Many Body Structure of Strongly Interacting Systems: Refereed and Selected Contributions from the Symposium "20 Years of Physics at the Mainz Microtron MAMI" 3540367535, 9783540367536

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Hartmuth Arenh¨ ovel [email protected] Hartmut Backe [email protected] Dieter Drechsel [email protected] J¨org Friedrich [email protected] Karl-Heinz Kaiser [email protected] Thomas Walcher [email protected] University of Mainz Institute for Nuclear Physics Johann-Joachim-Becher-Weg 45 55128 Mainz, Germany The articles in this book originally appeared on the internet (www.eurphysj.org) as open access publication of the journal The European Physical Journal A – Hadrons and Nuclei Volume 28, Supplement 1 ISSN 1434-601X c SIF and Springer-Verlag Berlin Heidelberg 2006 

ISBN-10 3-540-36753-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-36753-6 Springer Berlin Heidelberg New York Library of Congress Control Number: 2006929544 This work is subject to copyright. All rights reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SIF and Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c SIF and Springer-Verlag Berlin Heidelberg 2006  Printed in Italy The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting and Cover design: SIF Production Office, Bologna, Italy Printing and Binding: Tipografia Compositori, Bologna, Italy Printed on acid-free paper

SPIN: 11804239 – 5 4 3 2 1 0

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Deutsche Forschungsgemeinschaft, Bonn Institut f¨ ur Kernphysik, Universit¨at Mainz

The European Physical Journal, www.eurphysj.org ACCEL Instruments GmbH, Bergisch Gladbach BRUKER, Wissembourg, France DANFYSIK A/S, Jyllinge, Denmark SIGMAPHI, Vannes, France SFAR STEEL, Creusot, France THALES Electron Devices, V´elizy, France

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The European Physical Journal A Volume 28

 Foreword

 Many

Body Structure of Strongly Interacting Systems

1 R.G. Milner The beauty of the electromagnetic probe 7 L.S. Cardman Physics at the Thomas Jefferson National Accelerator Facility 19 W.U. Boeglin Few-nucleon systems at MAMI and beyond 29 D. Rohe A1 and A3 Collaboration

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Supplement 1

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2006

71 M. Vanderhaeghen Two-photon physics

81 M. Ostrick Electromagnetic form factors of the nucleon Experiments at MAMI

91 H. Schmieden Photo- and electro-excitation of the Δ-resonance at MAMI

101 S. Kowalski Parity violation in electron scattering

107 F.E. Maas Parity-violating electron scattering at the MAMI facility in Mainz The strangeness contribution to the form factors of the nucleon

Experiments with polarized 3 He at MAMI 39 M. Schwamb Few-nucleon systems (theory)

117 N. d’Hose Virtual Compton Scattering at MAMI

49 H.-W. Hammer Nucleon form factors in dispersion theory

129 H. Merkel Experimental tests of Chiral Perturbation Theory

59 S. Scherer Chiral perturbation theory

139 W. Hillert The Bonn Electron Stretcher Accelerator ELSA: Past and future

Success and challenge

VI 149 A. Jankowiak The Mainz Microtron MAMI —Past and future

197 M. El-Ghazaly et al. X-ray phase contrast imaging at MAMI

161 A. Thomas The Gerasimov-Drell-Hearn sum rule at MAMI

209 B.A. Mecking Twenty years of physics at MAMI —What did it mean?

173 R. Beck Experiments with photons at MAMI

185 W. Lauth et al. Coherent X-rays at MAMI

 Author

index

Eur. Phys. J. A 28, s01, VII VIII (2006) DOI: 10.1140/epja/i2006-09-022-5

EPJ A direct electronic only

Foreword

This volume contains the proceedings of the Symposium on Twenty Years of Physics at the Mainz Microtron (MAMI), which was held at the Johannes Gutenberg-Universit˜ at Mainz, October 19-22, 2005. The Symposium marks the retirement of several members of the Institut f˜ ur Kernphysik whose work has been devoted primaryly too scientiflc research at MAMI over many years. It was the primary aim of the Symposium to review past and current activities in the fleld of hadronic structure investigations with the electroweak interaction. However, the Symposium also gave an outlook on the physics with the MAMI upgrade, a double-sided mictrotron that is expected to provide a high-quality beam of up to 1.5 GeV later this year. The Institut f˜ ur Kernphysik was founded in the early 1960s by the late Hans Ehrenberg who served as its director for more than two decades. He provided the Institute with a 350 MeV pulsed linear electron accelerator, which became available in 1966 for studies of charge and magnetization distributions in nuclei and nucleons as well as photonuclear investigations in collaboration with the Max-Planck-Institut f˜ ur Chemie. Hans Ehrenberg knew about the importance of having excellent facilities for performing outstanding physics from his earlier studies at Bonn and Stanford, with the later Noble Prize winners Paul and Hofstadter, respectively. Therefore, he dedicated great efiort in I) building up a perfect infrastructure of mechanics, electronics, vacuum and computer workshops, and II) attracting a young accelerator physicist, Helmut Herminghaus, to the Institute. In the late 1960s it became common wisdom that the next accelerator generation had to provide a high dutyfactor in order to perform coincidence experiments for detailed studies of hadronic physics. Helmut Herminghaus had conceived a blueprint for such a device in 1975, a three-stage racetrack microtron (RTM). Shortly after a physics program around this RTM was worked out and the proposal was sent to the sponsoring agencies. The project received the support of the University and the State of Rheinland-Pfalz and sometime later also of the federal agencies. In the fall of 1978, the state minister was informed by the federal minister of research and technology (BMFT) that the project had been discussed with the German Science Council, the Deutsche Forschungsgemeinschaft (DFG), the MaxPlanck-Gesellschaft, and members of the scientiflc community. As a result these representatives agreed to support the proposal in order to I) demonstrate that also a large-scale research facility can be realized at a university, II) withstand a further emigration of such research from the universities, and III) flnd a constructive solution that could serve as a model for university research. As a matter of fact such a solution was found in the following years. However, it has to be said that the full flnancial support would never have arrived if the RTM had not been designed stage by stage, and each time delivered in perfect shape (often to the surprise of outside experts) by Helmut Herminghaus and his crew of physicists and technicians. The flrst stage of the RTM (14 MeV) went into operation already in May 1979, the second stage (183 MeV) followed in 1983, and the last stage was ready for the experiments in the fall of 1990. At present the microtron delivers a continuous beam of an intensity of about 100 μA for unpolarized and 40 μA for polarized electrons with a polarization degree of about 80 %. Its energy close to 1 GeV provides the perfect resolution to study the distributions of charge, magnetization, and strangeness inside the nucleon and light nuclei, the threshold production of the Goldstone bosons pion and eta, the polarizabilities of nucleons and pions, and the excitation of the most prominent nucleon resonance, the Δ(1232). Since the physics with the flrst two stages of the RTM was summarized already at an earlier workshop ( Physics with MAMI A ), the present Symposium concentrates on the achievements of the years with the 855 MeV stage (MAMI B). The organizers also decided to invite as speakers, with a few exceptions, young colleagues who have made a career with their work at MAMI. It remains to say thank you to many people and institutions for continuing support. We are grateful to all the colleagues from the Institute, the postdocs, Ph.D. and younger students who contributed to the MAMI project.

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Special thanks go to the people in the workshops and in the administration without whose efiorts the project could never have succeeded. We are grateful to the colleagues from the neighbouring Institut f˜ ur Physik for their work on polarized beams and targets, for the TAPS detector brought to Mainz by the Gie en group, to the Bonn/Bochum group for the polarized H2 -target, and to many other German institutions for active engagement and various detection devices, notably Darmstadt, Erlangen, G˜ ottingen, and T˜ ubingen. Our thanks go to the foreign colleagues who have participated in the project from the very beginning, notably to our Scottish colleagues who built the photon tagger with the support of their SERC, the groups from Pavia sponsored by the INFN, from Saclay supported by the CEA/DAPNIA and from Orsay supported by the CNRS. We appreciate common experimental and theoretical work with physicists from various other places in Europe, e.g. Amsterdam (NIKHEF), Basel, Genova, Gent, Lljubljana, Trento and several Russian universities and institutions, and from overseas, e.g., Jefierson Lab, MIT, Florida State University, University of Nagoya, George Washington University, and TRIUMF. Finally, in view of the upgrade two more collaborations have developed in recent years. The Crystal Ball Collaboration has shipped its detector from the Brookhaven National Lab to Mainz, and the KAOS detector is being installed in Mainz with the help of the GSI Darmstadt. Last but not least we are grateful to the members of the international Program Advisory Committee and of numerous evaluation and expert committees for their invaluable scientiflc advice and moral uphold. Concerning the institutions we flrst and foremost thank our Physics Faculty, the Johannes Gutenberg-Universit˜ at and the State of Rhineland-Palatinate for continued and coherent support. We are extremely grateful to the state and to the federal ministries (BMFT, BMBW, BMBF) who flnanced the construction of the new accelerator and experimental halls as well as the large spectrometers via the university construction program (HBFG). Our special thanks go to the Deutsche Forschungsgemeinschaft that backed up the project by means of Collaborative Research Centers (SFB 201, CRC 443) whose resources were of the utmost importance to sustain our postdoc and PhD program. Finally, we received recent support by the European networking activities via the I3HP/Transnational Access program. Last but not least the organizers are grateful to the speakers of this Symposium for summarizing the various achievements with MAMI and related research, and for bringing back memories of the past. Though retirees enjoy the latter aspects very much, there is no reason to engage in retrospection: The double-sided microtron is expected to yield its 1.5 GeV electron beam later this year, and we wish our colleagues and their students all the success in the years to come!

Mainz, April 1, 2006

Hartmuth Arenh˜ ovel Hartmut Backe Dieter Drechsel J˜org Friedrich Karl-Heinz Kaiser Thomas Walcher The Editors

Eur. Phys. J. A 28, s01, 1 5 (2006) DOI: 10.1140/epja/i2006-09-001-x

EPJ A direct electronic only

The beauty of the electromagnetic probe R.G. Milnera MIT-Bates Linear Accelerator Center, Laboratory for Nuclear Science, Massachusetts Institute for Technology, Cambridge, MA 02139, USA / Published online: 15 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Precision experiments using the electromagnetic probe have recently produced important new data on fundamental properties of the nucleon, e.g. charge, magnetism, shape, polarizability, spin and sea quark structure. These experiments have been made possible by a new generation of high duty factor electron accelerators, advances in spin polarization technology (beams, targets and recoil polarimeters), and the development of unique, optimized detector systems. In this contribution, the role of multiple photon exchange in electron scattering from the proton and the role of sea quarks in nucleon structure are highlighted. PACS. 13.40.Gp Electromagnetic form factors 13.60.-r Photon and charged-lepton interactions with hadrons 13.60.Fz Elastic and Compton scattering 14.20.Dh Protons and neutrons

1 Introduction Understanding the structure of the nucleon in terms of the fundamental constituents of the Standard Model, the quarks and gluons of Quantum Chromodynamics (QCD), is a major research area in Physics. The ultimate goal is to test QCD with precision measurements and ab initio calculations. Over the last decade, experimentalists have made substantial progress in determination of the quark and gluon distributions at high energies (ECM ∼ 100 GeV) and measurement of fundamental properties of the nucleon at low energies (ECM ∼ 1 GeV). Theorists are starting to produce full QCD Monte Carlo simulations (albeit with heavy pion masses) of nucleon structure using advanced computers [1]. The experimental study of the structure of the proton and of atomic nuclei is best carried out using the pointlike electroweak probe, the best understood interaction in Nature. Intense beams of highly polarized electrons have become available at energies of 0.5 to 6 GeV at high duty factor. Highly polarized proton, deuteron and 3 He targets have been developed as well as e– cient polarimeters for detection of recoil polarization. Optimized experiments utilizing uniquely designed detectors have been carried out. New data and insights have been obtained in measurement of the following properties of the nucleon: – The proton and neutron charge and magnetism through spin-dependent elastic electron scattering at Mainz [2], Bates [3], NIKHEF [4] and JLab [5]. Precise measurements of all four of the nucleon elastic a

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form-factors have been carried out. In particular, the relatively small neutron electric form-factor has been determined to better than 7% over the range 0.1 < Q2 < 2 (GeV/c)2 . – The shape of the proton through study of electroexcitation of the π 0 at the Δ(1232)-resonance at low Q2 ∼ 0.1 (GeV/c)2 using out-of-plane detection at Bates and Mainz [6]. It has been established that the proton shape is slightly non-spherical. A chiral extrapolation [7] of lattice QCD calculations [8] is in good agreement with the data. – The electric and magnetic polarizabilities of the proton through measurement of Virtual Compton Scattering from the proton at Mainz [9] and JLab [10] and using out-of-plane detection at Bates [11]. – The quark and gluon contributions to the spin structure of the proton using deep inelastic scattering at HERMES/DESY [12], JLab [13], COMPASS/CERN [14] and RHIC-spin [15]. – The role of strange quarks in the long distance magnetic and electric charge distribution of the proton at Bates, Mainz and JLab [16,17]. There are hints of a non-zero strange quark magnetic moment of the proton but these need to be conflrmed by more precise experiments. Here I concentrate on two areas of research where important results have recently been obtained.

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GpE /GpM

Fig. 1. The Jefierson Lab data [18] on the ratio showing the discrepancy between the recoil polarization (solid circles) and the Rosenbluth (other symbols) techniques.

2 Evidence for multiple photon effects in elastic electron scattering from the proton Essentially all electron scattering experiments to study proton and nuclear structure to date have been analyzed in terms of single photon exchange. The flne structure coupling constant α ∼ 1/137 is small enough that leading order has been adequate. There are a few speciflc examples where multiple photon exchange is known to be significant, e.g. in comparison of electron and positron scattering in kinematics where the single photon exchange cross-section is small, or in radiative processes. Thus, it came as a surprise when the Jefierson Lab Hall A recoil polarization measurements of electron-proton elastic scattering at momentum transfers of about 2 (GeV/c)2 [18] showed a substantial deviation from the data obtained over several decades with the Rosenbluth technique [19], which is based on precise cross-section measurements. This discrepancy has been interpreted as the efiect of multiple photon exchange in the elastic electron-proton crosssection [20]. The cross section for elastic electron-proton scattering in the one-photon exchange approximation can be written in terms of the pointlike Mott cross-section, the Sachs form factors GpE and GpM and the electron scattering angle θ as     p2 Gp2 + τ G dσ dσ θ p2 2 E M = + 2τ GM tan , · dΩ dΩ M ott 1+τ 2 where τ = Q2 /4M 2 . Figure 1 shows the recoil polarization determination of GpE /GpM (solid circles) as a function of momentum transfer Q2 . The Rosenbluth data (all other data points) are believed to be uncorrected for the efiects

Fig. 2. The quark and gluon momentum distributions at Q2 = 10 (GeV/c)2 as a function of parton momentum x as determined by the ZEUS experiment [23] at the HERA electronproton collider. Note that the sea quark momentum xS and the gluon momentum xg distributions are divided by a factor of 20.

of multiple photon exchange and so give an incorrect determination at higher Q2 , i.e. above about 1 (GeV/c)2 . This multiple photon exchange contribution to elastic electron-proton scattering can be conflrmed by precise comparison of electron-proton with positron proton elastic scattering or by measurement of the asymmetry Ay in scattering of unpolarized electrons from a vertically polarized proton target [21]. If conflrmed, this is a very signiflcant result.

3 Role of sea quarks in nucleon structure QCD tells us that the nucleon comprises three valence quarks and a sea of quark-antiquark pairs. From the earliest days of nuclear physics, these sea quarks in the form of mesons, have been viewed as playing an important role in the long distance structure of the nucleon e.g. the magnitude and sign of the proton and neutron magnetic moments. In addition, the most successful hadronic theoretical descriptions of light nuclei incorporate meson exchange between nucleons as an essential element of nuclear binding. This meson cloud structure to the nucleon has generally been accepted but has lacked both a rigorous theoretical underpinning and a deflnitive quantitative basis from experiment. The role of valence quarks in nucleon structure has been studied extensively. The efiects of sea quarks and gluons are relatively poorly determined, in large part because they require high center-of-mass energy, and are a major focus of interest for the future [22]. One of the important contributions over the last decade has been the experimental measurement of deep inelastic scattering at high energies to determine the efiects of the sea quarks and

R.G. Milner: The beauty of the electromagnetic probe

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Fig. 4. The proton charge elastic form-factor with the smooth contribution subtracted in the parameterization of Friedrich and Walcher [24].

A 2% dip in the parameterization is obvious at Q2 ∼ 0.1 0.2 (GeV/c)2 , which coincides with the location of the peak in the neutron charge elastic form-factor GnE . In the absence of realistic QCD calculations, it is hard to deflnitively state that this structure at low Q2 is due to the meson cloud structure of the nucleon. However, it is a physically plausible explanation.

4 BLAST Experiment at MIT-Bates

Fig. 3. Comparison of the gluon and sea distributions from the ZEUS-S NLO QCD flt for various Q2 values [23] as measured at the HERA electron-proton collider.

gluons. In particular, data taken by experiments at the HERA electron-proton collider [23] have for the flrst time allowed a determination of the gluon momentum distribution in the proton, as shown in flg. 2. The QCD evolution of HERA data [23] shows a signiflcant sea contribution at low Q2 , in contrast to the gluon contribution which vanishes, as seen in flg. 3. This supports the point of view of a strong role for sea quarks at low Q2 . At low energies, electron scattering experiments determine the elastic electric and magnetic form factors of the proton and neutron. Friedrich and Walcher have postulated that the Q2 dependence of the elastic form factors in the region 0.1 to 0.5 (GeV/c)2 may be sensitive to the meson cloud structure of the nucleon and have produced parameterizations of world data which suggest that there may be experimental support for this ansatz [24]. They flt the measured four form factors with a parameterization which consists of a smooth contribution and a bump contribution. Figure 4 shows the world’s data for the proton elastic form factor plotted as a function of momentum transfer Q2 , where the smooth contribution is subtracted.

A new set of precision measurements of the low Q2 elastic form factors of the proton and neutron have been carried out using the South Hall Ring (SHR) at the MIT-Bates Linear Accelerator Center. The Bates Large Acceptance Spectrometer Toroid (BLAST) was constructed [25] to detect scattered electrons, protons, neutrons and pions in the scattering of longitudinally polarized electrons with an energy of 850 MeV from polarized targets of hydrogen and deuterium. The polarized internal gas target technique offers minimal systematic uncertainties and a high statistics sample of data were taken by the BLAST experiment over an eighteen month period from late 2003 to mid 2005. The BLAST data are under analysis and will be able to provide new and independent experimental constraints of the Friedrich-Walcher ansatz. The polarized protons and deuterons (both vector and tensor) were produced using an Atomic Beam Source (ABS) [26], which was located in the substantial and spatially varying magnetic fleld of the BLAST toroid. The target spin state was alternated every flve minutes by switching the flnal RF transition immediately before the target to ensure equal target densities for each of the three states (vector +, vector −, tensor −). The electrons scattered from the polarized protons and deuterons in a cylindrical, windowless aluminum target tube 600 mm long, 15 mm in diameter and with a wall thickness of 50 μm. The polarized target was tuned and monitored using a Breit-Rabi system which continuously sampled the atomic polarization of a small fraction of the incoming beam from the ABS. The vector polarizations of both the proton and deuteron was typically 0.75. Data were taken with stored electron beam intensities up to 225 mA.

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Fig. 5. A schematic layout of the BLAST experiment at MITBates.

tion of a Monte Carlo simulation which uses Arenh˜ ovel’s theory [28] as well as a realistic description of the experiment. At low pm , the scattering is dominated by the Sstate in deuterium and the asymmetry is very close to that for scattering from a free proton. These data can be used to determine the product of beam and target vector polarization. At high pm , the scattering is dominated by the D-state in the deuteron, where both proton and neutron spins are anti-aligned with respect to the nuclear spin. Thus, the scattering asymmetry changes sign. The pm range of the data extend out to 500 MeV/c. BLAST data on the four elastic form factors of the proton and neutron are expected to be published in 2006. In addition, a sizable data set on electron scattering from tensor polarized deuterium was acquired with BLAST.

5 Conclusion

Fig. 6. The vector asymmetry AVed in quasielastic (e, e p) scattering from vector polarized deuterium as a function of missing momentum pm for 0.1 < Q2 < 0.2 (GeV/c)2 , as measured by the BLAST experiment [27].

The polarized electron beam originated from a GaAs polarized electron source and the storage ring was fllled with alternating electron polarizations approximately every half hour. The longitudinal beam polarization at the target was maintained using a Siberian Snake solenoid system. The beam polarization was continuously monitored using a laser Compton backscattering polarimeter, located upstream of the injection point in the SHR. The average beam polarization over the BLAST data taking period was 0.65. The BLAST (see flg. 5) consisted of eight copper coils which provided a 0.4 Tesla toroidal magnetic fleld. For these measurements it was instrumented with symmetric detectors in the horizontal plane: three drift chambers for momentum, angle and position determination; plastic scintillators for triggering and time of flight, and Cerenkov detectors for pion rejection. In addition, large plastic scintillators for neutron detection were arranged on one side. The background rate for scattering from the target cell was measured and found to be negligible. Figure 6 shows a fraction of the BLAST data acquired in quasielastic (e, e p) scattering from vector polarized deuterium [27]. The scattering asymmetry AVed is plotted as a function of the missing momentum (pm ) of the proton in the nucleon. The solid curve is the predic-

The electromagnetic probe provides a beautiful and precise means to study strongly interacting matter. We are fortunate to witness great advances in accelerator and experimental technology so that the full power of the electromagnetic probe can be exploited to study hadronic matter. The two examples discussed above indicate that new insight into Nature is being provided by the elementary elastic electron nucleon scattering reaction, particularly with spin polarization techniques. The role of the sea quarks/meson cloud in nucleon structure continues to be a subject of signiflcant interest. Precision determination of the elastic form factors at low momentum transfers from BLAST may conflrm the ansatz of Friedrich and Walcher. Conflrmation of a dip in the proton electric and magnetic form factors as well as the neutron magnetic form factor at Q2 ∼ 0.15 (GeV/c)2 will not deflnitively quantify the role of the meson cloud but it will demand of theorists a convincing explanation. I note that the recent G0 data [29] on the linear combination of the electric and magnetic strange form factors of the proton suggest a Q2 dependence at similar values of Q2 to that of the dip. Is this signiflcant? Clearly, more precise data are needed. The determination of GnE as a function of Q2 by many laboratories over a decade has clearly been a triumph for the fleld of electromagnetic nuclear physics. With the BLAST data, it is expected that this quantity will be determined to better than ±5% at low momentum transfers. It is anticipated that this will quantitatively constrain the meson cloud contribution to the charge distribution of the neutron. The experimental and theoretical contributions at MAMI, particularly by our flve distinguished colleagues who are honored here, have been important to the signiflcant progress made worldwide. It has been a pleasure and a privilege to be part of this unique celebration. I congratulate Profs. H. Arenh˜ ovel, H. Backe, D. Drechsel, J. Freidrich, K.-H. Kasier, and Th. Walcher on their distinguished careers and I wish them every success in the next phase of their lives.

R.G. Milner: The beauty of the electromagnetic probe The author would like to acknowledge discussions with A.M. Bernstein, T.W. Donnelly, R. Miskamen, A.H. Mueller, J.W. Negele, and C.N. Papanicolas. In addition, the author would like to acknowledge that the BLAST experiment is the fruit of a dedicated collaboration over an extended period of time. In particular, an outstanding cohort of graduate students is playing an essential role. The author’s research is supported by the United States Department of Energy under the Cooperative Agreement DE-FG02-94ER40818.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

R.G. Edwards et al., Phys. Rev. Lett. 96, 052001 (2006). D. Rohe et al., Phys. Rev. Lett. 83, 4257 (1999). T. Eden et al., Phys. Rev. C 50, R1749 (1994). I. Passchier et al., Phys. Rev. Lett. 82, 4988 (1999). R. Madey et al., Phys. Rev. Lett. 92, 122002 (2003). N. Sparveris et al., Phys. Rev. Lett. 94, 122003 (2005). V. Pascalutsa, M. Vanderhaeghen, Phys. Rev. Lett. 95, 232001 (2005). C. Alexandrou et al., Phys. Rev. Lett. 94, 021601 (2005). J. Roche et al., Phys. Rev. Lett. 85, 708 (2000). G. Laveissiere et al., Phys. Rev. Lett. 93, 122001 (2004). P. Bourgeois et al., submitted to Phys. Rev. Lett. (April 2006). A. Airapetian et al., Phys. Rev. D 71, 012003 (2005). X. Zheng et al., Phys. Rev. C 70, 065207 (2004).

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14. E.S. Ageev et al., Phys. Lett. B 633, 25 (2006). 15. J. Kiryluk (MIT) for the STAR collaboration, Proceedings of PANIC 2005, October 2005, Santa Fe, New Mexico; K. Boyle (Stony Brook) for the PHENIX collaboration, Proceedings of PANIC 2005, October 2005, Santa Fe, New Mexico, to be published by the American Institute of Physics. 16. S. Kowalski, these proceedings. 17. F. Maas, these proceedings. 18. V. Punjabi et al., Phys. Rev. C 71, 055202 (2005). 19. I.A. Qattan et al., Phys. Rev. Lett. 94 (142301) (2005). 20. P.A.M. Guichon, M. Vanderhaeghen, Phys. Rev. Lett. 91, 142303 (2003); P.G. Blunden, W. Melnitchouk, J.A. Tjon, Phys. Rev. Lett. 91, 142304 (2003). 21. A.A. Afanasev et al., Phys. Rev. D 72, 013008 (2005). 22. A. Deshpande, R. Milner, R. Venugopalan, W. Vogelsang, Annu. Rev. Nucl. Part. Sci. 55, 165 (2005). 23. S. Chekanov et al., Phys. Rev. D 67, 012007 (2002). 24. J. Friedrich, Th. Walcher, Eur. Phys. J. A 17, 607 (2003). 25. BLAST Technical Design Report August 10th, 1997. 26. D. Cheever et al., Nucl. Instrum. Methods A 556, 410 (2006). 27. A. Maschinot, MIT PhD Thesis 2005 (unpublished). 28. H. Arenh˜ ovel, W. Leidemann, E.L. Tomusiak, Phys. Rev. C 52, 1232 (1995); 46, 455 (1992); Z. Phys. A 331, 123 (1988); 334, 363 (1989). 29. The G0 Collaboration, Phys. Rev. Lett. 95, 092001 (2005).

Eur. Phys. J. A 28, s01, 7 17 (2006) DOI: 10.1140/epja/i2006-09-002-9

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Physics at the Thomas Jefferson National Accelerator Facility L.S. Cardmana Thomas Jefierson National Accelerator Facility, 12000 Jefierson Avenue, Newport News, VA 23606, USA and University of Virginia, Department of Physics, 382 McCormick Rd., P.O. Box 400714, Charlottesville, VA 22904-4714, USA / Published online: 31 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. The Continuous Electron Accelerator Facility, CEBAF, located at the Thomas Jefierson National Accelerator Facility, is devoted to the investigation of the electromagnetic structure of mesons, nucleons, and nuclei using high energy, high duty-cycle electron and photon beams. Selected experimental results of particular interest to the MAMI community are presented. PACS. 29.17.+w Electrostatic, collective, and linear accelerators 25.20.-x Photonuclear reactions 25.30.Bf Elastic electron scattering 25.30.Dh Inelastic electron scattering to speciflc states

1 Personal Comments It is an honor and a pleasure to be here to celebrate the achievements of MAMI and the distinguished careers of Professors Arenh˜ ovel, Backe, Drechsel, Friedrich, Kaiser, and Walcher. We are all deeply aware of the extent to which the science we do builds on the achievements of those who have gone before us, and on the insights and hard work of our colleagues working in the fleld today. One of my very earliest memories as a scientist, dating from the days when I was a young graduate student, is that of attending Photonuclear Physics Boot Camp (otherwise known as the Photonuclear Gordon Conference) and learning from (and with) many of those retiring today. Thomas (Walcher) was one of the very flrst scientists I ever knew beyond the boundaries of my own laboratory. He came to visit us (at Yale), and I went and visited him and his colleagues at Darmstadt. It has been a great pleasure to follow his distinguished career in science, from lowQ2 electron scattering to hadronic physics at CERN and beyond, and flnally to the leadership role he has played at MAMI for many years. Hartmuth (Arenh˜ ovel) has been the keeper of the flame of all knowledge about the deuteron, and a worthy successor to Gregory Breit. You should know that I was a graduate student at Yale, and it was one of Professor Breit’s missions in life to convince any and all who would listen that the deuteron was the essence of nuclear physics, and that until we understood the deuteron, we did not understand anything. I think it is fair to call Hartmuth the Gregory Breit of my generation; he has made so many contributions. a

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Dieter (Drechsel) has always been one of those people I have looked to as the source of the big picture in nuclear physics. He has provided us with deep insights, a sense of direction, and an understanding of what is really important. He has also been an inspiring example here at Mainz of the tremendous beneflts to everyone of having a close collaboration between theory and experiment. J˜org (Friedrich) has taught us all how to analyze and interpret electron scattering data with minimal prejudice (and, therefore, maximal honesty). It is a delight to see the same rigorous approach that was so successful in the study of nuclei and their excited states now being applied to nucleon structure. Karl-Heinz (Kaiser) and his mentor, Helmut Herminghaus, taught the world how to build superb continuouswave (cw) electron accelerators efiectively and e– ciently. Karl-Heinz, in particular, through the design and construction of the double-sided microtron, is leaving the Institute well positioned for another generation of superb experiments. In conclusion, on behalf of so many people I have worked with in nuclear physics, I want to thank each of you for your many contributions to our fleld, and to express the hope we all share that for each of you retirement is a formality, not a reality, and that you will continue to be active for years to come.

2 Research at Jefferson Laboratory The Thomas Jefierson National Accelerator Facility, also called Jefierson Lab (or JLab), operates the Continuous Electron Beam Accelerator Facility (CEBAF). CEBAF is a cw electron accelerator capable of delivering three electron beams for simultaneous experiments in the

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three experimental areas. Originally designed for 4 GeV, its present maximum energy is 5.7 GeV. The CEBAF user community consists of about 2000 physicists; more than half of them are actively involved in the experimental program. In addition to its main mission, JLab contributes to the development and use of Free Electron Lasers, to medical imaging, and to community outreach programs. The intellectual and technical foundations for the construction of CEBAF were provided by the scientiflc successes of earlier electron accelerators (the generation that included Saclay, MIT-Bates, NIKHEF, and, to some extent, SLAC), and by the enhanced research opportunities provided by cw electron beams as demonstrated at facilities such as MAMI. CEBAF is a large, international laboratory with a broad research program; it has been in operation for some seven and a half year now. What are the goals of CEBAF’s research program? Basically, we aim to understand strongly-interacting matter. How are the hadrons constructed from the quarks and gluons of QCD, and how does the nucleon-nucleon force arise from the strong interaction? We further aim to identify the limits of our understanding of nuclear structure by using the high precision attainable with the electromagnetic probe and the possibiltiy of extending investigations to very small distance scales. A speciflc issue that motivated the construction of CEBAF was our desire to gain insight into the question of where the description of nuclei based on nucleon and meson degrees-of-freedom fails and the underlying quark degrees-of-freedom must be taken into account. One can ultimately characterize all of this as trying to understand QCD, not in the perturbative regime accessible at very high energies and very short distance scales, but in the strong interaction regime relevant to most of the visible matter in the Universe. To make progress in these areas, there are other critical issues that must be addressed, such as the mechanism of conflnement, the dynamics of the quark interaction, and how chiral symmetry breaking occurs. To provides some shape and structure to the discussion of the experiments, the CEBAF program can be organized into half a dozen broad thrusts. This presentation will concentrate on two of them: – How are the nucleons made from quarks and glue? – Where are the limits of our understanding of nuclear structure

3 How are the nucleons made from quarks and glue? Among the most interesting puzzles in physics today are: why there is this efiective degree-of-freedom in QCD, the nucleon; and how something as complicated as the residual QCD interaction between quarks in nucleons can be characterized by a rather simple N-N potential? To provide experimental insights that will help us solve the flrst of these puzzles, the Jefierson Lab research community has mounted an array of investigations in three broad areas:

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– What are the spatial distributions of the u, d, and s quarks in the hadrons? – What is the excited state spectrum of the hadrons, and what does it reveal about the underlying degreesof-freedom? – What is the QCD basis for the spin structure of the hadrons?

3.1 What are the spatial distributions of the u, d, and s quarks in the hadrons? Elastic electron scattering has provided most of our information on the spatial distributions of the quarks in the nucleons. The data on the four electromagnetic structure functions of the nucleon, GE and GM for both the proton and the neutron, available just prior to the start of experiments at CEBAF is shown in flg. 1. The magnetic form factors of the proton and the neutron were known reasonably well, but the electric form factors were not. The electric form factor of the proton had not been determined accurately enough to distinguish between a wide range of theories based on rather difierent physics. First results on the electric form factor of the neutron were available from Bates, Mainz, and NIKHEF, but these data were limited to moderate momentum transfers and, therefore, not sensitive to the details of the distribution of charge inside the neutron. The measured form factor was consistent with the r.m.s. radius derived from neutron-electron scattering. The present status of the nucleon form factors is shown in flg. 2. The measurements of the polarization transfer from the incident electron to the elastically recoiling proton have shown that the electric and magnetic form factors for the proton difier substantially. The systematic difierences between the polarization transfer data and the Rosenbluth results for GE /GM are likely

L.S. Cardman: Physics at the Thomas Jefierson National Accelerator Facility

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due to two-photon exchange efiects modifying the results. Theoretical estimates suggest that the modiflcations are much smaller for the polarization transfer data than for the Rosenbluth data, so the former are likely to be more directly interpretable in terms of the nucleon form factors. The electric form factor of the neutron has now been measured up to a Q2 of 1.5 (GeV/c)2 using polarization transfer techniques, and the data taken with difierent methods agree quite well. The theoretical interpretation of the data is summarized in flg. 3. The theories that describe the data reasonably well reveal two key aspects of nucleon structure: the importance of the pion cloud, and the importance of incorporating the relativistic motion of the quarks into the theoretical description of the nucleon. When one looks at these form factors in a phenomenological way with minimum prejudice [2], what emerges is some of the clearest evidence we have for the nucleon’s pion cloud (see flg. 4). Similar results have been obtained using a difierent approach to model-independent analysis [3] of nucleon form factors. We plan to extend the proton form factor data to ∼ 9 (GeV/c)2 , where we may see evidence for a difiraction minimum. The neutron form factor will also be extended to ∼ 5 (GeV/c)2 . Further extensions of a factor of two are planned with the 12 GeV Upgrade. Such extensions have

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historically proven to be important, and we expect these data will provide further insight and sensitivity for completing our understanding of how to construct nucleons from quarks and gluons.

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The strange quark form factors have become an interesting area of study that is both analogous and complementary to the classical electromagnetic form factors. By using the weak component of the electro-weak interaction we access the weak neutral current form factor, which can be interpreted very elegantly in terms of the strange quark distribution. Because there are no valence strange quarks, this measurement provides a unique window on the sea quark distribution. The strange form factors can also be expected to provide us with interesting experimental insights into nucleon structure: by combining the electromagnetic and the weak neutral current form factors we should be able to separate the spatial distribution of the u, d, and s quarks. Figure 5 shows the world’s data on the strange proton form factor taken at forward angles as a function of Q2 . One sees data from the A4 experiment at Mainz [6,7,8], from HAPPEx I and II (JLab Hall A) [9,10,11], and from G0 (JLab Hall C) [13]. These di– cult experiments would be impossible without highly polarized electron beams from magniflcently stable accelerators. The fact that the data from difierent laboratories lie roughly on a smooth curve gives one confldence that the experimenters are doing it right. The flrst thing that strikes you about the data is that the form factor is rather small. This is to be expected, as all of the strange quarks emerge as quark-antiquark pairs popping in and out of the vacuum, and to get a flnite form factor there must be some kind of a polarizing efiect

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separating them spatially. Even at the highest momentum transfers reached experimentally we are averaging over a distance scale that is roughly the size of the nucleon, so it is not too surprising that the result is small. There is an intriguing suggestion in the data for something that one would call vaguely pion-cloud like behavior, but it is fair to say that the statistical signiflcance of this efiect is not very high. The data taken at forward angles includes a mixture of electric and magnetic form factors. At Q2 = 0.1 (GeV/c)2 we have data at both forward and backward scattering angles, so we can separate these efiects (see flg. 6). The data favor a positive value of GsM , which is at variance with most of the theoretical models. Experiments are in progress that will reduce the size of the error ellipse at this Q2 value by a about a factor of 3, and additional experiments planned at both MAMI and JLab will permit separations at other Q2 values. A broad, world-wide efiort will provide the results we want. Another interesting experiment is the measurement of the pion form factor. The pion is the simplest QCD bound system, the positronium of QCD. One expects that the pion form factor will provide us with evidence for the

L.S. Cardman: Physics at the Thomas Jefierson National Accelerator Facility 0.6

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transition of the strong interaction from the perturbative (QED-like) to the strong (conflnement) regime at the lowest possible momentum transfer. These data also constrain phenomenological models of the pion. Measuring the pion form factor is not simple. At low Q2 , one can scatter pions ofi atomic electrons, but a bowling ball does not transfer energy to a ping-pong ball e– ciently, and even with very high energy pions this experiment cannot reach high momentum transfers. To reach higher momentum transfer in the absence of a free pion target, one must scatter electrons ofi virtual pions inside a proton and extrapolate the data to the pion pole. The world’s data (see flg. 7) is beginning to distinguish between difierent theoretical approaches. With the 12 GeV Upgrade, we expect to extend the data out to a momentum transfer of 6 (GeV/c)2 and to be able to infer the distance scale for the onset of perturbative behavior. 3.2 What is the excited state spectrum of the hadrons? If one looks at several decades worth of data on nucleon resonances and tries to use a simple quark model to classify the states in terms of the excitation in units of hω and the angular momentum of the three quarks, the states that have been identifled so far flt nicely into this scheme, but there are many states that have been predicted but have not been found. It is an interesting fact that one can explain all of the states that have been seen so far by assuming that the nucleon and its excited states are a diquark-quark system. Since most of the data have been obtained from pion-induced reactions, and many of the missing states are predicted not to couple to pions, it is also possible that the missing states may have been overlooked for experimental reasons. In atomic spectroscopy the line spacing is large compared to the line width, and measuring the complete spectrum is relatively straightforward. In nucleon spectroscopy, the strong interaction causes the width to be comparable to the spacing. Identifying weak states and

Fig. 8. W-dependence of the scattered electron rate for the p(e, e )X reaction. CLAS data taken at 4 GeV primary beam energy. The energies of the known excited states are shown in black, while those of the states missing in the simple quark model description are shown in red.

extracting the internal quark structure from the measured cross sections is a di– cult task. The problem can be seen easily in flg. 8, which shows the inclusive electron scattering cross spectrum from the proton for a 4 GeV electron beam. With a modern electron accelerator and a large acceptance detector one can obtain data on the transition form factors over a large Q2 -range [1 → 4 (GeV/c)2 ] in a single shot. There is plenty of cross section in the region where the missing states (shown in red) have been predicted, but extracting their individual strengths from the data is a real challenge. The combination of cw electron beams and modern, large solid angle detectors provides important advantages for addressing this problem experimentally. If one looks at the same data set of flg. 8 but uses the information on the energy and momentum of the flnal state proton measured in coincidence with the inelastically scattered electron, it is straightforward to infer the missing mass associated with the decay of the excited state (see flg. 9). One can see clearly from the raw missing mass spectrum that the missing states do not couple to pions, but rather to the η and ω. With the further information on the angular correlations of those decay particles relative to the momentum transfer axis one flnally has the information necessary to decompose the spectrum of flg. 8, and learn just what is there. This efiort naturally begins with the Δ(1232), which decays predominantly into pion and nucleon. Figure 10 shows a comparison of separated structure functions from CLAS data for the p(e, e p)π o reaction with theoretical flts and results from previous experiments. The Δ → γ ∗ N-transition is characterized by three multipoles: the electric quadrupole E, the magnetic dipole M , and the scalar multipole S. As we examine this transition as a function of momentum transfer we expect that difierent aspects of the excitation will become apparent. At large distance scales (corresponding to low momentum transfers) we should see the efiect of the pion cloud, while at large momentum transfer (corresponding to short distances) we will eventually reach the limit given by pQCD

Missing States

The European Physical Journal A

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Fig. 10. Angular distributions of the separated structure functions for the p(e, e p)π o reaction in the Delta region [16].

where REM = E/M → 1, and we further expect that the S/M ratio RSM will become constant. Results from an early experiment at JLab and data from MAMI and Bates, all in the low-Q2 regime show the efiect of the pion cloud clearly (see flg. 11). As a function of Q2 , REM remains small and negative at high Q2 with a trend toward 0 and a possible sign change. RSM continues to rise in magnitude with Q2 . No trend is seen towards Q2 independence. We can only conclude that even at Q2 of 10 (GeV/c)2 we are far from the pQCD regime. Pion cloud models describe the data well (fltted to low and high-Q2 points). Unquenched Lattice QCD gives the correct signs and approximate magnitudes. One of the most interesting examples of the impact of the pion cloud and of the value of measuring the transition form factors for nucleon excitation is the Roper

resonance. According to the constituent quark model the N∗ (1440)P11 state is an N = 2 radial excitation of the nucleon. However, the properties of this state such as its mass and photocouplings are not well described by this model. The new CLAS data (see flg. 12) seem to explain this puzzle. At low momentum transfer, what one is measuring is dominated by the pion cloud. As you start squeezing down the distance scale, what emerges is the underlying quark structure of the Roper, which is, in fact, roughly consistent with a radial excitation. Investigation of nucleon excitation through the measurement of the transition form factors is now slowly moving up in excitation energy. Most of this analysis is at a preliminary stage, and what is really needed is a coherent study of many channels at many values of momentum transfer in a consistent (and comprehensive) analysis. It will be a long time before we have all the answers. As we search through this data, we are coming across intriguing evidence for states that have been missing . For example, there is evidence for a possible new N* state near 1840 MeV visible in the Λ photo- and electroproduction data. In the forward hemisphere, one sees a nice peak from a known N* state at 1.7 GeV; in the

L.S. Cardman: Physics at the Thomas Jefierson National Accelerator Facility 2

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backward hemisphere (see flg. 13), one sees an additional unexpected structure. A detailed analysis shows that the angular distribution can be flt nicely with the addition of a new P11 state at 1840 MeV with a width Γ = 140 MeV to the known D13 (1870) and D13 (2170) states. Intriguingly, a P11 state at 1840 MeV is consistent with the symmetric quark model and SU (6) × O(3) symmetry, but is inconsistent with diquark-quark symmetry. I feel obliged to bring you up to date on the pentaquark (or lack thereof). There was a great deal of excitement for a while about what appeared to be evidence for a 5-quark state. There have three experiments at JLab pushing to substantially higher statistics, both in the γp and the γd channel, and for virtual photons as well. No evidence for a 5-quark state has been found in the flrst analyses of these new data.

3.3 What is the QCD basis for the spin structure of the hadrons? In addition to the investigation of the spatial distributions of charge and magnetization in the nucleon and its excited state spectrum, the third important experimental focus is the nucleon’s spin structure. The flrst thing to look at is the spin structure function of the valence quarks at highx. The data for the proton was reasonable; the new CLAS data with somewhat tighter error bars are conflrming the old results and improving the overall accuracy. There were no data of any statistical signiflcance for the neutron above an x of 0.3. The 3 He experiment at JLab has provided three new data points (see flg. 14). The new data, when folded into a global analysis of the parton distribution functions (PDF), show that the theoretical prejudices used in earlier analyses were wrong; in particular we now know that Δd/d stays negative at high x. One can make predictions with a minimum of theoretical prejudice for the integrals of the spin structure functions at the two extremes of distance scales. In the limit of extremely small distances (i.e. for Q2 → ∞), assuming only isospin symmetry and current algebra (or the operator product expansion within QCD), Bjorken showed that

Fig. 14. (Left) Spin structure function of the neutron, An 1, derived from 3 He data [19]; and (Right) spin structure function of the proton Ap1 [20].

Fig. 15. GDH integral as a function of the upper limit νmax [21].

the difierence between the proton and the neutron integrals is related to the neutron β-decay coupling constant, with a small Q2 dependent correction due to the running of the coupling constant. For very large distance scales (i.e. for Q2 → 0), there is a slightly less rigorous set of assumptions (Lorentz invariance, gauge invariance, unitarity, and the dispersion relation applied to the forward Compton amplitude) that can be used to show that the difierence between the helicity 23 and 12 total cross sections is related to the nucleon anomalous magnetic moment (this is the GDH sum rule). There has been a lovely set of data taken at ELSA and MAMI that have determined the GDH integral as a function of the upper photon energy integration limit (see flg. 15). The experiments were technically challenging [21], requiring the combination of polarized electrons, a polarized target, and large-acceptance detectors. Theoretical analysis and interpretation of these data show that the GDH sum rule is satisfled at the 5% level. The efiort has also provided us with a better understanding of the physics of the reactions contributing to the integral. These data, and the precision with which they have deflned the GDH integral at the photon point, provide the foundation for our studies of the Q2 evolution of the moment of the nucleon’s spin structure functions. As one looks at the evolution of the moment of the proton spin structure function with Q2 , one expects to see

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of very low momentum transfer with high precision to test the predictions of χPT. The flrst signiflcant measurement of the Q2 -dependence of the Bjorken integral (see flg. 17) was made for Q2 = (0.05 − 2.5) (GeV/c)2 . Remarkably, pQCD-based Q2 evolution matches the data down to a Q2 of about 0.7 (GeV/c)2 . Deur et al. [23] have made an interesting interpretation of the Q2 -dependence of the Bjorken integral in terms of an efiective strong-coupling constant αef f (Q2 ) (see flg. 18). Again, there is evidence for a transition occurring around Q2 = 1 (GeV/c)2 .

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the anomalous magnetic moment of the proton in the long wavelength limit, whereas at inflnite Q2 the Bjorken sum rule is valid. In the regime close to the long wavelength limit, chiral perturbation theory (χPT) allows us to make predictions. The transition between the two extremes is an important piece of information on how the nucleon is put together, and how nucleon structure emerges from the parton soup. We have data now, mainly from JLab, on the evolution of the structure function’s integral for the proton (see flg. 16) and the neutron approaching the GDH sum rule limit at Q2 = 0, and approaching the Bjorken limit at a surprisingly low momentum transfer of about 1 (GeV/c)2 . Several experiments at JLab are investigating the region

As described above, experiments at Jefierson Lab are providing essential new insights into nucleon structure. In a very similar way, the precision, spatial resolution, and interpretability of experiments performed using electromagnetic probes are being used to address long-standing issues in nuclear physics, including speciflcally nucleon-nucleon correlations and the identiflcation of the limits of our understanding of flnite nuclei. 4.1 Correlations in nuclei Nucleon-nucleon correlations have been a subject of great interest since the beginnings of the fleld. In his fabled bible on nuclear physics, Hans Bethe estimated that these correlations should be of scale a third of what one observes in nuclear physics, and indeed they are. However, flnding clear, interpretable evidence for these correlations has been a real challenge to experimentalists. The previous generation of (e, e p) experiments carried out at Saclay, NIKHEF, and Mainz explored the spectral function strength for low-lying shells. Only about 2/3 of the strength anticipated from a simple shell model was found. However, the interpretability of these measurements was limited by the uncertainties introduced by the

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L.S. Cardman: Physics at the Thomas Jefierson National Accelerator Facility

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Fig. 19. Ratio of inelastic scattering cross sections ofi nuclei relative to 3 He as a function of Bjorken-x [24].

corrections necessary for the flnal-state interactions of the knocked-out protons. A new approach to nucleon-nucleon correlations avoids this problem by comparing the ratio of inelastic electron scattering ofi 4 He, 12 C, and 56 Fe to 3 He in a kinematical regime where the scattering is basically from the quarks within the nucleons, and the scattering from the nucleons as coherent objects is highly suppressed. These data (see flg. 19) tell us that at any given moment the number of correlated nucleons in 4 He, 12 C, and 56 Fe is ∼ 0.3, ∼ 1.2, and ∼ 6.7, respectively. So about 10% of the time a nucleon is involved in a nucleon-nucleon correlation. The measurements further show that three-nucleon correlations are clearly present (at x > 2), and about an order of magnitude smaller than two-nucleon correlations. Another approach [25] to the study of correlations is to search explicitly for the strength that was identifled as missing in the last generation of (e, e p) experiments. We are using the (e, e p) reaction at high momentum transfers and high missing energies, a region that was simply not accessible at the lower-energy, high duty-factor facilities previously available. The missing strength was, indeed, found (see flg. 20), and agrees rougly with the predictions of Correlated Basis Function theory (although the momentum distribution is not described correctly in detail). In a third study correlated pairs have been measured directly in the 3 He(e, e pp)n reaction. In this experiment, the absorption of the virtual photon kicks out a proton, and the opening angle of the remaining pair shows a backto-back peak. One can infer from the data the shape of the pair momentum distribution. Similar, though somewhat less direct, information can be obtained from examining the 3 He(e, e p)X reaction at very high missing momentum. Signiflcant strength above what is predicted by PWIA has been observed (see flg. 21). The quantitative understanding of the results is work in progress.

Fig. 21. Efiective nucleon density for the 3 He(e, e p)X reaction as a function of missing momentum. 2bbu stands for two-body breakup [26].

4.2 The limits of our understanding of finite nuclei One of the key issues that motivated the construction of CEBAF was our desire to gain insight into the question of where the description of nuclei based on nucleon and meson degrees of freedom fails and the underlying quark degrees-of-freedom must be taken into account. Data on the elastic scattering from the deuteron and highenergy photodisintegration, together with accurate theoretical calculations, are providing the answers. We begin with the elastic scattering form factors for the deuteron. The theory is in an advanced state: we use the best ab initio calculation of the structure of the deuteron with a potential VN N determined from a flt to N-N phase shifts, and then add exchange currents and relativistic corrections. The data set for the deuteron elastic form factors demonstrate the technical accomplishments of modern accelerators and equipment: elastic e-D scattering has been measured down to cross sections characteristic of ν-scattering! The data for the electric and the magnetic form factors, and for the tensor polarization (see flg. 22) demonstrate that conventional nuclear theory works up to Q2 of

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about 2 (GeV/c)2 , i.e. the nucleon-based picture is still valid at distance scales of about one half the size of a nucleon. Why, we do not know; none of us expected it to work that well before the experiments were undertaken. The shape of the deuteron derived from the form factor data is also shown in flg. 22; one can see clearly that the nucleon spins are aligned end-to-end (resulting in a dumbell -shaped distribution) rather than anti-parallel (which would have yielded a donut shape). The photodisintegration of the deuteron was one of the flrst experiments done in nuclear physics (at energies of only a few MeV) and also one of the most recent ones (now at energies approaching 6 GeV). The reaction probes internal nucleon momenta well beyond those accessible in electron scattering because of the momentum mismatch between the photon and the nucleon. In a parton-based description of the reaction, one expects the cross section to scale like s−11 , where s is the CM energy squared. The data (see flg. 23) demonstrate that s−11 scaling of the cross section is reached at photon energies which change with the proton center-of-mass angle. The transition occurs consistently at a transverse momentum of about (1.0 − 1.3) GeV/c, which shows that below ∼ 0.2 fm the nucleon-meson description of the deuteron is no longer valid, and a parton-based description is more appropriate. A more recent experiment [28] using CLAS has extended these data to include angular distributions for a broad range of energies; the data is described by a quark-gluon string model.

Fig. 23. Cross sections for deuteron photodisintegration. The energies associated with a transverse momentum of 1.37 GeV/c are indicated with a blue arrow in each panel [29].

5 Summary The CEBAF accelerator at JLab is fulfllling its scientiflc mission to understand how hadrons are constructed from the quarks and gluons of QCD, to understand the QCD basis for the nucleon-nucleon force, and to explore the transition from the nucleon-meson to a QCD description. Its success is based on the flrm foundation of experimental and theoretical techniques developed world-wide over the past few decades, on complementary data provided by essential lower-energy facilities, such as MAMI, and on the many insights provided by the scientists we are gathered here to honor. It is a pleasure to acknowledge the assistance of Bernhard Mecking in the development of this article, and thoughtful comments on the manuscript from Volker Burkert, Kees de Jager, John Domingo, and Rolf Ent. This work has been supported through The Southeastern Universities Research Association, Inc., which operates the Thomas Jefierson National Accelerator Facility under Contract No. DE-AC05-84150 with the U.S. Department of Energy.

References 1. C. Hyde-Wright, C.W. de Jager, Annu. Rev. Nucl. Part. Sci. 54, 217 (2004) and references therein. 2. J. Friedrich, T. Walcher, Eur. Phys. J. A 17, 607 (2003). 3. J.J. Kelly, AIP Conf. Proc. 698, 393 (2004). 4. T.M. Ito et al., Phys. Rev. Lett. 92, 102003 (2004). 5. D.T. Spayde et al., Phys. Lett. B 583, 79 (2004). 6. F.E. Maas et al., Phys. Rev. Lett. 94, 152001 (2005). 7. F.E. Maas et al., Phys. Rev. Lett. 95, 022002 (2004). 8. F. Maas, these proceedings.

L.S. Cardman: Physics at the Thomas Jefierson National Accelerator Facility 9. 10. 11. 12. 13. 14. 15. 16.

K.A. Aniol et al., Phys. Rev. Lett. 96, 022003 (2006). K.A. Aniol et al., Phys. Lett. B 635, 275 (2006). K.A. Aniol et al., Phys. Lett. B 509, 211 (2001). K.A. Aniol et al., Phys. Rev. C 69, 065501 (2004). D.S. Armstrong et al., Phys. Rev. Lett. 95, 092001 (2005). T. Horn, private communication for the Fπ collaboration. V.D. Burkert, Eur. Phys. J. A 17, 303 (2003). L.C. Smith, Invited Talk, Japan-US Workshop on Electromagnetic Meson Production and Chiral Dynamics, Osaka, Japan (April 2005). 17. CLAS Collaboration (V.D. Burkert), Int. J. Mod. Phys A 20, 1531 (2005). 18. I. Aznauryan talk at N*2005, Tallahasse, FL (October 2005), to be published. 19. X. Zheng et al., Phy. Rev. Lett. 92, 012004 (2004).

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20. CLAS Collaboration (K.V. Dharmawardane et al.), submitted to Phys. Rev. Lett. (2006). 21. A. Thomas, these proceedings. 22. A. Deur, 13th Int. workshop on Deep Inelastic Scattering (DIS2005), AIP Conf. Proc. 792, 969 (2005). 23. A. Deur, V. Burkert, J.P. Chen, W. Korsch, arXiv:hepph/0509113. 24. K. Egiyan et al., Phys. Rev. Lett. 96, 082501 (2006). 25. D. Rohe et al., Phys. Rev. Lett. 93, 182501 (2004). 26. F. Benmokhtar et al., Phys. Rev. Lett. 94, 082305 (2005). 27. M. Garcon, J.W. Van Orden, Adv. Nucl. Phys. 26, 293 (2001) and references therein. 28. M. Mirazita et al., Phys. Rev. C 70, 014005 (2004). 29. E.L. Schulte et al., Phys. Rev. Lett. 87, 102302 (2001).

Eur. Phys. J. A 28, s01, 19 27 (2006) DOI: 10.1140/epja/i2006-09-003-8

EPJ A direct electronic only

Few-nucleon systems at MAMI and beyond W.U. Boeglina Physics Department, Florida International University, University Park, Miami, FL 33199, USA / Published online: 10 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Few-body systems provide a testing ground for models of the NN interaction, reaction mechanisms and for models of nuclei. An overview of results of coincidence experiments on the deuteron, 3 He and 4 He obtained in the last 20 years at MAMI and at other facilities, covering a wide range of momentum and energy transfers, is presented. PACS. 25.10.+s Nuclear reactions involving few nucleon systems 25.30.Fj Inelastic electron scattering to continuum

1 Introduction Few-body systems are ideal to investigate fundamental problems in nuclear physics such as the ground state and continuum wave functions, the importance of correlations and the structure of the electromagnetic current operator. In addition, interaction efiects such as meson exchange currents (MEC), and isobar conflgurations (IC) can be studied. At large momentum transfers, one hopes to be able to explore the transition from the regime where observables are best described by nucleon/meson degrees of freedom to the regime where quark/gluon degrees of freedom are the most e– cient, in describing the interaction. The few-body systems presented here consist of the deuteron and the 3 He and 4 He nuclei. They range from a loosely bound system, such as the deuteron, to a very tightly bound one such as 4 He. In contrast to complex nuclei, the structure of few-body systems can nowadays be calculated with high precision using realistic nucleonnucleon interaction potentials. Very successful methods for the calculation of bound and continuum state wave functions include the solution of Faddeev-Yakubovsky [1] equations, Variational Monte Carlo [2], and other MonteCarlo based calculations [3,4]. Many laboratories have contributed to the study of few-body systems in the last twenty years (in alphabetical order): ALS (Saclay, France), ELSA (Bonn, Germany), Jefierson Lab or JLAB (Newport News, VA, USA), MAMI (Mainz, Germany), MIT-Bates (Middleton, USA), NIKHEF (Amsterdam, The Netherlands), and SLAC (Stanford, CA, USA). Recent electron accelerators such as the Mainz Microtron and Jefierson Lab (CEBAF) provide very high-intensity, continuous wave (CW) beams. These have made coincidence experiments possible that a

e-mail: [email protected]

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explore new, previously inaccessible kinematical regions with very high statistical precision. I will therefore focus on the study of few-body systems using knock-out reactions such as (e, e N) and (e, e NN).

2 Short overview of the (e, eN) reaction Treating the incoming and scattered electrons as plane waves, and applying the one-photon exchange approximation, the (e, e p) reaction can be viewed schematically as

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shown in flg. 1. In the case of the Plane-Wave Impulse Approximation (PWIA) the virtual photon is absorbed by a bound nucleon having a certain initial momentum pi . The struck proton subsequently leaves the nucleus with a flnal momentum pf . The residual system may remain in its ground or in an excited state and has a recoil momentum pm . In the following, the term missing momentum will be used synonymously with recoil momentum. Within PWIA the following relation between initial and missing momentum is valid: pm = −pi . The transferred energy ω is divided between the kinetic energy of the ejected nucleon, its separation energy, and the kinetic and, possibly, excitation energy of the residual system. The missing momentum pm and missing energy Em are deflned as follows: Momentum conservation : q = pf + pm , Energy conservation : E m = ω − T p − Tr . Here Tp is the kinetic energy of the ejected nucleon, and Tr is the kinetic energy of the recoiling system, calculated from pm under the assumption that the undetected (A1)-system remains in its ground state. Figure 2 shows the electron scattering plane, deflned by the incoming and scattered electron momenta, and the reaction plane, deflned by the flnal nucleon momentum and the momentum transfer. The cross section in the one photon exchange limit can be written as [5,6,7] d5 σ = σM ott (vL RL + vT RT + dωdΩe dΩp + vLT RLT cos φ + vT T RT T cos 2φ), where Ri are the response functions containing matrix elements of the charge and current operators. These, in turn, provide the nuclear structure information. The vi are kinematical factors depending on the electron kinematics only, and σM ott is the Mott cross section describing the scattering of relativistic electrons by a point charge. If one neglects the interaction of the outgoing nucleon with the recoiling system, one obtains the plane-wave impulse approximation (PWIA) which permits a factoriza-

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tion of the (e, e N) cross section into an elementary (ofishell) electron nucleon cross section [8] and the spectral function describing the probability of flnding a nucleon with a given initial momentum and missing energy. Integrating the spectral function over the missing energy leads to the momentum distribution. Final state interactions (FSI), MEC, and IC remove this simple relation and therefore pi = −pm (flg. 1).

3 Studies of the deuteron Early (e, e p) experiments were limited in luminosity by the duty factor of the available electron accelerators. Cross sections could be measured for large missing momenta (pm ≈ 0.5 (GeV/c)) only at relatively small momentum transfer (Q2 ≈ 0.1 (GeV/c)2 ) or for large Q2 only at relatively small (pm < 0.2 (GeV/c)). Experiments have been carried out at all facilities mentioned above. More recent experiments, carried out in the last ten years, beneflted from the availability of high duty cycle beams at Jefierson Lab, MAMI, NIKHEF(AmPS), and at MIT-Bates(SHR). In general, the various experiments can be separated into those that explored the D(e, e p)n cross section over a large range of missing momenta and those that extracted individual response functions. 3.1 Cross section measurements at low Q2 These experiments explored the D(e, e p)n cross section over a wide range of missing momenta at small to medium

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momentum transfers [9,10,11,12]. The focus of these measurements was the exploration of the momentum distribution within the plane-wave impulse approximation. It has been found, however, that with increasing recoil momentum FSI and, related to the corresponding energy transfer, MEC and IC contributions increase dramatically. Figure 3 shows the D(e, e p)n cross section measured at MAMI [11] and H. Arenh˜ ovel’s calculation that includes FSI, MEC, and IC [13]. One can see that the cross section is well reproduced up to pm = 350 MeV/c. At higher pm there are signiflcant discrepancies between experiment and theory. This occurs in a kinematical region where large virtual delta excitation contributions are expected. Since many experiments have measured the D(e, e p)n cross section at missing momenta below 300 MeV/c it is interesting to compare how well these results agree with each other in order to learn how accurately the D(e, e p)n reaction is known experimentally. As the various experiments have been carried out at difierent kinematical settings, I used Arenh˜ ovels calculation [13] as a reference to take into account FSI. MEC and IC are also included, however they tend to contribute less than FSI. The result is shown in flg. 4 where the ratio between the experimental and the theoretical cross sections is shown. It is interesting to note that the various experiments agree quite well among each other, while the experimental cross sections seem to be systematically smaller than the calculated cross sections by about 10%. The reason for this discrepancy needs further investigation but one has to keep in mind that most experiments quote systematic errors around 5%.

3.2 Structure function separations Experiments to extract various response functions of the D(e, e p)n reaction have been carried out at most electron accelerators. The major results of these experiments will be presented below. In general, all published re-

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sponse function separations have been limited to missing momenta below 200 MeV/c. RL and RT have been determined at NIKHEF [17,18], Saclay [14], and at MITBates [15]. Figure 5 shows a comparison of the various results for overlapping kinematics. The left panel shows the ratio to H. Arenh˜ ovels calculation [19,20]. The longitudinal response has been found to deviate up to 20% from the calculation depending on the missing momentum. The transverse response, depending on the missing momentum, deviates up to 10%. For missing momenta below 50 MeV/c the longitudinal response has been found to be about 20% smaller than the calculation in both, the Saclay and the Bates experiments. The transverse response has been found to be about 4% smaller than the calculation for the Saclay data and in agreement (within the error bars) for the Bates experiment. In contrast the longitudinal response found at NIKHEF is much larger than the MIT-Bates result, while the transverse responses are in agreement. At MAMI RL and RT have been extracted for missing momenta up to 350 MeV/c [21]. An example of the result of this experiment is shown in flg. 6. Unfortunately at small recoil momenta problems with the target lead to uncertainties in determining the absolute cross section making it impossible to compare the results to the low pm results discussed above. To summarize, the experimental knowledge of RL and RT is limited to low recoil momenta and there are disagreements among difierent experiments and also when compared to modern calculations. Currently there exists no experimental program to address these problems. The response function RLT is sensitive to relativistic contributions to the electromagnetic current operator [22] and FSI and has the advantage that it is particularly easy to extract. This is due to the fact that the electron kinematics are flxed and only the proton scattering

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and has the additional advantage that the absolute cross section normalization cancels in the ratio. An overview of experimental results is shown in flg. 7 together with the result of a determination of ALT at MAMI [23]. Other recent ALT measurements have been published in references [16] and [24]. A determination of RT T requires proton detection out of the electron scattering plane. This has been achieved at MIT-Bates using the Out-Of-Plane (OOPS) spectrometer [24] system and at NIKHEF [28] using the HADRON detectors. For an overview of results see [25].  , can be obtained An additional response function, RLT using out-of-plane detection of the proton and measuring the helicity dependence of the cross section with polarized electrons. This has been carried out at MIT-Bates [29] using OOPS.

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angle is changed (in the electron scattering plane) in such a way, that the reaction plane varies between φ = 0◦ and φ = 180◦ (see flg. 2). The cross section difierence obtained from these two measurements is then proportional to RLT . A quantity closely related to RLT is the left-right asymmetry σ ◦ − σ 0◦ ALT = 180 σ180◦ + σ0◦

D(e, e p)n cross sections have been obtained at SLAC for high Q2 but low recoil momenta (pm < 0.2 GeV/c) [30]. Recently, experiments have been carried out at Jefierson Lab in Hall A (experiment E01-020) as well as in Hall B using CLAS (experiment E94-019). The goal of the Hall A experiment is to test the Generalized Eikonal Approximation (GEA) description of FSI [31] in the D(e, e p)n reaction while the goal of the Hall B experiment is to use the GEA description of the D(e, e p)n reaction in the search for evidence of color transparency. Within the GEA, FSI are described by a series of small-angle scatterings of the outgoing nucleon. This approximation, which is typically valid for nucleon energies of 1 GeV and above, has been successfully applied in high-energy nucleon scattering. However it has never been tested for the D(e, e p)n reaction. Another goal of the Hall A experiment is the determination RLT for missing momenta up to 0.5 GeV/c where relativistic efiects are expected to be very large and RLT is sensitive to details of the current operator. The GEA predicts a characteristic dependence of the strength of FSI on the angle of the recoiling neutron with respect to the momentum transfer and on the value of the missing momentum. For angles around 80◦ FSI efiects are predicted to be maximal. For pm = 0.2 GeV/c a reduction of the cross section by about 30 40% is predicted and for pm = 0.4 GeV/c and pm = 0.5 GeV/c an increase of the cross section by more than a factor of two is predicted. The location of the extremum of the rescattering contributions give additional information about the details of the rescattering process such as the importance of the Fermi motion of the bound nucleons. In order to address these questions in Hall A, the D(e, e p)n cross section has been measured for Q2 = 0.8, 2.1 and 3.5 (GeV/c)2 and missing momenta between pm = 0 and pm = 0.5 GeV/c. A very preliminary result is shown in flg. 8 where the observed yield is compared to a Monte Carlo calculation using the PWIA for the cross section. Clearly the predicted angular dependence has been observed. A detailed analysis is currently in progress.

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Fig. 9. The missing energy spectrum for the 3 He(e, e p) reaction on the quasi-free peak. The 2-body breakup peak is clearly separated from the 3-body continuum. All strength above about 25 MeV is entirely due to the radiative tail [38].

4 3 He and 4 He studies The breakup of the 3,4 He nuclei can lead to a 2-body flnal state like in the deuteron or it can lead to a 3- and even 4-body flnal state for 3 He and 4 He, respectively. Only recently, with the advent of high computing power and efflcient computational techniques, can the continuum flnal state be calculated accurately. These nuclei are the simplest systems in which to study short range correlations.

Fig. 10. The asymmetry ALT and the response function RLT measured at MAMI for the 4 He(e, e p)3 H reaction [39] together with calculations with and without the inclusion of MEC [40].

4.1 Low Q2 experiments 

3

Similar to deuterium, early (e, e p) experiments on He and 4 He explored the cross section with the goal to obtain information on the momentum distribution [33,34]. The same problems as described in section 3.1 are encountered here. In addition to examining the momentum distribution one has also studied the missing-energy spectrum. Early experiments on 3 He by Marchand et al. [33] showed a structure in the missing energy spectrum that is shifting with increasing recoil momentum in agreement with the kinematics of scattering ofi a nucleon pair. After the absorption of the virtual photon by one member of the pair, the struck nucleon is observed and the other partner of the pair recoils with the negative initial momentum of the struck nucleon. To investigate the nucleon knock-out reaction in detail on these nuclei, a high-precision measurement of the (e, e p) cross section for 3,4 He has been carried out at MAMI. The goal was to determine the longitudinal and the transverse response in parallel kinematics close to the quasi-free peak and for missing energies up to 70 MeV/c.

Results of these measurements can be found in references [35,36]. As an example, flg. 9 shows the missing energy spectrum obtained on the quasi-free peak. No additional strength can be observed above a missing energy of about 25 MeV and the dependence of the cross section on the polarization of the virtual photon is the same as the one within PWIA. Hence, besides an overall reduction of the experimental cross section that is most likely due to FSI, no additional efiects have been observed. By contrast, an RL /RT separation carried out at Saclay [37] for missing momenta above 250 MeV/c found very large deviations of the data from the calculations for the 2-body breakup, even when FSI and MEC are included. Cross sections have also been obtained for the 3body breakup region. Again no additional dependence on the virtual photon polarization has been found beyond PWIA [36]. For the 2-body breakup of 4 He the interference response function, RLT has been extracted at MAMI [39] for missing momenta from 150 MeV/c up to 300 MeV/c at a momentum transfer Q2 ≈ 0.33 (GeV/c)2 . The results

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Fig. 12. As in flg. 9 but calculations by Schiavilla et al. [43]. MEC contribute very little to the cross section while a sizeable contribution to ALT for 300 ≤ pm is observed. At missing momenta above 800 (MeV/c) double rescattering dominates the cross section. Fig. 11. Left: The cross section for the 2-body breakup of 3 He for Q2 = 1.55 (GeV/c)2 [42] (pm < 0 corresponds to φ = 0 and pm > 0 corresponds to φ = 180 ) Right: the extracted ALT ratio. The calculations are Glauber-based by J.M. Laget (see [42]).

A detailed study of the 3,4 He electrodisintegration in quasi-free kinematics has been performed in Hall A at Jefierson Lab. The (e, e p) cross section has been measured in parallel kinematics to allow for a RL /RT separation and in perpendicular kinematics in order to determine the ALT asymmetry. The energy and 3-momentum transfers have been kept constant at ω = 0.84 (GeV) and at q = 1.502 (GeV/c) (Q2 = 1.55 (GeV/c)2 ) and cross sections have been measured up to pm = 1 GeV/c. At these large momentum and energy transfers Glauber based calculations are expected to be valid. In fig. 11 the experimental cross sections for the 2-body breakup are shown. Negative values of pm correspond to cross sections measured at φ = 0◦ while positive pm -values are cross sections measured at φ = 180◦ . Full Glauberbased calculations with modern 3-body wave functions from realistic potentials provide an excellent description of the experimental data up to missing momenta of 150 MeV/c. The same calculations give a good description of the observed cross sections up to missing momenta of 750 MeV/c. Some deviations between experiment and the calculation can be observed for φ = 0◦ between 250 and 500 MeV/c. It is also evident that FSI play a major role for recoil momenta above 300 MeV/c. In Laget’s calculation MEC and IC are found to contribute at most ≈ 25% which is in agreement with the expectation that these contributions diminish with increasing momentum transfer. Another recent Glauber-based calculation by R. Schiavilla et al. [43], where the full spin and isospin dependence of the underlying NN amplitudes is retained, reproduces the experimental 2-body breakup cross section very well (fig. 12). Rescattering efiects have been found to be important over the full range of recoil momenta studied and,

d d p [ b/MeVsr2]

4.2 High-Q2 experiments

d 5 /d

have been compared to calculations by R. Schiavilla et al. [40,41], which show the need for MEC to improve the agreement with the experiment (fig. 10).

10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 0

100

200

300

400

pm [MeV/c]

500

600

700

Fig. 13. The cross section for the 2-body breakup of 3 He for Q2 = 1.55 (GeV/c)2 [42] compared to a calculation by Kaptari et al. [44].

especially, double rescattering is important to improve the agreement between experiment and theory for recoil momenta above 750 MeV/c. A third calculation by Kaptari et al. [44], based on GEA, can also reproduce the measured cross sections up to pm = 700 MeV/c reasonably well. In this calculation MEC and IC contributions have not been included (flg. 13). This again supports the expectation that with increasing momentum transfer, MEC and IC contributions to the cross section decrease. In addition to 2-body breakup, the 3-body breakup reaction has also been observed at missing energies up to a 140 MeV [45]. As in the Saclay experiment at low momentum transfer, a broad bump has been observed whose location shifts to increasing missing energy with increasing missing momentum (flg. 14). The peak location is kinematically in agreement with the breakup of a nucleon pair. The observed nucleon momentum is the sum of the momentum transfer and the initial momentum in the pair and

W.U. Boeglin: Few-nucleon systems at MAMI and beyond

25

Fig. 15. Proton efiective momentum distributions in 3 He extracted from the 3-body breakup reaction (open circles) compared to the 2-body contribution (triangles) [45]. The integration in missing energy ranges from threshold up to 140 MeV. Calculations are by J. M. Laget [46, 32].

Fig. 14. Missing energy spectra for the 3-body beakup of 3 He in Hall A at JLAB [45], the kinematics is the same as in reference [42]. The arrows indicate the expected location of the peak for the breakup of a nucleon pair. The width is a consequence of the center-of-mass motion of the pair. The solid line includes FSI and MEC. The calculations are by J.M. Laget (see [45]).

the recoiling nucleon is the partner. The third nucleon in helium is a spectator. Calculations by J. M. Laget [46] show that FSI within the active nucleon pair contribute strongly to the observed structure while flnal state interactions between the pair nucleons and the spectator nucleon seem to be much less important. Integrating over missing energy and dividing by the electron-nucleon cross section [8] results in the (PWIA) efiective momentum distribution for the 3-body breakup. This can then be compared to the efiective momentum distribution from the 2-body breakup channel. The result is shown in flg. 15. With increasing recoil momenta the momentum distribution is dominated by the 3-body breakup process. However at this point one cannot experimentally distinguish contributions due to FSI within the nucleon pair from contributions of shortrange correlations. Maybe separated response functions will give more information to address this question. 4.3 Search for short-range correlations in 3 He The detection of nucleon pairs should allow one to study nucleon correlations in a direct way. These triple-

coincidence experiments can only be carried out with CW electron beams. 3 He(e, e NN) experiments have been carried out at NIKHEF with the pulse stretcher ring AmPS, at MAMI and at Jefierson Lab. Again, one of the fundamental problems is to isolate the various reaction processes that can lead to the emission of nucleon pairs in addition to initial state correlations. The most prominent processes are flnal state interactions and 2-body currents such as MEC and IC. In an experiment carried out at NIKHEF [47,48], 3 He(e, e pp)n cross sections were measured for momentum transfers around 400 MeV/c and at an energy transfer of about 220 MeV. The experimental results were compared to continuum Faddeev calculations performed with various modern potentials including oneand two-body currents. The relative momenta of pair nucleons was between 500 and 800 MeV/c. It was found that calculations performed with only a one-body hadronic current operator show a fair agreement with the data for small missing (neutron) momenta. This can be interpreted as a direct knock-out of a proton pair. With increasing missing momentum, contributions from MEC and IC also increase. FSI between the pair nucleons depend on the relative emission angle. Decreasing relative emission angle leads to increasing FSI. At MAMI 3 He(e, e pn)p cross sections have been measured and are currently being analyzed. Another study of the 3 He(e, e pp)n reaction has been carried out with CLAS at Jefierson Lab [49]. Via a Dalizplot of the kinetic energies of the observed nucleons divided by the energy transfer, various reaction mechanisms could be selected. The location where both nucleons have the smallest energy fraction corresponds to the process where the virtual photon is absorbed by the uncorrelated nucleon. The observed (small energy fraction) nucleons are then the pair nucleons. This interpretation is supported by

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The European Physical Journal A

Tp1

1

(a)

(b)

Data Cut Data Phase Space

Counts

150

0.5

100

50

0 0

0.5

1 Tp2

0 -1

-0.5

0

0.5 cos(

1 )

pp

Fig. 16. a) Dalitz plot for the lab frame Tp1 /ω versus Tp2 /ω for events with pN > 0.25 GeV/c [49]. b) The cosine of the p-p lab frame opening angle. Open circles for events with small proton momenta (Tp < 0.2 · ω) and closed circles show all data.

e θe q

P

y

pf P

P

Fig. 18. Polarization transfer experiments on 4 He from Mainz [55] and JLAB [56]. The best agreement is obtained for a modifled nucleon form factor within the quark-meson coupling model [57, 58]. 1.05

x

z

e Fig. 17. Deflnition of polarization variables for (e, e N) polarization transfer experiments. e represents the polarized electron beam, e the scattered electron and θe the electron scattering angle. q the virtual photon, pf the flnal nucleon momentum and Px , Py , Pz the polarization of the ejected nucleon.

the relative angular distribution of the observed nucleon pair that shows a pronounced peak at 180◦ (in the Lab frame) corresponding to nucleon emission back-to-back (flg. 16). As before [47,48] one flnds strong FSI within the nucleon pair and only small contributions due to twobody currents and rescattering of the struck nucleon with the pair.

5 Polarization transfer experiments Spin degrees of freedom open up a new, large set of observables that make it possible to address a variety of difierent questions in nuclear physics. Spin observables lead to interference terms between difierent reaction amplitudes. This in turn makes it possible to study very small amplitudes when their efiect is enhanced by a large one. An important application of this is the determination of the nucleon form factors where polarized electrons are scattered ofi an unpolarized target and the polarization of the struck nucleon is determined in a polarimeter [50]. → − − For the (→ e , e N ) reaction on a free nucleon one obtains (see flg. 17) GE P  (Ei + Ef ) tan θe /2 , = − x · GM Pz 2M

(P x /P z )/(Px /Pz )PWIA

1.00

0.95

0.90

4

He EXP H EXP OPT(no CH EX) OPT OPT+MEC

1

0.85

0.80 0.0

0.5

1.0

2

1.5

2

2.0

2.5

3.0

Q [(GeV/c) ] Fig. 19. Comparison of the 4 He polarization transfer experiment [56] and the calculation by Schiavilla et al. [59]. No nucleon form factor modiflcations have been included. OPT: Only one-body currents included and p3 H FSI described by an optical potential with and without (no CH-EX) charge exchange. OPT+MEC: Full optical potential including one- and two-body currents. The error bars in the calculation are due to the Monte Carlo method used and are similar for all calculations shown.

where GE and GM are the electric and magnetic Sachs form factors, Px and Pz are the nucleon polarization, Ei and Ef the incident and scattered (or flnal) electron energy, M the nucleon mass and θe the electron scattering angle. This method has been used extensively to determine the neutron form factor and the high-Q2 behavior of the ratio of the electric to the magnetic form factor of the proton [51,52,53,54] (see contributions by D. Rohe and M. Ostrick). The same process can also be measured in nuclei. Experiments carried out at MAMI [55] and at

W.U. Boeglin: Few-nucleon systems at MAMI and beyond

JLAB [56] on 4 He showed that the ratio R=

(Px /Pz )He (Px /Pz )H

is reduced by about 10% (flg. 18). Some calculations based on the quark meson coupling model [57,58] suggest a possible modiflcation of the nucleon form factors inside nuclei to account for this observation. Another calculation by Schiavilla et al. [59] uses realistic wave functions for the bound state that include correlation efiects. One- and twobody currents are included and spin and isospin dependences in the flnal state interaction including charge exchange have been taken into account. The calculation can reproduce the experimental data without the need of form factor modiflcations as can be seen in flg. 19. New highprecision experiments on 4 He are planned at Jefierson Lab in Hall A to improve and extend the available data.

6 Summary and conclusion In the last 20 years much progress has been made in the knowledge of the structure of few-body systems. The availability of high quality CW beams made it possible to measure coincidence cross sections over a wide range of kinematical variables which were inaccessible before. In parallel, theoretical progress together with increasing computational power has resulted in sophisticated models that agree very well with the data. Coincidence data on 3 He have enjoyed a lot of attention and the analysis of recent data taken at Jefierson Lab is still in progress. The importance of a detailed understanding of flnal state interactions, MEC, and IC is crucial in order to extract information on the short-range structure of light nuclei. While many new single-arm data on the deuteron have been obtained, available coincidence data and especially the lack of new, high-precision response function determinations are disappointing. Triple coincidence experiments on the He nuclei are expected to provide new data on the structure of correlations. However, these experiments are very complex to carry out, analyze, and interpret. Several experiments are being currently analyzed at MAMI and JLAB. The (upgraded ) Mainz Microtron will continue to play a leading role in nuclear physics. I would like to thank H. Arenh˜ ovel, H. Backe, D. Drechsel, J. Friedrich, K-H. Kaiser and Th. Walcher for making MAMI such a success, for their contributions to our fleld and for giving me the opportunity to carry out research at MAMI. I wish you all the best in the future. This work was supported in part by the Department of Energy, DOE grant DE-FG02-99ER41065.

References 1. A. Nogga, H. Kamada, W. Gl˜ ockle, Nucl. Phys. A 689, 357 (2001).

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2. R. Schiavilla, V.R. Pandharipande, R.B. Wiringa, Nucl. Phys. A 449, 219 (1986). 3. J. Carlson, Phys. Rev. C 36, 2026 (1987). 4. B.S. Pudliner et al., Phys. Rev. Lett. 74, 4396 (1995). 5. A.E.L. Dieperink, T. de Forest, Annu. Rev. Nucl. Sci. 25, 1 (1975). 6. S. Frullani, J. Mougey, Adv. Nucl. Phys. 14, 1 (1984). 7. S. Bo– , C. Giusti, F.D. Pacati, Phys. Rep. 226, 1 (1993). 8. T. de Forest, Nucl. Phys. A 392, 232 (1983). 9. M. Bernheim et al., Nucl. Phys. A 365, 349 (1981). 10. S. Turck-Chieze et al., Phys. Lett. B 142, 145 (1984). 11. K.I. Blomqvist et al., Phys. Lett. B 429, 33 (1998). 12. P.E. Ulmer et al., Phys. Rev. Lett. 89, 062301 1 (2002). 13. H. Arenh˜ ovel, W. Leidemann, L. Tomusiak, Phys. Rev. C 52, 1232 (1995). 14. J.E. Ducret et al., Phys. Rev. C 49, 1783 (1994). 15. D. Jordan et al., Phys. Rev. Lett. 76, 1579 (1996). 16. Kasdorp et al., Few-Body Syst. 25, 115 (1997). 17. M. van der Schaar et al., Phys. Rev. Lett. 66, 2855 (1991). 18. M. van der Schaar et al., Phys. Rev. Lett. 68, 776 (1992). 19. W. Fabian, H. Arenh˜ ovel, Nucl. Phys. A 314, 253 (1979). 20. H. Arenh˜ ovel, private communication (2001). 21. R. Boehm, Thesis, University of Mainz (2001). 22. S. Jeschonnek, J.W. Van Orden, Phys. Rev. C 62, 044613 (2000). 23. W.U. Boeglin, private communication (2005). 24. Z.-L. Zhou et al., Phys. Rev. Lett. 87, 172301 (2001). 25. Z.-L. Zhou et al., Proceedings of the MIT-Bates Workshop (1998). 26. F. Ritz, H. G˜ oller, Th. Wilbois, H. Arenh˜ ovel, Phys. Rev. C 52, 1232 (1995). 27. E. Hummel, J.A. Tjon, Phys. Rev. C 49, 21 (1994). 28. A. Pellegrino et al., Phys. Rev. Lett. 78, 4011 (1997). 29. S.M. Dolflni et al., Phys. Rev. C 60, 064622 (1999). 30. H.J. Bulten et al., Phys. Rev. Lett. 74, 4775 (1995). 31. L.L. Frankfurt, M.M. Sargsian, M.I. Strikman, Phys. Rev. C 56, 1124 (1997). 32. J.M. Laget, Phys. Lett. B 609, 49 (2005). 33. C. Marchand et al., Phys. Rev. Lett. 60, 1703 (1988). 34. E. Jans et al., Nucl. Phys. A 475, 687 (1987). 35. R.E.J. Florizone et al., Phys. Rev. Lett. 83, 2308 (1999). 36. A. Kozlov et al., Phys. Rev. Lett. 93, 132301 (2004). 37. J.M. Le Gofi et al., Phys. Rev. C 55, 1600 (1997). 38. R.E.J. Florizone, Thesis, MIT (1999). 39. K. Aniol et al., Eur. Phys. J. A 22, 449 (2004). 40. R. Schiavilla et al., Phys. Rev. Lett. 54, 835 (1990). 41. J. Forest et al., Phys. Rev. C 54, 646 (1996). 42. M. Rvachev et al., Phys. Rev. Lett. 94, 192302 (2005). 43. R. Schiavilla et al., Phys. Rev. C 72, 064003 (2005). 44. L.P. Kaptari, C. Ciofl degli Atti, Phys. Rev. C 71, 024005 (2005). 45. F. Benmokhtar et al., Phys. Rev. Lett. 94, 082305 (2005). 46. J.M. Laget, Few-Body Syst., Suppl. 15, 171 (2003). 47. D.L. Groep et al., Phys. Rev. Lett. 83, 5443 (1999). 48. D.L. Groep et al., Phys. Rev. C 63, 014005 (2000). 49. R.A. Niyazov et al., Phys. Rev. Lett. 92, 52303 (2004). 50. R. Arnold et al., Phys. Rev. C 23, 363 (1981). 51. M.K. Jones et al., Phys. Rev. Lett. 84, 1389 (2000). 52. O. Gayou et al., Phys. Rev. C 64, 038202 (2001). 53. O. Gayou et al., Phys. Rev. Lett. 88, 092301 (2002). 54. T. Pospischil et al., Eur. Phys. J. A 12, 125 (2001). 55. S. Dietrich et al., Phys. Lett. B 500, 47 (2001). 56. S. Strauch et al., Phys. Rev. Lett. 91, 052301 (2003). 57. D.H. Lu et al., Phys. Lett. B 417, 217 (1998). 58. D.H. Lu et al., Phys. Rev. C 60, 068201 (1999). 59. R. Schiavilla et al., Phys. Rev. Lett. 94, 072303 (2005).

Eur. Phys. J. A 28, s01, 29–38 (2006) DOI: 10.1140/epja/i2006-09-004-7

EPJ A direct electronic only

Experiments with polarized 3He at MAMI A1 and A3 Collaboration D. Rohea Departement f¨ ur Physik und Astronomie, Universit¨ at Basel, Klingelbergstr. 82, 4056 Basel, Switzerland / c Societ` Published online: 11 May 2006 –  a Italiana di Fisica / Springer-Verlag 2006

at MAMI have already a long tradition. The A3 Collaboration Abstract. Experiments with polarized 3 He started in 1993 with the aim to measure the electric form factor of the neutron. At this time MAMI was

were possible. Some years before this pilot experiment the second accelerator where experiments with 3 He the development of the apparatus to polarize 3 He in Mainz started. There are two techniques which allow

for to polarize sufficient large quantities of 3 He. Both techniques will be compared and the benefit of 3 He

at MAMI will be given nuclear physics will be discussed. A review of the experiments done so far with 3 He and the progress in the target development, the detector setup and the electron beam performance will be pointed out. PACS. 13.40.Gp Electromagnetic form factors – 13.88.+e Polarization in interactions and scattering – 25.70.Bc Elastic and quasielastic scattering – 29.25.Pj Polarized and other targets

1 Introduction  has gained increasing interest due to its Polarized 3 He special spin structure described below, but also due to the fact that the Schr¨ odinger equation for the three-body system can be exactly solved by means of the Faddeev formalism [1,2]. Furthermore it is the only polarized target which tolerates currents of several μA compared to  3 target. This helps to compensate ≈ 100 nA for a ND the smaller thickness of the gas target. The gas target has the advantage that it is almost not diluted by unpolarized  3 target. carrier material as it is the case for the ND With the availability of highly polarized 3 He of several bars and the delivery of polarized continuous electron beams of high intensity, spin-dependent quantities can be studied, which show a large sensitivity to the underlying nuclear structure and reaction mechanism. Since in 3 He the protons reside with high probability in the S-state, the spin of 3 He is essentially carried by the neutron [3]. This property of the 3 He-spin structure can be best exploited  e, e n) with restriction to in the quasielastic reaction 3 He(  e, e ) small missing momenta as well as in inclusive 3 He( near the top of the quasielastic peak. In such kinematics  the 3 He-target has been used extensively as polarized neutron target to measure the magnetic [4,5,6] and electric [7, 8,9,10,11] form factors of the neutron, Gmn and Gen . Combining the theoretical calculation with the data gives insight into the three-body system and the nuclear a

e-mail: [email protected]

 Final-state interactions (FSI) and mestructure of 3 He. son exchange currents (MEC) can be probed and studied under different kinematical conditions. There are also re does not appear as neuactions and kinematics where 3 He    appears 3 tron target. In the He(e, e p)d reaction, e.g., 3 He as a polarized proton target [12]. Such a measurement will also be presented in these proceedings. The usual Faddeev calculations include FSI and MEC, but they are carried out non-relativistically. It was shown in [13] that in particular relativistic kinematics plays an important role already at Q2 = 0.67 (GeV/c)2 (see sect. 3.2). Less important is the relativistic treatment of the current operator and of the 3 He ground state. A relativistic ground-state wave function became only recently available with the development of a Lorentz boosted nucleon-nucleon potential. It was constructed with the condition to give the same NN phase shifts with the relativistic Lippmann-Schwinger equation as the non-relativistic potential when used with the Schr¨ odinger equation [14]. The problem of the relativistic version of the Faddeev calculation is that it can treat only the interaction between the two nucleons which are not directly involved in the reaction (= spectators). We hope that further ongoing theoretical work will lead to a full relativistic treatment of the three-body system. Experimental data will support such an effort. On occasion of the symposium this contribution aims  at giving a review of experiments performed with 3 He at MAMI in the last 20 years. The huge progress made

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The European Physical Journal A

during this time in the development of the target, the performance of the polarized electron beam and the improvement of the detector setup will be demonstrated. The different objectives for the experiments will be presented. Finally upcoming experiments at MAMI in the near future are briefly presented.

2 Polarization methods For nuclear target applications two methods are in use, metastable-exchange optical pumping (MEOP) [15] and spin-exchange optical pumping (SEOP) [16]. Both methods were already developed in the 60’ies but became only efficient in use with the development of laser light sources of sufficient power and proper frequency band width. MEOP is used for the Mainz target whereas at, e.g., Jefferson Lab the SEOP technique is applied. Both techniques will be shortly explained and the advantages of each method discussed. It should be mentioned that there is a third method to polarize 3 He. Here high magnetic fields and low temperatures are needed which leads to polarizations of 38% in solid 3 He [17]. Due to the low heat conductivity of the solid 3 He this method is not suitable for nuclear-physics experiments with electron beams. In MEOP as well as in SEOP angular momentum is transferred to the atomic electrons by resonant absorption of circularly polarized light and subsequent re-emission of unpolarized light. A magnetic field of 5–30 G defines the quantization axis. In MEOP an atomic transition in 3 He is pumped whose lower level is the metastable 23 S1 state. It is reached by a weak gas discharge (a fraction of 10−6 atoms is excited). Therefore this method works only at low pressures of about 1 mbar which also guarantees a sufficiently long lifetime of the 23 S1 state. With moderate laser power of about 10–20 W and for large gas quantities of 20 liter at 1 mbar a polarization up to 80% can be reached in a minute. The formerly used LNA-Laser (λ = 1083 nm, ≈ 10 W) is nowadays replaced by two ytterbium fiber lasers (15 W each). Due to hyperfine coupling the electronic polarization results in a corresponding alignment of the nuclear spin. Subsequent collisions between polarized 3 He∗ -atoms in the first excited metastable state and unpolarized 3 He-atoms in the ground state transfer the nuclear polarization to the ground state 3 He. The process of metastable-exchange collisions is fast and has a large cross section (10−15 cm2 ). Therefore this method is quite efficient. The drawback of this method is that it can only be applied at low pressures of about 1 mbar. In SEOP an alkali-vapor (usually Rb) is optically pumped by the circularly polarized light provided by a Tisapphire laser or by diode lasers tuned to the D1 -resonance line of 795 nm. Once the Rb is polarized, the polarization is transferred to the 3 He via spin-exchange collisions. The spin-exchange mechanism proceeds via the hyperfine interaction between the 3 He nucleus and the Rb valence electron. This can induce both species to flip their spin. Because this interaction is weak the cross section for this process is small (10−24 cm2 ). Therefore optically thick Rb

vapor and large laser power (> 40 W) are needed to polarize the gas in a target cell of 10 bar within 20 h to 50%. The advantage is that no further compression stage is required and a compact design is possible. To avoid radiation trapping in the optical thick Rb vapor which occurs when unpolarized resonant fluorescence light is emitted and afterwards reabsorbed, 50–100 mbar nitrogen is added. The addition of a fraction of 10−2 N2 leads to ≈ 5 (10)% contribution to the scattering rate from a proton (neutron) and therefore to an effective dilution of the polarization observables. Except for experiments in a storage ring the mass  from MEOP is density of a few mbar of polarized 3 He too low for a nuclear physics experiment. Therefore one or two mechanical compression stages1 are necessary to reach pressures of up to 6 bars. Up to now three different polarizers were in use for nuclear physics experiments at MAMI. The first one was the Toepler compressor which uses 17.6 kg mercury as a piston. The pressure achieved in the 100 cm3 target cell was 1 bar and the polarization could be increased from 38% to 49% from 1993 to 1995. The target cell was filled in a continuous flow (0.1 bar l/h) with polarized gas and the polarization loss from the low pressure pumping cell to the target was 30%. The increase of the polarization was achieved by coating the target cell with cesium to reduce the relaxation of the polarization due to collisions with the container material (glass). The Toepler compressor was developed for the first measurement of the electric form factor of the neutron Gen which is described below. Nowadays the compression stage is replaced by one titanium piston which allows a production rate of 1.5 bar l/h. The polarization losses are negligible and the target cell is filled with 5 bar and 75% polarization. This is a great improvement and increases the performance of the nuclear physics experiment significantly.

