thus, a new cooling method was developed (Koike et al. 2015) with a mechanical pulse-tube refrigerator connected

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Fig. 3 Illustration of the Hyperball (left) and Hyperball2 (right) arrays. (Figures from Tamura 2010). See text for details

to each of the Ge detectors, reaching a crystal temperature of 72 K. During the experiment, the temperature of all the Ge detectors was kept 20 K lower than the LN2 case, and their neutron damage was greatly suppressed. Since the pulsetube refrigerator needs to be operated in vertical attitude, the Ge detectors in the Hyperball-J array are arranged vertically at the top and the bottom of the target position, as shown in Fig. 4b. In Hyperball-J, BGO counters surrounding each Ge detector were replaced with PWO (PbWO4 ) scintillation counters, of which the signal is by more than one order of magnitude faster than that of BGO.

Experimental Setup In the experiments, the K − (or π + ) beam particle was identified and momentumanalyzed by a magnetic spectrometer at the end of the secondary beam line and irradiated to an experimental target, and emitted π − (or K + ) is identified and momentum-analyzed by another magnetic spectrometer with a large acceptance. For both of the beam particle and the emitted particle, particle identification, particularly between kaon and pion, is quite important because misidentification causes serious contamination of beam scattering events and kaon decay-in-flight events. γ -ray detectors (a Ge detector array or NaI counters) are placed around the target and measure γ rays in coincidence with the reaction particles. Hypernuclear mass is obtained as the missing mass of the (K −, π − ) or the (π +, K + ) reactions. To reduce background in the γ -ray spectrum and to get useful information for level scheme reconstruction, a good resolution in the hypernuclear mass spectrum is preferable, but it is often limited by the energy loss straggling of the kaon and the pion in the target material because a thick (10–20 g/cm2 ) target is required to gain the yield of hypernuclear events.

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An Example: J-PARC E13 Experiment Figure 4 shows a typical experimental setup at J-PARC K1.8 beam line. In this experiment (J-PARC E13), 4 He and 19  F hypernuclei were studied via γ ray spectroscopy. A 30 GeV proton beam was bombarded to a production target (a 6 cm-thick golden rod), and secondary particles were transported via several beam lines to experimental areas (Agari et al. 2012a). The K1.8 beam line (Agari et al. 2012b), equipped with double-stage electrostatic separators, delivered a high-purity (K/π = 2–3) and high-intensity (3 × 105 per beam spill) K − beam of 1.5 GeV/c momentum. The momentum of each beam particle was precisely analyzed by the K1.8 beam spectrometer (Takahashi et al. 2012) composed of QQDQQ configuration with tracking detectors (a set of scintillating fiber detectors (BFT) at the upstream of the beam spectrometer and a pair of 3 mm-pitch drift chambers (BDC3, BDC4) at the downstream). The beam particle was identified via time of flight between segmented plastic counters placed upstream (BH1) and downstream (BH2) of the beam spectrometer. The beam was irradiated at the experimental target (cryogenic liquid helium and CF4 targets), and emitted particles were detected with the SKS spectrometer (Takahashi et al. 2012) consisting of a superconducting dipole magnet (SKS) and a set of counters. Just upstream and ˇ downstream of the target, aerogel Cerenkov counters (BAC1, SAC1, SAC2) were placed to discriminate between kaons and pions for (K −, π − ) reaction trigger. Drift

SFV

SksMinus spectrometer

SAC3 SMF

SKS magnet Iron block S-

TOF

Hyperball-J Ge

SDC3,4

SDC1,2 SP0

liquid He cryostat

PWO

SP0

SDC1

SDC2

SKS magnet

SAC1

Hyperball-J

Target

BH2 Q13

BC3,4

BAC1,2

K

BH2 BAC1 BAC2

Q12

Target

SAC1

pulse-tube cooler 20 cm

KD4

Mass slit

Q11 Q10

Beam line spectrometer BFT

0

5m

BH1

Fig. 4 Setup of a γ -ray spectroscopy experiment at J-PARC (E13) (Yamamoto et al. 2015; Yang et al. 2018). Right panel shows a setup around the target with the Ge detector array, Hyperball-J. See text for details

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(b)

(a) K

> 3.5 deg. bound region

highly unbound region

4000 empty target

2000

(normalized)

e+ (511) 76

100 50

counts / 8 keV

counts / 0.5 MeV

6000

103 150

(i) highly unbound region

Ge (564) 74 Ge (596) 56 Fe (847)

0

20

10

20 10 0 30 Excitation energy [MeV]

40

Al (1014) 70 Ge (1039)

(ii) bound region

150 100 50 0

(iii) bound region Doppler corrected

150 0

24

4

100

He : 1+

0+ (1406)

50 0

500

1000

1500

E [keV]

2000

2500

Fig. 5 (a) Missing mass spectrum of the 4 He(K −, π − )4 He reaction in the J-PARC E13 experiment. A peak corresponds to production of 4 He bound states (1+ , 0+ ). (b) γ -ray spectra in coincidence with the 4 He(K −, π − ) reaction; (i) shows spectrum for the highly unbound region of 4 He given in (a), and (ii) and (iii) show spectra for the bound region of 4 He given in (a) without   and with the event-by-event Doppler-shift correction. A γ -ray peak was observed in (iii), which is assigned as the 4 He(1+ → 0+ ) transition (Yamamoto et al. 2015)

chambers are located at upstream (SDC1, SDC2) and downstream (SDC3, SDC4) of the SKS magnet for measuring particle trajectory. A set of plastic counters (TOF) behind SDC4 was used to measure time of flight of the emitted particle. The other detectors (PI0, MFV) were used to reject muons and pions from beam K − decay in flight. Figure 5a shows a missing mass spectrum of the 4 He(K −, π − ) reaction. The peak corresponds to the production of the bound states (0+ and 1+ ) of 4 He hypernucleus. The 0+ state is known to be the ground state, and its mass (the  binding energy) is known from old emulsion experiments (Juriˇc et al. 1973). The 0+ state is produced without flipping a baryon spin when a neutron is converted to a , while the 1+ state is produced with flipping the spin. These states with the configuration of 0sn−1 0s are called “substitutional state,” in which a neutron in a particular orbit of the target nucleus is replaced by a  in the same orbit as the neutron. Population of substitutional states, characterized by no angular momentum transfer ( L = 0), has a particularly large cross section at forward angles with a low momentum transfer (q < 100 MeV/c). The elementary K − p → π − reaction is known to have a sizable spin-flip amplitude at >10◦ for K − momenta of 1.1–1.5 GeV/c, and the observed peak is expected to contain the excited 1+ state events that are not separated from the 0+ state events in the mass spectrum. Figure 5b shows γ -ray spectra in coincidence with the (K −, π − ) reaction. The spectra (i) and (ii) are obtained when the highly unbound region and the bound region of 4 He given in Fig. 5a are selected, respectively. In the spectrum (i), background γ rays from ordinary nuclei from the surrounding material are observed. After a Doppler-shift correction was applied to the spectrum (ii) for event-by-event basis, the spectrum

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(iii) was obtained. It exhibited a peak at 1.4 MeV. Here, the Doppler-shift corrected energy is calculated by assuming that the γ ray is emitted immediately after the production of the hypernucleus before it slows down in the target material as Eγcorrected = Eγmeasured 

1 1 − β2

(1 − β cos θ ),

where the recoil velocity β of the produced hypernucleus and the γ -ray emission angle θ with respect to the recoil direction are obtained from the measured K − and π − momentum vectors at the reaction point, as well as the direction of the γ -ray given by the positions of the reaction point and the Ge detector that detected the γ ray. Appearance of the narrow peak in the Doppler-shift corrected spectrum (c) shows that the lifetime of the γ -ray emitting excited state is much shorter than the slowing down time of the recoiling hypernucleus in the target material. The observed peak is ascribed to an M1 transition of 1+ → 0+ of 4 He. The expected lifetime of the excited state (∼0.1 ps) is consistent with the success of the Dopplershift correction. See Yamamoto et al. (2015) for details. The observation of the 1.4 MeV γ ray for the M1 transition of 4 He(1+ → 0+ ) brought a large impact on the study of charge symmetry breaking in N interaction as described in section “Study of Charge Symmetry Breaking (CSB).”

Study of N Interaction Spin-Spin Interaction from s-shell Hypernuclei As described in section “Study of N Interaction” (1), one of the most important motivations of hypernuclear γ -ray spectroscopy is to clarify spin-dependent N interactions (Eq. 1) via  hypernuclear structure. In s-shell  hypernuclei, the energy spacing of the hypernuclear doublet (see Fig. 2) is expected to be determined almost only by the spin-spin interaction term, Vσ (r) s  s N , which corresponds to the difference of the spin triplet and the spin singlet N interaction in nuclear matter. Experimentally, the A = 4 hypernuclei (4 H and 4 He) have been long known to have a bound state with a ∼1.1 MeV excitation energy through γ -ray measurement in the 1970s (Bedjidian et al. 1976, 1979). By coupling a  with 3 H(1/2+ ) and 3 He(1/2+ ), the 4 H and 4 He hypernuclei have (1+ , 0+ ) doublet states. The ground states of 4 H and 4 He    were known to have S(spin)= 0 from their weak decay properties, and the excited 1+ states were identified via γ transitions (Observation of the 4 He(1+ → 0+ ) transition was reported to be 1.15 ± 0.04 MeV (Bedjidian et al. 1979) but was later found to be incorrect by the J-PARC E13 experiment as described in the previous section). As for the other s-shell hypernuclei (3 H and 5 He), the 3 H ground state is known to have 1/2+, and the doublet partner (3/2+ ) of this state is presumed to be unbound due to a very small  binding energy (B = 0.13 ± 0.05 MeV) of the 1/2+ state, while the 5 He does not have a doublet because of the 0+ core nucleus, 4 He.

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Fig. 6 (a) Three-body N N N interaction induced by N N - N coupling (Fujita-Miyazawa’s three-body force). (b) Three-body N N interaction induced by N -N coupling

It was discussed, however, that N-N coupling interaction may significantly affect the 4 H(1+ , 0+ ) doublet spacing, and thus, the 1.1 MeV energy spacing cannot be a measure of the N spin-spin interaction strength (Gibson and Lehman 1988). According to Akaishi et al. (2000), the N -N coupling interaction, which is expected to be strong in the S = 1 N pair, induces NN three-body force (3BF) as shown in Fig. 6b and gives a large attraction only to the 0+ state. It is to be noted that, since the - mass difference (∼80 MeV) is much smaller than the -N mass difference (∼300 MeV), the effect of NN 3BF from the diagram in Fig. 6b is much larger than that of NNN 3BF from Fig. 6a.

p-shell Hypernuclei Revealing Spin-Dependent N Interactions The N -N coupling interaction is expected to affect hypernuclear doublet spacings much weakly for p-shell hypernuclei than for s-shell hypernuclei. Thus, measurement of the hypernuclear doublet spacings in p-shell hypernuclei was long awaited until the end of the 1990s, when we first measured the 7 Li(3/2+, 1/2+ ) spacing by observing a γ transition with Ge detectors (Tamura et al. 2000). In addition, p-shell hypernuclei also enable us to investigate other terms (the two spinorbit terms and the tensor term) of the spin-dependent N interactions. A−5 In the framework of the pN s shell model, excitation energies of lowlying levels of p-shell hypernuclei can be expressed in terms of four matrix elements, namely, radial integrals of the four spin-dependent effective interaction terms in Eq. 1 integrated with a s pN wavefunction. They are denoted as = pN s |Vσ |pN s , S = pN s |V |pN s , SN = pN s |VN |pN s , and T = pN s |VT |pN s (Dalitz and Gal 1978; Millener et al. 1985). Here, the radial wavefunction for pN s is assumed to be common all over the p-shell hypernuclei. Therefore, excitation energies of p-shell hypernuclear levels can be described only with the four parameters of the radial integrals above as their linear combination. Since shell-model calculations give coefficients of the linear combination, these four parameters can be phenomenologically determined from various p-shell hypernuclear excitation energy data. Then, the parameter values are compared with predictions based on theoretical baryon-baryon interaction models via G-matrix calculations. Since the hypernuclear doublet is split due to the interactions that depend on the ’s spin (s  in Eq. 1), the values of , S , and T can be derived from the spacing of

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the doublet. On the other hand, the nucleon-spin-dependent interaction (VN l N s N ) does not change the doublet spacing but can change the excitation energy of the core nucleus, and the SN value can be derived from change of the excitation energy of the core nucleus. Each parameter can be well determined from specific level spacings of p-shell hypernuclei having a particularly large contribution of the parameter; as described later, , S , and T can be well determined from 7 Li(3/2+, 1/2+ ), 9 16 + + − −  Be(3/2 , 5/2 ), and  O(1 , 0 ), respectively, and SN from core excitation 7 energies in  Li. The other hypernuclear data are used to cross-check the determined parameter values. As a result, this method successfully reproduced almost all the level energy data of p-shell  hypernuclei within accuracies of a few tens of keV as described in section “Consistency Test for the Spin-Dependent Interactions.” On the other hand, hypernuclear level energies have been calculated directly starting from the theoretical Y N interaction models and compared with the experimental data in order to test the validity of the interaction models (Motoba 1998; Halderson 2008; Yamamoto et al. 2010). Here, the theoretical Y N interactions in the free space are converted to effective interactions via G-matrix method and used for shell-model calculations. Another approach based on variational methods has also been developed. Directly from the theoretical Y N interactions, level energies of light hypernuclei have been calculated with precise few-body calculation methods (Hiyama et al. 2010). Here, for p-shell hypernuclei, few-body calculations with a cluster model have also been successfully performed. In addition, by modifying some components of the interaction so as to reproduce the data, properties of the N spin-dependent interaction have been quantitatively extracted as shown below. 7 Li 

As shown in the illustration in Fig. 7 left, the ground state of 6 Li(1+ ) has a structure of almost pure S = 1, L = 0 configuration in the LS coupling scheme in the shell model (or an α cluster plus deuteron-like 1+ pn pair in the cluster model). In the case of L = 0, the spin-dependent part of the N interaction potential in the Hamiltonian is written in terms of the spin operator of the core nucleus, S c , as A−1  i=5

s VN = i

A−1 

Vσ (r) s  s Ni = Vσ (r) s 

i=5

A−1 

s Ni = Vσ (r) s  S c .

i=5

Then, the spin-dependent part of the hypernuclear energy is

E s = ψH Y |

A−1 

s VN |ψH Y = ψH Y |Vσ (r) s  S c |ψH Y i

i=5

= J |s  S c |J pN s |Vσ (r)|pN s = J |

1 2 [J − s 2 − S 2c ] |J , 2

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Fig. 7 Low-lying level schemes of 6 Li and 7 Li, shown together with γ transitions observed with Ge detectors as well as the spin-parities of the states assigned from the γ -ray data (Tamura et al. 2000; Ukai et al. 2006)

because the spin of the hypernuclear state is given as J = s  + S c . Therefore, E s = 12 [J (J + 1) − Sc (Sc + 1) − 12 32 ] = 12 and − for the J = 3/2+ and 1/2+ doublet states (Sc = 1) of 7 Li, respectively. The energy spacing of the 7 Li ground-state spin doublet is thus determined only by the N spin-spin interaction as E[ 7 Li(3/2+ − 1/2+ ) ] = 32 . With a realistic shell-model wavefunction of 6 Li, it is described as (Ukai et al. 2006) E[ 7 Li : 3/2+ −1/2+ ] = 1.461 +0.038S +0.011SN −0.285T +,

(2)

showing again a dominant contribution of the spin-spin interaction. Here,  denotes an effect of the N -N coupling, which is estimated to be  = 0.072 MeV by Millener (Ukai et al. 2006) based on the NSC97f interaction (Rijken et al. 1999). On the other hand, as shown in Fig. 7, the first excited state 6 Li (3+ ) has a structure of S = 1, L = 2 or an α cluster plus deuteron-like 1+ pn pair rotating with L = 2 angular momentum. This 6 Li (3+ ) state is unbound, and the γ transition of 6 Li(3+ → 1+ ) has not been observed. When combined with a , however, this 3+ state becomes bound, and the core γ transition of 6 Li(3+ → 1+ ) is able to be observed in the hypernucleus. Such a property called “glue-like role of ” is characteristic in light hypernuclei with a loosely bound core nucleus. The 6 Li (3+ ) state is split into the doublet of 7 Li(7/2+, 5/2+ ), and, as illustrated in Fig. 7 right, the energy spacing is determined by both of the N spin-spin and

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-spin-dependent spin-orbit interactions as E[ 7 Li(7/2+ − 5/2+ ) ] = 76 + 7 14 3 S − 5 T (Dalitz and Gal 1978), and with a realistic shell-model wavefunction as (Ukai et al. 2006) E[ 7 Li : 7/2+ − 5/2+ ] = 1.296 + 2.165S + 0.020SN − 2.377T + , (3) with  = 0.074 MeV. If excitation energies of all the three states, 7 Li(7/2+ , 5/2+ , 3/2+ ), are measured, the VN parameter for the nucleon-spin-dependent interaction can be reliably extracted from the change of the excitation energy of the core level 6 Li(3+ ), by taking the spin-weighted centroid energy of each doublet, as E(7/2+ , 5/2+ ) − E(3/2+ , 1/2+ ) = Ecore − 0.05 + 0.07S + 0.74SN − 0.17T ,

(4)

with the 6 Li(3+ ) excitation energy of Ecore = 2.186 MeV. The 7 Li structure was investigated at KEK-PS (E419) and then at BNL-AGS (E930). In the KEK experiment, 7 Li(π +, K + )7 Li reaction with 1.05 GeV/c π + beam was used, employing the K6 beam line and the SKS spectrometer (Takahashi + + et al. 2012). In this reaction, the three bound states of 1/2+ gs , 5/2 , and 1/2 (T = 1) are expected to be mainly produced directly via a large non-spin-flip amplitude of the π + n → K + reaction, while the 3/2+ state, the -spin-flip partner of the + 1/2+ gs state, can be populated via a γ transition from the 1/2 (T = 1) state. The Ge detector array, Hyperball, was installed around a 7 Li metal target. By selecting the 7 Li bound-state region in the missing mass spectrum, γ -ray spectra before and after  the event-by-event Doppler-shift correction (Fig. 8a and b, respectively) exhibited

Fig. 8 γ -ray spectrum when the 7 Li bound-state region is gated in the 7 Li(π + , K + ) missing mass spectrum, (a) before and (b) after the event-by-event Doppler-shift correction. (Figure from Tamura et al. 2000)

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four γ -ray peaks attributed to 7 Li transitions. The narrow peak at 2.05 MeV in Fig. 8a is assigned as the E2(5/2+ → 1/2+ ) transition in 7 Li from the γ -ray energy close to the level spacing of (3+ , 1+ ) of the core nucleus 6 Li, as well as from the narrow γ -ray width indicating a lifetime of the γ -ray emission longer than ∼3 ps. This transition was reported in a previous experiment with NaI counters at BNLAGS (May et al. 1983), but the KEK experiment precisely determined the excitation energy of the 5/2+ state as 2050.4 ± 0.6 ± 0.7 keV. On the other hand, the broad peak observed at ∼0.69 MeV in Fig. 8a turns narrow by the Doppler-shift correction as shown in Fig. 8b, indicating that the peak comes from a fast ( 10 deg) with a large momentum transfer, the p3/2 state is mainly produced via L = 2, as shown in Fig. 14 bottom. Therefore, the energy difference of the γ -ray peak position between the forward angle events for (p1/2 ) → (s1/2 ) and the backward angle events for (p3/2 ) → (s1/2 ) shows the spin-orbit splitting of E(p1/2 ) − E(p3/2 ) = 152 ± 54 ± 36 keV. Since the nucleon’s spin-orbit splitting in ordinary nuclei in this mass region is given, for

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− − Fig. 14 Top: 13  C γ -ray spectra for three different ranges of the scattering angle in the (K , π ) C calculated by reaction. Bottom: differential cross sections of the 1/2− and 3/2− states of 13  Motoba. (Figure from Ajimura et al. 2001)

example, by E(13 C : 3/2− − 1/2− gs ) = 3.68 MeV, the ’s spin-orbit splitting is found to be more than 20 times smaller than the nucleon’s. Since the core state of 12 C(0+ ) is not a pure S = L = 0 state, the spin-orbit splitting is determined by not only the -spin-dependent spin-orbit term but also the other spin-dependent interaction terms in Eq. 1. Millener estimated the splitting − − as E[ 13  C: [1/2 − 3/2 ] = 107 keV (Millener 2001), with the aid of NSC97f interaction and the above four parameters for pN s wavefunction. The observed 13 C spin-orbit splitting is thus consistently explained with the 9 Be and other   p-shell hypernuclear data.

Consistency Test for the Spin-Dependent Interactions All the γ -ray data of p-shell hypernuclei were analyzed by Millener (2011). It was found that the spin-doublet spacing energies for A ≥ 10 are well reproduced by a slightly smaller value of = 0.33 MeV than the original value of = 0.43 MeV obtained from 7 Li. Thus, the parameter set that fits the data was determined as = 0.430, S = −0.015, SN = −0.390, T = 0.030 (MeV) for A ≤ 9 = 0.330, S = −0.015, SN = −0.350, T = 0.0239 (MeV) for A ≥ 10.

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Table 1 Values of the N effective interaction parameters (in MeV) for p-shell hypernuclei calculated from Nijmegen BB interaction models (NSC97f, ESC04a, ESC08a) through G-matrix calculation compared with experimentally obtained values (Exp.) (Millener 2011) Exp. (fit-djm) NSC97f ESC04a ESC08a

0.438 0.421 0.381 0.146

S −0.008 −0.149 −0.108 −0.074

SN −0.414 −0.238 −0.236 −0.241

T 0.031 0.055 0.013 0.055

By using this parameter set, almost all the doublet spacing energies were reproduced in accuracies within 30 keV as shown in Table 1 in Millener (2011), although the − − unobserved 10  B(2 , 1 ) spacing ( |2 = (2Jup + 1)−1 | < ψlow || [gc J + (g − gc )J  ] ||ψup > |2 2  3 3 1/2 Jlow Jc (2Jlow + 1) = (gc − g )2 Jup 1/2 1 4π 2 =

3 2Jlow + 1 (gc − g )2 , 8π 2Jc + 1

(9)

with gc , g , Jc and J  representing the effective g-factors of the core nucleus and the  and their spins, respectively. The spatial components of the wavefunctions for the lower and upper members of the doublet, ψlow and ψup (with spin Jlow and Jup ), are identical in the weak coupling limit. The M1 transition rate is given as (M1) =

16π 3 BR(M1) = Eγ B(M1), τ 9

with the lifetime of the upper member of the doublet, τ , and the transition energy, Eγ . Here, BR(M1) is the branching ratio of the  spin-flip M1 transition from the upper member, and, for the ground-state doublet, BR(M1) = 1 except for the case with a doublet spacing smaller than ∼0.2 MeV, where the weak decay of the upper state cannot be neglected. From Eq. 9, τ is estimated to be around 10−13 –10−10 s for Eγ = 1 − 0.1 MeV. This range of the lifetime may be measured with DSAM because the recoil momentum of the hypernucleus produced in the (K − , π − ) and (π + , K + ) reactions is around 0–0.4 GeV/c, corresponding to the stopping time of the orders of 10−13 –10−11 s for p-shell hypernuclei. Although spin-flip M1 transitions between ground-state spin-doublet states were 12 observed for 7 Li, 11  B, and  C, meaningful values of the B(M1) were not obtained, because their transition lifetimes and the stopping time of the recoiling hypernucleus are largely different, by more than a factor of 10, which prevented application of DSAM. As described in section “Study of Impurity Effects,” DSAM works

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when the transition lifetime and the stopping time are within the same order of magnitude. By estimating the lifetimes of spin-flip M1 transitions of various p-shell hypernuclei using Eq. 9 and the measured or expected transition energies, together with their stopping time calculated in various target materials from the stopping power (dE/dx) data, it was found that the 7 Li(3/2+ → 1/2+ ) lifetime of ∼0.5 ps and the stopping time of 2–3 ps for 7 Li produced via 1.1 GeV/c (K − , π − ) reaction and slowed down in Li2 O crystal (with a density of 2.01 g/cm3 ) are within the same order. According to a simulation, the lifetime can be determined in 6% accuracy in a reasonable beam time (35 days) at J-PARC with the K1.1 beam line, the SKS spectrometer, and the Hyperball-J array (Tamura et al. 2012, 2016). The proposed experiment (E63) will run in the near future.

Summary Hypernuclear γ -ray spectroscopy with Ge detectors has significantly changed the field of hypernuclear physics. With dedicated Ge detector arrays (Hyperball, Hyperball2, and Hyperball-J), 28 γ transitions from  hypernuclei in the mass range 19 15 16 12 of A = 4 − 19 (4 He, 7 Li, 9 Be, 11  B,  C,  N,  O,  F) were observed at KEK, 13 BNL, and J-PARC. Together with  C (3 transitions) and 4 H (1 transition) data measured with NaI counters, level schemes of ten hypernuclei were reconstructed in total, including the spin-doublet states called hypernuclear fine structure. Precise data of the energy levels have revealed the strengths of the spin-spin, spin-orbit, and tensor components of the N interaction, quantitatively shown that these -spin-dependent N interactions are much weaker, by one order of magnitude or more, than the spin-dependent NN interactions. It was also found that almost all the level energies of p-shell hypernuclei are quantitatively well reproduced with the four spin-dependent interaction parameters and the spindoublet spacings in s- and sd-shell hypernuclei are also consistently reproduced. In addition, observation of the 4 He γ ray established existence of a large effect of charge symmetry breaking in  hypernuclei. The B(E2) value for the 7 Li core transition measured via γ spectroscopy confirmed the theoretically predicted “shrinking effect” of hypernuclei. Measurement of a B(M1) value for a -spin-flip transition will be carried out to investigate possible modification of ’s magnetic moment in a nucleus, and further γ -ray studies for heavier hypernuclei will help us to clarify NN three-body force relevant to understand the high-density matter in neutron stars.

References K. Agari et al., Prog. Theor. Exp. Phys. 2012, 02B008 (2012a) K. Agari et al., Prog. Theor. Exp. Phys. 2012, 02B009 (2012b) S. Ajimura et al., Phys. Rev. Lett. 86, 4255 (2001); H. Kohri et al., Phys. Rev. C 65, 034607 (2002)

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Y. Akaishi et al., Phys. Rev. Lett. 84, 3539 (2000) H. Akikawa et al., Phys. Rev. Lett. 88, 082501 (2002) K. Aoki et al., (2021) arXiv:2110.04462v1 [nucl-ex]; F. Sakuma et al., EPJ Web. Conf. 271, 11001 (2022) M. Bedjidian et al., Phys. Lett. 62B, 467 (1976) M. Bedjidian et al., Phys. Lett. 83B, 252 (1979) W. Brückner et al., Phys. Lett. B 79, 157 (1978) R.H. Dalitz, A. Gal, Ann. Phys. 116, 167 (1978) A. Esser et al., Phys. Rev. Lett. 114, 232501 (2015) Y. Fujiwara, M. Kohno, K. Miyagawa, Y. Suzuki, Phys. Rev. C 70, 047002 (2004) Y. Funaki et al., Phys. Lett. B 773, 336 (2017) A. Gal, Phys. Lett. B 744, 352 (2015) A. Gal, E.V. Hungerford, D.J. Millener, Rev. Mod. Phys. 88, 035004 (2016) D. Gazda, A. Gal, Phys. Rev. Lett. 116, 122501 (2016) B.F. Gibson, D.R. Lehman, Phys. Rev. C 37, 679 (1988) D. Halderson, Phys. Rev. C 77, 034304 (2008) T. Hasegawa et al., Phys. Rev. Lett. 743, 224 (1995) O. Hashimoto, H. Tamura, Prog. Part. Nucl. Phys. 57, 564 (2006) E. Hiyama et al., Phys. Rev. C 53, 2075 (1996) E. Hiyama et al., Nucl. Phys. A 639, 173c (1998) E. Hiyama, M. Kamimura, K. Miyazaki, T. Motoba, Phys. Rev. C 59, 2351 (1999) E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Phys. Rev. Lett. 85, 270 (2000) E. Hiyama et al., Phys. Rev. C 65, 011301 (2001) E. Hiyama et al., Prog. Theor. Phys. Suppl. 185, 106 (2010) and references there in K. Hosomi et al., Prog. Theor. Exp. Phys. 2015, 081D01 (2015) H. Hotchi et al., Phys. Rev. C 64, 044302 (2001) M. Isaka et al., Phys. Rev. C 83, 044323 (2011) M. Isaka, M. Kimura, Phys. Rev. C 92, 044326 (2015) M. Juriˇc et al., Nucl. Phys. B 52, 1 (1973) T. Koike et al., Nucl. Instrum. Meth. A 770, 1 (2015) M. May et al., Phys. Rev. Lett. 47, 1106 (1981) M. May et al., Phys. Rev. Lett. 51, 2085 (1983) M. May et al., Phys. Rev. Lett. 78, 4343 (1997) D.J. Millener, Nucl. Phys. A 691, 93 (2001) D.J. Millener, Nucl. Phys. A 754, 48c (2005) D.J. Millener, J. Phys. Conf. Ser. 312, 022005 (2011) D.J. Millener, A. Gal, C.B. Dover, R.H. Dalitz, Phys. Rev. C 31, 499 (1985) G.A. Miller, A.K. Opper, E.J. Etephenson, Ann. Rev. Nucl. Part. Sci. 56, 253 (2006) Y. Miura et al., Nucl. Phys. A 754, 75c (2005); Y. Miura, Ph.D. thesis, Tohoku University, 2005 K. Miwa et al., J-PARC Proposal E86 (2021). http://www.j-parc.jp/researcher/Hadron/en/pac_ 2107/pdf/P86_2021-14.pdf T. Motoba, Nucl. Phys. A 639, 135c (1998) T. Motoba, H. Band¯o, K. Ikeda, Prog. Theor. Phys. 70, 189 (1983) S.N. Nakamura et al., J-PARC Proposal P84 (2021). http://www.j-parc.jp/researcher/Hadron/en/ pac_2107/pdf/P84_2021-08.pdf H. Nemura, Y. Akaishi, Y. Suzuki. Phys. Rev. Lett. 89, 142504 (2002) A. Nogga, Nucl. Phys. A 914, 140 (2013) and references there in A. Nogga, H. Kamada, W. Gloeckle, Phys. Rev. Lett. 88, 172501 (2002) Th.A. Rijken, Y. Yamamoto, Phys. Rev. C 73, 044008 (2006) Th.A. Rijken, V.G.J. Stokes, Y. Yamamoto, Phys. Rev. C 59, 21 (1999) Th.A. Rijken, M.M. Nagels, Y. Yamamoto, Nucl. Phys. A 835, 160 (2010) T. Takahashi et al., Prog. Theor. Exp. Phys. 2012, 02B010 (2012) H. Tamura, Nucl. Phys. A 754, 58c (2005) H. Tamura, Prog. Theor. Phys. Suppl. 185, 315 (2010)

68 High-Precision γ -Ray Spectroscopy of  Hypernuclei

2517

H. Tamura, EPJ Web. Conf. 271, 06001 (2022) H. Tamura et al., J-PARC proposal E63 (2016). https://j-parc.jp/researcher/Hadron/en/pac_1601/ pdf/P63_2016-2.pdf H. Tamura et al., Phys. Rev. Lett. 84, 5963 (2000) H. Tamura et al., Nucl. Phys. A 835, 422 (2010) H. Tamura et al., Nucl. Phys. A 881, 310 (2012) K. Tanida et al., Phys. Rev. Lett. 86, 1982 (2001) M. Ukai et al., Phys. Rev. Lett. 93, 232501 (2004) M. Ukai et al., Phys. Rev. C 73, 012501 (2006) M. Ukai et al., Eur. Phys. J. A 33, 247 (2007) M. Ukai et al., Phys. Rev. C 77, 054315 (2008) A. Umeya, T. Motoba, Nucl. Phys. A 954, 242 (2016) M.T. Win, K. Hagino, Phys. Rev. C 78, 054311 (2008) M.T. Win, K. Hagino, T. Koike, Phys. Rev. C 83, 014301 (2011) Y. Yamamoto, T. Motoba, Th.A. Rijken, Prog. Theor. Phys. Suppl. 185, 72 (2010) T.O. Yamamoto et al., Phys. Rev. Lett. 115, 222501 (2015) S.B. Yang et al., Phys. Rev. Lett. 120, 132505 (2018)

 Hypernuclei

69

Tomofumi Nagae

Contents From  Hypernuclei to  Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  − Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Narrow Width Puzzle of  Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bound States of  Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Nucleus Potential in Medium-Heavy Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of  Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2520 2521 2523 2527 2529 2532 2532

Abstract

Bound states of a  hyperon with a nucleus suggest several new concepts in hypernuclear research. First of all, such  hypernuclei should be unstable against strong interactions through the conversion process of N → N . Therefore, the bound states, if they exist, should have finite conversion widths. In the early days of  hypernuclear studies, it was believed that the width would prohibit spectroscopic investigation of  hypernuclei. Another novel aspect is the isospin dependence of  hypernuclei. This played an essential role in forming at least one bound state system, 4 He. The quasi-free  production shape analysis for Si to In targets suggested the -nucleus potential was repulsive for medium-heavy target nuclei. The  − is in the unique position of being the lightest negatively charged baryon. This provides a unique experimental means to probe the N interactions by forming  − atoms. In high density hadronic matter, the large

T. Nagae () Department of Physics, Kyoto University, Kitashirakawa, Kyoto, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_32

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Fermi energy of electrons could be absorbed by the  − s while retaining the charge neutrality. At this moment, we have information on the N interaction in normal nuclear matter. Information for higher density and, therefore, the shortrange part of the interaction will be subjects for the future.

From  Hypernuclei to  Hypernuclei +

Basic properties of  hyperons with spin-parity J P = 12 are listed in Table 1. The mass of the  hyperons is larger than the mass of the  (1115.683 ± 0.006 MeV/c2 ) by approximately 75 MeV/c2 . The mass difference is smaller than the mass of the pion, so that the  ± hyperons cannot decay via strong decay processes such as  ± → π ± due to energetics. The weak decay processes  ± → Nπ are the dominant decay modes in vacuum by changing the strangeness quantum number. Neutral  0 decays via the electromagnetic process of  0 → γ , much more rapid than any weak decay process. As we know already, the  hyperon can only decay through the weak decay processes of mesonic decay, such as  → Nπ , or the non-mesonic decay mode, such as  + N → N + N . The lifetimes of the  hypernuclei are typically ≈200 ps, and have small natural widths. Thus, the high-resolution spectroscopy of  hypernuclei makes sense to investigate. On the other hand, when the  hyperon is embedded in a nucleus, the strong decay process of N → N channel opens without changing the strangeness quantum number as indicated in (−1) + N(0) = (−1) + N(0). The numbers in parentheses are the strangeness, and there is no change of strangeness before/after the  −  conversion. This process connects  hypernuclei and  hypernuclei, and it’s the origin of  −  mixing effects. In ordinary nuclei, there are discussions of (1232) − N mixing effects with the mass difference of ≈300 MeV/c2 in spinisospin excitation modes. The situation is similar, or even more complex, because the  −  mass difference is significantly smaller in the S = −1 hypernuclei. Turning to the three-body force in ordinary nuclei, the Fujita-Miyazawa-type three-body force (Fujita and Miyazawa 1957) is induced through N − NN coupling. In the case of the S = −1 hypernuclei, N − N mixing induces a three-body force with one pion exchange (Fig. 1). In recent years, there has been significant discussion about the short-range properties of those three-body forces as they pertain to high density nuclear matter equation of states.

Table 1 Basic properties of  hyperons: mass, charge, lifetime in cτ , and main decay modes (GeV/c2 )

Mass Charge cτ Main decay modes

+ 1189.37 ± 0.07 ±e 2.404 cm π + n, π 0 p

0 1192.642 ± 0.024 0 2.22 ×10−11 m γ

− 1197.449 ± 0.03 ∼e 4.434 cm π −n

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N N N Δ N NN (a)

NΛ N Σ

N Λ N (b)

Fig. 1 (a) So-called Fujita-Miyazawa diagram (Fujita and Miyazawa 1957) for the three-body force in ordinary nuclei in which a N is converted into a . (b) In the case of  hypernuclei, a  hyperon is converted into a  hyperon. The dashed lines are the pion propagators

The S = −1 three-body force could have a large density dependence and enhance the short-range part of the potential. Because a negatively charged  exists, a large Fermi gas pressure of electrons can be absorbed by  − production as e− p →  − . In neutron stars, we are faced with the “hyperon puzzle.” At 2–3 times the normal nuclear matter density, ρ0 , which could be realized in the core of a neutron star, we believe hyperons (s) should appear, given the mass of the hyperon and its binding energy in nuclear matter. However, once the hyperons appear in the neutron star, the equation of state gets softened so that a neutron star mass greater than 1.5–2× M cannot be sustained. This is a natural consequence of the neutron star property. On the other hand, from astronomical observations, we know there exist neutron stars heavier than 2 times M (Demorest et al. 2010; Antoniadis et al. 2013). This is the “hyperon puzzle,” whether the maximum neutron star mass could exceed 2 M . Nuclear physics including the information from strangeness nuclear physics tells us the maximum mass cannot exceed 2 M , while astronomical observations tell us it should be larger than 2 M . Something is missing in our understanding of high density hadronic matter.

 − Atoms From the experimental viewpoint to investigate the octet-baryon interactions, baryons with strangeness brought us unique opportunities as compared with ordinary nuclei (nuclei made of protons and neutrons) as have charmed baryons. In the former case, we have not observed any negatively charged nuclei, so far. In terms of valance quarks, the proton has uud and the neutron has udd. In the case of charmed ++ + 0 baryons, + c has udc, and c has three charge states of c , c , and c . We don’t have any negative charmed baryons, either. This is because the c quark has a +2/3e charge like the u quark. In the strangeness sector, the s quark has a −1/3e charge, and we have negative baryons such as  − and − , and because they have rather long lifetimes, exotic atoms can form,  − atoms and − atoms.

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As compared with ordinary atoms composed of a nucleus and electrons, a  − stops in materials, is captured into a high exotic-atom orbit , and then cascades down through lower orbits to one close to the ground state by emitting X-rays. Because of the difference in scale between the size of an ordinary atom orbit and that of the exotic-atom orbit (roughly the ratio of me /m ∼0.5/1190≈1/2300), the exoticatom orbits close to the ground state would overlap with the wave function of the nucleus, where nuclear absorption of the  − by the nucleus through the N → N transition takes place, immediately. The energy of the last orbit X-ray will be influenced by the strong interaction. The X-ray energy is shifted (Fig. 2) due to the

(a)

103

n=3

n=9

n=4 102

Shift (eV)

10

n=10 n=5

1

n=6

10-1

10-2 0

(b)

10

20

30

40

50

60

80

90

Z 103

n=3

n=4

102

Width (eV)

70

n=9

10 n=10

n=5 1

n=6

10-1

10-2 0

10

20

30

40

50

60

70

80

90

Z

Fig. 2  − atom X-rays shifts and widths as functions of atomic number Z (Batty et al. 1997)

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real part of the optical potential, and a finite width (Fig. 2) develops, caused by the absorption process arising from the imaginary part of the optical potential (Dover et al. 1989). A systematic study of  − atom X-rays was carried out in the 1970s at CERN, RAL, and BNL from O, Mg, Al, Si and S (Backenstoss et al. 1975; Batty et al. 1978). These data were analyzed with an optical potential model Vopt (r) = teff ·ρ(r) having a potential form proportional to the nuclear density distribution of: V (r) =

V0  , 1 + exp r−R a

(1)

where R = r0 × A1/3 is nuclear radius, A is mass number, and r0 = 1.2 fm. Absorption of the  − takes place through the reaction processes of  − + p →  + n. From the fitting with such an optical potential, the following optimum parameter ranges were obtained: − Re(V ) = 25 − 30 MeV,

(2)

−Im(V ) = 10 − 15 MeV.

(3)

This suggested that the width of the bound state, if one exists, would be Γ ≈ 2ImV (0) = 20–30 MeV. Thus, it would be hard to observe a peak structure of the bound state. That is, a spectroscopic study would be impossible (Dover et al. 1989).

Narrow Width Puzzle of  Hypernuclei From this perspective, it was a big surprise that the CERN, Saclay, and Heidelberg group reported narrow peak structures in a 9 Be spectrum (Bertini et al. 1980). The widths were as narrow as 8 MeV, or smaller (Fig. 3). One thing should be noted that the peaks were in the unbound region, where we expect there should be a contribution from quasi-free  production. This was the  hypernuclear width puzzle in the 1980s. After the shutdown of the CERN PS that delivered the proton beam to produce the secondary kaon beams, the experimental investigations were moved to the BNL Alternate Gradient Synchrotron (AGS) and to the KEK 12-GeV Proton Synchrotron (PS). Typical spectra claiming the existence of narrow peak structures are shown in Fig. 4; other examples are compiled in Dover et al. (1989), as well. In the BNL AGS, a series of upgrades were made for kaon rare decay experiments. In one of four primary beam lines (A, B, C, D), the LESB II beam line was installed together with the Moby-Dick spectrometer (Fig. 5) in the C-line. The K − beam intensity at the LESB II improved as the beam intensity of the AGS improved. The AGS was usually operated at 29 GeV. The energy difference between the BNL AGS and KEK-PS was a serious handicap for the KEK-PS in the

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Fig. 3 Hypernuclear mass spectra obtained in the 9 Be(K − , π − ) reaction at a kaon momentum of 720 MeV/c (Bertini et al. 1980)

K − intensity. Therefore, two different experimental programs were conducted in BNL and KEK for  hypernuclear investigations. The in-flight (K − , π − ) reaction was used at BNL taking full advantage of the high-intensity K − beams. In the kinematics of the (K − , π − ) reaction, a so-called magic momentum at which the hyperon recoil momentum becomes zero is ≈300 MeV/c. Because the K − beam intensity drops rapidly below 600 MeV/c due to the short lifetime of the K − , it was better to operate with the incident momentum ≥600 MeV/c. The stopped (K − , π − ) reaction was used at KEK. The high efficiency of the formation probability of ≈10−4 K − hyperfragment formation produces a sufficient yield of hyperfragments. Further, in the K − stopped absorption, we have a rather large branch into , that is,  :  ≈ 15 : 85 (Table 2). In the early stage of  hypernuclear production, there were two misunderstandings experimentally: too much reliance on the recoilless condition after the success of  hypernuclei and  →  tagging to enhance the S/N (signal/noise). The recoilless condition worked well to selectively produce substitutional states, so that in p−shell hypernuclei states with a (pn−1 , p ), configuration was preferentially produced. They were highly excited states of  hypernuclei. Sometimes the tagging counters to count the multiplicity of the charged particles in the final state were

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Fig. 4 Hypernuclear mass spectra obtained in the 12 C(K − , π − ) reaction (Bertini et al. 1984). Narrow structures of 5 MeV width were claimed to exist at around 275 MeV

installed surrounding the experimental target. They were, in fact, effective in reducing the K − decay in-flight backgrounds such as K − → μ− νμ and π − π 0 in which decay modes only one charged particle is emitted in the final state. However, they are outside the acceptance of the Moby-Dick spectrometer (Fig. 5) in the case of -hypernuclei. The real backgrounds were Kμ3 (μ− π 0 νμ ), Ke3 and Kπ 3 (π − π + π − ). The first two decay modes emit one charged particle. However, the Kπ 3 is a serious background that we cannot remove simply based on the charge multiplicity. On the contrary, it was found that having the spectrometer at a forward finite scattering angles of 4◦ was effective to reduce decay-in-flight backgrounds with application of reaction vertex cuts. In the signal of  hypernuclei production, it was believed that the  + N → +N conversion process produced the p+π − and p. Thus, tagging the conversion, namely, tagging of the , would enhance the S/N. However, near the binding threshold of the , there exists a tail of quasi-free s, and a lot of s are produced as background. Therefore, the tagging for the conversion particles tends to just reduce the statistics of the data and not improve the S/N in many cases, considering the tagging counters do not fully cover the final states.

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Fig. 5 Moby-Dick spectrometer installed at LESB II beam line of the BNL-AGS

Table 2 Branching ratios for hyperon production in the stopped K − absorption reactions. The Rm stands for the branching ratio into multi-nuclear absorptions

Ratios/target R(π 0 ) R(π − ) R( + π − ) R( − π + ) R( 0 π 0 ) R( 0 π − ) R( − π 0 ) Rm

H 4.9 9.7 14.9 34.9 21.4 7.1 7.1

D 5. 10. 30. 22. 23. 5. 5. 0.01

4 He

12 C

Ne 6.2 4.4 3.4 12.6 8.7 6.7 37.3 37.7 37.7 10.9 16.8 20.4 21.2 25.7 27.6 5.9 3.3 2.1 5.9 3.3 2.1 0.16 0.19 0.23

Finally, at the BNL-AGS, a definitive answer to the question of whether narrow  bound states exist resulted from much better statistics as shown in Fig. 6. The previous narrow peak structures were found to be a broad quasi-free component above the binding threshold.

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Fig. 6 A direct comparison of the CERN (previous) and BNL (present) results for 9 Be (K − .π − ) reaction. The histogram represents the BNL measurement, and the CERN data shows the symbols with statistical error bars (Bart et al. 1999)

Fig. 7 Excitation energy spectra and cross sections of (K − , π ± ) reactions for targets of 4 He, 6 Li, and 9 Be (Bart et al. 1999)

In the BNL-AGS E774 experiment, not only for the 9 Be but also 6 Li, no narrow structures in the unbound regions were confirmed by the better statistics (Fig. 7). Therefore, experimentalists reached a conclusion that there were no narrow states in the unbound region as originally expected. Here, a historical misstep took place by a number of theorists. They tried to shut down the road to  hypernuclei completely. Namely, many theoretical experts tried to kill the possible existence of bound states of  hypernuclei.

Bound States of  Hypernuclei A new measurement of heavy  − atomic X-rays was carried out in 1993 for W and Pb targets (Powers et al. 1993). From a fitting in a wide mass-number region of 16 O to 208 Pb, it was suggested that a nonlinear term in the nuclear density was

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needed to achieve better χ 2 values. For example, the following density-dependent (DD) optical potential form was chosen:  μ 2μVopt (r) = − 4π 1 + m 



    ρ(r) α ρ(r) α × b0 + B0 ρ(r) + b1 + B1 δρ(r) , ρ(0) ρ(0) (4) where μ is the  − -nucleus reduced mass. ρ(r) = ρn (r) + ρp (r) and δρ(r) = ρn (r) − ρp (r). The existence of bound states of  hypernuclei was expected from the -atom X-ray data, as was discussed already. However, by changing the shape of the potential, there was a possibility to change from an attractive potential to a repulsive potential. A key was to include the density dependence. Nonlinearity of the shape in the nuclear density could produce a shallow attractive potential near the surface of the nucleus and a strongly repulsive potential inside of the nucleus. In fact, introduction of such a nonlinear term provided a fitting to the X-ray data with a better χ 2 . Contrary to the belief of some, the experimental data did not imply the exclusion − of bound states. There was evidence of a  bound state in the 4 He(Kstop , π −) spectrum at KEK. A bound state in the four-body I = 1/2 and S = 0 system was theoretically modeled by Harada et al. (1990). The difference in the π − and π + spectra as shown in Fig. 8 suggested an isospin dependence of the -nucleus interactions (Hayano et al. 1989). The (K − , π + ) reaction can only produce I = 3/2 states with S = 0, while the (K − , π − ) reaction can produce both I = 1/2 and 3/2 states with S = 0. Therefore, a bound state in the I = 1/2 and S = 0 configuration was inferred. Thus, it was suggested that the -nucleus potential U has a large isospin-dependent term (a so-called Lane term, W (r)), with an isospin dependence of the form τc · τ , where τC is the isospin of the core nucleus: τc · τ /A. U (r) = U0 (r) + W (r)

(5)

However, there were some discussions as to whether the bound state peak position was, in fact, in the bound region. In the energy region of the bound state peak structure a contribution from quasi-free  production exists in the larger excitation region, and a contribution of stopped  − decay ( − → π − n background) exists in the lower excitation region, so that it was not so easy to correctly extract the peak position of the bound state in such large backgrounds. Another theoretical question raised was whether the enhancement was due to a cusp effect at the N → N threshold. The cusp structure was observed in many reactions at the threshold in various spectrum shapes. In the theoretical analyses − of 4 He(Kstop , π − ), there also existed ambiguities regarding the K − absorption process: whether the K − is absorbed from atomic s orbits or from p orbits.

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Fig. 8 The π − spectrum plotted in the hypernuclear mass scale, M(4 He)-M(4 He) (Hayano et al. 1989). Binding energy scales for  + and  0 are indicated in the figure (a). The π + spectrum is shown in the figure (b)

Therefore, new data were obtained at the BNL AGS employing the in-flight (K − , π ∓ ) reactions at the K − incident momentum of 600 MeV/c (Nagae et al. 1998). The incident momentum was so chosen that the K − decay in-flight background was minimized. The simple distorted wave impulse approximation (DWIA) was applied (Fig. 9) in the analysis. The obtained spectra of the 4 He(K − , π ∓ ) are shown in Fig. 9. A peak was observed below the binding threshold in the (K − , π − ) spectrum but not in the (K − , π + ) spectrum, which suggests a bound state of I = 1/2 and S = 0. From the fitting to the data, the binding energy was estimated to be 4.4 ± 0.3 ± 1 MeV with a width of 7.0 ± 0.7+1.2 0.0 MeV. Harada analyzed the data with the DWIA as shown in Fig. 10 obtaining the pole position of -1.1-i6.2 MeV, which produces an asymmetric peak structure corresponding to a binding energy of about 3.7 MeV and a width of 10 MeV as observed in the experimental spectrum.

 Nucleus Potential in Medium-Heavy Nuclei After the establishment of the 4 He bound state, one thing remaining for experimentalists to confirm was whether the -nucleus potential was repulsive for A ≥ 20, as was suggested by many theorists based on density-dependent optical potentials.

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Fig. 9 Excitation energy spectra of 4 He(K − , π − ) and 4 He(K − , π + ) reactions at 600 MeV/c K − momentum measured at 4◦ (Nagae et al. 1998)

Fig. 10 Calculated excitation energy spectra of 4 He(K − , π − ) and 4 He(K − , π + ) reactions at 600 MeV/c K − momentum (Harada 1998)

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Fig. 11 A comparison with the calculated spectra of the pπ = 1.20 GeV/c at 6◦ (Harada and Hirabayashi 2005)

28 Si(π − , K + )

reaction at

While in the case of an attractive potential we could observe bound state peaks, in the case of a repulsive potential, we needed theoretical help that was provided by Morimatsu-Yazaki’s Green’s function method (Morimatsu and Yazaki 1994). Simply speaking, the quasi-free bump structure should be pushed out in the repulsive case while it would be pulled in for the attractive case. We could rely on Morimatsu-Yazaki’s Green’s function method to obtain a reliable excitation energy spectrum shape in a wide energy region from the binding threshold to a high-momentum quasi-free tail. The  − quasi-free spectra of the π − + p → K + +  − reactions on Si, C, Ni, In, and Bi targets were measured in KEK E438 at the pion incident momentum of 1.2 GeV/c (Saha et al. 2004; Noumi et al. 2001). The large momentum acceptance of the SKS spectrometer was a large benefit for the measurements. A theoretical analysis by Harada and Hirabayashi (2005) is shown for the 28 Si(π − , K + ) reaction (Fig. 11). They obtained a best fit to the experimental data with a +30 MeV repulsion for the real part of the optical potential. The double-differential cross section for the (π − , K + ) reaction at a K + forward direction angle θK in lab frame is expressed by: d 2σ = dEK dΩK



dσ dΩ

S(ω, q),

(6)

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 dσ  where S(ω, q) is the strength function and dΩ is the elementary cross section with Fermi averaging. The strength function contains the information of bound states and continuum states.

Summary of  Hypernuclei  hypernuclei provided a new paradigm of investigations in the field of strangeness nuclear physics. The -hypernuclei exist as unstable bound states, and the coupling between N − N plays an important role. The negative charge state of the  − provides a unique opportunity to form exotic  − atoms, which provides information about the optical potential parameters. Nevertheless, it should be noted that this information is limited to the region near the surface of the nucleus. The isospin dependence of the -nucleus optical potential is also important to help us understand the N interaction. As it happened, the N interaction is of a repulsive nature(+30 MeV), which is peculiar compared with  hypernuclei (30 MeV attractive) and ordinary nuclei (attractive by 50 MeV). Currently, it appears that  hypernuclear spectroscopy cannot be addressed to get information about the N interaction, as was possible using the -hypernuclear γ -ray spectroscopy. It may be, therefore, that probing the short-range part of the − N interaction must be uniquely approached through  ± p scattering experiments.

References J. Antoniadis et al., A massive pulsar in a compact relativistic binary. Science 340, 6131 (2013). https://doi.org/10.1126/science.1233232 [arXiv:1304.6875 [astro-ph.HE]] G. Backenstoss, T. Bunaciu, J. Egger, H. Koch, A. Schwitter, L. Tauscher, Intensity measurements on sigma – hyperonic and kaonic atoms. Z. Phys. A 273, 137 (1975). https://doi.org/10.1007/ BF01435833 S. Bart, R.E. Chrien, W.A. Franklin, T. Fukuda, R.S. Hayano, K. Hicks, E.V. Hungerford, R. Michael, T. Miyachi, T. Nagae et al., Sigma hyperons in the nucleus. Phys. Rev. Lett. 83, 5238–5241 (1999). https://doi.org/10.1103/PhysRevLett.83.5238 C.J. Batty, S.F. Biagi, M. Blecher, S.D. Hoath, R.A.J. Riddle, B.L. Roberts, J.D. Davies, G.J. Pyle, G.T.A. Squier, D.M. Asbury, Measurement of strong interaction effects in sigma atoms. Phys. Lett. B 74, 27–30 (1978). https://doi.org/10.1016/0370-2693(78)90050-3 C. J. Batty, E. Friedman and A. Gal, Strong interaction physics from hadronic atoms. Phys. Rept. 287, 385–445 (1997). https://doi.org/10.1016/S0370-1573(97)00011-2 R. Bertini et al. [Heidelberg-Saclay-Strasbourg], Hypernuclei with  particles. Phys. Lett. B 90, 375–378 (1980). https://doi.org/10.1016/0370-2693(80)90952-1 R. Bertini et al. [Heidelberg-Saclay],  hypernuclear states in (K − , π ± ) reactions on 12 C. Phys. Lett. B 136, 29–32 (1984). https://doi.org/10.1016/0370-2693(84)92049-5 P. Demorest, T. Pennucci, S. Ransom, M. Roberts, J. Hessels, Shapiro delay measurement of a two solar mass neutron star. Nature 467, 1081 (2010). https://doi.org/10.1038/nature09466 [arXiv:1010.5788 [astro-ph.HE]] C.B. Dover, D.J. Millener, A. Gal, On the production and spectroscopy of  hypernuclei. Phys. Rep. 184, 1–97 (1989). https://doi.org/10.1016/0370-1573(89)90105-1

69  Hypernuclei

2533

J. Fujita, H. Miyazawa, Pion theory of three-body forces. Prog. Theor. Phys. 17, 360–365 (1957). https://doi.org/10.1143/PTP.17.360 T. Harada, Calculation of the 4 He bound state in the 4 He(K − , π − ) reaction at 600-MeV/c. Phys. Rev. Lett. 81, 5287–5290 (1998). https://doi.org/10.1103/PhysRevLett.81.5287 T. Harada, Y. Hirabayashi, Is the Sigma-nucleus potential for Sigma- atoms consistent with the 28 Si(π − , K + ) data? Nucl. Phys. A 759, 143–169 (2005). https://doi.org/10.1016/j.nuclphysa. 2005.04.025 T. Harada, Y. Akaishi, S. Shinmura, H. Tanaka, Structure of the 4 He hypernuclear bound state. Nucl. Phys. A 507, 715–730 (1990). https://doi.org/10.1016/0375-9474(90)90178-O R.S. Hayano, T. Ishikawa, M. Iwasaki, H. Outa, E. Takada, H. Tamura, A. Sakaguchi, M. Aoki, T. Yamazaki,  hypernuclear bound state observed in stopped K − reaction on 4 He. Nuovo Cim. A 102, 437 (1989). https://doi.org/10.1007/BF02734862 O. Morimatsu, K. Yazaki, A Green’s function method for hadrons in nuclei. Prog. Part. Nucl. Phys. 33, 679–728 (1994). https://doi.org/10.1016/0146-6410(94)90051-5 T. Nagae, T. Miyachi, T. Fukuda, H. Outa, T. Tamagawa, J. Nakano, R.S. Hayano, H. Tamura, Y. Shimizu, K. Kubota et al., Observation of a 4 He bound state in the 4 He(K − , π − ) reaction at 600-MeV/c. Phys. Rev. Lett. 80, 1605–1609 (1998). https://doi.org/10.1103/PhysRevLett.80. 1605 H. Noumi, P.K. Saha, D. Abe, S. Ajimura, K. Aoki, H.C. Bhang, T. Endo, Y. Fujii, T. Fukuda, H.C. Guo et al., Sigma nucleus potential in A = 28. Phys. Rev. Lett. 89, 072301 (2002) [erratum: Phys. Rev. Lett. 90, 049902 (2003)]. https://doi.org/10.1103/PhysRevLett.89.072301 R.J. Powers, M. Eckhause, P.P. Guss, A.D. Hancock, D.W. Hertzog, D. Joyce, J.R. Kane, W.C. Phillips, W.F. Vulcan, R.E. Welsh et al., Strong interaction effect measurements in Sigma hyperonic atoms of W and Pb. Phys. Rev. C 47, 1263–1273 (1993) P.K. Saha, H. Noumi, D. Abe, S. Ajimura, K. Aoki, H.C. Bhang, K. Dobashi, T. Endo, Y. Fujii, T. Fukuda et al., Study of the Sigma-nucleus potential by the (pi-, K+) reaction on medium-toheavy nuclear targets. Phys. Rev. C 70, 044613 (2004). https://doi.org/10.1103/PhysRevC.70. 044613 [arXiv:nucl-ex/0405031 [nucl-ex]]

Experimental Aspect of S = –2 Hypernuclei

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypernuclei and Nuclear Emulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S = −2 Experiments Using Nuclear Emulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principles of Doubly Strange Hypernuclei Production and Event Filtering . . . . . . . . . . . E176 (KEK-PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E373 (KEK-PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E07 (J-PARC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scanning and Analysis Method in the E07 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Event Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Range-Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimating the Number of the Event by At-Rest Captured − Hyperon with Multiple Coulomb Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge Measurement of Tracks in the Emulsion Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-Ray Microscopy Instead of Optical Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall-Scanning Method Searching in Whole Volume of the Emulsion . . . . . . . . . . . . . Detected Samples of S = −2 Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double- Hypernucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Hypernucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absorbing Elements and Trapping Probability of Strangeness by − Capture at Rest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double- Hypernuclei and the H Dibaryon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Prospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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K. Nakazawa () Faculty of Education, Graduate School of Engineering, Gifu University, Gifu, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_33

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Abstract

The study of doubly strange (S = −2) hypernuclei, those possessing two units of strangeness, has seen significant progress because of advances in nuclear emulsion techniques in the past 35 years. Continual developments have been made not only in the emulsions themselves but also in the scanning techniques and the analysis methods. So far, 47 samples of doubly strange hypernuclei have been detected. Among them, the strong interaction energy between two  hyperons estimated from data of double- hypernuclei obtained in the 1960s was found to be misleading with the observation of the Nagara event in 2001, 6 He. In 2015, the existence of a which uniquely identified the hypernucleus   hypernucleus was confirmed by the Kiso event, in which a − hyperon was bound in a nucleus more deeply than would be expected due to the Coulomb force alone. Based upon binding energies in several double- hypernuclei, there seems to be a linear dependence on mass number (A = 6 ∼ 13). Regarding  hypernuclei, the inner level structure appears to be p (∼1 MeV) and s (6 ∼ 8 MeV) orbits for 15  C. Scanning emulsions by only skilled humans for a half century was upgraded to the hybrid-emulsion method that relies on support by electronic detectors to identify high-probability events. Due to great progress in electronic hardware and software, overall-scanning is now possible in which one can obtain images of the whole volume of an emulsion sheet and machine learning techniques can be applied to analyze images to pick out target objects. Because it is expected that one might soon detect 1 × 103 doubly strange hypernuclei, the time is right to discuss S = −2 physics in more detail.

Introduction Nuclei containing particles with a strange quantum number of −2 are labeled doubly strange hypernuclei. If information about the  interaction from  hypernuclei (S = −1) and the  interaction from double- hypernuclei and the  interaction (S = −2) from  hypernuclei along with the mass measurements of doubly strange hypernuclei is collected, then one has the beginning of a unified understanding of the baryon-baryon (B-B) interactions in the octet baryon scheme, consisting of spin-parity = 1/2+ baryons interacting under SU(3) flavor symmetry. [Note that the  hyperons should be included in the model, but only one  hypernucleus has been observed experimentally.] Because the mass difference between the N and  systems is only 28 MeV, these interactions can provide information on the mixed state of the two systems. Here, the presence or absence of the theoretical H dibaryon (uuddss) is also involved, as there is no repulsive core in the B-B interaction with S = −2 and I = J = 0. Furthermore, the B-B interactions are linked to the behavior of hyperons in dense nuclear matter, that is, hyperons are likely to appear in neutron stars, being the densest matter in the stellar universe, and test our understanding of the internal structure (modeled as the nuclear equation of state) of neutron stars.

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However, the short lifetimes (≈10−10 s) of the  and  make it extremely difficult to derive the interaction from their scattering experiments. Therefore, the current approach to obtaining their interactions from the production and decay of hypernuclei with S = −2 embedded in the nucleus, i.e., doubly strange hypernuclei, has been adopted. Nuclear emulsion sheets, an emulsion with submicron spatial resolution, have been the most suitable for their detection. However, after two cases of double- hypernuclei were reported in 1963 and 1966, there were no further reports of their detection for two decades. In the 1970s, the method requiring special skills to detect them with the human eye through a microscope was no longer feasible. In the late 1970s, attempts were initiated in Japan to efficiently detect rare reactions by combining the emulsion with electronic detectors by tagging them with particles emitted from the reaction. The search for charm particles started by tagging with high-momentum muons. The method of combining the emulsion with an electronic detector is called the emulsion-counter hybrid-emulsion method. In 1987, this hybrid-emulsion method was applied in the doubly strange hypernuclear detection experiment (E176 [KEK-PS: High Energy Accelerator Research Organization – Proton Synchrotron, Japan]) to tag K+ particles from (K− , K+ ) reactions searching for reactions that reliably produced − particles. This method was subsequently applied to doubly strange hypernuclei experiments, leading to the E373 [KEK-PS] experiment and next to the E07 [J-PARC: Japan Proton Accelerator Research Complex] experiment. By now 47 examples of doubly strange hypernuclei have been accumulated, including candidates for double- hypernuclei and -hypernuclei with uniquely identified nuclide cores. They have revealed weak attractive nuclear forces between  and  and between  and the nucleus. It has been suggested that there would likely be large numbers of doubly strange hypernuclei lying undetected in the emulsion used so far, which could not be tagged by electronic detectors. The development of searching and analyzing techniques has advanced, and there are now good prospects for being able to conduct an “overallscanning” procedure. Overall-scanning makes searching the entire volume of the emulsion for doubly strange hypernuclei with three or more branch points possible, which are a geometric characteristic of the formation and decay of doubly strange hypernuclei, without relying on electronic detector information. This chapter reviews the history of emulsion methods leading to the current detection of doubly strange hypernuclei, the experiments, the development of analytical methods, the detected doubly strange hypernuclei, the characteristics of the nuclear forces in the S = −2 world, and the near-term future with overallscanning.

Hypernuclei and Nuclear Emulsion The charged particles, which were recorded and first observed in the emulsion, were α-rays by Suekichi Kinoshita, who was studying in the UK in 1910 (Kinoshita 1910). Later in Europe, the emulsion produced by Ilford Ltd. could also record

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single-charged particles, leading to the detection of the sequential decay of π + → μ+ → e+ by Powell and his colleagues (Lattes et al. 1947). In 1952, a hypernucleus was first detected in emulsion on board a balloon (Danysz and Pniewski 1953). It was determined that cosmic rays (assumed to be protons) reacted with a heavier component (Ag or Br) in the emulsion. The ejected nucleus was determined to stop and decay, because of the change in the thickness of the track. In addition to the craftsmanship of estimating the nuclide of the daughter particles after decay of the ejected nucleus, from the thickness and length of the track, a hypernucleus was hypothesized to have appeared because it contained a particle similar to the V01 (now a ) particle identified by comparing its lifetime with that of the V01 particle, which had been discovered earlier. This  particle can be injected deep into the nucleus, because it is not subject to the Pauli exclusion principle within a nucleus composed of a number of protons and neutrons. This was expected to allow more detailed observations of nuclear forces, such as the exchange of heavy mesons comprising the short-range force, the interactions between quarks in different baryons, and the level structure depending on the binding energy of the  particle (B ) in the hypernucleus. Subsequently, as accelerator energies improved, hypernuclear experiments with emulsions using K− beams were actively conducted. The weakness of the emulsion is that it contains a variety of targets, including H, C, N, O, Ag, and Br. However, the early craftsmanship turned this weakness into an advantage; multiple types of hypernuclei could be produced in a single-beam irradiation experiment. Thus, by the 1970s, about 36,000 hypernuclear decays had been detected via the accompanying π − , revealing B from the lightest 3 H (2 H + ) hypernucleus to 15 N (14 N + ) (Davis 2005; Juriˇc et al. 1973). A double- hypernucleus was reported by Danysz et al. in 1963, which indicated the interaction energy between two Λ hyperons (B ) ∼ 4 MeV, although no unique interpretation could be determined (Danysz et al. 1963a, b). A second double- hypernucleus was reported by Prowse in (1966). This second event was 6 He, and its B reported as   was consistent with that presented by Danysz et al. 6 He has the basic structure of a 4 He core, in which both protons and neutrons are  in the ground state, with two  particles also in the ground state. This was later also called “Lambpha” (Band¯o et al. 1990). Therefore, the theoretical understanding was built on the assumption that B = 4 ∼ 5 MeV due to both events. An attempt has begun to understand in a unified way the interactions between relatively light baryons (the baryon octet) comprised of three types of u, d, and s valence quarks, ranging from protons and neutrons with no s quarks to − and 0 particles with two s quarks. However, after that, research efforts reached its limits and no new double- hypernuclei were detected until the E176 experiment [KEK] reported in 1991 (Aoki et al. 1991). In Japan, Kazuno et al. successfully recorded the positron of μ+ → e+ decay and the minimum ionizing particles with a charge of 1 in a nuclear emulsion of type ET7A produced by Fuji Photo Film Laboratory in 1957 (Imaeda et al. 1957). The ET7A was subsequently used to study cascade showers of high-energy gamma rays and the phenomenon of multiparticle production in high mountain locations (Akashi et al. 1975).

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τ

Fig. 1 Lifetime of X-particles. The dotted and dashed lines show lifetimes of charged and neutral X-particles, respectively. The measured lifetimes were (1 ∼ 1.5) × 10−12 s and (0.3 ∼ 0.4) × 10−12 s for charged and neutral X-particles, respectively

Niu et al. found one example of an X-particle (now a charm particle) in the reaction of cosmic rays in an aircraft-borne ECC (emulsion cloud chamber: the emulsion instead of electrodes in a cloud chamber) (Niu et al. 1971). Niu had established the F-Laboratory at Nagoya University in 1972. When J/Ψ was discovered in 1974, 16 examples of X-particle were already detected, and differences in the lifetimes of neutral and charged X-particles were observed as shown in Fig. 1 (Hoshino et al. 1975). Later, once the accelerator beam energy reached sub-TeV, Niu and his colleagues irradiated improved versions of the ECC at FNAL (Fermi National Accelerator Laboratory) and CERN (European Organization for Nuclear Research) to accumulate X-particles. The E531 experiment began at FNAL in November 1978 to determine the difference between the lifetimes of neutral and charged X-particles. E531 was the first experiment to use the “emulsion-counter hybrid method,” in which electronic detectors were added to assist in the search for more X-particles in the emulsion, and irradiation with neutrino beams was employed. High-momentum muons (pμ > 4 GeV/c) were tagged to search for neutrino reaction points and the charm particles emitted from them. A paper in 1986 (Ushida et al. 1986) showed results comparable to current lifetimes. The F-Laboratory group and its former members have continuously promoted ultrafast emulsion sheet readouts (in a microscope system) for bottom mesons, neutrino oscillations (reactions), ντ , seeing through pyramid with muon-radiography techniques, galactic gamma rays, dark matter, and so on. They continue to exceed the limits of emulsion sheet use as high position-resolution tracking detector for a variety of subjects. They are also currently involved in emulsion gel production. Meanwhile, the author, who trained in the emulsion analysis at the F-Laboratory, joined the Faculty of Education at Gifu University, Japan, in 1987, when the E176 experiment using nuclear emulsion began as the first doubly strange

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Table 1 Mol% of the composition of the nuclear emulsion. 12 C, 14 N, and 16 O are derived from gelatin 1H

12 C

14 N

16 O

32 S

80 Br

109 Ag

127 I

38.9

20.1

5.7

11.0

0.2

11.9

12.0

0.2

hypernuclear experiment in Japan. After a few years for analysis of E176, the discussion on an experiment that could be expected to have even higher statistics of doubly strange hypernuclei was initiated in parallel with compiling data that continued the detection of doubly strange hypernuclei in E176. However, the AgBr crystals in the emulsion (ET7B type) at that time had a thin tortoise shell-like shape and a size variation of 0.24 ± 0.078 μm. There were concerns about measuring the sequential decay of doubly strange hypernuclei with sub-μm accuracy, so with the help of Fujifilm Corporation, the ET7C (0.26 ± 0.023 μm) that is a dice with eight dropped corners and of uniform size was followed by the ET7D (0.176 ± 0.015 μm). The composition (mol%) of the current emulsion are listed in Table 1. With the use of ET-7C and ET-7D emulsion, the E373 experiment started in 1995 and succeeded to detect the first uniquely identified 6 He, in 2001 (Takahashi et al. 2001). In 2015, the double- hypernucleus,  presence of a  hypernucleus was confirmed while developing overall-scanning technique (Nakazawa et al. 2015). The emulsion has been used as a detector to record the “small sparkling f irework” aspect of the continuous decay of doubly strange hypernuclei from its formation (Using the emulsion, particle lifetimes produced in high-energy reactions can be measured down to 10−13 s. Hypernuclear mass measurements have reached a determination accuracy of less than 0.2 MeV when no neutrons is emitted in the decay. In the presence of neutron emission, the accuracy is not better than 0.2 MeV. The composition of the emulsion and the size and shape of the AgBr crystals affect the sensitivity to charged particles as well as their charge recognitions to be carried out under the most uniform developed silver grains (requiring >25 grains/0.1 mm)).

S = −2 Experiments Using Nuclear Emulsion Principles of Doubly Strange Hypernuclei Production and Event Filtering To produce doubly strange hypernuclei, a beam of K− particles with one s quark is used, and a quasi-free reaction with protons (p) in the target produces − and K+ particles as follows: ¯ + p (uud) → − (dss) + K+ (u¯s ). K− (us) Two s quarks must be distributed by bringing the produced − particles to rest in the emulsion and then absorbing them into a nucleus in the emulsion listed in

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Table 1. To efficiently detect such capture at-rest events, the following conditions are required: (1) − particle production must be tagged by the identification of a K+ particle produced via the reaction, (2) To nominate stopping − particles, the tagged events should have a highmomentum K+ particle and be confirmed that a − particle candidate does not escape from the emulsion stacks, with electronic detectors. (3) The − particle is tracked with high-precision position information. For item (1), a KURAMA (USHIWAKA in E176) dipole magnet was used as a spectrometer. Regarding item (2), the events with higher-momentum K+ particle (pK+ ≥ 1 GeV/c) were targeted to get lower-momentum − particles, to enhance stopping in the emulsion, emitted backward in the center-of-momentum system. Figure 2 shows the pK+ spectrum of E176 (Aoki et al. 2009), where the momentum of the incident K− particles was set to 1.66 GeV/c. In order to ensure that no − particles escaped the emulsion stack, track detectors were placed downstream of the stack. The stack of the emulsion sheets was driven by a system moving perpendicular to the beam direction in order to uniformly irradiate the emulsion sheets with a narrow beam.

E176 (KEK-PS) The primary goal of the E176 experiment was to confirm the existence of double- hypernuclei, where two  particles decay sequentially from − at-rest absorption events. This was because there was theoretical prediction of the existence of an H (uuddss) dibaryon decaying via the weak interaction (Jaffe 1977). The presence or Fig. 2 K + momentum distribution in the E176 experiment. Solid and dotted lines are experimental data and simulation results assuming quasi-free reactions, respectively. Both lines are normalized to the number of entries above 1 GeV/c. The experimental peak at 0.5∼0.7 GeV/c was consistent with ∗− particle production

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Fig. 3 A stack of dried emulsion sheets and semiconductor track detectors (SSD) near the target in the E176 experiment. The plane perpendicular to the beam contains X and Y

absence of double- hypernuclei is closely related to the mass of the H dibaryon and is discussed in the section “Double- Hypernuclei and the H Dibaryon”. The schematic of the E176 experiment using the emulsion (Aoki et al. 1994) as a target for − particle production and a detector of double- hypernuclei is shown in Fig. 3. The emulsion sheets were 23 × 23 cm2 in size, made of 70- μm-thick transparent polystyrene (PS) film coated with 550- μm-thick emulsion on both sides (thick-type emulsion). Thirteen stacks were prepared, where each stack consisted of 42 thick-type sheets. For uniform beam painting of the sheet, the stack was driven perpendicular (in the X and Y directions) to the beam direction (Z) by a so-called emulsion mover with an accuracy of 10 μm in the X and Y directions. The changeable emulsion sheets (CS) were replaced 10–12 times during stack irradiation to reduce the recording density of the particle tracks and to detect the K+ particles detected by the downstream SSD (silicon strip detector). In each time a CS was attached, X-rays of up to 67 keV were irradiated from downstream through a collimator and aligned the stack. A total of about 6 × 107 reactions with K− particles were recorded on the K2 beamline at the KEK-PS with a K/pi ratio of about 0.3. The analysis covers 796 K+ particles with pK+ ≥ 1 GeV/c in Fig. 2, which were detected in CS and then traced in the upstream stack to reach the (K− , K+ ) reaction points. The stopping points were determined by following the particle tracks from the reaction point to downstream. As a result, 52 events were observed in which charged particles were emitted from the absorption point (called σ -stop; see the section “Event Categorization”), which ensured that the negatively charged particles were absorbed by the nuclei in the emulsion. Based on the azimuthal angle between the K+ and − particle candidates (Aoki et al. 2009), the number of − at-rest +0.0 capture events was estimated to be 77.6 ± 5.1−12.2 , where the first and second errors were statistical error and errors originated by  − stopping contamination, respectively. Regarding σ - and ρ-stop events (without emission of any charged

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particles except for Auger electron(s)/see the section “Event Categorization”), the +0.0 number of stopping events was estimated to be 52 ± 0.0+0.0 −5.0 and 25.6 ± 5.1−7.2 for σ - and ρ-stop, respectively (The 52 σ -stop events were caused by absorption of negatively charged − particles. Assuming a quasi-free reaction, the angle (azimuthal angle) between K+ and − projected onto the plane perpendicular to the beam should ideally be 180◦ . Besides, as can be seen in Fig. 2, the larger the pK+ , the more likely it is to be a quasi-free reaction. Therefore, in σ -stops and ρ-stops, the search was conducted for a constant ratio of σ -stops to ρ-stops at different lower limits of azimuth, while changing the lower limit of pK+ . Finally, the number of ρ-stop events for all σ -stop 52 was determined based on ratios with azimuth angles >120◦ . and pK+ > 1.105 GeV/c. Errors are statistical errors). In the section “Absorbing Elements and Trapping Probability of Strangeness by − Capture at Rest,” the absorbing elements in the emulsion will be discussed as well as how many s quarks are trapped via − hyperon at-rest capture.

E373 (KEK-PS) The E373 experiment was conceived with the aim of detecting ten times more events than the earlier E176 experiment. As shown in Fig. 4, the target was changed from the emulsion to a diamond block (20 [X] × 20 [Y] × 30 [Z: beam direction] mm3 ) with high density and a large effective proton number (Alburger and May 2000), and sheets of scintillating fiber bundles (SciFi bundle), where one fiber had a diameter of 30 μm, were used to predict the − particle trajectory in the emulsion stack (Ichikawa et al. 1998). The size of the emulsion sheet is 25 × 24.5 cm2 . One emulsion stack consists of one thin-type emulsion sheet (100- μm-thick emulsion on both sides of a 200- μm-thick transparent PS film) to detect the − particle candidates being placed upstream of 11 thick-type emulsion sheets (500- μm-thick emulsion on both sides of 40- μm-thick transparent PS film). To measure the range of charged particles ejected from the stack, sheets made of 0.5 mm square fiber were assembled into scintillating fiber blocks (SciFi blocks) as shown in Fig. 5. The SciFi bundle and the stack were narrowed by an upstream block (U-block) and a downstream block (D-block). In particular, the U-block was fabricated to minimize the dead zone between the U-block and the target (Takahashi et al. 2002). The position of the emulsion stack and the beam coordinates via the emulsion mover was calibrated by radiating X-rays through the fiber bundle from upstream immediately before or after the stack exchange. A total of 95 stacks of the emulsion were irradiated with 1.4 × 1010 K− particles at an incident momentum of 1.66 GeV/c. The K/π ratio of the beam was about 1/3 at beam intensity of 1 × 104 K− /spill. As with E176, 5.1 × 103 − particle candidates were selected based on pK+ ≥ 1 GeV/c and the location of the reaction point as well as a high ionization loss of − particle candidates in the SciFi bundle, where K+ particles in E373 are near minimum ionization. The − particle candidates in the thin-type emulsion sheet were automatically scanned around the position predicted by the fiber bundle, and then noise tracks were removed by humans. This work was not so difficult, unlike

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Ξ ΛΛ Λ

Fig. 4 Setup near the target (diamond: 12 C) in the E373 experiment (Ahn et al. 2013)

Fig. 5 SciFi blocks of the E373 experiment. (a) U-block where the target was installed in the center space of the U-block, (b) D-block

that of humans who sat in front of a microscope and measured everything manually in the E176 experiment. Position calibration between the emulsion sheets was carried out to an accuracy of about 20 μm using several thick and black tracks with high ionization losses. In the automatic detection of − particle candidates on the upstream surface of the thick-type emulsion sheet after they were detected on the thin-type sheet, three or four tracks at similar angles to the − particle candidates were sometimes picked up; then, it was necessary to check by a human. This also occurred in connections between thick-type emulsion sheets. Because the angles of the candidates selected by human verification were known, the tracks in the thick-type emulsion sheet were followed by driving the microscope stage and lens barrel with a certain amount of pressing the keyboard, where a human check was still needed to avoid switching to tracks with similar angles. In this way, about 650 σ -stop events (Theint et al. 2019) via at-rest stopping of − particles with the hybrid-emulsion method were successfully detected, as discussed in the section “Estimating the Number of the Event by At-Rest Captured − Hyperon with Multiple Coulomb Scattering”.

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E07 (J-PARC) An experiment was proposed and approved as E964 [BNL-AGS: Brookhaven National Laboratory – Alternating Gradient Synchrotron] to search for more doubly strange hypernuclei (ten times more statistics than in E373 at K/π ∼ 1) immediately after the 11 September 2001 terrorist attacks in the USA. However, E964 was cancelled (May 2006) when DOE suspended fixed target experimental expenditure under the War on Terror. Therefore, an application was submitted to J-PARC, where beam K− particles are supplied with high purity (K/π ≥ 4), and was approved as E07 (January 2007) (Imai et al. 2006). The setup around the target of the E07 experiment is shown in Fig. 6. The setup is similar to E373, but the SciFi blocks used in E373 were removed and the SciFi bundle was replaced by silicon strip detectors (SSDs). The size of the diamond target was changed to be 50 mm (W) × 30 mm (H) × 30 mm (T) due to noncircular beam profile, typical size of 7.6 mm (W) and 5.3 mm (H) within one standard deviation (σ ). Upstream was placed Hyperball-X consisting of six germanium semiconductor detectors. The Hyperball-X measures energy shifts of X-rays in the absorption of − particles, in which strong interactions are at work, from the X-rays emitted in transitions (Fujita et al. 2022). Only events with at-rest absorption of − particles in the emulsion sheets after following − are able to be selected; thus, it was expected to reduce background. The size of an emulsion sheet was 34.5 (X) × 35.0 (Y) cm2 . Thicknesses of the emulsion layers and PS-film were same as thick- and thin-type sheets of E373, but the stack consisted of 11 thick-type sheets sandwiched by two thin-type sheets. In such a setup, a total 1.1 × 1011 K− beam particles of 82% purity (K/π = 4.6) were irradiated in 2016 and 2017 on 118 stacks made from 2.1 tons of emulsion gel, which were also driven by the new emulsion mover to give

Ξ

Λ

ΛΛ

Fig. 6 Near-target setup of the E07 experiment. The stack consists of 11 thick-type emulsion sheets sandwiched between 2 thin-type emulsion sheets to get better connection of − tracks between the SSD and the emulsion stack

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Fig. 7 Cut region to select K+ event which is shown as the area enclosed by the solid line. The slope of line was determined to select a 3 σ region in each momentum. The right ombre scale presents counts in the area of 0.005 [(GeV/c)2 ] (X) × 0.0085 [GeV/c] (Y)

the beam density up to 1.0 × 106 particles/cm2 on the emulsion sheet. The beam K− momentum was set to 1.8 GeV/c with a higher cross section for − particle production than the past experiments (E176 and E373). The masses of outgoing particles were obtained by analyses of magnetic spectrometers, e.g., beam-line spectrometer (Takahashi et al. 2012) for incoming beam K− particles and KURAMA spectrometer for outgoing K+ particles with momentum resolutions (p/p) of 3.3 × 10−4 (FWHM) and 2.7 × 10−2 (FWHM), respectively. In Fig. 7, the plots show momentum and mass2 for K+ region. After selection of the area enclosed by the solid line in Fig. 7, missing mass of (K− , K+ ) reactions is obtained, and the events for production of − particles are nominated. Thus, the cut for the K+ momentum was widely taken as 0.9 ∼ 1.45 GeV/c, and instead the conditions for − particle production and its stopping in the emulsion sheet were strictly imposed as follows: (1) (2) (3) (4)

selecting the SSD hits with large energy deposit, examining consistency between track angle and clustering size in SSD, examining consistency among the energy deposits through four SSD layers, examining consistency of the vertex point (checking impact parameter) between K+ and − , (5) examining kinematic constraints of the p (K− , K+ )− reaction (checking angle residual between − candidate and missing momentum), (6) selecting the target volume in the vertex fitting (checking reaction point inside the target), and (7) examining the other particle tracks from the vertex (removing particles from − decay, e.g., p and/or π − ). An accumulation of 1 × 104 events with at-rest captured − particles was expected in the emulsion sheets from these selections (Ekawa 2019a). Tracking of − particle

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candidates started in May 2018, when photographic development of the emulsion sheets was completed, and after nearly 2 years, the first round of tracking using the hybrid-emulsion method was completed; then about 2.2 × 103 σ -stop events have been detected.

Scanning and Analysis Method in the E07 Experiment Event Scanning Optical Microscope There are two types of optical microscopes for scanning: the EXPERT-G1 [Uniopt Co., Ltd.] and the LMIC [Nikon Corp.]. The former, in particular, was developed specifically for this study. The stage can be driven by a stepping motor over a range of 35 × 35 cm2 , and its hardware positioning accuracy is 10 μm. The optical axis direction has a working distance of 25 mm, and the lens barrel is also driven by a stepping motor. The positions of the stage and the barrel are monitored by linear and rotary encoders, respectively. By dividing the sine wave from the encoder into 20 segments, the stage and the barrel are positioned with an accuracy of 0.5 and 0.1 μm, respectively. A 5 W white LED is used as the light source in the Köhler illumination system. Microscope images in 8 bits are captured with a 512 [X] × 440 [Y] pixel CCD camera and digitalized on an image processing board. One pixel corresponds to 0.25 and 0.12 μm for 50× and 100× objective lenses (Objective lenses were supported by Chiyoda Optical instruments CO., LTD., Tokyo), respectively.

Position Alignment Between SSD and the Top Emulsion Sheet To detect − capture at-rest events, the − particle candidates recorded in the upstream SSD must be detected in the upstream thin-type sheet and then connected to the downstream thick-type sheet until the candidates stop in the emulsion. To align the positions of the − particle candidates in the sheet moved by the emulsion mover, where the SSDs were fixed, the four corners of the stack were irradiated with anti-protons (p) of 1.8 GeV/c at a density of 104 /cm2 . Alignment is achieved by the position distribution of the p on the SSD. The ps recorded on the thin-type sheet result in the same pattern as in the SSD by shifting the X-Y of the detected position of the thin-type sheet. On the other hand, grid marks with a diameter of 50 μm were printed into the upstream surface of all sheets at 10 mm intervals (34 × 34 points) in the X-Y direction; the center of the sheet was temporally set by using center of coordinates of all the grid marks. Figure 8 shows the displacements from the provisional coordinates at the best pattern fit, indicating that the SSD and the upstream sheet were aligned to an accuracy of approximately 20 μm. The displacements also made the provisional coordinates connection to the coordinates of the SSD.

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Fig. 8 Result of the p pattern matching between the SSD and the top emulsion sheet. The peaks show the displacements of positions of p tracks in the SSD to the top emulsion sheet. (a) is an X-Y plot, (b) and (c) show the histograms of differences for X and Y directions, respectively (Ekawa 2019a)

Scanning of − Candidate Tracks Predicted by SSD The − candidates of the predicted position (x, y) and angle (θx , θy ) of the SSD are analyzed in the upstream thin-type sheet. At that time, if the distance between the SSD and the thin-type sheet is set to be D, the predicted position on the sheet becomes (x + D tan θx , y + D tan θy ), then D is optimized. The − candidates were then probed in an area of 400 × 400 μm2 centered on the predicted position, and tracks satisfying the condition that the angular differences ( tan θ =| tan θSSD − tan θeml. |) did not exceed 0.08 were hypothesized as − candidates, recorded on the SSD. The θSSD and θeml. are the space angles in the SSD and the emulsion, respectively, for the − candidates. The position and angle deviations of the − candidates with respect to the SSD are shown in Fig. 9. It can be seen that the search area and the tolerance conditions for angular misalignment are reasonable. The tracks that fulfilled the condition of  tan θ ≤ 0.08 were followed further downstream to thick-type sheets. Position Alignment for Sheet by Sheet of the Emulsion The alignment accuracy of about 20 μm between the emulsion sheets in the previous section “Position Alignment Between SSD and the Top Emulsion Sheet” is not much different from the accuracy in the earlier E373. For automated − particle tracking, the accuracy should be improved by at least an order of magnitude. To accomplish this, instead of the antiproton beam of the previous section, it was decided to perform pattern matching using about 100 vertically recorded beam-like tracks within one microscope field of view (0.13 [X] × 0.11 [Y] mm2 ). First, the boundary between the areas, where black charged particle tracks were recorded on the emulsion layer, and the exterior layer without any tracks was distinguished by the brightness (luminance) of the microscope image. The beamlike tracks were then captured at the lower surface of the upstream ith sheet and at the upper surface of the downstream (i + 1)th sheet, and it was done to match the patterns of the tracks’ positions on both sheets. This beam-pattern matching method

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Fig. 10 Offsets in X and Y directions around correct positions. The right ombre scale presents counts in the area of 0.5 (X) × 0.5 (Y) μm2

µ

Fig. 9 (a) and (b) are distributions of position residuals between predicted points by the SSD and detected points in the top thin-type sheet for X and Y coordinates, respectively. The right ombre scale defines the counts in the areas of 0.02 (X) × 0.02 (Y) mm2 for (a) and 0.005 (X) × 0.005 (Y) for (b)

µ

gave us position alignment with an accuracy of approximately 1 μm (Soe et al. 2017). Figure 10 shows a sample that the pattern was matched when the beam-like tracks were offset by about −10 μm in both X and Y between the two sheets. Based on the luminance distribution of the overlapping areas of the patterns, the positions could be aligned with an accuracy of 2.10 ± 0.13 μm (FWHM) and 2.52 ± 0.12 μm (FWHM) in X and Y, respectively.

Tracking in the Emulsion Sheet As mentioned in the previous section “Position Alignment for Sheet by Sheet of the Emulsion”, the − following for the E373 emulsion required constant monitoring by humans to ensure that the following system did not change the track being followed to false tracks that have similar angles. To eliminate mis-tracking without human support, a method recognizing tracks was developed with image processing in the E07 experiment.

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Fig. 11 The flow of image processing method for the tracks. The size of each image is 145 (X) × 125 (Y) μm2 . The track to be followed is represented inside a parallelogram in each image. (a), the microscope image; (b), Gaussian blurred image of (a); (c), subtracted image from (b) to (a); (d), binarized image based on a certain value for brightness threshold

To recognize − candidate tracks, the image was processed as in Fig. 11. Figure 11a shows an image taken by an optical microscope. A Gaussian blurred image was obtained as in Fig. 11b, and then the image (Fig. 11c) of a uniform background was produced by subtracting (a) from (b). The binarization was applied for (c) under a threshold of brightness value, and then an enhanced track image could be obtained as shown in Fig. 11d. When the upstream and downstream emulsion sheets are aligned with nearly 1 μm accuracy, candidate − particle tracks from the upstream sheet can be uniquely detected on the upper surface of the downstream sheet, where detected track can then be displayed in the center of the monitor of the microscope image. The focal plane is then driven in 3 × cos θ [μm] steps and ten images are taken, where θ is the zenith angle of the − particle candidate relative to the sheet as mea1 sured on the upstream sheet, with the relationship tan θ = {(dx/dz)2 + (dy/dz)2 } 2 . If the microscope images are acquired while moving the X-Y stage according to the angle of the track (dx/dz, dy/dz), the track to be followed can always be observed at the center of the monitor (Fig. 12 (1)). A narrow rectangle surrounding the center positions in the ten images is processed to extract the followed − candidate track (Fig. 12 (2)). If there are at least six images containing the track among ten images, the track extraction is considered successful. From the (at least) six images, new position and angle candidates are calculated (Fig. 12 (3)) as a mean to acquire the ten images in the next depth. Such a process was repeated and the track following

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μ

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θ μ

θ

μ

μ

Ξ−

Fig. 12 The − candidate track can be detected on the surface region of the emulsion sheet guided by position alignment with an accuracy about 1 μm. The microscope stage and lens barrel are then driven in the direction guided by the track angle, and the track images are captured and processed simultaneously. See detail in the text

proceeded automatically until the number of images having the track, nimg , fell below 6 out of 10 (Fig. 12 (4)). The track following stops at or near the absorption point of the − candidate. The reason for stopping is that the track is no longer linear but wiggles due to the energy becoming lower; that means the track does not appear in the center of the monitor and nimg is less than 6. The track also cannot be followed in cases of in-flight decay or secondary reactions. When tracking was no longer possible, images were examined in a range from 40 μm upstream to 80 μm downstream based upon the position of the previous image and left as a superimposed image. Approximately 400 − candidate tracks in one emulsion sheet could be followed with 99.5% efficiency in about 8 h without human supervision and nearly three sheets were followed per day with one microscope. The superimposed images were inspected and classified by humans as described in the next section (Soe et al. 2017).

Event Categorization The features of the − candidate endpoints are classified by topology. If the − candidate decays inflight before stopping, it is classified as “Decay” and its shape is shown in Fig. 13 (1). If the − candidate also collides with a nucleus in the emulsion and shows broken fragments before stopping, the shape, shown in Fig. 13 (2), is classified as “Secondary interaction.” In these two cases, the tracks near the end of the − candidate are linear. If fragment(s) are ejected at the − candidate rest point, but the interaction is found to be caused by a beam, it is classified as “Beam interaction,” as shown in Fig. 13 (3). If the − candidate is followed to the bottom sheet in the stack, but the candidate goes further downstream (escaping the stack), it is classified as “Through,” as in Fig. 13 (4). As the − candidate loses kinetic energy due to ionization in the emulsion, the effects of Coulomb scattering by the nuclei in the emulsion are observed and the flight track becomes wobbly. Finally, when it comes to rest, the nucleus absorbs

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Ξ-

Ξ-

Ξ-

πσ

Ξ-

Ξ-

Λ

ρ

Ξ-

Ξ-

Ξ-

Fig. 13 Eight categorizations of features at the end point of the − candidate track based upon topology

the − particle candidate and forms a compound nucleus via a − atom and decays. If at least one charged fragment is ejected from the decay, its topology is shown as in Fig. 13 (5) and classified as “σ -stop.” If the − candidate is absorbed by lighter components in the emulsion (12 C,14 N,16 O), the Coulomb barrier in the nucleus is low (∼4.2 MeV) for the ejected positively charged fragment(s). Therefore, if even one of the fragments has a range of 3∼31 μm, the − particle is considered to be absorbed by a light component as introduced in Aoki et al. (2009). In the case that the fragment decays further, it is likely to contain hyperons within it. The case with no charged fragments is called a “ρ-stop” as shown in Fig. 13 (6), and the cases with or without Auger electron(s) are classified as “ρ-stop w/ Auger” or “ρ-stop w/o Auger,” respectively. The “ρ-stop w/o Auger” is mostly background such as a proton, since a positively charged particle is never absorbed by nuclei with positive charge. In the case that there are more than three vertices, including the captured point of − particle and the decay point of the fragments, it is most likely that a doubly strange hypernucleus has been produced. If one fragment decays and one of its daughter nuclei further decays, the parent fragment contains two  particles and decays sequentially, i.e., the parent is probably a double- hypernucleus, then the parent fragment is categorized as “Double-” (Fig. 13 (7)). On the other hand, if two fragments are emitted from the absorption point of the − , it is very likely that two single- hypernuclei with one  particle each have been produced and the event is called “Twin hypernucleus” (Fig. 13 (8)). The twin hypernucleus event provides information on how strongly the absorbed − particle is bound to the nucleus. If the − particle is more deeply bound than the atomic orbit, this is the birth of a  hypernucleus.

Range-Energy Calibration The relationship between the range and kinetic energy of charged particles, with the range and charge of the charged particle as R and Z, respectively, is:

70 Experimental Aspect of S = −2 Hypernuclei

R=

3 M β λ(β) + MZ 2 C Z ( ), Z Z2

2553

(1)

where M is the mass ratio of the charged particle to the proton. The first term represents the range without the electron capture effect, i.e., without neutralization of low-energy positive charged particles like nuclei. The second term accounts for the range extension due to neutralization, where CZ is an empirical function of β and Z for various nuclei considered in Barkas et al. (1958). The λ(β) is an expected range of a proton at velocity of β in a nonstandard (our) emulsion and is expressed by the equation: rd − 1 r(ds − d) λs (β) λs (β) = + , λ(β) rds − 1 rds − 1 λw (β)

(2)

where λs (β) and ds are the proton range for β and the density (3.815 g/cm3 ) for the standard emulsion (Ilford G5), respectively. λw (β) is the proton range in water, and r and d are the increasing ratio for volume to weight due to moisture and the density of our emulsion, respectively. When one measures the range, R, of a charged particle, one knows a suitable set of β, which corresponds to kinetic energy, and λ(β) from Eq. (1), if one uses the mass and the charge, Z, of known particles, then one also gets the d of our emulsion by Eq. (2). With the use of the d, one can calculate the kinetic energy via any measured ranges based upon the assumption of the mass and Z. In some emulsion sheets, when it is necessary to know the kinetic energy for a particular charged particle, one takes alpha particles (Z = 2) from the decay of the natural isotope, 228 Th, in the emulsion as shown in Fig. 14a. One measures the ranges of the alphas having unique kinetic energies as 5.423 MeV (Track #1) and 8.785 MeV (Track #2) from the decays of 228 Th and 212 Po, which correspond to the detached (#1) and the longest track (#2) in Fig. 14a, respectively, in the Thorium series. One also measures the ranges of Tracks #3 (4.784 MeV) and #4 (7.687 MeV) in the Uranium series as shown in Fig. 14b to estimate systematic errors derived by the range-energy calibration. Such α-decay events used to be detected by humans watching at a monitor displaying microscopic images, which is a very time-consuming task. The detection speed was, however, reduced to 1/15 by developing the overall-scanning method equipped with a high-speed microscope drive and machine learning, as described in the section “Future Prospect”. The detected events were recorded as dozens of tomographic images, and a human measured the range of the α particles on the monitor screen. When the emulsion sheet is photographically developed, the AgBr, which accounts for about 80% by weight, is almost released into the fixing solution, so that all the material except for the Ag that remains as tracks becomes gelatin. Thus, the sheet shrinks in the direction of thickness more than before development. The shrinkage rate that minimizes the standard deviation of the range distribution of α particles (which is approximately 1/2) was obtained by changing the shrinkage rate.

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μ

μ

Fig. 15 Distribution of track length of the α from 212 Po after calibrated by 150 α tracks

μ

Fig. 14 Superimposed images of alpha tracks in the decays of the (a) Thorium series and (b) Uranium series. In the image of (a), 5 α-tracks show the decay sequence as 228 Th →224 Ra (+ #1) →220 Rn →216 Po →212 Pb → (β-decay) →212 Po →208 Pb (+ #2). The image (b) shows 4 α-tracks as 226 Ra →222 Rn (+ #3) →218 Po →214 Pb → (β-decay) →214 Po →210 Pb (+ #4). The reason why Tracks #1 and #3 do not associate with each vertex is that 224 Ra and 222 Rn move in the emulsion due to thermal motion within their long lives, where the both half-lives are over 3 days

α

μ

Figure 15 shows a sample distribution of the range of the 8.785 MeV α particle. In this sample, the most probable range is 50.24 μm with its mean error of 0.10 μm, which is calculated from one standard deviation of 1.22 μm divided by (number 1 of α) 2 . The obtained density was 3.545 ± 0.011 g/cm3 . It was found that 150 α particles are enough to achieve an error in kinetic energy ( −20 MeV. In the E07 experiment, however, the condition has been changed to | B − B− − Ex. |< 5 MeV within 3σ based on the information from the Nagara event, which will be introduced in the section “The Nagara Event”, where B is a weak 1∼2 MeV. The parameter of Ex. is the excitation energy of A−2 Z and A−1  Z. In the case of twin hypernucleus, the B− value can be obtained as:

⎛ −

M( N) + M( ) − B− = 14

A2 1 M(A  Z1 ) + M( Z2 ) + M

⎝



k 

⎞ ml ⎠ + KE  ,

l=1 A2 A1 A2 1 where M(A  Z1 ) and M( Z2 ) are each mass of  Z1 and  Z2 , respectively, and they are already known from the past experiments. If the mass of a single-   hypernucleus, M(A  Z ), is not known, it is assumed that one can estimate its B   from B s of the same nuclides of the known single- hypernucleus, A  Z . In the case where the − is bound to the atomic 3D or higher orbit and a  atom is formed, there is little effect of the strong force, only Coulomb force, so B− has no ambiguity according to the difference of theories. The B− values in atomic 3D level are 0.13, 0.17, and 0.23 MeV for 12 C, 14 N, and 16 O, respectively. In the process without a neutral particle, e.g., a neutron(s) or π 0 , kinematic fitting was applied to minimize the error of the B or the B− . The Lagrange multipliers method was used under the constraints of the kinetic energy and three elements of

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momentum vectors. The practical applications for double- hypernuclei and twin hypernucleus are detailed in Ekawa (2019a) and Hayakawa (2019), respectively.

X-Ray Microscopy Instead of Optical Microscope In some events from past experiments, it was difficult to recognize the production and/or decay points of the doubly strange hypernuclei due to overlapping tracks, even with the high position resolution of the emulsion. Attempts have been made to recognize the production and/or decay points by swelling a small part of the emulsion, slicing it thinly, and observing them from a direction perpendicular to the normal optical axis as introduced in the section “The Demachi-Yanagi Event.” However, such observations tend to lose the original information of the event; then careful and precise measurements of the positions and angles of the tracks are essential before the above process, which has not been practical. Therefore, a nondestructive technique was developed with hard X-ray instead of visible light. The energy of the X-ray used is 8 keV at SPring-8 (the world’s largest synchrotron radiation facility), and in principle a position resolution of 70 nm can be expected in comparison with 270 nm by optical microscope using the maximummagnification objective lens. By using a track in the emulsion, the resolutions were checked by the sharpness of the edge of a track. By using images from X-ray and optical microscopes, brightness values perpendicular to track direction were measured, which means along the X-axis at each point of the Y-axis in Fig. 24. Those data have been fitted by the Eq. (5), where the term of ax + b is given by the background and the remaining term represents the peak structure of a Gaussian shape of the track.

−(x − μ)2 f (x) = α tanh β × exp 2σ 2

+ ax + b.

(5)

The sharpness was evaluated with the length on the Y-axis between 10% and 90% of maximum brightness calculated by fitted peaks. By using the X-ray microscope,

μ

μ

Fig. 24 Images of a track observed with X-ray (a) and optical microscopy (b). The Y-axis was defined in the direction of the track. The profile of the track was obtained from the change in brightness along the X-axis, a direction perpendicular to the Y-axis at a certain Y-value. (Reproduced from Kasagi et al. (2022) ©The Author(s) 2022, with kind permission of the European Physical Journal (EPJ))

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a better resolution of 0.20 ± 0.04 μm was obtained, compared with 0.47 ± 0.06 μm by the optical microscope. It was also demonstrated that vertical rotation to the ground axis enables threedimensional (stereo) imaging of events, using five α-particles associated with the decay of the natural radioisotope 228 Th in the emulsion, as shown in Fig. 25. The spatial resolution in the depth direction was evaluated to be about 0.28 μm by the stereo-imaging with ± 45◦ of the sample, where the resolution is ∼2.5 times better than 0.7 μm by optical microscope. The stereo-imaging was applied to a doubly strange hypernuclear candidate event, for which the production point was difficult to obtain clear recognition. The production point was deduced with the accuracy of ±0.04 μm, although the accuracy was 3μm by optical microscope. Then, the production mode was identified uniquely and a measurement error of B− was found to be ±0.86 MeV better than the ±3 MeV by the optical microscope (Kasagi et al. 2022).

Fig. 25 Images of an α-decay event obtained by stereo-imaging with X-ray microscopy. The number above each image denotes the rotation angle of the sample event. The tracks in the event are numbered from 1 to 5. (Reproduced from Kasagi et al. (2022) ©The Author(s) 2022, with kind permission of the European Physical Journal (EPJ))

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μ

Fig. 26 Images near the stopping point of the same α track from the α-decay event by optical microscope (a) and X-ray microscope (b). In the image of (b), it is seen for developed silver grains constituting the track

It is expected that about 103 doubly strange hypernuclei will be detected by a search for the whole volume of the emulsion currently on hand. In the research on doubly strange hypernucleus, it is a great advantage to be able to use X-ray microscopy to observe events whose production and decay points are difficult to recognize, which are expected to account for about 30% of all detected events. Moreover, as shown in Fig. 26, developed silver grains constituting a thick track like α-particles can be observed near the stopping point using an X-ray microscope. If the measurement of density of silver grains (G.D. [grains/100 μm]) can be developed to identify the charges of charged particles emitted in the production and decay of doubly strange hypernuclei, it is expected to be the new method with G.D. measurement that can replace the charge identification method described in the section “Charge Measurement of Tracks in the Emulsion Sheet”.

Overall-Scanning Method Searching in Whole Volume of the Emulsion With the hybrid-emulsion method, the tracks of − particle candidates can be followed that may have been stopped in the emulsion sheets predicted by counterinformation. However, the detection efficiency of the (K− , K+ ) reaction with the hybrid method has been at most 30%. This is due to the decay and reconstruction failures of the outgoing K+ particles. − particles are produced not only by p  (K− , K+ )− reactions but also by  n (K− , K0 )− reactions in the diamond target and the emulsion stack. Without the outgoing K+ particle, the − particles cannot be tagged, but they stay and some of them are stopped in the emulsion sheets. Such − particles produce doubly strange hypernuclei. If one can scan the whole volume

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Fig. 27 Imaging method of the overall-scanning method. The figure shows the emulsion sheet and the optical system in a cross-sectional view. The driving of the emulsion sheet and the objective lens is shown as the time lapse from (a) to (d). The field of view (FOV) was 130 × 110 μm. In about 3 s from (a) to (c), the images (shadow area) were captured and preprocessed, and processed images were stored

of the emulsion sheet for typical topologies in the formation for the production and decay of doubly strange hypernuclei, it was expected to obtain ten times as many doubly strange hypernuclei, which has been the case using the hybrid method. Therefore, development of the “overall-scanning” method to search for doubly strange hypernuclei in the whole volume of the emulsion sheets was initiated in 2007. The key technologies for the overall-scanning method are high-speed imaging and image recognition. In image capture, the emulsion sheet was continuously moved horizontally at a constant speed, while the objective lens was moved up and down vertically in the microscope stage, as shown in Fig. 27. Microscope images were analyzed for frequency filtering and line-segment detection, and then the vertex-like shapes were extracted, if they have three or more clearly associated end points of line segments. The selected candidates were visually sorted, and events with multiple vertices were analyzed in detail under the microscope. It has been confirmed that Nagara event can be detected by this image recognition flow. The overall-scanning method based on line-segment detection was tested with the microscope optics (50× objective lens [N.A.= 0.85] and CCD camera [0.2 M pixels, 100 fps]) using E373 emulsion sheet until 2013. Scanning in the emulsion volume of 1.46 cm3 detected, for the first time, a  hypernucleus as the Kiso event, which will be introduced in the section “The Kiso Event,” among approximately 8 M images. As a test application of the overall-scanning, the method was used for detection of alpha decay events which are useful calibration sources for the range-energy relation as mentioned in the section “Range-Energy Calibration.” The detection of α decay events was a difficult and time-consuming job for humans under the microscope; thus, the overall-scanning method is expected to be very helpful for the calibration. After upgrading the microscope system (20× objective lens [N.A.= 0.35, FOV= 1140 × 200 μm], CMOS camera [0.7 M pixels, 800 fps.], and Piezo actuator [Piezosystem MIPOS600SGD]), the detection was performed as follows (Yoshida et al. 2017): (1) Scanning with 20× lens ⇐ area 30 × 50 × 0.5 mm3 /12.2 h, (2) Image processing ⇒ picked up 3.1 ×104 vertex-like objects/10.0 h,

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(3) Scanning with 50× lens ⇐ 3.1 ×104 vertex-like objects/61.0 h, (4) Secondary image processing ⇒ picked up 1.2 ×104 vertex-like objects/2.6 h, (5) Human eye-check of (4) on the PC monitor ⇒ picked up 386 events like α-decay/1.7 h. By the human eye-check on the above (5), the total number of 1.2 ×104 objects was classified into α-decays (3.8 ± 0.2%), hadronic interactions (78.6 ± 0.8%), black objects on the surface (8.7 ± 0.3%), and cross-tracks (8.0 ± 0.3%). By using a microscope for 6.0 h, human checked 386 events picked up in the above (5). The α decay was confirmed in 162 and 50 events for the Thorium and Uranium series, respectively, where S/N reached 1.2 (= 212/174). The overall-scanning was very effective in studying the range-energy relation using only several hours with human effort. By moving away from line-segment detection, now, machine learning for checking track patterns is employed to detect α and 3 H decay events more smoothly and effectively, which will be introduced in the last section “Future Prospect”.

Detected Samples of S = −2 Hypernuclei To produce doubly strange hypernuclei, − particles must be absorbed by the nuclei in the emulsion. The events of double- hypernucleus showing production and decay of the 1960s were detected in a small number of − stopping events, which can only be described as sheer luck. Starting in 1987, E176 (KEK-PS) +0.0 recorded 77.6 ± 5.1−12.2 of − stopping events, and 650 ± 11 events were also − recorded for at-rest  absorption in the emulsion of the subsequent E373 (KEKPS) experiment. The E07 (J-PARC) experiment recorded about 3 × 103 stopping − candidate events. The number of doubly strange hypernuclei (including candidates) detected was 4, 10, and 33 in E176, E373, and E07, respectively.

Double- Hypernucleus A double- hypernucleus is a nucleus that contains two  hyperons. Two  hyperons are produced via the reaction of a − with a proton inside a nucleus absorbing the − particle. If the double- hypernucleus is emitted as a fragment from a − compound nucleus, three or more successive branching points are observed from the production to the decays, because two s in the double- hypernucleus decay sequentially. As mentioned in the section “Event Categorization,” if the mass is uniquely known in the decay of the first , i.e., the decay of the double- hypernucleus, it is known for the binding energy of − , B− , at the production of the double- hypernucleus. However, this has not been succeeded, because the uncertainty in B− is so far larger than that obtained for twin hypernucleus.

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Early Days Before the E373 Experiment Figure 28 shows a photograph of the event reported by Danysz et al. in 1963 (Danysz et al. 1963a, b). The − particles produced by irradiating the emulsion with K− particles (1.5 GeV/c) at CERN lost kinetic energy due to ionization, came to rest, and were captured by the nuclei in the emulsion. As shown in Fig. 28, tracks are crowded near the capture point. The nuclide was estimated from the thickness of the track of the emitted charged particles, the kinetic energy was obtained from their track length, and the following two candidates were selected as cases for conserving momentum and energy. The value of B in both cases was in the range of 3 ∼ 5 MeV as listed below, where the errors include the error of B of single- hypernucleus and the measurement errors of the range and angle of the emitted tracks. 1.

10 8  Be( Be +  + ) 9  Be

→

9 1 −  Be + H + π ,

→ 4 He + 4 He + 1 H + π − .

B = 17.5 ± 0.4 MeV, B = +4.5 ± 0.4 MeV. 2.

11 9  Be( Be +  + ) 10  Be

→

10 1 −  Be + H + π ,

→ 4 He + 4 He + 2 H + π − .

B = 19.0 ± 0.6 MeV, B = +3.2 ± 0.6 MeV. A second double- hypernucleus was reported by Prowse in (1966). The photograph and the track data were not presented in the publication. The emulsion was irradiated with 5 GeV K− particles, and somehow an incident − particle from outside came to rest in the emulsion. There were then three candidates accepted kinematically, and among them only the following case provides a consistent result

Ξ

π π

μ

Fig. 28 Photograph (left) and schematic drawing (right) of the production and the sequential decay of a double- hypernucleus. A − particle stopped and was absorbed at point A, and the double- hypernucleus (Track #1) and one charged particle (#3) are emitted. Two  particles inside the double- hypernucleus decay sequentially at points B and C. Tracks #4 and #6 are π − particles. (This photo was published in Nucl. Phys., 49, M. Danysz et al., ‘THE IDENTIFICATION OF A DOUBLE HYPERFRAGMENT’, 212 ©Elsevier (1963a))

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for B with that presented in 1963 ((Although any error value was not presented in Prowse (1966), but errors were denoted in Dalitz et al. 1989)): − +12 C → 6  He 5  He

→

6 4 7  He( He +  + ) +

Li,

5 1 −  He + H + π ,

→ 4 He + 1 H + π − ,

B = 10.9 ± 0.5 MeV, B = +4.6 ± 0.5 MeV. Another example of a double- hypernucleus (Event #15-03-37) detected in the E176 experiment is shown in Fig. 29. Although it not so easy to separate points B and C in the photograph, photographs from other angles are shown in Aoki et al. (1991). Track lengths, angles, and comments are listed in Table 2. Total visible energy release, Evis , obtained by ranges of all the charged particles was 69.0 ± 0.7 MeV. Since Q-values of the reaction − + p →  +  and the

π

Ξ

Fig. 29 Photograph and schematic of a double- hypernucleus detected in E176. No scale, but Track #5 is 5.2- μm-long. Tracks #1 and #2 are emitted after − stopping at point A. Track #1 decays to Tracks #3 and #4 at point B, and Track #3 decays with three charged particles at point C. (This photo was published in Nucl. Phys. A 828, S. Aoki et al., ‘Nuclear capture at rest of − hyperons’, 123 ©Elsevier (2009)) Table 2 Experimental data for track lengths and angles for the event of #15-03-37 (Aoki et al. 2009) The main part, ≈90%, of the large error of the length of Track #4 was estimated by range straggling. The zenith angles (θ) were measured with respect to the beam direction, which is parallel to the optic axis of a microscope. Azimuthal angles (φ) of Tracks #2, #4, #6, and #7 were expressed as Tracks #1, #3, and #5 to be zero at vertices A, B, and C. Then, the sets of Tracks #1 and #2 and Tracks #3 and #4 seemed to be colinear within measurement errors for two tracks in each set. Track #4 was found to be a π − due to the σ -stop topology of the end point. The density of the emulsion was 3.385 ± 0.032 g/cm3 Point A B C

Track no. #1 #2 #3 #4 #5 #6 #7

Length [μm] 3.9 ± 0.4 200.4 ± 3.6 1.9 ± 0.4 24096.5 ± 763.5 5.2 ± 0.4 85.7 ± 1.6 70.8 ± 1.3

θ [deg.] 62.4 ± 7.0 108.8 ± 1.9 25.5 ± 17.0 149.9 ± 1.3 115.8 ± 4.6 13.9 ± 0.3 92.3 ± 0.9

φ [deg.] 0.0 ± 7.0 177.0 ± 0.2 0.0 ± 17.0 176.0 ± 0.2 0.0 ± 4.6 34.4 ± 0.3 149.3 ± 0.3

Double- hyper. Single- hyper. π−

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decay of  → p + π − are 28.0 ± 0.2 MeV and 37.8 ± 0.1 MeV, respectively, the value of total visible energy was greater than the sum of two Q-values and strongly supported the production and decay of a double- hypernucleus. By checking the case of twin hypernucleus formation, it was found that the case was rejected because of an inconsistent Q-value with the decay of Track #3. Finally, two cases were accepted for formation of double- hypernucleus (Aoki et al. 1991): 1. point A − + 12 C → point B

10  Be

point C

10 B

→

10 3  Be + H, 10 −  Be + π ,

→ 1 H + 3 He + 4 He + 2n, etc.,

B = 8.5 ± 0.7 MeV, B = −4.9 ± 0.7 MeV. 2. point A − + 14 N → point B

13  B

point C

13 C

→

13 1  B + H + n, 13 − C+π ,

→ 3 He + 4 He + 4 He + 2n, etc.

B = 27.6 ± 0.7 MeV, B = + 4.9 ± 0.7 MeV, where B− was not taken into account for the calculation of B at the formation point in the 1990s. Although no discussion of the above results was made by B , some theorists objected that the B in the case 1 was inconsistent with a previous result of 13 4∼5 MeV for the 10  Be. They recommended just the case of  B formation with the following process: point A − + 14 N →

14 ∗  C

+n →

point B

13  B

point C

13 C

→

13 1  B + H + n,

13 − C+π ,

→ 3 He + 4 He + 4 He + 2n, etc.

∗ If they took a 14  C formation process, the B could be +4.8 ± 0.7 MeV, which was in good agreement with the previous results (Dover et al. 1991). However, the interpretation of the above three cases was significantly changed by the detection of the Nagara event. The interpretation and physical quantities after the change are described in the next section on the Nagara event.

The Nagara Event About 6 months after the beam irradiation of the E373 experiment, when the analysis finally started to progress smoothly, the second detection of a double- hypernucleus was received (The author received an e-mail with a photograph of the event from a first-year master’s student on the evening of January 18, 2001).

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μ

Ξ π 6 He). The photograph was a superimposed image taken at various Fig. 30 The Nagara event ( focusing positions of the microscope. The name will be decided after discussion among all the students in the laboratory. For this event, “Nagara,” a clear stream in Gifu, Japan, was chosen

Table 3 Track lengths and angles with comments for each, from the Nagara event. The errors are just measurement errors. In the first paper reporting the Nagara event (Takahashi et al. 2001), there were several typos; the corrected values were presented with underline in Ahn et al. (2013). The density of the emulsion layer was 3.619 ± 0.025 g/cm3 Point A

B

C

Track no. #1 #2 #3 #4 #5 #6 #7 #8

Length [μm] 8.1 ± 0.3 3.2 ± 0.4 88.6 ± 0.5 ¯ 9.1 ± 0.3 82.1 ± 0.6 13697 742.6 ± 0.6 5868 ± 20

θ [deg.] 44.9 ± 2.0 57.7 ± 5.2 156.2 ± 0.5 77.7 ± 1.6 122.8 ± 1.0 81.0 ± 0.5 ¯ ¯ 138.5 ± 0.2 52.2 ± 1.2

φ [deg.] 337.5 ± 1.8 174.9 ± 2.9 143.0 ± 1.0 115.9 ± 0.8 284.2 ± 0.7 305.5 ± 0.2 ¯ 322.1 ± 0.3 123.7 ± 0.7

Double- hyper.

Single- hyper. Stopped in PS film π− Stopped in D-block Before stopped

As shown in Fig. 30, the position of the double- hypernucleus formation and the two positions of sequential decay were clearly distinguishable. The range and angle information for each track is listed in Table 3 with some comments. 6 He With the constraint of B − B− < 20 MeV, the unique process of  was decided by conservation of momentum in three dimensions and the energy at the production (point A) and decay (point B) of Track #1, as denoted below: − + 12 C → 6  He

→

6 4 3  He(#1) + He(#2) + H(#3), 5 1 −  He(#4) + H(#6) + π (#5).

The absence of neutral particle emission from the points A and B was supported by good coplanarities: −0.002 ± 0.030 and 0.003 ± 0.013, respectively. They were → → → → → → → calculated as (− r2 × − r3 ) · − r1 and (− r5 × − r6 ) · − r4 , where − ri is the unit vector of the #ith Track. The equations of B and B were obtained as a function of the B− : B = 7.13 + 0.87(±0.19)B− MeV,

(6)

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B = 0.89 + 0.87(±0.20)B− MeV.

(7)

The numerical values of the coefficient would be changed by modifying the mass of the − several years later, which will be described below. Although B− was assumed to be 0.13 MeV, which is the atomic 3D level energy of − in 12 C, for the first time the B and the B had been uniquely measured to be 7.25 ± 0.19 +0.18 +0.18 −0.11 MeV and 1.01 ± 0.20 −0.11 MeV, respectively (Takahashi et al. 2001; For this detection of the Nagara event, the paper of E176 (Aoki et al. 1991) received the 7th Outstanding Paper Award of Japan Physics Society for being the starting point to develop the study of doubly-strange hypernuclei). In any case, it was confirmed that the - interaction was attractive but weak, B ∼ 1 MeV, which is significantly less than the 4∼5 MeV previously reported. Thus, the three former events had to be re-analyzed. The interpretation of the first event presented by Danysz et al. (1963a, b) was modified to be consistent with the B of the Nagara event as follows (Davis 2005): 2 − + 12 C → 10  Be + H + n, 10  Be

→ 9 Be∗ + 1 H + π − , 9  Be

→ 4 He + 4 He + 1 H + π − .

With the above interpretation of an excited state of 9 Be by 3.05 MeV (Hashimoto and Tamura 2006), the B and the B were revised to be 14.7 ± 0.4 MeV and 1.3 ± 0.4 MeV, respectively (Ahn et al. 2013). Regarding the second event of 6  He (Prowse 1966) by D.J. Prowse, the B of +4.6±0.5 MeV was completely at odds with the 1.01 ± 0.20 +0.18 −0.11 MeV from the Nagara event. Even before the detection of the Nagara event, the correctness of the second event had been questioned, because no photographic image was presented (just a schematic drawing was published) and “· · · · · · not measured angles. No independent study or analysis, nor indeed observation, of the event is known” (Dalitz et al. 1989). For the event in E176, the following interpretation was accepted (Aoki et al. 2009): − + 14 N → 13  B

→

13 1  B(#1) + H(#2) + n, 13 ∗ C

+ π − (#4).

Under the assumption of B− being 0.17 MeV for − in the 3D atomic state in 14 N, B and B were revised to be 23.3±0.7 MeV and 0.6±0.8 MeV, respectively. The mass and its error have been updated for the masses of , − and π − in 2008 (Amsler et al. 2008). These revisions were taken into account in obtaining the above results. Regarding the Nagara event, B and B were originally obtained with respect to B− in equations (6) and (7). With the revision of the masses as noted

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above, the equations became: B = 6.79 + 0.91B− (±0.16) MeV,

(8)

B = 0.55 + 0.91B− (±0.17) MeV.

(9)

With these revised equations, the B and the B values are 6.91 ± 0.16 MeV 6 He under the assumption of B − = and 0.67 ± 0.17 MeV, respectively, for   0.13 MeV (Ahn et al. 2013).

The Demachi-Yanagi Event As shown in the photo of Fig. 31, it was clear that the Demachi-Yanagi event is associated with an off-vertex track (Track #4). However, it was not clear from the difference of the second (decay of double- hypernucleus) and the third (decay of single- hypernucleus) points. Thus, the developed slicing method was applied, in which a small piece of the emulsion is cut (1 × 1 cm) locating the event in the center, swelling the piece with water containing 10% glycerin, slicing the piece perpendicular to the surface with width of 0.5 mm, and comparing the sliced part from the side direction against the original image. Analyzing those images, which were presented in Ichikawa (2001) and Ahn et al. (2013), the time sequence of the event was found to be in the order of A → B → C, as shown in the schematic drawing of Fig. 31. The measured lengths and angles data for each track were listed in Table 4. From the relationship between the ionization and length, Track #4 was determined to be a proton. Taking into account the constraint of B −B− < 20 MeV, the following three processes at the production point were in agreement with the B of the Nagara event shown in Table 5. For the mode (1), a kinematic fitting was made, and the values of B and B were obtained to be 11.90 ± 0.13 MeV and −1.52 ± 0.15 MeV, respectively, where the B− was assumed to be 0.13 MeV for the − captured by12 C. The value of −1.52 ± 0.15 MeV was not inconsistent with the ∗ 8 ∗ Nagara result, if one assumes an excitation energy of ∼3 MeV for 10  Be like Be and 9 Be∗ . The event topology suggested that the mode (1) was the most probable due to back-to-back emission of Tracks #1 and #2 without emission of any neutral particle,

Ξ

μ

Fig. 31 Photograph of superimposed image and a schematic drawing of the Demachi-Yanagi event (Ahn et al. 2013). The event was named after the first station of a private train company near Kyoto University

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K. Nakazawa

Table 4 Lengths and angles for each track with comments for the Demachi-Yanagi event. Because Track #4 passed through the base film, the length of Track #4 in the base was converted to that in the emulsion. The density of the emulsion layer was 3.59 ± 0.02 g/cm3 Point A B C

Length [μm] 0.25 4.2 + − 0.07 287.0 ± 1.9 0.3 3.7 + − 0.1 23575 11.3 ± 0.7 8.4 ± 0.7

Track no. #1 #2 #3 #4 #5 #6

θ [deg.] 124.5 ± 5.1 57.7 ± 1.1 32.6 ± 7.8 94.2 ± 0.4 9.4 ± 3.4 7.2 ± 2.5

φ [deg.] 343.2 ± 3.5 159.4 ± 0.5 145 ± 10 209.8 ± 0.3 341 ± 20 21 ± 19

Double- hyper. Single- hyper. Proton

Table 5 Acceptable production modes for the Demachi-Yanagi event having consistent B with the Nagara event. By the kinematic fitting for mode (1), the value of B − B− becomes −1.62 ± 0.15 MeV Modes

Target

#1

#2

neutral

B − B− (MeV)

B − B− (MeV)

(1)

12 C

10 Be 

3H

−

0.24 11.46+ − 0.13

0.25 −1.96+ − 0.15

(2)

12 C

11 Be 

1H

1n

22.31+1.70 −1.09

1.75 4.09+ − 1.18

(3)

14 N

13 N 

1H

1n

3.16 27.81+ − 2.02

3.17 5.07+ − 2.03

π μ

Ξ

Fig. 32 Photograph of superimposed image and a schematic drawing of the Mikage event. The event was named after a historical street near Kyoto University, Japan

although the modes (2) and (3) could not be rejected. The difference of the two B values in the events of Danysz et al. (14.7±0.4 MeV) and Demachi-Yanagi (11.90 ± ∗ 0.13 MeV) probably is due to the excitation energy of 10  Be , 2.8 ± 0.4 MeV (Ahn et al. 2013).

The Mikage Event In the Mikage event, as shown in Fig. 32 with track data listed in Table 6, the particle of Track #6, which was a decay daughter of single- hypernucleus (Track #3), escaped upstream from the stack. The particle of Track #3 was identified to be a π − by the relationship between the energy loss and track length in the emulsion. Because the trajectory extrapolated for Track #3 was recorded in the SciFi U-block, its kinetic energy was extracted to be 49.1 ± 1.7 MeV. Regarding Track #4, a proton was the most probable particle following the measurement of energy loss and the track length.

70 Experimental Aspect of S = −2 Hypernuclei

2573

Table 6 Lengths and angles for each track with comments for the Mikage event. The length of Track #6 was the one measured in the emulsion that escaped from the stack. The density of the emulsion layer was 3.580 ± 0.071 g/cm3 Point A B C

Track no. #1 #2 #3 #4 #5 #6

Length [μm] 2.9 ± 0.3 3.5 ± 0.8 1.8 ± 0.4 19044.7 ± 10.6 4.5 ± 0.6 >12975.6

θ [deg.] 96.7 ± 1.7 94.5 ± 2.1 74.6 ± 11.3 78.2 ± 0.1 54.2 ± 2.0 49.6 ± 0.1

φ [deg.] 109.0 ± 0.7 247.6 ± 2.1 44.7 ± 3.5 183.1 ± 0.1 87.3 ± 2.1 23.8 ± 0.1

Double- hyper. Single- hyper. Proton π − (49.1 ± 1.7 MeV)

If one assumed that the single- hypernucleus decayed at rest, no possible interpretation was found corresponding to a large energy release with a minimum energy of 68.6 MeV at point C. Therefore, the single- hypernucleus must have decayed in flight, and one obtained three interpretations consistent with the B of the Nagara event: 6 (1)− + 12 C → He(#1) + 6 Li(#2) + n,

6  He

→3 H(#3) + 1 H(#4) + 2n.

1 (2)− + 12 C →11  Be(#1) + H(#2) + n,

11  Be

→9 Li(#3) + 1 H(#4) + n.

3 (3)− + 14 N →11  Be(#1) + He(#2) + n,

11  Be

→9 Li(#3) + 1 H(#4) + n.

A single- hypernucleus produces a mesonic decay with a π − . In a heavier hypernucleus like 9 Li rather than 3 H, the mesonic-decay rate is small, being less than 20%; thus, the case (1) was the most probable. Taking 0.13 MeV for B− in the atomic 3D level of 12 C for case (1) into account, the values of B and B were 10.01 ± 1.71 MeV and 3.77 ± 1.71 MeV, respectively (Ahn et al. 2013).

The Hida Event The Hida event, in which the (K− , K+ ) reaction occurred in the SciFi bundle downstream of the diamond target, is shown in Fig. 33 with a photograph and a schematic drawing. Tracks #6 and #7 escaped the stack and went into the SciFi D- and U-blocks, respectively. The lengths and angles are listed in Table 7. Based upon length data in the emulsion and in the D-block, the particle of Track #6 was determined to be a proton. The particle of Track #7 was also a proton, based upon measurement of ionization in the emulsion and its brightness in the U-block. Both of the decays of the double- (Track #1) and single- (#4) hypernuclei were non-mesonic. The charge of the single- hypernucleus was 2 or more as determined for Tracks #7 and #8, while the charge of the double- hypernucleus was 4 or more as determined for Tracks #4, #5, and #6. Because the − was absorbed in (12 C,14 N or 16 O), the number of charge was 5, 6, or 7, respectively, at point A. Two tracks (#2 and #3) were emitted at point A, so that the charge of the

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K. Nakazawa

Ξ

μ

Fig. 33 Photograph of superimposed image and a schematic drawing of the Hida event. The event was named after a scenic region surrounded by mountains in northern Gifu, Japan Table 7 Lengths and angles for each track with comments for the Hida event. The lengths of Tracks #6 and #7 were the ones measured in the emulsion but escaping from the stack. Coplanarities were not confirmed at points A and B with their being −0.434 ± 0.073 and −0.449 ± 0.006, respectively. The density of the emulsion layer was 3.684 ± 0.092 g/cm3 Point A

B

C

Track no. #1 #2 #3 #4 #5 #6 #7 #8

Length [μm] 2.3 ± 0.2 53.5 ± 0.2 4.7 ± 0.6 7.7 ± 0.1 74.9 ± 0.4 >5847.9 ± 11.3 >6728.8 ± 8.2 96.0 ± 0.4

θ [deg.] 53.3 ± 4.0 100.7 ± 0.2 91.7 ± 3.2 4.1 ± 0.5 74.6 ± 0.1 147.6 ± 0.1 31.0 ± 0.1 107.2 ± 0.1

φ [deg.] 116.0 ± 5.8 7.0 ± 0.1 256.7 ± 1.0 95.1 ± 6.2 180.5 ± 0.1 51.2 ± 0.1 115.3 ± 0.1 356.6 ± 0.1

Double- hyper.

Single- hyper. Proton, went into D-Block Proton, went into U-Block

double- hypernucleus should be 4 or 5, and the case for the absorption by 12 C was not possible. It was checked for the production processes of  Be and  B following absorption by 14 N or 16 O. All acceptable processes included neutron(s) emission under the constraint of | B − B− |< 20 MeV. With kinematic analysis, B and B in  B production were quite different from that of 13  B (E176), more than 3 σ . Regarding  Be production, the following two processes agreed with the B value of the Nagara event within 3 σ : − + 14 N →

12 1 1  Be + H(#2) + H(#3) + n,

− + 16 O →

11 1 4  Be + H(#2) + He(#3) + n.

11 The values of B for 12  Be and  Be were 22.48 ± 1.21 MeV and 20.83 ± 1.27 MeV assuming B− values for the 3D atomic level of 0.17 and 0.23 MeV, 11 respectively. To obtain B for 12  Be, it was necessary to use B ( Be), for 11 which there was no experimental data. For the B ( Be), 10.24 MeV for the value B (11  B) was assumed. Then B values of 2.00±1.21 MeV and 2.61±1.34 MeV 11 were obtained for 12  Be and  Be, respectively (Ahn et al. 2013).

70 Experimental Aspect of S = −2 Hypernuclei

2575

Ξ μ

π

Fig. 34 Photograph of superimposed image (left) and a schematic drawing (right) of the Mino event. The event was named after Mino, a granary in the southern part of Gifu, Japan. (Reproduced from Ekawa et al. (2019b) ©The Author(s) 2019) Table 8 Lengths and angles for each track with comments for the Mino event. The density of the emulsion layer was 3.486 ± 0.013 g/cm3 Point A

B

C

Track no. #1 #3 #4 #2 #5 #6 #7 #8 #9

Length [μm] 2.1 ± 0.2 17.5 ± 0.2 65.7 ± 0.5 50.6 ± 0.3 122.1 ± 0.2 >23,170 5.0 ± 0.2 116.7 ± 0.2 >7378

θ [deg.] 83.7 ± 8.9 121.9 ± 1.9 41.7 ± 1.7 90.2 ± 2.0 61.4 ± 1.8 106.2 ± 0.6 31.1 ± 2.8 100.3 ± 1.9 147.4 ± 0.3

φ [deg.] 256.1 ± 5.3 48.2 ± 1.3 166.7 ± 2.1 306.3 ± 1.3 347.0 ± 1.5 147.7 ± 0.4 297.0 ± 4.0 144.2 ± 1.3 355.7 ± 0.5

Double- hyper.

Single- hyper. Stopping in SSD

Passed through SSD

The Mino Event The Mino event detected in the E07 experiment is shown in Fig. 34. Track lengths and angles with comment are listed for each track in Table 8. → → → Because the coplanarity given by (− r7 × − r8 ) · − r9 at point C was 0.001 ± 0.043, no neutral charged particle was initially assumed to be emitted. Under the constraints that (a) the angle of Track #9 would agree with back-to-back direction for the summed momentum vector of Tracks #7 and #8 within 3 σ , and (b) the conservation of momentum and energy were required within 3 σ in the kinematic fitting, then, the case of decay at point C was accepted to be: 5  He

→ 4 He(#7) + 1 H(#8) + π − (#9).

The cases involving neutron emission will be discussed later. Point B was also coplanar with a coplanarity of 0.007±0.019 from Tracks #2, #5, 5 6 and #6; the decay mode was only allowed for the process of 13  B → He(#2) + 2 6 He(#5) + H(#6) in the cases of no neutral particle emission. Although He should decay into 6 Li, e− , and ν e in 806.7 ms (half-life), no track was emitted from the end point of Track #5. Therefore, the decay modes were checked for all nuclide combinations with neutron(s) at point B and the cases of multiple neutron emissions were possible. Under the kinematic analysis, neutrons were assumed to have same momentum; thus, the kinetic energy of neutrons and the length of Track #6 were

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given to be minimum and maximum, respectively. At point B, the cases remained for the decay of  Be or  B for Track #1 satisfying maximum length of Track #6, which stopped in the SSD. At point A, 17 cases were possible for the production of  Be or  B with or without neutron(s) for the value of B − B− to be less than 20 MeV within 3 σ . By considering the coplanarity of 0.000 ± 0.099 at point A, which is likely in the case of no neutron(s) emission, then the B would not be so large as the 11 12 Nagara event; production was considered for a 10  Be,  Be, or  Be. Therefore, the production and the decay would be as follows:   11 12 4 3 2 1 − + 16O → 10  Be,  Be,  Be (#1)+ He(#3)+ H, H, H (#4),  11 12 5 1 2 3 1 (10  Be,  Be,  Be) →  He (#2) + H, H, H (#5) + H) (#6) + xn, 5 He → 4 He(#7) + 1 H(#8) + π − (#9). 

Taking into account 0.23 MeV for − in 16 O, the values of B and B are obtained as listed in Table 9. These results are valid under the current condition for the constraint of | B − B− − Ex. |< 5 MeV, which was discussed in the section “Mass Reconstruction”. In the case of neutron(s) emission at point C, there were two cases to be considered: 3 H

→ 1 H + 1 H + π − + n,

4 H

→ 2 H + 1 H + π − + n,

This requires that  H was produced at point B from the decay of a  Li hypernucleus, 9 Li or 10  Li, that was produced at point A. However, the above decay processes with a neutron emission are very rare, appearing in fewer than 30 and 5 samples for 3 H and 4 H among 2000 detected events, respectively (Bertrand et al. 1970). Thus, the probability is very unlikely for emission of neutron(s) at point C. 11 12 Following the kinematic fitting for 10  Be,  Be, and  Be at point A, the 11 production of  Be was found to be the most likely, due to the chi-square value being a minimum as shown in Table 10. The Mino event was confirmed to be the Table 9 Possible results for B and B of the Mino event. Under the assumption of the − captured in atomic 3D orbit on 16 O, i.e., B− = 0.23 MeV, those values are presented. The 12 B (11  Be) of 8.2 ± 0.5 MeV for  Be was obtained by extrapolating known B values of  Be nuclei. Systematic errors were estimated by the mass error of − (0.07 MeV) and the error of binding errors of B (A−1  Be) Nuclide 10 Be  11 Be 

B (MeV) 15.05 ± 0.09 ± 0.07 19.07 ± 0.08 ± 0.07

B (MeV) 1.63 ± 0.09 ± 0.11 1.87 ± 0.08 ± 0.36

12 Be 

13.68 ± 0.08 ± 0.07

(−2.7± 0.08 ± 1.0)

B (A−1  Be) (MeV) 6.71 ± 0.04 (Davis 1986) 8.60 ± 0.07 ± 0.16 (Gogami et al. HKSJLab E05-115 Collaboration 2016) 8.2 ± 0.5

70 Experimental Aspect of S = −2 Hypernuclei Table 10 χ 2 and p-values at point A by kinematic fitting, where degrees of freedom are 3

2577 χ2

Process − − −

+

16 O

+

16 O

+

16 O

p-value (%)

→

10 Be +4He +3H 

11.5

0.93

→

11 Be +4He +2H 

7.28

6.35

→

12 Be +4He +1H 

11.3

1.02

μ

Ξ

Fig. 35 Photograph of superimposed image (left) and a schematic drawing (right) of the D001 event, the first detected double- hypernucleus in the E07 experiment Table 11 Lengths and angles for each track with comments for the D001 event. The density of the emulsion layer was 3.635 ± 0.014 g/cm3 Point A B C

Track no. #1 #3 #2 #4 #5 #6

Length [μm] 4.1 ± 0.3 5.5 ± 0.4 1.1 ± 0.2 5728.6 ± 5.9 1.8 ± 0.5 1762.6 ± 2.6

θ [deg.] 53.6 ± 3.6 140.3 ± 2.1 33.7 ± 15.5 108.9 ± 2.4 50.6 ± 6.7 55.1 ± 1.9

φ [deg.] 82.7 ± 2.8 188.4 ± 2.7 153.3 ± 20.2 1.2 ± 2.0 187.4 ± 9.1 91.5 ± 1.6

Double- hyper. Single- hyper.

production and decay of 11  Be, as the most probable case, which led to the B and B being 19.07 ± 0.14 MeV and 1.87 ± 0.37 MeV, respectively. The B value is not consistent with the B of the Nagara event (0.67 ± 0.17 MeV) by more than 3 σ (Ekawa et al. 2019b; The paper Ekawa et al. (2019b) was awarded the 26th JPS Outstanding Paper Award (2021) for the first demonstration that the B changed due to differences in the core).

The D001 Event An image and a schematic drawing of the D001 event are shown in Fig. 35. The data for lengths and angles of tracks are summarized in Table 11. At point A, the point at which the − was absorbed at rest, two production modes 8 Li and 10 Be were possible within 3 σ as listed in Table 12 under the constraint of   of | B − B− − Ex. |< 5 MeV. Assuming B− was captured in the atomic 3D level, the set of (B , B ) was (17.5 ± 1.46 MeV, 6.34 ± 1.46 MeV) or (15.05 ± 8 Li or 10 Be, respectively. Regarding point B, the 2.78 MeV, 1.63±2.78 MeV) for   8 Li and 10 Be with neutron(s) emission acceptable decay modes are presented for   in Nyaw et al. (2020). The two types of double- hypernucleus are discussed in the section “Characteristics of Double- Hypernuclei”.

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Table 12 Acceptable production modes for the D001 event at point A Cases

Target

#1

#3

Neutral

B − B− (MeV)

B − B− (MeV)

(1)

12 C

8  Li

4 He

1n

17.50 ± 1.46

6.34 ± 1.46

(2)

14 N

10 Be 

4 He

1n

15.05 ± 2.78

1.63 ± 2.78

Table 13 Published data for B and B . An atomic 3D absorption of − is assumed for all cases. Those data from Danysz et al. and Hida are included based on consistent interpretations with the Nagara event. The daughter single- hypernucleus for E176 and Danysz et al. is the excited state. The difference of B (B ) values between Demachi-Yanagi and Danysz et al. would be the excitation energy of ≈2.8 MeV of 10  Be. The errors are separated into statistical and systematic ones for the Mino event in the text Ev. name

Nuclide Target B (MeV) B (MeV) Comments

Nagara

6  He

12 C

6.91 ± 0.16

0.67 ± 0.17

Uniquely identified

Danysz et al.

10 Be 

12 C

14.7 ± 0.4

1.3 ± 0.4

10 Be 

E176

13 B 

14 N

23.3 ± 0.7

0.6 ± 0.8

13 B 

Demachi-

10 Be∗ 

12 C

11.90 ± 0.13 −1.52± 0.15

6  He

12 C

10.01 ± 1.71 3.77 ± 1.71

11 Be 

12 C

22.15 ± 2.94 3.95 ± 3.00

11 Be 

14 N

23.05 ± 2.59 4.85 ± 2.63

11 Be 

12 C

20.83 ± 1.27 2.61 ± 1.34

12 Be 

14 N

20.48 ± 1.21 (2.00 ± 1.21) Assumed 10.24 MeV for B (11  Be)

10 Be 

16 O

15.05 ± 0.11 1.63 ± 0.14

11 Be 

16 O

19.07 ± 0.11 1.87 ± 0.37

12 Be 

16 O

13.68 ± 0.11 −2.7± 1.0

8  Li

12 C

17.50 ± 1.46 6.34 ± 1.46

10 Be 

14 N

15.05 ± 2.78 1.63 ± 2.78

Yanagi Mikage

Hida

Mino

D001

→9 Be∗ (Ex. = 3.0 MeV)

∗ →13  C (Ex. = 4.9 MeV)

Most probable (topology) ∗ Ex. ≈ 2.8 MeV for 10  Be

Most probable (mesonic decay)

Most probable (χ 2 minimum)

Likely by B

Characteristics of Double- Hypernuclei The data for B and B of the published double- hypernuclear events as above are summarized in Table 13. The nuclei absorbing − are discussed in terms of these results except for the Hida event, where the D001 event is counted to be the absorption by 14 N due to its similar B with Danysz data. Based on the molar ratio among 12 C, 14 N, and 16 O which is 4:1:2 (see Table 1), it is reasonable that the number of nuclei in the production of double- hypernuclei is 4, 2, and 1 for 12 C, 14 N, and 16 O, respectively. The B of the most probable interpretations are shown in Fig. 36. Although the values of B are small, a linear mass number (A) dependence is seen for B . The discrepancy of B between Danysz et al. and Demachi-Yanagi is nearly

70 Experimental Aspect of S = −2 Hypernuclei

2579

ΛΛ

ΛΛ ΛΛ

ΛΛ

ΛΛ

ΛΛ

Fig. 36 Relationship between B and mass number A based on nominated double- hypernuclei. The Demachi-Yanagi event is based upon the excited state of 10  Be. If the D001 event assuming 10  Be is taken into account, the linearity between B and to A is seen Fig. 37 The plot of B values of double- hypernuclei versus the mass number A

Δ

ΛΛ

ΛΛ

ΛΛ

ΛΛ

ΛΛ ΛΛ

2.8 MeV, which probably is due to the excitation energy of 10  Be. Regarding the 10 D001 event, its B value for the case  Be is consistent with that of Danysz 8 Li is not rejected due to a lack of et al. result. However, the B for the case of  candidates for the A = 8 nucleus. The plot of B in Fig. 37 seems to show that the value of B increases a little as the mass A becomes larger.

 Hypernucleus The binding energy, B− , the energy due to the − binding to the absorbing nucleus, determines the existence of  hypernucleus, if the B− is greater than the energy produced by Coulomb force. As mentioned in the section “Mass

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K. Nakazawa

Reconstruction,” B− cannot be measured in double- hypernuclei, if the nuclides are not uniquely identified in the decay. On the other hand, the B− can be measured in an event where two single- hypernuclei are ejected (twin hypernucleus), after the decay of the − absorbing nucleus, because the masses are known for almost all decay-daughter particles. At present, unknown single- hypernuclei cannot, of course, be taken into account. To search for  hypernuclei, one requires that the B− value be positive within 3σ . In this section, some of the twin hypernucleus events detected so far are presented and the characteristics of the parent  hypernuclei will be discussed.

Two Events of Twin Hypernucleus from the E176 Experiment Photographs of two candidates of  hypernucleus are shown in Fig. 38 (Aoki et al. 2009). Track lengths and angles are listed in Tables 14 and 15 for the event numbers of 10-09-06 and 13-11-14, respectively. These events indicated that − particles with two strange quarks were indeed absorbed by emitting two single- hypernuclei (Aoki et al. 1993, 1995). Kinematic analysis showed that both − particles were absorbed by 12 C, and then 4 H (Track #1) and 9 Be (Track #2) were likely emitted. On 10-09-06, if

Ξ

π π

Ξ

Fig. 38 Superimposed images of  hypernucleus candidates detected in the E176 experiment. In both events, − particles were captured at rest at point A. Both of Track #1s decayed into two charged particles of Tracks #3 and #4 at point B. Track #2 decayed with three (#5, #6, and #7) or two (#5 and #6) charged particles for the event number of 10-09-06 (a) or 13-11-14 (b), respectively, at point C. There are no scales in the photos, but the length of Track #2 was 5 and 3 μm in (a) and (b), respectively. (These photos were published in Nucl. Phys. A 828, S. Aoki et al., ‘Nuclear capture at rest of − hyperons’, 221 ©Elsevier (2009) (Aoki et al. 2009)) Table 14 Lengths and angles for each track with comments for the 10-09-06 event. The density of the emulsion layer was 3.385±0.032 g/cm3 , which was the same density in the case of the E176 double- hypernucleus Point A B C

Track no. #1 #2 #3 #4 #5 #6 #7

Length [μm] 116.1 ± 2.2 4.7 ± 0.6 39992.8 ± 1195.8 9.5 ± 0.6 21796.0 ± 411.9 100.9 ± 3.2 9.2 ± 1.2

θ [deg.] 101.4 ± 2.0 76.9 ± 10.0 13.1 ± 0.5 170.7 ± 0.9 91.5 ± 1.0 160.4 ± 2.0 47.3 ± 8.0

φ [deg.] 324.8 ± 0.9 147.9 ± 5.0 222.2 ± 1.0 46.7 ± 1.9 160.4 ± 0.5 122.3 ± 2.5 320.0 ± 5.0

Single- hyper. Single- hyper. π− Fast proton (1 H)

70 Experimental Aspect of S = −2 Hypernuclei

2581

Table 15 Lengths and angles for each track with comments for the 13-11-14 event. The density of the emulsion layer was 3.385± 0.032 g/cm3 , which was the same density in the case of the E176 double- hypernucleus Point A B C

Track no. #1 #2 #3 #4 #5 #6

Table 16 B− values [MeV] obtained by two events from the E176 experiment. Degrees of freedom for χ 2 was 3

Length [μm] 64.2 ± 1.4 3.1 ± 0.4 34099 8.6 ± 0.5 68.5 ± 1.6 37.4 ± 1.3

θ [deg.] 123.6 ± 2.2 −56.3± 7.7 159.9 ± 0.4 −21.6± 10.2 117.9 ± 3.0 −61.2± 7.0

φ [deg.] 175.0 ± 0.4 0.0 ± 7.7 175.3 ± 0.4 0.0 ± 10.2 179.9 ± 0.2 0.0 ± 7.0

Single- hyper. Single- hyper. π−

Cases

10-09-06 (χ 2 = 0.4)

4 H + 9 Be  

0.82 ± 0.17

3.89 ± 0.14

4 H∗ 

−0.23 ± 0.17

2.84 ± 0.15

4 H + 9 Be∗  

−

0.82 ± 0.14

4 H∗ 

−

−0.19 ± 0.15

+ +

9  Be

9 ∗  Be

13-11-14 (χ 2 = 1.3)

both 4 H and 9 Be were emitted in the ground state, the B− was inferred as 0.54 ± 0.20 MeV (Aoki et al. 1993). Regarding the 13-11-14 event, the B− was 9 0.62 +0.18 −0.19 MeV, where  Be was emitted in the first excited state (Aoki et al. 1995). The energy of the atomic 3D orbit in which − particles are bound to 12 C by Coulomb force alone is 0.13 MeV, so it was argued that both events are bound in the nuclear 2p orbit with nuclear force added, and U0 would be ∼16 MeV, and so on (Yamamoto et al. 1994), where the U0 is the real part of a WoodsSaxon potential. Then, based on the small probability in a theoretical calculation for − absorption in the nuclear 2p orbit, a publication claimed, “These events are clearly atomic 3D orbital absorption and are useless as a source of information on  hypernuclei” (Batty et al. 1999). Subsequently, mainly due to the update of the − particle mass by about 0.4 MeV, the B− value in both events was updated to 0.82 MeV using kinematic fitting, as shown in Table 16 (Aoki et al. 2009).

The Kiso Event As mentioned in the section “Overall-Scanning Method Searching in Whole Volume of the Emulsion,” a twin hypernucleus event was detected among 8 million images in the R&D for the overall-scanning method using the E373 emulsion sheet. A superimposed image and schematic drawing of the event are shown in Fig. 39. Lengths and angles of tracks are listed in Table 17. A notable feature of the Kiso event is that the daughter particle (Track #3) of one (Track #1) of the two single- hypernuclei decays into two particles (Tracks #7 and #8) with the similar energy (length). This is due to 8 Be∗ →4 He +4 He, and the β-ray emission from the decay point (D) has narrowed down the daughter nucleus to be a 8 He or a 8 Li. Although Tracks #1 and #2 are colinear, the − absorption point

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Ξ

μ

Fig. 39 A superimposed image from photographs (left) and a schematic drawing (right) of the Kiso event. This event was named the Kiso after the majestic Kiso River, Japan. (Reproduced from Nakazawa et al. (2015) ©The Author(s) 2015) Table 17 Lengths and angles for each track with comments for the Kiso event. The total length was 77.1 ± 0.3 μm from point B to C. The density of the emulsion layer was 3.486 ± 0.013 g/cm3 Point A B C D

Track no. #1 #2 #3 #4 #5 #6 #7 #8

Length [μm] 8.0 ± 0.3 69.1 ± 0.5 13.3 ± 0.4 ± > 4990.7 2492.0 ± 3.9 37.3± 0.7 6.7 ± 0.3 5.8 ± 0.3

θ [deg.] 133.0 ± 3.0 40.4 ± 0.9 102.3 ± 2.3 145.0 ± 0.9 43.1 ± 1.3 131.9 ± 1.3 49.6 ± 4.2 131.0 ± 4.5

φ [deg.] 13.2 ± 3.2 193.1 ± 1.2 340.4 ± 1.6 85.4 ± 1.3 191.8 ± 1.5 29.2 ± 1.3 132.6 ± 4.3 318.9 ± 4.7

Single- hyper Single- hyper Out of the emulsion stack

4 He 4 He

from 8 Be from 8 Be

A was clear due to an auger electron emission. Point A reappeared as the point to balance momentum of Tracks #1 and #2 between points B and C, where the length is 77.1 ± 0.3 μm. The production and decay sequence, together with the kinematic analysis, was found to be: point A point B

− + 14 N → 10  Be

10 5  Be(#1) +  He(#2),

→ 8 Li(#3) + 1 H(#4) + n, 8

point D

Li → 8 Be∗ (2+ ) + e− (+ν e ), 8

point C

5  He

Be∗ (2+ ) → 4 He(#7) + 4 He(#8),

→ 1 H(#5) + 2 H(#6) + 2n, etc.

The B− value was obtained to be 4.38 ± 0.25 MeV or 1.11 ± 0.25 MeV whether was in the ground or excited state. Although it remained indefinite whether was the ground or excited state, it was far larger than B− = 0.17 MeV in the atomic 3D level for − particles in 14 N, making it the first detection of a  hypernucleus (15  C) (Nakazawa et al. 2015; The publication Nakazawa et al. (2015) earned the 22nd JPS Outstanding Paper Award (2017)). Recently, the level energies of the ground and excited states of 10  Be have been measured with improved accuracy (Gogami et al. HKSJLab E05-115 Collaboration

10 Be  10 Be 

70 Experimental Aspect of S = −2 Hypernuclei

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2016), and the B− value of the Kiso event has been updated to 3.87 ± 0.21 MeV (for the ground state) or 1.03 ± 0.18 MeV (for the first excited state) (Hiyama and Nakazawa 2018).

μ

The Kinka Event The top view of the Kinka event along the optic axis is seen in Fig. 40, and track information is listed in Table 18. In this event, there was about 1.4 μm indeterminacy in the at-rest absorption point A of − particle, as seen in Fig. 40a and c; then the length from point B to C was also listed in Table 18 as Tracks #1 + #2. On the end point of Track #3, a β-decay was seen in Fig. 40b. In the cases of absorption by either 14 N or 16 O, the B− was 8∼13 MeV if the − was absorbed by 14 N and −13∼0 MeV for 16 O absorption after taking into account the 1.4 μm ends shown in Fig. 40c.

Ξ

Ξ μ

μ

Fig. 40 A photograph of the Kinka event. (a) Two single- hypernuclei (Tracks #1 and #2) were emitted from point A. The particle of Track #1 decayed into three charged particles at point B. (b) The particle of Track #3 produced a β-decay at point D. (c) A magnified image around the − absorption point A, which was indefinite about 1.4 μm. (Kinka was named after Mt. Kinka, which commands a panoramic view of the Nobi Plain in the center of the Japanese archipelago. Reproduced from Kinbara et al. (2019) ©The Author(s) 2019). Table 18 Track lengths and angles of the Kinka event. The track length was measured as Tracks #1 + #2 in the photo from top view of Fig. 40a. The length of each #1 and #2 was measured from the side view of Fig. 41a. The dried emulsion density was 3.71 ± 0.08 g/cm3 Point A

B

D

Track no. #1 #2 #1 + #2 #3 #4 #5 #6 #7

Length [μm] 4.34 ± 0.20 5.74 ∓ 0.20 10.08 ± 0.17 75.39 ± 0.40 >6110 7.79 ± 0.40 4338.4 ± 1.6 1185.3 ± 0.9

θ [deg.] 95.2 ± 2.3 81.2 ± 1.3

φ [deg.] 183.4 ± 2.7 1.4 ± 1.1

79.0 ± 0.4 72.8 ± 0.4 88.6 ± 1.2 149.6 ± 0.3 122.6 ± 0.2

305.2 ± 0.2 103.4 ± 0.9 249.3 ± 1.9 254.1 ± 0.9 313.4 ± 0.1

Single- hyper Single- hyper β-decay at the end point Out of the emulsion stack

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Ξ

Fig. 41 (a) A superimposed image of photographs from the side after swelling and slicing as was the Demachi-Yanagi event. (b) A magnified image of (a). (c) Schematic drawing of the event topology after the identification of the − absorption point of A. (Reproduced from Yoshimoto et al. (2021) ©The Author(s) 2021)

Therefore, a tiny part of the emulsion sheet was cut out, swollen, and sliced along the optical axis direction where it is normally monitored and observed from the “side” to identify the absorption point, which was similar to the treatment for the Demachi-Yanagi event. In Fig. 41c, the absorption point was determined based on (b), which is a magnified image of (a). The particle for Track #2 was identified as only possible for 5 He by checking all decay modes of single- hypernucleus. Track #3 emits a low-energy electron at point D. The energy corresponding to the range of the electron (∼60 keV) is greater than the energy of a β in 3 H ( 28 MeV, excluding double-, twin-, and single- hypernucleus events, are shown in Fig. 45. The Q value is 80 MeV for the  − absorption ( − + p →  + n). Thus, even though the Q value for the case of the − absorption is smaller, a distribution is seen for Evis that is larger than the  − absorption in Fig. 45a. In Fig. 45b, the distribution of − absorption events is

Table 21 Light or heavy element absorption according to the presence of short track(s) and Auger electron(s) for σ -stop events Short track Yes Yes No No Total

Auger Yes No Yes No

25 155 103 134 417

E373 (%) (6.3 ± 1.2) Light (36.9 ± 3.4) (43.2 ± 3.8) (24.6 ± 2.7) Heavy (32.3 ± 3.1) (56.8 ± 4.6)

4 18 17 13 52

E176 (%) ( 8 ± 4) Light. (35 ± 9) (42 ± 11) (33 ± 9) Heaavy (25 ± 8) (58 ± 23)

Table 22 Number of events classified according to topology, presence of short track(s), and visible energy release in light nuclei (C, N, O) absorption of a − hyperon. The Evis is the total visible energy release given by charged tracks assumed to be 1 H Signal Double- hypernucleus Twin single- hypernucleus Single- hypernucleus σ -stop events with Evis > 28 MeV Total (− absorbed by light elements)

E373 9 2 28 88 180

E176 1 2 7 8 22

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larger than that of the decay of the heavy single- hyperfragment produced with a high-energy beam. A  hyperon in heavy nuclei tends to produce non-mesonic decay ( + N → N + N) and a Q value of 176 MeV. It was concluded that events with Evis above 160 MeV were caused by non-mesonic decay of two  hyperons, and then the events having a such large Evis were from the production of heavy doubly strange hypernuclei. The frequency of heavy doubly strange hypernucleus production was 3/26 (about 10%) for − absorption in the emulsion in the E176 experiment (Aoki et al. 2009). For the E373 experiment, the same condition as above was applied for events showing heavy-element absorption (Theint et al. 2019), and the data are summarized for the number of events in Table 23. In Table 24, the trapping probabilities are summarized for light or heavy element absorption of a − hyperon in the emulsion.

Ξ Ξ Σ

Fig. 45 The left vertical axis for both (a) and (b) shows the entry number of − capture events. The right vertical axes are counts of  − captures and heavy hyperfragments for (a) and (b), respectively. (a) Evis distribution in the cases of − captures (solid line) by heavy elements in the emulsion and  − captures (dotted line) (Anderson et al. 1963), where both captures occurred by heavy elements in the emulsion. The number of events for  − captures is normalized to that of − capture events with Evis to be more than 80 MeV. (b) Evis distributions for − captures (sloid line) by heavy elements in the emulsion and for of heavy single- hyperfragments (dotted line) (Lagnaux et al. 1964) Table 23 Number of events classified according to absence of short track and Evis in heavy nuclei (Ag, Br) absorption of a − hyperon

Signal σ -stop with Evis > 160 MeV (Double-) σ -stop with Evis > 28 MeV Total (− absorbed by heavy elements)

E373 10 111 237

E176 3 19 30

Table 24 Trapping probabilities of  hyperons in σ -stop events in the E373 experiment. The parenthetical values show the results of E176 Trappingl 2 At least 1

Light elements (C, N, O) 5.0 ± 1.7% (4.8%) 69.4 ± 8.1% (48%)

Heavy elements (Ag, Br) 4.2 ± 1.4% (1.7%) 51.1 ± 5.7% (36%)

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Double- Hypernuclei and the H Dibaryon In the baryon-baryon interaction in the SUf (3) singlet state, it was hypothesized for an attractive force that a 6-quark bound state (uuddss) could be formed as an exotic state with baryon number (B = 2), in particular an H dibaryon having a mass of 2M() − 81 MeV/c2 (2.15 GeV/c2 ) (Jaffe 1977). It is strongly related to the confinement of quarks in the baryon, which can also be studied through tetraquarks (qq¯qq¯ : B = 0) and pentaquarks (qqqq¯q : B = 1). If the mass of the H is smaller than two  particles [M(H) < 2M()], then even if a  pair is formed by absorbing a − particle, it will immediately be converted into an H particle via the strong interaction and no sequential decay of a double- hypernucleus should be observed. On the other hand, if sequential decay is observed, the mass of the H could be narrowed to 2M() − B < M(H) < 2M() (= 2231.37 MeV/c2 ). 10 The events found in the early days, 13  B (Aoki et al. 2009) and  Be (Davis 2005), provided a lower mass limit for the H dibaryon of 2208.1 ± 0.7 MeV/c2 and 2216.7 ± 0.4 MeV/c2 , respectively. The Nagara event lowered the H mass gap to 6.91 ± 0.16 MeV/c2 with 2M() value, so that the H mass limit became 2224.46 MeV/c2 (Ahn et al. 2013). It is now suggested that the H may exist not as a ground state but as a resonant state at 10∼15 MeV/c2 greater than 2M() (Yoon et al. 2007), and a high-statistics experiment (E42) is underway at J-PARC (E42 (J-PARC) “Search for H-Dibaryon with a Large Acceptance Hyperon Spectrometer” (Available at: https://j-parc. jp/researcher/Hadron/en/pac_1201/pdf/KEK_J-PARC-PAC2011-06.pdf. Date last accessed 21 July 2022)).

Future Prospect As previously mentioned, E07 irradiated the emulsion sheets with an intense K− beam. In the hybrid-emulsion method, only about 30% of the (K− , K+ ) events could be tagged, and no  n (K− , K 0 )− reactions could be observed. It is estimated for that about ten times more doubly strange hypernucleus can be detected if the whole volume of the emulsion sheets was probed via overall-scanning rapidly and reliably, as mentioned briefly in the section about the Kiso event of E373. The hardware for reading the sheets has been upgraded from the 0.2 megapixel CCD camera (50 fps) in E373 to a 4 megapixel CMOS camera (180 fps), and the speed is being increased by an optical system loading objective lens with a piezo actuator mounted on a lens barrel driven by a stepping motor (The driving software for CMOS Camera and Piezo actuator are managed by Dr. M. Yoshimoto). On the software side, an object detection method based on machine leaning has been developed, replacing the method that picks up a vertex as a cross-point of detected lines of tracks. First, three-dimensional test events were created with Geant4, and those events were transformed into teaching images, where the transformation was made to mimic a microscopic image in the emulsion sheet with GAN

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σ

(generated adversarial networks) (https://arxiv.org/abs/1406.2661, http://arxiv.org/ abs/1711.11585). Using those images, a machine learning model was developed, based on Mask R-CNN (region-based convolutional neural networks) (https:// arxiv.org/abs/1703.06870), which is effective for object detection. This model has been applied to actual emulsion images to detect the events. Gifu University is responsible for the hardware improvement, while Saito High Energy Nuclear Research Laboratory at RIKEN is responsible for the software development. The performance of event detection with machine learning was evaluated in the detection of alpha decay events being used for range-energy calibration in the emulsion sheets. This improved the previous 34% detection efficiency (S/N ∼ 0.2) (Yoshida et al. 2021) to 80% efficiency (S/N ∼ 0.2) (Kasagi et al. 2023). Currently, the detection of 3 H is being vigorously pursued with machine learning. The hypertriton 3 H (pn [= d]+) is not understood consistently in terms B and lifetime, even though it is the lightest hypernuclear bound state. Figure 46 shows the first event of 3 H production and decay detected with machine learning (Kasagi et al. 2022). The shape of the decay point (T-shaped) was learned by machine learning. The B of 3 H is measured with high precision using an emulsion, and the lifetime is still being measured using a 6 Li ion beam at GSI (Gesellschaft für Schwerionenforschung: Gesellschaft for Heavy Ion Research) WASA-FRS experiment (Saito et al. 2021). Based on our experience with 3 H detection, a machine learning model suitable for doubly strange hypernuclei searches will be created, and mass detection by overall-scanning will be attempted in the near future.

π

Λ

μ

π

Λ

μ

Fig. 46 Formation and decay of 3 H in the emulsion. The intersection of the three tracks is the decay point of 3 H, where the short 3 He track and the thin π − track are emitted in opposite directions. The source of the 3 H track is an at-rest absorption point of wobbly charged particle (thought to be a K− ). (Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Review Physics, New directions in hypernuclear physics, T.R. Saito et al., Copyright, (2021))

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References J.K. Ahn et al., Phys. Rev. C 88, 014003 (2013) M. Akashi et al., Proceedings of 14th ICRC, München, vol. 7 (1975), p. 2549 D.E. Alburger, M. May, Nucl. Instrum. Methods A 443, 27 (2000) C. Amsler et al., Particle Data Group, Phys. Lett. B 667, 1 (2008) B. Anderson, O. Skjeggestad, D.H. Davis, Phys. Rev. 132, 2281 (1963) S. Aoki et al., Prog. Theor. Phys. 85, 1287 (1991) S. Aoki et al., Prog. Theor. Phys. 89, 493 (1993) S. Aoki et al., KEK Preprint 94-163 (1994) A. Aoki et al., Phys. Lett. B335, 45 (1995) S. Aoki et al., Nucl. Phys. A 828, 191 (2009) H. Band¯o, T. Motoba, J. Žofka, Int. J. Mod. Phys. A 5, 4021 (1990) W.H. Barkas et al., Il Nouvo Cimento 8, 194 (1958) C.J. Batty, E. Friedman, A. Gal, Phys. Rev. C 59, 295 (1999) D. Bertrand et al., Nucl. Phys. B 16, 77 (1970) R.H. Dalitz et al., Proc. R. Soc. A 426, 1 (1989) M. Danysz, J. Pniewski, Phil. Mag. 44, 348 (1953) M. Danysz et al., Nucl. Phys. 49, 121 (1963a) M. Danysz et al., Phys. Rev. Lett. 11, 29 (1963b) D.H. Davis, Contemp. Phys. 27, 91 (1986) D.H. Davis, Nucl. Phys. A 754, 3c (2005) C. Dilworth, S.J. Goldsack, L. Hilschberg, Nuove Cimento 11, 113 (1954) C.B. Dover, A. Gal, Ann. Phys. 146, 309 (1983) C.B. Dover et al., Phys. Rev. C 44, 1905 (1991) H. Ekawa, Ph.D thesis, Kyoto University, unpublished (2019a). https://www-nh.scphys.kyoto-u. ac.jp/articles/thesis/dron/d-ekawa.pdf H. Ekawa et al., Prog. Theor. Exp. Phys. 2019, 021D02 (2019b) E. Friedman, A. Gal, Phys. Lett. B 820, 136555 (2021) M. Fujita et al., Nucl. Instrum. Methods A 1042, 167439 (2022) T. Gogami et al. [HKS(JLab E05-115) Collaboration], Phys. Rev. C 93, 034314 (2016) P.A.M. Guichon, A.W. Thomas, K. Tsushima, Nucl. Phys. A 814, 66 (2008) Y.T. Harada, Y. Hirabayashi, Phys. Rev. C 103, 024605 (2021) O. Hashimoto, H.Tamura, Prog. Part. Nucl. Phys. 57, 564 (2006) S. Hayakawa, Ph.D thesis, Osaka University, unpublished (2019). https://ir.library.osaka-u.ac.jp/ repo/ouka/all/72648/30645_Dissertation.pdf S.H. Hayakawa et al., Phys. Rev. Lett. 126, 062501 (2021) E. Hiyama, K. Nakazawa, Annu. Rev. Nucl. Part. Sci. 68, 131 (2018) K. Hoshino et al., Proceedings 14th ICRC, München, vol. 7 (1975), p. 2442 A. Ichikawa, Ph.D thesis, Kyoto University, unpublished (2001) (Available at: https://www1.gifuu.ac.jp/~physics/J-PARC/Docs/Ichithesis.pdf) A. Ichikawa et al., Nucl. Instrum. Methods A 417, 220 (1998) A. Ichikawa et al., Phys. Lett. B 500, 37 (2001) K. Imaeda, M. Kazuno, I. Odagiri, Lecture Notes for the Annual Meeting of the Japanese Physical Society 12.2(0), 21 (1957), in Japanese (Available at: https://www.jstage.jst.go.jp/ article/jpsgaiyoi/12.2/0/12.2_21/_pdf/-char/ja. Date last accessed 21 July 2022) K. Imai, K. Nakazawa, H. Tamura, Proposal for K1.8 beam line at J-PARC (2006) (Available at: https://j-parc.jp/researcher/Hadron/en/pac_0606/pdf/p07-Nakazawa.pdf. Date last accessed 21 July 2022) R.L. Jaffe, Phys. Rev. Lett. 38, 195 (1977) M. Juriˇc et al., Nucl. Phys. B 52, 1 (1973) A. Kasagi et al., to be submitted to Nucl. Instr. Meth. A (2023)

2594

K. Nakazawa

A. Kasagi et al., Eur. Phys. J. A 58, 190 (2022) A. Kasagi et al., Supl. Rev. Mex. Fis. 3, 0308122 (2022) K. Khaustov et al., Phys. Rev. C 61, 054603 (2000) S. Kinbara et al., Prog. Theor. Exp. Phys. 2019, 011H01 (2019) S. Kinoshita, Proc. R. Soc. A 83, 432 (1910) M. Kohno, Phys. Rev. C 100, 024313 (2019) J.P. Lagnaux et al., Nucl. Phys. 60, 97 (1964) C. Lattes, H. Muihead, G. Occhialini, C. Powell, Nature 159, 694 (1947) P.M. Lin et al., Bull. Soc. Photogr. Imag. Jpn. 32(1), 10 (2022) A.S. Mondal et al., IL Nuove Cimento A 54, 333 (1979) M.M. Nagels, T.A. Rijken, Y. Yamamoto, Phys. Rev. C 99, 044003 (2019) K. Nakazawa et al., Prog. Theor. Exp. Phys. 2015, 033D02 (2015) K. Niu et al., Prog. Theor. Phys. 46, 1644 (1971) A.N.L. Nyaw et al., Bull. Soc. Photogr. Imag. Jpn. 30(2), 22 (2020) H. Polinder, J. Haidenbauer, U.G. Meißner, Phys. Lett. B 653, 29 (2007) D.J. Prowse, Phys. Rev. Lett. 17, 782 (1966) T.R. Saito et al., Nat. Rev. Phys. 3, 803 (2021) K. Sasaki et al., Nucl. Phys. A 998, 121737 (2020) R. Shyam, K. Tsushima, arXiv:1901.06090 [nucl-th] M.K. Soe et al., Nucl. Instrum. Methods A 848, 66 (2017) T. Sun, E. Hiyama, H. Sagawa, H.-J. Schulze, J. Meng, Phys. Rev. C 94, 064319 (2016) H. Takahashi et al., Phys. Rev. Lett. 87, 212502 (2001) H. Takahashi et al., Nucl. Instrum. Methods A 483, 689 (2002) T. Takahashi et al., Prog. Theor. Exp. Phys. 2012, 02B010 (2012) T. Tamagawa et al., Nucl. Phys. A 691, 234c (2001) A.M.M. Theint et al., Prog. Theor. Exp. Phys. 2019, 021D01 (2019) N. Ushida et al., Phys. Rev. Lett. 56, 1771 (1986) M. Yamaguchi et al., Prog. Theor. Phys. 105, 627 (2001) Y. Yamamoto et al., Prog. Theor. Phys. 117, 281 (1994) C.J. Yoon et al., Phys. Rev. C 75, 022201 (2007) J. Yoshida et al., Nucl. Instrum. Methods A 847, 86 (2017) J. Yoshida et al., Nucl. Instrum. Methods A 989, 164930 (2021) M. Yoshimoto et al., Prog. Theor. Exp. Phys. 2021, 073D02 (2021) P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020), p. 540

Theoretical Studies in S = −1 and S = −2 Hypernuclei

71

Emiko Hiyama and Benjamin F. Gibson

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Three-Body nnΛ System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Gaussian Expansion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results for nnΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonant State of nnΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superheavy Hydrogen Λ Hypernucleus 6Λ H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Neutron-Rich He Isotope Λ Hypernucleus 7Λ He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of S = −2 Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Double-Λ Hypernucleus 10 ΛΛ Be . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Double-Λ Hypernucleus 11 ΛΛ Be . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s-Shell Double-Λ Hypernuclei and ΛΛ − Ξ N Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of Ξ Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the Ξ Hypernucleus 7Ξ H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the Ξ Hypernucleus 10 Ξ Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the s-Shell Ξ Hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2596 2597 2598 2600 2603 2604 2606 2607 2609 2610 2612 2613 2616 2617 2619 2621 2622

Abstract

The present status of our understanding of S = −1 and S = −2 hypernuclei is discussed. In the S = −1 sector, including ΛN − ΣN coupling is essential to complete our modeling of the fundamental Y N interaction. For this purpose, several neutron-rich Λ hypernuclei such as nnΛ, 6Λ H, and 7Λ He are discussed. E. Hiyama () Department of Physics, Tohoku University, Sendai, Japan e-mail: [email protected] B. F. Gibson Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_34

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11 In the S = −2 sector, observation of 6ΛΛ He, 10 ΛΛ Be, and ΛΛ Be is introduced together with their theoretical modeling. The need to investigate ΛΛ − Ξ N mixing is explored. Also in the S = −2 sector, how the observation of 15 ΞC led to confirming the attractive nature of the Ξ -nuclear interaction is examined. Furthermore, a discussion is provided regarding how our understanding of the Ξ N interaction has benefitted from investigations of s-shell Ξ hypernuclei and p-shell Ξ hypernuclei with A = 7 and 10.

Introduction In hypernuclear physics, it is important to study the structure of S = −1 and S = −2 hypernuclei to accomplish two major goals: The first is to understand the baryon-baryon interaction in a unified way, and the second is to explore the interesting phenomena in many-body baryon systems that differ from the physics of conventional, non-strange many-nucleon systems. As a result, one seeks to extrapolate our knowledge of nuclear systems to model the inner part of neutron stars, which may include various hyperons such as Λs and Ξ s. In fact, recently, neutron stars with masses twice a solar mass have been observed (Antoniadis et al. 2013; Cromartie et al. 2020). To understand the structure of neutron stars, it is essential to comprehend the nuclear physics of these extraordinary galactic objects. To understand the strangeness components of the baryon-baryon interactions, hyperon-nucleon (Y N) and hyperon-hyperon (Y Y ) scattering experiments would be most useful. However, due to the short lifetimes of the hyperons, it is difficult to perform such experiments. In fact, there exist only about 40 Y N scattering data in contrast with the more than 4,000 nucleon-nucleon (NN ) scattering data. There are no Y Y scattering data. Using this small number of Y N and no Y Y scattering data along with symmetry hypotheses to connect to the NN sector, several Y N and Y Y potential models have been proposed, which understandably exhibit significant ambiguity. For instance, as discussed in  Chap. 67, “What Is Hypernuclear Physics and Why Studying Hypernuclear Physics Is Important”, the spin-doublet states of 4Λ H and 4Λ He were calculated with several Nijmegen models and Jülich models and found to be inconsistent with the experimental data. This inconsistency implies that the model spin-spin interactions have a large degree of uncertainty (see Fig. 3 of  Chap. 67, “What Is Hypernuclear Physics and Why Studying Hypernuclear Physics Is Important”). With the advent of accurate, powerful fewbody calculational methods such as the Gaussian expansion method (Kamimura 1988; Hiyama et al. 2003), it has become possible to obtain useful information about the Y N and Y Y interactions from the study of the structure of S = −1 and S = −2 hypernuclei. Indeed, following the strategy shown in Fig. 8 of Chapter 1, spin-dependent ΛN interactions have been obtained. In addition, the spin-orbit force of the ΛN interaction that arises in 9Λ Be and 13 Λ C was discussed in Chapter 1. In the S = −1 sector, another important Y N interaction issue is ΛN − ΣN coupling. As mentioned in Chapter 1, extensive studies of this hyperon mixing effect have been carried out for s-shell Λ hypernuclei such as 3Λ H, 4Λ H, and 4Λ He (Afnan and Gibson 1989; Miyagawa et al. 2000; Gibson et al. 1972; Gibson and

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Lehman 1979; Carlson 1999; Hiyama et al. 2002; Nogga et al. 2002). However, ΛN − ΣN coupling is still not understood. Further study of this ΛN − ΣN mixing effect is necessary. For example, investigation of neutron-rich Λ hypernuclei is needed. Because the number of core nuclei with different total isospin is large, it is expected that the ΛN − ΣN mixing matrix element in neutron-rich Λ hypernuclei will be variable. Experimentally, observations of nnΛ (Rappold et al. 2013), 6 H (Agnello et al. 2012a, b), and 7 He (Nakamura et al. 2013) have been reported. Λ Λ However, the existence of nnΛ and 6Λ H should be explored further owing to an inconsistency between the reported data and theoretical calculations. Therefore, the current status of nnΛ and 6Λ H together with 7Λ He will be discussed. Once one obtains more definitive information on ΛN − ΣN coupling, construction of realistic Y N interaction models can proceed. As a next step, one can address the S = −2 sector Y Y interaction. Because it is not possible to perform Y Y scattering experiments, the only option is to investigate double-Λ hypernuclei and Ξ hypernuclei. In 1963, the first observation of a doubleΛ hypernucleus, 10 ΛΛ Be, was reported by Danysz et al. (1963). After that event was observed, two additional double-Λ hypernuclei, 6ΛΛ He (Prowse 1966) and 13 ΛΛ B (Aoki et al. 1991), were reported. However, it was difficult to confirm the reports of these two double-Λ hypernuclei. In 2001, 6ΛΛ He was observed with no ambiguity (Takahashi et al. 2001); the binding energy of this double-Λ hypernucleus was reported to be 6.91 ± 0.16 MeV with respect to the α + Λ + Λ three-body breakup threshold. By means of this observation, one could infer information about the 1 S0 ΛΛ interaction. Following this experiment, several other double-Λ hypernuclei, 11 along with an excited state of 10 ΛΛ Be and ΛΛ Be (Nakazawa and Takahashi 2010), were observed in emulsion data, which provided new isotopes and their binding energies. To identify the spin and parity of the observed double-Λ hypernuclei, it is necessary to compare the data with theoretical calculations. To illustrate this process, some experimental data together with relevant theoretical calculations are discussed. Regarding the Ξ N interaction, for many years, no information about this interaction was available due to a lack of observed Ξ hypernuclei. In 2015, the 14 first observation of a Ξ hypernucleus, 15 Ξ C ( N+Ξ ), was reported; this was dubbed the KISO event (Nakazawa et al. 2015). From this observation of a bound Ξ hypernucleus, it was inferred that the Ξ N interaction was attractive. Following this event, additional Ξ hypernuclei have been observed (Hayakawa et al. 2021; Yoshimoto et al. 2021). Therefore, it has become possible to discuss the spin dependence of the Ξ N interaction. To aid in this endeavor, several theoretical calculations have been performed, some of which will be reviewed in this chapter.

The Three-Body nnΛ System Investigation of neutron-rich Λ hypernuclei is important in order to extract information about ΛN − ΣN coupling. In 2013, the HYPHI collaboration (Rappold et al. 2013) reported evidence of a bound nnΛ system, which would be a unique neutral

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E. Hiyama and B. F. Gibson

Λ hypernucleus, because this system contains no protons. However, no binding energy was reported by the HypHI collaboration. Thus, theoretical calculations were undertaken to address whether this system should exist as the 3Λ n bound state. A three-body calculation utilizing a variational method was performed with an effective ΛN interaction having no ΛN − ΣN coupling; it was concluded that no bound state existed (Downs and Dalitz 1959). A Faddeev calculation including ΛN − ΣN coupling was then performed, and the result was no bound state exists for the nnΛ system (Garcilazo et al. 2007). These two conclusions contradicted the HYPHI reported evidence. Neither of the two calculations discussed the consistency of a bound nnΛ with other observed systems such as 3Λ H, 4Λ H, and 4Λ He. Hence, the possibility of the existence of a nnΛ bound state, using a realistic Y N interaction model with ΛN − ΣN coupling explicitly taken into account that is consistent with other s-shell Λ hypernuclei, is discussed. For this purpose, the NSC97f interaction (Rijken et al. 1999), which reproduces the observed binding energies of 3Λ H, 4Λ H, and 4Λ He, is employed. The calculation was performed using the Gaussian expansion method as outlined in the following.

The Gaussian Expansion Method The Gaussian expansion method was introduced by Kamimura (1988) to carry out high-precision, non-adiabatic three-body calculations of muonic molecules and muon-atomic collisions. It has been developed by his collaborators and has been extensively applied to various types of three- and four-body systems. Recently, the method was used to perform up to five-body calculations. To demonstrate the accuracy of this method, a benchmark calculation for the four-nucleon bound state using a realistic NN interaction was performed (Kamada et al. 2001) using seven different few-body methods such as the Faddeev-Yakubovsky equations, a no-core shell model, etc., and agreement among the seven calculation schemes was perfect. The method is used here in the case of the nnΛ three-body system together with 3Λ H for simplicity of illustration. The total three-body wave functions for nnΛ and 3Λ H are described as a sum of amplitudes for all rearrangement channels (c = 1 − 3) of Fig. 1 in the LS coupling scheme:

ΨJ M (nnΛ,3Λ H) = ×

 

3   



A

Y =Λ,Σ c=1 n,N L,I sS,ttY

   (c) (c) [φnl (rc ) ψN L (Rc )]I [χ 1 (N1 )χ 1 (N2 )]s χ 1 (Y ) 2

 × [η 1 (N1 )η 1 (N2 )]t ηtY (Y ) 2

2

2

2

S JM

  T

.

(1)

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Fig. 1 Jacobi coordinates for the nnΛ three-body system. Anti-symmetrization of the two neutrons is to be performed

Here, A is the two-nucleon anti-symmetrization operator, and the χ s and ηs are the spin and isospin functions, respectively, with the isospin tY =0(1) for Y = Λ(Σ). T is the total isospin, 0 for 3Λ H and 1 for 3Λ n. J is the total spin, 1/2+ , for both hypernuclei. In addition, to investigate the contribution of the 3 S1 state in the Y N interaction, the binding energy for the J = 3/2+ state of 3Λ H is calculated. The 2 (c) (c) functional form of φnl (rc ) is taken to be φnl (rc ) = r  e−r/rn Ym (ˆr), where the Gaussian range parameters are chosen to satisfy a geometrical progression (rn = r1 a n−1 ; n = 1 ∼ nmax ), and similarly for φN L (R). Three basis functions were verified to be sufficient for describing both the short-range correlations and the longrange tail behavior of the few-body systems (Kamimura 1988; Hiyama et al. 2003). As noted above, the Y N interaction employed in the three- and four-body calculations is the NSC97f potential (Rijken et al. 1999). For the NN interaction, we employed the AV8 potential (Pudliner et al. 1997). The Y N interaction is represented as 2S+1

VN Y −N Y  (r) =



(2S+1 VNCY −N Y  e−(r/βi )

2

i

+

2S+1

VNT Y −N Y  S12 e−(r/βi )

+

2S+1

VNLSY −N Y  LS e−(r/βi ) )

2

2

T = 1/2 ,  2 2S+1 UN Σ−N Σ (r) = (2S+1 UNCΣ−N Σ e−(r/βi ) for

i

+

2S+1

UNT Σ−N Σ S12 e−(r/βi )

+

2S+1

UNLSΣ−N Σ LS e−(r/βi ) )

for

T = 3/2 ,

2

2

(2)

with Y, Y  = Λ or Σ. Here, C, T , and LS mean central, tensor, and spin-orbit terms with two-range Gaussian forms. The potential parameters are given in Table 1 of Hiyama et al. (2002). The interaction reproduces the observed binding energy of 3Λ H: the calculated Λ binding energy, BΛ = 0.19 MeV, is consistent with the experimental data

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Table 1 Calculated binding energy of EnnΛ , in the case that the 1 S0 component is multiplied by the factor x. The scattering length ann , the effective range reff , the energy of the di-neutron system εnn , and the 3 H energy E3 H are also listed for each value of x x 1.0 1.13 1.35

ann (fm) –23.7 25.1 6.88

reff (fm) 2.78 2.40 1.96

εnn (MeV) Unbound −0.066 −1.269

E3 H (MeV) −7.77 −9.75 −13.93

EnnΛ (MeV) Unbound Unbound −1.272

Fig. 2 Calculated Λ-separation energy for 3Λ H with (a) 3 VNT Λ−N Σ × 1.00, (b) 3 VNT Λ−.N Σ × 1.10, and (c) 3 VNT Λ−N Σ × 1.20. The energy is measured with respect to the npΛ three-body breakup threshold

[BΛ (3Λ H = 0.13 ± 0.05 MeV]. Furthermore, the calculated energies of 4Λ H and 4Λ He are 2.33 and 2.28 MeV, respectively.

Results for nnΛ To appreciate the results for nnΛ, it is necessary to discuss the consistency with the binding energies of 3Λ H, 4Λ H, and 4Λ He. The NSC97f potential reproduces the energies of these Λ hypernuclei. The calculated energies of 3Λ H and the A = 4 hypernuclei are shown in (a) of Figs. 2 and 3. Using the NSC97f potential, the calculated nnΛ energy is indicated in (a) of Fig. 4, which is unbound with respect to the nnΛ breakup threshold. Next, one should examine the possibility to have a bound 3Λ n system while maintaining consistency with the binding energies of 3 H, 4 H, and 4 He. In nnΛ, the nn pair is in the spin-singlet state (s = 0, spins Λ Λ Λ antiparallel), while in the 3Λ H system, the np pair is in the spin-triplet state (s = 1, spins parallel). The difference in the spin value of [NN ]s=1or0 leads to different

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Fig. 3 Calculated Λ-separation energy of the ground state in 4Λ He with (a) 3 VNT Λ−N Σ × 1.00, (b) 3V T 3 T 3 N Λ−.N Σ × 1.10, and (c) VN Λ−N Σ × 1.20. The energy is measured with respect to the He + Λ breakup threshold Fig. 4 Calculated Λ-separation energy for 3Λ n with (a) 3 VNT Λ−N Σ × 1.00, (b) 3 VNT Λ−.N Σ × 1.10, and (c) 3 VNT Λ−N Σ × 1.20. The energy is measured with respect to the nnΛ three-body breakup threshold

contributions of the ΛN spin-spin interaction to the doublet splitting. In the 3Λ n system, the 1/2+ ground state involves Y N spin-singlet and spin-triplet interactions, while in the 3Λ H system, the 1/2+ ground state is dominated by the Y N spin-singlet interaction. Therefore, one can tune the spin-triplet state of the Y N interaction in a manner that does not affect the binding energy of 3Λ H significantly. To accomplish this, one can multiply the strength of the tensor part of the ΛN − ΣN coupling by a factor x, because the tensor part of the ΛN − ΣN coupling acts only in the spin-triplet state of the Y N interaction. The calculated energy of 3Λ n with T = 1 and J π = 1/2+ is illustrated in Fig. 4. In Fig. 2 are shown the binding energies of the J = 1/2+ and 3/2+ states in 3Λ H. The energy of 3Λ n in Fig. 4a is obtained using the Y N interaction which reproduces the binding energy of the ground state in 3Λ H, the J = 1/2+ state. In Fig. 4b and 4c, one multiplies the tensor part of the ΛN − ΣN coupling by the factors 1.10 and 1.20.

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In Fig. 4b, one still obtains no bound state in the nnΛ system. When 3 VNT Λ−N Σ is multiplied by 1.20, then we obtain a very weakly bound state (−0.054 MeV) with respect to the nnΛ three-body breakup threshold. In order to judge whether the adjusted 3 VNT Λ−N Σ is physically reasonable, the calculated binding energies of the ground and the excited states in 3Λ H are shown in Fig. 2. In Fig. 2a the binding energy of the ground state, J = 1/2+ , is in good agreement with the observed data. As one should anticipate, one finds the energy of 3Λ H becomes deeper with increasing strength of 3 VNT Λ−N Σ . When the strength of 3 VNT Λ−N Σ is multiplied by the factor 1.20, which leads to a bound state in the nnΛ system, the energy of 3Λ H is over bound (−0.7 MeV) compared with the observed data (−0.13 MeV). To further investigate the reliability of the employed Y N interaction, one can calculate the binding energies of 4Λ H and 4Λ He. In these two Λ hypernuclei, one also sees evidence of the important effect of charge symmetry breaking (CSB) in the ΛN interaction. The CSB effects appear in the ground state and excited state differences ΔCSB = BΛ (4Λ He)−BΛ (4Λ H), the experimental values of which are 0.35±0.06 and 0.24 ± 0.06 MeV, respectively. The Λ-separation energies of the ground states in 4 He and 4 H using the present Y N interaction are 2.28 and 2.33 MeV, respectively, Λ Λ which do not reproduce the CSB effect. (To investigate CSB in greater detail, it is proposed to measure the Λ-separation energy of 4Λ H at Mainz and JLab.) Here, the purpose is not to explore the CSB effect in A = 4 Λ hypernuclei. Therefore, the average value of these hypernuclei is adopted. That is, as experimental data, BΛ = 2.21 and 1.08 MeV are used for the ground state and the excited state, respectively. In addition, other parts of the Y N potential, such as 3 VNCΛ−N Λ , 3 VNCΛ−N Σ , etc., were adjusted. However, no modification of the Y N potential produced a bound 3Λ n while maintaining consistency with the binding energies of the A = 3 and A = 4 Λ hypernuclei. In Fig. 3 are illustrated the average binding energies of the A = 4 hypernuclei. In the case of Fig. 3a, the calculated ground state energy reproduces the data nicely, while the excited state is less bound than the observed data. Then, as was done in the case of 3Λ H, 3 VNCΛ−N Σ was tuned. As shown in Fig. 3b and 3c, increasing the strength of 3 VNCΛ−N Σ means that both the 0+ and 1+ states become over bound by 1–3 MeV. There exists another possibility to generate a bound state of nnΛ, by tuning the strength of the T = 1 nn potential. It has been suggested that, if this channel supports a bound state (i.e., a di-neutron), then it may be possible to describe the anomalies in neutron-deuteron elastic scattering and the deuteron breakup reaction above the threshold (Witala and Gloeckle 2012). (However, one should note that pp scattering is well described by standard methods which do not admit a bound di-proton. Thus, the hypothesis of a bound di-neutron suggests strong CSB in the NN spin-singlet interaction.) This nn spin-singlet channel does not contribute to the binding energy of 3Λ H, because the deuteron core nucleus has spin 1. On the other hand, the 1 S0 state of the nn pair contributes to the binding energy of 3Λ n. It also contributes to the energy of 3 H. The observed energies of 3 H and 3 He are −8.48

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and −7.72 MeV. Then, it is interesting to ask what is the energy of nnΛ as one tunes the strength of the nn T = 1,1 S0 , together with the predicted binding energy of 3 H. In Table 1 are illustrated the properties of the 1 S0 state when multiplied by a factor x: scattering length, effective range, energy of the di-neutron, and energy of 3 n. When one multiplies the 1 S component by the factors 1.13 and 1.35, a bound 0 Λ state in the di-neutron system results. However, the nn interaction multiplied by a factor of 1.13 does not produce a bound state in 3Λ n. When the 1 S0 component is multiplied by a factor of 1.35, a bound state in 3Λ n results. However, as one increases the factor x in the 1 S0 component, that is, to 1.13 and 1.35, which produces a dineutron bound state, the energy of 3 H is over bound compared with the observed data. Thus, one does not obtain a bound state in 3Λ n while maintaining consistency with the observed data for 3 H, unless one introduces a large repulsive nnp threebody force. It is known that one needs a small (about 0.7 MeV) attractive npp threebody force to obtain agreement with the 3 He binding energy; our model value for the 3 He binding energy is −7.12 MeV. Thus, the hypothesis of a bound di-neutron would require a very large CSB in the NNN three-body force, which is not easily understood. In this approach, it is difficult to generate a bound state of the nnΛ system while maintaining consistency with the observed binding energies of the A = 3 and A = 4 Λ hypernuclei.

Resonant State of nnΛ Although several authors have pointed out the difficulty of supporting a bound 3Λ n, some authors emphasized that there is a possibility that a resonant state might exist. For instance, Afnan and Gibson (2015) suggested such a resonant state based upon a Faddeev calculation. The Y N potentials were approximated as rank-one separable potentials parameterized to reproduce the scattering lengths and effective ranges of four realistic Y N potential models: Chiral (Haidenbauer et al. 2013), ND (Nagels et al. 1975), Jülich04 (Haidenbauer and Meissner 2005), and NSC97f (Rijken et al. 1999). The trajectory of the resonance pole of the nnΛ system as a function of a multiplicative scaling parameter s in the strength of the Y N potential is shown in Fig. 5. Pole positions are plotted for 0.25 increments in s. One sees a similar tendency of the trajectory for the four Y N interactions. For a scaling parameter of s = 1.0, the energy poles using ND, Chiral, NSC97f, and Jülich04 are (−0.154 − 0.753i), (−0.114 − 0.782i), (−0.120 − 0.782i), and (−0.097 − 0.758i) MeV, respectively. That is, the calculation results indicate that sub-threshold resonant states would exist for the model nΛ interactions parameterized to reproduce the pΛ scattering lengths of the four realistic YN potentials that were modeled. However, if the strength of the nΛ interaction is enhanced by no more than  5%, a 3Λ n physical resonance is predicted. (In the calculation, a significantly larger value of s, one

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Fig. 5 Trajectories of the nnΛ resonance pole as one evolves the strength of the nΛ Yamaguchi potentials with a scale factor s in increments of 0.25. The energy Re[E] is measured with respect to the nnΛ three-body breakup threshold

greater than 1.3, would be required to achieve a bound state pole with I m(E) = 0.) Motivated by such theoretical results, an experiment using the 3 H(e, e K + )nnΛ reaction was performed at JLab [E12-17-003], seeking to provide a constraint on the nΛ scattering length based upon the likely existence of an nnΛ resonance (or the less likely discovery of a 3Λ n bound state). The analysis is ongoing (Suzuki et al. 2022; Pandy et al. 2022). Future analysis of nnΛ may provide useful information on ΛN − ΣN coupling.

Superheavy Hydrogen Λ Hypernucleus 6Λ H Another neutron-rich Λ hypernucleus, 6Λ H, was observed in the FINUDA experiment (Agnello et al. 2012a, b). The reported Λ-separation energy, BΛ , was 4.0 ± 1.1 MeV. The core nucleus is 5 H, which was called “superheavy hydrogen,” since this nucleus contains a proton and four neutrons. Thus, 6Λ H is called “superheavy hydrogen Λ hypernucleus.” Before this measurement, some theoretical calculations had appeared: Dalitz et al. predicted that 6Λ H was a bound state based upon a shell model calculation (Dalitz and Levi-Setti 1963). In a calculation by Akaishi and Yamazaki (2002), it was suggested that coherent ΛN − ΣN coupling played an important role in the binding energy of this hypernucleus.

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It should be noted that the ground state of the core nucleus 5 H is observed as a resonant state with a broad width, E = 1.7 ± 0.3 MeV and Γ = 1.9 ± 0.4 MeV (Korsheninnikov et al. 2001), with respect to the tnn three-body breakup threshold. In this case, the wave function of 5 H is very diffuse. Therefore, we emphasize that BΛ depends crucially on the spatial size of the core nucleus. These were pointed out in Hiyama et al. (2013). Thus, it is essential to reproduce the observed data of 5 H in modeling the structure of 6Λ H. With this in mind, 5 H and 6Λ H were studied within the framework of tnn and tnnΛ three- and four-body cluster models, respectively (Hiyama et al. 2013). For the model interactions, the AV8 potential represented the N N interaction. An effective single-channel Y N interaction was used to simulate the basic features of the Nijmegen model NSC97f, where the ΛN − ΣN coupling effects are renormalized into effective ΛN interactions. The tΛ interaction was obtained by folding the ΛN G-matrix interaction derived from the Nijmegen model F with the density of the triton cluster. The potential parameters were tuned so as to reproduce the observed data for BΛ in 4Λ H within a tΛ two-body cluster model. However, because it is difficult to reproduce the observed data of 5 H, a three-body force was introduced. The detailed Hamiltonian can be found in Hiyama et al. (2013). In Table 2 are listed the calculated energies of 5 H and 6Λ H. First, the observed data of 5 H, E = 1.60 MeV and Γ = 2.44 MeV, were reproduced as seen in Table 2a. For 6Λ H, from the ΛN spin-spin interaction, one has 0+ and 1+ spindoublet states, where the 0+ state is the ground state. In this chapter, the 0+ ground state is discussed. When the energy of 5 H is reproduced, the ground state of 6Λ H is obtained as a resonant state at −0.87 MeV with respect to the tnnΛ breakup threshold, having a width Γ = 0.23 MeV, which is located 1.13 MeV above the lowest threshold of 4Λ H+n + n. Tuning the strength of the tnn three-body force, the calculated energy of 5 H was adjusted to lie at the lower edge of the error band of the observed energy of 5 H. It is hard to obtain any bound state of 6Λ H. This is inconsistent with the FINUDA data (Agnello et al. 2012a, b). Alternatively, if the three-body force in tnn is tuned so as to make 6Λ H match a binding of −2.07 MeV with respect to the tnnΛ breakup threshold, then the resonance energy of 5 H is 1.17 MeV with Γ = 0.91 MeV, which is inconsistent with the data for 5 H (Korsheninnikov et al. 2001).

Table 2 Calculated energies and decay widths of 5 H and 6Λ H in the case of interactions which (a) reproduce the observed data of 5 H and (b) make 6Λ H bound. The energies are measured with respect to the tnn and tnnΛ three- and four-body breakup thresholds for 5 H and 6Λ H, respectively

(a) (b) Exp.

E(5 H(1/2+ )) (MeV) 1.60 1.17 1.7 ± 0.3

Γ (5 H) (MeV)

E(6Λ H(0+ )) (MeV)

Γ (6Λ H) (MeV)

2.44 0.91 1.9 ± 0.4

−0.87 −2.07 −2.3 ± 1.1

0.23 − −

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E. Hiyama and B. F. Gibson

Based upon these calculations, one cannot reproduce simultaneously the observed data for 6Λ H together with data for 5 H. Here, it should be reiterated that ΛN − ΣN coupling is not explicitly treated in the model. Since the total isospin of 5 H is 3/2, it is possible that coupling plays an important role, especially working as an effective ΛNN three-body force in 6Λ H. To investigate this possibility, it is necessary to perform a coupled channel calculation taking into account the tnnΛ and t (3 He)N N Σ channels. This is a subject for future research. After the FINUDA experiment, a search for 6Λ H was undertaken at J-PARC using the double-charge exchange (π − , K + ) reaction (Sugimura et al. 2014). The conclusion was that there is no peak around the observed energy region suggested by the FINUDA experiment (Agnello et al. 2012a, b). That is, the two experiments are inconsistent with one another. Therefore, before proceeding theoretically, it is highly desirable to confirm experimentally whether 6Λ H exists as a bound state.

The Neutron-Rich He Isotope Λ Hypernucleus 7Λ He In 2013, a neutron-rich He isotope Λ hypernucleus 7Λ He was observed in an (e, e K + ) experiment at JLab (Nakamura et al. 2013). The extracted Λ separation energy of the ground state was reported to be 5.68 ± 0.03(stat.) ± 0.25(sys.) MeV (Nakamura et al. 2013). This observation is important for the study of ΛN − ΣN coupling, especially the charge symmetry breaking (CSB) component. Before this measurement, in Hiyama et al. (1996, 2009), the energy spectrum of 7 He was predicted within the framework of a 5 He + n + n three-body model and Λ Λ the α + Λ + n + n four-body cluster model. The core nucleus 6 He is known to be a typical neutron halo nucleus. That is, the observed two-neutron separation energy is only 0.975 MeV. With the addition of a Λ particle, the resultant Λ hypernucleus 7 He should bind the two halo neutrons more tightly in the ground state. The 6 He Λ core has observed excited states: the 2+ state is 0.827 MeV above the α+n+n threebody threshold with decay width Γ = 0.113 MeV (Tilley et al. 2002). In 2012, the transfer reaction experiment p(8 He, t)6 He showed indication of a second 2+ state at Ex = 2.6 ± 0.3 with Γ = 1.6 ± 0.4 MeV. When a Λ particle is added to such a resonant state, due to the attraction of the ΛN interaction, the resultant states should be more stable. The calculation of 7Λ He within the framework of an α + Λ + n + n four-body cluster model was formulated in Hiyama et al. (2009). The two-body AV8 N N interaction was employed. Also, the two-body αN, αΛ, and ΛN interactions were chosen so as to reproduce accurately the observed properties of each of the αN, αΛ, αNN, and αΛN subsystems. For 6 He, the calculated 0+ and 2+ 1 states are consistent with the observed data. However, the calculated energy and width of the 2+ 2 are about 1 MeV higher and broader than seen in the observed data. Turning to 7Λ He, the calculated Λ separation energy BΛ of the ground state is 5.36 MeV, which is compatible with the observed data, 5.68 ± 0.03(stat.) ± 0.25(sys.) MeV (Nakamura et al. 2013).

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E(MeV)

3.0 2.0

+

2.81 (4.63)

2

2

+

0

0.96 (0.14)

21

α+n+n +

−2.0

−1.03

0

0.07 (1.01)

α+n+n+Λ

2

0.03 (1.13)

+

−4.0 6 −6.0

+

5/2 2 + 3/2

−4.65 5/2

1

5 ΛHe+n+n 6 He+n Λ

+

He

−4.73 3/21 +

−6.39

1/2

7

Λ He

Fig. 6 Calculated energy spectra of 6 He and 7Λ He. The energies are measured with respect to the α + n + n and α + Λ + n + n three- and four-body breakup thresholds. The values in parenthesis are decay widths in MeV + The calculated 3/2+ 1 and 5/21 states are obtained as bound states at E = −4.73 and E = −4.65 MeV with respect to the α + n + n + Λ four-body breakup threshold, respectively. The experiment to produce 7Λ He was performed again with five times more statistics (JLab experiment E05-115), and the first excited state + (3/2+ 1 or 5/21 ) was observed for the first time (Gogami et al. 2016). The observed exp BΛ = 3.65 ± 0.20(stat.) ± 0.11(sys.) MeV (Gogami et al. 2016). The averaged theoretical BΛ = 3.66 MeV was in very good agreement with the data. As shown in Fig. 6, the second 3/2+ and 5/2+ resonant states are predicted to be at 0.03 and 0.07 MeV with Γ ∼ 1 MeV, with respect to the α + n + n + Λ four-body breakup threshold, respectively. They have narrower widths than the 6 corresponding 2+ 2 state of the He core nucleus, due to the gluelike role of the Λ particle. In this work, the ΛN − ΣN coupling is not included. Recently, a no-core shell model calculation was performed using Chiral effective field theorybased N N and Y N interactions for the He isotope Λ hypernuclei with A = 4 to 10 (Wirth and Roth 2018), in which an induced Y NN three-body force was included. Theoretically, discussions of ΛN − ΣN coupling effects in neutron-rich Λ hypernuclei have been initiated. However, further experimental data are necessary for significant progress to occur.

Structure of S = −2 Hypernuclei In order to fully understand the baryon-baryon interaction, it is important to obtain information about interactions in the S = −2 sector, i.e., ΛΛ and Ξ N interactions. These interactions are important for the study of the equation of state

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in neutron stars. However, we have little knowledge of these interactions due to the lack of S = −2 scattering data. Even in the future, it would be impossible to perform scattering experiments at J-PARC. Thus, it is essential to obtain information about these interactions from the study of the structure of S = −2 nuclei such as double-Λ hypernuclei and Ξ hypernuclei. Experimentally, in the 1960s, two 6 double-Λ hypernuclei, 10 ΛΛ Be (Danysz et al. 1963) and ΛΛ He (Prowse 1966), were observed. However, the validity of the latter event was questioned. In 1991, a new double-Λ hypernucleus was observed (Aoki et al. 1991). However, there were two interpretations of this event: one was an observation of 10 ΛΛ Be, and the other was an B. Thus, it was difficult to uniquely identify the hypernucleus. observation of 13 ΛΛ In 2001, an observation of 6ΛΛ H in the epic KEK-E373 experiment was reported. This was labeled the NAGARA event; the two-Λ removal energy BΛΛ was found to be 6.91 ± 0.16 MeV (Takahashi et al. 2001). From this uniquely identified event, it was clear that the ΛΛ interaction was less attractive than the ΛN interaction. About the same time, another important event, the Demachi-yanagi event, was observed and identified as 10 ΛΛ Be (Nakazawa and Takahashi 2010). The extracted BΛΛ was reported to be 11.90 ± 0.13 MeV. However, it was hard to determine whether this event should be interpreted as the ground state or an excited state of the hypernucleus. In 2017, an emulsion experiment at J-PARC was performed, for which the data analysis has just begun. It is expected that dozens of emulsion events corresponding to double-Λ hypernuclei will be found. However, it will be difficult to determine the spin and parity of such events or even to determine whether an observed event corresponds to a ground state or an excited state. For this reason, it will be crucial to compare the data with theoretical calculations for proper characterization of the observed states. Therefore, the following strategy is suggested to identify the states: (1) Using the ΛΛ interaction based on, for example, the Nijmegen potential (Nagels et al. 1975), one can perform a three-body cluster model calculation of α + Λ + Λ. (2) One can then compare the result with data from the NAGARA event and adjust the strength of the interaction so as to reproduce the data. (3) Using the tuned interaction, one can either predict the level structure of unobserved double-Λ hypernuclei or analyze the level structure of an observed event such as the Demachi-yanagi event (Nakazawa and Takahashi 2010). In 2010, another double-Λ hypernucleus was observed, known as the HIDA event (Nakazawa and Takahashi 2010). However, again, there were two possible interpretations of the event: One is an observation of 11 ΛΛ Be, whose BΛΛ is 20.83 ± 1.27 MeV, and the other is an observation of 12 Be, whose BΛΛ is 22.48±1.21 MeV. ΛΛ Based upon the above strategy, it was requested that the event should be interpreted theoretically. In the next subsection, the theoretical interpretation of this event together with the Demachi-yanagi event will be discussed.

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Regarding the Ξ N interaction, as is the case with the ΛΛ interaction, it is important to study the structure of Ξ hypernuclei. However, until recently, there were no observed bound Ξ hypernuclei. Only a few experimental data indicated that the Ξ -nucleus interaction is attractive. For instance, for the (K − , K + ) reaction on a 12 C target at BNL, the observed cross section exhibited no peak (Khaustov et al. 2000). To extract information about the Ξ -nucleus interaction, a fit to the data using a Ξ -nucleus Woods-Saxon potential (W = V0 /[1 + exp(r − R)/a]) assumption suggested that V0 should be ≈ − 14 MeV. However, it was difficult to confirm this conclusion due to the lack of data on Ξ hypernuclei. 14 − In 2015, the Ξ hypernucleus 15 Ξ C ( N + Ξ ) was observed as a bound state (Nakazawa et al. 2015), which was called the KISO event. For this event, there were two possible values of the Ξ removal energy, BΞ = 4.38 ± 0.25 MeV or 1.11 ± 0.25 MeV. From this event, it was confirmed that the Ξ -nucleus interaction was definitely attractive. As mentioned above, an emulsion experiment was performed at J-PARC. Several events corresponding to the Ξ hypernucleus 15 Ξ C, such as IBUKI, MINO, etc., have been observed (Hayakawa et al. 2021; Yoshimoto et al. 2021). These observations have initiated the second phase of the study of the Ξ N interaction: What terms should be included in the Ξ N interaction? For this purpose, what kinds of Ξ hypernuclei should be produced experimentally? To answer these questions, several Ξ hypernuclei will be explored.

The Double-Λ Hypernucleus 10 Be ΛΛ To interpret the Demachi-yanagi event, with BΛΛ = 11.90±0.13 MeV, in Nakazawa and Takahashi (2010), a four-body cluster model ααΛΛ calculation was performed. The ΛΛ interaction was tuned so as to reproduce the BΛΛ [6ΛΛ He] of the NAGARA event. The calculated level structure was shown in Fig. 6. + Figure 7 shows the level structure of 8 Be, 9Λ Be, and 10 ΛΛ Be. The calculated 2 state energy is 11.90 MeV, which is consistent with the Demachi-yanagi value of 11.90 ± 0.13. Therefore, one can interpret the Demachi-yanagi event as the 2+ 10 excited state of 10 ΛΛ Be. The result for the ground state of ΛΛ Be is 14.74 MeV. The earlier emulsion experiment conducted by Danysz et al. (1963), based on the pionic exp 10 9 + − + decay of 10 ΛΛ Be →ΛBe(1/2 ) + p + π , yielded a value of BΛΛ [ΛΛ Be(0 )] = 17.7 ± 0.4 MeV. This value was used for a long time and implies a strongly attractive interaction. However, the authors suggested the possibility of another 9 + + + − decay, 10 ΛΛ Be(0 ) →ΛBe(3/2 , 5/2 ) + p + π . In this case, the value of exp 10 BΛΛ [ΛΛ Be(0+ )] would be 14.6 ± 0.4 MeV, which was obtained by using 3.05 MeV as the excitation energy of 9Λ Be(3/2+ , 5/2+ ). This modified value is consistent with the above calculated value of 14.74 MeV. Thus, one is able to reconcile the modern Demachi-yanagi event together with the older data of Danysz et al. (1963).

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Fig. 7 Calculated energy spectra of 8 Be, 9Λ Be, and 10 ΛΛ Be. The energies are measured with respect to the α + α, α + α + Λ, and α + α + Λ + Λ breakup thresholds

The Double-Λ Hypernucleus 11 Be ΛΛ As noted previously, a double-Λ hypernucleus was observed in the HIDA event (Nakazawa and Takahashi 2010). Two possible interpretations for this double-Λ hypernucleus were reported: One is 11 ΛΛ Be with BΛΛ = 20.83 ± 1.27 MeV, and the other is 12 ΛΛ Be with BΛΛ = 22.48 ± 1.21 MeV. Moreover, whether the event is an observation of a ground state or an excited state was unclear. One was able to explain the HIDA event in Hiyama et al. (2010) by assuming that this was a 11 ΛΛ Be

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hypernucleus and utilizing an ααnΛΛ five-body cluster model. Here, it should be noted that the core nucleus 9 Be is well described in an ααn three-body cluster model (Arai et al. 2003). Therefore, it should be possible to model the structure change of 9 Be to 11 Be by the addition of the two Λs as a five-body problem of ααnΛΛ. ΛΛ Such a calculation is quite challenging for the following reasons: (i) there exist three species of particles – α, Λ, and neutron; (ii) five different kinds of interactions (ΛΛ, Λn, Λα, nα, and αα) are involved; and (iii) one must take into account the Pauli principle between the two α particles and between the α and neutron. This calculation became possible because of the development of the Gaussian expansion method that was introduced previously. It should be emphasized, before going into the five-body calculation, that all the interactions are determined so as to reproduce the observed binding energies of the two- and three-body subsystems: αα, αn, αΛ, ααΛ, αΛΛ, and αnΛ. As a result, the calculated energies of the A = 10 four-body subsystems, 10 Λ Be (ααnΛ) Be, are reproduced without the need to introduce additional parameters. The and 10 ΛΛ 9 Be are shown in Fig. 8. energy spectra of 11 Be together with the core nucleus ΛΛ 10 In 11 ΛΛ Be, we have four bound states below the ΛΛ Be + n breakup threshold. It is − interesting to see that the order of the 5/2 and 1/2+ states inverts from 9 Be to 11 Be. This is because the energy gain due to the addition of the Λ(s) is larger in ΛΛ the compactly structured (5/2− ) state than in the loosely coupled (1/2+ ) state. The same type of theoretical prediction was reported in Hiyama et al. (2000) for 13 ΛC based on an αααΛ four-body cluster model. The core nucleus 12 C exhibits states + + with various characteristics: the 0+ 1 and 21 states have shell-like structure, the 02 − − is a well-developed α-cluster state, and the 31 and 11 states have intermediate or mixed character. With the addition of a Λ to 12 C, one predicts a level inversion for − − 13 12 the 0+ 2 , 11 , and 31 states between C and Λ C. As seen in Fig. 8, the calculated BΛΛ of the ground state in 11 ΛΛ Be is 18.23 MeV, while for the excited states, the BΛΛ values are calculated to be less than 15.5 MeV. Therefore, the observed HIDA event can be interpreted to be the ground state. When the calculated binding energy is compared with the experimental value of 20.83 MeV with a large uncertainty of σ = 1.27 MeV, one can say, at least, that the result does not contradict the data within 2σ . In 2017, the J-PARC E07 experiment was performed, and analysis is underway. Recently, an event involving the Be isotope double-Λ hypernucleus, called the MINO event, was reported (Nakazawa and Takahashi 2010). The extracted BΛΛ is 19.07 ± 0.11 MeV, and it is hypothesized that this event is an observation of the ground state of 11 ΛΛ Be. This is less bound than the HIDA event. Comparing with the theoretical calculation (Hiyama et al. 2010), the theoretical result is compatible with the new data of the MINO event. From these data, one can extract information on the 1 S0 component of the ΛΛ interaction. To understand further the ΛΛ interaction, such as p-wave, ΛΛ − Ξ N coupling, and so on, one needs more experimental data on double-Λ hypernuclei. Thus, further analysis of the J-PARC E07 experiment is awaited.

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Fig. 8 Calculated energy spectra of 9Λ Be and 11 ΛΛ Be. The energies are measured with respect to the α + α + n and α + α + n + Λ + Λ breakup thresholds

s-Shell Double-Λ Hypernuclei and ΛΛ − Ξ N Coupling The ΛΣ mass difference being much smaller than the NΔ mass difference has meant that ΛN − ΣN coupling has played a much larger role in s-shell Λ hypernuclei than has NN − ΔN coupling in conventional, non-strange s-shell nuclei. The ΛΛ − Ξ N mass difference is even smaller, which suggests that ΛΛ − Ξ N coupling should play a significant role in ΛΛ s-shell hypernuclei. [Note that one should consider ΛΛ−Ξ N −ΣΣ mixing, but the ΣΣ coupling is neglected in this discussion because of the large ΛΛ − ΣΣ mass difference.] The role of ΛΛ − Ξ N coupling in s-shell ΛΛ hypernuclei is a concept that has been around for many years; see, for example, Gibson (1994). A decade earlier, Aerts and Dover (1984) had examined the Ξ d → nΛΛ reaction in the context of the (Ξ − d)atom →

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nH, where H is the H dibaryon. Following that, Carr et al. (1998) investigated the Ξ − d → nΛΛ reaction in a Faddeev formalism including ΛΛ − Ξ N coupling as a means of gaining insight into the ΛΛ final-state interaction. (This is analogous to the nd → nnp breakup experimental measurement of the proton energy spectrum to determine the nn scattering length.) No such measurement has since been carried out. In Gibson (1994), the role of ΛΛ − Ξ N coupling in 6ΛΛ He and in 4ΛΛ H was discussed based upon the binding energy of the reported Prowse event for 6ΛΛ He. Following the publication of the NAGARA event for 6ΛΛ He, which indicated a much smaller value of BΛΛ than Prowse had reported, Afnan and Gibson (2003) demonstrated that ΛΛ − Ξ N coupling in 6ΛΛ He would play a negligible role. The question then remained as to whether one could utilize a 4ΛΛ H bound state to constrain the ΛΛ − Ξ N coupling. Although Filikhin and Gal (2005) and Nemura et al. (2003) did not agree about the possible existence of a 4ΛΛ H bound state, a fully coupled channel approach to the doubly strange s-shell hypernuclei (Nemura et al. 2005) suggested that bound states of 4ΛΛ H and 5ΛΛ H should be particle stable. Most recently, in a pionless effective field theory approach, Contessi et al. (2019) claim > that a rather large value of the ΛΛ scattering length (∼1.5 fm) is required to bind 4 H, thereby questioning its particle stability, because such a large ΛΛ scattering ΛΛ length appears inconsistent with the data from the NAGARA event. In contrast, the particle stability of the isodoublet 5ΛΛ H – 5ΛΛ He is claimed to be robust. There is a major caveat regarding this most recent theoretical result: no ΛΛ − Ξ N coupling is included in the model calculations. That is, one knows (i) that ΛN − ΣN coupling plays a major role in the structure of the s-shell Λ hypernuclei and (ii) that the Ξ + N − 2Λ mass difference in ΛΛ − Ξ N coupling is significantly smaller than the Σ −Λ mass difference in ΛN −ΣN coupling. Therefore, until one has experimental data on bound states of s-shell double-Λ hypernuclei (in addition to 6ΛΛ He), there are no firm constraints on ΛΛ − Ξ N coupling.

Structure of Ξ Hypernuclei Since the first observation of the bound Ξ hypernucleus 15 Ξ C in 2015 (Nakazawa et al. 2015), it has become possible to extract information about the Ξ N interaction from the structure of Ξ hypernuclei. The Ξ N interaction may be formulated as follows: VΞ N (r) = V0 (r)+σΞ ·σN Vσ (r)+τΞ ·τN Vτ (r)+(σΞ ·σN )(τΞ ·τN )Vσ τ (r).

(3)

From the observation of 15 Ξ C, one could confirm that the VΞ N interaction should be attractive. Next, one needs to understand the individual terms formulated in Eq. (3). As a first step, it is important to obtain information on V0 , which is the spin- and isospin-averaged component. This term is important for the study of the EOS of neutron stars. To obtain this term, αΞ (5Ξ H) and ααΞ (9Ξ Li) three- and

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four-body cluster systems are well suited. The reason is that the αΞ interaction can be derived by folding the Ξ N G-matrix interaction with the wave function of the α. The spin- and isospin-dependent parts vanish in a folding procedure involving a spin- and isospin-saturated system such as the α. To model this term, one should use realistic forces. Several realistic Ξ N interactions have been proposed. Some proposed potential models suggest a repulsive Ξ N interaction, and some proposed potentials, such as the Extended Soft Core 04 potential (ESC04) (Rijken and Yamamoto 2006a, b) and ND (Nagels et al. 1975), are attractive. As mentioned previously, recent experimental data indicate that the Ξ -nucleus interaction is attractive: observation of the (K − , K + ) reaction spectrum on a 12 C target can be reproduced by assuming an attractive Ξ -nucleus WoodsSaxon(WS) potential with a depth of ≈ 14 MeV (Khaustov et al. 2000). Using this WS potential, an s-state system bound by 2.2 MeV with respect to the 11 B + Ξ twobody breakup threshold is predicted. Moreover, the observation of 15 Ξ C confirms that the Ξ N interaction is attractive. To explore the αΞ (5Ξ H) and ααΞ (9Ξ Li) systems, ESC04 and ND potentials were employed. The partial-wave contributions of the two potentials are provided in Table 3. The UΞ values are found to differ between ESC04 and ND, because the odd-state contributions in the former are far more attractive than those in the latter. It should also be noted that the spin and isospin dependence between ESC04 and ND is significantly different. The two potentials should be tested in a study of the 11 B + Ξ system (12 Ξ Be) with its binding energy of 2.2 MeV before proceeding with calculations of αΞ (5Ξ H) and ααΞ (9Ξ Li). In Hiyama et al. (2008), the 12 Ξ Be calculation is based on an ααtΞ cluster model. In the case of the 12 C(K − , K + ) reaction, (T , J π ) = (1, 1− ) states are produced, because the Tz component is transformed by ΔTz = 1 on the T = 0 target. For this calculation, Ξ N G-matrix interactions are derived from ESC04 and ND in nuclear matter, where the imaginary parts arise from the energyconserving transitions from the Ξ N to ΛΛ channels in the nuclear medium. The α(t)Ξ interactions are obtained by folding the Ξ N G-matrix interaction with the density of the α(t). The αα and αt potentials reproduce reasonably well the lowlying bound states and low-energy scattering phase shifts of the αα and αt systems. The calculated binding energies of the (T , J π ) = (1, 1− ) state using the ESC04 and ND potentials are 2.24 MeV with Γ = 3.95 MeV (kF = 1.055fm−1 ) and 2.23 MeV with Γ = 1.38 MeV (kF = 1.025fm−1 ), respectively.

Table 3 Partial-wave contributions to the Ξ potential depth UΞ (ρ0 ) in nuclear matter at normal density ρ0 in the case of ESC04 and ND Model ESC04 ND

T 0 1 0 1

1S 0 6.3 7.2 −3.0 −4.1

3S 1 −18.3 −1.7 −0.5 −4.2

1P 1 1.2 −0.8 −2.1 −3.0

3P 0 1.5 −0.5 −0.2 0.0

3P 1 −1.3 −1.2 −0.7 −3.1

3P 2 −1.9 −2.5 −1.9 −6.5

UΞ

ΓΞ

−12.1

12.7

−29.5

0.8

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In this approach, one obtains appropriate kF parameter values for the effective Ξ N interactions based upon ESC04 and ND, which are consistent with the experimental observation of 12 Ξ Be. Therefore, these two potentials can be applied next to αΞ (5Ξ H) and ααΞ (9Ξ Li). However, here, it is important how one chooses the kf parameters for the G-matrix effective Ξ N interaction. The parameter kF specifies the nuclear matter density in which the G-matrix interactions are constructed. It is plausible that a corresponding value in a finite system is obtained from an average density. In the calculation of the A = 12 Ξ hypernucleus 12 Ξ Be considered here, to obtain the Ξ binding energy, the ND and ESC04 kF values are ≈ 1.0 → 1.1 fm−1 , which are reasonably close. Considering that the model value is kF = 1.055 fm−1 for ESC04 and 1.025 fm−1 for ND, it is a modest change to take kF = 0.9 fm−1 in the A = 4 to 6 systems. In Hiyama et al. (1997), the structure of 5Λ He, 9Λ Be, and 13 C using ΛN G-matrix interactions with the k parameter chosen to be around F Λ 0.9 fm−1 for 5Λ He and 9Λ Be and to be around 1.1 fm−1 for 13 Λ C obtained results consistent with the experimental data. Therefore, consider kF values for αΞ (5Ξ H) and ααΞ (9Ξ Li) of kF = 0.9, 1.055 and 1.3 fm−1 for ESC04 and kF = 0.9, 1.025 and 1.3 fm−1 for ND. The kF = 1.3 fm−1 is included only to demonstrate the kF dependence of the results. Table 4 summarizes the calculated energies for the αΞ and ααΞ systems with three kF values of ESC04 and ND, respectively. As shown in Table 4, there is one bound state for the αΞ two-body system in the case of kF = 0.9 fm−1 using either ESC04 or ND. When one more α particle is added to the αΞ system, creating the ααΞ three-body system, one sees that system becomes more stable, i.e., more bound. It is significant to see that the decay width Γ , related to the ΛΛ − Ξ N coupling term, is dependent on the Ξ N interaction employed: ESC04 gives a larger decay width than ND, which means that the ΛΛ − Ξ N coupling in ESC04 is larger than that in ND. In this way, it is possible to extract information about the spin- and isospin-independent parts of the Ξ N interaction from the binding energies of the αΞ and ααΞ two- and threebody systems. In reality, unfortunately, there are no corresponding nuclear targets to produce these systems via the (K − , K + ) reaction. Then, as secondary candidates

Table 4 The calculated energies of the 1/2+ state in (a) the αΞ system and (b) the ααΞ system using ESC04 and ND (a) kF (fm−1 ) E(MeV) Γ (MeV) (b) kF (fm−1 ) E(MeV) Γ (MeV)

αΞ (ESC04) 0.9 1.055 −1.36 −0.26 2.64 0.86 ααΞ (ESC04) 0.9 1.055 −4.81 −2.23 5.01 2.89

1.3 −0.14 0.15 1.3 −0.83 1.18

αΞ (ND) 0.9 −0.57 0.16 ααΞ (ND) 0.9 −2.87 0.58

1.025 −0.32 0.06

1.3 −0.15 0.004

1.025 −1.82 0.3

1.3 −0.79 0.06

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to obtain information on the spin- and isospin-independent terms, the structures of 7 H(αnnΞ ) and 10 Li(ααnΞ ) are well suited, and it is possible to produce these Ξ Ξ hypernuclei by using the 7 Li(K − , K + ) and 10 B(K − , K + ) reactions.

Structure of the Ξ Hypernucleus 7Ξ H In the calculation of this system, αn and nn interactions are generated so as to reproduce the low-energy properties of the αn and nn systems. The same αΞ potentials are used as in the case of the αΞ and ααΞ systems. One must consider the (T , J π ) = (3/2, 1/2+ ) state. The basic question is whether this state is bound. The 6 He core is composed of an α and two weakly bound (halo) neutrons. Due to the weak strength of the Ξ n interaction, the binding between 6 He and the Ξ is, to a large extent, determined by the αΞ interaction. The calculated energies in the 1/2+ ground state are illustrated in Fig. 9 as a function of kF , for the two Ξ N potential models without the imaginary part of the αΞ interaction. These 1/2+ states are composed of the ground state 0+ configuration of 6 He coupled with an 0s-state Ξ particle. The Coulomb interaction between the α and Ξ is taken into account. In the figure, the dashed lines show the positions of threshold energies of α + n + n + Ξ , 6 He + Ξ , and 5Ξ H(αΞ )cal + n + n, respectively. One should be aware that the 5Ξ H(αΞ )cal + n + n threshold energy depends on the kF value of the Ξ N interactions that are used. This situation is unavoidable, because the calculated energies for 5Ξ H must be used instead of the unmeasured experimental value. It is seen that in the case with kF = 0.9 fm−1 for ESC04, the lowest threshold is 5 H(αΞ ) 6 cal + n + n, while in the other cases, the He + Ξ threshold is lower than Ξ 5 the Ξ H(αΞ )cal + n + n threshold. However, in all kF cases with ND, the lowest threshold is 6 He + Ξ . The order of the 5Ξ H(αΞ )cal + n + n and 6 He + Ξ thresholds is determined by the competition between the αΞ correlation and the α − (nn) correlation. The results for αΞ using ESC04 are more bound than those using ND. However, in 7Ξ H, the energy difference between ESC04 and ND becomes smaller as shown in Fig. 9. The decay widths using ESC04 are 2.64 MeV for kF = 0.9, 1.15 MeV for kF = 1.055, and 0.31 MeV for kF = 1.3, respectively. Those using ND are 0.27 MeV for kf = 0.9, 0.15 MeV for kF = 1.025, and 0.032 MeV for kF = 1.3, respectively. With the large range of values for kF , one notes that a bound state of 7Ξ H exists with both the ESC04 and ND Ξ N interactions. The decay widths are strongly dependent on the Ξ N interaction employed. This means that an experimental finding of a 7Ξ H bound state should indicate the existence of an αΞ bound state in which the even-state spin-independent part of the Ξ N interaction is substantially attractive. This conclusion is almost independent of the interaction model.

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Fig. 9 Calculated energy spectra of 7Ξ H for three kF values using (a) ESC04 and (b) ND. The energies are shown when the imaginary part of the αΞ interaction is switched off. The energies are measured from the α + n + n + Ξ breakup threshold. The dashed lines represent thresholds

Structure of the Ξ Hypernucleus 10 Li Ξ As mentioned above, it is possible to have a bound state for the ααΞ system. Then, it is also possible to have a bound state of ααΞ n(10 Ξ Li), because the interaction between the Ξ and a p-orbit neutron is weakly attractive. The ground state of the core nucleus 9 Be is 3/2− , which is bound by about 1.57 MeV with respect to the α + α + n threshold. When a Ξ particle is added to this nucleus, the lowest T = 1 doublet states are J π 2− and 1− states. The calculated energies of ααΞ n(10 Ξ Li) are shown in Fig. 10.

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Fig. 10 Calculated energy spectra of 10 Ξ Li for three kF values using (a) ESC04 and (b) ND. The energies are shown when the imaginary part of the αΞ interaction is switched off. The energies are measured from the α + α + n + Ξ breakup threshold. The dashed lines represent thresholds

π = 2− , 1− ) are found to be fairly Although the binding energies of 10 Ξ Li (J sensitive to the choice of the kF values, especially in the case of ESC04, it is considered that the results with kF ∼ 1.0 fm−1 are acceptable. It is interesting to note that the 9Ξ Li(ααΞ )cal + n threshold comes below the 9 Be + Ξ threshold in most cases. It is reasonable that the lowest breakup threshold is 9Ξ Li(ααΞ )cal + n, because the value of kF in the A = 10 system is likely to be ∼ 1.0 fm−1 , similar to 10 that in the 12 Ξ Be system. It is notable that the binding energies of Ξ Li measured from the 9Ξ Li(ααΞ )cal + n thresholds are similar to each other for both ESC04 (2.6 MeV) and ND (2.7 MeV). We expect such structure, a valence neutron coupled to the 9Ξ Li hypernucleus, because the lowest threshold is 9Ξ Li(ααΞ )cal + n. However, if the kF value becomes by chance much larger, then the Ξ particle is coupled to the ground state of 9 Be, because the lowest threshold is 9 Be + Ξ .

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Table 5 The calculated energies of the 1− and 2− states in the 10 Ξ Li system for three values of kF using ESC04 and ND Ξ N interactions. The values in parenthesis are energies when the imaginary part of the αΞ interactions is switched off. The energies are measured from the α + α + n + Ξ threshold (fm−1 )

2−

kF E(MeV)

1−

Γ (MeV) E(MeV) Γ (MeV)

10 Li(ESC04) Ξ

0.9 −7.99 (−8.35) 5.87 −7.48 (−7.84) 5.72

10 Li(ND) Ξ

1.055 −4.83 (−5.16) 3.63 −4.42 (−4.77) 3.44

1.3 −2.87 (−3.13) 1.71 −2.64 (−2.89) 1.50

0.9 −5.83 (−5.85) 0.75 −5.98 (−5.99) 0.77

1.025 −4.42 (−4.43) 0.42 −4.53 (−4.53) 0.43

1.3 −2.92 (−2.92) 0.10 −2.97 (−2.97) 0.10

The 2− (1− ) state is dominated by the 33 S1 (31 S0 ) component of the two-body nΞ interaction. As demonstrated in Fig. 10, the 33 S1 interaction for ESC04 (ND) is more (less) attractive than the 31 S0 interaction. Therefore, the 2− (1− ) state of 10 Ξ Li becomes the ground state in the case of ESC04 (ND), as shown in Fig. 10. More detailed results are given in Table 5, which also lists the calculated values of the conversion widths Γ . As seen in Table 5, the decay widths Γ calculated with ESC04 are much larger than those for ND, primarily because the 11 S0 Ξ N − ΛΛ coupling interaction in ESC04 is far stronger than that in ND. In the (K − , K + ) reaction, if the spin-nonflip transition dominates, then the − 2 state of 10 Ξ Li is selectively excited. As shown in Fig. 10, one can likely expect the existence of a bound state with the predicted Ξ binding energy of BΞ = 3.26 MeV (ESC04, kF = 1.055 fm−1 ) or 2.85 MeV (ND, kF = 1.025 fm−1 ). The system can be produced by the (K − , K + ) reaction on 10 B, which is suitable to investigate the αΞ interactions, that is, the spin-independent terms of even and odd-state Ξ N interactions.

Structure of the s-Shell Ξ Hypernuclei As can be seen from the above discussion, the A = 7 and 10 Ξ hypernuclei are important in order to extract information about the spin- and isospin-averaged Ξ N interaction, V0 , where V0 is formulated as V0 (R) = [V

11 S

0

(R) + 3V

13 S

1

(R) + 3V

31 S

0

(R) + 9V

33 S

1

(R)]/16.

(4)

Next, we need to obtain information for each partial wave of the Ξ N interaction. The optimal Ξ hypernuclei are the s-shell systems such as NNΞ and NNNΞ . Among these two hypernuclei, the NNΞ system was studied by the Faddeev method (Garicilazo 2016; Garcilazo and Valcarce 2016) with an effective Ξ n potential based upon the ESC08c model (Rijken et al. 2013).

