Understanding Quarks 1536195286, 9781536195286

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PHYSICS RESEARCH AND TECHNOLOGY

UNDERSTANDING QUARKS

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PHYSICS RESEARCH AND TECHNOLOGY

UNDERSTANDING QUARKS

BENJAMIN HOUDE EDITOR

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Library of Congress Cataloging-in-Publication Data Names: Houde, Benjamin, editor. Title: Understanding quarks / Benjamin Houde, editor. Description: New York : Nova Science Publishers, [2021] | Series: Physics research and technology | Includes bibliographical references and index. Identifiers: LCCN 2021021030 (print) | LCCN 2021021031 (ebook) | ISBN 9781536195286 (paperback) | ISBN 9781536196764 (adobe pdf) Subjects: LCSH: Quarks. | Particles (Nuclear physics) Classification: LCC QC793.5.Q252 U53 2021 (print) | LCC QC793.5.Q252 (ebook) | DDC 539.7/2167--dc23 LC record available at https://lccn.loc.gov/2021021030 LC ebook record available at https://lccn.loc.gov/2021021031

Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface

vii

Chapter 1

Experimental Study of the Long - Range Quark Lepton Strong Interaction in Solids V. G. Plekhanov

Chapter 2

Measurement of the Associated Production of Top-Quark Pairs with a Higgs Boson A. Gutiérrez-Rodríguez, A. González-Sánchez and M. A. Hernández-Ruíz

Chapter 3

The Quarks: Invention and Discovery A. Gutiérrez-Rodríguez, M. A. Hernández-Ruíz, F. Ramírez-Sánchez and F. Mireles-García

109

Chapter 4

Strange Quark Matter and Exotic Dark Matter Hidezumi Terazawa

125

Index

1

93

137

PREFACE This book includes four chapters about quarks, which are a type of elementary particle and a fundamental constituent of matter. Chapter One is devoted to spectroscopy study of long-range quark-lepton strong interaction in solids as well as the origin of the mass of elementary particles. Chapter Two studies the production of top-quark pairs with a Higgs boson. Chapter Three begins with a brief historical outline of the development of the events that led to our understanding of quarks and analyzes the three families of quarks. Chapter Four discusses strange quark matter and strange quark stars as candidates for dark matter based on the Bodmer-Terazawa-Witten hypothesis. Chapter 1 would like to discuss the strong nuclear interaction – the heart of the Quantum Chromodynamics (QCD) which is the part of Standard Model (SM) - from point of view different from usual discussion in the accelerator high energy physics. Precision spectroscopy has long standing record of providing insights into fundamental physics. Chapter 1 is devoted to spectroscopy study of long - range quark - lepton strong interaction in solids as well as the origin of the mass of elementary particles. The authors have studied the low - temperature (reflection and luminescence) spectra of the LiH (without strong interaction in hydrogen nucleus) and LiD (with strong interaction in deuterium nucleus) single crystals which are different by term of one neutron from each other. The experimental observation of isotopic shift (0.103 eV) of the phononless free excitons emission line in LiD crystals is a direct non – accelerator manifestation of the long - range strong nuclear interaction such conclusion is made to the fact that the gravitation, electromagnetic and weak interaction are the same in both kind of crystals, it only emerges the strong interaction in the deuterium nucleus. There is a common place in SM of modern physics that the strong nuclear force does not

viii

Benjamin Houde

act on leptons. Our experimental results show the violation of this strong conclusion. According to the proposed model the main mechanism of long range quark – lepton interaction is their magnet - like long - range interaction. Since the isotope effect is a direct manifestation of the mass effect in microphysics, it is therefore natural expect here the origin of mass in Nature. The hypothesis formulated by theoretists about the complex form of the dependence of the mass acquisition by current quarks and gluons in the transitions to the momentum infrared limit is possibly reflected in the nonlinear dependence of the energy of strong nuclear interaction on the distance between nucleons in the deuterium nucleus measured by us. Throughout Chapter 1, the importance of studying the isotope effect is emphasized not only for the origin of the mass in the Universe, but also in search for new physics beyond SM. Top-quark physics is a very active area of research. On this topic, Chapter 2 studies the associated production of top-quark pairs with a Higgs boson through the process ee  tth . For this study, the authors consider the energies and luminosities of a future Compact Linear Collider (CLIC). The top-quark is a fermion and couples to all known bosons: the W  and Z bosons, the photon, the gluon and the Higgs boson. These coupling strengths are well predicted by the Standard Model. Thus, measurement of the associated production of top-quarks pairs with a Higgs boson provides an important test of the Standard Model. All mentioned processes have been observed by the ATLAS and CMS Collaboration at the Large Hadron Collider (LHC). The most important result is the direct observation of the tth process by these collaborations. This is particularly interesting, as it allows to probe the Yukawa sector of the mass generation through the Higgs mechanism. Chapter 3 presents the most relevant events of the process of invention and discovery of quarks, beginning with a brief historical outline of the development of the events that led to their understanding, and considering that the present consensus within the Standard Model accepts only three families of quarks as well as of leptons, the authors continue by analyzing these three families one by one. Emphasis is done in the study of the first family, where the constituents of the matter that are known; protons, neutrons and electrons are created. The study of unstable particles, mesons and other exotic particles, lies within the realm of the second and third families, the authors also studied and analyzed the experiments that led to the discovery of the top quark, since this particle is of particular interest in the current study of particle physics due to its implications in the Physics beyond the Standard Model.

Preface

ix

New forms of matter such as strange quark matter and strange quark stars as candidates for dark matter are discussed in Chapter 4 in some detail, based on the so-called “Bodmer-Terazawa-Witten hypothesis” assuming that they are stable absolutely or quasi-stable (decaying only weakly). Chapter 4 consists of the following Sections: Section I. Introduction, Section II. Exotic Nuclei and Strange Stars, and Section III. Strange Quark Matter as Dark Matter.

In: Understanding Quarks Editor: Benjamin Houde

ISBN: 978-1-53619-528-6 c 2021 Nova Science Publishers, Inc.

Chapter 1

E XPERIMENTAL S TUDY OF THE L ONG R ANGE Q UARK - L EPTON S TRONG I NTERACTION IN S OLIDS V. G. Plekhanov∗ Fonoriton Science Laboratory, Garon Ltd., Tallinn, Estonia

Abstract In this chapter I would like to discuss the strong nuclear interaction - the heart of the Quantum Chromodynamics (QCD) which is the part of Standard Model (SM) - from point of view different from usual discussion in the accelerator high energy physics. Precision spectroscopy has long standing record of providing insights into fundamental physics. Our chapter is devoted to spectroscopy study of long - range quark - lepton strong interaction in solids as well as the origin of the mass of elementary particles. We have studied the low - temperature (reflection and luminescence) spectra of the LiH (without strong interaction in hydrogen nucleus) and LiD (with strong interaction in deuterium nucleus) single crystals which are different by term of one neutron from each other. The experimental observation of isotopic shift (0.103 eV) of the phononless free excitons emission line in LiD crystals is a direct non - accelerator manifestation of the long - range strong nuclear interaction such conclusion is made to the fact that the gravitation, electromagnetic and weak interaction are the same in both kind of crystals, it only emerges the strong interaction ∗

Corresponding Author’s Email: [email protected].

2

V. G. Plekhanov in the deuterium nucleus. There is a common place in SM of modern physics that the strong nuclear force does not act on leptons. Our experimental results show the violation of this strong conclusion. According to the proposed model the main mechanism of long - range quark - lepton interaction is their magnet - like long - range interaction. Since the isotope effect is a direct manifestation of the mass effect in microphysics, it is therefore natural expect here the origin of mass in Nature. The hypothesis formulated by theoretists about the complex form of the dependence of the mass acquisition by current quarks and gluons in the transitions to the momentum infrared limit is possibly reflected in the nonlinear dependence of the energy of strong nuclear interaction on the distance between nucleons in the deuterium nucleus measured by us. Throughout the chapter, the importance of studying the isotope effect is emphasized not only for the origin of the mass in the Universe, but also in search for new physics beyond SM.

1.

INTRODUCTION

The exploration of subatomic physics started in 1896 with Becquerel’s discovery of radioactivity; since then it has been a constant source of surprises, unexpected phenomena, and fresh insights into the law of Nature. In years that followed, the phenomenon was extensively investigated, notably by the husband and wife team of Pierre and Marie Curie and by Ernest Rutherford and his collaborators, and it was established that there were three distinct types of radiation involved. These were named (by Rutherford) α -, β -, and γ - rays. We know now that α -rays are bound states of two protons and two neutrons (the nuclei of helium atoms), β - rays are electrons and γ - rays are high energy photons. The energies of the α -, β -, and γ - particles are of the order of 0.1 MeV up to 10 MeV, whereas energies of the orbital electrons are of the order of electron volts. Also, α - particles were found to be barely able to penetrate a piece of paper, whereas β - particles could penetrate a few millimeters of aluminium, and the γ - rays could penetrate several centimeters of lead. At about the same time as Becquerel’s discovery, J.J. Tomson in 1897 was first to definitely establish the nature of free electrons. The view of the atom at that time was that it consist of two components, with positive and negative electric charges, the latter now being the electrons. In his experiments on α scattering by gold, Rutherford showed that α - particles encountered a very small positively charged central nucleus. To explain the results of these experiments

Experimental Study of the Long - Range Quark - Lepton ...

3

Rutherford formulated a planetary model, where the atom was likened to a planetary system, with the electrons (the ”planets”) occupying discrete orbits about a central positively charged nucleus (”Sun”) (see, e.g., [ 1 - 3]). These are positively charged nucleus, about 105 times smaller than an atom (see below Fig. 1) - about 1 fm in size. The other is a number of electrons just compensating for the positively charged nucleus, which fill out essentially the rest of the atom. Because photons of a definite energy would be emitted when electrons moved from one orbit to another, this model could explain the discrete nature of observed electromagnetic spectra when excited atoms decayed. In the simplest case of hydrogen, the nucleus is a single proton with electric charge +e, where - e is the magnitude of charge on the orbited a single electron. Heavier atoms were considered to have nuclei consisting of several protons. Examples are carbon and nitrogen, with masses of 12.0 and 14.0. However, it could not explain why chlorine has a mass of 35.5 in these units. At about the same time, the concept of isotopism was conceived by Soddy. Isotopes are atoms whose nuclei have different masses, but the same charge. The explanation of isotopes was done by Chadwick [4]. He had discovered the neutron and in so doing had produced almost final ingredient for understanding nuclei. Iwanenko and Heisenberg formulated a modern model, according to which all nuclei consist of a positively charged proton and a neutral neutron. By the early 1930s, the 19th century view of atoms as indivisible elementary particles had been replaced and a larger group of physically smaller entities now enjoyed this status; electrons, protons and neutrons. To these we should add two electrically neutral particles: the photon (γ) and the neutrino (ν). The photon was postulated by Planck in 1900 to explain black - body radiation, where the classical description of electromagnetic radiation led to results incompatible with experiments. The neutrino was postulated by Fermi in 1930 to explain the apparent non - conservation of energy observed in the decay products of some unstable nuclei where β - rays are emitted, the so - called β - decay. Particle physics is the study of the fundamental constituents of matter and their interactions. However, which particles are regarded as fundamental has changed with time as physicists’ knowledge has improved. The best theory of elementary particles we have at present is called, rather prosaically, the Standard Model (SM) (see, e.g., [5, 6]). The Stanard Model attempts to explain all the phenomena of particle physics in terms of the properties and interactions of a small number of particles of three distinct types: two spin - 1/2 families of fermions called leptons and quarks, and one family of spin -1 bosons - called

4

V. G. Plekhanov

gauge bosons - which act as ’force carriers’ in the theory. In addition at least one spin - 0 particle, called Higgs boson, is postulated to explain the origin of mass within the theory, since without it all the particles in the SM are predicted to have zero mass. All the particles of the SM are assumed to be elementary; they are treated as point particles, without internal structure or excited states. Modern physics distinguishes three fundamental properties of atomic nuclei: mass , spin (and related magnetic moment), volume (surrounding field strength), which are the source of isotope effect. The stable elementary particles (electrons, protons and neutrons) have intrinsic properties. Some of these properties such as mass and electrical charge are the same for macroscopic objects. Some are purely quantum mechanical and have no macroscopic analog. Spin is an intrinsic angular momentum associated with elementary particles. The spin angular moment of an electron, measure along any particular direction, can only take on the values ~/2 or -~/2. The nuclear magnetic moment associates with nuclear spin and produces the magnetic interaction with its environment. Among the prime premises of quantum physics is the principle of indistinguishability of elementary particles. All electrons are by definition ’the same’ and hence, after two (or more) electrons interact, it is possible to say ’who is who’. The same applies to other elementary particles and even identical atoms (e.g., two 70 Ge atoms are indistinguishable). Not so for isotopes. Due to their mass difference, various isotopes of the same chemical element (e.g., 70 Ge and 76 Ge) are classical distinguishable particles. Within the framework of statistical thermodynamics, this classical distinguishability of isotopes renders isotopic mixtures a prime illustrative tool for a discussion of the Gibbs paradox. The latter refers to the (alleged) discontinuity of entropy upon the mixture of two slightly different species. Our present knowledge of physical phenomena suggests that there four types of forces between physical bodies [6]: 1) gravitational; 2) electromagnetic; 3) strong; 4) weak. Both the gravitational and the electromagnetic forces vary in strength as the inverse square of the distance and so able to influence the state of an object even at very large distances whereas the strong and the weak forces fall off exponen-

Experimental Study of the Long - Range Quark - Lepton ...

5

tially and so act only at extremely short distances. The strong forces does not act on leptons (electrons, positrons, muons and neutrinos), but only on protons and neutrons (more generally, on baryons and mesons - this is the reason for the collective name hadrons). It holds protons and neutrons together to form nuclei, and is insignificant at distances greater than 10−15 m. Its macroscopic manifestations are restricted up to now to radioactivity and the release of nuclear energy. The three forces which are relevant to elementary particles can be recognized in the three kinds of radioactivity: α - radiation is caused by the strong force, β radiation by the weak force, and γ - radiation by the electromagnetic force. The characteristics of these forces are summarized in Table 1. Table 1. The fundamental interactions forces

Force Strong (nuclear) Electromagnetic Weak (nuclear) Gravitational

Known Forces (Gauge bosons) Particle/quantum Relative strength∗ Mass (GeV) Gluon 1 0.14(?) Photon 7 · 10−3 none W± , Z bosons 10−5 80 - 90 Graviton (tentative) 6 · 10−39 none

Range (meters) 10−15 infinitive 10−17 infinitive

*) The unit of strength is hc/2 where h is Planck constant and c is the speed of light.

This table given for the strength and range of the forces come from a comparison of the effects they produce on two protons. We should add that the weak force does not appear to be particularly weak on this reckoning: the reason for its very short range (see Table 1) rather than its intrinsic strength. Since the protons and neutrons which make up the nucleus are themselves considered to be made up of quarks are considered to be held together by the color force, the strong force between nucleons may be considered to be a residual color force. In the strong nuclear interaction - the heart of the Quantum Chromodynamics (QCD) which is the part of SM, therefore the base exchange is the gluon which mediates the forces between quarks. The modern quantum mechanical view of the three fundamental forces (all except gravity) is that particles of matter (fermions = neutrons, protons, electrons) do not directly interact with each other, but rather carry a charge, and exchange virtual particles (gauge bosons = photons, gluons, gravitons) which are the interaction carriers or force mediators. The facts, summarized in the modern nuclear physics allow to draw several

6

V. G. Plekhanov

conclusions in regard to nuclear forces, most notably that the binding energy of a nucleus is proportional to the number of nucleons and that the density of nuclear matter is approximately constant. This lead to conclude that nuclear forces have a ”saturation property”. It seems from the last conclusion it is enough to change the number of neutrons in nucleus to change strength of nuclear force. But the last one constitutes the main ideas of the isotope effect. By the way we should note that the adiabatic approximation doesn’t hold in isotope mass effect. For comparison, protons are as large as ≈ 10−15 m. Leptons and quarks have spin 1/2, i.e., they are fermions. In contrast the atoms, nuclei and hadrons, no excited states of quarks or leptons have so far been observed. Thus they are to be elementary particles. In contrast to leptons, hadrons, as already mentioned above, have an internal structure and therefore they are not fundamental constituents of matter. Hadrons are not elementary particles. We should repeat that while accelerator - based experiments search directly for new particles over many orders of the mass and couplings, precision low - energy measurements may also reveal indirectly in a complementary approach the existences new phenomena, for example neutron - electron interaction in solids.

2.

T HE ATOM AND ITS C ONSTITUENTS

Although the discovery of the electron and of radioactivity marked the beginning of a new era in the investigation of matter, the existence of atoms was not yet generally accepted. The reason was simple: nobody was able to really picture these building block of matter, the atoms. We are all very well know, that the first building block of the atom to be identified was the electron. Although our knowledge of electricity dates back to ancient times, the concepts of the electron as a particle is much more recent. Perhaps the first experimental evidence for the existence of the electron as a charged particle comes from experiments in 1880s by Thomas Alva Edison (see, e.g., [1]). Further experimentation was J.J. Thomson. He was able to produce electrons as beams of free particles in discharge tubes. By deflecting them in electric and magnetic fields, he could determine their velocity and the ratio of their mass and charge. These are now known as electrons. He had in other words found a universal constituent of matter. In 1899 E. Rutherford investigated the types of radiation that were emitted by radioactive atoms and categorized these as α - particles (which were later shown to be the nuclei of 4 He atoms), β - particles (which were shown to be the same as Thomson’s electrons) and γ - rays (which were later shown to be

Experimental Study of the Long - Range Quark - Lepton ...

7

form electromagnetic radiation).

Figure 1. Structure within the atom. If the protons and neutrons in this picture were 10 cm across, then the quarks and electrons would be less than 0.1 mm in size and the entire atom would be about 10 km across (after http://www.lbl.gov/abc/wallchart/). Such chemically identically but differing mass nuclei were called isotopes. Since atoms are electrically neutral, the number of electrons must equal the nuclei’s positive charge Z. Thus, the mass and the charge of the nuclei must be two independent quantities to explain Rutherford’s observation. The explanation of isotopes had to wait 20 years until a classic discovery by Chadwick in 1932 [4]. He had discovered the neutron (n) and in so doing had produced almost the final ingredient for understanding nuclei. After this discovery, there was no longer any doubt that the building blocks of nuclei are protons and neutrons (collectively called nucleons) (see, e.g., [3]). Rutherford had shown that α - particles that are emitted by some radioactive atoms were heavy particles (compared to the electron) and that they carried a positive charge with a magnitude of twice the electron charge. At bombarding nitrogen with α - particles (4 He) Rutherford observed positively charged particles with an usually long range, which must have been ejected from the atom as well. He had indeed observed the reaction

8

V. G. Plekhanov 14

N + 4 He −→ 17 O + p

(1)

in which the nitrogen nucleus is converted into an oxygen nucleus, by the loss of a proton (from the Greek word meaning ’first’). The hydrogen nucleus could therefore be regarded as an elementary constituent of atomic nuclei. The neutron was also detected by bombarding with α - particles Rutherford’s method of visually detecting and counting particles by their scintillation on a zinc sulfide screen is not applicable to neutral particles. Chadwick in 1932 found appropriate experimental approach. In his experiments Chadwick was shown that the mass of the neutral radiation particle was similar to that of the proton Chadwick named this particle the ’neutron’ [4]. With these discoveries, the building blocks of the atom had been found. Modern equipment permits the investigation of the forces binding the nuclear constituents, i.e., the proton and the neutron. These forces were evidently much stronger than the electromagnetic forces holding the atom together, since atomic nuclei could only be broken up bombarding them with highly energetic α particles.

3.

T HE STRUCTURE OF ATOMIC N UCLEUS

The modern concept of the atom emerged at the beginning of the 20th century, the particular as a result of Rutherford’s experiments. An atom is composed of a dense nucleus surrounded by an electron cloud. The nucleus itself can be decomposed into smaller particles. The primary aim of nuclear physics is to understand the force between nucleons, the structure of nuclei and how nuclei interact with each other and with other subatomic particles. As say above, an atom consists of an extremely small, positively charged nucleus (see Fig. 1) surrounded by a cloud of negatively charged electrons. As can we see from Fig. 1 the nucleus is less than one ten - thousandth the size of atom, the nucleus contains more than 99.9% of the mass of the atom. Nucleus is central part of an atom consisting of A - nucleons, Z - protons and N - neutrons (Fig. 2). The atomic mass of the nucleus is equal Z + N. A given element can have many different isotopes, which differ from one other by the number of neutrons contained in the nuclei [2, 3]. Modern physics distinguishes three fundamental properties of atomic nuclei: mass, spin (and related magnetic moment) and volume (surrounding field strength) which are source of isotopic effect. In neutral atom, the number of electrons orbiting the nucleus

Experimental Study of the Long - Range Quark - Lepton ...

9

Figure 2. Atomic nomenclature (after [3]). equals the number of protons in the nucleus. The basic properties of the atomic constituents can be read in Table 2. The masses of proton and neutron are almost the same, approximately 1836 and 1839 electron mass (me ), respectively. Table 2. The basic properties of the atomic constituents (after [3]) Particle Proton Neutron Electron

Charge e 0 -e

Mass (u) 1.007276 1.008665 0.000549

Spin (}) 1/2 1/2 1/2

Magnetic Moment (JT−1 ) 1.411·10−26 - 9.66·10−27 9.28·10−24

Apart from electric charge, the proton and neutron have almost the same properties. Chemical properties of an element are determined by the charge of its atomic nuclei, i.e., by the number of protons (electrons). It should be added that although it is true that neutron has zero net charge, it is nonetheless composed of electrically charged quarks (see below) in the same way that a neutral atom is nonetheless composed of protons and electrons. As such, the

10

V. G. Plekhanov

neutron experiences the electromagnetic interactions. Inside a nucleus, neutrons and protons interact with each other and are bound within the nuclear volume under competing influences of attractive nuclear and repulsive electromagnetic forces. The protons and neutrons bound together in nuclei by so - called strong force. A first identification of a particle is usually made by measuring of its mass, m. In principle, the mass can be found → from Newton’s second law by observing the acceleration, − a , in a force field, → − F: m=

˛− ˛ ˛→˛ ˛F˛ . → − a

| |

(2)

Equation (2) is not valid relativistically, but the correct generalization poses no problems. We only note that with mass we always mean rest mass. The rest masses of subatomic particles vary over a wide range. The photon has zero rest mass. The lightest massive particles are the neutrinos with rest masses less than 1 eV/c2 ; the electron is the next lightest particle with mass, me , of about 10−17 g ≈ 0.51 MeV/c2 (see Table 2). Then comes the muon with a mass of about 200me (see e.g., [7]). Many particles with strange and wonderful properties have masses that lie between about 270 times the electron mass to several orders of magnitude higher [6]. Nuclei, which of course are also subatomic particles, start with proton, the nucleus of the hydrogen atom, with a mass of about 2000me (see Table 2). The masses consequently vary by a factor of over a billion. The measurement of nuclear masses occupies the extremely important place in the development of nuclear physics. Mass spectrometry (see, e.g., [1 - 3]) was the first technique of high precision available to the experimenter, and since the mass of nucleus increases in a regular way with the addition of one proton or neutron. In mass spectrometers, a flux identical nuclei (ions), accelerated to a certain energy, is directed to a screen (photoplate in the first samples of mass - spectrometer) where it makes a visible mark. Before striking the screen, this flux passes through magnetic field, which is perpendicular to velocity of the nuclei. As a result, the flux is deflected to certain angle. The greater mass, the smaller is the angle. Thus, measuring the displacement of the mark from the center of the screen, we can find the deflection angle and then calculate the mass. The example of a mass - spectrum of a different isotopes of krypton is shown in Fig. 3. From the relative areas of the peaks it can be determine the

Experimental Study of the Long - Range Quark - Lepton ...

11

abundance of the stable isotopes of krypton (for details see [3]). At the present time there are many different methods of the measurement of nuclei mass.

Figure 3. A mass-spectrum analysis of krypton. The ordinates for the peaks at mass positions 78 and 80 should be divided by 10 to show these peaks in their true relation to the others (after [2]). We should remind that the binding energy E of a nucleus is the difference in mass energy between nucleus A Z X and its constituent Z protons and N neutrons (see, Fig. 2). As we probe ever deeper into the constituent of matter, the binding energy becomes ever greater in comparison with rest energy of the bound system. For example, in hydrogen atom, the binding energy of 13.6 eV constitutes only 1.4x10−8 of total rest energy of the atom. On the other hand, in a simple nucleus, such as deuterium, the binding energy of ' 2.2 MeV is 1.2x10−3 of the total mass energy. At more deeper level, three massive quarks make up a nucleon. In this case as will be shown below, the binding energy of the quarks in a nucleon would be a fraction greater than 0.99 of the total mass of the quarks ( for more details see, e.g., [5 - 7]). A second property that is essential in classifying particles is the spin or intrinsic angular momentum. Spin is a purely quantum mechanical property, and it is not easy to grasp this concept at first. As an introduction we therefore begin to discuss the orbital angular momentum which has a classical meaning. → Classically, the orbital angular momentum of a particle with momentum − p is

12

V. G. Plekhanov

defined by − − → → L =→ r x− p,

(3)

→ where − r is the radius vector connecting the center of mass of the particle to the point to which the angular momentum is referred. Classically, orbital angular momentum can take any value. Quantum mechanically the magnitude → − of L is restricted to certain values. Moreover, the angular momentum vector can assume only certain orientation with respect to a given direction. The fact that such a spatial quantization exists appears to violate intuition, and it follows logically from the postulates of quantum mechanics (see, e.g., [8]). In quan→ − → tum mechanics, − p replaced by the operator -i~ (∂/∂x, ∂/∂y, ∂/∂z) ≡ -i~ ∇ and the orbital angular momentum consequently also becomes an operator whose z component, for instance, is given by:   ∂ ∂ ∂ Lz = -i~ x ∂y - y ∂x = -i~ ∂ϕ , (4) where ϕ is the azimuthal angle in polar coordinates. With a particle having → − → orbital angular momentum L , a current and thus a magnetic moment vector − µ can be associated (see [9]). → − → − µ = πr2 i I , (5) → − with I a unit vector, vertical to circular motion. For electron (or proton m = mp ) one has, in magnitude − → ev µ = πr2 2πr =

→ e − 2m ( L ).

(6)

e and 2mc in Gaussian units. Returning to a quantum mechanical description of orbital motion and thus of the magnetic moment description, one has the relation between operators

µbl =

µ d l,z =

e b 2m L,

(7)

e b 2m Lz

(8)

The eigenvalue of the orbital, magnetic dipole operator, acting on the orbital e eigenfunctions Ym l then becomes me µc r) = l,z Yl (b

me e b r) 2m Lz Yl (b

=

me e~ r), 2m ml Yl (b

(9)

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13

where l − orbital quantum number. The quantum numbers l and ml must be integers, and for given value of l, ml can assume the 2l + 1 values from -l to + l. e~ is called magneton. We may recall from atomic physics that the quantity 2m For atomic motion we use the electron mass and obtain the Bohr magneton µB = 5.7884 · 10−5 eV/T. Putting in the proton mass we have the nuclear magneton µN = 3.1525 · 10−8 eV/T. Note that µN  µB owing to the difference in the masses, thus, under most circumstances atomic magnetism has much larger effects than nuclear magnetism. Ordinary magnetic interaction of matter (ferromagnetism, for example) are determined by atomic magnetism We can write or the eigenvalue µN µl = gl lµN ,

(10)

where gl is the g - factor associated with the orbital angular momentum l. For protons gl = 1, because neutrons have no electrical charge, we can use Eq. (10) to describe the orbital motion of neutrons if we put gl = 0. We have thus been considering only the orbital motion of nucleons. Protons and nucleons, like electrons as above mentioned also have no classical analog but which we write in the same form as Eq. (10) µs = gs sµN ,

(10’)

where s = 1/2 for protons, neutrons and electrons (see Table 2). The quantity gs is known as the spin g - factor and is calculated by solving a relativistic quantum mechanics equation [8]. For free electrons, the experimental values are far from the expected value for point particles: for proton - gs = 5.585691 ± 0.0000022 and neutron - gs = 3.8260837 ± 0.0000018 [9]. Table 3 gives some representative values of nuclear dipole moments according [9]. For the electron gs factor turns out to be almost - 2 and at the present time of introducing intrinsic ~/2 spin electron this factor was not understood (see, e.g., [10]) and had be taken from experiment [9]. Dirac gave a natural explanation or this fact using the famous Dirac relativistic equation [8]. For a Dirac point electron this should be exact but small deviations given by a=

|g| - 2 2 ,

(11)

were detected, giving the result aexp = 0.001159658(4) e−

(12)

14

V. G. Plekhanov

Table 3. Sample values of nuclear magnetic dipole moments (after [11]) Nuclide n p 2 H(D) 17 O 57 Fe 57 Co 93 Nb

µ(µN ) - 1.9130418 + 2.7928456 + 0.8574376 - 1.89379 + 0.09062293 + 4.733 + 6.1705

Detailed calculations in quantum electrodynamics (QED) (see, e.g., [12]) and the present value give ath e− =

1 2

α π




- 0.328479


α 2 π

+


α 3 π ,

(13)

where α = e2 /~c, and the difference

exp th −6 5 (ath e− - ae− )/ae− = (2 ± 5) · 10 , which means 1 part in 10 .

Further we shortly return to the space quantization. The wave function of a particle with definite angular moment can then chosen to be an eigenfunction of →2 − L and Lz (see, e.g., [8, 10]) − →2 L Ψlm = l (l + 1)~2 Ψlm , (14) Lz Ψlm = m~Ψlm .

(15)

The equation (14) states that the magnitude of the angular moment is quantized and restricted to values [l (l + 1)]1/2 ~. The equation (14) states that the component of the angular moment in a given direction, called z by general agreement, can assume only values m~. The spatial quantization is expressed in a vector diagram (the details see [13]).

Experimental Study of the Long - Range Quark - Lepton ...

15

Table 4. Quarks characteristics

Charge Isospin Spin

4.

z I I3 s

u

d

+2/3

-1/3 1/2 -1/2 1/2

+1/2 1/2

p (uud) 1 +1/2 1/2

n (udd) 0 1/2 -1/2 1/2

QUARK STRUCTURE OF N UCLEONS

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the best-known of which are protons and neutrons. They are the only particles in the Standard Model to experience the strong interaction in addition to the three other fundamental interactions, also known as fundamental forces. Due to a phenomenon known as color confinement, quarks are never in isolation; they can only be found within hadrons [6]. For this reason, much of what is known about quarks has been drawn from observations of the hadrons themselves. There are six different types of quarks, known as flavors: up (symbol: u), down (d), charm (c), strange (s), top (t) and bottom (b) Up and down quarks have the lowest masses of all quarks, and thus are generally stable and very common in the universe. The other quarks are much more massive, and will rapidly decay into the lighter up and down quarks. Hence nucleons are built up out of at least three quarks. The proton has two u - quarks and one d - quark, while the neutron has two d quarks and one u - quark (Table 4). The proton and the neutron form an isospin doublet (I = 1/2). This is attributed to the fact that u - and d - quarks form an isospin doublet as well. The three quarks that determine of the nucleons are called valence quarks (see, e.g., [5]). As well as these, so called ’sea’ quarks, virtual quark - antiquark pairs, also exist in the nucleon. As well as u - and d - quarks, further types of quark - antiquark pairs are found in the ’sea’. The six quark types can be arranged in doublets (called families or generations [6]) according to their increasing mass:       u c t . d s b

16

V. G. Plekhanov

The quarks of the top row have charge zq = +2/3, those of the bottom row zq = -1/3. The c, b, and t quarks are so heavy that they play a very minor role in most nuclear experiments. The ’effective valence quarks’ are called constituent quarks. In interpreting deep inelastic scattering, usually neglected the rest masses of the bare u - and d - quarks. This is justified, since they are small: mu = 1.5 - 5 MeV/c2 , md = 3 - 9 MeV/c2 (see e.g., [14]). These masses are commonly called current quark masses. However, these are not the masses obtained from hadron spectroscopy; e.g., from calculations of magnetic moments and hadron excitation energies. The constituent quark masses are much larger (300 MeV/c2 ). The constituent quark masses must be mainly due to the cloud of gluons and ’sea’ quarks [15]. The masses for all quark flavors are compiled in Table 5. Table 5. The approximate masses of the quarks in GeV/c2 and their electric charges Q in units of e Flavor up down charm strange top bottom

Symbol u d c s t b

Free mass (5.6 ± 1.1) · 10-3 (9.9 ± 1.1) · 10-3 1.35 ± 0.05 0.199 ± 0.033

Constituent mass 0.33 0.33 1.5 0.5 180 4.5

Charge Q 2/3 - 1/3 2/3 - 1/3 2/3 - 1/3

The d - quark is heavier than the u - quark, which can be easily understood as follows. The proton (uud) and the neutron (ddu) are isospin symmetric; i.e., they transform into each of the under interchange of the u - and d - quarks. Since the strong interaction [16] is independent of quark flavor, the neutron - proton mass difference can only be due to the intrinsic quark masses and to the electromagnetic interaction between them [5, 6, 13]. If it assumes that the spatial distribution of the u - and d - quarks in the proton corresponds to the distribution of d - and u - quarks in the neutron, then it is easily seen that the Coulomb energy must be higher in the proton [16]. Despite this the neutron is heavier than the proton which implies that the mass of the d - quark is larger (see, also [13]). As we have mentioned earlier, hadrons have an internal structure and are

Experimental Study of the Long - Range Quark - Lepton ...

