Selected Papers of K C Chou (20th Century Physics) (World Scientific Series 20th Century Physics)

SE~ECTED PHPERS OF H. C. CHOU

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Vol. 22 A Quest for Symmetry -

Selected Works of Bunji Sakita

edited by K. Kikkawa, M. Virasoro and S. R. Wadia

Vol. 23 Selected Papers of Kun Huang (with Commentary) edited by B. -F. Zhu

Vol. 24 Subnuclear Physics - The First 50 Years: Highlights from Erice to ELN by A. Zichichi edited by O. Barnabei, P. Pupillo and F. Roversi Monaco Vol. 25 The Creation of Quantum Chromodynamics and the Effective Energy by V. N. Gribov, G. 't Hooft, G. Veneziano and V. F. Weisskopf edited by L. N. Upatov

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In Celebration of His 70th Birthday

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Vol. 36 Selected Papers (1945-1980) with Commentary by Chen Ning Yang

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World Scientific Series in 20th Century Physics

Vol. 42

SE~ECTEO PRPERS OF

H. C. CHOU editor

Yue-Liang Wu Chinese Academy ofSciences, China

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v

Foreword

Kuang-Chao Chou (Guang-Zhao Zhou) is a world-renowned theoretical physicist. Before I met him in the early nineteen seventies, I have already known his many significant research contributions for more than a decade from the late nineteen fifties. His published papers have won uniformly high praises by the international scientific community and his articles are always written with depth and elegance. Kuang-Chao was the originator of the helicity amplitude analysis. That concept was introduced by him in 1957, followed by a full mathematical analysis of this very effective tool to his work in high energy physics research. In 1959, he perfected an important theorem in T and P violations, which states that under CPT invariance, while T reflection in not conserved, the decay branching ratios of particle and antiparticle to different states can be different even though their total decay widths are the same. In addition to his work on resonances, Kuang-Chao also pioneered the use of nuclear absorption to detect the weak magnetism in weak interactions, as well as a large body of theoretical analysis on the important photo-nuclear reactions through the use of dispersion theory. In 1960, Kuang-Chao analytically demonstrated the partial conservation of axial current (PCAC), which has been regarded as one of the cornerstones in particle physics. For this fundamental contribution, he was recognized internationally as one of the founding fathers of PCAC. In the nineteen eighties after the Cultural Revolution, under the organization and guidance by Kuang-Chao, the younger generation of Chinese scientists were able to make many significant research works on grand unification theory, CP violation, non-linear sigma model effective Lagrangian theory, spontaneous symmetry breaking in super-symmetry, topological aspects of quantum field theory and its relation to anomaly. Many of these results (for example in topological aspects of quantum field theory and its relation to anomaly) have already attracted recognition from the international as well as Chinese physics communities. In statistical physics and condensed matter physics, the research group directed by Kuang-Chao systemized the Green's function formalism in non-equilibrium statistical mechanics. Their method was applied to the studies of laser, plasma, critical dynamics, random quenching, etc.

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In addition to the remarkable achievement made by Kuang-Chao in many fields of theoretical physics, he has been one of the most important scientists in the development of science and technology in China and to the promotion of the international science exchange and cooperation. Kuang-Chao Chou served as the director of Institute of Theoretical Physics at the Chinese Academy of Sciences, the Dean of the Science School of Tsinghua University, the President of Chinese Academy of Sciences, the Chairman of the China Association for Science and Technology, the Executive Vice-President of China Commission for Promoting International Science and Technology and the vice Chairman of China Association for Peace and Disarmament. He is an Academician of the Chinese Academy of Sciences and has also been elected as Foreign Associate of the US National Academy of Sciences, Fellow of the Third World Academy of Science, and Foreign Member of USSR Academy of Sciences, and the European Academy of Arts, Sciences and Humanities.

T

1>.

vii

Introduction

Zhou Guang-Zhao's scientific career is divided into several periods:

I. Early period (around 1955-1956). He was working in China. II. DUBNA period (1957-1960). Zhou was prolific in this period, publishing many papers in the JETP. I was in the USA at the time and had studied several of his papers, especially his work on PCAC. In the USA he was famous as the most brilliant young theorist in Dubna. III. National defence period (1961-1979). Zhou returned to China and ceased publication in scientific journals. But I had seen in a museum display in Qin-Hai Province the cover page of the design report of the first Chinese atomic bomb. It bears the names of Zhou and Deng Jia-Xian. Rumors say Zhou had made crucial contributions to that design, using analytic methods without the use of computers. IV. Later period (1979-around 1987). Zhou was at CERN, later at the VPI in the USA for several years, then back to China. In this period Zhou resumed his interest in academic research and was active in several frontier area of theoretical physics, often with young collaborators. V. Recent period (1987). Zhou is busy with Academy of Science matters after the late 1980s. But he still keeps up his interest in physics research.

I had first met Zhou at a dinner in Beijing hosted by Premier Zhou En-Lai in the early 1970s, almost forty years ago. During these forty years we have become close friends. I am deeply impressed by how he had transformed himself in the 1980s from a research physicist into an influential and highly respected policy maker and administrator. His success is partly rooted in his genes, of course, but I believe, also very much in his principled Chinese cultural background. Zhou is a first rate physicist: broad, powerful and very quick in grasping new ideas. His style of doing physics reminds me of that of Landau, Salam, and of Teller. But in personal relationship Zhou is a perfect Confucian gentleman, without

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the aggressive edges that characterize so many famous US, European and Russian theoretical physicists.

C. N. Yang

ix

Introduction

Professor Kuang-Chao Chou (Guang-Zhao Zhou) was born in Changsha, Hunan Province in 1929. Upon graduated from the Department of Physics at Tsinghua University in 1951, he enrolled as a postgraduate student in theoretical physics at Peking University where he became as a faculty member in 1954. He then immediately carried out several research works on nuclear physics and particle physics published in Chinese journals [lc,2c,3c]. In the period 1957-1960, Prof. Kuang-chao Chou worked as a researcher at the Joint Institute for Nuclear Research (JINR), Dubna in the Soviet Union, where he published more than 30 papers and made significant contributions in several important subjects concerning symmetries. His first paper at JINR was the issue on a symmetry property of the new Gell-Mann theory [1]. He then paid special attention to the spins and parities of particles [2,3,6], which enabled him to establish the relativistic theory of reactions for the polarized particles [7,8] and especially for the massless polarized particles [13], where he introduced, for the first time, the concept of helicity amplitude and its corresponding mathematical description, which has been shown to be very powerful in analyzing high energy scattering amplitude. He studied at the first time the particle-antiparticle asymmetry (CP violation) in the hyperon decays [10,14,23]' where he proved an important theorem in CP violation, i.e., when CPT is invariant but T reflection is not conserved, the decay branching ratios of particle and anti-particle to different states may be different although total decay widths are the same. He also noticed in the early time the possible symmetry properties for the 7l'-K system [22]. Its group symmetry was realized late on as the SU(3) symmetry by Cell-Mann in 1961. The renowned important paper written by Kuang-Chao Chou was on the pseudovector current and lepton decays of baryons and mesons [26], which has been an initial paper for proving the theorem of partial conservation of axial current (PCAC) in a simple and concise way, and it has been reputed to be an outstanding contribution in the research of hadron physics. He also conducted some interesting works concerning charge symmetry properties [11,18], and a series research works on dispersion relation, photo-nuclear reaction and scattering processes of mesons and hyperons [16,20,23-25,27,28,30-34,9c,10c,l1c]. He

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also calculated at the first time the mass difference between neutron and proton [4c]. With collaborators, he also studied the capture of muon in nucleus [5c,6c,7c,8c]. It was a very productive period for Prof. Kuang-chao Chao, his works immediately attracted special attentions worldwide, for instance, his most significant and fundamental work on PCAC [26] motivated the renowned paper on "Dynamical model of elementary particles based on analogy with superconductivity" by Y. Nambu and G. Jona-Lasinio (Phys. Rev. Vol. 122, No.1 and Vol. 124, No.1, 1961), for which Y. Nambu won the Nobel prize in 2008. Due to those remarkable achievements, Prof. Kuang-chao Chou was highly praised by the international scientific community and became a world-renowned theoretical physicist. The period 1960's was a golden time for particle physics. It was also the most creative period for Prof. Kuang-chao Chou carrying out the important initial works in the frontiers of particle physics. Nevertheless, it was in this important time, Prof. Kuang-chao Chou made an alternative and unusual choice that he decided to give up his research interest and the advanced working and living conditions at JINR when he learnt that the project of atomic Bomb in China met some hard problems in its first stage of design and all the relevant experts from Soviet Union had to leave China. In 1961, Prof. Kuang-chao Chou returned back to China and took part in the theoretical study and design of the atomic Bomb, which was known to be the most mysterious project. He then analyzed carefully all formalisms used in the computations and checked in detail all the numerical calculations. Eventually, he solved the hard problems caused in the design of atomic Bomb and made a decisive and right judgment based on the first physical principle. It was this breakthrough, made by Prof. Kuang-chao Chou, which played a crucial rule for the success of the first atomic Bomb (1964) in China. During the period 1961-1978, Prof. Kuang-chao Chou made important contributions in the development of atomic energy, applied physics and computational mathematics, which included the high temperature and high density physics, explosion mechanics, mechanics of radiative fluid, neutron physics, plasma physics and computational mechanics. In 1978, Institute of Theoretical Physics (ITP) was approved by the vice Premier Deng Xiao-ping to be established at the Chinese Academy of Sciences (CAS), Prof. Kuang-chao Chou joined the Institute of Theoretical Physics. In the period 19791987, he was able to work again on the frontiers of theoretical physics. In the early 1980s, he worked as a guest researcher and professor in the Virginia Polytechnic Institute, USA and the European Organization for Nuclear Research. Prof. Kuang-chao Chou made a number of important contributions in multitude research areas of theoretical physics. In the field of high energy particle physics, under his leadership, some significant achievements have been yielded at several interesting subjects, which included the gauge field theories [3541,43,45,49,52,12c,13c,14c,15c], nonlinear sigma models in symmetric coset and

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curved spaces [46,47,50,51,56,64,68]' U(l) anomaly and chiral dynamical theories [42,44,48,53,57,73]. The most important observations by Prof. Kuang-chao Chou and his collaborators were the gauge invariance and anomaly-free condition [58,16c,17c] of the Wess-Zumino-Witten effective action and the topological origin of gauge anomalies [60]. In the paper [59], it was firstly pointed out that Wittens effective Lagrangian for chiral field missed some important terms. Thus the topological properties of gauge fields were extensively explored in several papers [62,63,67,18c]. Late on, a simplified derivation of Chern-Simons cochain was realized [71] and it has been adopted to obtain the general Chern-Simons characteristic classes [72,19c,20c]. In particular, several interesting physical applications have been investigated, which included the derivation of the anomalous term in Virasoro algebra [69], the effective action of sigma model anomalous with external gauge fields [70], the possible origin of B-vacuum etc. At the same time, Prof. Kuang-chao Chou also paid special attention to the phenomenology of weak interactions. In particular, an interesting observation was made on the correlation between the top quark mass and direct CP violation in kaon decays [65]. As a consequence, a heavy top quark with mass much larger than W-boson mass was predicted and emphasized, and a fourth generation quark was also motivated [66] at that time. In addition to the gauge field theories and particle physics, Prof. Kuang-chao Chou (Guang-Zhao Zhou) has also made important contributions in the fields of statistical physics and condensed matter physics. With collaborators, the closed time path Green's functions has systematically been developed [79,80,21c,22c] and applied to the critical dynamics [81,23c,24c] and statistical physics [82,83,84,85,27c,28c]. A nice review article was published in Phys. Rep. 118, 1 (1985) which has widely been cited internationally. It has been shown that the method of closed time path Green's functions allows one to treat, in a unified way, both equilibrium and nonequilibrium systems. The method has also been applied to the random system [86,87,90] and the disorder electron system [88] as well as the Bohr-Sommerfield quantization for the fractional quantum Hall effect system [93]. The other interesting works conducted the spontaneous symmetry breaking [92] and time reversal invariance [95,25c,26c] in the nonequilibrium system as well as the influence functional [94]. In 1980s, Prof. Kuang-chao Chou has led, as the director of ITP-CAS, several key research directions at ITP, which concern: quantum field theory and particle physics, nuclear physics and plasma physics, statistical physics and condensed matter physics, gravity and astrophysics. For his outstanding contributions to theoretical physics, applied physics, atomic energy and basic research sciences in various aspects, and also for his great capability of leadership, in 1987, Prof. Kuang-chao Chou was pointed by the government to serve as the president of Chinese Academy of Sciences (CAS). Once again, he has to give up his own research interest and to

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make alternative contributions for leading the development of sciences and technology at the CAS. In 1996, Prof. Kuang-chao Chou was considering to return back to the scientific research. He presented in 1995 the opening address for the 17th International Lepton-Photon Conference which was regarded as the most influenced International Conference on particle physics held in Beijing. After that conference, he was actually thinking about the most hot topics in particle physics, which involved CP violation, origin of mass, neutrino physics, supersymmetric theories, low energy phenomena of grand unification theories [74-77] and a general discussion on CP and CPT violating observable in meson decays[78]. It has been shown that the 18 unknown quantities in the standard model of particle physics can well be predicted by a minimal set of 5 parameters in a class of supersymmetric grand unification model [74], about ten relations among CP violating-phase, masses and mixing angles of quarks and leptons have been obtained, which remain consistent with the experimental data. In 1997, Prof. Kuang-chao Chou would wish to realize his desire of doing theoretical research. Nevertheless, it was unexpected that he was elected to be the vice Chairman of Standing Committee of the National People's Congress of China and pointed again by the government to serve as the Chairman of the China Association for Sciences and Technology. Until 2006, he was honored as the honorary Chairman of the Chinese Association of Sciences and Technology. Since 2006, he has been invited to serve as the member of International Advisory Committee and the honorary Chairman of the Scientific Council for the Institute of Theoretical Physics at the Chinese Academy of Sciences. In this volume, the most papers written by Prof. Kuang-chao Chou (GuangZhao Zhou) are printed. All the collected papers have been arranged into four parts according to the subjects of research areas and the languages of publishing journals. Part I (in English) and Part III (in Chinese) are the papers on Field Theories, Particle Physics and Nuclear Physics, Part II (in English) and Part IV (in Chinese) are the papers on Statistical Physics and Condensed Matter Physics. From the published papers, it is seen how Prof. Kuang-chao Chou caught up the frontiers of theoretical physics in various periods and carried out the creative research works with initial ideas and motivations, as well as how he has worked in different key research directions of theoretical physics and made significant contributions to various interesting research subjects and interdisciplinary areas. Due to the outstanding research works and significant contributions, Prof. Kuang-chao Chou has earned numerous national and international awards which include: National Prize first order in Natural Science, Prize first order from Qiushi Foundation of HongKong, Gian Carlo Wick Commemorative gold medal from the World Federation of Scientists, the Chinese Meritorious Service Medals for Nuclear Scientists and Satellite Pioneers. He was elected in 1980 to be the Academician of

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Chinese Academy of Sciences. He has also been elected as Foreign Associate of the US National Academy of Sciences, Fellow of the Third World Academy of Science, and Foreign Member of USSR Academy of Sciences, Czechoslovak Academy of Sciences, Bulgarian Academy of Sciences, Romania Academy of Sciences, Mongolian Academy of Sciences, the European Academy of Arts, Sciences and Humanities, Membre fondateur Academie Francophone d'Ingenieurs. He has been honored to be the honorary PhD of several universities which include the City College of New York in USA, McGill University in Canada, Chinese University of Hong Kong. He also served as the Executive Vice-President of China Commission for Promoting International Science and Technology, the vice Chairman and honorary Chairman of China Association for Peace and Disarmament. He was conferred Commendatore dellOrdine Ai Merito Della Republica Italiana in 1993. The editor would like to thank Dr. Shan-Gui Zhou and Dr. Ling-Mei Cheng for their invaluable assistance in preparing this collection. The assistance of Ms. Lian-Zi Wei and Ms. Ji Zhang was also very helpful for completing and publishing this volume. I would like to thank Prof. K.K. Phua for providing many useful suggestions in publishing this volume. Yue-Liang Wu

K avli Institute for Theoretical Physics China Key Laboratory of Frontiers in Theoretical Physics Institute of Theoretical Physics, Chinese Academy of Sciences

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Contents

Foreword (T. D. Lee)

v

Introduction (C. N. Yang)

vii

Introduction (Y. L. Wu)

ix

Part I l.

2.

3.

4.

5.

6.

7.

8.

9.

Field theory, particle physics and nuclear physics

Concerning a symmetry property of the new Gell-Mann theory, Zh. Eksp. Teor. Fiz. 33 (1957) 1058. Soviet Physics JETP 6 (1958) 815. Spins and parities of the hyperfragment H\ and of the K meson, (with M. I. Shirokov), Zh. Eksp. Teor. Fiz. 33 (1957) 1072. Soviet Physics JETP 6 (1958) 828. Spins and parities of the AH4 hyperfragment and K-meson, (with M. I. Shirokov), Nuc!. Phys. 6 (1958) 10. On non-uniqueness of nucleon-nucleon scattering phase shifts, (with L. G. Zastavenko and R. M. Ryndin), Nuc!. Phys. 6 (1958) 669. Phase indeterminacies in nucleon-nucleon scattering, (with L. G. Zastavenko and R. M. Ryndin), Zh. Eksp. Teor. Fiz. 34 (1958) 526. Soviet Physics JETP 7 (1958) 363. On the determination of the relative parities of elementary particles, Zh. Eksp. Teor. Fiz. 34 (1958) 1027. Soviet Physics JETP 34 (1958) 710. The relativistic theory of reactions involving polarized particles, (with M. I. Shirokov), Zh. Eksp. Teor. Fiz. 34 (1958) 1230. Soviet Physics JETP 34 (1958) 85l. Selection rules in reactions involving polarized particles, Zh. Eksp. Teor. Fiz. 35 (1958) 783. Soviet Physics JETP 35 (1959) 543. The Shapiro integral transformation, (with L. G. Zastavenko), Zh. Eksp. Teor. Fiz. 35 (1958) 1417. Soviet Physics JETP 35 (1959) 990.

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10. A note on the decay of the ~:-hyperon and its antiparticle, Nuc!. Phys. 9 (1958/59) 652. 11. The universal Fenni interaction and the capture of muons in hydrogen, (with V. Maevskii), Zh. Eksp. Tear. Fiz. 35 (1958) 1581. Soviet Physics JETP 35 (1959) 1106.

12. Charge symmetry properties and representations of the extended Lorentz group in the theory of elementary particles, (with V. I. Ogievetskii), Zh. Eksp. Tear. Fiz. 36 (1959) 264. Soviet Physics JETP 36 (1959) 179. 13. Reactions involving polarized particles of zero rest mass, Zh. Eksp. Tear. Fiz. 36 (1959) 909. Soviet Physics JETP 36 (1959) 642. 14. Some symmetry properties in processes of antihyperon production with annihilation of antinucleons, Zh. Eksp. Tear. Fiz. 36 (1959) 938. Soviet Physics JETP 36 (1959) 663. 15. On the problem of investigating the interaction between 1t mesons and hyperons (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 37 (1959) 283. Soviet Physics JETP 37 (1960) 199. 16. Electromagnetic mass of the K meson, (with V. I. Ogievetskii), Zh. Eksp. Tear. Fiz. 37 (1959) 866. Soviet Physics JETP 37 (1960) 616. 17. Dispersion relations for the scattering of y quanta by nucleons, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 37 (1959) 1714. Soviet Physics JETP 37 (1960) 1213. 18. Charge symmetry properties and representations of the extended Lorentz group in the theory of elementary particles, (with V. I. Ogievetsky), Nuc!. Phys. 10 (1959) 235. 19. Integral transformations of the I. S. Shapiro type for particles of zero mass, (with L. G. Zastavenko), Zh. Eksp. Tear. Fiz. 38 (1960) 134. Soviet Physics JETP 11 (1960) 97. 20. Scattering of gamma-ray quanta by nucleons near the threshold for meson production, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 38 (1960) 201. Soviet Physics JETP 11 (1960) 147.

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21. On the production of an electron-positron pair by a neutrino in the field ofa nucleus, (with A. M. Badalyan), Zh. Eksp. Tear. Fiz. 38 (1960) 664. Soviet Physics JETP 11 (1960) 477. 22. Possible symmetry properties for the n-K system, Zh. Eksp. Tear. Fiz. 38 (1960) 1015. Soviet Physics JETP 11 (1960) 730. 23. On the decay of ~ hyperons, Zh. Eksp. Tear. Fiz. 38 (1960) 1342. Soviet Physics JETP 11 (1960) 966. 24. Dispersion relations and analysis of the energy dependence of cross sections near thresholds of new reactions, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 39 (1960) 112. Soviet Physics JETP 12 (1961) 82. 25. Inelastic final-state interactions and near-threshold singularities, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 39 (1960) 364. Soviet Physics JETP 12 (1961) 258. 26. On the pseudovector current and lepton decays of baryons and mesons, Zh. Eksp. Tear. Fiz. 39 (1960) 703. Soviet Physics JETP 12 (1961) 492. 27. Elastic scattering of gamma quanta by nuclei, (with L. I. Lapidus) Zh. Eksp. Tear. Fiz. 39 (1960) 1056. Soviet Physics JETP 12 (1961) 735. 28. The elastic scattering of y rays by deuterons below the pion-production threshold, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 39 (1960) 1286. Soviet Physics JETP 12 (1961) 898. 29. On the pion-pion resonance in the p-state, (with Ho Tso-Hsiu), Zh. Eksp. Tear. Fiz. 39 (1960) 1485. Soviet Physics JETP 12 (1961) 1032. 30. On the role of the single-meson pole diagram in scattering of gamma quanta by protons, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 41 (1961) 294. Soviet Physics JETP 14 (1962) 210. 31. Low-energy limit of the yN-scattering amplitude and crossing symmetry, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 41 (1961) 491. Soviet Physics JETP 14 (1962) 352.

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32. On the K +N~A(~)+y process, (with L. I. Lapidus), Zh. Eksp. Teor. Fiz. 41 (1961) 1310. Soviet Physics JETP 14 (1962) 932. 33. Scattering of photons by nucleons, (with L. I. Lapidus), Zh. Eksp. Teor. Fiz. 41 (1961) 1546. Soviet Physics JETP 14 (1962) 1102. 34. A suggested experiment to determine the spin ofYl * and the parities of A-~, A-Y1 *, and ~-Y1 *, (with Su Zhao-bin and Gao Chong-shou), Scientia Sinica 12 (1963) 451. 35. The pure gauge fields on a coset space, (with Tu Tung-sheng and Yean Tu-Nan), Scientia Sinica 22 (1979) 37. 36. Soliton-soliton scattering and the semi-classical approximation for the problem of scattering in three-dimensional space, (with Dai Yuan-ben), Scientia Sinica 22 (1979) 281. 37. The non-topological soliton with a non-Abelian internal symmetry, (with Zhu Zhong-yuan, Dai Yuan-ben and Wu Yong-shi), Scientia Sinica 23 (1980) 40. 38. On the vacuum of the pure gauge fields on coset, Scientia Sinica 23 (1980) 431. 39. A model of electro-weak interaction in SU(3) X U(1) gauge theory, (with Gao Chong-shou), Scientia Sinica 23 (1980) 566. 40. Electro-weak theory in SU(3), (with Gao Chong-shou), Chinese Science Bulletin 25 (1980) 21. 41. Disorder parameter and duality, (with Xian Ding-chang), Chinese Science Bulletin 25 (1980) 635. 42. Axial U(l) anomaly and chiral symmetry-breaking in QCD, AlP. Conf. Proc. 72 (1981) 621. 43. Possible SU(4)cxSU(3)fXU(l) model, (with Chong-shou Gao), Phys. Rev. D 23 (1981) 2690. 44. On the quantization and the renormalization of the pure-gauge fields on the coset-space, (with Ruan Tu-nan), J Univ. Sci. Tech. China 11 (1981) 15. 45. The U(l) anomalous Ward identities and chiral dynamics, Chinese Science Bulletin 27 (1982) 147. 46. New non-linear cr model on symmetric spaces, (with Song Xing-chang), Commun. Theor. Phys. 1 (1982) 69. 47. The Hn sigma-models and self-dual SU(n) Yang-Mills fields in four-dimensional Euclidean space, (with Song Xing-chang), Commun. Theor. Phys. 1 (1982) 185.

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48. On the dynamical problems and the Fermion spectra in the Rishon model, (with Dai Yuan-ben), Commun. Theor. Phys. 1 (1982) 725. 49. Wilson loop integral and string wave functional, (with Li Xiao-yuan), Scientia Sinica, Ser. A 25 (1982) 264. 50. Backlund transformation, local and nonlocal conservation laws for nonlinear cr models on symmetric coset spaces, (with Song Xing-chang), Scientia Sinica, Ser. A 25 (1982) 716. 51. Local conservation laws for various nonlinear cr models, (with Song Xing-chang), Scientia Sinica, Ser. A 25 (1982) 825. 52. Composite gauge bosons in a nonabelian theory, (with Swee-Ping Chi a and Charles B. Chiu), Phys. Lett. 109 B (1982) 457. 53. Massless baryons and anomalies in chiral (QCD)z, (with D. Amati and S. Yankielowicz), Phys. Lett. 110 B (1982) 309. 54. On the determination of effective potentials in supersymmetric theories, (with D. Amati), Phys. Lett. 114 B (1982) 129. 55. Koba-Nie1sen-Olesen scaling and production mechanism in high energy collisions, (with Liu Lian-sou and Men Ta-chong), Phys. Rev. D 28 (1983) 1080. 56. Nonlinear cr model on multidimensional curved space with certain cylindrical symmetry, (with Song Xing-chang), Commun. Theor. Phys. 2 (1983) 971. 57. Kac-Moody algebra for two dimensional principal chiral models, (with Song Xing-chang), Commun. Theor. Phys. 2 (1983) 1391. 58. On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action, (with Guo Han-ying, Wu Ke and Song Xing-chang), Phys. Lett. 134B (1984) 67. 59. On Witten's effective Lagrangian for chiral field, (with Guo Han-ying, Wu Ke and Song Xing-chang), Commun. Theor. Phys. 3 (1984) 73. 60. The topological origins of Gauge anomalies, (with Guo Han-ying, Wu Ke and Song Xing-chang), Commun. Theor. Phys. 3 (1984) 125.

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352

360

xx 61. On the dynamical symmetry breaking for the N = I pure supersymmetric 365 Yang-Mills model, (with Dai Yuan-ben and Chang Chao-hsi), Commun. Theor. Phys. 3 (1984) 221. 62. The unified scheme of the effective action and chiral anomalies in any even 374 dimensions, (with Han-ying Guo, Xiao-yuan Li, Ke Wu and Xing-chang Song), Commun. Theor. Phys. 3 (1984) 491. 382 63. Symmetric and asymmetric anomalies and effective Lagrangian, (Han-ying Guo, Ke Wu and Xing-chang Song), Commun. Theor. Phys. 3 (1984) 593. 64. On the two dimensional non-linear cr model with Wess-Zumino term (I) 393 classical theory, (with Dai Yuan-ben), Commun. Theor. Phys. 3 (1984) 767. 65. CP violation from the standard model, 397 Proceedings of Europhysics Topical Conference on Flavor Mixing in Weak Interactions, (Ettore Majorana International Science Series: Physical Sciences, Vol. 20), 609. 424 66. Top quark mass and the fourth generation of quark, (with Wu Yue-liang and Xie Yan-bo), Chinese Physics Letters 1 (1984) 47. 67. Anomalies of arbitrary gauge group and the its reduction group, Einstein and 426 Lorentz anomalies, (with Han-ying Guo and Ke Wu), Commun. Theor. Phys. 4 (1985) 91. 68. On the two dimensional non-linear cr model with Wess-Zumino term (II) 437 quantum theory, (with Dai Yuan-ben), Commun. Theor. Phys. 4 (1985) 123. 69. Derivation of the anomalous term in Virasoro algebra by topological method, 442 (with Wu Yue-liang and Xie Yan-bo), Commun. Theor. Phys. 5 (1986) 359. 70. Effective action of sigma model anomalous with external gauge fields, 448 (with Yue-liang Wu and Yan-bo Xie), Modern Phys. Lett. A 1 (1986) 23. 71. A simplified derivation of Chern-Simons co chain and a possible origin of 453 9-vacuum term, (with Wu Yue-liang and Xie Yan-bo), Commun. Theor. Phys. 7 (1987) 27. 72. The general Chern-Simons characteristic classes and their physical 464 applications, (with Wu Yue-liang and Xie Yan-bo), Commun. Theor. Phys. 8 (1987) 341. 73. Signature for chiral-symmetry breaking at high temperatures, 491 (with Lay-nam Chang and Ngee-pong Chang), Phys. Rev. D 43 (1991) 596.

xxi

74. CP violation, fermion masses and mixings in a predictive SUSY S0(10)x~(48)xU(1) model with small tanf3, (with Y. L. Wu), Phys. Rev. D 53 (1996) R3492. 75. Low energy phenomena in a model with symmetry group SUSY S0(10)x~(48)xU(1), (with Wu Vue-liang), Science in China, Ser. A, 39 (1996) 65. 76. A solution to the puzzles of CP violation, neutrino oscillation, fermion masses and mixings in an SUSY GUT model with small tanj3, (with Vue-liang Wu), Nuc!. Phys. B (Proc. Supp!.) 52A (1997) 159. 77. A possible unification model for all basic forces, (with Wu Vue-liang), Science in China, Ser. A, 41 (1998) 324. 78. Searching for rephase-invariant CP- and CPT-violating observables in meson decays, (with W. F. Palmer, E. A. Paschos and Y. L. Wu), Euro. Phys. J C 16 (2000) 279.

Part II

Statistical physics and condensed matter physics

79. Renormalization of the closed time path Green's functions in nonequilibrium statistical field theory, (with Su Zhao-bing), Chin. Phys. 1 (1981) 635. 80. Dyson equation and Ward-Takahashi identities of the closed time path Green's function, (with Su Zhao-bing), Chin. Phys. 1 (1981) 645. 81. Closed time path Green's functions and critical dynamics, (with Zhao-bin Su, Bai-lin Hao and Lu Yu), Phys. Rev. B 22 (1980) 3385. 82. On theory of the statistical generating functional for the order parameter (1) general formalism, (with Su Zhao-bin, Hao Bai-lin and Yu Lu), Commun. Theor. Phys. 1 (1982) 295. 83. On theory of the statistical generating functional for the order parameter (II) density matrix and the field-theoretical structure of the generating functional, (with Su Zhao-bin, Hao Bai-lin and Yu Lu), Commun. Theor. Phys. 1 (1982) 307. 84. On theory of the statistical generating functional for the order parameter (III) effective action formalism for the order parameter, (with Su Zhao-bin, Hao Bai-lin and Yu Lu), Commun. Theor. Phys. 1 (1982) 389.

497

501

514

518 524

533

535

545

559

582

594

606

xxii

85. On an approximate form of the coupled equations of the order parameter with the weak electromagnetic field for the ideal superconductor, (with Su Zhao-bin), Commun. Theor. Phys. 1 (1982) 669. 86. On a dynamic theory of quenched random system, (with Su Zhao-bin and Yu Lu), Commun. Theor. Phys. 2 (1983) 1181. 87. A dynamical theory of the infinite range random Ising model, (with Su Zhao-bin and Yu Lu), Commun. Theor. Phys. 2 (1983) 1191. 88. A dynamical theory of random quenched system and its application to infinite-ranged Ising model, (with Su Zhao-bin and Yu Lu), Proceedings of the First Asia-Pacific Physics Conference, (World Scientific Pub. Co.), 474. 89. Symmetry and Ward identities for disordered electron system, (with Lin Jian-cheng and Shen Yu), Commun. Theor. Phys. 3 (1984) 139. 90. Does Parisi's solution of the Sherrington-Kirkpatrick model locate on the absolute maximum of the free energy? (with Shen Yu), Commun. Theor. Phys. 3 (1984) 263. 91. Equilibrium and nonequilibrium formalisms made unified, (with Zhao-bin Su, Bai-lin Hao and Lu Yu), Phys. Rep. 118 (1985) 1. 92. Spontaneous symmetry breaking and Nambu-Goldstone mode in a non-equilibrium dissipative system, (with Zhao-bin Su), Prog. Thea. Phys. Supp. 86 (1986) 34. 93. The canonical description and Bohr-Sommerfeld quantization for the fractional quantum Hall effect system, (with Zhao-bin Su, Han-bin Pang and Yan-bo Xie), Phys. Lett A 123 (1987) 249. 94. Influence functional and closed-time-path Green's function, (with Zhao-bin Su, Liao-yuan Chen and Xiao-tong Yu), Phys. Rev. B37 (1988) 9810. 95. Time reversal invariance and its application to nonequilibrium stationary states, (with Z. B. Su), Thirty Years Since Parity Nonconservation: A Symposium for T. D. Lee, (Birkhauser Verlag, 1988) 117.

621

632

641

652

667

677

682

812

821

826

829

xxiii

Part III

1.

~~, *-\LT~J:I~~~J:I Field theory, particle physics and nuclear physics

xXfJflH*TjJ8~i:1it,

845

847

*JE!'¥:tlt, 1955 fFm 11 ~m 4 ,jt)jm 299-316 In. A discussion on two-range nuclear force, Acta Physica Sinica 11 (1955) 299 (in Chinese). 2.

3.

4.

5.

6.

7.

8.

il'T:J:h~ ~ -1*jJ1ft~~,

865

~t~:A'¥'¥:t~ § ~f4~AA, 1955 fFm 1 Mm 53-66 In. Three-body potentials in meson fields, Acta Scicentiarurn Natura/urn Universitis Pekinesis 1 (1955) 53 (in Chinese). E8 Vlix.9J EEt. ~~J 1JflHll* 1~ x.9J EEt. ~:J:h 1Jf¥~ll ~ 1J¥t;:, ~t*:k'¥'¥* §~H'¥Jt&, 1956 fFm 2 Mm 211-218 In. Derivation of Maxwell equations from the microscopic Lorentz equations, Acta Scicentiarurn Natura/urn Universitis Pekinesis 2 (1956) 211 (in Chinese). ~T*DJPJ:T~JPJ::!i&, *JI=Jj['¥tlt, 1959 fFm 15 ~m 5 ,jt)jm 269-276 In. Mass difference between neutron and proton, Acta Physica Sinica 15 (1959) 269 (in Chinese). ~~ffl*~~t9]i 1'FJ:IHJIHtEqlil'TfrJffi: Tt~.L~1*gjt, (~ B. lbIf~1I:1JT~), 4PJE! '¥: tit , 1959~ 15~m7WHfB77 -388In. The universal Fermi theory of weak interaction and the capture of muon in atomic nuclei, (with B. Maevskii), Acta Physica Sinica 15 (1959) 377 (in Chinese). 3 11 il'Tfr He Jffi: T~.L~l$gjt, (~ **~, ~)*~), W£!~tlt, 1960 fFm 16 ~m 2 WHj~ 61-69 In. Muon capture in He 3 nucleus, (with Chu Chia-chen and Peng Hong-an), Acta Physica Sinica 16 (1960) 61 (in Chinese).

*

~TIl1¥gjt~lf?X:;z)JjHJF:I:~l31JilJN~t1%, (~ ~~'T), W£!~tlt, 1960 fFm 16 ~m 2 Mm 70-75 In. On the sign of the effective pseudoscalar term in 11 capture reactions, (with Huang Nien-ning), Acta Physica Sinica 16 (1960) 70 (in Chinese). 11 il'T~~Jffi:Tt*~r:Utst~&fP~:;Z&, (~ ~j[;:L1'q, W;E!~tlt, 1960 ~m 16 ~m 2 ,jt)j~ 76-80 In. The effect of recoil on the scattering of Il-mesons by light nuclei, (with Dai Yuan-ben), Acta Physica Sinica 16 (1960) 76 (in Chinese).

879

887

895

907

916

922

xxiv

7t frT~Tii~tiF~ 7t frTEt<J~1[t~~~, (~ ~7c*), ~W,¥;J=It, 1960 if 16 ~~ 5 :JtJ1~ 252-262 1ft. Dispersion relations for pion production in pion-nucleon collisions, (with Dai Yuan-ben), Acta Physica Sinica 16 (1960) 252 (in Chinese). 10. {ft~ 7t frT1tY:~tttflt~1HjtE p ~;t!;*Et<J-1'-~~1J¥tEt<JJti5(,

9.