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Fig. 11 Calculated Ξ N phase shifts 11 S1 , 13 S1 , 31 S0 , and 33 S1 using ESC08 (Nagels et al. 2015)

The ESC08c (Nagels et al. 2015) calculated phase shifts of each partial wave are shown in Fig. 11. As seen in the figure, the 13 S1 channel is weakly attractive, and the 11 S0 and 31 S0 channels are repulsive in ESC08c. The most outstanding feature is the strong attraction in 33 S1 that supports a bound state of 1.59 MeV. In Garicilazo (2016) and Garcilazo and Valcarce (2016), two bound states of NNΞ are found, BΞ = 13.5 MeV with (T , J π ) = (1/2, 3/2+ ) and BΞ = 0.012 MeV with (1/2, 1/2+ ) with respect to the d + Ξ threshold. Furthermore, one bound state is obtained at 1.33 MeV with (T , J π ) = (3/2, 1/2+ ) with respect to the D ∗ + N threshold. It should be noted that the ESC08c potential has a number of fitted parameters, and it is difficult to uniquely fit all parameters due to a lack of Ξ N scattering experiments. In the 2010s, a potential based on first principles lattice QCD simulations, the HAL QCD Ξ N potential (HAL QCD), was proposed (Sasaki et al. 2020). The potential has no free parameters, and therefore, the energy spectrum and the potential are predictions. The phase shifts for each partial wave are shown in (Fig. 12). Using the Gaussian expansion method, a NNΞ three-body calculation was performed. However, it was difficult to avoid a bound state for (T , J π ) = (1/2, 1/2+ ), (1/2, 3/2+ ). Let us consider the NNNΞ four-body system. Four-body calculations with (T , J π ) = (0, 0+ ), (0, 1+ ), (1, 0+ ), and (1, 1+ ) states using the ESC08c (Nagels et al. 2015) and HAL QCD potentials were performed (Hiyama et al. 2020). With the ESC08c potential, the (T , J π ) = (0, 0+ ) state was unbound with respect to the 3 H/3 He + Ξ threshold, while the states with (T , J π ) = (0, 1+ ), (1, 0+ ), and (1, 1+ ) were bound by 10.20, 3.55, and 10.11 MeV, respectively. The physics of

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Fig. 12 Calculated Ξ N phase shifts 11 S1 , 13 S1 , 31 S0 , and 33 S1 using HAL QCD (Sasaki et al. 2020)

these states is: The dominant Ξ N pair interaction in the (T , J π ) = (0, 0+ ) system is the repulsive 11 S0 channel, which prevents binding of this system. In contrast, the dominant Ξ N pair interactions in the (T , J π ) = (1, 1+ ) and (0, 1+ ) systems are the 33 S1 and 13 S1 channels, so that the binding energies of these NNNΞ systems are large. When the HAL QCD potential is used, only the (T , J π ) = (0, 1+ ) system has a shallow bound state, namely, at 0.36 MeV with respect to the 3 H/3 He+Ξ threshold. The HAL QCD potential is moderately attractive in 11 S0 , while it is either weakly attractive or repulsive in the other channels. Although the binding energy of the N N N Ξ system with (T , J π ) = (0, 1+ ) is strongly dependent on the Ξ N potential employed, the NNNΞ system of (T , J π ) = (0, 1+ ) is a possible candidate for a bound Ξ hypernucleus. The decay widths of this system are 0.05 MeV and 0.43 MeV using the HAL QCD potential and the ESC08c potential, respectively. To produce NNNΞ states experimentally, heavy ion reactions at GSI and at the CERN LHC would be useful. If there exists a bound NNNΞ (0, 1+ ), it decays into d + Λ + Λ or a possible double-Λ hypernucleus 4ΛΛ H through Ξ N − ΛΛ coupling. On the other hand, to produce the NNNΞ (1, 0+ ) and NNNΞ (1, 1+ ) states as predicted by ESC08c, the (K − , K + ) reaction with a 4 He target would be useful.

Summary In this chapter, the structure of S = −1 and S = −2 hypernuclei was discussed. In the S = −1 sector since the 1990s, we have obtained experimental data which are related to the study of the Y N interaction. Thus, extracting information about Y N interactions was advanced by comparing hypernuclear data and theoretical calculations. At present, a primary open issue is fully understanding ΛN − ΣN

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coupling. At the J-PARC facility is planned the production of numerous neutronrich Λ hypernuclei introduced in this chapter. On the other hand, in the S = −2 sector there are still many open questions pertaining to the study of the Y Y and Y N interactions, e.g., the p-wave contribution to the ΛΛ interaction, ΛΛ − Ξ N coupling, the spin- and isospin-averaged Ξ N interaction, etc. To address such questions, producing multiple S = −2 hypernuclei is essential. In this chapter, some needed S = −2 hypernuclei were introduced. After these are produced at existing or new experimental facilities, the Y Y interactions can be explored as was accomplished for the Y N interaction in the S = −1 sector. Thus, many interesting phenomena in the S = −2 sector remain to be investigated.

References A.T.M. Aerts, C.B. Dover, Phys. Rev. D 29, 433 (1984) I.R. Afnan, B.F. Gibson, Phys. Rev. C 40, R7 (1989) I.R. Afnan, B.F. Gibson, Phys. Rev. C 67, 017001 (2003) I.R. Afnan, B.F. Gibson, Phys. Rev. C 92, 054608 (2015) M. Agnello et al., Phys. Rev. Lett. 108, 042501 (2012a) M. Agnello et al., Nucl. Phys. A 881, 269 (2012b) Y. Akaishi, T. Yamazaki, in Physics and Detectors for DAΦNe. Frascati Physics Series, vol. 16, SIS INFN-LNF, Rome (1999), p. 59; S. Shinmura, K.S. Myint, T. Harada, Y. Akaishi, J. Phys. G 28, L1 (2002) J. Antoniadis et al., Science 340, 1233232 (2013) S. Aoki et al., Prog. Theor. Phys. 85, 1287 (1991) K. Arai, P. Descouvemont, D. Baye, W.N. Catford, Phys. Rev. C 68, 014319 (2003); K. Arai, Phys. Rev. C 69, 014309 (2004) J. Carlson, AIP Conf. Proc. 224, 198 (1999) S.B. Carr, I.R. Afnan, B.F. Gibson, Phys. Rev. C 57, 2858 (1998) L Contessi, M. Schäfer, N. Barnea, A. Gal, J. Mares, Phys. Lett. B 797, 134893 (2019) H.T. Cromartie et al., Nat. Astron. 4, 72 (2020) R.H. Dalitz, R. Levi-Setti, Nuovo Cimento 30 489 (1963) M. Danysz et al., Phys. Rev. Lett. 11, 29 (1963) B.W. Downs, R.H. Dalitz, Phys. Rev. 114, 593 (1959) I.N. Filikhin, A. Gal, Phys. Rev. Lett. 89, 2002502 (2005) H. Garicilazo, Phys. Rev. C 93, 024001 (2016) H. Garcilazo, A. Valcarce, Phys. Rev. C 93, 034001 (2016) H. Garcilazo, A. Valcarce, T. Fernandez-Carames, Phys. Rev. C 76, 034001 (2007) B.F. Gibson, Phys. Rev. C 49, R1768 (1994) B.F. Gibson, D.R. Lehman, Nucl. Phys. A 239, 308 (1979) B.F. Gibson, A. Goldberg, M.S. Weiss, Phys. Rev. C 6, 741 (1972); Phys. Rev. C 8, 837 (1973) T. Gogami et al., Phys. Rev. C 94, 021302(R) (2016) J. Haidenbauer, U.-G. Meissner, Phys. Rev. C 72, 044005 (2005) J. Haidenbauer et al., Nucl. Phys. A 915, 24 (2013) S.H. Hayakawa et al., Phys. Rev. Lett. 126, 062501 (2021) E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Phys. Rev. C 53, 2075 (1996) E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Prog. Theor. Phys. 97, 881 (1997) E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Phys. Rev. Lett. 85, 270 (2000) E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Phys. Rev. C 66, 024007 (2002)

71 Theoretical Studies in S = −1 and S = −2 Hypernuclei

2623

E. Hiyama, Y. Kino, M. Kamimura, Prog. Part. Nucl. Phys. 51, 223 (2003) E. Hiyama, Y. Yamamoto, T. Motoba, T.A. Rijken, M. Kamimura, Phys. Rev. C 78, 054316 (2008) E. Hiyama, Y. Yamamoto, T. Motoba, M. Kamimura, Phys. Rev. C 80, 054321 (2009) E. Hiyama, M. Kamimura,Y. Yamamoto, T. Motoba, Phys. Rev. Lett. 104, 212502 (2010) E. Hiyama, S. Ohnishi, M. Kamimura, Y. Yamamoto, Nucl. Phys. A 908, 29 (2013) E. Hiyama, K. Sasaki, T. Miyamoto, T. Doi, T. Hatsuda, Y. Yamamoto, T.A. Rijken, Phys. Rev. Lett. 124, 092501 (2020) H. Kamada et al., Phys. Rev. C 64, 044001 (2001) M. Kamimura, Phys. Rev. A 8, 621 (1988) K. Khaustov et al., Phys. Rev. C 61, 054603 (2000) A.A. Korsheninnikov et al., Phys. Rev. Lett. 87 (2001) 092501 K. Miyagawa, H. Kamada, W. Gl’ockle, H. Yamamura, T. Mart, C. Bennhold, Few-Body Syst. Suppl. 12, 234 (2000) M.M. Nagels, T.A. Rijken, Y. Yamamoto, arXiv:1504.02634 (2015) M.M. Nagels, T.A. Rijken, J.J. de Swart, Phys. Rev. D 12, 744 (1975); 15, 2547 (1977) S.N. Nakamura et al., Phys. Rev. Lett. 110, 012502 (2013) K. Nakazawa, H. Takahashi, Prog. Theor. Phys. Suppl. 185, 335 (2010) K. Nakazawa et al., Prog. Theor. Exp. Phys. 2015, 033D022 (2015) H. Nemura, Y. Akaishi, K.S. Myint, Phys. Rev. C 67, 051001(R) (2003) H. Nemura, S. Shinamura, Y. Akaishi, K.S. Myint, Phys. Rev. Lett. 94, 202502 (2005) A. Nogga, H. Kamada, W. Gloeckle, Phys. Rev. Lett. 88, 172501 (2002) B. Pandy et al., Phys. Rev. C 105, L051001 (2022) D.J. Prowse, Phys. Rev. Lett. 17, 782 (1966) B.S. Pudliner, V.R. Pandharipande, J. Carlson, S.C. Pieper, R.B. Wiringa, Phys. Rev. C 56, 1720 (1997) C. Rappold et al., Phys. Rev. C 88, 041001(R) (2013) T.A. Rijken, Y. Yamamoto, [arXiv:nucl-th/0608074] (2006a) T.A. Rijken, Y. Yamamoto, Phys. Rev. C 73, 044008 (2006b) T.A. Rijken, V.G.J. Stoks, Y. Tamamoto, Phys. Rev. C 59, 21 (1999) T.A. Rijken, M.M. Nagels, Y. Yamamoto, Few-Body Syst. 54, 801 (2013) K. Sasaki et al. (HAL Collaboration), Nucl. Phys. A 998, 121737 (2020) H. Sugimura et al., Phys. Lett. B 729, 39 (2014) K.N. Suzuki et al., Prog. Theor. Exp. Phys. 2022(1), 013D01 (2022) H. Takahashi et al., Phys. Rev. Lett. 87, 212502 (2001) D.R. Tilley et al., Nucl Phys. A 708, 3 (2002) R. Wirth, R. Roth, Phys. Lett. B 779, 336–341 (2018) H. Witala, W. Gloeckle, Phys. Rev. C 85, 064003 (2012) M. Yoshimoto et al., Prog. Theor. Exp. Phys. 2021, 073D02 (2021)

Theoretical Study of Deeply Bound Pionic Atoms with an Introduction to Mesonic Nuclei

72

Satoru Hirenzaki and Natsumi Ikeno

Contents Introduction to Physics of Mesonic Atoms and Mesonic Nuclei . . . . . . . . . . . . . . . . . . . . . . Pion–Nucleus Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of Deeply Bound Pionic Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hadron Reactions Proposed for the Pionic Atom Formation . . . . . . . . . . . . . . . . . . . . . . . . . (d,3 He) Reactions for the Formation of the Deeply Bound Pionic Atoms . . . . . . . . . . . . . . . Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

A brief review of the study of the deeply bound pionic atoms and an introduction to the physics of the mesonic atoms and mesonic nuclei are given in this chapter. In the meson–nucleus bound systems considered here, the real meson exists inside and/or very close to the surface of the nucleus. The strong interaction plays the dominant role in the systems and determines the structures and properties of the mesonic atoms and mesonic nuclei. This chapter begins with the introductory remarks on the basic physics of the meson–nucleus bound systems, then, followed by the detailed explanation of the theoretical studies of the structure and formation of the deeply bound pionic atoms. The interests and importance are also mentioned for the investigation of the aspects of the strong interaction symmetry at finite density by the meson–nucleus bound systems. This

S. Hirenzaki () Department of Physics, Nara Women’s University, Nara, Japan e-mail: [email protected] N. Ikeno Department of Agricultural, Life and Environmental Sciences, Tottori University, Tottori, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_35

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chapter is written for beginners and is expected to be helpful for the readers to start the studies of this subject.

Introduction to Physics of Mesonic Atoms and Mesonic Nuclei In this chapter, a brief review is given for the studies of the REAL meson and nucleus bound systems, so-called mesonic atoms and mesonic nuclei. In the mesonic atoms, the negatively charged meson exists in the atomic orbit around the nucleus instead of the electron, while in the mesonic nuclei, the real meson exists inside the nucleus forming the bound states by the attractive strong interaction. Especially, the deeply bound pionic atoms (see also Yamazaki et al. 2012), in which pion exists on the surface of, even inside, the nucleus, are explained in detail. The interests of the other mesonic nuclear states are also mentioned briefly. Pion and other mesons are known to play important roles in the hadron–nuclear physics, the physics of the strong interacting particles. For example, the long- and middle-range parts of the nucleon–nucleon interaction have been attributed to the virtual meson exchange. Actually, hadrons are the practical degrees of freedom of the low-energy strong interaction world which can be observed directly in experiments. Because of the quark confinement, one cannot observe exactly the isolated quarks which are, together with gluons, the degrees of freedom of the elementary theory of the strong interaction quantum chromodynamics (QCD). Thus, the various studies of the aspects of the strong interaction have been carried out by means of the hadronic processes such as the relativistic heavy ion collisions for the studies of the phase structure of QCD and the new state of matter, quark–gluon plasma (see, e.g., Yagi et al. 2005). Meson and nucleus bound systems, mesonic atoms and mesonic nuclei, are one of the most appropriate objects to study the strong interaction phenomena in quite precise and exclusive manner since the bound systems are quasi-static states with relatively long life time and have the welldefined quantum numbers as the usual atomic states of electrons. Thus, in general, it can be expected to perform the spectroscopic studies of the systems and to deduce the exclusive information on the strong interaction from the meson–nucleus bound systems. In this sense, the studies of the meson bound systems are complementary to other studies using the dynamical processes such as the relativistic heavy ion collisions which have possibilities, on the other hand, to reach higher energy and/or density regime. The study of the meson–nucleus bound systems has long history as the study of the exotic atoms (Batty et al. 1997; Friedman and Gal 2007). Exotic atoms are the bound states of the negatively charged particles except for electron. The μ− – atom is one of the exotic atoms of lepton, for example. Hadronic atoms are the Coulombassisted nuclear bound states of the negatively charged hadrons such as π − −, K − −, and p¯ – atoms, which are usually populated by injecting the low-energy hadrons into the matter and observed by detecting the emitted X-ray in the deexcitation processes of the atomic bound states as shown in Fig. 1. The precise spectroscopic studies of

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Energy

X ray

Nuclear absorption

Nucleus

Fig. 1 A schematic figure of pionic states in atomic orbits. Deexcitation processes of pionic atoms accompanied by the X-ray emissions and the nuclear absorption process are also shown. The width of the deeply bound state is indicated by the hatched area

the meson–nucleus bound systems have been used to deduce the information on the nuclear structures and to obtain the information on the meson–nucleus interaction by observing the energies, widths, and decay processes of the states. For example, the p¯ – atoms have been used to probe the nuclear surface by the pp ¯ pair annihilation followed by the pion production processes. The definite quantum numbers of the states are also important to deduce the exclusive information on the meson–nucleus interactions and to describe them as the appropriate potential terms. The studies of meson–nucleus interaction, however, have been performed independently for each meson in the early stage of the research, which means that the existing symmetry behind the phenomena in mesonic atoms and mesonic nuclei has not been well investigated. The X-ray cascade ceases at the last orbital where the sizes of the electromagnetic transition width Γrad and the nuclear absorption width Γabs are comparable Γrad ∼ Γabs . Hadrons hardly reach to the deeper atomic bound states than the last orbital because of the nuclear absorption. Thus, the deeply bound states deeper than the last orbital cannot be observed by the X-ray cascade. For the pionic atoms, this experimental result led to the physicists to the wrong understanding that the deeper bound pionic states than the last orbital did not exist, at least as the discrete levels, in spite of the fact that the calculated results with the standard theoretical tools at that time showed that the level spacing ΔE is larger than the level width even for the deeply bound pionic states such as 1s in Pb. In the 1980s, the existence of the deeply bound atomic states of pion, which cannot be populated by the deexcitation processes neither observed by the Xray spectroscopy, was predicted theoretically (Friedman and Soff 1985; Toki and Yamazaki 1988; Toki et al. 1989). Because of the nuclear absorption, pions disappear in the deexcitation processes and do not reach to the deeply bound states such as 1s state in heavy nuclei. After many theoretical and experimental attempts,

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the deepest bound 1s and 2p pionic states in Pb isotopes were finally observed by the (d,3 He) reaction (Gilg et al. 2000; Itahashi et al. 2000; Yamazaki et al. 2012). The successful discovery of the deeply bound pionic atoms stimulated and motivated the physicists to develop the research of other nuclear bound states of hadrons including the systems of neutral mesons. Neutral mesons can form the bound states only inside the nucleus by the strong interaction as mesonic nuclei. At the same time, the bound states are begun to be recognized seriously as the valuable laboratory to study the aspects of the symmetry breaking and restoration of the strong interaction at finite density. In the studies of the meson–nucleus systems in these days, one of the most interesting subjects is to investigate the aspects of the strong interaction symmetry at finite nuclear density. In vacuum, the effects of the dynamical and the explicit chiral symmetry breaking and the UA (1) anomaly are believed to play the crucial role to realize the complex properties of hadrons. The dynamically broken chiral symmetry is expected to be restored at finite density which can be confirmed experimentally by observing the reduction of the chiral order parameter qq ¯ condensate qq ¯ in the environments of the finite nuclear density like inside the nucleus (Hatsuda and Kunihiro 1994; Vogl and Weise 1991). Thus, it can be expected to deduce the information on the partial restoration of the dynamically broken chiral symmetry by observing the properties of hadrons inside the nucleus with helps of the theoretical relations connecting the in-medium hadron properties to the qq ¯ condensate. For example, the complex mass spectrum of the meson shown in Fig. 2 is also considered to be understood quantitatively by taking into account the effects of the dynamical and explicit chiral symmetry breaking and the UA (1) anomaly in vacuum (see, e.g., Hatsuda and Kunihiro 1994). On the other hand in the chiral symmetry restoration limit, all mesons in Fig. 2 are expected to degenerate and have the same mass. Thus, as a possibility, by observing the various meson masses in nucleus, one can study the aspects of the strong interaction symmetry at finite density and the consequences of the partial restoration of the chiral symmetry, namely, the reduction of the chiral order parameters qq. ¯ It should be mentioned here that each meson changes its properties uniquely at finite nuclear density and has different features – even the mass degeneracy is expected in the chiral symmetry restoration limit. The systematic studies and understandings of these unique features of the various mesons are quite necessary to reveal the symmetry existing behind the hadrons, for which the precise information on the meson properties and interaction at finite density is mandatory. As for pion, after the first clear observation of the 1s pionic states in Sn (Suzuki et al. 2004), the reduction of the chiral order parameter qq ¯ at finite nuclear density was discussed using the naive extensions of the Tomozawa (Tomozawa 1966)–Weinberg (Weinberg 1966) and Gell-Mann–Oakes–Renner (GOR) (GellMann et al. 1968) relations to the finite density. The isovector π N scattering length b1free in vacuum is related to the pion decay constant fπ by the Tomozawa–Weinberg relation as Tπisovector = −4π ε1 b1free = N

ω , 2fπ2

(1)

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Fig. 2 A schematic figure of the mass spectrum of the light pseudoscalar mesons. The observed complex spectrum shown in the left is believed to be realized by the dynamical and explicit chiral symmetry breaking and by the UA (1) anomaly. In the right, the expected spectrum in the chiral symmetry restoration limit with mq = 0 and qq ¯ = 0 is shown

where Tπisovector is the isovector π N scattering amplitude, ω the pion energy, and N mπ ε1 = 1+ with the pion and nucleon masses mπ and M. The pion decay constant M fπ is related to the qq ¯ condensate in vacuum qq ¯ by the GOR relation as m2π fπ2 = −2mq qq, ¯

(2)

where mq is the average of the current quark masses of u-quark and d-quark mu + md . In Suzuki et al. (2004), the strength of the s-wave isovector optical mq = 2 potential term expressed by the b1 parameter in Eq. (11) was determined by the data of the 1s pionic states as the in-medium π N scattering length and was used to obtain the in-medium pion decay constant fπ∗ by the Tomozawa–Weinberg relation. Then, the value of fπ∗ was used to deduce the value of the in-medium quark condensate qqρ by the GOR relation neglecting the pion mass change as b1free qq ¯ ρ f ∗2 .  π2  b1 qq ¯ fπ

(3)

The obtained results in Suzuki et al. (2004) are the indication of the reduction of the qq ¯ condensate at ρ = 0.6ρ0 as qq ¯ ρ=0.6ρ0  0.78qq ¯ compared with the vacuum value qq. ¯ The theoretical foundations of the extension of the Tomozawa–Weinberg relation to the finite density region are given in the linear density approximation in Kolomeitsev et al. (2003) by the chiral perturbation theory and in Jido et al. (2008) by a correlation function analysis. The foundations of the extension of the GOR relation are given in Hatsuda and Kunihiro (1985), within

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the Nambu–Jona-Lasinio model, and in Pisarski and Tytgat (1996) and Thorsson and Wirzba (1995) by the chiral perturbation theory. The latest study of the deeply bound pionic atoms is reported in Nishi et al. (2018, 2022). For the K − – nucleus bound states, it is very interesting that there exist both kaonic atoms and kaonic nuclei because of the attractive electromagnetic and optical potentials. In Fig. 3, the schematic figure is shown for the structures of the kaon bound states. The major part of the density of the atomic state distributes around the nuclear surface and that of the nuclear state exists deep inside the nucleus. The existence of the nuclear state is considered to affect the formation spectrum of the kaon bound states (Yamagata et al. 2005, 2006). The experimental studies of the atomic and nuclear states of kaon are actively performed recently (Yamaga et al. 2020; Hashimoto et al. 2022). The interesting feature of the kaonic systems is the existence of the Λ(1405) resonance in the K − – proton channel just below the threshold. The properties of kaon in nucleus are strongly affected by those of the Λ(1405) resonance at finite density. The Λ(1405) resonance is interpreted as a dynamical resonance in the meson–baryon coupled system as proposed in Dalitz et al. (1967) and supported by the studies with the SU(3) chiral Lagrangian (Kaiser et al. 1995; Oset and Ramos 1998), by which the in-medium Λ(1405) properties are also investigated. Thus, the studies of the K − – atoms and K − – nucleus are closely related to the studies of Λ(1405) generated dynamically by the meson–baryon chiral interaction and its inmedium modifications. The η meson is also expected to form the η – nucleus bound states (Haider and Liu 1986). Since the η meson has no charge, the existence of the η mesonic nuclear states are expected. The η meson also has the N ∗ (1535) resonance in the

|φ|2

Nuclear state

Atomic state

Atomic state

ReV

Fig. 3 A schematic figure of the atomic and nuclear states in the kaon–nucleus system. The density distribution of the nuclear state is shown in the solid line and that of the atomic state in the dashed line. The Coulomb and the Coulomb+Optical potentials are also shown in the dashed and the solid lines, respectively. The expected energy levels of the atomic and nuclear states are shown by the solid bars. The nuclear radius is indicated by the vertical dotted line

Coulomb pot.

Nuclear state

Coulomb + Optical pot.

r [fm]

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η– nucleon channel at the threshold region, which is a candidate of the chiral partner of the nucleon. The properties of the η bound states are also strongly influenced by the in-medium properties of the N ∗ (1535) resonance. In Detar and Kunihiro (1989) and Jido et al. (2000, 2001), the N ∗ (1535) resonance is considered to form the chiral multiplet with nucleon and the reduction of the mass difference between nucleon, and N ∗ (1535) is concluded. On the other hand, in Garcia-Recio et al. (2002) and Inoue and Oset (2002), the N ∗ (1535) resonance is interpreted as a resonance generated by the meson–baryon coupled channel and predicted to have much different in-medium properties obtained in Detar and Kunihiro (1989) and Jido et al. (2000, 2001). Thus, the study of the η – nucleus states is closely related to reveal the nature of N ∗ (1535) and to investigate the aspects of the chiral symmetry of baryon sector. Recently, the d + d fusion reaction is studied for the formation of η – 4 He bound states. The new constraints to the η – 4 He interaction are reported in Ikeno et al. (2017) and Skurzok et al. (2018). The η – 3 He systems and their non-mesonic decays are also investigated (Skurzok et al. 2020). The η(958) (η ) meson is, on the other hand, very unique and has the exceptionally large mass in the light pseudoscalar meson multiplet which is believed to have a close connection to the UA (1) anomaly. Thus, the η and nucleus bound systems are uniquely expected to provide the information on the UA (1) anomaly effects at finite density. There exists only limited experimental information obtained by observing the η properties at finite density (Tanaka et al. 2016, 2018; Friedrich et al. 2016; Nanova et al. 2016). The results of the theoretical studies have also been reported. For example, in Kunihiro (1989), the effects of the UA (1) anomaly on the η properties at finite temperature are studied using the Nambu–Jona-Lasinio model with the Kobayashi–Masukawa–’t Hooft (KMT) term (Kobayashi and Maskawa 1970; Hooft 1976). The studies with the linear σ model are also performed (Pisarski and Wilczek 1984; Sakai and Jido 2013). Theoretical studies conclude the possible change of the η properties at finite density. The formations of the η – nucleus bound states are proposed firstly in the serious manner in Nagahiro and Hirenzaki (2005) for the study of the UA (1) anomaly at finite density. We stress here that the investigation of the symmetry breaking and its restoration hidden behind the complex properties of hadrons requires the systematic studies of the various hadron properties in the different circumstances like in the finite density and/or temperature.

Pion–Nucleus Interaction In this section, it is explained briefly how to take into account the medium effects to the π meson based on the π N interaction and to obtain the pion–nucleus interaction called the optical potential. For the detailed explanation for the formulation of the pion–nucleus optical potential, please see the references Ericson and Weise (1988) and Oset et al. (1982).

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=

+

+

+

+

...

= Fig. 4 The diagrammatical expression of the Dyson equation for the in-medium pion propagator D(q μ ) shown as the thick solid line, where q μ is the four-momentum carried by pion. The pion free propagator is shown as the short-dashed line and the pion self-energy as the solid circle

Let us start with the free pion propagator D 0 (q μ ) in vacuum written as i D 0 (q μ ) =

i , q 2 − m2π + iε

(4)

where q μ is the four-momentum carried by pion and mπ is the pion mass. The Dyson equation is considered to include the effects of the π N interaction in the nuclear matter and to obtain the pion propagator D(q μ ) at finite nuclear density. The Dyson equation can be written as i D(q μ ) = i D 0 + i D 0 (−i Π )i D 0 + i D 0 (−i Π )i D 0 (−i Π )i D 0 + . . . = i (D 0 + D 0 Π D 0 + D 0 Π D 0 Π D 0 + . . .) = iD 0 (q μ ) + iD 0 (q μ )Π (q μ )D(q μ ),

(5)

where Π indicates the pion self-energy which describes the effects of the pion– nucleon interaction. The Dyson equation Eq. (5) can be expressed diagrammatically as Fig. 4. The full pion self-energy Π is defined as the complete sum of all contributions of the one-particle irreducible diagrams of the π N interaction, which are defined as the diagrams that cannot be divided into two connected diagrams by cutting one internal line of the pion propagator. Π (q μ ) = Σ Π irreducible .

(6)

In Fig. 5 the simple example of the irreducible diagrams is shown for the π N interaction expressed by the 3-point π N N vertex. For other types of the π N interaction such as those expressed by the π π N N 4-point vertex, one has other series of the irreducible diagrams. Some examples of the actual functional form of the simple self-energies can be found in Ericson and Weise (1988). And an introductory guide for the practical calculations can also be found in Oset (1982). The Dyson equation Eq. (5) can be rewritten as i D(q μ ) =

i i i D 0 (q μ ) = = 2 . 1 − D 0 (q μ )Π (q μ ) (D 0 (q μ ))−1 − Π (q μ ) q − m2π − Π (q μ ) (7)

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

=

+

2633

+

+

...

Fig. 5 The diagrammatical expression of the pion self-energy Π for the π N interaction expressed by the 3-point π N N vertex. The short-dashed lines indicate the pion free propagator, and the solid lines indicate the in-medium nucleon particle and hole propagators. The small solid circle indicates the points where the external pion propagator lines are supposed to be connected

Thus, the expression of the in-medium pion propagator D(q μ ) can be obtained by implementing the pion self-energy Π in the denominator as in Eq. (7). It should be noticed that the infinite iteration of the self-energy Π is taken into account in the propagator D(q μ ). The pion Hamiltonian H for the equation of motion is written as the inverse operator of the propagator as H = D −1 = q 2 − m2π − Π (q μ ),

(8)

and the Klein–Gordon (KG) equation with the medium effects Π in the coordinate space is written as   −∇ 2 + μ2 + Π (Eπ , −i ∇, ρ(r)) φ(r) = [Eπ − Vem (r)]2 φ(r),

(9)

where μ is the pion–nucleus reduced mass and Eπ is the complex eigen energy i of the bound state which can be expressed as Eπ = μ − B − Γ with the 2 binding energy B and the width Γ of the bound state. The KG equation is solved to investigate the structure of the pionic atoms. The local density approximation is used to obtain the self-energy in the coordinate space. Vem (r) indicates the electromagnetic interaction between π − and the nucleus, which will be explained later and implemented into the equation of motion as the time component of the photon vector potential. The strong interaction between pion and the nucleus is described by the pion self-energy Π in the coordinate space which is related to the pion–nucleus optical potential Vopt . The energy dependence of Π is usually neglected in the study of the pionic atoms since the relativistic energy of the atomic state is always close to the pion mass. The standard optical potential is developed in Ericson and Ericson (1966) and called Ericson–Ericson potential which is expressed as

2634

S. Hirenzaki and N. Ikeno

Π(−i ∇, ρ(r)) = 2μVopt (r) = −4π [b(r) + ε2 B0 ρ 2 (r)] + 4π ∇ · [c(r) + ε2−1 C0 ρ 2 (r)]L(r)∇,

(10)

b(r) = ε1 [b0 ρ(r) + b1 [ρn (r) − ρp (r)]],

(11)

with

c(r) =

ε1−1 [c0 ρ(r) + c1 [ρn (r) − ρp (r)]],

−1  4 , L(r) = 1 + π λ[c(r) + ε2−1 C0 ρ 2 (r)] 3

(12) (13)

μ μ and ε2 = 1+ with the nucleon mass where ε1 and ε2 are defined as ε1 = 1+ M 2M M. The distributions of the center of the proton and neutron are written as ρp (r) and ρn (r), and the distribution of the nucleon ρ(r) is defined as ρ(r) = ρp (r) + ρn (r). The parameters bs and cs indicate the strength of the s- and p-wave π N interaction. The p-wave potential terms with cs and C0 are accompanied by the derivative operators and are called as nonlocal part, while other s-wave terms with bs and B0 are called as local part. The terms with B0 and C0 are higher-order contributions and describe the effects of the pion interaction with two nucleons including the pion absorption processes. The Lorentz–Lorenz correction L(r) takes into account the effects of the short-range correlations of nucleons. All potential parameters, in principle, are considered to be obtained theoretically by the calculations of pion selfenergy using the data of π N interaction in vacuum. However, some of the potential parameters are usually determined by fitting the atomic data since the theoretical potentials have difficulties to reproduce the experimental data satisfactorily without phenomenological pieces. Some parts of the phenomenological pieces added to the potential are considered to be understood as the effects of the renormalization factor introduced later in Eq. (17) (Kolomeitsev et al. 2003). In Table 1, a parameter set determined in Seki and Masutani (1983) is shown as an example. It should be noticed that there exist the correlations between the potential parameters as reported in Seki and Masutani (1983) which lead to the concept of the nuclear effective density probed by the meson in the bound states (Yamazaki and Hirenzaki 2003; Ikeno et al. 2011). As for the electromagnetic interaction Vem , the Coulomb interaction between π − and the nucleus with finite charge distribution ρch (Toki et al. 1989) is considered.

Table 1 The pion optical potential parameters obtained by Seki and Masutani (1983)

b0 = −0.0283 m−1 π c0 = 0.223 m−3 π B0 = 0.042 i m−4 π λ = 1.0

b1 = −0.12 m−1 π c1 = 0.25 m−3 π C0 = 0.10 i m−6 π

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

2635

The vacuum polarization effects are also included in Vem in this chapter (Nieves et al. 1993; Ikeno et al. 2015). The electromagnetic interaction Vem is written as e2 Vem (r) = − 4π ε0



ρch (r  )Q(|r − r |)  dr , |r − r |

(14)

where the function Q(r) is defined as 2 e2 Q(r) = 1 + 3π 4π ε0



∞

du e 1

−2me ru

  2 1 (u − 1)1/2 1+ 2 , 2u u2

(15)

with the electron mass me . For the nuclear densities used to evaluate the potentials, the charge density distribution in Fricke et al. (1995) is adopted for 207 Pb, and the prescription in Nieves et al. (1993) is used to deduce the distributions of the center of the proton and neutron with the appropriate normalization to the number of each particle. For 123 Sn, the density distributions obtained in Terashima et al. (2008) are used, and they are parametrized for interpolation in Nishi et al. (2022). s with the Seki–Masutani The local s-wave part of the optical potential Vopt parameter set in Table 1 and the Coulomb potential Vem are shown for 207 Pb in Fig. 6. The real part of the local optical potential and the electromagnetic potential is shown in Fig. 6a. The isovector interaction (b1 term) is also shown separately since the strength of this term is used to deduce the information on the symmetry of the strong interaction as described later. The real part of the whole local potential including the local optical potential and the electromagnetic potential is shown in the same figure as the thick solid line. It is found that the attractive electromagnetic potential and the repulsive local optical potential make the dip structure in the real part of the potential around the nuclear surface. The deeply bound pion is expected to distribute around the dip structure of the potential. The imaginary part of the local potential is expressed by the functional form of the square of the density ρ 2 indicating the two nucleon processes and is shown in Fig. 6b. The imaginary part of the potential takes into account the pion absorption to the nucleus, in other words, the decay (or the finite life time) of the pionic atoms due to the processes such as ppπ − → pn and pnπ − → nn followed by the backmπ or to-back two-nucleon emission from the nucleus with the kinetic energy ∼ 2 by the decay of the nucleus because of the energetic nucleons. The contribution of the pnπ − → nn process is considered to be expressed by the term proportional to ρn ρp and is shown separately in Fig. 6b. It would be good to be mentioned here that the decay of the pionic atoms by the one-nucleon process is possible in the pionic hydrogen by the π − p → π 0 n process because of the lighter mass of the neutral pion. This process is, however, (almost) forbidden in other heavier pionic atoms because of the energy cost of the nucleus due to the p → n conversion inside the nucleus.

2636

S. Hirenzaki and N. Ikeno 30

Re Vopt

(a) Real Part

20

Isovector interaction

Re V [MeV]

10

0

Re V -10

-20

Vem

-30 0

2

4

6

8

10

12

14

r [fm] 30

(b) Imaginary Part

20

Im V [MeV]

10

pn pair absorption 0

-10

Im V

-20

-30 0

2

4

8

6

10

12

14

r [fm]

Fig. 6 The local part of the π -207 Pb potential is shown as the functions of the radial coordinate r. (a) The real part of the local s-wave optical potential and the electromagnetic interaction is shown. The isovector part of the s-wave optical potential is also shown separately. The sum of the optical and the electromagnetic potential is shown as the thick solid line. The electromagnetic potential without vacuum polarization effects which corresponds to putting Q(r) = 1 in Eq. (15) cannot be distinguished from the line of Vem in this figure. (b) The imaginary part of the local s-wave optical potential is shown. The contribution of the pn pair absorption of pion in this potential is also shown separately

Structure of Deeply Bound Pionic Atoms In this section, the theoretical results of the structure of the deeply bound pionic atoms are explained. In Fig. 7, the calculated energy levels of the pionic atoms in

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

2637

Fig. 7 The calculated energy levels of the pionic atoms in 207 Pb and 123 Sn. The dotted lines indicate the energies obtained by the electromagnetic potential Vem . The solid lines indicate the energies by the optical potential Vopt and the electromagnetic potential Vem . The level widths due to the absorptive part of the optical potential are indicated by the hatched area

207 Pb

and 123 Sn are shown. It is found that the level spacing between the deeply bound levels such as 1s, 2p, and 2s is sufficiently larger than the level widths of these states. Thus, the existence of the discrete levels can be expected which can be, in principle, observed by the appropriate experimental method. It is also found from the calculated levels of both nuclei that the deeply bound levels are significantly shifted upward to the lighter bound direction by the strong interaction from the levels obtained only by the electromagnetic potential. This repulsive shift is tend to be larger for the deeper bound states and for the states with smaller angular momentum. Namely, the shift is larger for the 1s state than the 2s state and is larger for the 2s state than 2p state. This repulsive shift can be expected from the potential shown in Fig. 6a where the local optical potential is repulsive and is considered to push outward the pion wave functions from the center of the nucleus. In Fig. 8, the calculated energy levels are shown for the 1s and 2p states in 123 Sn with the several combinations of the potential terms to understand the contribution of each potential term to the energy shifts. It is found that for the 1s state, the electromagnetic level shown in (a) is largely shifted repulsively by the s-wave interaction terms as shown in (b) and (c) and is shifted a little back to the attractive direction by adding the nonlocal terms as in (e). Hence, the 1s state is suited to observe the effects of the s-wave interaction because the strong interaction shift is

2638

-1600

-3000

← 1s

-3500

-1800

-4500

-2000

-5000

2p →

-5500

-2200

Energy (2p) [keV]

-4000 Energy (1s) [keV]

Fig. 8 The calculated energy levels with the several combinations of the potential terms are shown in the solid lines for 1s state and dotted lines for the 2p state in 123 Sn. The level widths are indicated by the hatched area. The potential terms included in the calculation for the energy levels are (a) the electromagnetic interaction Vem ; (b) Vem and the isoscalar s-wave interaction (b0 term); (c) Vem , b0 term, and the isovector s-wave interaction (b1 term); (d) Vem and the whole part of the local potential (bs and B0 terms); and (e) the full potential (Vem and Vopt )

S. Hirenzaki and N. Ikeno

-6000 -2400 -6500

π-

123

Sn -2600

-7000

(a)

(b)

(c)

(d)

(e)

dominated by the s-wave interaction terms. On the other hand, the energy shift of the 2p state is shown to have the different details from that of the 1s state. As shown in Fig. 8, the repulsive shift due to s-wave interaction shown in (b) and (c) is largely canceled by the shift by the p-wave interaction in (e), and, as the consequence, the total shift remains to be small. Hence, the observation of the 2p state is considered to be suited to deduce the balance of the effects between the s-wave and the p-wave interactions. This different sensitivity of the states with the different angular momentum is also shown perturbatively using the Coulomb wave function (Ericson and Weise 1988). In Fig. 9, the calculated pion density distributions are shown for the 1s, 2s, and 2p states in 207 Pb and 123 Sn. It is observed in the figure that the large fraction of the pion distribution is located inside the nucleus for the bound states calculated only with the electromagnetic interaction. For the 1s state in 207 Pb, for example, pion exists inside the nucleus with about 50% probability due to the attractive electromagnetic interaction. The pion distributions are, then, pushed outward by introducing the repulsive optical potential as shown in the figure and are located around the nuclear surface for 1s state case. This behavior of the pion densities shown in Fig. 9 and the effects of the optical potential can be understood by considering the potential shape shown in Fig. 6. The pion distributions of the ground states obtained by the total potential are located just in the dip structure of the real potential shown in Fig. 6 as naturally expected. The repulsive effect of the real part of the optical potential, in other words, can be considered to play the important role to have the discrete levels in the deeply bound region. The imaginary part of the potential expressing the decay and/or finite life time of the pionic atoms takes into account the absorption of pion into nucleus

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . . 0.14

0.14

0.12

0.12

0.1

0.1 -

0.08 0.06

207

Pb

0.06

1s

0.04

0.04

0.02

0.02

-1 2 2

0 0.1 0.08

-

0.08

Pion Density |R nl(r) | r [fm ]

2 2

-1

Pion Density |R nl(r) | r [fm ]

2639

2s

0.06 0.04 0.02 0

1s

0.1 0.08

2s

0.06 0.04 0.02 0

0.1

0.1 0.08

2p

Sn

0

0.08 0.06

123

2p

0.06

0.04

0.04

0.02

0.02

0

0 0

5

10

15 r [fm]

20

25

30

0

5

10

15 r [fm]

20

25

30

Fig. 9 Pion density distributions of 1s, 2s, and 2p atomic states in 207 Pb and 123 Sn. The distributions obtained with the optical and electromagnetic potentials are shown by the solid lines and those with only electromagnetic potential by the dashed lines. The vertical lines indicate the nuclear surface, which are defined by the radius parameter of the two-parameter Fermi distribution for 207 Pb and as the radius parameter of the distribution of 124 Sn in Fricke et al. (1995) for 123 Sn

as mentioned above. The strength of the imaginary potential |ImV | at center of the nucleus is around 10 MeV which corresponds to the level width Γ = 2|ImV |  20 MeV. Thus, if the deeply bound pion exists at the nuclear center, the level width Γ is too large to form the discrete bound states. Because of the repulsive real part of the potential, the pion distribution is shifted outward to the nuclear surface, and the level width becomes significantly smaller as shown in Fig. 7 to form the discrete level even for the deepest 1s state in Pb. Actually if the real part of the optical potential is switched off, the calculated level width of the 1s state in 207 Pb is 7.2 MeV with the binding energy around 11 MeV. The size of the width is close to the half of Γ estimated at the nuclear center and seems consistent to the pion distribution shown in Fig. 9 where around half of the pion density exists inside the nucleus for the “Vem -only” potential case. The size of the width is too large to form the discrete level structure, and thus, the repulsive nature of the optical potential is one of the important pieces to allow the existence of the discrete deeply bound pionic levels.

2640

S. Hirenzaki and N. Ikeno

The residual interaction effects reported in Hirenzaki et al. (1999), and NoseTogawa et al. (2005) are also considered to study the structure of the deeply bound pionic atoms precisely in the nucleus with a neutron hole. The residual interaction effects are caused by the differences of the interaction between pion and nucleus with the different neutron-hole state, which are not taken into account in the optical potential. Though the effects are not large, they cannot be neglected for the studies of the deeply bound pionic atom in very high accuracy. The residual interaction effects to the binding energies and widths can be evaluated by the calculation of the matrix elements of the residual interaction and the diagonalization of the matrix (Hirenzaki et al. 1999; Nose-Togawa et al. 2005). Finally, as one of the recent interesting topics, the possibility of the precise determination of the value of the pion–nucleon sigma term σπ N by the observables of the deeply bound pionic atoms is explained briefly. The pion–nucleon sigma term σπ N is defined as the nucleon matrix element of the u- and d-quark mass terms of the QCD Hamiltonian as σπ N =

mq ¯ N|uu ¯ + dd|N, 2M

(16)

where M is the nucleon mass and mq the average of the current quark masses of the u- and d-quarks. This term explicitly expresses the contribution of the quark mass terms, namely, the explicit chiral symmetry breaking effects, to the nucleon mass. In addition, this sigma term is also important to consider the in-medium modification of the qq ¯ vacuum condensate because σπ N is proportional to the coefficient of the leading term of the low-density expansion of qq ¯ ρ . The value of σπ N reported by the various research groups is compiled in Yamanaka et al. (2018) and Gupta et al. (2021) and is found to be distributed in the range of σπ N = 30 ∼ 60 MeV. Thus, the determination of σπ N by the deeply bound pionic atom data is considered to be very important. For this purpose, the s-wave isovector π N interaction parameter b1 is expressed in the explicit density-dependent form as (Weise 2000, 2001)  −1 σπ N b1 (ρ) = b1free 1 − 2 2 ρ , mπ fπ

(17)

with the σπ N term, the pion mass mπ , and the pion decay constant fπ . The  −1 σπ N factor 1 − 2 2 ρ in Eq. (17) is the wave function renormalization for mπ fπ the in-medium pion, which is introduced to formulate the pion optical potential from the π N amplitude with the energy dependence implied by the breaking of the chiral symmetry (Ericson 1987; Kolomeitsev et al. 2003). The wave function renormalization factor including σπ N provides the explicit density dependence of the b1 parameter in the pion optical potential. Thus, the σπ N value can be deduced from the pionic atom data as reported in Friedman and Gal (2019, 2020), where the σπ N value is determined to be σπ N = 57 ± 7 MeV using the existing pionic atom data.

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

2641

The observables of the deeply bound pionic states are naturally expected to be more sensitive to the value of σπ N term because of the larger overlap of the pion distributions with the nucleus and expected to be used to determine the σπ N value more precisely. In Fig. 10, the σπ N value dependence of the potential parameter and the deeply bound pionic atom observables are shown. In Fig. 10a, the σπ N term dependence of the b1 parameter in Eq. (17) is shown for the finite nuclear density ρ = ρ0 and ρ = ρeff , where ρ0 is the normal nuclear density ρ0 = 0.17 fm−3 and ρeff is the effective nuclear density probed by the pionic atoms (Yamazaki and Hirenzaki 2003; Ikeno et al. 2011) defined as ρeff = 0.6ρ0 . It is found that the absolute value of b1 increases with the σπ N value and the potential becomes more repulsive for the larger σπ N value. The b1 value at ρeff is consistent with the constant b1 value obtained by Seki–Masutani (Seki and Masutani 1983) at σπ N ∼ 60 MeV which is close to the value obtained in Friedman and Gal (2019, 2020), while the results obtained in Suzuki et al. (2004) indicate the σπ N value σπ N ∼ 45 MeV. In Fig. 10b and c, the binding energies and widths of the deeply bound 1s and 2p pionic atom states in 207 Pb are shown as the functions of the σπ N value. It is found that the binding energy of the 1s state shifts about 11 keV for the 1 MeV change of the σπ N value. The 2p binding energy also shifts about 7 keV for the 1 MeV change of σπ N . The widths of the 1s and 2p states also depend on σπ N and change their values sensitively to the value of σπ N . By combining these sensitivities of the observables to the σπ N value and the expected accuracy of the experimental observations, the best observables can be found to determine the σπ N value quite precisely by the study of the deeply bound pionic states (Ikeno et al. 2022). The experimental studies of the deeply bound pionic states, on the other hand, have been performed at RIBF/RIKEN recently with improved conditions. The new data of the binding energies and widths and the formation spectra of the pionic atoms with high quality are reported for the Sn isotopes (Nishi et al. 2018). Thus, we believe that the precise studies of the strong interaction at finite density by the deeply bound pionic states will increase their importance in future and provide the quite important contributions to determine the values of the key quantities like σπ N .

Hadron Reactions Proposed for the Pionic Atom Formation Many experimental methods were proposed for the observations of the deeply bound pionic states. Some of the reactions are mentioned here as examples, which are (n, p) (Toki and Yamazaki 1988; Toki et al. 1989; Iwasaki et al. 1991; Nieves and Oset 1990), (n, d) (Toki et al. 1991; Trudel et al. 1991; Hirenzaki and Toki 1998), (d,3 He) (Toki et al. 1991; Hirenzaki et al. 1991), (p, 2p) (Tsunoda 1995; Matsuoka et al. 1995), (γ , p) (Hirenzaki and Oset 2002), (π, γ ) (Nieves and Oset 1992), and (π, p) (Kaufmann et al. 1992) reactions. Here, in the first five reactions (n, p), (n, d), (d,3 He), (p, 2p), and (γ , p), the π mesons are produced dynamically by consuming the energy of the initial state and are expected to be trapped into the atomic orbits. Within these five reactions, the masses of the projectile and ejectile are almost same in the (n, p) reaction. Thus, the momentum transfer is

2642

S. Hirenzaki and N. Ikeno

-0.09 (a)

-0.1 Ueff

-0.11 b1 [mS-1]

-0.12 -0.13

Seki-Masutani

U0

-0.14 -0.15 -0.16 -0.17 7500

(b)

1000

207

Pb

900

mBS(1s)

7300

800

*S(1s)o

7200

*S [keV]

BS [keV]

7400

700

7100

600 (c)

5500

800

207

Pb 700

mBS(2p)

5400

600

5350

*S(2p)o

*S [keV]

BS [keV]

5450

500

5300 5250

400 25

30

35

40

45

50

55

60

65

VS N [MeV] Fig. 10 (a) The σπ N value dependence of the b1 parameter defined in Eq. (17) is shown for ρ = ρ0 by the solid line and for ρ = ρeff (≡ 0.6ρ0 ) by the dashed line, where ρ0 is the normal nuclear density ρ0 = 0.17 fm−3 . The constant b1 parameter value in Table 1 (Seki and Masutani 1983) is also shown by the dotted line. (b) The binding energy (Bπ ) and the width (Γπ ) of the pionic 1s state in 207 Pb are plotted as the functions of the σπ N value. (c) Same as (b) except for 2p state in 207 Pb

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

2643

relatively large in (n, p) which causes difficulties for obtaining the sufficiently large formation rate of the pionic atoms. In other four reactions (n, d), (d,3 He), (p, 2p), and (γ , p), the momentum transfer can be small because of the larger mass of the ejectile than projectile. In these reactions, on the other hand, the daughter nucleus has a nucleon hole in the final state which could make the reaction spectra more complex because of the contributions from different hole states which cannot be distinguished in the inclusive reactions. In the reactions (π, γ ) and (π, p), the π mesons are injected as the projectiles into the nucleus and the reaction Q values are relatively small. In the (π, p) reaction, the daughter nucleus has a hole as other one-nucleon pick-up reactions as mentioned above. In the (π, γ ) reaction, the direct capture process of the pion into the atomic orbit is considered, and the observation of the emitted photon is proposed. This reaction is different from the traditional X-ray spectroscopy which includes the deexcitation processes of atomic states suffering from the nuclear absorption effects before reaching the deep states. Finally the (d,3 He) reaction is adopted as the most suitable method to populate and observe the deeply bound pionic atoms. The (d,3 He) reaction is one of the one-nucleon pick-up reactions as shown in Fig. 11 schematically. The momentum transfer of the (d,3 He) reaction can be tuned to be almost zero (recoilless) at the incident deuteron kinetic energy Td = 500 MeV as shown in Fig. 12 where the

Fig. 11 Schematic diagram for the (d,3 He) reactions for the formation of the pionic atoms with a neutron-hole state in the daughter nucleus

250 Momentum transfer q [MeV/c]

Fig. 12 Momentum transfers in the (d,3 He) reactions on heavy nuclei for Q = −140 MeV as the functions of the incident deuteron kinetic energy. The solid line indicates the momentum transfer for the forward reaction and the dashed line for the reaction with the 3 He angle in the laboratory frame θ = 2◦ case

200

150

100

Q = -140 MeV 50

0 400

0 deg. 2 deg. 500

600

700

800 900 Td [MeV]

1000

1100

1200

2644

S. Hirenzaki and N. Ikeno

large formation rate is expected. In addition, the projectile (d) and the ejectile (3 He) are both charged and stable particles which have advantages in principle to use the high-intensity primary beam and to perform the high-resolution spectroscopy. In the (d,3 He) reaction, the deeply bound pionic atoms were discovered firstly as clear peaks in the spectra, while the only unresolved some extra strengths were observed in subthreshold region in the (n, d) (Trudel et al. 1991) and (p, 2p) (Matsuoka et al. 1995) reactions which were not enough to prove the existence of the deeply bound pionic atoms as the discrete states. It may be helpful to give a few comments here concerning on the possible complexities of the inclusive spectra due to the inseparable contributions from the nucleon-hole states in the daughter nucleus in the (d,3 He) reaction. Actually, the number of the subcomponent, the combination of the nucleon-hole state and the pionic bound state in the final state, reaches typically order of 100. However, because of the matching condition between the momentum transfer and the angular momentum transfer of the reaction, only the selected combinations of the hole and the pionic states contribute dominantly to the spectrum as explained later which makes the interpretation and identification of the observed peaks quite clear and unambiguous.

(d,3 He) Reactions for the Formation of the Deeply Bound Pionic Atoms The effective number approach is used to evaluate the formation rate of the deeply bound pionic atoms in the (d,3 He) reactions (Toki et al. 1991; Hirenzaki et al. 1991), in which the formation spectra are expressed as the product of the elemental pion production cross section of the d + n → 3 He + π − reaction and the effective number of the nucleon participating to the reaction. The theoretical basis of this factorization and the expression of the effective number is given in Ikeno et al. (2011). The application of the effective number approach is extended to the variety of nuclei in Umemoto et al. (2000) and to the calculations of the formation spectra at finite angles in Ikeno et al. (2011). The basis of the effective number approach is explained briefly according to the formulation in Ikeno et al. (2011). The formation cross section of the deeply bound pionic atoms in the (d,3 He) reactions is written as dσ =

V2 1 V |Sf i |2 dpHe , vrel V T (2π )3

(18)

f

where vrel is the velocity of the projectile in the laboratory frame and pHe the momentum of the emitted 3 He. The nucleus is assumed to be heavy enough and to remain at rest in the reaction in the laboratory frame. The S-matrix of the reaction Sf i is expressed as

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

MHe 1 iEHe t ∗ 1 iEπ t ∗ = dtdr χHe (r) e φπ (r)iT (pd , pn , pHe , pπ ) √ e EHe V 2Eπ



Md 1 −iEd t Mn −iEn t × χd (r) e ψn (r), (19) √ e Ed V En



Sf i

2645

where Md , MHe , and Mn are the masses of deuteron, 3 He, and neutron participating ∗ (r) are the wave functions of in the (d,3 He) reaction and e−iEd t χd (r) and eiEHe t χHe 3 the projectile (d) and ejectile ( He), respectively. The wave functions of the bound neutron in the target and pion in the atomic state are written as e−iEn t ψn (r) and eiEπ t φπ∗ (r). The relativistic energy of each particle is indicated as Ed , EHe , En , and Eπ . The meson production amplitude T (pd , pn , pHe , pπ ) depending on the four-momenta of each particle will be replaced by the experimental data of the elementary process later. The standard manipulation for the cross-section formula leads to the expression of the formation cross section of the deeply bound pionic states in the laboratory frame as 

d 2σ dEHe dΩHe  = f



1 Md Mn MHe |pHe | 1 Γ |T (pd , pn , pHe , pπ )|2 |pd | 2π ΔE 2 + Γ 2 /4 2(2π )2 En Eπ

 2 ∗ ∗ × drχHe (r)φπ (r)ψn (r)χd (r) ,

(20)

where the Lorentz distribution function is included instead of the δ-function for the energy conservation to consider the width of the atomic states Γ . The effective number is defined as the part of the cross-section Eq. (20) as Neff =

2  dr χ ∗ (r)[φ ∗ (r) ⊗ ψj (r)]J M χd (r) , n π He

(21)

JM

where π and jn indicate the angular momenta of the pion bound state in the daughter nucleus and the neutron bound state in the target nucleus which couple to form the total angular momentum quantum numbers J and M in the final state. To show the angular momentum of each bound particle explicitly, the indexes of φπ and ψn are changed to π and jn here. The neutron wave functions are calculated by solving the Schrödinger equation with the effective potential determined in Koura and Yamada (2000). The spatial part of the wave functions of the projectile and the ejectile written as χd and χHe includes the distortion effects to these particles which are evaluated by the eikonal approximation (Hirenzaki et al. 1991). The amplitude T (pd , pn , pHe , pπ ) is then replaced by the experimental data of the elementary d + n → 3 He + π − process. The cross section for the elementary process is written using the same amplitude T (pd , pn , pHe , pπ ) as

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dσ = ×

1 Md Mn |T (pd , pn , pHe , pπ )|2 (2π )4 δ 4 (pd + pn − pHe − pπ ) vrel Ed En MHe dpHe 1 dpπ . EHe (2π )3 2Eπ (2π )3

(22)

Following the standard manipulation of the cross-section formula again, the expression of the cross section of the elementary process is obtained in the “target neutron rest frame” as 

dσ dΩHe =

 ele

|T (pd , pn , pHe , pπ )|2 Md Mn MHe (2π )2 λ1/2 (s, Md2 , Mn2 )



|pHe |2 , Eπ |pHe | + EHe (|pHe | − |pd | cos θdHe ) (23)

where λ(s, Md2 , Mn2 ) is the Källen function defined as λ(a, b, c) = a 2 + b2 + c2 − 2ab − 2bc − 2ca. The amplitude T (pd , pn , pHe , pπ ) can be evaluated by the observed elementary cross section using Eq. (23) and can be replaced by the experimental data (Toki et al. 1991; Hirenzaki et al. 1991). Finally, the formation spectra of the pionic atoms in (d,3 He) reactions are expressed in the laboratory frame using the effective  Neff and the observed elementary cross section in  number dσ the laboratory frame of the d + n → 3 He + π − reaction as dΩHe ele 

d 2σ dEHe dΩHe



 =

dσ dΩHe





ele ph

K

1 Γ Neff , 2π ΔE 2 + Γ 2 /4

(24)

where the sum is taken over the all combinations of the pion bound states and neutron-hole states considered in the final states. The energy gap ΔE is defined as ΔE = EHe + Eπ − Ed − En with the neutron energy En in the target nucleus and the energy of the pion bound state Eπ . En is defined as En = Mn − Sn (jn ) with the neutron separation energy Sn from the single-particle level indicated by jn and the neutron mass Mn . And Eπ is defined as Eπ = mπ − B(π ) with the binding energy B(π ) of the bound state indicated by π and  the pion mass mπ . dσ , the experimental As for the experimental elementary cross section dΩHe ele data of d + p → t + π + are used assuming the charge symmetry (Toki et al. 1991; Hirenzaki et al. 1991). In the present calculations, the same elementary cross sections as in Ikeno et al. (2011) are used including the finite scattering angle cases. The kinematical factor K is introduced to correct the difference of the kinematics between the elementary process d + p → 3 He + π and the reaction with the nuclear target A(d, 3 He) and is defined as

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

 K=

|pA He | En Eπ |pHe | EnA EπA

  EHe |pHe | − |pd | cos θdHe 1+ , Eπ |pHe |

2647

(25)

where all energies and momenta are evaluated in the laboratory frame and the superscript “A” indicates the energies and the momentum evaluated in the kinematics of the nuclear target case (Ikeno et al. 2011). In order to obtain the realistic sizes and shapes of the formation spectra for the reliable predictions, it is quite valuable to use the experimental information in a few points. As already mentioned above, the data for the elementary pion production process are used. At the recoilless kinematics in the laboratory (target rest) frame where the (d,3 He) reaction for the pionic atom formation is performed, the momentum transfer between the projectile and ejectile in the projectile rest frame is not small, and the theoretical treatment of the elementary process is difficult. Thus, it is worth using the experimental information for the elementary process for evaluating the realistic magnitude of the formation rate. In addition, the observed information on the nuclear responses to the one-nucleon pick-up processes is taken into account. Since the nuclei are the complex many-body systems, each of them shows the characteristic features to the one-nucleon pick-up processes, which require significant efforts to understand them sufficiently in general. Hence, the experimental spectroscopic factors and the energies are used for the excited levels produced by the one-nucleon pick-up reactions on the target nucleus, as described in Umemoto et al. (2000). This prescription is shown to be reliable by predicting the pionic atom formation spectrum of the 206 Pb(d,3 He) reaction correctly, which is clearly different from that of 208 Pb(d,3 He) (Hirenzaki and Toki 1997). To compare the theoretical results with the data of the 3 He spectra of the inclusive (d,3 He) reactions around the pion production threshold, it is necessary to evaluate the quasi-elastic (QE) pion production processes A Z(d,3 He)A−1 Z + π − and A Z(d,3 He)A−1 (Z − 1) + π 0 which also contribute to the (d,3 He) cross sections above the pion production threshold. The theoretical evaluation of the QE processes is explained in detail in Hirenzaki and Toki (1998). The wave functions of proton, which are necessary for the evaluation of the π 0 production, are obtained using the effective potential for proton in Koura and Yamada (2000) as in the same manner with the neutron wave functions. The correction factor K is also included in the evaluation of the QE contributions. In Fig. 13, the calculated forward spectrum of the 208 Pb(d,3 He) reactions at Td = 500 MeV is shown for the formation of the pionic atoms with the contributions from the QE π − and π 0 production. The single-particle levels included in the calculation are 6 neutron-hole states (3p1/2 , 2f5/2 , 3p3/2 , 1i13/2 , 2f7/2 , 1h9/2 ), 5 protonhole states (3s1/2 , 2d3/2 , 1h11/2 , 2d5/2 , 1g7/2 ), and 18 pionic states (1s ∼ 6f ) to evaluate the spectra for the 208 Pb target case. The neutron separation energies and neutron spectroscopic factors of 208 Pb in Itahashi et al. (2000) are adopted. The proton separation energies for the π 0 production are taken from Koura and Yamada (2000) and Parkinson et al. (1969), and the proton spectroscopic factors are assumed to be 1. It is found that the contribution from the QE π 0 production is

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S. Hirenzaki and N. Ikeno 350

d2σ/dΩ dE [μb/sr/MeV]

300

208

Pb(d,3He): 0 deg.

2p⊗p3/2

250 3p⊗p3/2

200

2p⊗p1/2

150 3p⊗p1/2

100 -

π QE

50 0

1s

π0 QE

-144

-142

-140

-138

-136

-134

-132

-130

Q [MeV]

Fig. 13 The calculated forward spectrum of the 208 Pb(d,3 He) reactions for the formation of the pionic atoms at Td = 500 MeV is shown as the thick solid line as a function of the reaction Q-value. The dominant subcomponents (n)π ⊗ (j )n and the quasi-elastic (QE) pion production contributions are also shown as indicated in the figure. The sum of the subcomponents 1s⊗p1/2 and 1s ⊗ p3/2 is shown as 1s in the figure. The vertical line indicates the π − production threshold Q = −140.2 MeV with the ground state of the daughter nucleus with a neutron hole. The π 0 production threshold with the ground state of the daughter nucleus with a proton hole is Q = −137.5 MeV and is 2.7 MeV different from that of π − because of the differences of the charged and neutral pion masses and the neutron and proton separation energies. Experimental energy resolution is assumed to be 150 keV

small. The contributions from the QE π − production is larger than that from π 0 and increases rapidly just above the threshold. This behavior of the π − QE contribution is due to the strongly attractive Coulomb interaction to π − . In both cases, the QE contributions are found to be well separated from those of the deeply bound pionic atom formation. As can be seen in Fig. 13, the selected subcomponents (2p, 3p)π ⊗ (p1/2 , p3/2 )n dominate the bound state formation spectrum because of the matching condition of the reaction written as Δ = q × R,

(26)

where Δ is the angular momentum transfer, q the momentum transfer, and R the nuclear radius. The matching condition is so effective in the (d,3 He) reactions for the pionic atom formation that the main subcomponents of the spectra are well understood by the condition. Thus, the momentum transfer of the reaction and the configuration of the neutron single-particle states in the target show us the pionic states which are populated remarkably in the spectrum. In the case of the reaction shown in Fig. 13, for example, since the incident energy is tuned to make the reaction almost recoilless q ∼ 0, the matching condition forces to make the

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

2649

angular momentum transfer small Δ ∼ 0, even for the large radius of the heavy nucleus R = 6 ∼ 7 fm. This requirement Δ ∼ 0 is satisfied for the combinations of pionic p states and neutron p states by making the sum of the angular momenta to be 0. Since the deeply bound pionic states such as 1s and 2p have the small angular momenta π = 0 and 1, the neutron s and p states in the valence shell are important to satisfy the matching condition for the formation of the deeply bound states for the recoilless kinematics. The reason why the classical expression of the matching condition Δ = q × R holds in the quantum mechanical calculation can be seen in the definition of the effective number in Eq. (21). The effective number is, simply to say, the evaluation of the overlap integral between the wave functions of the particles participating in the reaction. Let’s suppose the total orbital angular momentum of the bound pion and neutron to be L which is equal to the angular momentum transfer Δ. Since the product of the wave functions of the unbound projectile and ejectile can be written by the multipole expansion, the integration of the angular coordinates in the effective number calculation projects out the components with total orbital angular momentum L from the multipole expansion. The radial part of the total wave function of the bound states φπ and ψjn distributes inside and the surface of the nucleus, while the distortion effects to the projectile and ejectile remove out the contribution of the unbound particle wave functions from the inside of the nucleus by the distortion factor in the eikonal approximation. Thus, the size of the spherical Bessel function jL (x) with the angular momentum L, which is from the total radial wave functions of the projectile and ejectile, at the nuclear surface x = qR determines the size of the radial integration of the effective numbers, and the larger value of jL (qR) leads to the consequences with the larger effective numbers. Namely, the property of the spherical Bessel function finally makes the connection between q, R, and L (= Δ) which provides the matching condition. In the early stage of the study of the deeply bound pionic atoms, the Pb nucleus was selected as the target as one of the heaviest stable nuclei to form the deepest bound pionic states. The selection of the dominant subcomponents due to the matching condition is, however, unexpectedly strong, and, thus, the first clear observation of the deeply bound pionic state was not the 1s state but the 2p state which made the clear peaks in the spectrum coupled to the neutron p3/2 and p1/2 states in the neutron valence shell in Pb in the recoilless kinematics (Gilg et al. 2000; Itahashi et al. 2000). In Fig. 14, the spectra of the (d,3 He) reactions for the 208 Pb (upper) and 124 Sn (lower) targets are shown for the pionic atom formation at the incident energy Td = 500 MeV where the reaction is recoilless at forward angles. The 208 Pb(d,3 He) spectra for the finite angles θ = 1 and 2 degrees of the emitted 3 He are shown in Fig. 14 (upper) together with the forward spectrum. As can be seen from Fig. 12, the momentum transfer is larger at the finite scattering angles and is around 50 MeV/c at θ = 2 degrees at Td = 500 MeV. Thus, because of the matching condition explained above, the subcomponents with larger angular momentum transfer Δ are expected to be more enhanced in the spectra at the finite angles. The calculated results in Fig. 14 (upper) show that the spectra at finite angles have the features as expected

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2 d σ/dΩ dE [μb/sr/MeV]

208

3

Pb(d, He)

0 deg. 1 deg. 2 deg.

300 250 200 150 100 50 0

-144

-142

-140

-138

-136

-134

-132

-130

Q [MeV]

2 d σ/dΩ dE [μb/sr/MeV]

250

200

124

Sn(d,3He)

0 deg. 1 deg. 2 deg.

150

100

50

0 -142

-141

-140

-139

-138

-137

-136

-135

Q [MeV]

Fig. 14 The calculated spectra of the (d,3 He) reactions are shown for the formation of the pionic atoms at Td = 500 MeV for 208 Pb (upper) and 124 Sn (lower) targets for the emitted 3 He angles θ = 0, 1, 2 degrees in the laboratory frame. The quasi-elastic pion production processes are included only for the 208 Pb(d,3 He) spectra. The vertical line indicates the π − production threshold Q = −140.2 MeV for Pb and Q = −141.3 MeV for Sn with the ground state of the daughter nucleus with one neutron hole. Experimental energy resolution is assumed to be 150 keV

and the subcomponents with larger Δ are more enhanced relatively than those in the spectra at θ = 0 degrees. The absolute sizes of the whole spectra, however, tend to be smaller for the finite angles because of the larger momentum transfer which makes, in general, the formation rate of the bound states smaller. In Fig. 14 (lower), the spectra of the same (d,3 He) reaction are shown for the 124 Sn target as Fig. 14 (upper). The QE contributions are neglected in Fig. 14 (lower). In 124 Sn, there is the s1/2 state in the valence neutron shell, and 5 neutronhole states (3s1/2 , 2d3/2 , 1h11/2 , 2d5/2 , 1g7/2 ) and 18 pionic states (1s ∼ 6f ) are included in the calculation for the (d,3 He) spectra. The neutron separation energies and the spectroscopic factors of 124 Sn in Schneid et al. (1967) and Umemoto et al.

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

2651

208

Pb(d,3He): Td = 1200 MeV

0.004

3p ⊗ i13/2

(d2σ/dΩ dE)/(dσ/dΩ)

ele

[1/MeV]

0.005

2p ⊗ i13/2

0.003

0.002 1s ⊗ i13/2

0.001

0 -144

-142

-140

-138

-136

-134

-132

-130

Q [MeV]

Fig. 15 The relative strength of the spectrum of the 208 Pb(d,3 He) reactions for the formation of the pionic atoms at Td = 1200 MeV is shown as the thick solid line as a function of the reaction Q-value at forward angle. The dominant subcomponents are also shown as indicated in the figure. The quasi-elastic pion production processes are neglected. The vertical line indicates the π − production threshold Q = −140.2 MeV with the ground state of the daughter nucleus with a neutron hole. Experimental energy resolution is assumed to be 150 keV

(2000) are used. It is found in Fig. 14 (lower) that the dominant subcomponent is 1s ⊗ s1/2 which makes the largest clear peak in the forward spectrum. At finite angles, the peak becomes lower and other subcomponents become relatively larger in the spectra. Thus, the (d,3 He) reactions at the recoilless kinematics on the target nucleus with the s1/2 neutron in the valence shell are considered to be one of the best choices to observe the deepest 1s pionic states as the dominant peak structure in the spectrum. Finally, it is shown how the spectrum changes for the even larger momentum transfer cases. In Fig. 15, the (d,3 He) reaction spectrum is shown in unit of the elementary cross section by the relative strength for the 208 Pb target at Td = 1200 MeV, where the momentum transfer q for the pionic atom formation is large even at the forward angles as q ≥ 200 MeV/c as shown in Fig. 12. In this case, the matching condition is satisfied for the subcomponents with the angular momentum transfer Δ ∼ 200 MeV/c × 6 fm ∼ 6 around the surface of Pb. As can be seen from Fig. 15, the shape of the spectrum is much different from that in Fig. 13 at the recoilless kinematics, and the i13/2 neutron state coupled to the deeply bound states with  = 0 and 1 has the dominant contributions to the spectrum of 208 Pb(d,3 He). At the end of this section, two related topics on the formation of the pionic atoms are explained briefly. First, the (d,3 He) reactions on the odd-neutron nuclear target are also studied for the formation of the deeply bound pionic atoms to avoid the existence of the neutron hole in the final states and to avoid the residual interaction effects to the structures of the pionic atoms (Ikeno et al. 2013). The possibility

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is shown to provide the pionic atoms in nucleus without neutron-hole states by the (d,3 He) reactions on the odd-neutron nuclear target by picking up the valence neutron in the reaction. In this case, the residual interaction effects do not bother us in the precise study of the pionic levels. Secondly, Green’s function method can be applied to evaluate the (d,3 He) spectra of the pionic atom formation. Green’s function method is better suited for the calculations of formation rate of the unstable bound states (Morimatsu and Yazaki 1985, 1988), while it requires the significantly burdensome numerical calculations. For the pionic atoms, the differences between the results obtained by the effective number approach and Green’s function method are shown to be small (Ikeno et al. 2015). In general, the application of Green’s function method is known to be more important for the studies of more unstable bound states with larger widths such as the mesonic nuclear states.

Summary and Conclusion In this chapter, we try to provide the readers a brief review of the deeply bound pionic atoms and the related topics. First, the introduction is given for the mesonic atoms and mesonic nuclei including K − –, η –, and η(958) – nucleus bound states in addition to the pionic atoms, then, followed by the explanation of the interests and motivations of the bound states in relation to the studies of the aspects of the symmetry of the strong interaction at finite density. The basic building blocks of the theoretical calculations of the structures and formation spectra of the deeply bound pionic atoms are explained in detail. The various reactions proposed for the formation of the deeply bound pionic atom are also mentioned very briefly. We hope this chapter helps the readers to be familiar with this subject and to start the studies of this field.

References C.J. Batty, E. Friedman, A. Gal, Phys. Rep. 287, 385–445 (1997). https://doi.org/10.1016/S03701573(97)00011-2 R.H. Dalitz, T.C. Wong, G. Rajasekaran, Phys. Rev. 153, 1617–1623 (1967). https://doi.org/10. 1103/PhysRev.153.1617 C.E. Detar, T. Kunihiro, Phys. Rev. D 39, 2805 (1989). https://doi.org/10.1103/PhysRevD.39.2805 T.E.O. Ericson, Phys. Lett. B 195, 116 (1987). https://doi.org/10.1016/0370-2693(87)91180-4 M. Ericson, T.E.O. Ericson, Ann. Phys. 36, 323–362 (1966). https://doi.org/10.1016/00034916(66)90302-2 T. Ericson, W. Weise, Pions and Nuclei (Clarendon Press, Oxford, 1988) G. Fricke, C. Bernhardt, K. Heilig, L.A. Schaller, L. Schellenberg, E.B. Shera C.W. de Jager, Atom. Data Nucl. Data Tabl. 60, 177–285 (1995). https://doi.org/10.1006/adnd.1995.1007 E. Friedman, A. Gal, Phys. Rep. 452, 89–153 (2007). https://doi.org/10.1016/j.physrep.2007.08. 002 E. Friedman, A. Gal, Phys. Lett. B 792, 340–344 (2019). https://doi.org/10.1016/j.physletb.2019. 03.036

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

2653

E. Friedman, A. Gal, Acta Phys. Polon. B 51, 45–54 (2020). https://doi.org/10.5506/APhysPolB. 51.45 E. Friedman, G. Soff, J. Phys. G 11, L37–L42 (1985). https://doi.org/10.1088/0305-4616/11/3/003 S. Friedrich, M. Nanova, V. Metag, F.N. Afzal, D. Bayadilov, B. Bantes, R. Beck, M. Becker, S. Böse, K.T. Brinkmann et al., Eur. Phys. J. A 52, 297 (2016). https://doi.org/10.1140/epja/ i2016-16297-4 C. Garcia-Recio, J. Nieves, T. Inoue, E. Oset, Phys. Lett. B 550, 47–54 (2002). https://doi.org/10. 1016/S0370-2693(02)02960-X M. Gell-Mann, R.J. Oakes, B. Renner, Phys. Rev. 175, 2195–2199 (1968). https://doi.org/10.1103/ PhysRev.175.2195 H. Gilg, A. Gillitzer, M. Knulle, M. Munch, W. Schott, P. Kienle, K. Itahashi, K. Oyama, R.S. Hayano H. Geissel et al., Phys. Rev. C 62, 025201 (2000). https://doi.org/10.1103/ PhysRevC.62.025201 R. Gupta, S. Park, M. Hoferichter, E. Mereghetti, B. Yoon, T. Bhattacharya, Phys. Rev. Lett. 127, 24 (2021). https://doi.org/10.1103/PhysRevLett.127.242002 Q. Haider, L.C. Liu, Phys. Lett. B 172, 257–260 (1986). https://doi.org/10.1016/03702693(86)90846-4 T. Hashimoto et al. [J-PARC E62], Phys. Rev. Lett. 128, 112503 (2022) https://doi:10.1103/ PhysRevLett.128.112503 T. Hatsuda, T. Kunihiro, Prog. Theor. Phys. 74, 765 (1985). https://doi.org/10.1143/PTP.74.765 T. Hatsuda, T. Kunihiro, Phys. Rep. 247, 221–367 (1994). https://doi.org/10.1016/03701573(94)90022-1 S. Hirenzaki, E. Oset, Phys. Lett. B 527, 69–72 (2002). https://doi.org/10.1016/S03702693(02)01165-6 S. Hirenzaki, H. Toki, Phys. Rev. C 55, 2719–2722 (1997). https://doi.org/10.1103/PhysRevC.55. 2719 S. Hirenzaki, H. Toki, Nucl. Phys. A 628, 403–416 (1998). https://doi.org/10.1016/S03759474(97)00599-X S. Hirenzaki, H. Toki T. Yamazaki, Phys. Rev. C 44, 2472–2479 (1991). https://doi.org/10.1103/ PhysRevC.44.2472 S. Hirenzaki, H. Kaneyasu, K. Kume, H. Toki, Y. Umemoto, Phys. Rev. C 60, 058202 (1999). https://doi.org/10.1103/PhysRevC.60.058202 G. ’t Hooft, Phys. Rev. D 14, 3432–3450 (1976) [erratum: Phys. Rev. D 18, 2199 (1978)]. https:// doi.org/10.1103/PhysRevD.14.3432 N. Ikeno, R. Kimura, J. Yamagata-Sekihara, H. Nagahiro, D. Jido, K. Itahashi, L.S. Geng, S. Hirenzaki, Prog. Theor. Phys. 126, 483–509 (2011). https://doi.org/10.1143/PTP.126.483 N. Ikeno, H. Nagahiro S. Hirenzaki, Eur. Phys. J. A 47, 161 (2011). https://doi.org/10.1140/epja/ i2011-11161-9 N. Ikeno, J. Yamagata-Sekihara, H. Nagahiro, S. Hirenzaki, PTEP 2013, 063D01 (2013). https:// doi.org/10.1093/ptep/ptt035 N. Ikeno, J. Yamagata-Sekihara, H. Nagahiro, S. Hirenzaki, PTEP 2015, 033D01 (2015). https:// doi.org/10.1093/ptep/ptv013 N. Ikeno, H. Nagahiro, D. Jido, S. Hirenzaki, Eur. Phys. J. A 53, 194 (2017). https://doi.org/10. 1140/epja/i2017-12381-7 N. Ikeno, T. Nishi, K. Itahashi, N. Nose-Togawa, A. Tani, S. Hirenzaki, arXiv:2204.09211 [nucl-th] (2022) T. Inoue, E. Oset, Nucl. Phys. A 710, 354–370 (2002). https://doi.org/10.1016/S03759474(02)01167-3 K. Itahashi, K. Oyama, R.S. Hayano, H. Gilg, A. Gillitzer, M. Knulle, M. Munch, W. Schott, P. Kienle, H. Geissel et al., Phys. Rev. C 62, 025202 (2000). https://doi.org/10.1103/PhysRevC. 62.025202 M. Iwasaki, A. Celler, O. Hausser, R.S. Hayano, R. Helmer, R. Henderson, S. Hirenzaki, K.P. Jackson, Y. Kuno N. Matsuoka et al., Phys. Rev. C 43, 1099–1105 (1991). https://doi. org/10.1103/PhysRevC.43.1099

2654

S. Hirenzaki and N. Ikeno

D. Jido, Y. Nemoto, M. Oka A. Hosaka, Nucl. Phys. A 671, 471–480 (2000). https://doi.org/10. 1016/S0375-9474(99)00844-1 D. Jido, M. Oka, A. Hosaka, Prog. Theor. Phys. 106, 873–908 (2001). https://doi.org/10.1143/PTP. 106.873 D. Jido, T. Hatsuda, T. Kunihiro, Phys. Lett. B 670, 109–113 (2008). https://doi.org/10.1016/j. physletb.2008.10.034 N. Kaiser, P.B. Siegel, W. Weise, Nucl. Phys. A 594, 325–345 (1995). https://doi.org/10.1016/ 0375-9474(95)00362-5 W.B. Kaufmann, P.B. Siegel, W.R. Gibbs, Phys. Rev. C 46, 1474–1482 (1992). https://doi.org/10. 1103/PhysRevC.46.1474 M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 44, 1422–1424 (1970). https://doi.org/10.1143/ PTP.44.1422 E.E. Kolomeitsev, N. Kaiser, W. Weise, Phys. Rev. Lett. 90, 092501 (2003). https://doi.org/10. 1103/PhysRevLett.90.092501 H. Koura, M. Yamada, Nucl. Phys. A 671, 96–118 (2000). https://doi.org/10.1016/S03759474(99)00428-5 T. Kunihiro, Phys. Lett. B 219, 363–368 (1989) [erratum: Phys. Lett. B 245, 687 (1990)]. https:// doi.org/10.1016/0370-2693(89)90405-X N. Matsuoka, T. Noro, K. Tamura, M. Yoshimura, M. Yosoi, A. Okihana, T. Yoshimura, Phys. Lett. B 359, 39–42 (1995). https://doi.org/10.1016/0370-2693(95)01059-Y O. Morimatsu, K. Yazaki, Nucl. Phys. A 435, 727–737 (1985). https://doi.org/10.1016/03759474(85)90185-X O. Morimatsu, K. Yazaki, Nucl. Phys. A 483, 493–513 (1988). https://doi.org/10.1016/03759474(88)90081-4 H. Nagahiro, S. Hirenzaki, Phys. Rev. Lett. 94, 232503 (2005). https://doi.org/10.1103/ PhysRevLett.94.232503 M. Nanova et al. [CBELSA/TAPS], Phys. Rev. C 94, 025205 (2016). https://doi.org/10.1103/ PhysRevC.94.025205 J. Nieves, E. Oset, Nucl. Phys. A 518, 617–638 (1990). https://doi.org/10.1016/03759474(90)90182-L J. Nieves, E. Oset, Phys. Lett. B 282, 24–30 (1992). https://doi.org/10.1016/0370-2693(92)90474I J. Nieves, E. Oset, C. Garcia-Recio, Nucl. Phys. A 554, 509–553 (1993). https://doi.org/10.1016/ 0375-9474(93)90245-S T. Nishi et al. [piAF], Phys. Rev. Lett. 120, 152505 (2018). https://doi.org/10.1103/PhysRevLett. 120.152505 T. Nishi et al. [piAF], arXiv:2204.05568 [nucl-ex] (2022) N. Nose-Togawa, H. Nagahiro, S. Hirenzaki, K. Kume, Phys. Rev. C 71, 061601 (2005). https:// doi.org/10.1103/PhysRevC.71.061601 E. Oset, Mesonic degrees of freedom in nuclei, in Proceedings of the 5th Topical School on Nuclear Physics (World Scientific, 1982), pp. 1–60 E. Oset, A. Ramos, Nucl. Phys. A 635, 99–120 (1998). https://doi.org/10.1016/S03759474(98)00170-5 E. Oset, H. Toki, W. Weise, Phys. Rep. 83, 281–380 (1982). https://doi.org/10.1016/03701573(82)90123-5 W.C. Parkinson, D.L. Hendrie, H.H. Duhm, J. Mahoney, J. Saundinos, G.R. Satchler, Phys. Rev. 178, 1976–1996 (1969). https://doi.org/10.1103/PhysRev.178.1976 R. D. Pisarski, M. Tytgat, Phys. Rev. D 54, R2989–R2993 (1996). https://doi.org/10.1103/ PhysRevD.54.R2989 R. D. Pisarski, F. Wilczek, Phys. Rev. D 29, 338–341 (1984). https://doi.org/10.1103/PhysRevD. 29.338 S. Sakai, D. Jido, Phys. Rev. C 88, 064906 (2013). https://doi.org/10.1103/PhysRevC.88.064906 E.J. Schneid, A. Prakash, B.L. Cohen, Phys. Rev. 156, 1316–1331 (1967). https://doi.org/10.1103/ PhysRev.156.1316

72 Theoretical Study of Deeply Bound Pionic Atoms with an. . .

2655

R. Seki, K. Masutani, Phys. Rev. C 27, 2799–2816 (1983). https://doi.org/10.1103/PhysRevC.27. 2799 M. Skurzok, P. Moskal, N.G. Kelkar, S. Hirenzaki, H. Nagahiro, N. Ikeno, Phys. Lett. B 782, 6–12 (2018). https://doi.org/10.1016/j.physletb.2018.04.047 M. Skurzok, S. Hirenzaki, S. Kinutani, H. Konishi, P. Moskal, H. Nagahiro, O. Rundel, Nucl. Phys. A 993, 121647 (2020). https://doi.org/10.1016/j.nuclphysa.2019.121647 K. Suzuki, M. Fujita, H. Geissel, H. Gilg, A. Gillitzer, R.S. Hayano, S. Hirenzaki, K. Itahashi, M. Iwasaki, P. Kienle et al., Phys. Rev. Lett. 92, 072302 (2004). https://doi.org/10.1103/ PhysRevLett.92.072302 Y.K. Tanaka et al. [n-PRiME/Super-FRS], Phys. Rev. Lett. 117, 202501 (2016). https://doi.org/10. 1103/PhysRevLett.117.202501 Y.K. Tanaka et al. [η-PRiME/Super-FRS], Phys. Rev. C 97, 015202 (2018). https://doi.org/10. 1103/PhysRevC.97.015202 S. Terashima, H. Sakaguchi, H. Takeda, T. Ishikawa, M. Itoh, T. Kawabata, T. Murakami, M. Uchida, Y. Yasuda, M. Yosoi et al., Phys. Rev. C 77, 024317 (2008). https://doi.org/10. 1103/PhysRevC.77.024317 V. Thorsson, A. Wirzba, Nucl. Phys. A 589, 633–648 (1995). https://doi.org/10.1016/03759474(95)00166-X H. Toki, T. Yamazaki, Phys. Lett. B 213, 129–133 (1988). https://doi.org/10.1016/03702693(88)91012-X H. Toki, S. Hirenzaki, T. Yamazaki, R.S. Hayano, Nucl. Phys. A 501, 653–671 (1989). https://doi. org/10.1016/0375-9474(89)90154-1 H. Toki, S. Hirenzaki, T. Yamazaki, Nucl. Phys. A 530, 679–697 (1991). https://doi.org/10.1016/ 0375-9474(91)90774-Z Y. Tomozawa, Nuovo Cim. A 46, 707–717 (1966). https://doi.org/10.1007/BF02857517 A. Trudel et al., TRIUMF E628 and TRIUMF progress report (1991) H. Tsunoda, Prog. Theor. Phys. 93, 393–402 (1995). https://doi.org/10.1143/PTP.93.393 Y. Umemoto, S. Hirenzaki, K. Kume, H. Toki, Phys. Rev. C 62, 024606 (2000). https://doi.org/10. 1103/PhysRevC.62.024606 U. Vogl, W. Weise, Prog. Part. Nucl. Phys. 27, 195–272 (1991). https://doi.org/10.1016/01466410(91)90005-9 S. Weinberg, Phys. Rev. Lett. 17, 616–621 (1966). https://doi.org/10.1103/PhysRevLett.17.616 W. Weise, Acta Phys. Polon. B 31, 2715–2726 (2000) W. Weise, Nucl. Phys. A 690, 98–109 (2001). https://doi.org/10.1016/S0375-9474(01)00934-4 K. Yagi, T. Hatsuda, Y. Miake, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 23, 1–446 (2005). T. Yamaga et al. [J-PARC E15], Phys. Rev. C 102(4), 044002 (2020). https://doi.org/10.1103/ PhysRevC.102.044002 J. Yamagata, H. Nagahiro, Y. Okumura, S. Hirenzaki, Prog. Theor. Phys. 114, 301–316 (2005) [erratum: Prog. Theor. Phys. 114, 905–905 (2005)]. https://doi.org/10.1143/PTP.114.301 J. Yamagata, H. Nagahiro, S. Hirenzaki, Phys. Rev. C 74, 014604 (2006). https://doi.org/10.1103/ PhysRevC.74.014604 N. Yamanaka et al. [JLQCD], Phys. Rev. D 98, 054516 (2018). https://doi.org/10.1103/PhysRevD. 98.054516 T. Yamazaki, S. Hirenzaki, Phys. Lett. B 557, 20–25 (2003). https://doi.org/10.1016/S03702693(03)00182-5 T. Yamazaki, S. Hirenzaki, R.S. Hayano, H. Toki, Phys. Rep. 514, 1–87 (2012). https://doi.org/10. 1016/j.physrep.2012.01.003

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stopped Pion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectroscopy by Sodium Iodide and Germanium Detectors . . . . . . . . . . . . . . . . . . . . . . . Transition-Edge Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffraction Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laser Spectroscopy of Pionic Helium Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reaction Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of Deeply Bound Pionic Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Reaction Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formation Cross Section and Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discovery of Deeply Bound Pionic Atoms in (d,3 He) Reactions . . . . . . . . . . . . . . . . . . . . . Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pionic Pb Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Achievements in Spectroscopy of Pionic Pb Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pionic Atoms and Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pionic Sn Isotopes and Isovector b1 Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reaction Spectroscopy with Improved Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deduction of Chiral Condensate in Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Perspectives in Meson-in-Nucleus Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Discussed in this chapter is the spectroscopy of pionic atoms to analyze their properties quantum mechanically. Pions are the lightest hadrons, and their properties have information on the low-energy quantum chromodynamics known for the non-perturbative nature. Starting with the X-ray spectroscopy, we overview the laser spectroscopy, the reaction spectroscopy, and the resultant deduction of the spontaneous breakdown of the chiral symmetry of the vacuum.

Introduction Although the masses are closer, pions and muons have different characters. Unlike muons, a pion is a hadron and has the strong interaction with the nucleus (Ericson and Weise 1988). One of the most unique features of the pionic atoms, a bound system of a negatively charged pion π − and a nucleus, is the co-existence of the Coulomb interaction and the strong interaction in the system. Attractive Coulomb interaction plays a major role in binding the two particles and in transitions between the orbitals. The bound π − is finally absorbed by a pair of nucleons, np or pp, in the nucleus with the strong interaction emitting the nucleons (Ozaki et al. 1960). The quantum levels of the pionic atoms reflect the strengths of the strong interaction as the level shifts from the pure Coulomb levels and the widths. For many of the 2p, 3d, and 4f levels, the signs of the level shifts are positive, which means the strong interaction is attractive. For the 1s levels, the signs are negative, which shows the repulsive strong interaction. The experimental data of the level shifts indicate the existence of two components in the strong interaction. The strong interaction is attractive in the momentum-dependent p-wave part and repulsive in the local s-wave part (Ericson et al. 1966). In the nucleus, the pion-nucleus s-wave interaction is modified due to the wave function renormalization of the medium effect (Kolomeitsev et al. 2003; Chanfray et al. 2003; Jido et al. 2008). This modification has been elaborately studied in terms of the partial restoration of the chiral symmetry of the vacuum based on the spectroscopic data (Hayano and Hatsuda 2010; Friedman and Gal 2019). Indeed, the spectroscopy of pionic atoms contributes to the understanding of the symmetry and the structure of the vacuum. For pionic atoms with relatively heavy nuclei, the π − wave function largely overlaps with the nucleus, and the pion works as a probe to investigate quantum-mechanically the structure of the vacuum (Biddle et al. 2020) and its modification in the nuclear matter regarded as a low-temperature and ultimately high-density medium (Friedman and Gal 2019; Suzuki et al. 2004; Ikeno et al. 2011a). Investigation of the material property of the vacuum is one of the central subjects of modern physics (Nambu and Jona-Lasinio 1961). During the decrease of the temperature of the universe, the chiral symmetry was broken, and the vacuum has a non-trivial structure of the chiral condensate (Hatsuda and Kunihiro 1994; Brown and Rho 1996). Efforts have been made to study the temperature and the

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density dependence of the vacuum and to draw the phase diagram (Kaiser et al. 2008; Goda and Jido 2013; Lacour et al. 2010; Hübsch and Jido 2021). Highenergy colliders have been bringing stimulating data about the quark-gluon-plasma phase of the vacuum (Aidala et al. 2019), and investigations have been performed to explore high-density baryonic matter (Friman et al. 2011). Theories have been developed rapidly for the high-temperature region of the diagram as the growth of the computational power in the lattice QCD (Fu et al. 2020). Despite the sign problem that prevents the lattice calculations from approaching the high-density regions (Fukushima and Hatsuda 2011), the first-principle calculations have been yielding exciting results. Spectroscopy of pionic atoms facilitates research based on the unambiguous data in the low-temperature and high-density region, which is a domain of the chiral symmetry. The spontaneous breakdown of the chiral symmetry is due to the non-perturbative nature of the low-energy QCD, which involves various phenomena such as the generation of the Nambu-Goldstone bosons (Pagels 1975), the confinement of the quarks (Wilson 1974), and the dynamical generation of the hadron masses (Higashijima 1984). The quantum states of the π − in the nuclei are described by the Klein-Gordon equation with the strong interaction potential (Ericson and Weise 1988). The phenomenological optical potential of Ericson-Ericson type formulation (Ericson et al. 1966) is constructed based on microscopic scattering of the pion and the nuclei and is known to describe many of the pionic atoms fairly well. The optical potential Uopt (r) consists of two parts, the s-wave part Us (r) and the p-wave part Up (r), and is formulated as 2μUopt (r) = Us (r) + Up (r)

(1)

Us (r) = −4π [1 {b0 ρ(r) + b1 ρ(r)} + 2 B0 ρ 2 (r)] Up (r) = 4π ∇[c(r) + 2−1 {C0 ρ 2 (r) + C1 ρ(r)ρ(r)}]L(r)∇ with b(r) = 1 {b0 ρ(r) + b1 ρ(r)}, c(r) = 1−1 {c0 ρ(r) + c1 ρ(r)},   −1 4 , L(r) = 1 + π ξ c(r) + 2−1 {C0 ρ 2 (r) + C1 ρ(r)ρ(r)} 3 ρ(r) = ρn (r) + ρp (r) ρ(r) = ρn (r) − ρp (r) where 1 = 1 + μ/M = 1.147, 2 = 1 + μ/2M = 1.073. μ is the pionnucleus reduced mass and M is the nucleon mass. b0 , b1 , and B0 are the s-wave parameters, and c0 , c1 , C0 , and C1 are the p-wave parameters. ξ denotes the

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Lorentz-Lorenz parameter of the short-range nucleon correlation. There are several sets of potential parameters determined by fitting the measured shifts and widths of pionic atoms. Table 1 tabulates representative parameter sets. The Ericson-Ericson parameterization works fine and reproduces most of the experimental data well although the b0 and ReB0 parameters are known to have a strong correlation (Seki and Masutani 1983). This chapter discusses the pionic atoms from the experimental side and consists of eight subsections. The pionic atom X-ray spectroscopy is discussed in section “X-ray Spectroscopy”, the recently developed method of laser spectroscopy in section “Laser Spectroscopy of Pionic Helium Atoms”, the novel method developed for direct excitation of the pionic atoms in nuclear reactions in section “Reaction Spectroscopy”, the discoveries in section “Discovery of Deeply Bound Pionic Atoms in (d,3 He) Reactions”, the study of the chiral symmetry in section “Pionic Atoms and Chiral Symmetry”, and future perspectives in section “Future Perspectives in Meson-in-Nucleus Experiments”.

X-ray Spectroscopy Stopped Pion Method The pion beams became available when the development of the accelerators made the primary proton beam energy exceed the pion production threshold. High-energy proton beams bombarded a production target to produce pions as the secondary beams. Figure 1 shows a schematic drawing of the pion beam facility at the Paul Scherrer Institute (PSI) (Grillenberger et al. 2021). The 590 MeV proton beam impinges on the graphite target and produces the secondary particles including the pions. The pion production cross sections at this energy are known well (Crawford et al. 1980). Pion beams with an intensity of >109 /s are available depending on the target thickness and the primary beam current. The momentum of pions ranges between 10 and 450 MeV/c. The produced charged pions are transferred through the secondary beamlines to reach the experimental areas and used in many experiments including spectroscopy of pionic atoms. For the spectroscopy of pionic atoms, the π − beam is injected into the target materials. The injected and stopped π − is captured by the nuclei emitting an Auger electron and trapped in a quantum orbital. The Auger transition rate becomes highest with the largest overlap between electron and π − wave functions (Backenstoss 1970). Since the pion masses are much larger than the electron mass, the radii of the pionic atoms are ∼250 times smaller. The Bohr energy levels and the radii scale as En ∝ −μZ 2 /n2 and rn ∝ n2 /(μZ), respectively, where n is the principal quantum number and Z is the charge of the nucleus (Gotta 2004). After initial capture processes, subsequent de-excitation processes involve several physical processes. The major two mechanisms are the Auger emission of electrons and radiative de-excitation. For lighter nuclei such as hydrogen or helium, other processes, i.e., Stark mixing and Coulomb de-excitation, also play important

SM-1 SM-2 Batty-1 Batty-2 Batty-3 Konijn-1 Konijn-2 Ericson-1 Ericson-2 Ericson-3

b0 (m−1 π ) −0.0283 0.03 −0.017 −0.023 0 0.024 0.025 −0.0192 −0.0192 −0.0178

b1 (m−1 π ) −0.12 −0.143 −0.13 −0.085 −0.125 −0.090 −0.094 −0.0873 −0.0873 −0.0873

c0 (m−3 π ) 0.223 0.21 0.255 0.21 0.261 0.272 0.273 0.2087 0.2087 0.2087

c1 (m−3 π ) 0.25 0.18 0.17 0.089 0.104 0.107 0.184 0.1779 0.1779 0.1779

ImB0 (m−4 π ) 0 0.042 −0.15 0.046 −0.0475 0.0475 −0.021 0.049 −0.14 0.055 −0.261 0.0552 −0.265 0.0546 −0.0489 0.0489 −0.0489 0.0489 −0.0489 0.0489

ReB0 0 0.11 0 0.118 −0.25 −0.26 −0.14 0.1988 0.1988 0.1988

ReC0

ImC0 (m−6 π ) 0.1 0.09 0.09 0.058 0.059 0.0640 0.105 0.0879 0.0879 0.0879

ImC1 (m−6 π ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.737 0 −0.737 0 −0.737 0

ReC1

1 1 1 1 0 0 1 1 1 1

ξ

Table 1 Representative parameter sets of Ericson-Ericson optical potential. For details see Itahashi et al. (2000). b0 parameter is nearly zero, and the dominant term in the s-wave potential is parameterized by the b1 parameter

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Fig. 1 Schematic layout of the accelerator and the pion beamlines of the High Intensity Proton Accelerator facility at PSI taken from Grillenberger et al. (2021) and slightly modified

roles (Gotta 2004). Initially, Auger emission is dominant, and subsequently, radiative de-excitation starts. The probability of the radiative de-excitation by electric dipole transition is described (Backenstoss 1970) as WX =

4e2 E 3 3h¯ 4 c3

| < ψf |r|ψi > |2 ,

(2)

where ψi and ψf are meson wave functions in the initial and final states, E is the transition energy, and r is the radius from the center of the nucleus. The difference of the orbital quantum number |l| is 1 for the selection rule. The factor E 3 enhances transitions with larger differences in the principal quantum numbers. Thus the radiative de-excitations enhance the population of the circular orbitals, which have the maximum orbital quantum number (l = n−1). One can measure the energy difference between two levels by detecting the X-rays. The radiative transition stops when the probability of the π − absorption by the nucleus exceeds that of the radiative transition. Absorption by a single nucleon is strongly suppressed, and the π − is absorbed by a pair of nucleons, mainly by np pairs. The lifetime of the pionic atoms is predominantly determined by the absorptive interaction strength and the overlap of the π − orbital and the nuclear density.

Spectroscopy by Sodium Iodide and Germanium Detectors The first conclusive observation of X-rays from the pionic atoms was reported for carbon, oxygen, and beryllium at Rochester (Camac et al. 1952). The level shifts were observed experimentally (DeBenedetti 1956). Figure 2 shows a typical

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Fig. 2 Typical experimental configuration of pionic atom X-ray spectroscopy at TRIUMF M13 channel taken from Olin et al. (1985). The incident beam of 90 MeV/c π − is incoming from the right side and stopped in the Pb target. The emitted X-rays from the pionic Pb atoms are measured by the Ge counter surrounded by Compton suppressor BGO crystals. (Reprinted from Olin et al. (1985), © (2022), with permission from Elsevier)

experimental setup for pionic atom X-ray measurement (Olin et al. 1985). The incident 90 MeV/c π − beam is identified by the beam defining counters S1 and S2 located on the beamline. A 1-cm-thick beryllium degrader is placed to decelerate the π − beam and trap the π − in the target material of 1 g/cm2 thick 208 Pb. Another counter S4 was located behind the target to identify the stopped π − events. The emitted X-rays are measured by the Ge detector located at the side of the beamline. Compton scattering X-rays are identified and suppressed by the BGO counter surrounding the Ge detector. Figure 3 depicts a typical spectrum (Olin et al. 1985). 5-4 transition X-ray of a pionic lead atom is observed as the peak structure. The measured spectrum was fitted by contributions from the formation cross sections of the pionic atoms and the background. Formation cross sections of the states are Lorentzian convoluted by the Gaussian of the spectral response function to reflect the natural width and the experimental resolution, respectively. Figure 4 summarizes many of the measured levels in X-ray spectroscopy (Batty et al. 1997; Friedman and Gal 2007). Sixty-one pionic states were compiled. The plot shows the magnitude of the level shifts from the pure Coulomb states. Here the pure Coulomb energy levels are calculated with the nuclear finite-size effects and other corrections such as vacuum polarization terms (Fullerton and Rinker 1976). Note that there is a different definition of the level shift where the Coulomb level energy is calculated with point charge distribution. The negative shifts indicate repulsive strong interaction and the positive shifts attractive interaction. Orbitals closer to the nucleus have relatively large shifts. Repulsive interaction is dominant for the 1s states, which demonstrates the s-wave interaction is the major part. In contrast, the attractive p-wave interaction is dominant for the higher orbitals. The experimental data were fitted by theoretical values calculated with the optical potential in Eq. 1. There are several parameter sets, which fairly well reproduce the

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Fig. 3 Typical pionic atom X-ray spectrum. Pionic 208 Pb 5-4 transition X-ray is measured and fitted by a Gaussian folded Lorentzian and background. (Taken from Olin et al. 1985). (Reprinted from Olin et al. (1985), © (2022), with permission from Elsevier) 100

Attractive

ΔE(4f)×4 ΔE [keV]

0 100

ΔE(3d)×3

Repusive 0 0 ΔE(2p)×2 -100 0 ΔE(1s) -100 0

20

60

40

80

Z

Fig. 4 Compilation of measured pionic atoms of nuclei with Z ≥ 8 in the X-ray spectroscopy. The eye-guide lines are also shown. The abscissa is the charge Z of the nucleus and the ordinate is measured energy shifts in keV unit. The shifts of 2p, 3d, and 4f states are multiplied by the factors indicated

experimental data. Note that optical potential in Eq. 1 may not apply to light nuclei Z ≤ 6. As depicted in Fig. 4, circular orbits were spectroscopically observed. There are so-called last-orbits in the transitions, where the nuclear absorption widths become

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large, and electromagnetic transitions are suppressed beyond the states. The 1s states were observed only for the lighter nuclei with Z < 20.

Transition-Edge Sensor To achieve higher spectral resolution, a novel detector was adopted for the measurement of the X-rays (Okada et al. 2016; Tatsuno et al. 2016). In 2014 pionic 12 C 4f -3d transition and the parallel transition 4d-3p X-rays were measured by a microcalorimeter array of superconducting transition-edge sensors (TES), which have significantly better resolution than existing scintillation or semiconductor X-ray detectors, such as HPGe detectors or silicon drift detectors (Ullom and Bennett 2015). The adopted TES has 240 pixels of microcalorimetric X-ray detectors and was for the first time applied for measurement of the X-rays in hadronic atom transitions. The resolution was evaluated to be 6.8 eV (FWHM) at 6.4 keV under the π − beam rate of 1.45 MHz, which is much better than the energy resolution of 165 eV (FWHM) with a silicon drift detector (Okada et al. 2016). Figure 5 shows the experimental setup. The experiment was performed at the π M1 beamline of PSI. Incident π − beam with the energy of 173 MeV/c was identified and stopped in the carbon target by the scintillation counters BC1-3, the carbon degrader, and a veto counter BC4. At the side of the target, a 240-pixel TES array was installed to measure the emitted X-rays. A 4 μm bismuth absorber (80% efficiency for 6.4 keV X-rays) was attached to each pixel with an effective area of 320 × 305 μm. The X-ray energy was converted to heat measured by a thermal sensor of TES (Okada et al. 2016). TES pixels were controlled to keep the critical temperature of 107 mK and 20–30% of their normal resistance of ∼10 m. In order to make independent energy calibration for the 240 pixels, the X-ray tube was used for the in-situ energy calibration by producing intense Cr and Co calibration X-rays constantly.

Fig. 5 Experimental configuration of pionic carbon X-ray spectroscopy with the TES microcalorimeter taken from Okada et al. (2016)

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Fig. 6 The measured spectrum of pionic 12 C by the TES microcalorimeter. (Taken from Okada et al. (2016) and slightly modified without losing accuracy). Parallel transitions of 4f − 3d and 4d − 3p are observed as a peak and a bump, respectively. Fe Kα lines are also observed. The fitting curve and its decompositions are also shown

Figure 6 depicts the measured X-ray spectrum (Okada et al. 2016). The parallel transitions of pionic carbon 4f − 3d and 4d − 3p are clearly observed as a peak structure and a bump aside, respectively. Together with the signal pionic carbon transition X-rays, characteristic X-rays of Fe are observed without synchronization with the incident π − beam. The Fe Kα1 (6.404 keV) and Fe Kα2 (6.391 keV) were used to enhance the calibration of the pionic 12 C X-ray energy. The absolute energy had an uncertainty of 0.1 eV level. After spectral decomposition by a fit, the transition energies were determined to be E(4f − 3d) = 6428.39 ± 0.13(stat.) ± 0.09(syst.) eV and E(4d − 3p) = 6435.76 ± 0.30(stat.) +0.11 −0.07 (syst.) eV. The ratio of their yields is I (4d − 3p)/I (4f − 3d) = 0.30 ± 0.03 (stat) ±0.02 (syst.). The measured 4d − 3p transition energy is consistent with including the strong interaction effect assessed by the theoretical estimation (Seki and Masutani 1983). The magnitude of the strong interaction effects on the energy level was estimated to be as large as 0.78 eV for the 3p level and 90%) were used. As a result, data quality, acceptance, resolution, and statistics were substantially improved (Vázquez et al. 2016; Del Grande et al. 2019, 2020). To compare with the FINUDA data, a Λp invariant mass by the AMADEUS group is shown in Fig. 5. In contrast to the FINUDA interpretation, the Λp invariant mass spectrum is quite smooth in a broader range where FINUDA is insensitive, as shown in the figure. This improved study clarified that the back-to-back correlation is relatively weak, indicating an insufficient selectivity to isolate signal from multinucleon absorption by the back-to-back event geometry. Before the nuclear reaction, K − s are captured in atomic orbits. The atomic orbit kaon wave function overlaps with entire nucleus, so the multi-nucleon absorption reaction naturally occurs. In fact, the fit result of the AMADEUS data based upon

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Fig. 4 The Λp invariant mass for Λp in back-to-back geometry (cos θΛp < −0.8) of the FINUDA experiment. (left) Data in count base. (right) Relative acceptance corrected spectrum. The fitting function is given by the red curve. (The figure is redrawn from the spectra given in Agnello et al. 2005)

Fig. 5 One of the AMADEUS results on (left) Λp invariant mass spectra and (right) opening angle between Λ and p. (Figure is taken from Del Grande et al. 2019)

a multi-nucleon absorption process gives good agreement with the data without assuming any K¯ bound state formation. As a result, the entire spectrum is well reproduced by the fitting. When the energy of the kaon mass is shared by more than two nucleons, the resulting Λp invariant mass to be observed becomes smaller. In the decay, if the observed Λ originates from a Σ 0 (Σ 0 → γ Λ 100%) or Σ–Λ conversion in nuclei, the Λp invariant mass is smeared even more, as shown in the decomposition. Therefore, the AMADEUS group concluded that the dominant background process in the kaon-at-rest experiment on medium-heavy nuclei is kaon multinucleon absorption that smoothly continues to the Λp formation threshold of 2.05 GeV/c2 . It is quite difficult to discriminate possible signal out from the background in the stopped kaon method. The back-to-back Λp-pair selection is not sufficiently selective to suppress the background processes.

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s¯ s-Pair Creation via p¯ Annihilation It would be more efficient to search for a kaonic nuclear bound state, if one could ¯ utilize s¯ s-pair creation channels (or KK) for the search. In that case, one could utilize widely available stable particle beams for the search experiment, without being limited to the kaon as an incident particle. However, the cross section would be much smaller compared to that in the kaon-induced reaction channel, due to the OZI-rule (Okubo 1963), which could be compensated by the stronger beam intensity. One example of such an experiment is OBELIX, which utilized the p-at-rest ¯ annihilation reaction (Bendiscioli et al. 2007, 2009). The OBELIX group searched for the “K − pp” state, via the p¯ + 4 He → “K − pp” +Ks0 n X channel, where X consists of meson(s) and/or γ (s). From that event set, they studied the Λp invariant mass spectrum, by assuming the “K − pp” decay to a Λp-pair. In this reaction, s¯ s quark-pair creation is denoted by the presence of a Ks0 (¯s d-meson). In the Λp invariant mass spectrum, the OBELIX group observed a narrow (≈0.03 GeV/c2 ) peak formation at the mass value ≈2.22 GeV/c2 . However, two problems plague this result. One is the ambiguity regarding the event selection. The OBELIX group cannot obtain a clear Λ selection in the Λ → π − p invariant mass, due to the limited resolution, which gives a rather serious ambiguity about the reaction dynamics. The other is the peak position. It is located at the Λpπ formation threshold, so that it could be a Λpπ cusp effect (Piano et al. 2010). Thus, a more detailed study is needed to remove those problems in order to reach a clear conclusion. Note that the in-flight p¯ annihilation channel could be a very promising reaction to access multi-kaon bound states, because a low-energy p¯ may produce two low ¯ pairs at once (Sakuma et al. 2009). momentum KK

s¯ s-Pair Creation via pp Collision As another attempt to utilize s¯ s-pair creation channels, the DISTO group analyzed pp collision data at Tp = 2.85 GeV to search for a “K − pp” signal in the ΛpK + final state (Yamazaki et al. 2010). The reaction can be written as pp → (ppK − K + ) → “K − pp + K + ,

(1)

in which “K − pp” is formed by emitting K + and decays to “K − pp” → Λp in the final state. Figure 6 (top-left) shows their first result at Tp = 2.85 GeV. As shown in the figure, the obtained Λp invariant mass has a huge peak at M(Λp) ≈ 2.265 GeV, below the mass threshold of a kaon and two protons to be bound, M(K − pp) (= mK + 2mp ∼ 2.37 GeV/c2 ). To be sensitive to the “K − pp” formation signal, they selected the scattered proton to have large angle so as to observe a pp head-on

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Fig. 6 The pp collision data at Tp = 2.85 GeV (left) and 2.5 GeV (right) by DISTO, with largeangle proton selection (top) and small-angle proton selection (bottom) for the pp → ΛpK + final state. The red hatched region is the reported “K − pp” candidate named as X(2265). Figure is redrawn to make vertical-axis scale to be common for top and bottom of each column, for easy comparison of the two. The data given in Kienle et al. (2012) is used for the redraw. Top left is essentially the same as high-PT (p) selected data published in Yamazaki et al. (2010). Red hatched region is the applied Gaussian fitting function for X(2265) with linear background shown in blue (see text, for more detail)

collision. The rejected spectrum is shown in the bottom left of the figure. By this selection, they sought to be sensitive to events in which the produced K¯ captures two nucleons in the reaction. Then the large-angle spectrum was fitted by assuming that the spectrum could be fitted with a Gaussian over the very simple linear background as shown in the figure. The DISTO group concluded that they found a distinct peak at the 26σ confidence level and named this resonance signal the X(2265). If this is a signal of the simplest kaonic nuclear state “K − pp,” then the binding energy is ≈100 MeV, and the natural decay width is ≈110 MeV. The formation cross section is indicated to be as large as ∼ mb or more, in spite of the fact that the s¯ s-creation channel should be strongly suppressed by the OZI-rule. However, the fit results are not easily understood. The background seems to be too steep for the rejected event to simply justify the linear background assumption commonly applied for both selected and rejected spectra. The peak position of 2.265 GeV is almost the same as the π ΣN threshold, but X(2265) appears to be completely unaffected, even though Λ(1405) decay to 100% π Σ.

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To confirm the resonance, the DISTO group conducted a confirmation experiment at Tp = 2.5 GeV. In this energy, (ppK − K + ) is generated ∼ at-rest in the CM frame. Thus, the objective is clear that an even clearer X(2265) signal should be found compared to the initial DISTO run, because the K¯ can capture pp more efficiently by kicking out the K + at TK + ≈ 80 MeV (corresponding to the energy mismatch due to BK¯ ) in the CM. The result is shown in the right panel of Fig. 6. In Kienle et al. (2012), a scenario was introduced to explain the total loss of the X(2265) signal at Tp = 2.5 GeV. The ¯ scenario is: The Λ(1405) ≡ “KN” should be formed in the reaction channel as a doorway. However, at Tp = 2.5 GeV, the Λ(1405) cannot be formed, because of the limited phase space for the Λ(1405) formation. Thus, the X(2265) cannot be created at Tp = 2.5 GeV, but which does not exclude the existence of the X(2265). This is difficult to understand. Within the strong interaction time frame, it is more natural to produce the final state as a quantum jump rather than a gradual formation of the system making a Λ(1405) the core of the formation. Thus, it seems ¯ unreasonable to ask that the Λ(1405) ≡ “KN”, which is essentially a member of the kaonic nucleus family, to act as a gatekeeper for access to other families. If the pp collision has sufficient energy to produce a “K − pp,” it should form spontaneously without requiring the intermediate existence of the Λ(1405). It is also true that, if the pp collision forms a K¯ so strongly (producing a K¯ in the nucleus) at Tp = 2.85 GeV, then part of the K¯ must be produced as a quasi-free particle above the binding threshold. This quasi-free K¯ can be absorbed by nearby two nucleons and should appear as an event concentration above M(K − pp), similar to the signal. As shown in the figure, the spectrum monotonically decreases its yield even above M(K − pp), giving no hint of such a quasi-free process. To check this result, a verification study was conducted, led by L. Fabbietti (Fabbietti et al. 2013; Agakishiev et al. 2015). The HADES group performed the follow-up experiment on pp collisions, focusing on the ΛpK + final state at the √ energy of Tp = 3.55 GeV ( sNN ∼ 3.20 GeV). The internal kinetic energy of (ppK − K + ) is ≈315 MeV, which is sufficiently above the threshold to produce the Λ(1405) in the reaction. The result is shown in Fig. 7. Their Λp invariant mass [denoted as IM(Λp)] does not have any anomaly where DISTO reported the X(2265) signal. They performed a detailed partial wave analysis on the data and obtained a very tight upper limit on the cross section of below 4.2 μb in the mass region of interest. Thus, the Λ(1405)-doorway scenario introduced by the DISTO group appears ruled out. Instead, they observed a similar event concentration in the Λp invariant mass spectrum at around 2.505 GeV/c2 . The major contribution of this mass-shifted event concentration was located along the ΛK + invariant mass axis at IM(ΛK + ) ≈ 1.71 GeV/c2 (Fabbietti et al. 2013) in the two-dimensional plot between IM(Λp) and IM(ΛK + ). Judged from DISTO’s original paper (Yamazaki et al. 2010), it is clear that the X(2265) forms a IM(ΛK + ) peak at the same mass (≈1.71 GeV/c2 ) identified by HADES. Thus, the HADES group concluded that the X(2265) would be simply formed by the p + p → p + N + (1710) → p + ΛK + process, having a Λp invariant mass cutoff due to the kinematic boundary both on the lower and upper sides of the X(2265).

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Fig. 7 The pp collision data at Tp = 3.55 GeV by HADES. Two-particle mass distributions for the ΛpK + final state are shown. (left) Two-dimensional plot between Λp and ΛK + invariant mass. (middle) Projection to I MΛp . (right) Projection to I MΛK + . The applied Bonn-Gatchina partial-wave analysis gives a very tight upper limit below 4.2 μ b in the DISTO region of interest [X(2265)] via the pp collision at Tp = 3.55 GeV. Figure is redrawn from data given in Fabbietti et al. (2013) for the two-dimensional plot and Agakishiev et al. (2015) for the projections to I MΛp and I MΛK + with the fit result (red hatched)

They also extended their study by compiling existing data on the K + formation yield as a function of the K + missing mass (corresponding to the “K − pp” mass). If the X(2265) should really exist, the K + yield should become large at around 2.21 GeV assisted by the “K − pp” formation channel, at around the region where the X(2265) signal starts (including a width of ΓX(2265) ∼ 110 MeV). As shown in Fig. 8 (Epple and Fabbietti 2015), there exists a subthreshold K + yield increase of about 30%, but the increases are observed only from 2.3 GeV [≈70 MeV below the kaon bound threshold M(K − pp)], which is inconsistent with the mass value of X(2265).

s¯ s-Pair Creation via the (π + , K+ ) Reaction In the J-PARC E27 experiment, the “K − pp” formation signal was reported through the s¯ s-pair creation channel using the π + beam on a deuteron target at pπ + = 1.69 GeV /c provided by the J-PARC K1.8 beam line. They conducted a d(π + , K + )X missing mass (denoted as MMd ) spectroscopic search by measuring the forward K + momentum using a large acceptance spectrometer, the SKS, which is commonly utilized for various hypernuclear studies. The missing mass MMd is the total energy of X in the center-of-mass (CM) frame of X in the d(π + , K + )X reaction. From the conservation law, X should contain two baryons and an s-quark. Thus, if the “K − pp” bound state formed in the reaction as X, the MMd spectrum should have a peak at the “K − pp” mass, if the formation yield is strong enough. As a result of the experiment, the E27 group reported a candidate observation in the MMd spectrum of the Σ 0 pK + final state at BK¯ ≈ 95 MeV with a width of ΓK¯ ≈ 160 MeV in the cross section of ≈3 μb/sr (Ichikawa et al. 2015). They found

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Fig. 8 Relative K + yield as a function of K + missing mass around X(2265) region of interest. The pp collision data sets at Tp = 2.54, 2.88 and 3.03 GeV are compiled. Subthreshold K + yield increase observed only around ∼70 MeV below the M(K − pp) threshold, but not around the X(2265) region reported in Yamazaki et al. (2010). (Figure is redrawn from the data given in Epple and Fabbietti 2015)

Fig. 9 (left) The E27 d(π + , K + )X missing mass (MMd ) spectrum for the Σ 0 pK + final state (equivalent to the Σ 0 p invariant mass. The red line shows the fitted result for the reported “K − pp” candidate. (right) The E27 inclusive d(π + , K + ) missing mass spectrum. Three dominant peaks correspond to the quasi-free production of Λ, Σ, and Y ∗ = Σ(1385)/Λ(1405), i.e., the single nucleon π + N → Y (∗) K + reaction, from left to right. (Figure is redrawn from the data given in Ichikawa et al. 2015)

a candidate, when they required the final state to be Σ 0 pK + , i.e., the observed formation channel is π + + d → “K − pp” +K + → (Σ 0 p) + K + , as shown in Fig. 9 (left). Like the DISTO’s X(2265) candidate, the fitting function applied did not considered the threshold effect of π ΣN , in spite of Λ(1405) → π Σ. Let us consider further this result in order to understand how conclusive it is given the analysis procedure, the fitting function utilized, and the resulting interpretation as the “K − pp” candidate. Unfortunately, the single nucleon π + n → Λ(1405) K +

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reaction do not efficiently contribute to form Kpp. The energy of the incident √ π + and the single nucleon reaction in the CM frame corresponds to sπ N about 2.02 GeV. Thus, the Λ(1405) is kicked in the forward direction at the momentum qΛ(1405) ≈ 0.80 GeV/c [‘p-Λ(1405)’ system is ≈ 100 MeV above the binding threshold], which makes dibaryon formation difficult. The situation is even worse for the Λ(1520) case [qΛ(1520) ≈ 1.27 GeV/c]. A chance to form “K − pp” may come through two nucleon coherent K¯ production, described as ¯ π + + NN → (KNN) + K +,

(2)

¯ in which the momentum transfer to (KNN) is somewhat smaller, qKN ¯ N ≈ 0.71 GeV/c (at BK¯ = 0). At this momentum transfer, it is still hard for the K¯ to stick to the N N pair, because the range of the Fermi-motion is pF ≈ 200 MeV/c (or the nuclear size ∼1 fm, cf., Planck constant). The momentum transfer becomes smaller for larger kaon binding energy. At BK¯ = 100 MeV, the momentum transfer becomes 560 MeV/c, which is still larger than the pF though. Let us compare with the inclusive d(π + , K + ) missing mass spectrum shown in Fig. 9 (right). The inclusive MMd spectrum is composed of three major peak structures. The spectral yield starts increasing from ∼2.05, ∼2.14, and ∼2.3 GeV/c2 from left to right. The threshold values are in good agreement with the hyperon production threshold of Λ, Σ, and Y ∗ = Σ(1385) / Λ(1405), respectively. Thus, they come from quasi-free hyperon production of π + N → Y N , leaving another nucleon as a spectator. By comparing with Σ 0 pK + spectrum given in Fig. 9 (left), a candidate was found in the tail component of the quasi-free Σ production, extended well above the K¯ binding threshold, and furthermore it covers the whole quasi-free Y ∗ formation region. Before considering the d(π + , K + ) missing mass spectrum itself, let us understand the key component of the E27 experimental setup and an experimental basis how to seek the “K − pp” signal events. Figure 10 shows the detector system surrounding the liquid deuteron target. A proton detector system, RCA, is introduced to select the “K − pp” formation signal. As is discussed, the single nucleon d(π + , K + ) reaction is expected to form a severe background, as is shown in Fig. 9 (right). When the “K − pp” state is formed, it can decay to Λp or Σ 0 p, and these two channels produce two energetic protons in the final state, which can trigger the RCA. On the other hand, a single nucleon reaction that originates a background event should be substantially suppressed, because one of the nucleons should be a spectator and have low momentum, so it wouldn’t be detected by the RCA. Thus, the “K − pp” signal could be selected by the two energetic proton tagging with the RCA. This is the basis for the E27 experiment. The RCA covers from 39◦ to 122◦ measured from the beam axis. The protons in this angular range were momentum-analyzed by means of the time-of-flight method, and the particle identification is applied by the two successive layers’ information of the RCA, which defines the proton detection threshold of >250 MeV/c.

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Fig. 10 The E27 experimental setup around the deuteron target system. (Figure is redrawn from Ichikawa et al. 2015)

The final state of the d(π + , K + )X reaction can be specified by utilizing the RCA information. The missing mass of the residuals (except for the two measured protons) can be calculated by subtracting the four momenta of the two protons from that of the remaining system X. In this manner, ΛpK + final states are analyzed by checking whether they are consistent with one pion missing from the Λ decay (Λ → π − p). Similarly, the Σ 0 pK + final state can be specified if the missing particles are one pion plus a gamma from the Σ 0 decay chain Σ 0 → γ Λ → γ π − p. Because of the finite resolution, the overlap of the residual missing mass (M(X − 2p)) between the two possible final states can be relatively large. Thus, to obtain Σ 0 p spectrum [Fig. 9 (left)], the MMd spectrum was weighted by the probability proportional to that estimated for each final state. ¯ The decay processes from (KNN) to Λp or Σ 0 p are almost the same, so it is + important to compare the ΛpK and Σ 0 pK + spectra. In both channels, the missing mass MMd can be calculated from the initial π + and the outgoing K + momenta. Figure 11 (left) shows a pp coincidence spectrum (in count base) and a missing mass spectrum for the ΛpK + final state (right), taken from the PhD thesis of Y. Ichikawa (2015). As shown in the figure, the overall MMd spectrum of the ΛpK + final state (right) is very different from that of the Σ 0 p final state (Fig. 9 (left)). In contrast to this fact, it is very similar to the raw two proton coincidence MMd spectrum in Fig. 11 (left). Moreover, the spectral shape of the two is quite similar to the inclusive one in the MMd region around 2.13 ∼ 2.3 GeV/c2 , where quasi-free Σ 0 would be dominantly generated as shown in Fig. 9 (right). Thus, it is necessary to understand how the ΛpK + final state is formed in the d(π + , K + ) reaction. A reasonable assumption is that the channel to originate such a spectrum is the quasi-free Σ 0 channel, in which the π + n → Σ 0 K + reaction leaves the p as a spectator (where the n and p are constituent nucleons in a deuteron), at least in the primary reaction. The reason why it can be detected as a ΛpK + final state – Σ-Λ conversion – will be discussed later. Let us consider how the RCA is triggered by the d(π + , K + ) reaction and what can be observed as a difference between the two spectra, Σ 0 pK + and ΛpK + . When the K + is measured at ≈ zero degrees, i.e., the K + vector is aligned with

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Fig. 11 (left) The E27 pp coincidence spectrum in count base, including both ΛpK + and Σ 0 pK + final states. Pion contamination evaluated by the E27 group was subtracted. (right) Missing mass spectrum for ΛpK + final state. (Figures are redrawn from the data given in Ichikawa 2015)

the incident π + beam axis, a reaction plane can be defined in which all the vectors (beam axis, Σ 0 , and p vectors) should exist. Thus, a direction orthogonal to the reaction plane (in which elements of all four momenta are zero) can be deduced. By omitting the deduced direction, the conservation rule of four momenta in the μ laboratory frame, i pi = 0 (i = π + , d (initial); Σ 0 , p and K + (final)), can be simplified as ⎞ ⎛ ⎞ ⎞ ⎛ 2 ⎞ ⎛ ⎞ ⎛ 2 md EΣ 0 m2π +pπ2 mp +pF2 m2K +pK ⎟ ⎜ ⎟ ⎝ pπ ⎠+⎝ 0 ⎠ = ⎝q − pF cos θp ⎠+⎜ ⎝ pF cos θp ⎠+⎝ pK ⎠ (3) −pF sin θp 0 0 pF sin θp 0 ⎛

for the total Σ 0 energy of EΣ 0 =

 m2Σ 0 +(pF2 −2pF q cos θp +q 2 ),

(4)

where θp is the emission angle of the proton (should be spherical); mπ , mΣ , mp , and mK are the masses of π + , Σ 0 , proton, and K + ; and pπ , pK , q, and pF are the momenta of the incident π + , the outgoing K + , the momentum transfer to the Σ 0 , and the Fermi-motion, respectively. From Eqs. 3 and 4, all the parameters can be fixed, when one specifies pF and θp . Thus, the sensitive kinematic region for the quasi-free Σ 0 formation channel can be plotted as shown in Fig. 12, by numerically solving the equations. In the figure, the two-proton-tagging sensitive region is estimated by the direction of the spectator proton and that of the Σ 0 , because the proton is much heavier than the γ and the

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Fig. 12 One nucleon π + d → Σ 0 pK + reaction kinematics (black lines) with 2p trigger condition (red lines), i.e., θΣ 0 and θp are in 39 ∼ 122◦ and pF > 0.25 GeV/c region

π − , so that the decay proton’s direction is sufficiently close to that of the initial Σ 0 in a rough estimate. The kinematic region, where both Σ 0 and p are in the RCA acceptance (39 ∼ 122◦ ), is hatched in red. In this acceptance region, two proton momenta are greater than the RCA threshold, including the one coming from the Σ 0 decay, although the momentum of the proton from the Σ 0 decay will be smeared by the small momentum kicks provided by the cascade decay, Σ 0 → γ Λ → γ π − p. As shown in the figure, the pp tagging becomes efficient MMd  2.15 GeV. This tagging efficiency threshold is in good agreement with the mass value where the Σ 0 p event appears in Fig. 9 (left). It is well known that the Fermi-momentum has a long tail, and a non-negligible yield exists even above pF > 250 MeV/c. In the quasi-free Σ 0 formation channel, the spectator proton starts firing the second layer of the RCA (identified as a proton) at this momentum and is identified as two proton hits. Therefore, it is consistent with nonexistence of the events below MMd ∼ 2.15 GeV/c2 in Fig. 9 (left). On the other hand, the MMd spectrum of the Λp final state is much different. The difference in the spectra can be simply understood, if Σ 0 p-Λp conversion happens after the quasi-free Σ 0 formation in a cascade manner (as is pointed out by T. Ichikawa in his thesis). This Σ-Λ conversion provides sufficient energy to the proton to reach the second layer of the RCA, so as to be identified as a proton. The conversion process could be very strong, because the Σ and Λ can mix in the nuclear medium by about 1∼2% due to the small (≈70 MeV) energy separation between the two masses. This Σ-Λ mixing is known to largely enhance the conversion process from Σ to Λ in the nuclear system (Gibson et al. 1972). Equations 3 and 4 are given in a generic manner for the reactions producing a K + at zero degrees. Thus, the origin of the pF term (relative motion of the two baryons) can be replaced with Fermi-motion (coming from nucleon kinetic energies in the

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nuclear potential) or any other energy sources transferred to the spectator proton, no matter whether it comes from “K − pp” decay or Σ 0 p-Λp conversion. The two MMd spectral shapes of pp coincidence and ΛpK + final states are quite similar as shown in Fig. 11 (left and right), simply damping as a function of MMd toward MMd ≈ 2.45 − 2.5 GeV/c2 . It is known that the cross section of π + p → Σ + K + (would be similar to π + n → Σ 0 K + ) is about ten times that of the π + n → pK − K + channel at the E27 incident energy (Pan et al. 1970). Thus, the small yield of the Fermi-tail in the high-momentum region will result in giving a non-negligible contribution to the region of interest for both Σ 0 pK + and ΛpK + final states. On the other hand, the E27 spectra for both Σ 0 pK + and ΛpK + final states reach well above the quasi-free Y ∗ and K¯ formation thresholds (∼2.37 GeV/c2 ). If those spectra are sensitive to the “K − pp” formation, then MMd for those two tagged protons must be sensitive to those quasi-free channels as well, for exactly the same reason that the Σ-Λ conversion process can be tagged by two proton hits in the RCA. Namely, the energy excess in Y ∗ or K¯ in the quasi-free process can be shared with the spectator nucleon as an internal kinetic energy of the pair, resulting in a high-energy Λp (or Σ 0 p) pair in the final state. However, that contribution was not considered in the E27 spectral fit for the Σ 0 pK + final state in Fig. 9 (left). Therefore, it would be too naïve to accept the E27 interpretation that the MMd > 2.15 GeV/c2 events in the Σ 0 pK + final state can be treated as a backgroundfree “K − pp” formation signal. To obtain a conclusive result, it is necessary to decompose the MMd spectra considering possible contamination or to suppress those from the formation reaction. There is also an independent report from the LEPS group, utilizing somewhat similar reaction channel of γ d → K + π − X to search for “K − pp,” but the result was negative (Tokiyasu et al. 2014).

In-Flight K− Reaction The Kaon absorption-at-rest method gives us a huge background in the region of interest due to multi-nucleon absorption processes. On the other hand, in the s¯ s-pair ¯ creation method, it is difficult to produce a K-meson degree of freedom (behaving as a quasi on-shell particle) in the reaction, compared to the ΛK + production channel in terms of energy cost (e.g., ≈1 GeV for pp → ppK − K + and ≈0.68 GeV for pp → pΛK + ). Therefore, a more effective method is required to search for the kaonic nuclear bound state. A breakthrough idea – nucleon knockout reaction – was proposed by Kishimoto (1999). Figure 13 (left) shows the elementary cross sections of quasi-elastic ¯ reactions (K¯ in the backward direction), which clarifies the most K − N → KN ¯ desirable K − momentum to be utilized. The cross sections of K − N → KN backward K¯ quasi-elastic scattering have a common maximum momentum at pK − ≈ 1 GeV/c. At this beam momentum, the center-of-mass (CM) energy of the K¯ √ ∗ meson and a nucleon, sKN ¯ , is about 1.8 GeV, where Y hyperon resonances having

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¯ quasi-elastic K¯ backward (nucleon forward) scattering Fig. 13 (Left) Elementary K − N → KN cross sections. All the cross sections peak near pK − ∼ 1 GeV/c for K − n → K − n, K − p → K¯ 0 n, ¯ CM energy √s ¯ is plotted on top, and pole and K − p → K − p (from top to bottom). The KN KN ∗ positions of Y s are indicated by dotted lines. (Figure redrawn from Kishimoto 1999.) (Right) K¯ ¯ scattering in the laboratory frame. As for reference, q in momentum (q) after the K − N → KN ¯ N and KN ¯ are also plotted to take into account for the reduced mass of K¯ in the system KN

¯ exist. Thus, large quasi-elastic K¯ backscattering large decay branches Y ∗ → KN processes are realized having a maximum in common momentum of pK − ≈ 1 GeV/c. ¯ , nucleon This quasi-elastic backward K¯ scattering, i.e., a K − N → KN knockout reaction is an ideal K¯ source to produce kaonic nuclear bound state [“K¯ ⊕ (A−1)N”], via the substitutional reaction from a nucleon to the K¯ in a nucleus, described as, K − + (A)N → “K¯ ⊕ (A−1)N ” + N ,

(5)

where (A)N is the target nucleus with atomic number A. ¯ reaction produces a K¯ in the backward When the primary K − N → KN ¯ direction, then the K’s momenta become very small (∼200 MeV/c or less) in the laboratory frame as shown in Fig. 13 (right). This momentum is comparable with that of the nucleon Fermi-motion. Therefore, an overlap integral between the backscattered K¯ and the residual nucleus (a spectator of the reaction, at-rest in the laboratory frame) becomes large. Because the overlap integral is proportional to the sticking probability, the large K¯ nuclear bound state formation is expected. Thus, this reaction can substitute a K¯ for a nucleon in a target nucleus quite efficiently. Viewed from the residual nucleus, K¯ is generated close to the Fermi-momentum ¯ -nuclear potential by which the K¯ can be captured. of nucleons which provide a KN ¯ Viewed from the scattered K, if the residual nucleus provides a sufficiently attractive

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¯ differential cross sections (at pK − = 1 GeV/c and θN ∼ 0), KN ¯ isospin, Table 4 K − N → KN scattering amplitude, and interference between charge modes charge mode K − n → K − n K − p → K¯ 0 n K − p → K − p dσ /dΩ [mb/sr] |I, I3 KN ¯ ¯

amplitude f KN ¯

f KN interference

∼ 4.7 |1, −1 f(IKN =1)

∼ 2.4 |0, 0 & |1, 0 mixture   ¯ ¯ KN − f(IKN =0) + f(I =1) /2

∼ 1.8 |0, 0 & |1, 0 mixture  KN  ¯ ¯ f(I =0) + f(IKN =1) /2

w/ K − p → K¯ 0 n

w/ K − n → K − n

only within the charge mode

¯

¯ KN-nuclear potential to form a bound state, then the reaction can provide an offshell K¯ which fits into the bound state. The energy-momentum mismatch can be readily covered by the knockout nucleon N . Naturally, this kaonic nuclear bound state formation reaction competes with the quasi-free kaon emission process (above the binding energy threshold). The yield ratio between the two processes can be given as a sticking probability of the backscattered K¯ and the residual nucleus. This sticking probability largely depends on the momentum transfer that the backscattered K¯ brings to the residual nucleus. As a result, the kaonic nuclear bound state signal is expected to be observed close to the quasi-free kaon formation process, separated by the binding energy, and the yield ratio of the two depend on the dynamics of the reaction. Another big advantage of the in-flight K − method is the presence of an s-quark ¯ in the beam within the K − . In this reaction, the existence of the K-meson in the reaction is naturally guaranteed, unlike other reaction channels. In the invariant mass ¯ study, the s-quark flow can be traced to identify in which reaction-stage the Kmeson could be absorbed, since s-quark is conserved in the strong interaction. In order to estimate the cross section of the reaction given in Eq. 5 on a specific nuclear target, let us overview the relationship between the K − N → ¯ ¯ elementary cross section (σ ) and the scattering amplitude (f KN KN ) as for the preparation. Table 4 summarizes the differential cross sections and their relationship with the scattering amplitudes. The cross sections given in Kishimoto (1999) and those resulting from simple extrapolations to θK¯ = π of the measurements given in Damerell et al. (1977); Jones et al. (1975) and Conforto et al. (1976) using the Legendre-polynomial fits are different, but let us utilize Kishimoto (1999) values for simplicity. ¯ I3 = −1), so the In the case of the K − n charge mode, isospin must be one (KN ¯ − − − K n → K n scattering amplitude is simply given as K n | f |K − n = f(IKN =1) . In the case of the K − p charge mode, there are two possible charge modes in the final state, i.e., K¯ 0 n | and K − p |. The initial |K − p can be decomposed as; √ |K − p ≡ (−|0, 0 KN ¯ + |1, 0 KN ¯ )/ 2.

(6)

¯ scattering happens for each isospin channel with the specific scattering The KN amplitude, so the amplitude becomes

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√ ¯ ¯ KN f |K − p = (−f(IKN ¯ + f(I =1) |1, 0 KN ¯ )/ 2. =0) |0, 0 KN

(7)

The amplitude of the two charge modes are; √ K¯ 0 n | ≡ (0, 0|KN ¯ + 1, 0|KN ¯ )/ 2 and √ K − p | ≡ (−0, 0|KN ¯ + 1, 0|KN ¯ )/ 2,

(8) (9)

so the scattering amplitudes for K − p → K¯ 0 n and K − p → K − p can be given ¯ ¯ ¯ KN KN − − as K¯ 0 n | f |K − p = (−f(IKN =0) + f(I =1) )/2 and K p | f |K p = (f(I =0) + ¯

f(IKN =1) )/2, respectively, as shown in the table 5. The relationship between the differential cross sections and the amplitude¯ ¯ ¯ ¯ ∗ KN KN KN 2 2 square, |f(IKN =0) | , |f(I =1) | and their interference term Re(f(I =0) f(I =1) ) can be summarized as;   ¯ 2 d |f(IKN =0) | /dΩ = d − σnn + 2(σpn + σpp ) /dΩ,   ¯ 2 d |f(IKN =1) | /dΩ = d σnn /dΩ, and

    ¯ ¯ ∗ KN d Re f(IKN /dΩ = d − σpn + σpp /dΩ, =0) f(I =1)

(10) (11) (12)

where σnn = σ(K − n→K − n ) , σpn = σ(K − p→K¯ 0 n ) , and σpp = σ(K − p→K − p ) . Thus, ¯

the amplitude parameters f(IKN ) can be given by the elementary cross-sections to evaluate a formation cross-section on a specific nuclear target. It should be noted that the K − n → K − n and K − p → K¯ 0 n scattering amplitudes can interfere between the two modes, before taking the square of the amplitude. This is because the initial and the final states in Eq. 5 are identical to form “K¯ ⊕ (A−1)N ". What may be the most suitable spectroscopic method to detect kaonic nuclear bound state formation? The missing mass spectroscopy and invariant mass spectroscopy have their pros and cons. In both invariant and missing mass spectroscopies, the resulting mass values are defined by the binding energy, so the two methods are equivalent. In the case of the missing mass, the mass of X in the reaction A(B, C)X can be simply determined by measuring the momentum of C, since the mass value of A is defined by target material, and the incident momentum of B can be determined by the beam momentum. The missing mass of X is insensitive to the successive reaction, whatever it may be. On the other hand, the momentum transfer to X is defined by the spectrometer’s angle, depending on the X mass value. To study the reaction dynamics depending on momentum transfer, such as form(structure)factor, the spectrometer system must be rotated. In the case of invariant mass spectroscopy, the mass and the momentum transfer can be independently measured at once, but one needs to measure all the decay particles emitted from X and also C to resolve the kinematics. To be exact, one missing particle is allowed from the X decay particles and C, because it can be determined kinematically utilizing four-momentum conservation.

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A combinatorial background become serious when the number of particles to be measured becomes larger. An erroneous identification may happen between C and one of the decay particles from X, although this may happen in the case of missing mass spectroscopy as well. Thus, a suitable method depends on the atomic number of the target nucleus.

Missing Mass Spectroscopy in the In-Flight K− Reaction on 12 C KEK PS-E548 was carried out, led by T. Kishimoto, as one of the last experiments at the KEK 12 GeV proton synchrotron before the J-PARC operation (Kishimoto ¯ nuclear potential to bind the et al. 2007). To provide sufficiently attractive KN ¯ a 12 C target was selected. Because 12 C is a medium mass backscattered K, nuclear target, the E548 group used missing mass spectroscopy and measured the 12 C(K − , N) reaction, both for neutron and proton emission at ≈ 0 degrees. To define the reaction vertex (a forward going nucleon starting point for the TOF measurement), a charged particle detection around the target is required. In this sense, the missing mass spectra are not entirely inclusive. However, the bias introduced by this charged-particle requirement should be minimal, because the primary reaction and successive decays of the hyperon produce multiple charged particles around the target. Figure 14 shows the results of E548. As shown in the figure, both (K − , n) and (K − , p) spectra have large yields below the K¯ binding threshold energy. The integrated yields below the threshold [BE (≡ BK¯ ) = 0 in the figure] are reported to be as large as ≈20 mb/sr. Above the threshold, quasi-free K¯ formation processes

Fig. 14 The KEK E548 semi-inclusive spectra for the 12 C(K − , n) reaction (left), and the 12 C(K − , p) reaction (right). The red curves represent the best fit spectra for potentials with Re(V ) = −190 MeV and I m(V ) = −40 MeV for 12 C(K − , n), and Re(V ) = −160 MeV and I m(V ) = −50 MeV for 12 C(K − , p), and dotted curves for Re(V ) = −60 MeV and I m(V ) = −60 MeV. The dot-dashed curves represent an estimated background process. (The figure is redrawn from the data given in Kishimoto et al. 2007)

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are seen in both spectra. The tail components below the threshold extended up to 100 MeV or more, having almost the same cross section compared with that for the quasi-free K¯ formation processes. Therefore, the spectra suggest a large formation ¯ yield of kaonic nuclear bound states in the A = 11 nucleus. However, due to the KN absorptive potential, a part of the quasi-free spectra contributes to the yield below the bound threshold, although this contribution is limited to ∼10 MeV, in general. Interestingly, a step-like structure was observed in the (K − , n) spectrum below the bound state threshold. This implies the formation of K¯ bound states in the ¯ potential provided by the carbon nuclei. The stepnucleus, assisted by a deep KN like structure cannot be explained by leakage resulting from the absorptive potential. However, it is difficult to decompose the spectrum, because it is not observed as an ensemble of peaks, which would indicate a composition from multiple states. On the other hand, such a structure is not clear in the case of the (K − , p) spectrum, so that the interpretation of the internal structure observed in these spectra is not simple. Therefore, the E548 group conducted a global spectral fit, based on Green’s function method to produce their spectra. The spectral shape of the fitting function of the Green’s function method is given in Fig. 14, and the χ 2 -values around the best fit optical potential values, Re(V ) vs. I m(V ), are plotted in Fig. 15. The imaginary part of the best fit parameters for (K − , n) and (K − , p) are almost the same and absorptive, at around I m(V ) ≈ 50 MeV. On the other hand, the real part Re(V ) are both deep, but rather different. For (K − , n), Re(V ) is deduced to be ≈190 MeV, while for (K − , p) (11 B core), it is ≈160 MeV, weaker than the potential for (K − , n) (11 C core). The difference in the real part of V (∼20%) may come from ¯ potential provided by the 11 B and 11 C cores, where the the difference in the KN number of protons and neutrons is different by ∼20% and results in spectral shape differences in Green’s functions utilized for these spectra. The background processes in the (K − , N) missing mass spectroscopy are ¯ relatively simple. When a meson is produced in addition to the K − N → KN

Fig. 15 The χ 2 contour plot of the fit to the E548 spectra. The spectral shape of Green’s function method having a complex potential parameter V is applied both to the 12 C(K − , n) reaction (left), and the 12 C(K − , p) reaction (right). (The figure is redrawn from the data given in Kishimoto et al. 2007)

74 Kaonic Nuclei from the Experimental Viewpoint

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knockout reaction, the missing mass should shift to the heavier side by the pion mass (mπ ≈ 140 MeV/c2 at minimum). Thus, it does not affect the analysis described above. Another possible background is the two nucleon kaon absorption forming a hyperon (Y ) without producing a meson, i.e., K − NN → Y N (Y = Λ or Σ). This reaction can be much weaker than the single nucleon reaction, because the de Broglie wave length of the incident kaon is shorter than the mean nucleon distance within nuclei at pK − = 1 GeV/c. When the N of the K − NN → Y N reaction (or decay product of Y ) is detected in the forward spectrometer, the (K − , N) missing mass should be much lighter by ≈mK + mp − mY (235 MeV even for Σ − ), where mY is the hyperon mass. The mass difference is very large, and the yield of the (K − , N) spectra at that mass region is quite weak in E548. Thus, these channels can be safely neglected again. The Y ∗ = Λ(1405) (instead of Y ) formation can also ¯ be omitted, because it is nothing but a K-nucleus formation. However, there is still a possible process which might form serious backgrounds in the region of interest. If Y ∗ = Σ(1385) is formed in the K − NN → Y ∗ N reaction, the (K − , N) missing mass affects the binding energy region of BK¯ ≈ 50 MeV. Thus, Σ(1385) may form hidden structures (peak/background) in the region of interest in the missing mass spectrum. The Σ(1385) (or π ΛN system) may form a nuclear bound state with the residual nucleus (Σ(1385)-hypernucleus) (Garcilazo and Gal 2010, 2013). It may also produce Σ(1385) as a quasifree particle. These two processes may contribute to the missing mass spectrum independently. However, missing mass spectroscopy cannot discriminate the two, from the kaonic nuclear state formation. The PS-E548 group’s report is very promising regarding the existence of kaonic nuclear bound states. About the concern on the possible bias on the semi-inclusive spectra, it is proven to be minimal by an independent experimental group at J-PARC (Ichikawa et al. 2020), in pure inclusive 12 C(K − , p) method. However, to be conclusive about the kaonic bound state formation, (1) decomposition of the spectral shape below the binding threshold, (2) characterization of the quantum number of each state would be required, and also (3) the Σ(1385) contribution to the region-of-interest would need to be evaluated.

Observation of the Simplest Kaonic Nuclear Bound State To provide a definite result as to the existence of kaonic nuclear bound states, the J-PARC E15 experiment was proposed, led by M. Iwasaki. The E15 was conducted as a Day-One experiment at the J-PARC kaon factory, where realization of an intense kaon beam is one of the key features. E15 focused on producing the simplest ¯ knockout reaction, and studied the kaonic nuclear bound state, “K − pp” by the KN reaction both in the formation channel via the3 He(K − , n)X missing mass method, and in the decay channel via the invariant mass method X → Λp. There are several reasons why “K − pp” was selected as a primary objective to be intensely researched. Firstly, “K − pp” must exist if a Λ(1405) is a molecule-like hadronic cluster, as is suggested by the kaonic hydrogen data. Secondly, one can

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fully utilize the advantage of four-momentum conservation, which allows one to deduce the reaction kinematics unambiguously with one missing particle, when the number of particles in the final state is minimal. In the case of the Λ(1405), it decays 100% to Σπ , but in the case of a kaonic nucleus, non-mesonic decay channels open, e.g., “K − pp” → Λp, which helps by reducing the number of particles to be detected. Thirdly, “K − pp” is rather easily identified by the invariant mass study through the “K − pp” → Λp decay mode, because the Λ decays in a charge mode Λ → pπ − (64%), so that the final state consists only of charged particles, ppπ − , which can be analyzed easily by magnetic spectrometer. Finally, one can reduce the number of possible combinations of particles to be studied. In the invariant mass study, one needs to specify the system that finally absorbs the K¯ meson. Because the s-quark is conserved in the strong interaction, the number of combinations to be studied in the Λpn final state is only two in a 3 He target, i.e., Λp or Λn. To realize both the missing mass study and the invariant mass study simultaneously, the E15 experimental setup (Fig. 16) was designed to have a forward spectrometer system and a cylindrical detector system (CDS) surrounding the liquid 3 He target cell. In the forward spectrometer, protons, neutrons, and beam K − s were separated by a dipole magnet, and the momenta of the protons and neutrons were measured by the time-of-flight (TOF) method. In the cylindrical spectrometer, charged particles were momentum analyzed by a cylindrical drift chamber, and a

Fig. 16 The E15 setup to obtain the missing mass and the invariant mass spectroscopy at the same time. The missing mass study is conducted using the time-of-flight method by the plastic scintillation counters located 15 m downstream of the target. The invariant mass study is realized using cylindrical spectrometer system (CDS) surrounding 3 He target. (Figure is taken from Hashimoto et al. 2015.)

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Fig. 17 The E15 semi-inclusive 3 He(K − , n)X missing mass spectrum. The dominant peak ¯ , smeared by the observed in the spectra comes from the backward K¯ scattering via K − N → KN Fermi-motion of N in the reaction, and forms a quasi-free backscattered K¯ peak in the spectrum. The ACDS ∼ 0.7 represents for the detection efficiency of a charged particle in CDS. The observed yield below the binding threshold is about 1 mb/sr in total. The inset is the same spectrum tagged by Ks0 → π + π − to ensure that the backscattered K¯ escape from residual nucleons without finalstate-interaction. (Figure is redrawn from data given in Hashimoto et al. 2015)

clear particle identification (π ± , K − , p, and d) was obtained by combining the TOF information within the CDS. The E15 group used a staged approach for the experiment by focusing on the missing mass study in the initial phase. Figure 17 shows the 3 He(K − , n) missing mass spectrum (Hashimoto et al. 2015). To define the reaction vertex (a forward going neutron starting point for the TOF measurement), a charged particle detection in the CDS is required. As shown in the figure, the cross section has a peak structure at ≈2.4 GeV/c2 , above the kaon binding threshold of ≈2.37 GeV/c2 (= mK +2mp ), and the width of this peak is about ≈0.07 GeV/c2 . The peak position and the width indicate that ¯ where the structure is formed by a quasi-free K¯ formation process, K − N → Kn, ¯ the K is recoiled to the backward direction. The large probability of the quasi-free process confirms the dominance of the neutron (or nucleon) knockout reaction. In the heavier missing mass region above ≈2.51 GeV/c2 (= mπ +mK +2mp ), the cross section starts increasing again, because of pion production in addition to the simple neutron knockout mechanism. Thus, below this energy, K¯ and two residual nucleons ¯ may form three-hadron-system as K − +3 He → (KNN ) + n. The large yield of this − nucleon-knockout-reaction confirms that the K reaction at pK = 1 GeV/c is very promising to be used to search for “K − pp.”

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Note, no clear structure formation was found below the threshold of the missing mass spectrum. However, a clear event excess was observed below the threshold, which extended to the lighter mass region of about 100 MeV, in contrast to the K¯ escape channel as seen in the insert in Fig. 17, which sharply end at the threshold energy. When the K¯ leaves the residual nucleus without reacting with other nucleons, the missing mass clearly appears above the threshold (within the detector resolution). The tail below the threshold is quite similar to the subthreshold excess observed in the KEK-PS E548 data, in spite of a much smaller atomic number than in the case of the 12 C target. As is the case in E548 data, a mass reduction of ¯ 100 MeV is much too large to be explained only by the imaginary part of the KN potential. It is more natural to consider that the structure formed below the threshold is masked due to the tail contamination from the quasi-free reaction process (part of quasi-free kaon reaction spectrum leaks into the region of interest). However, the signal-to-noise ratio can be drastically improved for a specific decay channel. As was discussed above, “K − pp” can decay to Λp, and this decay mode can be efficiently analyzed by the invariant mass method. In this method, the K¯ escape channels are naturally omitted, and the quasi-free kaon above the mass threshold can only contaminate through the Λp conversion channel, so that the signal-to-noise ratio can be drastically improved. Following this procedure, the invariant mass was measured to search for a “K − pp” signal by the Λp-pair, although the initial run is quite limited in statistics (Sada et al. 2016). For the Λp invariant mass study, Λp (decaying to π − pp) detection in the CDS (Fig. 18) was required, and a neutron in the K − +3 He → ¯ (KNN) + n → (Λp) + n is identified. By utilizing four-momentum conservation,

Fig. 18 A schematic figure of the E15 CDS for the charged particle analysis. The CDS covers ≈50% of the target solid angle. The particle identification is made by the time-of-flight measurement in the CDS. (Figure is taken from Sada et al. 2016)

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the presence of a neutron in the final state (thus, Λpn final state) is ensured, without necessitating its detection in the forward neutron counter. It should be noted that the event selection in the Λpn final state can be contaminated by a Σ 0 pn final state, because the mass separation of the two channels is not very large (≈70 MeV), and it is difficult to detect a γ from the Σ 0 decay. However, the fraction is only at the 10% level. The Σ − pp final state may enter through the mis-identification of π − p in the final state as a Λ at about 7%, but this is even less of a problem, because the resulting Λp invariant mass is randomly distributed over the entire kinematic region. To see whether there is any kinematic anomaly in the Λp-pair kinematics, the correlation between Λ and p was studied for the Λpn final state. Figure 19 shows the result of Λp-pair’s correlation study. When the Λp is treated as a pair, the reaction kinematics of

Fig. 19 Correlation of the Λp-pair in between the Λp invariant mass (denoted as Minv.Λp ) and neutron emission angle (θnCM ) of the K − +3 He → (Λp) +n reaction, observed in the initial phase run of E15. The Λp is measured in the CDS, and the neutron is identified through the missing mass of the reaction. (top) Event projection to the Λp invariant mass axis. (right) Projection to the neutron emission angle axis. (left-bottom) An event correlation between the two axes. Near the kaon binding threshold, event concentration is clearly identified at θnCM ∼ 0◦ (corresponding to the lower momentum transfer to the Λp-pair). (Figure is redrawn from the data given in Sada et al. 2016)

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¯ K − +3 He → (KNN) + n → (Λp) + n

(13)

can be considered as a two body reaction. Thus, the kinematics can be specified by only two parameters. Therefore, events are plotted in the two-dimensional plane consisting of the Λp invariant mass and the neutron emission angle in the centerof-mass (CM) of the Λpn system. Although the statistics are limited, the Λp invariant mass spectrum drastically changed from the 3 He(K − , n) missing mass spectrum. As shown in the figure (top), a very interesting event concentration was observed around the binding threshold in the Λp invariant mass spectrum. More interestingly, about half of the events are located below the binding threshold, so that it cannot be explained by quasi-free kaon production. On the other hand, it is clear that the quasi-free kaon formation yield above the binding threshold is substantially suppressed. As shown in the figure (right), the event concentration is formed at forward neutron emission angle (θnCM ∼ 0◦ ). This indicates that the doorway reaction channel, which originates the event concentration, is the neutron knockout reaction, ¯ as is expected for “K − pp” formation. It is also quite interesting K − N → Kn that the neutron emission angle seems to be not very forward peaked for the event concentration (left-bottom), which indicates that the missing mass spectroscopy at θnCM ∼ 0◦ is not adequate to study the full reaction dynamics. In fact, the event concentration extends up to cos θnCM ≈ 0.8, which is ≈ 40◦ in the CM.

Confirmation of the Event Concentration Near M(K− pp) To study the origin of this event concentration, the E15 group proceeded to conduct a dedicated run for the Λp invariant mass spectroscopy. Figure 20 shows the result of the higher statistics data (Ajimura et al. 2019; Yamaga et al. 2020). In the figure, the vertical axis of the plot in terms of the neutron emission angle (θnCM ) is replaced by the momentum-transfer to the Λp-pair (qX , the synthetic momentum of the Λp-pair in the laboratory frame). The horizontal axis (denoted as mX ) is same as that of Fig. 19, i.e., the invariant mass of Λp-pair. The event concentration found in the initial phase given in Fig. 19 (at θnCM ∼ 0◦ ) becomes definitive in Fig. 20 (corresponding to the lower momentum-transfer limit (minimum qX ) to the Λp-pair). The thin black dotted line represents the kinematic boundary for the K − +3 He → (Λp) + n reaction. It is now evident that the event concentration found in the initial phase is composed by two independent subcomponents, one located below the kaon binding threshold (thick vertical dotted line) and the other located above that binding threshold. There is a third event concentration at the upper-right boundary of the figure. The blue dotted curve represent for the ‘invariant’ mass line for the Λp-pair, assuming that it is originated from the quasi-free K¯ absorption by two spectator nucleons at rest, which is given as

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Fig. 20 The invariant mass and momentum-transfer (mX , qX ) correlation of the Λp-pair (denoted as X) in 3 He(K − , Λp)n reaction. The statistics is improved substantially compared to the spectrum shown in Fig. 19 (the vertical axis is replaced from θnCM by the momentum transfer to the Λp-pair, i.e., qX ). Three event concentrations were found in the (mX , qX ) plane. It is clear that two of them, near the kaon binding threshold, corresponding to event concentrations found in Sada et al. (2016), should be attributed to different origins. (Figure is redrawn from the data given in Yamaga et al. 2020)

 mQF−KA ¯ (qX ) =

 2, 4m2N +m2K¯ +4mN m2K¯ +qX

(14)

where mN is the nucleon mass in the nucleus and qX is the K¯ momentum before the absorption. As it is given in the equation, the ‘invariant’ mass calculated from the ¯ kinetic energy must be converted Λp-pair is not independent of qX , because the K’s into internal kinetic energy between the Λ and p. As shown in the figure, the latter two event concentrations exist on the mQF−KA ¯ curve. Thus, these can be naturally interpreted as the quasi-free K¯ absorption ¯ ¯ reaction. The one in the process (QF -KA), cascading from primary K − N → Kn ¯ lower left is for the backscattered K, and the one in the upper right is the same for the ¯ The lower-left QF -KA ¯ event concentration at the kinematic forward scattered K. boundary has a small qX (θn ∼ 0) distribution. This is consistent with the missing mass spectra given in Fig. 17, which exhibits a quasi-free K¯ formation dominance at ¯ may have a contribution from direct two nucleon θn ≈ 0◦ . The upper-right QF -KA − ¯ K absorption, K (p p) → Λp. Because of the direct capture of a kaon, the internal energy between the Λ and p becomes large, resulting in events appearing at the upper limit of the Λp invariant mass, and the momentum transfer to the Λp-pair is about the incident kaon beam momentum (smeared by the Fermi-motion).

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In contrast, the event concentration below the kaon binding threshold (at ≈2.32 GeV/c2 ) has a much longer tail in the qX direction. Thanks to the existence of the long tail, it is clear that the event concentration centroid in mX direction is independent of the qX , even on the higher momentum side. This is the most essential requirement that a bound state must fulfill, having a specific/quantum energy. The signal is located at about the center of the kinematic boundaries in the lower qX region (θn ≈ 0◦ ), where a spurious event concentration is most difficult to form. Well below the region of interest (≈2.13 GeV/c2 ), there is a possible background process coming from the two-nucleon absorption reaction, i.e., K − (p n) → Σ 0 n followed by Σ 0 ps → Λp conversion, where ps is the spectator proton in the primary reaction. However, as shown in the figure, the yield is not strong, although the efficiency is not very large, especially in the small qX region. It is reasonable that the process is weak in the E15 data, since the primary reaction of this process (nonmesonic two-nucleon absorption) is very weak at pK − = 1 GeV/c. As is shown in the upper-right region of the (mX , qX ) plane, there is an event concentration partly coming from K − (p p) → Λp but that K −+(p p) reaction is already weaker than the signal in the primary reaction stage. As a similar process, K − (p n) → Σ 0 (1385)n followed by Σ 0 (1385)p → Λp conversion, should be considered. In terms of mX , it could be more serious due to the resulting mX location being rather close to the signal, because the Σ 0 (1385) ¯ binding threshold. However, unlike resonance appears ≈50 MeV/c2 below the KN Σ-Λ conversion, the mass difference between the Σ 0 (1385) and the Λ is much larger, so they are well decoupled, and the conversion strength must be much weaker than in the case of Σ-Λ conversion. Moreover, if this is the dominant origin of the event concentration in the region of interest, the event concentration should have a definitive qX dependence, similar to the quasi-free kaon absorption of Eq. 14. The mX coming from Σ 0 (1385)-Λ conversion can be written as  m

QF−Σ ∗A

(qX ) =

m2N +m2Σ ∗ +2mN



2. m2Σ ∗ +qX

(15)

The mX value estimated by mQF−Σ ∗A at qX ∼ 600 MeV/c is located well above the binding threshold, while the signal has no qX dependence. Thus, this process can be safely rejected as a primary origin of the event concentration below mX < M(K − pp). Therefore, the event concentration below mX < M(K − pp) should come from a quasi-bound state formation, which decays into Λp. Because the data consists of the minimum number of particles Λpn without any meson, all other complex background processes can be excluded. Based upon these considerations, the E15 group introduced two-dimensional functions to be applied for the data in the (mX , qX ) plane for the fitting, as shown in Fig. 21. The PWIA based fitting function for the S-wave resonance (fK¯ : left) is introduced as a Breit-Wigner formula in the mX direction having a reaction form factor (structure factor) of Gaussian shape as

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Fig. 21 Fit functions applied to the (mX , qX ) two-dimensional fitting of the E15 data. (left) “K − pp” signal component. (middle) Quasi-free K¯ formation and its absorption by two spectator ¯ nucleons (QF -KA). (right) Broad distribution, which most probably comes from an inaccurate analysis of (Λp) + n to the (Λn) + p event dynamics (see text, for more detail). The vertical ¯ and the black dashed line is the kaon binding threshold M(K − pp), blue dashed curve is QF -KA, dashed curve is the kinematical boundary of the Λpn final state. Contours are in logalistic scale. (The figure is redrawn from the data given in Yamaga et al. 2020)

fK¯ (mX , qX ) =

¯ AK 0

 2  2  ΓK¯ /2 qX × exp − 2  2  Q2K¯ mX − mK¯ + ΓK¯ /2

(16)

¯

where AK 0 is the normalization factor and mK¯ , ΓK¯ , and QK¯ are the mass, decay width, and reaction form factor for “K − pp,” respectively. For the fitting function to represent the quasi-free kaon absorption process (fF : middle) and the broad background distribution (fB : right), empirical formulae were introduced (Yamaga et al. 2020). The origin of the broad background distribution of the Λpn final state is not clear. One of the natural background sources could be due to the ambiguity involving the initial knockout nucleon. In the analysis, neutron knockout reaction, K − +3 ¯ He → (KNN ) + n → (Λp) + n, is predominantly assumed, but it is possible that a proton is knockout in the primary reaction instead of a neutron, K − +3 He → ¯ (KNN) + p → (Λn) + p. When the Λp-pair is analyzed for this reaction, then meaningful kinematic correlations are destroyed, and the events appear widely over the kinematically allowed region, resulting in an event distribution resembling that of phase space. Thus, the E15 group applied an empirical formula for the broad background distribution for fB . The three functions (fK¯ , fF , and fB ) were then multiplied by (1) the Lorentz invariant Λpn three-body phase space ρ(mX , qX ) (density of final states) and (2) experimental acceptance A(mX , qX ) evaluated by simulation. And the sum A(mX , qX )ρ(mX , qX )

 i

fi (mX , qX ),

(17)

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M. Iwasaki

¯ F and B, was applied to fit the data on (mX , qX ) bins using a where i = K, maximum-likelihood method on the data in the count base, whose statistical error can be safely assumed to follow a Poisson distribution. After the fitting, acceptance corrections were applied both to the data and to the fitting function, to avoid spurious fitting in the procedure for bins having a small number of counts. Figure 22 shows the invariant mass (left) and the momentum-transfer distributions of the signal region (right). In these spectra, the signal region is selected by a window of momentum transfer 0.3 < qX < 0.6 GeV/c for the qX spectrum and 2.27 < mX < 2.37 GeV/c2 for the mX spectrum (cf., Fig. 20). The fitted model functions are in good agreement with data also in these sliced spectra. As shown in the figure, the contribution of the other final states (Σ − pp and Σ 0 pn) is considered in the (mX , qX ) two-dimensional fitting. The fitting gives a BW pole with BK¯ ≈ 40 MeV, having a width ΓK¯ ≈ 100 MeV and a Gaussian form-factor parameter of QK¯ ≈ 400 MeV/c. The observed total cross section decaying into Λp is ≈ 10 μb (Ajimura et al. 2019; Yamaga et al. 2020). As expected from the reaction mechanism, the Λpn final state data are fully explained by the reaction process in which the K¯ behaves as a quasi-particle until the final stage of the reaction, when it is finally absorbed by two nucleons. The binding energy is relatively large, reaching ∼8% of the K¯ mass (≈500 MeV/c2 ); thus, the relationship to the in-medium hadron mass modification is noteworthy. On the other hand, the decay width is very wide and variable. Thus, the decay width is expected to lead to further research on the time scale of inmedium quark annihilation. The “K − pp” binding energy obtained appear to be consistent with an increase in K + yield in pp collisions. As shown in Fig. 8 (Epple and Fabbietti 2015),

Fig. 22 (left) Invariant mass of the Λp-pair of K − +3 He → (Λp) + n in the momentumtransfer region of interest (0.3 < qX < 0.6 GeV/c). (right) Momentum transfer to the Λp-pair in the invariant mass region of interest (2.27 < mX < 2.37 GeV/c2 ). (The figure is redrawn from Yamaga et al. 2020)

74 Kaonic Nuclei from the Experimental Viewpoint

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subthreshold K + yield increase is observed in pp collisions. This subthreshold K + yield increase in pp-collision could be partly formed with the assistance of the formation of “K − pp.” Although, this phenomenon can also be explained by pp → pΣ 0 (1385)K + channel. ¯ potential The system could be compact, because of the strongly attractive KN in the IKN = 0 channel and the deduced large binding energy. In fact, the form¯ factor parameter is as large as ≈400 MeV/c. A Fourier transformation of the form-factor function yields the wave function in spatial coordinates. Thus, the large form-factor parameter implies that the “K − pp” wave function could be very small in spatial size for the system, rK¯ ∼ 0.6 fm, even when the reduced mass of the K¯ in the system (≈4/5 mK ) is considered. This value is quite anomalous compared to the scale for normal nucleon separation twice (0.8 ∼ 0.9 fm). Even though the model fitting function is based on a PWIA, the implied spatial size is surprising from the viewpoint of our conventional understanding of nuclei. It should be emphasized that the measured form factor is a reaction form factor, in which the reaction dynamics is convoluted (simplified as a δ-function in the PWIA). Thus, a more detailed theoretical study is needed.

Dibaryon-“X” Candidates and Internal Isospin-Spin Configurations Let us reconsider the possible signal candidates as dibaryon-“X”, having s-quark (strangeness −1) and charge +1, locating just below the K meson binding energy threshold M(K − pp), and decays to Λp-pair in the strong interaction. The ground state of the dibaryon-“X” candidates (internal orbital angular momentum zero) can be uniquely obtained by the Clebsch-Gordan coefficients. Table 5 summarizes the ground state candidates of dibaryon-“X” and their isospin-spin structures by focusing on the nucleon-pair’s symmetry (cf., Table 3 for the first four rows). In I (J P ), the isospin (I ) is fixed to be I = 12 , because the decay branch to Λ 3 (I = 0) and proton (I = 12 ) is observed in E15, so the IKN ¯ N = 2 possibility was removed (cf., Table 3 (left column)). As it is described, I (J P ) = 12 (0−) (first two rows in Table 5) is given from Table 3 (middle column) with antisymmetric nucleon spin configuration (SN N = 0). As for a dibaryon-“X” candidate, Table 3 (right column) with nucleon symmetric spin (SN N = 1) configuration is also possible, whose I (J P ) is 12 (1−) (successive two rows in Table 5). ¯ The reason why I (J P ) = 12 (0−) is the most natural “KNN” candidate than 1 − P I (J ) = 2 (1 ) is not simply because of minimal isospin-spin configuration, but ¯ couples to I ¯ = 0 channel more strongly than I (J P ) = 1 (1−). In internal KN KN 2 the I (J P ) = 12 (0−) state, K¯ couples at 75% probability to IKN ¯ = 0 channel, and only 25% to IKN ¯ = 1 channel, as it is given in the Table 5. On the other hand, in ¯ coupling probabilities are opposite. Note that the I (J P ) = 12 (1−) state, the KN ¯ I ¯ = 0 attraction is the reason why Dalitz et al. suggested the the strong KN KN possibility of Λ(1405) to be a K¯ bound hadronic cluster.

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¯ coupling to I ¯ = 0 channel Table 5 The internal structure of dibaryon-“X” and the KN KN ¯ (I =0) |2 |KN ¯ (I =1) |2 Isospin-spin configuration of |KNN ¯ Dibaryon-“X” I (J P ) |KN or |Σ ∗ N     1 − 1 √ −K − {p, p}+ K¯ 0 {p, n} ⊗ ↑, ↓ “K − pp 3/4 1/4 2 (0 ) 2 3    1 √ + K¯ 0 {n, n} − K − {p, n} ⊗ ↑, ↓ “K¯ 0 nn” 2 3     1 − 1 ¯0 √ K p, n ⊗ ↑, ↑ “K¯ 0 d” 1/4 3/4 2 (1 ) 2 2     1 − √ “K d” K − p, n ⊗ ↑, ↑ 2 2√    1 + √1 “Σ ∗+ n” – – 2[Σ ∗+ , n]−[Σ ∗0 , p] ⊗ ↑3 , ↑ 2 (2 ) 6 √    √1 [Σ ∗0 , n]− 2[Σ ∗− , p] ⊗ ↑3 , ↑ “Σ ∗− p” 6

[A, B] : commutator = AB − BA, {A, B} : anti-commutator = AB + BA ↑3 : spin 32 , ↑ : spin 12 , Σ ∗ ⊗ ↑ = N ⊗ ↑3 ≡ 0

1 This coupling strength can be simply verified as follows. When IKN ¯ N = 2 is written as a system composed by (IK¯ ⊗ IN ) = 0 and IN = 12 (i.e., Table 2 (left) ¯ ⊗ Table 1 (right)), then √NN-pair’s isospin-symmetry-mixed KNN state is given, whose mixing ratio is 3 : 1 between I (J P ) = 12 (0−) and 12 (1−), by decomposing the nucleon isospin symmetry with commutator [A, B] = AB − BA and anticommutator {A, B} = AB+BA. The procedure is same for (IK¯ ⊗IN ) = 1 coupling. Note that the spin symmetry difference does not affect to the IKN ¯ coupling, because the spin configuration is automatically defined by the NN-pair’s isospin symmetry. Thus, I (J P ) = 12 (0−) is the most natural candidate as for the ground state and as for an experimentally observed state. However, other dibaryon-“X” possibilities listed in the Table 5 are not entirely excluded. To consider other possibilities, we need to know the formation yields of these channels compared to that of the most probable I (J P ) = 12 (0−), I3 = + 12 state. The last two rows of the Table 5 are for the newly added “Σ(1385)N ” candidate ¯ of dibarion-“X,” in addition to the “KNN ” candidates. As for the ground state, the isospin must be 12 . Thus, I (J P ) of “Σ(1385)N ” is uniquely fixed to be 12 (2+ ). Because isospin of Σ(1385) is one, nucleons’ symmetry should be antisymmetric (by treating Σ(1385)s are nucleon’s family). Thus, the spin of “Σ(1385)N ” must be symmetric. The spin of Σ(1385) is 32 , so the “Σ(1385)N ” spin must be two. The parity must be positive, since intrinsic parities of Σ(1385) and N are both positive. Then the isospin-spin configuration is fixed as it is given at the bottom of the table, ¯ coupling according to the Clebsch-Gordan coefficients. Naturally, there is no KN in this state.

Relative Formation Yields of the Dibaryon-“X” Candidates In order to evaluate the cross section of a particular dibaryon-“X” candidate, it is ¯ around the θ ¯ ∼ π , necessary to know the differential cross section of K − N → KN K ¯ and the K sticking probability to the two spectator nucleons (NN ) in 3 He. These

74 Kaonic Nuclei from the Experimental Viewpoint

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¯ two quantities may depend on (m, q) [mass and momentum transfer to the (KNN)system], and it is not yet well understood how they change depending on (m, q). Instead, let us estimate the relative-formation-yields by simply assuming the yield is proportional to the differential cross section at θK¯ = π . The relative-formation-yields of the dibaryon-“X" candidates can be estimated from the Clebsch-Gordan coefficients and the known differential cross sections + summarized in Table 4. The 3 He is almost pure I (J P ) = 12 ( 12 ) state (Wu et al. 1990). For the estimation, one can specify any coordinate system, so let’s take the coordinate system aligned to the spin-up direction. For a spin-up 3 He, there are 3! = 6 independent permutations of nucleons defined by the nucleon symmetry, which can be described as 3

1  He↑ := √ ij k ai aj ak 6 ij k  1  = √ n↑ (p↑p↓−p↓p↑)−p↑(n↑p↓−p↓ n↑)−p↓(p↑n↑−n↑ p↑) . (18) 6

where ij k is the totally antisymmetric tensor and ai = {p↑, p↓ and n↑} for i = + {1, 2 and 3}, respectively. This is an irreducible representation of I (J P ) = 12 ( 12 ) state, and exactly the same representation (except for sign) can be derived using the Clebsch-Gordan coefficients both from |IN N = 1, SN N = 0 ⊗ |IN = 12 , SN = 12 and |IN N = 0, SN N = 1 ⊗|IN = 12 , SN = 12 by noting the nucleons’ permutation √ [A ⊗ (BC) = (ABC − BAC + BCA)/ 3, where A, B, and C are nucleons with spins]. Equation 18 consists of three terms, each of which is described as the product of a particular state of one nucleon and the isospin-spin states of other nucleons that can couple to that state. Thus, each of the three terms can be considered as giving three independent final states with 1/3 probability each, in the K − N → ¯ reaction, by replacing the nucleon outside the parentheses by K. ¯ By noting KN ¯ (i) the nucleon outside of an inner parenthesis to be replaced by K, (ii) the middle term is an isospin-spin mixed state which can be decomposed to two isospin-spin √ orthogonalized terms, and (iii) renormalized by the number of nucleons (× 3), Eq. 18 can also be written as 3

 1 √ He(3N )↑ := √ 2 n↑ |I|1, +1 , S|0,0 NN − p↑ |I|1, 0 , S|0,0 NN 2  √ +p↑ |I|0,0 , S|1,0 NN − 2 p↓ |I|0,0 , S|1,+1 NN , (19)

where 3 He(3N ) ↑ represents for a spin-up helium-3 nucleus normalized by the number of nucleons and |I|Y , S|Z X represents for an X state having isospin I = |Y and spin S = |Z . As it is clear from the equation, the second and the third terms are the decomposition of the isospin-spin mixed term in Eq. 19. The

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former two terms of Eq. 19 (SN N = 0) may generate I (J P ) = 12 (0−) states via ¯ reaction, since the spin of the NN-pair is zero. On the the S-wave K − N → KN other hand, the latter two terms (SN N = 1) result in I (J P ) = 12 (1−) via the S-wave ¯ reaction (or 1 (2+) via P -wave for the “Σ ∗ N ” formation). K − N → KN 2 ¯ For example, let us consider the formation amplitude of the (KNN )-system (having quantum number same as “K − pp”) formation probability via the neutron knockout reactions K − n → K − n in S-wave can be simply evaluated using Eq. 19. Namely, by substituting a neutron outside of parentheses in Eq. 19 with K − , the reaction can be written as K

−

  − + He(3N ) ↑ → K pp +n ↑ := K |I|1, +1 , S|0,0 NN + n↑ 3

−

    2 1 + n↑ , (20) |I| 1 , + 1 , S|0,0 ¯ = √ |I| 3 , + 1 , S|0,0 ¯ + KNN KNN 3 2 2 3 2 2 ¯ ¯ I ¯ = 1 scattering amplitude of f KN at the KN KN (I =1) (cf. Table 5). Obviously, the latter part of the equation has the same quantum number as the “K − pp” state, thus, √ ¯ K − n → K − n reaction on 3 He gives an amplitude of 2/3 f(IKN =1) . The scattering ¯ reactions can be evaluated in the same amplitudes for other primary K − N → KN manner, by noting the amplitude of K − n → K − n and K − p → K¯ 0 n channels can interfere between the two, since the initial state and the final state are the same (cf. Eq. 5). The evaluated scattering amplitudes and the estimated relative yields ¯ (based on the assumption described at the beginning of this section) for the “KNN ” candidates are summarized in upper four rows of Table 7. The scattering amplitude evaluation of the “Σ(1385)N ” candidate is slightly complicated. The “Σ(1385)N ” I (J P ) = 12 (2+ ) state can only be generated through ¯ KNN having the same quantum number, i.e., I (J P ) = 12 (2+ ) channel. For the generation of this channel, the orbital angular momentum  of K¯ must be one at the minimum, to flip the parity of the system by (−1) term. To form J = 2 with K¯ in  = 1, only the latter part of the Eq. 19 [having NN -pair’s spin SN N = 1 (S|1, m terms)] can contribute to the formation reaction. Thus, the formation reaction of ¯ KNN I (J P ) = 12 (2+ ) can be written as

     KN ¯ N, I (J P ) = 1 2+ , I3 , J3 = m + m :=  2 |I| 1 , I3 , J|2, m+m ¯ 2

KN N



= |I| 1 , I3 , Y1m ¯ ⊗ |I|0,0 , S|1, m N N , 2

K

(21)

with possible spin combinations of m and m , where the I3 is the third component of ¯ N system and the Y m is the P -wave ( = 1) spherical function of the isospin of KN 1 ¯ having a third component of the orbital angular momentum of m (the K¯ spin K, ¯ is omitted, since it is zero). In Eq. 21, the isospin of KNN before the conversion

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is fixed to 12 ; thus, it is sufficient to consider the angular momentum coupling of 1 ⊗ 1 = 2. Because directions of P -wave K¯ scattering are arbitral, a factor of 5/9 must be applied as an average of all the possible m, m combinations. 1 1 + ∗ P ¯ N I¯ From this KN ¯ N = 2 system, “Σ N ” I (J ) = 2 (2 ) KN N = 2 , JKN ∗ ¯ state can be generated via the KN → Σ conversion reaction, which happens only 1 ¯ through IKN = 1 channel. From Table 5, KNN IKN ¯ ¯ N = 2 , SN N = 1 system ¯ combinations. Thus, may couple to IKN ¯ = 1 channel by 3/4 of all the possible KN ¯ → Σ ∗ conversion efficiency of 3/4 should be applied to the “Σ(1385)N ” a KN formation √ √ cross section. In Table 6, the square-root of the two reduction factors (i.e., 5/(2 3)) is applied as for an amplitude reduction factor for the “Σ(1385)N ” formation. Note that there are two unknown factors to evaluate the relative yields to compare with that of the “K − pp.” One is the P -wave / S-wave scattering ratio εP/S ≡ ¯ )/S(K − N → KN ¯ ), and another is the sticking probability P (K − N → KN ∗ ¯ ratio between Σ N and KNN to form a nuclear bound state [rstk(Σ ∗ N/KNN ¯ ) ], ¯ ¯ including KN → Σ(1385) conversion cross section within the (KNN)-system, as summarized at the last two rows of the table. A large discrepancy of the (I3 = − 12 )/(I3 = + 12 ) ratio in between I (J P ) = 1 − 2 (0 ) and other candidates is dominantly coming from the interference between K − n → K − n and K − p → K¯ 0 n charge-mode amplitudes in the formation reaction (cf. fourth column of Table 6), which is defined by the isospin-spin

Table 6 Evaluated scattering amplitudes and estimated relative yields of dibaryon-“X” via the cascade reaction; K − + 3 He → “X” + N (N in the forward direction) followed by “X” → Λp (N = n) or “X” → Λn (N = p) ¯ Dibaryon-“KNN ”

I (J P )

I3

“K − pp”

1 − 2 (0 )

+ 12

“K¯ 0 nn” “K¯ 0 d”

− 12 1 − 2 (1 )

“K − d” Dibaryon-“Σ ∗ N ” “Σ ∗+ n” “Σ ∗− p”

I (J P ) 1 + 2 (2 )

¯ |KNN amplitude   ¯ ¯ 1 KN √ − f(IKN =0) + 5f(I =1) 2 6  KN  ¯ ¯ 1 √ f(I =0) + f(IKN =1)

− 12

2 6 √   ¯ ¯ √3 − f KN + f KN (I =0) (I =1) 2 2 √   ¯ ¯ √3 f KN + f KN (I =1) 2 2 (I =0)

I3

¯ (I =1) ⊗ |N amplitude |KN

+ 12

+ 12 − 12

√   ¯ ¯ √5 − f KN + f KN (I =0) (I =1) 4 2 √   ¯ ¯ √5 f KN + f KN (I =1) 4 2 (I =0) −

σ (J P , I3 ) /σ (0− , + 12 ) 1a 0.06 0.68b 0.51b σ (J P , I3 ) /σ (0− , + 12 ) 0.28 εP/S rstk(Σ ∗ N/KNN ¯ ) 0.21 εP/S rstk(Σ ∗ N/KNN ¯ )

normalization channel to be I (J P ) = 12 (0 ), I3 = + 12 bK ¯ sticking probability is assumed to be same as I (J P ) = 1 (0−) 2 ¯ )/S(K − N → KN ¯ ) (see text, for more detail ) εP/S ≡ P (K − N → KN ∗ ¯ rstk(Σ ∗ N/KNN ¯ ) : the ratio of sticking probabilities between “Σ N ” and “KNN ", ¯ → Σ(1385) conversion probability in (KN ¯ N )-system including KN a

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symmetry and the configuration of the initial (3 He) and final (“X”) states. Thus, the yield ratio between the isospin partners (I3 = − 12 )/(I3 = + 12 ) is a good indicator to discriminate I (J P ) = 12 (0−) out from other candidates.

From Observation to Discovery Based upon the kaonic hydrogen X-ray measurements, an S-wave kaonic nuclear bound state is expected to exist in nuclei. For the longstanding question – whether one really exists – the E15 data provided clear experimental evidence for its existence in the two nucleon system, the “K − pp.” To make the E15 experimental observation into a discovery, further consideration is essential, because (1) this is a totally new form of nuclear matter based upon a meson as a quasi (≈ on-shell) particle, and (2) this is quite unstable as a nuclear system more than unstable nuclei. Even if the observed peak in the E15 experiment ¯ originates from a single quantum state having “KNN ” internal structure, there are four key issues to be considered. First there is the question about the decay branch. As described, the signal was detected in a non-mesonic Λp decay mode in E15. However theoretically, the major decay branch is expected to be the mesonic mode (Sekihara et al. 2013). Thus, the mesonic decay of “K − pp” must be investigated. The total quasi-free K¯ production is observed to be ≈10 mb/sr as shown in Fig. 17. On the other hand, the formation yield observed in the Λpn final state is only ≈10 μb; thus, the cross section decaying to non-mesonic channels including ΣN will be well below 100 μb. The sticking probability of a K¯ to NN is not known, but if we assume the sticking probability is similar to that of hypernuclear formation (about few %), the total “K − pp” formation cross section via in-flight K − reaction could be more than 100 μb. Thus, the total non-mesonic branch can be relatively weak, and the mesonic decay can be the major decay branch. Therefore, the mesonic decay branch study is very important, and the result from the E15 data analysis will soon be available. Second is the question about the isospin partner. The “K − pp” signal was found by measuring the Λp invariant mass; thus, the isospin of the observed state must be I = 12 , and the third component of isospin must be I3 = + 12 . If this is actually a nuclear bound state, then isospin symmetry must be fulfilled, and the symmetry requires the presence of the isospin partner with I3 = − 12 . Thus, the existence of the isospin partner “K¯ 0 nn” at ≈ same energy (mass) must be sought and confirmed experimentally. The method to search for the existence of the isospin partner is rather straight forward, because the reason why the isospin partner cannot be detected in E15 is clear. First of all, the invariant mass study to observe “K¯ 0 nn” must be conducted on the Λn-pair spectrum, instead of the Λp-pair. For the Λn-pair analysis, a neutron detector is a must, and it is much more difficult to detect neutrons compared to charged particles. Alternatively, it is possible to treat the neutron as a missing particle, as in the method actually used in the E15 data analysis. However, this

74 Kaonic Nuclei from the Experimental Viewpoint

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method requires detection of all the particles other than one missing particle. From the kinematics, “K¯ 0 nn” signal event emits a proton in the forward direction at a relatively large angle via nucleon knockout (K − p → K − p ) in the formation reaction, where the E15 spectrometer does not have sensitivity. Moreover, the relative yield of “K¯ 0 nn” is much weaker than that of the “K − pp”, if I (J P ) of the observed state is the natural candidate of 12 (0− ), as it is summarized in Table 6. Thus, a substantial improvement in an enhanced neutron detection efficiency or larger spectrometer acceptance, or both, is necessary. Third is the most difficult question, the absolute assignment of I (J P ). For the ¯ state having “KNN ” internal structure, there is a possibility for I (J P ) = 12 (1−), although it is unnatural to treat that to be a ground state, because IKN ¯ = 0 coupling is much weaker than I (J P ) = 12 (0−). The most simple indicator is the relative yield of the isospin partner, (I3 = − 12 )/(I3 = + 12 ), as it is described in the previous section. If the ratio is small as it is shown in the Table 6, the I (J P ) = 12 (1−) possibility can be excluded and “Σ ∗ N ” bound state possibility as well. It would be more preferable if one can assign the I (J P ) in an absolute manner, without assuming the formation reaction dynamics (in the cross section evaluation, it is ¯ , cascading to K¯ + (NN) → “KNN ¯ assumed to be K − N → KN ” reaction). To realize the absolute assignment, it is important to measure Λ and p spin-spin ¯ correlation of “KNN ” → Λp decay channel, which will be described in the next section. The final question can be the effect of isospin-mixing between Σ(1385) and ¯ Λ(1405) ( ≡ “KN”) in nuclei, because the mass difference between the two is relatively small. However, this mixing will be very small, in contrast to the case in hypernuclei where Σ-Λ mixing is non-negligible. Because the parity of the two ¯ is different, and “KNN ” can couple to “Σ ∗ N ” only through K¯ exchange in P -wave.

Toward an Absolute Determination of the Spin-Parity The I (J P ) is a key quantum number which characterizes the quantum state. Thus, a definitive experimental assignment is required. The J P determination is also important in terms of the form factor of the system. In the following, how to assign I (J P ) experimentally is discussed. If the observed state can be treated as a point-like particle, the J P determination is not very difficult, at least for the non-zero J case. This is because the angular distribution of the formation reaction or the decay products produce a specific angular pattern defined by J . In the present case, however, “K − pp” cannot be a point-like particle nor a hadron. Instead, it is a molecule-like hadronic cluster having finite size, and the most probable J is zero. As it is described, there is a one-to-one correspondence between the momentumtransfer qX and the neutron emission angle θn for the given mX , in the Λpn final state. Thus, the momentum-transfer distribution depends not only on the form (structure) factor but also depends on J . In the E15 analysis, a form-factor study

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¯ was conducted by postulating the signal to be from a “KNN ” I (J P ) = 12 (0− ) state, so that J = 0 was assumed. If J is not zero, then a spin polarization will be generated by the reaction dynamics, which is reflected in the θn distribution of the formation reaction and modifies the qX dependence. In other words, measurement of a physical quantity, which is sensitive only to the internal spin configuration, is needed to determine the I (J P ) of the “K − pp” signal. Because the signal is observed in the “K − pp” → Λp decay channel, the Λ-p spin-spin correlation αΛp can be measured, to determine whether the spins are parallel or anti-parallel (αΛp : +1 for parallel, and −1 for anti-parallel). The αΛp is defined by (i) the internal spin configuration of the initial J P , and (ii) the decay dynamics is defined by J P ; thus, inversely, αΛp can be utilized to deduce J P . The αΛp values and related quantum numbers are summarized in Table 7 for each I (J P ) state of the dibaryon-“X”. For example, let us consider the Λp decay of I (J P ) = 12 (0−) state (the first row of the table). The intrinsic parities of Λ and p are both positive, so the parity of the final state is given as (+1)Λ (+1)p (−1)LΛp , where LΛp is the orbital angular momentum between Λ and p. For Λp final state to be negative parity, LΛp must be one. To be J = 0 as a whole, SΛp = 1 is required, i.e., the Λp decay must be in P -wave, and the spins of Λ and p are both antiparallel to the P -wave angular momentum vector (Λ and p spins are parallel). Therefore, αΛp is +1 for I (J P ) = 1 − 2 (0 ). For the αΛp measurement in the Λp decay channel, one needs to know how spinspin correlation can be measured experimentally. Figure 23 shows the experimental basis of the measurement. Because the decay asymmetry of Λ → pπ − (αΛ ) is large in the spin direction, the most probable Λ spin orientation can be measured by the

Table 7 Λp decay asymmetry of the dibaryon-“X” and proton polarization function fpol Dibaryon-“X”

I (J P ) LΛp

“K − pp “K¯ 0 d”

1 − 2 (0 ) 1 − 2 (1 )

1 1

“Σ ∗+ n”

1 + 2 (2 )

2

SΛp 1 0 1 0 1

BR

αΛp

αΛp

1 1/3 2/3 2/5 3/5

+1 − 31 + 32 − 52 + 53

+1 + 13 + 15

a Proton polarization fpol 2

3 sin θM /(8π ) 1/(4π )b 3(1 + cos2 θM )/(16π ) 1/(4π )b  3 (3 cos2 θM − 1)2 + 10 sin2 θM cos2 θM + sin4 θM /(32π )

pol

αΛp

+0.88 +0.16 +0.04

a proton polarization referring to the direction of motion of the proton, normalized to unity, see text for more detail b spherical LΛp : orbital angular momentum between Λ and p SΛp : synthetic spin of Λ and p BR: branching ratio to antiparallel spin (SΛp = 0) or to parallel (SΛp = 1) α Λp : SΛp -averaged αΛp without considering proton direction of motion θM : polar angle from the direction of motion to the spin pol αΛp : proton polarization averaged αΛp ; a component sensitive to the pC scattering (without considering the “X”-motion in laboratory frame)

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Fig. 23 Illustration of the Λ-p spin-spin correlation (αΛp ) measurement method. The most probable Λ spin can be measured by the Λ → π − p decay axis, and that of proton can be measured by the pC nuclear quasi-elastic scattering. Nuclear quasi-elastic scattering is sensitive to transverse spin component of proton; thus the αΛp shall be analyzed in azimuthal angle φ around “K − pp” → Λp decay axis. Thus, spin-spin correlation is illustrated for a proton spin polarization-averaged pol asymmetry, αΛp , instead of direct spin-spin correlation αΛp . For simplicity, the motion of “K − pp” (i.e., qX ) in a laboratory frame and a resulting proton spin depolarization effect are not considered

decay axis. On the other hand, the proton scatters asymmetrically with the nucleus with respect to the spin direction (αpC : for carbon polarimeter), proportionally to the proton-spin-component orthogonal to the direction-of-motion. This means that the proton spin azimuthal orientation around the direction-of-motion can be observed. In this manner, the proton spin can only be measured in the azimuthal angle around the direction-of-motion (the quantum-axis is a hidden parameter in the experiment). pol pol Thus, let us evaluate αΛp around the direction-of-motion of proton. The αΛp , a proton spin-polarization-averaged αΛp , can be measured, which is given by an integral considering the proton-spin component orthogonal to the direction-ofmotion (i.e., a component sensitive to the pC scattering) as;

pol

αΛp =



⎛ sin θM ⎝



⎞ αΛp (SΛp ) fpol (θM ; SΛp )⎠ dΩM ,

(22)

SΛp

where fpol is the proton polarization function referring to the direction-of-motion of the proton and ΩM is the solid angle of the quantum-axis in referring to the direction-of-motion. In the equation, the sin θM term represents for the proportionality of the asymmetric scattering to the proton-spin-component orthogonal to the direction-of-motion. In the case of I (J P ) = 12 (0−) state, the relationship between the proton spin and the direction of motion can be given as follows. When we take Λp synthetic-

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spin direction as a quantum-axis, the angular momentum J = 0 is realized as J = |L|1,−1 ⊗ S|1,+1 Λp = |Y1−1 ⊗ [↑, ↑]/2 Λp , where L|1,−1 is the angular momentum between Λ and p, and S|1,+1 is the spin configuration of the system. Thus, the Λp-decay-axis to be observed distributes in |Y1−1 |2 = 3 sin2 θM /(8π ), where θM is the dibaryon’s decay-axis, so the direction of motion of the proton distributes around the ecliptic plane. This means that the proton spin is roughly orthogonal to the direction of motion for I (J P ) = 12 (0−) state. For the I (J P ) = 12 (1−) state, the evaluation becomes slightly complicated. In the P -wave decay (LΛp = 1), both SΛp = 0 and 1 are allowed to be J = 0. In the decay, K¯ 0 must be absorbed by neutron in P -wave at the spin flipping and non-flipping probability ratio of 2 : 1 according to the Clebsch-Gordan coefficient (cf., Table 5 I (J P ) = 12 (1−)). The spin non-flipping term’s synthetic spin should be equal to the initial state, so as a whole, state become J = |L|1,0 ⊗ S|1,+1 Λp . On the other hand, the spin flipping term results in a half-and-half mixture of SΛp = |1, 0 and |0, 0 , i.e., J = |L|1,0 ⊗ S|1,0 Λp + |L|1,0 ⊗ S|0,0 Λp . Thus, the intensity ratio between SΛp = 0 and 1 is 1 : 2, respectively. For the |L|1,0 ⊗ S|0,0 Λp term, the spin does not have any specific orientation by definition. For the SΛp = 1 term, the synthetic-spin and the decayaxis (by which proton direction-of-motion is defined) of |L|1,0 ⊗ S|1,+1 Λp term is clearly aligned. However, that of |L|1,0 ⊗ S|1,0 Λp term is not clear. They are both on/around the ecliptic plane, but the informations orthogonal to the quantum-axis has no specific meaning in the quantum-physics. Thus, let’s take synthetic Λp spin as a quantum-axis, instead. Then the relationship between the synthetic-spin and direction-of-motion can be simply given as, J = |L|1,0 ⊗ S|1,+1 Λp = |Y10 ⊗ [↑, ↑]/2 (both are aligned to quantum-axis)

and J = |L|1,−1 ⊗ S|1,+1 Λp = |Y1−1 ⊗ [↑, ↑]/2 (spin-orthogonal to the most probable direction of the proton direction-of-motion) at the equal probability, to form initial J = 1 by Λp. Thus, the angular distribution of the two can be given as it is summarized in Table 7. In the case of I (J P ) = 12 (2+) states (“Σ ∗ N "), the evaluation is similar to I (J P ) = 12 (1−) case. However, the decay must be in D-wave (LΛp = 2) in this case, because the initial spin of the state is J = 2. Then, the D-wave shall be applied to Σ ∗ (SΣ ∗ = 32 ) giving spin flip/non-flip Λ (SΛ = 12 ) in the final state. In both I (J P ) = 12 (1−) and 12 (2+), proton spin directions of the SΛp = 1 component are rather parallel (or antiparallel) to the direction of motion, in contrast to the I (J P ) = 12 (0−) case (roughly orthogonal), as shown in the table. Thus, the pol

resulting αΛp strength for these states is much smaller than the actual αΛp value. It should be noted that the observable asymmetry is further smeared by the following two asymmetry parameters. One is the Λ weak-decay asymmetry parameter αΛ (≈0.72 (Ireland et al. 2019)), and the other is asymmetric pC scattering parameter αpC (depends on the proton momentum, and it is ≈0.3 around “X”→ Λp decay). The Larmor precession of Λ and p spins in the magnetic field of the spectrometer system, before the decay of Λ and pC scattering, affects the measurement, although it can be corrected by event-by-event basis. In the

74 Kaonic Nuclei from the Experimental Viewpoint

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measurement, dibaryon-“X” is formed having a finite transfer momentum. The boost of a system weakly distorts the initial Λ and p spin polarization function in referring to the dibaryon-“X” decay axis. These effects should be properly treated to reach a definitive result.

Perspective on the Systematic Kaonic Nuclear Bound State Study The E15 result is the first definitive evidence of the existence of a dibaryon state having s-quark (strangeness −1) and located just below the kaon binding threshold (M(K − pp) = 2.37 GeV/c2 ). This indicates the K¯ meson can form a quantum state in nuclei [as a quasi (≈ on-shell) particle] for a limited period of time before being absorbed by nearby nucleons. This state appears to be a natural extension of the Λ(1405) resonance to be an molecule-like hadronic cluster, i.e., a q − ud nucleon is coupled with a s q¯ meson forming an “s q-qud” ¯ system that becomes an isospin-zero √ ¯ Λ(1405) resonance composed by “KN” [= (−K − p+ K¯ 0 n)/ 2] hadron composite state [molecule-like hadronic cluster]. This is supported by kaonic hydrogen X-ray data. However, it is astonishing that a meson still behaves as a particle without being absorbed in nuclei. Furthermore, the data indicated a significant mass reduction of the system [in-medium mass modification]. The data also suggest that the system may be spatially very compact [a hadron property at high density]. The current standard scenario of the origin of hadron mass is known to be spontaneous chiral-symmetry breaking, originated in Nambu’s theorem (Nambu and Jona-Lasinio 1961). It is believed that the positive-parity (scalar) qq -pair ¯ is spontaneously condensed in vacuum at some point, so as to stabilize the “space” in terms of energy, when the universe is cooling down (analogous to the Higgs condensation mechanism). Like the Higgs condensation gives mass to the elementary particles, the qq ¯ condensation may give mass to the hadron (Higashijima 1984). The qq ¯ condensation is considered to be a function of temperature (energy density) and matter density (Fukushima and Hatsuda 2010). At the high density in low-temperature region, a new phase of QCD-vacuum – color super-conductive phase – is expected, and hadron-quark crossover scenario bridging to hadronic phase to quark phase is discussed (Masuda et al. 2013) in relation with [massive neutron star]. Thus, the physics beyond the normal nuclear density is expected to be rich in physics. The kaonic nuclear bound state study may lead us to answer the fundamental question, How hadron mass is generated in QCD-vacuum? (Klimt et al. 1990; Nagahiro et al. 2013), What is the structure of the vacuum? (Nambu and Jona-Lasinio 1961; Hatsuda and Kunihiro 1994), and Physics beyond normal nuclear density (Alford et al. 2008). The physics at the density region beyond the normal nucleon density has never been studied at the quantum equilibrium. Thus the study might also give vital information on short-range NN repulsion, which may lead us to answer the key question, such as Why does a two solar mass neutron star do not collapse to the black hole by own gravity? What this experimental data shows may not necessarily fit within the current physical picture of hadrons.

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¯ By a systematic study of kaonic (K-mesonic) nuclear states in light nuclei (including further study of the Λ(1405)), one may disentangle the relationship ¯ in nuclei, short-range NN repulsion, and inbetween the binding energy of Ks medium mass modification of hadrons. One may proceed further to study physics at nuclear densities much beyond that of normal nuclear matter under quantum equilibrium conditions. That will lead to a better understanding of the fundamental ¯ and NN interactions) and an understanding of nature of strong interactions (KN high-density baryonic matter. To realize this, it would be important to approach medium-heavy nuclear systems in a step-by-step manner, so as not to lose the way along the path. Acknowledgments I appreciate intensive discussions that were deeply related to this paper with Dr. Takumi Yamaga. I would like to thank Dr. Tadashi Hashimoto for various suggestions and also Dr. Rie Murayama, Dr. Kenta Itahashi, and Dr. Fuminori Sakuma. I’m grateful to Dr. Benjamin F. Gibson, Dr. Hideyuki Sakai, Professor Emiko Hiyama, Professor Tetsuo Hyodo, and Professor Hiroyuki Noumi for valuable comments. Finally, I would like to thank Ms. Harumi Iwasaki for continuous support in various aspects of this work.

References R. Aaij et al., Phys. Rev. Lett. 112, 222002 (2014) R. Aaji et al., Phys. Rev. Lett. 115, 072001 (2015) G. Agakishiev et al., Phys. Lett. B 742, 242–248 (2015) M. Agnello et al., Phys. Rev. Lett. 94, 212303 (2005) S. Ajimura et al., Phys. Lett. B 789, 620–625 (2019) Y. Akaishi, T. Yamazaki, Phys. Rev. C 65, 044005 (2002) M.G. Alford et al., Rev. Mod. Phys. 80, 1455 (2008) M. Bazzi et al., Phys. Lett. B 704, 113 (2011) G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) G. Bendiscioli et al., Nucl. Phys. A 789, 222 (2007) G. Bendiscioli et al., Eur. Phys. J. A 40, 11 (2009) P.M. Bird et al., Nucl. Phys. A 404, 482 (1983) B. Conforto et al., Nucl. Phys. B 105, 189 (1976) S.K. Choi et al., Phys. Rev. Lett. 91, 262001 (2003) R.H. Dalitz, S.F. Tuan, Ann. Phys. 3, 307 (1960) C.J.S. Damerell et al., Nucl. Phys. B 129, 397 (1977) J.D. Davies et al., Phys. Lett. 83B, 55 (1979) R. Del Grande et al., Eur. Phys. J. C 79, 190 (2019) R. Del Grande et al., Phys. Scripta 95(8), 084012 (2020) S. Deser et al., Phys. Rev. 96, 774 (1954) E. Epple, L. Fabbietti, Phys. Rev. C 92, 044002 (2015) L. Fabbietti et al., Nucl. Phys. A 914, 60 (2013) K. Fukushima, T. Hatsuda, Rep. Prog. Phys. 74, 014001 (2010) A. Gal et al., Rev. Mod. Phys. 88, 035004 (2016) H. Garcilazo, A. Gal, Phys. Rev. C 81, 055205 (2010) H. Garcilazo, A. Gal, Nucl. Phys. A 897, 167–178 (2013) B.F. Gibson et al., Phys. Rev. C 6, (1972) 741. J.M.M. Hall et al., Phys. Rev. Lett. 114, 132002 (2015) T. Hashimoto et al., Prog. Theor. Exp. Phys. 2015, 061D01 (2015)

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T. Hatsuda, T. Kunihiro, Phys. Rep. 247(5–6), 221 (1994) K. Higashijima, Phys. Rev. D 29, 1228–1232 (1984) T. Hyodo, M. Niiyama, Prog. Part. Nucl. Phys. 120, 103868 (2021) T. Hyodo, W. Weise, arXiv: nucl-th 2202.06181 (2022) Y. Ichikawa, Ph.D dissertation submitted to Kyoto University (Available via SCPHYS, 2015). https://www-nh.scphys.kyoto-u.ac.jp/articles/thesis/dron/d-ichikawa.pdf Y. Ichikawa et al., Prog. Theor. Exp. Phys. 2015, 021D01 (2015) Y. Ichikawa et al., Prog. Theor. Exp. Phys. 2020, 123D01 (2020) D.G. Ireland et al., Phys. Rev. Lett. 123, 182301 (2019) T.M. Ito et al., Phys. Rev. C 58, 2366–2382 (1998) M. Iwasaki et al., Phys. Rev. Lett. 78, 3067 (1997) M. Izycki et al., Z. Phys. A 297, 11 (1980) M. Jones et al., Nucl. Phys. B 90, 349 (1975) P. Kienle et al., Eur. Phys. J. A 48, 183 (2012) T. Kishimoto, Phys. Rev. Lett. 83, 4701 (1999) T. Kishimoto et al., Prog. Theor. Phys. 118, 181 (2007) S. Klimt et al., Nucl. Phys. A 516, 429–468 (1990) M. Mai, Eur. Phys. J. ST 230(6), 1593–1607 (2021) A. Martin, Nucl. Phys. B 179, 33 (1981) K. Masuda et al., Prog. Theor. Exp. Phys. 2013(7), 073D01 (2013) U.G. Meißner, Symmetry 12, 981 (2020) K. Miyahara, T. Hyodo, Phys. Rev. C 93(1), 015201 (2016) H. Nagahiro et al., Phys. Rev. C 87, 045201 (2013) Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); Y. Nambu, G. Jona-Lasinio, Phys. Rev. 124, 246 (1961) Y. Nogami, Phys. Lett. 7, 288 (1963) S. Ohnishi et al., Phys. Rev. C 95, 065202 (2017) S. Okubo, Phys. Lett. 5, 165–168 (1963); G. Zweig, CERN Report No. 8419/TH412, preprint 8182/TH.401/412, 1964 (1963); J. Iizuka, Prog. Theor. Phys. Suppl. 37–38, 21–34 (1963) Y. Pan et al., Phys. Rev. D 2, 449 (1970) S. Piano et al., Nucl. Phys. A 835, 35 (2010) J. Révai, N.V. Shevchenko, Phys. Rev. C 90, 034004 (2014) Y. Sada et al., Prog. Theor. Exp. Phys. 2016, 051D01 (2016) F. Sakuma et al., Letter of Intent to J-PARC PAC (Available via J-PARC, 2009). http://j-parc.jp/ researcher/Hadron/en/pac_0907/pdf/LOI_Sakuma.pdf M. Sato et al., Phys. Lett. B 659, 107 (2008) T. Sekihara et al., Nucl. Phys. A 914, 338 (2013) N.V. Shevchenko, Nucl. Phys. A 890–891, 50–61 (2012) N.V. Shevchenko, J. Révai, Phys. Rev. C 90(3), 034003 (2014) T. Suzuki et al., Phys. Rev. C 76, 068202 (2007) A.O. Tokiyasu et al., Phys. Lett. B 728, 616–621 (2014) O. Vázquez Doce et al., Phys. Lett. B 758, 134–139 (2016) Y. Wu et al., Phys. Rev. Lett. 64, 1875 (1990); Y. Wu et al., Phys. Rev. Lett. 66, 242 (1991) T. Yamaga et al., Phys. Rev. C 102, 044002 (2020) T. Yamazaki et al., Phys. Rev. Lett. 104, 132502 (2010) H. Yim et al., Phys. Lett. B 688, 43 (2010)

Theory of Kaon-Nuclear Systems

75

Tetsuo Hyodo and Wolfram Weise

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Λ(1405) Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K− p Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¯ Interaction and Few-Body Kaonic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KN Kaonic Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-Body Physics and Kaons in Baryonic Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2750 2751 2761 2764 2769 2773 2778 2779

Abstract

The strong interaction between an antikaon and a nucleon is at the origin of various interesting phenomena in kaon-nuclear systems. In particular, the interaction in the isospin I = 0 channel is sufficiently attractive to generate a ¯ threshold. Based on this quasibound state, the Λ(1405) resonance, below the KN ¯ picture, it may be expected that the KN interaction also generates quasibound states in kaon-nuclear systems, sometimes called kaonic nuclei. At the same ¯ quasibound picture of the Λ(1405) is also related to the discussion time, the KN of hadronic molecules in hadron spectroscopy. Here an overview is presented on the theoretical studies developed for kaon-nucleon and kaon-nuclear systems. We start from the modern understanding of the Λ(1405) resonance. We then discuss

T. Hyodo () Tokyo Metropolitan University, Hachioji, Japan e-mail: [email protected] W. Weise Physics Department, Technical University of Munich, Garching, Germany e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_38

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¯ interaction and various aspects of few-body kaonic nuclei. Heavier kaonthe KN nuclear systems are examined from the viewpoint of nuclear many-body physics, with focus on the properties of antikaons in nuclear matter. Related topics, such as the K − p momentum correlation functions in high-energy collisions and the studies of kaonic atoms, are also discussed.

Introduction

Fig. 1 Masses of the lowest-lying hadrons in the strangeness S = 0 and S = ±1 sectors

mass

The kaons (K = K + , K 0 ) and antikaons (K¯ = K − , K¯ 0 ) are the lightest pseudoscalar mesons with strangeness (Zyla et al. 2020). Their nature is closely related to the symmetry breaking pattern in low-energy QCD. In the SU (3) octet of the light pseudoscalar mesons, kaons figure as flavor partners of the pions, the Nambu-Goldstone (NG) bosons associated with the spontaneous breaking of chiral symmetry in QCD. While the smallness of the pion mass has its origin in the almost vanishing up- and down-quark masses (mu  2 MeV, md  5 MeV at a renormalization scale of 2 GeV), kaons are relatively massive because of the more sizable strange quark mass (ms  0.1 GeV) reflecting the pronounced explicit chiral symmetry breaking in the strangeness sector. These features are illustrated in Fig. 1 where the masses of the lowest-lying hadrons are plotted: the kaons are not as light as the pions, but at the same time not as heavy as the ordinary hadrons other than the NG bosons. This intermediate nature of the kaons leads to various interesting phenomena of nonperturbative QCD at low energies. One such aspect is the possible formation of bound states of nuclei with antikaons ¯ Because the K¯ has isospin I = 1/2, there are two independent components (K). ¯ interaction, I = 0 and I = 1, as in the nuclear force. Moreover, of the KN ¯ interactions share a common feature. The NN interaction in the the NN and KN I = 0 (3 S1 ) channel is sufficiently attractive to form the deuteron bound state, and the I = 1 (1 S0 ) channel is also known to have an attractive scattering length. In a similar way, despite the otherwise obviously different physics of the NN and

Λ N ρ

¯ K*, K* K, K¯

π S=0

S=±1

75 Theory of Kaon-Nuclear Systems Table 1 Qualitative features ¯ s-wave of the N N and KN interactions

2751

NN ¯ KN

I =0 d (bound state) Λ(1405) (quasibound state)

I =1 Attractive Attractive

¯ systems, the KN ¯ interaction in the I = 0 channel also has a quasibound KN state below the threshold, the Λ(1405), and the I = 1 interaction is likewise considered to be attractive (see Table 1). A qualitative difference, though, is that ¯ system couples strongly to the lower-energy the deuteron is stable, whereas the KN π Σ and π Λ channels. Consequently, the Λ(1405) as a quasibound state imbedded in the π Σ continuum has a short lifetime, i.e., a large decay width, Γ  50 MeV. Nonetheless the question was raised whether a self-bound system can be obtained by replacing one of the nucleons in nuclei by an antikaon. In fact, shortly after the prediction (Dalitz and Tuan 1959, 1960) and the experimental observation of the Λ(1405) (Alston et al. 1961), possible bound kaon-nuclear systems have been discussed (Nogami 1963). Recent interest in the kaon-nuclear systems was triggered by the work of Akaishi and Yamazaki (2002) who proposed the existence of possible deeply bound kaonic nuclei with a narrow width. Since then, many theoretical and experimental investigations have been devoted to kaon-nuclear systems with these issues in mind. This chapter introduces the theoretical framework for kaon-nuclear systems, starting with the Λ(1405) as a prototype example. We then describe few-body and many-body kaon-nuclear systems, including a discussion of kaon properties in nuclear matter. We also refer here to several review articles covering related topics (Ramos et al. 2001; Friedman and Gal 2007; Hyodo and Jido 2012; Gal et al. 2016; Tolos and Fabbietti 2020; Meißner 2020; Mai 2021; Hyodo and Niiyama 2021).

The Λ(1405) Resonance The Λ(1405) resonance is nominally located just 27 MeV below the K − p threshold (Mp + mK − = 1432 MeV). Its properties are therefore closely linked to the low¯ interaction. The Λ(1405) is the lowest-lying excited baryon in the energy KN strangeness S = −1 and isospin I = 0 sector, rated as a four-star resonance in PDG (Zyla et al. 2020). Studies of the Λ(1405) go back to 1959 when Dalitz and ¯ threshold (Dalitz and Tuan Tuan theoretically predicted a resonance below the KN ¯ 1959, 1960). Based on the coupled-channels unitarity of the KN-π Σ system and ¯ the empirical KN scattering length, they deduced the existence of a resonance in the π Σ scattering amplitude. Shortly after the prediction, experimental evidence for the Λ(1405) was reported in the π Σ invariant mass distribution of the K − p → π π π Σ reaction at 1.15 GeV (Alston et al. 1961). Since then a large amount of theoretical and experimental investigations have been devoted to clarify the nature of the

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Λ(1405) (see Hyodo and Jido 2012; Meißner 2020; Mai 2021; Hyodo and Niiyama 2021). In particular, recent developments have firmly established the following basic properties of the Λ(1405): • Eigenenergy: The generalized eigenenergy of an unstable state is expressed by the position of the resonance pole of the scattering amplitude in the complex energy plane (the real and imaginary parts, respectively, correspond to the mass and half width of the resonance). The pole positions of the Λ(1405) have been pinned down (Ikeda et al. 2011, 2012; Guo and Oller 2013; Mai and Meißner 2015), thanks to accurate constraints from the precise measurement of kaonic hydrogen by the SIDDHARTA collaboration (Bazzi et al. 2011, 2012). • Spin and parity: Experimental determination of the spin and parity has been carried out by the CLAS collaboration in the photoproduction γ p → K + Λ(1405) (Moriya et al. 2014). The result confirms the expected quantum numbers of J P = 1/2− . Current interest on the Λ(1405) is focused on its internal structure. In conventional constituent quark models, the negative parity excited baryons are described by the internal excitation of three quarks in the confining potential. In a systematic study of negative parity baryons using such a quark model (Isgur and Karl 1978), it is shown that the mass of the Λ(1405) deviates significantly (by ∼100 MeV) from the standard quark model prediction, in contrast to other negative parity states which are well described by the quark model. This suggests that the Λ(1405) has a more exotic internal structure, unlike that of a simple three-quark state. In particular, the static quark model picture lacks the dynamical aspect of the excited baryons which couple to the meson-baryon continuum channels. An alternative ¯ molecule induced by the attractive picture of the Λ(1405), as a weakly bound KN ¯ KN interaction, is in line with Dalitz et al. (1967). This picture is also supported by a recent analysis of the compositeness (Kamiya and Hyodo 2016, 2017) and ¯ interaction. In this sense, emphasizes once again the attractive nature of the KN the internal structure of the Λ(1405) is important, not only in hadron spectroscopy to search for nonconventional configurations of hadrons but also for the deeper ¯ interaction and its implications for kaon-nuclear physics. understanding of the KN ¯ threshold, the Λ(1405) As a resonance in π Σ scattering located close to the KN requires a dynamical treatment in terms of coupled-channel meson-baryon scattering. The pseudoscalar π and K¯ are combined with the respective 1/2+ baryons in s wave to form the J P = 1/2− baryonic system with strangeness S = −1. The lowest-lying pseudoscalar mesons are regarded as Nambu-Goldstone bosons associated with the spontaneous breaking of Nf = 3 chiral symmetry. Therefore their low-energy s-wave interactions with the octet baryons are constrained by the Weinberg-Tomozawa theorem (Weinberg 1966; Tomozawa 1966). Based on this observation, a series of works (Kaiser et al. 1995; Oset and Ramos 1998; Oller and Meißner 2001) developed a successful theoretical framework, called chiral SU (3) dynamics, in which the coupled-channel meson-baryon scattering amplitude satisfying the unitarity condition is constructed in a systematic manner.

75 Theory of Kaon-Nuclear Systems

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+

= T

V

V

T

Fig. 2 Feynman diagrams for the scattering equation. Solid (dashed) lines represent the baryon (meson) propagator. Open (shaded) blobs correspond to the interaction kernel V (scattering amplitude T ), and the meson-baryon intermediate loop stands for the loop function G

Denoting the meson-baryon channels by the indices i, j , the scattering amplitude takes a matrix form, Tij , in channel space. In the s-wave ( = 0) channels, Tij is a function of the total meson-baryon energy W . In chiral SU (3) dynamics, Tij is then given by the solution of a coupled-channel Lippmann-Schwinger scattering equation Tij = Vij + Vik Gk Tkj ,

(1)

where the summation over the repeated indices is implicit. In Fig. 2, Feynman diagram representation of the scattering equation is shown. The kernel Vij represents the meson-baryon interaction, constructed so that it systematically satisfies the chiral symmetry constraints (Ecker 1995; Bernard et al. 1995; Pich 1995; Bernard 2008; Scherer and Schindler 2012). Furthermore Gk is the loop function representing the propagation of the meson and baryon in the intermediate state with index k. Substituting Tkj on the right-hand side of Eq. (1) iteratively, one finds Tij = Vij + Vik Gk Vkj + Vik Gk Vkl Gl Vlj + · · · ,

(2)

where the multiple scatterings in coupled channels are iterated to all orders in Tij . This guarantees the unitarity of the scattering amplitude. The same strategy is applied in the study of the nuclear force within the framework of chiral effective field theory (Epelbaum et al. 2009; Machleidt and Entem 2011). In the following, we present the properties of Vij and Gi . Chiral perturbation theory (Ecker 1995; Bernard et al. 1995; Pich 1995; Bernard 2008; Scherer and Schindler 2012) classifies the interaction kernel V by the chiral order in terms of a low-momentum scale p. The chiral order, denoted by O(pn ), refers to an expansion in powers of p/Λχ where Λχ ∼1 GeV is the characteristic scale of the spontaneous breaking of chiral symmetry. The terms with smaller chiral orders dominate in the low-energy region, p  Λχ ; thus low-energy phenomena can be described by a finite number of terms. In meson-baryon scattering, the leading order (LO) terms start at O(p) and involve couplings of the SU (3) pseudoscalar meson and J P = 1/2+ baryon octets through their vector and axial vector currents. For example, the dominant WeinbergTomozawa (WT) term in the K − proton elastic scattering channel derives from the following piece of the chiral interaction Hamiltonian:

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δHWT (K − p → K − p) = −

i 2f 2



d 3 x ψ¯ p (x)γ μ ψp (x) K + (x)

↔ − K (x), ∂μ

(3)

where ψp and K − are the proton and antikaon fields, respectively. Note that the WT term is completely determined by the pseudoscalar meson decay constant, f  0.1 GeV. The interaction vanishes in the limit of zero meson four-momentum, q μ = (E, q), a feature characteristic of Nambu-Goldstone bosons. The s-wave interaction is governed by the time component of the vector current. In the limit of a pointlike, static nucleon, the leading WT term is therefore a contact interaction proportional to E/f 2 , with the meson energy E = m2 + q2 . At threshold the WT interaction kernels in K − p and K − n channels have the form VWT (K − p) = 2 VWT (K − n) ∝ −

mK . f2

(4)

For K + p and K + n scattering, the corresponding terms are repulsive, of equal magnitude as in Eq. (4) but with opposite sign. The important point to note here is ¯ interaction close to threshold is much stronger than the s-wave that the leading KN π N interaction. It scales with the kaon mass mK (rather than the small pion mass mπ ) and reflects the stronger explicit chiral symmetry breaking characteristic of the strange quark. The meson-baryon interaction up to and including the next-to-leading order (NLO) terms of O(p2 ) is schematically written as V = VWT + VBorn + VNLO + · · · .       O (p1 )

(5)

O (p2 )

The corresponding Feynman diagrams are shown in Fig. 3. The ellipsis stands for the O(p3 ) and higher-order contributions. In each interaction term, there are lowenergy constants (LECs), whose strength cannot be determined from symmetry arguments alone. The precision of the calculation can be increased by introducing higher-order terms, but this requires a sufficient amount of experimental data to determine all the LECs. The dominant contribution in the leading order O(p1 ) terms is the WeinbergTomozawa term VWT (Fig. 3, first term in the right-hand side). In the chiral dynamics framework, the pertinent low-energy theorems are automatically built in, and the low-energy limit of V reduces to VWT satisfying the Weinberg-Tomozawa theorem. As mentioned, VWT has no LEC apart from the pseudoscalar meson decay constant. The sign and strength of the interaction are determined by flavor SU (3) symmetry. The Born terms VBorn are constructed by the s- and u-channel baryon exchange diagrams (Fig. 3, second and third terms). The strength parameters of the mesonbaryon-baryon three-point vertices in these diagrams are given by the axial vector coupling constants of the participating baryons, satisfying Goldberger-Treiman

75 Theory of Kaon-Nuclear Systems

=

+

2755

+

+

+ ···

V

Fig. 3 Feynman diagrams for the interaction kernel V . Black dots represent the O (p) vertices, and black square stands for the O (p 2 ) vertex. The first term in the right-hand side is VWT , the second and third terms are VBorn , the fourth term is VNLO , and ellipsis shows the O (p 3 ) contributions

relations (Goldberger and Treiman 1958). Their values reflect the internal structure of the baryons. While VBorn is counted as O(p), the main contribution of this term is in p-wave, and the s-wave projected part appears at higher order than VWT in the nonrelativistic expansion (Weinberg 2013). Thus, in the low-energy limit, VBorn is much smaller than VWT , and the meson-baryon interaction is model-independently given by the leading Weinberg-Tomozawa term, thanks to chiral symmetry. When confronted with high-precision data, such as the measurement of kaonic hydrogen by the SIDDHARTA collaboration (Bazzi et al. 2011, 2012), the precision of the theoretical framework is increased by introducing the higher-order terms. This is achieved by including the NLO terms VNLO at O(p2 ) (Ikeda et al. 2011, 2012; Guo and Oller 2013; Mai and Meißner 2015) (fourth terms in Fig. 3). The scattering equation Eq. (1) is in general an integral equation, with the offshell structure of the interaction kernel Vij being integrated over. In practical applications the equation is usually reduced into algebraic form by using the on-shell factorization. This formulation preserves the unitarity condition. It is also justified by the N/D method (see Oset and Ramos 1998; Oller and Meißner 2001; Hyodo and Jido 2012; Mai 2021). Because the leading order VWT is a contact interaction, the momentum integration in the loop function involves an ultraviolet divergence. This divergence is usually tamed by dimensional regularization, and the finite part of the loop function Gi is specified by a subtraction constant which plays the role of an ultraviolet cutoff. The one-loop diagram is counted as O(p3 ), and therefore the renormalization procedure of the meson-baryon scattering is completed at O(p3 ). In the unitarized amplitude in Eq. (1) with the O(p2 ) interaction kernel (V = VWT + VBorn + VNLO ), the subtraction constants need to be determined by fitting experimental data. As mentioned, VWT is completely determined by chiral symmetry. The Born terms contain the axial vector coupling constants of the octet baryons, conventionally denoted by D and F , which are determined by the semi-leptonic decays of hyperons. Thus, when the O(p1 ) interactions VWT and VBorn are used, the only free parameters are the subtraction constants. In the strangeness S = −1 coupled-channel scattering matrix, there are six isospin states ¯ (KN, π Λ, π Σ, ηΛ, ηΣ, KΞ ), and six subtraction constants can be used to fit the experimental data. In VNLO , there are seven LECs, which should be determined also by the experimental data in the O(p2 ) calculations, in addition to the subtraction constants. In the current situation, the LECs up to O(p2 ) terms can be determined by experimental data, but in order to perform O(p3 ) calculations, the data base is not (yet) sufficient.

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Best fit with NLO ETW

1

2

0.5

1.5

0

1

-0.5

0.5

Best fit with NLO ETW

Im F f (K − p → K − p) [fm]

Re F(K − p → K − p) [fm]

1.5

0

-1 1340

1360

1380

√

1400

1420

1440

1340

1360

s [MeV]

1380

√

1400

1420

1440

s [MeV]

Fig. 4√Real and imaginary parts of the K − p → K − p forward scattering amplitude, F = T /8π s, from chiral SU (3) dynamics at NLO (solid curve) and constrained by the SIDDHARTA kaonic hydrogen measurements. The dot at threshold marks the corresponding deduced scattering length. Uncertainties are represented by the gray band. Also shown (dashed curve) for comparison ¯ − π Σ − π Λ) is an effective Weinberg-Tomozawa amplitude in a reduced three-channel (KN scheme using values fπ = 92.4 MeV, fK = 109 MeV of the meson decay constants and three subtraction constants as fit parameters. (Adapted from Ikeda et al. (2012))

Figure 4 shows the result of a chiral SU (3) dynamics calculation of the K − p forward scattering amplitude (Ikeda et al. 2012), featuring the prominent emergence ¯ quasibound state imbedded in the π Σ continuum. This of the Λ(1405) as a KN amplitude has served as input to many investigations of K¯ interactions in more complex systems. Poles of the scattering amplitude Tij in the complex energy plane represent the generalized eigenenergies of unstable resonances (Hyodo and Niiyama 2021). Usually there is a single pole for a resonance state, whose position z in the complex energy plane is related to the mass MR and the width ΓR as MR = Re z,

ΓR = −2 Im z.

(6)

It was first pointed out in Oller and Meißner (2001) that there are actually two poles with I = 0 and S = −1 in the Λ(1405) energy region. This fact has been confirmed in many subsequent studies (Meißner 2020; Mai 2021; Hyodo and Niiyama 2021), including the cloudy bag model (Fink et al. 1990), meson-exchange model (Haidenbauer et al. 2011), dynamical coupled-channel model (Kamano et al. 2014, 2015), and Hamiltonian effective field theory (Liu et al. 2017). One pole ¯ threshold with a narrow decay width. A second pole is is located near the KN found around the π Σ threshold with relatively broad width. Namely, the nominal “Λ(1405)” is not a single resonance. It is realized as a superposition of two eigenstates in the same energy region. In fact, in the latest version of PDG (Zyla et al. 2020), the lower energy pole is called Λ(1380) as a new two-star resonance, and Λ(1405) is reserved for the higher energy pole around 1420 MeV. Here we note that the two resonance poles in the complex energy plane do not necessarily exhibit

75 Theory of Kaon-Nuclear Systems

2757

Fig. 5 The absolute value of the scattering amplitude in the complex energy z plane. (Adapted from Hyodo and Jido 2012)

two peaks in the projected spectrum on the real axis. The shape of the spectrum depends on the relative phase of the residues of the poles. In the Λ(1405) case, there is only one broad and asymmetric peak structure on the real axis (see Fig. 5). In other words, the Λ(1405) spectrum is produced by the contributions from both poles. The origin of the two poles can be understood by the properties of the WeinbergTomozawa interaction (Jido et al. 2003; Hyodo and Weise 2008). The meson-baryon channel can be expressed either in the physical basis (such as K − p, K¯ 0 n, . . . ), ¯ ¯ in the isospin basis (KN(I = 0), KN(I = 1), . . . ), or in the SU (3) basis (1(I = 0), 8(I = 0), . . . ). The basis transformation can be done through the SU (2) Clebsch-Gordan coefficients and the SU (3) isoscalar factors. Representing the interaction in the isospin basis and in the SU (3) basis is useful in order to study the symmetry underlying the interaction. The coupling strength of VWT is SU (3) symmetric; hence VWT becomes a diagonal matrix in the SU (3) basis where the mixing of different representations is absent. Therefore, in the SU (3) symmetric limit of hadron masses, the coupled-channel equation reduces to a set of independent single-channel problems (Hyodo et al. 2006, 2007). There are four channels with I = 0 (1, 8, 8 , and 27). Three of those are shown to be sufficiently attractive to generate bound states in the SU (3) limit (Jido et al. 2003). It is found that one of the poles evolves into the Λ(1670) resonance. The other two poles move toward the Λ(1405) energy region, along with the SU (3) breaking toward physical masses. In the same way, there are four channels in the isospin basis (π Σ, ¯ ¯ , and KΞ KN, ηΛ, and KΞ ), and VWT is attractive in the diagonal π Σ, KN ¯ channels. In the absence of off-diagonal couplings, the KN attraction generates a ¯ threshold, and the π Σ attraction creates a resonance bound state below the KN above the π Σ threshold (Hyodo and Weise 2008). By introducing the channel

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T. Hyodo and W. Weise

¯ couplings gradually, these eigenstates evolve into the two poles between the KN and π Σ thresholds. In this way, the origin of the two poles can be understood by the two attractive components in the Weinberg-Tomozawa term, which are modelindependently constrained by the low-energy theorem of chiral symmetry. Now we explain how the pole positions in PDG are determined, based on Ikeda et al. (2011, 2012), Guo and Oller (2013), and Mai and Meißner (2015). In these works of NLO chiral SU (3) dynamics, a systematic uncertainty analysis was performed with inclusion of the SIDDHARTA kaonic hydrogen measurement. The experimental data base for this analysis can be classified as follows (The K − p correlation function data was not available when Ikeda et al. (2011, 2012), Guo and Oller (2013), Mai and Meißner (2015) were published, but the data in Acharya et al. (2020a) is shown to be consistent with the scattering amplitude of Ikeda et al. (2011, 2012), as we describe below (Kamiya et al. 2020).): (i) (ii) (iii) (iv)

Elastic and inelastic total cross sections of K − p scattering, Threshold branching ratios (Tovee et al. 1971; Nowak et al. 1978), Energy-level shift and width of the kaonic hydrogen (Bazzi et al. 2011, 2012), π Σ invariant mass distributions in various reactions (Ahn 2003; Niiyama et al. 2008; Prakhov et al. 2004; Zychor et al. 2008; Agakishiev et al. 2013; Moriya et al. 2013; Lu et al. 2013).

√ The total cross section σij from channel j to i at the energy W = s is derived from Tij as (here we follow the convention in Ikeda et al. 2011, 2012): σij (W ) =

qi |Tij (W )|2 , qj 16π W 2

(7)

where qi is the magnitude of the three-momentum in channel i. The threshold branching ratios are calculated by the combination of the cross section at the K − p threshold, σij (W = mK − + Mp ). The energy shift ΔE and width Γ of kaonic hydrogen are related to the K − p scattering length aK − p through the improved Deser formula (Meißner et al. 2004), and aK − p can be calculated from the scattering amplitude as aK − p =

 TK − p,K − p (W )    8π W W =m

.

(8)

K − +Mp

In this way, data sets (i)–(iii) are related to the two-body scattering amplitude Tij and can be used as direct constraints on Tij . In contrast, the π Σ spectra cannot be calculated in terms of the corresponding scattering amplitude Tij only. Elastic π Σ scattering experiments are practically not possible, so the π Σ spectra can only be measured in production reactions, where particles other than π Σ exist in the final state. Consider, for example, the simplest case of a three-body final state, such as it

75 Theory of Kaon-Nuclear Systems

2759

appears in the photoproduction reaction, γ p → K + π Σ. Let channel i denote the π Σ subsystem in the final state. The distribution of the invariant π Σ mass MI can be written as  2    dσi (MI )  ∝ Tij (MI )Gj (MI )Cj  . (9) dMI  j  The coefficients Cj represent the weight of the initial channel j , which reflect various aspects of the reaction, such as the kinematics (initial energy, scattering angle of K + , etc.). In order to extract the information of the two-body scattering amplitude Tij , one needs to estimate the Cj s in some way. In addition, Eq. (9) does not include the final state interactions of other hadron pairs (π K and KΣ), so the full three-body dynamics is missing. Therefore, in comparison with the direct constraints (i)–(iii), using the π Σ spectra (iv) requires special care. In Ikeda et al. (2011, 2012), the meson-baryon scattering amplitude was constructed using NLO chiral SU (3) dynamics together with the constraints (i)–(iii). A systematic uncertainty analysis was performed to determine the subtraction constants and the LECs in the NLO term, achieving an accuracy χ 2 /d.o.f = 0.96. This indicates that the “puzzle” caused by the DEAR experiment (Beer et al. 2005), a possible inconsistency of the scattering data and an earlier kaonic hydrogen measurement (Borasoy et al. 2005a, b), has been resolved. Even when the interaction is restricted to VWT only, the overall description is reasonable (χ 2 /d.o.f = 1.12). This means that the scattering amplitude is essentially determined by VWT , while NLO corrections are required as a further improvement to deal with the accurate SIDDHARTA data. It also turns out that the SIDDHARTA result reduces the uncertainty of the subthreshold extrapolation of the scattering amplitude, in comparison to a similar analysis that only uses the data (i) and (ii) (Borasoy et al. 2006). In Guo and Oller (2013), in addition to (i)–(iii), the cross sections of K − p → ηΛ (Starostin et al. 2001), π Λ phase shift at Ξ − mass (Chakravorty et al. 2003; Huang et al. 2004), and the π Σ mass distributions in the Σ + (1660) → π + π − Σ + reaction (Hemingway 1985) and in the K − p → π 0 π 0 Σ 0 reaction (Prakhov et al. 2004) were included in the analysis. The effect of the SU (3) breaking in the meson decay constants was discussed in detail. In Mai and Meißner (2015), the π Σ spectra in the photoproduction data by CLAS (Moriya et al. 2013) was used in addition to (i)–(iii). It was shown that some solutions allowed by (i)–(iii) were rejected by the CLAS data, and two solutions were finally obtained. The results of the pole positions in Ikeda et al. (2011, 2012), Guo and Oller (2013), Mai and Meißner (2015) are shown in Table 2 and plotted in Fig. 6. In all those analyses, two poles are found in ¯ -π Σ systems. The position of the Λ(1405) the energy region of the coupled KN ¯ pole (near the KN threshold) has small ambiguity and converges to the region around 1420 MeV. This reflects the strong constraint from the SIDDHARTA data at the K − p threshold. On the other hand, the pole position of the Λ(1380) (near the π Σ threshold) shows sizable ambiguities in different analyses. It is therefore desirable to determine that pole position quantitatively in future studies.