3 Experiments with polarized 3 He 3.1 The electric form factor of the neutron 3.1.1 Motivation and techniques Form factors describe the contribution from the inner structure of a scatterer to the interaction. For spin-1/2 particles there are two form factors determining the electromagnetic response, the magnetic and the electric form factor. They are related via a Fourier transformation to the magnetic and to the charge distribution (see sect. 3.1.5), respectively. A form factor independent of the momentum transfer q to the particle would indicate a point-like distribution, hence any q-dependence points to an underlying substructure. The electric form factor Gen of the neutron is particularly sensitive to its internal structure because it is not obscured by the total charge as in the case for the proton. The substructure of the nucleon is determined by the (sea-)quarks and the gluons. 1

In the Hermes experiment the cell was cooled down to 25 K to achieve a compression factor of 3.5.

A1 and A3 Collaboration (D. Rohe): Experiments with polarized 3 He at MAMI 0.08

Gen

0.06 0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5 2

0.6

0.7

0.8

0.9

1

2

Q (GeV/c)

Fig. 1. The data points have been determined from elastic electron-deuteron scattering [19] using the Paris potential. The curves show how much the extracted Gen -values would vary if the data had been analyzed by other than the Paris NN potential (Nijmegen, Argonne V14, Paris, Reid Soft Core from top to bottom) and demonstrate the model dependence of the resulting Gen .

Therefore Gen is a particularly suitable test case for our understanding of the quark degrees of freedom and a constraint for models. QCD would be the first choice to calculate form factors but it is still limited due to the computer power available. Often approximations (quenched lattice) are applied to avoid the computationally expensive part. Recently a full lattice QCD could reproduce the trend of the data [18]. With the extension of the data base in the last few years the theoretical interest also increases and a large variety of models and model-based fits were developed. This was not the case in 1987 when the first Gen measurement at MAMI was planned. The data base was scarce and in particular the error bars exceeded 100%. The reason: Gen is difficult to measure, as its value is small, roughly a factor 10 smaller than the magnetic form factor Gmn . The nucleon form factors enter the (quasi)elastic cross section quadratically, so the magnetic scattering amplitude dominates by far. Therefore an LT separation in the reaction (e, e n) leads to unreasonably large errors. A further complication comes from the fact that there exists no free neutron target of sufficient density. Thus the contribution of the neutron to scattering off the deuteron or 3 He have been employed. The extraction of this contribution, however, requires to account for the nuclear structure and, for elastic scattering off the deuteron, for the large contribution of the proton electric form factor. In 1990 the best data were measured by Platchkov and collaborators [19] using elastic scattering on the deuteron. An LT separation gives the longitudinal and the transverse structure functions A(Q2 ) and B(Q2 ). A(Q2 ) depends quadratically on the charge and quadrupole form factors of the deuteron. The charge form factor dominates for small Q2 (< 0.4 (GeV/c)2 ). It is proportional to (Gep + Gen )2 and therefore contains an interference term Gep times Gen which increases the sensitivity to Gen . On the other hand the contribution from Gep 2 had to be removed which increases the uncertainty in Gen . The main drawback is that the extraction of Gen from A(Q2 ) requires the removal of the contribution from the deuteron structure via a calculation, which depends on the chosen NN potential. This introduces a large model dependence of about 50% as shown in fig. 1 [19]. Analyzing the data

31

with the modern NN potentials would lead to a smaller model uncertainty. The treatment of MEC, which makes a significant correction, introduces further uncertainties. The systematic errors described above can be significantly reduced by exploiting the quadrupole form factor of the deuteron instead of A(Q2 ). The contribution from two-body currents is relatively small and the sensitivity of Gen to the chosen NN potential is reduced [20]. However, at low Q2 the statistical error of FC2 is large because the monopole form factor FC0 dominates the T20 data. Thus, the analysis using A(Q2 ) becomes superior for Q2 < 0.4 (GeV/c)2 . A method which is much less model dependent exploits the observables measurable in a double-polarization experiment. In exclusive reactions it is a sensitive tool to measure Gen . Here the longitudinally polarized electron beam scatters quasi-elastically on deuterons or 3 He, which are either polarized or where the polarization of the knockedout neutron is detected [21] (see contribution of M. Ostrick to this symposium). The asymmetry with respect to the electron helicity contains an interference term Gen times Gmn which amplifies Gen by Gmn . The sensitivity to Gen is largest in the perpendicular asymmetry A⊥ , where the direction of the target spin is perpendicular to the momentum transfer (or the polarization of the scattered neutron is perpendicular to its momentum, respectively). In contrast the parallel asymmetry A does not depend on form factors (for Gen small) and therefore can serve as normalization. Measuring the asymmetry has the advantage that no absolute cross section measurements are required which avoids the effort (and systematic errors) of determining absolute efficiencies, solid angles and luminosity. The electron-target asymmetry is obtained via Aexp =

N + /L+ − N − /L− , N + /L+ + N − /L−

(1)

where L+ (L− ) are the integrated charge and N + (N − ) the number of events for positive (negative) electron helicity. The electron helicity is flipped every second randomly. In general, the asymmetry A can be decomposed according to the direction of the target spin which is given by the angles θS and φS with respect to the momentum transfer q and the scattering plane A = A⊥ sin θS cos φS + A cos θS .

(2)

Before the asymmetry obtained in the experiment can be compared to theory it has to be corrected for the polarization of the electron beam Pe and the target PT as well as for a dilution factor V : 1 Aexp . A= (3) Pe PT V The dilution factor V can come from the scattering on unpolarized carrier material in the target or scattering on the target container (background). Also charge exchange p → n in the shielding in front of the hadron detector con are almost untributes to V because the protons in 3 He polarized. The corrected asymmetry A contains the electromagnetic form factors but also depends on the reaction

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mechanism involved (see below). For scattering on a free neutron with polarization Pn one has  2 τ (1 + τ ) tan(θ/2)Gen Gmn 1 , (4) A⊥ = Pe Pn G2en + G2mn (τ + 2τ (1 + τ ) tan(θ/2))  1 τ 1 + τ + (1+τ )2 tan2 (θ/2) tan(θ/2)G2mn A = . 2 Pe Pn G2en + G2mn (τ + 2τ (1 + τ )) tan(θ/2) Gen is determined best from the ratio of the asymmetries A⊥ and A , A⊥ Gen ∝ , (5) A Gmn

3.1.2 Form factor measurements in the experimental hall A3 at MAMI

Fig. 2. Setup of the Gen experiment in the A3 hall at MAMI. Electrons are detected in the segmented lead glass detector in coincidence with neutrons in the plastic scintillator array. 0.08

3

He(e,e’)

D(e,e’n)

0.06

Gen

instead of A⊥ alone. This has several advantages. The product of the polarizations, Pe · PT in eq. (3), drops out in the ratio, thereby the systematic error introduced with the two measurements of absolute polarizations is considerably reduced. Furthermore the dilution factor V cancels. In addition, theoretical corrections accounting for the nu are reduced in the ratio. Some corclear structure in 3 He rections like the polarization of the neutron, Pn , are affecting both asymmetries in the same way and therefore also cancel in the ratio. For a bound, moving neutron Pn is  measured. usually smaller than the polarization PT of 3 He It is a function of the initial momentum of the neutron and therefore its mean value depends on the detector acceptances.

0.04 0.02 0 0

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The setup for the Gen experiment in the A3 experimental hall is shown in fig. 2. Electrons were detected in a calorimeter of 256 closely packed lead glass counters (4 × 4 cm) with a solid angle of 100 msr in a distance of 1.9 m. The shower produced by an electron extends over about 10 modules. Therefore the energy summed over clusters of detectors was used leading to an energy resolution ΔE/E = 20% FWHM. This moderate energy resolution was sufficient to separate the inelastic contribution from the quasielastically scattered events. The inelastic events, mainly resulting from π-production in the Δ-resonance, have vanishing asymmetry. In case of an admixture this would dilute the asymmetry. In front of the ˇ calorimeter a focusing air Cerenkov detector was placed which suppresses background from electrons scattered on the exit or entrance windows of the target cell or the beam line. Further it serves to discriminate photons and pions from electrons. The neutrons were detected in a plastic scintillator array which covers 250 msr and therefore the entire Fermi cone. It consisted of two walls and could also be used as a neutron polarimeter for the Gen -measurement using the reaction D(e, en) [22,23]. The overall detector thickness of 40 cm yields a neutron detection efficiency of n = 32%. The neutron detector was shielded with 5 cm lead on the front and surrounded by 1 m concrete against electromagnetic background.

Fig. 3. (Uncorrected) result from the first measurement of Gen

[7]. Two double-polarization experiments at MAMI using 3 He performed at Bates in the same time period are also shown [24, 25, 26]. For comparison the data already shown in fig. 1 are displayed as well.

In this setup the already mentioned Toepler compres with polarizations insor was used to produce 1 bar 3 He creasing from 38% to 49% from 1993 to 1995. At the same time the electron polarization could be increased from 30 to 50% by changing the cathode from a bulk to a strained layer GaAsP. Keeping in mind that the √ statistical error of the asymmetry decreases with (Pe PT T )−1 (T : measurement time) both improvements enhance the performance of the experiment significantly. With this setup Gen was measured at Q2 = 0.31 (GeV/c)2 [7,8]. The pilot experiment of Meyerhoff et al. in 1993 [7] did only use a quarter of the detector setup shown in fig. 2. Its result is shown in fig. 3 together with other double-polarization experiments performed at Bates  e, e ) [24,25] and D(e, en) [26]. at the same time using 3 He( In fig. 3 the uncorrected results are shown. In the mid of the 90’ies it was not clear that the measured value for Gen  as polarized neutron target needs a large corusing 3 He rection accounting for FSI. No exact Faddeev calculation

A1 and A3 Collaboration (D. Rohe): Experiments with polarized 3 He at MAMI

33

was available and the diagrammatic approach of Laget [27, 28] indicated a negligible correction. Later it was shown by the Bochum-Krakow group that a correction of about 30% had to be applied on Gen at Q2 = 0.35 (GeV/c)2 [9]. 3.1.3 Non-PWIA contributions An experimental measure for non-PWIA contributions is the target analyzing power Ay . For coplanar scattering Ay is identical to zero in PWIA due to the combination of time reversal invariance and hermiticity of the transition matrix [29]. Thus, a non-zero value of Ay signals FSI and MEC effects and its measurement provides a sensitive check of the calculation of these effects. For an unpolarized beam and the target spin aligned perpendicular to the scattering plane the target analyzing power can be measured 1 N↑ − N↓ , (6) Aoy = PT N ↑ + N ↓ − → where N ↑ , (N ↓ ) are the normalized 3 He(e, e N ) events for target spin aligned parallel (antiparallel) to the normal of the scattering plane. This quantity was measured for   the reactions 3 He(e, e p) and 3 He(e, e n) at Q2 = 0.37 and 2 0.67 (GeV/c) [11] using the experimental setup in the A1 spectrometer hall at MAMI described below. For this the target spin, aligned perpendicular to the scattering plane, was reversed every 2 minutes. Contrary to the determination of Gen , dilution effects do not cancel for Ay in eq. (6) and have to be determined. The main contribution for  3 He(e, e n) comes from charge exchange in the 2 cm lead shielding in front of the hadron detector. This factor was determined using hydrogen as target. Then the recoil proton was tagged with the elastically scattered electrons in the spectrometer and the number of neutrons detected in the scintillator were counted. The correction from charge exchange in the lead shielding is 10 to 15% for (e, e n). The corrected experimental result [11] for the reaction  3 He(e, e n) is shown in fig. 4 together with the data point measured at NIKHEF [30]. Furthermore the calculation

Ay

0.4 0.2 0 0

0.2

0.4 2 2 Q (GeV/c)

0.6

0.8

e n) as Fig. 4. Target asymmetry in the reaction 3 He(e, 2 a function of Q measured at MAMI [11] (circles) and at NIKHEF [30] (square). Shown is the result of the nonrelativistic Faddeev calculation including FSI and MEC (solid) and FSI only (dashed). The dot-dashed line (in green, close to zero) is obtained when neglecting charge-exchange by setting Gep = Gen = 0 in the calculation. The dotted curve represents the result from a diagrammatic approach [27].

Fig. 5. Electron momentum spectrum measured in spectrometer A for two electron helicities. The target spin was aligned parallel to q. The solid line corresponds to a Monte Carlo simulation.

from the Bochum-Krakow group is shown including FSI (dashed) and FSI plus MEC contributions (solid). The effect from MEC is small as expected for quasifree kinematics. A more detailed examination revealed that the large contribution from FSI at small Q2 comes mainly from (e, e p) followed by charge exchange. The full calculation is in good agreement with the data. Also shown is an early result from Laget [27]. Clearly it underestimates the effect from non-PWIA reaction mechanisms. Figure 4 confirms that the FSI contribution and thus the theoretical correction to Gen gets smaller with increasing Q2 . This is expected from simple arguments like the decreasing of charge-exchange cross section with Q2 and the shortening of the interaction time during the reaction at higher momentum transfer. 3.1.4 Form factor measurements in the A1 spectrometer hall at MAMI To avoid a large theoretical correction the Gen measurement was extended to higher Q2 . A pilot experiment was performed already in 1997 in the A1 spectrometer hall at a Q2 of 0.67 (GeV/c)2 followed by a second experiment in 2000 to double the statistics. Using the same setup data on Ay were taken (see sect. 3.1.3). Both the target and detector setup were considerably improved compared to the experiment in the A3 hall. The scattered electrons were detected in the magnetic spectrometer A which has a focal plane detector consisting ˇ of two drift chambers, a scintillator array and a Cerenkov detector. It has a momentum acceptance of 20% and a solid angle of 28 msr. Due to its high resolution and the ˇ efficient pion rejection in the Cerenkov the inelastic contribution can be well separated from the quasielastic region. Further the spectrometer serves to determine the direction of the momentum transfer q with good precision. The angle between q and the target spin direction has to be precisely known for the extraction of Gen from A⊥ . A momentum spectrum of the scattered electrons with an incident energy of 855 MeV is shown in fig. 5. The two spectra belong to different electron helicities and target spin parallel to the momentum transfer. The thick

The European Physical Journal A

solid line represents a Monte Carlo simulation using simple kinematical relations valid in Born approximation. The good agreement supports that the contribution from nonquasielastic events is negligible. The hadron detector was placed in the direction of q. It consists of four layers with five scintillator bars each. In front of the detector two layers of ΔE detectors discriminate protons and neutrons. In 160 cm distance from the target the detector covers a solid angle of about 100 msr. The entire detector was shielded with 10 cm lead except for an opening towards the target where a reduced shield of 2 cm was used. In addition a lead collimator in front of the detector helps to suppress background produced in the downstream beam line.  was enclosed in a rectangular The entire 3 He-target box of 2 mm thick μ-metal and iron except for a cutout towards the opening angle of the spectrometer. The box served as an effective shield for the stray field of the magnetic spectrometers. Three independent pairs of coils inside the box provided a homogeneous magnetic guiding field of ≈ 4 · 10−4 T. With additional correction coils a relative field gradient of less than 5·10−4 cm−1 was achieved. The setup also allowed for an independent rotation by remote control of the target spin in any desired direction with an accuracy of 0.1◦ . The polarization of 3 He was monitored with Adiabatic Fast Passage (AFP) using the technique described in ref. [31] which measures the magnetic field of the oriented spins. Since the AFP-technique destroys part of the polarization (≈ 0.1–0.2%) and since it cannot be used during data-taking due to spin-flipping, it is used only about once in 4 h. Therefore Nuclear Magnetic Resonance (NMR) monitored continuously (≈ every 10 min) the relative polarization and served mainly as online control of the polarization. The systematic error of the absolute polarization is estimated to be 4% and the uncertainty in the relaxation time is 2 h. In the first (second) beam time in 1997 (2000) the averaged target polarization was 32% (36%).  The 3 He-target consisted of a spherical glass container (diameter 9 cm) with two cylindrical extensions sealed with oxygen-free 25 μm Cu windows. The Cu windows were positioned outside of the acceptance of the spectrometer (∼ 5 cm) and shielded with Pb blocks to minimize background from beam-window interactions. The 3 He-target was polarized via metastable optical pumping to a typical polarization of 0.5 and compressed to an operating pressure of 5 bar with a two-stage titanium compressor [32]. Then the target cell was transported in a portable magnetic field to the target pivot. The relaxation time of the polarization due to contact with the surface is increased by careful cleaning and coating with cesium to 80 h. The relaxation time is reduced to about 40 h due to the dipole-dipole interaction between the 3 He-atoms at  by the electron high pressure and due to ionization of 3 He beam. The latter process leads to the creation of 3 He+ 2 and loss of polarization by transfer of angular momentum to the rotational degrees of freedom. An electron current of 10 μA was used with a polarization of 75–80%; the latter was measured with a Moeller polarimeter installed a

0.08

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0.06 0.05

Gen

34

0.04 0.03 0.02 0.01 0 0

0.5

1 2 2 Q (GeV/c)

1.5

2

Fig. 6. Gen extracted from quasielastic scattering of polarized

The data are taken from refs. [33,

and 3 He. electrons from D, D 22, 23, 8, 34, 11, 35, 36, 37]. The dashed line represents the Galster fit [38] and the solid curve the result of [39]. For some of the experimental data points the correction due to the reaction mechanism beyond PWIA is indicated by the size of the arrows.

few meters upstream of the target pivot in the A1 threespectrometer hall. Nowadays currents of more than 20 μA are routinely provided. Combining the measurement of Ay (sect. 3.1.3) and the Faddeev calculation of the Bochum-Krakow group one estimates a correction to Gen at Q2 of 0.67 (GeV/c)2 of (3.4±1.7)%. At this Q2 a relativistic calculation is already needed (see sect. 3.2). The corrected Gen -value is shown in fig. 6 together with all published results from doublepolarization experiments (apart from the very first ones which may be regarded as results of feasibility studies).  Measurements using polarized deuterium instead of 3 He (indicated by a circle in fig. 6) or detecting the polarization of the knocked-out neutron in a polarimeter (squares) were performed at NIKHEF, Jlab and MAMI. Also indicated in fig. 6 are the theoretical corrections applied to the Gen -value extracted from the data. Clearly the correction decreases with increasing Q2 . The dashed line in fig. 6 is the so called Galster fit, determined from the data available in the 70’ies. Most of these data were obtained using elastic electron-deuteron scattering. As mentioned in sect. 3.1.1 this method implies a large model dependence. The Galster fit was obtained by using data up to Q2 = 0.8 (GeV/c)2 with large statistical uncertainty [38]. For this fit the dipole form is modified in such a way that the slope at small Q2 known from n-e scattering could be reproduced. Surprisingly this fit still gives a good description of the actual data set up to Q2 = 0.8 (GeV/c)2 .

3.1.5 Charge distribution of the neutron A new fit to the present data set was provided by Friedrich and Walcher [39] using a phenomenological model of the nucleon. In this model a superposition of two dipoles for the smooth part and two Gaussians to account for a possible bump is used as fitting function. Their result is shown by the solid line in fig. 6. Here a bump

A1 and A3 Collaboration (D. Rohe): Experiments with polarized 3 He at MAMI

35

3.2 Test of the theory at Q2 = 0.67 (GeV/c)2

Fig. 7. Charge distribution (weighted with the radius squared) of the neutron decomposed into the contributions from the bare proton po , the pion cloud and the bare neutron no (picture taken from [39]).

at Q2 ≈ 0.2 (GeV/c)2 describes the data set best, but more precise data are needed for a firm confirmation. It is remarkable that also in Gmn , Gep and Gmp this bump appears in the same Q2 region. Due to the large magnetic moments of the nucleons and the charge of the proton it is only visible if the form factor data are divided by the dipole form factor. A physically motivated fit decomposes the neutron into a bare neutron no and a polarization part: n = (1 − bn )no + bn (po + π − ).

(7)

The bare neutron consists of three quarks with form factors assumed to be of the dipole form. In the polarization part the neutron exists as a bare proton po surrounded by a pion cloud. The form factor of the pion is constructed from the spatial distribution of the harmonic oscillator wave function in a p-state. With six free parameters a good description of both Gen and Gep is achieved. According to this fit the neutron exists to 90% as bare neutron and to bn ≈ 10% as proton with pion cloud. From such a fit the charge distribution of the neutron can be obtained by a Fourier transformation,  ∞ 1 sin(Qr) 2 ρ(r) = Q dQ, Gen (Q2 ) (8) 2π 2 0 Qr where Q is the momentum transfer in the Breit frame, i.e. a frame with no energy transfer to the nucleon. The behavior at high Q2 mainly determines the charge distribution deep inside the nucleon at small radii r. In fig. 7 the result for the charge distributions ρ(r) r2 of the neutron (black line) as well as for the three components in eq. (7) are shown. It is remarkable that the pion contribution extends as far as 2 fm (maximum at ≈ 1.5 fm). In contrast, the authors of ref. [40] separated the contribution of the two-pion continuum and found a peak at a distance of only 0.3 fm. The maximum of the pion cloud in the model of Friedrich and Walcher corresponds to the Compton wavelength of the pion (λ = 1.43 fm) determining the range of the nuclear force in the Yukawa model. This confirms that in this model only one pion is taken into account by construction. This consideration might resolve part of the disagreement because in ref. [40] two-pion contributions are considered.

The Faddeev calculation mentioned so far is fully nonrelativistic and it was not clear at which Q2 relativistic effects would become non-negligible. On the other hand, the contribution from non-PWIA reaction mechanisms to  e, e n) are small at high Q2 as the asymmetries in 3 He( shown in sect. 3.1.4. More sensitive are the asymmetries  e, e p). These asymmetries A and A⊥ in the reaction 3 He( are expected to be small because the two protons are most of the time in the S-state. In this case the asymmetries vanish, unless one resolves the different exit channels (see sect. 3.3). Therefore comparing the experimental result to the theory provides a sensitive test to effects from reaction  structure. Both might mechanisms as well as from the 3 He need a relativistic treatment. There are several ingredients in the Faddeev calculation which might be treated relativistically or non-relativistically. This includes the 1-body current operator, the T -matrix element describing the FSI, the kinematics and  ground state wave function. It should be menthe 3 He tioned that up to now in the relativistic description only the interaction between the spectator nucleons, i.e. the ones which are not involved in the primary reaction, can be included. This is called FSI23 or rescattering term of first order. At the moment there are no exact calculations available for 3 He which can treat MEC and full FSI at high  ground state beQ2 . The relativistic treatment of the 3 He came only recently available with the development of a Lorentz boosted NN potential. In ref. [14] such a potential was obtained and used in a relativistic 3N-Faddeev equation for the bound state to calculate the triton binding energy. The results presented below are still based on an exact but non-relativistic 3 He ground state. A newer calculation prepared for a recent proposal to measure Gen at Q2 of 1.5 (GeV/c)2 shows that the difference is small. The dependence on the NN interaction was studied with a calculation which employs the CD-Bonn NN potential [41] instead of the AV18 NN potential [42]. The difference in the result is negligible. It should be mentioned that the potential approach is not strictly valid when the center-of-mass energy of the 3N-system, E3N , is well above the pion production threshold. E3N can be obtained for the 3-body breakup via  E3N = (MHe + ω)2 − |q|2 − 2Mp − Mn . (9) However, in quasi-elastic kinematics the focus is mostly on the region of phase space, where one of the nucleons is struck with a high energy and momentum and leaves the remaining two-nucleon system with a rather small internal energy. Thus this approximation, which has to be made also in other calculations for 3 He and deuterium [43,44], might not be too serious.  e, e p) were taken siThe data on the reaction 3 He( multaneously to the measurement of Gen at Q2 = 0.67 (GeV/c)2 . Protons were selected in the hadron detector by requiring hits in two consecutive ΔE detectors. The background in the coincidence time spectrum, determined from the time difference between the first bar

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Fig. 8. Experimental results of A (left) and A⊥ (right) for the central region of the quasi-elastic peak as a function of the scattering angle of the knocked-out proton. The result of the full (PWIA) calculation is shown with solid (dashed) line. The results of three full calculations, however with a non-relativistic current (dot), or with only a (v/c)2 correction (dot-dot-dashed), or with non-relativistic kinematics (dot-dash) are also shown.

in the hadron detector and the scintillator plane of spectrometer A, was negligible. In order to study the effect of FSI on the asymmetries in different kinematic regions, the quasi-elastic peak is divided into two regions of ω. One region covers the peak and therefore emphasizes low nucleon momenta whereas the other region covers the low ω tail sensitive preferentially to high nucleon momenta. The events in each of the two regions are summed over the entire acceptance of the out-of-plane angle of electron and proton and over the electron scattering angle in a range from 75.8◦ to 81.8◦ . In fig. 8 the parallel and perpendicular asymmetries in the central region of the quasielastic peak are shown as a function of the scattering angle of the proton. They are compared to the theory which contains the two-body (2BB) and three-body breakup (3BB). The 3BB channel is integrated over the first 26 MeV. As can be seen from the figures the PWIA calculation (dashed line) clearly disagrees with the data. From the calculations which include FSI23 only, the one with non-relativistic kinematics (dotdashed line) cannot describe the experimental results. Relativistic (solid line) or non-relativistic (dots) treatment of the current operator does not make a large difference. The calculation taking into account relativistic kinematics and FSI23 provides a good description of the data. Both ingredients are important to achieve agreement with the experimental results.  3.3 Structure of 3 He In the experiment described in the previous section it was not possible to separate the 2BB and 3BB channel due to the limited resolution of the hadron detector. For a  spectromebetter understanding of the structure of 3 He ter B was taken for proton detection. The kinematics was limited to the central region of the quasielastic momentum distribution at Q2 of 0.31 (GeV/c)2 . Each hour the target spin was turned to measure the parallel, perpendic-

ular, antiparallel and antiperpendicular asymmetry alternately with the purpose to reduce the systematic errors. The target cell was of the same kind as already used for the Gen measurement (see sect. 3.1.4). A new polarizer was used consisting of one-stage titanium compressor [45] with effectively no polarization loss during the transfer from the low pressure gas reservoir to the target container. The 3 He was optically pumped with two Ytterbium fiber lasers each providing 15 W on the resonance transition (1083 nm). With this setup an initial target polarization of 70 to 75% could be achieved. Averaged over the beam time period and accounting for relaxation a target polarization PT of (49.8 ± 0.3(stat.) ± 2(syst.))% was obtained. From the measured kinematic variables in the two spectrometers, the missing energy is reconstructed according to (10) Em = E − Ee − Tp − TR . Here, E (Ee ) is the initial (final) electron energy and Tp is the kinetic energy of the outgoing proton. TR is the kinetic energy of the (undetected) recoiling (A–1)-system, which is reconstructed from the missing momentum under the assumption of 2BB. The resulting Em distribution reconstructed from the data is shown in fig. 9 as thick solid line. The resolution is limited mainly by the properties of the target cell and not by the resolution of the spectrometers. The FWHM of 1 MeV allows a clear separation of the Em -regions where only 2BB or 2BB and 3BB contribute. The Em -region from 4.0 to 6.5 MeV is interpreted as pure 2BB. This cut was chosen to avoid any contribution from the 3BB-channel (starting at 7.7 MeV) considering the experimental Em resolution. In agreement with ref. [46], the yield of the 3BB is negligible beyond 25 MeV. Therefore the cut for the 3BB-channel was made from 7.5 to 25.5 MeV in the Em spectrum. Because the 3BB resides on the radiation tail of the 2BB, the latter has to be accounted for in the analysis of the 3BB-region of the measured spectrum. To this end, the tail was calculated in a Monte Carlo simulation which accounts for

37

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A1 and A3 Collaboration (D. Rohe): Experiments with polarized 3 He at MAMI

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internal and external bremsstrahlung, ionization loss and experimental energy resolution adjusted to the experimental distribution. The simulated 2BB distribution is shown as thin red line in fig. 9. Subtracting this from the data leads to the distribution belonging to the 3BB channel which is also shown in fig. 9. The ratio of the Monte Carlo simulation of the 2BB to the experimental data in the region of the 3BB is denoted by a23 . For the region 7.5 < Em < 25.5 MeV it amounts to a23 = 0.434 ± 0.002(stat.) ± 0.015(sys.). Then the asymmetry A3BB for the 3BB-channel is extracted from the asymmetry A2+3BB in the 3BB region by accounting for the contribution from the radiation tail A2+3BB − A2BB a23 . 1 − a23

−||

||

⊥ −⊥ ⊥

target spin direction

Fig. 9. Experimental Em distribution (thick line) and the simulation of the 2BB (thin red line). The difference is shown as thick black line 3BB.

A3BB =

||

(11)

All asymmetries are corrected for target and electron polarization. In fig. 10 the parallel and perpendicular asymmetries A3BB and A2BB are compared to two calculations of the Bochum-Krakow group. One uses PWIA only (dotdashed), the other accounts for full FSI and MEC (solid line). The effect of MEC is negligible in this kinematics. The data integrated over the total detector acceptance are in good agreement with the calculation including FSI. The calculation shows that the FSI contribution is small in the 2BB while it is large in 3BB. This suggests that the main contribution of FSI results from the rescattering term which does not exist in the 2BB, and not from direct FSI. This was also confirmed by further examination of the theoretical result by Golak [47]. In the 2BB channel the spins of the neutron and proton in the recoiling deuteron are coupled to one, therefore they are parallel. Consequently, in a simplified picture, the spin of the second (knocked-out) proton must be antiparallel  Correct to the deuteron spin and thus to the spin of 3 He. coupling of the spins 1 and 1/2 to 1/2 leads to 33% polarization of the knocked-out proton relative to that of the  polarized 3 He-target. This is precisely what is observed as

Fig. 10. Comparison of the data to the theoretical calculation for the 2BB and 3BB for the four target spin directions (anti)parallel (, –; left panel) and (anti)perpendicular (⊥, – ⊥; right panel). In addition the combined sum for the parallel and perpendicular position is shown ( and ⊥, respectively). To facilitate the comparison, all 2BB (3BB)data are shown with positive (negative) sign. PWIA: dot-dashed lines. Full calculation including FSI and MEC: solid lines. Statistical errors point up, systematic uncertainties point down. For the 2BB the size of the error bars is smaller than the symbols.

 A2BB . In the 2BB channel, the polarized 3 He-target can thus be interpreted as a polarized proton target. For the 3BB channel the situation is different. In PWIA the asymmetry is almost zero for the 3BB which reflects the fact that the two protons, which are dominantly in the S-state and thus have opposite spin orientation, now contribute equally to the knock-out reaction. The inclusion of FSI, however, leads to an asymmetry, which is larger and opposite in sign compared to the 2BB. The main effect comes from the np t-matrix (rescattering term). Since different spin combinations of the singlet and  target cannot be intriplet np t-matrix contribute, the 3 He terpreted as a polarized proton target in the 3BB channel.

4 Summary and outlook In this contribution a review of the experiments with po performed at MAMI was given. The effort to larized 3 He build a machine to polarize 3 He started already in 1987.  was performed The first experiment at MAMI with 3 He in the experimental hall A3 to measure the electric form factor of the neutron. Experiments with the same purpose at higher Q2 followed, using improved target and detector setups in the three-spectrometer hall A1. With the new detector setup a better discrimination of inelastic events from the ones quasielastically scattered is possible. The performance of the target was steadily improved due to the development of new polarizers. This resulted in a more dense target (5 bar) with higher polarization (PT = 50%). In addition the electron source was improved using a

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strained layer crystal. This led to nowadays available currents of 20 μA with an electron polarization of 75%. Parallel to the experiments the non-relativistic Faddeev calculation was developed by the Bochum-Krakow group. One of the first applications was the calculation of the correction of Gen at Q2 = 0.35 (GeV/c)2 due to FSI which leads to a deviation of the asymmetry mea from that for a free neutron. The influence sured on 3 He of FSI was also confirmed by measuring the target asymmetry Ay where the beam is unpolarized and the target spin perpendicular to the scattering plane. This quantity is particularly sensitive to FSI and MEC contributions. In PWIA it vanishes. Good agreement between data and theory was found. Another experiment concentrated on the question when a relativistic calculation is needed and which ingredients need to be treated relativistically. For this the reac e, e p) was investigated at Q2 = 0.67 (GeV/c)2 . tion 3 He( It turned out that the kinematics has to be treated relativistically already at this Q2 . On the other hand, a relativistic current operator is much less important. At present a relativistic calculation is only possible in PWIA and with FSI23 included.  To become more sensitive to the inner structure of 3 He   3 the 2BB and 3BB channels in the reaction He(e, e p) were separated. Also here the theoretical calculation is in good agreement with the data. It is interesting that in the 2BB channel, which is almost unaffected by FSI at the kine target can be matics of the present experiment, the 3 He considered as a polarized proton target with the proton  By contrast, the 3BB chanspin opposite to that of 3 He.  cannot be nel is largely affected by FSI. In this case 3 He interpreted as polarized proton target. All these reactions considered so far were not sensitive to MEC because the kinematics were chosen to correspond to the top of the quasielastic peak and the Q2 was sufficiently high. At Q2 < 0.2 (GeV/c)2 MEC contribute significantly to the reaction and modify the asymmetries. Since MEC are not so well understood as compared to FSI it is planned to study kinematics which are sensitive to MEC. The data taken to measure Gen at Q2 = 0.25 (GeV/c)2 are affected by MEC in some kinematical regions covered by the detector acceptance. With the upgrade of MAMI to MAMI-C the Gen measurement will be pushed to Q2 = 1.5 (GeV/c)2 . For this a new hadron detector is under construction which should have a higher neutron detection efficiency.  with (poThere are also plans to use polarized 3 He  larized) photons in the A2 experimental hall. Then 3 He would be used as a polarized neutron target to measure the Gerasimov-Drell-Hearn sum rule. For this a new target setup is needed which is already under consideration. Finally I want to thank Karl-Heinz Kaiser for the excellent beam quality at MAMI and for his effort to adjust and setup the beam for our sensitive experiments. Then I want to thank J¨ org Friedrich and Thomas Walcher for their support and ad-

vice as well as Hartmuth Arenh¨ ovel, Hartmut Backe and Dieter Drechsel for the good atmosphere in the institute.

References 1. J. Golak et al., Phys. Rev. C 65, 064004 (2002). 2. R.W. Schulze, P.U. Sauer, Phys. Rev. C 48, 38 (1993). 3. B. Blankleider, R.M. Woloshyn, Phys. Rev. C 29, 538 (1984). 4. H. Gao et al., Phys. Rev. C 50, R546 (1994). 5. W. Xu et al., Phys. Rev. Lett. 85, 2900 (2000). 6. W. Xu et al., Phys. Rev. C 67, 012201(R) (2003). 7. M. Meyerhoff et al., Phys. Lett. B 327, 201 (1994). 8. J. Becker et al., Eur. Phys. J. A 6, 329 (1999). 9. J. Golak et al., Phys. Rev. C 63, 034006 (2001). 10. D. Rohe et al., Phys. Rev. Lett. 83, 4257 (1999). 11. J. Bermuth et al., Phys. Lett. B 564, 199 (2003). 12. P. Achenbach et al., Eur. Phys. J. A 25, 177 (2005). 13. C. Carasco et al., Phys. Lett. B 599, 41 (2003). 14. H. Kamada, W. Gl¨ ockle, J. Golak, Ch. Elster, Phys. Rev. C 66, 044010 (2002). 15. G.K. Walters, F.D. Colgrove, L.D. Schearer, Phys. Rev. Lett. 8, 439 (1962). 16. M.A. Bouchiat, T.R. Carver, C.M. Varnum, Phys. Rev. Lett. 5, 373 (1960). 17. D.G. Haase et al., Nucl. Instrum. Methods A 402, 341 (1998). 18. M. G¨ ockeler et al., Nucl. Phys. A 755, 537 (2005). 19. S. Platchkov et al., Nucl. Phys. A 510, 740 (1990). 20. R. Schiavilla, I. Sick, Phys. Rev. C 64, 041002(R) (2001). 21. R.G. Arnold, C.E. Carlson, F. Gross, Phys. Rev. C 23, 363 (1981). 22. C. Herberg et al., Eur. Phys. J. A 5, 131 (1999). 23. M. Ostrick et al., Phys. Rev. Lett. 83, 276 (1999). 24. C.E. Jones-Woodward et al., Phys. Rev. C 44, 571 (1991). 25. A.K. Thompson et al., Phys. Rev. Lett. 68, 2901 (1992). 26. T. Eden et al., Phys. Rev. C 50, R1749 (1994). 27. J.M. Laget, Phys. Lett. B 273, 367 (1991). 28. J.M. Laget, Phys. Lett. B 276, 398 (1992). 29. H.E. Conzett, Nucl. Phys. A 628, 81 (1998). 30. H.R. Poolman, PhD Thesis, Vrije Universiteit te Amsterdam, 1999. 31. E. Wilms et al., Nucl. Instrum. Methods A 401, 491 (1997). 32. R. Surkau et al., Nucl. Instrum. Methods A 384, 444 (1997). 33. I. Passchier et al., Phys. Rev. Lett. 82, 4988 (1999). 34. H. Zhu et al., Phys. Rev. Lett. 87, 081801 (2001). 35. R. Madey et al., Phys. Rev. Lett. 91, 122002 (2003). 36. G. Warren et al., Phys. Rev. Lett. 92, 042301 (2004). 37. D.I. Glazier et al., Eur. Phys. J. A 24, 101 (2005). 38. S. Galster et al., Nucl. Phys. B 32, 221 (1971). 39. J. Friedrich, Th. Walcher, Eur. Phys. J. A 17, 607 (2003). 40. H.-W. Hammer, D. Drechsel, Ulf-G. Meißner, Phys. Lett. B 586, 291 (2004). 41. R. Machleidt, F. Sammarruca, Y. Song, Phys. Rev. C 53, 1483 (1996). 42. R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, Phys. Rev. C 51, 38 (1995). 43. A. Deltuva et al., Phys. Rev. C 70, 034004 (2004). 44. H. Arenh¨ ovel, W. Leidemann, E. Tomusiak, Phys. Rev. C 46, 455 (1992). 45. E.W. Otten, Europhys. News 35, 16 (2004). 46. R.E.J. Florizone et al., Phys. Rev. Lett. 83, 2308 (1999). 47. J. Golak, private communication, 2005.

Eur. Phys. J. A 28, s01, 39 48 (2006) DOI: 10.1140/epja/i2006-09-005-6

EPJ A direct electronic only

Few-nucleon systems (theory) M. Schwamba Institut f˜ ur Kernphysik, Johannes Gutenberg-Universit˜ at, D-55099 Mainz, Germany / Published online: 9 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. An overview over present achievements and future challenges in the fleld of few-nucleon systems is presented. Special emphasis is laid on the construction of a unifled approach to hadronic and electromagnetic reactions on few-nucleon systems, necessary for studying the borderline between quark-gluon and efiective descriptions. PACS. 13.40.-f Electromagnetic processes and properties induced reactions 25.20.-x Photonuclear reactions

1 Introduction One of the most challenging topics in modern physics deals with the structure of atomic nuclei and their constituents. Despite the large efiorts in the last decades, our present understanding of hadronic systems is still far from being satisfactory. The non-Abelian gauge structure of the underlying fundamental theory quantum chromodynamics (QCD) leads to enormous complications in practical applications. Therefore, one uses in conventional nuclear physics not the fundamental quarks and gluons of QCD, but nucleons, isobars and mesons as relevant degrees of freedom (d. o. f.). These so-called efiective approaches are presently still the most promising ones for reaching a quantitative understanding of hadronic physics at low and intermediate energies below about 1 GeV excitation energy. A well-known example for the success of this effective picture is the quantitative understanding of N N scattering data below pion threshold in terms of mesonexchange mechanisms between two interacting nucleons (for a pedagogical introduction, see [1]). On the other side it is clear that the efiective description will break down at some su– ciently high energy/momentum transfer. Moreover, it is presently not clear whether a clear cut borderline exists or whether even at relatively small energies quark and gluon degrees of freedom manifest themselves in speciflc reactions and observables. It is obvious that for a detailed study of such fundamental questions a profound understanding of few-nucleon systems is inevitably necessary because the corresponding theoretical treatment is naturally the most cleanest one. Moreover light nuclei, especially the deuteron and 3 He, a

e-mail: [email protected]

21.45.+v Few-body systems

25.30.-c Lepton-

may serve as efiective neutron targets so that a better understanding of few-nucleon systems may also lead to a better understanding of neutron properties. Concerning the test of efiective theories, electromagnetic (e. m.) reactions have always been at the forefront in nuclear structure investigations. The electromagnetic interaction is well known from classical electrodynamics and is weak enough to allow a perturbative treatment in terms of the flne structure constant α ≈ 1/137. In this work, selected examples of present achievements in the fleld of few-nucleon systems are presented. We concentrate ourselves mainly on the two-nucleon system which deserves special attention because it has the same relevance in nuclear physics as the H-atom in atomic physics. However, also some recent progress in the description of more complex few-nucleon systems is presented.

2 The two-nucleon system 2.1 Introduction Although the two-nucleon system is the simplest few-nucleon system, it is far from being trivial. Even if we restrict ourselves to energies below the two-pion threshold, this becomes obvious by noting that quite a large number of difierent reactions is possible like N N -scattering Compton scattering e. m. deuteron breakup photopionproduction elastic electron scattering Bremsstrahlung pionic reactions

NN → NN, γd → γd, γd → N N, ed → e N N , γd → πd, γd → πN N , ed → e d, N N → γN N , πd → πd, πd  N N , N N → πN N .

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(a)

d

A

(b)

(c)

d

d

Fig. 1. Diagrammatic illustration of possible destinations of an electromagnetically produced pion. Further discussion in the text.

Thus the two-nucleon system ofiers a great variety of possible interaction mechanisms worthwhile to be studied. A very important point for the forthcoming discussion is the fact that these difierent reactions cannot be treated independently. First of all, they are linked by unitarity as becomes obvious by considering the corresponding optical theorems like Im T (N N → N N ) ∼ σtot (N N → N N, πd, πN N, . . .), (1) Im T (πd → πd) ∼ σtot (πd → N N, πd, πN N, . . .), (2) Im T (γd → γd) ∼ σtot (γd → N N, πd, πN N, . . .), (3) where the left sides are understood to be evaluated in forward direction. This means, for example, that the forward Compton scattering amplitude is related to all possible reactions with a photon and a deuteron in the initial state. If the restriction to energies below the two-pion threshold is dropped, of course also additional channels like 2π-, Kand η-production have to be considered. Therefore, as a consequence of unitarity, a unified description of all possible reactions is necessary. Before we outline such an approach in some detail, let us try to understand the connection of the difierent above-mentioned reactions from a more intuitive point of view without referring to formal arguments based on unitarity. For that purpose, let us consider the three diagrams depicted in flg. 1. In all of them, a photon is absorbed by a deuteron producing a real or virtual pion. The three diagrams difier with respect to their flnal state: In diagram (a) the pion leaves the two-nucleon system, whereas in diagrams (b) and (c) it is absorbed by one of the two outgoing nucleons. Despite the close relationship of the three diagrams, their physical interpretation is completely difierent: diagram (a) is a contribution to photopionproduction on the deuteron, diagram (b) a part of the meson-exchange currents (MEC) to deuteron photodisintegration, and diagram (c) contributes to the anomalous magnetic moment of the hit nucleon. This simple example illustrates that single-particle properties, pion production mechanisms and meson-exchange currents are closely related. This fact again underlines the

above-mentioned necessity for a unifled approach to the difierent possible reactions in the one- and two-nucleon sector. Needless to say that such a consistent picture is in principle also required for more complex nuclei. In a flrst step, we may restrict ourselves to the twonucleon system for energies up to the Δ-region so that a basically nonrelativistic treatment should be su– cient and channels with at most one asymptotically free pion need solely to be studied. Despite these simple boundary conditions, the construction of such a unifled approach is far from being trivial and in fact not successfully realized till now. In most existing approaches, only one or two reactions of interest are selected and the rest is just ignored. Moreover, simplifying approximations are used in order to reduce the numerical complexity. To be more precise, let us return to the difierent diagrams in flg. 1. In the meson-exchange contribution (b), a proper description of the propagation of the intermediate πN N -system (cut A) requires for a given invariant energy W of the system the numerical evaluation of the exact free retarded propagator G0 (z) = (z − HN (1) − HN (2) − Hπ )

−1

z = W ± i, (4) where HN (i) and Hπ describe the kinetic energy operators for nucleon i and the pion, respectively. Although this expression looks quite simple, its structure is quite nontrivial: It is nonlocal and due to its energy dependence non-Hermitean. Moreover, G0 (z) has poles beyond pion threshold leading to logarithmic singularities known from three-body scattering theory [2,3]. Intuitively, they describe the possibility that beyond pion threshold the produced pion must not necessarily be reabsorbed by one of the nucleons but may become onshell as indicated in diagram (a). Therefore, the singularities link N N - to πN N -scattering as required by the optical theorems (1) through (3) and their correct treatment is inevitably necessary. Due to these features of G0 , it is obvious that its numerical implementation is rather involved. Therefore, in most of the approaches an approximative treatment, the so-called static limit is used by assuming that the nucleons are inflnitely heavy during the meson exchange (cut A in diagram (b) of flg. 1) so that in consequence no energy transfer occurs. The resulting static propagator =− Gstat 0

1 Hπ

,

(5)

is local, energy independent and regular. Due to these nice features, which lead to large numerical simpliflcations, it is even nowadays very popular and used for example in stateof-the-art high precision N N -potentials like AV18 [4] or CD-Bonn [5,6]. The static limit works well below pion threshold but we will see that this approximation fails at higher energies. Intuitively, this is not very surprising: Due to the lack of singularities in (5), the pion is frozen inside the hadronic system and therefore no longer a dynamic degree of freedom. Finally, let us make a comment on the treatment of diagram (c) in flg. 1. In conventional approaches, it is just

M. Schwamb: Few-nucleon systems (theory)

N

N

N

N

41

the scattering equation for continuum states. The latter reads for a given invariant energy W as follows: |Ψ

(±)

= ±iG(W ± i)|φ

(P W )

,

(6)

(P W )

Fig. 2. Graphical illustration of meson-nucleon-nucleon vertices VXN (left) and VN X (right) which serve as the basic ingredients for the hadronic interaction.

denotes a plane-wave state (i.e. either a where |φ noninteracting N N -, πd- or πN N -system) and G the full propagator G(z) =

1 1 = , z−H z − H0 − V

(7)

containing the potential V and the kinetic energy operator H0 . The full propagator can be rewritten in terms of the scattering amplitude T (z) = V + V G0 (z)T (z) with G0 (z) = (z − H0 )

−1

(8) according to Fig. 3. Graphical illustration of the one-boson-exchange potential (left) and mesonic loop contributions to the nucleon self-energy. Both terms are generated by the second iteration of the XN -vertices depicted in flg. 1.

neglected by arguing that one uses physical nucleons as relevant efiective degrees of freedom which already contain the correct anomalous magnetic moments so that the additional consideration of diagram (c) would lead to double counting. However, one has to take into account that the loop in diagram (c) is energy dependent so that its simulation by a current governed by a constant anomalous magnetic moment might be a rather crude approximation. Moreover, due to the occurrence of the retarded propagator (4), the loop can become complex so that its neglect violates the optical theorem (3) beyond pion threshold. In order to solve these problems, a careful distinction between so-called bare and physical nucleons is necessary. This conceptual complication is in most cases just circumvented by neglecting diagram (c).

2.2 Model structure In this section, we present the general structure of our approach developed within the past years. It is suitable to study all hadronic and electromagnetic reactions on the two-nucleon system for energies up to the Δ-resonance region with at most one asymptotic free pion. In order to keep the discussion as transparent as possible, technical aspects are mostly avoided. The interested reader is referred to [7,8,9,10] concerning further details. In order to treat a meson X as a dynamic degree of freedom, one has to work within a Hilbert space H where X is treated explicitly. Consequently, one has to allow for transitions between the N N - and the XN N -sector. In our approach, they are generated by conventional XN vertices VXN and VN X = (VXN )† (X ∈ {π, ρ, ω, σ, . . .}), see flg. 2, known from N N -potential theory [1]. They serve as the basic ingredients of the hadronic interaction V in the Schr˜ odinger equation for the deuteron bound state and

G(z) = G0 (z) + G0 (z)T (z)G0 (z).

(9)

It contains, therefore, the interaction V up to inflnite order. The second order terms of (8), which are depicted in flg. 3, consist flrst of all of a one-boson-exchange potential (OBEP) of the type V OBEP (z) = VN X (1)G0 (z)VXN (2) + (1 ↔ 2),

(10)

and a contribution to the nucleon self-energy, see equation (11) below. Consequently, the possibility of meson production and annihilation as well as the structure of the N N -force is based on the same XN -vertex VXN . This allows therefore to construct the desired unifled approach. The price we have to pay is at least twofold. First of all, the N N interaction is more complex as conventional ones because it has to be treated in the exact retarded, energy dependent manner. In our explicit realization, we use the parametrization of the Elster potential [11], which just consists of the diagrams of flg. 3 with inclusion of π-, ρ-, ω-, σ-, δ- and η-exchange. The free parameters of the corresponding vertices (cutofis, coupling constants) are fltted to the N N -scattering phase shifts for energies up to the pion threshold. As a second complication, the mesonic loop diagrams V self (z) = VN X (1)G0 (z)VXN (1) + (1 ↔ 2)

(11)

appear, depicted on the right-hand side of flg. 3. In order to avoid any double counting, we have to distinguish therefore bare from physical nucleons. Whereas the flrst ones are the basic d. o. f. of our Hilbert space, the latter contain, among other things, the loop contributions (11). This distinction requires a proper renormalization procedure in order to formulate the model in a self-consistent manner, see [8] for more details. Its neglect leads to a severe violation of unitarity beyond pion threshold [11]. Next, we introduce the electromagnetic interaction. It is done by using the canonical gauge invariance preserving method of minimal substitution, and typical prototypes of resulting currents are depicted in flg. 4. More

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FSI in pion production

NN−interaction X

Fig. 4. Examples for current contributions in the one- and twonucleon sector: left: one-body current; middle: meson-exchange current (MEC); right: electromagnetic loop correction. Δ

Fig. 7. The amplitude T (12) contributes both to the N N interaction as well as to flnal state interactions (FSI) in pion production.

Neglecting three-body forces, T X has the form   T X (z) = V π + V N + V π + V N G0 (z)T X (z),

Δ

(12)

π

Δ

Δ

Fig. 5. πN -scattering in the P33 channel.

Fig. 6. Diagrammatic representation of the amplitude T X (12).

details can be found in [9,10]. In practice, gauge invariance (as well as unitarity) is not exactly fulfllled due to some technical reasons: Whereas the Elster potential is treated in a completely relativistic manner concerning the vertices, the vertex structure in the corresponding MEC is presently treated only nonrelativistically within a p/MN expansion. Moreover, MEC of at least fourth order in the πN -coupling constant are necessary to preserve gauge invariance exactly [9]. Their handling is technically very complicated and, therefore, presently neglected. These violations of gauge invariance and unitarity occur fortunately only at higher order in the 1/MN -expansion. The model discussed so far is only suitable for energies below the pion threshold. In order to allow for higher energies, nuclear resonances must necessarily be incorporated. In the present approach we restrict ourselves to the Δ which is again considered as a bare particle (Δ) with vanishing decay width. Similar to [12], its coupling to the πN -system is generated by a suitable πN Δ-vertex VΔπ whose parametrization is flxed by studying πN -scattering in the P33 -channel, see flg. 5. In a similar manner, the electromagnetic transition γN → Δ is flxed once for all by considering photopionproduction on the nucleon in the M1+ (3/2)-multipole [9]. As next step, the Δ has to be introduced in the two-nucleon system. This is performed nonperturbatively within a N N -N Δ coupled-channel approach, see [8] for more details. Last but not least, in addition the possibility of mutual interactions within the πN N -system needs to be considered. This can be tackled using standard three-body techniques for the relevant amplitude T X depicted in flg. 6.

where V describes the N N -interaction in the presence of a spectator pion, and V N the πN -interaction in the presence of a spectator nucleon. As indicated in flg. 7, T X (z) contributes simultaneously to the N N -scattering amplitude as well as to flnal state interactions (FSI) in pion production processes. For technical reasons we intend to parametrize these interactions in terms of suitable separable realizations [13, 14]. In the present realization, solely the so-called πdchannel, i.e. V π in the 3 S1 /3 D1 N N -channel is considered [8]. A more complete treatment of T X (z) is under construction. Moreover, the approach discussed so far is presently only realized for N N -scattering [8] and electromagnetic deuteron breakup [9,10,15]. An extension to photopionproduction as well as elastic πd-scattering will be available soon. Our proposed model is deflnitely a very promising one for studying simultaneously all possible hadronic and electromagnetic reactions up to the Δ-region with at most one asymptotic free pion. Concerning alternatives to our approach, we only mention here the presently most popular one, namely efiective fleld theory (EFT) which is based on the spontaneously broken chiral symmetry of QCD. EFT starts from the most general efiective Lagrangian which is consistent with the symmetries of QCD and therefore more involved than the Lagrangians used in our approach. On the other hand one has to recognize that our approach is nonperturbative whereas EFT performs a simultaneous expansion in small external momenta and quark masses. It is therefore a perturbative treatment in terms of an expansion parameter Q/Λ with Q ∼ mπ and Λ ∼ 1 GeV, the chiral symmetry breaking scale. In contrast to our approach, it is presently applicable only in quite a small energy domain like N N -scattering up to pion threshold (see [16,17] and references therein), low momentum elastic electron deuteron scattering [18] or electropionproduction near threshold [19]. 2.3 Deuteron breakup in the Δ-region Next, we turn to the results of our approach for a selected choice of reactions, starting with deuteron photodisintegration. Despite its simplicity, this reaction has posed severe problems for theoreticians until the middle of the 90s. This becomes obvious from flg. 8, where experimental data for the total cross section in the Δ-region

M. Schwamb: Few-nucleon systems (theory)

Fig. 8. The total cross section σtot of deuteron photodisintegration as a function of the photon energy. Results from Tanabe and Ohta [20] (dotted), Wilhelm and Arenh˜ ovel [21] (dashed) and Schwamb and Arenh˜ ovel [9, 10] (full). Experimental data from [22] (), [23] (◦) and [24] (•).

Fig. 9. Difierential cross sections of deuteron photodisintegration in the center-of-mass frame for a laboratory photon energy of 260 MeV (left) and 440 MeV (right). Notation of the curves as in flg. 8.

is compared with the most sophisticated models available at that time, namely the unitary three-body approach of Tanabe and Ohta [20] as well as the model of Wilhelm and Arenh˜ ovel [21]. Similar to our treatment, realistic N N interactions are used and a dynamical treatment of the Δ is incorporated. Moreover, a considerable conceptual improvement in comparison to earlier work was the fact that no free parameters occur in the photodisintegration channel because similar as in our approach all of them have been flxed in advance by considering πN - and N N scattering as well as photopionproduction on the nucleon. From flg. 8 it becomes obvious that the theory clearly fails in describing the data. The predicted total cross sections are too small and a dip structure around 90◦ occurs in the difierential cross section at higher energies which is not present in the data, see flg. 9. These problems were very severe ones because deuteron photodisintegration is the simplest photonuclear reaction on a nucleus. In the past decade, we have made considerable efforts to solve this problem [7,9,10] reaching now an almost quantitative description of the total cross section in the Δ-region, see flg. 8. Furthermore, also the description of the difierential cross section is considerably improved.

43

This success turned out to be the combined result of various independent improvements compared to our starting point [21]. Apart from the additional incorporation of dissociation currents, the πd-channel and conceptual improvements in the description of the γN → Δ-transition, retardation efiects both in the hadronic interaction as well as in the MEC turn out to be very important. The latter have been partially neglected in [20,21] by using the static Paris and Bonn-OBEPR potentials, respectively, and corresponding static MEC. This result clearly indicates that even in breakup reactions of nuclei, where no asymptotic free pions occur, the latter must be treated in a dynamic manner for energies beyond pion threshold. In a recent extension of this work, we have studied the role of retardation in deuteron electrodisintegration [15]. Neglecting polarization efiects the difierential cross section for this reaction in the one-photon-exchange approximation is determined by four structure functions, two diagonal ones fL and fT and two interference ones fLT and fT T [25,26]. They are functions of the squared three-momentum transfer q 2 , the flnal state kinetic energy Enp = W − 2MN , and the angle θ between q and the proton momentum in the flnal neutron-proton center-of-mass system. It turns out that retardation leads to dramatic changes in the structure functions fL and fLT for excitation energies beyond the pion threshold whereas the other structure functions fT and fT T are much less afiected. This is illustrated in flg. 10 for a suitable kinematics in the Δ-region which has been studied at NIKHEF [27]. It turns out that especially the recoil charge contribution (right panel in flg. 10) is very important. This mechanism is not present in conventional static approaches due to an implicitly applied wave function renormalization procedure [28, 29] whose aim is to construct orthonormalized baryonic wave functions. This concept breaks down beyond pion threshold, where the pion can become onshell and must be necessarily included in the hadronic wave functions. This fact, already discussed in [9], clearly indicates that a static treatment is only a poor approximation in reactions on the deuteron beyond pion threshold. It would of course be very important to perform experimental checks of these predictions.

2.4 The deuteron as effective neutron target The precise knowledge of elementary particle properties is very important for a better understanding of their internal structure. With respect to the neutron as one of the most important particles, its flnite lifetime forces us to consider few nucleon systems like the deuteron or 3 He as alternative efiective neutron targets. The basic question is, whether for a speciflc neutron property of interest a speciflc reaction on the deuteron exists where the neutron contribution is dominant and nuclear background efiects from Fermi motion, MEC, FSI, etc. are small or at least under control. As a flrst example, let us consider the neutron form factors GEn and GM n . The magnetic form factor GM n

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π

π

Fig. 10. Results for deuteron electrodisintegration taken from [15]. Left panel: the structure functions fL , fT , fLT and fT T for the kinematics of the NIKHEF experiment [27], i.e. Enp = 280 MeV, q 2 = 2.47 fm 2 . Notation of the curves: dashed: static approach; full: retarded approach. The additional dash-dotted curves represent the results of the retarded approach where the Coulomb monopole contribution of the recoil charge operator, depicted on the right, is switched ofi.

}effect G En Fig. 11. Relevant diagrams in deuteron electrodisintegration.

of the neutron can be determined from electron backscattering ofi the deuteron in quasi-free neutron kinematics. In this speciflc kinematics, the momentum of the virtual photon in the laboratory frame is completely transferred to the neutron whereas the spectator proton is at rest in the flnal state. These conditions lead to the rule of thumb Enp /MeV = 10 q 2 /fm−2 . Compared to GM n , the electric neutron form factor GEn is more di– cult to measure. Various possibilities to measure GEn have been discussed in [30]. It turned out that the cleanest determination is obtained in double polarization observables in deuteron electrodisintegration, → → → → i.e. d( e , e n)p or d ( e , e n)p. The relevant diagrams contributing to this reaction are depicted in flg. 11. In the following, we restrict ourselves to the reaction → → d( e , e n)p. In the Born approximation (PWBA), i.e. neglecting FSI, MEC as well as isobars, and neglecting in addition the D state of the deuteron, it turns out that in quasi-free neutron kinematics the polarization component  Px in the scattering plane perpendicular to the photon momentum is directly proportional to GEn so that one has a linear relation between the observable and the quantity of interest:  (13) Px ∼ GEn GM n .



Fig. 12. The polarization Px of the outgoing neutron in the scattering plane perpendicular to the photon momentum as a function of the proton scattering angle θ for a squared photon four-momentum of Q2 = 1 GeV2 , a squared three-momentum transfer q 2 = 25.67 fm 2 and a kinetic energy of the outgoing nucleons of Enp = 250 MeV. The neutron scattering angle θn is given by θn = 180 − θ. Notation of the curves: dotted: PWBA with GEn = 0; dash-dotted: PWBA with GEn = 0; dashed: full static calculation based on Bonn-OBEPR potential (GEn = 0); solid: full retarded calculation based on Elster potential (GEn = 0). In quasi-free neutron kinematics (θ = 180 ), one  readily recognizes the sensitivity of Px to GEn as well as its insensitivity to nuclear structure efiects like FSI, MEC and resonance contributions.

Moreover, it turns out that in quasi-free neutron kinematics the role of the background efiects of FSI, MEC and isobars is under control and almost model independent, see flg. 12 as an illustrative example. This allows therefore a very clean interpretation of the existing data (consider [31] and the references therein), so that we can conclude that the deuteron is a very e– cient efiective neutron target with respect to the extraction of GEn .

M. Schwamb: Few-nucleon systems (theory)

We now turn to a second example where the use of the deuteron as an efiective neutron target would be highly desirable. It deals with the investigation of the GerasimovDrell-Hearn sum rule (GDH) for various hadronic targets [32,33]. This sum rule links the anomalous magnetic moment of a particle to the energy weighted integral over the spin asymmetry of the absorption cross section. In detail it reads for a particle of mass M , charge eQ, anomalous magnetic moment κ and spin S  ∞ 2 dω   P  A  2 2 e σ I GDH = (ω ) − σ (ω ) = 4π κ S, ω M2 0 (14) where σ P/A (ω  ) denote for a given photon momentum ω  the total absorption cross sections for circularly polarized photons on a target with spin parallel (P ) and antiparallel (A) to the photon spin. This sum rule gives therefore a very interesting relation between a ground state property (κ) of a particle and its whole excitation spectrum. Apart from the general assumption that the integral in (14) converges, its derivation is based solely on flrst principles like Lorentz and gauge invariance, unitarity, crossing symmetry and causality of the Compton scattering amplitude of a particle. Consequently, a check for various targets, both from the experimental as well as from the theoretical point of view, would be very important. Inserting the known anomalous magnetic moments of proton and neutron into (14), one obtains quite large GDH sum rule values, i.e. IpGDH = 204.8 μb for the proton and InGDH = 233.2 μb for the neutron. On the other side, the deuteron has a small anomalous magnetic moment κd = −0.143 n.m. resulting in a very small GDH sum rule value of IpGDH = 0.65 μb. Whereas GDH measurements on proton targets can be directly performed (consider [34] and references therein), no free neutron target exists and one may try to extract InGDH from deuteron measurements. In contrast to the extraction of GEn , this task is however much more complicated. First of all, let us recall that for the extraction of the electric neutron form factor one speciflc reaction (e.g. deuteron electrodisintegration) in one speciflc kinematics (the quasi-free one) is su– cient. On the other hand, concerning the GDH sum rule one has to determine total inclusive cross sections, i.e. contributions in all possible kinematics from very difierent reactions like γN → πN, ππN, ηN, . . .

(15)

for the nucleon, and γd → N N, πN N, πd, ππN N, ηN N, . . .

(16)

for the deuteron have to taken into account. These complications become even more serious if one considers the sum of the proton and neutron value compared to the deuteron value. If one assumes that the meson production on the deuteron is dominated by the quasi-free production on the nucleons bound in the deuteron, one would expect that IdGDH should be roughly IpGDH +InGDH . This assumption is however wrong by more than two orders of magnitude. Consequently, concerning the GDH

γ

45 π

t γNπ

π

γ t Nγ π

N

d

T NN

d N

N N

(a)

(b)

γ

N t γNπ

d

T

πN

π N

(c)

Fig. 13. Considered diagrams for single pion production. (a) impulse approximation (IA), (b) incorporation of N N -flnal state interaction (N N -FSI), (c) incorporation of πN -flnal state interaction (πN -FSI).

sum rule the deuteron reaction cannot be considered just as an incoherent sum of the proton and the neutron reaction. In order to obtain the small deuteron GDH value, strong anticorrelation efiects between the difierent possible channels for the deuteron must occur which are not present in the elementary case. This cancellation is a challenge for any theoretical framework since it requires the above-mentioned unifled consistent treatment of hadronic and electromagnetic properties for the difierent possible channels in a wide energy region. In the past years, considerable efiorts have been undertaken in order to obtain a more quantitative understanding of the GDH sum rule on the deuteron [35,36,37]. In the presently most sophisticated approach [37], besides deuteron photodisintegration also coherent and incoherent single and double pion production as well as η-production are considered. At the moment, the aforementioned retarded approach is only available for the breakup channel. Concerning incoherent single pion production, the considered mechanisms in our present realization are depicted in flg. 13. For the elementary production operator, the MAID model [38] is used, allowing one to extend the calculation up to photon energies of 1.5 GeV. Moreover, flnal state interactions are perturbatively taken into account up to the flrst order in the corresponding πN - and N N scattering amplitudes. For coherent pion production, the model of [39] is used taking into account pion rescattering by solving a system of coupled equations for the N N , N Δ- and N N π-channels. It is partially similar to our approach discussed in section 2.2. However, no retardation concerning the N N -interaction and the corresponding MEC is presently taken into account. For double-pion production the evaluation is based on a traditional efiective Lagrangian approach similar to the one in [40]. It is presented in great detail in [41]. Although this treatment of the GDH sum rule on the deuteron is presently the most sophisticated one, we are aware of speciflc shortcomings. The most serious one is the use of difierent approaches for the difierent reactions. In order to obtain a more unifled picture, work is in progress to adopt the discussed retarded approach not only to the

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400 300

(ω) [μb]

γd → pn

-500

100 0 -100

-600 1

10

ω [MeV]

100

500

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γd → π-pp (IA+FSI) γd → π-pp (IA) γn → π p 500

1000 ω [MeV]

(ω) [μb]

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GDH

-40

0 -20

Id

-20

γd → π+nn (IA+FSI) γd → π+nn (IA) + γp → π n

-60 1500

500 150

γd → π0 π0 np (IA+FSI) γd → π0 π0 np (IA) γN → π0 π0 N

(ω) [μb]

(ω) [μb]

GDH

Id

1000 ω [MeV]

20

-80

Id

Id

1000 ω [MeV]

1500

- +

γd → π π np (IA+FSI) γd → π- π+ np (IA) γN → π- π+ N

100

GDH

10

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(ω) [μb]

γd → ηpn (x5) γN → ηN (x5)

1000

0

20

γd → π0np γd → π0d γN → π0N

200

Id

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GDH

(ω) [μb] Id

-400

GDH

-200

500

Bonn r-space retarded potential

-100

50

0 0 500

500 50

-

γd → π0 π pp (IA+FSI) γd → π0 π pp (IA) γn → π0 π p

40

30

1000 1500 ω [MeV]

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+

γd → π0 π nn (IA+FSI) + γd → π0 π nn (IA) + γp → π0 π n

30 20

Id

GDH

20

Id

GDH

(ω) [μb]

40

2000

(ω) [μb]

50

1000 1500 ω [MeV]

10

10

0

0 500

1000 1500 ω [MeV]

2000

500

1000 1500 ω [MeV]

2000

Fig. 14. Contributions of various channels to the flnite GDH integral (17) as a function of the upper integration limit for deuteron disintegration, single- and double-pion and ηproduction on nucleon and deuteron. For the neutral charge channels π 0 , η, π 0 π 0 , and π π + , the nucleon integrals are the sum of proton and neutron integrals. See [37] for more details.

breakup channel, but at least also to single-pion production. In order to present the results in a transparent way, we introduce for convenience the flnite GDH integral as deflned by  ω dω   P  GDH σ (ω ) − σ A (ω  ) , I (ω) = (17)  ω 0 for which the results for photodisintegration, single and double pion and η-production are exhibited in flg. 14. With respect to the photodisintegration channel, at very low energies a very large negative contribution arises from the M 1 transition to the resonant 1 S0 state which can only be reached if the spins of photon and deuteron are antiparallel. Sizeable difierences especially in the Δ-region occur between our retarded approach and an older static evaluation [35] which was based on the Bonn-OBEPR potential. Concerning single pion production, we show in flg. 14 the results both in IA and with inclusion of flnal state interactions (labeled as IA+FSI) together with the corresponding results for the elementary reactions. One notes besides a positive contribution from the Δ-resonance another one above a photon energy of about 600 MeV from D13 (1520) and F15 (1680). For charged pion production

FSI efiects are in general quite small. The same is true also for η-production. But FSI is nonnegligible for incoherent neutral pion production due to the non-orthogonality of the flnal state wave in IA to the deuteron bound state wave function, see [42] for more details. Please note moreover the signiflcant difierences between the deuteron and the corresponding nucleon values for I GDH (ω). This feature occurs also in double-pion production where the largest contribution is coming from the π − π + -channel. Here the inclusion of FSI, where only N N -rescattering is presently taken into account, is quite small. The contributions of various channels to the flnite GDH integral (17) for nucleon and deuteron are listed in table 1. While for the neutron the total sum is about 8 % lower than the sum rule value, it is too large by about 28 % for the proton. Concerning the deuteron, each of the difierent channels (apart from η-production) produces very large contributions. Due to the large cancellation of the photodisintegration and the meson production channels, the sum of all contributions is quite small (27.31 μb). This is still somewhat too large compared to the theoretical value of 0.65 μb. However, one should keep in mind that our approach still needs to be improved due to several shortcomings as indicated above. The strong cancellation between the regions at low and high energies is a fascinating feature clearly demonstrating the decisive role of the pion as a manifestation of chiral symmetry governing strong interaction dynamics in these two difierent energy regions. With respect to meson production channels on nucleon and deuteron, the difierent behaviour of the corresponding spin asymmetries indicates that a direct experimental access to the neutron spin asymmetry from a deuteron measurement by subtracting the one of the free proton is not possible. On the other hand, the measurement of the spin asymmetry for the difierent channels on the deuteron presents itself a stringent test of our present theoretical understanding of two-nucleon physics. Therefore, the experimental program at facilities like MAMI and ELSA concerning the GDH sum rule on the deuteron is very important for further progress in that fleld.

3 More complex few-nucleon systems Till now, we have concentrated ourselves solely on the twonucleon system. The present situation in the three-nucleon system is outlined in great detail in [43] and therefore not discussed here. Concerning even more complex fewnucleon systems, we want to present here merely some recent highlights obtained with the Lorentz integral transform method (LIT) [44]. The basic question in this context is, up to which mass number A and energy/momentum transfer precise microscopic calculations for the electromagnetic response can be performed. The most fundamental observable in this fleld is deflnitely the total inclusive cross section σtot . In conventional scattering theory, an economic method to calculate σtot is to apply the optical

M. Schwamb: Few-nucleon systems (theory)

47

Table 1. Contributions of various channels to the flnite GDH integral (in μb), integrated up to 0.8 GeV for photodisintegration, 1.5 GeV for single pion and η-production and 2.2 GeV for double pion production on nucleon and deuteron, see [37] for further details.  np π ππ η Sum rule value neutron proton deuteron

−381.52

138.95 176.38 263.44

82.02 93.93 159.34

A

A

... Fig. 15. Top panel: diagrammatic representation of the direct Compton scattering amplitude for an A-nucleon system. Only one-body currents are depicted for the sake of simplicity. Bottom panel: graphical illustration of a selected choice of contributing mechanisms to the imaginary part of the Compton scattering amplitude in (18).

theorem, here to Compton scattering (see flg. 15) σtot (γA → X; W ) ∼ lim Im T (γA → γA; W + i, θ = 0) →0

(18) with W as invariant energy of the reaction. In order to obtain the imaginary part, one has to know very precisely the pole structure of the intermediate virtual states between photon absorption and emission which requires a careful numerical treatment of the occurring singularities. It is obvious that with increasing mass number A and increasing energy/momentum transfer this task becomes more and more complicated and flnally practically impossible. An elegant solution of this problem has been proposed about a decade ago by the Trento group [44]. The essential idea is to perform flrst of all an integral transform of σtot according to  σtot (W ) (19) L(σR , σI ) = dW (W − σ)2 with σ = σR + iσI , where σR , σI can be treated as free parameters. After some algebra, using the completeness relation of the flnal states, it turns out that L(σR , σI ) has the same formal structure as the optical theorem for Compton scattering (18), i.e. L(σR , σI ) ∼ Im T (γA → γA; σR + iσI , θ = 0).

(20)

The essential difierence between (18) and (20) lies in the argument W + i versus σR + iσI . In the optical the-

−5.77 −8.77 −13.95

215.20 261.54 27.31

233.16 204.78 0.65

orem, the quantity  has to be treated as inflnitesimal small yielding in consequence the above-mentioned complicated pole structure. Its counterpart in the LIT, σI , is flnite and at our disposal. It can, at least in principle, be chosen arbitrarily. This has far reaching consequences, because the pole structure in (19) vanishes for σI flnite. This yields enormous numerical simpliflcations, because one needs only bound state techniques, avoiding in consequence the calculation of A-body scattering states. In order to obtain the desired inclusive cross section σtot , one has of course to perform a numerical inversion of the LIT. Recently, a variety of difierent reliable inversion methods has been presented [45] so that this problem is very well under control. An important cross check for the inversion is that the resulting cross section should be independent of the parameter σI so that the LIT method is completely parameter free. Due to these features, it is not very surprising that the LIT has been applied with considerable success to microscopic calculations of quite a few electroweak cross sections of various nuclei ranging form A = 2−7 like inclusive electron scattering (see e.g. [46,47]) and total photoabsorption cross sections (see e.g. [48,49,50]). In the meantime, it has also been extended to exclusive reactions [51, 52], photopionproduction on the deuteron [53,54] as well as weak processes [55]. This list of applications shows that the LIT approach constitutes an important progress opening up the possibility to carry out ab initio microscopic calculations not only for reactions on the classical fewbody systems (deuteron, three-body nuclei) but also for reactions on more complex nuclei.

4 Summary and outlook The study of reactions on few-nucleon systems is of particular importance for testing present theoretical frameworks in terms of efiective degrees of freedom. Of speciflc interest are electromagnetic reactions above pion threshold where a unifled approach needs to be constructed. Few-nucleon systems are moreover of importance as efiective neutron targets, for example with respect to the extraction of the electric neutron form factor GEn . The situation turns out to be much more complicated with respect to the study of the GDH sum rule on the neutron, where in contrast to GEn no selection of the pure quasi-free kinematics is possible and where many difierent reaction channels have to be taken into account. Nevertheless, the planned measurements of the GDH spin asymmetry on the deuteron

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and 3 He at MAMI will lead to very stringent tests of our present knowledge of nucleon and nuclear structure. Additional measurements are also desirable for electromagnetic reactions on more complex few-nucleon systems (A ≥ 4) where nowadays for the flrst time purely microscopic calculations with the help of the Lorentz integral transform method are possible. Summarizing, the study of few-nucleon systems is a very active fleld both from the experimental as well as theoretical point of view. The expected progress will be very important for the future development of hadronic physics in general. This is dedicated to the occasion of the retirement of H. Arenh˜ ovel, H. Backe, D. Drechsel, J. Friedrich, K-H. Kaiser and Th. Walcher. It has been supported by the Deutsche Forschungsgemeinschaft (SFB443). I would like to thank H. Arenh˜ ovel for his careful reading of the manuscript and for various stimulating discussions.

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

References 1. R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989). 2. E.W. Schmid, H. Ziegelmann, The Quantum-Mechanical Three-Body Problem (Pergamon Press, Oxford and Friedrich Vieweg & Sohn, Braunschweig, 1974). 3. W. Gl˜ ockle, The Quantum-Mechanical Few-Body Problem (Springer Verlag, Berlin 1983). 4. R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, Phys. Rev. C 51, 38 (1995). 5. R. Machleidt, F. Sammarrucca, Y. Song, Phys. Rev. C 53, 1483 (1996). 6. R. Machleidt, Phys. Rev. C 63, 024001 (2001). 7. M. Schwamb, H. Arenh˜ ovel, P. Wilhelm, Th. Wilbois, Phys. Lett. B 420, 255 (1998). 8. M. Schwamb, H. Arenh˜ ovel, Nucl. Phys. A 690, 647 (2001). 9. M. Schwamb, H. Arenh˜ ovel, Nucl. Phys. A 690, 682 (2001). 10. M. Schwamb, H. Arenh˜ ovel, Nucl. Phys. A 696, 556 (2001). 11. Ch. Elster, W. Ferchl˜ ander, K. Holinde, D. Sch˜ utte, R. Machleidt, Phys. Rev. C 37, 1647 (1988). 12. H. P˜ opping, P.U. Sauer, X.-Z. Zhang, Nucl. Phys. A 474, 557 (1987). 13. J. Haidenbauer, W. Plessas, Phys. Rev. C 30, 1822 (1984); Phys. Rev. C 32, 1424 (1985). 14. S. Nozawa, B. Blankleider, T.-S.H. Lee, Nucl. Phys. A 513, 459 (1990). 15. M. Schwamb, H. Arenh˜ ovel, Phys. Lett. B 588, 49 (2004). 16. R. Machleidt, D.R. Entem, J. Phys. G 31, S1235 (2005). 17. E. Epelbaum, W. Gl˜ ockle, U.-G. Meissner, Nucl. Phys. A 747, 362 (2005). 18. M. Walzl, U.-G. Meissner, Phys. Lett. B 513, 37 (2001). 19. H. Krebs, V. Bernard, U.-G. Meissner, Eur. Phys. J. A 22, 503 (2004).

37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.

H. Tanabe, K. Ohta, Phys. Rev. C 40, 1905 (1989). P. Wilhelm, H. Arenh˜ ovel, Phys. Lett. B 318, 410 (1993). J. Arends et al., Nucl. Phys. A 412, 509 (1984). G. Blanpied et al., Phys. Rev. C 52, R455 (1995); Phys. Rev. C 61, 024604 (2000). R. Crawford et al., Nucl. Phys. A 603, 303 (1996). W. Fabian, H. Arenh˜ ovel, Nucl. Phys. A 314, 253 (1979). H. Arenh˜ ovel, W. Leidemann, E.L. Tomusiak, Phys. Rev. C 46, 455 (1992). A. Pellegrino et al., Phys. Rev. Lett. 78, 4011 (1997). M. Gari, H. Hyuga, Z. Phys. 277, 291 (1976). H. Arenh˜ ovel, Czech. J. Phys. 43, 207 (1993). H. Arenh˜ ovel, W. Leidemann, E.L. Tomusiak, Z. Phys. A 331, 123 (1988); 334, 363 (1989). D.I. Glazier et al., Eur. Phys. J. A 24, 101 (2005). S.B. Gerasimov, Yad. Fiz. 2, 598 (1965) (Sov. J. Nucl. Phys. 2, 430 (1966)). S.D. Drell, A.C. Hearn, Phys. Rev. Lett. 16, 908 (1966). H. Dutz et al., Phys. Rev. Lett. 93, 032003 (2004). H. Arenh˜ ovel, G. Kre , R. Schmidt, P. Wilhelm, Phys. Lett. B 407, 1 (1997). E.M. Darwish, H. Arenh˜ ovel, M. Schwamb, Eur. Phys. J. A 17, 513 (2003). H. Arenh˜ ovel, A. Fix, M. Schwamb, Phys. Rev. Lett. 93, 202301 (2004). D. Drechsel, O. Hahnstein, S.S. Kamalow, L. Tiator, Nucl. Phys. A 645, 145 (1999). P. Wilhelm, H. Arenh˜ ovel, Nucl. Phys. A 593, 435 (1995); 609, 469 (1996). J.A. Gomez Tejedor, E. Oset, Nucl. Phys. A 600, 413 (1996). A. Fix, H. Arenh˜ ovel, Eur. Phys. J. A 25, 115 (2005). A. Fix, H. Arenh˜ ovel, Phys. Rev. C 72, 064005 (2005). J. Golak, R. Skibinski, H. Witala, W. Gl˜ockle, A. Nogga, H. Kamada, Phys. Rep. 415, 89 (2005). V.D. Efros, W. Leidemann, G. Orlandini, Phys. Lett. B 338, 130 (1994). D. Andreasi, W. Leidemann, Ch. Reiss, M. Schwamb, Eur. Phys. J. A 24, 361 (2005). V.D. Efros, W. Leidemann, G. Orlandini, Phys. Rev. Lett. 78, 432 (1997). V.D. Efros, W. Leidemann, G. Orlandini, E.L. Tomusiak, Phys. Rev. C 69, 044001 (2004). V.D. Efros, W. Leidemann, G. Orlandini, Phys. Rev. Lett. 78, 4015 (1997). S. Bacca, M. Marchisio, N. Barnea, W. Leidemann, G. Orlandini, Phys. Rev. Lett. 89, 052502 (2002). S. Bacca, H. Arenh˜ ovel, N. Barnea, W. Leidemann, G. Orlandini, Phys. Lett. B 603, 159 (2004). A. La Piana, W. Leidemann, Nucl. Phys. A 677, 423 (2000). S. Quaglioni, W. Leidemann, G. Orlandini, N. Barnea, V.D. Efros, Phys. Rev. C 69, 044002 (2004). Ch. Reiss, W. Leidemann, G. Orlandini, E.L. Tomusiak, Eur. Phys. J. A 17, 589 (2003). Ch. Reiss, H. Arenh˜ ovel, M. Schwamb, Eur. Phys. J. A 25, 171 (2005). D. Gazit, N. Barnea, Phys. Rev. C 70, 048801 (2004).

Eur. Phys. J. A 28, s01, 49 57 (2006) DOI: 10.1140/epja/i2006-09-006-5

EPJ A direct electronic only

Nucleon form factors in dispersion theory H.-W. Hammera Helmholtz-Institut f˜ ur Strahlen- und Kernhysik (Theorie), Universit˜ at Bonn, Nussallee 14-16, D-53115 Bonn, Germany / Published online: 11 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Dispersion relations provide a powerful tool to analyse the electromagnetic form factors of the nucleon in both the space-like and the time-like regions with constraints from other experiments, unitarity, and perturbative QCD. We give a brief introduction into dispersion theory for nucleon form factors and present flrst results from our ongoing form factor analysis. We also calculate the two-pion continuum contribution to the isovector spectral functions drawing upon the new high statistics measurements of the pion form factor by the CMD-2, KLOE, and SND collaborations. PACS. 11.55.Fv Dispersion relations neutrons

13.40.Gp Electromagnetic form factors

1 Introduction The electromagnetic form factors of the nucleon ofier a unique window on strong interaction dynamics over a wide range of momentum transfers [1,2]. At small momentum transfers, they are sensitive to the gross properties of the nucleon like the charge and magnetic moment, while at high momentum transfers they encode information on the quark substructure of the nucleon as described by QCD. Their detailed understanding is important for unraveling aspects of perturbative and nonperturbative nucleon structure. The form factors also contain important information on nucleon radii and vector meson coupling constants. Moreover, they are an important ingredient in a wide range of experiments from Lamb shift measurements [3] to measurements of the strangeness content of the nucleon [4]. With the advent of the new continuous beam electron accelerators such as CEBAF (Jefierson Lab.), ELSA (Bonn), and MAMI (Mainz), a wealth of precise data for space-like momentum transfers has become available [5]. Due to the di– culty of the experiments, the time-like form factors are less well known. While there is a fair amount of information on the proton time-like form factors [6,7, 8,9,10], only one measurement of the neutron form factor from the pioneering FENICE experiment [11] exists. It has been known for a long time that the pion plays an important role in the long-range structure of the nucleon [12]. This connection was made more precise using dispersion theory in the 1950’s [13,14]. Subsequently, Frazer and Fulco have written down partial-wave dispersion relations that relate the nucleon electromagnetic structure to pion-nucleon (πN ) scattering and predicted a

e-mail: [email protected]

14.20.Dh Protons and

the existence of the ρ-resonance [15,16]. Despite this success, the central role of the 2π continuum in the isovector spectral function has often been ignored. H˜ ohler and Pietarinen pointed out that this omission leads to a gross underestimate of the isovector radii of the nucleon [17]. They flrst performed a consistent dispersion analysis of the electromagnetic form factors of the nucleon [18] including the 2π continuum derived from the pion form factor and πN -scattering data [19]. In the mid-nineties, this analysis has been updated by Mergell, Mei ner, and Drechsel [20] and was later extended to include data in the time-like region [21,22]. Recently, the new precise data for the neutron electric form factor have been included as well [23]. Using chiral perturbation theory (ChPT), the longrange pionic structure of the nucleon can be connected to the Goldstone boson dynamics of QCD [24]. The nonresonant part of the 2π continuum is in excellent agreement with the phenomenological analysis [25] and the ρmeson contribution can be included as well [26,27,28]. It is well known that vector mesons play an important role in the electromagnetic structure of the nucleon, see, e.g., refs. [15,29,30,31,32,33], and the remaining contributions to the spectral function have usually been approximated by vector meson resonances. A new twist to this picture was recently given by Friedrich and Walcher [34]. They interpreted the form factor data based on a phenomenological flt with an ansatz for the pion cloud using the idea that the proton can be thought of as virtual neutron-positively charged pion pair. A very long-range contribution to the charge distribution in the Breit frame extending out to about 2 fm was found and attributed to the pion cloud. This was shown to be in conflict with the phenomenologically known 2π continuum and ChPT by Hammer, Drechsel, and Mei ner [35]. We will address this conundrum in more detail in sect. 9.

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jem μ p’

where F (t) is a generic form factor. In the case of the electric and Dirac form factors of the neutron, GnE and F1n , the expansion starts with the term linear in t and the normalization factor F (0) is dropped.

p

Fig. 1. The nucleon matrix element of the electromagnetic current jμem .

In this paper we give a brief introduction into dispersion theory for nucleon form factors and present preliminary results from our ongoing form factor analysis. We also calculate the two-pion continuum contribution to the isovector spectral functions drawing upon the new high statistics measurements of the pion form factor by the CMD-2, KLOE, and SND collaborations. Finally we address the question of the range of the pion cloud and give an outlook on future work.

2 Definitions The electromagnetic (em) structure of the nucleon is determined by the matrix element of the current operator jμem between nucleon states as illustrated in flg. 1. Using Lorentz and gauge invariance, this matrix element can be expressed in terms of two form factors, 

F2 (t)  em  ν p |jμ |p = u(p ) F1 (t)γμ + i σμν q u(p), (1) 2M 

where M is the nucleon mass and t = (p − p) the fourmomentum transfer. For data in the space-like region, it is often convenient to use the variable Q2 = −t > 0. The functions F1 (t) and F2 (t) are the Dirac and Pauli form factors, respectively. They are normalized at t = 0 as (2)

with κp = 1.79 and κn = −1.91 the anomalous magnetic moments of protons and neutrons in nuclear magnetons, respectively. It is convenient to work in the isospin basis and to decompose the form factors into isoscalar and isovector parts, 1 p (F + Fin ), 2 i

Fiv =

1 p (F − Fin ), 2 i

(3)

where i = 1, 2. The experimental data are usually given for the Sachs form factors GE (t) = F1 (t) − τ F2 (t), GM (t) = F1 (t) + F2 (t),

Based on unitarity and analyticity, dispersion relations relate the real and imaginary parts of the electromagnetic (em) nucleon form factors. Let F (t) be a generic symbol for any one of the four independent nucleon form factors. We write down an unsubtracted dispersion relation of the form  ∞ Im F (t ) 1 F (t) = dt , (6) π t0 t − t − i where t0 is the threshold of the lowest cut of F (t) (see below) and the i deflnes the integral for values of t on the cut.1 Equation (6) relates the em structure of the nucleon to its absorptive behavior. The imaginary part Im F entering eq. (6) can be obtained from a spectral decomposition [13,14]. For this purpose it is most convenient to consider the em current matrix element in the time-like region (t > 0), which is related to the space-like region (t < 0) via crossing symmetry. The matrix element can be expressed as Jμ = N (p)N (p)|jμem (0)|0 

F2 (t) σμν (p + p)ν v(p), = u(p) F1 (t)γμ + i 2M

2

F1p (0) = 1, F1n (0) = 0, F2p (0) = κp , F2n (0) = κn ,

Fis =

3 Dispersion relations and spectral decomposition

(4)

where τ = −t/(4M 2 ). In the Breit frame, GE and GM may be interpreted as the Fourier transforms of the charge and magnetization distributions, respectively. The nucleon radii can be deflned from the low-t expansion of the form factors,

(5) F (t) = F (0) 1 + t r2 /6 + . . . ,

(7)

where p and p are the momenta of the nucleon and antinucleon created by the current jμem , respectively. The fourmomentum transfer in the time-like region is t = (p + p)2 . Using the LSZ reduction formalism, the imaginary part of the form factors is obtained by inserting a complete set of intermediate states as [13,14] Im Jμ =

 π (2π)3/2 N p|JN (0)|λ Z

(8)

λ

× λ|jμem (0)|0 v(p) δ 4 (p + p − pλ ), where N is a nucleon spinor normalization factor, Z is the nucleon wave function renormalization, and JN (x) = J † (x)γ0 with JN (x) a nucleon source. This decomposition is illustrated in flg. 2. It relates the spectral function to on-shell matrix elements of other processes. The states |λ are asymptotic states of momentum pλ which are stable with respect to the strong interaction. They must carry the same quantum numbers as the current jμem : I G (J P C ) = 0− (1−− ) for the isoscalar current and I G (J P C ) = 1+ (1−− ) for the isovector component of 1

The convergence of an unsubtracted dispersion relation for the form factors has been assumed. We could have used a once subtracted dispersion relation as well since the normalization of the form factors is known.

H.-W. Hammer: Nucleon form factors in dispersion theory

51

60

N

KLOE CMD-2 SND

50

λ λ

jμem

30 30

N

20

Fig. 2. The spectral decomposition of the nucleon matrix element of the electromagnetic current jμem .

jμem . Furthermore, they have zero net baryon number. Because of G-parity, states with an odd number of pions only contribute to the isoscalar part, while states with an even number contribute to the isovector part. For the isoscalar part the lowest mass states are: 3π, 5π, . . ., KK, KKπ, . . .; for the isovector part they are: 2π, 4π, . . .. Associated with each intermediate state is a cut starting at the corresponding threshold in t and running to inflnity. As a consequence, the spectral function Im F (t) is difierent from zero along the cut from t0 to ∞ with t0 = 4 (9) Mπ2 for the isovector (isoscalar) case. The spectral functions are the central quantities in the dispersion-theoretical approach. Using eqs. (7,8), they can in principle be constructed from experimental data. In practice, this program can only be carried out for the lightest two-particle intermediate states (2π and KK) [19, 36,37]. The longest-range, and therefore at low momentum transfer most important pion cloud contribution comes from the 2π intermediate state in the isovector form factors. A new calculation of this contribution will be discussed in the following section.

4 Two-pion continuum In this section, we re-evaluate the 2π contribution in a model-independent way [38] using the latest experimental data for the pion form factor from CMD-2 [39], KLOE [40], and SND [41]. We follow ref. [42] and express the 2π contribution to the isovector spectral functions in terms of the pion charge form factor Fπ (t) and the P -wave ππ → N N am1 plitudes f± (t). The 2π continuum is expected to be the dominant contribution to the isovector spectral function from threshold up to masses of about 1 GeV [42]. Here, we use the expressions qt3 1 √ Fπ (t)∗ f+ (t), M t q3 1 Im GvM (t) = √t Fπ (t)∗ f− (t), (9) 2t  where qt = t/4 − Mπ2 . The imaginary parts of the Dirac and Pauli Form factors can be obtained using eq. (4). Im GvE (t) =

40

|Fπ (t)|

2

40

0.5

0.6

10 0

0

0.2

0.4

0.6

0.8

2

t [GeV ] Fig. 3. The pion electromagnetic form factor Fπ (t) in the timelike region as a function of the momentum transfer t. The diamonds, squares, and circles show the high statistics data from the CMD-2 [39], KLOE [40], and SND [41] collaborations, respectively. The dashed, solid, and dash-dotted lines are our model parametrizations. The inset shows the discrepancy in the resonance region in more detail. 1 The P -wave ππ → N N amplitudes f± (t) are tabulated in ref. [42]. (See also ref. [43] for an unpublished update that is consistent with ref. [42].) We stress that the representation of eq. (9) gives the exact isovector spectral functions for 4Mπ2 ≤ t ≤ 16Mπ2 , but in practice holds up to t  50Mπ2 . Since the contributions from 4π and higher intermediate states is small up to t  50Mπ2 , Fπ (t) and 1 the f± (t) share the same phase in this region and the two quantities can be replaced by their absolute values.2 The updated pion form factor is shown in flg. 3. The diamonds, squares, and circles show the high statistics data from the CMD-2 [39], KLOE [40], and SND [41] collaborations, respectively. The dashed, solid, and dashdotted lines are our model parametrizations which are of the Gounaris-Sakurai type [20,30]. The form factor shows a pronounced ρ-ω mixing in the vicinity of the ρ-peak. There are discrepancies between the three experimental data sets for the pion form factor [41]. The discrepancies in the ρ-resonance region are shown in more detail in the inset of flg. 3. Since we are not in the position to settle this experimental problem, we will take the three data sets at face value. We will evaluate the 2π continuum given by eq. (9) for all three sets and estimate the errors from the discrepancy between the sets. Using the new high statistics pion form factor data [39, 1 40,41] and the amplitudes f± (t) tabulated in ref. [42], we obtain the spectral functions shown in flg. 4 [38]. We show the spectral functions weighted by 1/t2 for GE (solid 2

We note that representation of eq. (9) is most useful for our purpose. The manifestly real functions J§ (t) = f§1 (t)/Fπ (t) also tabulated in ref. [42] contain assumptions about the pion form factor which leads to inconsistencies when used together with the updated Fπ (t).