(~ 1PJf'FR*), ~W,¥;J=It, 1961

938

17 ~tg 3 :JtJ1tg 133-134 1ft. A suggestion of experiment to detect the p-resonance in 7t-7t scattering, (with Ho Tso-Hsiu), Acta Physica Sinica 17 (1961) 133 (in Chinese). 11. 7t+p- A + n +K &L\IZ~*-JliI1XJ~j(L\IZ~®(Et<Jli}f~, (~ yj:*iJj(, r'2b*%), 940 ~W,¥;J=It, 1963 iFm 19 ~tg 10 WU!~ 649-672 JJr. Investigation of the correlation between resonance effect and near threshold effect in the 7t+p~A+7t+ K reaction, (with Su Zhao-bin and Gao Chong-shou), Acta Physica Sinica 19 (1963) 649 (in Chinese). 12. ~;J;JIlmM*Q,*;J;JIlmMEt<Jt§li 1'FmZ.rar1~Jxtf$tt~? (~ r\§J*%), 964 r\§J~Mmw~t~~w, 1980 iF~ 4 ~~ 4 M~ 536-538 1ft. Is there any symmetry between flavour and colour gauge interactions? (with Gao Chong-shou), Phys. Energ. Forties Phys. Nuc!. 4 (1980) 536 (in Chinese). 13. SU(3)9=' Et<J~~ It~-*~lli!, (~ r\§J*%), 967 r\§J~Mmw~t~~w, 1980 iFm 4 ~m 5 Mtg 609-622 1ft. Unified electro-weak model in SU(3), (with Gao Chong-shou), Phys. Energ. Forties Phys. Nucl. 4 (1980) 609 (in Chinese). 14. ~~q£f5{7t:m:T1tEt<J-fl19:%A, (~ J3'l'J~t~, ~jC[!]WJ, t±*~), 981 9='OO;r4'¥11*::k,¥,¥;J=It, 1980 iFtg 10 ~tg 2 Mtg 26-35 1ft. A general form of path integral quantization, (with Yin Hong-jun, Ruan Tu-nan and Du Dong-sheng), J. Univ. Sci. Tech. China 10 (1980) 26 (in Chinese). 15. 3tf" SU(3)XU(1)$£T;rQ~R:~~ 1t~-*jUt~Et<J-®i1i~, (~ ~*%), 991 r\§J~~~:EJIl.~~~W, 1981 iF~ 5 ~~ 1 :JtJ1~ 39-46 1ft. Some discussions on the SU(3)xU(l) electro-weak unified model ofleptons and quarks, (with Gao Chong-shou), Phys. En erg. Forties Phys. Nucl. 5 (1981) 39 (in Chinese). 16. Wess-Zumino-Witten 1f~j(1'Fm:m:Et<J;J;JIlm/G~tt, 999 (~ $~¥J.~, ~-ar, *1]'*), r\§J~~~W~~~W, 1984 iFtg 8 ~tg 2 M~ 252-255 1ft. On the gauge invariance ofWess-Zumino-Witten effective action, (with Guo Han-ying, Wu Ke and Song Xing-chang), Phys. Energ. Forties Phys. Nuc!. 8 (1984) 252 (in Chinese).

*-T

iFm

927

xxv 17.

1~xtf$'&'ffi;fI]1f~j(1iLf\;:I:, (l::j !J~¥J(~, ~PJ, J'i'ij~MmlIl::j~4mlJ. 1984:tFm 8 ~~ 4 :ltJ1~ 508-512 ~.

xtf$JX'ffi,

*11'*),

Symmetrical anomaly, unsymmetrical anomaly and effective Lagrangian, (with Guo Han-ying, Wu Ke and Song Xing-chang), Phys. Energ. Forties Phys. Nuc!. 8 (1984) 508 (in Chinese). 18. {± ~ 1~ ~1t at 2 a"rf=r ~j( 1-'F ffl ;fl] ~tiI.& 'ffi , (l::j .$~¥J(~, */J\~, ~PJ, *11'*), J'i'ijIlMmlIl::j~4mlI, 1985 :tFm 9 ~m 2 M~ 252-256 ~. Effective action and chiral anomalies in any even dimensional space, (with Guo Han-ying, Li Xiao-yuan, Wu Ke and Song Xing-chang), Phys. Energ. Forties Phys. Nuc!. 9 (1985) 252 (in Chinese). 19. tl'-iJ!Hl"J Chern-Simons lJ!;f1]'f::1fJi¥J@ffl, (l::j ~-ffi~, i~f~~), J'i'ij~~4mL!l::jt~4mlI, 1986 :tFm 10 ~m 2 177-185 ]}:t. The general Chern-Simons cochain and their application, (with Wu Vue-liang and Xie Yan-bo), Phys. Energ. Forties Phys. Nucl. 10 (1986) 178 (in Chinese). 20. -t~:8'JTi¥JLJll~":-TbZ.;1t@ffl, (l::j ~-ffi~, i~f~~), i\N:i~~4m1Il::j~4m1I, 1986:tFm 10 ~~ 3 M~ 283-287 ~. A new co-boundary operator and its applications, (with Wu Vue-liang and Xie Yan-bo), Phys. Energ. Forties Phys. Nuc!. 10 (1986) 283 (in Chinese).

Mm

Part IV

~jHt-~l1E.!:§i;~Hf~*~l1E

1004

1009

1014

1023

1029

Statistical physics and condensed matter physics

21. l~fijffJlX*~JE'ffi~i¥J Goldstone t~, (l::j ~*(JJ

~.) = ~ ;; I. m

~ yl:' (k) y{::,+ (ko) RI.; I'r' I,.;

I'. r

(3)

The values of s and f are determined from the rule for adding angular momenta, and are as follows: for s = 0 (singlet) != J; for s = 1 (triplet) f =j, J:I: 1. For a given value of J, the quan-

tities R~,f'S' form a symmetric (reversibUity of the motion) four-rowed matrix R! satisfying the condition

~

Yi:' (k) Yl~t (ko) R!!~ ~""

(6)

I ... I' •• '

(7)

The matrix L(I)j is a unitary, antisymmetric, Hermitian matrix. Therefore R(t)j satisfies E~. (4) and has the same symmetry properties as R. Therefore the elements of R(t)j may be considered as the elements of a new scattering matrix which leads to the same cross section as M. All this al?plies as well to the matrices R(2)j = L(2)j x RjL(Z)J+ and R(I)j =L(I)jRJL(I)j+, corresponding to the second and third transformation of (2). The matrix L(i)j has the form

=V2/+1 (

o

0

0

0

±Vi+1 -YT

=FYi::f- 1 =FYI ) Vi -VJ+T

±VT VT+T

a.

(2)

i-

where

(1)

Here at and a2 are the Pauli matrices'for the two nucleons, and lEo and It denote unit vectors along the incident and scattered nucleon directions. In order to obtain the transformation of interest, note that the scattering cross section for unpolarized nucleons = Y.Sp MM+, is invariant to an exchange of M (It, lEo; O:t, crz) with one of the following matrices Ml = (~lk) M (~lko). M. = (~.k) M (~.ko). M. = (~lk) (~.k) M (~~ko) (~.ko)·

(5)

-Io 0 0 1

L (S)} _ -

(

0

0 0

o

0

1

-21+1

-

2l'7V"+1i

2j+l

0 0

0 0

• (8)

0) 0

_ 2 Jf7V+T)

2j+l

,

_1_

2/+t

The first columns (row) correspond to singlet states, the remaining to triplet states in the following ordering of f: J, j + 1, j - 1. The upper sign corresponds to superscript 1, and the lower sign to superscript 2. Inasmuch as the operators (crl,tIt) in contradistinction to (crtlt)(atlt) do not commute with the square of the total spin operator Y. (crt + cra)I, the matrices Mt and Mz lead to Singlet-triplet transitions. Therefore the first two transformations cannot take place in the case of identical nucleon collisions where singlet-triplet transitions are forbidden by the Pauli principle. This also is true for n-p scattering if isotopiC invariance holds. The operators (crill) represent operators which

21 364

LETTERS TO THE EDITOR

rotate the spin of the i-th nucleon by an angle 1r about the direction q. This ailows to determine the transformation properties of various spin characteristics when Rj is replaced by R (3}j For example. this exchange leads to a change in the sign of the polarization P which takes place in the collision of unpolarized nucleons. We finally remark that changing the sign of all the phase shifts (taking the complex conjugate of Rj) leaves the cross section unchanged. and changes the Sign of Po. Thus a simultaneous application of this transformation with the transformation Rj into R(3)j leaves unchanged the cross section as well as the polarization. Therefore the two sets of elements of R obtained from one another by means of the indicated transformation. cannot be distinguished through the Simplest polarization experiments (double scattering). IS. Minami. Progr. Theor. Phys. 11. 213 (1954). Kawaguchi. and Minami. Progr. Theor. Phys. 12. 355 (1954). 3 R. Ryndin and la. SmorodinskU. Dokl. Akad. Nauk SSSR 103. 69 (1955). 2 Hayakwa.

Translated by M. A. Melkanoff 101

22 710

LETTERS TO THE EDITOR (1)

AHe i

with the assumption that the spins of and K are 1 and 0, respectively. This process is completely described by a spinspace matrix M (n, n') which gives the amplitude of the diverging wave.' The most general form of the matrix M (n, 11') is M (n, n') = a + be [D x D'l,

(2)

when the product I of the parities of all four particles is equal to + I, and M (n, n')

=

aan

+ ben',

(3)

when I = - 1. Here n and n' are unit vectors parallel to the momenta of the incident and emerging particles, respectively; a and b are certain functions of the energy and of the angle between n and n'. The density matrix Pi of the initial state has the form Pl=A(1

+cP),

(4)

where P is the polarization vector of the incident particles, If the reaction takes place at threshold, or if we select only the AHe i particles emitted forward (i.e., n Hn'), then we can neglect the second term in Eq. (2). At the threshold, Eq. (3) takes the form M = aan, and for nun' we have M = (a + b ) an. The polarization vector P' of the AHe i particle in the final state is calculated by the formula I P' = Sp (Mr'IM+c) / Sp (M~IM+).

(5)

Substituting Eqs. (2), (3), and (4) into (5), we get

ON THE DETERMINATION OF THE RELATIVE PARITIES OF ELEMENTARY PARTICLES CHOU HUAN-CHAO (CHZHOU GUAN-CHZHAO) Joint Institute of Nuclear studies Submitted to JETP editor January 16, 1958 J.Exptl. Theoret.Phys. (U.S.S.R.) 34, 1027-1028 (April, 1958) SINCE parity is not conserved in the weak interactions, it is of great interest to determine the parities of elementary particles by means of the strong interactions. We consider below several reactions which can be used for the determination of the relative parities of strange particles. We have in mind the following type of reaction:

P'=P,

when I

=

+1.

(6)

P' = (2(1>n) n - P),

when I

= -I.

(7)

If the parity is not conserved in the decay of the AHe i , then from the angular asymmetry of the decay one can measure the direction of polarization of the AHe i and distinguish between the possibilities (6) and (7). We emphasize that the Incident beam must be polarized, and in such a way that the polarization vector is neither parallel nor perpendicular to the direction n. The other reactions of this general type are as follows: );I + He' ~ .\He'+ ,,± at threshold, or when the .,He' emerges forward; (8)

E± p (n)

+ He' -> A +

He' + ,," at threshold;

+ He' --+ A + He' + Ii- + (/(") at threshold.

(8') (1')

In the last two reactions He" can be replaced by

any other nucleus with spin 0 (for example, C 12 ).

23 LETTERS TO THE EDITOR We note that, in the reactions (1) and (1'), the polarization vectors of the AHe i and A depend only on the vectors P and n and the relative parities of K and A. Therefore, a case of the reaction (I') can he simply added in with the cases of reaction (1). The writer expresses his gratitude to M. 1. Shirokov and L. G. Zastavenko for valuabl~ advice and a discussion of the results. I L. Wolfenstein and J. Ashkin, Phys. Rev. 85, 947 (1952); R. Oehme, Phys. Rev. 98, 147 (1955).

Translated by W. H. Furry 213

711

24 SOVIET PHYSICS JETP

VOLUME 34 (7), NUMBER 5

NOVEMBER, 1958

THE RELATIVISTIC THEORY OF REACTIONS INVOLVING POLARIZED PARTICLES CHOU KUANG-CHAO and M. 1. SHIROKOV Joint Institute for Nuclear Research Submitted to JETP editor December 6, 1957 ,J. E:xptl. Theoret. Phys. (U.S.S.R.) 34, 1230-1239 (May, 1958)

It is shown that, in the rest system, the relativistic formulas for the angular distribution and the polarization vectors and tensors for a reaction of the type a + b - c + d are essentially thc samc as the nonrolativistic formulas, if thc spin of a particle is defined as its internal angular momentum around its center of mass. The square of this internal angular momentum is Lorentz invariant. The spins of the particles are arbitrary, and their rest masses are nonvanishing. The main difference from the nonrelativistic case is that the description of the spin state is not the same in different Lorentz reference systems. Therefore for cascades of reactions (for example, for experimcnts on double scattering) corrections must be applied to the nonrelativistic formal theory. The relativistic changes in the angular correlations are indicated [or successive reactions of the type 7r + P - Y + K, Y - N + 7r.

INTRODUCTION FOR reactions of the type a " b - c "d, formal theories are known which express the angular distribution and the state of polarization of the products of a reaction in terms of the states o[ polarization of the incident beam and the target and unknown parameters which are the elements of.the S matrix for the process a " b - c "d. The simplest example is the well-known formula for the [unction f ( 0) which appears in the expression

'f (r)

'C."

c'"

+ f (Ii) Col" Ir

theory of the scattering of spiniess particles is essentially relativistic. In order to obtain the angular distribution in any Lorentz system, one has only to transform (J (e) = 1f (e) 12 from the centerof-mass system into the desired system by using known formulas. What is therefore required for the relativistic generalization is the definition of the relativistic spin operator. The spiro operator which we introduce satisfies all the requirements which can be demanded in terms of the concept of the spin as the intrinsic angular momentum of a particle.* To obtain the relativistic formulas we use the method of Coester and Jauch in a form presented in a paper by one of us. 3 We emphasize that in this method one needs only the ability to describe the state of a free particle possessing spin; we do not need relativistic equations (for free particles) like the Dirac equation (which plays an essential part in Stapp's relativistic theory' of the scattering of particles with spin It2).

[or the wave function of a stationary scattering process of particles of spin zero. In this case the unknown parameters are called the scattcring phase shifts. These theories are based on the use of the conservation laws (in particular the law of the conservation of the total angular momentum). Coester and Jauch 1 were the first to obtain formulas for the ,mgular distribution and polarization in the case of particles a, b, c, d of arbitrary spins; their 1. THE CONSERVED PHYSICAL QUANTITIES IN starting point was the explicit formulation of the A RELATIVISTIC THEORY conservation laws in terms of the diagonal propThe conservation laws are simply an expression erty of the S matrix with respect to the conserved of the fact that the physical processes in an isolated quantities. These same formulas have been obsystem must not depend on the means used to detained by Simon and Welton, but by a different method (cf., e.g., Ref. 2). *Iu. M. Shirokov has informed us that he has employed this These formulas are nonrelativistie, but only besame description of the spin state (which was obtained by him cause the spin state of the particles is described in from the theory of the irreducible representations of the inhoma. the Pauli approximation (so that it has the same geneous Lorentz group) to the formulation of a similar relativisappearance in all Lorentz reference systems). The tic theory of polarization and correlation effects. 851

25 852

CHOU KUANG-CHAO and M. 1. SHIROKOV

scribe it, in particular on the choice of the reference system. Here it is of course assumed that space-time is homogeneous and isotropic (we can suppose, however, that this assumption is contained in the concept of an isolated system). In quantum mechanics this fact is expressed by the requirement that the S matrix of a physical process must commute with ten operators, the infinitesimal displacements of the origin of the space and time coordinates providing operators P Ii' and infinitesimal space-time rotations giving operators MtLv • The fact that an operator commutes with the S matrix means that the S matrix is diagonal with respect to the eigenvalues of this operator, * and consequently that the corresponding physical quantity is conserved, i.e., remains unchanged in all internal processes. Four conservation laws have the clear physical meaning of the conservation of the total momentum and energy P tL {px , P y , P z , iP o }. Of the six other operators MtLV ' three operators Mk( (k = 1, 2, 3) do not have an immediate physical meaning' and instead of M tLV we shall introduce six other operatbrs, which have the physical meaning of the coordinates of the center of mass of the physical system and its total angular momentum around its center of mass. The properties of the center of mass follow directly from its conceptual meaning: the motion of the system as a whole can be characterized primarily (in the very first approximation) as the motion of a material point with mass equal to the rest mass (or energy) of the system and with momentum equal to the total momentum P of the system. The center of mass of an isolated system must therefore move uniformly in a straight line. Furthermore, in quantum mechanics we must require that the center of mass R actually be the coordinate operator of a certain particle, i.e., in particular, that the well known commutation relations hold between Rx , Ry , R z , P x , P y , P z . We can obtain such an operator R in the following way. It is well known that the following are the commutation relations which must be satisfied by the operators Pi and MtLv (cf. Ref. 6 and Sec. 3 of Ref. 5): [Pi, Pi] = 0; [Pi, E] = 0; [M i , Pd = i.iihP.;

(1.1)

*Let us write out AS - SA = 0 as a matrix product. In doing so, we choose a representation in which the operator A is diagonal (Le., we label matrix elements with its eigenvalues). Then

A"S,,- SimAml = (Ai i.e.. Sil must be equal to zero if i

*

A,) Si/

l.

=

0,

(1.2)

IN i , PJl

[M i , EJ = 0; [M i , M i ] = i.iikM.; tOijE; [Ni , E] = iP . ; [M . , Nil = isiikll'k; (1.3)

=

(1.4)

Notations: [A, B 1 = AB - BA; i, j, k take the values 1, 2, 3; and {M 1 , M 2 , M 3 } = {M 23 , M 31 , M 12 }; ilv'j = M i4 ;

Eijk is a tensor antisymmetric in all its indices, with E123 = 1. It is understood that we confine ourselves to those state vectors I/!o which describe states with a definite rest mass m o, i.e., for which PtLPtLl/!o = - m3l/!o, or (Po - E) I/!o = 0 (the time displacement operator Po is equivalent to the factor E). We note that since [Ni, El = iPi the average value of Ni is a linear function of the time. Therefore Ni is "conserved" in the sense that internal processes have no effect on this time dependence. We introduce three new operators R x , R y , Rz , for which

(from which if follows that [Ri, E 1 = iPi IE). We represent M in the form Mk = ~.'ijRiPj

+ 10,

i.f

and find from Eq. (1.1) that [Ji, Pj 1 = o. If we require that Rx , Ry • R z be the components of a spatial vector (as must indeed be the case), i.e., that [Mj, Rj J = iEijkRk, then [Ji, Rj J = 0, and J x , J y , Jz also form a three-dimensional vector. It then follows from Eq. (1.2) that [Ji, Jj 1 = i€ijkJk· In like fashion, representing N in the form

1/, (RiE + ER .. ) + Ki == R;£ - iPi /2E + Ki, we find that [Ki, Pj 1 = O. Therefore also [Ki, E 1 Ni =

0, and the average value of Ki is constant in time. We now pose the problem of expressing M and N in terms of the operators R and J which we have introduced. For M this has already been done. It remains only to express the spatial polar vector K in terms of P and J. It can be shown that if Rx, By. R z are the first three components of any four-vector [i.e., for example. [N, R 1 ,: if i;o' j). then K cannot be expressed in terms of J and P only so as to satisfy all the commutation relations for N. This means that if K is constructed from P and J alone, then [R, P lk and Jk are not the spatial components of a fourdimensional tensor of the second rank. =

°

26 THEORY OF REACTIONS INVOLVING POLARIZED PARTICLES The simplest K (namely one linear in J x , J y , Jz) satisfying Eqs. (1.3) and (1.4) has the form

= O.

K = [PJI(E ±mr'

(cf. Refs.. 7 and 6). * Since the problem has a solution, it follows that: (1) R is • conserved" (in the sarne sence as N), since it can be expressed in terms of the conserved operators M",v (see Appendix). R can be called the center-of-mass operator. It is the same as definition (e) of the center-of-mass in the papers of Pryce. 8 (2) J is also conserved and, what is particularly important for our purpose, J2 = J~ + J~ + J~ is Lorentz invariant, since [N, JZ) = O. We emphasize that this is true for arbitrary K = K(P, J).

2. USE OF THE CONSERVATION LAWS. RELATIVISTIC DEFINITION OF THE SPIN OF A PARTICLE The four conservation laws for the total momentum and energy and the three conservation laws for the center of mass can be expressed very simply. The argument is usually carried through in the Lorentz system of reference Ks in which the ( conserved) total momentum is zero (the socalled center-of-mass system). The origin of the coordinate system can be taken at the center of mass (more precisely, at the pOint given by the average value of the operator Rt for the particles a and b (or c and d). Then J is the total angular momentum. Since the commutation relations between J x , J y , Jz are the same as for the total angular momentum or for the Pauli spin matrices, the eigenvalues of .1 2 and jz are respectively equal to hZJ(J+ 1) and M=J, J-l, ... , -J. The conservation law for J is expressed by the fact that the S matrix is diagonal with respect to the eigenvalues of .1 2 and jz: ( ... J'M'IS/ .. . JM) = ( ... ISiMI ... )ill'Jil'WM

853

and also the fact that ( ... I SJM I ... ) is independent of the value of M, which follows from [Jx • S)

(2)

*We have not been able to show that no other K's exist. Beginning with different considerations, L. G. Zastavenko has evidently proved the uniqueness of K. We are grateful to him for a discussion of this question. tThe conservation law for R means something more than the conservation of the average value. The requirement [R, S] - 0 means that if the system is in a state with a definite value of R (we note that in the interaction representation the wave function for the external behavior of the system does not change with the time), internal processes in the system will not take it out of this state. This property does not get used explicitly. but an operator R of this kind is required for the definition of the conserved angular momentum J of the system (and of the spin of a particle. see below).

To find, for example, the angular distribution of particles c and d, we need to know the explicit expression of the elements of the S matrix in the representation of the particle momenta. In order to express these elements in terms of the elements (2), we must first of all enumerate the remaining variables of a complete set (denoted in Eq. (2) by dots), which commute with each other and with JZ and Jz. The initial and final states of the process a + b - c + d are states of systems consisting of two free nOninteracting particles. From the meaning of the S matrix, its elements are the transition amplitudes between such states. Therefore to label the elements of the S matrix we must take as our variables a complete set of quantum mechanical quantities describing the free particles a and b or c and d. The total angular momentum J (in the system Ks) is represented in the form J = 11 + h. where 1 is the total angular momentum of a Single particle in Ks. The procedure stated in Sec. 1 for obtaining the conserved angular momentum relative to the center of mass can be applied to a system of arbitrary physical nature (for example. to a system of fields). One needs only to know a concrete representation of the operators P and M. Therefore it is natural to apply this procedure to an "elementary" particle, whose physical nature is in general unknown (by the definition of "elementary"). Besides the coordinates r of the center of mass and the momentum p we then get just one conserved external characteristic of the particle. its angular momentum 8 relative to its center of mass r. In defining the spin s of a particle, we are only fixing precisely the concept of the spin as the intrinsic angular momentum of the particle. Accordingly, J = [ r x pI + I, and in the system Ks, in which PI = -P2 = p, we get J

= [r, x ptl + 5, + [r 2 xp.1 + 52

~[(r, -") x

pI

(3)

+ 8, + '" "I + 8, + "'.

In the system Ks we can now proceed in complete formal analogy with the nonrelativistic treatment to introduce the total spin operator 8 = 81 + 82 (the square of which. however, is not a Lorentzinvariant quantity). The eigenfunctions of the square of this quantity, S2, and its component Sz can be expressed in terms of the products I/>ilml x l/>i zm 2 of the eigenfunctions of the squares and components of the operators 81 and 82 (the eigenvalues of si are denoted by h2id i l + Since

1».

27 854

CHOU KUANG-CHAO and M. 1. SHIROKOV

the commutation relations for s, st> S2 have the usual form, [sx, Sy 1 = is z and so on, the coefficients in these expressions will be the well known Clebsch-Gordan coefficients (ili2mlm21 i li 2sm), which are also the transfurmation functions for the transformation from the representation in the variables it> i 2 , mt> m2 into the iI' i 2, 5, m -representation (and inversely). A similar meaning attaches to the coefficient (lsjJ.m IlsJM ). We can now take as the variables denoted by dots in Eq. (2) 5, l, and the absolute values of the momenta in Ks (or the total energy of the system, which in Ks is equal to an invariant, the rest mass of the system). 3. FORMULAS FOR THE CROSS-SECTION AND THE POLARIZATION VECTOR AND TENSORS. RELATIVISTIC ROTATION OF THE SPIN We can now express the elements of the S matrix in the representation of the momenta of the particles and their spin components in terms of the elements (Pc, s', l', J', M'lsl'Pa, s, l, J, M). The transformation function from the representation in the variables p, s, l, J, M into the representation of the momenta and the spin components is the product of three transformation functions. We can write out the transformation as follows (cf. Ref. 9):

X (i,.i dl1l,l1ld

I i,ids'm') (/'s'!,-'I/l' I

/'s'JN/)

(s'/'I

(4) ",J,np"}1 sl)

We have used Eq. (2) and the law of the conservation of the total energy. Pc and Pa are the momenta of particles c and d and a and b respectively in K; J c , 0 and approaching zero as p - 00 as l/p uniformly in p. We now evaluate fmaps (p). To do this we make use Df the series

Here t = p. n/p = cos Ii and t" = p. n" /p = cos Ii", while the angle a(p, t) is defined [see Eqs. (A.2) to (AA)]. Since R3(ECA Y OF THE X-HYPERON AND ITS ANTIPARTICLE CHOU KUANG-CHAO Joint blstilute of Nuclear Research, Dubna, USSR

Received 17 July 1958

Abstract: This note examines the information which may be derived from mea.surements of the branching ratios and angular correlation of E- and ~-decays.

1. Branching Ratio Recently Okubo in a very interesting paper 1) proposed an experiment on the measurement of the branching ratio in the decay of the };+ hyperon and its antiparticle ~+. A difference between their branching ratios will indicate the violation of invariance both under charge conjugation and under time reversal. To obtain his results Okubo utilized the lowest order perturbation theory for the weak interaction. In view of the great importance of this experiment as a test of the validity of invariance under time reversal we shall show in ~his note that Okubo's results are quite general. They can be derived from the Liiders-Pauli theorem together with the requirement of unitarity of the S-matrix. The probability of the decay process is characterized by the following elements of the R-matrix (i,

i,l. mlRli, m)

where i denotes the total isobaric spin of the nucleon-meson state; i the total angular momentum of the system which is equal to the spin of the I-particle; m the eigenvalue of the operator i.; 1 the orbital angular momentum of the nucleon-meson state; and R is connected with the scattering matrix S by the relation S = l-iR. Okubo's derivation rests on the relation (7) in his paper 1) which can be expressed in the form (i, i.l,

mlRli, m>c = Io(-l)l exp

(2i",.

181

=

g~>~~2 X l>f'f'~ = ig; (P'roP

+ /i'ron) 'P~'

(20)

= 't'

;Oc

-;0,

(14)

we similarly obtain the unique expression Indeed, under this transformation: F: <j>~ = exp (ii) h,

L = ig (~<j>,'P' - ~,<j>1') = 2ig (p'r,n'P' -+- "'r,P'f')'

Y~" . exp ( - ii.) <j>'>I' (15)

The three-parameter trnnsformation (10) is isomorphic to a rotation in the isobaric space. Indeed, if we form the conventional eight-component nucleon field iJ!N

=

(~~),

we have, under the transformation

(21)

We may thus assume that

, where /J = ± i is the eigen- transformation function (3). value of the operator f. Since f is invariant Let us denote the total angular momentum of under the proper Lorentz group, I/o remains unthe system by J, and the total angular momentum changed by a Lorentz transformation. Therefore of particle 0/ by JO/. We suppose at first that the state vectors in different coordinate systems differ rest mass of particle 2 is not zero. In this case only by a phase factor h can be represented as a sum of an orbital anguL lar momentum la and a spin S2. 1 We introduce Ip, ",)-Ip, ",)' = "I.. {L, p)\L-lp. "'). (2) a new vector J in the following way: where L is a Lorentz transformation; "'/J ( L, p) (4) J = j, + J, = j + 5., j = j, + I,. satisfies the group relation The spin operator of particle 2 commutes with the "I.. (Llo p) "I" (L" L;lp) = "I.. (L,L .. pl. operator J, by definition. Since J and S2 satisfy The explicit form of the factor "'/J depends on the the usual commutation relations for angular mochoice of the coordinate system in which the total menta, J also satisfies these relations. angular momentum is quantized. If we choose the In the center-of-mass system a complete set of axis of quantization of the total angular momentum operators for the two-particle system consists of in such a way that the z axis is parallel to p and the following operators: the x axis to [vxpl, where v is the velocity of J', J., ji, Elo p. (5) the new coordinate system relative to the old for the Lorentz transformation L, then by means of If particles 1 and 2 have definite parities and parity the formalism developed in reference 9 it can easis conserved in the reaction under conSideration, then it is better to replace the operator f 1 in the ily be shown that in this case "'1/0 = 1. * From Eq. (2) we easily find the transformation set (5) by the space-inversion operator I. function between the state vectors of a 'Y -particle To express observable quantities in terms of in two different coordinate systems 1 and 2: diagonal elements of the R matrix (R = S - I, where S is the scattering matrix), we must find (Plo ....1 P.,I'.) the eigenfunctions of the complete set of operators (5) in the representation of the momenta and spins • =VPl/P,'6' (P'I-~ ail 0..,,,,, i = 1,2,3, (3) of the free particles. We write these eigenfunc1=1 tions in the center-of-mass system in the form where aij is the matrix of the Lorentz transfor(6) ,=Co

,o",+""MOpp" IlIiJ.l

(7)

where k is the unit vector in the direction of the Z axis, and C is a constant which depends on J, j, and 1l0!. The relation (7) follows from the equation Jz = ~I + ~2' if the z axis is parallel to n. 2. Under a three-dimensional rotation the functions (6) transform in the following way (like spherical functions): (n, =

L so that 1)T = (- 1 )1l1+1l2, and using the relation

pi 1. M,

-1'1' -1'-2.

1"" 1",.

C"' Vi? (2jf 1)'/' PVV----;;~

I'll. M. j. 1'.1''>

.J", ""

J Clll15zl-tl DIJ.,+J.lI,M

C({n)

0

pp.:J

(16)

I t

where

If the rest masses of both particles are zero, we cannot introduce the operator j. In this case we use the following complete set of operators:

(17) The eigenfunctions of the operators (17) are found in a similar way, and have the form

52 REACTIONS INVOLVING POLARIZED PARTICLES OF ZERO REST MASS (n, 1'" 1'., pi J, =

M, I'~, I'~, p')

Y"R (2J + 1)'1' J pYV ~ D•• +."

2d

M

(g.)

, o.,••' o.,.,oop"

(18)

The complete sets (5) and (18) can also be used for a system of two particles witb nonvanishing rest masses. which is usually described by tbe set J Z, J z , (1-, S1. p, where 1 is tbe orbital angular momentum of tbe relative motion, and S = 11 + 12 is the sum of tbe spins of tbe two particles. Between tbe eigenfunctions of tbese sets of operators tbere exist unitary transformations (transformation functions)

645

J2, J z • (1-, S2. p. where 1 is tbe orbital angular momentum and S is tbe total spin of tbe two particles. We get the following results: 1. y + b - c + d. p' (qcXcl!dXd; nc) = [N,/(41t)') [(2id- I) (2id

(J,/;S~,

X }; Y;c XcOdXd

Jq'X'.

x <S;I;",'I RJ'I i.t.",.)'

+ I) 2-' J,I;S;)

Y"yx,. ..

(2i.

+ w'l'l'

<S;I;", I RJ'I j,/,,,,,)

x. (J,j,t,.

JqX. J,M,)

x D~'x (gcg:;') p (qyXyq.X •• ny),

(21)

where

(J. M, j, p.. pIJ'. M'.I. S. p') opp' (21 + 1)'/. (2S + 1)'/,cf~s. W (lSji" Ji,)

= oJJ'OMM'

and (J. M, 1'" 1'•• pIJ'. M', I. S, p') = OJJ,OMM'Op.' {(2/+ 1)/(2J

+ 1)}·I'C7.~:t::cft~:,~".. (22)

where W (abed. ef) is the Racah coefficient.

3. GENERAL FORMULAS FOR THE ANGULAR DISTRIBUTIONS AND THE POLARIZATION VECTORS AND TENSORS FOR THE REACTIONS a + b - 0 + d AND a - 0 + d

Y Oy'

x

Let us first consider the density matrix of tbe y -particles, . Since ~y takes only tbe two values ± i, we cannot construct tbe stensors from in the ordinary way. Fano 12 has shown tbat for photons it is convenient to use tbe stokes parmeters. This idea is easily generalized and is adopted in the present work. The Stokes parameters are related to tbe density matrix of tbe y -particles in tbe following way:

where

p

(qy, Xy) are the Stokes parameters,

(20), do not transform like a vector under rotations. The physical interpretation of the stokes parameters given in Fano's paper for photons is also correct for any y -particles. We shall not repeat it. In this paper tbe Stokes parameters will for convenience also be called s -tensors. The calculations of the s -tensors of the reaction-product particles is made by tbe metbod of M. I. Shirokov.s We use the notations introduced in reference 5. If the masses uf particles 1 and 2 in the initial and final states of tbe reaction are not zero. we use the complete set of operators

L (2J, + I) (2J, + I) (2j, + I) (2i. + I) (2J + I) (2q + 1))'/'

(J,i,t,. JqX. J.i,t.) =

F II (J't J't 1 _ I) -- (_I)i.+i,+h+I'C h2Iy . 1 It 2 ,!, /,i'fi1iy ,

II.

~y = iyO"y, and ~y = iya'y. We emphasize that p (1. Xy) defined by Eq.

yO. ' .

= (-

I)'·-Iy ' ' ' ' N,

= (21th)' R' (V'p~P~l-' (P.;

Pd)'.

The sum is taken over q'. X', J'. li, Si. J z• jz, t 2• J. q. X. qy. Xy. qb. Xb· 2. a + b - y + d.

lZ. 82. h. tlo

=

[N,j(41t}'] (2 (2id

x

+ I) (2i. -i- If' (2i. + If']'"

LY;yXyqdXd

(Jli;t;. Jq'/, J,i~t,)

x r. + p + 0

+ E± --+ ~o + E±

IT+

+ p --+;t' +;to + p

bl~PX"PI(PI('"

b, =oPx (PI(")"

(Pxpx,,)' - ph (I - rid, c,=o(p"P KA ) ' - ph(l-ph)· Cl =0

2. Recently Chew 3l1d Low, and Okun' and Pomeranchuk,3 have suggested that peripheral collisions be studied as a method for determ:.ning interactions between unstable particles. We shall assume that this method can be used for the determination of the scattering amplitude for 1: ( A) + IT - 1: (A ) + 7r through studies of the processes l; + N - ~ (A) + N + IT and A + N - ~ + N + IT. The key point of

(20)

and E-+p--+r.o+P+IT-, and A+p--+r.o+n+IT+, and ~-+p--+IT-+ITo+p etc.

b) Near the pole the amplitudes for the reactions E± (A)

b,

(18)

al=op~,,+ph=a,

ITO

under rotations in the isotopic space. Similarly it can be shown that the amplitudes for the following pairs of processes are equal:

IT+

(17)

+ ~ Vbi-a,c,} ,

corresponds to the pole term, whose residue is proportional to the amplitude for IT - A (~) scattering. Assuming that in the physical region near the pole the reaction 1: + N - ~ (A) + N + IT is determined by the process (19), one can extrapolate its amplitude into the nonphysical region and separate out the residue of the pole term. To estimate the effect of other terms in the physical region near the pole, we shall formulate certain rules that must be fulfilled if the contribution of the pole term in actually predominant in this region. a) In the region near the pole the amplitude of the reaction l; + + p - ~ + + p + ITo is equal to that of the reaction 1: - + p - ~ - + p + ITo. This rule follows from the invariance of the virtual process

A + p --+ r.' + n + ITO

+ ~Vb:-alcl}'

201

the method is that the amplitude for the reaction 1: + N -1: (A) + N + 7r, regarded as a function of (PN-PN)2, where PN(PN) is the four-"Vector momentum of the nucleon in the initial (final) state, has a pole in the nonphysical region, (PN -PN)2 = 11 2 (Il is the mass of the IT meson). It is shown that the virtual process

L+

<min{VI-pl('O; ~. /J.t. ... Xh' 01

max{O; a2 ~-.!..Vb2-a,c,}

sel) + iR .. [( ak,) (s~e) - (ak~) (see')!

=

t

iR .. (ak~) (sA - (ake) (e's,)I,

(15)

where R I• ~. and R5 describe the electric, and ~, R,. and Rs describe the magnetic transitions; e and e' are the polarization vectors before and after the colliSion; s = k x e. s' = k' x e', where k and k' are unit vectors in the direction of the momentum of the 'Y quantum before and after the scattering; the label "c" refers to the center-of-mass system. The expression for oK containing the terms not higher than those linear in the energy of the 'Y quanta l2 ,13 can be written in the form oK = - e'M-I (ee') + ieM-' (21'- - e 12M) Ve (a [e'x

en

+ 21'-"'e (a {sc"s~IH ieM-'I'-'e \(ek',)(as;)- (e'k,)(as,)]. (16) With. the help of the relation (as') (ek') - (as) (e'k) 2(a(e'x~l)

= -

+ (ak') (es') -

(17)

(ak) (e's),

we can bring (16) into the form (15). Here R~

= -e"l M, R~ = 0, R~ = - 2(e/ 2M)'v" R: = -21'-"''' R: =

o.

R~ = (e/ M) I~v,.

(18)

We write the scattering amplitude in the Breit system in the form (15), and obtain from a comparison of (15) and (5) the relations R,sin"O = E(oK,cos9 -\- oK,) I M -ko(oK,COSO R,sin"O = - E (oK, cos 0

+ oK,),

+ .... ,) I M + ko (oK, + oK, cos 9),

R.=kgoK ,/2M, R.= -kgoK ,/2M.