17

thus not elementary at all. It is a generic term used to designate mesons, baryons and their antiparticles. Mesons are mainly composed of quark ani - antiquark (not necessary of the same kind - flavor); they have a spin of 0 or 1, obey the Bose - Einstein statistics and for this reason are called bosons. Baryons are structures predominantly formed from three quarks, and are fermions of spin 1/2 or 3/2. Hundreds of hadrons have been produced, observed and identified, and their properties (mass, spin, charge, lifetime) determines (see e.g., [1, 17]). A small selection of such particles with relatively small masses are shown according [18] in Table 6. We see that such particles as π + , π- , and π0 have almost identical masses and moreover, all particles of the same spin and parity have very similar masses. We should also note that particles which mainly decay through electromagnetic interaction, signaled by the production of photons, have a mean lifetime in the range 10-20 - 10-16 s, whereas particles that decay through weak forces have a mean lifetime generally superior to 10-10 s. The lowest mass baryons are the proton and the neutron. They are the ground states of a rich excitation spectrum of well - defined energy (or mass) states. In this respect, baryon spectra have many parallels to atomic and molecular spectra. Yet, there is an important difference. The energy (or mass) gaps between individual states are of the same order of magnitude as the nucleon mass. These gaps are then relatively much larger than those of atomic or molecular physics. Consequently these states are also classified as individual particles with corresponding lifetimes. Like the proton and neutron, other baryons are also composed of three quarks, since quarks have spin 1/2, baryons have half integer spin. Many short - lived particles were thus detected, including excited states of the nucleon. This led to the conclusion that nucleons themselves are composed of smaller structures. This conclusion was extended to all known hadrons (see, e.g., [19]). Further we briefly discuss the properties of mesons. Hadrons composed of quark - antiquark pairs are called mesons. Mesons, as we can see from Table 6, have integer spin: their total spin results from a vector sum of the quark and antiquark spins, including a possible integer orbital angular momentum contribution. Mesons eventually decay into electrons, neutrinos and/or photons; there is no meson number conservation, in contrast to baryon (lepton) number conservation [1, 20]. This is understood in the quark model: mesons are quark antiquark combinations |qqi and so the number of quarks minus the number antiquarks is zero. Hence any number of mesons may be produced or annihilated.

18

V. G. Plekhanov

The lightest hadrons are the pions (see Table 6). Their mass, about 140 MeV/c2 , is much less than that of the nucleon. They are found in three different charge states: π- π0 , π+ . As we see above pions have spin 0. It therefore natural to assume that they are composed of two quarks, or more exactly, of a quark and an antiquark: this is the only way to build the three charge states out of quarks. The pions are the lightest systems of quarks. Hence, they can only decay into the even lighter leptons or into photons (see, e.g., [6, 21]. Table 6. Low - lying hadrons Hadrons Mesons π± π0 η K± 0 K0 , K K0S K0L Baryons p n Λ0 Σ+ Σ0 ΣΞ0 Ξa

I(Jp )a

Massb

Mean life (s)

Decay modes

1(0- ) 1(0- ) 0(0- ) 1/2(0-) 1/2(0-)

139.6 135.0 548.8 493.7 497.7

2.6 · 10-8 0.8 · 10-16 0.8 · 10-18 1.2 · 10-8

µ± ν γγ γγ, 3πo , π + π - π 0 µ± ν, π ± π o 50%K0S, 50%K0L π + π − , 2π 0 3π 0 , π + π - π 0 , π ± e± ν, π ± µ± ν

0.9 · 10-10 5.2 · 10−8 1/2(1/2+) 1/2(1/2+) 0(1/2+) 1(1/2+) 1(1/2+) 1(1/2+) 1/2(1/2+) 1/2(1/2+)

938.3 939.6 1115.6 1189.4 1192.5 1197.4 1314.9 1321.3

¿ 1031 yrs 917 2.6 · 10-10 0.8 · 10-10 7 · 10-20 1.5 · 10-10 2.9 · 10−10 1.6 ·10-10

stable pe- ν pπ - , nπ 0 pπ 0 , nπ + Λγ nπ − Λπ 0 Λπ -

Isospin, spin, parity; b In units of MeV/c2 .

Pions have the following quark structure:     |π+ i = ud , |π - i = |udi, π 0 = √1 |uui - dd . (16) 2 0  The π is a mixed state of |uui and dd . The above expression includes the correct symmetry and normalization. The pion mass is considerably smaller than the constituent quark mass describe above. This is another indication that the interquark interaction energy has a substantial effect on hadron masses (the more details see below).

Experimental Study of the Long - Range Quark - Lepton ...

5.

19

T HE QUARK - G LUON INTERACTION

It is well - known that including the color property, a kind of quark charge, the Pauli principle may be asserting. As was shown above, the quantum number color can assume three values, which called red, blue and green. Accordingly, antiquark carry the anticolors anti - red, anti - blue, and anti - green. Now the three u - quarks (for example in |∆++i = |u ↑ u ↑ u ↑i) may be distinguished [17]. Thus, a color wave function antisymmetric under particle interchange can be constructed, and we so have antisymmetric for the total wave function. We all know too that the interaction binding quarks into hadrons is called the strong interaction. As we mentioned above such a fundamental interactions always connected with a particle exchange. For the strong interaction, gluons are the exchange particles that couple to the color charge of quarks. This is analogous to the electromagnetic interaction in which photons are exchanged between electrically charged particles. The modern experimental findings (see, e.g., [22]) led to the development of a field theory called quantum chromodynamics - short QCD. As its name implies, QCD is modelled upon quantum electrodynamics QED [23]. In both, the interaction is mediated by exchange of a massless field particle with JP = 1- (a vector boson). Each gluon carries one unit of color and one of anticolor. It should appear, then, that there should be nine species of gluons - rr, rb, rg, br, bb, bg, gr, gb, gg. Such nine - gluon theory is perfectly possible in principle, but it would describe a world very different from our own [18]. In terms of group theory of color SU(3) symmetry (on which QCD is based [5, 24]), the 3 x 3 color combinations form two multiplets of states: a ’color octet’:  |1i    |2i  |3i   |4i

√ = (rb + br)/ √2 = -i(rb - br)/√ 2 = ( rr - bb)/ √2 = (rg + gr)/ 2

|5i |6i |7i |8i

√ = -i(rg - gr)/√ 2 = (bg + gb)/ √2 = -i(bg - gb)/ 2 √ = (rr + bb - 2gg)/ 6

and a ’color singlet’ √ |9i = (rr + bb + gg)1/ 3.

      

(17)

(18)

If the singlet gluon existed, it would be a common and conspicuous as photon. May be the ninth gluon is the photon. Confinement requires that all naturally occurring particles be color singlets, and this explain why the octet gluons

20

V. G. Plekhanov

never appear as free particles [14, 16]. But |9i is a color singlet, and if it exists as a mediator it should also occur as a free particle. Moreover, it could be exchanged between two color singlets (a proton and a neutron, say) giving rise to a long - range force strong coupling, whereas in fact up to present time we know that the strong force is of very short range. Therefore, there are evidently only eight kinds of gluons. We should add because gluons are massless, they mediate a force of infinite range ( the same as electrodynamics). In this since the force between two quarks is actually long range. However, confinement, and the absence of a singlet gluon, conceals this from us. A singlet state (such as the neutron) can only emit and absorb a singlet (such as the pion), so individual gluons cannot be exchanged between a proton and neutron. That’s why the force we observe is of short range. If the singlet gluon existed, it could be exchanged between singlets, and the strong force would have a component of infinite range. By their exchange the eight gluons mediate the interaction between particles carrying color charge (quarks), i.e., not only quarks but also the gluons themselves. This is an important difference to the electromagnetic interaction, where the photon field quanta have no charge, and therefore cannot couple with each other. In analogy processes of QED (emission and absorption, pair production and annihilation); emission and absorption of gluons (Fig. 4a ) take place in QCD, as do production and annihilation of quark - antiquark pairs (Fig. 4b ). In addition, however three or four gluons can couple to each other in QCD (Fig. 4c,d ).

Figure 4. The fundamental interaction diagrams of the strong interaction: emission of a gluon by a quark (a), splitting of a gluon into quark - antiquark pair (b), and ’self - coupling’ of gluons (c, d).

Experimental Study of the Long - Range Quark - Lepton ...

21

Further we briefly consider some main properties of group SU(3). The set of unitary 3x3 matrices with det U = 1 form the group SU(3). The generators may be taken to be any (32 - 1 = 8) linearly independent traceless Hermitian 3x3 matrices (see e.g., [5]). Since it is possible to have only two of these traceless matrices diagonal, this is the maximum number of mutually commuting generators. This number is called the rank of the group [24], so that SU(3) has rank 2, and SU(2) has rank 1. According to the theory of groups, the fundamental representation of SU(3) is a triple. The three color charges of a quark R, G, and B form the fundamental representation of an SU(3) symmetry group. In this representation, the generators are 3x3 matrices, which is called by λi , where i = 1, 2 ....8. For λi , the convention is to adopt Gell - Mann matrices [19], which are defined as 0

1 0 1 0 1 0 1 0 1 0 0 -i 0 1 0 0 0 0 1 λ1 = @ 1 0 0 A λ2 = @ i 0 0 A λ3 = @ 0 -1 0 A λ4 = @ 0 0 0 A 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 -i λ5 = @ 0 0 0 A λ6 = @ 0 0 1 A λ7 = @ 0 0 -i A λ8 = 0 i 0 0 1 0 i 0 0 0

1 √

1 1 0 0 @ 0 1 0 A 3 0 0 -2 0

(19)

with simultaneous eigenvectors 

 1 R= 0  0



 0 G= 1  0



 0 B =  0 . 1

(20)

0

The eight λi s satisfy the following commutations relations:

h

h

λi λj 2, 2

λi λj 2, 2

i

i

= ifijk λ2k

(21a)

= 13 δ ij + idijk λ2k ,

(21b )

where fijk is called the structure constant of SU(3) with {a, b} = ab + ba and fijk , dijk are totally antisymmetric or symmetric with respect to i j k permutations and have the values listed in Table 7. Below we briefly consider the interaction of a quark and anti - quark, in QCD. We shall assume that they have different flavors, so the only diagram (in lowest order is one in Fig. 8., representing for instance , u + d −→ u + d . The amplitude of this process is given (see Fig. 5) by (see e.g., [21]

22

V. G. Plekhanov Table 7. Structure constant of SU(3) f123 = 1 f147 = f246 = f√257 = f345 = f516 = f637 = f458 = f678 = 23 d118 = d228 = d338 = - d888 = √13 d146 = d157 = d256 = d344 = d355 = 12 d247 = d366 = d377 = - 12 1 d448 = d558 = d668 = d779 = - 2√ 3

M=

-gs 1 4 q2

1 2

 
[u(3)γ µ u(1)] v(2)γ µ v(4) c∗3 λα c1 (c∗2 λα c4 )

(22)

Here summation over α implied. This is exactly what we have for electron - positron scattering in QED, except that we use gs (of course), and we have in addition the ’color’ factor f=

1 4

(c∗3 λα c1 ) (c∗2 λα c4 ) .

(23)

The potential describing the qq interaction is, therefore, the same as that acting in electrodynamics between two opposite charges, only with α replaced by fαs : Vqq (r) = - f (αsr~c) .

(24)

Now, the color factor itself depends on the color state of the interacting quarks. from a quark and an antiquark we can make a color octet (17) and a color singlet (18) (all members of which yield the same f). A typical octet state (17) is rb (any of the others would do just as well) Here the incoming quark is red, and outgoing quark must also be red and the antiquark antiblue. Thus 

   1 0 c1 = c3 =  0 , c2 = c4 =  1  0 0

(25)

and hence 

   1 0 α f = 14 [(100) λα  0 ][(010) λα  1 ] = 14 λα 11 λ22 . 0 0

(26)

Experimental Study of the Long - Range Quark - Lepton ...

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Figure 5. The quark - antiquark interaction. A glance at the λ matrices (23) reveals that the only ones with entries in the 11 and 22 positions are λ3 and λ8 . So we can write f=

1 4



  √  √  (22) (11) (22) 1 λ(11) λ + λ λ = [(1) (-1) + 1/ 3 1/ 3 ] = - 61 . 3 3 8 8 4

(27)

√ The color singlet state is (22) [(rr + bb + gg)1/ 3] If the incoming quarks are in the singlet state the color factor is a sum of three terms:

1 √ 4 3

f9 = 82 0 13 2 0 13 2 0 13 1 0 0 < = 4c∗ λα @ 0 A5 (100) λα c4 + 4c∗ λα @ 1 A5 (010) λα c4 + 4c∗ λα @ 0 A5 (001) λα c4 3 3 : 3 ; 0 0 1 (28)

The outgoing quarks are necessary also in the singlet state, and we get nine terms in all, which can be written compactly as follows: f =

1 √1 √1 (λ λ ) 4 3 3 ij ij

=

α α 1 12 Tr(λ λ )

(29)

in last formula summation over i and j, from 1 to 3, implied in the second expression. Now Tr(λα λα ) = 2δαβ ,

(30)

so, with the summation over α, Tr(λα λα ) = 16

Evidently, then, for the color singlet

(31)

24

V. G. Plekhanov f=

4 3

(32)

Putting equations (28) and (33) into equation (25), we conclude that quark -antiquark potentials are Vqq (r) = -

4 (αs ~c) 3 r

(color singlet)

(33)

Vqq (r) =

1 (αs ~c) 6 r

(color octet).

(34)

and

from the signs we see that the force is attractive in the color singlet but repulsive for the octet. This helps to explain why quark - antiquark binding (to form mesons) occurs in the singlet configuration but not in the octet (which would have produced colored mesons) (see, [25]).

6.

P HENOMENOLOGY OF THE N UCLEAR F ORCE

As is well - known the nuclear force is that force which binds together the nucleons in the nucleus. Historically, the nuclear force was called the Strong Nuclear Force and was considered to be one of the four forces, along with the gravitational force, the electromagnetic force and the weak nuclear force [13]. Soon after the discovery of quarks, the force which holds the quarks together in a nucleon was called the Cromo (hadron) Dynamic Force. This chromo dynamic force, which is a sub - nucleon force, is considered to be much larger than the nuclear force. Later, the chromo dynamic force was redefined as the strong nuclear force, which is now considered to be a sub - nucleon force responsible for the behavior and interactions of sub - nucleon particles. The force which hold the nucleons together in a nucleus was renamed as the Nuclear Force. to add to this confusion even more, there is a model of the nuclear force, which is called the residual chromo dynamic force. Because of this model, it is presumed that the nuclear force is simply a subset of the strong nuclear force. Because of this assumption, it is still claimed that there are only four forces in nature: the strong nuclear force, the electromagnetic force, the gravitational force, and the weak nuclear force. So The strong nuclear force has two parts, the chromo dynamic force which is sub - nucleon force, and the nuclear force, which is that force holding the nucleons together in a nucleus. In this part of our chapter we

Experimental Study of the Long - Range Quark - Lepton ...

25

use the term nuclear force, which will be used to describe that force which holds together the nucleons in a nucleus. Our present view is that nuclear force may be divided into three parts, as illustrated schematically in Fig. 6 (see, also [13]). The long - range part (r > 2 fm) is dominated by one - pion exchange. If exchanges of a single pion are important, there is no reason to exclude similar processes involving two or more pions and mesons heavier than pions. The range of interaction associated with these more massive bosons is shorter, and for this reason, the intermediate range part of the nuclear force (1fm < r < 2 fm) comes mainly from exchanges of single heavier mesons and two pions. The hard core in the interaction (r . 1 fm) is made of heavy meson exchanges, multipion exchanges, as well as QCD effects (the more detail see [1, 3]). Now we turn to the task of understanding the strength and the form of the nuclear force from the structure of the nucleons and the strong interaction of the quarks inside the nucleons. Naturally in the following discussion we will employ qualitative arguments. The structure of the nucleon will be approached via the nonrelativistic quark model where the nucleons are built out of three constituent quarks. The nuclear force is primarily transmitted by quark - antiquark pairs, which we can only introduce ad hoc through plausibility arguments. We should repeat that a consistent theory of the nuclear force, based upon the interaction of quarks and gluons, does not yet exist [1, 3, 13, 20]. We begin with the short distance repulsive part of the nuclear force (see Fig. 6) and try to construct analogies to better understood phenomena. We remember that atoms repel each other at short distances is a consequence of the Pauli principle. The electron clouds of both atoms occupy the lowest possible energy levels and if the clouds overlap then some electrons must be elevated into excited states using the kinetic energy of the colliding atoms. Hence we observe a repulsive force at short distances. The quarks in a system of two nucleons also obey the Pauli principle, i.e., the 6 quark wave function must be totally antisymmetric. It is, however, possible to put as many as 12 quarks into the lowest l = 0 state without violating the Pauli principle, since the quarks come in three colors and have two possible spin (↓, ↑) and isospin (u - quark, d - quark) directions. The spin - isospin part of the complete wave functions must be symmetric since the color part is antisymmetric and, for l = 0, the spatial part is symmetric. We thus see that the Pauli principle does not limit the occupation of the lowest quark energy levels in the spatial wave function, and so the fundamental reason for the repulsive core must be sought elsewhere. The

26

V. G. Plekhanov

Figure 6. Schematic diagram showing the different parts of a nucleon - nucleon potential as a function of distance r between two nucleon. The hard core radius is around o.4 fm and it takes energy ¿ 1 GeV to bring two nucleons closer than (twice) this distance. The main part of the attractions lies at intermediate ranges, at radius ∼ 1 fm, and is believed to be dominated by the exchange of scalar mesons. The long - range part, starting at around 2 fm, is due to the single - pion exchange. real reason is the spin - spin interaction between the quarks [26, 27, 28, 29]. The potential energy then increases if two nucleons overlap (see below Fig. 7) and all 6 quarks remain in the l = 0 state since the number of quark pairs with parallel spins is greater for separated nucleons. Of course the nucleon - nucleon system tries to minimize its ’chromomagnetic’ energy of maximizing the number of antiparallel quark spin pairs. But this is incompatible with remaining in an l = 0 state since the spin - flavor part of the wave function must be completely symmetric. The colormagnetic energy can be reduced if at least two quarks are put into the l = 1 state. The necessary excitation energy is comparable to the decrease in the chromomagnetic energy, so the total energy will in any case

Experimental Study of the Long - Range Quark - Lepton ...

27

increase if the nucleons strongly overlap. Hence the effective repulsion at short distances is in equal parts a consequence of an increase in the chromomagnetic and the excitation energies. If the nucleons approach each other very close (r = 0) one finds in a non - adiabatic approximation that there is an 8/9 probability of two of the quarks being in a p state (see, e.g., [30, 31]. This configuration express itself in the relative wave function of the nucleons through a node at 0.4 fm. This together with the chromomagnetic energy causes a strong, short range repulsion. The nuclear force may be described by a nucleon - nucleon potential which rises sharply at separation less than 0.8 nm [1, 3]. Further we turn to the attractive part of the nuclear force. Again we will pursue analogies from atomic physics. As we know the bonds between atoms are connected to a change in their internal structure and we expect something similar from the nucleons bound in the nucleus. Indeed a change in the quark structure of bound nucleons compared to that of their free brethren has been observed in deep inelastic scattering off nuclei (EMC effect [32, 33]). It is clear upon a moments reflection that the nuclear force is not going to be well described by an ionic bond [35 - 37]: the confining forces are so strong that it is not possible to lend a quark from one nucleon to another. A van der Waals force, where the atoms polarize each other and then stick to each other via the resulting dipole - dipole interaction can also not serve us as a paradigm (see, also [34, 35]). A van der Waals force transmitted by the exchange of two gluons (in analogy to two photon exchange in the atomic case) would be too weak to explain the nuclear force at distances where the nucleons overlap and confinement does not forbid gluon exchange. At greater separations gluons cannot exchanged because confinement. Although color neutral gluonic states (glueballs) could still be exchanged, none which are light enough have ever been experimentally observed (see however below experimental manifestation of the strong interaction in solids). The only analogy left to us to explain the nuclear force is a covalent bond [37], such as that which is, e.g., responsible for holding the H2 molecule together [36]. Here the electrons of the two H atoms are continually swapped around and can be ascribe to both atoms. The attractive part of the nuclear force is strongest at distances around 1 fm and indeed reminds us of the atomic covalent bond. To simplify what follows, let us assume that the nucleon is made up of a two quark system (diquark [38, 39]) and a quark (Fig. 7). Such description has proven to be very successful in describing many phenomena [5, 6]. In this connection very important the hypothesis formulated by theoretics

28

V. G. Plekhanov

about the complex form of the dependence of the mass acquisition by current quarks and gluons in the transitions to the momentum infrared limit is possibly reflected in the nonlinear dependence of the energy of strong nuclear interaction on the distance between nucleons in the deuterium nucleus measured by us ( the details see [13, 38, 39]).

Figure 7. Quark configurations in a covalent bond picture. At large separations, when the nucleons just overlap, we may understand them as each being diquark - quark systems. The most energetically favorable configuration is that where a u - and d quark combine to form a diquark with spin 0 and isospin 0. The alternative spin 1 and isospin 1 diquark is not favored. The covalent bond is then expressed by the exchange of the single quarks. Since the nuclear attraction is strongest at distances of the order of 1 fm [9, 33], we do not to worry about confinement effects. The covalent bond contribution to this force can be worked out analogously to the molecular case. However, the depth of the potential that is found in this way is only about one third of the experimental value [40, 23]. In fact quark exchange is less effective to be exchanged the quarks must have the same color, and there is only a 1/3 probability of this. The contribution of direct quark exchange sinks still further if one takes the part of the nucleon wave function into account where the diquarks have spin 1 and isospin 1. Thus the covalent bond concept, if it is directly transferred from molecules to nuclei, does not give us a good quantitative description of what is going on in nuclei. It should be noted that this is not a consequence of confinement, but rather of direct quark

Experimental Study of the Long - Range Quark - Lepton ...

29

exchange being suppressed as a result of the quarks having three different color charges. One more the degree of freedom is an effective quark - quark exchange which may be produced by color neutral quark - antiquark pairs (see quarks). These pairs are continually being created from gluons and annihilated back into them again. This quark - antiquark exchange actually plays a larger role in the nucleon - nucleon interaction than does the simple swapping of two quarks. It must be stressed that this exchange of color neutral quark - antiquark pairs does not only dominate at great separations where confinement only allows the exchange of color neutral objects but also at relatively short distances. One may thus understand the nuclear force as a relativistic generalization of the covalent strong force via which the nucleons finally exchange quarks (the more details see [1, 41]).

7.

D EUTERON

As demonstrated earlier (see, e.g., [42]) most low - energy electron excitations in LiH (LiD) crystals are large - radius excitons [43]. LiH (LiD) crystals with a lattice of NaCl type, whose parameters are close to cubic crystals, are dielectrics with band gap of Eg = 4.992 eV (Eg = 5.090 eV for LiD at 2 K [42]). These crystals have an identical electronic structure [44]. The energy band structure of these substances is also identical [42]. All three kinds of forces - gravitational, electromagnetic and weak are also the same for compounds above. The difference between these substances consists out at one neutron in the nucleus of deuteron [1, 41, 45, 46]. Since the low - energy of the optical spectra of studied crystals is associated with deuterium, below we should briefly consider some peculiarities of the physics of deuteron. The deuteron is a very unique nucleus in many respects [9]. It is only loosely bound, having a binding energy much less than the average value between a pair of nucleons in all the other stable nuclei [8]. Partly because of the small binding energy, the deuteron has no excited states; all observations on the deuteron are made on the ground state. The main characteristics of the deuteron are shown by Table 8. Parity, or space reflection, transformation is the operation whereby all three coordinate axes in the Cartesian system change sign. In Cartesian coordinates, this means x −→ -x, y −→ -y, z −→ - z; in spherical coordinates, r −→ r, θ −→ π - θ, φ −→ φ + π. If a system is left unchanged by the parity operation, then we expect that no one of observable properties should change as a result of

30

V. G. Plekhanov Table 8. Ground state properties of deuteron Ground State Property Binding energy, EB Spin and Parity, Jπ Isospin, T Magnetic dipole moment, µd Electric quadrupole moment, Qd Radius, rd

Value 2.22457312(22) MeV 1+ 0 0.857438230(24) µN 0.28590(30) efm2 2.1413 (4) fm∗)

*) From CODATA, Review Modern Physics, 88, 0035009 (2015).

the reflection. Since the values we measure for the observable quantities depend on |ψ|2 then we have the following reasonable assertion: if V(r) = V(- r) then |ψ (r)|2 = |ψ (- r)|2 . For the deuteron, it is the parity is positive. Let us see what we can learn from this piece of experimental information. For this purpose, it is useful to separate the wave function into product of three parts: the intrinsic wave function of the proton, the intrinsic wave function of the neutron, and the orbital wave function for the relative motion between the proton and the neutron. Since a proton and a neutron are just two different states of a nucleon, their intrinsic wave functions have the same parity. As a result, the product of their intrinsic wave functions has positive parity, regardless of the parity of the nucleon. This leaves the parity of the deuteron to be determined solely by the relative motion between the two nucleons. → − For states with a definite orbital angular momentum L , the angular dependence in the wave function is given by spherical harmonics YLM (θ, φ). Under an inversion of the coordinate system, spherical harmonics transform according relation L

YLM (θ, φ) −→ Parity −→ YLM (π - θ, π + φ) = (- 1) YLM (θ, φ)

(35)

The parity of YLM (θ, φ) is therefore (- 1)L . The fact that the deuteron parity is positive implies that the orbital angular momentum must be even. → − The measured spin of the deuteron is I = 1. Besides that we know that → − deuteron has its parity is even [1]. The total angular momentum l of deuteron should be like → − − → → − → − I = Sn + Sp + l ,

(36)

Experimental Study of the Long - Range Quark - Lepton ... 31 → − → − where Sn and Sp are individual spins of the neutron and proton. The orbital → − momentum l of the nucleons as they move about their common center of mass → − is l . Since we know that the parity of the deuteron is even and the parity associated with orbital motion is determined by (-1)L we are able to rule out some options the observed even parity allows us to eliminate the combination of spins that include l = 1, leaving l = 0 and l = as possibilities (see also [1, 9]). For deuteron, only two nucleons are involved therefore we can assume further that the total magnetic moment is simply the combination of the neutron and proton magnetic moment  
µD = µn + µp = gSn µN Sn /~ + gSp µN Sp /~,

(37)

where the gyromagnetic ratio for a free nucleon is gn = - 3.826085 µN for a neutron and gp = 5.585695 µN for a proton. Here we have assumed that the structure of a bound nucleon inside a nucleus is the same as in its free state (see, however [13, 26, 27, 48]). Taking into account the maximum values of spins (+ ~2 ) we get µD = 21 µN (gsn + gsp ) = 0.879804 µN .

(38)

The observed value is 0.8574376µN in good but not quite exact agreement with calculated one. Usually this tiny discrepancy describes to take into account a small mixture of D state (l = 2) in the deuteron wave functions. Calculating the magnetic moment of deuteron from this wave function gives that deuteron is 96 per cent l = 0 (s orbit) and 4 percent (l = 2 ) (d orbit). Independent evidence for this fact comes from the observation that the deuteron has a small, but finite, quadrupole moment Q = 0.29 fm 2 (see, also [9, 46]). The last one points to the tensor character of the nucleon - nucleon interaction (the more details see, e.g., [1]). Nuclear magnetic dipole and electric quadrupole have a similar importance in helping us to interpret the deuteron structure (see, also [13]). The deviation of the actual deuterium moment from the S state moment can be explained if it assumed that the deuteron ground state is a superposition of S and D states. Part of the time, the deuteron has orbital angular momentum L = 2. Independent evidence for this act comes from the observation, as was noted above, the deuteron has a small, but finite, quadrupole moment (see, also [4]).

32

V. G. Plekhanov

As is well - known, the electric quadrupole moment measures the deviation of a charge distribution from sphericity [45‘]. For a spherical nucleus the expectations value of the squares of distance from the center to the surface along x -, y -, and z - directions are equal each other

2  2  2 2  x = y = z = r .

(39)

as a result, the expectation value of r2 = x2 + y2 + z2 is

2 2 

 r = x + y2 + z 2 = 3 z2

(40)


eq = e 3z2 - r2 .

(41)

 in the spherical case. Here r2 is the mean square radius of the orbit. The electric quadrupole moment eQ, which measures the lowest order departure from a spherical charge distribution in a nucleus, is defined in terms of the difference between 3z2 and r2 :

For a spherical nucleus hQi = 0. For a cigar shaped (prolate) nucleus, the charge is concentrated along z axis, and Q is positive. The quadrupole moment of a disk shape (oblate) nucleus Q is negative [45]. A positive quadrupole moment of Q = 0.29 fm2 according experiment indicates that the deuteron is slightly elongated the z axis, like an olive (prolate). Quantum mechanical definition of electric quadrupole moment for a single proton [9] is described by:
R eQ = e Ψ∗ 3z2 - r2 Ψdt.

(42)

Thus, if the quadrupole moment is not equal to zero, then the eigenfunction of the ground state of the deuteron assigns a probability of 0.04 to funding a 3 D1 state and a probability of a 0.96 to funding a 1 S1 state [46]. The last one points to the tensor character of the nucleon - nucleon interaction (for more details see, e.g., [13, 41, 46]).

7.1.

Nuclear Spin and Magnetic Moment

Much of what we know about nuclear structure comes from studying not the strong nuclear interaction of nuclei with their surroundings, but instead the much weaker electromagnetic interaction. That is, the strong nuclear interaction establishes the distribution and motion of nucleons in the nucleus, and we

Experimental Study of the Long - Range Quark - Lepton ...

33

probe that distribution with the electromagnetic interaction. In doing so, we can use electromagnetic fields that have less effect on the motion of nucleons then the strong force of nuclear environment, thus our measurements do not seriously distort the object we are trying to measure. As said above, many nuclei show an intrinsic angular momentum or spin [9]. This is always a multiple of ~ for nuclei of even mass number and always an odd multiple of ~/2 for nuclei of → − odd mass number. The spin in ~ units is indicated by the vector I . We must → − remind here that I has the properties of a quantum mechanical angular mo→ − mentum vector [8, 9]. The vector I can be considered the sum of the orbital and intrinsic contribution (spin) to the angular moment  − A − → − → → − − P → I = li + → si = L + S ,

(43)

i=1

→ − → where li and − si orbital moment and spin, respectively of the i - nucleon. The → − quantum number I has the usual connection with the vector I Ii = mi ~ (mi = I, I - 1, · · · · ·· -I +1, -I).

(44)

→ − → Analogous picture we have for the total (orbital) ( li (e) and − si (e) ) electronic angular moment Z − → − P → → J = ( li (e) + − si (e) ).

(45)

i=1

→ − → − → − Sum of the I and J vectors determines the vector F − − → → − → F = I + J.

(46)

→ − − → − → We should underlined that all indicated vectors F , I , J obey to usual quantum mechanics rules for angular momentum (see, e.g., [9]). Associated with the spin is a magnetic moment. Classically, the magnetic dipole moment µ arises from the motion of charged particles, and we can regard µ as a means to characterize a distribution of currents whose effect on the surroundings (that is, on the other moving charges) we call magnetic (see, also [47]). When we go to the quantum limit we find a similar relationship [9] with one distinctly nonclassical addition: the intrinsic angular momentum (spin) contributes to the magnetic moment also.

34

V. G. Plekhanov

Below we briefly review the classical electromagnetism [47] that leads to → →
− magnetic dipole moments. The calculation of the vector potential A − r at → the observation point − r by summing (integrating) over all the currents in the sample:
− → → A −r =

0 →’ − µ0 R j( r )dv 0 − → − 4π |→ r - r |

(47)

and then, as was shown above, the magnetic field follows directly from
− → − → −
− → → B −r = ∇ x A → r .

(48)

Following some mathematical manipulations, which can be found in standard text on electromagnetism (see, e.g., [47]), we can rewrite the vector potential as
− → → A −r =

µ0 1 R 4π [ r

 − − → 0 j → r dv’ +

1 r3

 R−
→ − 0 −r - − → → j → r r 0 dv’ + · · · · ·]

(49)

− − µ0 → µ x→ r , 4π r3

(50)

which can be written
− → → A −r =

where R →0 − → − 0
− → µ = 21 − r x j → r dv’.

(51)

The leading nonvanishing term is characterized by the magnetic dipole moment µ of the current distribution. Going over to the quantum limit, the charge →0
2 density is e Ψ − r , and it is entirely consistent with quantum mechanics to write this as µ=

e 2m

R


→ −r 0
dv’, Ψ∗ − r 0 lΨ →

(52)

where l is angular moment. If the wave function corresponds to a state of definite lz, then only the z component of the integral (52) is nonvanishing and µz =

e 2m

R


→ −r 0
dv’ Ψ∗ − r 0 lz Ψ →

(53)

and µz =

e~ 2m ml

(54)

Experimental Study of the Long - Range Quark - Lepton ...