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4

spectral function [1/Mπ ]

52

0.06

Im F iV

Im F iS 2ImGE/t

2

2ImGM/t

0.04

2

ω

ρ

S’’ ρπ

ππ

S’

t

ρ’ ρ’’

ρ’’’

t

φ , KK

0.02

0 0

20 2 t [Mπ ]

40

Fig. 4. The 2π spectral function using the new high statistics data for the pion form factor [39, 40, 41]. The spectral functions weighted by 1/t2 are shown for GE (solid line) and GM (dashdotted line) in units of 1/Mπ4 . The previous results by H˜ ohler et al. [42] (without ρ-ω mixing) are shown for comparison by the green lines.

line) and GM (dash-dotted line). The previous results by H˜ohler et al. [42] (without ρ-ω mixing) are given for comparison by the gray/green lines. The general structure of the two evaluations is the same, but there is a difierence in magnitude of about 10%. The difierence between the three data sets for the pion form factor is very small and indicated by the line thickness. The difierence in the form factors is largest in the ρ-peak region (cf. flg. 3), but this 1 (t) region is suppressed by the ππ → N N amplitudes f± which show a strong fall-ofi as t increases. The spectral functions have two distinct features. First, as already pointed out in [15], they contain the important contribution of the ρ-meson with its peak at t  30Mπ2 . Second, on the left shoulder of the ρ, the isovector spectral functions display a very pronounced enhancement close to the two-pion threshold. This is due to the logarithmic singularity on the second Riemann sheet located at tc = 4Mπ2 − Mπ4 /M 2 = 3.98Mπ2 , very close to the threshold. This pole comes from the projection of the nucleon Born graphs, or in modern language, from the triangle diagram. If one were to neglect this important unitarity correction, one would severely underestimate the nucleon isovector radii [17],  6 ∞ dt Im Gvi (t), (10) r2 vi = π 4Mπ2 t2 where i = E, M . In fact, precisely the same efiect is obtained at leading one-loop accuracy in relativistic chiral perturbation theory [44,45]. This topic was also discussed in heavy baryon ChPT [25,27] and in a covariant calculation based on infrared regularization [26]. Thus, the most important 2π contribution to the nucleon form factors can be determined by using either unitarity or ChPT (in the latter case, of course, the ρ contribution is not included).

Fig. 5. Illustration of the spectral function used in the dispersion analysis. The vertical dashed line separates the well-known low-mass contributions (2π, KK, and ρπ continua as well as the ω pole) from the efiective poles at higher momentum transfers.

5 Spectral functions As discussed above the spectral function can at present only be obtained from unitarity arguments for the lightest two-particle intermediate states (2π and KK) [19,36,37]. The ρπ continuum contribution can be obtained from the Bonn-J˜ ulich model [46]. The remaining contributions can be parametrized by vector meson poles. On one hand, the lower mass poles can be identifled with physical vector mesons such as the ω and the φ. In the the case of the 3π continuum, e.g., it has been shown in ChPT that the nonresonant contribution is very small and the spectral function is dominated by the ω [25]. The higher mass poles on the other hand, are simply an efiective way to parametrize higher mass strength in the spectral function. For our current best flt, the spectral function includes the 2π, KK, and ρπ continua from unitarity and the ω pole. In addition to that there are a number of efiective poles at higher momentum transfers in both the isoscalar and isovector channels. The spectral function then has the general structure ¯

Im Fis (t) = Im FiK K (t) + Im Fiρπ (t)  πaVi δ(MV2 − t), +

i = 1, 2,

(11)

V =ω,s1 ,...

Im Fiv (t) = Im Fi2π (t)  + πaVi δ(MV2 − t),

i = 1, 2,

(12)

V =v1 ,...

which is illustrated in in flg. 5. The vertical dashed line separates the well-known low-mass contributions to the spectral function from the efiective poles at higher momentum transfers. In our flts, we also include the widths of the vector mesons. The width and mass of the ω are taken from the particle data tables while the masses and widths of the efiective poles are fltted to the form factor data. We have performed various flts with difierent numbers of efiective poles and including/excluding some of the continuum contributions. In sect. 7, we will discuss preliminary results of this ongoing efiort.

H.-W. Hammer: Nucleon form factors in dispersion theory

6 Constraints

7 Fit results

The number of parameters in the flt function is reduced by enforcing various constraints. The flrst set of constraints concerns the low-t behavior of the form factors. First, we enforce the correct normalization of the form factors, which is given in eq. (2). Second, we constrain the neutron radius from a low-energy neutron-atom scattering experiment [47,48]. Perturbative QCD (pQCD) constrains the behavior of the nucleon em form factors for large momentum transfer. Brodsky and Lepage [49] flnd for t → −∞, Fi (t) → (−t)

53

−(i+1)

 −γ −t ln , Q20

i = 1, 2,

(13)

where Q0  ΛQCD . The anomalous dimension γ depends weakly on the number of flavors, γ = 2.148, 2.160, 2.173 for Nf = 3, 4, 5, in order. The power behavior of the form factors at large t can be easily understood from perturbative gluon exchange. In order to distribute the momentum transfer from the virtual photon to all three quarks in the nucleon, at least two massless gluons have to be exchanged. Since each of the gluons has a propagator ∼ 1/t, the form factor has to fall ofi as 1/t2 . In the case of F2 , there is additional suppression by 1/t since a quark spin has to be flipped. The power behavior of the form factors leads to superconvergence relations of the form 

∞

Im Fi (t) tn dt = 0,

We now discuss some preliminary flt results that are representative for the current status of the analysis. We present results for a flt with 4 efiective isoscalar poles and 3 efiective isovector poles whose residua, masses, and widths are fltted to the data. In flg. 6, we show the results for all four form factors compared to the world data for the form factors. Our data basis is taken from ref. [34] and in addition also includes the new data that have appeared since 2003 (see ref. [5]). The results for GnM , GpE , GpM are normalized to the phenomenological dipole flt: −2  Q2 2 , (15) GD (Q ) = 1 + 2 mD where m2D = 0.71 GeV2 . The dash-dotted line gives the result of ref. [23], while the the solid line indicates our present best flt. The new flt leads to an improved description of the form factor data compared with ref. [23]. In particular, the rapid fall-ofi of the JLab polarization data for GpE [51,52] is now described. The χ2 per degree of freedom is 0.84. Note that we do not obtain a pronounced bump structure in GnE as observed in ref. [34]. We will come back to this question in sect. 9 and discuss the modiflcations in the spectral function required to produce this structure. The stability constraint requires to use the minimum number of poles required to describe the data [50]. In the future, we plan to further reduce the number of efiective poles in order to improve the stability.

(14)

t0

with n = 0 for F1 and n = 0, 1 for F2 . The asymptotic behavior of eq. (13) is obtained by choosing the residues of the vector meson pole terms such that the leading terms in the 1/t-expansion cancel. The logarithmic term in eq. (13) was included in some of our earlier analyses [20,21,23] but has little impact on the flt. The particular way this constraint was implemented, however, lead to an unphysical logarithmic singularity of the form factors in the time-like region. In order to be able to include the data for the form factors at large time-like momentum transfers, the logarithmic constraint is not enforced in the current analysis. The number of efiective poles in eqs. (11, 12) is determined by the stability criterion discussed in detail in [50]. In short, we take the minimum number of poles necessary to flt the data. For the preliminary results discussed in the next section, we took 4 efiective isoscalar poles and 3 effective isovector poles whose residua, masses, and widths are fltted to the data. The number of free parameters is strongly reduced by the various constraints (unitarity, normalizations, superconvergence relations), so that we end up with 19 free parameters in the preliminary flt presented in the next section. Our general strategy is to reduce the number of parameters even further without sacriflcing the quality of the flt.

Table 1. Nucleon radii in fm extracted from the flt in flg. 6. p rE p rM n rE n rM

[fm] [fm] [fm] [fm]

This work 0.84...0.857 0.85...0.875 −0.12...−0.10 0.86...0.88

Ref. [23] 0.848 0.857 −0.12 0.879

Recent determ. 0.886(15) [53, 54, 55] 0.855(35) [54, 56] −0.115(4) [48] 0.873(11) [57]

In Table 1, we give the nucleon radii extracted from our flt. The neutron radius is included as a soft constraint in our flt and therefore not a prediction.3 The other nucleon radii are generally in good agreement with other recent determinations using only low-momentum-transfer data given in the table. Our result for the proton radius, however, is somewhat small. This was already the case in the dispersion analyses of refs. [20,23]. We speculate that the reason for this discrepancy lies in inconsistencies in the data sets. In this type of global analysis all four form factors are analyzed simultaneously and both data at small and large momentum transfers enter. This can be an advantage or a disadvantage depending on the question at hand. Another possible reason for the discrepancy is 2γ physics which was neglected in the data analysis of most older experiments [58]. 3

A soft constraint is not implemented exactly but deviations from the constraint are penalized in the χ2 of the flt.

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0.1

1.2 GM /(μnGD)

0.08 1

n

GE

n

0.06 0.04

0.8

0.02 0

0

0.5

1 1.5 2 2 Q [GeV ]

0.6 0.01

2

10

GM /(μpGD)

1.2

1

1

p

p

GE /GD

1.5

1 0.1 2 2 Q [GeV ]

0.5

0 0.01

1 0.1 2 2 Q [GeV ]

10

0.8

0.6 0.01

0.1

1 10 2 Q [GeV ]

100

2

p p Fig. 6. The nucleon electromagnetic form factors for space-like momentum transfer. The results for G n M , GE , GM are normalized to the dipole flt. The dash-dotted line gives the result of ref. [23], while the the solid line indicates our preliminary best fit.

8 Time-like data We have also performed flrst flts that include data in the time-like region. The extraction of these data is more challenging than in the space-like region. At the nucleonantinucleon threshold, the electric and magnetic form factors are equal by deflnition: GM (4M 2 ) = GE (4M 2 ), while one expects the magnetic form factor to dominate at large momentum transfer. Moreover, the form factors are complex in the time-like region, since several physical thresholds are open. Separating |GM | and |GE | unambiguously from the data requires a measurement of the angular distribution, which is di– cult. In most experiments, it has been assumed that either |GM | = |GE | (which should be a good approximation close to the two-nucleon threshold) or |GE | = 0 (which should be a good approximation for large momentum transfers). Most recent data have been presented using the latter hypothesis. The time-like data were already included in the dispersion analyses of refs. [21,22]. The proton magnetic form factor up to t ≈ 6 GeV2 was well described by these analyses. Data at higher momentum transfers were not included. The data for the neutron magnetic form factor are

from the pioneering FENICE experiment [11]. They have been analyzed under both the assumption |GE | = |GM | and |GE | = 0. The latter hypothesis is favored by the measured angular distributions [11]. Neither data set could be described by the analysis [22]. In flg. 7, we show the current status of the analysis of the time-like data for the magnetic form factors. For the proton magnetic form factor, data up to momentum transfers t ≈ 15 GeV2 have been included [6,7,8,9,10]. Our preliminary flt gives a good description in the threshold region but starts to deviate signiflcantly around t ≈ 5 GeV 2 . The data for t ≥ 10 GeV2 are well described. This seems to be due to a slight inconsistency in the data around 5 GeV2 and for t ≥ 10 GeV2 . This question deserves further attention. The status for the neutron form factor is the same as in the previous analysis [22]: Neither of the two data sets from ref. [11] can be described. Even though we are not yet in the region where perturbative QCD is applicable, it comes as a surprise that the neutron form factor is larger in magnitude than the proton one. Perturbative QCD predicts asymptotically equal magnitudes. In any case, there is interesting physics in the time-like nucleon form factors

H.-W. Hammer: Nucleon form factors in dispersion theory

55

1.2

p

4πr ρ(r) [1/fm]

|GM (t)|

0.2

0

5

10

15

2

t [GeV ] 0.8

n

v

v

0.8

GE

0.4

0.2

3.5

4

5

4.5

5.5

0.4

0.0 0.0

0.5

1.0 r [fm]

1.5

2.0

Fig. 8. Pion cloud contribution to the nucleon charge density. The lines show the result of Friedrich and Walcher [34], while the bands give the result of ref. [35]. Only the long-range contributions for r > 1 fm are meaningful for the comparison of the two results.

|GM (t)|

0.6

0 3

GM

2

0.4

6

2

t [GeV ] Fig. 7. Current status of our analysis of the magnetic form factors in the time-like region compared to the world data [6, 7, 8, 9, 10, 11]. The solid line gives our preliminary best fit, while the vertical dotted line indicates the two-nucleon threshold.

and new precision experiments such as the PANDA and PAX experiments at GSI would be very welcome.

9 Pion cloud of the nucleon Friedrich and Walcher (FW) recently analysed the em nucleon form factors and performed various phenomenological flts [34]. Their flts showed a pronounced bump structure in GnE which they interpreted using an ansatz for the pion cloud based on the idea that the proton can be thought of as a virtual neutron-positively charged pion pair. They found a very long-range contribution to the charge distribution in the Breit frame extending out to about 2 fm which they attributed to the pion cloud. While naively the pion Compton wave length is of this size, these flndings are indeed surprising if compared with the pion cloud contribution due to the 2π continuum contribution to the isovector spectral functions discussed in sect. 4. As was shown by Hammer, Drechsel, and Mei ner [35], these latter contributions to the long-range part of the nucleon structure are much more conflned in coordinate space and agree well with earlier (but less systematic) calculations based on chiral soliton models, see, e.g., [59]. In the dispersion-theoretical framework, the longest-range part of the pion cloud contribution to the nucleon form

factors is given by the 2π continuum the lowest-mass intermediate state including only pions. Note that a onepion intermediate state is forbidden by parity. The nonresonant part of the 2π continuum can be calculated in ChPT [27] while the full continuum can be obtained from experimental data and unitarity as discussed in sect. 4. The pion cloud corresponds to the nonresonant part of the 2π continuum excluding the ρ. Consequently, the ρ contribution has to be subtracted from the full 2π continuum.4 The error in this subtraction was estimated using three difierent methods for the separation of the contributions [35]. The charge distribution can then be obtained from the nonresonant part of the 2π continuum by Fourier transformation. This leads to the relation: √  40Mπ2 1 e−r t v v ρi (r) = , (16) dt Im Gi (t) 4π 2 4Mπ2 r where i = E, M . The 2π contribution from t ≥ 40Mπ2 is small and can be neglected [35]. The result for the pion cloud contribution to the nucleon charge density is shown in flg. 8. The lines show the result of FW [34], while the bands give the result of > 1 fm ref. [35]. Only the long-range contributions for r ∼ should be compared since the separation of the shortrange part into resonant and nonresonant contributions is arbitrary. In comparison with ref. [34], the 2π continuum contribution to the charge density is generally much smaller at distances beyond 1 fm, e.g., by a factor of 3 for ρvE (r) at r = 1.5 fm. We emphasize that this result is obtained from independent physical information that determines the 2π continuum (pion form factor and ππ → N N amplitudes, cf. sect. 4) and not from form factor flts. As a consequence, it remains to be shown how the proposed long-range pion cloud can be reconciled with what 4 Note that this separation is not unique. It is only meaningful for the long-range part. The separation of the short-range part is model- and even representation-dependent.

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n

0.1

2

GE (Q )

0.05

0 0

explicitly calculated in heavy baryon ChPT and no enhancement was found. Moreover, the inelasticity from four pions in ππ scattering and four-pion production in e+ e− annihilation at low momentum transfer are known to be small [42,60,61].

10 Summary & outlook

0.2

0.6

0.4 2

0.8

1

2

Q [GeV ] Fig. 9. The neutron electric form factor at low momentum transfer: present flt with additional low-mass strength (dashed line) compared to the flts of Friedrich and Walcher [34] (doubledash-dotted line). For comparison, the flts of sect. 7 (solid line) and ref. [23] (dash-dotted line) are also shown.

is known from dispersion relations and ChPT. In order to clarify this issue, we have performed various flts in order to understand what structures in the spectral function are required to reproduce the bump in GnE . We flnd that the structure can only be reproduced if additional low-mass < 1 GeV2 is alstrength in the spectral function below t ∼ lowed beyond the 2π, KK, and ρπ continua and the ω pole. In the flts of sects. 7 and 8 such strength was explicitly excluded. In flg. 9, we show the neutron electric form factor at low momentum transfer. The flt of FW [34] is given by the double-dash-dotted line, while the present flt with additional low-mass strength is given by the dashed line. For comparison, we show also the flt of ref. [23] (dashdotted line) and the flt from sect. 7 (solid line). The flt with additional low-mass strength shows a clear bump structure around Q2 ∼ 0.3 GeV2 . This structure requires three additional low mass poles: two isoscalar poles at Ms2 = 0.13 GeV2 , 0.54 GeV2 and one isovector pole at Mv2 = 0.30 GeV2 . In principle, vector meson dominance works well for t ≤ 1 GeV2 and one should be able to interpret these poles as physical vector mesons. However, no such vector mesons are known in this region. This raises the question of whether the efiective low-mass poles can be interpreted as something else? One possible solution would be to interpret the poles as efiective poles mimicking some continuum contribution. It is interesting to note that the three low-mass poles happen to come out at the thresholds of the 3π, 4π and 5π continua and are located in the correct isospin channel. Maybe these higher-order pion continua are more important than previously thought and have a threshold enhancement similar to the 2π continuum that is accounted for by the efiective poles? Even though this scenario has a certain appeal, it appears unlikely given the current state of knowledge. In ref. [25], the threshold behavior of the 3π continuum was

Dispersion theory simultaneously describes all four nucleon form factors over the whole range of momentum transfers in both the space-like and time-like regions. It allows for the inclusion of constraints from other physical processes, unitarity, and ChPT and therefore is an ideal tool to analyze the form factor data. We have presented preliminary results for our new dispersion analysis that is currently carried out in Bonn. The spectral function has been improved and contains the updated 2π continuum [38], as well the KK [36,37] and ρπ continua [46]. Our preliminary best flt gives a consistent description of the world data in the space-like region. The understanding of the time-like form factors is more di– cult and a future challenge for theorists and experimentalists alike. As part of this ongoing theoretical program, many things remain to be done: The stability constraint requires to use the minimum number of poles. Our strategy for the future is to successively reduce the number of poles without sacrifying the quality of the flt. Furthermore, the description of the time-like data needs to be improved. In previous experiments, the separation of GE and GM could only be carried out under overly simplifying assumptions. New data, such as planned for the PANDA and PAX experiments at GSI, are therefore called for. Other improvements concern the quantiflcation of theoretical and systematic uncertainties in the analysis, the inclusion of perturbative QCD corrections beyond superconvergence (leading logarithms etc.), and the inclusion of two-photon physics. The latter point might require to analyze the cross section data directly. Last but not least, the consequences of the new data for the strange vector form factors of the nucleon need to be worked out. This work was done in collaboration with M.B. Belushkin, D. Drechsel, and Ulf-G. Mei ner. M.J. Ramsey-Musolf has contributed in earlier stages of the project. The work was supported in part by the EU I3HP under contract number RII3CT-2004-506078 and the DFG through funds provided to the SFB/TR 16 Subnuclear Structure of Matter and SFB 443 Many Body Structure of Strongly Interacting Systems . I would like to thank Hartmuth Arenh˜ ovel, Hartmut Backe, Dieter Drechsel, J˜ org Friedrich, Karl-Heinz Kaiser, and Thomas Walcher for a very stimulating and enjoyable time in Mainz. I have had many personal interactions with them through scientiflc discussions and/or through lectures and seminars I attended as a student. In particular, I want to thank my PhD advisor Dieter Drechsel from whom I have learned much about physics and research.

H.-W. Hammer: Nucleon form factors in dispersion theory

References 1. H. Gao, Int. J. Mod. Phys. E 12, 1 (2003); 12, 567 (2003)(E) (arXiv:nucl-ex/0301002). 2. C.E. Hyde-Wright, K. de Jager, Annu. Rev. Nucl. Part. Sci. 54, 217 (2004) (arXiv:nucl-ex/0507001). 3. Th. Udem et al., Phys. Rev. Lett. 79, 2646 (1997). 4. D.H. Beck, B.R. Holstein, Int. J. Mod. Phys. E 10, 1 (2001) (arXiv:hep-ph/0102053). 5. M. Ostrick, these proceedings and references therein. 6. E835 Collaboration (M. Ambrogiani et al.), Phys. Rev. D 60, 032002 (1999). 7. BES Collaboration (M. Ablikim et al.), Phys. Lett. B 630, 14 (2005) (arXiv:hep-ex/0506059). 8. CLEO Collaboration (T.K. Pedlar et al.), Phys. Rev. Lett. 95, 261803 (2005) (arXiv:hep-ex/0510005). 9. BABAR Collaboration (B. Aubert et al.), Phys. Rev. D 73, 012005 (2006) (arXiv:hep-ex/0512023). 10. R. Baldini, E. Pasqualucci, in Chiral Dynamics: Theory and Experiment, edited by A.M. Bernstein, B.R. Holstein, Lect. Notes Phys., Vol. 452 (Springer, Heidelberg, 1995). 11. A. Antonelli et al., Nucl. Phys. B 517, 3 (1998). 12. H. Fr˜ ohlich, W. Heitler, N. Kemmer, Proc. R. Soc. A 166, 155 (1938). 13. G.F. Chew, R. Karplus, S. Gasiorowicz, F. Zachariasen, Phys. Rev. 110, 265 (1958). 14. P. Federbush, M.L. Goldberger, S.B. Treiman, Phys. Rev. 112, 642 (1958). 15. W.R. Frazer, J.R. Fulco, Phys. Rev. Lett. 2, 365 (1959). 16. W.R. Frazer, J.R. Fulco, Phys. Rev. 117, 1609 (1960). 17. G. H˜ ohler, E. Pietarinen, Phys. Lett. B 53, 471 (1975). 18. G. H˜ ohler et al., Nucl. Phys. B 114, 505 (1976). 19. G. H˜ ohler, E. Pietarinen, Nucl. Phys. B 95, 210 (1975). 20. P. Mergell, U.-G. Mei ner, D. Drechsel, Nucl. Phys. A 596, 367 (1996) (arXiv:hep-ph/9506375). 21. H.-W. Hammer, U.-G. Mei ner, D. Drechsel, Phys. Lett. B 385, 343 (1996) (arXiv:hep-ph/9604294). 22. H.-W. Hammer, in Proceedings of the e+ e Physics at Intermediate Energies Conference, edited by Diego Bettoni, eConf C010430, W08 (2001) (arXiv:hep-ph/0105337). 23. H.-W. Hammer, U.-G. Mei ner, Eur. Phys. J. A 20, 469 (2004) (arXiv:hep-ph/0312081). 24. V. Bernard, N. Kaiser, U.-G. Mei ner, Int. J. Mod. Phys. E 4, 193 (1995) (arXiv:hep-ph/9501384). 25. V. Bernard, N. Kaiser, U.-G. Mei ner, Nucl. Phys. A 611, 429 (1996) (arXiv:hep-ph/9607428). 26. B. Kubis, U.-G. Mei ner, Nucl. Phys. A 679, 698 (2001) (arXiv:hep-ph/0007056). 27. N. Kaiser, Phys. Rev. C 68, 025202 (2003) (arXiv:nuclth/0302072). 28. M.R. Schindler, J. Gegelia, S. Scherer, Eur. Phys. J. A 26, 1 (2005) (arXiv:nucl-th/0509005). 29. J.J. Sakurai, Ann. Phys. (NY) 11, 1 (1960). 30. G.J. Gounaris, J.J. Sakurai, Phys. Rev. Lett. 21, 244 (1968). 31. M. Gari, W. Kr˜ umpelmann, Z. Phys. A 322, 689 (1985). 32. E.L. Lomon, Phys. Rev. C 64, 035204 (2001) (arXiv:nuclth/0104039).

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33. S. Dubnicka, A.Z. Dubnickova, P. Weisenpacher, J. Phys. G 29, 405 (2003) (arXiv:hep-ph/0208051). 34. J. Friedrich, T. Walcher, Eur. Phys. J. A 17, 607 (2003) (arXiv:hep-ph/0303054). 35. H.-W. Hammer, D. Drechsel, U.-G. Mei ner, Phys. Lett. B 586, 291 (2004) (arXiv:hep-ph/0310240). 36. H.-W. Hammer, M.J. Ramsey-Musolf, Phys. Rev. C 60, 045205 (1999); 62, 049903 (2000)(E) (arXiv:hepph/9812261). 37. H.-W. Hammer, M.J. Ramsey-Musolf, Phys. Rev. C 60, 045204 (1999); 62, 049902 (2000)(E) (arXiv:hepph/9903367). 38. M.A. Belushkin, H.-W. Hammer, U.-G. Mei ner, Phys. Lett. B 633, 507 (2006) (arXiv:hep-ph/0510382). 39. CMD-2 Collaboration (R.R. Akhmetshin et al.), arXiv:hep-ex/9904027; Phys. Lett. B 527, 161 (2002) (arXiv:hep-ex/0112031); 578, 285 (2004) (arXiv:hepex/0308008). 40. KLOE Collaboration (A. Aloisio et al.), Phys. Lett. B 606, 12 (2005) (arXiv:hep-ex/0407048). 41. M.N. Achasov et al., J. Exp. Theor. Phys. 101, 1053 (2005) (arXiv:hep-ex/0506076). 42. G. H˜ ohler, Pion-Nucleon Scattering, Landolt-B˜ ornstein Vol. I/9b, edited by H. Schopper (Springer, Berlin, 1983). 43. E. Pietarinen, A calculation of ππ → N N amplitudes in the pseudophysical region, University of Helsinki Preprint Series in Theoretical Physics, HU-TFT-17-77, unpublished. 44. J. Gasser, M.E. Sainio, A. Svarc, Nucl. Phys. B 307, 779 (1988). 45. U.-G. Mei ner, Int. J. Mod. Phys. E 1, 561 (1992). 46. U.-G. Mei ner, V. Mull, J. Speth, J.W. van Orden, Phys. Lett. B 408, 381 (1997) (arXiv:hep-ph/9701296). 47. S. Kopecky et al., Phys. Rev. Lett. 74, 2427 (1995). 48. S. Kopecky, M. Krenn, P. Riehs, S. Steiner, J.A. Harvey, N.W. Hill, M. Pernicka, Phys. Rev. C 56, 2229 (1997). 49. S.J. Brodsky, G.P. Lepage, Phys. Rev. D 22, 2157 (1980). 50. I. Sabba-Stefanescu, J. Math. Phys. 21, 175 (1980). 51. Jefierson Lab Hall A Collaboration (M.K. Jones et al.), Phys. Rev. Lett. 84, 1398 (2000) (arXiv:nucl-ex/9910005). 52. Jefierson Lab Hall A Collaboration (O. Gayou et al.), Phys. Rev. Lett. 88, 092301 (2002) (arXiv:nuclex/0111010). 53. R. Rosenfelder, Phys. Lett. B 479, 381 (2000) (arXiv:nuclth/9912031). 54. I. Sick, Phys. Lett. B 576, 62 (2003) (arXiv:nuclex/0310008). 55. K. Melnikov, T. van Ritbergen, Phys. Rev. Lett. 84, 1673 (2000) (arXiv:hep-ph/9911277). 56. I. Sick, private communication. 57. G. Kubon et al., Phys. Lett. B 524, 26 (2002) (arXiv:nuclex/0107016). 58. P.A.M. Guichon, M. Vanderhaeghen, Phys. Rev. Lett. 91, 142303 (2003) (arXiv:hep-ph/0306007). 59. U.-G. Mei ner, Phys. Rep. 161, 213 (1988). 60. J. Gasser, U.-G. Mei ner, Nucl. Phys. B 357, 90 (1991). 61. G. Ecker, R. Unterdorfer, Eur. Phys. J. C 24, 535 (2002) (arXiv:hep-ph/0203075).

Eur. Phys. J. A 28, s01, 59 70 (2006) DOI: 10.1140/epja/i2006-09-007-4

EPJ A direct electronic only

Chiral perturbation theory Success and challenge S. Scherera Institut f˜ ur Kernphysik, Johannes Gutenberg-Universit˜ at Mainz, J.J. Becher Weg 45, D-55099 Mainz, Germany / Published online: 12 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Chiral perturbation theory is the efiective fleld theory of the strong interactions at low energies. We will give a short introduction to chiral perturbation theory for mesons and will discuss, as an example, the electromagnetic polarizabilities of the pion. These have recently been extracted from an experiment on radiative π + photoproduction from the proton (γp → γπ + n) at the Mainz Microtron MAMI. Next we will turn to the one-baryon sector of chiral perturbation theory and will address the issue of a consistent power counting scheme. As examples of the heavy-baryon framework we will comment on the extraction of the axial radius from pion electroproduction and will discuss the generalized polarizabilities of the proton. Finally, we will discuss two recently proposed manifestly Lorentz-invariant renormalization schemes and illustrate their application in a calculation of the nucleon electromagnetic form factors. PACS. 11.10.Gh Renormalization 11.30.Rd Chiral symmetries 13.40.-f Electromagnetic processes and properties 13.40.Gp Electromagnetic form factors 13.60.Fz Elastic and Compton scattering 13.60.Le Meson production

1 Introduction Chiral perturbation theory (ChPT) [1,2,3,4] is the efiective fleld theory (EFT) [5] of the strong interactions at low energies. The central idea of the EFT approach was formulated by Weinberg as follows [1]: . . . if one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry principles. In the context of the strong interactions these ideas have flrst been applied to the interactions among the Goldstone bosons of spontaneous symmetry breaking in quantum chromodynamics (QCD). The efiective theory is formulated in terms of the asymptotically observed states instead of the quark and gluon degrees of freedom of the underlying (fundamental) theory, namely QCD. The corresponding EFT mesonic chiral perturbation theory has been tested at the twoloop level (see, e.g., [6,7] for a pedagogical introduction). A successful EFT program requires both the knowledge of the most general Lagrangian up to and including the given order one is interested in as well as an expansion scheme for observables. Due to the vanishing of the Goldstone boson masses in the chiral limit in combination with their a

e-mail: [email protected]

vanishing interactions in the zero-energy limit, a derivative and quark-mass expansion is a natural scenario for the corresponding EFT. At present, in the mesonic sector the Lagrangian is known up to and including O(q 6 ), where q denotes a small quantity such as a four-momentum or a pion mass. The combination of dimensional regularization with the modifled minimal subtraction scheme of ChPT [2] leads to a straightforward correspondence between the loop expansion and the chiral expansion in terms of momenta and quark masses at a flxed ratio, and provides a consistent power counting for renormalized quantities. In the extension to the one-nucleon sector [4] an additional scale, namely the nucleon mass, enters the description. In contrast to the Goldstone boson masses, the nucleon mass does not vanish in the chiral limit. As a result, the straightforward correspondence between the loop expansion and the chiral expansion of the mesonic sector, at flrst sight, seems to be lost: higher-loop diagrams can contribute to terms as low as O(q 2 ) [4]. This problem has been eluded in the framework of the heavy-baryon formulation of ChPT [8,9], resulting in a power counting analogous to the mesonic sector. The basic idea consists in expressing the relativistic nucleon fleld in terms of a velocity-dependent fleld, thus dividing nucleon momenta into a large piece close to on-shell kinematics and a soft residual contribution. Most of the calculations in the onebaryon sector have been performed in this framework (for an overview see, e.g., [10]) which essentially corresponds to

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a simultaneous expansion of matrix elements in 1/m and 1/(4πFπ ). However, there is price one pays when giving up manifest Lorentz invariance of the Lagrangian. At higher orders in the chiral expansion, the expressions due to the 1/m corrections of the Lagrangian become increasingly complicated [11,12]. Moreover, not all of the scattering amplitudes, evaluated perturbatively in the heavy-baryon framework, show the correct analytical behavior in the low-energy region [13]. In recent years, there has been a considerable efiort in devising renormalization schemes leading to a simple and consistent power counting for the renormalized diagrams of a manifestly Lorentz-invariant approach [14,15,16,17,18,19,20,21]. In the following we will highlight a few topics in chiral perturbation theory which have been subject of experimental tests at the Mainz Microtron MAMI.

is a result of the flnite quark masses of the u, d and s quarks. This explicit symmetry breaking in terms of the quark masses is treated perturbatively. The symmetries as well as the symmetry breaking pattern of QCD once the quark masses are included are mapped onto the most general (efiective) Lagrangian for the interaction of the Goldstone bosons. The Lagrangian is organized in the number of the (covariant) derivatives and of the quark mass terms [1,2,3,25,26,27,28,29,30,31] Lπ = L2 + L4 + L6 + · · · , where the lowest-order Lagrangian is given by L2 =

(2) 1

  F2 Tr Dμ U (Dμ U )† + χU † + U χ† . 4

(3)

Here, 

φ U (x) = exp i F

2 Chiral perturbation theory for mesons 2.1 The effective Lagrangian and Weinberg’s power counting scheme The starting point of mesonic chiral perturbation theory is a chiral SU (Nl )L × SU (Nl )R × U (1)V symmetry of the QCD Lagrangian for Nl massless (light) quarks: L0QCD =

Nl  l=1

1 (qR,l iD / qR,l + qL,l iD / qL,l ) − Gμν,a Gaμν . (1) 4

In eq. (1), qL,l and qR,l denote the left- and right-handed components of the light quark flelds. Here, we will be concerned with the cases Nl = 2 and Nl = 3 referring to massless u and d or u, d and s quarks, respectively. Furthermore, we will neglect the terms involving the heavyquark flelds. The covariant derivative Dμ qL/R,l contains the flavor-independent coupling to the eight gluon gauge potentials, and Gμν,a are the corresponding fleld strengths. The Lagrangian of eq. (1) is invariant under separate global SU (Nl )L/R transformations of the left- and righthanded flelds. In addition, it has an overall U (1)V symmetry. Several empirical facts give rise to the assumption that this chiral symmetry is spontaneously broken down to its vectorial subgroup SU (Nl )V × U (1)V . For example, the low-energy hadron spectrum seems to follow multiplicities of the irreducible representations of the group SU (Nl ) (isospin SU (2) or flavor SU (3), respectively) rather than SU (Nl )L × SU (Nl )R , as indicated by the absence of degenerate multiplets of opposite parity. Moreover, the lightest mesons form a pseudoscalar octet with masses that are considerably smaller than those of the corresponding vector mesons. According to Coleman’s theorem [22], the symmetry pattern of the spectrum reflects the invariance of the vacuum state. Therefore, as a result of Goldstone’s theorem [23,24], one would expect 6 − 3 = 3 or 16 − 8 = 8 massless Goldstone bosons for Nl = 2 and Nl = 3, respectively. These Goldstone bosons have vanishing interactions as their energies tend to zero. Of course, in the real world, the pseudoscalar meson multiplet is not massless which



 ,

φ=

√ + 0 √π − 2π0 , 2π −π

is a unimodular unitary (2 × 2) matrix containing the Goldstone boson flelds. In eq. (3), F denotes the piondecay constant in the chiral limit: Fπ = F [1 + O(m)] = 92.4 MeV. When including the electromagnetic interaction, the covariant derivative is deflned as Dμ U = ∂μ U + ieAμ [Q, U ], where Q = diag(2/3, −1/3) denotes the quark charge matrix. We work in the isospin-symmetric limit mu = md = m. The quark masses are contained in χ = 2Bm = M 2 , where M 2 denotes the lowest-order expression for the squared pion mass and B is related to the quark condensate qq 0 in the chiral limit. The next-toleading-order Lagrangian contains 7 low-energy constants li [2]

 1 μν † R L μν R μν L4 = l5 Tr(fμν U fL U ) − Tr(fμν fL + fμν fR ) 2  R μ l6 L D U (Dν U )† + fμν (Dμ U )† Dν U + · · · , (4) +i Tr fμν 2 where we have displayed those terms which will be relevant for the discussion of Compton scattering below. In that case, the fleld strength is given by R L = fμν = −e(∂μ Aν − ∂ν Aμ )Q. fμν

In addition to the most general Lagrangian, one needs a method to assess the importance of various diagrams calculated from the efiective Lagrangian. Using Weinberg’s power counting scheme [1] one may analyze the behavior of a given diagram calculated in the framework of eq. (2) under a linear re-scaling of all external momenta, pi → tpi , and a quadratic re-scaling of the light quark masses, m → t2 m, which, in terms of the Goldstone boson masses, corresponds to M 2 → t2 M 2 . The chiral dimension D of a given diagram with amplitude M(pi , m) is deflned by M(tpi , t2 m) = tD M(pi , m), 1

(5)

In the following, we will give equations for the two-flavor case.

S. Scherer: Chiral perturbation theory

61

where, in n dimensions, D = nNL − 2Iπ +

∞  k=1

= 2 + (n − 2)NL +

π 2kN2k ∞ 

π 2(k − 1)N2k

(6) (7)

2 Fig. 1. One-loop contribution to the pion self-energy. The number 2 in the interaction blob refers to L2 .

k=1

≥ 2 in 4 dimensions.

Here, NL is the number of independent loop momenta, Iπ π the number of internal pion lines, and N2k the number of vertices originating from L2k . A diagram with chiral dimension D is said to be of order O(q D ). Clearly, for small enough momenta and masses diagrams with small D, such as D = 2 or D = 4, should dominate. Of course, the rescaling of eq. (5) must be viewed as a mathematical tool. While external three-momenta can, to a certain extent, be made arbitrarily small, the re-scaling of the quark masses is a theoretical instrument only. Note that, for n = 4, loop diagrams are always suppressed due to the term 2NL in eq. (6). In other words, we have a perturbative scheme in terms of external momenta and masses which are small compared to some scale (here 4πF ≈ 1 GeV). Figures 1 and 2 show contributions to the pion selfenergy with D = 4 · 1 − 2 · 1 + 2 · 1 = 4 and D = 4 · 4 − 2 · 5 + 2 · 2 = 10, respectively. As a speciflc example, let us consider the contribution of flg. 1 to the pion self-energy. Without going into the details, the explicit result of the one-loop contribution is given by (see, e.g., [6]) Σloop (p2 ) =

4p2 − M 2 Iπ (M 2 , μ2 , n) = O(q 4 ), 6F 2

where the dimensionally regularized integral is given by

 2  M M2 R + ln + O(n − 4). (8) Iπ (M 2 , μ2 , n) = 16π 2 μ2 In eq. (8), R is deflned as 2 R= − [ln(4π) − γE + 1], n−4

2

2

Fig. 2. Four-loop contribution to the pion self-energy.

every one of the inflnite number of interactions allowed by symmetries, the so-called non-renormalizable theories are actually just as renormalizable as renormalizable theories. According to Weinberg’s power counting of eq. (6), one-loop graphs with vertices from L2 are of O(q 4 ). The conclusion is that one needs to adjust (renormalize) the parameters of L4 to cancel one-loop inflnities. In doing so, one still has the freedom of choosing a suitable renormalization condition. For example, in the minimal subtraction scheme (MS) one would flx the parameters of the counterterm Lagrangian such that they would precisely absorb the contributions proportional to 2/(n − 4). In the  emmodifled minimal subtraction scheme of ChPT (MS) ployed in [2], the seven (bare) coe– cients li of the O(q 4 ) Lagrangian of (4) are expressed in terms of renormalized coe– cients lir as li = lir + γi

R , 32π 2

(10)

where the γi are flxed numbers. (9)

with n denoting the number of space-time dimensions and γE = −Γ  (1) being Euler’s constant. Note that both factors the fraction and the integral each count as O(q 2 ) resulting in O(q 4 ) for the total expression as anticipated. In other words, when calculating one-loop graphs, using vertices from L2 of eq. (3), one generates inflnities (so-called ultraviolet divergences). In the framework of dimensional regularization these divergences appear as poles at space-time dimension n = 4, since R is inflnite as n → 4. The loop diagrams are renormalized by absorbing the inflnite parts into the redeflnition of the flelds and the parameters of the most general Lagrangian. Since L2 of eq. (3) is not renormalizable in the traditional sense, the inflnities cannot be absorbed by a renormalization of the coe– cients F and B. However, to quote from ref. [32]: . . . the cancellation of ultraviolet divergences does not really depend on renormalizability; as long as we include

2.2 Electromagnetic polarizabilities of the pion In the framework of classical electrodynamics, the electric and magnetic polarizabilities α and β describe the response of a system to a static, uniform, external electric and magnetic fleld in terms of induced electric and magnetic dipole moments. In principle, empirical information on the pion polarizabilities can be obtained from the differential cross section of low-energy Compton scattering on a charged pion   2  2 dσ ω e e2 1 + z2 = dΩlab ω 4πMπ 4πMπ 2  

ωω (α + β)π+ (1 + z)2 + (α − β)π+ (1 − z)2 − 2 +··· ,

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where z = q·q  and ω  /ω = [1+ω(1−z)/Mπ ]. The forward and backward difierential cross sections are sensitive to (α + β)π+ and (α − β)π+ , respectively. The predictions for the charged pion polarizabilities at O(q 4 ) [33] result from an old current-algebra low-energy theorem [34] απ+ = −βπ+ = 2

γ

γ

π π

1 l6 − l 5 e2 4π (4πFπ )2 Mπ 6

+

+

p

n

Fig. 3. The reaction γp → γπ n contains Compton scattering on a pion as a sub diagram in the t channel, where t = (pn − pp )2 . +

= (2.64 ± 0.09) × 10−4 fm3 , which relates Compton scattering on a charged pion, γπ + → γπ + , in terms of a chiral Ward identity to radiative charged-pion beta decay, π + → e+ νe γ. The linear combination l6 −l5 of scale-independent low-energy constants [2] is flxed using the most recent determination of the ratio of the pion axial-vector form factor FA and the vector form factor FV via the radiative pion beta decay [35]: γ=

1 FA (l6 − l5 ) = = 0.443 ± 0.015. 6 FV

A two-loop analysis (O(q 6 )) of the charged-pion polarizabilities has been worked out in [36,37]2 : (α + β)π+ = (0.3 ± 0.1) × 10−4 fm3 , (α − β)π+ = (4.4 ± 1.0) × 10−4 fm3 .

(11) (12)

The degeneracy απ+ = −βπ+ is lifted at the two-loop level. The corresponding corrections amount to an 11% (22%) change of the O(q 4 ) result for απ+ (βπ+ ), indicating a similar rate of convergence as for the ππ-scattering lengths [2, 38]. The efiect of the new low-energy constants appearing at O(q 6 ) on the pion polarizability was estimated via resonance saturation by including vector and axial-vector mesons. The contribution was found to be about 50% of the two-loop result. However, one has to keep in mind that [36,37] could not yet make use of the improved analysis of radiative pion decay which, in the meantime, has also been evaluated at two-loop accuracy [39,40]. Taking higher orders in the quark mass expansion into account, Bijnens and Talavera obtain (l6 − l5 ) = 2.98 ± 0.33 [39], which would slightly modify the leading-order prediction to απ+ = (2.96 ± 0.33) × 10−4 fm3 instead of απ+ = (2.7 ± 0.4) × 10−4 fm3 used in [36,37]. Accordingly, the difference (α − β)π+ of (12) would increase to 4.9 × 10−4 fm3 instead of 4.4 × 10−4 fm3 , whereas the sum would remain the same as in eq. (11). As there is no stable pion target, empirical information about the pion polarizabilities is not easy to obtain. For that purpose, one has to consider reactions which contain the Compton scattering amplitude as a building block, such as, e.g., the Primakofi efiect in high-energy References [36, 37] use (l6 − l5 ) = 2.7 ± 0.4 instead of 2.64 ± 0.72 which was obtained in ref. [2] from γ = 0.44 ± 0.12. Correspondingly, this also generates a smaller error in the O(q 4 ) prediction απ+ = (2.7 ± 0.4) × 10 4 fm3 instead of (2.62 ± 0.71) × 10 4 fm3 . 2

pion-nucleus bremsstrahlung, π − Z → π − Zγ [41], radiative pion photoproduction on the nucleon, γp → γπ + n [42, 43], and pion pair production in e+ e− scattering, e+ e− → e+ e− π + π − [44,45,46,47]. The results of the older experiments are summarized in table 1. The potential of studying the influence of the pion polarizabilities on radiative pion photoproduction from the proton was extensively studied in [48]. In terms of Feynman diagrams, the reaction γp → γπ + n contains real Compton scattering on a charged pion as a pion pole diagram (see flg. 3). In the recent experiment on γp → γπ + n at the Mainz Microtron MAMI [43], the cross section was obtained in the kinematic region 537 MeV < Eγ < 817 cm ◦ MeV, 140◦ ≤ θγγ  ≤ 180 . The values of the pion polarizabilities have been obtained from a flt of the cross section calculated by difierent theoretical models to the data rather than performing an extrapolation to the t-channel pole of the Chew-Low type [49,50]. Figure 4 shows the experimental data, averaged over the full photon beam energy interval and over the squared pion-photon centerof-mass energy s1 from 1.5 Mπ2 to 5 Mπ2 as a function of the squared pion momentum transfer t in units of Mπ2 . For such small values of s1 , the difierential cross section is expected to be insensitive to the pion polarizabilities. Also shown are two model calculations: model 1 (solid curve) is a simple Born approximation using the pseudoscalar pion-nucleon interaction including the anomalous magnetic moments of the nucleon; model 2 (dashed curve) consists of pole terms without the anomalous magnetic moments but including contributions from the resonances Δ(1232), P11 (1440), D13 (1520) and S11 (1535). The dotted curve is a flt to the experimental data.

Table 1. Previous experimental data on the charged pion polarizability απ+ . 4

fm3 ]

Reaction

Experiment

απ+ [10

π Z → π Zγ γp → γπ + n γγ → π + π

Serpukhov [41] Lebedev Phys. Inst. [42] PLUTO [44] DM 1 [45] DM 2 [46] MARK II [47]

6.8 ± 1.4 ± 1.2 20 ± 12 19.1 ± 4.8 ± 5.7 17.2 ± 4.6 26.3 ± 7.4 2.2 ± 1.6

4

dσ/ds1dt (nb/μ )

S. Scherer: Chiral perturbation theory 1.2 1

0.8 0.6 0.4 0.2 0 -10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

t/μ

2

σ (nb)

Fig. 4. Difierential cross section averaged over 537 MeV < Eγ < 817 MeV and 1.5 Mπ2 < s1 < 5Mπ2 . Solid line: model 1; dashed line: model 2; dotted line: flt to experimental data. 14

In particular, the analysis of ref. [35] suggests an inadequacy of the present V − A description of the radiative beta decay, which would also re ect itself in an inadequacy of the ChPT description in its present form. What remains to be understood is why the dispersion sum rules give such a dramatically difierent result from the ChPT calculation where the higher-order terms have been estimated from resonance saturation by including vector and axial-vector mesons. Clearly, the model-dependent input deserves further study. In this context, a full and consistent one-loop calculation of γp → γπ + n including the Delta resonance [52] would be desirable. For a discussion of the so-called generalized pion polarizabilities see [53,54,55,56]. 2.3 Future perspectives at MAMI

12 10 8 6 4 2 0

63

550

600

650

700

750

800

Eγ (MeV)

Fig. 5. The cross section of the process γp → γπ + n integrated over s1 and t in the region where the contribution of the pion polarizability is biggest and the difierence between the predictions of the theoretical models under consideration does not exceed 3 %. The dashed and dashed-dotted lines are predictions of model 1 and the solid and dotted lines of model 2 for (α − β)π+ = 0 and (α − β)π+ = 14 × 10 4 fm3 , respectively.

The kinematic region where the polarizability contribution is biggest is given by 5Mπ2 < s1 < 15Mπ2 and −12Mπ2 < t < −2Mπ2 . Figure 5 shows the cross section as a function of the beam energy integrated over s1 and t in this second region. The dashed and solid lines (dashed-dotted and dotted lines) refer to models 1 and 2, respectively, each with (α − β)π+ = 0 ((α − β)π+ = 14 × 10−4 fm3 ). By comparing the experimental data of the 12 points with the predictions of the models, the corresponding values of (α − β)π+ for each data point have been determined in combination with the corresponding statistical and systematic errors. The result extracted from the combined analysis of the 12 data points reads [43] (α − β)π+ = (11.6 ± 1.5stat ± 3.0syst ± 0.5mod ) × 10−4 fm3 (13) and has to be compared with the ChPT result of, say, (4.9 ± 1.0) × 10−4 fm3 which deviates by 2 standard deviations from the experimental result. On the other hand, the application of dispersion sum rules as performed in [51] yields (α − β)π+ = (10.3 ± 1.9) × 10−4 fm3 . Both the precision measurement of radiative pion beta decay [35] and of radiative pion photoproduction indicate that further theoretical and experimental work is needed.

With the setup of the Crystal Ball detector, a dedicated η physics program will be possible at MAMI. In the reaction γ + p → p + η, 107 etas will be produced per day. The main physics objectives will be the investigation of neutral decay channels. In the framework of SU (3)L × SU (3)R symmetry the decay process η → π 0 γγ is closely related to γγ → π 0 π 0 . At O(q 4 ), the amplitude is given entirely in terms of oneloop diagrams involving vertices of O(q 2 ). The prediction for the decay width was found to be two orders of magnitude smaller [57] than the measured value. The pion loops are small due to approximate G-parity invariance whereas the kaon loops are suppressed by the large kaon mass in the propagator. Therefore, higher-order contributions must play a dominant role in η → π 0 γγ. Even at O(q 6 ) difierences of a factor of two are found for the decay rate and spectrum [57,58,59,60,61,62,63,64] although the most recent result for the decay width of Γ (η → π 0 γγ) = (0.45 ± 0.12) eV agrees with the original prediction (0.42 ± 0.20) eV of ref. [57]. The decay η → π 0 π 0 π 0 is a sensitive test of isospin symmetry violation with the transition amplitude being proportional to the light quark mass difierence (mu − md ) [65,66]. Moreover, the electromagnetic interaction was shown to produce only a small contribution [67]. As a flnal example for allowed decays we refer to the rare eta decay η → π 0 π 0 γγ [68,69]. On the other hand, in the forbidden decays such as η → π 0 π 0 and η → 4π 0 one will investigate (P, CP ) violation which may be connected to the so-called θ term in QCD. As a flnal example we would like to point at the potential of investigating the γπ + → π + π 0 amplitude in the γp → nπ + π 0 reaction. This would allow for an alternative test of the Wess Zumino Witten action [70,71] in terms of the F3π amplitude (see [72] for a recent overview).

3 Chiral perturbation theory for baryons 3.1 The power counting problem The standard efiective Lagrangian relevant to the singlenucleon sector contains, in addition to eq. (2), the most

64

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general πN Lagrangian [4,11,12], (1)

(2)

LπN = LπN + LπN + · · · .

(14)

Due to the additional spin degree of freedom LπN contains both odd and even powers in small quantities. In order to illustrate the issue of power counting in the baryonic sector, we consider the lowest-order πN Lagrangian [4], expressed in terms of bare flelds and parameters denoted by subscripts 0,   ◦ 1 gA 0 (1) μ a μ a LπN = Ψ0 iγμ ∂ − m0 − γμ γ5 τ ∂ π0 Ψ0 + · · · , 2 F0 (15) where Ψ0 and π0 denote a doublet and a triplet of bare nucleon and pion flelds, respectively. After renormaliza◦ tion, m, gA , and F refer to the chiral limit of the physical nucleon mass, the axial-vector coupling constant, and the pion-decay constant, respectively. In sect. 2.1 we saw that, in the purely mesonic sector, contributions of n-loop diagrams are at least of order O(q 2n+2 ), i.e., they are suppressed by q 2n in comparison with tree-level diagrams. An important ingredient in deriving this result was the fact that we treated the squared pion mass as a small quantity of order q 2 . Such an approach is motivated by the observation that the masses of the Goldstone bosons must vanish in the chiral limit. In the framework of ordinary chiral perturbation theory Mπ2 ∼ m which translates into a momentum expansion of observables at flxed ratio m/p2 . On the other hand, there is no reason to believe that the masses of hadrons other than the Goldstone bosons should vanish or become small in the chiral limit. In other words, the nucleon mass entering the pion-nucleon Lagrangian of eq. (15) should not be treated as a small quantity of, say, order O(q). Naturally the question arises how all this afiects the calculation of loop diagrams and the setup of a consistent power counting scheme. Our goal is to propose a renormalization procedure generating a power counting for tree-level and loop diagrams of the (relativistic) EFT for baryons which is analogous to that given in sect. 2.1 for mesons. Choosing a suitable renormalization condition will allow us to apply the following power counting: a loop integration in n dimensions counts as q n , pion and fermion propagators count as q −2 and q −1 , respectively, vertices derived from L2k (k) and LπN count as q 2k and q k , respectively. Here, q generically denotes a small expansion parameter such as, e.g., the pion mass. In total this yields for the power D of a diagram in the one-nucleon sector the standard formula D = nNL − 2Iπ − IN + = 1+(n−2)NL +

∞  k=1

≥ 1 in 4 dimensions,

∞ 

π 2kN2k +

k=1 π 2(k−1)N2k +

∞ 

kNkN

(16)

k=1 ∞ 

(k−1)NkN (17)

k=1

π , and NkN denote the number of where NL , Iπ , IN , N2k independent loop momenta, internal pion lines, internal

Fig. 6. One-loop contribution to the nucleon self-energy. The (1) number 1 in the interaction blobs refers to LπN .

nucleon lines, vertices originating from L2k , and vertices (k) originating from LπN , respectively. According to eq. (17), one-loop calculations in the single-nucleon sector should start contributing at O(q n−1 ). For example, let us consider the one-loop contribution of flg. 6 to the nucleon self-energy. According to eq. (16), the renormalized result should be of the order D = n · 1 − 2 · 1 − 1 · 1 + 1 · 2 = n − 1.

(18)

We will see below that the corresponding renormalization scheme is more complicated than in the mesonic sector. An explicit calculation yields [21] ◦ 2  3gA 0 (p/ + m)IN + M 2 (p/ + m)IN π (−p, 0) Σloop = − 4F02  (p2 − m2 )p/ 2 2 2 − [(p − m + M )I (−p, 0) + I − I ] , Nπ N π 2p2

where the relevant loop integrals are deflned as  i dn k Iπ = μ4−n , (19) n 2 (2π) k − M 2 + i0+  i dn k 4−n , (20) IN = μ (2π)n k 2 − m2 + i0+  i dn k IN π (−p, 0) = μ4−n n 2 (2π) [(k − p) − m2 + i0+ ] ×

1 . k 2 − M 2 + i0+

(21)

 renormalization scheme of ChPT [2,4] Applying the MS indicated by r one obtains 

2 3gAr M2 r (p/ + m) + · · · = O(q 2 ), − Σloop = − 4Fr2 16π 2 where M 2 is the lowest-order expression for the squared  pion mass. In other words, the MS-renormalized result does not produce the desired low-energy behavior of eq. (18). This flnding has widely been interpreted as the absence of a systematic power counting in the relativistic formulation of ChPT. 3.2 Heavy-baryon approach One possibility of overcoming the problem of power counting was provided in terms of heavy-baryon chiral perturbation theory (HBChPT) [8,9] resulting in a power

S. Scherer: Chiral perturbation theory

counting scheme which follows eqs. (16) and (17). The basic idea consists in dividing nucleon momenta into a large piece close to on-shell kinematics and a soft residual contribution: p = mv + kp , v 2 = 1, v 0 ≥ 1 (often v μ = (1, 0, 0, 0)). The relativistic nucleon fleld is expressed in terms of velocity-dependent flelds, Ψ (x) = e−imv·x (Nv + Hv ), with +imv·x 1

+imv·x 1

(1 + v/)Ψ, Hv = e (1 − v/)Ψ. 2 2 Using the equation of motion for Hv , one can eliminate Hv and obtain a Lagrangian for Nv which, to lowest order, reads [9] Nv = e

(1) LπN = Nv (iv · D + gA Sv · u)Nv + O(1/m).

The result of the heavy-baryon reduction is a 1/m expansion of the Lagrangian similar to a Foldy-Wouthuysen expansion with a power counting along eqs. (16) and (17). 3.3 Pion electroproduction near threshold and the axial radius

MμA,i = N (pf )|Aμi (0)|N (pi ) ,  = d 4 x eiq·x N (pf )|T [J μ (0)Aνi (x)] |N (pi ) , Mμν JA,i  μ MJP,i = d 4 x eiq·x N (pf )|T [J μ (0)Pi (x)] |N (pi ) , where the subscripts A, J and P refer to axial-vector current, electromagnetic current and pseudoscalar density and i refers to the ith isospin component of the axialvector current or the pseudoscalar density, respectively. The so-called Adler-Gilman relation [74] provides the chiral Ward identity μ μ qν Mμν JA,i = imMJP,i + 3ij MA,j

(22)

relating the three Green functions. In the one-photonexchange approximation, the invariant amplitude for pion electroproduction can be written as Mi = −ieμ Mμi , where μ = euγμ u/k 2 is the polarization vector of the virtual photon and Mμi the transition-current matrix element: Mμi = N (pf ), π i (q)|J μ (0)|N (pi ) . (23) The relation between the Adler-Gilman relation, eq. (22), and pion electroproduction is established in terms of the Lehmann-Symanzik-Zimmermann reduction formula, m lim (q 2 − Mπ2 )MμJP,i Mπ2 Fπ q2 →Mπ2 1 = 2 lim (q 2 − Mπ2 )(3ij MμA,j − qν Mμν JA,i ). Mπ Fπ q2 →Mπ2

Mμi = −i

2 1

2

2 1

1

1

1

Fig. 7. One-loop contributions leading to a modiflcation of the ( ) k2 dependence of E0+ .

At threshold, the center-of-mass transition current can be parameterized in terms of two s-wave amplitudes E0+ and L0+ eM |thr =

4πW iσ⊥ E0+ (k 2 ) + iσ L0+ (k 2 ) , mN

where W is the total center-of-mass energy, σ = σ · kk and σ⊥ = σ − σ . The contribution from pion loops (see flg. 7) has been analyzed in [75] and leads to a modiflcation of the k 2 de(−) pendence of the electric dipole amplitude E0+ [at O(q 3 )] (−) E0+ (k 2 )

As an example illustrating the strength of the EFT approach we consider pion electroproduction γ ∗ (k) + N (pi ) → π i (q) + N (pf ) near threshold (for an overview, see ref. [73]) and the extraction of the nucleon axial radius. To that end we introduce the Green functions

65

  k2 k2 2 egA 1 1+ κ r = + + v 8πFπ 4m2N 2 6 A   2  k Mπ2 + · · · , + f 8π 2 Fπ2 Mπ2

(24)

where κv = 3.706 is the isovector anomalous magnetic moment of the nucleon and rA is the axial radius. The flrst line corresponds to the traditional expression obtained in the framework of the partially conserved axial-vector current hypothesis (see, e.g., [76]). The second line generates the modiflcation    2  Mπ2 k k2 12 = 1 − + ··· . f 8π 2 Fπ2 Mπ2 128Fπ2 π2 The reaction p(e, e π + )n has been measured at MAMI at an invariant mass of W = 1125 MeV (corresponding to a pion center of mass momentum of |q ∗ | = 112 MeV) and photon four-momentum transfers of Q2 = 0.117, 0.195 and 0.273 GeV2 [77]. Using an efiective-Lagrangian model and a dipole form as an ansatz for the axial form factor GA , an axial mass of MA = (1.077 ± 0.039) GeV was extracted which has to be compared with the average of neutrino scattering experiments MA = (1.026 ± 0.021) GeV. Deflning MA = MA + ΔMA , the difierence between the two results can nicely be explained in terms of the additional k 2 dependence of eq. (24) yielding ΔMA = 0.056 GeV. In the meantime, the experiment has been repeated including an additional value of Q2 = 0.058 GeV2 [78] and is currently being analyzed. Recently, there have been claims that pion electroproduction data at threshold cannot be interpreted in terms

The European Physical Journal A

of GA [79]. However, as was shown in [80], using minimal coupling alone does not respect the constraints due to chiral symmetry. In the framework of the most general Lagrangian, this can be seen by considering the b23 term of the O(q 3 ) Lagrangian [11], (3)

Leff =

1 b23 Ψ γ μ γ5 [Dν , f−μν ]Ψ + · · · 2(4πF )2

3

HBChPT O(p ): Electric polarization in the nucleon induced by the field Ex 1.5

1

0.5

(25)

with f−μν = −2(∂μ aν − ∂ν aμ ) + 2i ([vμ , aν ] − [vν , aμ ]) i + [τ · π, ∂μ vν − ∂ν vμ ] + · · · . F

y (fm)

66

0

-0.5

-1

-1.5

The Lagrangian of eq. (25) is of a non-minimal type and the three terms contribute to the axial-vector matrix element, the JA Green function and pion electroproduction relevant to the Adler-Gilman relation. As a result it was conflrmed that threshold pion electroproduction is indeed a tool to obtain information on the axial form factor of the nucleon (see [80] for details). 3.4 Virtual Compton scattering and generalized polarizabilities As a second example, let us discuss the application of HBChPT to the calculation of the so-called generalized polarizabilities [81,82]. The virtual Compton scattering (VCS) amplitude TVCS is accessible in the reaction e− p → e− pγ. Model-independent predictions, based on Lorentz invariance, gauge invariance, crossing symmetry, and the discrete symmetries, have been derived in ref. [83]. Up to and including terms of second order in the momenta |q | and |q  | of the virtual initial and real flnal photons, the amplitude is completely specifled in terms of quantities which can be obtained from elastic electron-proton scattering and real Compton scattering, namely mN , κ, 2 GE , GM , rE , αp and βp . The generalized polarizabilities (GPs) of ref. [82] result from an analysis of the residual piece in terms of electromagnetic multipoles. A restriction to the lowest-order, i.e. linear terms in ω  leads to only electric and magnetic dipole radiation in the flnal state. Parity and angular-momentum selection rules, charge-conjugation symmetry, and particle crossing generate six independent GPs [82,84,85]. The flrst results for the two structure functions PLL − PT T / and PLT at Q2 = 0.33 GeV2 were obtained from a dedicated VCS experiment at MAMI [86]. Results at higher four-momentum transfer squared Q2 = 0.92 and Q2 = 1.76 GeV2 have been reported in ref. [87]. Additional data are expected from MIT/Bates for Q2 = 0.05 GeV2 aiming at an extraction of the magnetic polarizability. Moreover, data in the resonance region have been taken at JLab for Q2 = 1 GeV2 [88] which have been analyzed in the framework of the dispersion relation formalism of ref. [89,90]. Table 2 shows the experimental results of [86] in combination with various model calculations. Clearly, the experimental precision of [86] already allows for a critical test of the difierent models. Within ChPT and the

-1.5

-1

-0.5

0

0.5

1

1.5

x (fm)

Fig. 8. Scaled electric polarization r 3 αi1 [10 3 fm3 ] [91]. The applied electric fleld points in the x-direction.

linear sigma model, the GPs are essentially due to pionic degrees of freedom. Due to the small pion mass the effect in the spatial distributions extends to larger distances (see also flg. 9). On the other hand, the constituent quark model and other phenomenological models involving Gau or dipole form factors typically show a faster decrease in the range Q2 < 1 GeV2 . A covariant deflnition of the spin-averaged dipole polarizabilities has been proposed in ref. [55]. It was shown that three generalized dipole polarizabilities are needed to reconstruct spatial distributions. For example, if the nucleon is exposed to a static and uniform external electric fleld E, an electric polarization P is generated which is related to the density of the induced electric dipole moments, Pi (r) = 4παij (r) Ej . (26) The tensor αij (r), i.e. the density of the full electric polarizability of the system, can be expressed as [55] αij (r) = αL (r)ri rj + αT (r)(δij − ri rj )  3ri rj − δij ∞ [αL (r ) − αT (r )] r2 dr , + r3 r where αL (r) and αT (r) are Fourier transforms of the generalized longitudinal and transverse electric polarizabilities αL (q) and αT (q), respectively. In particular, it is important to realize that both longitudinal and transverse polarizabilities are needed to fully recover the electric polarization P. Figure 8 shows the induced polarization inside a proton as calculated in the framework of HBChPT at O(q 3 ) [91] and clearly shows that the polarization, in general, does not point into the direction of the applied electric fleld. Similar considerations apply to an external magnetic fleld. Since the magnetic induction is always transverse (i.e., ∇ · B = 0), it is su– cient to consider βij (r) = β(r)δij [55]. The induced magnetization M is given in terms of the density of the magnetic polarizability as M(r) = 4πβ(r)B (see flg. 9).

S. Scherer: Chiral perturbation theory

67

Table 2. Experimental results and theoretical predictions for the structure functions PLL − PT T / and PLT at Q2 = 0.33 GeV2 and  = 0.62. ∗ makes use of symmetry under particle crossing and charge conjugation which is not a symmetry of the nonrelativistic quark model. PLL − PT T / [GeV Experiment [86] Linear sigma model [92] Efiective Lagrangian model [93] HBChPT [94] Nonrelativistic quark model [95]

β(q2) (10−4 fm3)

p 2 1

2

fm )

−4 2

4πr β(r) (10

0

0.2

0.4

2

0.6

2

−q (GeV )

8

]

23.7 ± 2.2stat. ± 4.3syst. ± 0.6syst.norm. 11.5 5.9 26.0 19.2|14.9

3

0

2

PLT [GeV

2

]

−5.0 ± 0.8stat. ± 1.4syst. ± 1.1syst.norm. 0.0 −1.9 −5.3 −3.2| − 4.5

In the following we will concentrate on one of several methods that have been suggested to obtain a consistent power counting in a manifestly Lorentzinvariant approach [14,15,16,17,18,19,20,21], namely, the so-called extended on-mass-shell (EOMS) renormalization scheme [21]. The central idea of the EOMS scheme con sists of performing additional subtractions beyond the MS scheme. Since the terms violating the power counting are analytic in small quantities, they can be absorbed by counterterm contributions. Let us illustrate the approach in terms of the integral  i dn k , H(p2 , m2 ; n) = (2π)n [(k − p)2 − m2 + i0+ ][k 2 + i0+ ]

2

where Δ = (p2 − m2 )/m2 = O(q) is a small quantity. We want the (renormalized) integral to be of the order D = n − 1 − 2 = n − 3. Applying the dimensional counting analysis of ref. [96] (for an illustration, see the appendix of ref. [97]), the result of the integration is of the form [21]

0

H ∼ F (n, Δ) + Δn−3 G(n, Δ),

-2

where F and G are hypergeometric functions and are analytic in Δ for any n. Hence, the part containing G for noninteger n is proportional to a noninteger power of Δ and satisfles the power counting. On the other hand F violates the power counting. The crucial observation is that the part proportional to F can be obtained by first expanding the integrand in small quantities and then performing the integration for each term [96]. This observation suggests the following procedure: expand the integrand in small quantities and subtract those (integrated) terms whose order is smaller than suggested by the power counting. In the present case, the subtraction term reads    dn k i  H subtr = n 2 + 2 + (2π) [k − 2p · k + i0 ][k + i0 ] p2 =m2

6

p

4

0

1

2

3

4

r (fm) Fig. 9. Generalized magnetic polarizability β(q 2 ) and density of magnetic polarizability β(r) for the proton. Dashed lines: contribution of pion loops; solid lines: total contribution; dotted lines: VMD predictions normalized to β(0) [55].

3.5 Manifestly Lorentz-invariant baryon chiral perturbation theory Unfortunately, when considering higher orders in the chiral expansion, the expressions due to 1/m corrections of the Lagrangian become increasingly complicated. Secondly, not all of the scattering amplitudes, evaluated perturbatively in the heavy-baryon framework, show the correct analytical behavior in the low-energy region. Finally, with an increasing complexity of processes, the use of computer algebra systems becomes almost mandatory. The relevant techniques have been developed for calculations in the Standard Model and thus refer to loop integrals of the manifestly Lorentz-invariant type.

and the renormalized integral is written as H R = H − H subtr = O(q) as n → 4. In the infrared renormalization (IR) scheme of Becher and Leutwyler [16], one would keep the contribution proportional to G (with subtracted divergences when n approaches 4) and completely drop the F term. Let us conclude this section with a few remarks. With a suitable renormalization condition one can also obtain a consistent power counting in manifestly Lorentz-invariant

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baryon chiral perturbation theory including, e.g., vector mesons [98] or the Δ(1232) resonance [52] as explicit degrees of freedom. Secondly, the infrared regularization of Becher and Leutwyler [16] may be formulated in a form analogous to the EOMS renormalization [99]. Finally, using a toy model we have explicitly demonstrated the application of both infrared and extended on-mass-shell renormalization schemes to multiloop diagrams by considering as an example a two-loop self-energy diagram [97]. In both cases the renormalized diagrams satisfy a straightforward power counting.

3.6 Applications The EOMS scheme has been applied in several calculations such as the chiral expansion of the nucleon mass, the pion-nucleon sigma term, and the scalar form factor [100], the masses of the ground-state baryon octet [101] and the nucleon electromagnetic form factors [102,103]. As an example, let us here consider the electromagnetic form factors of the nucleon which are deflned via the matrix element of the electromagnetic current operator as N (pf ) |J μ (0)| N (pi ) =

 iσ μν qν N 2 μ N 2 u(pf ) γ F1 (Q ) + F (Q ) u(pi ), N = p, n, 2mN 2 where q = pf − pi is the momentum transfer and Q2 ≡ −q 2 = −t ≥ 0. Figure 10 shows the results for the electric and magnetic Sachs form factors GE = F1 − Q2 /(4m2N )F2 and GM = F1 + F2 at O(q 4 ) in the momentum transfer region 0 GeV2 ≤ Q2 ≤ 0.4 GeV2 without explicit vectormeson degrees of freedom [102]. The O(q 4 ) results only provide a decent description up to Q2 = 0.1 GeV2 and do not generate su– cient curvature for larger values of Q2 . The perturbation series converges, at best, slowly and higher-order contributions must play an important role. Including the vector-meson degrees of freedom along the lines of refs. [98,99] generates the additional diagrams of flg. 11. The results for the Sachs form factors including vector-meson degrees of freedom are shown in flg. 12. As expected on phenomenological grounds [104], the quantitative description of the data has improved considerably for Q2 ≥ 0.1 GeV2 . The small difierence between the two renormalization schemes is due to the way how the regular higher-order terms of loop integrals are treated. Note that on an absolute scale the difierences between the two schemes are comparable for both GpE and GnE . Numerically, the results are similar to those of ref. [104]. Due to the renormalization condition, the contribution of the vector-meson loop diagrams either vanishes (infrared renormalization scheme) or turns out to be small (EOMS). Thus, in hindsight our approach puts the traditional phenomenological vector-meson dominance model on a more solid theoretical basis.

Fig. 10. The Sachs form factors of the nucleon in manifestly Lorentz-invariant chiral perturbation theory at O(q 4 ) without vector mesons. Full lines: results in the extended on-mass-shell scheme; dashed lines: results in infrared regularization. The experimental data are taken from ref. [105].

Fig. 11. Feynman diagrams involving vector mesons (double lines) contributing to the electromagnetic form factors up to and including O(q 4 ).

S. Scherer: Chiral perturbation theory

69

Physics at the Mainz Microtron MAMI and express my best wishes for the future.

References

]

[

]

[

]

/μ

/μ

[

[

]

Fig. 12. The Sachs form factors of the nucleon in manifestly Lorentz-invariant chiral perturbation theory at O(q 4 ) including vector mesons as explicit degrees of freedom. Full lines: results in the extended on-mass-shell scheme; dashed lines: results in infrared regularization. The experimental data are taken from ref. [105].

4 Summary Chiral perturbation theory is a cornerstone of our understanding of the strong interactions at low energies. Mesonic chiral perturbation theory has been tremendously successful and may be considered as a full-grown and mature area of low-energy particle physics. The apparent con ict between the determination of the O(q 4 ) lowenergy constants (l6 − l5 ) from radiative pion beta decay, on the one hand, and the polarizability measurement, on the other hand, certainly requires additional work, in particular, from the theoretical side. The impact on baryonic chiral perturbation theory due to the investigation of electromagnetic reactions at MAMI such as elastic electron-nucleon scattering, (virtual) Compton scattering and the electromagnetic production of pions cannot be overestimated. The possibility of a consistent manifestly Lorentz-invariant approach in combination with the rigorous inclusion of (axial-) vectormeson degrees of freedom and of the Δ(1232) resonance open the door to an application of ChPT in an extended kinematic region. I would like to thank the organizers Hartmuth Arenh˜ ovel, Hartmut Backe, Dieter Drechsel, J˜ org Friedrich, Karl-Heinz Kaiser and Thomas Walcher of the symposium 20 Years of

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Eur. Phys. J. A 28, s01, 71 80 (2006) DOI: 10.1140/epja/i2006-09-008-3

EPJ A direct electronic only

Two-photon physics M. Vanderhaeghena Physics Department, The College of William & Mary, Williamsburg, VA 23187, USA and Theory Group, Jefierson Lab, 12000 Jefierson Ave, Newport News, VA 23606, USA / Published online: 15 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. It is reviewed how Compton scattering sum rules relate low-energy nucleon structure quantities to the nucleon excitation spectrum. In particular, the GDH sum rule and recently proposed extensions of it will be discussed. These extensions are sometimes more calculationally robust, which may be an advantage when estimating the chiral extrapolations of lattice QCD results, such as for anomalous magnetic moments. Subsequently, new developments in our description of the nucleon excitation spectrum will be discussed, in particular a recently developed chiral efiective fleld theory framework for the Δ(1232)-resonance region. Within this framework, we discuss results on N and Δ masses, the γN Δ transition and the Δ magnetic dipole moment. PACS. 25.20.Dc Photon absorption and scattering 12.39.Fe Chiral Lagrangians netic form factors 13.40.Em Electric and magnetic moments

1 Introduction Sum rules for Compton scattering ofi a nucleon ofier a unique tool to relate low energy nucleon structure quantities to the nucleon excitation spectrum [1]. E.g., the Gerasimov, Drell, Hearn (GDH) sum rule (SR) [2] relates a system’s anomalous magnetic moment to a weighted integral over a combination of doubly polarized photoabsorption cross sections. Impressive experimental programs to measure these photoabsorption cross-sections for the nucleon have recently been carried out at ELSA and MAMI (for a review see ref. [3]). Such measurements provide an empirical test of the GDH SR, and can be used to generate phenomenological estimates of electromagnetic polarizabilities via related SRs. The GDH SR is particularly interesting because both its left- and right-hand-sides can be reliably determined, thus providing a useful veriflcation of the fundamental principles (such as unitarity and analyticity) which go into its derivation. At the present time, it has been established that the proton sum rule is satisfled within the experimental precision, while the case is still out for the neutron. After a lightning review of the GDH and related sum rules in sect. 2, I discuss a recently proposed linearized version of the GDH sum rule [4,5]. When applying this new sum rule to the nucleon in the context of chiral perturbation theory, it allows for an elementary calculation (to one loop) of quantities such as magnetic moments and polarizabilities to all orders in the heavy-baryon expansion. The a

e-mail: [email protected]

13.40.Gp Electromag-

chiral behavior of the nucleon magnetic moment allows to make a link with lattice QCD calculations. Subsequently, the nucleon excitation spectrum is discussed in sect. 3. Many Compton scattering sum rules, such as the GDH sum rule, are dominated by the Δ(1232) resonance. I discuss a recently proposed relativistic chiral efiective fleld theory as a new systematic framework to both extract resonance properties from the experiment and to perform a chiral extrapolation of lattice QCD results for those resonance properties.

2 Sum rules in Compton scattering 2.1 Derivation of forward Compton scattering sum rules The forward-scattering amplitude describing the elastic scattering of a photon on a target with spin s (real Compton scattering) is characterized by 2s + 1 scalar functions which depend on a single kinematic variable, e.g., the photon energy ν. In the low-energy limit each of these functions corresponds to an electromagnetic moment charge, magnetic dipole, electric quadrupole, etc. of the target. In the case of a spin-1/2 target, such as the nucleon, the forward Compton amplitude is generally written as ∗

∗

T (ν) = ε  · ε f (ν) + i σ · (ε  × ε) g(ν) ,

(1)

where ε, ε  is the polarization vector of the incident and scattered photon, respectively, while σ are the Pauli matrices representing the dependence on the target spin. The

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crossing symmetry of the Compton amplitude of eq. (1) means invariance under ε ↔ ε, ν ↔ −ν, which obviously leads to f (ν) being an even and g(ν) being an odd function of the energy : f (ν) = f (−ν), g(ν) = −g(−ν). The two scalar functions f (ν), g(ν) admit the following low-energy expansion: e2 + (αE + βM ) ν 2 + O(ν 4 ), 4πM e2 κ2 g(ν) = − ν + γ0 ν 3 + O(ν 5 ), 8πM 2

f (ν) = −

(2) (3)

and hence, in the low-energy limit, are given in terms of the target’s charge e, mass M , and anomalous magnetic moment (a.m.m.) κ. The next-to-leading order terms are given in terms of the nucleon electric (αE ), magnetic (βM ), and forward spin (γ0 ) polarizabilities. In order to derive sum rules (SRs) for these quantities one assumes the scattering amplitude is an analytic function of ν everywhere but the real axis, which allows writing the real parts of the functions f (ν) and g(ν) as a dispersion integral involving their corresponding imaginary parts. The latter, on the other hand, can be related to combinations of doubly polarized photoabsorption crosssections via the optical theorem,

ν σ1/2 (ν) + σ3/2 (ν) , (4) Im f (ν) = 8π

ν σ1/2 (ν) − σ3/2 (ν) , Im g(ν) = (5) 8π where σλ is the doubly-polarized total cross-section of the photoabsorption processes, with λ specifying the total helicity of the initial system. Averaging over the polarization of initial particles gives the total unpolarized cross-section, σT = 21 (σ1/2 + σ3/2 ). After these steps one arrives at the results (see, e.g., [1] for more details):  ∞ σT (ν  ) ν2 dν  , f (ν) = f (0) + 2 (6) 2 2π 0 ν − ν 2 − i  ∞ ν Δσ(ν  ) g(ν) = − 2 ν  dν  , (7) 4π 0 ν 2 − ν 2 − i with Δσ ≡ σ3/2 − σ1/2 , and where the sum rule for the unpolarized forward amplitude f (ν) has been oncesubtracted to guarantee convergence. These relations can then be expanded in energy to obtain the SRs for the different static properties introduced in eqs. (2), (3). In this way we obtain the Baldin SR [6,7]:  ∞ σT (ν) 1 dν, (8) αE + βM = 2π 2 0 ν2 the GDH SR: e2 κ2 1 = 2M 2 π

 0

∞

Δσ(ν) dν, ν

a SR for the forward spin polarizability:  ∞ Δσ(ν) 1 γ0 = − 2 dν, 4π 0 ν3

(9)

(10)

and, in principle, one could continue in order to isolate higher-order moments [8]. Recently, the helicity difierence Δσ which enters the integrands of eqs. (9) and (10) has been measured. The flrst measurement was carried out at MAMI (Mainz) for photon energies in the range 200 MeV < ν < 800 MeV [9, 10], and was extended at ELSA (Bonn) [11] for ν up to 3 GeV. This difierence, shown in flg. 1, fluctuates much more strongly than the total cross section σT . The threshold region is dominated by S-wave pion production, and therefore mostly contributes to the cross section σ1/2 . In the region of the Δ(1232) with spin J = 3/2, both helicity cross sections contribute, but since the transition is essentially M 1, we flnd σ3/2 /σ1/2 ≈ 3. As seen from flg. 1, σ3/2 also dominates the proton photoabsorption cross section in the second and third resonance regions.

σ3/2-σ1/2 (μb)

72

600 500 400 300 200 100 0 -100 0

200 400 600 800 1000 1200 1400 1600 1800

ν (MeV) Fig. 1. The helicity difierence σ3/2 (ν) − σ1/2 (ν) for the proton. The calculations include the contribution of πN intermediate states (dashed curve) [12], ηN intermediate state (dotted curve) [13], and the ππN intermediate states (dashed-dotted curve) [14]. The total sum of these contributions is shown by the full curves. The MAMI data are from ref. [9, 10] and the ELSA data from ref. [11].

2.2 Linearized GDH sum rule Recently, it was shown that by taking derivatives of the GDH sum rule with respect to the a.m.m. one can obtain a new set of sum-rule like relations with intriguing properties [4,5]. To derive such sum rules. one begins by introducing a classical (or trial ) value of the particle’s a.m.m., κ0 . At the Lagrangian level this amounts to the introduction of a Pauli term for the spin-1/2 fleld : LPauli =

iκ0 ψ σμν ψ F μν , 4M

(11)

M. Vanderhaeghen: Two-photon physics Proton magnetic moment

4 3

Neutron magnetic moment

0

IR 

1 SR

2 2  1 HB 0

0.2

0.4 0.6 mΠ 2 GeV2 

0.8

1

3

0

0.2

0.4 0.6 mΠ 2 GeV2 

0.8

1

Fig. 2. Chiral behavior of proton and neutron magnetic moments (in nucleon magnetons) to one loop compared with lattice data (solid circles). SR (dotted lines): one-loop relativistic result based on eq. (12), IR (blue long-dashed lines): infrared-regularized relativistic result, HB (green dashed lines): leading non-analytic term in the heavy-baryon expansion. Red solid lines: single-parameter flt based on the SR result, see refs. [4, 5]. The open diamonds represent the experimental values at the physical pion mass.

where F μν is the electromagnetic fleld tensor and σμν = (i/2)[γμ , γν ] is the usual Dirac tensor operator. At the end of the calculation, κ0 is set to zero, but in the evaluation of the absorption cross sections the total value of the a.m.m. is κ = κ0 + δκ, with δκ denoting the loop contribution. It was shown in ref. [4,5] that this yields the SR :  ∞ 4π 2 αem dν , (12) κ = Δσ  (ν)|κ0 =0 2 M ν 0 where Δσ  (ν) is the derivative of an absorption cross section w.r.t. the trial a.m.m. value κ0 . The striking feature of this sum rule is the linear relation between the a.m.m. and the (derivative of the) photoabsorption cross section, in contrast to the GDH SR where κ appears quadratically. Although the cross-section quantity Δσ  (ν) is not an observable, it is very clear how it can be determined within a speciflc theory. Thus, for example, the flrst derivative of the tree-level cross-section with respect to κ0 , at κ0 = 0, in QED was worked out in ref. [4], yielding Schwinger’s one-loop result. It is noteworthy that this result is reproduced by computing only a (derivative of the) tree-level Compton scattering cross-section and then performing an integration over energy. This is deflnitely much simpler than obtaining the Schwinger result from the GDH SR directly [15], which requires an input at the one-loop level. The SR of eq. (12) can furthermore be applied to study the magnetic moment and polarizabilities of the nucleon in a relativistic chiral EFT framework [4,5]. In particular it allows to study the chiral extrapolation of these quantities, as shown in flg. 2 for the magnetic moments. One sees that the SR calculation, strictly satisfying analyticity, is better suited for the chiral extrapolation of lattice QCD results than the usual heavy-baryon expansions or the infraredregularized relativistic theory.

3 Nucleon excitation spectrum The sum rules for Compton scattering ofi the nucleon are dominated by its flrst excited state the Δ(1232) reso-

73

nance, as is apparent from flg. 1. Through the sum rules, the Δ therefore plays a preponderant role in our understanding of low-energy nucleon structure. This justifles a dedicated efiort to study this resonance. High-precision measurements of the N -to-Δ transition by means of electromagnetic probes became possible with the advent of the new generation of electron scattering facilities, such as BATES, MAMI, and JLab, many measurements being completed in recent years [16,17,18,19]. The electromagnetic nucleon-to-Δ (or, in short γN Δ) transition is predominantly of the magnetic dipole (M 1) type. In a simple quark-model picture, this M 1 transition is described by a spin flip of a quark in the s-wave state. Any d-wave admixture in the nucleon or the Δ wavefunctions allows for the electric (E2) and Coulomb (C2) quadrupole transitions. Therefore by measuring these one is able to assess the presence of the d-wave components and hence quantify to which extent the nucleon or the Δ wave-function deviates from the spherical shape, i.e., to which extent they are deformed [20]. The γN Δ transition, on the other hand, was accurately measured in the pion photo- and electro-production reactions in the Δ-resonance energy region. The E2 and C2 transitions were found to be relatively small at moderate momentumtransfers (Q2 ), the ratios REM = E2/M 1 and RSM = C2/M 1 are at the level of a few percent. Traditionally, the resonance parameters are extracted by using unitary isobar models [21,22,23,24,25,12,26], which in essence are unitarized tree-level calculations based on phenomenological Lagrangians. However, at low Q2 the γN Δ-transition shows great sensitivity to the pion cloud , which until recently could only be comprehensively studied within dynamical models [27,28,29,30, 31,32], which unlike the isobar models include quantum efiects due to pion loops. With the advent of the chiral efiective fleld theory (χEFT) of QCD [33,34] and its extensions to the Δ(1232) region [35,36,37,38,39,40], it has become possible to study the nucleon and Δ-resonance properties in a profoundly difierent way. Recently, flrst relativistic χEFT studies were performed of the γN Δ-transition in pion electroproduction [41,42] and of the Δ(1232) magnetic dipole moment (MDM) in the radiative pion photoproduction [43]. The advantages over the previous dynamical approaches are apparent: χEFT is a low-energy efiective fleld theory of QCD and as such it provides a flrm theoretical foundation, with all the relevant symmetries and scales of QCD built in consistently. The χEFT of the strong interaction is indispensable, at least at present, in relating the low-energy observables (e.g., hadron masses, magnetic moments, form factors) to ab initio QCD calculations on the lattice. On the other hand, χEFT can and should be used in extracting various hadronic properties from the experiment. The χEFT fulfllls both of these roles in a gratifying fashion. The following sections review recent progress in the χEFT in the Δ-resonance region that has been obtained for the N and Δ masses [44], the γN Δ transition [41,42], and the Δ MDM [43].

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3.1 Chiral effective field theory in the Δ(1232) region Starting from the efiective Lagrangian of chiral perturbation theory (χPT) with pion and nucleon flelds [45], the Δ is included explicitly in the so-called δ-expansion scheme [39]. In the following, the Lagrangian L(i) is organized such that superscript i stands for the power of electromagnetic coupling e plus the number of derivatives of pion and photon flelds. Writing here only the terms involving the spin-3/2 isospin-3/2 fleld Δμ of the Δ-isobar gives:1 (1)

LN Δ = Δμ (iγ μνα Dα − MΔ γ μν ) Δν  ihA  + N Ta γ μνλ (∂μ Δν ) Dλ π a + H.c. 2fπ MΔ HA − εμνρσ Δμ T a (∂ρ Δν ) ∂σ π a , (13) 2MΔ fπ ie(μΔ − 1) (2) LN Δ = Δμ Δν F μν 2MΔ   3iegM + N T3 ∂μ Δν F μν + H.c. 2MN (MN + MΔ )  ehA  N Ta γ μνλ Aμ Δν ∂λ π a + H.c. , (14) − 2fπ MΔ −3e (3) N T 3 γ5 [gE (∂μ Δν ) LN Δ = 2MN (MN + MΔ )  igC α + γ (∂α Δν − ∂ν Δα ) ∂μ F μν + H.c., (15) MΔ where MN and MΔ are, respectively, the nucleon and Δisobar masses, N and π a (a = 1, 2, 3) stand for the nucleon and pion flelds, Dμ is the covariant derivative ensuring the electromagnetic gauge-invariance, F μν and F μν are the electromagnetic fleld strength and its dual, Ta are the isospin 1/2 to 3/2 transition matrices, and T a are the generators in the isospin 3/2 representation of SU (2), satisfying T a T a = 5/3. The coupling constants are given by : fπ = 92.4 MeV, hA  2.85 is obtained from the Δresonance width, ΓΔ = 0.115 GeV, and for HA the largeNc relation HA = (9/5)gA is adopted, with gA  1.267 the nucleon axial-coupling constant. Note that the electric and the Coulomb γN Δ couplings (gE and gC , respectively) are of one order higher than the magnetic (gM ) one, because of the γ5 which involves the small components of the fermion flelds and thus introduces an extra power of the 3-momentum. The MDM μΔ is deflned here in units of [e/2MΔ ]. Higher electromagnetic moments are omitted, because they do not contribute at the orders that we consider. (1) Note that LΔ contains the free Lagrangian, which is formulated in [46] such that the number of spin degrees of freedom of the relativistic spin-3/2 fleld is constrained to the physical number: 2s + 1 = 4. The N to Δ transition couplings in eqs. (13,14,15) are consistent with these constraints [47,48,49]. The γΔΔ coupling is more subtle since 1

Here we introduce totally antisymmetric products of γmatrices: γ μν = 12 [γ μ , γ ν ], γ μνα = 12 {γ μν , γ α } = iεμναβ γβ γ5 .

in this case constraints do not hold for su– ciently strong electromagnetic flelds, see, e.g., [50]. In extracting the Δ MDM, it is therefore assumed that the electromagnetic fleld is weak, compared to the Δ mass scale. The inclusion of the Δ-resonance introduces another light scale besides the pion mass in the theory, the resonance excitation energy: Δ ≡ MΔ − MN ∼ 0.3 GeV. This energy scale is still relatively light in comparison to the chiral symmetry breaking scale ΛχSB ∼ 1 GeV. Therefore, δ = Δ/ΛχSB can be treated as a small parameter. The question is, how to compare this parameter with the small parameter of chiral perturbation theory (χPT),  = mπ /ΛχSB . In most of the literature (see, e.g., refs. [35,36,37,38, 40]) they are assumed to be of comparable size, δ ≈ . This, however, leads to a somewhat unsatisfactory result because obviously the Δ-contributions are overestimated at lower energies and underestimated at the resonance energies. To estimate the Δ-resonance contributions correctly, and depending on the energy region, one needs to count δ and  differently. A relation  = δ 2 was suggested and explored in [39], and is referred to as the δ-expansion. The second power is indeed the closest integer power for the relation of these parameters in the real world. In refs. [44,41,42,43] this relation was used for power-counting purposes only, and was not imposed in the actual evaluations of diagrams. Each diagram is simply characterized by an overall δ-counting index n, which tells us that its contribution begins at O(δ n ). Because of the distinction of mπ and Δ the counting of a given diagram depends on whether the characteristic momentum p is in the low-energy region (p ∼ mπ ) or in the resonance region (p ∼ Δ). In the low-energy region the index of a graph with L loops, Nπ pion propagators, NN nucleon propagators, NΔ Δ-propagators, and Vi vertices of dimension i is    iVi + 4L − NN − 2Nπ − NΔ ≡ 2nχPT − NΔ , n=2 i

(16) where nχPT is the index in χPT with no Δ’s [45]. In the resonance region, one distinguishes the one-Δ-reducible (OΔR) graphs [39]. Such graphs contain Δ propagators which go as 1/(p − Δ), and hence for p ∼ Δ they are large and all need to be included. This gives an incentive, within the power-counting scheme, to resum Δ contributions. Their resummation amounts to dressing the Δ propagators so that they behave as 1/(p − Δ − Σ). The self-energy Σ begins at order p3 and thus a dressed OΔR propagator counts as 1/δ 3 . If the number of such propagators in a graph is NOΔR , the power-counting index of this graph in the resonance region is given by n = nχPT − NΔ − 2NOΔR ,

(17)

where NΔ is the total number of Δ-propagators. A word on the renormalization program, as it is an indivisible part of power counting in a relativistic theory. Indeed, without some kind of renormalization the loop

graphs diverge as ΛN , where Λ is an ultraviolet cutofi, and N is a positive power proportional to the powercounting index of the graph. Also, contributions of heavy scales, such as baryon masses, may appear as M N . The renormalization of the loop graphs can and should be performed so as to absorb these large contributions into the available low-energy constants, thus bringing the result in accordance with power counting [51]. To give an example, consider the one-πN -loop contribution to the nucleon mass. For the πN N vertex, the power counting tells us that this contribution begins at O(m3π ). An explicit calculation, however, will show (e.g., [45]) that the loop produces O(m0π ) and O(m2π ) terms, both of which are (inflnitely) large. This is not a violation of power counting, because there are two lowenergy constants: the nucleon mass in the chiral limit, M (0) , and c1N , which enter at order O(m0π ) and O(m2π ), respectively, and renormalize away the large contributions coming from the loop. The renormalized relativistic result, up to and including O(m3π ), can be written as [44]: (0)

MN = MN − 4 c1N m2π (18)    5/2 2 3 gA mπ m2π − m3π 4 1 − arccos 2 2 (8πfπ ) 4MN 2MN  3 mπ 17mπ + − 16MN 2MN    2  4  mπ mπ mπ mπ + + 30 − 10 ln , 8MN MN MN MN and one can easily verify that the loop contribution begins at O(m3π ) in agreement with power counting. Likewise, the Δ mass has also been calculated in relativistic χEFT see ref. [44] for details. The mπ dependence of the nucleon and Δ-resonance masses are compared with lattice results in flg. 3. One of the two parameters in eq. (18) is constrained by the physical nucleon mass value at mπ = 0.139 GeV, while the other parameter is flt to the lattice data shown in the flgure. (0) This yields : MN = 0.883 GeV and c1N = −0.87 GeV−1 . As is seen from the flgure, with this two-parameter form for MN , a good description of lattice results is obtained up to m2π  0.5 GeV2 . Analogously to the nucleon case, one low-energy constant for the Δ is flxed from the physical value of the Δ mass, while the second parameter is flt to (0) the lattice data shown in flg. 3, yielding : MΔ = 1.20 GeV and c1Δ = −0.40 GeV−1 . As well as for the nucleon, this two-parameter form for MΔ yields a fairly good description of the lattice results up to m2π  0.5 GeV2 . 3.2 γNΔ transition The γN Δ transition is usually studied through the pion electroproduction process. The pion electroproduction amplitude to NLO in the δ expansion, in the resonance region, is given by graphs in flg. 4(a) and (b), where the shaded blobs in graph (a) include corrections depicted in

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(b)

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Fig. 4. Diagrams for the eN → eπN reaction at NLO in the δ-expansion. Double lines represent the Δ propagators.

flg. 4(c f). The hadronic part of graph (a) begins at O(δ 0 ) which here is the leading order. The Born graphs (b) contribute at O(δ). The one-loop vertex corrections of flg. 4(e) and (f) to the γN Δ-transition form factors have been evaluated in two independent ways in refs. [41,42], to which we refer for details. At NLO there are also vertex corrections of the type (e) and (f) with nucleon propagators in the loop replaced by the Δ-propagators. However, after the appropriate renormalizations and Q2  ΛχSB Δ, these graphs start to contribute at next-next-to-leading order. The vector-meson diagram, flg. 4(d), contributes to NLO for Q2 ∼ ΛχSB Δ. It was included efiectively in refs. [41,42] by giving the gM -term a dipole Q2 -dependence, in analogy to how it is usually done for the nucleon isovector form factor. The resonant pion photoproduction multipoles are used to determine the two low-energy constants: gM and gE , the strength of the M 1 and E2 γN Δ transitions. In flg. 5, we show the result of the χEFT calculations (3/2) for the pion photoproduction resonant multipoles M1+ (3/2) and E1+ , around the resonance position, as function of the total c.m. energy W of the πN system. These two multipoles are well established by the MAID [12] and SAID [53] partial-wave solutions which allow us to flt

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Fig. 5. Multipole amplitudes M1+ (top panels) and E1+ (bottom panels) for pion photoproduction as function of the invariant mass W of the πN system. Dashed curves: Δ contribution without the γN Δ-vertex corrections, (i.e., flg. 4(a) without flg. 4(e, f)). Dotted curves: adding the Born contributions, flg. 4(b), to the dashed curves. Solid curves: complete NLO calculation, includes all graphs from flg. 4. In all curves the low-energy parameters are chosen as : gM = 2.9, gE = −1.0. The data point are from the SAID analysis (FA04K) [53] (red circles), and from the MAID 2003 analysis [12] (blue squares).

the two low-energy constants of the chiral Lagrangian of eqs. (14,15) as : gM = 2.9, gE = −1.0. As is seen from flg. 5, with these values the NLO results (solid lines) give a good description of the energy dependence of the resonant multipoles in a window of 100 MeV around the Δresonance position. Also, these values yield REM = −2.3 %, in a nice agreement with experiment [16]. The dashed curves in flg. 5 show the contribution of the Δ-resonant diagram of flg. 4(a) without the NLO vertex corrections flg. 4(e, f). For the M1+ multipole this is the LO contribution. For the E1+ multipole the LO contribution is absent (the gE coupling is of one order higher than gM ). Hence, the dashed curve represents a partial NLO contribution to E1+ therein. Upon adding the nonresonant Born graphs, flg. 4(b), to the dashed curves one

obtains the dotted curves in fig. 5. These non-resonant contributions are purely real at this order and do not affect the imaginary part of the multipoles. One sees that the resulting calculation is flawed because the real parts of the resonant multipoles now fail to cross zero at the resonance position and hence unitarity, in the sense of Watson’s theorem [55], is violated. The complete NLO calculation, shown by the solid curves in fig. 5, includes in addition the vertex corrections, fig. 4(e, f), which restore unitarity exactly. Watson’s theorem is satisfied exactly by the NLO, up to-one-loop amplitude given the graphs in fig. 4. Figure 6 shows the NLO results for difierent virtual photon absorption cross sections (for definitions, see ref. [42]) at the resonance position, and for Q2  0.127 GeV2 , where recent precision data are available. Besides the low-energy constants gM and gE , which were fixed from the resonant multipoles in fig. 5, the only other low-energy constant from eq. (15) entering the NLO electroproduction calculation is gC . The main sensitivity on gC enters in σLT . A best description of the σLT data in fig. 6 is obtained by choosing gC = −2.36. The theoretical uncertainty due to the neglect of higher-order efiects was estimated in ref. [42]. We know that they must be suppressed by at least one power of δ (= Δ/ΛχSB ) as compared to the NLO and two powers of δ as compared to the LO contributions. These error estimates are shown by the bands in fig. 6. One sees that the NLO χEFT calculation, within its accuracy, is consistent with the experimental data for these observables. Figure 7 shows the Q2 dependence of the ratios REM and RSM . Having fixed the low energy constants gM , gE and gC , this Q2 dependence follows as a prediction. The theoretical uncertainty here (shown by the error bands) was also estimated in ref. [42] over the range of Q2 from 0 to 0.2 GeV2 . One sees that the NLO calculations are consistent with the experimental data for both of the ratios.