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Table 2 Resonance poles of Λ(1405) and Λ(1380) in PDG (Zyla et al. 2020) (Ikeda et al. 2011, 2012) NLO (Guo and Oller 2013) Fit II (Mai and Meißner 2015) solution #2 (Mai and Meißner 2015) solution #4

Λ(1405) [MeV] +3 1424+7 −23 − i26−14 +8 1421+3 − i19 −2 −5 +2 1434+2 −2 − i 10−1 +2 1429+8 − i 12 −7 −3

Λ(1380) [MeV] +19 1381+18 −6 − i81−8 +24 1388+9 − i114 −9 −25 +17 1330+4 − i 56 −5 −11 +12 1325+15 −15 − i 90−18

0 -20

Im z [MeV]

-40 -60 -80 -100 -120 -140

1320

1360

1400

Re z [MeV]

1440

Fig. 6 Resonance poles of Λ(1405) and Λ(1380) in the complex energy (z) plane. Triangles, squares, crosses, and circles show the results in Ikeda et al. (2011, 2012) (NLO) (Guo and Oller 2013) (Fit II) (Mai and Meißner 2015) (solution #2), and (Mai and Meißner 2015) (solution #4). Dotted lines represent the threshold energies of meson-baryon channels: π 0 Σ 0 , π − Σ + , π + Σ − , K − p, K¯ 0 n from left to right. (Adapted from Hyodo and Niiyama 2021)

The spin of the Λ(1405) was considered to be consistent with 1/2 in the past experiments (Engler et al. 1965; Thomas et al. 1973; Hemingway 1985), but its parity had not been determined. In 2014, the direct experimental determination of the parity of the Λ(1405) was performed in Moriya et al. (2014) with the CLAS photoproduction data of γ p → K + Λ(1405). Consider the decay angular distribution of the Λ(1405) → π − Σ + decay. In the unpolarized two-body decay via the strong interaction, the angular dependence is determined only by the spin,  and that of the Λ(1405) independent of the parity. If the polarization of the Σ + (Q)   with respect to P , as (P ) are specified, then the difference arises in the direction of Q shown in the left panel of Fig. 7. For instance, when the initial state is J P = 1/2−  is independent of the decay angle as in with an s-wave decay, the direction of Q + panel (a). On the other hand, for a 1/2 state with a p-wave decay, the direction  rotates around the P vector as in panel (b). Thus, taking the z axis in the of Q direction of P , the z component of the Σ polarization Qz is constant for 1/2− , while it changes sign depending on the decay angle for 1/2+ . Experimentally, the Λ(1405) polarization P is determined by the reaction plane specified by the photon

75 Theory of Kaon-Nuclear Systems

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Fig. 7 Upper figure: schematic illustration of the polarization transfer of Y ∗ → Y π decay with Y ∗ having J P = 1/2− (a) or J P = 1/2+ (b). Lower figure: z-component of the Σ + polarization Qz as a function of the decay angle cos θΣ + . Solid line, average of the experimental data; dotted line, prediction for p-wave decay; dashed line marks the case without polarization. (Adapted from Moriya et al. 2014)

 is given by the weak decay of the Σ + . The and the K + . The Σ + polarization Q experimental result for the total energy 2.65 < W < 2.75 GeV and the scattering angle of K + in the center-of-mass system, 0.70< cos θKcm. + 0 is the monopole condensate (η2 < 0 describes a phase without the condensate). The potential (56), which respects the dual gauge invariance (50), can also be derived from the worldline representation of the monopole partition function (Bardakci and Samuel 1978) with the interaction term λ appearing naturally due to contact intersections of the monopole worldlines. Instead of the single fugacity ζ in two spatial dimensions, we have now two parameters: η responsible for monopole mass and condensate and λ which determines the monopole self-interaction. (ix) Grouping all three terms together, we arrive to the Lagrangian of the dual Abelian Higgs model which describes the dynamics of magnetic monopoles: L∗AHM ≡ LC + Lφ + LV =

2  # #2 1 2 Cμν + #Dμ φ # + λ |φ|2 − η2 , (57) 4

where ∗AHM stands for the dual Abelian Higgs model. This dual model defines the confining properties of the monopole vacuum. It is the 3+1 dimensional analog of the 2+1 dimensional sine-Gordon model (20) relevant for compact QED in two spatial dimensions.

Breaking the Scale Symmetry The dual AHM model (57) has two massive scales corresponding to the mass of the monopole field φ and the mass of the dual gauge boson Cμ , respectively: √ mφ = 2 λη,

mC =

√

2gmon η,

(58)

where the gauge boson Cμ gets its mass via the dual Meissner mechanism. The generation of the mass scale in the confining phase is well seen in the numerical simulations (Stack and Wensley 1992; Jersak et al. 1999). In Yang-Mills theory, the breaking of the appearance of the mass scale is associated with the scale (trace) anomaly which leads to the mass gap generation. In the dual superconductor picture, it is the monopole condensation that leads to the mass gap generation (58): the scale symmetry is broken spontaneously as a result of breaking of the dual symmetry in the dual Abelian Higgs model (57). Thus, in the dual superconducting picture, the quantum trace anomaly is seen as a spontaneous symmetry breaking of the dual symmetry group.

Confining String in the Dual Superconductor Picture Duality The dual model (57) is a relativistic analog of the Ginzburg-Landau model of superconductivity. In a superconductor, electric charges (Cooper pairs) are condensed, and magnetic fluxes are squeezed due to the Meissner effect. A background magnetic field tends to destroy superconductivity. If we would put a monopole and an antimonopole in a middle of a sufficiently voluminous superconductor, then the

79 QCD Vacuum as Dual Superconductor: Quark Confinement and Topology

2921

magnetic flux emanating from the monopole and sinking into the antimonopole will be squeezed into a narrow string which would take a straight line between these two objects. This string is called the Abrikosov vortex. Due to the conservation of the magnetic flux, the string will remain spanned between the monopoles as we move them further from each other. As the energy density per unit length of the vortex (the string tension, σ ) remains a constant quantity for a sufficiently long string, the monopoles and antimonopole would feel a linear confining potential (1) at a sufficiently large separation R between them. Thus, the monopoles and antimonopoles are confined inside a superconductor. The dual model (57) uses exactly this property just with the interchange of magnetic with electric and vice versa, hence the name “the dual superconductor picture of confinement.” Electrically charged particles will be confined in the vacuum due to a monopole condensate which exhibits the dual Meissner effect and maintains the dual Abrikosov string that carries the electric flux (with the structure of vacuum monopole currents shown in Fig. 1). Structure of Confining String According to the dual superconducting picture, the confining string in the cU(1) vacuum is an exact counterpart of the usual Abrikosov string in a standard superconductor. The only difference consists in the replacement of the words “electric” ↔ “magnetic”. The masses (58) determine the characteristic length scales of the solutions: the size of the vortex core in terms of the monopole condensate is set by the correlation length of the monopole condensate ξ = m−1 φ , while the

penetration length λ = m−1 C gives us the typical width of the electric flux of the string. At large distance R between the static electrically charge sources, R ξ and R λ, the transverse string profile acquires axially symmetric form that possesses along the line that connects the test sources. Working in the cylindrical coordinates, (ρ, θ, z), we get that the electric field of the dual string has only a z component along the axis that connects two sources while the magnetic current j mon appears to circumvent the string, thus generating the electric field via a dual analog of the Ampère law, as illustrated in Fig. 1: E = Ez (ρ) ez ,

j mon = jθmon (ρ)eθ ,

φ = f (ρ)eiθ.

(59)

The vortex string solution is a topologically protected field configuration which is characterized by mapping of the phase of the monopole field φ ∝ einθ to the manifold at the spatial infinity in the transverse (ρ, θ ) plane. The mapping S 1 → S 1 is characterized by the nontrivial first homotopy group, π1 (S 1 ) = Z, with group elements labeled by the integer mapping n ∈ Z. Hereafter we discuss the elementary string (59) with the winding number n = 1. The profile functions (59) for the electric field Ez , the monopole current jθmon , and the Higgs condensate f depend only on the radius ρ. The electric field of the flux tube takes its maximum at the center of the string ρ = 0 and vanishes at spatial infinity ρ → ∞; the monopole current vanishes both at the center of the string and

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far from it while reaching its maximum in the middle; the monopole condensate vanishes at the center of the string and approaches the vacuum value f → η far from the string. The profile functions that describe the string solution (59) are solutions of a system of nonlinear equations which are not known in a closed analytical form (but can be easily obtained numerically if required). Away from the vortex core, ρ ξ , the monopole condensate approaches its vacuum value, the system can be linearized, and the solutions for the electric field and the monopole density can be given in terms of the modified Bessel functions of the second kind, Ka (x): Ez = m2C K0 (mC ρ),

jθmon = m3C K1 (mC ρ).

(60)

Notice that the string profile in terms of electric field and the monopole currents (60) share a single unknown parameter, mC , thus drastically restricting the solution. The penetration length λ = m−1 C dictates the string width encoded in the profiles (60). Nevertheless, the numerical results of the string profile (Koma et al. 2004), supported by improved high-accuracy calculations (Panero 2005), were found to be in a very good agreement with the analytical prediction, thus proving the validity of the dual superconductor picture of the infrared properties of the model. The simulations were done at the confining region slightly below the critical coupling βc where the non-exact (but, from a practical point of view, extremely accurate) continuum scaling is achieved. The existence of approximate continuum limit allowed the authors of Koma et al. (2004) and Panero (2005) to calculate the mass of the dual gauge boson, mC  4r0−1 , in terms of the Sommer scale r0 . The Sommer scale (Sommer 1994) provides a computationally cheap and practical way to set a reliable physical length on the lattice. It is defined as a distance r = r0 at which the force F (r) = dV /dr between the static test charges satisfies r 2 F (r) = 1.65. In the QCD phenomenology, r0  0.5 fm, which would correspond to the mass of the dual gauge boson mC  1.6 GeV. Short Summary Vacuum of (3+1)d compact Abelian model is a dual superconductor which (i) confines electric charges by squeezing their electric flux into the Abrikosov vortex via the dual Meissner effect and (ii) generates the mass gap via spontaneous breaking of the dual (gauge) symmetry.

Dual Superconductor Mechanism in Yang-Mills Theory Abelian Monopoles in Pure Gluon Vacuum? There are plenty of experimental and first-principle numerical evidences that the quark confinement in QCD appears as a result of the formation of the stringylike object, the QCD string, in the vacuum between the color sources (Brambilla et al. 2014). In addition to the observation of the linear potential between the static

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chromoelectric sources, the numerical simulations allow us to see, literally, the stringy object in between the test particles (Bali et al. 1995). All known facts, both phenomenological and numerical, point out to the single possibility that we do have a genuine chromoelectric string in the vacuum and which is responsible for the quark confinement in the colorless hadron states. The dual superconductor mechanism in Yang-Mills theory is invoked to explain the formation of the confining string (’t Hooft 1976; Mandelstam 1976). The idea is essentially the same as in compact Abelian gauge models discussed earlier: certain “chromomagnetic” monopoles condense in the gluonic vacuum, and a “chromoelectric” string between the test color sources is formed. The only problem of the monopole-based confining mechanism in QCD is that there are no natural monopole solutions in Yang-Mills theory. The monopoles simply do not appear as topological solutions in gluonic vacuum due the triviality of the second homotopy group of the vacuum manifold, π2 [SU (Nc )] = 1. From the first sight, this simple fact undermines the very idea of the dual superconducting mechanism in Yang-Mills theory. However, the absence of topologically stable monopole configurations in Yang-Mills theory does not mean that they do not appear in the gluonic configurations in the statistically significant, dynamical sense. While the formation of a stable condensate out of intrinsically unstable, topologically unprotected field configurations might sound unrealistic and doubtful, we would like to support this claim with a well-known example: the ordinary superconductivity. The Bardeen-Cooper-Schrieffer superconducting mechanism (Bardeen et al. 1957a, b) proceeds via the formation of weakly bound Cooper electronic pairs which are not protected topologically. While the condensate of Cooper pairs determines a stable ground state in the superconducting regime, individual Cooper pairs can be broken. How to identify these monopole-like structures in a gluonic field given the absence of topological arguments in favor of their existence? We know that monopole solutions appear in non-Abelian gauge theories with matter fields. The simplest relevant example is the SU(2) Georgi-Glashow model (Georgi and Glashow 1972) which possesses the gluon field coupled to a scalar triplet field φ a , a = 1, 2, 3. The triplet χˆ = χ a σ a transforms in an adjoint representation of the gauge group: SU (2) :

χˆ → Ω † χˆ Ω,

(61)

where Ω ∈ SU (2) is the gauge transformation matrix and σ a are the Pauli matrices. In the phase where the scalar field condenses, χ a  = 0, the original symmetry is spontaneously broken down to an Abelian subgroup, SU (2) → U (1). The residual Abelian group is a compact group since it is a subgroup of the original compact SU(2) group. Thus, the Georgi-Glashow model in the broken phase should possess a compact electromagnetic gauge field determined in the group space by the direction of the field χ. ˆ For a diagonal χ, ˆ the compact gauge field is identified with the Cartan A3μ field.

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A compact gauge field should necessarily admit Abelian singularities. In the context of the Georgi-Glashow model, this monopole singularity is known as the ’t Hooft-Polyakov monopole solution (’t Hooft 1974; Polyakov 1974). Now, let us apply this approach to Yang-Mills theory without matter fields.

Abelian Projection in Non-Abelian Gauge Theory No Matter Fields Available? Abelian Projection: “It does not Matter” The idea of the “Abelian projection” ’t Hooft (1981) proposes that in the absence of a genuine matter field χ a in Yang-Mills theory, its role can be played by a composite operator, χ a = χ a [A], constructed from the original gauge field Aaμ . In other words, the field χ a should be chosen in such a way that under the action of the SU (2) group, the gauge transformation of the gluon field SU (2) :

i Aˆ μ → Ω † Aˆ μ Ω − Ω † ∂μ Ω, g

(62)

generates the adjoint transformation (61) of the composite field. Then we can identify the Abelian monopoles as ’t Hooft-Polyakov monopoles in the effective “Georgi-Glashow model” that possesses Aaμ as a gauge field and χ a as an adjoint scalar field. Thus, we have just found a way to identify the Abelian monopoles in any given gluonic configuration: in a gauge, where the operator χˆ becomes a diagonal matrix, the Cartan component of the gauge field should be identified with a compact gauge field. This construction can be generalized to SU (Nc ) Yang-Mills theory with arbitrary number of colors Nc ≥ 2: the gauge group is projected down to its maximal Cartan subgroup, SU (Nc ) → [U (1)]Nc −1 . A detailed review of the subject can be found in Ripka (2004). The singularities in these fields are then the desired monopole solutions. The operator χˆ helps us to project the non-Abelian group to an Abelian subgroup, hence the term “Abelian projection.” An Abelian projection, basically, fixes an “Abelian gauge” which is determined by the condition of the diagonalization of the operator χˆ . The Abelian projection method raises immediately a few objections (welljustified remarks) about its applicability to Yang-Mills theories.

A Composite Matter Field = An Artificial (Unphysical) Matter Field? The use of a composite field to identify topological-like defects in the models with otherwise topologically trivial homotopy groups does not necessarily represent an artificial construction. Consider, for example, the electroweak theory with a scalar doublet (Higgs) field Φ = (φ1 , φ2 )T which transforms in the fundamental representation of the SU (2)W gauge group: Φ → ΩΦ. Since the condensate φ = 0 breaks the group completely, SU (2)W → 1, the vacuum manifold is π2 trivial, and no topologically stable monopoles could ever exist (we ignore here the hypergauge sector U (1)Y which does not play any role in our discussion).

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However, we can construct a composite adjoint field, χ a = Φ † σ a Φ, which transforms as an adjoint field (61). Then we can use the new field χ a to identify the ’t Hooft-Polyakov monopoles “embedded” into the electroweak group. These objects are known as celebrated Nambu monopoles (Nambu 1977). Despite an artificial character of their construction, they play an important role in the electroweak phenomenology related to electroweak sphalerons and baryon-number-violating transitions (Achucarro and Vachaspati 2000). First-principle simulations show that the Nambu monopoles densely populate the hot electroweak phase and thus can provide an important ingredient to the physics of Early Universe (Chernodub et al. 1998).

Stability of Abelian Monopoles? The original ’t Hooft-Polyakov monopoles are topologically stable configurations ! " only in the broken vacuum, with φˆ = 0. However, we know from experimental data on hadronic spectrum that the color symmetry in QCD is unbroken. So, the monopole is not topologically stabilized. What are the benefits of this construction then? The answer is threefold: (a) Given an operator χˆ = χ[A], ˆ the procedure allows us to identify trajectories of Abelian monopoles in any gluon configuration Aaμ . A gauge transformation of a gluon configuration (62) would not change the monopole worldlines, thus implying a gauge invariance of the construction (however, the position of ˆ to be the monopole worldlines depends on the Abelian projection operator φ, discussed below). (b) The magnetic charge of the monopole is quantized. (c) The monopole worldlines are closed so that the magnetic charge is conserved. These properties allow us to provide a useful link between the properties of vacua of Yang-Mills theory and compact electrodynamics. We briefly sketch the proof of these three statements, referring the reader to Kronfeld et al. (1987a, b), and Chernodub and Polikarpov (1997) for further discussions. The Abelian subgroup is identified by a diagonalization of φˆ → φˆ diag in every space-time point via the SU(2) gauge transformation (61). In the diagonalized form, the theory still holds invariance under the U (1) Cartan subgroup of the SU(2) group as it cannot be fixed by the diagonalization condition: φˆ diag (x) → ΩU† (1) (x)φˆ diag (x)ΩU (1) (x) ≡ φˆ diag (x),  ΩU (1) (x) = eiσ3 α ≡

 0 eiα(x) , 0 e−iα(x)

Representing the gauge field in the matrix form,

α ∈ [0, 2π ).

(63) (64)

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 Aˆ μ =

A3μ A+ μ 3 , A− −A μ μ

(65)

one finds that the components of the non-Abelian gauge field Aˆ transform under the residual Abelian group (64) as follows: 1 ∂μ α, g

aμ → aμ −

aμ = A3μ

(66)

1 2 A+ μ = Aμ + iAμ ,

A†μ → e2iα A†μ ,

+ † A− μ ≡ (Aμ ) ,

(67)

where we immediately recognize that the real-valued field aμ plays the role of the compact Abelian gauge field while the complex vector fields A± μ as Abelian vector matter fields that carry a double charge. Since the gauge transformation (64) is compact, the field strength must contain monopole singularities. Under a singular SU(2) gauge transformation (62), the nonAbelian field-strength tensor Fˆμν evolves as follows: mon [Ω], Fˆμν [A] → Fˆμν [A(Ω) ] = Ω † Fˆμν [A]Ω + Fˆμν

(68)

where the singular part of the transformation is as follows: i mon Fˆμν [Ω] = − Ω † (x)[∂μ ∂ν − ∂ν ∂μ ]Ω(x). g

(69)

If the transformation matrix is a singular function of the coordinate, then, in general, the derivatives applied to this matrix do not commute. After the diagonalization, the monopole singularity appears in the Abelian part of strength tensor: ph

mon fμν = ∂μ aν − ∂ν aμ = fμν + fμν ,

(70)

similarly to Eq. (3). The magnetic charge of the monopole can be determined via the Gauss law by calculating the net magnetic flux B emanating by magnetic monopole. ph The contribution of the photon part fμν is vanishing, while the monopole strength tensor gives us the following Kronfeld et al. (1987a, b): m=

1 4π

 S2

B d σ = −

i 4πg

 S2

dσμν εμναβ ∂α Tr[Ω † ∂β Ω] = 0, ±

1 1 , ± , . . . (71) 2g g

Mathematically, the quantization of the magnetic charge (71) arises due to the topological reasons as the surface integrals in the above equation count the integervalued winding number of SU (2) over the sphere S 2 surrounding the monopole.

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Physically, the quantization appears on the Dirac condition imposed on the invisibility of the Dirac string. Notice that the quantization in non-Abelian theory is formulated as e4π iggmon = 1 peculiar to the non-Abelian gauge theory. The electric charge is encoded in the gauge transformation (67). The magnetic flux carried by the Dirac string singularity is determined by the last integral in Eq. (71). Finally, the identity ∂μ jμmon ∝ εμναβ ∂μ ∂ν ∂α Tr[Ω † ∂β Ω] ≡ 0, valid for all singular matrices Ω, implies that the magnetic worldlines are closed.

Too Many Composite Matter Fields to Choose from? The choice of the operator φˆ cannot be specified on general principles in YangMills theory. Therefore, there is an infinite amount of operators that can be used to identify the Abelian projection. This remark raises the question of the projection independence of the approach: whether it is a universal feature or one should find a special operator χˆ to uncover a dual superconducting mechanism. This objection is kinematically correct. For example, one can show analytically that the Polyakov Abelian projection – the one which uses an untraced Polyakov loop operator as φˆ – possesses only static monopoles in the continuum limit. These monopoles cannot be responsible for the quark confinement since their contribution to the interquark potential cannot lead to the area law (Chernodub 2004). However, the dual superconductor mechanism involves also a dynamical aspect of the problem which may complement these kinematical properties. The number of “bad” choices of the projection can have a measure zero at the space of all projections. In addition, one could consider Abelian extremization gauges that are defined by extremization of a certain functional which is invariant under Abelian transformations (in general, this type of a gauge cannot be reduced to a maximization to an ˆ A famous extremization gauge, the “Maximal Abelian” (MA), adjoint operator φ). is determined by the maximization of the functional Kronfeld et al. (1987a): max R[Aˆ Ω ],

Ω∈SU (2)

ˆ =− R[A]



2 d 4 x |A+ μ|

(72)

where the matter field A+ μ is defined in Eq. (67). The meaning of the MA functional (72) determines the overall success of this gauge: the gauge transformations are used make the gauge field Aˆ μ , Eq. (65), as diagonal as possible. Both the gauge functional (72) and the corresponding differential condition for the extremum (∂μ ± 2igA3μ )A± = 0,

(73)

determine an Abelian gauge as they are invariant under the U (1) gauge transformations (67). We discuss subtleties related to the projection independence and related gauge invariance of the superconducting mechanism in the next section.

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Dual Superconductor in Lattice Yang-Mills Theory Analytical Theory: A Brief Summary Before we start discussing numerical results, let’s summarize our knowledge about the dual superconductivity we got so far. We know the model example in compact Abelian gauge theory in four space-time dimensions and know how the monopole worldlines should behave in the confining phase (an infinitely large percolating cluster) and in the deconfining phase (infinite number of small infrared clusters). We also know the monopole order parameter. We derived the effective dual superconducting model (the relativistic analog of Ginzburg-Landau model). Finally, we understood how to identify the Abelian monopoles in pure non-Abelian gauge theory, and we know the associated difficulties and ambiguities. Now, we are well equipped to apply this knowledge to Yang-Mills theory and check whether the dual superconductor scenario is realized in this theory. The first step in this direction is to check whether the Abelian monopoles have something to do with the physics of confinement.

Abelian and Monopole Dominance The dominance of Abelian gauge fields and Abelian monopole currents in confining properties of Yang-Mills theory are very well-established phenomena. While the effects of the Abelian dominance in hadronic spectrum has been discussed analytically long ago (Ezawa and Iwazaki 1982a, b), most of the results are obtained in numerical first-principle simulations of lattice QCD, and they have a nature of a numerical observation. These observations allow us to figure out the mechanism(s) of the complicated non-Abelian phenomena by comparison of the results of the numerical simulations (“numerical experiments”) with the predictions coming from the analytical calculations. Needless to say that a direct analytical derivation of these phenomena from the Yang-Mills partition function is missing although there are various promising analytical approaches (Kondo 1998a, b; Kondo and Shinohara 2000) (see also the review Kondo et al. 2015). The importance of monopole degrees of freedom for confinement in SU(2) lattice gauge theory within a dual superconducting scenario has been first noted (Kronfeld et al. 1987a) and an approximate effective action for monopoles constructed in Smit and van der Sijs (1991). The signatures of Abelian dominance in saturating the string tension of non-Abelian confining string have been first observed in Suzuki and Yotsuyanagi (1990) and subsequently confirmed in Hioki et al. (1991). The correlation of monopole dynamics with the string tension, exhibiting the renormalization scaling in terms of extended monopoles of larger size, has been established in Ivanenko et al. (1990). The monopole dominance in the string tension has been found in SU(2) QCD in the MA gauge (Shiba and Suzuki 1994; Stack et al. 1994). In the excellent

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agreement with our experience in compact U (1) gauge models, the string tension is well reproduced only by monopole contributions, whereas photons alone provide a Coulomb-like part in the Abelian static potential. A careful systematic study of the Abelian components of the Yang-Mills theory in the Maximal Abelian gauge has confirmed the factorization of the Abelian potential into monopole and photon contributions (with the former giving rise to the Abelian string tension). The Abelian string tension matches its non-Abelian counterpart within accuracy of 5 The profiles of the Abelian electric flux and the monopole currents around string induced by the quark sources both in SU(2) and SU(3) gauge theories agree with the structure of the dual Abrikosov string (Matsubara et al. 1994; Singh et al. 1993; Cea and Cosmai 1995). The quantitative structure of the confining string in the transverse plane which defined the relative thickness of the (chromo)electric flux and the size of the circulating monopole loops depends on the type of the dual superconductivity which is discussed below.

Condensation of Monopoles The dual superconducting picture of QCD vacuum relies on the monopole condensation in the confinement phase. Here we briefly summarize the results in non-Abelian gauge theory following our experience in the compact Abelian gauge theory in 3+1 dimensions (discussed in Lattice Compact U(1) Gauge Model). Similar to the case of compact electrodynamics, the monopole worldlines form a single large percolating cluster in the confining phase of the theory (Ivanenko et al. 1990; Hart and Teper 1998; UKQCD et al. 1999; Chernodub and Zakharov 2003; Bornyakov et al. 2003). The long loops reflect the presence of the monopole condensate which can be probed via asymptotic behavior (48) of the percolating correlator (47). The percolating cluster disappears above the finite-temperature phase transition in agreement with the dual superconducting picture (Kitahara et al. 1995; Ejiri et al. 1995; Damm and Kerler 1997). The monopole condensation can also be revealed by the energy-entropy balance of monopole trajectories (46) following the ideas of Banks et al. (1977). The effective monopole actions, reconstructed from the original gluonic configurations using an inverse Monte Carlo procedure, also confirm the monopole condensation in the confining phase of Yang-Mills theory (Shiba and Suzuki 1995; Arasaki et al. 1997; Kato et al. 1998). The monopole condensate breaks spontaneously the dual (gauge) symmetry in the gauge model. The spontaneous breaking reflects itself in a Mexican hat-type potential for the monopole field or, equivalently, as an expectation value of a disorder parameter describing condensation of monopoles in thermodynamic limit of the system. The first approach has been used to reveal the monopole condensation in numerical simulations of lattice Yang-Mills theory in Chernodub et al. (1997) using the Fröhlich-Marchetti monopole creation operator Frohlich and Marchetti (1987). Later, the second approach has been used to show the condensation using a

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different kind of monopole operator (Di Giacomo et al. 2000) (with pathologies and improvements discussed in Bonati et al. 2012) Summarizing, all independent numerical methods show the presence of the condensate of Abelian monopoles in confinement phase of non-Abelian gauge theory.

Type of Dual Superconductivity in Yang-Mills Theory Similarly to the vacuum of compact electrodynamics, the dual superconductivity in the non-Abelian vacuum is characterized by two mass parameters (58). For SU(2) gauge theory, their ratio provides us with the Ginzburg-Landau parameter: =

mφ λ = , ξ mC

(74)

which defines the type of the dual superconductivity. For the SU(3) gauge theory, the√generalization √ is straightforward (for example, Battelli and Bonati 2019). If  < 1/ 2 ( > 1/ 2), then the condensation is of the first (second) type implying that the parallel vortex strings attract (repel) each other (Tinkham 2004). In a superconductor of type I, an external magnetic field below a critical value Bc is always expelled from the bulk of the superconducting medium. The field penetrates the medium only at the distance of the penetration length λ, hence its name. Above the critical field, B > Bc , the superconductivity is destroyed. A superconductor of type II possesses two critical values, Bc1 and Bc2 . Similarly to a type I superconductivity, magnetic field weaker than the first critical field, B < Bc1 , is expelled from the superconductor, while a field stronger than the second critical field, B > Bc2 , destroys the superconductivity. In a mixed phase, Bc1 < B < Bc2 , the external field can penetrate the medium in the form of Abrikosov flux tubes, without destroying superconductivity globally. The question of the superconductivity type is interesting property of the confining vacuum by itself since it is not fixed by the dual superconducting scenario. From general arguments, one could suggest two scenarios: (i) A single non-Abelian excitation (e.g., the ground-state, scalar 0++ glueball) determines both √ the mass of the dual gauge field and the monopole mass. Then,  ≡ 1 > 1/ 2 implies a type II superconductivity. (ii) The correlation lengths are determined by the quantum numbers of the excitations: the ground-state scalar glueball determines the mass of the scalar monopole field, while the lightest vector glueball is responsible for the mass of the dual vector gauge boson. In this case, the dual superconductivity parameter is given by the ratio of the glueball masses: 0++ gl = −+ = 0



0.628(7) for SU (2), 0.645(7) for SU (3),

(75)

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where we took the glueball masses from the recent calculations (Athenodorou and Teper 2021). These numbers are worth comparing with the superconductivity threshold (Tinkham 2004): 1 c = √ = 0.7071 . . . , 2

(76)

implying that the glueball-inspired argument (75) implies a weak type I dual superconductivity with gl /c  0.9 close to the border between two types of superconductors for SU (2) and SU (3) gauge theories. Curiously, many lattice simulations of SU(2) and SU(3) gauge theories indicate that the vacuum type of the confinement phase is near to the border between the type I and the type II dual superconductivity (Bali et al. 1998; Gubarev et al. 1999; Koma et al. 2003; Haymaker and Matsuki 2007; Chernodub et al. 2005; D’Alessandro et al. 2007) being close to our second scenario. Moreover, a recent gauge-invariant study gives gl /c  0.9 within error bars (Ishiguro et al. 2022), thus supporting our glueball-based phenomenological argument. However, the type of the vacuum superconductivity is not fixed definitely as the value of  may depend on the methods and assumptions adopted in different studies, ranging from  = 0.243(88) (Cea et al. 2012) to  = 1.8(6) in Battelli and Bonati (2019).

Abelian Monopoles in Yang-Mills Theory as Physical Objects One could get curious whether the Abelian monopole current represent a physical degree of freedom in non-Abelian vacuum. We already asked this question on a theoretical basis. Now, we have a number of signatures suggesting the monopoles are physical objects (the condensation of elementary monopoles, the structure of confining string in terms of monopole trajectories, the monopole dominance in confining properties, and, as we will see slightly below, the chiral symmetry breaking). However, the Abelian monopole is a part of the gluon ensemble, and, most probably, a single monopole trajectory corresponds to a larger gluonic structure that has a gauge-invariant nature. Basically, the monopole trajectory is a tip of a large gluonic iceberg which is seen only due to the Abelian projection. However, if a monopole trajectory corresponds to a physical object, then the monopole position should correlate with gauge-invariant physical quantities (e.g., with the gluonic action) and exhibit renormalization-invariant properties. The local correlation of monopoles with the gluonic action has indeed been demonstrated in Bakker et al. (1998). A detailed investigation of the correlators led to an estimation of the monopole radius, Rmon  0.04 fm in Bornyakov et al. (2002). A clear local relations of the monopole currents with the chiral condensate have been found recently lattice in QCD in Ohata and Suganuma (2021) (we will discuss this question in the next section).

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The physical size of the Abelian monopoles aphys (β, n) = na(β) can be changed using the lattice blocking procedure as well as the variation of the lattice coupling β which determines the physical length of the lattice spacing a = a(β) according to the non-Abelian scaling (42) (Ivanenko et al. 1990). Remarkably, the SU(2) monopole action shows a perfect renormalization scaling described by the single parameter aphys (and not with n or β independently) implying that the lattice Abelian monopoles probe physical features of the non-Abelian gluonic vacuum (Shiba and Suzuki 1995). A similar analysis can also be done for the SU(3) gauge theory (Arasaki et al. 1997).

Gauge Invariance of Dual Superconductivity Most quoted numerical results on the Abelian- and monopole-based confinement mechanism have been obtained in the Maximal Abelian gauge. The success of this gauge apparently comes from its definition: it serves to maximize the role of the Abelian (Cartan) subgroup of the Yang-Mills group, thus effectively suppressing the off-diagonal gluons (Kronfeld et al. 1987a, b). Numerical simulations do indeed confirm the dynamical inhibition of the offdiagonal gluons in the MA gauge. The off-diagonal gluons gain a heavy mass of the order of 1 GeV (Suganuma et al. 1998) which makes them inessential for the infrared properties of the theory (Amemiya and Suganuma 1999). The leading role in the long-distance physics is determined by the Abelian degrees of freedom, while the off-diagonal gluons play a rather supportive role. However, the off-diagonal gluons are important for the short-distance physics by creating, at the same time, a “non-Abelian core” of an Abelian monopole similarly to the non-Abelian code of ’t Hooft-Polyakov monopoles. The off-diagonal gluons compensate the divergence of the Abelian monopole field and make its action finite (Ichie and Suganuma 2000). The monopole core has a finite size (Bornyakov et al. 2002). There are other Abelian projections in which the Abelian degrees of freedom and monopoles appear to be less pronounced in description of the confining (Kronfeld et al. 1987b; Chernodub et al. 1995) and chiral (Woloshyn 1995) properties in lattice Yang-Mills theory. As we have mentioned, one can analytically show the irrelevance of the Abelian monopoles determined in the Polyakov gauge for the confining properties in the continuum limit (Chernodub 2004). These examples, however, do not imply that the dual superconducting mechanism is a gauge- or projection-dependent and, therefore, an unphysical property of the theory. On the contrary, the local correlations of the Abelian monopoles, determined with the help of the Maximal Abelian gauge, with the gauge-invariant quantities (such as gluonic action and the quark condensate), imply that the Abelian monopoles defined within this particular prescription (gauge) correctly capture the gauge-invariant degrees of freedom of QCD. The Maximal Abelian projection should be considered as a tool, which takes the form of a gauge fixing, that allows us to identify relevant gauge-invariant

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degrees of freedom in the gluonic sector. There are convenient gauges where the nonperturbative gluonic degrees of freedoms are well visible via Abelian monopoles (and there other, off-diagonal degrees of freedom are inhibited), and there are also inconvenient gauges, where these degrees of freedom are hindered by the artifacts. Besides the Abelian projection method, there are also explicitly gauge-invariant approaches to the confinement problem which utilize the ideas of the dual superconductor picture without relying on a specific gauge fixing (Shibata et al. 2007; Kato et al. 2015; Suzuki et al. 2008; Ishiguro et al. 2022). These promising works incorporate the notions of Abelian and monopole dominance as well as (the violation of) non-Abelian Bianchi identity, ascribing the confining properties of SU (Nc ) Yang-Mills theory to a subset of gluonic degrees of freedom either via performing a decomposition of the gluon field into new variables or averaging overall degrees of freedoms without any restrictions. The former approach selects the relevant confining degrees of freedom from the start. The latter method averages out overall degrees of freedom, and what is left after the averaging is the gaugeinvariant non-Abelian essence of confinement property.

Monopoles and Chiral Symmetry Breaking One of the puzzles of QCD is the correlation between the confining phenomena of the gluonic vacuum with the chiral properties of quarks. This correlation has a phenomenological nature as a direct “kinematical” link between color confinement and chiral symmetry breaking is absent. The chiral and confining properties can only be related to each other dynamically, provided there is an agent in the QCD vacuum which drives them both. Such an agent could be a (chromo)magnetic monopole: if the monopoles are responsible for the chiral symmetry breaking, then the dual superconductor picture could explain the mysterious link between chiral and confining phenomena at the same time. In this section, we briefly discuss analytical and numerical results which support this point of view. The analytical model of the dual superconductivity, adopted to the SU (3) color group, has been developed earlier in Maedan and Suzuki (1989) and Maedan et al. (1990). The dynamical consequences of the interactions of the dual fields with light quarks have been first studied in Suganuma et al. (1995). Using the Schwinger-Dyson formalism, one can show that the dual Meissner effect, generated by the monopole condensation, leads to the dynamical breaking of the chiral symmetry. In addition, the physical poles in the light-quark propagator disappear, as a consequence of the quark confinement realized within the dual Ginzburg-Landau model. These findings support the close relation between the color confinement and the chiral symmetry breaking, both of which are generated by the monopole condensation. Numerically, the Abelian dominance in the chiral symmetry breaking has been found in Miyamura (1995) and Woloshyn (1995): the chiral condensate, calculated with the help of the Abelian field in the MA gauge, was found to be very close to the

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chiral condensate calculated with the help of the full non-Abelian fields in quenched SU(2) gauge theory. In addition, pseudoscalar and vector meson correlators were reproduced only by Abelian gauge fields in the region of small quark masses relevant for the chiral symmetry breaking. The Abelian field in the Abelian gauge can be further decomposed into the photon and monopole components. Our previous experience in the gauge field sector tells us that a dominant contribution to infrared quantities should come from monopoles while the photonic fields should provide us only perturbative corrections. And, indeed, in the quark section, Monte Carlo simulations with staggered lattice fermions reveal that monopoles in Maximal Abelian gauge largely reproduce the chiral condensate values of the full non-Abelian gauge theories with both SU(2) and SU(3) color groups (Lee et al. 1996), as expected in the dual superconductor mechanism. Monte Carlo simulations also show that the global properties of chiral condensate and Abelian monopoles are statistically correlated with each other. Moreover, the chiral condensate is enhanced locally by magnetic field generated by the monopoles (Ohata and Suganuma 2021; Suganuma and Ohata 2021). Why the monopoles are responsible for the chiral symmetry breaking in nonAbelian gauge theories? Traditionally, the role of the agents of chiral symmetry breaking is carried by the instantons, the classical solutions of Euclidean YangMills theory (Schäfer and Shuryak 1998). The instanton possesses a non-Abelian topological charge and hosts an exact zero fermion mode. It is often assumed that the gluonic vacuum in hadronic phase can be described as an interacting instanton gas or liquid. In this system, the fermionic zero modes of individual instantons overlap and produce a finite density of near-zero-mode states with ungapped spectral function, limλ→0 ρD (λ) = 0. The latter property leads straightforwardly to the breaking of the chiral symmetry (Diakonov and Petrov 1986). While the simplest instanton-gas picture of gluon vacuum cannot explain the color confinement (Diakonov et al. 1989), instantons can be related to (at least, a part of) Abelian monopoles in Abelian gauges. In this case, the monopole picture has a chance to incorporate both confinement and the chiral symmetry breaking within the original Yang-Mills theory. The local correlations between instantons and Abelian monopoles in the MA gauge have been observed both analytically and numerically (Chernodub and Gubarev 1995; Suganuma et al. 1996, 1998; Brower et al. 1997; Fukushima et al. 1997; Sasaki and Miyamura 1999). Since the instanton carries topological charge density, then a nearby monopole must acquire an electric charge and become a dyon via the Witten effect (Witten 1979). This phenomenon has indeed been observed numerically in non-Abelian configurations (Chernodub et al. 1999; Ilgenfritz et al. 2005). A historical overview of the subject, (inter)relations with other topological field configurations, as well as further discussions of the chiral symmetry breaking within the monopole picture can be found in the comprehensive review (Kondo et al. 2015). The existence of the local correlation of monopoles with lumps of chiral condensate suggests another intriguing connection of gluonic monopole configurations with the chiral symmetry breaking: the magnetic field produced by monopole catalysis generates the local chiral condensate around monopoles via the magnetic

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catalysis phenomenon (Klevansky and Lemmer 1989; Klimenko 1992; Shovkovy 2013). Since the worldlines of condensed monopoles form the infrared cluster, the lumps of quark condensates generated by monopoles in spatially distinct points are also spatially correlated, thus producing a non-disordered chiral condensate relevant to the low-temperature phase of QCD. Interestingly, this non-topological scenario does not explicitly rely on connection between monopoles, which produce confinement of quarks, with instantons, that generate the chiral condensate (Ohata and Suganuma 2021; Suganuma and Ohata 2021).

Conclusions In our review, we described in detail the confinement phenomenon in the Abelian models (where we think that we have a complete understanding of the system) and checked how the known Abelian features can be reutilized in the non-Abelian context (where we know that our understanding is far from being complete). We notice that most of the advances in the non-Abelian theory have a sense of a numerical “experiment” in which the results of first-principle numerical simulations are confronted with an analytical theory. The dual superconductor model describes adequately the phenomenon of quark confinement. The structure of the confining string in Yang-Mills theory resembles closely the structure of the dual Abrikosov vortex. In high-temperature phase, the monopole condensate evaporates and quarks get deconfined in the quark-gluon plasma regime. There are signatures that the monopole condensation generates also another important QCD phenomenon, the dynamical breaking of chiral symmetry. The monopoles capture gauge-invariant non-Abelian structures that are responsible for quark confinement highlighting that the dual superconducting property of the QCD vacuum is a gauge-invariant phenomenon.

References A. Achucarro, T. Vachaspati, Phys. Rept. 327, 347 (2000). hep-ph/9904229 R. Akhoury, V.I. Zakharov, Phys. Lett. B 438, 165 (1998). hep-ph/9710487 C. Allton, M. Teper, A. Trivini, JHEP 07, 021 (2008). 0803.1092 K. Amemiya, H. Suganuma, Phys. Rev. D 60, 114509 (1999). hep-lat/9811035 N. Arasaki, S. Ejiri, S.-I. Kitahara, Y. Matsubara, T. Suzuki, Phys. Lett. B 395, 275 (1997). heplat/9608129 A. Athenodorou, M. Teper, JHEP 12, 082 (2021). 2106.00364 M. Baig, E. Dagotto, J.B. Kogut, A. Moreo, Phys. Lett. B 242, 444 (1990) B.L.G. Bakker, M.N. Chernodub, M.I. Polikarpov, Phys. Rev. Lett. 80, 30 (1998). hep-lat/9706007 G.S. Bali, K. Schilling, C. Schlichter, Phys. Rev. D 51, 5165 (1995). hep-lat/9409005 G.S. Bali, V. Bornyakov, M. Muller-Preussker, K. Schilling, Phys. Rev. D 54, 2863 (1996). heplat/9603012 G.S. Bali, C. Schlichter, K. Schilling, Prog. Theor. Phys. Suppl. 131, 645 (1998). hep-lat/9802005 T. Banks, R. Myerson, J.B. Kogut, Nucl. Phys. B 129, 493 (1977) K. Bardakci, S. Samuel, Phys. Rev. D 18, 2849 (1978)

2936

M. Chernodub

J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 106, 162 (1957a) J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957b) N. Battelli, C. Bonati, Phys. Rev. D 99, 114501 (2019). 1903.10463 C. Bonati, G. Cossu, M. D’Elia, A. Di Giacomo, Phys. Rev. D 85, 065001 (2012). 1111.1541 V.G. Bornyakov et al., Phys. Lett. B 537, 291 (2002). hep-lat/0103032 V.G. Bornyakov, P.Y. Boyko, M.I. Polikarpov, V.I. Zakharov, Nucl. Phys. B 672, 222 (2003). hep-lat/0305021 V.G. Bornyakov, M.I. Polikarpov, G. Schierholz, T. Suzuki, S.N. Syritsyn, Nucl. Phys. B Proc. Suppl. 153, 25 (2006). hep-lat/0512003 V.G. Bornyakov, I. Kudrov, R.N. Rogalyov, Phys. Rev. D 105, 054519 (2022). 2101.04196 N. Brambilla et al., Eur. Phys. J. C 74, 2981 (2014). 1404.3723 R.C. Brower, K.N. Orginos, C.-I. Tan, Phys. Rev. D 55, 6313 (1997). hep-th/9610101 N.E. Brunk, R. Twarock, ACS Nano 15, 12988 (2021) P. Cea, L. Cosmai, Phys. Rev. D 52, 5152 (1995). hep-lat/9504008 P. Cea, L. Cosmai, A. Papa, Phys. Rev. D 86, 054501 (2012). 1208.1362 M.N. Chernodub, Phys. Rev. D 69, 094504 (2004). hep-lat/0308031 M.N. Chernodub, F.V. Gubarev, JETP Lett. 62, 100 (1995). hep-th/9506026 M.N. Chernodub, M.I. Polikarpov, Abelian projections and monopoles, in NATO Advanced Study Institute on Confinement, Duality and Nonperturbative Aspects of QCD (1997), pp. 387–414. hep-th/9710205 M.N. Chernodub, V.I. Zakharov, Nucl. Phys. B 669, 233 (2003). hep-th/0211267 M.N. Chernodub, M.I. Polikarpov, A.I. Veselov, Phys. Lett. B 342, 303 (1995). hep-lat/9408010 M.N. Chernodub, M.I. Polikarpov, A.I. Veselov, Phys. Lett. B 399, 267 (1997). hep-lat/9610007 M.N. Chernodub, F.V. Gubarev, E.-M. Ilgenfritz, A. Schiller, Phys. Lett. B 443, 244 (1998). heplat/9807016 M.N. Chernodub, F.V. Gubarev, M.I. Polikarpov, JETP Lett. 69, 169 (1999). hep-lat/9801010 M.N. Chernodub, F.V. Gubarev, M.I. Polikarpov, V.I. Zakharov, Phys. Lett. B 475, 303 (2000). hep-ph/0003006 M.N. Chernodub et al., Phys. Rev. D 72, 074505 (2005). hep-lat/0508004 K.G. Chetyrkin, S. Narison, V.I. Zakharov, Nucl. Phys. B 550, 353 (1999). hep-ph/9811275 A. D’Alessandro, M. D’Elia, L. Tagliacozzo, Nucl. Phys. B 774, 168 (2007). hep-lat/0607014 G. Damm, W. Kerler, Phys. Lett. B 397, 216 (1997). hep-lat/9702012 S. Das Gupta, A.Z. Mekjian, Phys. Rev. C 57, 1361 (1998) T.A. DeGrand, D. Toussaint, Phys. Rev. D 22, 2478 (1980) D. Diakonov, V.Y. Petrov, Nucl. Phys. B 272, 457 (1986) D. Diakonov, V.Y. Petrov, P.V. Pobylitsa, Phys. Lett. B 226, 372 (1989) A. Di Giacomo, B. Lucini, L. Montesi, G. Paffuti, Phys. Rev. D 61, 034503 (2000). heplat/9906024 S. Ejiri, S.-I. Kitahara, Y. Matsubara, T. Suzuki, Phys. Lett. B 343, 304 (1995). hep-lat/9407022 J.W. Essam, Rep. Prog. Phys. 43, 833 (1980) Z.F. Ezawa, A. Iwazaki, Phys. Rev. D 25, 2681 (1982a) Z.F. Ezawa, A. Iwazaki, Phys. Rev. D 26, 631 (1982b) J. Frohlich, P.A. Marchetti, EPL 2, 933 (1986) J. Frohlich, P.A. Marchetti, Commun. Math. Phys. 112, 343 (1987) M. Fukushima et al., Phys. Lett. B 399, 141 (1997). hep-lat/9608084 H. Georgi, S.L. Glashow, Phys. Rev. Lett. 28, 1494 (1972) M. Gopfert, G. Mack, Commun. Math. Phys. 82, 545 (1981) J. Greensite, Prog. Part. Nucl. Phys. 51, 1 (2003). hep-lat/0301023 L. Gross, Commun. Math. Phys. 92, 137 (1983) F.V. Gubarev, E.-M. Ilgenfritz, M.I. Polikarpov, T. Suzuki, Phys. Lett. B 468, 134 (1999). heplat/9909099 A.H. Guth, Phys. Rev. D 21, 2291 (1980) A. Hart, M. Teper, Phys. Rev. D 58, 014504 (1998). hep-lat/9712003 R.W. Haymaker, Phys. Rep. 315, 153 (1999). hep-lat/9809094

79 QCD Vacuum as Dual Superconductor: Quark Confinement and Topology

2937

R.W. Haymaker, T. Matsuki, Phys. Rev. D 75, 014501 (2007). hep-lat/0505019 S. Hioki et al., Phys. Lett. B 272, 326 (1991); [Erratum: Phys. Lett. B 281, 416 (1992)] H. Ichie, H. Suganuma, Nucl. Phys. B 574, 70 (2000). hep-lat/9808054 E.M. Ilgenfritz, B.V. Martemyanov, M. Muller-Preussker, A.I. Veselov, Phys. Rev. D 71, 034505 (2005). hep-lat/0412028 B.L. Ioffe, Phys. Atom. Nucl. 66, 30 (2003). hep-ph/0207191 A. Irback, C. Peterson, Phys. Rev. D 36, 3804 (1987) K. Ishiguro, A. Hiraguchi, T. Suzuki, Phys. Rev. D 106, 014515 (2022). 2207.04436 K.R. Ito, Nucl. Phys. B 205, 440 (1982); [Erratum: Nucl. Phys. B 215, 136 (1983)] T.L. Ivanenko, A.V. Pochinsky, M.I. Polikarpov, Phys. Lett. B 252, 631 (1990) T.L. Ivanenko, A.V. Pochinsky, M.I. Polikarpov, Phys. Lett. B 302, 458 (1993) J. Jersak, T. Neuhaus, P.M. Zerwas, Nucl. Phys. B 251, 299 (1985) J. Jersak, T. Neuhaus, H. Pfeiffer, Phys. Rev. D 60, 054502 (1999). hep-lat/9903034 M. Karliner, G. Mack, Nucl. Phys. B 225, 371 (1983) S. Kato, N. Nakamura, T. Suzuki, S. Kitahara, Nucl. Phys. B 520, 323 (1998) S. Kato, K.-I. Kondo, A. Shibata, Phys. Rev. D 91, 034506 (2015). 1407.2808 S.-I. Kitahara, Y. Matsubara, T. Suzuki, Prog. Theor. Phys. 93, 1 (1995). hep-lat/9411036 S.P. Klevansky, R.H. Lemmer, Phys. Rev. D 39, 3478 (1989) K.G. Klimenko, Z. Phys. C 54, 323 (1992) Y. Koma, M. Koma, E.-M. Ilgenfritz, T. Suzuki, Phys. Rev. D 68, 114504 (2003). hep-lat/0308008 Y. Koma, M. Koma, P. Majumdar, Nucl. Phys. B 692, 209 (2004). hep-lat/0311016 K.-I. Kondo, Phys. Rev. D 57, 7467 (1998a). hep-th/9709109 K.-I. Kondo, Phys. Rev. D 58, 105016 (1998b). hep-th/9805153 K.-I. Kondo, T. Shinohara, Phys. Lett. B 491, 263 (2000). hep-th/0004158 K.-I. Kondo, S. Kato, A. Shibata, T. Shinohara, Phys. Rep. 579, 1 (2015). 1409.1599 A.S. Kronfeld, M.L. Laursen, G. Schierholz, U.J. Wiese, Phys. Lett. B 198, 516 (1987a) A.S. Kronfeld, G. Schierholz, U.J. Wiese, Nucl. Phys. B 293, 461 (1987b) F.X. Lee, R.M. Woloshyn, H.D. Trottier, Phys. Rev. D 53, 1532 (1996). hep-lat/9509028 S. Maedan, T. Suzuki, Prog. Theor. Phys. 81, 229 (1989) S. Maedan, Y. Matsubara, T. Suzuki, Prog. Theor. Phys. 84, 130 (1990) S. Mandelstam, Phys. Rep. 23, 245 (1976) Y. Matsubara, S. Ejiri, T. Suzuki, Nucl. Phys. B Proc. Suppl. 34, 176 (1994). hep-lat/9311061 O. Miyamura, Nucl. Phys. B Proc. Suppl. 42, 538 (1995) Y. Nambu, Nucl. Phys. B 130, 505 (1977) H. Ohata, H. Suganuma, Phys. Rev. D 103, 054505 (2021). 2012.03537 M. Panero, JHEP 05, 066 (2005). hep-lat/0503024 M.E. Peskin, Ann. Phys. 113, 122 (1978) M.I. Polikarpov, L. Polley, U.J. Wiese, Phys. Lett. B 253, 212 (1991) L. Polley, U.J. Wiese, Nucl. Phys. B 356, 629 (1991) A.M. Polyakov, JETP Lett. 20, 194 (1974) A.M. Polyakov, Nucl. Phys. B 120, 429 (1977) G. Ripka Dual Superconductor Models of Color Confinement, vol. 639 (Springer-Verlag Berlin Heidelberg 2004). hep-ph/0310102 S. Sasaki, O. Miyamura, Phys. Rev. D 59, 094507 (1999). hep-lat/9811029 T. Schäfer, E.V. Shuryak, Rev. Mod. Phys. 70, 323 (1998). hep-ph/9610451 H. Shiba, T. Suzuki, Phys. Lett. B 333, 461 (1994). hep-lat/9404015 H. Shiba, T. Suzuki, Phys. Lett. B 343, 315 (1995). hep-lat/9406010 H. Shiba, T. Suzuki, Phys. Lett. B 351, 519 (1995). hep-lat/9408004 A. Shibata et al., Phys. Lett. B 653, 101 (2007). 0706.2529 I.A. Shovkovy, Lect. Notes Phys. 871, 13 (2013). 1207.5081 V. Singh, D.A. Browne, R.W. Haymaker, Phys. Lett. B 306, 115 (1993). hep-lat/9301004 J. Smit, A. van der Sijs, Nucl. Phys. B 355, 603 (1991) R. Sommer, Nucl. Phys. B 411, 839 (1994). hep-lat/9310022 J.D. Stack, R.J. Wensley, Nucl. Phys. B 371, 597 (1992)

2938

M. Chernodub

J.D. Stack, R.J. Wensley, Phys. Rev. Lett. 72, 21 (1994). hep-lat/9308021 J.D. Stack, S.D. Neiman, R.J. Wensley, Phys. Rev. D 50, 3399 (1994). hep-lat/9404014 H. Suganuma, H. Ohata, Universe 7, 318 (2021). 2108.08499 H. Suganuma, S. Sasaki, H. Toki, Nucl. Phys. B 435, 207 (1995). hep-ph/9312350 H. Suganuma, A. Tanaka, S. Sasaki, O. Miyamura, Nucl. Phys. B Proc. Suppl. 47, 302 (1996). hep-lat/9512024 H. Suganuma, H. Ichie, A. Tanaka, K. Amemiya, Prog. Theor. Phys. Suppl. 131, 559 (1998). hep-lat/9804027 T. Suzuki, I. Yotsuyanagi, Phys. Rev. D 42, 4257 (1990) T. Suzuki, K. Ishiguro, Y. Koma, T. Sekido, Phys. Rev. D 77, 034502 (2008). 0706.4366 G. ’t Hooft, Nucl. Phys. B 79, 276 (1974) G. ’t Hooft, in Proceedings of the EPS International, ed. by A. Zichichi (1976), p. 1225 G. ’t Hooft, Nucl. Phys. B 190, 455 (1981) M. Tinkham, Introduction to Superconductivity (Courier Corporation, New York, 2004) UKQCD, A. Hart, M. Teper, Phys. Rev. D 60, 114506 (1999). hep-lat/9902031 R.J. Wensley, J.D. Stack, Phys. Rev. Lett. 63, 1764 (1989) H. Wittig, Int. J. Mod. Phys. A 12, 4477 (1997). hep-lat/9705034 Witten, (1979). https://doi.org/10.1016/0370-2693(79)90838-4 R.M. Woloshyn, Phys. Rev. D 51, 6411 (1995). hep-lat/9503007

Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-Landau Theory

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Hiroshi Toki

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Vortex and Monopole in Abelian Gauge of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . The QCD Lagrangian Toward the Dual Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . Linear Potential in the DGL Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Symmetry Breaking in the DGL Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meson Spectra in Pion Channel in the DGL Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The String Tension and Ring Solution in the DGL Theory . . . . . . . . . . . . . . . . . . . . . . . . . . Confinement-Deconfinement Phase Transition in the DGL Theory . . . . . . . . . . . . . . . . . . . . Monopoles and Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hadrons in the DGL Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual Ginzburg-Landau Theory in Quark-Hadron Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The fundamental theory of strong interaction is quantum chromodynamics (QCD), which provides dynamics of quarks and gluons. These ingredients are confined in hadrons, which are colorless particles. The original Lagrangian of QCD has the approximate chiral symmetry with very small quark masses for up and down quarks, which is spontaneously and dynamically broken to provide masses of hadronic scale to quarks and create pions as Nambu-Goldstone bosons. These fundamental physics (confinement and chiral symmetry breaking) should have a strong impact on the dynamics of hadrons and nuclei. In this chapter, the QCD Lagrangian is introduced first, and efforts to construct the

H. Toki () Research Center for Nuclear Physics (RCNP), Osaka University, Toyonaka, Osaka, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_20

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QCD physics in terms of low-energy effective theory with the dual GinzburgLandau (DGL) Lagrangian are presented. The model studies on linear potential, glueballs, pion spectrum, etc. in association with color confinement and chiral symmetry breaking are discussed in detail using the DGL Lagrangian. Future applications of the DGL theory to construct hadrons and nuclei are described for the challenge.

Introduction The components of the nucleus are protons and neutrons, and these nucleons interact with each other through nucleon-nucleon (NN) interaction to form nucleus. Traditionally, the nuclear many-body problem was treated with the assumption that the masses of protons and neutrons are those experimentally measured at around 940 MeV, and the interactions between nucleons have been obtained from the experimentally measured nucleon-nucleon scattering data. Lots of knowledge on the nucleus were accumulated, and many-body theories were developed to analyze a large number of experimental data. The behaviors of nuclei are then influenced largely by the features of the NN interaction, which has a strong intermediate tensor interaction and a short-range repulsion. It is understood that the strong tensor interaction is caused by the pion exchange and the short-range repulsion originates from the quark structure of the nucleon. Hence, it is important to understand why the pion exchange is important for the nuclear physics and why the short-range repulsion is so strong. To this end, it is important to notice that the nucleon is made of quarks and gluons, whose dynamics is governed by quantum chromodynamics (QCD), which is a non-abelian gauge theory with three color charges for quarks and eight gluons (Han and Nambu 1965). The QCD Lagrangian consists of almost massless colored quarks (up and down quarks), and the gauge invariance provides the vector coupling with colored gluons. It is well known that the quarks and gluons are confined in hadrons and the chiral symmetry is dynamically broken. Hence, the quarks have large masses confined in hadrons, and the pions appear as Nambu-Goldstone bosons. As the consequence, the nucleons have the large mass and interact with each other through the small mass pion exchange. It is well known that perturbative QCD works nicely at high energy and the dynamics of quarks and gluons can be described well by QCD. The QCD coupling constant is small at high energy, but becomes strong at low energy (Gross and Wilzcek 1973; Politzer 1973). Hence, the non-perturbative aspect of QCD dynamics sets in at low energy below 1 GeV. There, the confinement and chiral symmetry breaking ought to be described using non-perturbative QCD. The non-perturbative aspect of QCD is studied numerically using lattice QCD by discretizing the QCD action (Wilson 1974). A lot of information has been accumulated for the confinement and chiral symmetry breaking, and lattice QCD is very powerful in getting observables as the hadron masses and other important quantities of hadrons. However, it does not tell the mechanism of confinement and chiral symmetry breaking using physical concepts. The fact that lattice QCD

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reproduces experimental data merely tells that the QCD is the theory of strong interaction. It would be nice if some Lagrangian with physically clear concepts could be constructed from the QCD Lagrangian at low energy and could be solved using the standard technology in quantum mechanics. It is important to challenge to construct the low-energy Lagrangian of QCD to be used for the low-energy hadron physics. To this end, there have been many efforts developed until now. Nielsen and Olesen introduced the GinzburgLandau Lagrangian to describe the string-like magnetic vortex in the standard superconducting material with the hope to describe the Veneziano model for hadrons and their interactions (Nielsen and Olesen 1973). Nambu proposed the dual version of the vortex for confinement of quarks with electric charges in the strong interaction (Nambu 1974). ’t Hooft showed that magnetic monopoles would appear in the abelian gauge by gauge fixing of the non-abelian gauge theory (Hooft 1981). Hence, at low energy, one promising non-perturbative technology is to introduce magnetic monopole in the abelian gauge and to condense the monopole fields as the dual Higgs mechanism for confinement of color electric charges. Such a model is called the dual Ginzburg-Landau (DGL) theory, which can be realized in a particular gauge called abelian gauge in QCD. The DGL theory is based on the existence of color magnetic monopoles, and its condensation provides the dual Meissner effect to confine charged quarks and gluons (Suzuki 1988; Maedan and Suzuki 1989; Suganuma et al. 1995a). In this chapter, the DGL Lagrangian is introduced following the scenario of confinement (Nambu 1974; Hooft 1981; Mandelstam et al. 1979), and its application on various phenomena is discussed. Here, the novel idea of magnetic vortex is introduced first, and the succeeding idea of the dual Meissner effect in the strong interaction is discussed. Following these ideas, the step of construction of the DGL Lagrangian is described. There are many publications on the dual Meissner effect for confinement and lattice simulations of magnetic monopoles and their condensation, and references are given in the  Chaps. 77 “Quantum Chromodynamics, Quark Confinement, and Chiral Symmetry Breaking: A Bridge Between Elementary Particle Physics and Nuclear Physics” and  79, “QCD Vacuum as Dual Superconductor: Quark Confinement and Topology” of Suganuma and Chernodub in this Handbook (Suganuma 2022; Chernodub 2022). The number of references is kept minimum in this chapter.

Magnetic Vortex and Monopole in Abelian Gauge of QCD Nielsen and Olesen studied the Ginzburg-Landau (GL) Lagrangian to describe string-like vortex with a hope to describe the Veneziano model in the field theoretical language (Nielsen and Olesen 1973). Their starting Lagrangian is the GL Lagrangian, which is written as 1 1 LGL = − Fμν F μν + |(∂μ + ieAμ )φ|2 + C2 |φ|2 − C4 |φ|4 . 4 2

(1)

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Here, Fμν = ∂μ Aν − ∂ν Aμ is the anti-symmetric tensor of the ordinary electromagnetic field Aμ , and the complex matter field φ with charge e couples with Aμ . This Lagrangian provides the London equation for superconductor, where the field Aμ acquires mass due to condensation of the matter field φ. ∂μ F μν = −MA2 Aν ,

(2)

2

where MA2 = e2 |φ|2 = e2CC42 . The superconductor material dislikes the electromagnetic field Aμ energetically and shows the superconducting property. As for the string-like vortex, the GL equations of motion are obtained by the Euler-Lagrange equation as (Nielsen and Olesen 1973) (∂μ + ieAμ )2 φ = 2C2 φ − 4C4 φ 2 φ ∗ , ∂μ F μν = j ν =

1 ie(φ ∗ ∂ ν φ − φ∂ ν φ ∗ ) + e2 Aν φ ∗ φ. 2

(3) (4)

The field tensor provides the magnetic flux Φ going through the surface element dσ μν   Φ = Fμν dσ μν = Aμ dx μ , (5) C

in the Minkowski space. Using the complex field φ = |φ|eiχ , Eq. (4) provides Aμ =

1 jμ 1 + ∂μ χ . 2 2 e e |φ|

(6)

The loop integral of the phase provides the flux as Φ = nΦ0 with Φ0 =

2π . e

(7)

Hence, the magnetic flux is quantized. To obtain the string-like solution, the axial symmetry is imposed to simplify the above equations. The potential is expressed as A(r) =

r × ez |A(r)|, r

(8)

for a static potential in the gauge A0 = 0 for the vortex in the z−direction. The axially symmetric equations are written as 1 d − r dr

    2 1 d 2 r |φ| + − e|A| − 2C2 + 4C4 |φ| |φ| = 0, dr r

(9)

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

d − dr



  1 d e (r|A|) + |φ|2 |A|e2 − = 0. r dr r

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(10)

Here, r is the radial direction from the center line of a vortex. There are two important scales for the behavior of the |φ| and |A| fields. They are expressed as the correlation length of a vortex.

1 = λ= e|φ| ξ=√

2C4 = m−1 A , e 2 C2

1 = m−1 φ . 2C2

(11) (12)

Here, the penetration length λ is the range of the magnetic field in the vortex, and the characteristic length ξ measures the distance of the condensed field φ in the vortex. One example of the behavior of the magnetic field and the condensed field in a vortex are provided in Fig. 1 of the Nielsen-Olesen paper (Nielsen and Olesen 1973). We shall provide a similar figure on a vortex later for the SU(3) color case in this chapter (Fig. 6). This vortex is infinitely long, and the radial behavior of the |A| and |Φ| fields is confined within certain distances characterized by λ and ξ , respectively. Hence, the Higgs mechanism in the GL Lagrangian provides a string-like vortex. Actually, this phenomenon is well known in the superconductor. Nielsen-Olesen proceeds to relate the vortex to the Nambu-Goto string action. It was Nambu who took the idea of the vortex to the confinement of quarks. Instead of the infinitely long vortex, Nambu considered an open-ended magnetic vortex, at both ends of which attached are particles with magnetic charges (Nambu 1974). Hence, the magnetic charges are connected by the magnetic vortex. Nambu discussed the case where the electric charge and magnetic charge coexist, which corresponds to the Dirac monopole theory (Dirac 1948): ∂ μ Fμν = −jν ,

(13)

∂ μ ∗Fμν = −kν .

(14)

This dual theory has very attractive characters for quark confinement and string phenomenology of strong interaction. Few years later, ’t Hooft verified that non-abelian gauge theory can provide magnetic charge in the abelian gauge (Hooft 1981). The abelian gauge is defined as a gauge fixing of some color variable X(x) to be diagonalized. In this case, at the degeneracy points of the eigenvalues of X(x), they behave as singular points of the residual abelian gauge field. Then, these degeneracy points originating from the color variable X(x) with the hedgehog characters become monopoles with respect to the abelian gauge field and provide world lines in the Minkowski space (Suganuma et al. 1995a).

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H. Toki

It is very interesting to say that the non-abelian gauge theory provides the physics world with color electric and color magnetic charges in the abelian gauge, where the interpretation of any phenomena in this gauge can be explained in the physics terms due to the abelian character of the gauge fields. All the ingredients to construct lowenergy effective theory of QCD for confinement were proposed to construct the dual Ginzburg-Landau theory with magnetic monopole fields and their condensation in the abelian gauge.

The QCD Lagrangian Toward the Dual Ginzburg-Landau Theory With all the necessary ingredients provided above, it is important to construct a low-energy effective theory of the non-abelian gauge theory, QCD. The step toward the DGL theory is nicely presented by Maedan and Suzuki (1989). The QCD Lagrangian density is written as the color gauge theory (Han and Nambu 1965). 1 μ ¯ LQCD = − tr (Fμν F μν ) + ψ(iγ Dμ − mψ )ψ. 2

(15)

Here, Fμν are the anti-symmetric tensors for the gluon fields Aμ , and the covariant derivative Dμ is written as Fμν = ∂μ Aν − ∂ν Aμ − ie[Aμ , Aν ], Dμ = ∂μ − ieAμ , Aμ = Aaμ T a = Aaμ

(16) λa 2

.

(17)

The quark fields ψ couple with the gluon fields Aμ through the gauge coupling with the coupling constant e. This Lagrangian is non-abelian, where the gluons make self-couplings. The perturbative treatment has been developed nicely, and it is well known that the running coupling constant becomes large at low energy, where non-perturbative technology has to be developed (Gross and Wilzcek 1973; Politzer 1973). The proposal of Nambu for color confinement due to the condensation of magnetic charge together with the appearance of magnetic monopole in abelian gauge fixing of ’t Hooft is very attractive for low-energy QCD (Nambu 1974; Hooft 1981). In addition, the choice of the abelian theory makes the interpretation of QCD phenomena possible using the physics analogies. Hence, it is useful to separate abelian gluons Aμ = (A3μ , A8μ ) from the non-diagonal gluons Cμi with i = 1, 2, 3, which are explicitly defined as the first step. These gluon fields couple with quarks a through the su(3) constant matrices, T a = λ2 (Maedan and Suzuki 1989). All the necessary notations are given here.

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

H = (H1 , H2 ) = (T3 , T8 ),

2945

(18)

1 1 1 E±1 = √ (T1 ± iT2 ), E±2 = √ (T4 ∓ iT5 ), E±3 = √ (T6 ± iT7 ). 2 2 2

(19)

We define here also the root vector  α for magnetic charge. [H, Eα ] =  α Eα ,

(20)

[H, E−α ] = − α E−α .

(21)

The gluon fields are written with the diagonal and non-diagonal gluon fields. Aμ = Aμ · H +

3 

(Cμ∗α Eα + Cμα E−α ).

(22)

α=1

Using these variables and coefficients, the QCD Lagrangian can be rewritten as LQCD = L1 + L2 .

(23)

The diagonal part and the non-diagonal part are written as 1 L1 = − fμν fμν , 4

(24)

1 ie |(Dα ∧ C α )μν + √ αβγ Cμ∗β Cν∗γ |2 2 2 α=1 3

L2 = −

+

(25)

3 3 ie  μν e2  (f ·  α )(C ∗α ∧ C α )μν + [ α (C ∗α ∧ C α )μν ]2 2 4 α=1

α=1

¯ μ ∂ − mψ + eγμ A · H)ψ + ψ(iγ μ

+e

3 

μ

¯ μ Eα ψ) + Cμα (ψγ ¯ μ E−α ψ)], [Cμ∗α (ψγ

α=1

where 3 8 (C ∧ D)μν = Cμ Dν − Cν Dμ , fμν = (fμν , fμν ), Dμα = ∂μ + ie( α · Aμ ). (26)

Up to here is rewriting of the QCD Lagrangian in terms of the abelian (diagonal) gluons and non-diagonal gluons, which behave as matter fields in the abelian gauge symmetry. In the abelian gauge by fixing the gauge partially, magnetic monopoles appear in general (Hooft 1981). The Bianchi identity is violated, and the derivative of the dual abelian field tensor should be written as

2946

H. Toki

∂μ∗ fμν

= k (x) = ν

N 



 ms

dτs

s=1

d ν x (τs ) δ 4 (x − xs (τs )). dτs s

(27)

Here, the magnetic charge m should be written as m=g



ξα  α ,

(28)

α

 with a constraint α  α = 0 and g = 4π e−1 . The magnetic monopoles live on the root vectors and move around the world line to provide the magnetic current kν . This magnetic current forces to introduce a modified field tensor as fμν = ∂μ Aν − ∂ν Aμ −

1 μναβ nα kβ . n·∂

(29)

Zwanziger proposed to write a local Lagrangian, which can treat both the electric and magnetic charges using the identity (Zwanziger 1971). Fμν =

1 {[n ∧ (n · F )]μν − ∗ [n ∧ (n ·∗ F )]μν }. n2

(30)

We then introduce the dual gauge field tensor [∂ ∧ B]μν = ∗ [∂ ∧ A]μν .

(31)

The Zwanziger form is written as L1 = L1 + L3 , 1 [n · (∂ ∧ A)]ν [n ·∗ (∂ ∧ B)]ν 2n2 1 + 2 [n · (∂ ∧ B)]ν [n ·∗ (∂ ∧ A)]ν 2n 1 1 − 2 [n · (∂ ∧ A)]2 − 2 [n · (∂ ∧ B)]2 , 2n 2n L3 = −kμ · Bμ .

L1 = −

(32) (33)

(34)

With this local Lagrangian in the Zwanziger form (33), the integration over the B fields reduces to the non-local form (29). As for the magnetic current, the introduction of magnetic monopole fields χα provides the magnetic current in the Minkowski space using the Samuel method (Bardakci and Samuel 1978). These considerations suggested the dual GinzburgLandau (DGL) theory to be written as a plausible theory of QCD at low energy in the abelian gauge, where the magnetic monopole is expressed as monopole fields

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

2947

and the condensation mechanism of the monopole fields should be introduced for the dual Meissner effect. Hence, the DGL Lagrangian is introduced here as (Suzuki 1988; Maedan and Suzuki 1989; Suganuma et al. 1995a) LDGL = L1 + L2 + Lg.f. +

3 

[|(∂μ + ig α · Bμ )χα |2 − λ(|χα |2 − v 2 )2 ], (35)

α=1

where Lg.f. is a gauge fixing term besides the monopole self-interaction term. Here, the monopole fields χα , α = 1, 2, 3 are not independent. Writing the complex fields as χα = |χα |eiφα ,

(36)

 the summation of the phase of the χ fields is zero: α φα = 0. When the magnetic monopole fields are condensed, the dual gauge fields Bμ become massive; √ √ mB = 3gv. At the same time, the monopole fields acquire masses as mχ = 2 λv. These two masses fix the size of the color electric vortex to be discussed later. The lattice QCD studies have been performed for the magnetic monopoles and their condensation in the QCD vacuum. The condensation of the magnetic monopoles was demonstrated in the maximal abelian gauge for the SU(3) case in the lattice QCD simulation (Arasaki et al. 1997). As for the non-diagonal gluons, it was demonstrated that they have large masses as 1.2 GeV in the maximal abelian gauge in lattice QCD (Amemiya and Suganuma 1999). Hence, hereafter the nondiagonal gluon part of the Lagrangian is dropped in the DGL Lagrangian. The DGL Lagrangian has to be used as a low-energy effective theory, and the short-range behavior obtained in the DGL theory has to be seen with caution.

Linear Potential in the DGL Theory The energy minimum condition causes the condensation of the monopole fields, and the mean field Lagrangian is written as (Suganuma et al. 1995a) 1 ¯ μ ∂ μ − eγμ Aμ · H − mψ )ψ + m2B B2μ . LDGL−MF = L1 + ψ(iγ 2

(37)

This is an important step for the dual Meissner effect, where the dual gauge fields Bμ (magnetic variables) acquire masses instead of the electric variables Aμ . Due to the dual Nielsen-Olesen vortex, the electric fields should be confined in the vortex. The information of the finite-size core in the vortex is lost by taking the mean field of condensed monopole fields for simplicity of formulation, but this feature will be recovered in the construction of the linear potential. Hence, with the mean field Lagrangian, the Bμ fields are integrated out first in favor of the Aμ fields for the gluon propagator for heavy quarks expressed by current jμ .

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H. Toki

LDGL−MF =

1 μ −1 A Dμν Aν + jμ Aμ . 2

(38)

Here, the gluon propagator is −1 Dμν = gμν ∂ 2 − (1 −

m2B n2 1 )∂μ ∂ν + Xμν , αe (n · ∂)2 + m2B n2

(39)

where Xμν is written as Xμν =

1 μαβ λνγ δ ε ε nα nγ ∂β ∂δ . n2 λ

(40)

The integration over the gluon fields provides the current-current correlation as

Lj −j

  m2B nμ nν ν n2  1 μ 1 gμν − 2 =− j gμν + j . 2 n ∂ 2 + m2B ∂ 2 + m2B (n · ∂)2

(41)

For the static quark and antiquark system, the current is given by the static color charges at two distant places. jμ = Qgμ0 [δ 3 (x − a) − δ 3 (x − b)].

(42)

From the current-current correlation, the potential between the two charges is written as V (r) = VYukawa (r) + Vlinear (r).

(43)

where r = a − b. The Yukawa term can be written as  VYukawa (r) = Q

2

=−

1 d 3k 1 (1 − eikr )(1 − e−ikr ) 3 2 (2π ) 2 k + m2B

(44)

Q2 e−mB r . 4π r

where a constant term is dropped. The linear term can be written as  Vlinear (r) = Q

2

m2B 1 d 3k [1 − cos(kr)] . 2 2 (2π )3 (n · k)2 k + mB

(45)

The vector n should be taken in the direction of r for axial symmetry; the integral is divided into the integral over the parallel and perpendicular directions

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

Vlinear (r) =

Q2 m2B 8π 2



∞

dkr [1 − cos(kr r)] kr2

−∞



∞ 0

dkT2

2949

1 kr2

+ kT2

+ m2B

.

(46)

The logarithmic divergence appears in the kT integral, which should be regulated by the coherent length m−1 χ of the monopole field due to the finite core size in the vortex. Hence, the integral becomes Vlinear (r) = =

Q2 m2B 8π 2



∞

−∞

m2χ + kr2 + m2B dkr [1 − cos(k r)] ln r kr2 kr2 + m2B

(47)

m2B + m2χ Q2 m2B r ln . 8π 2 m2B

This linear potential corresponds to the vortex energy (Nielsen and Olesen 1973). There is a phenomenological Cornell potential used for the analysis of heavy mesons VCornell = −

2 eC + kC r. 3π

(48)

The parameters of the DGL theory is fixed so as to reproduce the Cornell potential e = 5.5, mB = 0.5 GeV, mχ = 1.26 GeV. The obtained potential is shown in Fig. 1.

Fig. 1 The quark-antiquark potential obtained in the DGL theory is shown by the solid curve (Suganuma et al. 1995a). The Cornell potential with eC = 2.0 and kC = 1.0 GeV/fm is shown by the dashed curve. (Taken from Suganuma et al. 1995a)

(49)

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H. Toki

The DGL results shown by the solid curve reproduces the Cornell potential. Here, one should be cautious with the presence of the non-abelian part for the short-range behavior of the potential. Hence, the mean field approximation of the magnetic monopole field with introduction of the appropriate cutoff of the momentum integration provides the confining potential at least the long-range part of the quark-antiquark interaction.

Chiral Symmetry Breaking in the DGL Theory Dynamical chiral symmetry breaking is another important phenomenon in lowenergy QCD. Seeing the construction of the DGL theory, the confinement is the driving force to construct the DGL Lagrangian. Hence, it is a challenge for the DGL theory to study chiral symmetry breaking. Chiral symmetry breaking is studied using the Schwinger-Dyson equation for the quark propagator in the rainbow approximation Sq−1 (pM ) = γμ pM + μ



d 4 kM 2 μ sc Q γ Sq (kM )γ ν Dμν (kM − pM ). i(2π )4

(50)

Here, Sq (pM ) denotes the quark propagator with the Minkowski momentum pM . The screening gluon propagator is adopted here to take into account the breaking of the linear potential due to light quark-antiquark production (Suganuma et al. 1995a) sc Dμν (k) = −

1 k2

  m2B kμ kν 1 n2 gμν + (αe − 1) 2 + 2 Xμν . (51) 2 k k k 2 − mB (n · k)2 +  2

To simplify the SD equation, the quark propagator is expressed as 2 Sq (pM )−1 = γμ pM − M(−pM ) + iη. μ

(52)

With this simplification of the full quark propagator, the SD equation provides the mass equation as  M(p ) = 2

d 4 k 2 M(k 2 ) D sc (k − p), Q 2 (2π )4 k + M 2 (k 2 ) μμ

(53)

where the Wick rotation has been made to write in the Euclidean variables. The screening gluon propagator becomes sc (k) Dμμ

2 = (n · k)2 +  2



m2B −  2

2 + 2 2 k k 2 + mB

 +

1 + αe 2 + . k2 k 2 + m2B

(54)

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

2951

This mass equation should provide the mass of the quark. All the details to perform the angular integration and the treatment of the large momentum behavior in the perturbative region are written in Suganuma et al. (1995a). The numerical calculations are performed for the quark mass. The numerical results for the case e = 5.5, ΛQCD = 200 MeV, mB = 0.5 GeV, and  = 80 MeV are shown in Fig. 2. The calculation is done for the positive p2 values in the spacelike region. Here, the dynamical quark mass becomes M(0) = 348 MeV. The lightquark confinement is characterized by the disappearance of physical poles in the quark propagator. This is demonstrated by extrapolating the mass function M(p2 ) into the time-like region of p2 < 0 and comparing with the on-shell condition M 2 (p2 ) = −p2 (the dashed line) in the time-like region in Fig. 2. The extrapolated mass function M(p2 ) does not meet the on-shell condition. This indicates quark confinement. Chiral symmetry breaking is characterized by the quark condensate qq ¯ and the pion decay constant fπ . The quark condensate is expressed by the quark mass function as  qq ¯ Λ=

Λ

d 4 kM Nc tr Sq (kM ) = − 2 4 i(2π ) 4π

 0

Λ

dk 2

k 2 M(k 2 ) k 2 + M 2 (k 2 )

at the renormalization point Λ. The pion decay constant is written as

Fig. 2 The quark mass M 2 (p 2 ) in the space-like (p 2 > 0) and time-like (p 2 < 0) regions in unit of ΛQCD . The extrapolation in the time-like region using the numerical results does not meet the on-shell condition, indicating quark confinement. (Taken from Suganuma et al. 1995a)

(55)

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H. Toki

Fig. 3 The chiral condensate is shown in the left figure as functions of the coupling constant e for various mB . The pion decay constant fπ is shown in the right figure. (Taken from Suganuma et al. 1995a)

fπ2

Nc = 4π 2

 0

∞

k 2 M(k 2 ) dk 2 [k + M 2 (k 2 )]2 2

  k 2 dM(k 2 ) 2 M(k ) − . 2 dk 2

(56)

They are shown as functions of the coupling constant e in Fig. 3. The condensates for three sets of the dual gauge field mB are shown to see how these chiral quantities are sensitive to the confinement. The strength of these chiral quantities depends largely on the dual gauge field mass mB , which indicates that chiral symmetry breaking is caused by the dual Meissner effect. The comparison with the case mB = 0 is interesting in this respect. Without the monopole condensation, chiral symmetry breaking is not strong enough. This result is interpreted as the quark confinement provides strong impact for the motion of quarks in the confined region.

Meson Spectra in Pion Channel in the DGL Theory Pions are the Nambu-Goldstone modes in the strong interaction due to chiral symmetry breaking. The DGL theory is able to provide chiral symmetry breaking due to the magnetic condensation in the abelian gauge of QCD. It is interesting to study the quark content of the pions using the DGL theory. The mesons are made of quark and antiquark. The internal structure can be described by the BetheSalpeter equation, where the quarks are described by the Schwinger-Dyson equation as described in the previous section. This program was carried out by Kusaka et al. (1999). The Bethe-Salpeter (BS) amplitude is calculated in the diagram shown in Fig. 4, where the quark propagator

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

2953

Fig. 4 The Bethe-Salpeter equation for the BS amplitude for pions

Fig. 5 The meson spectra in pion channel obtained in the DGL theory. (Taken from Kusaka et al. 1999)

is obtained by solving the SD equation. The BS amplitude was solved order-byorder method. All the details are written in Kusaka et al. (1999). The results of pion-channel meson spectra are shown in Fig. 5. The calculated results are expressed by black dots, which were connected by dashed line. The experimental masses are shown by cross marks with error bars. The Regge slopes are clearly observed.

2954

H. Toki

The String Tension and Ring Solution in the DGL Theory The SU(3) color gauge theory corresponds to the real case, and the parameters of the model can be compared with the experimental results. The string solution was studied for the case of the SU(3) color DGL theory by Koma et al. (1999). In the DGL theory, after integrating out the A fields with the quark current as the source term jμ , the DGL Lagrangian is written in the following form: LDGL = − +

2  1 1 ∂μ Bν − ∂ν Bμ − μναβ nα jβ 4 n·∂ 3 

(57)

[|(∂μ + igα · B)χα |2 − λ(|χα |2 − v 2 )2 ].

α=1

Here, the color current is written as jμ = Qg μ0 [δ 3 (x − a) − δ 3 (x − b)].

(58)

¯ and then all Without loss of generality, the edge color of the vortex is taken as R-R, the values for the root vectors are set. In this case, the Lagrangian is rewritten for the open string-vortex as L = −

11 1 (∂μ BνR − ∂ν BμR )2 + 2|(∂μ + igBμR )χ R |2 + 2λ(|χ R |2 − v 2 )2 . (59) 34 2

Hence, by the redefinition of the fields and the coefficients, BμR =

√ 1 2 ˆ λ = 2λˆ , v = √ v, ˆ 3Bμ , χ R = φ, g = √ g, 3 2

(60)

the resulting Lagrangian is written as 1 2 ˆ L = − (∂μ Bν − ∂ν Bμ )2 + |(∂μ + i gB ˆ μ )φ|2 − λ(|φ| − vˆ 2 )2 . 4

(61)

The calculation of the vortex follows the calculation of Nielsen and Olesen discussed before. The results are shown in Fig. 6. The vortex structure is obtained for the Ez field and the magnetic monopole field φ. The open string may provide the ring solution by changing the boundary condition. This ring solution corresponds to a glueball without valence quarks. The configuration is shown in Fig. 7. The resulting energy of this ring solution is around MR = 1.6 GeV.

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

2955

Fig. 6 The color electric field Ez and the magnetic monopole field |φ| are plotted as functions of the distance in the radial direction. The φ field approaches the asymptotic value at large r. (Taken from Koma et al. 1999) Fig. 7 The ring solution of the electric vortex. (Taken from Koma et al. 1999)

Confinement-Deconfinement Phase Transition in the DGL Theory The confinement-deconfinement phase transition can be described in the DGL theory. The DGL Lagrangian of the dual gauge part together with the monopole fields is written as (Ichie et al. 1995) 1 LDGL = − (∂μ Bν − ∂ν Bμ )2 4 +

(62)

3  [|(∂μ + ig a · Bμ )χa |2 − λ(|χa |2 − v 2 )2 ]. a=1

To treat finite temperature, the DGL partition function is introduced as the generating functional in the imaginary-time formalism

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H. Toki

 Z[J ] =

   3  4 2 Dχa DBμ exp − d x(LDGL − J |χa | ) .

(63)

a=1

After taking into account the fluctuation around the mean field value for the monopole fields, the effective potential (the free energy) at finite temperature is written as     ∞ T − k 2 +m2B /T 2 Veff (χ¯ ; T ) = 3λ(χ¯ − v ) + 3 2 dkk ln 1 − e π 0     ∞ 3 T − k 2 +m2χ /T 2 + . dkk ln 1 − e 2 π2 0 2

2 2

(64)

The parameters of the DGL Lagrangian were set to λ = 25, v = 0.126 GeV, g = 2.3.

(65)

These values provide the masses of the dual gauge fields and the monopole fields as mB = 0.5 GeV, mχ = 1.26 GeV

(66)

at T = 0. These parameter values provide the effective potential at finite temperature as shown in Fig. 8. At zero temperature, the effective potential shows an absolute minimum at finite monopole condensate χ . The minimum moves toward

Fig. 8 The effective potential at finite temperature in the DGL theory. The energy minimum is indicated by ×, which appears at finite χ up to T = 0.45 GeV and moves to χ = 0 above T = 0.5 GeV. (Taken from Ichie et al. 1995)

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

2957

Fig. 9 The monopole mass mχ and the dual gauge mass mB as functions of temperature for the modified λ value. These masses drop to zero at Tc = 0.28 GeV as shown in the left figure. In the right figure, the string tension is compared with the lattice QCD results shown by the dots as function of the temperature. (Taken from Ichie et al. 1995)

small χ , and around T = 0.45 MeV, the minimum jumps from finite χ value to χ = 0. The critical temperature of this phase transition is Tc = 0.49 GeV, and it is a first-order phase transition. This corresponds to the confinement-deconfinement phase transition at finite temperature. The DGL Lagrangian is constructed at zero temperature. It is plausible that the monopole-monopole interaction is weakened due to the temperature effect. Hence, the interaction is reduced as  λr = λ

Tc − aT Tc

 .

(67)

These parameters are chosen to reproduce the critical temperature at Tc = 0.28 GeV as a = 0.89. With this modification, the change of the masses for mχ and mB is shown in Fig. 9. They drop to zero at the critical temperature. It is interesting to see that the string tension so obtained is compared with the lattice QCD results in the right figure. The agreement is very good.

Monopoles and Instantons In QCD, the instanton is another topological object. The non-abelian action provides the instanton solution. According to the Atiyah-Singer index theorem (Atiyah and Singer 1968), a chiral zero mode of the Dirac operator arises in the presence of the instanton. In the instanton vacuum, overlapping of the would-be zero modes leads to a non-zero chiral condensate and chiral symmetry breaking. Since the DGL theory provides chiral symmetry breaking due to the monopole condensation, it is natural

2958

H. Toki

to consider if the instantons are the sources of the color magnetic monopoles. In fact, the one instanton configuration in the regular gauge is given as Aμ =

1 a xν σa η . e νμ x 2 + ρ 2

(68)

a is called the self-dual ’t Hooft symbol Here, ημν

a ημν =

1 a tr(σa τμ τν† ) = −ηνμ = 2i

 aμν for μ, ν = 1, 2, 3 . −δaν for μ = 4 δaμ for ν = 4

(69)

Here, σ is the spin 2 × 2 Pauli matrix and τμ = (σ, −i). We write here the fourth component of the gauge field A4 =

1 xa σa . e x2 + ρ2

(70)

Hence, A4 becomes a hedgehog configuration. The abelian gauge fixing using this A4 field produces the monopole in the abelian space, as shown in Fig. 10. This observation is very interesting to imagine that the source of monopole fields is instantons. This relation was studied by several groups (Suganuma et al. 1995b; Chernodub and Gubarev 1995; Fukushima et al. 1997). It is then very natural to ask if the QCD ground states have many instantons and anti-instantons. This study was performed by Fukushima et al. using multi-instanton gas (Fukushima et al. 1997). With a reasonable density of instanton gas, they show

Fig. 10 Examples of monopole trajectory (world line) in two-instanton systems in an abelian gauge (Suganuma et al. 1995b). The monopole trajectory passes through the instanton center denoted by the circle

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

2959

the probability distribution of monopole strings (world lines) as a function of their length. High-density instanton gas provides a clustering of very long monopole strings, which is the indication of monopole condensation.

Hadrons in the DGL Theory Many studies have been performed using the DGL Lagrangian as discussed up to here. The confinement is highly difficult to treat, while chiral symmetry breaking is easier to treat following the NJL Lagrangian, since the color dynamics is not treated except for the number of degrees of freedom. It is difficult to consider both the confinement and chiral symmetry breaking to describe hadrons, and it has been resorted to take the quark soliton model, where the chiral symmetry plays an important role (Diakonov et al. 1997). The chiral quark soliton model Lagrangian is written as ¯ μ γ μ − MU γ5 )ψ, LCQM = ψ(i∂

(71)

where the chiral field is U = eiπ ·τ and U γ5 = eiπ ·τ γ5 =

1 − γ5 † 1 + γ5 U (x) + U (x). 2 2

(72)

The quark soliton model describes best the baryon properties by taking the hedgehog ansatz for the pion field Uc (r) = ein

a τ a P (r)

(73)

where r = |x| and na = x a /r. There are many studies on the SU(3)f baryons using this method, and the baryon properties are nicely reproduced. In these studies, it turned out that the pion fields in the hedgehog configuration play an important role. Particularly, the hedgehog description naturally explains the second excited state of nucleon being the delta state, which has S = 3/2 and T = 3/2. This fact comes from the conserved quantity of the hedgehog ansatz, the grand-spin G = J + T = L + S + T, where S is the spin and T is the isospin. If the ground state has Gπ = 0+ , then S + T has to be zero. In this case, only S = 1/2, T = 1/2, and S = 3/2, T = 3/2 states are obtained by projection to good spin and isospin states, and S = 1/2 and T = 3/2 states cannot be created from the Gπ = 0+ ground state. The quark soliton model takes into account the chiral properties of the strong interaction. However, it lacks the apparent dynamics of quark confinement. Since the confinement is nicely described by the DGL theory, it is interesting to take the DGL Lagrangian instead. Although the confinement is caused by the dual Meissner effect, chiral symmetry breaking part is not obvious in the DGL theory. However, it has been demonstrated that chiral symmetry breaking takes place in the DGL theory

2960

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due to the condensation of the monopole fields. Hence, the Nambu-Goldstone boson should appear in the form of massless pions. The good start of the Lagrangian is to take first the monopole condensation, and the Lagrangian is written in terms of the Aμ fields and the quark fields. In order to construct hadrons, the spherical symmetry has to be imposed for the static configurations. Due to the success of the quark soliton model, the hedgehog ansatz should be taken for the quark side, keeping the quantum number G = L + S + T. It is a challenge to describe nucleons and deltas using the DGL theory, since it has all the necessary ingredients for low-lying hadrons.

Dual Ginzburg-Landau Theory in Quark-Hadron Matter The NJL model has been used for the discussion of the change of hadron properties in nuclear matter. The NJL model is written in terms of nearly zero mass quarks interacting chiral symmetrically by the four point coupling terms (Nambu and Jona-Lasinio 1961). The quark property is nicely described by the mean field approximation, and the pions appear as Nambu-Goldstone modes. The NJL Lagrangian is written as ¯ μ ∂ μ − m)ψ + LNJL = ψ(iγ

G ¯ 2 + (ψiγ ¯ 5 τ ψ)2 ] [(ψψ) 2

(74)

Chiral symmetry breaking can be easily realized in this Lagrangian by taking the ¯ finite mean field for ψψ. ¯ μ ∂ μ − m − M)ψ + LNJL−MF = ψ(iγ

G ¯ 5 τ ψ)2 (ψγ 2

(75)

where ¯ M = −Gψψ

(76)

For the strong coupling of G, the vacuum expectation value of the quark scalar density becomes negative finite, which indicates chiral symmetry breaking. The residual interaction for the γ5 term coupling provides the massless pion field. This is the mechanism of chiral symmetry breaking in the vacuum, where quarks have finite masses and the pion mass becomes small. In nuclear matter, the ground state consists of nucleons, and they occupy low momentum states up to the Fermi surface. The description of nucleons in terms of quarks has not been performed, and it is customary to fill low momentum states up to the Fermi surface of quarks. Hence, the ground state of finite density matter is written as

80 Quark Nuclear Physics for Hadrons and Nuclei in the Dual Ginzburg-. . .

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ai |0 = 0, i > F,

(77)

ai† |0

(78)

= 0, i ≤ F.

Here, F denotes the momentum states with the Fermi momentum. The mean field approximation for finite density ground state is expressed by the quark condensation as ¯ 0|ψψ|0 = 12



F 0

d 3k MF  − 12 (2π )3 M 2 + k 2 F



Λ 0

d 3k MF  (2π )3 M 2 + k 2

(79)

F

Here, 12 = 2 × 2 × 3 is the two spin states, two isospin states, and three color states. The quark mass MF is the quark mass in medium to be determined by energy minimization. Hence, the chiral condensate decreases due to the finite Fermi sea, and the mass of quarks and therefore the mass of the nucleons decrease with density. This result has been used for the modification of the chiral condensate and hadron properties in nuclear medium. There is an expectation that the chiral symmetry is restored at large density, and the NJL results are in the line of this expectation. However, the NJL model lacks the confinement mechanism, and the present result should be treated as a temporary knowledge. If confinement is the important physics to be included in the description of nuclear matter, the use of the DGL Lagrangian should be tried instead of the NJL Lagrangian. To this end, certainly the light hadrons including nucleons have to be constructed first in the DGL theory.

Summary and Perspectives The QCD Lagrangian was introduced as the fundamental theory of strong interaction for quarks and gluons. Due to the difficulties of solving the Lagrangian at low energy, the dual Ginzburg-Landau theory was proposed as an alternative of QCD, which had the fundamental properties as confinement and chiral symmetry breaking. The DGL Lagrangian was able to describe the confinement potential and the quark mass as well as the pion mass spectrum. The NJL Lagrangian was investigated for many applications as the meson spectrum, chiral symmetry breaking at finite temperature and density. Although the NJL Lagrangian was good for the chiral symmetry, it lacks the confinement mechanism. Hence, it was proposed to use the DGL Lagrangian instead of the NJL Lagrangian. It would be important to study hadrons and nuclear many-body systems using the QCD-motivated model with confinement. This subject is completely open for the community to describe hadrons and nuclei.

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References K. Amemiya, H. Suganuma, Off-diagonal gluon mass generation and infrared Abelian dominance in the maximally Abelian gauge in lattice QCD. Phys. Rev. D60, 114509 (1999) N. Arasaki, S. Ejiri, S. Kitahara, Y. Matsubara, T. Suzuki, Monopole action and monopole condensation in SU(3) lattice QCD. Phys. Lett. B395, 275–282 (1997) M. Atiyah, I. Singer, The Index of elliptic operators. I. Ann. Math. 87, 484–530 (1968) K. Bardakci, S. Samuel, Local field theory for solitons. Phys. Rev. D18, 2849–2860 (1978) M. Chernodub, QCD vacuum as dual superconductor: quark confinement and topology. Handbook of Nuclear Physics (2022), Springer M.N. Chernodub, F.V. Gubarev, Instantons and monopoles in maximal Abelian projection of SU(2) gluodynamics. JETP Lett. 62, 100–104 (1995) D.I. Diakonov, Y. Petrov, P.V. Pobylitsa, M.V. Polyakov, C. Weiss, Unpolarized and polarized quark distributions in the large-Nc limit. Nucl. Phys. D56, 4069–4083 (1997) P.A.M. Dirac, The theory of magnetic poles. Phys. Rev. 74, 817–830 (1948) M. Fukushima, S. Sasaki, H. Suganuma, A. Tanaka, H. Toki, D. Diakonov, Clustering of monopoles in the instanton vacuum. Phys. Lett. B399, 141–147 (1997) D.J. Gross, F. Wilzcek, Ultraviolet behavior of non-Abelian gauge theories. Phys. Rev. Lett. 30, 1343–1346 (1973) M.Y. Han, Y. Nambu, Three triplet model with double SU(3) symmetry. Phys. Rev. 139(B), 1006– B1010 (1965) G. ’t Hooft, Topology of the gauge condition and new confinement phases in non-Abelian gauge theories. Nucl. Phys. B190, 455–478 (1981) H. Ichie, H. Suganuma, H. Toki, QCD phase transition at finite temperature in the dual GinzburgLandau theory. Phys. Rev. D52, 2944–2950 (1995) Y. Koma, H. Suganuma, H. Toki, Flux-tube ring and glueball properties in the dual GinzburgLandau theory. Phys. Rev. 60, 074024 (1999) K. Kusaka, T. Sakai, H. Toki, Bethe-Salpeter approach for mesons in the pion channel within the Dual Ginzburg-Landau theory. Prog. Theor. Phys. 101, 722–747 (1999) S. Maedan, T. Suzuki, An infrared effective theory of quark confinement based on monopole condensation. Prog. Theor. Phys. 81, 229–240 (1989) S. Mandelstam, Charge-monopole duality and the phases of non-Abelian gauge theories. Phys. Rev. D19, 2391–2409 (1979) Y. Nambu, Strings, monopoles, and gauge fields. Phys. Rev. D10, 4262–4268 (1974) Y. Nambu, G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. Phys. Rev. 122(1), 345–358 (1961) H.B. Nielsen, P. Olesen, Vortex-line models for dual strings. Nucl. Phys. B61, 45–61 (1973) H.D. Politzer, Reliable perturbative results for strong interactions? Phys. Rev. Lett. 30, 1346–1349 (1973) H. Suganuma, Quantum chromodynamics, quark confinement and chiral symmetry breaking. Handbook of Nuclear Physics (2022), Springer H. Suganuma, S. Sasaki, H. Toki, Color confinement, quark pair creation and dynamical chiralsymmetry breaking in the dual Ginzburg-Landau theory. Nucl. Phys. B435, 207–240 (1995a) H. Suganuma, K. Itakura, H. Toki, O. Miyamura, Correlation between instantons and QCDmonopoles in the Abelian gauge, in International Workshop on Non-Perturbative Approaches to Quantum Chromodynamics (PNPI Press, 1995b), pp.224–238. arXiv:hep-ph/9512347 [hep-ph] T. Suzuki, A Ginzburg-Landau type theory of quark confinement. Prog. Theor. Phys. 80, 929–934 (1988) K.G. Wilson, Confinement of quarks. Phys. Rev. D10, 2445–2459 (1974) D. Zwanziger, Local-Lagrangian quantum field theory of electric and magnetic charges. Phys. Rev. D3, 880–891 (1971)

Quark Nuclear Physics with Heavy Quarks

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Nora Brambilla

Contents The Role of Heavy Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heavy-Light Mesons, Quarkonia, Baryons with Two or More Heavy Quarks . . . . . . . . . . . The Potential and the Phenomenology of Quarkonium . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonrelativistic Effective Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonrelativistic QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential Nonrelativistic QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weakly Coupled pNRQCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strongly Coupled pNRQCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectra, Transitions, Decays and Production, and SM Parameters Extractions . . . . . . . . Confinement and Low-Energy QCD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BOEFT and X Y Z Exotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pNRQCD at Finite T, Open Quantum System, and Quarkonium in Medium . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2964 2966 2970 2975 2976 2978 2979 2984 2987 2988 2989 2993 2998 2999

Abstract

Heavy quarks have been instrumental for progress in our exploration of strong interactions. Quarkonium in particular, a heavy quark-antiquark nonrelativistic bound state, has been at the root of several revolutions. Quarkonium is endowed with a pattern of separated energy scales qualifying it as special probe of complex environments. Its multiscale nature has made a description in quantum field theory particularly difficult up to the advent of nonrelativistic effective field theories. We will focus on systems made by two or more heavy quarks.

N. Brambilla () Physik-Department, Technische Universität München, Garching, Germany Institute for Advanced Study, Technische Universität München, Garching, Germany Munich Data Science Institute, Technische Universität München, Garching, Germany e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_26

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After considering some historical approaches based on the potential models and the Wilson loop approach, we will introduce the contemporary nonrelativistic effective field theory descriptions, in particular potential nonrelativistic QCD which entails the Schödinger equation as zero order problem, defines the potentials as matching coefficients, and allows systematic calculations of the physical properties. The effective field theory allows us to explore quarkonium properties in the realm of QCD. In particular, it allows us to make calculations with unprecedented precision when high order perturbative calculations are possible and to systematically factorize short from long range contributions where observables are sensitive to the nonperturbative dynamics of QCD. Such effective field theory treatment can be extended at finite temperature and in presence of gluonic and light quark excitations. We will show that in this novel theoretical framework, quarkonium can play a crucial role for a number of problems at the frontier of our research, from the investigation of the confinement dynamics in strong interactions to the study of deconfinement and the phase diagram of nuclear matter, to the precise determination of standard model parameters up to the emergence of exotics X Y Z states of an unprecedented nature.

The Role of Heavy Quarks Heavy quarks and bound states of heavy quarks, primarily quarkonium, a bound state of a heavy quark and a heavy antiquark, have been historically instrumental to construct the theory of strong interactions and continue today to be at the forefront of our research as a golden probe of the strong dynamics. The discovery of heavy quarks drastically changed the standard model (SM) of particle physics. This happened in the November revolution of 1974 when two labs on opposite sides of the USA announced discovery of a new particle, a fact that helped the acceptance of the standard model of particle physics (Aubert et al. 1974; Augustin et al. 1974). The new particle was the first example of quarkonium: the J /ψ, the lowest excitation made by a charm and an anticharm. The J /ψ appeared as an unprecedented sharp peak, tall and narrow, 3 GeV in mass and 90 KeV in width, at variance with the typical width of several tens and hundreds of MeV of the hadrons discovered up to that time, i.e., strongly interacting light quark composite particles. The J /ψ discovery represented the confirmation of the quark model, the discovery of the charm quark, the confirmation of the GIM mechanism (Glashow et al. 1970) (the mechanism through which flavor-changing neutral currents are suppressed in loop diagrams), and the first discovery of a quark of large mass moving nonrelativistically. It triggered additional searches and in few years the higher excitations of charmonium were discovered as well as bottomonia (1977), i.e., bound states of bottom and antibottom, Bc (1998), and the top (1995). States made by top and antitop decay weakly before forming a proper bound state, however still leaving their signature in the form of an enhancement of the cross section at threshold (Brambilla et al. 2005).

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The J /ψ discovery was the confirmation of quantum chromodynamics (QCD) (Fritzsch et al. 1973), the quantum field theory describing strong interactions. QCD has a well-defined behavior in the ultraviolet (UV) region at large energy and a fundamental coupling constant αs = g 2 /4π running from small values at large energy to large values at small energy. This encodes the properties of asymptotic freedom (quarks are free at high momentum transfer) and confinement (quarks are confined in color singlet hadrons at low energy) (Wilson 1974). As we will see, confinement becomes manifest in the case of heavy quarks, where one can write the color singlet quark-antiquark interaction potential in terms of a so called Wilson loop (Brambilla and Vairo 1999). Confinement emerges in an area law of the Wilson loop and correspondingly in a linear potential growing with the distance between the quarks (Creutz 1977). An emergent scale ΛQCD parametrizes the importance of the nonperturbative corrections, i.e., the contributions that cannot be calculated in an expansion in αs and is mirrored in the hadron spectrum, the mass of the proton being proportional to ΛQCD . We consider a quark to be heavy when its mass m is larger than the scale ΛQCD : this qualifies as heavy the charm, bottom, and top quarks. These fundamental features of QCD find the best realization in the J /ψ and in quarkonium in general. The small width can be explained by the fact that J /ψ is the lowest energy level and can decay only via annihilation, which makes available in the process a large energy, of order of two times the mass of the charm (about 2 GeV). The annihilation width is then proportional to αs2 (2mc ) which is small due to asymptotic freedom, since mc is bigger than ΛQCD . On the other hand, when theorists set up to investigate the structure of the energy levels of charmonium and bottomonium, they noticed that it can be reproduced by using the Schrödinger equation a static potential superposition of an attractive Coulomb contribution (with ¯ and a term linear in the the appropriate SU (3) color factor for a singlet QQ) distance: the famous Cornell potential (Eichten et al. 1978, 1980). Such form of the potential has been later confirmed by nonperturbative calculations performed using computational lattice QCD (Creutz 1977; Brambilla and Vairo 1999; Rothe 1992; Bali 2001; Campostrini et al. 1987). In the following, we will address the importance of heavy quarks for quark nuclear physics. We will focus on systems made by two or more heavy quarks and discuss how they play a crucial role for a number of problems at the frontier of our research, from the investigation of the confinement dynamics in QCD to the study of deconfinement and the phase diagram of nuclear matter, to the precise determination of standard model parameters up to the emergence of exotics X Y Z states of an unprecedented nature (Brambilla et al. 2011, 2014, 2005; Andronic et al. 2016; Chapon et al. 2022). In particular, we will conclude that our progress in these strong interaction topics is connected to a broad sweep of physical problems in settings ranging from astrophysics and cosmology to strongly correlated systems in particle and condensed matter physics as well as to search of physics beyond the standard model. Heavy quarks had and have a key role to address the phenomenon of CP (charge conjugation and parity) violation, which is crucial to understand the asymmetry

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between matter and antimatter that exists in the universe, by testing and constraining the Cabibbo-Kobayashi-Maskawa matrix entries entering B and D mesons decays and mixing and by identifying new physics contributions; see, e.g., the reviews Buchalla et al. (2008), Neubert et al. (1996), Chang et al. (2017), and Gershon and Gligorov (2017). The Babar and Belle experiments at the B factories (Kou et al. 2019; Bevan et al. 2014) have been constructed to this aim but turned out to be also formidable heavy mesons machines giving a great boost to out knowledge of heavy quark systems and their strong interaction dynamics. The review is organized as follows. First, we will discuss the physics characteristics of systems made by heavy quarks. Then, we will summarize what have been historically the pioneering approaches and the phenomenological models. After that, we will explain that to address in quantum field theory a nonrelativistic multiscale system like quarkonium, it is necessary resort to nonrelativistic effective field theories (NREFTs). We will show how to construct NREFTs up to arriving at the simplest possible version, called potential nonrelativistic QCD (pNRQCD) which implements the Schrödinger equation as the zero order problem. Then, we will show how combining NREFTs and lattice we can get systematic and under control predictions on a number of physical processes and observables like the spectrum, decays, transitions, and production. In this framework, quarkonium becomes a unique laboratory for the study of strong interactions from the highenergy to the low-energy scales. The NREFT allows us to explore quarkonium properties in the realm of QCD. In particular, it allows us to make calculations with unprecedented precision when high order perturbative calculations are possible and to systematically factorize short from range contributions where observables are sensitive to the nonperturbative dynamics of QCD. Finally, we will address some of the most interesting open problems in relation to quarkonium, i.e., the nonequilibrium propagation of quarkonium in medium which calls for introducing open quantum system on top of NREFTs and the new exotic stares X Y Z discovered at the accelerator experiments.

Heavy-Light Mesons, Quarkonia, Baryons with Two or More Heavy Quarks Heavy quarkonia are systems composed by a heavy quark and a heavy antiquark with mass m larger than the “QCD confinement scale” ΛQCD , so that αs (m)  1 holds. We have that mc ∼ 1.5 GeV and mb ∼ 5 GeV. Of course the quark masses are scheme-dependent objects and their actual value depends on the considered scheme and scale, only the pole mass being scale independent; see, e.g., the review Brambilla et al. (2005). From the quarkonia spectra, see Figs. 1, 2, and 3, it is evident that the difference in the orbital energy levels is much smaller than the quark mass. It scales like mv 2 , while fine and hyperfine separations scale like mv 4 . Here v is the heavy quark velocity ( v = |v|) in the rest frame of the meson in ¯ v 2 ∼ 0.3 for cc¯ systems. This is the same units of c, and v 2 ∼ 0.1 for the bb, scaling of the hydrogen atom if one identifies v with the fine structure constant αem .

81 Quark Nuclear Physics with Heavy Quarks

MeV 500

2967

Υ(4S)

Υ(3S)

BB threshold ψ (3S) Χb(2P) ψ (3770)

Υ(2S)

η c (2S)

0

Χb(1P)

Χc(1P)

hc(1P)

J /Ψ

Υ(1S) −500

DD threshold

ψ (2S)

η c (1S)

S states

P states

Fig. 1 The experimental quarkonium energy levels (bb¯ and cc) ¯ as known in the eighties (plotted as relative to the spin-average of the χb (1P ) and χc (1P ) states to be able to present the two sectors in the same plot). The states are identified by names and mass in parenthesis, Υ being the vector states (with L = 0) in bottomonium, J /ψ and ψ the vector states in charmonium sector, η the pseudoscalar states, and χ and h states with L = 1 and different total angular momentum. The first strong decay thresholds are also shown

Therefore quarkonia are nonrelativistic systems. Being nonrelativistic, quarkonia are characterized by a hierarchy of energy scales: the mass m of the heavy quark (hard scale), the typical relative momentum p ∼ mv (in the meson rest frame) corresponding to the inverse Bohr radius r ∼ 1/(mv) (soft scale), and the typical binding energy E ∼ mv 2 (ultrasoft scale). Of course, for quarkonium there is another scale that can never be switched off in QCD, i.e., ΛQCD , the scale at which nonperturbative effects become dominant. A similar pattern of scales emerge in the case of baryons composed of two or three heavy quarks (Brambilla et al. 2005) and for the just discovered state X(6900) (Aaij et al. 2020) made by two charm and two anticharm quarks. The pattern of nonrelativistic scales makes all the difference between heavy quarkonia and heavy-light mesons, which are characterized by just two scales: m and ΛQCD . Being a multiscale system, heavy quarkonium is probing different energy regimes of the strong interactions, from the hard region, where an expansion in the coupling constant is possible and precision studies may be done, to the low-energy region, dominated by confinement and the many manifestations of nonperturbative dynamics. In addition, the properties of production and absorption of quarkonium

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Fig. 2 The spectrum of states in the cc¯ sector as of 2019, taken from Brambilla et al. (2020b). Thin solid lines represent the states established experimentally and dashed lines are for those that are claimed but not (yet) established (a state is regarded as established if it is seen in different modes). States whose quantum numbers are undetermined are not shown. States in the plot are labeled according to the PDG primary naming scheme that superseded the X Y Z notation; for further details see Brambilla et al. (2020b). Dashed lines show some relevant thresholds that open in the considered mass range; here D1 stands for D1 (2420) and D2∗ for D2∗ (2460). Thresholds with hidden strangeness or involving broad states are not shown. The states shown in the two columns to the right are isovectors containing a cc ¯ pair; they are necessarily exotic. The just discovered state X(6900) (Aaij et al. 2020) made by two charm and two anticharm quarks would be out of the scale of the figure. More states have been observed after 2019

in a nuclear and hot medium are crucial inputs for the study of QCD at high density and temperature, reaching out to cosmology. On the experimental side, the diversity, quantity, and accuracy of the data collected in the last decades is impressive and includes clean and precise samples of quarkonia spectra decay, transition and production processes, including the discovery of exotics X Y Z states at tau-charm (BES experiment) and B factories (Babar and Belle experiments), hadroproduction at Fermilab Tevatron and the Large Hadron Collider (LHC) experiments at CERN, production in photon-gluon fusion at DESY, photoproduction at Jlab, heavy ions production and suppression at RHIC, NA60, and LHC. New data are coming also from the upgraded experiments BELLE II and BESIII (Brambilla et al. 2011, 2005;

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Fig. 3 The spectrum of states in the cc ¯ sector as of 2019, taken from Brambilla et al. (2020b). Thin solid lines represent the states established experimentally and dashed lines are for those that are claimed but not (yet) established (a state is regarded as established if it is seen in different modes). States whose quantum numbers are undetermined are not shown. States in the plot are labeled according to the PDG primary naming scheme that superseded the X Y Z notation; for further details, see Brambilla et al. (2020b). Dashed lines show some relevant thresholds that open in the considered mass range; here D1 stands for D1 (2420) and D2∗ for D2∗ (2460). Thresholds with hidden strangeness or involving broad states are not shown. The states shown in the two columns to the right are isovectors containing a cc ¯ pair; they are necessarily exotic

Andronic et al. 2016; Chapon et al. 2022; Yuan and Olsen 2019) and more will come in the future from Panda at FAIR and the Electron Ion Collider (EIC) (Yuan and Olsen 2019; Barucca et al. 2021). On the theoretical side, the last few decades have seen the construction of new nonrelativistic effective field theories and new developments in computational lattice QCD which supply us with a systematic calculational framework in quantum field theory. All this make quarkonium a golden probe of strong interactions. From now on, we will concentrate on the study of systems with two or more heavy quarks. We will outline the rich interplay of theoretical advancement and experimental success and its implication on our control of strong interactions inside the standard model of particle physics. Before, however, I will summarize the models that have been used in the past to describe the quarkonium properties.

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The Potential and the Phenomenology of Quarkonium You find in Figs. 2 and 3 our present knowledge about the states in the charmonium and in the bottomonium sector. We will consider in the following quarks of equal mass for simplicity (in the case of Bc one should take into account that the masses are different). The quark and the antiquark spins combine to give the total spin S = S1 + S2 which combines with the orbital angular momentum L to give the total angular momentum J. The resulting state solution of a Schrödinger equation is denoted by n2S+1 LJ where n−1 is the number of radial nodes of the wave function. As usual, to L = 0 is given the name S, to L = 1 the name P , to L = 2 the name D, and so on. The experimental resonances (see Figs. 2 and 3) are classified via the J PC quantum numbers, P = (−1)L+1 being the parity number and C = (−1)L+S the C-parity. Strong decay thresholds are marked by horizontal dashed lines that represent the energy necessary to decay in a couple of heavy-light mesons. In the charmonium sector, only 10 states of the states presented in Fig. 2 have been discovered before 1980 and no one between 1980 and 2002. In 2003 the new revolution started with the discovery of the X(3872) (Choi et al. 2003) and a number of new states above and below the strong decay threshold (Brambilla et al. 2011). Many of the states discovered at or above threshold presented exotic features and have been initially termed X Y Z states (Brambilla et al. 2020b). The first tool used to describe quarkonium in the 1980s has been the quark model with some notions of QCD incorporated. In particular, constituent heavy quark masses have been considered and a static potential called Cornell potential (Eichten et al. 1978, 1980) has been used in a Schrödinger equation, successfully reproducing the quarkonia levels measured at that time. The Cornell potential is flavor independent and has the form V0 (r) = −

κ + σ r + const, r

(1)

r being the modulus of the quark-antiquark distance. The parameters κ should be identified with 43 αs , corresponding to the one gluon exchange that should dominate at small distances due to asymptotic freedom, and the string tension σ corresponds to a constant energy density related to confinement and originating a potential growing with the interquark distance at large distances. A fit to the states gave κ = 0.52 and σ = 0.182 GeV2 . Since then, several different phenomenological forms of the static potential have been exploited; see Lucha et al. (1991) and Brambilla and Vairo (1999) for a review. In order to explain the fine and hyperfine structure of the quarkonium spectrum, however, spin-dependent relativistic corrections to the ¯ v 2 ∼ 0.3 static potential have to be considered. Moreover, for v 2 ∼ 0.1 for the bb, for cc¯ systems, one expects relativistic corrections of order 20∇ · 30% for the charmonium spectrum and up to 10% for the bottomonium spectrum and also spinindependent but momentum-dependent corrections have to be considered, arriving at a phenomenological Hamiltonian of the type

81 Quark Nuclear Physics with Heavy Quarks

H =

 j =1,2

 mj +

pj2 2mj

−

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pj4 8m3j

 + V0 + VSD + VVD .

(2)

The 1/m2 spin-dependent VSD and velocity-dependent VVD potentials were initially derived in the 1980s from the semirelativistic reduction of a Bethe-Salpeter (BS) (Salpeter and Bethe 1951) equation for the quark-antiquark connected amputated Green functions or, equivalently at this level, from the semirelativistic reduction of the quark-antiquark scattering amplitude with an effective exchange equal to the BS kernel. Several ambiguities are involved in this procedure, due on the one hand to the fact that we do not know the relevant confining nonperturbative Bethe-Salpeter kernel and on the other hand due to the fact that we have to get rid of the temporal (or energy Q0 , Q = p1 − p1 being the momentum transfer) dependence of the kernel to recover a potential (instantaneous) description. It turned out that, at the level of the approximation involved, the spin-independent relativistic corrections at the order 1/m2 depend on the way in which Q0 is fixed together with the gauge choice of the kernel (Brambilla 1992; Lucha et al. 1991; Brambilla and Vairo 1999). The Lorentz structure of the kernel was also not known. On a phenomenological basis, the following ansatz for the kernel was intensively studied (Brambilla and Vairo 1999)  μ  I (Q2 ) = (2π )3 γ1 γ2ν Pμν Jv (Q) + Js (Q)

(3)

in the instantaneous approximation Q0 = 0. Notice that the effective kernel above was taken with a pure dependence on the momentum transfer Q. But, of course, the dependence on the quark and antiquark momenta could have been more complicated. The vector kernel Jv (Q) above would correspond to the one gluon exchange (with Pμν depending on the adopted gauge) while the scalar kernel Js (Q) would account for the nonperturbative interaction. 1 4 1 σ 1 Taking Jv = − 2 and Js = − 2 4 , reproduce the Cornell potential and 2 3 2π Q π Q the corrections in the nonrelativistic reduction of such kernel in the instantaneous approximation would give a form for the VSD and the VVD . The confining part of the kernel was usually chosen to be a Lorentz scalar in order to match the data on the fine separation on the P states (Schnitzer 1978). If one however takes a kernel which is not a pure convolution kernel, as it is to be expected in interactions generated at higher orders, and deals more appropriately with the instantaneous approximation, one can get quite different relativistic corrections, especially for the velocity-dependent part (Brambilla 1992). We conclude that this type of description is model dependent and does not allow for further progress. In particular, the parameters of the model cannot be related to the underlying field theory and the systematics of the model cannot be estimated. A more systematic procedure to relate the potential to QCD has been based on the so-called Wilson loop approach. In order to obtain information on the structure of nonperturbative corrections to the potential, it is useful to define the potential in

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terms of gauge invariant objects suitable for a direct lattice QCD evaluation. Indeed lattice QCD is one of the best methods to extract nonperturbative information from QCD; for some reviews, see Kronfeld (2012), Rothe (1992), Kogut (1983), Bali (2001), and Karsch (2002). Let us see how this works in the case of the static potential. Let us consider a locally gauge invariant quark-antiquark color singlet state (for more details, see Brambilla and Vairo (1999) and Lucha et al. (1991): δlj lj |φαβ  ≡ √ ψ¯ αi (x)U ik (x, y, C)ψβk (y)|0 3

(4)

where i, j, k, l are color indices (that will be suppressed in the following), |0 denotes the QCD ground state and the Schwinger string line has the form   U (x, y; C) = P exp ig



x

Aμ (z) dz

μ

(5)

,

y

where Aμ = Aaμ , a = 1, 8 is the gluon vector potential of QCD in the fundamental representation and g the QCD coupling constant and the integral is extended along the path C. The operator P denotes the path-ordering prescription which is necessary due to the fact that Aμ are non-commuting matrices. Let us see how the quark-antiquark potential can be extracted from the quark-antiquark Green function constructed with such color singlet states: G(T ) = φ(x, 0)|φ(y, T ) = φ(x, 0)| exp (−iH T )|φ(y, 0).

(6)

Inserting a complete set of energy eigenstates ψn with eigenvalues En and making a Wick rotation, we find  G(−iT ) = φ(x, 0)|ψn  ψn |φ(y, 0) exp (−En T ) n

→ φ(x, 0)|ψ0  ψ0 |φ(y, 0) exp (−E0 T )

for

T →∞

(7)

which gives the Feynman-Kac formula for the ground state energy log G(−iT ) . T →∞ T

E0 = − lim

(8)

The only condition for the validity of Eq. (8) is that the φ states have a non-vanishing component over the ground state. This is precisely the way in which hadron masses are computed on the lattice. Of course, to maximize the overlap with the ground state in consideration, appropriate operators may be used. If the φ state denotes a state of two exactly static particles interacting at a distance r, then the ground state energy is a function of the particle separation, E0 ≡ E0 (r), and gives the potential of the first adiabatic surface. It is possible to obtain an explicit analytic form of the quark-antiquark Green function for infinitely heavy quarks (m → ∞, static limit)

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Fig. 4 The static Wilson loop. It contains the interaction of a static quark-antiquark pair created at a time t = 0 and annihilated at a subsequent large time T . Initial and final states are made gauge invariant by the present of Schwinger line given in Eq. (5)

x1

t=T

x2

r y1

t=0

y2

and for large temporal intervals (T → ∞); see Brambilla and Vairo (1999). We consider that at a time t = 0 a quark and an antiquark pair is created and that they interact while propagating for a time t = T at which they are annihilated. Then (xj = (xj , T ), yj = (yj , 0), see Fig. 4) we obtain m→∞

Gβ1 β2 α1 α2 (T ) −→ δ 3 (x1 − y1 )δ 3 (x2 − y2 )(P+ )β1 α1 (P− )α2 β2 ×e−2mT Tr Pe

ig



Γ0

dzμ Aμ (z)



(9)

with P± ≡ (1 ± γ4 )/2. The integral in Eq. (9) extends over the circuit Γ0 which is a closed rectangular path with spatial and temporal extension r = |x1 − x2 | and T , respectively, and has been formed by the combination of the path-ordered exponentials along the horizontal (=time fixed) lines, coming from the Schwinger strings, and those along the vertical lines coming from the static propagators (see Fig. 4). The brackets in (9) denote the QCD vacuum expectation value, which in Euclidean space is 1 f [A] ≡ Z



DAf [A]e−



d 4 xLE YM

.

(10)

From Eq. (9), it is clear that the dynamics of the quark-antiquark interaction is contained in ig dzμ Aμ (z) Γ0 . (11) W (Γ0 ) = Tr Pe This is the famous static Wilson loop (Wilson 1974). In the limit of infinite quark mass considered, the kinetic energies of the quarks drop out of the theory, the quark Hamiltonian becomes identical with the potential, while the full Hamiltonian contains also all types of gluonic excitations. According to the Feynman-Kac formula, the limit T → ∞ projects out the lowest state, i.e., the one with the “glue” in the ground state. This has the role of the quark-antiquark potential for pure mesonic states. We will see in the section over exotics that the excited states have the role of the potential in the case of hybrid states, i.e., states with gluons

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contributing to the binding. Now, comparing Eq. (9) with Eq. (7) and considering that the exponential factor exp (−2mT ) just accounts for the fact that the energy of the quark-antiquark system includes the rest mass of the pair, we obtain V0 (r) ≡ E0 (r) = − lim

T →∞

1 log W (Γ0 ). T

(12)

The heavy quark degrees of freedom have now completely disappeared, and the expectation value in (12) can be evaluated in the pure Yang-Mills theory or in unquenched QCD (i.e., taking into account sea light quarks) (Wilson 1974; Brown and Weisberger 1979). Notice that the potential is given purely in terms of a gauge invariant quantity (the Wilson loop precisely). In this way, we have reduced the calculation of the static potential to a well-posed problem in field theory: to obtain the actual form of V0 , we need to calculate the QCD expectation value of the static Wilson loop. The static Wilson loop is the low-energy domain of QCD is dominated by an area law, i.e., can be approximated at leading order at large r as W (Γ0 ) ∼ exp{−σ rT } (i.e., the area of the loop in the exponent multiplied by the string tension), as shown in the strong coupling expansion (Rothe 1992; Creutz 1977, 1980; Bali 2001; Brambilla and Vairo 1999). In the high-energy domain, the first perturbative contribution comes from the one gluon exchange and therefore Eq. (12) would reproduce in these limits the Cornell potential of Eq. (1), which appears to be a simple superposition of these two behaviors. We will come back to the area law behavior of the Wilson loop in the section on studies of confinement. The Wilson loop formalism have been used to obtain the form of spin-dependent VSD potential in the famous papers of Eichten and Feinberg (1981) and Gromes (1984) and later to obtain the form of spin-independent, momentum-dependent VV D potential in the classical works (Barchielli et al. 1988, 1990). These works express the relativistic corrections to the potential in terms of generalized static Wilson loops containing field strengths (i.e., chromoelectric and chromomagnetic fields) insertion in the quark lines. In Brambilla et al. (1994), a nonstatic Wilson loop has been used to obtain the relativistic corrections to the static three-quarks potential. Up to the establishment of the NREFT called potential nonrelativistic QCD in the 1990s (Brambilla et al. 1999, 2000, 2005), the Wilson loop approach has been the best theory founded method and was used to calculate the potentials on the lattice (see, e.g., the review Bali 2001). However, this method was providing only part of the QCD result. The Wilson loop relativistic potential corrections were missing all the nonanalytic term in the mass, ln m/μ terms, that were instead obtained in a direct purely perturbative QCD one loop calculation (Gupta et al. 1982). Moreover, QCD in the static quark antiquark configuration can still change color by emission of gluons. This introduces a new dynamical scale in the evaluation of the potential that should be taken into account as it became evident in the seminal paper of Appelquist et al. (1977) that identified infrared divergences in the three loops perturbative fixed order calculation of the static Wilson. Moreover, already in Lucha et al. (1991) the question was raised about why potential models where working so well. It took a while and the

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development of nonrelativistic effective field theories to answer this question and to be able to calculate systematically the potentials from QCD. All these problems are addressed and solved in pNRQCD.

Nonrelativistic Effective Field Theories As we mentioned, the study of quarkonium in the last few decades has witnessed two major developments: the establishment of nonrelativistic effective field theories (NREFTs) and progress in lattice QCD calculations of excited states and resonances, with calculations at physical light-quark masses. Both allow for precise and systematically improvable computations that are largely model-independent. It is precisely this advancement in the understanding of quarkonium and quarkonium-like systems inside QCD that makes today quarkonium exotics as particularly valuable (see the section on Exotics). In fact, today that we are confronted with a huge amount of high-quality data, which have provided for the first time uncontroversial evidence for the existence of exotic hadrons, by using modern theoretical tools that allow us to explore in a controlled way these new forms of matter we can get a unique insight into the low-energy dynamics of QCD. The appearance of a hierarchy of scales calls for the application of effective field theory (EFT) methods. However, “heavy quark effective theory” (HQET) (Neubert 2004; Georgi 1991), the EFT description of heavy-light mesons, where only an ultraviolet mass scale m and an infrared mass scale ΛQCD appear, is not suitable for the description of heavy quarkonia, since HQET is unable to describe the dynamics of the binding. The existence of several physical scales makes the theoretical description of quarkonium physics more complicated. All scales get entangled in a typical amplitude involving a quarkonium observable. For example, quarkonium annihilation and production take place at the scale m, quarkonium binding takes place at the scale mv, which is the typical momentum exchanged inside the bound state, while very low-energy gluons and light quarks (also called ultrasoft (US) degrees of freedom) live long enough that a bound state has time to form and, therefore, are sensitive to the scale mv 2 . Ultrasoft gluons are responsible for phenomena similar to the Lamb shift in hydrogen atom. This pattern of scales has made the description in quantum field theory particularly difficult. The solution is to take advantage of the existence of the different energy scales to substitute QCD with simpler but equivalent NREFTs. A hierarchy of NREFTs, see Fig. 5, may be constructed by systematically integrating out modes associated with highenergy scales not relevant for the quarkonium system. Such integration is made in a matching procedure that enforces the equivalence between QCD and the EFT at a given order of the expansion in v. The EFT Lagrangian is factorized in matching coefficients, encoding the high-energy degrees of freedom and lowenergy operators. The relativistic invariance is realized via exact relations among the matching coefficients (Manohar 1997; Brambilla et al. 2003; Heinonen et al. 2012; Berwein et al. 2019). The EFT displays a power counting in the small parameter

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Fig. 5 The hard, soft, and ultrasoft scales of quarkonium and the corresponding NREFTs that can be constructed. NRQCD follows from the integration of the gluons and quarks at the hard scale, pNRQCD follows from integration of gluons at the soft scale. If the soft scale is bigger than ΛQCD , then the matching from NRQCD to pNRQCD is perturbative; otherwise, it is nonperturbative

LONG−RANGE QUARKONIUM

SHORT−RANGE QUARKONIUM

QCD m

perturbative matching

perturbative matching μ

mv

NRQCD μ

mv 2

non−perturbative matching

perturbative matching

pNRQCD

v; therefore, we are able to attach a definite power of v to the contribution of each EFT operators to the physical observables. We will introduce in the next sections nonrelativistic QCD (NRQCD) (Bodwin et al. 1995; Caswell and Lepage 1986; Lepage et al. 1992) and potential nonrelativistic QCD (pNRQCD) (Pineda and Soto 1998; Brambilla et al. 2000); see the review Brambilla et al. (2005). It turns to be instrumental to combine NREFTs and lattice. In fact, on the one hand, the NREFT is pulling out scales from observables in a controlled way, separating them, and delivering results for calculations at large scales including the logs resummation. On the other hand, the low-energy contributions that the EFT has factorized are often nonperturbative and can be evaluated on the lattice. This greatly reduces the complexity of the problem allowing the lattice to target the nonperturbative part directly, appropriately defined in the EFT in terms of gauge invariant, purely gluonic objects, a big simplification with respect to an ab initio lattice calculation of an observable. The latter is much more difficult because it still contains all the physical scales of the problem. In this framework also, quenched lattice calculations can be pretty useful because at least at higher energy the flavor dependence is accounted for by the EFT matching coefficients. We founded the TUMQCD lattice collaboration precisely to address these calculations directly inside the EFT (https://einrichtungen.ph.tum.de/T30f/tumqcd/index.html).

Nonrelativistic QCD Nonrelativistic QCD (NRQCD) (Caswell and Lepage 1986; Bodwin et al. 1995), follows from QCD integrating out the scale m. As a consequence, the effective Lagrangian is organized as an expansion in 1/m and αs (m): LNRQCD =

 cn (αs (m), μ) n

mn

× On (μ, mv, mv 2 , . . .),

(13)

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where cn are Wilson coefficients that contain the contributions from the scale m. They can be calculated via a well-defined procedure call matching, see, e.g., Brambilla et al. (2005), in which one calculates in perturbation theory Green functions or amplitudes in QCD and in NRQCD, expand in 1/m and insert the difference between the two calculations (i.e., the nonanalytic terms in the scale that has been integrated out, the mass) inside the matching coefficients of the EFT. In fact since in NRQCD we expand in the mass no contributions of the form log(m/μ) can be generated in calculations in the EFT. The On are local operators of NRQCD; the matrix elements of these operators contain the physics of scales below m, in particular of the scales mv and mv 2 and also of the nonperturbative scale ΛQCD . Finally, the parameter μ is the NRQCD factorization scale. The low-energy operators On are constructed out of two or four heavy quark/antiquark fields plus gluons. At the lowest order in the 1/m expansion, the NRQCD Lagrangian density is   D2 D2 ψ + χ † iD 0 − χ LNRQCD = ψ † iD 0 + 2m 2m 1 a μν a − Fμν F , 4

(14)

where ψ is the Pauli spinor field that annihilates the fermion and χ is the Pauli spinor field that creates the antifermion; iD 0 = i∂0 − gA0 and iD = i∇ + gA. At order 1/m2 bilinear terms in the quark (antiquark) field containing the chromoelectric and chromomagnetic fields, as well covariant derivatives and spin, start to appear. These are of the same form that one would obtain from a Foldy-Wouthuysen transformation but in addition are multiplied by matching coefficients that contain the UV behavior of QCD. In addition, four fermion terms start to appear at order 1/m2 with matching coefficients containing both real and imaginary parts. The presence of imaginary parts describe the decays of quarkonium via annihilation of hard gluons that have been integrated out from the theory. The quarkonium state |H  in NRQCD is expanded in the number of partons ¯ ¯ ¯ qq |H  = |QQ + |QQg + |QQ ¯ + ···

(15)

where the states including one or more light partons are shown to be suppressed ¯ by powers of v. In the |QQg, for example, the quark-antiquark are in a color octet state. The NRQCD Lagrangian has been extensively used on the lattice to calculate quarkonium spectra and decays. On the other hand, NRQCD has been deeply impactful on the study of quarkonium production at the LHC putting forward a factorization formula for the inclusive cross section for the direct production of the quarkonium H at large transverse momentum written as a sum of products of NRQCD matrix elements and short-distance coefficients:

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σ [H ] =



σn Kn4fermions 

(16)

n

where the σn are short-distance coefficients and the matrix elements Kn4fermions  are vacuum-expectation values containing four-fermions operators in color singlet ¯ angular momentum state quantum and color octet configurations identified by QQ ¯ pair plus anything numbers. They contain in the middle a projector over the QQ and some Schwinger lines with a particular path to make them gauge invariant. The factorization for production is proved only at order NNLO and in some cases (Nayak et al. 2005; Bodwin et al. 2020). It is different from the NRQCD factorization of inclusive decays that are proven at all orders. In such case, the low-energy part contains quarkonium expectation values of color singlet and color octet four quark operators without the projector in the middle. The matrix elements (LDMEs, long distance matrix elements) Kn4fermions  contain all of the nonperturbative physics ¯ pair into a quarkonium state and they are associated with the evolution of the QQ universal. Their NRQCD definition does not allow a lattice calculation and their extractions from collider data is still challenging (Brambilla et al. 2011; Chapon et al. 2022; Andronic et al. 2016; Bodwin et al. 2013; Chung 2020). Differently from HQET, the power counting of NRQCD is not unique, due to the fact that many physical scales are still dynamical (mv, mv 2 , ΛQCD ). This still complicates bound state calculations as the soft and US scale can get entangled.

Potential Nonrelativistic QCD Quarkonium formation happens at the scale mv. The suitable NREFT is potential nonrelativistic QCD, pNRQCD (Pineda and Soto 1998; Brambilla et al. 2000, 2005), which follows from NRQCD by integrating out the scale mv ∼ r −1 . The pNRQCD description directly addresses the bound state dynamics, implements the Schödinger equation as zero order problem, properly defines the potentials as matching coefficients, and allows to systematically calculate relativistic and retardation corrections. In this way even quantum mechanics can also be reinterpreted as a pNREFT. Moreover, since the scale mv has been integrated out, the power counting of pNRQCD is less ambiguous than the one of NRQCD. The soft scale, proportional to the inverse quarkonium radius r, may be either perturbative (mv ΛQCD ) or nonperturbative (mv ∼ ΛQCD ) depending on the physical system under consideration. We do not have any direct information on the radius of the quarkonia, and thus the attribution of some of the lowest bottomonia and charmonia states to the perturbative or the nonperturbative soft regime may be ambiguous, but it is likely that the lowest bottomonium and possibly also the lowest charmonium states have small enough radii that the scale mv is in fact still

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perturbative. In a case with ΛQCD  mv 2 , both scales are still perturbative and the system would be similar to a Coulombic system. For such a quarkonium, we would have – similar as for the hydrogen atom – v ∼ αs (mv). However, none of the bb¯ and cc¯ states satisfy this condition. In all realistic quarkonia (bb¯ and cc), ¯ the ultrasoft scale is nonperturbative. Only for t t¯ threshold states the ultrasoft scale may be considered still perturbative. In Fig. 5, we schematically show the various scales. The short-range quarkonia are small enough to allow for a perturbative treatment of the scale mv, while for the long range quarkonia already this scale requires a nonperturbative treatment. In the next section, we will present weakly coupled and strongly coupled pNRQCD, discuss the form of the potential, and briefly address the countless phenomenological applications of this that has become the standard treatment for quarkonium. On the one hand, pNRQCD allows us systematic and precise determinations of processes at high-energy collider experiments and the definition and computation of quantities of large phenomenological interest, and on the other hand, the systematic factorization enables us for studies of confinement. The lowenergy nonperturbative factorized effects depend on the size of the physical system: when ΛQCD is comparable or smaller than mv, they are carried by correlators of chromoelectric or chromomagnetic fields nonlocal or local in time; when ΛQCD is of order mv, they are carried by generalized static Wilson loops, nonlocal in space, with insertion on chromoelectric and chromomagnetic fields. The EFT allows us to make model-independent predictions, and we can use the power counting to attach an error to the theoretical predictions. The nonperturbative physics in pNRQCD is encoded in few low-energy correlators that depend only on the glue and are gauge invariant: these are objects in principle ideal for lattice calculations. Even more interesting is that this EFT description allows modifications to be used to describe exotics states (BOEFT) and the nonequilibrium evolution of quarkonium in medium (pNRQCD at finite temperature with open quantum system) as we will address in the last two sections.

Weakly Coupled pNRQCD When mv ΛQCD , we speak about weakly coupled pNRQCD because the soft scale is perturbative and the matching from NRQCD to pNRQCD may be performed in perturbation theory. The lowest levels of quarkonium, like J /ψ, Υ (1S), Υ (2S) . . . , may be described by weakly coupled pNRQCD, while the radii of the excited states are larger and presumably need to be described by strongly coupled pNRQCD. All this is valid for states away from strong-decay threshold, i.e., the threshold for a decay into two heavy-light hadrons. The effective Lagrangian is organized as an expansion in 1/m and αs (m), inherited from NRQCD, and an expansion in r (multipole expansion) (Brambilla et al. 2000, 2005)

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 Lweak pNRQCD

=

 3

d 3r

d R

  cn (αs (m), μ) n



× Ok (μ , mv , . . .), 2

k

mn

Vn,k (r, μ , μ) r k (17)

where R is the center of mass position and Ok are the operators of pNRQCD. We do not show explicitly the part of the Lagrangian involving gluons and light quarks, as this part coincides with the QCD one. The matrix elements of the operators above depend on the low-energy scale mv 2 and μ , where μ is the pNRQCD factorization scale. The Vn,k are the Wilson coefficients of pNRQCD that encode the contributions from the scale r and are nonanalytic in r. The cn are the NRQCD matching coefficients as given in (13). The degrees of freedom, which are relevant below the soft scale, and which ¯ states (a color-singlet S and a color-octet O = appear in the operators Ok , are QQ a Oa T state, depending on r and R) and (ultrasoft) gluon fields, which are expanded in r as well (multipole expanded and depending only on R). pNRQCD makes apparent that the correct zero order problem is the Schrödinger equation. Looking at the equations of motion of pNRQCD, we may identify Vn,0 = Vn with the 1/mn potentials that enter the Schrödinger equation and Vn,k =0 with the couplings of the ultrasoft degrees of freedom, which provide corrections to the Schrödinger equation. These nonpotential interactions, associated with the propagation of low-energy degrees of freedom start to contribute at NLO in the multipole expansion. They are typically related to nonperturbative effects and are carried by purely gluonic correlator local or nonlocal in time: they need to be calculated on the lattice. In particular at the NLO in the multipole expansion at leading order in the expansion in 1/m, the pNRQCD Lagrangian density is   p2 weak − Vs (r) + . . . S LpNRQCD = Tr S† i∂0 − m

 p2 † − Vo (r) + . . . O +O iD0 − (18) m    VB (r)  † Tr O r · E O + O† Or · E , +gVA (r)Tr O† r · E S + S† r · E O + g 2 where R ≡ (x1 + x2 )/2, S = S(r, R, t) and O = O(r, R, t) are the singlet and octet quark-antiquark composite nonrelativistic fields respectively. All the gauge fields in Eq. (19) are evaluated in R and t. In particular E ≡ E(R, t) and iD0 O ≡ ¯ static i∂0 O − g[A0 (R, t), O]. Vs0 and Vo0 are the singlet and octet heavy Q-Q potentials. Higher-order potentials in the 1/m expansion and the center-of-mass kinetic term are not shown here and are indicated by the dots. At higher order in the multipole expansion and in the 1/m expansion, more operators appear containing US chromomagnetic and chromoelectric fields but they are suppressed in the power counting in v. We define

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αV (r) Vs0 (r) ≡ −CF s , r 

αVo (r) CA 0 Vo (r) ≡ − CF 2 r

(19)

which shows that the singlet potential is attractive and the octet one is repulsive. VA and VB are the matching coefficients associated in the Lagrangian (19) to the leading corrections in the multipole expansion. Both the potentials and the coefficients VA and VB have to be determined by matching pNRQCD with NRQCD at a scale μ smaller than mv and larger than the US scales. Since, in particular, μ is larger than ΛQCD the matching can be done perturbatively. At the lowest order in the coupling constant, we get αVs = αVo = αs , VA = VB = 1. In order to have the proper free-field normalization in the color space, we define 11c S≡ √ S Nc

Ta O ≡ √ Oa, TF

(20)

where TF = 1/2, Nc = 3.

The Perturbative QCD Potential For system with a perturbative soft scale, the potentials can be calculated in perturbation theory, i.e., no nonperturbative quantities enter the potential (Brambilla et al. 2005). The potentials can be calculated at all order in the perturbative expansion in αs via a well-defined matching procedure that entails to compare appropriate Green functions in NRQCD and in pNRQCD calculated in perturbation theory up to the desired order of the perturbative expansion, order by order in 1/m and order by order in the multipole expansion. The difference between the two is encoded in the matching coefficients. On the pNRCD side, lower energy scales may be expanded in the loop integral and give no contribution in the matching in dimensional regularization. The singlet static potential is calculated by matching the relevant NRQCD Green function (the static Wilson loop) and the pNRQCD singlet Green function see Fig. 6. At three loops, an ultrasoft (US) divergence is emerging at fixed order calculation that can be regularized resumming series of diagrams containing the US scale which is the difference between the singlet and the octet potential, see Brambilla et al. (1999, 2005).

= QCD

+

+ ... pNRQCD

Fig. 6 The matching of the static potential. On the right side are the pNRQCD fields: simple lines are singlet propagator, double lines are octet propagators, circled-crosses are the singlet-octet vertices of Eq. (19), and the wavy line is the US gluon propagator

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This solves the problem raised by Appelquist et al. (1977); it explains how this divergence cancels between the potential (log rμ) and the contribution of the US chromoelectric correlator log((Vo − VS )/μ) (second contribution in the r.h.s. of Fig. 6) leaving a term in log αs in the static energy and qualifies the potential as a matching coefficient of an EFT which is depending on the US scale. Being matching coefficients of the effective field theory, the potentials undergo renormalization, develop a scale dependence, and satisfy renormalization group equations, which allow to resum large US logarithms; see, e.g., Pineda and Soto (2000) and Brambilla et al. (2007, 2009). In particular one obtains the complete expression for the static potential, at 3 loops, up to the relative order αs3 ln μr in coordinate space (αs is in the MS scheme) as:  αs (r) αVs (r, μ) = αs (r) 1 + (a1 + 2γE β0 ) (21) 4π     2

2 αs (r) CA3 αs3 (r) π 2 2 + γE (4a1 β0 + 2β1 ) + + 4γE β0 + a2 ln rμ , + 3 12 π 16 π 2 where βn are the coefficients of the beta function, and a1 and a2 are the constants at 1 and 2 loops, respectively. We emphasize that this US contribution to the static potential would be zero in QED. From Eq. (22) , it is clear that αVs depends on the US scale and as such is not a short-distance quantity. The evaluated terms clarify the long-standing issue of how the perturbative static potential should be defined at higher order in the perturbative series. In perturbation theory, the static potential does not coincide with the static Wilson loop starting from three loops. It should be emphasized that the separation between soft and US contributions is not an artificial trick but a necessary procedure if one wants to use the static potential in a Schrödinger-like equation in order to study the ¯ states of large but finite mass. In that equation, the kinetic dynamics of Q-Q ¯ term of the Q-Q system is US and so is the energy. Since the US gluons interact ¯ system, their dynamics is sensitive to the energies of the (nonwith the Q-Q static) system and hence it is not correct to include them in the static potential. When calculating a physical observable, the μ dependence must cancel against μdependent contributions coming from the US gluons. Finally, it is worth mentioning that the static potential suffers from IR renormalons ambiguities with the following structure δVs ∼ C + C2 r 2 + . . . The constant C is known to be cancelled by the IR pole mass renormalon (Brambilla et al. 2005). The static singlet potential is currently know at NNNLL (Brambilla et al. 2009), the only unknown at four loops is the constant term.

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The static energy in the perturbative regime has the form E0 (r) = Vs0 (r; μ) + δUS (r, Vs0 , Vo0 , . . . ; μ),

(22)

where δUS (r, Vs , Vo , VA , VB , . . . ; μ) (δUS for short) contains contributions from the ultrasoft gluons. As mentioned, Vs (r; μ) and Vo (r; μ) do not depend on μ up to N2 LO (Brambilla et al. 2000). The former coincides with E0 (r) at this order and the latter may be found in Kniehl et al. (2005). The fact that the μ dependence of δUS must cancel the one in Vs (r; μ) is the key observation that leads to a drastic simplification in the calculation of the log αs terms in E0 (r). So, for instance, the logarithmic contribution at N3 LO, which is part of the three-loop contributions to Vs (r; μ), may be extracted from a one-loop calculation of δUS (Brambilla et al. 1999, 2000) and the single logarithmic contribution at N4 LO, which is part of the four-loop contributions to Vs (r; μ), may be extracted from a two-loop calculation of δUS . Notice that we denote Nn LO, contributions to the potential of order αsn+1 and Nn LL, contributions of order αsn+2 logn−1 αs . The general form of the relativistic corrections to the singlet potential in the center of mass is (we drop the index s for simplicity): V (r) = V (0) (r) + (2)

V (1) (r) V (2) + 2 + ··· , m m

(23)

(2)

V (2) = VSD + VSI , 1  2 (2)  VL2 (r) 2 p , Vp2 (r) + L + Vr(2) (r), 2 r2 (2)

(2)

VSI = (2)

(2)

(2)

(24)

(2)

VSD = VLS (r)L · S + VS 2 (r)S2 + VS12 (r)S12 (r), S = S1 +S2 , L = r×p, S1 = σ 1 /2, S2 = σ 2 /2, and S12 (ˆr) ≡ 3ˆr ·σ 1 rˆ ·σ 2 −σ 1 ·σ 2 . We see that differently from what discussed previously, the corrections to the (2) potential start at order 1/m. The potential proportional to VLS may be identified (2) with the spin-orbit potential, the potential proportional to VS 2 with the spin-spin (2)

potential and the potential proportional to VS12 with the spin tensor potential. The above potentials read at leading (non-vanishing) order in perturbation theory (see, e.g., Brambilla et al. 2005): V (1) (r) = − Vr(2) (r) = (2)

VLS (r) =

4π (3) 3 αs δ (r ) , 2αs r3

,

(2)

2αs2 r2

(25)

,

4αs Vp(2) 2 (r) = − 3r ,

VL(2) 2 (r) =

16π αs (3) 9 δ (r ) ,

VS12 (r) =

VS 2 (r) =

(2)

2αs 3r

, (26)

αs 3r 3

. (27)

These potentials and the static octet potential have been calculated at higher order in perturbation theory see, e.g., Brambilla et al. (2005), Pineda (2012), and Kniehl et al.

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(2005). Once one has obtained the potentials as matching coefficients of pNRQCD, one can start doing calculations inside the EFT obtaining, for example, the energy levels of quarkonium (Brambilla et al. 1999; Kniehl et al. 2002). Nonperturbative effects in the form of local or time nonlocal chromoelectric correlators start to appear at order αs5 , i.e., at NNNLO (Brambilla et al. 2005; Pineda 2012): their contribution is suppressed. The case of baryons made by three heavy quarks or two heavy quarks and a light quark has also been addressed in pNRQCD (Brambilla et al. 2005) and the static three quark potential has been calculated at one loop accuracy, showing some interesting features (Brambilla et al. 2010).

Strongly Coupled pNRQCD When mv ∼ ΛQCD , we speak about strongly coupled pNRQCD because the soft scale is nonperturbative and the matching from NRQCD to pNRQCD is nonperturbative, i.e., it cannot be performed within an expansion in αs . In this case, since we work at a nonperturbative scale, only color singlet degrees of freedom ¯ states, hybrids QQg ¯ states, and glueballs remain dynamical and they include QQ (and if we consider light quarks as part of the binding, color singlet states formed by a heavy quark and heavy antiquark and light quarks, as in the case of tetraquarks). Since the physics is nonperturbative, we need to use input coming from the lattice to construct the EFT. In particular, we can use the lattice evaluation of the gluonic static ¯ pair: they have been calculated since long (Juge et al. 2003; Bali energies of a QQ and Pineda 2004) and recently updated (Müller et al. 2019). Such lattice calculations use insertions of gluonic fields at the initial and final Schwinger strings, inside a generalized static Wilson loop, to select some given symmetries. The gluonic static energies, EΓ in Fig. 8 are classified according to representations of the symmetry group D∞ h , typical of diatomic molecules, and labeled by Λση (see Fig. 7): Λ is the rotational quantum number |ˆr · K| = 0, 1, 2, . . . , with K the angular momentum of the gluons, that corresponds to Λ = Σ, Π, Δ, . . . ; η is the CP eigenvalue (+1 ≡ g (gerade) and −1 ≡ u (ungerade)); σ is the eigenvalue of reflection with respect to a

Fig. 7 Quarkonium hybrid symmetries

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Fig. 8 Hybrid static energies, EΓ , in lattice units. (From Juge et al. 2003)

¯ axis. The quantum number σ is relevant only for Σ plane passing through the QQ states. In general there can be more than one state for each irreducible representation of D∞ h : higher states are denoted by primes, e.g., Πu , Πu , Πu , . . . . Notice that this set of static energies will be fundamental later to address the exotics (Fig. 8). ¯ states below threshold, we are Since at this moment we are dealing with QQ interested merely in the Σg+ static energy (that in this case coincides with the static ¯ potential) and in the information that such curve develops a gap of singlet QQ order ΛQCD at a distance r ∼ Λ−1 QCD : therefore all the hybrid static energies can be integrated out as pNRQCD follows from integrating out all degrees of freedom with energy up to mv 2 . The quarkonium singlet field S is now the only low-energy dynamical degree of freedom in the pNRQCD Lagrangian (up to US pions) which reads (Brambilla et al. 2001, 2005; Pineda and Vairo 2001):  LpNRQCD =

 3

d R

 p2 − VS (r) S d r S i∂0 − 2m 3

†

(28)

and lends support to potential models in this particular regime. Indeed Eq. (28) originates a Schödinger equation governed by the singlet potential.

The Nonperturbative QCD Potential The difference with the phenomenological potential models is that now the singlet potential Vs (r) = V0 + V (1) /m + V (2) /m2 is the QCD potential calculated in

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the matching obtained in Brambilla et al. (2001) and Pineda and Vairo (2001) and display novel characteristics. These potentials are nonperturbative and are given in terms of generalized Wilson loops. However relevant differences emerge also with respect to the Wilson loop approach. The general decomposition of the potential is of the form of Eq. (23). It features a contribution already at order 1/m (Brambilla et al. 2001) given in terms of a novel expectation value of a static Wilson loop with insertion of a chromoelectric field. The potential at order 1/m2 appears factorized in the product of NRQCD matching coefficients, carrying contribution in the log(m/μ) and generalized Wilson loops with chromomagnetic and chromoelectric insertions. This solves the problem of the incompatibility of the Wilson loop potentials and the perturbative calculation. Moreover, additional generalized Wilson loops contribution of novel type emerge see Pineda and Vairo (2001) and Brambilla et al. (2005) for all the explicit expressions. Spin effects in quarkonium are appearing only at order 1/m2 and are therefore suppressed in the spectrum and in the transitions: we will see that things are different for exotics. Some of these generalized Wilson loops have been calculated on the lattice (only quenched) (Bali et al. 1997; Bali 2001; Koma et al. 2006, 2007) but some contributions at order 1/m2 are still to be calculated. The EFT then lends a clean definition and an interpretation of the static Wilson loops measured on the lattice as actual potentials in this regime, together with a prescription to use them to calculate observables. Once the nonperturbative potentials are given in terms of generalized Wilson loops, one can use a model of low-energy QCD to evaluate them. A minimal area law, for example, gives for the σ potentials of Eq. (23)–(24) a linear term for V (0) (r) = σ r and VL(2) 2 (r) = − 6 r and (2)

(2)

nonzero contributions to Vr (r) and VLS (r). The velocity-dependent relativistic nonperturbative contribution corresponds to the angular momentum and the energy of the flux tube between the quark and the antiquark while the one in the spindependent part is relevant to obtain agreement with the fine separations of the quarkonia multiplets (as it was pursued previously using a scalar confinement kernel in the Bethe-Salpeter equation). These findings are in agreement with what is obtained from the lattice calculations of the Wilson loops. Using a QCD effective string model, add several further subleading corrections; see Brambilla et al. (2014) and Perez-Nadal and Soto (2009). Using these potentials, all the masses for heavy quarkonia below threshold can be obtained by solving the Schrödinger equation with such potentials. Lorentz invariance is still there in the form of exact relations among potentials and it has been observed on the lattice. Decays are described by calculating the imaginary parts of the potentials (Brambilla et al. 2003) where nonperturbative contributions enter in the form of gauge invariant time nonlocal chromoelectric and chromomagnetic correlators that have still to be calculated on the lattice. In summary, strongly coupled pNRQCD factorizes low-energy nonperturbative contributions in terms of generalized gauge invariant Wilson loops, opening the way to a systematic study of the confinement mechanism and systematic applications to quarkonium spectrum and decay as we will see in the dedicated section.

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The three heavy quark static potential has been studied on the lattice using Wilson loops (Takahashi et al. 2001, 2002).

Spectra, Transitions, Decays and Production, and SM Parameters Extractions Together with lattice, pNRQCD is the theoretical framework that is nowadays mostly used for calculations and predictions of quarkonium properties. The power counting of the EFT allows to attach an error to each prediction. In the regime in which the soft scale is perturbative, pNRQCD enables precise and systematic higher order calculations on bound state allowing the extraction of precise determinations of standard model parameters like the quark masses and αs . For example, based on Eq. (22), it has been possible to use lattice calculations of the static energy with 2+1 flavor and the NNNLO pNRQCD calculation of the static energy, including the US log resummation, to extract a precise determination of αs at rather low energy and run it at the mass of the Z, obtaining αs (MZ ) = 0.11660+0.00110 −0.00056 , which is a competitive extraction made at a pretty high orders of the perturbative expansion (Bazavov et al. 2012, 2019; Brambilla et al. 2009). This method of αs extraction is now used by several groups, see, e.g., Ayala et al. (2020), Takaura et al. (2019), and the force, defined in terms of a single chromoelectric insertion in the Wilson loop, could be used as well (Brambilla et al. 2005; Vairo 2016). In the same way, one can extract precise determinations of the bottom and charm masses using the experimental measurements of the mass of the lowest states and comparing it to the formula for the energies in pNRQCD at NNNLO, which depends on the mass in a given scheme. The renormalon ambiguity cancels between the mass and the static potential and a pretty good determination is possible; see, e.g., Peset et al. (2018), Kiyo and Sumino (2014) and references therein. Moreover, the energy levels of some of the lowest quarkonia states have been obtained at high orders in perturbation theory using weakly coupled pNRQCD showing that the constant separation on the energy levels may be generated in this way; see, e.g., Brambilla et al. (2002), Brambilla and Vairo (2000), and Peset et al. (2016). Electromagnetic M1 and E1 transitions have been calculated in pNRQCD; see, e.g., Brambilla et al. (2006, 2012), Pineda and Segovia (2013), and Segovia et al. (2019). There are so many results that it is impossible to discuss all of them here and we refer you to some reviews Pineda (2012), Brambilla et al. (2005, 2011, 2014, 2005) For what concern production, it is very promising that by factorizing the quarkonium productions cross section at lower energy in pNRQCD (Brambilla et al. 2020b, 2021a, b), one can rewrite the octet NRQCD LDMEs, which are the nonperturbative unknowns, in terms of product of wave functions and gauge invariant low-energy correlators that depend only on the glue and not the on flavor quantum numbers. This allows us to reduce by half the number of nonperturbative unknowns and promises to have a great impact in the progress of the field.

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Confinement and Low-Energy QCD Models Strongly coupled pNRQCD realizes a scale factorization encoding the low-energy physics in generalized gauge invariant Wilson loops, i.e., Wilson loops with insertions of chromoelectric and chromomagnetic fields. Such objects no longer depends on the heavy quark degrees of freedom and on the quark flavor. It turns out that this is a systematic method to study the QCD confinement properties and put them directly in relation to the quarkonium phenomenology. Indeed lattice QCD seems more suitable to ask “what” instead of “why,” and to understand the mechanism underlying confinement, it may be useful to build models of lowenergy QCD and compare to the lattice results: the interface is offered by the EFT. We have seen that the area law emerging in the static Wilson loop at large distance is responsible of confinement; this in turn corresponds to the formation of a chromoelectric flux tube between the quark and the antiquark that sweeps the area of the Wilson loop; see Fig. 9. This effect is originated by the nonperturbative QCD vacuum that could be imagined as a disordered medium with whirlpools of color on different scales, thus densely populated by fluctuating fields whose amplitude is so large that they cannot be described by perturbation theory. A QCD vacuum model can be established by making an assumption on the behavior of the Wilson loop that gives the static potential. The relativistic corrections that involve insertions of gluonic fields in the Wilson loop follow then via functional derivative

Fig. 9 The origin of the linear potential between the static quark and antiquark may be traced back to a flux tube: a string of gluon energy between the quark pair. Here we present the historical picture of the action density distribution between a static quark antiquark couple in SU (2) at a physical distance of 1.2 fm. (From Bali et al. 1995)

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with respect to the quark path; see Brambilla et al. (1994); Brambilla and Vairo (1999). Then, one can notice, for example, that the part proportional to the square of the angular momentum in the of the Vvd potential obtained in strongly coupled pNRQCD takes into account the energy and the angular momentum of the flux tube, which is something that could not be obtained in any Bethe-Salpeter approach with a confinement scalar convolution kernel. Lattice simulations of the action density or the energy density between the static quark and the static antiquark show indeed a chromoelectric flux tube configuration; see Fig. 9 from Bali et al. (1995) and more recent calculations in Bicudo et al. (2018), Yanagihara and Kitazawa (2019), and Baker et al. (2022). The mechanism underlying confinement and flux tube formation has been investigated since long on the lattice (Greensite 2003) using the Wilson loops and the ’t Hooft Abelian projection, to identify the roles of magnetic monopoles (Amemiya and Suganuma 1999; Sasaki et al. 1995) and center vortices; see, e.g., the review Brambilla et al. (2014). In the continuum, several models of low-energy QCD have been used to explain the flux tube formation ranging from the dual Meissner effect and a dual Abelian Higgs model picture, dual QCD (Baker et al. 1991), the stochastic vacuum (Dosch and Simonov 1988), the flux tube model (Isgur and Paton 1985), and an effective QCD string description. Each of these models can be used to obtain analytic estimates of the behavior of the generalized Wilson loops for large distance, which in turn give the static potential and the relativistic corrections V1 , Vsd , Vvd as function of r; see Baker et al. (1996, 1998), Brambilla and Vairo (1997), Brambilla et al. (2014), and Perez-Nadal and Soto (2009). Similar nonperturbative configurations leading to confinement can be studied analyzing the Wilson loop in case of baryons with three or two heavy quarks (Nawa et al. 2007; Soto and Tarrús Castellà 2021).

BOEFT and X Y Z Exotics Exotic states, i.e., states with a composition different from a quark-antiquark or three quarks in a color singlet, have been predicted before and after the advent of QCD. In the last decades, a large number of states, either with a manifest different composition (with isospin and electric charge different from zero) or with other ¯ exotic characteristics, have been observed in the sector with two heavy quarks QQ, at or above the quarkonium strong decay threshold at the B-factories, tau-charm, and LHC and Tevatron collider experiments (Brambilla et al. 2011, 2020b, 2005; Brambilla). These states have been termed X, Y, Z in the discovery publications, without any special criterion, apart from Y being used for exotics with vector quantum numbers, i.e., J P C = 1− . Meanwhile, the Particle Data Group (PDG) has proposed a new naming scheme (Tanabashi et al. 2018) that extends the scheme used for ordinary quarkonia, in which the new names carry information on the J P C quantum numbers; see Brambilla et al. (2020b) for more details. Some of these exotics have quantum numbers that cannot be obtained with ordinary hadrons. In this case, the identification of these states as exotic is straightforward. In other

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cases, the distinction requires a careful analysis of experimental observations and theoretical predictions. Of course all hadrons should be color singlets but allowing ¯ and QQQ in the two heavy quarks sector calls for combinations beyond the QQ ¯ ¯ ¯ and so on tetraquarks like QQq q, ¯ QQq¯ q, ¯ pentaquarks like QQqqq, hybrids QQg, (Ali et al. 2017). We have observed these exotics up to now only in the sector with two heavy quarks likely due to the fact that the presence of the two heavy quarks stabilizes them. Some of the discovered states have an unprecedentely small width even if they are at or above the strong decay threshold. XY Z states offer us unique possibilities for the investigation of the dynamical properties of strongly correlated systems in QCD: we should develop the tools to gain a solid interpretation from the underlying field theory, QCD. This is a very significant problem with trade off to other fields featuring strong correlations and a pretty interesting connections to heavy ion physics, as propagation of these states in medium may help us to scrutinize their properties. Since the new quarkonium revolution, i.e., the discovery of the first exotic state, the X(3872) at BELLE in 2003 (Choi et al. 2003), a wealth of theoretical papers, appeared to supply interpretation and understanding of the characteristics of the exotics. Many models are based on the choice of some dominant degrees of freedom and an assumption on the related interaction Hamiltonian. An effective field theory molecular description of some of these states particularly close to threshold was also put forward; see, e.g., Guo et al. (2018), Braaten and Lu (2007), Braaten and Kusunoki (2004), and Fleming and Mehen (3872). A priori the simplest system consisting of only two quarks and two antiquarks (generically called tetraquarks) is already a very complicated object, and it is unclear whether or not any kind of clustering occurs in it. However, to simplify the problem it is common to focus on certain substructures and investigate their implications: in hadroquarkonia, the heavy quark and antiquark form a compact core surrounded by a light-quark cloud; in compact tetraquarks, the relevant degrees of freedom are compact diquarks and antidiquarks; in the molecular picture two color singlet mesons are interacting at some typical distance; for a review see Brambilla et al. (2020b). Discussions about exotics therefore often concentrate on the “pictures” of the states, like the tetraquark interpretation against the molecular one (of which both several different realizations exist). However, as a matter of fact all the light degrees of freedom (light quarks, glue, in different configurations) should be there in QCD close or above the strong decay threshold: it is a result of the strong dynamics which one sets in and which configuration dominates in a given regime. ¯ Even in an ordinary quarkonium or in a heavy baryon, which has a dominant QQ or QQQ configuration, subleading contributions of the quantum field theoretical Fock space may contribute, which have additional quark-antiquark pairs and active gluons. However, in the most interesting region, close or above the strong decay threshold, where the X Y Z have been discovered, the situation is much more complicated: there is no mass gap between quarkonium and the creation of a heavylight mesons couple, nor to gluon excitations; therefore many additional states built on the light quark quantum numbers may appear (Brambilla). Still, m is large scale and a first scale factorization is applicable so that nonrelativistic QCD is

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still valid. Then, if we want to introduce a description of the bound state similar to pNRQCD, making apparent that the zero order problem is the Schrödinger equation, we can still count on another scale separation. Let us consider bound states of two nonrelativistic particles and some light d.o.f., e.g., molecules in ¯ states) or tetraquarks (QQq¯ ¯ q states) in QCD: QED or quarkonium hybrids (QQg electron/gluon fields/light quarks change adiabatically in the presence of heavy quarks/nuclei. The heavy quarks/nuclei interaction may be described at leading order in the nonrelativistic expansion by an effective static energy (or potential) Eκ between the static sources where κ labels different excitations of the light degrees of freedom. A plethora of states can be built on each on the static energies Eκ by solving the corresponding Schrödinger equation, see Figs. 10 and 11. This picture corresponds to the Born-Oppenheimer (BO) approximation (Juge et al. 2003; Braaten et al. 2014a, b). Starting from pNRQED/pNRQCD, the BO approximation can be made rigorous and cast into a suitable EFT called Born-Oppenheimer EFT (BOEFT) (Berwein et al. 2015; Brambilla et al. 2018, 2019, 2020a, b; Oncala and Soto 2017; Soto and Tarrús Castellà 2020) which exploits the hierarchy of scales ΛQCD mv 2 , v being the velocity of the heavy quark.

Fig. 10 Pictorial view of electronic static energies in QED, labeled by a collective quantum number κ

Fig. 11 Pictorial view of the gluonic (or hybrid) static energies, EΓ , in QCD. The collective quantum number κ has been detailed in Λση as explained in the section on strongly coupled pNRQCD. It corresponds to the actual lattice results in Fig. 8

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Nora Brambilla

Fig. 12 Mass spectrum of neutral exotic charmonium states obtained by solving the BOEFT coupled Schrödinger equations. The neutral experimental states that have matching quantum numbers are plotted in solid blue lines. In the figure T stay for the total angular momentum. H1 is the first H1 radial excitation of H1 . The multiplets have been plotted with error bands corresponding to a gluelump mass uncertainty of 0.15 GeV. (Figure taken from Brambilla et al. 2020b)

In Berwein et al. (2015), we have obtained the BOEFT that describes hybrids. In particular we have obtained the static potentials and the set of coupled Schrödinger equations, solved them and produced all the hybrids multiplets, see Fig. 12, for the case of the two first static energies Σu− and Πu . Such static energies are degenerated at short distance where the cylindrical symmetry gets subdue to a O(3) symmetry and are then labeled by the quantum number of a gluonic operator 1+− called a gluelump. The hybrid static energies are described by a repulsive octet potential plus the gluelump mass in the short-distance limit and the O(3) symmetry is broken at order r 2 of the multipole expansion. In the long-distance regime, the static energies display a linear behavior. The gluelump correlator can be calculated on the lattice to determine the gluelump mass. It is depending on the scheme used (the scheme dependence cancels against the analogous dependence in the quark mass and in the octet static potential) but it is of the order of 800 MeV. The hybrid multiplets Hi are constructed from the solution of the Schrödinger equations in correspondence of their J P C quantum numbers. The coupling between the different Schrödinger equations is induced by a nonadiabatic coupling, known in the Born-Oppenheimer description of diatomic molecules, induced by the noncommutation between the kinetic term and a projector of the cylindrical symmetry in the BOEFT Lagrangian. The degeneracy of the static energies at small distance induces a phenomenon called Λ doubling, removing the degenerations between multiplets of opposite parity. This phenomenon is known in molecular physics but with smaller size. This and the structure of the multiplets differ from what is obtained in models for the hybrids, cf. Berwein et al. (2015). We used lattice input on the hybrid static energies and on the gluelump mass.

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In Brambilla et al. (2019, 2020a), we obtained the spin-dependent potentials at order 1/m and 1/m2 in the quark mass expansion and we could calculate all the hybrids spin multiplets. Notice that on the one hand, it is seldom to find the spin interaction considered in models and on the other hand, it would be different. In fact the 1/m contributions couple the angular momentum of the gluonic excitation with the total spin of the heavy-quark-antiquark pair. These operators are characteristic of the hybrid states and are absent for standard quarkonia. Among the 1/m2 operators besides the standard spin-orbit, total spin squared, and tensor spin operators, respectively, which appear for standard quarkonia, three novel operators appear. So interestingly, differently from the quarkonium case, the hybrid potential gets a first contribution already at order Λ2QCD /m. Hence, spin splittings are remarkably less suppressed in heavy quarkonium hybrids than in heavy quarkonia: this will have a notable impact on the phenomenology of exotics. We extracted the nonperturbative low-energy correlators appearing in the factorization fixing them on lattice data on the masses of charmonium hybrids and we could then predict all the bottomonium hybrids spin multiplets that are more difficult to evaluate on the lattice. In this same framework, it is also possible to calculate hybrids decays and quarkonium/hybrids mixing (Oncala and Soto 2017). The BOEFT may be used to describe also tetraquarks, double heavy baryons, and pentaquarks (Brambilla et al. 2018; Soto and Tarrús Castellà 2020). In the case of tetraquarks, a necessary input of the theory is the calculation of the lattice generalized Wilson loops with appropriate symmetry and light quark operators, so that besides the quantum number κ also the isospin quantum numbers I = 0, 1 have to be considered. The BOEFT approach reconciles the different pictures of exotics based on tetraquarks, molecules, hadroquarkonium, . . . In fact in the plot of a static energy as a function of ¯ q¯ or QQg, ¯ we will have different regions: for short distance a r for a state with QQq hadroquarkonium picture would emerge, then a tetraquark (or hybrid) one and when passing the heavy-light mesons line, avoided cross level effects should have to be taken into account and a molecular picture would emerge. However QCD would dictate, through the lattice correlators and the BOEFT characteristics and power counting, which structure would dominate and in which precise way. In addition production and suppression in medium may be described in the same approach (Brambilla et al. 2021a, c).

pNRQCD at Finite T, Open Quantum System, and Quarkonium in Medium Quarkonium is a special probe also for deconfinement, besides confinement. A prediction of QCD is that at a certain value of temperature (or energy density), hadronic matter undergoes a transition to a deconfined state of quarks and gluons called the quark-gluon plasma (QGP). Lattice QCD have shown that such transition takes place at a critical temperature around 150 MeV (Karsch 2002; Borsanyi et al. 2020). Experimentally heavy ion collisions make it possible to study strongly interacting matter under extreme conditions in the laboratory, recreating the QGP

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that should have primordially existed microseconds after the Big Bang. These experiments have therefore a great importance also for cosmology and allows to investigate the nuclear matter phase diagram, i.e., how nuclear matter change by varying the temperature and the chemical potential; see Brambilla et al. (2014) and Andronic et al. (2016). In heavy ions experiments at the LHC at CERN and at the RHIC at BNL, the produced matter is characterized by small net baryon densities and high temperature (Karsch 2002). Heavy quarks are good probes of the QGP. They are produced at the beginning of the collision and remain up to the end. The heavy quark mass m introduces a large scale, whose contribution may be factorized and computed in perturbation theory. Low-energy scales are sensitive to the temperature T , and even if nonperturbative, they may be accessible via lattice calculations (for what concerns equilibrium physics). As we discussed, quarkonia are special hard probes as they are multiscale systems. In the hot QCD medium also, the thermal scales of the quark-gluon plasma (QGP) are emerging: the scale related to the temperature (π )T , the Debye mass mD ∼ gT related to the (chromo) electric screening, and the scale g 2 T related to the (chromo)magnetic screening. In a weakly coupled plasma, the scales are separated and hierarchically ordered, in a strongly coupled plasma mD ∼ T . To address QCD at finite T calculations, EFTs have been developed too to resum contributions related to the thermal scales and to address IR sensitivities. In real time, the Hard Thermal Loop EFT (HTL) (Braaten and Pisarski 1990, 1992) is taking care of integrating out the temperature scale. Heavy quarkonium dissociation has been proposed long time ago as a clear probe of the quark-gluon plasma formation in colliders through the measurement of the dilepton decay-rate signal. In (Matsui and Satz 1986), this was related to the screening of the quark-antiquark interaction due to Debye mass, and it was suggested that this would have manifested in an exponential screening term exp(−mD r) in the static potential. One of the key quantities measured in experiments is the nuclear modification factor RAA = Y (P bP b)/(Ncoll Y (pp)) where Y (P bP b) and Y (pp) are the quarkonium yield in PbPb and in pp collisions, respectively. RAA is a measure for the difference in particle production in pp and nucleus-nucleus collisions. Since higher excited quarkonium states have larger radius, then the expectation was that, as the temperature increases, quarkonium would dissociate subsequently from the higher to the lower states giving origin to sequential melting. In order to study quarkonium properties in a thermal bath at a temperature T , the quantity to be determined is the quarkonium potential V which dictates, through the Schrödinger equation, the real-time evolution of the wave ¯ pair in the medium. This has been investigated for years using function of a QQ many phenomenological assumptions, spanning from the internal energy to the free energy, either the average free energy or the singlet one which is gauge dependent. pNRQCD has given us the possibility to define in QCD what is this potential: it is the matching coefficient of the EFT that results from the integration of all the scales above the scale of the binding energy. In a series of papers (Brambilla et al. 2008, 2010, 2011, 2013; Escobedo and Soto 2008; Brambilla 2021), a pNRQCD at finite T description has been constructed. One has to proceed integrating out all scales up to the binding energy and if the temperature is higher than the binding

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energy, then also the temperature has to be integrated out using HTL EFT. When T is bigger than the energy, the potential depends on the temperature, otherwise not. Thermal effects appear in any case in the nonpotential contributions to the energy levels. We assumed that the bound state exist for T  m and 1/r ≥ mD , we worked in the weak coupling limit and we consider all possible scale hierarchies (Brambilla et al. 2008). We found that the thermal part of the potential has a real part (roughly described by the free energy) and an imaginary part. The imaginary part comes from two effects: the Landau damping (Laine et al. 2007; Escobedo and Soto 2008; Brambilla et al. 2008), an effect existing also in QED, and the singlet to octet transition, existing only in QCD (Brambilla et al. 2008). Which one dominates depends on the ratio between mD and E. In the EFT, one could show that the imaginary part of the potential related to the Landau damping comes from inelastic parton scattering (Brambilla et al. 2013) and the singlet to octet transition from gluon dissociation (Brambilla et al. 2011). The existence of the imaginary part, first realized in Laine et al. (2007), changed our paradigm for quarkonium suppression as the state dissociates for this reason well before that the conventional screening becomes active (Laine et al. 2007; Escobedo and Soto 2008). The pattern of thermal corrections is pretty interesting (Brambilla et al. 2008): when T < E, thermal corrections are only in the energy; for T > 1/r, 1/r > mD or 1/r > T > E, there is no exponential screening and T dependent power like corrections appear; if T > 1/r, 1/r ∼ mD , we have exponential screening but the imaginary part of the static potential is already bigger than the real one and dissociation already happened. Once the potential has been calculated, the EFT gives the systematic framework to obtain the thermal energies: in Brambilla et al. (2010) the first QCD calculation of the thermal contributions to the Υ (1S) mass and width at order mαs5 at LHC below the dissociation temperature of about 500 MeV was performed. This calculation is very important because it gives the parametric T dependence of this observables. The width goes linear in T in the dominant term, and this has been confirmed by lattice calculations of the spectrum (Aarts et al. 2011). These findings in the EFT in perturbation theory have inspired many subsequent nonperturbative calculations of the static potential at finite T . In particular the Wilson loop at finite T has been calculated on the lattice (Rothkopf et al. 2012; Rothkopf 2020) finding hints of a large imaginary parts. These calculations are pretty challenging and refining of the extraction methods are currently in elaboration (Bala et al. 2022). Free energies defined by Polyakov loop correlators have been always very prominent in QCD at finite T ; see, e.g., the reviews Bazavov and Weber (2021) and Ghiglieri et al. (2020). The Polyakov loop correlators of a single heavy quark and of a quark-antiquark pair have been calculated both in perturbation theory using pNRQCD to resum scales contributions in Berwein et al. (2016, 2017), Brambilla et al. (2010) and on the lattice to obtain these quantities fully nonperturbatively in Bazavov et al. (2016, 2018). In particular, the Polyakov loop has been computed up ¯ free energy has been computed at short distances to order g 6 ; the (subtracted) QQ 7 ¯ free energy has been up to corrections of order g (rT )4 , g 8 ; the (subtracted) QQ computed at screening distances up to corrections of order g 8 ; the singlet free energy has been computed at short distances up to corrections of order g 4 (rT )5 , g 6 ;

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the singlet free energy has been computed at screening distances up to corrections of order g 5 (Berwein et al. 2016, 2017; Brambilla et al. 2010). From the lattice simulations and from comparison to the perturbative results, some interesting information can be obtained (Bazavov et al. 2016, 2018): lattice calculations are consistent with weak coupling expectations in the regime of application of the weakly coupled resummed perturbation theory which confirms the predictive power of the EFT; the crossover temperature to the quark-gluon plasma is 153 + 6.5 − 5 MeV as extracted from the entropy of the Polyakov loop; the screening sets in at rT ∼ 0.3 − 0.4 (observable dependent), consistent with a screening length of about 1/mD ; asymptotic screening masses are about 2mD (observable dependent); ¯ free energy has been obtained (Bazavov the first determination of the color octet QQ et al. 2018). The free energies turn out not to be the objects to be used as a potential in the Schrödinger equation even if the singlet free energy may provide a good approximation of the real part of the static potential. These are all results in thermal equilibrium. However, the evolution of quarkonium in the QGP is an out of equilibrium process in which many effects enter: the hydrodynamical evolution of the plasma and the production, dissociation, and regeneration of quarkonium in the medium, to quote the most prominent ones. It is necessary therefore to introduce an appropriate framework to describe the realtime nonequilibrium evolution of quarkonium in the QGP medium. This has been realized in Brambilla et al. (2017, 2018, 2019) where an open quantum system (OQS) framework rooted in pNRQCD at finite T has been developed that is fully quantum, conserves the number of heavy quarks, and considers both color singlet and color octet quarkonium degrees of freedom. For a review of open quantum system approach, see Akamatsu (2015, 2022), Sharma and Tiwari (2020), Yao (2021), and references therein. We consider the density matrix associated to our system. We distinguish between the environment (QGP), characterized by the scale T , and the subsystem of system (quarkonium) characterized by the scale E. We identify the inverse of E with the intrinsic time scale of the subsystem: τS ∼ 1/E and the inverse of π T with the correlation time of the environment: τE ∼ 1/(π T ). If the medium is in thermal equilibrium, or locally in thermal equilibrium, we may understand T as a temperature, otherwise is just a parameter. The medium can be strongly coupled. Then we trace the density matrix over the environment and we are left with a color singlet and color octet density matrix that can be written in terms of the pNRQCD fields, working in the close time path formalism. The evolution of the system is characterized by a relaxation time τR that is estimated by the inverse of the color singlet self-energy diagram in pNRQCD at finite T . We select quarkonia states with a small radius (Coulombic) for which 1/r π T , ΛQCD and we consider π T E. In this framework, in Brambilla et al. (2018), a set of master equations governing the time evolution of heavy quarkonium in a medium can be derived. The equations follow from assuming the inverse Bohr radius of the quarkonium to be greater than the energy scale of the medium, and model the quarkonium as evolving in the vacuum up to a time t = t0 at which point interactions with the medium begin. The equations express the time evolution of the density matrices of the heavy

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quark-antiquark color singlet, ρs , and octet states, ρo , in terms of the color singlet and octet Hamiltonians, hs = p2 /M −CF αs /r +. . . and ho = p2 /M +αs /(2Nc r)+ . . ., and interaction terms with the medium, which, at order r 2 in the multipole expansion, are encoded in the self-energy diagrams of the EFTs. These interactions ¯ pair induced by the medium, its account for the mass shift of the heavy QQ ¯ color singlet states decay width induced by the medium, the generation of QQ ¯ from QQ color octet states interacting with the medium and the generation of ¯ color octet states from QQ ¯ (color singlet or octet) states interacting with the QQ medium (recombination terms). Both Landau damping and singlet to octet effects ¯ are contained in the self-energy. The leading order interaction between a heavy QQ field and the medium is encoded in pNRQCD in a chromoelectric dipole interaction, which appears at order r/m0 in the EFT Lagrangian. The approach gives us master equations, in general non-Markovian, for the out of equilibrium evolution of the color singlet and color octet matrix densities. The system is in non-equilibrium because through interaction with the environment (quark-gluon plasma) singlet and octet quark-antiquark states continuously transform in each other although the total number of heavy quarks is conserved. ¯ Assuming that any energy scale in the medium is larger than the heavy QQ binding energy E, in particular that τR τE , we obtain a Markovian evolution while the chosen hierarchy of scales implies τs τE qualifying the regime as quantum Brownian motion. In this situation, we can reduce the general master equation to a Lindblad form. In this case the properties of the QGP are encoded in two transport coefficients: the heavy quark momentum diffusion coefficient, κ, and its dispersive counterpart γ which are given by time integrals of appropriate gauge invariant correlators at finite T given by the integral of gauge invariant finite T QCD correlators of chromoelectric fields: κ=

g2 6Nc

γ = −i



∞

ds 0

g2 6Nc



  E a,i (s, 0), E a,i (0, 0) ,

∞

ds

  E a,i (s, 0), E a,i (0, 0) .

(29) (30)

0

They come from the pNRQCD self-energies that in this regime could be factorized between this contribution and the bound state dependent part. In the case of a nonperturbative QGP, these objects are nonperturbative and should be evaluated on the lattice at a given temperature in an extended window of temperatures (Brambilla et al. 2020). In this way one could use lattice to give input to a nonequilibrium papers Brambilla et al. (2021b, c) we have solved the Lindblad equation using the highly efficient quantum trajectory method and realistically implementing a 3+1D dissipative hydrodynamics code (Omar et al. 2022), and we have obtained the bottomonium (Υ (1S), Υ (2S), Υ (3S)) nuclear modification factors, and the anisotropic flows; see Fig. 13 for a comparison between LHC data and our results on RAA . The computation does not rely on any free parameter, as it depends only on the two transport coefficients that have been evaluated independently in lattice

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Fig. 13 The nuclear modification factor RAA of the Υ (1S), Υ (2S), and Υ (3S) as a function of Npart compared to experimental measurements from the ALICE (Acharya et al. 2021), ATLAS (Lee 2020 Quark Matter), and CMS (Sirunyan et al. 2019) collaborations. The bands in the theoretical curves indicate variation with respect to κ(T ˆ ) (left) and γˆ (right). The central curves represent the central values of κ(T ˆ ) and γˆ , and the dashed and dot-dashed lines represent the lower and upper values, respectively, of κ(T ˆ ) ≡ κ/T 3 and γˆ ≡ γ /T 3 , see text for the definition of these parameters. (Figure taken from Brambilla et al. 2021b)

QCD. Our final results, which include late-time feed down of excited states, agree well with the available data from LHC 5.02 TeV PbPb collisions. Notice that the bands in the Fig. 13 depend on the undetermination with which the nonperturbative transport coefficients κ and γ , which are properties of the QGP, are presently known. More precise experimental data can help to narrow down their range. In this way we can use quarkonia with small radii as a diagnostic and investigation tool of the characteristics of the strongly coupled QGP. To apply the same description to charmonium would require to modify the master equations considering terms beyond leading order in the quarkonium density. Further work in a similar approach at the level of the Boltzmann equation has been done in Yao et al. (2021) and Yao and Mehen (2019).

Outlook The great progress of the last few decades in the construction of nonrelativistic effective quantum field theories and the progress in lattice QCD calculation allows to treat heavy quark bound states systematically in QCD. In this way, quarkonium becomes a golden probe of strong interactions at zero and finite temperature. Moreover, quarkonium is the prototype of a nonrelativistic multiscale system. Systems of such kind are ubiquitous in matter and in fields of physics, from particle to nuclear physics, to condensed matter and to astro and cosmological applications and play a key role in several open challenges at the frontier of particle physics. Therefore, all the progress obtained in this framework holds the promise to have an impact in a number of relevant contemporary problems.

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For example, combining the techniques of nonrelativistic effective field theories and the open quantum framework, we can address the nonequilibrium evolution of heavy dark matter pairs in the early universe. On the other hand, the XYZ exotics bear similarities to atomic and molecular physics, and results obtained in BOEFT can be used in those fields (Brambilla et al. 2017, 2018) as well as maybe extended to the consideration of strongly correlated system in condensed matter at finite temperature. In summary, the physics of heavy quarks not only has been and is extremely important for our advancement in nuclear and particle physics but has the promise to impact also many other nearby fields.

References R. Aaij et al. [LHCb], Sci. Bull. 65(23), 1983–1993 (2020). https://doi.org/10.1016/j.scib.2020.08. 032 [arXiv:2006.16957 [hep-ex]] G. Aarts, C. Allton, S. Kim, M.P. Lombardo, M.B. Oktay, S.M. Ryan, D.K. Sinclair, J.I. Skullerud, JHEP 11, 103 (2011). https://doi.org/10.1007/JHEP11(2011)103 [arXiv:1109.4496 [hep-lat]] S. Acharya et al. [ALICE], Phys. Lett. B 822, 136579 (2021). https://doi.org/10.1016/j.physletb. 2021.136579 [arXiv:2011.05758 [nucl-ex]] Y. Akamatsu, Phys. Rev. D 91(5), 056002 (2015). https://doi.org/10.1103/PhysRevD.91.056002 [arXiv:1403.5783 [hep-ph]] Y. Akamatsu, Prog. Part. Nucl. Phys. 123, 103932 (2022). https://doi.org/10.1016/j.ppnp.2021. 103932 [arXiv:2009.10559 [nucl-th]] M.T. AlFiky, F. Gabbiani, A.A. Petrov, Phys. Lett. B 640, 238–245 (2006). https://doi.org/10.1016/ j.physletb.2006.07.069 [arXiv:hep-ph/0506141 [hep-ph]] A. Ali, J.S. Lange, S. Stone, Prog. Part. Nucl. Phys. 97, 123–198 (2017) K. Amemiya, H. Suganuma, Phys. Rev. D 60, 114509 (1999). https://doi.org/10.1103/PhysRevD. 60.114509 [arXiv:hep-lat/9811035 [hep-lat]] A. Andronic, F. Arleo, R. Arnaldi, A. Beraudo, E. Bruna, D. Caffarri, Z.C. del Valle, J.G. Contreras, T. Dahms, A. Dainese et al., Eur. Phys. J. C 76(3), 107 (2016). https://doi.org/10.1140/epjc/ s10052-015-3819-5 [arXiv:1506.03981 [nucl-ex]] T. Appelquist, M. Dine, I.J. Muzinich, Phys. Lett. B 69, 231–236 (1977). https://doi.org/10.1016/ 0370-2693(77)90651-7 J.J. Aubert et al., Phys. Rev. Lett. 33, 1404–1406 (1974). https://doi.org/10.1103/PhysRevLett.33. 1404 J.E. Augustin et al., Phys. Rev. Lett. 33, 1406–1408 (1974). https://doi.org/10.1103/PhysRevLett. 33.1406 C. Ayala, X. Lobregat, A. Pineda, JHEP 09, 016 (2020). https://doi.org/10.1007/JHEP09(2020)016 [arXiv:2005.12301 [hep-ph]] M. Baker, V. Chelnokov, L. Cosmai, F. Cuteri, A. Papa, (2022) [arXiv:2111.06618 [hep-lat]] M. Baker, J.S. Ball, F. Zachariasen, Phys. Rep. 209, 73–127 (1991). https://doi.org/10.1016/03701573(91)90123--4 M. Baker, J.S. Ball, N. Brambilla, G.M. Prosperi, F. Zachariasen, Phys. Rev. D 54, 2829–2844 (1996) [erratum: Phys. Rev. D 56, 2475 (1997)]. https://doi.org/10.1103/PhysRevD.56.2475 [arXiv:hep-ph/9602419 [hep-ph]] M. Baker, N. Brambilla, H.G. Dosch, A. Vairo, Phys. Rev. D 58, 034010 (1998). https://doi.org/ 10.1103/PhysRevD.58.034010 [arXiv:hep-ph/9802273 [hep-ph]] D. Bala, O. Kaczmarek, R. Larsen, S. Mukherjee, G. Parkar, P. Petreczky, A. Rothkopf, J.H. Weber, Static quark-antiquark interactions at nonzero temperature from lattice QCD. Phys. Rev. D 105(5), 054513 (2022)

3000

N. Brambilla

G.S. Bali, Phys. Rep. 343, 1–136 (2001). https://doi.org/10.1016/S0370-1573(00)00079-X [arXiv:hep-ph/0001312 [hep-ph]] G.S. Bali, A. Pineda, Phys. Rev. D 69, 094001 (2004). https://doi.org/10.1103/PhysRevD.69. 094001 [arXiv:hep-ph/0310130 [hep-ph]] G.S. Bali, K. Schilling, C. Schlichter, Phys. Rev. D 51, 5165–5198 (1995). https://doi.org/10.1103/ PhysRevD.51.5165 [arXiv:hep-lat/9409005 [hep-lat]] G.S. Bali, K. Schilling, A. Wachter, Phys. Rev. D 56, 2566–2589 (1997). https://doi.org/10.1103/ PhysRevD.56.2566 [arXiv:hep-lat/9703019 [hep-lat]] A. Barchielli, E. Montaldi, G.M. Prosperi, Nucl. Phys. B 296, 625 (1988) [erratum: Nucl. Phys. B 303, 752 (1988)]. https://doi.org/10.1016/0550-3213(88)90036-3 A. Barchielli, N. Brambilla, G.M. Prosperi, Nuovo Cim. A 103, 59 (1990). https://doi.org/10.1007/ BF02902620 G. Barucca et al. [PANDA], Eur. Phys. J. A 57(6), 184 (2021). https://doi.org/10.1140/epja/s10050021-00475-y [arXiv:2101.11877 [hep-ex]] A. Bazavov, J.H. Weber, Prog. Part. Nucl. Phys. 116, 103823 (2021). https://doi.org/10.1016/j. ppnp.2020.103823 [arXiv:2010.01873 [hep-lat]] A. Bazavov, N. Brambilla, X. Garcia Tormo, i, P. Petreczky, J. Soto, A. Vairo, Phys. Rev. D 86, 114031 (2012). https://doi.org/10.1103/PhysRevD.86.114031 [arXiv:1205.6155 [hep-ph]] A. Bazavov, N. Brambilla, H.T. Ding, P. Petreczky, H.P. Schadler, A. Vairo, J.H. Weber, Phys. Rev. D 93(11), 114502 (2016) [arXiv:1603.06637 [hep-lat]] A. Bazavov et al. [TUMQCD], Phys. Rev. D 98(5), 054511 (2018) [arXiv:1804.10600 [hep-lat]] A. Bazavov et al. [TUMQCD], Phys. Rev. D 100(11), 114511 (2019). https://doi.org/10.1103/ PhysRevD.100.114511 [arXiv:1907.11747 [hep-lat]] M. Berwein, N. Brambilla, J. Tarrús Castellà, A. Vairo, Phys. Rev. D 92(11), 114019 (2015) M. Berwein, N. Brambilla, P. Petreczky, A. Vairo, Phys. Rev. D 93(3), 034010 (2016) [arXiv:1512.08443 [hep-ph]] M. Berwein, N. Brambilla, P. Petreczky, A. Vairo, Phys. Rev. D 96(1), 014025 (2017) [arXiv:1704.07266 [hep-ph]] M. Berwein, N. Brambilla, S. Hwang, A. Vairo, Phys. Rev. D 99(9), 094008 (2019). https://doi. org/10.1103/PhysRevD.99.094008 [arXiv:1811.05184 [hep-th]] A.J. Bevan et al. [BaBar and Belle], Eur. Phys. J. C 74, 3026 (2014). https://doi.org/10.1140/epjc/ s10052-014-3026-9 [arXiv:1406.6311 [hep-ex]] P. Bicudo, N. Cardoso, M. Cardoso, Phys. Rev. D 98(11), 114507 (2018). https://doi.org/10.1103/ PhysRevD.98.114507 [arXiv:1808.08815 [hep-lat]] G.T. Bodwin, E. Braaten, E. Eichten, S.L. Olsen, T.K. Pedlar, J. Russ, (2013) [arXiv:1307.7425 [hep-ph]] G.T. Bodwin, E. Braaten, G.P. Lepage, Phys. Rev. D 51, 1125–1171 (1995) [erratum: Phys. Rev. D 55, 5853 (1997)]. https://doi.org/10.1103/PhysRevD.55.5853 [arXiv:hep-ph/9407339 [hep-ph]] G.T. Bodwin, H.S. Chung, J.H. Ee, U.R. Kim, J. Lee, Phys. Rev. D 101(9), 096011 (2020). https:// doi.org/10.1103/PhysRevD.101.096011 [arXiv:1910.05497 [hep-ph]] S. Borsanyi, Z. Fodor, J.N. Guenther, R. Kara, S.D. Katz, P. Parotto, A. Pasztor, C. Ratti, K.K. Szabo, Phys. Rev. Lett. 125(5), 052001 (2020). https://doi.org/10.1103/PhysRevLett.125. 052001 [arXiv:2002.02821 [hep-lat]] E. Braaten, M. Kusunoki, Phys. Rev. D 69, 074005 (2004) E. Braaten, M. Lu, Phys. Rev. D 76, 094028 (2007) E. Braaten, R.D. Pisarski, Nucl. Phys. B 337, 569–634 (1990). https://doi.org/10.1016/05503213(90)90508-B E. Braaten, R.D. Pisarski, Phys. Rev. D 45(6), R1827 (1992). https://doi.org/10.1103/PhysRevD. 45.R1827 E. Braaten, C. Langmack, D.H. Smith, Phys. Rev. D 90(1), 014044 (2014a) E. Braaten, C. Langmack, D.H. Smith, Phys. Rev. Lett. 112, 222001 (2014b) N. Brambilla, [arXiv:2111.10788 [hep-lat]] N. Brambilla, Nuovo Cim. A 105, 949–958 (1992). https://doi.org/10.1007/BF02730836 N. Brambilla, PoS HardProbes2020, 017 (2021)

81 Quark Nuclear Physics with Heavy Quarks

3001

N. Brambilla, A. Vairo, (1999) [arXiv:hep-ph/9904330 [hep-ph]] N. Brambilla, A. Vairo, Phys. Rev. D 55, 3974–3986 (1997). https://doi.org/10.1103/PhysRevD. 55.3974 [arXiv:hep-ph/9606344 [hep-ph]] N. Brambilla, A. Vairo, Phys. Rev. D 62, 094019 (2000). https://doi.org/10.1103/PhysRevD.62. 094019 [arXiv:hep-ph/0002075 [hep-ph]] N. Brambilla, A. Vairo, Phys. Rev. D 71, 034020 (2005). https://doi.org/10.1103/PhysRevD.71. 034020 [arXiv:hep-ph/0411156 [hep-ph]] N. Brambilla et al. [Quarkonium Working Group], (2005). https://doi.org/10.5170/CERN-2005005 [arXiv:hep-ph/0412158 [hep-ph]] N. Brambilla, P. Consoli, G.M. Prosperi, Phys. Rev. D 50, 5878–5892 (1994). https://doi.org/10. 1103/PhysRevD.50.5878 [arXiv:hep-th/9401051 [hep-th]] N. Brambilla, A. Pineda, J. Soto, A. Vairo, Phys. Rev. D 60, 091502 (1999). https://doi.org/10. 1103/PhysRevD.60.091502 [arXiv:hep-ph/9903355 [hep-ph]] N. Brambilla, A. Pineda, J. Soto, A. Vairo, Phys. Lett. B 470, 215 (1999). https://doi.org/10.1016/ S0370-2693(99)01301--5 [arXiv:hep-ph/9910238 [hep-ph]] N. Brambilla, V. Leino, O. Philipsen, C. Reisinger, A. Vairo, M. Wagner, (2005) [arXiv:2106.01794 [hep-lat]] N. Brambilla, A. Pineda, J. Soto, A. Vairo, Nucl. Phys. B 566, 275 (2000). https://doi.org/10.1016/ S0550-3213(99)00693-8 [arXiv:hep-ph/9907240 [hep-ph]] N. Brambilla, A. Pineda, J. Soto, A. Vairo, Phys. Rev. D 63, 014023 (2001). https://doi.org/10. 1103/PhysRevD.63.014023 [arXiv:hep-ph/0002250 [hep-ph]] N. Brambilla, Y. Sumino, A. Vairo, Phys. Rev. D 65, 034001 (2002). https://doi.org/10.1103/ PhysRevD.65.034001 [arXiv:hep-ph/0108084 [hep-ph]] N. Brambilla, D. Gromes, A. Vairo, Phys. Lett. B 576, 314–327 (2003). https://doi.org/10.1016/j. physletb.2003.09.100 [arXiv:hep-ph/0306107 [hep-ph]] N. Brambilla, D. Eiras, A. Pineda, J. Soto, A. Vairo, Phys. Rev. D 67, 034018 (2003). https://doi. org/10.1103/PhysRevD.67.034018 [arXiv:hep-ph/0208019 [hep-ph]] N. Brambilla, A. Pineda, J. Soto, A. Vairo, Rev. Mod. Phys. 77, 1423 (2005). [arXiv:hepph/0410047 [hep-ph]] N. Brambilla, A. Vairo, T. Rosch, Phys. Rev. D 72, 034021 (2005). https://doi.org/10.1103/ PhysRevD.72.034021 [arXiv:hep-ph/0506065 [hep-ph]] N. Brambilla, Y. Jia, A. Vairo, Phys. Rev. D 73, 054005 (2006). https://doi.org/10.1103/PhysRevD. 73.054005 [arXiv:hep-ph/0512369 [hep-ph]] N. Brambilla, X. Garcia i Tormo, J. Soto, A. Vairo, Phys. Lett. B 647, 185–193 (2007). https://doi. org/10.1016/j.physletb.2007.02.015 [arXiv:hep-ph/0610143 [hep-ph]] N. Brambilla, J. Ghiglieri, A. Vairo, P. Petreczky, Phys. Rev. D 78, 014017 (2008). https://doi.org/ 10.1103/PhysRevD.78.014017 [arXiv:0804.0993 [hep-ph]] N. Brambilla, A. Vairo, X. Garcia i Tormo, J. Soto, Phys. Rev. D 80, 034016 (2009). https://doi. org/10.1103/PhysRevD.80.034016 [arXiv:0906.1390 [hep-ph]] N. Brambilla, J. Ghiglieri, A. Vairo, Phys. Rev. D 81, 054031 (2010). https://doi.org/10.1103/ PhysRevD.81.054031 [arXiv:0911.3541 [hep-ph]] N. Brambilla, M.A. Escobedo, J. Ghiglieri, J. Soto, A. Vairo, JHEP 09, 038 (2010). https://doi.org/ 10.1007/JHEP09(2010)038 [arXiv:1007.4156 [hep-ph]] N. Brambilla, J. Ghiglieri, P. Petreczky, A. Vairo, Phys. Rev. D 82, 074019 (2010) [arXiv:1007.5172 [hep-ph]] N. Brambilla, S. Eidelman, B.K. Heltsley, R. Vogt, G.T. Bodwin, E. Eichten, A.D. Frawley, A.B. Meyer, R.E. Mitchell, V. Papadimitriou et al., Eur. Phys. J. C 71, 1534 (2011). https:// doi.org/10.1140/epjc/s10052-010-1534-9 [arXiv:1010.5827 [hep-ph]] N. Brambilla, M.A. Escobedo, J. Ghiglieri, A. Vairo, JHEP 12, 116 (2011). https://doi.org/10.1007/ JHEP12(2011)116 [arXiv:1109.5826 [hep-ph]] N. Brambilla, P. Pietrulewicz, A. Vairo, Phys. Rev. D 85, 094005 (2012). https://doi.org/10.1103/ PhysRevD.85.094005 [arXiv:1203.3020 [hep-ph]] N. Brambilla, M.A. Escobedo, J. Ghiglieri, A. Vairo, JHEP 05, 130 (2013). https://doi.org/10.1007/ JHEP05(2013)130 [arXiv:1303.6097 [hep-ph]]

3002

N. Brambilla

N. Brambilla, S. Eidelman, P. Foka, S. Gardner, A.S. Kronfeld, M.G. Alford, R. Alkofer, M. Butenschoen, T.D. Cohen, J. Erdmenger et al., Eur. Phys. J. C 74(10), 2981 (2014). https:// doi.org/10.1140/epjc/s10052-014-2981-5 [arXiv:1404.3723 [hep-ph]] N. Brambilla, M. Groher, H.E. Martinez, A. Vairo, Phys. Rev. D 90(11), 114032 (2014). https:// doi.org/10.1103/PhysRevD.90.114032 [arXiv:1407.7761 [hep-ph]] N. Brambilla, M.A. Escobedo, J. Soto, A. Vairo, Phys. Rev. D 96(3), 034021 (2017) [arXiv:1612.07248 [hep-ph]] N. Brambilla, V. Shtabovenko, J. Tarrús Castellà, A. Vairo, Phys. Rev. D 95(11), 116004 (2017). https://doi.org/10.1103/PhysRevD.95.116004 [arXiv:1704.03476 [hep-ph]] N. Brambilla, G. Krein, J. Tarrús Castellà, A. Vairo, Phys. Rev. D 97(1), 016016 (2018) N. Brambilla, M.A. Escobedo, J. Soto, A. Vairo, Phys. Rev. D 97(7), 074009 (2018) [arXiv:1711.04515 [hep-ph]] N. Brambilla, W.K. Lai, J. Segovia, J. Tarrús Castellà, A. Vairo, Phys. Rev. D 99(1), 014017 (2019) N. Brambilla, M.A. Escobedo, A. Vairo, P. Vander Griend, Phys. Rev. D 100(5), 054025 (2019). [arXiv:1903.08063 [hep-ph]] N. Brambilla, V. Leino, P. Petreczky, A. Vairo, Phys. Rev. D 102(7), 074503 (2020). https://doi. org/10.1103/PhysRevD.102.074503 [arXiv:2007.10078 [hep-lat]] N. Brambilla, W.K. Lai, J. Segovia, J. Tarrús Castellà, Phys. Rev. D 101(5), 054040 (2020a) N. Brambilla, H.S. Chung, D. Müller, A. Vairo, JHEP 04, 095 (2020b). https://doi.org/10.1007/ JHEP04(2020)095 [arXiv:2002.07462 [hep-ph]] Copy to ClipboardDownload N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.P. Shen, C.E. Thomas, A. Vairo, C.Z. Yuan, Phys. Rep. 873, 1–154 (2020b). https://doi.org/10.1016/j.physrep.2020.05.001 [arXiv:1907.07583 [hep-ex]] N. Brambilla, H.S. Chung, A. Vairo, JHEP 09, 032 (2021a). [arXiv:2106.09417 [hep-ph]] N. Brambilla, H.S. Chung, A. Vairo, Phys. Rev. Lett. 126(8), 082003 (2021b). https://doi.org/10. 1103/PhysRevLett.126.082003 [arXiv:2007.07613 [hep-ph]] N. Brambilla, M.Á. Escobedo, M. Strickland, A. Vairo, P. Vander Griend, J.H. Weber, Phys. Rev. D 104(9), 094049 (2021b) [arXiv:2107.06222 [hep-ph]] N. Brambilla, M.Á. Escobedo, M. Strickland, A. Vairo, P. Vander Griend, J.H. Weber, JHEP 05, 136 (2021c) [arXiv:2012.01240 [hep-ph]] L.S. Brown, W.I. Weisberger, Phys. Rev. D 20, 3239 (1979). https://doi.org/10.1103/PhysRevD. 20.3239 G. Buchalla, T.K. Komatsubara, F. Muheim, L. Silvestrini, M. Artuso, D.M. Asner, P. Ball, E. Baracchini, G. Bell, M. Beneke et al., Eur. Phys. J. C 57, 309–492 (2008). https://doi.org/ 10.1140/epjc/s10052-008-0716-1 [arXiv:0801.1833 [hep-ph]] M. Campostrini, K.J.M. Moriarty, J. Potvin, C. Rebbi, Phys. Lett. B 193, 78–84 (1987). https://doi. org/10.1016/0370-2693(87)90459-X W.E. Caswell, G.P. Lepage, Phys. Lett. B 167, 437–442 (1986). https://doi.org/10.1016/03702693(86)91297-9 P. Chang, K.F. Chen, W.S. Hou, Prog. Part. Nucl. Phys. 97, 261–311 (2017). https://doi.org/10. 1016/j.ppnp.2017.07.001 [arXiv:1708.03793 [hep-ph]] E. Chapon, D. d’Enterria, B. Ducloue, M.G. Echevarria, P.B. Gossiaux, V. Kartvelishvili, T. Kasemets, J.P. Lansberg, R. McNulty, D.D. Price et al., Prog. Part. Nucl. Phys. 122, 103906 (2022). https://doi.org/10.1016/j.ppnp.2021.103906 [arXiv:2012.14161 [hep-ph]] G. Chiladze, A.F. Falk, A.A. Petrov, Phys. Rev. D 58, 034013 (1998). https://doi.org/10.1103/ PhysRevD.58.034013 [arXiv:hep-ph/9804248 [hep-ph]] S.K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003) H.S. Chung, (2020). https://doi.org/10.1142/9789811219313_0064 M. Creutz, Phys. Rev. D 15, 1128 (1977). https://doi.org/10.1103/PhysRevD.15.1128 M. Creutz, Phys. Rev. D 21, 2308–2315 (1980). https://doi.org/10.1103/PhysRevD.21.2308 M. Creutz, K.J.M. Moriarty, Phys. Rev. D 26, 2166 (1982). https://doi.org/10.1103/PhysRevD.26. 2166 H.G. Dosch, Y.A. Simonov, Phys. Lett. B 205, 339–344 (1988). https://doi.org/10.1016/03702693(88)91675-9

81 Quark Nuclear Physics with Heavy Quarks

3003

E. Eichten, F. Feinberg, Phys. Rev. D 23, 2724 (1981). https://doi.org/10.1103/PhysRevD.23.2724 E.J. Eichten, C. Quigg, Phys. Rev. D 49, 5845–5856 (1994). https://doi.org/10.1103/PhysRevD.49. 5845 [arXiv:hep-ph/9402210 [hep-ph]] E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.M. Yan, Phys. Rev. D 17, 3090 (1978) [erratum: Phys. Rev. D 21, 313 (1980)]. https://doi.org/10.1103/PhysRevD.17.3090 E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.M. Yan, Phys. Rev. D 21, 203 (1980). https:// doi.org/10.1103/PhysRevD.21.203 M.A. Escobedo, J. Soto, Phys. Rev. A 78, 032520 (2008). https://doi.org/10.1103/PhysRevA.78. 032520 [arXiv:0804.0691 [hep-ph]] M.Á. Escobedo, Phys. Rev. D 103(3), 034010 (2021). https://doi.org/10.1103/PhysRevD.103. 034010 [arXiv:2010.10424 [hep-ph]] W. Fischler, Nucl. Phys. B 129, 157–174 (1977). https://doi.org/10.1016/0550-3213(77)90026-8 S. Fleming, T. Mehen, Phys. Rev. D 85, 014016 (2012) H. Fritzsch, M. Gell-Mann, H. Leutwyler, Phys. Lett. B 47, 365–368 (1973). https://doi.org/10. 1016/0370-2693(73)90625-4 H. Georgi, (1991) HUTP-91-A039 T. Gershon, V.V. Gligorov, Rept. Prog. Phys. 80(4), 046201 (2017). https://doi.org/10.1088/13616633/aa5514 [arXiv:1607.06746 [hep-ex]] J. Ghiglieri, A. Kurkela, M. Strickland, A. Vuorinen, Phys. Rep. 880, 1–73 (2020) [arXiv:2002.10188 [hep-ph]] S.L. Glashow, J. Iliopoulos, L. Maiani, Phys. Rev. D 2, 1285–1292 (1970). https://doi.org/10.1103/ PhysRevD.2.1285 J. Greensite, Prog. Part. Nucl. Phys. 51, 1 (2003). https://doi.org/10.1016/S0146-6410(03)900123 [arXiv:hep-lat/0301023 [hep-lat]] D. Gromes, Z. Phys. C 26, 401 (1984). https://doi.org/10.1007/BF01452566 F.K. Guo, C. Hanhart, U.G. Meißner, Q. Wang, Q. Zhao, B.S. Zou, Rev. Mod. Phys. 90(1), 015004 (2018) S.N. Gupta, S.F. Radford, W.W. Repko, Phys. Rev. D 26, 3305 (1982). https://doi.org/10.1103/ PhysRevD.26.3305 J. Heinonen, R.J. Hill, M.P. Solon, Phys. Rev. D 86, 094020 (2012). https://doi.org/10.1103/ PhysRevD.86.094020 [arXiv:1208.0601 [hep-ph]] N. Isgur, J.E. Paton, Phys. Rev. D 31, 2910 (1985). https://doi.org/10.1103/PhysRevD.31.2910 S. Joosten, Z.E. Meziani, PoS QCDEV2017, 017 (2018). https://doi.org/10.22323/1.308.0017 [arXiv:1802.02616 [hep-ex]] K.J. Juge, J. Kuti, C. Morningstar, Phys. Rev. Lett. 90, 161601 (2003). https://doi.org/10.1103/ PhysRevLett.90.161601 [arXiv:hep-lat/0207004 [hep-lat]] F. Karsch, Nucl. Phys. A 698, 199–208 (2002). https://doi.org/10.1016/S0375-9474(01)01365-3 [arXiv:hep-ph/0103314 [hep-ph]] Y. Kiyo, Y. Sumino, Nucl. Phys. B 889, 156–191 (2014). https://doi.org/10.1016/j.nuclphysb.2014. 10.010 [arXiv:1408.5590 [hep-ph]] B.A. Kniehl, A.A. Penin, V.A. Smirnov, M. Steinhauser, Nucl. Phys. B 635, 357–383 (2002). https://doi.org/10.1016/S0550-3213(02)00403-0 [arXiv:hep-ph/0203166 [hep-ph]] B.A. Kniehl, A.A. Penin, Y. Schroder, V.A. Smirnov, M. Steinhauser, Phys. Lett. B 607, 96–100 (2005). https://doi.org/10.1016/j.physletb.2004.12.024 [arXiv:hep-ph/0412083 [hep-ph]] J.B. Kogut, Rev. Mod. Phys. 55, 775 (1983). https://doi.org/10.1103/RevModPhys.55.775 Y. Koma, M. Koma, H. Wittig, Phys. Rev. Lett. 97, 122003 (2006). https://doi.org/10.1103/ PhysRevLett.97.122003 [arXiv:hep-lat/0607009 [hep-lat]] Y. Koma, M. Koma, H. Wittig, PoS LATTICE2007, 111 (2007). https://doi.org/10.22323/1.042. 0111 [arXiv:0711.2322 [hep-lat]] E. Kou et al. [Belle-II], PTEP 2019(12), 123C01 (2019) [erratum: PTEP 2020(2), 029201 (2020)]. https://doi.org/10.1093/ptep/ptz106 [arXiv:1808.10567 [hep-ex]] A.S. Kronfeld, Ann. Rev. Nucl. Part. Sci. 62, 265–284 (2012). https://doi.org/10.1146/annurevnucl-102711-094942 [arXiv:1203.1204 [hep-lat]]

3004

N. Brambilla

M. Laine, O. Philipsen, P. Romatschke, M. Tassler, JHEP 03, 054 (2007). https://doi.org/10.1088/ 1126--6708/2007/03/054 [arXiv:hep-ph/0611300 [hep-ph]] S. Lee [ATLAS Collaboration], Quark Matter 2020. https://indico.cern.ch/event/792436/ contributions/3535775/ G.P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, K. Hornbostel, Phys. Rev. D 46, 4052–4067 (1992). https://doi.org/10.1103/PhysRevD.46.4052 [arXiv:hep-lat/9205007 [hep-lat]] W. Lucha, F.F. Schoberl, D. Gromes, Bound states of quarks. Phys. Rept. 200, 127–240 (1991) A.V. Manohar, Phys. Rev. D 56, 230–237 (1997). https://doi.org/10.1103/PhysRevD.56.230 [arXiv:hep-ph/9701294 [hep-ph]] T. Matsui, H. Satz, Phys. Lett. B 178, 416–422 (1986). https://doi.org/10.1016/03702693(86)91404-8 L. Müller, O. Philipsen, C. Reisinger, M. Wagner, Phys. Rev. D 100(5), 054503 (2019). https://doi. org/10.1103/PhysRevD.100.054503 [arXiv:1907.01482 [hep-lat]] K. Nawa, H. Suganuma, T. Kojo, Phys. Rev. D 75, 086003 (2007). https://doi.org/10.1103/ PhysRevD.75.086003 [arXiv:hep-th/0612187 [hep-th]] G.C. Nayak, J.W. Qiu, G.F. Sterman, Phys. Rev. D 72, 114012 (2005). https://doi.org/10.1103/ PhysRevD.72.114012 [arXiv:hep-ph/0509021 [hep-ph]] M. Neubert, (2004). https://doi.org/10.1142/9789812773579_0004 [arXiv:hep-ph/0512222 [hepph]] M. Neubert, Int. J. Mod. Phys. A 11, 4173–4240 (1996). https://doi.org/10.1142/ S0217751X96001966 [arXiv:hep-ph/9604412 [hep-ph]] H.B. Omar, M.Á. Escobedo, A. Islam, M. Strickland, S. Thapa, P. Vander Griend, J.H. Weber, Comput. Phys. Commun. 273, 108266 (2022). https://doi.org/10.1016/j.cpc.2021.108266 [arXiv:2107.06147 [physics.comp-ph]] R. Oncala, J. Soto, Phys. Rev. D 96(1), 014004 (2017) G. Perez-Nadal, J. Soto, Phys. Rev. D 79, 114002 (2009). https://doi.org/10.1103/PhysRevD.79. 114002 [arXiv:0811.2762 [hep-ph]] C. Peset, A. Pineda, M. Stahlhofen, JHEP 05, 017 (2016). https://doi.org/10.1007/ JHEP05(2016)017 [arXiv:1511.08210 [hep-ph]] C. Peset, A. Pineda, J. Segovia, JHEP 09, 167 (2018). https://doi.org/10.1007/JHEP09(2018)167 [arXiv:1806.05197 [hep-ph]] A.A. Petrov, J. Phys. Conf. Ser. 9, 83–86 (2005). https://doi.org/10.1088/1742-6596/9/1/013 A. Pineda, Prog. Part. Nucl. Phys. 67, 735–785 (2012). https://doi.org/10.1016/j.ppnp.2012.01.038 [arXiv:1111.0165 [hep-ph]] A. Pineda, J. Segovia, Phys. Rev. D 87(7), 074024 (2013). https://doi.org/10.1103/PhysRevD.87. 074024 [arXiv:1302.3528 [hep-ph]] A. Pineda, J. Soto, Nucl. Phys. B Proc. Suppl. 64, 428–432 (1998). https://doi.org/10.1016/S09205632(97)01102-X [arXiv:hep-ph/9707481 [hep-ph]] A. Pineda, J. Soto, Phys. Lett. B 495, 323–328 (2000). https://doi.org/10.1016/S03702693(00)01261-2 [arXiv:hep-ph/0007197 [hep-ph]] A. Pineda, A. Vairo, Phys. Rev. D 63, 054007 (2001) [erratum: Phys. Rev. D 64, 039902 (2001)]. https://doi.org/10.1103/PhysRevD.64.039902 [arXiv:hep-ph/0009145 [hep-ph]] H.J. Rothe, World Sci. Lect. Notes Phys. 43, 1–381 (1992). https://doi.org/10.1142/8229 A. Rothkopf, Phys. Rep. 858, 1–117 (2020). https://doi.org/10.1016/j.physrep.2020.02.006 [arXiv:1912.02253 [hep-ph]] A. Rothkopf, T. Hatsuda, S. Sasaki, Phys. Rev. Lett. 108, 162001 (2012). https://doi.org/10.1103/ PhysRevLett.108.162001 [arXiv:1108.1579 [hep-lat]] E.E. Salpeter, H.A. Bethe, Phys. Rev. 84, 1232–1242 (1951). https://doi.org/10.1103/PhysRev.84. 1232 S. Sasaki, H. Suganuma, H. Toki, Prog. Theor. Phys. 94, 373–384 (1995). https://doi.org/10.1143/ PTP.94.373 H.J. Schnitzer, Phys. Rev. D 18, 3482 (1978). https://doi.org/10.1103/PhysRevD.18.3482 J. Segovia, S. Steinbeißer, A. Vairo, Phys. Rev. D 99(7), 074011 (2019). https://doi.org/10.1103/ PhysRevD.99.074011 [arXiv:1811.07590 [hep-ph]]

81 Quark Nuclear Physics with Heavy Quarks

3005

R. Sharma, A. Tiwari, Phys. Rev. D 101(7), 074004 (2020) [arXiv:1912.07036 [hep-ph]] A.M. Sirunyan et al. [CMS], Phys. Lett. B 790, 270–293 (2019). https://doi.org/10.1016/j.physletb. 2019.01.006 [arXiv:1805.09215 [hep-ex]] J. Soto, J. Tarrús Castellà, Phys. Rev. D 102(1), 014012 (2020) J. Soto, J. Tarrús Castellà, Phys. Rev. D 104, 074027 (2021). https://doi.org/10.1103/PhysRevD. 104.074027 [arXiv:2108.00496 [hep-ph]] T.T. Takahashi, H. Matsufuru, Y. Nemoto, H. Suganuma, Phys. Rev. Lett. 86, 18–21 (2001). https:// doi.org/10.1103/PhysRevLett.86.18 [arXiv:hep-lat/0006005 [hep-lat]] T.T. Takahashi, H. Suganuma, Y. Nemoto, H. Matsufuru, Phys. Rev. D 65, 114509 (2002). https:// doi.org/10.1103/PhysRevD.65.114509 [arXiv:hep-lat/0204011 [hep-lat]] H. Takaura, T. Kaneko, Y. Kiyo, Y. Sumino, JHEP 04, 155 (2019). https://doi.org/10.1007/ JHEP04(2019)155 [arXiv:1808.01643 [hep-ph]] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98(3), 030001 (2018) A. Vairo, Mod. Phys. Lett. A 31(34), 1630039 (2016). https://doi.org/10.1142/ S0217732316300391 K.G. Wilson, Phys. Rev. D 10, 2445–2459 (1974). https://doi.org/10.1103/PhysRevD.10.2445 R. Yanagihara, M. Kitazawa, PTEP 2019(9), 093B02 (2019) [erratum: PTEP 2020(7), 079201 (2020)]. https://doi.org/10.1093/ptep/ptz093 [arXiv:1905.10056 [hep-ph]] X. Yao, Int. J. Mod. Phys. A 36(20), 2130010 (2021). https://doi.org/10.1142/ S0217751X21300106 [arXiv:2102.01736 [hep-ph]] X. Yao, T. Mehen, Phys. Rev. D 99(9), 096028 (2019). [arXiv:1811.07027 [hep-ph]] X. Yao, W. Ke, Y. Xu, S.A. Bass, T. Mehen, B. Müller, Quarkonium production in heavy ion collisions: from open quantum system to transport equation. Nucl. Phys. A 1005, 121854 (2021) X. Yao, T. Mehen, Quarkonium semiclassical transport in Quark-Gluon Plasma: factorization and quantum correction. JHEP 2, 062 (2021) C.Z. Yuan, S.L. Olsen, Nat. Rev. Phys. 1(8), 480–494 (2019). https://doi.org/10.1038/s42254-0190082-y [arXiv:2001.01164 [hep-ex]]

Generalization of Global Symmetry and Its Applications to QCD-Related Physics

82

Yuya Tanizaki

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry ⇒ Topological Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Symmetry = Topological Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Confinement-Deconfinement Phases and One-Form Symmetry . . . . . . . . . . . . . . . . . . . . . . Fradkin-Shenker’s (Non-)complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gauging of Generalized Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ’t Hooft Anomaly and Anomaly Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physics of θ Angle: Application of the Z[1] N Symmetry for YM Theory . . . . . . . . . . . . . . . . Applications to 4d QCD with Fundamental Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Symmetry has been a powerful tool to uncover nonperturbative aspects of strongly coupled quantum field theories (QFTs), including the dynamics of quantum chromodynamics (QCD). Recently, the notion of symmetry is drastically extended as people noticed that the essential feature of the conservation law or of the Ward-Takahashi identity is the presence of topological operators. Therefore, it is convenient to define generalized symmetry as the topological operators in studying strongly coupled QFTs. Here, the development of generalized

Y. Tanizaki () Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_24

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symmetry will be briefly reviewed including its applications to QCD-related physics.

Introduction Solving QFTs is a difficult task. For example, although gluons and quarks are natural degrees of freedom describing high-energy behaviors, they do not appear in low-energy physics, and instead, one can only observe the color-singlet hadrons. Our theoretical methods are not yet powerful enough to give controllable analytical computations of such physics, and then it would be natural to ask if one can give any exact statements based on QFTs. Here, let us pay attention to “symmetry.” According to the Noether theorem, symmetry provides a conservation law, and one can obtain fruitful theorems as its outcomes: • Selection rules and spectral degeneracy: The states should form the (projective) representation of the symmetry. Especially when the unbroken symmetry exists, the particle spectrum should follow its representation, and the scattering amplitudes obey the selection rule. • Landau’s criterion of phases (Landau 1937): When the symmetry-breaking patterns are different, there has to be a phase transition separating those two points as long as the symmetry is preserved. • Nambu-Goldstone theorem (Nambu 1960; Goldstone 1961): When continuous symmetry is spontaneously broken, there have to be massless particles created/annihilated by the broken current. • Anomaly matching condition (’t Hooft 1980; Frishman et al. 1981): When the symmetry exists, one can couple the background gauge field, but the background gauge invariance may not be maintained. This ’t Hooft anomaly is preserved by any local, symmetric perturbations, especially by the renormalization group (RG) flow. Of course, symmetry has played a crucial role in the study of strongly coupled QFTs, including that of QCD, for almost a century, and one might guess that there is nothing new here. Surprisingly, this guess turns out to be false, and people start to find many new “symmetries” that have been missed. More precisely speaking, it turns out that there exists a useful generalization about the notion of symmetry: The symmetry is extended in a way that the above theorems still hold and they are applicable to much wider class of QFTs. The purpose of this chapter is to give an idea about generalized symmetry and to provide some examples of its applications. The key message is that symmetry is nothing but the data of topological defect operators in various codimensions. In the following, let us first explain why this is a nice generalization of the conventional symmetry and then discuss its applications to 4d gauge theories. Unless explicitly stated, we shall restrict our attention to the local, unitary, and relativistic QFTs in Euclidean signature.

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Symmetry ⇒ Topological Defects In this section, let us give a motivation to regard the topological defects as generalized symmetries. For this purpose, it would be convenient to remind ourselves of the conventional continuous symmetry. Let us then assume that we have a QFT obtained by the classical action S[φ] with the fundamental field φ(x) and S[φ] is invariant under the continuous symmetry φ(x) → eiα φ(x): S[eiα φ] = S[φ]. Let us make the transformation parameter α spacetime dependent, α → α(x); then the classical action changes as  S[eiα(x) φ(x)] = S[φ(x)] + i

∂μ α j μ + O(α 2 ),

(1)

where j μ is the Noether current. When φ is the solution of the classical equation of motion, ∂μ αj μ should vanish for any infinitesimal α(x), and it derives the Noether conservation law: ∂μ j μ (x) = 0.

(2)

So far, everything is classical. The Noether theorem can be extended to the QFT counterpart, and it can be obtained by repeating the above analysis in the pathintegral formalism. That is the Ward-Takahashi (WT) identity, which claims that (2) holds as an operator identity: ∂μ j μ (x)O1 (x1 ) · · · On (xn ) =

n 

δ(x − xm )O1 (x1 ) · · · δOm (xm ) · · · On (xn ),

m=1

(3) where δOm denotes the infinitesimal transformation of the local operator Om . This shows that the correlation functions j μ (x)O1 (x1 ) · · · On (xn ) of the current j μ must have the specific singularity at the coincident points, x = xm , and this strongly constrains the possible forms of those correlators. Both the Nambu-Goldstone theorem and the anomaly matching condition can be obtained from this WT identity. This motivates us to generalize the notion of symmetry by maintaining the essential feature of the WT identity. This can be done by noticing that the WT identity is equivalent to the topological nature of the Noether charge of closed (d − 1)-manifolds (Gaiotto et al. 2015)  Q(Md−1 ) =

j,

(4)

Md−1

where the notation of differential forms is used for later convenience, j = 1 gμν j μ dx ν , and j is the Hodge dual of j , i.e., j = (d−1)! εμν1 ···νd−1 j μ dx ν1 ∧ · · · ∧ ν d−1 dx , on the  flat space. When one takes Md−1 as the spatial slice of the constant time, Q = j 0 dd−1 x, this coincides with the usual Noether charge.

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The extended operator Q(Md−1 ) is called topological if and only if the following operator identity holds:  ), Q(Md−1 ) = Q(Md−1

(5)

 where Md−1 and Md−1 are related by some continuous deformations, and let us show it for the above Noether operator. The existence of a continuous deformation implies the presence of the d-dimensional volume Vd such that ∂Vd = Md−1 ∪    (−Md−1 ), where −Md−1 is the orientation reverse of Md−1 . As a result, one finds  )= Q(Md−1 ) − Q(Md−1

 

 Md−1 ∪(−Md−1 )

=

j

dj Vd

= 0.

(6)

Stokes’ theorem is used to obtain the second line, and the last equality follows from the Noether conservation law as d  j ∝ ∂μ j μ dd x. It would be obvious that one can reverse the logic: When the Noether operator, Q(Md−1 ) = Md−1 j , is topological for any Md−1 , one obtains the local conservation law, ∂μ j μ = 0. In this sense, the essence of the conservation law or the WT identity is nothing but the topological nature of Q(Md−1 ). This allows us to generalize the WT identity so that it holds for not only continuous symmetry but discrete symmetry: A d-dim QFT has an internal symmetry G if and only if: • there exists a topological operator Ug (Md−1 ) for g ∈ G and codim-1 closed manifolds Md−1 , • their fusion rule obeys the group law, Ug1 (Md−1 )Ug2 (Md−1 ) = Ug1 g2 (Md−1 ), • the local operators transform as the representation of G when Ug (Md−1 ) surrounds it. For the continuous symmetry, one can obtain such Ug (Md−1 ) by taking the exponential of Q(Md−1 ), Ueiα (Md−1 ) = exp(iαQ(Md−1 )), with appropriate local counterterms. We note that the above definition can also apply to the discrete symmetry, such as the Z2 symmetry of the Ising model or real φ 4 theory. To see this explicitly, let us consider the d-dim classical Ising model

Z=

 {sx }

⎛ exp ⎝

 x,x  

⎞ J sx sx  ⎠ ,

(7)

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where sx = ±1 denotes the classical Z2 spin. This obviously has the Z2 global symmetry, sx → −sx . The question is how to construct the corresponding topological defect, and one can define it as the defect operator U (Md−1 ): When the bond x, x   is cut by the codim-1 surface Md−1 , the sign of J is flipped in the Boltzmann weight, i.e., Jx,x   =

J −J

(x, x   is not cut by Md−1 ), (x, x   is cut by Md−1 ).

(8)

In general, when there exists a lattice regularization such that symmetry has the onsite action, one can construct the topological defect in the same manner for both continuous and discrete G. As the topological nature of Ug (Md−1 ) gives the WT identity when G is continuous, one obtains its discrete version of the WT identity by having the Ug (Md−1 ) for discrete groups G.

Generalized Symmetry = Topological Defects In the above discussion, we have found that conventional symmetry gives topological defects: Symmetry ⇒ Topological defects. Then, it would be natural to ask if the converse is also true: Toplogical defects ⇒ Symmetry? The answer to this question is “No.” When studying QFTs, one often encounters topological defects that are unrelated to any conventional symmetries. Even in such a situation, those topological defects play important roles as their topological feature implies conservation laws, and hence topological defects and conventional symmetries are equally useful. Nothing prevents us from enlarging our notion of symmetry, so one can think of topological defects as generalized symmetries. In this regard, the conventional symmetry is specified by the codim-1 topological defects that obey the group multiplication. Among the three keywords, “codim-1,” “topological,” and “group multiplication,” the topologicalness is the most important condition for symmetry. The new kind of “symmetry” can be obtained by replacing the other conditions. One direction of generalization is to consider topological defects defined on general codimension: • p-form symmetry (Gaiotto et al. 2015): Topological defects are defined on the codim-(p +1) closed manifolds, Ug (Md−p−1 ). Charged operators are defined on p-dim closed manifolds, W (Cp ). – When p = 0, this reduces to the conventional symmetry. – When p = 1, the charged operators are line operators. This includes the center symmetry of gauge theories as special cases. • n-group symmetry (Sharpe 2015; Benini et al. 2019; Cordova et al. 2019; Tanizaki and Unsal 2020; Hidaka et al. 2021a, b): (Nontrivial) mixture of p-form symmetries with p = 0, 1, . . . , n − 1.

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Another kind of generalization is obtained by replacing the group-like fusion rule with more general ones: • Non-invertible symmetry (or categorical symmetry): Ta be topological

Let c (M defects; then the fusion rule is Ta (Md−1 )Tb (Md−1 ) = c Nab d−1 )Tc (Md−1 ), c (M where Nab d−1 ) is a c-number that can topologically depend on Md−1 . c = 1 for c = a · b and N c = 0 for others, – When a, b, c, . . . lives in G and Nab ab it reduces to the ordinary group-like symmetry. – In general, when one has a topological defect Ta , its inverse does not necessarily exist. This is why it is called “non-invertible.” – In 2d QFTs, the finitely generated non-invertible symmetry is mathematically characterized by the fusion category, and the general theory is well developed (Bhardwaj and Tachikawa 2018; Thorngren and Wang 2019; Komargodski et al. 2021). Therefore, this class of symmetry is also called “categorical.” – In higher dimensions, people just started to observe some examples of noninvertible symmetries (Nguyen et al. 2021; Koide et al. 2022; Choi et al. 2022; Kaidi et al. 2022; Hayashi and Tanizaki 2022). An interesting observation is c becomes the partition function of (d − 1)-dim that the fusion coefficient Nab topological QFTs instead of just a number. – There also exists the non-invertible higher-form symmetry (Rudelius and Shao 2020; Nguyen et al. 2021). The Hilbert space of QFT should form a representation of generalized symmetry, so they provide the selection rule. The notion of spontaneous breaking is also well defined, and thus one can make the distinction between phases of matters by applying Landau’s criterion. Furthermore, these symmetries can be gauged as in the case of ordinary symmetries. When there is an obstruction for the gauging procedure, it provides important RG-invariant information as in the case of the anomaly matching condition.