(I +

COSO)oK._~~

(2M

I

QI

< Qmax =

+ mw) (6M' + 9Mm. 4- 4m!) 4M (M + m.)' m!;::::: 3m~.

where m7r is the mass of the 7r meson. Regarding the forward scattering amplitude. one usually restricts oneself to two dispersion relations for the functions RI + R2 and ~ + R, + 2R5 + 2Rs. It is seen from formulas (20) that for () = 0 we actually have four dispersion relations for RI + ~. ~, R" and Rs + Rs separately. 2. The retarded causal amplitude for the scattering of photons can be written in the form a (P') N~~I u(P)= -2~'i(pop~/ M')'I. x ~d'ze-i"lml a. __ [

where ~, (p, u) is the wavefunction utilized in I; the notation D~(k) is also explained there. In the derivation of formulas (10) and (11) the following equation was used:

rf)' (k)-' D' (Lpn)D' (n)j",,, where () (p, n) is defined in Appendix B. We shall compare the expressions (8) and (9), obtained above with (10) and (11).* From the function fpmn which transforms according to the (p, m) representation we go over to the function 'P -p-mn, which transforms according to the ( - p, - m) representation. We make use of the fact that the representations (p, m) and (- p, - m) are equivalent. Therefore the function 'P-p,-m,n may be obtained from the function fpmn by the following unitary transformation:

)

~ U pm (n,

Upm (n,

k)fp"'k 1m (k),

k) 9_ p, -m,

ndD. (n),

4n j! piP

-I-ip,~

A pm

x[I-(nk)l-l-iP~Q",(n, k) 'Y(p, m),

""

~ dp ~ dD. (n) p-HiP!'

(23)

A.m

x[l-(nk)l-l "'Q",(n, k)9_ p ,_m,

(24)

Since I 1pm:,,I'

I ((J' -+(4r:)'01 ,

4m')

and Qm (n, k) = rim. (n. k) ( - l)+m

(25)

(cf. Appendi~ C), (23) and (24) differ from (10) and (11) only in that the integral over p in (23) and (24) is taken between the limits from - 00 to 00. In particular, for m = 0 each irreducible representation (p, 0) occurs twice in the expansion under consideration in contrast to the case M .. O.

(14)

APPENDIX A (15)

DEFINITION OF THE FUNCTION Upm(n, k)

where U,m(n, k) =

X

_ ..!.. \" d'p

n -

(13)

M-·O

f,mk =

Cfl_ p , -m,

'1' (p, m) =

___ .>o""_o(_I),+,,,/mo(p.o),

?_,. _"'. n =

We then obtain

r r

(I (I

~

+1 ~~ I

21

+ 1 + ip! 2) - I + 1 _ ip 12) D"

I

-m

(n) D,,,, (k).

(16)

·It is proved later that (10) and (11) are not equivalent to (8) and (9) (and are therefore incorrect). We emphasize that this by nO means indicates that the results of the present paper contradict those of I; it merely means that the formal manipulation which leads to (10) and (11) is not justified.

It may be easily shown that from the fundamental relations (14), (15) and the transformation law for fpmk and 'P-p,-m,n the following functional equation for Upm(n, k) may be obtained: ·In particular, for m = 0 we obtain the foUowing simple integral representation for the Il-function:

[ef. also formulas (A.4) and (A.6»).

79 99

INTEGRAL TRANSFORMATIONS OF THE I. S. SHAPIRO TYPE Upm

(5- 1

1

n, 5- k)

XI [I (l + 1)-1'(/' + I)-ip] + (_)I-/,+,X/,[I' (I' + I)

= Upm (o,k)[K(o)K(k)/K (S-' O)K(S-lk)]-I-ip.'\l X

exp{im['l'(S,nl+'I'(S,k)J)

- I (I + I)-ip] (21 + 1)/(2/' + I) = O.

(A.1)

(the notations K(D)/K(S-I D ) and cp (S, D) are defined in I). Since the functions D~m (k) for 1 = 1m I, 1m 1+ I, . .. and for fixed m form a complete ~ystem, U may be represented in the following form (A.2) On taking in formula (A.!) for S the pure rotation S = R we obtain 9n taking into account formula (l.9b), I,

Further, we take in (A.!) for S the infinitesimal pure Lorentz transformation L:

From this it follows that

+ I) r (l + I + ip /2) / r (I + I -

X I = C (21

ip /2).

On utilizing the unitarity condition (17) already mentioned in the main text we obtain. finally. formula (16). In order to obtain formula (20) we note that the function Qpm (0, k)

== [1- (nk)]HiPI·Upm.(o,

k)

(A.4)

satisfies the same functional equation (A.!) which is satisfied also by Upm(D. k). only we must set in it 1 +ip/2 = O. From this it follows that (I

«m

(

n, k) -- A pm

V

21

1 (I

"-'

+ 1)1 +

I -Do-m ( ) D' (k) n om •

1~lml

In order to find Ap. we set k = -D in (A.4). Cln = (n+ep)!!l + (oep)J.

Since Vi (n) = R.('!'+ ,,/2) R,(O),

It can then be easily seeD that

DI (-n)

K (Clo) / K (D) = 1 + (cpO). Since according to (l.9d). I

then

/m.IL. k)D,;m (L -Ik) = ~ D~y (R (L, k» D~m (k)

we obtain from (A.l) and (A.2)

}J X ;15~_m (n) D!m (k)

=

[K (n) K (k) / K (L -I n) K (L -I k)]'~ ip/'

x}J XIl5~y(R (L, oJ) 15~_m (n) D~~ (R (L, kJ) D~m (k). (A.3) The parameter of the rotation R occurring in the above is defined by formula (A.4). I: DI (R(L, k»

=

O-iIH.)."" l-i(H~),

~

= R.('!' -I 3rr/2)R I (rr-&),

=

[kx:;;].

[DI (n)-'D I (-o)J-m.m = [RI(-&)R.(!t)R,(rr-6)]~m.m im imn = [R, (- rr) R, (")]~m. on = e- " (- 1)'1 (_ l(t-m e-

Since ~

_~_~ (_ 1)I+m

1;:01",11(1+ 1)

=

~ !'.,l.lml

(_1_ (_I)I+m+~(_I)I+m) = l+1

(_1)2on~

1

~ (- 1)1+"1 (21 -i- I) r (l + I + ip I 2) /

r (I + I - i:-

We then obtain from formula (A.3)

}JX;15~._m (n)D!m(k)

= [1- (l

x}J XI {[I - i (HI [n

x

x

+ ip/ 2) ep (n + k)J

'I'll] D' (o)} •. _m

([I-i(HI[k x'I'J)JD I (k)) •.

m.

Here we must express the cyclic components of the vectors D and k in terms of the generalized spherical harmonics OhO(D). Oho(k). and we must then eliminate products of the D -functions in accordance with the following rule Db. (n) D;d (n)

= (I lac I LM)

1 affects the characteristics of the elastic scattering, including also effects for ,,< 1 (deviation from the Powell formula, or from Eq. (1.16) for y < 1). The deviation from monotonic variation in Eqs. (24) and (25) is characterized by a sharp drop from the value of the function at 1 in the region ,,< 1 (with an infinite derivative at 1) and a slow drop in the region " > 1 (with a finite derivative at 1). 5. In the range of energies 330 - 500 Mev (2.2 < " < 3.34) the quantity I EI12 is represented in the form

,,=

(v;-I)'I·_(v2 _1)'I••

1t

and

.

, (.... -1)'/. I(V~-Ij'/'+(V'-I)'I" tan-l(v2-t)/r _ _ _ _ ln

v>1

_ __

tan- l

{v'

(v' _ 1)

(25)

viv' = I

+ 2~

v'

": v' [}(v: - I )'/'+ (v: _ I )'/, (v' -

(1 - v')'h

and

'I

- , (v'--I)/'ln

tan-'

l

(V;-I('+(V'-I)'I" (v:-I)"'-(v'-I)'I, (v; - 1) i (I - v')

I)]

. •

v

needs only to study the cross sections Io (II, II) at II = 45', 90', and 135 with sufficient accuracy to find the energy dependence of the difference 0

/.(45°) +- lu(135°) -/0 (90°). It is interesting to note the energy dependence of the polarization of the recoil nucleon. Below the meson-production threshold the imaginary parts of the quantities R t , .•• , Rs vanish in the e 2 approximation, the right member of Eq. (13) is zero, and there is no polarization of the recoil nucleon. Below threshold, in virtue of invariance under time reversal, the cross section for scatter-

87 SCATTERING OF GAMMA-RAY QUANTA BY NUCLEONS quantities are quite appreciable, but rather severe requirements are imposed on the procedures for experimental studies, expecially as regards resolution in energy, since the widths of the dips in question are of the order of 5 to 10 Mev. The treatment given in the present paper shows that the effects near threshold are sometimes masked by the strong energy dependence of the scattering amplitudes. Therefore it seems that the most favorable conditions for the experimental study of such effects should be found at small energies, and also for the interaction of particles with small spins. In the case of y-N scattering, besides the contribution of the "peak" amplitudes R t and R 3, there are large effects from other amplitudes, particularly from R •. The effects of these" smearingout" factors may be smaller in the scattering of y rays by helium nuclei (cr other spinless nuclei), since in this case the transition matrix will have the form AI =

R; (ee') -+ R; (ss').

A treatment of the scattering of y -ray quanta by deuterons near the threshold for the photodisintegration of the deuteron, where local effects will evidently be large, will be presented in another paper. From the point of view of the general effect of some processes on others it is interesting to analyze the photodisintegration of the deuteron in the energy range near and below the threshold for meson production. Noting the results of the calculations on the y-N scattering, we can evidently suppose that the well known" resonance" energy dependence of the cross section for the photodisintegration of the deuteron is due to meson-production processes above threshold and can be treated by a method using dispersion relations. It is commonly assumed that at quite high y -ray energies the y-N scattering cross sections will be almost entirely due to inelastic processes, Le.,

15

to the imaginary parts of the amplitudes. In this connection it may be very interesting to study y-N scattering, and especially the polarization of the recoil nucleons, near the thresholds of reactions of the production of new particles, such a, "(

:~

N

->

Y -'-- K,

and a number of other processes. In this case the difficulties associated with the size of the cross section and the low energy of the recoil nucleon may very probably be smaller. The writers are deeply grateful to B. Pontecorvo and Ya. Smorodinskil for helpful discussion 1 E. Wigner, Phys. Rev. 73, 1002 (1948). A.1. Baz', JETP 33,923 (1957), Soviet Phys. JETP 6, 709 (1958). G. Breit, Phys. Rev. 107, 1612 (1957) A. 1. Baz' and L. B. Okun', JETP 35, 757 (1958), Soviet Phys. JETP 8, 526' (1958). R. K. Adair, Phys. Rev. 111, 632 (1958). 2 L. 1. Lapidus and Chou Kuang-Chao, JETP 37, 1714 (1959), Soviet Phys. JETP 10, 1213 (1960 3H. A. Tolhoek, Revs. Modern Phys. 28, 277 (1956). • L. 1. Lapidus, JETP 34, 922 (1958), Soviet Phys. JETP 7, 6S8 (1958). 5 L. Wolfenstein and J. Ashkin, Phys. Rev. 85, 947 (1952). L. Wolfenstein, Ann. Rev. Nuclear Sci. 6, 43 (1956).

6Watson, Keck, Tollestrup. and Walker. Phys. Rev. 101, 1159 (1956). 7 E. Fermi. Supp!. Nuovo cimento 2, 17 (1955). (Russian Trans!.. IlL. 1956). 8 M. Cini and A. Stroffolini, Nuclear Phys. 5, 684 (1958). 9

T. Akiba and J. Sato, Progr. Theoret. Phys.

19. 93 (1958). G. Chew, Proc. Annual Internat. Conf. on High Energy Physics at CERN. 1958, p.93.

Translated by W. H. Furry 33

88 477

LETTERS TO THE EDITOR

ON THE PRODUCTION OF AN ELECTRONPOSITRON PAIR BY A NEUTRINO IN THE FIELD OF A NUCLEUS A. M. BADALYAN and CHOU KUANG-CHAO

could be expected that the cross section for this process would be smaller than that for scattering, since it contains the factor (Ze 2 )2, and the phase volume gives an additional numerical factor (211')-2. On the other hand, the phase volume is proportional to since there are three particles in the final state. This process is described by two second-order diagrams. The calculation of the contributions of the two diagrams to the cross section leads to extremely cumbersome formulas. We shall, however, get the right order of magnitude for the total cross section if we confine ourselves to the contribution of one diagram. The differential cross section for the process then has the form

w1,

Submitted to JETP editor November 26, 1959 J. Exptl. Theoret. Phys. (U.S.S.R.) 38, 664-665 (February, 1960) PRESENT experimental possibilities have allowed a rather close approach to a measurement of the cross section for scattering of a neutrino by an electron. t This process is a very important one for testing the theory of the universal weak interaction. In the laboratory system, in which the electron is at rest, and for incident neutrino energy Wt »m, the cross section for scattering of a neutrino by an electron is (1)

i.e., a linear function of Wt. There is another process, II + Z - II + Z + e+ + e-, for which the laboratory system coincides with the center-of-mass system. On one hand, it

d~ =

16g' (le')'

2

w}w:!e-;-t_

dp_dp_Jk, (k,k.) q,,0+ 1(0) =

f (1t+ + 1(0 _itO

+ 1(+) = f (It- + 1(0 ---> ,,0 +

1(-)

= o.

3) The K + K - n1l' annihilation process proceeds only through the isoscalar state. To obtain experimental verification of these selection rules, one can study the angular distribution of the products in the reaction K + N - K + N + 11', for which the one-meson term in the cross section is proportional to

A"(AI+I'-")-llf(:+ 1(--->: + 1()I",.

POSSIBLE SYMMETRY PROPERTIES FOR THE 71-K SYSTEM

(3)

where A' is the square of the nucleon momentum transfer. Expression (3) has a maximum for At =".' in the physical region. Z A measurement of CHOU KUANG CHAO the form of this maximum would provide information on the amplitudes f ("Jr + K -11' + K). Joint Institute for Nuclear Research According to the theory of Okun' and PomeranSubmitted to JETP editor January 27, 1960 chuk3 and Chew and Mandelstam' the scattering phase shifts in high angular momentum states are J. Exptl. Theoret. Phys. (U.S.S.R.) 38, 1015-1016 determined by diagrams with the smallest number (March, 1960) of exchanged 11' mesons. If the K+ and KO have the same parity then the K + N - K + N scatterTHE Hamiltonian describing the 71'-K system has ing phase shifts in high angular momentum states the form are determined by diagrams with two mesons ex(1) changed. Consequently a phase shift analysis of the process K + N - K + N would give certain where H1I' is the pion Hamiltonian including the 11'71' information about the amplitudes f (11' + K - 11' + K). Interaction, HK is the K -meson Hamiltonian, and A violation of these selection rules would imply g is the coupling constant of the 71'7I'KK int-eraction.! that the Hamiltonian contains te~ms with derivaIt is assumed in (1) that the 71'-meson and K-meson tives of the form interactions with baryons can be neglectl'!\.+p+W

(6)

(7)

(8)

If the K meson is a pseudoscalar particle, the

spin structure of the T matrix has the form (ApKITlpp)=A.da,+a,. k)

+ B,\ {(a + CA {(al l -

a,. k)

+ i ([a,a,J k)}

a,. k) - i ([a 1a,J k»).

(10)

where T = q2/2J.1 is the kinetic energy of the K meson with respect to the center of mass of the A-N system. If the protons in the initial state are polarized (with polarization vector p). the polarization vector of the A particle in the final state, P A'. will be

- i CAI'J (kP) k + [i AA - C,\!'

-1 A.\ + C"I'J P.

(11)

The expression for the polarization of the nucleon in the final state differs from (11) by the sign in front of C A-

Let us look at the unitarity condition (ApKIT-r+lpp)

The admissible energy for the final state of reaction (6) in the c.m.s. does not exceed 80 Mev, so that we may assume that the particles which are formed are in an s state. Let us represent the S matrix element in the form (t\pK' I 5 I pp) = - 2nio (E, - E,) (/l.pK' IT i pp).

A\ - C.\I' + 21 C.\I'].

3. ELASTIC FINAL-STATE INTERACTION

below the threshold of the reaction p+p->EO+p+W.

[I AA + CAl' +!

P" [1 AA+ CAI'+ I AA-CAI' + 2ICAI'J = 2 [i AA+ CAi' (4)

The phase volume of the final state is expressed in terms of Py and q as follows: dJ =

x (2mA[L)'/' [T (T max - T)J'/'

= Zrri

L; (.\pK ITin) (n I r+ I pp) 0 (Ei

- En).

(12)

where 1n> is a possible intermediate state lymg on the same energy surface as the initial state. Let us assume that in the region of energy considered the imaginary part of the T matrix is related mainly to strong interaction in the A- p system. Then we may neglect on the right side of (12) all intermediate states except for ApK states, and approximately replace <ApKI T I A'p'K' > by <ApITIA'p'> , which are small for this reaction, but are necessary in other cases, complicates the expressions but

does not change the fundamental result.

103 INELASTIC FINAL-STATE INTERACTIONS where (0'" rr/2)

l, = A~ (pt.';4Tt) O;·A (PE = 0)

tan2

0a

+ A~ bal COS

2

C~ = C~ (pt.j4Tt) o~·" (P E = U) tan2 01 +C~ bl l cos 2

0,. 01'

(25)

The relation (24) is valid when the kinetic energy T of the K meson is less than E'. For T > E' the production of a real l; particle becomes impossible, and we must replace Pl; by ikl;' where kl; =,j 2ml;(T - E'), T > E', .so that the term which depends linearly on kl; appears in the real part of the reaction amplitude. The presence of terms proportional to Pl; (T < E') and kl;( T > E') causes the derivative with respect to the energy to become infinite both in the energy spectrum of the K mesons and in the energy dependence of the polarization of 11. particles (and nucleons). The order of magnitude of these anomalies is given by (24) and ,(25), and their shape depends on the relative sign of A~, A~, b 3,t and O. All four cases of anomalies which have been discussed in the literature for binary reactions can also occur in this present case. All of the expressions in Secs. 2, 3, and 4 were gi ven for the production of particles in pp collisions. It is not difficult to generalize them to the case of np collisions. This is done in the Appendix. We also discuss there the case of a scalar K particle. We note that, in the general case also, the quantities which replace AA and CA have terms which are directly related to the final-state interaction, as well as terms which are not caused by it. We emphasize that the expressions obtained in the present section refer to interaction in an s state of the final system. The relatively large mass difference of the 11. and l; hyperons makes it difficult to apply the theory of inelastic interaction to the analysis of reaction (1), but this does not change the basic assertion that there is a non-monotonic behavior in the spectrum and the causes for its occurrence. It was shown earlier 9 that the direct analytic continuation Pl; - ikl; can not be carried out when there is a resonance in the neighborhood of the threshold. In this case, it is necessary to make use of dispersion relations. Since the analytic behavior of the reaction amplitude as a function of w is not known, we have not carried out such an analysis. However, even if such a resonance occurs, we may expect non-monotonic variation with energy for a relative energy of the

261

11.- N pair equal to the threshold for the new channel. If l; and 11. have opposite parities, the first term of the expansion in (22) starts with P~ and only the second derivative with respect to the energy becomes infinite. Consequently, the study of threshold anomalies in the energy spectrum of K mesons with sufficiently high accuracy may prove important for determining the relative parity of the l; and 11. particles. 5. DISCUSSION

Thus, endothermic inelastic interactions of the type C + D - E + F in the final state of the reaction A + B - a + C + D can give rise to nonmonotonic variations with energy in the spectrum of the particles a, whose form can be determined from the condition of analyticity and unitarity of the S matrix. To investigate these singularities experimentally requires, of course, good accuracy and high energy resolution, but as a result of discovering them and studying them one can obtain information concerning the interaction of unstable particles, their spins and parities. Earlier we have treated the production of hyperons and K mesons in NN collisions. We mention various other processes in which similar anomalies can occur whose study may give information concerning the interaction of unstable particles. In the spectrum of mesons from the reaction

,,+

(26)

in the neighborhood of the threshold for (27) there will occur an anomaly whose magnitude and character will be related to ~p scattering at low energies via the reaction amplitude (27). In the spectrum of protons from the process for production of 7l" mesons by K mesons (28)

an anomaly may occur for an energy corresponding to the threshold for the reaction (29)

if there exist forces leading to such a reaction. If one attempts to construct a Lagrangian for the 1f K interaction and does not consider interactions containing derivatives, the expression obtained

104 262

L. 1. LAPIDUS and CHOU KUANG-CHAO Lint

= g('P'.. ''P~) ('P~'

.'l.pK") = do (tll' -. AnK'),

forPllk.

(A. 7)

(A.8)

These relations are obtained on the assumption

mesons from the re-

p~d

iklcr l "cr,ll. (A.3)

-~- 1't'+

p+p-n+p+~

near the threshold for n +-

cr"k)

P.dnp->.\pK")-~Pdnp--+,\nK")

near the threshold for ITO

Hd(cr ,

Under the assumptions made earlier we can take account of final-state interaction by setting

:< II A,\ + c"":

in the neighborhood of the threshold for the reaction

(A.2)

tl '.1\".

tl+I'->.\

-:-1t

0

~~

*The scattering lengths for low energies of the 1T o.p system differ from those obtained on the assumption of isotopic invariance because of the presence of non~monotonicities which violate isotopic invariance and are related to the reaction An estimate using dispersion relations gives a correction - 5%.

105 INELASTIC FINAL-STATE INTERACTIONS that one need only consider the s wave in the final state. They can be used for an experimental check of this assumption. B. PRODUCTION OF A SCALAR K MESON IN NN COLLISIONS In this case I.\NK

AA.

'.\N K I Tol N N)

A.\~' .4~\ (p., a,r' e'" sin 0"

=

B, = HO\ (p, a 3

B" (. Let us examine the matrix element of the commutator 1 = (012G.N,,+ i +(M o - M)~T.NIN).

(N I P. (0) I Y> = UN {gAyT.T.

From symmetry properties we have 1,= iA-.r.UN.

i3Au N = -2i(01'1(0)IN),

where

+ (M -

+ i~y [(P N -

py)

x T. -T. (.oN - py)] T. + ify (p y - PN).T.) U y, (31)

Multiplying Eq. (31) on the left by the matrix TY$, we get

'1 (0) = iG. (~,,) T.N

opInion that the universality of the weak Interactions evidently does not extend to strange-particle decays. Nevertheless, It is reasonable to assume the existence of a limited universality [a lepton current in the form (2)8]. In what follows we assume that the K meson is pseudo scalar and the V and A Interactions exist for the lepton decays of strange particles. In this case the Hamiltonian for the weak decays of strange particles is of the form (1). FollOwing the example given in Sec. 2 for the pseudoscalar theory with pseudoscalar couplIng, we can construct the pseudovector current In such a form that a dispersion relation without subtraction holds for the matrix element ' Generally speakIng, the matrix element for hyperon decay consists of three terms:

M.) N

is the current of the nuC'l.eon field. It Is known that the matrix element < 0 111 (0) 1N> Is equal to zero, and therefore 1=0. Thus we have shown that In the ordinary pseudoscalar theory there exists the pseudovector current (29), which satisfies all the necessary requirements. l! the pseudovector current is of the ordinary form

then the matrix element of the commutator is not zero, and In general there is no dispersion relation without subtraction. Even In this case there is hope that the G.T. result is valid. This question will be discussed In the Appendix.

4. LEPTON DECAYS OF HYPERONS AND K MESONS The experimental limit for the probabilities of lepton decays of A and E hyperons is an order of magnitude smaller than the theoretical value calculated on the hypothesis that the effective coupling constants In hyperon decays are equal to those In

(32)

from which we have (N I 0(0) I Y) = i (N

= i

[(MN

liV.IY>

+ My) gAY + fy5] u-NT.Uy,

(33)

where s = - (py - PN)2. Repeating one after another the arguments presented In Secs. 2 and 3, we easily get the following equation: [(MN

+ My) gAY + fy5] (34)

where GKY is the renormalized coupling constant for the KYN Interaction, and FK is a constant parameter associated with the decay of K mesons. We have further =-q.F K /V2q •.

(35)

We can determine FK from data on the lifetime for the decay K-" + II. In Eq.'(34) Ty(s) is a function that Is analytic in the region

I s I < (mK + 2m)'.

(36)

Let us denote by TN the kInetic e!lergy of the nucleon recoil in the rest system of the hyperon. Expressing s in terms of TN, we get (37) In the present case the values of s that correspond to fJ and " decays are very close together, as compared with the distance between the s given by Eq. (37) and s = (mK + 2m)2, Therefore with good accuracy we can replace Ty (s) by a constant ay.

110 496

CHOU KUANG-CHAO

Thus we have [(MN

Substituting (46) in (45), we get

+ My)gAY + tysl =

-

GKyFKmk / (-s

+ mk) + ay.

(38) The relation (38) can be used to test the universality of the pseudovector current in lepton decays of strange particles. Applying the dispersion theory of Goldberger and Treiman, we find for the function fy: fy = - GKYFK j(- S

+ mk) + T~ (s),

(39)

where Ty (s) is a function analytic in the region (36), which with good accuracy can be replaced by a constant aY. Substituting Eq. (39) in Eq. (38), we get (MN+My)gAy=-OKyFK+ay-Say.

(40)

The relation (40) is a generalization of the formula of Goldberger and Treiman for the decay of straqge particles. The experimental data on the llfetimes of K and IT mesons show that Fie« F~. Therefore It can be seen from a comparison of Eqs. (40) and (16) that to accuracy ay- sa'y g~y Is

Il;y[(;'"

of the

Comparing (47) and (16), one sees that to accuracy ay-say

(~)'=(~ gA~ MN-My

FKGKY)' =5C(G Ky )'

oft

f1f,G1f,

'

where C is of the order of unity. Therefore in the case of the scalar K meson the small probability of lepton decay of hyperons could be explained only by having the coupling constant GKY for the KYN interaction be smaller than the pion-nucleon constant G lT • We note that A and ~ can have different relative parities. Let us consider this case. For simplicity we call the K particle a scalar, if the relative parity of K and A N is positive, and a pseudoscalar if it is negative. In the case of the pseudoscalar K meson, Eq. (40) holds for the decay of A particles, and Eq. (47) holds for the decay of ~ particles, if we write in the form (43). In the case of the scalar K meson, conversely, Eq. (47) holds for the decay of A particles and Eq. (40) for ~ particles, if we write in the form (32). We note that the relations (38) and (45) can be used for the determination of the renormalized coupling constants GKY, if precise experiments are made on the decays of strange particles. The writer expresses his hearty gratitude to Professor M. A. Markov, Ya. A. Smorodinskir, and Chu Hung- Yliang, and also to Ho Tso-Hsiu and V. I Ogievetskil for their interest in this work and a discussion of the results.

1',.).I"y'

.lI y ) I!". I '[,'"'1 ""lIy'

APPENDIX In the usual theory the pseudovector current has the form

,;,:) ••

(43)

From this we have

= il(M N

(47)

(42)

iaaVa = () (x).

The matrix element

= a - ia (S[pxq])/2M' + aq'/2M'

+ bp' + h (Sp)(Sp) + e (pq) /2M'

+ if' (S [pxqj) /M.

From Eq. (43) it follows that

p,] (38)

*ThIs Is the most general expression for an arbitrary S, if we are Dot concerned with terms with energy dependence higher

than linear,

PlO=Eq=Vq'+M',

Y"

The summation in Eqs. (36) and (37) is taken over the spins of the particles involved in the ;reaction. Let us consider the case in which the states I q > and so on are eigenstates of a system with spin S. For the calculation of Eqs. (36) and (37) we need the expression for the current matrix < P21 j I PI> in the low-energy region to accuracy vic, and for < q' I jo I q> to accuracy v 2/c 2• It turns out that these matrix elements can be determined with the required accuracy on the basis of general principles. Since j and io are Hermitian operators and the interaction is invariant under three-dimensional rotations and time reversal, the most general form of the matrix element of the current is, in the approximation in question, (p,1 j I p,) = (e/2M) (p,

p,=q,

p, = p +

'

(q-q'Ji/J-q')' 100 Mev, the dominant contribution is that of photodisintegration with I' :>, 75 Mev. For the other amplitudes a more detailed analysis of the photodisintegration 2 S = i(2n)W/ (2n)'6 14)(q. (z,.) , r 2",.

The S-matrix element of the pole diagram is

x (q'ql/

-q-q')

(0) 10),

' (- i =

A~

Q) = (P'P) = P".

(P'p') = (P', P -

(A.I0)

=

-

o

f'(~~A) 6(v-~)

e' (I

+),) •

~u

A. =

(v- Q 1M) u(p) YsK.u (p).

+2P" M

KP"}

Ii (P')i>' 1-i(P-KH Ml u(p) = U (p) - i (P2 + (PK.)) + i (,(Pi O.

Substituting (14) in (12) we get

1',0 =

. (Koo~ 00) .

K

T;I( = (I -

+

where

then we can write

P'

+

:t 2ph21,

n - :t2~'rp1/[ 1 :t 2ph 2]

=

b = ,,~2pI:![ 1

(10)

= (I - in')-'

933

PROCESS

rA'\

lEA)

r,\~

hI:

~

,

=

(19)

(~AK' ~"K)'

It is easy to verify that in this case X is simply a complex number

X

=

a'

-+-

(20)

ib"

where a

,

= n -

R 'I

nppy' 1

+1r"

_, '/.p,T I

b'

PY p ,

I'

= ,,~py' 1

I

+ 'r"

1/ T

Py'~

(21)

From (12). (13), and (19)-'(21) it follows that IKK

="P K (a'

+ ib') t.~', T~K = n':'p~' (b~K)'I'e'>'AKt.~', ,,'Vi< (bh)'I'e"I:Ki\~', (22)

T~I( =

where n'V/li/:'l(e"AK,= (A I (I n'I'p'flitK"'r:.K == (E I (1 1'" = 1 -

inpK (a '

iy')_'~'T IK), iy,)-,~'TI K),

+ ib'),

(23)

and the quantities bAK and bEK are related with b by the equation bAK'+ bEK = b. If we represent the matrices ~ and 1) in the form

(14)

£ = (~AI(, £r:.I() ,

1]

=

'lAA ( 'lAE

'1~A),

'1';1;

(24)

132 934

L. 1. LAPIDUS and CHOU KUANG-CHAO

then the matrix elements TYAK and Tyl:K become

T~h.K = "p~}.p'kt..~' [SAK + i'lAh. ,,'V~b~'Ke"AK (25)

(26) To simplify matters we introduce new symbols

+ i'l~'J:,,"'p~'(b'('ei1::J, + iT)~En','Jp~ (bO)'/:en.!:), =tl,'~p~'~ [~~K + iT]~A:tl::p~ (b~J()'/~/AhK

a~, = rr.'/'p~~ [~~K cx~ = :tl/lp~'E [~~}( :J.~ =

photoproduction of mesons and hyperons. In the present paper we confine ourselves to a generalization of the Kroll-Ruderman theorem for photoproduction of pions near threshold. [,] Let us assume that the A and l: hyperons have a positive relative parity and that the K meson is pseudoscalar. If the created particles have low energies account of the electric dipole radiation is suffiCient. The generalized Kroll-Ruderman theorem states that, accurate to m7l' /M "" 15%, the matrix for the electric dipole transition is determined completely by the pion-hyperon coupling constant. Let us write the Hamiltonian of the pion-hyperon interaction in the form

-T i'l~~ ,,';'P~' (bh )'/'e"'EK J.

xb =

:tl/~p~~ (E~l(

+ ifJ~h. :t1."p~1 (b~J()"I/),/\.I\

J{ =

+ i'li,,!t'."p~· (b~K)'/·eo"KJ. a i = !t'/'P~i:!~iK + i'l'~f>.n"·p~(b~K)'VA/I.K

'l~,,-

(27)

with which the cross sections of the processes (1) can be written in the following form:

1(-

+p

-

AO

+. I

1 2:0 + • \ 1 2:0 + y J K- + n --> 2:- + Y)

K,0 + p --> 2:+ + Y

Cross section: 2.-rm K / EKk

z:ttnx

Exk

C1~

I'

~±"K; .

1- ~~

Ex k I Z1ttnx

a~

I

V3 +:.l:

~o

-

I'

+ 2~'

(11)

(T, - T,Y.

F.(vo) = (T,

M) (7//41"1

Re(R,

=

(10)

Consequently, nonsubtracted dispersion relations for the amplitude Rj + R2 violate the requirements of relativistic and gauge invariance on which the long wavelength limit is based. Let us remark that possible sum rules involving the square of the magnetic moment are not in direct contradiction with the long wavelength limit when nonsubtracted dispersion relations are assumed for F 2 (v). As can be seen from Eqs. (6) and (8), of particular importance here is the contribution of the resonant state, proportional to I M312. The result is unchanged if one takes into account the (numerically important) contribution from photoproduction in S states, which decreases the effective contribution of I Ms12. The sum rule for the square of the magnetic moment is very sensitive to the ratio of the photoproduction amplitudes E z and Mil.' For certain ratios (for example for E2 = M,[5J) one can arrive at a contradiction. At the present time. however. the analysis of photoproduction is not sufficiently precise to permit the assertion that the experimental data are in contradiction with the sum rule. An increase in the accuracy of the photoproduction analysis. aimed at obtaining information about the amplitudes E 2• Mz and Es. would be most welcome. The fact that unsubtracted dispersion relations give rise to definite sum rules may be of particular interest in certain processes. Thus. in the case of 11"11" scattering analogous considerations (applied to dispersion relations at Q2 = 0) lead to the conclusion that the S-state scattering lengths ao and a2 are positive at low energies. The same holds for 1I"K and KK scattering. 5. If in addition to the functions introduced

(13)

one concludes that F 5,6 (v) are odd functions of v and contain no poles, whereas F7 (v) is an even function of v with a second order pole. As v - 00 F5.6.'~V-'/.,

so that the dispersion relations for these functions need no subtractions. These dispersion relations may turn out to be useful since when photoproduction in states with J ::s % is taken into account the angular dependence of the amplitudes Ri (v. QZ ) in the barycentric frame takes the form (cf.W )

+ 21£, cos 9 +fm. -+- c (1S,m.). R, = mi - m, + 2m, cos 9 + +;e, -:- C(m.IS,), R, = i€, -IS.

R. = - 1£, -

R.

C (m.If.),

=~ -

m. - C (If.m,),

(14)

and is characterized by eight functions of energy ll",a, ml,l,a, C (1S.m.) C (m.IB.). which can be ex~ pressed in terms of Ri (v. 0) and Ri (v, 0). It follows from Eq. (14) that if we restrict ourselves to contributions from states with J::S %

R', = R:,

= 21S,

(il cos a/ilQ')Q'~= -

41S~/M'v~,

so that (R.

+ R,Y

= (R,

+ R,),

=

IR. +

R,

+ 2 (R. -+-

R.)I'.(15)

In the long wavelength limit[t4J

(R.

+ R.Y

= -

2e'/M'v

+0

(I),

+ R,)' = - e' 13 + 2 (I + A)'1/2M' + 0 (v), (R. + R, + m. + ml)' (R.

= - e' (2A' - 2A -

1)/2M'

+0

(v).

(16)

The fact that Eq. (15) is in contradiction with the long wavelength limit (16) means that the restriction to states with J::S % is not a good approximation even in the low energy region. The crossing symmetry conditions introduce kinematic corrections of the order of viM. which corresponds to inclusion of states with higher values of J. The carrying out of the analysis with this high a precision requires the introduction of additional functions of energy and disCUSSion of a larger number *The prime denotes differentiation with respect to Q' and subsequent passage to Q' ~ O.

136 SCATTERING OF PHOTONS BY NUCLEONS of dispersion relations. Introduction of the Low diagram does not resolve the indicated contradiction. All estimates of the amplitudes given here were obtained with the neglect of Rj (", 0). 6. The results of the calculations of the amplitudes Ri ("0) at Q2 = 0 are shown in the figures. The energy of the photons "0 is given in units of the threshold energy "t = 150 Mev, and the values of the amplitudes in units of eo/Mc 2. For the calculation of the forward differential scattering cross section ~ (0') ~

i R. -r- R, \' + i R, -j R, + 2R. + 2R. \'

the amplitudes Rj + R2 and R3 + R( + 2R5 + 2~ are sufficient. To estimate D j ("0) use was made of the data on the total cross section for the interaction of photons with protons, including the second maximum and the cross section for pion pair production. The dependence of Aj ("0) is shown in Fig. 1. Previously we have neglected contributions from the energy region above 500 Mev. The result of estimating the amplitude Rj + R2 is shown in Fig. 2. The main difference between this and previous results appeared in the region 1 < "0 < 2, where as a consequence of a cancellation between the long wavelength limit and dispersion terms the value of Dj ("0) is significantly decreased. Let us note that this is precisely the energy region that is sensitive to a change in Aj (vo)' The second maximum in Aj ("0) corresponds to the second maximum in photoproduction.