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Figure 8 shows the mπ -dependence of the ratios REM and RSM and compares them to lattice QCD calculations. The recent state-of-the-art lattice calculations of REM and RSM [58] use a linear, in the quark mass (mq ∝ m2π ), extrapolation to the physical point, thus assuming that the non-analytic mq -dependencies are negligible. The thus obtained value for RSM at the physical mπ value displays a large discrepancy with the experimental result, as seen in flg. 8. The relativistic χEFT calculation, on the other hand, shows that the non-analytic dependencies are not negligible. While at larger values of mπ , where the Δ is stable, the ratios display a smooth mπ dependence, at mπ = Δ there is an inflection point, and for mπ ≤ Δ the non-analytic efiects are crucial. One also notices from flg. 8 that there is only little difierence between the χEFT calculations with the mπ dependence of MN and MΔ accounted for, and an earlier calculation [41], where the ratios were evaluated neglecting the mπ -dependence of the masses. Figure 8 also shows a theoretical uncertainty of the ratios REM and RSM taken over the range of m2π from 0 to 0.15 GeV2 . The mπ dependence obtained from χEFT clearly shows that the lattice results for RSM may in fact be consistent with experiment.

3.3 Δ(1232) magnetic dipole moment Although the Δ(1232)-isobar is the most distinguished and well-studied nucleon resonance, such a fundamental property as its magnetic dipole moment (MDM) has thusfar escaped a precise determination. The problem is generic to any unstable particle whose lifetime is too short for its MDM to be measurable in the usual way through spin precession experiments. A measurement of the MDM of such an unstable particle can apparently be done only indirectly, in a three-step process, where the particle is flrst produced, then emits a low-energy photon which plays the role of an external magnetic fleld, and flnally decays. In this way the MDM of Δ++ is accessed in the reaction π + p → π + pγ [59,60] while the MDM of Δ+ can be determined using the radiative pion photoproduction (γp → π 0 pγ  ) [61]. A flrst experiment devoted to the MDM of Δ+ was completed in 2002 [62]. The value extracted in this experiment, μΔ+ = 2.7+1.0 −1.3 (stat.) ± 1.5(syst.) ± 3(theor.) (nuclear magnetons), is based on theoretical input from the phenomenological model [63,64] of the γp → π 0 pγ  reaction. To improve upon the precision of this measurement, a dedicated series of experiments has recently been carried out by the Crystal Ball Collaboration at MAMI [65]. These experiments achieve about two orders of magnitude better statistics than the pioneering experiment [62]. The aim of the investigation within χEFT was to complement these high-precision measurements with an accurate and model-independent analysis of the γp → π 0 pγ  reaction. The optimal sensitivity of the γp → π 0 pγ  reaction to the MDM term is achieved when the incident photon energy is in the vicinity of Δ, while the outgoing photon

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energy is of order of mπ . In this case the γp → π 0 pγ  amplitude to next-to-leading order (NLO) in the δ-expansion is given by the diagrams of flg. 9(a - c), where the shaded blobs, in addition to vertices from eqs. (13,14,15), contain the one-loop corrections shown in flg. 9(d - f). The contributions to μΔ of diagrams (e) and (f ) in flg. 9 have been calculated in ref. [43], to which we refer for technical details. The evaluation of these loop diagrams also allows to quantify the mπ dependence of μΔ which can be used to compare with lattice QCD results. As all lattice data for μΔ at present and in the foreseeable future are for larger than the physical values of mπ , their comparison with experiment requires the knowledge of the mπ -dependence for this quantity. Figure 10 shows the pion mass dependence of real and imaginary parts of the Δ+ and Δ++ MDMs, according to our one-loop calculation. Each of the two solid curves has a free parameter, the counterterm μΔ (2) from LN Δ , adjusted to agree with the lattice data at larger values of mπ . As can be seen from flg. 10, the Δ MDM develops an imaginary part when mπ < Δ = MΔ − M , whereas the real part has a pronounced cusp at mπ = Δ.

For μΔ+ , the curve is in disagreement with the trend of the recent lattice data, which possibly is due to the quenching in the lattice calculations. The dotted line in flg. 10 shows the result [4] for the magnetic moment for the proton. One sees that μΔ+ and μp , while having very distinct behavior for small mπ , are approximately equal for larger values of mπ . We next discuss the χEFT results for the γp → π 0 pγ  observables. The NLO calculation of this process in the δexpansion corresponds with the diagrams of flg. 9. This calculation completely flxes the imaginary part of the γΔΔ vertex. It leaves μΔ as only free parameter, which en(2) ters as a low energy constant in LN Δ . Thus the real part of μΔ+ is to be extracted from the γp → π 0 pγ  observables, some of which are shown in flg. 11 for an incoming photon energy Eγlab = 400 MeV as function of the emitted photon energy Eγ c.m. . In the soft-photon limit (Eγ c.m. → 0), the γp → π 0 pγ  reaction is completely determined from the bremsstrahlung process from the initial and flnal protons. The deviations of the γp → π 0 pγ  observables, away from the soft-photon limit, will then allow to study the sensitivity to μΔ+ . It is therefore very useful to introduce the ratio [64]: dσ 1 R≡ · Eγ , (19) σπ dEγ where dσ/dEγ is the γp → π 0 pγ  cross section integrated over the pion and photon angles, and σπ is the angular integrated cross section for the γp → π 0 p process weighted with the bremsstrahlung factor, as detailed in [64]. This ratio R has the property that in the soft-photon limit, the low energy theorem predicts that R → 1. From flg. 11 one then sees that the χEFT calculation obeys this theorem. This is a consequence of gauge-invariance which is maintained exactly throughout the calculation, also away from the soft-photon limit. The χEFT result for R shows clear deviations from unity at higher outgoing photon energies, in good agreement with the flrst data for this process [62]. The sensitivity of the χEFT calculation to the μΔ is a very promising setting for the dedicated second-generation experiment which has recently been completed by the Crystal Ball Coll. at MAMI [65]. It improves upon the statistics of the flrst experiment (flg. 11) by at least two orders of magnitude and will allow for a reliable extraction of μΔ+ using the χEFT calculation presented here. Besides the cross section, the γp → π 0 pγ  asymmetries for linearly and circularly polarized incident photons have also been measured in the recent dedicated experiment [65]. They are also shown in flg. 11. The photon asymmetry for linearly polarized photons, Σ, at Eγ = 0 exactly reduces to the γp → π 0 p asymmetry. It is seen from flg. 11 that the χEFT calculation is in good agreement with the experimental value. At higher outgoing photon energies, the photon asymmetry as predicted by the NLO χEFT calculation remains nearly constant and is very weakly dependent on μΔ . It is an ideal observable for a consistency check of the χEFT calculation and to test that the Δ diagrams of flg. 9 indeed dominate the

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It was discussed here how Compton scattering sum rules relate low-energy nucleon structure quantities to the nucleon excitation spectrum, with special emphasis on the GDH sum rule. I demonstrated the utility of taking derivatives of the GDH sum rule, in order to convert it to forms which are sometimes more calculationally robust. In particular it was shown how it allows to estimate the chiral extrapolations of lattice QCD results for anomalous magnetic moments of nucleons. Subsequently, new developments in our description of the nucleon excitation spectrum were discussed. In particular I reviewed recent work on a χEFT framework for the Δ(1232)-resonance region. This framework plays a dual role, in that it allows for an extraction of resonance parameters from observables and predicts their mπ dependence. In this way it may provide a crucial connection of present lattice QCD results obtained at unphysical values of mπ to the experiment. This was demonstrated here explicitely for the N and Δ masses, the γN Δ transition and the Δ magnetic dipole moment. As the next-generation lattice calculations of these quantities are on the way [69], such a χEFT framework will, hopefully, complement these efiorts. I am grateful to my colleagues in Mainz for the unique culture of cross-fertilization between experiment and theory. On the subject of two-photon physics, I like to thank in particular Dieter Drechsel and Barbara Pasquini, for the many collaborations. I also like to acknowledge Vladimir Pascalutsa for very fruitful recent collaborations on the χEFT in the Δ-resonance region. This work is supported in part by DOE grant no. DEFG02-04ER41302 and contract DE-AC05-84ER-40150 under which SURA operates the Jefierson Laboratory.

References process. Mechanisms involving π-photoproduction Born terms followed by πN rescattering have been considered in model calculations [63,64]. In the δ-counting they start contributing at next-next-to-leading order and therefore will provide the main source of corrections to the present NLO results. The asymmetry for circularly polarized photons, Σcirc , (which is exactly zero for a two-body process due to reection symmetry w.r.t. the reaction plane) has been proposed [64] as a unique observable to enhance the sensitivity to μΔ . Indeed, in the soft-photon limit, where the γp → π 0 pγ  process reduces to a two-body process, Σcirc is exactly zero. Therefore, its value at higher outgoing photon energies is directly proportional to μΔ . One sees from flg. 11 (lower panel) that our χEFT calculation supports this observation, and shows sizeably difierent asymmetries for difierent values of μΔ . A combined flt of all three observables shown in flg. 11 will therefore allow for a very stringent test of the χEFT calculation, which can then be used to extract the Δ+ MDM.

1. D. Drechsel, B. Pasquini, M. Vanderhaeghen, Phys. Rep. 378, 99 (2003). 2. S.B. Gerasimov, Sov. J. Nucl. Phys. 2, 430 (1966) (Yad. Fiz. 2, 598 (1966)); S.D. Drell, A.C. Hearn, Phys. Rev. Lett. 16, 908 (1966). 3. D. Drechsel, L. Tiator, Annu. Rev. Nucl. Part. Sci. 54, 69 (2004). 4. V. Pascalutsa, B.R. Holstein, M. Vanderhaeghen, Phys. Lett. B 600, 239 (2004). 5. B.R. Holstein, V. Pascalutsa, M. Vanderhaeghen, Phys. Rev. D 72, 094014 (2005). 6. A.M. Baldin, Nucl. Phys. 18, 310 (1960). 7. L.I. Lapidus, Sov. Phys. JETP 16, 964 (1963). 8. B.R. Holstein, D. Drechsel, B. Pasquini, M. Vanderhaeghen, Phys. Rev. C 61, 034316 (2000). 9. GDH and A2 Collaborations (J. Ahrens et al.), Phys. Rev. Lett. 84, 5950 (2000). 10. GDH and A2 Collaborations (J. Ahrens et al.), Phys. Rev. Lett. 87, 022003 (2001). 11. GDH Collaboration (H. Dutz et al.), Phys. Rev. Lett. 91, 192001 (2003).

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Eur. Phys. J. A 28, s01, 81 90 (2006) DOI: 10.1140/epja/i2006-09-009-2

EPJ A direct electronic only

Electromagnetic form factors of the nucleon Experiments at MAMI M. Ostricka Physikalisches Institut, Universit˜ at Bonn, 55115 Bonn, Germany / Published online: 12 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Elastic form factors are of fundamental importance for the understanding of microscopic spatial structures. In case of the proton and the neutron, charge and magnetic form factors can be studied in elastic electron scattering. Techniques to accelerate polarised continuous electron beams, the availability of polarised targets as well as modern concepts and instrumentation for coincidence experiments and recoil polarimetry had an enormous impact on these measurements. The developments and experiments at the Mainz Microtron MAMI will be discussed in a general context. PACS. 13.40.Gp Electromagnetic form factors 13.85.Dz Elastic scattering interactions and scattering 25.30.Bf Elastic electron scattering

1 Introduction Elastic and inelastic scattering experiments at difierent energy scales and with difierent projectiles provide essential insight into microscopic structures in terms of excitation spectra or spatial and momentum distributions of constituents. Form factors measured in elastic scattering are in particular determined by the ability of a system to absorb a momentum without excitation and, therefore, re ects the wave function of constituents in the ground state. Nonrelativistically, elastic form factors are momentum representations of spatial densities like mass or electroweak charge densities. In case of spherical symmetry, an expansion close to the static limit of zero momentum transfer is given by the integral quantity, e.g. the total charge Z, and the corresponding mean square radius: q2  2  r + O(q 4 ). (1) 6 The flrst evidence for such flnite size efiects of atomic nuclei was found by Lyman, Hanson and Scott in electron scattering experiments at a 20 MeV betatron [1]. Their pioneering work together with the flrst electron-protonscattering experiments by Hofstadter [2] mark the beginning of a fruitful era of using electromagnetic probes to analyse nuclear and subnuclear hadronic structures. The electromagnetic coupling is weak enough to allow a perturbative treatment and strong enough for precise measurements even at higher values of momentum transfer. Compared to pointlike fermions, the contributions of charge and total magnetic moment to the vector current of − F (|→ q |) = Z −

a

e-mail: [email protected]

13.88.+e Polarization in

protons and neutrons are modifled by charge (GE,p , GE,n ) and magnetic form factors (GM,p , GM,n ) which depend on the square of the 4 momentum transfer Q2 . The unpolarised cross section for electron scattering ofi nucleons can be expressed in leading order as an incoherent sum of the response to longitudinal and transverse polarisation components of the exchanged virtual photon. In case of elastic scattering these responses are equal to G2E (Q2 ) and τ G2M (Q2 ), respectively: σM  2 2 dσ = GE (Q ) + τ G2M (Q2 ) . dΩ (1 + τ )

(2)

Here τ = Q2 /4M 2 ,  = [1 + 2(1 + τ ) tan2 ϑ2e ]−1 describes the photon polarisation and σM is the Mott cross section for a pointlike particle. In principle, both form factors can thus be determined by studying the  dependence of the cross section at flxed values of Q2 . This technique, known as Rosenbluth or LT separation, has been intensively used in single arm experiments at low duty cycle machines. It requires measurements at difierent scattering angles and beam energies as well as the knowledge of absolute luminosities. This Rosenbluth separation technique is intrinsically limited if one of the two summands in the unpolarised cross section, G2E or τ G2M , is small compared to the other one. In this case, only the dominating form factor can be extracted reliably. For the proton, since two decades the cross section measurements of Simon et al. at the Mainz 300 MeV Linac provide the most precise data for GE,p at low Q2 allowing an accurate determination of the proton charge radius [3]. However, at higher values of Q2 the transverse part, σT ∼ τ G2M , is getting more and more dominant and a separation of GE,p sufiers from statistical as well as

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from systematic uncertainties. In single-arm experiments at high momentum transfer up to Q2 = 30 GeV2 only the proton magnetic form factor, GM,p , has been determined precisely at the SLAC accelerator [4]. Measurements of neutron form factors, GM,n and GE,n , are even more di– cult due to the lack of a free neutron target and the smallness of GE,n (see sect. 4.3). The results of these unpolarised, single-arm electron scattering experiments have usually been summarised in the past in terms of the approximate scaling relation GE,p ≈ GM,p /μp ≈ GM,n /μn and the approximate dipole form  2 1 GM,p ≈ μp GD = 1 + Q2 /m2

sp

ec

tro

m

et

er

A

(4)

2 Coincidence experiments to measure GM,n Lacking a free neutron target Ehrenberg and Hofstadter for the flrst time used inclusive electron scattering ofi light nuclei in quasi free kinematics to measure the neutron magnetic form factor [6]. In general, this procedure requires a separation of the longitudinal RL ∼ G2E,p + G2E,n and the transverse cross section RT ∼ G2M,p + G2M,n and a subsequent subtraction of the proton contribution. In addition to this complex procedure the influence of nuclear binding introduces model dependences even in case of the deuteron. At momentum transfers above 1 2 (GeV/c)2 the efiect of GM,n is large enough to employ this technique reliably [7]. The subtraction of the substantial proton contribution can be avoided by measuring the scattered neutron in coincidence with the electron [8]. In order to achieve acceptable signal-to-noise ratios in such coincidence experiments, electron beams with a high duty factor are essential. The di– culty is then shifted to the experimental task to calibrate and monitor the neutron detection e– ciency. In addition, the sensitivity to nuclear structure can be signiflcantly reduced by measuring the cross section ratio σ (d(e, e n)) σ (d(e, e p))

n’/p’

(3)

with the phenomenological parameter m2 = 0.71 GeV2 . Indispensable prerequisites for more detailed analyses and interpretations are precise measurements of all four nucleon form factors over a larger range in momentum transfer. To overcome the intrinsic limitations discussed above, techniques are required to accelerate polarised continuous electron beams together with polarised targets and an experimental instrumentation for coincidence experiments and recoil polarimetry. These technological and conceptual developments allow to exploit the full potential of electromagnetic probes and are characteristic for the modern era of electron scattering. These developments and form factor measurements at the Mainz Microtron MAMI will be discussed in the next sections. A recent review on form factors and their measurements in general can be found in [5].

Rd =

e’

(5)

hadron−detector Fig. 1. Setup to measure Rd at MAMI.

for scattering from neutrons and protons in quasi-free kinematics. In plane-wave-impulse-approximation spectral functions and the elementary electron nucleon scattering factorise so that the dependence on nuclear wave functions cancels in the ratio. Higher-order corrections like flnal state interactions (FSI) or meson exchange currents (MEC) are small (∼ 2%) and calculable [9,10]. Measurements of the ratio Rd thus provided a significant break-through in the knowledge of GM,n at low Q2 and have been pioneered at NIKHEF [11], ELSA [12] and MAMI [13,14]. The setup at MAMI is shown in flg. 1. The electrons are detected in a magnetic spectrometer with a solid angle of 28 msr, a momentum acceptance of 20% and a resolution Δp/p ≤ 10−4 (spectrometer A [15]). In coincidence with the electron, the scattered nucleons are identifled as proton or neutron in a well shielded scintillator array. As d(e, e n) and d(e, e p) yields are measured simultaneously, the ratio Rd is independent of fluctuations in luminosity and acceptance of the electron detector. The main experimental di– culty is the absolute calibration and monitoring of the neutron detection e– ciency which enters directly in the ratio Rd . A calibration using the kinematically complete p(n, p)n reaction to tag neutrons requires measurements under difierent experimental conditions at a facility providing intense neutron beams (e.g. [16]). Considerable care has to be taken to monitor efiective detection thresholds and to ensure portability of the measured e– ciencies [13,14]. In contrast, Bruins et al. [12] used the p(γ, π + n) to calibrate their neutron detector in situ. However, reactions from electroproduction p(e, π + n)e , where the exact kinematical correlation is lost in the 3 body flnal state, may lead to an underestimation of the detection e– ciency. This has been suggested [17,18] as origin of the 10% discrepancy in the extracted values for GM,n which

M. Ostrick: Electromagnetic form factors of the nucleon

83

reaction plane

y

pe’

x

pn

z

pe

d(e,e’n), Bates 93 Rn, NIKHEF 94 Rn, ELSA 95 Rn, MAMI 98, 02 3 He(e,e’), Bates 94 3 He(e,e’), Jlab 00 3 He(e,e’), Jlab 03

2

q

ΦR

pp

ϑnq electron scattering plane Fig. 3. Reference frame and kinematics of the d(e, e n) reaction. 2

Q /(GeV/c)

Fig. 2. The neutron magnetic form factor in units of μn GD as function of Q2 measured in coincidence and polarisation experiments [8, 11, 12, 13, 14, 19, 20, 21].

are summarised in flg. 2 in units of the empirical dipole expression (eq. (3)). An alternative method to determine GM,n is provided by inclusive scattering of polarised electrons from polarised 3 He in quasi-elastic kinematics [19,20,21]. Results obtained with this technique at Bates and Jefierson Lab are included in flg. 2 as well as absolute d(e, e n) cross section measurements from Bates [8]. Recently, new measurements of Rd at Q2 values up to 5 (GeV/c)2 have been completed at Jefierson Lab [22]. The large solid angle covered by the CLAS spectrometer allows to perform the e– ciency determination simultaneously with the Rd measurement by tagging neutrons in the p(e, e π + )n reaction where both, scattered electron and π + are detected. Preliminary results show, that GM,n follows the dipole approximation up to Q2 = 5 (GeV/c)2 within 10%.

3 Double-polarisation observables Experiments using polarised electron beams in combination with polarised nucleons either in the initial or flnal state ofier possibilities to measure interferences between longitudinal and transverse amplitudes which do not appear in the unpolarised cross section. This is particularly interesting in cases where one part is completely dominating and unpolarised cross section measurements are not su– cient to separate additional small amplitudes. Furthermore, most polarisation observables are insensitive to absolute luminosities and other experimental calibration factors. In electron-neutron scattering for example, the smallness of the electric form factor GE,n compared to the dominant magnetic form factor makes a reliable Rosenbluth separation impossible. As mentioned above, the situation is similar for protons at high momentum transfer

where the contribution of GE,p to the unpolarised cross section is kinematically suppressed. The increased sensitivity of double polarisation observables to GE,n and GE,p at high Q2 has already been pointed out more than 40 years ago [23,24,25]. For the ideal case of scattering longitudinally polarised → − − electron ofi free nucleons, N (→ e , e N ), the components of the recoil polarisation are given by  2τ (1 − ) GE GM Px = −Pe , (6)  G2E + τ G2M (7) Py = 0 , √ 2 2 1 −  τ GM Pz = P e . (8)  G2E + τ G2M They are equivalent to cross section asymmetries with respect to the beam helicity for the difierent nucleon spin orientations in the scattering from polarised targets: Ax = Px ;

Ay = 0;

Az = − P z .

(9)

The x and z direction are deflned by the electron scatter− ing plane with z given by the momentum transfer → q (see flg. 3). In Px and Ax both form factors enter linearly which increases the sensitivity compared to the unpolarised cross section, if G2E  τ G2M . In case of the neutron, the free e-n scattering has again to be approximated by the quasi-free scattering ofi light nuclei (2 H, 3 He) and one has to pay attention to nuclear binding and rescattering efiects. In leading order, spectral functions cancel in the polarisation and asymmetry components being ratios of cross sections. However, higher order efiects like FSI and MEC as well as influences of Fermi-motion on the projections of polarisation components have to be taken into account. Polarised 3 He can be used as an efiective polarised neutron target because in its ground state the two protons are dominantly in the s-state with the spins coupled to zero. Thus the spin of the 3 He is predominantly carried by the neutron. Additional d-wave components, meson exchange currents and flnal state interactions have recently been

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transverse polarisation

ϑnq = 0 and has to be taken into account in the determination of GE,n (see section 4.3).

0.2

Φ R=180

o

0 2

2

Q =0.32 GeV /c −0.2

0

Φ R =0

2

5

10

0.2

o

15

Φ R=180

o

0 2

Q =0.12 GeV −0.2

0

5

2

/c

Φ R =0

2

10

o

15

ϑnq in deg Fig. 4. Dependence of transverse neutron polarisation in the → → d(− e ,e − n ) reaction on the neutron kinematics for two difierent values of Q2 . ϑnq is the angle between the neutron recoil momentum and momentum transfer. The dashed curves indicate the in uence of the Fermi-motion, the solid curves are results of a calculation by Arenh˜ ovel [27, 28] including FSI and further higher-order contributions.

analysed at low Q2 within full three body calculations [26]. The dominant correction, that has to be applied in analy→ − −  e , e n) experiments, originates from scattering ses of 3 H e(→ ofi protons followed by a charge-exchange reaction simulating quasifree n(e, e n) events (see sect. 4.3). − − In the d(→ e , e → n ) reaction, the neutron recoil momen− tum → p n in general deviates from the direction of momentum transfer q due to Fermi motion as indicated by the angles ϑnq and ΦR in flg. 3. At flnite angles ϑnq the transverse polarisation Pt of the recoiling neutron gets admixtures from the Pz component. This is demonstrated in flg. 4, where Pt is plotted as a function of the angle ϑnq for the two extreme situations ΦR = 0◦ and ΦR = 180◦ . The dashed curves indicate the admixture of Pz due to this purely kinematical efiect which averages out if the detector acceptance is completely symmetric in ΦR . − − In studying the details of the d(→ e , e → n ) reaction, Arenh˜ ovel et al. have shown that meson exchange and isobar currents have a negligible efiect in quasifree kinematics as does the choice of the N N -potential so that there is essentially no dependence on the deuteron wave function [27,28]. However, at momentum transfers below Q2 = 0.25 (GeV/c)2 a strongly rising influence of flnal state interactions, especially of charge exchange reactions, is found (solid curves in flg. 4). This leads to a shift in the observed polarisation component Px even in the case

4 Double-polarisation experiments The realisation of double-polarisation experiments described in the previous section demands the technically sophisticated combination of continuous, high intensity, polarised electron beams with polarised targets or recoil polarimetry. At MAMI such experiments have been performed since the beginning of the 1990s.

4.1 Polarised electrons at MAMI In 1992 a spin-polarised electron beam was accelerated through MAMI for the flrst time. The electron source was based on photoemission of GaAsP illuminated by circularly polarised laser light [29]. The helicity sign of the laser and consequently of the electron beam was flipped at a rate of 1 Hz by reversing the high voltage of a Pockels cell. In order to have longitudinally polarised electrons at the experiment the spin precession in the magnetic flelds of accelerator and transfer beamlines has to be compensated. As there are no depolarising resonances in a microtron, this can be done at low energies before accelerator injection or by slightly tuning the beam energy [30, 31]. In the beginning, a polarisation of about 30-35% at beam currents of 5-10 μA has been achieved. The use of strained layer GaAs cathodes increased the polarisation signiflcantly [32]. The lower quantum e– ciency could be compensated by increasing the laser power and the transfer e– ciency into the accelerator. The beam polarisation is measured and monitored using the spin dependence of Mott , Moeller and Compton scattering. Today, high intensity (up to 80 μA), highly polarised (Pe ∼ 80%) beams with sophisticated monitoring systems are available which allow to measure even tiny parityviolating asymmetries [33,34].

4.2 Polarisation transfer to protons In one of the early double polarisation experiments at − − MAMI the spin transfer to protons in the p(→ e , e → p) → → − − and d( e , e p ) reactions has been analysed for the flrst time [35]. The transverse polarisation component Px (eq. (7)) is accessible experimentally by measuring an asymmetry in the azimuthal angular distribution (Φ ) of protons scattered in a carbon analyser: A(Φ ) = aT · sin Φ = Pe · ApC · Px · sin Φ .

(10)

The analysing power ApC of inclusive proton-carbon scattering is known and has frequently been used for spin analyses at proton facilities. With the number of events N ± (Φ ) for both helicity states of the electron beam, this

M. Ostrick: Electromagnetic form factors of the nucleon

85

Px in percent 1.2

Pospischil et al., MAMI(2001) Milbrath et al., Bates (1999) Jones et al., JLab (2000)

G Ep / (G Mp / μ p)

40 30 20 10 0

Px 0

1

0.9

eff

2

Simon et al., Mainz (1980)

1.1

4

6

θ

8

10

12

0.35

c.m. np

Fig. 5. Transverse proton polarisation Px measured in → → → → p(− e ,e − p ) (square) and d(− e ,e − p ) (circles) at Q2 = 0.3 (GeV/c)2 [35]. In case of the deuteron target the observed dependence on the angle θnp of the proton-neutron relative momentum is shown. The line is a calculation of Arenh˜ ovel et al. [27, 28].

0.4 0.45 Q 2 (GeV 2/c 2)

0.5

Fig. 6. The ratio μp GE,p /GM,p from polarisation transfer measurements at low Q2 [36, 37, 38]. The dotted line corresponds to the exact scaling behaviour of eq. (4), the solid line is a flt to Rosenbluth separated data [3].

azimuthal asymmetry A(Φ ) can be determined through the ratio 1 − A(Φ ) = 1 + A(Φ )

N + (Φ ) · N − (Φ + π) , N − (Φ ) · N + (Φ + π)

(11)

which is insensitive to detector e– ciencies and luminosity fluctuations. The detector system was completely non magnetic consisting out of a segmented lead-glass calorimeter for the electrons and a scintillator hodoscope including the carbon analyser for the proton detection and spin analysis. In the kinematics chosen for the experiment the dependence of Px on GE,p is weak and influences of binding and rescattering efiects can be tested. The measured polarisation transfer Px at Q2 = 0.3 (GeV/c)2 is shown in flg. 5. No signiflcant difierence in the spin transfer between free protons and protons bound in deuterons was observed at Q2 = 0.3 (GeV/c)2 in quasifree kinematics. In case of the deuteron target the results are in agreement with calculations of Arenh˜ ovel et al. [27,28]. The full power of measuring the polarisation transfer to protons has been demonstrated in experiments using polarimeters consisting out of tracking detectors in front and behind a carbon analyser sitting in the focal plane of a magnetic spectrometer. If the spin precession along the proton path is accurately taken into account, in principle all three polarisation components of the recoiling proton are accessible. In case of elastic scattering the ratio Px /Pz is directly proportional to the ratio of the charge and magnetic form factors: √ Px 2 GE  · = . (12) Pz G τ (1 + ) M

Q 2 /(GeV/c) 2 Fig. 7. The ratio μp GE,p /GM,p from polarisation transfer measurements (triangles, [38, 39]) compared to results extracted from Rosenbluth separation [40, 41].

Experimental calibration factors as the absolute value of the beam polarisation or the efiective analysing power of the polarimeter cancel in this ratio. The proton charge form factor has been measured using this technique at Bates [36] and MAMI [37] at low Q2 (flg. 6) as well as in Hall A at Jefierson Lab [38,39] (flg. 7). The measurements at JLab covered a Q2 range from 0.5 to 5 (GeV/c)2 and enormously influenced our knowledge about GE,p . The observed linear decrease of the ratio μp GE,p /GM,p at Q2 > 1 (GeV/c)2 contradicts the previously assumed approximate scaling behaviour (eq. (4)) and is in clear disagreement with the results obtained by LT separations of unpolarised cross sections. An experimental origin of this discrepancy has recently been excluded by a new dedicated Rosenbluth extraction which is in agreement with the earlier results [41]. The efiect of GE,p in the cross section is so small, that unknown  dependent corrections to the

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one-photon exchange approximation may have a similar size and could disturb the Rosenbluth extraction. In general, two-photon exchanges are suppressed by the electromagnetic coupling αem ≈ 1/137 and they are partly taken into account in radiative corrections [42]. However, there exist contributions which depend on the hadronic structure and on intermediate excited states of the nucleon. Calculations of these corrections are model dependent and they have been neglected in the past. A recent discussion of two-photon efiects and the hadronic physics involved can be found in [43].

Electron Detector 16x16 array of leadglass−detectors ~ 100 msr ΔΩ _ target

e

4.3 Measurements of GE,n /GM,n

concrete shielding

Magnet

Θ’ Φ ’ plastic−scintillators

Neutron−Detector and Polarimeter

A in %

The vanishing charge of the neutron makes any small electric interaction enormously di– cult to detect. Attempts to measure flrst moments of a charge distribution originate in the idea of Fermi and Marshal to study the scattering of thermal neutrons ofi atomic electrons [44]. These experiments have been reflned and a negative mean square charge radius close to the so called Foldy term 3κ/2Mn2 = −0.126 fm2 has been established [45,46]. However, a precise determination is still sufiering from systematic uncertainties due to the dominating nuclear scattering amplitude [47,48]. At Q2 > 0 the smallness of G2E,n compared to τ G2M,n makes a reliable Rosenbluth separation impossible. Finite values for GE,n have been extracted from the deuteron structure function A(Q2 ), measured in elastic electrondeuteron scattering. A(Q2 ) provides sensitivity to GE,n through the mixed term GE,p · GE,n in the square of the isoscalar form factor (GE,n + GE,p )2 . However, the necessary unfolding of the deuteron wave function introduces substantial model dependences ([49,50] and flg. 10). → − −  − − n )p experiments e , e → The flrst 3 H e(→ e , e n)pp and D(→ at MAMI have been performed with one common large solid angle detector system (flg. 8) [51,52,53,54]. The scattered electrons were detected in a segmented lead-glass calorimeter with an energy resolution δE/E ∼ 25% su– cient to suppress inelastic events from π-production. Only the electron angles entered the reconstruction of the 3body flnal state, which became kinematically complete through the measurement of the neutron time of flight and hit position in arrays of plastic scintillators well shielded by concrete and lead. − − In the d(→ e , e → n ) reaction, the polarisation of the recoiling neutron perpendicular to its momentum can be analysed using the detection process itself. n-p-scattering as well as inelastic processes, e.g., 12 C(n, n p)11 B, which contribute to a neutron detection in a plastic scintillator, provide reasonable analysing power Aeff (Θn , Tn ). The resulting asymmetry in the azimuthal angular distribution, N (Φ ), of the detected neutrons can be observed through the hit distribution in a second scintillator wall (flg. 8) and analysed via eq. (11). In front of the analyser a dipole magnet has been installed which allows to avoid an external calibration of the efiective analysing power through a controlled precession

1m

10

5

0

0

50

100 Φ ’ in

150

degree

Fig. 8. Detector-setup for the flrst double polarisaton experiments at MAMI and a typical asymmetry in the azimuthal angular distribution of neutrons detected in both scintillator walls calculated via eq. (11).

M. Ostrick: Electromagnetic form factors of the nucleon

a T (χ) in %

0.12

A eff= 20%

87

GE,n

,

D(e,e n), Bates ,

D(e,e n), MAMI, A3

0.1

3

,

He(e,e n), MAMI, A1

5

3

,

He(e,e n), MAMI, A3

0.08

,

D(e,e n), NIKHEF

A eff= 8%

0

0.06

0.04

χ0

−5

0.02

0

−10 −100

−50

0

50

100

χ in degree Fig. 9. Measured azimuthal asymmetries a (χ) for various precession angles χ with two difierent cuts on the analysing reaction leading to difierent analysing powers but leaving χ0 unchanged.

of the neutron spin. This technique has become standard in modern neutron polarimeters. After precession by the angle  μn · B(l)dl (13) χ= 2 βn c L the transverse neutron polarisation as well as the resulting azimuthal asymmetry a⊥ become a superposition of x and z components: a⊥ (χ) = Pe Aeff (Px cos χ − Pz sin χ) = a0 · sin(χ − χ0 ).

(14)

One immediately flnds that the angle χ0 of the zero crossing a⊥ (χ0 ) = 0 is directly related to the ratio Px /Pz (eq. (12)) and depends neither on the analysing power of the polarimeter nor on the polarisation of the electron beam, Pe Aeff Px tan χ0 = · . (15) Pe Aeff Pz Measured asymmetries for various precession angles χ are shown in flg. 9. Kinematic cuts on the analysing reaction change the amplitude, i.e. the efiective analysing power, but not the zero crossing angle χ0 . To extract values for GE,n binding efiects mainly due to flnal state interactions have been taken into account according to calculations of H. Arenh˜ ovel [27,28]. At low momentum transfer (Q2 = 0.12 (GeV/c)2 ) a correction of almost 100% is required which drops rapidly to 8% at Q2 = 0.35 (GeV/c)2 (see flg. 4, [54]). → − −  e , e n) reaction the 3 He gas is In case of the 3 H e(→ polarised by optical pumping a metastable excited state which then transfers the polarisation to the ground state. In the flrst experiments, after optical pumping, the gas was compressed to 1bar in the target cell [51,52]. Today typical target polarisations of about 50% at pressures up to 5 bar are achieved [55]. For difierent orientations of the

0

0.1

0.2

0.3

0.4

0.5

0.6 2

0.7

0.8 2

Q / (GeV/c)

Fig. 10. First GE,n results from double-polarisation observables. The arrows indicate the influence of few-body efiects mainly due to flnal state interactions. The shaded area represents the model dependence of GE,n values extracted from elastic D(e, e ) experiments [50] The dashed line is the parametrisation of Galster et al. [49].

target spin cross section asymmetries with respect to the beam helicity are measured. The arrays of plastic scintillators serve in this case as neutron detector and time of ight spectrometer only, not as polarimeter. In 1998 no full 3 body calculations were avail→ − −  able and the flrst 3 H e(→ e , e n) experiments have been analysed under the assumption of quasifree scattering from a neutron with no higher order efiects taken into account [51,52]. The initial 50% discrepancy between → − −  − − n ) and 3 H e(→ e , e n) experiments around Q2 = d(→ e , e → 2 0.3 (GeV/c) has been resolved in full 3-body calcula→ − −  e , e n) reaction including flnal state tions of the 3 H e(→ interactions [26]. At higher Q2 , binding and rescattering efiects are expected to decrease signiflcantly as for deuterium [55]. A detailed discussion of polarised 3 He targets and their use in experiments at MAMI can be found in [56]. Figure 10 summarises results for GE,n obtained in the flrst double-polarisation experiments. The arrows indicate the necessary corrections due to flnal state interactions. → − −  → − − − − In the meantime, d (→ e , e n), d(→ n ) and 3 H e(→ e , e → e, e n) reactions have been measured several times with reflned techniques. At MAMI the use of magnetic spectrometers to detect the scattered electron improved background suppression and kinematical reconstruction considerably [57,55,58]. In particular, the direction of the mo− mentum transfer vector → q , which deflnes the relevant coordinate system for the spin analysis, can be reconstructed precisely. At Jefierson Lab the Q2 range has been extended up to 1.5 (GeV/c)2 [59,60]. Below Q2 = 1 (GeV/c)2 , new preliminary data obtained with the Blast-detector at Bates have recently been shown [61]. Furthermore, GE,n has been extracted from an analysis of the deuteron quadrupole form factor FQ obtained from recent tensor polarisation measurements in elastic electron deuteron scattering [62]. Compared to the

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The European Physical Journal A

r 2 ρ(r) fm

G E,n 0.1

D(e,e’n)

0.08

0.015

Bates MAMI JLAB

3

MAMI

0.01 D(e,e’)D

NIKHEF JLAB

D(e,e’n)

He(e,e’ n)

0.005

0.06

0 0.04

−0.005

0.02 0

0

0.2

0.4

0.6

0.8

1

1.2

0

1.4

Q 2 / (GeV/c)

2

Fig. 11. Present status of GE,n measurements compared to the parametrisation of Galster et al. ([49], dashed line) and Friedrich, Walcher ([63], solid line). The data are from experiments using polarised Deuterium [64, 65, 59], polarised 3 He [55, 26, 58], recoil polarisation [66, 54, 60], and from an analysis of the elastic deuteron quadrupole form factor [62].

above-mentioned older analyses of the elastic deuteron structure functions A [50], the model uncertainties are reduced by the direct use of FQ . Taking all these novel approaches together, a consistent picture of the charge form factor of the neutron is starting to arise (see flg. 11). Results obtained with different targets and in difierent reactions are in fair agreement with each other even though at low Q2 substantial corrections due to rescattering are unavoidable. The data roughly follow a phenomenological parametrisation given by Galster et al. already in 1971 [49]. However, the accuracy is reaching a level at which deviations from such a simple, smooth behavior start to become signiflcant [63].

5 Interpretation With modern experimental techniques, for the flrst time all elastic nucleon form factors, including the neutron charge form factor, have been measured precisely over a flnite range in momentum transfer. Both magnetic form factors, GM,p and GM,n , follow the dipole approximation within 10% up to Q2 = 5 (GeV/c)2 . The scaling relation (eq. (4)) is violated considerably for the proton electric form factor. The almost linear decrease of the ratio GE,p /GM,p at Q2 > 1 (GeV/c)2 , as revealed by spin transfer measurements at JLab, indicates that the charge density of the proton is signiflcantly softer than its magnetisation density. Friedrich and Walcher emphasised local deviations from a smooth shape in the Q2 dependence of all four form factors. By fltting the available data with an ansatz given by the sum of a Gaussian and two dipoles discribing the smooth part, local minima in GE,p , GM,p and GM,n around Q2 = 0.25 (GeV/c)2 with a width of approx-

1

2

3

4

5

r/fm Fig. 12. Neutron charge distribution obtained from Fourier transforms of GE,n by Friedrich and Walcher (solid line [63]) and the Galster parametrisation (dashed line).

imately 0.2 (GeV/c)2 as well as a corresponding bump in GE,n are clearly revealed [63]. Nonrelativistically, if the Compton wavelength of a system is negligible compared to its size λC = h/M c   r2 , form factors can be measured over a su– ciently large range in momentum transfer in order to calculate spatial densities by a Fourier transform without relativistic efiects becoming important. This is the case for atomic nuclei and detailed information about nuclear charge distributions has been obtained from electron scattering [67]. Presently, also the extraction of mass or neutron densities are discussed [68,69].  Although for nucleons λC ≈ 0.25 r2 , a similar interpretation of GE (Q2 ) and GM (Q2 ) as momentum representations of spatial charge and magnetisation densities is still possible in the Breit frame of vanishing energy transfer. Figure 12 shows the corresponding charge distribution of the neutron as calculated by Friedrich and Walcher from their flts. In coordinate space, the structures observed around Q2 = 0.25 (GeV/c)2 influence the long distance tail (r ∼ 1.5 2 fm) of charge and magnetisation densities and may be interpreted as resulting from a pion cloud surrounding a bare nucleon. Within this picture, the data for all four form factors can be described by an intuitive phenomenological ansatz consisting of dipole functions for the constituent quarks together with a p-shell harmonic oscillator behaviour of the pion cloud [63]. Another method to analyse and interpret form factors, which does not directly refer to a particular model for nucleon structure, is based on dispersion relations in Q2 . They provide a mathematical framework to connect experimental date in spacelike (Q2 > 0) as well as in timelike (Q2 < 0) regions with spectral functions describing the spectrum of virtual intermediate states, through which a photon can couple to a nucleon [70]. Already early form factor data have been analysed systematically in this framework and it has been established that the spectral functions can be approximated by poles due to

M. Ostrick: Electromagnetic form factors of the nucleon

the existence of vector mesons and their coupling to nucleons. The prediction of the ρ(770) meson was an early success of this approach [71]. Reflned analyses demonstrated the importance of non resonant multi-pion intermediate states [72,73,74]. Twoand tree-pion systems are the lightest possible intermediate states. They provide a link to pion-nucleon scattering and to model-independent predictions from chiral perturbation theory. The analysis method based on dispersion relations as well as the influence of recent data on the spectral functions is discussed by H.W. Hammer [75].

6 Conclusions In facilities like the Mainz Microtron MAMI, high intensity, polarised, continuous electron beams in the energy range relevant to study phenomena at hadronic scales are available and can be combined with polarised targets and sophisticated detector systems for coincidence experiments and polarimetry. Electromagnetic form factors are signiflcant observables, directly related to the spatial structure of the nucleon. For the flrst time, all four nucleon form factors have been measured with a precision su– cient to identify local structures in the Q2 dependence at a few percent level. In the near future, measurements of GE,p /GM,p and GE,n will be extended to higher values of Q2 at Jefierson → − −  Lab. Below Q2 = 2 (GeV/c)2 , new 3 H e(→ e , e n) as well as  absolute p(e, e ) cross section measurements are planned at MAMI [76,77]. Besides electromagnetic form factors, a deeper understanding of the elastic nucleon response also includes the weak vector and axial-vector currents. The nucleon axial form factor at low Q2 has recently been measured in pion electroproduction at MAMI [78] and present-day experiments in parity-violating electron scattering provide access to the two weak vector form factors, which will allow a flavour decomposition of the charge and magnetisation distributions in the nucleon [34,69]. Also beyond elastic scattering, the experimental techniques discussed above help to fully exploit the properties of electromagnetic probes for studies of the much poorer known structure and dynamics of resonances in inelastic processes [79,80]. I would like to thank the organisers of the symposium 20 Years of Physics at the Mainz Microtron MAMI, Hartmut Arenh˜ ovel, Hartmut Backe, Dieter Drechsel, J˜ org Friedrich, Karl-Heinz Kaiser and Thomas Walcher and express all the best wishes for the future.

References 1. E.M. Lyman, A.O. Hanson, M.B. Scott, Phys. Rev. 84, 626 (1951). 2. R. Hofstadter, R.W. McAllister, Phys. Rev. 98, 217 (1955). 3. G.G. Simon et al., Nucl. Phys. A 333, 381 (1980). 4. L. Andivahis et al., Phys. Rev. D 50, 5491 (1994).

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5. C.E. Hyde-Wright, K. de Jager, Annu. Rev. Nucl. Part. Sci. 54, 217 (2004). 6. H.F. Ehrenberg, R. Hofstadter, Phys. Rev. 110, 544 (1958). 7. A. Lung et al., Phys. Rev. Lett. 70, 718 (1993). 8. P. Markowitz et al., Phys. Rev. C 48, R5 (1993). 9. W. Fabian, H. Arenh˜ ovel, Nucl. Phys. A 314, 253 (1979). 10. M. Schwamb, these proceedings. 11. H. Anklin et al., Phys. Lett. B 336, 313 (1994). 12. E.E.W. Bruins et al., Phys. Rev. Lett. 75, 21 (1995). 13. H. Anklin et al., Phys. Lett. B 428, 248 (1998). 14. G. Kubon et al., Phys. Lett. B 524, 26 (2002). 15. K.I. Blomqvist et al., Nucl. Instrum. Methods A 403, 263 (1998). 16. J. Arnold et al., Nucl. Instrum. Methods A 386, 211 (1997). 17. J. Jourdan, I. Sick, J. Zhao, Phys. Rev. Lett. 79, 5186 (1997). 18. E.E.W. Bruins et al., Phys. Rev. Lett. 79, 5187 (1997). 19. H. Gao et al., Phys. Rev. C 50, R546 (1994). 20. W. Xu et al., Phys. Rev. Lett. 85, 2900 (2000). 21. W. Xu et al., Phys. Rev. C 67, 012201 (2003). 22. W.K. Brooks, J.D. Lachniet, Nucl. Phys. A 755, 261 (2005). 23. A.I. Akhiezer et al., Sov. Phys. JETP 6, 588 (1958). 24. N. Dombey, Rev. Mod. Phys. 41, 236 (1969). 25. R.G. Arnold, C.E. Carlson, F. Gross, Phys. Rev. C 23, 363 (1981). 26. J. Golak et al., Phys. Rev. C 63, 034006 (2001). 27. H. Arenh˜ ovel et al., Z. Phys. A 331, 123 (1988). 28. H. Arenh˜ ovel et al., Phys. Rev. C 52, 1232 (1995). 29. K. Aulenbacher et al., Nucl. Instrum. Methods A 391, 498 (1997). 30. K.H. Stefiens et al., Nucl. Instrum. Methods A 325, 378 (1993). 31. V. Tioukine et al., contribution to the 8th European Particle Accelerator Conference (EPAC 2002), Paris, France, 3-7 June 2002. 32. P. Drescher et al., Nucl. Instrum. Methods A 381, 169 (1996). 33. A. Jankowiak, these proceedings. 34. F. Maas, these proceedings. 35. D. Eyl et al., Z. Phys. A 352, 211 (1995). 36. B.D. Milbrath et al., Phys. Rev. Lett. 80, 452 (1998). 37. T. Pospischil et al., Eur. Phys. J. A 12, 125 (2001). 38. M.K. Jones et al., Phys. Rev. Lett. 84, 1398 (2000). 39. O. Gayou et al., Phys. Rev. Lett. 88, 092301 (2002). 40. J. Arrington, Phys. Rev. C 69, 022201 (2004). 41. I.A. Qattan et al., Phys. Rev. Lett. 94, 142301 (2005). 42. L.W. Mo, Y.S. Tsai, Rev. Mod. Phys. 41, 205 (1969). 43. M. Vanderhaeghen, these proceedings. 44. E. Fermi, L. Marshal, Phys. Rev. 72, 1139 (1947). 45. S. Kopecki et al., Phys. Rev. C 56, 2229 (1997). 46. Yu.A. Alexandrov et al., Phys. Part. Nucl. 30, 29 (1999). 47. H. Leeb, C. Teichtmeister, Phys. Rev. C 48, 1719 (1993). 48. Yu.A. Alexandrov, Phys. Rev. C 49, 2297 (1994). 49. S. Galster et al., Nucl. Phys. B 32, 221 (1971). 50. S. Platchkov et al., Nucl. Phys. A 510, 740 (1990). 51. M. Meyerhofi et al., Phys. Lett. B 327, 201 (1994). 52. J. Becker et al., Eur. Phys. J. A 6, 329 (1999). 53. M. Ostrick et al., Phys. Rev. Lett. 83, 276 (1999). 54. C. Herberg et al., Eur. Phys. J. A 5, 131 (1999). 55. D. Rohe et al., Phys. Rev. Lett. 83, 4257 (1999).

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The European Physical Journal A D. Rohe, these proceedings. D.I. Glazier et al., Eur. Phys. J. A 24, 101 (2005). J. Bermuth et al., Phys. Lett. B 564, 199 (2003). G. Warren et al., Phys. Rev. Lett. 92, 042301 (2004). R. Madey et al., Phys. Rev. Lett. 91, 122002 (2003). R. Alarcon et al., contribution to the 16th International Spin Physics Symposium (SPIN 2004), Trieste, Italy, 1016 Oct 2004. R. Schiavilla, I. Sick, Phys. Rev. C 64, 041002 (2001). J. Friedrich, Th. Walcher, Eur. Phys. J. A 17, 607 (2003). I. Passchier et al., Phys. Rev. Lett. 82, 4988 (1999). H. Zhu et al., Phys. Rev. Lett. 87, 081801 (2001). T. Eden et al., Phys. Rev. C 50, 1749 (1994). J. Friedrich, N. Voegler, Nucl. Phys. A 373, 219 (1982). B. Krusche, Eur. Phys. J. A 26, 7 (2005). S. Kowalski, these proceedings.

70. G. Hohler et al., Nucl. Phys. B 114, 505 (1976). 71. W.R. Frazer, J.R. Fulco, Phys. Rev. Lett. 2, 365 (1959). 72. P. Mergell, U.G. Meissner, D. Drechsel, Nucl. Phys. A 596, 367 (1996). 73. H.W. Hammer, U.G. Meissner, D. Drechsel, Phys. Lett. B 385, 343 (1996). 74. H.W. Hammer, D. Drechsel, U.G. Meissner, Phys. Lett. B 586, 291 (2004). 75. H.W. Hammer, these proceedings 76. M.O. Distler (contact person) et al., Experiment MAMI A1-2/2005. 77. M.O. Distler, W. Heil, D. Rohe (contact persons) et al., Experiment MAMI A1-1/2005. 78. A. Liesenfeld et al., Phys. Lett. B 468, 20 (1999). 79. R. Beck, these proceedings. 80. H. Schmieden, these proceedings.

Eur. Phys. J. A 28, s01, 91 100 (2006) DOI: 10.1140/epja/i2006-09-010-9

EPJ A direct electronic only

Photo- and electro-excitation of the Δ-resonance at MAMI H. Schmiedena Physikalisches Institut, Universit˜ at Bonn, Germany / Published online: 15 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Over the last decade accurate experiments at MAMI played an essential role to improve our understanding of the nucleon to Δ(1232) transition. Originally to a large extent motivated through intra quark hyperflne interactions anticipated in QCD-inspired quark models they showed that pionic degrees of freedom are essential. The meson cloud is mainly responsible for the observed quadrupole excitation strength and afiects the magnetic dipole transition strength as well. PACS. 13.60.Le Meson production resonances with S = 0

13.40.-f Electromagnetic processes and properties

14.20.Gk Baryon

1 Introduction At very high energy and momentum transfers in deep inelastic lepton scattering, proton and neutron reveal their substructure of pointlike, almost massless spin-1/2 constituents, the quarks, and of gluons as the exchange bosons mediating the color force between them. In this regime of asymptotic freedom Quantum Chromodynamics is well established as the basic underlying theory of strong interaction. However, at momentum transfers corresponding to the nucleons’ size, the nonlinear strong couplings prohibit the solution of the QCD fleld equations using perturbation theory. Hence, basic properties such as mass, size and excitation spectrum are only qualitatively [1] understood and remain still a domain of models [2]. To prove that QCD provides also the correct theory at the conflnment scale is the challenge of Lattice calculations, currently stepping beyond quenched approximations [3] by realistic light quark vacuum polarisation and chiral quark actions [4,5]. Analogously to atomic spectroscopy at the threshold to the era of quantum mechanics, baryon spectroscopy serves today as a tool to improve our understanding of the inner dynamics of the nucleon. Generally, the high level density of excited states in combination with their short lifetimes, and thus large natural widths, provides an annoying experimental obstacle. However, there are two states which can be experimentally almost exclusively prepared, outstanding in cross section and well separated in mass from their neighbours. The N (1535)S11 negative-parity partner of the nucleon which selectively couples to the η-nucleon flnal state. Its mass puts it at the very edge of the energy range accessible with MAMI B. Contrary, the decuplet ground state Δ(1232)P33 couples almost entirely into the a

e-mail: [email protected]

Fig. 1. Photoexcitation of the N → Δ transition. Top: M1 photon generating a spin-flip transition; bottom: E2 or C2 photon introducing angular momentum L = 2.

π-nucleon channel, perfectly fltting the present nominal MAMI energy and thus most suited for a detailed study. From the viewpoint of the inner quark dynamics the nucleon to Δ(1232) transition is very interesting. In the SU (6) symmetric constituent quark model it corresponds to a pure spin-flip of one of the quarks, yielding the spin 3/2 of the Δ(1232). Electromagnetic excitation requires thus an M1 magnetic dipole photon as schematically depicted in flg. 1 (top). Parity and angular momentum conservation would alternatively allow the absorption of an L = 2 photon, coupling together with the nucleons spin 1/2 to J = 3/2, cf. flg. 1 (bottom). However, this requires L = 2 quadrupole components in one or both of the nucleons and deltas quark wave functions, in analogy to the deuteron in nuclear physics. As in the latter case, the quadrupole components can be associated with a spherical deformation of the system. They originate from

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tensor parts in the interaction of the constituents which, in QCD-motivated models, are attributed to the color hyperflne interaction among the quarks [6,7]. While most of our knowledge about the nucleon excitation spectrum stems from pion-nucleon scattering experiments, those are insensitive to the quadrupole strength in the nucleon to Δ(1232) transition. Due to the positive parity of both nucleon and Δ, the angular momentum ΔL = 1 magnetic transition is generated in π-N scattering, but not the ΔL = 2 one. The latter requires photons which provide positive parity with both ΔL = 1 and ΔL = 2. For photon energies up to 2000 MeV, flg. 2 shows the total absorption cross section of real photons on the proton. The Δ(1232) resonance exhibits almost isolated around Eγ = 340 MeV, whereas the second and third resonance regions are composed of numerous overlapping resonances which, in the total absorption cross section, cannot be separated. Due to their short life time, the direct detection of the resonances is precluded. Since the Δ(1232) almost exclusively decays into pion and nucleon, pion photo- and electro-production ofi the nucleon provide the major experimental tools for the investigation of its properties.

t=nxl y

q

e scattering plane

x

e’

0

z

n=qxp

p

l

(lab)

reaction plane

In its rest frame, the recoiling hadronic system decays back to back into π 0 and proton. Θπ and Θp denote the pion and proton angles with respect to q in the hadronic cm frame. The reaction plane, which is given by l = pcm p , cm n = q × pπ , and t = n × l, is tilted against the electron scattering plane by the angle Φ. In one photon exchange approximation the flvefold differential cross section

2 Kinematics and cross section of pion photoand electro-production factorizes into the virtual photon flux,

0.5

tot

/mb

0.4

0.3

0.2

0.1

0.0

Γ =

α E  kγ 1 , 2π 2 E Q2 1 − 

200

400

600

800

1000

1200

1400

1600

1800

2000

E /MeV Fig. 2. World data of the real photon total absorption cross section as a function of the photon energy [8]. The open and full circles represent the data from MAMI [9] and NINA [10]. The parameterization after [11] is shown as the solid line.

(1)

(2)

and d2 σv /dΩπcm , the virtual photon cm cross section. α denotes the flne structure constant, kγ = (W 2 − m2p )/2mp the real photon equivalent laboratory energy for the excitation of the target with mass mp to the cm energy W , and  = [1+(2|q|2 /Q2 ) tan2 ϑ2e ]−1 the photon polarisation parameter. Without target or recoil polarisation, the virtual photon cross section is given by [12] d2 σv = λ · [RT + L RL + c+ RLT cos Φ + dΩπcm RT T cos 2Φ + Pe c− RLT  sin Φ].

0.6

(cm)

Fig. 3. Kinematics of pion electroproduction. The vectors of the reaction plane are in the rest frame of the Δ.

d5 σ d2 σv =Γ cm dEe dΩe dΩπ dΩπcm

The kinematics of pion electroproduction are shown in flg. 3 at the example of the e+p → e+p+π 0 reaction. The electron is scattered by the laboratory angle ϑe and the exchanged virtual photon transfers the difierence between incoming and scattered electron energies and momenta, ω = E − E  and q = k − k , respectively. Q2 = −qμ q μ > 0 represents the invariant mass of the virtual photon. The electron scattering plane is spanned by the unit vectors z = q = q/|q|, y = k × k , and x = y × z.

p

(3)

The ratio λ = |pπcm |/kγcm is determined by the pion cm momentum pπcm and kγcm = (mp /W )kγ . The structure functions, Ri , parameterize the response of the hadronic system to the various polarisation states of the photon fleld, which are described by the transverse and longitu2 dinal polarisation,  and L = ωQ2 , also contained within cm  the factors c± = 2L (1 ± ). The degree of longitudinal electron polarisation is denoted by Pe . In lowest order, the tree graphs depicted in flg. 4 contribute to the cross section in the Δ(1232) resonance region. For π 0 production the pion pole and the contact term vanish, because the photon cannot couple to the chargeless pion. Thus, usually the π 0 channel is chosen to tag the Δ(1232) intermediate state. Nevertheless, the unambiguous tagging is hampered by the inevitable background

H. Schmieden: Photo- and electro-excitation of the Δ-resonance at MAMI

93

Table 1. Photon and pion multipoles up to Lγ = 2. γ-N system Lγ

γ-multipole

J

lπ

0

C0

1/2

1

1/2

0

E0+ /S0+

3/2

2

E2 /S2

1/2

1

M1

3/2

1

M1+

3/2

1

E1+ /S1+

5/2

3

E3 /S3

3/2

2

M2

5/2

2

M2+

Fig. 4. Lowest-order graphs of single-pion electroproduction. E1/C1

from non-resonant π production (cf. flg. 4). Therefore, to extract the small quadrupole admixture, a full multipole analysis would be desirable in analogy to π-N scattering. However, in pion production experiments with real and, in particular, virtual photons this is much more di– cult to achieve, because more invariant amplitudes need to be independently determined.

π-N system

1 M1

E2/C2 2 M2

π-multipole S1

parity + − +

+ −

2.1 Multipole decomposition The response functions R of eq. (3) can be decomposed into pion multipoles of the flnal state. In S and P wave approximation this yields [12,13]: RL = λ2 [ |S0+ |2 + 4|S1+ |2 + |S1− |2 ∗ ∗ −4e{S1+ S1− } + 2 cos Θ e{S0+ (4S1+ + S1− )} ∗ S1− }) ], +12 cos2 Θ (|S1+ |2 + e{S1+ 1 RT = |E0+ |2 + |2M1+ + M1− |2 2 1 + |3E1+ − M1+ + M1− |2 2 ∗ +2 cos Θ e{E0+ (3E1+ + M1+ − M1− )}

+ cos2 Θ |3E1+ + M1+ − M1− |2

RT T

1 − |2M1+ + M1− |2 2  1 − |3E1+ − M1+ + M1− |2 } , 2

3 1 = 3 sin2 Θ |E1+ |2 − |M1+ |2 2 2

 ∗ ∗ −e{E1+ (M1+ − M1− ) + M1+ M1− } ,

∗ RLT = − sin Θ λ e{ S0+ (3E1+ − M1+ + M1− ) ∗ ∗ −(2S1+ − S1− )E0+ ∗ +6 cos Θ (S1+ (E1+ − M1+ + M1− ) ∗ +S1− E1+ ) }, ∗ RLT  = sin Θ λ m{ S0+ (3E1+ − M1+ + M1− ) ∗ ∗ −(2S1+ − S1− )E0+ ∗ +6 cos Θ (S1+ (E1+ − M1+ + M1− ) ∗ +S1− E1+ ) }.

(4)

as electric/coulombic (E/C) or magnetic (M ). The pion multipoles AIlπ ± are characterized through their magnetic, electric or scalar (longitudinal) nature, A = M, E, S (L), the isospin, I, and the pion-nucleon relative angular momentum, lπ . The coupling of lπ with the nucleon spin to the total angular momentum, J, is indicated by ±. Omission of the isospin index as in eqs. (4-8) indicates amplitudes in the pπ 0 charge channel throughout this paper. In general, the unambiguous tagging of a resonance requires the multipoles to be determined in the appropriate isospin channel. However, at the resonance position of the Δ(1232) the pπ 0 amplitudes provide a very good approximation to the separated isospin components. Both methods, simultaneous measurement of π 0 and π + production and sole π 0 production, have been exploited.

3 Experimental methods and results (5)

(6)

(7)

The sensitivity to the N → Δ(1232) quadrupole amplitudes is directly illustrated by the S and P wave multipole decomposition of the cross section in eqs. (4 8). All structure functions contain E1+ or S1+ amplitudes, either quadratically or as bilinear combinations. The smallness of the quadrupole amplitudes makes a determination of their square against the leading |M1+ |2 very di– cult, e.g., a Rosenbluth separation of RL and RT for the extraction of |S1+ |2 . It is much more promising to exploit interferences with the dominating M1+ multipole. Thus, the experiments based on unpolarised electron beams extracted the quadrupole over dipole ratios 0

EMRπ = (8)

λ = ωcm /|qcm | is a current conservation factor related to the deflnition of the amplitudes. The photon and pion multipoles up to order Lγ = 2 are summarized in table 1. According to their parity the photon multipoles are classifled

and 0

CMRπ =

∗ M1+ } e{E1+ |M1+ |2

(9)

∗ e{S1+ M1+ } |M1+ |2

(10)

from measurements of the RT T and RLT type structure functions.

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3.1 Real photon experiments With real photon beams, i.e. Q2 = 0, the longitudinal parts of the cross section vanish. Of eqs. (4 8) thus only RT and RT T contribute and the cross section is often written in the form [12]

2  d σ d2 σ = · [1 − Pγ Σ cos 2Φ], (11) dΩπcm dΩπcm 0 d2 σ [ dΩ cm ]0 π

where denotes the unpolarised cross section and Pγ is the degree of linear polarisation of the photon beam. The photon-beam asymmetry Σ represents the ratio RT T /RT and thus enters the cross section with the cos 2Φ modulation characteristic for the directional sensitivity against the plane of linear polarisation. This plane is intrinsically flxed by kinematic constraints in electron scattering, usually the detection of the flnal-state electron under a certain angle. Contrary, real photon beams from electron bremsstrahlung are a homogeneous superposition of all polarisation directions. The net polarisation consequently vanishes. To obtain linear polarised photon beams it is necessary to flx the electron scattering plane in the bremsstrahlung process. This can be achieved by ofi-axis tagging or, as exploited at MAMI, through coherent bremsstrahlung ofi a diamond crystal. Under certain kinematic conditions depending on the crystal alignment relative to the electron beam the crystal lattice, similar to the M˜ ossbauer efiect, coherently takes the bremsstrahlung recoil instead of individual nuclei. This generates intensity peaks in the otherwise  1/Eγ distribution of the bremsstrahlung spectrum. Within the energy region of the coherent peak the crystal orientation determines a particular electron scattering plane. Hence, the photon beam is linearly polarised and the degree of linear polarisation is related to the intensity excess over the incoherent spectrum. According to eq. (11), linear polarised beams with Pγ = 0 enable the contribution of the beam asymmetry Σ to the observed cross section. Its size is determined by ∗ M1+ } + · · · interferences of the type |M1+ |2 + 6 e{E1+

Fig. 5. The cylinder-symmetric DAPHNE detector (Figure courtesy of MAMI-A2 collaboration).

which provide a very high sensitivity to the EMR. In order to exploit those experimentally, the cos 2Φ azimuthal modulation of the cross section needs to be determined. A detector with cylinder symmetry is ideally suited for this purpose. The flrst experiments of the A2 collaboration at MAMI consequently used the DAPHNE setup to detect the recoil protons from the γ + p → π 0 + p reaction, schematically depicted in flg. 5 [14,15]. Results of such measurements are shown in flg. 6. Exploiting the cleanly measured azimuthal modulation (left part of flgure) the photon beam asymmetries (right part) can be determined as a function of polar angle over the entire energy region of the Δ(1232) resonance. In conjunction with the simultaneously measured difierential cross sections it is possible to perform a detailed mulipole analysis. Results are depicted in flg. 7 [15,16]. Furthermore, including the results of the reaction γ + p → π + + n with linearly polarised photon beam enables an isospin decomposition. At the resonance position it provides EMR results in agreement with the π 0 channel alone. As flnal result [15] the electric quadrupole to magnetic dipole ratio

Fig. 6. Left: Measured relative cross section of the reaction γ + p → π 0 + p using linearly polarised photon beam as a function of the azimuthal angle Φ at flxed polar angle Θπcm = 90 . Right: Polar angle dependence of the extracted photon beam asymmetry.

Fig. 7. Multipole flts based on measurements of cross section and photon asymmetry using linearly polarised real photon beams [16].

H. Schmieden: Photo- and electro-excitation of the Δ-resonance at MAMI

∗ e{E1+ M1+ } = (−2.5 ± 0.1stat ± 0.2syst ) % |M1+ |2

95

(12)

was obtained. The beam asymmetry had also been extracted using the large acceptance TAPS photon detector, covering the full polar angular range for the π 0 decay photons [17]. Despite the coplanar detector arrangement, Σ could be determined through the deliberate rotation of the photon polarisation plane.

3.2 Virtual photon experiments Due to the nonzero mass of the exchange photon, in electroproduction also longitudinal pieces contribute, in total four structure functions with unpolarised electron beam and a flfth one with longitudinally polarised beam. If either a polarised target is provided in the initial state or the flnal state proton polarisation is determined, then 13 more structure functions enter the cross section, i.e. a total of 18. Not all of those are independent. In addition to the EMR now also the CMR (cf. eq. (10)) is accessible. As in the photoproduction case, sufflcient sensitivity is only obtained through interferences of the S1+ amplitude with the dominating M1+ .

Fig. 8. Three-Spectrometer-Setup of the MAMI-A1 collaboration. Spectrometer B (left) is lifted out of plane by 10 degrees (see text). Picture courtesy of Markus Weis.

3.2.1 Unpolarised electroproduction According to the discussion of sect. (3.1), in electron scattering a linearly polarised photon fleld is already obtained without the need to polarise the electrons. Thus, the EMR can be determined similar to real photon experiments with a large acceptance detector [18,19]. The kinematic focussing at large Q2 enables full centre-of-mass coverage using a relatively small acceptance in the laboratory, e.g. of magnetic spectrometers [20]. At smaller Q2 the required angular range must be covered by subsequent settings of the spectrometers. This technique has been used for 0 the MAMI experiments in order to determine the CMRπ from the longitudinal-transverse interference part of the cross section. Since RLT comes along with a cos Φ azimuthal modulation, cf. eq. (3), it is su– cient to measure the recoil protons of the p(e, e p)π 0 reaction within the electron scattering plane, i.e. within the laboratory floor plane. At each π 0 centre-of-mass polar angle θπcm 0 , the cosine term exhibits through a cross section asymmetry in measurements left (Φ = 0◦ ) and right(Φ = 180◦ ) of the direction of three-momentum transfer, q: ρLT (θπcm 0 ) :=

dσv (Φ = 0◦ ) − dσv (Φ = 180◦ ) . dσv (Φ = 0◦ ) + dσv (Φ = 180◦ )

(13)

According to eq. (3), it is related to the partial responses via ρLT (θπcm 0 ) =

c+ RLT . RT + L RL + RT T

(14)

The sensitivity to S1+ can be seen from the decomposition of eq. (14) into the leading S and P partial waves. At the Δ(1232)-resonance position the asymmetry cm ρLT (θπcm 0 )  f (θπ 0 )

∗ ∗ {(S0+ + 6S1+ cos θπcm 0 )M1+ } (15) 2 |M1+ |

is approximated. Thus measurements of ρLT in the forward (θ1 ) and backward cm-hemisphere (θ2 = π − θ1 ) allow the extraction of S1+ /M1+ and, simultaneously, of S0+ /M1+ , i.e. the main contribution of inherent non resonant background: ∗ {S1+ M1+ } = f1 (θ1,2 ) · [ρLT (θ1 ) − ρLT (θ2 )] + C1 , (16) |M1+ |2 ∗ {S0+ M1+ } = f0 (θ1,2 ) · [ρLT (θ1 ) + ρLT (θ2 )] + C0 . (17) |M1+ |2

The functions f denote kinematic factors, C0 and C1 contributions of multipoles beyond the simple approximation. Using the 3-Spectrometer-Setup of the MAMI-A1 collaboration shown in flg. 8, the asymmetries ◦ ρLT (θπcm 0 = 160 ) = (12.18 ± 0.27stat ± 0.82sys ) %, cm ρLT (θπ0 = 20◦ ) = (−11.68 ± 2.36stat ± 2.36sys ) %

have been measured [21]. They are shown in flg. 9 along with difierent calculations. The prediction of Sato and Lee [22] describes the asymmetries quite well, to a lesser

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Fig. 9. Left-right asymmetries ρLT measured in π 0 production with unpolarised electron beam compared to model predictions from MAID2003 [23] (dotted), DMT2001 [24, 25] (dashed), Sato/Lee [22] (dashed dotted). The full curve represents a MAID2003 re-flt [21]. The inner and outer errors are purely statistical and quadratically summed statistical and systematical, respectively.

extent the calculation within the dynamical model of Kamalov and Yang [25]. Also the standard MAID2003 parametrisation [23] provided only moderate agreement, in particular at backward pion angle. This was resolved through a re-flt including also polarisation data as described in the following subsect. 3.2.2. Using the MAID re-flt, the multipole ratios of eqs. (16) and (17) were determined to ∗ M1+ } {S1+ = (−5.45 ± 0.42) %, |M1+ |2 ∗ {S0+ M1+ } = (2.56 ± 2.25) %. |M1+ |2

(18) (19)

The S0+ result is compared to older measurements in 0 flg. 10. It practically excludes that a large negative CMRπ found by ref. [26] at Q2 = 0.12 (GeV/c)2 can be explained by the particular contribution of a large negative S0+ /M1+  −10 % [27] in forward pion kinematics. 3.2.2 Polarised electron beams: 5th structure function RLT  , the 5th structure function of eq. (3), measures the imaginary part of the same interference which RLT and ρLT contain the real part of. This is obvious from comparison of the S- and P -wave decompositions in eqs. (7) and (8). Experimentally, RLT  can be extracted through the cross section asymmetry with regard to the ip of beam helicity between ±1: dσ + − dσ − dσ + − dσ − c− RLT  sin Φ . = RT + L RL + c+ RLT cos Φ + RT T cos 2Φ

ρLT  =

(20) (21)

Fig. 10. Result for the S0+ strength from π 0 electroproduction at low Q2 (black cross). The open sympols represent older measurements from DESY [28] and NINA [29].

Fig. 11. Schematics of the M¿ller polarimeter of the MAMIA1 collaboration.

The measurement requires longitudinally polarised electron beam and, since RLT  enters the cross section with sin Φ (cf. eq. (3)), out-of-plane detection of the recoil protons or the scattered electrons of the p(e, e p)π 0 reaction. Intense and highly polarised electron beams are routinely available at MAMI [30]. The degree of longitudinal polarisation at the beam axis in the spectrometer hall is accurately measured by a M¿ller polarimeter, where the beam electrons are scattered ofi polarised electrons. Those are provided in an iron foil magnetised to saturation within the 4 Tesla fleld of a superconducting solenoid. Symmetric scattering kinematics is selected by momentum-speciflc detection of the outgoing electron pair behind a magnetic dipole fleld. The setup is schematically shown in flg. 11. A cm angular acceptance of δΘ/Θ  20 % around Θcm = 90◦ makes the influence of the Fermi motion of the bound electrons, the so-called Levchuk effect [31], negligible [32]. Out-of-plane detection capability is provided by spectrometer B, which can be tilted up to 10◦ [33] as visible in flg. 8. Using this setup the beam-helicity asymmetry was determined at the energy of the Δ(1232) resonance and Q2 = 0.2 (GeV/c)2 [34]. The result is shown in flg. 12. The calculations within the dynamical models of Sato and Lee [22] as well as of DMT2001 [25], and the MAID2000 parametrisation [23] all were found to disagree

H. Schmieden: Photo- and electro-excitation of the Δ-resonance at MAMI

97

Fig. 13. Recoil polarisation components in parallel kinematics of the reaction p(e, e p)π 0 . For visual clarity only, the π 0 is drawn slightly sideways. The x, y, and z components are deflned relative to the electron scattering plane.

Fig. 12. Beam-helicity asymmetry ρLT  measured in π 0 production with longitudinally polarised electron beam. The dotted curve represents the original MAID2003 calculation [23]. The dashed-dotted and dashed curves are the results of the dynamical models of Sato-Lee [22] and Kamalov-Yang [25], respectively. The full curve is the MAID re-flt of ref. [21]. Errors are purely statistical.

with this measurement. However, rescaled by a factor 0.75 MAID described the asymmetry very well. The later reflt of real and imaginary parts of the S1+ and S0+ π 0 amplitudes in MAID2003 on basis of the MAMI ρLT and ρLT  data provided a very satisfactory description of the polarisation data, cf. the full curve in flg. 12, and of the unpolarised measurement discussed in the preceding section as well [21]. This underlines the absolute need to understand the physical background amplitudes before the small quadrupole contributions can be reliably extracted. Ideally, a complete experiment with respect to a multipole decompostion is required.

3.2.3 Double Polarisation Experiments: Recoil proton polarimetry

A complete experiment in the above sense would require to measure more than 11 independent observables over the energy range of the Δ(1232) and the full angular range. Even with application of Watson’s theorem [35,36], which below 2π 0 threshold relates real and imaginary parts of the amplitudes, this presently is out of reach in pion electroproduction. In order to get as close as possible and thus minimise the model dependence in the extraction of the quadrupole amplitudes, it is mandatory to measure double polarisation observables. In parallel kinematics of the p(e, e p)π 0 reaction (cf. flg. 13) each of the three cartesian components of the proton polarisation, in addition to the unpolarised cross section σ0 , only depends on one speciflc structure func-

tion [37]:  t σ0 Px = λ · Pe · 2L (1 − )RLT ,  n σ0 Py = λ · 2L (1 + )RLT ,  σ0 Pz = λ · Pe · 1 − 2 RTl T  .

(22) (23) (24)

According to flg. 13 the axes are deflned relative to the electron scattering plane and the virtual photon direction. This is the natural choice, since in parallel kinematics the recoil polarisation is completely determined by the angular momentum transfer from the photon fleld. Decomposition up to S and P partial waves, σ0 Px = Pe c− λ e{(4S1+ + S1− − S0+ )∗ (M1+ − M1− − E0+ + 3E1+ )},

(25)

∗

σ0 Py = −c+ λ m{(4S1+ + S1− − S0+ ) (M1+ − M1− − E0+ + 3E1+ )}, (26) σ0 Pz = Pe c0 |M1+ |2 + |M1− |2 + 9|E1+ |2 + |E0+ |2 ∗ ∗ + e{6E1+ (M1+ − M1− ) − 2M1+ M1−

∗ −2E0+ (M1+ − M1− + 3E1+ )} , (27) reveals the high sensitivity to the quadrupole amplitudes. In particular, since σ0 is dominated by |M1+ |2 , Px pro0 vides√a rather direct measure of the CMRπ . The factor 2 c0 = 1 −  entering Pz is related to the circular polarisation of the photon fleld. Except the small contribution of RL to σ0 , in parallel kinematics Pz is entirely determined by kinematic constants, i.e. the transfer of circular photon polarisation to the protons, since RTl T  = RT . The transverse components Px and Py measure real and imaginary parts of the same interferences, similar to ρLT and ρLT  but in difierent combinations and with difierent weights of amplitudes. Thus, 0 the model dependence in the extraction of the CMRπ is difierent from unpolarised measurements. Morever, also the composition of systematic errors is completely difierent due to the orthogonal experimental technique. The flrst N → Δ double-polarisation experiment ever has been performed at the 3 Spectrometer setup of MAMI [38]. Central to the experiment is the measurement of proton polarisation behind the focal plane of one of the spectrometers [39,40]. This is based on inclusive proton

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carbon scattering. Due to the strong spin-orbit coupling, the transverse polarisation components, Pn and Pt , generate an azimuthal modulation of the unpolarized cross section, σ0C : σ C = σ0C (Θs , T ) [1 + AC (Θs , T ) (Pt sin Φ − Pn cos Φ)] . (28) The relative strength is determined by the known [41,42] analyzing power, AC . The two polarisation components accessible at the focal plane must be traced back to the target through the spectrometer magnets. This complication on the one hand provides, on the other hand, the opportunity to determine the otherwise inaccessible longitudinal polarisation component. A separation of Px , Py and Pz is achieved since x and z components are odd under beam-helicity reversal, while Py is even (cf. eqs. (22) (24)). At the energy of the Δ(1232) resonance and a momentum transfer of Q2 = 0.121 (GeV/c)2 the three polarisations were simultaneously measured to

0

Fig. 14. CMRπ from recoil polarisation (open diamond) [38] in comparison to unpolarised measurements [18, 19, 26, 28, 29, 44, 45] at low Q2 . The curves show model calculations MAID2003 [23] (solid), DMT2001 [25] (dashed) and Sato/Lee [22] (dashed dotted).

momentum conservation Px /Pe = (−11.4 ± 1.3stat ± 1.4syst ) %, Py = (−43.1 ± 1.3stat ± 2.2syst ) %, Pz /Pe = (56.2 ± 1.5stat ± 2.6syst ) %.

Based on the Px result, the CMR

(29) (30) (31)

χ2x + χ2y =

[46]: 1 χz (1 − χz ). L

(32)

was extracted using the MAID parametrisation [38]. Within the errors this agrees with the alternative method of extraction using Px /Pz , which experimentally provides the advantage that the magnitudes of both analysing power and beam polarisation drop out. Contrary, the ratio is afiected by a larger uncertainty in spin precession of the z-component compared to Px . To flrst order the latter would not precess at all in a homogeneous vertical-bend dipole fleld. 0

The result for the CMRπ is depicted in flg. 14 along with unpolarised measurements. Keeping in mind that 0 the large negative CMRπ of ref. [26] (full triangle tip down) is practically excluded by the S0+ results discussed in sect. 3.2.1, very convincing agreement is observed between the various measurements. A recent very high statistics measurement of the angular distributions of recoil polarisation [43] found, just outside the range of flg. 14, 0 CMRπ = (−6.61±0.18) % at Q2 = 1 GeV/c2 . This agreement extends also to the positive S0+ /M1+ ratio determined from ρLT . The three polarisation components measured in parallel kinematics are model independently related by angular

(33)

The reduced polarisations are deflned by t 1 RLT  · Px = , Pe c− R T + L R L n 1 RLT χy = · Py = , c+ R T + L R L 1 RTl T  · Pz = . χz = Pe c0 R T + L R L

χx =

π0

∗ {S1+ M1+ } = (−6.4 ± 0.7stat ± 0.8syst ) % |M1+ |2

1

(34) (35) (36)

The consistency relation seems hardly fulfllled by the measured recoil polarisations. The experimental result for eq. (33) is 3.9 ± 0.4stat ± 0.4syst = 7.9 ± 0.7stat ± 1.2syst .

(37)

Despite the non-linear error propagation on the r.h.s. of eq. (33), the probability for such a flnding is only a few percent. While MAID2000 fulfllls the consistency relation, there is a discrepancy with the Py measurement. The MAID-reflt mentioned above slightly improves this situation as shown in flg. 15, where the MAMI recoil polarisation data [38] are compared to the MAID versions 2000, 2003 and the re-flt. Nevertheless, the role of nonresonant background, which particularly shows up in Py , still remains unresolved. Another measurement of Py [48] is in agreement with the MAMI data but has a large error. Unfortunately, ref. [43] gives no explicit values of the measured polarisations. 1 Despite the occurence of L in eq. (33), the polarisation relation is nevertheless frame independent. The longitudinal polarisation parameter is also contained in the deflnition of the factors c§ of the reduced polarisations and thus could be eliminated. A similar relation holds in elastic electron nucleon scattering, where χy vanishes [47].

H. Schmieden: Photo- and electro-excitation of the Δ-resonance at MAMI

99

5 -20 -25

-10

(%)

-30 -35

Pzsp / Pe

(%)

-5

Pysp

(%)

0

Pxsp / Pe

70

-40 -45

65 60 55

-50

-15

-55 0

0.1 0.2 0.3 0.4 Q2 (GeV2/c2)

50 0

0.1 0.2 0.3 Q2 (GeV2/c2)

0.4

0

0.1 0.2 0.3 Q2 (GeV2/c2)

0.4

Fig. 15. Components of recoil polarisation measured at the energy of the Δ(1232)-resonance in the reaction p(e, e p)π 0 [38]. Curves show the MAID versions 2000 (full), 2003 (dashed), and the re-flt (dashed-dotted) on basis of ρ LT and ρLT  [21] discussed in the text. Note the suppressed zero middle and right.

In principle, from the reduced polarisations it is possible to determine the ratio of longitudinal to transverse response, RL /RT , without the need of a classical Rosenbluth separation. According to ref. [46] this can be achieved in three difierent ways, using either – only the longitudinal component, χz (cf. also [49]), – the quadratic sum of the transverse components, χ2x + χ2y , or – all three reduced polarisations. However, as a re ection of the almost violated consistency relation the results obtained from the MAMI measurements vary signiflcantly. The smallest value, RL /RT = (4.7 ± 0.4stat ± 0.6syst ) %, is extracted from the quadratic sum χ2x + χ2y of the transverse reduced polarisations and +2.9 the largest one, RL /RT = (12.2 +1.7 −1.6stat −2.7syst ) %, from χz alone. This presently prohibits a reliable extraction of RL /RT but stresses the importance of a simultaneous measurement of all polarisation components with further improved accuracy.

4 Interpretation The MAMI experiments towards the quadrupole strength in the N → Δ(1232) transition yield very consistent results. The flrst order physical background contributions like mS0+ are now much better under control. Remaining uncertainties and inconsistencies seem related to the contribution of higher partial waves especially in the imaginary parts of interferences as in ρLT  and Py . Though important for our understanding of the non-resonant processes, they are very much suppressed in the discussed observables with high sensitivity to EMR and CMR. At least in the vicinity of the photon point, a reliable extraction of the EMR and CMR is possible with a remaining relative model uncertainty of the order ≤ 10 %. Thus it is evident that the experimental results are an order of magnitude larger than expected from the quark model calculations.

Obviously, pionic degrees of freedom which in a microscopic picture would be responsible for e.g. the mS0+ contribution (flg. 10) play an important role. This is made particularly transparent within the dynamical models [24,22]. Such calculations can be split into the so-called bare and dressed parts, which represent the nucleon/delta core, and the pion cloud, respectively. This is also attempted within dispersion relation approaches [50]. While, in general, such a separation sufiers from an unitary ambiguity [51], the models yield consistent results. E.g., the full model of ref. [24] describes the data while the bare calculation yields a very small and positive CMR. Moreover, 3/2 the full calculation yields an M1+ amplitude in agreement with the experimental data. In contrast, the bare calcu3/2 lation gives only about two thirds of M1+ and a CMR which is an order of magnitude too small. This is similar to quark model calculations without pion degrees of freedom [52,53], which also underestimate the quadrupole strength by a factor of ten and get only about 60 % of the experimental M1+ . Despite the yet unsatisfactory statistics of dynamic lattice calculations [54], those are in qualitative agreement with the experimental results and hence further support the important role of the pion cloud. As outlined in the introduction the quadrupole transition strength can be interpreted in terms of a deformation of the baryons involved. Buchmann and Henley flnd opposite deformations of equal strength for nucleon and Δ [55]. Also the quadrupole moments for the nucleon core and the pion cloud are of opposite sign. However, the core appears almost spherical and the deformation due to the pion cloud is an order of magnitude stronger, very similar to the observation within the dynamical model. Buchmann and Henley quote total quadrupole moments in the range of Qnucl = ±(0.113 . . . 0.5) fm2 for nucleon (upper sign) and Δ (lower sign). It is interesting to note that, relative to the size of the objects, the corresponding deformation Qnucl 2 > 0.043 πrN

of the nucleon

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compared to Qdeut = 0.07 πrd2

of the deuteron

would be of similar magnitude.

5 Summary and outlook The MAMI experiments on pion photo- and electroproduction provided cornerstones for our understanding of the N → Δ(1232) transition. At the photon point and at low Q2 they unambiguously demonstrated that the quadrupole strength is an order of magnitude larger than expected in QCD-motivated constituent quark models. Often a precursor for similar experiments at other laboratories, polarisation techniques played a key role. They have already been extended to polarised target measurements and to energies slightly above the Δ(1232) [56]. In view of the higher resonances which become accessible with the increased energy of MAMI C, their potential seems not at all exhausted yet. Of particular interest will be the determination of Q2 -slopes of transition amplitudes to Roper, N (1440)P11 , and N (1535)S11 resonances. Since related to the spatial extention of the systems, new insights into the debated hybrid or molecular structure of such states can be expected. Extension of the experimental techniques to associated strangeness production seems relatively straightforward, provided that Kaons can be unambiguously identifled and su– ciently high beam energy is available. In particular through polarisation data of unprecedented accuracy MAMI has the potential to continuously challenge our understanding of the many body structure of the nucleon. With all the virtues of the electromagnetic probe it can provide experimental key observables also for the comparison to the progressivly improving Lattice calculations. It is a special pleasure for me to thank the retirees Hartmuth Arenh˜ ovel, Hartmut Backe, Dieter Drechsel, J˜ org Friedrich, Karl-Heinz Kaiser and Thomas Walcher who, through their invaluable individual contributions over the last two decades, made it possible to turn MAMI into the big success it has become. I also greatfully acknowledge the support of D. Elsner in preparing talk and written manuscript of this overview.

References 1. F. Wilczek, hep-ph/0201222, MIT CTP 3236. 2. A. Thomas, W. Weise, The Structure of the Nucleon (Wiley VCH 2001). 3. S. Aoki et al., Phys. Rev. Lett. 84, 238 (2000). 4. Q. Mason et al., Phys. Rev. Lett. 95, 052002 (2005). 5. J.W. Negele, hep-lat/0509101 and references therein. 6. A. de Rujula, H. Georgi, S.L. Glashow, Phys. Rev. D 12, 147 (1975). 7. S.L. Glashow, Physica A 96, 27 (1979). 8. Compilation by J. Ahrens, Mainz (2000). 9. M. MacCormick et al., Phys. Rev. C 53, 41 (1996). 10. T.A. Armstrong et al., Phys. Rev. D 5, 1640 (1972).

11. D. Babusci et al., Phys. Rev. C 57, 291 (1998). 12. D. Drechsel, L. Tiator, J. Phys. G: Nucl. Part. Phys. 18, 449 (1992). 13. G. Kn˜ ochlein, D. Drechsel, L. Tiator, Z. Phys. A 352, 327 (1995). 14. R. Beck et al., Phys. Rev. Lett. 78, 606 (1997); 79, 4515 (1997) (E). 15. R. Beck et al., Phys. Rev. C 61, 035204 (2000). 16. O. Hanstein, D. Drechsel, L. Tiator, Nucl. Phys. A 632, 561 (1998). 17. R. Leukel, doctoral thesis, Mainz (2001). 18. K. Joo et al., Phys. Rev. Lett. 88, 122001 (2002). 19. R.W. Gothe, Prog. Part. Nucl. Phys. 44, 185 (2000). 20. V.V. Frolov et al., Phys. Rev. Lett. 82, 45 (1999). 21. D. Elsner et al., Eur. Phys. J. A 27, 91 (2006). 22. T. Sato, T.-S.H. Lee, Phys. Rev. C 63, 055201 (2001). 23. D. Drechsel, O. Hanstein, S.S. Kamalov, L. Tiator, Nucl. Phys. A 645, 145 (1999); http://www.kph.uni-mainz.de/ MAID/. 24. S.S. Kamalov, S.N. Yang, Phys. Rev. Lett. 83, 4494 (1999). 25. S.S. Kamalov, S.N. Yang, D. Drechsel, L. Tiator, Phys. Rev. C 64, 032201 (2001). 26. F. Kalleicher et al., Z. Phys. A 359, 201 (1997). 27. H. Schmieden, Proceedings of NSTAR2001, edited by D. Drechsel, L. Tiator (World Scientiflc, 2001) p. 27. 28. J.C. Alder et al., Nucl. Phys. B 46, 573 (1972). 29. R. Siddle et al., Nucl. Phys. B 35, 93 (1971). 30. K. Aulenbacher, these proceedings. 31. L.G. Levchuk, Nucl. Instrum. Methods A 345, 496 (1994). 32. P. Bartsch, diploma thesis KPH 11/96, Mainz (1996). 33. K.I. Blomqvist et al., Nucl. Instrum. Methods A 403, 263 (1998). 34. P. Bartsch et al., Phys. Rev. Lett. 88, 142001 (2002). 35. K.M. Watson, Phys. Rev. 95, 228 (1954). 36. E. Fermi, Suppl. Nuovo Cimento 2, 17 (1955). 37. H. Schmieden, Eur. Phys. J. A 1, 427 (1998). 38. Th. Pospischil et al., Phys. Rev. Lett. 86, 2959 (2001). 39. Th. Pospischil et al., Nucl. Instrum. Methods A 483, 713 (2002). 40. Th. Pospischil et al., Nucl. Instrum. Methods A 483, 726 (2002). 41. E. Aprile-Giboni et al., Nucl. Instrum. Methods 215, 147 (1983). 42. M.W. McNaughton et al., Nucl. Instrum. Methods 241, 435 (1985). 43. J.J. Kelly et al., Phys. Rev. Lett. 95, 102001 (2005); nuclex/0509004. 44. K. B˜ atzner et al., Nucl. Phys. B 76, 1 (1974). 45. N.F. Sparveris et al., Phys. Rev. Lett. 94, 022003 (2005). 46. H. Schmieden, L. Tiator, Eur. Phys. J. A 8, 15 (2000). 47. Th. Pospischil et al., Eur. Phys. J. A 12, 125 (2001). 48. G.A. Warren et al., Phys. Rev. C 58, 3722 (1998). 49. J.J. Kelly, Phys. Rev. C 60, 054611 (1999). 50. I.G. Aznauryan, S.G. Stepanyan, Phys. Rev. D 59, 054009 (1999). 51. P. Wilhelm et al., Phys. Rev. C 54, 1423 (1996). 52. N. Isgur, G. Karl, R. Koniuk, Phys. Rev. D 25, 2394 (1982). 53. S.S. Gershtein, G.V. Dzhikiya, Sov. J. Nucl. Phys. 34, 870 (1981). 54. C. Alexandrou et al., Phys. Rev. Lett. 94, 021601 (2005); hep-lat/0509140. 55. A.J. Buchmann, E.M. Henley, Phys. Rev. C 63, 015202 (2000). 56. A. Thomas, these proceedings.