Confinement-Deconfinement Phases and One-Form Symmetry In gauge theories, confinement/deconfinement of test charges is the characteristic feature that discriminates gapped phases (Wilson 1974; ’t Hooft 1978). In the context of finite-temperature gauge theories, this criterion is rephrased as the behavior of the Polyakov loop or the Wilson loop wrapping along the imaginary-time circle. Since the finite-temperature pure gauge theories enjoy the symmetry acting on Polyakov loops, it has been known that the finite-temperature confinementdeconfinement criterion reduces to standard Landau’s criterion of spontaneous breaking, and this symmetry has been called the center symmetry. However, I would claim that this “center symmetry” was quite mysterious. It suddenly appears if we compactify the spacetime to a torus, but it does not exist for the noncompact spacetime Rd or the compact spacetime without nontrivial cycles

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such as the sphere S d . Although we are dealing with the “local” QFTs, should we care about the “global” structure of the spacetime on which QFT lives to identify the symmetry? It is the one-form symmetry that resolves this uneasiness. To be specific, let us consider the d-dim pure SU(N ) Yang-Mills theory. The Yang-Mills (YM) action is S[a] =

1 g2

 tr[F ∧ F ],

(9)

where F = da+ia∧a is the field strength of the SU(N ) gauge field a = ai,μ T i dx μ . We shall see that there exists the codim-2 topological defect Uk (Md−2 ) that obeys the ZN fusion rule (Kapustin and Seiberg 2014; Gaiotto et al. 2015) Uk (Md−2 )Uk  (Md−2 ) = Uk+k  (Md−2 ),

UN (Md−2 ) = 1,

(10)

and it can detect the N -ality of the Wilson line 

2π i k|R|link(Md−2 , C) WR (C), Uk (Md−2 )WR (C) = exp N

(11)

 where WR (C) = trR [exp(i C a)] denotes the Wilson loop of the representation R, |R| denotes its N -ality, and link(Md−2 , C) is the linking number between Md−2 and C. The easiest way to construct Uk (Md−2 ) is to consider Wilson’s lattice gauge formulation (see Appendix A of Tanizaki and Ünsal 2022) S[U ] = K

 p

⎛ ⎝trP

⎞ U + c.c.⎠ ,

(12)

∈p

 where U ∈ SU(N ) is the link variable, p denotes each plaquette, and P ∈p denotes the path-ordered product of the link variables around the plaquette p. In the case of the Ising model, one replaces the hopping term, or the nearest-neighbor coupling J , as in (8), and one can use the same trick here. That is, one should multiply ZN phases Bp to the (inverse) coupling K that depend on plaquettes, and it defines ⎞ ⎛

 2π i ⎝e N Bp trP S[U , Bp ] = K U + c.c.⎠ , (13) p

∈p

and Bp =

0 k

(p is not pierced by Md−2 ), (p is pierced by Md−2 ).

(14)

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Due to the closedness of Md−2 , Bp satisfies the flatness condition, that is, one can i  check exp( 2π p∈c Bp ) = 1 for any cubes c. Correlation functions that include N Uk (Md−2 ) are then computed as Uk (Md−2 ) · · ·  =

1 Z

 DU exp(S[U , Bp ]) · · · ,

(15)

where · · · refers to other operator insertions. One can readily check that Uk (Md−2 ) has the desired topological nature and detects the N -ality of test charges: Especially, the topological nature comes out of the redefinition of link variables by ZN phase rotations. This confirms the ZN one-form symmetry of the pure SU(N ) YM theory, which we denote as Z[1] N . The spontaneous breakdown of Z[1] N can be defined as the natural generalization of the off-diagonal long-range order (ODLRO), and it coincides with Wilson’s criterion of quark confinement for this case. That is, • Wfd (C) obeys the area law ⇔ Z[1] N is unbroken. • Wfd (C) obeys the perimeter law ⇔ Z[1] N is spontaneously broken. Here, we only consider the case of the fundamental Wilson loop for simplicity, but one can discuss the spontaneous breaking to a nontrivial subgroup by considering the ones with higher-dimensional representations.

Fradkin-Shenker’s (Non-)complementarity It has been known that the Abelian-Higgs model has three phases, Coulomb, confinement, and Higgs phases. When the Higgs particle has the minimal electric charge, the confinement and Higgs phases are connected smoothly, while if the Higgs particle has a larger U (1) charge, those gapped phases are separated by the phase transition (Fradkin and Shenker 1979; Banks and Rabinovici 1979). The continuity between confinement and Higgs regimes for the minimal U (1) charge is often referred to as Fradkin-Shenker’s complementarity. Let us reinterpret this phenomenon from the viewpoint of the one-form symmetry. For this purpose, let us quickly review the results of Fradkin and Shenker (1979). The lattice model considered is given by S=β

 x,μ

cos(∂μ θ + Naμ ) + K



cos(fμν ),

(16)

p

where exp(iaμ (x)) ∈ U (x) is the U (1) link variable and φ(x) = exp(iθ (x)) is the Higgs field with the strict constraint |φ(x)| = 1. Here, the spacetime dimension is 4. One can formally write the continuum theory as

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  1 4 2 2 2 2 2 S= d x |fμν | +(|(∂μ +iNaμ )φ| +g(|φ| − v ) +(monopoles), (17) 2e2 with K ∼ e12 , β ∼ v 2 . In the strong coupling K  1, monopoles are proliferated, and one can naturally expect confinement of charges. On the other hand, in the weak couplings with the deep Higgs potential, K  1 and β  1, the system is in a Higgs phase, i.e., it is gapped, and the electric flux is screened. When K  1 and β  1, nothing condenses, and the system is in a Coulomb phase. When K = 0 or β = ∞, the model enjoys significant simplification, and one can precisely evaluate the partition function. As a result, when N = 1, the confinement and Higgs regimes turn out to be smoothly connected. For N ≥ 2, they are separated even in this limit, and it is a natural guess that the phase transition always separates these two regimes. It can be checked that the theory (12) enjoys the Z[1] N symmetry, and its order parameters are Wilson loops. One can easily check its presence by repeating the discussion for the pure YM theory, but let us take another perspective here. Consider the 4d pure Maxwell theory, S = 2e12 f ∧ f ; then the Maxwell equation becomes  1 d  2 f = 0, e

 d

1 f 2π

= 0.

(18)

The first one is the equation of motion, while the second one is the Bianchi identity. As both equations are written in the form of the “conservation law,” one can define the topological operators   Uα(e) (M2 ) = exp iα

M2

1 f e2

,

   1 Uα(m) (M ) = exp iα f . (19)  2 2π M2

These are generators of the electric and magnetic one-form symmetries that act on Wilson and ’t Hooft loops, respectively. When one introduces the matter fields, Maxwell equations (18) are modified, and these continuous symmetries are explicitly broken. The magnetic symmetry is completely broken due to the presence of lattice monopoles, but the electric one still maintains the Z[1] N symmetry as the charge of the Higgs particle is quantized to N in the unit of the elementary charge. In the confinement regime, Z[1] N is unbroken as the Wilson loops obey the area law, while it is spontaneously broken in the Higgs regime. When N ≥ 2, this shows that confinement and Higgs regimes should be separated by phase transitions. On the other hand, when N = 1, Z1 is a trivial group, and thus no one-form symmetry discriminates the confinement and Higgs regimes. When the system is gapped and Z[1] N is spontaneously broken, the low-energy field theory should contain the ZN topological QFT, and thus the ground state is described by a topologically ordered state. In this regard, Fradkin-Shenker’s noncomplementarity for N ≥ 2 gives an example of quantum phase transition between topologically trivial and nontrivial states.

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Gauging of Generalized Symmetry When a QFT enjoys the continuous symmetry generated by the Noether current j μ , one can introduce a gauge field Aμ and consider the partition function  Z[A] =

  Dφ exp S[φ] + i Aμ j μ + counter terms .

(20)

This can be regarded as the generating functional for correlation functions of Noether current, and this procedure is called background gauging. Naively, this partition function is invariant under the gauge transformation of A: Since the Noether conservation law implies 

 (Aμ + ∂μ λ)j =

 Aμ j −

μ

μ

 λ∂μ j = μ

Aμ j μ ,

(21)

one would obtain Z[A + dλ] = Z[A].

(22)

When this gauge invariance actually holds, one can perform the path integral for gauge fields A, and the gauge theory coupled to the original QFT is obtained. We note that this procedure is not restricted to continuous symmetry: It can be performed for discrete symmetry, for higher-form and n-group symmetries, and even for non-invertible symmetry. To get an idea about how this can be done, let us consider the case of the ordinary discrete symmetry G, in particular, the Z2 symmetry of the Ising model (7). Since this is already a lattice theory, it is natural to introduce the Z2 link variable U = Ux,x   ∈ {±1}, and one has

Z[{U }] =



⎛ exp ⎝J

{sx }



⎞ sx Ux,x   sx  ⎠ .

(23)

x,x  

As a result, the Z2 symmetry is promoted to the local gauge redundancy: sx → (−1)λx sx ,

Ux,x   → (−1)λx +λx  Ux,x   .

(24)

One needs to take the continuum limit to find the relativistic QFT from the lattice model. This operation is usually done in the weak-coupling regime at the lattice scale, so it is natural to require that the discrete gauge fields have to be flat:

∈p

U = 1 ∈ Z2 .

(25)

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By comparing this with the definition of the Z2 symmetry defects U (Md−1 ) given in (8), {U } with the flatness condition is nothing but the specific network of the Z2 symmetry defects. In general, the (flat) gauge field A for the discrete symmetry G can be identified as the network of the symmetry defects Ug (Md−1 ). Its gauge transformation recombines the given network, and the gauge invariance requires that the partition function Z[A] is invariant under any recombination. This idea has a straightforward generalization to any kind of symmetries: For the case of one-form symmetry, the background gauge field is the same with a network of the codim-2 topological defects. When the gauge invariance holds, all the possible networks can be summed consistently, and one obtains the dynamically gauged model.

’t Hooft Anomaly and Anomaly Matching Gauging the symmetry is useful not only to construct new QFTs starting from a given one but also for studying the dynamics of the original QFT. Especially, the violation of background gauge invariance provides the RG-invariant data (’t Hooft 1980), and it constrains the possible dynamics of strongly coupled QFTs. Let us consider a d-dim QFT with the symmetry G. Here, G can refer to the higher group, and let us formally denote its background gauge field as A. If the partition function has the following type of anomaly, it is called the ’t Hooft anomaly of G (Kapustin and Thorngren 2014; Wen 2013): Z[A + δλ] = exp(iA [A, λ])Z[A],

(26)

where A [A, λ] is a d-dim local functional of the gauge field A and its gauge parameter λ that cannot be eliminated by adding local counterterms. Empirically, when a d-dim QFT has an ’t Hooft anomaly, there always exists a classical (d + 1)-dim topological G-gauge theory Sd+1 [A] that reproduces the same anomaly on the boundary. This is the generalization of Callan-Harvey’s anomaly inflow (Callan and Harvey 1985): When a d-dim QFT is defined on Md , let us take a (d + 1)-manifold Nd+1 such that ∂Nd+1 = Md , and we also extend the G-gauge field A on Md to the one on Nd+1 . By regarding our d-dim QFT as the boundary of the (d + 1)-dim classical topological theory, the following partition function, ZMd [A] exp(i Sd+1,Nd+1 [A]),

(27)

is fully gauge invariant. In the modern condensed-matter physics, this (d + 1)-dim classical topological action is regarded as the partition function of a (d + 1)-dimensional symmetryprotected topological (SPT) state with the symmetry G. Unlike the intrinsic topological orders, the SPT states do not have anyons, and, moreover, the ground state is always unique for any closed spatial manifolds. This is indeed almost trivial as it can be continuously connected to the trivial gapped vacuum when arbitrary

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(including G-breaking) perturbations are introduced. Still, it is nontrivial as one cannot reach the trivial gapped state without encountering quantum phase transitions as long as the symmetry G is preserved. This nontriviality can be understood by creating the boundary; then the symmetry G is realized in an anomalous way. This is the d-dim ’t Hooft anomaly discussed above, and there must be some boundary degrees of freedom that cancels the anomaly to be consistent with the gauge invariance. This is the physical picture of the anomaly-inflow argument. This physical picture is helpful to have a clear understanding of the anomaly matching condition. Let us consider the (d + 1)-dim G-SPT with the boundary and also assume the boundary condition is carefully chosen so that the d-dim QFT of our interest is realized on the boundary. We then take the low-energy limit to perform the nonperturbative RG transformation. Since the (d + 1)-dim SPT is uniquely gapped, nothing changes under the RG flow so the RG transformation acts nontrivially only on the boundary theory. However, the full G-gauge invariance is the exact feature of the system, and thus the d-dim low-energy theory should cancel the same anomaly inflow from the bulk. This clearly tells the anomaly matching between the ultraviolet (UV) and infrared (IR) theories AUV [A, λ] = AIR [A, λ].

(28)

Especially, the presence of ’t Hooft anomaly shows that the IR theory must be nontrivial since QFTs with the unique, gapped ground state cannot have nontrivial ’t Hooft anomalies. In the relativistic context, the low-energy effective theory must contain gapless excitations, topological degrees of freedom, and/or spontaneous breakdown of G to match the nontrivial anomaly. The correspondence between d-dim anomaly and (d + 1)-dim SPT states also provides a useful tool to classify the possible form of anomalies. Because of the simplicity of the SPT state, its classification is now promoted to the rigorous mathematical theory with the help of the algebraic topology (Kapustin 2014; Freed and Hopkins 2021; Yonekura 2019a). This provides a useful tool to understand the nonperturbative anomalies that have been missed in conventional techniques.

Physics of θ Angle: Application of the Z[1] Symmetry for YM N Theory In 4d SU(N ) YM theory, the renormalizable terms consist of the YM kinetic term and the topological term, and one gets the θ vacua by summing up all the topological sectors of π3 (SU(N ))  Z  Zθ =

   1 θ Da exp − 2 tr[F ∧ F ] + i 2 tr[F ∧ F ] . g 8π

(29)

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The effect of the θ term on low-energy physics is quite nontrivial and a difficult problem. In the ’t Hooft large-N limit, N → ∞ with fixed g 2 N , the θ dependence of the ground-state energy density can be computed as (Witten 1980) E(θ ) − E(0) = −

χtop 2 1 ln(Zθ /Z0 ) = θ + O(θ 4 /N 2 ). vol 2

(30)

However, the partition function must satisfy the 2π periodicity by definition of the topological charge, Zθ+2π = Zθ , and thus this cannot be the complete story as this expression violates the 2π periodicity. If there are meta-stable YM vacua whose lifetimes are exponentially long in large-N , this issue can be resolved by considering the multi-branch structure (Witten 1980): E(θ ) = min Ek (θ ), k∈Z

Ek (θ ) = E0 (θ − 2π k).

(31)

This suggests that the YM theory has two vacua at θ = π as a result of spontaneous CP breaking. We can get an intuitive scenario for the multi-branch vacua by taking an Abelian gauge (’t Hooft 1981). In such a gauge, the resultant U (1) gauge fields have various singularities that can be regarded as monopoles, vortices, etc., and one can think of the confining vacuum as the result of proliferated monopoles. However, the particlelike spectrum contains not only monopoles but also dyons, which have both electric and magnetic charges. When one turns on the θ angle, the magnetically charged particles acquire the fractional electric charge due to the Witten effect (Witten 1979), and the role of monopole and dyon should be exchanged when one dials θ : 0 → 2π . Therefore, by regarding the original vacuum as the monopole condensing state and another vacuum as the dyon condensing state, one can naturally interpret the above level-crossing phenomenon as the first-order quantum phase transition between these states. Although this explanation is intuitive and tells a lot of physics behind it, it is unclear if the phase transition is really there. Since both monopoles and dyons are magnetically charged particles, the Wilson loop obeys the area law for both condensations. This seems to imply that the vacua at θ ∼ 0 and θ ∼ 2π belong to the same phase, and then it is reasonable to ask if there is a continuous path in the space of couplings that smoothly connect these vacua. Moreover, if such a path exists in the coupling space, we would have no way to exclude that it is the θ parameter as the actual dynamics is strongly coupled. When one assumes confinement, it can be rigorously proven that the vacua at θ and θ + 2π should be distinct as the Z[1] N -SPT states (Gaiotto et al. 2017). Since the [1] pure SU(N ) YM theory has the ZN symmetry, we can introduce its background gauge field. As we have discussed earlier, the background gauge field for Z[1] N is nothing but the network of codim-2 topological defects, and it is equivalent to the plaquette ZN gauge field, Bp , in (13). Let us denote it as B formally and call it the ZN two-form gauge field because the Poincare dual of the codim-2 surface is a two-form.

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When a codim-2 surface in the network wraps a nontrivial 2-cycle of the spacetime, it is equivalent to take the ’t Hooft twisted boundary condition (’t Hooft 1979) along that cycle. Under such a twisted boundary condition, the topological charge is fractionally shifted (van Baal 1982): 

 1 N tr[F ∧ F ] ∈ B ∧ B +Z. 8π 2 8π 2   

(32)

∈(1/N )Z

As a result, one obtains the following exact relation for the YM partition function (Gaiotto et al. 2017): 

N Zθ+2π [B] = exp i 4π



B ∧ B Zθ [B].

(33)

This can be regarded as a generalization of anomaly, and it requires the presence of the quantum phase transition when θ is changed by 2π (for more detailed discussions, see also Tanizaki and Kikuchi 2017; Kikuchi and Tanizaki 2017; Tanizaki and Sulejmanpasic 2018; Karasik and Komargodski 2019; Cordova et al. 2020a, b). Two confining states given by the monopole and dyon condensation have different partition functions when one introduces the background gauge field B by N the contact term ∼ 4π B ∧ B, so they are different as Z[1] N -SPT states. Let us assume that CP at θ = π is spontaneously broken to satisfy the anomaly matching condition. This allows us to introduce the CP domain wall that connects these two vacua. Since these vacua are different Z[1] N -SPT states, the anomaly-inflow argument predicts the SU(N )1 Chern-Simons theory on the boundary (Gaiotto et al. 2017). Although the test quark is assumed to be confined on the bulk, the deconfinement necessarily occurs on the wall (see also Sulejmanpasic et al. 2017; Komargodski et al. 2018).

Applications to 4d QCD with Fundamental Quarks As generalized symmetry has a beautiful application to the physics of pure YM theory, it is natural to ask if one can apply it to the case of QCD with fundamental quarks. Before that, the following question needs to be considered: Is there any generalized symmetries in QCD with fundamental matters? In the case of QCD, the new kind of symmetry has not been found, unfortunately. Despite this fact, many new anomalies are recently found for QCD in the chiral limit using the technique of one-form gauging (Tanizaki and Kikuchi 2017; Shimizu and Yonekura 2018; Gaiotto et al. 2018; Tanizaki et al. 2017, 2018; Tanizaki 2018; Yonekura 2019b). Without having one-form symmetries, its technique turns out to be useful to obtain the refined information of the conventional symmetry.

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Why are new anomalies found without having new symmetries? The key point is the correct identification of the symmetry group. When one writes down the chiral symmetry of massless QCD, it is often written as SU (Nf )L × SU (Nf )R × U (1)V , where Nf is the number of massless flavors. However, since the vectorlike symmetry should be U (Nf )V  (SU (Nf )diag × U (1)V )/ZNf instead of SU (Nf )diag × U (1)V , one should divide the above group by the common ZNf center, and the symmetry group becomes [SU (Nf )L × SU (Nf )R × U (1)V ]/ZNf . This is, however, not yet complete. Since the global symmetry must act nontrivially on physical local operators, one must further divide the above group by the common center with the gauge group, and the correct symmetry group of massless QCD is given by G=

SU (Nf )L × SU (Nf )R × U (1)V , ZNf × ZNc

(34)

where Nc is the number of color. To detect the ’t Hooft anomaly, one should introduce the background G-gauge field, and here there exists a huge difference depending on whether one takes into account the correct quotient structure of G or not. One must introduce not only the one-form gauge fields but also the two-form gauge fields to properly gauge G, and the two-form gauge fields play the pivotal role to find the new anomaly. For example, when one pays attention to the vector-like subgroup and the discrete axial symmetry, SU (Nf )diag × U (1)V × (ZNf )χ ⊂ G, ZNf × ZNc

(35)

its background gauge field gives the anomalous violation of the baryon-number current (Tanizaki 2018) μ

∂μ Jbaryon =

Nf Bf ∧ dAχ , (2π )2

(36)

where Bf is the ZNf two-form gauge field to describe the ZNf quotient and Aχ is the discrete chiral gauge field. We note that this anomaly vanishes if Bf is turned off so this baryon-number anomaly for the discrete chiral symmetry can be detected only if one takes into account the quotient structure. Such a discrete anomaly of massless QCD has an interesting application to the QCD phase diagram with imaginary chemical potentials. In the case of the baryon imaginary chemical potential, the one-loop Polyakov-loop potential predicts that the high-temperature phase has the Roberge-Weiss phase transition (Roberge and Weiss 1986). At the Roberge-Weiss phase transition point, there exists the Z2 symmetry acting on the Polyakov loop, and one can properly define the deconfinement as the breaking of Z2 symmetry (Shimizu and Yonekura 2018). One can instead consider the case of flavor-dependent imaginary chemical potential (Kouno et al. 2012; Iritani

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et al. 2015), and then it can realize the full ZN symmetry acting on the Polyakov loop (Cherman et al. 2017). In this way, the imaginary chemical potentials provide a playground of QCD with fundamental quarks with the well-defined confinementdeconfinement transition with the Polyakov-loop order parameter. The discrete anomaly discussed above turns out to persist in the S 1 -compactified spacetime with the suitable choice of the imaginary chemical potentials (Shimizu and Yonekura 2018; Tanizaki et al. 2017, 2018). The anomaly matching condition then tells that phases with the special value of imaginary chemical potentials must be always nontrivial, and it predicts that the chiral restoration temperature must be above the deconfinement temperature: Tchiral ≥ Tdeconf .

(37)

One can prove this inequality as long as Nc and Nf have a nontrivial common divisor, gcd(Nc , Nf ) > 1. There exists an interesting consistency check with previous literature about this inequality. When one takes the ’t Hooft-type large-Nc limit, i.e., Nc → ∞ with fixed Nf , the effect of the quark boundary condition is just a subleading correction in large Nc . Therefore, the above inequality should be true for thermal large-Nc QCD, and then we can compare it with holographic QCD (Sakai and Sugimoto 2005a, b). This has been studied in Aharony et al. (2007) using the D4-D8-D8 model. In the standard Sakai-Sugimoto setup where the D8 and D8 branes are set antipodally, the chiral restoration and deconfinement occur at the same temperature, Tchiral = Tdeconf . When the separation of those flavor branes is changed, those two transition temperatures can take different values, but it turns out that the chiral restoration always occurs after the deconfinement, Tchiral > Tdeconf . In both cases, the result is consistent with the discrete anomaly matching condition.

Summary We have seen that the notion of symmetry is drastically extended, and it provides a useful tool to analyze strongly coupled QFTs. For this generalization, it is important to notice that the presence of topological defects implies the conservation law, and we can equally use it as we have used conventional symmetry. This extends the systematic characterization of phases of matter by its spontaneous breaking, and we can also perform its gauging to find the new anomaly matching condition. In the case of QCD with fundamental quarks, we have currently found no generalized symmetries. Despite this fact, the techniques developed for higher symmetries turn out to be useful for getting more information about conventional chiral symmetry. The correct identification of the global symmetry provides the new nonperturbative anomaly, and it gives an interesting constraint on the phase diagram with imaginary chemical potentials. Lastly, it should be emphasized that the scope of generalized symmetries is still rapidly growing. Honestly, I am quite astonished by the fact that symmetries of

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QFTs are so fruitful and we newly discover them in this twenty-first century. It is my great pleasure if this chapter gives a useful introduction for readers to this exciting and developing research field. Acknowledgments The work of Y. T. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI grant numbers, 22H01218 and 20K22350, and by the Center for Gravitational Physics and Quantum Information (CGPQI) at Yukawa Institute.

References O. Aharony, J. Sonnenschein, S. Yankielowicz, A holographic model of deconfinement and chiral symmetry restoration. Ann. Phys. 322, 1420–1443 (2007). https://doi.org/10.1016/j.aop.2006. 11.002 T. Banks, E. Rabinovici, Finite temperature behavior of the lattice Abelian Higgs model. Nucl. Phys. B 160, 349–379 (1979). https://doi.org/10.1016/0550-3213(79)90064-6 F. Benini, C. Córdova, P.-S. Hsin, On 2-group global symmetries and their anomalies. JHEP 03, 118 (2019). https://doi.org/10.1007/JHEP03(2019)118 L. Bhardwaj, Y. Tachikawa, On finite symmetries and their gauging in two dimensions. JHEP 03, 189 (2018). https://doi.org/10.1007/JHEP03(2018)189 C.G. Callan Jr., J.A. Harvey, Anomalies and fermion zero modes on strings and domain walls. Nucl. Phys. B250, 427–436 (1985). https://doi.org/10.1016/0550-3213(85)90489-4 A. Cherman, S. Sen, M. Unsal, M.L. Wagman, L.G. Yaffe, Order parameters and color-flavor center symmetry in QCD. Phys. Rev. Lett. 119(22), 222001 (2017). https://doi.org/10.1103/ PhysRevLett.119.222001 Y. Choi, C. Cordova, P.-S. Hsin, H.T. Lam, S.-H. Shao, Noninvertible duality defects in 3+1 dimensions. Phys. Rev. D 105(12), 125016 (2022). https://doi.org/10.1103/PhysRevD.105. 125016 C. Cordova, T.T. Dumitrescu, K. Intriligator, Exploring 2-group global symmetries. JHEP 02, 184 (2019). https://doi.org/10.1007/JHEP02(2019)184 C. Cordova, D.S. Freed, H.T. Lam, N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications I. Sci. Post Phys. 8(1), 001 (2020a). https://doi.org/10.21468/ SciPostPhys.8.1.001 C. Cordova, D.S. Freed, H.T. Lam, N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications II. Sci.Post Phys. 8(1), 002 (2020b). https://doi.org/10.21468/ SciPostPhys.8.1.002 E.H. Fradkin, S.H. Shenker, Phase diagrams of lattice gauge theories with Higgs fields. Phys. Rev. D 19, 3682–3697 (1979). https://doi.org/10.1103/PhysRevD.19.3682 D.S. Freed, M.J. Hopkins, Reflection positivity and invertible topological phases. Geom. Topol. 25, 1165–1330 (2021). https://doi.org/10.2140/gt.2021.25.1165 Y. Frishman, A. Schwimmer, T. Banks, S. Yankielowicz, The axial anomaly and the bound state spectrum in confining theories. Nucl. Phys. B177, 157–171 (1981). https://doi.org/10.1016/ 0550-3213(81)90268-6 D. Gaiotto, A. Kapustin, N. Seiberg, B. Willett, Generalized global symmetries. JHEP 02, 172 (2015). https://doi.org/10.1007/JHEP02(2015)172 D. Gaiotto, A. Kapustin, Z. Komargodski, N. Seiberg, Theta, time reversal, and temperature. JHEP 05, 091 (2017). https://doi.org/10.1007/JHEP05(2017)091 D. Gaiotto, Z. Komargodski, N. Seiberg, Time-reversal breaking in QCD4 , walls, and dualities in 2 + 1 dimensions. JHEP 01, 110 (2018). https://doi.org/10.1007/JHEP01(2018)110 J. Goldstone, Field theories with superconductor solutions. Nuovo Cim. 19, 154–164 (1961). https://doi.org/10.1007/BF02812722

3024

Y. Tanizaki

Y. Hayashi, Y. Tanizaki, Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly. JHEP 08, 036 (2022). https://doi.org/10.1007/JHEP08(2022)036 Y. Hidaka, M. Nitta, R. Yokokura, Global 3-group symmetry and ’t Hooft anomalies in axion electrodynamics. JHEP 01, 173 (2021a). https://doi.org/10.1007/JHEP01(2021)173 Y. Hidaka, M. Nitta, R. Yokokura, Topological axion electrodynamics and 4-group symmetry. Phys. Lett. B 823, 136762 (2021b). https://doi.org/10.1016/j.physletb.2021.136762 T. Iritani, E. Itou, T. Misumi, Lattice study on QCD-like theory with exact center symmetry. JHEP 11, 159 (2015). https://doi.org/10.1007/JHEP11(2015)159 J. Kaidi, K. Ohmori, Y. Zheng, Kramers-Wannier-like duality defects in (3+1)D gauge theories. Phys. Rev. Lett. 128(11), 111601 (2022). https://doi.org/10.1103/PhysRevLett.128.111601 A. Kapustin, Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology (2014) A. Kapustin, N. Seiberg, Coupling a QFT to a TQFT and duality. JHEP 04, 001 2014. https://doi. org/10.1007/JHEP04(2014)001 A. Kapustin, R. Thorngren, Anomalies of Discrete Symmetries in Various Dimensions and Group Cohomology (2014) A. Karasik, Z. Komargodski, The bi-fundamental gauge theory in 3+1 dimensions: the vacuum structure and a cascade. JHEP 05, 144 (2019). https://doi.org/10.1007/JHEP05(2019)144 Y. Kikuchi, Y. Tanizaki, Global inconsistency, ’t Hooft anomaly, and level crossing in quantum mechanics. Prog. Theor. Exp. Phys. 2017, 113B05 (2017). https://doi.org/10.1093/ptep/ptx148 M. Koide, Y. Nagoya, S. Yamaguchi, Non-invertible topological defects in 4-dimensional Z2 pure lattice gauge theory. PTEP 2022(1), 013B03 (2022). https://doi.org/10.1093/ptep/ptab145 Z. Komargodski, T. Sulejmanpasic, M. Unsal, Walls, anomalies, and deconfinement in quantum antiferromagnets. Phys. Rev. B97(5), 054418 (2018). https://doi.org/10.1103/PhysRevB.97. 054418 Z. Komargodski, K. Ohmori, K. Roumpedakis, S. Seifnashri, Symmetries and strings of adjoint QCD2 . JHEP 03, 103 (2021). https://doi.org/10.1007/JHEP03(2021)103 H. Kouno, Y. Sakai, T. Makiyama, K. Tokunaga, T. Sasaki, M. Yahiro, Quark-gluon thermodynamics with the Z(N(c)) symmetry. J. Phys. G39, 085010 (2012). https://doi.org/10.1088/09543899/39/8/085010 L.D. Landau, On the theory of phase transitions. I. Zh. Eksp. Teor. Fiz. 11, 19 (1937) Y. Nambu, Quasiparticles and gauge invariance in the theory of superconductivity. Phys. Rev. 117, 648–663 (1960). https://doi.org/10.1103/PhysRev.117.648 M. Nguyen, Y. Tanizaki, M. Ünsal, Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond N -ality. JHEP 03, 238 (2021). https://doi.org/10.1007/ JHEP03(2021)238 A. Roberge, N. Weiss, Gauge theories with imaginary chemical potential and the phases of QCD. Nucl. Phys. B275, 734–745 (1986). https://doi.org/10.1016/0550-3213(86)90582-1 T. Rudelius, S.-H. Shao, Topological operators and completeness of spectrum in discrete gauge theories. JHEP 12, 172 (2020). https://doi.org/10.1007/JHEP12(2020)172 T. Sakai, S. Sugimoto, Low energy hadron physics in holographic QCD. Prog. Theor. Phys. 113, 843–882 (2005a). https://doi.org/10.1143/PTP.113.843 T. Sakai, S. Sugimoto, More on a holographic dual of QCD. Prog. Theor. Phys. 114, 1083–1118 (2005b). https://doi.org/10.1143/PTP.114.1083 E. Sharpe, Notes on generalized global symmetries in QFT. Fortsch. Phys. 63, 659–682 (2015). https://doi.org/10.1002/prop.201500048 H. Shimizu, K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition. Phys. Rev. D97(10), 105011 (2018). https://doi.org/10.1103/PhysRevD.97.105011 T. Sulejmanpasic, H. Shao, A. Sandvik, M. Unsal, Confinement in the bulk, deconfinement on the wall: infrared equivalence between compactified QCD and quantum magnets. Phys. Rev. Lett. 119(9), 091601 (2017). https://doi.org/10.1103/PhysRevLett.119.091601 G. ’t Hooft, On the phase transition towards permanent quark confinement. Nucl. Phys. B 138, 1–25 (1978). https://doi.org/10.1016/0550-3213(78)90153-0

82 Generalization of Global Symmetry and Its Applications to QCD-Related Physics

3025

G. ’t Hooft, A property of electric and magnetic flux in nonabelian gauge theories. Nucl. Phys. B153, 141–160 (1979). https://doi.org/10.1016/0550-3213(79)90595-9 G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, in Recent Developments in Gauge Theories. Proceedings, Nato Advanced Study Institute, Cargese, 26 Aug–8 Sep 1979, vol. 59 (1980), pp. 135–157. https://doi.org/10.1007/978-1-4684-7571-5_9 G. ’t Hooft. Topology of the gauge condition and new confinement phases in nonabelian gauge theories. Nucl. Phys. B190, 455–478 (1981). https://doi.org/10.1016/0550-3213(81)90442-9 Y. Tanizaki, Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking. JHEP 08, 171 (2018). https://doi.org/10.1007/JHEP08(2018)171 Y. Tanizaki, Y. Kikuchi, Vacuum structure of bifundamental gauge theories at finite topological angles. JHEP 06, 102 (2017). https://doi.org/10.1007/JHEP06(2017)102 Y. Tanizaki, T. Sulejmanpasic, Anomaly and global inconsistency matching: θ-angles, SU (3)/U (1)2 nonlinear sigma model, SU (3) chains and its generalizations. Phys. Rev. B98(11), 115126 (2018). https://doi.org/10.1103/PhysRevB.98.115126 Y. Tanizaki, M. Unsal, Modified instanton sum in QCD and higher-groups. JHEP 03, 123 (2020). https://doi.org/10.1007/JHEP03(2020)123 Y. Tanizaki, M. Ünsal, Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications. PTEP 2022, 04A108 (2022). https://doi.org/10.1093/ ptep/ptac042 Y. Tanizaki, T. Misumi, N. Sakai, Circle compactification and ’t Hooft anomaly. JHEP 12, 056 (2017). https://doi.org/10.1007/JHEP12(2017)056 Y. Tanizaki, Y. Kikuchi, T. Misumi, N. Sakai, Anomaly matching for phase diagram of massless ZN -QCD. Phys. Rev. D97, 054012 (2018). https://doi.org/10.1103/PhysRevD.97.054012 R. Thorngren, Y. Wang, Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases (2019) P. van Baal, Some results for SU(N) gauge fields on the hypertorus. Commun. Math. Phys. 85, 529 (1982). https://doi.org/10.1007/BF01403503 X.-G. Wen, Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders. Phys. Rev. D88(4), 045013 (2013). https:// doi.org/10.1103/PhysRevD.88.045013 K.G. Wilson, Confinement of quarks. Phys. Rev. D10, 2445–2459 (1974). https://doi.org/10.1103/ PhysRevD.10.2445 E. Witten, Dyons of charge eθ/2π . Phys. Lett. B86, 283–287 (1979). https://doi.org/10.1016/ 0370-2693(79)90838-4 E. Witten, Large N chiral dynamics. Ann. Phys. 128, 363 (1980). https://doi.org/10.1016/00034916(80)90325-5 K. Yonekura, On the cobordism classification of symmetry protected topological phases. Commun. Math. Phys. 368(3), 1121–1173 (2019a). https://doi.org/10.1007/s00220-019-03439-y K. Yonekura, Anomaly matching in QCD thermal phase transition. JHEP 05, 062 (2019b). https:// doi.org/10.1007/JHEP05(2019)062

Chiral Magnetic Effect: A Brief Introduction

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Contents Chirality of Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chirality of Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Magnetic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Magnetic Effect as a Probe of Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Magnetic Effect in Heavy Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Broader Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Chirality of fermions is linked to topology of gauge fields by the index theorem and chiral anomaly. While the chiral anomaly is traditionally associated with the short-distance behavior in quantum field theory, recently, it has been realized that it also affects the macroscopic behavior of systems with chiral fermions. In particular, the local chiral imbalance in the presence of a magnetic field induces non-dissipative transport of electric charge (“the chiral magnetic effect,” CME). In heavy ion collisions, there is an ongoing search for this effect at Relativistic Heavy Ion Collider at BNL and Large Hadron Collider at CERN, with results from a dedicated isobar run presented very recently by the STAR Collaboration. An observation of CME in heavy ion collisions could shed light on the mechanism of baryon asymmetry generation in the Early Universe. For the case of an Abelian gauge theory, the CME has been already discovered in Dirac

D. E. Kharzeev () Center for Nuclear Theory, Stony Brook University, Stony Brook, NY, USA Brookhaven National Laboratory, Upton, NY, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_25

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and Weyl semimetals possessing chiral quasi-particles. This observation opens a path toward new applications, including chiral qubits.

Chirality of Fermions The chiral asymmetry (i.e., the asymmetry between the right and left) of fundamental interactions was predicted in 1956 by T.D. Lee and C.N. Yang (Lee and Yang 1956) and established shortly afterward through the observation of angular asymmetry in the β decay of 60 Co nuclei by C.S. Wu (Wu et al. 1957). Nuclear reactions thus allowed to discover the chiral asymmetry of fundamental interactions that may be at the origin of chiral asymmetry of life and the Universe. Indeed, already in 1874, Louis Pasteur wrote (Pasteur 1874): “The universe is asymmetric and I am persuaded that life, as it is known to us, is a direct result of the asymmetry of the universe or of its indirect consequences.” Life is asymmetric, and the concept of chirality – the distinction between the left and right or between an object and its reflection in the mirror – is ingrained in our DNA. The DNA double helix is chiral; moreover, DNA molecules also form knots (Arsuaga et al. 2005) – and most of the complex knots are chiral as well. In fundamental interactions, the notion of chirality can be seen as emerging from the symmetries of spacetime. Indeed, relativistic equations that describe the dynamics of fundamental particles correspond to different representations of the Lorentz symmetry group of Minkowski space. In particular, the solutions of Dirac equation transform under Lorentz group as bispinors ((0, 12 ) ⊕ ( 12 , 0)) that consist of left-handed ψ − and right-handed ψ + Weyl spinors:  ψ− . ψ= ψ+ 

(1)

For the fermions described by these Weyl spinors, the chirality is defined as the projection of spin on momentum, with a minus sign for antifermions. In Dirac theory, it is an eigenvalue of γ 5 matrix, with projectors on the right and left chiral states given by P+ =

1 1 (1 + γ 5 ); P − = (1 − γ 5 ). 2 2

(2)

A Dirac spinor ψ can thus be decomposed into right- and left-handed chiral states by ψ + = P + ψ and ψ − = P − ψ. ˆ Dirac equation for massless fermions is i Dψ ≡ iγ μ ∂μ ψ = 0, and Dirac operator can also be decomposed into right- and left-handed components: ˆ + ; D − ≡ i DP ˆ −. D + ≡ i DP

(3)

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For massless fermions, chirality is conserved even when the interactions with gauge fields (e.g., photons and gluons) are included on the classical level – so a lefthanded fermion is expected to always stay left-handed. However, quantum chiral anomaly (Adler 1969; Bell and Jackiw 1969) allows for a chirality transmutation – a left-handed fermion can become right-handed if it interacts with a configuration of gauge field that can change its own chirality. While the chirality of a fermion is now a familiar concept, how does one characterize the chirality of a gauge field?

Chirality of Gauge Fields The notion of gauge field chirality can be explained by using the example of DNA knots mentioned above. Indeed, let us consider a knot of magnetic flux. The chiral invariant of this gauge field configuration is known as magnetic helicity:  CS[A] =

d 3 x A · B,

(4)

where A is the vector gauge potential and B = ∇ × A is magnetic field. Using analogy between the gauge potential and velocity v, we can easily understand that (4) detects chirality – indeed, when A → v, magnetic field gets replaced by vorticity B → ω = 12 ∇ × v, and a non-zero scalar product v · ω implies a helical motion. For an Abelian gauge theory (such as electrodynamics), magnetic helicity coincides with a more general three-form derived by Chern and Simons (1974) in differential geometry to characterize the global topology of a manifold with a Lie algebra valued one-form A over it:   1 CS[A] = Tr F ∧ A − A ∧ A ∧ A , 3

(5)

where the curvature (field strength tensor) is defined as F = dA + A ∧ A. In physical terms, the invariant (4) corresponds to the chirality of the knot made of magnetic flux – for example, a torus would be characterized by CS = 0, a right-handed trefoil knot by CS = +1, and a left-handed one by CS = −1. Chern-Simons p-form can be defined for any odd spacetime dimension p. We happen to live in 3 + 1-dimensional spacetime with even p = 4 – but it does not mean that Chern-Simons invariant is irrelevant. What this means is that in our four-dimensional spacetime, Chern-Simons invariant can change in time, with nonvanishing exterior derivative dCS[A] = Tr [F ∧ F],

(6)

which is known as the Chern-Pontryagin invariant. In a more familiar for physicist notation, this relation implies that

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∂ CS[A] = −2 ∂t

 d 3 x E · B,

(7)

which can be obtained from (4) in the Coulomb gauge where the electric field E ˙ Therefore, a change of topology of is related to the gauge potential by E = −A. the gauge field in time quantified by (7) gives rise to an electric field parallel to the magnetic one. What is the effect of parallel electric and magnetic fields on chirality of chiral fermions? Can this electric field drive a current?

Chiral Anomaly Charged particles moving in an external magnetic field B experience a Lorentz force Fm = e v × B. If the projection of their velocities on the direction of magnetic field is equal to zero, this force results in the motion along closed cyclotron orbits with ω = ∇ × v = 0, but v · ω = 0. Even if the charge has a non-zero velocity component along ω, but there is no external force directed along B, then we can always choose a frame in which v · B = v · ω = 0, so the motion of the charge is not helical. However, if there is a force applied along the direction of B, e.g., a Lorentz force Fe = e E from an electric field E parallel to B, then the motion is helical in any inertial frame. In quantum theory, charged particles occupy quantized Landau levels, and for massless fermions, the lowest Landau level (LLL) is chiral and has a zero energy, due to a cancellation between a positive kinetic energy and negative Zeeman energy of the interaction between magnetic field and spin. This cancellation can be seen as resulting from supersymmetry of Dirac Hamiltonian in magnetic field (AlvarezGaume 1983; Cooper et al. 1988). Therefore, on the LLL, the spins of positive (negative) fermions are aligned along (against) the direction of magnetic field. All excited levels are degenerate in spin and are thus not chiral. More formally, this can be seen as a consequence of the Atiyah-Singer index theorem (Atiyah and Singer 1968) that relates the analytical index of Dirac operator to its topological index. Simply stated, this theorem relates the number of zero modes of Dirac operator acting on a manifold M to the topology of this manifold. The analytical index of Dirac operator (3) is given by the difference in the number of zero energy modes with right (ν+ ) and left (ν+ ) chirality: ind D = dim kerD + − dim kerD − = ν+ − ν− ,

(8)

where ker D is the subspace spanned by the kernel of the operator D, i.e., the subspace of states that obey D + ψ = 0 or D − ψ = 0; see (3). Fora two-dimensional manifold M, the topological index of this operator is equal 1 to 2π M tr F, and the Atiyah-Singer index theorem thus states that ν+ − ν− =

1 2π

 Tr F. M

(9)

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Performing analytical continuation to Euclidean (x, y) space (with B along the z-axis), we thus find that the number of chiral zero modes is given by the total magnetic flux through the system (Aharonov and Casher 1979). For positive fermions with charge e > 0, we have ν− = 0, and the number of right-handed chiral modes from (9) is given by ν+ =

eΦ , 2π

(10)

which is just the number of LLLs in the transverse plane; we have included an explicit dependence on electric charge e. For negative fermions, ν+ = 0 and ν− = eΦ/(2π ). Let us assume for simplicity that the charge chemical potential is equal to zero, μ = 0. It is clear that for this neutral system, ν+ = ν− and the system in its ground state possesses zero chirality. Let us now turn on an external electric field E  B. The dynamics of fermions on the LLL is (1 + 1) dimensional (with a spatial direction along B), and we can apply the index theorem (9) to the (z, t) manifold; for positive fermions of right (“+”) chirality 1 ν+ = 2π

 dzdt eE, ν− = 0,

(11)

and for negative fermions of left (“−”) chirality 1 ν+ = 0, ν− = − 2π

 dzdt eE.

(12)

These relations can be understood from a seemingly classical argument (Nielsen and Ninomiya 1983): the positive charges are accelerated by Lorentz force along the electric field E and acquire Fermi momentum pF+ = eEt. The density of states in one spatial dimension is pF /(2π ), and so the total number of positive fermions with positive chirality is ν+ = 1/(2π ) dzdt eE, in accord with (11). The same argument applied to negative fermions explains (12). While the notion of acceleration by Lorentz force is classical, in assuming that it increases the Fermi momentum, we have made an assumption that there is an infinite tower of states that are accelerated by the Lorentz force. This tower of states does not exist in non-relativistic theory; however, it is a crucial ingredient of the relativistic quantum theory of Dirac and is a consequence of the geometry of Minkowski spacetime. Indeed, the Casimir invariant of Lorentz group is particle mass squared, m2 = pμ pμ = E 2 − p2

(13)

given by the square of the momentum four-vector pμ = (E,p). The solution of (13) for energy necessarily contains the negative branch, E = ± p2 + m2 , implying the

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existence of (antiparticle) states with negative energies. These negative energy for fermions imply the relevance of chirality, because the bispinor (1) in Dirac basis assumes the form   1 ψ+ + ψ− ψDirac = √ + − . 2 ψ −ψ

(14)

In the non-relativistic case, the lower (negative energy) component is vanishingly small, and chirality is irrelevant. However, as soon as the lower “small component” starts to become important in the relativistic domain, the chirality comes into play. Let us now get back to the fermions in parallel electric and magnetic fields. Multiplying the density of states in longitudinal direction pF /(2π ) by the density of states eB/(2π ) in the transverse direction, we find from (11) and (12) that in (3 + 1) dimensions ν+ − ν− = 2 ×

e2 4π 2

 d 2 x dz dt E · B =

e2 2π 2

 d 2 x dz dt E · B,

(15)

where the factor of 2 is due to the contributions of positive and negative fermions. This relation represents the Atiyah-Singer theorem for U (1) gauge theory in (3 + 1) dimensions, so we could use it directly instead of relying on dimensional reduction of the LLL dynamics. Note that the quantity on the r.h.s. of (15) is nothing but the derivative of Chern-Simons three-form; see (6) and (7). The relation (15) can also be written in differential form in terms of the axial current (equal to the difference of vector currents of right- and left-handed fermions) ¯ μ γ 5 ψ = Jμ+ − Jμ− Jμ5 = ψγ

(16)

as (Adler 1969; Bell and Jackiw 1969) ∂ μ Jμ5 =

e2 E · B. 2π 2

(17)

The chiral anomaly equation (17) is an operator relation. In particular, we can use it to evaluate the matrix element of transition from a pseudoscalar excitation of interacting Dirac vacuum (a neutral pion) into two photons. This can be done by using on the l.h.s. of (17) the partial conservation of axial current (PCAC) relation that replaces the divergence of axial current by the interpolating pion field ϕ ∂ μ Jμ5 Fπ Mπ2 ϕ,

(18)

where Fπ and Mπ are the pion decay constant and mass. Taking the matrix element of (17) between the vacuum and the two-photon states 0|∂ μ Jμ5 |γ γ  then yields the amplitude of the π 0 → γ γ decay which is a hallmark of chiral anomaly. However,

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the chiral anomaly has much broader implications when the classical gauge fields are involved, as we will now discuss.

Chiral Magnetic Effect Let us now observe that the derivation of chiral anomaly presented above implies the existence of non-dissipative electric current in parallel electric and magnetic fields. Indeed, in a globally neutral system, the (vector) electric current ¯ μ ψ = Jμ+ + Jμ− Jμ = ψγ

(19)

contains equal contributions from positive charge, positive chirality fermions flowing along the direction of E (which we assume to be parallel to B), and negative charge, negative chirality fermions flowing in the direction opposite to E: Jz = 2 ×

e2 e2 E·Bt = E · B t. 2 4π 2π 2

(20)

In constant electric and magnetic fields, this current grows linearly in time – this means that if we define conductivity σ through Ohm’s law J = σ E, it becomes divergent, and the resistivity ρ = 1/σ vanishes. This means that the current (20) is non-dissipative, similar to what happens in superconductors! One can also write down the relation (20) in terms of the chemical potentials μ+ = pF+ and μ− = pF− for right- and left-handed fermions, which for massless dispersion relation are given by the corresponding Fermi momenta pF+ = eEt and pF− = −eEt. It is convenient to define the chiral chemical potential μ5 ≡

1 (μ+ − μ− ) 2

(21)

related to the density of chiral charge ρ5 = J05 ; for small μ5 , it is proportional to ρ5 , μ5 χ −1 ρ5 where χ is the chiral susceptibility. The relation (20) then becomes (Fukushima et al. 2008) J=

e2 μ5 B. 2π 2

(22)

It is important to realize that unlike a usual chemical potential, the chiral chemical potential μ5 does not correspond to a conserved charge – on the contrary, the nonconservation of chiral charge due to chiral anomaly is necessary for the chiral magnetic effect (CME) described by (22) to exist. Indeed, a static magnetic field cannot perform work, so the current (22) can be powered only by a change in the chiral chemical potential – it is thus a non-equilibrium effect.

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Another way to see this is to consider a power (Basar et al. 2014) of the current (22): P = d 3 x E J ∼ μ5 d 3 x E B. For a constant μ5 , it can be both positive and negative, in clear conflict with energy conservation. In particular, one would be able to extract energy from the ground state of the system with a constant μ5 = 0! On the other hand, if μ5is dynamically generated through the chiral anomaly (17), it has the same sign as d 3 x E B, and the electric power is always positive, as it should be. It is important to note that in the latter case, the chirally imbalanced state with μ5 = 0 is not the ground state of the system and relaxes to the true ground state through the anomaly by generating the CME current. For parallel E and B, the CME relation (22) is a direct consequence of the Abelian chiral anomaly for classical background fields. However, this relation is valid also when the chiral chemical potential is sourced by non-Abelian anomalies (Fukushima et al. 2008), coupling to time-dependent axion field (Wilczek 1987), or is just a consequence of a non-equilibrium dynamics (Kharzeev et al. 2018). The relation (22) was first introduced in 1980 in a pioneering paper by Vilenkin (1980b) – however, he considered the case of a constant μ5 , motivated by parity violation in weak interactions. In this case, as we have discussed above, the system is in the true ground state, and the electric current cannot exist in equilibrium (Vilenkin 1980a) – so while the relation (22) was known for some time, turning it into a real physical effect required understanding the role of chiral anomaly; see review (Kharzeev 2014) for a detailed discussion and additional references. The same is true in condensed matter applications when the nodes of dispersion relations of leftand right-handed fermions are located at different energies, E+ (p = 0) = E− (p = 0). In this case, contrary to some early claims, the CME current does not appear (Basar et al. 2014). For the CME to emerge, the chiral asymmetry has to appear in the occupancy of the left- and right-handed states.

Chiral Magnetic Effect as a Probe of Topology The main motivation behind the early work on CME has been the detection of topological transitions in QCD matter; see Kharzeev (2006). To see how CME can be utilized for this purpose, let us note that the CME relation (22) can be proven also for the case of non-Abelian plasma containing chiral fermions in an external Abelian magnetic field B (Fukushima et al. 2008). In this case, the chiral charge (and the corresponding chiral chemical potential) is created by non-Abelian anomaly due to the instanton transitions (Belavin et al. 1975) between topological sectors marked by different Chern-Simons numbers (6) and the electric current flows along (or against) the direction of B. This presents a unique opportunity to get a direct experimental access to the study of non-Abelian topological fluctuations (Kharzeev 2006; Kharzeev et al. 2008; Kharzeev and Zhitnitsky 2007). Topological transitions in non-Abelian electroweak plasma violate the baryon number conservation and may be responsible for the baryon asymmetry of our Universe (Kuzmin et al. 1985; Rubakov and Shaposhnikov 1996) provided that the expanding Early Universe is out of equilibrium, as stipulated by Sakharov’s criteria (Sakharov 1967).

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The only non-Abelian theory with topological transitions that are accessible to experiment is quantum chromodynamics (QCD). Its chiral fermions are quarks that are confined into hadrons. According to the Atiyah-Singer index theorem (Atiyah and Singer 1968) discussed above, topological transitions in QCD should be accompanied by the change in chirality of quarks. They can thus be responsible for spontaneous breaking of U+ (3)×U− (3) chiral symmetry in QCD, emergence of the corresponding Goldstone bosons (pions, kaons, and η meson), and a flavor-singlet η meson with a large mass resulting from the anomaly. While there is a significant evidence for the prominent role of topological transitions (instantons, sphalerons, etc.) in the structure of QCD vacuum and the properties of hadrons (see Schäfer and Shuryak (1998) for a review), such transitions have never been observed in experiment. The CME, with its directly detectable electric current, makes such an observation possible. Indeed, consider a QCD plasma is created in a high-energy heavy ion collision. Because the colliding ions possess positive electric charges, the produced plasma, at least during its early moments, is embedded into a very strong magnetic field, with strength which is on the order of a typical QCD scale (Kharzeev et al. 2008), eB ∼ Λ2QCD . In such a strong magnetic field (possibly the strongest in the present Universe), electromagnetic interactions of quarks are comparable in strength to the strong ones. In addition, the rate of topological “sphaleron” (Klinkhamer and Manton 1984) transitions in hot QCD plasma is high, and this should result in the creation of chirally imbalanced domains (“P-odd bubbles” (Kharzeev et al. 1998)) characterized by non-zero value of the chiral chemical potential. This means that all conditions for the CME are met, and there should be an electric current (22) propagating through the QCD plasma. This CME current should create an electric dipole moment oriented along the direction of magnetic field. As this is a nonequilibrium phenomenon, this electric dipole would eventually disappear when the system approaches global equilibrium – but the rapid expansion of the quark-gluon plasma should still allow to observe it, similar to the case of the expanding Early Universe (Sakharov 1967).

Chiral Magnetic Effect in Heavy Ion Collisions The magnetic field produced by the colliding ions is directed perpendicular to the reaction plane of the collision; therefore, the CME current (22) is orthogonal to the reaction plane as well. The magnetic field is strongest during the early moments of the collision; the sphaleron transitions are thus expected to induce the electric charge separation relative to the reaction plane early in the evolution of the produced matter (Kharzeev 2006). Because the QCD plasma is rapidly expanding (similar to the Early Universe), this initial electric charge separation cannot be fully scrambled by final state interactions and survives till the moment when the plasma cools down and undergoes “freeze-out” into hadrons. Since the electric charge is conserved throughout the hadronization, the azimuthal distribution of produced hadrons should

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retain the charge asymmetry relative to the reaction plane, and this asymmetry can be directly detected in experiment. Of course, QCD does not violate parity P symmetry globally, and the sign of chiral imbalance in the produced matter should fluctuate event-by-event – we are thus dealing with a “local P-violation” (Kharzeev et al. 1998, 2008). The experimental signature of CME is thus a dynamical enhancement of out-of-plane fluctuations of charge asymmetry, relative to the in-plane fluctuations that are not affected by CME and originate from backgrounds. The experimental observable sensitive to this effect was proposed by Voloshin (2004) almost immediately after the idea to search for CME in heavy ion collisions was put forward (Kharzeev 2006). In terms of the azimuthal angles of positive φ + and negative φ − hadrons, and the azimuthal angle of the reaction plane ΨRP , the proposed “γ correlator” observable is Voloshin (2004) 

γ +− ≡ cos φ + + φ − − 2ΨRP (23) +  −  +  −  = cos φ − ΨRP cos φ − ΨRP − sin φ − ΨRP sin φ − ΨRP , where the sum is performed over charged hadrons in a given event. This correlator can be averaged over many events, and so can be measured with a very high statistical precision. The CME results in the electric charge separation relative 

− to the reaction +−Ψ plane and should produce either sin φ > 0, sin φ − ΨRP < 0, or RP



 sin φ + − ΨRP < 0, sin φ − − ΨRP > 0, depending on the

 sign of the chiral imbalance in a given event. In both cases, sin φ + − ΨRP sin φ − − ΨRP < 0, and the correlator γ +− should be positive. Of course, the background fluctuations would also contribute to the γ correlator, but γ +− is the difference of in-plane and out-of-plane fluctuations, so the backgrounds that do not depend on the reaction plane should cancel out. Therefore, the only backgrounds that do survive in γ +− should depend on the reaction plane and are thus expected to be proportional to the “elliptic flow” that is defined as the second Fourier harmonic of the hadron azimuthal angle distribution (Voloshin 2004). In hydrodynamical description of heavy ion collisions, the elliptic flow results from the pressure anisotropy generated by the almond-shape geometry of quark-gluon matter produced in an off-central collision. It is important to note that γ +− is expected to scale with the inverse of hadron multiplicity, for both the signal and background contributions (Kharzeev 2006; Voloshin 2004); see Kharzeev et al. (2016) for a review and detailed discussion. STAR Collaboration at Relativistic Heavy Ion Collider (RHIC) at BNL performed the measurements of γ correlators, and other CME observables, over a range of collision energies and for different colliding ions (Abelev et al. 2009, 2010; Adamczyk et al. 2013). The ALICE (Abelev et al. 2013) and CMS (Sirunyan et al. 2018) Collaborations at the LHC extended these studies to higher energies; see Kharzeev et al. (2016) and Kharzeev and Liao (2021) for reviews and compilations of published results. For all studied heavy ion collisions, non-zero and positive

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γ +− was measured. The correlators γ ++ and γ −− were also non-zero and negative, as expected for CME. However, large background contributions proportional to elliptic flow have also identified, and the problem of separating the possible signal from background has assumed the center of the stage. Separation of the signal from background was the motivation behind the proposal to use the isobar collisions (Voloshin 2010). The idea is that since the isobars have the same mass number, they have approximately the same size and shape, and thus the elliptic flow, and the backgrounds driven by it, should be nearly identical. On the other hand, the difference in the electric charge should create a difference in the produced magnetic field and thus in the CME which is proportional to it. The measurement of γ correlators, and other CME observables, in isobar collisions would thus allow to isolate the CME signal from the background. 96 A dedicated high statistics run to measure the CME observables in 96 44 Ru 44 Ru 96 96 and 40 Zr 40 Zr collisions was performed by STAR Collaboration at RHIC in 2018 (Abdallah et al. 2022). The data from 3.8 billion collision events were recorded, and a blind analysis of the data, unprecedented in heavy ion physics, was performed. The expectation has been that since the electric charge of Ru is higher than that of Zr, the ratio of CME observables, e.g., γ Ru /γ Zr , would exceed 1 if CME were present and be equal to 1 if CME were to be absent. In all scenarios, one expected to find γ Ru /γ Zr ≥ 1. Surprisingly, the STAR analysis revealed that the ratio γ Ru /γ Zr was significantly below 1! This certainly did not fit the CME expectations and in fact any theory expectation, CME-based or not. Basing on these findings, and the predefined 96 96 96 criteria assuming the identical backgrounds in 96 44 Ru 44 Ru and 40 Zr 40 Zr collisions, STAR Collaboration justifiably concluded that “no CME signature that satisfies the predefined criteria has been observed” (Abdallah et al. 2022). While the original CME expectations for the isobar run have certainly not been met, the puzzling observation of γ Ru /γ Zr < 1 still begs for an explanation and may affect the outcome of the investigation. A detailed investigation is still ongoing, but the post-blinding analysis made by STAR (Abdallah et al. 2022) has already revealed that the key to the solution is a very significant difference in 96 96 96 hadron multiplicity in 96 44 Ru 44 Ru and 40 Zr 40 Zr collisions observed by STAR in centrality cuts relevant for the CME analysis. As mentioned above, the γ correlator scales, in the good first approximation, with inverse multiplicity, and the multiplicity 96 96 96 measured in 96 44 Ru 44 Ru is significantly higher than in 40 Zr 40 Zr collisions in the same centrality cut. This explains the surprisingly low ratio γ Ru /γ Zr < 1. If one establishes a new baseline given by the measured ratio of inverse multiplicities, the observed γ Ru /γ Zr ratio in fact exceeds the baseline by (1 − 4) σ , depending on the details of the analysis. A theoretical analysis performed recently Kharzeev et al. (2022) took into account also the observed differences in the elliptical flow, as well as the differences in the transverse flow predicted by hydrodynamics that stem from the shape difference between the isobars. It was concluded that the STAR data is consistent with the CME contribution of (6.8 ± 2.6)%.

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More detailed experimental and theoretical analyses of the isobar data are clearly necessary and are still ongoing, but it has already become clear that the isobar run data represent an important milestone in the hunt for CME in heavy ion collisions. The final conclusions on the presence or absence of CME will have to be based however on the joint analysis of both isobar and symmetric heavy ion collisions. It will also be very important to extend the CME studies to lower collision energies, where topological fluctuations can be strongly enhanced (Ikeda et al. 2021) due to proximity to the critical point in the QCD phase diagram. This program is planned for the beam energy scan at RHIC and can also be performed at future heavy ion facilities, such as NICA and FAIR.

Broader Connections CME is a macroscopic quantum phenomenon driven by the chiral anomaly; it is a direct probe of change in the gauge field topology. As a result, it induces a variety of novel phenomena involving chiral fermions in quark-gluon plasma, the Early Universe (Brandenburg et al. 2017), astrophysics (Gorbar and Shovkovy 2022), and condensed matter physics. In the latter case, the CME has by now been firmly established (Li et al. 2016; Xiong et al. 2015) through the studies of magnetotransport in Dirac and Weyl semimetals (Armitage et al. 2018) and continues to find new applications. The emerging new frontier in condensed matter physics is chiral photonics, including the studies of chiral magnetic photocurrents (Kaushik et al. 2019) and other chiral phenomena. Among potential future directions, let us mention the use of CME for controlling “chiral qubits” (Kharzeev and Li 2019) in quantum processors. It is clear that the chiral asymmetry of the Universe has many farreaching consequences that we are just beginning to uncover. This work was supported by the US Department of Energy, Office of Science, grant numbers DE-FG88ER40388 and DE-SC0012704 and Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2 QA), under contract number DE-SC0012704.

References √ M. Abdallah et al., Search for the chiral magnetic effect with isobar collisions at sN N = 200 GeV by the STAR Collaboration at the BNL relativistic heavy ion collider. Phys. Rev. C 105(1), 014901 (2022). https://doi.org/10.1103/PhysRevC.105.014901 B.I. Abelev et al., Azimuthal charged-particle correlations and possible local strong parity violation. Phys. Rev. Lett. 103, 251601 (2009). https://doi.org/10.1103/PhysRevLett.103. 251601 B.I. Abelev et al., Observation of charge-dependent azimuthal correlations and possible local strong parity violation in heavy ion collisions. Phys. Rev. C 81, 054908 (2010). https://doi.org/ 10.1103/PhysRevC.81.054908 √ B. Abelev et al., Charge separation relative to the reaction plane in Pb-Pb collisions at sN N = 2.76 TeV. Phys. Rev. Lett. 110(1), 012301 (2013). https://doi.org/10.1103/PhysRevLett.110. 012301

83 Chiral Magnetic Effect: A Brief Introduction

3039

L. Adamczyk et al., Fluctuations of charge separation perpendicular to the event plane and local √ parity violation in sN N = 200 GeV Au+Au collisions at the BNL relativistic heavy ion collider. Phys. Rev. C 88(6), 064911 (2013). https://doi.org/10.1103/PhysRevC.88.064911 S.L. Adler, Axial vector vertex in spinor electrodynamics. Phys. Rev. 177, 2426–2438 (1969). https://doi.org/10.1103/PhysRev.177.2426 Y. Aharonov, A. Casher, The ground state of a spin 1/2 charged particle in a two-dimensional magnetic field. Phys. Rev. A 19, 2461–2462 (1979). https://doi.org/10.1103/PhysRevA.19. 2461 L. Alvarez-Gaume, Supersymmetry and the Atiyah-Singer index theorem. Commun. Math. Phys. 90, 161 (1983). https://doi.org/10.1007/BF01205500 N.P. Armitage, E.J. Mele, A. Vishwanath, Weyl and dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90(1), 015001 (2018) J. Arsuaga, M. Vazquez, P. McGuirk, S. Trigueros, D.W. Sumners, J. Roca, DNA knots reveal a chiral organization of DNA in phage capsids. Proc. Natl. Acad. Sci. 102(26), 9165–9169 (2005) M.F. Atiyah, I.M. Singer, The Index of elliptic operators. 1. Ann. Math. 87, 484–530 (1968). https://doi.org/10.2307/1970715 G. Basar, D.E. Kharzeev, H.-U. Yee, Triangle anomaly in Weyl semimetals. Phys. Rev. B 89(3), 035142 (2014). https://doi.org/10.1103/PhysRevB.89.035142 A.A. Belavin, A.M. Polyakov, A.S. Schwartz, Y.S. Tyupkin, Pseudoparticle solutions of the yang-mills equations. Phys. Lett. B 59, 85–87 (1975). https://doi.org/10.1016/0370-2693(75) 90163-X J.S. Bell, R. Jackiw, A PCAC puzzle: π 0 → γ γ in the σ model. Nuovo Cim. A 60, 47–61 (1969). https://doi.org/10.1007/BF02823296 A. Brandenburg, J. Schober, I. Rogachevskii, T. Kahniashvili, A. Boyarsky, J. Frohlich, O. Ruchayskiy, N. Kleeorin, The turbulent chiral-magnetic cascade in the early universe. Astrophys. J. Lett. 845(2), L21 (2017). https://doi.org/10.3847/2041-8213/aa855d S.-S. Chern, J. Simons, Characteristic forms and geometric invariants. Ann. Math. 99, 48–69 (1974). https://doi.org/10.2307/1971013 F. Cooper, A. Khare, R. Musto, A. Wipf, Supersymmetry and the dirac equation. Ann. Phys. 187(1), 1–28 (1988) K. Fukushima, D.E. Kharzeev, H.J. Warringa, The chiral magnetic effect. Phys. Rev. D 78, 074033 (2008). https://doi.org/10.1103/PhysRevD.78.074033 E.V. Gorbar, I.A. Shovkovy, Chiral anomalous processes in magnetospheres of pulsars and black holes. Eur. Phys. J. C 82(7), 625 (2022). https://doi.org/10.1140/epjc/s10052-022-10604-6 K. Ikeda, D.E. Kharzeev, Y. Kikuchi, Real-time dynamics of Chern-Simons fluctuations near a critical point. Phys. Rev. D 103(7), L071502 (2021). https://doi.org/10.1103/PhysRevD.103. L071502 S. Kaushik, D.E. Kharzeev, E.J. Philip, Chiral magnetic photocurrent in dirac and weyl materials. Phys. Rev. B 99(7), 075150 (2019). https://doi.org/10.1103/PhysRevB.99.075150 D. Kharzeev, Parity violation in hot QCD: why it can happen, and how to look for it. Phys. Lett. B 633, 260–264 (2006). https://doi.org/10.1016/j.physletb.2005.11.075 D.E. Kharzeev, The chiral magnetic effect and anomaly-induced transport. Prog. Part. Nucl. Phys. 75, 133–151 (2014). https://doi.org/10.1016/j.ppnp.2014.01.002 D. Kharzeev, A. Zhitnitsky, Charge separation induced by P-odd bubbles in QCD matter. Nucl. Phys. A 797, 67–79 (2007). https://doi.org/10.1016/j.nuclphysa.2007.10.001 D.E. Kharzeev, Q. Li, The Chiral Qubit: quantum computing with chiral anomaly, (2019), p. 3 D.E. Kharzeev, J. Liao, Chiral magnetic effect reveals the topology of gauge fields in heavy-ion collisions. Nat. Rev. Phys. 3(1), 55–63 (2021). https://doi.org/10.1038/s42254-020-00254-6 D. Kharzeev, R.D. Pisarski, M.H.G. Tytgat, Possibility of spontaneous parity violation in hot QCD. Phys. Rev. Lett. 81, 512–515 (1998). https://doi.org/10.1103/PhysRevLett.81.512 D.E. Kharzeev, L.D. McLerran, H.J. Warringa, The effects of topological charge change in heavy ion collisions: ‘Event by event P and CP violation’. Nucl. Phys. A 803, 227–253 (2008). https:// doi.org/10.1016/j.nuclphysa.2008.02.298

3040

D. E. Kharzeev

D.E. Kharzeev, J. Liao, S.A. Voloshin, G. Wang, Chiral magnetic and vortical effects in highenergy nuclear collisions—A status report. Prog. Part. Nucl. Phys. 88, 1–28 (2016). https://doi. org/10.1016/j.ppnp.2016.01.001 D. Kharzeev, Y. Kikuchi, R. Meyer, Chiral magnetic effect without chirality source in asymmetric Weyl semimetals. Eur. Phys. J. B 91(5), 83 (2018). https://doi.org/10.1140/epjb/e201880418-1 D.E. Kharzeev, J. Liao, S. Shi, Implications of the isobar run results for chiral magnetic effect in heavy ion collisions (2022), p. 4 F.R. Klinkhamer, N.S. Manton, A saddle point solution in the Weinberg-Salam theory. Phys. Rev. D 30, 2212 (1984). https://doi.org/10.1103/PhysRevD.30.2212 V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov, On the anomalous electroweak baryon number nonconservation in the early universe. Phys. Lett. B 155, 36 (1985). https://doi.org/10.1016/ 0370-2693(85)91028-7 T.D. Lee, C.-N. Yang, Question of parity conservation in weak interactions. Phys. Rev. 104, 254–258 (1956). https://doi.org/10.1103/PhysRev.104.254 Q. Li, D.E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosic, A.V. Fedorov, R.D. Zhong, J.A. Schneeloch, G.D. Gu, T. Valla, Observation of the chiral magnetic effect in ZrTe5. Nat. Phys. 12, 550–554 (2016). https://doi.org/10.1038/nphys3648 H.B. Nielsen, M. Ninomiya, Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal. Phys. Lett. B 130, 389–396 (1983). https://doi.org/10.1016/0370-2693(83)91529-0 L. Pasteur, On the use of fungi against phylloxera.[paris] acad. des sci. Compt. Rend 79, 1233–34 (1874) V.A. Rubakov, M.E. Shaposhnikov, Electroweak baryon number nonconservation in the early universe and in high-energy collisions. Usp. Fiz. Nauk 166, 493–537 (1996). https://doi.org/10. 1070/PU1996v039n05ABEH000145 A.D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe. Pisma Zh. Eksp. Teor. Fiz. 5, 32–35 (1967). https://doi.org/10.1070/ PU1991v034n05ABEH002497 T. Schäfer, E.V. Shuryak, Instantons in QCD. Rev. Mod. Phys. 70, 323–426 (1998). https://doi. org/10.1103/RevModPhys.70.323 A.M. Sirunyan et al., Constraints on the chiral magnetic effect using charge-dependent azimuthal correlations in pPb and PbPb collisions at the CERN Large Hadron Collider. Phys. Rev. C 97(4), 044912 (2018). https://doi.org/10.1103/PhysRevC.97.044912 A. Vilenkin, Cancelation of equilibrium parity violating currents. Phys. Rev. D 22, 3067–3079 (1980a). https://doi.org/10.1103/PhysRevD.22.3067 A. Vilenkin, Equilibrium parity violating current in a magnetic field. Phys. Rev. D 22, 3080–3084 (1980b). https://doi.org/10.1103/PhysRevD.22.3080 S.A. Voloshin, Parity violation in hot QCD: how to detect it. Phys. Rev. C 70, 057901 (2004). https://doi.org/10.1103/PhysRevC.70.057901 S.A. Voloshin, Testing the chiral magnetic effect with central U+U collisions. Phys. Rev. Lett. 105, 172301 (2010). https://doi.org/10.1103/PhysRevLett.105.172301 F. Wilczek, Two applications of axion electrodynamics. Phys. Rev. Lett. 58, 1799 (1987). https:// doi.org/10.1103/PhysRevLett.58.1799 C.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes, R.P. Hudson, Experimental test of parity conservation in β decay. Phys. Rev. 105, 1413–1414 (1957). https://doi.org/10.1103/PhysRev. 105.1413 J. Xiong, S.K. Kushwaha, T. Liang, J.W. Krizan, M. Hirschberger, W. Wang, R.J. Cava, N.P. Ong, Evidence for the chiral anomaly in the dirac semimetal na3bi. Science 350(6259), 413–416 (2015)

Crossover Between Quark Nuclear Matter and Condensed-Matter Physics

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Tomáš Brauner and Naoki Yamamoto

Contents Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phases of Nuclear and Quark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meson Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Color Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BCS–BEC Crossover and Quark–Hadron Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Order and Higher-Form Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Soliton Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Transport Phenomena of Quark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Berry Phase of Quark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Magnetic Effect and Related Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

We review quantum many-body phenomena in nuclear and quark matter which share similarities or universalities with condensed-matter physics. The focus is on the quantum phases of quark nuclear matter and topological transport phenomena due to the Berry phase of quark matter. The former includes nuclear superfluidity/superconductivity via the Bardeen–Cooper–Schrieffer (BCS) mechanism, Bose–Einstein condensation (BEC) of mesons, color superconductivity of quarks, BCS–BEC crossover phenomena and quark–hadron

T. Brauner Department of Mathematics and Physics, University of Stavanger, Stavanger, Norway e-mail: [email protected] N. Yamamoto () Department of Physics, Keio University, Yokohama, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_28

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continuity, chiral soliton lattice of mesons, and the application of the notions of topological order and higher-form symmetry. The latter includes the so-called chiral magnetic effect and related phenomena. Their potential phenomenological implications as well as the close connections with a number of condensed-matter systems are also discussed.

Introduction and Summary The field of physics is divided by a hierarchy of scales from microscopic to macroscopic into particle physics, nuclear physics, atomic and molecular physics, condensed-matter physics, astrophysics and cosmology, and related disciplines. However, physical phenomena at different scales are often related to each other. It is a remarkable aspect of physics that quantum many-body systems described by different microscopic Hamiltonians or Lagrangians at various scales share emergent similar (and sometimes even universal) phenomena. One of the wellknown and important examples is the Nambu–Jona-Lasinio (NJL) model (Nambu and Jona-Lasinio 1961), which provides a mechanism for the origin of mass of matter in particle and nuclear physics, based on the analogy with superconductivity in condensed-matter physics. The underlying concept there is spontaneous symmetry breaking; see, e.g., Brauner (2010) for a review. Another example is the skyrmion (Skyrme 1961), which was originally introduced by Skyrme in the context of particle and nuclear physics to describe baryons in terms of the pion field with a nontrivial topological configuration. The concept of the skyrmion was later generalized to various condensed-matter systems, such as magnetic skyrmions realized in magnetic crystals with potential applications to spintronics; see, e.g., Tokura and Kanazawa (2021) for a recent review. In this way, different fields of physics are closely connected to one another through crossovers without boundaries. This chapter gives a review of quantum many-body phenomena in nuclear and quark matter described by the theory of the strong interaction, quantum chromodynamics (QCD), which share similarities or universalities with those appearing in condensed-matter physics. Typical exotic phases in condensed-matter physics are superfluidity, superconductivity due to the Bardeen–Cooper–Schrieffer (BCS) mechanism, and Bose–Einstein condensation (BEC). It will be described how these quantum phases naturally emerge in quark nuclear matter, including their potential phenomenological implications. Sometimes quark nuclear matter shows richer structures owing to the color and flavor degrees of freedom as well as ultrarelativistic effects. Recent developments in this topic, such as the topological order and higherform symmetry, chiral soliton lattice (CSL) of mesons, Berry phase, and related topological transport phenomena like the so-called chiral magnetic effect (CME) in quark matter, are also covered. It should be remarked that just the single theory of QCD exhibits this variety of effects depending on different extreme conditions at finite densities and/or under certain external fields. There are also other topics

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in quark nuclear matter that will not be covered in this review, e.g., the chiral phase transition and the QCD critical point (see Fukushima and Hatsuda 2011 and Stephanov 2004 for reviews), inhomogeneous chiral condensates (see Buballa and Carignano 2015 for a review), and non-Fermi liquid behaviors (Schäfer and Schwenzer 2004) and the Kondo effect (Hattori et al. 2015) in quark matter. Throughout this chapter, we use the natural units  = c = kB = 1.

Phases of Nuclear and Quark Matter This section discusses quantum phases of nuclear and quark matter at finite baryon and/or isospin densities and at zero or low temperature: nuclear superfluidity/superconductivity, meson condensation, color superconductivity, BCS–BEC crossover, quark–hadron continuity, the possibility of topological order, and chiral soliton lattice of mesons in a magnetic field or rotation.

Nuclear Matter At sufficiently low baryon density and zero temperature, the physical degrees of freedom carrying baryon charge are nucleons with the mass mN ≈ 939 MeV, and they form nuclear matter. The situation is however different between pure QCD without the electromagnetic interaction and real QCD with the electromagnetic interaction (Halasz et al. 1998), and each case will be separately considered below. Let us start with the pure QCD at finite baryon density. As nucleons are fermions and follow the Pauli principle like electrons in solids, they form a Fermi surface. In the absence of the Coulomb interaction, the energy of the nuclear matter with a given atomic number A is minimized when the numbers of protons and neutrons are equal, N = Z = A/2. This is the symmetric nuclear matter. The binding energy of nuclear matter per nucleon, Bpure , in this case is dominated by the volume energy term in the empirical Weizsäcker–Bethe formula and is given by Bpure ≈ 16 MeV. To describe the phase transition in this pure QCD as a function of the baryon chemical potential μB , recall that the ground state at finite μB can be found from  = HQCD − μB NB is minimized, where HQCD is the QCD the condition that HQCD Hamiltonian (in the vacuum) and NB is the number of baryons. When μB > mN − Bpure , adding one nucleon increases HQCD by mN − Bpure and NB by one, and so the  change of HQCD is negative. This means that it is energetically favorable to excite nucleons in this regime, where the ground state is the symmetric nuclear matter with finite baryon density. On the other hand, when μB < mN − Bpure , excitation  , and so the ground state remains the QCD vacuum of nucleons increases HQCD without any baryon number density. Therefore, there is a first-order phase transition at μB = μ0 ≡ mN − Bpure , where the baryon number density jumps from zero to the density of nuclear matter n0 ≈ 0.16 fm−3 . Here note that it is a highly nontrivial fact from the viewpoint of microscopic QCD that while the QCD functional integral

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at finite μB is different from that at μB = 0, the physical quantities are exactly the same as long as μB < mN − Bpure . This is known as the baryon Silver Blaze problem (Cohen 2003). The scenario above would be modified once the effects of the Coulomb interaction are taken into account in realistic QCD. With increasing μB in this case, what happens first is the formation of solid iron (with electrons to ensure the electric neutrality) that has the largest binding energy per nucleon Breal ≈ 8 MeV at A ≈ 56. This is the first-order phase transition at μB = μ0 ≡ mN − Breal , where the density jumps from zero to the density of atomic iron. When μB is increased further, there should eventually be a phase transition to neutron matter (Z  A), in a way similar to the transition to symmetric nuclear matter in pure QCD. At sufficiently low density of such neutron matter, one can focus on the s-wave neutron–neutron interaction, characterized by the scattering length a ≈ −18.5 fm and the effective range reff ≈ 2.7 fm. Here, a and reff are defined through the effective range expansion of the phase shift δ(k) as a function of momentum k: k cot δ(k) = −

1 1 + reff k 2 + O(k 3 ). a 2

(1)

A remarkable property of this interaction is the presence of the hierarchy |a|  reff . As proposed by Bertsch (2001), one can consider an idealized limit kF |a| → ∞ and kF reff → 0, where kF is the Fermi momentum. This so-called unitary limit, known from cold atom systems, possesses a nonrelativistic version of the scale (and conformal) invariance. The neutrons in this regime are thus a field (approximately) described by the nonrelativistic conformal field theory (Nishida and Son 2007), which is called the “unnucleus” (Hammer and Son 2021) following Georgi’s terminology of the “unparticle” sector described by the relativistic conformal field theory (Georgi 2007). The properties of fermions in this regime are universal, independent of microscopic details. Therefore, the cold atom systems at unitarity may give (semi)quantitative insights to such universal aspects of the neutron matter or “unnuclear” matter; see, e.g., Gandolfi et al. (2015) and Ohashi et al. (2020) for reviews.

Nuclear Superfluidity and Superconductivity From the scattering phase shifts as a function of the laboratory energy, the phase shift in the 1 S0 channel is positive at low energy and that in the 3 P2 channel is positive at higher energy. Now recall that a positive phase shift means an attractive interaction. According to the BCS theory stating that any attractive interaction near the Fermi surface induces the Cooper pairing of fermions, one can expect that nucleons pair in the spin-singlet 1 S0 state at low density and spin-triplet 3 P2 state (with a small admixture of 3 F2 due to the tensor force) at higher density. Since neutrons are charge neutral and protons are charged particles, the neutron pairing and proton pairing lead to the neutron superfluidity and proton superconductivity, respectively.

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There have been a number of works to estimate the pairing gap Δ of the neutron superfluidity and proton superconductivity; see, e.g., Dean and HjorthJensen (2003), Gandolfi et al. (2015), and Gezerlis et al. (2014) for reviews. In the low-density regime kF |a|  1, the s-wave pairing gap Δ is obtained based on the standard BCS theory and using the bare nucleon–nucleon (NN) interaction as

ΔBCS =

8 kF2 −π/(2kF |a|) e . e2 2mN

(2)

The critical temperature Tc is related to the gap (at T = 0) as Tc =

eγ ΔBCS , π

(3)

where γ 0.577 is the Euler–Mascheroni constant. It was shown by Gor’kov and Melik-Barkhudarov (1961) that inclusion of the screening effect of interactions by the medium reduces the pairing gap as ΔGMB =

1 ΔBCS . (4e)1/3

(4)

However, there still remain large uncertainties on the magnitude of Δ at higher density region and the maximum value of Tc . This is mainly because as the density is increased, the noncentral parts of the NN interaction and three- and higherbody interactions become more important, which makes the many-body calculations difficult. As an order of magnitude, the 1 S0 pairing gap is found to be ∼1 MeV, and 3 P pairing gap is 0.1 MeV. 2 These nuclear superfluidity and superconductivity are expected to be realized in the inner crust of neutron stars and have phenomenological relevance to the physics there. Firstly, the neutron superfluid can lead to vortex pinning of the superfluid in the inner crust, which could explain the sudden increase of the rotational frequency of pulsars called glitches; see, e.g., Shapiro and Teukolsky (1983). It is also relevant to the cooling of neutron stars. While the BCS gap Δ itself suppresses the cooling due to the processes concerning thermal excitation of gapped particles by a factor ∼ e−Δ/T (Shapiro and Teukolsky 1983), the nuclear pairing also triggers an additional neutrino (ν) emission process called the “pair breaking and formation” (PBF) process (Flowers et al. 1976): when neutron–neutron or proton–proton pairing is formed, its binding energy is released in the form of ν–¯ν pairs. The observation of the rapid cooling of the neutron star in Cassiopeia A (Heinke and Ho 2010) might be explained by the increase of the neutrino emission due to such a PBF process after the onset of the neutron 3 P2 superfluid phase transition (Page et al. 2011; Shternin et al. 2011).

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Meson Condensation The phenomenon of BEC was predicted theoretically nearly a century ago. Soon afterward, it was linked to superfluidity of helium. However, it took seven decades and a dramatic advance in low-temperature physics before a clear experimental evidence for BEC, using engineered dilute atomic gases, was achieved in the mid1990s. A reader wishing to learn more about this fascinating subject is advised to consult Pethick and Smith (2008). In spite of its deep link to condensed-matter physics, BEC may be expected to occur in any physical system where interactions allow a description in terms of well-defined bosonic quasiparticles. Assuming the transition to the BEC phase to be of second order, it can be located by tuning the thermodynamic conditions to the point where the excitation energy of a quasiparticle drops to zero. This signals an instability that must be accompanied by a restructuring of the ground state. The lightest excitations of the QCD vacuum are the pseudoscalar mesons. The natural candidates for BEC in QCD are therefore the pions, followed by the kaons. Since pions form an isospin triplet, it is natural to expect π + to condense if the chemical potential for isospin, μI , exceeds the pion mass, mπ , or π − to condense if μI < −mπ . The kaons furnish an isospin doublet and carry a unit of strangeness, coupled to the chemical potential μs . The K + is then expected to condense if μs + μI /2 > mK , K 0 if μs − μI /2 > mK , and similarly for K − and K¯ 0 . For meson BEC to occur in nature, one needs a mechanism to generate sufficiently high chemical potential. A natural laboratory where this might be possible are the neutron stars. Were it only for strong interactions, the energetically most favored state of nuclear matter with fixed baryon chemical potential μB would be a Fermi sea composed of equal numbers of protons and neutrons. The long-range Coulomb interaction, however, does not allow a thermodynamically stable uniform state of such matter. It favors states that are electrically neutral. This generates an effective chemical potential, which was proposed a half-century ago to lead to BEC of π − , compensating for the electric charge of protons. Due to the nature of the pion–nucleon interaction, the condensation was predicted to occur in the p-wave. Kaon condensation in dense nuclear matter was suggested as a realistic possibility in the late 1980s. The p-wave condensation of kaons would be favored at baryon densities above roughly 2n0 by the kaon–nucleon interactions. This might provide a route to the formation of strange nuclear matter with strangeness-to-baryon-number ratio close to one.

Theoretical Approaches to Meson Condensation It was understood soon after the original proposal that many properties of meson condensates are dictated by the approximate chiral symmetry of QCD. Nowadays, the main analytic tool of choice is therefore the low-energy effective field theory of QCD: the chiral perturbation theory (ChPT). The question whether meson condensation may occur in a realistic dense electrically neutral nuclear medium remains open. However, if one merely assumes the existence of a sufficiently high

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chemical potential, then ChPT provides a systematic framework to study meson condensation within a controlled approximation. The use of ChPT for QCD with two light quark flavors and nonzero μI was pioneered by Son and Stephanov (2001a, b). Their work was extended to three quark flavors and nonzero μs by Kogut and Toublan (2001). This setting makes it possible to study the competition of pion and kaon condensation. Only very recently have these analyses been furthered beyond the leading order of ChPT. Apart from increasing precision due to loop corrections, this allows to include the effects of nonzero temperature. The state of the art amounts to the next-to-leading order of ChPT for two-flavor QCD at zero (Adhikari et al. 2019) and nonzero (Adhikari et al. 2021) temperature and three-flavor QCD at zero temperature (Adhikari and Andersen 2020), all including nonzero μI . A next-to-leading-order ChPT analysis of three-flavor QCD with nonzero μI , μs and temperature is not available yet. An alternative to ChPT is to use models with the same global symmetry as QCD. The advantages of such chiral models include often simplified computation and the possibility to study the interplay of meson and quark or baryon degrees of freedom and hence get closer to the goal of understanding electrically neutral nuclear matter. The linear sigma model was used early on to investigate the Higgs mechanism and electromagnetic superconductivity in the charged pion condensate. It was shown that π − BEC is a type II superconductor featuring stable vortex excitations (Harrington and Shepard 1976, 1977). The NJL model is more popular than the linear sigma model thanks to its ability to break the chiral symmetry dynamically. To the best of our knowledge, the first to consider meson condensation using the NJL model were Toublan and Kogut (2003) for two and Barducci et al. (2005) for three quark flavors. Within the NJL model, it is easy to include the effects of nonzero temperature on the meson condensates. One can also straightforwardly implement the constraint of electric charge neutrality. This turns out to eliminate pion BEC from the phase diagram of two-flavor quark matter unless the pion mass is tuned to values of roughly 10 MeV or below (Abuki et al. 2009). The ultimate theoretical approach to study meson condensation is, of course, lattice simulation of full QCD. This is, however, formidably difficult for nonzero μB due to the sign problem. In contrast, two-flavor QCD with nonzero μI is free of the sign problem. The state of the art of lattice studies of QCD at nonzero μI and temperature includes the phase transitions between pion BEC and the vacuum and quark–gluon plasma phases for physical pion masses (Brandt et al. 2018). The reader is referred to Mannarelli (2019) for a recent pedagogical review, including the early works on meson condensation in nuclear matter, ChPT, and model approaches and comparison thereof with lattice data.

Meson Condensation in Chiral Perturbation Theory Next follows a brief review of the mathematical setup that underlies our current understanding of meson condensation within ChPT. The leading-order Lagrangian of ChPT for the two lightest quark flavors reads

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L =

 fπ2  tr Dμ ΣD μ Σ −1 + m2π (Σ + Σ −1 ) , 4

(5)

where fπ and mπ are the pion decay constant and mass. Their empirically determined values are approximately fπ ≈ 92 MeV and mπ ≈ 140 MeV. Moreover, Σ ∈ SU(2) is a matrix-valued field carrying the pion degrees of freedom. Finally, the covariant derivative of Σ, Dμ Σ ≡ ∂μ Σ − I Lμ Σ + I ΣRμ , ensures simple transformation properties under the transformation, Σ → UL ΣUR−1 , from the chiral group (UL , UR ) ∈ SU(2)L × SU(2)R . The Lμ and Rμ are the corresponding gauge fields. These are typically treated as a fixed background but may also become dynamical in case QCD is coupled to the electroweak sector of the Standard Model. Nonzero chemical potential such as μI can be implemented as a static temporal background, Lμ = Rμ = (1/2)δμ0 μI τ3 , where τ3 is the third Pauli matrix. By using the residual symmetry left intact by the chemical potential, the ground state of the Lagrangian (5) can be sought in the form Σ = 1 cos θ + I τ1 sin θ , where θ is a parameter. For μI < mπ , the ground state is the QCD vacuum with θ = 0. The charged pion BEC is realized for μI > mπ , whereby cos θ = (mπ /μI )2 . The condensate carries nonzero isospin density, nI = fπ2 μI sin2 θ . A detailed analysis shows that the spectrum of masses of the three pions in the BEC phase is mπ 0 = μI ,

 mπ ± = μI 1 + 3 cos2 θ.

(6)

See Son and Stephanov (2001b) for further details. The extension of the setup to the next-to-leading order is described, for instance, in Adhikari et al. (2019).

Color Superconductivity As already mentioned, an arbitrarily weak attractive coupling in a many-fermion system leads to an instability of the Fermi surface with respect to formation of bound pairs. This was the key element of the BCS theory of superconductivity. The same mechanism however also underlies the superfluidity of fermionic liquids such as 3 He or the neutron component of dense nuclear matter. From the microscopic point of view, the only difference between superfluids and superconductors is whether the bosonic Cooper pairs are electrically neutral or charged. A thorough discussion of the physics of superfluids and superconductors can be found, e.g., in Leggett (2006) and Svistunov et al. (2015). In deconfined dense quark matter, interactions between quarks are mediated by gluon exchange. It was noticed long ago that these interactions may be attractive for some combinations of quantum numbers of quarks, thus leading to Cooper instability. This should be contrasted with metallic superconductivity in condensedmatter systems where the fundamental Coulomb interaction between electrons is repulsive and the attractive interaction is induced by the mediation of phonons. This idea was revived in the late 1990s, when it was shown that nonperturbative

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instanton-mediated interactions between quarks might lead to Cooper pairing of quarks with phenomenologically interesting energy gap of the order of 100 MeV. While these first works focused on the Cooper instability of the quark Fermi surface, the competition with chiral condensation, present in the QCD vacuum, was analyzed soon afterward. This led to first sketches of the QCD phase diagram from low to high baryon densities and at nonzero temperature. A comprehensive list of references can be found in some of the available reviews on the subject such as Alford (2001) and Alford et al. (2008).

The Zoo of Color-Superconducting Phases The basic quantity characterizing color-superconducting order is the diquark condensate, that is, the expectation value of the local quark–quark operator, αβ

β

α Δij,ab ≡ qia qj b ,

(7)

where q denotes collectively the set of quark fields, α, β are SU(3) color triplet indices, i, j are SU(3) flavor triplet or SU(2) flavor doublet indices, and a, b are Dirac spinor indices. There is a variety of candidate color-superconducting phases with different color, flavor, and Dirac spinor structures of the diquark condensate (7). Which concrete pairing pattern is eventually favored depends on external conditions. At very high baryon density when all three light quark flavors are active and their masses are negligible, the energetically most favored pairing pattern is the color–flavor locked (CFL) phase (Alford et al. 1999b). In this phase, the order parameter (7) is dominated by αβ

Δij,ab ∝ ΔCFL (Cγ5 )ab εαβC εij C ,

(8)

where ΔCFL is a gap parameter and the presence of Cγ5 (with C being the charge conjugation operator) ensures that the Cooper pair carries positive parity and zero spin. The CFL order parameter breaks the gauge color and global chiral symmetries of QCD according to SU(3)c × SU(3)L × SU(3)R × U(1)B → SU(3)c+L+R × Z2 .

(9)

The residual continuous symmetry corresponds to a simultaneous color and flavor unitary rotation of both left- and right-handed quarks. The CFL phase is a baryon superfluid and an electromagnetic insulator. The first property is a direct consequence of spontaneous breaking of U(1)B . The second property follows from the fact that the unbroken group SU(3)c+L+R contains a gauged U(1) subgroup whose gauge boson includes an admixture of the photon. The low-energy spectrum of the CFL phase thus contains two massless modes: the Nambu–Goldstone (NG) boson of U(1)B and the massless pseudo-photon. The lowest-lying gapped states are the eight pseudoscalar mesons of the spontaneously broken chiral symmetry, just like in the QCD vacuum.

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Unlike superconductors in condensed-matter systems, the CFL is a non-Abelian superconductor in that the unbroken symmetry in (9) is non-Abelian. Because of this fact, the CFL allows for stable non-Abelian vortices, which are 1/3 quantized superfluid vortices carrying color magnetic flux tubes; see Eto et al. (2014) for a review. A number of properties of the CFL phase are accessible by ab initio semianalytic calculation. Thus, the energy gap ΔCFL can be derived in an asymptotic expansion as a function of the QCD coupling g and the quark chemical potential, μ ≡ μB /3 (Pisarski and Rischke 2000; Schäfer and Wilczek 1999b; Son 1999), 512π 4 ΔCFL 1/3 5 μ 2 g

 5/2   2 π2 + 4 3π 2 . exp − 1/2 − 3 8 2 g

(10)

Note here that the parametric dependence on g is different from the one in the usual 2 BCS theory, Δ/μ ∼ e−c/g with c being some constant. This originates from the fact that the color superconductor is a relativistic superconductor where the colormagnetic interaction is no longer negligible compared with the color-electric one and survives as a long-range attractive interaction in the static limit. Similarly, the critical temperature at which the CFL order parameter melts is Tc =

21/3 eγ ΔCFL , π

(11)

which differs from the prediction of the BCS theory (3) by the factor 21/3 (Schmitt et al. 2002). Finally, the low-lying meson excitations of the CFL phase are described by an effective theory similar to ChPT, with effective couplings calculable within the weak-coupling expansion of QCD (Son and Stephanov 2000). As the baryon density decreases, the mass of the strange quark starts playing a role, imposing stress on the CFL pairing (Alford et al. 1999a). As long as the stress is small, the CFL state may respond to it by developing a kaon condensate (Bedaque and Schäfer 2002; Schäfer 2000a). At lower densities still, the pairing of all three quark flavors with equal Fermi momenta may not be possible to maintain. The CFL phase gives way to other pairing structures. The simplest among these is the twoflavor color superconductor (2SC), in which αβ

Δij,ab ∝ Δ2SC (Cγ5 )ab εαβ3 εij 3 .

(12)

The s-quarks are left out, and only two colors of the u,d-quarks participate in the pairing. The 2SC phase is not a baryon superfluid, owing to the presence of an unbroken global U(1) symmetry containing an admixture of baryon number. Like the CFL phase, it is an electromagnetic insulator, thanks to an unbroken gauge U(1) symmetry, containing an admixture of the photon. Other possibilities to resolve the stress on Cooper pairing of quarks of different flavors include various crystalline color-superconducting phases; see Anglani et al.

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(2014) and Casalbuoni and Nardulli (2004) for dedicated reviews. Yet another possibility is pairing of quarks of the same flavor, which may occur via colorspin locking (Schäfer 2000b) or other spin-one pairing patterns (Schmitt 2005). For a number of such alternative patterns for color superconductivity, weak-coupling results for the gap at zero temperature similar to (10), and for the critical temperature similar to (11), are available (Schmitt et al. 2002).

BCS–BEC Crossover and Quark–Hadron Continuity The weak-coupling approximation, implicit to the BCS theory of superconductivity, assumes that the spatial extent of Cooper pairs is much larger than the average distance between the elementary fermionic constituents (be it electrons, atoms, or quarks). If one now gradually increases the attractive coupling between the fermions, the size of Cooper pairs will diminish. Eventually, one will reach a qualitatively different regime in which the Cooper pairs are tightly bound difermion “molecules” of size much smaller than the average distance between elementary fermions. In this regime, the thermal phase transition to the superfluid state is adequately described as BEC of the preformed bound pairs. The transition between the BCS-like and BEC-like superfluid regimes is smooth and is known as the BCS–BEC crossover. A comprehensive account of the BCS–BEC crossover physics is given in Zwerger (2012). For reviews that also address aspects of BCS–BEC crossover in nuclear and quark matter, see He et al. (2013), Ohashi et al. (2020), and Strinati et al. (2018). Let us now describe in some detail the BCS–BEC crossover in dense quark matter. There are different approaches how to probe the presence of the crossover. The arguably most intuitive one is to extend the definition of the diquark order parameter (7) to the coordinate-dependent correlation function αβ

β

α Δij,ab (x) ≡ qia (x)qj b (0) .

(13)

This gives one access to the spatial profile of the Cooper pairs. For understanding the thermodynamics of the crossover, it is important to assess the relative contributions of fermionic and bosonic quasiparticles to the total baryon density. In the BCS limit, the total baryon number is dominated by the Fermi sea of unpaired quarks, which is largely insensitive to the strength of the pairing interaction. The superconducting properties of the ground state depend on the pairing gap, which is found by solving a gap equation. The critical temperature for the thermal phase transition to the BCS state is likewise determined by the gap via (3). In the BEC limit, on the other hand, the gap equation fixes the binding energy of the bosonic molecules. The thermodynamics of the system is however driven by the interactions among these molecules. The thermal phase transition corresponds to

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BEC of the composite bosons, and the transition temperature is accordingly largely fixed by their density. This replaces (3) with 2π Tc = mB



nB ζ (3/2)

2/3 ,

(14)

where mB is the mass of the bosonic molecules and nB their density. In order to properly account for the BEC side of the crossover, it is thus imperative to include the effect of bosonic degrees of freedom. If one merely wishes to compute the critical temperature for the thermal transition to the BCS or BEC phase, the contribution of bosonic molecules to the total density can be included using the Nozières–Schmitt-Rink theory (Nozières and Schmitt-Rink 1985). This led among others to the discovery of a second crossover in the strong-coupling limit (Nishida and Abuki 2005). Through this crossover, the BEC phase dominated by bosonic molecules turns into a relativistic BEC state, where the bound pairs are nearly massless and the thermodynamic observables receive large contributions from both bosonic particles and antiparticles. Below the critical temperature, one needs detailed understanding of the spectral function of pair fluctuations. This gives us yet another tool to localize the BCS– BEC crossover. Namely, on the BCS side of the crossover, a bound bosonic state appears below the critical temperature as dictated by the Goldstone theorem. On the BEC side of the crossover, preformed bound states exist already above the critical temperature. While many early studies were done using simplified relativistic fermionic models, the question whether a BCS–BEC crossover actually happens in colorsuperconducting quark matter remains open. There is no doubt that the high-density limit features the BCS pairing regime. The ground state is tuned toward BEClike pairing with decreasing density. Actually reaching the BEC regime before the transition to a non-superconducting state, however, seems to require an unphysically strong coupling (Kitazawa et al. 2008). The BCS–BEC crossover can nevertheless be expected with confidence in other corners of the QCD phase diagram, in particular at zero baryon chemical potential μB and nonzero isospin one, μI . Indeed, it is already known that for μI slightly above the pion mass mπ, the ground state of QCD will be a BEC of charged pions. ¯ On the other hand, at very large μI , the u-quarks and d-antiquarks will form Fermi seas of equal sizes. The ground state will be a superconductor featuring ud¯ Cooper ¯ 5 u , has the same quantum pairs. The order parameter in this BCS-like phase, dγ + numbers as π . One therefore expects the transition between the BEC and BCS states to be a smooth crossover (Brandt et al. 2018).

Quark–Hadron Continuity The above discussion suggests that the low-density phase of hadronic matter and the high-density phase of quark matter may be continuously connected. This idea was first put forward in the context of QCD with three degenerate quark flavors (Schäfer

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and Wilczek 1999a). For this to be possible, the two phases must have the same symmetries. On the hadron side, chiral symmetry is spontaneously broken in the β β usual way by the chiral condensate, q¯αia qj b ∝ δji δba δα . On the quark side, one has the CFL ground state with its symmetry-breaking pattern (9). Note that it is only the global symmetry part of (9) that matters. The gauge part is not observable. The chiral symmetry of QCD is broken in the hadron phase and in the CFL phase in exactly the same way. It is the U(1)B symmetry that appears problematic, since the CFL state is a baryon superfluid. The problem was bypassed by Schäfer and Wilczek (1999a) by proposing that the hadronic phase features a BEC of light dibaryons. The matching of symmetries of the hadron and quark phases does not preclude the existence of a first-order phase transition separating the two. In fact, this has long been the default scenario predicted by various model approaches to QCD. A mechanism how to avoid the first-order phase transition was proposed by Hatsuda et al. (2006). If one describes the flavor-singlet chiral condensate of the hadron phase by a single real parameter σ , and the CFL diquark condensate by a single complex parameter Δ, then the competition of the two orders can be captured by the Ginzburg–Landau (GL) free energy density  F =

   a 2 c 3 b 4 α † β † 2 σ − σ + σ + Δ Δ + (Δ Δ) − γ σ Δ† Δ + λσ 2 Δ† Δ. 2 3 4 2 4 (15)

The cubic c and γ terms originate from the axial anomaly. Both are positive and of the same order of magnitude (Yamamoto et al. 2007). Positive γ favors coexistence of the chiral and CFL order parameters. Within the parameter space of the GL theory (15), one can then smoothly connect a “hadron-like” state with large σ and small Δ and a “CFL-like” state with large Δ and small σ . To what extent such a smooth crossover can be realized in the QCD phase diagram however depends on the mapping of the parameters in (15) to the thermodynamic variables of QCD. This problem was addressed by Abuki et al. (2010) and Powell and Baym (2012), who used respectively the NJL model and its extension coupled to the order parameter for deconfinement, the Polyakov loop. These works confirm the conjectured presence of a new axial-anomaly-induced critical point in the phase diagram of QCD, which prevents the first-order transition separating the hadron and quark phases from reaching the chemical potential axis. At low temperatures, the transition may therefore be smoothed out. Moreover, the above works find evidence for a low-density coexistence phase where the CFL order is BEC-like. Hence, the crossover from hadronic to quark matter simultaneously realizes a BEC–BCS crossover for baryon superfluidity. The quark–hadron continuity conjecture remains under active investigation. Sogabe and Yamamoto (2017) observed that the new anomaly-induced critical point exhibits a novel dynamical universality class that does not belong to the conventional classification of critical phenomena. More recently, Alford et al. (2019) extended the discussion of the continuity of spectra in the hadron and quark phases from quasiparticles to vortex excitations. The possible continuity between

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superfluid neutron matter and two-flavor quark matter was studied by Fujimoto et al. (2020). From a phenomenological viewpoint, equations of state for neutron stars from this quark–hadron continuity scenario are compatible with the recent observations of the neutron star radii and neutron star masses ∼2M ; see Baym et al. (2018) for a review.

Topological Order and Higher-Form Symmetry In the conventional Landau paradigm, the phases of matter are classified by certain local order parameter characterizing the patterns of broken or unbroken symmetries. However, there also exist condensed-matter systems in which phases cannot be classified in terms of the Landau paradigm, such as the fractional quantum Hall effect and the toric code model (Kitaev 2003). It turned out that these phases can be understood in terms of the so-called topological order (Wen 1990). Originally, the topological order was characterized by the ground-state degeneracy which depends on the topology of the compact manifold of a system. From the recent viewpoint of generalized global symmetries (or higher-form symmetries) (Gaiotto et al. 2015), which is the generalization of the notion of symmetries for point-like objects to extended objects like strings and branes, a class of topological orders can be further interpreted as the spontaneous breaking of discrete higher-form symmetries. In this sense, the topological order can be regarded as an extension of the Landau paradigm to nonlocal order parameters. Then, one can ask whether such a new classification of phases could also be applied to nuclear and quark matter to detect novel quantum phases uncovered so far. One specific question addressed by Hirono and Tanizaki (2019) is whether the quark–hadron continuity discussed above persists even in terms of topological order beyond the conventional Landau paradigm. It is instructive to first review a simple condensed-matter system that exhibits the spontaneous breaking of discrete higher-form symmetries and that possesses a topological order: the s-wave superconductor (Hansson et al. 2004; Nussinov and Ortiz 2009). The relativistic version of the low-energy action is 

S=

d4 x

 v2 1 2 , |∂μ φ − kAμ |2 − 2 Fμν 2 4e

(16)

where the complex scalar field Φ for the Cooper pairing is decomposed as Φ = √v eiφ , k = 2 is the charge of Φ in units of the electric charge e, Aμ is the U(1) 2 gauge field, and Fμν is the field strength. Here, the amplitude fluctuation of Φ is ignored as it is heavy and is irrelevant to the low-energy dynamics. One can dualize this theory by introducing a two-form gauge field Bμν such that 2π v 2 εμνρσ (∂ σ φ − kAσ ) = ∂μ Bνρ + ∂ν Bρμ + ∂ρ Bμν . By further taking the lowenergy limit to ignore the kinetic terms of Aμ and Bμν , one arrives at the topological field theory known as the BF theory (Horowitz 1989):

84 Crossover Between Quark Nuclear Matter and Condensed-Matter Physics

SBF =

d4 x

k μνρσ ε Bμν Fρσ . 8π

3055

(17)

This theory has only nonlocal observables: the Wilson loop of an electrically charged particle and the world sheet of a magnetically charged vortex,   μ W (C) = exp i Aμ dx , C

  1 μν V (S) = exp i , Bμν dS S 2

(18)

respectively, where C is a one-dimensional closed loop and S is a two-dimensional closed surface. This BF theory has the emergent electric Zk one-form symmetry and magnetic Zk two-form symmetry, under which Aμ → Aμ + k1 aμ and Bμν → Bμν +

1 aμ and bμν are flat connections satisfying the conditions C aμ dx μ ∈ k bμν , where

2π Z and S 21 bμν dS μν ∈ 2π Z. These two symmetries are spontaneously broken in the ground state as characterized by the nonlocal order parameters W (C) = 1 and V (S) = 1, which follow from the fact that both Aμ and Bμν are gapped and can be integrated out at low energy. One can also compute the correlation function W (C)V (S) = e

2π i k Link(C,S)

V (S) ,

(19)

which can be seen as the symmetry transformation of V (S) generated by W (C), where “Link(C, S)” stands for the linking number of C and S. From (19) together with V (S) = 1, one can show that when the system is put on a compact manifold with a nontrivial topology such that the Wilson loop and vortex world sheet can wrap its subspaces, then the system has ground-state degeneracy – a hallmark of topological order. In the context of the CFL phase in high-density QCD, Hirono and Tanizaki (2019) derived a BF -type low-energy effective theory, where nonlocal objects are the color Wilson loop and the world sheet of the color magnetic vortex. These objects satisfy a relation similar to (19) with k = 3 as derived in Cherman et al. (2019). The theory then has an emergent Z3 two-form symmetry generated by the Wilson loop. However, the difference from the case of the s-wave superconductor above is that because of the presence of the superfluid phonon associated with the U(1)B symmetry breaking in (9), the potential between vortices at distance R becomes proportional to log R, leading to the confinement of vortices. Hence, the Z3 two-form symmetry is not spontaneously broken, and there is no topological order. It is for this reason that Hirono and Tanizaki (2019) concluded that the quark–hadron continuity is still a consistent scenario in terms of topological order. On the other hand, Cherman et al. (2019) argued that the Z3 particle–vortex braiding phase itself is sufficient to distinguish the hadron phase and CFL phase. Therefore, there has not yet been a final word on the validity of the quark–hadron continuity scenario.

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Chiral Soliton Lattice CSL denotes a relatively novel class of phenomena, characterized by spatially modulated order tunable by an external field, carrying topological charge. It was discovered decades ago in condensed-matter physics; see Kishine and Ovchinnikov (2015) for a review and references therein. In the past decade, it has however found a number of realizations in quark matter under the influence of external fields. The low-energy physics giving rise to CSL is captured by the following universal Lagrangian: L =

f2 (∂μ φ)2 + m2 f 2 cos φ + αB · ∇φ. 2

(20)

Here φ is a dimensionless pseudoscalar field and B a divergence-less background axial vector field that φ is coupled to. Finally, f , m, and α are constant parameters. The first two terms in (20) represent a low-energy effective field theory of a spontaneously broken U(1) symmetry; φ is the angular variable on U(1). The symmetry is weakly explicitly broken by the parameter m. Thanks to the assumption ∇ · B = 0, the last term in (20) is a surface term, measuring winding of the phase φ in the direction of B. This term therefore does not affect the equation of motion for φ but gives different contributions to the total energy of field configurations in different topological sectors. In the exact symmetry limit (m → 0), the Hamiltonian associated with (20) can easily be minimized explicitly when the external field B is uniform. The ground state carries a constant gradient of φ, ∇φ =

αB , f2

(21)

indicating uniform winding in the direction of B. Since states in which the value of φ differs by an integer multiple of 2π are equivalent, the ground state is periodic with the period  = 2πf 2 /(αB). Away from the exact U(1) symmetry limit, the ground state may still be spatially modulated. Suppose that the (uniform) external field B is oriented in the direction of the positive z-semiaxis. The general solution of the corresponding one-dimensional equation of motion is, up to spatial translation, given implicitly by

cos

φ(zm/k) = sn(zm/k, k), 2

(22)

where k (0 ≤ k ≤ 1) is the so-called elliptic modulus and sn is one of Jacobi’s elliptic functions. The actual value of k is obtained by minimization of the Hamiltonian and is determined by

84 Crossover Between Quark Nuclear Matter and Condensed-Matter Physics

π αB E(k) = , k 4mf 2

3057

(23)

where E is the complete elliptic integral of the second kind. This implies that the ground state is periodically modulated provided the external field B exceeds the critical value BCSL =

4mf 2 . πα

(24)

The period of the optimal CSL solution is given in terms of its modulus k by =

2kK(k) , m

(25)

where K is the complete elliptic integral of the first kind. The CSL ground state spontaneously breaks translations in the direction of B. This implies via Goldstone theorem the existence of a gapless excitation in the spectrum, corresponding to the phonon of the crystalline order. A detailed analysis shows that the spectrum of single-particle excitations consists of two bands. The lower band is gapless as expected. It is possible to derive an exact dispersion relation for the CSL phonon in terms of elliptic integrals. The leading contribution to the energy ω(p) valid in the limit long wavelength, or small momentum p, reads  ω(p)2 = px2 + py2 + (1 − k 2 )

K(k) E(k)

2 pz2 + O(pz4 ).

(26)

Physical Realizations of CSL In condensed-matter physics, CSL is realized, for instance, in cholesteric liquid crystals and chiral magnets, both of which feature helical order. In particular, the low-energy physics of chiral magnets placed in an external magnetic field perpendicular to the chiral axis can be described by an effective one-dimensional Lagrangian of the type (20). Here φ measures the orientation of the local magnetization in the plane transverse to the chiral axis, m2 is proportional to the external magnetic field, and B represents an intrinsic anisotropy due to the so-called Dzyaloshinskii–Moriya (DM) interaction. In this realization, it is therefore m that is tunable, whereas B is fixed and determined by the structure of the crystal lattice of the given magnetic material. The limit m → 0 leads to a perfect helical ordering. This is gradually deformed by increasing the magnetic field. Above certain critical value of m determined by (24), the aligning effect of the external field wins over the distorting effect of the DM interaction. The ground state is then a perfect (spatially uniform) ferromagnet. In QCD, the low-energy effective theory of the type (20) is naturally realized in sufficiently strong external magnetic fields. These induce Landau-level quantization of the spectrum, which makes charged pions heavy. Maintaining the neutral pion π 0

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as the sole low-energy degree of freedom reduces the Lagrangian (5) of ChPT to the first two terms of (20). The precise mapping is fπ → f , mπ → m, and π 0 → fπ φ. The last term in (20) is required to match the chiral anomaly of ChPT in the presence of a background magnetic field B and chemical potential (Son and Stephanov 2008). When both the baryon and the isospin chemical potential are included, the parameter α takes the value α=

μB μI + . 4π 2 8π 2

(27)

The special case of nonzero baryon chemical potential was the first realization of CSL within QCD (Brauner and Yamamoto 2017). It is well known that uniform rotation has a similar effect on particle dynamics as uniform magnetic field. The case of QCD under uniform rotation (and no magnetic field) was investigated by Huang et al. (2018). Here B is interpreted as the angular velocity. A detailed anomaly matching shows that α = μB μI /(2π 2 ). In this case, the kinetic term for φ differs somewhat from (20) due to the effects of the background rotation. A CSL-like state with a spatially modulated condensate of η mesons has also been proposed to arise in the color-superconducting CFL phase of quark matter under rotation (Nishimura and Yamamoto 2020). Here α is proportional to μ2B . When other degrees of freedom than the U(1) mode φ are included, the physics of CSL is no longer universal but becomes dependent on the specific realization of the CSL ground state. In QCD, the next-to-lightest degrees of freedom are the charged pions. These can be taken into account by extending the basic Lagrangian (20) to the full two-flavor ChPT (5). It has thus been predicted that in strong magnetic fields, the effects of the chiral anomaly will outweigh Landau level quantization and lead to a condensation of charged pions, distorting the CSL state (Brauner and Yamamoto 2017). A similar effect was studied very recently in the context of rotating quark matter and predicted to lead to exotic CSL-like phases (Eto et al. 2021). Further interesting phenomena arise when the basic Lagrangian (20) is coupled to a dynamical electromagnetic field by adding the usual Maxwell kinetic term −(1/4)Fμν F μν as well as the anomaly-induced axion coupling of φ to the electromagnetic field, proportional to φεκλμν Fκλ Fμν . Here the CSL ground state leads to an intriguing nonrelativistic helicity-dependent deformation of the photon spectrum (Brauner and Kadam 2017; Yamamoto 2016). While nonrelativistic gapless modes, such as magnons in ferromagnets, can be understood as the socalled type-B NG modes and the counting rule of type-B NG modes associated with the spontaneous breaking of usual (0-form) symmetries is well known, this nonrelativistic photon in the CSL provides the first example of a type-B NG mode of higher-form symmetries. Finally, a fixed CSL-like background may also be emulated by using a material constructed out of alternating layers of ordinary and topological insulators. The resulting polarization-dependent deformation of the photon spectrum was proposed by Ozaki and Yamamoto (2017) to have potentially interesting applications to optoelectronics.

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Topological Transport Phenomena of Quark Matter In this section, we describe the topological transport phenomena due to the Berry phase of quark matter.

Berry Phase of Quark Matter In condensed-matter physics, the geometric phase known as the Berry phase (Berry 1984) has profound effects on the dynamics of electrons including their transport properties; see, e.g., Xiao et al. (2010) for a review. When the Hamiltonian of a system depends on generic three-dimensional parameters n = (n1 , n2 , n3 ) and the Hilbert space of the wave function un is restricted to certain hypersurface S in the n space, then one can introduce a Berry connection An ≡ iu†n ∇ n un and Berry curvature Ω n ≡ ∇ n × An on S. For a closed loop C in the n space, the Berry phase is defined as

An · dn =

θ (C) = C

Ω n · dS n ,

(28)

SC

where SC is a two-dimensional surface whose boundary is C. A prototype example where a nontrivial Berry phase emerges is a spin-1/2 particle in a magnetic field B (Berry 1984). The Hamiltonian of this system is H = σ · B, where the coefficient is set to be unity for simplicity. With keeping the magnitude |B| fixed and varying the orientation of B slowly, the Berry curvature in the parameter B space is given by ΩB =

B , 2|B|3

and

1 2π

|B|>0

Ω B · dS B = 1.

(29)

This Berry curvature takes the form of the “magnetic field” of a magnetic monopole at the origin in the parameter B space. This geometric property is a consequence of the fact that any quantum state of this spin-1/2 particle has a one-to-one correspondence with a point on the sphere S 2 . In the context of high-energy physics, ultrarelativistic elementary particles such as quarks and neutrinos possess a Berry curvature due to their intrinsic property of chirality (or helicity). Consider the (Weyl) Hamiltonian of right- or left-handed chiral fermions, H = χ σ · p, where p is momentum and χ = ±1 is the chirality. Note that this is mathematically equivalent to the Hamiltonian of the spin-1/2 particle in the magnetic field above under the replacement χ p ↔ B. Hence, chiral fermions have a Berry curvature, which takes the form of the “magnetic field” of a magnetic monopole in momentum space (Son and Yamamoto 2012; Stephanov and Yin 2012),

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Ωp = χ

p , 2|p|3

and

1 2π

|p|>0

Ω p · dS p = χ .

(30)

In particular, the monopole charge is equal to the chirality of chiral fermions. The condition of the adiabaticity for the Berry phase to be well defined is satisfied, e.g., when chiral fermions form a Fermi surface at finite chemical potential μ  T and when one is interested in the low-energy dynamics close to the Fermi surface, |p| ≈ μ; the Berry curvature on the Fermi surface is purely a topological property of Fermi liquids (Haldane 2004). As is well known in condensed-matter physics, the presence of the Berry curvature affects the dynamics of relativistic chiral fermions. It was also shown that the energy shift due to the magnetic moment of chiral fermions (with charge e) is expressed by the Berry curvature as (Chen et al. 2014; Son and Yamamoto 2013) εp = |p|(1 − eB · Ω p ).

(31)

One can then write down the semiclassical equations of motion for charged chiral fermions taking into account the effects of the Berry curvature as x˙ =

∂εp + p˙ × Ω p , ∂p

 ∂εp . p˙ = e E + x˙ × B − ∂x

(32)

Here, the “Lorentz force” p˙ ×Ω p in momentum space in the first equation is known as the anomalous velocity in condensed-matter physics (Sundaram and Niu 1999). These Berry curvature corrections are relevant to quark matter at high density where the quark mass is negligibly small. In particular, as will be described in more detail below, the Berry phase is an essential ingredient to satisfy the anomaly matching condition in the kinetic framework.

Chiral Magnetic Effect and Related Phenomena The Berry phase of relativistic chiral fermions leads to unusual transport phenomena, such as the CME, which is a current along the magnetic field, and the chiral vortical effect (CVE), which is a current along the vorticity (or global rotation). The CME and CVE were derived earlier based on quantum field theory in Fukushima et al. (2008), Nielsen and Ninomiya (1983), Vilenkin (1979, 1980), and Landsteiner et al. (2011), respectively, and within the framework of relativistic hydrodynamics combined with the second law of thermodynamics (Son and Surówka 2009). For a review on the chiral transport phenomena including the CME and CVE, see, e.g., Landsteiner (2016). To consider the generic (non)equilibrium many-body problem of chiral fermions, one can formulate a kinetic theory by introducing its distribution function in phase space denoted by np (x). The kinetic equation for np reads

84 Crossover Between Quark Nuclear Matter and Condensed-Matter Physics

dnp ∂np ∂np ∂np = + x˙ · + p˙ · = C[np ], dt ∂t ∂x ∂p

3061

(33)

where x˙ and p˙ follow the equations of motion (32) and C[np ] is the collision term. This is the chiral kinetic theory (Son and Yamamoto 2012; Stephanov and Yin 2012). In this kinetic theory, physical quantities such as the particle number current and energy–momentum tensor receive Berry curvature corrections. For example, the particle number and current are given by nχ =

d3 p √ ωnp , (2π )3

jχ =

d3 p √ ˙ p, ωxn (2π )3

(34)

where the invariant phase space of the Hamilton dynamics is modified by the factor √ ω = 1 + eB · Ω p due to the Berry curvature corrections (Duval et al. 2006; Xiao et al. 2005). Although the chiral kinetic theory using the Berry curvature above does not have manifest Lorentz covariance, one can also formulate a Lorentz-covariant chiral kinetic theory by introducing an additional frame vector (Chen et al. 2015), which is also applicable to higher temperatures by including the contribution of antiparticles. It was further shown that such a chiral kinetic theory can indeed be derived from the underlying quantum field theory; see, e.g., Hidaka et al. (2017). By using (33) and (32) for np in local thermal equilibrium, one obtains the corrections to the continuity equation and particle number current proportional to B: ∂t nχ + ∇ · j χ = χ

e2 E · B, 4π 2

jχ = χ

eμ B. 4π 2

(35)

The former is the chiral anomaly (Adler 1969; Bell and Jackiw 1969), and the latter is the CME for chiral fermions. Note that both of these coefficients are topologically quantized by the monopole charge (30). In particular, the CME is a topological transport phenomenon similar to the two-dimensional integer quantum Hall effect in condensed-matter systems, where the Hall conductance is topologically quantized by integer multiples of e2 / h. Similarly, the CVE can also be derived from the chiral kinetic theory (Chen et al. 2014, 2015; Stephanov and Yin 2012). For Dirac fermions, adding or subtracting the contributions of right- and lefthanded fermions in (35), one obtains ∂t n5 + ∇ · j 5 =

e2 E · B, 2π 2

j=

eμ5 B, 2π 2

j5 =

eμ B, 2π 2

(36)

where n5 = nR − nL , j 5 = j R − j L , μ = (μR + μL )/2, and μ5 = (μR − μL )/2. The second relation in (36) is the CME for Dirac fermions, and the third one is called the chiral separation effect (CSE) (Metlitski and Zhitnitsky 2005; Son and Zhitnitsky 2004). Note that the CME should be regarded as a nonequilibrium

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current rather than a ground-state (or equilibrium) current, as any persistent vector current in the thermodynamic limit is prohibited by the generalized Bloch theorem based on the gauge symmetry (Yamamoto 2015). On the other hand, that the CSE is a persistent axial current is not prohibited by the Bloch theorem, and, in fact, the CSE can be understood as the spin polarization of relativistic Dirac matter. While the spin polarization itself exists in nonrelativistic condensed-matter systems where the magnetic susceptibility depends on microscopic details, the coefficient 1/(2π 2 ) in (36) in Dirac matter has a connection to the chiral anomaly and is exact independently of interactions. The chiral transport phenomena of quarks may be potentially realized in quark– gluon plasmas created in heavy ion collision experiments; see Kharzeev et al. (2016) for a review. There, gluon configurations with E a · B a = 0 (with E a and B a being color electromagnetic fields) can generate a chiral charge according to the QCD anomaly. Also, two charged ions moving with nearly the speed of light at finite impact parameter lead to a strong magnetic field and vorticity. Hence, this system has the essential ingredients for the CME and CVE to occur. The chiral transport phenomena can also be realized in condensed-matter systems called Dirac/Weyl semimetals, where electrons as Dirac/chiral fermions appear emergently close to the band touching points; see Armitage et al. (2018) for a review. In these materials, a nonzero μ5 is generated by applying electromagnetic fields with E · B = 0 according to the chiral anomaly relation in (36). The value of μ5 can be determined by the balance between the chiral charge pumping proportional to E · B due to the chiral anomaly and the chiral charge relaxation proportional to 1/τ due to the inter-Weyl node scattering, where τ is the corresponding mean free time. For E  B, the conductivity σ has an additional contribution from CME with quadratic dependence on B ≡ |B| (Son and Spivak 2013):

σ (B) = σ0 +

e4 B 2 τ , 4π 4 g(εF )

(37)

where σ0 is the conductivity at B = 0 and g(εF ) is the density of states at the Fermi surface. This B-dependence of the magnetoconductivity has been experimentally observed, e.g., in Huang et al. (2015), Li et al. (2016), signaling the presence of the chiral anomaly and CME in condensed-matter systems. For the chiral transport phenomena in the context of high-energy astrophysics and cosmology, such as neutron stars, core-collapse supernovae, and the early universe, see Kamada et al. (2022) for a review. Acknowledgments This work was supported in part by the grant no. PR-10614 within the ToppForsk-UiS program of the University of Stavanger and the University Fund and by the Keio Institute of Pure and Applied Sciences (KiPAS) project at Keio University and JSPS KAKENHI Grant No. 19K03852.

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References H. Abuki, R. Anglani, R. Gatto, M. Pellicoro, M. Ruggieri, Phys. Rev. D79, 034032 (2009). arXiv:0809.2658 [hep-ph] H. Abuki, G. Baym, T. Hatsuda, N. Yamamoto, Phys. Rev. D81, 125010 (2010). arXiv:1003.0408 [hep-ph] P. Adhikari, J.O. Andersen, Phys. Lett. B804, 135352 (2020). arXiv:1909.01131 [hep-ph] P. Adhikari, J.O. Andersen, P. Kneschke, Eur. Phys. J. C79, 874 (2019). arXiv:1904.03887 [hepph] P. Adhikari, J.O. Andersen, M.A. Mojahed, Eur. Phys. J. C81, 173 (2021). arXiv:2010.13655 [hepph] S.L. Adler, Phys. Rev. 177, 2426 (1969) M.G. Alford, Ann. Rev. Nucl. Part. Sci. 51, 131 (2001). arXiv:hep-ph/0102047 M.G. Alford, J. Berges, K. Rajagopal, Nucl. Phys. B558, 219 (1999a). arXiv:hep-ph/9903502 M.G. Alford, K. Rajagopal, F. Wilczek, Nucl. Phys. B537, 443 (1999b). arXiv:hep-ph/9804403 M.G. Alford, A. Schmitt, K. Rajagopal, T. Schäfer, Rev. Mod. Phys. 80, 1455 (2008). arXiv:0709.4635 [hep-ph] M.G. Alford, G. Baym, K. Fukushima, T. Hatsuda, M. Tachibana, Phys. Rev. D99, 036004 (2019). arXiv:1803.05115 [hep-ph] R. Anglani, R. Casalbuoni, M. Ciminale, N. Ippolito, R. Gatto, M. Mannarelli, M. Ruggieri, Rev. Mod. Phys. 86, 509 (2014). arXiv:1302.4264[hep-ph] N.P. Armitage, E.J. Mele, A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018). arXiv:1705.01111 [cond-mat.str-el] A. Barducci, R. Casalbuoni, G. Pettini, L. Ravagli, Phys. Rev. D71, 016011 (2005). arXiv:hepph/0410250 G. Baym, T. Hatsuda, T. Kojo, P.D. Powell, Y. Song, T. Takatsuka, Rep. Prog. Phys. 81, 056902 (2018). arXiv:1707.04966 [astro-ph.HE] P.F. Bedaque, T. Schäfer, Nucl. Phys. A697, 802 (2002). arXiv:hep-ph/0105150 J.S. Bell, R. Jackiw, Nuovo Cim. A60, 47 (1969) M.V. Berry, Proc. R. Soc. Lond. A392, 45 (1984) G.F. Bertsch, Int. J. Mod. Phys. B 15, 10, iii (2001), ed. by R.F. Bishop B.B. Brandt, G. Endrodi, S. Schmalzbauer, Phys. Rev. D97, 054514 (2018). arXiv:1712.08190 [hep-lat] T. Brauner, Symmetry 2, 609 (2010). arXiv:1001.5212 [hep-th] T. Brauner, S. Kadam, JHEP 03, 015 (2017). arXiv:1701.06793 [hep-ph] T. Brauner, N. Yamamoto, JHEP 04, 132 (2017). arXiv:1609.05213 [hep-ph] M. Buballa, S. Carignano, Prog. Part. Nucl. Phys. 81, 39 (2015). arXiv:1406.1367 [hep-ph] R. Casalbuoni, G. Nardulli, Rev. Mod. Phys. 76, 263 (2004). arXiv:hep-ph/0305069 J.-Y. Chen, D.T. Son, M.A. Stephanov, H.-U. Yee, Y. Yin, Phys. Rev. Lett. 113, 182302 (2014). arXiv:1404.5963 [hep-th] J.-Y. Chen, D.T. Son, M.A. Stephanov, Phys. Rev. Lett. 115, 021601 (2015). arXiv:1502.06966 [hep-th] A. Cherman, S. Sen, L.G. Yaffe, Phys. Rev. D100, 034015 (2019). arXiv:1808.04827 [hep-th] T.D. Cohen, Phys. Rev. Lett. 91, 222001 (2003). arXiv:hep-ph/0307089 D.J. Dean, M. Hjorth-Jensen, Rev. Mod. Phys. 75, 607 (2003). arXiv:nucl-th/0210033 C. Duval, Z. Horvath, P.A. Horvathy, L. Martina, P. Stichel, Mod. Phys. Lett. B20, 373 (2006). arXiv:cond-mat/0506051 M. Eto, Y. Hirono, M. Nitta, S. Yasui, Prog. Theor. Exp. Phys. 2014, 012D01 (2014). arXiv:1308.1535 [hep-ph] M. Eto, K. Nishimura, M. Nitta, JHEP 08, 305 (2022). arXiv:2112.01381 [hep-ph] E. Flowers, M. Ruderman, P. Sutherland, Astrophys. J. 205, 541 (1976)

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Y. Fujimoto, K. Fukushima, W. Weise, Phys. Rev. D101, 094009 (2020). arXiv:1908.09360 [hepph] K. Fukushima, T. Hatsuda, Rep. Prog. Phys. 74, 014001 (2011). arXiv:1005.4814 [hep-ph] K. Fukushima, D.E. Kharzeev, H.J. Warringa, Phys. Rev. D78, 074033 (2008). arXiv:0808.3382[hep-ph] D. Gaiotto, A. Kapustin, N. Seiberg, B. Willett, JHEP 02, 172 (2015). arXiv:1412.5148 [hep-th] S. Gandolfi, A. Gezerlis, J. Carlson, Ann. Rev. Nucl. Part. Sci. 65, 303 (2015). arXiv:1501.05675 [nucl-th] H. Georgi, Phys. Rev. Lett. 98, 221601 (2007). arXiv:hep-ph/0703260 A. Gezerlis, C.J. Pethick, A. Schwenk, arXiv:1406.6109 [nucl-th] (2014) L.P. Gor’kov, T.K. Melik-Barkhudarov, Sov. Phys. JETP 13, 1018 (1961) A.M. Halasz, A.D. Jackson, R.E. Shrock, M.A. Stephanov, J.J.M. Verbaarschot, Phys. Rev. D58, 096007 (1998). arXiv:hep-ph/9804290 F. Haldane, Phys. Rev. Lett. 93, 206602 (2004). arXiv:cond-mat/0408417 H.-W. Hammer, D.T. Son, Proc. Nat. Acad. Sci. 118, e2108716118 (2021). arXiv:2103.12610 [nucl-th] T.H. Hansson, V. Oganesyan, S.L. Sondhi, Ann. Phys. 313, 497 (2004). arXiv:cond-mat/0404327 B.J. Harrington, H.K. Shepard, Phys. Lett. B64, 71 (1976) B.J. Harrington, H.K. Shepard, Phys. Rev. D16, 3437 (1977) T. Hatsuda, M. Tachibana, N. Yamamoto, G. Baym, Phys. Rev. Lett. 97, 122001 (2006). arXiv:hepph/0605018 K. Hattori, K. Itakura, S. Ozaki, S. Yasui, Phys. Rev. D92, 065003 (2015). arXiv:1504.07619 [hepph] L. He, S. Mao, P. Zhuang, Int. J. Mod. Phys. A28, 1330054 (2013). arXiv:1311.6704 [condmat.quant-gas] C.O. Heinke, W.C.G. Ho, Astrophys. J. Lett. 719, L167 (2010) Y. Hidaka, S. Pu, D.-L. Yang, Phys. Rev. D95, 091901 (2017). arXiv:1612.04630 [hep-th] Y. Hirono, Y. Tanizaki, Phys. Rev. Lett. 122, 212001 (2019). arXiv:1811.10608 [hep-th] G.T. Horowitz, Commun. Math. Phys. 125, 417 (1989) X. Huang et al., Phys. Rev. X5, 031023 (2015). arXiv:1503.01304 [cond-mat.mtrl-sci] X.-G. Huang, K. Nishimura, N. Yamamoto, JHEP 02, 069 (2018). arXiv:1711.02190 [hep-ph] K. Kamada, N. Yamamoto, D.-L. Yang, (2022). arXiv:2207.09184 [astro-ph.CO] D.E. Kharzeev, J. Liao, S.A. Voloshin, G. Wang, Prog. Part. Nucl. Phys. 88, 1 (2016). arXiv:1511.04050 [hep-ph] J.-I. Kishine, A.S. Ovchinnikov, Solid State Phys. 66, 1 (2015) A.Y. Kitaev, Ann. Phys. 303, 2 (2003). arXiv:quant-ph/9707021 M. Kitazawa, D.H. Rischke, I.A. Shovkovy, Phys. Lett. B663, 228 (2008). arXiv:0709.2235 [hepph] J.B. Kogut, D. Toublan, Phys. Rev. D64, 034007 (2001) K. Landsteiner, Acta Phys. Polon. B47, 2617 (2016). arXiv:1610.04413 [hep-th] K. Landsteiner, E. Megias, F. Pena-Benitez, Phys. Rev. Lett. 107, 021601 (2011). arXiv:1103.5006 [hep-ph] A.J. Leggett, Quantum Liquids (Oxford University Press, Oxford, 2006) Q. Li, D.E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosic, A.V. Fedorov, R.D. Zhong, J.A. Schneeloch, G.D. Gu, T. Valla, Nat. Phys. 12, 550 (2016). arXiv:1412.6543 [cond-mat.str-el] M. Mannarelli, Particles 2, 411 (2019). arXiv:1908.02042 [hep-ph] M.A. Metlitski, A.R. Zhitnitsky, Phys. Rev. D72, 045011 (2005). arXiv:hep-ph/0505072 Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961) H.B. Nielsen, M. Ninomiya, Phys. Lett. B130, 389 (1983) Y. Nishida, H. Abuki, Phys. Rev. D72, 096004 (2005). arXiv:hep-ph/0504083 Y. Nishida, D.T. Son, Phys. Rev. D76, 086004 (2007). arXiv:0706.3746 [hep-th] K. Nishimura, N. Yamamoto, JHEP 07, 196 (2020). arXiv:2003.13945 [hep-ph] P. Nozières, S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985) Z. Nussinov, G. Ortiz, Ann. Phys. 324, 977 (2009). arXiv:cond-mat/0702377

84 Crossover Between Quark Nuclear Matter and Condensed-Matter Physics

3065

Y. Ohashi, H. Tajima, P. van Wyk, Prog. Part. Nucl. Phys. 111, 103739 (2020) S. Ozaki, N. Yamamoto, JHEP 08, 098 (2017). arXiv:1610.07835 [cond-mat.mes-hall] D. Page, M. Prakash, J.M. Lattimer, A.W. Steiner, Phys. Rev. Lett. 106, 081101 (2011). arXiv:1011.6142 [astro-ph.HE] C. Pethick, H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2008) R.D. Pisarski, D.H. Rischke, Phys. Rev. D61, 074017 (2000). arXiv:nucl-th/9910056 P.D. Powell, G. Baym, Phys. Rev. D85, 074003 (2012). arXiv:1111.5911 [hep-ph] T. Schäfer, Phys. Rev. Lett. 85, 5531 (2000a). arXiv:nucl-th/0007021 T. Schäfer, Phys. Rev. D62, 094007 (2000b). arXiv:hep-ph/0006034 T. Schäfer, K. Schwenzer, Phys. Rev. D70, 054007 (2004). arXiv:hep-ph/0405053 T. Schäfer, F. Wilczek, Phys. Rev. Lett. 82, 3956 (1999a). arXiv:hep-ph/9811473 T. Schäfer, F. Wilczek, Phys. Rev. D60, 114033 (1999b). arXiv:hep-ph/9906512 A. Schmitt, Phys. Rev. D71, 054016 (2005). arXiv:nucl-th/0412033 A. Schmitt, Q. Wang, D.H. Rischke, Phys. Rev. D66, 114010 (2002). arXiv:nucl-th/0209050 S.L. Shapiro, S.A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (Wiley, New York, 1983) P.S. Shternin, D.G. Yakovlev, C.O. Heinke, W.C.G. Ho, D.J. Patnaude, Mon. Not. R. Astron. Soc. 412, L108 (2011). arXiv:1012.0045 [astro-ph.SR] T.H.R. Skyrme, Proc. R. Soc. Lond. A260, 127 (1961) N. Sogabe, N. Yamamoto, Phys. Rev. D95, 034028 (2017). arXiv:1609.09214 [nucl-th] D.T. Son, Phys. Rev. D59, 094019 (1999). arXiv:hep-ph/9812287 D.T. Son, B.Z. Spivak, Phys. Rev. B88, 104412 (2013). arXiv:1206.1627 [cond-mat.mes-hall] D.T. Son, M.A. Stephanov, Phys. Rev. D61, 074012 (2000). arXiv:hep-ph/9910491 D.T. Son, M.A. Stephanov, Phys. Rev. Lett. 86, 592 (2001a). arXiv:hep-ph/0005225 D.T. Son, M.A. Stephanov, Phys. Atom. Nucl. 64, 834 (2001b). arXiv:hep-ph/0011365 D.T. Son, M.A. Stephanov, Phys. Rev. D77, 014021 (2008). arXiv:0710.1084 [hep-ph] D.T. Son, P. Surówka, Phys. Rev. Lett. 103, 191601 (2009). arXiv:0906.5044 [hep-th] D.T. Son, N. Yamamoto, Phys. Rev. Lett. 109, 181602 (2012). arXiv:1203.2697 [cond-mat.meshall] D.T. Son, N. Yamamoto, Phys. Rev. D87, 085016 (2013). arXiv:1210.8158 [hep-th] D.T. Son, A.R. Zhitnitsky, Phys. Rev. D70, 074018 (2004). arXiv:hep-ph/0405216 M.A. Stephanov, Prog. Theor. Phys. Suppl. 153, 139 (2004). arXiv:hep-ph/0402115 M.A. Stephanov, Y. Yin, Phys. Rev. Lett. 109, 162001 (2012). arXiv:1207.0747 [hep-th] G.C. Strinati, P. Pieri, G. Röpke, P. Schuck, M. Urban, Phys. Rep. 738, 1 (2018). arXiv:1802.05997 [cond-mat.quant-gas] G. Sundaram, Q. Niu, Phys. Rev. B59, 14915 (1999). arXiv:cond-mat/9908003 B. Svistunov, E. Babaev, N. Prokof’ev, Superfluid States of Matter (CRC Press, Boca Raton, 2015) Y. Tokura, N. Kanazawa, Chem. Rev. 121, 2857 (2021). D. Toublan, J.B. Kogut, Phys. Lett. B564, 212 (2003). arXiv:hep-ph/0301183 A. Vilenkin, Phys. Rev. D20, 1807 (1979) A. Vilenkin, Phys. Rev. D22, 3080 (1980) X.G. Wen, Int. J. Mod. Phys. B4, 239 (1990) D. Xiao, J.-R. Shi, Q. Niu, Phys. Rev. Lett. 95, 137204 (2005). arXiv:cond-mat/0502340 D. Xiao, M.-C. Chang, Q. Niu, Rev. Mod. Phys. 82, 1959 (2010). arXiv:0907.2021 [cond-mat.meshall] N. Yamamoto, Phys. Rev. D92, 085011 (2015). arXiv:1502.01547 [cond-mat.mes-hall] N. Yamamoto, Phys. Rev. D93, 085036 (2016). arXiv:1512.05668 [hep-th] N. Yamamoto, M. Tachibana, T. Hatsuda, G. Baym, Phys. Rev. D76, 074001 (2007). arXiv:0704.2654 [hep-ph] W. Zwerger, The BCS–BEC Crossover and the Unitary Fermi Gas. Lecture Notes in Physics, vol. 836 (Springer, Berlin/Heidelberg, 2012)

Hadrons, Quark-Gluon Plasma, and Neutron Stars

85

Akira Ohnishi

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hadrons and Hadron-Hadron Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hadrons with Light Quarks (u, d, s) and Flavor SU(3) Symmetry . . . . . . . . . . . . . . . . . . . Quark-Quark Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hadron Structure and Exotic Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exotic Hadron Production from Heavy-Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hadron-Hadron Interaction Models and Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Femtoscopic Study of Hadron-Hadron Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quark-Gluon Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QCD Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . History and Evidence of QGP Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QCD Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron Stars and Nuclear Matter Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass and Radius of Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Matter EOS and Symmetry Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperon Puzzle and Dense QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

A review is given on hadron physics, quark-gluon plasma (QGP), and neutron stars from a viewpoint of QCD matter, i.e., the nuclear and quark(-gluon) matter. In the hadron physics section, after a short overview of hadrons, the SU(3)f

A. Ohnishi () Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_27

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symmetry, and basic concepts of hadron physics, recent developments on exotic hadrons and hadron-hadron interactions are discussed. In the QGP section, a short overview of the QCD phase transition and the history and evidence of QGP formation is given, followed by discussions on two of the unsolved problems: the early thermalization puzzle and the QCD phase diagram. In the neutron star section, after a review of mass and radius of neutron stars, the symmetry energy and hyperon puzzle, which are the current problems of neutron stars in view of the nuclear matter, are discussed.

Introduction The subjects of hadrons, quark-gluon plasma, and neutron stars are extensively studied recently. In hadron physics, many exotic states have been observed, and their structures are discussed intensively. In high-energy heavy-ion collision experiments in these two decades, it is probable that the quark-gluon plasma is produced, and it is found to have the strongly coupled nature. Neutron stars are attracting much attention at present. In addition to the discovery of massive neutron stars having around 2 M , gravitational wave observation from binary neutron star mergers and the new X-ray telescope provide information on neutron star radii. While these subjects have been studied separately to a large extent, all of them are necessary to elucidate the dense nuclear matter phases and the equation of state. Thus, in this chapter, these subjects are discussed mainly from the nuclear matter point of view.

Hadrons and Hadron-Hadron Interactions Nuclei are made of nucleons (protons and neutrons), and nucleons interact via nuclear force, whose long-range part is described by pion exchange (Yukawa 1935). Color-singlet strongly interacting elementary particles such as nucleons and pions are called hadrons. Hadrons are classified as baryons and mesons, which consist of three quarks (baryons) like nucleons and a quark and an antiquark (mesons) like pions (Gell-Mann 1964). While the elementary particles in QCD are quarks and gluons, they do not appear in vacuum due to the color confinement. Quarks acquire the constituent quark masses due to the spontaneous chiral symmetry breaking. These two nonperturbative effects lead to the appearance of hadrons as composite particles of constituent quarks or as the Nambu-Goldstone bosons of the spontaneous chiral symmetry breaking. In this section, a review is given on some features of hadrons. An emphasis is put on the two recently developing subjects, exotic hadrons and hadron-hadron interactions.

85 Hadrons, Quark-Gluon Plasma, and Neutron Stars

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Hadrons with Light Quarks (u, d, s) and Flavor SU(3) Symmetry Because of the composite particle nature, there are a variety of hadrons. For example, 215 mesons and 168 baryons are listed in the table summarized by the Particle Data Group (Zyla et al. 2020). A part of the variety comes from the quark flavors. Table 1 shows the hadrons in the ground state baryon octet (J π = 1/2+ ) and the ground state pseudoscalar meson octet (J π = 0− ) consisting of u, d and s quarks and their antiparticles. Octet baryons include nucleons N (= p, n) as well as , (=  + ,  0 ,  − ), and  (= 0 , − ) hyperons. Pseudoscalar octet mesons include pions π (= π + , π 0 , π − ), ¯ (= K− , K ¯ 0 ), and η. While the mass differences kaons K (= K+ , K0 ), antikaons K are not very small in these octets, a large part of these differences are explained by the strange-quark mass as will be discussed later. On the Iz -Y plane with Iz and Y = B + S being the z component of the isospin and the hypercharge, respectively, both baryons and mesons are located on the vertex and center of the same hexagon as shown in Fig. 1. This similarity owes to the same SU(3)f transformation of an antiquark q¯ and a flavor antisymmetric diquark D. Three quarks (u, d, and s) have relatively small current quark masses, mu,d  Table 1 Baryon octet and pseudoscalar meson octet. (Masses are taken from Zyla et al. 2020) Mesons

Baryons Mass Iz S Content (MeV) Hadron Q I p +1 1/2 +1/2 0 uud 938.27 n 0 1/2 −1/2 0 udd 939.57 √  0 0 0 −1 uds−dus 1115.68 + 0

+1 1 0 1

− 0 −

−1 1 −1 −1 dds 0 1/2 +1/2 −2 uds −1 1/2 −1/2 −2 dds

+1 0

2

−1 uus 1189.37 √ −1 uds+dus 1192.64 2

1197.45 1314.86 1321.71

Name Q I π+ +1 1

Iz +1

S 0

Content ud¯

π0

0

1

0

0

π− K+ K− K0 ¯0 K

−1 +1 −1 0 0

1 1/2 1/2 1/2 1/2

−1 +1/2 −1/2 +1/2 −1/2

uu−d ¯√ d¯ 2

0 1 1 −1 −1

η

0

0

0

0

Fig. 1 Baryon octet and pseudoscalar meson octet in the (Iz , Y ) plane

Mass (MeV) 139.57 134.98

d u¯ u¯s s u¯ d s¯ s d¯

139.57 493.68 493.68 497.61 497.61

¯ uu+d ¯ √d−2s s¯ 6

543.86

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ΛQCD and ms  ΛQCD , and then the flavor SU(3) symmetry (SU(3)f symmetry) approximately holds. The SU(3)f transformation of a quark (an antiquark) is given as q → q  = U q (q¯ → q¯  = qU ¯ † ) with q = (u, d, s)T and U ∈ SU(3). For an SU(3) matrix U , the relation εij k Ujj  Ukk  = εi  j  k  Ui† i tells that a flavor antisymmetric diquark Di = εij k qj qk is transformed as an antiquark, D → D  = DU † . Thus, the meson matrix Mij = q¯j qi and the baryon matrix Bij = Dj qi are transformed in the same way. ⎛ 1 M=√ 2

8  a=1

π0 √ ⎜ 2

Mi λ i = ⎜ ⎝ ⎛

1 B=√ 2

8  a=1

0 √ ⎜ 2

Bi λ i = ⎜ ⎝

+ π− K−

+ − −

π+

η8 √ 6

 √ 6

0

π −√ + 2 ¯ K0

η8 √ 6

+ 0

 −√ + 2

0

6

⎞

p

 √ 6

⎞

⎛ uu ¯ ⎟ ⎜ 0 ⎟ ∼ K ⎠ ⎝ud ¯ 2η 8 √ us ¯ − K+

⎛

⎞ ¯ s¯ u du ¯ s¯ d ⎟ dd ⎠, ¯ s¯ s ds

[ds]u [su]u [ud]u

(1)

⎞

⎟ ⎜ ⎟ n ⎟ ⎠ ∼ ⎝[ds]d [su]d [ud]d ⎠ , 2 [ds]s [su]s [ud]s −√ 6

(2) M → M  = U MU † , B → B  = U BU † ,

(3)

where λi is a Gell-Mann matrix and [qi qj ] = qi qj − qj qi = εij k Dk is a flavor antisymmetric diquark, e.g., [ud] = ud − du = Ds . (Here γ matrices are omitted for simplicity.) The meson matrix is transformed as M → M  = U MU † , where the matrix element is given as Mij = q¯j qi . Among the nine matrix elements of a 3 × 3 matrix, the trace is invariant under the transformation (flavor singlet), and the other eight components form the meson octet. In reality, the SU(3)f symmetry is not exact, so the flavor singlet meson (η0 ) and the octet meson (η8 ) mix, and η and η appear as the energy eigenstates after diagonalization. It should be also noted that the instanton-induced interaction (also known as the Kobayashi-Maskawa-’t Hooft interaction (Kobayashi and Maskawa 1970)) from the U(1)A anomaly works repulsively for η0 . Octet baryons consist of a flavor antisymmetric diquark (D) and a quark (q), and the baryon matrix is transformed as that of mesons, B → B  = U BU † (Bij = Dj qi = εj kl qk ql qi ). The octet baryon matrix is traceless. The trace of the above matrix elements reads Bii = εj kl qk ql qj , which is totally antisymmetric in flavor. Since the quark colors need to be totally antisymmetric and three quark spins cannot be totally antisymmetric, there is no totally flavor antisymmetric (flavor singlet) baryons in the low-lying s-wave states. Hadron masses in the same SU(3)f multiplet are the same if the quark masses are the same, but the mass difference of the strange-quark s and the u, d quarks causes the mass splitting among the multiplets. Gell-Mann and Okubo obtained the mass formula in a multiplet, called the Gell-Mann-Okubo (GMO) mass formula (GellMann 1961). For baryon masses, the GMO formula reads

85 Hadrons, Quark-Gluon Plasma, and Neutron Stars



MB = a0 + a1 Y + a2 I (I + 1) − Y 2 /4 ,

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(4)

where a0 , a1 , and a2 are the parameters depending on the multiplet and the isospin breaking is ignored. This formula is obtained by assuming that the SU(3) f ¯ 8 , B] , and symmetry breaking terms in the Lagrangian is caused by λ8 , Tr B[λ  ¯ 8 , B} . Equation (4) gives MN = a0 +a1 +a2 /2, M = a0 , M = a0 +2a2 , Tr B{λ and M = a0 − a1 + a2 /2. These masses reproduce the data at the precision of 1 % or better. The GMO formula implies the relation (MN + M )/2 = (3M + M )/4, which is also well satisfied. For decuplet baryons which include (1232)(I = 3/2, S = 0),  ∗ (1385)(I = 1, S = −1), ∗ (1530)(I = 1/2, S = −2), and (1672)(I = 0, S = −3), the GMO formula predicts equal spacing, M10 = a0 + 2a2 + (a1 + 3a2 /2)Y . By using this relation, the baryon was predicted to exist by Gell-Mann and was discovered later in the experiment (Barnes et al. 1964). For mesons, squared mass appears in the Lagrangian, so the GMO formula 2 = a +a Y +a I (I + 1) − Y 2 /4 . For octet mesons, should be understood as MM 0 1 2 a1 = 0 because the masses of K and K¯ are the same, so one finds Mπ2 = a0 + 2a2 , MK2 = a0 + a2 /2, and Mη28 = a0 . Thus, one can predict the η8 mass from the masses of pions and kaons as Mη28 = (4MK2 − Mπ2 )/3, which gives a clue to calculate the flavor octet-singlet η8 -η0 mixing.

Quark-Quark Interactions Masses of hadrons should be understood as a consequence of quark-gluon dynamics. Since quarks (and antiquarks) are the main constituents of most hadrons, it is necessary to know quark-quark interactions which are obtained after integrating out gluon degrees of freedom. There are several origins of the quark-quark interaction, such as confinement, gluon exchange, and the chiral and U(1)A symmetries. Confinement potential Confinement potential between a quark and an antiquark is well studied using quarkonia. A quarkonium is a bound state of a heavy ¯ The quarkonium spectra tell us the details quark Q and a heavy antiquark Q. of the confinement potential, since the nonrelativistic approximation and quantum mechanical treatment are expected to work for heavy quarks. The most famous one is the Cornell potential (Eichten et al. 1975): V Cornell (r) = −

4 αs (r) + σ r. 3 r

(5)

The first term is a Coulomb potential from the one-gluon exchange as described below, and the r dependence of αs = g 2 /4π is taken into account in describing the quarkonium spectrum precisely. The confinement potential, the second linear term in r, implies the string or flux-tube picture of quarkonia and requires nonperturbative methods to understand. The string tension σ takes a value of around 1 GeV/fm. The

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¯ potential from the Wilson loop Cornell potential (5) has been examined in the QQ

W (r, τ ) ∝ exp [−V (r)τ ] from the lattice QCD calculations (Bali et al. 1995). In addition to heavy-quark systems, the flux-tube picture and the string tension around σ ∼ 1 GeV/fm seem to be valid also for hadrons including light quarks from the Regge trajectories. For a series of hadrons with the angular momentum interval of J = 2 but with the same quantum numbers other than J , squared masses show the linear dependence on the angular momentum J (Regge trajectory), J = α0 + α  MJ2 with α  ∼ 1 GeV−2 (Chew and Frautschi 1962). This relation suggests the string or flux-tube picture of hadrons. Let us consider the simple cases where the string, with a length  and a string tension of σ , is rotating with the speed of light at the end points. Then the mass and the angular momentum of this string are obtained as 

/2

MJ = 2 0

σ dr π  = σ , J = 2 2 2 1 − v (r)



/2 0

r×

σ dr 1 − v 2 (r)

v(r) =

π 2 σ , 8 (6)

where v(r) = r/(/2). These mass and angular momentum of a string have the relation of J = MJ2 /(2π σ ). Thus, the Regge trajectories of hadrons suggest the string tension of σ = α  /2π ∼ 0.8 GeV/fm, which is comparable to that from the analyses of quarkonium. Since the Regge trajectories are found also in baryons, one can expect that the confinement potential has a linear dependence on the sum of Y-type flux-tube lengths as demonstrated by the lattice QCD calculation (Takahashi et al. 2002). One-Gluon Exchange One of the important quark-quark interactions is the onegluon exchange (OGE) potential (De Rujula et al. 1975): VijOGE

    4σi · σj αs (i) (j ) 1 π 1 1 (λ · λ ) = − δ(r ij ) + 2+ + ··· , 4 rij 2 3Mi Mj Mi2 Mj (7)

Nc2 −1 (i) (j ) λa λa is the color factor and the · · · part contains where (λ(i) · λ(j ) ) ≡ a=1 momentum dependent terms, LS force terms, and tensor force terms. The color factor is found to be −16/3 for a color-singlet q q¯ pair and −8/3 for a colorantitriplet qq pair in a  baryon. The color factor can be calculated from the color-singlet condition, ( ni=1 λ(i) )2 = n(λ(1) )2 + n(n − 1)(λ(1) · λ(2) ) = 0 with n being the number of constituents and the normalization of the Gell-Mann matrix, Tr λ2a = 2, for each a. Thus, the first term proportional to 1/rij is nothing but the ¯ systems given in Eq. (5). It is Coulomb part of the confinement potential in QQ important to find that the spin-spin part of the OGE potential, sometimes called the color-magnetic interaction, is attractive for spin zero pairs (J = 0, σi · σj = −3) and repulsive for spin one pairs (J = 1, σi · σj = 1). Thus, by assuming s-wave spatial wave functions, the spin-spin interaction is attractive for pseudoscalar

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mesons J π = 0− and is repulsive for vector mesons J π = 1− . For quark pairs antisymmetric in color in a baryon, the spin-spin interaction is attractive for pairs antisymmetric in spin (J = 0) and then also in flavor and is repulsive for pairs symmetric in spin (J = 1) and then also in flavor. This feature explains the reason why the masses of decuplet baryons, which contain spin one quark pairs, are larger than those in the octet baryons. Spontaneous broken chiral symmetry Together with the color confinement, the spontaneous chiral symmetry breaking is one of the important nonperturbative aspects of QCD. The QCD Lagrangian is given as LQCD = q(iγ ¯ μ Dμ − m) −

1 Tr Gμν Gμν 2

= q¯L (iγ μ Dμ )qL + q¯R (iγ μ Dμ )qR − q¯L mqR − q¯R mqL −

1 Tr Gμν Gμν , 2 (8)

where Dμ = ∂μ + igAμ , Aμ = Aaμ Ta , Gμν = [Dμ , Dν ]/ig = ∂μ Aν − ∂ν Aμ + ig [Aμ , Aν ]. The QCD Lagrangian is invariant under the chiral transformation,  qL,R → qL,R = UL,R qL,R , with qL,R = (1 ∓ γ5 )q/2 and UL,R ∈ SU(Nf ), when quarks are massless. The chiral symmetry is relevant also to hadron spectra. In the case where the chiral transformation with Nf = 2 is given as UL,R = ¯ and θˆ a πa (πa ≡ qiγ ¯ 5 τa q) are transformed as σ → σ  = exp (∓iθ a τa /2), σ ≡ qq a a a  ˆ ˆ ˆ σ cos θ + θ πa sin θ and θ πa → θ πa = −σ sin θ + θˆ a πa cos θ (θ = |θ|, θˆ a = θ a /θ ), while the components of π perpendicular to θ are unchanged. Since σ and πa mix in the chiral transformation, there should be a scalar meson having the same mass as those of pions. However, there is no such a scalar meson. The idea to respect both the chiral symmetry of the QCD Lagrangian and the observed hadron spectra is to introduce the spontaneous chiral symmetry breaking, which is proposed by Nambu and Jona-Lasinio (NJL) (Nambu and Jona-Lasinio 1961). Let us explain how the chiral symmetry is spontaneously broken in the NJL model (Nf = 2) (Nambu and Jona-Lasinio 1961; Goldstone 1962; Hatsuda and Kunihiro 1994), where quarks interact through the chiral invariant four-fermion interaction term, (qq) ¯ 2 + (qiγ ¯ 5 τa q)2 : LNJL = q(iγ ¯ μ ∂μ − m)q +

G2 (qq) ¯ 2 + (qiγ ¯ 5 τa q)2 , 2 2Λ

(9)

respects the chiral symmetry. By using the Wick rotation and the HubbardStratonovich transformation, the Lagrangian in the Euclidean spacetime with auxiliary fields (σ, π ) is given as

 Λ2 2 LE,AF = q¯ −iγμ ∂μ + m + G(σ + iγ5 τ · π) q + σ + π2 , 2

(10)

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where μ runs over (x, y, z, τ ) and γτ = iγ0 . In the mean field approximation, σ = constant and πa = 0, one can calculate the free energy density Feff by using the summation over the Matsubara frequencies, and the result is found to be Feff = −

df /2 T Λ2 2 T  log Zβ = σ − log ωn2 + Ek2 V 2 V n,k

Λ2 2 σ − df = 2



Λ

d 3k (2π )3



 Ek k2 1 + , 2 3Ek eEk /T + 1

(11)

where df = 4Nc Nf denotes the number of fermion degrees of freedom, Ek = √ k 2 + M 2 with M = m+Gσ , Λ is the three-momentum cutoff, and ωn = π T (2n+   1) is the Matsubara frequency. A formula n log ωn2 + E 2 = 2 log[cosh(E/T )] is used from the first to the second line. The first term in the integral shows the fermion zero point energy, and the second term shows the pressure from particles whose distribution is given by the Fermi-Dirac distribution function. The spontaneous symmetry breaking is understood from the shape of Feff as a function of σ . In the chiral limit in the vacuum, (m, T ) = (0, 0), the free energy density is found to be a function of x = Gσ/Λ:   vac Feff df df x 2 G2c df x2 = − + − 1 + O(x 4 log x), (12) I (x) = − 2 Λ4 2G2 16π 2 16π 2 G2   √  1  2 1 + x 4π k 2 dk  2 1 1+ I (x) = x + k2 = 1 + x 2 (2 + x 2 )−x 4 log |x| (2π )3 16π 2 0     |x| 1 x4 2 6 = 2 1+x + 1 + 4 log + O(x ) , 8 4 8π

(13)

where G2c = 8π 2 /df . When the coupling is strong enough, G > Gc , the free energy density shows the double well structure as a function of σ , which takes a finite value σ = σvac in the energy minimum state, i.e., the √ physical vacuum. With the π degrees of freedom, Feff is given as a function of σ 2 + π 2 and shows the wine bottle shape on the (σ, πa ) plane, since √ the free energy is chiral invariant and it is always possible to rotate (σ, π ) to ( σ 2 + π 2 , 0). With finite but small quark masses, the wine bottle is slightly tilted in the σ direction, and the vacuum is uniquely identified. Thus, the spontaneous chiral symmetry breaking takes place. As a result of the spontaneous chiral symmetry breaking, the chiral condensate

qq ¯ = −Λ2 σ/G also takes a finite value, and quarks acquire constituent mass, M = Gσvac , in vacuum. The free energy is flat in the π direction around the vacuum, and the pion mass is zero in the chiral limit. Therefore, pion is a Nambu-Goldstone (NG) boson (Goldstone 1962), which appears as a massless particle associated with the spontaneous symmetry breaking.

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Chiral quark model Once the chiral symmetry is spontaneously broken, massless NG bosons appear and will dominate the low-energy phenomena in the chiral limit. The chiral effective field theory (chiral EFT) (Weinberg 1979) is based on the symmetry of QCD, the chiral symmetry, and the deviation from the chiral and low-energy limit can be systematically evaluated. Then it can be regarded as a theory based on first principles. Let us consider the expectation value of q¯Rj qLi in vacuum, which is a matrix of flavor indices, i and j . When all the quarks are massless (chiral limit), the chiral condensates are the same for different flavors, ¯ = ¯s s, and the pseudoscalar condensate should be zero in vacuum,

uu ¯ = dd and then one finds 0|q¯Rj qLi |0 = −Λ3 δij in vacuum, where Λ has dimensions of mass and is finite after the spontaneous breaking of chiral symmetry. In the chiral limit, one can redefine the quark field as qL → LqL and q¯R → q¯R R † , and the above δij is replaced by a matrix, (LR † )ij ≡ Σij . The unitary matrix Σ can be parameterized as Σ(x) = exp (2iπ(x)/f ), π(x) = πa (x)Ta ,

(14)

where f is a constant and Ta denotes the SU(Nf ) generator normalized as Tr(Ta Tb ) = δab /2. When π(x) is constant, it connects different vacua and then π(x) should represent the pseudoscalar NG bosons, corresponding to long-wavelength spacetime-dependent rotations of the condensate. Since only NG bosons are relevant at very low energies in the chiral limit, the low-energy effective Lagrangian is given as a function Σ(x) as Lchiral limit =

 1 f2  Tr ∂μ Σ † ∂ μ Σ = ∂μ πa ∂ μ πa + O((πa /f )4 ). 4 2

(15)

Finite quark masses are taken into account in the form of the external field (Gasser and Leutwyler 1984), L2 =

 f2  Tr ∂μ Σ † ∂ μ Σ + χ † Σ + χ Σ † , 4

χ = 2B(s + ip),

(16)

where the external field χ is assumed to transform as χ → Lχ R † . There are two low-energy constants (LECs), the pion decay constant f and the quark condensate B in the chiral limit, which are given in the leading order in mq as f = fπ = 93 MeV and 0 | uu ¯ | 0 = −f 2 B. With s = Mq = diag(mu , md , ms ), the pseudoscalar meson masses to the leading order in mq are found to be 2 2 ˆ MK Mπ2 + = Mπ2 0 = 2mB, + = (mu + ms )B, MK 0 = (md + ms )B,

Mη28 =

2 1 (m ˆ + 2ms )B, m ˆ ≡ (mu + md ). 3 2

(17)

The Lagrangian Eq. (16) is O(p2 ) in the chiral counting rule, where Σ = O(p0 ), Dμ Σ = O(p1 ), and s, p = O(p2 ). The third counting rule takes account of

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the fact that the quark masses are proportional to the NG boson mass squared. In including the higher-order terms in the chiral expansion, one writes down the most general effective Lagrangian containing those terms to a given order, and the corresponding coefficients, LECs, are determined by using the experimental data. For more discussions and recent developments in the chiral EFT, readers are referred to Epelbaum et al. (2009), Machleidt and Entem (2011), Haidenbauer et al. (2013), and Oller et al. (2000). For the discussion of quark-quark interactions, it is useful to consider the chiral quark model (Manohar and Georgi 1984), which can be regarded as a quark-meson version of the chiral EFT. One can introduce a new field ξ , defined as Σ = ξ ξ and then ξ = eiπ/f , which transforms as ξ → Lξ U † (x) = U (x)ξ R † . The matrix U is an implicit function of (L, R) and the π fields. A set of color triplet Dirac fermions ψ transforming as ψ → U ψ are defined by their left and right components by using ξ and the quark fields as ψL = ξ † qL and ψR = ξ qR (Ecker 1995), which transform as ψL → U ξ † L† LqL = U ψL and ψR → U ξ R † RqR = U ψR . Then the following effective Lagrangian is invariant under the chiral transformation: ¯ μ γ5 Aμ ψ ¯ + gA ψγ ¯ μ (iDμ + Vμ )ψ − mψψ L = ψγ f2 Tr ∂μ Σ † ∂ μ Σ − 4  1 † ξ ∂ μ ξ + ξ ∂μ ξ † , Vμ = 2 +

1 Tr Gμν Gμν , 2  i  † ξ ∂ μ ξ − ξ ∂μ ξ † , Aμ = 2

(18) (19)

where Dμ = ∂μ + igGμ is the covariant derivative in QCD, and the vector and axial vector currents, Vμ and Aμ , transform as Vμ → U Vμ U † + U ∂μ U † and Aμ → U Aμ U † . Hence the dressed quarks ψ can exchange NG bosons through Vμ and Aμ ¯ = m(q¯L ΣqR + q¯R Σ † qL ). and can have masses, mψψ

Hadron Structure and Exotic Hadrons Most of the hadrons are made of q q¯ pairs (mesons) and qqq (baryons), and low-lying baryons consist of a quark and a diquark, as explained in the previous subsections. Between q and q¯ (or qq), the confinement potential operates by forming flux-tubes between the constituents, and other quark-quark interactions modify the hadron masses according to the quantum numbers. Since the spontaneous chiral symmetry breaking generates the constituent quark masses and nonrelativistic treatments can work (Manohar and Georgi 1984), one can investigate the hadron structure quantum mechanically as long as the number of constituents (quarks and antiquarks) is kept. Actually the quantum mechanical models of hadron structure, the quark models of hadrons (Isgur and Karl 1977), have been successful in describing low-lying hadron spectra. In quark models, use the quadratic confinement potential between quarks, and the one-gluon exchange potential is taken into account, and relativistic effects are also considered (Isgur and Karl 1977). It should

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Fig. 2 Expected structure of exotic hadrons

be noted that ground state hadron masses and the static quark potential are well explained by the lattice QCD calculations (Aoki et al. 2009). During the last decades, many exotic hadron candidates, hadronic states deviating from the quark model predictions, have been observed. These findings started from the discovery of Ds (2317) (Aubert et al. 2003) and X(3872) (Choi et al. 2003) and continued on to the discovery of Pc (4380) (Aaij et al. 2015) and Tcc (Aaij et al. 2022). While a ground state hadron for a given set of quantum numbers can be excited in spin, isospin, angular momentum, and radial coordinate degrees of freedom (normal excitation), the energy required to produce a q q(q ¯ = u, d) pair is not very different from those in normal excitation. Thus, one can expect the existence of multiquark hadrons as schematically shown in Fig. 2. For example, Ds (2317) and X(3872) are considered to contain additional q q¯ pair in addition to the minimum configurations, such as c¯s q q[D ¯ s (2317)] and ccq ¯ q[X(3872)], ¯ since they are found to decay into Ds (2317) → Ds+ π 0 and X(3872) → π + π − J /ψ. Tcc is a doubly charmed meson, and the minimum configuration is already a tetraquark state, ud c¯c, ¯ a manifestly exotic (flavor exotic) hadron candidate. The minimum configurations is uud for Pc (4380) from the quantum numbers, but the mass is too heavy not to include cc¯ pair and Pc (4380) decays into J /ψp. Then Pc (4380) is considered to be a cryptoexotic but an almost manifestly exotic hadron candidate with quark contents of ccuud. ¯ Multiquark hadrons A flavor antisymmetric diquark (D) is a plausible constituent of a baryon as already mentioned in the previous subsection. A diquark which is antisymmetric in spin, color, and flavor and has positive parity (J π = 0+ , 3¯ c , 3¯ f ) is called “a good diquark” (Selem and Wilczek 2005), and can be a constituent of multiquark hadrons (Jaffe 1977). One of the examples is the scalar meson nonet (J π = 0+ ). In order to form a scalar meson by q q, ¯ p-wave relative momentum ¯ can form a scalar is necessary. By comparison, a diquark-antidiquark pair (D D) meson in s-wave. If this is the case, the σ meson (f0 (500)) is made of the

1400 π Pseudoscalar (J =0 ) 1200 η’ 1000 800 − η 600 K, K 400 π 200 0 1 0 -1 Iz

A. Ohnishi

Mass (MeV)

Mass (MeV)

3078 1400 1200 1000 800 600 400 200 0

f0(980)

a0 κ, − κ

σ Scalar (Jπ=0+)

-1

0 Iz

1

Fig. 3 Left: Pseudoscalar and scalar meson nonet. Right: Masses of quarks, antiquarks, and diquarks

¯ the κ meson (K ∗ (700)) with [ud]-[¯s q], diquark-antidiquark pair ([ud]-[u¯ d]), ¯ and 0 a0 and f0 (980) mesons with [sq]-[¯s q]. ¯ This choice explains the reason why the pseudoscalar mesons form a triangle in the (Iz , M) plane while scalar mesons form a reversed triangle (Jaffe 1977) as shown in Fig. 3. There is only one light diquark ([ud]) and two heavier diquarks ([su], [ds]) consisting of u, d and s quarks, so one light scalar meson and four heavier scalar mesons are expected to appear. It should be remembered that these diquark-antidiquark configurations mix with the naïve q q¯ configurations and two pseudomeson states. The KMT interaction also causes the effects to reduce the mass of the σ meson. Several other hadrons are predicted to contain diquarks. One of the examples is the doubly charmed tetraquark state, Tcc , first proposed in Zouzou et al. (1986) and recently found by the LHCb collaboration (Aaij et al. 2022). With the quark content ¯ u¯ and d¯ tend to form a good antidiquark [u¯ d], ¯ and the remaining cc will of ccu¯ d, π + make a spin symmetric pair, e.g., J = 1 , to be color antitriplet. The energy gain ¯ is larger than the excitation energy in cc, since the color-magnetic interaction in [u¯ d] is inversely proportional to the quark mass product, 1/mi mj . The tetraquark state with J π = 1+ cannot decay into DD due to the angular momentum conservation, and the width should be small as observed. The mass of Tcc was predicted by using the color-magnetic interaction as MTcc = 3786 MeV (Cui et al. 2007), about 100 MeV below the DD ∗ thresholds, and MTcc = 3882.2 MeV (Karliner and Rosner 2017), around the DD ∗ thresholds. The former (latter) is obtained with the parameter determined by fitting the X(3872) (cc ) mass. The data show the Tcc mass is obtained using the Breit-Wigner fit as MTcc = MD ∗+ MD 0 − (273 ± 61) keV = 3874.83 MeV, slightly below the D ∗+ D 0 threshold. The theoretical uncertainty of ∼100 MeV in mass implies that further data are needed to determine the Hamiltonian for the constituent quarks and diquarks. Multiquark hadrons containing diquarks are expected to be compact, since the confinement potential operates to keep colored objects close. This also applies to other exotic hadrons including gluons as constituents such as glueballs (gg) and gluon-q q¯ states.

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Hadronic molecules A part of the exotic hadron candidates around the threshold are expected to be hadronic molecules (Guo et al. 2018). A significant part of the exotic hadron candidates are found around the hadron-hadron thresholds, where the wavelength in the corresponding channel is long and molecule-like bound or resonance states tend to appear if the interaction is attractive. This is analogous to the clustering structure, which appears around the thresholds in atomic nuclei (Ikeda et al. 1968). A hadronic-molecule state has a larger size than normal hadrons and multiquarks, since the confinement potential does not operate among the color-singlet constituent hadrons. The size depends on the binding energy from the corresponding threshold. For a two-body s-wave pure hadronic molecule, the asymptotic wave function is √ given as φ(r) ∝ exp (−r/R)/r (R = 1/ 2μB with μ and B being the reduced mass and the binding energy). Hadronic-molecule and multiquark components generally mix, and the probability of finding the hadronic-molecule state, compositeness, is 0 ≤ X ≤ 1. The famous example is the deuteron as first discussed by Weinberg (Weinberg et al. 1965) and extended later (Hyodo et al. 2012). The relation between the scattering length (a0 ) and the binding energy (B) is found to be related with the compositeness X as  2X a0 = R + O(m−1 π ). 1+X 

(20)

From the deuteron binding energy (B = 2.2 MeV), the size parameter is found to be R = 4.3 fm, which is comparable with the 3 S1 (I = 0, J = 1) pn scattering length (a0 = 5.4 fm). Thus, the compositeness of a deuteron is close to unity, and then a deuteron is found to be a composite particle of a proton and a neutron rather than a six quark state. Enhancement from kinematic effects Some of the observed invariant mass peaks may be generated also by kinematical effects such as the threshold cusps or the triangle singularities (Guo et al. 2020). Cusps appear at thresholds. Since the slope of the scattering amplitude with respect to the energy changes discontinuously at channel thresholds, the invariant mass spectrum of the lower-energy channel shows the cusp behavior where the slope of the line shape changes discontinuously. The triangle singularity arises from triangle diagrams, when all the three lines can take the on-shell condition simultaneously (Szczepaniak 2015). Readers are referred to Guo et al. (2020) for more discussions. At present, understanding the structure of exotic hadron candidates and categorizing those into compact multiquark states (including glueballs and quark-gluon hybrids), hadronic molecules, and peaks from kinematical effects together with predicting and observing new exotic hadron states are the challenges in hadron physics.

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Exotic Hadron Production from Heavy-Ion Collisions In order to categorize the physical exotic hadron states, the size of the candidate would be the key quantity. Here multiquark states and hadronic molecules are considered. As already mentioned, multiquark states will be compact, while hadronic molecules will have sizes depending on the binding energies which are generally larger than normal and multiquark hadrons. The size of a hadronic molecule can be evaluated if the interaction between the constituent hadrons is known or the form factor is measured, but most of the hadrons are unstable and scattering experiments are difficult. In deducing the size of exotic hadrons, production yields from heavy-ion collisions would be helpful. It was proposed that the production yields of hadronic molecules (compact multiquarks) are comparable with (smaller than) those of normal hadrons, provided that the coalescence model is the underlying mechanism of hadron production (Cho et al. 2011). It is well known that the production yields of ground state normal hadrons from high-energy heavy-ion collision are explained by the statistical model (Becattini et al. 2006; Andronic et al. 2018). As reviewed in the next section, hadron gas with a large volume is created in the final stage of high-energy heavy-ion collisions, and abundant hadrons are produced simultaneously and nearly statistically. Thus, highenergy heavy-ion collisions can play the role of hadron factories. The production yield in the statistical model is given as Nhstat

gh VH = (2π )3



4πp2 dp γh−1 exp (Eh /TH ) ± 1

(21)

, nb +n ¯

b where gh is the degeneracy of the hadron h and γh = γcnc +nc¯ γb exp [(μB B + μs S)/TH ] represents the fugacity for a hadron with the baryon number B, strangeness S and containing nc (nc¯ ) (anti-)c quarks and nb (nb¯ ) (anti-)b quarks. Hadrons are assumed to be produced statistically at the volume VH and temperature TH . Since charm and bottom quarks are mostly produced in the initial hard scatterings, their numbers are much larger than the chemical equilibrium values as represented by γc and γb (γc,b > 1). The coalescence model may provide the underlying mechanism of the statistical production of hadrons (Sato and Yazaki 2003). While the statistical model meets successes in explaining ground state s-wave hadron production yields, it does not necessarily explain the excited hadron yields. For example, the statistical model overestimates the yield of (1520)(J π = 3/2− ), which requires at least one quark to be excited to the p-wave state in the quark model. By taking the intrinsic wave function information into account, the observed (1520) yield can be understood in the coalescence model (Kanada-En’yo and Muller 2006). The hadron production yield in the coalescence model is given as

Nhcoal = gh

   1 pi · dσi d 3 pi f (x , p ) f W (x1 , · · · , xn ; p1 , · · · , pn ), i i i gi (2π )3 Ei i

(22)

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where xi , pi , dσi , gi , and fi are the spacetime coordinate, the four momentum, the space-like hypersurface element, the degeneracy, and the phase space distribution function of the ith constituent, respectively, and f W is the Wigner function of the produced hadron. With several assumptions such as the Bjorken expansion of constituents in the longitudinal direction, thermal distribution in the transverse directions, and nonrelativistic harmonic oscillator wave functions of the constituents, the hadron production yields in the coalescence model are approximately obtained as Nhcoal 

n n−1  Nj (4π )3/2  (2li )!! (2T /ω)L gh V (Mω)2/3 , (2li + 1)!! (4π )3/2 (1 + 2T /ω)n+L−1 gj V (mj ω)3/2 j =1

i=1

(23) where li is the orbital angular momentum of the ith Jacobi coordinate, L =  n−1 j are the yields, degeneracy, and mass, respectively, of the i=1 li ; Nj , gj , and m  j th constituents; M = ni=1 mi ; and ω denotes the harmonic oscillator frequency. Coalescence is assumed to take place at the volume V and temperature T . When the number of constituents n is large, additional factor appears and generally suppresses the production yield. For each orbital angular momentum li , there appears a suppression (penalty) factor of [(2T /ω)/(1 + 2T /ω)]li × (2li )!!/[(2li + 1)!!]. In the case of loosely bound hadronic molecules, the frequency ω is small, the number of constituents is not large, and then the penalty becomes less severe. The left panel of Fig. 4 shows the ratio of yields in the coalescence and statistical models, RhCS = Nhcoal /Nhstat . Parameters are taken from Cho et al. (2011), while the binding energy of Tcc is updated by the recent data (Aaij et al. 2022). The ratios for normal hadrons are in the normal band, 0.2 ≤ RhCS ≤ 2, which supports the idea that the coalescence can be the underlying mechanism of statistical production of hadrons. The ratios for hadronic molecules are also in the normal band, while the production of compact multiquark states is generally suppressed in the coalescence model. A closer look suggests that the ratios of very loosely bound hadronic molecules, for example, Tcc and X(3872), are around RhCS  2. The enhancement of loosely bound hadronic-molecule production yield from coalescence may explain the enhancement of X(3872) in heavy-ion collisions. Very recently, CMS collaboration found that the ratio of production yields of X(3872) and ψ(2S) is small in pp collisions, RX/ψ (pp) = 0.1, but large in Pb+Pb collisions, RX/ψ (AA) = 1.08 (Sirunyan et al. 2022). The right panel of Fig. 4 shows the invariant mass spectra. The prompt production yields of ψ(2S) and X(3872) can be evaluated by subtracting the yields from the b-quark decay. The production yield ratio of exotic and normal hadrons may be a better quantity than RhCS , since it is measurable and can give an estimate of RhCS by assuming that the normal hadron yield is consistent with the statistical model estimate, CS ∼ R Rex ex/n × exp {[M(ex) − M(n)]/T }, where ex and n denote exotic and normal CS hadrons. In the present example, RX(3872) (AA) ∼ 3 from RX/ψ , which is even larger CS than the estimate RX(3872)  1.6 based on Cho et al. (2011). A part of this difference may come from the melting of ψ(2S) in Pb+Pb collisions. A transport model

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1

2

3 Mass (GeV)

4

Pc+(4450)

+

Pc (4380)

1

Tcc

+

Θ+

0

Zc (3900)

H++ c

Θcs

−2

2021 update

5

6

50

15 < p < 50 GeV/c T Cent. 0-90% |y| < 1.6

35

3.65

3

ψ (2S)

2

150

1 0

100

30

−1

3.7

3.75

σX(3872) = 4.7 MeV/c2

3.8

3.85

b-enriched (l xy > 0.1 m mm) (GeV/c2) J/ ψ ππ

25

3.9

−2 −3 3.95

data total fit background

20 15 10 5 0

4 3

2 1 0 −1 −2 −3

Pull

10

ΔΔ

10−1

X(3872)

Entries / (5 MeV/c2)

100

CMS Inclusive

200

Entries / (5 MeV/c2)

1

Z+(4430)

Tcc ΛcNN

N

ΛcN ΩΩ DbarNN X(3872)

N

*bar

D

NN

bar

bar

D

K

NΩ

Ds(2317)

H

KN bar

Λ(1405)

Deuteron

K

f0,a0

1

1.7 nb-1 (PbPb 5.02 TeV)

Normal bar qq /qqq multiquark Hadronic

Pull

coal stat RCS /N h =N

10

K(1460)

LHC (5.02 TeV, Scenario 1)

3.65

3.7

3.75

3.8

3.85

mJ/ ψππ (GeV/c2)

3.9

3.95

4

Fig. 4 Left: Hadron production yield ratio of the coalescence model and the statistical model (Cho et al. 2011) updated with Tcc mass. Right: Invariant mass distribution of mμμπ π in Pb+Pb collisions, for the inclusive (upper) and b-enriched (lower) samples. (Taken from Sirunyan et al. 2022)

calculation using coalescence to form hadrons also predicts enhanced production of hadronic molecules compared to the tetraquarks, while another model using statistical formation predicts suppressed hadronic molecules (Zhang et al. 2004). These results support the hadronic-molecule nature of X(3872) and the coalescence formation of hadrons.

Hadron-Hadron Interaction Models and Theories Interactions among hadrons are the basic inputs to study many-body systems consisting of hadrons such as nuclei including hypernuclei and mesic nuclei, nuclear matter including neutron star matter, and hadronic molecules. Nucleon-nucleon (NN ) interactions have been clarified by using the rich NN scattering data. By comparison, hadron-hadron (hh) scattering data other than NN are limited, since most of the hadrons are unstable and cannot be used as the target. Therefore, there have been a lot of hh interaction models proposed so far. For example, in the case of the hyperon-nucleon (Y N) interactions, boson exchange models (Rijken et al. 1999; Haidenbauer and Meissner 2005) and quark cluster models (Oka et al. 1988; Takeuchi et al. 2000) provide basic pictures of baryon-baryon (BB) interactions. Recently, hh interactions from the first principles, the chiral effective field theory (Haidenbauer et al. 2013) and the lattice QCD (Ishii et al. 2007, 2012; Inoue et al. 2011; Gongyo et al. 2018; Sasaki et al. 2020), have been developed and are successfully applied to hadron many-body problems such as hypernuclei (Gal et al. 2016). In the following, some of the interaction models and theories, which have derived Y N interactions in addition to those of NN, are reviewed.

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Boson exchange models In boson exchange models, the one pion exchange potential (OPEP) or other pseudoscalar Nambu-Goldstone boson (kaons and η meson) exchange dominates the outer region, e.g., r ≥ 2 fm for OPEP. The baryon-baryonmeson (BBM) coupling constants are obtained by the SU(3)f symmetry. From the matrix representation of mesons and baryons, Eqs. (1) and (2), the SU(3)f invariant pseudoscalar meson-baryon coupling terms can be written as   √   √ ¯ 5 {M, B} − 2gF Tr Biγ ¯ 5 [M, B] , LBBM = − 2gD Tr Biγ

(24)

where [A, B] = AB − BA and {A, B} = AB + BA. The 12 pseudoscalar BBM coupling constants are given only by the 2 parameters, g = (gD + gF ) = gN π and the F /(F + D) ratio α = gF /(gD + gF ) (de Swart 1965), √ gN π = g, gπ = 2gα, gπ = 2g(1 − α)/ 3, gπ = −gN K = −g(1 − 2α), √ √ gN K = gη = −g(1 + 2α)/ 3, gK =gN η = −g(1 − 4α)/ 3, gK =−g, √ gη = −gη = −2g(1 − α)/ 3. (25) The spin-flavor SU(6) symmetric value of the F /(F +D) ratio, α = 2/5, together with the NN π coupling constant (gN π = g) determined from NN scattering data can be used as a reasonable starting point. For the middle range, scalar meson exchange and/or the meson pair exchange causes the attraction. The short-range part of the hh interactions is modeled by the exchange of vector mesons and pomerons, or a phenomenological repulsive core is introduced. High precision meson exchange potentials reproduce the NN scattering phase shift at the quality of χ 2 /dof  1, and the Y N potentials have been applied to hypernuclei (Gal et al. 2016). On the other hand, there are several shortcomings in boson exchange models. First, the middle- and short-range parts of BB interactions depend on the model treatment, and the prediction power is not very high for BB interactions when experimental information is not enough. One of the examples is the σ meson or correlated pion pairs. The scalar-isoscalar σ meson is introduced to generate middle-range attraction in the Nijmegen models (Rijken et al. 1999), while its exchange is replaced by the uncorrelated and correlated π π exchange in the Jülich potential (Haidenbauer and Meissner 2005). The former treatment is simpler, while the ambiguity can be suppressed in the latter. It should be noted that the short-range part of the potential is still treated phenomenologically. There are some differences in the observables, which need to be considered as the theoretical systematic uncertainty. Second, the N antisymmetric spin-orbit potential in boson exchange models is not large enough to explain the small LS splitting in hypernuclei (Akikawa et al. 2002). The antisymmetric spin-orbit (ALS) potential, VALS (r)L · (σ 1 − σ 2 ), is considered to cancel the symmetric LS potential for hyperons. In the boson exchange models, the ALS potential cannot be large enough to cancel the symmetric LS potential (Takeuchi et al. 2000).

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Quark models In quark models of BB interactions, one starts from the interaction Hamiltonian for quarks, which consists of the phenomenological confinement potential (Liberman 1977), the OGE (one-gluon exchange) potential (De Rujula et al. 1975) with explicit SU(3)f breaking. The BB scattering can be described by that of (3q)-(3q) by using the resonating group method (RGM) in the quark cluster models, which explains the mechanism of having the repulsive core of the N N interaction: The color-magnetic interaction favors mixed symmetry states of quarks, there appears a forbidden state in the NN s-wave state (Neudachin et al. 1977), and then the relative s-wave function of NN has to have a node in order to be orthogonal to the forbidden state. The OGE potential and the node explain the repulsive core (Oka and Yazaki 1981). Later on, it becomes possible to describe also the long-range part of the BB interactions (Y N as well as NN interactions) by including effective meson exchange between quarks or between baryons (Oka et al. 1988). The quark model BB potential models overcome some of the problems in the boson exchange potential models: the mechanism to generate the repulsive core (Oka and Yazaki 1981) and the strength of the antisymmetric LS force (Takeuchi et al. 2000). Yet some unsatisfactory points still remain. One of them is the double counting of the meson and quark exchange. The NG bosons appear as a result of spontaneous breaking of the chiral symmetry and can be regarded as almost elementary particles, but other mesons such as scalar and vector mesons are well described in quark models including diquarks. Thus, one needs to introduce the form factors carefully to the scalar and vector meson exchange between quarks or baryons. Another concern is on the chiral symmetry. The quark model BB interactions are obtained by using the 3q structure of baryons, and the constituent quark masses are assumed. Then the chiral symmetry before the spontaneous breaking is not seriously taken into account. This problem is answered, at least in part, by the chiral effective field theory discussed below. Chiral effective field theory (Chiral EFT) As explained before, from the chiral transformation of quarks, ψ → U ψ, in the chiral quark model, octet baryons consisting of three dressed quarks transform as B → U BU † , and the following covariant derivative is chiral invariant:  ¯ μ Dμ B) = Tr(Biγ ¯ μ ∂μ B) + Tr Biγ ¯ μ [Vμ , B] . Tr(Biγ

(26)

Thus, the couplings of the NG bosons and octet baryons are given without introducing additional parameters and called the Weinberg-Tomozawa interaction (Weinberg 1979; Tomozawa 1966). This coupling gives the invariant scattering amplitude of the target hadron T and the NG boson in the SU(3) representation of α as Mα =

1 p·q ω

F T · F Ad α + O((m/MT )2 )  − 2 Cα,T , 2 f 2MT 2f

(27)

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where MT , p, and F T (m, q and F Ad ) are the mass, momentum, and the SU(3) generator of the target hadron T (the NG boson). The second equality holds for a baryon target, Cα,T = − 2F T · F Ad α , and ω denotes the energy of the NG boson. The chiral EFT has been applied to NN (Epelbaum et al. 2009; Machleidt and Entem 2011) as well as Y N and Y Y interactions (Haidenbauer et al. 2013). These interactions have been widely utilized to describe normal nuclei, hypernuclei, and nuclear matter. For the NN interactions, the next-to-next-to-next-to-leadingorder (N3 LO) interactions reproduce the experimental data with the precision comparable with the phenomenological potentials. For the Y N interactions, the next-to-leading-order (NLO) interactions are obtained and have been examined via the femtoscopy as discussed below. For meson-meson and meson-baryon systems, theoretical frameworks based on the chiral Lagrangian with and without resonances and unitarity in coupled channels have been developed (Oller et al. 2000). Lattice QCD Hadron-hadron interactions have been also investigated by the nonperturbative first-principles calculation, the Monte Carlo simulation of the lattice QCD, in several approaches. One of the approaches is to use the finite volume method proposed by Luscher (1986). On a finite volume lattice, the phase shift (δ(k)) is related with the energy shift of a two-particle system on the L3 × Lt lattice (EL ) as k cot δ(k) =

1  1 , 2 − (kL/2π )2 πL |n| 3

 EL = 2 k 2 + m2 − 2m,

(28)

n∈Z

where the masses of the two particles m are assumed to be the same. This relation is powerful in a sense that an experimental observable, δ(k), is related with an observable on the lattice, EL . This method has been applied successfully to the I = 2 π π scattering (Sharpe et al. 2003). It is also applied to extract the NN scattering lengths in the quenched QCD simulations (Fukugita et al. 1995) and in the (2+1)-flavor QCD simulations with the mixed action (Beane et al. 2006). Another approach is the so-called HAL QCD method (Ishii et al. 2007), where the hadron-hadron potential is extracted from the Nambu-Bethe-Salpeter (NBS) amplitude:   R(r, t − t0 ) ≡ 0 | T {φα (x + r, t)φβ (x, t)}J¯αβ (t0 ) | 0 → A0 ψαβ (r) e−E0 (t−t0 ) ,

tt0

(29)

where J¯αβ is the source of the two-hadron (αβ) state, φα,β shows the field operator, and A0 = E0 | J¯αβ | 0. When the time difference between t and t0 is large, higher energy states are more strongly suppressed with increasing t as exp[−En (t − t0 )], and eventually only the ground state wave function remains. The NBS amplitude at large t satisfies the Schrödinger equation:

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 (∇ 2 + kn2 )ψn (r) = 2μ

dr  U (r, r  ) ψn (r  ),

(30)

where r is the relative coordinate of the two particles and kn is related to the discrete energy eigenvalues En = kn2 /(2 μ). The nonlocal potential U (r, r  ) is assumed to be energy independent and Hermite. Then by the derivative expansion of

2 1   2 ˆ ˆ U , U (r, r ) = δ(r − r ) V0 (r) + 2 {V2 (r), (p/μ) } + V2 (r) L + · · · , one can obtain the local potential. The idea to use the NBS amplitude with the Schrödinger equation was mentioned in Luscher (1986), and the NN potential was numerically obtained in the quenched lattice QCD simulation by Ishii et al. (2007). Later on, this method, referred to as the HAL QCD method, has been applied to various hadron-hadron interactions such as the BB interactions in the SU(3) limit (Inoue et al. 2011), and N potentials (Gongyo et al. 2018), N- coupled-channel potentials (Sasaki et al. 2020), and so on. In these two methods, finite volume method and HAL QCD method, the NBS amplitude at large t − t0 is used implicitly or explicitly. One of the serious problems of studying multibaryon systems on the lattice is the signal-to-noise (S/N ) ratio, which significantly decreases with the increase of the time difference as exp[−(2mN − 3mπ )t] (In obtaining the energy or the NBS amplitude of the NN systems, the signal becomes smaller as exp(−2mN t) with t being the difference of the source and measurement (sink) time on the lattice. By comparison, the variance of the observable on the lattice decays as exp(−6mπ t), since the squared operator of the signal, (NN)† (NN), contains six quarks and six antiquarks and the 6π contributions contaminate.). Thus, in order to evade this S/N ratio problem, it is desired to perform the measurement at small t. In the finite volume method, the diagonalization method (Luscher and Wolff 1990) is useful. By using several source and sink operators, the four-point functions G(t) form a matrix, and the eigenvalues of G(t0 )−1/2 G(t)G(t0 )−1/2 with t0 being the reference time are found to be a single exponential rather than multi exponential functions. In the HAL QCD method, the time-dependent method (Ishii et al. 2012) is powerful. Even if several eigenstates are superposed, the wave functions satisfy the same (imaginary-)time-dependent Schrödinger equation: 

∂ ∇2 1 ∂2 − − + ∂t mN 4mN ∂t 2



 R(r, t) = 2μ

dr  U (r, r  ) R(r  , t),

(31)

for the N N system. Since it is assumed that there are no other particles than two nucleons, it is desired to take the time difference as mπ t > 1. The N (Gongyo et al. 2018) and N- (Sasaki et al. 2020) potentials from the lattice QCD calculation have been examined via the femtoscopy, and both of these are found to be reasonable.

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Femtoscopic Study of Hadron-Hadron Interactions Contrary to the importance of the hadron-hadron interactions, experimental information other than NN interactions are scarce. Recent development of the femtoscopic study of hadron-hadron interactions has opened a way to access hadronhadron interactions, for which the scattering experiments are not possible. Femtoscopy is a category of studies using the momentum correlation functions, which are related to the source function and the wave functions as  Nij (p i , p j ) Cij (q) = (32)  dr S(r)|ϕq (r)|2 , Ni (p i )Nj (p j ) where Nij (Ni,j ) is the two-particle (one-particle) finding probability with an appropriate measure so that Cij = 1 for a nonidentical free particle pair, S(r) is a normalized source function in the relative coordinate, and ϕq (r) is the relative wave function in the pair rest frame with the spatial relative momentum q in the asymptotic region of the final state. Historically, this formula, referred to as the Koonin-Pratt (KP) formula (Koonin 1977), has been used for identical particle pairs having the quantum statistics to deduce the source size of a star as proposed by Hanbury-Brown-Twiss (Hanbury Brown and Twiss 1956) or a hot region to produce pions proposed by Goldhaber-Goldhaber-Lee-Pais (Goldhaber et al. 1960), so the deviation from unity of the correlation function is generally called the HBT or HBTGGLP effects. By comparison, one can guess the wave function squared averaged over the source function, when the source function is known. Since the wave function reflects the interaction strength, one can access the hadron-hadron interactions. For example, the correlation function is analytically evaluated as (Lednicky and Lyuboshits 1982): CLL (q) = 1 +

2Ref (q) Im f (q) |f (q)|2 F2 (2x) + F3 (reff /R), F1 (2x) − √ R 2R 2 πR (33)

F1 (x) =

1 x



x 0

1 − e−x , x 2

dt et

2 −x 2

,

F2 (x) =

x F3 (x) = 1 − √ , 2 π

(34)

under the assumptions that there is no Coulomb potential, only the s-wave function is modified by the interaction, the  source function in the relative coordinate is a Gaussian S(r) = exp −r 2 /4R 2 /(4π R 2 )3/2 , and the asymptotic wave function, ϕq = eiq·r − j0 (qr) + sin(qr + δ(q))/qr, is used. The scattering amplitude is given as f (q) = (eiδ(q) − 1)/2iq, and the phase shift at low energies is parameterized as q cot δ(q) = −1/a0 +reff q 2 /2+O(q 4 ) (The nuclear and atomic physics convention is used here for the sign of the scattering length, δ(q)  −a0 q + O(q 2 ), so the negative a0 shows attractive potential and the positive a0 means the interaction is repulsive or there is a bound state.). The left panel of Fig. 5 shows the correlation function in the LL model at R/a0 = ±5, ±1 and ±0.3 and reff = 0. The correlation

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Fig. 5 Left: The correlation function in the Lednicky-Lyuboshits analytical model for reff = 0 and R/a0 = ±5, ±1 and ±0.3. Middle: Comparison of the − p correlation function data and the calculated results using the N- coupled-channel potential from lattice QCD (Sasaki et al. 2020). (Taken from Kamiya et al. 2022). Right: The K − p correlation function data from Pb+Pb ¯ potentials. (Taken from Acharya collisions in comparison with calculated results using several KN et al. 2020a)

function at low q is strongly enhanced when the absolute value of the scattering length |a0 | is larger than the source size R, and it is enhanced for negative a0 independent of the source size. With positive a0 > 0, by comparison, the correlation function is suppressed for a0  R because of the repulsion or the existence of a bound state. The s-wave asymptotic wave function reads qr ϕ0 (r) → sin(qr + δ)  sin[q(r − a0 )], there appears a node in the scattering wave function at r  a0 , and then the wave function squared is suppressed on average. The correlation functions of various pairs have been measured and utilized to access various hadron-hadron interactions (Acharya et al. 2005; Fabbietti et al. 2021), such as p (Adams et al. 2015),  (Adamczyk et al. 2015), p− (Acharya et al. 2005, 2019), p (Acharya et al. 2005; Adam et al. 2019), K− p (Acharya et al. 2020a), pD − (Acharya et al.), and others (Acharya et al. 2020b). In the following, two typical examples of femtoscopic studies of hadron-hadron interactions are shown. The first one is the p− correlation function. The S = −2 baryon-baryon interaction is relevant to the existence of the H dibaryon state (Jaffe 1977), as mentioned before in this section, and to the hyperon puzzle in neutron stars, as will be discussed later. The p− correlation function has been measured experimentally at LHC (Acharya et al. 2005, 2019), and the S = −2 N- coupled-channel potential has been theoretically measured on the lattice QCD (Sasaki et al. 2020). In order to evaluate the p− correlation function, one needs to extend the Koonin-Pratt formula to include effects of the coupled-channel, Koonin-PrattLednicky-Lyuboshitz-Lyuboshitz (KPLLL) formula (Kamiya et al. 2020, 2022; Lednicky 1998), and the Coulomb potential, and the source functions of the p− pair and other coupled channels are also necessary. The middle panel of Fig. 5 shows the comparison of the p− correlation function data from high-multiplicity pp events at the collision energy of 13 TeV (Acharya et al. 2005, 2019) and the calculated results (Kamiya et al. 2022) using the N- coupled-channel potential from lattice QCD at almost physical quark masses (Sasaki et al. 2020), which

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are found to agree well. Since the source functions are carefully parameterized by the ALICE collaboration (Acharya et al. 2005, 2019), this agreement implies that the baryon-baryon potential from lattice QCD (Sasaki et al. 2020) is examined ¯ to be reasonable. The second example is the K − p correlation function. The KN interaction is considered to be attractive due to the attractive Weinberg-Tomozawa ¯ bound state, (1405). The (1405) is the interaction and to form a I = 0 KN lightest negative parity baryon, which cannot be explained in the naïve quark model and is considered to be a hadronic-molecule state (Hyodo and Jido 2012). The K − p correlation function data from pp show strong enhancement at low q, as expected ¯ from the correlation function calculation (Kamiya et al. 2020) with the KN-π  coupled-channel potential from the chiral SU(3) dynamics (Ikeda et al. 2012). In AA(Pb + Pb) collisions, the correlation function shows a dip, as shown in the right panel of Fig. 5. These are in agreement with the existence of a bound state ¯ potential. Furthermore, the K − p scattering length deduced by the attractive KN from C(q) in AA collisions (Acharya et al. 2020a) is consistent with that from K − p atoms (Bazzi et al. 2011). While the femtoscopic analysis is based on the asymptotic wave functions with the Coulomb potential and further studies are necessary, it seems that the correlation functions are very useful to access hadronhadron interactions quantitatively.

Quark-Gluon Plasma During the first microsecond after the big bang, a plasma of quarks and gluons, the quark-gluon plasma (QGP), existed and formed hadrons including nucleons, which form the visible matter in our universe. In this section, the basic properties of the QGP and the QCD phase transition are reviewed.

QCD Phase Transition Due to the nature of a non-Abelian gauge theory, QCD has asymptotic freedom, the interaction becomes weaker, and quarks and gluons behave as free particles at large energy scales. By comparison, nonperturbative treatment is necessary, when the energy scale is small. Important nonperturbative effects are the color confinement and the spontaneous chiral symmetry breaking, as discussed in the previous section. Quarks and gluons cannot be observed by themselves, and quarks acquire large constituent masses and the mass of the Nambu-Goldstone particles is small. Based on these features, one expects the transition from the hadronic matter to the quarkgluon matter at high temperatures and/or densities. The nonperturbative feature of QCD also causes the change of the vacuum. The perturbative vacuum has a higher energy density (ε) and a lower pressure (P ) than the physical vacuum by B, where B, the so-called bag constant, is the energy density required to create the perturbative vacuum. In the bag model (Chodos et al. 1974), one can evaluate ε and P of QGP as

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εQGP = εqg + B, PQGP = Pqg − B,

(35)

where εqg and Pqg are the energy density and pressure of the quarks and gluons, respectively. If the interaction and the quark masses are ignored, the StefanBoltzmann law gives Pqg = dqg

π 2T 4 7 , εqg = 3Pqg , dqg = × 4Nc Nf + 2(Nc2 − 1), 90 8

(36)

where T is the temperature. When the hadron gas is modeled as the free massless pion gas, the pressure and energy density are given as PH = dπ π 2 T 4 /90 and εH = 3PH where dπ = (Nf2 − 1) is the degrees of freedom of Nambu-Goldstone bosons (dπ = 3 for Nf = 2). At zero T , the perturbative vacuum has a negative pressure and is crushed by the physical vacuum. With increasing T , PQGP increases faster (dqg > dπ ) and eventually catches up PH . The transition temperature from the hadron gas to QGP at μq = 0 is obtained from the condition, PQGP = PH , as Tc4 =

B 90 . 2 π dqg − dπ

(37)

With the bag constant B = (220 MeV)4 , the transition temperature is estimated to be Tc  160 MeV (Yagi et al. 2005). While this estimate looks too simple, it gives a good guess as shown below. Let us also consider the transition in the NJL model (Nambu and Jona-Lasinio 1961) as in the previous section. In addition to the vacuum contribution given in Eq. (12), the free energy density includes the contribution from particles at finite temperature and density. By using the hightemperature expansion technique, the free energy density in the chiral limit is obtained in the mean field approximation as (Kapusta and Gale 2006; Ohnishi 2016) c2 (T , μ) 2 c4 (T , μ) 4 m + m + O(m6 ), (38) 2 24        df 3df 3 2 πT 2 2 ν μ γE − 1 − log T + 2 μ − Tc , c4 = −H , c2 = 24 2Λ T π 4π 2 (39) Feff (m; T , μ) = Feff (0; T , μ) +

H ν (ν) =

∞  ν 2 

π

l=1

2 , (2l − 1)[(2l − 1)2 + (ν/π )2 ]

(40)

where Tc2 = 3Λ2 (1 − 8π 2 /G2 df )/π 2 is the critical temperature at μ = 0, γE = 0.577 · · · is the Euler’s constant, and m = Gσ is the constituent quark mass in the chiral limit. At zero baryon density, the spontaneously broken chiral symmetry is restored at T = Tc , and this transition is of the second order. It should be noted that the m4 log m terms disappear from the cancellation in the vacuum and

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4

3

2

stout HISQ ( -3p)/T4 p/T4 s/4T 4

1

T [MeV] 0 130

170

210

250

290

330

370

Fig. 6 Left: The order of the finite T phase transition (Columbia plot). (Taken from de Forcr and D’Elia 2017). Right: The equation of state from lattice QCD. (Taken from Bazavov et al. 2014)

particle contributions. When the quark masses are finite, the finite T phase transition becomes the crossover. These orders are the same as those in the lattice QCD results with Nf = 2 (Laermann and Philipsen 2003). The finite T QCD phase transition has been studied extensively by using the lattice QCD (Bazavov et al. 2014; Aoki et al. 2006; Borsanyi et al. 2014). The left panel of Fig. 6 shows the Columbia plot, the order of the finite T phase transition as a function of the mu,d and ms . There are two regions where the first order phase transitions take place, the upper-right corner and the lower-left corner. In these corners, the pure Yang-Mills field theory having the Z(3) symmetry (mu,d = ms = ∞) and the SU(3) chiral symmetric theory with axial anomaly (mu,d = ms = 0) having the SU(3)L × SU(3)R × Z(3)A symmetry (Pisarski and Wilczek 1984) are realized. On the left edge (mu,d = 0), the SU(2) chiral symmetry is exact, and the second order phase transition is expected in analogy to the O(4) σ model. Between the first order regions, the transition is a smooth crossover, and the transition for physical quark masses with Nf = 2 + 1 was proven to be in the crossover region (Aoki et al. 2006). The transition temperature is defined as the pseudocritical temperature given as the peak of the chiral susceptibility for light (u, d) quarks and is found to be Tc = 154 ± 9 MeV (Bazavov et al. 2012). The equation of state from the HotQCD collaboration (shown as HISQ) (Bazavov et al. 2014) and the Wuppertal-Budapest group (shown as stout) (Borsanyi et al. 2014) are found to agree as shown in the right panel of Fig. 6. Hence the finite T QCD phase transition should take place at around the above Tc .

History and Evidence of QGP Formation People have been trying to create QGP in terrestrial experiments and finally produced it at the last moment of the twentieth century. These challenges started with

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the LBL-Bevalac experiments in the 1970s and continued with JINR-Nuclotron, GSI-SIS, BNL-AGS, CERN-SPS, BNL-RHIC, and CERN-LHC accelerator experiments. The RHIC and LHC are the colliders and many researchers admitted the formation of QGP. At present, further researches are being conducted at RHIC and LHC and will be continued in the future accelerators such as GSI-FAIR, JINRNICA, and (possibly) J-PARC-HI. At the beginning of this journey, it was expected that a large fraction of the collision energy would be converted to thermal energy of matter a la Landau and that QGP would be produced at incident energies reachable in fixed target accelerators. However, in actual high-energy nuclear collisions, nuclei go through with less stopping and almost boost invariant matter is formed a la Bjorken and only a part of the collision energy is converted to heat. Thus, collider experiments such as RHIC and LHC were needed to confirm the formation of QGP. Figure 7 schematically shows the stages during high-energy heavy-ion collisions. Before the heavy-ion collisions at high energies, small x gluons in nuclei below the momentum Qs (the saturation scale) are saturated due to the self-interaction of gluons, and the classical coherent Yang-Mills field configurations called the color glass condensate (CGC) (Gribov et al. 1983; McLerran and Venugopalan 1994) are generated by fast (large x) partons. At the contact of two nuclei, the CGC is converted to the soft classical Yang-Mills field configuration, which is almost boost invariant and has longitudinal color electromagnetic fields (McLerran and Venugopalan 1994). The initial state of classical Yang-Mills (CYM) field together with fast partons and jets is called glasma. The glasma is considered to be converted to the (almost) equilibrated matter consisting of quarks and gluons, which can be regarded as the QGP. After hydrodynamical evolution, the QGP hadronizes and produces a hadron gas, which eventually freezes out and emits hadrons. One may naïvely expect that the QGP is realized as an almost free gas of partons, which can be described in perturbative QCD (pQCD). If this is the case, however, the thermalization time is long, and the formation of an equilibrated hot QGP was questioned even at RHIC. For example, before the RHIC experiments, people

Fig. 7 Expected stages during high-energy heavy-ion collisions

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talked as “If some miracle happens and thermalization occurs within 1fm/c, fluid dynamics can be applied.” And a miracle happened. The dynamical evolution of the created matter is described well by the fluid dynamics under the early thermalization assumption. The created matter has several remarkable features (Arsene et al. 2005; Back et al. 2005; Adams et al. 2005; Adcox et al. 2005) such as the jet quenching, collectivity, perfect fluidity, statistical hadron production, and Debye screening, which imply the formation of a strongly coupled QGP (sQGP). Jet quenching In high-energy pp or pp¯ collisions, hard (high-momentum transfer) scattering between partons (quarks, antiquarks, or gluons) inside p and/or p¯ causes a bunch of hadrons in narrow cones around the primary parton directions. Since a parton has color and cannot exist in the free form asymptotically, it needs to fragment into hadrons, which are emitted around the direction of the parton as long as the parton momentum is large. A narrow cone of hadrons is called a jet. When a parton goes through the color deconfined QCD matter, the parton loses energy via the radiation of gluons or the scattering with color charges in the medium a la Bethe-Bloch. When the energy of the parton becomes small, it cannot produce a jet. This phenomenon is called the jet quenching. The jet quenching is observed in AA collisions, where a jet is observed in the direction of the high-momentum trigger hadron as in pp collisions but it disappears in the opposite direction. In deuteronnucleus (dA) collisions, by comparison, the jet quenching is not observed. The top-left panel of Fig. 8 shows the azimuthal angle correlation of the trigger and other particles from pp, d+Au, and Au+Au collisions (Adams et al. 2003), as an example of the jet quenching. From the jet quenching, one can conclude that color charges are distributed in the created matter with a significant volume in AA collisions. The energy loss of partons is mainly caused by the induced gluon radiation. Ordinary mechanism of the energy loss of charged particles is the elastic scattering of the particle with electrons in matter, but this mechanism is not enough to explain the observed jet quenching. Collectivity Collective flows have been widely used to understand the degree of thermalization and the stiffness of the equation of state. In heavy-ion collisions at collider energies, strong elliptic flow is seen, where the elliptic flow is defined as v2 = cos 2φ with φ being the azimuthal angle. In noncentral collisions, the participants (the overlapping region of two nuclei) take an almond shape in the front view. Provided that the created matter is equilibrated, the pressure gradient is stronger in the short axis (in-plane) directions (±x or φ = 0, π ), particles are predominantly produced in-plane, and then positive elliptic flow will be observed. If the created matter is far from equilibrium and there are almost no secondary interactions, the angular distribution will be the same as that in the primary interactions, isotropic in the azimuthal angle, and then one expects a small elliptic flow. Thus, the strength of the elliptic flow shows the degree of thermalization. √ Actually, below the SPS energy (158 A GeV, sN N = 17.6 GeV), the elliptic flow

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Fig. 8 Experimental signals of (s)QGP formation. Top left panel: Two-particle azimuthal angle distributions from pp, d+Au, and Au+Au collisions, showing the jet quenching. (Taken from Adams et al. 2003). Top right: The ratio of the elliptic flow to the eccentricity (v2 /ε) at SPS and RHIC energies, showing that the hydrodynamic limit is reached at RHIC. (Taken from Alt et al. 2003). Middle left: The elliptic flows of identified particles as functions of the transverse kinetic energy, showing the quark number scaling. (Taken from Afanasiev et al. 2007). Middle right: The elliptic flow as a function of the transverse momentum in comparison with hydrodynamic model results with η/s = 0.08( 1/4π ), 0.16 and 0.24. (Taken from Luzum and Romatschke 2008). Bottom: Hadron production yields from Pb+Pb collisions at LHC in comparison with statistical hadronization model results. (Taken from Andronic et al. 2018)

normalized by the spatial eccentricity ε = (y 2 − x 2 )/(y 2 + x 2 ), is found to be smaller than the fluid dynamical predictions. At RHIC and LHC, v2 /ε data seem to reach the hydrodynamic limit (Alt et al. 2003), as shown in the top-right panel of Fig. 8.

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Another interesting feature is the quark number scaling which is found for the elliptic flows of identified particles: q

v2h (pT )  Nq v2 (pT /Nq ),

(41)

q

where v2h and v2 are the elliptic flow of a hadron and a quark, respectively, and Nq denotes the sum of quark and antiquark numbers in a hadron. In the case that a hadron with the transverse momentum pT is formed via the coalescence of quarks and antiquarks having the momenta p T /Nq , the azimuthal angle distribution of h at a given pT is given as  

  q,i dN h q,i ∝ 1 + 2v2 (pT /Nq ) cos 2φ  1 + 2 v2 (pT /Nq ) cos 2φ , pT dpT dφ i

i

(42) q,i

where v2 represents the elliptic flow of the ith constituent, a quark or an antiquark, the elliptic flows are assumed to be small compared to unity, and other Fourier coefficients (vn , n = 2) are omitted for simplicity. When one further assumes that the elliptic flow is common to quarks and antiquarks, the quark number scaling in Eq. (41) is obtained. This scaling is found to describe the elliptic flow of identified particles well as shown in the middle-left panel of Fig. 8 and is considered to be one of the evidences that hadrons are formed from quarks and antiquarks via coalescence. Perfect Fluidity As mentioned above, the elliptic flow is well described by the fluid dynamics. In addition, the shear viscosity is evaluated to be in the range of η/s = (1-3)/4π , where η and s denote the shear viscosity and the entropy density, respectively. The middle-right panel of Fig. 8 shows the elliptic flow at 4π η/s = 1, 2 and 3. This ratio is extremely small and is expected to be close to the lower bound. For example, the kinetic-theory arguments lead to the estimate of the shear viscosity, whose minimum value would be constrained by the uncertainty principle (Danielewicz and Gyulassy 1985): η≈

1 1 (n p λ)i  n, 3 3

(43)

i

where ni , pi , and λi are the local density, average  momentum, and momentumdegradation mean free path of quanta i and n = i ni is the total density of quanta. The lower bound of η is obtained by requiring λi ≥ p−1 i , since it is meaningless to consider the mean free path less than the minimum spatial spread from the uncertainty principle. For free massless particles, the entropy per particle is approximated as s/n ≈ 4, and then the lower-bound ratio would be (η/s)lower bound ≈ 1/12. Another estimate of the lower bound is given by the

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superstring theory (Kovtun et al. 2005). From the AdS/CFT correspondence, it was found that the ratio takes a value of 1/4π for supersymmetric gauge theories having gravity duals. Since the gauge/gravity correspondence is realized in the strong coupling limit (g 2 Nc → ∞, g → 0, Nc → ∞) and η/s is expected to be a decreasing function of the coupling, the above value is conjectured to be the lower bound (Kovtun et al. 2005). The almost perfect fluidity observed in heavy-ion collisions (RHIC Scientists Serve Up Perfect Liquid 2005) was considered to be the realization of this conjecture and motivated later works using the gauge/gravity correspondence. Statistical hadron production As mentioned in the previous section and as shown in the bottom panel of Fig. 8, hadron production yield in high-energy heavy-ion collisions is well described by statistical models (Becattini et al. 2006; Andronic et al. 2018). This implies that the system reaches an equilibrium state at chemical freezeout, at the latest. The temperature deduced by the statistical model is close to the phase transition temperature predicted by the lattice QCD calculations, suggesting that the system chemically freezes out immediately after the hadronization from QGP. It should be noted, however, that there are several data suggesting that statistical hadron production is not perfect. For example, the yield of hadrons with finite orbital angular momentum may not be explained in the statistical model as discussed in the previous section on (1520)(J π = 3/2− ) (Kanada-En’yo and Muller 2006). Another example is the quark number scaling in the elliptic flow, Eq. (41). The statistical model results would be realized in dynamical models via the CooperFrye formula (Cooper and Frye 1974), in which the amount of produced hadrons is determined from the continuity of the conserved currents [energy-momentum currents (T μν ) and conserved charge (B, S, Q) currents] on the freeze-out hypersurface. In this treatment, the collective flows at zero conserved charge densities, i.e., at very high energies, should be given by a common function of the velocity, independent of the quark content of the hadron, while the data favor the quark number scaling. The third example is the yield of short-lived hadron resonances, such as K ∗ (890). When the lifetime of the resonance is comparable to or shorter than that of the hadronic stage, the production yield would be suppressed from the statistical model estimate (Adams et al. 2005) but can be explained by considering the decay and reformation using a master equation (Cho and Lee 2018). These examples suggest that the coalescence is a promising underlying mechanism of statistical hadron production and dynamical evolution process after hadronization needs to be considered. Debye screening The J /ψ suppression has been proposed as a signal of QGP formation (Matsui and Satz 1986). The confinement potential of heavy-quarkantiquark pairs is expected to be screened in a deconfined state. This screening causes the mass shift of quarkonia (Hashimoto et al. 1986) or dissociation of the quarkonium states (Matsui and Satz 1986). The J /ψ suppression has been actually

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observed at SPS (Abreu et al. 2000), RHIC (Adare et al. 2017), and LHC (Abelev et al. 2014). The nuclear modification factor of J /ψ (RAA (J /ψ)) decreases with increasing the number of participants Npart (in more central collisions). It is also ¯ states. interesting that the sequential suppression is observed for the bottomium (bb) The excited states, Υ (2S) and Υ (3S), are more suppressed than the ground state, Υ (1S) (Chatrchyan et al. 2012). Since the excited states have larger size rQ , they are more sensitive to the confinement potential at large r, where the Debye screening will work more strongly. The above observations suggest that the confinement potential is screened, but the conclusion on the screening effects in QGP is not clearly drawn. The suppression comes also from the cold nuclear matter effects, and the regeneration of J /ψ, recombination of c and c¯ produced in primary collisions, seems to modify the centrality and pT dependence of the nuclear modification factor (RAA ) at LHC. Readers are referred to Andronic et al. (2016) for more discussions. These features imply that the created matter is deconfined (colored) matter consisting of partons (quarks, antiquarks, and gluons), shows collectivity, has a small viscosity close to the lower bound, and is chemically equilibrated in the hadronization stage at the latest. The existence of the Debye screening is still under debate, but it is promising. Hence while the created matter in the early stage may not be a plasma in a naïve sense, weakly coupled and thermalized gas, it is definitely a new form of matter and is named strongly coupled QGP (sQGP). There are further interesting features of the QGP, which are not yet fully understood but have attracted much attention. For example, challenges and perspectives are described in detail in Brambilla et al. (2014). Among them, the early thermalization and the QCD phase diagram are discussed in the following two subsections.

Early Thermalization The early thermalization is a puzzle that is not yet solved. The thermalization time estimated in perturbative QCD is considerably longer than phenomenologically estimated one obtained from the analyses using fluid dynamics, τth = (0.6 − 1.0) fm/c (Heinz and Kolb 2002). In high-energy nuclear collisions, the initial distribution of gluons just after the contact of two nuclei is given by the saturation scenario (Gribov et al. 1983; McLerran and Venugopalan 1994), where the small x gluons in nuclei below the momentum Qs (saturation scale) are saturated due to the self-interaction of gluons, form classical coherent field configuration called the color glass condensate (CGC), and are converted to the soft classical fields having longitudinal color electromagnetic fields (McLerran and Venugopalan 1994). A perturbative QCD estimate of the thermalization time in the saturation scenario is given, for example, in Baier et al. (2001), where the bottom-up thermalization picture is proposed. In this picture, first the soft gluons are emitted and form a thermal bath, and then the thermal bath draws energy from the hard gluons. Full thermalization is achieved when the primary hard gluons have lost energy

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at τ ∼ α −13/5 Q−1 s (Baier et al. 2001). This thermalization time has stronger α dependence than a naïve estimate, in which τ is inversely proportional to the elastic cross section, τ ∝ α −2 . This is because the inelastic particle production processes are important for thermalization (Wong 1996). The saturation scale is estimated to be Qs ≈ 1 GeV at RHIC and (2−3) GeV at LHC. Together with a typical coupling α ≈ 0.3, the thermalization time is expected to be τ ≈ (2 − 5) fm/c. In order to understand the early thermalization, the dynamics of the classical Yang-Mills (CYM) field is expected to play some role. For example, there are instabilities in the Yang-Mills field (Arnold et al. 2005; Weibel 1959), which may promote thermalization. By using the classical statistical approach, where the initial fields are allowed to fluctuate and the classical equation of motion is solved, it was shown that the ratio of longitudinal to transverse pressure becomes closer to unity at the time of the order of 1 fm/c when the coupling constant is g  0.5 (Epelbaum and Gelis 2013), as shown in the left panel of Fig. 9. In a similar framework but in a nonexpanding geometry, the shear viscosity is found to scale as η ∝ 1/g 1.49±0.39 at weak coupling (Matsuda et al. 2020), where the obtained value of η/ε3/4 is found to be consistent with that evaluated in Epelbaum and Gelis (2013), as shown in the right panel of Fig. 9. This coupling dependence is weaker than that in the leading-order perturbation theory, η ∝ 1/(g 4 log g), but is consistent with that of the “anomalous viscosity,” η ∝ 1/g 1.5 , which appears for particles moving in the strong disordered field (Asakawa et al. 2007). While the classical field does not reach the quantum equilibrium, the classical field evolution is consistent with that with the Boltzmann equation at large occupation numbers (Mueller and Son 2004). These observations suggest the importance of the interplay of the background (classical) field and particles in the early stage of high-energy heavy-ion collisions, where the “particles” are not necessarily hard. Actually, the effective kinetic theory (EKT) for the particle distribution function is proposed as the bridge between saturated gluon fields and fluid dynamics (Kurkela and Zhu 2015). A new concept, hydrodynamization, at which hydrodynamics becomes applicable even before thermalization and pressure isotropization (Strickland 2014), was also introduced in Kurkela and Zhu (2015).

Fig. 9 Left: Time evolution of the transverse and longitudinal pressure normalized by the energy density. (Taken from Epelbaum and Gelis 2013). Right: The shear viscosity normalized by ε3/4 as a function of the coupling. (Taken from Matsuda et al. 2020)

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Hence, the early thermalization puzzle seems to urge theorists to consistently understand the dynamics of the classical (background) field, particles, and fluid dynamics.

QCD Phase Diagram Elucidating the QCD phase diagram is another grand challenge of high-energy heavy-ion collision physics (Fukushima and Hatsuda 2011). In the high baryon density region, a transition from nuclear matter to quark matter is expected to take place. For example, when the distances between nucleons become sufficiently small, it will be difficult to identify the nucleon to which a quark belongs (percolation). The free energy density expression in the NJL model, Eq. (38), shows that the curvature of the free energy density changes sign at T 2 + 3μ2 /π 2 = Tc2 , which suggests the phase boundary is an ellipse in the (T , μ) phase diagram. The left panel of Fig. 10 shows the expected phase diagram in the 1980s (Baym 1984). The finite density phase transition at zero temperature was found to be the first order (Asakawa and Yazaki 1989) for some of the reasonable parameter sets (Hatsuda and Kunihiro 1994). Since the transition is known to be the crossover at zero baryon density (Aoki et al. 2006), there should be a critical point (CP), which connects the crossover and the first order phase boundaries. In addition, the color flavor locked (CFL) phase of the color superconductor (CSC) is known to be the ground state of the Nf = 2 + 1 QCD at very high densities (Barrois 1984; Alford et al. 2008). The updated phase diagram including the hadronic matter, QGP, and CSC is shown in the middle panel of Fig. 10 (Fukushima and Hatsuda 2011). Unfortunately, except for the existence of the above three forms of matter, not much is convincing. For the nuclear liquid-gas phase transition, one of the convincing observations was the plateau in the caloric curve, which is the temperature as a function of the excitation energy per nucleon, T (E ∗ /A), obtained from the fragment yield double ratio (Pochodzalla et al. 1995). Other phase boundaries and CPs shown in the figure are conjectured or expected, but are not confirmed by experiments nor examined in ab initio calculations. Yet detecting the CP and/or the phase boundary at high densities is relevant to the neutron stars and the binary neutron star mergers, where the cold dense quark matter may emerge as shown in the right panel of Fig. 10. In order to access the dense QCD, one needs more data from the ongoing beam energy scan programs (BES) (Adamczyk et al. 2017), the future facilities such as the FAIR (Spiller and Franchetti 2006) and NICA (Sissakian et al. 2009), and the planed program such as the J-PARC-HI (Sako et al. 2014). Critical Point One of the keys in the QCD phase diagram is the CP. Around the CP, the fluctuation of the chiral condensate becomes large. Since the chiral condensate couples with the baryon number density at finite density, the fluctuation of the latter is also enhanced (Stephanov et al. 1999; Fodor and Katz 2004; Allton et al. 2005; Borsanyi et al. 2013). Specifically, the higher-order cumulant ratios of the net-baryon number have been attracting attention, since higher-order cumulants diverge more rapidly with increasing coherence length and explicit

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Fig. 10 QCD Phase diagram. (Top and middle figures are taken from Baym (1984) and Fukushima and Hatsuda (2011), respectively). The bottom panel shows the expected phase diagram in the (T , ρ, δ) space with δ being the isospin asymmetry

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volume dependence cancels by taking the ratio. The nth order cumulant is defined as χμ(n) = ∂ n (P /T 4 )/∂ μˆ n = ∂ n (log Z )/∂ μˆ n /V T 3 with μˆ = Nc μ/T . The cumulant ratios are related with the skewness S and the kurtosis κ as χμ(3) /χμ(2) = Sσ and (4) (2) χμ /χμ = κσ 2 , which have been measured in the BES program (Adam et al. 2021; Adamczyk et al. 2014) as shown in the left panel of Fig. 11. One observes the nonmonotonic colliding energy dependence in κσ 2 . This ratio takes unity for Poisson distribution and the Skellam distribution (The Skellam distribution is the probability distribution of the difference of two independent variables following the Poisson distribution.). Provided that the freeze-out line lies close to but below the CP, the above nonmonotonic dependence can be explained qualitatively (Borsanyi et al. 2013; Stephanov 2011), as shown in the right panel of Fig. 11. In order to confirm the existence of the CP, it will be necessary to develop dynamical models which include the fluctuations given in the equilibrium theories.

1.0

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HADES

Net-proton High Moments

Directed flow slope and softening of EOS The nonmonotonic colliding energy dependence is observed also in the directed flow, v1 = cos φ. The left panel of Fig. 12 shows the directed flow slope dv1 /dy of protons, antiprotons, and net protons as functions of the colliding energy (Adamczyk et al. 2014). Let us here concentrate on the proton flow. The data shows the change of the slope at the √ balance colliding energy, sN N ≈ 10 GeV, which seems to show the softening of the equation of state (EOS) (Rischke et al. 1995) and may signal the first order phase transition. At low energies, proton directed flow slope is positive (positive flow), reflecting the larger pressure in the participants from the kinetic pressure and repulsive potential at high densities as examined by transport models (Bass et al. 2000). At high energies, the negative slope (negative flow or antiflow) emerges by the combination of geometry and stopping in transport models (Snellings et al. 2000) or by the particle emission from the tilted matter in hydrodynamical models (Adil et al. 2010). Then one expects that the balance energy is explained by the hybrid models of the hadronic transport and fluid dynamics, but most of

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(2)

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Fig. 11 Left and middle: Cumulant ratios, χμ /χμ = Sσ and χμ /χμ = κσ 2 , observed in Au+Au collisions. (Taken from Adam et al. 2021). Right: Calculated κσ 2 as a function of the temperature on the freeze-out (T , μ). (Taken from Borsanyi et al. 2013)

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dv 1/dy|

y=0

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Fig. 12 Left: Directed flow slope of p, ¯ p, and net p as a function of the colliding energy. (Taken from Adamczyk et al. 2014). Right: Calculated directed flow at several colliding energies. (Taken from Nara and Ohnishi 2022)

the models using a single EOS fail (Steinheimer et al. 2014). Very recently, a hadronic transport model using a single EOS with vector potential effects is found to explain the energy dependence of the v1 slope for the first time as shown in the right panel of Fig. 12 (Nara and Ohnishi 2022). The mechanism of the slope sign change is the competition of the abovementioned two origins: the positive flow in the compression stage and the negative flow from the tilted matter. In order to realize both using a single EOS, hadronic matter needs to be strongly coupled. It should be noted that, however, the hadronic transport approach in Nara and Ohnishi (2022) does not explain the hadronic production yields. Further studies including both hadronic transport and fluid dynamics as well as further experiments will be necessary to understand the heavy-ion collisions in the nonmonotonic region, √ 3 GeV ≤ sN N ≤ 30 GeV. Sign Problem in Finite Density QCD If the lattice QCD simulations can be applied to finite density matter, it is very helpful to elucidate the properties of dense matter. However, there is the sign problem in the Monte Carlo simulation of lattice QCD at finite baryon chemical potential (de Forcrand 2009). In lattice QCD, the fermion determinant appears in the statistical weight of the MC sampling. The fermion matrix D(μ) =D + m − μγ0 has the γ5 hermiticity:   γ5 D(μ)γ5 = [D(−μ∗ )]† , det[D(μ)] = det D(−μ∗ )

∗

.

(44)

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Then det D is real for zero and pure imaginary μ, and an even number of degenerate quarks (same mass and μ) yield a nonnegative factor in the weight. For finite real μ, the fermion determinant is generally complex, and the probability interpretation fails. It is, in principle, possible to perform the reweighting:

Opq



eiθ Opq 1 ˆ

O = , DU O(U ) e−Seff = Z

eiθ pq   1 = DU O(U ) e−Re Seff , Zpq = DU e−Re Seff , Zpq

(45) (46)

where Z is the partition function, Seff = SU − log det D is an effective action with SU being the Yang-Mills action, and θ = −Im Seff . Since the phase quenched weight exp(−Re Seff ) is real, one can perform the MC simulation provided that the average phase factor (APF), eiθ pq , is not zero. The APF is, however, an exponentially decreasing function of the volume. Since the APF is a ratio of the partition function and the phase quenched one, it is given as the free energy difference: eiθ pq = Z /Zpq = exp(−V f/T ) with f = f − fpq ≥ 0, where f is expected to depend on the volume only weakly at large volume. Hence an exponentially large number of configurations are needed to reduce the statistical error of the APF. This is the sign problem in a strict sense (The sign problem in a broader sense contains numerical problems related with the strong cancellation in the integral of rapidly oscillating functions.). While the sign problem exists, there are many attempts to study finite density QCD using lattice QCD, such as the reweighting method (Fodor and Katz 2004), the Taylor expansion in μ/T (Allton et al. 2005), the analytic continuation from the imaginary chemical potential (de Forcr and Philipsen 2003), and the canonical ensemble method (Hasenfratz and Toussaint 2008). These are the methods to utilize the lattice configurations having the real weight, configurations with complex weights are not sampled directly, and it seems that the results are reliable at small chemical potential, μ/T ≤ (1 − 2). In this decade, approaches based on complexified variables have been developed, such as the complex Langevin method (CLM) (Parisis and Wu 1981; Aarts et al. 2010), the Lefschetz thimble method (LTM) (Witten 2011), and the path optimization method (POM) (Mori et al. 2017). The CLM is based on the stochastic quantization. By solving the complex Langevin equation for complexified variables, z˙ j = −∂S/∂zi + ηj (t) with ηj being the white noise satisfying ηj (t)ηk (t  ) = 2δ(t − t  ), one can generate configurations and calculate observables from the ensemble average. In the CLM, there is no sign problem, the numerical cost is low, and then it has been applied to 3+1D finite density QCD. Figure 13 shows the comparison of the plaquette averages in the CLM and in the reweighting method at small μ/T (Fodor et al. 2015) (left) and the quark number at very weak coupling (β = 2Nc /g 2 = 20) but at large μ/T in the CLM (Tsutsui et al. 2022) (right).