1105

For estimating real parts of the amplitudes, other than Rj + R2, which require much more detailed experimental data on photoproduction, we limit ourselves to the energy region up to 300 Mev. For the amplitude R j + R2 it turns out to be possible t6 go much further, although with increaSing energy the indeterminacy in the contribution from photoproduction of pairs (and larger numbers) of pions becomes appreciable. In a number of papers[j5,16] the yp scattering at 300-800 Mev has been looked upon as a diffraction process with Re Ri « 1m Ri' The experimental study of yp scattering in the region of the second resonance is of interest as a sensitive method of investigation of the maximum itself. If, ignoring all Re R i , we restrict ourselves to the imaginary parts of the amplitudes alone and consider only the contribution proportional to I E31 2, then we find immediately from Eq. (7) that

R, ' - R.

=

R.

=

R.

=

0,

R, ,lmR, =-:2lmR,=2",!E 31 ', whereas the differential cross section[6] is equal to 6

(q)

= -:- R; (7 + 3 cos'S) =

+R; (7 +

3 cos'S),

(17)

in agreement with the results of Minami. [t6] The same result for the form of the angular distribution remains valid if in Eq. (7) only M3 (Rj - R 2, R3 - R() is different from zero. If simultaneously E3 and M3 (with Re Ri = 0) are different from zero then we have u (0)

'0'

-;,

(R:

+ R;) (7 + 3 cos' 0) + 10 R,R. cos O.

(18)

However, as our estimates indicate, the quantities Re (Rj + R2 ) are large in the region of the second resonance and cannot be ignored. From this point of view the second resonance differs drastically from the resonance, in whose energy region

%, %

I

;

J

5

"

b

7

Re (R. -:- R,) --:-~ 1m (R 1

The results of the calculations for R3 ± ~, R3 + R( + 2R5 + 2~ and ~ + ~ are shown in Figs. 2-4. In the evaluation of dispersion inte-

FIG. 1

5Jl

.J{/

'er,'''')

"l"IR,."'.2. . ,.\ . \ -2

e'lHe'

./

~

"/H,'

-J

FIG. 2

+ R,).

FIG. 3

137 1106

L. I. LAPIDUS and CHOU KUANG-CHAO dispersion relations (6) are not sufficient. Let us consider the function F(v) =w-'(v)dv

~ v(v'-V~)

,

(27)

'/

where, according to B, Eq. (2), 1jl (0) = - e' (2

1m1\'

and e' (

v~

K(vo) = "

2v ) M'

+ K(vo) +

4v: Re F (M/2) M(v.+M/2) ; (24)

(25)

'/

Since 0,

=

Re F (M/2) cannot be determined from Eq. (24), and this quantity enters as a free parameter, which must be determined starting from the experimental data. Under the restriction to photoproduction in the states with J:S % only we get 1m F (v)

=

-IM,j')

y {~(lEII'-1 MIl')

+ 2(IE.I'-1 Mal') (I + } w-: M)

+ i-~(IE.I·-IM'I')( 1 + i- w-: M)}.

The results of estimating Re (R5

-

Be) at

(29) ~

= 0 for Re F ( M/2) = 0 are shown in Fig. 4. Es-

7J 1m F (v) [t M' + 2Mv t ]dv v-v. + M'- 2Mvv+ v. VI' K (M/2)

(wIM)'{w[-i"(IEol'

(28)

+ Re (E;M. - M;Es)] + My (M + ~tl [3 (lE.j" -IM,jO) + +(IE,I' -IMol") + Re(E;M.-M;Ea))}.

FIG. 6

Re F (vo) = -M 1-

(v) =

+ A.)/2M.

(25')

In Fig. 7 are shown the results of estimating Re (Rt - ~) with the help of Eq. (24) when the contribution proportional to Re F ( M/2) is ignored.

timates of the quantities Rs ± Be and Rs - Re, which playa dominant role in the differential cross section for Vo ;::. I, do not differ appreciably from those obtained previously.[8] The results here obtained are of interest from the point of view of the study of the energy dependence of amplitudes near the threshold of a new reaction. [8] In that case all estimates can be carried out to the end. Let us call attention to the dependence of the amplitude Re (Rt + R2)' whose value continues to fall off also above threshold. This result indicates that a sharp energy dependence of the imaginary .parts of the amplitudes above threshold may also for other processes lead to a displacement of the near-threshold minimum (or maximum) of the cross section relative to the reaction threshold. In Figs. 5 and 8-11 are shown the results of the calculations, with the help of Ri (v, 0), of angular distributions

1=0

for the angles 8 = 90, 135, 139 and 180·, and also of the total elastic scattering cross section a,/41f.

FIG. 7

For an estimate of Rs - Re at Q2 = 0, as can be seen from B, Eq. (4), it is sufficient to consider the function 1jl (vo) =

x

-i- v~ [T~ + ~ (T I + T.)l' = (~)"

{~(R. - R.) + w.~ M [RI

-

R. - (R, - R.)l} , (26)

for which the dispersion relation has the form

=

Bo + B./2

and of the polarization of recoil nucleons for 8 = 90·. The experimental data are summarized in[to] and[U]. The coefficient

B. ('\10)=2 [ I R. + R.IO_j R. - R.I"] is near to zero in the entire energy region Vo ~ 2. The experimental data, apparently, indicate that the quantity Re (Rs - Be) is positive. We were not able to achieve this by introducing Re F ( M/2) .. O. The requirement that Re (Be -He) be positive leads to large (negative) values for Re F ( M/2), which at the same time Significantly

139 L. I. LAPIDUS and CHOU KUANG-CHAO

1108

the dispersional analysis and experimental data is obtained. In the region 1 < 110 < 1.3. which is particularly sensitive to dispersion effects. it is apparently necessary to take into account contributions from higher states. for which it is necessary to have information on pion photoproduction in a larger energy region. 1 Gell-Mann. Goldberger. and Thirring. Phys. Rev. 95. 1612 (1954). M. L. Goldberger. Phys. Rev.

99. 979 (1955). FIG. 8. Energy dependence of the coefficients in the angular distribution. The experimental points are from ['.".17]'

~ (el/Mcz;Z

6

]I( eZ

a

I

I

(1958).

«R. H. Capps. Phys. Rev. 106. 1031 (1957); 108.

fMc')'

1032 (1957).

b

;f

H

I

·f Jf

J

2 I

'~!4-dJ o

---"L

o.s

1.0

I.S

\

1,J) .~

t.P 0

as

I

(5

1.0

FIG. 9. Energy dependence of the scattering cross section: a - for (J ~ 135°. b - for (J - 139°. The experiments! data are from[·, lO l l1J.

us 180

r ....

LL:s~:::::::=-.:2.J'E!!.0o

•,

0.1 o.s o.J

0

2A. A. Logunov. Dissertation. Joint Inst. for Nucl. Research (1959). 3 M. Cini and R. Stroffolini. Nucl. Phys. 6. 684

FIG. 10. Differential cross sections at different photon energies (indicated on the curves).

·o.J-0.5 -1J.7 -/ r.asH

FIG. 11. Polarization of recoil protons.

increases the contribution of I RI - R212 to the cross section and does not lead to an improvement in the agreement with the experimental data. It is necessary to remark that outside the region 1 < "0 < 1.3 a satisfactory agreement between

5 T. Akiba and I. Sato. Progr. Theor. Phys. 19. 93 (1958). 6 L. 1. Lapidus and Chou Kuang-chao. JETP 37. 1714 (1959) and 38. 201 (1960). Soviet Phys. JETP 10. 1213 (1960) and 11. 147 (1960). 7 M. Jacob and J. Mathews. Phys. Rev. 117. 854 (1960). 8

M. Gell-Mann and M. L. Goldberger. Proc.

1954 Glasgow Conf. on Nucl. and Meson Physics. Pergamon Press. London-N. Y. (1954). 9 Hyman. Ely. Frisch. and Wahlig. Phys. Rev. Lett. 3. 93 (1959). 10 Bernardini. Hanson. Odian. Yamagata. Auerbach. and Filosofo. Nuovo cimento 18. 1203 (1960). II F. E. Low. Proc. 1958 Ann. Intern. Conf. on High Energy Physics at CERN. p. 98. 12Glasser. Seeman. and Stiller. Bull. Amer. Phys. Soc. 6. 1 (1961). 13 L. I. Lapidus and Chou Kuang-chao. JETP 41. 294 (1961). Soviet Phys. JETP 14. 210 (1962). U L. I. Lapidus and Chou Kuang-chao. JETP 41. 491 (1961). Soviet Phys. JETP 14. 352 (1962) 15 Y. Yamaguchi. Progr. Theor. Phys. 12. 111 (1954). S. Minami and Y. Yamaguchi. Progr. Theor. Phys. 17. 651 (1957) . 16 S. Minami. Photon-Proton Collision at 250800 Mev (preprint). 17 Govorkov. Gol'danskii. Karpukhin. Kutsenko. Pavlovskaya. DAN SSSR 111. 988 (1956). Soviet Phys. "Doklady" 1. 735 (1957). Gol'danskll. Karpukhin. Kutsenko. and Pavlovskaya. JETP 38. 1695 (1960). Soviet Phys. JETP 11. 1223 (1960). V. V. Pavlovskaya. Dissertation. Phys. Inst. Acad. Sci. (1961).

Translated by A. M. Bincer 261

140

451

PI-IYSICS

A Suggested Experiment to Determine

th.

Spin of

Y:

and the Parities

of

,1 - I, ,1 - Yf and I - Y:

rn this nOte we propose an experiment In observe the final state ,1 - " resonance and lhe cusp arising from the near tbreshold effect of the l:-production, and from this to determine the spin of yt' (]).~) and the relative parities of .1 - J:, .1 - y~ and !: - y~ (P(,l - S). PCl and Pl.); - Yn). The centre oi mass energy of the initial " - P system ~hould be around 1900 MeV (this correspond. to 1305 MeV of the incident pion ill the laboratory system). Choose the evellts in which the energy of the kaon lies below 90 MeV and observe the correlation between resonance and ~ne cusp. This energy region will iust cover the y~ resonance due to Y ~ and the cusp due to

yn

141

452 I-production. Furthermore, since the energy of the knon is reiatively 10\\' and much below the mass of K* (MK .. -BBOMcV,QK*-250MeV), only the I-wave of kaon need. to be considered. Obviously, the cusp will appear in the J 1/2 state with Pd. = 0 or Pd. = t according as peA - X) is even or odd. Since tbe p()sition of the 1:: -". threshold lies below the A _ " resonance only by 55 MeV, and the background of the rC."nOlle" i. rather small (around 1 : 2 -+ 3 to the slope of this resonance). Thc cusp can only be observed when it appears in the same panilll wave state as tbe reso.nallce, and til is will lead w a useful information about the A - Z relative parity and the spin of Yi. Moreover, the relative angular momentum Pr~ of the A - n system can be determined from -' the angular disttlDucion and polarization measurements. From the above observations the spin of yt and the relative parities can bl uetermineu. We list below the diffcrent. con. elusion corrcspClnding to all possible experimental situations.

=

(1) Ii the cusp it observed togcther wid, the rc.~s()nallc('\ then we have Jr~ = ] :'2, P(I.· - y~) and if lhe P\"~ c,;n further he determined (sec (4) below), then peA - I)=

= -,

(-/}.~. (2") If only the resonance i. observed, and if the Jl't, P\'~ can further be determined (sec (3) and (4) hclow), then

i.

when

Jl'''' =

1/2, we shlllJ have P(I.-

Yf)=+. P(A':X)=(-/y~+I: ii. when J1' ,. = 3/2, we may have P(.1-

= -. since the Prt = 2 ~t:ltC can be excluded in this energy.

Yt)

(3) rized:

i.

When the initial if

nucleon is unpola-

r/u is isotropic and the hydIJ".,dE k

peron is unpolarized, then we have ii. if

1/2;

--~~I!.--. . . .- - a cos! 8 +bcosO + c and dIJ,h,dL1:

da

P II

Jrt =

_ sin 20 a" .. X a; ,

d!J"."dE k

=

In.b

then

we

y. nrl

e

Jy't 3/2, where cos = Di • D,h.. a;, and Q d.. being respectively tbe incident direction of ".-meson, the direction of the A-:r relative motion, and the solid angle of the .1-11' system. have

DA.",

(4) When the initial nucleon is polarized and its polarization is orthogonal to Di, then

in the case of Jr~ = 1/2: , do i. when P II -.-.- - , ' - = P • have Pl't

"

dQd",dL:.k

= II; h

II. W en

P

A

d() d!J",.dEk -

COlIst..

pO'

-~aA,,\nA'"

we

P

j

we have Py~ = I, where P

and P" ate the polariz:ttions of the initial nudeon :llld the final hyperon respectively. It will be noted that (1) and (3) ate two indepenuent observations for J},~; (2) and (4) arc ohservations On rei. ative parities. Two of the authors (Su Zhao-bin and Gao Chong-shou) arc indebted to Dr. Chou Kuangchao lor his kindly guidance, an'; .lls() to Prof. Hu Ninr; for his interest and support. Su Zhao-bin ("~) Gao Chong.-sh()11 (;flj~r;(n Chou KUan!Hhao UIiI:l'dD Peking Unit'f1TSUY

Jan.

7, 1963

142

SCIENTIA

Vol. XXlI No. 1

SINICA

.T::mnnr.'· 1979

THE PURE GAUGE FIELDS ON A COSET SPACE CHOU KUANG-CHAO

(nil7tE)

Tu TUNG-SHENG

(Institute of Theoretical Physics, Academia Sinica)

(t1:*~)

(In.sti~ltte

of High Energy Physics. ..:l.cademia Sinica)

AND YEAN TU-NAN

(~mm)

(University of Science and Technology of Chi·na) Received August 18, 1978.

ABSTRACT

The concept of the pure gauge fields on a coset space is introduced. By using gaug" fields on subgroup H, pure gauge fields on coset space GIH and the induced representation, a local gauge invariant Lagrangian theory on group G is constructed. The application of this theory to SUo X SUo gauge theory, the a model and the non·trivial t.opological property of the pure gauge field are discussed.

I.

INTRODUCTION

Since the emerging of the non-Abelian gauge field theory unifying weak anti electromagnetic interactionUl , the properties of the non-Abelian gauge field have been extensively investigated and the important progress has been made[·I. ~o\.s is well known, even with the symmetry of the vacuum spontaneously broken and the Goldstone bosons absorbed through the Higgs mechanism, the non-Abelian gauge field theory can still be renormalized. Though in the unitary gauge, only physical particles appear, the theory is not obviously renormalizable. While in the Landau gauge the theory is manifestly renormalizable, it still contains fictitious particles, which seem to destroy unitarity. The theory is in fact both unitary and renormalizable as can be shown simply in the R, gauge. The worst divergent diagrams are cancelled with each other. The undesirable effects of the fictitious particles are also cancelled. JUany low energy hadronic experiments showed that hadrons have not only SU3 (or BU.) symmetry but also SUs X SUo (or SU3 X SU.) chiral symmetry[31. The pion is a pseudo-Goldstone boson resulting from the spontaneous breaking of the chiral symme~ry. It was proved in [4] that when the chiral group is broken in Goldstone mode, the effective Lagrangian involving the pion field still possesses chiral symmetry in the approximation of the lowest order in breaking parameter and pion momentum. In this case, the pion field offers a non-linear realization[51 of the chiral group. If the pion field is taken as an elementary field offering a non-linear realization of the chiral group, the resultant Lagrangian has complicated non-linear terms. This Lagrangian gives some results which in the lowest order perturbation theory agree with experiments. But the theory is not renormalizable[OI. The linear a model can be constructed,

143 38

SCIENTIA SINICA.

Vol. XXII

using linear representation of the chiral group. This· model is renormalizable, but the pion field does not apparently have the characteristic of a non-linear representation of the chiral group. In this model, it is difficult to get results in the lowest order perturbation expansion in agreement with experiments. If a gauge degree of freedom like that of the non-Abelian gauge field theory is introduced to make the theory gauge invariant, the theory might be manifestly renorrnalizable in one gauge (which is to be called the renormalizable gauge), while in another gauge (which is to be called the physical gauge) good physical results might be easily extracted by tree approximation.

For this purpose we introduce the concept of the pure gauge scalar fields on the coset space. In the case the global topological properties are trivial, the pure gauge scalar fields are the manifestation of the gauge degrees of freedom and can be eliminated by choosing a suitable gauge. By using the pure gauge scalar fields the renormalizable gauge can be connected with the physical gauge. If the subgroup is U(l), the gauge field on which is the electromagnetic field, and if monopole exists, the monopole and its electromagnetic field can be described in terms of the pure gauge scalar fields. In this way we can avoid introducing singular strings in the expression describing the vector potential of the E. M. fields" of the monopole. Further the equation of motion of the monopole and the electromagnetic fields might be derived from the Lagrangian.

The plan of this paper is as follows: In section II we review briefly the induced representation of group G on its subgroup H, introduce the concept of the pure gauge field on the coset space, and discuss its transformation properties. In section III we discuss a physical system which is local gauge invariant with respect to the subgroup H. Using the pure gauge fields on the coset space we construct a local gauge invariant Lagrangian with respect to the whole group G. In section IV, for the sake of illustration, we construct an SUI X SUI gauge invariant theory, which coincides with linear a model in the renormalizable gauge and with the non-linear chiral model in the physical gauge except a few additional terms which account for the renormalizability. In section V, we discuss how to describe monopoles by means of pure gauge fields on the coset space. In section VI, we give the coset element parametrization with respect to the subgroup U... X Uft-.. of the group Uft. The corresponding expressions for the pure gauge fields on the coset space are also given.

II.

THE INDUCED REPRESENT.l.TION ON THE SUBGROUP A...'i[D THE

PURE

GAUGE FIELDS ON THE COSET SPACE

Consider a transformation group G underlying a physical system. Let H denote one of the subgroups of G, and g and h designate any element in G and H respectively, namely,

G={···g···},

H={···h···}.

tP denotes any representative element of the coset space with respect to subgroup H, namely,

144 PURE GAUGE FIELDS ON COSET SPACE

No .. l

39

GIH = {.. ·cP···}. According to the theory of Lie group, for any element g E G, there is the unique left coset decomposition,

g

=

(g', llsity: (54)

1'2

= ~

gl(XZ - X:,y,

where cp(x) belongs to some representation of the 8U(2) group and X is a real scalar field. The equations of motion corresponding to (54) are

(56)

According to the discussions in the above section, we shall concern ourselves only with soliton solutions of the form, cp(x, t)

=

lex, t)

= l(x),

g(O)e+i01l"cpc(x),

(57)

180 48

Vol. XXIII

SCIENTIA. SINICA.

where 13 is diagonal. The isotopic-spin of tbis solution is

~

.. , 00(13; + 13/)e-i..crJl-r·j)'fd3XIp!(X)

J. =

X [a(0)-11.a(0) ]iiq>Kx),

where 13; is the eigenvalue of 13 corresponding to 1p~(X). Since 5., being conserved, must be independent of t, the above equation can be reduced to

j d3XIp:i(X)[g(OyI1.u(0)]iiq>~(X).

J. = ~ 2001 3; From the relation

rr(0)1.u(O)

= ~

.,

R..,]., ,

where R is a matrix of rotation determined by g(O), and the fact that only nonzero diagonal element'!! are those of 13 , the equation of J. can be rewritten as

J.

~

.

=

2oonR. 3

j d3XIp:j(X)Ip~(x).

Define the total isotopic-spin as

J

== .jJ.Ja = ~

200n

i

which i'!! evidently independent of yeO). (57) is

E

=

f

d3x {VIp:(x) • Vlp,(x)

+

!

f d 3xlp:i(X)cp!(x),

(58)

-

The energy of the field corresponding to

(VX)Z

+

VI

+

V2

+ oozlp:(x)1~1p,(x)},

(59)

which is also indt>pendent or !leO). Substitnting (57) into (55), equations satisfied by Ip, and X nre found to be

{ - V2 + { _ VzX

+

a( a~, ") Ip, q>,

ooz

n} Ip!(x) =

ay, + aY2} = o. ax ax

0, (60)

It can be seen from (58)-(60) that solutions of the form (57) with different !l(0),8 havc the same energy, isotopic-spin and equations of motion. Therefore, we can consider only the solution with g(O)~l as the representative of these solutions. Furthermore, the energy, isotopic-spin and equations of motion are all invariant under arbitrary rotntions ill the subspace spIlnned by cp~(x) and q>;i(X). Therefore, we need only to di'lCUS.'!! the case in which q>!(x) = 0 for 13; < O. It is conveni('ut to introduce dimensionles.'!! quantities by .

cp,(x)

=

g!' R(p),

X(x) =-

!:... A(p), g

181 !'ici.l

NONTOPOLOGICAL SOLITON WITH

NONABELIA~

INTER);"AL SY1fMETRY

49

'",u'

V=-.

= IX",

'In

p =

!t=gX.,

,U%,

(58-00) can now be written as $

E= £

gl

Jd

3

p

= ~~~ ~ (l3pB+I~B,

(61)

g •

1Y} + 1.. ",$,

{VpB+VpB + 1..(V pAY + I(z.A1B+B +..!.. (..P -

2

lr -

V~ +

8

:!.K1B+B

+ 1.. (.1 2 2

J) 1.:1 J

=

2

(62)

O.

.

(68)

It can be easily proved tlUlt (63) can be obt:lined from tIlt' yariatioll of respect to A and B at fixed $. 1.

(6~)

"'ith

"Free" Solutions-SolutiolM Independent of Spflce Coordinates

Putting A(p) = const:mt and B(p) = constant inside the cubic box of the VOIWlU~ V in Eq. (63) the condition for non-vanishing B' ir; fOlmd to be

It follows thnt other eomponents must be zero. has

Denoting this solution by B(i), one

and

Evidently, corresponding to each 131 , there is a solution with the total isotopic-spin

and the energy

In the limit Y

-+

co, 8* • B

-+

0, A. -+ 1 and ",13,

-

m. we haye (li4)

This formula tells ns that for fixed .~ "free" solutions corresponding 1:0 llifferi'llt 13,'8 differ in their energies. The larger the 131 the lower the energy. The solution with the least energy has only one component corresponding to 1 3max different from zero.

182

50

SI~ICA

SCIENTIA

Vol. XXIII

Are these solutions all classically stable? '1'o.answer this question, one must calculate the second variation of the energy. It is easy t{) prove that

Jd plJ!+HIJ! + ~~ [J d p(IJ!+IiB + B+nlJ!) 3

3

(lllE}, = ;

r.

(65)

where

_1- Vl + K2B+B + H=

(

2

3iP -1 4

p

'

(lili)

2KzAB, qf=

(8A) 8B '

(67)

Substituting in(65) the "free" solution of A and B and taking the limit y--OO, we find that only the diagonal terms of H do not vanish. The eigenvalues of H are continuous spectra starting from 1- (3A.z - 1) and (I[Z.d. z - vZn) respecth·ely. If for ~ome solu4

tion BwCl =F 13max) we choose ...LO 8 Bill { ..,=0 llA.

thpn we have (llZE), =

i= i =F

for for

[3m.. , [3

m,.,

= 0,

.!:.. \. cl3p(8B(;»(vl n gZ.

v Z1i max)8B(l)


nH and RI = 0 ¥ j ~ nH'

to the first order in

As a nonlinear realization of the group G, the functions RI •• tmd kj,. satisfy the following reaJiza.tion of the commutation relations:

195 No.4

VACUUM OF PURE GAUGE FIELDS ON COSET

433

(2.7)

and (2.8)

where (2.9)

and

!i/.m

are the structure constants of the group G.

Consider now a system with fields CP(z) which form the basis of a linear representation of the group G. In. the absence of the gauge fields the La.,"Tangian has the form (2.10) which is assumed to be global invariant, and we must introduce gauge fields. In. I we have constructed a gauge field B,. on the group G which consists of a vector gauge field 1,. on the subgroup and a pure gauge field cPo(:I;) on the coset, B,.(x) = cPo(x)(a,.

+ 1,.(z»cPo1(z),

cPo(z) E G/H.

(2.11)

When 1,.(x) transform under g(x) E G in the following nonlinear Way, g(x): 1,.(x) -1:(x) = h(g, cPo)(a,.

it is easily verified that

13,.

+ 1,.(x»h- (g, cPo), 1

(2.12)

transform as a usual gauge fields under the group G, (2.13)

With the help of these gauge fields we can construct a local invariant La.,oorangian from (2.10), £t'(CP(x),

D~B)CP(X), F~~» =

£t'o(CP(x),

D~)CP(x»

-

1 4 a tr

{m.~)F(B)I'P},

(2.14)

where (2.15)

In. (2.14) a is the normalization constant determined by the condition,

tr {liid

=

(2.16)

aBjI..

In. the La.,oorangian (2.14) we can choose CP(x), l,.(x) and ao/(x), l = nH + 1, •• " nG, which parametrize the coset function cPo(x) as independent dynamical variables. The La.,"Tangian (2.14) is invariant under the infinitesimal gauge transformation y(n = 1

+ iliSi

with the corresponding changes of the fields,

BCP(z)

=

ilkCP(z)Sk,

BA~(z) = fiilA~(z)kt.k("o)gk - a,.(h,.k(ao)gk), 6lJo1(z) = iRJ.r.("o)sr.,

i

=

nH

+ 1, "', ?lG.

/-1," ',nR }

(2.17)

196 Vol XXIII

SCIENTIA SlNIeA

In (2.17), A.~(x) are the components of the gauge field l' .fJ.,.

=

.

I

AI"

A

(2.18)

-~A .. Il'

The operator which produces this gaug~ transformation can be written as (2.19)

T(g(x» = exp {iftPxI(g(x),x)}.

For infinitesimal gauge transformation g(x) in (2.5), the generator [(O(x), x) is equal to

[(1

+ ii/gi,

x)

=

iP~(x)I/ctP(x)g/c(x)

- P ,,1(X) f milA!(x)hm./c(cio)gle(x)

•

.

+ ,iPaol(X)Rl.k(cio)g/c(x) +

P ,,1(X )ap(hl.k(aO)gk(X»,

•

(2.20)

where P~(x), P a,leX) and P ,,1(X) are respectively the conjugate momentums of the quantized field., $(x), aOl(X) and 1;(x). The commutation relations are [tP(x), P~(Y)]%D=IID = i8 (x - y), [aOl(X), P aom (Y)]"D=II. = i8/ m 83(x - y), 3

)

[.J~(x), P"::(Y)]"D9IO = i8,.p8/ m 8 3(x - y). Actually these quantized fields are not independent. There are constrained conditions due to the gauge invariance of the La,,'"'l"angian. We shall regard these constraints as weak conditions which are satisfied only when a physical state is aeted upon. As we have mentioned in I that the derivative terms a,.ao/(x) appear only in aql os momentum Paol(x) must be related to P~(x) through the constraint conditions, D~D)~(X),

PaD/eX) = p~(x)ll(ao)cP(x),

l = llu

+ 1, ... , nG,

(2.22)

where I/(ao) is determined by the relation, cPo(x)al'cPo1(x)

= Il(aO)al'ao/(x).

(2.23)

From the group relation, (2.24) and (2.23), we get for arbitrary g E G, li(cP(g, cPo)aI'Ri(g, cPo)

=

cP(g, cPo)al'cP-1(g, cPo) = gcPo(h-1(g, cPo)al'h(g, cPo)cP01g-1 + gli(cPo)g-lal'aoi

+

ga,.u-l.

(2.25)

Equating terms proportional to al'§/c(x) on both sides of (2.25) we get li(ao)Ri.1«ao)

=

cPOllcP01hl.1«aO) -lie.

(2.26)

Using this relation we can rewrite the constraint equations (2.22) in the following form, (2.27)

197 :No.4

VACUUM OF PURE GAUGE FIELDS O:N COSET

435

When the generator (2.20) acts upon a physical state, it can be simplified with the help of the constraints (2.27). Thus we obtain 1(1

+ il"g" , x)lphys)

= [(h(l

+ il";,, , cPo),

x) Iphys) ,

(2.28)

where l(h(l

+ il;;i, cPo), x) = iP",l,, we have

jr""(h)lnL, nR) = InL + 1, nR + 1), } jr""(cf»lnL' nR) = If'lL + 1, nR -1).

(3.18)

Let us now introduce two winding numbers corresponding to the gauge transformations h and c/> respectively: (3.19)

200 438

Vol. XXIII

SCIENT!.-\' SINICA.

In terms of these new winding numbers we can write the state InL,nR) as In, nc) and have

T+(h) In, nc) T+(e/»ln,ll e )

= =

In + 1, ?Ie), } In,ne + 1).

The eigenstates of the operators T(lL) and T(ef»

(3.20)

can be written as

le,ec) = ~ e.s:p{i(ne + 11eec)} In, l1e). This is the e-vacuum states for the present e.s:ample. Like the first e.s:ample, T(e/» is equivalent to T(Tt(e/>, e/>o)) when a physical state is acted upon. Since T(lt) cannot change the winding number nc, we conclude that Bc can be normalized to zero for physical states. In general cases, if H is a semi-simple Lie group, '%J(H) = 0, one can proye with the help of the e.s:act sequencers] -,.. '%)(H)

~ ,,"3(G) ~ -:r;(GjH) ~ -:riH) ~ "zCH)-1/

o that (3.21)

When '%;(GjH) -'-;- 0, the purc gauge fields 011 the coset will huye topolog·icul nontrivial yacnum. states. Ou account of the constraint conditions (2.30) the Bc-yacuum introduced on the coset has the yalue ee = O. Therefore it does not cause CF non-conservation. Before the conclusion of the present section we shall briefly discuss the problem of vacuum ttmneling". .As an illustration we choose G = SU(2) and H to be the identity element. In this simple case the coset is the ,vhole group. In the temporal gauge it is well known rl ] that the change of the winding number between the future and the past vacuum states is equal to the second Chern class or the Pontryagin number, q

=

v(t

= +00) -

v(t

= -00)

= __1_ rd\J; tr {jr(B)*jr(S)I'~} 32~a

J

,..

,

(3.22)

In our case the field

is a pure gauge field.

On those points where g(z) is regular we always have

Pes of the gauge bosons and the fermions. After spontaneous symmetry breaking foul' heavy Higgs with two neutral ones Xa, cpo and a charged pair X:I: remain. On account of the conservation of weak strangeness, V:!:, U:!::!: and X:!: mesons can be produced only in pairs and the lightest of these mesons might be a stable particle.

204 No.5

SU(3) xU(1) MODEL OF ELECTRO-WEAK I~TERACTION

567

The paper i~ organized as follows: In Sec. II tlie transformation properties of fields are studied. A representation of the SU(3) algebra, which mixes the rs and the internal group generators, i~ used for fermions. In Sec. m, the mas'! spe~tnlln of "Vector gauge bosons is obtained through Higgs mechanism and the conservation of weak strangeness i'! e"tablished. The eletro-weak interaction of the fermions i!'! clir!cussecl in Sec. IV. Deviation in neutral CUl'rents from the Weinberg-~alam model is obtained. Finally we "hull di~cuss the results obtained. II, TRANSFORMATION PROPERTIES

The generators Ii, i = 1,"',8 of the SU(3) group can be decomposed into two sets and There are many possibilities. One possible choice which will be used in the following- is It = 1, :3, 8, 4, 6 and a = ~, ;j, i. Other choices giyC similar results. Define

t

t.

f\..

l~r;} =

'"

eI.

"

I

.•

l~t. =

,...

(~.l)

II'.

where e commute!'! with 1; and satisfies the relation 13 1 = 1. One easily yernies that the Lie algebra for I~') i'! the same as that for t. Four possible choices for e : e = 1, -1, rs, -1'5 will be u!':ed below. The goeneratol' of the r(1) g-auge group will be u,'noted b~· j'"'.

+

Besides the con~;eryution of yarious fermion number!'! there is another global F(I) sYlllmetr~- whose generator will be denoted by .~. Thi'l global U(1) will combine with an Abelian subl,.'"'l'oup in SU(:3) X [\1) to giYe a new conseryation number Sw after spontaneous symllletry breaking. This new quantum number HII' is called weak strangeness in I. Each g"cllC'ratioll (11" C), (l'I" .!L) or (l'r ... ) of leptons forllls half of It triplet in SF(:l) with r = o. Quarks of each color and isotopic doublet form half of a r = 2/iJ triplet plus half of a r=~/a singlet in "C(:~). ')'ran~fornmti()n propertirs fo1' fermion!'! are

c/J --~ c/J' = rtS)(;i(Z) )eiYU(r)ei.~~c/J;

for triplet:

II.~U.'

for singlet:

=

(2.2)

eiYU(r)eiS'iu..

where (2.3)

= +, -, oj, -oj stand for e = +1, -1, +1'5, -1'5 rt>specth-ely. In Eq. (2.2) = 1, .. "'~, e(x) and 1/ are group parameter" for the local 8["(3) X F(I) and the global CO) respecth-el.v. The !7enerator .g will take yalues l.. rs _l.. 2 for lepton 0

Here e

gj(x), j

1 "'lt t rIp e , 6

6 • ' let. for quark trIplet and - -1 1's - -5 f 01' qual'ksmg 6 2 6

rs - -;3

There are nine gauge fields A.~(z), j Define

= 1, ... , S of

Sr(8) and BI,(r,) of UO).

(2.4)

205 SCIE~TIA

568

Vol. XXIII

SINICA

Then :A~' transforms under SU(3) in the following way, 1~"---- :A~"

= F<e'(§j(x))(al' +

A~·'(x))U1 with Y = 0 and q>2 with Y = transform a.';!

(2.5)

- 1 are used. They (2.6)

= - 1.- for

The generator S

3

q>1' S

=

5 for q>2 and S

3

=

0 for gauge yector bo:;uns.

The covariant derivatives for the fermions and the Higgs are easily constructed. are

The~r

for fermion triplet:

D,,'"

for fermion singlet:

a" + :A~) + ;

(

Du'lt ,

( a".

=

a/, +

D"q> -- (

for Higgs triplet:

III.

=

"(+' A,.

U'l-BI') cP;

~ ) + -2i( J, J:Bu .

(~.7)

U;

+ 2,i",,) g YBI' q>.

SYMMETRY BRE.\KING, CONSERVATION OF CHARGE .... ND WE.\K STR.\NGENESS

The vacuum expectation yalnes of the Higgs are taken to be

(3.1)

s.nllllletr~·

One local F(1) and one /!lobal F(1) are

remain unbroken.

Their generators (3.2)

for charge: .. 811'

for weak strangeness:

2

1\

,..

= ,.; 3 18 + }' +

A

(3.3)

S,

which are conseryed quantum numbers. After spontaneous symmetry breaking all physical particles are eigenstates of charge Q and ,veal;: strangeness § w • 1 . For triplet fermiolls we must replace' is by 2 1'5.1.s and get from (3.3): 1 ~

,:sw

1

= ,.;_ 3

.

1'5.1.8

+

-1 1'5 6

+

"1 Y = -,) 1'5

-

(

1

)

+ Y.

(3.4)

-1

We use the quantum number Sw to classify all three components in a triplet. This implies that the he1icity components with the same Sw be either (L, L, R) or CR,

206 Xo.5

SU(3) XU(l) MODEL OF ELECTRO·WEAK .INTERACTION

569

R, L). 'l'he now observed leptons and quarks are then chosen to be

(3.5)

The other half components are weak strange fermion.'!, which will get heuyy masses by a suitable choice of additional Higgs fields and an additional singlet lepton field. The problem of mass spectrum for fermions will be discussed elsewhere. The charges and the weak strangeness of the particles participating in lo\v-energy 'Weak interactions are given in Table 1. Table 1

The mass terms of the vector gauge bosons are easily derived as,

~ rll ulI (W+W-+Y+v--+! 2

+;

g2IulI2(v+r-

+ ..!. III vll l ( 4

+ r-++c--) ZO

1

ZOl)

.) 3

+ _._1_ Z'o)~,

(3.6)

Slll'P

where

v:l: = .)2"':>'_' _1_(.~ + ..1') , U:!::I:

= .)2 _1_ ( 4.' + iA7) . , Z'G = - sincp

(~

(.1 3

-

./3.4.')) +

coscpB,

(3.7)

and .

Slncp

= / 'V

g

gZ

+ g'2

'

coscp

=

g'

./ g2

+ g'2

.

(3.8)

From (3.6) and (3.7) we obtain: (i)

The photon field A = cos cp

(~

(A 3 -

./3.1

quired: (li)

The maJ!Ses of the charged vector bosons are

8

))

+

sin cpB is massless as re-

207 570

SCIENTL~

SINICA

Vol.

Xxnr

(a.!)) m~ =m~

Let us introduce

11

+

m~.

new parameter (:UO)

which will be useful in the following. The Z and Z' mesons are not eigenstates of the mass matrix. Let the tllle neu· tral vector bosons be Z, and Z2, which are related to Z and Z' b~' it rotation.

+

Z = cosaZ, Z'

sinaZ2'

(:Ul)

= - sinaZ, + eosaZ2'

Diagonizing the mass lllutrL"': we have

,

mz

, [1 + -'v 14 (1 -

=

nlz

=

, nlz, [tg'a

1

m~.

+

-1

4

L'

-1'3. t!? ' a )~] . SIll(y, C)

=

A -I( C)1. tr {T (la(Y) exp

{ie f A/y)dyi})}.

(30)

c

The 1.(y) in Eq. (30) represents the 1. at point y of loop C. Generally speaking, ci>(y, C) is a functional of loop C. However, it has been proved[61 that ci>(y, C) is invariant under parrallel displacement along the direction of loop C, though it does not possess this property along other directions.