35

with lz = ml ~. The quantum number ml has a maximum value of +l, and thus the magnetic moment µ is µ=

e~ 2m l.

(55)

As was shown above, putting in (55) the proton mass for m, we get a nuclear magneton µN µN =

e~ 2mp

= 3.15245 · 10−8 eV/T.

(56)

Taking into account the intrinsic spin, which has no classical analog, we make a simple extension of Eq. (55) µ = (gl l + gs s) µN /~,

(57)

where g factors gl and gs account for orbital and intrinsic (spin) contributions to µ. Their values can be adjusted as needed for individual particles: gl = 1 for protons and gs must be measured for ”free protons” in which l does not contribute to µ (see, also [13, 26, 27, 50), gs is measured to be 5.5856912 for protons. For neutrons, which are uncharged, we can get gl = 0 and gs is measured to be -3.8260837. There is no single theory that allows to evaluate Eq. (61) to calculate µ because the interaction between nucleons are strong and the relative spin orientations are not sufficiently well known (for details see [13, 26, 27, 44, 50]). It is very important, that modern view of magnetism can be described as a purely relativistic effect [5].

7.2.

Hyperfine Interaction

Inside an atom the nuclear moment can magnetically interact with the electronic moment but only very weakly. This leads to energy splitting which are even smaller than the fine structure (spin - orbit interaction) and is known as hyperfine structure [51]. The weak magnetic interaction between the nucleus and the electrons are known as hyperfine interaction. Their origin is rather complex [53, 54] , but the essential principle can be understood in the following simple picture → [52]. Consider a nuclear magnetic moment − µ N which sits in a magnetic field → − B elec which is produced by the motion and the spin of all the electrons. This → − → − → produces an energy term - − µ N B elec . Now B elec is expected to be proportional → − to the total angular momentum of all the electrons, J , so that the Hamiltonian for the hyperfine interaction (hf) can be written as

36

V. G. Plekhanov →− → b hf = A− H I J

(58)

Here I is, as early, the nuclear angular moment and A is a parameter which can be determined from experiment and measures the strength of the hyperfine interaction [1]. The precise form of hyperfine interaction is a complicated for a general atom [51], but can be considered out in detail for a single electron atom interacting with point nucleus. Below we treat the our task in outline only to illuminate the basic mechanism of the interaction. There are two types of magnetic interaction [52]: magnetic dipolar and Fermi contact interaction. The dipolar interaction between a nuclear moment → − → → − → b dipol µ N = gN µI I and electronic moment − µ e = ge µ B − s is given by H or in form

b dipol = H

b dipol = H

µ0 4πr3



µ0 ge gN µB µI 4πr3

µeµI -

3 r2

h− →− → I S -


−−
 → − → µe − r → µI → r

3 r2

−  − i →− →− S→ r I→ r .

b dipol ∝ (1 - 3cos2 θ)IzSz /r3 H

(59)

(60) (61)

and hence leads to an interaction energy which is proportional to R

−
2 → d3 − r Ψ → r (1 - 3cos2 θ)mS mI /r2

(62)

an expression which vanishes in the case of an s orbital. This is essentially because a dipole field averages to zero over a spherical surface (see, also [45]). The spherically symmetric s orbital wave function average to zero. The integral does not vanish however for an unpaired electron in an orbital with l i 0 such as a p orbital. Above treatment has also only considered the spin part of the electronic moment, the orbital moment of the electron also produces a magnetic →− − → field of the nucleus which gives a term proportional I L . The second mechanism is the Fermi contact interaction, which in contrast to the previous mechanism which vanished for an s orbital, vanishes for every orbital except an s orbital. The nucleus is not course a point dipole, we further assume that nucleus is a sphere of uniform magnetization M = µI /V, where V is a finite nucleus volume. The magnetic flux density inside this sphere is then µ0 M, but we must first substract the demagnetization field which for sphere is µ0 M/3 (here µ0 is a magnetic permeability of free space µ0 = 4π · 10−7 Hm−1 ) leaving a flux density equal 2µ0 M/3. Thus an electron which ventures inside the nuclear sphere would experience a magnetic flux density

Experimental Study of the Long - Range Quark - Lepton ... − → B =

→ 2µ0 − µI 3V ,

37 (63)

→ so that the energy cost can be obtained by multiplying this by − µe |Ψ (0)|2 V, the amount of electronic moment which is in the nucleus. Note that |Ψ (0)|2 V is the probability of finding the electron inside the nucleus [55]. Thus the energy cost Econtact is then Econtact =



2µ0 3



→ →− − 2µB gN I S ,

(64)

→ − → − where µe = 2µB S and µI = gN µN I . We repeat that only s - electrons have any → amplitude at − r = 0 and so only they show this effect. The hyperfine splitting caused by the Fermi contact interaction for s - electrons tends to be much larger than the hyperfine splitting due to the magnetic dipolar effect for electrons l i 0. As was shown above (Eq. (46)) the total angular momentum of the atom, i.e., of the combination of the nucleus and the electron is given by − − → → − → F = I + J.

(46’)

From this equation, we have −2 − → → → − →− − → F = J2+ I 2+2I J

(65)

→− − → and since the hyperfine interaction takes the form A I J , the expected value of this energy is D − →− →E AI J =

A 2

[F (F + 1) - I (I - 1) - J (J + 1)].

(66)

The hyperfine interaction can be added as a perturbation, and it splits the electronic levels up into different hyperfine structure levels. Each of these levels has a degeneracy of 2F + 1, so that these states can be split up into their different mF values by applying a magnetic field. The energy separation between adjacent levels E(F) and E (F - 1) is therefore given by E(F) - E(F - 1) = AF.

(67)

Thus splitting is proportional to F, the larger of the quantum numbers of the adjacent levels being considered (the more details see [52, 55]) lie at the heart of the phenomenon of long range magnetic order. Exchange interactions, arising because charges of the same sign cost energy when they are close together and save energy when they are apart [56].

38

7.3.

V. G. Plekhanov

The Origin of the Fundamental Forces

As we have mentioned earlier, there are just four fundamental forces in nature: electromagnetic, weak, strong and gravitational. To each of these forces there belongs a physical theory [6]. The classical theory of gravity of course, famous Hook ’s law of universal gravitation. A completely satisfactory quantum theory of gravity has yet to be worked out; for the moment most physicists assume that gravity is simply too weak to play a significant role in elementary particle physics. Indeed the ratio of electromagnetic force and gravitational interaction between two point - like particles (e.g., protons) have the value: Fel /Fgr ' 1039 . This is very small value. As far as we know, the physical theory that describes electromagnetic forces is called electrodynamics. It was given its classical formulation by Maxwell in 19 century. The quantum theory of electrodynamics was perfected by Tomonaga, Feynman, and Schwinger in the 1940s (see, e.g., [10, 23]. The first theory of the weak forces was presented by Fermi in 1933, which account for nuclear β - decay were unknown to classical physics [1, 57]. As for the strong forces, beyond the pioneering work of Yukawa in 1934 [58] there really was no theory until the emergence of chromodynamics in the mid seventies [5, 25]. Ever since Maxwell unified the theory of electricity and magnetism, the unification of the gravitational and electromagnetic fields had become the dream of the physics community. The first attempt was to combine the theory of gravity and electromagnetism by Wilson [59] about 100 years ago, using Eddington’s fake experimental data on the bending of a light beam by the Sun [60]. After this work, a number of papers [61 - 65] were performed on this topic. Wang’s work [64] deserves close attention, which briefly out lines the history of the issue. Both the gravitational and the electromagnetic forces vary in strength as the inverse square of the distance and so able to influence the state of an object even at very large distances. Gravitational is important for the existence of stars, galaxes, and planetary systems as well as for our daily life [5]. Electromagnetism is the force that acts between electrically charged particles (atoms, molecules, condensed matter), when nuclear physics developed, two new short - ranged forces joined the ranks. These are the strong nuclear force, which acts between nucleons (proton, neutron etc.) and the weak force, which manifests itself in nuclear β - decay (see, e.g., [1]). Each force is characterized by its own constant of interaction, which differ by many orders of magnitude. The latter cannot serve as a basis for the division of forces with different names for them,

Experimental Study of the Long - Range Quark - Lepton ...

39

which may have the same nature of origin. Indeed, a single theory which when applied to different physical structures can manifest itself in widely differing strengths so as to look like different interactions. Consider the chemical force between neutral atoms. Its strength is enormously weak compared to Coulomb force. We could have described it as a new phenomenon and ascribed to it a new small coupling constant. But it is a weak manifestation of electromagnetism when applied to neutral atoms which are composite. Once more, consider the α - decay process. It could be described by a new coupling constant between the parent nucleus, the daughter nucleus and the α - particle [66]. It is generally very weak, but again it is a weak manifestation [67] of electromagnetism, namely the tunnel effect [66]. Or, consider as a third case, the ratio of the electric force between two electrons with the magnetic force between their magnetic moments, assumed to be pointlike, at the distance r we have Vmagn Velect

∼

µ2 /r3 e2 /r

∼ 1/4m2 r2

(68)

This ratio is of the order of α-2 at the distance of Bohr radius r = 1/mα, but of the order of α-2 ∼ 104 at the nuclear distance r = α/m = 2.8 fermis. Thus electromagnetism can manifest itself very weakly (chemical force, α - decay), or very strongly (magnetic force at short distance) (see, also [68]), or simply electromagnetically (Coulomb). A very interesting statement was published by the author of the paper [62]. ....Instead of giving different names to the different manifestations of the same force, describing them by new fields and new coupling constants, one should try to identify the different manifestations of the single force and the different situation to which it is applied..... It is difficult to disagree with this statement, since there is nothing to unify, so far as the nature of force is the same and we should not separated the forces in the first place. It is clear that the unification of the electromagnetic, weak and strong interactions is possible on the basis of magnetic or spin interactions...... We emphasize that above estimates were made in the model of point particles, whereas Landau [69] already more than half of century ago, underlined that the physics of elementary particles should be based not on a point model, but on a spatially blurred (see below). However, even careful model allows us to conclude that the effective interaction potential between two electrons for example, is such that at large distances it is essentially the usual Coulomb potential, but widely separated from it, at the nuclear distance, it can be a very strong attractive magnetic force followed at still shorter distances by a repulsive core (see Fig. 6 in

40

V. G. Plekhanov

paragraph 6). Thus the magnetic forces show the remarkable phenomenon that it is possible for the same particles to have one type of interaction at lower energies and another type of qualitatively and quantitatively different interaction at higher energies, or shorter distances. In such a way we can identify the strong interaction with spin forces. As was indicated above unification of gravitational force with other fundamental forces has been a dream of physicists during the 20th century. However unification through general relativity and quantum field theory proves hopeless. Wang [64] developed a classical theory of unifications of the gravitational and electromagnetic forces ( see, also [70]). His theory is mathematically rigorous and complete without any ad hoc hypothesis. It is very interesting to compare Hook’s (Newton’s) law of gravitation − → F = - G m1r2m2 br

to Coulomb’s law

(69)

− → q q F = - k1 1r2 2 br

(70)

it looks strikingly similar. We do not see in physics such identical form of interactions considering the fundamentally and complexity of the two kind interactions. Moreover, both equations are static having no time dependence. True enough, it knows that Eq. (70) is only an approximation when the magnetic force is not present. We all know that Hook’s (Newton’s) law does not have anything similar to the magnetic force. In his theory Wang postulates that Hook’s (Newton’s) law of gravitation needs a dynamic term to explain the propagation of gravitational wave and answer the historical question of action - at - distance. Wang’ theory takes exactly the same form of Maxwell’s theory of electrodynamics, which predicts that the gravitational wave propagates with the same speed of light [70]. Following to Wang, consider the Lorentz force between two → → charges q1 and q2 , their velocities are respectively − v 1 and − v 2 , separated by a → − displacement r : − → q q  F = k1 1r2 2 br +

→ 1 − v2 c2


 − x → v 1 x br =

q1 q2 4πε0 r2

 br +

→ 1− v2 c2


 − x → v 1 x br

(71)

where k1 is the Coulomb constant, br is the unit displacement vector, and ε0 the electric susceptibility of the vacuum, and c the speed of light: c=

√ 1 ε0 µ0 .

(72)

Experimental Study of the Long - Range Quark - Lepton ...

41

According to Wang the first term of Eq. (71) is the electrostatic force, while the second term is the dynamic (magnetic) force. As it is well - known the dynamic term of the electromagnetic force is vitally important for the spectacle of Maxwell’s theory of electrodynamics [47], and for working of electrical engineering. A complete theory of interaction must contain a dynamic term if we want to describe the propagation of the interaction. But there is no dynamic term in Hook’s (Newton’s) law of gravitation. It is a drastic difference between Hook’s gravitational force and electromagnetic force. Taking dynamic term into account, Wang obtains next achievements; 1. The first achievements of his theory is a logical and natural explanation of the propagation of the gravitational force, which is nothing different from propagation of electromagnetic wave. 2. The second achievement of Wang’s theory is the revelation of the essence of the inverse - square law that governs both the electrodynamic and gravitational interactions. It demonstrates that the inverse square law is the result of the conservation of the total static and dynamic fluxes as expressed in Gauss’ Law and Wang’s law. The Gauss’ law says that the mass is conserved, and Wang’s law says that the total momentum transmitted into space is conserved. According to Wang estimation the dynamic term of the gravitational force is about a factor of 3 x 10−14 smaller than the static term. This is why the dynamic term of gravitation escaped detection by scientists. We should repeat that the Wang theory was considered in point - like particle model. Thus, all fundamental forces have the electromagnetic origin. On the other hand, we should note that taking into account the spatial extent of, for example, an electron has been going on for more than century ago starting with the classic paper of Compton [71]. At the beginning XX century it was propose a new model for the electron with a ring shaped geometry where unitary charge moves around the ring generating a magnetic field. The electron behaves not only as the unit of electric charge but also as the unit of magnetic charge or magneton [2]. Compton explained his effect more consistently in the ring model than in its spherical model. The origin of the strong interaction is still highly controversial. Returning to the classical description in the model of the Yukawa potential [58], it is necessary to note the following [65]. The most important feature Yukawa’s forces is that they have a small range (∼ 10−15 m). The central dogma of atomic

42

V. G. Plekhanov

physics after Yukawa’s paper that proton - electron attraction could be explained in terms of classical electrostatic theory, while the strong force effects were essentially new and inexplicable. So, far the best theoretical guess is the Yukawa potential, but it is a static potential not dependent on velocities of the nucleons. A static force is not a complete one because it can not explain the propagation of the nuclear interaction. Moreover, phenomenological Yukawa potential cannot be directly verified experimentally. We should note that nowadays in text books and elsewhere the separation of electromagnetic and strong interaction tacitly assumed. It is very strange up to present time we do not even know the strong force very well. And what is more we have some contradiction taking into account that the forces between quarks must be long - range, because the gluons have zero mass. But the force between colorless hadrons is short - range, when the distance between hadrons is more than nuclear size [65]. We can see that the border of the nuclear size transforms long - range interaction in the short - range one. It is very old question which up to present time has not any theoretical explanation. Returning to the ring model of the electron it is important to add that this model supported by Schr¨odinger’s Zitterbewegung. Both the phenomenon and its name were conceived by Erwin Schrodinger who, in 1930, published the paper Ueber die kraeftefreie Bewegung in der relativistischen Quantenmechanik [72], in which he was analyzed to the Dirac equation [8] and identified a term called the Zitterbewgung. It is well - known that Dirac equation (see above) of the electron has three sets of independent dynamical variables: position of → − → → → the charge − x , velocity of the charge ddtx = c− α , and momentum − p . Position and velocity, can be specified simultaneously, they commute, but position and momentum do not. Namely, this gives rise to the remarkable Schr¨odinger phenomena of Zitterbewegung. In consequence, the electron velocity is not a constant of the motion also in absence of external fields. Such an effect must be of a quantum nature as it does not obey Newton’s first law of classical motion. Schr¨odinger calculated the resulting time dependence of the electron velocity and position concluding that, in addition to classical motion, they experience very fast periodic oscillations which he called Zitterbewegung (ZB). Schr¨odinger’s idea stimulated numerous theoretical investigations but no experiments since the predicted frequency ω Z = 2mc2 /~ = 1.6 · 1021 s−1

(73)

and he interpreted this as a fluctuation of electron with the radius which is equal

Experimental Study of the Long - Range Quark - Lepton ...

43

to the Compton wavelength λC = c/ω0 = ~/mc = 1.9 · 10−13 m.

(74)

In 1952, Huang [73] provided a classical interpretation of the Dirac equation in which the ZB is the mechanism that causes the electron’s angular momentum (spin) λC (mc) = ~/2 (see, also [74]). According to these investigations this angular momentum is the cause of the electron’s magnetic moment. Numerous investigations of this phenomena have showed that the ZB is a the electron’s oscillatory motion that is hidden in the Dirac equation. The Helical Electron Model (see Fig. 8) assumes that the electron’s charge is concentrated in a single infinitesimal point called the center of charge that rotates at the speed of light around a point in space called the center of mass (see also [75]).

Figure 8. Helical electron model. The electron’s helical motion can be deconstruct into two orthogonal components; a rotational motion and translational motion. The velocities of rotation and translation are not independent; they are constrained by the electron’s tangential velocity that is constant and equal to the speed of light [76]. As noted above, when the electron is at rest, its rotational velocity is equal to the speed of light. As the translational velocity increases, the rotational velocity must decrease. At no time can the translational velocity exceed the speed of light (for the details see [74]). Unitary charge flows through the ring (helical motion) at the speed of light, generating an electric current and associated magnetic field. In the paper [77] was calculated the magnetic field at the center of the ring, resulting in a magnetic field of 30 million Tesla. It is very huge value of magnetic

44

V. G. Plekhanov

field. For comparison, the magnetic field of the Earth is 0.000005 Tesla, and the largest artificial magnetic field created by man is only 90 Tesla. It is necessary to add, that if the electron moves at a constant velocity, the particle’s trajectory is a cylindrical helix. The electron’s helical motion can be interpreted as a wave motion with a wavelength equal to the helical pitch and a frequency equal to the electron’s natural frequency. In a toroidal solenoid, any magnetic flux is confined within the toroid. By analogy we can assume the all subatomic particles have the same structure as the electron, differing mainly by their charge and mass. The radius of a nucleon is equal to its reduced Compton wavelength. As is well - known the Compton wavelength is inversely proportional to an object’s mass, so for subatomic particles, as mass increases, size decreases [71]. The current of a nucleon is about 2000 times the current of an electron, and the radius is about 2000 times lower. This results in a magnetic field at the center of the nucleon’s ring that is about four million times bigger than that of the electron [77]. As we all know, magnetic forces are dominant over very small distances (see, also [78]), but their influence decreases rapidly with respect to electrical forces as distant expand Fmag ∼

1 , R3

while Felec ∼

1 . R2

(75)

In this model the magnetic field inside the neutron’s ring is huge, outside the ring, the magnetic field decays much faster (79) than the electric field. The comparison of our non accelerator experimental results with this model will be

8. 8.1.

E XPERIMENTAL MANIFESTATION OF STRONG INTERACTION Energy Band Structure

As it was briefly noted in introductory part the macroscopic manifestations of the strong interaction are restricted up to now to radioactivity and the release of nuclear energy [13, 41, 46]. The present review describes the new way of the strong interaction study, literally we have taken into account the tight binding

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between nuclear and condensed matter physics. Literally this idea was used in our experiments, where we have measured the low - temperature optical spectra of solids with the isotope effect. Photoluminescence is the optical radiation emitted by a physical system (in excess of the thermal equilibrium blackbody radiation) resulting from excitation to a nonequilibrium state by irradiation with light. Three process can be distinguished: (i) creation of electron - hole pairs (excitons); (ii) radiative recombination of electron - hole pairs (excitons); and (iii) escape of the recombination radiation from the sample. Since the exciting light is absorbed in creating electron - hole pairs (excitons), the greatest excitation of the sample is near surface; the resulting carriers (excitons) distribution is both immunogenicity and nonequilibrium. In attempting to regain homogeneity and equilibrium, the excess carriers will diffuse away from the surface while being depleted by both radiative and nonradiative recombination process. Most of the excitation of the crystal is thereby restricted to a region within a diffusion length (or absorption length) of the illuminated surface. It follows that recombination radiation most readily escapes through the nearby illuminated surface. Consequently, the vast majority of photoluminescence experiments are arranged to examine the light emitted from the irradiated side of the sample. Photoluminescence is also rapidly evolving into a major basic research tool comparable to absorption (reflection) measurements in importance. Two reasons for this stand out as significant. First is the sensitivity of the luminescence technique. It often happens that feature which are just discernible in absorption will completely dominate the luminescence spectra. The converse is also sometimes true, making luminescence and absorption (reflection) complementary techniques. Second is the simplicity of data collection. A disadvantage of photoluminescence technique is the increased remoteness of the raw data from the physical phenomena of principal interest. A major purpose of the present part of our book is to provide to introduction to the theoretical and experimental aspects of the photoluminescence of crystal with isotope composition. On the experimental side our purpose is to bring together all the important independent empirical results and take overview of the collection of findings to establish what is actually known and what needs additional experimental or theoretical work. To attain these goals, it is intended that this background be studied in close conjunction with the cited literature (see [3] and references therein). It is not intended as a self contained treatise. Concluding this paragraph we should underlined, that if the structure of spectra of the fundamental reflection (absorption) depends on the

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internal degrees of freedom of Wannier - Mott exciton, then the structure and shape of the luminescence spectrum are determined primarily by its external degrees of freedom. The latter is associated with the translational motion of large - radius exciton as whole, with the translation mass M = me + mh , here me and mh are effective masses of electron and hole (for details see [45]). We should remind that in a solid one deals with a large number of interacting particles, and consequently the problem of calculating the electronic wave functions and energy levels is extremely complicated. It is necessary to introduce a number of simplifying assumptions. In the first place we shall assume that nuclei in the crystalline solid are at rest. In an actual crystal this is of course never the case, but the influence of nuclear motion on the behavior of electrons may be treated as a perturbation for the case in which they are assumed to be at rest. Even with above assumption, however, we are still with a many – electron problem which can be solved only be approximative methods. In the case of solids, the most important approximative method which has been applied extensively is the so – called one – electron approximation. In this approximation the total wave function for the system is given by a combination of wave functions, each of which involves the coordinates of only one electron. In other words, the field seen by a given electron is assumed to be that of the fixed nuclei plus some average field produced by the charge distribution of all other electrons. The difference between a good conductor and a good insulator is striking. The electrical resistivity of a pure metal may be as low as 10−10 ohm-cm at a temperature of 1 K, apart from the possibility of superconductivity. The resistivity of good insulator may as high as 1022 ohm-cm. The understand the difference between insulators and conductors, we shall use the band – gap picture ( Fig. 9 below). The possibility of band gap is the most important property of solids. The simple electron structure of lithium hydride (Li+ and H - ions, having 1s2 configurations) (combined with the negligibly small spin-orbital interaction) is very helpful for calculating the band structure: all electron shells can easily be taken into account in the construction of the electron potential. The first calculations of band structure of lithium hydride were carried out as early as 1936 by Ewing and Seitz [79] using the Wigner-Seitz cell method. This method consists essentially in the following. The straight lattice is divided into polyhedra in such a way that the latter fill the entire space; inside each polyhedron is an atom forming the basis of the lattice (Wigner-Seitz cells). The potential inside each cell is assumed to be spherically symmetrical and coinciding with the potential

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Figure 9. Various possibilities to present the band-structure of homogenous, undoped semiconductor (insulator). 1 - The dispersion relation i.e., the energy E as a function of the wavevector ~k, 2 - The energy regions of allowed and forbidden states as a function of a space coordinate x and, 3 - the density of states (all curves are schematic ones). of free ion. This approximation works well for ions with closed shells. The radial Schr¨odinger equation in the coordinate function Rl (~r) is solved within each selected cell, the energy being regarded as a parameter. Then the Bloch function is constructed in the form of expansion Ψ~k (~r) =

  ~k Ylm (θ, ϕ)Rl (~r, E), C lm m=−l

P∞ Pl l=0

(76)

where ~r, θ, ϕ are the spherical coordinates (with respect to the center of the cell); Ylm are spherical functions. The coefficients Clm (~k) and the energy E(~k) are found from conditions of periodicity and continuity on the boundaries of the cell. If ~r1 and ~r2 are the coordinates of two points on the surface of Wigner-Seitz ~ l , then the boundary conditions are cell, linked by the translation vector R   ~ l Ψ~ (~r1 ), Ψ~k (~r2 ) = exp i~k R k

(77)

  ~ l ∇n Ψ~ (~r1 ), ∇n Ψ~k (~r2 ) = exp −i~k R k

(78)

and

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V. G. Plekhanov

where ∇n is the gradient normal to the surface of the cell. We see that the method of cells only differs from the problem of free atom in the boundary conditions. Owing to the complex shape of the cell, however, the construction of boundary conditions is a very complicated task, and this method is rarely used nowadays. The method of plane associated waves (PAW) was used for calculating the band structure and the equation of state for LiH was used in Perrot [80]. According to this method, the crystal potential is assumed to be spherically symmetrical within a sphere of radius ~rs described around each atom, and constant between the spheres (the so-called cellular muffin-tin (MT) potential). Inside each sphere, like in the Wigner-Seitz method, the solutions of Schr¨odinger equation have the form of spherical harmonics; outside the spheres they become plane waves. Accordingly, the basis functions have the form   P Ψ~k (~r) = exp i~k~r θ (~r - ~rs ) + alm Ylm (θ, ϕ) Rl (E, ~r) θ (~rs - ~r),

(79)

where θ(x) =1 at x ≥ 0, and θ(x) = 0 at x < 0. The coefficients alm can be easily found from condition of sewing on the boundary of the sphere. This is an important advantage of the PAW method over the method of cells. The calculations of Perrot [80] are self-consistent, and the local potential is used in the Cohn-Sham form. The correlation corrections were neglected. The method of CorringiCohn-Rostocker (CCR method), or the method of Green’s functions, was used for calculating the band structure of LiH in Zavt et al. [81] (only concerned with the valence band) and in Kulikov [82]. Calculation of band structure of LiH in Grosso and Paravicini [83] was based on the wave function used in the method of orthogonalized plane waves (OPW) of the form   P D   E Ψ~k (~r) = exp i~k~r - c exp i~k~r | Xc Xc (~r) ,

(80)

D   E where Xc are the atomic functions of state of the skeleton; exp i~k~r | Xc is the integral of overlapping of plane wave with skeleton function (see also [84]). The method of linear combination of local basis functions was applied to the calculation of band structure of LiH in Kunz and Mickish [85]. This method is based on constructing the local orbitals for the occupied atom states, based on certain invariant properties of the Fock operator. The main feature of local orbitals is that they are much less extensive than the atom orbitals. Importantly, the correlation correction is taken into account in Kunz and Mickish [85]. Owing to

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the high polarizability of hydrogen molecules, the correlation effect in lithium hydride is exceptionally strong. Yet another calculation of band structure of LiH was carried out in Zavt et al. [81] using the so-called method of extended elementary cell [86]. This approach is based on the semiempirical techniques of the theory of molecules, and is much similar to the cluster calculations. Let us add that the cluster is selected in such a way that the quasi-molecular wave function transforms in accordance with the group symmetry of certain wave vectors in the Brillouin zone. This methods only yields the energy values at the points of high symmetry. We ought to mention also Hama and Kawakami [87], where, in connection with the study of high pressure effects on the transition NaCl CsCI in lithium hydride, the band structure and the equation of state of the latter are analyzed in detail. The calculated band structures of LiH are compared in Fig. 10. We see that the overall picture given by various methods is generally the same, despite the vast spread of the transition energy values (see Table 9).

Figure 10. Band structure crystal as calculated in Ref. [85] (a); [80] (b); [83] (c). Looking at the structure of the valence band we see that it is very similar to the s - band in the method of strong bond (see also [86]). This is surprising, given the strong overlap of the anion s - functions in lithium hydride. The wave functions in this band are almost entirely composed of the Is states of hydrogen ion. Different authors place the ceiling of the band either at point X or at point W of the Brillouin zone. Although in all cases the energies of the states X1

50

V. G. Plekhanov

and W1 differ little ( ≤ 0.3 eV), the question of the actual location of the top of the valence band may be important for the dynamics of the hole. Different calculations also disagree on the width of the valence band. Table 9. Calculated energy values of some direct optical transitions in LiH reduced to the experimental value of Eg = 5.0 eV Transition K1 - K3 W1 - W3 L1 - L’2 W1 - W’2 X1 - X’5 K1 - K4 L1 - L’3 Γ1 - Γ15

1 6.9 8.0 9.2 12.6 12.9 14.7 19.7 24.5

2 7.5 7.9 9.6 14.9 13.8 16.1 20.9 25.3

3,4 6.5 7.3 9.0 12.2 13.6 15.0 20.7 33.3

5 6.4 7.4 9.1

For example, the width of the valence band in LiH without correlation is, according to Kunz and Mickish [85], Ev = 14.5 eV, and the value of Ev is reduced to one half of this when correlation is taken into account. This shows how much the polarization of crystal by the hole affects the width of the valence band Ev . According to Perrot [80], the width of the valence band in LiH is 5.6 eV. The density of electron states in the valence band of LiH was measured in Betenekova et al. [88] and Ichikava et al. [89]. In Betenekova et al. the measurements were carried out with a magnetic spectrometer having the resolution of 1.5 eV, whereas the resolution of hemispherical analyzer used in Ichikawa et al. [89] was 1.1 eV. From experimental data, the width of the valence band is 6 eV according to Betenekova et al., and 6.3 eV according to Ichikawa et al. Observe the good agreement with the calculated value of Ev in this theory. Let us add also that the measured distribution of the electron density of states in the valence band of LiH exhibits asymmetry typical of s-bands (for more details see Betenekova et al. [88] and Ichikawa et al. [89]). The lower part of the conduction band is formed wholly by p-states and displays an absolute minimum at point X which corresponds to the singlet symmetry state X4 . The inversion of order of s and p-states in the spirit of LCAO method may be understood as the result of the s-nature of valence band. Mixing of s-states of the two bands leads

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51

to their hybridization and spreading, which changes the sequence of levels (see also [90] and references therein). If we compare the structure of the conduction band with the p-band of the method of strong bond (see, also [90]), we see that the general structure and the sequence of levels are the same except for some minor details (the location of L3 level, and the behavior of E(~k) in the neighborhood of Γ15 ). In other words, the lower part of the conduction band in lithium hydride is very close to the valence p-band of AHC. The direct optical gap in LiH according to all calculations is located at X point and corresponds to the allowed transition X1 -X4 . The indirect transition W1 -X4 ought to have a similar energy. According to the above calculations, the energies of these transitions differ by 0.03-0.3 eV. The different values of Eg for LiH obtained by different authors are apparently due to the various methods used for taking into account the exchange and correlation corrections (see above). As follows from Table 9, the transitions at critical points in the low-energy region form two groups at 7 - 9 and 13-15 eV. Measurements of reflection spectra in the 4 - 25 eV range at 5 K (Kink et al. [91]) and 4 - 40 eV at 300 K throw new light on the results of calculations (see also review by Plekhanov [92]). The singularities occurring at 7.9 and 12.7 eV in reflection spectra are associated in the above papers with the interband transitions W1 -W4 and X1 -X5 respectively. From the standpoint of dynamics of quasi-particles, an important consequence of such band structure is the high anisotropy of the tensor of effective mass of electrons and (especially) holes. The estimated mass of electron in the neighborhood of X4 is, according to Kunz and Mickish [85], (me )x ' 0.3m0 in the direction X - Γ, and (me )y = (me )z ' 0.8 m0 in the direction X - W. Similarly, the mass of hole in the neighborhood of X1 is X - Γ in the direction (me )x ' 0.55 m0 and about the same in the neighborhood of W1 . It is assumed that the transverse components of mh are greater by several orders of magnitude (Zavt et al. [81]). Note also that, according to Baroni et al. [84], the estimated masses of carriers are: mel = 0,121; met = 0,938; mhl = 0,150; mht = 4,304 me , where the subscripts l and t denote, respectively, the longitudinal (in the direction Γ X) and the transverse (in the direction X - W) components. This high anisotropy of masses of electron and hole ought to have resulted in the high anisotropy of the reduced (1/µ = I/me + 1/mh ) and the translation (M = me + mh ) masses of exciton. This, however, is not the case. Moreover, the study of Plekhanov and Altukhov [93] reveals that with a good degree of confidence one may assume that in the energy range E ≤ 40 meV the exciton band is isotropic and exhibits parabolic dispersion (me = 0.04m and mh = 0.15m). As was shown below, the

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studies of exciton - phonon luminescence of free excitons and resonance Raman scattering of light in LiH crystals [86] reveal that the kinetic energy of excitons in these crystals is greater than Eb by an order of magnitude exactly because of the very small masses of electron and hole. The last one may indicate that in the metallic phase of hydrogen at the high pressure [86] we can expect the Dirac character of the electronic excitations [94]. We should add that isotope substitution will be very useful method for renormalization of the band - gap in graphene - future semiconducting material [95].