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Fig. 13 Left: Comparison of plaquette averages calculated in the CLM and reweighting from the μ = 0 ensemble. (Taken from Fodor et al. 2015). Right: Quark number as a function of μ/T on a 83 × 128 lattice in comparison with the free quark limit. (Taken from Tsutsui et al. 2022)

These results show the reliability of the CLM at small μ/T or at weak coupling. The problem of the CLM is that there are regions in the (T , μ) plane where the results converge to wrong ones, when the magnitude distribution of the drift term has a long tail than the exponential decay (Aarts et al. 2010). Careful monitoring is necessary to rely on the results. In the LTM, integration path is shifted from the real axis to  the so-called thimble, Jσ , defined by the holomorphic flow equation, z˙ j = ∂S/∂zj from the fixed point zσ , ∂S/∂z|z=zσ = 0. (Note that in the case where the number of integration variables is larger than one, the integration path and the real axis need to be rephrased as the integration manifold and the real space, but one variable words are used here for simplicity.) Since Im S is constant on one thimble, the sign problem is weakened. Even if the integration path is shifted from the real axis, the partition function and observables are invariant due to Cauchy’s theorem, as long as the shift is a continuous deformation from the real axis and does not pass through the pole of the integrand. The remaining sign problem in the LTM comes from the integration measure (Jacobian) and competition of different thimbles (multimodal problem). The latter may be evaded by the tempering procedure (Fukuma and Umeda 2017). The POM utilizes the abovementioned invariance of the partition function with respect to the continuous shift of the integration path. The integration path can be chosen to weaken the sign problem, for example, to enhance the APF. This is a kind of optimization problem, and one can use optimization methods that are used in various fields such as the neural network. Ineffective optimization due to the degeneracy by gauge degrees of freedom is a problem in POM, but optimization using gauge-invariant inputs (Namekawa et al. 2022) or the use of gauge-covariant lattices (Tomiya and Nagai 2021) may be efficient to evade this problem. The subjects discussed in the previous and this subsections are those to be investigated further. Theoretically, these are related to the real-time evolution of quantum field and finite fermion density, both of which have the sign problem, so ideas are necessary to overcome.

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Neutron Stars and Nuclear Matter Equation of State Neutron stars are supported by the pressure from baryons and quarks. Since the density in the inner core may reach ρ = (5 − 10)ρ0 , they provide a unique playground of cold dense matter physics. In this section, an overview of the masses and radii of neutron stars are given. Next, the neutron star matter EOS is discussed.

Mass and Radius of Neutron Stars One can access the neutron star properties through the observables such as masses, radii, moments of inertia, and tidal deformabilities. These observables do not directly show the EOS of dense matter but are related to that via the TolmanOppenheimer-Volkoff (TOV) equation (Tolman 1934) and its extension. The TOV equation is a hydrostatic equation of spherical stars with the general relativity effects: dP (ε/c2 + P /c2 )(M + 4π r 3 P /c2 ) dM = −G , = 4π r 2 ε/c2 , dr dr r 2 (1 − 2GM/rc2 )

(47)

where P and ε are the pressure and the energy density, respectively, and M is the mass inside the radius r. These three are functions of r to be obtained by solving the equation, but there are only two equations. Thus, one additional equation, the EOS, P = P (ε), is required. By solving the TOV equation with a given energy density at center, ε(r = 0) = εc , until the pressure reaches zero, one obtains the mass and radius as a function of εc . The left panel of Fig. 14 compares the mass-radius (MR) relation from several neutron star matter EOSs and some of the observed neutron star masses, PSR J1614-2230 (1.97 ± 0.04M ) (Demorest et al. 2010) (later updated to be 1.928 ± 0.017M (Fonseca et al. 2016), and 1.908+0.016 −0.016 M Arzoumanian et al. 2018), PSR 1913+16 (Hulse-Taylor pulsar (Hulse and Taylor 1975), M1 = 1.442 ± 0.003M Taylor and Weisberg 1989), and PSR J1903+0327 (1.667 ± 0.021M ) (Freire et al. 2011). One finds that there are several EOSs whose MR curves do not reach the observed mass of 1.97M . This is because there is a maximum mass for each EOS, where the mass does not increase with increasing central energy density. Above this critical central energy density, neutron stars are unstable and collapse to black holes. Thus, the discovery of massive neutron stars provide a severe test of theoretically proposed EOSs. For example, the magenta and green curves are the EOSs including strangeness, hyperons, antikaons, or strange-quark matter. The dense matter EOS becomes soft when hyperons appear, and the two solar mass neutron stars cannot be supported. This problem is called the hyperon puzzle and will be discussed later. The maximum mass of neutron stars are updated from the 2010 value (1.97M ) (Demorest et al. 2010) to 2.01 ± 0.04M (PSR J0348+0432) in

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Fig. 14 Neutron star mass-radius relation in 2010 (Left) (Demorest et al. 2010) and in 2021 (Right) (Burgio et al. 2021)

2013 (Antoniadis et al. 2013) and 2.14+0.10 −0.09 M (PSR J0740+6620) in 2019 (Cromartie et al. 2019) (revised in 2021 to 2.08 ± 0.07M Miller et al. 2021). The latter is close to the upper limit of neutron star masses, Mmax  2.17 M (Margalit and Metzger 2017), (2.15−2.25)M (Shibata et al. 2017), or 2.01+0.04 −0.04 ≤ Mmax /M ≤ +0.17 2.16−0.15 (Rezzolla et al. 2018), deduced from the gravitational wave observation from the binary neutron star merger event of GW170817 (Abbott et al. 2017). Neutron star masses are measured by using the radial velocity (Hulse and Taylor 1975), perihelion shift together with the Einstein delay (Taylor and Weisberg 1989), or recently the Shapiro delay (Demorest et al. 2010; Cromartie et al. 2019). It should be noted that these mass measurement methods require a companion, so one can measure the mass precisely only in binaries. Compared with the masses, it is much more difficult to measure the neutron star radii, since one needs to resolve the 10 km size from 10,000 to 100,000 light year distance from the Earth. One of the ideas is to use the X-ray bursters. Mass accretion from companion occasionally induces explosive hydrogen and/or helium burning, and the surface temperature becomes much higher. There are three methods using the X-ray bursters. One can guess the temperature from the photon spectrum by assuming the Stefan-Boltzmann law. Then the radius is obtained from the flux as ! R∞ =

F D2 , σSB T 4

(48)

 where R∞ = R/ 1 − 2GM/Rc2 , F and D are the photon flux and the distance of the pulsar, respectively, and σSB is the Stefan-Boltzmann constant (Guillot et al. 2013). This method needs the distance to the pulsar, and the flux from the pulsar is affected by the Galactic absorption. Using this method, the neutron star radii are deduced to be R = 9.1+1.3 −1.5 km (Guillot et al. 2013), but the reanalysis using

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the similar method (but with additional source) gives R = 11.09+0.38 −0.36 km or R = +0.39 11.04−0.35 km (Baillot d’Etivaux et al. 2019), where two sets show different Markov chain Monte Carlo runs. The second method is based on the assumption that the Eddington limit is reached. In the Eddington limit, the radiation pressure balances the gravity, where the radius is expected to be stationary for a while. The balance equation leads to the estimate 2 R∞ =

2GMcmN NN , σT σSB T 4 Ne

(49)

where σT is the Thomson cross section (photon-charged particle cross section) and NN and Ne are the numbers of nucleons and electrons, respectively. In this method, the distance to the pulsar is not necessary. On the other hand, one needs the ratio NN /Ne , which takes one for hydrogen atmosphere and a half for light elements. Using this method, for example, the radius of a neutron star in the low-mass X-ray binary 4U 1820-30 was estimated to be R = 9.1 ± 0.4 km (Guver et al. 2010). Updates are summarized in Özel and Freire (2016), where the neutron radii are found to be in the range R = (10−11.5) km for masses of (1.17−2.0)  M . The third one is to measure the redshift of the absorption lines, Eobs = Esurf 1 − 2GM/Rc2 , which is almost the direct observation of M/R. This method was proposed in the Hitomi satellite, but was not applied because of the accident. Recently, two new tools have been added for the measurement of the neutron star radii, the tidal deformability from the binary neutron star mergers (Abbott et al. 2017, 2019) and the Neutron Star Interior Composition Explorer (NICER) (Miller et al. 2021, 2019; Riley et al. 2021). The right panel of Fig. 14 shows the MR regions obtained from the tidal deformability and NICER (Burgio et al. 2021). The tidal deformability λ is the first order coefficient of the quadrupole deformation Qij with respect to the tidal force Eij (Hinderer et al. 2010): Qij = −λ Eij , λ =

Λ Λ (GM/c2 )5 = (CR)5 , C = GM/Rc2 , G G

(50)

where Λ is the dimensionless version of the tidal deformability and C is the compactness. The tidal deformability Λ is related to the Love number k2 as Λ = 2k2 /3C 5 , and one needs to solve a set of the first order differential equation simultaneously with the TOV equation in order to obtain k2 . Since the tidal deformability is closely related with the neutron star radius as Λ ∝ R 7.5 (Annala et al. 2018), one can guess the neutron star radius with an uncertainty of around 0.5 km if Λ(1.4 M ) is determined. One can extract the tidal deformability from the gravitational wave form in the inspiral phase, which is found to be Λ < 800 (Abbott et al. 2017) (R1.4 < 13.6 km Abbott et al. 2017; Annala et al. 2018), +420 Λ = 222+420 −138 (R1.4 = (8.9 − 13.2) km) (De et al. 2018), and Λ = 300−230 (R1.4 = (10.5 − 13.3) km) (Abbott et al. 2019), where R1.4 is the radius of 1.4 M mass neutron stars. The tidal deformability is important, since it relies on the first order

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perturbation and is model independent. While the relation between Λ and R1.4 depends on the EOS, one can directly calculate Λ for a given EOS. By comparison, the NICER measures the energy-dependent X-ray wave form from hot spots, and the mass and radius of a neutron star are obtained by using a Bayesian inference approach. In Miller et al. (2019), the mass and radius of +1.24 PSR J0030+0451 are obtained to be M = 1.44+0.15 −0.14 M and R = 13.02−1.06 km. In Miller et al. (2021), (M, R) of PSR J0740+6620 is obtained (updated) by combining the data of NICER and the X-ray Multi-Mirror (XMM-Newton) as (M, R) = (2.08 ± 0.07M , 12.35 ± 0.75 km). Since the observations of many neutron stars are possible, further data will be obtained in the near future. There are several other methods to deduce the neutron star radii, which are mentioned in Özel and Freire (2016).

Nuclear Matter EOS and Symmetry Energy Of the five regions of neutron star matter (atmosphere, envelope, crust, outer core, and inner core), the EOS of neutron star matter in the core ρ  ρ0 /2 is discussed here, where uniform matter is expected to emerge. In the outer core, ρ0 /2  ρ  2ρ0 , the neutron star matter is considered to be uniform and composed of neutrons with small fraction of protons, electrons, and muons. While the EOS of symmetric nuclear matter (SNM) is constrained by the nuclear masses and excitation spectra, the EOS of pure neutron matter (PNM) is not known well experimentally. Hence one of the main motivations of studying neutron-rich nuclei is to determine the symmetry energy S(ρ), which is the leadingorder coefficient in the asymmetry parameter expansion of the energy per nucleon, W = E/A: W (u, x) = W (u, 0) + x 2 S(u) + O(x 4 ), x =

ρn − ρp ρ , u= , ρn + ρp ρ0

(51)

where u is the density normalized by the saturation density and x is the asymmetry. The symmetry energy can be also defined as the difference of the energy per nucleon in PNM and SNM, S(u) = W (u, 1) − W (u, 0), provided that the O(x 4 ) terms are negligible. It is useful to consider the Taylor expansion around the saturation density, ρ = ρ0 (u = 1): K (u − 1)2 + O[(u − 1))3 ], 18 Ksym L (u − 1)2 + O[(u − 1)3 ]. S(u) = S0 + (u − 1) + 3 18

W (u, 0) = E0 +

(52) (53)

The symmetry energy slope parameter, L, is known to be relevant to the neutron star radii (Steiner et al. 2010; Lattimer and Lim 2013). Since the pressure of cold nuclear matter is obtained from W (u, x) as

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  K + x 2 Ksym ∂W Lx 2 2 ∂W P =ρ = ρ0 u = ρ0 (u − 1) + + O[(u − 1)2 ], ∂ρ ∂u 9 3 (54) 2

P is dominated by the symmetry energy slope parameter L around the saturation density. Hence by using (M, R) of neutron stars, the symmetry energy parameters have been deduced to be in the range 28.5 MeV ≤ S0 ≤ 34 MeV and 41 MeV ≤ L ≤ 51 MeV (Steiner et al. 2010; Lattimer and Lim 2013). The symmetry energy parameters, (S0 , L), have been investigated extensively both from the theoretical and experimental sides in nuclear physics. Experimental constraints have been combined and summarized on the (S0 , L) plane in Lattimer and Lim (2013) and Tews et al. (2017). The constraints include those from nuclear mass (Kortelainen et al. 2010), skin thickness of Sn isotopes (Chen et al. 2010), the dipole polarizability of 208 Pb (Roca-Maza et al. 2013) giant dipole resonance (GDR) (Trippa et al. 2008), isotope diffusion in heavy-ion collisions (Tsang et al. 2009), and isobaric analog states and isovector skin (IAS+R) (Danielewicz et al. 2017), as shown in the left panel of Fig. 15. The constraint from the unitary gas conjecture (UGC) (Tews et al. 2017) and recent report from the PREX experiment (Abrahamyan et al. 2012; Adhikari et al. 2021; Reed et al. 2021) are also shown. These constraints are obtained from the correlation between the observables and (S0 , L) in theoretical calculations such as the mean field models. The right panel of Fig. 15 shows the correlation between the L and the neutron skin thickness of 208 Pb (Roca-Maza et al. 2011). Each point shows the result of one mean field model. The thickness rnp is found to be strongly correlated with L as rnp = 0.101 + 0.00147L. Thus, if one measures the thickness with some uncertainties, the allowed region of L is determined.

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Fig. 15 Left: Symmetry energy parameter constraints from various measurements and the unitary gas conjecture (Tews et al. 2017), updated with the PREX constraint. Right: The correlation between L and the neutron skin thickness of 208 Pb. (Taken from Roca-Maza et al. 2011)

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One notes that the white pentagon area around the center in the left panel of Fig. 15 satisfies six constraints out of seven, and one may consider that the symmetry energy parameters are well determined to be in that pentagon and satisfy 29.7 MeV ≤ S0 ≤ 32.2 MeV and 42.5 MeV ≤ L ≤ 62.4 MeV. It should be noted that, however, there is some tension between this area and new data. First the IAS+R (Danielewicz et al. 2017) constraint suggests slightly larger values of (S0 , L). Furthermore, the parity-violation measurement in electron scattering on nuclei (lead(Pb) Radius EXperiment, PREX) reported the asymmetry, which corresponds to the neutron skin thickness of 208 Pb as R = Rn − Rp = 0.283 ± 0.071 fm as the combined result of the PREX-1 (Abrahamyan et al. 2012) and PREX-2 (Adhikari et al. 2021). This skin thickness suggests a stiff symmetry energy, L = 106 ± 37 MeV (Reed et al. 2021), which is outside of the pentagon. The PREX experiment is a purely electroweak measurement and there is no model dependence to obtain the asymmetry, while further works would be necessary for serious estimate of the skin thickness. For example, the strange-quark contributions in nucleons may cause modifications of the form factor at the 1% level (Reinhard et al. 2013). All the experimental data discussed above rely on data relevant to densities around ρ0 or lower. In order to probe the symmetry energy at densities above ρ0 , pion production from neutron-rich nuclear collisions was measured by the Sπ RIT collaboration (Jhang et al. 2021). Since pions are produced in the early dense stage and smaller symmetry energy allows larger asymmetry leading to more frequent π − production, the π − /π + ratio is expected to be sensitive to the symmetry energy (Li 2002). In the experiment, the double ratio of produced pions (π − /π + ) from 132 Sn +124 Sn and 108 Sn +112 Sn collisions at E/A = 270 MeV was reported to be DR ≡ [Y (π − )/Y (π + )]132+124 /[Y (π − )/Y (π + )]108+112 = 2.42 ± 0.05. Theoretical model predictions using EOS with 45.6 MeV ≤ L ≤ 152 MeV show that the double ratio is mildly sensitive to L, and all model results underestimate the double ratio. Further studies are necessary to probe the symmetry energy above ρ0 using nuclear collisions.

Hyperon Puzzle and Dense QCD From hypernuclear experiments, hyperons are found to feel potential in nuclear matter at ρ0 as U (ρ0 )  −30 MeV, U (ρ0 )  +30 MeV, and U (ρ0 )  −15 MeV (Ishizuka et al. 2008). Based on these potential strengths, one can obtain neutron star matter EOSs including hyperons in a standard manner, such as the relativistic mean field (RMF) (Ishizuka et al. 2008; Glendenning and Moszkowski 1996), the Skyrme-Hartree-Fock mean field (Lanskoy and Yamamoto 1997), the Brückner-Hartree-Fock theory (Baldo et al. 2002), and the variational method (Togashi et al. 2016). In most of theoretical calculations before 2010, hyperons are found to appear in neutron stars above ρ = (2 − 4) ρ0 . The EOS is softened by hyperons, and the maximum mass of neutron stars is reduced to be Mmax = (1.3 − 1.6) M from that without hyperons, Mmax = (1.5 − 2.7) M . The

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left panel of Fig. 16 shows the comparison of the neutron star mass as a function of the central density obtained using the nucleonic EOS (Mmax = 2.17 M ) (Sugahara and Toki 1998) and hyperonic EOS (Mmax = 1.63 M ) (Ishizuka et al. 2008). In this case, hyperons reduce the maximum mass by 0.54 M . The above neutron star maximum mass with hyperons is smaller than 2 M reported in 2010 (Demorest et al. 2010). Thus, the discovery of 2 M neutron stars ruled out most of the EOSs with hyperons and implies that something else is necessary. This problem is called the hyperon puzzle. It should be noted that the same problem applies to EOS with kaon condensation (Kaplan and Nelson 1994) or with the strange-quark matter (Weber 2004). There are several mechanisms proposed so far to solve the hyperon puzzle: repulsive three-body interactions, transition to quark matter, and modified gravity (Cheoun et al. 2013). The first two mechanisms related to quark-nuclear physics are explained here. The first one is to introduce stiffer hyperon potentials in nuclear matter at high densities than that from the naïve two-body hyperon-nucleon interactions. A naïve choice of vector coupling in RMF is (xσ , xω ) ≡ (gσ  /gωN , gσ  /gσ N )  (2/3, 2/3) from the SU(6) symmetry or the quark counting rule (Ishizuka et al. 2008; Glendenning and Moszkowski 1996). Since the SU(6) symmetry is broken, it is possible to take, for example, xω  1 within the SU(3)f symmetry (Weissenborn et al. 2012). A stronger vector repulsion at high densities can support massive neutron stars, while this choice may not be supported by hadron interaction models. Since three-nucleon (NNN) force needs to be introduced in the first-principles calculations of nuclei (Carlson et al. 2015) and nuclear matter (Togashi and Takano 2013; Friedman and Pandharipande 1981; Brockmann and Machleidt 1984; Akmal et al. 1998), it would be reasonable to introduce the repulsive three-body Y NN , Y Y N , Y Y Y interactions (Baldo et al. 2002; Gerstung et al. 2020; Petschauer et al. 2017; Doi et al. 2012) or a change in the nature of hadrons at finite density (Rikovska-Stone et al. 2007). Yet experimental information is scarce, there

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is much uncertainty, and one needs to introduce some assumptions to give threebody interactions involving hyperons. For example, in Gerstung et al. (2020), the  potential in nuclear matter has been obtained in the chiral EFT framework using the decuplet baryon dominance assumption on the NNLO diagrams generating the three-baryon forces (Petschauer et al. 2017). It was found that there is a low-energy constant range where the  baryons do not appear in pure neutron matter. The middle panel of Fig. 16 compares the  single particle energy at zero momentum (μ (ρ) ≡ M + U (ρ)) in comparison with the neutron chemical potential μn in neutron star matter. While μn overtakes μ and  will be generated at ρ/ρ0 ≥ 2.5 only with N interaction, the NN three-body repulsion pushes up μ and avoids the  appearance in neutron stars. In both phenomenological and first-principles approaches, there are parameters introduced and tuned to support 2 M neutron stars, and they should be examined in experiments or in ab initio calculations. Three-body interactions among baryons are in principle possible to examine by using lattice QCD calculations (Doi et al. 2012), but it requires huge computer resources and several years or more are necessary to finalize. In experiments, high precision measurement of hypernuclear binding energies in a wide range of mass region may help to determine the three-baryon forces and is planned using the future facility (Takahashi 2019). Femtoscopic study of hyperon-nucleus interactions and hyperon collective flows may be also helpful. Another promising mechanism to solve the hyperon puzzle is to assume the transition to quark matter. Provided that the QCD phase transition in cold dense matter is a smooth crossover, the EOS is not necessarily softened and can be compatible with 2 M neutron stars (Annala et al. 2018; Masuda et al. 2013; Kurkela et al. 2014; Baym et al. 2018). The basic idea to construct the crossover EOS is to interpolate the EOSs in low-density nuclear matter and high-density quark matter, by using, for example, the squared speed of sound, cs2 = dP /dε (Tews et al. 2018):

cs2 (ε)

dP 2 1 , cs  = dε 3



pf Mn

2 (non-rel. free PNM), cs2 =

1 (massless free), 3 (55)

where pf is the Fermi momentum. The nuclear matter EOS at low density is relatively well known, and the quark matter EOS will be approximated as massless free fermion gas in the high-density limit. Then one can interpolate cs2 as schematically shown in the left panel of Fig. 17. Once the squared speed of sound is given, cs2 (ε), the pressure is calculated by integrating cs2 with respect to the energy density and one obtains the EOS, P = P (ε). The interpolation and extrapolation using cs2 or others have been used by many groups. Let us comment on some of the findings in these studies. The piecewise polytropes, P (ρ) = Ki ρ Γi (ρi−1 ≤ ρ ≤ ρi ), are used to deduce the EOS via the analysis of probability distributions of (M, R) from the X-ray bursts (Read et al. 2009; Ozel et al. 2010). In Steiner et al. (2010), by using the Bayesian method with

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Fig. 17 Left: Conceptual sketch of interpolation of the speed of sound. Middle: Speed of sound in the quarkyonic transition model. (Taken from McLerran and Reddy 2019). Right: EOS of neutron star matter. (Taken from Annala et al. 2018)

the prior limits of nuclear matter parameters such as the incompressibility, symmetry energy parameters have been obtained from the (M, R) distributions from the X-ray bursts. Extrapolation to higher densities from the chiral EFT EOS was considered by using cs2 in Tews et al. (2018). In Annala et al. (2018), Masuda et al. (2013), Kurkela et al. (2014), and Baym et al. (2018), the interpolation of the low-density and high-density EOSs has been performed. The right panel of Fig. 17 shows the EOS from the interpolation of the nuclear matter EOS and the pQCD EOS with the constraint Mmax ≥ 2.0 M (Annala et al. 2018). The green area shows the region with Λ < 400 as suggested by the gravitational wave observation (Abbott et al. 2019). This region seems to agree with the EOS from deep learning of the (M, R) distribution, tidal deformability, and NICER data (Fujimoto et al. 2021). At very high densities, cs2 (ε) is predicted to converge to 1/3 from below in the quarkyonic model (McLerran and Reddy 2019; Zhao and Lattimer 2020) and in the NJL model (Masuda et al. 2013; Baym et al. 2018). By comparison, it is found to converge from above from the deep learning (Fujimoto et al. 2021). It should be remembered that the first order phase transition in neutron star matter is not denied yet, and if it is the case, the third family of compact stars (quark stars) in addition to two families (white dwarfs and neutron stars) may exist (Bonanno and Sedrakian 2012; Bejger et al. 2017). In this case, the third family is expected to have considerably smaller radii than the second branch (neutron stars). However, the observation by NICER seems to suggest the radii of neutron stars are similar for M = 1.4 M and M = 2.08 M neutron stars. For more discussions and recent developments on the neutron star matter EOSs, readers are referred to recent reviews (Oertel et al. 2017; Lattimer 2021).

Summary Physics of hadrons, quark-gluon plasma (QGP), and neutron stars are reviewed. In these subjects, there are many unsolved problems including existence and structure of exotic hadron states, early thermalization and QCD phase diagram, and symmetry energy and hyperon puzzle. Hadron (and parton) interactions and their density

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dependence are the keys to answer these problems, and these are also the important subjects in quark-nuclear physics. This work is supported in part by the Grants-in-Aid for Scientific Research from JSPS (No. JP19H01898 and No. JP21H00121) and by the Yukawa International Program for Quark-Hadron Sciences.

References R. Aaij et al. [LHCb], Phys. Rev. Lett. 115, 072001 (2015). https://doi.org/10.1103/PhysRevLett. 115.072001, [1507.03414]; Phys. Rev. Lett. 122, 222001 (2019). https://doi.org/10.1103/ PhysRevLett.122.222001, [1904.03947] R. Aaij et al. [LHCb], Nat. Phys. 18, 751 (2022).https://doi.org/10.1038/s41567-022-01614-y, [2109.01038] G. Aarts, E. Seiler, I.O. Stamatescu, Phys. Rev. D 81, 054508 (2010). https://doi.org/10.1103/ PhysRevD.81.054508, [0912.3360]; K. Nagata, J. Nishimura, S. Shimasaki, Phys. Rev. D 94, 114515 (2016). https://doi.org/10.1103/PhysRevD.94.114515, [1606.07627] B.P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 119, 161101 (2017). https://doi. org/10.1103/PhysRevLett.119.161101, [1710.05832] B.P. Abbott et al. [LIGO Scientific and Virgo] Phys. Rev. X 9, 011001 (2019). https://doi.org/10. 1103/PhysRevX.9.011001, [1805.11579] B.B. Abelev et al. [ALICE], Phys. Lett. B 734, 314 (2014). https://doi.org/10.1016/j.physletb.2014. 05.064, [1311.0214] S. Abrahamyan et al., Phys. Rev. Lett. 108, 112502 (2012). https://doi.org/10.1103/PhysRevLett. 108.112502, [1201.2568] M.C. Abreu et al. [NA50], Phys. Lett. B 477, 28 (2000). https://doi.org/10.1016/S03702693(00)00237-9 S. Acharya et al. [ALICE], Phys. Rev. D 106, 052010 (2022). https://doi.org/10.1103/PhysRevD. 106.052010, [2201.05352] S. Acharya et al. [ALICE], Nature 588, 232 (2020); [erratum: Nature 590, E13 (2021)]. https://doi. org/10.1038/s41586-020-3001-6, [2005.11495] S. Acharya et al. [ALICE], Phys. Rev. Lett. 123, 112002 (2019). https://doi.org/10.1103/ PhysRevLett.123.112002, [1904.12198] S. Acharya et al. [ALICE], Phys. Rev. Lett. 124, 092301 (2020a). https://doi.org/10.1103/ PhysRevLett.124.092301, [1905.13470]; Phys. Lett. B 822, 136708 (2021). https://doi.org/10. 1016/j.physletb.2021.136708, [2105.05683] S. Acharya et al. [ALICE], Phys. Lett. B 805, 135419 (2020b). https://doi.org/10.1016/j.physletb. 2020.135419, [1910.14407];(pΣ 0 ), Phys. Rev. Lett. 127, 172301 (2021). https://doi.org/10. 1103/PhysRevLett.127.172301, [2105.05578];(pφ), Phys. Lett. B 790, 22 (2019). https://doi. org/10.1016/j.physletb.2018.12.033, [1809.07899]; (KS0 K ± ), Phys. Lett. B 774, 64 (2017). https://doi.org/10.1016/j.physletb.2017.09.009, [1705.04929](KS0 K ± ) L. Adamczyk et al. [STAR], Phys. Rev. Lett. 112, 032302 (2014). https://doi.org/10.1103/ PhysRevLett.112.032302, [1309.5681] L. Adamczyk et al. [STAR], Phys. Rev. Lett. 112, 162301 (2014). https://doi.org/10.1103/ PhysRevLett.112.162301, [1401.3043] L. Adamczyk et al. [STAR], Phys. Rev. Lett. 114, 022301 (2015). https://doi.org/10.1103/ PhysRevLett.114.022301, [1408.4360]; S. Acharya et al. [ALICE], Phys. Rev. C 99, 024001 (2019). https://doi.org/10.1103/PhysRevC.99.024001, [1805.12455]; Phys. Lett. B 797, 134822 (2019). https://doi.org/10.1016/j.physletb.2019.134822, [1905.07209] L. Adamczyk et al. [STAR], Phys. Rev. C 96, 044904 (2017). https://doi.org/10.1103/PhysRevC. 96.044904, [1701.07065]; X. Luo, Nucl. Phys. A 956, 75 (2016). https://doi.org/10.1016/j. nuclphysa.2016.03.025, [1512.09215]

85 Hadrons, Quark-Gluon Plasma, and Neutron Stars

3115

J. Adam et al. [STAR], Phys. Lett. B 790, 490 (2019). https://doi.org/10.1016/j.physletb.2019.01. 055, [1808.02511] J. Adam et al. [STAR], Phys. Rev. Lett. 126, 092301 (2021). https://doi.org/10.1103/PhysRevLett. 126.092301, [2001.02852] J. Adams et al. [STAR], Phys. Rev. Lett. 91, 072304 (2003). https://doi.org/10.1103/PhysRevLett. 91.072304, [nucl-ex/0306024] J. Adams et al. [STAR], Nucl. Phys. A 757, 102 (2005). https://doi.org/10.1016/j.nuclphysa.2005. 03.085, [nucl-ex/0501009] J. Adams et al. [STAR], Phys. Rev. C 71, 064902 (2005). https://doi.org/10.1103/PhysRevC.71. 064902, [nucl-ex/0412019] J. Adams et al. [STAR], Phys. Rev. C 74, 064906 (2006). [nucl-ex/0511003]; V.M. Shapoval, B. Erazmus, R. Lednicky, Y.M. Sinyukov, Phys. Rev. C 92, 034910 (2015). [1405.3594]; J. Adamczewski-Musch et al. [HADES], Phys. Rev. C 94, 025201 (2016). https://doi.org/10. 1103/PhysRevC.94.025201, [1602.08880] A. Adare et al. [PHENIX], Phys. Rev. Lett. 98, 232301 (2007). https://doi.org/10.1103/ PhysRevLett.98.232301, [nucl-ex/0611020]; L. Adamczyk et al. [STAR], Phys. Lett. B 771, 13 (2017). https://doi.org/10.1016/j.physletb.2017.04.078, [1607.07517] K. Adcox et al. [PHENIX], Nucl. Phys. A 757, 184 (2005). https://doi.org/10.1016/j.nuclphysa. 2005.03.086, [nucl-ex/0410003] D. Adhikari et al. [PREX]. Phys. Rev. Lett. 126, 172502 (2021). https://doi.org/10.1103/ PhysRevLett.126.172502, [2102.10767] A. Adil, M. Gyulassy, T. Hirano, Phys. Rev. D 73, 074006 (2006). https://doi.org/10.1103/ PhysRevD.73.074006, [nucl-th/0509064]; P. Bozek, I. Wyskiel, Phys. Rev. C 81, 054902 (2010). https://doi.org/10.1103/PhysRevC.81.054902, [1002.4999] S. Afanasiev et al. [PHENIX], Phys. Rev. Lett. 99, 052301 (2007). https://doi.org/10.1103/ PhysRevLett.99.052301, [nucl-ex/0703024] H. Akikawa et al., Phys. Rev. Lett. 88, 082501 (2002). https://doi.org/10.1103/PhysRevLett.88. 082501 A. Akmal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C 58, 1804 (1998). https://doi.org/10. 1103/PhysRevC.58.1804, [nucl-th/9804027] M.G. Alford et al., Rev. Mod. Phys. 80, 1455 (2008). https://doi.org/10.1103/RevModPhys.80. 1455, [0709.4635]; E.V. Shuryak, Phys. Rept. 61, 71 (1980). https://doi.org/10.1016/03701573(80)90105-2 C.R. Allton et al., Phys. Rev. D 71, 054508 (2005). https://doi.org/10.1103/PhysRevD.71.054508, [hep-lat/0501030] C. Alt et al. [NA49], Phys. Rev. C 68, 034903 (2003). https://doi.org/10.1103/PhysRevC.68. 034903, [nucl-ex/0303001] A. Andronic et al., Eur. Phys. J. C 76, 107 (2016). https://doi.org/10.1140/epjc/s10052-015-38195, [1506.03981] A. Andronic et al., Nature 561, 321 (2018). https://doi.org/10.1038/s41586-018-0491-6, [1710.09425] E. Annala et al., Phys. Rev. Lett. 120, 172703 (2018). https://doi.org/10.1103/PhysRevLett.120. 172703, [1711.02644] J. Antoniadis et al., Science 340, 6131 (2013). https://doi.org/10.1126/science.1233232, [1304.6875] Y. Aoki et al., Nature 443, 675 (2006). https://doi.org/10.1038/nature05120, [hep-lat/0611014] S. Aoki et al. [PACS-CS], Phys. Rev. D 79, 034503 (2009). https://doi.org/10.1103/PhysRevD.79. 034503, [0807.1661]; A. Bazavov et al. [MILC], Rev. Mod. Phys. 82, 1349 (2010). https://doi. org/10.1103/RevModPhys.82.1349, [0903.3598] P.B. Arnold et al., Phys. Rev. Lett. 94, 072302 (2005). https://doi.org/10.1103/PhysRevLett. 94.072302, [nucl-th/0409068]; T. Lappi, L. McLerran, Nucl. Phys. A 772, 200 (2006). https://doi.org/10.1016/j.nuclphysa.2006.04.001, [hep-ph/0602189]; P. Romatschke, R. Venugopalan, Phys. Rev. D 74, 045011 (2006). https://doi.org/10.1103/PhysRevD.74.045011, [hepph/0605045]; K. Fukushima, F. Gelis, Nucl. Phys. A 874, 108 (2012). https://doi.org/10.1016/j. nuclphysa.2011.11.003, [1106.1396]

3116

A. Ohnishi

I. Arsene et al. [BRAHMS], Nucl. Phys. A 757, 1 (2005). https://doi.org/10.1016/j.nuclphysa.2005. 02.130, [nucl-ex/0410020] Z. Arzoumanian et al. [NANOGrav]. Astrophys. J. Suppl. 235, 37 (2018). https://doi.org/10.3847/ 1538-4365/aab5b0, [1801.01837] M. Asakawa, K. Yazaki, Nucl. Phys. A 504, 668 (1989). https://doi.org/10.1016/03759474(89)90002-X M. Asakawa, S.A. Bass, B. Muller, Phys. Rev. Lett. 96, 252301 (2006). https://doi.org/10.1103/ PhysRevLett.96.252301, [hep-ph/0603092]; Prog. Theor. Phys. 116, 725 (2007). https://doi.org/ 10.1143/PTP.116.725, [hep-ph/0608270](inspire) B. Aubert et al. [BaBar], Phys. Rev. Lett. 90 (2003), 242001. https://doi.org/10.1103/PhysRevLett. 90.242001, [hep-ex/0304021] B. B. Back et al. [PHOBOS], Nucl. Phys. A 757, 28 (2005). https://doi.org/10.1016/j.nuclphysa. 2005.03.084, [nucl-ex/0410022] R. Baier et al., Phys. Lett. B 502, 51 (2001). https://doi.org/10.1016/S0370-2693(01)00191-5, [hep-ph/0009237] N. Baillot d’Etivaux et al., Astrophys. J. 887, 48 (2019). https://doi.org/10.3847/1538-4357/ab4f6c, [1905.01081] M. Baldo, G.F. Burgio, H.J. Schulze, Phys. Rev. C 61, 055801 (2000). https://doi.org/10.1103/ PhysRevC.61.055801, [nucl-th/9912066]; S. Nishizaki, T. Takatsuka, Y. Yamamoto, Prog. Theor. Phys. 108, 703 (2002). https://doi.org/10.1143/PTP.108.703 G.S. Bali, K. Schilling, C. Schlichter, Phys. Rev. D 51, 5165 (1995). https://doi.org/10.1103/ PhysRevD.51.5165, [hep-lat/9409005] V.E. Barnes et al., Phys. Rev. Lett. 12, 204 (1964). https://doi.org/10.1103/PhysRevLett.12.204 B.C. Barrois, Nucl. Phys. B 129, 390 (1977). https://doi.org/10.1016/0550-3213(77)90123-7; D. Bailin, A. Love, Phys. Rept. 107, 325 (1984). https://doi.org/10.1016/0370-1573(84)90145-5; M.G. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. B 422, 247 (1998). https://doi.org/10.1016/ S0370-2693(98)00051-3, [hep-ph/9711395]; Nucl. Phys. B 537, 443 (1999). https://doi.org/10. 1016/S0550-3213(98)00668-3, [hep-ph/9804403]; M. Kitazawa et al., Prog. Theor. Phys. 108, 929 (2002); [erratum: Prog. Theor. Phys. 110, 185 (2003)]. https://doi.org/10.1143/PTP.108. 929, [hep-ph/0207255] S.A. Bass et al., Prog. Part. Nucl. Phys. 41, 255 (1998). https://doi.org/10.1016/S01466410(98)00058-1, [nucl-th/9803035]; Y. Nara et al., Phys. Rev. C 61, 024901 (2000). https:// doi.org/10.1103/PhysRevC.61.024901, [nucl-th/9904059]; M. Isse et al., Phys. Rev. C 72, 064908 (2005). https://doi.org/10.1103/PhysRevC.72.064908, [nucl-th/0502058]; W. Cassing, E.L. Bratkovskaya, Nucl. Phys. A 831, 215 (2009). https://doi.org/10.1016/j.nuclphysa.2009. 09.007, [0907.5331] G. Baym, Nucl. Phys. A 418, 433C (1984). https://doi.org/10.1016/0375-9474(84)90566-9 G. Baym et al., Rept. Prog. Phys. 81, 056902 (2018). https://doi.org/10.1088/1361-6633/aaae14, [1707.04966] A. Bazavov et al., Phys. Rev. D 85, 054503 (2012). https://doi.org/10.1103/PhysRevD.85.054503, [1111.1710] A. Bazavov et al. [HotQCD], Phys. Rev. D 90, 094503 (2014). https://doi.org/10.1103/PhysRevD. 90.094503, [1407.6387] M. Bazzi et al. [Siddharta], Phys. Lett. B 704, 113 (2011) https://doi.org/10.1016/j.physletb.2011. 09.011, [1105.3090] S.R. Beane et al., Phys. Rev. Lett. 97, 012001 (2006). https://doi.org/10.1103/PhysRevLett.97. 012001, [hep-lat/0602010] F. Becattini et al., Phys. Rev. C 64, 024901 (2001). https://doi.org/10.1103/PhysRevC.64.024901, [hep-ph/0002267]; J. Cleymans et al., Phys. Rev. C 73, 034905 (2006). https://doi.org/10.1103/ PhysRevC.73.034905, [hep-ph/0511094]; F. Becattini, J. Manninen, M. Gazdzicki, Phys. Rev. C 73, 044905 (2006). https://doi.org/10.1103/PhysRevC.73.044905, [hep-ph/0511092] M. Bejger et al., Astron. Astrophys. 600, A39 (2017). https://doi.org/10.1051/0004-6361/ 201629580, [1608.07049]

85 Hadrons, Quark-Gluon Plasma, and Neutron Stars

3117

L. Bonanno, A. Sedrakian, Astron. Astrophys. 539, A16 (2012). https://doi.org/10.1051/00046361/201117832, [1108.0559] S. Borsanyi et al., Phys. Rev. Lett. 111, 062005 (2013). https://doi.org/10.1103/PhysRevLett.111. 062005, [1305.5161] S. Borsanyi et al., Phys. Lett. B 730, 99 (2014). https://doi.org/10.1016/j.physletb.2014.01.007, [1309.5258] N. Brambilla et al., Eur. Phys. J. C 74, 2981 (2014). https://doi.org/10.1140/epjc/s10052-0142981-5, [1404.3723] R. Brockmann, R. Machleidt, Phys. Lett. B 149, 283 (1984). https://doi.org/10.1016/03702693(84)90407-6; G.Q. Li, R. Machleidt, R. Brockmann, Phys. Rev. C 45, 2782 (1992). https:// doi.org/10.1103/PhysRevC.45.2782 R. Hanbury Brown, R.Q. Twiss, Nature 178, 1046 (1956). https://doi.org/10.1038/1781046a0 G.F. Burgio et al., Prog. Part. Nucl. Phys. 120, 103879 (2021). https://doi.org/10.1016/j.ppnp.2021. 103879, [2105.03747] J. Carlson et al., Rev. Mod. Phys. 87, 1067 (2015). https://doi.org/10.1103/RevModPhys.87.1067, [1412.3081] S. Chatrchyan et al. [CMS], Phys. Rev. Lett. 109, 222301 (2012); [erratum: Phys. Rev. Lett. 120, 199903 (2018)]. https://doi.org/10.1103/PhysRevLett.109.222301, [1208.2826] L.W. Chen et al., Phys. Rev. C 82, 024321 (2010). https://doi.org/10.1103/PhysRevC.82.024321, [1004.4672] M.K. Cheoun et al., JCAP 10, 021 (2013). https://doi.org/10.1088/1475-7516/2013/10/021, [1304.1871]; A.V. Astashenok, S. Capozziello, S.D. Odintsov, Phys. Rev. D 89, 103509 (2014). https://doi.org/10.1103/PhysRevD.89.103509, [1401.4546] G.F. Chew, S.C. Frautschi, Phys. Rev. Lett. 8, 41 (1962). https://doi.org/10.1103/PhysRevLett.8.41 S. Cho, S.H. Lee, Phys. Rev. C 97, 034908 (2018). https://doi.org/10.1103/PhysRevC.97.034908, [1509.04092] A. Chodos et al., Phys. Rev. D 9, 3471 (1974). https://doi.org/10.1103/PhysRevD.9.3471 S. Cho et al. [ExHIC], Phys. Rev. Lett. 106, 212001 (2011). https://doi.org/10.1103/PhysRevLett. 106.212001, [1011.0852]; Phys. Rev. C 84, 064910 (2011). https://doi.org/10.1103/PhysRevC. 84.064910, [1107.1302]; Prog. Part. Nucl. Phys. 95, 279 (2017). https://doi.org/10.1016/j.ppnp. 2017.02.002, [1702.00486] S.K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003). https://doi.org/10.1103/PhysRevLett. 91.262001, [hep-ex/0309032] F. Cooper, G. Frye, Phys. Rev. D 10, 186 (1974). https://doi.org/10.1103/PhysRevD.10.186 H.T. Cromartie et al. [NANOGrav]. Nature Astron. 4, 72 (2019). https://doi.org/10.1038/s41550019-0880-2, [1904.06759] Y. Cui et al., HEPNP 31, 7 (2007). [hep-ph/0607226] P. Danielewicz, M. Gyulassy, Phys. Rev. D 31, 53 (1985). https://doi.org/10.1103/PhysRevD.31. 53 P. Danielewicz, P. Singh, J. Lee, Nucl. Phys. A 958, 147 (2017). https://doi.org/10.1016/j. nuclphysa.2016.11.008, [1611.01871] S. De et al., Phys. Rev. Lett. 121, 091102 (2018); [erratum: Phys. Rev. Lett. 121, 259902 (2018)]. https://doi.org/10.1103/PhysRevLett.121.091102, [1804.08583] P. de Forcrand, PoS LAT2009, 010 (2009). https://doi.org/10.22323/1.091.0010, [1005.0539] P. de Forcrand, M. D’Elia, PoS LATTICE2016, 081 (2017). https://doi.org/10.22323/1.256.0081, [1702.00330] P. de Forcrand, O. Philipsen, Nucl. Phys. B 642, 290 (2002). https://doi.org/10.1016/S05503213(02)00626-0, [hep-lat/0205016]; M. D’Elia, M.P. Lombardo, Phys. Rev. D 67, 014505 (2003). https://doi.org/10.1103/PhysRevD.67.014505, [hep-lat/0209146] P. Demorest et al., Nature 467, 1081 (2010). https://doi.org/10.1038/nature09466, [1010.5788] A. De Rujula, H. Georgi, S.L. Glashow, Phys. Rev. D 12, 147 (1975) J.J. de Swart, Rev. Mod. Phys. 35, 916 (1963); [erratum: Rev. Mod. Phys. 37, 326 (1965)]. https:// doi.org/10.1103/RevModPhys.35.916

3118

A. Ohnishi

T. Doi et al. [HAL QCD]. Prog. Theor. Phys. 127, 723 (2012). https://doi.org/10.1143/PTP.127. 723, [1106.2276] G. Ecker, Prog. Part. Nucl. Phys. 35, 1 (1995). https://doi.org/10.1016/0146-6410(95)00041-G, [hep-ph/9501357] E. Eichten et al., Phys. Rev. Lett. 34, 369 (1975); [erratum: Phys. Rev. Lett. 36, 1276 (1976)]. https://doi.org/10.1103/PhysRevLett.34.369 T. Epelbaum, F. Gelis, Phys. Rev. Lett. 111, 232301 (2013). https://doi.org/10.1103/PhysRevLett. 111.232301, [1307.2214] E. Epelbaum, H.W. Hammer, U.G. Meissner, Rev. Mod. Phys. 81, 1773 (2009). https://doi.org/10. 1103/RevModPhys.81.1773, [0811.1338] L. Fabbietti, V. Mantovani Sarti, O. Vazquez Doce, Ann. Rev. Nucl. Part. Sci. 71, 377 (2021). https://doi.org/10.1146/annurev-nucl-102419-034438, [2012.09806] Z. Fodor, S.D. Katz, JHEP 04, 050 (2004). https://doi.org/10.1088/1126-6708/2004/04/050, [heplat/0402006] Z. Fodor et al., Phys. Rev. D 92, 094516 (2015). https://doi.org/10.1103/PhysRevD.92.094516, [1508.05260] E. Fonseca et al., Astrophys. J. 832, 167 (2016). https://doi.org/10.3847/0004-637X/832/2/167, [1603.00545] P.C.C. Freire et al., Mon. Not. Roy. Astron. Soc. 412, 2763 (2011). https://doi.org/10.1111/j.13652966.2010.18109.x, [1011.5809] B. Friedman, V.R. Pandharipande, Nucl. Phys. A 361, 502 (1981). https://doi.org/10.1016/03759474(81)90649-7 Y. Fujimoto, K. Fukushima, K. Murase, JHEP 03, 273 (2021). https://doi.org/10.1007/ JHEP03(2021)273, [2101.08156] M. Fukugita et al., Phys. Rev. D 52, 3003 (1995). https://doi.org/10.1103/PhysRevD.52.3003, [heplat/9501024] M. Fukuma, N. Umeda, PTEP 2017, 073B01 (2017). https://doi.org/10.1093/ptep/ptx081, [1703.00861] K. Fukushima, T. Hatsuda, Rept. Prog. Phys. 74, 014001 (2011). https://doi.org/10.1088/00344885/74/1/014001, [1005.4814] A. Gal, E.V. Hungerford, D.J. Millener, Rev. Mod. Phys. 88, 035004 (2016). https://doi.org/10. 1103/RevModPhys.88.035004, [1605.00557]; E. Hiyama, Few Body Syst. 53, 189 (2012). https://doi.org/10.1007/s00601-011-0296-8 J. Gasser, H. Leutwyler, Ann. Phys. 158, 142 (1984). https://doi.org/10.1016/00034916(84)90242-2 M. Gell-Mann, Synchrotron Laboratory Report CTSL-20 (California Institute of Technology, 1961). https://doi.org/10.2172/4008239; S. Okubo, Prog. Theor. Phys. 27, 949 (1962). https:// doi.org/10.1143/PTP.27.949 M. Gell-Mann, Phys. Lett. 8, 214 (1964) https://doi.org/10.1016/S0031-9163(64)92001-3; G. Zweig, Part of Developments in the Quark Theory of Hadrons, vol. 1. 1964–1978, 22–101. CERN-TH-412 D. Gerstung, N. Kaiser, W. Weise, Eur. Phys. J. A 56, 175 (2020). https://doi.org/10.1140/epja/ s10050-020-00180-2, [2001.10563] N.K. Glendenning, S.A. Moszkowski, Phys. Rev. Lett. 67, 2414 (1991). https://doi.org/10.1103/ PhysRevLett.67.2414; J. Schaffner, I.N. Mishustin, Phys. Rev. C 53, 1416 (1996). https://doi. org/10.1103/PhysRevC.53.1416, [nucl-th/9506011]; K. Tsubakihara et al., Phys. Rev. C 81, 065206 (2010). https://doi.org/10.1103/PhysRevC.81.065206, [0909.5058] G. Goldhaber, S. Goldhaber, W.Y. Lee, A. Pais, Phys. Rev. 120, 300 (1960). https://doi.org/10. 1103/PhysRev.120.300 J. Goldstone, Nuovo Cim. 19, 154 (1961). https://doi.org/10.1007/BF02812722; J. Goldstone, A. Salam, S. Weinberg, Phys. Rev. 127, 965 (1962). https://doi.org/10.1103/PhysRev.127.965 S. Gongyo et al. [HAL QCD], Phys. Rev. Lett. 120, 212001 (2018). https://doi.org/10.1103/ PhysRevLett.120.212001, [1709.00654]; T. Iritani et al. [HAL QCD], Phys. Lett. B 792, 284 (2019). https://doi.org/10.1016/j.physletb.2019.03.050, [1810.03416]

85 Hadrons, Quark-Gluon Plasma, and Neutron Stars

3119

L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rept. 100, 1 (1983). https://doi.org/10.1016/03701573(83)90022-4; F. Gelis et al., Ann. Rev. Nucl. Part. Sci. 60, 463 (2010). https://doi.org/10. 1146/annurev.nucl.010909.083629, [1002.0333] S. Guillot et al., Astrophys. J. 772, 7 (2013). https://doi.org/10.1088/0004-637X/772/1/7, [1302.0023] F.K. Guo et al., Rev. Mod. Phys. 90, 015004 (2018). [1705.00141] F.K. Guo, X.H. Liu, S. Sakai, Prog. Part. Nucl. Phys. 112, 103757 (2020). https://doi.org/10.1016/ j.ppnp.2020.103757, [1912.07030] T. Guver et al., Astrophys. J. 719, 1807 (2010). https://doi.org/10.1088/0004-637X/719/2/1807, [1002.3825] J. Haidenbauer, U.G. Meissner, Phys. Rev. C 72, 044005 (2005). https://doi.org/10.1103/ PhysRevC.72.044005, [nucl-th/0506019] J. Haidenbauer, S. Petschauer, N. Kaiser, U.G. Meissner, A. Nogga, W. Weise, Nucl. Phys. A 915, 24 (2013). https://doi.org/10.1016/j.nuclphysa.2013.06.008, [1304.5339]; J. Haidenbauer, U.G. Meißner, A. Nogga, Eur. Phys. J. A 56, 91 (2020). https://doi.org/10.1140/epja/s10050-02000100-4, [1906.11681] A. Hasenfratz, D. Toussaint, Nucl. Phys. B 371, 539 (1992). https://doi.org/10.1016/05503213(92)90247-9; S. Ejiri, Phys. Rev. D 78, 074507 (2008). https://doi.org/10.1103/PhysRevD. 78.074507, [0804.3227]; A. Li, A. Alexandru, K.F. Liu, Phys. Rev. D 84, 071503 (2011). https:// doi.org/10.1103/PhysRevD.84.071503, [1103.3045]; A. Nakamura, S. Oka, Y. Taniguchi, JHEP 02, 054 (2016). https://doi.org/10.1007/JHEP02(2016)054, [1504.04471] T. Hashimoto, K. Hirose, T. Kanki, O. Miyamura, Phys. Rev. Lett. 57, 2123 (1986). https://doi.org/ 10.1103/PhysRevLett.57.2123 T. Hatsuda, T. Kunihiro, Phys. Rept. 247, 221 (1994). https://doi.org/10.1016/03701573(94)90022-1, [hep-ph/9401310]; U. Vogl, W. Weise, Prog. Part. Nucl. Phys. 27, 195 (1991). https://doi.org/10.1016/0146-6410(91)90005-9; V. Bernard, U.G. Meissner, I. Zahed, Phys. Rev. D 36, 819 (1987). https://doi.org/10.1103/PhysRevD.36.819 U.W. Heinz, P.F. Kolb, Nucl. Phys. A 702, 269 (2002). https://doi.org/10.1016/S03759474(02)00714-5, [hep-ph/0111075] T. Hinderer et al., Phys. Rev. D 81, 123016 (2010). https://doi.org/10.1103/PhysRevD.81.123016, [0911.3535] R.A. Hulse, J.H. Taylor, Astrophys. J. Lett. 195, L51 (1975). https://doi.org/10.1086/181708 T. Hyodo, D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012). https://doi.org/10.1016/j.ppnp.2011.07. 002, [1104.4474] T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C 85, 015201 (2012). https://doi.org/10.1103/PhysRevC. 85.015201, [1108.5524] K. Ikeda, N. Takigawa, H. Horiuchi, Prog. Theor. Phys. Suppl. E68, 464 (1968) Y. Ikeda, T. Hyodo, W. Weise, Nucl. Phys. A 881, 98 (2012). https://doi.org/10.1016/j.nuclphysa. 2012.01.029, [1201.6549]; K. Miyahara, T. Hyodo, W. Weise, Phys. Rev. C 98, 025201 (2018). https://doi.org/10.1103/PhysRevC.98.025201, [1804.08269] T. Inoue et al. [HAL QCD], Phys. Rev. Lett. 106, 162002 (2011). https://doi.org/10.1103/ PhysRevLett.106.162002, [1012.5928] N. Isgur, G. Karl, Phys. Lett. B 72, 109 (1977); Phys. Rev. D 18, 4187 (1978); Phys. Rev. D 19, 2653 (1979); [erratum: Phys. Rev. D 23, 817 (1981)]. https://doi.org/10.1103/PhysRevD. 19.2653; S. Capstick, N. Isgur, AIP Conf. Proc. 132, 267 (1985). https://doi.org/10.1103/ PhysRevD.34.2809; S. Godfrey, N. Isgur, Phys. Rev. D 32, 189 (1985). https://doi.org/10.1103/ PhysRevD.32.189 N. Ishii, S. Aoki, T. Hatsuda, Phys. Rev. Lett. 99, 022001 (2007). https://doi.org/10.1103/ PhysRevLett.99.022001, [nucl-th/0611096]; S. Aoki, T. Hatsuda, N. Ishii, Prog. Theor. Phys. 123, 89 (2010). https://doi.org/10.1143/PTP.123.89, [0909.5585] N. Ishii et al. [HAL QCD], Phys. Lett. B 712, 437 (2012). https://doi.org/10.1016/j.physletb.2012. 04.076, [1203.3642] C. Ishizuka et al., J. Phys. G 35, 085201 (2008). https://doi.org/10.1088/0954-3899/35/8/085201, [0802.2318]

3120

A. Ohnishi

R.L. Jaffe, Phys.Rev. D 15, 267 (1977). https://doi.org/10.1103/PhysRevD.15.267 G. Jhang et al. [Sπ RIT and TMEP], Phys. Lett. B 813, 136016 (2021). https://doi.org/10.1016/j. physletb.2020.136016, [2012.06976] Y. Kamiya et al., Phys. Rev. Lett. 124, 132501 (2020). https://doi.org/10.1103/PhysRevLett.124. 132501, [1911.01041] Y. Kamiya et al., Phys. Rev. C 105, 014915 (2022). https://doi.org/10.1103/PhysRevC.105.014915, [2108.09644] Y. Kanada-En’yo, B. Muller, Phys. Rev. C 74, 061901 (2006). https://doi.org/10.1103/PhysRevC. 74.061901, [nucl-th/0608015] D.B. Kaplan, A.E. Nelson, Phys. Lett. B 175, 57 (1986). https://doi.org/10.1016/03702693(86)90331-X; G.E. Brown et al., Nucl. Phys. A 567, 937 (1994). https://doi.org/10.1016/ 0375-9474(94)90335-2, [hep-ph/9304204]; V. Thorsson, M. Prakash, J.M. Lattimer, Nucl. Phys. A 572, 693 (1994); [erratum: Nucl. Phys. A 574, 851 (1994)]. https://doi.org/10.1016/ 0375-9474(94)90407-3, [nucl-th/9305006]; T. Muto, T. Tatsumi, Phys. Lett. B 283, 165 (1992). https://doi.org/10.1016/0370-2693(92)90001-K J.I. Kapusta, C. Gale, Finite-temperature Field Theory: Principles and Applications (Cambridge, 2006). https://doi.org/10.1017/CBO9780511535130 M. Karliner, J.L. Rosner, Phys. Rev. Lett. 119, 202001 (2017). https://doi.org/10.1103/ PhysRevLett.119.202001, [1707.07666] M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 44, 1422 (1970). https://doi.org/10.1143/PTP.44. 1422; M. Kobayashi, H. Kondo, T. Maskawa, Prog. Theor. Phys. 45, 1955 (1971) https://doi.org/ 10.1143/PTP.45.1955; G. ’t Hooft, Phys. Rev. D 14, 3432 (1976); [erratum: Phys. Rev. D 18, 2199 (1978)] https://doi.org/10.1103/PhysRevD.14.3432; Phys. Rept. 142, 357 (1986). https:// doi.org/10.1016/0370-1573(86)90117-1; M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 163, 46 (1980). https://doi.org/10.1016/0550-3213(80)90389-2 S.E. Koonin, Phys. Lett. B 70, 43 (1977). https://doi.org/10.1016/0370-2693(77)90340-9; S. Pratt, Phys. Rev. D 33, 1314 (1986). https://doi.org/0.1103/PhysRevD.33.1314 M. Kortelainen et al., Phys. Rev. C 82, 024313 (2010). https://doi.org/10.1103/PhysRevC.82. 024313, [1005.5145] P. Kovtun, D.T. Son, A.O. Starinets, Phys. Rev. Lett. 94, 111601 (2005). https://doi.org/10.1103/ PhysRevLett.94.111601, [hep-th/0405231] A. Kurkela, Y. Zhu, Phys. Rev. Lett. 115, 182301 (2015). https://doi.org/10.1103/PhysRevLett. 115.182301, [1506.06647] A. Kurkela et al., Astrophys. J. 789, 127 (2014). https://doi.org/10.1088/0004-637X/789/2/127, [1402.6618] E. Laermann, O. Philipsen, Ann. Rev. Nucl. Part. Sci. 53, 163 (2003). https://doi.org/10.1146/ annurev.nucl.53.041002.110609, [hep-ph/0303042] D.E. Lanskoy, Y. Yamamoto, Phys. Rev. C 55, 2330 (1997). https://doi.org/10.1103/PhysRevC.55. 2330 J.M. Lattimer, Ann. Rev. Nucl. Part. Sci. 71, 433 (2021). https://doi.org/10.1146/annurev-nucl102419-124827 J.M. Lattimer, Y. Lim, Astrophys. J. 771, 51 (2013). https://doi.org/10.1088/0004-637X/771/1/51, [1203.4286] R. Lednicky, V.V. Lyuboshitz, V.L. Lyuboshitz, Phys. Atom. Nucl. 61, 2950 (1998); J. Haidenbauer, Nucl. Phys. A 981, 1 (2019). https://doi.org/10.1016/j.nuclphysa.2018.10.090, [1808.05049] R. Lednicky, V.L. Lyuboshits, Sov. J. Nucl. Phys. 35, 770 (1982); [Yad. Fiz. 35, 1316 (1981)] B.A. Li, Phys. Rev. Lett. 88, 192701 (2002). https://doi.org/10.1103/PhysRevLett.88.192701, [nucl-th/0205002] D.A. Liberman, Phys. Rev. D 16, 1542 (1977) M. Luscher, Nucl. Phys. B 354, 531 (1991). https://doi.org/10.1016/0550-3213(91)90366-6; Commun. Math. Phys. 104, 177 (1986). https://doi.org/10.1007/BF01211589; Commun. Math. Phys. 105, 153 (1986). https://doi.org/10.1007/BF01211097

85 Hadrons, Quark-Gluon Plasma, and Neutron Stars

3121

M. Luscher, U. Wolff, Nucl. Phys. B 339, 222 (1990). https://doi.org/10.1016/05503213(90)90540-T M. Luzum, P. Romatschke, Phys. Rev. C 78, 034915 (2008); [erratum: Phys. Rev. C 79, 039903 (2009)]. https://doi.org/10.1103/PhysRevC.78.034915, [0804.4015] R. Machleidt, D.R. Entem, Phys. Rept. 503, 1 (2011). https://doi.org/10.1016/j.physrep.2011.02. 001, [1105.2919] A. Manohar, H. Georgi, Nucl. Phys. B 234, 189 (1984). https://doi.org/10.1016/05503213(84)90231-1 B. Margalit, B.D. Metzger, Astrophys. J. Lett. 850, L19 (2017). https://doi.org/10.3847/2041-8213/ aa991c, [1710.05938] K. Masuda, T. Hatsuda, T. Takatsuka, Astrophys. J. 764, 12 (2013). https://doi.org/10.1088/0004637X/764/1/12, [1205.3621] H. Matsuda et al., Phys. Rev. D 102, 114503 (2020). https://doi.org/10.1103/PhysRevD.102. 114503, [2007.06886] T. Matsui, H. Satz, Phys. Lett. B 178, 416 (1986). https://doi.org/10.1016/0370-2693(86)91404-8 L. McLerran, S. Reddy, Phys. Rev. Lett. 122, 122701 (2019). https://doi.org/10.1103/PhysRevLett. 122.122701, [1811.12503] L.D. McLerran, R. Venugopalan, Phys. Rev. D 49, 2233 (1994). https://doi.org/10.1103/PhysRevD. 49.2233, [hep-ph/9309289] M.C. Miller et al., Astrophys. J. Lett. 887, L24 (2019). https://doi.org/10.3847/2041-8213/ab50c5, [1912.05705] M.C. Miller et al., Astrophys. J. Lett. 918, L28 (2021). https://doi.org/10.3847/2041-8213/ac089b, [2105.06979] Y. Mori, K. Kashiwa, A. Ohnishi, Phys. Rev. D 96, 111501 (2017). https://doi.org/10.1103/ PhysRevD.96.111501, [1705.05605]; PTEP 2018, 023B04 (2018). https://doi.org/10.1093/ptep/ ptx191, [1709.03208] A.H. Mueller, D.T. Son, Phys. Lett. B 582, 279 (2004). https://doi.org/10.1016/j.physletb.2003.12. 047, [hep-ph/0212198] Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961). https://doi.org/10.1103/PhysRev.122.345; Phys. Rev. 124, 246 (1961). https://doi.org/10.1103/PhysRev.124.246 Y. Namekawa et al., Phys. Rev. D 105, 034502 (2022). https://doi.org/10.1103/PhysRevD.105. 034502, [2109.11710] Y. Nara, A. Ohnishi, Phys. Rev. C 105, 014911 (2022). https://doi.org/10.1103/PhysRevC.105. 014911, [2109.07594] V.G. Neudachin, Y.F. Smirnov, R. Tamagaki, Prog. Theor. Phys. 58, 1072 (1977) M. Oertel et al., Rev. Mod. Phys. 89, 015007 (2017). https://doi.org/10.1103/RevModPhys.89. 015007, [1610.03361] A. Ohnishi, J. Phys. Conf. Ser. 668, 012004 (2016). https://doi.org/10.1088/1742-6596/668/1/ 012004, [1512.08468]; see also the note on high-temperature expansion. https://www2.yukawa. kyoto-u.ac.jp/~akira.ohnishi/Education/HighTExp.pdf M. Oka, K. Yazaki, Prog. Theor. Phys. 66, 556 (1981). https://doi.org/10.1143/PTP.66.556; ibid. 572 (1981). https://doi.org/10.1143/PTP.66.572; A. Faessler et al., Phys. Lett. B 112, 201 (1982). https://doi.org/10.1016/0370-2693(82)90961-3 M. Oka, K. Shimizu, K. Yazaki, Nucl. Phys. A 464, 700 (1987). https://doi.org/10.1016/03759474(87)90371-X; U. Straub et al., Nucl. Phys. A 483, 686 (1988). https://doi.org/10.1016/ 0375-9474(88)90092-9; Z.Y. Zhang et al., Nucl. Phys. A 578, 573 (1994). https://doi.org/10. 1016/0375-9474(94)90761-7; Y. Fujiwara, Y. Suzuki, C. Nakamoto, Prog. Part. Nucl. Phys. 58, 439 (2007). https://doi.org/10.1016/j.ppnp.2006.08.001, [nucl-th/0607013] J.A. Oller, E. Oset, A. Ramos, Prog. Part. Nucl. Phys. 45, 157 (2000). https://doi.org/10.1016/ S0146-6410(00)00104-6, [hep-ph/0002193] F. Özel, P. Freire, Ann. Rev. Astron. Astrophys. 54, 401 (2016). https://doi.org/10.1146/annurevastro-081915-023322, [1603.02698] F. Ozel, G. Baym, T. Guver, Phys. Rev. D 82, 101301 (2010). https://doi.org/10.1103/PhysRevD. 82.101301, [1002.3153]

3122

A. Ohnishi

G. Parisi, Y.S. Wu, Sci. Sin. 24, 483 (1981); J.R. Klauder, Acta Phys. Austriaca Suppl. 25, 251 (1983). https://doi.org/10.1007/978-3-7091-7651-1_8; E. Seiler, D. Sexty, I.O. Stamatescu, Phys. Lett. B 723, 213 (2013).https://doi.org/10.1016/j.physletb.2013.04.062, [1211.3709] S. Petschauer et al., Nucl. Phys. A 957, 347 (2017). https://doi.org/10.1016/j.nuclphysa.2016.09. 010, [1607.04307] R.D. Pisarski, F. Wilczek, Phys. Rev. D 29, 338 (1984). https://doi.org/10.1103/PhysRevD.29.338 J. Pochodzalla et al., Phys. Rev. Lett. 75, 1040 (1995). https://doi.org/10.1103/PhysRevLett.75. 1040 J.S. Read et al., Phys. Rev. D 79, 124032 (2009). https://doi.org/10.1103/PhysRevD.79.124032, [0812.2163] B.T. Reed et al., Phys. Rev. Lett. 126, 172503 (2021). https://doi.org/10.1103/PhysRevLett.126. 172503, [2101.03193] P.G. Reinhard et al., Phys. Rev. C 88, 034325 (2013). https://doi.org/10.1103/PhysRevC.88. 034325, [1308.1659] L. Rezzolla, E.R. Most, L.R. Weih, Astrophys. J. Lett. 852, L25 (2018). https://doi.org/10.3847/ 2041-8213/aaa401, [1711.00314] RHIC Scientists Serve Up Perfect Liquid, Press release from Brookhaven National Laboratory (2005) T.A. Rijken, V.G.J. Stoks, Y. Yamamoto, Phys. Rev. C 59, 21 (1999). https://doi.org/10.1103/ PhysRevC.59.21, [nucl-th/9807082] J. Rikovska-Stone et al., Nucl. Phys. A 792, 341 (2007). https://doi.org/10.1016/j.nuclphysa.2007. 05.011, [nucl-th/0611030]; T. Miyatsu, S. Yamamuro, K. Nakazato, Astrophys. J. 777, 4 (2013). https://doi.org/10.1088/0004-637X/777/1/4, [1308.6121] T.E. Riley et al., Astrophys. J. Lett. 918, L27 (2021). https://doi.org/10.3847/2041-8213/ac0a81, [2105.06980] D.H. Rischke et al., Acta Phys. Hung. A 1, 309-322 (1995). [nucl-th/9505014]; J. Brachmann et al., Phys. Rev. C 61, 024909 (2000). https://doi.org/10.1103/PhysRevC.61.024909, [nuclth/9908010]; L.P. Csernai, D. Rohrich, Phys. Lett. B 458, 454 (1999). https://doi.org/10.1016/ S0370-2693(99)00615-2, [nucl-th/9908034] X. Roca-Maza et al., Phys. Rev. Lett. 106, 252501 (2011). https://doi.org/10.1103/PhysRevLett. 106.252501, [1103.1762] X. Roca-Maza et al., Phys. Rev. C 88, 024316 (2013). https://doi.org/10.1103/PhysRevC.88. 024316, [1307.4806]; A. Tamii et al., Phys. Rev. Lett. 107, 062502 (2011). https://doi.org/10. 1103/PhysRevLett.107.062502, [1104.5431] H. Sako et al., Nucl. Phys. A 931, 1158 (2014). https://doi.org/10.1016/j.nuclphysa.2014.08.065 K. Sasaki et al. [HAL QCD], Nucl. Phys. A 998, 121737 (2020). https://doi.org/10.1016/j. nuclphysa.2020.121737, [1912.08630] H. Sato, K. Yazaki, Phys. Lett. B 98, 153 (1981). https://doi.org/10.1016/0370-2693(81)90976-X; R.J. Fries et al., Phys. Rev. Lett. 90, 202303 (2003). https://doi.org/10.1103/PhysRevLett.90. 202303, [nucl-th/0301087]; V. Greco, C.M. Ko, P. Levai, Phys. Rev. Lett. 90, 202302 (2003). https://doi.org/10.1103/PhysRevLett.90.202302, [nucl-th/0301093] A. Selem, F. Wilczek, Proceedings of the Ringberg Workshop on New Trends in HERA Physics 2005: Ringberg Castle, Tegernsee, Germany, 2–7 Oct, 2005, p. 337 https://doi.org/10.1142/ 9789812773524_0030, [hep-ph/0602128] S.R. Sharpe, R. Gupta, G.W. Kilcup, Nucl. Phys. B 383, 309 (1992). https://doi.org/10.1016/05503213(92)90681-Z; S. Aoki et al. [CP-PACS], Phys. Rev. D 67, 014502 (2003). https://doi.org/ 10.1103/PhysRevD.67.014502, [hep-lat/0209124]; T. Yamazaki et al. [CP-PACS], Phys. Rev. D 70, 074513 (2004). https://doi.org/10.1103/PhysRevD.70.074513, [hep-lat/0402025] M. Shibata et al., Phys. Rev. D 96, 123012 (2017). https://doi.org/10.1103/PhysRevD.96.123012, [1710.07579] A.M. Sirunyan et al., [CMS], Phys. Rev. Lett. 128, 032001 (2022). https://doi.org/10.1103/ PhysRevLett.128.032001, [2102.13048] A.N. Sissakian et al., [NICA], J. Phys. G 36, 064069 (2009). https://doi.org/10.1088/0954-3899/ 36/6/064069

85 Hadrons, Quark-Gluon Plasma, and Neutron Stars

3123

R.J.M. Snellings et al., Phys. Rev. Lett. 84, 2803 (2000). https://doi.org/10.1103/PhysRevLett.84. 2803, [nucl-ex/9908001] P. Spiller, G. Franchetti, Nucl. Instrum. Meth. A 561, 305 (2006). https://doi.org/10.1016/j.nima. 2006.01.043; T. Ablyazimov et al., [CBM], Eur. Phys. J. A 53, 60 (2017). https://doi.org/10. 1140/epja/i2017-12248-y, [1607.01487] A.W. Steiner, J.M. Lattimer, E.F. Brown, Astrophys. J. 722, 33 (2010). https://doi.org/10.1088/ 0004-637X/722/1/33, [1005.0811] J. Steinheimer et al., Phys. Rev. C 89, 054913 (2014). https://doi.org/10.1103/PhysRevC.89. 054913, [1402.7236]; V.P. Konchakovski et al., Phys. Rev. C 90, 014903 (2014). https://doi. org/10.1103/PhysRevC.90.014903, [1404.2765]; Y.B. Ivanov, A.A. Soldatov, Phys. Rev. C 91, 024915 (2015). https://doi.org/10.1103/PhysRevC.91.024915, [1412.1669]; Y. Nara et al., Phys. Rev. C 94, 034906 (2016). https://doi.org/10.1103/PhysRevC.94.034906, [1601.07692]; Y. Akamatsu et al., Phys. Rev. C 98, 024909 (2018). https://doi.org/10.1103/PhysRevC.98. 024909, [1805.09024] M.A. Stephanov, Phys. Rev. Lett. 107, 052301 (2011). https://doi.org/10.1103/PhysRevLett.107. 052301, [1104.1627] M.A. Stephanov, K. Rajagopal, E.V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998). https://doi.org/10. 1103/PhysRevLett.81.4816, [hep-ph/9806219]; Phys. Rev. D 60, 114028 (1999). https://doi.org/ 10.1103/PhysRevD.60.114028, [hep-ph/9903292]; Y. Hatta, T. Ikeda, Phys. Rev. D 67, 014028 (2003). https://doi.org/10.1103/PhysRevD.67.014028, [hep-ph/0210284]; M.A. Stephanov, Prog. Theor. Phys. Suppl. 153, 139 (2004). https://doi.org/10.1142/S0217751X05027965, [hepph/0402115]; R.V. Gavai, S. Gupta, Phys. Rev. D 71, 114014 (2005). https://doi.org/10.1103/ PhysRevD.71.114014, [hep-lat/0412035]; M. Asakawa, S. Ejiri, M. Kitazawa, Phys. Rev. Lett. 103, 262301 (2009). https://doi.org/10.1103/PhysRevLett.103.262301, [0904.2089]; B. Friman et al., Eur. Phys. J. C 71, 1694 (2011). https://doi.org/10.1140/epjc/s10052-011-16942, [1103.3511] M. Strickland, Acta Phys. Polon. B 45, 2355 (2014). https://doi.org/10.5506/APhysPolB.45.2355, [1410.5786] Y. Sugahara, H. Toki, Nucl. Phys. A 579, 557 (1994). https://doi.org/10.1016/03759474(94)90923-7; H. Shen et al., Prog. Theor. Phys. 100, 1013 (1998). https://doi.org/10.1143/ PTP.100.1013, [nucl-th/9806095] A.P. Szczepaniak, Phys. Lett. B 747, 410 (2015). https://doi.org/10.1016/j.physletb.2015.06.029, [1501.01691]; M. Bayar et al., Phys. Rev. D 94, 074039 (2016). https://doi.org/10.1103/ PhysRevD.94.074039, [1609.04133] T. Takahashi [J-PARC HUA] (2019). [1906.02357] T.T. Takahashi, H. Suganuma, Y. Nemoto, H. Matsufuru, Phys. Rev. D 65, 114509 (2002). https:// doi.org/10.1103/PhysRevD.65.114509, [hep-lat/0204011] S. Takeuchi et al., Prog. Theor. Phys. Suppl. 137, 83 (2000). https://doi.org/10.1143/PTPS.137.83 J.H. Taylor, J.M. Weisberg, Astrophys. J. 345, 434 (1989). https://doi.org/10.1086/167917 I. Tews et al., Astrophys. J. 848, 105 (2017). https://doi.org/10.3847/1538-4357/aa8db9, [1611.07133] I. Tews et al., Astrophys. J. 860, 149 (2018). https://doi.org/10.3847/1538-4357/aac267, [1801.01923]; I. Tews, J. Margueron, S. Reddy, Phys. Rev. C 98, 045804 (2018). https://doi. org/10.1103/PhysRevC.98.045804, [1804.02783] H. Togashi, M. Takano, Nucl. Phys. A 902, 53 (2013). https://doi.org/10.1016/j.nuclphysa.2013. 02.014, [1302.4261] H. Togashi et al., Phys. Rev. C 93, 035808 (2016). https://doi.org/10.1103/PhysRevC.93.035808, [1602.08106] R.C. Tolman, Proc. Nat. Acad. Sci. 20, 169 (1934). https://doi.org/10.1073/pnas.20.3.169; Phys. Rev. 55, 364 (1939). https://doi.org/10.1103/PhysRev.55.364; J.R. Oppenheimer, G.M. Volkoff, Phys. Rev. 55, 374 (1939). https://doi.org/10.1103/PhysRev.55.374 A. Tomiya, Y. Nagai (2021) [2103.11965] Y. Tomozawa, Nuovo Cim. A 46, 707 (1966). https://doi.org/10.1007/BF02857517

3124

A. Ohnishi

L. Trippa, G. Colo, E. Vigezzi, Phys. Rev. C 77, 061304 (2008). https://doi.org/10.1103/PhysRevC. 77.061304, [0802.3658] M.B. Tsang et al., Phys. Rev. Lett. 102, 122701 (2009). https://doi.org/10.1103/PhysRevLett.102. 122701, [0811.3107] S. Tsutsui et al., PoS LATTICE2021, 533 (2022). [2111.15095] F. Weber, Prog. Part. Nucl. Phys. 54, 193 (2005). https://doi.org/10.1016/j.ppnp.2004.07.001, [astro-ph/0407155] E.S. Weibel, Phys. Rev. Lett. 2, 83 (1959). https://doi.org/10.1103/PhysRevLett.2.83; S. Mrówczy´nski, Phys. Lett. B 214, 587 (1988); [Erratum: Phys. Lett. B 656, 273 (2007)]. https://doi.org/10.1016/0370-2693(88)90124-4; P. Romatschke, M. Strickland, Phys. Rev. D 68, 036004 (2003). https://doi.org/10.1103/PhysRevD.68.036004, [hep-ph/0304092]; G.K. Savvidy, Phys. Lett. B 71, 133 (1977). https://doi.org/10.1016/0370-2693(77)90759-6; N.K. Nielsen, P. Olesen, Nucl. Phys. B 144, 376 (1978). https://doi.org/10.1016/05503213(78)90377-2; H. Fujii, K. Itakura, Nucl. Phys. A 809, 88 (2008). https://doi.org/10.1016/j. nuclphysa.2008.05.016, [0803.0410]; J. Berges et al., Phys. Rev. D 85, 034507 (2012). https:// doi.org/10.1103/PhysRevD.85.034507, [1111.2751]; S. Tsutsui et al., Phys. Rev. D 91, 076003 (2015). https://doi.org/10.1103/PhysRevD.91.076003, [1411.3809] S. Weinberg, Phys. Rev. Lett. 17, 616 (1966). https://doi.org/10.1103/PhysRevLett.17.616; Phys. A 96, 327 (1979). https://doi.org/10.1016/0378-4371(79)90223-1 S. Weinberg, Phys. Rev. 137, B672 (1965). https://doi.org/10.1103/PhysRev.137.B672 S. Weissenborn, D. Chatterjee, J. Schaffner-Bielich, Phys. Rev. C 85, 065802 (2012); [erratum: Phys. Rev. C 90, 019904 (2014)]. https://doi.org/10.1103/PhysRevC.85.065802, [1112.0234] E. Witten, AMS/IP Stud. Adv. Math. 50, 347-446 (2011). [1001.2933]; M. Cristoforetti et al. [AuroraScience], Phys. Rev. D 86, 074506 (2012). https://doi.org/10.1103/PhysRevD. 86.074506, [1205.3996]; H. Fujii et al., JHEP 10, 147 (2013). https://doi.org/10.1007/ JHEP10(2013)147, [1309.4371]; A. Alexandru et al., JHEP 05, 053 (2016). https://doi.org/10. 1007/JHEP05(2016)053, [1512.08764] S.M.H. Wong, Phys. Rev. C 54, 2588 (1996). https://doi.org/10.1103/PhysRevC.54.2588, [hepph/9609287] K. Yagi, T. Hatsuda, Y. Miake, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 23, 1 (2005) H. Yukawa, Proc. Phys. Math. Soc. Jap. 17, 48 (1935); (Prog.Theor.Phys.Suppl. 1, 1 (1955)) H. Zhang et al., Phys. Rev. Lett. 126, 012301 (2021). https://doi.org/10.1103/PhysRevLett.126. 012301, [2004.00024] T. Zhao, J.M. Lattimer, Phys. Rev. D 102, 023021 (2020). https://doi.org/10.1103/PhysRevD.102. 023021, [2004.08293] S. Zouzou et al., Z. Phys. C 30, 457 (1986) P.A. Zyla et al. [Particle Data Group], PTEP 2020, 083C01 (2020). https://doi.org/10.1093/ptep/ ptaa104

Section XII Cosmic and Galactic Chemical Evolution Toshitaka Kajino

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Equation of State in Neutron Stars and Supernovae Kohsuke Sumiyoshi, Toru Kojo, and Shun Furusawa

Contents Introduction: Matter in the Cosmos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Neutron Stars and Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observations and the Properties of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Structure of Neutron Star and Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Properties of Dense Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideal Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Matter Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition Inside Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matter in Core-Collapse Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of Matter and Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Statistical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collapse of the Fe Core and Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Core Bounce Toward the Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of Nuclear Physics on Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Birth of Proto-neutron Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formation of Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matter in Merger of Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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K. Sumiyoshi () National Institute of Technology, Numazu College, Numazu, Shizuoka, Japan e-mail: [email protected]; [email protected] T. Kojo Department of Physics, Tohoku University, Sendai, Japan e-mail: [email protected] S. Furusawa College of Science and Engineering, Kanto Gakuin University, Kanazawa-ku, Yokohama, Kanagawa, Japan Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_104

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Quarks in Neutron Star Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Free Quark Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quark Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid Hadron-Quark Equations of State: First- or Second-Order Phase Transitions . . . . . Hadron-Quark Mixed Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid Hadron-Quark Equations of State: Crossover Scenario . . . . . . . . . . . . . . . . . . . . . . . Three-Window Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Window Model in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interactions in Strongly Correlated Quark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diquark Pairings in Color Superconductivity (CSC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quarkyonic Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Neutron stars and supernovae provide cosmic laboratories of highly compressed matter at supranuclear saturation density which is beyond the reach of terrestrial experiments. The properties of dense matter are extracted by combining the knowledge of nuclear experiments and astrophysical observations via theoretical frameworks. A matter in neutron stars is neutron-rich and may further accommodate non-nucleonic degrees of freedom such as hyperons and quarks. The structure and composition of neutron stars are determined by equations of state of matter, which are the primary subject in this chapter. In case of supernovae, the time evolution includes several dynamical stages whose descriptions require equations of state at finite temperature and various lepton fractions. Equations of state also play essential roles in neutron star mergers which allow us to explore new conditions of matter not achievable in static neutron stars and supernovae. Several types of hadron-to-quark transitions, from first-order transitions to crossover, are reviewed, and their characteristics are summarized.

Introduction: Matter in the Cosmos Exploration to the world at extreme conditions is one of the most fascinating themes in science. Expedition to the high density and temperature of matter in the universe goes far beyond the experimental ranges attained on the Earth. Neutron stars and supernovae are such cosmic laboratories where new phases of matter, including hyperons and quarks, may be realized. It is thrilling, at the same time, to envisage the exotic matter from the information of nuclei at the limiting condition. Experimental studies of exotic nuclei are powerful tools to extend the knowledge on the hot and dense matter with theoretical models. This endeavor has been made for decades, and the expedition is rapidly advancing the frontier with modern technology of nuclear experiments and astrophysical observations. The purpose of this chapter is to provide the basic knowledge of hot and dense matter in compact objects, i.e., neutron stars, supernovae, and neutron star mergers in the universe, and is to delineate the relation between these astrophysical

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objects and the properties of dense matter in quantum chromodynamics (QCD). Our discussion starts with the overview of the compact objects and proceeds to the examination of conditions such as density, temperature, and composition, realized in the compact objects. These variables specify equations of state of matter which play crucial roles in determining the structural and dynamical aspects of neutron stars. The conditions realized in supernovae or neutron star mergers are reviewed. A matter in the similar conditions may be also studied by laboratory experiments, i.e., heavy ion collisions and unstable nuclei, and such similarity encourages the interplay between nuclear physics and astrophysics. Near the nuclear saturation density, nuclear many-body theories give important constraints on the structure of neutron stars, and they have been implemented in the analyses of neutron star observables. Heavy neutron stars may accommodate matters at densities several times greater than the saturation density, where the appearance of hyperons and quarks may change the overall trend of nuclear equations of state. The latter part of this chapter examines the characteristics of nuclear and quark matters and then classifies several types of hadron-to-quark matter transitions. In this review, the natural unit, c =  = 1, is used unless otherwise stated.

Properties of Neutron Stars and Supernovae Neutron stars are highly compact objects with a mass of 1–2M (M : solar mass) and a radius of ∼10 km, which means the average mass density, mN n (mN  939 MeV: nucleon mass), of ∼7 × 1014 g/cm3 (Shapiro and Teukolsky 1983). This density is about twice as high as the nuclear density ρ0 = mN n0 = 3 × 1014 g/cm3 (n0 = 0.16 fm−3 ) for a canonical neutron star with the mass ∼1.4M and is even higher for more massive neutron stars with ∼2M , possibly including a matter beyond the purely nucleonic regime. The mass and radius of a neutron star are determined by equations of state which reflects the properties of strongly correlated matter. The competition between pressure and energy density of a matter is essential; the energy density induces the gravitational attraction toward the center, while pressure increase in such compression prevents the matter from collapsing. If the energy density dominates over pressure, massive matter collapses to a black hole. A matter having a large (small) P (ρ) at a given energy density is called stiff (soft), and stiff equations of state allow the existence of very massive neutron stars. The mass threshold dividing neutron stars and black holes and how stiff a neutron star matter can be are key questions in this chapter. As the name suggests, the interior of neutron stars is neutron-rich; the proton fraction in nucleon density, Yp = np /n, is ∼0.1 at n ∼ n0 , and the positive charges are neutralized by charged leptons (Shapiro and Teukolsky 1983). The temperature of interior can be inferred to be ∼108 K (∼0.01–0.1 MeV), which is much smaller than the Fermi momenta of nucleons pF ∼ 300 MeV. In this sense, neutron stars are regarded as cold. In Fig. 1, typical environment is schematically shown in the phase diagram. This highly neutron-rich and cold dense matter at n > n0 is not achieved in terrestrial experiments such as heavy ion collisions where the matter is

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Temperature [MeV]

Proton fraction 0.5

Black hole formation

102 Nucleon gas

101

Protoneutron star

ProtoBlack hole neutron star formation

0.2 Nuclei

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10-2 n0

Nuclear matter

n0

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Number density

0.1 O

Neutron star

10-4 n0

10-2 n0

n0 Number density

Fig. 1 Typical environment of neutron stars and supernovae is schematically shown in the plane of number density-temperature and number density-proton fraction. Light blue thick lines show the conditions of neutron stars from the surface to the center. The temperature of cold neutron stars is essentially zero. The proton fraction of neutron star decreases as the density increases and becomes very small (neutron-rich) around n0 but at higher density increases again due to the growth of the symmetry energy. Shaded areas show the evolving conditions at the central region in the supernova core during the dynamics of collapse, bounce, and the birth of proto-neutron star. Red-shaded areas show the conditions in the case of black hole formation due to the failure of explosion. The phase boundaries for nuclei, nucleons, and quarks-gluons are sketched by dotted and dashed lines

isospin symmetric and inevitably accompanies heat which results in temperature of 1011 –1012 K (∼10–100 MeV). In this respect, neutron stars are quite unique. Meanwhile in supernovae that give the birth of neutron stars, the conditions (temperature and isospin fraction) similar to heavy ion experiments are realized (Hempel et al. 2015; Oertel et al. 2017). Supernova explosions occur at the end of stellar life after ∼107 years, leaving the compact objects (neutron stars and black holes) as remnants at the center as shown in Fig. 2. As a result of gravitational collapse of the massive star, a proto-neutron star is born after the bounce-back of the central core just above the nuclear density. The success or failure of the propagation of shock wave, which is launched by the core bounce, determines the fate of the massive star; the supernova explosion leaves a neutron star or a black hole. During supernova explosion, a hot and dense matter from dilute to dense conditions, with the number density ranging from 10−10 to 100 fm−3 , the temperature of T = 1 − 50 MeV, and the proton fraction (proton density per baryon density) Yp = 0.3 − 0.4, appears in the dynamical process as shown in Fig. 1. Note that the matter is not yet fully neutron-rich in the proto-neutron star due to the existence of neutrinos. If the collapse to a black hole happens, the density and temperature may become higher than those in proto-neutron stars, and equations of state with even broader range of (n, Yp , T ) contribute. In the context of matter in quantum chromodynamics (QCD), cold neutron stars and supernovae offer information for the QCD phase diagram expanded in a density (chemical potential)-temperature plane. The astrophysical observations cover the

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Birth of compact object Supernova neutrino

Massive star Fe core

Explosion

Neutron star

Proto-neutron star

Launch of shock wave

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Failure Black hole

Fig. 2 Schematic diagrams of the evolution of supernova cores from massive stars to compact objects. Starting with the gravitational collapse of the Fe core, the central core is compressed, and neutrinos are trapped inside due to interactions with hot and density matter. The central core bounces back by a matter around the nuclear saturation density, and the shock wave is launched. A proto-neutron star, which is hot and contains abundant neutrinos, is born at the center. If the shock wave successfully propagates through accreting outer layers of the Fe core, it leads to the supernova explosion and the birth of neutron star. If it fails, the central object collapses to the black hole

domain not explored by terrestrial experiments and ab ininio lattice QCD MonteCarlo simulations. The lattice QCD has been very powerful tools to study the QCD vacuum and the high-temperature domain, but its application to finite density is prevented by the sign problem.

Observations and the Properties of Matter It should be useful to glance at how the abovementioned properties have been inferred from observations (Shapiro and Teukolsky 1983; Haensel et al. 2007). Below, some basic observations are mentioned. Neutron stars can contain large magnetic fields and emit strong pulses of electromagnetic waves. Such pulse emitters are called pulsars. The pulses arriving at the Earth is periodic as neutron stars are rotating. The extremely precise and stable frequency of signal with a very short period of ∼1 ms indicates that the rotating object is very compact; otherwise, an object with the large radius R and high frequency ω would lead to the velocity of ∼ωR exceeding the light velocity. There are many pulsars which are discovered in a binary composed of a neutron star and another star which can be a neutron star or a white dwarf or a black hole (Lorimer 2008). Such systems are useful to estimate the neutron star mass, as the Kepler’s third law can be applied to the binary motion (Lattimer 2012; Özel et al. 2012; Kiziltan et al. 2013). The most precise mass measurement further utilizes the general relativistic time delay on pulses passing gravitational fields around the companion star (Demorest et al. 2010; Antoniadis et al. 2013). The heaviest neutron star discovered is PSR J0740+6620 with the

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mass 2.08 ± 0.07M (Cromartie et al. 2020; Fonseca et al. 2021) which put tight constraints on the lower bound of the stiffness of equations of state. The measurements on neutron star radii are more difficult than the mass measurements (Fortin et al. 2015; Özel and Freire 2016). The analyses of X-ray bursts on the neutron star surface lead to the estimate of ∼10 km, but they contain several uncertainties such as modeling and the distance between the Earth and neutron stars (Watts et al. 2016). In this respect, dramatic progress took place from the first detection of gravitational waves from the neutron star merger event, GW170817 (Abbott et al. 2018). The gravitational wave patterns are very sensitive to the (tidal) deformation of each neutron star just before the collision, since such deformation induces additional gravitational attraction between neutron stars, accelerating the merging process. The tidal deformation strongly depends on the neutron star radii; a larger radius leads to a larger tidal deformability. For a 1.4M neutron star, the upper bound is R < 13.4 km (Annala et al. 2018; Most et al. 2018). The LIGO collaboration yielded the estimate R = 11.9+1.4 −1.4 for neutron stars in GW170817 (Abbott et al. 2018). Another important constraint was set by the NICER measurement of X-rays from hotspots of rotating neutron star surfaces (Miller et al. 2019, 2021; Riley et al. 2021). It keeps track of the hotspots and examine the Doppler shift of the X-ray spectra from moving hotspots, taking gravitational lensing into account. The measurements of the rotation frequency ω, the surface velocity ∼Rω, and the gravitational lensing related to M/R in principle allow us to extract M and R separately. The NICER analyses have been done for 1.4M and ∼2.1M neutron stars, and the two radii turn out to be rather similar, 12.45±0.65 km for a 1.4M neutron star and 12.35±0.75 km for a 2.08M neutron star (Miller et al. 2021). The birth places of neutron stars are considered to be supernova explosions (Bethe 1990). In fact, pulsars are found in the supernova remnant as in the case of Crab pulsar in the remnant of supernova in 1054. Supernovae are transient phenomena with a bright display for months and disappearance afterward. A few events of supernovae occur on the average per century in a galaxy, but more than 1000 cases are recorded per year in the monitoring surveys. The process of ending in the stellar life includes the matter evolution through nuclear fusion, dissociation, electron captures, and subsequent neutrino production, as well as thermal production of neutrino-antineutrino pairs, affecting the composition of heavy elements (Arnett 1996). From observation of light curves and spectra with modeling of explosive nucleosynthesis, the energy of explosion is estimated to be ∼1051 erg. In the case of supernova observed in 1987, neutrino bursts from supernova SN 1987A were detected with average energy ∼10 MeV for ∼10 s at the terrestrial detectors (Hirata et al. 1987; Bionta et al. 1987). The detection of supernova neutrinos vindicates the birth of neutron star, which has a condition at high density and temperature. The total energy of neutrinos is evaluated to be ∼3 × 1053 erg (Sato and Suzuki 1987; Burrows 1988; Suzuki 1994), which is well in accord with the gravitational binding energy of neutron stars, supporting the scenario of supernova explosion driven by neutrinos. In fact, neutrinos play an essential role in the explosion mechanism and control properties of hot and dense matter. At the terrestrial detectors such as Super-

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Kamiokande and IceCube, a burst of supernova is being monitored to find neutrinos from astrophysical phenomena. In the next nearby supernova, the improvement of the detectors allows us to detect ∼1000 times more neutrinos than found in SN 1987 A (Scholberg 2012; Hyper-Kamiokande Proto-Collaboration et al. 2018; Suwa et al. 2019). The detailed information of the energy spectra of neutrinos as a function of time will be used to reveal the explosion dynamics and the properties of compact object. The supernova explosion is also a target of the observation of gravitational waves, which is generated by the time-dependent quadrupole energy distribution and thus carries the information of multidimensional dynamics, at the detectors such as LIGO, VIRGO, and KAGRA (Kotake et al. 2006; Ott 2009; Kotake 2013). These observations are rapidly progressing. The gravitational wave detection and the NICER measurements have just begun. When the gravitational detector reaches the design sensitivity, the detection of gravitational waves from neutron star mergers should be daily events. The NICER is collecting more photons from neutron stars and improving the statistics and at the same time is increasing the number of targets. Supernova events in our galaxy may happen within several decades and should dramatically improve our understanding of supernova matter as well as the properties of neutrinos (Ando et al. 2005). The diffuse supernova neutrino background, which has accumulated from the neutrino bursts from the collapse of massive stars, will be detected in the coming years (Horiuchi et al. 2009). Our aim in the following sections is to summarize basic aspects of neutron star and supernova matter and to prepare ourselves for the expected and unexpected new discoveries.

The Structure of Neutron Star and Equation of State The most basic quantities in neutron star observations are the mass-radius (M-R) relations of neutron stars. As mentioned, the structure is determined by pressure vs. energy density, P (ρ), which enters Tolman-Oppenheimer-Volkoff (TOV) equation, a general relativistic version of the Newton equation for gravity:     dP (r) P (r) GM(r)ρ(r) 4π r 3 P (r) 2GM(r) −1 1 + =− 1 + 1 − , dr ρ(r) M(r) r r2 (1) where the last three factors characterize the general relativistic effects and P (r) and ρ(r) are the pressure and density at the radial position, r (Shapiro and Teukolsky 1983). The mass M(r) inside the radial coordinate r is obtained by integrating the mass shell at r: dM(r) = 4π r 2 ρ(r) . dr

(2)

Actual computations begin with setting the central density nc at r = 0, and then an equation of state is used to prepare P and ρ at given nc . With these initial conditions and equations of state P (ρ), the differential equations 1 and 2 are integrated until the

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pressure reaches zero, P (r = R) = 0, which defines the radius R of a neutron star. The neutron star mass is defined as M(r = R). In this way, for a given central density nc , one can obtain M(R; nc ). Repeating this procedure with changing nc , one obtains the collection of such data forming a M-R curve. There is oneto-one correspondence between M-R curves and equations of state; in principle, the equation of state can be directly extracted from observations. In reality, there are only a few examples where M and R are determined simultaneously. In this situation, some guides from theories should be supplemented. From the center to the surface, the density drastically drops from ∼2 − 5n0 to ∼10−10 n0 , and apparently it seems a formidable task to extract equations of state from such complex structure. Actually details of dilute matter do not have significant impacts on the M-R relations, since dilute matter does not have large energy and its domain is highly compressed by the gravity; dilute matter with n  n0 forms only a thin shell of ∼0.5 km at most. A possible exception to this discussion is very light neutron stars with M ∼ 0.1–0.5M for which dilute matter is loosely compressed, but such neutron stars have not been discovered perhaps due to the absence of the formation process for such light neutron stars. With these considerations, one can concentrate on equations of state for n  0.1n0 for discussions of M-R curves (the dilute matter part is crucial to discuss phenomena near the neutron star surfaces or at finite temperature as the loosely bound matter is quite active) (Page et al. 2006; Haensel et al. 2007). It is very important to note that the shapes of M-R curves are strongly correlated with the stiffness at several fiducial densities (Fig. 3). Stiff (soft) equations of state lead to larger (smaller) radii and larger (smaller) maximum masses. From low mass to the mass ∼1.0M , M-R curves first show the shrinkage of R and then radically change the direction toward the vertical direction with small changes in R. This bending occurs with the central density around ∼n0 . This reflects that the compressed matter begins to observe repulsive forces in nuclear matter and matter can no longer be squeezed easily. The location of the bending sets the overall radius of neutron stars. Thus, neutron star radii are typically sensitive to equations of state at 1–2n0 for wide range in M. The exceptions to this rule are equations of state with first-order phase transitions; the associated radical softening induces kinks in M-R curves, although up to now the signature of such kink structure has not been found for the interval 1.4–2.1M . Beyond ∼1.4M , the core density typically exceeds 2–3n0 , entering the regime beyond purely nucleonic regime (see discussions below), and around ∼2M , the density may reach 4–7n0 . The existence of 2M neutron stars requires the high-density matter at 4–7n0 to be very stiff. In addition to observational constraints on M and R, there are constraints on the interplay between low- and high-density equations of state. The first is the causality constraint, dP /dρ = cs2 ≤ 1, which demands the sound velocity cs to be smaller than the light velocity. Another constraint is the thermodynamic stability, d 2 ρ/dn2 ≥ 0. For example, one cannot combine extremely soft lowdensity equations of state and extremely stiff high-density ones, since dP /dρ must grow too rapidly from low to high densities. With these constraints, it becomes theoretically more challenging if the maximum masses of neutron stars are larger

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Fig. 3 M-R curves as functions of the central density nc . For a small M neutron stars have large radii due to loosely bound crust. When the central density reaches nc = 1 ∼ 2n0 , the dilute matter is highly compressed, and the size of a neutron star is characterized by a matter beyond the saturation point. The curves go up with small variation in the radii. The exception is equations of state with the first-order transitions which lead to kinks in the M-R curves. The top right figure illustrates how the causality constrains the relation between low- and high-density behaviors of P (ρ)

and the radii are smaller than the presently available constraints. The low- and highdensity equations of state constrain each other. The following section starts with discussions of nuclear matter which has been studied intensively. Matters in neutron stars and supernovae are not isospin symmetric, so it is necessary to discuss equations of state as functions of density n and proton fraction Yp . The effects of temperatures are discussed for applications to supernova matter. These discussions are given for neutron stars with the masses up to ∼1.4M and the core densities up to 2–3n0 . For 2M neutron stars, the core density is higher, and more hypothetical arguments are needed. Discussions related to quark matter are postponed to the final part of this review.

Basic Properties of Dense Matter Ideal Fermi Gas In order to understand the conditions in astrophysical phenomena, it is helpful to recall the basic properties of ideal gas of electrons and nucleons (Lang 1980; Shapiro and Teukolsky 1983; Weiss et al. 2004). Simple relations of the thermodynamics for the degenerate gas of fermions provide the energy scale in neutron stars and nuclei. The number density of fermion gas is given by n = gpF3 /3π 2 where pF is the Fermi momentum and g is the number of degree of freedom for

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spin. For example, at the initial stage of supernovae, electrons in Fe cores with the 1 number density n and Yp = 0.46 have pF = 10 MeV (n/10−5 fm−3 ) 3 which is much greater than the electron mass me  0.5 MeV. Another characteristic scale is the nucleon Fermi momenta in symmetric matter (Yp = 0.5) which is evaluated to 1

be pF = 263 MeV (n/n0 ) 3 . The nucleons become relativistic when pF  mN or n  50n0 . For a gas in the relativistic limit, the energy density scales as ρ ∝ pF4 ∝ n4/3 . Using the relation μ = ∂ρ/∂n (μ: chemical potential) and the thermodynamic relation P = μn − ρ, one can write ρrela (n) = cn4/3 → Prela = ρrela /3 .

(3)

This regime is relevant for electrons in Fe cores or neutron stars. Meanwhile, if fermions with the masses m are nonrelativistic (m pF ), the energy density behaves as ρ ∼ c1 mn + c2 n5/3 /m with c1 , c2 being some constants. In this case, ρNR (n) = c1 mn + c2

n5/3 m

→ PNR =

2 n5/3 5/3 c2 ∝ ρNR m−8/3 . 3 m

(4)

It is to be noted that the energy density mn is much larger than the pressure; a nonrelativistic gas is soft. In order for purely nucleonic descriptions to achieve a large neutron star mass of M , substantial repulsions among nucleons must be added to increase the pressure. In order to parameterize the effects of interactions and the evolution from nonrelativistic to relativistic regimes of dense matter, one often uses a polytrope form, P = Cρ Γ , with Γ called adiabatic index, and changes Γ at some fiducial densities (Read et al. 2009). It has been known that a piecewise polytrope with the proper choice of Γ and the fiducial densities can cover a wide class of realistic equations of state.

Nuclear Matter The structure of neutron stars, especially the overall radii, is very sensitive to nuclear equations of state near the saturation density n0 (Lattimer and Prakash 2001). Nuclear matter here is considered to be uniform and infinitely spread in space. The saturation density n0 is close to the density inside of heavy nuclei, and one can infer the properties of nuclear matter from laboratory experiments for finite size nuclei. But the removal of finite size effects introduces uncertainties (Atkinson et al. 2020) which would change the neutron star radii by ∼0.5–1 km; the pressure of nuclear matter is basically small; thus, even small corrections from interactions affect the predictions for neutron stars. Hence, the precise determination of nuclear matter properties still remains an important problem. Below the basic properties of nuclear matter near saturation density are summarized. It is usually sufficient to evaluate contributions from nuclear interactions by neglecting the Coulomb energy among

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Fig. 4 Energy per nucleon of nuclear matter and neutron matter (left) and symmetry energy (right) are shown as a function of number density. Approximate curves at the nuclear density are plotted by dashed lines with parabolic and linear expressions

protons because the matter with leptons is locally charge neutral in most cases. The interactions among leptons are negligible, and leptons can be added separately as an ideal gas. Shown in Fig. 4 are typical behaviors of energies per nucleons with the mass subtracted, ρ/n − m, in symmetric matter (Yp = 0.5) and pure neutron matter (Yp = 0). In symmetric matter, there is a minimum at n0 = 0.16 fm−3 with ρ/n − m  −16 MeV. This is called the saturation point. Using the relation P = μn−ρ = n2 ∂(ρ/n)/∂n, one can conclude the pressure at n0 is zero; no external pressure is needed to maintain the finite matter, meaning that the matter is self-bound and stable against compression or expansion. This saturation property brings the stability of various nuclei at a constant density, and the density greater than n0 is achieved only by substantial external pressure. Meanwhile, pure neutron matter is not self-bound. Thus, neutron stars are bound by the gravitational forces which require macroscopic amount of materials as the source. Thus, neutron stars cannot be arbitrarily light. In applications to astrophysical phenomena, it is necessary to know the energy at various densities and proton fractions. The exploration to dense matter is often made by a simple form of the energy function, EN (n, Yp ) = ρ/n − m, as K(Yp ) EN (n, Yp ) = EN (n0 , Yp = 0.5) + 18



n − n0 n0

2 + S(n)(1 − 2Yp )2 + · · · (5)

in terms of the expansion around n = n0 and Yp = 0.5. The coefficient K(Yp ) is called the incompressibility, the curvature at n0 for a given Yp . The coefficient S(n) is called the symmetry energy which, in conventional literatures, is defined to be the second derivative of E(n, Yp ) with respect to Yp at Yp = 0.5. This definition does not directly coincide with E(n, Yp = 0) − E(n, Yp = 0.5) but is chosen for

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the practical limitation that nuclear experiments cannot access the matter at Yp ∼ 0 where nuclei are unstable. In practice, available theoretical calculations suggest keeping only the leading order of the expansion, which starts with (1 − 2Yp )2 , is the good approximation around n0 . The symmetry energy near the nuclear density is expressed by S(n) = S(n = n0 ) + 3nL0 (n − n0 ) using the slope parameter, L. These quantities are extracted to be K(Yp = 0.5) = 220 − 260 MeV (Shlomo et al. 2006; Stone et al. 2014; Garg and Colò 2018), S(n = n0 ) = 24 − 36 MeV, and L = 30 − 90 MeV from the analyses of experimental data of nuclei for masses, radii, and excitations (Tsang et al. 2012; Lattimer and Lim 2013; Horowitz et al. 2014; Özel and Freire 2016; Li et al. 2018, 2019). These uncertainties are converted into the uncertainties in overall neutron star radii with ΔR ∼ 0.5–2 km.

Nuclear Matter Theories The direct calculations of infinite nuclear matter have played important roles in the neutron star context (Burgio et al. 2021). It can be applied to a matter with arbitrary Yp , and especially pure neutron matter calculations are cleaner than symmetric matter due to fewer parameters in calculations. The most systematic approach is based on microscopic two- and three-nucleon forces which are constrained by nuclear two-body scattering below the pion production threshold, spectra of light nuclei, and a deuteron scattering off a proton that is sensitive to three-nucleon forces. Either traditional potential models (Stoks et al. 1994; Machleidt et al. 1996), which are based on the meson exchange picture, or chiral effective theory (Holt and Kaiser 2017; Drischler et al. 2021a), which is based on momentum expansion, is used to characterize the nuclear forces (Machleidt and Sammarruca 2020). These nuclear forces are then used in many-body framework such as variational (Akmal et al. 1998; Togashi et al. 2017), quantum-Monte Carlo (Carlson et al. 2015), or manybody perturbation theories (Drischler et al. 2019). The modern calculations are consistent with the abovementioned experimental estimates, but the pressure, which is sensitive to fine details, still has the uncertainties of ∼30% at n0 (see, e.g., Fig. 1 in Kojo et al. 2022). These uncertainties grow with density, and in addition the validity to truncate many-body forces beyond three-body forces becomes questionable at n  2n0 . Thus, a phenomenological modeling is often used beyond n ∼ 2n0 together with 2M constraints. Taking the microscopic calculations as the lowdensity constraints, typical radii of 1.4M neutron stars are 11.5–13 km. The thermodynamic conditions in core-collapse supernovae and neutron star mergers vary over a wide range of density, temperature, and proton fraction. To construct data of the equations of state for the astrophysical simulations, phenomenological approaches are useful. The Skyrme-type interactions, which are represented as expansions of the effective interaction in powers of momenta and density-dependent three-body contributions, are the most well-known phenomenological modeling (Dutra et al. 2012). In the Skyrme-type models with the SLy4 parameters (Chabanat et al. 1998), the energy density of nuclear matter is expressed as

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ρ(n, Yp , T ) =

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τp τn + + (a + 4bx(1 − x)) n2 ∗ 2mn 2m∗p    + cj + 4dj x(1 − x) n1+δj + (1 − x)nmn + xnmp , (6) j =0,1,2

1 1 = + α1 nn + α2np , 2m∗n 2mn

(7)

1 1 = + α1 np + α2nn , ∗ 2mp 2mp

(8)

where a, b, cj , dj , and δj are parameters of the Skyrme forces, τn and τp are the kinetic energy densities of neutrons and protons, respectively, and mn and mp are the masses of neutrons and protons, respectively. The first and second terms in Eq. (6) correspond to the nonrelativistic kinetic energy densities of neutrons and protons, respectively. The third term represents two-nucleon interactions, and the summation over j approximates the effects of many-body or density-dependent interactions. The last two terms in Eq. (6) express the rest masses of neutrons and protons, respectively. The parameters α1 and α2 for effective masses, m∗n and m∗p , are chosen to reproduce observables of uniform nuclear matter together with a, b, cj , dj , and δj . Another phenomenological model of nuclear matter energy is the relativistic mean-field theory, in which nuclear interactions are described by the exchange of mesons. Up to this time, many parameter sets in the relativistic mean field theory have been adopted to construct equations of state for astrophysical simulations, e.g., DD2 (Typel et al. 2010), SFHx, and SFHo (Steiner et al. 2013). They are subject to constraints from terrestrial experiments and astrophysical observations (Stone 2021). For example, the Lagrangian with a parameter set TM1e (Shen et al. 2020) is (MN : nucleon mass) LRMF =



 ψ¯ i iγμ ∂ μ − (MN + gσ σ )

i=p,n



gρ −γμ gω ωμ + τa ρ aμ ψi 2 1 1 1 1 + ∂μ σ ∂ μ σ − m2σ σ 2 − g2 σ 3 − g3 σ 4 2 2 3 4 2 1 1 1  − Wμν W μν + m2ω ωμ ωμ + c3 ωμ ωμ 4 2 4 1 a aμν 1 2 a aμ − Rμν R + mρ ρμ ρ 4 2   + Λv gω2 ωμ ωμ gρ2 ρμa ρ aμ ,

(9)

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where ψ, σ , ω, and ρ denote nucleons, scalar-isoscalar mesons, vector-isoscalar mesons, and vector-isovector mesons, respectively, and Wμν = ∂ μ ων − ∂ ν ωμ and a = ∂ μ ρ aν − ∂ ν ρ aμ + g  abc ρ bμ ρ cν . Nucleon-meson interactions are expressed Rμν ρ as Yukawa couplings, and isoscalar mesons (σ and ω) interact with themselves. In the TM1e parameter set, the masses of mesons, mσ , mω , and mρ , and the coupling constants, gσ , gω , gρ , g2 , g3 ,c3 , and Λv , are determined to reproduce both properties of uniform nuclear matter at the saturation density (Oertel et al. 2017) and finite nuclei (Sugahara and Toki 1994; Bao and Shen 2014). In the mean field theory, mesons are assumed to be classical and are replaced by their ensemble averages. The Dirac equations for nucleons are quantized, and the free energies are evaluated based on their energy spectra. The meson fields and Dirac equations are self-consistently solved (Sumiyoshi and Toki 1994). For astrophysical applications, some microscopic models have been constructed with realistic interactions determined using the nucleon–nucleon scattering data. The variational method (Togashi and Takano 2013) for Schrodinger’s equation is based on the realistic two-body nuclear potentials, which are adjusted to account for the data, and on a three-body potential. The Dirac–Brückner–Hartree–Fock theory (Katayama and Saito 2013) also employs the bare nuclear interactions. In contrast to variational method with a three-body potential, the Dirac–Brückner–Hartree– Fock theory reproduces nuclear saturation properties starting from two-body forces by solving the Bethe–Salpeter equation, single-particle self-energy, and the Dyson equation (Brockmann and Machleidt 1990).

Composition Inside Neutron Stars In neutron star matter, the charge neutrality and beta-equilibrium conditions introduce considerable asymmetry in the isospin density (Shapiro and Teukolsky 1983). The proton fraction is strongly correlated with the symmetry energy which characterizes the energy cost from the isospin asymmetry. Neutron star matter at n0 has a small proton fraction of Yp ∼ 0.1, which is obtained by minimizing the total energy density, ρtotal (n) = ρ(n, Yp ) + ρe (ne ), with respect to the proton fraction, where ρe (ne ) and ne = Yp n are the electron energy density and number density. At higher density muons can also contribute. The condition at the minimum corresponds to the relation of the chemical equilibrium among particles, μn = μp + μe , which states the balance of the Fermi energy of neutrons versus those of protons and electrons, n ↔ p + e− , as shown in the left panel of Fig. 5. The set of these conditions is called as the beta-equilibrium. The composition and proton fraction change considerably from the surface to the core of a neutron star, where the density changes from ∼10−10 n0 to ∼n0 as shown in Figs. 6 (Heiselberg 2002; Sumiyoshi 2018) and 7 (Chamel and Haensel 2008; Shen et al. 2020). Near the surface, nucleons form a nucleus, and electrons are localized around it. The most stable nuclei are 56 Fe, and the corresponding proton fraction is Yp  0.46. This dilute regime begins to be modified beyond 10−8 n0 , where it is energetically more favorable to reduce the number of electrons by the process

86 Equation of State in Neutron Stars and Supernovae

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Fig. 5 Schematic diagram of the occupied energy states of the degenerate particles in the neutron star matter. Each particle fills the state up to the chemical potential (Fermi energy) under the betaequilibrium (left). When the neutron chemical potential exceeds the hyperon mass, neutrons are converted to hyperons to reduce the total energy and fulfill the chemical equilibrium with hyperons (right) (Sumiyoshi 2018)

Fig. 6 Schematic diagram of the composition and the phase of matter in the interior of neutron star (Heiselberg 2002; Sumiyoshi 2018)

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Fig. 7 Proton fraction of the neutron star matter as a function of number density (Courtesy of H. Shen) (Shen et al. 2020) (left) and species of nuclei in the neutron star matter in the nuclear chart (Courtesy of H. Koura) (Chamel and Haensel 2008; Sumiyoshi 2018) (right)

p + e− → n + νe . Accordingly, the proton fraction decreases, and a number of neutron-rich nuclei, as indicated in the right panel of Fig. 7, appear. This region is called the outer crust (Chamel and Haensel 2008). Around 10−3 n0 , there are too many neutrons, and they begin to drip out of nuclei. The matter consists of nuclei, neutrons, and electrons in the region called the inner crust (Chamel and Haensel 2008). Further compression of nuclei to densities beyond 5 × 10−2 n0 merges them into the pasta structure, which has huge nuclei with various nonspherical shapes (Oyamatsu 1993). Above the nuclear density, ∼0.5n0 , the nuclei are dissolved into neutrons and protons, and the matter becomes uniform. In this regime, the cost associated with the symmetry energy dominates over the cost of having electrons; thus, Yp grows as density does. Deep inside the central core at n  2n0 , there may be exotic phases with hyperons or quarks. Hyperons contain strange quarks. The appearance of these new particles is controlled by the condition of the chemical equilibrium. The chemical potential of neutrons, μn , increases as the density goes up. When μn exceeds the mass of hyperons, mΛ , for example, neutrons can be converted to hyperons because it reduces the Fermi energy of neutrons (right panel of Fig. 5). Allowing the appearance of new particles at a given density usually softens equations of state, as it increases the energy density by Δρ ∼ mn, but the associated increase in pressure is much smaller, ΔP ∼ n5/3 /m. Shown in Fig. 8 are examples of M-R curves for equations of state with and without hyperons, based on equations of state calculated in variational methods (Togashi et al. 2016). The maximum mass of ∼2.2M for purely nucleonic equations of state drops to ∼1.6M after including hyperons together with reasonable Y N forces. The small maximum mass incompatible with the 2M constraints suggests the necessity of some additional mechanisms. One possibility is the existence of strong Y NN repulsions that increase the density at which hyperons appear, as

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Fig. 8 (left) Gravitational mass of neutron stars constructed by the equation of state based on the cluster variational method (Togashi et al. 2016) is shown for a choice of repulsive ΛΛ interaction as functions of the radius in the left panel. The models for hyperonic matter are shown by red solid and dashed lines with and without three-body force for hyperons, respectively. The model without hyperons is plotted by black dot-dashed line. The onset of Λ mixture is marked by symbols. Number fractions of the neutron star matter for a choice of repulsive ΛΛ interaction with threebody force are shown as functions of the density in the right panel. The solid red lines show the number fraction of Λ (Courtesy of H. Togashi)

shown in Fig. 8. Clearly, it is important to examine the properties of Y N and Y NN interactions (Vidaña 2018; Tolos and Fabbietti 2020). The experimental examination has difficulties since hyperons are unstable with respect to the weak decays and cannot be studied as in NN scattering experiments. One possible way to study Y N interactions is to inject hadrons with strangeness into nuclei, create hypernuclei, and then study the spectroscopy (Hiyama and Nakazawa 2018). Another method, which has been developed in the last 10 years, is to measure Y N interactions in lattice QCD (Iritani et al. 2019; Sasaki et al. 2020). The Y NN forces have not been determined yet. The composition of matter is important not only for M-R relations but also for nonmechanical aspects of neutron stars, such as cooling curves or chemical reactions (Yakovlev and Pethick 2004; Page et al. 2004; Potekhin et al. 2015). The neutron stars cool down through the emission of neutrinos in dense matter, and its cooling rate strongly depends on the matter composition. The fastest cooling mechanism in npeμ matter is the direct Urca processes in which n → p + e− + ν¯ e and p + e− → n + νe processes produce neutrinos escaping from the neutron star cores (Lattimer et al. 1991). The condition for these processes to occur is Yp  0.1, as can be derived from the energy and momentum conservations. If this condition is not met, the modified Urca processes, n + n → n + p + e + ν¯ e and n + p + e → n + n + νe , are the next candidates for cooling mechanisms. The modified Urca is much slower than the direct one as the former requires two thermally excited nucleons, whose population is suppressed by the Boltzmann factor, to interact. Due to this large difference in cooling time scale for the Yp  0.1 and Yp  0.1 cases, nuclear equations of state with the large symmetry energy

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can be differentiated from the others (Page and Applegate 1992). The complication is that typical nuclear equations of state lead to Yp  0.1 only at sufficiently high densities where hyperons or quarks may appear. They open new channels for the fast cooling. Another important effect is the pairing gap which suppresses the abundance of thermally excited particles and thus neutrino production (Chamel and Haensel 2008). In what follows, the fast cooling requires sufficiently high density. The observed cooling curves seem to be consistent with the modified Urca, but it remains to be shown whether the core density reaches the density threshold for the direct Urca or not; for now the cooling curves and neutron star masses are not measured simultaneously.

Matter in Core-Collapse Supernovae Evolution of Matter and Neutrinos The properties of hot and dense matter in core-collapse supernovae drastically change in the dynamical situations of collapse, bounce, and explosion (Oertel et al. 2017). Typical conditions of the evolution of number density, temperature, and electron fraction (Ye = Yp ) at the center in the supernova core are shown in Fig. 9. The density and temperature increase rapidly in the collapse and reach high values at the core bounce. The electron fraction Ye decreases and remains ∼0.3. It is useful to understand the time scale of changes of conditions and to examine the response of matter and neutrinos in the evolving environment. The matter composed of nucleons, nuclei, electrons, positrons, and photons is treated as a fluid component. The dynamical time scale, Tdyn , in fluid motion

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Fig. 9 Time evolution of the number density, temperature, and number fractions of electrons and leptons during the core collapse and bounce is shown in the left, middle, and right panels, respectively. The quantities at the center are taken from the numerical simulation with a 11M star. The time is measured from the timing at the core bounce. The lepton and electron fractions are shown by thick-solid and solid in the right panel, respectively (Furusawa et al. 2017a; Nakazato et al. 2021)

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of free fall under the gravitational force can be estimated to be Tdyn ∼ 5 × 10−3 s (n/10−4 fm−3 )−1/2 (Shapiro and Teukolsky 1983). It takes ∼0.2–0.3 s from the start of the gravitational collapse to the core bounce. This is much longer time scale than in strong and electromagnetic interactions. Therefore, the nuclear and electromagnetic processes can proceed fast enough to achieve the thermal and chemical equilibrium. Equations of state at a certain density, temperature, and electron fraction are necessary to perform numerical simulations. Neutrinos are treated separately by solving the equations of neutrino transfer, which describes the propagation and reaction of neutrinos in matter. The time scale of weak reactions can be comparable or even longer than the dynamical time scale. For example, the time scale of the electron capture on free proton can be estimated to be Te−cap ∼ 3 × 102 s (n/10−6 fm−3 )−5/3 (Ye /0.46)−2/3 (Xp /10−4 )−1 (Shapiro and Teukolsky 1983; Suzuki 1994) where Xp is the mass fraction of free protons and the degenerate relativistic electron gas is assumed. It is not possible to assume the equilibrium for weak reactions, being different from the case of cold neutron stars. It is mandatory to follow the time evolution of the electron fraction of matter by tracking the modifications of composition through weak reactions. This requires detailed descriptions of neutrino reactions in matter as a source of changes. It is also necessary to provide the weak reaction rates, which largely depend on the target particles and environment of the matter.

Nuclear Statistical Equilibrium At temperature of a few MeV achieved in supernova matter, fusing N neutrons and Z protons into a nuclide (N, Z) is balanced by the photo-breakup reaction of the nuclide (N, Z) into N free neutrons and Z free protons. In this nuclear statistical equilibrium, the compositions of nuclear matter (number densities of nucleons and all nuclei) are determined by minimizing the free energy of a model (Mazurek et al. 1979; Hempel and Schaffner-Bielich 2010). The free energy density of nuclear matter consists of contributions from nucleons and various nuclei as  f = fp + fn + fN (N, Z), (10) N,Z

where fp and fn are free energy densities of protons and neutrons, respectively. The free energy density of nuclei, fN (N, Z), is expressed as fN (N, Z) = nN (N, Z){M(N, Z) + Ft (N, Z)}

(11)

where nN (N, Z) represents the nuclear number density, M(N, Z) represents nuclear mass, and Ft (N, Z) represents the free energy of translational motion. The number densities of nuclei under the nuclear statistical equilibrium for the given ρ, T , and Yp are obtained by minimizing the model free energy with respect

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to the variational parameters. In the minimization, the following mass and charge conservations are imposed: np + nn +



(N + Z)nN = n,

(12)

ZnN = Yp n,

(13)

N,Z



np +

N,Z

where np and nn are the number densities of free protons and neutrons, respectively. By introducing Lagrange multipliers α and β for the above constraints, the free energy minimization as a function of np and nn leads to (i = p, n) ⎡ ⎛ ⎞  ∂ ⎣ f − α ⎝np + nn + (N + Z)nN − n⎠ ∂ni N,Z

⎛ −β ⎝np +



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(14)

N,Z

Thus, the Lagrange multipliers are expressed with the chemical potentials of protons and neutrons, μp and μn , as α=

∂f = μn , ∂nn

(15)

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∂f ∂f − = μp − μn . ∂np ∂nn

(16)

The differentiation with respect to nN leads to the chemical potentials of nuclei, μ(N, Z) = ∂f/∂nN , as μ(N, Z) = Nμn + Zμp .

(17)

Here, nucleons and nuclei are treated as the ideal Boltzmann gases with their constant masses as     nN (N, Z) f (N, Z) = nN (N, Z) M(N, Z) + T ln − T , (18) g(N, Z)nQ (N, Z)  nQ (N, Z) =

M(N, Z)kB T 2π

3/2 .

(19)

The partial differential of Eq. 18 with respect to nN (N, Z) and Eq. (17) leads to nN (N, Z) as follows:

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 Zμp + Nμn − M(N, Z) nN (N, Z) = g(N, Z) nQ (N, Z) exp , T

(20)

where g(N, Z) is degeneracy factor of the nucleus. Calculations of M(N, Z) depend on equations of state (Furusawa and Nagakura 2023).

Collapse of the Fe Core and Weak Interactions Environment in supernovae evolves from a gravitationally collapsing massive star to the birth of a neutron star (Bethe 1990; Janka 2012; Oertel et al. 2017). The gravitational collapse starts from the central Fe core in the final stage of a massive star with the mass greater than ∼10M . The central number density and temperature is typically ∼10−5 fm−3 and 1 MeV in the Fe core, which has a mass of ∼1.5M and radius of ∼104 km, supported by pressure of electrons. The main composition is nuclei around 56 Fe which has a proton fraction, Yp = Z/A = 26/56 = 0.46, as the final product of nuclear fusion reactions to the most bound nuclei. Above the temperature ∼1 MeV, the nuclear reactions among nucleons and nuclei proceed fast enough to maintain the nuclear statistical equilibrium. The matter becomes neutron-rich in the collapse of the central core (Fig. 9 right). The electron captures proceed due to the high Fermi energy of electrons, and the electron fraction decreases. The distribution of nuclei is shifted to the neutronrich side, being away from the stability line. Figure 10 displays the nuclei that are abundant during the collapse. The nuclear reactions during the collapse depend on the properties of neutron-rich nuclei and the associated response to electron captures. At the initial stage, the neutrinos produced by the electron captures freely escape from the central core. They carry away the lepton number which is originally carried by electrons. As the electron captures proceed and neutrinos fly away, the electron fraction in the core decreases. Further collapse of the central core, however, leads to higher densities of matter and the neutrino trapping due to the frequent scattering of neutrinos on nuclei. The neutrino transport eventually proceeds with the diffusion process. The time scale for the diffusion, Tdiff , in the central core is estimated to be Tdiff ∼ 4 × 10−2 s (n/10−4 fm−3 )(A/56) based on the neutrino mean free path (Shapiro and Teukolsky 1983; Suzuki 1994). It becomes longer than the dynamical time scale, Tdyn ∼ 5 × 10−3 s at the density, ρ = 10−4 fm−3 (Shapiro and Teukolsky 1983; Suzuki 1994). Therefore, the neutrinos cannot escape anymore and are trapped inside the central core beyond this density. In matter compressed together with neutrinos, the trapped neutrinos can be treated as degenerate leptons. This state of matter is often called as the supernova matter, which is in thermal and chemical equilibrium. The supernova matter is parameterized by the lepton fraction, YL , instead of the electron fraction, Ye . After the neutrino trapping, the lepton fraction remains constant during the gravitational

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proton number Z

60 50 40 30 20

electron capture data drip lines stable nuclei average nuclei 99.9% of XH

10 0 0

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Fig. 10 The nuclear species for which the data of electron capture rate is available (green squares). Black circles represent stable nuclei, and solid thin black lines display the neutron and proton drip lines. A dashed red line indicates the trajectory of average neutron and proton numbers of heavy nuclei. Nuclei inside the solid thick red line appear in the center of collapsing cores and account for the top 99.9% of total mass fraction of heavy nuclei at any time step (Furusawa et al. 2017b)

collapse and after the core bounce (Fig. 9 right). The lepton fraction at this stage is important to determine the size of the bounce core to launch the shock wave. Since there is no escape of particles to carry energy, the matter is compressed under the adiabatic condition. The entropy per baryon remains constant after the neutrino trapping during the gravitational collapse. The rise of the temperature is determined by the properties of the matter along the adiabatic curve. The temperature increases moderately even though the density increases dramatically (Fig. 9 middle).

Core Bounce Toward the Explosion The gravitational collapse with the neutrino trapping halts suddenly just above the nuclear density, n0 . The pressure abruptly increases above the nuclear density, and the equation of state becomes stiff as shown in Fig. 11. This is mainly due to the repulsive contribution of nuclear forces. The adiabatic index, which corresponds to the slope of pressure versus density, becomes large ∼2 above the value ∼4/3 for lepton gas during the collapse. This stiffening brings to a sudden stop of the compression of matter at the center and leads to bouncing back of the inner core. The shock wave is launched at the core bounce and starts propagating toward the outside as shown in Fig. 12. The central density is slightly above the nuclear density, and the temperature is around 10–20 MeV. The electron and lepton fractions are ∼0.29 and

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∼0.35 due to the neutrino trapping with neutrino fraction ∼0.06 in this example. Note that the matter is not so neutron-rich having the proton fraction ∼0.29. This is the starting point of the explosion although there are a number of obstacles to overcome. The initial position of the shock wave is located in the middle of the Fe core. The shock wave must reach the surface of the Fe core by propagating against the free fall of matter of the outer part. It slows down also due to the loss of energy by the dissociation of Fe-group nuclei. The shock wave eventually stalls during the propagation inside the Fe core. This is the second stage of the explosion typically seen in many simulations, especially under the spherical symmetry, which leads to the failure of explosion. In the present understanding of the explosion mechanism, it is considered that the neutrino heating mechanism assists the revival of the stalled shock wave (Bethe and Wilson 1985; Bethe 1990). The trapped neutrinos in the central part are gradually emitted to the outside, and a part of outgoing neutrinos are absorbed by the material in the heating region just behind the stalled shock wave. The absorption of neutrinos heats the matter, increasing the internal energy and pressure to push the shock wave outwardly. The matter below the shock wave is composed of nucleons and light nuclei such as deuterons and α particles. Figure 13 displays the mass fractions of the nuclei. The deuterons are abundant at R ∼ 10–50 km above the surface of the proto-neutron star, while α particles are available at R ∼ 100–300 km around and inside the shock wave. The nuclear matter that consists of nucleons and light

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nuclei also appears in heavy ion collisions and equations of state at low densities, 0.01n0  n  0.2n0 , and may be experimentally constrained (Qin et al. 2012; Hempel et al. 2015). The success of explosion requires an additional factor to raise the efficiency of the neutrino heating mechanism (Janka et al. 2012, 2016; Burrows 2013; Kotake 2013; Janka 2017a). In many of recent multidimensional simulations, the hydrodynamical instabilities assist the revival of the shock wave by giving an enough time for the neutrino heating. The convective motion, for example, brings the matter hovering in the heated region behind the shock wave to absorb enough neutrinos. The combination of the neutrino heating and the hydrodynamical instability is believed to be the major scenario of the explosion mechanism although the outcome of the (non-)explosion widely depends on the numerical simulations and microphysics.

Influence of Nuclear Physics on Supernovae Nuclear physics plays an important role to understand the initial stage of the shock wave during the collapse and bounce. The initial shock energy at the core bounce is estimated from the gravitational binding energy of the bounce core to be ini 2 Eshock ∼ GMbounce /Rbounce where Mbounce and Rbounce is the mass and radius of the bounce core. The mass of the bounce core is determined by the Chandrasekhar mass, MCh = 1.5(YL /0.5)2 M , supported by the pressure of leptons (Shapiro and Teukolsky 1983; Suzuki 1994). The lepton fraction, YL , is determined by the amount of electron captures and neutrino trapping during the collapse. The radius of the bounce core is determined by equations of state above the saturation density. For successful supernova explosions, high compression of a matter and the associated quick increase in pressure are favored. Equations of state which are soft at low density meet such condition, but too soft ones are incompatible with the known properties of nuclei and neutron stars (Baron et al. 1985; Takahara and Sato 1988). There are also intriguing possibilities that a strong quark-hadron first-order phase transition triggers the second bounce to launch the shock wave for the successful explosion (Fischer et al. 2011). After the launch of shock waves, the nuclear physics is influential to the propagation of shock wave. One of the key factors is the mass of the bounce core; for a larger mass, the shock wave propagation is less disturbed by the energy loss associated with dissociation of the Fe nuclei (Janka et al. 2012). Another important physics is the neutrino energy at the emission region. Efficient heating of the shock waves favors energetic neutrino fluxes, which can be produced by high compression and high temperature as achieved by soft equations of state. Here the composition of matter is also very important, as the neutrino reactions strongly depend on target particles. Modeling of the nuclear statistical equilibrium and nuclear weak interactions has a significant impact on proto-neutron star masses and shock wave evolution to the same degree as the stiffness of the equation of state (Nagakura et al. 2019b).

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Birth of Proto-neutron Star After the launch of shock waves, the central part of the core settles down to the quasi-hydrostatic configuration, and the compact object is born at the center ∼0.3 s after the core bounce. It is called proto-neutron star which contains a plenty of neutrinos and antineutrinos of all three flavors (Burrows and Lattimer 1986; Suzuki 1994; Pons et al. 1999). Those neutrinos are produced by the weak reactions including the thermal processes such as the pair creation. It is also hot (T  10 MeV) and less neutron-rich (Ye  0.3) with a large radius ∼50 km as compared to cold neutron stars as shown in the red lines in Fig. 14. The lepton fraction including neutrinos is determined during the collapse. Proto-neutron stars cool down by emission of neutrinos. The temperature and electron fraction decrease since the neutrinos carry away the internal energy and lepton number as shown in Fig. 14. The evolution proceeds gradually over the time scale of ∼20 s through the diffusion of neutrinos in the matter. The proto-neutron star becomes compact with radii 15 km, and the density becomes high. The proton fraction decreases to a small value, Yp  0.1, which is determined by the betaequilibrium without neutrinos. The proto-neutron star turns into a cold neutron star over the time scale of minutes. The supernova neutrinos contain various information to probe inside the supernova core (Burrows 1988; Suzuki 1994; Janka 2017b; Müller 2019). The total energy of neutrinos can be used to derive the binding energy of the neutron star (∼1053 erg). The average energy roughly reflects the temperature at the emission region. The time duration of neutrino emission is related with the diffusion time scale determined by the density. For example, soft equations of state lead to high energies of neutrinos and a long duration due to high density and temperature. Note that the neutrino emission is closely related with the explosion mechanism. The dynamics due to nonspherical motion of matter such as the convection and/or

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rotation leads to rapid variations of neutrino emission in time and, therefore, may be probed by neutrinos (Müller 2019). Observation of supernova neutrinos may provide the information on the properties of neutrinos through the phenomena of neutrino oscillations (Kotake et al. 2006; Duan and Kneller 2009; Mirizzi et al. 2016). The neutrinos may affect also the explosive nucleosynthesis, which takes place in the outer layers, by changing the composition (Woosley et al. 1990). The supernova explosion is a target of the observation of gravitational waves, which brings the information of multidimensional dynamics (Kotake et al. 2006; Ott 2009; Kotake 2013). The time variation of the quadrupole moment of the matter distribution is a source of the variation of space-time metric in the Einstein equation of general relativity. The proto-neutron star is excited by fluid motions to have oscillations with the eigenfrequencies of the gravitational wave (Ferrari et al. 2003; Sotani and Takiwaki 2016). Simultaneous detection of the neutrinos and gravitational waves from the nearby supernova will help to reveal the explosion dynamics in the era of the multi-messenger astronomy (Yokozawa et al. 2015).

Formation of Black Hole The fate of the massive stars depends on the properties of the Fe core such as the compactness of density profile (Heger et al. 2003; O’Connor and Ott 2011). If the shock waves fail to revive, the accretion of matter continues, and the mass of the proto-neutron star keeps growing. When it goes beyond the maximum mass supported by the supernova matter, the dynamical collapse occurs, and a black hole is formed. A massive star fades away as a failed supernova (Kochanek et al. 2008; Adams et al. 2017). In Fig. 15, examples of the condition of matter in the evolution of supernova cores are shown. In the case of black hole formation, the density and

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temperature become very high and can be even beyond 100 MeV and 10 fm−3 due to the mass increase. The condition is more extreme than the case of proto-neutron star where the density stops growing at some point. This may open up possibilities of the appearance of the exotic particles such as hyperons and quarks. Note that the electron fraction is not so low (Fig. 15 right) due to the neutrino trapping, which tends to suppress the appearance of exotic particles at the birth of proto-neutron star. Nevertheless, the evolution of density and temperature in the black hole formation can be extreme enough to yield hyperons and quarks. Since the energy release of the accreting matter is efficient and the temperature becomes high, the luminosity and average energy of neutrinos increase rapidly toward the black hole formation. The typical duration of neutrino emission is ∼1 s under the spherical symmetry. These features may be distinguished from the ordinary supernova neutrinos. They can be used to probe equations of state at extreme conditions (Sumiyoshi et al. 2006).

Matter in Merger of Neutron Stars A merger of a neutron star binary offers a lot of information of neutron stars, from the static properties of cold neutron stars to dynamical processes of warm and dense matter after the collisions. Figure 16 displays the time evolution. There are a variety of signals including gravitational waves, electromagnetic waves, and neutrinos, and hence neutron star mergers are important targets in the multi-messenger astronomy. So far gravitational waves have been detected in several events (including a black hole) since 2015. As for binary neutron star mergers, the clearest is GW170817 with the total mass of Mtot  2.74+0.04 −0.01 M detected by a LIGO and Virgo (Abbott et al. 2017); in this event, gravitational waves at frequencies (cs )2 ; the sound velocity is smaller in quark matter.

(27)

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Hadron-Quark Mixed Phases Another possibility is a mixed phase of hadronic and quark matters where the matter contains nonuniform domains of various kinds (pastas) (Glendenning 1992) (Fig. 19). To characterize such phases, we first recall that neutron star matter contains baryon number and charge densities as conserved quantities. Hence, the pressure may contain two chemical potentials, P (μ, μQ ). Imposing the neutrality condition nQ = 0, the charge chemical potential μQ is determined as μQ (μ), so the pressure is written as P (μ). In discussions of first-order phase transitions in the last section, Ph (μ) and Pq (μ) are compared assuming the conditions nhadron = 0 Q quark

and nQ

= 0, i.e., each phase is separately charge neutral. Then, μhadron (μ) = Q

quark

μQ (μ) in general. In contrast, in mixed phases, the chemical equilibrium conditions are imposed for quark = μQ . As a consequence, hadronic all components, μhadron = μquark and μhadron Q and quark matter are not separately charge neutral, but only the mixture is charge quark (μ) + (1 − χ )nQ (μ) = 0 where χ neutral. This is achieved as nQ = χ nhadron Q characterizes the volume fraction. The charge density has nonuniform distributions in space. At low density, the finite size domains of a quark matter emerge in a hadronic matter, and the domain grows with density changing the shapes (Alford et al. 2001; Ju et al. 2021; Maruyama et al. 2007; Nakazato et al. 2008). The quark matter is differentiated from hadronic domains by sharp surfaces, and the finite size domains are located in a periodic way. Eventually, the quark matter domains become bigger than the hadronic ones; the latter are immersed in a quark matter. Eventually, the hadronic domain disappears, and the system is entirely described by a pure quark matter. In this nonuniform description, the P vs. ρ relations do not contain jumps in

Fig. 19 Maxwell (left) vs. Gibbs (right) construction of hybrid equations of state

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ρ, as the volume fraction χ interpolates pure hadronic and quark matter. The sound velocity decreases in the interval of mixed phases.

Hybrid Hadron-Quark Equations of State: Crossover Scenario A transition from hadronic to quark matter can be also a crossover with the derivatives of P (μ) being continuous. To examine this possibility, it is instructive to discuss a finite temperature equation of state (at μ = 0) which includes a transition from a hadronic matter to a quark-gluon plasma (QGP). Lattice MonteCarlo simulations, first principle calculations of QCD, have established that the transition is a crossover which begins to occur around Tc  155 MeV and continues to T ∼ 2–3Tc (Aoki et al. 2006). At low temperature, quarks and gluons are confined within hadrons, and the system is dominated by a dilute hadron resonance gas (HRG). At higher temperature, those hadrons are thermally excited and begin to overlap around T  Tc . Then quarks and gluons should gradually become natural degrees of freedom to characterize the system. Traditionally, many works used a hybrid hadron-QGP model in which two phases are separated by a first-order phase transition (Fig. 20). The modeling is similar to the previous section. The simplest version of such hybrid models is to combine an ideal pion gas equation of state for hadronic matter Pπideal and a free quark and gluon gas with a bag constant for QGP equations of state. Neglecting quark masses, for three flavors (dQGP = 47.5) (Yagi et al. 2005), ideal PQGP (T ) = dQGP

T4 −B. 90

(28)

Fig. 20 A schematic description of normalized pressure, P /T 4 , in hadronic and QGP phases. (Left) A pion gas vs. a bag model for a QGP. The transition temperature is determined from the crossing point of two pressure curves. (Right) A more realistic comparison, HRG vs. pQCD gas, together with the lattice result. HRG and pQCD are consistent with the lattice results at low temperature (T  Tc )and high temperature (T  2Tc ), respectively, but neither of them describe the intermediate region

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ideal Again the bag constant is necessary to describe the phase transition; if B = 0, PQGP would be always greater than Pπideal , and no phase transition would happen. Besides the orders of phase transitions, it turned out that the above hybrid descriptions contain at least two problems. First, one cannot approach to T  Tc with the pion gas descriptions; near Tc equations of state are not dominated by pions but by other massive resonances, with the masses, mH , much greater than T . Those massive contributions should be suppressed by a Boltzmann factor e−mH /T , but the number of states grows fast with T to compensate such suppression effects (Hagedorn 1965). Including resonances up to E ∼ 2.5 GeV, the HRG model, PHRG , reproduces the lattice results up to T  Tc very well. Second, extrapolating the ideal gas model for QGP down to ∼Tc substantially overestimates the pressure and entropy, as a consequence of neglecting confinement; more realistically, as T approaches Tc from above, quarks and gluons should be trapped into hadrons with reduction of pressure and entropy. The present perturbative QCD estimates of the pressure, PpQCD , seem valid down to ∼2Tc (Ghiglieri et al. 2020). A description more conservative than direct comparisons of the extrapolated equations of state is to limit the use of HRG and QGP equations of state to the domain of applicability and then to consider possible interpolations taking the hadronic equations of state at Tc and QGP equations of state at 2–3Tc as boundary conditions. For the finite temperature case, one can simply take smooth curves for the interpolation. For finite μ cases, the orders of transitions (first or second or crossover) are not established; case studies are necessary to prepare for future empirical determinations.

Three-Window Modeling A three-window modeling for dense QCD matter explicitly limits the domain of each model (Masuda et al. 2013a, b) (Fig. 21). The three window refers to a nuclear (hadronic) matter at low density (n  2n0 ), quark matter at high density (n  5n0 ), and a domain intermediate between the low- and high-density regimes (2n0  n  5n0 ). In the following, first, the physical picture is outlined, and then a practical modeling is discussed.

Fig. 21 A three-window modeling of a unified equation of state covering from nuclear to quark matter domains

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In a nuclear matter regime, a matter is dilute, and nucleons do not have many chances to interact. Nuclear interactions are mediated by only few meson (or quark) exchanges. Here one can use various nuclear equations of state, e.g., in Akmal et al. (1998), Togashi et al. (2017), Steiner et al. (2013), Typel et al. (2010), and Drischler et al. (2021b), which reproduce the nuclear laboratory experiments. The dilute regime ends when many-body forces become sizable. This is supposed to occur around n ∼ 2n0 (Akmal et al. 1998). A good measure to examine the convergence of many-body forces is the relative magnitudes of contributions from two- and three-body forces to the energy density. For example, in case of contact interactions, the contributions from N -body forces to the energy density are supposed to grow as ∼nN , increasing rapidly with n. This raises questions whether nucleons remain reasonable effective degrees of freedom, as baryon interactions are mediated by quark exchanges (Fukushima et al. 2020). Meanwhile, the density is not high enough to trust quark matter descriptions as baryons do not largely overlap. The problem here is the identification of proper degrees of freedom which is the starting point of any reliable calculations. Beyond ∼5n0 , baryons begin to overlap, and our descriptions become simplified, as quarks become the natural effective degrees of freedom. This domain is regarded as quark matter, irrespective of the existence of sharp hadron-quark phase transitions. But quark matter in this regime is by no means weakly interacting. The state-of-the-art pQCD calculations, including N2 LO and N3 LO soft contributions (Gorda et al. 2021a, b), have clarified that the domain of the weak coupling picture is n  40n0 . In order to reconcile the quark pictures with such large αs corrections, presumably one must include non-perturbative effects to strongly renormalize physical parameters, e.g., effective masses and couplings in quark matter descriptions. Such strong renormalizations are rather common in hadron physics; quark models for hadrons (De Rujula et al. 1975; Hatsuda and Kunihiro 1994), reproducing many hadron properties, are based on constituent quarks whose effective masses differ from those in the QCD Lagrangian. As noted before, the most theoretically challenging is the description of the intermediate regime between ∼2n0 and ∼5n0 . Fortunately, however, it is this domain where neutron stars of M  1–2M can give the most powerful constraints. The M-R relations of neutron stars have the one-to-one correspondence with the equations of state so that a better measurement of M-R more precisely determines equations of state (Lattimer and Prakash 2001).

Three-Window Model in Practice One of possible ways to implement these pictures is to adopt the following phenomenological modeling (Kojo et al. 2015; Baym et al. 2018). The first step is to choose the densities nL and nU at which nuclear and quark matter descriptions, respectively, are terminated. For instance, nL = 2n0 and nU = 5n0 . Then,

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nuclear equations of state at n ≤ nL and quark equations of state at n ≥ nU are prepared. In practice, one may choose nuclear equations of state based on the modern nuclear forces and many-body calculations. Meanwhile, quark equations of state must be calculated by using some constituent quark models developed for hadron spectroscopy. Finally, the nuclear and quark equations of state are phenomenologically interpolated. For interpolating functions, one can choose, for instance (Kojo et al. 2015; Baym et al. 2018) (see also, e.g., Ayriyan et al. 2021),

Pinter (μ) =

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cj μj ,

(29)

j =0

where the coefficients of the polynomials are fixed by demanding matching conditions up to the second derivatives of P ,  dl Ph dl Pinter  = (dμ)l μL (dμ)l

   

μL

,

 dl Pq dl Pinter  = (dμ)l μU (dμ)l

   

, (l = 0, 1, 2) . (30) μU

The μL and μU are extracted from the conditions n(μL ) = nL and n(μU ) = nU , respectively. Six boundary conditions are available to determine all the coefficients. In the abovementioned procedures, one can construct a unified equation of state. But not all of those equations of state are physical. Physical equations of state must satisfy the conditions of the thermodynamic stability (χ = d2 P /dμ2 ≥ 0) and the causality (cs2 ≤ 1). In order to make interpolated equations of state physical, a proper combination of nuclear and quark equations of state must be chosen (Baym et al. 2019; Kojo et al. 2022). In general, the causality tends to be violated when a nuclear equation of state is softer and a quark equation of state is stiffer, because such soft-to-stiff combination requires the pressure grows rapidly as a function of energy density, accompanying a large slope, dP /dρ = cs2 . In the context of neutron star physics, the nuclear equation of state up to ∼2n0 is strongly correlated with the radii of ∼1.4M neutron stars (Lattimer and Prakash 2001), while the quark equation of state is correlated with the maximum mass of neutron stars which must be larger than 2M . If one assumes the first-order phase transitions in the interpolated domain, it becomes more difficult to satisfy the causality constraint. During the first order, P is constant and ρ grows. After the phase transition is over, the dP /dρ must grow even more steeply to achieve the stiffness necessary to pass the 2M constraints. Of course, weak first-order transitions are still possible, but such small transitions may be treated as a small perturbation to the crossover scenarios. Thus, the three-window model made of smooth interpolating functions may be taken as a baseline to discuss more detailed phase structures of matter.

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Interactions in Strongly Correlated Quark Matter The question is: What kind of quark matter can generate equations of state which are stiff enough to pass the 2M constraints. As noted before, noninteracting massless quarks can give a very stiff equation of state, so starting with a free relativistic quark equation of state and then adding masses and interactions as corrections may be a good strategy. Before moving to the quark cases, it is useful to address how the extrapolation of a purely nucleonic model would give a stiff equation of state. Here, the following parametrization is considered: ρ(n) = mN n + a

n5/3 + bnα , mN

(31)

where the first is the mass density, the second is the kinetic energy, and the third is an interaction with a, b positive constants. Noting μ = ∂ρ/∂n and P = μn − ρ, we find P =

2 n5/3 + b(α − 1)nα . a 3 mN

(32)

The 2M constraint requires P and ρ to be the same order. This is never achieved unless we have very strong repulsions; for b = 0, P /ρ ∼ n2/3 /m2N , very small unless n1/3 ∼ mN or n ∼ 50n0 . This regime is not achievable in neutron star cores. Thus, in the following, we neglect the kinetic energy term. Next, it is instructive to assume α > 1 and consider the regime where the interaction terms dominate over the rest mass energy. Then, (Fig. 22, left)

Fig. 22 A squared sound velocity cs2 as a function of n/n0 . (Left) Purely nucleonic descriptions for free nucleons, plus two-body repulsion, and plus three-body repulsion. The repulsions are assumed to be the contact type. (Right) The behaviors typical in crossover models arranged to satisfy nuclear and 2M constraints. Stiffening is typically more rapid than in purely nucleonic models

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P ∼ (α − 1)ρ → cs2 ∼ (α − 1) .

(33)

For contact interactions, the dominance of two-body forces with α ∼ 2 leads to cs2 ∼ 1, while three-body forces with α ∼ 3 lead to cs2 ∼ 2 with violation of the causality. This trend indicates that, while many-body repulsions help equations of state to achieve necessary stiffness, their dominance in equations of state would violate the causality. As density increases, terms with the highest powers of n dominate over the others so that one eventually has to stop using models with more than twobody forces or must assume the couplings to strongly decrease at high density. For example, time-honored Akmal-Pandharipande-Ravenhall (APR) equation of state violates the causality at ∼5n0 before reaching the maximum mass (Akmal et al. 1998). For a relativistic quark matter (Alford et al. 2005), it is useful to consider a simple parametrization: ρ(n) = an4/3 + bnα ,

(34)

where the first is the relativistic kinetic energy and the second describes interactions or mass energies. As before, one can calculate μ and use P = μn − ρ. It is possible to derive a useful expression by eliminating an4/3 terms in favor of ρ to reach (Kojo et al. 2015)   4 ρ +b α− nα , P = 3 3

(35)

where n is a function of ρ. The b = 0 case leads to the conformal limit, P = ρ/3, and cs2 = 1/3 (Fig. 22, right). It is important to note that the effects of interactions enter as the product of b and (α − 4/3); hence, not only the sign of b but also the density dependence is important to judge whether interactions stiffen or soften equations of state. For α > 4/3, the repulsive interactions (b > 0) stiffen equations of state; an example is a contact quark N(≥ 2)-body repulsion characterized by b > 0 and α = N . It is typical to discuss stiffening based on repulsions, but it should be kept in mind that terms with α > 4/3 should not be extrapolated to very high density, as they would dominate over the kinetic energy and contradict with the asymptotic-free nature of QCD at short distance. In order to find stiffening terms that have the natural high-density limit, one can consider α < 4/3 where the attractive interactions (b < 0) stiffen equations of state. Terms with the powers of n less than 4/3 must accompany some mass scales other than n. A possible term is the mass term from the expansion of the quark energies, E ∼ p(1 + m2q /p2 + · · · ) (mq : quark mass), but it yields ρmass ∼ +m2q n2/3 , softening equations of state. The bag constant, with b > 0 and α = 0, also softens equations of state. Yet there are still other possibilities that the dynamical scale of QCD, ΛQCD ∼ 200–300 MeV, appears in attractive correlations, leading to an

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Fig. 23 The quark Fermi sea with non-perturbative correlations near the Fermi surface. (Left) The diquark pairing in the S-wave, color-, spin-, and flavor-antisymmetric channels. The Fermi surface is diffused over the momentum scale of the energy gap ∼ Δ. (Right) Three-particle or baryonic correlations discussed in quarkyonic matter scenario

energy density ρdyn ∼ −Λ2QCD n2/3 . This case stiffens equations of state. The factor n2/3 indicates that the desired term is related to the non-perturbative dynamics near the Fermi surface whose area is ∼4πpF2 ∼ n2/3 . This brings our attention to the physics near the Fermi surface in quark matter. Those include, e.g., pairing effects in color superconductivity (CSC), baryonic correlations in quarkyonic matter, and so on, which will be discussed in the following (Fig. 23).

Diquark Pairings in Color Superconductivity (CSC) The CSC is one of very popular scenarios in the high-density limit. Alford et al. (2008) is recommended as an extensive review. Here, only the basic idea is mentioned. The color superconductivity is triggered by the condensations of quark-quark pairs (diquarks) as those of electron-electron pairs in usual superconductivity (Schrieffer 1999). The energetic costs to populate quarks near the Fermi sea are small, and attractive diquark correlations trigger the macroscopic appearance (condensates) of diquark pairs. They reorganize the ground state and open a gap Δ in a quark spectrum which reflects the energy required to excite a quark. The appearance of the gaps drastically changes the thermal and transport properties of quark matter since quarks cannot be activated by weak perturbations. In QCD, gluon exchanges offer various attractive channels for quark-quark interactions, as quarks have color, flavor, spin, and orbital degrees of freedom. As ¯ Diquarks can have for colors, quarks (antiquarks) belong to the representation 3 (3). ¯ the representations 3 ⊗ 3 = 3 ⊕ 6, where the former (latter) is an antisymmetrized (symmetrized) color state. A diquark in 3¯ has the smaller color charge than two separated quarks, producing less color-electric fields with smaller energetic costs. Thus, it is natural to consider 3¯ channels. For relative orbital wave functions, the

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S-wave channel can utilize the entire Fermi surface to make diquark pairs. Now the Fermi statistics demands the leftover spin-flavor combinations to be symmetric. To proceed further, it is important to consider the color-magnetic interactions, 1 · λ  2 )( ∼ −c(λ σ1 · σ2 ), arising from a one-gluon exchange with c being positive, λ being the Gell-Mann matrix for colors, and σ being the Pauli matrix for spins  2 ) ¯ < 0, so the spin should 1 · λ (De Rujula et al. 1975). For the 3¯ color states, (λ 3 be singlet to get an energetic benefit. Accordingly, the flavor states are fixed to antisymmetric states 3¯ (1) in three (two) flavor theories. In three flavors, the phase with the abovementioned diquark condensates is called color-flavor-locked (CFL) phase (Alford et al. 1999). The diquark contributions to the energy density are ρΔ ∼ −Δ2 (4πpF2 ) ∼ −Δ2 n2/3 .

(36)

If Δ depends on n only weakly, the diquark condensates stiffen equations of state. In the high-density limit, the magnitude and density dependence of Δ can be rigorously estimated within the weak coupling framework (Son 1999): Δ ∼ μq gs−5 exp



3π 2 − √ 2gs

 .

(37)

The gap Δ is essentially determined by two ingredients: (i) the strength of interactions and (ii) the phase space available for quantum fluctuations (or intermediate states appearing in the self-energy calculations). The high momentum transfer processes, for which αs is small, are weak in interactions, but there are many intermediate states to enhance the magnitudes. Meanwhile, small momentum transfer processes have stronger coupling, but not much phase space is available. At high enough densities, the large momentum processes dominate, and the estimate of Δ can be done within the weak coupling regime. Such high-density estimate, however, is questionable in applications to the NS physics. The Fermi surface is not large enough to be dominated by weak coupling processes; a typical momentum transfer is pF < 1 GeV, and hence αs is not small. The estimates of Δ or the existence of the CSC at strong coupling is not established. But one can gain some insights from two-color QCD, a cousin of our three-color QCD. Unlike the three-color case, the lattice Monte-Carlo simulations at finite density are doable in two-color QCD. Various quantities, e.g., phase structures and equations of state, have been computed (Boz et al. 2020; Iida et al. 2020; Bornyakov et al. 2022). The lattice simulations showed the existence of diquark condensed phase as model calculations predicted. What is quite remarkable is that, even at n  10n0 , the melting temperature of diquark condensates is high, Tc  100 MeV, and is insensitive to density at least to ∼50–100n0 . In näive BCS estimate, the Tc is related to the gap as Δ  175 MeV. The value is comparable to ΛQCD , suggesting the non-perturbative physics persist to high density close to, or even above, the domain where pQCD has been trusted.

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Quarkyonic Matter In addition to diquarks, one can also consider baryonic correlations near the Fermi surface. This should be increasingly important in the lower density regime. Diquarks, which have colors, cannot survive in dilute regime because of the confinement and must pick up a quark to get color neutralized. This consideration draws our attention to the concept of a quarkyonic matter (McLerran and Pisarski 2007). The quarkyonic matter was originally discussed in the limit of a large number of colors (large Nc ) with gs2 Nc = O(1) fixed (’t Hooft 1974). In this large Nc limit, the fermion loops of O(1/Nc ) are negligible compared to the gluon loops, so that the gluons are unaffected by the quark dynamics. This picture leads to an interesting consequence at high density. As μ increases, the baryonic matter appears at μq = μ/Nc ∼ Mq ∼ MB /Nc where Mq is the constituent quark mass. Increasing μq further to μq ΛQCD , baryons eventually overlap and quarks form a Fermi sea. This looks paradoxical in the large Nc limit: while baryons largely overlap, gluons must remain the same as in vacuum and should confine any colored objects. More precisely, the gluons are unaffected until the screening effects become strong enough to dominate over non-perturbative effects. The Debye mass reaches the nonperturbative scale as 1/2

m2D ∼ gs2 μ2q ∼ Λ2QCD → μq ∼ Nc ΛQCD .

(38) 1/2

Large Nc emphasizes the possibility of a window ΛQCD  μq  Nc ΛQCD , where a seemingly paradoxical matter, a quark matter with confinement, is realized. There are several scenarios to resolve the paradox. The first is to just assume that baryons remain effective degrees of freedom, entirely hiding quarks from the 1/2 dynamics until the condition μq ∼ Nc ΛQCD is met. The constituents of the Fermi sea are color-singlet, so is the resulting Fermi sea. But the description is not necessarily natural; the kinetic energy ∼pF2 /mN ∼ 1/Nc is much smaller than the interaction energy ∼Nc (Witten 1979). This raises questions concerning the choice of proper degrees of freedom. On the contrary, the quarkyonic matter scenario does not demand the Fermi sea to be made of confined objects but demand only the Fermi sea as a whole to satisfy the color-singlet condition. In this picture, one may take quarks as natural degrees of freedom to describe the bulk part of the Fermi sea, as in conventional quark matter pictures at weak coupling. Meanwhile, quarks near the Fermi surface are subject to soft gluon exchanges as the phase space for the final states is open. Such small momentum transfer does not change much the location of quarks in momentum space, but the coupling is very strong and confining. An efficient way to include such confining effects is to choose baryons as effective degrees of freedom near the quark Fermi surface. A quarkyonic matter describes a quark matter with a baryonic Fermi surface.

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Fig. 24 The relation between the baryon (B) and quark (fq ) occupation probabilities for given momenta. In dilute baryonic matter, quarks are packed within a baryon and have broad distribution in momenta; none of momentum states are saturated, and quark Pauli blocking effects are minor. As baryon density increases, eventually quark states at low momenta are saturated. Beyond this regime, baryons must occupy higher momenta but with small occupation probabilities not to violate the quark Pauli principle. The quarks eventually establish conventional Fermi sea

In the context of neutron star physics, the quarkyonic matter picture explains stiffening of matter in the transition from a nuclear to a quark matter (McLerran and Reddy 2019; Jeong et al. 2020). The essential ingredient is the quark Pauli blocking constraint on baryonic degrees of freedom. To see how it works, it is useful to consider the occupation probability of quark states, fq (p; n), at a given density (Kojo 2021, Fig. 24). At very high density, it should describe a usual quark Fermi sea, fq ∼ θ (pF − p). The question is: How does it evolve from nuclear to quark matter? In a single nucleon, quarks are localized within the domain of ∼1 fm3 and thus occupy momentum states from zero to ∼ΛQCD . The occupation probability for each state is much less than 1. With such small occupation probability, the quark Pauli blocking is not important in a dilute baryonic matter. But as baryon density increases, such small occupation probability from each baryon is accumulated, and at n ∼ Λ3QCD , the low momentum states begin to get saturated, approaching to the usual fq ∼ θ (pF − p)-type distribution. In this quark saturation regime, the quark Pauli blocking constraint must be taken into account in baryonic descriptions. The above idea is naturally implemented in the quarkyonic matter description. At low density, there is no quark Fermi sea, and only a baryonic Fermi surface is available. At densities of the quark saturation, a quark Fermi sea is established, pushing the baryonic Fermi surface to the high momentum domain. The resulting baryons, with quarks collectively moving in the same directions, have high momenta and are relativistic. The pressure is naturally large. This situation should be contrast to baryons in dilute regime where quarks do not have the common orientation; the forces from quarks cancel within a baryon, without yielding much pressure. Before and after the quark saturation, the pressure increases rapidly, but the energy density changes only smoothly. In fact, even pure nuclear and pure quark matters do not

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have much difference in the energy densities: they have ρ ∼ mN n ∼ Nc ΛQCD n and ∼Nc n4/3 , respectively, which are the same order at n ∼ Λ3QCD . This rapid increase in P with gentle change in ρ leads to a peak in sound velocity.

References B.P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 119, 161101 (2017). arXiv:1710.05832 [gr-qc] B.P. Abbott, R. Abbott, T.D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R.X. Adhikari, V.B. Adya, Phys. Rev. Lett. 121, 161101 (2018). arXiv:1805.11581 [gr-qc] S.M. Adams, C.S. Kochanek, J.R. Gerke, K.Z. Stanek, X. Dai, MNRAS 468, 4968 (2017). arXiv:1609.01283 [astro-ph.SR] A. Akmal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C 58, 1804 (1998). arXiv:nuclth/9804027 M.G. Alford, K. Rajagopal, F. Wilczek, Nucl. Phys. B 537, 443 (1999). arXiv:hep-ph/9804403 M.G. Alford, K. Rajagopal, S. Reddy, F. Wilczek, Phys. Rev. D 64, 074017 (2001). arXiv:hepph/0105009 M. Alford, M. Braby, M.W. Paris, S. Reddy, Astrophys. J. 629, 969 (2005). arXiv:nucl-th/0411016 M.G. Alford, A. Schmitt, K. Rajagopal, T. Schäfer, Rev. Mod. Phys. 80, 1455 (2008). arXiv:0709.4635 [hep-ph] S. Ando, J.F. Beacom, H. Yüksel, Phys. Rev. Lett. 95, 171101 (2005). arXiv:astro-ph/0503321 [astro-ph] E. Annala, T. Gorda, A. Kurkela, A. Vuorinen, Phys. Rev. Lett. 120, 172703 (2018). arXiv:1711.02644 [astro-ph.HE] J. Antoniadis, P.C.C. Freire, N. Wex, T.M. Tauris, R.S. Lynch, M.H. van Kerkwijk, M. Kramer, C. Bassa, V.S. Dhillon, T. Driebe, J.W.T. Hessels, V.M. Kaspi, V.I. Kondratiev, N. Langer, T.R. Marsh, M.A. McLaughlin, T.T. Pennucci, S.M. Ransom, I.H. Stairs, J. van Leeuwen, J.P.W. Verbiest, D.G. Whelan, Science 340, 448 (2013). arXiv:1304.6875 [astro-ph.HE] Y. Aoki, G. Endrodi, Z. Fodor, S.D. Katz, K.K. Szabo, Nature 443, 675 (2006). arXiv:heplat/0611014 D. Arnett, Supernovae and Nucleosynthesis: An Investigation of the History of Matter from the Big Bang to the Present. Princeton University Press, Princeton, New Jersey (1996) M.C. Atkinson, W.H. Dickhoff, M. Piarulli, A. Rios, R.B. Wiringa, Phys. Rev. C 102, 044333 (2020). arXiv:2001.07231 [nucl-th] A. Ayriyan, D. Blaschke, A.G. Grunfeld, D. Alvarez-Castillo, H. Grigorian, V. Abgaryan, Eur. Phys. J. A 57, 318 (2021). arXiv:2102.13485 [astro-ph.HE] L. Baiotti, L. Rezzolla, Rept. Prog. Phys. 80, 096901 (2017). arXiv:1607.03540 [gr-qc] S.S. Bao, H. Shen, Phys. Rev. C 89, 045807 (2014). arXiv:1405.3837 [nucl-th] E. Baron, J. Cooperstein, S. Kahana, Phys. Rev. Lett. 55, 126 (1985) A. Bauswein, O. Just, H.-T. Janka, N. Stergioulas, Astrophys. J. Lett. 850, L34 (2017). arXiv:1710.06843 [astro-ph.HE] A. Bauswein, N.-U.F. Bastian, D.B. Blaschke, K. Chatziioannou, J.A. Clark, T. Fischer, M. Oertel, Phys. Rev. Lett. 122, 061102 (2019). arXiv:1809.01116 [astro-ph.HE] G. Baym, T. Hatsuda, T. Kojo, P.D. Powell, Y. Song, T. Takatsuka, Rept. Prog. Phys. 81, 056902 (2018). arXiv:1707.04966 [astro-ph.HE] G. Baym, S. Furusawa, T. Hatsuda, T. Kojo, H. Togashi, Astrophys. J. 885, 42 (2019). arXiv:1903.08963 [astro-ph.HE] H.A. Bethe, Rev. Mod. Phys. 62, 801 (1990) H.A. Bethe, J.R. Wilson, ApJ 295, 14 (1985) R.M. Bionta, G. Blewitt, C.B. Bratton, D. Casper, A. Ciocio, R. Claus, B. Cortez, M. Crouch, S.T. Dye, S. Errede, G.W. Foster, W. Gajewski, K.S. Ganezer, M. Goldhaber, T.J. Haines, T.W. Jones, D. Kielczewska, W.R. Kropp, J.G. Learned, J.M. Losecco, J. Matthews, R. Miller,

3172

K. Sumiyoshi et al.

M.S. Mudan, H.S. Park, L.R. Price, F. Reines, J. Schultz, S. Seidel, E. Shumard, D. Sinclair, H.W. Sobel, J.L. Stone, L.R. Sulak, R. Svoboda, G. Thornton, J.C. van der Velde, C. Wuest, Phys. Rev. Lett. 58, 1494 (1987) S. Blacker, N.-U.F. Bastian, A. Bauswein, D.B. Blaschke, T. Fischer, M. Oertel, T. Soultanis, S. Typel, Phys. Rev. D 102, 123023 (2020). arXiv:2006.03789 [astro-ph.HE] V.G. Bornyakov, N.V. Gerasimeniuk, V.A. Goy, A. Nakamura, R.N. Rogalyov, (2022). arXiv:2203.04909 [hep-lat] T. Boz, P. Giudice, S. Hands, J.-I. Skullerud, Phys. Rev. D 101, 074506 (2020). arXiv:1912.10975 [hep-lat] C. Breu, L. Rezzolla, Mon. Not. R. Astron. Soc. 459, 646 (2016). arXiv:1601.06083 [gr-qc] R. Brockmann, R. Machleidt, Phys. Rev. C 42, 1965 (1990) G.F. Burgio, H.J. Schulze, I. Vidaña, J.B. Wei, Prog. Part. Nucl. Phys. 120, 103879 (2021). arXiv:2105.03747 [nucl-th] A. Burrows, ApJ 334, 891 (1988) A. Burrows, Rev. Mod. Phys. 85, 245 (2013). arXiv:1210.4921 [astro-ph.SR] A. Burrows, J.M. Lattimer, ApJ 307, 178 (1986) J. Carlson, S. Gandolfi, F. Pederiva, S.C. Pieper, R. Schiavilla, K.E. Schmidt, R.B. Wiringa, Rev. Mod. Phys. 87, 1067 (2015). arXiv:1412.3081 [nucl-th] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, Nucl. Phys. A 635, 231 (1998) N. Chamel, P. Haensel, Living Rev. Relativ. 11, 10 (2008). arXiv:0812.3955 [astro-ph] H.T. Cromartie, E. Fonseca, S.M. Ransom, P.B. Demorest, Z. Arzoumanian, H. Blumer, P.R. Brook, M.E. DeCesar, T. Dolch, J.A. Ellis, R.D. Ferdman, E.C. Ferrara, N. Garver-Daniels, P.A. Gentile, M.L. Jones, M.T. Lam, D.R. Lorimer, R.S. Lynch, M.A. McLaughlin, C. Ng, D.J. Nice, T.T. Pennucci, R. Spiewak, I.H. Stairs, K. Stovall, J.K. Swiggum, W.W. Zhu, Nat. Astron. 4, 72 (2020). arXiv:1904.06759 [astro-ph.HE] P.B. Demorest, T. Pennucci, S.M. Ransom, M.S.E. Roberts, J.W.T. Hessels, Nature 467, 1081 (2010). arXiv:1010.5788 [astro-ph.HE] A. De Rujula, H. Georgi, S.L. Glashow, Phys. Rev. D 12, 147 (1975) C. Drischler, K. Hebeler, A. Schwenk, Phys. Rev. Lett. 122, 042501 (2019). arXiv:1710.08220 [nucl-th] C. Drischler, J.W. Holt, C. Wellenhofer, arXiv e-prints (2021a). arXiv:2101.01709 [nucl-th] C. Drischler, J.W. Holt, C. Wellenhofer, Ann. Rev. Nucl. Part. Sci. 71, 403 (2021b). arXiv:2101.01709 [nucl-th] H. Duan, J.P. Kneller, J. Phys. G Nucl. Phys. 36, 113201 (2009). arXiv:0904.0974 [astro-ph.HE] M. Dutra, O. Lourenco, J.S. Sa Martins, A. Delfino, J.R. Stone, P.D. Stevenson, Phys. Rev. C 85, 035201 (2012) V. Ferrari, G. Miniutti, J.A. Pons, MNRAS 342, 629 (2003). arXiv:astro-ph/0210581 [astro-ph] T. Fischer, I. Sagert, G. Pagliara, M. Hempel, J. Schaffner-Bielich, T. Rauscher, F.K. Thielemann, R. Käppeli, G. Martínez-Pinedo, M. Liebendörfer, ApJS 194, 39 (2011). arXiv:1011.3409 [astro-ph.HE] E. Fonseca, H.T. Cromartie, T.T. Pennucci, P.S. Ray, A.Y. Kirichenko, S.M. Ransom, P.B. Demorest, I.H. Stairs, Z. Arzoumanian, L. Guillemot, A. Parthasarathy, M. Kerr, I. Cognard, P.T. Baker, H. Blumer, P.R. Brook, M. DeCesar, T. Dolch, F.A. Dong, E.C. Ferrara, W. Fiore, N. Garver-Daniels, D.C. Good, R. Jennings, M.L. Jones, V.M. Kaspi, M.T. Lam, D.R. Lorimer, J. Luo, A. McEwen, J.W. McKee, M.A. McLaughlin, N. McMann, B.W. Meyers, A. Naidu, C. Ng, D.J. Nice, N. Pol, H.A. Radovan, B. Shapiro-Albert, C.M. Tan, S.P. Tendulkar, J.K. Swiggum, H.M. Wahl, W.W. Zhu, ApJ 915, L12 (2021). arXiv:2104.00880 [astro-ph.HE] M. Fortin, J.L. Zdunik, P. Haensel, M. Bejger, A&A 576, A68 (2015). arXiv:1408.3052 [astroph.SR] S. Fujibayashi, Y. Sekiguchi, K. Kiuchi, M. Shibata, ApJ 846, 114 (2017). arXiv:1703.10191 [astro-ph.HE] Y. Fujimoto, K. Fukushima, K. Hotokezaka, K. Kyutoku, (2022). arXiv:2205.03882 [astro-ph.HE] K. Fukushima, T. Kojo, W. Weise, Phys. Rev. D 102, 096017 (2020). arXiv:2008.08436 [hep-ph] S. Furusawa, H. Nagakura, Prog. Part. Nucl. Phys. 129, 104018 (2023). arXiv:2211.01050 [nuclth]

86 Equation of State in Neutron Stars and Supernovae

3173

S. Furusawa, H. Togashi, H. Nagakura, K. Sumiyoshi, S. Yamada, H. Suzuki, M. Takano, J. Phys. G Nucl. Phys. 44, 094001 (2017a). arXiv:1707.06410 [astro-ph.HE] S. Furusawa, H. Nagakura, K. Sumiyoshi, C. Kato, S. Yamada, Phys. Rev. C 95, 025809 (2017b). arXiv:1701.08414 [astro-ph.HE] U. Garg, G. Colò, Prog. Part. Nucl. Phys. 101, 55 (2018). arXiv:1801.03672 [nucl-ex] J. Ghiglieri, A. Kurkela, M. Strickland, A. Vuorinen, Phys. Rept. 880, 1 (2020). arXiv:2002.10188 [hep-ph] N.K. Glendenning, Phys. Rev. D 46, 1274 (1992) T. Gorda, A. Kurkela, R. Paatelainen, S. Säppi, A. Vuorinen, Phys. Rev. Lett. 127, 162003 (2021a). arXiv:2103.05658 [hep-ph] T. Gorda, A. Kurkela, R. Paatelainen, S. Säppi, A. Vuorinen, Phys. Rev. D 104, 074015 (2021b). arXiv:2103.07427 [hep-ph] P. Haensel, A.Y. Potekhin, D.G. Yakovlev, Neutron Stars 1: Equation of State and Structure, vol. 326 (2007) R. Hagedorn, Nuovo Cim. Suppl. 3, 147 (1965) T. Hatsuda, T. Kunihiro, Phys. Rept. 247, 221 (1994). arXiv:hep-ph/9401310 A. Heger, C.L. Fryer, S.E. Woosley, N. Langer, D.H. Hartmann, ApJ 591, 288 (2003). arXiv:astroph/0212469 [astro-ph] H. Heiselberg, arXiv e-prints , astro-ph/0201465 (2002). arXiv:astro-ph/0201465 [astro-ph] M. Hempel, J. Schaffner-Bielich, Nucl. Phys. A 837, 210 (2010). arXiv:0911.4073 [nucl-th] M. Hempel, K. Hagel, J. Natowitz, G. Röpke, S. Typel, Phys. Rev. C 91, 045805 (2015). arXiv:1503.00518 [nucl-th] T. Hinderer, B.D. Lackey, R.N. Lang, J.S. Read, Phys. Rev. D 81, 123016 (2010). arXiv:0911.3535 [astro-ph.HE] K. Hirata, T. Kajita, M. Koshiba, M. Nakahata, Y. Oyama, N. Sato, A. Suzuki, M. Takita, Y. Totsuka, T. Kifune, T. Suda, K. Takahashi, T. Tanimori, K. Miyano, M. Yamada, E.W. Beier, L.R. Feldscher, S.B. Kim, A.K. Mann, F.M. Newcomer, R. van, W. Zhang, B.G. Cortez, Phys. Rev. Lett. 58, 1490 (1987) E. Hiyama, K. Nakazawa, Ann. Rev. Nucl. Part. Sci. 68, 131 (2018) J.W. Holt, N. Kaiser, Phys. Rev. C 95, 034326 (2017) S. Horiuchi, J.F. Beacom, E. Dwek, Phys. Rev. D 79, 083013 (2009). arXiv:0812.3157 [astro-ph] C.J. Horowitz, E.F. Brown, Y. Kim, W.G. Lynch, R. Michaels, A. Ono, J. Piekarewicz, M.B. Tsang, H.H. Wolter, J. Phys. G Nucl. Phys. 41, 093001 (2014). arXiv:1401.5839 [nucl-th] Y.-J. Huang, L. Baiotti, T. Kojo, K. Takami, H. Sotani, H. Togashi, T. Hatsuda, S. Nagataki, Y.-Z. Fan, Phys. Rev. Lett. 129, 181101 (2022) Hyper-Kamiokande Proto-Collaboration, K. Abe, K. Abe, H. Aihara, A. Aimi, R. Akutsu, C. Andreopoulos, I. Anghel, L.H.V. Anthony, M. Antonova, Y. Ashida, V. Aushev, M. Barbi, G.J. Barker, G. Barr, P. Beltrame, V. Berardi, M. Bergevin, S. Berkman, L. Berns, T. Berry, S. Bhadra, D. Bravo-Berguño, F.D.M. Blaszczyk, A. Blondel, S. Bolognesi, S.B. Boyd, A. Bravar, C. Bronner, M. Buizza Avanzini, F.S. Cafagna, A. Cole, R. Calland, S. Cao, S.L. Cartwright, M.G. Catanesi, C. Checchia, Z. Chen-Wishart, J.H. Choi, K. Choi, J. Coleman, G. Collazuol, G. Cowan, L. Cremonesi, T. Dealtry, G. De Rosa, C. Densham, D. Dewhurst, E.L. Drakopoulou, F. Di Lodovico, O. Drapier, J. Dumarchez, P. Dunne, M. Dziewiecki, S. Emery, A. Esmaili, A. Evangelisti, E. Fernandez-Martinez, T. Feusels, A. Finch, A. Fiorentini, G. Fiorillo, M. Fitton, K. Frankiewicz, M. Friend, Y. Fujii, Y. Fukuda, D. Fukuda, K. Ganezer, C. Giganti, M. Gonin, N. Grant, P. Gumplinger, D.R. Hadley, B. Hartfiel, M. Hartz, Y. Hayato, K. Hayrapetyan, J. Hill, S. Hirota, S. Horiuchi, A.K. Ichikawa, T. Iijima, M. Ikeda, J. Imber, K. Inoue, J. Insler, R.A. Intonti, A. Ioannisian, T. Ishida, H. Ishino, M. Ishitsuka, Y. Itow, K. Iwamoto, A. Izmaylov, B. Jamieson, H.I. Jang, J.S. Jang, S.H. Jeon, M. Jiang, P. Jonsson, K.K. Joo, A. Kaboth, C. Kachulis, T. Kajita, J. Kameda, Y. Kataoka, T. Katori, K. Kayrapetyan, E. Kearns, M. Khabibullin, A. Khotjantsev, J.H. Kim, J.Y. Kim, S.B. Kim, S.Y. Kim, S. King, Y. Kishimoto, T. Kobayashi, M. Koga, A. Konaka, L.L. Kormos, Y. Koshio, A. Korzenev, K.L. Kowalik, W.R. Kropp, Y. Kudenko, R. Kurjata, T. Kutter, M. Kuze, L. Labarga, J. Lagoda, P.J.J. Lasorak, M. Laveder, M. Lawe, J.G. Learned, I.T. Lim, T. Lindner, R.P. Litchfield, A. Longhin, P. Loverre, T. Lou, L. Ludovici, W. Ma, L. Magaletti, K. Mahn,

3174

K. Sumiyoshi et al.

M. Malek, L. Maret, C. Mariani, K. Martens, L. Marti, J.F. Martin, J. Marzec, S. Matsuno, E. Mazzucato, M. McCarthy, N. McCauley, K.S. McFarland, C. McGrew, A. Mefodiev, P. Mermod, C. Metelko, M. Mezzetto, J. Migenda, P. Mijakowski, H. Minakata, A. Minamino, S. Mine, O. Mineev, A. Mitra, M. Miura, T. Mochizuki, J. Monroe, D.H. Moon, S. Moriyama, T. Mueller, F. Muheim, K. Murase, F. Muto, M. Nakahata, Y. Nakajima, K. Nakamura, T. Nakaya, S. Nakayama, C. Nantais, M. Needham, T. Nicholls, Y. Nishimura, E. Noah, F. Nova, J. Nowak, H. Nunokawa, Y. Obayashi, H.M. O’Keeffe, Y. Okajima, K. Okumura, Y. Onishchuk, E. O’Sullivan, L. O’Sullivan, T. Ovsiannikova, R.A. Owen, Y. Oyama, M.Y. Pac, V. Palladino, J.L. Palomino, V. Paolone, W. Parker, S. Parsa, D. Payne, J.D.Perkin, C. Pidcott, E. Pinzon Guerra, S. Playfer, B. Popov, M. Posiadala-Zezula, J.M. Poutissou, A. Pritchard, N.W. Prouse, G. Pronost, P. Przewlocki, B. Quilain, E. Radicioni, P.N. Ratoff, F. Retiere, C. Riccio, B. Richards, E. Rondio, H.J. Rose, C. Rott, S.D. Rountree, A.C. Ruggeri, A. Rychter, R. Sacco, M. Sakuda, M.C. Sanchez, E. Scantamburlo, M. Scott, S.M. Sedgwick, Y. Seiya, T. Sekiguchi, H. Sekiya, S.H. Seo, D. Sgalaberna, R. Shah, A. Shaikhiev, I. Shimizu, M. Shiozawa, Y. Shitov, S. Short, C. Simpson, G. Sinnis, M.B. Smy, S. Snow, J. Sobczyk, H.W. Sobel, Y. Sonoda, R. Spina, T. Stewart, J.L. Stone, Y. Suda, Y. Suwa, Y. Suzuki, A.T. Suzuki, R. Svoboda, M. Taani, R. Tacik, A. Takeda, A. Takenaka, A. Taketa, Y. Takeuchi, V. Takhistov, H.A. Tanaka, H.K.M. Tanaka, H. Tanaka, R. Terri, M. Thiesse, L.F. Thompson, M. Thorpe, S. Tobayama, C. Touramanis, T. Towstego, T. Tsukamoto, K.M. Tsui, M. Tzanov, Y. Uchida, M.R. Vagins, G. Vasseur, C. Vilela, R.B. Vogelaar, J. Walding, J. Walker, M. Ward, D. Wark, M.O. Wascko, A. Weber, R. Wendell, R.J. Wilkes, M.J. Wilking, J.R. Wilson, T. Xin, K. Yamamoto, C. Yanagisawa, T. Yano, S. Yen, N. Yershov, D.N.Yeum, M. Yokoyama, T. Yoshida, I. Yu, M. Yu, J. Zalipska, K. Zaremba, M. Ziembicki, M. Zito, S. Zsoldos, arXiv e-prints, (2018). arXiv:1805.04163 [physics.ins-det] K. Iida, E. Itou, T.-G. Lee, JHEP 01, 181 (2020). arXiv:1910.07872 [hep-lat] T. Iritani et al. (HAL QCD), Phys. Lett. B 792, 284 (2019). arXiv:1810.03416 [hep-lat] H.-T. Janka, Annu. Rev. Nucl. Part. Sci. 62, 407 (2012). arXiv:1206.2503 [astro-ph.SR] H.-T. Janka, Neutrino-driven explosions, in Handbook of Supernovae (Springer Nature, Cham, Switzerland, 2017), p. 1095. ISBN 978-3-319-21845-8 H.-T. Janka, Neutrino Emission from Supernovae. in Handbook of Supernovae (Springer Nature, Cham, Switzerland, 2017), p. 1575. ISBN 978-3-319-21845-8 H.-T. Janka, F. Hanke, L. Hüdepohl, A. Marek, B. Müller, M. Obergaulinger, Prog. Theor. Exp. Phys. 2012, 01A309 (2012). arXiv:1211.1378 [astro-ph.SR] H.-T. Janka, T. Melson, A. Summa, Annu. Rev. Nucl. Part. Sci. 66, 341 (2016). arXiv:1602.05576 [astro-ph.SR] K.S. Jeong, L. McLerran, S. Sen, Phys. Rev. C 101, 035201 (2020). arXiv:1908.04799 [nucl-th] M. Ju, X. Wu, F. Ji, J. Hu, H. Shen, Phys. Rev. C 103, 025809 (2021). arXiv:2102.12276 [nucl-th] D. Kasen, B. Metzger, J. Barnes, E. Quataert, E. Ramirez-Ruiz, Nature 551, 80 (2017). arXiv:1710.05463 [astro-ph.HE] T. Katayama, K. Saito, Phys. Rev. C 88, 035805 (2013). arXiv:1307.2067 [nucl-th] B. Kiziltan, A. Kottas, M. De Yoreo, S.E. Thorsett, ApJ 778, 66 (2013). arXiv:1011.4291 [astroph.GA] C.S. Kochanek, J.F. Beacom, M.D. Kistler, J.L. Prieto, K.Z. Stanek, T.A. Thompson, H. Yüksel, ApJ 684, 1336 (2008). arXiv:0802.0456 [astro-ph] T. Kojo, Phys. Rev. D 104, 074005 (2021). arXiv:2106.06687 [nucl-th] T. Kojo, P.D. Powell, Y. Song, G. Baym, Phys. Rev. D 91, 045003 (2015). arXiv:1412.1108 [hep-ph] T. Kojo, G. Baym, T. Hatsuda, ApJ 934, 46 (2022). arXiv:2111.11919 [astro-ph.HE] K. Kotake, C. R. Phys. 14, 318 (2013). arXiv:1110.5107 [astro-ph.HE] K. Kotake, K. Sato, K. Takahashi, Rep. Prog. Phys. 69, 971 (2006). arXiv:astro-ph/0509456 [astroph] K.R. Lang, Astrophysical Formulae. A Compendium for the Physicist and Astrophysicist., second corrected and enlarged edition. (Springer, Berlin/Heidelberg/New York, 1980) J.M. Lattimer, Annu. Rev. Nucl. Part. Sci. 62, 485 (2012). arXiv:1305.3510 [nucl-th]

86 Equation of State in Neutron Stars and Supernovae

3175

J.M. Lattimer, Y. Lim, ApJ 771, eid 51 (2013). arXiv:1203.4286 [nucl-th] J.M. Lattimer, M. Prakash, Astrophys. J. 550, 426 (2001). arXiv:astro-ph/0002232 J.M. Lattimer, C.J. Pethick, M. Prakash, P. Haensel, Phys. Rev. Lett. 66, 2701 (1991) B.-A. Li, B.-J. Cai, L.-W. Chen, J. Xu, Prog. Part. Nucl. Phys. 99, 29 (2018). arXiv:1801.01213 [nucl-th] B.-A. Li, P.G. Krastev, D.-H. Wen, N.-B. Zhang, Euro. Phys. J. A 55, 117 (2019). arXiv:1905.13175 [nucl-th] D.R. Lorimer, Living Rev. Relativ. 11, 8 (2008). arXiv:0811.0762 [astro-ph] R. Machleidt, F. Sammarruca, Eur. Phys. J. A 56, 95 (2020), arXiv:2001.05615 [nucl-th] R. Machleidt, F. Sammarruca, Y. Song, Phys. Rev. C 53, R1483 (1996). arXiv:nucl-th/9510023 B. Margalit, B.D. Metzger, Astrophys. J. Lett. 850, L19 (2017). arXiv:1710.05938 [astro-ph.HE] T. Maruyama, S. Chiba, H.-J. Schulze, T. Tatsumi, Phys. Rev. D 76, 123015 (2007). arXiv:0708.3277 [nucl-th] K. Masuda, T. Hatsuda, T. Takatsuka, Astrophys. J. 764, 12 (2013a). arXiv:1205.3621 [nucl-th] K. Masuda, T. Hatsuda, T. Takatsuka, PTEP 2013, 073D01 (2013b). arXiv:1212.6803 [nucl-th] T.J. Mazurek, G.E. Brown, J.M. Lattimer, ApJ 229, 713 (1979) L. McLerran, R.D. Pisarski, Nucl. Phys. A 796, 83 (2007). arXiv:0706.2191 [hep-ph] L. McLerran, S. Reddy, Phys. Rev. Lett. 122, 122701 (2019). arXiv:1811.12503 [nucl-th] M.C. Miller, F.K. Lamb, A.J. Dittmann, S. Bogdanov, Z. Arzoumanian, K.C. Gendreau, S. Guillot, A.K. Harding, W.C.G. Ho, J.M. Lattimer, R.M. Ludlam, S. Mahmoodifar, S.M. Morsink, P.S. Ray, T.E. Strohmayer, K.S. Wood, T. Enoto, R. Foster, T. Okajima, G. Prigozhin, Y. Soong, ApJ 887, L24 (2019). arXiv:1912.05705 [astro-ph.HE] M.C. Miller, F.K. Lamb, A.J. Dittmann, S. Bogdanov, Z. Arzoumanian, K.C. Gendreau, S. Guillot, W.C.G. Ho, J.M. Lattimer, M. Loewenstein, S.M. Morsink, P.S. Ray, M.T. Wolff, C.L. Baker, T. Cazeau, S. Manthripragada, C.B. Markwardt, T. Okajima, S. Pollard, I. Cognard, H.T. Cromartie, E. Fonseca, L. Guillemot, M. Kerr, A. Parthasarathy, T.T. Pennucci, S. Ransom, I. Stairs, ApJ 918, L28 (2021). arXiv:2105.06979 [astro-ph.HE] A. Mirizzi, I. Tamborra, H.T. Janka, N. Saviano, K. Scholberg, R. Bollig, L. Hüdepohl, S. Chakraborty, Nuovo Cimento Riv. Ser. 39, 1 (2016). arXiv:1508.00785 [astro-ph.HE] E.R. Most, L.R. Weih, L. Rezzolla, J. Schaffner-Bielich, Phys. Rev. Lett. 120, 261103 (2018). arXiv:1803.00549 [gr-qc] E.R. Most, L.J. Papenfort, V. Dexheimer, M. Hanauske, S. Schramm, H. Stöcker, L. Rezzolla, Phys. Rev. Lett. 122, 061101 (2019). arXiv:1807.03684 [astro-ph.HE] B. Müller, Annu. Rev. Nucl. Part. Sci. 69, 253 (2019). arXiv:1904.11067 [astro-ph.HE] H. Nagakura, K. Sumiyoshi, S. Yamada, ApJ 880, L28 (2019a). arXiv:1907.04863 [astro-ph.HE] H. Nagakura, S. Furusawa, H. Togashi, S. Richers, K. Sumiyoshi, S. Yamada, ApJS 240, 38 (2019b). arXiv:1812.09811 [astro-ph.HE] K. Nakazato, K. Sumiyoshi, S. Yamada, Phys. Rev. D 77, 103006 (2008). arXiv:0804.0661 [astroph] K. Nakazato, K. Sumiyoshi, H. Togashi, PASJ 73, 639 (2021). arXiv:2103.14386 [astro-ph.HE] E. O’Connor, C.D. Ott, ApJ 730, eid70 (2011). arXiv:1010.5550 [astro-ph.HE] M. Oertel, M. Hempel, T. Klähn, S. Typel, Rev. Mod. Phys. 89, 015007 (2017). arXiv:1610.03361 [astro-ph.HE] C.D. Ott, Classical Quantum Gravity 26, 063001 (2009). arXiv:0809.0695 [astro-ph] K. Oyamatsu, Nucl. Phys. A 561, 431 (1993) F. Özel, P. Freire, ARA&A 54, 401 (2016). arXiv:1603.02698 [astro-ph.HE] F. Özel, D. Psaltis, R. Narayan, A. Santos Villarreal, ApJ 757, 55 (2012). arXiv:1201.1006 [astroph.HE] D. Page, J.H. Applegate, ApJ 394, L17 (1992) D. Page, J.M. Lattimer, M. Prakash, A.W. Steiner, ApJS 155, 623 (2004). arXiv:astro-ph/0403657 [astro-ph] D. Page, U. Geppert, F. Weber, Nucl. Phys. A 777, 497 (2006). arXiv:astro-ph/0508056 [astro-ph] J.A. Pons, S. Reddy, M. Prakash, J.M. Lattimer, J.A. Miralles, ApJ 513, 780 (1999). arXiv:astroph/9807040 [astro-ph]

3176

K. Sumiyoshi et al.

A.Y. Potekhin, J.A. Pons, D. Page, Space Sci. Rev. 191, 239 (2015). arXiv:1507.06186 [astroph.HE] L. Qin, K. Hagel, R. Wada, J.B. Natowitz, S. Shlomo, A. Bonasera, G. Röpke, S. Typel, Z. Chen, M. Huang, J. Wang, H. Zheng, S. Kowalski, M. Barbui, M.R.D. Rodrigues, K. Schmidt, D. Fabris, M. Lunardon, S. Moretto, G. Nebbia, S. Pesente, V. Rizzi, G. Viesti, M. Cinausero, G. Prete, T. Keutgen, Y. El Masri, Z. Majka, Y.G. Ma, Phys. Rev. Lett. 108, 172701 (2012) D. Radice, L. Dai, Eur. Phys. J. A 55, 50 (2019). arXiv:1810.12917 [astro-ph.HE] V. Ravi, P.D. Lasky, Mon. Not. R. Astron. Soc. 441, 2433 (2014). arXiv:1403.6327 [astro-ph.HE] J.S. Read, B.D. Lackey, B.J. Owen, J.L. Friedman, Phys. Rev. D 79, 124032 (2009). arXiv:0812.2163 [astro-ph] L. Rezzolla, E.R. Most, L.R. Weih, Astrophys. J. Lett. 852, L25 (2018). arXiv:1711.00314 [astroph.HE] T.E. Riley, A.L. Watts, P.S. Ray, S. Bogdanov, S. Guillot, S.M. Morsink, A.V. Bilous, Z. Arzoumanian, D. Choudhury, J.S. Deneva, K.C. Gendreau, A.K. Harding, W.C.G. Ho, J.M. Lattimer, M. Loewenstein, R.M. Ludlam, C.B. Markwardt, T. Okajima, C. PrescodWeinstein, R.A. Remillard, M.T. Wolff, E. Fonseca, H.T. Cromartie, M. Kerr, T.T. Pennucci, A. Parthasarathy, S. Ransom, I. Stairs, L. Guillemot, I. Cognard, ApJ 918, L27 (2021). arXiv:2105.06980 [astro-ph.HE] M. Ruiz, S.L. Shapiro, A. Tsokaros, Phys. Rev. D 97, 021501 (2018). arXiv:1711.00473 [astroph.HE] K. Sasaki et al. (HAL QCD), Nucl. Phys. A 998, 121737 (2020). arXiv:1912.08630 [hep-lat] K. Sato, H. Suzuki, Phys. Rev. Lett. 58, 2722 (1987) K. Scholberg, Annu. Rev. Nucl. Part. Sci. 62, 81 (2012). arXiv:1205.6003 [astro-ph.IM] J. Schrieffer, Theory of Superconductivity. Advanced Books Classics (1999) S.L. Shapiro, S.A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (WILEY-VCH Verlag GmbH and Co. KGaA, CRC Press, Boca Raton, Florida, 1983) H. Shen, F. Ji, J. Hu, K. Sumiyoshi, ApJ 891, 148 (2020). arXiv:2001.10143 [nucl-th] M. Shibata, K. Hotokezaka, Ann. Rev. Nucl. Part. Sci. 69, 41 (2019). arXiv:1908.02350 [astroph.HE] M. Shibata, E. Zhou, K. Kiuchi, S. Fujibayashi, Phys. Rev. D 100, 023015 (2019). arXiv:1905.03656 [astro-ph.HE] S. Shlomo, V.M. Kolomietz, G. Colò, Euro. Phys. J. A 30, 23 (2006) D.T. Son, Phys. Rev. D 59, 094019 (1999). arXiv:hep-ph/9812287 H. Sotani, T. Takiwaki, Phys. Rev. D 94, 044043 (2016). arXiv:1608.01048 [astro-ph.HE] A.W. Steiner, M. Hempel, T. Fischer, ApJ 774, 17 (2013). arXiv:1207.2184 [astro-ph.SR] V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen, J.J. de Swart, Phys. Rev. C 49, 2950 (1994). arXiv:nucl-th/9406039 J.R. Stone, Universe 7, 257 (2021) J.R. Stone, N.J. Stone, S.A. Moszkowski, Phys. Rev. C 89, 044316 (2014). arXiv:1404.0744 [nuclth] Y. Sugahara, H. Toki, Nucl. Phys. A 579, 557 (1994) K. Sumiyoshi, in Journal of Physics Conference Series, vol. 861 (2017), p. 012028 K. Sumiyoshi, The Puzzle of Supernova Explosions Unraveled by Nuclear Physics (Kyoritsu Shuppan Co., Ltd., Tokyo, Japan, 2018) K. Sumiyoshi, H. Toki, ApJ 422, 700 (1994) K. Sumiyoshi, S. Yamada, H. Suzuki, S. Chiba, Phys. Rev. Lett. 97, 091101 (2006). arXiv:astroph/0608509 [astro-ph] K. Sumiyoshi, K. Nakazato, H. Suzuki, J. Hu, H. Shen, ApJ 887, 110 (2019). arXiv:1908.02928 [astro-ph.HE] K. Sumiyoshi, S. Fujibayashi, Y. Sekiguchi, M. Shibata, ApJ 907, 92 (2021). arXiv:2010.10865 [astro-ph.HE] Y. Suwa, K. Sumiyoshi, K. Nakazato, Y. Takahira, Y. Koshio, M. Mori, R.A. Wendell, ApJ 881, 139 (2019). arXiv:1904.09996 [astro-ph.HE]

86 Equation of State in Neutron Stars and Supernovae

3177

H. Suzuki, in Physics and Astrophysics of Neutrinos, ed. by M. Fukugita, A. Suzuki (Springer, Tokyo, 1994), p. 763 M. Takahara, K. Sato, ApJ 335, 301 (1988) K. Takami, L. Rezzolla, L. Baiotti, Phys. Rev. Lett. 113, 091104 (2014). arXiv:1403.5672 [gr-qc] G. ’t Hooft, Nucl. Phys. B 72, 461 (1974) H. Togashi, M. Takano, Nucl. Phys. A 902, 53 (2013). arXiv:1302.4261 [nucl-th] H. Togashi, E. Hiyama, Y. Yamamoto, M. Takano, Phys. Rev. C 93, 035808 (2016). arXiv:1602.08106 [nucl-th] H. Togashi, K. Nakazato, Y. Takehara, S. Yamamuro, H. Suzuki, M. Takano, Nucl. Phys. A 961, 78 (2017). arXiv:1702.05324 [nucl-th] L. Tolos, L. Fabbietti, Prog. Part. Nucl. Phys. 112, 103770 (2020). arXiv:2002.09223 [nucl-ex] M.B. Tsang, J.R. Stone, F. Camera, P. Danielewicz, S. Gandolfi, K. Hebeler, C.J. Horowitz, J. Lee, W.G. Lynch, Z. Kohley, R. Lemmon, P. Möller, T. Murakami, S. Riordan, X. RocaMaza, F. Sammarruca, A.W. Steiner, I. Vidaña, S.J. Yennello, Phys. Rev. C 86, 015803 (2012). arXiv:1204.0466 [nucl-ex] S. Typel, G. Röpke, T. Klähn, D. Blaschke, H.H. Wolter, Phys. Rev. C 81, 015803 (2010). arXiv:0908.2344 [nucl-th] I. Vidaña, Proc. R. Soc. Lond. Ser. A 474, 20180145 (2018). arXiv:1803.00504 [nucl-th] A.L. Watts et al., Rev. Mod. Phys. 88, 021001 (2016). arXiv:1602.01081 [astro-ph.HE] A. Weiss, W. Hillebrandt, H.C. Thomas, H. Ritter, Cox and Giuli’s Principles of Stellar Structure (Cambridge Scientific Publishers, Cambridge, 2004) E. Witten, Nucl. Phys. B 160, 57 (1979) E. Witten, Phys. Rev. D 30, 272 (1984) S.E. Woosley, D.H. Hartmann, R.D. Hoffman, W.C. Haxton, ApJ 356, 272 (1990) K. Yagi, T. Hatsuda, Y. Miake, Quark-Gluon Plasma: From Big Bang to Little Bang. Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology (Cambridge University Press, Cambridge, 2005) D.G. Yakovlev, C.J. Pethick, ARA&A 42, 169 (2004). arXiv:astro-ph/0402143 [astro-ph] T. Yokozawa, M. Asano, T. Kayano, Y. Suwa, N. Kanda, Y. Koshio, M.R. Vagins, Astrophys. J. 811, 86 (2015) Y.-W. Yu, L.-D. Liu, Z.-G. Dai, Astrophys. J. 861, 114 (2018). arXiv:1711.01898 [astro-ph.HE]

Galactic Chemical Evolution, Astronomical Observation from Metal-Poor Stars to the Solar System

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constraints on Nucleosynthesis Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Observations of Stars, Stellar Ejecta, and Explosive Events . . . . . . . . . . . . . . . . . . Very Metal-Poor Stars to Constrain First Stars and Explosive Events . . . . . . . . . . . . . . . . Binary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enrichment History in the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solar-System Abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stellar Abundances for a Variety of Ages and Structures . . . . . . . . . . . . . . . . . . . . . . . . . . Observational Constraints on the Chemical Enrichment Histories of Galactic Stellar Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Milky Way Bulge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Milky Way Disk System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Milky Way Halo and Its Substructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Group Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abundance Analysis: Methods and Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of Elemental Abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isotope Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The nucleosynthesis in the universe is probed by observations of evolved stars and explosive events such as novae and supernovae that are yielding elements. The nucleosynthesis processes are also examined by chemical abundances of very metal-poor stars and binary component that record yields provided

W. Aoki () · M. N. Ishigaki National Astronomical Observatory, Mitaka, Japan e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_105

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by individual processes. Observations of stars in different populations of the Milky Way Galaxy and surrounding dwarf galaxies provide different kinds of constraints on the understanding of a variety of nucleosynthesis processes through the history of the universe. This chapter also provides a brief overview of the observational technique to determine chemical abundances of stars.

Introduction Nuclear processes inside stars produce a variety of elements, which are ejected through mass loss in the late stages of stellar evolution. This includes light elements like C, N, and O, and also heavy elements (heavier than iron-peak elements) that are formed by the so-called s-process. Massive stars also produce α-elements during their evolution, and majority of metals up to iron-peak elements are synthesized and ejected through core-collapse supernovae. Another type of supernovae, called type Ia supernovae, is caused by explosive thermonuclear reactions due to merging binary white dwarfs or mass accretion into a white dwarf in a binary system. Binary systems including a white dwarf also sometimes show explosions called novae, by nuclear reactions at the surface following mass accretion from its companion star. Some specific elements including Li are produced from such phenomena. Mergers of binary neutron stars are recently regarded as the promising site of the r-process. Spectroscopic observations of mass-losing stars and explosive events are the most direct ways to examine these nucleosynthesis processes and the resulting yields. Useful information has been obtained from optical and infrared spectra of evolved low-mass stars, i.e., red giants and asymptotic giant branch (AGB) stars, in which material produced by internal processes, e.g., CNO cycle, 3α-reaction, and s-process, appears at their surface atmospheres. Timely observations of nova explosions, supernova explosions, and neutron star mergers also provide unique information to constrain the products of these explosive phenomena. A limitation of such observations is that the number of elements measured for such explosive events and relatively complicated stellar atmospheres is not large, and the accuracy of abundance determination is not sufficiently high in general. More detailed and accurate chemical abundances are measurable for more stable stars, i.e., main-sequence stars and red giants. However, a majority of stars in the solar neighborhood have similar chemical compositions to the Sun, which are determined by numerous nucleosynthesis reactions through the long history of the universe. In some special conditions, however, stars could have been affected by specific processes. The chemical compositions of very metal-poor stars, which are low-mass stars formed in the early phase of chemical evolution, should be determined by a small number of processes that occurred before the star formation and, hence, could record the yields of massive stars having short lifetime and supernova explosions. Another case is binary stars: nucleosynthesis products of AGB stars and possibly more massive objects could be transferred to the companion stars with lower mass in binary systems. Observations of the companion stars, which are main-sequence or red giant stars, enable us to determine precise abundance patterns.

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On the other hand, chemical compositions of stars are results of the numerous nucleosynthesis processes in general. For instance, the chemical composition of the Sun and the Solar System reflects the composition of the gas cloud of some specific part of the Milky Way 4.6 billion years ago. Similarly, chemical compositions of individual stars are regarded as records of chemical evolution in some place of the Milky Way Galaxy, which is connected to the understanding of stellar populations and substructures of the Galaxy.

Constraints on Nucleosynthesis Yields Direct Observations of Stars, Stellar Ejecta, and Explosive Events CNO Cycle, 3α-Process, S-Process in Low-Mass Evolved Stars The surface chemical compositions (abundances) in stars do not show significant changes in the main-sequence phase, as found for the Sun, whereas the composition near the central core changes through hydrogen burning. An exception is lithium, which shows significant depletion even in main-sequence stars. When the star evolves to red giants, abundance ratios of light elements could slightly change by the extended convective layer in the stellar envelope (the so-called first dredge-up). Low-mass and intermediate-mass stars gradually lose matter from the surface by stellar wind (stellar mass loss). The stars finally lose the envelope, evolving into white dwarfs through the “post-AGB” stage, in which elements produced inside the stars emerge in the surface atmosphere and extended circumstellar matter. The energy of stellar radiation is provided by hydrogen burning at the central core during the main-sequence phase. The CNO cycle is the dominant energy source in stars with mass larger than two solar masses. The abundances of C, N, and O isotopes in lower-mass stars are also affected by the CNO cycle even though it has minor contributions to the energy compared to the pp-chain reaction. The material affected by the CNO cycle partially appears when a star evolves to red giant phase by the deep convection in the envelope. The products of the CNO cycle are observable in abundances of elements and isotopes (e.g., 12 C/13 C). (e.g., Clegg et al. 1981; Smiljanic et al. 2009). In red giants, lower 12 C/13 C and C/N ratios are seen than in a main-sequence stars (estimated from the solar-system material). It should be noted that the quite low ratios found for low-mass (M 4 M ), which are affected by the hot bottom burning (HBB) process (Herwig (2005), and references therein), the surface composition changes toward the equilibrium values of the CNO cycle with a reduced C/O ratio. Some s-process elements might also be enriched in such stars. From the observational point of view, optical and near-infrared spectra are dominated by molecular absorption lines formed in the cool surface atmospheres of AGB stars (∼3000 K), e.g., from CO, TiO, and C2 , depending on chemical composition and temperature. While the abundances of C, N, O, and some other elements are determined via molecular lines (see, e.g., Lambert et al. 1986), analyses of atomic spectra are difficult. However, detailed analyses based on the spectrum synthesis technique provide useful results on the surface abundances of heavy elements in cool AGB stars. Molecular absorption features such as ZrO are also applied to determine abundances of neutron-capture elements and their isotope ratios (e.g., Lambert et al. 1995). It should be noted that such elements in the surface atmosphere of AGB, as well as red giants, are not necessarily supplied into interstellar matter. It depends on the stellar mass loss, which is a long-standing problem in stellar physics. Larger variations of C, N, and O abundances and excesses of heavy s-process elements are also found in post-AGB stars, which are in the evolutionary stage from AGB stars to white dwarfs losing their envelopes. The duration of this evolutionary stage is very short (tens to thousands of years). The surface of such objects becomes warmer and molecular absorption becomes weaker as this transition proceeds. A difficulty in the analysis of spectra is that the atmospheric structure is unstable and quite complicated compared to that of main-sequence and red giant stars. Useful results have been obtained by observations with large telescopes and detailed analyses of high-resolution spectra. Planetary nebulae are formed following the post-AGB phase by the ejected material that is ionized by the central star evolving to a hot white dwarf. Since the nebulae are the ejected material from stars, the spectra of nebulae provide direct information on the contributions of low-mass stars to the chemical enrichment of the Galaxy. The light elements (e.g., C and O) in planetary nebulae are accessible by X-ray observations (see, e.g., Murashima et al. 2006), while neutron-capture elements can be studied by optical spectroscopy.

Li (7 Be) in Novae Lithium is a unique element that has a variety of origins in the universe. It is produced by the Big Bang nucleosynthesis (BBN), interstellar spallation reaction, stellar interiors, and explosive events like novae and supernovae. This element is destroyed by the temperature of stellar interior (>2.5 and 2.0 million K for 7 Li and 6 Li, respectively) and, hence, useful as a probe of stellar mixing and evolution. Although lithium productions in many sites and phenomena mentioned above have been studied theoretically and examined by chemical evolution models (e.g., Prantzos 2012), direct evidence for each process had not been obtained by observations until recently. Tajitsu et al. (2015) have reported detection of 7 Be, which decays to 7 Li with a half-life of 53 days, in the ejecta of a nova explosion (Fig. 1).

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Fig. 1 Absorption spectra observed for two nova explosions caused by the ejecta (Tajitsu et al. 2016). Both nova explosions show two distinctive velocity components of the absorption features with about −1400 and −800 km s−1 that are shown in the left and right panels, respectively. The absorption lines of the 7 Be doublet are shown by red and blue lines. Detection of the doublet feature confirms the identification of this unstable isotope. The half-life of 7 Be, forming 7 Li, is about 53 days, indicating that the isotope should be produced by the explosion events

This first discovery was followed by further detection of 7 Be in other novae (Tajitsu et al. 2016; Molaro et al. 2016) as well as detection of 7 Li (Izzo et al. 2015). These observations indicate that nova explosions could be the major source of lithium (7 Li) in the current Milky Way. It should be noted that, for such studies, high-resolution spectra covering the 7 Be doublet around 3130 Å are required, which are available by observations with large telescopes in high elevation.

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Metals Produced by Massive Stars and Supernova Explosions Supernova explosions play dominant roles in metal production in the universe. Metals supplied by individual supernovae are estimated by measurements of flux variations (light curves) for several months and spectra. Supernovae show peak of the light several tens days after the explosion, depending on the types (mechanisms) of the event. The light curves of supernovae are governed by radioactive nuclei produced in the explosions, in particular 56 Ni that finally decays to 56 Fe. The amount of Fe in the ejecta is estimated by the brightness of the explosions. More detailed information on the synthesis of elements is available by spectroscopy in the late phase of the supernovae (Fig. 2). A large number of supernovae

Fig. 2 Optical spectra of a variety of types of supernovae (Filippenko 1997). Hydrogen Balmer lines are observed in a type II supernova (b), which is a core-collapse supernova of a star still having hydrogen envelope. Type Ib and Ic supernovae, (d) and (c) in this figure, respectively, are also core-collapse supernovae, but their progenitor stars would have lost hydrogen-rich envelope and even helium-rich layer, respectively, before the explosion. Absorption spectra of metals are identified in these spectra. A type Ia supernova (a) is an explosive thermonuclear reaction of a white dwarf in a binary system whose mass exceeds the Chandrasekhar limit (1.4 M ) due to mass accretion or merging event of two white dwarfs

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are detected every year by recent wide-field observations e.g., Yasuda et al. (2019). Spectroscopic follow-up observations are also conducted using large telescopes, revealing variations of supernovae types and ejecta.

R-Process in Merging Events of Binary Neutron Stars Measurements of heavy neutron-capture elements in explosive events by optical spectroscopy to measure spectral lines, as well as by observations of γ -ray from radioactive nuclei, are desired to identify the sites of the r-process and to determine the production yields. Measurements of heavy elements in absorption spectra of supernova remnants were attempted, resulting in no clear detection (Wallerstein et al. 1995). Merging binary neutron stars are another candidate of the r-process site. A merging event was identified for the first time by detection of gravitational wave in 2017 (GW170817). Follow-up multiwavelength observations, including optical and infrared ones, provide useful constraint on the heavy elements produced by the event. Follow-up optical and infrared observations for have been providing evidence of the production of heavy elements in merging events of binary neutron stars and useful constraint on the spices and their amount produced (Tanaka et al. 2017). The excess of infrared in light curves is a signature of large opacity of lanthanide elements, though the amount of these elements, compared to lighter neutron-capture elements, is not as large as that expected from models and Galactic chemical evolution. Spectroscopic features found in the near-infrared range (Smartt et al. 2017) is identified to be Sr rather than lanthanides (Fig. 3), supporting the efficient production of light neutron-capture elements in this event (Watson et al. 2019). Another feature in 13,000–15,000 Å is suggested to be caused by lanthanides (La and Ce) (Domoto et al. 2022). The excess in the light curve at around 100 days after the event is suggested to be due to the decay of 254 Cf, though this interpretation is in controversy. Further observations for gravitational wave events are obviously required. Given the fact that GW170817 is exceptionally bright (the event occurred relatively close to us), future follow-up program requires deeper observations using larger telescopes.

Very Metal-Poor Stars to Constrain First Stars and Explosive Events Another approach to investigate the nucleosynthesis yields is to measure elemental abundances of early generation stars in the universe. Low-mass stars have lifetime as long as, or longer than, the age of the universe (13.8 Gyr). Hence, low-mass stars formed in the early universe still survive in the Milky Way Galaxy. These objects are currently found as “metal-poor” stars in the solar neighborhood, reflecting the low amount of heavy elements in the early universe.

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Fig. 3 Optical and near-infrared spectra of the gravitational event GW170817 (Watson et al. 2019). The changes of spectra since the event are shown from the top to the bottom. The absorption feature of around 8000 Å, shifting from short to long wavelengths with time, is identified by singly ionized strontium (Sr II) in this study, taking account of the velocity shift of the ejecta from the explosion. The absorption of Sr II of around 4100 Å lines is also suggested for the top spectrum

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The elements heavier than He in very metal-poor stars should have been provided by a limited number of nucleosynthesis events (ideally a single event), in particular supernova explosions of early (first) generations of massive stars. An advantage of this approach is that detailed and reliable abundance ratios are obtained from high-resolution spectra of main-sequence and red giant stars with low metallicity, which are stable objects whose atmospheres are well modeled (Fig. 4). The most extreme examples are stars having Fe abundances less than 10−5 of the solar value, which are found by wide-field surveys (e.g., SDSS/Segue (Yanny et al. 2009), SkyMapper (Keller et al. 2014)) and high-resolution spectroscopy with large telescopes (e.g., Frebel and Norris 2015; Aoki et al. 2014). Such observations are useful to constrain the mass distribution of first generations of massive stars that is a long-standing problem in astrophysics. Detailed chemical abundances of metal-poor stars are also applied to constrain r-process and the Big Bang nucleosynthesis.

First Stars’ Masses and Explosion Mechanisms Since the evolution timescale of massive stars is short, the elemental abundances of very metal-poor stars are mostly determined by massive star evolution and supernova explosions. Taking account of the dependence of elemental abundance patterns produced by supernovae on their progenitor masses, abundance studies of very metal-poor stars are also able to estimate the masses of early generations of stars (e.g., Heger and Woosley 2010; Ishigaki et al. 2018). Figure 5 shows examples of elemental abundance patterns calculated by supernova nucleosynthesis models compared with those obtained for four very metal-poor stars based on the analysis of high-resolution spectra. The metallicity of the four stars is extremely low, indicating that the elemental abundance patterns are determined by a single supernova explosion that occurred before the formation of these low-mass stars. Interestingly, the abundance patterns of the four stars show a large variation, suggesting that progenitors are quite different from each other. R-Process and Another Neutron-Capture Process in the Early Universe Observations of early generation stars also contribute to revealing the nature of the r-process and identifying the sites. Searches for very metal-poor stars and followup detailed measurements of chemical compositions have revealed that abundance ratios of heavy neutron-capture elements show large star-to-star scatter, and a small fraction of extremely metal-poor stars have large excesses of r-process elements like Eu. This indicates that these heavy elements are produced by quite rare events, and gas clouds from which stars are formed were not well mixed in the early universe (e.g., Sneden et al. 2008). An interesting discovery is a so-called ultrafaint dwarf galaxy of just thousands of solar masses in which most stars seem to be r-process-enhanced (Ji et al. 2016; Roederer and Lawler 2012). This suggests that the progenitor gas cloud from which the stars in this galaxy were formed was polluted by a very efficient r-process event, which could be a merger of binary neutron stars. Moreover, the r-process-enhanced stars found in the Milky Way could be formed in such small galaxies and later accreted forming the Galactic halo structure.

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Fig. 4 High-resolution spectra of very metal-poor stars (Frebel and Norris 2015). The metallicity of the object is presented in each panel. The number of metal absorption lines (e.g., Fe) detected around 3860 Å significantly decreases in stars with lower metallicity, and no line is seen in the most metal-poor object shown in the bottom panel. There are prominent doublet absorption lines of ionized Ca (Ca II H and K lines). Even these lines are quite weak in the most metal-poor object. On the other hand, there are many absorption lines in 3870–3980 Å, which are identified as CH molecules, in the bottom two objects, indicating significant excesses of carbon compared to Fe and other heavy metals in these objects

The detailed abundance patterns determined for individual objects provide strong constraints on the understanding of r-process nature (Fig. 6). A remarkable result obtained for r-process-enhanced stars is that abundance patterns of neutron-capture elements of these stars are very similar to that of the r-process component in solar-

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Fig. 5 Elemental abundances determined for the four extremely metal-poor stars (red circles and blue triangles), presented as a function of atomic number, compared to the predictions by supernova nucleosynthesis models (lines) (Nomoto et al. 2013). The abundance patterns are significantly different between stars. Comparison with supernova models assuming different progenitor mass, explosion energy, and other parameters provides useful constraint on the nature of the progenitor stars, which could be metal-free first-generation massive stars

system material (Sneden et al. 1996). The agreement is in particular evident in elements from the second to the third peaks corresponding to the neutron magic numbers 82 and 126 (Ba-Pt). We note that the measurements of the elements at the second r-process peak (Te-Xe) is very limited because there is no useful spectral lines in the optical range (Roederer and Lawler 2012). This result indicates that the r-process produces neutron-capture elements with similar abundance patterns in every event. This phenomenon is called the universality of the r-process, and has a large impact on understanding of the process, because it could be a strong constraint on r-process models that predict more or less variation of abundance ratios depending on model parameters like electron fraction and entropy. It should be noted that the abundance patterns of light neutron-capture elements (e.g., Sr, Y) are also similar between such Eu-enhanced stars, but they show some deviation from the solar r-process pattern (Sneden et al. 2008). Moreover, there are many metal-poor stars that have low abundances of heavy neutron-capture elements like Eu, but have large excess of light ones. The origins of light neutron-capture elements in early Galaxy are still in controversy. Detailed abundance patterns of metal-poor stars with large excesses of light neutron-capture elements also provide useful constraints on modeling the corresponding processes (e.g., Honda et al. 2006).

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Fig. 6 Abundance ratios of neutron-capture elements, presented as a function of atomic number, compared with the r-process and s-process abundance patterns estimated from solar-system material (Sneden et al. 2008). The abundance pattern of this star (CS 22892–052), an extremely metal-poor star with large excess of neutron-capture elements, is well explained by the r-process. From the low metallicity of this object, the abundance pattern of neutron-capture elements of this object is expected to be determined by a single r-process event. The agreement between the abundance pattern of this object and that of the r-process component of solar-system material, which should be produced by many r-process events, suggests that the abundance pattern produced by individual r-process events is quite homogeneous

Primordial Li Lithium is a unique element that was produced in the Big Bang nucleosynthesis (BBN) in addition to hydrogen and helium. The Li abundance determined by the BBN is expected to be recorded in most metal-poor stars to which contributions of other processes (e.g., nova explosions) are negligible. Indeed, almost constant Li abundance ratios have been found in main-sequence stars with very low metallicity (Spite and Spite 1982), which has been regarded as useful information to support the Big Bang theory.

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The Li production in the Big Bang is, however, quantitatively a problem that is yet solved. The standard BBN models with cosmological parameters determined by observations of cosmic microwave background (CMB) radiation predict log ε(Li)∼ 2.7, whereas the Li abundance measured for metal-poor main-sequence turn-off stars is log ε(Li)∼2.2 (Fig. 7). The discrepancy is not explained by measurement errors estimated for known error sources including non-LTE and 3D effect of stellar atmospheres in calculation of spectral line formation and is called “lithium problem.” Recent observations extended to the lowest metallicity range ([Fe/H]< −3) suggest systematically low Li abundances compared to less metal-poor stars with little scatter (e.g., Matsuno et al. 2017). This is a strong constraint on the mechanism that causes the lithium problem.

Binary Systems Another approach to measure the products of individual nucleosynthesis events is to observe stars in binary systems that have experienced significant mass transfer between the pairs. A large fraction of stars, or possibly a majority of stars, belong to binary systems. In the case that the separations of the components are sufficiently short, significant mass exchange occurs when the primary star evolves to an AGB star. The surface material of the primary star is accreted into the surface of the secondary component. After the primary star evolves to a very faint white dwarf star, the (formerly) secondary star, which is a main-sequence star or a red giant, is observed in the optical spectrum. Such a star is a target to measure the chemical abundances of the material produced by AGB stars. Similarly, the so-called lowmass X-ray binary (LMXB) could be a target to study the yields of massive stars and supernovae. In this case, the (formerly) primary star is a massive star, which has evolved to a neutron star or a black hole through a supernova explosion.

C- and S-process Elements Recorded by Mass Transfer Events As mentioned in section “CNO Cycle, 3α-Process, S-Process in Low-Mass Evolved Stars,” the products in AGB nucleosynthesis are estimated by observations of their surface chemical compositions, but the uncertainty is large in general because of the complicated molecular spectra of cool AGB stars. On the other hand, detailed information on the abundance patterns of heavy elements produced by AGB stars are obtained by analyses of a binary companion that is affected by mass accretion from the primary AGB star. In these cases, the target objects are main-sequence or red giant stars, but they are distinguished from normal stars by excesses of carbon and heavy s-process elements such as Ba. Such stars with high metallicity similar to the Sun are known as Ba stars (Bidelman and Keenan 1951) that have anomalously strong absorption features of Ba. Periodic variations of the radial velocity have been found for many Ba stars, supporting the scenario that these stars experienced accretion of s-process-enhanced

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Fig. 7 Lithium abundances of metal-poor stars (circles, squares, triangles, and asterisks) and metal-rich stars (small dots) as a function of metallicity ([Fe/H]). The solid line indicates a prediction of the Li abundances by a chemical evolution model (Prantzos 2012) that includes different sources of this element: the Big Bang nucleosynthesis (BBN) adopting WMAP cosmological parameters, Galactic cosmic-ray reactions (GCR), neutrino process in supernovae (NN), and processes in low-mass stars including novae (STARS). The near-constant Li abundances found in metal-poor stars are lower than the prediction of the model, which is determined by the BBN model. This discrepancy is yet fully explained by BBN models and stellar models on Li depletion. In the metal-rich range, the major contributor to the Li abundances is low-mass stars including novae. This is supported by the Li and 7 Be observations for nova explosions

material from a primary AGB star in a binary system (McClure 1984; McClure and Woodsworth 1990). Mass accretion from AGB stars has larger impacts on metal-poor stars. Such objects are classified into CH stars (Keenan 1942) or subgiant CH stars (Bond 1974). Excesses of heavy elements have also been detected in such objects, and the binarity

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has been confirmed by radial velocity monitoring (McClure 1984; Preston and Sneden 2001). Large surveys of metal-poor stars based on the weakness of calcium absorption lines have detected a number of carbon-enhanced objects, (e.g., Beers et al. 1992). Follow-up high-resolution spectroscopy has been made to determine detailed chemical compositions including neutron-capture elements, e.g., Aoki et al. (2007). Whereas there are a fraction of carbon-rich objects that do not exhibit radial velocity variations nor excess of s-process elements, many others (with [Fe/H]> −3) have similar properties to CH stars. The detailed abundance patterns determined for these carbon-rich stars are useful constraints on the s-process models. Figure 8 shows examples of comparisons between models with different model parameters and observations. The abundance ratios of some elements including Eu are not well explained by s-process; hence, contributions of r-process are also assumed. Another interpretation that these abundance patterns are reproduced by neutroncapture process with intermediate neutron density (called i-process: Cowan and Rose 1977) is also suggested.

Enrichment History in the Universe Chemical abundances of stars discussed in the previous section are special cases in which products of individual nucleosynthesis processes are well recorded (e.g., very metal-poor stars and binary stars affected by mass transfer). In general, stellar chemical abundances are records of yields of many processes accumulated through the history of a portion of the universe. Detailed chemical compositions of stars are, however, still useful to constrain nucleosynthesis processes through comparisons with nucleosynthesis models and chemical evolutions models. The most detailed chemical composition is obtained for the Sun and the Solar System, in which isotope ratios are also available for most elements.

Solar-System Abundances The solar abundances are a record of numerous nucleosynthesis processes in the long history of the Milky Way before the formation of the Sun at around 4.6 billion years ago. The elemental abundances of solar-system material are determined from chemical analysis of meteorites and analysis of the solar spectrum. Recent measurements by the two methods exhibit fairly good agreement for heavy neutroncapture elements in general (Asplund et al. 2009). In particular, the recent updates of atomic line data, including the effect of isotope shift and hyperfine splitting, have been making large contribution to the improvements of solar atmospheric abundances (e.g., Lawler et al. 2009). An advantage of the analysis of meteorites is that it can determine isotope ratios, which provide much stronger constraints on the models of neutron-capture processes than elemental abundance ratios. The abundance ratios of major metals (e.g., C, O, alpha-elements, iron-peak elements) are used as the benchmark of the nucleosynthesis models of massive stars

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Fig. 8 Abundance ratios of neutron-capture elements determined for a very metal-poor star showing large excesses of s-process elements. (Taken from Sneden et al. 2008). Stars having similar excess of s-process elements are found in carbon-enhanced very metal-poor stars, many of which show variations of radial velocities, indicating that the objects belong to binary systems. The lines show model predictions assuming different contributions of r-process in addition to the s-process by the progenitor AGB star

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and supernova explosions. The heavy neutron-capture elements with mass number (A) larger than ∼70 in the solar-system material originated from both the s- and r-processes. Since the abundances of s-process yields are basically determined by reactions along stable nuclei, modeling of this process is better established than that of the r-process. There are dozens of nuclei which are never produced by the r-process because they are shielded by other stable nuclei from β-decay of neutron-rich unstable nuclei (“s-only” nuclei). Models of the s-process are calibrated by the abundance of these s-only nuclei, which enable us to estimate the fraction of abundances produced by the s-process for other nuclei (Kappeler et al. 1989). The r-process component of each isotope is obtained by subtracting the s-process component from the abundance of each isotope for solar-system material. It should be noted that a small error in the estimate of the s-process component could result in a large error in the obtained r-process fraction for elements to which s-process contributions are dominant. An example is 208 Pb, for which the s-process component is estimated to be as large as 80% in solar-system material. This makes it difficult to constrain the models for production of heaviest stable nuclei by the r-process (Arlandini et al. 1999).

Stellar Abundances for a Variety of Ages and Structures Distributions of abundance ratios of individual elements for a large sample of stars in the Milky Way Galaxy are useful constraints on chemical evolution models. Chemical evolution models help understand the contributions of individual processes as a function of time and/or metallicity. For instance, abundance ratios of α and iron-peak elements as a function of metallicity are used as an indicator of contributions of core-collapse supernovae and type Ia supernovae, which reflect the timescale of chemical evolution. Comparisons of elemental abundance ratios determined for many stars with chemical evolution models (Fig. 9) enable us to estimate contributions of nucleosynthesis processes and their astrophysical sites for individual elements (Kobayashi et al. 2020). For such studies, we need knowledge of stellar populations in the Milky Way Galaxy and surrounding galaxies, which is reviewed in the next section.

Observational Constraints on the Chemical Enrichment Histories of Galactic Stellar Populations As mentioned earlier, elemental abundances in stars with a wide range of ages serve as a fossil record of the Galactic formation and evolution. A group of stars that exhibit a characteristic spatial distribution, kinematics, and chemical compositions is often called a “stellar population.” The Milky Way, which is a typical (barred) spiral galaxy in the local universe, consists of multiple stellar populations, each of which consists of stars likely sharing a common birth epoch and environment.

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Fig. 9 Abundance ratios of Mg (upper) and Mn (lower) as a function of metallicity ([Fe/H]) determined for Galactic field stars compared with predictions of chemical evolution models (Kobayashi et al. 2020). High Mg/Fe ratios found in metal-poor stars are determined by yields of core-collapse supernovae, whereas the decrease of this trend with increasing metallicity is explained by contributions of type Ia supernovae to Fe abundance. Mn abundance ratios show an opposite trend, but is also explained by contributions of two types of supernovae including metallicity-dependent yields in type Ia supernovae

Metallicities expressed in [Fe/H] or [M/H] are often used as a proxy for an age of a stellar population, since Fe abundances increase as the system becomes more chemically evolved. As shown in Fig. 9, abundance ratios, such as [X/Fe] for an element X, as a function of [Fe/H] are commonly used to distinguish the dominant astrophysical sources that have produced particular elements at a characteristic timescale. Over the last few decades, our knowledge about the stellar populations in our Galaxy have been expanding rapidly, thanks to dedicated surveys from the ground and space to measure spatial, kinematic, and chemical distribution of millions of stars in the Milky Way and its dwarf satellite galaxies (Ivezi´c et al. 2012; Gaia

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Fig. 10 Chemical map of the Milky Way revealed by Gaia Data Release 3 (Gaia Collaboration et al. 2022). The color coding represents average [M/H] (abundance of all the elements heavier than hydrogen and helium) at each sky pixel. It can be seen that the spatial locations occupied by the bulge, disk, and halo populations show distinct metallicities

Collaboration et al. 2022) (Fig. 10). Those datasets allow us to test theories of stellar nucleosynthesis and galaxy formation. In this subsection, we discuss what we have learned about chemical compositions in various stellar populations in our Galaxy and the Local Group of galaxies from these surveys.

The Milky Way Bulge The bulge in the Milky Way often refers to the region within about 10◦ of the Galactic center. Compared to large bulges in some of external galaxies, the Milky Way bulge is categorized as a “pseudo” bulge (Barbuy et al. 2018). In contrast to the large “classical” bulges, a pseudo bulge is characterized by a more flatter stellar distribution and rotationally supported and relatively small bulge-to-total ratio 0, which might suggest the presence of multiple stellar populations. For a large fraction of the bulge stars, abundance ratios of elements synthesized in massive (>8 M ) stars such as oxygen or magnesium are enhanced compared to the solar ratio ([O/Fe]∼ 0.5) at [Fe/H]< −1. At higher metallicities,

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the ratios are decreased when increasing [Fe/H], as expected from the delayed metal ejection from type Ia supernovae. The observed oxygen and alpha-element abundance ratios suggest that shortlived massive stars are mainly responsible for synthesizing metals in the bulge. This suggests that the bulge is formed within a short period of time, possibly as short as 1–3 Gyrs. Observed chemistry of bulge stars is known to be similar to the old stars in the Galactic thick disk (McWilliam and Rich 1994; Griffith et al. 2021). The similarity of the chemistry between bulge and disk stars supports a scenario that the Milky Way bulge was formed through the dynamical instability of the Galactic disk (Bland-Hawthorn and Gerhard 2016).

The Milky Way Disk System The Milky Way disk system comprises ∼1011 M of stars and is the major stellar component in our Galaxy. Understanding the formation history of the Milky Way disk system is crucial not only to reveal the environment in which our Solar System was formed but also to put constraints on the formation theories of spiral galaxies in general. Multiple observational evidences suggest that the Milky Way disk system consists of at least two stellar populations, the thin disk and the think disk, that are characterized by different spatial distributions, kinematics, and chemical compositions (Gilmore and Reid 1983; Juri´c et al. 2008). The formation of the disk system, which exhibits such a complex mixture of stellar populations, has been actively studied in the past few decades, but a clear consensus has not been obtained. Both internal dynamical effects, such as the bar, spiral arms, or orbital migration, and external perturbation, such as mergers of dwarf galaxies, have been proposed to explain the observed properties of the disk system. To observationally constrain the origin of the Galactic disk, their observed chemical composition has played an important role. Thanks to high-resolution spectroscopic observations targeted at Galactic disk stars, it is now feasible to build a large statistical sample of elemental abundance distributions as a function of Galactocentric radius and distance from the Galactic plane (Fig. 11). Those observations show that the thick disk stars are more metal-poor with [Fe/H]∼ −0.5 and enhanced in [α/Fe], where “α” stands for the elements whose dominant isotope consists of the alpha particle (the helium nucleus) such as O or Mg, compared to the thin disk stars. While spatial and kinematic distributions of thin and thick disk stars largely overlap, the chemical signature is found to be the best discriminator of a stellar population formed with similar age (Bovy et al. 2012). As can be seen in Fig. 11, the chemically identified thin disk and thick disk populations suggest that the thick disk population is absent at Galactocentric radius beyond ∼11 kpc and, therefore, found to exhibit a shorter scale length. On the other hand, the thin disk stars can be identified out to larger Galactic radii beyond ∼11 kpc and show a negative metallicity gradient. These observations are crucial to examine the validity of cosmological simulations of disk galaxy formation.