Eur. Phys. J. A 28, s01, 101 106 (2006) DOI: 10.1140/epja/i2006-09-011-8

EPJ A direct electronic only

Parity violation in electron scattering S. Kowalskia Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA / Published online: 26 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Parity-violating electron scattering has been a very useful tool for probing the structure of neutral currents and providing detailed information on electroweak form factors. A pioneering SLAC measurement in the mid-70s provided an important early test of the Standard Model. Modern electron accelerators provide high-intensity (> 100 μA), CW beams with polarizations as high as 85%. Experiments such as SAMPLE, A4, HAPPEX and G0 have exploited these capabilities and obtained new information on electroweak strange form factors in the Q2 range of 0.1 1.0 (GeV/c)2 . That activity continues. Other experiments are designed to provide stringent tests of the Standard Model. E-158 at SLAC recently measured the weak charge of the electron. Qweak is a challenging new experiment at JLAB which is designed to measure the weak charge of the proton. This will probe for physics beyond the Standard Model corresponding to energy scales of more than 5 TeV. PACS. 12.15.-y Electroweak interactions distributions

14.20.Dh Protons and neutrons

1 Introduction The electromagnetic probe is one of our most important tools for probing nucleon and nuclear structure and dynamics. Modern accelerators use electron scattering to probe distances of less than 1 fm where signatures of quark degrees of freedom are expected to be observable. A new generation of electron accelerators, 0.5 6 GeV, with CW capability and intense polarized beams have become operational over the past decade. This has enhanced our capability for doing coincidence experiments and other measurements by more than two orders of magnitude. During this time, longitudinally polarized electrons have emerged as a very important new tool to study nucleon and nuclear structure. They have provided precise new information on nuclear electromagnetic form factors and other nuclear structure functions. An important class of experiments using polarized electrons has been the study of parity violation. Parityviolating electron scattering involves the scattering of longitudinally polarized electrons from an unpolarized target. A change in counting rate resulting from a reversal of the beam helicity is a signal of a parity non-conserving effect. At high energies the SLAC parity violation experiment provided a crucial understanding of electro-weak processes and a precise measurement of the weak mixing angle, sin2 θW . At low energies the flrst parity violation experiments were carried out at MIT-Bates and Mainz. a

e-mail: [email protected]

21.10.Gv Mass and neutron

These were designed as tests of the Standard Model. During the past decade there has been an extensive program of such parity violating experiments at MIT-Bates, JLAB and Mainz. The goal of these parity-violating experiments currently is three-fold: – study the strangeness content of the nucleon, – measure the neutron density distribution in a heavy nucleus and – sensitive tests of the Standard Model. In this paper, we will discuss and report on the recent progress of parity violation experiments in these three broad physics areas. Results and future prospects of the difierent experiments will be presented. The Mainz A4 experiment will not be discussed since it is the subject of another contribution to these proceedings.

2 Physics program 2.1 Nuclear structure: strange quarks The proton is made up of three valence quarks, uud, and a sea of gluons and qq pairs all of which contribute to its electromagnetic properties at short distance scales. It was realized [1] that a measurement of the parity-violating asymmetry arising from the interference between the electromagnetic and neutral current amplitudes would allow us to extract the contributions of strange quarks to the

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ground state charge and magnetization distributions (e.g., magnetic moment) of the nucleon. There have been many theoretical estimates of strange quark contributions to nucleon properties. These include both phenomenological models and lattice-gauge calculations. Separation of strange quark contributions to nucleon currents was developed by Kaplan and Manohar [2]. The parity-violating asymmetry [3] for scattering longitudinally polarized electrons from a proton can be written as −GF √ · AP V = 4πα 2 

p Z p  2 e εGE GE + τ GpM GZ M − ε (1 − 4sin θW )GM GA · . ε(GpE )2 + τ (GpM )2

(1)

The three terms in AP V arise as a result of the interference between the electromagnetic and weak interactions. The terms contain bilinear products of electromagnetic and weak form factors. GpE,M are the form factors associated with the distribution of the protons charge and e magnetism. The weak form factors GZ E,M and GA contain s contributions from strange quarks, GE,M . The kinematic factors depend upon speciflcs of an experiment. They are chosen to enhance the relative sensitivity of individual terms to AP V . The flrst two terms are important at forward angles and the last two at backward angles: τ = Q2 /4Mp2 ,

−1 ε = 1 + 2(1 + τ )tan2 (θe /2) and

ε =

 (1 − ε2 )τ (1 + τ ) .

Fig. 1. A schematic view of one module of the SAMPLE experimental apparatus. Ten mirror-phototube pairs are placed asymmetrically about the beam axis.

(2) (3) (4)

1. SAMPLE SAMPLE [4] at MIT-Bates was the flrst experiment to use parity violation as a probe for strange quarks in the proton. A longitudinally polarized electron beam of 200 MeV was incident on a liquid hydrogen or deuterium target. The polarized electrons were produced using a bulk GaAs crystal resulting in an average polarization of 36%. The linac produces a pulsed beam of 25 μs duration at a repetition rate of 600 Hz. The beam current was 40 μA. The helicity of the beam is changed randomly pulse-by-pulse and in addition a half-wave plate can be inserted to change the overall sign of the helicity. The scattered electrons were detected in a large solid angle (1.5 sr) air Cerenkov detector spanning angles between 130◦ and 170◦ . At backward angles SAMPLE is mostly sensitive to GsM and GeA . Figure 1 shows one of the 10 detector modules which are placed A symmetrically about the beam axis. The Cherenkov light is focused by an ellipsoidal mirror unto a phototube. The integrated light is proportional to the scattered electron rate of about 108 s−1 in a beam pulse. A shutter located in front of the phototube could be closed providing a measurement of the background originating from neutrons and charged particles. Tight control of helicity-correlated efiects on the properties of the beam

Fig. 2. Uncertainty bands of GsM vs. GeA at Q2 = 0.1 (GeV/c)2 for the SAMPLE experiment in both hydrogen and deuterium. Also shown is the uncertainty band of the theoretical expectation for GeA .

was critical to the success of the experiment. Several feedback systems were used to minimize such efiects. These included energy, beam position, angle, and intensity. All parity experiments implement similar feedback controls. Experiments were carried out at 200 MeV on both hydrogen and deuterium targets. In addition a measurement at 125 MeV was also made on a deuterium target. The two targets in principle allow a separation of GsM and GeA . The results for both hydrogen and deuterium are summarized

S. Kowalski: Parity violation in electron scattering

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in flg. 2. Using a theoretical prediction for GeA (T = 1) one can extract a value for GsM : GsM (Q2 = 0.1 (GeV/c)2 ) = 0.37 ± 20 ± 0.26 ± 0.07. (5) The experiment favors a small positive value for the magnetic moment contribution, μs , which is at variance with most theoretical predictions. 2. HAPPEX HAPPEX at JLAB was the second experiment designed to look for strange quarks using parity violating electron scattering. The flrst measurement was carried out at Q2 = 0.48 (GeV/c)2 . A 3.2 GeV beam of polarized electrons was incident on a liquid hydrogen target. The scattered electrons were detected in the pair of high resolution spectrometers in Hall-A at a scattering angle of 12.5◦ . Beam currents up to 100 μA were used. The Cherenkov light from a lead-lucite sandwich was integrated over the duration of the helicity window. Beam helicity was reversed at 30 Hz. Beam polarization was 38% in the flrst run and improved to 70% in the second run. It was measured with both Moeller and Compton polarimeters. Tight control of helicity correlated systematics was similar to those used in the SAMPLE experiment. The experiment is sensitive to both GsE and GsM . They extracted [5] a linear combination of strange form factors, GsE + 0.392GsM = 0.014 ± 0.020 ± 0.010.

(6)

In 2004, a second HAPPEX measurement was carried out at Q2 = 0.10 (GeV/c)2 . Measurements were made on both 1 H and 4 He targets. Helium is a special target since the nuclear spin I = 0. It has sensitivity only to GsE . Combining both the hydrogen and helium results allows a direct extraction of GsE and GsM . A 3.0 GeV polarized electron beam was incident on the target. The scattered electrons were detected in the pair of high resolution spectrometers in Hall-A at a scattering angle of 6◦ . A pair of septum magnets in front of the spectrometers made this feasible. Beam currents of up to 35 A were used. Total absorption detectors were used to detect the scattered electrons. The experiment on hydrogen yielded [6] the strange form factor combination, GsE + 0.08GsM = 0.030 ± 0.025 ± 0.006.

(7)

The measurement on 4 He yielded [7], GsE = −0.038 ± 0.042 ± 0.010.

Fig. 3. The four AP V measurements at Q2 = 0.1 (GeV/c)2 are shown, with shaded bands representing 1-sigma combined statistical and systematic uncertainty. Also shown is the combined 95% C.L. ellipse from all four measurements.

Fig. 4. The combination GsE + ηGsM for the HAPPEX, A4 and G0 experiments. Also shown at Q2 = 0.6 (GeV/c)2 the projected uncertainty for a future HAPPEX-III measurement.

In a future experiment HAPPEX is planning a measurement at Q2 = 0.6 (GeV/c)2 . Figure 4 shows the expected uncertainty for this measurement. We expect to run this experiment in 2008-09. 3. G0

(8)

Both measurements are consistent with zero. The combined hydrogen and helium HAPPEX results were consistent with no evidence for strange quarks at Q2 = 0.1 (GeV/c)2 . The results are summarized in flg. 3. Following the above Phase-I measurements, a second experiment on both hydrogen and helium was completed in 2005. These results, not yet available, when analyzed are projected to have error bars which are 1/3 those obtained so far. This will be a very signiflcant improvement.

G0 is another ambitious JLAB parity violation experiment. Its goal is to search for evidence of strange quarks in the proton. It is designed to make both forward and backward angle measurements in the Q2 range, 0.2 1.0 (GeV/c)2 . G0 is an 8 sector superconducting toroidal spectrometer. A 3.0 GeV electron beam was incident on a liquid hydrogen target. The beam current was 40 μA. The detector is designed to detect the recoil protons (flg. 5). It is segmented along the focal plane (16) so that each segment

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Rn . The electromagnetic coupling to the protons is given by QpEM = 1 while the neutrons couple by QnEM = 0. In contrast the weak coupling of the protons is given by QpW ≈ 1 − 4 sin2 θW and the neutrons by QnW ≈ 1. The neutrons have a very strong weak coupling to the electromagnetic probe. The resulting parity-violating asymmetry is given by 2 −GF Q2 FW (Q ) (9) AP V = 4πα√2 Fγ (Q2 ) ,

where FW (Q2 ) = (1 − 4 sin2 θW )Fp (Q2 ) − N Fn (Q2 ),

(10)

and Fγ (Q2 ) = ZFp (Q2 ) Fig. 5. Layout of the G0 spectrometer system. Shown are the superconducting toroidal magnet and the segmented scintillator detector array.

measures a difierent Q2 range. The beam duty factor was reduced to 6% allowing timing measurements to be used to separate background from the recoiling protons. The detector operates in counting mode. The A4 experiment at Mainz also operates in counting mode. All other electron parity violation experiments operate in integrating mode. The results [8] are shown graphically in flg. 4 together with the results from other experiments. The results indicate a non-zero, Q2 -dependent, strange quark contribution. They cover a much broader Q2 range than the other experiments. The uncertainties for the higher Q2 points are quite large as a result of signiflcant unanticipated background contributions. G0 is planning to make backward angle measurements in 2006. At backward angles, scattered electrons are detected instead. Each measurement will be at a single Q2 . They expect to make the flrst measurements at approximately 0.2 and 0.6 (GeV/c)2 .

2.2 Neutron densities Charge radii of nuclei are well known. For example, the proton radius of the lead nucleus, Rp = 5.490 ± .002 fm. This is very precise. A very interesting question is, What is the neutron radius in Pb ? Experimentally, Rn is rather poorly known. Perhaps it is known to 5% at best, where the best estimates come from theory. Models of a neutron star posit a solid crust over a liquid core. The lead nucleus is believed by some to have a neutron skin. Both a possible neutron skin and a neutron star crust would be made of neutron rich matter at similar densities. A measurement of Rn to 1% accuracy would have a major impact on nuclear theory, neutron star structure and atomic parity violation. Currently atomic parity violation experiments are limited in accuracy by our knowledge of the neutron radius in a heavy nucleus. Donnelly, Dubach and Sick [9] suggested the possibility of using parity violating electron scattering to measure

(11)

A measurement of AP V to 3% would provide a measurement of Rn to 1%. An experiment has been approved at JLAB to carry out such a measurement. An 850 MeV, 50 μA polarized beam would be incident on a lead target sandwiched between diamond sheets for cooling. Electrons scattered through 60 would be detected in the pair of HallA high-resolution spectrometers. At Q2 = 0.01 (GeV/c)2 the parity-violating asymmetry, AP V = 0.5 ppm. This is a very challenging experiment. It will require control of helicity correlated systematics to much better than 15 ppb. It appears to be feasible. 2.3 Standard Model tests Parity violating electron scattering has been used to test the Standard Model (SM) since the pioneering SLAC experiment in the mid-70s. Collider experiments at the Zpole provide our best measure of the weak mixing angle, sin2 θW . There remains much interest in exploring the running of sin2 θW from the Z-pole to Q2 = 0. At Q2 = 0, atomic parity experiments give results consistent with the running of sin2 θW . Other experiments testing the SM include 12 C(e, e), 9 Be(e, e), NuTeV, Moeller scattering and Qweak . All parity violating experiments can be described in terms four fundamental quark coupling constants,

  GF LP V = − √ eγμ γ5 e C1u uγ μ u + C1d dγ μ d + 2    μ 5 μ 5 + eγμ e C2u uγ γ u + C2d dγ γ d . (12) The SM makes predictions for the constants. These constants in turn can be written in terms of isoscalar and isovector combinations at the hadronic vertex: α = −C1u + C1d = −(1 − 2 sin2 θW ),

(13)

2

β = −C2u + C2d = −(1 − 4 sin θW ),

(14)

γ = −C1u − C1d = 2/3 sin θW ,

(15)

δ = −C2u − C2d = 0,

(16)

2

sin2 θW = 0.23120 ± 0.00015.

(17)

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Fig. 7. Schematic layout of the Qweak experiment showing the target, collimators, shielding, toroidal magnet and detector system. Fig. 6. Results for the SLAC and Bates parity violation experiments in the space of the coupling constants α and γ.

1.

12

C / 9 Be

12

C is a spinless and isoscalar nucleus and elastic electron scattering is described by a single form factor. The parityviolating asymmetry may be written at the tree level [10, 11] as √ 2 −1 3 (18) AP V = γ 2 GF Q ( 2 πα) . A parity violation experiment was carried out at MITBates [12] on 12 C. The results are shown in fig. 6 together with the results of the SLAC experiment [13] on deuterium. The results of both experiments are consistent with the predictions of the SM.

2. E-158 A recent experiment testing the SM was E-158 at SLAC. It is a purely leptonic process involving Moeller scattering. The goal of the experiment was to measure the weak charge of the electron, e = g sin θW , using parity violation. In this experiment 50 GeV electrons were incident on a liquid hydrogen target. At Q2 = 0.027 (GeV/c)2 the parity violating asymmetry is expected to be 150 ppb. A quadrupole spectrometer was used to focus the scattered Moeller electrons while at the same time defocusing e-p scattering events. The flux of scattered electrons was integrated for each beam burst. The experiment yielded an asymmetry, AP V = −175 ± 30 ± 20 ppb.

(19)

The extracted weak mixing angle is totally consistent with predictions. The result [14] is shown plotted in fig. 8. 3. Qweak A major new initiative is under development at JLAB. The goal of the Qweak experiment is to measure the pro-

Fig. 8. Calculated running of the weak mixing angle in the Standard Model. Data points are from the atomic parity violation experiment on Cs, the NuTeV experiment, the Moeller experiment (E-158) at SLAC and from experiments at the Z0 pole. Also shown are the anticipated error bars for Qweak .

ton‘s weak charge, QpW = 1 − 4 sin2 θW , to the highest precision possible. The SM makes a firm prediction of QpW based on the running of the weak mixing angle sin2 θW from the Z0 pole to low energies, corresponding to a 10 σ efiect in our experiment. Fig. 8 shows the SM prediction for sin2 θW together with existing data and the expected precision for this experiment. This parity violating experiment is in the semi-leptonic sector. This is in contrast to E-158 which is in the purely leptonic sector. The measurement will be carried out using a 1.2 GeV electron beam at a scattering angle of 9◦ and a momentum

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transfer Q2 = 0.03 (GeV/c)2 . The 180 μA polarized beam will be incident on a 35 cm liquid hydrogen target. An eight sector toroidal spectrometer is being constructed for this measurement. The scattered electrons will be detected by quartz Cherenkov detectors operating in integrating mode. A schematic layout of the spectrometer is shown in fig. 7. The measurement will take 2200 hours and will determine the proton‘s weak charge with 4% statistical accuracy. Figure 8 shows the projected quality of the Qweak results in the context of other existing data.

3 Summary Parity-violating high-energy electron scattering is an important probe of nucleon and nuclear structure. Several physics areas are currently under investigation. There have been many experiments investigating the importance of strange quarks to the structure of the proton. Results to date indicate that the contribution of strange quarks to nucleon structure are relatively small or vanishing. A proposed experiment on Pb would measure the neutron radius to 1%. Qweak is a challenging Standard Model test which probes 5 TeV energy scales.

Such demanding measurements have been made possible by important advances in accelerator technology. We now have high intensity CW beams with beam polarization of 85%. Control of helicity correlated beam properties allows measurement of asymmetries to an accuracy approaching 10’s ppb.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

R.D. McKeown, Phys. Lett. B 219, 140 (1989). D. Keyslan, A. Manohar, Nucl. Phys. B 310, 527 (1988). M.J. Musolf et al., Phys. Rep. 239, 1 (1994). D.T. Spangle et al., Phys. Lett. B 583, 79 (2004). K.A. Aniol et al., Phys. Rev. C 69, 065501 (2004). The HAPPEX Collaboration (K.A. Aniol et al.), Phys. Lett. B 635, 275 (2006), nucl-ex/0506011. The HAPPEX Collaboration (K.A. Aniol et al.), Phys. Rev. Lett. 96, 022003 (2006), nucl-ex/0506010. D.S. Armstrong et al., Phys. Rev. Lett. 95, 092001 (2005). T.W. Donnelly et al., Nucl. Phys. A 503, 589 (1989). G. Feinberg, Phys. Rev. D 12, 3575 (1975). J.D. Walker, Nucl. Phys. A 285, 345 (1977). P.A. Souder et al., Phys. Rev. Lett. 65, 694 (1990). C.Y. Prescott et al., Phys. Lett. B 84, 524 (1979). P.L. Anthony et al., Phys. Rev. Lett. 95, 081601 (2005).

Eur. Phys. J. A 28, s01, 107–115 (2006) DOI: 10.1140/epja/i2006-09-012-7

EPJ A direct electronic only

Parity-violating electron scattering at the MAMI facility in Mainz The strangeness contribution to the form factors of the nucleon F.E. Maasa For the A4-Collaboration Institut de Physique Nucl´eaire, CNRS/IN2P3, Universit´e de Paris Sud, 15, rue Georges Clemenceau, F-91406 Orsay CEDEX, France / c Societ` Published online: 29 May 2006 –  a Italiana di Fisica / Springer-Verlag 2006 Abstract. A measurement of the weak form factor of the proton allows a separation of the strangeness contribution to the electromagnetic form factors. The weak form factor is accessed experimentally by the measurement of a parity violating (PV) asymmetry in the scattering of polarized electrons on unpolarized protons. We performed such measurements with the setup of the A4-experiment at the MAMI accelerator facility in Mainz. The role of strangeness in low energy nonperturbative QCD is discussed. The A4experiment is presented as well as the results on the strangeness form factors which have been measured at two Q2 -values. The plans for backward angle measurements at the MAMI facility are presented. PACS. 12.15.-y Electroweak interactions – 11.30.Er Charge conjugation, parity, time reversal, and other discrete symmetries – 13.40.Gp Electromagnetic form factors – 25.30.Bf Elastic electron scattering

1 Introduction In 1965, the discovery of weak interaction violating the symmetry transformation of parity P : x → −x [1,2] has opened a new door in studying the weak force. In contrast electromagnetic and strong interaction conserve parity and therefore parity non-conserving observables are an important tool to uniquely identify weak interaction. Parityviolating correlation observables like pseudo-scalars allow one in electron scattering to access the weak neutral current and separate it from the electromagnetic current and the associated photon exchange. Such an observable is for example the cross section asymmetry in scattering righthanded versus left-handed electrons off an unpolarized target. The first pioneering parity-violation (PV) experiment in electron scattering at high energy at SLAC [3], and at low energies at the MIT Bates accelerator [4] and at the Mainz linac [5] were intended to study the standard model. These experiments were ground breaking in developing techniques for measuring the small (order 10−4 ) asymmetries with order 10−5 errors. Today, parity-violating observables are studied in many areas of physics, from atomic physics to nuclear physics, from hadron physics to high-energy physics [6]. The availability of a polarized high-intensity continuous electron beam at the Mainz Microtron MAMI was important to increase the sensitivity in measuring a parityviolating asymmetry by two orders of magnitude to a level of 0.1 ppm. The standard model and its parameters are a

e-mail: [email protected]

tested to an extent that the weak neutral current of the nucleon can be used to study nucleon structure. An experimental program to measure the strangeness contribution to the form factors at TJLab and at the MAMI facility has emerged. In this contribution to the proceedings we will focus on the experimental activity of the Mainz program and the results achieved with the A4 experiment. The parity program at MIT-Bates and at TJLab will be reviewed in another contribution to these proceedings (see the contribution of S. Kowalski).

2 Strangeness in the nucleon Quantum chromodynamics (QCD) is the quantum field description of strong interaction. Quarks carrying color charge interact via the emission of gluons. The gluon is like the photon a spin-1 particle and it couples like the photon to the vector current of quarks. Similar to Quantum Electrodynamics (QED) an expansion of the interaction in a power series of the coupling constant αs = αQCD is possible. The renormalization of the Lagrangian leads in the case of QED to a running coupling αQED (Q2 ), which rises slowly from the low-energy value of αQED ≈ 1/137 to the Z-mass where it reaches αQED (Q2 ≈ m2Z ) ≈ 1/128 [7]. The gluons carry color charge and this coupling leads to a very much different running of the coupling αQCD (Q2 ), which is depicted in fig. 1. For scales Q2 > 1 (GeV/c)2 , αQCD (Q2 ) becomes small and perturbative treatment of QCD (“perturbative QCD”) is possible [8] with the well

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Fig. 1. Running coupling in QED (αQED ) and QCD (αQCD ) up to one loop. World data suggest that ΛQCD ≈ 250 MeV analyzed in the M S-renormalization scheme using 5 quark flavors [7].

known “asymptotic freedom” at higher scales [9]. An associated energy scale appears at low energies, ΛQCD , at which αQCD (Q2 ) diverges. Any perturbative description breaks down, the theory is here not solvable, this is the regime of nonperturbative QCD. Virtual excitations of pairs of quarks and antiquarks play an important role in the range of nonperturbative QCD. Understanding QCD in this regime is closely linked to the structure of the matter surrounding us like proton and neutron. The successful description of a wide variety of observables by the concept of effective, heavy (≈ 350 MeV) constituent quarks, which are not the current quarks of QCD, is still a puzzle. There are other equivalent descriptions of hadronic matter at low energy scales in terms of effective fields like chiral perturbation theory (χPT) or Skyrme-type soliton models. The effective fields in these models arise dynamically from a sea of virtual gluons and quark-antiquark pairs. In this context the contribution of strange quarks plays a special role since the nucleon has no net strangeness, and any contribution of strange quarks to nucleon structure observables is a pure sea-quark effect and can not be clouded by valence quark effects. Due to the heavier current quark mass of strangeness (ms ) as compared to up (mu ) and down (md ) with ms ≈ 140 MeV  mu , md ≈ 5–10 MeV, one expects a suppression of strangeness effects in the creation of quark-antiquark pairs. On the other hand, the strange quark mass is in the range of the mass scale of QCD (ms ≈ ΛQCD ) so that the dynamic creation of strange sea quark pairs could still be substantial. The consequence of the presence of virtual strangeness quark pairs on different matrix elements has been studied: For example, the strange density of the vacuum ¯ ss is around 20 % suppressed as compared to u- and d-quark condensates: ¯ ss / ¯ q q = 0.8±0.3 [10], which supports the mechanism indicated above. The strange scalar density in ¯ |¯ the nucleon N ss|N is studied in connection with the pion nucleon sigma term. The scalar strangeness content ¯ of the nucleon is defined as y = 2 ¯ p|¯ ss|p / p|¯ uu + dd|p , and a recent evaluation of πN scattering length data yields

a value of y = 0.46 corresponding to a contribution of the strange scalar density to the nucleon mass of 220 MeV [11]. This contribution from scalar density to the nucleon mass is most likely cancelled by other contributions like potential energy and kinetic energy to an extend that the net strangeness contribution to the nucleon mass might be small [12]. The strangeness contribution to unpolarized nucleon structure functions has been determined in deep inelastic neutrino scattering. One obtains the momentum fraction of strange quarks as compared to u- and d-quarks: ¯ κ = x(s(x) + s¯(x)) / x(¯ u(x) + d(x)) ≈ 0.5 [13,14]. This corresponds to the fact, that at a scale of Q2 = 5 (GeV/c)2 the strange sea carries about 3% of the nucleons momentum. In a dynamical QCD evolution model one starts with some unpolarized quark structure function at a scale of μ = 0.3 (GeV/c)2 and evolutes it to the scale of Q2 = 5 (GeV/c)2 where finally the measured parton distributions have been reproduced [15]. In this approach one finds, that the observed value of κ ≈ 0.5 is compatible with a vanishing strange sea contribution at the low scale  μ. Information on the axial charge N |¯ sγμ γ 5 s|N and on the strangeness contribution to the spin of the nucleon comes from the interpretation of deep inelastic scattering data and suggests a sizeable contribution of the strange quarks of Δs(Q2 = 1 (GeV/c)2 ) = −4.5 ± 0.7 to the nucleon spin from a next-to-leading order perturbative QCD analysis of the available world data set including higher twist effects [16].

3 Strangeness contribution to the form factors of the nucleon Parity-violating (PV) electron scattering off nucleons provides experimental access to the strange quark vector current in the nucleon N |¯ sγμ s|N which is parameterized in the electromagnetic form factors of proton and neutron, GsE and GsM [17]. Recently the SAMPLE-, HAPPEX-, A4- and G0-collaborations have published experimental results. A direct separation of electric (GsE ) and magnetic (GsM ) contribution at forward angle has been impossible so far, since the measurements have been taken at different Q2 -values. The experimental details of the SAMPLE, HAPPEX and G0 collaboration are discussed in a different contribution to these proceedings (see the contribution of S. Kowalski). A determination of the weak vector form factors of ˜ p and G ˜ p ) is done by measuring the PV the proton (G E M asymmetry in the scattering of longitudinally polarized electron off unpolarized protons. It allows the determination of the strangeness contribution to the electromagnetic form factors GsE and GsM . The weak vector form ˜p factors G E,M of the proton can be expressed in terms of the known electromagnetic nucleon form factors Gp,n E,M and the unknown strange form factors GsE,M using isospin symmetry and the universality of the quarks in weak and p n ˜p electromagnetic interaction G E,M = (GE,M − GE,M ) − p 2 s 4 sin θW GE,M − GE,M . The interference between weak

F.E. Maas: Parity-violating electron scattering at the MAMI facility in Mainz (A4)

AV arises from the Z0 coupling to the proton vector current and contains the electromagnetic nucleon form factors Gp,n E,M . A possible strangeness contribution to the proton electromagnetic vector form factors has been separated into As . The coupling to the proton axial current is presented by AA and contains the neutral cur˜ p . θe is the scattering anrent weak axial form factor G A gle of the electron in the laboratory and Q2 the negative square of the four momentum transfer. τ = Q2 /(4Mp2 ) and  = [1+2(1+τ ) tan2 (θe /2)]−1 represent purely kinematical factors with Mp the proton mass. Gμ and α represent the Fermi coupling constant as derived from muon decay and the fine structure √ constant respectively. a denotes the factor (Gμ Q2 )/(4πα 2). In order to average A0 = AV + AA over the acceptance of the detector and the target length, we take values for the electromagnetic form factors Gp,n E,M from a parametrization (version 1, page 5) by Friedrich and Walcher [19] and assign an experimental error of 3 % to GpM and GpE , 5 % to GnM , and 10 % to GnE . For evaluating A0 , electromagnetic internal and external radiative corrections to the asymmetry and energy loss due to ionization in the target have been calculated and they reduce the expected asymmetry for our kinematics by 1.3 %. Electro-weak quantum corrections have been applied in the M S renormalization scheme according to [20] and are contained in the factors ρeq , with sˆ2Z = sin2 θˆW (MZ )M S = 0.23120(15) [21]. The electro-weak quantum corrections to AA [22] are applied and absorbed ˜p . in the value of G A The largest contribution to the uncertainty of A0 ˜p , comes from the uncertainty in the axial form factor G A p the electric form factor of the proton GE , and the magnetic form factor of the neutron GnM . For our experimental program at backward angles, the uncertainty stemming ˜ p will be more important in the separation of Gs from G E A and GsM . Using a different target like deuterium will allow ˜ p , Gs and Gs . The parity admixture in the to separate G E M A ground state of deuterium is negligible [23].

4 The A4 experimental setup and analysis The A4 experiment at MAMI has been developed and build in order to measure a small (order ppm) parity-

helicity correlated measurement of: beam current, beam energy beam position beam angle luminosity: monitor system pol. electron source P=80%, I=20μA MAMI E=855MeV ΔE/E=10-6

Moellerpolarimeter (A1)

(Z) and electromagnetic (γ) amplitudes leads to a PV asymmetry ALR (ep) = (σR − σL )/(σR + σL ) in the elastic scattering cross section of right- and left-handed electrons (σR and σL respectively), which is given in the framework of the Standard Model [18]. ALR (ep) is of order parts per million (ppm). The asymmetry can be expressed as a sum of three terms, ALR (ep) = AV + As + AA .   GpE GnE + τ GpM GnM   2 AV = −aρeq (1 − 4ˆ κeq sˆZ ) − , (1) (GpE )2 + τ (GpM )2 Gp Gs + τ GpM GsM As = aρeq E p E2 , (2) (GE ) + τ (GpM )2  √ ˜p (1 − 4ˆ s2Z ) 1 − 2 τ (1 + τ )GpM G A AA = a . (3) p 2 p 2 (GE ) + τ (GM )

Comptonlaser back scatter polarimeter

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Fig. 2. Measurement principle of the A4 experiment. The polarized electrons from the source are accelerated in the MAMI accelerator to a maximum energy of 855 MeV. Scattered electrons from the 10 cm hydrogen target are detected in the homogeneous 1022 channel PbF2 -Cherenkov calorimeter. The sensitive measurement and stabilization of all electron beam parameters is crucial for the sensitivity of the experiment.

violating cross section asymmetry in the scattering of polarized electrons off unpolarized protons. It is complementary to other experiments since for the first time counting techniques have been used in a PV electron scattering experiment. Possible systematic contributions to the experimental asymmetries and the associated uncertainties are of a different nature as compared to previous experiments, which use analogue integrating techniques. Figure 2 shows the measurement principle of the A4 experiment. The polarized 570.4 and 854.3 MeV electrons were produced using a strained layer GaAs crystal that is illuminated with circularly polarized laser light [24]. Average beam polarization was about 80 %. The helicity of the electron beam was selected every 20.08 ms by setting the high voltage of a fast Pockels cell according to a randomly selected pattern of four helicity states, either (+P − P − P + P ) or (−P + P + P − P ). A 20 ms time window enabled the histogramming in all detector channels and an integration circuit in the beam monitoring and luminosity monitoring systems. The exact window length was locked to the power frequency of 50 Hz in the laboratory by a phase locked loop. For normalization, the gate length was measured for each helicity. Between each 20 ms measurement gate, there was an 80 μs time window for the high voltage at the Pockels cell to be changed. The intensity I = 20 μA of the electron current was stabilized to better than δI/I ≈ 10−3 . An additional λ/2-plate in the optical system was used to rotate small remaining linear polarization components and to control the helicity correlated asymmetry in the electron beam current to the level of < 10 ppm in each five minute run. From the source to the target, the electron beam develops fluctuations in beam parameters such as position, energy and intensity which are partly correlated to the reversal of the helicity from +P to −P . We have used a system of microwave resonators in order to monitor beam

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Fig. 3. Floor plan of the MAMI accelerator with the three race track microtrons (RTM). The A4 experiment is located in experimental halls 3 and 4. Feedback stabilization systems for energy, position, angle and current had been developed for the A4 experiment.

current, energy, and position in two sets of monitors separated by a drift space of about 7.21 m in front of the hydrogen target. In addition, we have used a system of 10 feed-back loops in order to stabilize current, energy [25], position, and angle of the beam. Figure 3 shows an outline of the MAMI floor plan with the location of the A4 experiment and the beam monitoring and stabilisation systems. The polarization of the electron beam was measured with an accuracy of 2 % using a Møller polarimeter which is located on a beam line in another experimental hall [26]. Due to the fact that we had to interpolate between the weekly Møller measurements, the uncertainty in the knowledge of the beam polarization increased to 4 %. The 10 cm high-power, high-flow liquid-hydrogen target was optimized to guarantee a high degree of turbulence with a Reynolds-number of R > 2 × 105 in the target cell in order to increase the effective heat transfer. For the first time, a fast modulation of the beam position of the intense CW 20 μA beam could be avoided. It allowed us to stabilize the beam position on the target cell without target density fluctuations arising from boiling. The total thickness of the entrance and exit aluminum windows was 250 μm. The luminosity L was monitored for each helicity state (R, L) during the experiment using eight waterCherenkov detectors (LuMo) that detect scattered particles symmetrically around the electron beam for small scattering angles in the range of θe = 4.4◦ − 10◦ , where the PV asymmetry is negligible. The photomultiplier tube currents of these luminosity detectors were integrated during the 20 ms measurement period by gated integrators and then digitized by customized 16-bit analogue-todigital converters (ADC). The same method was used for

Fig. 4. The electron beam enters from the left and hits the hydrogen target at about 2.2 m above ground. Scattered electrons are detected with the 1022 channel PbF2 -calorimeter, which covers a scattering angle of 30◦ < θe < 40◦ and an azimuthal angle of 360◦ . Part of the detector has been left out in the drawing for better visibility.

all the beam parameter signals. A correction was applied for the nonlinearity of the luminosity monitor photomultiplier tubes. From the beam current helicity pair data I R,L and luminosity monitor helicity pair LR,L data we calculated the target density ρR,L = LR,L /I R,L for the two helicity states independently. To detect the scattered electrons we developed a new type of a very fast, homogeneous, total absorption calorimeter consisting of individual lead fluoride (PbF2 ) crystals [27,28]. Figure 4 shows a CAD-drawing of the calorimeter together with the hydrogen target. The material is a pure Cherenkov radiator and has been chosen for its fast timing characteristics and its radiation hardness. This is the first time this material has been used in a large scale calorimeter for a physics experiment. The crystals are dimensioned so that an electron deposits 96 % of its total energy in an electromagnetic shower extending over a matrix of 3 × 3 crystals. Together with the readout electronics this allows us a measurement of the particle √ energy with a resolution of 3.9 %/ E and a total dead time of 20 ns. For the data taken at 854.3 MeV only 511 out of 1022 channels of the detector and the readout electronics were operational, for the 570.4 MeV data all

F.E. Maas: Parity-violating electron scattering at the MAMI facility in Mainz (A4)

Counts

x 102 1800 1600 1400 1200 1000 800 600 400 200 0 0

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Fig. 6. The dashed histogram shows a raw energy spectrum of accepted particles from the hydrogen target as read directly from the hardware memory of the readout electronics of the lead fluoride calorimeter. For the solid black curve, the raw spectrum has been corrected for the differential nonlinearity of the ADC, i.e. for measured variations of the ADC channel width. The elastic scattering peak position, the π 0 -production threshold and the Δ-resonance position are indicated as well as the lower and upper cut position for the extraction of NeR and NeL as described in the text. Fig. 5. Design drawing of the A4 readout electronics. The upper part contains the analog sum, trigger and veto circuits together with the digitization. In the lower part, the histogramming and the VMEbus access is done. The system is about 3.5 m high.

the 1022 channels were installed. The particle rate within the acceptance of this solid angle was ≈ 50 × 106 s−1 . Due to the short dead time, the losses due to double hits in the calorimeter were 1 % at 20 μA. This low dead time is only possible because of the special readout electronics employed. The signals from each cluster of 9 crystals were summed and integrated for 20 ns in an analogue summing and triggering circuit and digitized by a transient 8-bit ADC. There was one summation, triggering, and digitization circuit per crystal. The energy, helicity, and impact information were stored together in a three dimensional histogram. Neighboring crystals have to go to neighboring electronics channels in the electronics resulting a ring shape. Analogue summation and digitization has been galvanically separated from histogramming and VMEbus access. Figure 5 shows a design drawing of the fast A4 experiment electronics. Figure 6 shows an energy spectrum of scattered particles. The number of elastic scattered electrons is determined from this histogram for each detector channel by integrating the number of events in an interval from 1.6 σE above pion production threshold to 2.0 σE above the elastic peak in each helicity histogram, where σE is the energy resolution for nine crystals. These cuts ensure a clean separation between elastic scattering and pion production or Δ-excitation which has an unknown PV cross section asymmetry. The linearity of the PbF2 detector system with respect to particle counting rates and possible effects

due to dead time were investigated by varying the beam current. We calculate the raw normalized detector asymmetry as Araw = (NeR /ρR − NeL /ρL )/(NeR /ρR + NeL /ρL ). The possible dilution of the measured asymmetry by background originating from the production of π 0 ’s that subsequently decays into two photons where one of the photons carries almost the full energy of an elastic scattered electron was estimated using Monte Carlo simulations to be much less than 1 % and is neglected here. The largest background comes from quasi-elastic scattering at the thin aluminum entrance and exit windows of the target cell. We have measured the aluminum quasi-elastic event rate and calculated in a static approximation a correction factor for the aluminum of 1.030 ± 0.003 giving a smaller value for the corrected asymmetry. Corrections due to false asymmetries arising from helicity correlated changes of beam parameters were applied on a run by run basis. The analysis was based on the five minute runs for which the counted elastic events in the PbF2 detector were combined with the correlated beam parameter and luminosity measurements. In the analysis we applied reasonable cuts in order to exclude runs where the accelerator or parts of the PbF2 detector system were malfunctioning. The analysis is based on a total of 7.3×106 histograms corresponding to 4.8 × 1012 elastic scattering events for the 854.3 MeV data and 4.8·106 histograms corresponding to 2·1013 elastic events for the 570.4 MeV data. For the correction of helicity correlated beam parameter fluctuations we used multi-dimensional linear regression analysis using the data. The regression parameters have been calculated in addition from the geometry of the

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precisely surveyed detector geometry. The two different methods agree very well within statistics. The experimental asymmetry is normalized to the electron beam polarization Pe to extract the physics asymmetry, Aphys = Aexp /Pe . We have taken half of our data with a second λ/2-plate inserted between the laser system and the GaAs crystal. This reverses the polarization of the electron beam and allows a stringent test of the understanding of systematic effects. The effect of the plate can be seen in fig. 7: the observed asymmetry extracted from the different data samples changes sign, which is a clear sign of parity violation if, as in our case, the target is unpolarized. Our measured result for the PV physics asymmetry in the scattering cross section of polarized electrons on unpolarized protons at an average Q2 value of 0.230 (GeV/c)2 is ALR (ep) = (−5.44 ± 0.54stat ± 0.26syst ) ppm for the 854.3 MeV data [29] and ALR (ep) = (−1.36 ± 0.29stat ± 0.13syst ) ppm for the 570.4 MeV data [30]. The first error represents the statistical accuracy, and the second represents the systematical uncertainties including beam polarization. The absolute accuracy of the experiment represents the most accurate measurement of a PV asymmetry in the elastic scattering of longitudinally polarized electrons on unpolarized protons.

Fig. 8. Top: The solid line represents all possible combinations of GsE + 0.225GsM as extracted from the work presented here at a Q2 of 0.230 (GeV/c)2 . The densely hatched region represents the 1-σ uncertainty. The recalculated result from the HAPPEX published asymmetry at Q2 of 0.477 (GeV/c)2 is indicated by the dashed line, the less densely hatched area represents the associated error of the HAPPEX result. Bottom: The solid lines represent the result on GsE + 0.106GsM as extracted from our new data at Q2 = 0.108 (GeV/c)2 presented here. The hatched region represents in all cases the one-σ-uncertainty with statistical and systematic and theory error added in quadrature. The dashed lines represent the result on GsM from the SAMPLE experiment [31]. The dotted lines represent the result of a recent lattice gauge theory calculation for μs [32]. The boxes represent different model calculations and the numbers denote the references.

5 Conclusion From the difference between the measured ALR (ep) and the theoretical prediction in the framework of the Standard Model, A0 , we extract a linear combination of the strange electric and magnetic form factors for the 570.4 MeV data at a Q2 of 0.108 (GeV/c)2 of GsE ± 0.106 GsM = 0.071 ± 0.036. For the data at 854.3 MeV corresponding to a Q2 value of 0.230 (GeV/c)2 we extract GsE + 0.225 GsM = 0.039 ± 0.034. Statistical and systematic error of the measured asymmetry and the error in the theoretical prediction of A0 been added in quadrature. In fig. 8 the results for the 570.4 MeV data are displayed.

F.E. Maas: Parity-violating electron scattering at the MAMI facility in Mainz (A4)

Fig. 9. A compilation of the world PV asymmetry data. The plot shows the difference (Aphys − A0 ) between the published measured PV physics asymmetries Aphys and the asymmetry A0 expected in the standard model assuming no strangeness. A significant difference is a direct sign for strangeness contribution. The blue points have been measured detecting the scattered electron, the red points denote the G0 experiment results where the proton has been detected under forward angles.

A recent very accurate determination of the strangeness contribution to the magnetic moment of the proton μs = GsM (Q2 = 0 (GeV/c)2 ) from lattice gauge theory [32] would yield a larger value of GsE = 0.076 ± 0.036 if the Q2 dependence from 0 to 0.108 (GeV/c)2 is neglected. The theoretical expectations for another quenched lattice gauge theory calculation [33], for SU (3) chiral perturbation theory [34], from a chiral soliton model [35], from a quark model [36], from a Skyrme-type soliton model [37] and from an updated vector meson dominance model fit to the form factors [38] are included in fig. 8. Recently the HAPPEX- and the G0-collaboration published measurements which are reviewed in a different contribution to these proceedings. Figure 9 shows a compilation of the world data in the Q2 -range up to 0.5 (GeV/c)2 . The plot shows the difference (Aphys − A0 ) between the published measured PV physics asymmetries Aphys and the asymmetry A0 expected in the standard model assuming no strangeness. A significant difference is a direct sign for strangeness contribution. The blue points have been measured detecting the scattered electron, the red points denote the G0 experiment results where the proton has been detected under forward angles. Combining the HAPPEX data at Q2 = 0.1 (GeV/c)2 and the extrapolated G0 data and with our previous data at this Q2 value gives further constraints on GsE and GsM . Figure 10 shows the present status of the world data at Q2 = 0.1 (GeV/c)2 in the GsE versus sGsM plane. The SAMPLE data have been measured at backward angle. A4, HAPPEX and G0 at forward angle. The HAPPEX He data have been measured using a 4 He target, which has no magnetic form factor due to the fact that the nucleus has spin 0. One can extract a value for both GsE and GsM separately: GsM = 0.62 ± 0.31 and GsE = −0.012 ± 0.029. While the combined value for GsE has a sensitivity which

113

Fig. 10. The combination of all available data at forward and backward angles at from H and He target constrains the possible values for GsE and GsM . The ellipse gives the two sigma contour plot.

is now reaching the accuracy of the electromagnetic form factors, the error on GsM is still very large.

6 Perspectives Further constrains on GsM can be expected from further measurements combining H and He data at forward angles, as planned by the HAPPEX collaboration. A different approach is to combine H and D data at backward angles. We are preparing a series of measurements of the parity-violating asymmetry in the scattering of longitudinally polarized electrons off unpolarized protons and deuterons under backward scattering angles of 140◦ < θe < 150◦ with the A4 apparatus at a Q2 of 0.23 (GeV/c)2 in order to separate the electric (GsE )and magnetic (GsM ) strangeness contribution to the electromagnetic form factors of the nucleon. We have changed the experimental setup. The detector has been put on a rotatable platform, construction work and reinstallation of detector and 2028 cables in the experimental hall had been finished by April 2005. We have found a large background stemming from photons coming from π 0 -decay. The energy range of those photons cover partly the same energy interval as the energy range of the elastic scattered electrons. Our electromagnetic shower calorimeter has the same response function for electrons as for photons. π 0 -production has its own unknown parity-violating asymmetry. In order to avoid pollution of our elastic signal with photons from π 0 decay, we have installed an electron tagging system of 72 additional plastic scintillators between scattering chamber and PbF2 -crystals. Figure 11 shows a schematic of the

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Fig. 11. The drawing shows a cut through the upper part of the A4 detector system. In the upper part, one sees the aluminum frame with seven PbF2 -crystals. The scintillator system for detecting electrons in coincidence with the calorimeter is located between the PbF2 -crystals and the vacuum chamber. Two concentrical rings contain 36 scintillator each. The electron rates are low enmough, so that one scintillator can cover 14 PbF2 -crystals, i.e. two aluminum frames.

setup. The scintillators are arranged in two rings symmetrically around the scattering chamber. Only charged particle traversing the scintillator produce an output signal. If a scintillator produces a signal, it is converted to a logical level which then serves for the short coincidence time of 25 ns as an additional bit in the histogramming circuit of the readout electronics. If the bit is set, and in addition a signal arrives from the PbF2 -calorimeter, the event is histogrammed into one region of the histogramming memory. If there is an event in the calorimeter without tagging bit from the scintillators, it is histogrammed into a different memory address range. Figure 12 shows 7 typical PbFs calorimeter spectra taken with the electron tagging scintillator system. The blue histograms show electron spectra where the additional scintillator system had given a coinci-

Run 28256: Chan 918 pol1 ProjX

1.99541e+01 EntriesMean Normal: 3.17202e+06

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EntriesIntegral Coinc: 7.18291e+05 1.58541e+06

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1.85214e+01 EntriesMean Normal: 2.72850e+06

Run 28256: Chan 921 pol1 ProjX

EntriesIntegral Coinc: 5.12944e+05 1.36421e+06

----------------------norm pol1 1 raw Kurtosis -6.04778e-01

----------------------norm pol1 1 raw Kurtosis -5.11082e-01

Beam conditions: T=300 s, I=19.9 muA norm pol0 1 raw

Expected elastic: 3.56634e+05

Beam conditions: norm pol0 1 T=300 raw s, I=19.9 muA

Expected elastic: 3.22953e+05

Expected elastic: 2.98658e+05

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1.93011e+01 EntriesMean Normal: 2.02822e+06

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EntriesIntegral Coinc: 3.30970e+05 7.12172e+05 ----------------------norm pol1 1 raw Kurtosis -2.00809e-01

Beam conditions: norm pol0 1 T=300 raw s, I=19.9 muA Expected elastic: 2.48168e+05

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Expected elastic: 2.72263e+05

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This work has been supported by the DFG in the framework of the SFB 201 and SPP 1034. We are indebted to K.H. Kaiser and the whole MAMI crew for their tireless effort to provide us with good electron beam. We are also indebted to the A1Collaboration for the use of the Møller polarimeter. My deep thanks got to H. Arenh¨ ovel, H. Backe, D. Drechsel, J. Friedrich, K.H. Kaiser and Th. Walcher. During the last ten years at the Institut f¨ ur Kernphysik it had been a pleasure to work with them. I want to thank them not only for their continuous encouragement but also for many stimulating discussions and also for keeping a lifely, scientific spirit at the institute.

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1.94098e+01 EntriesMean Normal: 2.60810e+06 EntriesIntegral Coinc: 5.24322e+05 1.30461e+06

----------------------norm pol1 1 raw Kurtosis -5.92825e-01

Beam conditions: norm pol0 1 T=300 raw s, I=19.9 muA

Expected elastic: 3.91186e+05

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1.95670e+01 EntriesMean Normal: 2.89210e+06 EntriesIntegral Coinc: 5.80886e+05 1.44674e+06

----------------------norm pol1 1 raw Kurtosis -2.86987e-01 Beam conditions: norm pol0 1 T=300 raw s, I=19.9 muA

dence signal. They exhibit a clear elastic peak with a good signal-to-background ratio. The count rate under the elastic peak corresponds to 80 %–90 % of the expected elastic count rate. At forward angles, this ratio had been 70 %. We expect at backward angles a higher efficiency since the radiative tail is less pronounced at the beam energy of 315.13 MeV. The black and red histograms (only partly shown) correspond to the large background from photons from π 0 -decay. For these signals the scintillator system had no coincident signal. The sum of both spectra would give the total spectrum. The system will be commissioned in December 2005. Data taking will start in January 2006. We plan to accumulate 1000 h with hydrogen target corresponding to an error in GsM of ±0.13, corresponding to a factor of three over the SAMPLE result and a factor of two improvement over the present combined world data at Q2 = 0.1 (GeV/c)2 . In order to improve the understanding on the systematical uncertainty coming from the axial form factor, we will also collect the same amount of statistics with a deuterium target.

0

20

40

60

80

100

Fig. 12. The blue histograms show electron spectra where the additional scintillator system had given a coincidence signal. They exhibit a clear elastic peak with a good signal-tobackground ratio. The black and red histograms (only partly shown) correspond to the large background from photons from π 0 -decay. Both spectra are taken always in parallel for better control of systematic effects.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

T.D. Lee, C.N. Yang, Phys. Rev. 104, 254 (1956). C.S. Wu et al., Phys. Rev. 105, 1413 (1957). T.R. Prescott et al., Phys. Lett. B 84, 524 (1979). P.A. Souder et al., Phys. Rev. Lett. 65, 694 (1990). J. Ahrens et al., Bucl. Phys. A 446, 377c (1985). S. Kox, D. Lhuillier, F. Maas, J. Van de Wiele (Editors), From Parity Violation to Hadronic Structure and More, Proceedings of PAVI2004, Eur. Phys. J. A 24, s02 (2005). T.R. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model, first paperback edition (with corrections) (Cambridge University Press, Cambridge, 1992). H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973). D.J. Gross, F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973). M. Jamin, Phys. Lett. B 538, 71 (2002). M.M. Pavan et al., PiN Newslett. 16, 110 (2002). X.-D. Ji, Phys. Rev. Lett. 74, 1071 (1995). M. Goncharov et al., Phys. Rev. D 64, 112006 (2001). M. Tzanov et al., Nutev neutrino dis., hep-ex/0306035, 2003. M. Gl¨ uck, E. Reya, A. Vogt. Eur. Phys. J. C 5, 461 (1998). E. Leader et al., Phys. Rev. D 67, 074017 (2003). D.B. Kaplan et al., Nucl. Phys. B 310, 527 (1988). M.J. Musolf et al., Phys. Rep. 239, 1 (1994). J. Friedrich, Th. Walcher, Eur. Phys. J. A 17, 607 (2003). W.J. Marciano, A. Sirlin, Phys. Rev. D 29, 75 (1984).

F.E. Maas: Parity-violating electron scattering at the MAMI facility in Mainz (A4) 21. S. Eidelmann et al., Review of particle properties, Phys. Lett. B 592, 1 (2004). 22. S.-L. Zhu et al., Phys. Rev. D 62, 033008 (2000). 23. G. K¨ uster, H. Arenh¨ ovel, Nucl. Phys. A 626, 911 (1997). 24. K. Aulenbacher et al., Nucl. Instrum. Methods A 391, 498 (1997). 25. M. Seidl et al., High precision beam energy stabilisation of the Mainz microtron MAMI, in Proceedings of the EPAC 2000 (2000) p. 1930. 26. P. Bartsch, Aufbau eines Møeller-Polarimeters f¨ ur die Drei-Spektrometer-Anlage und Messung der Helizit¨ atsasymmetrie in der Reaktion p(e,e p)π 0 im Bereich der Δ-Resonanz, Dissertation Mainz, 2001. 27. F.E. Maas et al., Proceedings of the ICATPP-7 (World Scientific, 2002) p. 758.

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28. P. Achenbach et al., Nucl. Instrum. Methods A 465, 318 (2001). 29. F.E. Maas et al., Phys. Rev. Lett. 93, 022002 (2004). 30. F.E. Maas et al., Evidence for strange quark contributions to the nucleon’s form-factors at q 2 = 0.108 (Gev/c)2 , nuclex/0412030, 2004. 31. D.T. Spayde et al., Phys. Lett. B 583, 79 (2004). 32. D.B. Leinweber et al., Phys. Rev. Lett. 94, 212001 (2005). 33. R. Lewis et al., Phys. Rev. D 67, 013003 (2003). 34. T.R. Hemmert et al., Phys. Rev. C 60, 045501 (1999). 35. A. Silva et al., Eur. Phys. J. A 22, 481 (2004). 36. V. Lyubovitskij et al., Phys. Rev. C 66, 055204 (2002). 37. H. Weigel et al., Phys. Lett. B 353, 20 (1995). 38. H.-W. Hammer et al., Phys. Rev. C 60, 045204 (1999).

Eur. Phys. J. A 28, s01, 117 127 (2006) DOI: 10.1140/epja/i2006-09-013-6

EPJ A direct electronic only

Virtual Compton Scattering at MAMI N. d’Hosea CEA-Saclay, DAPNIA/SPhN, F91191 Gif-sur-Yvette Cedex, France / Published online: 17 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Virtual Compton Scattering (VCS) ofi the proton is a recent fleld of investigation of the nucleon structure. VCS at threshold gives access to the Generalized Polarizabilities (GPs) of the proton. The qualities of both the beam and the high-resolution spectrometers available at the Mainz Microtron MAMI allowed us to perform at flrst such delicate experiments. This paper deals with difierent experiments dedicated to the GPs measurements. They are realized without and with polarization, below and just above pion threshold. PACS. 13.60.Fz Elastic and Compton scattering production reactions

1 Virtual Compton Scattering and polarizabilities One of the main challenges of hadronic physics in the regime of strong (non-perturbative) QCD is to identify the relevant degrees of freedom of the nucleon. Though the small distance structure is rather well described by point-like quarks and gluons, its structure at larger distance is not so well understood. There exist many models ranging from constituent quark models to chiral models. Polarizabilities are one of the fundamental observables to describe the internal structure of the nucleon and they have been investigated with real Compton scattering (RCS) since the early 1950s. As the light scattering on atmospheric atoms which gives the well known Rayleigh efiect for blue skies and red sunsets through oscillation of the electrons inside the atoms, real Compton scattering sheds light on the nucleon structure. This is clearly illustrated in a common deflnition of the electric polarizability αE in a non relativistic approach at the flrst-order perturbation for an applied electric dipole moment D: αE = 2ΣN  =

N

| N  |Dz |N |2 . EN  − E N

In this formula N  indicates each nucleon resonance. The polarizability is then sensitive to all the excitation spectrum of the nucleon (even if the low energy of the perturbating photon does not allow the real formation of the nucleon resonances). The Mainz laboratory has a long tradition in this fleld. Several experiments have been dedicated to the determination of proton, neutron or pion polarizabilities. Today a

e-mail: [email protected]

14.20.Dh Protons and neutrons

25.30.Rw Electro-

the world global average of the electric (αE ) and magnetic (βM ) polarizabilities on the proton is based on an experimental study investigated at MAMI with the tagged real photon beam [1]. αE =(11.9 ± 0.5(stat.) ∓ 1.3(syst.) ± 0.3(mod.))10−4 fm3 , βM =(1.2 ± 0.7(stat.) ± 0.3(syst.) ± 0.4(mod.))10−4 fm3 . We can note the small size of the polarizabilities which reveals the feature that the nucleon is strongly bound. For comparison the electric polarizability of the hydrogen atom is of the order of the atomic volume, and the electric polarizability of the proton αE is only 0.05 per cent of its volume. Furthermore the magnetic polarizability is still smaller, one tenth of the electric polarizability. Virtual Compton scattering (VCS) ofi the proton refers to the reaction γ ∗ p → pγ, where γ ∗ stands for an incoming virtual photon of four-momentum squared Q2 . This reaction is experimentally accessed through photon electroproduction ep → epγ. The corresponding Feynman diagram is indicated in flg. 1. In the 1960s the VCS appeared as a rather unwanted contribution to radiative corrections to electron scattering on a proton [2]. It was mentioned as proton Bremsstrahlung. In 1974 Arenh˜ ovel and Drechsel [3], from the Institut f˜ ur Kernphysik at Mainz, considered the VCS for the flrst time as a good way to measure generalized polarizabilities (GPs). Only in 1995 with the new generation of facilities of high duty cycle to investigate exclusive reactions, and with new theoretical concepts it regained interest. The general theoretical framework for VCS at threshold has been extensively described by Guichon et al. [4,5] and the Mainz theoretical group conducted by Drechsel [6, 7]. VCS reaction at threshold means that the produced

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e’

e

γ* ( q ) p

q’ small

γ ( q’ )

N , N* , Δ , ...

p’

Fig. 1. The VCS graph for the proton.

photon has a small enough momentum or that its electric (E) and magnetic (M) flelds look constant over the size of the nucleon. In the following the three-momenta absolute values of the virtual and real photons in the photon-proton center of mass (c.m.) system are noted q and q , respectively, and vary independently, this is in contrast with real Compton scattering where q = q . Here the low momentum q of the produced real photon deflnes the size of the electromagnetic (EM) perturbation, while the momentum of the virtual photon q (or the four-momentum squared Q2 ) sets the scale of the observation of the nucleon internal structure. In the low momentum regime the reaction can be interpreted as electron scattering on a nucleon placed in a quasi-constant applied EM fleld [5]. The induced motion of the nucleon as a whole can be eliminated thanks to a low-energy theorem [8], so one is left with the deformation, due to the applied fleld, of the nucleon internal currents δJ μ (r) and the electron scattering measures its Fourier transform δJ μ (Q). To lowest order in αQED , δJ μ (Q) is linear in the applied fleld and the 6 coe– cients of proportionality are the GPs [4,5,6,7]. When Q2 = 0 two of them reduce to the usual polarizabilities αE and βM measured in real Compton scattering. Analogously to the form factors for elastic scattering, which describes the charge and magnetization distributions, VCS gives access to the deformation of these distributions by an external EM fleld, and will yield valuable information about the non perturbative structure of the nucleon. This can be illustrated by a very naive picture of the polarizabilities which are the results of an electromagnetic perturbation applied to the nucleon components. An electric fleld moves positive and negative charges inside the proton in opposite directions. The induced electric dipole moment is proportional to the electric fleld, and the proportionality coe– cient is the electric polarizability αE which measures the rigidity of the proton. A magnetic fleld acts difierently on the quarks and the pion cloud. The quarks (of spin 1/2) align their magnetic moment parallel to the magnetic fleld giving the strong magnetic excitation of the Δ(1232) resonance. The pions are at low energy, an essential element of the structure of the nucleon notably at its surface giving the famous representation of a pion cloud surrounding the nucleon. The pions (of spin 0) distributed at the surface of the proton, will generate eddy currents.

When the magnetic fleld is applied, they are modifled in such a way that the induced magnetic moment is antiparallel to the magnetic fleld (Lenz law). Quarks and pions thus give rise to two difierent contributions: para and diamagnetic or resonant and non resonant contributions to the magnetic polarizability βM . At present the GPs have been calculated in the framework of various theoretical models [9,10,11,12,13,14,15] yielding quite difierent results with regard to both their absolute value and their Q2 dependence. Figure 2 presents difierent theoretical predictions as functions of Q2 : – Non relativistic constituent quark model (CQM) is based on the assumption that baryons are composed of three massive quarks moving within a harmonic oscillator conflning potential and additional hyperflne interactions. One of its success is to explain most of the observed nucleon resonance mass spectrum. Calculations have been performed in this framework by Guichon, Liu and Thomas [4,9] and Pasquini, Scherer and Drechsel [10]. – Phenomenological approach can be realized with an efiective Lagrangian model (ELM). Such a calculation has been performed by Vanderhaeghen [11] which includes the efiects of all the flrst nucleon resonances and π 0 exchange in the t channel. These two flrst kinds of model describe well all the resonant contributions, but not the non-resonant one. Their limitation is that they have no relationship to chiral symmetry. This is an important property of QCD which governs much of low-energy hadron physics. The pion is the Goldstone boson of spontaneously broken chiral symmetry, and plays a very special and major role at low energy. The two next groups of calculations respect chiral symmetry. – A simple model to describe interaction of Dirac particles with a chiral fleld is the linear sigma model (LSM) in the limit of an inflnite sigma mass. Though this model is not a very realistic description of the nucleon, nevertheless it fulfllls all the relevant symmetries like Lorentz, gauge and chiral invariance. A complete calculation of all the one-loop diagram contributions (for the photon interaction with a nucleon-pion system) has been performed by Metz and Drechsel [12]. – Chiral perturbation theory (ChPT) is a very systematic and consistent approach with a most general Lagrangian based on QCD symmetries. Heavy-baryon chiral perturbation theory allows for a systematic perturbative expansion in powers of small parameters (no- ted p) as quark masses, inverse of hadron masses or external momenta. Hemmert, Holstein, Kn˜ ocklein and Scherer [13] have performed a third order O(p3 ) calculation for all the GPs while Kao and Vanderheaghen [16] have performed a fourth order O(p4 ) calculation but only for the spin polarizabilities which exclude predictions for αE and βM . Nevertheless previous calculations for αE (Q2 ) and βM (Q2 ) have been realized at Q2 = 0 by Bernard, Kaiser, Schmidt and Meissner [17] including all terms to order O(p4 )

N. d’Hose: Virtual Compton Scattering at MAMI Table 1. prediction at Q2 = 0 in the heavy baryon chiral perturbation theory. Calculation at O(p3 ) [13] αE = 12.5 × 10 4 fm3 βM = 1.25 × 10 4 fm3

119

k’

k

q’

=

Calculation at O(p4 ) [17] αE = 10.5 × 10 4 fm3 βM = 3.5 × 10 4 fm3

p

p’

k’ k

6 5 4 3 2 1 0 -1 -2

3 -4

0

0.2

0.4 0.6 0.8 2 2 Q (GeV )

q’

+

k’ k’

+

k

q’ q

βM (10 fm )

-4

3

αE (10 fm )

14 12 10 8 6 4 2 0

q’ k

p

p’ (a)

0

0.2

0.4 0.6 0.8 2 2 Q (GeV )

Fig. 2. Evolution of the electric and magnetic polarizabilities with Q2 . Experimental results [1] at Q2 = 0 (with only statistical errors) and flve theoretical predictions CQM [10], ELM [11], LSM [12], ChPT [13] and DR [15] are reported. See the text for comments.

and also Δ(1232)-resonance contribution via counter terms. They found that in the case of the magnetic polarizability, a large positive contribution from the Δ(1232)-resonance is largely canceled by a negative pionic contribution, which gives a rather small resulting value. The results at Q2 = 0 are reported in table 1. Another consistent and unifled approach for RCS and VCS has been given by the Mainz theoretical group through the dispersion relation formalism (DR) [14,15] which connects the low energy nucleon structure quantities as polarizabilities to the nucleon excitation spectrum. A more detailed description will be given in the next section as this formalism is also used to extract the GPs. In flg. 2 we observe a relatively sharp fall-ofi of the electric polarizability with increasing momentum transfer Q2 , while we can remark for the HBChPT, DR and ELM models a rise of the magnetic polarizability at very low transfer and then a decrease at larger transfer. This remarkable efiect has its origin in the dominance of diamagnetism caused by the pion cloud at long distance and the dominance of paramagnetism due to a quark core at short distance. It is thus clear that the GPs are sensitive to the respective role of quark and pion degrees of freedom and as such they are very valuable new observables to compare theory with experiment.

2 The unpolarized experiments at threshold A pioneer VCS experiment has been realized at MAMI at Q2 = 0.33 GeV2 [18] during the years 1995-6-7, and then two experiments have been performed in two other

p

p’ (b)

p

p’ (c)

Fig. 3. The p(e, e p)γ reaction. The initial, flnal electron and initial, flnal proton quadri-momenta are k, k and p, p respectively. The flnal photon quadri-momentum is q . In the one photon exchange approximation, a) and b) correspond to the Bethe-Heitler (BH) process. c) corresponds to the Virtual Compton Scattering (VCS) process. We note q the quadrimomentum of the virtual photon exchanged in the VCS process, that is q = k − k (Q2 = −q 2 ).

complementary kinematical regimes at Jefierson Lab. (Q2 = 1 and 2 GeV2 ) [19] in 1998 and MIT-Bates (Q2 = 0.05 GeV2 ) [20] in 2000. They are long and delicate experiments and they rely on a careful analysis of the data. 2.1 Theoretical framework As it was mentioned the general theoretical framework for VCS is extensively presented in the following references [4, 5] and [6,7]. Only the relevant points for the analysis of these experiments will be discussed here. In the reaction ep → epγ, the flnal photon can be emitted either by the electrons, referring to the Bethe-Heitler (BH) process, or by the proton, giving access to the VCS process (see flg. 3). The BH process dominates and interferes strongly with the VCS process. The amplitude is the sum of the BH, Born and Non-Born amplitudes. The two last-named refer to the proton radiation: the Born amplitude depends only on the static properties of the proton (charge, mass) and elastic form factors, while the Non-Born amplitude contains dynamical internal structure information in terms of generalized polarizabilities. The difierential cross section has the form d5 σ exp = (1) dklab [dΩe ]lab [dΩp ]CM   (2π)−5 klab q √ × M ≡ φq M, 32m klab s wherein klab , klab are the moduli of the incoming and outgoing electron momentum, respectively. The relevant kinematical variables of the problem are q and q previously deflned;  the virtual photon polarization; θ and ϕ the two spherical angles indicating the CM real photon direction on a globe with the virtual photon as a pole. θ is the CM angle between the real and virtual photons while ϕ represents the angle between the two electron plane and the photon-proton plane. In the precedent formula φ stands

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The European Physical Journal A

for a phase space factor. M is the coherent sum of the difierent amplitudes: 2 1   BH M= + T V CS  , (2) T 4 spin =

2 1   BH T + T Born + T N onBorn  . 4 spin

EM transition

T N onBorn = b1 (q, , θ, ϕ)q + O(q2 ).

(4)

Consequently in the low energy limit of the flnal photon, the cross section is independent of the dynamical nucleon structure [8], and can be evaluated using only the known BH and Born amplitudes. This can be summarized by the following equation: d5 σ exp (q, q , , θ, ϕ) = d5 σ BH+Born (q, q , , θ, ϕ)

(5)

2

+ φq Ψ0 (q, , θ, ϕ) + φO(q ) where d5 σ is a notation for the difierential cross section d5 σ/dklab [dΩe ]lab [dΩp ]CM . Ψ0 (q, , θ, ϕ) is the leading term in the expansion in powers of the real photon momentum q . It corresponds to the interference between −1 the term of order q in the BH+Born amplitude and the 1 leading order term of order q in the Non-Born amplitude. It contains the dynamical internal structure information of the proton, parametrized by 6 generalized polarizabilities given for the electric and magnetic dipole radiation of the outgoing real photon. We note L(L ) the initial (flnal) photon orbital angular momentum, ρ(ρ ) the type of multipole transition (0 for Coulomb, 1 for Magnetic, 2 for Electric), and S the type of the transition at the nucleon side (non-spin-flip S = 0 and spin-flip S = 1). Assuming that the emitted real photon has low energy, we may use the dipole approximation (L = 1). For a dipole transition in the flnal state, parity and angular momentum conservations lead to 10 GPs presented in table 2. Crossing symmetry and charge conjugation invariance provide 4 relations between the 10 GPs and we are left with 6 independent GPs: 2 scalar (S = 0) and 4 spin-dependent (S = 1) polarizabilities, functions of q (or equivalently Q2 = Q2 |q =0 . See footnote 1 ). The choice of 6 GPs is a arbitrary, and can be realized for example by the 6 surrounded GPs in table 2. In an unpolarized measurement, Ψ0 (q, , θ, ϕ) can be written as Ψ0 (q, , θ, ϕ) = v1 (θ, ϕ, q)(PLL (q) − PTT (q)/) + v2 (θ, ϕ, q)PLT (q) Q2 = Q2 |q =0 = 2m · ( the proton mass. 1



VCS GPs

RCS polarizabilities Q2

The low energy theorem (LET) from Low [8] states that in an expansion in powers of the real photon energy q (but flxed arbitrary q), the flrst term of the amplitudes −1 (well-known infraT BH and T Born is of the order q red divergence), while the flrst term of T N onBorn is of the 1 order q : b−1 (q, , θ, ϕ) T BH + T Born = + O(q0 ), (3) q



Table 2. List of the 10 GPs with the corresponding electromagnetic transitions. Their relation with the polarizabilities obtained in real Compton scattering are indicated. 6 GPs are independent. Our choice is a priori arbitrary, and is realized by the 6 surrounded GPs.

(6)

m2 + q2 − m) where m stands for

M(C0

M 1)S=1

P (11,00)1

→

M

(C2

M 1)S=1

P

(11,02)1

→

(M 1

M 1)S=0

P

(11,11)0

→

(11,11)1

P P (11,2)1

→ →

M

M M(C2,E2

M 1)S=1

M(C1

E1)S=1

P (01,01)1

→

M

E1)S=0

(01,01)0

→

P (01,12)1 P (01,1)1 P (01,1)0

→ → →

(M 1

(C1

M 1)S=1

P

M(M 2 E1)S=1 M(C1,E1 E1)S=1 M(C1,E1 E1)S=0

0

−



0 8 4π (γ2 27 e2 

−

+ γ4 )

8 4π (βM ) 3 e2

0 = 0

−

 0

−

2 3 2 3

4π (αE ) e2 4π (γ3 ) e2

= 0 0 =

where v1 (θ, ϕ, q), v2 (θ, ϕ, q) are known kinematical factors. PLL (q), PTT (q), PLT (q) are structure functions related to the GPs(q) with some kinematical factors: √ (7) PLL = −2 6mGE P (01,01)0 !√ " 2 (01,12)1 (11,11)1 PTT = 3GM q 2P −P /q0 # 3 mq PLT = GE P (11,11)0 2 Q √   3Q q2 (11,02)1 (11,00)1 √ GM P P + + 2 q 2 where m stands for the proton mass, GE and GM denote the form factors evaluated at Q2 and q0 is the CM virtual photon energy at q = 0. (See footnote 2 .) The two structure functions PLL (q) − PTT (q)/ and PLT measured in an unpolarized VCS experiment are the sum of two contributions: one coming from the scalar or spin-independent polarizabilities and another one coming from the spin-dependent polarizabilities (which vanishes at Q2 = 0). 2.2 The MAMI experiment With the high luminosity and high duty cycle provided by the 855 MeV Mainz Microtron MAMI it was possible to investigate the measurement of the small photon electroproduction cross section at threshold. Absolute cross sections d5 σ exp [18] have been measured at Q2 = 0.33 GeV2 using the three-spectrometer facility [21] of the A1 collaboration at MAMI (see flg. 4). The scattered electron and the recoiling proton were detected in coincidence with two of the high-resolution magnetic spectrometers. The photon production process was selected by a cut on the missing mass 2

q0 = m −



m2 + q 2 .

N. d’Hose: Virtual Compton Scattering at MAMI

121

0.7 0.6 0.5 0.4 0.3 0.2 -100

0

-100

-100

0

-100

0

0.2

0.1 0.09 0.08 0.07 0.06 -100

0

0

Θγγ (deg)

Fig. 4. The 3 high-resolution spectrometer facility of the A1 collaboration at MAMI.

1600 1400 1200 1000 800 600 400 200 0 -5000

0

5000

10000

15000

20000 25000 2 2 Mx (MeV )

Fig. 5. Missing mass spectrum obtained for the setup at q = 111.5 MeV/c.

around zero, which was possible thanks to the excellent resolution of the facility (momentum resolution of 10−4 and angular resolution better than 3 mrad) (see flg. 5). The aim of this flrst VCS experiment below pion threshold was to measure the flve-fold difierential cross sections in a wide photon angular range, at 5 values of the photon momentum q : 33.6, 45, 67.5, 90, and 111.5 MeV/c (presentation in flg. 6). The 3 other kinematical variables were held flxed, namely the virtual photon momentum, q = 600 MeV/c (Q2 = 0.33 GeV2 ), the virtual photon polarization  = 0.62. The out-of-plane angle ϕ range is

Fig. 6. Difierential cross sections for the reaction ep → epγ as a function of θ for flxed q, , ϕ and for flve values of the real photon momentum q . The known part of the cross section d5 σ BH+Born , is presented by the solid lines. The experimental data points d5 σ exp deviate from the solid lines as q increases the efiect of the proton polarizabilities. The dotted lines represent the expected cross sections with the efiect of the polarizabilities measured by the two structure functions deduced from this experiment.

determined by the acceptance of the two spectrometers around 0◦ and 180◦ . The spherical angles θ and ϕ are deflned such that ϕ = 0◦ corresponds to the half plane containing the electron momenta. To ease the presentation the data are plotted with θ ranging from −180◦ to +180◦ ; the negative values corresponding in fact to ϕ = 180◦ . The wide range of θ from −141◦ to +6◦ covers the backward direction relative to the incoming and outgoing electrons. Here, the VCS contributions are dominant because the electron radiations (BH) are emitted predominantly in the electron directions. The cross sections d5 σ BH+Born are presented by the solid lines in flg. 6. At small photon momentum q = 33.6 MeV/c the agreement between the radiatively corrected data and d5 σ BH+Born is excellent, and the deviation from this known cross section increases when q increases, as expected from the efiect of the proton polarizabilities. In order to determine accurately the polarizabilities, a careful analysis of possible systematic errors on the deviation is of particular importance. First the BH and Born contributions rely on the knowledge of the proton form factors. Consequently we also measured the absolute elastic scattering cross section for each kinematic setting of the VCS experiment. These measurements validate the use of the form factor parametrization from H˜ ohler [22] at a precision better than ±1%. Second the radiative corrections, which are of the order of 20% of the cross section, have been evaluated by Vanderhaeghen et al. [23]. The systematic uncertainties are estimated to equal ±2% for the calculation performed to order α4 in the VCS cross section. Third the luminosity and the detector e– ciencies are controlled within an accuracy

5

The European Physical Journal A 5 BH+Bo

(d σ-d σ

0.5 0.25 0 -0.25 0.5 -0.5 0.25 0 -0.25 0.5 -0.5 0.25 0 -0.25 -0.5

,

,

-2

)/q = Ψ0 + q Ψ1 + ... (in GeV )

0.5 0.25 0 -0.25 -0.5

Ψ0 /v2 (GeV-2)

122

10 7.5 5 2.5 0 -2.5 -5 -7.5 -10 -0.1

0

0.1

0.2

0.3

0.4 0.5 v1/v2 (Θγγ)

Fig. 8. Compilation of the complete data set (for the 14 angles) of Ψ0 /v2 as a function of v1 /v2 . The data are reasonably well aligned; the errors indicated are statistical only. This allows to extract the two structure functions PLL − PTT / and PLT with statistical errors and the χ2 given. ,

q (MeV/c) Fig. 7. (d5 σ − d5 σ BH+Born )/φq studied as a function of the real photon momentum q for the 14 measured scattering angles θ. The intercept at origin is Ψ0 . In a flrst method it is determined at each scattering angle θ by the mean value in the investigated real photon momentum range (solid line). The dash-dotted, dashed and dotted lines show evolutions in the framework of the 2nd, 3rd and 4th methods, respectively.

of ±1%, the solid angles are determined within an accuracy of ±2% using a Monte Carlo [24] simulation which reproduces perfectly the missing mass spectra. All these uncertainties are constant over the angular range of the real photon and are controlled by the fairly good agreement between the radiatively corrected data and the predicted BH and Born cross section at small q . However small imperfections in the spectrometer optic calibration which could provide distortion of the angular distributions are estimated to give a variation of cross section of ±2.5%. Figure 7 shows the behavior of (d5 σ − d5 σ BH+Born )/  φq as a function of the real photon momentum q for the 14 measured scattering angles θ. The goal is to determine the intercept at origin (noted Ψ0 in eq. (5)), and this flgure illustrates the basic di– culty of this experiment that is the increase of the statistical errors when q decreases. Four methods are then considered in the following. First method based on the LET: As is apparent in flg. 7, there is no strong evolution with the real photon momentum. Therefore we make the hypothesis that there is no q dependence in (d5 σ − d5 σ BH+Born )/φq . Ψ0 is then determined at each scattering angle θ by the mean value of the data at the 5 photon energies.

Figure 8 presents the complete data set (for the 14 angles) of Ψ0 /v2 as a function of v1 /v2 (cf. eq. (6)). The data are reasonably well aligned, which suggests that the higher-order terms in the expansion of the cross section (cf. eq. (5)) are not so important. This good alignment for a wide angular range is also indicative of the consistency of the experimental data. We extract the two structure functions PLL − PTT / and PLT as the slope and intercept of a linear flt to the data (according to eq. (6)) [18]. Second method: We make the hypothesis of a linear evolution with the real photon momentum for each angle which is fltted to the data. The result is indicated by the dash-dotted line in flg. 7. Third method: The q evolution is supposed to be governed by the interference between the complete BH and Born amplitudes considered at all order in the q expansion (complete eq. (3)) and the Non-Born amplitude truncated at the flrst order in the q expansion (truncated eq. (4)). The only parameters are a priori the 6 generalized polarizabilities contained in the flrst and only term of the considered Non-Born amplitude. They are adjusted with a best flt on the complete set of 14 × 5 data. In order to have a better convergence, the polarizability P (01,01)0 (Q2 = 0.33 GeV2 ) is flxed by the result obtained in real Compton scattering scaled by the electric form factor and P (11,02)1 (Q2 = 0.33 GeV2 ) is flxed at 0 (it corresponds to the quadrupolar deformation of the N-Δ transition which is expected to be very small). The result of the flt for the 4 remaining polarizabilities is presented in table 3. This third method, mainly realized to justify a rather flat q evolution of (d5 σ − d5 σ BH+Born )/φq presented by the dashed points in flg. 7, allows one to

N. d’Hose: Virtual Compton Scattering at MAMI Table 3. Results for the polarizabilities extracted in the third method. These results are compared to the heavy-baryon chiral perturbation theory (HBChPT) predictions [13]. PLL PTT PLT

P (01,01)0 P (11,11)1 P (01,12)1 P (11,11)0 P (11,00)1 P (11,02)0

Third Method −0.0626 fixed +0.0048 ± 0.0034 −0.0123 ± 0.0026 −0.0384 ± 0.0186 −0.157 ± 0.070 0. fixed

HBChPT −0.056 +0.001 −0.008 −0.034 −0.096 +0.003

Units fm3 fm3 fm4 fm3 fm2 fm4

determine some spin polarizabilities with reasonable precision (notably P (01,12)1 ). Fourth method using Dispersion Relations: This method was used after the publication [18] of the flrst VCS MAMI experiment at the sight of the other experiments of JLab and MAMI where a rather flat q evolution was not so obviously conflrmed by the data. This method is based on the formalism of Dispersion Relations (DR) [14,15] for the invariant VCS amplitudes and works below pion threshold as well as in the flrst resonance region. Assuming analyticity, crossing symmetry and an appropriate high-energy behavior, unsubtracted dispersion relations relate the real part of VCS amplitudes to an integral over the virtual photon energy of a function of their imaginary part. The imaginary part of a VCS amplitude is given by the sum of πN intermediate states, computed from γ ∗ N → πN data (in the phenomenological MAID2000 analysis [25]), plus higher order contributions beyond πN . Moreover asymptotic contributions have also to be considered for two VCS amplitudes (F1 and F5 ) which cannot fulflll unsubtracted dispersion relation framework. The t-channel π 0 exchange and the knowledge of the Fπ0 γγ form factor flx the asymptotic contribution to F5 and determine completely the spin-dependent GPs. The asymptotic contribution of the amplitude F1 related to the polarizability P (11,11)0 or βM (Q2 ) originates from the t-channel ππ intermediate states. In a phenomenological analysis, this continuum is parametrized through the exchange of a scalar-isoscalar particle in the t-channel, i.e. an efiective σ -meson which gives rise to a diamagnetic contribution. The asymptotic part and the dispersive contributions beyond πN are estimated using a dipole parametrization of the difierence: πN βM (Q2 ) − βM (Q2 ) =

πN )Q2 =0 (βM − βM 2 (1 + Q /Λ2β )2

(8)

The mass scale Λβ is a free parameter related to the diamagnetism distribution inside the nucleon. It can be extracted from a flt to the VCS data at difierent Q2 values. Though unsubtracted dispersion relation is valid for the amplitude F2 related to the polarizability P (01,01)0 or αE (Q2 ), it is particularly relevant to wonder about the quality of the saturation of the subtracted dispersion integrals by πN intermediate states only. For this purpose

123

Table 4. The structure functions determined in the MAMI experiment using the four methods and compared to model predictions at Q2 = 0.33 GeV2 and  = 0.62. The errors are statistical only, except for the flrst method where two systematic errors are indicated in brackets. The prediction for DR model is given for 2 values of Λα and Λβ close to the values determined experimentally. Q2 = 0.33 GeV2  = 0.62 method 1 [18] (χ2 = 1.4) method 2 (χ2 = 1.3) method 3 (χ2 = 1.7) method 4 (χ2 = 1.5) HBChPT [13] LSM [12] ELM [11] NRQCM [9] NRQCM [10] DR [15]

PLL − PTT /ε (GeV 2 ) 23.7 ± 2.2 (±4.3 ± 0.6) 23.7 ± 8.1

PLT (GeV 2 ) −5.0 ± .8 (±1.4 ± 1.1) −7.8 ± 3.0

33.6 ± 11.7

−6.5 ± 4.2

23.2 ± 3.0 (Λα = 1.6 ± 0.2) 26.3 10.9 5.9 11.0 14.7 22.0 (Λα = 1.4 GeV)

−3.2 ± 2.0 (Λβ = 0.5 ± 0.2) −5.7 0.2 −1.9 −3.5 −4.5 −5.5 (Λβ = 0.5 GeV)

a dipole parametrization has also been proposed: πN αE (Q2 ) − αE (Q2 ) =

πN )Q2 =0 (αE − αE 2 (1 + Q /Λ2α )2

(9)

The mass scale Λα is the second free parameter of the DR formalism which can be extracted from a flt to the VCS data at difierent Q2 values. The evolution with the real photon momentum q of the MAMI VCS data obtained at Q2 = 0.33 GeV2 is relatively sensitive to the choice of the free parameter values: Λα = 1.6 ± 0.2 GeV and Λβ = 0.5 ± 0.2 GeV. The corresponding evolution with the real photon momentum q is presented by the dotted line in flg 7. This prediction is rather close to the evolution of method 3 given by the interference between the complete BH+Born amplitude and the truncated Non-Born amplitude, except for θ close to 0◦ . This kinematical point is only sensitive to PLT and this indicates difierent results for this observable in the framework of these 2 methods. Results: Table 4 presents the two structure functions in the framework of the four methods and compares them to theoretical predictions presented in the introduction: the heavy-baryon chiral perturbation theory calculation (HBChPT) [13], the linear sigma model (LSM) [12], the effective Lagrangian model (ELM) [11], two non-relativistic constituent quark models (NRQCM) [9,10] and the dispersive relation approach (DR) [15]. The three errors for the flrst method are, respectively, statistical error on the data, systematic error on photon angular distributions, and systematic error on the normalization. For the other methods only statistical errors are reported. The prediction for the DR model is given for 2 values of Λα and Λβ close to the values determined experimentally.

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80

80

60

60

40

40

20

20

0

2 0 2 4 6 8 −10 −12

0

0

0.2

0.2

0.4 Q (GeV2)

0

2

0.4 Q2 (GeV2)

2 0 2 4 6 8 −10 −12

dently all the scalar and spin-dependent polarizabilities, it is necessary to perform a double polarization experiment. 2.3 The two other unpolarized experiments at JLab and MIT-Bates

0

0.2

0.4 Q2 (GeV2)

0

0.2

0.4 Q2 (GeV2)

Fig. 9. Comparison of the unpolarized structure functions determined in the VCS MAMI experiment at Q2 = 0.33 GeV2 and in the RCS results [1] with the predictions of the DR formalism [15] (left panel) and of the O(p3 ) HBChPT [13] (right panel). The upper panels give the result for PLL − PTT / and the lower panel for PLT . The contributions of the scalar GPs are indicated by the dashed (or dotted) lines and the total contributions of the scalar and spin-dependent GPs are indicated by the solid (or dashed-dotted) lines. The DR prediction for the scalar GP αE (Q2 ) is calculated for Λα = 1.4 GeV (upper left panel). Two values of Λβ are used to calculated the scalar GP βM (Q2 ) contribution (lower left panel) in order to show the sensitivity. The contributions for Λβ = 0.4 and 0.6 GeV are used in the dotted and dashed lines, respectively. Figure extracted from [15].

The JLab experiment E93-050 [19] has been performed in the Hall A of the Thomas Jefierson National Accelerator Facility at Q2 = 0.9 and 1.8 GeV2 . The values of  are 0.95 and 0.88, respectively. Data cover the region below the pion threshold and the resonance region up to √ s = 2 GeV. The experimental analysis of the complete experiment is presented in ref. [19]. The Bates experiment 97-03 [20] has been performed at Q2 = 0.05 GeV2 and  = 0.90. Measurements have been done in-plane and at 90◦ out-of-plane, using the OOPS spectrometers. The experiment covers a limited range in polar angle θ around 90◦ , so the structure functions are determined from the ϕ-dependence of the cross section. Data analysis is still in progress and only preliminary results [20] can be presented. This experiment represents a laboratory achievement, having made the flrst use of the high duty factor beam in the South Hall Ring and of the full OOPS system.

4

10

2 5

0

The flrst experimental method for which the systematic errors have been carefully studied, gives the two structure functions PLL − PTT / and PLT presented in ref. [18]. The three other methods conflrm the large values of the two structure functions. The 4th method in the DR approach is particularly in good agreement with the 1st method for the value of PLL − PTT / and gives a slightly smaller value for PLT . Only the heavy-baryon chiral perturbation theory calculation (HBChPT) [13] and the dispersive relation approach (DR) [15] predict large values for these two structure functions and seem relevant for the description of the MAMI VCS experiment (see flg. 9). The structure functions measured in an unpolarized VCS experiment are the sum of two contributions (cf. eq. (7)) : one scalar related to the electric αE and magnetic βM polarizabilities measured in RCS and one spindependent. The last contribution vanishes at Q2 = 0. Figure 9 indicates that the efiect of the spin GPs is much smaller in the DR calculation [15] than in the O(p3 ) HBChPT [13]. To go beyond, that is to measure indepen-

0 -2 0

0.5

1

1.5 2 2 2 Q (GeV )

0

0.5

1

1.5 2 2 2 Q (GeV )

Fig. 10. Compilation of the data on electric αE (Q2 ) (left) and magnetic βM (Q2 ) (right) GPs of the proton. Data points at Q2 = 0 are from ref. [1]. The other points are the analyses of MIT-Bates [20] (still preliminary), of MAMI [18] and difierent analyses of JLab [19]. JLab points are slightly shifted in abscissa for better visibility. The inner error bars are statistical; the outer ones are the total error. The curves show calculations in the DR model with the values of Λα and Λβ obtained in each experiment (Λα , Λβ ) = (0.70,0.63) GeV in the JLab experiment (solid curve), (1.60,0.50) GeV in the MAMI experiment (dashed curve) and (0.60,0.51) in the MIT-Bates experiment (dotted curve). Note that the DR predictions for βM (Q2 ) for the MIT-Bates and MAMI experiments are on the same dotted curve. Figure done thanks to Helene Fonvieille [26].

N. d’Hose: Virtual Compton Scattering at MAMI

d 5σ (nb/GeV sr 2)

4 2 0 -2 -4 -6 -8

125

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

0.2

0.4 2 2 Q (GeV )

Fig. 11. PTT evaluated using the DR model [15] (solid line), the O(p3 ) HBChPT [13] (dotted line) and the O(p4 ) HBChPT [16] (dashed line).

We present in flg. 10 the world results for the electric αE (Q2 ) and magnetic βM (Q2 ) GPs deduced from the MAMI, JLab and MIT-Bates experiments. The value of the electric αE (Q2 ) and magnetic βM (Q2 ) GPs can determined directly by the coe– cients Λα and Λβ obtained in the DR analysis and using eq. (9) and eq. (8) or indirectly by the structure functions PLL − PTT / and PLT determined in the LET analysis in which the spin GPs contributions are evaluated in the DR model and subtracted using eq. (7). The agreement between these 2 methods was reasonably controlled in the JLab experiment (see the difierent results at Q2 = 0.9 and 1.8 GeV). The direct determination allows us to use also the VCS data in the resonance region (see in flg. 10 the result at Q2 = 0.9 GeV2 with the smallest statistical error.) The curves in flg. 10 are calculated using the DR model and the difierent values of Λα and Λβ obtained in each experiment. By deflnition all the DR predictions (see eqs. (9) and (8)) are constrained to go through the experiment RCS point at Q2 = 0. The fact that there is no unique DR curve going through all the data points, especially for the electric polarizability, does not invalidate the model. It simply means that the dipole parametrization of eqs. (9) and (8) does not hold over the entire Q2 range. Another fact to be aware of is the model-dependency introduced in this flgure by transforming the structure functions into GPs. The spin-dependent GPs are evaluated using the DR model, and as it has been pointed in flg. 9, this evaluation is quite smaller than in the O(p3 ) HBChPT [13]. It is clear that measurements of individual scalar and spin-dependent GPs are necessary to go further. We can note in flg. 11 the very difierent predictions for PTT using the DR model [15] or the O(p3 ) HBChPT [13] or else the O(p4 ) HBChPT [16]. An extraction of PTT can be achieved in a further experiment at MAMI at the same Q2 = 0.33 GeV2 , but with an other value of . We can take the beneflt of the next 1.5 GeV energy of the beam to access a new value of  to have a comfortable lever arm for a longitudinal-transverse separation.

0.05

0.1

0.15

0.2

0.25

0.3

,

q (GeV) Fig. 12. VCS difierential cross section as a function of the real photon energy q in the MAMI kinematics below and above the pion threshold. The VCS MAMI data are reported. The BH+Born contribution is given by the dash-dotted line. Predictions for the total cross section are given in the DR approach [15] using a flxed value of Λα = 1 GeV and for three values of Λβ : 0.6 GeV (solid line); 0.7 GeV (dotted line) and 0.4 GeV (dashed line). Fig extracted from ref. [15].

3 Single polarized experiments above pion threshold Figure 12 shows the DR predictions for photon energies ranging from threshold to the Δ (1232)-resonance region. The deviation from the BH+Born prediction rises strongly after pion threshold. When crossing the pion threshold, the VCS amplitude acquires an imaginary part due to the coupling to the πN channel. Therefore single polarization observables become non-zero above pion threshold. A particularly relevant observable is the electron single spin asymmetry (SSA) which is obtained by ipping the electron beam helicity. For VCS this observable is mainly due to the interference of the real BH+VCS amplitude with the imaginary part of the VCS amplitude. As the SSA vanishes in-plane, its measurement requires an out-of-plane experiment. Such an experiment has been proposed at MIT-Bates [27] and is being realized at MAMI [28] thanks to one of the spectrometers of the A1 collaboration moving out-of-plane. In flg. 13, the SSA is presented√ for a kinematics in the Δ (1232) region, corresponding to s = 1.2 GeV. The DR calculation shows that the SSA is quite sizeable, and it is mainly sensitive to the imaginary part of the VCS amplitude, displaying only a rather weak dependence on the GPs (obtained for the difierent values of Λα and Λβ ). Therefore it provides an excellent cross-check of the dispersive input (MAID 2000) in the DR formalism for VCS, in particular by comparing at the same time the pion and photon electroproduction channels through the Δ excitation. The MAMI analysis is still in progress.

4 Double polarized experiments at threshold A double-polarization VCS experiment is also presently being realized at MAMI. The theoretical framework of

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The European Physical Journal A

e’

γ*

e

−0.05

e→Ycm ϕ

γ’

0

p’

θ p

→

eXcm e→Zcm

Fig. 14. Kinematics for the p(e, e p )γ reaction. −0.1

−0.15 0

10

20

30

40

50

60

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Fig. 13. Electron single-spin asymmetry (SSA) for the VCS MAMI kinematics as a function of the photon scattering angle. The full dispersion results are shown for the values of Λα = 1 GeV and Λβ = 0.6 GeV (solid curve); Λα = 1 GeV and Λβ = 0.4 GeV (dashed curve); Λα = 1 GeV and Λβ = 0.7 GeV (dotted curve) and Λα = 1.4 GeV and Λβ = 0.4 GeV (dashdotted curve). Figure extracted from ref. [15].

such experiment has been developed by Vanderhaeghen and Guichon [5,29]. Below pion production threshold, the VCS amplitude is purely real, and all single-polarization observables are zero. So only double polarization experiment observables can disentangle the GPs. For a polarized electron of pure helicity state h = ±1/2 (longitudinal polarization ξe = 2hk/me ), we can measure the average polarization P of the recoil proton along the 3 vectors (excm , eycm , ezcm ) associated to the virtual photon direction in the photon-proton CM as deflned in flg 14. We can deflne for the 3 axes: P · e(i) = (10)     2hk 2hk 5 5 d σ ξe = me , ξp = e(i) − d σ ξe = me , ξp = −e(i)     2 hk 5 d5 σ ξe = 2hk me , ξp = e(i) + d σ ξe = me , ξp = −e(i) 5

=

Δd σ(h, i) 2 · d5 σ

We obtain a similar low energy prediction as in eq. (5): Δd5 σ(h, i) = Δd5 σ BH+Born (h, i) + φq ΔΨ0 (h, i) + φO(q2 ).