220 No.8

KEXUE TO::-i"GBAO

639

By comparing (24) with (28), it can be seen that there is some sort of duality property between B(C') and A.CC). But this duality property is not as perfect as that in the Abelian case because of the fact that lex) and t:/>(y, C) iLre not completely similar in character. This may not be as surprising as it appears since it is not always possible to find out an appropriate dual potential(7] in the non-Abelian gauge field theory. After the submission of this work for publication, we noticed that Mandelstam[B] had considered some problems similar to that discussed in the present paper. REFERENCES

[ 1] [ 2] [ 3] [4,]

[5] [ 6]

[ 7] [8]

't Rooft G., N1I.c1. Phys., B138 (1978), l. Halpern, M. B., Phys. Re1l., 019 (1979), 517. Yoneya, T., Nucl. Phys., Bl« (1978), 195; Englert, F. & Windey, P., Phys. Rep., 49 (1979), 173. 't Hooft G., Nucl. Phys., B79 (1974), 276; Polyakov, A. M., JETP Lett., 2(J (1974), 20; Ezawa, Z. F. & Tze, R. C., Nucl. Phys., BI00 (1975), 1; ~-±, ;§;!l!i*, ~f8+, «~llI!~f[l:o, 25 (1976), 514; Arafune, J., Frennd, P. G. O. & 000001, C. J., .1. Jlath. Phys .• 16 (1975), 433. Corrigan, E. & Olive, D., Nucl. P1IYs., B110 (1976), 237. Nambu, Y., Phys. Lett., 80B (1979), 372; Corrigan, E. & Ifussla.cher, B., Physl Lett., 81B (1979), 181; Gervais, J. L. & Neveu, A., Phys. Lett., 80B (1979), 255. Gn Chao·hao & C. N. Yang, Sci.. Sin., 18 (1975), 483; Brandt, R. A. & Neri, F., Nucl. Phys., BI4S (1978), 22l. 1Iandelstam, S., Phys. Rev., 019 (1979), ~391.

221 Reprinted with permission from AlP. Con! Proc. 72 (1980) 621. Copyright 1981 American Institute of Physics.

621 AXIAL UCI) ANOMALY AND CHIRAL SYMMETRY-BREAKING IN QCD K. C. Chou* Virginia Polytechnic Institute, Blacksburg, VA It is well known that the absence of the AIlJ anomaly is necesl sary for the corresponding gauge field theory to be renormalizable . This condition places severe restrictions on the choice of the possible gauge group,and the representation for fermions, as we have just heard from A. Zee.

I would like to report, on the

other hand, some consequences of the presence of ABJ anomaly in certain global current conservation equations.

This is a work done

in collaboration with L. N. Chang. One important question in QCD is the origin of chiral symmetry breaking.

This problem is related to the structure of the theory

for large distances,where perturbation theory cannot be used. It has long been conjectured that topologically non-trivial gauge field configurations play some significant role in explaining both confinement and chiral symmetry breaking

2

3 4 Recently, Coleman and Witten , and Veneziano , have analyzed the question within the context of liN

c

expansion.

In particular,

Coleman and Witten argue that the axial anomaly and confinement already imply chiral symmetry breaking.

Their argument makes

no essential use of the non-trivial topological configurations, but relies instead on the absence of analyticity structure in the axial vector vertex, brought about by the anomaly. reaches the same conclusion, using the liN

c

Veneziano

expansion and by

*On leave from Institute for Theoretical Physics, Peking.

222 622 consideration of the fluctuation of topological charge density in the pure Y-M field sector. In this talk I want to point out that their conclusion on the necessity of chiral symmetry breaking can be obtained without recourse to the liN

c

expansion, if proper attention is paid to

the quantum fluctuations in the topological charge density. We start by recalling that in the presence of non-trivial topological configurations, one could incorporate the 8-vacuum caused by the resultant phenomenon of tunneling by augmenting the conventional Lagrangian with an additional term

2

vex)

~ Fi *F illV D 3211"2 llV ' II

*F

2

1 ].J\i

d

II

(1)

gAll

FOP

£ ].J\iCp

In equation (1), 8 is the parameter characterizing the topological structure of the vacuum, while J (x) are external a sources coupled to various combinations of quark currents 0 (x). a Since we are interested in chiral symmetry breaking, our attention will be focused on the densities Ci(l±YS)q

==

O± and Cil(1± YS)q

==

O~.

In the following 8 will be considered as a function of x in some intermediate steps of calculation. The Lagrangian of (1) has an apparent D(N ) x D(N ) chiral f f symmetrY,with N flavours, if we set Ja(x) f

0.

However, due to

the ABJ anomaly in the axial current, 8 will change to 8 +/2N f ~S under the abelian chiral phase transformation q(x)

~ exp[iY5~5]q(x).

The best way to study the chiral symmetry structure of (1)

223 623 when J(x) #

° is

through the effective generating functional,

which we shall now define.

The generating functional, W, for

the connected Green's functions implied by (1) can be expressed as W[J,B]

tn

-i

=

Z = JDqDqDA

Z 4

II

exp{i!d x c£(x) + /l,i..(x)]

(2)

Here/l,~x) includes the gauge fixing and compensating terms

necessary to give meaning to the A integration. II

The classical

fields Vex) can now be defined by direct differentiation

v

oW

(x)

(3)

As a result of the axial anomaly, and the formal invariance properties of (1), the generating functional W has to satisfy a Ward identity of the form

a

II

oW oj +(x) ll-

= iJ

±

oW + N ~ oJt(x) - f oB(x)

(4)

-

This Ward identity can be satisfied by any functional with the following local invariance property

B(x) - ~ ~5(x)J

where

~5(x)

(5)

is an arbitrary function of x.

We define now the generating functional r by using a Legendre transformation on the sources of the scalar currents

o±

224 624

f[U±(x), JW±(x), O(x)] W[J,(x), J ,(x), O(x)] -

+ Jd

\J:'"

4

(J+(x)U+(x) + J (x) U_(x»)

x

(6)

Then the Ward identity for the axial UCI) symmetry implies that r is invariant under the local transformation

J + ~-

8(x)

J +(x) +3

7

w-

~N

w~Nf

8(x) - 12N

7

f

(7)

f,S(x)

~s(x)

Or in other words, r is a functional of the form +i

L

r[U±e

8

Nf

, J +(x) ~-

I

+ 3 (N

-

~

f

e)]

(8)

Note that the classical fields U±(x), which are the vacuum expectation values of the corresponding quark bilinear fields, can be determined through the relation

(9)

Any nontrivial solution to (9) when J

±

= 0,

J

~±

=0

and e(x)

= cQnstant,

would signal the existence of spontaneous chiral symmetry breaking. Since U

+

~

U

-

+if'f U

±

+ Ue-

1N"f

*

we can write them in the form

.'L LIT

(10)

where n' can be interpreted as the vacuum expectation value of the n'-meson field which corresponds to the axial U(l) pseudo Goldstone field in chiral dynamics.

225 625

fiNf

Notice that r is an even function of 8 - ---f- n' as a consequence 11

of (8), and the symmetry under space reflection. that the CP conserving solution of eq. (9) at J.

It is easily proved =

0, J+

-11

= 0,

e(x) = e is f

n'

e

11

12N

(11)

f

Thus all the physical quantity evaluated at this point will be

e

independent and CP conserving. Nevertheless, the nth order derivative of r with respect to

8 when n' is kept fixed is the Green's function of nv(x) 's where diagrams with one particle lines of nand U are removed.

::~ In"~

(_i)n fd4xl···d4xnI.P.r.

It has been shown by one of us the n' meson mass.

5

that

02~ I11 'u 08

(12)

is proportional to

This result is a generalization of the result

given first by Witten

6

in the leading liN

c

expansion approximation.

The point we wish to emphasize is that (12) can be nonvanishing only if U 1 0, so that if any of the moments defined in (12) were to be nonzero, chiral symmetry would be spontaneously broken.

This is the main conclusion of my talk.

Now the right

hand side of (12) represents a quantum correlation function

of

the topological charge density, and there is no general reason for (12) to vanish for n even.

The case for n odd can be

excluded, of course, in the chiral limit when CP is a good symmetry.

We therefore argue that QCD will. in general, induce

the spontaneous symmetry breaking of flavor

chiral symmetry.

226 626 The picture we are presenting may therefore be summarized as follows:

Owing to the presence of instantons, the QCD vacuum

acquires an additional parameter O.

In the absence of any

external spontaneous chiral symmetry breaking, like those induced by Higgs couplings, the chiral phases of the quarks will automatically refer themselves to 8. consequence of the axial anomaly.

This is the direct

However, large scale quantum

fluctuations of the topological charge density requires such phases to be defined globally, which can only occur if the chiral symmetry is spontaneously broken.

Hence quantum corrections to

topologically non-trivial gauge configurations induce spontaneous chiral symmetry breaking.

REFERENCES 1.

D. J. Gross and R. Jackiw, Phys. Rev. D6, (1972) 477.

C. P.

Korthals Altes and M. Perrottet, Phys. Lett. 39B, (1972) 546. 2.

G. 't Hooft, Phys. Rev. D14 (1976) 3432. Rev. Lett.

~

(1977) 121.

D. G. Caldi, Phys.

C. G. Callan, R. F. Dashen and

D. J. Gross, Phys. Rev. D17 (1978) 2717. 3.

S. Coleman and E. Witten, Phys. Rev. Lett. 45 (1980) 100.

4.

G. Veneziano, CERN TH-2872 (1980).

5.

K. C. Chou, ASITP-80-005 (1980).

6.

E. Witten, Nucl. Phys. B156 (1979) 269.

227 PHYSICAL REVIEW D

1 JUNE 1981

VOLUME 23, NUMI:IER 11

Possible SU(4), X SU(3)J X U(l) model Chong-Shou Gao StQnford Linear Accelerator Center, Stanford University, Stanford, Calljornia 94305 and Department of Physics. Beijing University. Beijing, People's Republic of China'

K uang-Chao Chou Institute a/Theoretical Physics, Academia Sinica, Beijing, People's Republic nlChina IReceived 14 October 1980) An anomaly-free model of strong and electroweak interactions involving leptons and quarks in the

SU(4), XSU(3)jxU(I) gauge theory is constructed. After spontaneolls symmetry breaking, it reduces to quantum chromodynamics for strong interactions and a broken SU(J) X U( 1) model for electroweak interactions. As a limiting case it gives the same results as those of the Weinberg-Salam model in the low-energy region. The Weinberg angle is bounded by sin'e" < 1/4 and becomes slightly less than 30' in the limiting case. Below the mass scale of SU(4), breaking there exists an inequality between the Weinberg angle and the strong coupling constant. which is consistent with experiments. A correction to the neutral current of the Weinbcrg·Salam model is suggested. A new conserved quantum number is introduced in this model and there exist several new fermions with masses lighter than 160 GeV. The Kobayashi-Maskawa expression of Cabibbo mixing for quarks may be obtained in the model, generalized to include several generations of fcrmions.

l INTRODUCTION

Recent neutrino-induced neutral-current experiments are in agreement with the expectations based on the Weinberg-Salam model. ' The Weinberg angle 8w is found to be sin28w = 0.230 ± 0.009, averaged over the various experiments. 2 Beyond the Weinberg-Salam model one may ask the following. (1) Is there any symmetry higher than SU(2)L x U(l) for the electroweak interaction? (2) Does sin28w being slightly less than t have special physical meaning? (3) How can the Weinberg-Salam model be unified with the strong interaction? If the answer to the first question is "no," the next problem to be solved is the grand unification of the Weinberg-Salam model with the strong interaction. In this way one may construct a model of grand unification, such as the SU(5) model suggested by Georgi and Glashow. 3 If one thinks that the answer to the first question is "yes," this leads to another question: What kind of higher symmetry might this be? There are several considerations which may become the motivations to choose the higher symmetry group: (i) left-right symmetry before spontaneous symmetry breaking, (ii) quark-lepton unification, and (iii) a sensible prediction for the empirical Weinberg angle. Among the many possibilities meeting these criteria is the SU(3)X U(i) group for the electroweak interactions. In a previous paper' a model with the SU(3)x U(i) gauge group was proposed. This model is left-right symmetric before spontaneous symmetry breaking and anomaly free. It gives the same

results as those of the Weinberg-Salam model in the low-energy region in a limiting case. The Weinberg angle is bounded by sin 2 8w "; t in this model and sin28w becomes slightly less than t in the limiting case. In this paper we discuss a way of unifying the strong and electroweak interactions by embedding this SU(3)x u(1) model into a larger one, SU(4)c xSU(3)x U(l), where the main results of Ref. 4, including the interesting property of sin 2 ew ,,; t, are preserved and several further consequences are obtained. Before discussing this model we will briefly analyze the construction of the SU(3)x U(l) group, which will be helpful in understanding the motivation of an extension to SU(4)XSU(3)XU(1). When one embeds the SU(2)LX U(l) model into an SU(3)x U(l) model, a naive requirement is that "L and e L correspond to the first two components of a left-handed triplet and e R correspond to the third component of the right-handed triplet of the SU(3) group. There are two pOssibilities. Case A. "L and e L belong to the representation ~L and e R belongs to the representation ~R of the SU(3) group. This case has been investigated by Lee, Weinberg, Shrock, Segre, Weyers, and many others" in detail. Case B. "L and e L belong to the representation l L while eR belongs to the representation ~j" the conjugate representation of~. This case is investigated in Ref. 4. These two cases lead to different consequences, as summarized in Table I. In the expressions for the charge operator, j 3 and j, are the third and the eighth generators of the SU(3) group, respect2690

© 19R1 TIle Americ;)n Physical Society

228 23

2691

POSSIBLE SU(4), X SU(3), X UtI) MODEL TABLE I. Comparison of the two possible schemes in the SU(3)x U(l) group for electroweak interactions. Case A

Case B

Charge operator

Q;i3 +(1/,r:nia+ f

Q;i3 -FJia+Y

Sin 201V

1 3 41+(3g'/g")

1 1 41+(g'/g")

Boundary

i 4>, + b, (trot> T4>, )2J + c(trot> ~ 4> D)(tr4> 1ot> A)

I=A

(3.2)

where a's, b's, c, d, e, andf> 0,

c' ot> D' ot> E' and ot> F' respectively. As shown in Appendix A,

230 23

2693

POSSIBLE SU(4)c X SU(3), X U(l) MODEL

D respectively. We make both v A and v D positive by suitable choice of the phases. We further introduce an additional term as

APPENDIX: THE FORM OF THE HIGGS POTENTIAL

d[ tp b, tp DltpB*1 i.1 tp AI E J>I + tp Ditp bltp B{r.} tp ~,E I>']

There are six Higgs multiplets introduced in this model. Their transformation properties are 4> A: (!.,~,o,-2), 4>B: (!.,~·,O,-2),

(A1)

tr1

,A.),

of

genr:rators

of

liIr:

group

C;.

Thr

I,'adrleev-['npllv

',l'ilielt sali:;fi~~

(;).2)

F;=Fs(¢(g,rpo),g1),h(g,1)o)(.3.,-L~J,,)h-l(g, n - 1) ,l( F ../) 6.( e i

.< B

J11 ( g) d

0 ( F D i'> 5 0:;., r

ri J 1> n . i

(6.5)

If the gauge conditiun IdA Jl1 8 (CPo - 1)8(FA)~A J[dcpdAJl18(FA)~A + iJiCPi(CP)}

exp{ iSB + iJiCPn

exp{iSA

= exp {iJiCPi [i:K] }ZA [K1IK=o' rl[

(6.6)

where cp.I (cp) = cP Igo I'1'0'" _ \. The formula (6.6) can be used to yield relationship between the Green's functions in the general gauge and that in the renormalizable gauge. On the mass shell the two generating functionals have no other differences except for changes in the renormalization constants of the external lines. So the S-matrix is the same when the gauge condition is changed. Thus we have demonstrated that the theory of pore-gauge field on coset space can be quantized and is renormalizable in arbitrary gauge provided that it is so in the gauge CPo = 1, FA = O.

References [1 J

ChOll

F\.lIilng-chao,

Til

Tung-sheng.

RUal1

Ttl-nan,

scielltia

Sinica,

XXH (1979), No.1, 37. [2 J [3] [4] [5

J

V. N. Popov and L. D, Faddecv. Phys. Letts, 25B (1967), 29. F. A. Bere3in, The method of second quantiza (Academic Press. :\c'" Yurk, 1966), 49. C. Becchi, A. Ruuct and R. Slora, Comm. Math. Phys., 42 (975), 127. R. \1/. Lee, :\lelhods in Field Theory (Session 28 of 1.1'.5 ·Houchos 1975), 79, and I he re f ercnccs quoted there.

*)(1'1j ffl F add ee v-P 0 pov t~ T55tnX: T RHjl: ~HJ) ~1~J!ire:#4(J9l1*t2ff(5} ii:-J-ft. ~1E SjJ 7 f:1I1 1~Ji';: B. R.

cp 0 =

s.

1, F .• = 0

J>f. ~K ~Jf HjJ .

~~"f~~(J~. F.l:!tll:~tt\Y W'ard-Slavnoytf~A. d!f-:fI~tEm1lt',*,H

"f ~ liT mJJ: It Jl~, rt; tE;fI~ fli1m m: TIlL ~ PT 1f! ~ it ag. j;3: nr dl

s ~€ ~41 fI~ mffi X

249 KEXUE

Vol. "27 No.2

TONGBAO

February 1982

THE U(l) ANOMALOUS WARD IDENTITIES AND CHIRAL DYNAMICS ZHOU GUANGZH.\O

(mIJ'tB)

(Institute of Theoretical Physics, Academia Sinica) Received August 20, 1980.

It is generally believed that the QCD Lagrangian has N1 conserved axial currents except for quark mass terms. In the world of zero bare quark masses for the first L flavours, the corresponding conservation laws are spontaneously broken and L' pseudoGoldstone bosons are thought to be generated. For three quark flavours the absence of the ninth light pseudoscalar in the real world is a well-known puzzle first pointed out by Glashow m and studied subsequently by Wienberg"\ Kogut and Susskind"] and many others'4- B]. In 1976 it was observed by G't Hooft'4] that instantons might help resolve the paradox through the anomaly in the axial U(l) channel. However, Crewther[5] has shown in a series of papers that instantons with integer Pontryagin number are impossible to satisfy the anomalous Ward identities in WKB approximation. Recently, a very interesting proposal based on the analysis of anomaly III the framework of the liNe approximation has been made by "\Vitten'6]. According to his opinion the U(l) problem can be solved at the next-to leading order of its liNe expansion, due to quantum fluctuations in the topological charge density. In this note we argue that the idea underlying Witten's proposal is a general one not necessarily connected with the liNe expansion. To P' order in the low energy limit both the mass and the wave function renormalization constant for the 1'(' meson are determined by the quantum fluctuations in the topological charge density. In thc world of zero bare quark masses, the Lagrangian has the form

st'

= - ..!.. F~.,Fal" + qil/Jq - 8(z)v(z) - J(z)O(z), 4

(1)

where

is the topological charge density. In Eq. (1) a set of hermitian composite fields O,(z) is introduced with J,(x) as their external sources. The Lagrangian (1) is invariant under U(L) X U(L) chiral group with L quark flavours, except for the source term J(x) O(x). We shall put 8(x) =8, J,(x)=O only at the final stage of the calculation_ The currents for the U(L) X U(L) symmetry are

250 148

KEXUE TONGBAO

.

JI" =

1 . 2 l'

.,.

i'J.yl"_,1,.q

_

Vol. 27

1·

(2)

J5'1 = qyI"Y5 -2l q 1 ,

where ii indicates the Gell-Mann matrices with

10

= .../ 2/ L.

We shall include the currents to the composite fields and write their external sources as Jl'i and JpSi respectively. Sometimes we use O(x) and J(x) to denote romposite fields other than the currents and their sources. The generating functional for the connected Green's functions is defined as

(3) where

OJ

indicates all the operators written in Heisenberg picture.

The vertex functional [[O(x), J~(x), JI'51(X)] is related to 1V [J(x), lui (x), JpS;( z)] by a IJegrendre transformation

[[O(z), Jl'i(Z), JpSi(Z)]

=

W[J(z), JI';(x), JpS;(z) 1

(4)

-1.f'(x)0;(:r;)d4x, where O,(x)

IS

determined by

O,(z) =~. lJJj(x)

(5)

In Eq. (4) the sources relating to the currents are not Legrendre-transformed. .Eq. (4) it is easily convinced that

Sf

8f

8W

F.rom

(6)

ojpSi (x)'

We assume that the composite fields O,(z) form a linear representation of the (!hiral group. The Ward identities for the generating functional have the forms 81'

81'

"11' "TV = iJ(x)Ti ~(y)' 8J I'lx) BJ x U

01~) = iJ(x)T5i

lJJ 1'5i X

SW

lJJ pSi

+ "'/2L(v(x)Oio'

(7)

In (7) , Ti and T5! are the representation matrices of the group generators. The axial U (1) anomaly term

V -2L (v( x»

.

can also be WrItten as

sW V -.2L -(-)-.

S8 z These Ward identities have been justified by Crewther and used in literature to study the U(l) problem.

It is more convenient to start from the Ward identities for the generating func"tional of the vertex functions. They have the form

251 KEXUE TONGBAO

No.2

149

(8)

(9) The Ward identities (8) and (9) are satisfied by any functionals of O(x), J~;(x), e(x) and are invariant under the following infinitesimal local gauge transformation

J "'; (x) and

O(x) - (1 + iakc)Ti + iaSi(x)TsJO(x), Jl'i(X) - Jl'i(X) - iJl'k(X)hiiCti(X) - Jp.six)dkiiaS/x) - BI'Cli(X),

(10)

Jpi(x) -J,,:;;(x) - iJplx)fkiia.(x) - Jl'k(x)dwas.(x) - BI'Clsi(X), e(x) - e(x) -J2LClso. Therefore the vertex functional formations.

IS

invariant under the above local gauge trans-

In the following we shall choose the composite field O(x) to be the '2Lz scalar and pseudoscalar biquark densities

q

Ai q and illY 5

2

1i q. 2

Tht'y l:an be representpd by

an L X L matrix

(11) and its hermitian conjugate, which transforms under [O(L)

X U(L)

into

(12) where Y L and VR are unitary matrices belongiug to the left-handed and right-handed U (L) groups respectively. The vacuum average yalue of fJ: will be denoted by U. For L = 3 one can construct only four SU(3)X S["(3) invariants from U and its hermitian conjugate U+. They are tr(UV+), tr(UV+VV+), det(VU+) and det V/det V+, the first three of which are automatically U(3) X U(3) invariants. Under the axial U(l), In ( (det U/det U+ 'j transform in the following way In (det U / det U+) - In (det U/ det [;-+) +i2 J 2LCtso and

e-

(13)

.2... In (det UI det U+) is an invariant. 2

Within the limit of long wavelength and low frequency, the vertex functional should be a function of these invariants only. Hence after putting the external sources· to zero and e(x) = e, we have

252 KEXUE TONGBAO

150

Vol. 21

r =F [tr(UU+),tr(UU+UU+),det(UU+),8 - ~ In(detUjdet U+)].

(14)

The symmetry U(L) X U(L) is broken spontaneously down to the parity conserving U (L) . It is argued in [9] that to the second order in energy momentum P of the pseudoscalar Goldstone bosons

uu+ bolds after a suitable normalization. masses we can always put

1-f2 2 •

=

(15)

In chiral symmetry limit of zero bare quark

(I6) "here

Jr.

is the 1( boson in this notation.

From Eq. (16) we see that 8 - i..ln(detUjdetU+) 2

=

8

+ ~~L t

(17)

1)'.

x

The gauge invariance conditions (10) and (13) then imply that the vertex functional for 7J' and e should be constructed from the inyariant 8(x)

+ ~2L

(18)

"I}'(x)

L

and the covariant derivative

(19) The effective Lagrangian L.u (x) is related to the vertex functional by

(20) and can be constructed from the invariance properties required by the Ward identities. it has the form To the order of

P"

rR e f _ f -1 ;;z, -

A.(8+~2L --"I}')D

1)

'Da' '1)

2

f"

+ 1f A. 2 " ~8

(8 + ~f""I},, 2L ')(a 8+ J 2L a ') D" ' f" ,,1] 1]

+ 1- f 2

- E

A. " 8

"

(8 + J f"2L 1]" ')(a 8+ Jf"2L a ,)2 ,,1]

(8 + J2L 1]'). f"

(21)

253 KEXUE TONGBAO

No.2

The wave function renormalization constant for the

z~, ;1nd the mass of the

7)'

= (A(e) meson is

+ .j2LA~eCe) + m',

151 1)'

meson is equal to

2LAe(e»-1

(22)

2L dIE Z ,

=

(23)

j!de2~·

?

From Eqs. (20) and (21) one obtains

r

J Be

(Br ) e-iP"rd"xle(x)=e x Be( 0

=

dIE _ PlnAeCe)

de 2

dZE

'rherefore, both dez

+

=

-i

r (T(iJ(x)iJ(o»)ee- iP .xd x 4

J

O(p4).

(24)

and Ae(8) are related to the quantum fluctuatuions of the topo-

.lugical charge density in the 8-vacuum. We also find from Eqs. (20) and (21) that

r

J BJ pSo =

r

t~

x Be( 0 )

e- iP-xd4x Ie(x)=e=-i

.l.... P'"f;A~eCe) 2

+

r (TcJ;o(x)iJ(o»)ee-iP-xd~x

J

(25)

O(P3),

-iP·xd4 I -·P'"f' x e(x)=e = t "

Blr

J BJpSo(x)B7!'(o) e =

iP'"fiA(e)

+

.jLj2A~8(e»

+

(26)

O(P3).

It is easily verified that in the liNe expansion

AeCe) ::::: O(ljNJ,

(27)

A~eCe) = O(ljND,

a.nd

f ~<e)

::::: f ~(1 + O(l/NJ).

Therefore to the leading order in liNe, Eq. (23) reduces to Witten's formula • = -2L (d- E)' = 1, m;, 2

z~,

f;

dez

(28) nO

Quarks, 0=0'

To the next order in liN. we may neglect A,. in Eq. (22) and find

(29) where A.(O) is related to the quantum fluctuations of the topological charge density b~r Eq. (24). It is interesting to note that in gener al Z'1 is not equal to f ,Jf ~'. A full analysis of the effective Lagrangian for low energy pseudoscalar mesons

254 152

KEXUE TONGBA.O

Vol. 27

with bare quark masses taken into account can be carried through without difficulty. Results on e periodicity and the like can be obtained in the general case in accord with those given by Di Vecchia, Veneziano and Crewther[··71. These questions will be discussed elsewhere. REFERENCES

( 1]

Glashow, S. L., in Hadro1l.ll and Their In.teractions, Academic Press Ine. New York, (1968). 83. l 2] Weinberg, S., PhY8. Rev., Dl1(1975), 3583. r 3] Kogut, J. & Susskind, It-. ibid., Dll (1975), 3594. [ 4] G't Hooft, ibid., 37 (1976), 8. [51 Crewther, R. J., Riv. Nuovo Cimento, 2(1979), 83; Phys. Lett., 70B(1977), 349; CERN, TH2791, (1979). [6] Witten, E., Nucl. Phy&., B156(1979). 269. r 7] Veneziano, G., ibid., B159(1979), 213; Di Vechia, P., Phys. Lett., 85B(1979), 357; Di Vecchia. P. & Veneziano, G., CERN TH-2814, (1980). [ 8] 1\ath, P. '" Arnowitt, R., CERN, TH·2818, (1980). [9] Lee, B. W., Chiral Dynamics, Gordon & Breach, New York, 1972.

255 Canlm. in Tlleor. Phys.

vol.

1, No. 1

(1982)

69-75

NEW NONLI NEAR a MODELS ON SYl"1[~ETR I C SPACES CHOU Kuang-chao ( J~!';; (Institute of Theoretical Physics, Academia Sinica) SONG Xing-chang ( 51: :H: ·(Institute of Theoretical Physics, Peking University) Received July 9, 1981. Abstract Two new nonlinear

a models,

defined on the symmetric coset

spaces GL(n,c)® GL.(n,c)/GL(n,c) and GL(n,c)/U(n) respectively, are

fo~lated

in this paper.

The latter may be useful in discus-

sing the four-dimensional Yang-Mills fields.

In previous papersl1,l], we have studied various classical nonli~ear a models taking values on symmetric coset spaces and have sho~vn generally how to use the duality symmetry to deduce the infinite sets of conserved currents, both local and nonlocal. In these papers, a unifying point of view is proposed to ~onnect varieties of integrable 0 models which have attracted our attention in recent years. The same point has also been noticed by Eichenherr and Forger(3] who have proved that, for compact global symmetry groups, the a model has the dual symmetry if and only if the field takes values in a symmetric space. The main feature of our formulation is to use the so-called "gauge-covariant current" ff,t. (3] as the central role to formulate the equation of motion, the duality symmetry and to deduce the conserved currents. The important advantage of this formulation is its unification and simplicity. In this note, we shall extend our study to other models, such as the GL(n,c) prinCipal model, and a new model Hn which takes values in the symmetric space G1(n,c) /U(n). For the purpose of application some of the notations used previously will be changed slightly. As mentioned previously(2,31, the U(n) principal field defined on the homogeneous space U(n)/{l} can be re-formulated as a o·model on symmetric space U(~ ®U(n)/U(n). Similarly we consider the model on symmetric space GL(n,c) ® GL (n,c)/GL(n,c). An element of the group G=GL(n,c)~GL(n,c) is expressed as (1) with both gL and rule+)is

gR

belonging to GL(n,c) independently.

The multiplication (2)

+)

++)

Xbe gR defined here is just the g; in the Ref. (2J. A factor -i appearing in Ref.(2] is omitted bere. J.l is suppressed.

Sometimes the Lorentz index

256 70.

CHOU Kuang-chao, SONG xing-chang

An element of the subgroup ~ E H = GL(n,c) is represented by . (3)

which is invariant with respect to the involution a (4)

Then G/H is a symmetric space with involution a, whose element is ~

= (

cp.

cpt-!).

(5)

Now any element of G can be decomposed into the product of a subgroup element and a left coset element g=
(9)

and (10)

we have (11)

.md then

257 New nonlinear cr models on symmetric spaces

-tTrJC}LJC'''=

71.

+r,. (~;E~LC1t.c) •

(29)

The involution operator which makes H invariant is (30)

Now define

259 New nonlinear cr models on symmetric spaces

n=~-I

Cli!

=

73.

(<j>-I
and

a....... J..II. -a,Jj''

1>-') cf .6p

(3.4)

'fH ,

whereas the covar:~nt derivative for Ku is (3.3 ' )

The hlaurer-Cartan integrability conditions (3.5a)

and (3.5b)

imply (3.6a)

t.,..

K,..

K" - 0 (i=l,2 ••• n) are eigenvalues of the positive definite hermitian matrix (K ~ K~)?t :: A • For four-dimensional Euclidean space (p=l,2,3,.4), Hp is still the SU(n) gauge field. By using Eq.(3.6b), the field equation (3.l0b) takes the form

269 192

CHOU Kuang-chao, SONG Xiang-chang

(3.14)

This equation together with Eqs.(3.6) describes the whole system of fourdimensional ~ models. Different from the two-dimensional case in which the gauge field ~ is sourceless[15], Eq.(3.6b) shows that H~ is not sourceless in four-dimensional case. Now we restrict ourselves to discussing the so-called self-dual En models, i.e., models in which the anti-self-dual part of the field strength vanishes. Then Eq. (3.6a) splits into the following form

Fr1=-{ I~

+s - Sine Cl

COSI Ay t Si"e AJ ;

= Co.s8 t-'t,-Sid 'r'fy;' Si.e Cy = c.oSSAS - Si,,8A y •

Sin.e c y .

277 200

CHOU Kuang-chao, SONG Xiang-chang

w'/= - ~ e ("'1 wt =

tJ'

T IN

('Ny t VI

Then the existence of W implies

J~

)

Jy )

Wyz - WZ!I = O. 1. e. , (4.25)

Expanding W in powers of para!lleter F; = tg e. an infinite number of non local continuity equations given by POhlmeyer[8] follows. Using an inductive procedure Ogielski et al.[22) have also constructed a set of nonlocal conservation laws. The following important fact can also be seen from Eq. (4.24). [nUke w- 1 Wu. which are local functionals of field q as in Eq.(4.23). w- 1 Wrr may involve some kind of compLicated integrals. This means that Cu and Crr will undergo nonlocal transformations under the rotation (4.18). So it is impossible to obtian the local conservation laws by applying to e u and Cli the "duality-like" symmetry linked with rotation (4.18). However. in spite of this nonlocal transformation property. the existence of W in Eq.(4.24) preserves the locality of the covariant derivatives of Cu and Cli' so that the Maurer-Cartan identities (4.14) and the self-dual condition (4.l7b) are still satisfied by themselves. This can be checked by direct calculation. From the above discussion we can see that great similarity exists between the four-dimensional Hn models and the classical Yang-Mills fields. Both theories deal with the same sort of field ~ or q which undergo the same kind of gauge transformations and are subject to the same set of constraint equations-Maurer-Cartan ider.tities, Eqs.(3.6) or (4.14). Their dynamical equations. the field equation (3.14) in the former and the self-dual equation (4.17) in the latter. are similar in form but different in content (since A~~H~). The maiD difference lies in the fact that ~ in the former is an exact scalar while in the latter ~ behaves as an ordinary scalar only for the transformations of the subgroup SG(2)xU(1). Rotations like the one considered in Eq.(4.18) induce local SL(n,c) transformations in internal space. Therefore the self-dual Sr(n) gauge fields contain more solutions than that of the corresponding Hn models. From the forlllal resemblance between Eqs.(3.14) and (4.17), we can see that the symmetry (3.20) holds and the scalar current j~ =

t,. Cwo 6

can also be defined in gauge theories as in Eq.(3.17). This current will be "quasi-conserved" (dujBU =0) if and only if DuB is trace orthogonal to Cu' Then similar to Eq.(3.18) we assume that DuB has the form

])y B =

a. C1 t b 8 CJ B + C, (6.

.])5 B=-Il.

e,l t c. {B, 'J}

ey-b B Cy sH,(S, Cjl-C.{s, Cy} •

278 The Hn sigma-models & self-dual SU(n} Yang-Mills fields in four-dimensional E.1Jclidean space

201

and the constraints imposed bi the compatibility condition are again

C, = o. Making the choice

a

= b = c2= 1,

expaning B(Y) in powers or Y

inserting it into Eq.(4.27a) and collecting the terms of the same order in Y, we obtain a set of equations to determine b k . Just like the situation in Eq. (3.28) the equations for b k (k ~1) yield non local solutions from which follows another set of nonlocal continuity equations. This result is similar to that given by Pohlmeyer[8]. Similar equations can 3.1so be derived from the parametric Backlund transformation.+) For those solutions of the self-dual SU(n) gauge theories which satisfy an additional condition

H

(and then FyE=O, by taking hermitian conjugate), we see that the compatibili"y condition gives C,

=

0

as in Eq.(3.19), but leaves a,b and c2 arbi'trary. Then following the sarne procedure as given in Eqs.(3.20) through (3.24), we obtain an infinite number of local conservation laws as in Eqs.(3.25) (with the replacement l\.u+D:z' 80: Ku+C u ' etc.) The self-dual Hn sigma-models discussed in the preceding section are the special cases for tbis kind of solutions.

V,

SUIrJr.a ry

In this paper the Hn a-models both in two-dimensional and four-~imensional Euclidean space are formulated, and the classical self-dual Yang-Mills fields are analysed in terms of the quantities defined in the ~ models. We have shown that a close resemblance exists between these two theories. In some twodimensional planes the self-dual gauge fields reduce to the En models in twodimensional space. And in whole four-dimensional Euclidean space, the selfdual gauge fields have a great similarity with the Hn models, partially in content and partially in form. The essential difference lies in the fact 'that the field ~ in the Hn models is an exact scalar under the spacial rotations oISO(~) while, in the self-dual gauge theories, • is a scalar only for the transforma+) From Eqs.(I4} of Ref.(9], the following Contizluity equation

D=aJ

tT

tt' J' t

i'-'lylt al t,. l 'Ii'}'

1"

r-' h)

..

can be derived, where q is a solution of the field equation and q' another solution lihich dependes 011 a par_ter Il i.JlIplicitly. Expanding q' in powers of y ('/'=£ ~(")yl"), yZ ~

-(l-;}/(l+))), tore obtain an infinite set of conservation laws. However the BT gives the solution of q(kJ (1c ~lJ being nonlocal functionals of q, ajJq, etc.

279 202

CHOU Kuang-chao, SONG Xiang-chang

tions of the subgroup SU(2)XU(1) aRd the rotations otaer than SU(2)xU(1) give rise to non-linear local SL(n,c) transformations on ~ in internal space. By using the known structures of two-dimensional Hn models as a guide, an infinite number of nonlocal conservation laws in matrix form has been obtained for both four-dimensional Hn models and the self-dual Yang-Mills fields. The "duality" symmetry often used in two-dimensional non-linear a-models gets a new feature of being coordinate dependent in four-dimensional self-dual gauge theories. Another set of symmetry can be defined in four-dimens~onal theories from which another set of infinite continuity equations in trace form,the analogy of the two-dimensional local conservation laws,can be cons~ructed. The meaning of this set of continuity equations has yet to be clarified. This set of equations turn out· to be local expression for a restricted set of solutions, e.g., the four-dimensional self-dual Hn mOdels. It might be worthwhile to study corresponding problems for the Yang-Mills theories in loop space form and to investigate the associated quantum theories.