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Fig. 11 Stellar density distributions in the α-element to iron ratios ([α/Fe] ) versus [Fe/H] planes at various locations in the Milky Way disk. The three rows correspond to different vertical distances (Z) from the Galactic plane. Six columns correspond to different Galactocentric radii (R). The high-α and low-α sequences, which presumably correspond to the thick and the thin disk populations, are clearly seen with various relative fractions among different locations. (Adopted from Hayden et al. 2015)

There are mounting evidences that the stellar migration plays an important role in shaping the chemical element distribution in the solar neighborhood (Jofré 2021; Nissen et al. 2020). This raises a new intriguing question that where in the Galactic disk the Sun, currently at ∼8 kpc from the Galactic center, was originally born. The answer to this question is crucial to constrain which astrophysical site is responsible for the chemical abundance pattern observed in the Sun and in the Solar System material.

The Milky Way Halo and Its Substructures The stellar halo predominantly consists of stars as old as 10–13 Gyrs and thus provides rich information on the nucleosynthesis and chemical evolution in the early universe. The extremely metal-poor stars, which encode nucleosynthesis in the very first stars in the universe (section “Very Metal-Poor Stars to Constrain First Stars and Explosive Events”), also predominantly belong to the stellar halo. Under the framework of the currently favored ΛCDM cosmology, a stellar halo is formed through hierarchical accretions of smaller stellar systems. Past observations of the stellar halo population have provided evidences in line with this theoretical prediction. The global structure of the stellar halo is approximately characterized as a power low density distribution with a total stellar mass of M ∼ 109 M (Deason et al. 2019). In the solar neighborhood, halo stars are extremely rear with a local fraction of less than 1% (Ivezi´c et al. 2012). Recent observations that can probe distant halo stars have demonstrated that the stellar halo in fact consists of numerous substructures, which are likely debris of accreted stellar systems, as predicted by the hierarchical formation scenario. For example, a direct

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evidence of a dwarf galaxy merger in the history of the Milky Way is the ongoing merger of the Sagittarius dwarf galaxy, currently plunging into the Galactic disk (Ibata et al. 1994). Tidal debris of the Sagittarius accretion event dominate the stellar populations in the outer part of the stellar halo (Sesar et al. 2017). Stellar orbital motion and stellar surface metallicity provide evidences of more ancient accretion events, since those information conserve over a long timescale, during which spatial information can be completely lost. Making use of both orbital velocities and metallicity of a large sample of individual halo stars in the solar neighborhood, the stellar halo is found to have at least two broadly overlapping stellar populations, the inner halo and the outer halo (Chiba and Beers 2000; Carollo et al. 2007, 2010). The dual nature of the stellar halo confirms that the halo stars have multiple origins, either formed in situ within the proto-Milky Way or in dwarf galaxies later merged to populate the Galaxy. As mentioned earlier, the Gaia satellite mission provides a more concrete picture of the stellar halo. Based on accurate astrometric data, it has been clearly discovered that the halo stars in the solar neighborhood largely originated from one dwarf galaxy merged with the Milky Way more than 10 Gyrs ago. The discovery of this past merger, called Gaia-Enceladus-Sausage event (Belokurov et al. 2018; Helmi et al. 2018), demonstrates the strength of the phase-space information of stars as the evidence to reconstruct the assembly history of the Milky Way. The other phase-space substructures have been discovered, whose contributions add up to the total stellar mass of the Milky Way halo (Naidu et al. 2020). To address the origin of each of these substructures, observational effort is ongoing to chemically characterize these substructures. Characteristic chemical compositions exhibited those substructures provide invaluable constraints on the nature (e.g., stellar masses, star formation history) of the building blocks of the stellar halo. It has already been demonstrated with a high-precision differential abundance analysis that some of the major substructures have distinct chemical abundance patterns, which indicates a unique chemical evolution history at the birthplace of stars in the substructures (Matsuno et al. 2021).

Local Group Galaxies The Local Group of galaxies consists of two large (M∗ ∼ 1012 M ) spiral galaxies, the Milky Way and the Andromeda galaxy (M31), and their associated dwarf (M∗ < 1010 M ) satellite galaxies. The dwarf satellite galaxies in the Local Group are roughly categorized as late-type and early-type galaxies (Tolstoy et al. 2009). The late-type galaxies, such as the Small Magellanic Cloud, are characterized by an irregular morphology and contain a significant fraction of gas to host ongoing star formation. The early-type galaxies, such as dwarf spheroidal (dSph) galaxies, are dominated by old stellar populations and are devoid of gas content, while showing a signature of being hosted by a massive dark matter halo with a mass >10– 1000 times larger than the stellar mass (Simon and Geha 2007). Because of their relative proximity to the Sun (within ∼150 kpc) where spectroscopic observations

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of individual stars are possible, the dSphs have provided unique observational constraints on the astrophysical nucleosynthesis sites (Tolstoy et al. 2009; Kirby et al. 2013; Simon 2019). Relatively luminous (> 107 L ) dSphs that were known prior to the modern photometric surveys are called classical dwarf galaxies. They are characterized by stars with a wide range of metallicities and ages. The characteristic trend of [α/Fe] vs [Fe/H] ratios is clearly seen as shown in the top panel of Fig. 12 (solid lines) for [Mg/Fe]. The difference of the dSph’s trend from that of the field Milky Way stars (gray dots) can be interpreted as the slower chemical evolution timescales in the dSphs due to the lower star formation rates, which result in the lower [Fe/H] at the onset of type Ia SNe (Tolstoy et al. 2009). Thanks to large-scale surveys, such as the Sloan Digital Sky Survey, a number of new dwarf satellites as faint as down to a few thousands of the solar luminosity

Fig. 12 Observed [Mg/Fe] (top) and [Ba/Fe] (bottom) ratios plotted against [Fe/H] for the Milky Way’s dSph satellite galaxies. The solid lines show the mean and the standard deviation of the abundance ratios for three classical dSphs (green, Sculptor; blue, Fornax; yellow: Sagittarius). The red and cyan points are for stars in UFDs. The gray dots show the abundances for field Milky Way stars. The classical dSphs show lower [Mg/Fe] ratios than the field Milky Way stars. Compared to the classical dSphs, which show clear trends with [Fe/H], the abundance ratios of UFDs are characterized by large scatter. (Adopted from Tolstoy et al. 2009)

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have been discovered (Belokurov et al. 2007; Bechtol et al. 2015). In the context of hierarchical galaxy formation scenario, these faintest Milky Way satellites called ultra-faint dwarf (UFD) galaxies are hosted by the smallest dark matter halos, which have been formed at the very beginning of the structure formation, prior to the epoch of cosmic reionization (Bromm and Yoshida 2011). This makes the UFDs the most promising objects that host stars with the most pristine chemistry. In fact, some of the UFDs hosts carbon-enhanced metal-poor stars, which might carry nucleosynthetic signatures of the very first stars in the universe (Frebel and Norris 2015). As introduced in the previous section, the UFDs also provide constraints on the astrophysical site of r-process because extreme r-process-enhanced stars are most frequently found in UFDs (Ji et al. 2016; Roederer et al. 2016). Along with our Galaxy and its satellite system, the Milky Way’s companion galaxy M31 provides another excellent laboratory of galaxy formation through measurements of resolved stars. Similar to what has been observed for the Milky Way stellar halo, rich substructures and tidal streams have been discovered in the stellar halo of M31, hinting at past mergers and tidal disruptions of infalling satellite galaxies (Tanaka et al. 2010; Ibata et al. 2014). A cutting-edge observation with the 10m Keck telescope has measured metallicity and [α/Fe] ratios in individual red giant branch stars for selected fields in the M31 halo, which provide constraints on the origin of each substructure (Escala et al. 2021). With next-generation widefield multi-object spectrographs, stellar chemical abundances will be an even more powerful probe for the formation mechanism of M31.

Abundance Analysis: Methods and Uncertainties In this section, a brief overview of the standard procedure to determine chemical abundances for stars and background information are given.

Sample Selection Detailed and reliable chemical abundances of stars are determined based on highresolution spectra. Hence, the target stars need to be relatively bright. The most reliable elemental abundances, and sometimes isotopic abundances, are determined for solar-type stars in the solar neighborhood. Such bright stars are compiled in star catalogues edited in the past century. Abundance studies are extended to relatively faint objects by using spectrometers mounted on large telescopes, but most of them are still objects in the Milky Way Galaxy. Some (intrinsically) bright red giants in the satellite dwarf galaxies around the Milky Way are also included in the targets. The number is, however, still small, and next-generation extremely large telescopes are desired for studies of chemical abundances of stars beyond the Milky Way Galaxy. The recent great progress in the understanding of Galactic stars is the huge dataset of stellar positions, distance, and proper motions (motions across the

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sky found for individual stars). Determination of distance has a large impact on determining stellar parameters, in particular surface gravity (see below), as well as on the understanding of the Milky Way structure. Combination of chemical abundances and information on stellar motion is very useful to study the formation processes of galaxies and, as a result, for understanding of origins of elements. As described in previous sections, metal-poor stars have particular importance in studies of chemical evolution and nucleosynthesis in the universe. Searches for stars that record the products of early generations of stars require a large survey of candidates of metal-poor stars, because such objects are quite rare. In the past two decades, continuous efforts of spectroscopic studies have provided chemical abundance data based on high-resolution spectra for several hundreds of very metalpoor stars (Cayrel et al. 2004; Cohen et al. 2004; Honda et al. 2004). Recent studies indicate that high-resolution, short-exposure spectra, called “snapshot” spectra, are very efficient to investigate the overall abundance distributions of metal-poor stars in which absorption lines are very weak in general (Barklem et al. 2005). This approach is adopted in more recent studies for stars found with spectroscopic survey projects like SDSS (Aoki et al. 2013) and LAMOST (Li et al. 2022). The data volume and their homogeneity are useful to study the statistics not only of chemical abundance ratios but also of other stellar properties like binary frequency and also to select extreme objects for further observations like the α-poor object SDSS J0018– 0939 (Aoki et al. 2014). Searches for metal-poor stars based on photometric data using narrowband filters have also been conducted with successes in discovering the most Fe-poor star (Keller et al. 2014) as well as abundance distributions of many metal-poor stars, e.g., Yong et al. (2021).

Determination of Elemental Abundances Chemical abundances of stars are obtained by analyses of absorption line spectra formed in stellar atmospheres (photospheres). The measurements are essentially done by comparisons of observed spectral lines with those calculated using model stellar atmospheres and spectral line data. A simple approach is the direct comparisons between observed spectra and calculated ones (called synthetic spectra). Another way is to define the strength of the absorption lines by so-called equivalent widths, and searches for the abundance of corresponding species that reproduce the observed equivalent widths. This section provides a brief overview of the standard procedure to determine chemical abundances of stars based on high-resolution spectra.

High-Resolution Spectra To determine accurate and reliable chemical abundances for stars, high spectral resolution is essential as individual spectral lines are distinguished. The velocity field of stellar photosphere is typically several km s−1 , as in the case of the Sun, caused by the thermal motion and turbulence. This corresponds to spectral resolution of R = c/v =50,000–100,000, where v is the velocity and c is the speed

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of light. This means that this level of resolving power R is sufficient to determine elemental abundances. Even with such sufficiently high resolving power, blending of spectral lines makes the abundance analysis complicated. For such cases, synthetic spectra are calculated including multiple spectral lines and compared with observed spectra to determine the abundances of the target element. If the blending includes lines by more than one element or molecular species, reliable abundance information for elements other than the target species is required. This makes the derived abundance relatively uncertain compared to the result obtained by analyses of well-separated spectral lines. When spectral lines are too crowded, the wavelength range to be analyzed is fully covered by spectral lines, and the continuum level of the light in which absorption lines are formed is not well determined. In such cases, the abundance results obtained by comparisons of synthetic spectra are particularly uncertain. Spectral lines are further broadened by stellar rotation, which makes the analysis of individual spectral lines more difficult. The rotational velocity of the Sun is about 2 km s−1 , which does not significantly affect the analysis. Rotational velocities of old stars like metal-poor stars are slow in general. On the other hand, many young stars, i.e., hot main-sequence stars, have high rotational velocities, which make spectral lines quite broad and shallow. Abundance measurements for such stars are very difficult.

Atomic and Molecular Line Data The abundance analysis based on absorption line spectra requires atomic or molecular line data including the wavelength, the excitation potential, and the transition probability. The databases of atomic lines, e.g., VALD (Kupka et al. 1999), are established for abundance analysis and calculations of model stellar atmospheres. The broadening parameters of spectral lines that present the line broadening by other particles are also important for analysis of relatively strong spectral lines, e.g., Barklem et al. (2000). Spectral line data obtained in laboratories are useful to obtain accurate chemical abundances. Atomic and molecular line data with high excitation potentials are, however, not sufficiently measured by experiment, and line data based on theoretical calculation are sometimes used for the analyses. Transition probabilities are sometimes calibrated as the abundance analysis for the solar spectrum reproduces the solar abundance. Updates of atomic lines are essential to obtain reliable chemical abundances, e.g., Lawler et al. (2009). The data of hyperfine splitting and isotope shifts are required for measurements of elemental abundances, because saturation of absorption feature is weakened if spectral lines split by these effects. Here the saturation effect means that strengths of absorption feature become insensitive to the elemental abundances associated with the absorption feature. Model Atmospheres To derive elemental abundances, spectral lines observed are compared with calculated ones. The calculation requires model stellar atmospheres (photosphere)

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Fig. 13 Examples of model stellar atmospheres (Gustafsson et al. 2008). The temperaturepressure relation is shown in this diagram for warm giants (Teff = 5000 K and log g = 3.0) and for cool giants (Teff = 3000 K and log g = 0.0). The models developed by four groups are compared

that describe the temperature structures as a function of depths (optical depths). Figure 13 shows the relation between temperature and gas pressure of model atmospheres. Stellar atmospheres have thickness of about 1000 km in solar-type stars and are much thicker in evolved stars (red giants and AGB stars). As the photosphere is sufficiently thin compared to the stellar radius in main-sequence stars, photosphere is treated as plane (plane parallel approximation). This is not the case for highly evolved red giants and supergiants, which have quite thick photosphere with low density. For such stars, model atmospheres with spherical geometry have been developed.

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Model stellar atmospheres describe the temperature structure as a function of depth, optical depth, mass, etc. The temperature decreases when decreasing depth (i.e., cooler in upper layers). Standard model atmospheres are calculated assuming hydrostatic equilibrium; chemical equilibrium including ions and molecules; and radiation equilibrium. These assumptions are well justified for main-sequence stars like the Sun and red giants in relatively early evolutionary phase. On the other hand, for model calculation for hot stars, active stars, and highly evolved cool stars, the above assumptions need to be reconsidered. A focus in studies of model atmospheres is the opacity used in radiative transfer. In particular, the approximation of line opacity for numerous atomic and molecular lines is essential. The opacity sampling is applied in recent model calculations (Gustafsson et al. 2008), for which comprehensive dataset of spectral lines like VALD (see above) is required. Whereas such 1D model atmospheres are well established for solar-type mainsequence stars and red giants (see Fig. 13), models including 3D effects have also been developed (Asplund 2005). The line profiles and sometimes the strengths are significantly impacted by this effect. The models also indicate large impact on the temperature structures of upper layers of metal-poor stars. Analysis of spectral lines using model atmospheres enables one to derive elemental abundances from (at least one) spectral line of species in some ionization/excitation status. However, taking account of inevitable errors in model atmospheres and parameters to select models to be used in the analysis (see below), abundance measurements based on analyses of spectral lines formed by major species of the ionization stage are robust. For instance, in photospheres of solartype stars and red giants, majority of Fe is singly ionized; hence, Fe abundances determined from the lines of singly ionized Fe (usually presented as Fe II) are more reliable than from neutral ones (Fe I). It should be noted, however, that the number of spectral lines useful for abundance measurements is larger for Fe I. As a result, Fe abundances are determined from Fe I lines in many cases. Measurements from the lines of both ionization stages are even preferable.

Stellar Parameters The parameters of standard model atmospheres mentioned above are the effective temperature, surface gravity, and chemical composition (The effective temperature 4 , where L and R are the luminosity and radius of Teff is defined as L = 4π R 2 σ Teff the star and σ is the Stefan-Boltzmann constant). The goal of chemical abundance studies is to determine the abundance of the target element. Hence, compositions of other elements, as well as the effective temperature and surface gravity, need to be determined a priori. Practically, the chemical composition is assumed to be scaled solar composition adopting metallicity ([Fe/H] in many cases) as the scaling factor. In the case that chemical composition is significantly different from the scaled solar one, such as carbon-enhanced stars, the effect needs to be included in the model atmospheres. Estimates of reliable effective temperature and surface gravity are an important issue for abundance studies. To estimate the effective temperature, flux density

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distribution, which is represented by color of the star, is frequently used. For this purpose, large efforts have been made for temperature-color relations for various types of stars, e.g., Casagrande et al. (2010). The surface gravity is determined by the following relation: log g = log g + log

M Teff L + 4 log − log , M Teff L

(1)

where M is the mass and L is the luminosity (M and L are solar values). Luminosity has been uncertain for most of Galactic field stars due to the uncertainties of distances. Recently, the situation is dramatically improved by the accurate astrometry data obtained with Gaia mission that have determined trigonometric parallaxes for more than one billion stars. Another approach is to determine all these parameters from spectral line analyses based on high-resolution spectra. The effective temperature is determined as the same elemental abundance (e.g., Fe abundance) is obtained from any spectral lines with different excitation potentials. Since the ionization status of an element is strongly dependent on the (electron) density of the atmosphere, and then on the surface gravity, the surface gravity is estimated as the same elemental abundance (e.g., Fe abundance) is obtained from spectral lines with different ionization stages (Fe I and Fe II). Such analysis requires a large number of spectral lines in different ionization stages for at least one element, and, hence, high-quality spectra and reliable model atmospheres.

Error Sources, Relative Analysis to Minimize Errors There are many sources of errors in chemical abundances obtained by analysis of spectral lines using model atmospheres. The errors in measurements of absorption line strengths include (1) errors of spectral data determined by the data quality, (2) uncertainty of the continuum level, and (3) contamination of other (unidentified) spectral lines. Errors of spectral line data (transition probability), as well as errors of other atomic and molecular data (e.g., partition function, dissociation energy of molecules), are also sources of abundance errors. Uncertainties of model atmospheres also affect the derived chemical abundances. They include (1)uncertainties due to the assumption of model calculations (e.g., one-dimensional geometry, local thermodynamic equilibrium, plane-parallel approximation); (2)uncertainties due to opacity used in model calculations, in particular the treatment of line opacity for cool stars; and (3)uncertainties of stellar parameters (Teff , log g). All these error sources need to be taken into consideration to determine elemental abundances. On the other hand, to determine abundance ratios, some of them are cancelled. For instance, for abundance ratios of two elements (e.g., Ni/Fe, La/Eu) having similar chemical properties (e.g., ionization potential), the impact of the uncertainties of stellar parameters is quite small. This is a reason that abundance pattern of lanthanide elements is well determined for some r-process-enhanced

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Fig. 14 Spectral lines of MgH molecules (Yong et al. 2003). Isotope ratios of 24 Mg, 25 Mg, and 26 Mg are estimated from these spectral lines by comparisons with synthetic spectra

stars. Moreover, relative abundance of some element between similar stars determined using the same set of spectral lines is quite reliable because uncertainties of spectral line data and model atmospheres are mostly cancelled. Indeed, the relative abundance studies for solar twins, which have very similar stellar parameters to the Sun, achieve errors as small as 1% in the abundance ratios with respect to solar ones (Meléndez et al. 2009).

Isotope Ratios Stronger constraints on nucleosynthesis can be provided by determination of isotope abundance ratios than elemental abundance ratios. Spectral lines of different isotopes are distinguished for light elements (e.g., Li, Be) or molecules (e.g., CH, CN, MgH, SiO, Fig. 14). Isotope ratios of a few lanthanide elements and Ba have been also determined for metal-poor stars, as spectral lines of some isotopes show detectable hyperfine splitting due to nuclear spin. For instance, Eu isotope ratios (151 Eu/153 Eu) are determined by detailed analysis of spectral line profiles for metal-poor stars with both r-process and s-process elements (Sneden et al. 2002; Aoki et al. 2003).

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References W. Aoki et al., ApJ 592, L67 (2003) W. Aoki et al., ApJ 655, 492 (2007) W. Aoki et al., AJ 145, 13 (2013) W. Aoki et al., Science 345, 912 (2014) C. Arlandini et al., ApJ 525, 886 (1999) M. Asplund, ARA&A 43, 481 (2005) M. Asplund et al., ARA&A 47, 481 (2009) B. Barbuy, C. Chiappini, O. Gerhard, ARA&A 56, 223 (2018) P.S. Barklem, N. Piskunov, B.J. O’Mara, A&AS 142, 467 (2000) P.S. Barklem et al., A&A 439, 129 (2005) K. Bechtol et al., ApJ 807, 50 (2015) T.C. Beers, G.W. Preston, S.A. Shectman, AJ 103, 1987 (1992) V. Belokurov et al., ApJ 654, 897 (2007) V. Belokurov et al., MNRAS 478, 611 (2018) W.P. Bidelman, P.C. Keenan, ApJ 114, 473 (1951) J. Bland-Hawthorn, O. Gerhard, ARA&A 54, 529 (2016) H.E. Bond, ApJ 194, 95 (1974) J. Bovy et al., ApJ 753, 148 (2012) V. Bromm, N. Yoshida, ARA&A 49, 373 (2011) D. Carollo et al., Nature 450, 1020 (2007) D. Carollo et al., ApJ 712, 692 (2010) L. Casagrande et al., A&A 512, A54 (2010) R. Cayrel et al., A&A 416, 1117 (2004) M. Chiba, T.C. Beers, AJ 119, 2843 (2000) R.E.S. Clegg, D.L. Lambert, J. Tomkin, ApJ 250, 262 (1981) J.G. Cohen et al., ApJ 612, 1107 (2004) J.J. Cowan, W.K. Rose, ApJ 212, 149 (1977) A.J. Deason, V. Belokurov, J.L. Sanders, MNRAS 490, 3426 (2019) N. Domoto et al., ApJ 939, 8 (2022) I. Escala et al., AJ 162, 45 (2021) A.V. Filippenko, ARA&A 35, 309 (1997) A. Frebel, J.E. Norris, ARA&A 53, 631 (2015) Gaia Collaboration et al., (2022) arXiv e-prints, arXiv:2206.05534 G. Gilmore, N. Reid, MNRAS 202, 1025 (1983) E. Griffith et al., ApJ 909, 77 (2021) B. Gustafsson et al., A&A 486, 951 (2008) M.R. Hayden et al., ApJ 808, 132 (2015) A. Heger, S.E. Woosley, ApJ 724, 341 (2010) A. Helmi et al., Nature 563, 85 (2018) F. Herwig, ARA&A 43, 435 (2005) S. Honda et al., ApJ 607, 474 (2004) S. Honda et al., ApJ 643, 1180 (2006) R.A. Ibata, G. Gilmore, M.J. Irwin, Nature 370, 194 (1994) R.A. Ibata et al., ApJ 780, 128 (2014) M.N. Ishigaki et al., ApJ 857, 46, 46 (2018) Ž. Ivezi´c, T.C. Beers, M. Juri´c, ARA&A 50, 251 (2012) L. Izzo et al., ApJ 808, L14, L14 (2015) A.P. Ji et al., Nature 531, 610 (2016) P. Jofré, ApJ 920, 23 (2021) M. Juri´c et al., ApJ 673, 864 (2008) F. Kappeler, H. Beer, K. Wisshak, Rep. Prog. Phys. 52, 945 (1989) P.C. Keenan, ApJ 96, 101 (1942)

3209

3210 S.C. Keller et al., Nature 506, 463 (2014) E.N. Kirby et al., ApJ 779, 102 (2013) C. Kobayashi, A.I. Karakas, M. Lugaro, ApJ 900, 179 (2020) F. Kupka et al., A&AS 138, 119 (1999) D.L. Lambert et al., ApJS 62, 373 (1986) D.L. Lambert et al., ApJ 450, 302 (1995) J.E. Lawler et al., ApJS 182, 51 (2009) H. Li et al., ApJ 931, 147 (2022) T. Matsuno et al., AJ 154, 52 (2017) T. Matsuno et al., (2021). arXiv e-prints, arXiv:2111.15423 R.D. McClure, ApJ 280, L31 (1984) R.D. McClure, A.W. Woodsworth, ApJ 352, 709 (1990) A. McWilliam, R.M. Rich, ApJS 91, 749 (1994) J. Meléndez et al., ApJ 704, L66 (2009) P. Molaro et al., MNRAS 463, L117 (2016) M. Murashima et al., ApJ 647, L131 (2006) R.P. Naidu et al., ApJ 901, 48 (2020) P.E. Nissen et al., A&A 640, A81 (2020) K. Nomoto, C. Kobayashi, N. Tominaga, ARA&A 51, 457 (2013) N. Prantzos, A&A 542, A67, A67 (2012) G.W. Preston, C. Sneden, AJ 122, 1545 (2001) I.U. Roederer, J.E. Lawler, ApJ 750, 76 (2012) I.U. Roederer et al., AJ 151, 82 (2016) B. Sesar et al., ApJ 844, L4 (2017) J.D. Simon, ARA&A 57, 375 (2019) J.D. Simon, M. Geha, ApJ 670, 313 (2007) S.J. Smartt et al., Nature 551, 75 (2017) R. Smiljanic et al., A&A 502, 267 (2009) C. Sneden et al., ApJ 467, 819 (1996) C. Sneden et al., ApJ 566, L25 (2002) C. Sneden, J.J. Cowan, R. Gallino, ARA&A 46, 241 (2008) M. Spite, F. Spite, Nature 297, 483 (1982) M. Spite et al., A&A 430, 655 (2005) A. Tajitsu et al., Nature 518, 381 (2015) A. Tajitsu et al., ApJ 818, 191, 191 (2016) M. Tanaka et al., ApJ 708, 1168 (2010) M. Tanaka et al., PASJ 69, 102, 102 (2017) E. Tolstoy, V. Hill, M. Tosi, ARA&A 47, 371 (2009) G. Wallerstein et al., ApJ 449, 688 (1995) D. Watson et al., Nature 574, 497 (2019) C. Wegg, O. Gerhard, M. Portail, MNRAS 450, 4050 (2015) B. Yanny et al., AJ 137, 4377 (2009) N. Yasuda et al., PASJ 71, 74, 74 (2019) D. Yong, D.L. Lambert, I.I. Ivans, ApJ 599, 1357 (2003) D. Yong et al., MNRAS 507, 4102 (2021)

W. Aoki and M. N. Ishigaki

Chemo-dynamical Evolution of Galaxies

88

Chiaki Kobayashi and Philip Taylor

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Galactic Chemical Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Metal Ejection Terms in GCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nucleosynthesis Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Galactic Terms in GCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Origin of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The [X/Fe]–[Fe/H] Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The First Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The First Chemical Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemodynamical Evolution of Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling of Baryon Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Various Big Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Galactic Archaeology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extra-galactic Archaeology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3212 3214 3215 3217 3219 3219 3224 3229 3231 3233 3235 3241 3243 3248 3252 3254

Abstract

Stars are fossils that retain the history of their host galaxies. Elements heavier than helium are created inside stars and are ejected when they die. From the spatial distribution of elements in galaxies, it is therefore possible to constrain

C. Kobayashi () Department of Physics, Astronomy and Mathematics, Centre for Astrophysics Research, University of Hertfordshire, Hatfield, UK e-mail: [email protected] P. Taylor Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT, Australia © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_106

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the physical processes during galaxy formation and evolution. This approach, Galactic archaeology, has been popularly used for our Milky Way Galaxy with a vast amount of data from Gaia satellite and multi-object spectrographs to understand the origins of sub-structures of the Milky Way. Thanks to integral field units, this approach can also be applied to external galaxies from nearby to distant universe with the James Webb Space Telescope. In order to interpret these observational data, it is necessary to compare with theoretical predictions, namely, chemodynamical simulations of galaxies, which include detailed chemical enrichment into hydrodynamical simulations from cosmological initial conditions. These simulations can predict the evolution of internal structures (e.g., metallicity radial gradients) as well as that of scaling relations (e.g., the mass-metallicity relations). After explaining the formula and assumptions, we will show some example results and discuss future prospects.

Introduction Explaining the origin of the elements is one of the scientific triumphs linking nuclear physics with astrophysics. As Fred Hoyle predicted, carbon and heavier elements (“metals” in astrophysics) were not produced during the Big Bang but instead created inside stars. The so-called α elements (O, Ne, Mg, Si, S, Ar, and Ca) are mainly produced by core-collapse supernovae, while iron-peak elements (Cr, Mn, Fe, and Ni) are more produced by thermonuclear explosions, observed as Type Ia supernovae (SNe Ia; Kobayashi et al. (2020a), hereafter K20). The production depends on the mass of white dwarf (WD) progenitors, and a large fraction of SNe Ia should come from near-Chandrasekhar (Ch) mass explosions (see Kobayashi et al. (2020b) for constraining the relative contribution between near-Ch and sub-Ch mass SNe Ia). Among core-collapse supernovae, hypernovae (1052 erg) produce a significant amount of Fe as well as Co and Zn, and a significant fraction of massive stars (20 M ) should explode as hypernovae in order to explain the Galactic chemical evolution (GCE; Kobayashi et al. 2006). Heavier elements are produced by neutron-capture processes. The slow neutroncapture process (s-process) occurs in asymptotic giant branch (AGB) stars (e.g., Busso et al. 1999; Herwig 2005; Cristallo et al. 2011; Karakas and Lattanzio 2014), while the astronomical sites of rapid neutron-capture process (r-process) have been debated. The possible sites are neutron-star (NS) mergers (NSMs, Wanajo et al. 2014; Just et al. 2015), magneto-rotational supernovae (MRSNe, Nishimura et al. 2015; Mösta et al. 2018; Reichert et al. 2021), magneto-rotational hypernovae (MRHNe, Yong et al. 2021), accretion disks/collapsars (Siegel et al. 2019), and common envelope jet supernovae (Grichener et al. 2021). Light neutron-capture elements (e.g., Sr) are also produced by electron-capture supernovae (ECSNe, Wanajo et al. 2013), ν-driven winds (Arcones et al. 2007; Wanajo 2013), and rotating massive stars (Frischknecht et al. 2016; Limongi and Chieffi 2018). The cycles of chemical enrichment are schematically shown in Fig. 1, where each cycle produces different elements and isotopes with different timescales.

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9

10 yr

MC

WD thermonuclear SN core collapse in a WD binary

6-8

10 yr r?

HN SNII

SNIa

AGB

Fe,Zn O,Mg r NS

Fe,Mn

NS

ISM C,N s

© Kobayashi 2019

Fig. 1 Schematic view of chemical enrichment in galaxies

In a galaxy, not only the total amount of metals, i.e., metallicity Z ≡ Σi>He mi /m, but also elemental abundance ratios, [X/Fe]≡ log NX /NFe /(NX, /NFe, ), evolve as a function of time. Therefore, we can use all of this information as fossils to study the formation and evolutionary histories of the galaxy. This approach is called Galactic archaeology, and several ongoing and future surveys with multi-object spectrographs (MOS; e.g., APOGEE, HERMES-GALAH, Gaia-ESO, DESI, WEAVE, 4MOST, MOONS, Subaru Prime Focus Spectrograph (PFS), and Maunakea Spectroscopic Explorer (MSE)) are producing a vast amount of observational data of elemental abundances. Moreover, integral field unit (IFU) spectrographs (e.g., SAURON, SINFONI, CALIFA, SAMI, MaNGA, KMOS, MUSE, HECTOR, and NIRSpec on JWST) allow us to measure metallicity and some elemental abundance ratios within galaxies. It is now possible to apply the same approach not only to our own Milky Way but also to other types of galaxies or distant galaxies. Let us call this extra-galactic archaeology. While the evolution of the dark matter in the standard Λ cold dark matter (CDM) cosmology is reasonably well understood, how galaxies form and evolve is still much less certain because of the complexity of the baryon physics such as star formation and feedback. The thermal energy ejected from supernovae to the interstellar medium (ISM) suppresses star formation, while the production of heavy elements in these supernovae enhances gas cooling. Since these processes affect each other, galaxy formation and evolution are complicated and have to be solved consistently with a numerical simulation. Feedback from central super-massive black holes (SMBHs) is also found to be very important for explaining the observed

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properties of galaxies (e.g., Croton et al. 2006; Springel et al. 2005; Hopkins et al. 2006; Dubois et al. 2012; Taylor and Kobayashi 2014). Since the 1990s, the development of high performance computers and computational techniques has made it possible to simulate the formation and evolutionary history of galaxies, not only of isolated systems (e.g., Burkert and Hensler 1987; Katz 1992; Steinmetz and Mueller 1994; Mihos and Hernquist 1996; Kawata and Gibson 2003; Kobayashi 2004) but also for cosmological simulations of individual galaxies (e.g., Navarro and White 1994) or of the galaxy population as a whole (e.g., Cen and Ostriker 1999; Kobayashi et al. 2007). Thanks to the public release of hydrodynamical simulation codes such as Gadget (Springel et al. 2001), RAMSES (Teyssier 2002), and AREPO (Springel 2010), it became easier to run galaxy simulations, and even simulation data are made public (e.g., EAGLE, Illustris). Needless to say, the simulation results highly depend on the input baryon physics, namely, nuclear astrophysics, for predicting chemical abundances. In this review we first discuss how the yields are constrained with observations using a simple chemical evolution model and then how chemical enrichment can be calculated in more sophisticated, chemodynamical simulations of galaxies.

Galactic Chemical Evolution Galactic chemical evolution (GCE) has been calculated analytically and numerically since the 1970s (e.g., Tinsley 1980; Prantzos et al. 1993; Timmes et al. 1995; Pagel 1997; Chiappini et al. 1997; Matteucci 2001; Kobayashi et al. 2000, hereafter K00) basically integrating the following equation: d(Zi fg ) = ESW + ESNcc + ESNIa + ENSM − Zi ψ + Zi,inflow Rinflow − Zi Routflow dt (1) where the mass fraction of each element i in gas-phase (fg denotes the gas fraction, or the gas mass in the system considered with a unit mass) increases via element ejections from stellar winds (ESW ), core-collapse supernovae (ESNcc ), Type Ia supernovae (ESNIa ), and neutron star mergers (ENSM ). It also decreases by star formation (with a rate ψ) and can be modified by inflow (with a rate Rinflow ) and outflow (with a rate Routflow ) of gas in/from the system considered. It is assumed that the chemical composition of gas is instantaneously well mixed in the system (called a one-zone model), but the instantaneous recycling approximation is not adopted nowadays (Matteucci 2021). The model with Rinflow = Routflow = 0 is called a closed-box model but is not realistic in any observed galaxies. The initial conditions are fg,0 = 1 (a closed system) or fg,0 = 0 (an open system) with the chemical composition (Zi,0 ) from the Big Bang nucleosynthesis (Zi,BBN ). External enrichment is often neglected, assuming Zi,inflow = Zi,BBN . In Eq. (1) the first two terms depend only on nucleosynthesis yields, while the third and fourth terms also depend on modeling of the progenitor binary systems, which is uncertain. The last

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three terms are Galactic terms, and should be determined from Galactic dynamics, but are assumed with analytic formulae in GCE models. The stars formed at a given time t have the initial metallicity, Z(t), which is equal to the metallicity of the ISM from which the stars form (Later, integrated metallicity of stellar populations in a galaxy will be described as Z∗ ).

The Metal Ejection Terms in GCE The ejection terms are given by integrating the contributions with various initial mass (m) and metallicity (Z) of progenitor stars. Hence, the first assumption is the initial mass function (IMF), φ. The IMF is often assumed to have time/metallicityinvariant mass spectrum normalized to unity at m ≤ m ≤ mu as: φ(m) ∝ m−x , 

mu m

m−x dm =

1 (m1−x − m1−x  ) = 1. 1−x u

(2) (3)

Salpeter (1955) found the slope of the power law, x = 1.35, from observation of nearby stars (see Tinsley (1980) for a different definition of the slope, α=x + 1). Kroupa et al. (1993) updated this result finding three slopes at different mass ranges, which are summarized as x = −0.7, 0.3, and 1.3 for m/M  0.08, 0.08  m/M  0.5, 0.05  m/M  150, respectively (Kroupa 2008). However, the author suggested that the slope of the massive end should be x = 1.7 in the solar neighborhood, which has been used in some GCE works (e.g., Romano et al. 2010). In most of results in this paper, the Kroupa IMF (with the massive-end slope x = 1.3) is adopted for a mass range from m = 0.01 M to mu = 120 M , while Chabrier (2003)’s IMF is often used in hydrodynamical simulations. Note that there are a few claims for an IMF variation from observations (e.g., van Dokkum and Conroy 2010; Gunawardhana et al. 2011; Cappellari et al. 2013). Then the ejection terms for single stars can be calculated as follows (The metals in Eq. (4) and Eq. (5) are called “unprocessed” and “processed” metals, respectively. Our supernova yield table contains “processed” metals only, the AGB yield table contains net yields, so the mass loss with star particle’s chemical composition should also be included with Eq. (4)):  ESW =

mu

mt

 ESNcc =

mu

mt

(1 − wm − pzi m,II ) Zi (t − τm ) ψ(t − τm ) φ(m) dm,

(4)

pzi m,II ψ(t − τm ) φ(m) dm.

(5)

The lower mass limit for integrals is the turning off mass mt at t, which is the mass of the star with the main-sequence lifetime τm = t. τm is a lifetime of a star with m

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and also depends on Z of the star. wm is the remnant mass fraction, which is the mass fraction of a WD (for stars with initial masses of m  8M ), a NS (for ∼8−20 M ) or a black hole (BH) (for 20 M ). pzi m,II denotes the nucleosynthesis yields of core-collapse supernovae, given as a function of m, Z, and explosion energy. The additional ejection terms from binary systems are more complicated. Suppose that the event rates are given, the enrichment can be calculated as: EIa = mCO pzi m,Ia Rt,Ia ,

(6)

ENSM = mNSM ejecta pzi m,NSM Rt,NSM .

(7)

Rt,x are the rates of SNe Ia and NSMs per unit time per unit stellar mass formed in a population of stars with a coeval chemical composition and age (simple stellar population, SSP) and called the delay-time distribution (DTD). For the DTDs, simple analytic formula were also proposed (Matteucci 2021, and references therein). However, the functions are significantly different from what are obtained from binary population synthesis (BPS), ignore metallicity dependence during binary evolution, and do not include the effects of supernova kicks for NSMs. De Donder and Vanbeveren (2004) was the first work that combined BPS to GCE. All of the matter in the WD (mCO ) is ejected from SNe Ia (except for a subclass called SNe Iax; see Kobayashi et al. 2015); for Ch-mass SNe Ia, mCO = 1.38 M . On the other hand, only a small fraction of matter is ejected from a NSM (mNSM ejecta ∼ 0.01 M ), depending on the mass ratios, the equation of state of the NSs, and the spin of BHs (Kobayashi et al. 2022). These are provided together with the nucleosynthesis yields (pzi m,Ia and pzi m,NSM ). For SNe Ia from the single-degenerate systems, Kobayashi et al. (1998) proposed another analytic formula based on binary calculation:  Rt,Ia = b

mp,u (Z)

max[mp, (Z), mt ]

1 φ(m) dm m



md,u (Z) max[md, (Z), mt ]

1 ψ(t − τm ) φd (m) dm. m (8)

The first integral is for the primary star, and the mass range is for the stars that can produce ∼1 M of C+O WDs in binaries, which is set to be ∼3 − 8 M . The second integral is for the secondary star, and the mass range depends on the optically thick winds from the WD, which is about ∼1 M and ∼3 M for the red-giant+WD systems and main-sequence+WD systems, respectively, depending on Z. The metallicity dependence on the SN Ia DTD is very important to reproduce the observed [α/Fe]–[Fe/H] relation in the solar neighborhood (see Kobayashi and Nomoto 2009, for the detailed parameters and the resultant DTDs). These parameters are for Ch-mass explosions. Equation (8) can also be used for SNe Iax and sub-Ch mass explosion triggered by slow H accretion (see Kobayashi et al. 2015, for the detailed parameters), but not for the double-degenerate systems (which are likely to cause sub-Ch mass explosions). DTDs of these subclasses of SNe Ia

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have been predicted by various BPS models, but none of these models can reproduce the observations because the total rate is too low and/or the typical timescale is too short (see Kobayashi et al. 2022, for more details). For NSMs, K20 used a metallicity-dependent DTD from a BPS (Mennekens and Vanbeveren 2014, 2016). Kobayashi et al. (2022) used various BPS models and also provided new analytic formulae that can reproduce the observed [Eu/(Fe,O)]–[(Fe,O)/H] relation only with NSMs, without the r-process associated with core-collapse supernovae. Currently, there is no BPS model that can explain the observation only with NSMs because the rate is too low and/or the timescale is too long (see Kobayashi et al. 2022, for more details). Other binary systems such as novae can also contribute GCE for some elements or isotopes (e.g., Romano et al. 2019), but the DTDs are very uncertain (see Kemp et al. 2022, for the nova DTDs from BPS) and the yield tables are not available.

Nucleosynthesis Yields Nucleosynthesis yields (pzi m,X ) are integrated over stellar lifetimes of single stars, or delay-time distributions for binaries, depending on Z (Eqs. 4, 5, 6, and 7). It is extremely important to take account of this metallicity dependence, though it is often ignored in hydrodynamical simulations. Because of this, it is also not possible to solve chemical evolution as a post-process. Chemical enrichment including all elements must be followed on-the-fly. For core-collapse supernovae, our yields were originally calculated in Kobayashi et al. (2006), three models of which are replaced in Kobayashi et al. (2011a) (the identical table was also used in Nomoto et al. 2013), and a new set with failed supernovae is used in K20. This is based on the lack of observed progenitors at supernova locations in the HST data (Smartt 2009) and on the lack of successful explosion simulations for massive stars (Janka 2012; Burrows and Vartanyan 2021). This resulted in a 20% reduction of both the net and oxygen yields (see Table 3 of K20). Supernova yields assumed E51 ≡ E/1051 ergs = 1 of explosion energy as observed for SN1987A, while hypernova yields included the observed mass dependence of the explosion energy; E51 = 10, 10, 20, 30 for 20, 25, 30, 40 M stars. These supernova and hypernova yield tables are provided separately, and it is recommended to assume εHN =50% hypernova fraction for stars with ≥20 M (Kobayashi et al. 2006). This fraction is expected to decrease at high metallicities due to smaller angular momentum loss and was assumed as εHN (Z) = 0.5, 0.5, 0.4, 0.01, and 0.01 for Z = 0, 0.001, 0.004, 0.02 in Kobayashi and Nakasato (2011). Woosley and Weaver (1995) provided a yield table as a function of mass and metallicity, which was tested with a GCE model by Timmes et al. (1995) and has been widely used in chemical evolution studies. It would be useful to note that stellar mass loss is not included in their pre-supernova evolution calculations. Another problem is that their core-collapse supernova models tend to produce more Fe than

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observed because of the relatively deep mass cut, which leads to [α/Fe]∼ 0 in the ejecta. Therefore, the theoretical Fe yields were divided by a factor of 2 (Timmes et al. 1995; Romano et al. 2010), which is obviously unphysical. This artificial reduction could mimic a shallower mass cut, but in that case the yields of other ironpeak elements formed in the same layer as Fe should also be reduced. For the first problem, Portinari et al. (1998) obtained C+O core masses from stellar evolution models with mass loss and adopted to the Woosley and Weaver (1995) supernova yields, which is not physically accurate either. It is also known that Mg production in the 40 M model with E51 = 1 is unreasonably small compared with that in other models. This Mg underproduction problem also exists in Portinari et al. (1998). Limongi and Chieffi (2018) also provided a yield table but as a function of stellar rotation (see Kobayashi (2022a) for the impact of stellar rotation in GCE). Note that these yields do not reproduce the observed elemental abundances in the solar neighborhood, namely, iron-peak elements. This is because this yield set does not have hypernovae, and mixing during explosion is not taken into account either. There are also pre-supernova yields in the literature that do not include explosive nucleosynthesis. These are not useful for GCE since during explosions iron-peak elements are produced and α element yields are also largely modified, although they can be used for studying isotopic ratios of light elements. The yields for AGB stars were originally calculated in Karakas (2010) and Karakas and Lugaro (2016), but a new set with the s-process is used in K20 with optimizing the mass of the partial mixing zone. In K20, the narrow mass range of super-AGB stars is also filled with the yields from Doherty et al. (2014); stars at the massive end are likely to become ECSNe, as is believed to have been the case for the progenitor of the Crab Nebula in 1054. At the low-mass end, off-center ignition of C flame moves inward but does not reach the center, which remains a hybrid C+O+Ne WD. This might become a subclass of SNe Ia called SN Iax. The contribution of SN Iax is negligible in the solar neighborhood but can be important for dwarf spheroidal galaxies (Kobayashi et al. 2015; Cescutti and Kobayashi 2017). There are other yields in the literature (see Romano et al. (2010, 2019) for comparison with GCE). For SNe Ia, the nucleosynthesis yields of near-Ch and sub-Ch mass models are newly calculated in Kobayashi et al. (2020b), which used the same 2D code as in Leung and Nomoto (2018, 2020) but with more realistic, solar-scaled initial composition. The initial composition gives significantly different (Ni, Mn)/Fe ratios, compared with the classical W7 model (Nomoto et al. 1997) or more recent delayed detonation model (Seitenzahl et al. 2013). When constraining the progenitors of SNe Ia from the observed Mn abundances in the solar neighborhood, it is important to use the latest yields of SNe Ia. See Blondin et al. (2022) for a complete comparison including many other yields. The r-process yields are taken from the following references in K20: 8.8 M model in Wanajo et al. (2013, 2D) for electron capture supernovae, 1.2 − 2.4 M models in Wanajo (2013) for ν-driven winds, 1.3 M +1.3 M model in Wanajo et al. (2014, 3D-GR) for neutron star mergers, and 25 M “b11tw1.00” model in Nishimura et al. (2015, axisymmetric MHD) for magnetorotational supernovae. See Kobayashi et al. (2022) for the GCE models including the neutron star merger yields

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from Just et al. (2015) with various masses of compact objects. These yields are uncertain depending on (1) initial conditions (e.g., M, Z, rotation, magnetic fields), (2) the quality of the base hydrodynamical simulations (three-dimensional general relativity or not), (3) neutrino physics, and (4) nuclear physics (e.g., reaction rates, fission modeling).

The Galactic Terms in GCE Star formation rate is usually modeled as ψ = τ1s fgn but often n = 1 is adopted, and thus it is proportional to the gas fraction. τs denotes a timescale of star formation, and 1/τs gives the efficiency of star formation. The inflow rate is often assumed t −t to be exponential as Rinflow = τ1i exp −t τi , but occasionally as Rinflow = 2 exp τi τi

in Pagel (1997) and in the solar neighborhood model of K20. The outflow rate is assumed to be proportional to the star formation rate as Routflow = τ1o fgn ∝ ψ, which is reasonable if the outflow is driven by supernova feedback. In addition, star formation is quenched (ψ = 0) at a given epoch tw , which corresponds to a Galactic wind driven by the feedback from active Galactic nuclei (AGN). The timescales are determined to match the observed metallicity distribution function (MDF) of stars in each system modeled. Figures 2 and 3 show the assumed star formation history and the resultant metallicity evolution and MDF. The parameter sets that have very similar MDFs give almost identical tracks of elemental abundance ratios (Fig. A1 of Kobayashi et al. 2020b). This means that, for the given MDF, elemental abundance tracks do not depend so much on the star formation history. Therefore, GCE can be used to constrain nuclear astrophysics, if the MDF is known, in Galactic archaeology. Without knowing the MDFs, star formation histories are unconstrained, and elemental abundance tracks can vary depending on the star formation histories. Provided that nuclear astrophysics is known, elemental abundance measurements can be used to constrain the star formation histories through GCE, which is the case for extra-galactic archaeology.

The Origin of Elements Using the K20 GCE model for the solar neighborhood, we summarize the origin of elements in the form of a periodic table. In each box of Fig. 4, the contribution from each chemical enrichment source is plotted as a function of time: Big Bang nucleosynthesis (black), AGB stars (green), core-collapse supernovae including SNe II, HNe, ECSNe, and MRSNe (blue), SNe Ia (red), and NSMs (magenta). It is important to note that the amounts returned via stellar mass loss are also included for AGB stars and core-collapse supernovae depending on the progenitor star mass (Eq. 4). The x-axis of each box shows time from t = 0 (Big Bang) to 13.8 Gyrs, while the y-axis shows the linear abundance relative to the Sun, X/X . The dotted lines indicate the observed solar values, i.e., X/X = 1 and 4.6 Gyr for the age

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Fig. 2 Star formation histories (panel a), age-metallicity relations (panel b), and metallicity distribution functions (panel c) for the solar neighborhood (blue solid lines), halo (green shortdashed lines), halo with stronger outflow (light-blue dotted lines), bulge (red long-dashed lines), bulge with outflow (olive dot-short-dashed lines) and thick disk (magenta dot-long-dashed lines). (Figure is taken from Kobayashi et al. (2020a); see the reference for the observational data sources and the model details)

of the Sun; the solar abundances are taken from Asplund et al. (2009), except for A (O) = 8.76, A (Th) = 0.22, and A (U) = −0.02 (§2.2 of K20 for the details). The adopted star formation history is similar to the observed cosmic star formation rate history, and thus this figure can also be interpreted as the origin of elements in the universe, which can be summarized as follows: • H and most of He are produced in Big Bang nucleosynthesis. As noted, the green and blue areas also include the amounts returned to the ISM via stellar mass loss in addition to He newly synthesized in stars. After tiny production in Big Bang

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Fig. 3 The same as Fig. 2 but for dwarf spheroidal galaxies, Fornax (red solid lines), Sculptor (green long-dashed lines), Sextans (blue short-dashed lines), and Carina (magenta dotted lines), and ultra-faint dwarf (UFD) galaxies (cyan dot-dashed lines). (Figure is updated from Kobayashi et al. (2020b); see the reference for the observational data sources and the model details. The observational data for the UFD is provided by A. Ji (priv. comm.))

nucleosynthesis, Be and B are supposed to be produced by cosmic rays (Prantzos et al. 1993), which are not included in the K20 model. • The Li model is very uncertain because the initial abundance and nucleosynthesis yields are uncertain. Li is supposed to be produced also by cosmic rays and novae, which are not included in the K20 model. The observed Li abundances show an increasing trend from very low metallicities to the solar metallicity, which could be explained by cosmic rays. Then the observation shows a decreasing trend from the solar metallicities to the super-solar metallicities, which might be caused by the reduction of the nova rate (Grisoni et al. 2019); this is also shown in theoretical calculation with binary population synthesis (Kemp et al.

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Fig. 4 The time evolution (in Gyr) of the origin of elements in the periodic table: Big Bang nucleosynthesis (black), AGB stars (green), core-collapse supernovae including SNe II, HNe, ECSNe, and MRSNe (blue), SNe Ia (red), and NSMs (magenta). The amounts returned via stellar mass loss are also included for AGB stars and core-collapse supernovae depending on the progenitor mass. The dotted lines indicate the observed solar values. (Figure is taken from Kobayashi et al. 2020a)

2022), where the nova rate becomes higher due to smaller stellar radii and higher remnant masses at low metallicities. • 49% of C, 51% of F, and 74% of N are produced by AGB stars (In extra-galactic studies, the N production from AGB stars is referred as “secondary,” which is confusing, and is a primary production from freshly synthesized 12 C. For massive stars, the N yield depends on the metallicity of progenitor stars for secondary production and can also be enhanced by stellar rotation for primary production (Kobayashi et al. 2011a)) (at t = 9.2 Gyr). For the elements from Ne to Ge, the newly synthesized amounts are very small for AGB stars, and the small green areas are mostly for mass loss. • α elements (O, Ne, Mg, Si, S, Ar, and Ca) are mainly produced by core-collapse supernovae, but 22% of Si, 29% of S, 34% of Ar, and 39% of Ca come from SNe Ia. These fractions would become higher with sub-Ch-mass SNe Ia (Kobayashi et al. 2020b) instead of 100% Ch-mass SNe Ia adopted in the K20 model. • A large fraction of Cr, Mn, Fe, and Ni are produced by SNe Ia. In classical works, most of Fe was thought to be produced by SNe Ia, but the fraction is only 60% in the K20 model, and the rest is mainly produced by HNe. The inclusion of HNe is very important as it changes the cooling and star formation histories of the universe significantly (Kobayashi et al. 2007). Co, Cu, Zn, Ga, and Ge are largely produced by HNe. In the K20 model, 50% of stars at ≥20 M are assumed to explode as hypernovae, and the rest of stars at >30 M become failed supernovae.

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• Among neutron-capture elements, as predicted from nucleosynthesis yields, AGB stars are the main enrichment source for the s-process elements at the second (Ba) and third (Pb) peaks. • 32% of Sr, 22% of Y, and 44% of Zr can be produced from ECSNe, which are included in the blue areas, even with the adopted conservative mass ranges; we take the metallicity-dependent mass ranges from the theoretical calculation of super-AGB stars (Doherty et al. 2015). Combined with the contributions from AGB stars, it is possible to perfectly reproduce the observed trends, and no extra light element primary process (LEPP) is needed (but see Prantzos et al. 2018). The inclusion of ν-driven winds in GCE simulation results in a strong overproduction of the abundances of the elements from Sr to Sn with respect to the observations. • For the heavier neutron-capture elements, contributions from both NS-NS/NSBH mergers and MRSNe are necessary, and the latter is included in the blue areas. Note again that the green areas include the mass-loss component, i.e., not newly produced but recycled. Note that Tc and Pm are radioactive. Since the Sun is slightly more metal-rich than the other stars in the solar neighborhood (see Fig. 2 of K20), the fiducial model in K20 goes through [O/Fe]=[Fe/H]= 0 slightly later compared with the Sun’s age. Thus, a slightly faster star formation timescale (τs = 4 Gyr instead of 4.7 Gyr) is adopted in this figure. The evolutionary tracks of [X/Fe] are almost identical. In this model, the O and Fe abundances go though the cross of the dotted lines, meaning [O/Fe] = [Fe/H] = 0 at 4.6 Gyr ago. This is also the case for some important elements including N, Ca, Cr, Mn, Ni, Zn, Eu, and Th. The remaining problems can be summarized as follows: • The contribution from rotating massive stars is not included in the K20 model, which can probably explain in the underproduction of C and F (Kobayashi 2022a). A binary effect during stellar evolution may also increase C. These elements could be enhanced by AGB stars as well, but the observed high F abundance in a distant galaxy strongly supported rapid production of F from Wolf-Rayet stars (Franco et al. 2021, at z = 4.4 observed with ALMA). • Mg is slightly under-produced in the model, although at low metallicity the model [Mg/Fe] is slightly higher than observed (see Fig. 7). This may be due to a NLTE effect (Lind et al. 2022) or due to a binary effect. Massive stars in binaries tend to have a smaller CO core with a higher C/O ratios (Brown et al. 2001), which could result in a higher Mg/O ratio. Observed Mg/O ratios suggest that this binary effect should not be important at low metallicities (0.1 Z ). • The underproduction of the elements around Ti is a long-standing problem since Kobayashi et al. (2006). It was shown that these elemental abundances can be enhanced by multidimensional effects (Sneden et al. (2016); see also K15 model in K20). This is due to the lack of 3D simulations that can capture the production of these (less abundant) elements. 2D nucleosynthesis calculation showed an enhancement of these elements (Maeda and Nomoto 2003; Tominaga 2009).

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• The s-process elements are slightly overproduced even with the updated sprocess yields. Notably, Ag is over-produced by a factor of 6, while Au is under-produced by a factor of 5. U is also over-produced. These problems may require revising nuclear physics modeling (see Kobayashi 2022a, for more discussion).

The [X/Fe]–[Fe/H] Relations Although the slopes in Fig. 4 show the different enrichment timescales of each element, it is easier to see the differences in the [X/Fe]–[Fe/H] diagrams (There were attempts to plot against a different element (e.g., O) to avoid the uncertainty of Fe yields (e.g., Cayrel et al. 2004), but it became rather hard to understand the plots). Figure 5 shows the [α/Fe]–[Fe/H] relation, which is probably the most important diagram in GCE. O is one of the α elements. At the beginning of the universe, the first stars (Population III stars) form and die, whose properties such as mass and rotation are uncertain and have been studied using the abundance patterns of the second-generation, extremely metal-poor (EMP) stars. Secondly, core-collapse supernovae occur, and their yields are imprinted in the abundance patterns of Population II stars in the Galactic halo. The [α/Fe] ratio is high and stays

Fig. 5 The [O/Fe]–[Fe/H] relations in the solar neighborhood for the models with Ch-mass SNe Ia only (solid line), 75% Ch plus 25% sub-Ch-mass SNe Ia (dashed line), and sub-Ch-mass SNe Ia only (dotted line). The observational data (filled circles) are high-resolution nonlocal thermodynamic equilibrium (NLTE) abundances from Zhao et al. (2016). (Figure is taken from Kobayashi et al. (2020b) with modification)

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roughly constant (There was a debate if the [O/Fe] ratio increases toward the lowest metallicity or not. UV OH line showed such an increase (e.g., Israelian et al. 1998) reaching [O/Fe] ∼1 at [Fe/H] ∼ − 3, which was later found to be due to 3D effects) with a small scatter. This plateau value does not depend on the star formation history but does on the IMF. Finally, SNe Ia occur, which produce more Fe than α elements, and thus the [α/Fe] ratio decreases toward higher metallicities; this decreasing trend is seen for the Population I stars in the Galactic disk. The contribution from SNe Ia depends on the progenitor binary systems, namely, the mass of the progenitor WDs. Sub-Ch mass explosions produce less Mn and Ni and more Si, S, and Ar than near-Ch mass explosions (Seitenzahl et al. 2013; Kobayashi et al. 2020b). Figure 5 shows the [O/Fe]–[Fe/H] relations with varying the fraction of sub-Ch-mass SNe Ia. Including up to 25% sub-Ch mass contribution to the GCE (dashed line) gives a similar relation as the K20 model (solid line), while the model with 100% sub-Ch-mass SNe Ia (dotted line) gives too low an [O/Fe] ratio compared with the observational data. For Ch-mass SNe Ia, the progenitor model is based on the single-degenerate scenario with the metallicity effect due to optically thick winds from WDs (Kobayashi et al. 1998). For sub-Ch-mass SNe Ia, the observed delay-time distribution is used since the progenitors are the combination of mergers of two WDs in double degenerate systems and low accretion in single degenerate systems; Kobayashi et al. (2015)’s formula are for those in single degenerate systems only. Because of this [α/Fe]–[Fe/H] relation, high-α and low-α are often used as a proxy of old and young ages of stars in galaxies, respectively. Note that, however, this relation is not linear but is a plateau and decreasing trend (called a “knee”). The location of the “knee” depends on the star formation timescale, with a higher metallicity for faster star formation history (Matteucci and Brocato 1990), e.g., [Fe/H] ∼ −0.5, −0.8, −1, −2 for the Galactic bulge, thick disk, thin disk, and satellite galaxies. Figure 6 shows the GCE model results for various environments. The bulge (with outflow), thick disk, and solar neighborhood (thin disk) models are taken from K20 with 100% Ch-mass SNe Ia, with the star formation histories in Fig. 2. The models for dwarf spheroidal galaxies are taken from Kobayashi et al. (2020b) adding 100% sub-Ch mass SNe Ia on top of the 100% of Ch mass SNe Ia, with the star formation histories in Fig. 3. Recently, a similar relation is also shown for M31 using planetary nebulae. Since Fe is not available, Ar is used as a significant fraction of Ar is produced by SNe Ia. The thin and thick disk dichotomy seems to exist, but not exactly the same as in the Milky Way, probably due to M31 experiencing an accretion of a relatively massive, gas-rich satellite galaxy (Arnaboldi et al. 2022). There are GCE models that try to explain both thin and thick disk observations with two infalls (Chiappini et al. 1997; Spitoni et al. 2019). However, more realistic, chemodynamical simulations show that a significant number of thickdisk stars are formed in satellite galaxies before they accrete onto the disk, which can be better approximated with two independent GCE models in this figure (also in Grisoni et al. 2017). Figure 7 shows the evolution of elemental abundance ratios [X/Fe] against [Fe/H] from C to Zn in the solar neighborhood. All α elements (O, Mg, Si, S, and Ca) show

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faster SFH

slower SFH more SNIa

Fig. 6 The [O/Fe]–[Fe/H] relations in the Galactic bulge (red long-dashed line), thick disk (green short-dashed line), thin disk (blue solid line), Fornax (magenta dot-short-dashed line), Sculptor (cyan dot-long-dashed line), and ultra-faint dwarf (UFD) galaxies (olive dotted line). The star formation history of each system is constrained from its MDF in Figs. 2 and 3 (see Fig. 5 for the observational data source for thin (navy open circles) and thick (maroon filled circles) disk stars)

the [α/Fe]–[Fe/H] relations, i.e., the plateau and decreasing trend from [Fe/H] ∼ −1 to higher metallicities. The odd-Z elements (Na, Al, and Cu) show an increasing trend toward higher metallicities due to the metallicity dependence of the corecollapse supernova yields. See K20 for more detailed discussion on the evolutionary trends of each element and Kobayashi (2022b) for the discussion with more recent observations for Cu and Zn. In the following we focus the role of each enrichment source. • The contribution to GCE from AGB stars (green dotted lines in Fig. 7) can be seen mainly for C and N and only slightly for Na, compared with the model that includes supernovae only (blue dashed lines). Hence, it seems not possible to explain the O-Na anticorrelation observed in globular cluster stars (e.g., Kraft et al. 1997) with AGB stars (but see Ventura and D’Antona 2009). Although AGB stars produce significant amounts of Mg isotopes, their inclusion does not affect the [Mg/Fe]-[Fe/H] relation. • The contribution from super-AGB stars (red solid lines) is very small; with super-AGB stars, C abundances slightly decrease, while N abundances slightly increase. It would be very difficult to put a constraint on super-AGB stars from the average evolutionary trends of elemental abundance ratios, but it might be

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Fig. 7 Evolution of the elemental abundances [X/Fe] from C to Zn against [Fe/H] for the models in the solar neighborhood, with only supernovae (without AGB and super-AGB stars, blue shortdashed line), with AGB without super-AGB stars (green dotted lines), with AGB and super-AGB stars (red solid line, fiducial model), with ECSNe (magenta long-dashed lines), and with SNe Iax (cyan dot-dashed lines). (Figure is taken from Kobayashi et al. (2020a) and see the reference for the observational data sources)

possible to see some signatures of super-AGB stars in the scatters of elemental abundance ratios. • With ECSNe (magenta long-dashed lines), Ni, Cu, and Zn are slightly increased. These yields are in reasonable agreement with the high Ni/Fe ratio in the Crab Nebula (Nomoto 1987; Wanajo et al. 2009). • No difference is seen with/without SNe Iax (cyan dot-dashed lines) in the solar neighborhood because the progenitors are assumed to be hybrid WDs, which has a narrow mass range in the adopted super-AGB calculations (Doherty et al. 2015, ΔM ∼ 0.1 M ). This mass range depends on convective overshooting, mass loss, and reaction rates. Even with the wider mass range in Kobayashi et al. (2015, ΔM ∼ 1 M ), however, the SN Iax contribution is negligible in the solar neighborhood, but it can be important at lower metallicities such as in dwarf spheroidal galaxies with stochastic chemical enrichment (Cescutti and Kobayashi 2017). Figure 8 shows the evolutions of neutron-capture elements as [X/Fe]-[Fe/H] relations.

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Fig. 8 The [X/Fe]-[Fe/H] relations for neutron-capture elements, comparing to the models in the solar neighborhood, with s-process from AGB stars only (blue long-dashed lines); with sprocess and ECSNe (light blue short-dashed lines); with s-process, ECSNe, and ν-driven winds (green dotted-long-dashed lines); with s-process, ECSNe, and NS-NS mergers (olive dotted lines); with s-process, ECSNe, and NS-NS/NS-BH mergers (orange dotted-short-dashed lines); and with s-process, ECSNe, NS-NS/NS-BH mergers, and MRSNe (red solid lines). (Figure is updated from Kobayashi et al. (2020a) with more observational data)

• As known, AGB stars can produce the first (Sr, Y, Zr), second (Ba), and third (Pb) peak s-process elements, but no heaver elements (navy long-dashed lines). The second-peak elements are under-produced around [Fe/H] ∼ −2, which is eased with chemodynamical simulations, and can also be reproduced better with rotating massive stars. However, the model with rotating massive stars results in overproduction of the first-peak elements (Kobayashi 2022a). • It is surprising that ECSNe from a narrow mass range (ΔM ∼ 0.15 − 0.2 M ) can produce enough of the first-peak elements; with the combination of AGB stars, it is possible to reproduce the observational data very well (cyan shortdashed lines). This means that no other light element primary process (LEPP), such as rotating massive stars, is required (but see Cescutti and Chiappini 2014). The elements from Mo to Ag seem to be overproduced, which could be tested with the UV spectrograph proposed for the VLT, CUBES. • Additional production from ν-driven winds leads to further overproduction of these elements in the model (green dot-long-dashed lines), but this should be studied with more self-consistent calculations of supernova explosions.

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• Neutron star mergers can produce lanthanides and actinides, but not enough (olive dotted lines); the rate is too low and the timescale is too long, according to binary population synthesis. This is not improved enough even if NS-BH mergers are included (orange dot-short-dashed lines). It is very unlikely this problem will be solved even if the mass dependence of the nucleosynthesis yields are included, unless the BH spins are unexpectedly high (Kobayashi et al. 2022). • In the GCE model with MRSNe (magenta solid lines), it is possible to reproduce a plateau at low metallicities for Eu, Pt, and Th, relative to Fe. However, even with including both MRSNe and neutron star mergers, the predicted Au abundance is more than ten times lower than observed. This underproduction is seen not only for the solar abundance but also for low metallicity stars although the observational data are very limited. UV spectroscopy with HST, or NASA’s future LUVOIR, is needed to investigate this problem further. In conclusion, an r-process associated with core-collapse supernovae, such as MRSNe, is required. The same conclusion is obtained with other GCE models and more sophisticated chemodynamical simulations (e.g., Haynes and Kobayashi 2019; van de Voort et al. 2020), as well as from the observational constraints of radioactive nuclei in the solar system (Wallner et al. 2021). See more discussion in the later section. It seems not to be easy to solve the missing gold problem with astrophysics; only Au yields should be increased since Pt is already in good agreement with the current model and Ag is rather overproduced in the current model. However, there are uncertainties in nuclear physics, namely, in some nuclear reaction rates and in the modeling of fission, which might be able to increase Au yields only, without increasing Pt or Ag. It may be hard to predict the only one stable isotope of 197 Au, while Pt has several stable isotopes. The predicted Th and U abundances are, after the long-term decays, to be compared with observations of metal-poor stars, and the current model does not reproduce the Th/U ratio either.

The First Galaxies The Atacama Large Millimeter/submillimeter Array (ALMA) has opened a new window to study elemental abundances and isotopic ratios of light elements at very high redshifts (e.g., Zhang et al. 2018; Franco et al. 2021), which provide us an independent constraint on stellar nucleosynthesis. In the early universe, stellar rotation may be important at low metallicities because weaker stellar winds result in smaller angular momentum loss. The impact of stellar rotation on stellar evolution and nucleosynthesis has been studied to explain Wolf-Rayet stars (e.g., Meynet and Maeder 2002; Hirschi 2007; Limongi and Chieffi 2018), and the observed N and 13 C abundances at low metallicities have been used to constrain the effect (Chiappini et al. 2006; see also Fig. 13 of Kobayashi et al. 2011a) (However, the N/O–O/H relation can be reproduced by inhomogeneous enrichment in chemodynamical simulations (Fig. 26)). As discussed in the previous section, the importance of jet-like supernova

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Fig. 9 Evolution of CNO (number) abundances in GCE models for the Milky Way (red solid and blue long-dashed lines) and for a submillimeter galaxy (green short-dashed and magenta dotted lines), which are calculated with the original K20 yields (red solid and green short-dashed lines) or the yields with Wolf-Rayet stars (blue long-dashed and magenta dotted lines; see text for the model details)

explosion for iron-peak elements (Co and Zn) and r-process elements (via MRSNe) also indicates the importance of stellar rotation, although how to transport the angular momentum in stellar envelopes to stellar cores is uncertain. Figure 9 shows the evolution of CNO abundances in GCE models for two representative galaxies with different star formation histories: the solar neighborhood in the Milky Way, which has continues star formation, and a submillimeter galaxy (SMG), which underwent a star burst. The SMG models use the same star formation history as in Franco et al. (2021), while the red solid lines are the same as in Fig. 8 and Fig. 10 of K20. As discussed, ∼50% C and ∼75% N come from low- and intermediate-mass AGB stars, respectively, at later times ([O/H] ∼ −1 for C). The rest comes from massive stars, either released before supernova explosions by stellar winds or ejected at supernova explosions. The N yield from massive stars increases, while the C yield decreases at higher metallicities. These two effects cause the increase of N/O and C/O, and the decrease of C/N, toward high metallicities. With rotation, both C and N yields are increased due to the interplay between the core He-burning and the H-burning shell, triggered by the rotation-induced instabilities (Limongi and Chieffi 2018). Limongi and Chieffi (2018)’s yields do not reproduce the observations of iron-peak elements because of their simple description of supernova explosions. Therefore, the C and N yields from Limongi and Chieffi (2018) are combined with the K20 yields in the models with WolfRayet stars. High abundance of CNO enhances the production of F in He convective shell, which can explain the high F abundance observed in a submillimeter galaxy at redshift z = 4.4 (Franco et al. 2021). The evolution of isotopic ratios is also largely affected by stellar rotation. Without rotation, 13 C and 25,26 Mg are produced from AGB stars (17 O might be overproduced in our AGB models), while other minor isotopes are more produced from metal-rich massive stars, and thus the ratios between major and minor isotopes (e.g., 12 C/13 C, 16 O/17,18 O) generally decrease as a function of time/metallicity (Figs. 17–19 of Kobayashi et al. (2011a) and Fig. 31 of K20; see also Romano et al. 2019). Isotopic ratios of CNO, Si, S, Cl, and Ar have been estimated for a couple of spiral galaxies

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around z ∼ 1 (Wallström et al. 2019, and the references therein), as well as for various sources in the Milky Way and nearby galaxies. With rotation, 13 C and 17,18 O are enhanced. The evolution of isotopic ratios and comparison to observations can be found in Fig. 7 of Kobayashi (2022a), which gives consistent results with Romano et al. (2019)’s GCE models. 12 C/13 C ratio became too high, while it is possible to explain the low 13 C/18 O ratio observed in submillimeter galaxies at redshift z ∼ 2 − 3 (Zhang et al. 2018) without changing the IMF. Stellar rotation also causes the weak-s process that forms s-process elements from the existing seeds at a much shorter timescale than AGB stars. This may help solving the underproduction of Ba at [Fe/H]  −1 (Fig. 8) but causes an overproduction of light s-process elements such as Sr, Y, and Zr. The impact of stellar rotation on nucleosynthesis is not well understood yet.

The First Chemical Enrichment Extremely metal-poor (EMP) stars have been an extremely useful relic in the Galactic archaeology. At the beginning of galaxy formation, stars form from a gas cloud that was enriched only by one or very small number of supernovae (Audouze and Silk 1995), and hence the elemental abundances of EMP stars can offer observational evidence of particular types of supernovae in the past. The expectation was that the first stars were so massive (∼140 − 300 M ) that they exploded as a pair-instability supernova (PISN, Barkat et al. 1967), which causes high [(Si,S)/O] ratios (e.g., Nomoto et al. 2013). However, after a half century of surveys, no star has been found with an elemental abundance pattern fitted by a PISN in the Milky Way. Instead, it is found that quite a large fraction of massive stars become faint supernovae (Umeda and Nomoto 2003) that give a high C/Fe ratio leaving a relatively large black hole (∼5 M ). If there were C-enhanced low-α stars, that would indicate black hole formation even from 10−20 M Pop III stars (Kobayashi et al. 2014). From faint supernovae, the ejected iron mass is very small but the explosion energy can be high; faint hypernovae can explain the Zn enhancement of CEMP stars (Ishigaki et al. 2018). Similar C enhancement might also be found for damped Ly-α system (DLA, Kobayashi et al. 2011b). Figure 10 shows the observed abundance pattern of a DLA at z = 2.34 with [Fe/H] −3, comparing to the nucleosynthesis yields of a faint supernova (red solid line), a faint hypernova (green dashed line), and a PISN (blue dotted line). In recent surveys, no DLA is found with an elemental abundance pattern consistent with a PISN either. A small number of EMP stars show a relatively low α abundance, which does not necessarily mean that the ISM from which the star formed was already enriched by SNe Ia. The reasons that could cause low α/Fe ratios were summarized in Kobayashi et al. (2014): (1) SNe Ia, (2) less-massive core-collapse supernovae (20 M ), which become more important with a low star formation rate, (3) hypernovae, although the majority of hypernovae are expected to give normal [α/Fe]

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Fig. 10 Elemental abundance patterns of the Population III supernovae. The red solid and green short-dashed lines show the nucleosynthesis yields of faint core-collapse supernovae from 25 M stars with mixing fallback. The blue dotted line is for pair-instability supernovae from 170 M stars. Stellar rotation is not included. The dots are for the metal-poor C-rich DLA (filled circles) and peculiar DLA (open circles). (Figure is taken from Kobayashi et al. (2011b) with modification)

ratios (∼0.4), and (4) PISNe, which could be very important in the early universe. Therefore, the [α/Fe] ratio is not a perfect clock. It is necessary to also use other elemental abundances (namely, Mn and neutron-capture elements) or isotopic ratios, with higher resolution (>40,000) multi-object spectroscopy on 8 m class telescopes (e.g., cancelled WFMOS or planned MSE, proposed HRMOS for VLT). EMP stars with r-process enhancement (called rII stars), namely, those with enhancement of the heaviest elements (actinide-boost stars), can offer confirmation of an r-process site. Recently, from the SkyMapper survey, Yong et al. (2021) found such a star with [Fe/H] = −3.5, the lowest metallicity known for rII stars. The abundance pattern is shown Fig. 11, which shows a clear detection of U and Th, the universal r-process pattern, normal α enhancement, but unexpectedly showing high Zn and N abundances ([Zn/Fe]= 0.72, [N/Fe]= 1.07). This abundance pattern cannot be explained with a model with a neutron star merger (orange dot-dashed line), but instead strongly supports a magnetorotational hypernova from a 25 M Pop III star (blue dashed line). There are a few magneto-hydrodynamical simulations and postprocess nucleosynthesis that successfully showed enough neutron-rich ejecta to produce the third-peak r-process elements as in the Sun (Winteler et al. 2012; Mösta et al. 2018). The predicted iron mass is rather small (Nishimura et al. 2015; Reichert et al. 2021), and the astronomical object may be faint. Yong et al. (2021)’s proposal is Fe-

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a

3.0

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SMSS 2003−1142

MRHN

NSM

[X/Fe]

2.0 1.0 0.0 −1.0 b

1.5

Δ [X/Fe]

1.0

Mg Si

Cr

Zn Co

Y

Mo

Ce Eu Ho Lu Ba Nd Tb Tm Os

U

0.5 0.0 −0.5 −1.0 −1.5

Al

Ca

20

Ni Mn

Zr Sr Ru

La Sm Dy Yb Pr Gd Er Hf

40 60 Atomic Number

Th

80

Fig. 11 The elemental abundance of an extremely metal-poor star SMSS J200322.54-114203.3, which has [Fe/H]= −3.5, comparing with mono-enrichment from a magneto-rotational hypernovae (blue dashed line and squares) and multi-enrichment including a neutron star merger (orange dot-dashed line and triangles). The lower panel shows the differences, i.e., the observed values minus model values. (Figure is taken from Yong et al. 2021)

producing, magnetorotational hypernova, which is more luminous. The r-process elements could also be formed in the accretion disk of a collapsar (Siegel et al. 2019), but the inter-ejecta that contains Fe and Zn must be ejected at the explosion as for a hypernova. As it is luminous, it might be possible to detect absorption features of Au and Pt in the spectrum of a transient in future.

Chemodynamical Evolution of Galaxies Thanks to the development of super computers and numerical techniques, it is now possible to simulate the formation and evolution of galaxies from cosmological initial conditions, i.e., initial perturbation based on Λ cold dark matter (CDM) cosmology. Star formation and gas inflow and outflow in Eq. (1) are not assumed but are, in principle, given by dynamics. The baryon cycle is schematically shown in Fig. 12. Due to the limited resolution, star-forming regions, supernova ejecta, and active Galactic nuclei (AGN) cannot be resolved in galaxy simulations, and

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Seed Formation

Star Formation

BH

Remnant WD/NS/BH

Growth Cooling

EO Fe

EO Fe SNIa

Z dependent

E

OE Fe

EO Fe

BH

E

E Fe

OE © Kobayashi 2014

Stellar Winds

E

Fe

SNII/HN

OE

BH

Feedback

E

E

Feedback

Fig. 12 Schematic view of chemodynamical evolution of galaxies

thus it is necessary to model star formation and the subsequent effects – feedback – introducing a few parameters. Fortunately, there are many observational constraints, from which it is usually possible to choose a best set of parameters. To ensure this, it is necessary to run the simulation until the present-day, z = 0, and reproduce a number of observed relations at various redshifts. Although hydrodynamics can be calculated with publicly available codes such as Gadget, RAMSES, and AREPO, modeling of baryon physics is the key and is different depending on the simulation teams/runs. These are various simulations of a cosmological box with periodic boundary conditions and containing galaxies with a wide mass range (e.g., 109−12 M in stellar mass) at z = 0. In order to study massive galaxies, it is necessary to increase the size of the simulation box (e.g., 100 Mpc), while in order to study internal structures, it is necessary to improve the spatial resolution (e.g., 0.5 kpc). The box size and resolution are chosen depending on available computational resources. On the other hand, zoom-in techniques allow us to increase the resolution focusing a particular galaxy, but this also requires tuning the parameters with the same resolution, comparing to a number of observations in the Milky Way. Needless to say, elemental abundances are the most informative. There are a few zoom-in simulations for Milky Way-type galaxies (e.g., Brook et al. 2012; Few et al. 2014; Grand et al. 2017; Buck et al. 2020; Font et al. 2020), but in most of the cases the input nuclear astrophysics are too simple to make use of elemental abundances.

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Modeling of Baryon Physics For the baryon physics, the first process to calculate is radiative cooling and photoheating usually by a uniform and evolving UV background radiation (Haardt and Madau 1996). It would be ideal to simulate radiative feedback self-consistently, but this is computationally very expensive and the available simulations are only for high redshifts. The cooling function depends on the chemical composition of gas, and a metallicity-dependent cooling function computed with the MAPPINGS III software (Sutherland and Dopita 1993; Kewley et al. 2019) is used in our code, assuming the [α/Fe]–[Fe/H] relation of the solar neighborhood (Fig. 13). Elementdependent cooling functions are also available (Wiersma et al. 2009). Usually, the simulation code finds candidate gas particles (or mesh) that can form stars in a given timestep and forms star particles with some certain mass. Our star formation criteria are: converging flow, (∇ · v)i < 0, rapid cooling, tcool < tdyn , Jeans unstable gas, tdyn < tsound .

(9) (10) (11)

Fig. 13 (left) Cooling functions without photoionization. Included processes are collisional excitation (long-dashed lines) of H0 (red) and He+ (cyan); collisional ionization (short-dashed lines) of H0 (red), He0 (green), and He+ (cyan); standard recombination (short-dash dotted lines) of H+ (yellow), He+ (cyan), and He++ (blue); dielectric recombination (long-dashed dotted line) of He+ (cyan); and free-free emission (dotted line). (right) Metallicity dependence of the cooling function. The solid, long-dashed, short dashed, dotted, short-dash dotted, and long-dash dotted lines correspond to [Fe/H] = 0, −1, −2, −3, −4, and −5, respectively. (Figures are taken from Kobayashi 2002)

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This still needs a parameter to define the star formation timescale relative to the dynamical timescale, as: tsf ≡

1 tdyn , c∗

(12)

where c∗ is a star formation timescale parameter. The parameter c∗ basically determines when to form stars following cosmological accretion and has a great impact on the size-mass relation of galaxies (Fig. 14). Alternatively, a fixed density criterion (e.g., n∗H = 0.1 cm−3 ) can be adopted (Katz 1992) although we found that it is better not to include it. More sophisticated analytic formulae are also proposed based on smaller-scale simulations, including the effects of turbulence and magnetic fields (Federrath et al. 2011). A fractional part of the mass of the gas particle turns into a new star particle (see Kobayashi et al. 2007, for the details). Note that an individual star particle has a typical mass of ∼105−7 M , i.e., it does not represent a single star but an association of many stars. The masses of the stars associated with each star particle are distributed according to an IMF. We adopt Kroupa IMF (with x = 1.3), which is assumed to be independent of time and metallicity, and limit the IMF to the mass range 0.01 M ≤ m ≤ 120 M . This assumption is probably valid for the resolution down to ∼104 M .

Fig. 14 Dependence of the star formation timescale parameter c∗ and the IMF slope x on the galaxy scaling relations: size-mass (upper panels) and mass-metallicity (lower panels) relations. A single slope like Salpeter IMF is assumed. The points show a set of chemodynamical simulations of early-type galaxies that undergo various merging histories. The solid and dashed lines indicate the observed relations. (Figure is taken from Kobayashi (2005) with modification)

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Kobayashi (2004) introduced a scheme to follow the evolution of the star particle as a simple stellar population (SSP), which is defined as a single generation of coeval and chemically homogeneous stars of various masses, i.e., it consists of a number of stars with various masses but the same age and chemical composition. The production of each element i from the star particle (with an initial mass of m0∗ ) is calculated using a very similar equation as Eq. (1): EZ (t) = m0∗ [ESW + ESNcc + ESNIa + ENSM ] .

(13)

With this SSP method, the identical equations for GCE, Eqs. (4, 5, 6, and 7) can used for ESW , ESNcc , EIa , and ENSM . Similarly, the energy production from the star particle is:   Ee (t) = m0∗ ee,SW RSW (t) + ee,SNcc RSNcc (t) + ee,SNIa RSNIa (t) ,

(14)

where the event rates of SWs (RSW ) and core-collapse supernovae (RSNcc ) can be calculated as:  RSW =

mu

mt

 RSNcc =

1 ψ(t − τm ) φ(m) dm, m

min[mSNcc,u ,mu ]

max[mSNcc, , mt ]

(15)

1 ψ(t − τm ) φ(m) dm. m

(16)

The lower and upper limits are the same as IMF upper limit for stellar winds (0.01 and 120 M ), while it is set to be ∼8 M (depending on Z) and 50 M for corecollapse supernovae. The energy production from SNe Ia is significant, and the rate is given by Eq. (8), while the energy production from NSMs is negligible. The energy per event (in erg) for SWs is ee,SW = 0.2 × 1051 (Z/Z )0.8 for >8 M stars; Type II supernovae, ee,SNII = 1 × 1051 ; hypernovae, ee,HN = 10, 10, 20, 30 × 1051 for 20, 25, 30, 40 M stars, respectively; and for SNe Ia, ee,SNIa = 1.3 × 1051 erg. We distribute this feedback metal mass and energy from a star particle j isotropically to a fixed number, NFB , of neighbor gas particles , weighted by the smoothing kernel as:   NFB  d(Zi,k mg,k )  = Wj Wkj EZ,j (t)/ dt j

(17)



and   NFB   duk = (1 − fkin ) Wj Wkj Ee,j (t)/ dt j



(18)

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where uk denotes the internal energy of a gas particle k and fkin denotes the kinetic energy fraction (Navarro and White 1993), i.e., the fraction of energy that is distributed as a random velocity kick to the gas particle. Note that in the first sum, the number of neighbor star particles is not fixed to NFB . To calculate these equations, the feedback neighbor search needs to be done twice in order to ensure proper mass and energy conservation: first to compute the sum of weights for the normalization and a second time for the actual distribution. The parameter NFB determines the average energy distributed to gas particles; a large value of NFB leads to inefficient feedback as the ejected energy radiatively cools away shortly, while a small value of NFB results in a only small fraction of matter affected by feedback. Alternatively, with good resolution in zoom-in simulations, feedback neighbors could be chosen within a fixed radius (rFB ), which affects a number of observations including the MDF (Fig. 12 of Kobayashi and Nakasato 2011). The parameter fkin has a great impact on surface brightness profiles and radial metallicity gradients in galaxies, and fkin = 0, i.e., purely thermal feedback, gives the best match to the observations (Fig. 15). Note that various feedback methods are proposed such as the stochastic feedback (Dalla Vecchia and Schaye 2008, used for EAGLE) and the mechanical feedback (Hopkins et al. 2018a, used for FIRE). Feedback modeling can have an impact on chemical evolution of galaxies (D. Ibrahim et al., in prep.). Figure 16 shows a test simulation of an isolated disk galaxy with supernova feedback. The initial condition is a rotating gas cloud with total mass of 1010 h−1 M , and non-isotropic infall of gas form a dense disc, where stars form. As a result of the energy input, the low-density hot-gas region expands in a bipolar flow (left panels) ejecting metals outside the galaxy. In the disk plane, the hot gas region expands and forms a dense shell where stars keep on forming. The energy from

Fig. 15 Dependence of the star formation timescale parameter c∗ and the kinetic feedback fraction fkin on the internal structure of galaxies: surface brightness profile (left panel) and the stellar metallicity radial gradient (right panel) for a chemodynamical simulation of an early-type galaxy with the initial perturbation. A single slope with x = 1.1 is adopted. (Figure is taken from Kobayashi (2004) with modification)

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these stars can quickly propagate to the surrounding low-density region. After the Galactic wind forms (middle panel), the gas density becomes so low at the center that star formation is terminated for a while. Because of radiative cooling, a part of the ejected, metal-enhanced gas, however, returns, again settling in the disk where it fuels new star formation, but this secondary star formation is not as strong as the initial starburst. Although some small bubbles are forming in the galaxy in this stage, not much gas and metals are ejected by them from the disk (right panels). However, Galactic winds become much weaker in more massive galaxies. Figure 17 shows the fraction of metal loss as a function of galaxy mass in a cosmological simulation including hypernova feedback. Most of metals are inside the galaxy at Mtot  1012 h−1 M . Equation (17) gives positive feedback that enhances radiative cooling, while Eq. (18) gives negative feedback that suppresses star formation. The mass ejection in Eq. (17) never becomes zero, while the energy production in Eq. (18) becomes small after ∼35 Myrs. Therefore, it is not easy to control star formation histories depending on the mass/size of galaxies within this framework. In particular, it is not possible to quench star formation in massive galaxies, since low-mass stars keep returning their envelope mass into the ISM for a long timescale, which will cool and keep forming stars.

Fig. 16 Time evolution of the formation of an isolated disk galaxy in a halo of mass of 1010 h−1 M . The black points show star particles, while the gas particles are color-coded according to their temperature. Each panel is 20 kpc on a side. The upper row shows face-on projections, and the lower row gives edge-on views. (Figure is taken from Kobayashi et al. 2007)

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Fig. 17 (Left) Spatial distribution of ISM (blue) and wind (red) particles at z = 0. (Right) Ejected metal fraction fZ,w = MZ,w /(MZ,g + MZ,∗ + MZ,w ) against total mass. (Figures are taken from Kobayashi et al. (2007) with modification)

Therefore, in order to reproduce observed properties of massive galaxies, additional feedback was required, and the discovery of the co-evolution of supermassive black holes and host galaxies (Magorrian et al. 1998) provided a solution – AGN feedback. Modeling of AGN feedback consists of (1) seed formation, (2) growth by mergers and gas accretion, and (3) thermal and/or kinetic feedback (see Taylor and Kobayashi (2014) for the details). In summary, we introduced a seeding model where seed black holes originate from the formation of the first stars, which is different from the “standard” model by Springel et al. (2005) and from most large-scale hydrodynamical simulations. In our AGN model, seed black holes are generated if the metallicity of the gas cloud is zero (Z = 0) and the density is higher than a threshold density (ρg > ρc ). The growth of black holes is roughly the same as in other cosmological simulations and is calculated with Bondi-Hoyle accretion, swallowing of ambient gas particles and merging with other black holes. Since we start from relatively small seeds, the black hole growth is driven by mergers at z  3 (Fig. 18), which could be detected by gravitational waves. On a very small scale, it is not easy to merge two black holes, and an additional time delay is applied in more recent simulations (P. Taylor et al. in prep.). Proportional to the accretion rate, thermal energy is distributed to the surrounding gas, which is also the same as in many other simulations. In more recent simulations, non-isotropic distribution of feedback area is used to mimic the small-scale jet (e.g., Mukherjee et al. 2018). There are a few parameters in our chemodynamical simulation code, but Taylor and Kobayashi (2014) constrained the model parameters from observations and determined the best parameter set: α = 1, ε f = 0.25, Mseed = 103 h−1 M , and ρc = 0.1 h2 mH cm−3 . Our black holes seeds are

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Fig. 18 Cosmic BH accretion rate density (solid line) and BH growth rate density (including BH formation, dashed line) with redshift. The difference of these rates shows the BH merger rate. (Figure is taken from Taylor and Kobayashi (2014) with modification)

indeed the debris of the first stars although we explored a parameter space of 101−5 h−1 M . This is not the only one channel for seeding, and the direct collapse of primordial gas and/or the collapse of dense stellar clusters (∼105 M , e.g., Madau and Rees 2001; Woods et al. 2019) are rarer but should be included in larger volume simulations. Nonetheless, our model can successfully drive large-scale Galactic winds from massive galaxies (Taylor and Kobayashi 2015a) and can reproduce many observations of galaxies with stellar masses of ∼109−12 M (Taylor and Kobayashi 2015b, 2016, 2017). Metals are ejected from galaxies to circumgalactic and intergalactic medium mainly by supernova-driven winds in low-mass galaxies (as in Fig. 17), while by AGN-driven winds by massive galaxies (Taylor et al. 2020). The fraction of metals ejected by AGN is only a few percent of the total produced metals, and thus it does not affect mass-metallicity relations (Taylor and Kobayashi 2015a). The movie of our fiducial run is available at https://www.youtube.com/watch?v=jk5bLrVI8Tw.

Various Big Simulations There are various large volume simulations available, but the input physics is very different, which can be summarized as follows. The differences in modeling of subgalactic physics largely affects the predicted galaxy properties, and the parameters must be calibrated using galaxy scaling relations (e.g., Fig. 14) and internal structures (e.g., Fig. 15) at z = 0, before predicting observables at higher redshifts using the simulation outputs. Figure 19 shows a comparison of some of these simulation to our simulation, which is run by a Gadget-3 based code (Taylor and Kobayashi 2014; in prep.). See also Wang et al. (2019) for the discussion on the black hole seed mass.

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Fig. 19 Cosmic star formation rate history of our 100 h−1 Mpc simulation, comparing to other cosmological simulations (see text for the details). The observational data (points) are taken from Madau and Dickinson (2014) and Driver et al. (2018). Note that other observations (e.g., Chary et al. 2007; Rowan-Robinson et al. 2016) show higher rates at high redshifts. The data of the other simulations are provided by R. Yates and C. Lovell (priv. comm.)

EAGLE (Schaye et al. 2015) – Star formation is calculated with a metallicity and  Z −0.64 −3 temperature-dependent density threshold (n∗H (Z) = 0.1 0.002 cm and T < 104.4 K). Star particle mass is determined from a pressure law, obtained from the observed Kennicutt-Schmidt, with two free parameters (A and n). A Chabrier IMF is adopted. Stochastic feedback from Dalla Vecchia and Schaye (2012) is assumed. SMBHs are grown from the seeds of 105 M , and the AGN feedback is thermal, independent of metallicity. The SN Ia rate decays exponential with a timescale of τ = 2 Gyr, with a fixed normalization ν. The simulation box size is up to 100 Mpc3 , which has 1.81 × 106 M and 0.7 kpc resolution for gas. IllustrisTNG (Pillepich et al. 2018) – It is run with a magneto-hydrodynamic code (Pakmor et al. 2011). Star formation and pressurization of the multi-phase are treated following Springel and Hernquist (2003), which includes a density threshold nH 0.1 cm−3 . A Chabrier IMF is adopted. Isotropic, kinetic (wind) feedback from Springel and Hernquist (2003) is assumed (instead of bipolar winds in Illustris). SMBHs are grown from the seeds of 8 × 105 M , and kinetic wind is assumed for AGN feedback in the case of low accretion rates as in Weinberger et al. (2017). SN Ia rate is calculated with a “simple DTD” (−1 slope) with a constant normalization N0 . The simulation box size is up to 302.6 Mpc3 , but a 110.7 Mpc3 run has 1.4×106 M and 0.19 kpc resolution for gas.

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SIMBA (Davé et al. 2019) – This uses H2 -based star formation rate from Krumholz and Gnedin (2011), which is based on the metallicity and local column density, assuming the efficiency of ε∗ = 0.02 for the H2 density. Stellar feedback is assumed to be kinetic, with decoupled wind particles, galaxy mass dependent (η(M∗ )), and metal loaded (dZ = fSNII ySNII / max[η, 1]). SMBHs are grown from the seeds of 1 × 104 h−1 M . AGN feedback is also assumed to be kinetic for both fast and slow accretion and purely bipolar. In addition, on-the-fly dust production and destruction are calculated. SNe Ia have two distinct delay times, 0 and 0.7 Gyr, with two constant rates. The simulation box size is up to 100 h−1 Mpc3 , which has 1.82 × 107 M and 0.5 h−1 kpc resolution for gas. HORIZON-AGN (Dubois et al. 2016) – This uses a density threshold of nH 0.1 cm−3 , thermal energy injection to model stellar feedback, and two modes of AGN feedback with thermal or bipolar outflow depending on the Eddington ratio. The simulation box size is 100 h−1 Mpc3 , which has 1 × 107 M , and the grid is adaptively refined down to 1 proper kpc. Magneticum (Dolag et al. 2017) – Multiphase model for star formation from Springel and Hernquist (2003), and isotropic winds with 350 km s−1 are adopted. Two modes of AGN feedback with different efficiencies of energy injection depending on the Eddington ratio are assumed. The simulation box size is 48 h−1 Mpc3 , which has 3.7 × 107 h−1 M and 0.7 h−1 kpc resolution for gas. Similarly, there are a series of cosmological zoom-in simulations of Milky Waytype galaxies. Auriga (Grand et al. 2017) uses similar code as Illustris but with n = 0.13 cm−3 and tsf = 2.2 Gyr, with resolution from 6 × 103 to 4 × 105 M for gas. FIRE-2 (Hopkins et al. 2018b) is run with the GIZMO code, which is also used for SIMBA, but with improved modeling of small-scale physics. AGN feedback is not included. ARTEMIS (Font et al. 2020) uses the same code as EAGLE with resolution of 2 × 104 h−1 M for gas.

Galactic Archaeology In the Local Group, i.e., in the Milky Way and dwarf spheroidal (dSph) galaxies, the spatial distribution of detailed elemental abundances can be obtained from high-resolution spectra of individual low-mass stars (Ages of stars can also be well estimated with asteroseismology recently). This has been done for more than half a century, and the average trends of the observations for many elements have been used to constrain the stellar physics (Timmes et al. 1995; Kobayashi et al. 2006; Romano et al. 2010). The recent Galactic archaeology surveys with mediumresolution MOS dramatically increased the sample and made possible to discuss the distributions of stars along the average trends, which requires more realistic, chemodynamical simulations of galaxies. Meantime, observations with different lines for each element (e.g., Israelian et al. 1998; Sneden et al. 2016) revealed NLTE

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and 3D effects in the stellar atmosphere. Detailed star-by-star analysis of wide range of high-resolution spectra (including UV) is still required to obtain absolute values of elemental abundances (Kobayashi 2022a, for the need of UV spectra). Figure 20 shows the edge-on and face-on maps of our simulated Milky Waytype galaxy, where V-band luminosities of star particles are calculated using stellar population synthesis models from Kodama and Arimoto (1997) and all detailed chemical enrichment is included. Although there were simulations of isolated galaxies, Kobayashi and Nakasato (2011) was the first chemodynamical simulations that showed the evolution elements from O to Zn in a Milky Way-type galaxy from cosmological initial condition. Our new simulation, which is run by a Gadget3 based code (Kobayashi, in prep.), includes all stable elements, and a fully cosmological initial condition is applied (Scannapieco et al. 2012). The spatial and mass resolutions are 0.5 kpc and 3×105 M , respectively. The movie of our fiducial run is available at https://star.herts.ac.uk/~chiaki/works/Aq-C-5-kro2.mpg Our simulated galaxy shows very similar maps of metallicity and [α/Fe] ratios as observed. Figure 21 shows the observations in Gaia DR3, which contains a half billion of stars. The data show too blue outer bulge in the metallicity map, and in the [α/Fe] map the patterns close to the Ecliptic Poles are artifacts. There are both vertical and radial metallicity gradients; metallicity is high on the plane and becomes higher toward Galactic center. On the other hand, the [α/Fe] ratios show a strong vertical gradient but no radial gradient except for the Galactic bulge; the majority of the bulge stars have high [α/Fe] ratios (Kobayashi and Nakasato 2011), which may depend on the feedback from Galactic center. There is no chemodynamical model that includes feedback from the central SMBH in the Milky Way. The formation of the basic structure is the same as described in Kobayashi and Nakasato (2011). The Galactic bulge formed by an initial star burst, by the assembly of gas-rich subgalaxies beyond z = 2. The disk formed with a longer timescale and has grown inside out; the disk was small in the past and becomes

Fig. 20 V-band luminosity map for the edge-on (left panels) and face-on (right panels) views at z = 0 in 30 kpc on a side, of our chemodynamical zoom-in simulation of a Milky Way-type galaxy

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Fig. 21 (Left panel) Metallicity map of Gaia DR3 released on 13 June 2022, in the range of [M/H]= −0.3 (blue) to 0.0 (red). (Right panel) The same as the left panel but for [α/Fe]= −0.02 (blue) to 0.12 (red). (Figures are taken from Gaia Collaboration et al. 2022)

larger at later times. In the ΛCDM cosmology, satellite galaxies keep accreting, which interact and merge with the main body, but there is no major merger after z = 2. Otherwise, it is not possible to keep the disk structure as in the Milky Way. These galaxy mergers also make the disk thicker. Approximately one third of thickdisk stars already formed in merging galaxies, which are disrupted by tidal force and accreted onto the disk plane. In the thin disk, star formation is self-regulated, and chemical enrichment takes place following cosmological gas accretion, radial flows, and stellar migrations; these physical processes are analyzed in detail in Vincenzo and Kobayashi (2020). Inhomogeneous enrichment in chemodynamical simulations leads to a paradigm shift on the chemical evolution of galaxies. As in a real galaxy, (i) the star formation history is not a simple function of radius, (ii) the ISM is not homogeneous at any time, and (iii) stars migrate, which are fundamentally different from one-zone or multi-zones GCE models. As a consequence, (1) there is no tight age-metallicity relation, namely, for stars formed in merging galaxies. It is possible to form extremely metal-poor stars at a later time, from accretion of nearly primordial gas, or in isolated chemically-primitive regions. (2) Enrichment sources with long time delays such as AGB stars, SNe Ia, and NSMs can appear at low metallicities. This effect can naturally explain the observed N/O-O/H relation (Vincenzo and Kobayashi 2018a) but is not sufficient to explain observed [Eu/(O,Fe)] ratios only with NSMs (Haynes and Kobayashi 2019). (3) There is a significant scatter in elemental abundance ratios at a given time/metallicity, as shown in Figs. 16–18 of Kobayashi and Nakasato (2011). Figure 22 shows the frequency distribution of [O/Fe] ratios in the solar neighborhood of our simulated galaxy. The same [O/Fe]-[Fe/H] relation as in Fig. 5 is seen: the plateau at [O/Fe] ∼ 0.5 caused by core-collapse supernovae and the decrease of [O/Fe] from [Fe/H] ∼ −1 to ∼0 due to the delayed enrichment from SNe Ia. The difference is that this chemodynamical model predicts not a line but a distribution (contours). A similar figure was also shown in Kobayashi and Nakasato (2011), which predicted a bimodal distribution of [O/Fe] ratio at a given [Fe/H], concluding that “this may be because the mixing of heavy elements among gas particles is not included in our model.” However, with the APOGEE survey, Hayden et al. (2015) clearly showed the bimodality of [α/Fe], which was in fact already seen in earlier

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Fig. 22 The [O/Fe]-[Fe/H] relation in the solar neighborhood of our simulated Milky Way-type galaxy at z = 0. Figure is the same as Fig. 10 of Kobayashi and Nakasato (2011) but with a newer simulation. The color indicates the number of stars in logarithmic scale. The observational data sources are: Bensby et al. (2004, open and filled triangles) for thickand thindisk stars, respectively, Spite et al. (2005, open circles), Zhao et al. (2016, filled circles), Amarsi et al. (2019, crosses)

works with a much smaller sample but careful analysis (Fuhrmann 1998; Bensby et al. 2003). The advantage of the APOGEE survey was its large dynamic range, and the change of the bimodality depending on the location within the galaxy is also clearly shown. Kobayashi (2016) showed a similar change in the simulated galaxy, and Vincenzo and Kobayashi (2020) showed a comparison to APOGEE DR16. Our simulated galaxy predicts this bimodality not only for [α/Fe] ratios but also for most of elements from He to U; some elements even show trimodality (Kobayashi (2022a); in prep.). At [Fe/H]  −1 Fig. 22 also shows a significant number of stars with low [α/Fe] ratios, which is caused by local enrichment from SNe Ia. Note that the SN Ia rate becomes almost zero from the stars below [Fe/H] = −1.1 in the adopted SN Ia model. Nonetheless, stars formed in less-dense regions can have low [α/Fe] caused by SNe Ia or by low-mass Type II supernovae (13 − 15 M ); this effect is important also for dSph galaxies. The number of such low-α stars increases with sub-Ch mass SNe Ia (Kobayashi, in prep.). The origin of the scatter/bimodality is discussed in Kobayashi (2014). In chemodynamical simulations, it is possible to trace back the formation place of star particles. As a result, stars formed in merging galaxies found to have old age, low metallicity, high [α/Fe] ratios, and relatively low [(Na,Al,Cu)/Fe] ratios, while stars formed in situ show a tighter age-metallicity relation and low [α/Fe] ratios. There is no age-metallicity relation for the stars that migrated, as expected. In

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cosmological zoom-in simulations, stars tend to migrate outward by a few kpc, which flatten the metallicity gradient by ∼0.05 dex/kpc. Radial flows steepen the gradient as chemically enriched gas moves inward, but the average velocity is found to be only ∼0.7 km/s (Vincenzo and Kobayashi 2020). Figure 23 shows the [Eu/Fe] ratios in the solar neighborhood of our simulated galaxy but with switching the r-process sites. With only AGB stars (panel a), Eu is not sufficiently produced. Different from one-zone GCE models, AGB contribution can be seen with a large scatter at low metallicity. With ECSNe or ν-driven winds (panel b), Eu production is not increased enough. With NSMs (panel c), it is possible to reproduce the solar Eu/Fe ratios at the solar metallicity, but the scatter is too large at [Fe/H]  0. Unlike one-zone GCE models, there are a small number of stars that have high [Eu/Fe] ratios at low metallicities due to the inhomogeneous enrichment. The scatter can be much reduced with MRSNe (panel d) as they occur at a very short timescale. Since only a small fraction of core-collapse supernovae produce both Eu and Fe, the small scatter still remains, consistent with observations. Very high [Eu/Fe] ratios were reported for ultrafaint dSphs (Ji et al. 2016), which might be caused by local enrichment with a NSM with no Fe production (and no Zn enhancement unlike Yong et al. (2021)’s star). The effect of supernova kick in binary neutron star systems may help (van de Voort et al. 2022). Although

Fig. 23 Distribution of [Eu/Fe] against [Fe/H] for the star particles in the solar neighborhood of our simulated Milky Way-type galaxy at z = 0. The panels in order show control, ECSNe+ν-driven winds, NSMs, and MRSNe. The observational data sources are Hansen et al. (2016, red squares), Roederer et al. (2014, orange circles), Zhao et al. (2016, green triangles), and Buder et al. (2018, GALAH DR2, cyan contours). The contours show 10, 50, and 100 stars per bin. The color bar shows the linear number per bin of simulation data. (Figure is taken from Haynes and Kobayashi 2019)

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supernova kick is included in the delay-time distributions from BPSs, it is difficult to include it in the subgalactic scale. In dSph galaxies, star formation takes place slowly, chemical enrichment proceeds inefficiently, and thus the inhomogeneous enrichment effect becomes even more important. Higher resolution of simulations with a new formalism/code will be required.

Extra-galactic Archaeology Elemental abundances and isotopic ratios provide additional constrains on the timescales of formation and evolution of galaxies. In galaxies beyond the Local Group, it is not usually possible to resolve stars or HII regions, and metallicities of stellar populations (Multiple lines should be used to break the age-metallicity degeneracy) or ISM are estimated from absorption or emission lines in integrated spectra. Thanks to MOS surveys since SDSS, the sample is greatly increased, and thanks to IFU surveys since SAURON, the spatial (projected) distributions of metallicities are also obtained. Note that the methods for obtaining the absolute values of physical quantities are still debated (Worthey et al. 1992; Conroy 2013; Maiolino and Mannucci 2019; Kewley et al. 2019). For stellar populations, α/Fe ratios have been used to estimate the formation timescale of early-type galaxies (Thomas et al. 2005; Kriek et al. 2016), while Fe is not accessible in star-forming galaxies, and instead CNO abundances can be used (Vincenzo and Kobayashi 2018b). The James Webb Space Telescope (JWST) will push these observations toward higher redshifts, and the wavelength coverage of which will allow us to measure CNO abundance simultaneously (Fig. 24). More elements are available for X-ray hot gas or quasar absorption line systems, although neutron-capture elements are out of reach. Isotopic ratios of light elements and some light element abundances (e.g., F) are also estimated with Atacama Large Millimeter Array (ALMA). Figure 25 shows the evolution of stellar luminosities and metallicities in a cosmological simulation. More massive galaxies tend to have higher metallicities, which is called mass-metallicity relations (MZR). As already discussed, the origin of these relations are mass-dependent Galactic winds by supernova feedback (Kobayashi et al. 2007; Taylor et al. 2020). More intense star formation also happens in the simulated massive galaxies. An IMF variation is also possible (Kobayashi 2010). Metal loss by AGN feedback is not significant (Taylor and Kobayashi 2015a). Various ways are possible to calculate the abundance ratio of the stellar population (or ISM) of a galaxy:

log X/Y = log

 N

MX /



 MY

, log

N

 N

M



   MX  MX / M , M log M, / MY MY N

N







   LMX  LMY  MX   MX L / L , or L log L, log / , log / M M MY MY N

N

N

N

N

N

N

(19)

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Fig. 24 (Left panel) The first NIRCam/JWST image of a cluster of galaxy SMAC0723 at z = 0.39 but showing much higher redshift galaxies too, released on 11 July 2022 (figure is taken from https://webbtelescope.org). (Right panel) Emission lines that can be obtained with NIRSpec/JWST as a function of wavelength and redshift

Fig. 25 The time evolution of our cosmological simulation in a periodic box 50 h−1 Mpc on a side. We show the projected stellar V -luminosity (upper panels) and gas metallicity log Zg /Z (lower panels)

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where N is the number of stars/star particles (or HII regions/gas particles). The first three are mass-weighted, but in order to compare with observational data, it should be weighted by luminosity L. For stellar populations, we use V-band luminosity where Mgb line (5143–5206 Å) is located, and these definitions give slightly different results. For the ISM, the star formation rate of each gas particle can be used for weighting since it correlates with Hα emissions. This weighting gives very different results as there are many non-star-forming gas particles in galaxies (see Fig. 19 of Kobayashi et al. 2007). It is not obvious which is the best definition. Within galaxies, central parts are more metal-rich than outskirts, which is called metallicity radial gradients. The origin of these gradients is the inside-out formation, where star formation starts from the center with a higher efficiency and more chemical enrichment cycles. The star formation is more intense at the center in the simulated galaxies. The star formation duration is not necessarily longer in the center if quenching happens also inside out by AGN feedback. Radial flows steepen the gradients, while galaxy mergers flatten the gradients (Kobayashi 2004). Redistribution of metals by AGN feedback may be important for gas-phase metallicity gradients (Taylor and Kobayashi 2017). Simulations predict a significant scatter of gradients at a given mass (Kobayashi 2004; Taylor and Kobayashi 2017), while the average of the gradients become steeper at higher redshifts (Pilkington et al. (2012); Kobayashi & Taylor, in prep.). The metallicity gradient is defined in a log-scale for stellar populations but in a linear scale for gas-phase, and their example values are: Δ log Z∗ /Δ log r ∼ 0.3

(20)

for nearby early-type galaxies and ΔZg /Δr ∼ 0.05 [dex/kpc]

(21)

for nearby disks, where r is the projected, galactocentric radius. These definitions give a good fit to slit observations of nearby galaxies (e.g., Zaritsky et al. 1994; Kobayashi and Arimoto 1999) that cover a large radius (r  2re ), where re is the effective radius at which a half of stellar luminosity is enclosed. Often, the central part (r  1 kpc) is excluded from the fitting because it shows a flattening due to a limited seeing in observations or due to a limited spatial resolution in simulations. The outer part also shows a sudden decrease in metallicity or an increase due to satellite galaxies, which is also excluded from the fitting. However, recent analysis of IFU surveys stellar metallicities is also measured against linear r, and it is important to clarify the definition. Our cosmological simulations also predict elemental abundances and isotopic ratios. The left panel of Fig. 26 shows theoretical predictions of CNO abundance ratios for simulated disk galaxies that have different star formation timescales. These galaxies are chosen from a cosmological simulation, which is run by a Gadget3-based code that includes detailed chemical enrichment (Kobayashi et al. 2007). In the simulation, C is mainly produced from low-mass stars (4 M ), N is from

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Fig. 26 (Left) Evolution of gas-phase CNO abundance ratios in our simulated disk galaxies with different star formation timescales, taken from a cosmological simulation. (Figure is taken from Vincenzo and Kobayashi 2018b). (Right) Evolution of the gas-phase N/O-O/H relation of our simulated galaxies in a cosmological simulation. The gray bar indicates the compilation of observational data. (Figure is taken from Vincenzo et al. 2019)

intermediate-mass stars (4 M ) as a primary process (Kobayashi et al. 2011a), and O is from massive stars (13 M ). C and N are also produced by massive stars; the N yield depends on the metallicity as a secondary process and can be greatly enhanced by stellar rotation (as for F). In the nearby universe, the N/O-O/H relation is known for stellar and ISM abundances, which shows a plateau (N/O ∼ −1.6) at low metallicities and a rapid increase toward higher metallicities. Damped Lyα systems at higher redshifts also roughly follow the same plateau. This relation was interpreted as the necessity of rotating massive stars by Chiappini et al. (2006). However, this should be studied with hydrodynamical simulations including detailed chemical enrichment, and Kobayashi (2014) first showed the N/O-O/H relation in a chemodynamical simulation. Vincenzo and Kobayashi (2018a) showed that both the global relation, which is obtained for average abundances of the entire galaxies, and the local relation, which is obtained for spatially resolved abundances from IFU data, can be reproduced by the inhomogeneous enrichment from AGB stars. Since N yield increases at higher metallicities, the global relation originates from the mass-metallicity relation of galaxies, while the local relation is caused by radial metallicity gradients within galaxies. Moreover, the right panel of Fig. 26 shows a theoretical prediction on the time evolution of the N/O-O/H relation, where galaxies evolve along the relation. Recent observation with KMOS on VLT confirmed a near redshift-invariant N/O-O/H relation (Hayden-Pawson et al. 2021, KLEVER survey). Figure 27 shows [α/Fe] ratios of stellar populations as a function of galaxy mass for our simulated galaxies with and without AGN feedback. Without AGN feedback (red diamonds), since star formation lasts longer in massive galaxies with a deeper potential well, [α/Fe] ratios become lower in massive galaxies, which is the opposite

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Fig. 27 Stellar [O/Fe] ratios as a function of central velocity dispersion, which is a proxy of galaxy mass, for our simulated galaxies with (black asterisks) and without (red diamonds) AGN feedback at z = 0. The arrows indicate the differences in two cosmological simulations from the same initial conditions. The solid lines indicate the observed relations. (Figure is taken from Taylor and Kobayashi 2015b)

compared with the observations (solid lines). With AGN feedback (black asterisks), star formation can be suppressed before the SN Ia enrichment becomes dominant in massive galaxies, so [α/Fe] ratios can stay high. However, the scatter of [α/Fe] ratios at a given galaxy mass is still larger than observed. [α/Fe] ratios become higher at higher redshifts in the simulations, but not as much as observed (e.g., Kriek et al. 2016). Some cosmological simulations claimed that they could reproduce the observed [α/Fe]-mass relation at z = 0, but with an ad hoc SN Ia model. Some even introduced a variation in the IMF or in the binary fraction. However, the [α/Fe] problem should be discussed with a chemodynamical model that is based on nuclear astrophysics and that can reproduce the observed elemental abundances both in the Milky Way and dSph galaxies. We do not have such a model yet.

Conclusions and Future Prospects Thanks to the long-term collaborations between nuclear and astrophysics, we have good understanding on the origin of elements (except for the elements around Ti and a few neutron-capture elements such as Au). Inhomogeneous enrichment is extremely important for interpreting the elemental abundance trends. It can reproduce the observed N/O-O/H relation only with AGB stars and supernovae (Fig. 26), but not the observed r-process abundances only with NSMs (Fig. 23); an r-process associated with core-collapse supernovae such as magneto-rotational hypernovae is required (Fig. 11), although the explosion mechanism is unknown. It is necessary to run chemodynamical simulations from cosmological initial conditions, including detailed chemical enrichment based on nuclear astrophysics.

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The impact of stellar rotation, binaries, and magnetic fields during stellar evolution and the multidimensional effects of supernova explosions in nucleosynthesis should be investigated further. If Wolf-Rayet stars are producing heavy elements on a very short timescale, it might be hard to find very metal-poor or Population III (and dust-free) galaxies at very high redshifts even with JWST. In that case the elemental abundances can be quite different in very high redshift galaxies (e.g., high (C,N)/O; Fig. 9). Although there is no observational evidence, pair-instability supernovae could also cause very different abundance pattern (e.g., high (Si,S)/O; Fig. 10). Finally, some metals are locked in a solid state – it is also important to calculate element-by-element dust formation, growth, and destruction, as well as the detailed chemical enrichment. Galactic archaeology is a powerful approach for reconstructing the formation history of the Milky Way and its satellite galaxies. APOGEE and HERMES-GALAH surveys have provided homogeneous datasets of many elemental abundances that can be statistically compared with chemodynamical simulations. Future surveys with WEAVE and 4MOST will provide more. Having said that, the number of EMP stars will not be increased so much in these surveys, and a target survey such as the SkyMapper EMP survey is also needed, in particular for constraining the early chemical enrichment from the first stars. It is important to increase spectral resolution and wavelength coverage (including UV), to obtain more accurate abundances of more elements (namely, neutron-capture elements). Reflecting the difference in the formation timescale, elemental abundances depend on the location within galaxies. Although this dependence has been explored toward the Galactic bulge by APOGEE, the dependence at the outer disk is still unknown, which requires 8m-class telescopes such as the PFS on Subaru telescope. Despite the limited spectral resolution of the PFS, α/Fe and a small number of elements will be available. The PFS will also be able to explore the α/Fe bimodality in M31; it is not yet known if M31 has a similar α/Fe dichotomy or not. The next step will be to apply the Galactic archaeology approach to external or distant galaxies. Although it became possible to map metallicity, some elemental abundances, and kinematics within galaxies with IFU, the sample and/or spatial resolution is still limited even with JWST. Integrated physical quantities over galaxies, or stacked quantities at a given mass bin, will also be useful, which can be done with the same MOS developed for Galactic archaeology (although optimal spectral resolutions and wavelength coverages are different). Spectroscopic surveys with 8m-class telescopes will be useful, and it is a matter of urgency to establish the analysis methods to obtain absolute values of metallicities and elemental abundances from the observational data. In addition, ALMA has opened a new window for elemental abundances and isotopic ratios in high-redshift galaxies. Extra-galactic archaeology will become popular in the coming years. Acknowledgments We thank D. Yong, F. Vincenzo, A. Karakas, M. Lugaro, N. Tominaga, S.-C. Leung, M. Ishigaki, K. Nomoto, L. Kewley, R. Maiolino, A. Bunker for fruitful discussion and V. Springel for providing Gadget-3. CK acknowledge funding from the UK Science and Technology Facility Council through grant ST/M000958/1, ST/R000905/1, ST/V000632/1. The work was also funded by a Leverhulme Trust Research Project Grant on “Birth of Elements.”

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References A.M. Amarsi, P.E. Nissen, Á. Skúladóttir, A&A 630, A104 (2019). https://doi.org/10.1051/00046361/201936265 A. Arcones, H.T. Janka, L. Scheck, A&A 467, 1227 (2007). https://doi.org/10.1051/0004-6361: 20066983 M. Arnaboldi, S. Bhattacharya, O. Gerhard et al., (2022). https://doi.org/10.1051/0004-6361/ 202244258 M. Asplund, N. Grevesse, A.J. Sauval, P. Scott, ARAA 47, 481 (2009). https://doi.org/10.1146/ annurev.astro.46.060407.145222 J. Audouze, J. Silk, ApJ Lett. 451, L49 (1995). https://doi.org/10.1086/309687 Z. Barkat, G. Rakavy, N. Sack, Phys. Rev. Lett. 18, 379 (1967). https://doi.org/10.1103/ PhysRevLett.18.379 T. Bensby, S. Feltzing, I. Lundström, A&A 410, 527 (2003). https://doi.org/10.1051/0004-6361: 20031213 T. Bensby, S. Feltzing, I. Lundström, A&A 415, 155 (2004). https://doi.org/10.1051/0004-6361: 20031655 S. Blondin, E. Bravo, F.X. Timmes, L. Dessart, D.J. Hillier, A&A 660, A96 (2022). https://doi.org/ 10.1051/0004-6361/202142323 C.B. Brook, G.S. Stinson, B.K. Gibson et al., MNRAS 426, 690 (2012). https://doi.org/10.1111/j. 1365-2966.2012.21738.x G.E. Brown, A. Heger, N. Langer et al., New Astron. 6, 457 (2001). https://doi.org/10.1016/S13841076(01)00077-X T. Buck, A. Obreja, A.V. Macciò et al., MNRAS 491, 3461 (2020). https://doi.org/10.1093/mnras/ stz3241 S. Buder, M. Asplund, L. Duong et al., MNRAS 478, 4513 (2018). https://doi.org/10.1093/mnras/ sty1281 A. Burkert, G. Hensler, in Nuclear Astrophysics, ed. by W. Hillebrandt, R. Kuhfuss, E. Mueller, J.W. Truran, vol. 287 (1987), p. 159. https://doi.org/10.1007/BFb0016576 A. Burrows, D. Vartanyan, Nature 589, 29 (2021). https://doi.org/10.1038/s41586-020-03059-w M. Busso, R. Gallino, G.J. Wasserburg, ARAA 37, 239 (1999). https://doi.org/10.1146/annurev. astro.37.1.239 M. Cappellari, R.M. McDermid, K. Alatalo et al., MNRAS 432, 1862 (2013). https://doi.org/10. 1093/mnras/stt644 R. Cayrel, E. Depagne, M. Spite et al., A&A 416, 1117 (2004). https://doi.org/10.1051/0004-6361: 20034074 R. Cen, J.P. Ostriker, ApJ Lett. 519, L109 (1999). https://doi.org/10.1086/312123 G. Cescutti, C. Chiappini, A&A 565, A51 (2014). https://doi.org/10.1051/0004-6361/201423432 G. Cescutti, C. Kobayashi, A&A 607, A23 (2017). https://doi.org/10.1051/0004-6361/201731398 G. Chabrier, Publ. Astron. Soc. Pac. 115, 763 (2003). https://doi.org/10.1086/376392 R. Chary, E. Berger, L. Cowie, ApJ 671, 272 (2007). https://doi.org/10.1086/522692 C. Chiappini, R. Hirschi, G. Meynet et al., A&A 449, L27 (2006). https://doi.org/10.1051/00046361:20064866 C. Chiappini, F. Matteucci, R. Gratton, ApJ 477, 765 (1997). https://doi.org/10.1086/303726 C. Conroy, ARAA 51, 393 (2013). https://doi.org/10.1146/annurev-astro-082812-141017 S. Cristallo, L. Piersanti, O. Straniero et al., ApJS 197, 17 (2011). https://doi.org/10.1088/00670049/197/2/17 D.J. Croton, V. Springel, S. White et al., MNRAS 365, 11 (2006). https://doi.org/10.1111/j.13652966.2005.09675.x C. Dalla Vecchia, J. Schaye, MNRAS 387, 1431 (2008). https://doi.org/10.1111/j.1365-2966.2008. 13322.x C. Dalla Vecchia, J. Schaye, MNRAS 426, 140 (2012). https://doi.org/10.1111/j.1365-2966.2012. 21704.x

88 Chemo-dynamical Evolution of Galaxies

3255

R. Davé, D. Anglés-Alcázar, D. Narayanan et al., MNRAS 486, 2827 (2019). https://doi.org/10. 1093/mnras/stz937 E. De Donder, D. Vanbeveren, New Astron. Rev. 48, 861 (2004). https://doi.org/10.1016/j.newar. 2004.07.001 C.L. Doherty, P. Gil-Pons, H.H.B. Lau, J.C. Lattanzio, L. Siess, MNRAS 437, 195 (2014). https:// doi.org/10.1093/mnras/stt1877 C.L. Doherty, P. Gil-Pons, L. Siess, J.C. Lattanzio, H.H.B. Lau, MNRAS 446, 2599 (2015). https:// doi.org/10.1093/mnras/stu2180 K. Dolag, E. Mevius, R.-S. Remus, Galaxies 5, 35 (2017). https://doi.org/10.3390/galaxies5030035 S.P. Driver, S.K. Andrews, E. da Cunha et al., MNRAS 475, 2891 (2018). https://doi.org/10.1093/ mnras/stx2728 Y. Dubois, J. Devriendt, A. Slyz, R. Teyssier, MNRAS 420, 2662 (2012). https://doi.org/10.1111/ j.1365-2966.2011.20236.x Y. Dubois, S. Peirani, C. Pichon et al., MNRAS 463, 3948 (2016). https://doi.org/10.1093/mnras/ stw2265 C. Federrath, S. Sur, D.R.G. Schleicher, R. Banerjee, R.S. Klessen, ApJ 731, 62 (2011). https:// doi.org/10.1088/0004-637X/731/1/62 C.G. Few, S. Courty, B.K. Gibson, L. Michel-Dansac, F. Calura, MNRAS 444, 3845 (2014). https:// doi.org/10.1093/mnras/stu1709 A.S. Font, I.G. McCarthy, R. Poole-Mckenzie et al., MNRAS 498, 1765 (2020). https://doi.org/10. 1093/mnras/staa2463 M. Franco, K.E.K. Coppin, J.E. Geach et al., Nat. Astron. 5, 1240 (2021). https://doi.org/10.1038/ s41550-021-01515-9 U. Frischknecht, R. Hirschi, M. Pignatari et al., MNRAS 456, 1803 (2016). https://doi.org/10. 1093/mnras/stv2723 K. Fuhrmann, A&A 338, 161 (1998) Gaia Collaboration, A. Recio-Blanco, G. Kordopatis et al., (2022). arXiv e-prints, arXiv:2206.05534 R.J.J. Grand, F.A. Gómez, F. Marinacci et al., MNRAS 467, 179 (2017). https://doi.org/10.1093/ mnras/stx071 A. Grichener, C. Kobayashi, N. Soker, (2021). https://doi.org/10.3847/2041-8213/ac4f68 V. Grisoni, F. Matteucci, D. Romano, X. Fu, MNRAS 489, 3539 (2019). https://doi.org/10.1093/ mnras/stz2428 V. Grisoni, E. Spitoni, F. Matteucci et al., MNRAS 472, 3637 (2017). https://doi.org/10.1093/ mnras/stx2201 M.L.P. Gunawardhana, A.M. Hopkins, R.G. Sharp et al., MNRAS 415, 1647 (2011). https://doi. org/10.1111/j.1365-2966.2011.18800.x F. Haardt, P. Madau, ApJ 461, 20 (1996). https://doi.org/10.1086/177035 T.T. Hansen, J. Andersen, B. Nordström et al., A&A 586, A160 (2016). https://doi.org/10.1051/ 0004-6361/201527235 M.R. Hayden, J. Bovy, J.A. Holtzman et al., ApJ 808, 132 (2015). https://doi.org/10.1088/0004637X/808/2/132 C. Hayden-Pawson, M. Curti, R. Maiolino et al., (2021). https://doi.org/10.1093/mnras/stac584 C.J. Haynes, C. Kobayashi, MNRAS 483, 5123 (2019). https://doi.org/10.1093/mnras/sty3389 F. Herwig, ARAA 43, 435 (2005). https://doi.org/10.1146/annurev.astro.43.072103.150600 R. Hirschi, A&A 461, 571 (2007). https://doi.org/10.1051/0004-6361:20065356 P.F. Hopkins, L. Hernquist, T.J. Cox et al., ApJS 163, 1 (2006). https://doi.org/10.1086/499298 P.F. Hopkins, A. Wetzel, D. Kereš et al., MNRAS 477, 1578 (2018a). https://doi.org/10.1093/ mnras/sty674 P.F. Hopkins, A. Wetzel, D. Kereš et al., MNRAS 480, 800 (2018b). https://doi.org/10.1093/mnras/ sty1690 M.N. Ishigaki, N. Tominaga, C. Kobayashi, K. Nomoto, ApJ 857, 46 (2018). https://doi.org/10. 3847/1538-4357/aab3de

3256

C. Kobayashi and P. Taylor

G. Israelian, R.J. García López, R. Rebolo, ApJ 507, 805 (1998). https://doi.org/10.1086/306351 H.-T. Janka, Annu. Rev. Nucl. Part. Sci. 62, 407 (2012). https://doi.org/10.1146/annurev-nucl102711-094901 A.P. Ji, A. Frebel, A. Chiti, J.D. Simon, Nature 531, 610 (2016). https://doi.org/10.1038/ nature17425 O. Just, A. Bauswein, R. Ardevol Pulpillo, S. Goriely, H.T. Janka, MNRAS 448, 541 (2015). https://doi.org/10.1093/mnras/stv009 A.I. Karakas, MNRAS 403, 1413 (2010). https://doi.org/10.1111/j.1365-2966.2009.16198.x A.I. Karakas, J.C. Lattanzio, PASA 31, e030 (2014). https://doi.org/10.1017/pasa.2014.21 A.I. Karakas, M. Lugaro, ApJ 825, 26 (2016). https://doi.org/10.3847/0004-637X/825/1/26 N. Katz, ApJ 391, 502 (1992). https://doi.org/10.1086/171366 D. Kawata, B.K. Gibson, MNRAS 340, 908 (2003). https://doi.org/10.1046/j.1365-8711.2003. 06356.x A.J. Kemp, A.I. Karakas, A.R. Casey, C. Kobayashi, R.G. Izzard, MNRAS 509, 1175 (2022). https://doi.org/10.1093/mnras/stab3103 L.J. Kewley, D.C. Nicholls, R.S. Sutherland, ARAA 57, 511 (2019). https://doi.org/10.1146/ annurev-astro-081817-051832 C. Kobayashi, PhD Thesis, University of Tokyo, 2002 C. Kobayashi, MNRAS 347, 740 (2004). https://doi.org/10.1111/j.1365-2966.2004.07258.x C. Kobayashi, MNRAS 361, 1216 (2005). https://doi.org/10.1111/j.1365-2966.2005.09248.x C. Kobayashi, Hunting for the dark: the hidden side of galaxy formation, in American Institute of Physics Conference Series, vol. 1240. ed. by V.P. Debattista, C.C. Popescu (2010), pp. 123–126. https://doi.org/10.1063/1.3458465 C. Kobayashi, in Setting the Scene for Gaia and LAMOST, vol. 298, ed. by S. Feltzing, G. Zhao, N.A. Walton, P. Whitelock (2014), pp. 167–178. https://doi.org/10.1017/S1743921313006339 C. Kobayashi, in The General Assembly of Galaxy Halos: Structure, Origin and Evolution, vol. 317, ed. by A. Bragaglia, M. Arnaboldi, M. Rejkuba, D. Romano (2016), pp. 57–63. https:// doi.org/10.1017/S1743921315009783 C. Kobayashi, (2022a). https://doi.org/10.1017/S1743921322001132 C. Kobayashi, Exp. Astron. (2022b). https://doi.org/10.1007/s10686-022-09862-9 C. Kobayashi, N. Arimoto, ApJ 527, 573 (1999). https://doi.org/10.1086/308092 C. Kobayashi, M.N. Ishigaki, N. Tominaga, K. Nomoto, ApJ Lett. 785, L5 (2014). https://doi.org/ 10.1088/2041-8205/785/1/L5 C. Kobayashi, A.I. Karakas, M. Lugaro, ApJ 900, 179 (2020a). https://doi.org/10.3847/1538-4357/ abae65 C. Kobayashi, A.I. Karakas, H. Umeda, MNRAS 414, 3231 (2011a). https://doi.org/10.1111/j. 1365-2966.2011.18621.x C. Kobayashi, S.-C. Leung, K. Nomoto, ApJ 895, 138 (2020b). https://doi.org/10.3847/1538-4357/ ab8e44 C. Kobayashi, N. Nakasato, ApJ 729, 16 (2011). https://doi.org/10.1088/0004-637X/729/1/16 C. Kobayashi, K. Nomoto, ApJ 707, 1466 (2009). https://doi.org/10.1088/0004-637X/707/2/1466 C. Kobayashi, K. Nomoto, I. Hachisu, ApJ Lett. 804, L24 (2015). https://doi.org/10.1088/20418205/804/1/L24 C. Kobayashi, V. Springel, S.D.M. White, MNRAS 376, 1465 (2007). https://doi.org/10.1111/j. 1365-2966.2007.11555.x C. Kobayashi, N. Tominaga, K. Nomoto, ApJ Lett. 730, L14 (2011b). https://doi.org/10.1088/ 2041-8205/730/2/L14 C. Kobayashi, T. Tsujimoto, K. Nomoto, ApJ 539, 26 (2000). https://doi.org/10.1086/309195 C. Kobayashi, T. Tsujimoto, K. Nomoto, I. Hachisu, M. Kato, ApJ Lett. 503, L155 (1998). https:// doi.org/10.1086/311556 C. Kobayashi, H. Umeda, K. Nomoto, N. Tominaga, T. Ohkubo, ApJ 653, 1145 (2006). https://doi. org/10.1086/508914 C. Kobayashi, I. Mandel, K. Belczynski et al., (2022). https://doi.org/10.3847/2041-8213/acad82

88 Chemo-dynamical Evolution of Galaxies

3257

T. Kodama, N. Arimoto, A&A 320, 41 (1997) R.P. Kraft, C. Sneden, G.H. Smith et al., AJ 113, 279 (1997). https://doi.org/10.1086/118251 M. Kriek, C. Conroy, P.G. van Dokkum et al., Nature 540, 248 (2016). https://doi.org/10.1038/ nature20570 P. Kroupa, Pathways through an eclectic universe, in Astronomical Society of the Pacific Conference Series, vol. 390, ed. by J.H. Knapen, T.J. Mahoney, A. Vazdekis (2008), p. 3 P. Kroupa, C.A. Tout, G. Gilmore, MNRAS 262, 545 (1993). https://doi.org/10.1093/mnras/262.3. 545 M.R. Krumholz, N.Y. Gnedin, ApJ 729, 36 (2011). https://doi.org/10.1088/0004-637X/729/1/36 S.-C. Leung, K. Nomoto, ApJ 861, 143 (2018). https://doi.org/10.3847/1538-4357/aac2df S.-C. Leung, K. Nomoto, ApJ 888, 80 (2020). https://doi.org/10.3847/1538-4357/ab5c1f M. Limongi, A. Chieffi, ApJS 237, 13 (2018). https://doi.org/10.3847/1538-4365/aacb24 K. Lind, T. Nordlander, A. Wehrhahn et al., A&A 665, A33 (2022). https://doi.org/10.1051/00046361/202142195 P. Madau, M. Dickinson, ARAA 52, 415 (2014). https://doi.org/10.1146/annurev-astro-081811125615 P. Madau, M.J. Rees, ApJ Lett. 551, L27 (2001). https://doi.org/10.1086/319848 K. Maeda, K. Nomoto, ApJ 598, 1163 (2003). https://doi.org/10.1086/378948 J. Magorrian, S. Tremaine, D. Richstone et al., AJ 115, 2285 (1998). https://doi.org/10.1086/ 300353 R. Maiolino, F. Mannucci, Astron. Astrophys. Rev. 27, 3 (2019). https://doi.org/10.1007/s00159018-0112-2 F. Matteucci, The Chemical Evolution of the Galaxy, vol. 253 (2001). https://doi.org/10.1007/97894-010-0967-6 F. Matteucci, Astron. Astrophys. Rev. 29, 5 (2021). https://doi.org/10.1007/s00159-021-00133-8 F. Matteucci, E. Brocato, ApJ 365, 539 (1990). https://doi.org/10.1086/169508 N. Mennekens, D. Vanbeveren, A&A 564, A134 (2014). https://doi.org/10.1051/0004-6361/ 201322198 N. Mennekens, D. Vanbeveren, A&A 589, A64 (2016). https://doi.org/10.1051/0004-6361/ 201628193 G. Meynet, A. Maeder, A&A 390, 561 (2002). https://doi.org/10.1051/0004-6361:20020755 J.C. Mihos, L. Hernquist, ApJ 464, 641 (1996). https://doi.org/10.1086/177353 P. Mösta, L.F. Roberts, G. Halevi et al., ApJ 864, 171 (2018). https://doi.org/10.3847/1538-4357/ aad6ec D. Mukherjee, G.V. Bicknell, A.Y. Wagner, R.S. Sutherland, J. Silk, MNRAS 479, 5544 (2018). https://doi.org/10.1093/mnras/sty1776 J.F. Navarro, S.D.M. White, MNRAS 265, 271 (1993). https://doi.org/10.1093/mnras/265.2.271 J.F. Navarro, S.D.M. White, MNRAS 267, 401 (1994). https://doi.org/10.1093/mnras/267.2.401 N. Nishimura, T. Takiwaki, F.-K. Thielemann, ApJ 810, 109 (2015). https://doi.org/10.1088/0004637X/810/2/109 K. Nomoto, ApJ 322, 206 (1987). https://doi.org/10.1086/165716 K. Nomoto, K. Iwamoto, N. Nakasato et al., Nucl. Phys. A 621, 467 (1997). https://doi.org/10. 1016/S0375-9474(97)00291-1 K. Nomoto, C. Kobayashi, N. Tominaga, ARAA 51, 457 (2013). https://doi.org/10.1146/annurevastro-082812-140956 B.E.J. Pagel, Nucleosynthesis and Chemical Evolution of Galaxies. Cambridge University Press, Cambridge, UK (1997) R. Pakmor, A. Bauer, V. Springel, MNRAS 418, 1392 (2011). https://doi.org/10.1111/j.1365-2966. 2011.19591.x K. Pilkington, C.G. Few, B.K. Gibson et al., A&A 540, A56 (2012). https://doi.org/10.1051/00046361/201117466 A. Pillepich, V. Springel, D. Nelson et al., MNRAS 473, 4077 (2018). https://doi.org/10.1093/ mnras/stx2656

3258

C. Kobayashi and P. Taylor

L. Portinari, C. Chiosi, A. Bressan, A&A 334, 505 (1998) N. Prantzos, C. Abia, M. Limongi, A. Chieffi, S. Cristallo, MNRAS 476, 3432 (2018). https://doi. org/10.1093/mnras/sty316 N. Prantzos, M. Casse, E. Vangioni-Flam, ApJ 403, 630 (1993). https://doi.org/10.1086/172233 M. Reichert, M. Obergaulinger, M. Eichler, M.Á. Aloy, A. Arcones, MNRAS 501, 5733 (2021). https://doi.org/10.1093/mnras/stab029 I.U. Roederer, G.W. Preston, I.B. Thompson et al., AJ 147, 136 (2014). https://doi.org/10.1088/ 0004-6256/147/6/136 D. Romano, A.I. Karakas, M. Tosi, F. Matteucci, A&A 522, A32 (2010). https://doi.org/10.1051/ 0004-6361/201014483 D. Romano, F. Matteucci, Z.-Y. Zhang, R.J. Ivison, P. Ventura, MNRAS 490, 2838 (2019). https:// doi.org/10.1093/mnras/stz2741 M. Rowan-Robinson, S. Oliver, L. Wang et al., MNRAS 461, 1100 (2016). https://doi.org/10.1093/ mnras/stw1169 E.E. Salpeter, ApJ 121, 161 (1955). https://doi.org/10.1086/145971 C. Scannapieco, M. Wadepuhl, O.H. Parry et al., MNRAS 423, 1726 (2012). https://doi.org/10. 1111/j.1365-2966.2012.20993.x J. Schaye, R.A. Crain, R.G. Bower et al., MNRAS 446, 521 (2015). https://doi.org/10.1093/mnras/ stu2058 I.R. Seitenzahl, G. Cescutti, F.K. Röpke, A.J. Ruiter, R. Pakmor, A&A 559, L5 (2013). https://doi. org/10.1051/0004-6361/201322599 D.M. Siegel, J. Barnes, B.D. Metzger, Nature 569, 241 (2019). https://doi.org/10.1038/s41586019-1136-0 S.J. Smartt, ARAA 47, 63 (2009). https://doi.org/10.1146/annurev-astro-082708-101737 C. Sneden, J.J. Cowan, C. Kobayashi et al., ApJ 817, 53 (2016). https://doi.org/10.3847/0004637X/817/1/53 M. Spite, R. Cayrel, B. Plez et al., A&A 430, 655 (2005). https://doi.org/10.1051/0004-6361: 20041274 E. Spitoni, V. Silva Aguirre, F. Matteucci, F. Calura, V. Grisoni, A&A 623, A60 (2019). https:// doi.org/10.1051/0004-6361/201834188 V. Springel, MNRAS 401, 791 (2010). https://doi.org/10.1111/j.1365-2966.2009.15715.x V. Springel, T. Di Matteo, L. Hernquist, MNRAS 361, 776 (2005). https://doi.org/10.1111/j.13652966.2005.09238.x V. Springel, L. Hernquist, MNRAS 339, 289 (2003). https://doi.org/10.1046/j.1365-8711.2003. 06206.x V. Springel, N. Yoshida, S.D.M. White, New Astron. 6, 79 (2001). https://doi.org/10.1016/S13841076(01)00042-2 M. Steinmetz, E. Mueller, A&A 281, L97 (1994) R.S. Sutherland, M.A. Dopita, ApJS 88, 253 (1993). https://doi.org/10.1086/191823 P. Taylor, C. Kobayashi, MNRAS 442, 2751 (2014). https://doi.org/10.1093/mnras/stu983 P. Taylor, C. Kobayashi, MNRAS 452, L59 (2015a). https://doi.org/10.1093/mnrasl/slv087 P. Taylor, C. Kobayashi, MNRAS 448, 1835 (2015b). https://doi.org/10.1093/mnras/stv139 P. Taylor, C. Kobayashi, MNRAS 463, 2465 (2016). https://doi.org/10.1093/mnras/stw2157 P. Taylor, C. Kobayashi, MNRAS 471, 3856 (2017). https://doi.org/10.1093/mnras/stx1860 P. Taylor, C. Kobayashi, L.J. Kewley, MNRAS 496, 4433 (2020). https://doi.org/10.1093/mnras/ staa1904 R. Teyssier, A&A 385, 337 (2002). https://doi.org/10.1051/0004-6361:20011817 D. Thomas, C. Maraston, R. Bender, C. Mendes de Oliveira, ApJ 621, 673 (2005). https://doi.org/ 10.1086/426932 F.X. Timmes, S.E. Woosley, T.A. Weaver, ApJS 98, 617 (1995). https://doi.org/10.1086/192172 B.M. Tinsley, Fundam. Cosm. Phys. 5, 287 (1980) N. Tominaga, ApJ 690, 526 (2009). https://doi.org/10.1088/0004-637X/690/1/526 H. Umeda, K. Nomoto, Nature 422, 871 (2003). https://doi.org/10.1038/nature01571

88 Chemo-dynamical Evolution of Galaxies

3259

F. van de Voort, R. Pakmor, R. Bieri, R.J.J. Grand, MNRAS 512, 5258 (2022). https://doi.org/10. 1093/mnras/stac710 F. van de Voort, R. Pakmor, R.J.J. Grand et al., MNRAS 494, 4867 (2020). https://doi.org/10.1093/ mnras/staa754 P.G. van Dokkum, C. Conroy, Nature 468, 940 (2010). https://doi.org/10.1038/nature09578 P. Ventura, F. D’Antona, A&A 499, 835 (2009). https://doi.org/10.1051/0004-6361/200811139 F. Vincenzo, C. Kobayashi, MNRAS 478, 155 (2018a). https://doi.org/10.1093/mnras/sty1047 F. Vincenzo, C. Kobayashi, A&A 610, L16 (2018b). https://doi.org/10.1051/0004-6361/ 201732395 F. Vincenzo, C. Kobayashi, MNRAS 496, 80 (2020). https://doi.org/10.1093/mnras/staa1451 F. Vincenzo, C. Kobayashi, T. Yuan, MNRAS 488, 4674 (2019). https://doi.org/10.1093/mnras/ stz2065 A. Wallner, M.B. Froehlich, M.A.C. Hotchkis et al., Science 372, 742 (2021). https://doi.org/10. 1126/science.aax3972 S.H.J. Wallström, S. Muller, E. Roueff et al., A&A 629, A128 (2019). https://doi.org/10.1051/ 0004-6361/201935860 S. Wanajo, ApJ Lett. 770, L22 (2013). https://doi.org/10.1088/2041-8205/770/2/L22 S. Wanajo, H.-T. Janka, B. Müller, ApJ Lett. 767, L26 (2013). https://doi.org/10.1088/2041-8205/ 767/2/L26 S. Wanajo, K. Nomoto, H.T. Janka, F.S. Kitaura, B. Müller, ApJ 695, 208 (2009). https://doi.org/ 10.1088/0004-637X/695/1/208 S. Wanajo, Y. Sekiguchi, N. Nishimura et al., ApJ Lett. 789, L39 (2014). https://doi.org/10.1088/ 2041-8205/789/2/L39 E.X. Wang, P. Taylor, C. Federrath, C. Kobayashi, MNRAS 483, 4640 (2019). https://doi.org/10. 1093/mnras/sty3491 R. Weinberger, V. Springel, L. Hernquist et al., MNRAS 465, 3291 (2017). https://doi.org/10.1093/ mnras/stw2944 R.P.C. Wiersma, J. Schaye, B.D. Smith, MNRAS 393, 99 (2009). https://doi.org/10.1111/j.13652966.2008.14191.x C. Winteler, R. Käppeli, A. Perego et al., ApJ Lett. 750, L22 (2012). https://doi.org/10.1088/20418205/750/1/L22 T.E. Woods, B. Agarwal, V. Bromm et al., PASA 36, e027 (2019). https://doi.org/10.1017/pasa. 2019.14 S.E. Woosley, T.A. Weaver, ApJS 101, 181 (1995). https://doi.org/10.1086/192237 G. Worthey, S.M. Faber, J.J. Gonzalez, ApJ 398, 69 (1992). https://doi.org/10.1086/171836 D. Yong, C. Kobayashi, G.S. Da Costa et al., Nature 595, 223 (2021). https://doi.org/10.1038/ s41586-021-03611-2 D. Zaritsky, R.C.J. Kennicutt, J.P. Huchra, ApJ 420, 87 (1994). https://doi.org/10.1086/173544 Z.-Y. Zhang, D. Romano, R.J. Ivison, P.P. Papadopoulos, F. Matteucci, Nature 558, 260 (2018). https://doi.org/10.1038/s41586-018-0196-x G. Zhao, L. Mashonkina, H.L. Yan et al., ApJ 833, 225 (2016). https://doi.org/10.3847/1538-4357/ 833/2/225

Cosmic Radioactivity and Galactic Chemical Evolution

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Roland Diehl and Nikos Prantzos

Contents Cosmic Gas and the Role of Radioactivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling Compositional Evolution of Cosmic Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yields of Stable and Radioactive Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Interstellar Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of Our Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recent Progress in Modelling Chemical Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radioactive Isotopes and Compositional Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Long-Lived Radioactivities and Nucleocosmochronology . . . . . . . . . . . . . . . . . . . . . The Content of 26 Al and 60 Fe in the Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radioactive Nuclei in Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Fe and 244 Pu in Sediments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solar System Formation and Short-Lived Radioactivities . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3262 3264 3264 3271 3281 3286 3300 3304 3308 3308 3309 3319 3326 3328 3333 3335

Abstract

The description of the tempo-spatial evolution of the composition of cosmic gas on galactic scales is called ‘modelling galactic chemical evolution’. It aims to use knowledge about sources of nucleosynthesis and how they change the composition of interstellar gas, following the formation of stars and the

R. Diehl () Max Planck Institut für extraterrestrische Physik, Garching, Germany e-mail: [email protected] N. Prantzos Institut d’Astrophysique, Paris, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_107

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ejection of products from nuclear fusion during their evolution and terminating explosions. Sources of nucleosynthesis are diverse: Stars with hydrostatic nuclear burning during their evolution shed parts of the products in planetary nebulae, winds, and core-collapse supernovae. Binary interactions are important and lead to important sources, such as thermonuclear supernovae and kilonovae. Tracing ejecta from sources, with their different frequencies and environments, through the interstellar medium and successive star formation cycles is the goal of model descriptions. A framework that traces gas and stars through star formation, stellar evolution, enriched-gas ejections, and large-scale gas flows is formulated. Beyond illustrating the effects of different assumptions about nucleosynthesis sources and gas recycling, this allows us to interpret the large amount of observational data concerning the isotopic composition of stars, galaxies, and the interstellar medium. A variety of formalisms exist, from analytical through semi-analytical, numerical, or stochastic approaches, gradually making descriptions of compositional evolution of cosmic matter more realistic, teaching us about the astrophysical processes involved in this complex aspect of our universe. Radioactive isotopes add important ingredients to such modelling: The intrinsic clock of the radioactive decay process adds a new aspect to the modelling algorithms that leads to different constraints on the important unknowns of star formation activity and interstellar transports. Several prominent examples illustrate how modelling the abundances of radioactive isotopes and their evolutions has resulted in new lessons; among these are the galaxy-wide distribution of 26 Al and 60 Fe, the radioactive components of cosmic rays, the interpretations of terrestrial deposits of 60 Fe and 244 Pu, and the radioactive decay daughter isotopes that were found in meteorites and characterise the birth environment of our solar system.

Cosmic Gas and the Role of Radioactivities The composition of gas in galaxies evolves over cosmic times. Our goal is to understand this evolution. This requires an understanding of the processes that are agents to change such composition, by bringing new materials into this system or eliminating some out of it. Nucleosynthesis sources are the agents that bring in new materials, in the form of freshly produced isotopes. A substantial fraction of newly produced isotopes are radioactive, as a characteristic result of the nuclearreaction processes within such nucleosynthesis sources. The study of radioactive materials, therefore, offers a direct link to the sources of nucleosynthesis. This direct link is particularly important for short-lived radioisotopes, where the radioactive decay occurs within the environment of one single nucleosynthesis event only, so that it can be attributed to this one ejection event of new nuclei and be exploited as a diagnostic of the processes within and around such a specific source. Examples are the 56 Ni and 44 Ti decays that have been shown to power light curves of supernovae in the cases of SN1987A, Cas A, and SN2014J through measurement of the characteristic γ -ray lines for these radioisotopes and emissions in varieties of

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other electromagnetic windows from these same sources. Radioactive decay times in this case must be longer than the intrinsic ejection times of sources of order a day and, on the other hand, shorter than event frequencies in connected regions of space, where 107...8 y (10–100 Myrs) may be a useful guideline. There is the second aspect that supports the role of radioactivity studies in cosmic compositional evolution: Radioactive decay offers an intrinsic clock, based on the fundamental physical properties, and is largely independent of environmental circumstances. This adds the perspective to investigate the aspects of time in compositional evolution in an independent way, making use of this clock, rather than referring to concepts such as metallicity or [Fe/H] or [α/Fe] proxies that are often used to represent time in these studies. The radioactive decay is an independent physical process that only depends on the existence of some amount of a specific radioactive isotope and not on the complex mixing and transport processes that are described below as characteristic for compositional evolution of interstellar gas in a galaxy. Therefore, inclusion of radioactive isotopes within the framework of compositional evolution that otherwise traces stable isotopes and elements adds an important consistency check on the modelling, because the link of ejected radioisotope abundances to observations is systematically different while sources may be identical for specific isotope groups. In particular, radioactive isotopes with decay times in the range of 0.1 to 100 Gyrs will be useful here, as they cover time spans that are characteristic of compositional evolution. For radioactive isotopes with intermediate decay times in the range of several to about 100 Myrs, attribution to single specific sources will not be possible in general, while on the other hand the compositional evolution effects are localised in time and space. Therefore, compositional evolution modelling can be set up in a more specific and localised manner of sources of nucleosynthesis, and temporal signatures are predominantly ejecta flow and radioactive decay. This allows interesting studies of the larger-scale surroundings of nucleosynthesis sources and the transport of ejecta away from these and towards a mixed state; this mixed state must often be assumed in galactic composition evolution modelling and can be studied through such specific radioisotopes. Examples are 60 Fe as found in the wider galaxy, in cosmic rays, and on oceanic crust deposits on Earth, and a set of radioactive isotopes identified to have been present in the early solar system, from measuring their characteristic daughter products in meteorites. This chapter first presents the fundamentals and concepts of modelling compositional evolution of gas in systems, such as galaxies, as have been developed over more than 40 years for interpreting stellar populations and their elemental abundances. Special aspects of radioactivities will be addressed along the way, where appropriate. A few recently paths towards improved concepts of modelling compositional evolution have emerged, as higher resolution in space, time, and source variety is being achieved. Then, the specifics of radioactive isotopes are discussed, with a few examples and their lessons on the various aspects of compositional evolution. This chapter concludes with prospects as well as limitations for future developments (see also Diehl et al. 2018, for a broader book review of astrophysics with radioactive isotopes).

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Modelling Compositional Evolution of Cosmic Gas Concept and Equations Nucleosynthesis events produce new isotopes, which are mixed with ambient gas and then end up in new generations of stars, which again lead to nucleosynthesis events (see Fig. 1). This cycle began from first stars (Population III stars) that were created from almost metal-free primordial gas and has since continued by forming stars until today. Stars with ages comparable to, or much younger than, the Sun (4.5 Gy) are called Population I. They are the only stellar population which contains massive (hence short-lived) stars, which are still observable today. Stars of the galactic halo typically are much older (>10 Gy) and are called Population II. Star formation, evolution, and nucleosynthesis all vary with changing composition, measured through the metal content or metallicity. Stars are formed across a wide range of masses, from 0.1 to 100 M . Stars evolve on different timescales, depending on the initial mass, and with different internal processes. Then, binary interactions in multiple systems change such evolution and lead to entirely new sources of nucleosynthesis. Different types of nucleosynthesis sources arise from a

Fig. 1 Illustration of the cycle of matter. Stars form from molecular clouds and eventually return gas enriched with nucleosynthesis products into interstellar space

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stellar population. It is the challenge of chemical evolution models to account for the complex and various astrophysical processes in a description suitably summarising the various complexities to represent the known astronomical constraints, hence enable comparing models to astronomical data. Radioactivities contribute to add new observables and constraints, together with the archaeological memory of metalpoor stars in our galaxy and various measurements of composition and abundances in specific regions and objects throughout the universe. Analytical treatments of compositional evolution have been developed decades ago, to relate the elemental-abundance distribution and their evolution in galaxies to the activity of star formation and its history (Clayton 1968; Cameron and Truran 1971; Truran and Cameron 1971; Audouze and Tinsley 1976; Tinsley 1980, and many others). The physical processes included herein then received more sophisticated treatments, such as allowing for gas flows in and out of a galaxy, for multiple and independent components of a galaxy, and for more complex histories of how different stellar components inject their products into the gas cycle (Clayton 1988; Matteucci et al. 1989; Pagel 1997; Chiappini et al. 1997; Prantzos and Silk 1998; Boissier and Prantzos 1999; François et al. 2004). A specific and useful standard description of what is called galactic chemical evolution can be found in Clayton (1985, 1988). This framework enables us to exploit the rich variety of astronomical constraints from messengers of cosmic nucleosynthesis (Diehl et al. 2022), to obtain a coherent and consistent description of the compositional evolution of cosmic gas in terms of astrophysical processes of nucleosynthesis processes in different sources and of the mixing and recycling of ejecta in interstellar gas, as shown in Fig. 1. Owing to the complexity of this entire system, different approximations and compromises have to be made to obtain a description which is useful and can help to learn from astronomical observations. One may, for example, investigate which description best represents the observed distribution of stars of different ages in the solar neighbourhood, to exploit the constraints of these specific characteristics of the observed stellar population. Or one may investigate which description best represents the variety of radioactivities inferred to have been present in the early solar system nebula. Comparison of the predictions of such a description with observational data and their uncertainties offers clues as to the plausibility of the model and its parameters. Alternatively, one may optimise parameters of the description using the observational data, and parameter fitting algorithms can thus use measurements within their statistical precision. This also can be used together with probability theory to judge acceptability, or failure, of a particular description. In such descriptions, a fundamental approach is to track the amounts and composition of the reservoirs of gas and stars in a galaxy over time. Key concepts herein are: • Gas is consumed by the process of star formation. • Stars as they evolve eventually return gas enriched with newly produced isotopes and lock up gas in compact remnant stars. • Gas (and stars) may be lost from the galaxy, or acquired from outside the galaxy.

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These processes are traced through relations among the different components. Mass conservation reads therefore: m = mgas + mstars + minfall + moutflow

(1)

which includes the mass in stars and in gas as well as infall and outflow terms.The populations of stars may – sometimes usefully – be subdivided into luminous (l) and inert (c for ‘compact remnants’) stars: mstars = ml + mc

(2)

Theoretical and/or empirical prescriptions for the astrophysical processes can be formulated to obtain a formalism linking these to different observational quantities: • The birth rate of stars is introduced either empirically or through theories of star formation as supported by observations. Theories link the birth rate to the (atomic, molecular, or total) gas content of a galaxy. • The theory of stellar evolution traces the fate of stars as it depends on their initial mass, from stellar birth to death and formation of compact stellar remnants. This allows us to track the stellar population as it changes over time. • The nucleosynthesis yields from stars, i.e., the amount of ejecta in specific isotopes in their different evolutionary phases, are obtained from nucleosynthesis models of stars and their explosions. These are introduced to trace the progressive enrichment of the star-forming gas with metals and are linked to stellar evolution through the time of ejecta release after star formation. • The evolving composition of interstellar gas in a galaxy as linked to the evolving stellar population is thus tracked as a function of time, providing the output of the model for compositional evolution. In practice, however, stellar ages are difficult to evaluate from observations. Therefore, a proxy for time is used that provides a better-defined link to observations: The Fe elemental abundance is relatively high and most easily observed and thus is used as a proxy for time; sometimes the abundance of O and sometimes of α-elements, in general, are used instead. Note that the recycling time from the Fe ejected in nucleosynthesis to its incorporation and observability in stellar abundances in principle also provides an offset in time between (theoretical) Fe production and (observed) Fe abundance. • The dynamics of the interstellar gas and its different phases may affect considerably the above scheme, as, e.g., the efficiency of star formation, the distribution of the newly produced metals until it ends up in star-forming gas, the preferential ejection of specific metals from the system, etc., all vary with such gas dynamics. This dynamics and the phase transitions, however, are poorly understood, which provides a substantial systematic limitation of all descriptions of galactic compositional evolution.

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The observational quantities constraining the models are: • Number counts or densities of stars with their respective observational characteristics, i.e., mass or luminosity, surface composition, location, and possibly kinematics • Abundances of elements or isotopes, in different locations and galaxy components (stars, possibly discriminated for different populations or origins, and interstellar gas, possibly discriminated in different phases) Within the framework provided by an adopted model of galactic compositional evolution, the various parameters listed above can be adjusted to satisfy observational constraints, in order to end up with a description of the system in the adopted model with consistent and plausible parameters. In such a description, a galaxy consists initially of gas of primordial composition, i.e., XH ∼ 0.75 for H and XH e ∼ 0.25 for 4 He, as well as trace amounts of D, 3 He, and 7 Li (abundances are given as mass fractions X for element or isotope i i, with Σi Xi =1). The gas is progressively turned into stars, as measured by the star formation rate (SFR) Ψ (t). Herein, the masses M of newly formed stars are constrained to follow a number distribution Φ(M), called the initial mass function (IMF) (In principle, the IMF may depend on time, either explicitly or implicitly (i.e., through a dependence on metallicity, which increases with time); in that case, one should adopt a star creation function C(t, M) (making the solution of the equations more difficult). In practice, however, observations indicate that the IMF does not vary with the environment, allowing to separate the variables t and M and adopt C(t, M) = Ψ (t)Φ(M)). Depending on its lifetime τM , the star of mass M, which was created at time t, reaches the end of its evolution at time t + τM . It returns a part of its mass to the interstellar medium (ISM), during and at the end of its evolution. Mass returns occur either through stellar winds as the star evolves or through core-collapse supernova explosions for massive stars. In the case of lowand intermediate-mass stars, wind is the only mass return, as stellar evolution ends up in a formation of white dwarfs. Massive stars, however, eject a significant part of their mass through a wind, either in the red giant stage (a rather negligible fraction) or in the Wolf-Rayet stage (an important fraction of their mass, in the case of the most massive stars). Ejected material can be enriched in elements and isotopes synthesised by nuclear reactions in the stellar interiors, while some fragile isotopes (such as deuterium D) have been destroyed during stellar evolution and are absent in ejected material. In general, the composition of ejected material differs from the material composition that formed the star. Thus, the gas in the interstellar medium is progressively enriched in elements heavier than H and also is enriched with radioactive isotopes produced by stellar and supernova nucleosynthesis. Their decay later during ejecta transport through the interstellar medium leads to interesting compositional changes as well. New stellar generations are successively formed

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from this interstellar gas, their composition being progressively more enriched in heavy elements, i.e., with an ever-increasing metallicity Z (where Z = ΣXi for all elements i heavier than He). In more simple versions of modelling, it is assumed that the ejecta from a nucleosynthesis source are immediately and efficiently mixed in the interstellar medium. This is the so-called instantaneous mixing approximation (This should not be confused with the (stronger) instantaneous recycling approximation (IRA) that was made in earliest simple models and that assumes ejecta return at the same time as stars were formed; this means stellar evolution until nucleosynthesis release is short enough to ignore its delay, which is approximately true for very massive stars). The interstellar medium is characterised at every moment by a unique composition Xi (t), which is also the composition of the stars formed at that time t. The surface composition of stars on the main sequence is not affected, by nuclear reactions in deeper layers and the core of the star. (An exception to that rule is fragile D, already burned in the pre-main sequence all over the star’s mass; Li isotopes are also destroyed and survive only in the thin convective envelopes of the hottest stars). Observations of stellar abundances reflect, in general, the composition of the gas at the time when those stars were formed. One may thus recover the compositional history of the system through observations of stars and their abundances. The modelling approach as sketched above can be quantitatively described by a set of integro-differential equations (see Tinsley 1980): The evolution of the total mass of the system m(t) is given by: dm = [f − o] dt

(3)

If the system evolves without any input or loss of mass, the right-hand member of Eq. 3 is equal to zero; this is the so-called closed box model, the simplest model of compositional evolution. The terms of the second member within brackets are optional and describe infall of extragalactic material at a rate f (t) or outflow of mass from the system at a rate o(t); both terms will be discussed in section “Gaseous Flows Into and Out of a Galaxy.” The evolution of the mass of the gas mG (t) of the system is given by: dmG = −Ψ + E + [ f − o ] dt

(4)

where Ψ (t) is the star formation rate (SFR) and E(t) is the rate of mass ejection by dying stars, given by:  E(t) =

MU

(M − CM ) Ψ (t − τM ) Φ(M) dM

(5)

Mt

where the star of mass M, created at the time t − τM , dies at time t (if τM < t) and leaves a compact object (white dwarf, neutron star, black hole) of mass CM , i.e., it

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ejects a mass M − CM into the ISM. The integral in Eq. 5 is weighted by the initial mass function of the stars Φ(M) and runs over all stars heavy enough to die at time t, i.e., the less massive of them has a mass Mt and a lifetime τM ≤ t. The upper mass limit of the integral MU is the upper mass limit of the IMF. Obviously, the total mass of stars mS (t) of the system (i.e., those still evolving and those that are compact remnants of stellar evolution) can be derived through: m = mS + mG

(6)

The evolution of the chemical composition of the system is described by equations similar to Eqs. 4 and 5. The mass of element/isotope i in the gas is mi = mG Xi , and its evolution is given by: d(mG Xi ) = −Ψ Xi + Ei + [ f Xi,f − oXi,o − λi mG Xi ] dt

(7)

i.e., star formation at a rate Ψ removes element i from the ISM at a rate Ψ Xi , while at the same time stars re-inject into the ISM that element at a rate Ei (t). If infall is assumed, the same element i is added to the system at a rate f Xi,f , where Xi,f is the abundance of nuclide i in the infalling gas (usually, but not necessarily, assumed to be primordial). If outflow takes place, element i is removed from the system at a rate oXi,o , where Xi,o is the abundance in the outflowing gas; usually, Xi,o = Xi , i.e., the outflowing gas has the composition of the average ISM, but in some cases, it may be assumed that the hot supernova ejecta (rich in metals) leave preferentially the system, in which case Xi,o > Xi for metals. Finally, the last optional term describes the radioactive decay of nucleus i with decay rate λi >0. The rate of ejection of a nuclear species i by sources of new isotopes is given by:  Ei (t) =

MU

Yi (M) Ψ (t − τM ) Φ(M) dM

(8)

Mt

where Yi (M) is the nucleosynthesis yield of isotope i, i.e., the mass ejected in the form of that element by the star of mass M. Note that Yi (M) may depend implicitly on time t, if it is metallicity dependent. The masses involved in the system of Eqs. 3, 4, 5, 6, 7, and 8 may be either physical masses – i.e., m, mG , mS , etc. are expressed in M and Ψ (t), E(t), c(t), etc. in M Gyr−1 – or reduced masses (mass per unit final mass of the system), in which case m, mG , mS , etc. have no dimensions and Ψ (t), E(t), c(t), etc. are in Gyr−1 . The latter possibility allows to perform calculations for a system of arbitrary mass and normalise the results to the known/assumed present-day mass of that system; note that instead of absolute mass, one may use volume or surface mass densities. Because of the presence of the term Ψ (t − τM ), Eqs. 7 and 8 can only be solved numerically, except if specific assumptions are made, such as the instantaneous recycling approximation (IRA). The integral 8 is evaluated over the stellar masses,

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properly weighted by the term Ψ (t − τM ). It is explicitly assumed in that case that all the stellar masses, created in a given place, release their ejecta in that same place (which is not a reality as stars migrate). There exists another formalism, more general and useful. In that case, the mass Ei (t) released at time t is the sum of the ejecta of stars born at various times t − t  , with different star formation rates Ψ (t  ) for all stellar masses M with lifetimes τM < t − t  . Instead of Eq. 8, the isochrone formalism, concerning instantaneous bursts of star formation or single stellar populations (SSP), is used, and Equation 8 is rewritten as  Ei (t) =

t τMU

Ψ (t  )dt 



Yi (M)dN dt 

 t−t 

(9)

where dN = Φ(M)dM is the number of stars between M and M +dM and Ψ (t  )dt  is the mass of stars (in M (use solar symbol!)) created in time interval dt  at time t  . The term (dN/dt  )t−t  represents the stellar death rate (by number) at time t of a unit mass of stars born in an instantaneous burst at time t −t  . The term Yi (M)dN/dt represents the corresponding rate of release of element i in M yr −1 . Expression 9 is equivalent to Expression 8. It naturally incorporates the metallicity dependence of the stellar yields and of the stellar lifetimes, both found in the term  Yi (M, Z) dN dt (t − t ). However, it represents a significant advantage over Eq. 8, since the latter cannot apply if stars are allowed to travel away from their birth places before dying, i.e., in realistic cases. In multi-zone simulations with stellar radial migration and in N-body+SPH simulations, the isochrone formalism allows one to account for the ejecta Ei (t, R) released in a given place of spatial coordinate R and at time t as the sum of the ejecta of stars born in various places R  and times t − t  , with different star formation rates Ψ (t  , R  ) for all stellar masses M with lifetimes τM < t − t  (see Lia et al. 2002; Wiersma et al. 2009; Kubryk et al. 2015b). Because of the presence of the term Ψ (t − τM ) in Eq. 8 and t − t  in 9, those equations (as well as the corresponding ones for the total mass of gas) have to be solved numerically. There exist analytical solutions, which require some specific assumptions to be made, e.g., the instantaneous recycling approximation (IRA), which assumes that τM ∼ 0 for all stars with lifetimes much smaller than the lifetime of the studied galactic system, e.g., massive stars. Although this may provide an acceptable approximation for the abundances of massive-star products at early times, it fails in general at late times, because these abundances are diluted to the amounts of hydrogen released lately by smaller mass stars. The solution of this set of equations requires three types of ingredients: • Properties of the nucleosynthesis sources: their link to star formation, their lifetimes τM before ejection of new nuclei, the masses of eventual compact remnant stars CM , and the yields Yi (M) of a particular species i. Those characteristics can be derived from the theory of the different sources of nucleosynthesis, i.e., stellar evolution, supernova explosions, or compact-star collisions, and their

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nucleosynthesis. They depend (to various degrees) on the initial metallicity Z of the star-forming gas. • Collective properties of stars affecting the nucleosynthesis sources: the star formation rate Ψ and the initial mass function Φ(M) of stars. Neither of these can be derived from first principles; rather, empirical prescriptions form our basis, as extracted from theoretical models and their often sparse observational support. • Gas flows into and out of the considered system, i.e., a galaxy: generic gas infall from the intergalactic medium, specific inflows from gas streams or galaxy collisions, and outflows in the forms of chimneys, gas streams, or a galactic wind. These should be derived self-consistently from the astrophysics of the system. At this point, it should be emphasised that all simple models, whether one-zone or multi-zone ones, adopt the instantaneous mixing approximation: the stellar ejecta are assumed to be immediately and thoroughly mixed with the interstellar gas, which has then a uniform composition at a given position and time. Obviously, there should exist some typical scales, both in space and time, characterising the mixing processes. Such scales appear in principle in hydrodynamical or SPH simulations, but the corresponding processes are often not resolved and therefore parametrised. Those scales are very poorly understood at present, but their impact is expected to be important, regarding, e.g., the abundance dispersion of stars in a given region or the issue of short-lived radionuclides incorporated by ‘last-minute’ events in the early solar system. Modelling of compositional evolution acquires different levels of sophistication in their respective treatment of the astrophysical processes in these three areas. Therefore, a discussion of those ingredients in more detail follows now, starting with the role of stars, then that of the interstellar medium and its components, and finally of processes inter-related between stars and gas.

Stars Star Formation Stars form from cold and dense components of interstellar gas. The efficiency with which such interstellar gas is turned into stars varies with the composition and properties of the gas. The entire process is still poorly understood (see Krumholz 2014; Krumholz et al. 2018, for recent reviews of the physical processes and their variations within a galaxy). But generally, this efficiency is of order percent. Star formation is formulated as acting on the mass of interstellar gas to produce a presumably universal spectrum of stellar masses (see Kroupa and Jerabkova 2019, for a recent review). The creation rate of stars or stellar birth rate B(m, t) = Ψ (t) · Φ(M) links the star formation rate Ψ (t) with the initial mass function Φ(M) and explicitly assumes those two key ingredients to be independent (which may not be true; see Bastian et al. 2010; Dib 2011; Kroupa and Jerabkova 2019, and discussion below).

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Star formation and the following energy and mass outputs of stars are the main drivers of galactic evolution. Yet, despite decades of intense observational and theoretical investigation (see Elmegreen 2002; Zinnecker and Yorke 2007; Vázquez-Semadeni 2015, and references therein), our understanding of the subject remains frustratingly poor. Observations of various tracers of star formation in galaxies provide some empirical estimates and, in particular, relative values, under the assumption that the IMF is the same everywhere (Kennicutt 1998). Notice that most such tracers concern formation of stars more massive than ∼2 M ; very little information exists for the star formation rate of low-mass stars, even in the Milky Way. Moreover, those tracers have revealed that star formation apparently occurs in different ways, depending on the type of the galaxy. In spiral galaxies, star formation occurs mostly in these spiral arms and apparently in a sporadic way. In dwarf galaxies (or otherwise gas-rich galaxies), it has been inferred to occur in a small number of bursts, separated by long intervals of inactivity. Luminous infrared galaxies (LIRGS) and starburst galaxies (as well as, most probably, elliptical galaxies in their youth) are characterised by a much higher current rate of star formation, possibly induced by the interaction (or merging) with another galaxy. There is no universally accepted theory to predict large scale star formation in a galaxy, given the various physical ingredients that may affect the star formation rate (e.g., density and mass of gas and stars, temperature and composition of gas, magnetic fields, and the frequency of collisions between giant molecular clouds, their fractions ending up in star-forming dense cores, external drivers such as galactic rotation or inflows and mergers, etc.) (see Li et al. 2005, 2006; Ostriker et al. 2010, for discussion and examples). Schmidt (1959) suggested that the star formation rate density Ψ is proportional to some power N of the density of gas mass mG : Ψ = ν mN G

(10)

Surprisingly, Kennicutt (1998) found that in normal spiral galaxies, the surface density of the star formation rate correlates with atomic rather than with molecular gas. This conclusion is based on average surface densities, i.e., the total star formation and gas amounts of a galaxy are divided by the corresponding surface area of the disk. In fact, Kennicutt (1998) finds a fairly good correlation between star formation rate density and total (i.e., atomic + molecular) gas density. This correlation extends over four orders of magnitude in average gas surface density ρS and over six orders of magnitude in average star formation rate surface density Ψ , from normal spiral galaxies to active galactic nuclei and starburst galaxies. It can be described as: Ψ ∝ Σ 1.4

(11)

i.e., N = 1.4. However, Kennicutt (1998) notes that the same data can be fitted equally well by a different exponent value of N , this time involving the dynamical

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timescale τdyn = R/V (R), where V (R) is the orbital velocity of the galaxy at the optical radius R: Ψ ∝

Σ τdyn

(12)

More recent observations of galaxies at higher spatial resolution have indicated that on the sub-kpc scale, a different relation appears between gas and star formation rate densities (Bigiel et al. 2008). The star formation rate appears to depend linearly on the molecular gas surface density, rather than the total gas surface density. H2 surface density can be obtained by semi-empirical prescriptions for the ratio for the ratio Rmol = H2 /HI (Blitz and Rosolowsky 2006). f2 =

Rmol Rmol + 1

(13)

The resulting radial profiles H2 (R) = f2 (R) ΣG (R) and HI(R) = [1 − f2 (R)]ΣG (R) compare favorably to the observed ones in the Milky Way and other galaxies (see, e.g., Appendix B in Kubryk et al. 2015a). The corresponding star formation rate  Ψ (R) = α f2 (R)

ΣG (R) M /pc2

 M /kpc2 /yr

(14)

with coefficient α properly adjusted, reproduces well the ‘observed’ star formation profile of the galaxy’s disk. This formulation is consistent with our starting point that stars are formed from gas, after all. However, it is not clear whether volume density ρ or surface density Σ of gas should be used in Eq. 10, while preference for the latter resulted from procedures of extragalactic observations (When comparing data with models for the solar neighbourhood, Schmidt (1959) used surface densities (Σ in M /pc2 ). But, when finding ‘direct evidence for the value of N ’ in his paper, he uses volume densities (ρ in M /pc3 ) and finds N = 2. Schmidt (1959) describes the distributions of gas and young stars perpendicularly to the galactic plane (z direction) in terms of volume densities ρGas ∝ exp(−z/ hGas ) and ρStars ∝ exp(−z/ hStars ) with corresponding scale heights (obervationally derived) hGas = 78 pc and hStars = 144 2 , that is N = 2). The pc∼2 hGas ; from that, Schmidt deduces that ρStars ∝ ρGas volume density is more ‘physical’ (denser regions collapse more easily), but the surface density is more easily measured in galaxies. It seems that the density of molecular gas should be used since stars are  formed from molecular gas, and not the total gas density. Obviously, since Σ = z ρ(z)dz, one has: Σ N = z ρ(z)N dz. In discussions of astrophysical processes of star formation, it is useful to consider the efficiency of star formation ε, i.e., the star formation rate per unit mass of gas. In the case of a Schmidt law with N = 1, one has: ε = ν = const., whereas in the

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case of N = 2 one has: ε = νmG . Typical star formation efficiencies are of order of a few percent.

The Masses of Stars The distribution of stellar masses as an ensemble of stars is formed is called the initial mass function (IMF). It appears that astrophysical processes regulate a universal outcome of the process of star formation, creating a continuum of stellar masses. But exceptions may occur in particular for the regime of very high-mass stars, where accumulation of mass occurs in parallel to the stellar evolution. As long as the astrophysics of star formation is not understood, the initial mass distribution cannot be calculated from first principles. It is mainly derived from observed distributions of stellar masses (the current mass distribution, often integrated over an entire galaxy, the integrated galactic mass function (IGMF)). Such a derivation, or extrapolation, is not straightforward, and important uncertainties remain. Based on observations of stars in the solar neighbourhood and accounting for various biases (but not for stellar multiplicity), Salpeter (1955) derived a local IMF in the mass range 0.3–10 M following a power-law function: Φ(M) =

dN = A M −(1+X) dM

(15)

with a slope X = 1.35. That slope is deduced from observations and appears to apply throughout a large variety of conditions. This “Salpeter IMF” is often used throughout the entire stellar mass range. However, it is clear now that there are fewer stars in the low-mass range (below 0.5 M ) than predicted by the Salpeter slope of X = 1.35. As reviewed by Kroupa (2002) (see also Fig. 2), a multi-slope power-law IMF may provide a good description, with X = 0.35 in the range 0.08 to 0.5 M . Alternatively, often a log-normal IMF below 1 M is used (Chabrier 2003, 2005). Observations of the stellar mass distribution in various environments, and, in particular, in young clusters (where dynamical effects are negligible) suggest that a Salpeter slope X = 1.35 describes the high-mass range well (Fig. 2). However, determination of the IMF in young clusters suffers from considerable biases introduced by stellar multiplicity and pre-main sequence evolution. For field stars in the solar neighbourhood, Scalo (1986) finds X = 1.7, i.e., a much steeper IMF than Salpeter. For many purposes of compositional evolution modelling, low-mass stars are ‘eternal’ and just lock-up stellar matter, which then is excluded from the recycling. Most important for the compositional enrichment is the mass distribution in mass range of high-mass stars with their rapid evolution, that is, at stellar masses above 1 M . Weidner et al. (2011) present a concept that links a stellar mass distribution as observed in clusters (and possibly controlled by the processes of star formation and feedback) to a galaxy-integrated IMF, which would apply for our description, i.e., the sum of the action from all clusters. They thus obtain a mass distribution, which is steeper than the stellar IMF. Although every single cluster is assumed here to have had the same birth function of stellar masses (say, an IMF with X = 1.35),

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Fig. 2 The exponent 1 + X of the power-law description of the initial mass distribution of stars, according to observations in various astrophysical environments; the dashed horizontal lines indicate average values in three selected mass ranges, with 1 + X = 2.35 being the classical Salpeter value. (From Kroupa 2002)

the maximum stellar mass MMAX,C within a cluster increases with the total mass of that cluster (Weidner et al. 2010, 2013). Observations also show that small clusters may have MMAX,C as low as a few M , whereas large clusters have MMAX,C up to 150 M . If this were just a statistical effect, the slope of the resulting galaxyintegrated IMF would also be X = 1.35. But if there is a physical reason for the observed MMAX,C vs MCluster relation, then the resulting galaxy-integrated IMF would necessarily be steeper (as a consequence of the steep decline of the cluster mass function with increasing cluster mass). This concept of a ‘universal’ initial mass function characterising the astrophysical processes, mediated by stellar evolution and observational biases, appears to capture best what is known now about the stellar mass distribution (Chabrier et al. 2014; Kroupa et al. 2013) (see however Dib and Basu 2018). The IMF is normalised to  Φ(M) =

MU

ML

Φ(M) M dM = 1

(16)

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Fig. 3 Top: Three initial mass functions – solid curve, Salpeter (power law in the whole mass range); dotted curve, Kroupa (multi-slope power law for M1 M ); dashed curve, Chabrier (log-normal for M1 M ). Bottom: Ratio of the three IMFs to the one of Salpeter

where MU is the upper mass limit and ML the lower mass limit. Typical values are MU ∼ 100 M and ML ∼ 0.1 M , and the results depend little on the exact values (if they remain in the vicinity of the typical ones). A comparison between three normalised IMFs, namely, the ‘reference’ one of Salpeter, one proposed by Kroupa (with the Scalo slope at high masses) and one by Chabrier (with the Salpeter slope at high masses), is made in Fig. 3. A useful quantity is the return mass fraction R  R =

MU

(M − CM ) Φ(M) dM

(17)

MT

i.e., the fraction of the mass of a stellar generation that returns to the interstellar medium. It depends on the IMF, on stellar evolution, and on the masses of the stellar remnants CM . For the three IMFs displayed in Fig. 3, one obtains estimates from the effects of stellar evolution of R ∼= 0.3 (Salpeter), 0.34 (Kroupa+Scalo), and 0.38 (Chabrier+Salpeter), respectively. This roughly means that about 1/3 of the mass gone into stars returns to the ISM.

The Lifetimes of Stars and Their Remnants The lifetime of stars, i.e., the timescale it takes from star formation to reaching the final stages, such as white dwarf formation and gravitational-collapse supernova, is a rapidly decreasing function of stellar mass (see Fig. 4). Its precise value depends on the various assumptions (about, e.g., mixing, mass loss, etc.) adopted in stellar evolution models (see Romano et al. 2005, for a compilation of various sets of

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Fig. 4 Left: Lifetimes of stars of solar metallicity from Schaller et al. 1992 (points), fitted by Eq. 18. Right: Ratio of stellar lifetimes at metallicity Z = Z /20 to those at Z = Z , from the same reference

stellar lifetimes) and, most importantly, on stellar metallicity. Low-metallicity stars have lower opacities and are more compact and hot than their high-metallicity counterparts; as a result, their lifetimes are shorter (see Fig. 4 right). However, in stars with M > 2 M , where H burns through the CNO cycles, this is compensated to some degree by the fact that the H-burning rate (proportional to the CNO content) is smaller, making the corresponding lifetime longer; thus, for M > 10 M , lowmetallicity stars live slightly longer than solar metallicity stars. Of course, these results depend strongly on other ingredients, such as stellar rotation. In principle, such variations in τM should be taken into account in models of compositional evolution; in practice, however, the errors introduced by ignoring them are smaller than the other uncertainties of the problem, related, e.g., to stellar yields or to the IMF (Metallicity dependent lifetimes have to be taken into account in models of the spectrophotometric evolution of galaxies, where they have a large impact. In galactic chemical evolution calculations, they play an important role in the evolution of s-elements, which are mostly produced by long-lived AGB stars of ∼1.5–2 M ). The lifetime of a star of mass M (in M ) with metallicity Z can be approximated by: τ (M) = 1.13 1010 M −3 + 0.6 108 M −0.75 + 1.2 106 yr

(18)

This fitting formula is displayed as solid curve in Fig. 4 (left). A Z star of 1 M , like the Sun, is bound to live for 11.4 Gyr, while a 0.8 M star, for ∼23 Gyr; the latter, however, if born with a metallicity Z ≤ 0.05 Z , will live for ‘only’ 13.8 Gyr, i.e., its lifetime is comparable to the age of the universe (Fig. 4, right). Stars of mass 0.8 M are thus the lowest mass stars that have ever come to their end of evolution

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Fig. 5 Left: Masses of stellar residues as a function of initial stellar mass, for stars of metallicity Z ; for massive stars (M>30 M ), the two curves correspond to different assumptions about mass loss, adopted in Limongi and Chieffi (2009) (solid curve) and Woosley and Heger (2007) (dotted curve), respectively. Right: Mass fraction of the ejecta as a function of initial stellar mass; the two curves for M > 30 M result from the references in the left figure

since the beginnings of star formation (are the heaviest stars surviving in the oldest globular clusters). Stellar evolution of single stars eventually leads to compact remnant stars (white dwarfs, neutron stars, or black holes, depending on the mass of the star), which locks up a part of stellar gas remaining at the end. Binary systems, however, open channels for recycling this locked-up stellar mass into the gas reservoir (see below). The masses of stellar residues are derived from stellar evolution calculations (see Fig. 5), and can be confronted to observational constraints. In the regime of low- and intermediate-mass stars (LIMS) (LIMS are defined as those stars evolving to white dwarfs. However, there is no universal definition for the mass limits characterising low- and intermediate-mass stars. The upper limit is usually taken around 8–9 M , although values as low as 6 M have been suggested in models with very large convective cores. The limit between low and intermediate masses is the one separating stars powered on the main sequence by the p-p chains from those powered by the CNO cycle and is ∼1.2–1.7 M , depending on metallicity.), i.e., for M/M ≤8–9, the evolutionary outcome is a white dwarf (WD), the mass of which (in M ) is (Weidemann 2000): CM (W D) = 0.08 M + 0.47

(MM0,i = M0,i 0 =0 1 =1 1 kpc. The total interstellar masses (including helium and metals) of the three gas components in the galactic disk are highly uncertain; estimates are in the range ∼(0.9 − 2.5) 109 M for the molecular component, ∼(0.65 − 1.1) 109 M for the atomic component, and ∼1.5 109 M for the ionised component. The total interstellar mass in the Galaxy is probably between ∼0.9 1010 M and ∼1.5 1010 M , representing ∼15%–25% of the baryonic mass in our galaxy, or ∼25%–35% of the total mass of the galactic disk. The dramatic density and temperature contrasts between the different phases of interstellar medium, and observed supersonic random motions, all suggest that the interstellar medium is highly turbulent and dynamic (see above and Fig. 9). Drivers are the powerful winds and the supernova explosions of massive stars. Interstellar turbulence manifests itself over a huge range of spatial scales, from 1010 cm up to 1020 cm; throughout this range, the power spectrum of the free-electron density in the local interstellar medium is consistent with a Kolmogorov-like power law (Armstrong et al. 1995; Elmegreen and Scalo 2004; Vázquez-Semadeni 2015). The origins of interstellar dust are complex and unclear (see Draine 2003, for a review). Dust formation is rather well modelled in AGB star envelopes (Sedlmayr and Patzer 2004). For more massive stars, this has not been achieved; Wolf-Rayet winds are complex, clumpy, and very energetic. Exploding supernova envelopes are

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Fig. 11 Azimuthally averaged surface densities of interstellar atomic, molecular, and ionised hydrogen as functions of galactic radius. The total gas (bottom) includes a 40% contribution by helium. Notice the change of scale at R = 2 kpc. (See Prantzos 2010, for details and references)

even more dynamic, and dust formation is only beginning to be explored (Sugerman et al. 2006; Cherchneff 2016). Interstellar dust is modified in size and composition on its journey through interstellar space (see Jones 2009, for a review). Once created in a ‘nucleation’, the size of the particle rapidly grows by condensation of interstellar molecules, growing considerable ice mantles. Interstellar shocks, but also the intense radiation near massive stars, can destroy particles, and thus reprocesses dust grains through partial or full evaporisation of ice mantles. Interstellar shocks enhance grain collisions and may incur sputtering of larger grains into smaller ones. Radiation from dust is a prime tracer for star-forming environments, as radiation from stars heats dust particles to higher temperatures than the typical ∼10 K of normal interstellar space; thermal emission is observed and studied through infrared telescopes.

Scales of Interstellar Medium Processes The interstellar medium is a key mediator for the outputs of nucleosynthesis sources, i.e., ejected matter, ionising radiation, and kinetic energy from winds and explosions. Such impact processes the interstellar medium between phases and states, which determine further star formation; this is called feedback and determines the evolution of normal disk galaxies (In active galaxies, the central supermassive black hole also plays a role and even dominates over the impact from massive stars for central regions and for the entire galaxies in late (largely

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processed) evolution such as at low redshifts). Turbulence generated by stellar winds and explosions determines how interstellar gas eventually forms stars or ceases to form new stars, thus driving galactic evolution on a more fundamental level. Feedback from nucleosynthesis sources occurs throughout a galaxy and influences its embedded objects. Exactly how matter spreads from nucleosynthesis sites into the next-generation stars will determine the chemical enrichment over a galaxy’s evolution (mixing). The major other drivers of galactic evolution are material inflows from extragalactic space through clouds, streams, or mergers, and a supermassive black hole in a galaxy’s centre. Compositional evolution of the universe at large involves mixing of material at different scales: the early phase of forming a star (before/until planets are being formed), stellar winds and explosions, clusters of coevolving stars, the disks of typical galaxies, and intergalactic space. It is possible to trace matter in its different appearances, i.e. as plasma (ionised atoms and their electrons), atoms and molecules, and dust particles. Different spatial scales can be characterised in more detail: (a) At the smallest scale, a stellar/planetary formation site evolves from decoupling of its parental interstellar cloud (i.e., no further material exchange with nucleosynthesis events in the vicinity) until the star and its planets have settled and overcome the disk accretion phase with its asteroid collision and jet phases. This phase may have a typical duration of ∼My. Issues here are how inhomogeneities in composition across the early solar nebula are smoothed out over the timescales at which chondrites, planetesimals, and planets form. (Chondrites are early meteorites and the most common meteorites falling on Earth (85%). Their name derives from the term chondrule, which are striking spherical inclusions in those rocks. The origin of those is related to melting events in solids of the early solar system, the nature of which is the study of cosmochemistry (Cowley 1995). Carbonaceous chondrites are 5% of all falling meteors and are believed to be the earliest known solid bodies within the solar system.) Inhomogeneities may have been created from (i) the initial decoupling from a triggering event or from (ii) the energetic-particle nuclear processing in the jet-wind phase of the newly forming star. Radioactive dating is an important tool in such studies. (b) The fate of the ejecta of a stellar nucleosynthesis event is of concern at the next-larger scale. Stellar winds in the late evolutionary stages of stars such as the asymptotic giant or Wolf–Rayet phases and also explosive events, novae, and two kinds of supernovae (according to their different evolutionary tracks) involve different envelope masses, ejection energies, and dynamics. The astronomical display of such injection of fresh nuclei into interstellar space is impressive throughout the early phases of the injection event (AGB stars form colourful planetary nebulae, massive-star winds form gas structures within the HII regions created by the ionising radiation of the same stars, and thus a similarly-rich variety of colourful filamentary structure from atomic recombination lines. Supernova remnants are the more violent version of the

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same processes.); however, no real mixing with ambient interstellar gas occurs, yet at this phase, ejected gas expands into the lower-pressure interstellar medium but decelerates upon collisional interaction with interstellar atoms and collisionless interactions with the magnetised plasma. This process is an important ingredient for the acceleration of cosmic rays. Once ejecta velocities have degraded to the velocity range of interstellar gas (∼100–few km s−1 ), the actual mixing process can become efficient. Cooling of gas in its different phases is a key process and also incurs characteristic astronomical signatures. (Hα radio emission, C[II] recombination in the IR, or FIR thermal emission of dust are important examples.) Radioactive isotopes are key sources of energy for the astronomical display (supernova light curves) and sensitive tracers of the nucleosynthesis conditions of these events. (c) Coevolving stellar groups and clusters provide an astrophysical object on the next larger scale. The combined action of stars, successively reaching their individual wind phases and their terminating supernovae, shapes the interstellar environment so that it may vary for each nucleosynthesis event. Giant HII regions and superbubbles are the signposts of such 10–100 pc-sized activity, which can be seen even in distant galaxies (Oey et al. 2007). The evolution of disks in galaxies is determined by the processes on this scale: formation of stars out of giant molecular clouds, as regulated by feedback from the massive stars, as it stimulates further star formation, or terminates it, depending on gas dynamics and the stellar population. This is currently the frontier of the studies of cosmic evolution of galaxies (Calzetti and Kennicutt 2009; Marasco et al. 2015). Cumulative kinetic energy injection may be sufficient to increase the size and pressure in a cavity generated in the interstellar medium, such that blowout may occur perpendicularly to the galactic disk, where the pressure of ambient interstellar gas is reduced with respect to the galactic disk midplane. This would then eject gas enriched with fresh nucleosynthesis product into a galaxy’s halo region through a galactic fountain. Only the fraction of gas below galactic escape velocity would eventually return on some longer timescale (>107 – 108 y), possibly as high-velocity clouds (HVCs). Long-lived (∼My) radioactive isotopes contribute with age dating and radioactive tracing of ejecta flows. (d) In a normal galaxy’s disk, large-scale dynamics is set by differential rotation of the disk and by large-scale regular or stochastic turbulence as it results from star formation and the incurred wind and supernova activity (see (c)): This drives the evolution of a galaxy. (Feedback from supermassive black holes is small by comparison but may become significant in AGN phases of galaxies and on the next-larger scale (clusters of galaxies, see (e)). Galaxy interactions and merging events are also important agents over cosmic times; their overall significance for cosmic evolution is a subject of many current studies). As a characteristic timescale for rotation, one may adopt the solar orbit around the galaxy’s centre of 108 years. Other important large-scale kinematics may be given by spiral density waves, sweeping through the disk of a galaxy at a characteristic pattern speed, and by the different kinematics towards the central galaxy region with its bulge, where a bar often directs gas and stellar orbits in a more radial trajectory

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and with a bar pattern speed that will differ from Keplerian circular orbits in general. Not only infalling clouds of gas from the galactic halo but also gas streams from nearby galaxies and from intergalactic space will add drivers of turbulence in a galaxy’s disk at this large scale. The mixing characteristics of the interstellar medium therefore will, in general, depend on the location, and on history within a galaxy’s evolution. Radioactive isotopes are part of the concerted abundance measurement efforts, which help to build realistic models of a galaxy’s chemical evolution. (e) On the largest scale, gas streams into and away from a galaxy are the mixing agents on the intergalactic scale. Galactic fountains thus offer alternative views on superbubble blowout. This may also altogether form a galactic wind ejected from galaxies (observed, e.g., in starburst galaxies, see Heckman et al. 1990). Galaxies are part of the cosmic web and appear in coherent groups (and clusters). Hot gas between galaxies can be seen in X-ray emission; elemental abundances can be inferred from characteristic recombination lines. Gas clouds between galaxies can also be seen in characteristic absorption lines from distant quasars, constraining elemental abundances in intergalactic space. The estimated budget of atoms heavier than H and He appears incomplete (the missing metals issue (Sommer-Larsen 2006), which illustrates that mixing on these intergalactic scales is not understood.)

Magnetic Fields The presence of interstellar magnetic fields in our galaxy was first revealed by the discovery that the light from nearby stars is linearly polarised. This polarisation is due to elongated dust grains, which tend to spin about their short axis and orient their spin axis along the interstellar magnetic field; since they preferentially block the component of light parallel to their long axis, the light that passes through is linearly polarised in the direction of the magnetic field. Thus, the direction of linear polarisation provides a direct measure of the field direction on the plane of the sky. This technique applied to nearby stars shows that the orientation of the magnetic field in the interstellar vicinity of the Sun is horizontal, i.e., parallel to the galactic plane, and that it makes a small angle 7◦ to the azimuthal direction (Heiles 1996). The magnetic field strength in cold, dense regions of interstellar space can be inferred from the Zeeman splitting of the 21-cm line of HI (in atomic clouds) and centimetre lines of OH and other molecules (in molecular clouds). It is found that in atomic clouds, the field strength is typically a few μG, with a slight tendency to increase with increasing density (Troland and Heiles 1986; Heiles and Troland 2005), while in molecular clouds, the field strength increases approximately as the square root of density, from ∼10 to ∼3,000 μG (Crutcher 1999, 2007). The magnetic field in ionised parts of the interstellar medium is generally probed with measures of Faraday rotation along its propagation path of radiation that originates from galactic pulsars and extragalactic radio sources. An important advantage of pulsars is that their rotation measure divided by their dispersion measure directly yields the electron-density weighted average value of B between them and the observer. Faraday rotation studies have provided interesting properties of interstellar

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magnetic fields: The interstellar magnetic field has uniform (or regular) and random (or turbulent) components; near the Sun, the uniform component is 1.5 μG, and the random component, ∼5 μG (Rand and Kulkarni 1989). The uniform field is nearly azimuthal in most of the galactic disk, but it reverses several times along a radial line (Rand and Lyne 1994; Han et al. 1999; Vallée 2005; Han 2006; Brown et al. 2007). These reversals have often been interpreted as evidence that the uniform field is bisymmetric (azimuthal wavenumber m = 1), although an axisymmetric (m = 0) field would be expected on theoretical grounds; see however Men et al. (2008) discussing counter arguments; the uniform field may have a more complex origin. The uniform field increases towards the centre of the galaxy, from 1.5 μG near the Sun to 3 μG at R = 3 kpc (Han 2006); this increase corresponds to an exponential scale length 7.2 kpc. In addition, the uniform field decreases away from the midplane, albeit at a very uncertain rate – for reference, the exponential scale height inferred from the rotation measures of extragalactic sources is ∼1.4 kpc (Inoue and Tabara 1981). The observed antisymmetric pattern inside the solar circle, combined with the detection of vertical magnetic fields near the galactic centre, led Han et al. (1997) to suggest that an axisymmetric dynamo mode with odd vertical parity prevails in the thick disk or halo of the inner galaxy. While plausible, the conclusion would be premature that the uniform galactic magnetic field is simply a dipole sheared in the azimuthal direction by the large-scale differential rotation. From the radio emission synchrotron map (Beuermann et al. 1985) (and assuming equipartition between magnetic fields and cosmic rays), Ferriere (1998) found a local value 5.1 μG, a radial scale length 12 kpc, and a vertical scale height near the Sun of 4.5 kpc, for the total magnetic field. The properties of the turbulent magnetic field are not well established. Rand and Kulkarni (1989) provided a first rough estimate for the typical spatial scale of magnetic fluctuations, ∼55 pc, although they recognised that the turbulent field cannot be characterised by a single scale. Later, Minter and Spangler (1996) presented a derivation of the power spectrum of magnetic fluctuations over the spatial range ∼(0.01–100) pc; they obtained a Kolmogorov spectrum below ∼4 pc and a flatter spectrum consistent with a 2D turbulence above this scale. In a complementary study, Han (2004) examined magnetic fluctuations at larger scales, ranging from ∼0.5 to 15 kpc; at these scales, they found a nearly flat magnetic spectrum, with a 1D power-law index ∼−0.37 (Fig. 12). The properties of the turbulent galactic magnetic field are poorly understood at present. However, its local and overall configurations are extremely important for understanding cosmic-ray propagation in the galaxy.

The Role of Massive-Star Groups All above considerations concern a larger (representative, or averaged) region of a galaxy. But the formation of massive stars occurs in groups. From a single parental giant molecular cloud groups of hundreds to thousands of massive stars will be born. Each group will be coeval, all stars born at the same time. But from the same giant molecular cloud, many such groups will be created within the few to tens of Myrs

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Fig. 12 Composite magnetic energy spectrum in our galaxy. The thick solid line is the large-scale spectrum. The thin solid and dashed/dotted lines give the Kolmogorov and 2D turbulence spectra, respectively, inferred from the Minter and Spangler (1996) study. The 2D turbulence spectrum is uncertain; it probably lies between the dashed [EB (k) ∝ k −2/3 ] and dotted [EB (k) ∝ k −5/3 ] lines. (From Han 2004)

before the cloud has been shredded and dissolved by the feedback of these young and active stars. The stellar content, interstellar gas enrichment, and dynamical state of regions, such as the ones resulting from the evolution of a single giant molecular cloud complex, are causally connected through the processes that occur as star formation is affected by feedback from the activity of the bright young massive stars and their winds and supernovae. On the larger scale of an entire galaxy, such regions can be viewed as random, because on such larger scale other physical processes, such as spiral waves, bars, and collisions with other galaxies, play the dominant role for regulating star formation, rather than stellar feedback. Therefore, the large-scale galactic average may be different from the evolutionary states, star-to-gas content, and composition of such local regions of massive-star groups from one such giant

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molecular cloud. This phenomenon had been called ‘beads on a string’ by Bruce Elmegreen, when discovering the flocculent appearance of spiral arms in tracers of massive stars in external galaxies (Elmegreen et al. 2003). Groups of massive stars are more straightforward in treatment of the astrophysical processes, as approximations in chemical evolution descriptions can be formulated more explicitly or are better in such restricted context. So, ages of stars are better constrained; recycling of newly formed nuclei will be minor as feedback produces them in a hot, non-star-forming phase, while shock interactions from locally created shocks will dominate the triggering of next-generation star formation from the cloud at other locations and times. The resulting picture here is that the feedback from coeval groups of massive stars will lead to stellar feedback and ejections from nucleosynthesis in an environment that has been shaped by neighbouring or more rapidly evolving stars.

Gaseous Flows Into and Out of a Galaxy A galaxy is clearly not an isolated system, and it is expected to exchange matter (and energy) with its environment. This is true even for seemingly isolated galaxies which are found away from galaxy groups. Most of the baryonic matter in the universe today (and in past epochs) is in the form of gas residing in the intergalactic medium (Fukugita and Peebles 2004), and part of it is slowly accreted by galaxies. Also, small galaxies are often found in the tidal field of larger ones, and their tidal debris (gas and/or stars) may be captured by the latter. In both cases, gaseous matter is accreted by galaxies. In the framework of the simple compositional evolution model, this is generically called infall (Gaseous flows in the plane of a galactic disk, due, e.g., to viscosity, are called inflows; for simple compositional evolution models, they also constitute a form of infall). On the other hand, gas may leave the galaxy, if it gets sufficient (1) kinetic energy or (2) thermal energy and (3) its turbulent velocity becomes larger than the escape velocity. Condition (1) may be met in the case of tidal stripping of gas in the field of a neighbour galaxy or in the case of ram pressure from the intergalactic medium. Condition (2) is provided by heating of the interstellar gas from the energy of supernova explosions, especially if collective effects (i.e., a large number of supernovae in a small volume, leading to a superbubble) become important. Finally, condition (3) is more easily met in the case of small galaxies, with shallow potential wells. Note that, since galaxies (i.e., baryons) may be embedded in extended dark matter (non-baryonic) haloes, a distinction could be made between gas leaving the galaxy but still remaining trapped in the dark halo and gas leaving even the dark halo. In the former case, gas may return back to the galaxy after ‘floating’ for some time in the dark halo and suffering sufficient cooling. In the framework of the simple compositional evolution model, all those cases are described generically as outflows. With the Gaia satellite and recent kinematic analysis of large numbers of stars in the galaxy, it became clear (Helmi 2020) that the stellar population seen in our galaxy includes components that originate in other galaxies, which have collided or merged with the galaxy over cosmic time. These streams of stars can be

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recognised from their peculiar kinematic behaviour as a group and also show distinct and different stellar abundance patters. They have led to a new picture of stellar populations and of the merger history of our galaxy (see Helmi 2020, for a review) that provides new challenges for compositional evolution modelling. The rate of infall or outflow is difficult to calculate from first principles. In the case of infall, this is possible, in principle, for a hydrodynamical model evolving in an appropriate cosmological framework. In the case of outflows, the interaction between stars and the star-forming properties of interstellar gas clouds, known as feedback, also requires detailed hydrodynamic modelling. No satisfactory models exist up to now for such complex processes. Note that, the treatment of feedback also affects the star formation rate of the system (by making gas unavailable for star formation, either by heating it or by pushing it out of the system altogether). In simple models of compositional evolution in galaxies, infall and outflow are treated as free parameters. These are adjusted as to reproduce observed features of the galaxy systems under study. Such features are the metallicity distributions of long-lived stars or the mass-metallicity relationship of external galaxies, which provide strong constraints on the history of the systems. Popular parametrisations are an infall exponentially decaying with time and an outflow proportional to the star formation rate (see Matteucci 2021, for references). Infall has been assumed to prefer inner galactic radii, to support the inferred insideout formation of the galactic disk. Recent recognition of a rather flat stellar density across large parts of the disk, with a change only outside the galactocentric radius of the Sun towards a density falloff, argues for the preferential infall in the outer parts of the disk, however (Lian et al. 2022).

Cosmic Rays The interstellar medium and its cosmic-ray content are closely intertwined through physical processes: (i) Cosmic rays are produced from interstellar medium properties such as magnetic turbulences. (ii) As cosmic rays propagate through the interstellar medium, they create waves that efficiently reduce their diffusion length, leading to localised cosmic-ray enhancements at the 100 pc scale. (iii) Cosmic rays are the only agents that can penetrate into dense molecular clouds, thus affecting the process of star formation directly in its final collapse. Therefore, some aspects of cosmic rays are now presented, as they need to be incorporated into compositional evolution models to properly account for the link between nucleosynthesis events that drive interstellar medium dynamics and morphology and the formation of stars that is the prerequisite for any nucleosynthesis. Despite more than a century of research since Victor Hess discovered cosmic rays (in 1911), the astrophysics of cosmic rays is far from understood: The sources cannot be clearly identified, a variety from pulsar wind nebulae through supernova shocks to interstellar shocks, gamma-ray burst jets, and jets from accretion onto supermassive black holes in active galaxies all coexist and are investigated (Blasi 2013). The process of acceleration also is unclear (see Marcowith et al. 2020, for a recent review), from pulsar wind nebulae (Amato 2020; Bykov et al. 2017), through second-order Fermi acceleration by irregular shock fronts, to magnetic reconnection

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in turbulent flows (Lazarian et al. 2020). Each of these finds support in astronomical observations, but no one such process can be proven to dominate nor to be irrelevant. Modelling cosmic-ray propagation requires not only a detailed knowledge about the morphology of the interstellar medium and its magnetic fields but also of processes on how cosmic rays excite waves and thus self-confine, so that largescale and steady-state approximations can be complemented by localised variations thereof. Commonly, cosmic-ray production is believed to be dominated by supernovae and in particular their remnants through diffusive shock acceleration (Ellison et al. 1997). But pulsar wind nebulae have been recognised to be important contributors towards higher energies (Aharonian et al. 2005). Some fraction of cosmic-ray origin is attributed to particle acceleration in pulsars, compact objects in close binary systems, and stellar winds. Particles that are accelerated within such sources escape from the acceleration sites, as their boundaries (mostly resulting from irregular magnetic fields at outer shocks) become transparent with increasing energies. Shock acceleration is viewed as a ‘universal’ acceleration mechanism for cosmic rays, working well on different scales and in different astrophysical environments (Jones and Ellison 1991; Ellison et al. 1997; Berezhko and Ellison 1999). Non-relativistic shocks may be formed where the pressure of a supersonic stream of gas drops to the much lower value of the environment. Relativistic shocks provide an ideal setting for Fermi acceleration, in particular, in dense clusters of massive stars (Bykov et al. 2011). Acceleration produces non-thermal and relativistic particles herein, and in astrophysical plasma jets. These are believed or known to exist from black hole accretion such as in active galactic nuclei and quasars, microquasars, and in γ ray burst jets. A considerable amount of energy in the jet flow may be transferred to the accelerated particles, and they thereby act back on the shocks, dynamically modifying their acceleration environment (Ellison 2001; Ellison et al. 2007). Cosmic rays are predominantly composed of protons and heavier nuclei up to iron, with a small fraction of leptons, i.e., electrons and positrons, of order percent at GeV energies. Their energy spectrum extends from the GeV to MeV energies defining their relativistic characteristics up to energies of 1021 eV. This is several orders of magnitude more energetic than what can be reached on terrestrial particle accelerator facilities. The general spectrum can be characterised by a power-law distribution with a power-law index near 3; characteristic breaks in the power-law index at energies ∼1015 and above 1018.5 eV indicate changes in either propagation characteristics or source origins. The spectrum of leptons characteristically steepens above a TeV, which reflects the larger energy losses for leptons towards high energies from ionisation, Coulomb scattering, bremsstrahlung, inverse Compton scattering, and synchrotron emission. Thus, above TeV energies, the cosmic-ray composition mostly consists of hadrons and nuclei, with an indication that heavier nuclei are least affected by energy losses at highest energies. Once injected into the interstellar medium, these cosmic-ray particles become an important part of it, collisions with ambient interstellar gas providing spallation nucleosynthesis products (Tatischeff et al. 2018), some heating energy, and ionisation. The energy density in cosmic rays corresponds roughly to 1 eV cm−3 , thus is

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comparable to the energy density of the interstellar radiation field, magnetic field, and turbulent motions of the interstellar gas. Because of the huge galactic volume of a galaxy and its irregular magnetic fields, the propagation of cosmic-ray particles occurs on complex parts preventing directional tracking and eventually leading to escape from the galaxy into intergalactic space after a typical timescale of 10 Myrs (shorter for highest energies). Several steps are involved between the production of the cosmic-ray nuclides in stellar interiors and their detection near Earth: 1. 2. 3. 4. 5.

Nucleosynthesis, most likely in stars and their explosions Ejection by stellar winds and explosions Fractionation and biases with ionisation and dust formation properties Acceleration of such primary nuclei, by some process that acts near a source Propagation through the interstellar medium, where those nuclei are partially fragmented (spallated through collisions with the ambient gas), giving rise to secondary cosmic-ray nuclei 6. Modulation by nearby magnetic structures and in particular the heliosphere 7. Detection of cosmic rays near Earth in specialised detectors Cosmic-ray transport through interstellar space within the galaxy has been studied with models of varying sophistication, which account for a large number of astrophysical observables (see the comprehensive review of Strong et al. 2007, and references therein). If a description of the composition is the only aim, less sophisticated models are sufficient, such as the leaky-box model. Herein, cosmic rays are assumed to fill homogeneously a cylindrical box – the galactic disk – and their intensity in the interstellar medium is assumed to be in a steady-state equilibrium between several production and destruction processes. The former involve acceleration in cosmic-ray sources and production in-flight through fragmentation of heavier nuclides, while the latter include either physical losses from the leakybox (escape from the galaxy) or losses in energy space (ionisation) and in particle space (fragmentation, radioactive decay, or pion production in the case of protonproton collisions). Most of the physical parameters describing these processes are well-known, although some spallation cross sections still suffer from considerable uncertainties. The many intricacies of cosmic-ray transport may be encoded in a simple parameter, the escape length Λesc (in g cm−2 ): it represents the average column density traversed by the nuclei of galactic cosmic rays before escaping the galactic leaky box. Determination of the confinement (or residence or escape) timescale of cosmic rays in the galaxy is a key issue, because τConf determines the power required to sustain the energy density of galactic cosmic rays. A conventional way to derive the propagation parameters is to fit the secondary/primary ratio of characteristic nuclei, e.g., B/C, from spallation along the cosmic-ray trajectory. The leaky-box model assumes galactic cosmic-ray intensities and gas densities to be uniform in the propagation volume (and in time). The abundance ratio of a secondary to a primary nucleus depends essentially on the escape parameter Λesc in the

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leaky-box model. Observations of Li-Be-B/CNO and Sc-Ti-V/Fe nuclei ratios in arriving galactic cosmic rays, interpreted in this framework, suggest a mean escape length of Λesc ∼7 g cm−2 . Such measurements, if interpreted within this model, can thus probe the average density of the confinement region. Results imply that galactic cosmic rays spend a large fraction of their confinement time in a volume of smaller average density than the one of the local gas, i.e., in the galactic halo. Low-energy cosmic-ray measurements 10 Gyr). The dominant component of the Milky Way is the so-called thin disk, a rotationally supported structure composed of stars of all ages (0–10 Gyr). A non-negligible contribution is in the thick disk, an old (>10 Gyr) and kinematically distinct entity identified by Gilmore and Reid (1983).

The Bulge and the Centre To first approximation, and by analogy with external galaxies, the bulge of the Milky Way can be considered as spherical, with a density profile either exponential or of Sersic-type (ρ(r) ∝r1/n with n >1). Measurements in the near-infrared (NIR), concerning either integrated starlight observations or star counts, revealed that the bulge is not spherical but elongated (Robin et al. 2012). Recent models suggest a triaxial ellipsoid, but its exact shape is difficult to determine (Rattenbury et al. 2007) because of the presence of a galactic bar. The mass of the bulge lies in the range

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Fig. 14 Surface densities of stars and gas, star formation rate, supernova rates, and scale heights of gas and stars as a function of galactocentric distance. Star profiles are from data of Table II, and the gas profile is from Dame (1993).Data for the star formation rate are from: Lyne et al. (1985) (open circles), Case and Bhattacharya (1998) (filled circles), McKee and Williams (1997) (open squares), and Guibert et al. (1978) (filled squares). The solid curve is an approximate fit, normalised to 2 M yr−1 for the whole galaxy. The same curve is used for the CCSN rate profile (third panel), normalised to 2 CCSN/century; the SNIa rate profile is normalised to 0.5 SNIa/century. (From Prantzos et al. 2011)

1–2 1010 M (Robin et al. 2003). By comparing colour-magnitude diagrams of stars in the bulge and in metal-rich globular clusters, Zoccali et al. (2003) find that the populations of the two systems are coeval, with an age of ∼10 Gyr. The innermost regions of the bulge, within a few hundred pc, are dominated by a distinct, disk-like component, called the nuclear bulge, which contains about 10% of the bulge stellar population (∼1.5 109 M ) within a flattened region of radius 230±20 pc and scale height 45±5 pc (Launhardt et al. 2002). It is dominated by three massive stellar clusters (nuclear stellar cluster or NSC in the innermost 5 pc,

89 Cosmic Radioactivity and Galactic Chemical Evolution

3303

Arches, and Quintuplex), which have a mass distribution substantially flatter than the classical Salpeter IMF. Finally, in the centre of the Milky Way, at the position of SgrA∗ source, lies the massive galactic black hole (MBH) with a total mass of ∼4 106 M (Gillessen et al. 2008).

The Disk(s) The galaxy is characterised by two disk components, a thin disk (scale height or order 100 pc), which includes the current and recent stellar populations, and a more extended thick disk (scale height of order kpc) that is believed to be the older remains of early star formation. The Sun is located in the thin disk of the Milky way, at a distance of R ∼8 kpc from the galactic centre (Groenewegen et al. 2008, and references therein), and currently offset from the midplane towards the Northern Hemisphere by z ∼25 pc (Juri´c et al. 2008; Siegert 2019). For the solar neighbourhood, local properties can, in general, be measured with greater accuracy than global ones; often these are our baseline. The total baryonic surface density of the solar cylinder, defined as a cylinder of radius 500 pc centred on the Sun’s position and extending perpendicularly to the galactic plane up to several kpc, is estimated to ΣT = 48.8 M pc−2 (Flynn et al. 2006), with ∼13 M pc−2 belonging to the gas. This falls on the lower end of the dynamical mass surface density estimates (from kinematics of stars perpendicularly to the plane), which amount to ΣD = 50–62 M pc−2 (Holmberg and Flynn 2004) or 57–66 M pc−2 (Bienaymé et al. 2006). Thus, the values for the baryon content of the solar cylinder, summarised in Table 3, should be considered rather as lower limits: the total stellar surface density could be as high as 40 M pc−2 . The density profiles of the stellar thin and thick disks can be satisfactorily fit with exponential functions, both in the radial direction and perpendicularly to the galactic plane. The recent SDSS data analysis of star counts, with no a priori assumptions as to the functional form of the density profiles, finds exponential disks with scale lengths as displayed in Table 3 (from Juri´c et al. 2008). The thin and thick disks cannot extend all the way to the galactic centre, since dynamical arguments constrain the spatial coexistence of such rotationally supported structures with the pressure-supported bulge. The exact shape of the central hole of the disks is poorly known (see, e.g., Freudenreich (1998) and Robin et al. (2003), for parametrisations), but for most practical purposes (i.e., estimate of the total disk mass), the hole can be considered as truly void of disk stars for disk radius R> m) nondegenerate (T >> μ) particles n, ρ, and P , take the simple form: ζ (3) gi T 3 (Boson) π2 3ζ (3) = gi T 3 (Fermion) 4π 2

ni =

ρi =

(5)

π2 gi T 4 (Boson) 30

7 π2 gi T 4 (Fermion), 8 30 ρi Pi = , 3 =

where ζ (3) = 1.20206 . . . is the Riemann zeta function of argument 3.

(6) (7)

91 Big Bang Thermodynamics and Cosmic Relics

3361

In the nonrelativistic limit (mi >> T ), the corresponding quantities are the same for either Bosons or Fermions, and the familiar Maxwell-Boltzmann statistics can be adopted:   mi T 3/2 exp [−(m − μ)/T ] (8) ni = gi 2π ρi = mi ni

(9)

Pi = ni T .

(10)

When solving the Friedmann Equation, one desires the total energy density of all particles in equilibrium. This can be expressed in terms of the photon temperature T as  ρ≡

  π2 geff T 4 = 30





i=all species

Ti T

4

gi 2π 2



∞ xi

 (u2 − xi2 )1/2 u2 du T 4, exp (u − yi ) ± 1 (11)

where u = E/T , xi ≡ mi /T and yi ≡ μi /T , and the (Ti /T ) factor accounts for the possibility that a species i may have a different temperature than the photons, as happens for neutrinos at late times as discussed below. Similarly, the pressure can be expressed as P =T



4



i=all species

Ti T

4

gi 6π 2



∞ xi

(u2 − xi2 )3/2 du . exp (u − yi ) ± 1

(12)

The quantity geff (T ) in Eq. (11) is the effective number of relativistic degrees of freedom in all particles. This is sometimes approximately given by a step function generated by assuming each species becomes relativistic when the temperature exceeds its rest mass: geff (T ) =

 i=Bosons

 gi

Ti T

4 +

7 8

 j =Fermions

 gj

Ti T

4 .

(13)

This approximation, however, is only valid when the temperature is well above the rest mass of all species in the sum. Figure 1 compares the actual values of geff from Eq. (11) with the step function approximation. It is useful, however, to evaluate Eq. (13) in various limits when building cosmological models. For example, in the limit that all standard model particles are relativistic, one has three families of quarks and leptons. Within each family, there are two quarks that come in three colors and two spin states. So, there are 2 · 3 · 2 = 12 states for each quark family. Each charged lepton (e.g., the electron) has two spin states. However, each neutrino has only one helicity state (there are only left-handed neutrinos in the standard model). So, each family has 15 states times three families times two when including antiparticles. So the total number of

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G. J. Mathews and G. Tang

Fig. 1 Numerical values of geff from Eq. (11) compared to histogram approximation that each species becomes relativistic when the temperature exceeds its rest mass

fermion states is 90. To this one must add the photon and eight gluons, each of which has two helicity states, plus the W ± and the Z with spin= 1 and three helicity states each. The spin= 0 Higgs boson has only one helicity state. So the total number of degrees of freedom in the standard model for temperatures well above the Higgs mass (say, 1 TeV) is geff =

7 · 90 + 18 + 9 + 1 = 106.75. 8

(14)

This value is often employed in models of the very early universe. This value, however, could be much larger. For example, in the minimal supersymmetric model, one has twice the number of degrees of freedom due to the supersymmetric partner particles. In addition, there are five Higgs particles, three neutral plus two charged, plus the partner Higgsinos, so the total number of degrees of freedom would be something like 221.5. At lower temperatures, relevant for the start of big bang nucleosynthesis at T ∼ 1 MeV, the total mass-energy density of the universe at the epoch of nucleosynthesis

91 Big Bang Thermodynamics and Cosmic Relics

3363

is only due to photons, neutrinos, and electrons (plus their antiparticles): ρ = ργ + ρνi + ρe ,

(15)

so the effective number of degrees of freedom (before e+ − e− pair annihilation) is just geff = 2 +

7 · (2 · 3 + 2 · 2) = 10.75. 8

(16)

The Friedmann Equation Simplified During the radiation-dominated epoch, writing the total energy density as in Eq. (11) leads to a particularly simple form for the Friedmann equation: H =

2 1/2 1/2 T (8/3)π G geff T 2 = 1.660geff mpl

(17)

where m2pl = 1/G is the Planck mass (Kolb and Turner 1990). In conventional units, this is 2 H = 0.207 geff TMeV (sec−1 ), 1/2

(18)

where TMeV is the temperature in units of MeV. The age t = 1/(2H ) is then −1/2 mpl T2

t = 0.301geff

−1/2

−2 ∼ 2.42 geff TMeV (sec).

(19)

−2 sec. So, for geff = 3.36 after the electron pairs annihilate, t ∼ 1.3TMeV Equation (19) can also be inverted to give the temperature vs. time in the radiation-dominated universe: −1/4

T ≈ 1.55geff



t 1 sec

−1/2 MeV.

(20)

Entropy The evaluation of geff thus depends upon the temperature and number of degrees of freedom for the various species. For much of the early universe, collision times are rapid enough such that all species are in equilibrium at the same temperature. However, for temperatures below ∼1 MeV, the neutrinos will have decoupled from the background plasma. Thus, when the e+ −e− pairs annihilate, the photons will be

3364

G. J. Mathews and G. Tang

heated, but the neutrinos remain unchanged. To determine the neutrino temperature, one generally introduces conservation of entropy density s, where s=

2π 2 s 3 g T . 45 eff

(21)

Here, in the histogram approximation, 



s ≈ geff

gi

i=Bosons

Ti T

3 +





7 8

gj

j =Fermions

Ti T

3 (22)

is the effective number of degrees of freedom contributing to the entropy. Now the total entropy is conserved for particles in equilibrium. So, one can write 

 s geff (aT )3

before

  s 3 = geff (aT )

(23)

.

after

Just before the e+ − e− pairs annihilate, the neutrinos have already decoupled from the background plasma; hence, they do not contribute to the entropy. Therefore, only s = 2 × 2 × (7/8) + 2 = 11/2, while electron pairs and the photon contribute. So, geff s just after annihilation only photons remain geff = 2. Hence, (aT )before = (aT )after



s ) (geff after s (geff )before

1/3

 =

4 11

1/3 .

(24)

Hence, at a given value for a, the photons are heated by the pair annihilation, and the neutrinos are unaffected. The neutrino temperature remains lower than the photon temperature by the same factor: 

Tν Tγ



 =

4 11

1/3 .

(25)

This factor enters in all processes below ∼0,1 MeV, including big bang nucleosynthesis and the photon decoupling epoch as discussed in later chapters.

Baryon-to-Photon Ratio A key quantity for BBN is the present ratio η of the baryon number density to the photon number density. This quantity relates to the value of Ωb h2 deduced by Planck (Planck Collaboration 2020). Specifically, η≡

nb ≈ 2.738 × 10−8 Ωb h2 = 6.11 ± 0.04 × 10−10 . nγ

(26)

91 Big Bang Thermodynamics and Cosmic Relics

3365

To clarify this, consider that the baryon number density is nb =

ρb , mb

(27)

where mb is the mean baryon mass. The photon number density is 3  2ζ (3)T 3 T nγ = = 410.7 cm−3 2.7255 K π2

(28)

where a present-day CMB background temperature of T = 2.7255(6) K is adopted from Fixen (2009). Now, the baryon density relates to the critical density defined below. ρb = Ωb ρcrit , with ρcrit =

3H02 = 1.05375(13) × 10−2 h2 MeV cm−3 . 8π G

(29)

A subtle point about Eq. (27), however, is that the mean baryon mass mb depends upon the composition (Steigman 2006). Assuming a composition of nearly pure hydrogen and helium at the end of BBN, one can write      1 mH e Yp mb = m H 1 − 1 − 4 mH = 937.11 − 6.683(Yp − 0.25) MeV,

(30)

where Yp is the primordial helium mass fraction. Combining these equations leads to the baryon-to-photon ratio: −8

η = 2.733 × 10



2.7255 Ωb h (1 + .00717Yp ) kT 2

3 .

(31)

If one adopts Yp = 0.25, the above numerical value of Eq. (26) is obtained. From Eq. (28), one must conclude that there about 400 photons per cm3 in the universe and there are roughly 2 billion photons per baryon. This number was fixed in the epoch of baryogenesis (Sakharov et al. 1967). However, exactly how this number arises is not yet fully understood (Bödeker and Buchmüller 2021).

Vacuum Energy Density and Dark Energy To evaluate the Friedmann equation, one must also include the energy density from dark energy. The nature and origin of dark energy remains a mystery. Although dark energy as a vacuum energy is a natural consequence of quantum field theory, the observed magnitude of dark energy is much smaller than the quantum prediction.

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G. J. Mathews and G. Tang

To illustrate this, consider that the vacuum in quantum field theory comprises a collection of quantum harmonic oscillators. The nth energy level of a quantum harmonic oscillator is just 1 En = (n + )hω. ¯ 2

(32)

The ground-state vacuum energy must be given by the sum over all frequency modes: E0 =

1 i

2

h¯ ωi .

(33)

One can perform the sum by considering the system to reside in a box of volume V = L3 . Boundary conditions are then placed on the oscillations such that for some integer ni : L = λi · ni =

2π · ni , ki

(34)

where ki is the wave number. The sum over frequency modes can then be replaced by an integral: E0 =

1 hL ¯ 3 2 (2π )3

 ωk d 3 k.

(35)

The vacuum energy density is then obtained by dividing out the L3 . ρvac =

1 h¯ 2 (2π )3

 k · 4π k 2 dk =

4 hk ¯ max . 16π 2

(36)

Clearly, this diverges for kmax → ∞. If kmax is limited to below the Planck energy of Epl = 1.2 × 1019 GeV, then for kmax = Epl /h, ¯ the vacuum energy would be obs = Ω ρ ρvac ∼ 1077 MeV h¯ −3 , or 1092 g cm−3 . The observed value of ρvac Λ crit ≈ −30 −3 6 × 10 g cm for h = 0.68. This implies a discrepancy of about a factor of 10121 between theory and observation. As of yet there is no explanation for this discrepancy. To the degree that supersymmetry is an exact symmetry (which it is not), there could be a cancellation between bosons and fermions. However, supersymmetry was broken at a high temperature. Perhaps, there is an explanation in string theory, or some other unification of general relativity and quantum mechanics. For now, one simply takes the observed value of ρvac and asks what is it? The simplest explanation is that there exists a nearly perfect cancellation of the vacuum energy such that only a small value remains. This would be a simple constant energy density or cosmological constant as appears in the Friedmann equation.

91 Big Bang Thermodynamics and Cosmic Relics

3367

Cosmic Quintessence Another possible interpretation of the dark energy, however, is that it is the result of a self-coupled scalar field which in this case is called “quintessence.” In quintessence models, the vacuum energy can become time-dependent. This arises from a scalar field Q which obeys a Klein-Gordon equation of motion: ¨ + 3H Q˙ + dV = 0. Q dQ

(37)

The potential V (Q) defines the quintessence model. Construction of the energy momentum tensor for a quintessence field leads to a simple relation for the density and pressure: ρQ =

˙2 Q + V (Q) 2

(38)

PQ =

˙2 Q − V (Q). 2

(39)

The equation of state parameter is then ωQ ≡

˙ 2 /2 − V (Q) PQ Q = 2 . ˙ /2 + V (Q) ρQ Q

(40)

When V (Q) dominates, this approaches −1 and hence acts like a cosmological constant. Many scalar field models have been suggested as possible candidates for quintessence (e.g., Brax and Martin 2000). An important property for all quintessence models is tracking. This guarantees that independent of the choice of initial conditions, the evolution converges to a tracker solution. As an example, a common choice is the inverted power law potential originally proposed by Ratra and Peebles (1988): V (Q) = M (4+α) Q−α .

(41)

It has been shown (Ratra and Peebles 1988) that such inverse power law potentials lead to a tracking solution that maintains the condition: d 2V 9 2 = (1 − ωQ )[(α + 1)/α]H 2 . 2 dQ2

(42)

Although quintessence models are of interest, there are presently significant constraints on the time dependence of dark energy such that a significant change in the dark energy over time seems unlikely.

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G. J. Mathews and G. Tang

Relic Abundances and Nonequilibrium Thermodynamics Although for the most part the universe can be described as a system in which all components are in thermal equilibrium, there are a number of episodes in which various species fall out of equilibrium and freeze out leaving a relic abundance. This includes, for example, the formation of cold dark matter, the relic abundance of neutrinos, big bang nucleosynthesis, and the epoch of photon last scattering leading to the cosmic microwave background. In this section, we overview the basic formalism for these processes.

The Boltzmann Equation The Boltzmann equation describes the rate of change of the abundance of a species as the difference between the rates for producing and destroying that species, regardless of the equilibrium state. For the simple case of particles 1+2 ↔ 3+4, the Boltzmann equation for the number density of species 1 in an expanding space-time is given by 1 d(n1 a 3 ) = dt a3



d 3 p1 d 3 p2 d 3 p3 d 3 p4 (2π )3 2E1 (2π )3 2E2 (2π )3 2E3 (2π )3 2E4

×(2π )4 δ 3 (p1 + p2 − p3 − p4 )δ(E1 + E2 − E3 − E4 )|M |2 ×{f3 f4 [1 ± f1 ][1 ± f2 ] − f1 f2 [1 ± f3 ][1 ± f4 ].

(43)

The left-hand side of this equation just expresses the conservation of the number of particles in an expanding volume if the right-hand side vanishes. The interactions are counted on the right-hand side. Here, the Dirac delta functions account for momentum and energy conservation, while the quantities fi , i = 1 − 4 refer to the occupation probabilities for particles fi , while the factors 1 ± f refer to the Pauli blocking (minus) for fermions or Bose enhancement (plus) for bosons. The matrix element M contains the fundamental physics in the scattering. Now imposing energy conservation in the integral over E and for E − μ >> T , so that the ±1 from quantum statistics can be ignored (Kolb and Turner 1990; Dodleson 2003), then the distribution functions can be written as f (E) → eμ/T e−E/T ,

(44)

  1 d(n1 a 3 ) n3 n4 n1 n2 (0) (0) = n1 n2 σA |v| (0) (0) − (0) (0) , dt a3 n3 n4 n1 n2

(45)

and Eq. (43) reduces to

(0)

where ni

is an equilibrium number density of species i defined as

91 Big Bang Thermodynamics and Cosmic Relics

(0)

ni

 ≡ gi

d 3 p −Ei /T e , (2π )3

3369

(46)

and the averaged annihilation cross section is σA |v| ≡

1



(0) (0)

n1 n2

d 3 p2 d 3 p3 d 3 p4 d 3 p1 e−(E1 +E2 )/T (2π )3 2E1 (2π )3 2E2 (2π )3 2E3 (2π )3 2E4

×(2π )4 δ 3 (p1 + p2 − p3 − p4 )δ(E1 + E2 − E3 − E4 )|M |2 .

(47)

Abundance of Weakly Interacting Dark Matter As discussed above, there is strong evidence for the existence of cold dark matter, and the most plausible candidate is in the form of a weakly interacting massive particle (WIMP). Such a particle could be in equilibrium with the cosmic background plasma at high temperature. It would be formed in equal numbers of matter-antimatter pairs that can annihilate to form light particles l at a rate equal to their thermal production. Denoting the dark matter particle as χ , then we have ¯ χ + χ¯ ↔ l + l.

(48)

At some point, however, the dark matter annihilation rate per particle will diminish below the cosmic expansion rate so that the dark matter freezes out at a relic abundance. For this circumstance, one can utilize Eq. (45) to deduce the relic abundance. The light particles remain in equilibrium with the background plasma, so (0)

nl = nl ,

(49)

  1 d(nχ a 3 ) (0) 2 2 = σA |v| (nχ ) − nχ . dt a3

(50)

so that Eq. (45) reduces to

At this point, it is common to replace the scale factor with the temperature since T ∝ a −1 . Then introducing an auxiliary variable: Yχ ≡

nχ , T3

(51)

Eq. (50) then reduces to   dYχ 2 − Yχ2 , = T 3 σA |v| YEQ dt

(52)

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G. J. Mathews and G. Tang

where (0)

YχEQ ≡

nχ . T3

(53)

This can be simplified further by replacing time with temperature by introducing a variable: x≡

mχ , T

(54)

where m is the mass of the particle. Using the chain rule plus the relation between time and temperature in a radiation-dominated universe, −1/2 mpl , T2

t = 0.301geff

(55)

and the expansion rate 1/2

H = 1.66geff

T2 . mpl

(56)

One can reduce Eq. (52) to   dYχ λ 2 EQ 2 = − 2 Yχ − (Y ) , dx x

(57)

with λ≡

m3χ σA |v|

(58)

H (mχ )

where H (mχ ) ≡ H (T = mχ ) so that H (T ) = H (mχ )/x 2 . Alternatively, one can write ΓA x dYχ =− EQ dx H Yχ EQ



Yχ EQ

Yχ

2

 −1 ,

(59)

where the annihilation rate is ΓA = nχ σA |v| . Equations (57) and (59) do not have an analytic solution. An illustration of the numerical solution for Yχ is shown in Fig. 2. The relic abundance of Yχ depends upon the reaction rate as labeled. The qualitative behavior of Yχ shown in Fig. 2 is easy to discern. At high temperature, the reaction rates are sufficiently rapid that the abundance Yχ remains EQ equal to Yχ and the term in brackets vanishes. However, as the reaction rate

91 Big Bang Thermodynamics and Cosmic Relics

3371

Fig. 2 Numerical solution to Eq. (59) showing the dependence of the final relic abundance on the annihilation rate, as labeled

becomes slower than the expansion rate, the value of Yχ will become fixed, while EQ Yχ continues to diminish due to the Boltzmann factor e−x in Eq. (46). Hence, at late times such that x >> 1, Eq. (57) becomes λYχ2 dYχ =− 2 . dx x

(60)

This can be integrated analytically from freezeout xf to x∞ to become 1 1 λ − = . Y∞ Yf xf

(61)

Since the value at freezeout Yf is generally about an order of magnitude larger than the value at infinity Y∞ , it is a reasonable approximation to have Y∞ ≈

xf . λ

(62)

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G. J. Mathews and G. Tang

Thus, the relic abundance can be specified once the freezeout temperature xf is specified. This can be determined from the condition nEQ χ (xf )σA |v| = H (xf )

(63)

and is usually xf ≈ 10. That is, freezeout occurs at a temperature of about one tenth of the rest mass of the particle. What remains is to deduce the present relic density. First, note that  ρχ = mχ Y∞ T03

a1 T1 a0 T0

3 ,

(64)

where a1 T1 denotes the epoch at which the DM abundance was locked in. Just as in the case of the neutrino temperature described in the previous section, the conservation of entropy can be used to deduce the (aT ) ratio. That is, from Eq. (22), we have 

a1 T1 a0 T0

3

 =

s (0)  geff s (1) . geff

(65)

If we adopt a mass of mχ ∼ 1 TeV such that all standard model particles are present when the DM abundance is locked in, then (a1 T1 /a0 T0 )3 = (3.91/106.75) ≈ 1/27. The present closure contribution from dark matter is then Ωχ =

mχ Y∞ T03 H (mχ )xf T03 ρχ = = . ρc 27ρc 27m2χ σA |v| ρc

(66)

Finally, inserting T = mχ into Equation (56), a rule of thumb for the dark matter contribution can be written as Ωχ = 0.3h−2



xf 10



g∗ (m]χ ) 100

1/2 

 10−37 cm2 . σA |v|

(67)

Note that since we are in units of c = 1, then the average σA |v| has units of cm2 . Equation (67) says that the dark matter component can be accounted for by a massive ∼TeV particle with an annihilation cross section suggestive of the weak interaction, hence the designation as a weakly interacting massive particle (WIMP) . There are several possible candidates for such a particle. Among them is the lightest supersymmetric particle, possibly a Higgsino, photino, or Zino. Such a particle would be stable against decaying to normal particles by R-parity. Another possibility is a neutralino made up of a linear superposition of the photino, the Zino, and the Higgsino. Examples of other models for WIMPS are from a universal extradimensional (Appelquist et al. 2001) model leading to the lightest Kaluza-Klein

91 Big Bang Thermodynamics and Cosmic Relics

3373

particle that is stabilized by KK-parity, or the little Higgs model (Arkani-Hamed et al. 2002) leading to the lightest R-odd particle, stabilized by R-parity.