8.2.

Excitons

The properties of the electron and the hole are both described by the band structure within the one - electron approximation. In this section we shall go beyond this approximation and consider the effects of electron - electron interaction on the absorption spectra [96]. To simplify the calculation we shall make the following assumptions. We shall include only the Coulombic part of the electron - electron interaction neglecting both exchange and correlation terms. Furthermore, the interaction between the excited electron in conduction band (see, Fig. 9) and those left behind in the now almost filled valence band will be replaced by an electron - hole interaction. Attraction between the electron and the hole causes their to be correlated and the resultant electron - hole pair is known as an exciton [97]. It has been more than eight decades since the introduction of quasiparticle exciton by Frenkel [97] and the extreme fertility of this idea has been demonstrated most powerfully. According to Frenkel the exciton is an electron excitation of one of the atoms (ions) of the crystal lattice, because of the translation symmetry, moves through the crystal in an electrically neutral formation. Since Frenkel the concept of an exciton has been developed in the papers of Peierls [98] and Slater and Schokley [99]. Problems concerning light absorption by solid state have been considered somewhat differently Wannier [100] and Mott [101]. According to the Wannier - Mott results the exciton is the state of an electron and hole bonded by the Coulomb force. The electron and hole in exciton state are spatially separated and their charges are screened. In the Frenkel papers the excitations localized on the lattice site were described thus, after the Wannier - Mott papers, the excitons became divided into the excitons on the Frenkel (small radius) excitons (for details see [102]) and the Wannier - Mott (large radius) excitons ( [43]). However, a description of the basic difference

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between these two models is absent [102 - 104]. The experimental discovery (see e.g., Gross [105]) of the Wannier - Mott exciton (see Fig.11) on the hydrogen-like absorption spectrum in the semiconducting crystals was the basis of a new subject - exciton physics (see also Agekyan [106]; Permogorov [107]). The influence of external perturbation (electrical and magnetic fields, uniaxial and hydrostatic deformation) on the optical spectra of the Wannier - Mott excitons (see e.g., Gross [105]) and their energetic characteristics (see also Cardona [108]) has been demonstrated repeatedly.

Figure 11. Discrete and continuous (hatched area) Wannier - Mott exciton energy spectrum taking into account its kinetic energy. The broken line connects to the dispersion of light in the medium. These investigations permitted high-accuracy measurements not only the exciton binding energy but also of their translational mass, values of effective masses of the electron and hole, their g - factors etc. Moreover, the detailed account of the photon - exciton interaction has led to the concept of polaritons (Pekar [109]). From the time of the experimental discovery of the Wannier Mott exciton the problem concerning the interaction of excitons and the crystal

54

V. G. Plekhanov

lattice has persisted for more than four decades (Haken [110]; Haug and Koch [111]). We should remind that briefly consideration of the effect of Coulomb attraction on the motion of electrons and holes (see, also [43]) in the vicinity of an M0 critical point of a direct band - gap semiconductor in three dimensions was done early [86, 112]. The energy spectrum of a Wannier - Mott excitons is shown in greater detail in Fig. 11. The above model of excitons based on electrons and holes with spherically symmetric parabolic dispersion is useful for understanding exciton effects on optical spectra. However, it is not accurate enough for quantitative interpretation of experimental spectra in diamond and zinc - blende - type semiconductors. Of the various attempts to calculate excitonic effects based on realistic band structures, we shall mention the paper [113] (see, also [114]).

8.3.

Experimental

The main experimental results were obtained on a device used already in the investigations [13, 42, 44]. The experimental equipment consisted mainly of an home - made immersion helium cryostat and two double prism (grating) ˚ monochromators (with reciprocal dispersion 10 A/mm for the wavelength λ = 220 nm) arranged at right angle. Spectra were excited by various lines of a 400 W deuterium lamp and a 120 W mercury arc lamp (see Fig. 12). The excitation light was felt on the entrance slit of the first monochromator and after leaving it passed through quartz windows of the immersion helium cryostat and irradiated of the crystals. The luminescence and scattering light from the samples was dispersed in another double prism (grating) monochromator and photoelectrically detected. A photoelectric registration of the optical signal was realized by a cooled photomultiplier for ultraviolet (UV) range of the spectrum and the subsequent high - sensitivity photon - counting system with a storage facilities (computer). the signal - to - noise ratio was approximately n · 102 where n = 2 ÷ 5 for different batches of crystals. The light scattering experiments were performed in the usual backscattering geometry, at temperatures of 2 to 300 K. To allow for the ≈ ω 4 law, a freshly cleaved of LiF crystal was used, whose dispersion of the refractive index in the spectral region 4 to 6 eV is small. The experiments were carried out in the same way for two (LiH and LiD) compounds, which differ by a term of one neutron.

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Figure 12. Scheme of experimental setup: 1 - lamp, 2 - lens, 3 - monochromator, 4 - He cryostat, 5 - photomultiplier, 6 - photon counter, 7 - personal computer. The single crystals of LiH and LiD were grown from the melt by the modified method of Bridgeman - Stockbarger (see, e.g., [86] and reference quoted therein). The crystals were synthesized from 7 Li metal and hydrogen of 99.7% purity and deuterium of 99.5% purity. The device of LIH (LiD) single crystals growth is shown in Fig. 2. ( in Ref. [95]). Virgin crystals fad a slightly blue - grew color, which can be attributed to nonstoichiometric excess of lithium present during the grown cycle. On annealing for several days (up to 20) at 5000 C under ∼ 3 atm of hydrogen or deuterium, this color could be almost completely eliminated. Developed special equipment which is allowed to prepare samples with a clean surface cleaving their in the bath of helium cryostat with normal or superfluid liquid helium (see e.g., [86] and references quoted therein). The samples with such surface allow to preform measurements during 15 hours. An improved of LiH (LiD) crystals were used in the present study. In spite of the identical structure of all free - exciton luminescence spectra, it is necessary to note a rather big variation of the luminescence intensity of the crystals from the different batches observed in experiment.

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F UNDAMENTALS SPECTRA OF L I H AND L I D C RYSTALS

In this paragraph we describe the non - accelerator experimental results obtained on two kind of crystals (LiH and LiD) which differ by the addition of one neutron. LiH + n = LiD Lithium hydride and lithium deuteride are ionic insulating crystals with simple electronic structure, four electrons per unit cell, both fairly well – described structurally (neutron diffraction) and dynamically (second – order Raman spectroscopy - see below) and through ab initio electronic structure simulation. Among other arguments, LiH and LiD are very interesting systems due to their extremely simple electronic and energy structure and to the large isotopic effects when the hydrogen ions are replaced by the deuterium ones. On the other hand, the light mass of the ions, specially H and D, makes that these solids have to be considered like quantum crystals, and consequently, described theoretically by quantum theory. As was said above in view of the high hygroscope of the investigated samples, the crystals were cleaved in liquid (superfluid) helium in the cryostat bath [86]. This makes possible to prepare samples with a clean surface. We found no changes in the free – exciton luminescence or resonance Raman scattering (RRS) spectra when a samples with such a surface was studied for period lasting 15 hours. In our experiments we used very simple and inexpensive devices, which is used home – made helium cryostat as well as standard monochromators and photon counting system with personal computer memory. The usual way to determine the optical properties of a solid is to shine monochromatic light onto an appropriate sample and then to measure the reflectance or transmittance of the sample as a function of photon energy. In the spectral region of greatest interest is generally quite high, so that often a negligible small fraction of the incident light is transmitted. It is the main reason, that the most experiments are measurements of the reflectivity. As is well known that isotopic substitution only affects the wavefunction of phonons; therefore, the energy values of electron levels in the Schr˝odinger equation ought to have remained the same [44]. This, however, is not so, since isotopic substitution modifies not only the phonon spectrum, but also the constant of electron-phonon interaction (see above). It is for this reason that the

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energy values of purely electron transition in molecules of hydride and deuteride are found to be different [115]. This effect is even more prominent when we are dealing with a solid [116]. Intercomparison of absorption spectra for thin films of LiH and LiD at room temperature revealed that the longwave maximum (as we know now, the exciton peak [44]) moves 64.5 meV towards the shorter wavelengths when H is replaced with D. For obvious reasons this fundamental result could not then receive consistent and comprehensive interpretation, which does not be little its importance even today. As will be shown below, this effect becomes even more pronounced at low temperatures (see, also [44, 45]). The mirror reflection spectra of mixed and pure LiD crystals cleaved in liquid helium are presented in Fig.13. For comparison, on the same diagram we have also plotted the reflection spectrum of LiH crystals with clean surface. All spectra have been measured with the same apparatus under the same conditions. As the deuterium concentration increases, the long-wave maximum broadens and shifts towards the shorter wavelengths. As can clearly be seen in Fig. 13, all spectra exhibit a similar long-wave structure. This circumstance allows us to attribute this structure to the excitation of the ground (Is) and the first excited (2s) exciton states. The energy values of exciton maxima for pure and mixed crystals at 2 K are presented in Table 10. The binding energies of excitons Eb , calculated by the hydrogen-like formula, and the energies of interband transitions Eg are also given in Table 10. The ionization energy, found from the temperature quenching of the peak of reflection spectrum of the 2s state in LiD is 12 meV. This value agrees fairly well with the value of ∆E2s calculated by the hydrogen-like formula. Moreover, Eb = 52 meV for LiD agrees well with the energy of activation for thermal quenching of free-exciton luminescence in these crystals [116]. Going back to Fig. 13, it is hard to miss the growth of ∆12 , [116], which in the hydrogen-like model causes an increase of the exciton Rydberg with the replacement of isotopes [44, 110]. When hydrogen is completely replaced with deuterium, the exciton Rydberg (in the Wannier-Mott model) increases by 20% from 40 to 50 meV, whereas Eg exhibits a 2% increase, and at 2 ÷ 4.2 K is ∆Eg = 103 meV. This quantity depends on the temperature, and at room temperature is 73 meV, which agrees well enough with ∆Eg = 64.5 meV as found in the paper of Kapustinsky et al [86]. The continuous change of the exciton Rydberg was earlier observed in the crystals of solid solutions A3 B5 and A2 B6 . Isotopic substitution of the light isotope (32 S) by the heavy one (34 S) in CdS crystals [114]) reduces the exciton Rydberg, which was attributed to the ten-

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Figure 13. Mirror reflection spectra of crystals cleaved in superfluid helium; LiH, curve 1; LiHx D1-x , curve 2; and LiD, curve 3 at 2 K. Light source without crystals, curve 4. tative contribution from the adjacent electron bands, which, however, are not present in LiH (for more details see [86, 114]).The single-mode nature of exciton reflection spectra of mixed crystals LiHx D1−x agrees qualitatively with the results obtained with the virtual crystal model (see e.g., [124]), being at the same time its extreme realization, since the difference between ionization potentials (∆ζ) for this compound is zero. According to the virtual crystal model, ∆ζ = 0 implies that ∆Eg = 0, which is in contradiction with the experimental results for LiHx D1-x crystals. As was shown above the change in Eg caused by isotopic substitution has been observed for many broad-gap and narrow-gap semiconductor compounds. All of these results are documented in Table 10, where the variation of Eg , Eb , are shown at the isotope effect. We should highlighted here that the most prominent isotope effect is observed in LiH crystals, where the dependence of Eb = f (CH ) is also observed and investigated. To end this section, let us note that Eg decreases by 97 cm−1 when 7 Li is replaced with 6 Li (see Table 10).

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Table 10. Values of the energy of maxima (in meV) in exciton reflection spectra of pure and mixed crystals at 2K, and energies of exciton binding Eb , band-to-band transitions Eg . Energy, meV E1s E2s Eb Eg

LiH 4950 4982 42 4992

LiH0.82 D0.18 4967 5001 45 5012

LiH0.40 D0.60 5003 5039 48 5051

LiD 5043 5082 52 5095

6 LiH

(78K) 4939 4970 41 4980

Under the continuous optical excitation a stationary population in the exciton states can be created. Due to the free motion in the crystal and the interaction with the crystal lattice the gas of free excitons is spread over some region of kinetic energy [43]. However, usually only the lowest n = 1S (ground state) exciton is populated (Fig. 11) at low temperatures [43, 44, 110]. The photoluminescence emission of free excitons can take place either in resonance with the exciton absorption line (a so - called zero - phonon luminescence) or can be shifted in energy due to the simultaneous creation of phonon (phonon - assisted luminescence). In the resonant excitons emission only excitons with → − small wavevectors of the order of the photon wavevector K ' 0 can participate. Phonon - assisted luminescence of free excitons in polar compounds is mainly due to the creation of longitudinal optical (LO) phonons [43, 44]. Since → the energy of LO phonons has a weak dependence on wavevector − q (see, e.g., [125]), the spectrum of emitted photons is simply related to the initial distribution of exciton energy. Fig. 14 shows the free excitons photoluminescence in LiH and LiD crystals at 2 K with photoexcitation above the intrinsic absorption edge [44]. The free exciton photoluminescence spectrum of LiH crystals cleaved in superfluid liquid helium consists of the narrow zero - phonon line and its wider phonon replicas corresponding to the radiative annihilation of excitons accompanied by excitation from one to five LO phonons. The zero - phonon emission line is almost in resonance with the reflection line in the exciton ground state (En = 1S = 4.950 eV for LiH crystals at 2 K [44]), which is due of the direct electron transition X1 - X4 of the first Brillouin zone [126]. Phonons replicas form an equidistant series to the red from resonance exciton emission line. The dif-

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Figure 14. Free exciton emission spectra at 2 K in LiH and LiD crystals cleaved in superfluid helium. ference in energies of these replicas, as early [116] is 140 meV which is close to the energy of LO phonons in the center of the Brillouin zone obtained in [114, 127 - 130]. The photoluminescence spectrum of a LiD crystals with a pure surface shown also in Fig. 14 is similar in many respects to the free excitons luminescence spectrum of LiH crystals. However, there are some differences. These differences are related with the next manifestations: 1. The short - wavelength shift as whole of the free excitons photoluminescence spectrum of LiD crystals on 103 meV relatively the spectrum of LiH crystals. 2. In the case of LiD crystals the energy difference between lines in the spectrum is on average 104 meV corresponding to the energy of the LO phonons in the Γ - point of the Brillouin zone [44, 127, 128, 129]. Firstly the zero - phonon emission line of free excitons in LiD crystals shifts to the short - wavelength side on 103 meV. These results directly show the violation of the strong conclusion (see, e.g., [5, 6]) that the strong force does not

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act on leptons. The second difference concludes in less value of the LO phonon energy, which is equal to 104 meV. When light is excited by photons in a region of fundamental absorption in mixed LiHx D1−x crystals at low temperature, line luminescence is observed (Fig. 15), like in the pure LiH and LiD crystals. As before [114], the luminescence spectrum of crystals cleaved in superfluid liquid helium consists of the relatively zero - phonon line and its wide LO replicas. For the sake of convenience, and without scarfing generality, Fig. 15 shows the lines of two replicas. Usually up to five LO repetitions are observed in the luminescence spectrum as described in detail in [86]. In Fig. 15 we see immediately that the structure of all three spectra is the same. The difference is in the distance between the observed lines, as well as in the energy at which the luminescence spectrum begins, and in the half - width of the lines.

Figure 15. Photoluminescence spectra of free excitons in LiH (1), LiHxD1−x (2) and LiD (3) crystals cleaved in superfluid helium at 2 K. Spectrometer resolution is shown. The simplest approximation, in which crystals of mixed isotopic composition are treated as crystals of identical atoms having the average isotopic mass, is referred to as virtual crystal approximation (VCA) [122]. Going beyond the VCA, in isotopically mixed crystals one would expect local fluctuations in local isotopic composition within some effective volume, such as that an exciton. As

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follows from Fig. 13, excitons in LiHxD1−x crystals display a unimodal character, which facilitates the interpretation of their concentration dependence. As will be shown in the discussion the concentration dependence of the power of strong nuclear interaction, i.e., dependence on the neutron concentration. As will be shown below in first approximation the mechanism of isotope shift will be connect with the neutron magnetic field of the deuterium nucleus (neutron) (see, also [131]). Because of the low intensity of scattered light, and thanks to the high resolution of modern spectroscopic instruments, the development of high sensitive techniques of detection of weak optical signals (photon counting mode, optical multichannel analyzers, optical linear arrays and the specialized systems [132]), the light scattering method has become one of the most common technique for the research. In the last five decades, inelastic light scattering spectroscopy has become one of the most powerful and widely used optical method for obtaining quantitative information about the interaction of light with matter. For example, it has been used to investigate resonant Raman scattering (RRS) in crystals. For a displacement of the excitation like frequency toward long wavelength by an amount compared to the exciton resonance, e.g., Ei h En = 1s (where En = 1s is the energy of the exciton ground state) intense light scattering is observed [129]. The spectrum of this scattering is shown in Fig. 16. As in the luminescence, the process of energy relaxation take place, mainly with emission of LO phonons. This is shown by the character of the structure in the scattering spectrum. Indeed, as for the case of the excitation well within the exciton zone (see Fig. 16), the energy difference between the peaks in the scattered spectrum equals the energy of the LO phonon in the Γ - point of Brillouin zone [127 - 129]. The relatively large half - width of the scattered peaks should be noted [130]. Additional investigations have shown that their half - width are always larger than that of the excitation line [133]. In addition to the LO(Γ) lines the RRS spectrum contains two more bands the maxima of which are displaced by twice the energy of the TO(Γ) and LO(X) phonons from the exciting line (for LiH ~ωTO(Γ) = 76 meV and ~ωLO(X) = 117 meV [13, 127]. More precise measurement of the dependence of the second - order Raman spectra for LiH [124] and LiD on the excitation energy (below En = 1s) is shown in Fig. 17. The long wavelength displacement of the excitation line frequency relatively exciton resonance a monotonic decrease the intensity of RRS spectrum as whole more than 60 fold in both LiH and LiD crystals (see, also Fig. 17).

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Figure 16. Resonant Raman scattering spectrum of a LiD crystals cleaved in liquid helium at the excitation E = 4.992 eV at 4.2K. The arrow 1s indicates the energy position of the ground exciton states. Another very interesting example is carbon. Carbon atom is built from 6 protons, A neutrons and 6 electrons, where A = 6 or 7, yield the stable isotopes 12 C and 13 C, respectively, and A = 8 characterizes the radioactive isotope 14 C [45]. The isotope 12 C , with nuclear spin I = 0, is the most common one in nature with 99% of all carbon atoms, whereas only ' 1% are 13 C with nuclear spin I = 1/2. There only traces of 14 C (10−12 of all carbon atoms) which transforms into nitrogen 14 N by β - decays [2]. Although 14 C only occurs rarely, it is important isotope used for historical dating (see, e.g., [112]). Carbon, one of the most elements in nature, still gives a lot surprises. It is found in many different forms - allotropes - from zero dimensional fullerene, one dimensional carbon nanotubes, two dimensional graphene and graphite, to three dimensional diamond - and the properties of the various carbon allotropes can vary widely [134, 135]. For instance, diamond is the hardest material, while graphite is one of the softest; diamond is transparent to the visible part of spectrum, while graphite is opaque; diamond is an electrical insulator, while graphite and graphene are a conductors. Very important is that all these different properties originate from the same carbon atoms, simply with different arrangements of the atomic structure. Below we describe the new phenomena of the carbon - isotope effect in diamond. Crystals 12 C and 13 C diamond differ only one neutron. Due to the indirect gap of Eg = 5.47 ± 0.005 eV (295 K), at K = 0.76 X, diamond has intrinsic phonon - assisted free exciton luminescence lines (see,

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Figure 17. The dependence of the shape and maximum frequency of the 2LO(Γ) Raman line on the excitation energy of LiD crystals. e.g., [114]). The change of the indirect gap of diamond between pure 12 C and 13 C crystals has been determined by Collins et al. [117]. The luminescence spectra of the natural (12 C) and synthetic (13 C) diamond at electron excitation were investigated by Collins et al. [117] (Fig. 19), Ruf et al. [118], Watanabe et al. [118]. Fig. 19 compares the free exciton luminescence for a natural diamond with that for a synthetic diamond. The peaks labeled A, B and C are due, respectively, to the recombination of a free exciton with the emission of transverse - acoustic, transverse - optic and longitudinal - optic phonons having wavevector ± kmin and quanta (in 12 C

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Figure 18. Structure of some representative carbon allotropes [diamond, graphene, fullerene (C60) and SWCNT]. [114, 133]. ~ωTA = 87 ± 2; ~ω TO = 141 ± 2; ~ωLO = 163 ± 1 meV.

(81)

Features B2 and B3 are further free exciton processes involving the above TO phonon with one and two zone - center optic phonons respectively. As we can see from Fig. 19 the isotope shift of the free exciton luminescence spectrum of 13 C diamond is equal 16.5 ± 2.5 meV [114]. The more detailed and quantitative investigation of Eg ∼ f(x), where x is the isotope concentration was done by Ruf et al. [118] (Fig. 20) and Watanabe et al. [119] (Fig. 21) where five samples of diamond with different concentrations x were studied. Watanabe et al. have concluded that the maximum change of the band gap due to substitution of 12 C by 13 C is ∆Eg = 15.4 meV. This value is in good agreement with the estimate of 16.5 ± 2.5 meV by Collins et al. [117] using a zero - point renormalization obtained from a fit of the experimental temperature dependence of the band gap (see, also [121]). It is clear that the lattice - dynamic properties of a crystal are directly affected by the atomic mass. To a first approximation, phonon behave like harmonic oscillators with frequencies [123]

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Figure 19. Spectra measured at 77K phonon - assisted free cathodoluminescence feature (A, B and C) and the phonon assisted bound - exciton feature (D) from natural semiconducting 12 C diamond and a 13 C synthetic diamond (after [117]). ω ∝ m−1/2 .

(82)

Crystals containing various isotopes are usually described within virtual crystal approximation (VCA) [122] where m in Eq. (82) is replaced by the average atomic mass m=

P ci mi .

(83)

i

It is obtained from the sum over the isotopes masses mi and concentration ci . In spite of VCA simplicity, the model describes the general feature of the lattice dynamics of mixed alkali - halide crystal sufficiently well (for details see [123]). Isotopes are ideal for lattice - dynamitic investigations of crystals. One can make use of the unique isotopic properties by varying the isotopic composition. Isotope substitution helps to disentangle the individual contributions of anharmonic and disorder - induced effects and to clarity the origin of phonon

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Figure 20. Cathodoluminescence spectra of isotopically modified diamond at 36 K. Intrinsic phonon - assisted recombination peaks are labeled in the top spectrum, those from boron - bound excitons in that at the bottom. The spectra are normalized to the intensity of the B peak and vertically offset for clarity (after [118]). broadening mechanisms. Due to the fact that substitution of the isotopic mass in semiconducting crystals a small variation of Eg [121], perturbation theory is applicable (excluding LiH crystals [136]). Raman spectroscopy is a powerful means to gain experimental access to phonons and their interaction and scattering mechanisms. All studies presented in this paragraph are restricted to stable, i.e., non - radioactive isotopes. About 300 stable and 1000 radioactive isotopes are known today. Some elements are isotopically pure (for example, Co), while others may contain numerous isotopic modifications (for example, Sn has 10 stable isotopes with atomic masses ranging from 112 to 124, while Xe has 23 isotopes, 9 of which are stable (see, Table 1 in [121]) [11]. In this part, the modern understanding of first - order Raman slight scattering spectra in isotopically mixed elementary and compound (CuCl, GaN, GaAs, GaP) semiconductors having a zinc blende structure is described. It is well -

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Figure 21. Luminescence spectra of free excitons in homoepitaxial diamond films grown from mixture of methane in hydrogen by means of a microwave plasma - assisted CVD. The spectra illustrate the effects of isotope composition 12 13 C1-x Cx ( x = 0.001; 0.247; 0.494; 0.740; and 0.987) mixed in the CVD gas phase. All spectra are normalized to the same height (after [119]). known that materials having a diamond structure (C, Si Ge, α - Sn) are characterized by the triply degenerate phonon states in the Γ - point of the Brillouin → − zone ( k = 0) (see, e.g., [137]). Isotope effect in light scattering spectra in Ge crystals was first investigated by Agekyan et al. [138]. A more detailed study of Raman light scattering spectra in isotopically mixed Ge crystals has been performed by Cardona and coworkers [121]. The cubic modifications of crystalline carbon, diamond, is characterized by a tetrahedral coordination underlying its structure dictated by sp3 bonding between the nearest neighbor atoms. Diamond has two atoms per primitive (Bravais) cell. The strong covalent bonding and the light mass of the constituent atoms result in a large frequency (see, Eq. (82)) for zone center, Raman active, triply degenerate F2 mode [125]. Its crystalline perfection and transparency [139] make diamond ideally suited for inelastic light scattering. First Raman

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study of the dependence the frequencies of phonons on the isotopic composition was conducted on diamond by Chrenko [141] in 1988. Several publications on diamond by other groups followed later [140 - 144]. Most clear results was obtained by Hanzawa et al. [ 140] (see Fig. 22).

Figure 22. First - order Raman scattering in isotopically mixed diamond crystals 12 13 Cx C1−x. The peaks A, B, C, D, E and F correspond to x = 0.989; 0.90; 0.60; 0.50; 0.30 and 0.001(after [140]). First - order Raman light scattering spectrum in diamond crystals also includes one line with maximum ωLTO (Γ) = 1332.5 cm−1 [143]. In Fig. 22 the first - order scattering spectrum in diamond crystals with different isotope concentrations is shown [140]. As was shown in [121], the maximum and width of the first - order scattering line in isotopically - mixed diamond crystals are nonlinearly dependent on the concentration of isotopes x. The maximum shift of this line is 52.3 cm−1 , corresponding to the two limiting values of x = 0 and x = 1. The effect of the isotopic 12 C to 13 C ratio on the first - and second order Raman scattering of light in the diamond has been investigated in [144]. As 13 C content is increased from the natural ratio (12 C/13 C = (1 - x)/x, where x = 0.011 to the almost pure 13 C (x = 0.987) the whole spectrum has shifted towards longer wavelength (Fig. 23) in good agreement with the expected M−0.5

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frequency dependence on the reduced mass M. For an approximately equal mix of the two isotopes, the authors reported that the feature seen in the above two phonon spectra were either broadened or unresolved.

Figure 23. The Raman spectra of natural and a 13 C diamond. The spectra show the dominant first - order Raman - active F 2g line and the significantly weaker quasi - continuous multiphonon features (after [144]). We should highlight that addition one next neutron in nuclei is called the change in more or less of energy of elementary excitations in solids

10.

P OSSIBLE MECHANISMS OF H ADRON - L EPTON INTERACTION

We should repeat that subatomic physics is distinguished from all other sciences by one feature; it is playground of three different interactions, and two of them act only when the objects are very close together. Residual strong interactions of the quarks holds the protons and the neutrons together to form nuclei. We should pay attention to very strong conclusion in modern physics; there are

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common place in modern physics that the strong force does not act on leptons (electrons, positrons, muons, neutrinos). Our experimental results show the violation of this strong conclusions. These experimental results are numerous reported on the different International Conferences [145 - 148]. This part is devoted to interpretation of our experimental results: The attempt to find the mechanism of renormalization the energy of elementary excitations in solids by strong nuclear interaction. As we remember that nucleus is central part of an atom consisting of A nucleons, Z - protons and N - neutrons. The atomic mass of the nucleus is equal Z + N (see Fig. 2 in paragraph 3). A given element can have many different isotopes, which differ from one other by the number of neutrons contained in the nuclei [6 - 8]. Modern physics distinguishes three fundamental properties of atomic nuclei: mass, spin (and related magnetic moment) and volume (surrounding field strength) which are source of isotopic effect (see, also [13]). All kinds of considered crystals with isotope substitution have an identical electronic structure. The energy band structure of these substances is also identical [41, 45]. All three kinds of forces - gravitational, electromagnetic and weak are also the same for compounds above. The difference between these substances consists out at one or two, the neutrons in the nucleus. The most simple is deuteron, which differs from hydrogen for one neutron. Much of what we know about nuclear structure comes from studying not the strong nuclear interaction of nuclei with their surroundings, but instead the much weaker electromagnetic interaction, That is, the strong nuclear interaction establishes the distribution and motion of nucleons in the nucleus, and we probe that distribution with the electromagnetic interaction. As is well - known from classical electrodynamics, any distribution of electric charges and currents produces electric and magnetic fields that vary with distance in a characteristic fashion (see, e.g., [1, 41]). It is customary to assign to the charge and current distribution an electromagnetic multipole moment associated with each characteristic spatial dependence - the 1/r2 electric field arises from the net charge, which we can assign as the zeroth or monopole moment; the 1/r3 electric field arises from the first or dipole moment; the 1/r4 electric field arises from the second of quadrupole moment and so on. The monopole electric moment is just the net nuclear charge Ze. The next nonvanishing moment is the magnetic dipole moment µ. A circular loop carrying i and enclosing area A has a magnetic moment of magnitude |µ| = iA; if the current is caused by a charge e, moving e with speed v in a circle of radius r (with period 2πr/v), then |µ| = (2πr/v) πr2

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e = evr 2 = 2m (l), where (l) is the classical angular momentum mvr [1, 2]. From e~ is called a magneton. In the case atomic physics we know that the quantity 2m m = me (me is electron mass) we have the Bohr magneton µB = 5.7884 · 10−5 eV/T and if m = Mp (Mp is proton mass) we have the nuclear magneton µN = 3.1225 · 10−8 eV/T. Above we have been considering only the orbital motion of nucleons. From quantum mechanics we know that protons and neutrons, like electrons, also have intrinsic or spin magnetic moments, which have not classical analog. As far as the most simple from studied sample was LiH (LiD) below we should briefly consider in the first step some peculiarities of the physics of deuteron. Nucleons can combine to make four different few - nucleon systems, the deuteron (p + n), the triton (p + 2n), the helion (2p + n) and the α - particle (2p + 2n). These particles are grouped together because they are all stable (apart from triton which has a half - life of about twelve years and so may be treated as a stable entity for most practical purposes), have no bound excited states (except the α - particle which has two excited states at about 20 and 22 MeV [46]), and are frequently used as projectively in nuclear reactions. Few - nucleon systems provide the simplest systems to study nuclear structure (see, e.g., [20]). The bright example of the last approval the results of paper by Wigner [149]. In 1933, Wigner pointed out that a comparison of the binding energies of the deuteron, the triton and the α - particle leads to the conclusion that nuclear forces must have a range of about 1 fm and be very strong. The argument goes of follows [149]. According Wigner the binding energies of the three nuclides are given in Table 11. Also listed are the binding energies per particle and per ”bond”. The increase in binding energy cannot be due only to the increased number of bonds. However, if the force has a very short range, the increase can be explained: The larger number of bonds pulls the nucleons together, and they experience a deeper potential; the binding energies per particle and per bond increase correspondingly. The deuteron provides important information about the nucleon - nucleon interaction. As was noted, the deuteron consists of a proton and a neutron and is the only bound state of two nucleons. Its binding energy is 2.2245 MeV and its total angular momentum J and parity are 1+ [2]. Since the intrinsic parities of the neutron and the proton are positive parity of the deuteron implies that the relative orbital angular momentum of the neutron and the proton must be even. If the orbital angular momentum L is a good quantum number, states with lower orbital angular momentum generally have lower energy than states with

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Table 11. Binding energies of 2 H, 3 H and 4 He

Nuclide 2 H 3H 4 He

Number of bonds 1 3 6

Binding enegy (MeV) Total Per particle Per bond 2.2 1.1 2.2 8.5 2.8 2.8 28 7 4.7

higher angular momentum, and so we expect the ground state of the deuteron to have orbital angular momentum L = 0, so that it is in an S state. Then, if the spins of the proton and the neutron in the deuteron are parallel, we expect the magnetic moment of the deuteron to be approximately the sum of the magnetic moments of the proton and neutron, namely µp + µn = (2.793 - 1.913)µN e~ is, as early, as nuclear magneton) [39]. If, however, = 0.880µN (where µN = 2m p the spins are anti - parallel , we expect it to be (2.793 + 1.913)µN = 4.706µN . Experimentally it is 0.857µN [8, 11] so the spins of the proton and neutron are parallel and so the total spin S of the deuteron is one, since J = L + S, J = 1. The small but definite difference between µd = 0.857µN and µp + µn = 0.880µN is due, as shown below, to tensor character of strong forces in deuteron. We thus conclude that the ground state of deuteron is a triplet S state. However this cannot the whole story because S states are spherically symmetrical and thus have no quadrupole moment [41]. This is contradict to experiments. Experimentally the deuteron has a positive quadrupole moment of 0.29 fm2 [46]. The deviation of the actual deuterium moment from the S state moment can be explained if it assumed that the deuteron ground state is a superposition of S and D states. Part of the time, the deuteron has orbital angular momentum L = 2. Independent evidence for this fact comes from the observation that, as was shown above, the deuteron has a small, but finite, quadrupole moment (see, also [9]). Shown above (see preceding paragraph), the electric quadrupole moment measures the deviation of a charge distribution from sphericity (see Eqs. 45 - 46). Thus, if the quadrupole moment is not equal to zero then the eigenfunction of the ground state of the deuteron assigns a probability of 0.04 to finding a 3 D1 state and a probability of a 0.96 to finding a 3 S1 state [1]. The last one points to the tensor character of the nucleon - nucleon interaction (the more details see, e.g., [20]).