(11)

So in such a complete experiment we can access 4 observables: Ψ0 = v1 (PLL − PTT /) + v2 PLT , z ΔΨ0 (h, z) = (4h) [v1z PTT + v2z PzLT + v3z P LT ],

(12)

⊥

⊥

⊥

⊥

x ⊥ x  x  ΔΨ0 (h, x) = (4h) [v1x P⊥ LT + v2 PTT + v3 P TT + v4 P LT ], y ⊥ y  y  ΔΨ0 (h, y) = (4h) [v1y P⊥ LT + v2 PTT + v3 P TT + v4 P LT ].

Only 6 structure functions are independent: PLL , PTT , z ⊥ PLT , PzLT , P LT and P LT . P⊥ LT is a combination of PTT

Fig. 15. Predictions for the deviation of the double polarization asymmetry from the BH+Born contribution using the DR model [15] (solid line), the O(p3 ) HBChPT [13] (dotted line) and the O(p4 ) HBChPT [31] (dashed line). ⊥

and PLL , P⊥ combination of PzLT and PLT , P TT a TT a z  combination of P LT and PLT . PLL = aP (01,01)0 , PTT = PLT = bP

c1 P (11,11)0

(13) (11,11)1

+ c2 P

(01,12)1

+ c3 [P

,

(11,00)1

+ d1 P (11,02)1 ],

PzLT =

c4 P (11,11)1 + c3 [P (11,00)1 + d1 P (11,02)1 ],

z PLT =

c5 P (11,11)1 + c6 [P (11,00)1 + d1 P (11,02)1 ],

⊥ PLT =

[d2 P (11,00)1 + d3 P (11,02)1 ].

 

The vij in eq. (12) are kinematical factors depending on θ and ϕ. If the out-of-plane angle ϕ remains close to 0◦ or 180◦ , we have the following approximations: v1 = v1z ∼ v1x ∝ sin θ; v2 ∼ v2z ∼ v2x ∼ constant; v3z ∼ v3x ∝ cos θ; v4x ∝ sin ϕ ∼ 0; ∀i, viy ∝ sin ϕ ∼ 0. The GPs can be extracted from the linear system above using the angular distributions of Ψ0 and the 3 ⊥ ΔΨ0 (h, i). Note that P LT can only be extracted by an out-of-plane measurement. Theoretical predictions [31] using the DR model, the O(p3 ) HBChPT and the O(p4 ) HBChPT give a few % deviation of the double polarization asymmetry from the BH+Born contribution (see flg. 15). Such a delicate experiment is being realized using the polarized electron beam available at MAMI and the measurement of the recoil polarization of the outgoing proton in a focal plane polarimeter [30], and the detection of the outgoing electron in the high resolution spectrometer

N. d’Hose: Virtual Compton Scattering at MAMI

moving out-of-plane. It is clearly a very challenging experiment, relying on a very delicate expertise of the complete apparatus and requiring high statistics and very reduced systematic errors.

5 Conclusion An ambitious program to reach the generalized polarizabilies of the proton has been undertaken at MAMI over the last ten years. The ultimate Grail is the separation between spin-independent and spin-dependent GPs which seems very promising for the study of the nucleon structure. All the results reported here are the fruits of the complete A1-VCS collaboration. I would like to acknowledge all the students, Luca Doria, Peter Janssens, Imad Bensafa, Jan Friedrich, Julie Roche, David Lhuillier, Dominique Marchand for which the work was essential to produce reliable results on these very meticulous experiments. I wish to underline the strong support and the synergy given by Helene Fonvieille, Harald Merkel, Michael Distler, Luc Van Hoorebeke, Gabriel Tamas, Robert Van de Vyver, J˜ org Friedrich, Thomas Walcher for this research. I am also very grateful to Dieter Drechsel, Pierre Guichon, Marc Vanderhaeghen, Barbara Pasquini, Stefan Scherer, Thomas Hemmert, Ulf Meissner for their pedagogical lectures and theoretical support in the data interpretation. It is clear that the success of the VCS MAMI experiments has its origin in the coherent efiort between excellent physicists as Karl-Heinz Kaiser, always concerned with the performance and the high quality of the electron facility, Thomas Walcher strongly supporting and managing all the efiorts for such a challenging experimental program, J˜ org Friedrich, expert of delicate and precise experiments and Dieter Drechsel, stimulating progress in our scientiflc knowledge.

References 1. 2. 3. 4.

V. Olmos de Leon et al., Eur. Phys. J. A 10, 207 (2001). Y.S. Tsai, Phys. Rev. 122, 1898 (1961). H. Arenh˜ ovel, D. Drechsel, Nucl. Phys. A 233, 153 (1974). P.A.M. Guichon, G.Q. Liu, A.W. Thomas, Nucl. Phys. A 591, 606 (1995). 5. P.A.M. Guichon, M. Vanderhaeghen, Prog. Part. Nucl. Phys. 41, 125 (1998).

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6. D. Drechsel, G. Kn˜ ochlein, A. Metz, S. Scherer, Phys. Rev. C 55, 424 (1997). 7. D. Drechsel, G. Kn˜ ochlein, A. Yu Korchin, A. Metz, S. Scherer, Phys. Rev. C 57, 941 (1998). 8. F.E. Low, Phys. Rev. 110, 974 (1958). 9. G.Q. Liu, A.W. Thomas, P.A.M. Guichon, Austral J. Phys. 49, 905 (1996). 10. B. Pasquini, S. Scherer, D. Drechsel, Phys. Rev. C 63, 025205 (2001). 11. M. Vanderhaeghen, Phys. Lett. B 368, 13 (1996). 12. A. Metz, D. Drechsel, Z. Phys. A 356, 351 (1996); 359, 165 (1997). 13. T.R. Hemmert, B.R. Holstein, G. Kn˜ochlein, S. Scherer, Phys. Rev. D 55, 2630 (1997); Phys. Rev. Lett. 79, 22 (1997); T.R. Hemmert, B.R. Holstein, G. Kn˜ ochlein, D. Drechsel, Phys. Rev. D 62, 014013 (2000). 14. B. Pasquini, M. Gorchtein, A. Metz, M. Vanderhaeghen, Eur. Phys. J. A 11, 185 (2001). 15. D. Drechsel, B. Pasquini, M. Vanderhaeghen, Phys. Rep. 378, 99 (2003). 16. C.W. Kao, M. Vanderhaeghen, Phys. Rev. Lett. 89, 272002 (2002). 17. V. Bernard, N. Kaiser, A. Schmidt, U. Meissner, Phys. Lett. B 319, 269 (1993); Z. Phys. A 348, 317 (1994). 18. J. Roche et al., Phys. Rev. Lett. 85, 708 (2000). 19. G. Laveissiere et al., Phys. Rev. Lett. 93, 122001 (2004). 20. J. Shaw, R. Miskimen, MIT-Bates Proposal 97-03, (1997) and P. Bourgeois, PhD Thesis. 21. K.I. Blomqvist et al., Nucl. Instrum. Methods A 403, 263 (1998). 22. G. H˜ ohler, E. Pietarinen, I. Sabba-Stefanescu, F. Borkowski, G.G. Simon, V.H. Walther, R.D. Wendling, Nucl. Phys. B 114, 505 (1976); private communication. 23. M. Vanderhaeghen, J.M. Friedrich, D. Lhuillier, D. Marchand, L. Van Hoorebeke, J. Van de Wiele, Phys. Rev. C 62, 025501 (2000). 24. P. Janssens, L. Van Hoorebeke et al., to be published in Nucl. Instrum. Methods. 25. D. Drechsel, O. Hanstein, S.S. Kamalov, L. Tiator, Nucl. Phys. A 645, 145 (1999). 26. H. Fonvieille, Proceedings of the Erice School of Nuclear Physics, 26th Course, Prog. Part. Nucl. Phys. 55, 198 (2005) and private communication. 27. N.I. Kaloskamis, C.N. Papanicolas, MIT-Bates proposal (1997). 28. N. d’Hose, H. Merkel, MAMI Proposal (2001). 29. M. Vanderhaeghen, Phys. Lett. B 402, 243 (1997). 30. Th. Pospischil et al., Nucl. Instrum. Methods A 483, 726 (2002). 31. C.W. Kao, B. Pasquini, M. Vanderhaeghen, Phys. Rev. D 70, 114004 (2004).

Eur. Phys. J. A 28, s01, 129 137 (2006) DOI: 10.1140/epja/i2006-09-014-5

EPJ A direct electronic only

Experimental tests of Chiral Perturbation Theory H. Merkela Institut f˜ ur Kernphysik, Johannes Gutenberg-Universit˜ at Mainz, D-55099 Mainz, Germany / Published online: 15 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Over the last decade, a series of dedicated experiments to test heavy baryon chiral perturbation theory was performed at MAMI. Photo production of neutral pions close to threshold with unpolarized and polarized photon beam was performed to separate the multipole amplitudes at threshold. The extension of this experiments to a modest photon virtuality of Q2 < 0.1 GeV2 /c2 was performed to extract additional longitudinal multipoles and to exploit the Q2 evolution predicted by theory. An out-of-plane measurement above π + threshold with polarized electron beam gave access to the imaginary part of the s-wave amplitude. Finally, by coherent photo and electro production from the deuteron the neutron amplitude could be extracted. PACS. 25.30.Rw Electroproduction reactions

13.60.Le Meson production

1 Introduction Chiral perturbation theory is an efiective fleld theory, which utilizes the symmetry properties of the QCD Lagrangian to extract observables at low momenta in a systematic fashion (see, e.g., S. Scherer, this issue). Meanwhile, a vast amount of observables can be calculated in the framework of chiral perturbation theory, leading to numerous ways to test this theory in experiment. This contribution, however, concentrates on a dedicated series of experiments, which were especially designed to test the predictions of Heavy-Baryon Chiral Perturbation Theory (HBChPT). The investigated reaction is the electromagnetic production of the Goldstone Boson of the theory, the pion. The choice of the electromagnetic probe ensures the precision, which is necessary to extract amplitudes very close to threshold. While the π + production is dominated by the charge (Kroll-Ruderman term), the π 0 production is the ideal testing ground for HBChPT. Over the last decade, several experiments on π 0 threshold production were performed at MAMI and at other electron accelerator laboratories. The formalism of HBChPT was developed in parallel with large success in predicting or fltting of the measured threshold observables, leading to an improving consistent picture. In this article, an overview over the existing experiments and their interpretation in terms of HBChPT is given.

a

e-mail: [email protected]

12.39.Fe Chiral Lagrangians

2 Neutral pion photo production Since chiral perturbation theory is an expansion in small momenta, only for a small region above threshold predictions can be derived. Close to threshold, the angular structure of the difierential cross section of photo production of pseudoscalar mesons can be expanded in q σ(θ) = A + B · cos θ + C · cos2 θ , k with q and k the pion and photon center of mass momentum and θ the pion center of mass production angle. The angular coe– cients can be further decomposed in s- and p-wave multipoles: 1 2 P2 + P 3 2 , 2 B = 2 · Re (E0+ P1 ∗ ) , 1 2 P2 + P 3 2 . C = P1 2 − 2 2 A = E0+ +

In this fleld, it is common to use the p-wave combinations P1 = 3E1+ + M1+ − M1− , P2 = 3E1+ − M1+ + M1− , P3 = 2M1+ + M1− . Not only the angular form of the multipoles is known, but also their energy dependence. The p-waves rise proportional to photon and pion CMS momentum q · k, while the form of the s-wave multipole is dictated by unitary, as will be discussed later. The flrst experiments on threshold π 0 photo production [1,2] were designed to test the Low-Energy Theorem

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series [5]. These authors calculated s- and p-waves and showed, that while E0+ and P3 are slow converging, the P1 and P2 multipoles were fast converging and should be a strong prediction of HBChPT. Figure 1 shows the total cross section of the last MAMI experiment [3] in comparison with former data from SAL [6] and MAMI [7]. The data were taken at the tagged photon beam of the MAMI A2 Collaboration (flg. 2). The reaction was identifled by the detection of the two decay photons of the pion in coincidence with the TAPS detector, an array of 504 BaF detector modules arranged in 7 blocks. With this setup, difierential cross sections were measured up to an incoming photon laboratory energy of 168 MeV. By fltting the angular coe– cients A, B, and C to the difierential cross section and extrapolating to threshold by using the known energy dependence, the multipole combinations Re E0+ , P1 , and P22 +P32 could be extracted. 2.1 Polarized photon asymmetry

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To further decompose P22 + P32 , an additional polarization observable has to be measured. The polarized photon asymmetry Σ is deflned as the asymmetry of the cross section in respect to the polarization plane of a polarized photon beam, i.e. Σ = σ0 (1 − Pγ Σ(θ) cos 2φ) am

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with φ the angle between polarization plane and production plane and Pγ the photon polarization. This asymmetry is proportional to the difierence P22 − P32 and allows the decomposition of the modulus of these two multipole amplitudes. At MAMI, a polarized photon beam was prepared by coherent Bremsstrahlung from a diamond crystal. The asymmetry was again determined by detecting the decay photons of the pion with the TAPS detector. Figure 3

0.4 0.3 0.2 504 BaF 2 Detector Modules Fig. 2. One BaF module and the complete setup of the photon spectrometer TAPS at the MAMI A2-Collaboration.

(LET) [4], which derived from general principles a value for the s-wave amplitude E0+ at threshold. These flrst experiments showed a serious discrepancy from the prediction of the low-energy theorem. The development of the formalism of Heavy-Baryon Perturbation Theory resolved this puzzle by showing, that the LET value is only the leading term in a slow converging

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shows the measured asymmetry, averaged over the energy range. Combining the unpolarized and polarized measurement, the four threshold amplitudes can be extracted. Table 1 summarizes these amplitudes. For E0+ and P1 , the values can be compared with the data from an experiment at SAL [6], which are in agreement within the error bar. The quoted values of HBChPT, which are based on a reflt of the low-energy constants [8] also successfully describe the data. For comparison, also the results of a dispersion relations calculation [9] is added. While the values for P2 and P3 are reasonable in the table, in flg. 3 the remaining small discrepancy is more pronounced due to the large sensitivity of Σ to the difierence of these two amplitudes. 2.2 Cusp effect of the s-wave amplitude While the p-wave amplitudes rise proportional to the momenta q ·k the energy dependence of the s-wave amplitude is given by the unitarity of the scattering matrix. Chiral perturbation theory alone predicts a roughly constant s-wave amplitude. The π 0 p amplitude, however, is an order of magnitude smaller than the π + n amplitude due to the sizable Kroll-Ruderman term. Above the π + n threshold, a photon coupling to a π + in the intermediate state with a following pion charge exchange in the flnal state (see flg. 4) leads to a large contribution to the swave amplitude. Figure 5 shows the real part of the E0+ amplitude in photo production extracted from the MAMI data [3]. The two difierent data sets are extracted from the same difierential cross section by the two assumptions of the p-waves rising proportional to q (open squares) or proportional to q · k (fllled squared). The flrst assumption is phenomenological valid over a larger energy range, while the second assumption corresponds to the prediction of chiral perturbation theory. The cusp efiect at the π + n threshold is clearly seen. Again, the dispersion relations calculation [9] is included

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(dashed line). The strength of this efiect is roughly given by the E0+ amplitude of the π + n production times the charge exchange scattering length. Only the real part of E0+ could be extracted by this experiment. For the imaginary part, an additional polarization experiment would be necessary, e.g., a measurement of the polarized target asymmetry.

3 Electroproduction While in photo production two amplitudes are given by a flt of low-energy constants to the data, the extension to electro production exploits additional predictive power of HBChPT. In addition to the dependence of the multipoles on the photon virtuality Q2 the two longitudinal p-wave multipoles P4 and P5 and the longitudinal s-wave multipole L0+ occur. The flve-fold difierential cross section is given by ! dσ(θ, φ) = Γ σT +  σL +  σT T cos 2φ dΩ  dE  dΩ  + 2(1 + ) σT L cos φ "  + h 2(1 − ) σT L sin φ with the virtual photon flux Γ , the transverse photon polarization  and the photon-proton center of mass angles θ and φ deflned as in photoproduction with respect to the virtual photon direction. The transverse cross section σT and the longitudinal cross section σL can be disentangled by varying , for the other structure functions the difierential cross section has to be measured over the corresponding angular range. The transverse-longitudinal interference σT L can be extracted by a polarized electron beam with helicity h.

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Fig. 6. The three-spectrometer setup of the A1 Collaboration at MAMI.

In the threshold region, the cross section can be further decomposed into s- and p-wave multipoles:  σT (θ) = p/kγ A + B cos θ + C cos2 θ ,  σL (θ) = p/kγ A + B  cos θ + C  cos2 θ , σT L (θ) = p/kγ (D sin θ + E sin θ cos θ) ,  σT T (θ) = p/kγ F sin2 θπ∗ , # Q2 σT L (θ) = p/kγ (G sin θ + H sin θ cos θ) , ω ∗2 where p/kγ is the phase space ratio of pion CM momentum and photon CM equivalent momentum, and the angular coe– cients are combinations of two s-wave and flve p-wave multipoles: 1 |P2 |2 + |P3 |2 , 2 = 2 Re (E0+ P1∗ ) , 1 |P2 |2 + |P3 |2 , = |P1 |2 − 2 = − Re (E0+ P5∗ + L0+ P2∗ ) , = − Re (P1 P5∗ + P4 P2∗ ) , 1 |P2 |2 − |P3 |2 , = 2

A = |E0+ |2 + B C D E F

A = |S0+ |2 + |P5 |2 , B  = 2 Re (S0+ P4∗ ) ,  C  = |P4 |2 − |P5 |2 ,  ∗ ∗ P5 + S0+ P2 , G = Im E0+ H = Im (P1 P5∗ + P4 P2∗ ) . 3.1 Unpolarized experiments A flrst electroproduction experiment at NIKHEF [10] estimated the s-wave cross section at threshold, a second experiment [11] tried to extract p-wave amplitudes in addition by measuring the pion emission angle in plane left and right from the virtual photon direction. While these pioneering works were in good agreement with ChPT, the predictive power was not full exploited, since the data has to be analyzed with assumptions from theory. A more complete experiment was performed at MAMI [12] at the three-spectrometer setup of the A1 collaboration. In electro production, the recoil proton is detected in coincidence with the scattered electron. Very close to threshold, the boost by the Lorentz transformation from the center-of-mass system to the laboratory system focuses the full solid angle of the center of mass system into a narrow cone.

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At MAMI, two high-resolution spectrometer of the A1 Collaboration were used for the detection of the scattered electron and the recoil proton (flg. 6). At a photon virtuality of Q2 = 0.1 GeV2 /c2 up to a center-of-mass energy of 4 MeV above threshold full coverage of the angular range was achieved. To separate transverse and longitudinal cross section, data were taken at three values of the photon polarization . For each bin in  and energy the difierential cross section was fltted to separate the structure functions σT and σL and the interference structure functions σT T and σT L . Similar to photo production, the angular structure of the structure functions was fltted with the assumption of only s- and p-waves contributing to the cross section. Figure 7 shows the result for the s-wave multipoles in comparison with the results from NIKHEF and with the calculations in HBChPT [13]. As can be seen, the extracted multipoles are in agreement within the error bars and can be fltted by HBChPT. Actually, this result was surprising, since at that point a photon virtuality of Q2 = 0.1 GeV2 /c2 was expected to be somewhat beyond the scope of HBChPT. Therefore, a further experiment was performed at a lower virtuality of Q2 = 0.05 GeV2 /c2 (ref. [14]). The same experimental technique was used as by the previous experiment, however the systematic errors are somewhat larger due to the lower outgoing proton momentum.

Figure 8 summarizes the result of this experiment in comparison with the real photon data and the data at Q2 = 0.1 GeV2 /c2 . While the photon point and the higher Q2 point was included in the flt of HBChPT, the middle Q2 point was not. As can be seen, there was a clear discrepancy which has to be clarifled. 3.2 Extended momentum range The flrst step was, to extend the momentum range to be more sensitive to the interference structure functions. At MAMI an extended experiment was performed at a four-momentum transfer of Q2 = 0.05 GeV2 /c2 [15]. Due to the extension of the momentum range to a centerof-mass energy of 40 MeV above threshold the full angular range cannot be measured in a single kinematical setting. Several settings along the in plane angular range (φ = 0◦ , 180◦ ) are required to separate the difierential cross section σ0 = σT + σL and the longitudinaltransverse interference structure function σLT . To separate the transverse-transverse interference σT T , which has a dependence on the out-of-plane angle of cos 2φ, the according out-of-plane acceptance is required. At MAMI, one of the high-resolution spectrometer (spectrometer B in flg. 6) can be tilted by up to 10◦ . By using this spectrometer as electron detector in extreme forward direction, this translates to 90◦ in the center-of-mass frame. Data were taken at 14 difierent kinematical settings and were combined to extract the structure functions. Figure 9 shows the structure functions versus the centerof-mass energy above threshold. For comparison, the calculation in HBChPT [13] was included (solid line). As

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stated before, this calculation overestimates the difierential cross section. Even more striking, the derivation of the transverse-transverse interference is considerable, which is proportional to the difierence of the p-wave multipoles P2 and P3 , which are reproduced in good quality in photoproduction. For comparison, the phenomenological model MAID [16] is included (dashed line). For the structure functions, this model is able to describe the data sets roughly. This model is basically a global flt to the existing data sets in photo- and electro-production and can be seen as check of the consistency of the data with the other existing data sets, however data at this photon virtuality and energy are scarth. The dynamical Dubna-MainzTaipeh model (DMT) [17] is shown as dashed-dotted line and is in good agreement with the data.

tional to the imaginary part of L0+ multiplied by the large Δ(1232) multipole M1+ . Since it is very small, one has to deflne the helicity asymmetry σ(h = 1) − σ(h = −1) σ(h = 1) + σ(h = −1)  2(1 − )σT L (θ) = σT (θ) + σL (θ) − σT T (θ)

AT L (θ = 90◦ , φ = −90◦ ) =

to reduce the systematic errors. The expected energy structure is given by the unitary cusp at the opening of the π + threshold. Figure 10 shows this asymmetry. Again, the calculations in HBChPT and the models MAID and DMT are included in the graph. This small asymmetry enhances the difierences between the models, and as can be seen clearly, only the DMT model is able to describe the data.

3.3 Helicity asymmetry By using polarized electrons, the flfth structure function σLT  can be extracted. This structure function has the multipole contents σT L (90◦ ) ∝ Im[L∗0+ (3 E1+ − M1+ + M1− ) − E0+ (2 L∗1+ − L∗1− )], i.e. it is basically propor-

4 Coherent production from the deuteron The π 0 production on the neutron is in principle a strong prediction of HBChPT, since by the π 0 production from

H. Merkel: Experimental tests of Chiral Perturbation Theory

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the proton nearly all low-energy constants are already flxed. Obvious, the lack of a free neutron target complicates the situation. The use of the deuteron as neutron target requires detailed modeling of the deuteron structure as flrst step. Since the Fermi momenta of the nucleons in the deuteron is of the same order as the momenta of a threshold experiment, the coherent production is the most promising channel to get the deuteron structure and flnal state interaction under control. By this choice, small cross sections are to be expected as pointed out flrst in ref. [18]. Due to the restrictions for the intermediate state of two neutrons in an s-wave due to Pauli blocking an exact cancelation between the π + n intermediate state in the s-wave for production from the proton and the rescattering graph by charge exchange on the spectator neutron occurs. This cancellation causes small s-wave cross sections on the one hand, but reduces the systematic uncertainties by the estimation of the flnal state interaction on the other hand.

In contrast to the technique of coherent photo production, in electro production the recoil deuteron is detected. This leads to clear and unique identiflcation of the reaction channel by the missing mass of the pion. Without this subtraction, also the extraction of p-waves is possible. While the theoretical advantage is obvious, the experimental challenge is large. In a threshold experiment at a four momentum transfer of Q2 = 0.1 GeV2 /c2 the deuteron has a kinetic energy of T ≈ 30 MeV, which makes it di– cult to achieve the necessary accuracy to identify the reaction. At MAMI, the spectrometer setup is optimized to detect low energetic particles. By using a thin LD2 target with 3 mm path length in the deuterium and connecting the spectrometer vacuum system with the vacuum system of the target chamber, the multiple scattering and energy loss of the deuterons could be minimized. Still, the detection e– ciency for deuterons is close to 80% (in comparison to 98% for protons) due to hadronic interactions of the deuteron. Figure 11 shows the achieved missing mass resolution of the MAMI experiment [21]. Again, a full transverselongitudinal separation was performed by measuring a three difierent values of the photon polarization  and the full center-of-mass angular range was covered up to a center-of-mass energy of 4 MeV above threshold. Since at that time only predictions for the s-wave amplitudes were available [22], a flt of the difierential cross

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section with the usual assumptions of s- and p-waves with the known energy dependence was performed to extract the threshold s-wave amplitudes. It has to be stressed, that this flt ignores completely the deuteron structure and flnal state interaction, leading to a large systematic errors in the extracted amplitudes. Since Ld is very large, for Ed only an upper limit could be extracted. The amplitudes

were extracted to |Ed | ≤ 0.42 · 10−3 /mπ , |Ld | = (0.50 ± 0.11) · 10−3 /mπ . The transverse multipole Ed was within agreement with the calculations as at the photon point, while the

H. Merkel: Experimental tests of Chiral Perturbation Theory

calculations overestimated the longitudinal multipole by a factor of 2. The situation improved a lot by a new calculation by the same group [23], which now used a complete multipole decomposition and extracted s- and p-waves. By this, the comparison has not to be done on the level of the questionable extracted s-wave amplitudes, but can be done directly with the difierential cross section. Figure 12 shows the result of their calculation in comparison with the MAMI data. The plot shows two kind of flts: the flrst was performed by flxing the Ld multipole to the extracted value of the data set, reducing the number of parameters to a single low-energy constant. This flt, however, does not describe the data set completely. The second flt was performed with two free parameters and describes the data.

The success of ChPT in the SU (2) sector encourages one to extend this program to the strangeness threshold. At MAMI a new kaon spectrometer is under construction, and kaon threshold production will be among the flrst experiments with this spectrometer.

References 1. 2. 3. 4. 5.

6. 7.

5 Summary and outlook While the data basis on pion threshold production is now quite large, there are still a number of unsolved problems. The strong Q2 -dependence of the total cross section (flg. 8) seems to be unlikely and might indicate the large systematic errors of the data sets as quoted by the authors. To resolve this puzzle, a dedicated experiment was performed by MAMI, where three difierent Q2 values were measured within one experiment and with emphasis on reducing the systematic error [25]. This experiment is under analysis, but preliminary data seems to indicate, that the Q2 = 0.1 GeV2 /c2 data set is somewhat high. Since that data set is included in the flt of HBChPT, a reflt of the theory seems to be necessary as soon as the new data set is published. All cited calculations are based on the heavy-baryon formalism. Meanwhile, better regularization schemes are on the market and an improvement of the calculations can be expected. Especially in electroproduction, where the pwaves are not yet calculated to the same order as in photo production, a considerable improvement is possible.

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8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

E. Mazzucato et al., Phys. Rev. Lett. 57, 3144 (1986). R. Beck et al., Phys. Rev. Lett. 65, 1841 (1990). A. Schmidt et al., Phys. Rev. Lett. 87, 232501 (2001). P. de Baenst, Nucl. Phys. B 24, 633 (1970); I.A. Vainshtein, V.I. Zakharov, Nucl. Phys. B 36, 589 (1972). V. Bernard, N. Kaiser, J. Gasser, U.-G. Mei ner, Phys. Lett. B 268, 291 (1991); V. Bernard, N. Kaiser, U.-G. Mei ner, Z. Phys. C 70, 483 (1996). J.C. Bergstrom et al., Phys. Rev. C 53, R1052 (1996). M. Fuchs et al., Phys. Lett. B 368, 20 (1996); A.M. Bernstein et al., Phys. Rev. C 55, 1509 (1997). V. Bernard et al., Eur. Phys. J. A 11, 209 (2001). O. Hanstein et al., Phys. Lett. B 399, 13 (1997). T.P. Welch et al., Phys. Rev. Lett. 69, 2761 (1992). H.B. van den Brink et al., Phys. Rev. Lett. 74, 3561 (1995). M.O. Distler et al., Phys. Rev. Lett. 80, 2294 (1998). V. Bernard, N. Kaiser, U.-G. Mei ner, Nucl. Phys. A 607, 379 (1996); 633, 695 (1998)(E). H. Merkel et al., Phys. Rev. Lett. 88, 012301 (2002). M. Weis, Doctorate Thesis, Mainz, 2003. D. Drechsel et al., Nucl. Phys. A 645, 145 (1999); S.S. Kamalov et al., Phys. Lett. B 522, 27 (2001). S.S. Kamalov et al., Phys. Rev. Lett. 83, 4494 (1999); Phys. Rev. C 64, 032201 (2001). M. Rekalo, E. Tomasi-Gustafsson, Phys. Rev. C 66, 015203 (2002). J.C. Bergstrom et al., Phys. Rev. C 57, 3203 (1998). S.R. Beane et al., Nucl. Phys. A 618, 381 (1997). I. Ewald et al., Phys. Lett. B 499, 238-244 (2001). V. Bernard, H. Krebs, U.-G. Mei ner, Phys. Rev. C 61, 58201 (2000). H. Krebs, V. Bernard, U.-G. Mei ner, Eur. Phys. J. A 22, 503-514 (2004). H. Krebs, V. Bernard, U.-G. Mei ner, Nucl. Phys. A 713, 405 (2003). J. Garc a Llongo, Diploma Thesis, Mainz, in preparation.

Eur. Phys. J. A 28, s01, 139 148 (2006) DOI: 10.1140/epja/i2006-09-015-4

EPJ A direct electronic only

The Bonn Electron Stretcher Accelerator ELSA: Past and future W. Hillerta Physikalisches Institut, Universit˜ at Bonn, Nussallee 12, 53115 Bonn, Germany / Published online: 16 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. In 1953, it was decided to build a 500 MeV electron synchrotron in Bonn. It came into operation 1958, being the flrst alternating gradient synchrotron in Europe. After flve years of performing photoproduction experiments at this accelerator, a larger 2.5 GeV electron synchrotron was built and set into operation in 1967. Both synchrotrons were running for particle physics experiments, until from 1982 to 1987 a third accelerator, the electron stretcher ring ELSA, was constructed and set up in a separate ring tunnel below the physics institute. ELSA came into operation in 1987, using the pulsed 2.5 GeV synchrotron as pre-accelerator. ELSA serves either as storage ring producing synchrotron radiation, or as post-accelerator and pulse stretcher. Applying a slow extraction close to a third integer resonance, external electron beams with energies up to 3.5 GeV and high duty factors are delivered to hadron physics experiments. Various photo- and electroproduction experiments, utilising the experimental set-ups PHOENICS, ELAN, SAPHIR, GDH and Crystal Barrel have been carried out. During the late 90’s, a pulsed GaAs source of polarised electrons was constructed and set up at the accelerator. ELSA was upgraded in order to accelerate polarised electrons, compensating for depolarising resonances by applying the methods of fast tune jumping and harmonic closed orbit correction. With the experimental investigation of the GDH sum rule, the flrst experiment requiring a polarised beam and a polarised target was successfully performed at the accelerator. In the near future, the stretcher ring will be further upgraded to increase polarisation and current of the external electron beams. In addition, the aspects of an increase of the maximum energy to 5 GeV using superconducting resonators will be investigated. PACS. 29.20.-c Cyclic accelerators and storage rings

1 The beginning Accelerator physics started at Bonn in 1952, when Wolfgang Paul, who received the Nobel price 1989 for his work on ion traps, accepted a call to Bonn and became a full professor at the physics institute of Bonn university. Paul, who had worked at the university of G˜ ottingen on a 6 MeV betatron together with his teacher Hans Kopfermann, was deeply impressed by the recently discovered principle of strong focusing in particle accelerators, applying transverse magnetic flelds with strong alternating fleld gradient [1,2,3]. He was thinking about building a strong focusing 100 MeV electron synchrotron at Bonn and made an application to the Deutsche Forschungsgemeinschaft (DFG). Although this application covered about 10 % of the total annual funding of the DFG, it was accepted, and Paul was even encouraged to increase the energy of the planned machine to 500 MeV.

a

Since 2001 in charge of the ELSA accelerator; e-mail: [email protected]

29.20.Lq Synchrotrons

29.27.Hj Polarised beams

2 The 500 MeV synchrotron In November 1953, Paul started, together with a small group of scientiflc assistants, PhD and diploma students, to work on the design and construction of the accelerator. At this time, there was no experience in building a strong focusing accelerator, and a lot of problems had to be overcome. Since at that time no electronic computer was available, a pendulum as a mechanical analogue was built in order to solve the difierential equations describing the particles’ motion. A fleld gradient of 10 T/m at the maximum energy of 500 MeV was proposed, actually the highest gradient ever used for a combined function synchrotron. After construction had started in 1954 and all of the nine magnets of type 1/2D-F -1/2D had already been installed, it turned out that the accelerator would operate just on top of a non-linear stop band, which had been discovered 1956 at CERN, and would never work. This problem could be solved by enlarging the gap between the defocusing and focusing sectors of the magnets, using 1 cm thick pieces of plywood. The complete radio frequency system, consisting of the transmitter, the ampliflers and the six accelerating cavities, was built in the framework of a diploma thesis [4]. Pure ceramics was used for the vacuum chamber, which

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Fig. 1. The Bonn 500 MeV strong focusing electron synchrotron.

was constructed from difierent tubes glued together with AralditTM . A van de Graafi accelerator generated a 3 MeV electron beam, which was injected into the synchrotron. The transfer beamline was calculated and set up by a PhD student [5], and a 60 kV electrostatic deflector served for particle injection. Its housing was flnally completely built out of AralditTM in order to overcome the beam deteriorating efiects of eddy currents, generated by the fringe flelds of the bending magnets. Figure 1 shows a photograph of the 500 MeV synchrotron and part of its injection beamline. The main parameters of the machine are given in table 1 (see also [6]). After overcoming a couple of additional throwbacks, the accelerator came into operation in 1958, being the flrst strong focusing synchrotron operational in Europe. This happened to some extent unexpectedly, and actually the institute was not prepared to perform scientiflc experiments at the accelerator at this time. So Paul sent a member of his machine group to the USA in order to learn about physics experiments at the weak focusing 1 GeV synchrotron of the California Institute of Technology. In the following years, a number of pilot experiments, mainly on the photoproduction of pions ofi protons and deuterons, were carried out. For the flrst time the recoil neutron polarisation was measured. Two rotating targets produced external photon beams by the process of bremsstrahlung. The energy of the external photon beam was measured by means of tagging counters using one bending magnet of the synchrotron as dispersive element to determine the momentum of the scattered electrons. The 500 MeV synchrotron was operated until 1984. The total operation time exceeded 100000 hours. More than 150 diploma and doctoral theses were carried out

Table 1. Main parameters of the 500 MeV synchrotron. Focusing type Number of basic periods Basic lattice Maximum fleld at design orbit Maximum fleld gradient Repetition frequency Circumference Orbit radius Revolution frequency No. of betatron oscillations per turn Gap height at orbit Momentum compaction factor Coils, no. of turns per magnet unit AC voltage / current DC voltage / current Accelerating frequency Peak voltage per turn Number of accelerating cavities Injection energy Injector type Vacuum chamber Pressure (4 oil difiusion pumps)

AG 9 1/2D-F -1/2D 1T 10 T/m 50 Hz 16.45 m 1.70 m 18.12 MHz 2.4 6 cm 0.16 98 11670 V / 215 A 100 V / 180 A 163.1 MHz 2.5 kV 6 3 MeV van de Graafi ceramics 10 6 Torr

based on the construction of the machine and scientiflc experiments at this accelerator.

3 The 2.5 GeV synchrotron In 1963, it became clear to the physicists performing experiments at the 500 MeV synchrotron that the

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Fig. 2. The Bonn 2.5 GeV electron synchrotron, close to its flnal set-up.

experimental facilities of the Bonn accelerator laboratory had to be extended in order to keep up with the development of high-energy physics worldwide. A new electron accelerator with higher energy was considered, which should again be a university machine where scientiflc assistants and students should participate in design and construction. At this time, the DESY synchrotron came close to operation, the Cambridge accelerator was already operating and it was decided to build the NINA synchrotron. In order to contribute to fllling the gap between these 6 GeV machines and the existing ones around 1 GeV, the Bonn group decided to build an electron synchrotron with a maximum energy of 2.5 GeV. At the end of 1963, the state government agreed to support the new accelerator and the design of the machine started. The general dimensions of the synchrotron were flxed by the limited area of approx. 30 × 60 m2 available between the two physics institute buildings. Therefore a simple magnetic structure with twelve combined function bending magnets based on a O/2-F D-O/2 lattice was chosen. In order to correct for chromatic efiects, an additional sextupole component was introduced in the focusing and defocusing sectors of the bending magnets. To reach a short construction time the pole proflle and the cross section of these AG magnets were calculated on a computer, using difierent two-dimensional relaxation methods which had been developed shortly before [7,8]. In order to avoid corrections of quadrupole and higher order flelds at the time of injection, it was intended to inject at a fleld of at least 0.01 T. From the beginning on a slow extraction of the electron beam was planned. In 1964, the design was completed and the major parts were ordered. In 1965, the accelerator building and the laboratories were constructed. Manufacturing of the magnets’ blocks was started based only on the performed computer simulations. In contrast to the usual design proce-

Table 2. Main parameters of the 2.5 GeV synchrotron. Focusing type Number of basic periods Basic lattice Field at 2.3 GeV Field index n Repetition frequency Circumference Orbit radius Revolution frequency No. of betatron oscillations per turn Gap height at orbit Maximum of momentum compaction Minimum of momentum compaction Coils, no. of turns per magnet unit Peak current at 1 T Vacuum chamber Accelerating frequency Voltage per turn RF peak power Number of accelerating cavities Injection energy Injector type Accelerating frequency (injector)

AG 12 O/2-F D-O/2 1.003 T −22.26, 23.26 50 Hz 69.6 m 7.65 m 4.3074 MHz 3.4 6 cm 1.647 m 0.81 m 36 1360 A ceramics 499.67 MHz 700 kV 40 kW 2 25 MeV linac 2998 MHz

dure at other labs, no magnet model was checked in advance. The efiects of the transition zones between the Fand D-sectors and the various identical blocks of each sector, which could not be modelled by two-dimensional computer simulations, were measured after the flrst magnet had been assembled, and were compensated by well designed exponentially shaped end blocks forming the magnet ends. At the end of 1966, almost all components were in place and in March 1967, the flrst electrons could be injected

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Fig. 3. Floor plan of the synchrotrons and the associated experiments.

and, without problems, accelerated up to 2.3 GeV. Acceleration was performed by means of two cylindrical RF cavity resonators, which had been developed and constructed in a special electroforming process by DESY, Hamburg [9]. A conventional television transmitter, built by Telefunken, with a modifled power amplifler was used as RF power source. A linear accelerator, manufactured by Varian (Paolo Alto, California), served as injector, delivering a pulsed 25 MeV beam with a current of 250 mA at an energy spread of ±0.5 %. In July 1967, the accelerated electrons could be extracted with an e– ciency of about 60 %. Slow beam extraction was performed by exciting a half integer horizontal betatron resonance with the help of the nonlinear magnetic fleld of a current strip. The main parameters of the 2.5 GeV synchrotron are listed in table 2 (see also [10]). Figure 2 shows a photograph of the machine, close to its flnal set-up. In its

flrst operation period from 1967 to 1984, the synchrotron served as accelerator for particle physics experiments and as source for synchrotron radiation. A broad spectrum of physics experiments has been performed at this machine, starting from photo- and electroproduction of pseudoscalar mesons ofi protons and deuterons, continuing with photoproduction of associated strangeness in KΛ/Σ 0 and flnally including the measurement of the recoil nucleon polarisation. Since 1970, polarised solid state targets (starting with polarised protons and later continuing with polarised neutrons and deuterons) became available and were also employed. A floor plan of the experimental hall is shown in flg. 3, presenting the experimental set-ups at the two synchrotrons. The total operation time of the 2.5 GeV synchrotron amounted to 85000 hours for its flrst operation period. In

W. Hillert: The Bonn Electron Stretcher Accelerator ELSA: Past and future

143

PHOENICS Magnetstromversorgung Synchrotron

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M

Dipole (horizontal) Dipole (vertical) Quadrupole Sextupole Combined-Function Magnet Solenoid Radio Frequency

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Fig. 4. Floor plan of the ELSA accelerator laboratory, indicating the flrst experimental set-up of the time period 1989 1997.

total 210 diploma and doctoral theses were carried out during this time.

4 The 3.5 GeV ELSA stretcher ring After more than ten years of experimental particle physics at the 2.5 GeV synchrotron, it turned out that the quality of the experimental data was severely limited by the low duty factor of the pulsed synchrotron, which amounts to about 5 % and could not be increased signiflcantly with this type of accelerator. Following the suggestions of other laboratories around the world, it was decided to build a pulse stretcher ring, using the 2.5 GeV synchrotron as injector [11]. A proposal was made in 1979, which was revised in 1981 and flnally accepted at the end of 1981. After several years of planning and designing, the construction of the new ELectron Stretcher Accelerator ELSA started in 1982. To avoid additional costs, the 500 MeV synchrotron was dismantled in 1984 in order to use the existing accelerator building for experiments at the stretcher ring. The new accelerator was placed in a separate tunnel system constructed below the physics institute. ELSA is a separated function machine of simple FODO-type, which provides radiation damping of the horizontal betatron oscillations necessary for beam storage and allows a wide range variation of the betatron tune. In addition to the dipoles and quadrupoles, a total number of twelve sextupoles were installed for correction of chromatic efiects

and for excitation of a third integer betatron resonance needed for slow beam extraction. Two long straight sections with vanishing dispersion are equipped with the accelerating cavities and beam injection and extraction elements. Depending on the maximum beam energy chosen, a single-cell resonator of DORIS type or two flve-cell resonators of PETRA type are driven by klystron-based transmitters, operating at a frequency of 500 MHz and delivering a maximum power of 40 kW and 250 kW, respectively. The vacuum system is based on thin wall (0.3 mm) oval tubes of stainless steel, whose rigidity is provided by 1 mm thick reinforcing ribs being brazed on the tubes. A floor plan of the ELSA facility is shown in flg. 4, representing the flrst experimental set-up (1989 1997). The main parameters of the stretcher ring are listed in table 3. ELSA can be operated in three difierent modes: the stretcher mode, the post accelerator mode and the storage mode. Stretcher mode: Single pulses from the booster synchrotron are injected into the stretcher ring at a maximum rate of 50 Hz. Using a slow extraction at a third integer tune, an external electron beam of constant intensity is obtained for the time between two injections. The maximum energy is limited to 1.6 GeV by the beam transfer from the synchrotron to the stretcher ring. Post accelerator mode: After injection of several pulses, the accumulated electron beam is accelerated to the required energy and then extracted slowly. The maximum energy obtainable is 3.5 GeV, limited by the dipole magnet

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Table 3. Main parameters of the 3.5 GeV stretcher ring ELSA. Focusing type Number of basic periods Basic lattice Field at 3.5 GeV Circumference Orbit radius Revolution frequency Number of dipoles Number of quadrupoles Number of sextupoles Gap height (dipole magnets) Momentum compaction factor Natural emittance (3.5 GeV) Natural energy width (3.5 GeV) Current (dipoles) at max. energy Vacuum chamber Accelerating frequency Energy loss per turn (3.5 GeV) Maximum RF power Accelerating cavities transmitter 1 Accelerating cavities transmitter 2

AG 16 F ODO 1.073 T 164.4 m 10.88 m 1.82 MHz 24 32 12 5 cm 6.3 % 0.9 mm mrad 0.09 % 3015 A stainless steel 499.67 MHz 1.22 MeV 250 kW 1 of type DORIS 2 of type PETRA

power supply and the RF generated acceleration voltage. The macroscopic duty factor depends on the ramping speed and the flat top time and scales inversely proportional to the external current. For typical operation parameters (3.2 GeV, 1 nA external current) a duty factor of about 60 70 % is achieved. Storage mode: When ELSA is operated as a synchrotron radiation source, a large number of pulses from the synchrotron are accumulated in ELSA. Then the energy is ramped slowly to the desired value and the beam is stored for hours. The beam lifetime depends on the circulating beam current and energy and amounts to 1 2 hours for typical operation parameters (2.3 GeV, 50 mA). Until 1994, the flrst set-up of the accelerator control and timing system did not allow for a fast ramping of the dipole magnets. ELSA was only operated in the stretcher and storage modes [12]. In 1990, experiments utilising the synchrotron radiation, which is emitted from the electrons when passing the bending dipole magnets of ELSA, started in a separate laboratory constructed close to the accelerator tunnel. Since that time, a total number of six bending magnet beamlines are used for X-ray lithography and radiation chemistry [13], molecular fragmentation and Auger- [14], photoemission- and X-ray absorption spectroscopy experiments [15,16] and time resolved studies [17]. For the flrst years, one method of beam extraction was based on scattering the circulating electrons ofi a carbon wire and collimating the external beam to the required beam dimensions. Since 1991, after suppressing the ripple of the main magnet power supplies by more than a factor of 100 with the help of active fllters, a slow resonance extraction is successfully applied, using the following technique:

Menu

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3 Hewlett Packard B2000/B2600 Operating system: HP-UX (Unix)

Menu program

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More than 12000 parameters in data base, 19 permanently running expert programs.

TCP/IP UDP, RPC Local data base

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Equipment control layer 32 VME computers with real time OS VxWorks (WindRiver) 5 PCs with Linux

TCP/IP, HDLC, BUEP64, RK512

Device interface layer 12 PLCs 57 MACS computers (monoboard computers developed in Bonn) IEC bus, analog and digital I/O... Power supplies etc.

Power supplies etc.

measuring instruments etc.

Equipment

Fig. 5. Hierarchical structure of the ELSA control system (status 2005).

A third integer resonance is driven by four extraction sextupoles, located in dispersion free sections and dividing the phase space into a stable triangle and an unstable surrounding area. With the additional help of four ironless quadrupoles, the horizontal betatron tune is slowly shifted to a value of 4 23 , thereby shrinking the area of the stable triangle. When applying this procedure, part of the electron beam becomes unstable, resulting in an increasing betatron amplitude. These unstable electrons move along the separatrix lines in phase space and are flnally extracted after crossing the septa of two extraction magnets. The major problem of the pure stretcher mode operation turned out to be the achievement of a homogenous fllling of ELSA with one injection from the synchrotron. Due to the circumference ratio of 3 : 7 of these two machines, a sophisticated three turn shaving extraction from the synchrotron had to be employed in order to obtain a complete fllling of ELSA. Applying this method, the phase space structure of the ELSA beam often varied along the closed orbit shortly after injection, even in case of careful adjustments. This structure was not equalized by radiation damping within the short extraction time and caused small scale intensity variations of the external beam, thus decreasing the duty factor. In 1994, a newly developed accelerator control system [18,19] was set up. This system is based on three Unix workstations and in total consists of more than 100 computers (status 2005) at difierent layers (see flg. 5). Its

W. Hillert: The Bonn Electron Stretcher Accelerator ELSA: Past and future

graphical user interface combines machine steering, diagnostics and data analysis in one integrated environment. In combination with the control system, a new accelerator timing system, based on programmable delay units, was also set up. The improved systems were the key to precise and flexible tracking of the main magnets on the energy ramp and enabled a fast ramping operation (maximum speed about 7 GeV/s) of ELSA. In addition they allow for joining of several injections from the synchrotron in ELSA with adjustable overlap, thus making a nearly perfect homogeneous fllling of the stretcher ring possible. Since the successful implementation of the post accelerator mode at the end of 1994, ELSA was no longer operated in the stretcher mode. Difierent experimental set-ups were supplied with an external electron beam. The PHOENICS experiment was performed until the end of 1996, utilising a polarised frozen spin target and carrying out pion and eta photoproduction ofi protons and deuterons (see [20] and references therein). The ELAN experiment (1988 1997) performed electroproduction and electrodisintegration experiments, detecting the scattered electrons with a magnetic spectrometer [21]. It was flnished in August 1997 and followed up by an experiment requiring a beam of circularly polarised photons and a polarised proton target in order to investigate the contributions to the GDH sum rule in the energy range accessible at ELSA (see [22] and references therein). The GDH experiment was carried out in the years 2000 2002 after ELSA had been successfully upgraded for the operation with polarised electrons (see sect. 5). The SAPHIR spectrometer was operated from 1991 to 1999, carrying out photoproduction experiments and concentrating on the detection of charged particles. A large variety of reactions has been studied with this detector, reaching from photoproduction of associated strangeness in KΛ/Σ 0 to the production of vector mesons ω, φ and ρ (see [23] and references therein). SAPHIR was disassembled in 1999 and followed up by a new experimental set-up (CB@ELSA), based on the 4π photon detector Crystal Barrel which had been moved in 1997 from LEAR/CERN to Bonn and equipped with an inner scintillating flber detector for charged particle detection and triggering. During the flrst data-taking period of the CB@ELSA experiment from 2001 to 2003, proton, deuteron and solid state targets were used in photoproduction experiments (see [24] and references therein). A linearly polarised photon beam, produced by coherent bremsstrahlung ofi a goniometer-aligned diamond crystal, was set up in 2002 for this experiment and has been routinely available since then.

5 Polarised electrons The production of polarised electron beams has been studied in Bonn for more than 35 years. Already in 1969, a source of polarised electrons, based on the photoionisation of polarised lithium atoms, was set up [25]. Beam

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intensity and polarisation were enlarged using circularly polarised UV laser light for photoionisation of unpolarised atoms. Based on this method (Fano efiect), two sources of polarised electrons were built in Bonn; one operating with caesium, the other with rubidium gas [26,27]. At the Bonn 2.5 GeV synchrotron, with the help of the rubidium source, polarised electrons were accelerated for the flrst time worldwide in a synchrotron [28]. The work on sources of polarised electrons was continued after the construction of ELSA, using the photoemission of GaAs crystals pumped with circularly polarised laserlight. After more than flve years of work setting up a suitable vacuum system and a pulsed titanium sapphire laser of su– cient power, another source, based on a GaAs-AlGaAs superlattice photocathode was brought into operation in 1997 [29]. This 120 keV source was used to investigate the efiects of depolarising resonances in the stretcher ring [30]. During these studies, it turned out that reliability, life time and beam transfer e– ciency of this source would be insu– cient for hadron physics experiments. The situation improved with the construction and assembling of a new source, adapted for the operation with a second linear accelerator, which had been moved from the university of Mainz to Bonn and installed at the 2.5 GeV synchrotron. This 50 keV source [31] is based on an inverted-geometry electron gun, operated in space charge limitation in order to suppress the spiking of the free running flashlamp-pumped titanium sapphire laser. Special care was taken in construction and set-up of the gun and the transfer beamline to reach and maintain a low base pressure (10−11 mbar) and extremely low partial pressures of poisoning gas species (10−14 mbar) by application of difierential pumping [32]. To improve the gun vacuum and consequently the lifetime of the photocathodes, heat cleaning and activation of the photocathodes are carried out in a load-lock system, which in addition allows to change crystals without breaking the vacuum of the gun (see flg. 6). The 50 keV source was operated from 2000 to 2003 with a Be-InGaAs/AlGaAs strained layer superlattice photocathode [33] for machine studies and the GDH experiment, emitting a peak current of 100 mA in rectangular 1 μs long electron pulses with about 80 % beam polarisation and demonstrating photocathode lifetimes of more than 2000 hours. To prevent depolarisation during acceleration in the circular accelerators due to spin precession around the guiding fleld of the dipole magnets, the electron spins, originally orientated longitudinally at the source, have to be rotated in order to point perpendicularly to the accelerator plane. This rotation is performed by a 90 degree electrostatic bend in the low energy beamline from the source to the linac. An additional Larmor precession in the linear accelerator, caused by its focusing solenoid lenses, is compensated by additional double solenoids in the injection beamline, which allow to vary the spin rotation angle while their focusing strengths remain flxed. After beam extraction out of the stretcher ring, a superconducting solenoid rotates the spin back into the accelerator

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Operation Parameters Acceleration voltage 48 kV Repetition rate 50 Hz Pulse length 1 s Pulse current 100 mA Beam polarisation 80 % Laser spot size (diameter) 8 mm -11 Pressure in gun chamber 10 mbar Beam lifetime > 2000 h

Fig. 6. Set-up of the inverted source of polarised electrons and the load-lock system. 1 L

J

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Px

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1

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Fig. 7. Spin transfer to the tagger of the GDH experiment.

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plane via Larmor precession. Longitudinal polarisation at the radiator target is obtained by Thomas precession of the spin in the bending flelds of the two dipole magnets of the external beamline located downstream of the solenoid. Due to the limited fleld strength of the solenoid and the flxed bending angle of the dipoles, the transformation of transverse to longitudinal polarisation is incomplete and a transverse component is always present at the radiator target (see flg. 7). Besides the incomplete spin transfer, severe beam depolarisation is caused by speciflc depolarising resonances appearing at certain magic energies in the circular accelerators. These resonances are caused by horizontal magnetic flelds, present in every quadrupole and combined function bending magnet and acting on the electrons if they are passing the magnet out of its central plane. Two difierent types of resonances have to be distinguished: Imperfection resonances are caused by vertical displacements of the beam or misaligned focusing elements and will depolarise the beam if the spin tune γa equals an integer number. In case of electrons this happens at energies which are integer multiples of 440.65 MeV. Intrinsic resonances are caused by the flnite vertical beam size originating from the vertical betatron oscillations. The resonance condition is therefore linked to

Beam Energy / MeV

Fig. 8. Depolarisation due to crossing of synchrotron sidebands, derived from numerical simulations [35].

the machine optics and depends on the vertical betatron tune Qz and the superperiodicity P of the accelerator: γa = kP ± Qz , where k is an integer and P = 2 for ELSA, but P = 12 for the booster synchrotron, respectively. When crossing a resonance, the depolarisation Pf /Pi depends on the crossing speed α and the resonance strength . Neglecting the in uence of the emission of synchrotron radiation, we obtain, applying the well known Froissart Stora formalism (see, e.g., [34,30]), a conservation of the polarisation for vanishing resonance strength and a spin ip √ (Pf /Pi = −1) for considerably strong values of  (|| / α  1). If additional depolarisation, originating from crossing of synchrotron side-bands, is included (see flg. 8), it turns out that a total spin flip cannot be observed at ELSA at energies higher than 1.6 GeV [36]. Therefore, it is not feasible to enhance the strengths of all resonances su– ciently in order to avoid depolarisation,

W. Hillert: The Bonn Electron Stretcher Accelerator ELSA: Past and future Horizontal orbit

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Source Source

Fig. 9. Fast tune jumping of all intrinsic resonances in ELSA.

but it is required to correct for all resonances strong enough to cause a signiflcant depolarisation. Due to the fast ramping speed and high superperiodicity, no signiflcant depolarisation is observed in the booster synchrotron if the beam is transferred to ELSA at energies below the third imperfection resonance at 1.32 GeV. A difierent situation shows up for the stretcher ring. It turned out from numerical simulations that at least nine resonances in ELSA are strong enough to produce signiflcant depolarisation. Three techniques are applied to avoid depolarisation: In case of intrinsic resonances, the crossing speed is enhanced with the help of two pulsed betatron tune jump quadrupoles [37], thereby shifting the vertical betatron tune by ΔQz ≈ 0.1 in a Δt2 ≈ 10 ms long triangular pulse with a rise time of Δt1 = 4 μs (see flg. 9). The strengths of the imperfection resonances are reduced with a dynamic correction of the closed orbit during the energy ramp. Vertical and horizontal displacements of the beam in the quadrupoles are measured by 28 monitor stations [38] and corrected with 19 horizontal and 21 vertical corrector magnets [39]. The following correction scheme is used: The beam is stored at the energies of the imperfection resonances and the optimal corrections are determined. Afterwards a linear interpolation between these corrections is applied during the energy ramp. The remaining distortions are generally smaller than 0.2 mm rms (see flg. 10). Further reduction of the resonance strengths is achieved by correction of speciflc harmonics of closed orbit distortions relevant for a single imperfection resonance. This method is based on an empirical determination of two amplitude factors for each resonance by parameter

Energy / GeV

Fig. 11. Achieved beam polarisation in the stretcher ring.

variation and measuring the polarisation response of the extracted beam with a M¿ller polarimeter. With all correction methods applied successfully, polarised electrons can be accelerated up to 3.2 GeV. At energies higher than 2.0 GeV some polarisation loss is observed due to incomplete resonance compensation (see flg. 11).

6 Future plans Starting in 2006, a new experimental set-up of CB@ELSA (see flg. 12) will come into operation, utilising a polarised beam and a polarised nucleon frozen spin target for photoproduction experiments in the framework of the SFB/TR 16 Subnuclear Structure of Matter , funded by the Deutsche Forschungsgemeinschaft (DFG). With this set-up stronger demands will be put on the quality of the external electron beam. In order to enhance the beam polarisation and the external current at maximum beam energy, a number of improvements are planned in the near future, the most important of which are the following: The 32 old quadrupole vacuum chambers will be replaced by new watercooled ones, equipped with improved monitor stations (capacitive pickups) and clearing electrodes for ion clearing. The existing 40 corrector magnets will be removed and 60 newly designed magnets will be installed, together with four-quadrant power supplies developed at the institute.

Eur. Phys. J. A 28, s01, 149 160 (2006) DOI: 10.1140/epja/i2006-09-016-3

EPJ A direct electronic only

The Mainz Microtron MAMI —Past and future A. Jankowiaka Institut f˜ ur Kernphysik, Universit˜ at Mainz, J.-J.-Becher Weg 45, D-55128 Mainz, Germany / Published online: 15 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. The Mainz Microtron MAMI is a cascade of three racetrack microtrons, delivering since 1991 a high-quality 855 MeV, 100 μA cw-electron beam for nuclear, hadron and radiation physics experiments. An energy upgrade of this facility to 1.5 GeV by adding a Harmonic Double-Sided Microtron (HDSM) as a fourth stage is well underway and flrst beam is expected during the flrst half of 2006. A detailed description of the multiple recirculation scheme with normal conducting accelerator structures, the basis for the reliable operation of MAMI, is given and the historical development from MAMI A to MAMI B is described. The natural advancement to MAMI C by realizing a polytron of the next higher order, the HDSM, is covered in the last section and a flrst glimpse into the future of MAMI is given. PACS. 29.20.-c Cyclic accelerators and storage rings 41.75.Lx Other advanced accelerator concepts 41.85.Lc Beam focusing and bending magnets, wiggler magnets, and quadrupoles

1 Introduction Since the late 1950s the electromagnetic probe had proven to be the most successful precision tool for investigating the internal structure of the atomic nucleus and hadrons. A great deal of the early experiments was done by high accelerating gradient (∼ 20 MeV/m) pulsed linacs, flrst with a duty factor (DF) around 0.1% and flnally (Saclay, Amsterdam, MIT [1]) up to 2%. However, because the capability of these latter machines for coincidence experiments was still limited, a strong demand came up in the early 1970s for continuous wave (cw, 100% DF) high quality electron beams in the few 100 MeV to multi GeV range (“Lindenberger und Pinkau Ausschuß” 1980 in Germany, “Livingston Report”, 1978 and “Barnes Report” 1980 in the USA ). Two difierent paths to satisfy this demand were discussed a) the pulse stretcher-ring [2], mostly designed for upgrading an existing pulsed injector, with the capability to reach energies in the some GeV regime, but always limited by the maximum achievable current, and b) the multiple recirculation of the beam through a linac. For this path b) again two methods were suggested: In the independent orbit recirculation scheme the beam is guided a few times through one or two linacs by a quite complex, but very flexible achromatic and isochronous return optics. Clearly the cw-gradient must be as high as possible and, therefore one would prefer using superconducting (sc) rf-technology. An advantage of these machines is their potential for energy upgrades e.g., since 1975 the limit for a stable cw-operation of sc-cavities increased from ca. 4 MeV/m to >20 MeV/m [3]. However, for a high beam a

e-mail: [email protected]

quality an extraordinary amplitude and phase stability of the rf-wave is necessary. The other possibility is to guide the beam by a few simple combined optical elements many times through a low gradient linac, and for this the racetrack microtron (RTM) scheme ([4,5,6]) lends itself. Rf-gradients of 1 MeV/m can be achieved very stable and reliable with quite low power (∼ 15 kW/m) by conventional but highly developed normal conducting (nc) accelerating structure technology. Further, with the high beam load by many recirculations one gets a high e– ciency. Because of the strong longitudinal focusing of the RTM the demands on the stability of the rf-wave are only moderate; however, realising the necessary excellent homogeneity of the magnetic fleld in the two 180◦ -magnets ΔB/B ≤ 10−4 is an ambitious task. With electron beams one has to aim for a precision of better than 1% of the very small cross sections in order to achieve meaningful physics results. Therefore, the accelerator must be expected to running reliably for up to 7000 hours a year, delivering a stable beam of excellent longitudinal and transverse emittance to the experiments. In addition, for measuring small interference efiects, beams of polarised electrons are crucially important. So in 1975 a detailed design study and the construction of an RTM-cascade MAMI was started at the Institut f˜ ur Kernphysik (IKPh) at the University of Mainz [7], with the goal to realise a world class accelerator facility, capable to deliver for the flrst time an excellent cw-electron beam of up to about 800 MeV. At the same time flrst considerations on the development of a polarised photocathode gun began.

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reversed fleld stripe of −20% B, so that each magnet acts like a πR long drift. In the horizontal plane the transformation through one magnet is given by the negative unit matrix. For transverse focusing two schemes were discussed: a) A strong focusing with quadrupoles on each return line, their strength adapted to the increasing beam energy to e.g. stay with constant beta-functions for each turn. b) A weak focusing with only two quadrupole-doublets on the linac axis with naturally decreasing strength ∼ 1/E 2 and therefore increasing beta-functions. Fig. 1. Basic setup of a racetrack microtron (RTM).

2 The racetrack microtron (RTM) The basic scheme of a racetrack microtron is depicted in flg. 1. For a phase coherent acceleration one has to fulflll two conditions (β = v/c = 1 assumed): a) The so called static coherence-condition for the length of the flrst complete circulation L1 = k · λrf =

2π(EInj + ΔE) + 2d, ecB

(1)

which must be an integer multiple of the rf-wavelength λrf and has to be adjusted either by the injection energy EInj or the distance d of the 180◦ -dipoles (magnetic fleld B). b) The dynamic resonance-condition for the increase in path length from turn to turn Li+1 − Li = 2πΔR = n · λrf ,

(2)

which must be also an integer multiple of the rf-wavelength and is fulfllled by setting the energy gain per turn to ecB · n · λrf . ΔE = (3) 2π Moreover, one has to consider, that the synchronous phase range for stable longitudinal motion is given by − 2 < n · π · tg(ϕs ) < 0,

(4)

and for practical reasons (individual correction steering in the dispersive section) that the distance of the return paths is 2ΔR = n · λrf /π. Evidently for an ample longitudinal stability range and low rf-power consumption n = 1 is the natural choice (−32.5◦ < ϕs < 0). As rf-power-source a 50 kW cwklystron just developed for industrial heating was available in 1975 (Thomson-CSF TH2075, νrf = 2449.5 MHz, λrf = 12.24 cm), so that one got 2ΔR = 3.9 cm, enough distance for introducing slender correction steerers on each return path tube. Concerning the beam optics the vertical defocusing in the fringe fleld of the 180◦ -magnets is compensated by a

It was decided to realise option b), because option a) has the disadvantage of introducing a strong transversal/longitudinal phase space coupling, and also the fabrication of very slender quadrupoles with low higher multipole content would have been di– cult. Due to pseudo ) the beam damping (emittance x,y = 1/(βγ) · normalised x,y size stays nearly constant with increasing number of turns; however the phase space ellipse gets flnally quite flat. The focusing of beams of all energies simultaneously in the same quadrupole doublets is the one reason for the so called Herminghaus rule [7], that the ratio of output to input energy of an RTM should not exceed a factor of about 10. The other is that at a given injection energy there is an upper limit for the magnetic fleld strength B, in order to allow for enough space for the flrst return of the beam backwards to pass aside the linac structures. In order to stay within reasonable limits with the necessary strengths of the return path correctors and also to avoid noticeable distortions of the dynamic resonance condition of eq. (3), the B-fleld of the end magnets must be homogeneous to about 10−4 . Because the homogeneity of the casted iron and the available mechanical manufacturing precision allows only an accuracy of some parts in 10−3 , an extensive mapping of the magnetic fleld distribution had to be done. Based on these measurements surface correction coils were constructed, flattening the magnetic fleld distribution by more than a factor of ten (flg. 2, [8]). A problem which had shown up at the flrst recirculators with super conducting accelerator structures ([9, 10]) was the regenerative beam blow up (BBU). If the bunched beam passes the accelerating structure slightly ofi axis, it excites a T M110 -like deflecting rf-mode at ca. 1.7fold the frequency of the accelerating T M010 -like mode. Its amplitude is proportional to its ofiset and the beam gets a small transversal kick (flg. 3). If positive feedback conditions for the next returns are given, this will lead to beam loss above a certain threshold current IsBBU . According to a short cavity model of this process [11] a worst case approximation is Ez · λBBU 1 1 , · · (R/Q)BBU · QBBU βf oc N · ln(Eout /Ein ) (5) where (R/Q)BBU is the shunt impedance / rf-quality factor of the BBU-mode, Ez the accelerating fleld strength, βf oc the average beta-function of the recirculation, N are ISBBU ∼

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Fig. 2. Example of a fleld map of a RTM 180 -dipole magnet before and after applying the surface correction coils.

the number of turns, Eout/in is the output/input beam energy, respectively. Numerical BBU-calculations together with rf-measurements on the MAMI biperiodic accelerating structures [12] showed, that by staggered T M110 -detuning of the linac sections the threshold current can be shifted distinctly above the maximum design current of 100μA.

3 From MAMI A to MAMI B Following these design principles MAMI was realised as a 3-RTM-cascade between 1979 and 1990. Since the skepticism that a machine with many recirculations could be build a proof of principle was requested. Therefore, a 14 MeV stage (MAMI A1) was built for testing and optimising the rf-structure and rf-control, the B-field correction by the surface coil technique and, quite advanced at that time, a complete computer control using steering algorithms. It was set into operation in March 1979 and later used as the first RTM (RTM1) of MAMI A and MAMI B. This machine was already used for first physics experiments [13] from November 1979 on. Only the klystron TH2075 and the Van-de-Graafi injector were bought, otherwise it consisted completely of in-house-made or used components as, e.g., the end magnets from DESY, Hamburg.

Originally, the second stage RTM2 was planned with an end energy of only 100 MeV with one klystron TH2075 feeding RTM1 and RTM2 [7]. However, because of a strong demand to surpass already with this machine distinctly the pion production threshold, it was decided to add a second klystron and accelerate in RTM2 by a factor of thirteen from 14 MeV to 180 MeV. The cost increase for the larger end magnets could be lowered by using the iron of the Heidelberg Cyclotron in use from 1943 to 1973 [14]. This setup (Van-de-Graafi + RTM1 + RTM2 ≡ MAMI A) delivered from July 1983 to October 1987 about 70% of its 18700 hours of beam time for hadron and nuclear physics experiments. The maximum achieved beam parameters were 187 MeV beam energy and a current of 65 μA. Its operation was funded since 1984 by a Collaborative Research Centre (SFB201, Medium energy physics with the electromagnetic interaction ) and the main components were bought via the HBFG funding. A somewhat weak point of MAMI A was the cheap Van-de-Graafi injector. Its maximum usable voltage of only 2.1 MV (β = 0.981) caused a migration of the operating phase in RTM1 from +15◦ to −22◦ , resulting in a reduced longitudinal acceptance and stability. This adverse efiect was enhanced by the high sensitivity of the stability of the high voltage of the Van-de-Graafi to any impact of background γ radiation. Because this background

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Fig. 3. Simplifled sketch of the regenerative beam blow up (BBU) mechanism.

Fig. 4. Annual operation time of MAMI, according to machine setup (tuning and development), polarised and unpolarised beam time, for the years 1991 to 2005. It should be noted that in 2001 a half year shutdown took place for the preparation of the beam lines for MAMI C. Of the remaining 4428 hours the MAMI was operating for 4277 hours, i.e. 97%.

radiation increased with increasing beam current, the voltage instabilities of the Van-de-Graafi were the reason for the limit of max. 65 μA beam current. Moreover, the mediocre vacuum conditions and bad accessibility of the high voltage terminal were evidently prohibitive for any operation of a GaAs-photocathode source of polarised electrons. Therefore, when transferring MAMI A as the injector for RTM3, the Van-de-Graafi was replaced by a 3.5 MeV linac designed and built in-house [15] with high energy stability (≤ 1 keV; β = 0.992). At this energy the phase migration in RTM1 is only −12◦ to −22◦ . The final scheme realised as MAMI B, set up in new halls from 1987 to 1990 with first operation in August 1990, is shown in fig. 12, and its main data are given in table 1. Apart from its excellent beam parameters (table 1), the machine showed an extraordinarily stable and reliable operation. The beam time over the years of operation since 1991, classified for machine setup and polarised and unpolarised operation for experiments, is shown in the histogram of fig. 4. The high e– ciency of MAMI is due on one side to the inherent properties of the RTM, but also to a considerable extent to a sophisticated monitor sys-

Fig. 5. Simplifled scheme of the measurement of the absolute beam energy (top) and beam energy fluctuations (bottom) in RTM3.

tem [16], allowing to computer control a wide diversity of parameters and feedback loops. Among the most important monitors are the low-Q rf-cavities on the linac axes of each RTM. They allow by injection of 10 ns diagnostic beam pulses during machine setup, actually realised as 10 ns blackouts in the cw-operation, to supervise the transverse positions and the phase and intensity of the beam for each recirculation individually at the entrance and exit of the linac. This information make quick and e– cient correcting actions possible by a machine model implemented in the computer control system [17]. The beam profile is viewed turn by turn and in the transfer lines between the RTMs via a synchrotron radiation camera system. Very helpful for fast tuning of the matching of the beam parameters is a synchrotron radiation camera with high magnification looking through the axes of the linacs of RTM2 and RTM3. All 51 (RTM2) and 90 (RTM3) turns have to overlap and any mismatch can easily be detected and globally corrected. A system of many T M110 -rf-cavities allows a control of the beam position in diagnostic-pulse and also cw mode (above 1 μA beam current) down to a few μm. The RTM configuration easily permitted in RTM3 the installation of two control setups extremely valuable for precision experiments (fig. 5). The distance between the return pipes is such that small 4 × 2.45 GHz = 9.8 GHz

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Table 1. Main parameters of MAMI. MAMI C comprises the injector linac, MAMI A, MAMI B and the HDSM being constructed. Injector General injection / extraction energy (total) number of turns total power consumption Rf-System energy resp. energy gain / turn frequency linac length (electrically) number of sections / klystrons power dissipation / beam power power consumption Magnet-System ux density (within the gap) gap height min./max. de ection radius iron / copper weigth of the magnets number of corrector magnets number of quadrupoles and solenoids power consumption Beam-Parameters energy spread (1σ) norm. emittance hor. / vert. (1σ) standard-energies for experiments a b

[GeV]

0.511/3.97 · 10

RTM1 3

3.97/14.86 · 10 18 92

RTM2 3

14.86/180 · 10 51 220

3

RTM3

HDSM

0.180 / 0.855 90 650

0.855 / 1.5 43 1400

[kW]

92

[MeV] [GHz] [m]

3.5 2.4495 4.93 3/1 33.2 / 0.35 90

0.599 2.4495 0.80 1/1 7.9 / 1.1 90

3.24 2.4495 3.55 2/2 48.4/16.6 180

7.50 2.4495 8.87 5/5 102.5 / 67.5 450

16.58-13.66 4.8990|2.4495 8.57|10.1 8/4|5/5 299 / 65 1000 a

[kW]

40 20 2

0.1026 6 0.129-0.482 4 / 0.2 72 2 2

0.5550 7 0.089-1.083 90 / 2.3 204 4 40

1.2842 10 0.467-2.216 900 / 11.6 360 4 200

1.53-0.95 8.5-13.9 2.23-4.60 1000 / 27.4 2 · 172 + 2 · 6 2·4 400

[keV] [π · 10 6 m]

1.2 0.05 / 0.04

1.2 0.07 / 0.07

2.8 0.25 / 0.13 180MeV

13 13 / 0.84 195-855MeV in steps of 15MeV

110 b 27 b / 1.2 b 0.855-1.5GeV in steps of ca. 15MeV

[kW] [kW] [T] [cm] [m] [t]

Including the power consumption of one matching section between RTM3 and HDSM. Simulation with SYTRACE, a particle tracking program including efiects of stochastic emission of synchrotron radiation photons.

- T M010 - and T M110 -resonators can be inserted there (T M110 -cavities at 720 and 855 MeV and T M010 -cavities at 315, 420, 510, 570 and 855 MeV). So flrstly, with the T M110 -position monitor and the known distance between this monitor and the linac axis, one can use one 180◦ dipole with its NMR-controlled fleld strength and precisely measured fleld map as a sensitive spectrometer with large 2.2 m bending radius to determine the absolute beam energy to ±2 · 10−4 (±140 keV at 855 MeV). Secondly, with an additional 9.8 GHz T M010 -cavity on the extraction path and monitoring its phase difierence to one of the return-pipe T M010 -resonators, one can precisely measure changes of the length of the last half turn after extraction of the respective energy. The corresponding energy change of the electron bunches is given by eq. (2) and the sensitivity reads 61.2 mm λrf /2 = = 8.16 mm/MeV, ΔE/turn 7.5 MeV

(6)

which corresponds to 96◦ phase per MeV at 9.8 GHz. With a resolution of 0.1◦ at 9.8 GHz energy changes of about 1 keV, corresponding to 1.2 · 10−6 at 855 MeV, are detected. A further increase of resolution seems not to be reasonable at the moment since fluctuations of the beam direction are producing signal levels of about the same amount. By feeding back the energy signal to the linac phase, it is possible to routinely provide this energy stability of ±1 keV during physics experiments. Of course, correct tuning and a su– cient stability of the longitudinal Q-value are key to the well functioning of the system [18]. This setup was of fundamental importance for the parityviolating electron scattering experiments, where the cross −5 section change with Ebeam and must be measured with a relative precision of 10−6 .

The development of the polarised source [19] after MAMI ran with a clean linac as injector is listed in the following time table: 1992 – 1995: First experiments for the A3-collaboration, beam current I = 5 μA, polarisation P = 30 − 40%. The 100 keV source was installed at ground level meaning a 14 m long beam line to the injector linac [20]. 1995: Introduction of strained layer cathodes, I = 2 μA, P = 75%. 1997: The source moves to the accelerator hall, which allows a much easier and reproducible injection of the beam into the linac. Installation of a 2f-prebuncher with 4.9 GHz [21], which allows for 145◦ longitudinal phase space acceptance instead of the design value of 40◦ [15], I = 10 μA, P > 75%. 1998: Introduction of the so called synchro-laser for pulsing the source, which allows for 90% transmission of the precious polarised electrons, I = 20 μA, P = 80%. 2001: Introduction of the mask activation technique, which strongly reduces losses of electrons starting at the cathode due to stray light and thus improves the vacuum conditions at the cathode. The charge extracted in one run was increased from 22 C to 115 C. From then on several weeks of continuous operation at high current I > 20 μA were possible. 2003 – 2005: A Wien fllter as spin-rotator at 100 keV electron energy directly behind the polarised source was installed. This allows for a much easier adjustment of the beam polarisation at the target for all experimental stations and all energies. Before, the spin direction at the experiments was controlled by tuning the MAMI end energy making use of the gyromagnetic

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Fig. 6. Detailed scheme of the Harmonic Double Sided Microtron (HDSM) for MAMI C.

anomaly of the electron. Currents of I > 30 μA with polarisations of P > 85% are now possible. Apart from steady improvements of the degree of polarisation and the lifetime of the photocathodes, the installation of the harmonic 2f-prebuncher at the injector linac and the rf-synchronised laser for up to 90% transmission e– ciency made it possible to increase the MAMI operation with polarised electrons from 20% to now 60% of the total beam time satisfying the demands of the experiments.

4 The Harmonic Double-Sided Microtron (HDSM) as the fourth stage of MAMI In 1999 a new Collaborative Research Centre (SFB443, Many-body structure of strongly interacting systems ) was founded, which for its second stage physics program demanded an electron beam of 1.5 GeV. The ideas for upgrading the MAMI energy had to consider as boundary conditions that the excellent beam quality and reliability of MAMI B must be preserved, that the new fourth stage had to flt into the existing buildings and that the research with the existing MAMI B had to go on without any longer shutdown periods. Moreover, considering the limited manpower capacity of the institute and the tight time schedule envisaged, it was evident that one had to base the new accelerator stage on the expertise of the institute. So very early the decision was taken to stay with the

well proven and tested technology applied for the RTMs: normal conducting rf-accelerator structures and iron core magnets with normal conducting excitation coils. The latter point clearly implied that one could not realise MAMI C as a fourth RTM. With iron core magnets one cannot increase the fleld strength very much beyond the 1.3 T of RTM3. The size and weight of the two such 180◦ end magnets would grow with the cube of the maximum energy, i.e. to formidable weight of 450 t×(1.5/0.855)3 = 2430 t each. However, the RTM with one linac is not the only possible microtron. Already since 1979 H. Herminghaus and K.H. Kaiser developed ideas and designs for higher order microtrons called Polytrons , as multi-turn recirculators with strong phase focusing ([22,23,24]). At the bicyclotron or Double-Sided Microtron (DSM) (see the scheme in flg. 6) the pole face area is reduced by a factor of (π − 2)/π compared to an RTM. Therefore, a 1500 MeV DSM has roughly the same magnets weight as an 850 MeV RTM. As the next step one must consider, however, that the dynamic coherence condition changes and for a DSM is given by ΔE/turn = n ·

ecB ecB · 2λrf = n · · λrf , 2(π − 2) π−2

(7)

and naturally one will take n = 1 for the lowest possible path lengthening of 2 · λrf /turn. With the parameters of RTM3 (B = 1.28 T, λrf = 0.1224 m) one would need

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Fig. 9. Longitudinal input and output phase space for the DSM- / HDSM-conflguration (left / right) with phasing errors of 3 @4.90GHz / 5 @4.90GHz.

Fig. 7. Field gradient perpendicular to the pole edge of the 90 dipoles of the HDSM for compensation of the vertical edgedefocusing.

Fig. 8. Development of the synchronous phases of both linacs in DSM conflguration for a phasing error between the two linacs of 3 .

ΔE = 41.1 MeV/turn. With the moderate well tested MAMI rf-gradient of 1 MV/m one would need 20 m long linacs, which would not flt into the existing buildings (flg. 12) and would moreover consume about four times the electric power of MAMI B. So it was evident that the frequency of the DSM had to be 4.90 GHz (λrf = 0.0612 m), with two about 10 m long linacs and with the other parameters similar to that of RTM3. Concerning the transverse optics, the 45◦ entrance and exit angles of the beam at the four 90◦ magnets are with their strong vertical defocusing a very critical point. A detailed investigation [25] was done for several quadrupole conflgurations on the dispersive paths (quadrupole triplet / two quadrupole doublets on each half recirculation). It showed, that this way of compensation would work only in

principle. To avoid strong distortions of the phase space, the quadrupoles must be extremely free of sextupole and other higher multipole errors and their individual setting and alignment for a dispersion free beam on the linac axis would be in practice a very cumbersome procedure. Therefore, it was decided to use the combined function solution, namely to introduce a magnetic fleld gradient perpendicular to the pole edge in the 90◦ dipoles, which compensates for the edge-defocusing in the complete range of beam energies. The corresponding fleld proflle is given in flg. 7. Due to the fleld decay from 100% to 60%, however, the mean fleld along the beam path decreases with increasing energy so that the necessary synchronous energy gain becomes lower and lower. As a consequence, the synchronous phase has to move away from the crest of the rf-waves in the linac in order to fulfll the dynamic coherence condition (eq. (7)). This happens automatically and smoothly by the longitudinal focusing if input energy and linac phases are optimised with respect to minimum phase oscillations. For an injection phase of −8◦ one will end at −34◦ for turn 43. This is well within the phase stable range of the DSM (−4 < n · π · tg(ϕs ) < 0, i.e. −51.9◦ < ϕs < 0◦ for n = 1), however, only for the ideally symmetric DSM, where one half turn can be considered as a machine period. If the symmetry is perturbed, e.g. by a wrong phase setting of one of the linacs, the DSM can be considered as spliting into two RTMs with a stable phase boundary of −32.5◦ . From flg. 8, assuming a phase error of only 3◦ between the two linacs, one sees that only the last ca. 10 turns are afiected by this stopband. However, more detailed tracking calculations showed that the acceptance of the DSM is distinctly diminished (flg. 9, left). Therefore, a standard DSM with gradient dipole magnets would have to be operated with extreme care for its symmetry, e.g. a phase change of 1◦ at 4.9 GHz would mean a change of a steering cables efiective length of only 0.1 mm. However, because the reason for this unstable region is flnally the too strong longitudinal focusing when the electrons slip down on the 4.9 GHz wave, a glance at the time table of the bunches in the two linacs of a DSM with sub-harmonic injection showed a way out of this di– culty (flg. 10). One can see, that in one of the linacs only every second bucket is populated by the recirculated bunches. So one could operate here with the MAMI B frequency

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Fig. 10. Simplifled scheme of the bunch arrival time in a DSM operating with a sub-harmonic injection frequency.

Fig. 11. Bunch phase migration in the DSM (phase scale shifted by 90 ) and HDSM conflguration. Clearly the instable area of the 4.9 GHz wave can be avoided only in the HDSM scheme.

of 2.45 GHz, and it turned out, that by an appropriate amplitude and phase relation between the 2.45 GHz and 4.9 GHz wave most of the necessary reduction of the energy gain can be overtaken by the low frequency linac with its less steep gradient (flg. 11). On the 4.9 GHz wave the bunches migrate now only from 0◦ to ca. −26◦ and so stay distinctly away from the dangerous −32.5◦ . This Harmonic Double-Sided Microtron (HDSM, [26]) shows a good longitudinal stability, e.g. a phase error of more than 5◦ between the two linacs is tolerable (flg. 9, right). Concerning the transverse optics, due to the moderate total energy gain factor of the HDSM of only 1.75, it is possible to stay with the horizontal and vertical beta-functions below 20 m during the acceleration process with just two quadrupole doublets on each linac axis. Tracking calculations taking into account the ef-

fect of emittance growth due to quantum fluctuations of synchrotron radiation1 [27], show that the normalised longitudinal/horizontal emittances increase only by a factor of 2/1.5. Therefore, the absolute emittance will stay nearly constant and the beam sizes on the linac axes are in the order of only some tenths of a millimetre. The flnal design parameters for the HDSM are given in table 1, its scheme in flg. 6 and its floor plan in flg. 12. The MAMI B frequency of 2.45 GHz was in a wellestablished industrial heating band, with many components available from the shelve, whereas at 4.9 GHz there existed nearly no high power rf-components. Therefore, here many developments had to be started. To have a quick start of production for the flve rf-sections needed for the 2.45 GHz linac, it was decided to make them as copies of the well tried ones at RTM3 [28] (with a slightly adapted length, 33 accelerating cells (AC) instead of 29). The contract was given to the INP of the Moscow State University collaborating with the klystron flrm TORIY, because of their great experience in developing and optimising the biperiodic on-axis coupled structures. However, because of many technical failures (vacuum leaks, contaminations in the sections resulting in severe multipacting efiects) only three sections were flnally delivered. The contract had to be terminated, and the remaining sections were produced by the company ACCEL Instruments (Bergisch Gladbach, Germany) [29] without greater di– culties, apart from also some multipactor problems. Concerning the 4.9 GHz accelerating structures more time was available because of the lack of a high-power klystron at the beginning. So an intensive optimising work was done at IKPh. For the quite tiny cavities the relative mechanical tolerances had to be relaxed for a promising industrial series production. This afiected mainly the cellto-cell coupling increasing it by a factor of two (k2.45 GHz = 4%, k4.9 GHz = 8.8%, [30]). As for their modifled geometry there remained uncertainties, e.g. concerning multipactoring, and a prototype was successfully built and full high power tests were performed at the IKPh [31]. The series production of these very well behaving structures was done without any di– cult problems again in cooperation with ACCEL. Concerning the high power rf-sources for MAMI C, the 2.45 GHz klystron posed no fundamental problem. The tube (TH2174) was delivered by THALES Electron Devices as an improved and modernised version with better electron beam focusing of the old TH2075 used at MAMI B. At 4.9 GHz a new klystron had to be developed, and offers were asked from THALES and CPI/Varian. No problems were expected, because the power-limiting curves given in the literature [32] asserted, that at this frequency cw-klystrons up to several 100 kW could be built. Quite ample speciflcations were given to the factories (60 kW cw for feeding two 4.9 GHz sections, ≥ 55% e– ciency and ≥ 47 dB gain), also with respect to a later energy upgrade 1 These efiects scale with E 5 7 and, therefore, with a beam energy in the order of 1 GeV their influence on the beam acceleration needs to be carefully investigated.

A. Jankowiak: The Mainz Microtron MAMI

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Fig. 12. Floor plan of MAMI C. The installation up to RTM3 (MAMI B) has not been changed essentially since 1990 (A2: Tagger / A4: Parity Violation / X1: Radiation Physics, till 2000 in the HDSM Hall).

of the HDSM. The contract was given to THALES. Considering both bidders would have caused an increase of the total costs of this system of more than 50%. But obviously the problems for fabricating a power klystron at this high frequency were underestimated. It took a long series of prototypes partly damaged by trivial technical failures and 27 months compared to the anticipated 12 months delivery time, before the desired tube was in house and could be used for the power tests of many other 4.9 GHz components (rf-structures, circulators, special waveguide components). These tests were successfully performed in 2003. However, the prototype TH2166-tube showed strong multipacting discontinuities on its transfer curve and was therefore not qualifled for precision operation at the accelerator. THALES could solve the multipacting problem

at the next prototype by Ti-coating the nose cones of the klystron resonators, but unfortunately through the higher surface resistivity thermal problems occurred. Finally, with a total delay of 26 months all tubes needed for MAMI C were delivered, fortunately, with a production guarantee for the TH2166 tube till 2010 by THALES. However, with somewhat reduced speciflcations (50 kW, ≥ 45% e– ciency and multipactor freeness only for somewhat restricted operating conditions), which is just safely adequate for the 1.5 GeV operation of the HDSM. All other high power components, e.g. the circulators by AFT and the water loads from Spinner, worked satisfactorily from the beginning. The two 30 kV / 27 A klystron power supplies built by BRUKER, Wissenbourg, were successfully operated in several longterm tests,

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Fig. 13. Measured vertical B-fleld of HDSM dipole No.2 normalised to the ideal fleld gradient (top), and the construction drawing of the corresponding correction coil (bottom). The dark quadratic area at the lower left of the fleld map is due to a piece of parallel pole faces necessary for an NMR-probe for precise fleld regulation.

Table 2. Main parameters of the HDSM dipoles. fleld strength [T] gap distance [mm] mech. length of front edge [m] usable length of front edge [m] iron weight [t] coils copper proflle outside [mm2 ] coils cooling duct diameter [mm] number of windings current/voltage [A/V] copper weight [t]

1.53 - 0.95 85 - 139 7 6.5 250 12 × 12 8 2 × 256 212/340 6.85

especially during the high power conditioning and testing of the fully installed and commissioned 2.45 GHz linac of the HDSM. Naturally, beside the task to design and build up the worldwide flrst 4.9 GHz cw linac, the manufacturing of the four 90◦ -bending magnets with fleld gradient presented the second highly critical challenge for the completion of the HDSM. The mechanical and magnetic design of these dipoles was completely done at IKPh. The main goal was to get magnets with excellent fleld quality at minimum size (existing halls) and iron consumption ([33,34]). The main parameters of these magnets are given in table 2. The call for tender started in 1999 and in 2000 the contract

was awarded to the French company USINOR2 . Aside the promising manufacturing capabilities of this company the main argument was, that only USINOR ofiered to produce the magnets essentially of only two symmetric pieces (upper and lower piece), which is clearly the favourable geometry to avoid any discontinuities perpendicular to the pole edge. The magnet pieces, each weighing 125 t, were casted out of high permeable iron and then machined at the company SFAR, a subcontractor of USINOR. This machining procedure for a high quality and precise surface of the partly concave pole pieces was worked out in close collaboration with IKPh. Due to the complicated pole geometry it was expected, that for the flnal fleld correction of the magnets to the 10−4 level not only symmetric, but also asymmetric fleld errors (resulting in unwanted fleld components in the plane of beam acceleration) must to be corrected. Based on the well proven concept of surface correction currents [35], a procedure had been developed which allows to extract the symmetric and asymmetric fleld components by a simultaneous measurement of the vertical magnetic fleld in and ±25 mm out of the midplane of the magnets, and to construct surface correction coils which compensate both errors simultaneously ([36,37]). The flrst magnet was delivered end of 2001 and all four magnets were flnally in place end of 2002. The contract for 2

Today SFAR STEEL (Le Creusot, France).

A. Jankowiak: The Mainz Microtron MAMI

the manufacturing of the excitation coils was awarded to the company SIGMAPHI (Vannes, France). They introduced a special bi-fllar winding technology, which allowed to realise optimum heat distribution within the coils, to avoid internal brazing and to choose reasonable power supply parameters [34]. Both guarantees a high reliability over the lifetime of the accelerator. Each magnet is fed by an individual, highly stabilised power supply (478 V, 260 A, short term/long term stability: 3 ppm/10 ppm) manufactured by DANFYSIK (Jyllinge, Denmark). By feeding back the reading of NMR-probes to the PS, the fleld of each magnet is stabilised to better than 10−5 . It took till September 2003 to flnish all magnet fleld measurements. To explore the capabilities of the magnets for a later energy upgrade, these measurements were not only done at the nominal fleld of 1.53 T but also at 1.64 T ∼ = 1.61 GeV and even 1.71 T ∼ = 1.67 GeV. In flg. 13 the measured fleld of dipole No.2 (at the nominal fleld of 1.53 T normalised to the ideal fleld gradient) is plotted. In the central area of the magnet the fleld deviations are already in the order of 10−4 , a clear proof of the excellent work done by USINOR/SFAR. As a further result of this high manufacturing precision, the analysis of the asymmetric fleld errors of the magnets showed, that the transverse components are well below 1 mT. A rough estimation of the influence of the resulting vertical beam deflections of 0.1 mrad to max. 0.35 mrad leads to an acceptable coupling of only a few percent between the horizontal and vertical phase spaces. So it was decided to do the flnal correction only for symmetric fleld errors, resulting in much simpler identical upper and lower correction coils. In the lower part of flg. 13 a sketch of one of this correction coils, manufactured by water jet cutting of a 3 mm thick aluminium plate, is shown. With these pairs of coils the desired fleld accuracy of 2 × 10−4 was easily achieved for all four magnets. One can clearly see, that most of the correction must be done near the corners of the magnet, because here quite large fleld decays exist. This behaviour was already predicted by TOSCA-simulations and are due to the triangular cut necessary to flt the magnets as far as possible into the corners of the accelerator hall. It turned out, that even at the design fleld level of 1.53 T this fleld decay leads to deflection errors of up to 2.2 mrad at low electron energies. Because it reaches far into the fringe fleld region, it cannot be corrected by surface correction coils alone. Therefore, at the entrance and exit corner of each dipole individually designed vertical iron shims attached to its front face are necessary. Together with the steering magnets on the return paths and the linac axis they will provide a proper angle and position correction of the beam [37]. Because with increasing fleld of the dipoles the fleld decay at the magnet corners gets much stronger, a later energy upgrade of the HDSM, based on the experiences gained during the operation at 1.5 GeV, will most probably require the construction and installation of a fully new set of correction coils and iron shims. Presently, all four dipole magnets are aligned and equipped with their individual set of correction coils and vacuum chambers. The 2.45 GHz linac is commissioned

Past and future

159

and ready for operation, whereas the installation of the 4.90 GHz linac, after the flnal delivery of the 10 needed accelerator sections, has just started. The next step is the installation of the two recirculation path vacuum systems and the completion of the injection and extraction beam lines. The flrst operation of the HDSM is expected in the flrst half of 2006. After a period of commissioning in diagnostic pulse mode with low beam power (10 ns, high-intensity bunch trains with a repetition rate of max. 10 kHz), very soon the flrst physics experiments will be started since all upgrades of the beams lines, the photon tagger and the spectrometers has been flnished. Many people worked together to realise the very successful operation of MAMI over the last 25 years. Here I just want to mention the certainly most important ones: Helmut Herminghaus, who is the intellectual father of MAMI and laid the strong foundations for this success story and KarlHeinz Kaiser, who overtook the responsibility for the MAMI operation and development in the early 1990s and set the guidelines for the future of MAMI: the design and installation of the Harmonic Double-Sided Microtron. The construction and operation of MAMI would not have been possible without Hans Euteneuer who is, amongst others, responsible for the developments in the rf-fleld. His great help for the preparation of this manuscript must be stressed as my last point.