References ~(1976)

1.

K. Pohlmeyer, Comm. Mach. phys.,

2.

M. Luscher and K. Pohlmeyer, Nucl. pnys.,

207. ~(1978)

V.E. Zaharov and A.V. Mikhailov, (Sov. Phys.) JcTP H. Eichennerr, Nucl. Phys., 8146(1978) 215; H. Eichenherr and M. Forger, Nucl. Phys.,

46.

~(1978)

~(1979)

~(1979)

1017;

544;

381;

V.L. Golo and A.M. Perelomov, Phys. Lett., B79(1978) 112; M. Dubois-Vio.1ette and Y. Georgelin, Phys. Lett., 3.

~(1979)

251.

A. d'Adda, M. Luscher and P. di Vecchia, HUcl. Phys., B146(1978) 63;

M. Luscher, Phys. Lett.,

~(1978)

465;

E. Witcen, Nucl. Phys., B149(1979) 285. 4.

A.A. Migdal, (Sov. Phys.) JETP £(1975) 743; E. Brezin, J. Zinn-Justin and J.C. Le Guillou, Phys. Rev., £!!(1976) 3615; 4976; S. Hikami, Prog. Theor. Phys.,

~(1979)

226;

A. Mckane and M. Stone, Oxford Univ. preprint (1979). 5.

A.M. Polyakov, Phys. Lett., B59(1975) 79;

6.

A.A. Belavin and A.M. polyakov, (Sov. Phys.) JETP Lett.,

7.

D. Gross and A. Neveu, Phys. Rev., DI0(1974) 3235;

E. Brezin, S. Hikami and J. Zinn-Justin, Nucl. Phys., B165(1980) 528. ~(1975)

245.

B. Brezin, and J. Zinn-Justin, Phys. Rev., Lett., 36(1976) 691; W. Bardeen, B. Lee and R. Shrock, Phys. Rev., £!!(1976) 985. 8.

E. Corrigan, D.B. Fairlie, R.G. Yates and P. Goddard, Comm. Math. Phys., 58(1978) 223; Y. Bribaye, D.B. Fairlie, J. Nuyts and R.G. Yates, Journ. Math. Phys., !!(1978) 2528; M.K. Prasad, A. Sinha and L.L. Chau wang, Phys. Rev. Lett., £(1979) 750;

9.

K. PohJueyer, Comm. Math. Phys., ZlJ1980} 317. For a review see L.L. Chau Wang, in Proc. of the 1980 Guangzhou Conf. on Theoretical Palticle Physics,

10.

~980,

pl082.

A.A. Belavin and V.E. zabkarov, Phys. Lett., !Z!(1978) 53.

280 The Hn sigma-TIIOdels

11.

&

self-dual SUrD) Yang-Hills fields in four-dimensional Euclidean sl=ace

A.M. Polyakov, Phys. Lett., BB2(1979} 249; S.Y. WU, Physica Energiae Fortis et Physica Nuclearis, I. Ya. Aref'eva Lett. Math. Phys.,

~(1979)

~(1979) 382;

241;

L.L. Chau Wang, talk at the xxth Int. Conf. on High Energy Physics, Wisconsin, July, 1980. 12.

M. Luscher and Pohlmeyer, Ref. (2); A.T. Ogielski, Phys. Rev.,

~(1980)

406.

13.

E.Brezin, C. Itzykson, J. Zinn-Justin and J.B. Zuber, Phys. Lett., B82(1979) 442:

14.

H. Eichenherr and M. Forger, Ref.(2};

15.

K. Pohlmeyer, Ref. (1):

H.J. de Vega, Phys. Lett., BB7(1979} 233. K.C. Chou and X.C. Song, ASITP preprint 80-008, to be published in Scientia Sinica. A.T. Ogielski, M.K. prasad, A. Sinha and L.L. Chau Wang, Phys. Lett.,

!!E..( 1980}

387:

K.C. Chou and X.C. Song, ASITP.preprint 81-009. 16.

H. Eichenherr, Phys. Lett., B90(1980} 121: K. Scheler, Phys. Lett., B93(1980} 331; F. Gursey and X.C. Tze, Ann. of Phys., !!!(1980} 29; K.C. Chou and X.C. Song, ASITP preprint 80-010. to be published in Scientia Sinica.

17.

M. Luscher, Nucl. Phys., B135(1978) 1.

18.

A.M. polyakov, Trieste preprint IC/77/122 (1977).

19.

D. Iagolnitzer, Phys. Rev.,

20.

A.B. Zamolodchikov and Al. B. Zamolodchikov, Hucl. Phys., B133(1978} 525.

~(1978)

1275.

21.

A.M. polyakov, Ref. (11).

22.

M.K. Prasad, A. Sinha and L.L. Chau Wang, Phys. Lett., B87(1979} 237.

23.

K.C. Chou and X.C. Song, ASITP preprint 81·-003, Commun. in Theor. Phys. 1 (1982) 69.

24.

C.N. Yang, Phys. Rev. Lett., 38(1977) 1377.

25.

K.C. Chou and X.C. Song, Ref. (14) and (16).

203

281 Commun. in Theor. Phys. (Beijing, China)

725-730

Vol. 1, No. 5 (1982)

ON THE DYNAMICAL PROBLEMS AND THE FERMION SPECTRA IN THE RISHON MODEL CHOU Kuang-chao ( JlIBOCD CERN

~d

Institute of Theoretical Physics, Academia Sinica Beijing, China

DAI Yuan-ben ( liUt:.fz ) Institute of Theoretical Physics, Academia Sinica

Received May 6, 1982

Abstract We propose to introduce a strong coupling

U(l.' y axial

gauge boson into the risbon model which may explain dynamical problems concerning preservation of chiral symmetries and existence of solutions to anomaly conditions.

Mechanisms through

which exotic particles gain masses are discussed.

It is found

that exotic particles such as color octet leptons and color sextet quarks may gain masses ranging from

10 1

to

10 3 •

Some

phenomenological aspects of these exotic particles are discussed.

Among various composite models proposed for leptons and quarks the rishon

model~] has several special interesting features including economy and fitting in well investigated gauge theories. Powever, there are embarrassing problems of dynamical nature for this model among which are the following, 1. Within leptons and quarks the color SU(3)c interaction is much weaker than the hyper-color SU(3)HC interaction. In the approximation Qc=O the theory is similar to QCD with 6 flavours. However, it is necessary to assume that some chiral symmetries are not broken by the SU(3)HC interaction in contrast to QCD wAere chiral symmetries are assumed broken spontaneousely. 2. There is no flavour number n independent solution to 't-Hooft anomaly conditions[2] for the Qc=O limit SU(3)HC gauge theory with SU(n)L x SU(n)R xU(l)v flavour sy~etry. On the other hand, it seems unlikely that the weak SU(3)c force can play an essential role in binding light fermions. 3. It is assumed that there are quarks in the configuration uL=(tLtL)VL but none in the configuration (tRtR)VL though space parts of these wave functions can have the same symmetry. Note also that the fundamental Lagrangian of the SU(3)HC x SU(3)c gauge theory has the symmetry t£o- tR with VL , VR fixed. 4. There are exotic particles (ttt) in 8 and 10 and (ttv) in 6* and 15 representations Of SU(3)c which are assumed to be heavy. How can it be achieved and how large are the masses of these particles? 5. In order to avoid contradiction with the observed long life time of the proton the scale A of the SU(3)HC interaction must be assumed to be

282 726

CHOU Kuang-chao and DAI Yuan-ben

larger than 10'GeV at least[3]. What can be the origin of the scale of the weak interaction GF1;o 300GeV i f A » G;l? In this note we shall suggest some ideas which may provide partial solutions to these problems. As a possible soluton of the problems 1 and 2 we propose to introduce an additional U(l)y gauge field interacting with the current

into the theory.

The local symmetry of the theory is assumed to be

We have used U(l)B_L to replace U(l)O in the original rishon model. The photon is assumed to emerge in the later stage when the B-L gauge boson is mixed with composite weak interaction gauge bosons in the low energy effective theory. Since contributions to U(l)y anomaly of t and V cancel each other ·the theory is renormalizable. The global symmetry of the theory is U(l)R x Zl:' where Zl~ is the unbroken discrete subgroup of U(1)x with

l)..= it~.r,s t

+ i. V~ l",s V.

We assume that the U(l)y gauge coupling constant gy is comparable to SU(3)nc coupling constant gHC. Since the. U(l)y interaction between the pair (tRt L } or (VRV L ) is repulsive, it may prevent the scalar composites condensating and thereby keep some of the chiral symmetries unbroken. In order to avoid contradicting the low energy phenomenology the U(l)y symmetry must be broken at a large scale A'. This can be realized by the condensate (VLVLVR )2 corresponding to a Major~na mass term of vR which is required also from phenomenology of the rishon model [4]. We assume that the scale A' lies within the range A> A' »G;I. To see that the U(l)y interaction prefers this condensation to (tt) or (VV) note thoat the average U(l)y interaction energy per rishon in a composite of n rishons is l.

I (2

:{!.

Z) ,

1y Tn" Y- ~ Yj

"

J

(1)

where y and Yi are Y charges of the composite and constituents respectively. This interaction energy is negative for the composites ,(VLVLVR )2 while positive for (tt) and (VV). The corresponding quantity for the SU(3)HC interaction is (2)

where C and C~ are the Casimir operators of SU(3)HC. From Eqs.(l), (2) and other considerations it seems plausible to assume that the condensate (VLVLVR ) 2 is fa~ored over other possible Y breaking condensates. This condensation also breaks U(l)B_L' U(l)R and Z12. Being vR Majorana mass term it transforms as (!, !*) under SU(3)CLXSU(3)CR. However, it is easy to see that effective interactions produced by this condensation in the rishon number conserving sector preserve Z12 and transform as representations of zero tri-

283 On

the Dynamical

P~oblems

and the Fermion Spectra in

th~

Rishon Model

727

ality with respect to both SU(3)~L and SU(3)2R such as 10'GeV and M > 40GeV does not contradict known experimental facts. The colored leptons in ! and quarks in £ discussed above can be produced in e+-e- or p-p collisions. They decay into ordinary leptons and quarks by emitting a gluon with a narrow decay width of the order of (15)

For h > 10'GeV, r is extremely small. Before decaying to ordinary leptons and quarks they should form exotic mesons or baryons. Low lying exotic baryons should have long life-time also. Low lying exotic mesons decay mainly into gluons, just like J/W or T. The colored lepton can also combine with a gluon to form a color singlet. In our opinion, the most feasible test of the rishon model is to search for such exotic particles in experiments which can be carried out with accelerators of the next generation. The above discussions have not provided an explanation of the problem of the scale of the weak interaction. However, in view of the complicated spectra -l- ,,300GeV can emerge from of the theory, it seems possible that a scale of GF the theory.

References 1.

H. Harari, Phys. Lett. 86B (1979) 83. M.A. Shupe, Phys. Lett. H. Harari and N. Seiberg, Phys. Lett.

!!!

~

(1979) 87.

(1981) 269, the Risbon MOdel, WIS-81/38.

2.

G.'t-Hooft, Cargese Lectures (1979).

3.

H. Harari, R.N. MOhapatra and N. Seiberg. Ref. TH-3123-CERN.

4.

H. Harari and N. Seiberg, Phys. Lett. 100B (1981) 41.

5.

s. King, Phys. Lett.

6.

J-H. Blairon, R. Brout, F. Englert and J. Greensite, Nucl. Phys. B180 (1981) 439. J. Greensite and J. Primack, Nucl. Phys. B180 (1981) 170. H. Kluberg-5tern, A. MOrel, O. Napoly and,B. Peterson, Nucl. Phys. B190 (1981) 504. J.R. Finger and J.E. Mandula, Quark Pair Condensation and Chiral 5YI1lll/etry Breaking in QeD. S. Weinberg, Color and Electroweak Forces as a So'Jrce of Quark and Lepton Masses. Dept. of Phys., University of Texas preprint. 1981.

7. 8. 9. 10.

~

(1981) 201.

C.B. Chiu and Y.B. Dai, Phys. Lett.

l£!! (1982) 341.

J. KOgut, M. S~one. H.N. WUld. J. Shiqe1llitsu. S.H. Shenksr and D.K. Sinclair. The Scale

of Chiral 5Y1lIllletry Breaking in OUantum Chromodyna1llics, ILL- (TH) -82-5.

287 SC I ~NT IA SIN leA (Series A)

Vol. XXV No.3

Mareb. 1982

WILSON LOOP INTEGRAL AND STRING WAVE FUNCTIONAL ZHOU GUANGZH'O n

(CHOU K UANG-CiiAO

EEl.llf a~ ,

(HI)

(3) where

f

=

1/1(ffi!,"fol'v) is a COIlstiLnt or at

As a result of these approximations,

Ale J would ubc'y

2

a A[C]

=

o:r.,.(s)axl'(s)

slo\vly Y u. we obtaiu

- a .8Y _ - A(e]. 8xl'(.~)

(~2)

.

where

8.9'"' (/x" ---=n"SI-.' . ds

(23)

OXI'(8)

and

I

_ a(x!',x") fJCcT,.r) ..

IIp.-

thcboundor,pDint of the :Ut"3. i?

II

S'~(s)

I

a(xI' , x") a(cT.-r)·

(24)

.

C11(,l!iting l<Jq. (10) against 1<Jq. (22), we han' vP~.(.r(s),

.

in the· ('on fining

yaCUlIDI.

Liiseher hns shown tllat lO •

. dx'·

J) = tls

o."i"

-(l,-_.-

i5 J"( S )

=

dx"

--all,." a",l}(.c)

=

iUJ(f).H,,(.r)] + (/~u..fl'·(J:) - b"p.~Ii(.J:)K'·(.e)B(.J:) + i!.'1 [Ii(.e). k,.(.r)] + l',t p..{ t:(.t), k,'C.J;)}.

(25)

Ii is "asily \,erifit'll ThaT for ill·hitrar.\- ClIlI:stallts II. b, 1.'1 auu !.',. tht' t:ousl'rvatioll ':lIl1llition- is satisfit'u. aml th,' iUT,>grauiliT.'- ,~t)JldiTiou for 1i(.J:) illlpost's a l!ullst.raint alHoUg thl's!' paramell'l's:

fib

( 1)

('I

=

1.

It

=

b

= (',

D.

=

+ c; +

C3

(:!(i)

1.

=

Then E,\. (:!.j) turns out to

btl

a/lex) = i[k Hd + iUi, K;1, } A

....

+

"

a.,B(J:) = i[J.J, II,,]

....

A

(27)

i[ n. Ii,).

This llleans

J(.c)

=

10 =

eOJlst,

anu then the fullowing normalization cunditiull mllst be satisfied:

trBZ(x) = trA2(.r) = CUllst.

(28)

Duality symmetry theJl implies that

j~(r)

= tr{Kir)B(7)}

(2~J )

is also a conserved current if B( 7) satisfies the following equations!

a~B(r)= i[B(r),H~]

+ i7[B(7),K~J,

(30a)

a/J(7) = i[B(r),H~]

+ i7-I[B(7),i~].

(30b)

299 720

SI~ICA (Scrie~

SCIENTIA

Now expanding B( y) ill powers of

A)

Vol. XXV

y-I

.

L: B. y-'.

B( y) =

(:31)

"=0

and substituting it into Eq. (aDa), one gets

[B.,Kd

-[B.-I.Hd -

=

ia}J.- I.

Evidently the first cot'fficient

Bo = satisfieos both Eq. (:32) with

II

(trKi{~)-tji:~

= 0 and the normalization condition

(2:;).

Explicit expressions for B.(n ~ 1) can be obtained from Eq. (:~::!) for C['l lIlodel or 0(3) chiral model. Fl)r thesp. models Eq. (2!l) f,riyt's un infinir,· num!J,.>r ilr local conserved currents When B( y) is expanded in powers of y-I as in E.!. (:31). aej.~

+

o.

a~j.~ =

j.~ = tr{KeB.L j.~

=

}

(:{..J: )

tr{K~B.-,},

with the first one equal to .

Jo~

;:.. )'" = (K'\. tr ~I1.~

.

Jo~

= I) .

Another set of conserved currents can be obtained if of y.

expanol~

OIl!'

T:r Y) irl

In general. Eq. (:32) does not lWI~I'ssaril:- ,\"ield lneal solmion for to the constraint equations,

8.,

Ilwiw.!

(:36) held for arbitrar.r

illtl~gpr III.

When the duality sYlllllletry with Y = -1 is applied to thi,; cast' Olll' will obtain t.he cllse ill which CI = -I,ll = b = C2 = O. (2)

CI =

C2

= 0,

·0

b = 1. Eq. (25) takes thl' form ...

a;B •

a~B

~

A

+

AI'" A

...

"

•

aK~

=

i[B, H;J

=

i[B, H~] - aK~

••

- a- BK.B, }

+

a-

I'"

BK~B.

(:17)

The llorlllalizat·ioll I;olldition (28) is compatibh' with Eq. (:37) if tht' addirional condition

(:38) is held simultalll-'ously.

For CPR models this conditiun coincides with Eq. (::!';).

The local cO!U;I'n"ed I;urreuts can be generated from Eq. (::!!I) with B( Y) satisfyimr the following equations:

a;B(y)

=

a~B( y) =

Expanding

+ i[B( 1'), H~] -

-i[B(y),Hd

Be y) in powers of

y-l

ayKe - a-lyB(y)KeB(y), ay-li(~

+

a-ly-1B( y )K~B( y ).

we get from Eq. (39a) that

(3!Ja) (a!Jb)

300 No. i

17·~rODELS

COXSERVATION LAWR FOR XOXLIXE.\R

7:!1

(40) (41}

It is t>vidl'nt that

Eo =

a is

it

solution of Eq. (4,0).

But this trivial solution iii Illlt,

uSl'ful sinct> it gives jol' zero identically.

n

For CPo models and their generalization, the C( 11 + m)!T( n) X III) mudds, or their orthogonal analogues, the Lie algebra valued function on coset spa\!t' ,an ~ tIt'composed in to two parts:

it

it =

Ki+) + k·-),

sueh that

j{.-> = (K'T»t. fCT>K'T)

=

!,' .j,:l)

k->K-·-j = O.

or

For The

r B'T>( r )lQ-lE(+>( r).

(-1:4)

with both l('+!k-) and j«-)K bt>lon~inl! to thl' Lil' algebra the subgroup. tIlt'sl' IllIMl,,!:s. E'I. (:l~l) split into TWO bran.:hl~", a Pl)siTiw Oil€' and anegati\,t' ont'o positi n' bral11'h of E". (:l!la) rl'ud"

a/J'Tl( r) = i[EiTl( r). H, REFERID;CES

[ 1]

Golo, V. r. &; Pt.>rt'IOIllOV, A. M., Letts. in lCath. PIIYS., 2 (1978), 4i7; PolY-.llcov, A. ~r .• in "Collective Effeet.q in COlll1en~etl Media", Proc. of 14t:, Winter School of Theoretical Physi.cal i'n Karpucz, Wroc·

lat', 1978.

[2]

Zaharov, V. E. & )Iikhnilov, A. Y., (SOli, Phys.) JETP, 47 (1978), 1017.

301 722

SCIENTIA SINICA. (Series A)

&; Pohhnl~·,·r. K .• X/(cl. PhYR .• 8137 (HI'i8). 41i. Eicheuhc·rr. R. oS: Po/'g,'I'. :\1.. ibid .• 81SS (1979). ,:81; BI'l'zill. E .• Itz)·kson. C. Zinn·.Juslill •. r. & Zuht'r, J. B .• Phy~. LPN., 828 (1979), 44:?; Ogil'l~ki. A. T .• Phy.•. Be,. .• D21 (If1S0). 40(1. Pohlm('yt'T. K. COmllll/lI. J/ath. P1I!ls .• 46 (197(i). 207. L'hereuni.k, 1. Y.. 1·/'for..lft/th. Ph.'l.•.• 38 (1979). 1211, Eichenherr. R .• PIt!ls. Lftt .• 908 (1980). l:!l. Scheler, K .• fbi,l., 938 (1!lSO). 3~1. Ogi('lski. A. T .. Pl-asnd. ),1. K .. Sinhu. A. &: C1mu '''nllg, L. L .• ibid., 918 (980). 387. Flume. R. &: .\re~"t't, S., -ibid •• 8S8 (1979). 353. Coleman. S .• WCfS ••J. &: Zumino. B .• Phys. Rei' .• 177 (1969). :?:?39; Callun. C. G. .11'. ('1 al.. ibid., 177 (I!l69). :!:?4'i; 8.11 Sill , A. &: Slrathuee, J .• ibid., 184 (1969). 1750.

r. 3] LllsehN.)1.

l -1] [5] [6]

[ i] [8] [ 9] [10] [11]

v.ol. XXV

302 Vol. XXV Xo. 8

SCI EN TI A

SIN I C A (Series A)

Al1gu~t

1982

LOCAL CONSERVA TION LAWS FOR VARIOUS NONLINEAR a-MODELS ZHOU GL'ANGZIlAO (CHOU Kt:.\NG-CHAO

(II/.~titl/tp

"f 7'heoreliClzl Ph!lliics,

fa! it B)

':/'clIl!p,mia Sinka. B.,j>;"!I)

AND SONG XINGCHANG

(;f:f:j"'*)

(Peki'Jlfl Uni!"er.~ity)

Up("iWd .Jannar:· !I, 1.981; re~i"ed Jllnl' :!';, 19t11.

ABSTIUCT

On the b:L·i~ of the ronnllmtion giYen in a prceeding paper, we I1erive an iniiuiti'l"e >;eries or conEerwu iO":11 I'urrpllt~ e:\"'Pli,'itl~- for the two-dimensional elassh,'al a-nIl1,lel~ ,'II th.· ",",'l!pJr.,. Grn~Sl1lnlln manifold C;(m + /I ) / [ ' ( m)r&U(n) :n:r1 for prill.'ipal ~hiral fiel'l.

rOn

III a previous Ps. tiellL'rally th,' La!!r:l1l!!i.all ,'''r rhe nonlinear C1-IlIUt.l1'! has the form

,,, = J.. .' p-'.I'. ""-"\..' .)"

.:.L.

whieh is iuyariallt undl'l' the I!lubal trallsforlllation of a '~OIIl[lilCT l.i.· ~l'IJIIP (; wilil,· 11l\:al-iIlYilriant under tht:: gaul!l· trall:sfllrlllatioll of a closed :subgroup J1 of (;. lIer,· ~. > uenotes the U invariant illll\~r product. For tlw fidd !l(.c) taking values ill U (!l(x) is equiva.lent to cP(x) in [1] lip to a ~!iLul!e trnllsformatiou). we define !j ...

= -i!l-I(.c)a,.!I(.r) = iI" + ft..

(2)

wh~l"L· H" allli K" are valued ill the Lie algebra of t111~ subgruup II awl its orthogonal eompl~lIlellt l:orre:spouding to the coset (;jH respectively. Both flu ami K,.. art:: lrerillitian when y( x) is unitary. Define the covariant deriYiltives as

D"y(x)

a,..y(;;) - ·iy(x)H,..(x).

=

(D,,!l(x) = a"y+(x)

+

-iH,,(.c)y+(.c) for y(x) uuitnr.,-).

then

K" =

-'iU-1(x)D"U(.c)( =iD"y(x)y(.c»).

(4)

SO H,. is tht:: composite gauge field whl'reas X,. is a vector field whil:h transforms covariantly under the gauge transformation. The Lagrangian (1) tan be written a.~

~ = l:.. st>Cond casf',

D"B(r;) = t. writtpll as

!L

=

-

1 i,K "" K"

2'"

=

1-

-

:!

=..l tr8 pa"l' = 2 'u

fr

D

VD'~!J

.11.

fr

DuoD";;. .

H"rp tht' covariant dprivative for .. fipld is defined as

D"z

=

a,,< -

-z:.• .I" ,

D"z

=

a"z + i.-l"z,

(34)

with tht' lJ(lIl) gaugt' field defined as

A"

= -.;za".,

For a matrix X transforming as D"X

zz, =

(A,,) •• == (II,,).;.

(:35)

the covariant derh'ative is defined as

al'x + i[A.".XJ.

The field equation is

[P, a,.8I'p]

== 0,

(34 )

307 830

Vo!. XXV

SCIEXTIA STXICA (Series A)

or

D"D"z

+

z{)"zD"z

=

O.

It can be shown after some algt·braic operatioll, that k:k~ = D;zD~z =

k:Dik~ Df,k~k;

Df,kii,Df,k~

=

(::l7a)

AZ.

= D~zDiD~z, = D~D~'ZD~z.

D~Df,zDiD~z

(:fib)

(:37 e) (37d)

- (Dii,zDf,zY,

and

D;AZ

=

Df,Df,z Diz

+

D;zD;D;z.

After a lengthy deduction, Eqs. (:31) can be rewritten as j~f,

= -t,A-3

[D~Diz

Dii,D;z - A'-

- L: i.;llb'(i.~ +

Df,D~zD;z/1-1D;zD;D;:

J

i.!,)-;(Df,D~zDf.z).JiDf,zD;D;z).;

"b

+

L

l;llb1(l.

+

lb)-l (D~A')b.(D;/l')b.J'

"b

For CP. model. A is a single compout"llt quantity rl"pt"llllill", I..nly I.·n g .Illll e:lll ht' taken equal to 1 by U~illg i! propt"r eOllforlllal transil'rnwtiuu:';. ~I) Uiil~ = U alit! Eqs. (:38) retiUet' tu (3!Ja) jz~ = A-l(Df,zD~z

+ D~zD;z).

(3!Jh)

Tl1t'n the l'('sult eoillcides with results given by Eichellherr:1J and by i-iehelt·r w . Xow we turn to discuss the CO'-1) principal chiral field[;J. G

_-\.11 d~IIlellt of grollp

= UL (.Y)0FlN) is expressed as a = (ar., YR),

yiYL = gtaR

=

1-

with the multiplication rule (-l.l )

Any element of A EH

(J

can be decomposed iuto a product of an dement

I .•

f

tht' subgroup,

= TJL+R(N), A

= (h, k)

with a left coset element 4' E G/ H,

Therefore g

= 4''' means (42)

308 LOCAL COXSERYATIOX L.-\WS FOR \'ARIOn; XOXLTXE.-\R (J·:.\lODELS

Xo.8

8m

Following Eq. (2), we define

·Zt" + ..x- ==

-ir/l-Iar/l = -!:(.p-Ia.p, .pO,p-I) = (H

+

K.

f{ -

Here

~ = (Il, Il).

..x- =

(Il, -K),

II = - ~ ( cjJ -lb.p + cpacp -I), l{ = - i.- Ccp -I Ocp - ,pa,p -I). ~

XI)W

K).

(43)

I

(48' )

~

the Cart an inller product of t\\'o f'1E:'mf'nts can be written as

paraml'tl'r (/ is pnt into 1), Bo

(I{~K~)-!/W; =

=

Th"'l1

han' thl'

WI'

f{~(I{;K: )-U

(;j;j )

with tilt' prlJpt>rries,

Bo/{;

=

I(~BQ = (/{;K~ )1.'2.

m

=

(5H)

1,

and the rE'l:urrence forlUulas similar to Eqs. (27) and OS) are r.") . _ (C t ,IOi; (B~!l\. iJ ~,

.1.;

+

(KB)' _ (I(C 1.)/ ;} - .

I_j

I.;

+

(iij)

T ),;

.'

I.j

(.")8 )

The zeroth consern,d current follows from Eq.

0,"».

jo~ = trK~Bo = fr(K;K~)Ti',

(5!) )

jo~ = 0,

and thp first currl'ut can bf' dffiuced from Eq. (5;).

(60) .il~

= O.

These results coincide with those of Ref. [6]. model, \~P can also obtain an equation

Similar to Eq. (:3fJ ) for (; raS!;lllann

(Ii I)

as requirt.' = ~Q>aAa. and and a new source 7)p. by means of the SU(3) local transformation U =exp( -igl/J/a). In particular,

A,.. = U\B" + /.Q(I/J)o"q,oj

11 12

ut,

oJ; '" Ut#pand 7) = UTiI' (10)

The original argument for the abelian case can be found for example in ref. (51. For the abolian case, sec ref. 161.

313 Volullle 10913, number 6

£-4 _~B~Jja/lv +R(gRB/lv) +iip(i~ -m -gRQ)-I Tlp

where (11)

Substituting Ihese into eg. (3), the lagrangian becomes

£= ~p(i~-

II March 1982

I'HYSICS LETTERS

//I .'

gQ) l/I p + ~pTlp + iipl/l p

+a 2tr[(B/l + LarV/l)(B/l + Laa/l rfl)] .

(\2)

The path integral measure is also changecYaccordingly:

CZ>= D[l/I ]D[iiI]D[A a]

=D[l/Ip]D[~p]D[Ba]D[¢a] 5W).

(13)

The 5(rpa) constraint in eq. (13) can be converted into a gauge condition on B-fields, for definiteness say the axial gauge condition: n/lB/l =O. where nil is a space· like four·vector. After some formal minipulation, the measure (13) can be written as

where ~(rpa) is a determinant involving only rp-fields. We record here that although the axial gauge is chosen explicitly. we could have also chosen the Lorentz gauge and obtained a gauge invariant measure which is also frame independent. Finally after integrating over l/I l' and ~ 1" the gen· erational function becomes

w~ JD/na]DW] ~W)5(IZ'Ba)exp(i

J

d4

x£l.

(J 5)

where

+ a2 tr(l.a L b) a/ll/la/lr:} .

It is to be noted that so long as Z 3 is nonzero, ¢ is not an independent field. It can always be transformed away through the redefinition of the field. On the other hand, in the limit Z 3 = 0, rp can no longer be transformed away. So among the three degrees of freedom for the massive vector boson. two of them are transferred to the massless vector boson and the remaining one to the scalar boson. Furthermore, the cou.pling term between jj~ and cff1 is proportional to zjl-. In the limit Z3 =0, cff1 completely decouples from B~. Also in this limit the a2 tr(La Lb)a/lcff1 a/lrpb term is the only term in eq. (18) which is not local gauge invariant. Owing to the rp-decoupling mechanism, the remaining effective theory has now a local gauge invariance. One can explicitly check that for the QeD theory with a local gauge symmetry, the generating [unc· tional after integrating over the fermion fields gives, in the strong-coupling limit. an expression which is identical to eq. (I 8) without the rp-term. Thus the four-fermion interaction theory defined by eq. (1) has at least in a formal sense generated a gauge boson. We now turn to the Noether current of eq. (2). From eq. (3),

x exp(i Jd 4x ~ A*Ab/l) X

+iip(i~ -m·gQrlT/p

+a2 tr[(B/l + Laa/l¢a)(B/l + Laa/lrpa)] . Using Z3 and gR defined earlier and jja we can rewrite the langrangian as

(16)

=lJaZ 31/2,

£ _ 1 -a -a/lv -. -I - -..l1/lV B +R(gRB/lvl+T/Jl(I~ -m -KRiI) TIp +a 2 tr[(ZY2 B/l +L aa/lrpaHzY2 B /l +Laa/lrpa)j. ( 17) Taking the limitK -400. which implies 7.3 40,

(18)

-~. exp(i (

aAo /l

.

d4x [\ji(i;l- m -g·f)j l/I +iil/l

+\1171)'

J

(19)

Integrating over'" and. iii, and taking the derivative with respect to A~, we obtain (~O)

We have used the Euler-Lagrange equation to arrive at eq. (20). The same result can also be obtained for· mally through integration by parts. In eq. (20) A~ is the field operator in the A /l sector with a fermion field having been averaged out. The relation of eq. (20) can be traced to stem from the original constraint associated with the auxiliary

459

314 Volume l09B, number 6

II March 1982

PHYSICS LETTERS

field AI" After our integrating over fermion fields and the identification of the massive bound state, this constraint is now promoted to a current·field identity. It can be shown thatj~ is still a conserved current, as it should be p . Eq. (20) also enables us to calculate the matrix element of j~ between two massive A-bound states. For instance, to lowest order in perturbation theory,

ever, the important point is that it is now decoupled from Rand the fermions, "'p' ~ On the other hand. in the effective theory of B and of the non-l/> sector, there is a new Nocther current:

"'p

,.al'

non

= __ g :,.

",1'1 Aa .,.

R 'l'p'

2

'l'p

+g fabc7Jb7JcI'V R

v

•

(25)

In general, the matrix elements of j~~~-

=N(min(p_, p'-)/max(p_., p,-rl'/2

X (p_p'j/2 /i(q_ - p. + p'-) /i(q+).

(IS)

The spectrum as well as the matrix elements found above are not a surprise. In order to see why, let us first recall that the light cone gauge .4 _ =0 implies that the right-handed quark field 1/1 _ = ~(I - 15) 1/1 remains free. Moreover, in two dimensions a system of free massless quarks moving in the same direction has zero mass. Had we chosen the opposite light-cone gauge condition.4+ =0 the roles would be reversed, 1/1+ would be free and the gauge invariant states (7) and (8) would appear as massless mesons and baryons constituted of free quarks. Their singlet current matrix elements are those of a free theory and being gauge-invariant quantities should coincide with those calculated before in the ..L =0 gauge. And, indeed, eqs. (14) and (15) represent free theory matrix elements. Moreover, BB will also appear as a massless bound state satisfying I which goes into the unphysical sheet [8]. Note that for N" I (i.e., A '" I) we obtain a free fermion theory. Thus for A 0;;;; I the intermediate states contributing to the current correlation function will be fermion-antifermion pairs. Therefore, in the limit In -+ 0 the anomaly is trivially saturated by the original quark. This apparent change of responsibility in going from A> I to A0;;;; I in the anomaly saturation has little meaning. As long as m =F 0 a free qq system is baSically different from a meson bound state, but for m -+ 0 they become indistinguishable. This is obviously a peculiarity of two dimensions. We see therefore that SU(Nh, which has a rich light spectrum for In 0, has a trivial chiral (m '" 0) limit. There, the left and right moving worlds are separated and each includes free mesons and baryons which are trivial composites of massless quarks and antiquarks moving parallely. As it is easy to visualize from this trivial picture, anomalies are saturated by mesons (collinear massless qq states!) while baryon-antibaryons

*"

I April 1982

do not contribute because they cannot be created by the quark currents from the vacuum. This trivial way of realizing chirality is clearly a characterization of two dimensions and is therefore of little help in understand· ing how chirality is realized or broken in four·dimensional gauge theories.

References II] G. 't Hooft. in: Proc. Cargcsc School (1979); sce also: Y. Frishman, A. Schwimmer. T. Banks and S. Yankiclowicz, Nucl. i'hys. BI77 (1981) 157. [2\ S. Colcman and E. Witten, Phys. Rev. Lett. 45 (1980) 100; G. VeneZiano, Phys. Lett. 95B (1980) 90. 13] G. 't Hooft, Nucl. Phys. B75 (1974) 461. 14] W. Biichmiiller, S.T. Love and R.D. Peccei, MPI·PAE/PTh. 70/81 (1981).

(5 J V. Baluni, Phys. Lett. 90B (1980) 407. [6] P.J. Steinhart. Nucl. Phys. BI76 (1980) 100; D. Amali and E. Rabinovici. Phys. Lett. 1018 (1981) 407. [7J S. Elitzur, Y. I'rishman and E. Rabinovici, Phys. Lett. 106B (1981) 403. 18J M. Kurowski and P. Weisz, NucI. Phys. B139 (1978) 455. (9J T. Banks. D. Horn and H. Neuberger, Nucl. Phys. BI08 (1976) 119. [ 10] S. Coleman, Erice lecture notes (1975).

320 Volume 114B, number 2,3

22 July 1982

PHYSICS LETfERS

ON THE DETERMINATION OF EFFECTIVE POTENTIALS IN SUPERSYMMETRIC THEORIES D. AMATI and Kuang-chao CHOU

I

C.ERN. Geneva. Switzerland

Received 8 April 1982

We propose·a renormalization procedure for dynamically generated effective actions. We show that. as expected, it leads to no spontaneous supersymmetIy breaking if this is Iinbroken at the tree level. We also understand why the usually adopted renormalization prescription has led in some models to an apparent supersymmetry breaking for an unacceptable negativeenergy vacuum state.

Application of the well-established l/N expansion to some supersymmetric models seemed to indicate the possibility of spontaneous dynamical breaking of supersymmetry violating general properties such as positivity of the ground-state energy or the index theorem tJ. This would shed negative light on l/N techniques which. nevertheless. seem applicable to supersymmetric theories. This puzzling result is well illustrated by Zanon's model {2l where a non-supersymmetric minimum was found for a negative value of the effective potential. In this note we wish to show that this apparent contradiction stems from an un appropriate renormalization procedure [3] adopted in the evaluation of effective potentials. Moreover. if the renormalization is cor· rectly performed. the effective potential. in terms of the dynamical fields. vanishes at the origin and is otherwise positive. thus confirming that supersymmetry will be preserved by radiative corrections if it is not broken at the tree level [4]. We shall use Zanon's model to illustrate the correct renonnalization procedure and fmd the loophole in that used in ref. [2]. The model consists of N + I chiral

supermultiplets tP and tPi' j = 1 • ...• N. described by the action *2

S=Jd4X d4 6 (ii>itPi+ii>tP)

-Jd x 2

d2 6 {[~mtP2 +~motPt + (g/..jiii)tPtPr] + h.c.}

The tPi fields appearing only bilinearIy may be integrated over. thus leading to an effective action depending only on tP. The well-known fact that these theories need only a wave function renormalization suggests a rescaling ifJ-+(NZ) 1/2 ifJ.

g-+Z- 1/2g •

m -+Z-I m .