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Nuclear magnetic dipole and electric quadrupole have a similar importance in helping us to interpret the deuteron structure. We should repeat that each nuclear state is assigned a unique ”spin” quantum number I, representing the total angular momentum (orbital + intrinsic) of all the nucleons in the nucleus. As is well - known, nuclear magnetic moment is the sum orbital and intrinsic spin moments, e.g., µ = (gl l + gs s) µN /~,

(84)

here the nuclear g - factor is dimensionless number gl = γ~/µN (where γ is gyromagnetic ratio [1,2]). In equation (84) gl and gs are the orbital and intrinsic contributions to µ. Their values as early are even to gl = 1 and gs = 5.5856912 for protons and gl = 0, and gs = - 3.8260837 for uncharged neutrons. Today we haven’t a single theory, which allows to calculate µ [26, 27, 49, 50]. Now relativistic quark models without gluonic or pion cloud degrees of freedom generally predict about 60 % of the proton’s spin should be carried by the quarks, with remaining 40 % in quark orbital and angular momentum. Today data and theory point to a consistent picture where the proton spin puzzle is a valence quark effect (the details see [26, 27, 49, 50]). Traditionally nuclear - electron interaction (in our case neutron - electron interaction) taking into account the solving Schr¨odinger equation using Born - Oppenheimer (adiabatic) approximation [150]. Since electrons are much faster and lighter than the nuclei by a factor nearly 2000 the electron charge can quickly rearrange itself in response to the slower motion of the nuclei, and this is the essence of the Born - Oppenheimer approximations. This approximation results the omission of certain small terms which result from the transformation. As was shown in the eigenvalue (energy) of the electronic Schr¨odinger equation (equation 6 in [151]) depends on the nuclear charges through the Coulomb potential, but doesn’t include any references to nuclear mass and it is the for the different isotopes. Besides this approximation, it is necessary to take into account a small contributions to isotopic shift through reduced electron mass µ = me · MN ucl me + MN ucl so far as MN ucl is different for the hydrogen and deuterium nucleus. In this case the contribution equals ∆E ' 6 meV. Contributions to isotope shift of the zero - phonon emission line in luminescence spectra of LiD crystals as well as Lamb shift and specific Coulomb potential approximately equal 1, and 1 meV, respectively. These estimations are forcing us to search for new models and mechanisms of neutron - electron interaction including the results of subatomic physics, e.g., hadron - lepton interaction. By the way we should re-

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mind that during 20 century the problem of neutron - electron interaction was numerous investigation (see, e.g., [152, 153]). In these papers was shown the dependence of the electron energy level on the strong interaction, however, all the scientific literature on subatomic physics states that the strong interaction do not affects leptons (electrons etc.). For the interpretation of our non - accelerator experimental results we should take into account the long range magnetic interaction which is a part of the electromagnetic field. Ordinary magnetic interaction of matter are determined by atomic magnetism [55]. Originally the hyperfine structure was taken to include those atomic effects (much smaller than the fine structure) that arise from the coupling between the electronic and nuclear angular moments. It is thus an internal effect in atoms, and we cannot switch it off or modify it except by changing the nuclear or electronic structure (going to excited states, for instance). − → The motion of the electrons produces a magnetic field Be at the nucleus, which interacts with the nuclear magnetic moment µN (see, e.g., [52]): − → → E=-− µ N · Be .

(85)

Typical energy differences of hyperfine multiplets are only about 10−7 eV (in case of the deuteron it is 3.16 · 10−7 eV). This value is by more than seven order less than we observe in experiments: the isotopic shift of the n = 1s in LiD excitons is equal to 0.103 eV. In view of such a discrepancy with the experimental value, it is perhaps reasonably to consider the possibility of some kind of manifestation of residual strong forces via anomalous magnetic moment of neutron, for example. How can the departure of the magnetic moments of the proton and the neutron from the ”Dirac values” be understood. Before quarks were introduced, the explanation of the anomalous magnetic moments of the nucleons was based on virtual mesons that are present in their structures [41]. At the present time it is clear that nucleons are composed primarily of three quarks, the proton has the composition (uud), the neutron (udd) [154, 155]. Nucleons contain not just one point particle and a meson cloud; three point particles reside there. The interaction among the quarks, as we all know, is transmitted by gluons; the force is weak at short distances (. 0.1 fm) and strong at large ones (% 0.5 fm). The anomalous magnetic moments of the nucleons are due to hadronic effects [48, 156 - 158 ], thus they cannot be computed to anywhere near the accuracy of the anomalous g factors for the leptons [25].

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Continuing the analysis of non - accelerator experimental observations, taking into account the theoretical works of Condon [152] and Foldy [153] we came to the unambiguous conclusion that the strong interaction affects the electronic excitations. Moreover, this fact indicates the long - range nature of the strong nuclear interaction. It is possible that this is a reflection of the long - range action of pions and gluons, especially in the infrared limit [68, 160, 161]. Returning to Fig. 15, we point out that measuring the optical characteristics (reflection and luminescence) of a number of mixed crystals LiHxD1−x made it possible to plot the dependence of the forth (energy) of strong interaction of a distance between nucleons in the deuterium nucleus (Fig. 24). The results presented are the first measurements in the world scientific literature. Let us add that, as can be seen from Fig. 24 the observed dependence of the strong interaction force on the distance between nucleons is a nonlinear function. The experimentally observed increase in the value of the strong interaction energy in the region r ≥ 2 fm (rD = 2.1314 fm [162]) agrees with qualitative picture in Fig. 6 of paragraph 2. Using the magnitude of the anomalous value of the magnetic moment of the neutron and the experimentally observed value of the isotopic shift energy at 0.103 eV, the value of the strong interaction constant was found in the paper [68] equal to 2.4680 which is very different from the results found from the data of accelerator technology (αs (Mz ) = 0.1198 ± 0.002 [155, 160]. It is well known that the coupling strength αs is the basic free parameter of QCD, the they of the strong interaction which is one of the four fundamental forces of nature. Confinement implies that the coupling strength αs , the analogue to the fine structure constant in QED, becomes large in the regime of large distance or low - momentum transfer interactions. (By the way in the world of quantum physics, ”large” distances ∆s correspond to ∆s i 1 fm, ”low” momentum transfers to Q h 1 GeV/c (see, e.g., [163]). Conversely, quarks and gluons are probed to behave like free particles, for short time intervals ( ”short” time intervals correspond to ∆t h 10-24 s.), in high - energy or short - distance reactions; they are said to be ”asymptotically free”, i.e., αs −→ 0 for momentum transfers Q −→ ∞ [164 - 167]. The value αs , at a given energy or momentum transfer scale Q, must be obtained from experiment. Determining αs at a specific energy scale q is therefore a fundamental measurement, to be compared with measurement of the electromagnetic coupling α. Taking into account that in our experiments we study strong force on very large distance (≈ 104 more than in nuclear physics) we may expect the very large value of αs (compare to the results of Fig. 6 Ref. [167]). Moreover, our experimental results allow to determine αs in rather large

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energy scale ( see Fig. 24). Thus, for the first time in a non - accelerator experiments, the maximum value of the strong nuclear interaction coupling αs was obtained.

Figure 24. Dependence of the strong interaction energy on the distance between nucleons in deuterim nucleus. The short range character of the strong interaction made difficult to find mechanism of the long - range interaction of the strong force, which was observed in the experiments. Naturally, the origin of Van der Waals or new type forces are in need of more quantitative not only experimental but also theoretical investigations of observed effects. Nevertheless, taking into account all the above considerations, we should dwell on the magneto - like mechanism of long - range action of hadrons and leptons (see, also [70]). This is facilitated by nonlocality of elementary particles to Landau [69]. In this connection we should add that the massless photon - like ninth gluon may be strongly interacts

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between hadrons (neutrons) and leptons (electrons) (see e.g., [15]). Thus, the tentative interpretation of describing experimental results don’t find consistent explanation at the change strong interaction leaving it to be another mystery of SM (see, also [25]). We should remind that intrinsic contradiction of SM is already well - known. Really, the Lagrangian of quantum chromodynamics (theory of the strong interaction) has the next form (see, e.g., [168]): L=i

X

a

Ψq (∇µ γ µ + imq )Ψaq - 41 Gnµν Gnµν ,

(86)

q

where n

∇µ = ∂µ - ig λ2 Anµ , Gnµν = ∂µ Anµ - ∂n Anµ + gfnmlAnµ Alν .

(87)

Ψaq and Anµ are quark and gluon fields, a=1,2,3,...8 are color indices , λn and fnml are Gell - Mann matrices and f symbols, mq - are bare (current) masses, q = u, d, s, c, ... different quarks. It is common place [15, 168] that the Lagrangian (86) contains the members which describe both free motion and interaction between quarks and gluons, which is defined by the strength couple g. Spacing of which it is necessary to remark that although the Lagrangian (86) possesses rather attractive peculiarities , its eigenstates are the quarks and the gluons which are not observed in free states. The observed hadrons in the experiments doesn’t eigenstates in quantum chromodynamics. It is obvious to expect that the modern theory of quantum chromodynamics should finally overcome these difficulties [45].

R EFERENCES [1] Henley E. M. and Garcia A., Subatomic Physics, 3rd Edn. (World Scientific, Singapore, 2007). [2] Krane K. S., Introductory Nuclear Physics (Wiley, New York - Chichester, 1988). [3] Plekhanov V. G., Isotopes in Condensed Matter (Springer, Heidelberg, 2013).

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[4] Chadwick J., The existence of neutron, Proc. Roy. Soc. (London) A 136, 692 - 708 (1932). [5] Cottingham W. N., Greenwood D. A., An Introduction to the Standard Model of Particle Physics (Cambridge University Press, Cambridge, 2007). [6] Perkins D. H., Introduction to High Energy Physics (Cambridge University Press, Cambridge, 2006). [7] Kronfeld A. S., Lattice gauge theory and the origin of mass, in, One Hundred Years of Subatomic Physics, ed by E. Henley and S. Ellis (World Scientific, Singapore, 2013). [8] Dirac P. A. M., The Principle of Quantum Mechanics (Oxford University Press, UK, 1958). [9] Edmonds A. R., Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1957). [10] Tomonaga S., The Story of Spin (The University of Chicago Press, Chicago and London, 1997). [11] Lederer C. M., Shirley V. S., Table of Isotopes (Wiley, New York, 1978). [12] Miloni P. W., The Quantum Vacuum (Academic Press, Boston - New York, 1994). [13] Plekhanov V. G., Modern View of the Origin of Isotope Effect (LAP, LAMBERT Academic Publishing, Saarbr¨ucken, Germany, 2018). [14] Leutwyler H., The mass of two lightest quarks, Mod. Phys. Lett. A 28, 136014 - 11 (2013). [15] Griffiths G., Introduction to Elementary Particles (Wiley - WCH, Weinheim, 2008). [16] Leutwyler H., On the history of the strong interaction, Mod. Phys. Lett., A 29, 143023 - 16 (2013). [17] Halzen F., Martin A. D., Quarks and Leptons (J. Wiley & Sons, New York, Chichester, 1984).

80

V. G. Plekhanov

[18] Tanabashi M. et al., The Review of Particle Properties, Phys. Rev. D98, 030001 - 93 (2018). [19] Gell - Mann M., Neeman Y., The Eightfold Way (Benjamin, Reading, Mass., 1964). [20] Martin B. R., Nuclear and Particle Physics (J. Wiley & Sond, Weinheim, Chichester, 2006). [21] Ho - Kim Q., Pham X - Y., Elementary Particles Introduction (Springer, Berlin, 1998). [22] Nagashima Y., Elementary Particle Physics (Vol. 2. Foundation of the Standard Model and Experiments) (Verlag, Wiley - VCH, Weinheim, Chichester, 2013). [23] Greiner W., Reinhart J., Quantum Electrodynamics, 3rd Ed.( Springer, Berlin - New York, 2003). [24] Elliott J. P. and Dawber P. G., Symmetry in Physics (MacMillan Press LtD, 1979). [25] Close F. E., An Introduction to Quarks and Partons (Academic Press, London - New York, 1979). [26] Bass S. D., The Spin Structure of the Proton (World Scientific, Singapore, 2007). [27] Aidala C. A., Bass S. D., Hasch D. et al., The spin structure of the nucleon, Rev. Mod. Phys. 85, 655 - 691 (2013). [28] Faessler A., Interaction between two baryons and the six quark model, Prog. Part. Nucl. Phys. 20, 151 -173 (1988). [29] Guichon P. A., Stone J. R., Thomas A. W., Quark - meson coupling (QMC) model for finite nuclei, nuclear matter and beyond, ibid 100, 262 - 297 (2018). [30] Faessler A., Fernandez F., L˝ubeek G., The nucleon - nucleon interaction and the role of the [42] orbital six - quark - symmetry, Nucl. Phys. A402, 555 - 568 (1982).

Experimental Study of the Long - Range Quark - Lepton ...

81

[31] Straub U., Zhang Z - Y., Br¨aer K. et. al., Hyperon - nucleon interaction in the quark cluster model, ibid A483, 686 - 710 (1988). [32] Aubert J. J., Bassompierre G., Becks K. H. et. al., The ratio of the nucleon structure functions F2N for iron and deuterium, Phys Lett., 23, 275 278(1983). [33] Hen O., Miller G. A., Piasetzky E. et. al., Nucleon - nucleon correlation, short - lived excitations, and the quarks, Rev. Mod. Phys. 89, 045002 63 (2017). [34] Plekhanov V. G., Macroscopic manifestation of the strong nuclear interaction in the optical spectra of solids, in, Proc. ISINN - 25, Dubna, Russia, 2018, 49 - 56. [35] Plekhanov V. G., Isotope effect is the paradigm of a non - accelerator study of the residual nuclear strong interaction, Edelweiss Chem. Sci. J. 3, 8 - 14 (2020). [36] see e.g., Eliashevich M. A., Atomic and Molecular Spectroscopy (Fizmatgiz, Moscow, 1962) (in Russian). [37] Harrison W. A., Electronic Structure and the Properties of Solids (W.H. Freeman & Co., San Francisco, 1980). [38] Qin S.-X., Roberts C. D., Schmidt S. M., Spectrum of light - and heavy baryons, Few - Body Systems 60, 26 - 44 (2019). [39] Baranov M. Y., Bedola M. A., Brooks W. K. et al., Diquark correlation in hadron physics: Origin, Impact and Evidence, ArXiv/ hep - ph/ 2008.07630 (2020). [40] Rosina M., Povh B., Contribution of the covalent and van der Waals force to the nuclear binding, Nucl. Phys. A572, 48 - 56 (1994). [41] Segre E., Nuclei and Particles: An introduction to Nuclear and Subnuclear Physics, Second Ed. (Addison Wesley Publishing Co., New York, 1977). [42] Plekhanov V. G., Isotope - induced energy - spectrum renormalization of the Wannier - Mott exciton in LiH crystals, Phys. Rev. B54, 3869 - 3877 (1996).

82

V. G. Plekhanov

[43] Knox R. S., Theory of Excitons (academic Press, New York, 1963). [44] Plekhanov V. G., Exciton spectroscopy of isotopic crystals, Opt. Spectr. (St. - Petersburg) 79, 715 - 734 (1995). [45] Plekhanov V. G., Hadron - Lepton Strong Interaction (Services for Science and Education, Cheshir, UK, 2021). [46] Carlson J., Schiavilla R., Structure and dynamics of few - nucleon systems, Rev. Mod. Phys. 70, 743 - 841 (1998). [47] Tamm I. E., The Theory of Electricity (GITTL, Moscow, 1957) (in Russian). [48] Clo¨et I. C., Miller G. A., Ron G., Neutron properties in the medium, Phys. Rev. Lett. 103, 082301 - 4 (2009). [49] Kelly J. J., Nucleon charge and magnetization densities from Sachs form factors, Phys. Rev. C66, 065203 - 065206 (2002). [50] Martin A., History and spin statistics, ArXiv/hep - ph/0209068 (2002). [51] Berestesky V. B., Lifshitz E. M. and Pitaevsky L. P., Relativistic Quantum Theory (Pergamon Press, New York, 1967). [52] Vonsovsky S. V., Magnetism (Science, Moscow, 1971) (in Russian). [53] Frenkel J. I., Elementare Theorie magnetischer und elektrischer Eigenschaften der Metalle beim absoluten Nullpunkt der Temperatur [Elementary theory of magnetic and electrical properties of metals at absolute zero temperature], Zs. Phys. 49, 31 - 45 (1928). [54] Heisenberg W., Zur Theorie der Ferromagnetism [On the theory of ferromagnetism], ibid, 49, 619 - 631 (1928). [55] Mattis D. C., The Theory of Magnetism (Springer, New York, 1981). [56] Griffiths D. J., Hyperfine splitting in the ground state of hydrogen, Am. J. Phys. 50, 698 - 703 (1982) [57] Povh B., Rith K., Scholz C., Zetsche F., Particles and Nuclei (Springer Verlag, Berlin - Heidelberg, 2006).

Experimental Study of the Long - Range Quark - Lepton ...

83

[58] Yukawa H., On the interaction of elementary particles, I, Proc. Phys. Mat. Soc. (Japan) 17, 48 - 57 (1935). [59] Wilson H. A., An electromagnetic theory of gravitation, Phys. Rev. 17, 54 - 59 (1921). [60] Dyson F. W., Eddington A. S. and Davidson C., A determination of the deflection of light by the Sun’s gravitational fields, from observations made at the total eclipse of May 29, 1919, Phil. Trans. Roy. Soc. (London) A 220, 291 - 333 (1920). [61] Sakharov A. D., Vacuum quantum fluctuation in curved space and the theory of gravitation, Sov. Phys. Dokl. 12, 1040 - 1041 (1968). [62] Barut F. O., Unification based on electromagnetism, Annalen Phys. (Leipzig) 498 , 83 - 92 (1986). [63] Buitrago J., Unified picture of electromagnetic and gravitational forces in two - spinor language, Results in Physics 16, 102859 - 4 (2020). [64] Wang Ling Jun, Unification of gravitational and electromagnetic fields, Edelweiss Applied Science and Technology, 4, 9 - 18 (2020). [65] Plekhanov V. G., Non - accelerator observation of the long - range strong nuclear interaction, J. Phys. and Opt. Sci., 2, 1 - 5 (2020). [66] Gamow G., Zur quantentheorie des atomkerns [To the quantum theory of the atomic nucleus], Zs. Physik 51, 204 - 212 (1928). [67] Fermi E., Tentatiro di una theoria dei raggi bets [Attempt of a theory of beta rays], Il Nuovo Cimento, 9, 1 - 5 (1934). [68] Plekhanov V. G. and Buitrago J. G., Evidence of residual strong interaction in nuclear atomic level via isotopic shif in LiH - LiD crystals, Prog. Phys. 15, 68 - 71 (2019). [69] Landau L. D., About fundamental problems, in, Selected Papers, Vol. 2 (Science, Moscow, 1969) 421 - 424 (in Russian). [70] Wang Ling Jun, Unification of Gravitational and Electromagnetic Forces (Schloar’s press, USA, 2019).

84

V. G. Plekhanov

[71] Compton A. H., The size and shape of the electron, Phys. Rev. 14, 247 259 (1919). ¨ [72] Schr¨odinger E., Uber die kraftfree bewegung in der relativischen Quantenmachanik, Sitzungsber [About the force-free movement in the relative quantum mechanics, session]. Preuss. Akad. Wiss. Phys. Math. KI. 24, 418 - 428 (1930). [73] Huang K., On the zitterbewegung of the Dirac electron, Amer. J. Phys. 20, 479 - 484 (1952). [74] Hestens D., The zitterbewegung interpretation of Quantum Mechanics, Found. Phys. 20, 1213 - 1232 (1990). [75] Sciama D. W., On the origin of intertia, Roy. Astron. Soc. 113, 34 - 42 (1953). [76] Vil’f F. J., Once More about the Spin of the Point - like Particle, and Dirac’s Equation (Editorial, URSS, Moscow, 2002) (in Russian). [77] Consa O., Helical Solenoid model of the electron, Prog. Phys. 14, 80 89 (2018). [78] Plekhanov V. G., Measurements of the wide value range of strong interaction coupling constant, SSRG - IJAP 6, 32 - 37 (2019). [79] Ewing D. H., Seitz F., On the electronic constitution of crystals: LiF and LiH, Phys. Rev. 50, 760 - 777 (1936). [80] Perrot F., Bulk properties of lithium hydride up to 100 Mbar, Phys. Stat. Sol. (b) 77, 517 - 525 (1976). [81] Zavt G. S., Kalder K. et al., Electron excitation and luminescence LiH single crystals, Fiz. Tverd. Tela (St. Petersburg) 18, 2724 - 2730 (1976) (in Russian). [82] Kulikov N. I., Electron structure, state equation and phase transitions insulator - metal in hydride lithium, ibid, 20, 2027 - 2035 (1978) (in Russian).

Experimental Study of the Long - Range Quark - Lepton ...

85

[83] Grosso G., Parravichini G. P., Hartree - Fock energy bands by the orthogonalized - plane - wave method: lithium hydride results, Phys Rev. B20, 2366 - 2372 (1979). [84] Baroni S., Parravichini G. P., Pezzica G., Quasiparticle band structure of lithium hydride, Phys. Rev. B32, 4077 - 4087 (1985). [85] Kunz A. B., Mikish D. J., Electronic structure of LiH and NaH, ibid, B11, 1700 - 1704 (1975). [86] Plekhanov V. G., Giant Isotope Effect in Solids (Stefan University Press, La Jola, 2004). [87] Hama J., Kawakami N., Pressure induced insulator - metal transition in solid LiiH, Phys. Lett. A126, 348 - 352 (1988). [88] Betenekova T. A., Shabanova I. N., Gavrilov F. F., The structure of valence band in lithium hydride, Fiz. Tverd. Tela 20, 2470 - 2477 (1978) (in Russian). [89] Ichikawa K., Susuki N., Tsutsumi K., Photoelectron spectroscopic study of LiH, J.Phys. Soc. Japan 50, 3650 - 3654 (1981). [90] http://physics.nist.gov/PhysRefData/Compositions/index.html. [91] Kink R. A., Kink M. F., Soovik T. A., Reflection spectra of lithium hydride crystals in 4 - 25 eV at 5K, Nuclear Instruments and Methods in Physics Research, A261, 138 - 140 (1987). [92] Plekhanov V. G., Isotopic and disorder effects in large exciton spectroscopy, Physics - Uspekhi 40, 553 - 579 (1997). [93] Plekhanov V. G., Altukhov V. I., Determination of exciton and exciton phonon interaction parameters via resonant secondary emission of insulators, Sov. Phys. Solid State 23, 439 - 447 (1981). [94] Plekhanov V. G., Nuclear technology of the creation of quantum dots in graphene, Sci. Transactient of SHI, Tallinn, 2011, p.p. 66 -71 (in Russian). [95] Plekhanov V. G., Isotopical band - gap opening in graphene, Universal J. Phys. & Apll. 10, 16 - 21 (2016).

86

V. G. Plekhanov

[96] Pankove J. I., Optical Processes in Semiconductors (Prentice - Hall, Inc., Englewood Cliffs, New Jersey, 1971). [97] Frenkel J. I., On the transformation light into heat in solids, I, Phys. Rev. 37: 17 - 44 (1931); On the transformation light into heat in solids, II, 37, 1276 - 1294 (1931). [98] Peierls R., Zur theorie der absorptionsspektren fester K¨orper, Ann. Phys. 31: 905 - 942 (1932). [99] Slater J. K. and Shockley W. H., Optical absorption by alkali halides, Phys. Rev. 50, 705 - 719 (1936). [100] Wannier G. H., Structure of electronic excitation levels in insulating crysals, Phys. Rev. 52, 191 - 197 (1937). [101] Mott N. F. and Littleton M. J., Conduction in polar crystals. I. Electrolytic conduction in solid salts, Trans. Faraday Soc. 34, 485 - 499 (1938); N.F. Mott, Conduction in polar crystals. II. The conduction band and ultra-violet absorption of alkali-halide crystals, Trans. Faraday Soc. 34, 500 - 505 (1938);R. W. Gurney and N. F. Mott, Conduction in polar crystals. III. On the colour centres in alkali-halide crystals, Trans. Faraday Soc. 34, 506 - 511 (1938). [102] Davydov A. S., Theory of Molecular Excitons (Science, Moscow, 1968) (in Russian). [103] Agranovich V. M., Theory of Excitons (Science, Moscow, 1976) (in Russian). [104] Agranovich V. M. and Ginzburg V. L., Crystallooptic and the Theory of Excitons (Science, Moscow, 1979) (in Russian). [105] Gross E. F., Selected Papers (Science, Leningrad,1976) (in Russian). [106] Agekyan V. F., Spectroscopic properties of semiconductor crystals with direct forbidden energy gap, Phys. Stat. Solidi (a) 43, 11 – 42 ( 1977). [107] Permogorov S., Hot excitons in semiconductors , Phys. Stat. Solidi (b) 68, 9–42 (1975).

Experimental Study of the Long - Range Quark - Lepton ...

87

[108] Cardona M., Modulation Spectroscopy (New York - London, Academic Press, 1969). [109] Pekar S. I., Crystal Optics and Additional Light Waves (Benjamin Cummings, Menlo Park, CA, 1983). [110] Haken H., Quantum Field Theory of Solids (North-Holland, Amsterdam, 1976). [111] Haug H. and Koch S. W., Quantum Theory of Optical and Electronic Properties of Semiconductors (World Scientific, London, 1993). [112] Plekhanov V. G., Application of Isotopic Effect in Solids (Springer, Heidelberg - Berlin, 2004). [113] Bastard G., Wave Mechanics Applied to Semiconductor Heterostructures (Halsted Press, New York, 1988). [114] Plekhanov V. G., Elementary excitations in isotope - mixed crystals, Phys. Reports 410, 1 - 235 (2005). [115] Herzberg G., Molecular Spectra and Molecular Structure of Diatomic Molecules (van Nostrand, New York, 1945). [116] Plekhanov V. G., Wannier - Mott excitons in isotope - disordered crystals, Rep. Prog. Phys. 61, 1045 - 1097 (1998). [117] Collins A. T., Lawson S. C., Davies G. and Kanda H., Indirect Energy gap of 13 C, Phys. Rev. Lett. 65, 891 - 893 (1990). [118] Ruf T., Cardona M., Pavone P. et al., Cathodoluminescence investigation of isotope effect in diamond, Solid State Commun. 105, 311 - 316 (1998). [119] Watanabe H., Koretsune T., Nakashina S. et al., Isotope composition dependence of the band - gap energy in diamond, Phys. Rev. B 88, 205420 -5 (3013). [120] Collins A. T., in: Prelas M. A., Popovichi G., Bigelow L. K. (eds.) Handbook of Industrial Diamonds and Diamond Films (Marcel Dekker, New York, 1998) Chap. 1, pp. 1 - 17.

88

V. G. Plekhanov

[121] Cardona M., Thewalt M. L. W., Isotope effects on the optical spectra of semiconductors, Rev. Mod. Phys. 77, 1173 - 1224 (2005). [122] Nordheim l., Zur Elektonentheorie der Metalle, Ann. Phys. (Leipzig) 401, 641 - 678 (1931). [123] Taylor D. W., Phonon response theory and the Infrared and Raman measurements, in: R.J. Elliott, I.P. Ipatova (eds.), Optical Properties of Mixed Crystals (Elsevier Sci. Pub. B.V., Amsterdam, 1988). [124] Plekhanov V. G., Conparative study of isotope and chemical effects on the exciton states in LiH crystals, Prog. Solid State Chem. 29, 71 - 177 (2001). [125] Bilz H., Kress W., Phonon Dispersion Relations in Insulators (Springer Verlag, Berlin - Heidelberg, 1979). [126] Baroni S., Pastori Parravichini G., Pezzica G., Quasiparticle band structure of lithium hydride, Phys. Rev. B 32, 4077 - 4082 (1985). [127] Verble J. L., Warren J. L., Yarnell J. L., Lattice dynamics of lithium hydride, ibid, 160, 980 - 989 (1968). [128] Plekhanov V. G., O’Konnel - Bronin A. A., Resonant Raman scattering in the wide - gap insulators of LiH and LiD, JETP Lett. (Moscow) 27, 413 416 (1978). [129] Dammak H., Antoshchenkova E., Hayoum M. and Finocchi F., Isotope effects in lithium hydride and lithium deuteride crystals by molecular dynamics simulations, J. Phys.: Condens. Matter 24, 435402 - 6 (2012). [130] Plekhanov V. G., Altukhov A. I., Light scattering in LiH crystals with LO phonon emission, J. Raman Spectr. 16, 358 - 365 (1985). [131] Plekhanov V. G., Phenomenology of the origin of isotope effect. Physical Sci. Intern. J. 18, 1 - 11 (2018). [132] Chang R. K., Long M. B., Optical multichannel registration, in M. Cardona, G. G¨untherodt (eds.) Light Scattering in Solids, Vol. 2 (Springer, Berlin, 1982) pp 238 - 276.

Experimental Study of the Long - Range Quark - Lepton ...

89

[133] Plekhanov V. G., Isotope effects in lattice dynamics, Phys. - Uspekhi 46, 689 - 711 (2003). [134] Field J. E., The Properties of Natural and Synthetic Diamond (Academic Press, New York, 1992). [135] Reich S., Thomsen C., Maultzsch J., Carbon Nanotubes: Basic Concepts and Physics Properties (Imperial College Press, London, 2004). [136] Plekhanov V. G., Experimental evidence of strong phonon scattering in isotope disordered systems: The case of LiH, Phys. Rev. B51, 8874 8878 (1995). [137] Yu P. Y. and Cardona M., Fundamentals of Semiconductors (Springer, Berlin, 2001). [138] Agekyan V. F., Asnin V. M., Kryukov A. M. et al., Isotope effect in germanium, Fiz. Tverd. Tela (St. - Petersburg) 31, 101 - 104 (1989) (in Russian). [139] Klark C. D., Dean P. J. and Harris P. V., Intrinsic edge absorption in diamond, Proc. Roy. Soc. 277, 312 - 329 (1964). [140] Hanzawa H., Umemura N. and Nisida Y. et al., Direct effects of nitrogen impurities, and 13 C isotope composition on the Raman spectrum in synthetic Ib diamond, Phys. Rev. B 54, 3793 - 3799 (1996). [141] Chrenko R. M., 13 C - doped diamond: Raman spectra, J. Appl. Phys. 63, 5873 - 5875 (1988). [142] Spitzer J., Etchegoin P. and Cardona M. et al., Isotopic - disorder induced Raman scattering in diamond, Solid State Commun. 88, 509 - 514 (1992). [143] Hass K. C., Tamor M. A. and Anthony T. R. et al., Lattice dynamics and Raman spectra of isotopically mixed diamond, Phys. Rev. B 45, 7171 7182 (1992). [144] Ramdas A. K. and Rodriguez S., Lattice vibrations and electronic excitation in isotopically controlled diamonds, Phys. Stat. Sol. (b) 215, 71 - 80 (1999).

90

V. G. Plekhanov

[145] Second International Conference on Astrophysics and Particle Physics, San Antonio, USA, November 13 - 15 November, 2017; Plekhanov V.G., Direct observation of the strong (nuclear) interaction in the optical spectra of solids, J. Astrophys. Aerospace Techn. 5 (N 3), 114 (2017). [146] Third International Conference on High Energy Physics, Roma, Italy, December 11 - 12, 2017; Plekhanov V.G., Direct observation of the strong nuclear interaction in the optical spectra of solids, J. Astrophys. Aerospace Techn. 5 (N 4), 46 (2017). [147] ISINN - 25, Dubna, Russia, May 22 - 26, 2017; Plekhanov V.G., Macroscopic manifestation of the strong interaction in the optical spectra of solids, Proseed ISINN - 25, Dubna, 49 - 56 (2018). [148] 5th International Conference on Theoretical and Applied Physics, Vienna, Austria, July 02 - 03 (2018); Plekhanov V.G., Spectroscopic manifestation of the strong nuclear interaction in solids, J. Phys. Chem. & Biophysics 8, 43 (2018). [149] Wigner E. P., On the mass defect of helium, Phys. Rev. 43, 252 - 257 (1933). [150] Born M., Kun H., Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954). [151] Plekhanov V. G., Measurements of the wide value range of strong interaction coupling constant, SSRG - IJAP, 6, 32 - 37 (2019). [152] Condon E. U., Note on electron - neutron interaction, Phys. Rev. 49, 459 - 461 (1936). [153] L. L. Foldy, Neutron - electron interaction, Rev. Mod. Phys. 30, 471 481 (1958). [154] Marciano W. J., Pagels H., Quantum Chromodynamics: A Review, Phys. Rept. 36, 137 - 203 (1978). [155] Brodsky S. J., Deshpande A. L., Gao H., at al., QCD and Hadron Physics, ArXiv/ hep - ph/ 1502.05728.