References 1. J. Haimson, Linear Accelerators, edited by P.M. Lapostolle, A.L. Septier (Amsterdam, 1970) p. 415. 2. D. Husmann, IEEE Trans. Nucl. Sci. NS-30, No. 4, 3252 (1983). 3. L. Harwood, Proceedings of PAC2003, Portland, OR, USA (2003) p. 586. 4. E.M. Moroz, Sov. Phys. Dokl. 1, 326 (1956). 5. A. Roberts, Ann. Phys. (N.Y.) 4, 115 (1958). 6. B.H. Wiik et al., Linear Accelerators, edited by P.M. Lapostolle, A.L. Septier (Amsterdam, 1970) p. 553. 7. H. Herminghaus et al., Nucl. Instrum. Methods 138, 1 (1976). 8. H. Herminghaus et al., Nucl. Instrum. Methods 187, 103 (1981). 9. P. Axel et al., IEEE Trans. Nucl. Sci. NS-24, No. 3, 1133 (1977). 10. O. Hanson, Charlottesville Conference Paper Q (1979). 11. H. Herminghaus et al., Nucl. Instrum. Methods 163, 299 (1979). 12. H. Euteneuer et al., Proceedings of LINAC84, Seeheim, Germany (1984) p. 394. 13. M. Begemann et al., Nucl. Instrum. Methods 201, 287 (1982). 14. U. Schmidt-Rohr, Die Deutschen Teilchenbeschleuniger (U. Schmidt-Rohr, Heidelberg, 2001) p. 144. 15. H. Euteneuer et al., Proceedings of EPAC88, Rome, Italy (1988) p. 550. 16. H. Euteneuer et al., Proceedings of LINAC92, Ottawa, Canada (1992) p. 356. 17. H.J. Kreidel, PhD Thesis, KPH 12/87, University of Mainz, Mainz, Germany (1987).

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18. M. Seidl, Proceedings of EPAC2000, Vienna, Austria (2000) p. 1930. 19. K. Aulenbacher et al., Journal AIP, Vol. 675 (2002) 1088. 20. H. Stefiens PhD Thesis, KPH 01/94, University of Mainz, Mainz, Germany (1994). 21. V.I. Shvedunov et al., Proceedings of EPAC96, Barcelona, Spain (1996) p. 1556. 22. K.H. Kaiser, Proceedings of the Conference on Future Possibilities for Electron Accelerators, Charlottesville, VA, USA (1979) V-1. 23. H. Herminghaus et al., Proceedings of LINAC81, Santa Fe, N.M., USA (1981) p. 260. 24. H. Herminghaus, Nucl. Instrum. Methods A 305, 1 (1991). 25. S. Ratschow, PhD Thesis, KPH 02/00, University of Mainz, Mainz, Germany (2000). 26. A. Jankowiak et al., Proceedings of EPAC02, Paris, France (2002) p. 1085. 27. J. Herrmann et al., Proceedings of PAC99, New York, USA (1999) p. 2915.

28. H. Euteneuer et al., Proceedings of LINAC86, Stanford, CA, USA (1986) p. 508. 29. H. Euteneuer et al., to be published in Proceedings of EPAC06, Edinburgh, GB (2006). 30. H. Euteneuer et al., Proceedings of EPAC00, Vienna, Austria (2000) p. 1954. 31. A. Jankowiak et al., Proceedings of LINAC04, L¨ ubeck, Germany (2004) p. 842. 32. G. Faillon et al., Proceedings of LINAC86, Stanford, CA, USA (1986) p. 122. 33. U. Ludwig-Mertin et al., Proceedings of EPAC98, Stockholm, Sweden (1998) p. 1931. 34. A. Thomas et al., Proceedings of EPAC02, Paris, France (2002) p. 2379. 35. H. Herminghaus, Proceedings of EPAC88, Rome, Italy (1988) p. 1151. 36. M. Seidl et al., Phys. Rev. STAB 5, 062402 (2002). 37. F. Hagenbuck et al., Proceedings of EPAC04, Lucerne, Switzerland (2004) p. 1669.

Eur. Phys. J. A 28, s01, 161 171 (2006) DOI: 10.1140/epja/i2006-09-017-2

EPJ A direct electronic only

The Gerasimov-Drell-Hearn sum rule at MAMI A. Thomasa Institut f˜ ur Kernphysik, Universit˜ at Mainz, 55099 Mainz, Germany / Published online: 23 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. An extended experimental program to investigate the Gerasimov-Drell-Hearn (GDH) sum rule and related partial reaction cross sections on proton and neutron has been carried out by the GDH collaboration at the electron accelerators MAMI (Mainz) and ELSA (Bonn). The GDH sum rule connects the helicity-dependent photoabsorption cross section with the anomalous magnetic moment of the nucleon. The GDH collaboration has measured the total cross section of circularly polarised photons with longitudinally polarised protons at MAMI and ELSA to check this sum rule experimentally. In addition partial reaction channels like pion, double pion and eta production were determined. This provides new information on the helicity-dependent excitation spectrum of the nucleon. With the help of partial wave analyses it is possible to extract new, complementary information on the broad, overlapping resonances in this energy region. The double polarisation observable E measured in this experiment enhances the smaller multipoles via interference terms. The analysis of our data provides a new possibility to study the photon couplings to the nucleon resonances, especially above the Δ resonance where many properties of the observed states (e.g., coupling constants, branching ratios, helicity amplitudes) are only poorly known. In this paper we present several new results from our measurements on polarised proton and deuteron targets. PACS. 16.60.Le Meson production

14.20 Baryon resonances with S = 0

1 Introduction The GDH sum rule connects static properties of the nucleon like the anomalous magnetic moment κ and the nucleon mass M , with the helicity dependent total absorption cross sections σ1/2 and σ3/2 , which are related to the dynamics of the excitation spectrum:  ∞ 2π 2 α 2 dν (σ3/2 − σ1/2 ). κ = (1) 2 M ν 0 Efiectively for the nucleon the lower integration limit is the π-production threshold energy and ν denotes the photon energy, α is the flne-structure constant. The GDH sum rule was derived under very general assumptions (Lorentz and gauge invariance, causality, relativity, unitarity and the no-subtraction hypothesis) in 1966 by Gerasimov, Drell and Hearn [1,2], but it was not checked experimentally until the pioneering experiment at MAMI in 1998. Some authors [3,4] have calculated the right hand side of eq. (1) using partial wave analyses of single π-photoproduction experiments and rough estimates for the double π contribution. They always obtained a discrepancy with the left side of eq. (1), which yields 205 μb for the proton. The interest in the GDH sum rule was renewed with the measurements of the spin structure functions for proton and neutron, because the GDH sum rule a

e-mail: [email protected]

25.20 Photoproduction

can be seen as an extrapolation from the Bjorken and the Ellis-Jafie sum rules [5] for deep inelastic lepton scattering to the real photon limit. Another characteristic property of the nucleon is the forward spin polarisability:  ∞ dν 1 γ0 = (σ3/2 − σ1/2 ). (2) 4π 2 0 ν 3 Due to the stronger weighting of the integrand with the third power of the inverse photon energy this integral converges at much lower energies. The predictions for γ0 by several theoretical calculations reveal serious discrepancies. Especially the results obtained by dispersion relations difier signiflcantly from the predictions of chiral perturbation theory. Nowadays, improved technologies for polarized photon beams and polarised targets allow us to check this sum rule performing a dedicated double polarisation experiment directly. The goal of the GDH collaboration is to measure the energy dependence of the helicity-dependent total absorption cross section as well as partial reaction channels on proton and neutron targets to determine the dominant contributions to eq. (1). In addition to checking the sum rule experimentally there is a strong motivation to carefully measure the integrand and the helicity dependence of the partial reaction channels such as single- or double-pion photoproduction, which provides completely

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Fig. 1. The Mainz Microtron MAMI. The GDH experiment was performed in the A2 tagger hall.

DAPHNE

polarized -beam

polarized target

Cerenkov

Star

-strip

1m

shower trigger plates

Fig. 2. Overview of the GDH experimental setup at MAMI. The polarised photon beam is coming along the axis of the 3 He/4 He refrigerator of the polarised frozen spin target.

new and up to now inaccessible information on partialwave amplitudes. Besides the measurements with the polarised proton target, we have also performed an extended series of experiments with polarised deuteron targets in order to extract information on the neutron and thus on the isospin dependence of the helicity structure.

2 Experimental setup The MAMI accelerator (flg. 1) with its source of polarised electrons, based on the photoefiect on a strained GaAs crystal, routinely delivers polarised beams with a maxi-

mum energy of 855 MeV. In our flrst successful data taking period in 1998 (flg. 3) we typically had a degree of polarisation of about 75%, in the 2nd period in the year 2003 for the neutron runs this has been improved to about 80%. A dedicated experimental apparatus (flg. 2) including a polarised solid target and a detector with full angular acceptance was installed in the A2 Hall. The photons were produced by bremsstrahlung in the A2-Glasgow-Mainz tagging facility (flg. 4), which flrstly determines the photon energy by the help of 352 scintillation counters with a resolution of approximately 2MeV at 855MeV primary beam energy and secondly measures the degree of polarisation

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163

Fig. 3. The time dependence of the electron beam polarisation during the flrst data-taking period in 1998.

Fig. 5. The helicity transfer from the incoming longitudinal polarised electron to the photons.

Fig. 6. The end part of the 3 He/4 He refrigerator is equipped with a superconduction holding coil.

Fig. 4. The A2-Glasgow-Mainz tagging facility.

of the electrons by detecting the asymmetry in the M¿ller process. This M¿ller polarimeter allows an online monitoring of the degree of electron beam polarisation. The statistical error is of the order of 2% for a measuring time of 4 hours (each point in flg. 3 represents 4 hours of data taking). The energy-dependent helicity transfer to the photon can be calculated reliably [6]. In order to achieve a high degree of photon beam polarisation we used 525MeV and 855MeV as primary electron beam energies. Figure 5 shows the degree of circular polarisation of the outgoing photons as function of longitudinal polarisation of the incoming electrons. A new solid state frozen spin polarised target [7], developed and operated by the universities of Bonn, Bochum, Mainz and Nagoya, was used. The tar-

get materials butanol (C4 H9 OH) or deuterated butanol (C4 D9 OD), chemically doped with paramagnetic radicals to allow the process of Dynamic Nuclear Polarisation , were cooled to a temperature of 50 mK in a 3 He/4 He dilution refrigerator. In this horizontal cryostat, which was developed and constructed in the Bonn polarised target group, a nearly full angular acceptance was achieved by the integration of a thin internal holding coil on the thermal radiation shields (see flg. 6). This major step forward in polarised target technology was possible by using the inner thermal radiation shield as coil holder and cooling it with liquid 3 He/4 He mixture from the still to a temperature of 1.2K [8]. The momentum threshold for the outgoing particles is determined by the thickness of the coil including a thermal radiation shield (coil holder), which was equivalent to 0.8mm copper. The polarising magnetic fleld of 2.5T was produced by an external superconducting solenoid. After 4 hours values for the degree of proton polarisation of 80% 85% were reached by irradiation with microwaves of a frequency near to the electron spin resonance (70GHz). The external solenoid was moved on a rail system and the polarisation was maintained in the frozen spin mode at 50 mK and 0.4 T (see flg. 7).

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Fig. 9. The DAPHNE detector.

Fig. 7. The experimental principle for the GDH frozen spin target. The 3 He/4 He refrigerator is stationary, the detector DAPHNE and the polarising solenoid are movable. The upper picture shows the set-up in the polarising mode, the lower one in the data-taking mode. A high-precision rail system was used to move the DAPHNE detector and the superconducting polarising magnet.

of 4π) particle identiflcation and has a moderate e– ciency for neutral particles. DAPHNE was developed by a collaboration of Saclay and Pavia for the investigation of photoreactions on light nuclei. As shown in flg. 9, DAPHNE surrounds the target and has a cylindrical symmetry. It consists of a vertex detector (three multi-wire proportional chambers, MWPC) for reconstructing the trajectories of charged particles surrounded by a hodoscope for their identiflcation and energy determination. The outer part forms a lead-aluminum-scintillator sandwich serving as a calorimeter for the detection of decay photons and protons. Below a primary beam energy of 700 MeV all emitted protons are retained in the detector providing a good energy determination. The threshold momentum for the detection of charged particles emitted from a target including holding coil amounts to 80 MeV/c for pions and 270 MeV/c for protons. Leptonic background in DAPHNE is suppressed by the selection of appropriate discriminator thresholds. In forward direction further detection components (silicon μ-strip detector, a Cerenkov detector, a scintillation counter array, the ring shaped STAR detector) have been added to expand the angular acceptance.

3 Results Fig. 8. The time dependence of the target polarisation during the flrst data-taking period in 1998.

The relaxation time for the proton spins was about 200 hours. Consequently we could take data for typically 2 days before repolarizing the target, for example three cycles of polarising the target and data taking during the relaxation of the target can be seen in flg. 8. The repolarizing time for the target (approximately 3 hours) was used to reflll the 400l liquid-helium bufier containers for the cryostat and the polarising magnet. The movement of the polarising solenoid and the detector took around 15 minutes. The cylindrical detector DAPHNE (Detecteur a grande Acceptance pour la Physique photoNucleaire Experimentale) [9] was especially designed for handling multi particle flnal states by provision of a large solid angle (94%

3.1 Results on the GDH-Integral and the forward spin polarizibility integral on the proton Figure 10 shows our data for the helicity difierence of the total cross section measured at MAMI in the flrst phase of the GDH experiment [10]. The data are compared to predictions from the partial wave analyses SAID and MAID. The negative values of the cross section difierence (σ3/2 − σ1/2 ) in the threshold region are due to the dominance of the E0+ multipole in the single π + production channel. The excitation of the Δ-resonance (P33 (1232)) prefers the σ3/2 cross section at energies around 300 MeV. It is excited by a strong M1+ transition with only 2.5% E1+ admixture. At higher energies the double pion production plays an important role and is not represented in the theoretical curves. In flg. 11 the experimental running GDH integral (right-hand side of eq. (1)) is displayed and compared to the model predictions. The integration starts at

800

running GDH integral (μb)

σ3/2 - σ1/2 (μbarn)

A. Thomas: The Gerasimov-Drell-Hearn sum rule at MAMI

MAINZ HDT (Nπ)

600

SAID (Nπ) UIM Nπ+Nππ+ρ)

400

200

165

250 This work

200

HDT SAID

150

UIM

100

50

0 0 -200 100

200

300

400

500

600

700

800

Eγ (MeV) Fig. 10. The total cross section difierence (σ3/2 − σ1/2 ) on 1 H is compared to previous results [11] (open circles) and to the predictions of the HDT [12], SAID [13] and UIM [14] analyses. Only statistical errors are shown.

-50 100

200

300

400

500

600

700

800

Eγ (MeV) Fig. 11. The running GDH integral obtained in this work starting at 200 MeV is compared to the model predictions. Only statistical errors are shown.

Table 1. Measured values of the GDH integral in various photon energy intervals (see text for references). ELSA MAMI combined

ν[GeV] 0.8 − 2.9 0.2 − 0.8 0.2 - 2.9

GDH Integral [μb] 27.5 ± 2.0 ± 1.2 226 ± 5 ± 12 253.5 ± 5 ± 12

Table 2. Measured value of the GDH integral and model predictions for the unmeasured energy intervals (see text for references). MAID SAID Exp.(MAMI+ELSA) Simula et al. Bianchi and Thomas combined GDH sum rule value

ν[GeV] < 0.2 < 0.2 0.2 2.9 > 2.9 > 2.9

GDH Integral [μb] −27.5 −28 253.5 ± 5 ± 12 −13 −14 211.5 213 205

Eγ = 200 MeV and the upper integration limit is taken as the running variable. The measured value of the GDH integral between 200 and 800 MeV amounts to 226 ± 5 (stat) ± 12 (sys) μb. Due to the ν −3 weighting in eq. (2), the γ0 running integral is almost saturated at Eγ = 800 MeV. The value of the γ0 integral between 200 and 800 MeV amounts to [−187 ± 8 (stat) ± 10 (sys)] · 10−6 fm4 . The data taken at ELSA [15,16] from 0.7 to 2.9 GeV (see flg. 12), together with the previously measured data at MAMI, cover a broader photon energy interval, and result

Fig. 12. The helicity dependent total photoabsorption cross section difierence in the second and third resonance region measured at ELSA.

in 254 μb for the GDH integral between 0.2 and 2.9 GeV, see table 1. In order to check the validity of the GDH sum rule the contribution of the missing low and high energy regions have to be added according to the existing models (see table 2): The unitary isobar model MAID2002 [14] gives a contribution of (−27.5 ± 3) μb for photon energies below 0.20 GeV [17]. The Regge approaches from [18,19] predict a negative contribution above 2.9 GeV of −14 μb and −13 μb, respectively. The combination of our experimental results from MAMI and ELSA with these predictions yields an estimate of (211.5 − 213) μb which within the experimental errors of ±5 (stat) ± 12 (sys) μb is consistent with the GDH sum rule value of 205 μb.

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Fig. 13. Helicity difierence Δσ = σ3/2 − σ1/2 and unpolarised total photoproduction cross section on the proton (combined data set from MAMI and ELSA).

A summary of the present status is shown in flg. 13, where the helicity difierence Δσ = σ3/2 −σ1/2 for the total cross section on the proton is compared to the unpolarised cross section. As the large background of non-resonant photoproduction is spin independent and therefore almost disappears in Δσ, the helicity difierence delivers valuable information to study the properties of nucleon resonances. 3.2 Results on partial reaction channels The DAPHNE detector with forward components was capable to measure besides the total absorption cross section also partial reaction channels (see flg. 14). We have published our results on the proton in a series of papers: – the total photoabsorption cross section for energies up to 800MeV [11] – the single pion production channels up to a photon energy of 450MeV [10] and [20] – the single π 0 production up to a photon energy of 800MeV [21] – the single π + production up to a photon energy of 800MeV [22] – the double pion production channels π 0 π + [23] and π 0 π 0 [24] – helicity dependent η production [25]. Besides the importance of this data to check the GDH sum rule and to measure the forward spin polarizibility as shown in the previous section, new information about the nucleon’s excitation spectrum can be extracted. This feature can be understood from the multipole decomposition of the γN → N π total cross section. In the following we use the pion multipole notation, where E and M denote the electric or magnetic character of the incoming photon and the indices l± describe the coupling of the pion angular momentum l and the nucleon spin to the total angular momentum J = l ± 1/2. In the Δ(1232)-resonance region the measured helicity asymmetry is sensitive to the E1+ ratio of the multipoles EM R = M . In the second res1+ onance region (500 MeV ≤ ν ≤ 900 MeV), where several overlapping states are present, e.g., P11 (1440), D13 (1520), S11 (1535), the helicity dependent observables are particularly sensitive to the behaviour of the electromagnetic

Fig. 14. Overview on the helicity dependent partial photoabsorption cross sections measured at MAMI.

multipoles which are responsible for the excitation of the D13 (1520)-resonance. Considering only partial waves with l ≤ 2, the total unpolarised cross section (σ) can then be written as (see, for instance, [26]): σ ∝ |E0+ |2 + |M1− |2 + 6|E1+ |2 + 2|M1+ |2 + 6|M2− |2 + 2|E2− |2 + . . . ,

(3)

while the helicity dependent total cross section Δσ31 = (σ3/2 − σ1/2 ), where the subscripts 3/2(1/2) correspond to the (anti)parallel γ-nucleon spin conflguration, is Δσ31 ∝ −|E0+ |2 − |M1− |2 − 3|E1+ |2 + |M1+ |2 − (4) ∗ 2 2 ∗ 6E1+ M1+ − 3|M2− | + |E2− | + 6E2− M2− + . . . Since the unpolarised cross section σ of eq. (3) is given by the sum of absolute squares of multipoles, only a few dominant partial waves can be confldently evaluated. On the other hand, the sensitivity to some the weaker multipoles is greatly enhanced by measuring the polarised cross section Δσ31 of eq. (4). In particular new interference terms appear, e.g., between E1+ and M1+ , which are directly related to the Δ(1232) or P33 (1232) excitation in the lowenergy region or between E2− and M2− , which are directly related to the D13 (1520) excitation. 3.3 The Δ(1232)-resonance region There has been an extended program in difierent laboratories to increase our knowledge about the EMR ratio for the Δ(1232)-resonance [27,28]. The determination of the double-polarisation observable E provides new, complementary information to clarify this question. In flgs. 15 and 16 we have compared our data with predictions from the multipole analysis MAID2000 using EM R values of −2.5%, 0% and +2.5%. The data are well reproduced with an EM R = −2.5% [20].

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Due to its cylindrical symmetry the DAPHNE detector is ideally suited to measure angular distributions of the outgoing particles, see for example flgs. 17 and 18 for π + production [20]. Consequently the angular dependence of the new double polarisation observable has delivered urgently required input for partial wave analyses.

3.4 The second resonance region Since DAPHNE also has a moderate e– ciency for neutral particle detection we can distinguish in our data all the contributing partial reaction channels in the 2nd resonance region. Figure 19 shows our results for the single π 0 production. The agreement of the MAID2000 partial-wave analysis with our data could be improved signiflcantly by changing the parametrisation of the multipoles E2− and M2− that drive the D13 -excitation by approximately 20%. In addition flg. 20 shows our preliminary results [29, 22] for single π + production at higher energies. Although the π + channel is mainly produced via intermediate D13 (1520) excitation, a cusp structure can be observed close to the η production threshold in the σ1/2 channel.

Fig. 17. Angular dependency for the GDH observable Δ31 = σ3/2 − σ1/2 for single π + production from detection threshold to 290MeV incoming photon energy. The curves were produced using the MAID and SAID partial wave analyses.

Combining the data from our double polarised experiment with the new beam asymmetry data from Grenoble and Yerewan will improve the knowledge on the higher, strongly overlapping resonances. A detailed discussion can be found in [21,25,22].

3.5 Double-pion production Additional channels are needed to disentangle the resonances at higher nucleon excitation energies. Double pion photoproduction is particularly important for the study of the second resonance region, where the P11 (1440) state with its unclear origin is located, since almost 50 % of the total photoabsorption cross section can be attributed to the N ππ channels. In the MAMI B energy range up to 800MeV we are presently analysing our data for the helicity difierence σ3/2 − σ1/2 of the double-pion production channel π + π − . The results for the π + π 0 and π 0 π 0 channels have already been published [23,24]. Comparing our results with predictions from difierent theoretical models [30,31] gives a new insight into the double pion production mechanism, specially the role of the D13 (1520), the P11 (1440) and the Δ(1700) are under discussion.

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Fig. 18. Angular dependency for the GDH observable Δ31 = σ3/2 − σ1/2 for single π + production for photon energies between 300 MeV and 450 MeV. The curves were produced using the MAID and SAID partial wave analyses.

Fig. 19. The asymmetry σ3/2 − σ1/2 for selected energies for single π 0 production compared to the MAID2000 (black) and SAID SP01 (grey) analyses. The dashed curve was produced using MAID2000 with a modifled parametrisation for the D13 (1520)-resonance.

A strong helicity dependence can be observed in our data (see flgs. 21 and 22), with a clear dominance of the σ3/2 over the σ1/2 cross section. This suggests a resonant behaviour due to the intermediate excitation of the D13 (1520)-resonance. However, the σ1/2 cross section is not negligible, with an indication of a resonance contribution, possibly from the P11 (1440) excitation. Future experiments with the recently installed Crystal Ball detector with its high e– ciency for neutral particles will deliver further insights in the production mechanism.

3.6 η photoproduction The helicity dependence of the η production has been measured for the flrst time at a center-of-mass angle of Θη∗ = 70◦ in the proton energy range from 780MeV to 790MeV and was reported in [25]. The results shown in flg. 23 demonstrate the importance of the S11 (1535) resonance for η production. The dσ )3/2 is small as expected for S-wave domicross section ( dΩ nance near threshold. Clearly, better statistics and a wider kinematical range are required to disentangle the small contributions from resonances other than the S11 (1535).

Fig. 20. Preliminary data for σ3/2 and σ1/2 for π + production. A pronounced cusp structure due to an interference with the η production process at threshold can be clearly observed in the σ1/2 cross section [22].

A. Thomas: The Gerasimov-Drell-Hearn sum rule at MAMI

Fig. 21. The helicity dependent cross-sections for π 0 π 0 photoproduction. The errors shown are statistical only.

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Fig. 23. The measured helicity dependent difierential crossdσ sections for η photoproduction. Left: Δσ13 = (dΩ )1/2 − dσ dσ dσ ( dΩ )3/2 . Right: ( dΩ )1/2 and ( dΩ )3/2 . The difierent lines show the predictions of the MAID analysis for four difierent cases including all resonances (continuous line), S11 (1535) only (dashed line), without S13 (1520) (dash-dotted line), without S11 (1650) (dotted line). The errors shown are statistical only.

Fig. 24. Preliminary helicity-dependent total cross section for the deuteron compared to theoretical predictions from ref. [32, 33, 34]. Fig. 22. The helicity dependent difierential cross-sections for π + π 0 photoproduction. The errors shown are statistical only.

With the new accelerator stage MAMI C and the multiphoton detector Crystal Ball in combination with TAPS as forward detector the data set will be improved signiflcantly.

3.7 Results on the GDH sum rule on the neutron In 1998 there has been done a pilot experiment to investigate the feasibility to measure the GDH sum rule for

the neutron. Since there is no free neutron target available, the Bonn frozen spin target had to be loaded with deuterated butanol (C4 D9 OD). First preliminary results are available for the deuteron for the total cross section (see flg. 24) and the partial → − → − − − γ d → nnπ + [35], based on channels → γ d → ppπ − and → a small subset of the data. The full data taking was carried out in the flrst half of 2003 using the Bonn frozen-spin target [7]. In the course of the experiment the degree of polarisation could be increased from 35% to more than 70% due to a new target material based on trityl-doped D-butanol [36]. This development by the Bochum polarised target group is a major

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Fig. 27. The new dilution refrigerator for the Crystal Ball detector. Fig. 25. Progression of the target polarisation during the experiment with the standard and the new target material (labeled Fin II).

Fig. 26. Crystal Ball detector with inner tracker and TAPS forward wall.

step forward in the polarised target technology. The progression of the target polarisation measured during the experiment with the two target materials is shown in flg. 25. The data analysis is in progress [37,29,38]. 3.8 Conclusions and outlook The international GDH collaboration, working successfully at the electron accelerators MAMI and ELSA, has provided data to check the GDH sum rule experimentally for the flrst time. A newly developed polarised solid state target with high angular acceptance and highly polarisable new target materials in combination with 4π detectors and highly polarised continuous photon beams made

the experiment possible. For the proton the sum rule has been verifled on a 10% level, the deuteron data are still being analysed. It is a challenge that requires a tremendous theoretical efiort to extract the neutron properties from the deuteron data. The investigation of the partial reaction channels led to an additional knowledge on the helicity dependence of the nucleon’s excitation spectrum. The double-polarisation observable E has been determined in a broad kinematical range. With the upgrade of the MAMI accelerator to 1.5 GeV a signiflcant part of the outgoing particles would escape from the DAPHNE detector. The new standard detector to be used with the tagged photon beam, the Crystal Ball (CB) detector completed by TAPS as a forward wall, has already taken data in the years 2004 and 2005 with unpolarised targets, and both unpolarised and polarised photon beams. The CB consists of 672 optically isolated NaI(Tl) crystals, 15.7 radiation lengths thick. The counters are arranged in a spherical shell with an inner radius of 25.3 cm and an outer radius of 66.0 cm. Charged particles can be measured by the central tracker consisting of a scintillator barrel and modifled DAPHNE cylindrical multi-wire proportional chambers (two layers only). In order to achieve a better particle identiflcation for charged particles, an inner barrel of plastic scintillators (PID-detector) has been included. The TAPS wall is composed of 522 BaF2 detectors [39] arranged in a hexagon. The apparatus is schematically shown in flg. 26. The high granularity, large acceptance and good energy resolution make this setup a unique instrument for the detection of multi-photon flnal states. The tagger will be upgraded to cope with the increased beam energy. Linearly and circularly polarised photons will be available. A polarised frozen-spin target is presently under development to allow for double polarisation experiments with the new facilities. The central part of this target will be a horizontal dilution refrigerator (see flg. 27). The data presented in this paper have been mostly produced by the international GDH collaboration. We gratefully

A. Thomas: The Gerasimov-Drell-Hearn sum rule at MAMI acknowledge the excellent support of the MAMI and ELSA accelerator groups. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 201, SFB 443, Schwerpunktprogramm 1034, and GRK683), the INFN-Italy, the FWO Vlaanderen-Belgium, the IWT-Belgium, the UK Engineering and Physical Science Council, the DAAD, JSPS Research Fellowship, and the Grant-in-Aid (Specially Promoted Research) in Monbusho, Japan.

References 1. S.B. Gerasimov, Yad. Fiz. 2, 598 (1965) (Sov. J. Nucl. Phys. 2, 430 (1966)). 2. S.D. Drell, A.C. Hearn, Phys. Rev. Lett. 16, 908 (1966). 3. I. Karliner, Phys. Rev. D 7, 2717 (1973). 4. A. Sandorfl, Phys. Rev. D 50, R6681 (1994). 5. Anselmino et al., Sov. J. Nucl. Phys. 49, 553 (1989). 6. H. Olsen, L.C. Maximon, Phys. Rev. 114, 887 (1959). 7. C. Bradtke et al., Nucl. Instrum. Methods A 436, 430 (1999). 8. H. Dutz et al., Nucl. Instrum. Methods A 340, 272 (1994). 9. G. Audit et al., Nucl. Instrum. Methods A 301, 473 (1991). 10. J. Ahrens et al., Phys. Rev. Lett. 87, 022003 (2001). 11. J. Ahrens et al., Phys. Rev. Lett. 84, 5950 (2000). 12. O. Hansein et al., Nucl. Phys. A 632, 561 (1998). 13. R.A. Arndt et al., Phys. Rev. C 66, 055213 (2002). 14. D. Drechsel et al., Nucl. Phys. A 645, 145 (1999). 15. H. Dutz et al., Phys. Rev. Lett. 91, 192001 (2003). 16. H. Dutz et al., Phys. Rev. Lett. 93, 032003 (2004).

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17. L. Tiator, Proceedings of GDH2002 (World Scientiflc, Singapore, 2003). 18. N. Bianchi, E. Thomas, Phys. Lett. B 450, 439 (1999). 19. S. Simula et al., Phys. Rev. D 65, 034017 (2002); private communication. 20. J. Ahrens et al., Eur. Phys. J. A 21, 323 (2004). 21. J. Ahrens et al., Phys. Rev. Lett. 88, 232002 (2002). 22. J. Ahrens et al., submitted to Phys. Lett. C. 23. J. Ahrens et al., Phys. Lett. B 551, 49 (2003). 24. J. Ahrens et al., Phys. Lett. B 624, 173 (2005). 25. J. Ahrens et al., Eur. Phys. J. A 17, 241 (2003). 26. D. Drechsel, Prog. Part. Nucl. Phys. 34, 181 (1995). 27. R. Beck, H.P. Krahn et al., Phys. Rev. Lett. 78 (1997). 28. G.S. Blanpied et al., Phys. Rev. Lett. 79, 4337 (1997). 29. T. Rostomyan, PhD Thesis (Gent) (2005). 30. J. Nacher, Proceedings of NSTAR2001 (World Scientiflc Pub. Co, 2001) p. 189. 31. M. Vanderhaeghen, H. Holvoet, private communication, Mainz 2001. 32. H. Arenh˜ ovel, The GDH for the deuteron, in Proceedings of GDH2000, Mainz, edited by D. Drechsel, L. Tiator (World Scientiflc, Singapore, 2001) p. 67. 33. A. Fix, private communication. 34. M. Schwamb, private communication. 35. M. Martinez, PhD Thesis, University of Mainz (in preparation). 36. St. Goertz et al., Nucl. Instrum. Methods A 526, 43 (2004). 37. O. Jahn, PhD Thesis, University of Mainz (2005). 38. S. McGee, PhD Thesis, Duke University (in preparation). 39. R. Novotny, IEEE Trans. Nucl. Sci. 38, 379 (1991).

Eur. Phys. J. A 28, s01, 173 183 (2006) DOI: 10.1140/epja/i2006-09-018-1

EPJ A direct electronic only

Experiments with photons at MAMI R. Becka Helmholtz-Institut f˜ ur Kern- und Strahlenphysik, Nussallee 14-16, 53115 Bonn, Germany / Published online: 26 May 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. A very successful experimental program with real photons has been achieved in 20 years of operation at the Mainz Microtron (MAMI) facility. The difierent detector setups, like DAPHNE, TAPS and the Crystal Ball are centered around the tagged photon facility the so-called Glasgow Tagger. From the rich spectrum of results only a few highlights will be discussed here, the proton polarizabilities, the pion polarizabilities, pion photoproduction close to the pion threshold and in the Δ(1232)-resonance region. PACS. 13.40.-f Electromagnetic processes and properties 13.60.Le Meson production and Compton scattering 14.20.Dh Protons and neutrons

1 Introduction Experiments with real photons at MAMI have been performed in the framework of the A2-collaboration. Monochromatic photons from bremsstrahlung tagging by the Glasgow Tagger [1,2] are used for all experiments. Polarized photon beams, linear and circular, are available as well as polarized targets. Groups from several institutions and countries (see [3]) have provided difierent detector components for example the photon spectrometer TAPS [4], the 4π charge particle tracking detector DAPHNE [5] and more recent the photon spectrometer Crystal Ball [6]. Many data have been taken on the proton and on light and complex nuclei including the total photon absorption, Compton scattering, meson production, break up reactions and multi pion production in the flnal state. The experimental work was based on 175 Diploma and PhD thesis, which are published in more than 100 refereed articles. Here only a few highlights of this experimental program can be adressed.

2 Low-energy Compton scattering Next to the size and the anomalous magnetic moment, the polarizability is a further property of a particle with a substructure. In the present of the electromagnetic flelds (E and B), electric dipole moments are induced and magnetic dipole moments may be oriented (paramagnetism) or induced according to Lenz’s rule (diamagnetism). The most precise determination of the proton polarizabilities comes from Compton scattering experiments. These measurements rely on a Low-Energy Theorem to establish a a

e-mail: [email protected]

13.60.Fz Elastic

unique relation between a low-energy expansion of the Compton-scattering cross section and the static polarizabilities α and β. For photon energies small compared to pion mass, this expansion reads [7,8]:     dσ dσ = dΩ dΩ    2 Point 

ω e α+β α−β (1 + z)2 + (1 − z)2 −ωω  (1) ω m 2 2 with z = cos(θγ ), where ω and ω  are the energies of dσ the incident and scattered photon, respectively; dΩ Point is the exact cross section for a structureless proton with an anomalous magnetic moment. The quantities α and β are the static polarizabilities. Low-energy Compton scattering from the proton in the energy range from 55 MeV to 165 MeV was measured using the TAPS detector set up at the photon beam at MAMI. The energy of the incident electron beam was chosen to be 180 MeV. The target consisted of a Kapton cylinder of 20 cm length fllled with liquid hydrogen. Data obtained from about 200 h of beam time were analysed [9, 10]. The scattered photons were detected with 6 blocks of the TAPS. Since the recoiling protons could not be detected, a single-particle trigger had to be used. Therefore, this minimum bias trigger included all kinds of background events for example cosmic ray events which have not been suppressed by an active shield and electromagnetic background from the beam collimation system and from the target itself. These sources of background were partially suppressed by time cuts and a missing-energy cut, which is deflned as the difierence between the measured incident photon energy (tagger) and the expected incident photon, as calculated from the measured scattered photon assuming Compton kinematics.

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rule constraint leads to the following result:

Fig. 1. Measusred difierential cross-sections in the lab system [9] compared with a dispersion relation calculation (solid line) [11].

The difierential cross sections obtained are plotted in flg. 1. The systematic errors of ±3% arise from uncertainties in the photon flux (±2%) and the target density (±2%) combined in quadrature. The efiective solid angles were determined with Monte Carlo simulations. Errors from uncertainties in the experiment geometry are estimated to be ±5%. With the help of the dispersion relation approach the electromagnetic polarizabilities of the proton can be extracted from the experimental cross-sections. The procedure used in the analysis was to take α and β as free parameters, and sometimes as well the constraint given by the Baldin sum rule. Using standard χ2 minimization, the result obtained, when fltting the MAMI/TAPS data alone without the sum rule constraint, is α = 11.9 ± 0.5(stat.) ∓ 1.3(syst.), β = 1.2 ± 0.7(stat.) ± 0.3(syst.).

(2) (3)

The Baldin sum rule obtained from this result, α + β = 13.1 ± 0.9 is in agreement with the value determined by the total photon absorption cross section. A flt to the existing low-energy Compton scattering data including the new MAMI/TAPS data and the sum

α = 12.1 ± 0.3stat. ∓ 0.4syst. ± 0.3mod. , β = 1.6 ± 0.4stat. ± 0.4syst. ± 0.4mod. ,

(4)

where the flrst error denotes the statistical, the second the systematic and the third the model-dependent one. The results are summarized in flg. 2 (contourplot) where the contours in the (α − β) plane for χ2min + 1 are plotted. In addition, the Baldin sum and the value obtained from the experiment by Zieger [12] are included.

3 Pion polarizability The pion polarizabilities characterize the dynamical deformation of the pion in the electromagnetic fleld. The values of the electric α and magnetic β pion polarizabilities depend on the rigidity as a composite particle and provide important information of internal structure. Very difierent values for the pion polarizabilities have been calculated in the past. All predictions agree, however, that the sum of the two polarizabilities of the π ± meson is very small. On the other hand, the values of the difierence of the polarizabilities are very sensitive to theoretical models. For example, investigations within the framework of the chiral perturbation theory (ChP T ) predict (α − β)π± ≈ 5.4 [13] in one-loop calculations and 4.4 ± 1.0 for two-loops [14] (all values of the polarizabilities are given in units of 10−4 fm3 ). The calculations in the extended Nambu-Jona

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Lasinio model with linear realization of chiral U (3) × U (3) symmetry [15] result in απ± = −βπ± = 3.0 ± 0.6. The application of dispersion sum rules (DSR) at flxed value of the Mandelstam variable u = μ2 for calculation of this parameter [16,17] leads to (α − β)π± = 10.3 ± 1.9. DSR at flnite energy [8] gave the similar result: (α − β)π± = 10.6. A calculation in the linear σ model with quarks and vector mesons included to one loop order predicted (α − β)π± = 20 [18]. An evaluation in the Dubna quark conflnement model [19] results in (α − β)π± = 7.05. Because there is no stable pion target, experimental information about the pion polarizabilities is not easy to obtain. One has to investigate reaction channels, like scattering high energy pion in the Coulomb fleld of a heavy nuclei or the radiative pion photoproduction. The scattering of high energy pions ofi the Coulomb fleld of heavy nuclei [20] has resulted in απ− = −βπ− = 6.8 ± 1.4 ± 1.2. This value agrees with prediction of the dispersion sum rules but is about 2.5 times larger than the ChP T result. The experiment of the Lebedev Institute on radiative pion photoproduction from the proton [21] has given απ+ = 20 ± 12. This value has large error bars and shows the largest discrepancy with regard to the ChP T predictions. The attempts to determine the polarizability from the reaction γγ → ππ sufier greatly from theoretical [22] and experimental [23] uncertainties. The most recent analysis of MARK II and Crystal Ball data [24] flnds no evidence for a violation of the ChP T predictions. However, even changes of polarizabilities by 100% and more are still compatible with the present error bars.

Fig. 4. Enlarged view showing the details of the TAPS conflguration.

The experiment discussed here has been performed at the continuous-wave electron accelerator MAMI B [25, 26] using Glasgow-Edinburgh-Mainz tagger photon facility [1,2]. The quasi-monochromatic photon beam covered the energy range from 537 to 819 MeV with an intensity ∼ 6 × 105 /s in the tagger channel for the lowest photon energy and average energy resolution of 2 MeV. The tagged photons entered a scattering chamber, containing a 3 cm diameter and 11.4 cm long liquid hydrogen target with Capton windows. The emitted photon γ  , π + meson, and the neutron were detected in coincidence. The experimental setup is shown in flg. 3. The photons were detected by the spectrometer TAPS [4], assembled in a special conflguration (flg. 4). The TAPS spectrometer consists of 528 BaF2 crystals. Each hexagonally shaped crystal is 250 mm long corresponding to 12 radiation lengths. All crystals were arranged into three big blocks. Two blocks (A, B) consisted of 192 crystals arranged in 11 columns and the third block (C) had 144 crystals arranged in 11 columns. These three blocks were located in the horizontal plane around the target at angles 68◦ , 124◦ , 180◦ with respect to the beam axis. Their distances to the target center were 55 cm, 50 cm and 55 cm, respectively. All BaF2 modules were equipped with 5 mm thick plastic veto detectors for the identiflcation of charged particles. The neutrons were detected by a wide aperture timeof-flight spectrometer (TOF) [27]. It consisted of 111 scintillation detector bars of 50 × 200 × 3000 mm3 and 16 counters (10 × 230 × 3000 mm3 ) which were used as veto detectors. The bars are made from NE110 plastic scintilator and each bar is read out on both ends by two 3 phototubes XP2312B. All bars were assembled in 8 planes

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Fig. 5. The difierential cross section of the process γp → γπ + n averaged over the full photon beam energy interval and over s1 from 1.5m2π to 5m2π . The solid and dashed lines are the predictions of model-1 and model-2, respectively, for (α − β)π+ = 0. The dotted line is a flt to the experimental data (see text).

of a special conflguration with 16 detectors in each, following one after another (flg. 3). Such a neutron detector allows to detect the neutrons in the energy region 10 100 MeV with e– ciency 30 50% and to determine their energy with a resolution ∼ 10% using the neutron time of flight and the angle of the neutron emission measured with a precision ∼ 2 3%. To detect the π + meson two two-coordinate multiwire proportional chambers (MWPC) and a forward scintillator detector (FSD), for getting a fast trigger signal, have been developed and constructed. The MWPC’s overlap angles in the laboratory system were θ ∼ = 2◦ − 20◦ , ◦ ◦ ◦ ϕ∼ 0 − 360 and were located under 0 with respect to = the beam direction. The cross section of the process γp → γπ + n has been calculated in the framework of two difierent models. In the flrst model (model-1) the contribution of all the pion and nucleon pole diagrams is taken into account using pseudoscalar pion-nucleon coupling [28]. In the second model (model-2), the nucleon and the pion pole diagrams without the anomalous magnetic moments of the nucleons, and in addition the contributions of the resonances Δ(1232), P11 (1440), D13 (1520), and S11 (1535) are included. To control the model dependence of the result the kinematic regions were limited to regions where the difierence between model-1 and model-2 does not exceed 3% when (α − β)π+ is constrained to zero. First, a kinematic region where the contribution of the pion polarizability is negligible, i.e. the region 1.5m2π ≤ s1 < 5m2π was analysed, where s1 is the squared pion-photon center-of-mass energy. In flg. 5, the experimental data for the difierential cross section, averaged over the full photon beam energy interval from 537 MeV up to 817 MeV and over s1 in the indicated interval, are compared to predictions of model1 (dashed curve) and model-2 (solid curve). The dotted curve is the flt of the experimental data in the region of −10m2π < t < −2m2π , where t is the squared pion mo-

0

550

600

650

700

750

800

Eγ (MeV) Fig. 6. The cross section of the process γp → γπ + n integrated over s1 and t in the region where the contribution of the pion polarizability is biggest and the difierence between the predictions of the theoretical models under consideration does not exceed 3%. The dashed and dashed-dotted lines are predictions of model-1 and the solid and dotted lines of model-2 for (α − β)π+ = 0 and 14 × 10 4 fm3 , respectively.

mentum transfer. As seen from this flgure, the theoretical curves are very close to the experimental data. This means that the dependence of the difierential cross section on the square of the four-momentum transfer t which is basically the kinetic energy of the neutron is well reproduced by using the mentioned GEANT simulations for the e– ciency. In a second step, the kinematic region where the polarizability contribution is maximal was investigated. This is the region 5m2π ≤ s1 < 15m2π and −12m2π < t < −2m2π . In the considered region of the phase space, the cross sections of the process γp → γπ + n integrated over s1 and t are calculated according to model-1 and model-2 for two difierent values of (α−β). The obtained experimental cross sections and their theoretical predictions for (α−β)π+ = 0 and 14 × 10−4 fm3 are presented in flg. 6. The error bars are the quadratic sum of statistical and systematic errors. For each model, we obtain (α − β)π+ = (12.2 ± 1.6stat ± 3.3syst ) × 10−4 fm3 (model − 1), (α − β)π+ = (11.1 ± 1.4stat ± 2.8syst ) × 10−4 fm3 (model − 2).

(5) (6)

Averaging over the results of the two models, the flnal result is obtained [29]: (α − β)π+ = (11.6 ± 1.5stat ± 3.0syst ± 0.5mod ) × 10−4 fm3 . (7)

4 Pion photoproduction in the threshold region The photoproduction of pions near threshold has been a topic of considerable experimental and theoretical activities over the past years, ever since the results of the experiments, performed in Saclay [30], Mainz ([31,32]) and

R. Beck: Experiments with photons at MAMI

q dσ(θ) = (A + B cos(θ) + C cos2 (θ)), dΩ k

(8)

where θ is the c.m. polar angle of the pion with respect to the beam direction and q and k denote the c.m. momenta of pion and photon, respectively. The coe– cients A = |E0+ |2 + |P23 |2 , B = 2 Re(E0+ P1∗ ) and C = |P1 |2 − |P23 |2 are functions of the multipole ampli2 = 12 (P22 + P32 ). Earlier measurements of tudes with P23 the total and difierential cross sections already allowed determination of E0+ , P1 and the combination P23 . In order to obtain E0+ and all three P -waves separately and to test the new LETs of ChPT, it is necessary to measure, in addition to the cross sections, the photon asymmetry Σ, dσ⊥ − dσ , (9) Σ= dσ⊥ + dσ

4.5

this work Ref. [3] Ref. [4]

4.0 3.5 3.0 / b

Saskatoon [33], were at variance with the prediction of a low energy theorem (LET), which was derived in the early 70s [34,35]. Being based on fundamental principles, this LET predicted the value of the S-wave threshold amplitude E0+ in a power series in μ = mπ /mN , the ratio of the masses of the pion and nucleon. The discrepancy could be explained by a calculation in the framework of heavy-baryon chiral perturbation theory (ChPT) [36], which showed that additional contributions due to pion loops in μ2 have to be added to the old LET. Reflned calculations within heavy-baryon ChPT [37] led to descriptions of the four relevant amplitudes at threshold by well-deflned expansions up to order p4 in the S-wave amplitude E0+ and p3 in the P -wave combinations P1 , P2 and P3 , where p denotes any small momentum or pion mass, the expansion parameters in heavy-baryon ChPT. To that order, three low-energy constants (LEC) due to the renormalization counter terms appear, two in the expansion of E0+ and an additional LEC bP for P3 , which have to be fltted to the data or estimated by resonance saturation. However, two combinations of the P -wave amplitudes, P1 and P2 , are free of low-energy constants. Their expansions in μ converge rather well leading to new LETs for these combinations. Therefore, the P -wave LETs ofier a signiflcant test of heavy-baryon ChPT. However, for this test the S-wave amplitude E0+ and the three P -wave combinations P1 , P2 and P3 have to be separated. This separation can be achieved by measuring the photon asymmetry using linearly polarized photons, in addition to the measurement of the total and difierential cross sections. The difierential cross sections can be expressed in terms of the S- and P -wave multipoles, assuming that close to threshold neutral pions are only produced with angular momenta lπ of zero and one. Due to parity and angular momentum conservation only the S-wave amplitude E0+ (lπ = 0) and the P -wave amplitudes M1+ , M1− and E1+ (lπ = 1) can contribute and it is convenient to write the difierential cross section and the photon asymmetry in terms of the three P -wave combinations P1 = 3E1+ + M1+ − M1− , P2 = 3E1+ − M1+ + M1− and P3 = 2M1+ + M1− . The c.m. difierential cross section is

177

2.5 2.0 1.5 1.0 0.5 0.0 144

147

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153 156 159 E / MeV

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165

168

Fig. 7. Total cross sections for π 0 photoproduction close to threshold with statistical errors (without systematic error of 5%) as function of incident photon energy (solid squares, this work ref. [38], open circles, ref. [33], open diamonds ref. [32]).

where dσ⊥ and dσ are the difierential cross sections for photon polarizations perpendicular and parallel to the reaction plane deflned by the pion and proton. The asymmetry is proportional to the difierence of the squares of P3 and P2 : Σ(θ) =

q dσ(θ) (P32 − P22 ) · sin2 (θ)/ . 2k dΩ

(10)

A measurement of the reaction p(γ , π 0 )p [39] was performed at the Mainz Microtron MAMI [40] using the Glasgow/Mainz tagged photon facility [1,2] and the photon spectrometer TAPS [4]. The MAMI accelerator delivered a continuous wave beam of 405 MeV electrons. Linearly polarized photons were produced via coherent bremsstrahlung in a 100 μm thick diamond radiator [41, 42] with degrees of polarization of up to 50%. The neutral pion decay photons were detected in TAPS [43], an array of 504 BaF2 detectors, which was built up around a liquid-hydrogen target. The total and difierential cross sections were measured over the energy range from π 0 threshold to 168 MeV. Figure 7 shows the results for the total cross section in comparison to ref. [33] and [32]. The results for the photon asymmetry are shown in flg. 8 in comparison to the values of ChPT [37] and to a prediction of a dispersion theoretical calculation (DR) by Hanstein, Drechsel and Tiator [44]. The photon asymmetry was determined from all the data between threshold and 166 MeV for which the mean energy was 159.5 MeV. The theoretical predictions are shown for the same energy. The values for the real and imaginary part of E0+ and the three P -wave combinations were extracted via two multipole flts to the cross sections and the photon asymmetry simultaneously. The two multipole flts difier in the energy dependence of the real parts of the P -wave combinations. For the flrst flt the usual assumption of a behaviour proportional to the product of q and k was adopted (qk-flt, χ2 /dof = 1.28). The assumption made

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Table 1. Results of both flts (qk-flt and q-flt) for Re E0+ at the π 0 - and π + -threshold (unit: 10 3 /mπ+ ), for the parameter β of Im E0+ (unit: 10 3 /m2π+ ) and for the three combinations of the P -wave amplitudes (unit: q · 10 3 /m2π+ ) with statistical and systematic errors in comparison to the predictions of ChPT [37, 45] (O(p3 )) and of a dispersion theoretical approach (DR, [44]). ChPT

DRa

−1.16 −0.43 2.78 9.14 ± 0.5 −9.7 ± 0.5 10.36 11.07

−1.22 −0.56 3.6 9.55 −10.37 9.27 9.84

This work 0

pπ E0+ (Ethr ) nπ + E0+ (Ethr ) β P1 P2 P3 P23

q-flt

−1.23 ± 0.08 ± 0.03 −0.45 ± 0.07 ± 0.02 2.43 ± 0.28 ± 1.0 9.46 ± 0.05 ± 0.28 −9.5 ± 0.09 ± 0.28 11.32 ± 0.11 ± 0.34 10.45 ± 0.07

−1.33 ± 0.08 ± 0.03 −0.45 ± 0.06 ± 0.02 5.2 ± 0.2 ± 1.0 9.47 ± 0.08 ± 0.29 −9.46 ± 0.1 ± 0.29 11.48 ± 0.06 ± 0.35 10.52 ± 0.06

Values of the P -wave combinations converted into the unit q · 10−3 /m2π+ .

0.0

0.3

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0.2

-0.4

+

0.4

-3

ReE0+ / 10 /m

a

qk-flta

0.1 0.0 -0.1 ChPT [8] DR [17] fit to the data

-0.2 this work

-0.3

0

20

40

60

80

100 120 140 160 180 / deg

cms 0

Fig. 8. Photon asymmetry Σ for π 0 photoproduction at 159.5 MeV photon energy with statistical errors (without systematic error of 3%) as a function of the polar angle θ (solid line: flt to the data) in comparison to ChPT [37] (dotted line) and DR [44] (dashed line).

for the second flt is an energy dependence of the P -wave amplitudes proportional to q (q-flt, χ2 /dof = 1.29). This is the dependence which ChPT predicts for the P -wave amplitudes in the near-threshold region, but at higher energies the prediction is in between the q and qk energy dependence. The results of both multipole flts for Re E0+ as a function of the incident photon energy are shown in flg. 9 and compared with the predictions of ChPT and of DR. The results for the threshold values of Re E0+ (at the π 0 - and π + -threshold), for the parameter β of ImE0+ and for the values of the threshold slopes of the three P -wave combinations of the qk-flt and the q-flt are summarized in table 1, for more details see [38]. For both flts the low-energy theorems of ChPT (O(p3 )) for P1 and P2 agree with the measured experimental results within their systematic and statistical errors. The experimental value for P3 is higher than the value of ChPT, which can be explained by the smaller total and difierential cross sections of ref. [32], used by ChPT to determine the dominant low-energy constant bP for this multipole [45]. A new fourth-order calculation in heavy-baryon ChPT by Bernard et al., introduced in [46] and compared

this work: q k-fit this work: q-fit

ChPT [8] DR [17]

-0.6 -0.8 -1.0 n

-1.2 -1.4 145

+

Ethr =151.4 MeV 150

155 E / MeV

160

165

Fig. 9. Results for Re E0+ with statistical errors as a function of incident photon energy Eγ for an assumed energy dependence of the P -wave amplitudes proportional to q · k (solid squares) and q (open squares) in comparison to ChPT [37] (dotted line) and DR [44] (dashed line).

to the new MAMI data presented in this letter, shows, that the potentially large Δ-isobar contributions are cancelled by the fourth-order loop corrections to the P -wave low-energy theorems. This gives confldence in the thirdorder LET predictions for P1 and P2 , which are in agreement with the present MAMI data. With the new value of bP [46], fltted to the present MAMI data, the ChPT calculation is in agreement with the measured photon asymmetry. In a recent work, pion photoproduction on the nucleon is evaluated by dispersion relation at constant t [47]. The extension to the unphysical region provides a unique framework to determine the low-energy constants of chiral perturbation theory by global properties of the excitation spectrum. See also the most recent work for pion production at threshold in the framework of covariant baryon chiral perturbation theory [48].

5 The γN → Δ(1232) transition and the E2/M1 ratio Low-energy electromagnetic properties of baryons, such as mass, charge radius, magnetic and quadrupole moments

R. Beck: Experiments with photons at MAMI

179

are important observables for any model of the nucleon structure. In various constituent-quark models a tensor force in the inter-quark hyperflne interaction, introduced flrst by de Rujula, Georgi and Glashow [49], leads to a d-state admixture in the baryon ground-state wavefunction. As a result the tensor force induces a small violation of the Becchi-Morpurgo selection rule [50], that the γN → Δ(1232) excitation is a pure M 1 (magnetic dipole) transition, by introducing a non-vanishing E2 (electric quadrupole) amplitude. For chiral quark models or in the Skyrmion picture of the nucleon, the main contribution to the E2 strength stems from tensor correlations between the pion cloud and the quark bag, or meson exchange currents between the quarks. To observe a static deformation (d-state admixture) a target with a spin of at least 3/2 (e.g. Δ matter) is required. The only realistic alternative is to measure the transition E2 moment in the γN → Δ 3/2 transition at resonance, or equivalently the E1+ partial wave amplitude in the Δ → N π decay. The experimental quantity of interest to compare with the difierent nucleon 3/2 3/2 models is the ratio REM = E2/M 1 = E1+ /M1+ of the electric quadrupole E2 to the magnetic dipole M 1 amplitude in the region of the Δ(1232)-resonance. In quark models with SU (6) symmetry, for example the MIT bag model, REM = 0 is predicted. Depending on the size of the hyperflne interaction and the bag radius, broken SU (6) symmetry leads to −2% < REM < 0 [51,52,53,54]. Larger negative values in the range −6% < REM < −2.5% have been predicted by Skyrme models [55] while results from chiral bag models [56] give values in the range −2% to −3%. The flrst Lattice QCD result is REM = (+3 ± 9)% [57] and a quark model with exchange currents yields values of about −3.5% [58].

Fig. 10. Photon asymmetries Σ in the Δ-resonance region (solid circles, this work ref. [64], open diamonds ref. [59] and crosses ref. [60]).

The determination of the quadrupole strength E2 in the region of the Δ(1232) resonance has been the aim of a considerable number of experiments and theoretical activities in the last few years. Experimental results have been published for the difierential cross section and photon asymmetry of pion photoproduction ofi the proton from the Mainz Microtron MAMI and the laser backscattering facility LEGS at Brookhaven National Laboratory, with the results REM = −(2.5 ± 0.2stat ± 0.2sys )% from the Mainz group [59] and REM = −(3.0 ± 0.3stat+sys ± 0.2mod )% from the LEGS group [60]. These new REM results have started intense discussions about the correct way to extract the E2/M 1 ratio from the new experimental data. In particular the large variation in the REM values obtained in theoretical analysis of these data at RPI [61] (REM = −(3.2 ± 0.25)%), VPI [62] (REM = −(1.5 ± 0.5)%) and Mainz [63] (REM = −(2.5 ± 0.1)%) was quite unsatisfactory. Since small difierences in the difierential cross section occur in the mentioned MAMI/DAPHNE and LEGS experiments, a new experiment on neutral pion photoproduction ofi the proton has been performed at the Mainz Microtron covering the full polar angle range of the pion. The new enlarged set of experimental results should allow a determination of REM more accurately.

Figure 10 shows the new results for the photon asymmetry for six difierent energies in the Δ-resonance region. For the flrst time this new experiment delivers data in the full polar angle range. The new results are in good agreement with the experimental data of MAMI/DAPHNE and LEGS. In addition, the photon asymmetries of all three experiments are compared to the dispersion theoretical analysis of Hanstein [63,65] and good agreement is found. The unpolarized difierential cross sections for the same six photon energies in the Δ-resonance region are shown in flg. 11. The new results are in agreement with the MAMI/DAPHNE, the LEGS data difier not only in the absolute values of the difierential cross section but show as well a difierent angular distribution. In addition, the results of the Hanstein analysis for the MAMI/TAPS data are shown. In the angular momentum expansion of the neutral pion photoproduction it is su– cient to take into account s- and p-waves, i.e. lπ = 0 or 1 only. The angular distributions for the unpolarized cross section dσ0 /dΩ, the parallel part dσ /dΩ (pion detected in the plane deflned by the photon polarization and the photon momentum vector), and perpendicular part dσ⊥ /dΩ can be expressed in the

0.8

E =280 MeV

E =300 MeV

0.6 0.4 0.2 0.0

MAMI / TAPS MAMI / DAPHNE LEGS (E =275 MeV) Hanstein (TAPS)

-0.2 -0.4 0.8

(E =298 MeV)

E =320 MeV

E =340 MeV

0.6 0.4 0.2 0.0 -0.2 (E =322 MeV)

-0.4 0.8

E =360 MeV

E =380 MeV

0.6 0.4 0.2 0.0 -0.2 -0.4

0

40

80 CMS 0

120

(Grad)

160

0

40

80 CMS 0

120

160

(Grad)

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25.0

3/2

0.04 0.02

15.0

-0.02 -0.04

10.0

-0.06

5.0

E =280 MeV

0.0 40.0

/ ( b/sr) d /d

15.0

0

E =300 MeV -0.08

LEGS (E =322 MeV)

-0.1

35.0 30.0

3/2

REM=(Im E1+ / Im M1+ ): 0 + Multipolanalysis (p -, n -data) Hanstein(DR)

0.0

20.0

R

/ ( b/sr) d /d

30.0

0.06

(E =298 MeV)

MAMI / TAPS MAMI / DAPHNE LEGS (E =280 MeV) Hanstein (TAPS)

35.0

R=REM 250

300

R: p -Data only MAMI/TAPS Hanstein(DR)

350

400

E / MeV

25.0

3/2

3/2

Fig. 12. The energy dependence of the ratio E1+ /M1+ is shown as solid diamonds. In addition, the energy dependence of R = C /(12A )is shown as solid squares.

20.0 10.0 5.0

E =320 MeV

E =340 MeV

3/2

/ ( b/sr) d /d

25.0 20.0

10.0

1+

15.0

5.0 E =360 MeV 0.0

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40

80 CMS 0

120

(Grad)

E =380 MeV 160

0

40

80 CMS 0

120

3/2

can be identifled with the ratio REM = E1+ /M1+ at the Δ(1232)-resonance (δ33 = 90◦ )  3/2 Im E1+  R  REM = . (16)  3/2 Im M 

0.0 30.0

160

(Grad)

Fig. 11. Difierential cross sections in the Δ-resonance region. MAMI/TAPS results are shown with statistical (1 2 %) and systematic errors (solid circles, this work ref. [64], open diamonds ref. [59] and crosses ref. [60]).

W =MΔ

This is the crucial point of our analysis [66]. This method ofiers the advantage of being independent of absolute normalization and insensitive to many systematic errors, because REM is extracted from the ratio of the coe– cients C and A fltted to the angular distribution of dσ /dΩ. Further, the following identity can be derived [64]: R=

◦ 1 C 1 C A + Σ(θ = 90 ) ≈ REM , = ◦ 12 A 12 1 − Σ(θ = 90 )

(17)

s- and p-wave approximation by the parameterization

which depends only on the shape (C/A) of the difierential cross section dσ/dΩ and the photon asymmetry Σ at θCM S = 90◦ . Using eq. (17), the ratio REM can be extracted [64]:

dσj (θ) q = (Aj + Bj cos(θ) + Cj cos2 (θ)), dΩ k

REM = (−2.4 ± 0.16stat. ± 0.24sys ).%

(11)

(18) 3/2

where q and k denote the center-of-mass momenta of the pion and the photon, respectively, and j indicates the parallel (), perpendicular (⊥) and unpolarized (0) components. The coe– cients Aj , Bj and Cj are quadratic or bilinear functions of the s- and p-wave amplitudes. In particular, dσ /dΩ is sensitive to the E1+ amplitude, because of interference with M1+ in the terms A = | E0+ |2 + | 3E1+ − M1+ + M1− |2 , B = 2 Re[E0+ (3E1+ + M1+ − M1− )∗ ], C = 12 Re[E1+ (M1+ − M1− )∗ ].

6 Future plans (12) (13) (14)

Furthermore, the ratio R=

1 C Re(E1+ (M1+ − M1− )∗ ) = 12 A | E0+ |2 + | 3E1+ + M1+ − M1− |2

According to the Fermi-Watson theorem the E1+ and have the same phase δ33 and the ratio quantity. As shown in flg. 12, this ratio is strongly dependent on the photon energy and varies from −8% at Eγ = 270MeV to +2% at Eγ = 420MeV. 3/2 M1+ partial waves 3/2 3/2 E1+ /M1+ is a real

(15)

The flrst round of experiments with the Crystal Ball is centered on the flrst measurement of the magnetic dipole moment of the Δ+ (1232)-resonance. The magnetic dipole moment, μb , provides us with a simple way for testing the validity of the theoretical hadron description in the nonperturbative sector of QCD. This includes quark soliton models, the standard quark models, various efiective Lagrangians and lattice QCD calculations. Our experimental

R. Beck: Experiments with photons at MAMI

181

technique takes advantage of the very short Δ lifetime by having the Δ radiatively decay to itself. This method has been successfully pioneered for the Δ++ using the reaction π + p → γ  Δ++ → γ  π + p [67]. We propose to determine μb [Δ+ (1232)] using radiative π 0 photoproduction: γp → Δ+ → γ  Δ+ → γ  π 0 p. A flrst pilot experiment with the TAPS calorimeγp → γ  π 0 p has been performed √ ter at MAMI for energies s = 1221−1331 MeV. Angular and energy difierential cross section have been determined for all particles in the flnal state in three bins of the excitation energy [68]. The theoretical aspects have been dealt with in detail already by the theory groups at MAMI [69] and Tuebingen [70]. μb can be determined from the difierential cross section dσ 5 /dΩγ dΩπ dEγ and from the asymmetry, Σ, for linearly polarized photons.

The new experimental apparatus is shown in flg. 13. The Crystal Ball with TAPS as the forward wall will be used for detection of photons and nucleons. In addition the polar and azimuthal angles of the outgoing proton for Θlab > 20◦ will be measured by the central tracker which is based on the DAPHNE cylindrical multiwire proportional chamber. The chamber will be inserted into the Crystal Ball beam cavity. The Crystal Ball was build at SLAC and used in J/ψ measurements at SPEAR and b-quark physics at DESY [71]. The CB is constructed of 672 optically isolated NaI(Tl) crystals, 15.7 radiation lengths thick. The counters are arranged in a spherical shell with an inner radius of 25.3 cm and an outer radius of 66.0 cm. The hygroscopic NaI is housed in two hermetically sealed evacuated hemispheres. Each crystal is shaped like a truncated triangular pyramid, 40.6 cm high, pointing towards the center of the Ball. The sides on the inner end are 5.1 cm long and 12.7 cm on the far end. Electromagnetic showers in

Fig. 13. The Crystal Ball detector and TAPS as forward wall. Meson_phLadd_v_IM2phot_px

Counts

The broad spectrum of MAMI bremsstrahlung photons from Eγmin ≈ 100 MeV to Eγmax ≈ 1500 MeV together with the 4π acceptance of the experimental apparatus allows the simultaneous survey of π 0 , 2π 0 , 3π 0 and η production at all energies and for the full angular range. Such measurements will be perform with LH2 and LD2 targets using linearly and circularly polarized pho ton beams. A unique frozen spin target fllled with 1 H, 2 or H will be used in the second stage of the experiment (MAMI-C). The target makes possible new high precision, high statistics measurements of the cross sections for the  → π 0 N and γ N  → π 0 π 0 N processes at incident phoγ N ton energies up to 1.5 GeV. In particular it provides a unique opportunity to measure the partial contributions to the GDH sum rule on a neutron target in the reactions γn → π 0 n and γn → π 0 π 0 n. Our measurements will also provide new information on the photon coupling of lowmass baryon and hyperon resonances. An incomplete list of other possible measurements includes: i) threshold photoproduction of π 0 and η at MAMI-B as well as η  , ω and Ks0 at MAMI-C with polarized and unpolarized beams and targets; ii) measurements of the N ∗ (1535) magnetic dipole moment using γp → γ  ηp; iii) a new measurement of the η mass.

Entries Mean RMS

900

4428 203.4 148.3

800 700 600 500 400 300 200 100 0

100

200

300

400

500 600 Inv.Mass, MeV

Fig. 14. Invariant mass of two-cluster events for beam photons with energy above 700 MeV after requiring the missing mass to be equal to the mass of proton. The peaks are due to π 0 → 2γ and η → 2γ decays.

the spectrometer are measured with an energy resolution σE /E ∼ 1.7%/(E (GeV))0.4 ; the angular resolution for photon showers at energies of 0.05 0.5 GeV is σθ = 2◦ 3◦ in the polar angle and σφ = 2◦ / sin θ in the azimuthal angle. High granularity and a large acceptance make the Crystal Ball a unique instrument for measuring reactions with multiphoton flnal states. The CB detects neutrons with an e– ciency of ≈ 35% at En = 150 MeV [72]. The flrst production run of the CB@MAMI program, a measurement of the photon asymmetry in π 0 photoproduction at threshold, was accomplished in July-August 2004. In October 2004 we have started a 600 hours long production run for the measurements of the Δ+ (1232) magnetic dipole moment. Figures 14 17 illustrate the

The European Physical Journal A

σtotal (γ p → η p), arb. units

182 7

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5

0.16 4

0.14 0.12

3

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0.08 0.06

1

0.04 0 620 640 660 680 700 720 740 760 780 800 820 ELab γ , MeV

Fig. 15. The total cross section of γp → ηp in arbitrary units from η → 2γ decay modes (solid circles) is compared to the γp → 3π 0 p total cross section (open circles). Above the η threshold the γp → 3π 0 p is mainly from η → 3π 0 decay. The total cross sections are relatively normalized at Eγ = 730 MeV. 0

pγ Σ(π p)

0.02 0

-1

-0.5

0

0.5

1 cos(ΘCM π )

Fig. 17. Preliminary results for the photon asymmetry times beam polarization shown as a function of cos Θπ0 in c. m. for Eγ = 375 MeV (triangles), 405 MeV (squares), and 436 MeV (circles), compared to MAID predictions. The MAID curves are normalized to the experimental data at cos Θπ = 0.

0.2 0.15 0.1 0.05 -0 -0.05 -0.1 -0.15 -0.2

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0

50

100 150 0 φ(π ), deg

Fig. 16. φ-dependence of the beam photon asymmetry times polarization, pγ × Σ(φ), for the reaction γp → π 0 p, where σ (φ) σ⊥ (φ) . The function shows clear cos(2φ) behavΣ(φ) = σ (φ)+σ⊥ (φ)  ior over the full angular range. The data covers the beam energy interval of 360 450 MeV and is integrated over Θπ0 .

quality of the data, showing some characteristic distributions. The invariant mass of two photons for an incident beam with energy above 700 MeV is shown in flg. 14. The two peaks of the spectra are due to the reactions γp → π 0 (γγ)p and γp → η(γγ)p. The two-gamma invariant mass is shown for events with the missing mass equal to

the mass of the proton. Figure 15 shows an excitation function for γp → η(γγ)p in arbitrary units. The total cross section for γp → 3π 0 p is shown on the same flgure for comparison. Below η threshold 3π 0 events are produced via sequential decay of resonances, while above the η-threshold most of the events are produced by the η → 3π 0 decay. The experimental setup made up of the Crystal Ball and TAPS is almost perfectly φ-symmetric. Together with the good quality polarized MAMI beam it allows high statistics, low systematics uncertainty, polarization measurements. Figure 16 shows the φ-dependence of the beam photon asymmetry, Σ(φ), for the reaction γp → π 0 p. The asymmetry is not corrected for the beam polarization. The data represent about 5 % of the statistics obtained in the course of our most recent μ(Δ(1232)) run. The photon asymmetry as a function of cos Θπ0 is shown in flg. 17 for beam photon energies of 375 MeV, 405 MeV, and 436 MeV in comparison with MAID [73]. The results are very preliminary. The measured distributions are not corrected for the beam photon polarization therefore the MAID curves are normalized to the data at cos Θπ = 0. The photon asymmetries obtained in our experiment show good agreement with the MAID evaluations for cos Θπ > 0 and slightly deviate in the backward angles. The difierence between our data and MAID gets larger at higher beam energies. I would like to thank the organizers Hartmuth Arenh˜ ovel, Hartmut Backe, Dieter Drechsel, Karl-Heinz Kaiser and Thomas Walcher of the symposium 20 Years of Physics at the Mainz Microtron MAMI .

R. Beck: Experiments with photons at MAMI

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Eur. Phys. J. A 28, s01, 185–195 (2006) DOI: 10.1140/epja/i2006-09-019-0

EPJ A direct electronic only

Coherent X-rays at MAMI W. Lautha , H. Backe, O. Kettigb , P. Kunz, A. Sharafutdinov, and T. Weber Institut f¨ ur Kernphysik der Universit¨ at Mainz, D-55099 Mainz, Germany / c Societ` Published online: 31 May 2006 –  a Italiana di Fisica / Springer-Verlag 2006 Abstract. Coherent radiation in the range from soft X-rays up to hard X-rays, produced by the lowemittance electron beam of MAMI, can be used for various applications. Novel types of interferometers have been developed for the measurement of the complex index of refraction of thin self-supporting foils. For the vacuum ultraviolet and soft X-ray region the interferometer consists of two collinear undulators, and a grating spectrometer. A foil placed between the undulators causes a phase shift and an attenuation of the oscillation amplitude. The complex index of refraction has been measured at the L2,3 -absorption edges of nickel. A novel method is described for the measurement of the X-ray magnetic circular birefringence. For the hard X-ray region the interferometer consists of two foils at which the 855 MeV electron beam produces transition radiation. Distinct interference oscillations have been observed as a function of both, the photon emission angle and the distance between the foils. The refractive index decrement δ(ω) of a 2 μm thick nickel sample foil has been measured at X-ray energies around the K absorption edge at 8333 eV and at 9930 eV with an accuracy of better than 1.5 %. The line width of parametric X radiation (PXR) was measured in backward geometry with a Si single-crystal monochromator. Upper limits of the line width of 42 meV, 50 meV, and 44 meV, have been determined for the (333), (444) and (555) reflections at photon energies of 5932 eV, 7909 eV, and 9887 eV, respectively. Small angle scattering of the electrons in the crystal leads to a stochastic frequency modulation of the exponentially damped wave train which results in the line broadening. To elucidate the quest if the production of PXR is a kinematical or a dynamical process the radiation from silicon single-crystal targets, emitted close to the electron direction, has been studied. The observed interference structures and the narrow-band radiation in forward direction shows that PXR is produced in a dynamical process. PACS. 07.60.Ly Interferometers – 78.20.Ci Optical constants (including refractive index, complex dielectric constant, absorption, reflection and transmission coefficients, emissivity) – 41.60.-m Radiation by moving charges – 41.50.+h X-ray beams and X-ray optics

1 Introduction Immediately after MAMI B became fully operational for nuclear physics experiments, in the early nineties also a research program was launched to explore the potential of the high-quality, low-emittance electron beam for applications. It was our conviction that it ought to be possible to use X-rays produced with the 855 MeV beam of MAMI in various fields of physics, material science, medicine and biology. Of course, attention focused at that time on the third-generation synchrotron radiation sources like ESRF, APS and Spring8 which were on the horizon and, in particular, on the production of brilliant soft X-ray flashes in a single pass of high-current electron bunches through an undulator by the process of self-amplified spontaneous emission (SASE). However, various other processes were

also considered at that time to be of interest for the production of soft and hard X-ray beams with external high-quality electron beams. The most important ones are schematically depicted in fig. 1. These are transition radiation (TR), channeling radiation (CR), parametric X-radiation (PXR), undulator radiation (UR), and SmithPurcell radiation (SPR). There are potential advantages of such X-ray sources over synchrotron radiation sources or free electron lasers on the basis of SASE. First of all, since accelerators may become relatively inexpensive in the future, they could meet the radiation requirements of research laboratories or hospitals on the spot. Secondly, the X-ray beam can be triggered and its time structure adapted to nearly any experimental requirement. In particular, the electron beam can easily be turned off if the X-ray beam is not used minimizing power consumption and radiation production in the beam dump.

a

e-mail: [email protected] Present address: Arcor AG & Co. KG, 65760 Eschborn, Germany. b

At MAMI with UR and TR brilliant photon beams can be produced with energies covering the range between

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Undulator Radiation

Transition Radiation

Parametric X-Radiation (PXR)

N

S

N

S

S

N

S

N

Channeling Radiation (CR)

Smith-Purcell Radiation (SP)

Fig. 1. Processes for the generation of coherent radiation with relativistic electrons.

some 100 eV and up to about 50 keV. Brilliance means that a large number of photons from a small source spot size down to the μm range are emitted with high directionality in space. In particular, the hard TR X-ray beam turned out to be comparable in photon flux and brilliance with second-generation synchrotron radiation sources [1]. Taking advantage of this fact a novel K-edge imaging system [2] was developed. In addition, X-ray phase contrast imaging has been accomplished. For the latter we refer to the contribution ref. [3] in this issue. SPR is generated when a beam of charged particles passes close to the surface of a periodic structure. This type of radiation has been investigated in the optical spectral range with the 855 MeV beam of MAMI [4]. A detailed discussion of the emitted photon number per electron led to the conclusion that a SPR source is not advantageous in comparison with an UR source for ultrarelativistic beam energies. At present experiments are being performed at the 3.4 MeV injector LINAC of MAMI to explore the generation of intense SPR in the THz region of the electromagnetic spectrum [5]. In channeling, the charged particle directions are closely aligned with an atomic row or with crystal planes, and their motion is governed by many correlated collisions with crystal atoms. As a result, the particles are steered along strings or planes and CR is emitted. Channeling experiments have been taken up at MAMI only very recently, see ref. [6]. They are connected with the feasibility of a crystalline undulator with positrons which was investigated recently in great detail [7]. In sect. 2 of this contribution our results obtained with a novel interferometry principle will be reviewed. The interferometer consists of two spatially separated, phasecorrelated radiation sources in the soft and hard X-ray ranges. For soft X-rays the radiation sources are undulators with small period length, for hard X-rays they are foils in which the electron beam produces TR. It will be shown in this section 2 that the optical properties of foils can be determined in the soft and hard X-ray region. If the electron beam strikes a crystal, it emits quasimonochromatic PXR close to a Bragg angle. This kind of radiation source is amazing for its compactness, since

production and monochromatisation of the radiation take place in the same crystal. The expected small spectral line width of PXR would promise an abundance of application possibilities, e.g. within the field of solid-state physics. However, line broadening by multiple scattering of the electrons in the crystal may spoil the superb line width. In sect. 3 our experiments addressing this question are reviewed, including fundamental aspects like the question whether the process of PXR production is of kinematical or of dynamical nature. The paper closes with a conclusion and an outlook.

2 X-Ray interferometry Resonant anomalous X-ray scattering plays an increasingly important role in many disciplines of physics, biology, and material sciences. Using the brilliant and tuneable X-ray beams from modern synchrotron radiation sources, it is now possible to fully exploit the information in the strong energy and polarisation dependence of the atomic scattering amplitude f (ω, q) = f0 (q) + f  (ω) + if  (ω) near absorption edges [8,9]. This microscopic description can be translated into a macroscopic description with the complex index of refraction n(ω) = 1 − δ(ω) + iβ(ω) by the relations for the refractive index decrement δ(ω) = (1/2) (ωp /ω)2 (f0 (0) + f  (ω))/Z and the absorption index β(ω) = (1/2) (ωp /ω)2 f  (ω)/Z. In these expressions Z is the atomic number, ωp the plasma frequency with ωp 2 = 4πr0 c2 na Z, r0 the classical electron radius, na the number of atoms per volume, and f0 (0) = Z neglecting relativistic corrections. The imaginary part of the scattering amplitude f  can be directly determined from the total photon cross section σ(ω) by employing the optical theorem: f  (ω) = ω σ(ω)/(4πr0 c). The total cross section is well approximated by the absorption cross section which can be measured by a transmission experiment. The real part f  can be calculated from f  by means of Kramers-Kronig dispersion relations. However, this method is suited for a relative comparison only, since it requires precise absorption data for all frequencies from zero to infinity [10]. If precise absolute values are needed, a direct measurement of f  (ω) is required. Direct measurements are based on X-ray interferometry with the Bonse Hard-X-ray interferometer [11], refraction through a prism [12,13], diffraction from perfect crystals and pendell¨ osung fringes [14,15], determination of the angle of total reflection [10,16], and Fresnel bi-mirror interferometry [17]. Whereas most of these methods are based on splitting of either wave amplitudes or wave fronts the novel type of interferometer which is described here uses two spatially separated coherent X-ray emitters. The basic idea of the interferometer will be explained by means of the schematic experimental setup shown in fig. 2. Relativistic electrons create two wave trains in source 1 and source 2, the relative distance Δ of which is given in leading order by Δ(θ, d) = 12 (γ −2 + θ2 )d. Here d is the distance between the sources, γ the Lorentz factor of the electron, and θ the observation angle with respect to the electron beam direction. The monochromator serves

W. Lauth et al.: Coherent X-rays at MAMI X-ray sources Sample S 2 foil S 1

e-

Monochromator

e-

Detector A2

T1

=Z = 847.3 eV

with sample

2000

A1

187

1500

T2

1000 500

'

d

0

Variable distance

as a Fourier analyser of the wave trains. The two resulting plane waves have a phase difference of Φ = ωv Δ(θ, d) (v is the velocity of the electron) and interfere in the detector, resulting in oscillations of the intensity I(d), if the distance d is varied. A sample foil placed between the two sources produces an additional phase shift and attenuation of wave 2. Consequently, both quantities, i.e. the refractive index decrement δ and the absorption index β, can be extracted from the measured interference oscillations I(d) with and without the foil between the sources. This holds independently of the nature of the emission process, provided that the produced X-rays remain coherent.

=Z = 854.3 eV

with sample

1000

counts

Fig. 2. Interferometry with spatially separated coherent X-ray emitters.

800 600 400 200 0 2000

with sample

=Z = 874.1 eV

1500 1000 500 0 210

220

230

240

250

260

d [ mm ] 2.1 The soft X-ray interferometer For photon energies in the range of about 100 eV up to 2 keV we use two identical undulators (period length LU = 12 mm, number of periods 10, undulator parameter K = 1.1) and a grating spectrometer. The recorded intensity with a foil between the undulators is given by ω

ω

I(d) = |A1 |2 + |A2 |2 e−2 c β(ω)t0 + 2|A1 | |A2 |e− c β(ω)t0 

 ω K2 × cos (1) Δ(θ, d) + δ(ω)t0 + 2 LU c 4γ with A2 being the amplitude of the upstream undulator, A1 that of the downstream one and t0 the thickness of the foil. The interferometer has been developed at the Mainz Microtron MAMI and its performance was demonstrated with measurements at the K absorption edge of carbon at 284 eV. Details of this experiment can be found elsewhere [18]. The visibility (coherence), defined by C = (Imax − Imin )/(Imax + Imin ) without sample foil, is close to its maximum value C = 1. No loss of coherence was observed over the scanning distance of 15 cm. Therefore, the optical constants δ and β could be extracted by a fit with simple cosine functions. Measurements were also performed at the L2 absorption and L3 -absorption edges of nickel at 871 eV and 855 eV, respectively. The experimental setup was similar with that described in ref. [18] with the following modifications: the third harmonics of the undulator was chosen as radiation, which was generated at an electron energy of 766.3 MeV. The radiation was analyzed with a variable line spacing (VLS) grating [19] with an energy

Fig. 3. Intensity oscillations as a function of the distance d between the undulators with and without the self-supporting 83.2 μg/cm2 nickel sample foil at three different photon energies at the L2,3 absorption edges. Note the change of sign of the phase shift at the photon energy of 854.3 eV.

resolution of 0.44 eV. Typical measured intensity oscillations are shown in fig. 3. The extracted optical constants at the Ni L2,3 are shown in fig. 4. A high accuracy has been reached, even in the region where β ≥ δ in which the optical constants can be determined from reflectivity measurements only with large uncertainties [20]. The remarkable fact about this measurement is the hight of the L3 resonance with its maximum value β = (5.64±0.24)·10−3 . This value corresponds to an imaginary scattering factor f  = 65.7 ± 2.8. In ref. [21] a mass attenuation coefficient μ = 2 · 104 cm2 /g was determined from which, with the relation μ = (4π)·β, an f  = 24 can be calculated with an estimated uncertainty of 10 %. The difference may originate from the better resolution in our experiment. Correlated with this L3 absorption resonance is a change of sign of the refractive index decrement δ(ω). It is interesting to notice that for δ(ω) < 0 the real part of the refractive index 1 − δ(ω) is larger than 1 and a monochromatic Cherenkov radiation with an energy of 855 eV will be emitted. This fact has been pointed out in the literature. The strong absorbtion line at the L3 edge of Ni is the result of an allowed dipole transition between the 2p3/2 core state and empty 3d valence states above the Fermi energy. This transition exhibits a strong X-ray Magnetic Circular Dichroisim (XMCD) effect which can

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+/- 90°

4

e-

-3

[ 10 ]

6

Undulator 1

2

E

Movable Gap

0 2

Beam tube

L2

L3

3 mm

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0 -2 -4 840

850

860

870

880

=Z [ eV ] Fig. 4. Absorption index β and refraction index decrement δ of an 83 μg/cm2 Ni sample foil as obtained from the analysis of the intensity oscillation measurements as shown in fig. 3. The photon energy resolution was 0.44 eV.

be used to probe the magnetic properties of the material [22]. With XMCD spectroscopy, pioniered by Sch¨ utz and coworkers [23], the difference in the absorption of left and right circular polarized light is measured. Connected with the XMCD is the X-ray Magnetic Circular Birefringence (XMCB) also known as Faraday Magneto-Optical Rotation (MOR). Both effects can be described with the complex index of refraction n± (ω) = 1 − δ± (ω) + iβ± (ω) for the two ± helicity states of the radiation. Thickness variation in the transmission measurements hampered the accuracy in the determination of β± (ω) with the XMCD spectroscopy, especially near the strong-absorption lines where the XMCD effect is the largest. It has been shown in ref. [24], that such thickness effects are less important for the measurement of the refractive index δ± (ω). However, this measurement is difficult because of the lack of polarization analyzers in the soft X-ray region. Our interferometer has been developed further to measure δ± (ω) without any polarization filter. The magnetized sample foil with the magnetization direction parallel or antiparallel to the electron beam axis was positioned between the two undulators, see fig. 5. Due to the XMCB and XMCD effect, the linear polarized light from the first undulator suffers a helicity-dependent phase shift and absorption resulting in elliptical polarized light. The major semi-axis of the ellipse is rotated by an angle ψ with respect to the plane in which the impinging linear polarized E field vector oscillates. The second undulator which can be rotated around the electron beam axis acts as the analyzer. The maximum visibility determines the rotation angle ψ of the ellipse on which the

Fig. 5. Experimental setup of the new undulator interferometer. The undulator 2 can be both moved along and rotated around the electron beam axis. These possibilities allow production of radiation with a well-defined polarization state between linear and circular. In addition, the undulator gap can be changed for an online variation of the photon energy via the undulator parameter K. Note that the electron beam traverses a vacuum tube of only 3 mm in diameter over a length of about 1 m.

10 5

\ [°]

G

-3

[ 10 ]

1m

0 -5 -10 840

L2

L3 850

=Z

860 [ eV ]

870

Fig. 6. Preliminary results of a measurement of the rotation angle ψ of the ellipse on which the E field vector rotates behind a 72.5 nm self-supporting Ni foil, which was illuminated with a linear polarized undulator radiation. Shown are measurements at the L3 and L2 absorption edges.

E field vector moves behind the foil. This angle is given ω t0 (δ+ (ω) − δ− (ω)). The first measurements of by ψ = 2c the angle ψ of a 72.5 nm magnetized Ni foil, placed in a magnetic field of 1.0 T, are shown in fig. 6. In this experiment the photons were detected in the energy dispersive plane of the grating spectrometer with a windowless CCD-detector [25]. The visibility for rotation angles between −105◦ and +105◦ of the second undulator was extracted from the interference oscillation as a function of the distance between the undulators. Furthermore, also the refractive index decrement β± (ω) can be extracted simultaneously from this measurement which allows the full determination of the helicity-dependent complex index of

W. Lauth et al.: Coherent X-rays at MAMI

189 447

60

335

row

40 223

20

111

0

0 0

20

40

60

80 100 120 140 160 180 200

col mn Fig. 8. Measured interference pattern at a fixed foil distance d = 475.0(2) μm. The central photon energy ¯ hω = 9929 eV is well above the K absorption edge of nickel. Grey levels indicate the intensities.

θ

Fig. 7. Experimental setup of the transition radiation interferometer. The monochromator is a flat silicon single crystal, cut with the (111) plane parallel to the surface, at a distance of 5.5 m from the foils. It acts as energy dispersive mirror. A silicon 1 × 3 cm2 pn CCD with active thickness of 300 μm and a pixel size of 150 × 150 μm2 was used as detector [26]. The CCD is located at a distance of 5.5 m from the monochromator crystal.

refraction n± (ω) without knowledge of the degree of the polarization of the light.

2.2 The hard X-ray interferometer When a charged particle traverses the interface of two media with different dielectric polarizability, transition radiation (TR) is emitted. The TR radiation is sharply peaked into forward direction with a characteristic opening angle of about 2/γ and features broadband characteristics with a cut-off energy at about 40 keV. It is well known that the TR amplitudes from two interfaces of a single foil interfere coherently. The same holds for the amplitudes from two or more foils (interfoil interference), see, e.g., ref. [27]. It was suggested more than fifteen years ago by Moran et al. [28] that the interfoil coherence of TR, generated by relativistic electrons, constitutes a new technique for the measurement of the refractive index decrement δ(ω) in the X-ray region. A number of experiments were performed in the soft and hard X-ray region in which more or less clear interference structures were observed [28,29, 30,31,32,1]. However, that δ(ω) really can be measured by such a technique has been demonstrated first by the X-ray interferometer described in this work. It consists of two foils at which the 855 MeV MAMI beam produces transition radiation, a single crystal spectrometer with a flat crystal in Bragg geometry, and a pn CCD X-ray detector, see fig. 7. Details of the experiment have been reported elsewhere [33,34,35]. Measurements have been performed on nickel around the K absorption edge at 8333 eV as well as around 9930 eV, well above the K absorption edge, where extended diffraction anomalous fine structures (EDAFS) in the dispersion spectra are supposed to be negligible. A

ω

μ Fig. 9. Examples of interference oscillations as a function of foil distance d at a fixed photon energy ¯ hω and a fixed observation angle θ as indicated. The ordinates are counts/(ms mrad2 ). The measurements (dots) are well met by the simulation calculation (full line).

typical example of a measured TR interference pattern is shown in fig. 8. There are two possibilities to extract δ1 of the downstream nickel foil from the TR interference patterns shown in fig. 8. In the first one the information is obtained from the interference oscillations as a function of the observation angle θ at fixed distance between the foils. Since the pn CCD detector was arranged horizontally the interference oscillations were observed essentially along the energy dispersive angular coordinate θx . Reliable results can be obtained if δ1 (ω) can be approximated with reasonable accuracy by a linear expansion as a function of the photon energy ¯hω. Close to an absorption edge such an approximation is not valid. In such a case, as a second possibility, δ1 (ω) can be extracted from the interference oscillations observed as a function of foil distance d, as shown in fig. 9. The rapid damping of the visibility after about three periods originates mainly from the small-angle scattering of the electrons in the upstream foil which was made of the low-Z element beryllium to minimize the effect. The results of the refractive index decrement δ1 (ω) around the K absorption edge of Ni are shown in fig. 10. Details of the analysis procedure can be found in refs. [34,36]. The measured δ1 (ω) agrees at the K absorption edge within the total error of Δδ1 /δ1 ≤ 1.5%, with the Kramers-Kronig transformation of β1 (ω) of refs. [37,38].

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divergence, the small-angle electron scattering in the crystal and the solid angle of the X-ray detector can all be neglected, and the crystal has a perfect lattice structure. An exponentially damped wave train is emitted of which the Fourier transform is a Lorentzian with width [54]

δ1 • 105

2.10 2.00 1.90 1.80 8300

8320

8340

8360

8380

–

hω [eV] Fig. 10. Refractive index decrement δ1 (¯ hω) at the K absorption edge as obtained from interference oscillations. The error bars represent pure statistical errors of the experiment. The full line is a Kramers-Kronig transform of the absorption data [37, 38].

In comparison, the measurements of δ1 (ω) by Bonse and co-workers [39,40] with Bonse-Hart interferometers [11] are systematically too low by about 1.5%. This small deviation may originate from a systematical error in the foil thickness measurements in those experiments.

3 Parametric X-radiation Parametric X-radiation (PXR) or quasi-Cherenkov radiation is produced when a relativistic electron traverses a single crystal, and the wave vector kv of the virtual photon associated with the electron field, the reciprocal lattice vector H of a specific crystal plane and the wave vector kr of the emitted X-ray nearly fulfil the well-known diffraction condition kv + H ∼ = kr . For ultra-relativistic electrons the wave vector kv nearly coincides with the electron velocity vector v and the emission process can also be understood as diffraction of virtual photons by the crystal. The production mechanism of PXR in such a medium with three-dimensional periodic permittivity was extensively studied both theoretically by Baryshevsky and Feranchuk [41,42], Garibyan and Yang [43,44], and TerMikaelian [45] and experimentally by a large number of researchers. For an overview of the theoretical and experimental work up to the year 1997 see, e.g., ref. [46] and also ref. [47,48,49,50] for the recent works. PXR features a sharp quasi-monochromatic X-ray beam close to the Bragg angle with a very narrow line width. The angular distribution consists of one peak above and one below the symmetry plane of the crystal. Their spatial widths are characterized by the angle θph = (1/γ 2 + (ωp /ω)2 )1/2 , with γ the Lorentz factor of the moving particle, ω the angular frequency of the emitted photon, and ωp the plasma frequency of the crystal, with h ¯ ω = 31 eV for Si. The theoretical description of the intensity distribution of PXR [51,52,45,53,54], suitably modified for self-absorption and multiple-scattering effects, has been tested for a broad range of electron energies extending from 3.5 MeV [55,56] to about 1 GeV [46]. It was found to be accurate within 12%. The line width Wnat of PXR is in essence determined by the photo absorption in the crystal, if the electron beam

Wnat = −

χ0 · ¯hω0 . 2 sin2 θ0

(2)

The quantity χ0 is the imaginary part of the mean dielectric susceptibility χ0 = χ0 + iχ0 . This line width Wnat is called “natural line width”. For example, the (444) reflection of silicon at h ¯ ω0 = 7908 eV yields χ0 = −3.74 · −7 10 [57]. With this value a natural line width Wnat = 1.48 meV results from eq. (2) for backward emission, i.e. for θ0  π/2. It is interesting to note that the corresponding Darwin-Prins curve has a width of WDP = 38.5 meV and is a factor of 26 broader, see also fig. 13. In view of the fact that such a narrow bandwidth source could be of extreme interest for many applications, a number of experiments were performed to determine the line width of PXR. Measurements at the low electron beam energy of 6.8 MeV result in a line width of 48 eV for a 55 μm thick diamond crystal at a photon energy of 8.98 keV [58]. This rather large line width originates from the multiple scattering of electrons in the crystal. With the critical absorber technique experiments have been performed at MAMI at an electron beam energy of 855 MeV [59]. Upper limits of the line width of 1.2 eV and 3.5 eV have been determined for the (111) and (022) reflections of silicon single crystals at photon energies of 4966 eV and 8332 eV. These limits originate mainly from geometrical line broadening effects. In backward geometry geometrical line broadening contributions, originating from the angular spread of the electron beam and small-angle scattering of the electrons in the crystal, minimize. To investigate whether under these conditions line widths as small as the Darwin-Prins values can be reached, experiments have been performed at MAMI which will be described in the next subsection. 3.1 Measurements of the line width of PXR Measurements of the line shape of PXR were done with the 855 MeV electron beam of MAMI [60]. The experimental setup is shown in fig. 11. The backward-emitted radiation of a reflection from an (nnn) plane was analyzed with a silicon single-crystal monochromator in Bragg geometry at the same (nnn) reflection. A vertical slit of 2 mm width in front of the analyzer crystal reduces the divergence of the X-rays to 0.07 mrad. The PXR radiation was separated from the backward-diffracted transition radiation (DTR) by tilting the target crystal to ψx /2 = 5 mrad for which PXR emission dominates in comparison to DTR. The measured line shapes are shown in fig. 12 for various reflections. The lines are convolutions of the PXR emission spectra, which are broadened by multiple scattering of the electrons, and the response function of the analyzer crystal. The data were analyzed assuming a Gaussian distribution for the broadened PXR line and the DarwinPrins curve of the analyzer crystal. The best fits are shown

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ψ 2, x 2

 H2

 H1

1.12m

Si-Drift

855 MeV

Detector

Electron Beam

15.3m Fig. 11. Experimental setup. The target crystal was cooled down to a temperature of T1 = 145 K, while the analyzer crystal was kept at room temperature T2 = 296 K. Since the lattice parameter a(T ) of the crystal diminishes by cooling, an enlarged photon energy and an enlarged deflection angle ψ2,x = 45.6 mrad at ψ1,x = 0 result. The angle is large enough for a lateral displacement of the Si-drift detector that it does not shadow the radiation to be analyzed. A 1 mm thick silicon single crystal with an area of 30 mm ×20 mm, cut with the (111) plane parallel to the surface, was used as analyzer crystal.