Neutrino Decoupling From the standpoint of nuclear physics, neutrino decoupling is, perhaps, the most relevant as the decoupling of the electron neutrinos fixes the neutron abundance during big bang nucleosynthesis. In this case, the relevant weak reactions are those converting neutrons to protons and the inverse: p + ν¯ e ↔ n + e+ p + e− ↔ n + νe n ↔ p + e− + ν¯ e .

(68)

Note that the reactions of νμ and ντ neutrinos decouple at a higher temperature ∼10 MeV than the electron neutrinos that decouple at a temperature around 1 MeV. However, this does not affect the neutrino temperature as discussed in Eq. (76), since there are no other annihilations affecting the photon temperature between 1 and 10 MeV. For this case, the relevant Boltzmann equation is that evolving neutrons and protons. As long as the weak reactions are more rapid than the expansion rate, the relative equilibrium ratio of neutrons to protons is just given to a good approximation by a Boltzmann factor involving the neutron-proton mass difference Δm = mn − mp = 1.293 MeV: 

(0)

nn

(0)

np

=

mn mp

3/2

e−Δm/T ≈ e−Δm/T .

(69)

It is useful to deduce the Boltzmann equation (45) in terms of the neutron mass fraction: (0)

Xn = XnEQ =

nn (0) nn

(0) + np

=

1 (0) (0) 1 + np /nn

.

(70)

Then Eq. (45) reduces to   dXn = λnp (1 − Xn )e−Δm/T − Xn . dt

(71)

The quantity λnp is the sum of the rates for the three weak reactions above. Each rate requires an integral over the phase space integral. The simplest is the neutron decay n → p + e− + ν¯ e . This can be written as

3374

G. J. Mathews and G. Tang

λn→p+e− +¯νe =

G2 1 2 = F3 (1 + 3gA )m5e λ0 , τn 2π

(72)

where τn = 879.4 ± 0.6 sec is the neutron lifetime (Zyla et al. 2020), GF is the Fermi coupling constant, and gA is the axial vector coupling constant, and 

q

λ0 =

dx x(x − q)2 (x 2 − 1)1/2 ≈ 1.636,

(73)

1

where q = Δm/me and x ≡ me /T . Similarly, the p + e− → n + νe rate can be written as λp+e− →n+νe =

1 τn λ0



∞

dx x

x(x − q)2 (x 2 − 1)1/2 . [1 + exp (xme /T )][1 + exp ((q − x)me /Tν )] (74)

An approximation to the total λnp can be written as (Bernstein 1988) λnp ≈

255 (12 + 6x + x 2 ). τn x 5

(75)

Comparing this rate to the expansion rate for a radiation-dominated universe, one obtains λnp ∼ H



T 0.8 MeV

3 ,

(76)

suggesting that the freezeout of electron neutrinos occurs at ∼0.8 MeV. Just as in the case of the dark matter freezeout, one can replace the time dependence of Eq. (71) with the temperature evolution variable x to obtain  xλnp −x dXn = e − Xn (1 − e−x ) , dx H (Δm)

(77)

where H (Δm) ≈ 1.13 sec−1 is the expansion rate when T = Δm. A numerical solution to Eq. (77) indicates that the neutron abundance freezes out at about 0.8 MeV corresponding to Xn ≈ 1/6, but neutrons will continue to decay until the onset of nucleosynthesis at T ≈ 0.1 MeV reducing the neutron mass fraction to Xn ≈ 0.12. As will be described in the chapter on BBN, most neutrons end up as 4 He, and since each neutron couples with one proton of approximately the same mass, the primordial mass fraction of 4 He will be Yp ≈ 2Xn ≈ 24%.

91 Big Bang Thermodynamics and Cosmic Relics

3375

Photon Decoupling and the CMB The last nonequilibrium freezeout to consider is the decoupling of photons from the primordial plasma leading to the formation of the cosmic microwave background (CMB). In this case, photons are at first tightly coupled to the background plasma by Compton scattering from free electrons. Once the free electrons are captured by protons to form atomic hydrogen, however, the photons can escape. Here, the relevant reaction is e− + p ↔ H + γ .

(78)

In equilibrium, Eq. (45) reduces to (0) (0)

ne np ne np = . (0) nH nH

(79)

Inserting the distribution functions for the nonrelativistic particles leads directly to the Saha equation:   gH me mp T −3/2 B/T nH = e , ne np ge gp 2π mH

(80)

where B = me + mp − mH = 13.6 eV is the electron binding energy for hydrogen. At this point, it is convenient to define the free electron fraction: Xe ≡

np ne = . ne + nH np + nH

(81)

Then the Saha equation can be rewritten in terms of this ionization fraction: √  3/2 1 − Xe T 2 2ζ (3) = eB/T , η √ b me Xe2 π

(82)

where ηb is baryon-to-photon ratio from Eq. (26). This is also an important parameter in big bang nucleosynthesis. One can use the Saha equation to infer the epoch of photon decoupling as illustrated in Fig. 3 that shows the free electron fraction as a function of temperature. The red line indicates the temperature at which the ionization is 50%. As the universe expands and cools, Fig. 3 shows that the ionization fraction plummets rapidly at a temperature of about Tdec ≈ 3700 K or about 0.3 eV. An important parameter is the redshift at photon decoupling, z∗ ≡ z(Xe = 0.5). Figure 3 gives z∗ ≈ Tdec /T0 ≈ 1300. The more detailed Planck analysis (Planck Collaboration 2020) deduced a value of z∗ = 1089.80 ± 0.21. The difference presumably arises from the contribution of 4 He and the departure from equilibrium.

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Fig. 3 Free electron fraction Xe as a function of temperature

Finally, a solution of the relic ionization fraction after decoupling requires a solution of the Boltzmann equation. For this case, we can write the rate of change of the free electron density as 2 n˙ e + 3H ne = σrec |v| [n2e − (n(0) e ) ]

(83)

where σrec is the recombination cross section σrec = 4.7 × 10−24 (1 eV/T)1/2 cm2 . The numerical solution to Eq. (83) gives Tf ≈ 0.25 eV, and Xe (∞) ≈ 2.7 × 10−5 (Ωb h)−1 . This value, however, increased as the epoch of star formation began. The subsequent epoch of reionization introduced the “dark ages” that endured until z ≈ 7 (Planck Collaboration 2020). Acknowledgments Work at the University of Notre Dame was supported by the U.S. Department of Energy under Nuclear Theory Grant DE-FG02-95-ER40934.

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References T. Appelquist, H.-C. Cheng, B.A. Dobrescu, Phys. Rev. D 64(3), 035002 (2001) N. Arkani-Hamed, A.G. Cohen, E. Katz, A.E. Nelson, J. High Energy Phys. 2002(7), 034 (2002) J. Bernstein, Kinetic Theory in the Expanding Universe (Cambridge University Press, London, 1988) D. Bödeker, W. Buchmüller, Rev. Mod. Phys. 93, 035004 (2021) P. Brax, J. Martin, Phys. Rev. D 61, 103502 (2000) S. Dodleson, Modern Cosmology (Academic Press, San Francisco, 2003) D.J. Fixen, Astrophys. J 707, 916 (2009) E.W. Kolb, M.S. Turner, The Early Universe (Addison-Wesley, Menlo Park, 1990) Planck Collaboration, A&A 641, A6 (2020) B. Ratra, P.J.E. Peebles, Phys. Rev. D 37, 3406 (1988) A.D. Sakharov, JETP Lett. 5, 24 (1967) G. Steigman, J. Cosmo. Astropart. Phys. 115, 357 (2006) P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) and 2021 update

Big Bang Nucleosynthesis: Nuclear Physics in the Early Universe

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Brian D. Fields

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory: Cosmological Light-Element Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helium Production in Early Universe and the Sun: A Comparison . . . . . . . . . . . . . . . . . . . . Nuclear Reactions and BBN Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Astronomical and Cosmological Observables for BBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light-Element Abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cosmic Microwave Background: An Independent Baryometer . . . . . . . . . . . . . . . . . Standard BBN: Concordance, Implications, and the Lithium Problem . . . . . . . . . . . . . . . . . BBN Probes of New Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3380 3381 3386 3387 3389 3389 3391 3392 3395 3397 3398

Abstract

We summarize the status of big bang nucleosynthesis (BBN), which describes the production of the lightest nuclides during the first 3 min of cosmic time. We emphasize the transformational influence of cosmic microwave background (CMB) experiments culminating today with Planck, which pins down the cosmic baryon density to exquisite precision. Standard BBN combines this with the standard model of particle physics and with nuclear cross section measurements – notably recent precision measurements of d(p, γ )3 He by the LUNA collaboration. These allow BBN to make tight predictions for the primordial light-element abundances, with the result that deuterium observations agree spectacularly with theoretical expectations, and helium observations are in good

B. D. Fields () Departments of Astronomy and Physics, University of Illinois, Urbana, IL, USA Illinois Center for the Advanced Study of the Universe, Urbana, IL, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_111

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agreement. This CMB/BBN concordance marks a profound success of the hot big bang, and BBN and the CMB together now sharply probe cosmology, neutrino physics, and dark matter physics back to times of around 1 s. But this success is tempered by lithium observations (in metal-poor halo stars) that are significantly discrepant with BBN+CMB predictions. Recent work strengthens the case that the resolution of this “lithium problem” could well lie in the stellar astrophysics of lithium depletion, while new physics solutions are possible but becoming ever more tightly constrained. We conclude with an outlook for how future CMB, astronomical, and laboratory measurements can better probe new physics and bring into focus the solution to the lithium problem.

Introduction Big bang nucleosynthesis (BBN) accounts for the production of the lightest nuclides – D, 3 He, 4 He, and 7 Li – throughout the universe during the first t ∼ 1 s to ∼3 min of cosmic time. Light-element abundance predictions rest on well-studied MeV-scale weak and nuclear physics and are in broad agreement with astronomical abundance measurements (Wagoner et al. 1967; Steigman 2007; Iocco et al. 2009; Pitrou et al. 2018; Fields et al. 2020). This concordance marks a great success of the hot big bang cosmology and stands as our earliest reliable probe of the universe, deep within the radiation-dominated era. The comparison between theory and observations also provides a measure of a fundamental cosmological parameter: the density of baryons in the universe today. BBN thus stands at the crossroad of nuclear physics, cosmology, and astrophysics. Nuclear physics plays a central role in BBN, which represents the first appearance of nuclear reactions in the universe. As we will see, BBN calculations require measurements of nuclear cross sections and the neutron lifetime at precisions unprecedented in nuclear astrophysics, and continued progress will demand a new generation of measurements. BBN also represents as a cornerstone of cosmology, representing the earliest cosmic probe relying on well-understood standard model physics and, as we will see, can test physics beyond the standard model. In astrophysics, BBN determines the baryonic content of the universe after 3 min, which represents the starting point for galactic chemical evolution (Timmes et al. 1995; Chiappini et al. 1997; Nomoto et al. 2013). That is to say, BBN sets the initial conditions for the formation of the first stars (Abel et al. 2002; Bromm and Larson 2004) and, more generally, establishes the domination of hydrogen and helium as the most abundant elements up to and including the present. This has profound effects for stellar and galactic evolution. BBN has enjoyed dramatic progress over the past two decades, largely due to advances in the cosmic microwave background (CMB). The CMB is cosmic blackbody radiation – the remains of the cosmic fireball – and is nearly perfectly isotropic, demonstrating that the early universe was almost perfectly homogeneous. But the small temperature fluctuations in the CMB are critical, as they are evidence

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of small density perturbations that were the seeds of cosmic structures today. These fluctuations also encode a wealth of cosmological parameters. The first acoustic peaks now measure the cosmic baryon density precisely and independently of BBN. A comparison of these two measures of cosmic baryons provides a strong and successful test of the basic hot big bang framework (Schramm and Turner 1998a; Cyburt et al. 2003). Moreover, BBN and the CMB now separately and jointly measure the total radiation content at nucleosynthesis, expressed as the effective number Nν of neutrino species; this opens a window to exotic physics. Yet despite these remarkable successes, BBN has for almost 20 years faced a 4 − 5σ discrepancy between 7 Li predictions and observations. This “lithium problem” has been studied intensively from all angles and is not yet resolved (Fields 2011). But as we will see, recent new evidence in the form of 6 Li isotope searches support the view that stellar evolution effects have systematically depleted lithium in essentially all halo stars. In this chapter we highlight role of nuclear physics in BBN and the broad impact that nuclear physics enables. This work draws heavily on recent work (Yeh et al. 2021, 2022; Fields et al. 2020; Cyburt et al. 2016) without which this review would not be possible. For broader discussion and more complete bibliography, there are many excellent reviews, e.g., Pitrou et al. (2018), Cyburt et al. (2016), Pisanti et al. (2008), and Steigman (2007).

Theory: Cosmological Light-Element Production BBN calculations follow weak and nuclear reactions in an expanding, cooling, homogeneous universe. Gamow and his students recognized that the early universe would be hot enough to host nuclear reactions and that at very early times and high temperatures T  1 MeV, baryons would only be the in the form of free neutrons and protons (Alpher et al. 1948). Indeed, Alpher and Herman (1948) realized that the blackbody photons present at early times should still be present and predicted that after cooling with cosmic expansion today the universe should have T ∼ 5 K; this anticipated the discovery of the cosmic microwave background (CMB) by more than a decade! Wagoner et al. (1967) made the first complete BBN calculation, and reaction studies began in Wagoner (1969). There has been continuous effort thereafter, in recent years with efforts from groups in Paris (Coc et al. 2004, 2012; Pitrou et al. 2018; Pisanti et al. 2021), Naples (Serpico et al. 2004; Pisanti et al. 2008, 2021), and our own group (Cyburt et al. 2004; Cyburt 2004; Fields et al. 2020). During BBN there is a thermal bath of particles that includes both relativistic (mc2  kT ) and nonrelativistic (mc2  kT ) species, which are also referred to as radiation and matter respectively. Temperatures during BBN span broadly from T ∼ 1 MeV to ∼100 keV in energy units, roughly corresponding to T9 = T /109 K ∼10 to 1. Under these conditions, photons and all three species of standard neutrinos are relativistic and thus count as radiation, while the baryons are always nonrelativistic and thus count as matter. Electron-positron pairs e± are relativistic at the start of our

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calculations but become nonrelativistic by the end, and so their properties must be calculated with care using the full Fermi-Dirac distribution. Cosmic expansion is quantified via the scale factor a(t) that measures the separation of particles comoving with the universe; this leads to Hubble’s law: galaxies move apart with speed v = H r, with r the separation and H = a/a ˙ the expansion rate (here and throughout a˙ = da/dt). The growth of the scale factor is set by the Friedmann equation that is the equation of motion for the cosmos:  2  a˙ 8π 8π 8π  Gρtot ≈ Gρrad = G ργ + ρe± + ρν = H2 = a 3 3 3

(1)

where ρtot is the total mass/energy density and ρrad = urad /c2 is the density in radiation. The mass density of cosmic baryons scales as ρ ∝ 1/a 3 , reflecting baryon number conservation and volume dilution. Because cosmic temperature drops with scale factor as T ∝ 1/a, that means that baryons – and matter generally – have ρm ∝ T 3 . On the other hand, thermal radiation has ρrad ∝ T 4 . The stronger temperature dependence for radiation means that at early times, the cosmic density is dominated by radiation. BBN lies deep within this radiation-dominated era; hence, only the radiation terms are important in Eq. (1). For radiation-dominated epochs, we have H 2 ∼ T 4 ∼ a −4 where we again use T ∝ 1/a. This gives a/a ˙ ∼ 1/a 2 , so a ∝ t 1/2 . Combining these expressions, we can find a relation between time and temperature: t ≈ 1s



1 MeV T

2 (2)

and so the BBN temperature range from 1 to 0.07 MeV spans from around 1 to 200 s. We see that cosmic element production spans the first 3 min (Weinberg 1977). Standard big bang nucleosynthesis (SBBN) embodies the most straightforward marriage of the particle content and interactions of the standard model of particle physics, with the best-fit Λ CDM cosmology. This includes cosmic radiation comprised of the photon/pair plasma plus additional radiation in the form of the three active neutrinos species (and their antiparticles). Their interactions with each other and with nucleons are those of the standard model of particle physics, and nucleons obey standard nuclear physics. There are negligible dynamical effects from weakly interacting nonrelativistic dark matter and from dark energy. Within SBBN, the number ratio of baryons to photons: η≡

nb = 6.03 × 10−10 nγ



Ωb h2 0.022

 (3)

is unchanged from BBN through recombination to today. This is because baryon conservation gives nb ∝ a −3 , while nγ ∝ T 3 for a blackbody distribution, so we have η ∝ (aT )−3 , and aT is constant (i.e., temperature redshifts) after e±

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annihilation. This represents the only free parameter of the theory. Moreover, the baryon-to-photon ratio is proportional to the present baryon density parameter Ωb h2 = ρb /ρcrit , where ρcrit = 3H 2 /8π G ≈ ρtot , and where H0 = 100 h km sec−1 Mpc−1 . Our story begins with T > 1 MeV, when weak reactions are fast compared to the expansion rate. This means that all neutrino species are thermally populated along with their antiparticles and in equilibrium with the baryon-photon plasma. Hayashi (1950) and Alpher et al. (1953) pointed out the importance of neutrinos in setting the neutron-to-proton ratio. At early times while the weak interactions are in equilibrium, these drive neutron-proton interconversion via: νe p ↔ e− n

(4)

ν¯ e n ↔ e+ p

(5)

as well as free neutron decay which at early times is slower than these. At T  1 MeV the neutrino interaction rates per nucleon Γν = nν σνN c  H is much larger than the expansion rate, so the n ↔ p reactions come into a thermal equilibrium. When this equilibrium holds the neutron-to-proton ratio: n 2 = e−(mn −mp )c /kT p

(6)

follows the Boltzmann expression for a two-level system (the nucleon) where the neutron plays the role of the excited state while the proton becomes the ground state. Equation 6 shows that the n/p ratio drops as the universe cools, as long as the reactions in Eq. (5) are effective. But the weak reactions fall out of equilibrium when the weak rate per nucleon ΓνN = σνN cnν ∼ T 5 becomes smaller than the expansion rate H ∼ T 2 . This occurs at the freezeout temperature T ∼ 1 MeV. At this point the neutron-to-proton ratio is essentially fixed to (n/p)f = e−Δm/Tf ∼ 1/7. The underabundance of neutrons relative to protons already foretells that the most abundant nuclide coming out of BBN will be ordinary hydrogen. At weak freezeout, the cosmic baryon density is around ρb ∼ 0.1 g/cm3 and drops thereafter. With these modest densities and short timescales, the buildup of elements in BBN thus requires 2-body reactions, starting with free n and p. The only possible first step is p(n, γ )d, but because of the large number of photons per baryon nγ /nb = 1/η ∼ 109 , the reverse reaction d(γ , n)p remains effective even when the temperature drops below the deuteron binding T < Bd = 2.22 MeV. The fraction of photons with energy > Bd is roughly fγ = nγ (> Bd )/nγ ∼ (Bd /T )2 e−Bd /T , and so deuterium production is only effective when nγ (> Bd )  nb , or fγ < η. This gives the temperature for deuterium emergence Td ∼ Bd / ln η−1 ∼ 0.1 MeV, corresponding to T ∼ 109 K, or a time of ∼100 s. Once this “deuterium bottleneck” is overcome, strong reactions proceed rapidly. The key reactions are well-established (Smith et al. 1993) and are illustrated in

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Fig. 1; many other reactions are in the BBN codes, but all of the others make minimal contributions to the main light elements produced: D, 3 He, 4 He, and 7 Li. The main result is to make the most tightly bound light nucleus, 4 He, which is thus the most favored thermodynamically. Because 4 He has equal numbers of neutrons and protons, and neutrons are rarer, their abundance sets the final 4 He production. That is to say, to an excellent approximation, all neutrons go into 4 He, and the resulting 4 He baryonic mass fraction is: Yp =

ρ(4 He) n/p ≈ 25% ≈ ρB 1 + n/p

(7)

where n/p ∼ 1/7 is the neutron-to-proton ratio just prior to the onset of nuclear reactions. The other main light species have much smaller abundances and are the results of incomplete burning to 4 He. A small amount of primordial 6 Li is also made, but at 6 Li/H ∼ 10−14 it is unobservably low. BBN produces even smaller amounts of heavier species such as 9 Be, 10,11 B (Thomas et al. 1993), and CNO (Iocco et al. 2007; Coc et al. 2012). The lack of heavy element production is largely due to the relative low densities

Fig. 1 Simplified BBN nuclear network, showing the 12 reactions important for determining lightelement abundances. Uncertainties in these reaction rates propagate to give the error bands in to give the error bands in Figure 2

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and short timescales during BBN, which require only two-body reactions and thus suppress the triple-α reaction that produces carbon and beyond in stars. The mass gaps at A = 4 and 8 also play a role (see Scherrer and Scherrer 2017). As we have seen, in SBBN, the baryon-to-photon ratio η is fixed and is the one free parameter determining light-element abundances. Figure 2 presents the socalled Schramm plot showing how the light-element predictions depend on η. We

D/H

4

He mass fraction

0.27 0.26 0.25 0.24 0.23

10−3

He/H

10−4

3

baryon density parameter ΩBh2 10−2

C M B

10−5

7

Li/H

10−9

10−10 10−10

baryon-to-photon ratio η = nb/nγ

10−9

Fig. 2 “Schramm plot” BBN theory predictions for light nuclide abundances versus the cosmic baryon-to-photon ratio η. Curve widths: 1σ theoretical uncertainties. Cyan vertical band: Planck CMB determination of ηCMB (Aghanim et al. 2020). Yellow boxes: light-element observations and corresponding η ranges. Hatched vertical band: D/H+Yp concordant range for ηBBN . (From BBN review in Particle Data Group et al. 2020)

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see that 4 He has a baryonic mass fraction always about Yp ≈ 25%, with a slow rise as we consider universes with larger η. Thus, the main elements produced by BBN are ordinary hydrogen and 4 He. The other light species are trace components left over due to incomplete burning when the strong reactions freeze out. The next most abundant species is deuterium, which drops sharply in Fig. 2: as we consider universes with larger η, which gives higher baryon density at fixed T and thus deuterium burning is more rapid and more complete. A similar story accounts for the drop in 3 He. Figure 2 shows that the 7 Li abundance is very low, 7 Li/H ∼ 10−10 to 10−9 . We also see that 7 Li declines to a minimum and then increases. This is because at low η, mass-7 is produced as 7 Li via t (α, γ )7 Li. But at high η, mass-7 is dominantly produced via 3 He(α, γ )7 Be, with the 7 Be decaying to 7 Li much later via electron capture. Figure 2 thus sums 7 Be+7 Li in presenting the observable 7 Li/H ratio. The high Coulomb barrier for these reactions, along with the need to first produce 4 He, leads to the low 7 Li abundance.

Helium Production in Early Universe and the Sun: A Comparison It is instructive to compare helium production in BBN with 4 He production in another well-studied site: the Sun. Both are central topics of study in nuclear astrophysics and involve major experimental campaigns to measure cross sections to high precision (see, e.g., Adelberger et al. (1998, 2011) for evaluations of reactions for solar neutrino production). There are several reactions in common between BBN and solar 4 He production, which therefore hold particular interest. These are: 1. d(p, γ )3 He 2. 3 He(α, γ )7 Be, and 3. 7 Li(p, α)4 He. These reactions play a key role in BBN as we have seen, but also in understanding neutrino production and physical condition in the solar core (see, e.g., reviews by Bahcall 1989; Haxton et al. 2013). As a result, progress in understanding these reactions benefits both fields. Indeed, the success of the standard solar model allows the Sun to serve as a probe of these reactions, constraining nuclear solutions to the lithium problem (Cyburt et al. 2004). Yet even as BBN and solar 4 He production are closely linked, there are important differences between these processes. One is that the central temperature of the Sun T ,c = 0.016×109 K (Bahcall et al. 2006) is constant on human timescales, whereas the expansion of the universe leads to a time-varying cosmic temperature during BBN (Eq. 2), spanning a much higher range TBBN ∼ (1 − 10) × 109 K. The Gamow window for BBN reactions thus corresponds to significantly higher energies than in the Sun. Consequently, BBN requires knowledge of the energy dependence of

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the relevant cross sections or equivalently S(E) factors. In contrast, solar reactions essentially require only S(0). Moreover, the higher BBN energies are much more accessible experimentally. Another major difference between BBN and solar 4 He production is qualitative: free neutrons are abundant at the outset of BBN, while they are absent in stellar hydrogen burning. As we have seen, as BBN begins, neutrons are populated via proton interactions with the abundant population of cosmological neutrinos. The action of nuclear reactions in BBN is thus effectively to incorporate (most of) these free neutrons into 4 He. By contrast, in stellar hydrogen burning, a weak reaction is needed to occur within the chain, dominantly pp → de+ νe for the Sun and stars with mass M  1.3 M , and the CNO cycle for stars of higher mass. We thus see that the experimental needs for BBN and the standard solar model reactions are closely related but not identical. Progress in each aids the other.

Nuclear Reactions and BBN Uncertainties Cosmology is now a precision science, and BBN must follow suit. As a result, considerable effort has gone into quantifying the BBN error budget. This is illustrated by the widths of the curves in Fig. 2, which reflect the uncertainties in each of the light-element predictions. These errors are dominated by uncertainties in the nuclear reaction rates and thus ultimately the reaction cross section for the key reactions appearing in Fig. 1. Fortunately, these cross section can be measured at energies relevant for BBN, in contrast to much of stellar nucleosynthesis where it is necessary to extrapolate to low energies that often are inaccessible or impractical to reach. Following Smith et al. (1993), each reaction is studied in energy space, with systematic and statistical errors separately evaluated. As the need for precision grows, systematic errors become ever more important, and it is critical that future measurements consider these carefully and report them as completely as possible. The cross section energy dependences have been quantified via polynomial fits for some groups (e.g., Pisanti et al. 2008; Cyburt et al. 2004), and in theory-based calculations in others (Coc et al. 2012; Descouvemont et al. 2004; Iliadis and Coc 2020). When using the same inputs, there is good agreement between groups; where differences arise, it is due to different treatments of the reaction rates and their errors. The cross sections and their uncertainties as a function of energy are then averaged over a Maxwell-Boltzmann distribution to derive thermonuclear rates and errors as a function of temperature. These are propagated in BBN codes using Monte Carlo methods. The nuclear reaction uncertainties link the evolution of different isotopes and lead to well-understood correlations among the light elements (Fiorentini et al. 1998). Scaling relations for light-element abundances with cosmological inputs and nuclear reactions are widely available (Yeh et al. 2021). Recent important studies include Coc et al. (2012), Pitrou et al. (2018), Cuoco et al. (2004), Pisanti et al. (2008), Cyburt et al. (2016) and Fields et al. (2020).

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In Fig. 3 we summarize the BBN uncertainty budget in the light-element abundances as a function of η. Plotted for each nuclide is the 1σ uncertainty divided by the mean abundance μ, i.e., the percent error. Overall, the predictions are quite tight. We see that the 4 He prediction error is far below 1%. The largest errors are for 7 Li, where the uncertainties are ≈14% for η values of interest; this largely reflects the uncertainties in the dominant mass-7 production reaction 3 He(α, γ )7 Be and the destruction channels 7 Be(n, p)7 Li and 7 Li(p, α)4 He. Well-targeted experiments can make an important impact on BBN. A recent example of this is the LUNA measurement of d(p, γ )3 He (Mossa et al. 2020). This reaction had been the dominant contributor to the uncertainty in the cosmic D/H abundance. Mossa et al. (2020) make careful measurements within and surrounding the BBN energy range, with tight control of statistical and systematic errors. As a result, the uncertainty for this reaction dropped by about a factor of 2. Consequently, the D/H uncertainty decreased significantly (Pisanti et al. 2021; Pitrou et al. 2021a, b; Yeh et al. 2021), as illustrated in Fig. 3. Future improvements in D/H now will require more precise measurements of d(d, n)3 He and d(d, p)t.

0.200

baryon density Ωbh2 10−2 This work FOYY

0.175 0.150

σ7 Li 7 Li

0.125 0.100 σ3 He 3 He

0.075

σD D

0.050 0.025 0.000 −10 10

10 ×

σYp Yp

baryon-to-photon ratio η = nb/nγ

10−9

Fig. 3 Fractional errors in the BBN light-element predictions shown in Fig. 2. For each light species, we plot the reaction uncertainty σ divided by the mean μ. We see that the 4 He abundance is quite well determined, while the largest uncertainties are for 7 Li whose production is dominated by just three reactions. Dotted curves are from Fields et al. (2020), while solid curves include improved d(p, γ )3 He measurements from Mossa et al. (2020). (Figure from Yeh et al. 2021)

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Astronomical and Cosmological Observables for BBN In this section we summarize the astronomical and cosmological observations used for BBN. We first describe the astronomical observations used to infer primordial light-element abundances. Afterward, we turn to the CMB.

Light-Element Abundances BBN predicts cosmic abundances of the light nuclides after ∼3 min. Because the early universe was almost perfectly homogeneous, these abundances were the same everywhere and remained so throughout the epoch of recombination when the present-day CMB photons began their journey to us. The cosmic composition only began to change later, when the first stars ignited (Karlsson et al. 2013; Frebel and Norris 2015). Subsequent generations of stars alter the light-element abundances from their primordial values, e.g., stars destroy deuterium while producing 4 He. Fortunately, nucleosynthetic processing of baryons through stars also leads to production of heavy elements, the so-called metals such as O and Fe, the mass fraction of which is sometimes collectively denoted Z. Thus, to infer the primordial abundances of light elements, one seeks systems that have suffered the least processing and thus have the lowest possible heavy element abundances. There is no such system where all of the light elements can be measured at once, so instead different species are measured where conditions are the best for that element. Values adopted here follow those of Fields et al. (2020) and Yeh et al. (2022). Deuterium The best sites for deuterium measurements are in gas clouds of highredshift protogalaxies that happen to lie along the sightlines to quasars. Neutral hydrogen atoms in the protogalaxies absorb the quasar light at their characteristic Lyman series wavelengths, most strongly in the Lyman-α transition; these systems are thus known as Lyα absorbers. These clouds are typically at redshift z ∼ 3 and show relatively low metallicity, ∼10−2 of solar, so deuterium destruction (“astration”) should be minimal and D/H close to primordial. Because the deuterium has a slightly different reduced mass μ than ordinary hydrogen, its Lyman series has an isotope shift of δλ/λ = −δμ/μ = −me /2mp = −82 km s−1 /c. This places the deuterium Lyman lines blueward of the main hydrogen feature, allowing for both D and H to be measured and D/H inferred. Observations of D/H use high-resolution spectra with multiple Lyman transitions to measure both H and D very precisely. A small fraction of systems allows for such measurements. Using systems with the highest-quality measurements (Pettini and Cooke 2012; Cooke et al. 2014, 2016, 2018; Riemer-Sørensen et al. 2015; Balashev

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et al. 2016; Riemer-Sørensen et al. 2017; Zavarygin et al. 2018), a weighed average gives:   D = (2.55 ± 0.03) × 10−5 H obs

(8)

a 1% determination! This precision will play a crucial role in the power of BBN as a probe of cosmology and new physics. Helium-3 Because 3 He is by far the subdominant helium isotope, it is challenging to measure. At present the only successful detections are in gas found in our own Milky Way galaxy (Bania et al. 2007). These are high-metallicity systems that have been substantially processed and thus are not appropriate for determining the primordial 3 He abundance (Bania et al. 2007; Olive et al. 1995). Consequently, we do not use these data and do not have an observational constraint on (3 He/H)obs (Vangioni-Flam et al. 2003). Helium-4 Helium is observed via its atomic emission in hot gas clouds. The noble gas helium has a high ionization potential and thus requires high temperatures to excite the atoms. The best such astrophysical sites for our purposes are in starforming clouds found inside low-metallicity dwarf galaxies. Over many years, many such systems have been found, and an increasingly sophisticated analysis procedure has been developed to infer the helium abundance (Izotov and Thuan 1998, 2004; Peimbert et al. 2002; Aver et al. 2015, 2021), traditionally expressed as the mass fraction Y = ρ(He)/ρtot , for systems with a range of metallicity Z. An extrapolation to zero metallicity gives a primordial abundance (Aver et al. 2022): Yp = 0.2488 ± 0.0033

(9)

with other recent studies giving similar results (Aver et al. 2021; Kurichin et al. 2021; Hsyu et al. 2020).

Lithium-7 Lithium is observed in the most primitive stars in our galaxy: metalpoor halo stars (extreme Population II). The neutral lithium line (a blended doublet) is seen in absorption of the stellar continuum spectrum and represents the lithium abundance in the outer atmosphere (photosphere) of the star. A longstanding concern is that convective motions in the star could carry the fragile 7 Li atoms to interior regions where they could be destroyed. But for a set of metal-poor stars around 1 M , these effects are shown to be minimized, and (Spite and Spite 1982, 1985) showed that these stars show a constant Li/H abundance over more than an order of magnitude in metallicity. This “Spite plateau” indicates that lithium is primordial. Ryan et al. (2000) found that the scatter in the plateau is very small and found an abundance:

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Li H



= (1.6 ± 0.3) × 10−10 .

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(10)

obs

While it is clear 7 Li is primordial, subsequent work has challenged whether the Spite plateau and thus Eq. (10) gives the primordial value. Sbordone et al. (2010) found that at extremely low metallicities, the Li/H dispersion becomes large, always scattering below the plateau. Indeed, stellar evolution theory implies that some lithium depletion must occur over time in stellar atmospheres, due to a combination of nuclear destruction in the interior, gravitational settling, and rotation (e.g., Michaud et al. 1984; Vauclair and Charbonnel 1998; Korn et al. 2006; Pinsonneault 1997).

The Cosmic Microwave Background: An Independent Baryometer The CMB we observe today is the remains of the “cosmic fireball” – nearly isotropic radiation that has spectrum in frequency or wavelength that very precisely follows the blackbody form (see, e.g., Peebles 1993). The near isotropy confirms the basic cosmological principle that undergirds modern cosmology. The precise Planckian form of the radiation tells us that the universe was once in thermal equilibrium and thus that matter was once in good thermal contact in order to reach such a state. This implies that the universe was once in a very hot, dense state; in other words, the CMB requires that the universe began in a big bang. The CMB represents a cosmic photosphere, the last scattering surface for the thermal photons we observe. Scattering was quenched – and photons were released – when electrons and protons combined to create neutral hydrogen atoms. Prior to this, free electrons were abundant and led to vigorous Thomson scattering of the CMB photons. Thus, the cosmic transition from ionized to neutral also was a transition from opaque to transparent, and the CMB is literally a picture of the universe at this so-called epoch of recombination at t ∼ 400,000 yr and a redshift z ∼ 1000. Another fundamental property of the CMB is the presence of small but significant (ΔT /Tavg ∼ f ew×10−5 ) temperature fluctuations across the sky. These reflect tiny density fluctuations that were present when the CMB photons began their journey to us. Prior to recombination, the cosmic plasma underwent oscillations that were driven by the excess gravity of these overdensities, with photon pressure providing a restoring force. These “acoustic” oscillations are imprinted on the CMB angular power spectrum and encode a wealth of cosmological parameters (Hu and Dodelson 2002). One of the parameters the CMB determines most robustly is the cosmic baryon density. The final analysis of the Planck team found ΩbCMB h2 = 0.02224 ± 0.00022 (Aghanim et al. 2020), which via Eq. (3) translates to a baryon-to-photon ratio:

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ηCMB = (6.090 ± 0.061) × 10−10

(11)

a measurement at the 1% level! This result is independent from the BBN determination of the same parameter and thus presents the opportunity to measure η two different ways and compare. The CMB also now measures the effective number Nν of neutrino species, again independently of BBN. Aghanim et al. (2020) found: NνCMB = 2.800 ± 0.294

(12)

where in Eq. (11) we use results from analyses that do not use information from BBN. Finally, the CMB is also sensitive to the cosmic helium abundance and thus measures Yp independently from BBN. Here the measurements are not yet competitive with astronomical determinations. We will show these results and comment in the promising future for these measurements.

Standard BBN: Concordance, Implications, and the Lithium Problem With light-element abundance theoretical predictions and observations in hand, we are now in a position to assess the status of BBN. Recall that SBBN has only one free parameter, the baryon-to-photon ratio η, but predicts four primordial abundances, of which three (D/H, Yp , and 7 Li/H) are accessible observationally. Thus, the theory is overdetermined. That is to say, each observed abundance (and its uncertainty) selects a range in η; these are illustrated in the yellow boxes of Fig. 2. The universe is homogeneous and has a single value for η, so all of the light-element η ranges should overlap. In Fig. 2 we see that the precision of the observed deuterium abundance leads to narrow range in η. For 4 He, the range in η is much larger, due largely to the fact that Yp changes very little with η. Indeed, the 4 He observations are consistent with D/H observations, but they are also consistent with the 7 Li observations. On the other hand, 7 Li and D are inconsistent – they select disjoint ranges of η. This mismatch points to a problem and requires a “tiebreaker.” The tiebreaker comes in the form of the CMB, which independently selects its own range in η, shown as the vertical cyan bar in Fig. 2. We see that the CMB measurement is in excellent agreement with D, and with Yp , but that 7 Li is an outlier (Cyburt et al. 2003; Cyburt 2004; Coc et al. 2004; Cuoco et al. 2004). This mismatch is the cosmological lithium problem. The precision of the CMB determination of η suggests a new way to assess BBN: we can use ηCMB as an input and then use BBN to predict the light-element abundances. These can then each be directly compared to light-element observations (Cyburt et al. 2002). We have performed this test via a likelihood analysis that

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P lanck baseline (Nν = 3) + BBN (a)

Likelihood

1.0

(b)

0.8 0.6 0.4 0.2 0.0 0.23

Yp

0

1

2

3

4

5

105 × D/H

(c)

1.0

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0.26

0.25

0.24

(d)

0.8 0.6 0.4 0.2 0.0

7

8

9

10

6

3

11

10 × He/H

12

1

2

10

3

4

10

7

5

6

7

× Li/H

Fig. 4 Comparison of BBN light-element abundance predictions and observations. Likelihood curves (normalized to peak at unity) for BBN+CMB predictions are shown in magenta. Primordial abundances inferred from astronomical observations are shown in yellow. The CMB determination of 4 He is shown in cyan. We see that D/H and 4 He both show concordance, marking a success of the hot big bang cosmology. Yet 7 Li shows a striking mismatch: this is the lithium problem. (From Fields et al. 2020)

takes into account the uncertainties in the CMB measurements and in the BBN predictions. The results appear in Fig. 4; SBBN+CMB predictions are shown in magenta, while the astronomical observations are shown in yellow. The CMB determination of Yp is shown in cyan. Panel (a) of Fig. 4 shows that the helium prediction is quite precise and is in good agreement with both the astronomical observations and the CMB measure of Yp . Panel (b) shows that the BBN+CMB prediction for deuterium is in excellent agreement with the observations – the likelihoods overlap almost perfectly! The remarkable agreement in D and the good agreement with 4 He reflect the concordance between the baryon density measured independently by BBN and the CMB. This concordance represents a nontrivial success of the hot big bang and links cosmological events at times of 1 s to those at 400,000 years.

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But this concordance is not complete. Panel (d) of Fig. 4 shows that the BBN+CMB predictions for 7 Li are a factor ∼4 higher than the observations and that the likelihoods have almost no overlap, corresponding to a ∼4 − 5σ discrepancy. This sharpens the statement of the lithium problem, which has lingered since the first WMAP CMB measurements gave unambiguous measures of ηCMB (Spergel et al. 2003; Cyburt et al. 2004). The lithium problem arises from the mismatch when combining standard cosmology, standard nuclear physics as summarized in Fig. 1, and the 7 Li observations in metal-poor stars that give Eq. (10). Solutions to this problem challenge these elements (see Fields 2011, for a review), and much effort has gone into identifying new physics that could lead to a reduction in mass-7 production. Notably, however, almost all new physics models that destroy 7 Be also change D/H. As deuterium observations have continued to become more precise, these ever more tightly constrain the ability of nonstandard physics to perturb light-element production. One class of solutions involves nuclear physics, in the form of reactions that enhance 7 Be destruction. The dominant BBN reactions shown in Fig. 1 are well studied in the laboratory at the relevant energies, so a nuclear solution must invoke other reactions. Cyburt and Pospelov (2009) proposed that a new, as-yet undiscovered, resonance in the 7 Be + d reaction could be sufficient to resolve the lithium problem; Chakraborty et al. (2010) followed up with a systematic study proposing two other reactions and their needed strengths, which were quite large. Subsequently, experimental searches shown that there are no such resonances (O’Malley et al. 2011; Hammache et al. 2013; Li et al. 2018), essentially closing off the nuclear solution to the lithium problem and strengthening the reliability of the BBN calculations. Stellar astrophysics provides well-motivated mechanisms that change a star’s surface lithium abundance with time; these include thermonuclear burning of lithium within and possibly below the convection zone, possibly enhanced via rotation, and gravitational settling via diffusion (Michaud et al. 1984; Pinsonneault 1997; Vauclair and Charbonnel 1998; Richard et al. 2002; Korn et al. 2006). These could resolve the lithium problem if the destruction reaches the needed factor ∼4. The lack of dispersion in the Spite plateau (Ryan et al. 2000) presented a challenge for models, but Sbordone et al. (2010) subsequently observed that Li dispersion becomes large at low metallicity. Recently, Wang et al. (2022) revoked the longstanding claims of 6 Li detections in some Spite plateau stars. Because 6 Li is more fragile than 7 Li, its apparent survival had been an argument against strong lithium destruction. 6 Li is not made appreciably in BBN but is an inevitable byproduct of cosmic ray interactions with interstellar gas, e.g., fusion αCR + αgas → 6,7 Li + . . ., and spallation reactions such as pCR + 16 Ogas →  + · · · with  ∈ (6,7 Li, 9 Be, 10,11 B) (Meneguzzi et al. 1971). Using models for 6,7 Li, Be, and B cosmic ray production in the galaxy, Fields and Olive (2022) used the (Wang et al. 2022) limits on 6 Li/H to estimate the 6 Li and 7 Li depletion in these Spite plateau stars; they found that the 7 Li destruction was sufficient to account for the discrepancy with BBN. If this can be confirmed with

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more limits and ideally detections of 6 Li, this would confirm that stellar depletion is the origin of the lithium problem. Putting the lithium problem aside for the moment, we turn the far-reaching implications of the BBN+CMB concordance. This agreement also allows for a combined BBN and CMB determination of η and thus of the cosmic baryon density: Ωb = (0.02224 ± 0.00020)h−2 ≈ 0.0445.

(13)

We see that cosmic baryon represents only about 4% of the cosmos today. Moreover, the CMB and other observations show the cosmic inventory of nonrelativistic matter to be Ωmatter = 0.315 ± 0.007 (Aghanim et al. 2020). We see that the matter content of the Universe far exceeds that in baryons: the universe contains non-baryonic dark matter, which dominates the cosmic matter content! The only standard model candidates for non-baryonic dark matter (for reviews see, e.g., Bergstrom et al. 1999; Bertone et al. 2005; Feng 2010) are neutrinos. But from a number of avenues (beta decay experiments, neutrino oscillations, cosmological effects), it is clear that standard model neutrinos have too low a mass to make an important contribution to Ωmatter , and thus non-baryonic dark matter must have its origin beyond the standard model and hence demands new physics. We can also compare the baryonic density to the density of the universe in luminous matter, i.e., stars. Using the cosmic luminosity density and mass-to-light ratios appropriate for stars gives Ωlum  (0.0020 ± 0.0006) h−1 (Fukugita et al. 1998; Bell et al. 2003). We see that Ωlum  Ωb which frames a second dark matter problem: most cosmic baryons are dark, i.e., not in stars. There are many possible reservoirs for dark baryons (Fukugita et al. 1998), but recent work shows that a substantial component is in hot, low-density intergalactic gas (Davé et al. 2001; Nicastro et al. 2018). Before closing, we note that panel (b) of Fig. 4 shows that the uncertainty in the predicted D/H is larger than that of the observed abundance by almost a factor of 4. This is a testament to the care and persistent effort in deriving the astronomical abundance but also reflects a pressing need for improvement of the theory. Thanks to the LUNA measurements of d(p, γ )3 He, the precision of the deuterium predictions is now dominated by the error budget of d(d, n)3 He and d(d, p)t reactions. Precise measurements of these cross sections are of the highest priority. Fortunately, these reactions have the same initial state, so it is possible to measure them both in the same experiment.

BBN Probes of New Physics BBN is one of the few arenas in nature where all four fundamental interactions participate in an essential way. Consequently, BBN not only probes the standard model and the standard cosmology but also provides a sensitive probe of new

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physics involving any of the fundamental interactions and/or new particle content. For recent reviews, see Malaney and Mathews 1993; Steigman 2007; Pospelov and Pradler 2010; Jedamzik and Pospelov 2009; Iocco et al. 2009; Fields 2011. The light elements, particularly 4 He, are sensitive to the expansion rate H 2 ≈ 8π Gρradiation /3 during BBN (Steigman et al. 1977; Cyburt et al. 2005; Mangano and Serpico 2011). Thus, BBN probes the density of any and all invisible cosmic radiation. This is usually quantified via the effective number Nν of (relativistic, lefthanded, nondegenerate) neutrino species: ρrad = ρEM + Nν ρ1ν ν¯ , where ρ1ν ν¯ = 7/8 π 2 Tν4 /15. The concordance of D and 4 He abundances with BBN+CMB predictions makes these nuclides excellent probes of new physics. The potential resolution of the lithium problem adds confidence to this approach. The case of changing Nν is the simplest and best-studied departure from SBBN. The basic physics is that adding relativistic degrees of freedom increases the expansion rate in Eq. (1). This leads to an earlier weak freezeout, and thus higher n/p ratio, which in turn raises the 4 He abundance (Eq. 7). The D/H abundance is also higher due to earlier freezeout of the strong reactions. These perturbations are observable and allow 4 He and D/H to probe new physics. Indeed, if we use BBN and the CMB in combination, the CMB determination of η leaves all light elements to probe Nν (Cyburt et al. 2002, 2005). Figure 5 shows constraints on Nν from BBN and the CMB separately and jointly. The tightest limit combines all information: BBN theory, the CMB, and D/H and Yp observations. This gives (Yeh et al. 2022) Nν = 2.898 ± 0.141

(14)

Fig. 5 Effective number Nν of neutrino species measured by BBN and the CMB independently and together, for different combinations of light-element abundances. We see that the CMB and BBN determinations agree remarkably well and have similar uncertainties. (From Yeh et al. 2022)

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and a full likelihood analysis gives a 2σ upper limit of Nν < 3.180. We see that this is fully consistent with the standard model Nν = 3 result. It is also substantially tighter than the CMB-alone result in Eq. (12). Moreover, the allowed departure from the standard model is quite small, which has important implications for dark matter and for new physics generally, which would appear as excursions in Nν . For instance, a new scalar particle in equilibrium at BBN would have Nν − 3 = 0.57, which is ruled out by Eq. (14); new fermions in equilibrium are even more strongly ruled out. In fact, BBN and the CMB can now probe Nν independently and to very similar precision. This means it is now possible to probe changes in Nν (and in η) that occur between the BBN and CMB epochs; recall that in the standard cosmology these should be the same. Yeh et al. (2022) present these limits and have shown that the standard model results hold remarkably well, with no need for departures from the conventional picture. This places important new constraints on a wide variety of nonstandard scenarios. Many new physics scenarios lead to perturbations that are not captured simply by a change in Nν . These are reviewed in Pospelov and Pradler (2010) and Jedamzik and Pospelov (2009) and include a wide range of possibilities. One class of scenarios leading to particularly rich physics during BBN involves dark matter decay that injects electromagnetic energy and possibly hadronic particles into the cosmic plasma (e.g., Jedamzik 2006; Kawasaki et al. 2008, 2018; Depta et al. 2019; Forestell et al. 2019; Hufnagel et al. 2018; Poulin and Serpico 2015; Berezhiani et al. 2013, to mention a very few). This leads to nonthermal reactions on light nuclides, and possibly even the production of bound states (e.g., Pospelov 2007; Cyburt et al. 2006; Kusakabe et al. 2010, 2014), all of which require nuclear reaction data beyond the ordinary thermonuclear rates in standard BBN. In sum, BBN has pioneered particle cosmology since the first constraints on extra neutrinos by Steigman et al. (1977). Recent progress led by the CMB revolution has sharpened these constraints and offered new ones. All of this is possible due to the well-understood nuclear physics of BBN.

Summary and Future Prospects As we have seen, BBN enjoys remarkable success. BBN theory and deuterium observations make precise prediction for the cosmic baryon density, and these agree with the independent determinations of the same quantity from the CMB. Moreover, both BBN and CMB show that the standard model works very well in the early universe, with no need to invoke new physics during or between these two cosmic epochs. This has important consequences for cosmology, dark matter, and physics beyond the standard model. On the other hand, BBN has faced the lingering challenge of the cosmological lithium problem: BBN+CMB predictions of primordial 7 Li are significantly larger than those inferred in ancient metal-poor stars. Thanks to careful measurements of nuclear reaction cross sections for both the dominant BBN reactions and possible

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resonantly enhanced channels, it is clear that there is not a nuclear physics solution to this problem. New physics solutions are tantalizing but are seriously constrained and often ruled out by very tight astronomical D/H measurements. But now there is new evidence in halo star non-detections of 6 Li, which imply that stellar depletion occurs for 7 Li and is likely significant. This supports the predictions of many stellar evolution models in which effects such as diffusion, rotation, and gravitational settling can reduce surface lithium over the lifetimes of Spite plateau stars. BBN is therefore in a healthy state as was often proclaimed by David Schramm (Schramm and Turner 1998b), in which there is close interplay among theory, experiment, and observation. With the possibility of a solution to the lithium problem, these are particularly interesting times. There is a clear path to future progress on all fronts: • Nuclear experiment: Precision measurements of d(d, n)3 He and d(d, p)t are critical to reduce the uncertainties in D/H predictions and bring them in line with the very precise astronomical D/H observations. And as lithium comes into a new light, its uncertainties become important to reduce, for which 3 He(α, γ )7 Be is largest contributor. • Nuclear theory: Ab initio calculations offer important guidance, and more such work will be of great use in the future. The most pressing is that d(p, γ )3 He calculations (Marcucci et al. 2016) show important differences with the recent LUNA measurements at BBN energies. We look forward to further studies to clarify this situation. • Astronomical observations and theory: More 6 Li observations in halo stars – especially 6 Li detections – would shed new light on both 6 Li and 7 Li depletion. Improvements in 4 He precision can increase the power of the BBN probe of new physics. On the theory side, an interplay between stellar evolution models and cosmic ray models for lithium production can help clarify the nature of lithium depletion and could help point to how lithium observations could be used to recover the primordial 7 Li abundance. • CMB observations: Future measurements by the CMB-Stage 4 experiment will sharpen both the CMB probe of Nν and of primordial 4 He. As progress occurs on each of these fronts, we have reason for optimism that the solution to the lithium problem will come into focus. We can look forward to BBN moving beyond, to renewed importance for cosmology, particle physics, and nuclear astrophysics. In this bright future, nuclear physics will continue to take center stage.

References T. Abel, G.L. Bryan, M.L. Norman, The formation of the first star in the universe. Science 295(5552), 93–98 (2002). https://doi.org/10.1126/science.295.5552.93 E.G. Adelberger et al., Solar fusion cross sections II: the pp chain and CNO cycles. Rev. Mod. Phys. 83, 195 (2011). https://doi.org/10.1103/RevModPhys.83.195

92 Big Bang Nucleosynthesis: Nuclear Physics in the Early Universe

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E.G. Adelberger et al., Solar fusion cross-sections. Rev. Mod. Phys. 70, 1265–1292 (1998). https:// doi.org/10.1103/RevModPhys.70.1265 N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters. Astron. Astrophys. 641, A6 (2020); [Erratum: Astron. Astrophys. 652, C4 (2021)]. https://doi.org/10.1051/0004-6361/ 201833910 R. A. Alpher, H. Bethe, G. Gamow, The origin of chemical elements. Phys. Rev. 73(7), 803–804 (1948). https://doi.org/10.1103/PhysRev.73.803 R.A. Alpher, R. Herman, Evolution of the universe. Nature 162(4124), 774–775 (1948). https:// doi.org/10.1038/162774b0 R.A. Alpher, J.W. Follin, R.C. Herman, Physical conditions in the initial stages of the expanding universe. Phys. Rev. 92(6), 1347–1361 (1953). https://doi.org/10.1103/PhysRev.92.1347 E. Aver, K.A. Olive, E.D. Skillman, The effects of He I λ10830 on helium abundance determinations. JCAP 2015(7), 011 (2015). https://doi.org/10.1088/1475-7516/2015/07/011 E. Aver, D.A. Berg, K.A. Olive, R.W. Pogge, J.J. Salzer, E.D. Skillman, Improving helium abundance determinations with Leo P as a case study. JCAP 2021(3), 027 (2021). https:// doi.org/10.1088/1475-7516/2021/03/027 E. Aver, D.A. Berg, A.S. Hirschauer, K.A. Olive, R.W. Pogge, N.S.J. Rogers, J.J. Salzer, E.D. Skillman, A comprehensive chemical abundance analysis of the extremely metal poor Leoncino Dwarf galaxy (AGC 198691). MNRAS 510(1), 373–382 (2022). https://doi.org/10. 1093/mnras/stab3226 J.N. Bahcall, Neutrino Astrophysics (1989). Cambridge University Press, Cambridge J.N. Bahcall, A.M. Serenelli, S. Basu, 10,000 standard solar models: a Monte Carlo simulation. ApJS 165(1), 400–431 (2006). https://doi.org/10.1086/504043 S.A. Balashev, E.O. Zavarygin, A.V. Ivanchik, K.N. Telikova, D.A. Varshalovich, The primordial deuterium abundance: subDLA system at zabs = 2.437 towards the QSO J 1444+2919. MNRAS 458(2), 2188–2198 (2016). https://doi.org/10.1093/mnras/stw356 T.M. Bania, R.T. Rood, D.S. Balser, The milky way 3-helium abundance. Space Sci. Rev. 130, 53–62 (2007). https://doi.org/10.1007/s11214-007-9144-z E.F. Bell, D.H. McIntosh, N. Katz, M.D. Weinberg, The optical and near-infrared properties of galaxies. I. Luminosity and stellar mass functions. ApJS 149(2), 289–312 (2003). https://doi. org/10.1086/378847 Z. Berezhiani, A. Dolgov, I. Tkachev, BBN with light dark matter. JCAP 02, 010 (2013). https:// doi.org/10.1088/1475-7516/2013/02/010 L. Bergstrom, S. Iguri, H. Rubinstein, Constraints on the variation of the fine structure constant from big bang nucleosynthesis. Phys. Rev. D 60, 045005 (1999). https://doi.org/10.1103/ PhysRevD.60.045005 G. Bertone, D. Hooper, J. Silk, Particle dark matter: evidence, candidates and constraints. Phys. Rep. 405(5–6), 279–390 (2005). https://doi.org/10.1016/j.physrep.2004.08.031 V. Bromm, R.B. Larson, The first stars. Ann. Rev. Astron. Astrophys. 42(1), 79–118 (2004). https://doi.org/10.1146/annurev.astro.42.053102.134034 N. Chakraborty, B.D. Fields, K.A. Olive, Resonant destruction as a possible solution to the cosmological lithium problem (2010). ArXiv e-prints C. Chiappini, F. Matteucci, R. Gratton, The chemical evolution of the galaxy: the two-infall model. ApJ 477(2), 765–780 (1997). https://doi.org/10.1086/303726 A. Coc, E. Vangioni-Flam, P. Descouvemont, A. Adahchour, C. Angulo, Updated big bang nucleosynthesis compared with Wilkinson microwave anisotropy probe observations and the abundance of light elements. ApJ 600, 544–552 (2004). https://doi.org/10.1086/380121 A. Coc, P. Descouvemont, K.A. Olive, J.-P. Uzan, E. Vangioni, The variation of fundamental constants and the role of A=5 and A=8 nuclei on primordial nucleosynthesis. Phys. Rev. D 86, 043529 (2012). https://doi.org/10.1103/PhysRevD.86.043529 A. Coc, S. Goriely, Y. Xu, M. Saimpert, E. Vangioni, Standard big bang nucleosynthesis up to CNO with an improved extended nuclear network. ApJ 744(2), 158 (2012). https://doi.org/10. 1088/0004-637X/744/2/158

3400

B. D. Fields

R.J. Cooke, M. Pettini, R.A. Jorgenson, M.T. Murphy, C.C. Steidel, Precision measures of the primordial abundance of deuterium. ApJ 781(1), 31 (2014). https://doi.org/10.1088/0004-637X/ 781/1/31 R.J. Cooke, M. Pettini, K.M. Nollett, R. Jorgenson, The primordial deuterium abundance of the most metal-poor damped lyman-α system. ApJ 830(2), 148 (2016). https://doi.org/10.3847/ 0004-637X/830/2/148 R.J. Cooke, M. Pettini, C.C. Steidel, One percent determination of the primordial deuterium abundance. ApJ 855(2), 102 (2018). https://doi.org/10.3847/1538-4357/aaab53 A. Cuoco, F. Iocco, G. Mangano, G. Miele, O. Pisanti, P.D. Serpico, Present status of primordial nucleosynthesis after WMAP: results from a new BBN code. Int. J. Mod. Phys. A 19, 4431– 4453 (2004). https://doi.org/10.1142/S0217751X04019548 R.H. Cyburt, M. Pospelov, Resonant enhancement of nuclear reactions as a possible solution to the cosmological lithium problem (2009). ArXiv e-prints R.H. Cyburt, B.D. Fields, K.A. Olive, Primordial nucleosynthesis with CMB inputs: probing the early universe and light element astrophysics. Astropart. Phys. 17, 87–100 (2002). https://doi. org/10.1016/S0927-6505(01)00171-2 R.H. Cyburt, B.D. Fields, K.A. Olive, E. Skillman, New BBN limits on physics beyond the standard model from 4 He. Astropart. Phys. 23, 313–323 (2005). https://doi.org/10.1016/j. astropartphys.2005.01.005 R.H. Cyburt, J. Ellis, B.D. Fields, K.A. Olive, V.C. Spanos, Bound-state effects on light-element abundances in gravitino dark matter scenarios. JCAP 11, 14–+ (2006). https://doi.org/10.1088/ 1475-7516/2006/11/014 R.H. Cyburt, Primordial nucleosynthesis for the new cosmology: determining uncertainties and examining concordance. Phys. Rev. D 70, 023505 (2004). https://doi.org/10.1103/PhysRevD. 70.023505 R.H. Cyburt, B.D. Fields, K.A. Olive, Primordial nucleosynthesis in light of WMAP. Phys. Lett. B567, 227–234 (2003). https://doi.org/10.1016/j.physletb.2003.06.026 R.H. Cyburt, B.D. Fields, K.A. Olive, Solar neutrino constraints on the BBN production of Li. Phys. Rev. D 69(12), 123519 (2004). https://doi.org/10.1103/PhysRevD.69.123519 R.H. Cyburt, B.D. Fields, K.A. Olive, T.-H. Yeh, Big bang nucleosynthesis: present status. Rev. Mod. Phys. 88, 015004 (2016). https://doi.org/10.1103/RevModPhys.88.015004 R. Davé, R. Cen, J.P. Ostriker, G.L. Bryan, L. Hernquist, N. Katz, D.H. Weinberg, M.L. Norman, B. O’Shea, Baryons in the warm-hot intergalactic medium. ApJ 552(2), 473–483 (2001). https:// doi.org/10.1086/320548 P.F. Depta, M. Hufnagel, K. Schmidt-Hoberg, S. Wild, BBN constraints on the annihilation of MeV-scale dark matter. JCAP 04, 029 (2019). https://doi.org/10.1088/1475-7516/2019/04/029 P. Descouvemont, A. Adahchour, C. Angulo, A. Coc, E. Vangioni-Flam, Compilation and Rmatrix analysis of big bang nuclear reaction rates. At. Data Nucl. Data Tables 88, 203–236 (2004). https://doi.org/10.1016/j.adt.2004.08.001 J.L. Feng, Dark matter candidates from particle physics and methods of detection. Ann. Rev. Astron. Astrophys. 48, 495–545 (2010). https://doi.org/10.1146/annurev-astro082708-101659 B.D. Fields, The primordial lithium problem. Annu. Rev. Nucl. Part. Sci. 61, 47–68 (2011). https:// doi.org/10.1146/annurev-nucl-102010-130445 B.D. Fields, K.A. Olive, Implications of the non-observation of 6 Li in halo stars for the primordial 7 Li problem (2022). https://arxiv.org/abs/2204.03167 B.D. Fields, K.A. Olive, T.-H. Yeh, C. Young, Big-Bang nucleosynthesis after planck. JCAP 03, 010 (2020); [Erratum: JCAP 11, E02 (2020)]. https://doi.org/10.1088/1475-7516/2020/03/010 G. Fiorentini, E. Lisi, S. Sarkar, F.L. Villante, Quantifying uncertainties in primordial nucleosynthesis without Monte Carlo simulations. Phys.Rev. D 58(6), 063506–+ (1998). https://doi.org/ 10.1103/PhysRevD.58.063506 L. Forestell, D.E. Morrissey, G. White, Limits from BBN on light electromagnetic decays. JHEP 01, 074 (2019). https://doi.org/10.1007/JHEP01(2019)074

92 Big Bang Nucleosynthesis: Nuclear Physics in the Early Universe

3401

A. Frebel, J.E. Norris, Near-field cosmology with extremely metal-poor stars. Ann. Rev. Astron. Astrophys. 53, 631–688 (2015). https://doi.org/10.1146/annurev-astro082214-122423 M. Fukugita, C.J. Hogan, P.J.E. Peebles, The cosmic baryon budget. ApJ 503(2), 518–530 (1998). https://doi.org/10.1086/306025 F. Hammache, A. Coc, N. de Séréville, I. Stefan, P. Roussel, S. Ancelin, M. Assié, L. Audouin, D. Beaumel, S. Franchoo, B. Fernandez-Dominguez, S. Fox, C. Hamadache, J. Kiener, A. Laird, B. Le Crom, A. Lefebvre-Schuhl, L. Lefebvre, I. Matea, A. Matta, G. Mavilla, J. Mrazek, P. Morfouace, F. de Oliveira Santos, A. Parikh, L. Perrot, A.M. Sanchez-Benitez, D. Suzuki, V. Tatischeff, P. Ujic, M. Vandebrouck, Search for new resonant states in 10 C and 11 C and their impact on the cosmological lithium problem. Phys. Rev. C 88(6), 062802 (2013). https://doi. org/10.1103/PhysRevC.88.062802 W.C. Haxton, R.G. Hamish Robertson, A.M. Serenelli, Solar neutrinos: status and prospects. Ann. Rev. Astron. Astrophys. 51(1), 21–61 (2013). https://doi.org/10.1146/annurev-astro081811-125539 C. Hayashi, Proton-neutron concentration ratio in the expanding universe at the stages preceding the formation of the elements. Prog. Theor. Phys. 5(2), 224–235 (1950). https://doi.org/10. 1143/ptp/5.2.224 T. Hsyu, R.J. Cooke, J.X. Prochaska, M. Bolte, The PHLEK survey: a new determination of the primordial helium abundance. Astrophys. J. 896(1), 77 (2020). https://doi.org/10.3847/15384357/ab91af W. Hu, S. Dodelson, Cosmic microwave background anisotropies. Ann. Rev. Astron. Astrophys. 40, 171–216 (2002). https://doi.org/10.1146/annurev.astro.40.060401.093926 M. Hufnagel, K. Schmidt-Hoberg, S. Wild, BBN constraints on MeV-scale dark sectors. Part II. Electromagnetic decays. JCAP 11, 032 (2018). https://doi.org/10.1088/1475-7516/2018/11/ 032 C. Iliadis, A. Coc, Thermonuclear reaction rates and primordial nucleosynthesis. ApJ 901(2), 127 (2020). https://doi.org/10.3847/1538-4357/abb1a3 F. Iocco, G. Mangano, G. Miele, O. Pisanti, P.D. Serpico, Path to metallicity: synthesis of CNO elements in standard BBN. Phys. Rev. D 75(8), 087304–+ (2007). https://doi.org/10.1103/ PhysRevD.75.087304 F. Iocco, G. Mangano, G. Miele, O. Pisanti, P.D. Serpico, Primordial nucleosynthesis: from precision cosmology to fundamental physics. Phys. Rep. 472, 1–76 (2009). https://doi.org/ 10.1016/j.physrep.2009.02.002 Y.I. Izotov, T.X. Thuan, The primordial abundance of 4He revisited. ApJ 500, 188–+ (1998). https://doi.org/10.1086/305698 Y.I. Izotov, T.X. Thuan, Systematic effects and a new determination of the primordial abundance of 4 He and dY/dZ from observations of blue compact galaxies. ApJ 602, 200–230 (2004). https:// doi.org/10.1086/380830 K. Jedamzik, Big bang nucleosynthesis constraints on hadronically and electromagnetically decaying relic neutral particles. Phys. Rev. D 74(10), 103509–+ (2006). https://doi.org/10. 1103/PhysRevD.74.103509 K. Jedamzik, M. Pospelov, Big bang nucleosynthesis and particle dark matter. New J. Phys. 11, 105028 (2009). https://doi.org/10.1088/1367-2630/11/10/105028 T. Karlsson, V. Bromm, J. Bland-Hawthorn, Pregalactic metal enrichment: the chemical signatures of the first stars. Rev. Mod. Phys. 85(2), 809–848 (2013). https://doi.org/10.1103/RevModPhys. 85.809 M. Kawasaki, K. Kohri, T. Moroi, A. Yotsuyanagi, Big-bang nucleosynthesis and gravitino. Phys. Rev. D 78, 065011 (2008). https://doi.org/10.1103/PhysRevD.78.065011 M. Kawasaki, K. Kohri, T. Moroi, Y. Takaesu, Revisiting big-bang nucleosynthesis constraints on long-lived decaying particles. Phys. Rev. D 97(2), 023502 (2018). https://doi.org/10.1103/ PhysRevD.97.023502

3402

B. D. Fields

A.J. Korn, F. Grundahl, O. Richard, P.S. Barklem, L. Mashonkina, R. Collet, N. Piskunov, B. Gustafsson, A probable stellar solution to the cosmological lithium discrepancy. Nature 442, 657–659 (2006). https://doi.org/10.1038/nature05011 O.A. Kurichin, P.A. Kislitsyn, V.V. Klimenko, S.A. Balashev, A.V. Ivanchik, A new determination of the primordial helium abundance using the analyses of H II region spectra from SDSS. Mon. Not. R. Astron. Soc. 502(2), 3045–3056 (2021). https://doi.org/10.1093/mnras/stab215 M. Kusakabe, T. Kajino, T. Yoshida, G.J. Mathews, New results on catalyzed BBN with a longlived negatively-charged massive particle. Phys. Rev. D 81, 083521 (2010). https://doi.org/10. 1103/PhysRevD.81.083521 M. Kusakabe, K.S. Kim, M.-K. Cheoun, T. Kajino, Y. Kino, G.J. Mathews, Revised big bang nucleosynthesis with long-lived negatively charged massive particles: updated recombination rates, primordial 9 Be nucleosynthesis, and impact of new 6 Li limits. Astrophys. J. Suppl. 214, 5 (2014). https://doi.org/10.1088/0067-0049/214/1/5 E.-T. Li, Z.-H. Li, S.-Q. Yan, J. Su, B. Guo, Y.-J. Li, Y.-B. Wang, G. Lian, S. Zeng, S.-Z. Chen, S.-B. Ma, X.-Q. Li, C. He, H.-B. Sun, W.-P. Liu, Measurement of the 2 H(7 Be, 6 Li)3 He reaction rate and its contribution to the primordial lithium abundance. Chin. Phys. C 42(4), 044001 (2018). https://doi.org/10.1088/1674-1137/42/4/044001 R.A. Malaney, G.J. Mathews, Probing the early universe: a review of primordial nucleosynthesis beyond the standard big bang. Phys. Rep. 229, 145–219 (1993). https://doi.org/10.1016/03701573(93)90134-Y G. Mangano, P.D. Serpico, A robust upper limit on Neff from BBN, circa 2011. Phys. Lett. B 701, 296–299 (2011). https://doi.org/10.1016/j.physletb.2011.05.075 L. E. Marcucci, G. Mangano, A. Kievsky, M. Viviani, Implication of the proton-deuteron radiative capture for big bang nucleosynthesis. Phys. Rev. Lett. 116(10), 102501 (2016). https://doi.org/ 10.1103/PhysRevLett.116.102501 M. Meneguzzi, J. Audouze, H. Reeves, The production of the elements Li, Be, B by galactic cosmic rays in space and its relation with stellar observations. Astron. Astrophys. 15, 337 (1971) G. Michaud, G. Fontaine, G. Beaudet, The lithium abundance – Constraints on stellar evolution. ApJ 282, 206–213 (1984). https://doi.org/10.1086/162193 V. Mossa, K. Stöckel, F. Cavanna, F. Ferraro, M. Aliotta, F. Barile, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C.G. Bruno, A. Caciolli, T. Chillery, G.F. Ciani, P. Corvisiero, L. Csedreki, T. Davinson, R. Depalo, A. Di Leva, Z. Elekes, E.M. Fiore, A. Formicola, Z. Fülöp, G. Gervino, A. Guglielmetti, C. Gustavino, G. Gyürky, G. Imbriani, M. Junker, A. Kievsky, I. Kochanek, M. Lugaro, L.E. Marcucci, G. Mangano, P. Marigo, E. Masha, R. Menegazzo, F.R. Pantaleo, V. Paticchio, R. Perrino, D. Piatti, O. Pisanti, P. Prati, L. Schiavulli, O. Straniero, T. Szücs, M.P. Takács, D. Trezzi, M. Viviani, S. Zavatarelli, The baryon density of the universe from an improved rate of deuterium burning. Nature 587(7833), 210–213 (2020). https://doi. org/10.1038/s41586-020-2878-4 F. Nicastro, J. Kaastra, Y. Krongold, S. Borgani, E. Branchini, R. Cen, M. Dadina, C.W. Danforth, M. Elvis, F. Fiore, A. Gupta, S. Mathur, D. Mayya, F. Paerels, L. Piro, D. Rosa-Gonzalez, J. Schaye, J.M. Shull, J. Torres-Zafra, N. Wijers, L. Zappacosta, Observations of the missing baryons in the warm-hot intergalactic medium. Nature 558(7710), 406–409 (2018). https://doi. org/10.1038/s41586-018-0204-1 K. Nomoto, C. Kobayashi, N. Tominaga, Nucleosynthesis in stars and the chemical enrichment of galaxies. Ann. Rev. Astron. Astrophys. 51(1), 457–509 (2013). https://doi.org/10.1146/ annurev-astro-082812-140956 K.A. Olive, R.T. Rood, D.N. Schramm, J. Truran, E. Vangioni-Flam, What is the problem with 3He? ApJ 444, 680 (1995). https://doi.org/10.1086/175640 P.D. O’Malley, D.W. Bardayan, A.S. Adekola, S. Ahn, K.Y. Chae, J.A. Cizewski, S. Graves, M.E. Howard, K.L. Jones, R.L. Kozub, L. Lindhardt, M. Matos, B.M. Moazen, C.D. Nesaraja, S.D. Pain, W.A. Peters, S.T. Pittman, K.T. Schmitt, J.F. Shriner Jr., M.S. Smith, I. Spassova, S.Y. Strauss, J.L. Wheeler, Search for a resonant enhancement of the 7 Be + d reaction and primordial 7 Li abundances. Phys. Rev. C 84(4), 042801 (2011). https://doi.org/10.1103/ PhysRevC.84.042801

92 Big Bang Nucleosynthesis: Nuclear Physics in the Early Universe

3403

Particle Data Group, P.A. Zyla, R.M. Barnett, J. Beringer, O. Dahl, D.A. Dwyer, D.E. Groom, C.J. Lin, K.S. Lugovsky, E. Pianori, D.J. Robinson, C.G. Wohl, W.M. Yao, K. Agashe, G. Aielli, B.C. Allanach, C. Amsler, M. Antonelli, E.C. Aschenauer, D.M. Asner, H. Baer, S.W. Banerjee, L. Baudis, C.W. Bauer, J.J. Beatty, V.I. Belousov, S. Bethke, A. Bettini, O. Biebel, K.M. Black, E. Blucher, O. Buchmuller, V. Burkert, M.A. Bychkov, R.N. Cahn, M. Carena, A. Ceccucci, A. Cerri, D. Chakraborty, R. Sekhar Chivukula, G. Cowan, G. D’Ambrosio, T. Damour, D. de Florian, A. de Gouvêa, T. DeGrand, P. de Jong, G. Dissertori, B.A. Dobrescu, M. D’Onofrio, M. Doser, M. Drees, H.K. Dreiner, P. Eerola, U. Egede, S. Eidelman, J. Ellis, J. Erler, V.V. Ezhela, W. Fetscher, B.D. Fields, B. Foster, A. Freitas, H. Gallagher, L. Garren, H.J. Gerber, G. Gerbier, T. Gershon, Y. Gershtein, T. Gherghetta, A.A. Godizov, M.C. GonzalezGarcia, M. Goodman, C. Grab, A.V. Gritsan, C. Grojean, M. Grünewald, A. Gurtu, T. Gutsche, H.E. Haber, C. Hanhart, S. Hashimoto, Y. Hayato, A. Hebecker, S. Heinemeyer, B. Heltsley, J.J. Hernández-Rey, K. Hikasa, J. Hisano, A. Höcker, J. Holder, A. Holtkamp, J. Huston, T. Hyodo, K.F. Johnson, M. Kado, M. Karliner, U.F. Katz, M. Kenzie, V.A. Khoze, S.R. Klein, E. Klempt, R.V. Kowalewski, F. Krauss, M. Kreps, B. Krusche, Y. Kwon, O. Lahav, J. Laiho, L.P. Lellouch, J. Lesgourgues, A.R. Liddle, Z. Ligeti, C. Lippmann, T.M. Liss, L. Littenberg, C. Lourengo, S.B. Lugovsky, A. Lusiani, Y. Makida, F. Maltoni, T. Mannel, A.V. Manohar, W.J. Marciano, A. Masoni, J. Matthews, U.G. Meißner, M. Mikhasenko, D.J. Miller, D. Milstead, R.E. Mitchell, K. Mönig, P. Molaro, F. Moortgat, M. Moskovic, K. Nakamura, M. Narain, P. Nason, S. Navas, M. Neubert, P. Nevski, Y. Nir, K.A. Olive, C. Patrignani, J.A. Peacock, S.T. Petcov, V.A. Petrov, A. Pich, A. Piepke, A. Pomarol, S. Profumo, A. Quadt, K. Rabbertz, J. Rademacker, G. Raffelt, H. Ramani, M. Ramsey-Musolf, B.N. Ratcliff, P. Richardson, A. Ringwald, S. Roesler, S. Rolli, A. Romaniouk, L.J. Rosenberg, J.L. Rosner, G. Rybka, M. Ryskin, R.A. Ryutin, Y. Sakai, G.P. Salam, S. Sarkar, F. Sauli, O. Schneider, K. Scholberg, A.J. Schwartz, J. Schwiening, D. Scott, V. Sharma, S.R. Sharpe, T. Shutt, M. Silari, T. Sjöstrand, P. Skands, T. Skwarnicki, G.F. Smoot, A. Soffer, M.S. Sozzi, S. Spanier, C. Spiering, A. Stahl, S.L. Stone, Y. Sumino, T. Sumiyoshi, M.J. Syphers, F. Takahashi, M. Tanabashi, J. Tanaka, M. Taševský, K. Terashi, J. Terning, U. Thoma, R.S. Thorne, L. Tiator, M. Titov, N.P. Tkachenko, D.R. Tovey, K. Trabelsi, P. Urquijo, G. Valencia, R. Van de Water, N. Varelas, G. Venanzoni, L. Verde, M.G. Vincter, P. Vogel, W. Vogelsang, A. Vogt, V. Vorobyev, S.P. Wakely, W. Walkowiak, C.W. Walter, D. Wands, M.O. Wascko, D.H. Weinberg, E.J. Weinberg, M. White, L.R. Wiencke, S. Willocq, C.L. Woody, R.L. Workman, M. Yokoyama, R. Yoshida, G. Zanderighi, G.P. Zeller, O.V. Zenin, R.Y. Zhu, S.L. Zhu, F. Zimmermann, J. Anderson, T. Basaglia, V.S. Lugovsky, P. Schaffner, W. Zheng, Review of particle physics. Prog. Theor. Exp. Phys. 2020(8), 083C01 (2020). https://doi.org/10.1093/ptep/ptaa104 P.J.E. Peebles, Principles of Physical Cosmology (1993). Princeton University Press, Princeton. https://doi.org/10.1515/9780691206721 A. Peimbert, M. Peimbert, V. Luridiana, Temperature bias and the primordial helium abundance determination. ApJ 565, 668–680 (2002). https://doi.org/10.1086/324601 M. Pettini, R. Cooke, A new, precise measurement of the primordial abundance of deuterium. MNRAS 425(4), 2477–2486 (2012). https://doi.org/10.1111/j.1365-2966.2012.21665.x M. Pinsonneault, Mixing in Stars. Ann. Rev. Astron. Astrophys. 35, 557–605 (1997). https://doi. org/10.1146/annurev.astro.35.1.557 O. Pisanti, A. Cirillo, S. Esposito, F. Iocco, G. Mangano, G. Miele, P.D. Serpico, PArthENoPE: public algorithm evaluating the nucleosynthesis of primordial elements. Comput. Phys. Commun. 178, 956–971 (2008). https://doi.org/10.1016/j.cpc.2008.02.015 O. Pisanti, G. Mangano, G. Miele, P. Mazzella, Primordial deuterium after LUNA: concordances and error budget. JCAP 2021(4), 020 (2021). https://doi.org/10.1088/1475-7516/2021/04/020 C. Pitrou, A. Coc, J.-P. Uzan, E. Vangioni, Precision big bang nucleosynthesis with improved Helium-4 predictions. Phys. Rep. 754, 1–66 (2018). https://doi.org/10.1016/j.physrep.2018.04. 005 C. Pitrou, A. Coc, J.-P. Uzan, E. Vangioni, A new tension in the cosmological model from primordial deuterium? MNRAS 502(2), 2474–2481 (2021a). https://doi.org/10.1093/mnras/ stab135

3404

B. D. Fields

C. Pitrou, A. Coc, J.-P. Uzan, E. Vangioni, Resolving conclusions about the early universe requires accurate nuclear measurements. Nat. Rev. Phys. 3(4), 231–232 (2021b). ISSN 2522-5820, https://doi.org/10.1038/s42254-021-00294-6 M. Pospelov, Particle physics catalysis of thermal big bang nucleosynthesis. Phys. Rev. Lett. 98(23), 231301–+ (2007). https://doi.org/10.1103/PhysRevLett.98.231301 M. Pospelov, J. Pradler, Big bang nucleosynthesis as a probe of new physics. Ann. Rev. Nucl. Part. Sci. 60, 539–568 (2010). https://doi.org/10.1146/annurev.nucl.012809.104521 V. Poulin, P.D. Serpico, Nonuniversal BBN bounds on electromagnetically decaying particles. Phys. Rev. D 91(10), 103007 (2015). https://doi.org/10.1103/PhysRevD.91.103007 O. Richard, G. Michaud, J. Richer, Models of metal-poor stars with gravitational settling and radiative accelerations. III. Metallicity dependence. ApJ 580(2), 1100–1117 (2002). https://doi. org/10.1086/343733 S. Riemer-Sørensen, J.K. Webb, N. Crighton, V. Dumont, K. Ali, S. Kotuš, M. Bainbridge, M.T. Murphy, R. Carswell, A robust deuterium abundance; re-measurement of the z = 3.256 absorption system towards the quasar PKS 1937-101. MNRAS 447(3), 2925–2936 (2015). https://doi.org/10.1093/mnras/stu2599 S. Riemer-Sørensen, S. Kotuš, J.K. Webb, K. Ali, V. Dumont, M.T. Murphy, R.F. Carswell, A precise deuterium abundance: remeasurement of the z = 3.572 absorption system towards the quasar PKS1937-101. MNRAS 468(3), 3239–3250 (2017). https://doi.org/10.1093/mnras/ stx681 S.G. Ryan, T.C. Beers, K.A. Olive, B.D. Fields, J.E. Norris, Primordial lithium and big bang nucleosynthesis. Astrophys. J. Lett. 530, L57–L60 (2000). https://doi.org/10.1086/312492 L. Sbordone, P. Bonifacio, E. Caffau, H.-G. Ludwig, N.T. Behara, J.I. González Hernández, M. Steffen, R. Cayrel, B. Freytag, C. van’t Veer, P. Molaro, B. Plez, T. Sivarani, M. Spite, F. Spite, T.C. Beers, N. Christlieb, P. François, V. Hill, The metal-poor end of the Spite plateau. I. Stellar parameters, metallicities, and lithium abundances. Astron. Astrophys. 522, A26+ (2010). https://doi.org/10.1051/0004-6361/200913282 R.T. Scherrer, R.J. Scherrer, Big bang nucleosynthesis with stable 8 Be and the primordial lithium problem. Phys. Rev. D 96(8), 083507 (2017). https://doi.org/10.1103/PhysRevD.96.083507 D.N. Schramm, M.S. Turner, Big-bang nucleosynthesis enters the precision era. Rev. Mod. Phys. 70, 303–318 (1998a). https://doi.org/10.1103/RevModPhys.70.303 D.N. Schramm, M.S. Turner, Big-bang nucleosynthesis enters the precision era. Rev. Mod. Phys. 70(1), 303–318 (1998b). https://doi.org/10.1103/RevModPhys.70.303 P.D. Serpico, S. Esposito, F. Iocco, G. Mangano, G. Miele, O. Pisanti, Nuclear reaction network for primordial nucleosynthesis: a detailed analysis of rates, uncertainties and light nuclei yields. JCAP 12, 10–+ (2004). https://doi.org/10.1088/1475-7516/2004/12/010 M.S. Smith, L.H. Kawano, R.A. Malaney, Experimental, computational, and observational analysis of primordial nucleosynthesis. ApJS 85, 219–247 (1993). https://doi.org/10.1086/ 191763 D.N. Spergel, L. Verde, H.V. Peiris, E. Komatsu, M.R. Nolta, C.L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S.S. Meyer, L. Page, G.S. Tucker, J.L. Weiland, E. Wollack, E.L. Wright, First-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: determination of cosmological parameters. ApJS 148, 175–194 (2003). https:// doi.org/10.1086/377226 F. Spite, M. Spite, Abundance of lithium in unevolved halo stars and old disk stars – Interpretation and consequences. Astron. Astrophys. 115, 357–366 (1982) M. Spite, F. Spite, The composition of field halo stars and the chemical evolution of the halo. Ann. Rev. Astron. Astrophys. 23, 225–238 (1985). https://doi.org/10.1146/annurev.aa. 23.090185.001301 G. Steigman, D.N. Schramm, J.E. Gunn, Cosmological limits to the number of massive leptons. Phys. Lett. B 66, 202–204 (1977). https://doi.org/10.1016/0370-2693(77)90176-9 G. Steigman, Primordial nucleosynthesis in the precision cosmology era. Annu. Rev. Nucl. Part. Sci. 57(1), 463–491 (2007). https://doi.org/10.1146/annurev.nucl.56.080805.140437

92 Big Bang Nucleosynthesis: Nuclear Physics in the Early Universe G.

3405

Steigman, Primordial Nucleosynthesis in the Precision Cosmology Era. Ann. Rev. Nucl .Part. Sci. 57, 463–491 (2007). https://doi.org/10.1146/annurev.nucl.56. 080805.140437 D. Thomas, D.N. Schramm, K.A. Olive, B.D. Fields, Primordial nucleosynthesis and the abundances of beryllium and boron. ApJ 406, 569–579 (1993). https://doi.org/10.1086/172469 F.X. Timmes, S.E. Woosley, T.A. Weaver, Galactic chemical evolution: hydrogen through zinc. ApJS 98, 617 (1995). https://doi.org/10.1086/192172 E. Vangioni-Flam, K.A. Olive, B.D. Fields, M. Cassé, On the baryometric status of 3 He. ApJ 585, 611–616 (2003). https://doi.org/10.1086/346232 S. Vauclair, C. Charbonnel, Element segregation in low metallicity stars and the primordial lithium abundance. Astrophys. J. 502, 372 (1998). https://doi.org/10.1086/305906 R.V. Wagoner, W.A. Fowler, F. Hoyle, On the synthesis of elements at very high temperatures. ApJ 148, 3–+ (1967). https://doi.org/10.1086/149126 R.V. Wagoner, Synthesis of the elements within objects exploding from very high temperatures. ApJS 18, 247 (1969). https://doi.org/10.1086/190191 E.X. Wang, T. Nordlander, M. Asplund, K. Lind, Y. Zhou, H. Reggiani, Non-detection of 6 Li in Spite plateau stars with ESPRESSO. MNRAS 509(1), 1521–1535 (2022). https://doi.org/10. 1093/mnras/stab2924 S. Weinberg, The First Three Minutes: A Modern View of the Origin of the Universe (1977). Andre Deutsch, London T.-H. Yeh, K.A. Olive, B.D. Fields, The impact of new d(p, γ )3 rates on big bang nucleosynthesis. JCAP 03, 046 (2021). https://doi.org/10.1088/1475-7516/2021/03/046 T.-H. Yeh, J. Shelton, K.A. Olive, B.D. Fields, Probing physics beyond the standard model: limits from BBN and the CMB independently and combined. JCAP 10, 046 (2022). https://doi.org/ 10.1088/1475-7516/2022/10/046 E.O. Zavarygin, J.K. Webb, V. Dumont, S. Riemer-Sørensen, The primordial deuterium abundance at zabs = 2.504 from a high signal-to-noise spectrum of Q1009+2956. MNRAS 477(4), 5536– 5553 (2018). https://doi.org/10.1093/mnras/sty1003

Inflation, Perturbations, and Structure Formation

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Grant J. Mathews and Guobao Tang

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shortcomings of the Standard Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Flatness Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Monopole Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Smallness Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Structure Formation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Initial Conditions Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inflation Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slow Roll Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slow Roll Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inflation Effective Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . End of Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Warm Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How Much Inflation Occurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Initial Conditions for Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacuum Fluctuations of the Inflation Field and the Primordial Power Spectrum . . . . . . . . . Power Spectrum of Primordial Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuations and the CMB Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CMB Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer Functions and Computation of the CMB Power Spectra . . . . . . . . . . . . . . . . . . . . . Formation of Large-Scale Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3408 3408 3408 3409 3410 3410 3412 3412 3412 3413 3414 3415 3416 3416 3417 3417 3418 3419 3420 3421 3423 3427 3428 3429 3432

G. J. Mathews () · G. Tang Department of Physics and Astronomy, Center for Astrophysics, University of Notre Dame, Notre Dame, IN, USA e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_112

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Abstract

This chapter reviews the basics of cosmic inflation and the formation of structure.

Introduction The epoch of photon decoupling fixes the temperature (and redshift) at the surface of last scattering. What remains, however, is to understand the residual fluctuations in the cosmic microwave background (CMB) at the level of ΔT ∼ 2 × 10−5 T

(1)

on an angular scale of ∼10 degrees (Planck Collaboration 2020a). On the largest scales, these fluctuations are believed to be the result of quantum fluctuations imprinted on the sky as the universe experienced a period of rapid accelerated expansion. Before going on to analyze the fluctuations in the CMB, a review of inflationary cosmology is in order.

Shortcomings of the Standard Big Bang There are six problems with the standard big bang that are addressed (at least in part) by an early epoch of accelerated expansion. These are as follows.

The Horizon Problem The horizon problem refers to the fact that the cosmic microwave background temperature is almost the same in all directions, even for regions that were not in apparent causal contact at the time the CMB was emitted. To see this, consider the angular size of a region in causal contact at the time the CMB was emitted for a flat cosmology. If the size of a region in causal contact is ∼H (z)−1 , the angle subtended on the sky is then θ=

1 , dA H (z)

(2)

while for a flat cosmology dA = dH /(1 + z), and for the Planck values of Ωm = 0.311, ΩΛ = 0.6889, one can numerically evaluate the angular distance to the surface of last scattering at z ≈ 1100 dA =

1 H0 (1 + z)

 0

1100

dz 3.15 . =  3 1/2 H0 (1101) [Ωm (1 + z ) + ΩΛ ]

(3)

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Then the expansion rate at that epoch is H (z) = H0 [Ωm (1 + z)3 + ΩΛ ]1/2 = H0 × 2.04 × 104 ,

(4)

and the angle subtended on the sky is θ ≈ 0.017 ≈ 1 degree. So, the size of a region in causal contact is only about a degree. However, regions 180◦ apart on the sky somehow know to be at the same temperature. That is the horizon problem. An epoch of inflation fixes this dilemma by expanding a region in causal contact to exceed the scale of the horizon today, so that all regions began in equilibrium.

The Flatness Problem To understand the flatness problem, consider separating out the curvature term from the Friedmann equation and representing the sum of the other closure parameters as Ωt , then one can write   H02 H02 Ωk Ωr Ωm ≡ Ω + + Ω − 1 = − Λ t H 2 (a) a 4 a3 a2H 2

(5)

As the universe expands, the quantity on the right-hand side grows continually during most of the evolution of the universe, while it was dominated by either matter or radiation. For example, in a nearly flat matter-dominated universe, |Ωt − 1| ∝ a ∝ t 2/3 , or for a radiation-dominated universe, ∝ a 2 ∝ t. So, neglecting the slight growth in the curvature term as the universe has begun to enter the Λ-dominated epoch, for the curvature to be Ωk < 0.01 today (Planck Collaboration 2020a), it was necessary that the curvature term to be extremely small in the past. As an illustration, consider how much the curvature term has grown since the epoch of nucleosynthesis at t = 1 sec. The epoch of radiation domination lasted from then until the epoch of matter-radiation equality at t ∼ 105 yr. During this interval, the curvature term would grow linearly in time corresponding to growth by a factor of ∼3 × 1012 . Then, in the matter-dominated epoch from 105 to 1010 yr, the curvature term would grow by another factor of (105 )2/3 ∼3 × 107 . So, for the curvature term to be Ωk < 0.01 today, then the curvature term must have been smaller by a factor of ∼10−20 at the start of the nucleosynthesis epoch. Indeed, extending the radiation-dominated term all the way back to the Planck epoch would require that the curvature be fine-tuned to better than a part in 1060 . An inflationary epoch fixes this problem by introducing an epoch of nearly constant vacuum energy 1/2 ΩΛ = 1. For this case H = H1 ΩΛ , where H1 is a constant and ΩΛ = 1. The scale factor then grows exponentially: a(t) = a1 eH1 t .

(6)

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As long as the number of e-folds H1 t are sufficiently large ∼50 − 60, the curvature term can be made small enough to account for the present value.

The Monopole Problem This was one of the original motivations for an epoch of inflation (Guth 1981). The problem (Peacock 1999) is that a general feature of grant unified theories (GUTs) is the existence of point-like defects that have a magnetic field configuration at infinity. These defects have the mathematical appearance of an isolated magnetic charge, that is, B=

μ0 M r, 4π r 2

(7)

where M denotes a magnetic charge. Indeed, there is a proof due to ’tHooft (1974) and Polyakov (1974) that almost any spontaneously broken non-Abelian gauge theory will lead to the existence of magnetic monopoles. Moreover, there is an argument due to Dirac (1931) that the existence of even one monopole in the universe implies that the monopole charge will be quantized with M=

2π h¯ n. μ0 e

(8)

A detailed discussion of the physics of magnetic monopoles is beyond the scope of this chapter. However, the reader is referred to Peacock (1999) for an introduction. The problem, of course, is that such magnetic monopoles have never been observed. An epoch of accelerated expansion occurring after or during the epoch of GUTs resolves this problem by diluting the number density of magnetic monopoles to a very small number (Guth 1981).

The Smallness Problem This is a problem that is related to inflation, but not solved by it. It is the fact that the cosmological constant has such a small value compared to the value expected from particle physics. There is a simple heuristic argument that the value of the vacuum energy should be enormous compared to the observed of ρvac = ΩΛ ρcrit ≈ 5 × 10−3 MeV cm−3 . Consider the uncertainty principle which says that even in the vacuum, there should be fluctuations in energy in a short enough timescale such that ΔEΔt = h. ¯ Imagine a particle and antiparticle pair of mass m such that ΔE = 2mc2 . The timescale during which this particle pair could appear is Δt = h¯ /(2mc2 ). Then even if restricted to standard model particles with m ≤ 125 GeV, this would correspond to an energy density of

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3 ρ = 2mc2 /(cΔt)3 = (2mc2 )4 /(hc) > 2 × 1053 MeV cm−3 . ¯

(9)

If one extends this to particles near the Planck scale of 1019 GeV, then the energy increases to ∼10121 MeV cm−3 , a discrepancy of 124 orders of magnitude. A slightly more rigorous argument from quantum field theory is the following. The vacuum is treated as a field of quantized harmonic oscillators which for a massless scalar field has energy levels:   1 En = n + hω. ¯ 2

(10)

The vacuum energy E0 is then a sum over all lowest frequency modes: E0 =

1

h¯ ωi .

2

i

(11)

The sum is evaluated, for example, within a box of length L for which L → ∞. Boundary conditions on the box are chosen such that the wavelength of each mode must vanish at the boundary, i.e., L = λi ni . Adopting the wave number k = 2π/λ then n=

Lk . 2π

(12)

and the total number of frequency nodes within the box becomes E0 =

1 h¯ L3 2 (2π )3

 ωk d 3 k,

(13)

where the integral over the wave number k is performed up to some maximum value kmax . Then the vacuum energy density becomes E0 1 h¯ = 3 L→∞ L 2 (2π )3



kmax

ρvac = lim

0

k · 4π k 2 dk =

4 h¯ kmax . 16π 2

(14)

If kmax is restricted to be below the Planck scale, kmax ∼ Epl /h¯ with Epl ∼ 1022 MeV, then one obtains a vacuum energy density of ρvac ∼ 1086 MeV4 (hc) ¯ −3 ∼ 10118 Mev cm−3 . Hence, there are ∼121 orders of magnitude between the observed value of 5 × 10−3 MeV cm−3 and the anticipated vacuum energy from quantum field theory. Although inflation does not really resolve this discrepancy, it does pose a situation in which the vacuum energy can begin with a value at or near the Planck energy density that can subsequently evolve to a smaller value.

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The Structure Formation Problem Decades ago it was realized that the structure observed today in galaxies and largescale structure needed a spectrum of initial fluctuations in order to account for the galaxies and large-scale structure observed at high red shift. As shall be seen later in this section, the inflationary epoch provides a primordial spectrum of fluctuations that provides the seeds of galaxy formation and large-scale structure. In this case, it is attributable to nearly scale-free quantum fluctuations that were unavoidably produced during the epoch of inflation. These fluctuations, plus the existence of cold dark matter, provided the needed seeds for galaxy and structure formation. They also place significant constraints on the nature of the inflationary epoch as discussed below.

The Initial Conditions Problem Until the 1980s, it was presumed that the initial conditions needed to produce the presently observed universe required a solution of the formidable problem of quantum gravity. A final motivation for inflation might be that the theory provides many of the needed conditions of the early universe at energy scales for which classical relativity (for the most part) can be applied, thus rendering the problem solvable with the tools at hand.

Inflation Basics The starting point for inflation is to postulate the existence of an unspecified scalar field φ referred to as the inflation field. The classical Klein Gordon Lagrangian density for this self-interacting inflation field is of the form L =

1 μ ∂ φ∂μ φ − V (φ), 2

(15)

where the choice of V (φ) defines the inflation model. The simplest choice is that of a quadratic potential of the form V (φ) =

1 2 2 m φ , 2

(16)

where m is the inflaton mass. Although this form has some motivation from its loose similarity with the Kahler potential in supergravity (Liddle and Lyth 1998), this potential, however, is now nearly ruled out by the Planck observation (Planck Collaboration 2020b) which, as shown below, requires a very low ratio of tensor to scalar perturbations in the CMB power spectra. Other forms of the potential are discussed below.

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Given the Lagrangian density in Eq. (15), one can construct the energymomentum tensor for the inflation field:   μν μ ν μν μ ν μν 1 α Tφ = ∂ φ∂ φ − g L = ∂ φ∂ φ − g (17) ∂ φ∂α φ − V (φ) 2 If the field is uniform, the spatial derivatives vanish, and the energy-momentum tensor becomes diagonal. In analogy with the perfect fluid energy-momentum tensor, one can identify the energy density with ρφ =

φ˙ 2 + V (φ), 2

(18)

φ˙ 2 − V (φ), 2

(19)

while the pressure term looks like pφ =

Note that even if the spatial derivative terms were included, they would be divided by a(t)2 which means that within a short time during inflation, they would be exponentially suppressed. Finally, an equation of motion for the scalar field is needed. This can be obtained μν from the covariant derivative of the energy momentum tensor T ν = 0 or directly from the relativistic Lagrange’s equations applied to the Lagrangian density. This leads to φ¨ + 3H φ˙ + ∇ 2 φ +

dV = 0, dφ

(20)

where ∇2 ≡

1 ∂2 . a2 ∂ 2x i

(21)

Assuming a uniform field then the more commonly used equation of motion is φ¨ + 3H φ˙ +

dV = 0. dφ

(22)

Slow Roll Approximation Equation (22) is analogous to that of a ball rolling down a hill with a friction term of ˙ Because H is a large number, this friction term damps the acceleration term 3H φ. ¨ φ to be small compared to the other two terms. This is the slow-roll approximation. In this case the equation of motion of the inflation field simplifies to

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G. J. Mathews and G. Tang

3H φ˙ = −

dV (φ) , dφ

(23)

The Friedmann equation for a universe dominated by an inflation field is just H2 =

  8π G 1 2 φ˙ + V (φ) , 3 2

(24)

which under the slow-roll condition can be reduced to H2 =

8π G 1 V (φ), V (φ) = 3 3m2pl

(25)

where here, it is convenient to introduce (Liddle and Lyth 1998) the reduced Planck mass mpl = (8π G)−1/2 , for which the reduced Planck energy is 2.436 × 1018 GeV. The utility of this will be apparent later as a means to identify the energy scales relevant to various inflation effective potentials. Note that this definition (Liddle and Lyth 1998) differs from √the usual definition of the Planck mass (Kolb and Turner 1990) by a factor of 1/ 8π .

Slow Roll Parameters Now the condition that the universe be accelerating is simply a¨ = H˙ + H 2 > 0, a

(26)

where the equation of the right-hand side follows from a simple application of the chain rule. Since it is likely that the H˙ term is negative, then the condition for acceleration is that the H˙ term be small compared to H 2 . This can be expressed ε≡−

    m2pl V  2 1 1 H˙ = =

1 dt H 2 V H2

(27)

where V  denotes differentiation dV /dφ. One can also differentiate this expression to obtain the criterion V  1,

(103)

where r ≡ |r − rj |. This distribution has a finite range of influence outside of which no mass from particle j is distributed. This reduces the computation time, since only

93 Inflation, Perturbations, and Structure Formation

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those particles with non-negligible contributions are included in the sum. Since each SPH particle has its own smoothing sphere, the density at a point r involves a sum over every smoothing sphere that contains that point. With this discretization of particles, the Euler momentum equation becomes  dvi mj =− dt j



 Pj Pi + 2 + Πij ∇i W ij , ρi2 ρj

(104)

where Πij is an artificial viscosity term. The equation for the internal energy density can then be written 1 dui = mj dt 2 j



 Pj Pi + 2 + Πij (vi − vj ) · ∇i Wij . ρi2 ρj

(105)

These equations are used to advance the fluid system in time. There are of course also the issues (Springe et al. 2001; Springel 2005) of the gravitational potential, heating and cooling flows, the role of black-hole formation and heating, star formation, chemical evolution, etc. All of which contribute to the formation and evolution on galaxies and large-scale structure (Mo et al. 2010). As an illustration, Fig. 3 shows a cosmological large-scale structure simulation (Zhao 2011) in a comoving 200 Mpc/hc box with a periodic boundary condition for a ΛCDM cosmology at the current epoch. This clearly illustrates the cosmic web of filaments, sheets, and voids that characterizes the large-scale structure of the universe.

Fig. 3 Simulated cosmic structure on a scale of 200 Mpc/hc in a ΛCDM cosmology at the current epoch from Zhao (2011)

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Acknowledgments Work at the University of Notre Dame was supported by the U.S. Department of Energy under Nuclear Theory Grant DE-FG02-95-ER40934. The authors acknowledge the help of Miguel Correra (UND) in the generation of the tensor-to-scalar ratios from monomial potentials.

References J.M. Bardeen, J.R. Bond, N. Kaiser, A.S. Szalay, Astrophys. J. 304, 15 (1986) P. Callin, eprint arXiv:astro-ph/0606683 (2006) M. Correra, M.R. Gangopadhyay, N. Jamanc, G.J. Mathews, Phys. Lett. B835 137510 (2022) P.A.M. Dirac, Proc. Roy. Soc. (Lond.) A 133, 60 (1931) S. Dodleson, Modern Cosmology (Academic Press, San Francisco, 2003) R.A. Gingold, J.J. Monaghan, Mon. Not. R. Astron. Soc. 181, 375 (1977) A.H. Guth, Phys. Rev. D23, 347 (1981) L. Kofman, A. Lnde, A.A. Starbinsky, arXiv:hep-ph:9704452 (1997) E.W. Kolb, M.S. Turner, The Early Universe (Addison-Wesley, Menlo Park, 1990) A. Lewis, S. Bridle, Phys. Rev. D66, 103511 (2002) A.R. Liddle, D.H. Lyth, Cosmological Inflation and Large Scale Structure (Cambridge University Press, Cambridge, 1998) A. Liddle, P. Parsons, J.D. Barrow, Phys. Rev. D50, 7222 (1994) A. Linde Phys. Rev. Lett. B129, 117 (1983) A. Linde, in Inflation and Quantum Cosmology (Academic Press, New York, 1990a) A. Linde, in Particle Physics and Inflationary Cosmology (Harwood, Chur, 1990b) L.B. Lucy, Astron. J. 82, 1013 (1977) G.J. Mathews, A. Snedden, L.A. Phillips, I.-S. Suh, J. Coughlin, A. Bhattacharya, X. Zhao, N.Q. Lan, Mod. Phys. Lett. A29, 1430012 (2014) G.J. Mathews, B. Rose, P. Garnavich, D. Yamazaki, T. Kajino, Astrophys. J. 827, 60 (2016) L. Mersini-Houghton, R. Holman, JCAP 2, 006 (2009) H. Mo, F. van den Bosch, S. White, Galaxy Formation and Evolution (Cambridge University Press, New York, 2010) B.W. O’Shea, K. Nagamine, V. Springel, L. Hernquist, M.L. Norman, Astrophys. J. Suppl. 160, 1 (2005) J.A. Peacock, Cosmological Physics (Cambridge University Press, Cambridge, 1999) Planck Collaboration, A&A 641, A6 (2020a) Planck Collaboration, A&A 641, A10 (2020b) A.M. Polyakov, JETP Lett. 20, 194 (1974) V. Springe, N. Yoshida, S.D.M. White, New Astron. 6, 51 (2001) V. Springel, MNRAS 364, 1105 (2005) G. ’tHooft, Nucl. Phys. B. 79, 276–284 (1974) X. Zhao, PhD Thesis, University of Notre Dame, 2011

Section XIV Evolution of Stars and Nucleosynthesis Ken’ichi Nomoto

Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction) Friedrich-Karl Thielemann

94

and Thomas Rauscher

Contents Describing Nuclear Composition Changes via Nuclear Reaction Rates . . . . . . . . . . . . . . . . Stellar Burning Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrogen Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helium Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbon, Neon, and Oxygen Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Silicon Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Burning in Explosive Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burning Timescales for Explosive He, C, Ne, O, and Si-Burning . . . . . . . . . . . . . . . . . . . Special Features of Explosive Si-Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The r-process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explosive H-Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Type of Reactions Are Needed in Explosive Burning? . . . . . . . . . . . . . . . . . . . . . . . Thermonuclear Rates and the Hauser-Feshbach Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . Particle Transmission Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . γ -Transmission Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Level Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Width Fluctuation Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross Section Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3436 3441 3441 3445 3447 3451 3454 3457 3459 3461 3461 3463 3463 3465 3467 3472 3474 3477 3478 3482 3483

F.-K. Thielemann () Department of Physics, University of Basel, Basel, Switzerland GSI Helmholtz Center for Heavy Ion Research, Darmstadt, Germany e-mail: [email protected] T. Rauscher Department of Physics, University of Basel, Basel, Switzerland Centre for Astrophysics Research, University of Hertfordshire, Hatfield, UK e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_115

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Abstract

This chapter will go through the important nuclear reactions in stellar evolution and explosions, going through the individual stellar burning stages and also explosive burning conditions. To follow the changes in the composition of nuclear abundances requires the knowledge of the relevant nuclear reaction rates. For light nuclei (entering in early stellar burning stages), the resonance density is generally quite low, and the reactions are determined by individual resonances, which are best obtained from experiments. For intermediate mass and heavy nuclei, the level density is typically sufficient to apply statistical model approaches. For this reason, while we discuss all burning stages and explosive burning, focusing on the reactions of importance, we will for light nuclei refer to the chapters by M. Wiescher, & deBoer & Reifarth, Chap. 95, “Experimental Nuclear Astrophysics” and P. Descouvement, Chap. 37, “Theoretical Studies of Low-Energy Nuclear Reactions”, which display many examples, experimental methods utilized, and theoretical approaches how to predict nuclear reaction rates for light nuclei. For nuclei with sufficiently high level densities, we discuss statistical model methods used in present predictions of nuclear reaction cross sections and thermonuclear rates across the nuclear chart, including also the application to nuclei far from stability and fission modes.

Describing Nuclear Composition Changes via Nuclear Reaction Rates Before going into more details about the nuclear reactions in evolving stars in the next sections, we present here a short overview of the tools used to calculate abundances changes via nucleosynthesis networks and the relevant reaction rates. Abundances of nuclei can defined by their mass fraction (Xi ), i.e. the percentage of their mass in species i (or density ρi ) with respect to the total mass (density) in two possible ways: ni mi ni Ai mu ni ρi = = = Ai ρ ρ ρ (ρ/mu ) ni mi ni ni Xi = NA = Ai mu NA = A i Mu , ρNA ρNA ρNA

Xi =

(1a) (1b)

with ni being the number density, ρ the overall mass density, NA the Avogadro 1 number, mu = 12 m(12 C) the nuclear mass unit, Ai the relative atomic mass, and Mu = NA mu the molar mass constant. Replacing the relative atomic mass Ai by Ai = Ni + Zi , i.e. the nuclear mass number given by the sum of neutrons and protons in a nucleus, introduces an error in the per-mille range due to the nuclear binding, which is usually neglected. The abundance of a nucleus, rather than its mass fraction, should not reflect its weight and is defined as Yi = Xi /Ai ≈ Xi /Ai , which can be expressed, utilizing Eq. (1), by Yi = ni /(ρNA )Mu = ni /(ρ/mu ).

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Until 2019 Mu had the value of 10−3 kg/mole or 1 g/mole in cgs units. This led Caughlan and Fowler (1988) and the following literature to omit this factor of 1 in the definition of the abundance (but also introducing the dimension of g/mole). According to the latest CODATA evalution (Table XXXI; Tiesinga et al. 2021) this is still close to being correct (numberwise) with a precision of 3×10−10 , but introduces somewhat confusing dimensions for the abundances. We will continue to utilize here the traditional expressions, but want to make clear that, when changing all terms ρNA – appearing in equations introduced here and later in this text – to ρ/mu , like in Yi = ni /(ρ/mu ) (as often done in more modern approaches, see e.g. Cowan et al. 2021), the expressions stay valid and lead for the mass fractions Xi = Ai Yi as well as the abundances Yi to dimensionless numbers. All mass fractions add up to unity: 

Xi =



i

Ai Yi = 1.

(2)

i

Astrophysical environments carry generally no charge, i.e., the total number of protons has to be equal to the number of electrons: 

Zi Yi = Ye =

i

ne , ρNA

(3)

with the electron fraction (Ye ) being on the one hand defined by the electron number density but alsoequal tothe number of protons per nucleon (i.e., protons plus neutrons) Ye = i Zi Yi / i Ai Yi . The individual abundances can be calculated with a nuclear reaction network based essentially on r, the number of reactions per volume and time that can be expressed, when targets and projectiles follow specific distributions dn, by  rij =

σ · | vi − vj |dni dnj .

(4)

The evaluation of this integral depends on the type of particles and distributions which are involved. For nuclei i and j in an astrophysical plasma, obeying a Maxwell–Boltzmann distribution, we find rij = ni nj σ vij where σ v is integrated over the relative bombarding energy and is only a function of temperature T :  σ vij (T ) =

8 π μij

1/2

(kT )−3/2



∞ 0

  E dE, Eσ (E) exp − kT

(5)

where μij is the reduced mass of target and projective i and j . For a reaction with photons, we have j = γ , i.e., in this case the projectile j is a photon. The relative velocity is the speed of light c, and the distribution dnj is the Planck distribution of photons. As the relative velocity between the nucleus and the photon is a constant (c) and the photodisintegration cross section is only dependent on the photon energy Eγ , the integration over dni can be easily performed, resulting in

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riγ = ni λiγ (T ) =



1 π 2 c2 h¯ 3  1

π 2 c2 h¯ 3

σi (γ ; Eγ )Eγ2 exp(Eγ /kT ) − 1 σi (γ ; Eγ )Eγ2 exp(Eγ /kT ) − 1

dEγ = ni λiγ (T )

dEγ .

(6)

Contrary to the reactions among nuclei or nucleons, where both reaction partners are following a Boltzmann distribution, this expression has only a linear dependence on number densities. The integral acts like an effective (temperature dependent) decay constant of nucleus i. Electron captures behave in a similar way as the mass difference between nucleons/nuclei and electrons is huge and the relative velocity is with high precision given by the electron velocity. This leads to  ri;e = ni

σe (ve )ve dne = λi;e (ρYe , T )ni .

(7)

This is an expression similar to that for photodisintegrations, but now we have a temperature and density-dependent “decay constant,” because the electron distribution has also a density dependence and can change to a degenerate Fermi gas. In principle neutrino reactions with nuclei would follow the same line, because the neutrinos, propagating essentially with light speed, would lead to a simple integration over the neutrino energies. However, as neutrinos, interacting very weakly, do not necessarily obey a thermal distribution for local conditions, their spectra depend on detailed transport calculations (see the chapters by Rrapaj & Reddy,  Chap. 100, “Neutrino Charged and Neutral Current Opacities in the Decoupling Region” and Fuller & Grohs,  Chap. 102, “Big Bang Nucleosynthesis” and Wang & Surman,  Chap. 103, “Neutrinos and Heavy Element Nucleosynthesis” and Wanajo,  Chap. 108, “Nucleosynthesis in Neutrino-Heated Ejecta and Neutrino– Driven Winds of Core-Collapse Supernovae: Neutrino-Induced Nucleosynthesis” and Obergaulinger,  Chap. 109, “Nucleosynthesis in Jet-Driven and Jet-Associated Supernovae”), leading to  riν = ni

σe (Eν )cdnν (Eν ) = λiν (transport)ni .

(8)

Finally, for normal decays, like beta- or alpha-decays or ground state fission with a half-life τ1/2 , we obtain a similar equation with a decay constant λi = ln2/τ1/2 (see the chapters by Rubio et al.,  Chap. 9, “Beta Decay: Probe for Nuclear Structure and the Weak Interaction” and Block et al.,  Chap. 11, “Heaviest Elements: Decay and Laser Spectroscopy” and Oberstedt & Oberstedt,  Chap. 20, “The Multi-humped Fission Barrier” and Kowal & Skalski,  Chap. 24, “The Fission Barrier of Heaviest Nuclei from a Macroscopic-Microscopic Perspective”) and ri = λi ni .

(9)

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In this case the change of the number density due to decay is n˙ i = −λi ni , with the solution ni = ni (0)e−λi t and ni (τ1/2 ) = 12 ni (0). The decay half-life of a nuclear ground state is a constant. Adding all these different kind of reactions, we can describe the time derivative of abundances Y = n/ρNA as the difference of production and destruction terms with a differential equation for each species. Both the production and destruction channels include particle-induced reactions, decays, photodissociation, electron capture, etc. For every nucleus i, the abundance is given by a differential equation Y˙i =



Nji λj Yj

+

j

j,k

i  Nj,k,l j,k,l

i  Nj,k

1 + Δj kl

1 + δj k

ρNA σ vj,k Yj Yk +

ρ 2 NA2 σ vj,k,l Yj Yk Yl ,

(10)

where the factors 1/(1 + δj k ) and 1/(1 + Δj kl ) prevent double counting of reactions in two- and three-body reactions, respectively. Δj kl has the value 0, 1, or 5, so that, dependent on the multiplicity of identical partners, the denominators are equal to 1!, 2!, or 3!. Knowing the cross section σ (E) of a nuclear reaction, σ v can easily be determined, provided that the participating nuclei obey Boltzmann statistics. In the following section, we want to utilize this method to predict abundance/composition changes. We will first go through the stellar burning stages identifying the important reactions, while we rely on the sections by M. Wiescher and P. Descouvement how they are determined experimentally or theoretically for light nuclei, before introducing statistical model methods for intermediate mass and heavy nuclei. One extreme case should however be mentioned, which occurs at sufficiently high temperatures when on the one hand, the Coulomb barriers can be overcome for capture reactions and on the other hand, the reverse photodisintegrations are efficient due to the high-energy photons of the corresponding Planck distribution. For a reaction i(p, γ )m, with i standing for nucleus (Z, A) and m for (Z + 1, A + 1), the relation between the forward rate σ vi;p,γ and the photodisintegration rate λm;γ ,p is given by gp Gi λm;γ ,p (T ) = Gm



μin kT 2π h¯ 2

3/2 exp(−Qi;p,γ /kT ) σ vi;p,γ ,

(11)

containing the reduced mass μin and the capture Q-value for the reaction, gp = 2 × (1/2) + 1 for the protons and the partition functions G for the nuclei involved. When utilizing that the difference between forward proton capture and backward photodisintegration flux for nucleus i, i.e., Y˙i = −ρNA σ vi;p,γ Yp Yi + λm;γ ,p Ym vanishes in chemical equilibrium. Making also use of the relation between σ vi;p,γ and λm;γ ,p , based on detailed balance, results in

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Ym Gm = ρNA Yp Yi 2Gi



Am Ai

3/2

2π h¯ 2 mu kT

3/2



Qi;p,γ exp kT

 .

(12)

Such a chemical equilibrium for a specific forward and reverse reaction can also apply for, e.g., neutron or even alpha-captures. When temperatures are sufficiently high to permit a whole sequence of proton and neutron captures up to nucleus (Z, A) or (Z, N ), a complete chemical equilibrium (also termed nuclear statistical equilibrium nuclear statistical equilibrium (NSE)) can be attained, linking neighboring nuclei via neutron or proton captures as in Eq. (12) up to nucleus (Z, N ). The approach to such NSE conditions in discussed in the following section on burning phases in stellar evolution. It is important to note that the reaction cross sections appearing in Eqs. (4), (5), (6), (7), and (8), as well as the reactivities σ v in Eq. (10) must account for the thermal excitation of nuclei in an astrophysical plasma and may differ from laboratory cross sections determining cross sections of target nuclei in their ground states only. In an astrophysical environment, nuclei are found in excited states with a probability given by the plasma temperature T and the excitation energies Ei of low-lying excited states (Ward and Fowler 1980). Thus, the actual stellar reaction rate r ∗ is given by a weighted sum of rates of reactions on individual excited states bombarded by a thermal (or nonthermal, in case of neutrinos) distribution of projectiles (Rauscher 2020), r ∗ = P0 r0 + P1 r1 + P2 r2 + . . . ,

(13)

with the individual population factor of an excited state given by Pi =

2Ji + 1 −Ei /(kT ) e . G

(14)

Here, G is the nuclear partition function of the target nucleus and Ji the spin of excited state i (i = 0 for the ground state). The reciprocity relation shown in Eq. (11) also only holds when such stellar rates are used. Instead of summing over individual contributions from several rate integrals as shown in Eq. (13), the form of a single integral (as in Eq. 4) can be retained by using a stellar cross section σ ∗ (E, T ) as defined in Eq. (26) (see also (Holmes et al. 1976; Rauscher 2020, 2022)) instead of σ in Eq. 4. The relative contribution to the stellar rate of reactions on nuclides in the ground state can be computed from r0 (15) X0∗ (T ) = ∗ , r G where r0 is the reaction rate calculated from reactions on the ground state only. Since the population of excited states depends on the excitation energies of available excited states, the impact of excited states is smaller (and thus X0 ≈ 1) for light nuclides with large level spacings but is non-negligible (X0 < 1) for intermediate

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and heavy nuclides even at such low temperatures kb T of a few keV, as encountered, e.g., in the s-process (Rauscher 2022). In hot, explosive astrophysical environments, X0 becomes tiny and most contributions to the stellar rate come from reactions on excited states.

Stellar Burning Stages Stellar burning stages are characterized by reactions involving increasingly larger nuclei (and charges). The necessary velocities (energies) to overcome the Coulomb barriers of heavier nuclei increase with charge Z and are related to the environment temperature. When one type of reaction sequence is started, the energy loss of a star (its luminosity) is equal to the total energy generation, once stable burning conditions have been attained. The burning temperature stays approximately constant until the fuel is exhausted. At that point decreasing pressure support will lead to a contraction, which again causes a temperature increase, due to the release of gravitational binding energy. At some critical temperature threshold, the nuclei of the next burning “fuel” can overcome the Coulomb barriers in the main reactions of this next burning stage. In massive stars the sequence of burning stages is given by the fuels 1 H, 4 He, 12 C, 20 Ne, 16 O, and 28 Si (see also the  Chap. 99, “Thermal Evolution of Neutron Stars” by K. Nomoto on the impact of reaction rate uncertainties on stellar evolution). We want to discuss the details and the main reaction sequences of all these burning stages in the following subsections.

Hydrogen Burning The PP-Cycles In a pure hydrogen gas, the reaction sequences in Table 1 occur at T ≈ 107 K. We give the individual sub-cycle name, the Q-value, and a lifetime for each reaction. Although the purely nuclear Q-value of the first pp-reaction is only 0.420 MeV, the effective Q-value is also 1.442 MeV as in the pep-reaction, because the positron annihilates with an electron in the plasma and produces 1.022 MeV in photons. In Table 1 The PP-cycles in hydrogen burning Cycle PPI (Pep-reaction)

PPII

PPIII PPIV (Hep reaction)

Reaction 1 H(p,e+ ν)2 H 1 H(pe− , ν)2 H 2 H(p,γ )3 He 3 He(3 He,2p)4 He 3 He(α, γ )7 Be 7 Be(e− , ν)7 Li 7 Li(p,α)4 He 7 Be(p,γ )8 B 8 B(e+ ν)8 Be∗ (α) 4 He 3 He(p,e+ ν)4 He

τ (years) 7.9×109 3.7×1012 5.9×10−8 1.4/Y3 1.1×106 2.9×10−1 4.3×10−5 1.8×102 3.5×10−8 3.7 ×107

Q(MeV) 0.420 1.442 5.493 12.859 1.586 0.861 17.347 0.135 18.078 19.795

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the case of a decay, lifetimes are related to the decay constant λ via τ = 1/λ, or half-life with λ = ln(2)/τ1/2 . After a lifetime τ passed in a reservoir of unstable nuclei, a fraction 1/e will remain. For reactions involving a projectile and target, the target lifetime τi is given by Y˙i = − τ1i Yi , containing the reaction rate (i.e., requiring a knowledge of density and temperature) and the projectile abundance. The table values were obtained assuming ρ = 100 g cm−3 and T = 1.5 × 107 K, based on reaction rate evaluations of Angulo et al. (1999) and Adelberger et al. (2011). The abundances are taken from an intermediate state of H-burning with YH = 0.5, YHe = 0.5/4 (i.e., half of the H is burned to He), while in a steady flow equilibrium the abundances of the intermediate nuclei are close to negligible. For most of the reactions in Table 1, the projectile is either 1 H or 4 He, i.e., the lifetimes can be determined with these known abundances; an exception is 3 He(3 He,2p)4 He. When taking typical steady flow values of the order Y3 = Y3 He ≈ 10−4 , this leads to a lifetime of about 104 years. In all cases (i.e., PPI, PPII, PPIII, or PPIV), the net reaction is the fusion of four protons into one alpha particle (4 He) with two β-decays or electron captures among the intermediate reactions. In the PPI-cycle, six hydrogen nuclei (protons) are required in total; two are finally released in the 3 He-3 He-reaction, together with one helium nucleus, i.e., the net reaction is 4 1 H→4 He. The slowest reaction (of relevance) in all the cycles is the first one listed. The pep-reaction is the slowest of all, because a three-body collision is a very rare event. It is, however, unimportant for H fusion, as the pp-reaction burns and produces the same nuclei, only the properties of the emitted neutrinos are different. The PPIV cycle or Hep reaction is mostly important for relatively high-energy neutrinos, but, as can be seen from the lifetime of the Hep reaction in comparison to 3 He(p,γ ), it carries also about one third of the latter PPII flux from 3 He to 4 He. The pp-reaction is very slow, because a (2 He) is unstable and breaks up immediately. during the very limited time of the existence of a diproton, a weak transition of one of the protons into a neutron occurs. The production of deuterium (2 H) is only possible during the very limited time of the existence of a diproton, when a weak transition of one of the protons into a neutron occurs. Due to its rarity, this reaction rate is based entirely on theoretical predictions. A cross section that small is not accessible by experiments. It should, nevertheless, be quite accurate, because only the fundamental constants of weak interaction theory enter. Thus a description of nuclear many body systems (nuclei), which may or may not be correct, is not required (see the detailed discussion in, e.g., Rolfs and Rodney 1988; Bahcall 1989; Adelberger et al. 2011). The other PP-cycle reactions are studied by experiments (see for historical reasons the detailed discussions in Rolfs and Rodney 1988), Caughlan and Fowler (1988) and updates in Angulo et al. (1999) and Adelberger et al. (2011). Typically, cross sections are measured at somewhat higher energies, where cross sections are larger and then extrapolated to the solar energies of interest (see e.g. Formicola et al. 2016). There existed some uncertainty in the reaction 7 Be(p,γ )8 B, but recent evaluations by Tursunov et al. (2021) provide nice agreement with the (Adelberger et al. 2011) rate. 7 Be is unstable against electron capture (EC), not β + -decay, due to a small Q-value. The terrestrial lifetime, τ = 77 d (half-life τ1/2 = 53 d), is caused

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by capture of a bound K-shell electron. For conditions in the center of stars, where atoms are completely ionized, electron capture has to occur with electrons from the continuum electron distribution, which leads to a lifetime of 106d for the conditions of Table 1 (Adelberger et al. 2011). The terrestrial lifetime is long enough to produce radioactive 7 Be targets, and several direct measurements of the 7 Be(p,γ )8 B reaction have been performed by radioactive ion beam facilities (see, e.g., Adelberger et al. 2011). In such a case, 7 Be, produced by a primary nuclear reaction, is accelerated in a second step onto an existing hydrogen gas target. Thus, the role of target and projectile is reversed and the reactions. After a burning cycle proceeds for a timescale in excess of all reaction timescales in the cycle, except for the longest one, an abundance equilibrium is attained. In general such a situation is called a steady flow equilibrium, where the slowest link determines the speed and all other reactions adjust to that reaction flow. The destruction of 1 H (and the buildup of 4 He) is caused, in temperature and densitydependent fractions, by the four PP-cycles, determined by the branch points at 3 He and 7 Be, which open the PPII-, PPIV-, and PPIII-cycles, respectively. Only the latter branching is temperature and density dependent, because a decay [7 Be(e− , ν)7 Li] and a capture reaction have a different density dependence. Knowing the hydrogen abundance at a given time (and the density and temperature for determining the reaction rate) automatically determines the steady flow abundances of 2 H and 3 He. From Table 1 and the destruction timescales, we see that only A = 1 and 4 can have non-negligible abundances, more than four orders of magnitude larger than those of any other cycle member. Mass conservation gives Y4 = (1 − Y1 )/4. As can be seen from Table 1, the much larger lifetime of 3 He in the Hep reaction by two to three orders of magnitude makes the PPIV cycle contribution unimportant for energy generation, but this sub-cycle can contribute the neutrinos with the highest energies.

The CNO-Cycles If heavier elements (CNO) are already present in the stellar plasma, another chain of reactions is possible, when temperatures are large enough to permit the penetration of the larger Coulomb barriers. These reactions can be faster at appropriate temperatures than the weak pp-reaction which involves a p(e+ ν)n transition. Such a sequence of reactions, involving the most abundant lightest nuclei beyond 4 He (C, N, and O), are known as the CNO-cycles. Most of the reactions listed in Table 2 are also studied experimentally, but again the extrapolation to low energies in the keV range is important to understand

Table 2 The CNO-cycles in hydrogen burning Cycle CNOI CNOII CNOIII CNOIV

Reaction sequence 12 C(p,γ )13 N(e+ ν)13 C(p,γ )14 N (p,γ )15 O(e+ ν)15 N(p,α)12 C 15 N(p,γ )16 O(p,γ )17 F (e+ ν)17 O(p,α)14 N 17 O(p,γ )18 F(e+ ν)18 O(p,α)15 N 18 O(p,γ )19 F(p,α)16 O

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the behavior at Gamow peak energies (see e.g. Adelberger et al. 2011). A major uncertainty has been the 17 O(p,α)/17 O(p,γ )-branching between the CNOII and CNOIII cycle, related mostly to a low-lying 65 keV resonance (see, e.g., Rolfs and Rodney 1988) which has recently been determined by Bruno et al. (2016) in a direct measurement. The CNOIV chain is not of great importance at all temperatures; the flux is more than a factor 100 smaller than in the CNOIII chain. At lower temperatures we have only one single chain without branchings, which is the CNOI chain. Therefore, the flux passing through each reaction in case of a steady flow equilibrium, C = ρNA i1 Yi Y1 , must be identical. 14 N(p,γ )15 O has the smallest reaction rate, which means that the larger rates for other reactions have to be balanced with smaller abundances and almost negligible abundances of the other CNO nuclei remain. In this case, when assuming initial solar abundances for all CNO-nuclei, a steady-flow equilibrium assembles all CNO nuclei in 14 N. This leads to Y14 ≈

XCN O , 14

(16)

Solar system abundance determinations have changed in recent years. While meteoritic measurements help to determine the relative abundances of nonvolatile elements, when requiring the ratio to H and He, also solar spectra have to be analyzed. Early 1D spherically symmetric approaches of the solar photosphere have been replaced by multidimensional 3D analyses (Anders and Grevesse 1989; Asplund et al. 2009). This led to a change for XCN O , i.e., the mass fraction of all CNO nuclei in a solar composition, from 1.4 × 10−2 to 9.33 × 10−3 (which is the mass fraction of all CNO nuclei in a solar composition with YC = 2.18 × 10−4 , YN = 5.73 × 10−5 , YO = 3.7 × 10−4 and XCN O = 12YC + 14YN + 16YO , see e.g. Amarsi et al. 2021). For solar CNO abundances (Y14 = 9.33 × 10−3 /14) and Y1 = 0.5, the CNOcycles dominate over the PP-cycles beyond T = 1.75 × 107 K. Under solar conditions the PP-chains dominate by more than one order of magnitude. As in the PP-cycles, the final product of H-burning is 4 He. Essentially all initial CNO abundances are transferred to 14 N, with none being burned to heavier nuclei. It is important to note that the CNO nuclei only act as a catalysts. A cycle similar to the CNO-cycle, requiring slightly higher temperatures, is the NeNaMg-cycle, shown in Table 3. For updates beyond the compilations mentioned above (Angulo et al. 1999; Adelberger et al. 2011), see the impressive recent activities in underground labs which attempt to measure the related cross sections down to the relevant stellar

Table 3 The NeNaMg-cycles in hydrogen burning Cycle NeNaMgI NeNaMgII

Reaction sequence 20 Ne(p,γ )21 Na(e+ ν)21 Ne(p,γ )22 Na (p,γ )23 Mg(e+ ν)23 Na(p,α)20 Ne 21 Na(p,γ )22 Mg(e+ ν)22 Na

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energies: i.e., ongoing investigations in LUNA (https://luna.lngs.infn.it/index.php/ new-about-us), CASPAR (https://caspar.nd.edu/), and JUNA (Liu et al. 2022), which avoid background noise and permit cross section measurements down to energies in the 50 keV region. This also applies for reactions of interest in the upcoming sections of He- and C-burning.

Helium Burning Main Reactions There is no stable A = 5 nucleus. Proton or neutron captures on 4 He cannot produce heavier nuclei in stellar environments. This is the reason why hydrogen burning does not proceed beyond 4 He. The ashes of hydrogen burning consist almost entirely of 4 He, except for minor amounts of 2 H, 3 He, 7 Li, and preexisting heavy elements. 3 He(4 He, γ )7 Be produces 7 Be and, after decay, 7 Li. In hydrogen burning Li is destroyed again via 7 Li(p, α)4 He; in helium burning the small abundance of 3 He left over from hydrogen burning is exhausted very fast. The only reaction possible among 4 He-nuclei is 4

He +4 He 8 Be,

which produces the unstable 8 Be nucleus, decaying on a timescale of 2.6×10−16 s via α-emission. However, increasing temperatures and densities leads to a large production of 8 Be, which decays immediately, but a tiny abundance remains in (chemical) equilibrium (where forward and backward reaction cancel), leading to (with 4 and 8 standing for 4 He and 8 Be 1 Y˙8 = ρNA 4, 4 Y42 − λ8 Y8 = 0 2

λ8 =

1 τ8

Γ8 τ8 = h¯ λ8 =

Γ8 . h¯

(17)

The life time of 8 Be is related to the width of the ground state via the Heisenberg uncertainty principle. The equilibrium abundance of 8 Be is consequently Y8 =

h¯ ρNA 4, 4 Y42 . 2Γ8

Another alpha-capture can follow, leading to the production of 12 C: Y˙12 = ρNA 4, 8 Y4 Y8 =

h¯ 2 2 ρ NA 4, 4 4, 8 Y43 2Γ8

≡

1 2 2 ρ NA 4, 4, 4 Y43 . 3!

(18)

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F.-K. Thielemann and T. Rauscher

The last line is the definition of the triple-alpha-rate, treating it as if it were a real three-body reaction. In the case of a collision of three identical particles, we would have a double counting of possible reactions by taking the full helium abundance. The overcount is given by the permutations of the three particles, which would all end up in the same final 12 C nucleus, whatever the reaction sequence is. The number of permutations among three particles is 3! and thus is included in the above definition. This leads to the triple-alpha-reaction rate: 4

He(2α, γ )12 C Q = 6.445 MeV

NA2 4, 4, 4 = NA2

3h¯ 4, 4 4, 8 . Γ8

(19)

A new feature is encountered in the triple-alpha reaction sequence which does not occur in regular two-body reactions. The second alpha-capture proceeds on a target in an unstable state with a natural width. If 8 Be was not produced at the resonance energy but in the low-energy tail, the energetics of the second alphacapture (with target 8 Be) would be changed with respect to resonances in 12 C. An appropriate approach (especially of importance at relatively low temperatures and high densities) which is also important in other applications, like explosive Hburning, has been outlined in a number of publications (see, e.g., Nomoto et al. 1985; Görres et al. 1995). A possible alpha-capture reaction on the product 12 C depends on T , with 12

C(α, γ )16 O Q = 7.161 MeV.

16 O(α, γ )20 Ne is blocked by a very small cross section, due to missing resonances at the appropriate energies in 20 Ne. Thus, the dominant ash of hydrogen burning (4 He) is burned in helium burning to 12 C and 16 O. For typical burning conditions of T = (1 − 2) × 108 K and ρ = 3 × 102 − 104 g cm−3 , the result is a mixture of 12 C and 16 O, with 16 O dominant at higher temperatures. This situation is, however, complicated by additional aspects of stellar evolution and overshadowed by a large experimental uncertainty of the 12 C(α, γ )16 O cross section (see, e.g., deBoer et al. 2017, and the last paragraph of the H-burning section). The effectiveness of the 12 C(α, γ )16 O reaction depends on the temperature of He-burning, i.e., the stellar mass and the cross section of the reaction itself. Since the temperature increases during He-burning, it is also important to note that the mixing of fresh He into the burning core, at the end of the core burning phase, can mimic a high reaction rate. Thus, the astrophysical treatment of convection is also important in this case. All of these effects contribute to the “effective” transformation of 12 C into 16 O. The C/O-ratio at the end of Heburning determines the further evolution of the star. Small carbon concentrations, or completely depleted carbon, can result in the case that either of the two burning stages of C-burning and Ne-burning is not occurring (for a detailed discussion see, e.g., Ojima et al. 2018, or the  Chap. 99, “Thermal Evolution of Neutron Stars” by K. Nomoto).

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Neutron Production in Helium Burning Essentially all material in CNO-nuclei, present in the initial stellar composition, is processed to 14 N during hydrogen burning. This leads to a side branch of helium burning: 14

N(α, γ ) 18 F(e+ ν) 18 O(α, γ ) 22 Ne.

This is the one reaction in helium burning, where protons are changed to neutrons, leading to a decrease of Ye . For a solar initial composition with 1.4% by mass in CNO nuclei (which turned into 14 N in hydrogen burning), an abundance of Y (22 Ne) = Y (14 N) = 0.014/14 = 1.0 × 10−3 is obtained. This nonsymmetric nucleus is characterized by N − Z = 2 and results in Ye = 0.499. Some of the 22 Ne undergoes another alpha-capture to produce 26 Mg. A smaller fraction (dependent on temperature) follows an endoenergetic reaction 22 Ne(α, n)25 Mg (Q = −0.482 MeV), i.e., only alpha energies (center of mass) beyond 0.482 MeV can induce this reaction. Such energies can be attained for T > 3.5 × 108 K. This is larger than the temperatures in helium burning discussed above. On the other hand, the energy threshold does not have to be at the center of the Gamow window of Eq. (5). The high-energy tail also gives a contribution. It is clear, however, that this neutron producing reaction is only active in late phases of core helium burning, which experience high temperatures. When helium burning occurs in burning shells and shell flashes (to be discussed later), additional features can occur. High temperatures can be attained, and in some cases hydrogen is mixed into the helium burning zones, as a result of thermal pulses (or helium shell flashes). This permits proton captures on 12 C, which produce 13 N, decaying to 13 C. The latter is also a very efficient neutron source as due to a strong (α,n)-reaction. Both neutron sources (22 Ne and 13 C) lead to neutron production and the buildup of heavy elements via neutron capture and β − -decay of unstable nuclei. If neutron capture timescales are longer than beta-decay timescales, this is called a slow neutron capture process (s-process), which will be discussed later in more detail. The important reactions are summarized in Table 4. As neutron capture cross sections increase with mass number, neutron captures occur preferentially on the most abundant heavy nucleus, i.e., 56 Fe. Some intermediate mass nuclei, which are produced in important quantities due to the 22 Ne neutron source and neutron capture chains based on it, are 21 Ne, 22 Ne, 25 Mg, 26 Mg, 36 S, 37 Cl, 40 K, and 40 Ar. The status of uncertainties in He-burning reactions, especially with respect to (α, n)- and (n, γ )-reactions, is discussed in Käppeler et al. (2011) and Febbraro et al. (2020), but see also the last paragraph of the H-burning section.

Carbon, Neon, and Oxygen Burning Carbon Burning We want to point out here that stars less massive than about 8M end their core evolution after helium burning. They become (electron-)degenerate and do not contract further due to the electron degeneracy pressure. Cores of more massive stars do contract, heat up, and induce the next burning stage. Among the ashes

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F.-K. Thielemann and T. Rauscher

of helium burning, the nucleus with the smallest charge is 12 C. At densities of ρ ≈ 105 g cm−3 and temperatures of T ≈ (6 − 8) × 108 K, the Maxwell-Boltzmann velocity distribution reaches energies sufficient to penetrate the Coulomb barrier in the 12 C+12 C reaction, which has two open particle channels:  12

C

12

C,

α p

 20

Ne

23 Na

4.62 MeV 2.24 MeV.

The products can interact with the released particles (protons and alphas): 23

Na(p, α)20 Ne

23

Na(p, γ )24 Mg

12

C(α, γ )16 O.

Besides these reactions, which dominate the energy generation, we also list in Table 5 the reactions whose fluxes are down by up to a factor of 100. The Q-value of the proton capture reaction on 12 C is 1.93 MeV. Eq. (6), when utilizing detailed balance, results in the fact that the photodisintegration reactions are faster than the inverse capture reactions if kT > 1/24Q. This means that in high temperature carbon burning, close to 8 × 108 K, the photodisintegration of 13 N will win over the β + -decay, and the reaction sequence (c) in Table 5 will not be of any importance. However, in low-temperature carbon burning, the photodisintegration Table 4 Major reactions in helium burning (a) Basic energy generation 4 He(2α, γ )12 C 12 C(α, γ )16 O[(α, γ )20 Ne] (b) Neutron sources 14 N(α, γ )18 F(e+ ν)18 O(α, γ )22 Ne 22 Ne(α,n)25 Mg 12 C(p,γ )13 N(e+ ν)13 C(α,n)16 O (c) High-temperature burning with neutron sources 22 Ne(n,γ )23 Ne(e− ν¯ )23 Na(n,γ )24 Na (e− ν¯ )24 Mg 20 Ne(n,γ )21 Ne(α,n)24 Mg Further s-processing via neutron captures and β-decays 24 Mg(n,γ )25 Mg etc. Production of heavy elements 56 Fe(n,γ )57 Fe(n,γ )58 Fe, etc. Table 5 Major reactions in carbon burning (a) Basic energy generation 12 C(12 C,α)20 Ne 12 C(12 C,p)23 Na 23 Na(p,α)20 Ne 23 Na(p,γ )24 Mg 12 C(α, γ )16 O (b) Fluxes > 102 ×(a) 20 Ne(α, γ )24 Mg 23 Na(α,p)26 Mg(p,γ )27 Al 20 Ne(n,γ )21 Ne(p,γ )22 Na (e+ ν)22 Ne(α,n)25 Mg(n,γ )26 Mg 21 Ne(α,n)24 Mg 22 Ne(p,γ )23 Na 25 Mg(p,γ )26 Al(e+ ν)26 Mg (c) Low-temperature, high-density burning 12 C(p,γ )13 N(e+ ν)13 C(α,n)16 O(α, γ )20 Ne 24 Mg(p,γ )25 Al(e+ ν)25 Mg 21 Ne(n,γ )22 Ne(n,γ )23 Ne(e− ν¯ )23 Na(n,γ )24 Na(e− ν)24 Mg + s-processing

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is slow, and we have the 13 C neutron source similar to hot helium burning, with the protons being supplied from the carbon fusion reaction. Only in this latter case an increase in the total neutron/proton ratio occurs, which can decrease Ye down to 0.4975. For burning conditions with higher temperatures, Ye from helium burning remains unchanged. Besides the uncertainties in the measurements of reactions overlapping with Table 4, one has to discuss the behavior of the dominant fusion reaction 12 C+12 C (see, e.g., for early and recent investigations Rolfs and Rodney 1988; Caughlan and Fowler 1988; Tang and Ru 2022; Heine et al. 2022; Monpribat et al. 2022).

Neon Burning The most abundant nuclei after carbon burning are 16 O, 20 Ne, and 24 Mg. The reaction 16 O(α, γ )20 Ne has a Q-value of only 4.73 MeV. The typical reaction Q-value for reactions among stable nuclei is about 8–12 MeV. This means, that in the temperature range of 1–2 MeV (following the relation kT > 1/24Q), alpha particles will be liberated by the photodisintegration of 20 Ne and can be used to rearrange nuclei: 20

20

Ne(γ , α)16 O

Ne(α, γ )24 Mg(α, γ )28 Si.

The typical conditions for neon burning in stars are T ≈ (1.2 − 1.4) × 109 K and ρ ≈ 106 g cm−3 . Stars in the mass range 8–10 M appear to not reach this stage to burn Ne in a stable fashion, but their cores become degenerate, and they contract to high densities at low temperatures. In that case the electron Fermi energies start to become important, and electron captures start to occur. The electrons turn degenerate with a Fermi energy for the completely degenerate nonrelativistic case and ne = (ρNA Ye )2/3 : EF =

h¯ 2 (3π 2 )2/3 ne2/3 2me

EF (ρYe = 107 g cm−3 ) = 0.75 MeV

(20)

EF (ρYe = 109 g cm−3 ) = 4.7 MeV. That means that electron captures, which are energetically prohibited e− + (Z, A) → (Z − 1, A) + ν with a negative Q-value, can become possible and lead to an enhanced “neutronization” of the astrophysical plasma, in addition to the role of beta decays and electron captures with positive Q-values. In degenerate Ne-O-Mg cores, electron captures on 20 Ne and 24 Mg cause the loss of degeneracy pressure support and introduce a collapse rather than only a contraction, which shortens all further burning stages on a collapse timescale (for detailed discussions see Kirsebom et al. 2019; Leung et al. 2020).

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More massive stars pass through the main neon burning reactions, as discussed above. In addition, many other reactions can occur with fluxes much smaller than for those reactions that dominate the energy generation. We present a list in Table 6. However, there is not any important change in Ye for these massive stars. For an overview discussing experimental and theoretical evaluation of alpha-induced reactions from Ne and Mg to Ca, see Rauscher et al. (2000).

Oxygen Burning The oxygen, produced in helium burning [4 He(2α, γ )12 C(α, γ )16 O], in carbon burning [12 C(α, γ )16 O with α  s from 12 C(12 C, α)20 Ne], and in neon burning [20 Ne(γ , α)16 O], is still unburned and the nucleus with the smallest charge. At ρ ≈ 107 g cm−3 and T ≈ (1.5 − 2.2) × 109 K, the Coulomb barrier for oxygen fusion can be overcome, leading to the major reactions: 16

O(16 O, p)31 P

7.676 MeV

O(16 O, α)28 Si

9.593 MeV

O(16 O, n)31 S(β + )31 P

1.459 MeV

16 16

31

(21)

P(p, α)28 Si etc. . . .

In high-density oxygen burning (ρ > 2 × 107 g cm−3 ), two electron capture reactions become important and lead to a decrease in Ye : 33

S(e− , ν)33 P

35

Cl(e− , ν)35 P.

Ye can decrease to 0.495–0.4825. We list in Table 7 the major and minor reaction sequences. Many individual reactions of intermediate mass nuclei in oxygen and silicon burning have been analyzed in the past. Such experiments usually find quite good agreement with statistical model (Hauser-Feshbach) calculations, which will be

Table 6 Major reactions in neon burning

(a) Basic energy generation 20 Ne(γ , α)16 O 20 Ne(α, γ )24 Mg(α, γ )28 Si (b) Fluxes > 102 ×(a) 23 Na(p,α)20 Ne 23 Na(α,p)26 Mg(α,n)29 Si 20 Ne(n,γ )21 Ne(α,n)24 Mg(n,γ )25 Mg(α,n)28 Si 28 Si(n,γ )29 Si(n,γ )30 Si 24 Mg(α,p)27 Al(α,p)30 Si 26 Mg(p,γ )27 Al(n,γ )28 Al(e− ν¯ )28 Si (c) Low-temperature, high-density burning 22 Ne(α,n)25 Mg(n,γ )26 Mg(n,γ )27 Mg(e− ν¯ )27 Al 22 Ne left from prior neutron-rich carbon burning

94 Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction) Table 7 Major reactions in oxygen burning

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(a) Basic energy generation 16 O(16 O,α)28 Si 16 O(12 O,p)31 P 16 O(16 O,n)31 S(e+ ν)31 P 31 P(p,α)28 Si(α, γ )32 S 28 Si(γ , α)24 Mg(α,p)27 Al(α,p)30 Si 32 S(n,γ )33 S(n,α)30 Si(α, γ )34 S 28 Si(n,γ )29 Si(α,n)32 S(α,p)35 Cl 29 Si(p,γ )30 P(e+ ν)30 Si electron captures 33 S(e− , ν)33 P(p,n)33 S 35 Cl(e− , ν)35 S(p,n)35 Cl (b) High-temperature burning 32 S(α, γ )36 Ar(α,p)39 K 36 Ar(n,γ )37 Ar(e+ ν)37 Cl 35 Cl(γ ,p)34 S(α, γ )38 Ar(p,γ )39 K(p,γ )40 Ca 35 Cl(e− , ν)35 S(γ ,p)34 S 38 Ar(α, γ )42 Ca(α, γ )46 Ti 42 Ca(α,p)45 Sc(p,γ )46 Ti (c) Low-temperature, high-density burning 31 P(e− ν)31 S 31 P(n,γ )32 P 32 S(e− , ν)32 P(p,n)32 S 33 P(p,α)30 Si

discussed in the next section. Here we still want to point to the tremendous experimental efforts undertaken for determining solid cross section data.

Silicon Burning When temperatures reach (3 − 4) × 109 K, every nucleus is connected to each other by reaction links which are open in both directions: (i) capture reactions due to high enough temperatures to overcome Coulomb barriers and (ii) photodisintegrations due to high enough temperatures and high-energy photons in the Planck distribution. This situation, initiated by photodisintegration reactions on dominant 28 Si, which produce protons, alpha particles and neutrons, finally leads to an equilibrium where the abundances depend only on the nuclear mass (binding energy), density, and temperature. Then the composition can be described by nuclear statistical equilibrium (NSE) and expressed by Eq. (22), which results from utilizing the chain of abundance ratios as given in Eq. (12) (an alternative derivation results from utilizing the chemical potentials of neutrons, protons, and nucleus (Z, N ) from a Maxwell-Boltzmann distribution and solving for the equation Zμp + Nμn = μZ,N ): YZ,N =

A3/2 GZ,N (ρNA )A−1 A 2



2π h¯ 2 mu kT

3/2(A−1) exp(BZ,N /kT )YnN YpZ .

(22)

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Electron captures, which occur on longer timescales than particle captures and photodisintegrations, are not in equilibrium and have to be followed explicitly. Thus, the NSE has to be solved for a given ρ(t), T (t), and Ye (t). This leads to two equations and a composition, resulting from Eq. (22) and a given Ye , determined by 

Ai Yi = Yn + Yp +

i

 i

Zi Yi = Yp +





(Z + N)YZ,N (ρ, T , Yn , Yp ) = 1

Z,N

Z YZ,N (ρ, T , Yn , Yp ) = Ye .

(23)

Z,N

In general, very high densities favor large nuclei, due to the high power of ρ A−1 , while very high temperatures favor light nuclei, due to (kT )−3/2(A−1) . In the intermediate regime, occurring in hydrostatic stellar evolution, exp(EB /kT ) favors tightly bound nuclei with the highest binding energies in the mass range A = 50−60 (of the Fe-group) but depending upon the given Ye . The width of the composition distribution is determined by the temperature. Ye is determined by β + -decays and electron captures in Si-burning. Close to the end of core silicon burning, we have Ye ≈ 0.46. However, this final stage of Si-burning, resulting in a complete chemical equilibrium or NSE, is initially hampered by slow reactions, especially due to small Q-values of reactions into closed neutron or protons shells. This leads initially to quasi-equilibrium groups (QSE), e.g., around 28 Si and a group centering on Fe or Ni nuclei, which are both within their groups in a complete equilibrium with abundance ratios as in Eq. (12), but the connections across the closed shells N = Z = 20 are hindered. Then the relative ratios of neighboring nuclei in each of the groups are in equilibrium, but the sum of abundances in each of the two groups is not identical to an NSE distribution. It is only approached by starting out with the QSE around Si dominating and, as a function of time, both – the Si and the Fe QSE groups – attain their final NSE values. This behavior has been discussed in Hix and Thielemann (1996). Here we present a few illustrations clarifying this background. While temperatures of T9 = 5 are in stellar evolution only reached at the end of Si-burning, we give here an example of a plasma consisting initially only of Si isotopes with a composition representing Ye = 0.498. We show the evolution at two points in time around 3 × 10−6 and 5.6 × 10−5 s. Light particles are created by photodisintegration reactions on Si, and heavier nuclei are produced by captures of these light particles. This means that initially these light particles and the heavier Fe-group are underabundant in comparison to a full NSE. This can be seen in the two panels of Fig. 1. Thielemann and Arnett (1985) made an early attempt for such Ye -conditions to identify the important (slow) reactions which connect the two groups (see Fig. 2). In a more extended study, Hix and Thielemann (1996) analyzed the nuclei connecting the two QSE-groups. We can see the nuclei positioned between the QSE groups for different conditions with respect to Ye , ρ, and T as well as simulation time t in Figs. 3 and 4. We notice that the deviation from equilibrium decreases as a

94 Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction)

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Fig. 1 Shown is the logarithmic ratio of equilibrium abundance predictions in comparison to results from full network calculations at two points in time when the mass fraction in the Si-group is still 99 and later 90%. Initially only the Si QSE-group is in full equilibrium (= 0), while the light nuclei created in photodisintegration reactions and the heavier Fe-group nuclei created by capture reactions are still underabundant in comparison to equilibrium. However, we see constant lines for

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function of time in the two panels of Fig. 3, being already very close to equilibrium in the second panel. The nuclei out of equilibrium between the two groups vary as a function of time and are relatively neutron-rich for Ye chosen to be 0.46. Fig. 4 shows that the time to reach equilibrium is quite dependent on the environment temperature and density. Especially this last subsection, Si-burning and also the previously discussed Oburning contain an overwhelming number of reactions of intermediate mass and heavy nuclei which need to be known and cannot all be provided by experiments. Thus, a reliable method for theoretical predictions is needed. The following section (after also going through explosive burning stages in the next subsection) will focus on the statistical model or Hauser-Feshbach approach that can be applied if a sufficiently large density of excited state levels in the compound nucleus exists.

Nuclear Burning in Explosive Environments Many of the hydrostatic burning processes discussed in earlier subsections can occur also under explosive conditions at much higher temperatures and on shorter timescales. The major reactions remain still the same in many cases, but often the beta-decay half-lives of unstable products are longer than the timescales of the explosive processes under investigation. This requires in general the additional knowledge of nuclear cross sections for unstable nuclei. Extensive calculations of explosive carbon, neon, oxygen, and silicon burning, appropriate for supernova explosions, have already been performed in the late 1960s and early 1970s with the accuracies possible in those days, before detailed stellar modeling became available (for present-day results, see, e.g., Curtis et al. 2019, and the chapters by Seitenzahl & Pakmor,  Chap. 107, “Nucleosynthesis and Tracer Methods in Type Ia Supernovae” and S. Wanajo,  Chap. 108, “Nucleosynthesis in Neutrino-Heated Ejecta and Neutrino-Driven Winds of Core-Collapse Supernovae: Neutrino-Induced Nucleosynthesis” and M. Obergaulinger,  Chap. 109, “Nucleosynthesis in Jet-Driven and Jet-Associated Supernovae” and A. Bauswein & H.-T. Janka,  Chap. 112, “Dynamics and Equation of State Dependencies of Relevance for Nucleosynthesis in Supernovae and Neutron Star Mergers”). Besides minor additions of 22 Ne after He-burning (or nuclei which originate from it in later burning stages), the fuels for explosive nucleosynthesis consist mainly of alpha particle nuclei like 12 C, 16 O, 20 Ne, 24 Mg, or 28 Si. Because the timescale of explosive processing is very short (a fraction of a second to several seconds), only few beta-decays can occur during explosive nucleosynthesis events (unless  Fig. 1 (continued) other QSE groups, i.e., the light group of n, p, and 4,6 He and the Fe-group. The Fe-group approaches equilibrium fast already when X(Si) = 0.9 while it will take a longer timescale by another order of magnitude until also the lighter nuclei have reached equilibrium. (Image reproduced with permission from Hix and Thielemann (1996), copyright by AAS)

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Fig. 2 The major reactions which connect the Si QSE-group and the Fe QSE-group during hydrostatic stellar Si-burning for Ye = 0.498 from an early analysis (Thielemann and Arnett 1985). Dependent on the stellar mass, this takes place for higher central temperatures (high mass stars) or slightly lower temperatures (massive stars in the lower mass interval >10 M which still experience central Si-burning). Members of the Si-group are indicated by a box, members of the Fe-group by a circle. (Image reproduced with permission from Thielemann and Arnett (1985), copyright by AAS)

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Fig. 3 The deviation from equilibrium decreases as a function of time, being already very close to equilibrium for t ≈ 5×10−5 s. But it can be realized as well that there exist still a (small) deviation between the two groups and that the nuclei out of equilibrium between the two groups vary as a function of time, increasing with Z (and being relatively neutron-rich for Ye = 0.46). In both cases nuclei close to the borders of the shell closures N = Z = 20 are the ones which still hinder to reach an equilibrium fast due to their slow reaction rates for reactions with small Q-values. (Image reproduced with permission from Hix and Thielemann (1996), copyright by AAS)

Fig. 4 Two results for Ye = 0.498 but at quite different temperatures and points in time. Similar deviations from equilibrium prevail for t ≈ 5×10−6 s and in contrast for 111 s. In both cases finally an NSE will be reached, but either for conditions close to explosive Si-burning or otherwise for conditions which are similar to the initial ignition conditions of hydrostatic Si-burning in massive stars. (Image reproduced with permission from Hix and Thielemann (1996), copyright by AAS)

highly unstable species are produced), resulting in heavier nuclei, again with N ≈ Z. However, a spread of nuclei around a line of N = Z is involved, and many reaction rates for unstable nuclei have to be known. Dependent on the temperature, explosive burning produces intermediate to heavy nuclei. Two processes encounter nuclei far from stability where either a large supply of neutrons or protons is available. In those cases, the r-process and the rp-process (i.e., explosive hydrogen burning), nuclei close to the neutron and proton drip lines can be produced and beta-decay timescales can be short in comparison to the process timescales. We will not discuss these nucleosynthesis processes in detail

94 Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction)

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here, but refer to the chapters by M. Wiescher,  Chap. 95, “Experimental Nuclear Astrophysics” and M. Pfützner & C. Mazzocchi,  Chap. 33, “Nuclei Near and at the Proton Dripline” for reactions involving proton-rich nuclei and the chapters by S. Goriely,  Chap. 110, “R-Process Nucleosynthesis in Neutron Star Merger Ejecta and Nuclear Dependences” and S. Wanajo,  Chap. 108, “Nucleosynthesis in Neutrino-Heated Ejecta and Neutrino-Driven Winds of Core-Collapse Supernovae: Neutrino-Induced Nucleosynthesis” for reactions involving neutron-rich nuclei in r-process environments. Independent of these individual aspects, for all the intermediate and heavy nuclei, we will present (after a short discussion of these explosive nucleosynthesis environment) theoretical methods to determine nuclear reaction cross sections and reaction rates. While most of the explosive burning processes are to some extent (with the exceptions mentioned above and below) “cousins” of their hydrostatic versions, involving just much shorter timescales and a more extended set of nuclei, this is different for Si-burning which we will discuss in a separate subsection. A possible connection between explosive Si-burning and the beginning of an r-process is pointed out as well.

Burning Timescales for Explosive He, C, Ne, O, and Si-Burning In stellar evolution, burning timescales are dictated by the energy loss timescales of stellar environments. Processes like hydrogen and helium burning, where the stellar energy loss is dominated by the photon luminosity, choose temperatures with energy generation rates equal to the radiation losses. For the later burning stages, neutrino losses play the dominant role among cooling processes and the burning timescales are determined by temperatures where neutrino losses are equal to the energy generation rate. These criteria led to the narrow bands in temperatures and densities discussed in previous subsections on hydrostatic burning stages in stellar evolution. Explosive events are determined by hydrodynamic equations which provide different temperatures or timescales for the burning of available fuel. We can generalize the question by defining a burning timescale, dependent on whether the key reaction is a fusion reaction or a photodisintegration, responsible for the destruction of the major fuel nuclei i: τi = |

Yi 1 |= ρNA < σ v > (T )Yf uel Y˙i or =

(24)

1 λγ (T )

These timescales for the fuels i ∈ H, He, C, and O are determined by the major fusion destruction reaction. They are in all cases temperature dependent and for a fusion process they are also density dependent. Ne- and Si-burning,

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Fig. 5 Burning timescales for fuel destruction of He-, C-, Ne-, O-, and Si-burning as a function of temperature, as defined in Eq. (24) and generalized to photodisintegrations and three-body reactions, accordingly. A 100 fuel mass fraction was assumed. When rewriting Eq. (24) for 4 He, the destruction of three identical particles has to be considered. The density-dependent burning timescales are labeled with the chosen typical density. They scale linearly for C- and O-burning and quadratically for He-burning. Notice that the almost constant He-burning timescale beyond T9 = 1 has the effect that efficient destruction on explosive timescales can only be attained for high densities. (Image reproduced with permission from Thielemann et al. (2011), copyright by Springer Nature)

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which are dominated by (γ , α) destruction of 20 Ne and 28 Si, have timescales only determined by the burning temperatures. The temperature dependencies are typically exponential, due to the functional form of the corresponding NA < σ v >. The density dependencies are linear for fusion reactions, as seen from Eq. (10), with the exception of He-burning. We have plotted these burning timescales as a function of temperature (see Fig. 5), assuming a fuel mass fraction of 1. The curves for (also) density-dependent burning processes are labeled with a typical density. Heburning has a quadratic density dependence; C- and O-burning depend linearly on density. If we take typical explosive burning timescales to be of the order of seconds (e.g., in supernovae), we see that one requires temperatures to burn essential parts of the fuel in excess of 4×109 K (Si-burning), 3.3×109 K (O-burning), 2.1×109 K (Ne-burning), and 1.9×109 K (C-burning). Beyond 109 K He-burning is determined by an almost constant burning timescale. We see that essential destruction on a timescale of 1 s is only possible for densities ρ > 105 g cm−3 . This is usually not encountered in He-shells of massive stars.

Special Features of Explosive Si-Burning Zones which experience temperatures in excess of 4.0–5.0×109 K undergo explosive Si-burning. For T >5×109 K, essentially all Coulomb barriers can be overcome, and a nuclear statistical equilibrium is established. Such temperatures lead to complete Si-exhaustion and produce Fe-group nuclei. The doubly magic nucleus 56 Ni, with the largest binding energy per nucleon for N = Z, is formed with a dominant abundance in the Fe-group in case Ye is larger than 0.49. Explosive Siburning can be divided into three different regimes: (i) incomplete Si-burning and complete Si-burning with either a (ii) normal or an (iii) alpha-rich freeze-out. Which of the three regimes is encountered depends on the peak temperatures and densities attained during the passage of supernova shock front (see Fig. 20 in Woosley et al. 1973, and Fig. 6 for applications to supernova calculations). One recognizes that SNe Ia and SN II (core-collapse supernovae) experience different regions of complete Si-burning. The most abundant nucleus in the normal and alpha-rich freeze-out is 56 Ni, in case the neutron excess is small, i.e., Ye is larger than 0.49. It is evident that the electron fraction Ye has a major imprint on the final composition. Nuclei with a Z/A-value corresponding to the global Ye show the largest abundances, combined with a maximum in binding energy. The even-Z-even-N nuclei show the largest abundances due to their large binding energies. In an alpha-rich freeze-out, final alpha-captures (or rather the shift of the upper QSE-group to heavier nuclei, based on the large 4 He abundance) play a dominant role for the less abundant nuclei, transforming, e.g., 56 Ni, 57 Ni, and 58 Ni into 60 Zn, 61 Zn, and 62 Zn and leaving trace abundances of 32 S, 36 Ar, 40 Ca, 44 Ti, 48 Cr, 52 Fe, 54 Fe, and 55 Co. In that case the major NSE nuclei 56 Ni, 57 Ni, and 58 Ni get depleted when the remaining alpha fraction increases, while all other species mentioned above increase. At high temperatures in complete Si-burning, before freeze-out or in a normal freeze-out,

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Fig. 6 Division of the ρp − Tp -plane into incomplete and complete Si-burning with normal and alpha-rich freeze-out. Contour lines of constant 4 He mass fractions in complete burning are given for levels of 1% and 10%. They coincide with lines of constant radiation entropy per gram of matter. For comparison also the maximum ρ − T -conditions for individual mass zones of type Ia and type II (core-collapse) supernovae are indicated. (Image reproduced with permission from Thielemann et al. (2011), copyright by Springer Nature)

the abundances are in a full NSE. An alpha-rich freeze-out occurs generally at low densities when the triple-alpha reaction, transforming 4 He into 12 C, is not fast enough to keep the He-abundance in equilibrium during the fast expansion and cooling in explosive events (see Fig. 6). This also leads to a slow supply of carbon nuclei still during freeze-out, leaving traces of alpha nuclei, which did not fully make their way up to 56 Ni. Incomplete Si-burning is characterized by peak temperatures of 4 − 5 × 109 K. Temperatures are not high enough for an efficient bridging of the bottle neck above the proton magic number Z = 20 by nuclear reactions. Besides the dominant fuel nuclei 28 Si and 32 S, we find the alpha-nuclei 36 Ar and 40 Ca being most abundant. Partial leakage through the bottle neck above Z = 20 produces 56 Ni and 54 Fe as dominant abundances in the Fe-group. Smaller amounts of 52 Fe, 58 Ni, 55 Co, and 57 Ni are encountered. In a superposition of (a) incomplete Si-burning, (b) complete Si-burning with alpha-rich freeze-out, and (c) explosive O-burning, supernova explosions can provide good fits to solar abundances. For recent features of the results of explosive burning in core-collapse supernovae, see, e.g., Curtis et al. (2019) and Ghosh et al. (2022), and in type Ia supernovae, see, e.g., Lach et al. (2020) and Leung and Nomoto (2021).

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The r-process The operation of an r-process is characterized by the fact that 10 to 100 neutrons per seed nucleus in the Fe-peak (or beyond in an alpa-rich freeze-out) have to be available to form all heavier r-process nuclei by neutron capture. This translates into a Ye = 0.15–0.3. Such a high neutron excess can only by obtained through capture of energetic electrons (on protons or nuclei) which have to overcome large negative Q-values. This can be achieved by degenerate electrons with large Fermi energies and requires a compression to densities of 1011 − 1012 g cm−3 , with a beta equilibrium between electron captures and β − -decays as found in supernova core collapse and the forming neutron stars. Another option is an extremely alpha-rich freeze-out in complete Si-burning with moderate and Ye ’s (even >0.42). After the freeze-out of charged particle reactions in matter which expands from high temperatures but relatively low densities, 70, 80, 90, or 95% of all matter can be locked into 4 He with N = Z. Figure 6 showed the onset of such an extremely alpha-rich freeze-out by indicating contour lines for He mass fractions of 1% and 10%. These contour lines correspond to T93 /ρ = const, which is proportional to the entropy per gram of matter of a radiation dominated gas. Thus, the radiation entropy per gram of baryons can be used as a measure of the remaining He mass fraction. In such high entropy conditions, all excess neutrons are left with the remaining mass fraction of Fe-group (or heavier) nuclei, and neutron captures can proceed to form the heaviest r-process nuclei. A recent general review of the functioning of an r-process, the nuclear physics input, astrophysical sites and related observations can be found in Cowan et al. (2021).

Explosive H-Burning We will not discuss here in detail explosive hydrogen burning which takes place in novae and X-ray bursts, as their nucleosynthesis contribution to our known abundance pattern is close to negligible. Novae, due to high temperatures attained in H-burning, burn H in the accreted material not via the well-known CNO-cycle but in the so-called hot CNO-cycle. It is characterized by the fact that in the branching between the slow beta-decay of 13 N and a further proton capture to 14 O, the proton capture wins because of a highly enhanced reaction rate for such conditions. In a similar way, hot CNO-type cycles for elements beyond Ne rearrange nuclei up to Mg and Si. When including ejecta of such nova explosions, a few not highly abundant isotopes up to Mg and Si can be considered to have a non-negligible contribution from novae (Jose 2016). In type I X-ray bursts, due to the explosive ignition of H- and subsequent Heburning reactions, matter from the hot CNO-cycle can even be transferred to heavier nuclei up to 100 Sn via the rp-process. This is an important stellar explosion, observed in X-rays, but the explosion energy is likely not sufficient to eject matter out of the high gravitational binding of the neutron star. A detailed discussion of novae and X-ray bursts can be found in Jose (2016), the evolution of knowledge on the nuclear

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reactions evolved over many years from Wiescher et al. (1986) over Schatz et al. (1998) to Cyburt et al. (2010), Schatz and Ong (2017), and Meisel et al. (2020). In addition to the abovementioned explosive burning processes, there exist other processes producing some isotopes beyond iron that are not accessible by neutron capture processes. They include the p-process or γ -process (occurring in as a byproduct of explosive Ne-burning which was discussed before) and the νp-process (Fig. 7), responsible for proton-rich stable nuclei up to A = 80–90. The p-process consists mainly of photo dissociation of existing heavy isotopes. This moves matter from nuclei previously produced by the s- and r-process to the proton-rich side of stability. The initial suggestion was that conditions in a hydrodynamic shock wave trigger this process when running through layers of an exploding star that contain already heavy nuclei from previous generations. An additional option to produce light p-nuclei is the so-called νp-process in proton-rich ejecta in supernova explosions (Fröhlich et al. 2006; Ghosh et al. 2022). For an overview of the processes and the relevant nuclear physics, see Rauscher et al. (2013).

Fig. 7 The nuclear chart with stable isotopes marked with black boxes, the gray region indicates nuclei that have been produced already in the laboratory, and the light blue region shows the exotic isotopes that rare isotope facilities will discover. The various explosive nucleosynthesis processes discussed in previous subsections are schematically indicated by color lines. Explosive C-, Ne-, O-, and Si-burning (unless happening under very neutron-rich conditions) involves stable and unstable nuclei close to stability. The s-process, taking place in hydrostatic He-burning, is also shown here, but it can to a large extent (with exceptions) be investigated with experimentally determined neutron capture cross sections of stable nuclei. (Courtesy of M. Jacobi and A. Arcones)

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What Type of Reactions Are Needed in Explosive Burning? After going through the various hydrostatic and explosive burning processes which need input for nuclear reaction cross sections, we attempt to give a short overview on the regions in the nuclear chart where such reaction cross sections and reaction rates are needed. A summary of all these processes and their actions in specific regions of the nuclear chart is given in Fig. 7. These include proton capture reaction in the rp-process and νp-process, photodisintegration reactions in the p- or γ -process, and neutron capture reactions in the r-process. The other explosive burning processes mentioned in the previous subsections work closer to stability, but also include to some extent unstable nuclei, for which charged particle and neutron capture reactions need to be predicted.

Thermonuclear Rates and the Hauser-Feshbach Formalism In the section on hydrostatic burning stages, we indicated that at least all the early burning stages rely on experimentally determined nuclear reaction cross sections. In late burning stages and explosive burning, a number of unstable nuclei are produced and a much larger amount of stable nuclei. It is thus advisable to look into specific experimental techniques or alternatively into reliable theoretical approaches which can provide the necessary information. Explosive burning in supernovae involves in general intermediate mass and heavy nuclei. Due to a large nucleon number, they have intrinsically a high density of excited states. A high level density in the compound nucleus at the appropriate excitation energy allows to make use of the statistical model approach for compound nuclear reactions (HauserFeshbach), which averages over resonances. Therefore, it is often colloquially termed that the statistical model is only applicable for intermediate and heavy nuclei, which gives repeatedly rise to misunderstandings. The only necessary condition for application of the statistical model formalism is a large number of resonances at the appropriate bombarding energies, so that the cross section can be described by average transmission coefficients. Whether we have a light or heavy nucleus is only of secondary importance. A high level density automatically implies that the nucleus can equilibrate in the classical compound nucleus picture. As the capture of an alpha particle leads usually to larger Q-values than neutron or proton captures, the compound nucleus is created at a higher excitation energy, and especially in the case of alpha-captures, it is often even possible to apply the Hauser-Feshbach formalism for nuclei as light as Li. Another advantage of alpha-capture is that the capture Q-values vary very little with the N/Z-ratio of a nucleus. This means that the requirements are equally well fulfilled near stability as at the proton or neutron drip lines, which makes the application very safe for all intermediate and heavy nuclei (and for light nuclei, the level density test has to be done individually). Opposite to the behavior for alpha-induced reactions, the reaction Q-values for proton or neutron captures vary strongly with the N/Z-ratio,

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leading eventually to vanishing Q-values at the proton or neutron drip line. For small Q-values, the compound nucleus is created at low excitation energies, and also for intermediate nuclei, the level density can be quite small. Therefore, it is not advisable to apply the statistical model approach close to the proton drip line for intermediate nuclei. For neutron captures close to the neutron drip line in rprocess applications, it might be still permissible for heavy and often deformed nuclei, which have a high level density already at very low excitation energies. On the other hand, this method is not applicable for neutron-rich light nuclei. Where predictions for nuclear reaction rates are needed to perform investigations in the related astrophysical processes has been outlined in Fig. 7. The statistical model approach has been employed in the calculation of thermonuclear reaction rates for astrophysical purposes by many researchers, starting with Truran et al. (1966), who only made use of ground state properties. Arnould (1972) pointed out the importance of excited states. Extended compilations have been provided (Holmes et al. 1976; Woosley et al. 1978; Thielemann et al. 1986; Rauscher and Thielemann 2000; Goriely et al. 2008, for updates see (https:// nucastro.org, https://reaclib.jinaweb.org and https://www-nds.iaea.org, including TALYS results)). The codes for the latter three entries are known under the names SMOKER, NON-SMOKER, and TALYS. More recently, the code SMARAGD (Rauscher 2010, 2011) has been developed as a successor to NON-SMOKER, with improvements in the treatment of charged particle and photon transmissions, in the nuclear level density, and with inclusion of recent experimental information on low-lying excited states. Reaction rate compilations obtained from calculations using NON-SMOKER and TALYS are presently the ones utilized in large-scale applications in all subfields of nuclear astrophysics, when experimental information is unavailable. SMARAGD is being used to analyze experimental data and for improvements of smaller sets of reaction rates (Cyburt et al. 2010) but a large-scale set of reaction rates has not been published yet. A high level density in the compound nucleus permits to use averaged transmission coefficients T , which do not reflect a resonance behavior, but rather describe absorption via an imaginary part in the (optical) nucleon-nucleus potential (for details see, e.g., Mahaux and Weidenmuller 1979). This leads to the well-known expression μν

σi (j, o; Eij ) =

π h¯ 2 /(2μij Eij ) μ (2Ji + 1)(2Jj + 1)

×

μ μ μ μ ν , J ν , πν )  Tj (E, J, π , Ei , Ji , πi )Toν (E, J, π , Em m m (2J + 1) Ttot (E, J, π ) J,π

(25) for the reaction i μ (j, o)mν from the target state i μ to the exited state mν of the final nucleus, with center of mass energy Eij and reduced mass μij . J denotes the spin,

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E the excitation energy, and π the parity of excited states. When these properties are used without subscripts, they describe the compound nucleus, subscripts refer to the participating nuclei in the reaction i μ (j,o)mν , and superscripts indicate the specific excited states. Experiments measure ν σi0ν (j, o; Eij ), summed over all excited states of the final nucleus, with the target in the ground state. Target states μ in an astrophysical plasma are thermally populated, and the astrophysical cross section σi∗ (j, o) is given by  σi∗ (j, o; Eij )

=

μ μ (2Ji

 μν μ + 1) exp(−Ei /kT ) ν σi (j, o; Eij ) .  μ μ μ (2Ji + 1) exp(−Ei /kT )

(26)

The summation over ν replaces Toν (E, J, π ) in Eq. (25) by the total transmission coefficient to all excited states in the final nucleus m To (E, J, π ) =

νm 

ν ν Tom (E, J, π , Em , Jmν , πmν )

ν=0 E−S  m,o

+ νm Em



Tom (E, J, π , Em , Jm , πm )ρ(Em , Jm , πm )dEm .

Jm ,πm

(27) Here Sm,o is the channel separation energy, and the summation over excited states above the highest experimentally known state νm is changed to an integration over the level density ρ. The summation over target states μ in Eq. (26) has to be generalized accordingly (Fig. 8). The important ingredients of statistical model calculations are the particle and γ -transmission coefficients T and the level density of excited states ρ. Therefore, the reliability of the of such calculations is determined by the accuracy with which these components can be evaluated. In the following we want to discuss the methods utilized to estimate these quantities and recent improvements.

Particle Transmission Coefficients The transition from an excited state in the compound nucleus (E, J, π ) to the state μ μ μ (Ei , Ji , πi ) in nucleus i via the emission of a particle j is given by a summation over all quantum mechanically allowed partial waves: μ μ μ μ Tj (E, J, π , Ei , Ji , πi )

=

J +s 

μ

Ji +Jj



l=|J −s| s=|J μ −Jj | i

μ

Tjls (Eij ).

(28)

F.-K. Thielemann and T. Rauscher

thermally populated states

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o j

i

m C

Fig. 8 Energetics for the reaction i(j, o)m and its inverse m(o, j )i for transitions through a resonance state (E, J π ) in the compound nucleus C

μ Here the angular momentum l and the channel spin s = Jj + Ji couple to J = l + s. The individual transmission coefficients Tl are calculated by solving the Schrödinger equation with an optical potential for the particle-nucleus interaction. While all early studies of thermonuclear reaction rates (Truran et al. 1966; Arnould 1972; Holmes et al. 1976; Woosley et al. 1978) employed optical square well potentials and made use of the black nucleus approximation, the other compilations utilize optical potentials from microscopic calculations. The resulting s-wave neutron strength function (the average ratio of the width of states populated by s-wave neutrons and the appropriate level spacing D for a 1 eV capture energy) can be expressed by S0 =< Γ o /D > |1eV = (1/2π )Tn(l=0) (1eV). A specific optical potential choice (Jeukenne et al. 1977) with corrections for the imaginary part and updated parameters for low energies (Lejeune 1980) is shown in Fig. 9 (utilized in the NON-SMOKER code). In a similar way, there exist a variety of optical potentials for protons (including Jeukenne et al. 1977, with the appropriate energetics; see also many more options in the TALYS package). Alpha particles have been treated in many cases with a phenomenological Woods-Saxon potential based on extensive data by McFadden and Satchler (1966). In general, for alpha particles and heavier projectiles, the best results can probably be obtained with folding potentials (see, e.g., Satchler and Love 1979; Mohr et al. 2020).

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10-3

S0

10-4

10-5

10-6 40

60

80

100

120

140 160 A (Target)

180

200

220

240

Fig. 9 Neutron strength function when utilizing a sophisticated optical potential (Jeukenne et al. 1977). An equivalent square well potential would result in a straight line. When utilizing a spherical potential a deviation from experiment, around A = 160 would show up, with an So -line similar to a sinus curve. This is avoided here by treating deformed nuclei in a very simplified way (without utilizing more sophisticated coupled channel calculations) by using an effective spherical potential of equal volume, based on averaging the deformed potential over all possible angles between the incoming particle and the orientation of the deformed nucleus. (Image reproduced with permission from Cowan et al. (1991), copyright by Elsevier)

γ -Transmission Coefficients At low energies photon captures on nuclei lead to relatively simple excited states, often involving only few particles in the nucleus. Beyond about 10 MeV, so-called giant resonances are populated which correspond to a collective motion involving many if not all particles of a nucleus. They occur as electric or magnetic multipole resonances and can be excited by capture of a photon or can decay by emitting a photon. The dipole E1 and M1 resonances dominate. In a macroscopic way, they can be understood by protons oscillating against neutrons (a) in the GoldhaberTeller mode where fixed proton and neutron density distributions move against each other or (b) in the Steinwedel-Jensen mode with density oscillations of neutrons against protons taking place within a fixed nuclear surface. In such “macroscopic” descriptions, the resonance energy and its width can be explained with the ingredients of macroscopic-microscopic nuclear mass models (e.g., Thielemann et al. 1986; Thielemann and Arnould 1983; Rauscher and Thielemann 2000). Extended collections of experimental data, showing essentially a Lorentzian distribution as a function of energy, have been provided in the past (e.g., Berman and Fultz 1975; Junghans et al. 2008); see for recent comprehensive reviews (Ishkhanov and

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Kapitonov 2021; Liang and Litvinova 2022), the latter reference corresponding to the  Chap. 57, “Model for Collective Vibration” by Liang and Litvinova in this Handbook, containing a detailed derivation within a microscopic (rather than phenomenological “macroscopic”) description of these giant resonances. Both types of options to choose are available in the TALYS code. In the more phenomenological approach, the E1 transitions are calculated on the basis of the Lorentzian representation of the giant dipole resonance (GDR). Within this model, the E1 transmission coefficient for the transition emitting a photon of energy Eγ in a nucleus A N Z is given by TE1 (Eγ ) =

2 2 E4 ΓG,i 8 NZ e2 1 + χ  i γ . 2 2 2 E2 2 2 3 A hc 3 mc (Eγ − EG,i ) + ΓG,i ¯ γ i=1

(29)

Here χ (= 0.2) accounts for the neutron-proton exchange contribution and the summation over i includes two terms which correspond to the split of the GDR in statically deformed nuclei, with oscillations along (i = 1) and perpendicular (i = 2) to the axis of rotational symmetry. Specific experimental results (Junghans et al. 2008) are shown in Fig. 10. As discussed above, the GDR resonance energy EG can be predicted within macroscopic (hydrodynamic) approaches based on the ingredients of macroscopicmicroscopic mass models (Thielemann and Arnould 1983; Rauscher and Thiele-

Fig. 10 Experimental strength functions of the giant dipole resonance fE1 (Eγ ) = (TE1 (Eγ )/2π )(Eγ )−3 for spherical and deformed nuclei from Junghans et al. (1998). For deformed nuclei the Lorentzian form splits into two sub-resonances (see text). The lines represent two choices for the description of the GDR width chosen by Junghans et al. (2008). This includes, however, also low-lying E1 strength, which will be discussed below in more detail. (Image reproduced with permission from Junghans et al. (2008), copyright by Elsevier)

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mann 2000), which gives excellent fits to the GDR energies and can also predict the split of the resonance for deformed nuclei, when making use of the deformation. In that case, the two resonance energies are related to the mean value by the following expression (Danos 1958): EG,1 + 2EG,2 = 3EG EG,2 /EG,1 = 0.911η + 0.089.

(30)

η is the ratio of (1) the diameter along the nuclear symmetry axis and (2) the diameter perpendicular to it. It can be obtained from the experimentally known deformation or mass model predictions (see Fig. 11). The width of the GDR ΓG is less understood but can also predicted in a phenomenological approach via the ingredients of macroscopic-microscopic nuclear mass models which satisfactorily reproduces the experimental data for spherical and deformed nuclei. The GDR width can be described as a superposition of a macroscopic width due to the viscosity of the nuclear fluid and a coupling to quadrupole surface vibrations of the nucleus (Thielemann and Arnould 1983; Rauscher and Thielemann 2000; Goriely 1998). The width is reduced near magic nuclei, because these closed shell nuclei act within the hydrodynamic picture in a stiff way and couple less to the quadrupole surface vibrations. Alternatively microscopic approaches have been utilized within the quasi-particle random phase approximation (QRPA) (see, e.g., besides a macroscopic approach similar to the above description, different options in the TALYS code based on Skyrme HatreeFock BCS, Skyrme Hartree-Fock-Bogoluibov, the temperature-dependent relatistic mean field or the Gogny Hartree-Fock-Bogoluibov model) or also the relativistic quasiparticle time blocking approximation (RQTBA) framework (Litvinova et al. 2009). For a general review, see Liang and Litvinova (2022).

Fig. 11 E1 giant dipole resonance energy for oscillations parallel (E01 ) and perpendicular (E02 ) to the rotational symmetry of deformed nuclei. For spherical nuclei both energies coincide. The energies are predicted with parameters from microscopic-macroscopic mass models (as utilized in the NON-SMOKER code), deformations taken from experiments or mass models as well. (Image reproduced with permission from Cowan et al. (1991), copyright by Elsevier)

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The total radiation width at the neutron separation energy for nuclei in experimental compilations can be calculated via the total photon transmission coefficients for E1 radiation from a compound nucleus state with energy E, spin J , and parity π is given by Tγ (E, J, π ) =

νC 

Tγν (E, J, π , ECν , JCν , πCν )

ν=0

+

E 

ν EmC

Tγ (E, J, π , EC , JC , πC )ρ(EC , JC , πC )dEC ,

(31)

JC ,πC

where the first term represents a summation over the known low-lying states ν up to an excitation energy Eω = Ecνc , while an integration involving the level density ρ is performed at higher excitation. Tγν (E, J, π , ECν , JCν , πCν ) is either zero if the E1 selection rules are violated or equal to TE1 (Eγ = E − ECν ) otherwise. With TE1 from Eq. (29), the average radiation width < Γγ (E, J, π ) > (not to be confused with the GDR width) is related to the total photon transmission coefficient by < Γγ (E, J, π ) > = Tγ (E, J, π )/(2πρ(E, J, π )). This way the nuclear level density (which will be discussed in the next subsection) enters in the average radiation width. Extensive sets of measured average radiation widths come from thermal s-wave (l = 0) neutron capture, and the relevant theoretical quantity to be compared with such experimental data is < Γγ >0 =

Ji + 1 Ji < Γγ (Sn , Ji +1/2, πi )> + < Γγ (Sn , Ji −1/2, πi ) >, 2Ji + 1 2Ji + 1 (32)

where Sn is the compound nucleus neutron separation energy and Ji (πi ) is the spin (parity) of the target nucleus. Utilizing the methods outlined above within a macroscopic-microscopic approach for the Lorentzian form of the E1 giant dipole resonance leads to a global agreement generally within a factor of 1.5 for nuclei experimentally accessible. This involves a correction to an energy-dependent width due to the fact that, for low-energy gamma-transitions which are part of the Giant resonance, the Lorentz curve is suppressed and, in addition, the GDR width increases with excitation energy. This led to ΓG (Eγ ) = ΓG ×(Eγ /EG )δ with δ = 0.5 (McCullagh et al. 1981), for a slightly different term (close to δ = 1), have a look at Goriely and Plujko (2019). An additional entity of the low-lying E1 response below the GDR energy was experimentally noticed (see, e.g., Goriely 1998, termed as Pigmy dipole resonance PDR) and led other authors (Junghans et al. 2008) to an artificial broadening of the GDR with a δ = 1.5 (see the broader curve in Fig. 10). Microscopically it can be well described by Skyrme+QRPA or RMF+cQRPA approaches; macroscopically it is interpreted as an oscillation of the neutron skin against the N = Z core. Further investigations into this topic have been undertaken theoretically and experimentally

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(e.g, Litvinova et al. 2009; Guttormsen et al. 2022). An approximate description for the E1 PDR within a phenomenological macroscopic approach, based on the neutron skin thickness, has been provided (van Isacker et al. 1992) and utilized (Goriely 1998). Recent developments include an advanced understanding of the PDR (with E1 and M1 contributions) and its total effect on capture cross sections (Markova et al. 2021; Guttormsen et al. 2022) as shown in Fig. 12. Various treatments are available for the M1 giant resonance in a similar functional form as discussed above in Eq. (29) but with typically smaller resonance energies and widths and a smaller normalization factor (see the M1 component in Fig. 12). This involves two M1 modes, the spin-flip and the scissor mode (the latter only present in deformed nuclei), for which empirical expressions have been derived in Goriely and Plujko (2019) in forms identical to Eq. (31), but with different scaling factors and typical resonance energies around 7–10 MeV, respective 2.5–4 MeV and corresponding widths of about 4 respective 1.5 MeV (for shell model based predictions, see, e.g., Loens et al. 2008; Sieja 2018). An additional effect relates to the fact that for small energies the Brink hypothesis (the transition strength as a function of energy is independent of the excitation energy of the state) is apparently broken at low energies, leading to a different behavior for absorption in comparison to de-excitation. This results, based on shell model calculations, in a constant limit toward lowest energies for the E1 strength and an upbend in case of the M1 strength (Goriely et al. 2018; Goriely and Plujko 2019). The effects can nicely be seen in Fig. 12, with the upbend at lowest energies, the two M1 modes, and an extended E1 PDR. While the low-lying E1 strength contributes only on the less than 10% level close to stability, for very neutron-rich nuclei with an extended neutron skin, it can lead to large enhancements, and the low-lying M1 strength and especially the upbend at lowest energies can have a quite sizable effect close to the neutron drip line.

Fig. 12 Total gamma ray strength function as a function of transition energy (see also Fig. 10) for two Sn isotopes. See especially the contribution of the low-lying strength in the E1 PDR, the two M1 spin flip and scissor mode contributions, and the upbend at lowest energies. (courtesy of M. Markova)

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Another effect has to be taken into account for alpha-capture reactions on selfconjugate (N = Z) nuclei, because due to isospin selection rules, γ -transitions between isospin I = 0 levels are forbidden. This leads to a suppression of corresponding alpha-capture cross sections which can be treated via a suppression of the appropriate γ -transitions (Rauscher et al. 2000).

Fission Along the valley of stability, actinide and transuranic nuclei become increasingly unstable as a result of fission in their ground states. This loss of stability arises because the disruptive Coulomb force, which increases as Z 2 , overcomes the cohesive surface tension, proportional to A2/3 . The nuclear mass, i.e., potential energy for a given (Z, A), as a function of deformation, has a local minimum at the ground state deformation. It first increases with increasing deformation until reaching a maximum at the saddle point, from which point onward fission is inevitable. This potential barrier (=fission barrier) can have very small tunneling transmission coefficients and therefore very long fission half-lives. While the simple liquid-drop approach to nuclear masses leads only to a single barrier plotted as a function of deformation, the barrier is in most cases split into a double-humped (or even multiple) barrier prescription, due to shell effects (Strutinsky 1967, 1968). The higher of the peaks is denoted the fission barrier, since states beyond that excitation energy can actually fission instantaneously. More generally, the mass (potential energy) can be evaluated not only along a deformation path but rather as a function of several deformation parameters. One possibility is the choice of utilizing the parameters elongation, neck diameter, left-fragment spheroidal deformation, right-fragment spheroidal deformation, and nascent-fragment mass asymmetry. Fig. 13 (top panel) shows such a potential energy contour plot for 232 Th (Möller et al. 2015b). Because the potential energy (=mass) as a function of deformation must be calculated for a particular choice of mass model, many theoretical efforts have been undertaken (see, e.g., Schunck and Regnier 2022, and references therein), and the results will be dependent on these mass models. Fig. 14 gives the results of Möller et al. (2015b). Investigations utilized in astrophysical applications developed from 1980 until present (see, e.g., Howard ´ and Möller 1980; Myers and Swia¸ tecki 1999; Mamdouh et al. 2001; Goriely et al. 2009; Vassh et al. 2019; Giuliani et al. 2020). For most of the actinides, the fission barrier development as a function of deformation can be described within the so-called double-humped fission barrier (Bjørnholm and Lynn 1980), and the fission probability can be calculated within the complete-damping formalism, making use of two Hill-Wheeler inverted parabolas barriers (Lynn and Back 1974), as utilized in a number of applications (Thielemann et al. 1983; Panov et al. 2005, 2010) (see Fig. 13, bottom panel); otherwise a more complex quantum mechanical approach for the penetration through the barrier potential has to be performed (Goriely et al. 2009).

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Energy Definitions

10

Energy (MeV)

Eldsad sad Epot

5

Bld Bf

Es+p Emic

0

Ezp

5 0

2 4 6 8 1/2 Nuclear Deformation [(Q2/b) ]

o

10 Z,N

β−

E, Ji,πj E

Tf

Tn

TB

TA

Tγ

z+1, N−2

EA

EB

Sn −Q

z+1, N−1 Fig. 13 top: The energy (or respective mass of the nucleus) as a function of the quadrupole moment Q2 , which measures the elongation of the nucleus due to deformation, based on macroscopic-microscopic fission potential energy calculations for 232 Th (Möller et al. 2015a). The dotted line corresponds to the macroscopic “liquid-drop” energy along a specified path; the solid line is the total macroscopic-microscopic energy along a sequence of different shapes. Q2 = 0 stands for a spherical shape. From the ground state toward larger deformations the total energy curve is given along the optimal fission path that includes all minima and saddle points, identified along this path in a five-dimensional deformation space (elongation, neck diameter, leftfragment spheroidal deformation, right-fragment spheroidal deformation, and nascent-fragment mass asymmetry). Bottom: similar plot of the potential energy as a function of deformation with a double-humped fission barrier, as the typical feature for actinide nuclei (Thielemann et al. 1983). The energetics is shown here for an application for beta-delayed fission. ([left] Image provided from the database of P. Möller, see https://t2.lanl.gov/nis/molleretal/, [right] image reproduced with permission from Thielemann et al. (1983), copyright by Springer Nature)

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130

Calculated Fission-Barrier Height

Proton Number Z

Bf(Z,N) (MeV) 1 2 3 4 5 6 7 8

120 110 100 90 80 130 140 150 160 170 180 190 200 210 220 230 Neutron Number N

Fig. 14 Fission barrier heights Bf based on macroscopic-microscopic fission potential energy calculations (Möller et al. 2015a). (Image provided from the database of P. Möller, see https://t2. lanl.gov/nis/molleretal/)

As discussed previously, excited states close to or above the fission barrier will have much shorter fission half-lives than ground states, and therefore processes like neutron capture or beta-decay, which populate excited states of the compound nucleus or daughter nucleus, will lead to increased probabilities for fission relative to the ground state. Therefore, neutron-induced and beta-delayed fission can play an important role in astrophysics, including the the r-process. The statistical model calculations discussed in this section can be extended to include a fission channel, in order to calculate cross sections for neutron-induced fission. The cross section for a reaction i 0 (j, o)m, from the target ground state i 0 to all excited states mν of a final nucleus with center of mass energy Eij and reduced mass μij , is again given by Eq. (26), with the summation taken over all final states ν. Therefore, the sum of Toν is again replaced by the expression for To . The ratio going into the fission channel is then To /Ttot with o = f , i.e., the outgoing channel is the fission channel. When making use of a double-humped fission barrier, or a more complex transmission calculations, the result and can be expressed in terms of the fission probability Pf (E, J, π ) = Tf (E, J, π )/Ttot (E, J, π ). With a value obtained for Pf and the calculated transmission coefficients To (o = f ) for all other channels, one can solve Pf = Tf /(Tf + o =f To ) for Tf to be used, e.g., for neutron-induced cross section predictions (or also beta-delayed fission). For applications to astrophysical questions, see, e.g., Martinez-Pinedo et al. 2007; Petermann et al. 2012; Erler et al. 2012; Eichler et al. 2015, 2019; Giuliani et al. 2018, 2020; Kullmann et al. 2022; Holmbeck et al. 2023.

Level Densities The present status of the theoretical understanding of nuclear level densities (one of the most important statistical nuclear properties), from empirical models to

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microscopic methods, has been given in an excellent review (Alhassid 2021). Many statistical model calculations use the level density description of the back-shifted Fermi gas (Gilbert and Cameron 1965): ρ(U, J, π ) = 1/2f (U, J )ρ(U ) √ √ 1 π exp(2 aU ) ρ(U ) = √ U 5/4 2π σ 12a 1/4 f (U, J ) = σ = 2

Θrigid h2 ¯

U a

(33)

2J + 1 exp(−J (J + 1)/2σ 2 ) 2σ 2 Θrigid =

2 mu AR 2 5

U = E − δ,

which assumes that positive and negative parities are evenly distributed and that the spin dependence f (U, J ) is determined by the spin cut-off parameter σ . The level density of a nucleus is therefore dependent only on two parameters: the level density parameter a and the backshift δ, which determines the energy of the first excited state. This backshift is related to the level spacing between the last occupied and the first unoccupied state in the single particle shell model and to the pairing gap, the energy necessary to break up a proton or neutron pair. Of these two effects, the pairing energy dominates except in the very near vicinity of closed shells. Within this framework, the quality of level density predictions depends on the reliability of systematic estimates of the level density parameter a and the backshift δ. Eq. (33) provides a valid functional form for reproducing level densities down to an excitation energy of U = 1–2 MeV, but it diverges for U = 0, i.e., E = δ. This causes problems if δ is a positive backshift. Ericson plots, which show the number of excited states as a function of excitation energy, reveal an almost linear behavior for ln(N ) = f (E) with an intercept at E = E0 = δ (Ericson 1959). This resulted in the ansatz ρ(U ) = exp(U/T )/T , commonly called the constant temperature formula. The two formulations can be combined for low and high excitation energies, with E0 = δ and T being determined by a tangential fit to the Fermi gas formula. This description is usually called the composite (Gilbert-Cameron) formula. The first compilation of a and δ was provided for a large number of nuclei by Gilbert and Cameron (1965). They found that the backshift δ is well reproduced by experimental pairing corrections. Theoretical predictions result in a/A ≈ 1/15 for infinite nuclear matter, but the inclusion of surface and curvature effects of finite nuclei enhances this value to 1/6–1/8. While in principle a can be dependent on temperature or excitation energy, such a dependence is minimal for excitation energies up to 3×A (MeV), and thus the zero temperature value for a can be used up to these energies, covering all values of interest within the present context. When comparing experimentally derived values of a with such a simple formulation, one notices large deviations due to shell effects (nuclei near closed shells have much smaller values for a). An empirical correlation with experimental shell corrections S(N, Z) was realized (Gilbert and Cameron 1965):

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a/A = c0 + c1 S(Z, N ),

(34)

where S(N, Z) is negative near closed shells. Many other functional dependencies have been proposed and more microscopic treatments have been carried out (see, e.g., Goriely et al. 2012). If one constructs level densities from single particle spectra, it means that large care has to be invested in predicting correct single particle spectra. There have been a number of compilations for a and δ or T and E0 , based on purely experimental level densities, but predictions for unstable nuclei or those for which experimental information is not available have to be based on theory, including the predictions for shell and pairing corrections. For a review of early investigations, see Cowan et al. (1991), utilizing the pairing corrections of a droplet nuclear mass model. Such an average behavior changes, however, within one unit of magic nucleon numbers, where the backshifts are often much larger. Deformed nuclei should, in principle, be treated differently, as a rotational band can be associated with each excited state. At high excitation energies, this treatment can, however, lead to double counting. Another option is to keep the same formalism as in Eq. (34) but performing an independent evaluation of the coefficients c0 and c1 for deformed nuclei, because only one parameter set for c0 and c1 cannot achieve a significantly better agreement with experimental level densities than found in Gilbert and Cameron (1965) with a deviations up to a factor of 10. By dividing the nuclei into three classes ((a) those within three units of magic nucleon numbers, (b) other spherical nuclei, (c) deformed nuclei)), an improved agreement was obtained (maximum deviations of less than a factor of 3 where experimental information at the neutron separation energy is available) (Cowan et al. 1991). However, this treatment was still very phenomenological and led Rauscher et al. (1997) to make use of an improved approach, considering that the energy dependence of the shell effects vanish at high excitation energies. Although, for astrophysical purposes only energies close to the particle separation thresholds have to be considered, an energy dependence can lead to a considerable improvement of the global fit. This is especially true for strongly bound nuclei close to magic numbers. An excitation energy-dependent description has been proposed for the level density parameter a (Ignatyuk et al. 1980): a(U, Z, N) = a(A)[1 ˜ + C(Z, N ) a˜ = αA + βA2/3

f (U ) ] U (35)

f (U ) = 1 − exp(−γ U ). The values of the free parameters a, b, and γ can be determined by fitting to experimental level density data. The shape of the function f (U ) permits the two extremes: (i) for small excitation energies, the original form of Eq. (34) is retained with S(Z, N ) being replaced by C(Z, N ), and (ii) for high excitation energies, a/A

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approaches the continuum value obtained for infinite nuclear matter (for details see Rauscher et al. 1997). The approach to the backshift δ could be improved further by utilizing the masses of neighboring nuclei (from experiment of mass models if those are not available) in order to obtain a more realistic pairing correction. When obtaining the pairing correction directly from the masses with 1 [Δn + Δp ] 2 1 Δn = [−M(A − 1, Z) + 2M(A, Z) − M(A + 1, Z)] 2 1 Δp = [−M(A − 1, Z − 1) + 2M(A, Z) − M(A + 1, Z + 1)] 2 δ=

(36)

better fits to the total level density could be achieved with experimental level densities at the neutron separation energy within a factor of 2 to 3, as utilized in the NON-SMOKER code. Parity-dependent level density descriptions, not using the factor 1/2 in Eq. (33), have also been utilized for tests in application to cross section predictions (Mocelj et al. 2007; Loens et al. 2008) and are the default option in the code SMARAGD.

Width Fluctuation Corrections In addition to the ingredients required for Eq. (25), like the transmission coefficients for particles and photons and the level densities, width fluctuation corrections W (j, o, J, π ) have to be employed. They define the correlation factors with which all partial channels of incoming particle j and outgoing particle o, passing through excited state (E, J, π ), have to be multiplied. This is due to the fact that the decay of the state is not fully statistical, but some memory of the way of formation is retained and influences the available decay choices. The major effect is elastic scattering: the incoming particle can be immediately re-emitted before the nucleus equilibrates. Once the particle is absorbed and not re-emitted in the very first (precompound) step, the equilibration is very likely. This corresponds to enhancing the elastic channel by a factor Wj . The NON-SMOKER code uses the description of Tepel et al. (1974), leading to Eqs. (37) and (38). In order to conserve the total cross section, the individual transmission coefficients in the outgoing channels have to be renormalized to Tj . The total cross section is proportional to Tj , and one obtains the  ) + T  (T  − T  )/T  . This can be (almost) solved condition Tj = Tj (Wj Tj /Ttot tot j tot j for Tj Tj =

Tj  . 1 + Tj (Wj − 1)/Ttot

(37)

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This approach requires an iterative solution for T  (starting in the first iteration with Tj and Ttot ), which converges fast. The enhancement factor Wj have to be known in order to apply Eq. (37); a general expression in closed form is very complicated and computationally expensive to employ. A fit to results from Monte Carlo calculations (Tepel et al. 1974) leads to Wj = 1 +

2 1/2

1 + Tj

.

(38)

For a general discussion of approximation methods, Gadioli and Hodgson (see, e.g., 1992). There exist further expressions for this renormalization (Hofmann et al. 1980; Moldauer 1980) which are utilized as options in the TALYS code. The NONSMOKER code makes use of Tepel et al. (1974) with Eqs. (37) and (38), which redefine the transmission coefficients in Eq. (26) in such a manner that the total width is redistributed by enhancing the elastic channel and weak channels over the dominant one. While this is only an approximation to the correct treatment, it could be shown that this treatment is quite adequate (Thomas et al. 1986).

Cross Section Applications Our discussion in the previous subsections reveals that one can expect to obtain good agreement with experiment for particle and photon transmission coefficients in comparison to experiments. On the other hand, level density predictions which are based on shell correction terms of of microscopic-macroscopic mass models show a statistical spread (a factor of 2 at the neutron separation energy). But all this applies to nuclei where experimental information is available and can clearly be larger for nuclei far from stability. Neutron capture cross sections at 30 keV (utilized for many astrophysical applications) of the NON-SMOKER code show agreement by less than a factor of 2, with a few exceptions for nuclei located specifically at magic numbers. This is a consequence of the still imperfect predictions of nuclear level densities. This includes the question whether the statistical Hauser-Feshbach formalism can be applied to all nuclei, also far from stability. The statistical model assumes that one can utilize a transmission behavior averaged over resonances. This requires that in the Gamow window for charged particle reactions or in the energy window of a thermal neutron distribution, a sufficient number of resonances are available in the compound nucleus. This energy window depends on the temperature in the plasma of reacting nuclei, because in astrophysical applications, the reaction rate σ v depends on an integration of the cross section over a thermal distribution of reactants as shown in Eq. (5). Requiring about ten levels in this thermal window leads to the following applicability criterion for neutron capture cross sections shown in Fig. 15. A major question is whether one would need to utilize direct capture predictions for the marginal regions in the nuclear chart, which include

94 Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction)

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Fig. 15 Required minimum temperatures for the applicability of the Hauser-Feshbach statistical model for neutron capture reactions, based on the request that about ten resonances (levels) exist in the thermal energy window when utilizing a specific level density description (Rauscher et al. 1997). While in most explosive nucleosynthesis application such conditions are fulfilled, r-process applications in the range of 1–3×109 K might be marginal close to the neutron drip line just above neutron shell closures. These predictions were done by utilizing the microscopic-macroscopic FRDM model (Möller et al. 1995); updates of this mass model and microscopic level density treatments might improve this situation. A similar plot for alpha-capture reactions (Rauscher et al. 1997) leads to the conclusion that no temperature restrictions apply, because alpha-capture Q-values do not experience such a dependence of the distance from stability. For proton-capture reactions restrictions also apply close to the proton-drip line up to Z ≈ 30, requiring temperatures in excess of 2.5×109 K (Rauscher 2011, 2020). (Image reproduced with permission of Rauscher et al. (1997), copyright by APS)

large uncertainties (Mathews et al. 1983; Rauscher et al. 1998; Goriely 1998), or a modification of the statistical model (Rauscher 2011). Figure 16 shows a comparison of Maxwellian-averaged neutron capture cross sections for two Sn isotopes as a function of environment temperature. One can see that (still close to stability) the predictions of NON-SMOKER and the various options in the TALYS code provide a quite reasonable prediction within a factor of 2. The enhanced gamma-strength function due to low-lying strength of the PDR, M1 contributions, and an upbend at lowest energies (see the subsection on γ -transmission coefficients), which can be included in the TALYS code, could, however, increase the difference between both codes for neutron-rich nuclei far from stability.

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Fig. 16 Maxwellian-averaged neutron capture cross section prediction for the Sn isotopes 120 and 124 as a function of the astrophysical environment temperature. Shown are the results based on experiments and the predictions of the NON-SMOKER as well as TALYS codes (for the latter indicates the spread, which results from the variety of options available). (Courtesy of M. Markova)

Fig. 17 Cross section prediction for the reactions 51 V(p, γ ) and 51 V(p, n), utilizing width fluctuation corrections (Tepel et al. 1974). Without this correction the capture cross section would be enhanced by a factor of 4 beyond the neutron channel opening at 1.6 MeV. (Image reproduced with permission from Cowan et al. (1991), copyright by Elsevier)

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Fig. 18 Neutron-induced fission cross sections predictions for a variety of actinide nuclei (Panov et al. 2010, 2005) in comparison to experiments from JENDL-3.3 (crosses Nakagawa et al. 2005), averaged by the code JANIS (black lines Soppera et al. 2011). It is clearly seen that the quality of theoretical fission barrier predictions (ETFSI, TF, HFB-14, Mamdouh et al. 2001; Myers ´ and Swia¸ tecki 1999; Goriely et al. 2009) affects the results strongly. (Image reproduced with permission from Panov et al. (2010), copyright by ESO)

At this point we also show some energy-dependent cross section evaluations for charged particle reactions involving intermediate mass nuclei which play an important role in silicon burning. Figure 17 indicates the importance of width fluctuation corrections when at a specific energy another reaction channel opens. Finally we want to present also some results of fission cross section predictions (Panov et al. 2010), which show that the procedure to predict these cross sections works quite well but is highly sensitive to the quality of fission barrier predictions, when applied for unstable nuclei without experimental determinations of the fission barriers. Figure 18 shows the prediction of neutron-induced fission cross sections as a function of energy, (a) in comparison to experiments and (b) when applying different theoretical fission barrier predictions; Fig. 19 shows the astrophysical reaction rates as a function of environment temperature, also indicating when neutron energies surpass the fission barrier energies.

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Fig. 19 Similar to Fig. 18, but for astrophysical neutron-induced fission reaction rates as a function of environment temperature (Panov et al. 2010), utilizing either only the target ground state or thermally populated targets, which have an effect on the opening of the fission channel. TALYS (2009) stands for Goriely et al. (2009), who include a thermal target population. (Image reproduced with permission from Panov et al. (2010), copyright by ESO)

Summary In the present Handbook chapter, we examined the nuclear reactions in evolving stars, starting with hydrostatic burning phases of stellar evolution and continuing through explosive burning in various astrophysical environments, taking place in supernovae or explosive binary events. Important reactions were listed in Table 1 through Table 7, going over to features being accompanied with the transition from dominating individual reactions to nuclear statistical equilibrium NSE in Fig. 1 through Fig. 4. Finally regions of the nuclear chart were identified, where nuclear reaction input is required for the different explosive nucleosynthesis processes. While the early phases of stellar burning, involving mostly light to intermediate mass nuclei, need to be tackled mostly by experimental approaches (see the  Chap. 95, “Experimental Nuclear Astrophysics” by M. Wiescher, J. deBoer and R. Reifarth), for intermediate and heavy nuclei – with a high density of levels at the bombarding energy – statistical model (Hauser-Feshbach) approaches can be applied. This topic has been followed in detail, including the treatment of particle channels, gamma-transitions, fission, and level density predictions, also for nuclei far from stability. While our focus has been on charged particle and neutron-induced reactions, we only mentioned the influence and importance of weak interactions in stellar environments, e.g., in late burning stages and in stellar explosions, but want to refer here to the chapters by G. Martinez-Pinedo and Robin, and Suzuki on electron capture reactions, and Rrapaj and Reddy, Fuller and Grohs, Wang and Surman, Famiano et al., Fröhlich, Wanajo, and Obergaulinger on the role of neutrino reactions in evolving stars and their explosive endpoints. Finally it should be mentioned, that the recent white paper on nuclear astrophysics (Schatz et al. 2022) is an enormous resource for presently ongoing

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experimental efforts from cross section measurements at astrophysical energies to investigations of nuclear properties with radioactive ion beam facilities far from stability. Experiments at underground facilities with the aim to measure at lowest possible energies, avoiding background noise, are ongoing, e.g., at LUNA (https://luna.lngs.infn.it/index.php/new-about-us), CASPAR (https://caspar. nd.edu/) and JUNA (Liu et al. 2022). On the theoretical side, extended compilations for reaction rate predictions are provided at several publically available databases (https://nucastro.org), (https:// reaclib.jinaweb.org), (https://www-nds.iaea.org) including results from the previously described codes NON-SMOKER and TALYS. All of these resources provide the input for the understanding of nuclear transmutation by reactions in evolving stars. Acknowledgments This article benefited from exchange and interactions within the European COST Action CA16117 Chemical Elements as Tracers of the Evolution of the Cosmos (ChETEC) and the International Research Network for Nuclear Astrophysics (IReNA). We also want to thank a large number of colleagues for communication and exchange related the present overview, to name a few: Michael Wiescher, Ken’ichi Nomoto, Gabriel Martinez-Pinedo, Elena Litvinova, Maria Markova, and Stephane Goriely.

References E.G. Adelberger, A. Garcia, R.G.H. Robertson, K.A. Snover, A.B. Balantekin, K. Heeger, RamseyMusolf MJ, D. Bemmerer, A. Junghans, C.A. Bertulani, J.W. Chen, H. Costantini, P. Prati, M. Couder, E. Uberseder, M. Wiescher, R. Cyburt, B. Davids, S.J. Freedman, M. Gai, D. Gazit, L. Gialanella, G. Imbriani, U. Greife, M. Hass, W.C. Haxton, T. Itahashi, K. Kubodera, K. Langanke, D. Leitner, M. Leitner, P. Vetter, L. Winslow, L.E. Marcucci, T. Motobayashi, A. Mukhamedzhanov, R.E. Tribble, K.M. Nollett, F.M. Nunes, T.S. Park, P.D. Parker, R. Schiavilla, E.C. Simpson, C. Spitaleri, F. Strieder, H.P. Trautvetter, K. Suemmerer, S. Typel, Solar fusion cross sections. II. The pp chain and CNO cycles. Rev. Mod. Phys. 83(1), 195–246 (2011). https://doi.org/10.1103/RevModPhys.83.195, 1004.2318 Y. Alhassid, Nuclear level densities: from empirical models to microscopic methods, in CompoundNuclear Reactions, ed. by J. Escher, Y. Alhassid, L.A. Bernstein, D. Brown, C. Fröhlich, P. Talou, W. Younes (Springer International Publishing, Cham, 2021), pp. 97–112 A.M. Amarsi, N. Grevesse, M. Asplund, R. Collet, The solar carbon, nitrogen, and oxygen abundances from a 3D LTE analysis of molecular lines. Astron. Astrophys. 656, A113 (2021). https://doi.org/10.1051/0004-6361/202141384, 2109.04752 E. Anders, N. Grevesse, Abundances of the elements: meteoritic and solar. Geochem. Cosmochem. Acta 53(1), 197–214 (1989). https://doi.org/10.1016/0016-7037(89)90286-X C. Angulo, M. Arnould, M. Rayet, P. Descouvemont, D. Baye, C. Leclercq-Willain, A. Coc, S. Barhoumi, P. Aguer, C. Rolfs, R. Kunz, J.W. Hammer, A. Mayer, T. Paradellis, S. Kossionides, C. Chronidou, K. Spyrou, S. degl’Innocenti, G. Fiorentini, B. Ricci, S. Zavatarelli, C. Providencia, H. Wolters, J. Soares, C. Grama, J. Rahighi, A. Shotter, M. Lamehi Rachti, A compilation of charged-particle induced thermonuclear reaction rates. Nucl. Phys. A 656(1), 3–183 (1999). https://doi.org/10.1016/S0375-9474(99)00030-5 M. Arnould, Influence of the excited states of target nuclei in the vicinity of the iron peak on stellar reaction rates. Astron. Astrophys. 19, 92 (1972) M. Asplund, N. Grevesse, A.J. Sauval, P. Scott, The chemical composition of the sun. Annu. Rev. Astron. Astrophys. 47, 481–522 (2009). https://doi.org/10.1146/annurev.astro.46.060407. 145222

3484

F.-K. Thielemann and T. Rauscher

N.H. Bahcall, Neutrino Astrophysics (Cambridge University Press, Cambridge, 1989) B.L. Berman, S.C. Fultz, Measurements of the giant dipole resonance with monoenergetic photons. Rev. Mod. Phys. 47(3), 713–761 (1975). https://doi.org/10.1103/RevModPhys.47.713 S. Bjørnholm, J.E. Lynn, The double-humped fission barrier. Rev. Mod. Phys. 52(4), 725–931 (1980). https://doi.org/10.1103/RevModPhys.52.725 C.G. Bruno, D.A. Scott, M. Aliotta, A. Formicola, A. Best, A. Boeltzig, D. Bemmerer, C. Broggini, A. Caciolli, F. Cavanna, G.F. Ciani, P. Corvisiero, T. Davinson, R. Depalo, A. Di Leva, Z. Elekes, F. Ferraro, Z. Fülöp, G. Gervino, A. Guglielmetti, C. Gustavino, G. Gyürky, G. Imbriani, M. Junker, R. Menegazzo, V. Mossa, F.R. Pantaleo, D. Piatti, P. Prati, E. Somorjai, O. Straniero, F. Strieder, T. Szücs, M.P. Takács, D. Trezzi, LUNA Collaboration, Improved direct measurement of the 64.5 keV resonance strength in the 17 O (p ,α )14 N reaction at LUNA. Phys. Rev. Lett. 117(14), 142502 (2016). https://doi.org/10.1103/PhysRevLett.117. 142502, 1610.00483 G.R. Caughlan, W.A. Fowler, Thermonuclear reaction rates V. At. Data Nucl. Data Tables 40, 283 (1988). https://doi.org/10.1016/0092-640X(88)90009-5 J.J. Cowan, F.K. Thielemann, J.W. Truran, The r-process and nucleochronology. Phys. Rep. 208, 267–394 (1991). https://doi.org/10.1016/0370-1573(91)90070-3 J.J. Cowan, C. Sneden, J.E. Lawler, A. Aprahamian, M. Wiescher, K. Langanke, G. MartinezPinedo, F.K. Thielemann, Origin of the heaviest elements: the rapid neutron-capture process. Rev. Mod. Phys. 93(1), 015002 (2021). https://doi.org/10.1103/RevModPhys.93.015002, 1901.01410 S. Curtis, K. Ebinger, C. Fröhlich, M. Hempel, A. Perego, M. Liebendörfer, F.K. Thielemann, PUSHing core-collapse supernovae to explosions in spherical symmetry. III. Nucleosynthesis yields. Astrophys. J. 870, 2 (2019). https://doi.org/10.3847/1538-4357/aae7d2 R.H. Cyburt, A.M. Amthor, R. Ferguson, Z. Meisel, K. Smith, S. Warren, A. Heger, Hoffman RD, T. Rauscher, A. Sakharuk, H. Schatz, F.K. Thielemann, M. Wiescher, The jina reaclib database: its recent updates and impact on type-I x-ray bursts. Astrophys. J. Suppl. 189(1), 240–252 (2010). https://doi.org/10.1088/0067-0049/189/1/240 M. Danos, On the long-range correlation model of the photonuclear effect. Nucl. Phys. 5, 23–32 (1958). https://doi.org/10.1016/0029-5582(58)90005-1 R.J. deBoer, J. Görres, M. Wiescher, R.E. Azuma, A. Best, C.R. Brune, C.E. Fields, S. Jones, M. Pignatari, D. Sayre, K. Smith, F.X. Timmes, E. Uberseder, The 12 C(α ,γ )16 O reaction and its implications for stellar helium burning. Rev. Mod. Phys. 89(3), 035007 (2017). https://doi.org/ 10.1103/RevModPhys.89.035007, 1709.03144 M. Eichler, A. Arcones, A. Kelic, O. Korobkin, K. Langanke, T. Marketin, G. Martínez-Pinedo, I. Panov, T. Rauscher, S. Rosswog, C. Winteler, N.T. Zinner, F.-K. Thielemann, The role of fission in neutron star mergers and its impact on the r-process peaks. Apj 808, 30 (2015). https://doi. org/10.1088/0004-637X/808/1/30 M. Eichler, W. Sayar, A. Arcones, T. Rauscher, Probing the production of actinides under different r-process conditions. Apj 879, 47 (2019). https://doi.org/10.3847/1538-4357/ab24cf T. Ericson, A statistical analysis of excited nuclear states. Nucl. Phys. 11, 481–491 (1959). https:// doi.org/10.1016/0029-5582(59)90291-3 J. Erler, K. Langanke, H. Loens, G. Martinez-Pinedo, P.G. Reinhard, Fission properties for rprocess nuclei. Phys. Rev. C 85, 025802 (2012). https://doi.org/10.1103/PhysRevC.85.025802 M. Febbraro, R.J. deBoer, S.D. Pain, R. Toomey, F.D. Becchetti, A. Boeltzig, Y. Chen, K.A. Chipps, M. Couder, K.L. Jones, E. Lamere, Q. Liu, S. Lyons, K.T. Macon, L. Morales, W.A. Peters, D. Robertson, B.C. Rasco, K. Smith, C. Seymour, G. Seymour, M.S. Smith, E. Stech, B.V. Kolk, M. Wiescher, New 13 C (α ,n )16 O cross section with implications for neutrino mixing and geoneutrino measurements. Phys. Rev. Lett. 125(6), 062501 (2020). https://doi.org/10.1103/ PhysRevLett.125.062501 A. Formicola, P. Corvisiero, G. Gervino, The nuclear physics of the hydrogen burning in the Sun. Eur. Phys. J. A 52(4), 73 (2016). https://doi.org/10.1140/epja/i2016-16073-6 C. Fröhlich, G. Martinez-Pinedo, M. Liebendörfer, F.K. Thielemann, E. Bravo, W.R. Hix, K. Langanke, N.T. Zinner, Neutrino-induced nucleosynthesis of A > 64 nuclei: the νp-process. Phys. Rev. Lett. 96, 142502 (2006). https://doi.org/10.1103/PhysRevLett.96.142502

94 Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction)

3485

E. Gadioli, P.E. Hodgson, Pre-Equilibrium Nuclear Reactions (Clarendon Press, Oxford, 1992) S. Ghosh, N. Wolfe, C. Fröhlich, PUSHing core-collapse supernovae to explosions in spherical symmetry. V. Equation of state dependency of explosion properties, nucleosynthesis yields, and compact remnants. Astrophys J 929(1), 43 (2022). https://doi.org/10.3847/1538-4357/ac4d20, 2107.13016 A. Gilbert, A.G.W. Cameron, A composite nuclear-level density formula with shell corrections. Can. J. Phys. 43, 1446 (1965). https://doi.org/10.1139/p65-139 S.A. Giuliani, G. Martínez-Pinedo, L.M. Robledo, Fission properties of superheavy nuclei for r -process calculations, Phys. Rev. C, 97(3), 034323 (2018). https://doi.org/10.1103/PhysRevC. 97.034323, 1704.00554 S.A. Giuliani, G. Martinez-Pinedo, M.R. Wu, L.M. Robledo, Fission and the r-process nucleosynthesis of translead nuclei in neutron star mergers. Phys. Rev. C 102(4), 045804 (2020). https:// doi.org/10.1103/PhysRevC.102.045804, 1904.03733 S. Goriely, Radiative neutron captures by neutron-rich nuclei and the r-process nucleosynthesis. Phys. Lett. B 436, 10–18 (1998) S. Goriely, V. Plujko, Simple empirical E 1 and M 1 strength functions for practical applications. Phys. Rev. C 99(1), 014303 (2019). https://doi.org/10.1103/PhysRevC.99.014303 S. Goriely, S. Hilaire, A.J. Koning, Improved microscopic nuclear level densities within the Hartree-Fock-Bogoliubov plus combinatorial method. Phys. Rev. C 78, 064307 (2008). https:// doi.org/10.1103/PhysRevC.78.064307 S. Goriely, S. Hilaire, A.J. Koning, M. Sin, R. Capote, Towards a prediction of fission cross sections on the basis of microscopic nuclear inputs. Phys. Rev. C 79, 024612 (2009). https://doi.org/10. 1103/PhysRevC.79.024612 S. Goriely, S. Hilaire, M. Girod, Latest development of the combinatorial model of nuclear level densities. in Journal of Physics Conference Series, vol. 337 (2012), p. 012027. https://doi.org/ 10.1088/1742-6596/337/1/012027 S. Goriely, S. Hilaire, S. Péru, K. Sieja, Gogny-hfb+qrpa dipole strength function and its application to radiative nucleon capture cross section. Phys. Rev. C 98, 014327 (2018). https://doi.org/ 10.1103/PhysRevC.98.014327, URL https://link.aps.org/doi/10.1103/PhysRevC.98.014327 J. Görres, M. Wiescher, F.K. Thielemann, Bridging the waiting points: the role of two-proton capture reactions in the rp process. Phys. Rev. C 51, 392 (1995) M. Guttormsen, K.O. Ay, M. Ozgur, E. Algin, A.C. Larsen, F.L. Bello Garrote, H.C. Berg, L. Crespo Campo, T. Dahl-Jacobsen, F.W. Furmyr, D. Gjestvang, A. Görgen, T.W. Hagen, V.W. Ingeberg, B.V. Kheswa, I.K.B. Kullmann, M. Klintefjord, M. Markova, J.E. Midtbø, V. Modamio, W. Paulsen, L.G. Pedersen, T. Renstrøm, E. Sahin, S. Siem, G.M. Tveten, M. Wiedeking, Evolution of the γ -ray strength function in neodymium isotopes. Phys. Rev. C 106(3), 034314 (2022). https://doi.org/10.1103/PhysRevC.106.034314 M. Heine, G. Fruet, S. Courtin, D.G. Jenkins, P. Adsley, A. Brown, R. Canavan, W.N. Catford, E. Charon, D. Curien, S. Della Negra, D. Duprat, F. Hammache, J. Lesrel, G. Lotay, A. Meyer, E. Monpribat, D. Montanari, L. Morris, M. Moukaddam, J. Nippert, Z. Podolyák, P.H. Regan, I. Ribaud, M. Richer, M. Rudigier, R. Shearman, N. de Séréville, C. Stodel, Direct Measurement of Carbon Fusion at Astrophysical Energies with Gamma-Particle Coincidences. in European Physical Journal Web of Conferences, vol. 260 (2022), p. 01004. https://doi.org/ 10.1051/epjconf/202226001004 W.R. Hix, F.K. Thielemann, Silicon burning. I. Neutronization and the physics of quasiequilibrium. APJ 460, 869 (1996). https://doi.org/10.1086/177016, astro-ph/9511088 H.M. Hofmann, T. Mertelmeier, M. Herman, J.W. Tepel, Hauser-Feshbach calculations in the presence of weakly absorbing channels with special reference to the elastic enhancement factor and the factorization assumption. Z. Phys. A Hadrons Nuclei 297(2), 153–160 (1980). https:// doi.org/10.1007/BF01421472 E.M. Holmbeck, T.M. Sprouse, M.R. Mumpower, Nucleosynthesis and observation of the heaviest elements, European Physical Journal A, 59(2), 28 (2023). https://doi.org/10.1140/epja/s10050023-00927-7, 2304.01850

3486

F.-K. Thielemann and T. Rauscher

J.A. Holmes, S.E. Woosley, W.A. Fowler, B.A. Zimmerman, Tables of thermonuclear-reaction-rate data for neutron-induced reactions on heavy nclei. At. Data Nucl. Data Tables 18, 305 (1976). https://doi.org/10.1016/0092-640X(76)90011-5 W.M. Howard, P. Möller, Calculated fission barriers, ground-state masses, and particle separation energies for nuclei with 76 ≤ Z ≤ 100 and 140 ≤ N ≤ 184. At. Data Nucl. Data Tables 25, 219 (1980). https://doi.org/10.1016/0092-640X(80)90005-4 A.V. Ignatyuk, K.K. Istekov, V.N. Okolovich, G.N. Smirenkin, Level density and fission probability in spherical and deformed nuclei, in Presented at the IAEA Symposium on Physics and Chemistry of Fission, pp. 14–18 B.S. Ishkhanov, I.M. Kapitonov, Giant dipole resonance of atomic nuclei. prediction, discovery, and research. Phys. Usp. 64(2), 141–156 (2021). https://doi.org/10.3367/ufne.2020.02.038725 J.P. Jeukenne, A. Lejeune, C. Mahaux, Optical-model potential in finite nuclei from Reid’s hard core interaction. Phys. Rev. C 16, 80–96 (1977). https://doi.org/10.1103/PhysRevC.16.80 J. Jose, Stellar Explosions: Hydrodynamics and Nucleosynthesis (CRC Press/Taylor and Francis, 2016). https://doi.org/10.1201/b19165 A.R. Junghans, M. de Jong, H.G. Clerc, A.V. Ignatyuk, G.A. Kudyaev, K.H. Schmidt, Projectilefragment yields as a probe for the collective enhancement in the nuclear level density. Nucl. Phys. A 629, 635–655 (1998). https://doi.org/10.1016/S0375-9474(98)00658-7 A.R. Junghans, G. Rusev, R. Schwengner, A. Wagner, E. Grosse, Photon data shed new light upon the GDR spreading width in heavy nuclei. Phys. Lett. B 670(3), 200–204 (2008). https://doi. org/10.1016/j.physletb.2008.10.055, 0810.2268 F. Käppeler, R. Gallino, S. Bisterzo, W. Aoki, The s process: Nuclear physics, stellar models, and observations. Rev. Mod. Phys. 83, 157–194 (2011). https://doi.org/10.1103/RevModPhys.83. 157 O.S. Kirsebom et al., Discovery of an exceptionally strong β-decay transition of 20 F and implications for the fate of intermediate-mass stars. Phys. Rev. Lett. 123, 262701 (2019). https:// doi.org/10.1103/PhysRevLett.123.262701 I. Kullmann, S. Goriely, O. Just, A. Bauswein, H.-T. Janka, Impact of systematic nuclear uncertainties on composition and decay heat of dynamical and disk ejecta in compact binary mergers, arXiv e-prints, arXiv:2207.07421 (2022). https://doi.org/10.48550/arXiv.2207.07421, 2207.07421 F. Lach, F.K. Röpke, I.R. Seitenzahl, B. Coté, S. Gronow, A.J. Ruiter, Nucleosynthesis imprints from different Type Ia supernova explosion scenarios and implications for galactic chemical evolution. Astron. Astrophys. 644, A118 (2020). https://doi.org/10.1051/0004-6361/ 202038721, 2010.14084 A. Lejeune, Low-energy optical model potential in finite nuclei from Reid’s hard core interaction. Phys. Rev. C 21(3), 1107–1108 (1980). https://doi.org/10.1103/PhysRevC.21.1107 S.C. Leung, K. Nomoto, Type Ia supernovae and their explosive nucleosynthesis: constraints on progenitors (2021). arXiv e-prints arXiv:2111.14795, 2111.14795 S.C. Leung, K. Nomoto, T. Suzuki, Electron-capture supernovae of super-AGB stars: sensitivity on input physics. APJ 889(1), 34 (2020). https://doi.org/10.3847/1538-4357/ab5d2f, 1901.11438 H. Liang, E. Litvinova, MOdel for Collective Vibration (Springer International Publishing, Cham, 2022), p in press E. Litvinova, H. Loens, K. Langanke, G. Martinez-Pinedo, T. Rauscher, P. Ring, F. Thielemann, V. Tselyaev, Low-lying dipole response in the relativistic quasiparticle time blocking approximation and its influence on neutron capture cross sections. Nucl. Phys. A 823, 26–37 (2009). https://doi.org/10.1016/j.nuclphysa.2009.03.009 W. Liu, Z. Li, J. He, X. Tang, G. Lian, J. Su, Y. Shen, Z. An, F. Chao, J. Chang, L. Chen, H. Chen, X. Chen, Y. Chen, Z. Chen, J. Cheng, B. Cui, X. Fang, C. Fu, L. Gan, B. Guo, Z. Han, X. Guo, G. He, J. He, A. Heger, S. Hou, H. Huang, N. Huang, B. Jia, L. Jiang, S. Kubono, J. Li, M. Li, K. Li, E. Li, T. Li, Y. Li, M. Lugaro, X. Luo, H. Ma, S. Ma, D. Mei, W. Nan, W. Nan, N. Qi, Y. Qian, J. Qin, J. Ren, C. Shang, L. Sun, W. Sun, W. Tan, I. Tanihata, S. Wang, P. Wang, Y. Wang, Q. Wu, S. Xu, Y. Yang, X. Yu, Q. Yue, S. Zeng, L. Zhang, H. Zhang, H. Zhang, L. Zhang, N. Zhang, P. Zhang, Q. Zhang, T. Zhang, X. Zhang, X. Zhang, W. Zhao, J. Zhou,

94 Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction)

3487

Y. Zhou, Commissioning of Underground Nuclear Astrophysics Experiment JUNA in China, in European Physical Journal Web of Conferences, vol. 260 (2022), p. 08001. https://doi.org/10. 1051/epjconf/202226008001 H. Loens, K. Langanke, G. Martinez-Pinedo, T. Rauscher, F.K. Thielemann, Complete inclusion of parity-dependent level densities in the statistical description of astrophysical reaction rates. Phys. Lett. B 666, 395–399 (2008). https://doi.org/10.1016/j.physletb.2008.07.073 J.E. Lynn, B.B. Back, Sub-barrier fission probability for a double-humped barrier. J. Phys. Math. Gen. 7(3), 395–409 (1974). https://doi.org/10.1088/0305-4470/7/3/011 C. Mahaux, H.A. Weidenmuller, Recent developments in compound-nucleus theory. Annu. Rev. Nucl. Part. Sci. 29, 1–31 (1979). https://doi.org/10.1146/annurev.ns.29.120179.000245 A. Mamdouh, J. Pearson, M. Rayet, F. Tondeur, Fission barriers of neutron-rich and superheavy nuclei calculated with the etfsi method. Nucl. Phys. A 679, 337–358 (2001) M. Markova, P. von Neumann-Cosel, A.C. Larsen, S. Bassauer, A. Görgen, M. Guttormsen, F.L. Bello Garrote, H.C. Berg, M.M. Bjørøen, T. Dahl-Jacobsen, T.K. Eriksen, D. Gjestvang, J. Isaak, M. Mbabane, W. Paulsen, L.G. Pedersen, N.I.J. Pettersen, A. Richter, E. Sahin, P. Scholz, S. Siem, G.M. Tveten, V.M. Valsdottir, M. Wiedeking, F. Zeiser, Comprehensive test of the Brink-Axel hypothesis in the energy region of the pygmy dipole resonance. Phys. Rev. Lett. 127(18), 182501 (2021). https://doi.org/10.1103/PhysRevLett.127.182501, 2012.11956 G. Martinez-Pinedo, D. Mocelj, N. Zinner, A. Kelic, K. Langanke, I. Panov, B. Pfeiffer, T. Rauscher, K.H. Schmidt, F.K. Thielemann, The role of fission in the r-process. Prog. Part. Nucl. Phys. 59, 199–205 (2007). https://doi.org/10.1016/j.ppnp.2007.01.018 G.J. Mathews, A. Mengoni, F.K. Thielemann, W.A. Fowler, Neutron capture rates in the r-process – The role of direct radiative capture. Astrophys. J. 270, 740–745 (1983). https://doi.org/10. 1086/161164 C.M. McCullagh, M.L. Stelts, R.E. Chrien, Dipole radiative strength functions from resonance neutron capture. Phys. Rev. C 23(4), 1394–1403 (1981). https://doi.org/10.1103/PhysRevC.23. 1394 L. McFadden, G.R. Satchler, Optical-model analysis of the scattering of 24.7 MeV alpha particles. Nucl. Phys. A 84, 177–200 (1966). https://doi.org/10.1016/0029-5582(66)90441-X Z. Meisel, S. George, S. Ahn, D. Bazin, B.A. Brown, J. Browne, J.F. Carpino, H. Chung, R.H. Cyburt, A. Estradé, M. Famiano, A. Gade, C. Langer, M. Matoš, W. Mittig, F. Montes, D.J. Morrissey, J. Pereira, H. Schatz, J. Schatz, M. Scott, D. Shapira, K. Smith, J. Stevens, W. Tan, O. Tarasov, S. Towers, K. Wimmer, J.R. Winkelbauer, J. Yurkon, R.G.T. Zegers, Nuclear mass measurements map the structure of atomic nuclei and accreting neutron stars. Phys. Rev. C 101(5), 052801 (2020). https://doi.org/10.1103/PhysRevC.101.052801, 2004.14101 D. Mocelj, T. Rauscher, G. Martinez-Pinedo, K. Langanke, L. Pacearescu, A. Faessler, F.K. Thielemann, Y. Alhassid, Large-scale prediction of the parity distribution in the nuclear level density and application to astrophysical reaction rates. Phys. Rev. C 75(4), 045805 (2007). https://doi.org/10.1103/PhysRevC.75.045805, nucl-th/0703033 P. Mohr, Z. Fülöp, G. Gyürky, G.G. Kiss, T. Szücs, Successful prediction of total α-induced reaction cross sections at astrophysically relevant sub-coulomb energies using a novel approach. Phys. Rev. Lett. 124(25), 252701 (2020). https://doi.org/10.1103/PhysRevLett.124.252701, 2006.03885 P.A. Moldauer, Statistics and the average cross section. Nucl. Phys. A 344(2), 185–195 (1980). https://doi.org/10.1016/0375-9474(80)90671-5 P. Möller, J.R. Nix, W.D. Myers, W.J. Swiatecki, Nuclear ground-state masses and deformations. At. Data Nucl. Data Tables 59, 185 (1995). https://doi.org/10.1006/adnd.1995.1002, https://t2. lanl.gov/nis/data/astro/molnix96/massd-FRDM1992.html P. Möller, A.J. Sierk, T. Ichikawa, A. Iwamoto, M. Mumpower, Fission barriers at the end of the chart of the nuclides. Phys. Rev. C 91(2), 024310 (2015a). https://doi.org/10.1103/PhysRevC. 91.024310 P. Möller, A.J. Sierk, T. Ichikawa, A. Iwamoto, M. Mumpower, Fission barriers at the end of the chart of the nuclides. Phys. Rev. C 91, 024310 (2015b). https://doi.org/10.1103/PhysRevC.91. 024310

3488

F.-K. Thielemann and T. Rauscher

E. Monpribat, S. Martinet, S. Courtin, M. Heine, S. Ekström, D.G. Jenkins, A. Choplin, P. Adsley, D. Curien, M. Moukaddam, J. Nippert, S. Tsiatsiou, G. Meynet, A new 12 C + 12 C nuclear reaction rate: Impact on stellar evolution. Astron. Astrophys. 660, A47 (2022). https://doi.org/ 10.1051/0004-6361/202141858, 2111.15224 ´ W.D. Myers, W.J. Swia¸ tecki, Thomas-fermi fission barriers. Phys. Rev. C 60, 014606 (1999). https://doi.org/10.1103/PhysRevC.60.014606 T. Nakagawa, S. Chiba, T. Hayakawa, T. Kajino, Maxwellian-averaged neutron-induced reaction cross sections and astrophysical reaction rates for kT = 1 keV to 1 MeV calculated from microscopic neutron cross section library JENDL-3.3. At. Data Nucl. Data Tables 91(2), 77–186 (2005). https://doi.org/10.1016/j.adt.2005.08.002 K. Nomoto, F.K. Thielemann, S. Miyaji, The triple alpha reaction at low temperatures in accreting white dwarfs and neutron stars. Astron. Astrophys. 149(2), 239–245 (1985) T. Ojima, Y. Ishimaru, S. Wanajo, N. Prantzos, P. François, Stochastic chemical evolution of galactic subhalos and the origin of r-process elements. Astrophys. J. 865, 87 (2018). https:// doi.org/10.3847/1538-4357/aada11 I.V. Panov, E. Kolbe, B. Pfeiffer, T. Rauscher, K.L. Kratz, F.K. Thielemann, Calculations of fission rates for r-process nucleosynthesis. Nucl. Phys. A 747, 633–654 (2005). https://doi.org/10.1016/ j.nuclphysa.2004.09.115 I.V. Panov, I.Y. Korneev, T. Rauscher, G. Martinez-Pinedo, A. Kelic-Heil, N.T. Zinner, F. Thielemann, Neutron-induced astrophysical reaction rates for translead nuclei. Astron. Astrophys. 513, A61 (2010). https://doi.org/10.1051/0004-6361/200911967 I. Petermann, K. Langanke, G. Martinez-Pinedo, I.V. Panov, P.G. Reinhard, F.K. Thielemann, Have superheavy elements been produced in nature? Eur. Phys. J. A 48, 122 (2012). https://doi.org/ 10.1140/epja/i2012-12122-6 T. Rauscher, code SMARAGD (2010). unpublished T. Rauscher, The Path to Improved Reaction Rates for Astrophysics. Int. J. Mod. Phys. E 20, 1071–1169 (2011). https://doi.org/10.1142/S021830131101840X T. Rauscher, Essentials of Nucleosynthesis and Theoretical Nuclear Astrophysics (IOP Publishing, 2020). https://doi.org/10.1088/2514-3433/ab8737 T. Rauscher, Stellar neutron captures at low and high temperatures. Eur. Phys. J. A 58, 214 (2022). https://doi.org/10.1140/epja/s10050-022-00866-9 T. Rauscher, F.K. Thielemann, Astrophysical reaction rates from statistical model calculations. At. Data Nucl. Data Tables 75, 1–351 (2000). https://doi.org/10.1006/adnd.2000.0834 T. Rauscher, F.K. Thielemann, K.L. Kratz, Nuclear level density and the determination of thermonuclear rates for astrophysics. Phys. Rev. C 56, 1613–1625 (1997). https://doi.org/10. 1103/PhysRevC.56.1613 T. Rauscher, R. Bieber, H. Oberhummer, K.L. Kratz, J. Dobaczewski, P. Möller, M.M. Sharma, Dependence of direct neutron capture on nuclear-structure models. Phys. Rev. C 57(4), 2031– 2039 (1998). https://doi.org/10.1103/PhysRevC.57.2031, nucl-th/9801005 T. Rauscher, F.K. Thielemann, J. Görres, M. Wiescher, Capture of /α particles by isospinsymmetric nuclei. Nucl. Phys. A 675(3–4), 695–721 (2000). https://doi.org/10.1016/S03759474(00)00182-2, nucl-th/0003057 T. Rauscher, N. Dauphas, I. Dillmann, C. Fröhlich, Z. Fülöp, G. Gyürky, Constraining the astrophysical origin of the p-nuclei through nuclear physics and meteoritic data. Rep. Prog. Phys. 76, 066201 (2013). https://doi.org/10.1088/0034-4885/76/6/066201 C. Rolfs, W. Rodney, Cauldrons in the Cosmos: Nuclear Astrophysics (University of Chicago Press, Chicago, 1988) G.R. Satchler, W.G. Love, Folding model potentials from realistic interactions for heavy-ion scattering. Phys. Rep. 55(3), 183–254 (1979). https://doi.org/10.1016/0370-1573(79)90081-4 H. Schatz, W.J. Ong, Dependence of x-ray burst models on nuclear masses. Astrophys. J. 844(2), 139 (2017). https://doi.org/10.3847/1538-4357/aa7de9, 1610.07596 H. Schatz, A. Aprahamian, J. Goerres, M. Wiescher, T. Rauscher, J.F. Rembges, F.K. Thielemann, B. Pfeiffer, P. Moeller, K.L. Kratz, H. Herndl, B.A. Brown, H. Rebel, rp-process nucleosynthesis at extreme temperature and density conditions. Phys. Rep. 294, 167–264 (1998). https://doi.org/ 10.1016/S0370-1573(97)00048-3

94 Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction)

3489

H. Schatz, A.D. Becerril Reyes, A. Best, E.F. Brown, K. Chatziioannou, K.A. Chipps, C.M. Deibel, R. Ezzeddine, D.K. Galloway, C.J. Hansen, F. Herwig, A.P. Ji, M. Lugaro, Z. Meisel, D. Norman, J.S. Read, L.F. Roberts, A. Spyrou, I. Tews, F.X. Timmes, C. Travaglio, N. Vassh, C. Abia, P. Adsley, S. Agarwal, M. Aliotta, W. Aoki, A. Arcones, A. Aryan, A. Bandyopadhyay, A. Banu, D.W. Bardayan, J. Barnes, A. Bauswein, T.C. Beers, J. Bishop, T. Boztepe, B. Côté, M.E. Caplan, A.E. Champagne, J.A. Clark, M. Couder, A. Couture, S.E. de Mink, S. Debnath, R.J. deBoer, J. den Hartogh, P. Denissenkov, V. Dexheimer, I. Dillmann, J.E. Escher, M.A. Famiano, R. Farmer, R. Fisher, C. Fröhlich, A. Frebel, C. Fryer, G. Fuller, A.K. Ganguly, S. Ghosh, B.K. Gibson, T. Gorda, K.N. Gourgouliatos, V. Graber, M. Gupta, W. Haxton, A. Heger, W.R. Hix, W.C.G. Ho, E.M. Holmbeck, A.A. Hood, S. Huth, G. Imbriani, R.G. Izzard, R. Jain, H. Jayatissa, Z. Johnston, T. Kajino, A. Kankainen, G.G. Kiss, A. Kwiatkowski, La Cognata M, A.M. Laird, L. Lamia, P. Landry, E. Laplace, K.D. Launey, D. Leahy, G. Leckenby, A. Lennarz, B. Longfellow, A.E. Lovell, W.G. Lynch, S.M. Lyons, K. Maeda, E. Masha, C. Matei, J. Merc, B. Messer, F. Montes, A. Mukherjee, M. Mumpower, D. Neto, B. Nevins, W.G. Newton, L.Q. Nguyen, K. Nishikawa, N. Nishimura, F.M. Nunes, E. O’Connor, B.W. O’Shea, W.J. Ong, S.D. Pain, M.A. Pajkos, M. Pignatari, R.G. Pizzone, V.M. Placco, T. Plewa, B. Pritychenko, A. Psaltis, D. Puentes, Y.Z. Qian, D. Radice, D. Rapagnani, B.M. Rebeiro, R. Reifarth, A.L. Richard, N. Rijal, I.U. Roederer, J.S. Rojo, J.S. K, Y. Saito, A. Schwenk, M.L. Sergi, R.S. Sidhu, A. Simon, T. Sivarani, Á. Skúladóttir, M.S. Smith, A. Spiridon, T.M. Sprouse, S. Starrfield, A.W. Steiner, F. Strieder, I. Sultana, R. Surman, T. Szücs, A. Tawfik, F. Thielemann, L. Trache, R. Trappitsch, M.B. Tsang, A. Tumino, S. Upadhyayula, J.O. Valle Martinez, M. Van der Swaelmen, C. Viscasillas Vázquez, A. Watts, B. Wehmeyer, M. Wiescher, C. Wrede, J. Yoon, R.G.T. Zegers, M.A. Zermane, M. Zingale, Horizons: nuclear astrophysics in the 2020s and beyond (2022). arXiv e-prints arXiv:2205.07996, 2205.07996 N. Schunck, D. Regnier, Theory of nuclear fission. Prog. Part. Nucl. Phys. 125, 103963 (2022). https://doi.org/10.1016/j.ppnp.2022.103963, 2201.02719 K. Sieja, Shell-model study of the M1 dipole strength at low energy in the A > 100 nuclei. Phys. Rev. C 98, 064312 (2018). https://doi.org/10.1103/PhysRevC.98.064312 N. Soppera, M. Bossant, E. Dupont, H. Henriksson, Y. Rugawa, Recent upgrades to the nuclear data tool JANIS. J. Korean Phys. Soc. 59, 1329–1332 (2011). https://doi.org/10.3938/jkps.59. 1329 V.M. Strutinsky, Shell effects in nuclear masses and deformation energies. Nucl. Phys. A 95(2), 420–442 (1967). https://doi.org/10.1016/0375-9474(67)90510-6 V.M. Strutinsky, Shells in deformed nuclei. Nucl. Phys. A 122(1), 1–33 (1968). https://doi.org/10. 1016/0375-9474(68)90699-4 X. Tang, L. Ru, The 12 C+12 C fusion reaction at stellar energies. in European Physical Journal Web of Conferences, vol. 260 (2022), p. 01002. https://doi.org/10.1051/epjconf/202226001002 J.W. Tepel, H.M. Hofmann, H.A. Weidenmüller, Hauser-Feshbach formulas for medium and strong absorption. Phys. Lett. B 49(1), 1–4 (1974). https://doi.org/10.1016/0370-2693(74)90565-6 F.K. Thielemann, W.D. Arnett, Hydrostatic nucleosynthesis – part two – core neon to silicon burning and presupernova abundance yields of massive stars. Astrophys. J. 295, 604 (1985). https://doi.org/10.1086/163403 F.K. Thielemann, M. Arnould, Average radiation widths and the giant dipole resonance width. in Nuclear Data for Science and Technology, ed. by K.H. Böckhoff (Springer Netherlands, Dordrecht, 1983), pp. 762–765 F.K. Thielemann, J. Metzinger, H.V. Klapdor, Beta-delayed fission and neutron emission: consequences for the astrophysical r-process and the age of the galaxy. Z. Phys. A: Hadrons Nuclei 309, 301–317 (1983). https://doi.org/10.1007/BF01413833 F.K. Thielemann, M. Arnould, J.W. Truran, Thermonuclear reaction rates from statistical model calculations. in Advances in Nuclear Astrophysics, ed. by E. Vangioni-Flam, J. Audouze, M. Casse, J.P. Chieze, J. Tran Thanh Van (1986), pp. 525–540 F.K. Thielemann, R. Hirschi, M. Liebendörfer, R. Diehl, Massive Stars and Their Supernovae. in Lecture Notes in Physics, vol. 812, ed. by R. Diehl, D.H. Hartmann, N. Prantzos (Springer, Berlin, 2011), pp 153–232

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J. Thomas, M.R. Zirnbauer, K. Langanke, 26 Al(n,p)26 Mg reaction: comparison between the Hauser-Feshbach formula and the exact random-matrix result for the cross section. Phys. Rev. C 33(6), 2197–2199 (1986). https://doi.org/10.1103/PhysRevC.33.2197 E. Tiesinga, P.J. Mohr, D.B. Newell, B.N. Taylor, CODATA recommended values of the fundamental physical constants: 2018∗ . Rev. Mod. Phys. 93(2), 025010 (2021). https://doi.org/10.1103/ RevModPhys.93.025010. https://ui.adsabs.harvard.edu/abs/2021RvMP...93b5010T J.W. Truran, C.J. Hansen, A.G.W. Cameron, A. Gilbert, Thermonuclear reactions in medium and heavy nuclei. Can. J. Phys. 44, 151 (1966). https://doi.org/10.1139/p66-011 E.M. Tursunov, S.A. Turakulov, A.S. Kadyrov, L.D. Blokhintsev, Astrophysical S factor and rate of 7 Be(p ,γ )8 B direct capture reaction in a potential model. Phys. Rev. C 104(4), 045806 (2021). https://doi.org/10.1103/PhysRevC.104.045806, 2105.11091 P. van Isacker, M.A. Nagarajan, D.D. Warner, Effect of the neutron skin on collective states of nuclei. Phys. Rev. C 45(1), R13–R16 (1992). https://doi.org/10.1103/PhysRevC.45.R13 N. Vassh, R. Vogt, R. Surman, J. Randrup, T.M. Sprouse, M.R. Mumpower, P. Jaffke, D. Shaw, E.M. Holmbeck, Y. Zhu, G.C. McLaughlin, Using excitation-energy dependent fission yields to identify key fissioning nuclei in r-process nucleosynthesis. J. Phys. G: Nucl. Part. Phys. 46, 065202 (2019). https://doi.org/10.1088/1361-6471/ab0bea R.A. Ward, W.A. Fowler, Thermalization of long-lived nuclear isomeric states under stellar conditions. Astrophys. J. 238, 266–286 (1980). https://doi.org/10.1086/157983 M. Wiescher, J. Gorres, F.K. Thielemann, H. Ritter, Explosive hydrogen burning in novae. Astron. Astrophys. 160, 56–72 (1986) S.E. Woosley, W.D. Arnett, D.D. Clayton, The explosive burning of oxygen and silicon. Astrophys. J. Suppl. 26, 231 (1973). https://doi.org/10.1086/190282 S.E. Woosley, W.A. Fowler, J.A. Holmes, B.A. Zimmerman, Semiempirical thermonuclear reaction-rate data for intermediate-mass nuclei. At. Data Nucl. Data Tables 22, 371 (1978). https://doi.org/10.1016/0092-640X(78)90018-9

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Michael Wiescher, Richard James deBoer, and René Reifarth

Contents Stellar Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Goals and Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Tools and Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stable Beam Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plasma Fusion Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radioactive Beam Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gamma-Induced Reaction Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Reaction Studies with Charged Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactions with Stable Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Reversal Studies (Photon Beam) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indirect or Surrogate Reaction Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactions with Neutron Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Cross Section Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiments at Reaction-Driven Neutron Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiments at White Neutron Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Studies with Reactor Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extrapolation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extrapolation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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M. Wiescher () · R. J. deBoer Department of Physics and Astronomy and the Joint Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, IN, USA e-mail: [email protected]; [email protected] R. Reifarth Goethe University, Frankfurt, Germany © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_116

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Abstract

This contribution describes the experimental challenges to investigate nuclear reaction cross sections for stellar burning processes in the laboratory and the theoretical needs to transform the experimental data into reaction rates to be used for simulating quiescent and explosive nuclear burning processes. This paper provides an overview of the scientific questions associated with charged particle and neutron-induced stellar reaction rates. It also provides a summary of a variety of experimental methods and techniques to obtain the cross section data. Two major phenomenological theories for extrapolating the laboratory data into the stellar energy regime are presented, and their advantages, shortcomings, and uncertainties are discussed.