Experimental Study of the Long - Range Quark - Lepton ...

91

[156] Saito K., Tsushima K., Thomas A. W., Nucleon and hadron structure changes in the nuclear medium and impact on observables, Prog. Part. Nucl. Phys. 58, 1 - 64 (2007). [157] Thomas A. W., Modification of hadron structure and properties in medium, ArXiv: nucl - th/ 1402.5218 (2014). [158] Guichon P. A. M., Stone J. R., Thomas A. W., Quark - meson coupling (QMC) model for finite nuclei, nuclear matter and beyond, Prog. Part. Nucl. Phys. 100, 262 - 297 (2018). [159] Plekhanov V. G., Non - accelerator study of the residual nuclear strong interaction, Int. J. Theor. and Compur. Physics, 1, 1 - 9 (2020). [160] Deur A., Brodsky S. J., Teramond G. F., The QCD running coupling, Prog. Part. Nucl. Phys. 90, 1 - 74 (2016). [161] Roberts C. D., Empirical consequuences of emergent mass, rXiv/ hep ph/ 2009.04011. [162] Mohr P. J., Newel D. B., Taylor B. N., CODATA recommended values of the fundamental physical constants: 2014, Rev. Mod. Phys. 88, 035009 6 (2015). [163] Hofstadter R. (ed.), Nuclear and nucleon structure (Benjamin, New York, 1963). [164] Pich A., Review of αs determinations, ArXiv: hep - ph/ 1303.2262 (2013). [165] Altarelli G., The QCD running coupling and its measurement, ArXiv: hep - ph/ 1303.6065. [166] Deur A., Brodsky S. J., de Teramond G. F., The QCD running coupling, Prog. Part. Nucl. Phys. 90, 1 - 74 (2016). [167] Bethke S., The 2009 world average αs , Eur. Phys. J. C64, 689 - 703 (2009). [168] Dremin I. M., Kaidalov A. B., Quantum chromodynamics and the phenomenology of strong interactions, Phys. - Uspekhi 49, 263 - 273 (2006).

In: Understanding Quarks Editor: Benjamin Houde

ISBN: 978-1-53619-528-6 c 2021 Nova Science Publishers, Inc.

Chapter 2

M EASUREMENT OF THE A SSOCIATED P RODUCTION OF T OP -Q UARK PAIRS WITH A H IGGS B OSON A. Guti´errez-Rodr´ıguez1,∗, A. Gonz´alez-S´anchez2 and M. A. Hern´andez-Ru´ız3 1 Unidad Acad´emica de F´ısica, Universidad Aut´onoma de Zacatecas, M´exico 2 Unidad Acad´emica de Ciencia y Tecnolog´ıa de la Luz y la Materia, Universidad Aut´onoma de Zacatecas, M´exico. 3 Unidad Acad´emica de Ciencias Qu´ımicas, Universidad Aut´onoma de Zacatecas, M´exico

Abstract Top-quark physics is a very active area of research. On this topic we study the associated production of top-quark pairs with a Higgs boson through the process e+ e− → tt¯h. For our study, we consider the energies and luminosities of a future Compact Linear Collider (CLIC). The top-quark is a fermion and couples to all known bosons: the W ± and Z bosons, the photon, the gluon and the Higgs boson. These coupling strengths are well predicted by the Standard Model. Thus, measurement of the associated production of top-quarks pairs with a Higgs boson provides an important test of the Standard Model. All mentioned processes ∗

Corresponding Author’s Email: [email protected].

94

A. Guti´errez-Rodr´ıguez, A. Gonz´alez-S´anchez et al. have been observed by the ATLAS and CMS Collaboration at the Large Hadron Collider (LHC). The most important result is the direct observation of the tt¯h process by these collaborations. This is particularly interesting, as it allows to probe the Yukawa sector of the mass generation through the Higgs mechanism.

1.

INTRODUCTION

One of the deepest questions on nature is, how elementary particles acquire their fundamental properties such as mass, electric charge, spin, among other. Understanding this, is useful to get some insight of the different ways of matter cohesion in the Universe. In the Standard Model (SM), in order to explain the wide gamut of masses of elementary particles, in the 1960s was proposed the mechanism of spontaneous symmetry breaking, whose associated particle, the Higgs boson was responsible for the mass [1, 2]. Even when the Higgs boson needs a associated mass, this is not predicted by the theory and has to be determined experimentally. After its discovery in 2012, at 126.0 ± 0.4 (estad.) ± 0.4 (sys.) GeV Ref. [3] at the Large Hadron Collider (LHC), the SM was considered temporary a complete theory since all the particle degrees of freedom that it contains theoretically have been found experimentally. Moreover, until now, there are not convincing deviations from the SM which could come directly from high energy particle experiments. However, we are far from having gotten the ultimate theory of nature. There are plenty of evidences pointing to the need of new physics coming from e.g., [4, 5, 6, 7]; neutrino oscillations, dark matter and baryon asymmetry in the Universe, the hierarchy problem, the strong CP-problem, sources of the flavour symmetry breaking problem, etc. Moreover, the ATLAS and CMS experiments at CERNs LHC have exhaustively investigated the properties of the Higgs boson [3]; they have measured its mass to be around 126.0 ± 0.4 (estad.) ± 0.4 (sys.) GeV Ref. [3] and found it has zero electric charge and spin. In spite of this, for the detected Higgs boson, we still have poor determinations of its global properties (e.g., self coupling, spin, parity). The top-quark is a fermion and couples to all known bosons: the W ± and Z bosons, the photon, the gluon and the Higgs boson. Meanwhile, the Higgs boson gets coupled to all the particles to which it gives mass, and therefore there would have several potential ways of decaying; directly, to two fundamental fermions or bosons, coupling to mass, or indirectly to massless particles such as

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photons, τ leptons, bottom-quark jets or gluons via massive loops. Masses are then fixed by the Yukawa couplings, which are the strength of the coupling to the Higgs field, proportional to masses [8]. The upper limit of strong interaction of Higgs particle is with the most massive particles. From the quantum mechanics, we now know that Higgs can fluctuate for a very short time into a top quark and a to antiquark rapidly annihilate each other into a massless photon pair. The probability for this process to occur depends on the interaction strength between the Higgs boson and top quarks. Therefore, its measurement allows us to indirectly infer the value of the Higgs-top coupling. Thus, measurement of the associated production of top-quarks pairs with a Higgs boson provide an important test of the validity limits of energy of the SM [9]. In the middle of this process, undiscovered heavy new-physics particles could likewise participate in this type of decay altering the result. This is why the Higgs boson is considered as a doorway to new physics [10]. This fact, is part of a good number of strong theoretical reasons to consider that the SM is just a low energy approximation of a more general theory (e.g., [11]). All mentioned processes of interaction and decay have been observed by the ATLAS [12] and CMS Collaboration at the LHC [13, 14]. The most important result is the direct observation of the tt¯h process by these collaborations. This is particularly interesting, as it allows to probe the top quark Yukawa sector of the mass generation through the Higgs mechanism. Already, indirect measurements of the Yukawa coupling between the Higgs boson and the top quark have been made by the ATLAS luminosity of 36.1 ± 0.8 f b−1 , at a centre-of-mass energy mode was found in 2017 with an observed (expected) significance relative to the background-only hypothesis of 4.2 (3.8) standard deviations. Combining data at 7, 8, and 13 TeV, the CMS Collaboration reported an significance of 5.2 (4.2) standard deviations [15]. In a short term, capable to performing e+ e− collisions at a center-ofmass energies ranging from 0.380 − 3 T eV and luminosities within L = 500 − 3000 f b−1 , the Compact Linear Collider (CLIC) [16, 17, 18] has the purpose of reaching high precision measurements of the properties of the Higgs boson, its self-interaction and the interaction with massive particles within the SM [19, 20, 21], and with new particles predicted by others extensions of the model [22]. Therefore, precise measurements of the Higgs-top quark coupling shall provide unprecedented sensitivity to new physics scenarios and phenomena, occurring at much higher energies than that of the collisions themselves. In summary, top quark pair production is useful due to a) precision measure-

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ments in the production and the decay phase, b) its new physics potential: its large mass give us unique features for the investigation of electroweak symmetry breaking and new physics; a larger coupling with Higgs, and an appropriate sector where new physics could manifest itself in production and decay, and c) is an essential tools for calibration: jet energy scale (b, light jets). In this chapter we focus on studying the production of Higgs boson h, and a top quark pair via the processes e+ e− → Z → tt¯h, which has already been assessed in an analytical and exact form [23]. Thus, these calculations can easily be implemented in searching for signatures of new physics by contrasting its predictions with those of two main improvements; the improved Born Approximation and a Top Triangle Correction [24, 25, 26]. With these goals, the remainder of this chapter is organized as follows: In Section 2, we give a brief review of the SM gauge and Yukawa sector. In Section 3, we present an analytical form of the transversal of the process and its variation with the centerof-mass energies reachable for the future CLIC collider. Here, we also quantify the expected number of events in the frame of exact calculations and first order corrections. Then, in Section 4 we extend the analysis to one more level loop. Our results and concluding remarks are presented in Section 5.

2.

B RIEF R EVIEW

OF THE

SM

The SM of elementary particles is based on the three fundamental facts; i) Quarks and leptons are fundamental particles, which appear in three chiral families. Quarks constitute the hadrons (e.g., proton, neutron, pions, etc.). Charged leptons, come together with their corresponding light neutrinos. ii) Quarks and leptons have interactions mediated by vector particles associated with gauge symmetries. There is one gauge field for each generator of the Lie algebras associated with the symmetries of the system. iii) The masses of the weak gauge bosons (W ± , Z), and the fermions, arise from the interactions of the particles with the Higgs field. A nice review can be found in [27, 28].

2.1.

Gauge and Fermions Sector

The complete Lagrangian of the standard model is the sum of the gauge, matter, Higgs, and Yukawa interactions, L = LGauge + LM atter + LHiggs + LY ukawa ,

(1)

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which is the sum of four terms, respectively; the kinetic energy of the gauge bosons and their self interactions, the kinetic energy and gauge interactions with fermions (which depend on the fermion quantum numbers), the potential of the Higgs field responsible for generating the masses of the gauge bosons and fermions, and the coupling terms of Yukawa between the Higgs field and fermions. The Yukawa interactions are therefore responsible for providing the charged fermions with mass. The Lagrangian for the gauge sector is given by [29, 30, 31] 1 1 a aµν W , (2) LGauge = − Bµν B µν − Wµν 4 4 a , and B Wµν µν being the field strength tensors for SU (2)L, U (1)Y . The electroweak symmetry breaking mechanism is introduced to the model to generate the masses of the bosons and fermions. This is achieved by introducing another field into the model, the Higgs field φ. So, the simplest and most general Lagrangian for the Higgs field, consistent with the spontaneous symmetry breaking SU (3)C × SU (2)L × U (1)Y → SU (3)C × U (1)EM , is LHiggs = (D µ φ)†(Dµ φ) − V (φ)

(3)

where V (φ) is the most general Higgs potential gauge invariant V (φ) = µ2 φ†φ − λ(φ†φ)2 .

(4)

The covariant derivative is given by [31] Dµ = ∂µ + igs tα Gαµ + i[gT aWµa + g1 Y Bµ ],

(5)

where gs , g and g1 are the SU (3)C , SU (2)L, and U (1)Y and couplings with tα , T a , and Y being their corresponding group generators. In the potential, the quadratic term is chosen such that the minimum of the potential lies not at zero, but on a circle of minima √ v hφ0 i = µ/ 2λ ≡ √ , 2

(6)

where φ0 is the lower (neutral) component of the Higgs doublet field. This equation defines the parameter v ≈ 246 GeV, the Higgs field vacuum expectation value. Once one makes the substitution of this φ value in the Higgs Lagrangian, it is find that the W and Z bosons have acquired masses

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1q 2 1 vg MZ = g + g12 v, (7) 2 2 from the interaction of the gauge bosons with the Higgs field. Since we know g and g1 , these equations determine the numerical value of v. Therefore, the Higgs sector of the theory, introduces two new parameters, the mass µ and λ, the latest of these being the Higgs-field self interaction. After spontaneous symmetry breaking, the scalar field can be written as MW =

1 φ= √ 2

0 v + φ0

!

,

(8)

and the potential V (φ) becomes: V =

1 4 1 λv + λv 2 h2 + λvh3 + λh4 , 4 4

(9)

the second term in the potential indicates that the Higgs (h) is Mh2 = 2v 2 λ. The cubic and quartic terms describe the 3- and 4-point Higgs self couplings, which will not figure in this analysis.

2.2.

Higgs Boson

The Higgs boson couples to all massive particles of the SM, however because the coupling is proportional to the mass of the fermions, and to the square of the mass of the bosons, the dominating channels production involve the heaviest particles, such as top and bottom quarks, and W and Z bosons and tauons. Recent confirmation on the strength coupling as a function of particle mass can be seen in Figure 1, as reported by the CMS collaboration [32, 33, 34]. Therefore, the most relevant modes are; h → τ τ¯, b¯b, c¯ c, γγ, ZZ, W W , gg. The top quark pair production associated with a Higgs boson is directly sensitive to the absolute value of the top Yukawa coupling yt . A direct determination of yt is not possible in measurements of the Higgs boson decays, due to the top quark is too heavy to allow the Higgs boson to decay into a pair of top quarks [36]. Before the measurement of the tt¯h process, yt was only accessible via quantum loops as present in the gluon-gluon fusion process or the Higgs boson decay into photons. As these loops may also contain non-SM particles,

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Figure 1. SM Higgs branching ratios as a function of the Higgs-boson mass derived at LHC. Figure from Ref. [35].

which could compensate deviations in the top quark Yukawa coupling, a direct measurement is needed. With a predicted SM cross-section of [37] +11.1 σtSM ¯ th = 507.1−46.6(scale) ± 18.3(PDF + αs )f b,

(10)

the cross-section is clearly smaller than for the tt¯ process, which is one of the main backgrounds for this analysis. This overwhelming background contribution makes this analysis especially challenging. The tt¯h process was observed in 2018 at the LHC [34]. As the cross-section of the tt¯h process, σt¯th (κt ) = κ2t σtSM ¯ th ,

(11)

depends on the square of the top quark Yukawa coupling, the measurement of this process allows to set limits on the magnitude of κt and probe whether this coupling is compatible with the SM prediction.

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3.

H IGGS B OSON P RODUCTION A SSOCIATED WITH A TOP Q UARK PAIR

Figure 2. Feynman diagrams for the Higgs boson production in association with tt¯ pair e+ e− → tt¯h. We now proceed to calculate the cross-section of the process e+ e− → tt¯h. The Feynman diagrams contributing to the process are shown in Figure 2. We present the transition amplitudes for the process e+ e− → (Z) → tt¯h, as well as the formulas for the Ii (x1 , x2 ): M1 = × × M2 =

h i −ig 2 mt α t t u ¯ (p )(k / + m )γ (g − g γ )v(p ) 3 t 5 4 v A 4v cos2 θW (k2 − m2t ) (gαβ − pα pβ /MZ2 ) h i (12) (p1 + p2 )2 − MZ2 − iMZ ΓZ

h

i

e u ¯(p2 )γ β (gve − gA γ5 )v(p1) ,

h i −ig 2 mt α t t 0 u ¯ (p )γ (g − g γ )(k / + m )v(p ) 3 5 t 4 0 v A 4v cos2 θW (k 2 − m2t )

Measurement of the Associated Production of Top-Quark Pairs ... × × M3 = × ×

(gαβ − pα pβ /MZ2 )

h

h

(p1 + p2 )2 − MZ2 − iMZ ΓZ i

e u ¯(p2 )γ β (gve − gA γ5 )v(p1 ) ,

(13)

i

i ig 2 m2Z h α t t u ¯ (p )γ (g − g γ )v(p ) 3 5 4 v A 2v cos2 θW (gαβ − pαpβ /MZ2 ) h

h

ih

(p1 + p2 )2 − MZ2 − iMZ ΓZ (p3 + p4 )2 − MZ2 − iMZ ΓZ i

e u ¯(p2 )γ β (gve − gA γ5 )v(p1 ) .

101

i

(14)

In these equations, p1 , p2 (p3 , p4 ) stands for the momentum of the positron, electron (top, antitop). k(k0 ) stands for the momentum of the virtual top (antie (g t ), are given in Ref. [3]. top). The coupling constants gVe (gVt ), gA A The total cross-section is then expressed analytically, free of approximations as [23] σZ (e e → tt¯h) = + −

G2F MZ4 s π2

Z

x+ 1

x− 1

Z

x+ 2

x− 2

e2 (gVe2 + gA ) [(s − MZ2 )2 + MZ2 Γ2Z ]



t2 t2 × (gVt2 + gA )I1 (x1 , x2 ) + gA



+gVt2 I4 (x1 , x2 ) + I6 (x1 , x2 )

6 X

Ii (x1 , x2 )

i=2



dx1 dx2 ,

(15)

± where the limits of integration x± 1 (x2 ) are given by

0 ≤ x± ≤ 1, 1

−x1 (1 − x1 ) ≤ x± ≤ x1 (1 − x1 ). 2

(16)

The explicit formulas for the integrants Ii (x1 , x2 ), i = 1, 2, ..., 6 corresponding to the cross-section of the process e+ e− → tt¯h are given by m2t 4πv 2 (1 − x1 )(1 − x2 )   m2 (2 − x1 − x2 )2 × (2 − x1 − x2 )2 − h s (1 − x1 )(1 − x2 )

I1 (x1 , x2 ) =

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A. Guti´errez-Rodr´ıguez, A. Gonz´alez-S´anchez et al. m2h 2m2t m2 ) + 4(2 − x1 − x2 − h ) s s s  2 2  2  4mt m (2 − x1 − x2 ) − h +2 , (17) (1 − x1 )(1 − x2 ) s s 

+ 2(1 − x1 − x2 − +

I2 (x1 , x2 ) = × + ×

m2t 2πv 2 (1 − x1 )(1 − x2 )

"

m2 (1 − x1 )(1 − x2 )(3 − x1 − x2 ) − h (1 − x1 )(1 − x2 ) s 

8m2t 2m2h 3m2t + 2(2 − x1 − x2 ) − + (2 − x1 − x2 ) s s s  # (2 − x1 − x2 ) 4(2 − x1 − x2 )  4m2t m2h  + − ,(18) 3 (1 − x1 )(1 − x2 ) s s 

I3 (x1 , x2 ) =

I4 (x1 , x2)



×

"

+

m2t MZ2

×



m2h s

+

2 MZ 2 s )

m2t 4m2h 12MZ2 − (2 − x1 − x2 )2 − s s s 4m2h − (2 − x1 − x2 )2 s

m2 M2 1 − x 1 − x2 − h + Z s s

!

!

#

,

(19)

m2

M2

πv2 s(1 − x1 − x2 − sh + sZ )2 " # m2h 4m2t + (1 − x1 )(1 − x2 ) − 2(1 − x1 − x2 ) + , s s

×

×

πv 2 (1 − x1 − x2 −

2MZ4

=

I5 (x1 , x2) =

2MZ2

2MZ mt (2 − x1 − x2 ) 2 πv (1 − x1 )(1 − x2 )(1 − x1 − x2 −

"

m2 (1 − x1 )(1 − x2 ) − h s

!

m2h s

+

1 − x 1 − x2 −

s

2 MZ s ) ! m2h

(20)

Measurement of the Associated Production of Top-Quark Pairs ... +

4m2h m2t 12MZ2 − + (2 − x1 − x2 )2 s s s

−

3MZ2 s

m2h 2(1 − x1 )(1 − x2 ) − s (2 − x1 − x2 )

!#

,

(21)

2MZ2 m2t

I6 (x1 , x2 ) =

πv 2 s(1 − x1 )(1 − x2 )(1 − x1 − x2 −

×

"

m2h 4m2t − −2 (2 − x1 − x2 ) s s

m2h s

+

2 MZ s )

!

2

#

− 2(1 − x1 )(1 − x2 ) + (2 − x1 − x2 ) .

3.1.

103

!

(22)

The Improved Born Approximation and the Top Triangle Correction

The Higgs boson production cross section presented above is exact as it is in a completely analytical form. However, such cross section can still be improved in two ways which we shall address next; by mean of the Improved Born Approximation [25], and by considering Higgs production described by d) in the Faynman diagrams of Figure 2, which includes a top triangle vertex correction. In the former of these, the propagators carry self interacting energy contributions from light and a heavy top quark. The propagator corrections can be implemented in the e+ e− amplitudes in terms of three 1PI (one-particle irreducible) self energy functions. The amplitude for e+ e− → f f¯ with dressed propagators is similar to the Born amplitude. Therefore, we shall shift the coupling constants to the mass of Z as; α(0) ≡ α → α(MZ2 ) =

α , 1 − ∆α

(23)

and the ρ parameter defined as the ratio of charged and neutral currents ρ=1→ρ= where ∆ρ =

1 , 1 − ∆ρ

3GF √ m2t , 8π 2 2

(24)

(25)

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and the parameter ρ and the coupling constant α are related via the mixing angle as: π ρMZ2 sin2 θ¯W cos2 θ¯W = α(MZ2 ) √ . (26) 2GF Therefore, the improved approximation of the cross-section, results; σIBA = σZ (1 + 3∆ρ),

(27)

which increases with mt . The mass dependence arises from the Yukawa coupling of the two Ztt¯ vertices in the fermionic loop coating the vector boson propagator, each contributing mq . The contribution from two quark flavours from the same SU (2) doublet is proportional to the difference in the square of their masses. An additional contribution, comes from a t-quark triangle at the vertex correction, which reduces the cross-section by a factor of (1 − 38 ∆ρ) [26]. Therefore, 8 1 σ1L → σZ (1 + 3∆ρ)(1 − ∆ρ) = σZ (1 + ∆ρ). (28) 3 3 Taking into account these two corrections we can see that σ1L and σIBA vary less than 1% and 10%, respectively with a mt = 174 GeV. In the next section we will compare the predictions of eqs. (15), (27) and (28) to scrutiny the observational levels of discrepancy and detectability of the number of useful interactions expected (events).

4.

R ESULTS

AND

C ONCLUSION

√ We have addressed the analysis of the Z cross-section σZ = σZ ( s) for the processes of simple Higgs boson production e+ e− → tt¯h in association with a top quark pair within the context of the SU (3)C × SU (2)L × U (1)Y model, and compared with the results considering one triangle vertex correction, and the improved Born Approximation. This is done adopting the parameters of √ the model, which will be used by the CLIC, that is s = 0.380 − 3 T eV and L = 500 − 3000 f b−1 . We adopted the up-today data of the model listed in Table 1. In Figure 3, we show the total cross-section σZ (e+ e− → tt¯h) as a func√ tion of the center-of-mass energy s for the SM, σIBA and σ1L for the improved Born approximation and the one loop vertex corrections, respectively.

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Table 1. Current experimental data [3] Observable sin2 θW mτ mb mt

Value 0.23149 ± 0.00016 1776.82 ± 0.16 M eV 4.6 ± 0.18 GeV 172.6 ± 0.9 GeV

Observable mW mZ ΓZ mh

Value 80.389 ± 0.023 GeV 91.1876 ± 0.0021 GeV 2.4952 ± 0.0023 GeV 126 ± 0.04 GeV

The general behaviour of three plots is similar, as shall be expected, and the √ cross-section decreases for large s. The promising case to detect deviations from the SM is provided by the 1L correction which present variations of up to 10%. The σZ cross-section increases with the collider energy, reaching a maximum at the resonance of the Z gauge boson. In Table 2, we present the tt¯h number of expected events for the center-of-mass en√ ergies of s = 1000, 2000, 3000 GeV , integrated luminosities L = 500, 1000, 2000, 3000 f b−1 . The possibility of observing the process e+ e− → tt¯h is promising as shown in Table 2, and it would be possible to perform precision measurements for the Higgs boson in the future high-energy and highluminosity linear e+ e− colliders experiments, mainly at high energies. Table 2. Expected number of detections of tt¯h for mt = 174GeV , L = 500, 1000, 2000, 3000 f b−1 for exact σZ and corrections σIBA and σ1L √

s (GeV ) 1000 2000 3000

L = 500; 1000, 2000; 3000 f b−1 σZ σIBA 481472; 962948 501123; 1002247 1925897; 2888845 2004494; 3006741 201714; 403428 209946; 419893 806857; 1210286 839786; 1259679 100168; 200336 102191; 204383 400673; 601010 408768; 613151

σ1L 486313; 972626 1945252; 2917879 203741; 407483 814966: 1222449 100569; 201139 402279; 603419

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7

tt h) (fb)

6

-



SM+IBA+Loop

(e+e-




5 

SM 

IBA 4

3 0

1000

2000

3000

4000

s [GeV]

Figure 3. Total cross-section σZ for the Higgs production via e+ e− → tt¯h as √ a function of the center-of-mass energy s for the SM, σIBA and σ1L for the improved Born approximation and the one loop vertex corrections, respectively.

The presented results definitely are inside the scope, although marginally, of detection in future experiment with improved sensitivity of the new generation of linear colliders. These results encourage the development of extended theories of elementary particles. The process e+ e− → tt¯h remains as important tool for the precision measurements of the top Yukawa coupling, and to study some, or all, the implications of the CP violation Higgs-top coupling [4]. As an application of this process, another point to study, is the importance of the Higgs-top coupling in the hierarchy problem [38] and a deeper understanding of the vacuum stability of the SM [5, 39].

ACKNOWLEDGMENTS A. G. R. and M. A. H. R. thank SNI and PROFEXCE (M´exico).

R EFERENCES [1] Higgs P. W., Phys. Lett. 12, 132 (1964).

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[2] Higgs P. W., Phys. Rev. Lett. 13, 508 (1964). [3] Zyla P. A., et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020). [4] Bezrukov F. and Shaposhnikov M., J. Exp. Theor. Phys. 120, 335 (2015) [Zh. Eksp. Teor. Fiz. 147, 389 (2015)]. [5] Degrassi, G., Di Vita, S., Elias-Mir, J. et al. J. High Energ. Phys. 2012, 98 (2012). [6] Ellis, J., J. Nature 481, 24 (2012). [7] Lee, M. H., arXiv:1907.12409v3, (2019). [8] Cortina G., Reviews in Phys. 1 (2016) 60. [9] Dawson S. and Reina L., Phys. Rev. D57, 5851 (1998). [10] Rappoccio S., Reviews in Phys. 4, 100027 (2019). [11] Kibble T. W. B., European Review Phys. 23, 1 (2014). [12] Aad G., et al. [ATLAS Collaboration], Phys. Lett. B726, 88 (2013). [13] Aad G., et al., [ATLAS Collaboration], Phys. Rev. D90, 052005 (2014). [14] Aaboud M., et al., Phys. Rev. D97, 072003 (2018). [15] Wang J., et al., Proceedings of Science, (CORFU2019) 023, Summer Institute 2019 School and Workshops on Elementary Particles and Gravity, (2019). [16] Accomando E., et al. [CLIC Physics Working Group Collaboration], arXiv: hep-ph/0412251. [17] Dannheim D., Lebrun P., Linssen L. et al., arXiv: 1208.1402 [hep-ex]. [18] Abramowicz H., et al., Eur. Phys. J. C77, 475 (2017). [19] Englert F. and Brout R., Phys. Rev. Lett. 13, 321 (1964). [20] Guralnik G., Hagen C., and Kibble T., Phys. Rev. Lett. 13, 585 (1964).

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[21] Kibble T., Phys. Rev. 155, 1554 (1967). [22] Khachatryan V., et al. (2015) CMS Collaboration, Note CMS-PAS-EXO15-004, CERN, Geneva. https://cds.cern.ch/record/2114808 [23] Ram´ırez-S´anchez F. et al., Advances in High Energy Phys. 218, 1 (2018). [24] Wlodek T., Ph. D. Thesis, Wizmann Institute of Science, Israel, 1996. [25] Consoli M. and Hollik W., Electroweak Corrections for Z Physics; CERN 89-08 Vol.1 (1989) 3. [26] Hioki Z., Phys. Letters B 224, 417 (1989). [27] D´ıaz-Cruz J. L., Rev. Mexicana de F´ısica 65(5), 419 (2019). [28] Tanabashi M., Phys. Rev. D 98, 030001 (2018). [29] Ferroglia A., Lorca A. and van der Bij J. J., Ann. Phys. 16, 563 (2007). [30] Rizzo T. G., arXiv:hep-ph/0610104. [31] Abdelhak Djouadi, Phys. Rept. 457:1-216, (2008). [32] Khachatryan V., et al., [CMS Collaboration], JHEP 1504, 025 (2015). [33] Khachatryan V., et al., [CMS Collaboration], Phys. Rev. D91, 052009 (2015). [34] [CMS Collaboration], Observation of ttH Production, Phys. Rev. Lett. 120, 231801 (2018). [35] Dittmaier S. et al. [LHC Higgs Cross Section Working Group], doi:10.5170/CERN-2011-002 arXiv:1101.0593 [hep-ph]. [36] Bezrukov F. and Sahposhnikov M., J. Exp. Theor. Phys. 120(3), 335 (2015). [37] de Florian D., et al., Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector, 2016. [38] ’t Hooft G., NATO Sci. Ser. B59, 135 (1980). [39] Sher M., Phys. Rept. 179, 273 (1989).

In: Understanding Quarks Editor: Benjamin Houde

ISBN: 978-1-53619-528-6 c 2021 Nova Science Publishers, Inc.

Chapter 3

T HE Q UARKS : I NVENTION AND D ISCOVERY A. Guti´errez-Rodr´ıguez1,∗, M. A. Hern´andez-Ru´ız2 , F. Ram´ırez-S´anchez3 and F. Mireles-Garc´ıa4 1 Unidad Acad´emica de F´ısica, Universidad Aut´onoma de Zacatecas, M´exico 2 Unidad Acad´emica de Ciencias Qu´ımicas, Universidad Aut´onoma de Zacatecas, M´exico 3 Unidad Acad´emica Preparatoria, Universidad Aut´onoma de Zacatecas, M´exico 4 Unidad Acad´emica de Estudios Nucleares, Universidad Aut´onoma de Zacatecas, M´exico

Abstract This work presents the most relevant events of the process of invention and discovery of quarks, we begin with a brief historical outline of the development of the events that led us to their understanding, and considering that the present consensus within the Standard Model accepts only three families of quarks as well as of leptons, we continue by analyzing these three families one by one. Emphasis is done in the study of the first family, where the constituents of the matter that we know; protons, neutrons and electrons are created. The study of unstable particles, mesons and other exotic particles, lies within the realm of the second and third families, we also studied and analyzed the experiments that led us to the discovery of the top quark, since this particle is of particular interest in ∗

Corresponding Author’s Email: [email protected].

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A. Guti´errez-Rodr´ıguez, M. A. Hern´andez-Ru´ız et al. the current study of particle physics due to its implications in the Physics beyond the Standard Model.

1.