(333)

-12

NPh [10 /e]

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(444)

1 0 0.3

100

6

8

10

10

1 0

2

4

=Z [keV] Fig. 13. Measured and calculated line widths (FWHM) for various (nnn) reflections for a silicon single crystal of 525 μm thickness. Open circles: measured PXR line width, full squares: natural PXR line widths for straight electron trajectories, open squares: Darwin-Prins width, triangles: line widths calculated with the analytical model of ref. [48], and stars: Monte Carlo simulations.

It has been discussed in ref. [48] that the small-angle scattering of the electron in the Coulomb potential of the crystal atoms results in a stochastic change of the electron direction which leads to a stochastic frequency modulation of the exponentially damped wave train of PXR. As a consequence, the PXR line broadens. The scattering distribution function was approximated by a Gaussian. This approximation may not be anymore valid if the electron enters the crystal close to a channeling axis or a channeling plane. In such cases Monte Carlo simulations of the scattering process must be applied to obtain the PXR wave train. However, it is interesting to notice that calculations according to ref. [48] as well as also Monte Carlo simulated line shapes, which are both shown also in fig. 13, are in good agrement with the measurements.

3 2

(111)

Slit

Aperture

(555)

ω

(444)

ψ 1, x 2

Line Width [meV]

(T1 = 145 K)

Analyzer Crystal (T2 = 296 K)

(333)

1000

Target Crystal

(555)

0.2 0.1

3.2 Measurement of forward-diffracted PXR

0.0 -0.4

-0.3

-0.2

-0.1

0.0

'=Z [eV] Fig. 12. Line shape measurements of PXR at ψ1,x /2 = 5 mrad for different reflections as indicated by (nnn) values; Δ¯ hω = hω − ¯ ¯ hω0 , with ¯ hω0 the energy of the DTR reflection.

in fig. 12 by the full lines. The resulting PXR widths are shown in fig. 13 together with the natural line width according to eq. (2) and the width of the Darwin-Prins curve. Only for the (333) reflex the observed line width is smaller than the width of the Darwin-Prins curve. It could not be excluded that imperfections of the analyzer crystal itself broaden the higher reflections. Therefore, the real PXR line could be somewhat smaller as the fit results shown in fig. 13.

Up to now it could not be decided experimentally whether PXR emission is a kinematical or a dynamical process. The reason has been discussed by Nitta in a recent paper [61]. He showed that the first-order approximation of the dynamical calculation gives the kinematical expression. Extremely accurate absolute intensity measurements would be required to figure out a difference. Baryshevsky [62] proposed to search for the predicted forwarddiffracted wave (FDPXR) which is associated to PXR and emitted close to the direction of the electron propagation. Similar proposals have also been communicated by Nasonov [63,64]. In ref. [65] the observation of narrow FDPXR structures from a 410 μm thick tungsten single crystal at photon energies of 28.3 and 40 keV is reported. At the Mainz Microtron MAMI experiments were performed for the search of the forward-diffracted wave (FDPXR) in single silicon crystals of various thicknesses [66].

PXR HA

z

O

ω0 ω

1

x′

Fig. 14. Schematic experimental setup for the search of FDPXR.

1000 500 0 1000

0

Events

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50

100

150

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200

0

1 0.5 0

θx

pn CCD Column 200

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50

100

150

200

Column

Fig. 15. Measurements at photon energy of 10.554 keV and target thickness of 58 μm (left) and 1000 μm (right). Shown are intensity distributions summed over all 64 rows of the pn CCD detector as a function of the column number. From the upper to the lower panel the rotation angle ψx of the target crystal was varied in steps δψx = 0.5894 mrad. Beam current: 53.5 nA, exposure time: 600 s. Left panel: Beam spot size about 500 μm (FWHM) horizontally and 434 μm (FWHM) vertically. The destructive interference fringes can clearly be recognized. Right panel, upper curves: Beam spot size about 500 μm (FWHM) horizontally and 434 μm (FWHM) vertically. Right panel, lower curves: Reduced beam spot size 114 μm (FWHM) horizontally and 200 μm (FWHM) vertically.

The basic idea of the experiment will be explained by means of fig. 14. A silicon single-crystal target was positioned in such a way that the PXR reflex at a photon energy ¯hω0 = 10.554 keV is located at twice the Bragg angle θ0 = 10.797◦ in the horizontal plane of drawing. The radiation in forward direction close to the electron direction was analyzed with a flat silicon single-crystal monochromator in combination with a pn CCD camera as a position-sensitive and energy-resolving photon detector. The quasi-monochromatic FDPXR peak energy matches with the energy of the analyzer crystal at only one specific

0

50 100 150 Column Number

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2 1.5 1 0.5 0

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50 100 150 Column Number

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Ne pixel 1012

x

Ne pixel 1012

H0

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Ne pixel 1012 

e- Beam

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Ne pixel 1012

ψx

θ0

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Ne pixel 1012 

The European Physical Journal A

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Ne pixel 1012

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0.1

0

0

50 100 150 Column Number

200

0

50 100 150 Column Number

200

0

50 100 150 Column Number

200

0.1

0

0.1

0

Fig. 16. Results of simulation at photon energy of 10.554 keV and target thickness of 58 μm (left) and 1000 μm (right). Shown are the number of photons ΔN per pixel and electron for one row of the pn CCD detector as a function of the column number. From the upper to the lower panel the rotation angle of the target crystal ψx was varied in steps of Δψx = 0.5894 mrad. The beam spot sizes of the experiment and scattering of the electron beam were taken into account. The residual interference oscillations originate from the interference of the remaining 4% amplitude created at the entrance interface with the amplitude at the exit interface of the crystal. However, these oscillations are smoothed out in a real experiment in which summation is made over several or all rows of the pn CCD detector.

observation angle θx . Since its reflecting power of the crystal monochromator exhibits energetically a narrow-band characteristics, quasi-monochromatic intensity structures emitted from the target crystal can be detected by this experimental arrangement. Experiments were performed with target crystals of varying thickness and for different photon energies. As an example the intensity distributions of the experiment with 58 μm and 1 mm crystal thicknesses are shown in fig. 15. The most striking features are the structures which move across the pn CCD detector if the rotation angle ψx of the target crystal around the vertical y-axis is varied. These structures are for the thin targets interferences of the radiation amplitudes created at the entrance and exit interfaces of the crystal which originate from a resonance in the dispersion surface of the electron in the crystal. The pronounced peak structures observed for the thick target which are clearly correlated to the interference structures of the thin targets are interpreted as FDPXR contributions to the smooth transition radiation background from the downstream interface of the target crystal. The interference structures can quantitatively be explained in the framework of the well-known TR production

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mechanism utilizing a generalized formation length for crystalline matter [66]. Within this model, which is based on the formalism described in ref. [67], the resonance is connected to forward-diffracted PXR (FDPXR). Calculated emission spectra of this model, including multiple scattering of the electrons, are displayed in fig. 16. The simulations are in good agreement with the experimental observation.

duction of PXR. For thin crystals pronounced interference structures in forward direction have been observed when about a Bragg condition was fulfilled. This interference is corroborated by the narrow FDPXR lines observed with a thick Si crystal for which essentially only the exit interface of the target crystal contributes to the observed intensity. Our experimental results show that PXR production is a dynamical rather than a kinematical process.

4 Conclusion and outlook

We thank A. Steinhof for substantial contributions in the redesign and reconstruction of the undulator interferometer and N. Clawiter, S. Dambach, Th. Doerk, M. El-Ghazaly, F. Hagenbuck, G. Kube, A. Rueda and D. Schroff for their help during the course of the experiments described in this contribution. This work has been supported by Deutsche Forschungsgemeinschaft DFG under contract BA 1336/1-4.

A novel interferometer for soft X-rays has been developed. Intensity oscillations with a high degree of coherence have been observed, not only at the K absorption edge of carbon as described in ref. [18] but with the third harmonics of the undulator radiation also at the L absorption edge of Ni around 865 eV. The polarized undulator radiation can be used to investigate the magneto-optical properties of 3d transition metals. First experiments to determine the X-ray magnetic circular circular dichroism look promising. The energy band between 50 eV and about 1500 eV, which can be covered with the interferometer at MAMI, allows the investigation of many elements throughout the periodic table with samples of masses as low as 10 ng. A novel X-ray interferometer has been developed for hard X-rays. The good agreement within experimental errors of our measurements with that of Bonse et al. [39,40] proves that it is fully operational. A simultaneous precision measurement of both, δ1 (ω) and β1 (ω) of the sample foil, should be possible if the foil thicknesses are optimized properly. A possible application of this type of interferometer originates from the transient state of matter which can be produced by multi GeV bunches of about 1 nC charge and a duration in the 400 fs range. Quite unusual properties are connected with such bunches if focused to a radius " of a few μm, e.g. [68]. A strong static electric field of E0 ∼ A is present at the periphery = 16/("μm) V/˚ of the charge distribution. In addition, the high electrical current of 2500 A produces a strong magnetic field B ∼ = 540/("μm) T. During the passage of such a bunch through a thin foil the matter is put in a transient state which is characterized by a high dielectric polarization in the presence of a strong magnetic field at probably a rather large non-equilibrium electron temperature. This state can, in principle, be investigated by our novel interferometer. PXR emission from silicon single crystal slabs has been investigated with the electron beam of MAMI. The results of the line width measurements in backward geometry shows that the PXR line widths in Si crystals are superior to the Darwin-Prins widths only for the (111) and (333) reflections. Line broadening effects due to multiple scattering spoil the predicted outstanding resolution for higher-order reflections. With crystals from low-Z materials, like diamond or LiH, the small-angle scattering is reduced and much narrower lines may be expected as for silicon single crystals. The radiation from silicon single crystals, emitted close to the electron beam direction, has been studied to elucidate the discussion of kinematical versus dynamical pro-

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64. N. Nasonov, V. Sergienko, N. Noskov, Nucl. Instrum. Methods Phys. Res. B 201, 67 (2003). 65. N. Aleinik, A.N. Baldin, E.A. Bogomazova, I.E. Vnukov, B.N. Kalinin, A.S. Kubankin, N.N. Nasonov, G.A. Naumenko, A.P. Potylitsyn, A.F. Sharafutdinov, JETP Lett. 80, 393 (2004) (Pis’ma Zh. Eksp. Teor. Fiz. 80, 447 (2004)). 66. H. Backe, A. Rueda, W. Lauth, N. Clawiter, M. ElGhazaly, P. Kunz, T. Weber, Nucl. Instrum. Methods Phys. Res. B 234, 138 (2005). 67. A. Caticha, Phys. Rev. A 40, 4322 (1989). 68. C.D. Back, D. Weller, J. Heidmann, D. Mauri, D. Guarisco, E.L. Garwin, H.C. Siegmann, Phys. Rev. Lett. 81, 3251 (1998).

Eur. Phys. J. A 28, s01, 197 208 (2006) DOI: 10.1140/epja/i2006-09-021-6

EPJ A direct electronic only

X-ray phase contrast imaging at MAMI M. El-Ghazalya , H. Backe, W. Lauth, G. Kubeb , P. Kunz, A. Sharafutdinov, and T. Weber Institut f˜ ur Kernphysik der Universit˜ at Mainz, D-55099 Mainz, Germany / Published online: 6 June 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. Experiments have been performed to explore the potential of the low emittance 855 MeV electron beam of the Mainz Microtron MAMI for imaging with coherent X-rays. Transition radiation from a microfocused electron beam traversing a foil stack served as X-ray source with good transverse coherence. Refraction contrast radiographs of low absorbing materials, in particular polymer strings with diameters between 30 and 450 μm, were taken with a polychromatic transition radiation X-ray source with a spectral distribution in the energy range between 8 and about 40 keV. The electron beam spot size had standard deviation σh = (8.6 ± 0.1) μm in the horizontal and σv = (7.5 ± 0.1) μm in the vertical direction. X-ray fllms were used as detectors. The source-to-detector distance amounted to 11.4 m. The objects were placed in a distance of up to 6 m from the X-ray fllm. Holograms of strings were taken with a beam spot size σv = (0.50 ± 0.05) μm in vertical direction, and a monochromatic X-ray beam of 6 keV energy. A good longitudinal coherence has been obtained by the (111) reflection of a flat silicon single crystal in Bragg geometry. It has been demonstrated that a direct exposure CCD chip with a pixel size of 13 × 13 μm 2 provides a highly e– cient on-line detector. Contrast images can easily be generated with a complete elimination of all parasitic background. The on-line capability allows a minimization of the beam spot size by observing the smallest visible interference fringe spacings or the number of visible fringes. It has been demonstrated that X-ray fllms are also very useful detectors. The main advantage in comparison with the direct exposure CCD chip is the resolution. For the Structurix D3 (Agfa) X-ray fllm the standard deviation of the resolution was measured to be σf = (1.2 ± 0.4) μm, which is about a factor of 6 better than for the direct exposure CCD chip. With the small efiective X-ray spot size in vertical direction of σv = (1.2±0.3) μm and a geometrical magniflcation of up to 7.4 high-quality holograms of tiny transparent strings were taken in which the holographic information is contained in up to 18 interference fringes. PACS. 87.59.Bh X-ray radiography 52.59.Px Hard X-ray sources 41.50.+h X-ray beams and X-ray optics 07.85.Fv X- and gamma-ray sources, mirrors, gratings, and detectors 07.85.Nc X-ray and gamma-ray spectrometers

1 Introduction The contrast in conventional absorption X-ray imaging is based on the difierence in the absorption of the materials constituting the sample. Thin samples of light elements, such as soft tissues and organic materials with Z ≤ 8, show a weak absorption contrast even at low X-ray energies, i.e., the big deflciency is that the conventional absorption radiography cannot distinguish between materials with similar attenuation coe– cients. For low-Z materials, however, a high contrast could be obtained if the phase shift of the X-rays introduced by the object could be exploited instead of the intensity of the transmitted wave. The enhancement of the contrast is attributed to the fact that, in a

Former PhD Scholarship Holder in the Long Term Mission System from the Arabic Republic of Egypt. b Present address: Deutsches Elektronen-Synchrotron DESY, Notkestra e 85, D-22603 Hamburg, Germany.

particular for low-Z materials, the phase shift for X-rays is higher than the absorption of the incident X-rays. Also, for the radiography based on the phase shift mechanism, the absorbed dose is considerably lower in comparison to the conventional absorption radiography, see, e.g., refs. [1, 2,3]. X-ray phase contrast imaging can be carried out with various methods, for an overview see the recent ref. [4]. In particular, it has been pointed out by Wilkins et al. [5] that a very simple experimental setup with a polychromatic X-ray source of good transverse coherence, i.e. a small micro-focused spot, is already su– cient. Information can be supplied by such a method on the sample morphology, i.e. its boundaries, interfaces and location of small features, see e.g. ref. [6,7,8]. If, in addition, the Xray source emits monochromatic X-rays, holograms can be taken. The experimental setup is similar to that of Gabor in-line holography [9]. In principle, such a setup

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is rather simple but a highly transverse and longitudinal coherent X-ray source of good intensity and also high spatial resolution detectors are required. Such sources are available at third generation synchrotron radiation sources like ESRF, APS, and SPRING8, and hard X-ray phase contrast imaging, in-line holography and microtomography have been accomplished at these facilities, see, e.g., refs. [10,11,12]. The work presented here exploits the potential of the low-emittance 855 MeV electron beam of the race track microtron MAMI to produce X-rays with very good transverse coherence. Our approach is based on transition radiation (TR) production in the X-ray region with a microfocused electron beam. In sect. 2 some features of the complex refraction index will be recalled with particular view on phase contrast methods. In sect. 3 the results of phase contrast imaging with a polymonochromatic X-ray beam from a TR foil stack with good transverse coherence will be presented. Section 4 deals with our approach toward a hard X-ray in-line holography using monochromatic Xrays. The paper closes with a conclusion.

2 Absorption versus phase shift When a parallel beam of X-rays penetrates matter, it suffers an attenuation and a phase shift. These macroscopic quantities are described by the complex refraction index of X-rays [13] n(ω) = 1 − δ(ω) + iβ(ω) .

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Fig. 1. Transmission of an electromagnetic wave through a piece of matter of thickness d and complex refraction index n = 1−δ+iβ. The transmitted wave is attenuated by the factor exp[−(ω/c)·β·d] and has sufiered a phase shift φm (d)−φv (d) = (ω/c) · δ · d with respect to the unperturbed vacuum wave.

hω = 30 keV, δ/β for polycarbonate (C14 H14 O3 ) is approximately 40 times larger than for nickel (Z  28). This clearly demonstrates that for low-absorbing materials the phase shift dominates in comparison with the attenuation. For a polycarbonate foil of 10 μm thickness and a photon energy hω = 12 keV, for which the complex refraction index parameters are δ = 1.826 · 10−6 and β = 1.573 · 10−9 , the phase shift is φ = 1.11 rad while the intensity attenuation is only 1 − exp[(−2ω/c) · β · d] = 1.91 · 10−3 . These considerations lead to the important conclusion that a high contrast combined with a low absorbed dose could be achieved by using the phase shift mechanism to produce a radiograph [5]. In the next section the question will be addressed how the phase shift can be exploited for radiography.

(1)

The real part [n(ω)] = 1 − δ(ω) describes the refraction of the wave of angular frequency ω in a material, the quantity δ(ω) gives the deviation of the refractive index of a material from unity (refraction index of vacuum). It is called the refractive index decrement. The imaginary part [n(ω)] = β(ω) specifles the attenuation of the X-rays in matter. It is called the absorption index. The transmission of an electromagnetic wave through a piece of matter of thickness d is illustrated schematically in flg. 1. The undisturbed wave propagation in x -direction is described by the expression Av = A0 ei(kv d−ωt) ;

!

(3)

where km = nω/c is the wave number in the medium. Equation (3) contains a phase factor exp(−iφ(d)) with φ(d) = φm (d) − φv (d) = (ω/c)δ · d the phase difierence between the wave in matter with phase φm , and in vacuum with phase φv . In addition, the wave sufiers an amplitude attenuation |Am |/|Av | = exp[−(ω/c) · β · d]. The ratio δ/β is drastically larger for a low-Z material in comparison with a high-Z material at photon energies in the order of 20 40 keV. For example, at a photon energy

3 Refraction contrast radiography 3.1 Basics In an ideal experiment a point source emanates a monochromatic wave and illuminates the sample. The Xray wavefront impinging on a sample will be deformed at the passage through the medium when its thickness or refractive index is inhomogeneous. In the framework of the eikonal approximation the wave vectors of the X-rays are normal to the equi-phase surfaces. In this picture of ray optics, i.e. for λ → 0, the deviation from the initial direction is due to refraction. For the sake of simplicity, in flg. 2 a one-dimensional object such as a string of radius R and a homogeneous refraction index n2 = 1 − δ2 + iβ2 is considered, which is embedded in a medium of refraction index n1 = 1 − δ1 + iβ1 . It is illuminated with a nearly parallel X-ray beam. The phase shift of the outgoing wave relative to the wave in vacuum is given by # ! z "2 4π o φ(z0 ) = (δ2 − δ1 )R 1 − , (4) λ R with zo the vertical coordinate at the object. The angular deviation α of the normal to the incoming wavefront is, in the eikonal approximation,   λ  ∂φ(z0 )  2(δ2 − δ1 ) zo = α= (5) 1 . 2π  ∂zo  R [1 − ( zRo )2 ] 2

M. El-Ghazaly et al.: X-ray phase contrast imaging at MAMI

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Fig. 2. Formation of a refraction contrast radiograph according to geometrical ray optics. Refracted X-rays slightly deviate from the initial propagation direction at the interfaces in accordance with Snell’s law of refraction. Since the refraction index for X-rays is slightly smaller than unity (about 10 6 ), X-rays are refracted in opposite manner to visible light, i.e. they are focused by a concave and defocused by a convex object. For tangentially incidence the X-rays encounter maximum deviation resulting in the formation of a contrast which enhances the visibility of the interfaces. The source-to-object distance is xso , the object-to-detector distance xod and the source-to-detector distance xsd = xso + xod .

The phase gradient diverges at zo = R. The rays deviate by a large angle from the original propagation direction even though (δ2 −δ1 ) is very small as in the case of X-rays, which leads to a loss of intensity at boundaries or an edge contrast. This explains why the radiograph looks like a direct image of contours of the details which constitute the sample. More generally, any rapid variation of the refraction index or the thickness of the sample may be imaged by the edge contrast which appears in the radiograph even when a polychromatic X-ray beam is used. In fact, the wave refracted by the sample interferes with the unperturbed wave. The difiracted wave and the unscattered wave form an interference pattern at the detector which is called a hologram. It is recorded by an image detector of high spatial resolution. For the polymer string with circular cross-section stretched along the horizontal yo -axis, the normalized electric wave fleld E(zd , λ)/E0 (zd , λ) at the detector plane can be calculated by means of the Fresnel-Kirchhofi integral. As shown in [14,15] the result is # E(zd , λ) xsd =1+ E0 (zd , λ) iλxso xod +R · −R



4π  iδ + β · R · exp − λ

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The quantities xso , xod , and xsd are deflned in flg. 2. The normalized intensity distribution perpendicular to the string direction, the zd -direction at the detector (δ) plane, In (zd , λ0 ) = |E(zd , λ0 )|2 /|E0 (zd , λ0 )|2 is shown in flg. 3 (a) for a monochromatic X-ray source with wavelength λ0 . In a real experiment the spectral distribution of the X-rays and the flnite beam spot size deteriorate

Fig. 3. Calculated interference patterns for a polymer string with a diameter of 30 μm, complex refraction index δ = 7.24 · 10 6 and β = 2.42 · 10 8 at an X-ray energy of 6 keV, source-to-object distance xso = 10.45 m, object-to-detector distance xod = 3.15 m. (a) Normalized intensity distribution |E(zd , λ0 )|2 /|E0 (zd , λ0 )|2 derived from eq. (6) for a point source and monochromatic X-rays with λ0 = 2.067 ” A, corresponding to an energy of 6 keV; (b) for a point source but a Gaussian spectral distribution around λ0 with standard deviation of σλ = 0.6 ” A; (c) for a Gaussian intensity distribution of the X-ray source spot with standard deviation σz = 6 μm and monochromatic X-rays. Convolutions according to eq. (7). Ideal detector resolution is assumed.

the visibility of the interference fringes as demonstrated in flg. 3 (b) and (c). Source spot size and spectral distribution have been taken into account as convolutions with the normalized intensity distributions g(zs ) of the beam spot and the spectral distribution f (λ) of the X-rays according to    xod (δ) In (zd ) = In zs , λ − λ0 ·g(zs )·f (λ)·dzs ·dλ . zd − xso (7) It can be seen from flg. 3 (b) that a few interference fringes remain visible, which resemble the structure of the string, even for polychromatic X-rays with a small longitudinal coherence length LL = λ2 /(2Δλ) ≈ λ, and/or a beam spot size σz = 6 μm (rms) which corresponds to a transverse coherence length LT = xsd · λ/(2πσz ) = 75 μm. Radiography based on these conditions will be called in the following refraction contrast radiography. For X-ray in-line holography, both a very good transverse coherence and a good longitudinal coherence of the X-ray beam are required since the holographic information is imprinted in the interference pattern as shown in flg. 3 (a).

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Fig. 4. Schematic diagram showing the experimental setup for refraction contrast radiography. The inset shows the calculated TR spectrum as function of the photon energy for which multiple scattering, electron beam divergence (0.6 mrad) and self absorption were taken into account. The foil stack consists of 30 polyimide foils with a thickness of 25 μm each, and spacings between the foils of 75 μm. It was optimized for a photon energy of 33 keV.

3.2 Experimental The principle of the refraction contrast radiography will be explained by means of flg. 4. The 855 MeV electron beam, with a Lorentz factor γ = 1673, produces in a transition radiation foil stack a polychromatic X-ray beam which propagates in the forward direction in a cone with a typical apex angle of 2/γ  1.2 mrad. The X-ray emission spectrum is shown in the inset of flg. 4. The polychromatic X-rays leave the vacuum system through a polyimide exit window of 120 μm thickness which is located at a distance of 5.88 m from the foil stack. The beam line is shielded by a concrete wall of 1 m thickness and 3.5 m height to reduce the background in the experimental area. The background originates from electrons which emitted a bremsstrahlung photon in the TR foil stack and left the beam line behind the bending magnet, as well as the background from the beam dump itself. The objects to be imaged are mounted in air at difierent distances from the target xso , and from the X-ray fllm xod , with 5.88 m < xso < 13 m and 0 m < xod < 7.12 m, respectively. The source-to-detector distance was xsd = 11.38 m. The Mamoray MR5 II PQ X-ray fllm produced by Agfa1 was used as position-sensitive detector. It is based on silver bromide with an emulsion thickness of df = 12 μm [16]. The exposed X-ray fllms were processed manually. The X-ray fllms were digitized with a Nikon fllm scanner Super CoolScan 4000 ED [17] which has a spatial resolution of 4000 dpi 2 corresponding to a pixel size of (6.35 × 6.35) μm2 . The primary quantity which is measured by an X-ray fllm is the photographic density Dp [18,16]. It is deflned with the basis-10 logarithm as Dp = log(i0 /i) with i0 1 2

the optical light intensity impinging on the fllm and i the intensity measured by the detector of the densitometer. From this primary quantity the so-called fog Df = log(i0 /i0f ) of an unexposed part of the fllm must be subtracted to obtain the density D = Dp − Df = log(i0f /i). The latter must be related to the exposure b(r) of the fllm, i.e. the energy per unit area dE(r)/dA deposited by the X-ray photons at a certain location r of the fllm. In a simple theoretical model [18] the photographic density can be described by

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(8)

The saturation density Dsat and b0 are characteristic quantities of the X-ray fllm. From eq. (8) the relative exposure is obtained as   Dsat b(r) = ln . (9) b0 Dsat − D(r) Since we are interested in normalized exposure ratios (b(r)/b0 )/(b/b0 ) with b the value without the object which can be replaced by a mean value on some position outside the domain of interest, the unknown quantity b0 cancels. The still unknown saturation density Dsat must, in principle, be determined. However, since the digitization devices to our disposal had only a depths of 8 bits the main restriction in the dynamical range is expected to originate from the digitization procedure and not from the dynamical range of the X-ray fllm. In view of the low digitization depth, the procedure we adapted to obtain the contrast Cref of a string as deflned by eq. (10) in the next subsection was the following. At flrst, domains on the X-ray fllm were selected in which the photographic density was assumed to be in the linear region. In this case eq. (8) reduces to D(r) = Dsat · b(r)/b0 and in the exposure ratio also the saturation density cancels. Thereafter, we determined the contrast Cref at various positions of the string for which the exposure varied due to the intensity proflle of the X-ray beam spot and selected the maximum value as the experimental contrast.

3.3 Results An extensive study of the contrast generation as a function of the object-to-detector distance xod has been performed for polyamide strings of difierent diameters. Figure 5 (a) shows a typical example for a polyamide string 3 with a diameter of about 270 μm. The calculated absorption contrast for such a polyamide string does not exceed about 1%. Therefore, no absorption contrast can be observed with the traditional contact radiography, i.e. for xod ∼ = 0, in accord with our measurements. By moving the object away from the detector, the imaging regime is changed from absorption radiography to phase contrast radiography and phase shift is the mechanism to produce the contrast. The contrast appears at the borders of the 3

Supplied by Goodfellow.

M. El-Ghazaly et al.: X-ray phase contrast imaging at MAMI

Fig. 5. Refraction-enhanced radiograph of a polyamide string with a diameter of 270 μm at an object-to-detector distance xod = 5.5 m and a source-to-detector distance xsd = 11.38 m. The electron beam current was 6 nA, the exposure time amounted to 60 s. X-ray source sizes were σh = (8.6 ± 0.1) μm and σv = (7.5 ± 0.1) μm in horizontal and vertical direction, respectively. (a) Radiograph, (b) intensity proflle for which 100 vertical pixels were added together to improve the statistics. (c) Normalized intensity proflle according to geometrical optics with the following parameters: fllm and scanner resolution σt =(10.0 ± 0.4) μm, and the wave optical contribuA. tion σw = λxsd xod /(2πxso ) = 2.3 μm with λ = 0.633 ” (d) Same as (c) on the basis of wave optics.

polyamide string where the density gradient reaches its maximum value. An edge contrast can be deflned as Cref =

Imax − Imin Imax + Imin

(10)

with Imax and Imin deflned in flg. 2. As can be seen from flg. 5 (b) the contrast amounts to Cref = 17.8%. The contrast Cref as a function of the object-to-detector distance xod is shown in flg. 6 as error bars for all measurements. 3.4 Discussion The most interesting feature of the radiograph shown in flg. 5 is that an edge enhancement or phase contrast can be observed with a polychromatic X-ray beam. This fact has been discussed in a number of papers also in connection with the interplay between refraction and difiraction [5, 19]. The general features of refraction contrast imaging will be discussed by means of flg. 6. It can be stated that the distance xod between object and detector must be at least as large that the wave optical spread of the difiracted X-rays becomes comparable with the detector resolution. Otherwise all interference fringes are blurred and the contrast is low. With increasing object-to-detector distance xod the contrast increases about linearly. However, at the same time the projected X-ray spot size on the detector

201

Fig. 6. Contrast Cref for a polyamide string of 270 μm diameter as a function of the object-to-detector distance xod . The source-to-detector distance xso = 11.38 m was kept constant. Error bars are measurements, crossed circles calculations on the basis of the wave optical model with a beam spot size σv = 7.5 μm and a total X-ray fllm resolution and scanner resolution σt = (10.0 ± 0.4) μm. Stars designate calculations according to geometrical optics.

plane increases and this worsens the contrast at larger xod distances. The maximum of the contrast is a function of the beam spot size and of the fllm resolution. But contrast is not the only flgure of merit. It must also be taken into account that with increasing xod the edge spread increases and the resolution deteriorates. The latter might be undesirable in case that resolution is of importance and nearby features must be resolved. Next, the question will be addressed whether the measured edge enhancement structures can be understood quantitatively in the framework of the wave optical and geometrical models. As has already been pointed out above, the refraction contrast is in a strict sense a wave optical phenomenon, however, with some care it can also be explained in the framework of geometrical optics. Such an approach might be a good approximation at experimental conditions in which interference patterns are smeared out, i.e., if the object is illuminated with polychromatic X-rays, or if the projected source size or the detector resolution are too large, respectively too bad. Both models have two free parameters which are the standard deviation of the beam spot size and the resolution of the X-ray fllm including the fllm scanner. The wave optical calculations were performed with the efiective X-ray spectrum which takes into account the transition radiation spectrum and the absorption characteristics of air and the X-ray fllm. The efiective Xray spectrum was approximated with 22 discrete values in the energy range between 8 and 30 keV. For the geometrical model it was su– cient to approximate the Xray spectrum by a delta-function at the mean photon energy hω = 19.6 keV, corresponding to a wavelength λ = 0.633 ”A, since model calculations showed, via the optical parameters, a rather weak energy dependence of the sharp edge structure. As can be seen, both the geometrical

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ages from the difierent veins in the radiograph are overlapping. Such three-dimensional structures may be disentangled by a holographic method some principles of which are sketched in the next section.

4 Toward hard X-ray in-line holography

Fig. 7. A refraction contrast radiograph of a part of green leaf of Rumex crispus. The radiograph was recorded by the MAMORAY MR5 II PQ (Agfa) X-ray fllm. The object-todetector distance was xod = 5.5 m at a source-to-object distance xso = 5.88 m. With these parameters the magniflcation was 1.94 times. The electron beam energy was 855 MeV, the electron beam spot size had standard deviations of σh = (8.6 ± 0.1) μm and σv = (7.5 ± 0.1) μm in the horizontal and vertical direction, respectively. The TR foil stack described in flg. 4 was used. The electron beam current was 6 nA, the exposure time 40 s.

and the wave optical model describe the general features of the measurement quite well. The rather good results for the contrast ratio Cref of the geometrical model, as shown for the example in flg. 5 (c), were obtained after a convolution with a Gaussian of standard deviation  σ=

xod xso

2 σs2 + σd2 + σp2 +

λxod xsd . 2πxso

(11)

This parameter takes into account projected source size σs , X-ray fllm resolution σd , pixel resolution of the  fllm scanner σp , and an additional term σw = λxso xod /(2πxsd ). This term accounts for the difiraction which is absent in the geometrical model. It was estimated from the second exponential in eq. (6). The argument is that for a flxed point zo at the object the typical spread in the detector plane is given by a region |zo xsd /xso − zd | < σw . For |zo xsd /xso − zd |  σw the exponential oscillates rapidly and the mean value in zd approaches zero. The real parameter σw may difier from the assumed one. However, the good agreement may be a consequence of the rather poor total fllm resolution $ 2 2 σt = σd + σp = 10 μm which is much larger as the wave optical contribution σw = 2.3 μm. As an example of the visualization of low-Z objects by the refraction contrast, in flg. 7 the image of a green leaf is shown. In the part labelled with (a) where the leaf is thinner than 1 mm, the visibility of a bundle of vascular tissue (veins) could be resolved with high contrast. In the middle part labelled by (b), the object is about 3 mm thick and contains a bundle of vascular tissue (veins). However, the identiflcation of an individual vein is di– cult since im-

In the preceding section it has been shown that the transition radiation (TR) X-ray source is well suited for refraction contrast imaging. This chapter deals with the investigation of the possibility of X-ray phase contrast imaging and hard X-ray in-line holography with monochromatic X-rays at MAMI. The good emittance of MAMI allows the preparation of a micro-focus which is a prerequisite of the required transverse coherence of the TR X-ray source. The longitudinal coherence can be achieved by a single-crystal monochromator. The basics of in-line holography, the experimental setup, the preparation of the micro-focused electron beam and the results obtained so far will be described in the following. 4.1 Basics A wave emanating from a point source may illuminate an object from which it is scattered. The wave amplitude E(r) = E0 (r) + Escat (r) can be split into the reference wave E0 (r) and a scattered wave Escat (r) = a(r) · E0 (r). The amplitude ratio can be written as E(r)/E0 (r) = 1 + a(r). The scattering amplitude a(r) contains the required information on the object. On a detector screen, such as an X-ray fllm or a CCD detector, the squared absolute values of the amplitudes |E(r)|2 and |E0 (r)|2 are measured from which the contrast image |E(r)|2 −|E0 (r)|2 can be obtained. By division through the reference wave |E0 (r)|2 the normalized contrast ratio, |E(r)|2 − |E0 (r)|2 = 2 [a(r)] + |a(r)|2 , |E0 (r)|2 (12) can be determined. The appearance of 2 [a(r)] = a(r) + a∗ (r) on the right-hand side of eq. (12) shows that the hologram contains also information on the real part of the scattering amplitude rather than only its absolute value squared |a(r)|2 which may be referred to as classical difiraction pattern of the complementary transmission function of the object [15]. Such classical difiraction patterns are observed in difiraction experiments in which the reference wave is absent, e.g., at difiraction on a slit which is the complementary to an opaque object as, e.g., an opaque wire. While the classical difiraction pattern is rather smooth, see flg. 8 (a), the holographic difiraction pattern oscillates rapidly, see flg. 8 (b). These oscillations have a rather small amplitude and can hardly be seen in a measurement of the hologram such is shown in flg. 8 (c). Much more pronounced oscillations are observed for transparent objects as polymer strings, see flg. 8 (d) and (e) which are maintained in the sum of the classical and the Inorm (r) =

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Micro focus Cooled slow scan CCD detector

Objects P

Fig. 8. Analysis of the calculated normalized contrast image in distinct patterns for a totally opaque tungsten wire, left column (a), (b) and (c), and for an approximately transparent polymer string, right column (d), (e) and (f). Both wires have the same diameter of 25 μm. The X-ray photon energy is 6 keV (λ = 2.067 ” A), the complex refraction index parameters are δW = 8.5 · 10 5 and βW = 1.1 · 10 5 and δP = 7.3 · 10 6 and βP = 2.55 · 10 8 for tungsten and polymer, respectively, at this energy. The source-to-object distance is xso = 1.92 m and the object-to-detector distance xod = 11.68 m. Panels (a) and (d) show the classical difiraction pattern |a(zd )|2 which is the difiraction pattern of the complementary object, (b) and (e) show the holographic difiraction pattern 2 [a(zd )] which come about by the interference between the wave front disturbed by the object and the reference wave emanating from the source, (c) and (f) show the normalized contrast images.

holographic difiraction pattern, see flg. 8 (f). These oscillations contain information on the distance between the object and the detector or the source, and via the refractive index decrement δ and the absorption β also on the bulk of the string. In addition, the hologram contains via the transverse coherence length also information on the beam spot size (ref. [20]). 4.2 Experimental The observation of interference patterns as shown, for instance, in flg. 8 requires both, a good transverse and a good longitudinal coherence which can be achieved with a microfocused and monochromatic X-ray beam. These requirements led to an experimental arrangement at MAMI

Fig. 9. Schematic experimental setup for X-ray in-line holography at MAMI. Shown are the TR foil stack, the single-crystal monochromator at a distance of 7.8 m from the target, and a CCD detector or an X-ray fllm at a distance of 5.8 m from the monochromator. The objects to be imaged can be positioned at distances of 1.88, 4.3, 7.47, 10.78, 12.71 and 13.6 m from the Xray source. All components are housed in a connected vacuum system to avoid self-absorption of the X-rays. The inset shows the calculated TR energy spectrum as function of the photon energy for which multiple scattering, electron beam divergence (0.8 mrad) and self-absorption were taken into account. The TR foil stack consists of 25 polyimide foils with a thickness of 12.5 μm which are spaced out by aluminium foils of 100 μm thickness, the latter with centric holes of 2 mm diameter for the passage of the electron beam. Beam Dump

to Nuclear Physics Experiments e - Beam

P X1 Hall e-- Beam

Focusing Quadrupoles RTM3

Radiator Chamber

Crystal Monochromator Polychromatic Transition Radiation

Monochromatic Transition Radiation

CCD detector

Fig. 10. Floor plan of the experimental area at MAMI. The electron beam is fed into the X1-beam line just behind the third stage of the race track microtron RTM3. During the course of the experiments the entrance to the X1 hall below is closed by a concrete door.

which is schematically depicted in flg. 9. The floor plan at MAMI is shown in flg. 10. A flat single crystal in Bragg geometry is used as monochromator. The objects to be imaged can be placed between the TR radiator and the monochromator crystal close to the TR source resulting in a magniflcation of the object of up to a factor of 7.4 or, alternatively, between monochromator and X-ray detector. The magniflcation may be of importance to compensate

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for a moderate detector resolution if, e.g., CCD-chips in a direct exposure mode are used, see below. For the preparation of a micro-focused electron beam, a low beam emittance in horizontal and vertical directions is of particular importance. The emittance of the MAMI electron beam in horizonal direction is bigger than the emittance in vertical direction because the electrons emit synchrotron radiation in the bending magnets of racetrack microtron 3. The horizontal emittance grows rapidly above an electron beam energy of 400 MeV, while the vertical emittance still decreases. As a compromise, a beam energy of 600 MeV was chosen for which the emittances are h = 2.3 μm mrad and v = 0.52 μm mrad in the horizontal and vertical directions, respectively. The polyimide foil stack to produce transition radiation is optimized for a high X-ray flux at a photon energy of 6 keV at the electron beam energy of 600 MeV. The calculated photon energy spectrum is shown in the inset of flg. 9. The flat silicon single crystal with its surface parallel to the (111) crystal plane acts as a mirror for the TR photons. However, the mirror is energy dispersive in the horizontal direction. The deviation ε of the photon energy, deflned by the equation hω = hωB (1 + ε), from the nominal Bragg energy √ 2π h2 + k 2 + l2 hc hωB = (13) a0 2 sin θB is approximately given by the expression [21] ε=

(χ0 ) θx − , tan θB 2 sin2 θB

(14)

where θx = θ − θB is the deviation from the nominal Bragg angle θB . The integers h, k, l are the Miller indices, a0 = 5.4309 ” A the lattice constant, and (χ0 ) the real part of the dielectric susceptibility χ0 . The Bragg angle for hωB = 6 keV amounts for the (111) reflection to θB = 19.25◦ . In in-line holography a scattered wave from the object interferes with an unscattered wave from the source. For an assumed transverse coherence length of LT = 250 μm, at a distance of 13.6 m a θx = 18.4 μrad results. The angular spread which originates from the beam spot size and the pixel resolution is less than 1 μrad and can be neglected. The angle θx corresponds, according to eq. (14), to a relative energy shift of 5.3 · 10−5 or Δhω = 0.32 eV. This means that two waves with slightly difierent energies must interfere which is only possible if the longitudinal coherence length LL is long enough. The width of the reflecting power ratio of the monochromator crystal is Δε = 1.4 · 10−4 corresponding to Δhω = 0.84 eV [21], and a longitudinal coherence length LL = 0.5λ2 /Δλ = 0.5λ/Δε = 0.74 μm results. This value is su– ciently large for all objects investigated in this work which had thicknesses in the sub-mm range, since at a refractive index decrement of δ = 1 · 10−6 the optical path difierence is less than 0.01 μm. Hard X-ray holography requires, like refraction contrast radiography, a two-dimensional resolving detector

with a large dynamic range and linear relationship between the incident radiation intensity and the response of the detector. Such conditions can be fulfllled by a chargecoupled device (CCD) or an X-ray fllm. For the current experiments the CCD system ANDOR DO-434 BN CCD [22] was used. It contains a back-illuminated CCD low-noise sensor from Marconi CCD47-10 [23] with 1024 × 1024 pixels of size 13 × 13 μm2 . The chip has a good quantum e– ciency over a wide spectral range. For X-rays of 6 keV energy it amounts to still about 45%. These features ofier the opportunity to use the CCD chip in the direct exposure mode in which the signal is generated by direct energy deposition of X-rays in the sensitive layer of approximately 10 μm thickness. Direct-exposure CCD camera chips have, compared with X-ray fllms, the big advantage that they have a good linearity over a wide dynamical range, a good signal-tonoise ratio, and that they are on-line capable. The latter fact is very important since contrast or normalized contrast images can easily be generated in which all parasitic background, originating not from the object, can be eliminated. The disadvantage of a moderate spatial resolution in comparison to an X-ray fllm can be alleviated by a geometrical magniflcation. In reality, however, the spatial resolution is larger than the pixel size because of so-called split events in which the deposited energy is shared by neighboring pixels. The Structurix D3 X-ray fllm from Agfa is a useful detector as well. The main advantage in comparison with the direct exposure CCD chip is its very good resolution. The standard deviation of the resolution was measured to be σf = (1.2 ± 0.4) μm, which is about a factor of 6 better than for the direct-exposure CCD chip. The main disadvantage of the X-ray fllm is the missing on-line capability with the consequence that the generation of normalized contrast images is rather involved. The procedure to obtain the intensity information from the photographic density is similar to that already described in sect. 3.2. The X-ray fllm was digitized with a fllm scanner (Nikon Coolscan LS 4000 [17]) and with an optical microscope equipped with a high-resolution 8-bit CCD camera (F-View XS [24]). From this system limitations are expected because the dynamical range cannot be better than the digitization depth of the ADC (1:256), while the X-ray fllm has a dynamical range which is more than a factor of 10 better (3.5 decades corresponding to 1:3160). Since, in addition, the illumination time was selected automatically by the scanner after the part of interest of the picture and the optical magniflcation were selected, the holograms were digitized at various positions of the string. Along the imaged strings the exposure is changing and the sector with the best contrast was selected for further analysis. 4.3 Measurements and discussion 4.3.1 Investigation of the transverse coherence in horizontal direction To study the coherence in the horizontal and vertical directions, radiographs of two polymer strings of the same

M. El-Ghazaly et al.: X-ray phase contrast imaging at MAMI

Fig. 11. Two radiographs of a polymer string of diameter 30 μm. In radiograph (a) the string is mounted vertically, in (b) horizontally. The X-ray source spot size was σh = (1.7 ± 0.1) μm in the horizontal and σv = (3.9±0.4) μm in the vertical direction. The electron beam spot size was checked with the wire scanner before and after the imaging in order exclude a possible shift of the beam spot. The source-to-object distance was xso = 4.3 m, and the object-to-detector distance xod = 9.61 m. Electron beam current 700 nA, exposure time 1.8 s, 50 frames added up.

Fig. 12. A background-corrected hologram (contrast image) of a polymer string with (150 ± 20) μm diameter, supplied by Goodfellow. Source-to-object distance xso = 1.88 m, source-todetector distance xsd = 13.91 m, corresponding to a magniflcation of 7.4 times, tilt angle of the object 46 , X-ray source spot size σh = (5.9 ± 0.1) μm, and σv = (2.6 ± 0.1) μm, electron beam current 600 nA, exposure time 8.1 s, 50 frames added up.

thickness of 30 μm were taken which were mounted horizontally and vertically. The radiographs are shown in flg. 11. Although the beam size in the horizontal direction σh = (1.7 ± 0.1) μm was smaller than σv = (3.9 ± 0.4) μm in the vertical direction, no interference patterns were observed for the vertically mounted polymer string. The maximum contrast, Cref = (Imax − Imin )/(Imax + Imin ) for the horizontally mounted string was 64%, while for the vertically mounted one it was only 11%. The only reasonable explanation for this observation is that the transverse coherence in the horizontal direction is deteriorated by the monochromator crystal. Obviously, in the energy dispersive direction (horizontally) an additional angular divergence is introduced by the crystal. Since the reason of this efiect could not be found, all experiments with strings described in this work were performed with horizontally mounted strings. Despite the moderate transverse coherence in horizontal direction holograms could be taken which show also interesting features in the horizontal direction. Figure 12 shows a contrast image of a polymer string with a diameter

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Fig. 13. Fringe visibility as a function of the X-ray source spot size. Shown are holograms of a polymer string with a diameter of 30 μm for an X-ray source spot size as measured with a wire scanner of (a) σh = (5.9 ± 0.1) μm, σv = (2.6 ± 0.1) μm, and (b) σh = (19.1 ± 0.7) μm, σv = (0.50 ± 0.05) μm. From the smallest discernible fringe spacings rmax = 50 μm (a) and 25 μm (b), with eq. (15) standard deviations σv = 2.4 μm (a), and 1.2 μm (b) result. Source-to-object distance xso = 1.88 m, object-to-detector distance xod = 12.03 m, corresponding to a geometrical magniflcation of 7.4 times. The angle between string and beam direction amounted to 46 . The electron beam current was 500 nA, exposure time 8.1 s per frame, 100 frames added up.

of (150±20) μm. According to inspection under an optical microscope the string has a nearly ideal cylindrical shape. No deformations or impurities inside the string could be observed. However, in the hologram inhomogeneities are clearly visible which may be air bubbles or impurity inclusions with a difierent density than the string material. The background correction assures that these inhomogeneities do not originate from dust particles on the monochromator crystal or the detector. 4.3.2 Optimization of the beam spot size The most important prerequisite for taking high-quality holograms is the minimization of the beam spot size. In the flrst step, the spot size was measured with a tungsten wire of (4.0±0.4) μm diameter which was scanned through the electron beam. In particular, the electrical current of the quadrupole doublet in flg. 9 was varied until the scan yielded the smallest spot size. In the next step holograms of polymer strings were taken with the CCD camera and the spot size was estimated from the smallest discernible fringe visibility estimated according to [25] σ = 0.31

xso rmax . xod

(15)

As already mentioned, a CCD chip allows fast on-line imaging, however, the resolution in the direct-exposure

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Fig. 14. (a) Hologram of a polyamide string with diameter of (150 ± 20) μm. Source-to-object distance xso = 1.88 m, objectto-detector distance xod = 11.73 m, corresponding magniflcation 7.24 times, X-ray source spot size σh = (19.1 ± 0.7) μm and σv = (0.50 ± 0.05) μm as measured with the tungsten wire. The X-ray fllm was digitized with an optical microscope with a magniflcation of 4 in order to maintain a good resolution. Therefore, only part of the hologram was in the fleld of view. (b) Intensity proflle with 200 rows added up. Notice that also this string was stretched horizontally.

mode is limited by the pixel size of 13 μm. To achieve a geometrical magniflcation, the object was placed at close distance to the X-ray source. In flg. 9 the possible positions of the objects are marked. An example of the spot size measurement by the fringe method is shown in flg. 13 which yielded after optimization a standard deviation σv = 1.2 μm. To exclude a possible influence of the moderate resolution of the CCD detector, high-quality holograms were taken with the high-resolution Structurix D3 X-ray fllm from Agfa. Figure 14 (a) shows a part of a hologram for a polyamide (Nylon) string. A large number of about 18 interference fringes can be seen, as demonstrated in flg. 14 (b). In this radiograph the main deterioration in the fringe visibility results from the X-ray spot size. The minimum discernible distance between two adjacent fringes is about 25 μm and the estimated X-ray source size is again about σv = 1.2 μm. In comparison with the wire scanner measurement which yielded a spot size σv = (0.50 ± 0.05) μm, the measured values with the fringe method either determined with the direct exposure CCD, or with the highresolution X-ray fllm, deviate signiflcantly. This deviation can be explained by the longitudinal depth of the foil stack which amounts to 2.8 mm. For the measured vertical emittance v = 0.52 μm mrad at the electron beam energy of 600 MeV, and a micro-focused electron beam spot size σv = (0.50 ± 0.05) μm, the corresponding divergence is 1.04 mrad. When the focus is exactly in the middle of the foil stack, the beam spread within the foil stack amounts to a standard deviation of 1.5 μm, in accord with the observation with the direct-exposure CCD chip and the X-ray fllm 4 . 4.3.3 Analysis of holograms for polyamide strings There are two possibilities to analyze holograms of strings. In the flrst one, calculations on the basis of the Fresnel4

The errors of the fringe method may be in the order of 20%.

Fig. 15. (a) Radiograph of a polyamide string of (150±20) μm diameter. Source-to-object distance xso = 4.3 m, object-todetector distance xod = 9.31 m, corresponding geometrical magniflcation 3.17 times, X-ray source size σh = (19.1±0.7) μm and σv = (0.50 ± 0.05) μm. The X-ray fllm was digitized with an optical microscope with a magniflcation of 4. (b) Intensity proflle, 200 rows added up. (c) Calculated intensity proflle of the radiograph. The spatial resolution of the X-ray source spot size, σv = (0.50±0.05) μm, of the fllm, σf = (1.2±0.4) μm, and the optical resolution, σsc,4 = (4.1±0.1) μm, were incorporated in the calculations.

Kirchhofi integrals can be performed in which assumptions about the density and morphology of the string are incorporated. The right solution can be found by a systematical trial and error method. An example is shown in flg. 15. Although a good overall agreement between measurement and calculation could be achieved with the assumption of a homogeneous density distribution within the string, in detail signiflcant difierences in the interference pattern close to the boundaries can be recognized. These difierences may indicate a density gradient at the periphery of the string. Reflnements of the string model are required to further investigate the origin of these difierences. The second possibility is based on reconstruction algorithms to flnd the phase proflle produced by the transparent object. One of these is the modifled Gerchberg-Saxton algorithm [26]. It is an iterative method with which the phase information can be found from two holograms taken

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207

ing cases may be of particular interest for the reconstruction of the phase proflle. However, this issue went beyond the scope of this explorative experimental work and is subject of ongoing investigations.

4.4 Conclusions

Fig. 16. Holograms of a polyamide string of 30 μm diameter taken at difierent positions: (a) xod = 0.9 m, (c) xod = 2.83 m, (d) xod = 9.31 m and (g) xod = 11.73 m, the latter at an angle of 45 . The corresponding intensity proflles are shown on the right-side in panels (b), (d), (f), and (h), 200 rows were added up. The X-ray fllm was digitized with an optical microscope, (a) and (b) with a magniflcation by a factor of 10, (c) and (d) with a magniflcation by a factor of 4.

at difierent distances between object and detector. Therefore, holograms of a polyamide string with a diameter of 30 μm were taken at difierent object-to-detector distances. The holograms are shown in flg. 16. The selected distances cover difierent imaging regimes. For the contact regime the contrast would be (1 − exp[−(4πβD)/λ]) = 4.26% and the absorption contrast of the polyamide string with a diameter of D = 30 μm can be neglected. In the near fleld region, at an object-todetector distance xod = 0.9 m, flg. 16 (a) and (b), the interference pattern produced by both edges have only very little overlap. √ The reason is that the size of the flrst Fresnel zone λxod = 13.6 μm is smaller than the diameter of the string. However, at the largest distance xod = 11.73 m, flg. 16 (g) and (h), one obtains for the flrst Fresnel zone √ λxod = 49.2 μm and the interference pattern from both edges do overlap. The resemblance between the original object and the radiograph is more or less lost. Both limit-

Phase contrast radiography has been accomplished with an external 855 MeV electron beam using broad-band transition radiation X-rays with a mean photon energy hω ≈ 20 keV and a micro-focus with standard deviations of typically σh = 8.6 μm and σv = 7.5 μm in the horizontal and vertical direction, respectively. In-line holograms of polymer strings were taken with a low-emittance 600 MeV electron beam using narrow-band transition radiation Xrays with a photon energy of hω = 6 keV and a micro-focus with a standard deviation of typically σv = 1.2 μm. Highquality holograms were obtained with high-resolution Xray fllms and a direct-exposure cooled CCD camera chip. The advantage of the former is the very good spatial resolution, that of the latter its on-line capability. An X-ray beam spot with micro-dimensions can be prepared directly with the micro-focused external electron beam via transition radiation production in a foil stack. Objects to be investigated can be placed in close distance to the small X-ray beam spot. This has the advantage that a large geometrical magniflcation of up to a factor of 10 can easily be achieved in our relatively small experimental area. The disadvantage of the transition radiation X-ray source is its contamination with high-energy bremsstrahlung photons. Typical electron beam charges required to capture a single image are about 0.3 μC for phase contrast radiographs with broad-band polychromatic X-rays and an Xray fllm as detector, and some nC for a cooled CCD chip. In-line holograms with narrow-band X-rays require about 500 μC for a high-resolution X-ray fllm, and 5 10 μC for the CCD detector. We thank F. Hagenbuck, and H.-K. Kaiser for signiflcant contributions in the early stage of the experiment, and Mrs. C. Koch-Brandt, Institut f˜ ur Biochemie, Universit˜ at Mainz, for making her fllm scanner device available to us. This work has been supported by Deutsche Forschungsgemeinschaft DFG under contract BA 1336/1-4.

References 1. F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palmaz, M. Di Michiel, R. Longox, A. Olivox, S. Panix, D. Pontoni, P. Poropat, M Prestx, A Rashevskyx, G. Trombay, A. Vacchix, E. Vallazza, F. Zanconati, Phys. Med. Biol. 43, 2845 (1998). 2. C.J. Kotre, I.P. Birch, Phys. Med. Biol. 44, 2853 (1999). 3. L.D. Turner, B.B. Dhal, J.P. Hayes, A.P. Mancuso, K.A. Nugent, D. Paterson, R.E. Scholten, C.Q. Tran, A.G. Peele, Opt. Expr. 12, 2960 (2004).

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4. F. Pfeifier, T. Weitkamp, O. Bunk, Ch. David, Nature Physics advance online publication www.nature.com/ naturephysics, published online: 26 March 2006; doi:10.1038/nphys265. 5. S.W. Wilkins, T.E. Gureyev, D. Gao, A. Pogany, A.W. Stevenson, Nature (London) 384, 335 (1996). 6. Xizeng Wu, Hong Liu, Med. Phys. 30, 2169 (2003). 7. T. Takeda, A. Momose, E. Ueno, Y. Itai, J. Synchrotron Rad. 5, 1133 (1998). 8. R.A. Lewis, Phys. Med. Biol. 49, 3573 (2004). 9. D. Gabor, Nature 161, 777 (1948). 10. P. Spanne, C. Raven, I. Snigireva, A. Snigirev, Phys. Med. Biol. 44, 741 (1999). 11. P. Cloetens, R. Barrett, J. Baruchel, J. Guigay, M. Schlenker, J. Phys. D 29, 133 (1996). 12. Z.W. Hu, B. Lai, Y.S. Chu, Z. Cai, D.C. Mancini, B.R. Thomas, A.A. Chernov, Phys. Rev. Lett. 87, 148101 (2001). 13. R.W. James, The optical Principles of the Diffraction of X-rays (Cornell University Press, 1965). 14. V. Kohn, I. Snigireva, A. Snigirev, Opt. Commun. 198, 293 (2001). 15. Mahmoud El Ghazaly, X-ray Phase Contrast Imaging at the Mainz Microtron MAMI, Dissertation, Institut f˜ ur Kernphysik, Universit˜ at Mainz, 2005.

16. B.L. Henke, J.Y. Uejio, G.F. Stone, C.H. Dittmore, F.G. Fujiwara, J. Opt. Soc. Am. B. 11, 1540 (1986). 17. http://www.filmscanner.info/NikonSuperCoolscan4000ED.html. 18. Georg Joos, Erwin Schopper, Grundriss der Photographie und ihrer Anwendungen besonders in der Atomphysik (Akademische Verlagsgesellschaft M. B. H., Frankfurt am Main, 1958). 19. Y. Hwu, H.H. Hsieh, M.J. Lu, W.L. Tsai, H.M. Lin, W.C. Goh, B. Lai, J.H. Je, C.K. Kim, D.Y. Noh, H.S. Youn, G. Tromba, G. Margaritondo, J. Appl. Phys. 86, 4613 (1999). 20. O. Chubar, A. Snigirev, S. Kuznetsov, T. Weitkamp, V. Kohn, Proceedings DIPAC 2001, ESRF, Grenoble, France. 21. A. Caticha, Phys. Rev. A 40, 4322 (1989). 22. http://www.andor-tech.com/germany/products/oem. cfm 23. http://www.data.it/support/data sheets/e2vtech/ 47-10back.pdf 24. http://www.olympus.pl/pliki/mikroskopy/dokumenty/ LM cameras ENG.pdf. 25. C. Raven, Microimaging and Tomography with High Energy Coherent Synchrotron X-Rays (Shaker Verlag, 1998). 26. R.W. Gerchberg, W.O. Saxton, Optik 35, 237 (1972).

Eur. Phys. J. A 28, s01, 209 219 (2006) DOI: 10.1140/epja/i2006-09-020-7

EPJ A direct electronic only

Twenty years of physics at MAMI —What did it mean? B.A. Meckinga Thomas Jefierson National Accelerator Facility, 12000 Jefierson Avenue, Newport News, VA 23606, USA / Published online: 7 June 2006

c Societa Italiana di Fisica / Springer-Verlag 2006 

Abstract. The development over the last twenty years of the physics program and the experimental facilities at the Mainz Microtron MAMI will be reviewed. Ground-breaking contributions have been made to the development of experimental techniques and to our understanding of the structure of nucleons and nuclei. PACS. 29.17.+w Electrostatic, collective, and linear accelerators 25.20.-x Photonuclear reactions 25.30.Bf Elastic electron scattering 25.30.Dh Inelastic electron scattering to speciflc states

1 Introduction The goal of nuclear physics is to study the properties of nuclei, and to understand these properties on the basis of the fundamental theory of the constituents making up the nucleus. Quantum-chromodynamics (QCD) has emerged as the leading candidate for the theory of hadronic interactions. Presently, QCD cannot be solved in the strongcoupling regime due to the lack of appropriate perturbative solution. The best hope in the near future is to use large-scale numerical calculations to approximate the space-time continuum by a discrete lattice in the framework of Lattice QCD (LQCD). Due to limitations in compute power, the lattice spacing is still fairly coarse, and the masses of the quarks used are much larger than their actual values. These limitations require signiflcant extrapolations which need to be constrained theoretically. For zero-mass quarks, QCD can be solved in a rigorous way via Chiral Perturbation Theory (χPT). Again, an extrapolation is required, this time from zero-mass quarks up to the actual values. An interesting recent development is the use of functional forms derived from χPT to extrapolate LQCD results. Before these extrapolations can be trusted, it is very important to verify the predictive power of χPT via experimental tests, in particular via pion photoproduction close to threshold. On the experimental side, valuable information on the properties of bound quark systems is still lacking. In particular, the knowledge of the spatial distribution of the charges and the currents inside the nucleon is not satisfactory, especially for the neutron. Our knowledge of the excited states of the nucleon is still insu– cient, e.g. what are the degrees-of-freedom governing the mass spectrum, a

e-mail: [email protected]

and what are the difierences in quark wave functions between the ground state and the excited states. Many of the issues mentioned above can be investigated using electron scattering. Electrons interact only with the charged constituents of the object under investigation. The interaction is described by Quantum-Electrodynamics (QED), and is su– ciently small to be handled with perturbative methods. In the past, the usefulness of the electromagnetic probe was limited by the technical features of the available electron accelerators. In particular, coincidence experiments and the operation of large acceptance detectors were hampered by the low duty-cycle of the electron beams. Dramatic progress in accelerator and detector technology has made it possible to overcome these limitations and to study electromagnetic processes with an accuracy that is no longer limited by technical problems. The Institut f˜ ur Kernphysik (Institute for Nuclear Physics) at the University of Mainz in Germany has been at the forefront of this development. For more than 20 years, the Institute has developed novel electron accelerators (the MAMI series of microtrons) and the corresponding experimental equipment, and has used those devices for ground-breaking research into the electromagnetic structure of nucleons and nuclei. On the occasion of the retirement of six key people (Hartmuth Arenh˜ ovel, Hartmut Backe, Dieter Drechsel, J˜org Friedrich, Karl-Heinz Kaiser, and Thomas Walcher) from the Institute, a symposium was held in October 2005 to review 20 years of Physics at MAMI and to commemorate their contributions. This paper will attempt to review the major contributions MAMI and its user community have made to the fleld, to identify the particular circumstances that made these contributions possible, and to speculate on their lasting impact.

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Fig. 1. Layout of the MAMI accelerators and the experimental areas. MAMI A is located in the RTM2 area, MAMI B in RTM3, the new MAMI C in the area labeled HDSM.

Table 1. MAMI microtron development. Year

Activity

1975

Proposal for a Race-Track Microtron (design by H. Herminghaus et al.) 14 MeV beam from MAMI A1 Preliminary Sonderforschungsbereich (SFB) established 183 MeV beam from MAMI A2 MAMI A operation with a total of 18,700 h Development of the 855 MeV MAMI B SFB 201 established First 855 MeV beam from MAMI B (flrst experiment by A2 Collaboration) MAMI B operation with a total of 82,843 h Sonderforschungsbereich 443 established Approval of 1.5 GeV HDSM (Harmonic Double-Sided Microtron, design by K.-H. Kaiser et al.) Installation of the four HDSM magnets Commissioning and begin of physics

1979 1982 1983 1983 - 87 1983 - 90 1984 1990 1990 - 2005 1999 2000

2001 - 03 2006

2 MAMI microtron development The history of the MAMI microtron development is summarized in table 1. The pioneering development of MAMI A and B at Mainz has established the microtron as a cost-

efiective way to build an accelerator capable of delivering a high quality electron beam [1]. The layout of the accelerator and its experimental areas is shown in flg. 1. The microtron design relies on sending the beam repeatedly through the same room-temperature accelerating structure with moderate energy gain per turn. Recirculation is achieved by two homogeneous 180◦ end-magnets. The size of these end-magnets for the last microtron stage, MAMI B, is evident from flg. 2. The perpendicular entry and exit of the electron orbits at the end-magnets results in simple and robust beam optics. Due to the continuous-wave nature of the radio-frequency power and the constant magnetic fleld, the quality of the beam is very high: an energy spread of δE/E = 1.5 × 10−5 and an emittance of  = 8 × 10−9 m is achieved routinely. A laser-driven polarized gun produces electron beams with 80% polarization. Parity violation experiments are possible since helicity-correlated changes in the beam parameters are very small: energy variations of δE/E ≤ 10−8 and position variations of δx ≤ 100 nm have been achieved. Particularly impressive is the high operational stability: overnight and during weekends, the MAMI microtrons are routinely operated by students. It has been the conventional wisdom in the accelerator community that the maximum energy of a microtron is limited to about one GeV since the construction of the end-magnets which increase rapidly in size with increasing energy becomes technically and flnancially impractical.

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What did it mean?

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Table 2. Three-spectrometer system parameters. Spectrometer

A

B

C

conflguration pmax [MeV/c] δΩ [msr] Θmin δp/p [%]

QSDD 665 28 18 20

D 810 5.6 7 15

QSDD 490 28 18 25

Fig. 2. The MAMI accelerator team standing in front of one of the MAMI B 180 end-magnets. The common accelerating section is located between the magnets on the right-hand side, the separated return paths are on the left-hand side.

9.0 MV / turn max gain

LINAC I (4.90GHz) n 1500MeV Extractio

43 recirculations io ct je In n eV 5M 85

B max=1.539 T

10 m

LINAC II (2.45GHz) 9.3 MV / turn max gain

Fig. 3. Layout of the MAMI C microtron.

The design and construction of the 1.5 GeV Harmonic Double-Sided Microtron (HDSM, design by K.-H. Kaiser et al.) [2] is poised to shatter that boundary. Building on the experience with the previous microtrons, the challenging HDSM design relies on two parallel accelerating sections joined by four inhomogeneous 90◦ end-magnets with a weight of 250 metric tons each (see flg. 3 for a layout). The strong vertical defocusing at the entrance and exit of the magnets is compensated by a radial gradient fleld. Meeting the microtron coherence condition within the conflned space of the existing experimental area forces the fundamental accelerating frequency to be twice the frequency of the MAMI B microtron. Phase stability considerations require to leave one of the two accelerating sections at the present MAMI B frequency. The installation of the four HDSM magnets has been completed, and commissioning is expected to start soon. As shown in table 1, important milestones parallel to the technical developments were the establishment of the Sonderforschungsbereiche ( Special Research Ini-

Fig. 4. Photograph of the three high-resolution magnetic spectrometers in the A1 area. Spectrometer A (red) is on the lefthand side, B (blue) in the center, and C (green) is on the right-hand side.

tiatives , abbreviated SFB), a funding scheme used by the German funding agency Deutsche Forschungsgemeinschaft (DFG) to support new initiatives for a limited period of time.

3 Experimental equipment at MAMI The broad physics program at MAMI requires an equally broad range of experimental equipment, from high-resolution magnetic spectrometers to large acceptance detectors. Most of the electron scattering instrumentation has been provided by the Institute, a large fraction of the equipment for the tagged photon experiments has been contributed by the user community. For electron scattering experiments, the three-spectrometer system ofiers an unprecedented combination of momentum resolution, solid angles, and momentum range. The parameters are given in table 2, a photograph is shown in flg. 4. For experiments with real photons, the Glasgow-Mainz bremsstrahlung tagging system, located in the A2 area, provides photons of known energy and flux. Circularly polarized photons can be obtained from the bremsstrahlung of polarized electrons, linearly polarized photons from an oriented crystal radiator. The detection equipment is focused on charged and neutral particle detection in a large

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Fig. 5. The DAPHNE large acceptance detector. Fig. 7. The Bonn frozen-spin polarized target during a repolarization cycle. The polarizing magnet is on the left-hand side, the pumping unit on the right-hand side.

In combination with the DAPHNE detector it has been used for studying the GDH sum rule on hydrogen and deuterium.

4 Selected experiments

Fig. 6. The Crystal Ball detector (left) and TAPS (right).

solid angle and energy range. The Saclay-built DAPHNE detector (Detecteur a grande Acceptance pour la Physique photo-Nucleaire Experimentale) uses a combination of proportional wire chambers and layers of scintillation counters and absorbers for charged particle and photon detection. A three-dimensional drawing is shown in flg. 5. The TAPS detector (original abbreviation for TwoArm Photon Spectrometer) can be arranged in difierent conflgurations. Its 528 BaF2 crystals give good energy resolution for photon detection. Charged particles can be identifled via the ratio of fast and slow scintillation light. The newest addition to the experimental arsenal is the Crystal Ball detector which has seen prior service at highenergy facilities like SPEAR, DORIS, and the BNL AGS. Its central detector consists of 672 NaI crystals, again optimized for photon detection. The electronic readout system has been modernized and equipped with 80 MHz flashADC’s. In the flrst experiment, the Crystal Ball will be used in combination with TAPS as a forward detector (see flg. 6 for a picture of the setup) to measure the Δ+ magnetic moment via the angular distribution of the decay photons in the Δ+ → Δ+ γ transition (from the high-mass tail of the Δ+ to its low-mass tail). Of particular importance for studying the spin degreesof-freedom of the nucleon has been the addition of the Bonn frozen-spin polarized H and D target (see flg. 7 for a picture). Its low magnetic holding fleld and open geometry make it an ideal match for large acceptance detectors.

The following sections will give examples for experiments that are characteristic for the MAMI physics program and that have had a large impact on the fleld. 4.1 Real Compton Scattering and the Polarizability of the Proton The electric and magnetic polarizabilities of the nucleon are static quantities that characterize the response of the system to external electric and magnetic flelds. Since the highest flelds that one can produce in the laboratory are much too weak to have a measurable influence, the best approach is to derive the polarizabilities from the energy and angular dependence of real photon scattering (RCS) at low photon energies. The experiments are challenging since the cross sections are very small and, above pion production threshold, there is a large background of photons from π 0 decays. The experimental results obtained with the TAPS detector and the bremsstrahlung tagging system [3] require signiflcant theoretical corrections and interpretation (see flg. 8 for an example). The cross section is dominated by scattering ofi the charge and the magnetic moment of the proton. In addition, the incident and outgoing photons can couple to an exchanged pion. Finally, the polarizabilities enter linearly only in the low-energy expansion. In practice, higher-order terms need to be incorporated. A dispersion relation analysis [4] of the entire body of Compton scattering data shows that the proton is a very stifi objects, i.e. it does not deform much under the influence of external static flelds.

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Dispersion relation N contribution asymptotic + N asymptotic contribution

MAMI data

Fig. 8. Photon energy dependence of the Compton scattering cross section at a scattering angle of 135 . The dispersion relation analysis is the thin black line marked DR.

DR analysis JLab LEX analysis

Fig. 9. Q2 -dependence of the generalized polarizabilities αE (Q2 ) and βM (Q2 ).

4.2 Generalized nucleon polarizabilities and virtual Compton scattering The concept of the nucleon polarizabilities can be extended to virtual photons. This leads to six general polarizabilities which are functions of the momentum transfer, Q, and can be accessed in the ep → e p(γ) reaction. The difierential cross section is dominated by the Bethe-Heitler process and by Born terms which can be calculated from QED and the known proton form factors. These contributions have to be subtracted from the measured cross sections. At MAMI, the process was measured by using two high-resolution magnetic spectrometers to detect the scattered electron and the recoiling proton, and using the missing-mass technique to identify the (undetected) photon in the flnal state [5]. From the MAMI data (see flg. 9), the generalized polarizabilities αE (Q2 ) and βM (Q2 ) have been determined using a dispersion theoretical analysis [4]. The data are well described by taking the asymptotic and the πN -contribution into account; this demonstrates the importance of the pion cloud contribution to the nucleon polarizability. 4.3 Near-threshold π 0 production The physics goal of this program is to test the predictions of χPT which is an exact representation of QCD for the limiting case mπ → 0. To compare to the real world, an extrapolation to physical pion mass required. This introduces low-energy constants into χPT which need to be determined from experiment. Once the low-energy constants are known, the predictive power of χPT can be tested. The experimental information required are precise π 0 and π + photo- and electro-production difierential cross sections close to threshold where χPT is expected to be valid. In addition, polarization data are necessary to separate the contributing multipoles. The experimental challenges are considerable: the cross sections are small, and

Fig. 10. Beam asymmetry for π 0 photoproduction as a function of the π 0 production angle.

they have very strong photon energy dependence. Also, the reaction close to threshold typically involves low-range particles which are di– cult to detect. The experimental setup for the π 0 -photoproduction part of the program [3] used the tagged photon beam and the TAPS photon spectrometer. Using an oriented crystal as a radiator, linearly polarized photons were produced to get the desired polarization information from the beam asymmetry. The TAPS crystals were arranged into 7 independent arrays covering the entire π 0 angular range. As an example for the results, flg. 10 shows the beam asymmetry as a function of the π 0 c.m.s. angle. The experimental results show that χPT has considerable predictive power. This flnding, in combination with other tests of χPT, has increased dramatically the confldence in the validity of χPT. The most important application in other areas is the use of χPT for extrapolating LQCD calculations. The compute power required for LQCD calculations depends on the pion mass as mπ−7 . Therefore, the

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Fig. 12. GDH integral as a function of the upper limit νmax . Fig. 11. LQCD results for the proton magnetic moment. The data points give the results of the calculations for difierent values of mπ 2 . The physical values of μp and mπ 2 are marked by the red cross. The short blue arrow indicates the typical χPT extrapolation range; the dotted blue arrow indicates the typical LQCD extrapolation range. The best theoretical extrapolation is given by the green solid line.

calculations are performed far away from the physical pion mass. The generally accepted technique to extrapolate to the physical pion mass is to use the functional form for the mπ -dependence given by χPT. As an example, flg. 11 shows the result of a LQCD calculation [6] for the magnetic moment of the proton as a function of mπ 2 . With the chiral extrapolation, the LQCD result gets close to the true value.

Table 3. GDH Results from ELSA and MAMI.

Source of Information

k [MeV]

IGDH [μb]

MAMI PRL 87 (2001) 022003 ELSA PRL 93 (2004) 032003 Low-energy extrapolation MAID (2003) High-energy extrapolation PLB 450 (1999) 439

200

800

226 ± 5 ± 12

800

2900

27.5 ± 2.0 ± 1.2

140

200

−27.5 ± 3

2900

∞

−14

Experimental sum

212 ± 5 ± 12

Theoretical GDH integral

205

experimentally) and above 2900 MeV were estimated theoretically and added to the experimental sum. For the MAMI GDH experiment [7], circularly polar4.4 Experimental test of the GDH sum rule ized tagged photons from the bremsstrahlung of polarized electrons were hitting a longitudinally polarized frozenThe GDH sum rule for the nucleon was derived in 1966 by spin target. The DAPHNE detector in combination with Gerasimov and, independently, by Drell and Hearn. The a forward scintillation counter and shower detector was sum rule is based on fundamental assumptions: Lorentz used to measure the total cross section via identifying and and gauge invariance, unitarity, and no-subtraction dis- adding up the individual reaction channels. In addition to persion relations. The sum rule links the weighted integral testing the GDH sum rule, this experimental technique over σ3/2 (k)−σ1/2 (k) to the anomalous magnetic moment also allows the measurement of the energy and angular of the nucleon. σ3/2 (k) and σ1/2 (k) are the total hadronic dependence of the helicity-separated difierential cross secphotoproduction cross sections for total helicity 3/2 and tions for the individual reaction channels that contribute 1/2, respectively, as a function of the photon energy k. The to the total cross section. integral over k extends from pion production threshold to From the MAMI and ELSA data, the GDH integral inflnity, the weight factor is 1/k. has been calculated. It is shown as a function of the upper This fundamental sum rule had never been tested due integration limit in flg. 12. Table 3 gives the partial inteto the lack of appropriate experimental facilities. The test gral for difierent energy ranges; in the table, the statistical of the sum rule requires a circularly polarized photon error is given flrst, the second entry is the systematic error. beam and a longitudinally polarized target, both with high Also listed are the contributions from photon energies bepolarization. In addition, a detector to measure the total low and above the range that was covered experimentally. Note that, because of the 1/k-weighting, the contribution cross section reliably needs to be available. A major efiort has been launched in Europe to test from threshold to 200 MeV is about the same as the conthe validity of the GDH sum rule. For the proton, the low- tribution from 800 to 2900 MeV. Within the errors, the energy part of the sum (200 800 MeV) has been measured flnal experimental value agrees well with the theoretical at MAMI, the contribution between 800 and 2900 MeV prediction, thus verifying the GDH sum rule. First preliminary data from a polarized deuteron tarhas been investigated at the ELSA accelerator in Bonn. Contributions below 200 MeV (that are di– cult to access get are available. Extracting the GDH sum for the neutron

B.A. Mecking: Twenty years of physics at MAMI

from these experimental data is a conceptual challenge since the GDH sum for the deuteron is dominated by the γD → pn process close to the 2.2 MeV threshold. Only with major theoretical support will there be a chance to extract the GDH integral for the neutron from the deuteron data. It is instructive to look at the impact the GDH effort has had on the community beyond just validating the GDH sum rule. To address this important physics question, a coherent efiort of several European groups was required, an experience that will likely lead to more collaboration in the future. On the physics side, thinking in terms of the sum rule created an awareness that there are no isolated physics problems: all energy regions are interrelated, sometimes in ways that are not immediately obvious. An important consequence of the technical advances required for the GDH sum rule tests is that the measurement of helicity observables has now become routine. These observables contain valuable information, e.g. on the excitation of the nucleon resonances. Beyond the total cross section, detailed information on the energy and angular dependence for the difierent decay channels will be required to make the best use of the information.

4.5 Nucleon electromagnetic form factors In elastic electron scattering, the response of a composite system is given by its electromagnetic form factors. These describe the probability that the system will stay intact after absorbing a virtual photon characterized by the four-momentum transfer Q. Form factors provide the ideal meeting ground for experiment and theory. Electric (due to the charges) and magnetic (due to the magnetization) form factors can be separated by measuring the angular dependence of the scattering cross section at constant Q2 (Rosenbluth separation). An intuitive picture of the nucleon is provided by the spatial distributions of charge and magnetization which can be obtained from the form factors via a Fourier transform. Separating the electric, Ge , and magnetic, Gm , form factors using the Rosenbluth separation technique is difflcult when Ge /Gm  1 which is always the case for the neutron. The solution is to use polarization transfer to measure the Ge /Gm ratio. In practice, there are two possibilities: using polarized electrons on a polarized target, or using polarized electrons on an unpolarized target and analyzing the recoil nucleon polarization. From the measured proton and neutron form factors (and assuming charge symmetry), the u and d-quark contributions to the nucleon form factor can be determined. Information on the s-quark contribution can be obtained by adding the results of parity violating scattering ofi the proton (parity violation is due to interference between the exchange of a virtual photon and a Z 0 ). To derive neutron form factors from either deuteron or 3 He data requires careful consideration of nuclear effects. Observables and kinematical conditions need to be selected to minimize nuclear corrections. Systematic cal-

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Fig. 13. Neutron polarimeter part of the Ge /Gm setup.

culations for all observables, including polarization, have been carried out by Arenh˜ ovel and collaborators [8]. For example, the calculations show that the asymmetry AVeD in the quasi-free e-n scattering of polarized electrons ofi a polarized deuterium target (with the deuteron spin perpendicular to the direction of the momentum transfer Q) is linearly dependent on Gne ; there is very little background. The remaining sensitivity to the reaction dynamics (flnal-state interactions, isobar conflgurations, meson-exchange currents, and relativistic efiects) and to the N N -potential can be quantifled, and corrections can be applied to the measured Gne value. MAMI has made signiflcant contributions to all nucleon form factors [9]. For the neutron, the ratio Ge /Gm has been measured in quasi-free e-n kinematics with two difierent polarization techniques: 1) using polarized electrons ofi an unpolarized D target and analyzing the recoil neutron polarization, and 2) using polarized electrons on a polarized 3 He target. The neutron magnetic form factor, Gnm , has been determined via the quasi-free eD → e n(p) reaction, and the strange quark contribution has been measured via parity-violating electron scattering ofi the proton [10]. A particularly interesting experimental technique was developed for measuring the ratio Gne /Gnm in the quasi-free D(e, e n)p scattering of polarized electrons ofi an unpolarized D target and analyzing the longitudinal and transverse polarization of the recoiling neutron. The setup of the neutron polarimeter is shown in flg. 13. The polarimeter is only sensitive to the transverse polarization components. To access the longitudinal polarization component, a magnet (which also duplicates as a sweeping magnet for charged particles) is used to precess the longitudinal component into the transverse plane. The experimental trick is now to precess the dominant longitudinal component just enough to compensate the small transverse component. By varying the magnetic fleld and thus the neutron precession angle, a setting can be found where the transverse neutron polarization disappears. Ge /Gm can then be directly calculated from the precession angle. The great advantage of this null measurement which was used at MAMI for the flrst time is that it is completely insensitive to the precise

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2 Fig. 14. Neutron electric form factor, Gn e , as a function of Q 2 in units of (GeV/c) . The red curve is the best flt to the world nucleon form factor data [11].

Fig. 16. Ratio of electric quadrupole, E2, to magnetic dipole, M 1, strength in the N → Δ(1232) transition as a function of Q2 .

nucleon and a polarisation part. By fltting all proton and the neutron form factor data, a picture emerges where the neutron spends a considerable part of its time as a system of a proton (located close to the center) and a π − (located on average about 1.5 fm away from the center); see flg. 15 for the charge distributions of neutron and proton. Similarly, the proton can be viewed as spending part of its time as a neutron-π + combination.

4.6 Excitation of the Δ(1232)-resonance

Fig. 15. Neutron (top) and proton (bottom) radial charge distributions from a Fourier analysis of the world nucleon form factor data [11]. The green solid lines show the pion cloud contribution. Note that for the neutron r 2 ρ(r) is plotted which emphasizes the contributions at large radii.

values of the beam polarization and the analyzing power of the neutron polarimeter. The MAMI measurements ofi D and 3 He, in combination with measurements from other laboratories, have now led to a consistent set of experimental data for the neutron electric form factor (see flg. 14). No generally accepted theoretical interpretation exists. Friedrich and Walcher [11] arrived at an intuitive interpretation by describing the nucleon as the sum of a bare

The physics motivation for studying the electromagnetic excitation of the nucleon resonances is to understand QCD in the strong coupling regime. The mass spectrum and the quantum numbers of the nucleon excited states need to be understood in terms of the relevant degrees of freedom and the wave function and the interaction of the constituents. The electromagnetic amplitudes for the N → N ∗ transition are sensitive to the difierence between the quark wave functions of the N and N ∗ . The spatial resolution of the probe can be tuned by varying the momentum transfer. The electric and magnetic parts of the transition can be separated using partial-wave analysis (PWA) techniques. The determination of the electromagnetic transition form factors for the N → N ∗ transition requires a large, high-quality data set covering a broad kinematical range in momentum transfer, excitation energy, decay modes (π, ππ, η, ρ, ω), and decay angles. Polarization information is especially useful since it is sensitive to the interference between overlapping resonances, or to the interference between a resonance and the background.

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Fig. 17. Setup to use quasi-monochromatic transition radiation for K-edge imaging. The sample containing the Mo foil is located in front of the pn-CCD detector.

For higher N ∗ resonances, the extraction of the transition form factor becomes quite involved: in the theoretical description, Born terms, unitarity, and channel coupling need to be taken into account. A full PWA is presently not possible due to lack of data; especially polarization data is missing. To compensate for the lack of experimental data, the analysis is often constrained by assuming the energy dependence of the resonance excitation. A particularly interesting example is the N → Δ(1232) transition. A full partial wave analysis is possible since the Δ(1232) is an isolated resonance, and the Watson theorem constrains the phases of the helicity amplitudes. The spin 1/2 → 3/2 transition is dominated by spin-flip M 1; however, non-zero E2 and C2 multipoles are possible. This would be a signature for a non-spherical charge distribution in the Δ(1232) or could be caused by the virtual photon coupling to a pion cloud. At MAMI, a major experimental efiort [3,12] was launched to determine the E2/M 1 ratio for the N → Δ(1232) transition starting at the real photon point and extending the measurements to virtual photons. For real photons, the experiments used the tagging system to produce linearly polarized photons and TAPS to detect the π 0 → γγ decay. For electroproduction, the three-spectrometer setup was used to measure the ep → e p(π 0 ) process. Tilting the proton spectrometer provided access to out-of-plane observables. The Q2 -dependence of the E2/M 1 ratio is shown in flg. 16. The results show that the E2/M 1 ratio is small (around 2%) and negative. This flnding is at variance with all models that consider constituent quarks, only. Models that explicitly include the pion cloud can explain the data.

applications, e.g. in the material sciences, in biology, and in medicine. The most powerful X-ray sources in the energy regime of interest (K-edge of oxygen at 0.53 keV to the K-edge of iodine at 33.16 keV) are dedicated synchrotron radiation facilities. Modern electron accelerators with their low emittance electron beams may ofier an attractive alternative. The interaction of these beams with matter have the potential of producing high brilliance X-rays with a tunable time structure. Processes of interest are transition radiation, parametric X-rays, undulator radiation, the Smith-Purcell efiect, and channeling radiation. An illustrative example is the use of the 855 MeV MAMI beam hitting a stack of 30 polyimide foils to produce hard X-rays via transition radiation [14]. Using a highly oriented pyrolytic graphite crystal, an X-ray beam of about 20 keV was prepared; its two-dimensional spatial distribution was measured in a pn-CCD detector (see flg. 17 for details). Due to the crystal monochromator, the X-ray beam had a correlation between position and energy. By synchronously changing the electron beam direction and the crystal position, the X-ray energy spectrum could be swept as a function of time. This technique avoids making the X-ray beam monochromatic with a slit system, and thus does not reduce the flux. Using the Xray beam for K-edge imaging as a demonstration project, a 2.5 μm Mo foil hidden in a 100 times thicker copper foil could be detected. Once perfected, this technique may be used to image the human lung using xenon (mixed with oxygen) as an absorber.

5 Experiment —Theory interplay at MAMI 4.7 Production of low-energy radiation In the context of the applied physics program [13], techniques were developed that make use of the high-quality MAMI electron beam for the production of high brilliance X-rays. In addition to clarifying fundamental aspects of radiation production, there are potential practical

As already pointed out in the discussion of the experimental program, there is a very close and efiective collaboration between the theorists working in the Institute and the experimentalists using MAMI. This tight coupling increases the impact of the MAMI experimental program in two important ways.

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Fig. 18. Photos of the retirees. Top row from left to right: Hartmuth Arenh˜ ovel, Thomas Walcher, and Karl-Heinz Kaiser; bottom row: Dieter Drechsel, J˜ org Friedrich, and Hartmut Backe.

First, it provides the necessary corrections and facilitates the physics interpretation for quantities for which the measurement strategy is clear and unambiguous. Typical examples are the calculations of the corrections to the measured Gne values, or the corrections to the GDH integral to account for the unmeasured angular and energy range. Second, theoretical support starting in the early phase of the experiment makes it possible to attack the determination of quantities for which due to the existence of strong competing channels the measurement and analysis strategy is not obvious. Typical examples include the extraction of the nucleon polarizability from γp → γp, the pion polarizability from γp → π + nγ, and the magnetic moment of the Δ+ (1232) from γp → pπ 0 γ. The tight coupling between theorists and experimentalists at MAMI is unusual and unique. It clearly has been a major contributor to the success of MAMI. The ingredients necessary for the tight coupling are not easy to identify, and cannot easily be transfered to other facilities. Very likely, the early history of the MAMI project and the personal inclination of the people involved on both sides have played an important role.

6 MAMI funding and operation Securing steady funding for operating a major electromagnetic nuclear physics facility within a university environment has been a real challenge. Contributions from the University of Mainz, the state of Rhineland-Palatinate, and from the Federal Government are required to keep the operation of MAMI flnancially viable. A large fraction of the federal funding is provided by the DFG in the framework of the SFB which is meant to support new ventures for a limited period of time, but not meant to support steady-state operation. The success of MAMI in this di– cult funding environment is testimony to the skills of the people in charge of the Institute. The MAMI physics program has been comprehensive and more characteristic for a national facility than for a university-based accelerator. No physics problem that was worth attacking has been left out, even when it required additions to the experimental equipment and improvements to the accelerator (e.g.: the parity violation program). International collaborators have played an important role at MAMI, contributing both ideas and experimental equipment. They obviously felt welcome at the Institute; the natural hospitality and curiosity of the population of

B.A. Mecking: Twenty years of physics at MAMI

Mainz and the surrounding areas have likely had a beneflcial efiect, too. MAMI has been an ideal training ground for students who could experience all stages of an experiment, from planning to analyzing and publishing the results. The students were also trained in developing and using sophisticated experimental equipment, a good preparation for those who went on to pursue careers outside of nuclear physics. In addition to educating students at MAMI, a generation of young researchers has been trained who have now gone out and successfully competed for faculty positions.

7 The legacy of 20 year of MAMI physics What this 20-year period at MAMI will be remembered for will depend on the range of interests of the person asking the question. The development of the single-sided microtrons, MAMI A and B, has already changed the textbooks on accelerators; the design and construction of the doublesided MAMI C microtron will complete the microtron development. The quality of the experimental data will not be surpassed for a long time. Note that very often the accuracy of the flnal answer is limited by the accuracy of the theoretical corrections. Better accuracy can only be achieved by an improvement of both the experimental results and the theoretical interpretation. Twenty years of MAMI physics have demonstrated that a tight coupling between theory and experiment can be mutually beneflcial. This coupling, which may not be easy to reproduce at other facilities, has been a major contributor to the success of MAMI. Finally, 20 years of MAMI physics have demonstrated that it is possible although with a lot of efiort to operate a major facility within the framework of a German university.

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8 Summary The development of the physics program and the experimental facilities at the Mainz Microtron MAMI over the last twenty years has been reviewed. MAMI and its user community have been working at the forefront of electromagnetic nuclear physics. Novel electron accelerators and experimental equipment were developed and have been used for ground-breaking research into the structure of nucleons and nuclei. The six people, Hartmuth Arenh˜ ovel, Hartmut Backe, Dieter Drechsel, J˜org Friedrich, Karl-Heinz Kaiser, and Thomas Walcher (flg. 18) who have retired (or are about to retire) have played key roles in this development. They have every right to be proud of what has been accomplished. I would like to take this opportunity to wish them Happy Retirement .

References 1. A. Jankowiak, these proceedings. 2. A. Jankowiak et al., prepared for the 8th European Particle Accelerator Conference (EPAC 2002), Paris, France, 3-7 June 2002. 3. R. Beck, these proceedings. 4. M. Vanderhaeghen, these proceedings. 5. N. d’Hose, these proceedings. 6. R. Young (Jefierson Lab), private communication. 7. A. Thomas, these proceedings. 8. H. Arenh˜ ovel et al., Z. Phys. A 331, 123 (1988). 9. M. Ostrick, these proceedings. 10. F. Maas, these proceedings. 11. J. Friedrich, T. Walcher, Eur. Phys. J. A 17, 607 (2003), arXiv:hep-ph/0303054. 12. H. Schmieden, these proceedings. 13. W. Lauth, these proceedings. 14. F. Hagenbuck et al., IEEE Trans. Nucl. Sci. 48, 843 (2001).

Author index Backe H. → El-Ghazaly M. Backe H. → Lauth W. Beck R.: Experiments with photons at MAMI 173 Boeglin W.U.: Few-nucleon systems at MAMI and beyond 19 Cardman L.S.: Physics at the Thomas Jefferson National Accelerator Facility 7

Maas F.E.: Parity-violating electron scattering at the MAMI facility in Mainz 107 Mecking B.A.: Twenty years of physics at MAMI —What did it mean? 209 Merkel H.: Experimental tests of Chiral Perturbation Theory 129 Milner R.G.: The beauty of the electromagnetic probe 1

d’Hose N.: Virtual Compton Scattering at MAMI 117

Ostrick M.: Electromagnetic form factors of the nucleon 81

El-Ghazaly M., Backe H., Lauth W., Kube G., Kunz P., Sharafutdinov A. and Weber T.: X-ray phase contrast imaging at MAMI 197

Rohe D. (A1 and A3 Collaboration): Experiments with polarized 3 He at MAMI 29

Hammer H.-W.: Nucleon form factors in dispersion theory 49 Hillert W.: The Bonn Electron Stretcher Accelerator ELSA: Past and future 139 Jankowiak A.: The Mainz Microtron MAMI —Past and future 149 Kettig O. → Lauth W. Kowalski S.: Parity violation in electron scattering Kube G. → El-Ghazaly M. Kunz P. → El-Ghazaly M. Kunz P. → Lauth W.

Scherer S.: Chiral perturbation theory 59 Schmieden H.: Photo- and electro-excitation of the Δ-resonance at MAMI 91 Schwamb M.: Few-nucleon systems (theory) 39 Sharafutdinov A. → El-Ghazaly M. Sharafutdinov A. → Lauth W. Thomas A.: The Gerasimov-Drell-Hearn sum rule at MAMI 161

101

Lauth W., Backe H., Kettig O., Kunz P., Sharafutdinov A. and Weber T.: Coherent X-rays at MAMI 185 Lauth W. → El-Ghazaly M.

Vanderhaeghen M.: Two-photon physics 71 Weber T. → El-Ghazaly M. Weber T. → Lauth W.