(1)

Fields. coupling constants and masses will now represent renormalized quantities in terms of which the integration of S over tPi leads to S eff =N(ZStf> - ~Iog det 9f).

(2)

where

(3) and det 9f

On leave from Institute of Theoretical Physics. Academia Sinica. Beijing. People's Republic of China. *! The index theorem states that no spontaneous breaking of supersymmetry could happen if the numbers of the zero energy fermionic and bosonic statcs are unequal. see ref. [I J.

= det CU det- I en

I

o 031-9163/82/0000-0000/$02.75 © 1982 North-Holland

t2

We have introduced a mass m for the fields tf>j. It avoids infrared problems. The final potential is infrared free so that m may be set to zero if one wishes to make a comparison with ref. [2).

129

321 Volume 114B, number 2,3

r~F C)L-0

.-~-mo

PHYSICS LEITERS

-0

-·2gA- mo

-2g°Fo

0

0

0

-2g°Ao- m

o

-]

-:~.---j -]

0

A (x) and F(x) being the lowest componen t and the auxiliary field of the supermultiplet 4>, respectively. The usual way to evaluate the exact formal expression of eq. (2) is to develop it around constant fields_ In so dOing, one has to regularize ultraviolet divergences. We adopt a Pauli-Villars prescription which suffices for the case under analysis, and call M the regulator mass. We now evaluate the log det term in eq. (2) through its liN expansion around constant fields A = a, F = f, X = O. We shall see a singular behaviour in M for M -+ 00 not only in the potential (Le., -Eerr for constant fields) but also in the coefficients of the A and X kinetic terms. We thus write

Eerr =(Z +CA)A*OA + (Z + Cx) ~ixO""a"x - V +R ,

+ VI'

(6)

where CA , Cx and VI are explicit functions of M,a and/. VI is given by

VI

=(1/641T2)[(a2 + fj)21n(a 2 + (3) + (a 2 - fj)2ln(a 2 - (3) - 2a4 ln a 2 - 2{321n(M2 + ( 2) - 3(32) ,

(7)

with a 2 = 12ga + mol2,

{3 =21gfl .

(8)

CA has the form

CA = (lgI2/81T 2 ) In [(M2 +( 2)/Q2) + terms regular in M . 130

(9)

(10)

Surely enough this is a legitimate renormalization condition in the sense that it eliminates all M2 singularities in eq. (5) and in particular in the potential V. But if the auxiliary field f is eliminated in terms of a by

av/af=o,

(11)

~hen

V= V(a,f(a)) acquires negative values, in particular for sufficiently large a. Moreover, if there is a stationary point,

dV(a,f(a»/da = 0,

(12)

away from the origin, it happens for negative values of V. This was the stationary point detected in ref. [2] and to which was assigned the responsibility of spontaneous symmetry breaking. In order to understand the reason for this apparent inconsistency with positivity of the ground-state energy implied by supersymmetry, let us analyze the kinetic terms of eq. (5). Using (9) and (10) we fmd, after settingM = 00 Z + CA = 1 + (lgI2/81T 2 ) In iJ. 2/a 2 + ... ,

(5)

where R is regular in M and contains derivatives in the fields A, F and X higher than those explicitly written in eq. (5). V will have an expression of the form V= -Zrf+maf+m*a*f*

The usual renormalization prescription [3] consists of determining Z by imposing a condition on Vat an arbitrary specific value of f and a that defines a renormalization scale iJ. 2 . This implies

Z = I - (lgI2/81T2) InM2/iJ. 2 . (4)

22 July 1982

(13)

an expression which becomes negative for large values of Q. The same happens for Z + Cx and it is possible to see that the negativity of both kinetic terms is correlated with the negativity of the potential. Thus the stationary point of ref. (2) is a property of a ghost potenti.al and cannot be identified with a stationary ground state. This rather unpleasant description may be circumvented by an alternative renormalization prescription showing clearly that supersymmetry is unbroken in this theory. A wave function renormalization controls the normalization of the corresponding kinetic term. We are thus naturally led to normalize it at the background field values (a andfin our notation) that we wish to consider. The minimum conditions on the potential thus obtained determine those values for which linear terms in the fluctuations are absent, thus allowing the identification of Q andfat the minimum with (A) and (F), respectively. We could determine Z from Z + CA = 1, but to avoid useless complications with the regular terms in

322 Volume 114B, number 2,3

PHYSICS LEITERS

m5

eq. (9), let us choose Z= 1 -(lgI2/87T2)lnM2/0i 2 ,

(14)

which eliminates from Z + CA' and therefore from the A kinetic term, the logarithmic term which was at the origin of its negative values for the choice of Z in eq. (10). Eq. (1) shows that if Z depends on a, the renormal· ized parameters g and m depend on a through Z. This dependence is the usual one, Le., g-2(0i) =g-2(Oio) + (1/87T 2) In 0i5/0i 2 .

(IS)

Eqs. (6), (7) and (14) imply

V= -lfI 2 +maf+m*a*r + O/647T2) [(0i2 + (j)2In(0i2 + (j) + (0i 2 - (j)2 In(0i 2 _ (j) _ 2a41n 0i2 _ 2(j21n 0i 2 _ 3(j2] . (16) It is easy to see that

v - f iW/af - r av/ar = Ifl2 + O/647T2){(0i4 _ (j2) In [(0i4 _ ;;;;. 0,

(j2)/a4] + (j2}

(17)

for all a and (j '" (X2 , the equality sign in (17) holding only for (j = O. It is then clear that on the line

av/af= av/ar =0 ,

(18)

which expresses f in terms of a, the potential V = V(a, f(a» is a positive function of a with its absolute min· imum at a = 0 where V vanishes. Therefore (,4) = {F} = 0 and supersymmetry is unbroken. We see therefore that with our renormalization prescription, we succeeded in describing the theory for all choices of the background field a with a posi. tive potential and in terms of fluctuations which have positive kinetic terms. On the other hand, in order to describe the same theory at different values of a, we found that the coupling constant depends on a, as described in eq. (I 5). If we call go the coupling constant at a =0, we find

g2(a)=g5/[l + (g5/ 87T2 )In m5/0i 2 ] ,

22 July 1982

(19)

which shows a Landau-type pole at 0i 2 = exp(87T 2/ ga) characteristic of a non.asymptotic free theory like the one analyzed here. To summarize, we have shown that the renormalization prescription of refs. [2] and [3] generates nega· tive potentials together with negative kinetic terms. Ghosts try to increase their potential energy so that a negative stationary point is energetically unfavourable as compared with a zero potential configuration. Thus, even in this language, we understand why supersymmetry is not broken in the model of ref. [2]. Moreover, we have shown how to avoid this pathological ghost interpretation through an alternative renormalization prescription that leads to bona fide fields with positive kinetic energy and with non·negative potentials as reo quired by supersymmetry. We may wonder why the generally adopted prescrip· tion of renormalizing the interaction at a fixed scale does not work for supersymmetric theories while it is perfectly applicable to usual theories as "Np4 [5]. In conventional theories the coupling constant is renor· malized independently of the wave function and thus any preSCription for the first one cannot influence the kinetic terms which are controlled by the second. In supersymmetry the only renormalization is the wave function one and therefore is determined by the kinetic terms. Or, at least if a definition of the interaction is introduced which leads to negative kinetic energy, it is impossible to appeal to another independent renor· malization to correct that sign. We wish to acknowledge fruitful discussions with L. Girardello, 1. Iliopoulos, R. Cahn, G. Veneziano and S. Yankielowicz.

References 11) E. Witten, Lecture notes at Trieste (1981); S. Ceeotti and L. GirardeUo, Phys. Lett. llOB (1982) 39. [2) D. Zanan, Phys. Lett. 100B (1981) 127. [3) M. Hug, Phys. Rev. D14 (1976) 3548; D16 (1977) 1733. [4) L. O'Raifearlaigh and G. Parravieini, Nue!. Phys. Bll1 (1976) 516; W. Lang, Nue!. Phys. B114 (1976) 123. [5) S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888.

131

323 PHYSICAL REVIEW D

I SEPTEMBER 1983

VOLUME 28, NUMBER 5

Koba-Nielsen-Olesen scaling and production mechanism in high-energy collisions Chou Kuang-chao Institute of Theoretical Physics, Academia Sinica, Beijing, China

Liu Lian-sou' and Meng Ta-chung Institutfiir Theorelische Physik der Freie Universitiit Berlin, Berlin, Gemany (Received 14 March 1983)

An analysis of the existing data on photoproduction and electroproduction of protons is made. Koba-Nielsen-Olesen (KNO) scaling is observed in both cases. The scaling function of the nondiffractive rp processes tums out to be the same as that for nondiffractive hadron-hadron collisions, but the scaling function for deep-inelastic e -p collisions is very much different from that for e -e + annihilation processes. Taken together with the observed difference in KNO scaling functions in e -e + annihilation and nondiffractive hadron-hadron processes these empirical facts provide further evidence for the conjecture: The KNO scaling function of a given collision process reflects its reaction mechanism. Arguments for this conjecture are given in terms of a semiclassical picture. It is shown that, in the framework of the proposed picture, explicit expressions for the above-mentioned KNO scaling functions can be derived from rather general assumptions.

I. INTRODUCTION

The recent CERN pp collider eXr!?ments,' in which Koba-Nielsen-Olesen (KNO) scaling has been observed, have initiated considerable interest 3,4 in studying the implications of this remarkable property. A physical picture has been proposed in an earlier paper4 to understand the KNO scaling in the above-mentioned experiments,' and in pp~ and e+e- reactions. 6 It is suggested in particular that the qualitative difference between the KNO scaling function in e + e - annihilation and those in nondiffractive hadron-hadron collisions is due to the difference in reaction mechanisms. In this paper we report on the result of a systematic analysis of high-energy 'YP and e -P data,,8 as well as that of a theoretical study of the possible reaction mechanisms of these and other related processes. We show the following. (A) KNO scaling is valid also in high-energy 'YP and e - P processes. The scaling functions for nondiffractive 'YP and low_Q2 (invariant momentum-transfer squared) e-p processes are the same as for nondiffractive hadronhadron collisions, but the scaling function for deepinelastic e-p collisions is very much different from that for e - e + annihilation processes. (B) The KNO scaling function for e - e + annihilation, 1/J(z) = 6z 2exp( _az 3), a'/3= T) , (I)

n

and that for nondiffractive hadron-hadron collisions, 1/J(z)= l6/5(3dexp( -6z)

KNO scaling functions mentioned in (A) can be understood in the framework of the proposed picture. II. KNO SCALING IN 'YP AND e -p PROCESSES

We studied photoproduction and electroproduction of protons at incident energies above the resonance region. We made a systematic analysis of the existing data,,8 and found that: there is KNO scaling in e -pas well as in 'YP processes. (See Figs. I and 2.) The KNO scaling function for nondiffractive 'YP processes and that for e-p at low momentum transfer are the same as that for nondiffraclive hadron-hadron collisions. (See Fig. I.) The KNO scaling function for deep-inelastic e -P collisions is very much different from that for e + e - annihilation processes. (See Fig. 2.) The similarity between the KNO scaling function in nondiffractive 'YP (and low-momentum-transfer e - p) and that in nondiffractive hadron-hadron processes is not very surprising. In fact, it shows nothing else but the wellknown fact 9 that real (or almost real) photons at high energies behave like hadrons. But does the difference in KNO scaling functions in e-e+ and deep-inelastic e-p processes indicate that the reaction mechanisms of these two kinds of processes are qualitatively different from each other? Before we try to answer this question, let us first examine in more detail the relatiOriship between KNO scaling functions and reaction mechanisms in e - e + annihilation and in nondiffractive hadron-hadron collisions.

(2)

(here z =n I( n ), n is the charged multiplicity and (n) is its average value), can be obtained from the basic assumptions of the proposed physical picture using statistical methods. (C) The similarities and differences between observed

III. e-e+ ANNIHILATION: FORMATION AND BREAKUP OF ELONGATED BAG

The KNO scaling function in e - e + annihilation processes is shown in Fig. 3. It is sharply peaked at n I (n ) = I (n is the mUltiplicity of the charged hadrons 1080

® 1983 The American Physical Society

324 1081

KOBA-NIELSEN-OLESEN SCALING AND PRODUCTION ...

0"2.8-8.0 (GeV/d 2

2 .S

.4

\ +

.3

\

2

\ \

N

Elf(GeV)

~

1.4-16 2.8-3.6 4.S-S.7 7.4-9.4

\

C

a: €

\ \

.05 .04 .03

\ \

.02

~

.01

0.3-0.S 0.5-0.7 1.8-2.2 2.2-2.8. •

.01

•

•

WinGeV 02,n (GeV/c)2

0

z",n/ FIG. 3. The scaled multiplicity distribution for e -e + annihilation processes. The experimental data are taken from Ref. 6. The curve is the scaling function given by Eq. (I).

325 CHOU KUANG-CHAO, LIU LIAN-SOU, AND MENG TA-CHUNG

1082

colilSlon

a typical violent collision

e-et

example

annihilation

coUis,on

-------

first stage after the collision

~q

before the

,/

/'

a typical gentle colllSiQtl non·d,ffracl'\le pp collision

p

----0 0--

-0

(0

G-

is~~.~:~~ collision

FIG. 4. Two qualitatively different types of high-energy collisions are illustrated; and one characteristic example in each case is given.

not seen in e - e + annihilation processes, where the decay of the virtual photon into a quark-antiquark (qq) pair is generally accepted to be true. In fact, it is envisaged that as q and ij of the original qij pair move apart (at almost the velocity of light) a color-electric field is developed, and a number of polarized pairs (secondary qij pairs) are formed between them. Now, since the gluon exchanges allow q (ij) of arbitrarily high subenergy to interact with finite probability, an "inside-outside" cascade l6 takes place and as a consequence only color-singlet hadrons are produced. Quantitative comparisons between experiments ll ,I2 and the Lund model,l7 which is a semiclassical model that incorporates all the relevant features of the Schwinger model,I4 have been made." The agreement seems to be very impressive. Is it possible to understand the observed KNO scaling behavior in e-e+ annihilation processes in models based on the Schwinger mechanism? We now show that this question-which does not seem to have been asked before--can be answered in the affirmative. In order to study mUltiplicity distributions in this framework, we need to know the relationship between the observed multiplicity of charged hadrons and the properties of the elongated bag. The following points are of particular importance in establishing this relationship: (a) Since the number of sub-bags at the final stage of a given event is nothing else but the total mUltiplicity of hadrons in that event, it seems plausible to assume that the total multiplicity of charged hadrons (n) is proportional to the final length (1) of the elongated bag in every

approximately the same transversed momentum with respect to the jet axis; (ii) The multiplicity of charged hadrons is distributed mainly around its average value (n) which is rather high at incident energies where KNO scaling has been observed (e.g., (n) "" 7 and 14 at Vs = 10 and 30 GeV, respectively). We note that the final length I is determined by the first breakup of the elongated bag. This is essentially a kinematical effect which can be readily demonstrated in terms of the one-dimensional Lund model 17 as shown in Fig. 5. The generalization from the one-dimensional string to a three-dimensional elongated bag does not influence the arguments used to reach this conclusion. The reason is: In the present model, the existence of a qq pair is not a sufficient, but a necessary condition for the breakup. (fJ) As the original quark-antiquark pair fly apart, their kinetic energy is converted into volume and surface energies. Secondary qq pairs are produced and the elongated bag begins to split when the bag reaches a certain length such that it is energetically more favorable to do so. Note that the collective effect due to color interaction is a substantial part of the bag concept. Hence, it is expected that the probability of bag splitting should depend on the global rather than the loeapo properties of the entire system. We shall assume, for the sake of simplicity, that the elongated bag is uniform in the longitudinal direction, and that the probability df Idl l for a bag of length I to break somewhere (at II, say, where 0 < II < /) is proportional to I, that is approximately proportional to the total energy U of the elongated bag. 21 This means [(1)=

f: ;(

df ="A.l dl l

dl l

(4) (5)

'

where A is a constant. It should be mentioned that we do

event,

n =("111 0 )1.

(3)

Here, 10 is the average length of the elongated hadron-bags in their "rest frames" and "1 is the inverse of the average Lorentz contraction factors of the hadrons along the jet axis. This means, we have assumed that "1110 depends only on the total c.m. energy Vs, provided that n is not too small compared to (n). Obviously, Eq. (3) is in accordance with the following empirical facts 6• 19 : (i) The overwhelming part of the produced hadrons are pions of

FIG. 5. The one-dimensional Lund model (see Ref. 17) is used to demonstrate that in e - e + annihilation processes the fi·

nal length of the elongated bag is determined by the first breakup. Here, t and x denote the time and space coordinates,

respec~

tively. Note that Ihe generalization from the one-dimensional string to a three-dimensional elongated bag does not influence the arguments used to reach this conclusion.

326 KOBA-NIELSEN-OLESEN SCALING AND PRODUCTION ... not know why the above-mentioned I dependence [Eq. (5)] should be linear. What we know for the moment is: By assuming a power behavior Ik for df Idll> the experimental data require k = I. (y) Having obtained the probability f(l) for an elongated bag of length I to break up, the density function for the I-distribution P (l) can be calculated in the following way: Consider N events, among which in N (l) of them the bag has reached the length without breaking and dN of them will break up in the interval (1,1 +dn, then

dN = - f(llN(l)dl .

(6)

It follows from Eqs. (4), (5), and (6)

dN P(l)=di =CI2exp( _Al 3 13) ,

(7)

where C is a normalization constant. The corresponding density function for multiplicity distribution Pin) is therefore [see Eq. (3)] (8)

The constants A and B are determined by the usual normalization conditions22 :

fo"'P(n)dn =2,

(9)

fo'" nP(n)dn =2(n) . From Eqs. (8), (9), and

(10),

(10)

we have

(n )P(n)=t/J(n/(n» ,

(II)

where t/J(z) is given by Eq. (1). Comparison with experiments 6 is shown in Fig. 3. The following should be pointed out: (a) The KNO scaling behavior is obtained as a direct consequence of Eqs. (8), (9), and (10). (b) The elongated-bag model, which is obviously consistent with the physical picture discussed in Ref. 4, is more specific and gives a better description (than the Gaussian approximation) of the existing data. (c) There is a discrepancy between model and data for z < o. 3. This is due to the fact that Eq. (3) is only a poor apProximation for n « (n ). (d) since the final length is determined by the first breakup of the elongated bag, the existence of intermediate states does not influence the observed multiplicity of charged hadrons. IV. NONDIFFRACTIVE HADRON-HADRON COLLISIONS: FORMATION AND DECAY OF THREE-FIREBALLS We now tum to Eq. (2) and show that it can be derived in the framework of the proposed picture under more general conditions than those mentioned previously. We recall that, according to this picture,' the dominating part of the high-energy inelastic hadron-hadron collision events are nondiffractive. The reaction mechanism of such processes can be summarized as follows: Both the projectile hadron (P) and the target hadron (n are spatially extended objects with many degrees of freedom. They go through each other during the interaction and distribute their energies in three distinct kinematical regions in phase space:

1083

the projectile fragmentation region R (P*), the target fragmentation region R (T*), and the central rapidity region R (e*). Part of these energies materialize and become hadrons. We denote these parts by Epo, E p , and Eco, respectively. They are the internal (or excitation) energies of the respective systems. The difference in reaction mechanisms of e - e + annihilations and nondiffractive hadron-hadron collisions is illustrated in Fig. 4. Let us consider the internal energy E j of the system i (i =p*,T*,e*) in a large number of collision events. Viewed from the rest frame of the system i, both the projectile (P) and the target (n before the collision are moving with a considerable amount of kinetic energy. The interaction between P and T causes them to convert part of their kinetic energies into internal energies of the systems P*, T*, and e*. Hence, each system i has two energy sources so that E j can be expressed as (12)

where E jp and E jT are the contributions from the source P and that from the source T, respectively. Note that the two sources are independent of each other, and that among the nine variables in Eq. (12) six of them are completely random. Let Fp(Ejp ) be the probability for the system i to receive the amount E jp from P, and FT(EjT ) is that for the system i to receive E jT from T, then the probability for the system i to obtain E 1P from P and E jT from T is the product Fp(Ejp)FT(EjT). Physically, it is very likely that the system i completely forgets its history as soon as the system is formed. This means, the probability for the system i to obtain E;p from P and E jT from T depends only on the sum Ejp+E,T . That is (13)

where E; and E jp and E jT are related to one another by Eq. (12). Hence

d

d

dE [lnFp(E,P )]= - dE [lnFT(Ej-E jp )] , jp 1P

(14)

that is Fp(E1P)=Apexp( -BE,p) ,

(15)

FT(Ej -Ejp)=ATexp[ -B(Ej -E,p)] ,

(16)

where A p, AT' and B are constants. In order to obtain the total probability P(Ej ) for the system i to be in a state characterized by a given energy Ej, without asking the question "How much of E j is contributed from P and how much of it from T/" we have to integrate over all the possible values of E jp and E'T under the condition given in Eq. (12). That is P(Ej =

)

f dEjpdEjT8(Ej-Ejp-EjT)Fp(Ejp)F.r(EiT)·

(17)

It follows from Eqs. (15), (16), and (17) P(Ej)=CEiexp(-BEj ) ,

(18)

where the constants Band C are determined by the normalization conditions

327 1084

CHOU KUANG-CHAO, LIU LIAN-SOU, AND MENG TA-CHUNG !P(Ej)dEj=I,

(19)

! EjP(Ej)dEj=(Ej }

(20)

Hence (EI }P(Ej )=4E;I(Ej )exp( -2E;I(Ej »

,

(2 I)

which is Eq. (11) of Ref. 4. The mUltiplicity (nND) distribution for nondiffractive hadron-hadron collisions given by Eq. (2) is obtained by taking into account (for details, see Ref. 4) n;l(nj})=E,I(Ej } , i=P*, T*, and C*"

(22)

and (23)

Note that z in Eq. (2) stands for nNo/( nNO)' It should be emphasized that the simple relationship, Ejlnj=constant (i=C*,P*,T*), is an idealization. In reality, fluctuation in nj for a given E j is expected. Such effects have been taken into account by assuming that the KNO distribution for e-e+ annihilation (which can be approximated by a Gaussian; see Ref. 4) is due to the fluctuation of n.. I (n .. ) about I, and that the fluctuations of nj about (nj) is of the same magnitude. These fluctuations are folded into the distributions obtained from the three-fireball model for hadron-hadron processes (a detailed discussion on this point is given in the preliminary version of our paper; see Ref. 20 of Ref. 4). Comparison between data and results of that calculation shows, however, that the effect is negligible in first-order approximation.

freedom (possibly a large number of colored gluons and sea quarks in addition to the colored valence quarks) such that various colorless objects can be formed in an excited proton, it seems natural to conjecture that deep-inelastic e --p processes take place as follows: The virtual photon in such collision processes interacts with a part of the proton gently in the sense that it "picks up" a certain amount of colorless matter in order to fragment. 2s Note that by picking up a certain amount of colorless matter from the proton, the virtual photon becomes a real physical object. The fragmentation products of this object are nothing else but the "current fragments" observed in lepton-nucleon reactions. 26 This conjecture can be readily tested experimentally. Because, if it is correct, we should see: First, the average multiplicity (n) does not depend on Q2 (the invariant momentum transfer). Second, (n) depends on W (the total energy of the hadronic system) in the same way as the average multiplicity in hadron-hadron collisions depends on v'S (the total c.m. energy). Third, the rapidity distribution in single-particle inclusive reactions shows a dip in the central rapidity region (near Yc.m. =0) at sufficiently high incident energies. This is because the center of the current fragments (formed by the virtual photon and the colorless matter it picked up from the photon) and that of the residue target (the rest of the target proton) move away from the central region in opposite directions. Fourth, the KNO scaling function is !/I(z)=4/3(4z>3exp( -4z).

(24)

This is because, according to the proposed picture4 the two fragmenting systems mentioned above act ind~dently, and the KNO scaling function of each system is [See Eq. (21)1

V. A POSSIBLE REACTION MECHANISM FOR DEEP-INELASTIC e -p PROCESSES

We now come back to the question raised at the end of Sec. II. According to the conventional picture23 for deepinelastic e - p collisions, one of the colored quarks inside the proton is hit so violently that it tends to flyaway from the rest (to which it is bounded by the confining forces). As a consequence quark-diquark jet structure is expected. 23 Hence, it is natural to believe that also in this case elongated bags lS or strings l7 are formed which hadronize. In fact, compared with the above-mentioned model for e - e + annihilation, the only difference would be that the bags, tubes, or strings end with quark and diquark, instead of quark and anti quark. If this were true, the KNO scaling function for deep-inelastic e -p processes would be the same as that for e - e + annihilation. The qUalitative difference in KNO scaling functions of e - e + and deep-inelastic e - p collisions is probably because the virtual photon in e -p processes behaves differently as that mentioned in the conventional picture. Once we accept that (a) the virtual photon in deepinelastic e - p collisions cannot fragment like a hadron in hadron-hadron collisions because it has an energy deficiency compared with its momentum,24 and (b) the proton is a spatially extended object with many internal degrees of

!/I(z)=4zexp(-2z) .

(25)

That is, the mechanism of e - p deep-inelastic scattering can be described as the formation and decay of two fireballs. Here we have assumed, by analogy with the nondiffractive hadron-hadron collision, that the average multiplicities of the two fireballs are equal. The first and the second points are well-known experimental facts. 27 In connection with the third point, we see that the rapidity distribution in neutrino-proton reactions at W > 8 GeV clearly shows the expected dip. (See, e.g., Fig. 10 of Ref. 26.) Corresponding data for electronproton reactions at comparable energies is expected to exhibit the same characteristic feature. La~t but not least, Fig. 2 shows that Eq. (24) (the fourth point mentioned above) is indeed in agreement with the data. The conclusion that there should be two fireballs in the intermediate stage of deep-inelastic e - p collisions can also be reached without referring to the properties of the virtual photons, provided that such collisions take place as follows: The pointlike electron goes through the spatially extended proton, gives part of its energy and momentum to a colorless subsystem of the proton al.d separates this subsystem from the rest. While the incident electron is only deflected due to the interaction, the two separated subsys-

328 KOBA-NIELSEN-OLESEN SCALING AND PRODUCTION ... terns of the proton become excited and subsequently decay. It should also be pointed out that, if this conjecture is correct, we expect to see only one central fireball in the e - e + ->e - e + X processes at sufficiently large momentum transfer. In that case the corresponding KNO scaling function should be the same as that given in Eq. (25). It would be very interesting to see whether this and other

'On leave from Hua-Zhong Teachers' College, Wuhan, and Peking University, Beijing, China. 'K. Alpgard el al., Phys. Lett. !!lTIl, 315 (1981); G. Amison el al., ibid. I07B, 320 (1981); 123B, 108 (1983); K. Alpgard ., al., ibid. illB. 209 (1983); and the papers cited therein. 2Z. Koba, H. B. Nielsen, and P. Olesen, NucL Phys. B4O,317 (1972). 3See, e.g., S. Barshay, Phys. Lett . .!l6ll, 197 (1982); T. T. Chou and C. N. Yang, ibid. llill, 301 (1982); Y. K. Lim and K. K. Phua, Phys. Rev. D 2.6, 1785 (1982); C. S. Lam and P. S. Yeung, Phys. Lett. l!2, 445 (1982); F. W. Bopp, Report No. SI-82-14, 1982 (unpublished). 4Liu Lian-sou and Meng Ta-chung, Phys. Rev. D n, 2640 (1983). 5See P. Slattery, Phys. Rev. D I, 2073 (1973); C. Bromberg el al., Phys. Rev. Lett. 11, 1563 (1973); D. Bogert el al., ibid. 11, 1271 (1973); S. Barish el al., Phys. Rev. D 2, 268 (1974); J. Whitmore, Phys. Rep. lOC, 273 (1974); A. Firestone el al., Phys. Rev. D !ll, 2080 (\974); W. Thome el al., Nucl. Phys. Bll2. 365 (1977); W. M. Morse el al., Phys. Rev. D 12, 66 (1977); R. L. Cool el al., Phys. Rev. Lett. ~, 1451 (1982); and the papers cited therein. liSee, e.g., R. FeIst, in Proceedings of Ihe 1981 Internalional Symposium on Leplon and Pholon Interaclions al High Energies, Bonn, edited by W. Pfeil (Physikalisches Institut, Universitiit Bonn, Bonn, 1981), p. 52 and the papers cited therein. 7R. Erbe el al., Phys. Rev. 175, 1669 (1968); R. Schiffer el al., Nucl. Phys. IDl!, 628 (1972); J. Ballam el al., Phys. Rev. D ~, 545 (1972); 1,3150 (1973); H. H. Bingham el al., ibid. .!!.,1277 (1973). See also H. Meyer, in Proceedings of 'he Sixth Internalional Symposium on Eleclron and Pholon Inleraclions al High Energy, Bonn, Germany, 1973, edited by H. Rollnik and W. Pfeil (North-Holland, Amsterdam, 1974), p. 175. By. Eckardt el al., Nucl. Phys. ~,45 (1973); J. T. Dakin el al., Phys. Rev. Lett . .N, 142 (1973); Phys. Rev. D .!!., 687 (1973); L. Ahrens el al., Phys. Rev. Lett. n, 131 (1973); Phys. Rev. D 2, 1894 (1974); P. H. Carbincius al., Phys. Rev. Lett. 32, 328 (1974); C. K. Chen el al., Nuel. Phys. I!.!ll, 13 (1978). 9J. D. Bjorken, in Proceedings oflhe Third Internalional Symposium on Eleclron and Pholon Inleraclions al High Energies (SLAC, Stanford, 1967), p. 109; H. T. Nieh. Phys. Rev. D 1. 3161 (1970). '0K. Goulianos el al., Phys. Rev. Lett. ~. 1454 (1982). liSee, e.g., D. Haidt, in Proceedings of Ihe 1981 Inlernalional Symposium on Leplon and Pholon Inleraclions al High Energies, Bonn, (Ref. 6), p. 558 and references given therein.

e'

1085

consequences of the proposed reaction mechanism will agree with future experiments. ACKNOWLEDGMENT This work was supported in part by Deutsche Forschungsgemeinschaft Grant No. Me-470-4/1.

I2See, e.g., W. Hofmann, Jets of Hadron •• Yol. 90 of Springer Tracls in Modern Physics. (Springer, Berlin, 1981) and papers cited therein. I3See, e.g., W. Hofmann (Ref. 12), pp. 25 and 47 and the papers cited therein. 14J. Schwinger, Phys. Rev. ill, 397 (1962); ill, 2425 (1962). ISA. Casher et al., Phys. Rev. D Ill. 732 (1974). 16J. D. Bjorken, in Curren I Induced Reactions, proceedings of the International Summer Institute on Theoretical Particle Physics, Hamburg, 1975, edited by J. G. KOrner, G. K. Kramer, and D. Schildknecht (Springer, Berlin, 1976). 178. Andersson et al., Z. Phys. C 1, 105 (1979). 18See, e.g., D. Fournier, in Proceedings of 'he 1981 Inle,nalional Symposium on Lepton and Pholon Interactions al High Energies, Bonn (Ref. 6) p. 91 and the papers cited therein. 19gee, e.g., Refs. 6, II, and 12 and the papers cited therein. 2OSee, e.g., Ref. 17. 21 In first-order approximation the energy of the elongated bag at a given instant is proportional to its length at that moment. This energy can be considered as the total potential energy of the qq system. We recall: Hadron-spectroscopy strongly suggests that the interaction inside a hadron can be described by a linear potential (the potential energy is directly proportional to the distance) between q and q, provided that they can be considered as a static source and sink of color flux. This is, e.g., the case when the quark-antiquark pairs are heavy (ci:. bli, etc.). Now, in the case of e -e + annihilation processes, since the primary q and q are always on the two ends of the elongated bag while they separate, it is always possible to envisage the existence of an instantaneous static source and a corresponding sink in each bag. Hence we can assume the existence of such linear potentials for all kinds of primary qqpairs. 22S ee, e.g., Refs. 1,5,6, and 10 and the papers cited therein. 23S ee, e.g., Hofmann (Ref. 12), p. 71 and papers cited therein. 24T. T. Chou and C. N. Yang, Phys. Rev. D~. 200S (1971). 25This colorless matter is not chargeless. It consists probably of a large number of sea quarks. Note also that the existence of such processes does not necessarily contradict the underlying picture of the quark-parton model which may be true to the impulse approximation. 26See, e.g., N. Schmitz. in Proceedings of Ihe 1979 Inlernational Symposium on Lepton and Photon Interactions al High Energies, Fermi/ab, edited by T. B. W. Kirk, and H. D. I. Abarbanel (Fermilah, Batavia, Illinois, 1980), p. 259. 27See, e.g., Chen el al. (the last paper of Ref. 8).

329 Co"",,,,". in Theor. Phys. (Beijillg, China)

Vol. ;:, No. 2 (1983)

97l-982

NONLINEAR a-MODEL ON MULTIDIMENSICNAL CURVED SPACE WITH CERTAIN CYLINDRICAL SYMMETRY CHOU Kuang-chao ( f.l1t~ Institute of

Tt.·~oretical

and

)

Physics, Academia Siflica, Beijing, China

SONG Xing-chang (

*tH: )

Institute of Theoretical Physics, Physics Department, Pekinq University Beijinq, China

Received September 20, 1£F2

Abstract It

dual transformation is fount! for a class of nonlinear

l'-

model definlJd on a multidimens':'or.a:' curt-"ad space k!ith cylind~ical symmetry.

T."'le syst-e:r. is ':'nllariar:t :.znc.er a proper

the dual transrormation and the ~n

~~neral

combination

of

coordinate transformation.

infinite number of nonlocal conser"-rltion laws as well as the

Kac-Mood'l alge!>ra follow directly from tile dual transformatior:. A Backlund transformation that cenerates new solutions from a siven one can also be constructed.

I.

Introduction

In recent years considerable progress has been madt' in th" i n\"est igtl t iC'n of the two-dimensional a-model or chiral model[l], which possesses a lot of rather interesting and mutually connected properties like the soliton solutions[2 1• the Backlund transformation[3], the infinite number of conservation laws ,fiJ associated with a hidden symmetry,G-8] and the close similaritr to thE' sE'H-dun] Yan~-Mills theory in four dimensions. Besides, it has also been shown that th~ chiral field equation in three-dimensional cylindrical symmetric case benrs ~ resemblance to the Ernst equation in yeneral relativity. Moreover, in the 50called super-unification theory an important role has been played by the nonlinl;;ur a-model though the four-dimension'al versio.m of which has not been fully investigated yet. Therefore it is meaninll'ful to extend our earlier work on two-dimensional nonlinear a-model to the case of higher dimensions with certain cylindrical symmetry. First let us recall some of the important results on the ordinary twodimensional nonlinear sigma model. These results are formulated in the K-form which has the advantage of uniformity and simplicity and will be used throu~hDut this paper. The Lagrangian for the ordinary two-dimensional non-linear sigma mode1 can be written in one of the following forms[Sl.

,4

~ =-fTr I ~ (~<J,) = 0

The compatibility condition for

C)l

nlJ

(1. Sa)

(l.Sb) (1. Sc)

gives for symmetric coset space

H" - 0, H, + U·t. · H,J ... -

~ K~-D,KJL=o

0

\)

[k",~] •

(1.6a) ( 1.6b)

.

These two equations are known as the Gauss-Coddazi equations. In light cone coordinates

.[l~=t -x

(1. 7)

Eqs.(l.Sa) and (1.6) can be written in the fOrm (1.

Sa)

(1.

8b)

Explicitly, the dual transformation

K(-I\i =(1(1

K~ -

K~ = r-IK~

H~-H~=H1

H~-H~=H~

(1.9a) (1.9b)

with y as an arbitrary constant. leaves the Lagrangian and the equations (1.8) invariant. From this transforma~ion an infinite nUlllber of conservation laws, both local and nonlocal. can be deduced[S]. Under the transformation (1.9), the physical field ." and/or q transforms as

91'%) -", 'T. [ep]) ,IX) ii."\ r, [f]) = ep'l r.:x) and g(n) are arbitrary functions of £ and n respectively. From Eq.(2.7) we see that the transformation (2.5) can be well defined if and only if

I\«("n)

n, i.e.,

A=f(E)-g(~).

can be decomposed into a sum of separative functions of

and

E:

Moreover it is evident that such a kind of decomposition

is not unique. If, 'for instance·, A-I'=fC nand A_=G(n) is a reasonable decomposition, then A+=f(;)+k and A_=g(n)+k (k const.) is also a possible choice.