Stellar Reaction Rates Stellar evolution is governed by various chains of nuclear reactions, which are primarily driven by the available fuel content in the stellar medium. Hydrogen burning is dominated by proton capture reactions, helium burning by alpha-induced reaction processes and carbon and oxygen burning by fusion between carbon and oxygen isotopes, while neon and silicon burning are correlated with photon-induced dissociation processes. These reactions release light proton and helium isotopes, which can again trigger fast capture reactions on the remaining composition of heavier isotopes. The various reaction chains release the energy stabilizing the star while also changing the abundance distribution in the stellar interior. The associated reaction sequences depend on the composition of the fuel material and the strength of the various reaction rates. These reaction rates are determined by integrating the reaction cross sections over the Maxwell Boltzmann distribution of the interacting particles at stellar temperatures. The quantity that needs to be determined or confirmed experimentally for a reliable reaction rate is the nuclear reaction cross section at energies that correspond to the stellar burning temperatures. The reaction rate per particle pair is given in the standard conventional mode (Fowler et al. 1975) as a function of temperature T :  σ v =

8 πμ

1/2

1 (kB T )3/2



∞

σ (E)Ee−E/kB T dE,

(1)

0

where kB is the Boltzmann constant and μ is the reduced mass. The total reaction then rate is simply r = NX NY σ v/(1 + δXY ),

(2)

with the number densities NX and NY of species X and Y as well as the Kronecker symbol δXY .

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The nuclear reaction cross section σ (E) represents the probability of a specific reaction taking place at a specific energy E. It is a quantity independent from experiment or environment, which has to be derived from experimental data and needs therefore to be corrected for the specifics of the experimental setup. To convert the experimental signal or yield Y (E) into an experiment-independent cross section requires a full understanding of detector efficiency, energy loss, and other target effects (Fowler et al. 1948). For angular distributions, corrections often depend on the position of the detector with respect to the beam axis. For detector efficiency corrections are often required for signal pile-up and summing in the detector and data acquisition system. Additionally, branching effects associated with the decay pattern of the reaction process need to be accounted for as discussed in more detail in section “Direct Reaction Studies.” The cross section depends on the specific characteristics of a nuclear reaction and reflects the quantum mechanical probability for the transition of the initial reaction system of the two interacting particles and the final system of the reaction products but also on the probability for tunneling through the Coulomb and orbital momentum barrier of the interacting particles. This is described by the penetrability P , which can be expressed in terms of the regular and irregular Coulomb functions F (ρ, η) and G (ρ, η), respectively: P =

F

(ρ, η)2

ρ . + G (ρ, η)2

(3)

Here the two dimensionless parameters are ρ = k · r, the dimensionless radius and the Sommerfeld parameter, η = (Z1 Z2 e2 μ)/(h¯ 2 k), where k is the wave number, Z1 and Z2 the electrical charge of the interacting particles, and e the elementary charge. The parameter μ represents the reduced mass of the reaction system. The tunneling probability declines steeply with energy for nuclear reactions between charged particles. An increasing orbital momentum  reduces the cross section further. For neutrons the tunneling probability only depends on the orbital momentum . For  = 0 s-wave neutrons, √ the cross section increases toward lower energies following the σ (E) ∼ 1/v ∼ μ/E law. For reactions with higher orbital momenta, the cross sections decrease toward lower energy (Wiescher et al. 1990). The study of the cross section as a function of energy, the so-called excitation function, requires its measurement over a wide energy range, which is typically done at low-energy particle accelerators. The effective energy range for stellar temperature conditions is defined by the overlap range of the Maxwell-Boltzmann distribution of the interacting particles and the cross section in that energy range. For a nonresonant cross section, this region is called the Gamow range, which resembles approximately the shape of a Gaussian function with the center E0 and width ΔE given by E0 ± ΔE = 0.122 · (Z12 Z22 μT92 )(1/3) ± 0.236 · (Z12 Z22 μT95 )(1/6) ,

(4)

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where Z1 and Z2 are the charge numbers of the interacting particles, μ is the reduced mass of the interacting particles, and T9 is the stellar temperature in units 109 Kelvin (Fowler et al. 1975). For each reaction, the Gamow range depends on the respective charge of the interacting nuclei and the temperature of the stellar burning environment. If the reaction cross section is characterized by low-energy resonances, the concept of the Gamow window is not valid, and the reaction rate is determined by the overlap between the Maxwell-Boltzmann distribution and the Breit-Wigner shapes of the resonances. For narrow resonances the efficient energy range corresponds to the resonance energies Er and widths Γ of these resonances.

Experimental Goals and Concepts The reaction cross section may be defined by a number of different reaction contributions. Nonresonant contributions are usually due to one-step quantum transitions between the initial configuration of two interacting nuclei and the final configuration of the reaction products, φf |H |φi . The transition strength is determined by the Hamiltonian H of the reaction mechanism, which depends on the strong interaction for nuclear fusion reactions, the electromagnetic interaction for radiative capture, or in some cases also on the weak interaction, such as the fusion of two protons to deuterium. The transition probability is smooth for these processes, and the energy dependence of the cross section is only determined by the Coulomb and orbital momentum barriers: σd (E) ∝ P (E) · φf |H |φi 2 .

(5)

In addition to these one-step reaction contributions, two-step processes may occur through the formation of a compound state, which corresponds to an excited state in the nucleus formed by the fusion of the initial projectile/target system. This excited state is short-lived and decays either by particle emission to the final configuration of reaction products or by γ emission to the ground state of the compound nucleus. This mechanism is reflected in a resonance structure in the Lorentzian energy dependence of the cross section, usually described by a BreitWigner curve at the resonance energy Er : σr (E) =

Γ1 Γ2 π , 2 k (E − Er )2 + Γ 2 /4

(6)

where k is the wave number, Γ1 and Γ2 are the partial widths of the incoming and outgoing channels, respectively, and Γ is the total width. The total width Γtot of a resonance depends on the lifetime τ of the populated excited state: Γtot = h/τ ¯ =



Γpart .

(7)

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The partial widths of the entrance channel Γin for a projectile with spin jp and target with spin jt and exit channel Γex depend on the respective transition probabilities feeding Γin ∝ P (E) · φc |H |φi 2 and depleting Γex ∝ P (E) · φf |H |φc 2 for the compound state. For a narrow resonances with spin J , the energy dependence in the partial widths due to the Coulomb and orbital momentum penetrability can be neglected, and the reaction rate will be directly proportional to the resonance strength: ωγ =

Γin Γex 2J + 1 · . (2ip + 1)(2jt + 1) Γtot

(8)

This simplifies Equation (1), and the reaction rate contribution of a single resonance i with resonance energy Ei is directly proportional to the resonance strength ωγi : 

2π NA σ v = NA · μkT

3/2

· h¯ 2 · ωγi · e−Ei /kB T .

(9)

For multiple narrow resonances, the reaction rate just corresponds to the sum of the single resonance contributions. The goal of experiments determining the narrow resonance contributions for a nuclear reaction therefore aims at an accurate determination of the resonance energies and the corresponding resonance strengths (Boeltzig et al. 2019). This approach is described in section “Direct Reaction Studies with Charged Particles.” For broad resonances, corresponding to states with short life times, the energy dependence of the partial widths needs to be taken into account if the resonance width is larger than the Gamow range as defined in Equation (4). For these conditions the narrow resonance approximation cannot be applied, but the reaction rate depends on the integration over the tails of the resonance within the Gamow range. In the case of broad overlapping resonances, the cross section is frequently influenced by interference effects between broad resonances or broad resonances and the direct capture components and the reaction rate are best calculated by using Equation (1) and adopting the R-matrix formalism (Lane et al. 1958) for mapping the cross section over a wide energy range. In this case the goal of the experiments is to map the cross section over a wide energy range and possibly also angle range to uniquely identify the broad resonance contributions, the direct capture transitions, and the interference pattern between these components. This approach is described in section “R-Matrix Phenomenology.” Resonant and nonresonant contributions as well as possible interference effects between these components determine the overall cross section behavior as a function of energy, and it is the challenge for experimentalist and theorist alike to determine this cross section behavior reliably at the very low-energy Gamow range (see Equation (4)). In most cases, this requires an extrapolation of the experimental data, taken at higher energies toward the lower energy range. For this extrapolation the concept of the astrophysical S-factor, or simply S-factor, has been introduced to overcome the computational challenges of the exponential decline of the cross

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section for charged particle reactions. The S-factor in the presently used form was introduced in 1952 by Salpeter, but the concept goes back to the early attempts by Bethe (1939) as well as Gamow and Teller (1938) in which the correlation between cross section and penetrability was first summarized. The expression S(E) = σ (E) · E · e2π η

(10)

introduces the S-factor as a cross section corrected for an approximate Coulomb term of an s-wave particle (orbital momentum  = 0) at a certain energy E. This means that the S-factor contains all information about the quantum mechanical components of the transition strength between the initial and the final nuclear configuration φf |H |φi 2 as well as the impact of the orbital momentum barrier for higher orbital momenta particles in the tunneling probability. The Hamiltonian H describes the nature of the interaction process facilitating the transition. In these early approximations, the S-factor was assumed to be a constant, not considering that possible near-threshold resonances could dramatically change the quantum mechanical transition strength. Figure 1 shows the low-energy S-factor for the 22 Ne(α, n)25 Mg reaction. The cross section measured by Jaeger et al. (2001) displays a mixture of narrow and broad resonances, where, in general, the resonance widths become smaller at lower energies, a result of the decreasing penetrability. To illustrate the effect of the

Fig. 1 An example of narrow and broad resonances in an experimentally measured S-factor of the 22 Ne(α, n)25 Mg reaction by Jaeger et al. (2001). The experimental data has some finite energy resolution, for which the R-matrix model has been convoluted (grey dashed line). The resolutionless R-matrix S-factor is shown by the blue solid line. The green dashed-dotted line indicates a Hauser-Feshbach calculation of the S-factor

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experimental resolution, an R-matrix calculation is shown both with and without resolution effects applied. In the present case, the experimental resolution is a few keV, determined, for the most part, by the energy loss of the beam through the target. For resonances with widths of 10’s or 100’s of keV, the resolution is nearly negligible, and the shape of the experimental yield is nearly the same as the theoretical calculation. However, for a very narrow resonance, in this case a good example is that at ≈700 keV, the true resonance width is only eV, and thus the observed yield is highly distorted from the theoretical shape. A reliable extrapolation of the data toward the very low energy range requires the understanding and implementation of all the possible reaction components with their respective energy dependence and interference possibilities. This also requires considering a number of other reaction features, which may occur in the experimentally unknown range of the excitation curve, such as unknown resonances, interference effects with subthreshold states, which produce deviations from the anticipated cross section due to variations in the nuclear potential parameters used for the extrapolation. All these features may have a substantial impact on the reliable extrapolation of laboratory cross section data. They need to be understood to correct the standard Sfactor formalism accordingly. For charged particle reactions, a direct measurement of the impact of these quantum factors is extremely challenging because of the rapid decline of the cross section toward the stellar energy range; it requires deep underground measurements to reduce the natural cosmic ray background in order to detect a statistically significant reaction signal. An alternative might be the so-called Trojan Horse Method (THM) (Tumino et al. 2021), which has been proposed as a surrogate for the actual reaction study, since as a particle transfer mechanism, it determines the nuclear transfer probability without the handicap of the Coulomb barrier in the capture of the fusion cross section. However, the THM data need to be translated into reliable reaction data, which again requires a reliable theoretical approach for the low-energy treatment of the aforementioned near-threshold processes. The extrapolation techniques have been considerably refined. Instead of the assumption of a constant S-factor at low energies or a polynomial fit to the lowest energy data, new computational techniques have been introduced. For reactions with a high resonance density, or level density, in the compound nucleus statistical models, sometimes associated with Monte Carlo simulations of reaction parameters, are frequently employed to fit laboratory cross section data and extrapolate the low energy range (Simon et al. 2015). For reactions with resolvable resonances, Rmatrix techniques, including multiple reaction channels, are being utilized to come to a reliable extrapolation by utilizing measurements from several combined data sets (Azuma et al. 2010). One of the main advantages of the R-matrix technique is its ability to take in a wide range of experimental data types. These data can have very different sources of systematic uncertainties that are hard to estimate; thus their comparison helps to determine if these uncertainties were well estimated. R-matrix calculations can also be used for nonresonant reactions, but in these cases simple potential models are often favored. In any case, detailed information on

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the nuclear reaction parameters such as level densities, nuclear deformation, nuclear potentials, nuclear radii, and other parameters are of crucial importance to come to a reliable extrapolation. These details will be discussed in section “R-Matrix Phenomenology.”

Experimental Tools and Technologies There is a broad range of experimental tools and techniques that can be used to determine the low-energy cross sections or reaction rates for nuclear processes between stable particles. These range from direct reaction studies at the traditional low-energy accelerator and high-intensity photon sources for photon-induced studies. Other ways for determining the cross section are based on the use of surrogate reactions, probing the reaction by indirect means, or more recently the use of laser plasma facilities to also take into account plasma effects on the reaction process. Similarly, a broad range of techniques has been developed for the study of neutroninduced reactions on light and heavy isotopes. These include reaction-induced neutron sources and spallation sources, using nuclear reactors to provide an intense neutron-rich environment. Finally, radioactive beam facilities are the main tool for investigating, with inverse kinematics techniques, reactions on short-lived particles expected for explosive nucleosynthesis environments.

Stable Beam Facilities Accelerators are the classical tool for reaction studies on stable isotopes. The goal is to measure the reaction cross section as a function of energy toward the energy range of the stellar environment for which the reaction is being investigated. These studies are important for the analysis of nucleosynthesis in a long-lived stellar environment and the study of the evolution of stars during their different burning phases toward their quiet end as white or brown dwarfs or in the case of massive stars, their violent end as supernovae. For the purpose of determining the reaction cross section, the beam is directed on a target of well-known composition to achieve maximum interaction probability between beam and target particles. A set of detectors is arranged around the target to measure the reaction products, as they are the experimental signature for the reaction probability or cross section. While these measurements are straightforward at laboratory energies, it is extremely challenging to reach the energy range corresponding to stellar temperatures and has been achieved in only a few cases. The limitation is primarily due the low event rate and radiation background in the detectors. In most cases, the cross section determination in the Gamow energy range relies on theoretical extrapolation of higher-energy data. The set of experimental data used today is based largely on direct reaction studies at accelerators but also, in a few cases, on so-called indirect studies, probing the nuclear structure of nuclear transition probabilities through surrogate techniques at higher energies to circumvent the challenges of the Coulomb barrier.

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Accelerators Low-energy accelerators, with high beam intensity and optimum energy resolution and calibration, have been the traditional tool of the nuclear astrophysics community in the study of charged particle reactions with stable beams. These accelerators were originally of the Cockroft-Walton or the Van de Graaff type, generating high voltages through an electric amplifier circuit or by up-charging an isolated terminal using an insulating rubber belt, respectively. More recent machines are of the Pelletron type that use a chain of insulated metal sections for the up-charge or a Dynamitron/Singletron type, with the high-voltage terminal being charged electrostatically by a high-frequency oscillating voltage, fueling a Cockroft-Waltonlike electronic circuit. While the Cockroft-Walton accelerator is limited in terminal voltage to about 400 kV and the Van de Graaff accelerator is limited in current, to the limited charging capacity of the belt or chain, the newer accelerator types have a broader energy range and can provide higher beam intensities. Modern types of machines reach milliampere currents in beam intensity. This in turn can be challenging for the experimentalist, since this level of intensity can be destructive for the target and may change the target composition. The energy stability and resolution are enormously improved, which is of particular importance for lowenergy studies with rapidly declining cross section. A reliable conversion of cross section into S-factor requires an exact knowledge of the beam energy at which the cross section has been determined. In either case, the accelerator and the beamoptical system produces a mono-energetic, intense beam, which is focused on the target material, which in turn is surrounded my radiation detectors to measure the intensity of the radiation products of the nuclear reaction between beam and target nuclei as discussed in section “Direct Reaction Studies with Charged Particles.” A limiting factor in the analysis of reaction data at low cross sections is the natural background in the detectors. This background is cosmogenic, produced by cosmic ray showers in the atmosphere and on the experimental environment. The cosmogenic radiation generate a background distributed over a large energy range between a 2 and 20 MeV. The second background component is radiogenic radiation, which is produced by the natural decay chains of long-lived actinides (238 U and 232 Th) as well as the decay of 40 K in the local environment. This background handicaps the data in the energy range of 0.3 and 3.0 MeV. In addition to these two natural background contributions, there is always an additional background component from beam-induced nuclear reactions with higher cross sections on chemical impurities in the target material. Limiting this background component depends on the chemical or physical target preparation techniques and the purity of the target backing material. These background components need to be removed, and for this purpose a number of complementary technologies have been developed. At accelerator experiments at the surface, the background is primarily reduced by so-called active shielding, relying on coincidence or timing techniques between reaction signals to identify and remove the background signals and expand the measurements toward lower energies. The background is then reduced to the level of random coincidences, which typically depends on the specific reaction under investigation. The coincidence requirement also reduces the actual event rate,

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setting another limit for the experiments. The passive reduction of background is by absorption through local shielding material surrounding the target detector assemble. This technique will not reduce the beam-induced background in the detector. A combination of passive and active shielding techniques is achieved in a deep underground environment.

Underground Accelerators A major breakthrough in pushing experiments into the astrophysically relevant energy range is the development of accelerators located in deep underground laboratories to reduce the cosmogenic background. This background component is mostly absorbed in the rock shielding, with the remaining level of cosmogenic background depending on the depth of the location. Depending on the rock composition, the low-energy radiogenic background may be increased and must be achieved by local shielding or again or by active suppression with the aforementioned coincidence techniques. The major challenge in underground experiments are the rapidly declining event rate toward lower energies and the beam-induced background generated by proton or alpha particle-induced nuclear reactions on target impurities such as boron, carbon, and fluorine. These reactions generate a high-energy background in neutron and gamma radiation depending on the level of material contamination. Even spurious amounts of the contaminating material can generate an appreciable background level, preventing measurements at very low energies. Despite this handicap, accelerators in an underground location are frequently far superior in the direct study of low-energy reaction processes compared to accelerators at above ground locations. But the issue remains that in most cases the observed cross sections will have to be extrapolated into the stellar energy range. Inverse Kinematic Experiments with Heavy Ion Beams An alternative technique is the use of heavy ion beams focused on a target material of light hydrogen or helium elements. This is the inverse kinematic technique, since the heavy reaction recoil particles move by momentum transfer within a small cone in forward direction. In light beam experiments, the reaction products (light particles or gammas) are collected over a small angular range defined by the experimental setup. This limits the geometrical efficiency in measuring the reaction products since only a small angle range is covered by the detector. In inverse kinematics experiments, most heavy recoil products are moving in forward direction in a narrow cone, defined by the kinematics. The recoils can be selected and counted with high efficiency, depending on the acceptance angle of the detector. The challenge in this arrangement is that the recoils have to be separated from the primary beam particles. In most cases the heavy recoil particles have different masses and different velocities in comparison to the initial particles; hence, the separation can be achieved by a combination of magnetic and electric fields. In single-pass experiments, a rejection ratio of the primary beam of more than fifteen orders of magnitude can be achieved. At a lesser rejection factor, the primary beam would remain as background in the final detection device. The use of gas-filled ion chambers, however, helps to

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distinguish between the primary beam and recoil particles by analyzing the different energy loss characteristics (Couder et al. 2008). A variant of the inverse kinematics approach is the use of ion storage rings (Litvinov et al. 2013; Lestinsky et al. 2016). The (heavy) ions will pass a thin gas target in the order of a million times per second. The energy loss in the target can be compensated after each passing using electron coolers. This ensures high luminosity at virtually constant beam energy. An additional advantage of using rings is that the charge state of the revolving ions is fixed, while the ions in single-pass experiments typically show a rather broad charge state distribution. The major challenge is the short beam live time at low energies. So far, the lowest energies reached are 5.5 MeV in the center of mass, which is the realm of the γ -process (Glorius et al. 2019).

Accelerators for Surrogate Reaction Studies Surrogate reaction is a generic label, which refers to a technique to directly probe the quantum mechanical reaction mechanism at high energies without being handicapped by the Coulomb barrier (Escher et al. 2012). Instead of measuring directly the nuclear reaction process in question at the desired low energies, this process is replaced by a transfer reaction mimicking the original initial and final quantum system. This is a model-dependent approach since it requires the translation of the observed reaction data into a low-energy reaction cross section, necessitating a thorough understanding of the Coulomb or other hindrance processes for deriving the exact reaction rate. There are no specifically designed accelerators for this approach, which is typically done at higher-energy tandem or cyclotron accelerators. The big advantage in this approach is that features can be studied in the excitation energy range near the threshold that remain inaccessible to the direct probe due to the aforementioned reasons of extremely slow cross section and high background conditions. The study of surrogate reactions therefore is an extremely important complementary approach, which helps to identify important features at low energies that are otherwise inaccessible. An interesting new development is the use of ion storage rings to apply the surrogate technique in inverse kinematics (Jurado et al. 2021).

Plasma Fusion Experiments Another important approach is the use of laser plasma fusion techniques to study nuclear reactions. In laser plasma fusion experiments, a large number of high-power laser beams are focused on a small gas-filled capsule. The capsule is compressed by radiation pressure reaching density and pressure conditions similar to the stellar environment for fractions of a second. During this short period of time, fusion reactions between components of the gas filling can occur and can be collected and subsequently analyzed. The environmental conditions in the plasma are monitored by the analysis of the emitted X-ray and neutron radiation during the compression or shot period. The advantage of such experiments is that the fusion actually takes place in a stellar like environment, which allows for the study of electron screening

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effects, the reduction of the Coulomb barrier between charged particles by the free electron cloud in a plasma, or also the contribution of thermally excited states to the fusion process. In accelerator-based experiments, these two factors are theory-based corrections in the reaction rate formalism. The big disadvantage of laser-confined plasma experiments is the short shot period of less than a nanosecond and the associated challenges in the efficiency in collecting and analyzing the products of the fusion reactions (Cerjan et al. 2018).

Radioactive Beam Facilities Radioactive facilities emerged in the 1990s (Boyd et al. 1983; Decrock et al. 1991) to study nuclear reactions on radioactive species, which can be produced in an explosive stellar environment of high temperature. Since the charged particleinduced reaction rates increase exponentially as a function of temperature, at sufficiently high temperatures in a stellar environment, these reactions become typically faster than the beta- or electron capture decay of the radioactive species. This depends on the reaction cross sections, which therefore are an important factor in the analysis of the onset of stellar explosions such as novae or X-ray bursts, or for the nucleosynthesis pattern in the expanding supernovae shock front. Only in a few cases of long-lived isotopes, such as 22 Na, 26 Al, 44 Ti, 60 Fe, and 151 Sm, can the measurement of these reactions be done with radioactive target material. This, however, has the great disadvantage that the radiation emitted by the target itself is a major background source in the detectors. For this reason two techniques for the production of radioactive heavy ion beams for inverse kinematic studies have been developed. These techniques distinguish themselves by the production, separation, and analysis of the radioactive beam particles. The ISOL technique is based on the use of spallation reactions triggered by high-energy protons on a lead or uranium target, while fragmentation facilities rely on the production of radioactive species by the fragmentation and separation of heavy ion beams impinging on light target material.

ISOL Beams The ISOL, or isotope separation online technique, relies on the production of radioactive species through the bombardment of a heavy mass material with high energetic light particles, causing spallation of the heavy material in multiple stable and radioactive spallation products. These products have to be collected by slow diffusion and subsequent chemical and physical separation techniques. Traditionally, the goal of these ISOL facilities was the study of the decay pattern of the radioactive products. With the onset of radioactive beam studies, the analyzed reaction products were collected, ionized, and reaccelerated for secondary reaction studies using the inverse kinematics technique, analyzing and counting the reaction products in a recoil separator. The technique has proved to be successful but is handicapped by the selectivity in the beam production process, since the diffusion

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time of the radioactive material can be so long that a large fraction of the ions decay before reaching the ion source for reacceleration. High chemical activity of some of the radioactive species also prevents, in some cases, the separation from the primary target material. A further challenge is the ionization of the radioactive species, which requires an ion source design optimized for the ionization process. The radioactive beams need to be reaccelerated to the desired energy range before hitting the secondary light ion helium or hydrogen gas target. The reaction products need to be collected and analyzed in terms of charge and mass distribution by a suitably large acceptance recoil separator system (Hutcheon et al. 2003).

Fragmentation Techniques Because of the challenges in producing an intense radioactive beam of arbitrary nature using the ISOL techniques, great effort was invested in the production of radioactive beams using the fragmentation technique. Here the heavy nuclei in the target are not being disintegrated by spallation, but instead the heavy beam particles at high velocity are being fragmented into multiple lower mass species when hitting lighter target nuclei. The fragmentation products need to be separated and analyzed to select the desired component for the planned experiment. This is done by charge, mass, and velocity selection through the electrical and magnetic field arrangement in the fragment separator, as well as by energy loss and stopping of ions in the so called wedges of the separator, taking advantage of the charge dependence of the energy loss processes. Because of momentum transfer, the fragmentation products move on with high velocity and need to be decelerated toward the desired low energy needed for nuclear astrophysics-related experiments. This can be done by natural energy loss in material or by de-acceleration in a reverse cyclotron device. The production of each beam species remains a major challenge and technical effort. So far only a very limited number of studies of real relevance for the field have been successful. The inverse kinematics technique needs to be utilized for direct nuclear astrophysics-related reaction studies; however, because of the production process for the radioactive species, the beam is of lesser quality in terms of emission and energy resolution than in the case of stable beam inverse kinematics studies. This, therefore, requires high-quality recoil separator techniques for the analysis of the reaction products. The radioactive beam particles must be collected and focused on the secondary light hydrogen or helium gas target system with the reaction products exiting with a wider angle and energy distribution. The recoil separator must be optimized for efficient collection and counting of the reaction yield (Tsintari et al. 2022).

Gamma-Induced Reaction Studies Photon- or gamma-induced reaction studies play an increasingly important role in the field of experimental nuclear astrophysics. For one, they allow the direct measurement of photodissociation processes such as that expected for neon and

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silicon burning during the last phase of stellar evolution and also direct studies of reactions in the p-process as pioneered by Zilges and Mohr (2000). In many cases, the method is used to study radiative proton (p, γ ) or alpha (α, γ ) capture by the time reversal (γ , p) and (γ , α) photonuclear reactions, respectively. This is often considered advantageous because, through the detailed balance theorem, the photonuclear cross section σγ ,α =

kα2 · σα,γ 2kγ2

(11)

is considerably higher than the capture cross section √ because of the ratio of the wave numbers for α particles at low energy, hk ¯ α = 2μEcm , and high-energy photons, hk = Eγ /c. The reaction products are produced and detected and energy ¯ γ = hω/c ¯ analyzed in a so-called active hydrogen or helium target device, where the reaction takes place and the heavy ion reaction products are identified on the basis of their energy loss in the target gas (Carnelli et al. 2014). The production of high-energy photons traditionally relies on the use of high-energy electron accelerators or Free Electron Lasers for the production of monoenergetic photon beams by Compton backscattering techniques (Scott Carman et al. 1996).

Neutron Sources The study of neutron-induced nucleosynthesis processes for the s- as well as the r-process requires the production of high intensity neutron beams at the energy E = kT corresponding to the temperature of the specific stellar environment. For s-wave neutrons, without the hindrance of the orbital momentum barrier, the cross section for radiative neutron capture follows the 1/v law. In this case, the neutron-capture cross section can be determined at any energy and extrapolated toward the stellar energy range. In other cases, however, the neutron-capture cross section is determined by a multitude of resonances whose contributions need to be determined. This requires the study of the neutron-capture cross section over a wider energy range corresponding to the temperature conditions of the specific stellar environment. In the following sections, the present techniques for the production of neutrons for neutron-capture studies at the laboratory will be summarized. This however does not cover techniques necessary for the investigation of neutroncapture reactions on short-lived isotopes, which are important for mapping the reaction trajectory and determining the final abundance distribution of i- and rprocess environments. These studies typically rely on theoretical modeling of the reaction cross sections using statistical model assumptions to determine the level density and strength distribution in the γ decay of the respective compound nucleus. New efforts concentrate on surrogate measurements probing the γ and also neutron strength distribution of nuclei along the projected reaction path. These attempts still depend, to a considerable extent, on nuclear model parameters. For that reason,

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new efforts are underway in developing concepts for neutron targets in storage ring experiments for a direct study of neutron capture on radioactive isotopes.

Reaction-Induced Sources To study nuclear reactions with higher-energy neutrons, a number of different techniques have been developed. One very successful technique relies on the use of low-energy reactions, producing neutrons within a limited energy range given by the energetics and kinematics conditions of the specific study. While exothermic reactions such as 2 H(d, n) and 2 H(t, n) produce neutrons at energies that are too high for a direct reaction study of neutron-induced reactions at astrophysical conditions, the endothermic 3 H(p, n) and 7 Li(p, n) reactions emerged as a powerful neutron sources, providing a neutron flux of up to 109 neutrons/cm2 /s in the 5 to 100 keV range. The initial goal was to study the impact of epithermal and fast fission neutrons on reactor materials (Müller et al. 1974), but the techniques were quickly adopted for the direct measurement of neutron-capture reactions at energies in the keV range. Meanwhile, the 7 Li(p, n) reaction has been established as a working horse for activation and time-of-flight (TOF) applications (Reifarth et al. 2014). Neutron Spallation Sources Neutron spallation sources rely on high-power proton beams in the GeV range impinging onto heavy target material such as mercury, tungsten, lead, and up to uranium. The incident protons disintegrate the nucleus through a range of inelastic nuclear reactions causing the emission of protons, neutrons, α-particles, and other isotope fragments, the so-called spallation process. While the charged particles are typically stopped in the target block, the neutrons produced in such a reaction diffuse out of the material with a wide energy distribution, due to multiple scattering events. The neutron flux produced in this kind of reaction depends on the type of target and the energy of the incident particles. The extracted neutrons can be used in a wide range of applications including the bombardment of secondary targets to obtain neutron-capture cross sections. For this purpose, the flight time of the neutrons is measured over a long flight path, ranging from a few meters to a few hundred meters. The choice of the flight path length depends on the desired goal of the experiment – high neutron flux or high neutron energy resolution. The reaction products of the neutron capture are often measured using 4π γ BaF2 detector arrays, which are less sensitive to neutron exposure. While there exist multiple neutron spallation sources for a wide range of applications from condensed matter to biophysics, the two premier facilities for the field of nuclear astrophysics are the n_Tof facility at CERN (Chiaveri et al. 2014) and LANSCE at Los Alamos National Laboratory (Nowicki et al. 2017). Photoneutron Sources Photoneutron sources for nuclear astrophysics applications are typically electron LINACs producing an intense electron beam that is impinged on a heavy target material like tantalum or tungsten to generate an intense bremsstrahlung flux, which in turn causes neutron emission by photon-induced (γ , n) reactions occurring in

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the target. Neutrons are produced in electron accelerators when the electron energy exceeds the threshold for photoneutron production (1.7 MeV for Be, 2.2 MeV for deuterium, and 6–10 MeV for most other nuclei). Electron accelerators include circular accelerators such as the betatron and synchrotron and also linear accelerators. The primary radiation hazard is X-rays produced by accelerated electrons striking the target or other material. However, these accelerators can produce large numbers of neutrons from a target due to (γ , n) reactions occurring in the target. Neutrons are produced when the electron energy exceeds the threshold for photoneutron production (1.7 MeV for Be, 2.2 MeV for deuterium, and 6–10 MeV for most other nuclei). For nuclear astrophysics purposes, higher energies are desired to achieve a high neutron flux. The most successful photoneutron source has been the ORELA facility in Oak Ridge (Guber et al. 2007), which was pioneering in the study of nuclear reactions for the s-process (Macklin and Allen 1971).

Nuclear Reactors Nuclear reactors are an obvious neutron source. The neutrons are primarily produced as fast neutrons in the fission process and to a far lesser extent by the β-delayed neutron decay of the fission products. Depending on the reactor design, a high neutron flux of up to 1015 neutrons/cm2 /s can be generated near the reactor core. Since most of the fission neutrons are thermalized by multiple scattering in the reactor fuel elements and water-cooling environment, only integral values for thermal and epithermal reaction cross sections can be extracted applying the cadmium difference method. Nevertheless, such measurements provide important benchmarks for neutron-capture studies in particular if assumptions about the energy dependence of the cross section can be made – usually the 1/v law. Using neutron guides, a fraction of this flux can be extracted for external neutroncapture and scattering studies. This technique is, however, limited to cross section measurements at or below thermal neutron energies, since neutrons with higher energies cannot be guided. Neutron Traps A major challenge for the understanding of neutron-induced nucleosynthesis processes is the study of neutron-induced reactions on radioactive isotopes. This is important for mapping the trajectory for the i-process in massive early stars (Roederer et al. 2016) as well as the n-process and r-process in explosive stellar environments from supernovae to merging neutron stars (Cowan et al. 2021). The presently used rates on radioactive isotopes are based primarily on Hauser Feshbach statistical model calculations with various assumptions about level densities and γ strength functions. Direct measurements have not been possible except by activation techniques on long-lived targets (Uberseder et al. 2009) or via TOF on “emissionless” isotopes (Abbondanno et al. 2004; Weigand et al. 2015). A new development is the idea of coupling a charged particle storage ring with neutron traps. The neutrons will be produced inside a reactor or by spallation reactions and thermalized

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Fig. 2 Schematics of a possible realization of a neutron target setup, which allows the investigation of neutron-induced reaction in inverse kinematics. The neutrons will be produced in a spallation process and trapped in moderating material like natural carbon. The ions under investigation or stored in a storage ring penetrating the moderator. The reaction detection occurs with particle detectors after the first dipole possibly using electric components to separate the reaction products from the unreacted beam (Reifarth et al. 2017)

via scattering in a moderator (heavy water or natural carbon). The moderated neutrons penetrate the heavy ion storage ring and serve as a neutron target for inverse kinematic neutron-induced reactions. The heavy ion recoils move forward due to momentum conservation and then are separated by the magnetic and electric field components of the ring design and monitored by particle detection methods (Reifarth and Litvinov 2014; Reifarth et al. 2017), Fig. 2. All important neutron-induced reaction channels can be investigated – capture, charged particle production and fission. While this is at present only a conceptual design, several laboratories have started to work on more detailed design plans with the goal to directly measure neutron-capture reactions on short-lived heavy ions in inverse kinematics with storage ring experiments.

Direct Reaction Studies with Charged Particles The determination of the reaction cross sections in the energy range of stellar burning is the key ingredient for the determination of the stellar reaction rates. However, the direct measurements of cross sections face serious handicaps. For

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charged particle reactions, the cross section is extremely low in the energy range of quiescent stellar burning, and a direct measurement is handicapped by a broad range of cosmogenic, radiogenic, and beam-induced background sources. For the study of reactions on radioactive beams in inverse kinematics, the beam intensity is often too limited to provide reliable cross section data for the desired energy range of stellar explosions. The study of neutron-induced reactions on radioactive particles is handicapped by the experimental challenge that both projectile and target are short-lived. In any of these cases, a combination of direct reaction studies and the application of surrogate techniques has been applied to address the challenges. In the following, we will discuss these in more detail.

Reactions with Stable Particles The study of low-energy nuclear reactions between light nuclei is important for the understanding of quiescent burning phases dictating the timescale, energy production, and nucleosynthesis patterns associated with the different phases of stellar evolution. For the understanding of hydrogen burning, proton-induced reactions such as (p, γ ) radiative capture and (p, α) nuclear reactions need to be studied at low energies. The Coulomb barrier typically limits the direct studies at energies corresponding to the stellar energy range, and the experimental data need to be extrapolated. For helium burning, typically α-induced capture reactions determine the energy production and nucleosynthesis, while (α, n) reactions serve as neutron sources for neutron-induced reaction patterns such as the s-process for the production of heavier elements beyond iron. These studies also aim at the understanding of proton and α-induced processes on higher mass isotopes in stellar neon and silicon burning. There is considerable interest in the measurement of the 12 C and/or 16 O fusion reactions that determine the carbon and oxygen burning phase of stellar evolution.

Direct Reaction Studies The traditional setup for light ion reaction studies is quite simple, a monoenergetic high-energy beam impinges the target of well-known stoichiometric composition to determine the energy loss of the projectile or the stopping power ε(E) = −N −1 · dE dx with N being the number density of target nuclei and dE the energy loss of the dx incoming particle per distance in the target. The yield in reaction products Y depends directly on the number of target nuclei and the cross section σ (E), integrated over the target thickness Δx  Y = 0

Δx



ΔE

σ (x) · N · dx = 0

σ (E) dE, ε(E)

(12)

with ΔE being the energy loss of the incoming projectile in the target. For thin targets, in which both cross section and stopping power have a negligible energy dependence, the equation simplifies to

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Ythin =

σ ΔE. ε

(13)

For a pronounced energy dependence of either stopping power or cross section, this approximation is not valid. For a smooth energy dependence, numerical integration over a fit of the two parameters is sufficient, but not in cases with rapid changes of the cross section over energy as one can observe in cases of single resonances. In the case of a single resonance with a peak of the cross section σ0 at the energy E0 , one has to integrate over the Breit-Wigner function with a Lorentzian energy shape dependence as described in Eq. (6). Unless the resonance is broad with a large width Γ , the stopping power ε(E) is approximately constant and the reaction yield will be λ2 ω Y (E0 ) = 4π



E0 E0 −ΔE

Γin · Γex 1 dE. ε(E) (E0 − E)2 + Γ 2 /4

(14)

The energy dependence of the reaction yield for a narrow resonance can then be described by the simplified expression for a “thick-target yield” (Fowler et al. 1948) Y (E0 ) =

     E0 − Er E0 − Er − ΔE λ2 ωγ arctan arctan , 2π εr Γ /2 Γ /2

(15)

with εr being the stopping power at the resonance energy Er and ωγ the resonance strength. For a narrow resonance, with ΔE ≥ 5 · Γ , the so-called thick-target yield Yres , measured on top of the resonance curve is directly proportional to the resonance strength: Yres =

λ2 · ωγ . 4π ε

(16)

A measurement of the yield on top of the thick-target yield curve therefore provides direct information about the resonance strength, which determines its contribution to the reaction rate. Determining a reaction yield Yres requires a detailed setup for the detector to measure either the characteristic γ or particle radiation Ydet produced in the reaction. This detector must be placed at a geometrically well-defined position and angle with respect to the target and beam axis, respectively. The efficiency of the detector needs to be determined by the use of either well-known, calibrated, radiation sources or nuclear reactions with well-known radiation emission characteristics. The measured reaction yield depends on the solid angle of the detector Ωdet , the efficiency of the detector material for detecting the radiation η, and also the angular distribution of the emitted radiation W (θ ), which is determined by the coupling of the particle spins and the orbital momenta of the interacting particles. Taking all this into account, the detection yield as a function of beam energy can be described by

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Yres =

Ydet · W (θ ). η · Ωdet

(17)

These procedures provide a direct extraction of nonresonant cross section terms by adopting the thin target yield formalism or a determination of the resonance strength based on the thick-target approach.

Inverse Reaction Studies (Gamma Measurements, Recoil Separators) Instead of bombarding a heavy ion target material with proton or alpha particles to study radiative capture processes, the inverse kinematics technique is based on bombarding a helium or hydrogen target gas with a heavy ion beam. For stable beam experiments, the advantage is that the total reaction yield can be measured by separating the heavy ion recoils from the primary heavy ion beam particles and counting them at a considerably higher efficiency than compared to the detection of the γ radiation or other light reaction products. A rejection ratio for the primary beam particles of at least 10−15 needs to be achieved to reduce the background from inelastically scattered primary beam particles. Using inverse kinematics, the heavy reaction recoil products will proceed in a small ejection cone from the target material. Due to charge exchange processes in the target material, they are ejected with a certain charge distribution, which depends on the energy of the primary particles. The beam needs to be refocused using quadrupole elements, before entering a magnetic field for charge separation and selection of a single charge state. A subsequent quadrupole dipole magnet system serves as a mass filter for this particular charge state since the mass of the recoil is higher than the mass of the primary beam particles. However, this selection is not sufficient to reach the desired rejection ratio since multiple scattered primary particles may still be present in the transmitted particle assembly. To improve the separation, a combination of magnetic and electric fields perpendicular to each other serve as a velocity filter, since the recoil particles have much lower kinetic energies than the primary beam particles. A subsequent sequence of additional mass and velocity filters will help in further reduction of multiple scattered beam particles from the recoils. The last step is the detection and identification of the particles by a detector array using a combination of time-of flight, energy loss, and total energy measurements with a sequence of ion gas or silicon detectors. The observed yield of the recoil particles Y (E) for the reaction process depends on a number of instrumental features of the recoil separator system. This includes the charge state fraction of the selected recoil particles fq , the transmission efficiency of the separator τsep , the efficiency of the focal plane detector arrangement ηdet , and finally the angular and energy acceptance of the separator system Ω(E) for the recoil particles emitting the target Nrecoil (E) Y (E) =

Nrecoil (E) . fq · τsep · ηdet · Ω(E)

(18)

A critical aspect of recoil separators is the acceptance angle Ω(E) for the recoil particles, which depends on the Q-value of the reaction and the energy and

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momentum spread of the emitted particles. If the acceptance angle is too small, only a fraction of the recoil particles will be able to transmit to the detector array. This means that the acceptance angle determines the counting efficiency of the device. This translated into a difference in the design for recoil separators that is primarily defined by the Q-values of the radiative capture reactions in question. Capture reactions of light particle on stable isotopes along the line of stability typically have higher Q-values than reactions on short-lived isotopes off stability, which have less binding energy. Kinematically, the recoil products from inverse kinematics experiments with heavy stable beam particles are typically emitted from the light target volume with a large emittance angle defined by the Q-value. This requires a large acceptance angle for the emitted recoils. Recoil separators designed for stable beam inverse kinematics studies include ERNA at CIRCE at the University of Caserta (Buompane et al. 2022) and St. GEORGE at the University of Notre Dame (Couder et al. 2008). The inverse kinematics method can be used for stable beam experiments; however, it is the only tool for directly measuring proton or alpha capture reactions on short-lived isotopes. Recoils from inverse kinematic experiments with heavy radioactive isotope beams are emitted with a small emittance angle and usually require a smaller acceptance cone. However, this generalized statement depends on the quality of the radioactive particle beams. The direct measurement of radiative capture processes on short-lived radioactive nuclei has been the main motivation for the development of inverse kinematic techniques with recoil separators from Dragon at TRIUMF (Hutcheon et al. 2003), RMS at RIKEN in Japan (Kubo 2003) to SECAR at FRIB (Tsintari et al. 2022). A logical extension for recoil separators are storage ring experiments as have been developed at GSI-Darmstadt in Germany (Glorius et al. 2019). The heavy ion beam, either from the ion source or produced by secondary reactions, is accelerated and injected into a storage ring, which consists of a sequence of several magnetic dipoles and a number of magnetic quadrupoles and sextupoles to maintain an orbital trajectory for the beam. A light ion target is positioned in the trajectory for the reaction studies. The heavy ion recoil products are typically separated from the orbital trajectory of the primary beam and after the first dipole. They can then be detected in or extracted from the ring to more advanced detection systems. Presently these systems are limited to 1 MeV/nucleon energies for the primary particles, which is too high for low-energy reaction studies except for time reverse measurements of p-process photo-disintegration measurements. To study stellar reactions with stable beams at a storage ring will be a challenging task since the quality and lifetime of the beam in the ring will be increasingly difficult to maintain toward lower energies.

Time Reversal Studies (Photon Beam) The use of time reversal reactions is an important tool in nuclear astrophysics, which is based on the detailed balance theorem and takes advantage of the difference in Q-value between the direct and reverse reaction channel. It is of particular importance for the study of photon-induced reactions of the p-process, where a

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direct measurement of the reaction rate of a (γ , x) reaction such as (γ , α), (γ , p), or (γ , n) can be substituted by the measurement of the reverse (α, γ ), (p, γ ), or (n, γ ) radiative capture processes. The photon-induced rate λγ can be directly correlated to the time reverse radiative capture rate if the ground state spins of the participating projectile j1 , target j2 , and recoil particle j3 are known   (2j1 + 1)(2j2 + 1) 2 (2π μkT )3/2 · λγ = 3 · σ12 ν · e−Q/kt , (2j3 + 1) h (1 + δ12 )

(19)

with μ being the reduced mass of the system and Q the reaction Q-value. This approach is limited to the study of ground state to ground state transitions; to explore photon-induced reaction branches to higher excited states, photon facilities such as e-Dalinac in Darmstadt, Germany (Sonnabend et al. 2011), HIγ S at TUNL (Scott Carman et al. 1996) or at ELI in Romania (Tanaka et al. 2020) will need to be utilized.

Indirect or Surrogate Reaction Studies In many cases direct reaction studies are not possible. For low-energy charged particle reactions, this is primarily due to the low reaction cross section associated with the Coulomb barrier; for radioactive beam studies, the limited beam intensity associated with the beam production mechanism is the main handicap. For neutroninduced reactions with radioactive species, it is the simple fact that projectile as well as target nuclei are short-lived. The best case would be the aforementioned design of a thermalized neutron target gas with an intense radioactive beam, but direct cross section measurements are being handicapped by limitations in both the intensity of the beam as well as the limited number of target particles in the neutron trap. These limitations require the use of alternative or surrogate techniques, which aim at the determination of critical reaction rate parameters. The latter depends on the actual reaction rate formalism applied to the reaction system. For reactions with very light nuclei, the reaction rate is frequently determined by a direct, nonresonant, reaction component, which depends on the transition matrix element between initial and final configurations as well as the treatment of the potential parameters. Toward higher mass systems, this nonresonant component is interspersed with single, wellseparated resonances, which can be treated by the single resonance Breit-Wigner formalism. For single, well-separated resonances, it is sufficient to determine the spin and parity, as well as the partial widths of the resonance state. Yet these resonances can be broad, causing interference with other broad resonances and the direct reaction component, which requires the introduction of R-matrix techniques. In the case of high level densities in the reaction components, the Hauser-Feshbach formalism is typically applied, which is based on the assumptions about the level density parameters and about the spin parity distribution, as well as on the

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determination of transmission parameters for the reaction channels in question. In cases of single resonance states, not accessible to direct measurement, the resonance strength can be extracted from the independent study of the partial widths of the entrance channel Γin , the exit channel Γex , and the total width Γtot as well as the spin value of the resonances state js as defined in equation (8). The resonance strength for low-energy resonances in charged particle reactions is typically determined by the partial width of the entrance channel since Γin

Γex ≈ Γtot . This relation is not given for higher-energy resonances where the partial width for the exit channel may be comparable to the one of the entrance channel, or if several exit channels are open, such as a particle and a γ channel in cases of (p, γ ) versus (p, α) or also (α, γ ) versus (α, n) channels. The strength of particle channels can be indirectly determined by single particle or α cluster transfer reactions. The particle width for a particular orbital momentum  is typically described in the framework of the R-matrix model, as described in section “R-Matrix Phenomenology,” and corresponds to the reduced width of the h¯ 2 state γ2 = C 2 S · θ2 of the resonance state times the Wigner limit μr 2 . The c

term C 2 S · θ2 describes the spectroscopic factor of the state times the square of the single particle wave function at the interaction radius rc as defined in Iliadis (1997). It contains the nuclear structure information of the single particle or cluster configuration of the quantum level in question. The indirect methods used for determining the strengths of very low-energy resonances or the contributions of high-energy tails of subthreshold levels are typically aimed at measuring the spectroscopic factor C 2 S, corresponding to the single particle or cluster component of the corresponding level, via single particle or cluster transfer reactions. The spectroscopic factor can be determined from the comparison of the experimental cross section to the theoretical cross section predicted by distorted wave Born approximation (DWBA) resulting from perturbation theory. The uncertainty here is in the assumptions and choice of the associated nuclear optical potentials for the nucleus as well as in the assumptions made for the nuclear radii and Coulomb potential to calculate reliably the penetrability through the Coulomb potential. Because of the steep decline of penetrability with energy, small variations in the model assumptions can translate into large uncertainties in the predictions of resonance strengths and reaction rates. The so-called Trojan Horse Method (THM) (Tumino et al. 2021) uses similar techniques by replacing the measurement of a low-energy charged particle reaction with the study of a quasi-free cross section of a suitable three-body process. While this approach has been very successful in studying possible reaction features at very low energies by essentially removing the limiting factor of the Coulomb barrier, it is nevertheless handicapped by model-dependent factors in converting the THM cross section back to actual low-energy reaction data. This requires a complete and reliable knowledge of the Coulomb functions at very low energies as well as a full understanding of the associated three-body interaction mechanisms involved. The THM method does not result in a prediction of an absolute cross section but needs

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to be normalized to direct cross section data. To do this, both the THM data and the direct reaction data need to have reliable information in the overlapping energy range of the respective cross section studies. Uncertainties in this normalization procedure translate directly into uncertainties in the low-energy reaction rate. Nevertheless, the THM is a very powerful approach, if not the only approach to explore the nuclear structure near the threshold, but a comprehensive method needs to be identified to merge the THM results with the results of direct reaction measurements to come to a comprehensive understanding of the reaction mechanism and all its contributions to the low temperature component of the reaction rates.

Reactions with Neutron Beams The by far most likely interaction of neutrons in a stellar plasma is elastic scattering. Neutrons can therefore assumed to be thermalized before any nuclear reaction occurs. It is therefore useful to introduce the Maxwellian-averaged cross section (MACS), which follows directly from Equation (1): σ  = with the most probable velocity vT = can then be simply written as

σ v vT

(20)

√ kB T /μ. The total reaction rate (see Eq. 2)

r = NX Nn vT σ 

(21)

with the neutron density Nn . The temperature regime of astrophysical nucleosynthesis is approximately 5 < kT /keV< 300. The experimental determination of the required MACS can therefore be achieved with integral measurements, provided the neutron energy distribution is tailored to the right energy regime or applying the time-of-flight technique requiring a pulsed neutron source.

Integral Measurements The neutron capture on a nucleus can be expressed as A

X + n →A+1 X∗

(22)

where A X stands for the isotope with mass A of the element X. The star in the reaction product symbolizes the fact that the nucleus will be in an excited state after the fusion with the neutron. If it de-excites via γ -emission, A+1

X∗ →A+1 X + γ ,

(23)

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the neutron is captured. The detection of those promptly emitted γ -rays is the main idea of the time-of-flight (TOF) method described in section “Differential Cross Section Measurements.” If the freshly produced nucleus A+1 X is radioactive, it will decay following the exponential decay law. The particles emitted during the delayed decay, e.g., A+1

X →A+1 Y ∗ + β − →A+1 Y + γ ,

(24)

can be detected after the neutron irradiation. This approach is called the activation technique – it always consists of two distinctly different phases: irradiation of the sample and detection of the freshly produced nuclei. There are several huge advantages of the activation technique over the TOF method. First, the neutron flux at the sample is typically about five orders of magnitude higher, because the sample can be very close to the neutron source, and the neutron production does not need to be pulsed. Second, the detection setup can be in a low-background environment with very sensitive equipment. An additional advantage is the low demands on the sample purity. Usually sample material with natural composition can be used. Very often, more than one isotope can be investigated simultaneously with one sample. The last years witnessed enormous progress in the field of data acquisition. The combination of traditional detectors with state-of-the-art electronics allows the processing of much higher count rates. Samples with higher intrinsic decay rate can therefore be used. The disadvantage is that the neutron energies are not known anymore at the time of the activity measurement. Only spectrum-averaged cross sections (SACS) can be determined; therefore it is called integral measurement:  SACS =

σ (E)Φ(E)dE  Φ(E)dE

(25)

The number of atoms produced during the activation (Nactivation ) can be written as Nactivation = Nsample Φn σ ta ,

(26)

where Φn is the energy-integrated neutron flux (cm−2 s−1 ). If the activation time (ta ) is short compared to the half-life time (t1/2 ) of the radioactive neutroncapture product, the freshly produced activity (Aactivation ) increases linearly with the activation time: Aactivation ≈ λNactivation =

ln 2 Nsample Φn σ ta t1/2

(27)

Small cross sections σ or small samples Nsample can therefore partly be compensated with longer activation time or increased neutron flux. The amount of nuclei, which decays before the activity measurement can be accounted for Reifarth et al.

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(2018). Samples smaller than 1 μg could be investigated with neutrons in the keVregime with this very sensitive method. If the half-life of the product is very long, it might not be feasible to determine the activity of the capture product. In some of those cases, it is possible to count the number of produced atoms with accelerator mass spectroscopy (Nassar et al. 2005). If, however, the half-life is very short, it might be necessary to repeatedly irradiate and count the decays. This can be carried out as an automated cyclic activation (Reifarth et al. 2008).

Differential Cross Section Measurements Neutron-induced cross sections usually show a strong resonant structure, caused by the existence of excited levels in the compound nucleus. The excitation function for a reaction can accordingly be divided into three regions: the resonance region, where the experimental setup allows to identify individual resonances; the unresolved resonance region, where the average level spacing is still larger than the natural resonance widths; and the continuum region, where resonances start to overlap. The border between the first two regions is determined by the average level spacing and by the neutron energy resolution of the experiment. The time-of-flight (TOF) method enables cross section measurements as a function of neutron energy. Neutrons are produced quasi-simultaneously by a pulsed particle beam, thus allowing one to determine the neutron flight time t from the production target to the sample where the reaction takes place. For a flight path L, the neutron energy is En = mn c2 (γ − 1)

(28)

where mn is the neutron mass and c the speed of light. The relativistic correction −1

γ = 1 − (L/t)2 /c2 can usually be neglected in the neutron energy range of interest in nucleosynthesis studies and Eq. (28) reduces to 1 En = mn 2

 2 L . t

(29)

The TOF method requires that the neutrons are produced at well-defined times. This is achieved by irradiation of an appropriate neutron production target with a fastpulsed beam from particle accelerators. The TOF spectrum measured at a certain distance from the target is sketched in Fig. 3. The essential features are a sharp peak at t = L/c, the so-called γ -flash due to prompt photons produced by the impact of a particle pulse on the target, followed by a broad distribution of events when the neutrons arrive at the sample position, corresponding to the initial neutron energy spectrum.

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Fig. 3 Schematic time-of-flight spectrum. The sharp peak at t = L/c is caused by prompt photons produced by the impact of a particle pulse on the target. Neutrons reach the measurement station at later times and give rise to a broad distribution depending on their initial energies

Neutron TOF facilities are mainly characterized by two features, the energy resolution ΔEn and the flux φ. The neutron energy resolution is determined by the uncertainties of the flight path L and of the neutron flight time t: ΔEn Δt 2 ΔL2 =2 + En t2 L2

(30)

The neutron energy resolution can be improved by increasing the flight path but only at the expense of the neutron flux, which scales with 1/L2 . The ideal combination of energy resolution and neutron flux is, therefore, always an appropriate compromise. The energy resolution is affected by the Doppler broadening due to the thermal motion of the nuclei in the sample, by the pulse width of the particle beam used for neutron production, by the uncertainty of the flight path including the size of the production target, and by the time resolution of the detector system.

Experiments at Reaction-Driven Neutron Beams Reaction-driven neutron beams have the advantage that it is possible to shape the spectrum based on target material, beam energy, thickness of the target, as well as the distance to the neutron-irradiated sample. The most widely used reaction for this purpose is 7 Li(p, n), because it is easy to use, and the neutron yield is high, even close to the neutron production threshold. Figure 4 shows an example of a spectrum, used as a workhorse for s-process studies, since it closely resembles a spectrum corresponding to kT ≈ 25 keV. It is ideally suited for activation experiments, since only small corrections to the actual stellar environment are necessary (Reifarth et al. 2018). Other spectra can be produced too; however, more corrections need to be applied (Reifarth et al. 2008). The use of other target materials with endothermic reactions offers the possibility of Maxwellian-like spectra corresponding to other temperatures, e.g., 3 H(p, n) with kT ≈ 52 keV (Käppeler et al. 1987) or 18 O(p, n) with kT ≈ 5 keV (Heil et al. 2005).

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Fig. 4 The 7 Li(p, n) reaction can be used to emulate a neutron spectrum corresponding to an activation in a neutron gas corresponding to kT ≈25 keV (Reifarth et al. 2009)

Because of its high neutron yield, the 7 Li(p, n) is also often used for time-offlight measurements. This requires a pulsed proton beam (Reifarth et al. 2002).

Experiments at White Neutron Sources White neutron sources are characterized by a neutron energy distribution covering several orders of magnitude – typically starting in the thermal regime (25 MeV) all the way up to hundreds of MeV. The number of neutrons per energy decade is roughly constant, which makes such sources unsuited for integral measurements. However, spallation and photoneutron sources, as the most prominent examples, can be pulsed and yield the highest neutron fluxes. This enables the investigation of neutron-induced cross sections with the TOF method even on radioactive isotopes (Esch et al. 2008; Damone et al. 2018).

Studies with Reactor Neutrons Reactor spectra cover a wide range of neutron energies; see Fig. 5. Imposing fast time structures on reactors via pulsing or chopping techniques cannot be done fast enough to enable the TOF technique in the astrophysically interesting energy regime. Experiments with astrophysical impact are therefore typically restricted to integral methods. In addition to the three unavoidable spectrum components (Fig. 5), sometimes cold or hot moderators are installed, which adds additional thermal components. The different components of the reactor spectrum can be separated using neutron

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Fig. 5 The typical reactor spectrum can be assumed to be a sum of three components: the fast component originating from the fission process, the epithermal component resulting from the moderation process, and finally the thermal component since most moderated neutrons will spend some time in the moderator of the reactor. Depending on the position and the particular reactor type, the ratio of the different components differs

guides, which only transport neutrons below a given cut-off energy or the cadmium difference method, since cadmium absorbs all neutrons below a given neutron energy. The integral measurements are then mostly used to constrain energydependent predictions. Reactor experiments often support the nuclear astrophysics activities indirectly. Because of the enormous neutron flux available, reactors can be used to produce radioactive sample material. Another application are relative measurements taking advantage of the fact that thermal neutron-induced reaction cross section are often well-known. The experiment is then carried out twice – once with keV neutrons and once with reactor neutrons (Reifarth et al. 2000). The study of γ -cascades following capture reactions provides valuable information about the nuclear structure of the product nuclei (Jentschel et al. 2017).

Extrapolation Theory Stars are cold, and the nuclear reactions that maintain stellar stability are characterized by nuclear processes at extremely low, near-threshold energies. Since, the energy production of the various stellar evolution phases depends critically on the charged particle interaction processes and therefore on the Coulomb barrier and the penetrability determining the reaction cross section. There have been a very

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limited number of reactions that have actually been measured in the Gamow range and most of the data need to be extrapolated from the experimental data range to the stellar energy range. This requires a comprehensive understanding of the underlying nuclear structure and nuclear reaction physics near the threshold. To amplify these specific nuclear features in the reaction cross section, the astrophysical S-factor has been introduced to remove the impact and uncertainty of the Coulomb barrier from the data by adopting an exponential function for approximating the Coulomb barrier. The remaining S-factor curve should reflect the nuclear structure and nuclear reaction features for a more reliable extrapolation process.

Extrapolation Techniques As described in section “Experimental Goals and Concepts,” the astrophysical S-factor is a convenient transformation that helps to implement low-energy extrapolations of the cross section by approximately removing the  = 0 Coulomb penetrability (Eq. (3)) as given by Eq. (10). An example comparison between the cross section and S-factor for the 12 C(α, γ )16 O reaction is shown in Fig. 6. Here the energy range of interest is ≈300 keV, but the lowest energy measurements are limited to ≈1000 keV. While the cross section spans nearly 13 orders of magnitude over this energy range, the S-factor range is reduced to only about 3 orders of magnitude. This highlights the nuclear components, that is, the portion

Fig. 6 Example of a cross section compared, (a), to its corresponding astrophysical S-factor, (b), for the 12 C(α, γ )16 O reaction. The experimental data is from Schürmann et al. (2005), and the R-matrix calculation is taken from deBoer et al. (2017). For charged particle-induced reactions, data must be measured at higher energies where the cross section, due to the Coulomb penetrability, is higher. Models are then required to extrapolated to the energy range of astrophysical interest, which is around 300 keV for the 12 C(α, γ )16 O reaction

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of the cross section that deviates from the Coulomb penetrability and thus needs to be well defined for an accurate low-energy extrapolation. That is, if only the Coulomb penetrability dominated at low energies, the S-factor would be a nearly constant function of energy. In the particular case of the 12 C(α, γ )16 O reaction, the S-factor increases sharply at low energies, indicating the presence of one or more subthreshold resonances. An important effect at very low energies is electron screening, where the atomic potential changes with the ionization state of the target material. Forward kinematics experiments usually rely on single ionized light particles interacting with atoms and therefore are exposed to full screening; inverse kinematic experiments often have heavy ion beams in a higher charge state interacting with light atoms, which translates into different screening conditions. Finally laser-driven plasma fusion is based on the interaction of completely ionized particle species, where the screening effect is due to an electron cloud surrounding positive ions in the plasma environment. While this is close to the situation in stellar burning, it operates on a much shorter timescale, introducing the question of whether a full plasma equilibrium situation has been achieved. These are corrections that have to be included to convert the extrapolated yield data to low-energy cross sections. Other threshold effects may be associated with the shape of the nuclear potentials and the impact on the capture and fusion process in the very low-energy regime. These considerations have led to new initiatives to replace the traditional linear or polynomial extrapolation methods with phenomenological techniques that seek to include possible threshold effects. The two extrapolation techniques are based on statistical models, such as the Hauser-Feshbach approach, which are typically applied to nuclear reactions involving medium mass nuclear systems that are characterized by high level densities. In the case of nuclear reactions involving nuclei with low level density or no levels in the energy range of interest, the R-matrix has emerged as the most appropriate approach for describing the reaction mechanism and extrapolating the observed cross section data toward the subCoulomb, low-energy, regime. In the following we will describe a short summary of these extrapolation techniques from the experimentalist’s point of view.

Hauser Feshbach Statistical Model The Hauser-Feshbach approach is being discussed in F.-K. Thielemann & T. Rauscher,  Chap. 94 “Nuclear Reactions in Evolving Stars (and Their Theoretical Prediction)”. The bulk of the nuclear reaction rates used in stellar modeling is based on this model. This is mostly true for reactions far off stability but frequently also for reactions of presumed lesser importance, where no experimental data are available. This also includes cases where the application of a statistical model is more than questionable given that the level density is low and a reliable statistical treatment is not justified (Coc et al. 2011). The application of the Hauser-Feshbach model is based on the assumption that a compound reaction between two nuclei j and i leads to the formation of a compound nucleus, which decays into particle o plus nucleus m assuming a high-level density ρ(E) for the associated target, compound and recoil nuclei. This mechanism is characterized by two probabilities for the entrance

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and the exit channel of the reaction; the transmission Tj (E, J, π ) for the entrance channel populating a compound nuclear state (E, J, π ) at excitation energy E, with spin J and parity π via all possible channel spin and angular momenta; and the transmission probability To (E, J, π ) for the decay into a particle or gamma channel. The cross section for such a reaction is described by σi (j, o) =

 Tj (ρ, E, J, π )To (ρ , E , J, π ) (1 + δij ) π . (2J + 1) Ttot (E, J, π ) kj2 (2Ii + 1)(2Ij + 1) J,π (31)

Equation (31) is usually referred to as the Hauser-Feshbach or statistical model formula, since it implies a purely statistical probability of the decay in the outgoing channel (see Gadioli et al. 1992). The role of the Hauser-Feshbach model in nuclear astrophysics is described in a different chapter. Here we concentrate only on the use of this model in extrapolating experimental data toward the low-energy regime. A reasonable agreement is found with experimental cross section determinations, provided that a high density of excited states (resonances) exists in the compound nucleus at the appropriate bombarding energy, but the accuracy of the model predictions depend sensitively on the quality of the transmission functions. The cross section or probability that a compound nucleus is being formed corresponds to the total transition probability T +  through the entrance barrier. This corresponds to the probability of populating a wide range of overlapping levels with all spin and parity values, corresponding to the populated excitation range in the Compound nucleus determined by the level density ρ(E). The total cross section for forming the compound nucleus is σi (j, o) =

(1 + δρ,i,j,π )  π (2J + 1)Tj (E, J ) 2 kj (2Ii + 1)(2Ij + 1) J,π

(32)

The populated excitation range in the Compound nucleus will decay either by gamma or particle emission (depending of Q values) in the exit channel populating another excitation rage associated with a different level density ρ (E) in the final nucleus configuration. The transition functions for the exit channel depend on the nature of the decay. For particle channels (neutron, alpha, proton), the strength function is split in a part describing the decay populating known final states at excitation energy E and spin, parity assignments of H π , and in a part for the decay populating an excited energy range in the final nucleus with a specific level density ρ (E ): Ti (ρ , E , J, π ) =

ω  ν=0

 Tiν E, J, π +

ε



ω J,π

Tiν (εν , J.π ) · ρiν (εν , J.π )dεν

(33)

The particle channel transmission factor is the sum over all transitions to all possible final state configurations, where the transition to the higher (unknown)

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excitation range is a statistical component, adopting a level density function ρiν , which is either based on experimental data or – if not available – on using the back-shifted Fermi-gas model, which in turn also depends on the deformation of the specific nucleus (Beard et al. 2014). For radiative capture reactions, the exit channel is the gamma decay of the populated excited states to the ground state of the compound nucleus, presumably by a γ cascade via multiple lower excited states, which again can be treated in a statistical manner. The total photon-transmission coefficient Tγ is primarily composed of the sum of electric (E1) dipole term with a smaller magnetic (M1) dipole component. Typically, Tγ (E1) is described phenomenologically in terms of the γ -strength function, which is described by a parametrized Lorentzian representation of the Giant Dipole Resonance (GDR) at the excitation energy EGDR with a width of ΓGDR relative to the cross section σGDR at the maximum of the resonance and an additive term taking into account a possible temperature T dependence of the resonance feature: TE1 (Eγ ) = 2π Eγ

σGDR ΓGDR 3π 2 h2 c2 ¯

·

Eγ Γ (Eγ ) 2 (Eγ2 − EGDR )2 + Eγ2 Γ (Eγ )2  0.7ΓGDR 4π 2 T 2 + 5 EGDR

(34)

This is a purely phenomenological formalism and a range of different model descriptions have been suggested in the literature (Beard et al. 2014). The Hauser-Feshbach modeling of nuclear reactions on isotopes with high level density therefore requires a careful study and determination of the collective model parameters of the associated nuclei, including deformation and level density spin distribution. A rich portfolio of direct and indirect measurements has been employed to study the collective parameters of the associated nuclei by using Coulomb excitation (Adrich et al. 2005) and surrogate reaction techniques (Escher et al. 2018) to map the contributions to the strength functions in equation (31). One of the techniques recently employed is the so-called OSLO method, which is in particular employed to determine neutron-capture reaction rates on radioactive nuclei, cot directly accessible to experiment (Larsen et al. 2019).

R-Matrix Phenomenology Phenomenological R-matrix has its origins in the early nuclear reaction theory of Kapur and Peierls (1938), which was later reformulated by Wigner and Eisenbud (1947), and then reviewed in detail in Lane et al. (1958). The framework is most readily applicable to compound nucleus type reactions, although it can, in general, describe any type of reaction. While this theory was developed originally in the 1930s, its general applicability and the lack of precise predictions from more fundamental theory have resulted in its continued and widespread to the present day. Despite its long-term use, the methodology continues to develop, incorporating

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more convenient parameterizations, methods for calculating cross section or spectra for new reaction types, and incorporating modern uncertainty analysis techniques. More recent reviews can be found in Descouvemont and Baye (2010) and deBoer et al. (2017), for example. While the original theory deals mostly with reactions of nuclei, the theory has been extended to other classes of reactions. For nuclear astrophysics, one important class is capture reactions. This is commonly implemented using perturbation theory and also includes an explicit direct capture mechanism model (Lane et al. 1958; Holt et al. 1978; Barker and Kajino 1991; Angulo and Descouvemont 2001) called external capture (EC). In conjunction with EC calculations, the implementation of asymptotic normalization coefficients (ANCs) in R-matrix has been described in detail in Mukhamedzhanov and Tribble (1999) and Mukhamedzhanov et al. (2001). ANCs are often determined from transfer reaction cross sections combined with a potential model calculation, providing a method for incorporating information from these types of studies into an R-matrix analysis. The main advantage of the ANC method is the reduced dependence of on the potential model parameters compared to the spectroscopic factor method. Yet some model uncertainty remains, and it is often somewhat challenging to quantify and remains the dominant source of uncertainty. As the general mathematical implementation of R-matrix theory is somewhat cumbersome, many codes exist to facilitate its use. These include, but are not limited to, the EDA code of Los Alamos National Laboratory (Dodder et al. 1972; Hale and Paris 2017), SAMMY from Oak Ridge National Laboratory (Larson 2008), and the AZURE2 code developed under the Joint Institute for Nuclear Astrophysics at the University of Notre Dame (Azuma et al. 2010). While the EDA and SAMMY codes are designed mainly for the analysis of neutron-induced reactions, AZURE2 is tailored for charged particle-induced reactions, in particular charged particle capture, and is one of the only open source and freely available codes. R-matrix theory parameterizes the cross section primarily in terms of nuclear levels, as it is most directly associated with the compound nucleus model. Each level has an associated energy (Eλ ) and reduced width amplitudes (γλc ) that correspond to each different possible population/decay channel. The reduced width amplitudes can then be related to the partial widths of unbound states or ANCs of particle bound states. Further, an alternative parameterization has been developed so that the energies and widths correspond directly to classical Breit-Wigner (or “observable”) parameters. Here λ is the level index and c is the channel index of Lane et al. (1958). The R-matrix is made up of elements: Rcc (E) =

 γλc γλc . Eλ − E

(35)

λ

Here the γλc are the values of overlap integrals of the internal and channel wave functions for a specific resonance and channel. Without specifying the underlying nuclear potential explicitly, these reduced width amplitudes cannot be calculated. Instead they, and the resonance energies, are treated as fit parameters and their values are determined by comparison with experimental data.

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The R-matrix itself has several useful features. It explicitly contains all of the unknown parameters in the phenomenological theory. Further, the transition matrix (Tcc ) can then be written in terms of the R-matrix, from which any experimental observable can be calculated, following general reaction theory methods, where the collision matrix is given by (Azuma et al. 2010) U =ρ 1/2 O −1 (1 − RL0 )−1 (1 − RL0 ∗ )I ρ −1/2 ,

(36)

and the bold font indicates matrices. Tcc = e2iωc δcc − Ucc ,

(37)

where ωc is the Coulomb phase shift. The main point to note is that the other matrices besides R (ρ, O, I , and L0 ) are explicitly calculable (Lane et al. 1958). Once the elements of the Tcc are obtained, any observable quantity can be calculated over the associated energy range. For example, the angle integrated cross section is given by σα,α =

2J + 1 π  J 2 |Tcc

| , 2 (2I + 1)(2I + 1) kα α α 1 2



(38)

J  ss

where the index α refers to a pair of particles, for example, 12 C+α or 15 N+p. Its important to note that the formal width of R-matrix theory 2 Γλc = 2Pc γλc

(39)

is not equal to the observed width (Brune et al. 2002) Γ˜λc =

1+



2Pc γ˜c2

c

2 γ˜λc

dSc ˜ dE (Eλ )

,

(40)

where dSc /dE is the derivative of the shift function (Lane et al. 1958) and Γ˜λc , γ˜λc , and E˜ λ correspond to “observable” R-matrix parameters. In practice, the denominator in Eq. (40) cannot be neglected and is in many cases some significant fraction of the formal width of Eq. (39). Using these types of approximations facilitates comparison with previous works and makes it so that level parameters and their uncertainties determined using other techniques, for example, distorted wave Born approximation (DWBA), can be used to further constrain an R-matrix fit. This is especially important in making extrapolations as will be discussed below. While the alternative R-matrix parameterization of Brune et al. (2002) has been widely adopted, other parameterizations have also been proposed (Ducru and Sobes 2022). Its flexibility in calculating the many different types of cross sections in a selfconsistent framework makes R-matrix theory a natural choice for data evaluation.

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Indeed, the EDA and SAMMY codes are used for the evaluation of many cross sections for the Evaluated Nuclear Data Library (ENDF) (Brown et al. 2018), which forms a data base used by the whole of nuclear physics and other fields that rely on nuclear cross sections. In addition to evaluations of neutron-induced reactions, which can often be measured directly over the energy regions that correspond to stellar burning temperatures, it is also used to evaluate and then extrapolate cross sections to low energies for charged particle reactions.

R-Matrix Uncertainties The estimation of uncertainties from an R-matrix evaluation can be challenging for a number of reasons. A complete uncertainty characterization requires a detailed knowledge of the uncertainties in the experimental data, the nuclear level structure, and the phenomenological model. It is often the case that at least some of these uncertainties are very challenging to quantify accurately. These issues are then amplified when extrapolations are made outside the energy range of the experimental data. From experiment to experiment, the details of reported experimental uncertainties differ greatly as there is no completely standardized method for uncertainty reporting. The more explicitly defined these uncertainties are, the better they can be propagated through the model, but, as discussed above, there can be many situations where the experimental conditions make quantify all of the uncertainties very challenging. Generally speaking, more modern experimental studies give more detailed uncertainty descriptions (certainly many exceptions exist), as it has become more in the modern experimental mindset that data will be used for repeated data evaluation long after its initial publication. Generally speaking, uncertainties are often divided into two broad categories, point-to-point and systematic. This is a product of the approximate statistical models that have been developed to implement these uncertainties in models like R-matrix. In particular, least square fitting procedures are the most widely used. The most common type of point-to-point uncertainty comes from statistics, but there may be other pseudo-random uncertainties that fall into this category. On the other extreme are systematic uncertainties that are common for all of the data in a data set. A common example is the target sample thickness. Of course almost all uncertainties fall somewhere in between these two extremes, and this is where approximations have to be made. On the model side, all phenomenological models have some types of inherent uncertainties that result from their very nature. For R-matrix, the two main model uncertainties are the choice of channel radius and background contributions. The channel radius defines the boundary between the internal and external regions of the model space. In the internal region, there is a short-range force that is negligible outside some radial distance (external region). In phenomenological R-matrix theory, this radius can in principle be any value, but in practice it is usually between

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3 and 8 fm. It is important to note that the channel radius does not correspond to a physical quantity like the nuclear radius, for example. It is simply a model parameter without physical meaning. Background contributions come from the fact that there are an infinite number of nuclear levels in a compound system; thus Eq. (35) contains a sum over levels (λ) that extends to infinity in the rigours theory. In practice this sum is heavily truncated to just a few, or even one! This truncation can still yield accurate results because each resonance is strongly peaked in energy, thus it only has a very strong contribution to the cross section over a limited energy range. In practice, experimental data also only cover some finite range in energy. Resonances are included individually in the calculation over the energy range of the data, but it is often the case that the tails of strong resonances, either just above the experimental energy range or the combined effects of far-away resonances, can have a significant influence on the cross section, especially in off-resonance regions. For close by, strong resonances, it may be necessary to include a background level close in energy above the experimental data. Very often a single background level can mimic the combined effects of several real levels. For the combined effect of many higher energy levels, the background level should be placed several MeV higher in energy than the experimental data range so that its tail contribution is approximately constant as a function of energy over the range of the data. In practice it can be very challenging to determine the J π of the background levels that are statistically significant. In addition, experimental resolution effects can often distort the shape of the experimental data in ways that are similar to the effect of higher-lying level contributions. Thus it is very important to properly account for these effects before adding background levels. The propagation of uncertainties through the R-matrix model has always been challenging. Part of this stems from the constraints inherent in the least squares minimization techniques that have very often been employed. This problem is being somewhat alleviated by the implementation of more general Bayesian uncertainty techniques. While Bayesian techniques certainly do not solve all of the issues of uncertainty quantification, they often make the implementation of many additional types of uncertainties easier in practice. This is often facilitated by the ability to place a prior on the probability of the R-matrix fit parameter. This may come from a previous R-matrix fit (see SAMMY (Larson 2008)) or may come from other types of measurements beyond those included directly in the R-matrix analysis. For example, the uncertainties on an ANC determined from a transfer measurement. Bayesian uncertainty analysis has become easier to implement in recent years because of its widespread use in the field of data science. For example, the code BRICK (Odell et al. 2022) was recently released for use with the AZURE2 R-matrix code, which acts as a communication routine between AZURE2 and an “off the shelf” Bayesian Markov Chain Monte Carlo (MCMC) sampler (Foreman-Mackey et al. 2013).

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R-Matrix Extrapolation As discussed in section “Stellar Reaction Rates,” the Coulomb penetrability (Eq. (3)) results in a sharp decrease in charged particle cross sections at low energies. The cross sections are very often so low that they cannot be measured directly in the laboratory, but they need to be determined for astrophysics calculations. The extrapolation to low energies depends greatly on the characterization of the level structure near the reaction threshold and an understanding of the significant reaction mechanisms that dominate the cross section. As the phenomenological model cannot predict which reaction mechanism will dominate, experimental data or more fundamental theory calculations are needed to guide the calculations. Understanding of the level properties near particle thresholds is gained by using reactions that can populate these states by means that bypass the small penetrability of the reaction of interest. This is done predominantly through transfer reactions, inelastic scattering, lifetime, and β-decay studies. These types of studies can often determine, or at least place limits on, the J π , energies, widths, or ANCs of these levels. One challenging aspect is that, especially for partial widths and ANCs, their determination can only be made by comparison to another phenomenological model, DWBA for example, which introduces model uncertainties that can be challenging to quantify. Nonetheless, a great deal of effort and success has been achieved in recent years in minimizing and characterizing these additional model uncertainties. The foremost example of improvement in this area is the replacement of spectroscopic factors by ANCs, where ANCs have less dependence on the potential model used for their determination. Further, improved experimental techniques, such as making subCoulomb transfer measurements (e.g., Brune et al. 1999; Avila et al. 2015) also decreases the model uncertainty on the ANC. ANCs play a key role in low-energy extrapolations because they characterize both the strength of subthreshold resonances and the direct component of radiative capture reactions (Mukhamedzhanov and Tribble 1999; Mukhamedzhanov et al. 2001). At higher energies, measurements of the reaction of interest can be made directly where the cross section is correspondingly larger. These measurements can often determine if resonant or direct reaction mechanisms dominate, usually by measurements of angular distributions. Most importantly, these measurements give the vital information of the interference pattern between resonances. Since contributions between different resonances of the same J π are indistinguishable, their individual contributions can either add or subtract from one another. In Rmatrix theory, the interference pattern is determined by the sign of the reduced width amplitudes of Eq. (35). Unfortunately, the only way to experimentally determine these interference signs is by direct measurement. This is often one of the largest sources of uncertainty in low-energy extrapolations. Figure 7 shows an example of an extrapolation of the S-factor at low energy, demonstrating how a lack of detailed knowledge of the resonance structure at low energies can lead to very different results. In practice, extrapolations with phenomenological R-matrix theory are most useful when broad resonances and direct capture components dominate the cross

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Fig. 7 An example of the importance of having a detailed knowledge of the nuclear level structure at low energies in the extrapolation of the 10 B(α, n)13 N reaction at low energies. The grey dashed line indicates an R-matrix extrapolation using the resonances reported in Van der Zwan and Geiger (1973); the blue solid line those subsequently reported in Liu et al. (2020). The (Liu et al. 2020) data strongly indicate the presence of a strong low-energy resonance that has not yet been fully characterized

section at low energies. For some reactions, this is not the case, as the cross section can instead be largely dominated by strong narrow resonances. In this case, the reaction rate can be calculated directly using Eq. (9). In cases where the cross section is a mixture of these components, the broad resonance and direct capture portions can be modeled with R-matrix where the rate from the R-matrix portion is determined by numerical integration of Eq. (1). The R-matrix and the narrow resonance components can then simply be summed to obtain the total reaction rate.

References U. Abbondanno, G. Aerts, F. Alvarez-Velarde, H. Alvarez-Pol, S. Andriamonje, J. Andrzejewski, G. Badurek, P. Baumann, F. Becvar, J. Benlliure, E. Berthoumieux, F. Calvino, D. Cano-Ott, R. Capote, P. Cennini, V. Chepel, E. Chiaveri, N. Colonna, G. Cortes, D. Cortina, A. Couture, J. Cox, S. Dababneh, M. Dahlfors, S. David, R. Dolfini, C. Domingo-Pardo, I. Duran, M. EmbidSegura, L. Ferrant, A. Ferrari, R. Ferreira-Marques, H. Frais-Koelbl, W. Furman, I. Goncalves, R. Gallino, E. Gonzalez-Romero, A. Goverdovski, F. Gramegna, E. Griesmayer, F. Gunsing, B. Haas, R. Haight, M. Heil, A. Herrera-Martinez, S. Isaev, E. Jericha, F. Käppeler, Y. Kadi, D. Karadimos, M. Kerveno, V. Ketlerov, P. Koehler, V. Konovalov, M. Krticka, C. Lamboudis, H. Leeb, A. Lindote, I. Lopes, M. Lozano, S. Lukic, J. Marganiec, S. Marrone, J. Martinez-Val, P. Mastinu, A. Mengoni, P.M. Milazzo, A. Molina-Coballes, C. Moreau, M. Mosconi, F. Neves, H. Oberhummer, S. O’Brien, J. Pancin, T. Papaevangelou, C. Paradela, Neutron capture cross section measurement of 151 Sm at the cern neutron time of flight facility (n-TOF). Phys. Rev. Lett. 93(16), 161103 (2004)

3530

M. Wiescher et al.

P. Adrich, A. Klimkiewicz, M. Fallot, K. Boretzky, T. Aumann, D. Cortina-Gil, U. Datta Pramanik, Th.W. Elze, H. Emling, H. Geissel, M. Hellström, K.L. Jones, J.V. Kratz, R. Kulessa, Y. Leifels, C. Nociforo, R. Palit, H. Simon, G. Surówka, K. Sümmerer, W. Walu´s, Evidence for pygmy and giant dipole resonances in 130 Sn and 132 Sn. Phys. Rev. Lett. 95, 132501 (2005) C. Angulo, P. Descouvemont, The 14 N(p, γ )15 O low-energy S-factor. Nucl. Phys. A 690(4), 755–768 (2001) M.L. Avila, G.V. Rogachev, E. Koshchiy, L.T. Baby, J. Belarge, K.W. Kemper, A.N. Kuchera, D. Santiago-Gonzalez, New measurement of the α asymptotic normalization coefficient of the 1/2+ state in 17 O at 6.356 MeV that dominates the 13 C(α, n)16 O reaction rate at temperatures relevant for the s process. Phys. Rev. C 91, 048801 (2015) R.E. Azuma, E. Uberseder, E.C. Simpson, C.R. Brune, H. Costantini, R.J. de Boer, J. Görres, M. Heil, P.J. LeBlanc, C. Ugalde, M. Wiescher, AZURE: an R-matrix code for nuclear astrophysics. Phys. Rev. C 81(4), 045805 (2010) F.C. Barker, T. Kajino, The 12 C(α, γ )16 O cross section at low energies. Aust. J. Phys. 44, 369–396 (1991) M. Beard, E. Uberseder, R. Crowter, M. Wiescher, Comparison of statistical model calculations for stable isotope neutron capture. Phys. Rev. C 90, 034619 (2014) H.A. Bethe, Energy production in stars. Phys. Rev. 55, 434–456 (1939) A. Boeltzig, A. Best, F.R. Pantaleo, G. Imbriani, M. Junker, M. Aliotta, J. Balibrea-Correa, D. Bemmerer, C. Broggini, C.G. Bruno, R. Buompane, A. Caciolli, F. Cavanna, T. Chillery, G.F. Ciani, P. Corvisiero, L. Csedreki, T. Davinson, R.J. deBoer, R. Depalo, A. Di Leva, Z. Elekes, F. Ferraro, E.M. Fiore, A. Formicola, Zs. Fülöp, G. Gervino, A. Guglielmetti, C. Gustavino, Gy. Gyürky, I. Kochanek, M. Lugaro, P. Marigo, R. Menegazzo, V. Mossa, F. Munnik, V. Paticchio, R. Perrino, D. Piatti, P. Prati, L. Schiavulli, K. Stöckel, O. Straniero, F. Strieder, T. Szücs, M.P. Takács, D. Trezzi, M. Wiescher, S. Zavatarelli, Direct measurements of low-energy resonance strengths of the 23 Na(p, γ )24 Mg reaction for astrophysics. Phys. Lett. B 795, 122–128 (2019) R.N. Boyd, L. Rybarcyk, M. Wiescher, H.J. Hausman, RISOAR: a facility for production of beams of radioactive ions. IEEE Trans. Nucl. Sci. 30(2), 1387–1390 (1983) D.A. Brown, M.B. Chadwick, R. Capote, A.C. Kahler, A. Trkov, M.W. Herman, A.A. Sonzogni, Y. Danon, A.D. Carlson, M. Dunn, D.L. Smith, G.M. Hale, G. Arbanas, R. Arcilla, C.R. Bates, B. Beck, B. Becker, F. Brown, R.J. Casperson, J. Conlin, D.E. Cullen, M.-A. Descalle, R. Firestone, T. Gaines, K.H. Guber, A.I. Hawari, J. Holmes, T.D. Johnson, T. Kawano, B.C. Kiedrowski, A.J. Koning, S. Kopecky, L. Leal, J.P. Lestone, C. Lubitz, J.I. Márquez Damián, C.M. Mattoon, E.A. McCutchan, S. Mughabghab, P. Navratil, D. Neudecker, G.P.A. Nobre, G. Noguere, M. Paris, M.T. Pigni, A.J. Plompen, B. Pritychenko, V.G. Pronyaev, D. Roubtsov, D. Rochman, P. Romano, P. Schillebeeckx, S. Simakov, M. Sin, I. Sirakov, B. Sleaford, V. Sobes, E.S. Soukhovitskii, I. Stetcu, P. Talou, I. Thompson, S. van der Marck, L. Welser-Sherrill, D. Wiarda, M. White, J.L. Wormald, R.Q. Wright, M. Zerkle, G. Žerovnik, Y. Zhu, ENDF/BVIII.0: the 8th major release of the nuclear reaction data library with CIELO-project cross sections, new standards and thermal scattering data. Nucl. Data Sheets 148, 1–142 (2018); Special Issue on Nuclear Reaction Data C.R. Brune, W.H. Geist, R.W. Kavanagh, K.D. Veal, Sub-coulomb α transfers on 12 C and the 12 C(α, γ )16 O S factor. Phys. Rev. Lett. 83(20), 4025–4028 (1999) C.R. Brune, Alternative parametrization of R -matrix theory. Phys. Rev. C 66, 044611 (2002) R. Buompane, A. Di Leva, L. Gialanella, G. Imbriani, L. Morales-Gallegos, M. Romoli, Recent achievements of the ERNA collaboration. Universe 8(2), 135–147 (2022) P.F.F. Carnelli, S. Almaraz-Calderon, K.E. Rehm, M. Albers, M. Alcorta, P.F. Bertone, B. Digiovine, H. Esbensen, J.O. Fernández Niello, D. Henderson, C.L. Jiang, J. Lai, S.T. Marley, O. Nusair, T. Palchan-Hazan, R.C. Pardo, M. Paul, C. Ugalde, Measurements of fusion reactions of low-intensity radioactive carbon beams on 12 C and their implications for the understanding of x-ray bursts. Phys. Rev. Lett. 112, 192701 (2014) C.J. Cerjan, L. Bernstein, L. Berzak Hopkins, R.M. Bionta, D.L. Bleuel, J.A. Caggiano, W.S. Cassata, C.R. Brune, D. Fittinghoff, J. Frenje, M. Gatu-Johnson, N. Gharibyan, G. Grim,

95 Experimental Nuclear Astrophysics

3531

C. Hagmann, A. Hamza, R. Hatarik, E.P. Hartouni, E.A. Henry, H. Herrmann, N. Izumi, D.H. Kalantar, H.Y. Khater, Y. Kim, A. Kritcher, Yu.A. Litvinov, F. Merrill, K. Moody, P. Neumayer, A. Ratkiewicz, H.G. Rinderknecht, D. Sayre, D. Shaughnessy, B. Spears, W. Stoeffl, R. Tommasini, Ch. Yeamans, C. Velsko, M. Wiescher, M. Couder, A. Zylstra, D. Schneider, Dynamic high energy density plasma environments at the National Ignition Facility for nuclear science research. J. Phys. G: Nucl. Part. Phys. 45(3), 033003 (2018) E. Chiaveri, S. Altstadt, J. Andrzejewski, L. Audouin, M. Barbagallo, V. Bécares, F. Beˇcváˇr, F. Belloni, E. Berthoumieux, J. Billowes, V. Boccone, D. Bosnar, M. Brugger, M. Calviani, F. Calviño, D. Cano-Ott, C. Carrapiço, F. Cerutti, M. Chin, N. Colonna, G. Cortés, M.A. CortésGiraldo, M. Diakaki, C. Domingo-Pardo, I. Duran, R. Dressler, N. Dzysiuk, C. Eleftheriadis, A. Ferrari, K. Fraval, S. Ganesan, A.R. García, G. Giubrone, M.B. Gómez-Hornillos, I.F. Gonçalves, E. González-Romero, E. Griesmayer, C. Guerrero, F. Gunsing, P. Gurusamy, A. Hernández-Prieto, D.G. Jenkins, E. Jericha, Y. Kadi, F. Käppeler, D. Karadimos, N. Kivel, P. Koehler, M. Kokkoris, M. Krtiˇcka, J. Kroll, C. Lampoudis, C. Langer, E. Leal-Cidoncha, C. Lederer, H. Leeb, L.S. Leong, R. Losito, A. Mallick, A. Manousos, J. Marganiec, T. Martínez, C. Massimi, P.F. Mastinu, M. Mastromarco, M. Meaze, E. Mendoza, A. Mengoni, P.M. Milazzo, F. Mingrone, M. Mirea, W. Mondalaers, C. Paradela, A. Pavlik, J. Perkowsk, The CERN n_TOF facility: neutron beams performances for cross section measurements. Nucl. Data Sheets 119, 1–4 (2014) A. Coc, S. Goriely, Y. Xu, M. Saimpert, E. Vangioni, Standard big bang nucleosynthesis up to CNO with an improved extended nuclear network. Astrophys. J. 744(2), 158 (2011) M. Couder, G.P.A. Berg, J. Görres, P.J. LeBlanc, L.O. Lamm, E. Stech, M. Wiescher, J. Hinnefeld, Design of the recoil mass separator St. George. Nucl. Instrum. Methods Phy. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 587(1), 35–45 (2008) J.J. Cowan, C. Sneden, J.E. Lawler, A. Aprahamian, M. Wiescher, K. Langanke, G. MartínezPinedo, F.-K. Thielemann, Origin of the heaviest elements: The rapid neutron-capture process. Rev. Mod. Phys. 93, 015002 (2021) L. Damone, M. Barbagallo, M. Mastromarco, A. Mengoni, L. Cosentino, E. Maugeri, S. Heinitz, D. Schumann, R. Dressler, F. Käppeler, N. Colonna, P. Finocchiaro, J. Andrzejewski, J. Perkowski, A. Gawlik, O. Aberle, S. Altstadt, M. Ayranov, L. Audouin, M. Bacak, J. Balibrea-Correa, J. Ballof, V. Bécares, F. Beˇcváˇr, C. Beinrucker, G. Bellia, A.P. Bernardes, E. Berthoumieux, J. Billowes, M.J.G. Borge, D. Bosnar, A. Brown, M. Brugger, M. Busso, M. Caamaño, F. Calviño, M. Calviani, D. Cano-Ott, R. Cardella, A. Casanovas, D.M. Castelluccio, R. Catherall, F. Cerutti, Y.H. Chen, E. Chiaveri, J.G.M. Correia, G. Cortés, M.A. CortésGiraldo, S. Cristallo, M. Diakaki, M. Dietz, C. Domingo-Pardo, A. Dorsival, E. Dupont, I. Duran, B. Fernandez-Dominguez, A. Ferrari, P. Ferreira, W. Furman, S. Ganesan, A. GarcíaRios, S. Gilardoni, T. Glodariu, K. Göbel, I. F. Gonçalves, E. González-Romero, T. D. Goodacre, E. Griesmayer, C. Guerr. 7 be(n,p)7 li reaction and the cosmological lithium problem: measurement of the cross section in a wide energy range at n_tof at cern. Phys. Rev. Lett. 121, 042701 (2018) R.J. deBoer, J. Görres, M. Wiescher, R.E. Azuma, A. Best et al., The 12 C(α, γ )16 O reaction and its implications for stellar helium burning. Rev. Mod. Phys. 89, 035007 (2017) P. Decrock, Th. Delbar, P. Duhamel, W. Galster, M. Huyse, P. Leleux, I. Licot, E. Liénard, P. Lipnik, M. Loiselet, C. Michotte, G. Ryckewaert, P. Van Duppen, J. Vanhorenbeeck, J. Vervier, Determination of the 13 N(p, γ )14 O reaction cross section using a 13 N radioactive ion beam. Phys. Rev. Lett. 67, 808–811 (1991) P. Descouvemont, D. Baye, The R-matrix theory. Rep. Prog. Phys. 73(3), 036301 (2010) D.C. Dodder, G.M. Hale, K. Witte, The Energy Dependent Analysis Code (Los Alamos National Laboratory, unpublished internal report, 1972) P. Ducru, V. Sobes, Definite complete invariant parametrization of R-matrix theory. Phys. Rev. C 105, 024601 (2022) J.E. Escher, J.T. Harke, F.S. Dietrich, N.D. Scielzo, I.J. Thompson, W. Younes, Compound-nuclear reaction cross sections from surrogate measurements. Rev. Mod. Phys. 84(1), 353–397 (2012)

3532

M. Wiescher et al.

J.E. Escher, J.T. Harke, R.O. Hughes, N.D. Scielzo, R.J. Casperson, S. Ota, H.I. Park, A. Saastamoinen, T.J. Ross, Constraining neutron capture cross sections for unstable nuclei with surrogate reaction data and theory. Phys. Rev. Lett. 121, 052501 (2018) E.-I. Esch, R. Reifarth, E.M. Bond, T.A. Bredeweg, A. Couture, S.E. Glover, U. Greife, R.C. Haight, A.M. Hatarik, R. Hatarik, M. Jandel, T. Kawano, A. Mertz, J.M. O’Donnell, R.S. Rundberg, J.M. Schwantes, J.L. Ullmann, D.J. Vieira, J.B. Wilhelmy, J.M. Wouters, Measurement of the 237 Np(n,γ ) cross section from 20 MeV to 500 keV with a high efficiency, highly segmented 4π BaF2 detector. Phys. Rev. C 77, 034309 (2008) D. Foreman-Mackey, D.W. Hogg, D. Lang, J. Goodman, emcee: the MCMC Hammer. Publ. Astron. Soc. Pac. 125(925), 306–312 (2013) W.A. Fowler, C.C. Lauritsen, T. Lauritsen, Gamma-radiation from excited states of light nuclei. Rev. Mod. Phys. 20, 236–277 (1948) W.A. Fowler, G.R. Caughlan, B.A. Zimmerman, Thermonuclear reaction rates, II. Ann. Rev. Astron. Astrophys. 13(1), 69–112 (1975) E. Gadioli, E. Gadioli, P. Edward Hodgson, Pre-Equilibrium Nuclear Reactions (Clarendon Press, Oxford, 1992) G. Gamow, E. Teller, The rate of selective thermonuclear reactions. Phys. Rev. 53, 608–609 (1938) J. Glorius, C. Langer, Z. Slavkovská, L. Bott, C. Brandau, B. Brückner, K. Blaum, X. Chen, S. Dababneh, T. Davinson, P. Erbacher, S. Fiebiger, T. Gaßner, K. Göbel, M. Groothuis, A. Gumberidze, G. Gyürky, M. Heil, R. Hess, R. Hensch, P. Hillmann, P.M. Hillenbrand, O. Hinrichs, B. Jurado, T. Kausch, A. Khodaparast, T. Kisselbach, N. Klapper, C. Kozhuharov, D. Kurtulgil, G. Lane, C. Lederer-Woods, M. Lestinsky, S. Litvinov, Yu.A. Litvinov, B. Löher, F. Nolden, N. Petridis, U. Popp, T. Rauscher, M. Reed, R. Reifarth, M.S. Sanjari, D. Savran, H. Simon, U. Spillmann, M. Steck, T. Stöhlker, J. Stumm, A. Surzhykov, T. Szücs, T.T. Nguyen, A. Taremi Zadeh, B. Thomas, S.Yu. Torilov, H. Törnqvist, M. Träger, C. Trageser, S. Trotsenko, L. Varga, M. Volknandt, H. Weick, M. Weigand, C. Wolf, P.J. Woods, Y.M. Xing, Approaching the Gamow window with stored ions: direct measurement of 124 Xe (p ,γ ) in the ESR storage ring. Phys. Rev. Lett. 122(9), 092701 (2019) K.H. Guber, T.S. Bigelow, C. Ausmus, D.R. Brashear, J.A. Harvey, P.E. Koehler, D. Wiarda, R.B. Overton, J.A. White, V.M. Cauley, Recent refurbishment of the Oak Ridge Electron Linear Accelerator neutron source, in International Conference on Nuclear Data for Science and Technology (2007), pp. 441–443 G.M. Hale, M.W. Paris, Neutron cross sections for carbon and oxygen from new R-matrix analyses of the 13,14 C and 17 O systems. EPJ Web Conf. 146, 02027 (2017) M. Heil, S. Dababneh, A. Juseviciute, F. Käppeler, R. Plag, R. Reifarth, S. O’Brien, Quasistellar spectrum for neutron activation measurements at kT = 5 keV. Phys. Rev. C 71, 025803 (2005) R.J. Holt, H.E. Jackson, R.M. Laszewski, J.E. Monahan, J.R. Specht, Effects of channel and potential radiative transitions in the 17 O(γ , n0 )16 O reaction. Phys. Rev. C 18, 1962–1972 (1978) D.A. Hutcheon, S. Bishop, L. Buchmann, M.L. Chatterjee, A.A. Chen, J.M. D’Auria, S. Engel, D. Gigliotti, U. Greife, D. Hunter, A. Hussein, C.C. Jewett, N. Khan, M. Lamey, A.M. Laird, W. Liu, A. Olin, D. Ottewell, J.G. Rogers, G. Roy, H. Sprenger, C. Wrede, The DRAGON facility for nuclear astrophysics at TRIUMF-ISAC: design, construction and operation. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 498(1), 190–210 (2003) C. Iliadis, Proton single-particle reduced widths for unbound states. Nucl. Phys. A 618(1), 166– 175 (1997) M. Jaeger, R. Kunz, A. Mayer, J.W. Hammer, G. Staudt, K.L. Kratz, B. Pfeiffer, 22 N e(α, n)25 Mg: the key neutron source in massive stars. Phys. Rev. Lett. 87, 202501 (2001) M. Jentschel, A. Blanc, G. de France, U. Köster, S. Leoni, P. Mutti, G. Simpson, T. Soldner, C. Ur, W. Urban, S. Ahmed, A. Astier, L. Augey, T. Back, P. Baczyk, A. Bajoga, D. Balabanski, T. Belgya, G. Benzoni, C. Bernards, D.C. Biswas, G. Bocchi, S. Bottoni, R. Britton, B. Bruyneel, J. Burnett, R.B. Cakirli, R. Carroll, W. Catford, B. Cederwall, I. Celikovic, N. CieplickaOry´nczak, E. Clement, N. Cooper, F. Crespi, M. Csatlos, D. Curien, M. Czerwi´nski, L.S. Danu, A. Davies, F. Didierjean, F. Drouet, G. Duchêne, C. Ducoin, K. Eberhardt, S. Erturk, L.M.

95 Experimental Nuclear Astrophysics

3533

Fraile, A. Gottardo, L. Grente, L. Grocutt, C. Guerrero, D. Guinet, A.L. Hartig, C. Henrich, A. Ignatov, S. Ilieva, D. Ivanova, B.V. John, R. John, J. Jolie, S. Kisyov, M. Krticka, T. Konstantinopoulos, A. Korgul, A. Krasznahorkay, T. Kröll, J. Kurpeta, I. Kuti, S. Lalkovski, C. EXILL—a high-efficiency, high-resolution setup for γ -spectroscopy at an intense cold neutron beam facility. J. Instrum. 12(11), P11003 (2017) B. Jurado, R. Reifarth, D. Denis-Petit, J. Glorius, C. Langer, Y.A. Litvinov, P.J. Woods, Direct and Indirect Measurements of Neutron Induced Cross Sections at Storage Rings. in 10th International Conference on Nuclear Physics at Storage Rings (STORI’17) (2021), p. 011001 F. Käppeler, A.A. Naqvi, M. Al-Ohali, Stellar krypton cross sections at kt = 25 and 52 kev. Phys. Rev. C 35, 936–941 (1987) P.L. Kapur, R.E. Peierls, The dispersion formula for nuclear reactions. Proc. Roy. Soc. (London) A 166, 277 (1938) T. Kubo, In-flight RI beam separator BigRIPS at RIKEN and elsewhere in Japan. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. At. 204, 97–113 (2003); 14th International Conference on Electromagnetic Isotope Separators and Techniques Related to their Applications. A.M. Lane, R.G. Thomas, R-matrix theory of nuclear reactions. Rev. Mod. Phys. 30, 257–353 (1958) N.M. Larson, Updated User’s Guide for Sammy: Multilevel R-Matrix Fits to Neutron Data Using Bayes’ Equations, vol. 10. Technical Report: ORNL/TM-9179/R8 (2008) A.C. Larsen, A. Spyrou, S.N. Liddick, M. Guttormsen, Novel techniques for constraining neutron-capture rates relevant for r-process heavy-element nucleosynthesis. Oak Ridge National Laboratory. Prog. Part. Nucl. Phys. 107, 69–108 (2019) M. Lestinsky, V. Andrianov, B. Aurand, V. Bagnoud, D. Bernhardt, H. Beyer, S. Bishop, K. Blaum, A. Bleile, At. Borovik, F. Bosch, C.J. Bostock, C. Brandau, A. Bräuning-Demian, I. Bray, T. Davinson, B. Ebinger, A. Echler, P. Egelhof, A. Ehresmann, M. Engström, C. Enss, N. Ferreira, D. Fischer, A. Fleischmann, E. Förster, S. Fritzsche, R. Geithner, S. Geyer, J. Glorius, K. Göbel, O. Gorda, J. Goullon, P. Grabitz, R. Grisenti, A. Gumberidze, S. Hagmann, M. Heil, A. Heinz, F. Herfurth, R. Heß, P.-M. Hillenbrand, R. Hubele, P. Indelicato, A. Källberg, O. Kester, O. Kiselev, A. Knie, C. Kozhuharov, S. Kraft-Bermuth, T. Kühl, G. Lane, Yu.A. Litvinov, D. Liesen, X.W. Ma, R. Märtin, R. Moshammer, A. Müller, S. Namba, P. Neumeyer, T. Nilsson, W. Nörtershäuser, G. Paulus, N. Petridis, M. Reed, R. Reifarth, P. Reiß, J. Rothhardt, R. Sanchez, M.S. Sanjari, S. Schippers, H.T. Schmidt, D. Schneider, P. Scholz, R. Schuch, M. Schulz, V. Shabaev, A. Simons, Physics book: cryring@esr. Eur. Phys. J. Spec. Top. 225(5), 797–882 (2016) Y.A. Litvinov, S. Bishop, K. Blaum, F. Bosch, C. Brandau, L.X. Chen, I. Dillmann, P. Egelhof, H. Geissel, R.E. Grisenti, S. Hagmann, M. Heil, A. Heinz, N. Kalantar-Nayestanaki, R. Knöbel, C. Kozhuharov, M. Lestinsky, X.W. Ma, T. Nilsson, F. Nolden, A. Ozawa, R. Raabe, M.W. Reed, R. Reifarth, M.S. Sanjari, D. Schneider, H. Simon, M. Steck, T. Stöhlker, B.H. Sun, X.L. Tu, T. Uesaka, P.M. Walker, M. Wakasugi, H. Weick, N. Winckler, P.J. Woods, H.S. Xu, T. Yamaguchi, Y. Yamaguchi, Y.H. Zhang, Nuclear physics experiments with ion storage rings. Nucl. Instrum. Methods Phys. Res. B 317, 603–616 (2013) Q. Liu, M. Febbraro, R.J. deBoer, S. Aguilar, A. Boeltzig, Y. Chen, M. Couder, J. Görres, E. Lamere, S. Lyons, K.T. Macon, K. Manukyan, L. Morales, S. Pain, W.A. Peters, C. Seymour, G. Seymour, R. Toomey, B. Vande Kolk, J. Weaver, M. Wiescher, Low-energy cross-section measurement of the 10 B(α, n)13 N reaction and its impact on neutron production in firstgeneration stars. Phys. Rev. C 101, 025808 (2020) R.L. Macklin, B.J. Allen, Fast neutron capture cross section facility. Nucl. Instrum. Methods 91(4), 565–571 (1971) A.M. Mukhamedzhanov, R.E. Tribble, Connection between asymptotic normalization coefficients, subthreshold bound states, and resonances. Phys. Rev. C 59, 3418–3424 (1999) A.M. Mukhamedzhanov, C.A. Gagliardi, R.E. Tribble, Asymptotic normalization coefficients, spectroscopic factors, and direct radiative capture rates. Phys. Rev. C 63, 024612 (2001)

3534

M. Wiescher et al.

R. Müller, F. Gönnenwein, F. Käppeler, A. Ernst, J. Scheer, Search for fission shape isomers in (n,f) reactions induced by neutrons of about 1 MeV. Phys. Lett. B 48(1), 25–27 (1974) H. Nassar, M. Paul, I. Ahmad, D. Berkovits, M. Bettan, P. Collon, S. Dababneh, S. Ghelberg, J.P. Greene, A. Heger, M. Heil, D.J. Henderson, C.L. Jiang, F. Käppeler, H. Koivisto, S. O’Brien, R.C. Pardo, N. Patronis, T. Pennington, R. Plag, K.E. Rehm, R. Reifarth, R. Scott, S. Sinha, X. Tang, R. Vondrasek, Stellar (n,γ ) cross section of Ni-62. Phys. Rev. Lett. 94, 092504 (2005) S.F. Nowicki, S.A. Wender, M. Mocko, The Los Alamos neutron science center spallation neutron sources. Phys. Proc. 90, 374–380 (2017); Conference on the Application of Accelerators in Research and Industry, CAARI 2016, 30 Oct – 4 Nov, 2016, Ft. Worth, TX D. Odell, C.R. Brune, D.R. Phillips, R.J. deBoer, S.N. Paneru, Performing Bayesian analyses With AZURE2 using BRICK: an application to the 7 Be system. Front. Phys. 10, 888476 (2022) R. Reifarth, Y.A. Litvinov, Measurements of neutron-induced reactions in inverse kinematics. Phys. Rev. ST Accel. Beams 17, 014701 (2014) R. Reifarth, K. Schwarz, F. Käppeler, The stellar neutron-capture rate of 34 s: the origin of 36 s challenged. Ap. J. 528, 573–581 (2000) R. Reifarth, M. Heil, F. Käppeler, F. Voss, K. Wisshak, F. Be˘cvá˘r, M. Krti˘cka, R. Gallino, Y. Nagai, The stellar neutron capture cross sections of 128,129,130 Xe. Phys. Rev. C (Nucl. Phys.) 66(6), 064603 (2002) R. Reifarth, M. Heil, C. Forssén, U. Besserer, A. Couture, S. Dababneh, L. Dörr, J. Görres, R.C. Haight, F. Käppeler, A. Mengoni, S. O’Brien, N. Patronis, R. Plag, R.S. Rundberg, M. Wiescher, J.B. Wilhelmy, The 14 C(n,γ ) cross section between 10 keV and 1 MeV. Phys. Rev. C 77, 015804 (2008) R. Reifarth, M. Heil, F. Käppeler, R. Plag, PINO-a tool for simulating neutron spectra resulting from the 7 Li(p,n) reaction. Nucl. Inst. Meth. A 608, 139 (2009) R. Reifarth, C. Lederer, F. Käppeler, Neutron reactions in astrophysics. J. Phys. G Nucl. Phys. 41(5), 053101 (2014) R. Reifarth, K. Göbel, T. Heftrich, M. Weigand, B. Jurado, F. Käppeler, Y.A. Litvinov, Spallationbased neutron target for direct studies of neutron-induced reactions in inverse kinematics. Phys. Rev. Accel. Beams 20, 044701 (2017) R. Reifarth, P. Erbacher, S. Fiebiger, K. Göbel, T. Heftrich, M. Heil, F. Käppeler, N. Klapper, D. Kurtulgil, C. Langer, C. Lederer-Woods, A. Mengoni, B. Thomas, S. Schmidt, M. Weigand, M. Wiescher, Neutron-induced cross sections – from raw data to astrophysical rates. Eur. Phys. J. Plus 133, 424 (2018) I.U. Roederer, A.I. Karakas, M. Pignatari, F. Herwig, The diverse origins of neutron-capture elements in the metal-poor star HD 94028: possible detection of products of i-process nucleosynthesis. Astrophys. J. 821(1), 37 (2016) E.E. Salpeter, Nuclear reactions in the stars. I. Proton-proton chain. Phys. Rev. 88, 547–553 (1952) D. Schürmann, A. Di Leva, L. Gialanella, D. Rogalla, F. Strieder, N. De Cesare, A. D’Onofrio, G. Imbriani, R. Kunz, C. Lubritto, A. Ordine, V. Roca, C. Rolfs, M. Romano, F. Schümann, F. Terrasi, H.-P. Trautvetter, First direct measurement of the total cross-section of 12 C(α, γ )16 O. Eur. Phys. J. A 26, 301–305 (2005) T. Scott Carman, V. Litveninko, J. Madey, C. Neuman, B. Norum, P.G. O’Shea, N. Russell Roberson, C.Y. Scarlett, E. Schreiber, H.R. Weller, The TUNL-FELL inverse Compton γ -ray source as a nuclear physics facility. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 378(1), 1–20 (1996) A. Simon, M. Beard, A. Spyrou, S.J. Quinn, B. Bucher, M. Couder, P.A. DeYoung, A.C. Dombos, J. Görres, A. Kontos, A. Long, M.T. Moran, N. Paul, J. Pereira, D. Robertson, K. Smith, E. Stech, R. Talwar, W.P. Tan, M. Wiescher, Systematic study of (α, γ ) reactions for stable nickel isotopes. Phys. Rev. C 92, 025806 (2015) K. Sonnabend, D. Savran, J. Beller, M.A. Büssing, A. Constantinescu, M. Elvers, J. Endres, M. Fritzsche, J. Glorius, J. Hasper, J. Isaak, B. Löher, S. Müller, N. Pietralla, C. Romig, A. Sauerwein, L. Schnorrenberger, C. Wälzlein, A. Zilges, M. Zweidinger, The Darmstadt High-Intensity Photon Setup (DHIPS) at the S-DALINAC. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators Spectrom. Detect. Assoc. Equip. 640(1), 6–12 (2011)

95 Experimental Nuclear Astrophysics

3535

K.A. Tanaka, K.M. Spohr, D.L. Balabanski, S. Balascuta, L. Capponi, M.O. Cernaianu, M. Cuciuc, A. Cucoanes, I. Dancus, A. Dhal, B. Diaconescu, D. Doria, P. Ghenuche, D.G. Ghita, S. Kisyov, V. Nastasa, J.F. Ong, F. Rotaru, D. Sangwan, P.-A. Söderström, D. Stutman, G. Suliman, O. Tesileanu, L. Tudor, N. Tsoneva, C.A. Ur, D. Ursescu, N.V. Zamfir, Current status and highlights of the ELI-NP research program. Matter Radiat. Extremes 5(2), 024402 (2020) P. Tsintari, R. Garg, G. Berg, J. Blackmon, K. Chipps, M. Couder, C. Deibel, N. Dimitrakopoulos, U. Greife, A. Hood, R. Jain, C. Marshall, Z. Meisel, S. Miskovich, F. Montes, G. Perdikakis, T. Ruland, H. Schatz, K. Setoodehnia, M. Smith, L. Wagner, SECAR: a recoil separator for nuclear astrophysics. EPJ Web Conf. 260, 11044 (2022) A. Tumino, C.A. Bertulani, M. La Cognata, L. Lamia, R.G. Pizzone, S. Romano, S. Typel, The trojan horse method: a nuclear physics tool for astrophysics. Annu. Rev. Nucl. Part. Sci. 71(1), 345–376 (2021) E. Uberseder, R. Reifarth, D. Schumann, I. Dillmann, C. Domingo Pardo, J. Görres, M. Heil, F. Käppeler, J. Marganiec, J. Neuhausen, M. Pignatari, F. Voss, S. Walter, M. Wiescher, Measurement of the 60 Fe(n, γ )61 Fe cross section at stellar temperatures. Phys. Rev. Lett. 102, 151101 (2009) L. Van der Zwan, K.W. Geiger, The 10 B(α, n)13 N cross section for α-energies from 1.0–5 MeV. Nucl. Phys. A 216(1), 188–198 (1973) M. Weigand, T.A. Bredeweg, A. Couture, K. Göbel, T. Heftrich, M. Jandel, F. Käppeler, C. Lederer, N. Kivel, G. Korschinek, M. Krtiˇcka, J.M. O’Donnell, J. Ostermöller, R. Plag, R. Reifarth, D. Schumann, J.L. Ullmann, A. Wallner, 63 Ni (n ,γ ) cross sections measured with DANCE. Phys. Rev. C 92(4), 045810 (2015) M. Wiescher, J. Gorres, F.-K. Thielemann, Capture reactions on 14 C in nonstandard big bang nucleosynthesis. ApJ 363, 340 (1990) E.P. Wigner, L. Eisenbud, Higher angular momenta and long range interaction in resonance reactions. Phys. Rev. 72, 29–41 (1947) A. Zilges, P. Mohr, Reactions with real photons in the vicinity of the neutron threshold. Prog. Part. Nucl. Phys. 44, 39–48 (2000)

Slow Neutron-Capture Process in Evolved Stars

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Raphael Hirschi

Contents Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Standard Weak s Process in Non-rotating Massive Stars . . . . . . . . . . . . . . . . . . . . . . . . . S Process Production During He-Core Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S Process Production During C-Shell Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The “Not So Weak” s Process in Rotating Massive Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotation-Induced Mixing and Production of Primary 22 Ne and 14 N . . . . . . . . . . . . . . . . . Impact of Rotation on the s Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative Contributions Between Helium and Carbon Burning and Total s Process Yields as a Function of Mass and Metallicity . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison Between Models and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production of Elements at the Sr and Ba Peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Very Low-Z Stars: The Case of CEMP-No Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production of p Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stellar Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key Nuclear Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The slow neutron-capture process (s process) contributes to the production of roughly half of the elements heavier than iron. The main component of the s process takes place in the late phases (AGB phase) of low-mass stars and is the topic of most reviews on the s process. This review thus focuses on the weak s

R. Hirschi () Astrophysics Group, Keele University, Keele, UK Institute for Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2023 I. Tanihata et al. (eds.), Handbook of Nuclear Physics, https://doi.org/10.1007/978-981-19-6345-2_118

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process taking place in massive stars with a special emphasis on the key role of rotation as well as key stellar and nuclear uncertainties. While the weak s process production at solar metallicity is limited to the first peak (Sr), rotation-induced mixing boosts this production at low metallicities to the second peak (Ba) and in extreme cases to the third peak (Pb). Rotation-boosted s process also opens the door to an enhanced production of p nuclei compared to classical non-rotating massive star model predictions.

Brief Introduction Fusion reactions and charged-particle captures produce elements from helium to iron. Beyond iron, the Coulomb repulsion inhibits charged-particle captures, and neutron captures dominate the production of heavier elements. Neutron capture processes are classified mainly according to their pace relative to β−decays. If neutron captures are faster, these are described as rapid (r process, e.g., Cowan et al. 1991; Freiburghaus et al. 1999; Nishimura et al. 2015; Thielemann et al. 2017; Arnould and Goriely 2020; Cowan et al. 2021; Kullmann et al. 2022; Arcones and Thielemann 2023), and if they are slower, they are described as slow (s process hereinafter; see e. g., Käppeler et al. 2011). The s process is subdivided into two main categories. The main s process takes place in the late phases of low-mass stars and produces elements up to lead (Pb) (see e. g., Arlandini et al. 1999; Lattanzio and Lugaro 2005; Herwig 2005; Käppeler et al. 2011; Piersanti et al. 2013; Battino et al. 2019; den Hartogh et al. 2019; Cseh et al. 2022; Lugaro 2023), whereas the weak s process takes place in massive stars during helium and carbon burning phases and generally produces elements up to the strontium (Sr) peak (linked to the N = 50 magic number). Reviews on the s process have generally had a stronger focus on the nuclear physics aspects and the main s process in low-mass stars (e.g., Käppeler et al. 2011). This review instead focuses on the weak s process in massive stars with a special emphasis on the key role of rotation and other stellar and nuclear uncertainties. The review is centered on the study of Frischknecht et al. (2016), who completed a large grid of massive star models covering a wide range of masses and metallicities and included the effects of rotation and complete s process nucleosynthesis.

The Standard Weak s Process in Non-rotating Massive Stars The classic view of the s process nucleosynthesis in (nonrotating) massive stars is that it occurs in He- and C-burning regions (see Fig. 1) of the stars, thanks to the activation of the neutron source 22 Ne(α, n), and produces only the low mass range of the s process elements, typically the elements with an atomic mass number up to about 90 (e.g., Käppeler et al. 2011; Prantzos et al. 1990; Baraffe et al. 1992; Baraffe and Takahashi 1993; Rayet and Hashimoto 2000; The et al. 2000, 2007;

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Fig. 1 Structure evolution (Kippenhahn) diagram of 25 M star with Z = Z and no rotation (model A25s0 in Table 1). Blue-shaded areas indicate convective regions. The gray-shaded area shows the mass expected to be locked into the remnant. Symbols (H, He, C, Ne, O) indicate the timing of the core burning phases from hydrogen to oxygen burning. The red vertical line marks the end of core helium burning, while the red horizontal line separates the carbon and helium burning shells. (This and following figures are taken from Frischknecht et al. 2016)

Pumo et al. 2010, and references therein). It has also been shown that in the regions where the s process occurs, the fact that, when the metallicity decreases, (1) the neutron source, mainly the 22 Ne(α, n) reaction, decreases; (2) the neutron seeds (Fe) also decreases; and (3) the neutron poisons as, for instance, 16 O remain independent of the metallicity implies that the s process element production decreases with the metallicity and that there exists some limiting metallicity below which the s process becomes negligible. This limit was found to be around Z/Z =10−2 (Prantzos et al. 1990). To explore the standard weak s process in more details, we will use characteristic s process quantities and the stellar models of Frischknecht et al. (2016), who computed a large grid of massive stars models over a wide range of metallicities. Table 1 lists these models as well as several characteristic quantities for s process in He burning (first letter indicates the metallicity of the model and the two digits after that indicate the initial mass of the models) with (code names ending in s4) and without (s0) rotation. Note that some of these quantities are averaged quantities over the convective core and integrated over the helium-burning phase, encompassing in one number complex processes varying both in space and in time. These quantities are useful in the sense that they allow through a unique number to see the importance of different phases and also to compare the outputs of different models. In a one-zone nucleosynthesis calculations, a useful quantity is the neutron (n) exposure defined as:

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Table 1 s process parameters at central He exhaustion. (Table taken from Frischknecht et al. 2016) Modela A15s0 A15s4 B15s0 B15s4 C15s0 C15s4 A20s0 A20s4 B20s0 B20s4 C20s0 C20s4 A25s0 A25s4 B25s0 B25s4 C25s0 C25s4 C25s4bh C25s5 C25s5bh D25s0 D25s4 D25s4bh D25s6 D25s6bh A40s4 B40s4 C40s4

τc b [mb−1 ] 1.52 2.93 0.883 3.06 0.0157 2.21 2.97 4.66 1.88 9.73 0.0286 6.55 4.42 5.63 2.65 12.1 0.0466 6.73 16.4 13.5 20.3 0.166 0.804 2.29 19.2 24.6 7.76 12.1 11.6

τ c [10−1 mb−1 ] 0.581 1.02 0.427 1.51 0.0561 1.07 0.971 1.43 0.761 4.07 0.0401 2.80 1.33 1.60 0.970 4.80 0.0829 2.94 7.15 5.73 8.67 0.0866 0.354 1.048 7.78 10.0 2.00 4.12 4.67

d

nc 0.77 1.60 0.53 2.55 0.04 2.18 1.52 2.57 1.13 9.85 0.05 5.87 2.42 3.13 1.64 12.7 0.08 5.77 23.1 16.5 31.8 6.31 14.0 16.5 33.5 48.5 4.05 10.6 10.4

n¯ en,max [cm−3 ] 3.04(5) 4.65(5) 2.32(5) 5.18(5) 2.85(3) 3.38(5) 5.17(5) 5.89(5) 4.11(5) 8.73(5) 6.00(3) 6.84(5) 5.85(5) 5.98(5) 4.99(5) 8.03(5) 9.36(3) 5.77(5) 8.02(5) 8.27(5) 1.01(6) 9.61(2) 1.39(5) 3.85(5) 6.77(5) 9.77(5) 3.77(5) 6.38(5) 6.12(5)

nn,c,max f,g [cm−3 ] 6.58(6) 1.17(7) 4.32(6) 1.07(7) 5.59(4) 7.33(6) 1.22(7) 1.54(7) 9.10(6) 2.22(7) 1.31(5) 1.69(7) 1.56(7) 1.72(7) 1.20(7) 2.31(7) 2.13(5) 1.53(7) 2.10(7) 2.26(7) 2.74(7) 2.24(4) 3.38(6) 8.60(6) 2.03(7) 2.76(7) 1.42(7) 2.13(7) 1.97(7)

ΔX(22 Ne)g 3.06(−3) 5.59(−3) 3.54(−4) 9.37(−4) 3.75(−6) 4.84(−4) 5.34(−3) 7.23(−3) 6.16(−4) 3.49(−3) 5.66(−6) 1.52(−3) 7.68(−3) 9.69(−3) 7.52(−4) 4.08(−3) 7.21(−6) 1.23(−3) 1.27(−3) 3.83(−3) 3.75(−3) 8.28(−7) 1.05(−4) 1.06(−4) 4.57(−3) 4.44(−3) 1.23(−2) 3.31(−3) 2.70(−3)

Xr (22 Ne)g 9.70(−3) 7.42(−3) 8.02(−4) 1.02(−3) 7.70(−6) 3.92(−4) 7.43(−3) 5.03(−3) 5.46(−4) 1.14(−3) 5.74(−6) 4.62(−4) 5.10(−3) 3.28(−3) 4.16(−4) 6.22(−4) 4.14(−6) 1.69(−4) 1.82(−4) 4.94(−4) 4.85(−4) 4.67(−7) 3.81(−5) 3.96(−5) 2.68(−4) 3.11(−4) 5.29(−4) 1.06(−4) 1.93(−5)

A series models have metallicity of Z = Z , B series Z = 10−3 , C series Z = 10−5 , and D series Z = 10−7 b Central neutron exposure calculated according to Eq. 1 c Neutron exposure averaged over He core (see Eq. 2) d Number of neutron capture per seed calculated according to Eq. 3, averaged over the He-core mass e Maximum of the mean neutron density f Maximum of the central neutron density g Values in brackets are the exponents (x(y) = x × 10y ) h This model was calculated with the same initial parameters as the model, on the line above, but with 17 O(α, γ ) reaction rate of CF88 divided by 10 a The

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 τ=

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tend−He

υT nn dt

(1)

tini−He

where tini−He and tend−He are the age of the star at the beginning and the end of the core He-burning√phase, respectively, nn is the neutron density and υT is the thermal velocity, vT = 2kT /mn with kT = 30 keV (mn is the neutron mass). The value of 30 keV is typical of the conditions at the end of the core He-burning phase. In multiple-zone simulations, as in stellar models, the neutron number density, nn , varies with time and the mass coordinate in the star. For the investigation of s process in convective zones, one can define a mean or effective neutron exposure:  τ  = nn (t)vT dt. (2) In Eq. 2, nn (t) is an average over the convective core. Such a global quantity has to be interpreted with caution since in reality the neutrons are captured locally during core He burning, near the center of the star and later the s process products are mixed outward. Another characteristic s process quantity is the average number of neutron captures per iron (Z = 26) seed (e.g., Käppeler et al. 1990): 209 

nc =

A=56

(A − 56) (Y (A) − Y0 (A))  , Y0 (A)

(3)

Z=26

where Y (A) and Y0 (A) are the final and the initial number abundance, respectively, of a nucleus with nuclear mass number A. Additionally, the core averaged (n¯ n,max ) and central (nn,c,max ) peak neutron density, the amount of 22 Ne burned during He burning (ΔX(22 Ne)), and the amount of 22 Ne left in the center at core He exhaustion (Xr (22 Ne)) are tabulated.

S Process Production During He-Core Burning Let us begin by discussing the solar metallicity models. At the beginning of the core He-burning phase, 14 N is converted into 22 Ne via a double α-capture and massive stars have initially in the He core about X(22 Ne) = 1.3 × 10−2 . The abundance of 22 Ne remains unchanged during a large fraction of the core He-burning phase. Only close to the end of central He burning, part of the 22 Ne is transformed into 25 Mg and 26 Mg via (α, n) and (α, γ ), respectively. When the temperatures for an efficient activation of 22 Ne(α, n)25 Mg are reached, some 22 Ne has already been destroyed

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by the (α, γ )26 Mg reaction. (e.g., when the core reaches T8 ≈ 2.8, at which point the (α, n)-channel starts to dominate, only X(22 Ne) = 10−2 to 5.0 × 10−3 is left in 15 to 40 M models). Because only a small helium mass fraction, X(4 He), is left when 22 Ne+α is activated (less than ten percent in mass fraction), the competition with other α-captures such as the 12 C(α, γ ) and 3α is essential at the end of He burning and will affect the s process efficiency in core He burning. This low amount of X(4 He), when the neutron source is activated, means also that not all of 22 Ne is burned and a part of it will be left for subsequent C-burning phase. This depends on the stellar core size. The more massive the core, the more 22 Ne is burned and the more efficient is the s process in core He burning, as can be seen from the increasing number of neutron captures per seed nc with initial mass in Table 1. During the late He burning stages, the bulk of the core matter consists of 12 C and 16 O, which are both strong neutron absorbers. They capture neutrons to produce 13 C and 17 O, respectively. 13 C will immediately recycle neutrons via 13 C(α, n) in He-burning conditions. On the other hand, the relevance of 16 O as a neutron poison depends on the 17 O(α, γ ) and 17 O(α, n) rates. In particular, the strength of primary neutron poisons like 16 O increases toward lower metallicities, because of the decreasing ratio of seeds to neutron poisons. The s process production in nonrotating models at various metallicities is shown in Fig. 2 (Z = Z ), Fig. 3 (Z = 10−3 ), Fig. 4 (Z = 10−5 ), and Fig. 5 (Z = 10−7 ). In combination with the values given in Table 1, we can clearly see the metallicity dependence of the s process in nonrotating massive stars, which we will call the standard s process in the rest of this paper. As mentioned above, the production of nuclei between A = 60 and 90 decreases with decreasing metallicity and mass. The decreasing production with decreasing metallicity is due to the secondary nature of both the neutron source (22 Ne(α, n)25 Mg) and the seeds (mainly iron) (Prantzos et al. 1990; Raiteri et al. 1992; Pignatari and Gallino 2008). During helium

Fig. 2 Isotopic overproduction factors (abundances over initial abundances) of 25 M models with solar metallicity after He exhaustion. The rotating model (A25s4, circles) has slightly higher factors than the nonrotating model (A25s0, diamonds)

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Fig. 3 Isotopic abundances normalized to solar abundances of 25 M models with Z = 10−3 after He exhaustion. The rotating model (B25s4, circles) has higher factors than the nonrotating model (B25s0, diamonds)

Fig. 4 Isotopic abundances normalised to solar abundances of 25 M models with Z = 10−5 after He exhaustion. The rotating model (C25s5, circles) has much higher factors than the nonrotating model (C25s0, diamonds)

burning, the neutron poisons are a mixture of secondary (mainly 20 Ne, 22 Ne and 25 Mg) and primary (mainly 16 O) elements. The s process production thus becomes negligible below Z/Z = 10−2 (Prantzos et al. 1990), which can be seen in the nonrotating models at Z = 10−5 and Z = 10−7 (C and D series). The decreasing production with decreasing mass is due to the fact that lower mass stars reach lower temperature at the end of He burning. Thus, less 22 Ne is burned during He burning (see Table 1). Shell He burning, similarly to the other burning shells, appears at higher temperatures (T8 ≈ 3.5–4.5) and lower densities (ρ ≈ 3–5.5 × 103 g cm−3 ) than the equivalent central burning phase and causes an efficient 22 Ne(α,n) activation for the

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Fig. 5 Isotopic abundances normalised to solar abundances of 25 M models with Z = 10−7 after He exhaustion. The rotating model (D25s6, circles) has slightly higher factors than the nonrotating model (D25s0, diamonds)

s process. However, the highest neutron densities are reached in most models only in the layers below the convective shell helium burning. Therefore, generally only a narrow mass range, extending over about 0.2 M in nonrotating models, at the bottom of the He shell is strongly affected by neutron capture nucleosynthesis. The contribution of the s process in the He shell remains modest for 25 M models (at most ∼5% of the total s process yields at solar metallicity). For less massive stars (15 to 20 M ), the He shell contributes more and should not be neglected (see also Tur et al. 2009).

S Process Production During C-Shell Burning Shell C burning occurs in the carbon-oxygen (CO) core (the part of the star that was in the convective He-burning core (see Fig. 1) after central C burning. In most massive star models, a large fraction of the He burning s process material is reprocessed (e.g., Pignatari et al. 2010). Temperatures and densities at the start of C shell burning show the same trend with stellar mass as the core burning conditions, i.e., the temperature increases and the density decreases with stellar mass. They vary between T9 ≈ 0.8, ρ ≈ 2 × 105 g cm−3 in 15 M models and T9 ≈ 1.3, ρ ≈ 8 × 104 g cm−3 in 40 M models. These temperatures are higher than in the central C burning, where T9 = 0.6 − 0.8. The efficiency of the s process during C-shell burning mainly depends on the remaining iron seeds and 22 Ne left after He burning, Xr (22 Ne) (see Table 1), in the CO core. All the remaining 22 Ne is burned quickly with maximal neutron densities between 6 × 109 and 1012 cm−3 . The time scale of this s process in this phase is of the order of a few tens of years in 15 M stars to a few tenth of a year in 40 M

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Fig. 6 Ratio of abundances after shell C burning to the abundances after core He burning, XC /XHe , in a nonrotating 25 M star at Z = Z (A25S0). It illustrates the modification of the abundances by s process in shell C burning

stars. A striking difference between the s processes in the He shell and in the C shell is the neutron density, which is much higher in the C shell than in the He shell. The activation of 22 Ne(α, n) at the start of C shell burning leads to a short neutron burst with relatively high neutron densities (typically nn ∼ 1010 − 1012 cm−3 , see The et al. 2000, 2007), compared to He burning (nn ∼ 105 − 107 cm−3 ). The ratio of abundances after shell C burning to the abundances after core He burning, XC /XHe , is plotted for the nonrotating 25 M model at Z = Z in Fig. 6. We can see an overproduction of most isotopes from Zn to Rb. The C shell contribution is strongest when there is enough iron seeds and 22 Ne (Xr (22 Ne)  10−3 ) left at the end of core He burning, which is the case in 15 to 25 M stars at solar Z. In the mass range A = 60 to 90, there are several branching points at 63 Ni, 79 Se, and 85 Kr, respectively. The high neutron densities modify the s process branching ratios, in a way that the neutron capture on the branching nuclei are favored over the β−decay channel (see, e.g., Pignatari et al. 2010, and references therein). Consequently, isotopic ratios like 63 Cu/65 Cu, 64 Zn/66 Zn, 80 Kr/82 Kr, 79 Br/81 Br, 85 Rb/87 Rb, and 86 Sr/88 Sr are lowered. Overall, stars with different initial masses show very different final branching ratios. For instance, stars with 15 M and with 20 M (without rotation) produce 64 Zn, 80 Kr, 86 Sr in the C shell, while in heavier stars, these isotopes are reduced compared to the previous He core. The impact of the high neutron densities during C shell can be seen in Fig. 6. It causes up to three orders of magnitude overproduction of some r process nuclei, such as 70 Zn, 76 Ge, 82 Se, or 96 Zr, compared to the yields of the “slower” s process during He burning. However, the production of r-only nuclei in carbon burning compensates only the destruction in the He core s process when looking at the final yields. Only for the 40 M model is 96 Zr weakly produced.

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During C burning, the main neutron poisons are 16 O, 20 Ne, 23 Na, and 24 Mg, which are all primary. Thus the C shell contribution to the s process will vanish at low metallicities even faster than during He burning ( Dh ). In the models, just above the convective He core, where K/Dh has typical values between 10 and 100 and where ∇μ is highest, the term including ∇μ reaches values up to 103 , which shows the strong inhibiting effect of μ-gradients on mixing. This can also be seen on the left hand side in Figs. 8, 9, and 10 at Mr ≈ 5 − 10 M , where K is the black dotted line and Dh is the blue dash-dotted line. The K/Dh ratio does not change significantly in the relevant regions in the course of central He burning. Regions of strong μ-gradients can be identified by steep slopes in the abundance of hydrogen and carbon on the right-hand side of these figures as discussed below. In rotating models, there are three different configurations of the stellar structure that may occur during central He burning. These cases are illustrated with the help

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Fig. 8 Diffusion coefficient profiles on the left hand side and abundance profiles on the right hand side during central He-burning, when a convective H-shell is present, inside the 25 M star with rotation at Z = 10−3 (B25S4). The shear diffusion coefficient (red continuous line) is responsible for the mixing between He core and H shell. The convective regions are represented by the gray shaded areas

Fig. 9 Diffusion coefficient profiles on the left hand side and abundance profiles on the right hand side during central He-burning, when a retracting convective H-shell is present, inside the 25 M star with rotation at Z = 10−3 (B25S4). The shear diffusion coefficient (red continuous line) describes the mixing between He core and H shell. The convective regions are represented by the gray shaded areas

of three evolutionary snapshots of a rotating 25 M Z = 10−3 star during central He burning: • Case (a): In the first configuration, shown in Fig. 8, the convective H burning shell (Mr ≈ 9 − 13 M ) rotates considerably slower than the regions below

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Fig. 10 Diffusion coefficient profiles on the left hand side and abundance profiles on the right hand side during central He-burning, when no convective H-shell is present, inside the 25 M star with rotation at Z = 10−3 (B25S4). The shear diffusion coefficient (red continuous line) describes the mixing between He core and H shell. The convective core is represented by the gray shaded areas

(the angular velocity Ω profile is plotted as an orange dashed line on the left hand side). The steep gradient of Ω at the lower boundary of the convective shell compensates for the inhibiting effect of ∇μ , which is strongest just below the convective shell where the gradient of hydrogen abundance is very steep. In this configuration, Dshear has values between 104 and 107 cm2 s−1 throughout the radiative region between the convective He-core and the H-shell zones, facilitating a strong production of primary nitrogen. • Case (b): this configuration shown in Fig. 9 is very similar to case (a), i.e., there is a convective H burning shell but with the important difference that the convective H shell is moving away from its lowest mass coordinate. The upward migration of the lower boundary leaves a shallow Ω-gradient behind, at Mr ≈ 9.5 M on the left-hand side in Fig. 9. In this case, the steep Ω-gradient and the μ-gradient do not coincide, and a region with low values of Dshear develops, i.e., Dshear between 10 and 104 cm2 s−1 . The mixing across the bottom of the convective shell is thus less efficient and abundance gradients are steeper below the convective shell (just below 10 M in the right panel of Fig. 9) • Case (c), shown in Fig. 10, is the case with no convective zone in the H-rich layers and only a moderate Ω-gradient across the H burning shell. At the mass coordinate, where abundance gradients are steepest (at 10 M in the right panel of Fig. 10), the shear diffusion coefficient is weakest, with Dshear between 1 and 103 cm2 s−1 . During helium burning, case (c) may follow case (b). In this situation, the Ω-gradient is even lower at the bottom of the H burning shell and Dshear has the lowest values. If there is no convective H burning shell, then case (c) is the only case the model goes through.

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The rotating models of Frischknecht et al. (2016) listed in Table 1 can be used to determine the mass and metallicity dependence of rotation-induced mixing. The solar metallicity 15, 20, and 25 M models, as well as the 15 M with subsolar Z, do not develop a convective zone at the inner edge of the hydrogen-rich layers during central He burning. Thus, mixing in these models corresponds to case (c). The subsolar Z models with 20, 25, and 40 M , as well as the 40 M Z = Z model, develop before the start of central He burning, a convective H shell where the H shell burning occurs. It shrinks and retreats when the convective He core grows. These models follow therefore the sequence: (a)–(b)–(c), but with a basic difference between the models at Z = 10−5 and those at higher metallicity. While the latter develop case (b) with a very low Dshear as soon as the convective shell starts to shrink, the former show strong angular momentum transport at the steep Ω-gradient, which is fast enough to follow the retreating convective zone and therefore develops rather a hybrid case between (a) and (b) when the convective shell shrinks. The mixing is thus strongest in Z = 10−5 models, followed by subsolar Z models with 20, 25, and 40 M and the 40 M Z = Z model, and finally followed by the Z 15, 20, and 25 M models and the sub-solar 15 M models. Table 2 summarizes the production of the 14 N and 22 Ne production and the effect of rotation on models across masses and metallicities. Mass factions ΔX(22 Ne) of burned 22 Ne during central He-burning (Since 22 Ne is produced and destroyed at the same time in rotating stars, we derived the amount of 22 Ne burned during central He burning from the sum of the 25 Mg and 26 Mg produced during this stage), Xr (22 Ne) of remaining 22 Ne after He burning, Xshell (22 Ne) of 22 Ne in the He-shell at the preSN stage, and the yields of 22 Ne and 14 N are tabulated for all models. ΔX(22 Ne) is the 22 Ne destroyed mainly by the (n,γ ) and α-capture channels, where the (α,n) channel is the s process neutron source in He burning. Xr (22 Ne) is the 22 Ne left in the He core ashes, and it will be destroyed mostly by the (p,γ ) and (α,n) channels during C burning (e.g., Pignatari et al. 2010). We can see from Table 2 (ΔX(22 Ne, burned)) that rotating models at all metallicities produce and burn significant amounts of 22 Ne. At solar metallicity, 22 Ne is predominantly secondary. At low metallicities, in the models including rotation, mixing is strong enough to produce a pocket of primary 14 N above the convective core, which is then converted to primary 22 Ne. The amount of primary 22 Ne in the convective He core at the end of He burning, when s process is activated, is between 0.1% and 1% in mass fractions. Considering a constant value of υini /υcrit = 0.4 at all metallicities, the primary 22 Ne in the He core decreases slightly with decreasing metallicity. There is, however, theoretical and observational support to consider a slight increase of υini /υcrit with decreasing metallicity as discussed in the previous section. Table 2 also includes models with 25 M and υini /υcrit = 0.4 at Z = Z and 10−3 , υini /υcrit = 0.5 at Z = 10−5 , and υini /υcrit = 0.6 at Z = 10−7 , which correspond to a slight increase of υini /υcrit with decreasing metallicity. Considering a slightly increasing initial rotation rate with decreasing metallicity, rotating models produce and burn a constant quantity of 22 Ne, around 0.5% in mass fraction, almost independent of the initial metallicity.

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Table 2 14 N and 22 Ne production and destruction. See text for explanations (Table taken from Frischknecht et al. 2016) Model A15s0 A15s4 B15s0 B15s4 C15s0 C15s4 A20s0 A20s4 B20s0 B20s4 C20s0 C20s4 A25s0 A25s4 B25s0 B25s4 C25s0 C25s4 C25s4bb C25s5 C25s5bb D25s0 D25s4 D25s4bb D25s6 D25s6bb A40s4 B40s4 C40s4

ΔX(22 Ne)a 3.06(−3) 5.59(−3) 3.54(−4) 9.37(−4) 3.75(−6) 4.84(−4) 5.34(−3) 7.23(−3) 6.16(−4) 3.49(−3) 5.66(−6) 1.52(−3) 7.68(−3) 9.69(−3) 7.52(−4) 4.08(−3) 7.21(−6) 1.23(−3) 1.27(−3) 3.83(−3) 3.75(−3) 8.28(−7) 1.05(−4) 1.06(−4) 4.57(−3) 4.44(−3) 1.23(−2) 3.31(−3) 2.70(−3)

(22 Ne)a

Xr 9.70(−3) 7.42(−3) 8.02(−4) 1.02(−3) 7.70(−6) 3.92(−4) 7.43(−3) 5.03(−3) 5.46(−4) 1.14(−3) 5.74(−6) 4.62(−4) 5.10(−3) 3.28(−3) 4.16(−4) 6.22(−4) 4.14(−6) 1.69(−4) 1.82(−4) 4.94(−4) 4.85(−4) 4.67(−7) 3.81(−5) 3.96(−5) 2.68(−4) 3.11(−4) 5.29(−4) 1.06(−4) 1.93(−5)

(22 Ne)a

Xshell 9.23(−3) 1.38(−2) 9.24(−4) 7.34(−3) 1.02(−5) 7.55(−3) 1.14(−2) 1.99(−2) 1.15(−3) 3.20(−2) 1.32(−5) 1.67(−2) 1.27(−2) 1.56(−2) 1.15(−3) 1.99(−2) 1.13(−5) 1.15(−2) 1.17(−2) 1.59(−2) 1.61(−2) 3.09(−7) 1.46(−2) 1.45(−2) 1.95(−2) 2.00(−2) 1.21(−2) 2.08(−2) 8.75(−3)

m(22 Ne)a [M ] 9.11(−3) 2.78(−2) 1.28(−3) 1.49(−2) 6.42(−5) 1.39(−2) 2.50(−2) 4.99(−2) 2.68(−3) 7.59(−2) 1.21(−4) 4.09(−2) 3.39(−2) 4.06(−2) 3.36(−3) 6.72(−2) 2.38(−4) 3.61(−2) 3.49(−2) 4.80(−2) 4.81(−2) 1.63(−4) 3.81(−2) 3.71(−2) 5.52(−2) 5.56(−2) 3.34(−2) 7.99(−2) 3.21(−2)

m(14 N)a [M ] 3.19(−2) 2.63(−2) 2.91(−3) 7.17(−3) 4.77(−5) 5.25(−3) 3.76(−2) 3.72(−2) 4.06(−3) 9.39(−3) 5.80(−5) 4.04(−3) 4.76(−2) 4.95(−2) 5.90(−3) 8.47(−3) 9.38(−5) 1.85(−3) 9.33(−4) 2.07(−3) 1.99(−3) 1.80(−5) 1.10(−2) 1.11(−2) 3.43(−3) 3.48(−3) 2.23(−2) 1.84(−2) 2.07(−3)

in brackets are the exponents (x(y) = x × 10y ) model was calculated with the same initial parameters as the model, on the line above, but with the 17 O(α, γ ) reaction rate of CF88 divided by 10 a Values b This

These results show that significant amounts of 22 Ne are expected to be produced in massive rotating stars over the entire range of masses and all metallicities. The convective He shell, which follows on the 14 N-rich zone, transforms most of this 14 N into 22 Ne. While the 22 Ne in the He shell of nonrotating model is purely secondary, in rotating models it is primary at the pre-SN stage and almost independent of metallicity. The 22 Ne is only partially destroyed during

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the He shell burning, and there is a mass fraction of X(22 Ne) between 0.7% and 3.2% in the He layers at the pre-SN stage. This is relevant for explosive neutron capture nucleosynthesis in He shell layers. This site was investigated by Blake and Schramm (1976), Truran et al. (1978), and Thielemann et al. (1979) as a possible r process scenario, but later on found to be unlikely (Blake et al. 1981). Instead, the explosive shell He burning in core-collapse supernovae is hosting the n process (e.g., Blake and Schramm 1976), with typical abundance signatures identified in presolar silicon-carbide grains of type X (e.g., Meyer et al. 2000; Zinner 2014). It will be worthwhile to explore in the future the impact of these large amounts of primary 22 Ne produced in rotating models at all Z, for explosive neutron capture nucleosynthesis. Carbon shell burning is the second efficient s process production site inside massive stars at solar metallicity (e.g., Raiteri et al. 1991b; The et al. 2007; Rauscher et al. 2002; Pignatari et al. 2010). One might think that rotation-induced mixing would occur in the same way as during He burning, mixing down some of the primary 22 Ne into the C shell and boosting the s process. However, the time scale of the secular shear mixing, which is still present between convective He and C shells, is of the same order as during central He burning. On the other hand, the burning time scales of Ne, O, and Si are at least 5 to 6 orders of magnitude smaller than that of He burning. This implies that the 22 Ne available to make neutrons via the 22 Ne(α,n) reaction in the convective C burning shell is what is left in the ashes of the previous convective He core, like in nonrotating models. Rotation, nevertheless, affects the CO core sizes and the 12 C/16 O ratio after He burning (e.g., Hirschi et al. 2004). This will indirectly affect all subsequent burning phases and their heavy element production.

Impact of Rotation on the s Process Rotation significantly changes the structure and pre-SN evolution of massive stars (Hirschi et al. 2004) and thus also the s-process production. Rotating stars have central properties similar to more massive nonrotating stars. In particular, they show typically 30% to 50% larger He cores and CO cores than the nonrotating models (see also Heger and Langer 2000; Chieffi and Limongi 2013). A 20 M star with rotation has thus a core size which is almost as large as the one of a 25 M nonrotating star. The higher core size means higher central temperatures at the same evolutionary stage, and consequently the 22 Ne+α is activated earlier. In these conditions the He core s-process contribution increases at the expense of the C shell contribution. Since in He burning conditions the amount of neutrons captured by light neutron poisons and not used for the s process is lower compared to C burning conditions, an overall increase of the s-process efficiency is obtained (see also Pignatari et al. 2010). At solar metallicity the difference between rotating and nonrotating stars is mainly found in the core size, but not in the amount of available 22 Ne. This becomes clear if one compares X(22 Ne) = ΔX(22 Ne) + Xr (22 Ne) of the A series models

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in Table 1. In mass fraction, X(22 Ne) ≈ 1.3 × 10−2 is available for α-captures, which is therefore mainly secondary. Similar values are obtained in both rotating and nonrotating models. The difference in s-process efficiency is therefore mainly due to the rotation-induced larger core size and the related impact on temperature (higher) and density (lower). The difference in the neutron exposure is due to higher fraction of burned 22 Ne. The difference in s processing between rotating and nonrotating stars saturates around 25 M (Langer et al. 1989). The saturation is caused by the exhaustion of 22 Ne from this mass upward (e.g., model A40s4 has burned 96% of available 22 Ne after He burning). In Fig. 2 the overproduction factors of 25 M models (A25s0 and A25s4) with solar metallicity after the end of He burning are shown. Model A25s4 (circles) shows only a moderate increase of the s process production with respect to A25s0 (diamonds). Both models produce heavy isotopes from iron seeds up to the Srpeak (A ≈ 90). In A25s0 model, 66% of Fe is destroyed and in A25s4 73%. The varying overproduction factors ( = 1) beyond A = 90 are the signature of a local redistribution of preexisting heavy nuclei. This figure therefore illustrates that not only the s-process quantities given in Table 1 are similar, but also the abundance pattern of rotating and nonrotating models at solar Z is almost identical. The difference in the efficiency is mostly caused by the larger core size in the rotating models. At subsolar metallicities the differences between rotating and nonrotating models are much more striking. Rotating models have much higher neutron exposures compared to nonrotating stars, which is due to the primary 22 Ne produced and burned during central He burning (see previous section). This is also illustrated by the 3 to 270 times higher amount of 22 Ne burned in rotating stars up to central He exhaustion, depending on the initial mass (or MCO ) and metallicity. The large production of neutrons by 22 Ne is partially compensated by the larger concentration of 25 Mg and 22 Ne itself, which become primary neutron poisons in rotating massive stars (Pignatari et al. 2008). Figures 3, 4, and 5 show the abundance normalized to solar in the CO core of 25 M stars with Z = 10−3 , Z = 10−5 and Z = 10−7 just after central He exhaustion, each for a rotating (circles) and a nonrotating model (diamonds). Going from Z = Z (Fig. 2) to Z = 10−3 and 10−5 (Figs. 3 and 4), the production of nuclei between A = 60 and 90 vanishes in the nonrotating models, which is what is expected from the combination of secondary neutron source, secondary seeds, and primary neutron poisons. The nonrotating model at Z = 10−7 (D25s0, diamonds in Fig. 5) is special with its small amount of primary 22 Ne. The rotating models at subsolar Z produce efficiently up to Sr (Z = 10−3 ), Ba (Z = 10−5 ), and finally up to Pb (Z = 10−7 ). At the same time, the consumption of iron seeds increases from 74% at Z = Z (A25s4) to 96% (B25s4), 97% (C25s4), and 99% (D25s6) at Z = 10−3 , Z = 10−5 , and Z = 10−7 , respectively. Also with the standard rotation rate υini /υcrit = 0.4, around 90% of initial Fe is destroyed in models with 25 M and Z < Z . Hence, already from the s process in He burning, one can conclude that the primary neutron source in the rotating models is sufficient to deplete all the seeds and the production is limited by the seeds (not the neutron source any more). The other stellar masses show similar trends with

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Z. It is interesting to look at the rotation dependence of the nonstandard s process production. At Z = 10−5 the faster rotating model (C25s5) does not produce more heavy isotopes beyond iron compared to the one with standard rotation (C25s4). Instead, what happens is that not only iron is depleted but elements up to Sr are partially destroyed (after being produced) and heavier elements like Ba are produced. Even at the lowest metallicities in a very fast rotating model (D25s6 and D25s6b, υini /υcrit = 0.6 instead of the standard 0.4), and thus with a larger primary neutron source, there is no additional production of s process elements starting from light element seeds like 22 Ne. Indeed, going from [Fe/H] = −3.8 (C25s4) to [Fe/H] = −5.8 (D25s4), the Sr yield decreases by a factor of ∼9, while the Ba yield increases by a factor of 5. Hence, the production is limited mainly by the iron seeds. The normalization to solar composition allows to compare the low Z models in Figs. 3, 4, and 5 to the solar Z models in Fig. 2 with respect to their total production. Model B25s4 produces overall similar amounts of heavy nuclei in the range A = 60–90 as models A25s0 and A25s4. A closer look reveals that the solar metallicity models produce higher amounts beyond Fe up to Ge. For isotopes of As, Se, Br, and Kr, A25s0, A25s4, and B25s4 produce similar amounts, while for Sr, Y, and Zr, B25s4 produces more. However, here one has to keep in mind that for the final picture, the shell C burning contribution has to be taken into account. The impact on GCE of these results has been discussed elsewhere (e.g., Cescutti et al. 2013). However, according to models A25s0, A25s4, and B25s4 compared to C25s5 (Fig. 4), rotating stars at Z = 10−5 (initial [Fe/H] = −3.8) probably do not contribute significantly to the s process chemical enrichment at solar Z, because the X/X values are only around 1 or lower for C25s5. This is confirmed for the model D25s6 in Fig. 5. For the Sr, Y, and Zr, a small contribution from rotating stars with Z between 10−3 (initial [Fe/H] = −1.8) and 10−5 can nevertheless be expected. Instead, for the nonrotating stars, the s process contribution is already negligible at 10−3 . Rotation only has a mild impact on the He shell contribution. Rotation-induced mixing widens the radiative zone where 22 Ne(α, n) is activated to about 0.4 M in rotating stars (compared to 0.2 M in nonrotating models). As explained in the previous section, the contribution to the total s process yields is therefore low in these models and only in the region of 5% for solar metallicity 25 M stars with and without rotation. For less massive stars, the He shell gains more weight and produces in 15 M models with rotation up to 50% of the total yields. In Fig. 11, the ratio of abundances after shell C burning to the abundances after core He burning, XC /XHe is plotted for the rotating 25 M model at Z = Z (A25s4). As in the nonrotating Z = Z model, the high neutron densities lower the s process branching ratios. Rotating models with 15 M still produce 64 Zn, 80 Kr, and 86 Sr in the C shell, while in 20 M and heavier stars, these isotopes are depleted due  to the large neutron densities favoring the neutron capture channel at the s process branching points 63 Ni, 79 Se, and 85 Kr (e.g., Pignatari et al. 2010). This effect mainly occurs at solar Z (or higher), but it is still relevant at lower metallicities to calculate the complete s process pattern.

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Fig. 11 Ratio of abundances after shell C burning to the abundances after core He burning, XC /XHe , in a rotating 25 M star at Z = Z (A25S4). It illustrates the modification of the abundances by s process in shell C burning

Relative Contributions Between Helium and Carbon Burning and Total s Process Yields as a Function of Mass and Metallicity In Fig. 12 the yields of 68 Zn of the three s process contributions (from top to the bottom: He core, C shell, and He shell burning yields; see Fig. 1 for the division between these three contributions) normalized to the total yields are displayed, for nonrotating stars on the left-hand side and rotating stars on the right-hand side. 68 Zn is chosen as a representative for the isotopes in range A = 60–80, because it is produced by the s process in all three phases. This figure allows to compare the contributions of the three different sites to the total yields. The following points can be seen. (1) In general, the contribution from He core burning (colors yellow to red in Fig. 12a and b) dominates over the other two phases overall. (2) Shell carbon burning is, compared to the other two sites, only efficient at solar metallicity (see Fig. 12c and d). The weak contribution at low-Z is due to the low amount of 22 Ne left, the smaller amount of seeds, and the primary neutron poisons, which have an increased strength toward lower Z in C shell conditions. The only massmetallicity range for which the C shell dominates is at solar Z with M  25 M for non-rotating models and with M  20 M for rotating models. Such a dominant contribution from C shell is not seen in older studies (e.g., The et al. 2007). This may be due to the high 22 Ne(α, γ ) rate of NACRE, which is in strong competition to the neutron source during central He burning and dominates for stars with M  20 M . This inhibition during He core burning is weaker for rotating stars since they have higher central temperatures. (3) Shell He burning contributes only a small fraction, typically 5% to the final yields (see Fig. 12e and f). The exceptions are the rotating 15 to 25 M stars at low Z and rotating 15 to 20 M stars at solar Z. It is the effect of decreasing contribution from the He core toward lower masses and the higher burning temperatures in the shell compared to the He core, which allows an

3556

R. Hirschi

0.0

0.0 1.05

0.75

−1.5

0.45 −2.0

−3.0 16

18

20 Mass M

22

0.75 0.60

−1.5

0.45 −2.0

0.30

−2.5

0.90

−1.0 log10 (Z/Z )

0.60

(b)

0.15

−2.5

0.00

−3.0

24

0.30 0.15 0.00

15

20

25 30 Mass M

35

40

0.0

0.0

1.05

1.05 0.90 0.75 0.60

−1.5

0.45 −2.0

−3.0 16

18

20 Mass M

22

0.90 0.75 0.60

−1.5

0.45 −2.0

0.30

−2.5

(d)

−1.0 mC−sh /mtot

−1.0

0.15

−2.5

0.00

−3.0

24

0.30 0.15 0.00 20

15

0.0

30 25 Mass M

35

40

0.0 1.05

0.75 0.60

−1.5

0.45 −2.0

0.30

−3.0

0.00

−3.0

22

24

0.60 0.45

0.15

20 Mass M

0.75

−2.0 −2.5

18

0.90

−1.5

−2.5

16

(f)

−1.0 log10 (Z/Z )

−1.0 log10 (Z/Z )

1.05 −0.5

0.90

mHe−sh /mtot

−0.5

(e)

15

mHe−sh /mtot

log10 (Z/Z )

−0.5

mC−sh /mtot

(c)

log10 (Z/Z )

−0.5

mHe−core /mtot

−0.5

0.90

−1.0 log10 (Z/Z )

1.05

(a)

mHe−core /mtot

−0.5

0.30 0.15 0.00 20

25 30 Mass M

35

40

Fig. 12 s process site yields of 68 Zn normalized to the total yields to illustrate the different relative contributions as a function of mass and metallicity Z, for He core without (a) and with rotation (b), for C shell without rotation (c) and with rotation (d), and the He shell without (e) and with rotation (f). The red circles display the location of the stellar models of Frischknecht et al. (2016) in the mass-metallicity space. The values in between the data points are interpolated linearly in log(m). Note that decayed yields are plotted in this figure

96 Slow Neutron-Capture Process in Evolved Stars

3557

efficient activation of 22 Ne(α, n) in the 15 M models. Additionally, the He shell is not limited by the diminished iron seeds consumed by s process in He core but occurs in a region still containing its initial iron content. In Fig. 13, the dependence of the total 68 Zn yields on the mass and metallicity is displayed for rotating stars with standard rotation rate (υini /υcrit = 0.4) on the righthand side and for nonrotating stars on the left-hand side. As mentioned above, 68 Zn is representative for the isotopes in range A = 60-80. A similar plot for the neutronmagic isotope 88 Sr is presented in Fig. 14 to show the dependence of the Sr-peak

−0.5

−5.8 log10 (m)

−8.2

log10 (Z/Z )

log10 (Z/Z )

−7.4

−2.0

−5.0 −5.8

−1.0

−6.6

−1.5

−4.2

−0.5

−5.0

−1.0

−3.4

0.0

−4.2

−6.6

−1.5

−7.4 −8.2

−2.0

−9.0

−9.0 −2.5

−2.5

−9.8 −10.6

−3.0 16

18

20 Mass M

22

log10 (m)

−3.4

0.0

−9.8 −10.6

−3.0 15

24

20

30 25 Mass M

35

40

Fig. 13 s process yields, m, of 68 Zn in M to illustrate the mass and metallicity dependence of the s process, without rotation on the left-hand side and with rotation on the right-hand side. The red circles display the location of the stellar models of Frischknecht et al. (2016) in the massmetallicity space. The values in between the data points are interpolated linearly in log(m) 0.0

−5

0.0

−5

−0.5

−6

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−6

log10 (Z/Z )

−9

log10 (m)

log10 (Z/Z )

−8

−1.5

−7

−1.0

−8

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log10 (m)

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−1.0

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−11 −3.0 16

18

20 Mass M

22

24

−12

−3.0 15

20

30 25 Mass M

35

40

−12

Fig. 14 s process yields, m, of 88 Sr in M to illustrate the mass and metallicity dependence of the s process, without rotation on the left-hand side and with rotation on the right-hand side. The red circles display the location of the stellar models of Frischknecht et al. (2016) in the massmetallicity space. The values in between the data points are interpolated linearly in log(m)

3558

R. Hirschi

production on rotation (86 Sr, 87 Sr, 89 Y, and 90 Zr show the same trends as 88 Sr). Several differences between the standard and rotation boosted s process can be seen: (1) Rotating models clearly produce more s process elements at all metallicities. (2) Whereas the s process production in nonrotating model decreases steeply with metallicity (dependence steeper than linear, e.g., Pignatari and Gallino 2008), the 68 Zn yields of rotating stars show a secondary-like behavior, going from reddish to blueish colors toward lower Z. While the 68 Zn yields of nonrotating stars drop by 5 orders of magnitude when the metallicity goes down by a factor 103 , the yields from rotating stars drop only by a factor 103 . The scaling with metallicity is thus much less steep for rotating models. (3) Furthermore, the Sr-peak isotopes do not show a secondary behaviour for stars with rotation and M > 15 M in the metallicity range between solar (log(Z/Z ) = 0) and about one hundredth (Z = 1.4 × 10−4 , log(Z/Z ) = −2) of solar metallicity, but they eject maximal absolute yields around one tenth of solar metallicity (dark red around log(Z/Z ) = −1) for 20 to 30 M stars (see Fig. 14; see also Fig. 7 in Choplin et al. (2018) showing how this significant production takes place for massive stars up to 150 M at Z = 0.001).

Comparison Between Models and Observations Production of Elements at the Sr and Ba Peaks Spectroscopic observations have shown a secondary trend of [Cu/Fe] (e.g., Bisterzo et al. 2005; Sobeck et al. 2008, and references therein), in agreement with s-process calculations which predicts that a major part of Cu come from the s process in massive stars (e.g., Pignatari et al. 2010). The same trend is expected for Ga, for which only few observations and upper limits are available from low-metallicity stars and not a real comparison can be made, and for Ge (see discussion in Pignatari et al. 2010). More data is available for Ge compared to Ga (Cowan et al. 2005), but the metallicity range of interest is still not fully covered by observations. As explained above, rotation does not change the secondary nature of the s process production of these elements. Travaglio et al. (2004) compared the spectroscopic observations of the Sr-peak elements Sr, Y, and Zr at different metallicities with the s process distribution in the solar system obtained from GCE calculations. They proposed that a lighter element primary process (LEPP) was responsible for both the observations and the missing s process abundances in the solar distribution. Later, Montes et al. (2007) compared the “stellar LEPP” signature at low metallicity with the “solar LEPP” in the solar system, concluding that while they are compatible, also explosive nucleosynthesis processes can be responsible for the same elemental signature in the early galaxy. While the existence of the solar LEPP has been recently questioned (Maiorca et al. 2012; Cristallo et al. 2015), one cannot exclude that an additional s process component is needed to contribute to its total amount. It is unlikely that the s process in fast-rotating massive stars is the solution to the LEPP, due

96 Slow Neutron-Capture Process in Evolved Stars

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to its secondary nature and its significance only at much lower metallicities for elements in the Sr mass region and heavier. On the other hand, Cescutti et al. (2013) and Barbuy et al. (2014) showed that s process in fast-rotating massive stars is compatible with observations at low metallicity (e.g., Hansen et al. 2013). Alternative or complementary theoretical scenarios proposed to explain the stellar LEPP are explosive nucleosynthesis components, mainly associated with neutrinodriven winds on top of the forming neutron star (e.g., Fröhlich et al. 2006; Qian and Wasserburg 2008; Farouqi et al. 2009; Arcones and Montes 2011). There is a lot of scatter in the production up to Ba in stellar models (see below and the comparison of models in Rizzuti et al. (2021)), which is strongly affected by nuclear uncertainties. Additionally, a scatter in Sr production is intrinsic to the rotation boosted s process, since a varying rotation rate would lead to a varying amount of primary 22 Ne and thus to a varying neutron exposure and s process production, respectively. Typically, the s process in massive stars produces only minor amounts of Ba and [Sr/Ba] around +2, with an upper limit of ≈ + 2.3. However, due to the seed limitation and the larger neutron capture per iron seed, the enhanced s process in fast-rotating massive stars can produce more significantly also elements at the Ba neutron-magic peak. On the other hand, as shown by Pignatari et al. (2013), the intrinsic nature of 22 Ne as a neutron source and neutron poison does not allow to efficiently feed also heavier elements along the s process path, up to Pb (see however predictions of Limongi and Chieffi 2018; Banerjee et al. 2019).

The Very Low-Z Stars: The Case of CEMP-No Stars At metallicities [Fe/H]-2 it is possible to observe a large number of “carbonenhanced metal-poor” (CEMP; [C/Fe]> 0.7 (Aoki et al. 2007) stars, which exhibit large excesses of carbon with values of [C/Fe] reaching more than 3.0 dex. At the same time, nitrogen, oxygen, and other elements are also largely overabundant. These stars are very old low-mass stars (about 0.8 M ) still surviving and exhibiting the particular nucleosynthetic products of the first stellar generations. CEMP stars were classified in CEMP-s, CEMP-r/s, and CEMP-r (e.g., Beers et al. 1992; Beers and Christlieb 2005), depending on the observed abundances of s-elements (mainly Ba) and r-elements (mainly Eu). Another group was identified: the CEMP-no stars, with much weaker overabundances of n-capture elements (typically [Ba/Fe]< 1). Nevertheless, a fraction of them contains measurable amounts of heavy s-elements. The CEMP-no stars clearly dominate for low-metallicity stars with [Fe/H] = |n >. F (Z, ω) is the Fermi function, and Se (ω) is the Fermi-Dirac distribution for electrons, where the chemical potential is determined from the density ρYe as shown in Eq. (1). In the case of β-decay, t+ is replaced by t− , t− |n >= |p >, and Φ ec in Eq. (4) is replaced by 

Qij

Φ β (Qij ) =

ωp(Qij − ω)2 F (Z + 1, ω)(1 − Se (ω))dω.

(6)

1

Transitions from the excited states of the parent nucleus are included with the partition function Wi as they can be important at high temperatures when the excitation energies are low. Because of the factors Se (ω) and 1 − Se (ω) in the integrals of Φ ec and Φ β , respectively, the e-capture (β-decay) rates increase (decrease) as the density and the electron chemical potential increase. Coulomb corrections to the rates are taken into account (Juodagalvis et al. 2010; Toki et al. 2013; Suzuki et al. 2016). The Coulomb corrections affect the thermodynamic properties of a high density plasma. The interaction of ions in a uniform electron background leads to the corrections to the equation of state of matter and modifies the chemical potential of the ions. The effects on the weak rates are mainly caused by the modification of the threshold energy, ΔQC = μC (Z − 1) − μC (Z),

(7)

97 Weak Interactions in Evolving Stars

3577

where μC (Z) is the Coulomb chemical potential of the nucleus with charge number Z (Slattery et al. 1982; Ichimaru 1993; Yakovlev and Shalybkov 1989). The Coulomb correction of the chemical potential of the ion with charge Z is given 2 by μC (Z) = kTf (Γ ), where Γ = Z 5/3 ( kTe ae ) with ae = ( 4π3ne )1/3 and ne the electron density is the Coulomb coupling parameter. For the strong-coupling regime (Γ > 1), results of Monte Carlo calculations for one-component plasma (Slattery et al. 1982) have been parametrized by a function of Γ (Ichimaru 1993). For the weak-coupling regime (Γ < 0.1), f (Γ ) is obtained by cluster-expansion method (Abe 1959). Γ 3/2 11 Γ3 3 f (Γ ) = − √ − [ ln(3Γ ) + 0.57721 − ] (Γ < 0.1) 2 4 12 3 f (Γ ) = −0.8980Γ + 3.8714Γ 1/4 − 0.8828Γ −1/4 − 0.8610lnΓ −2.5269 (Γ > 1)

(8)

The first term for the weak-coupling regime is the Debye-Huckel limit, and the second term is the next-order correction. For the intermediate coupling (0.1 < Γ < 1), interpolation of weak and strong couplings is usually taken, but this leads to a rather complex formula (Ichimaru 1993). Instead, a simple√parametrization proposed in Yakovlev and Shalybkov (1989), f (Γ ) = −Γ 3/2 / 3 + β/γ Γ γ , is used here for 0 < Γ < 1. For small Γ , it becomes the Debye-Huckel limit and smoothly connected to the strong-coupling region at Γ = 1 with β = 0.2956 and γ = −1.9885. The correction of the ion chemical potential is negative, and its magnitude increases as Z increases. This results in an enhancement of the threshold energy, and the e-capture (β-decay) rates are reduced (enhanced) by the Coulomb effects. Another Coulomb correction to the rates comes from the reduction of the electron chemical potential. The amount of the reduction is evaluated by using the dielectric function obtained by relativistic random phase approximation (RPA) (Itoh et al. 2002). This correction also leads to a slight reduction (enhancement) of e-capture (β-decay) rates. The e-capture and β-decay rates for the 23 Na-23 Ne and 25 Mg-25 Na pairs are shown in Fig. 1. Here, all the GT transitions from the states with excitation energies below 2 MeV are taken into account. We see from Fig. 1 that both e-capture and β-decay processes take place simultaneously almost independent of the temperature at a certain density. Such a density is called a ‘Urca density’, where cooling of the core occurs efficiently as the core evolves rather slowly in the density region. The Urca densities for the A = 25 and 23 pairs are log10 (ρYe ) = 8.8 and 9.0, respectively. The cooling of the core takes place first by the Urca process for the 25 Mg-25 Na pair and then for the 23 Na-23 Ne pair. The neutrino-energy-loss rates and γ -ray heating rates are also evaluated and tabulated together with the rates with fine grids for log10 (ρYe ) = 8.0–11.0 and log10 T = 7.0–9.65 (Suzuki et al. 2016).

3578

T. Suzuki

Fig. 1 Electron-capture and β-decay rates for the (a) A = 23 and (b) A = 25 Urca nuclear pairs obtained with the USDB with the Coulomb effects. Solid and dashed curves show e-capture and β-decay rates, respectively, as functions of density log10 (ρYe ) for temperatures, log10 T = 8.0–9.2 in steps of 0.2

Fig. 2 The evolution of the central temperature Tc of the 8.8 M star as a function of time in units of years. During t = 0 to 4 yr, the central density of the core increases from log10 (ρ) = 9.0 to 9.4. Cooling of the ONeMg core of the 8.8 M star by the nuclear Urca processes of the 25 Mg-25 Na and 23 Na-23 Ne pairs is shown by the lower curve. The time changes of the amounts of 25 Mg and 23 Na (mass fractions X ) are also shown. The upper curve shows the case with the rates of Oda c et al. (1994) without using the fine grids for densities and temperatures. (From Toki et al. 2013)

Cooling of the ONeMg Core by Nuclear Urca Processes The cooling of the ONeMg core is realized by the Urca processes of the nuclear pairs with A = 23 (23 Na-23 Ne) and A = 25 (25 Mg-25 Na). The time evolution of the central temperature in the 8.8 M star is shown in Fig. 2.

97 Weak Interactions in Evolving Stars

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Two distinct drops of the temperature due to A = 25 Urca cooling up to around log10 ρc = 9.1 (t = 1 yr) and due to A = 23 Urca cooling between log10 ρc = 9.15 and 9.25 (t = 2 yr and t = 2.5 yr) are noticed. When the temperature drops take place, the abundances of 25 Mg and 23 Na also drop due to the e-captures eventually dominating over β-decays. After the Urca cooling, the 8.8 M star evolves to double e-captures on 24 Mg and 20 Ne that leads to heating of the core and ignition of oxygen deflagration and becomes an e-capture SN (ECSN). If the contraction of the core is fast enough, it will collapse; otherwise thermonuclear explosion may occur. The final fate, collapse or explosion, is determined by the competition between the contraction of the core due to e-captures on post-deflagration material and the energy release by the propagation of the deflagration flames. This subject is discussed in the next section. Cases for progenitor masses of 8.2 M , 8.7 M , 8.75 M and 9.5 M are also investigated up to the ignition of the oxygen deflagration in Jones et al. (2013). The 8.2 M star is found to end as an ONe WD. The 8.75 M and 8.7 M stars evolve to the double e-capture processes and would become ECSN. The 9.5 M star evolves to Fe-CCSN. More detailed discussion is found in Jones et al. (2013). After the cooling of the ONeMg core caused by the nuclear Urca processes for the A = 25 and 23 pairs, successive e-capture reactions are expected to be triggered at higher densities corresponding to the Q-values at late stages of the evolution. The products of e-capture and β-decay rates for various nuclear pairs are shown in Fig. 3. The peak position for each pair is determined by the Q-value of each transition. The Urca densities for the pairs, 24 Na-24 Mg, 25 Ne-25 Na, 23 F-23 Ne, and 27 Na-27 Mg increase in this order. For the pair 20 F-20 Ne, GT transition cannot occur between the ground states as the ground state of 20 F is 2+ , while that of 20 Ne is 0+ . In this case, the contributions from the forbidden transitions between the ground states are taken into account (Martínez-Pinedo et al. 2014). The peak at lower density at log10 (ρYe ) ≈9.2 corresponds to the transition between 20 Ne (2+ ) and 20 F (2+ ), while the peak at higher density at log10 (ρYe ) ≈9.7 corresponds to the transition between 20 Ne (0+ ) and 20 F (1+ ). The peak at log10 (ρYe ) ≈9.5 corresponds to the second-forbidden transition between the ground states. The heating of the cores occurs by double

Fig. 3 Products of electron-capture and β-decay rates for the nuclear pairs, 25 Ne-25 Na, 23 F-23 Ne, 27 Na-27 Mg, 24 Na-24 Mg, and 20 F-20 Ne for a temperature log10 T = 8.75 as functions of density log10 (ρYe ). (From Suzuki et al. 2016)

3580

T. Suzuki

e-capture reactions on 24 Mg and 20 Ne due to the pairing effects, that is, by successive e-capture reactions on 24 Na and 20 F following the capture reactions on 24 Mg and 20 Ne, respectively. The Urca process for the nuclear pair, 25 Ne-25 Na, occurs just after the ignition of the e-capture on 20 Ne by the forbidden transition.

Electron-Capture on 20 Ne and Heating of the ONeMg Core When the core is compressed and the core mass becomes close to the Chandrasekhar mass, the core undergoes exothermic electron captures on 24 Mg and 20 Ne that release enough energy to cause thermonuclear ignition of oxygen fusion and an oxygen-burning deflagration. The final fate of the core, whether collapse or explosion, is determined by the competition between the energy release by the propagating oxygen deflagration wave and the reduction of the degeneracy pressure due to electron captures on the post-deflagration material in nuclear statistical equilibrium (NSE). As the energy release by double electron captures in A = 20 and 24 nuclei is about 3 and 0.5 MeV per a capture, respectively, the heating of the core due to γ emissions succeeding the reactions, 20 Ne (e− , νe ) 20 F (e− , νe ) 20 O, is important in the final stage of the evolution of the core. We discuss the effects of the forbidden transition in the e-capture processes in 20 Ne on the evolution of the final stages of the high-density electron-degenerate ONeMg cores.

Electron-Capture Rates for the Forbidden Transition 20 Ne (0+ g.s. ) → 20 F (2+ ) g.s. The weak rates for A=20 pairs, 20 Ne-20 F and 20 F-20 O, were evaluated in MartínezPinedo et al. (2014), Suzuki et al. (2016), and Takahara et al. (1989). In the case of the 20 Ne-20 F pair, the transitions between the ground states are forbidden, and 20 F the main GT contributions come from transitions between 20 Ne (0+ g.s. ) and (1+ , 1.057 MeV) and those between 20 Ne (2+ , 1.634 MeV) and 20 F (2+ g.s. ). The rates were obtained with only the GT transitions in Takahara et al. (1989). The contribution from the second-forbidden transition between the ground states was also included in Martínez-Pinedo et al. (2014) and Suzuki et al. (2016) by assuming that the transition is an allowed GT one with the strength determined to reproduce log ft = 10.5, which was the experimental lower limit value for the β-decay, 20 F 20 + (2+ g.s. ) → Ne (0g.s. ) (Ajzenberg-Selove 1987). However, this is not a satisfactory prescription as the strengths for forbidden transitions generally depend on lepton energies contrary to the case for allowed transitions. Recently, a new log ft value for the β-decay was measured: log = 10.89±0.11 (Kirsebom et al. 2019a). Here, we evaluate the weak rates for forbidden transitions in proper ways by using the multipole expansion method of Walecka (1975) as well as the method of Behrens-B¨uhring (Behrens and Bühring 1971). Electrons are treated as plane waves in the method of Walecka, while in the method of Behrens-B¨uhring, electrons are

97 Weak Interactions in Evolving Stars

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treated as distorted waves in a Coulomb potential, and coupling terms between the transition operators and the Coulomb wave functions are taken into account. The latter method is more accurate, but its formula is rather complex for primers. We start from the method of Walecka that is easier to handle, and compare it with that of Behrens-B¨uhring clarifying their differences. The difference in the calculated rates between the two methods proves to be small for the present case with Z = 10 as far as the conserved-vector-current (CVC) relation for the electric quadrupole (E2) operator is fulfilled. The e-capture rates for finite density and temperature are given as (Walecka 1975; O’Connell et al. 1972), λ

ecap

V 2 gV2 c (T ) = ud π 2 (hc) ¯ 3

σ (Ee , T ) =



∞

σ (Ee , T )Ee pe cSe (Ee )dEe Eth

 (2Ji + 1)e−Ei /kT  σf,i (Ee ) G(Z, A, T ) i

G(Z, A, T ) =



f

(2Ji + 1)e−Ei /kT ,

(9)

i

where Vud = cos θC is the up-down element in the Cabibbo-Kobayashi-Maskawa quark mixing matrix with θC the Cabibbo angle; gV = 1 is the weak vector coupling constant; Ee and pe are electron energy and momentum, respectively; Eth is the threshold energy for the electron capture; and Se (Ee ) is the Fermi-Dirac distribution for electron. The cross section σf,i (Ee ) from an initial state with Ei and Ji to a final state with excitation energy Ef and angular momentum Jf is evaluated with the multipole expansion method (Walecka 1975; O’Connell et al. 1972). G2F F (Z, Ee )W (Eν )Cf,i (Ee ) 2π   1 mag ˆ ˆ · q))[| J ||Ji |2 Cf,i (Ee ) = dΩ( {(1 − (ˆν · q)(β f ||TJ 2Ji + 1 σf,i (Ee ) =

J ≥1

+

| Jf ||TJelec ||Ji |2 ]

− 2qˆ · (ˆν − β)Re Jf ||TJ ||Ji Jf ||TJelec ||Ji ∗ }  2 ˆ ˆ + {(1 − νˆ · β + 2(ˆν · q)(β · q))| J F ||LJ ||Ji | mag

J ≥0

+ (1 + νˆ · β)| Jf ||MJ ||Ji |2 − 2qˆ · (ˆν + β)Re Jf ||LJ ||Ji Jf ||MJ ||Ji ∗ }),

(10)

where q = ν − k is the momentum transfer with ν and k the neutrino and electron momentum, respectively, and qˆ and νˆ are the corresponding unit vectors and

3582

T. Suzuki

β =k/Ee . GF is the Fermi coupling constant, F (Z, Ee ) is the Fermi function, and E2

W (Eν ) is the neutrino phase space factor given by W (Eν ) = 1+Eνν/MT , where Eν = Ee + Q + Ei − Ef is the neutrino energy with Ei and Ef the excitation energies of initial and final states, respectively, and MT is the target mass. The Q value is determined from Q = Mi −Mf , where Mi and Mf are the masses of parent and daughter nuclei, respectively. The Coulomb, longitudinal, transverse magnetic mag and electric multipole operators with multipolarity J are denoted as MJ , LJ , TJ and TJelec , respectively. In the multipole expansion formula, transition matrix elements of the Coulomb multipole with J =0 correspond to allowed Fermi transition. The transition matrix  2 elec,5 2 elements of the axial electric dipole operator, T1 ≈ FA (q ) k 3 j0 (qrk )  1 k , and the axial longitudinal dipole operator, L5 ≈ F (q 2 )  Y 0 (Ωk )σ k t+ A k 1 3 k , where F (q 2 ) is the nucleon axial-vector form factor and j0 (qrk ) Y 0 (Ωk )σ k t+ A FA (0) = gA , correspond to allowed GT transition. In case of first-forbidden transitions, the axial Coulomb and longitudinal multipoles contribute for 0− and 2− , and axial electric and vector magnetic quadrupoles contribute in addition for 2− , while for 1− there are contributions from the Coulomb, longitudinal, and electric dipoles from weak vector current and the axial magnetic dipole from weak axialvector current. For second-forbidden transitions, 0+ ↔ 2+ , the transition matrix elements for Coulomb, longitudinal, and electric transverse operators from the weak vector current as well as axial magnetic operator from the weak axial-vector current with multipolarity J = 2 contribute to the rates.

M2 (q) =



k F1V (q 2 )j2 (qrk )Y 2 (Ωk )t±

k

  1 2 V 2 F1 (q )( j1 (qrk )[Y 1 (Ωk ) × ∇ k ]2 L2 (q) = M 5 k  3 k j3 (qrk )[Y 3 (Ωk ) × ∇ k ]2 )t± + 5   q 3 ∇k 2 elec V 2 F1 (q )( j1 (qrk )[Y 1 (Ωk ) × ] T2 (q) = M 5 q k  2 ∇k 2 k j3 (qrk )[Y 3 (Ωk ) × ] )t± − 5 q  q k μV (q 2 )j2 (qrk )[Y 2 (Ωk ) × σ k ]2 t± + 2M k  mag,5 k T2 = FA (q 2 )j2 (qrk )[Y 2 (Ωk ) × σ k ]2 t± , k

(11)

97 Weak Interactions in Evolving Stars

3583

where M is nucleon mass and μV is the nucleon magnetic form factor (Kuramoto et al. 1990). In the low momentum transfer limit, matrix elements of these operators can be described by using the matrix elements defined by (Schopper 1966)  1 Jf || rk2 C 2 (Ωk )||Ji

x= √ 2Ji + 1 k  1 ∇k 2 y= √ ] ||Ji

Jf || rk [C 1 (Ωk ) × M 2Ji + 1 k  1 u= √ gA Jf || rk2 [C 2 (Ωk ) × σ k ]2 ||Ji

2Ji + 1 k

(12)

 mag,5 4π with C λ = 2λ+1 Y λ . Matrix elements of M2 (q), L2 (q), T2elec , and T2 are described by x, y, and u, respectively.  Cf,i (Ee ) ≡ C(k, ν), where k = |k| = Ee2 − m2e and ν =Eν , becomes (Suzuki 2022) C(k, ν) =

1 2 4 45 x (k

− 43 βk 3 ν +

10 2 2 3 k ν

− 43 βkν 3 + ν 4 ) +

√ 2 6xy(βk 3 − 53 k 2 ν + 53 βkν 2 − ν 3 ) + 45 + 15 y 2 (k 2 + ν 2 + 43 βkν) +

1 2 4 45 u (k

+ 2βk 3 ν +

2 2 2 15 y (k

10 2 2 3 k ν

− 2βkν + ν 2 )

+ 2βkν 3 + ν 4 )

2 yu(βk 3 + 53 k 2 ν + 53 βkν 2 + ν 3 ). − 15

(13)

The first, second, and third terms in Eq. (13) correspond to the Coulomb, the longitudinal, and interference of the Coulomb and the longitudinal form factors, respectively. The next terms proportional to y 2 , u2 , and yu denote the transverse electric, the axial magnetic form factors, and their interferences, respectively. The matrix element y is related to x due to the conservation of the vector current V± (CVC). Here, we follow the discussion in Suzuki (2022). The longitudinal and the transverse E2 operator in the long-wavelength limit can be expressed as (de Forest and Walecka 1966)  L2 (q) =

2 elec i  q 2 rk2 T2 (q) = − ∇ · V±,k Y 2 (Ωk ). 3 q 15

(14)

k

The CVC relation leads to ∇ · V± = −

∂ρ± = −i[H, ρ± ] ∂t

(15)

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 where ρ± = F1V (q 2 ) k δ(r − rk )t± is the time component of V± and H is the total Hamiltonian of the nucleus. In the low momentum transfer limit, we obtain  Jf ||L2 (q)||Ji =

Ef − Ei 2 Jf ||T2 (q)||Ji = − Jf ||M2 (q)||Ji

3 q

(16)

with Ei and Ef are the energies of the initial and final states, respectively. These relations are equivalent to Ef − Ei y=− √ x. 6h¯

(17)

When the electromagnetic interaction is added to the CVC relation (Eichler 1963; Blin-Stoyle 1973), the energy difference is modified to include the isovector part of the electromagnetic interaction rotated into the ± direction in isospin space, that is, the Coulomb energy difference and the neutron-proton mass difference (Fujita 1962a, b), ΔE ≡ Ef − Ei ± VC ∓ (mn − mp ).

(18)

For the transition 20 Ne (0+ , g.s.) → 20 F (2+ , g.s.), this is just the excitation energy of 20 Ne (2+ , T = 1. 10.274 MeV), the analog state of 20 F (2+ , T = 1, g.s.). 20 + The electron-capture rates for the forbidden transition, 20 Ne (0+ g.s. ) → F (2g.s. ), are evaluated with the USDB shell-model Hamiltonian (Brown and Richter 2006) within sd-shell. Calculated shape factor and e-capture rates for the forbidden transition obtained with the CVC relation (Suzuki 2022) are shown in Fig. 4. Here, the quenching factor for the axial-vector coupling constant gA is taken to be q = 0.764 (Richter et al. 2008). In the multipole expansion method of Walecka (1975), leptons are treated as plane waves, and effects of Coulomb interaction between electron and nucleus are taken into account by Fermi function. However, in forbidden transitions, Coulomb distortion of electron wave functions needs more careful treatment. This has been done with explicit inclusion of Coulomb wave functions (Behrens and Bühring 1971; Bühring 1963; Behrens and Janecke 1969). The shape factor for the e-capture rates for the second-forbidden transition in 20 Ne is given as   ν2 ν Ee ν 1 2 Ee 2  C(k, ν) = {[y + x( − ) − u( + ) + 3ξ( x − u1 )]2 3 3 3 5 3 5 3 3 1   ν Ee ν 2 Ee 1 2 k2 2 x − u) } + {[y + x( − ) − u( + ) + ( 9 3 3 3 5 3 5 3   2  2 3ξ 1 ( x − u2 )]2 + ( − u)2 } + 5 3 2 25 3

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Fig. 4 (a) Shape factors as functions of electron energy and (b) e-capture rates for the second− 20 + forbidden transition, 20 Ne (0+ g.s.s ) (e , νe ) F (2g.s. ) as functions of density log10 (ρYe ). The rates are evaluated with the Coulomb (screening) effects at log10 T = 8.6. The dashed and dash-dotted curves are obtained by shell-model calculations with the USDB by the multipole expansion method of Walecka without and with the CVC for the evaluation of the transverse E2 transition matrix elements, respectively. The solid curve is obtained by the method of Behrens-B¨uhring with the CVC for the transverse E2 matrix elements. The short-dashed and dotted curves are results of GT prescription, in which the transition is treated as allowed GT one with the strength determined to give log f t = 10.89 and 10.5, respectively, for the β-decay

  2 2 k2ν 2 2 2 k4 2 2 2 ν4 ( x − u) + x + ( x + u)2 , + 50 3 3 27 3 50 3 3

(19)

with ξ = αZ 2R . Here, R is the radius of a uniformly charged sphere approximating the nuclear charge distribution, and α is the fine structure constant. The electron radial wave functions are solved in a potential of a uniformly charged sphere whose radius is the nuclear radius, leading to the modification of the matrix elements, x and u. The modified matrix elements, x1 , x2 , u1 and u2 , are given in Behrens and Bühring (1971). They are reduced about by 25±3% compared with x and u. When the terms with ξ are neglected, Eq. (19) becomes equivalent to Eq. (13) in 2

ν the limit of me =0.0 except for a term, ( 27 + 2

k2 75 )(

2 3x

− u)2 , which is higher order

in ( mh¯ecc )2 compared with other terms and negligible. As we see from Fig. 4a, the shape factors obtained by the multipole expansion method depend on electron energies, while those of the GT prescription are energy independent. When the CVC relation is used for the evaluation of the transverse E2 matrix elements, the shape factor is enhanced especially in low electron energy region. The difference between the Behrens-B¨uhring (BB) denoted as USDB (BB, CVC) and the Walecka methods, USDB (CVC), is insignificant. The e-capture rates

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obtained with the CVC are also found to be enhanced compared with those without the CVC relation for the transverse E2 matrix element, denoted as USDB, by an order of magnitude at log10 (ρYe ) ≥ 9.6. The longitudinal form factor is always evaluated with the CVC relation in the method of Walecka. The difference between the BB and the Walecka methods with the CVC relation is rather small. The rates by the GT prescription with log f t = 10.5, which was adopted in Martínez-Pinedo et al. (2014), are close to the rates with the CVC at log10 ρYe < 9.6, while they become smaller beyond log10 ρYe = 9.6. Note that the optimum log ft value for a constant shape factor determined from likelihood fit to the experimental β-decay spectrum is equal to 10.46 (see Table III of Kirsebom et al. 2019b). The e-capture rates have been also evaluated in Kirsebom et al. (2019a, b) by the method of Behrens-B¨uhring (BB) with the use of the CVC relation for the transverse E2 matrix element. In Kirsebom et al. (2019b), the experimental strength of B(E2) for the 20 Ne (2+ , 10.273 MeV) is used for the evaluation of transition 20 Ne (0+ g.s. ) → the matrix element x in Eq. (12), which results in a reduction of x by about 27% compared to the calculated value. The rates obtained in Kirsebom et al. (2019a, b) are essentially the same as those denoted by USDB (BB, CVC) in Fig. 4 except for this point, and they are close to each other. The rates in Suzuki et al. (2019) correspond to those denoted as USDB obtained without the CVC relation. The calculated total e-capture rates on 20 Ne including the contributions from both the GT and the second-forbidden transitions (Suzuki 2022) are shown in Fig. 5 for the cases of log10 T = 8.6 and 8.8. The effects of the second-forbidden transition are found to be sizable at log10 ρYe ≈ 9.4–9.7 for log10 T ≤ 8.8. The rates obtained with the CVC relation, denoted by USDB (CVC) and USDB (BB, CVC), are found to be

Fig. 5 Total e-capture rates on 20 Ne with the Coulomb (screening) effects at log10 T =(a) 8.6 and (b) 8.8 as functions of density log10 (ρYe ). The curves denote the same cases as in Fig. 4

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enhanced by an order of magnitude compared with those without the CVC relation (USDB) as well as those denoted by GT (log f t = 10.89) at log10 (ρYe ) ≈9.6 at log10 T = 8.4–8.6. 20 + For the second-forbidden β-decay transition, 20 F (2+ g.s. ) → Ne (0g.s. ), the shape ΔE factors Cf,i (Ee ) are obtained by changing signs of ν, y = − √ x and u in C(k, ν) 6h¯ in Eqs. (13) and (19) for the method of Walecka (1975) and Behrens and Bühring (1971), respectively. Note that ΔE is now negative (ΔE = −10.274 MeV). The log ft values for the β-decay are obtained from f t = ln2 λIβ , where λβ is the β-decay rate for the transition and I is the phase space integral. Calculated log f t value for the β-decay from 20 F (2+ g.s. ) is log f t = 10.70 (10.65) for the Walecka (BehrensB¨uhring) method (Suzuki 2022), which is close to the experimental value, log f t =10.89±0.11 (Kirsebom et al. 2019a).

Heating of the ONeMg Core and Evolution Toward Electron-Capture Supernovae Heating of ONeMg cores by double e-capture reactions on 20 Ne in late stages of star evolution is discussed. The oxygen ignition occurs in the central region of the core at T ∼ 109 K, within ∼100 km of the center, initiated by heating due to e-capture on 20 Ne. Here, the  ignition is defined as the stage where the nuclear energy generation exceeds the thermal neutrino losses. The central density at the oxygen ignition is denoted as ρc,ign . Subsequent oxygen burning grows into the thermonuclear runaway, that is, oxygen deflagration, when the timescale of temperature rise gets shorter than the dynamical timescale. The central density when the oxygen deflagration starts is denoted as ρc,def . Note that at the oxygen ignition, the heating timescale by local oxygen burning is estimated to be ∼107−8 s, which is larger than the dynamical timescale, that is, the timescale of Ye reduction by e-capture on proton at ρc ∼1010 g cm−3 , by 8–9 orders of magnitude (Zha et al. 2019), and the thermonuclear runaway of the local oxygen burning does not take place yet. Thus ρc,def is not the same as ρc,ign , but usually higher than ρc,ign . Further evolution of the core depends on the competition between the nuclear energy release by the oxygen deflagration and the reduction of the degeneracy pressure by e-capture in the NSE ash (Nomoto and Kondo 1991; Timmes and Woosley 1992; Jones et al. 2016; Leung and Nomoto 2019). Recent multidimensional simulations of the oxygen deflagration show that the competition depends sensitively on the value of ρc,def . If ρc,def is higher than a certain critical density ρcr , the core collapses to form a neutron star (NS) due to e-capture (Fryer et al. 1999; Kitaura et al. 2006; Radice et al. 2017), while if ρc,def < ρcr thermonuclear, energy release dominates to induce partial explosion of the core (Jones et al. 2016). For the value of ρcr , log10 (ρcr /g cm−3 ) = 9.95–10.3 and 9.90–9.95 have been obtained by the two-dimensional (2D; Nomoto and Leung 2017; Leung and Nomoto 2019; Leung et al. 2020) and three-dimensional (3D; Jones et al. 2016) hydrodynamical simulations, respectively. There exists a large uncertainty in the

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treatment of the propagation of the oxygen deflagration (Timmes and Woosley 1992) as well as the e-capture rates (Seitenzahal et al. 2009). The value of ρc,def is also subject to uncertainties involved in the calculation of the final stage of the core evolution. Evaluated value of log10 (ρc,def /g cm−3 ) is currently in the range of 9.9–10.2 depending on the treatment of convection (Schwab et al. 2015, 2017; Takahashi et al. 2019). Oxygen burning forms a convectively unstable region, which will develop above the oxygen-burning region. A smaller value for ρc,def ≈ 109.95 (Schwab et al. 2017) was obtained without convection. The evolution of the 8.4 M star from the main sequence until the oxygen ignition in the degenerate ONeMg core has been studied (Zha et al. 2019) using the MESA code (Schwab and Rocha 2019). The weak rates of Suzuki et al. (2019) including the second-forbidden transition for the e-capture on 20 Ne (denoted as USDB in Figs. 4 and 5) have been used. Here, we follow the discussion by Zha et al. (2019). The core evolves through complicated processes of mass accretion, heating by e-capture, cooling by Urca processes, and Ye change. It has been investigated how the location of the oxygen ignition (center or off-center) and the Ye distribution depend on the input physics and the treatment of the semiconvection and convection. There are two extreme criteria for the convective stability, the Schwarzschild criterion and the Ledoux criterion (Miyaji and Nomoto 1987; Kippenhahn et al. ∂lnT ∂T 2012). The Schwarzschild criterion for stability is expressed as ∂lnP < Tp ( ∂P )s , where the right side is evaluated with adiabatic condition (with constant entropy). The Ledoux criterion takes into account inhomogeneous chemical composition, which enhances the stability. The semiconvective region is treated as convectively unstable (stable) when using the Schwartzschild (Ledoux) criterion. The evolution of the ONeMg core in late stages after the e-capture on 24 Mg is shown in Fig. 6 for several convection models. The central temperature increases as the density increases due to the double e-captures on 24 Mg and 20 Ne. While the cooling of the core caused by the nuclear Urca process for the 25 Na -25 Ne pair occurs between the two double e-capture processes, the core temperature continues to increase up to about 108.8 K after the e-capture on 24 Mg, for the Schwarzschild criterion cases, as the second-forbidden transition in 20 Ne takes place just before the ignition of the Urca process (see Fig. 3). The β-decay Q-value for the 25 Ne (1/2+ , g.s.) → 25 Na (3/2+ , 0.895 MeV) transition is 7.161 MeV, which is a bit larger than the β-decay Q-value of 7.024 MeV for the 20 F (2+ ) → 20 Ne (0+ ) transition. The temperature drop due to the Urca process for the A = 25 pair is seen at higher density around log10 (ρc ) = 9.8. The heating of the core by the e-capture on 20 Ne increases the central temperature again from log10 (ρc ) ∼9.9. The curves in Fig. 6 end at the oxygen ignition since the MESA code cannot go further than this point. When the Schwarzschild criterion for the convective stability is applied, the oxygen ignition takes place at the center. The convective energy transport delays the oxygen ignition until log10 (ρc,ign /g cm−3 ) ∼10.0 is reached, and the convective mixing makes Ye in the convective region as high as 0.49. When the Ledoux criterion for the convective stability is applied, the second-forbidden transition does not ignite oxygen burning at the related threshold density but decrease the central Ye to ∼0.46 during the core contraction. The oxygen ignition takes place

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Fig. 6 Evolution of the accreting ONeMg core in the central density-temperature plane for different treatments of the convection. Here “L” stands for the Ledoux criterion, and “S” stands for the Schwarzschild criterion. See Zha et al. (2019) for the additional model nomenclature. (From Zha et al. 2019)

when the central density reaches log10 (ρc,ign /g cm−3 ) = 9.96–9.97. Even with the Ledoux criterion, the oxygen ignition creates the convectively unstable region, and the convective mixing forms an extended region with Ye ∼0.49 above the oxygen ignited shell. For both the convective stability criteria, the convective energy transport would slow down the temperature increase, and the thermonuclear runaway to form a deflagration wave is estimated to occur at log10 (ρc,def /g cm−3 )> 10.10, by extrapolation of the curves in Fig. 6 till the temperature of the oxygen burning, log10 (T/K) ≈9.3. This estimate is consistent with log10 (ρc,def /g cm−3 ) ≈10.2 obtained with the semiconvective mixing (Takahashi et al. 2019). Then, to examine the final fate of the ONeMg core, 2D hydrodynamical simulation for the propagation of the oxygen deflagration wave has been performed based on the above simulation of the evolution of the star until the oxygen ignition. Three cases of Ye distributions, three locations of the oxygen ignition (center, offcenter at rign = 30 and 60 km), and various central densities at log10 (ρc,def /g cm−3 ) =9.96–10.2 are used for the initial configurations at the initiation of the deflagration (see Zha et al. 2019 for the details). The explosion-collapse bifurcation analysis is shown for certain initial configurations of the deflagration in Fig. 7. The evolutions of the central density and Ye as functions of time are shown. The explosion-collapse bifurcation diagram is shown in Fig. 8 for the accretion mass rate of M˙ = 10−6 M . The critical density for the explosion-collapse bifurcation is found to be at log10 (ρcrt /g cm−3 ) =10.01. The deflagration starting from log10 (ρc,def /g cm−3 ) > 10.01 (< 10.01) leads to a collapse (a thermonuclear explosion). Since log10 (ρc,def ) is estimated to be above 10.1 and exceeds this critical value, the ONeMg core is likely to collapse to form a NS irrespective of the

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Fig. 7 The ρc,def dependence of the Ledoux models for mass accretion rate M˙ = 10−6 M yr−1 . (Left panel) Central density evolution of models with log10 (ρc,def /g cm−3 ) =9.96, 9.98, and 10.00. Initial value of Ye =0.49 and oxygen ignition take place at 30 km from the center. The time lapse of ∼0.1 s is the time for the flame to arrive at the center to trigger the first expansion. The collapsing model shows a monotonic increase of the central density after the early expansion, while the other two exploding models show a turning point after which the star expands due to the energy input by oxygen deflagration. (Right panel) The same as the left panel but for the central Ye . (From Zha et al. 2019)

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Fig. 8 Explosion-collapse bifurcation diagram as a function of ρc,def and the initial Ye distribution for two Ledoux models and one Schwartzschild model for the case of mass accretion rate of 10−6 M yr−1 . Here “E” and “C” stand for “explosion” and “collapse,” respectively. Left (right) symbols (“E” or “C”) for L− no− mix and S− ρ− mix cases correspond to the oxygen ignition at the center (off-center at 30 km from the center). (From Zha et al. 2019)

central Ye and ignition position, although further studies of the convection and semiconvection before the deflagration are needed in the future by improving the stellar evolution modeling. It would be interesting to see if the present conclusion remains valid for the rates with the CVC relation discussed in the first subsection, USDB (CVC) and USDB (BB, CVC), which are enhanced compared with the rates used here around log10 (ρYe ) = 9.6 at log10 T < 8.8. A preliminary study shows that the present conclusion would likely to remain though the location of the oxygen ignition is pushed away from the center by ∼30 km (Zha, private communications). Jones et al. (2016) with the rates GT (log ft = 10.50) and Kirsebom et al. (2019b) with the rates USDB (BB, CVC), on the other hand, obtained the opposite conclusion in favor of thermonuclear explosion by assuming that the effects of convection and semiconvection would be small. Investigations whether the convective energy transport is efficient enough to delay the ignition and the start of the oxygen deflagration wave to densities above the critical density for collapse were left for the future. In the case of thermonuclear explosions, the oxygen deflagration results in a partial disruption of the ONeMg core with an ONeFe WD left behind (Jones et al. 2016). The turbulent mixing by the flame allows the ejecta to consist of both Fe-peak elements and ONe-rich fuel. Ejecta can be rich in neutron-rich isotopes such as 48 Ca, 50 Ti, 54 Cr, 60 Fe, and 66 Zn, which are overproduced relative to their solar abundances (Jones et al. 2019). A substantial enrichment was reported for 54 Cr and 50 Ti with 54 Cr/52 Cr and 50 Ti/48 Ti ratios ranging from 1.2 to 56 and from 1.1 to 455 times the solar values, respectively, in the presolar oxide grains from the Orguil CI meteorite (Nittler et al. 2018). The enrichment of 54 Cr and 50 Ti obtained in ejecta in the thermonuclear explosion simulation (Jones et al. 2016) is found to be consistent with the most extreme grain, 237 , among the enriched grains.

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It is not easy to find clear evidence for thermonuclear ECSN (tECSN) or collapsing ECSN (cECSN). In tECSN, ONeFe WD is expected to be formed as a remnant. Information on its mass-radius relation could assign ONeFe WD (Jones et al. 2019). The progenitor of SN2018zd, which proved to eject relatively small amount of 56 Ni and faint X-ray radiation, has been suggested as a massive AGB star that collapsed by ECSN (Zhang et al. 2020). In Hiramatsu et al. (2021), SN2018zd is shown to have strong evidence for or consistent with various indicators of e-capture supernovae, that is, progenitor identification, circumstellar material, chemical composition, explosion energy, light curve, and nucleosynthesis. Theoretically, we need to understand more clearly the evolution from the oxygen ignition (at the end of the MESA calculations) till the beginning of the deflagration by taking into account the semiconvection and convection. More observational and theoretical studies are desirable to be done to draw a definite conclusion on the final fate of the stars with 8–10 M , whether cECSN or tECSN. Acknowledgments The author would like to thank K. Nomoto, S. Zha, S.-C. Leung, and H. Toki for the collaboration of the present work. This work was supported in part by JSPS KAKENHI grant No. JP19K03855. He also thanks NAOJ Japan for accepting him as a visiting researcher during the period of the work.

References R. Abe, Prog. Theor. Phys. 21, 475 (1959). https://doi.org/10.1143/PTP.21.475 F. Ajzenberg-Selove, Nucl. Phys. A 475, 1 (1987). Data extracted from the ENSDF database, version (March 17, 2014), National Nuclear Data Center. https://www.nndc.bnl.gov/ H. Behrens, W. Bühring, Nucl. Phys. A 162, 111 (1971). https://doi.org/10.1016/03759474(71)90489-1 H. Behrens, J. Janecke, Numerical Tables for Beta-Decay and Electron Capture. LandoltBornstein, New Series, Group I, vol. 4 (Springer, Berlin, 1969) R.J. Blin-Stoyle, Fundamental Interactions and the Nucleus (Amsterdam, North-Holland, 1973) E. Bravo, D. Garcia-Senz, Mon. Not. R. Astron. Soc. 307, 984 (1999). https://doi.org/10.1046/j. 1365-8711.1999.02694.x B.A. Brown, W.A. Richter, Phys. Rev. C 74, 034315 (2006). https://doi.org/10.1103/PhysRevC.74. 034315 B.A. Brown, B.H. Wildenthal, Phys. Rev. C 28, 2397 (1983). https://doi.org/10.1103/PhysRevC. 28.2397 B.A. Brown, B.H. Wildenthal, Ann. Rev. Nucl. Part. Sci. 38, 29 (1988). https://doi.org/10.1146/ annurev.ns.38.120188.000333 W. Bühring, Nucl. Phys. 40, 472 (1963). https://doi.org/10.1016/0029-5582(63)90290-6 T. de Forest Jr., J.D. Walecka, Adv. Phys. 15, 1 (1966). https://doi.org/10.1080/ 00018736600101254 J.Z. Eichler, Physics 171, 463 (1963). https://doi.org/10.1007/BF01377868 C. Fryer, W. Benz, M. Herant, S.A. Colgate, Astrophys. J. 516, 892 (1999). https://doi.org/10.1086/ 307119 J. Fujita, Phys. Rev. 126, 202 (1962a). https://doi.org/10.1103/PhysRev.126.202 J. Fujita, Prog. Theor. Phys. 28, 338 (1962b), https://doi.org/10.1143/PTP.28.338 G.M. Fuller, W.A. Fowler, M.J. Newman, Astrophys. J. Suppl. 42, 447 (1980). https://doi.org/ 10.1086/190657; Astrophys. J. 252, 715 (1982). https://doi.org/10.1086/159597; Astrophys. J. Suppl. 48, 279 (1982). https://doi.org/10.1086/190779

97 Weak Interactions in Evolving Stars

3593

J. Gutierrez, E. Garcia-Berro, I. Iben Jr., J. Isern, J. Labay, R. Canal, Astrophys. J. 459, 701 (1996). https://doi.org/10.1086/176934 D. Hiramatsu, D.A. Howell, S.D. Van Dyk et al., Nat. Astron. 5 (2021). https://doi.org/10.1038/ s41550-021-01384-2 S. Ichimaru, Rev. Mod. Phys. 65, 255 (1993). https://doi.org/10.1103/RevModPhys.65.255 N. Itoh, N. Tomizawa, M. Tamamura, S. Wanajo, Astrophys. J. 579, 380 (2002). https://doi.org/10. 1086/342726 S. Jones, R. Hirschi, K. Nomoto, T. Fischer, F.X. Timmes, F. Herwig, B. Paxton, H. Toki, T. Suzuki, G. Martínez-Pinedo, Y.H. Lam, M.G. Bertolli, Astrophys. J. 772, 150 (2013). https://doi.org/10. 1088/0004-637X/772/2/150 S. Jones, F.K. Ropke, R. Pakmor, I.R. Seitenzahl, S.T. Ohlmann, P.V. Edelmann, Astron. Astrophys. 593, A72 (2016). https://doi.org/10.1051/0004-6361/201628321 S. Jones, F.K. Ropke, C. Fryer et al., Astron. Astrophys. 622, A74 (2019). https://doi.org/10.1051/ 0004-6361/201834381 A. Juodagalvis, K. Langanke, W.R. Hix, G. Martínez-Piedo, J.M. Sampaio, Nucl. Phys. A 848, 454 (2010). https://doi.org/10.1016/j.nuclphysa.2010.09.012 R. Kippenhahn, A. Weigert, A. Weiss, Stellar Structure and Evolution, 2nd edn. (Springer, Berlin, 2012) O.S. Kirsebom et al., Phys. Rev. Lett. 123, 262701 (2019a). https://doi.org/10.1103/PhysRevLett. 123.262701 O.S. Kirsebom et al., Phys. Rev. C 100, 065805 (2019b). https://doi.org/10.1103/PhysRevC.100. 065805 F.S. Kitaura, H.-T. Janka, W. Hillebrandt, Astron. Astrophys. 450, 345 (2006). https://doi.org/10. 1051/0004-6361:20054703 T. Kuramoto, M. Fukugita, Y. Kohyama, K. Kubodera, Nucl. Phys. A 512, 711 (1990). https://doi. org/10.1016/0375-9474(90)90232-B K. Langanke, G. Martínez-Pinedo, At. Data Nucl. Data Tables 79, 1 (2001). https://doi.org/10. 1006/adnd.2001.0865 S.-C. Leung, K. Nomoto, Pub. Astr. Soc. Aust. 36, e006 (2019). https://doi.org/10.1017/pasa.2018. 49 S.-C. Leung, K. Nomoto, T. Suzuki, Astrophys. J. 889, 34 (2020). https://doi.org/10.3847/15384357/ab5d2f G. Martínez-Pinedo, Y.H. Lam, K. Langanke, R.G. Zegres, C. Sullivan, Phys. Rev. C 89, 045806 (2014). https://doi.org/10.1103/PhysRevC.89.045806 S. Miyaji, K. Nomoto, Astrophys. J. 318, 307 (1987). https://doi.org/10.1086/165368 S. Miyaji, K. Nomoto, K. Yokoi, D. Sugimoto, Pub. Astr. Soc. Jpn. 32, 303 (1980). https://ui. adsabs.harvard.edu/abs/1980PASJ...32..303M L.R. Nittler, O’D. Alexander, N. Lau, J. Wang, Astrophys. J. Lett. 856, L24 (2018). https://doi.org/ 10.3847/2041-8213/aab61f K. Nomoto, Astrophys. J. 277, 791 (1984). https://doi.org/10.1086/161749; Astrophys. J. 322, 206 (1987). https://doi.org/10.1086/165716 K. Nomoto, M. Hashimoto, Phys. Rep. 163, 13 (1988). https://doi.org/10.1016/03701573(88)90032-4 K. Nomoto, Y. Kondo, Astrophys. J. Lett. 367, L19 (1991). https://doi.org/10.1086/185922 K. Nomoto, S.-C. Leung, in Handbook of Supernovae, vol. I, ed. by A.W. Alsabti, P. Murdin (Springer, Berlin, 2017), p. 483 J.S. O’Connell, T.W. Donnelly, J.D. Walecka, Phys. Rev. C 6, 719 (1972). https://doi.org/10.1103/ PhysRevC.6.719 T. Oda, M. Hino, K. Muto, M. Takahara, K. Sato, At. Data Nucl. Data Tables 56, 231 (1994). https://doi.org/10.1006/adnd.1994.1007 D. Radice, A. Burrows, D. Vartanyan, M.A. Skinner, J.C. Dolence, Astrophys. J. 850, 43 (2017). https://doi.org/10.3847/1538-4357/aa92c5 W.A. Richter, S. Mkhize, B.A. Brown, Phys. Rev. C 78, 064302 (2008). https://doi.org/10.1103/ PhysRevC.78.064302

3594

T. Suzuki

H. Schopper, Weak Interactions and Nuclear Beta Decays (Amsterdam, North-Holland, 1966) J. Schwab, K.A. Rocha, Astrophys. J. 872, 131 (2019). https://doi.org/10.3847/1538-4357/aaffdc J. Schwab, E. Quataert, L. Bildsten, Mon. Not. R. Astron. Soc. 453, 1910 (2015). https://doi.org/ 10.1093/mnras/stv1804 J. Schwab, L. Bildsten, E. Quataert, Mon. Not. R. Astron. Soc. 472, 3390 (2017). https://doi.org/ 10.1093/mnras/stx2169 I.R. Seitenzahal, D.M. Townsley, F. Peng, J.W. Truran, At. Data Nucl. Data Tables 95, 96 (2009). https://doi.org/10.1016/j.adt.2008.08.001 W.L. Slattery, G.D. Doolen, H.E. DeWitt, Phys. Rev. A 26, 2255 (1982). https://doi.org/10.1103/ PhysRevA.26.2255 T. Suzuki, Prog. Part. Nucl. Phys. 126, 103974 (2022). arXiv:2205.09262, https://doi.org/10.1016/ j.ppnp.2022.103974 T. Suzuki, M. Honma, H. Mao, T. Otsuka, T. Kajino, Phys. Rev. C 83, 044619 (2011). https://doi. org/10.1103/PhysRevC.83.044619 T. Suzuki, H. Toki, K. Nomoto, Astrophys. J. 817, 163 (2016). https://doi.org/10.3847/0004-637X/ 817/2/163 T. Suzuki, S. Zha, S.-C. Leung, K. Nomoto, Astrophys. J. 881, 64 (2019). https://doi.org/10.3847/ 1538-4357/ab2b93 M. Takahara, M. Hino, T. Oda, K. Muto, A.A. Wolters, P.W.M. Glaudemans, K. Sato, Nucl. Phys. A 504, 167 (1989). https://doi.org/10.1016/0375-9474(89)90288-1 K. Takahashi, K. Sumiyoshi, S. Yamada, H. Umeda, T. Yoshida, Astrophys. J. 871, 153 (2019). https://doi.org/10.3847/1538-4357/aaf8a8 F.X. Timmes, S.E. Woosley, Astrophys. J. 396, 649 (1992). https://doi.org/10.1086/171746 H. Toki, T. Suzuki, K. Nomoto, S. Jones, R. Hirschi, Phys. Rev. C 88, 015806 (2013). https://doi. org/10.1103/PhysRevC.88.015806 J.D. Walecka, in Muon Physics, vol. II, ed. by V.W. Hughes, C.S. Wu (Academic, New York, 1975) D.G. Yakovlev, D.A. Shalybkov, Sov. Sci. Rev. E Astrophys. Space Phys. 7, 311 (1989). https://ui. adsabs.harvard.edu/abs/1989ASPRv...7..311Y S. Zha, S.-C. Leung, T. Suzuki, K. Nomoto, Astrophys. J. 886, 22 (2019). https://doi.org/10.3847/ 1538-4357/ab4b4b J. Zhang, X. Wang, V. Jozsef et al., Mon. Not. R. Astron. Soc. 498, 84 (2020). https://doi.org/10. 1093/mnras/staa2273

Stellar Evolution and Nuclear Reaction Rate Uncertainties

98

Ken’ichi Nomoto and Wenyu Xin

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactions and Nuclear Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics of Stellar Evolution . . . . . . . . . . . . . . . . . . .