INTRODUCTION

In this chapter we try to illustrate the great adventure of the human being in the last 120 years in his eagerness to investigate the fundamental structure of matter. For this we will focus on the discovery of quarks, first giving a brief description of the experimental path that led to the intuition of their existence, and finally the discoveries of these extraordinary particles which are the building blocks of the Standard Model (SM) [1, 2, 3]. Man has always had admiration and curiosity for the constant revelations of the Universe in which we live, our origins and the mysteries of life. In this sense, the history of the search and discovery of elementary particles is parallel to the history of the modern human being, to his philosophical and scientific questions. This quest to try to explain nature begins with the Greeks approximately 2,500 years ago and continues until the beginning of the 20th century. At the beginning of this century the progress on the understanding of the matter accelerates dramatically due to the studies of Dalton and Avogadro. Greek philosophers such as Thales of Miletus, Anaximenes and Heraclitus suggested that everything was made up of a three elements”, that is to say water, air and fire. On the other hand, Empedocles proposed a model with four elements, adding the earth. Later, Democritus, one of the founders of the atomist theory, introduced the idea that everything was made up of invisible and indivisible particles, that is, atoms. Undoubtedly, the revolutionary thought of the Greeks of that time has been the guide that has forged the scientific investigation of the structure of matter to this day. The idea that there are elementary blocks that interact through an exchange of energy” has essentially not changed since then. However, the language that describes these interactions is formed by the most modern theories of physics and mathematics such as the theory of relativity, quantum physics, field theory, group theory, etc. It was not until the twentieth century that Rutherford with his famous experiment had a scientific approach to the true composition of matter of the universe. Rutherford’s atomic model established that the atom is not indivisible, but consists of a central nucleus with positive electrical charge and a cloud of electrons that revolve around it. We now know that the atomic nucleus is made up of protons and neutrons. These three elementary particles (protons, neutrons and

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electrons) passed to four, when in the 1930s Chadwick discovered the neutrino, bombarding a beryllium disc with alpha particles from a polonium source, neutrinos were previously proposed by W Pauli. It has been 60 years since the Big Bang emerged as the standard cosmological model, and at present it is known that about 13,800 million years ago, all matter and radiation were compressed and concentrated at a point of immeasurable temperature and density: the Big Bang, the moment of initial expansion proposed by Hubble and in which we continue immersed. The primordial form in which the Universe was just a few millionths of a second after the Big Bang occurred, is a plasma known as the quark-gluon. A few moments later the Universe becomes a soup” of elementary particles-quarks, gluons, electrons, neutrinos and photons among others. With the high temperatures, the quarks moved freely, overcoming any attraction and colliding at very high speeds, generating quark-antiquark pairs in their collisions that destroyed each other. When the temperature drops enough, the quarks will lose that energy and the strong force, transmitted by the gluons, will keep them together forming protons and neutrons. After that, in a less hot stage nuclei are formed. In those instants, the laws of physics that have governed until now the world were no longer valid. At that moment there is an infinitely dense and infinitely hot Universe in which everything that exists, forces, particles and energy, is so compressed that the three spatial dimensions in which we move are undifferentiated. In this Universe, so different from the one we know, the four forces that govern nature and that explain all the interactions that we know (the strong force, the weak force, the electromagnetic force and the gravitational force) constituted a single universal force. The present consensus, according with the SM, is that quarks along with leptons: electrons, muons, taons and neutrinos, are the fundamental constituents of visible matter and are the smallest particles man has managed to identify. The quark model is a classification scheme for hadrons in terms of their valence quarks, the quarks and antiquarks give rise to the quantum numbers of the hadrons. The quark model in its modern form was developed by Murray Gell-Mann [4] and independently by George Zweig [5]. The idea arose to explain the regularities of hadrons states, their charges and spins could be readily explained (and even predicted) by simply combining the then known up u, down d and strenge s quarks. It is appropriate to mentioned that Murray Gell-Mann, won the Nobel Prize

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in Physics in 1969 for his contribution to the elementary particles physics. He classified the subatomic particles (Quark Model) and brought order to the universe. In addition, he introduced principles that allowed subatomic particles to be classified with a criterion that Gell-Mann named the eight-fold path”. Murray Gell-Mann also coined the term quark” in 1963 for one of the fundamental components of matter, taking inspiration from a novel by James Joyce. Quark confinement in hadrons is perhaps one of their most extraordinary feature, it has never been possible to isolate a quark, there are no free quarks in nature, they are always found in groups of three, as in protons and neutrons, where they are linked by gluon exchange, or in groups of two, as in π mesons and K mesons. Their ability to unite is due to the fact that they experience the action of the strong nuclear force. Since quarks are considered Fermions that obey the Paulis Exclusion Principle that states that two of them cannot occupy the same physical state simultaneously. Some arrangements of quarks apparently violated this principle. This problem was resolved by the introduction of color. In this theory of strong interactions color has nothing to do with the colors of the everyday world but rather represents a property of quarks that is the source of the strong force. Color plays the role analogous to that of electric charge in the electromagnetic force, and just as charge implies the exchange of photons between charged particles, so does color involve the exchange of mass less particles called gluons among quarks. The colors red, green, and blue are ascribed to quarks, and their opposites, antired, antigreen, and antiblue, to antiquarks. Since combinations of quarks must contain mixtures of these imaginary colors the idea is that they must cancel out one another, with the resulting particle being neutral or colorless. For this reason the work done by Murray Gell-Mann to study this phenomenon is called Quantum Chromodynamics (QCD). Currently, six types of quarks can be distinguished with their corresponding antiquarks and they are classified, in groups of two, called families or generations; up (u) and down (d), which are considered to belong to the first family or generation, charm (c) and strange (s) to the second family and top (t) and bottom (b) to the third family, particle physicists have named them without any physical reason. In his original article [4], Murray Gell-Mann did not name them and only used the letters u, d, and s to represent them. On the other hand, George Zweig [5] used the letters p0 , n0 and Λ0 . It was not until the 1970s that quarks were called by their current name. For more details see Refs. [7, 8, 9].

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The six types of quarks are characterized by their electrical charge, mass, flavor, color and spin. Quark up, electric charge 2/3 and discovered in 1964. Quark down, electric charge −1/3 and discovered in 1968. Quark charm, electric charge 2/3 and discovered in the year 1971. Quark strange, electric charge −1/3 and discovered in the year 1964. Quark top, electric charge 2/3 and discovered in 1995. Quark bottom, electrical charge −1/3 and discovered in 1995. Further information regarding this particles will be given in the next sections. Varieties of quarks such as s, c, b, and t are very unstable and it took them only a fraction of a second after the Big Bang to disappear from the universe, but today particle physicists can recreate and study them in particle accelerators. The varieties of quarks u and d currently exist in nature in a stable way, and are distinguished between them by their electric charge and their mass. The model of quarks originally had three quarks; up, down and strange [4], carrying the quantum numbers isospin up, isospin down and strangeness (quantum number introduced by Gell-Mann and Nishijima) respectively. The Glashow-Iliopoulos-Maiani mechanism (GIM) [10] predicted a fourth quark (charm). The Cabbibo-Kobayashi-Maskawa mechanism [11] predicts a third generation of quarks, top and bottom. In our Universe, all the macroscopic forms of matter can be traced back to a basic building block of Nature interacting by four fundamental forces: strong (or nuclear), electromagnetic, weak and gravitational. The SM of elementary particle physics presents an elegant and simple description of the interactions between the fundamental constituents of nature: leptons and quarks. These appear naturally as a consequence of the symmetry of the systems against a set of transformations. The properties of the interaction are completely determined by the symmetry group and its intensity is given by the couplings constants. It is worth mentioning that the formulation of the SM. is marked by two important events, the first originates from the combination of fundamental theories such as quantum mechanics, electromagnetism and special relativity that allowed the formulation of the Dirac equation and the second event derives from the quantization of the fields which leads to the formulation of the Quantum Field Theory (QFT). The SM has the virtue of being theoretically consistent, and aside from neutrino masses, has not been contradicted by any experiment carried out to date, so in many respects it has an excellent correspondence with the experimental measurements. Although, despite this paramount success, the SM is not complete in its current form. There are strong conceptual as well as experimental indications for

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the existence of new physics and fundamental understanding of these phenomena demands physics Beyond the SM. Fig. 1 describes the dimensional scheme of the atom, while Fig. 2 illustrate table of the Standard Model of Fundamental Particles and Interactions: The quarks and leptons of the three families are presented plus the vector bosons that mediate the interactions of the strong, weak and electromagnetic forces, the Higgs boson is also presented [12].

Figure 1. Dimensional scheme of the atom [12].

1.1.

The Standard Model

The SM of elementary particle physics presents an elegant and simple description of the interactions between the fundamental constituents of nature: leptons and quarks. These appear naturally as a consequence of the symmetry of the systems against a set of transformations. The properties of the interaction are completely determined by the symmetry group and its intensity is given by the couplings constants. It is worth mentioning that the formulation of the SM is marked by two important events, the first originates from the combination of fundamental theories such as quantum mechanics, electromagnetism and special relativity that allowed the formulation of the Dirac equation and the second

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Figure 2. The Standard Model of Fundamental Particles and Interactions [12]. event was derives from the quantization of the fields and that leads to the formulation of the Quantum Field Theory. The SM of elementary particle physics has the virtue of being theoretically consistent and has not been contradicted by any experiment carried out to date, and in many respects it has an excellent correspondence between the predictions of the SM and the experimental measurements. Fig. 3 illustrates and schematize the fundamental constituents of matter, as well as the fundamental interactions between the particles described by the Standard Model [12]. In particle physics, a generation or family is a division of the elementary particles. Between generations, particles differ by their flavor quantum number and mass, but their strong and electric interactions are the same. Considering the mass difference between individual quarks and hence, between families, we can conclude that the three generations couple differently to the Higgs sector and this is telling us something, but we don’t really know what yet. The great importance of this quark model is that it is a huge simplification of nature, as it reduces the multitude of known hadrons to a few fundamental

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quarks. According to these ideas, in the next sections the three different families in which quarks are classified are analyzed. The document is organized as follows: Section 2 deals with the first family of quarks, that is, the u (up) and d (down) quarks. Section 3, considers the second family of quarks corresponding to the c (charm) and s (strange) quarks. Section 4, takes care of the third family of quarks, that is, the t (top) and b (bottom) quarks, and the conclusions are given in Section 5.

Figure 3. Summary of interactions between particles described by the Standard Model [12].

2.

F IRST FAMILY

OF

QUARKS , u ( UP)

AND

d ( DOWN )

As we mentioned before, all hadrons consist of quarks: baryons by three quarks; anti-baryons by three antiquarks, and the mesons by a quark and an antiquark. In addition, all the quantum numbers of a hadron are obtained by adding the corresponding quantum numbers of its constituent quarks, so that all allowed combinations of quarks originate previously known hadrons, without any other combination being able to do so. This is the lightest of the three families, and consists of the electron, the electron neutrino, the up quark, and the down quark. The other two families are progressively more massive. The proton consists of two quarks u and a quark d; its electric charge is;

The Quarks: Invention and Discovery Q=

2 3

+

2 3

−

1 3

= 1,

B=

1 3

+

1 3

+

1 3

= 1,

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its baryonic number

and the strangeness S = 0 + 0 + 0 = 0. The lepton number L for the proton is evidently zero. The charge Q, the baryon number B, and the strangeness S are more than simply labels; they are conserved additive quantum numbers in strong interactions. That is, a particle reaction (e.g., π − p → π + n or n → pe− ν¯) can only occur if the sum of these quantum numbers in the initial state equals that in the final state. It is worth mentioning that strangeness is violated in weak interactions and that the hipercharge Y is defined as the sum of the baryon number plus strangeness Y = B + S. Feynman theory of neutron beta decay is a V − A theory and produces surprisingly good results. But since nucleons (proton and neutron) are composite particles, made out of quarks, a more sophisticated theory is needed, one that will take into consideration the observed generational mixing of quarks associated with the weak interaction. Moreover, since a neutron is composite particle, the presence of spectator quarks must also be included in the theory Fig. 4. The previous ideas led physicists to consider arrangements of two particles in a way similar to the Isospin theory of Heisenberg, where he observed that a proton and a neutron had similar characteristics except for their electric charge, in the same way a quark up and a quark down have similar characteristics except for their charge. These two quarks are considered to belong to the first family or generation.

3.

SECOND FAMILY AND c ( CHARM )

OF

QUARKS , s ( STRANGE )

This family is heavier than the previous one and consists of the muon, the muon neutrino, the charm quark, and the strange quark. The second neutrino, the muon neutrino νµ , was detected experimentally in 1962, confirming the theoretical forecast. There were, then, four leptons: the electron e− , the muon µ− ,

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Figure 4. Neutron β decay process, n → pe− ν¯ with spectator quarks. the electron neutrino νe and the muon neutrino νµ . So why not four quarks?. Physicist are always trying to find symmetries in nature, or trying to explain them. If in fact there was a certain analogy between quarks and leptons, such as truly elementary particles, the asymmetry of three quarks versus four leptons made no sense. The simplest way to approach this problem was to assume the existence of a fourth quark. As we mentioned before, in 1970 American physicists Sheldon Glashow, Greek physicists John Iliopoulos and Italian physicists Luciano Maiani developed a theory (The GIM mechanism) [10] that predicted the existence of a fourth quark, called quark c or quark charme, with a charge (2/3)e. That fourth quark, was discovered (simultaneously at Stanford SLAC and Lawrence Berkeley BNL) experimentally in 1976, indirectly, through the discovery of a hadron called the J/Ψ (gypsy) particle, that meson was a combination of this entirely new charm quark and its correspondent antiquark. To this combination of charm − anticharm quarks physicists gave it the name of charmonium. Although J/Ψ ’s idea as a meson was strange, it was Gerard ’t Hooft one of the first to suspect that it could be a charm quark-antiquark pair. This particle revolutionized particle physics, as it did not fit among the known ones. It didn’t seem to be a resonance of another already discovered particle, and if it was new, it was extremely strange. The J/Ψ particle had a half-life that, although it

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seemed short, was very long compared to what was expected to be. Soon it was discovered that J/Ψ ’s unusual life was due to a phenomenon called asymptotic freedom, the fact that quarks interact more feebly at small distances than at large ones, unlike the other particles with other forces. It turns out that the typical distance at which an object that interacts strongly vibrates is inversely proportional to its mass. Thus, the quarks up and down in protons and neutrons are further away than the charm quarks within a J/Ψ. Being closer, they feel less the strong force, and their disintegrations take place at a slower rate, thus explaining gypsy’s long life. After the discovery of the fourth quark, the charm quark, and following the same idea as for the first generation, quarks; strange and charm were considered to belong to the second family, though the mass ratio between them is slightly higher than the first family, they can steel be considered a family, since, besides their charge, both are very similar and have the same characteristics.

4.

T HIRD FAMILY OF QUARKS , t ( TOP ) AND b ( BOTTOM )

In 1973, Makoto Kobayashi and Toshihide Maskawa predicted the existence of a third generation of quarks to explain observed CP violations in kaon decay. The names top and bottom were introduced by Haim Harari in 1975, to match the names of the first generation of quarks (up and down) reflecting the fact that the two were the up” and down” component of a weak isospin doublet [15]. This is the heaviest of the three families and consists of the taon τ − , the taon neutrino ντ , the top quark t, and the bottom quark b. In 1977, Fermilab researchers announced the discovery of the fifth quark: the quark bottom, by the E288 experiment team, led by Leon Lederman. This strongly suggested that there must also be a sixth quark, the top, to complete the pair. It was known that this quark would be heavier than the bottom, requiring more energy to create in particle collisions, but the general expectation was that the sixth quark would soon be found. However, it took another 18 years before the existence of the top was confirmed. The sixth quark, the quark top, previously postulated, was also found by the experimental physicists of Fermilab, in 1995, sometimes also referred to as the truth quark, is the most massive of all observed elementary particles. Because of its enormous mass, the top quark is extremely short-lived with a

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predicted lifetime of only 5×10−25 s. As a result, top quarks do not have time before they decay to form hadrons as other quarks do, which provides physicists with the unique opportunity to study the behavior of a bare” quark. Because the top quark is so massive, its properties allowed indirect determination of the mass of the Higgs boson. Considering its previous characteristics, the top quark’s properties are extensively studied as a means to discriminate between competing theories of new physics beyond the SM. The top quark is the only quark that has been directly observed due to the fact that it decays faster than the hadronization time. It derives its mass from its coupling to the Higgs Boson. This coupling yt (Yukawa coupling) is very close to unity; in the SM of particle physics, it is the largest (strongest) coupling at the scale of the weak interactions and above. Like all other quarks, the top quark is a fermion with spin 1/2 and participates in all four fundamental interactions: gravitation, electromagnetism, weak interactions, and strong interactions. It has an electric charge of +2/3 e. The antiparticle of the top quark is the top antiquark ¯t, which obviously only differs in that some of its properties have the same magnitude but opposite sign. The top quark interacts with gluons of the strong interaction and is typically produced in hadron colliders via this interaction. However, once produced, the top (or antitop) is able to emmit a Higgs boson and it will decay only through the weak interaction. It decays to a W boson and either a bottom quark (most frequently), a strange quark, or, on the rarest of occasions, a down quark [15]. After the discovery of the Higgs boson the focus is on the precise measurement of its properties, in particular couplings to fermions and gauge bosons. The top quark is the most strongly-coupled SM particle to the Higgs boson. Considering that top quarks are so heavy, Higgs particles, as we said before, can be radiated off by them and could be produced copiously in future high energy positron-electron (e+ e− ) colliders [16, 17], Fig. 5. This process can be used to measure the Higgs-top quark coupling. The measurement of the Higgs couplings to other fundamental particles provides one of the crucial tests of the Higgs mechanism. However, this coupling to fermions is quite difficult to measure directly. A promising and nearly unique method, though experimentally not easy, is provided by the Higgs brems-strahlung process e+ e → (γ, Z) → tt¯h off heavy top quarks in high-energy e+ e− colliders [18]. These processes are analyzed in detail in [19, 20]. Nothing in the SM rules out more families of higher masses, but experimental evidence suggests that three is all there is.

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Figure 5. Higgs production process, e+ e → (γ, Z) → tt¯h.

C ONCLUSION The historical events in the development of the discovery of quarks were thoroughly analyzed. In nature it is the particles of the first family that play the most important role, they interact with each other in the same way as the four particles of the second or third families (e.g., in ordinary radioactive processes a quark u is transformed into another d and vice versa), but the latter, which have a much shorter half-life, disintegrate very fast, through weak interaction, resulting in particles of the first family. On the other hand, while there are known interactions between quarks of different families, the same is not true of leptons, nor are there any known transitions from leptons in to quarks, nor from quarks in leptons. There are new theories that construct a model of quarklepton unification at the TeV scale based on an SU (4) gauge symmetry, while still having acceptable neutrino masses and enough suppression in flavor changing neutral currents. An approximate U (2) flavor symmetry is an artifact of family-dependent gauge charges leading to a natural realization of the CKM mixing matrix [14]. However, after all these considerations, there is no doubt whether quarks are, in fact, elementary particles, or whether, on the contrary, as has happened many times in the course of the history of Physics, new discoveries could emerge highlighting the existence of a substructure within both, quarks and lep-

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tons. Fortunately, quantum mechanics and the theory of relativity completely rule out such a possibility. In fact, different experimental methods lead to an upper diameter limit for the electron of 10−15 m., and this same limit value can also be assigned to the remaining leptons and quarks. In the hypothetical case that quarks and leptons had constituent parts, they should be confined to a region of space ∼ 10−15 m in diameter, and according to Heisenbergs uncertainty principle this would yield a position uncertainty of ∆x < 10−15 m, giving a momentum which speed would violate the relativity principle. Thus, a search of approximately 30 years was completed, from the proposal of Gell-Mann and Zweig, in 1964, to the discovery of the quark top in 1995.

ACKNOWLEDGMENTS We acknowledge support from CONACyT, SNI and PROFEXCE (M´exico).

R EFERENCES [1] Glashow S. L., Nucl. Phys. 22, 579 (1961). [2] Weinberg S., Phys. Rev. Lett. 19, 1264 (1967). [3] Salam A., Elementary Particle Theory, Ed. N. Svartholm (Almquist and Wiskell, Stockholm, 1968) 367. [4] Gell-Mann Murray, Phys. Lett. 8, 214 (1964). [5] Zweig George, An SU3 model for strong interaction symmetry and its breaking, Part I, CERN Reports No. 8182/TH.gg01, 1964 (unpublished). [6] Lipkin Harry J., Phys. Rev. Lett. B13, 590 (1964). [7] Leutwyler H., Phys. Lett. B48, 431 (1974). [8] Leutwyler H., Nucl. Phys. B76, 413 (1974). [9] Wilson Kenneth G., Phys. Rep. 23, 331 (1976). [10] Glashow S. L., Iliopoulos J. and Maiani L., Phys. Rev. D2, 1285 (1970). [11] Kobayashi M. and Maskawa T., Prog. Theor. Phys. 49, 652 (1973).

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[12] https://en.wikipedia.org/wiki/Particle physics. [13] https://www.bnl.gov/world/, Brookhaven National Laboratory. [14] Greljo A. and Stefanek Ben A., Phys. Lett. B782, 131 (2018 ). [15] https://en.wikipedia.org/wiki/Top-quark. [16] Baer H., et al., The International Linear Collider - Technical Design Report, vol. 2: Physics. International Linear Collider, ILC-REPORT-040 (2013). [17] Aicheler M., et al., A multi-TeV linear collider based on CLIC technology: CLIC Conceptual Design Report. CERN European organization for nuclear research, CERN-2012-007. [18] Djouadi A., Kalinowski J. and Zerwas P. M., Z. Phys. C54, 255 (1992). [19] Ram´ırez-S´anchez F., Guti´errez-Rodr´ıguez A., Gonz´alez-S´anchez A., and Hern´andez-Ru´ız M. A., J. Phys. G: Nucl. Part. Phys. 43, 095003 (2016). [20] Ram´ırez-S´anchez F., Guti´errez-Rodr´ıguez A., Gonz´alez-S´anchez A., and Hern´andez-Ru´ız M. A., Adv. High Energy Phys. 2018, 8523854 (2018).

In: Understanding Quarks Editor: Benjamin Houde

ISBN: 978-1-53619-528-6 c 2021 Nova Science Publishers, Inc.

Chapter 4

S TRANGE Q UARK M ATTER AND E XOTIC D ARK M ATTER∗ Hidezumi Terazawa† Center of Asia and Oceania for Science(CAOS), Tokyo, Japan and Midlands Academy of Business & Technology(MABT), Leicester, United Kingdom This Chapter is dedicated to the late Professor Masatoshi Koshiba, the founder of neutrino astronomy, who has been the life-long tutor of the present author since 1963. ∗ Some of the contents of this Chapter have already been presented in the contributed paper to the XXVI International Seminar, “Nonlinear Phenomena in Complex Systems”, Minsk, May 2124, 2019, published in the Proceedings, edited by V.I. Kuvshinov et al., Nonlinear Phenomena in Complex Systems 22:4 (2019) 311, and in the two Chapters (Chapter I. Quark Matter and Strange Stars and Chapter III. Dark Energy, Dark Matter, and Strange Stars) of the book, Terazawa H., “Quark Matter: From Subquarks to the Universe” (Nova Science Publishers, New York, 2018), https:// novapublishers.com/shop/quark-matter-from-subquarks-to-the-universe/. Also, this is a revised and further up-dated version of the two papers, “Strange Quark Matter and Strange Quark Stars”, H. Terazawa, Nonlinear Phenomena in Complex Systems 19 (2016) 147, and “Exotic Nuclei and Strange Stars”, H. Terazawa, Nonlinear Phenomena in Complex Systems 18 (2015) 25, which are the extended and up-dated versions of the Chapters I and III in the Proceedings of the 12th International Conference & School “Foundation & Advances in Nonlinear Science”, Minsk, Belarus, 2004, edited by Kuvshinov V.I. and G.G. Krylov (Belarusian State University, Minsk, 2005), p.84, and in the Proceedings of the International Conference on New Trends in HighEnergy Physics, Yalta, Crimea (Ukraine), 2005, edited by Bogolyubov P.N., P.O. Fedosenko, L.L. Jenkovszky, and Yu.A. Karpenko (Bogolyubov Institute for Theoretical Physics, Kiev, 2005), p.259, arXiv:1304.5655v2[physics.gen-ph] 15 Apr 2015. † Corresponding Author’s Email: [email protected].

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Hidezumi Terazawa Abstract New forms of matter such as strange quark matter and strange quark stars as candidates for dark matter are discussed in some detail, based on the so-called “Bodmer-Terazawa- Witten hypothesis” assuming that they are stable absolutely or quasi-stable (decaying only weakly). This Chapter consists of the following Sections: Section I. Introduction, Section II. Exotic Nuclei and Strange Stars, and Section III. Strange Quark Matter as Dark Matter.

1.

INTRODUCTION

In the twentieth century, the atomism became one of the most important principles in physics: matter consists of molecules, a molecule consists of atoms, an atom consists of a nucleus and electrons, a nucleus consists of nucleons, a nucleon consists of quarks, and, perhaps, a quark consists of subquarks, the most fundamental constituents of matter [1]. In the twenty- first century, it has become one of the ultimate goals in physics to find the substructure of fundamental particles such as quarks, leptons, and gauge and Higgs bosons. It is, however, still interesting to find new forms of matter which differ from ordinary molecules, atoms, nuclei, hadrons, quarks, and leptons. Recently, such exotic forms of matter as carbon nanofoams [2] and pentaquarks [3] have been found experimentally. In the near future, more exotic forms of matter such as hexalambdas [4] and color-balls [5] would be found. In this paper, I am going to discuss these new forms of matter such as strange quark matter (superhypernuclei) and strange quark stars (super-hypernuclear stars) as candidates for dark matter in some detail, based on the so-called “Bodmer-Terazawa-Witten hypothesis” assuming that they are stable(absolutely) or semi-stable (decaying only weakly).

2.

E XOTIC N UCLEI

AND

STRANGE STARS

A super-hypernucleus is a nucleus which consists of many strange quarks as well as up and down quarks. In 1979, I proposed the quark-shell model of nuclei in quantum chromodynamics(QCD), presented the effective two-body potential between quarks in a nucleus, pointed out violent breakdown of isospin invariance and importance of U-spin invariance in superheavy nuclei, and predicted possible creation of “super-hypernuclei” in heavy- ion collisions at high

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energies, based on the natural expectation that not only the Fermi energy but also the Coulomb repulsive energy is reduced in such nuclei [4]. A similar idea was presented independently and almost simultaneously by Chin and Kerman, who called super-hypernuclei “long-lived hyperstrange multiquark droplets” [6]. Five years later in 1984, the possible creation of such super-hypernuclear matter in bulk (or a much larger scale of mass number and space size) in the early Universe or inside neutron stars was discussed in detail in QCD by Witten, who called super-hypernuclear matter “quark nuggets” [7] while the properties of super-hypernuclei were investigated in detail in the Fermi gas model by Farhi and Jaffe, who called super-hypernuclei “strange matter ”[8]. In a series of papers published in 1989 and 1990, I reported an important part of the results of my investigation on the mass spectrum and other properties of superhypernuclei in the quark-shell model [4]. Let Nu , Nd , and Ns be the number of u’s, that of d’s, and that of s’s, respectively. Then, a nucleus of (Nu , Nd , Ns ) has the atomic number, the mass number, and the strangeness given by Z = (2Nu − Nd − Ns )/3, A = (Nu + Nd + Ns )/3, and S = −Ns . By noting not only a possible similarity between the effective nuclear potential and the effective quark one but also an additional three color-degrees of freedom, I have predicted that the magic numbers in the quark-shell model are three times the famous magic numbers in the nucleon-shell model (Z, A − Z = 2, 8, 20, 28, 50, 82, 126, ...), i.e. Nu , Nd, Ns = 6, 24, 60, 84, 150, 246, 378, .... Therefore, the magic nuclei such as 42 He and 16 8 O are doubly magic and superstable also in the quark-shell model since Nu = Nd = 6 for 42 He and Nu = Nd = 24 for 16 8 O. What is new in the quark-shell model is the expectation that not only certain exotic nuclei with a single magic number such as the “dideltas”, Dδ ++++ with Nu = 6 and Dδ −− with Nd = 6, and the “diomega”, Dω −− with Ns = 6, but also certain super-hypernuclei with a triple magic number such as the “hexalambda”, Hλ with Nu = Nd = Ns = 6, and the “vigintiquattuoralambda”, V qλ with Nu = Nd = Ns = 24, may appear as quasi-stable nuclei. In fact, in the MIT bag model [9], one can easily estimate the mass of Hλ to be as small as 6.3GeV, which is smaller than 6mΛ (∼ = 6.7GeV ). However, in the quark-shell model, there is no qualitative reason why the “dihyperon” or “H dibaryon”, H with Nu = Nd = Ns = 2, should be quasi-stable or even stable. I have made many other predictions including a sudden increase of the K/π ratio due to production of super-hypernuclei in heavy-ion collisions at high energies [4]. In 1990, Saito et al. found in cosmic rays two abnormal events with the

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charge of Z = 14 and the mass number of A ∼ = 370 and concluded that they may be explained by the hypothesis of super-hypernuclei [10]. In order to find whether these cosmic ray events are really super-hypernuclei as suggested by the cosmic ray experimentalists, I investigated how the small charge-to-massnumber ratio of Z/A is determined when super-hypernuclei are created. In the paper published in 1991, I have shown that such a small charge of 3 ∼ 30 may √ be realized as Z ≤ 2A/3(∼ = 15.7 for A = 370) if the super-hypernuclei are created spontaneously from bulk super-hypernuclear matter due to the Coulomb attraction [11]. Therefore, the most likely explanation for the abnormal events seems to be that they are, at least, good candidates for super-hypernuclei as suggested by Saito et al. [10]. However, in the other paper published in 1993, I have suggested the second most likely explanation that they may be “technibaryonic nuclei” or “technibaryon-nucleus atoms” [12]. A technibaryon is a baryon which consists of techniquarks in a bound state due to the technicolor force [13]. A technibaryonic nucleus is a nucleus which consists of nucleons and a technibaryon. A technibaryon-nucleus atom is an atom which consists of a negatively charged technibaryon and an ordinary nucleus in a bound state due to the Coulomb force. The technibaryon mass can be expected to be about 2TeV either from scaling of the baryon mass with the color and technicolor dimensional parameters ΛC and ΛT C [14] or from my estimation of the techniquark mass to be about 0.5 ∼ 0.8T eV from the PCDC (Partially-Conserved-Dilation-Current) anomaly sum rule for quark and lepton masses [15]. The mass value of about 0.4TeV obtained for the abnormal cosmic ray events is much smaller than the expected values for technibaryonic nuclei or technibaryon-nucleus atoms. However, this value would not be excluded since a large experimental error in determining the masses might be involved. In this respect, note that the abnormal cosmic ray event found in 1975 by Price et al. may be better explained by a technibaryonic nucleus or technibaryon-nucleus atom since their later analysis might indicate the charge of Z ∼ = 46 and the mass number of A ≥ 1000 [16]. Also note that the abnormal cosmic ray event found in 1993 by Ichimura et al. may be better (but much less better) explained by a technibaryonic nucleus or technibaryon-nucleus atom since they reported the charge of Z ≥ 32 ± 2 [17]. More recently, I have proposed the third most likely explanation for the abnormal cosmic ray events that they may be “color-balled nuclei” [18]. A colorball is a color-singlet bound state of an arbitrary number of gluons [5] or of “chroms”, Cα (α = 0, 1, 2, 3), which are the most fundamental constituents of

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quarks and leptons (called “subquarks” in a generic sense) with the color quantum number and which form quarks and leptons together with a weak-isodoublet of subquarks (called “wakems”), wi (i = 1, 2), in the unified composite model of all fundamental particles and forces [1]. A color-balled nucleus is a nucleus which consists of nucleons and a color-ball. The color-ball of (C0 C1 C2 C3 ) is not only electromagnetically neutral but also weakly neutral. However, it strongly interacts with any hadrons due to the van der Waals force induced by the color-singlet state of (C1 C2 C3 ) as baryons, the color-singlet states of three quarks. Its mass may be very large as scaled by the subcolor energy scale ΛSC (of the order of, say, 1TeV) [19] and its size may be very small as scaled by 1/ΛSC (∼ 1/1T eV ) but it may be absolutely stable. This extremely exotic particle (which we may call “primitive hydrogen”) may provide us not only the third most likely explanation for the abnormal cosmic ray events but also another candidate for the missing mass or dark matter in the Universe. Already in 1970, Itoh and, independently in 1971, Bodmer pointed out the possibility that super-hypernuclei (which the latter called “collapsed nuclei”) may exist on a large scale [20]. Bodmer even suggested then that they may explain the missing mass in the Universe. For the last two decades, “strange stars” consisting of super-hypernuclear matter have been investigated in great detail not only theoretically but also experimentally [20]. If the possible identification of the recently discovered unusual x-ray burster GRO J1744-28 as a strange star by Cheng, Dai, Wei, and Lu is right [21], the existence of super-hypernuclear matter or “strange matter” has already been discovered by astrophysicists as a gigantic super-hypernucleus or “strangelet”, the super- hyperstar or “strange star” in the Universe, before being discovered by high-energy experimentalists in heavy-ion collisions. I must also mention that not only the recent possible identification of the x-ray pulsar Her X-1 as a strange star claimed by Li, Dai, and Wang and of the x-ray burster 4U 1820-30 proposed by Bombaci [22] but also the more recent possible identification of the newly discovered millisecond x-ray pulsar SAX J1808.4-3658 suggested by Li, Bombaci, Dey, and van den Heuvel [23] seems to be just as reasonable as that of GRO J1744-28 by Cheng et al. [21]. Even more recently, NASA’s Chandra X-ray Observatory has found two stars, RXJ185635-3757 which is too small (about 11.3km) and 3C58 which is too cold (less than 1 million degrees in Celsius), being most likely strange stars [24]. As for more recent searches for strange stars, see the reviews by Weber in Ref. [20]. In September, 1999, just before the BNL RHIC was about to open up a

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high- energy range of order hundred GeV/nucleon for heavy-ion-heavy-ion colliding beams, a rumor shocked the whole world [25]. It said that if a negatively charged stable strangelet were produced by RHIC experiments, it would convert ordinary matter into strange matter, eventually destroying the Earth. However, I argued that there is no danger of such a “disaster ” at RHIC since the most stable configuration of strange matter must have positive electric charge thanks to the fact that the up quark is lighter than the down and strange quarks [4]. This liberation from such a horrible fear in the “disaster story” would remind us of the fact that the very existence of ordinary matter depends on the mass difference between the proton and neutron which depends on that between the up and down quarks (which further depends on that between the subquarks, w1 and w2 ).

3.

STRANGE QUARK MATTER

AS

DARK MATTER

In concluding this paper, I would like to emphasize that, although there had not yet appeared a clear indication of exotic nuclei such as super- hypernuclei found in high-energy experiments, there are so many candidates for strange quark matter and strange quark stars reported by the cosmic-ray experimentalists and by the astrophysicists. Also, I must add that not only the recent discovery of “triaxially deformed nuclei” [26] but also that of “superheavy hydrogen” [27], 5 H and 7 H, would be something exotic in low-energy nuclear physics. Finally, let me propose simple solutions to the most intriguing problem in current cosmology, the problem of dark matter: The problem of the dark matter can be simply solved by either one of the following possibilities: 1) It consists of exotic matter such as strange quark matter in the so-called Bodmer-Terazawa-Witten hypothesis that super-hypernuclear matter consisting of almost equal numbers of up, down, and strange quarks is stable [4]. In fact, many observations of the possible candidates for strange stars consisting of strange quark matter have recently been reported by astronomical experiments [28], inducing many extensive theoretical investigations on strange stars [29]; 2) It consists of “color-balls”, the color-singlet complex objects consisting of an arbitrary number of gluons [5]; 3) It consists of axions [30]; 4) It consists of WIMPSs (Weakly Interacting Massive Particles) such as superpartners of the SM (Standard Model) neutrinos, γ, and Z in supersymmetry [31]; and 5) It consists of primordial black holes [32]. For the last two decades, there have been many extensive experimental searches for axions and WIMPs, how-

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ever, all in vain. It seems to me that the exotic possibilities such as 1) strange quark matter, 2) color balls, or 5) black holes would be more likely than the intriguing possibilities such as 3) axions or 4) WIMPs. Future experiments would tell us which one of these possibilities is a right answer to one of the most mysterious questions in current cosmology.

ACKNOWLEDGMENTS The author would like to thank Mrs. Nadya Columbus and Ms. Stella Mottola for inviting the author to participate on their publishing program of their hardcover edited collection tentatively entitled: Understanding Quarks, and to thank Professor Masaki Yasu`e for correcting the original manuscript. He also thanks the late Professor Masatoshi Koshiba, the founder of neutrino astronomy and the life-long tutor of the author, to whom this Chapter is dedicated, for many useful helps and valuable advices which the author has received since 1963.

R EFERENCES [1] For a classical review, see Terazawa H., in Proc. 22nd International Conf. on High Energy Physics, Leipzig, 1984, edited by A. Meyer and Wieczorek E. (Akademie der Wissenschaften der DDR, Zeuthen, 1984), Vol.I, p.63. For a more recent review, see Terazawa H., in Proc. International Conf. “New Trends in High-Energy Physics”, Alushta, Crimea, 2003, edited by Bogolyubov P.N., Jenkovszky L. L., and Magas V. K. (Bogolyubov Institute for Theoretical Pysics, Kiev, 2003), Ukrainian J. Phys. 48, 1292 (2003). For a even more recent review, see Terazawa H., in Proc. XXII International Conf. on New Trends in High Energy Physics, Alushta (Crimea), 2011, edited by Bogolyubov P. N. and Jenkovszky L. L. (Bogolyubov Institute for Theoretical Physics, Kiev, 2011), p.352, arXiv:1109.3705v5[physics.gen-ph], 10 Feb 2012. For our latest papers on unified subquark models of all fundamental particles and forces, see Terazawa H. and Yasue M., J. Mod. Phys. 5, 205(2014), arXiv:1401.3562v8 [hep-ph], 5 Apr 2014; Nonlinear Phenomena in Complex Systems 19, 1 (2016), arxiv:1505.00172v2[hep-ph] 9 Sep 2015; and Terazawa H., ibid. 20, 327 (2017). For the latest review, see Terazawa H., “Quark Matter: From Subquarks to the Universe” (Nova Science Publishers, New York,

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Hidezumi Terazawa 2018), https://novapublishers.com/shop/quark-matter- from-subquarks-tothe-universe/.

[2] For a brief review, see, for example, Giles J., Nature Science Update, 23 March 2004, http://www.nature.com/nsu/040322/040322-5.html. See also Minot E.D. et al., Phys. Rev. Lett. 90, 156401 (2003), arXiv:condmat/0211152v3[cond-mat.mes-hall] 4 Feb 2003; Nature 428, 536 (2003), arXiv:cond-mat/0402425v2[cond-mat.mes-hall] 5 Mar 2004. [3] For the first experimental evidences, see Nakano T. et al. (LEPS Collaboration), Phys. Rev. Lett. 91, 0120002(2003), and Barmin V. V. et al (DIANA Collaboration), Phys. Atom. Nucl. 66, 1715(2003); Yas. fiz. 66, 1763(2003). [4] For the prediction of a super-hypernucleus, Hλ (hexalambda), with B = 6 and S = −6 at m ∼ = 5.6 ∼ 6.3GeV in the MIT bag model and at m ∼ = 7.0GeV in the quark-shell model, see Terazawa H., INSReport-336 (INS, Univ. of Tokyo) May, 1979; J. Phys. Soc. Jpn. 58, 3555(1989); 58, 4388(1989); 59, 1199(1990). For a classic review on super-hypernuclei, see Terazawa H., in Proc. 2nd Conf. on Nuclear and Particle Physics, Cairo, 1999, edited by Comsan N. M. H. and Hanna K. M. (Nuclear Research Center, Atomic Energy Authority, Cairo, 2000), p.28. For a more recent review, see Terazawa H., in Proc. International Conf. on New Trends in High-Energy Physics, Yalta, Crimea (Ukraine), 2005, edited by Bogolyubov P. N., Fedosenko P. O., Jenkovszky L. L., and Karpenko Yu. A. (Bogolyubov Institute for Theoretical Physics, Kiev, 2005), p.259, arXiv:1304.5655v2 [physics.gen-ph] 15 Apr 2015. For the latest review, see Terazawa H., “Quark Matter: From Subquarks to the Universe” (Nova Science Publishers, New York, 2018), https://novapublishers.com/shop/from-subquarks- to-the-universe/. [5] For reviews, see Terazawa H., in Proc. International Conf. (VIIIth ‘Blois Workshop’) on Elastic and Diffractive Scattering, Protvino, 1999, edited by Petrov V. A. and Prokdin A. (World Scientific, Singapore, 2000), p.153; in Proc. IX Annual Seminar “Nonlinear Phenomena in Complex Systems”, Minsk, 2000, edited by Kuvshinov V. I. et al. (Institute of Physics, National Academy of Sciences of Belarus, Minsk, 2000), Nonlinear Phenomena Complex Systems 3:3, 293(2000). Recently, Hooper et al. have pointed out that the observed 511keV emission from the galactic bulge

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by INTEGRAL (International Gamma-Ray Astrophysics Laboratory), and previously by Compton Gamma-Ray Observatory, could be due to very light (1-100MeV) annihilating dark matter particles. It can be taken as the first observation of very light color-balls. See Hooper D. et al., Phys. Rev. Lett. 93, 161302(2004). For related works, see Boehm C. et al., Phys. Rev. Lett. 92, 101301 (2004); Picciotto C. and Pospelov M., Phys. Lett. B605, 15(2005); Hooper D. and Wang L. T., Phys. Rev. D 70, 063506(2004); Casse M. et al., astro-car/0404490. See also Oaknin D. H. and Zhitnitsky A. R., Phys. Rev. Lett. 94, 101301(2005), in which they discuss the possibility that it can be naturally explained by the supermassive very dense droplets (strangelets) of dark matter. [6] Chin S. A. and Kerman A. K., Phys. Rev. Lett. 43, 1292(1979). [7] Witten E., Phys. Rev. D 30, 272(1984). [8] Farhi E. and Jaffe R. L., Phys. Rev. D 30, 2379(1984); 32, 2452 (1985). [9] Chodos A., Jaffe R. L., Johnson K., Thorn C. B., and Weisskopf V. F., Phys. Rev. D 9, 3471(1974); Chodos A., Jaffe R. L., Johnson K., and Thorn C. B., ibid.10, 2599(1974); DeGrand T., Jaffe R. L., Johnson K., and Kiskis J., ibid.12, 2060(1975); Jaffe R. L., ibid. 15, 267, 281(1977). For the prediction of a stable “di-lambda ”, H, in the MIT bag model, see Jaffe R. L., Phys. Rev. Lett. 38, 195, 612(E)(1977). [10] Saito T., Hatano Y., Fukuda Y., and Oda H., Phys. Rev. Lett. 65, 2094 (1990); Kasuya M., Saito T., and Yasue M., Phys. Rev. D 47, 2153(1993). For reviews, see Mori K. and Saito T., in Proc. 24th International Cosmic Ray Conference, Rome, 1995 (Rome, 1995), Vol. 11, p. 878; Saito T., ibid., Vol. 11, p.898. [11] Terazawa H., J. Phys. Soc. Jpn. 60, 1848(1991). [12] Terazawa H., J. Phys. Soc. Jpn. 62, 1415(1993). [13] Weinberg S., Phys. Rev. D 19, 1277(1979); Susskind L., ibid. 20, 2619 (1979). For the earlier related proposal, see B´eg M. A. B. and A. Sirlin, Ann. Rev. Nucl. Sci. 24, 379(1974). [14] See, for example, Farhi E. and Susskind L., Phys. Rep. 74, 277(1981).

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[15] Terazawa H., Phys. Rev. Lett. 65, 823(1990). For the earlier works on the PCDC sum rule later leading to the QCD sum rule in a generic sense, see also Terazawa H., Phys. Rev. Lett. 32, 694(1974); Phys. Rev. D 9, 1335(1974); 11, 49(1975); 12, 1506(1975). [16] Price P.B. et al., Phys. Rev. Lett. 35, 487(1975); Phys. Rev. D 18, 1382(1978). [17] Ichimura M. et al., Nuovo Cimento A106, 843(1993). [18] Terazawa H., J. Phys. Soc. Jpn. 69, 2825(2000); in Proc. IX Annual Seminar “Nonlinear Phenomena in Complex Systems”, Minsk, 2000, edited by Babichev L. and Kuvshinov V. (Institute of Physics, National Academy of Sciences of Belarus, Minsk, 2000), Nonlin. Phenom. Complex Syst. 9, 280(2000). [19] ’t Hooft G., in Recent Developments in Gauge Theories, edited by ’t Hooft G. (Plenum, New York, 1980), p.135; Terazawa H., Prog. Theor. Phys. 64, 1763(1980). [20] Itoh N., Prog.Theor.Phys. 44, 291(1970); Bodmer A. R., Phys. Rev. D 4, 1601(1971). For a classical review, see Weber F., Schaab Ch., Weigel M. K., and Glendenning N. K., Report No. LBL-37264, UC-413 (LBL, Berkeley, 1995), in Proc. Ringer Workshop, Tegernsee, Germany, 1995, presented at Conference: C95-03-06. See also Glendenning N. K., Compact Stars, Nuclear Physics, Particle Physics, and General Relativity, 2nd ed. (Springer-Verlag, New York, 2000). For a recent review, see Weber F., arXiv:astro-ph/0407155v2 27 Sep 2004, published in Prog. Part. Nucl. Phys. 54, 193(2005). For a more recent review, see Weber F. et al., arXiv: 1210.1910[astro-ph.SR] 6 Oct 2012, published in the Proceedings of the IAU Symposium 291, edited by van Leeuwen J. (International Astronomical Union, 2013), p.61. For recent papers on strange quark matter in compact stars, see, Blaschke D. and Chamel N., Astrophy. Space Sci. Libr. 457, 337(2017), arXiv:1803.01836v2 [nucl-th] 6 Jul 2018, and many references therein. For some latest papers on theories of strange quark matter and strange quark stars, see, for example, Xia Cheng-Jun, APS Conf. Proc. 2127(2019)1, 020029, arXiv:1904.08347v1[nucl-th] 17 Apr 2019, and many references therein.

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[21] Chen K. S., Dai Z. G., Wei D. M., and Lu T., Science 280, 407(1998). See also Chen K. S. and Dai Z. D., Phys. Rev. Lett. 77, 1210(1996). [22] Li X.-D., Dai Z.-G., and Wang Z.-R., Astron. Astrophys. 303, L1(1995); Bombaci I., Phys. Rev. C 55, 1587(1997). See also Dey M., Bombaci I., Dey J., Rey S., and Samanta B. C., Phys. Lett. B438, 123(1998). [23] Li X.-D., Bombaci I., Dey M., Dey J., and van den Heuvel E. P. J., Phys. Rev. Lett. 83, 3776(1999). For a recent review on quark matter in compact stars, see Bombaci I., in Proc. 11th Marcel Grossmann Meeting on General Relativity, Berlin, 2006, edited by Kleinert H. and Jantzen R. T., (World Scientific, 2008), p.605, arXiv:0809.4228v1[gr-qc] 24 Sep 2008. [24] For a review, see http://www1.msfc.nasa.gov/NEWSROOM/news/releases /2002/ 02-082.html, April 10, 2002 and Seife C., Science 296, 238(2002). For RXJ185635-3754, see Pons J.A. et al., Astrophys. J. 564, 981(2002); Drake J. J. et al., ibid. 572, 996 (2002); Walter F. M. and Lattimer J., ibid. 576, L145(2002). For 3C58, see Slane P. et al., Astrophys. J. 571, L45(2002); Yakovlev D. G. et al., Astron. Astrophys. 389, L24(2002). [25] Marburger J., http://www.pubaf.bnl.gov/pr/bnlpr091799.html, September 17, 1999; http://www.bnl.gov/bnlweb/rhicreport.html, October 6, 1999. [26] Odegard S. W. et al., Phys. Rev. Lett. 86, 5866(2001). For a recent theoretical analysis of the triaxially deformed nuclei, see Chen Q. B. et al., arXiv: 2003.04065v1[nucl-th] 9 Mar 2020. [27] Korsheninnikov A. A. et al., Phys. Rev. Lett. 87, 092501 (2001); 90, 082501(2003). For the recent observation of excited states in 5 H and 7 H, see Golovkov M. S. et al., Phys. Rev. Lett. 93, 262501(2004), and Bezbakh A. A. et al., ibid. 124, 022502(2020), respectively. [28] For a recent review, see Weber F. et al., in Ref. [20]. [29] For a recent review, see Biswas S. et al., arXiv:1409.8366v5[nucl-th] 1 Oct 2015. [30] Peccei R. D., Quinn H. R., Phys. Rev. Lett. 38, 1440(1977). For some recent theoretical investigations on axions as the dark matter, see Sadnik Y. V. and Flambaum V. V., Mod. Phys. Lett. A32, 1740004(2017),

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Hidezumi Terazawa arXiv:1506.08364v1[hep-ph] 28 Jun 2015; Roberts B. M. et al., AXIONWIMP 2015, 120, asrXiv:1511.04098v1[physics.atom-ph] 12 Nov 2015; and references therein. For a latest review, see Yang Q., Mod. Phys. Lett. A32, 1740003(2017), arXiv:1509.00673v3[hep-ph] 26 Feb 2017. Recently, however, Frampton has disputed whether axions exist. See Frampton P. H., Mod. Phys. Lett.A31, 1650093(2016), arXiv:1510.00400v7[hepph] 16 Mar 2016. For some latest searches for dark matter axions, see Backes K. M. et al., Nature 590, 238(2021); Rogers K. K. and Peiris H. V., Phys. Rev. Lett. 126, 071302(2021). For a latest study yielding the strongest constrains on dark matter properties, see Nadler E. O. et al. (DES Collaboration), Phys. Rev. Lett. 126, 091101(2021).

[31] Miyazawa H., Prog. Theor. Phys. 36, 1266(1966); Gol’fand Yu. A. and Likhtman L. P., ZhETF Pis. Red. 13, 452(1971)[JETP Lett.13, 323(1971)]; Volkov D. V. and Aklov V. P., ibid.16, 621(1972) [ibid16, 438(1972)]; Phys. Lett. 46B, 109(1973); Wess J. and Zumino B., Nucl. Phys. B70, 39(1974). Recently, Frampton has queried arguments for WIMPs which arise from electroweak supersymmetry. See Frampton P. H., in Ref. [30]. For a latest review on searches for feebly- interacting particles, including WIMPs as well as axions, see Agrawal P. et al., arXiv:2102.12143v1[hepph] 24 Feb 2021. [32] Recently, Frampton has argued that dark matter constituents must uniquely be primordial black holes if they constitute all dark matter. See, for the details, Frampton P. H., in Ref. [30]; Chapline G. F. and Frampton P. F., JCAP 11, 042(2016), arXiv:1608.04297v3[gr-qc] 3 Nov 2016; and Coriano C. and Frampton P. H., arXiv:2012.13821v2[astro-ph.GA] 2 Feb 2021. Very lately, however, Coriano, Frampton, and Kim have suggested that ultralight axions are more suited than primordial black holes to be constituents of dark matter. See Coriano C., Frampton P. H., and Kim J. E., arXiv:2102.11826v1[hep-ph] 23 Feb 2021. For an earliest proposal of primordial black holes as dark matter constituents, see Chapline G. F., Nature 253, 251(1975). See also Terazawa H., Nonlinear Phenomena Complex Systems 18, 25(2015), and references therein.

INDEX A absorption spectra, 52, 57 accelerator, vii, 1, 6, 44, 56, 75, 76, 77, 81, 83, 91 aluminium, 2 amplitude, 21, 37, 103 anisotropy, 51 annealing, 55 annihilation, 20, 59 antiparticle, 120 asymmetry, 50, 94, 118 ATLAS, viii, 94, 95, 107 atomic nucleus, 83, 110 atomism, 126 atoms, 2, 3, 4, 6, 7, 25, 27, 38, 39, 52, 61, 68, 75, 110, 126, 128 azimuthal angle, 12

B backscattering, 54 band gap, 29, 46, 65 baryon, 17, 94, 117, 128 baryons, 5, 17, 80, 81, 116, 129 basic research, 45 beams, 6, 130 Belarus, 125, 132, 134 bending, 38 beryllium, 111

Big Bang, 111, 113 binding energies, 57, 72 binding energy, 6, 11, 29, 53, 72 black hole, 130, 131, 136 blackbody radiation, 45 bonds, 27, 72, 73 boson(s), viii, 3, 4, 5, 17, 19, 25, 93, 94, 95, 96, 97, 98, 99, 104, 105, 114, 120 breakdown, 126 building blocks, 7, 8, 110

C candidates, vii, ix, 126, 128, 130 carbon, 3, 63, 65, 68, 126 carbon atoms, 63 CERN, 108, 122, 123 charm, 15, 16, 112, 113, 116, 117, 118, 119 chemical, 4, 39, 88 chlorine, 3 clarity, 66, 67 classical electrodynamics, 71 cluster model, 81 collaboration, 98 collisions, 95, 111, 126, 127, 129 color, 5, 15, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 55, 78, 112, 113, 126, 127, 128, 129, 130, 131, 133 complexity, 40 composition, 45, 61, 66, 68, 69, 75, 87, 89, 110

138

Index

compounds, 29, 54, 58, 59, 71 computer, 54, 55, 56 conduction, 50, 51, 52, 86 conductor, 46 configuration, 24, 27, 28, 130 confinement, 15, 20, 27, 28, 29, 112 consensus, viii, 109, 111 conservation, 3, 17, 41 constituents, viii, 3, 6, 8, 9, 109, 111, 113, 114, 115, 126, 128, 136 construction, 46, 48 contradiction, 42, 58, 78 correlation, 48, 49, 50, 51, 52, 81 cosmic rays, 127 Coulomb energy, 16 coupling constants, 39, 101, 103 covalent bond, 27, 28, 68 crystalline, 46, 68 crystals, vii, 1, 29, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 71, 74, 76, 81, 82, 83, 84, 85, 86, 87, 88

D dark matter, vii, ix, 94, 126, 129, 130, 133, 135, 136 dark matter particles, 133 data collection, 45 decay, 3, 15, 17, 18, 38, 39, 95, 96, 98, 117, 118, 119, 120 deformation, 53 degenerate, 68 detection, 41, 62, 106 deuteron, 29, 30, 31, 32, 71, 72, 73, 74, 75 deviation, 31, 32, 73 dielectrics, 29 diffraction, 56 diffusion, 45 dipole moments, 13, 14, 34 Dirac equation, 42, 43, 113, 114 direct observation, viii, 94, 95 disaster, 130 discontinuity, 4 disorder, 66, 85, 89 disordered systems, 89

dispersion, 47, 51, 53, 54 displacement, 10, 40, 62 distribution, 16, 32, 33, 34, 45, 46, 50, 59, 71, 73

E electric charge, 2, 3, 9, 16, 41, 71, 94, 113, 116, 117, 120, 130 electric current, 43 electric field, 44, 71 electrical properties, 82 electricity, 6, 38 electromagnetic, vii, 1, 3, 4, 5, 7, 8, 10, 16, 17, 19, 20, 24, 29, 32, 33, 38, 39, 40, 41, 42, 71, 75, 76, 83, 111, 112, 113, 114 electromagnetic fields, 33, 38, 83 electromagnetic origin, 41 electromagnetism, 34, 38, 39, 83, 113, 114, 120 electron state, 50 electronic structure, 29, 56, 71, 75 electron(s), viii, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 17, 22, 25, 27, 29, 35, 36, 37, 39, 41, 42, 43, 44, 45, 46, 50, 51, 52, 53, 54, 56, 57, 58, 59, 63, 64, 71, 72, 74, 75, 78, 84, 90, 101, 109, 110, 111, 116, 117, 118, 120, 122, 126 elementary particle, vii, 1, 3, 4, 5, 6, 15, 38, 39, 77, 83, 94, 96, 106, 110, 111, 112, 113, 114, 115, 118, 119, 121 emission, vii, 1, 20, 59, 60, 62, 64, 74, 85, 88, 132 energetic characteristics, 53 energy, vii, viii, 1, 2, 3, 5, 6, 10, 11, 17, 18, 25, 26, 27, 28, 29, 30, 35, 36, 37, 44, 46, 47, 49, 50, 51, 52, 53, 54, 56, 57, 59, 60, 61, 62, 63, 64, 70, 71, 72, 74, 75, 76, 77, 81, 85, 86, 87, 94, 95, 96, 97, 103, 104, 105, 106, 110, 111, 119, 120, 127, 129, 130 equilibrium, 45 equipment, 8, 54, 55 evidence, 6, 31, 73, 89, 120

Index excitation, 16, 17, 26, 27, 45, 52, 54, 57, 59, 62, 63, 64, 84, 86, 89 exciton, 46, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 81, 85, 88

F fermions, 3, 5, 6, 17, 94, 96, 97, 98, 120 ferromagnetism, 13, 82 Feynman diagrams, 100 field theory, 19, 110 films, 57, 68 flavor, 16, 17, 26, 113, 115, 121 flavour, 94 force, vii, 2, 4, 5, 6, 8, 10, 20, 24, 25, 27, 28, 29, 38, 39, 40, 41, 42, 52, 72, 75, 76, 81, 84, 111, 112, 128, 129 formation, 52 formula, 23, 57 freedom, 29, 46, 74, 94, 119, 127 fullerene, 63, 65 fundamental forces, 5, 15, 38, 40, 41, 76, 113

G GaP, 67 gauge invariant, 97 gauge theory, 79 general relativity, 134, 135 geometry, 41, 54 gluons, viii, 2, 5, 16, 19, 20, 25, 27, 28, 29, 42, 75, 76, 78, 95, 111, 112, 120, 128, 130 graphite, 63 gravitation, vii, 1, 38, 40, 41, 83, 120 gravitational field, 83 gravitational force, 24, 40, 41, 83, 111 gravity, 5, 38 growth, 55, 57

139 H

hadrons, 5, 6, 15, 16, 17, 18, 19, 42, 77, 78, 96, 111, 112, 115, 116, 120, 126, 129 Hamiltonian, 35 heavy particle, 7 helium, 2, 54, 55, 56, 57, 58, 59, 60, 61, 63, 90 Higgs boson, vii, viii, 4, 93, 94, 95, 96, 98, 100, 103, 104, 105, 114, 120, 126 Higgs field, 95, 96, 97, 98 Higgs particle, 95, 120 history, 38, 79, 110, 121 homogeneity, 45 hybridization, 51 hydrogen, vii, 1, 3, 8, 10, 11, 49, 52, 53, 55, 56, 57, 68, 71, 74, 82, 129, 130 hyperfine interaction, 35, 36, 37 hypothesis, vii, viii, ix, 2, 27, 40, 95, 126, 128, 130

I indirect measure, 95 initial state, 117 insulators, 46, 85, 88 integration, 101 inversion, 30, 50 ionization, 57, 58 ionization potentials, 58 ions, 10, 46, 47, 52, 56 irradiation, 45 isolation, 15 isospin, 15, 16, 25, 28, 113, 119, 126 isotope, viii, 2, 4, 6, 45, 52, 57, 58, 62, 63, 65, 68, 69, 71, 74, 87, 88, 89

K krypton, 10, 11

140

Index L

Large Hadron Collider, viii, 94 lens, 55 lepton, vii, viii, 1, 2, 17, 75, 117, 128 Lie algebra, 96 lifetime, 17, 120 light, 27, 38, 43, 45, 51, 52, 53, 54, 56, 57, 61, 62, 68, 69, 81, 83, 86, 96, 103, 133 light beam, 38 light scattering, 62, 68 lithium, 46, 49, 51, 55, 56, 84, 85, 88 low temperatures, 57, 59 luminescence, vii, 1, 45, 46, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 74, 76, 84 luminosity, 95, 105

M magnetic field, 6, 10, 34, 35, 37, 41, 43, 44, 53, 62, 71, 75 magnetic moment, 4, 8, 12, 16, 31, 33, 35, 39, 43, 71, 72, 73, 74, 75, 76 magnetism, 13, 35, 38, 75 magnetization, 36, 82 magnitude, 3, 7, 10, 12, 14, 17, 38, 51, 52, 71, 76, 99, 120 mass, vii, viii, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 28, 31, 33, 35, 41, 42, 43, 44, 46, 51, 53, 56, 61, 65, 66, 67, 68, 70, 71, 72, 74, 79, 90, 91, 94, 95, 96, 97, 98, 99, 103, 104, 105, 106, 112, 113, 115, 119, 120, 127, 128, 129, 130 massive particles, 10, 95, 98 matter, vii, viii, ix, 3, 5, 6, 11, 13, 15, 38, 45, 62, 75, 94, 96, 109, 110, 111, 112, 113, 115, 125, 126, 127, 128, 129, 130, 132 measurement(s), viii, 6, 10, 11, 33, 45, 50, 53, 55, 56, 62, 76, 88, 91, 93, 95, 98, 99, 105, 106, 113, 115, 120 mesons, viii, 5, 17, 24, 25, 26, 75, 109, 112, 116 metals, 82 mixing, 104, 117, 121

models, 53, 74, 131 modifications, 67, 68 molecular dynamics, 88 molecules, 28, 38, 49, 57, 126 momentum, viii, 2, 4, 11, 12, 13, 17, 28, 30, 31, 33, 35, 37, 41, 42, 43, 72, 73, 74, 76, 101, 122 muons, 5, 71, 111

N neutral, 3, 7, 8, 9, 27, 29, 39, 52, 97, 103, 112, 121, 129 neutrinos, 5, 10, 17, 71, 96, 111, 130 neutron stars, 127 neutrons, viii, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 35, 63, 70, 71, 72, 74, 78, 109, 110, 111, 112, 119 nitrogen, 3, 7, 8, 63, 89 nonequilibrium, 45 nonlocality, 77 nuclear charge, 71, 74 nuclear matter, 6, 80, 91 nuclei, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 27, 28, 29, 32, 33, 46, 70, 71, 74, 80, 91, 111, 126, 127, 128, 129, 130, 135 nucleons, viii, 2, 5, 6, 7, 8, 13, 15, 17, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 38, 42, 71, 72, 74, 75, 76, 77, 117, 126, 128, 129 nucleus, vii, viii, 1, 2, 3, 5, 6, 8, 9, 10, 11, 24, 25, 27, 28, 29, 31, 32, 35, 36, 37, 39, 62, 71, 74, 75, 76, 77, 110, 126, 127, 128, 129 nuclides, 72

O one dimension, 63 optical properties, 56 orbit, 3, 31, 32, 35 oscillators, 65 overlap, 25, 26, 27, 28, 49 oxygen, 8

Index P parallel, 26, 73, 110 parity, 17, 18, 29, 30, 31, 72, 94 particle collisions, 119 particle mass, 98 particle physics, viii, 3, 110, 115, 118 phase transitions, 84 phonons, 56, 59, 60, 62, 64, 65, 67, 69 photoluminescence, 45, 59, 60 photons, 2, 3, 5, 17, 18, 19, 59, 61, 95, 98, 111, 112 physical phenomena, 4, 45 physical structure, 39 physics, vii, viii, 1, 2, 3, 4, 5, 8, 10, 13, 17, 27, 29, 38, 39, 40, 42, 45, 53, 70, 71, 72, 75, 76, 81, 85, 90, 93, 94, 95, 96, 110, 111, 112, 113, 114, 115, 120, 123, 125, 126, 130, 131, 132, 136 pions, 18, 25, 76, 96 Planck constant, 5 plane waves, 48 planets, 3 polar, 12, 59, 86 polarizability, 49 polarization, 50 polonium, 111 positron(s), 5, 22, 71, 101, 120 present value, 14 principles, 112, 126 probability, 27, 28, 32, 37, 73, 95 probe, viii, 11, 33, 71, 94, 95, 99 propagation, 40, 41, 42 propagators, 103 protons, viii, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 35, 38, 63, 70, 71, 72, 74, 109, 110, 111, 112, 119 purity, 55

141

quantum chromodynamics (QCD), vii, 1, 5, 19, 20, 21, 25, 76, 78, 90, 91, 112, 126, 127, 134 quantum dot, 85 quantum electrodynamics (QED), 14, 19, 20, 22, 76 quantum field theory, 40 quantum mechanics, 12, 13, 33, 34, 72, 84, 95, 113, 114, 122 quantum theory, 38, 56, 83 quark matter, vii, ix, 126, 130, 131, 134, 135 quarks, vii, viii, 2, 3, 5, 6, 7, 9, 11, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 42, 70, 74, 75, 76, 78, 79, 81, 93, 95, 98, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 126, 129, 130, 132 quartz, 54

R radiation, 2, 3, 5, 6, 7, 8, 45, 111 radioactive isotopes, 67 radius, 12, 26, 29, 32, 39, 42, 44, 46, 48, 52, 71 Raman light scattering, 68, 69 Raman spectra, 62, 70, 89 Raman spectroscopy, 56, 67 reactions, 72, 76 recombination, 45, 64, 67 refractive index, 54 relativity, 40, 110, 122 renormalization, 52, 65, 71, 81 resolution, 50, 61, 62 response, 74, 88 ring model, 41, 42 room temperature, 57 rules, 33, 120

Q S quanta, 20, 64 quantization, 12, 14, 113, 115

scalar field, 98 scattering, 2, 16, 22, 27, 52, 54, 56, 62, 63, 67, 69, 88, 89

142

Index

Schrödinger equation, 47, 48, 74 semiconductor(s), 47, 54, 58, 67, 86, 88 single crystals, vii, 1, 55, 84 solid solutions, 57 solid state, 52 special relativity, 113, 114 species, 4, 19 spectroscopy, vii, 1, 16, 62, 82, 85 speed of light, 5, 40, 43 spin, 3, 4, 6, 8, 11, 13, 17, 18, 25, 26, 28, 30, 33, 35, 36, 39, 40, 43, 46, 63, 71, 72, 73, 74, 80, 82, 94, 113, 120, 126 standard deviation, 95 standard model, vii, viii, 1, 3, 79, 80, 93, 94, 109, 110, 114, 115, 130 stars, vii, ix, 38, 126, 129, 130, 134, 135 strange star, 129, 130 strong force, 5, 10, 20, 29, 33, 38, 42, 60, 71, 73, 75, 76, 77, 112, 119 strong interaction, vii, 1, 15, 16, 19, 20, 25, 27, 39, 41, 42, 44, 70, 75, 76, 77, 78, 79, 81, 83, 84, 90, 91, 95, 112, 117, 120, 122 structure, 4, 6, 8, 16, 18, 21, 25, 27, 29, 31, 32, 35, 37, 44, 45, 46, 47, 48, 49, 51, 52, 55, 56, 57, 61, 62, 63, 67, 68, 71, 72, 74, 75, 76, 80, 81, 84, 85, 88, 91, 110 substitution, 52, 56, 57, 58, 65, 66, 67, 71, 97 superconductivity, 46 superfluid, 55, 56, 58, 59, 60, 61 supersymmetry, 130, 136 suppression, 121 symmetry, 18, 19, 21, 49, 50, 52, 80, 94, 96, 97, 98, 113, 114, 121, 122

temperature, vii, 1, 45, 46, 57, 61, 65, 82, 111 temperature dependence, 65 thermal quenching, 57 thermodynamics, 4 top quark, viii, 95, 96, 98, 99, 103, 104, 109, 119, 120 total energy, 26 transformation(s), 29, 74, 86, 113, 114 translation, 43, 46, 47, 51, 52 transparency, 68

U unification, 38, 39, 40, 121 unique features, 96 universe, 15, 110, 112, 113, 125, 132

V vacuum, 40, 97, 106 valence, 15, 16, 48, 49, 50, 51, 52, 74, 85, 111 vector, 12, 14, 17, 19, 33, 34, 40, 47, 96, 104, 114 velocity, 6, 10, 42, 43, 44

W water, 110 wave vector, 49 wavelengths, 57 weak interaction, vii, 1, 117, 120, 121

T techniques, 45, 49, 62 technology, 76, 85, 123

Z zinc, 8, 54, 67