Then Eqs.(2.6a,b) give

the function of the parameter

y

as the functional of

feE)

and

g(n)

and

k (2.8)

The relation between

A(F.,n)

and

A'(E,n)

is also determined by Eq.(2.4a') (2.9a) (2.9b)

Substituting the expressions for

y2

and

A

given in Eqs.(2.7) and (2.8) into

333 Nonljnear a-Hodel on HUltjdjmensional Curved Space with Certajn Cyljndrical

Symmetr~

975

(2.9), we obtain by integration (2.10) The integration constant

C

can be determined by the boundary condition

Y-J,

Therefore

t:.'-D.,

c-f··

(2.11)

(2.10' ) Therefore we see that, under the condition (2.7). the dual-like tran~forma­ tion for the cylindrically symmetric non-linear si~ma mode) is specified by Eqs.(2.S), (2.8) and (2.10') with k-1=tl as the group parameter. tl=O (k"''''') corresponds to the identical transformation y=1, and two transformations with parameters CI. and tl' give another transformation with tl"=tl+tl':

A("~.~l

'"

=f'1'-St~"'-fi~i'=

AC1,')

0('

=f'Ci'-9(~,';l)=JzlJi+i +j~+i

)2

(4.4)

,

then it follows by a simple calculation that

/- t r -r-=T+Y

I

I-tY -2.-=--:s=r

(4.5)

Therefore Eqs,(4.3) become (4.3' ) This is just the pair of equations ~iven by Mikhailov and Yarimecruk l9 ] wrich has been used to construct the soliton-like solutions. It is well known that, besides beinF consIdered as the li~ht cone coordinates of the variables t and r as ~iven in Eq.(1.7), (~,n) can also be used as the complex combination of two Euclidean coordin::.tes, say. z and r

.fl1 =t:c-Y

,

(4.6)

Therefore, the sallie equations in case (iv), can be used equally well to descritle the static axially symmetric model in 3-dimensional space. In this case our form of the linear scatterinF: problelll associated with t.he sirma model, Eq. (4.3), coinsides with that given by Bais and Sasaki l101 . The spectral parameter s given in Ref.IIO) is related to u we used above by the formuh\

cc=iZ.[2S .

(4.7)

For the static axially symmetric model, by 'changing the variables from to (r,z), and introducin~ w=i~ instead of c into Eq.(4.4) we can show that w=c.ocr, ~ ;J1.)=l\.-~ -rj T"- (7I.-rt (4.8) (~,n)

with

(4.9)

Kow it iti easy to find that (4.10)

Passing to the variables

(r,z), Eq.(4.3) can be rewrjtten in the form (4.11)

where

w given by Eq.(4.8) 1s a function of the coordinates

rand

z

and of

337 Nonlinear a-MOdel on Multidimensional Curved Space with Certain Cylindrical Symmetry

979

the spectral parameter ).. The solut ion of Eq. (4.11) may be considered as a functional of M as well as a function of (r,z), i.e., (4.12)

From this point of view the differentiation with respect to the a compound one:

(r,z)

must be (4.13a)

(4.13b) These two differentiation operators have been denoted as Dr (D 2 ) and Dz(D 1 ) respectively by Belinsky and Zakharov in their paper on the gra\'ity field .. equaticn[II]. From our formulation it is straightforward to show the commutath'ity of these two operators:

(Dr' DiI ]

== [d,. , d.} =0

(4.14)

,

and the equivJleRce of the pair of equations[ll]

Dr 'IIr'= T

y J,

+ U) .T.

y'+w'

(4.15)

to the linearzation system of equations we have obtained in the last section. \Veemphasizehere once again that in our formulation, " is a transformation functional which carries one field into another satisfyin~ a similar equation of motion. As another example of utilizat.ion of the general coordinate transformation we give here the Backlund transformation of our sigma model. Suppose we start with a model in the standarc form

and cp(I;,n) is a solution of this model. By using- the dual-like transformation, ~'(I;,n) as defined in Eq.(3.2a) is a solution to the model witr tpe reduced metric ,

~1;(Il

LlrV\) = ~-t

(4.16)

- T-r

It is possible to carry the reduced metric back into dinate transformation (c.t)

Ll'-~=i-~

6

by means of a coor(4.17)

Therefore we obtain another solution (4.18) corresponding to the oriyinal 6=I;-n. This is the Backlund tranbformation we have obtained~ Evidently, it is just the combined transformation

338 980

CHOU Kuang-chao and SONG Xing-chang

which leaves the equation of motion (2.4) (and hence the invariant.

La~rancian

(2.2»

V. Kac-Moody algebra In Sectiol,;; II and III we have shown that by taking- cx=k- l as the parameter, the dual-like transformations as given by Eqs.(2.5), (2.12) and (2.14) form an one parameter Abelian yroup A(l): (5.1) and that by choosin~ the boundary condition properly the set of the transformation functiona1s {w} becomes a nonlinear realization for this ~oup (5.2) From Eq.(5.2) we have VI-OI; CP'IOI.~

.,1) =-V-l(

01;

l principal chiral model [1-8] !Jas developed rapidly ovpr the past few Yl.!~rs. Some beautiful mathematical results and a great deal of insight into the nature of the hidden symmetry inherent in two dimensional integrable systems have been achieved. In spite of all these developments, the conserved currents obtained in Refs. (1-8] consist of only half of the Kac-Moody algebra. T~e older derivation [1-6] uses two different values of the parameters 1 and 1', which makes the whole procedure too cumbersome. In papers [7-8] the regular Riemann-Hilbert (R-H) tran",formaticns were used to "demonstrate the: algebraiC structure of the linearized equations in a much simpler and more elegant way. However, no relation between the transformation of the 1 inenl'izerl equations and that of the basic field was given in Refs. [7-S]. In the present note a Darboux transformation is constructed that forms a nonlinear representati.on of the chiral group containintr .a continuous parall'eter t. The R-H kernel H(t) is assumed to be analytic inside an annular regicn a~ltl~b in the complex plane" and expanded into a Laurent series. That part of the Laurent expansion analytiC at t=±l is shown to be relaT~d to the Darboux transformation whose generators form a I

is of the same form as

(t ,j jX) =-1:t j E; (j jX ) dE; + 1; t,i'1 (j jX

(3.3)

!l(t,j). Le.,

)d~,

(2.4)

the new current

(3.5) The relation (3.1) is often called

will be both curvatureless and conserved. Darboux transformation.

IV. Riemann-Hilbert transformations Solution

lP(t,j)

of the linearized equatjons (2.6) c:ln in genE'ral have

singularities representing solitons in the complex t-plane other than To find the desired transformation function with a fundamental solution analytic in an

annul~r

lP(t,j)

xCt,j)

to the linearized

region in the complex t-plane.

t=±l.

in Eq.C3.1), I.e start s~'stem

(2.6) which if.;

In the following it is

useful to construct a function H(t,j,u(t» =r1(t,j)u(t)W(t,j) with

u(t)

(4.1)

a group element independent of space-time and analytic in the same

annular region in the complex t-plane.

For an infinitl"sirr.al group transforma-

tion u(t) =1+w(t), we have

p. ( t ,,j , u (t ) )

(4.2)

= ~.. ,i , II Ct

))•

h ( t , j , II" ( t ) ) = 1/1-1 ( t , j )11 ( t ) If; ( t ,j ) • By assuming the annular region in which

as I t I~b

h(t.,j,w(t»

is analytic to be

with O, h_(t,j,w o )]_+[h_Ct,.i,w 1 >, hCt,j,w c )]_ Therefore the infinitesimal transformations form a Lie algebra. explicit let us take

To be more (5.7)

with

Ia

the generators of the group

G and form the co~mutator (5.8)

where k and 1 are positive or negative integers. With this choice of wet) it is easily seen that our transformation generates an algebra isomor~hic to the Kac-Moody algebra GxC(t- 1 ,t). After this work was finished, we learned that the authors of Refs. [7] and [8] have given similar proof to the same problem. We decide to publish our res~lts because of the siwplicity of our approach. In our proof neither a~xi­ liary quantities such as G(R.',R.)=

r

G(m,n)R.,mR. n =R.':2.{R.'_2.

(.1/.,)-1

p.)}

m,n=O

nor transformations corresponding to different values of

t

are needed.

VI. Transformation of the basic field Putting

t~m

in Eq.(2.7) we obtain immediately Q(",j)=j.

(6.1)

Substituting Eq.(6.1) into Eq.(2.6) and using Eq.(2.1) we find (6.2) with v a constant element of the group G. Therefore the corresponding transformation for the field q(x) is nothing but x(m,j,u), i.e., q ~ qx(m,j,u):: qX(j,u).

(6.3)

Furthermore, in the limit t+m, the gauge transformation (3.3) for the potential Q(t,j) reduces to the one for the current j j (j

,u)=X- 1 (j ,u)dX (j ,u)+x -1 (j ,U)jX (j, u).

(6.4)

According to Eq.(5.3) the function X(t,j,u) forms a nOlllinear representation of the group G for any fixed value of- t, in particular, for t=m. So the same is true of the transformation on the basic field q(x) (6.5) By virtue

of Eqs. (4.10) and (4.5). only the zero power term in the

347 Kac-Moody Algebra for Two Dimensional Principal Chiral Models

expansion of

b(t,j,w)

appears in

1397

X(j,u)=X(~,j,u)

X (j,u)=t+h. (,j,w).

(6.6)

For particular choice with

w(u)=wala kept fixed and k integers, we change the notation from to h(t,j,k) etc. and denote h(t,j,k=O) by A(t,j)

h(t,j,~)

(6.7)

Expanding

A(t,j)

into a Laurent series +~

A(t,j)=

r

n An(j)t ,

(6.8)

we get h(t,j ,k)=I.

.\n(j )t,,+k.

(Ci. 8' )

n=-cm

from which it follows that

x(j ,0) =1+ h. (j • 0) =1+ A• ( j ) • x(j ,k)=l+h D (j ,k )=l+A_k (j).

(6.9) (6.9' )

Tberefore for any fixed value of integer k the transformation X(j,k) on the field q(x) is generated by only one term A_k(j) in the expansion of A(t,j). Multiplying t-:< on A_k(j) and sun.ming over k, we see that A(t,j) itself generates a t-dependent transformation on the field q(x), i.e., (6.10)

q -... q+qA(t,j).

This is nothing but the transformation considered in Refs. [2], [5] and [7], from which the hidden symmetry is extracted. These authors have c~osen a particular class of fundamental solutions such that W(t,j) is analytic in a circle with the center at the origin t=o. In t~is case the negative powers disappear in the expansion of A(t,j). Fence in their treatmen! the transformation corresponding to k-positive integer is trivial and the nontrivial algebra associated with its transformation consists of only half of the Kac-Moody algebra.

VII. Sorre comments 1.

If the analytic region of lO.Assuming ~:~Y>o (This is certainly true for sufficiently large ~ at which the renormalized coupling constant g(~) is small). it is easy to see that the minimum of Eq.(34) is (35)

at L=O

and

(36 )

.p=0.

That isin this case the supersymmetry and chiral symmetry are both unbroken. In case II), ;tlle' effective Lagrangian is 3

Yeff=-S(gy)-1(SCF+s;F)+Cf(Sc3~~)F 3

1

+Cf[(S+)3+~lF+C [(s S+)mlD CdC c and

th~

(37)

scalar potential is (38)

wherE' (39) ~v

From aL-O, we obtain (40)

and (41 )

There is one minimum for the V(¢,L)I

L=L

of Eq.(41) (ymin(¢,L)=O) at 0.:

(and L. 2 =0).

(42 )

372 CHOU Kuang-ehao, DAI Yuan-ben and CHANG Chao-hsi

228

For ;:Cil~O ,there is another minimum (Vmin(~,L)=O) at 28

3+'1

-1

~=~I=(yg C f

)

-cy-

(and L02 =0) •

(43)

Both the degenerate minima are reached at L Q 2=0, with one of them at ~=O and the other at ~~O. This means the supersymmetry is unbroken no matter which minimum is taken and the chiral symmetry may be broken (when ~-~1) and may not be broken (when ~=~o). If the non-renormalization theorem[3] valid for arbi trary order in perturbation theory can be used here we should expect Cf=O.i.e •• case I), the chiral symmetry as well as supersymmetry is unbroken as shown above. More generally, the solutions of Eqs.(24) and (25) may have terms containing D acting on Se or Se+ but we would not consider them at this moment in this paper as is done in Ref.[2]. In addition, the solution of a' +b' Eq.(25) may also have terms of the form Cdi(Se~Se ~+h·c.) provided ai+b i =

~ , but the, can only exchange Cd (See Eq.(3l»

in, the scalar potential

sothey too _ill not affect the conclusion. Therefore the scalar potential Eqs.(3l) and (38) are quite general. Finally, results obtained in Ref.[9] indicate that in SSYM theory, apart from the renormalization made in Ref.[9], the superfield V may need additional non-linear renormalization. As this additional renormalization is equivalent to a gauge transfQrma~ion we can expect that it d~~s not affect the gauge invariant quantity discussed here. In summary. we conclude that in consistency with Witten I s index theorem the supersymmetry is unbroken for the N-l pure Yang-Mills supersymmetric model but the equation obtained here admits of both, chiral symmetry preserving and chiral symmetry breaking, solutions.

References J. G.veneziano and S. rankielowiez, Phgs.LBtt., 1l1!J19tf2)23J. 2.r.R.!aglor, G.P8neziano and S.rankielowicz, BU~.Phys.,~(J983)493; N.B.Pesldn, ,preprint SLAC-Pub.-306J; A.Davis, N.Dine and N.Se1berg, Phgs.Lett., ~(J983)487; H.P.Nilles, Phgs.Lett., ~(J983)J03. 3.B.Zum:tno, BUcl.Phys., !!!!,(J975)535;

P.flest, BUel.Phgs .., !!.!E2JJ976)2J9;

".Lang,

MICl.l'hys., BU4(J976)J23;

N.'.t.Grium, II.Rocek and ".Siegel, NUcl.PhYs., ~(t979)429.

4. J.fless and J.Bagger, ·supers!PJlllMltry and Supergravlty'!, Princeton Lectrzre Notes(t98t). 5.R.l'Ukada and Phgs. Rev.,

r.JraZlllllol,

Phgs.Rev.Lett., 1l.,(J980}J142;

~(J980}485.

6. S.Ferrara and B.Zlllllino, BUcl.Phgs., !!l(J975}207;

373 an the Dynamical Symmetry Breaking fOr the

H.'

_ _ _~229

PYre supersymmetric Yang-Mills MOdel

~.CUrtright,

L.F.Abbott,

Phys.Lett., M.~.Grisaru

~(1977)185;

and H.J.Schnitzer, Phys.Rev.,

~(1977)2995;

Phys.Lett.,

~(977)161;

M.Grisaru, in "Recent Developments in Gravitation" rCarge se 1980) eds.M.Levy and S.Deser. 7. J.C.Collins, A.Duncan and S.D.Joglekar, Phys. Rev., 8. Y.Kazama, Preprint KIlNS 683 HE 9. O.Piguet

~d

('1'H)

~(1977)438.

83/09.

K. Sibold, Nucl.Phys., BI97(1982)257, 272.

374 COJ/IDW2 •. in Theoz. Ph?s.

(Beijing, China)

Vol.3, NO.4 (1984)

491-498

THE UNIFIED SCHEME OF THE EFFECTIVE ACTION AND CHlRAL ANOMALIES IK ANY EVEN DIMENSIONS Kuang-chao CHOU{ JIlti ), Han-ying GUO( tlSi(~ Xiao-yuan LI( f/}' ) and Ke WO( ~ if) Institute of Theoretical Phgsics, Academia Sinica, P.O.Box 2735, Beijing, China

and Xing-chang SONG t ( ;IHf* ) Institute for fheoretical Physics, State university of Ne., York at Stong BrooJc, Stony BrooJc, Ne., York 11794, USA

Received April 29, 1984

Abstract

_thad,.

Based on the flei1 homomorplliSlll a unified scheme in which all the important topological properties of the pseudoscalar GOldstone boson and gauge fields in even dimensional space are described in one remarkablg compact fpzm is given. fhase properties include the effective action, the skgZl/lion anomalous current, Abelian anaIIIalll, spIIIIIetric and as!IIIINtric non-Abelian c:hiral anomalies, and anomalll free conditions.

A year ago E. Witten discussed the global aspec~s of current algebra[1] and proposed that the Wess-Zumino chiral effective action[2] can be described in a new mathematical framework. He pointed out that this action obeys a priori quantization law. analogous to Dirac's quantization of magnetic charge, and incoporates in current algebra both perturbative[3] and non-perturbative anomalies[4]. Applications to the standard weak interaction model require an arbitrary subgroup of global flavorsymm.t~ieato be gauged. However. the standard road to gauging global symmetry of the Wess-Zumino action is not available since its topological nature is unknown. In Witten's original work the author did this by resorting to the trial and error Noether method. In their series of works[5-8] CHOU-GUO-WO-SONG suggested that one can as usual introduce the minimal gauge coupling in 5-dimensional space first. then the gauge invariant Wess-Zumino action can be obtained by a series of nontrivial but systematic mathematical ma~ipulations.[5] In the processes a deep connection among the manifestly gauge invariant 5-forms. the Chern-Simons secondary topological invariants. anomaly free condition and the gauge invariant Wess-Zumino action has beeu found. Remarkably. the structure of symmetriC chiral anomalies[9] can be determined by studying the gauge transformation properties of the Chern-Simons. secondary topological invariants without having to evaluate any Feynman diagram[5,10]. Furthermore it has been shown[6] that th~ Bardeen's t Pezmanent address: Depar1:lllent of Phgsics, Peking university, Beijing, ChiDa.

375 492

Kuanrr-chao CHOU, Han-ying GUO, Xiao-yuan !.I ,Ke WU and Xinq-chanq SONR

asymmetric non-Abelian anomalies[ll] and the corresponding counterterms can be obtained if one does Goldstone boson expansions in the gauge invariant effective Wt.'ss-Zumino action functional.The relation between the symmetric and asymmetrlc non-Abelian anomalies has also been found[7,l2).Recently,it has been !'!uggested[l3) that the gauge invariant Wess-Zumino actior. functional in 2ndimensional space can b" obtain.,.,:l by properly constructing the m:.nifestly gauge invaraint (2n+l)-forms which sa.i3.t'y tht: Abelian anomalous Ward identity in (2n+2)-di~ensional space. The problem about the uniqueness of the gauge-invariant Wess-Zumino effective action has also been discussed. [l41 I~ this note, we would like to suggest a unified scheme of the gauge invariant Wess-Zumino effective action functional, Abelian and non-Abelian chiral anomalies, the anomaly free condition and the r.lanifestly gauge invariant generalized skyrmion anomalous current[15] in any even dimensional space. The description is based on the Weil homomorphism method in differential geometry[lG). It turns out that the Abelian anomaly in (2n+2)-dimensional space, the gauge invariant Wess-Zumino effective action, the symmetric and asymmetric non-Abelian anomalies and anomaly free conditions in 2n-dimensional space can all be given in a remarkably compact form. The formalism further reveals the intrinsic connection among these objects. We start by considering the theory with a chiral SU(N)!.xSU(N)R symmetry which is spontaneously broken down to the diagonal group SU(N). Under an SU(N)!. xSU(N)R transformation by unitary matrices (gL,gR)' U(x) transforms as (1)

In order to obtain a gauge invariant action functional under the gauge group the'rauge covariant derivative

H~SUCR)LXSU(N)R'

(2)

should be introduced, where AL(RJ the field strength 2-form is

is~tbe

gauge connection 1-form of RL(RJ' and

(3)

respectively. Under the gauge transformation

(4)

We have

376 The unified Sche1/le' of the Effective Action ad Cbiral Anomalies in 1In!l Even Dimensions

493

(5)

In particular, if we define n=u-1Du=u-l(d+AL)U-~R=UAL-AR

where

U

(6 )

AL(x) is the "gauge transformed" gauge field l-Iorm (7)

then, under the gauge transformation (Eq.(4»

we have

( 8')

where the "gauge transformed" field strength 2-form UFL(X) is defined as (9)

Eq.(8) implies that the gauge transformation property of UAL is exactly the same as those of AR' and n is the gauge covariant I-form. u Let us consider the interpolation between AR and AL '

o DEL

Kuaog-chao Cllou

Institute of Theoretical Physics Academia Sinica Beijing, China IHTRQDUCTION Two decades have passed since the first observation of CP violation in kaon decayl.

The subject is still not well under-

stood and the progress is rather slow compared with what has been achieved in the other branches of weak interactions.

As we all know now, nature has chosen the standard SU(2) x U(l) gauge model to describe physics at an energy scale below 100 GeV.

Both W and ZO bosons have already been seen

within the error predicted by the theory2.

It is therefore of

great interest to accomodate CP violation in gauge theories which seem to be the most promising ways from a theoretical point of view.

As Kobayashi and Maskawa 3 (K-M) first pointed out, CP violation can occur in the standard model through complex phases in mass matrix with more than two generations.

For three

generations favored by the present experiment there is only one phase causing CP violation.

Could this single phase be suffi-

cient to explain all the CP violation effects? welcome if it could.

It is certainly

However, it can not be answered a priori.

The origin of CP violation is closely related to that of the 609

398 masses and the number of generations, which in turn are described by physics at much higher energy scales.

It would not be a

surprise if some new ingredients had to be added to solve the CP problem.

We shall wait and see.

Since there were excellent review papers not long ag0 4 , it is unnecessary for me to repeat all the known results to you. What I would like to report is a recent analysis of CP violation in the K-M model after the measurement of the unexpected long lifetime of the b-quarks. The outline of this talk is as follows: I.

Parametrization of the K-M matrix;

II.

Physics of £ and £' in kaon systems;

III. Neutral particle-antiparticle mixing and CP violation in BO-BO system;

IV.

CP violation in partial decay rates of particles and antiparticles;

V.

Concluding remarks.

I. PARAMETRIZATION OF THE K-M MATRIX

For three generations of quark the K-M matrix containing three angles and one phase is usually expressed in the following form

V

=

=

610

Vud

Vus

Vcd

V cs

V td

V ts

Vub \

V)

V:b

(I.I)

399 where ci(si), i = 1,2,3, are the cosine (sine) of the angle 6i·

The Cabbibo angle 61 is determined to be~ s

1

=

•

227+. 0104 -.0110

0.2)

Recent measurements on b quark lifetime and the branching ratio rb+u/rb+c have put stringent bounds on the matrix elements IVCb/VUbl.

Their values can be found in the talks given by

Lee-Franzini and Kleinknecht in this conference. IVCbl = 0.0435 ± 0.0047 ,

(1. 3)

IVub IV cb I ~ O. 119

0.4)

•

Both IVcbl and IVub/Vcbl have been reduced from the 1983 values 5 and The fact that IVcbl is small and of the order of s1 2 can be used to simplify the K-M matrix.

In a first order approxima-

tion where ReVij are correct to order s1 3 and ImVij to order SIS

we have c1 V

=

s1 c 1 -s2 s3 e

-s1

s2+ s 3e

-s1 s2

i6

i6

0.5) -e

i6

Writing Vts and Vcb in the _following form V ts

=

Vcb

= s3

s2 + s3 e

i6

=

Ivtsle

i6 ts

0.6)

i6

=

I Vcb I e iticb

0.7)

and + s2 e

611

400 it is easily proved that V

.~

ts

.. el. u V

*

(1.8)

cb

Hence we obtain (1.9)

and (1.10)

One can now redefine the phases of the band t quarks by a transformation (l.ll)

(1.12)

+

and get from Eqs. (1.5)-(1.12) a form first suggested by Wolfenstein6

IV cbl

V ..

-io

-s s e t s

1 2

(1.13 )

1

The phases 0ts' Ocb are related to 0 by the following relations: (1.14)

(1.15)

612

401 The advantage of the present form for the K-M matrix is that ImVij is always proportional to a common factor (1.16)

which is the appropriate parameter measuring CP violation effects in various processes.

A similar but rigorous representation of the K-M matrix was obtained recently by Chau and Keung7 •

Since both s2 and s3 are proportional to IVCbl, it is more convenient in numerical calculations to scale it out. We write s3 :: a

Iv cb I =.!.... sl Iv ub I

(l.17)

(1.18)

From the experimental bound Sq. (1.4) and the value of sl we find a


10- 2 (-0.81»

VcbV~.(b+e) + Vubv~.(d+e)

+

1- •

No. of Event. Heeded

0

UI

VCbV~d(a+e)

K-J/,

Ie

C8

+ VubV~d(d+e) +

"Penluia

V* (d-e)

us ca

1>10- 1 (7. )"10-")

-1"10-'

]>10- 3

1.]>10' 3

(4.1"10- )

0.»10 2

S"10- 3

2.P10

2 "IO-~

].5-105

( 1.8"10- 2 )

(1.4-10")

)

9

V·bV. (;) I

as 0.19 (6.2-10- 2 )

_1>10- 2 (7.2"10-')

-4 _10- 2

1.7_10- 3

2.1"10-"

2.7>10 6

(-0.86)

(1"10- 3 )

(8.S-IO-')

(1.6-10 3 )

-0.17

-2.2_10- 2

1.1'10-'

2.1"10'

(-0.R6)

(-S.7-10- 3 )

(4.8-10-')

2.8"10'

-1.6'10-"

-l.l-IO-~

1.8"10- 3

(_2>10- 3 )

(-1.7 xIO-',

0.4 x I0- 3 )

4.5-10 9 0.4"\0"

421

needed to observe the CP violation effects is large, of the order 10 5-10 6 •

ACKNOWLEDGMENT This talk is the result of a collaboration with Wu Yue-liang and Xie Yan-bo. Discussions with Profs. Li Xiao-yuan, Chu Chen-yuan and L.-L. Chau have helped enormously in improving my understanding of the problem. I would like to thank Mrs. Isabell for her kindness and support in typing the manuscript. REFERENCES 1.

J.H. Christenson, J.W. Cronin, V.L. Fitch and R. Turlay, Phys. Rev. Lett. 13 (1964) 138.

2.

G. Arnison et ale , Phys. Lett. 126B (1983 ) 398; ibid 129B (1983) 273; P. Bagnaia et ale , Phys. Lett. 129B (1983) 130.

3.

M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.

4.

For a recent review, see L.-L. Chau, "Quark Mixing in Weak Interactions", Phys. Rep. 95, No.1 (1983).

5.

L.-L. Chau and W.-Y. Keung, preprint BNL-23811 (1983).

6.

L. Wolfenstein, Phys. Rev. Lett. 51 (1984) 1945.

7.

L.-L. Chau and W.-Y. Keung, BNL pre print (1984).

8.

C. Itzykson, M. Jacob and G. Mahoux, Nouvo Cim. Supple 5 (1967) 978.

9.

J.S. Hagelin, Phys. Lett. 117B (1982) 441.

10.

T. Inami and C.S. Lim, Prog. Theor. Phys. 65 (1981) 297.

11.

N. Cabibbo and C. Martinelli, TH-3774 CERN (1983); R.C. Brower, G. Maturana, M.B. Gavela and R. Gupta, HUTP-84/A004 NUB # 2625 (1983).

12.

P.H. Ginsparg, S.L. Glashow and M.B. Wise, Phys. Rev. Lett. 50 (1983) 1415.

13.

F.J. Gilman and J.S. Hagelin, preprint SLAC-PUB-3226 (1983).

14.

P.H. Ginsparg and M.B. Wise, Phys. Lett. 127B (1983) 265.

633

422

15.

Pham Xuan-Yen and Vu Xuan-Chi, preprint, PAR LPTHE 82/28 (1983).

16.

K.C. Chou, Y.L. Wu and Y.B. Xie, preprint ASITP-84-005 (1984).

17.

J.S. Hagelin, Nucl. Phys. B193 (1981) 123; B. Carter and A.I. Sanda, Phys. Rev. Lett. 45 (1980) 952; Phys. Rev. D23 (1981) 1567; L.I. Bigi and A.I. Sanda, Nucl. Phys. B193 (1981) 85; Ya. I. Azimov and A.A. Iogansen, Yad. Fix. 33 (1981) 383, [Sov. J. of Nucl. Phys. 33 (1981) 205].

18.

L.I. Bigi and A.I. Sanda, preprint NSF-ITP 83-168 (1983); L. Wolfenstein, preprint CMU-HEG 83-9; NSF-ITP-83-146 (1983); E.A. Paschos and U. TGrke, preprint NSF-ITP-83-168 (1983).

19.

A. Pais and S.B. Treiman, Phys. Rev. D12 (1975) 2744.

20.

L.L. Chau and W.Y. Keung, Phys. Rev. D29 (1984) 592.

21.

J. Bernabeu and C. Jarlskog, Z. Phys. C8 (1981) 233; L.L. Chau and H.Y. Cheng, BNL preprint (1984).

634

423

DISCUSSION PAVLOPOULOS: Does a t-quark mass of 60-70 GeV lead to a hopelessly small E '/E? CHOU: No, if ~ is around 0.33, E'/E can reach .01 for rot ~ 60-70 GeV. HITLIN: If IEI/ELis small and sin~ is of the appropriate value, then the t quark ss could be greater than the W or Z mass. Could enough tt pairs be produced at the SPS collider to make the decay t + W + b a plausible explanation of the recently reported events at the SPS Collider? CHOU: We better keep in mind such possibilities.

635

424 CHINESE PHYSICS LETTERS

Nov. 1984

Vol.1. No.2

TOP QUARK MASS AND THE FOURTH GENERATION OF QUARK CHOU Kuang-chao, WU Yue-liang, ~IE ran-bo (Institute of Theoretical Physics, Academia Sinica, Beijing) (Received 9 August 1984) To explain CP violation in Kaon system in the light of the recently measured b-lifetime within the franework of three generations of quarks, the top quark mass has to be greater than 50GeV if the current algebra value of the factor ~ is adopted and [e:'/E\ < 0.01. In this letter a fourth generation of quark is considered which can fit the present experimental data on CP violation, and KL+~+~- decay rate for top quark mass is around40GeV. The mass of the new charge 2/3 quark is predicted to be over 100 GeV.

The unexpected long lifetime of quark in the picosecond rangJlJ has changed the

whole picture on the CP violation and the mass

difference

In the Kobayashi-Maskawa theor~ with three genera-

in the KL-KS system.

tions of quark the mixing angles &J. and 63

determined from the b decays

are found to be very small[3]. Consequently, the contribution due to top quark exchange to the box diagram of the small unless

top quark mass

is

large.

KO-KG

mass matrix element

is

For top quark mass less than 1

TeV there will be no appreciable effect to the Kr. -Ks

mass difference by

top quark exchange. All theoretical caculation of the box diagram contains matrix element relating the KO and i{0 states to the product of quark operators.

Current

algebra evaluation[4\ of this matrix element differs from its vaccuum insertion value by a factor of BK=0.33±O.17. In a

previous worJJ5] we have said the

following:

New sources of CP

violation besides the single K-M phase might exist if IlIt around 40GeV,

[E'/E l:ii

ticle of mass

around 40GeV,

0.005 and BK:ii a

is found to be

0.6. Just a few weeks ago a new par-

candidate of the top quark,

was announced

by UAL group in CERrJ6J • The measurement of E'/e: has also reduced its upper bound[7J.

Now we

are facing the possibility that something new might be

needed in order to explain all the existing experimental data.

The simp-

lest possibility is to add a new generation of quark. Standard model with four generations of Oakes[B] .

been

studied by

He pointed out that a relation Mt' =Mt=40GeV exists if 8uras an-

alysiJ9I on KL -I.:

2735~

.~cademia.

;'-5- ~

S.:..nica,

3.;ijinq, C!i.i::a

Received June 24, 1985

Abstract In this paper, it is shown that the cohomology of r;:eneralized secondary classes, the F~ddeev type cohcrnolc~y a~= the generalized gaug~ ~~ansfQrmation can be easily obtai~ed by expandi~g the Chern form accordi:2q to t.r,e d,=gre~ of t ..1c :o=ns ~n its submanifolds and usi~g :~~ c1csed ;:roye=~~' of the C..'-:er:-: =C~:::. =-t is a.2so snot"n :::..:It.a -vacuum ~e.,:,,~ ':'.10'1 t.=:e =::ec'tit·-c :~g':a,,'1q~ar: arises when gauge rield ~n c~c grcu~ ~anif=ld it; pre.c:er..t.

I. Introduction Rece~tly, H.Y. Guo et al.(11 have introduce~ higher ordpr charac'teristic classe~ and cocycles y;hich lfeneraliz~ 1:!~" Weyl homodescribed by the coboundary of the Chern class. About the same ti~e Faddeev[2] has constructed another kind of higher order co~rcles based on the group manifold, which ha~e been elaborated in aef.[3]. In chis paper, a simclified deriv~tio~ of Cher~-Sirnons cochain is given. \'ie expand &lie Chern form according to 'the degree of the forms in its submanifolds. Only by usj.ng the closed property of the Chern form, can one obtain a Chern-Simons cochain which represents a descent relation between both submanifolds. In some special cas~s, one .can easily obtain famil.iar results gh·.;!i. in Refs. [1,2,3]. III Sec.II we show the ca.se of the c0ho!'lolcgy of th;: generalized secondary classes. In Sec.III, we ~ive the Factdeev type cohomology. In Sec. IV, the generalized gauge transformaticn and the Chern form gaugG potential are discussed. There the P. po,rameter ill the f.-vacuum is interpreted as a line integral of gauge potential in the group manifold. morp~yism

454 CHOU Kuang-chao, rvu Yue-liang and XIE Yan-bo

28

II. Generalized Secondary Characteristic Class The Chern density and the Chern-Simons secondary class are now familiar to theoretical physicists. In a 2n-dimensional space with gauge potential A=A,,(x)dxi.' ,..

(2.1)

and curvature (2.2)

F=dA+A 2 the Chern density is defined as

(2.3)

which is a 2m-form.

Using Bianchi identity

dF=[F,A]

(2.4)

it is easily verified that

~(F)

is closed (2.5)

d;1 2 ::1 ( F) =0

and can be expressed as (2.6)

where 1

n~m_l(A,F)=mLdtn2m_l(A,F~-1) 2

2

F t =tdA+t A

,

I

(2.7)

is the Chern-Simons secondary class. In a recent paper[l] the authors introduced generalized secondary characteristic classes relating to the higher order cochain :l.rod cohomology. We shall show in this note that their results can be easily dedaced from Eq.(2.5) by the general property that the density is a closed form. Consider a manifold consisting of two submanifolds, one of which is the ordinary space manifold of dimension Nl with coordinates x~, u=1,2 .•. N1 , The other submanifold is one of parameters ~1,i=1,2,.,.Nz, The exterior differential operators are Cher~

455 A Simplified Derivation of Chern-Simons Cochain and a Possible Origin of a-vacuum Term

29

(2.8)

and

The gauge potential ..4 can be decomposed into two parts

(2.9) where

1

and

(2.10)

j

The curvature .7 now becomes .9=(d x +d: )(A+B)+(A+B):=F+G+~I

(2.11)

,

'>

~vhere

I

G=d.B+BZ

and

(2.12)

J

The Chern denisty

(2.13) satisfies Eq.(2.5), i.e.,

By expanding ':2n(:?) according to the degree of the forms in d:'.4, we

have 2n

n2n c,9)= L

m=O

where

n2n _ m ,m(A,B)

m in f,i.

1'2 2n _ m ,m(A,B)

(2.15)

,

is a form of degrees 2n-m in

,.

x~

and of degrees

456 30

CHOU Kuang-chao, WU Yue-liang and XIE :'"a.n-bo

Substituting Eq.(2.15) into Eq.(2.14) and comparing the degrees of the form, we obtain

(2.16)

As a special case, we choose

1

B(x,O=O ,

A(x,O=A{O)(x)+~.;i(A(i)(X)_A{O)(x»

,

f

(2.17)

~

then G=O,

I

\:2.18)

By a change of the notation

M-H.

}

(2.19)

Eq.(2.16) can be easily seen to be just the theorem 1 proved in Ref.[1].

The cohomology and generalized secondary characteristic

classes then follow

by merely integrating over simplexes in the

submanifold .;.

III. Faddeev Type Cohomology In this section we show that the cohomology of gauge groups in Faddeev's approach I2 ] can be deduced as a special case of Eqs.(2.6) and (2.7)

(3.1)

457 A Simplified Derivation of Chern-Simons Cochain and a Possible Origin of 6-vaccum Term

31

Expanding the Chern-Simons secondary class according to the of the form in d~i 2n

;:;

n-

1 (A ,B

)=E n~n_m m-l (A,B). m=l'

degre~~

(3.2)

Substituting Eqs.(2.15) and (3.2) into Eq.(3.1) and comparing the degrees of the forms, we have

n2 n, 0 =d x n~

~n-

I~

2n-m,m

~.

-d

=d

1,0 '

nO

x 2n-m-1,m

(3.3)

+d 11 0 ~

2n-m,m-l ,

,.,0

"0,2n- ('0,2n-1

As a special case, we choose B(X,;)=U-l(X,~)d~U(x,~) , t,

A(x,~)=U-l(X,;)(A(x)+dx)U(X,;)

where

U(X,~)

,

1j

(3.4)

is an element belonging to the gauge group G.

Then

}f=O, G=O,

(::.5)

F=U-l(X,~)F(x)U(x,~)

,

F(x)=d x A(x)+A 2 (x) •

In this special case the sequence Eqs.(3.3) become

(3.6)

d~n~,2n_l=O

•

Eqs.(3.6) are just the results obtained in Ref.[3].

Therefore one

458 32

CHOU Kuang-chao, WU Yue-liang and XIE Yan-bo

can choose (3.7)

where

(3.8)

The Faddeev type cohomology then follows by integrating over the simplexes in the submanifold s consisting of points pi

=~":)~l ,..,.~2 ... ,..,:i-1~! ,...., '-,:i+1 • ... ). =(1,1, ... ,1,0,1, ••. ),

(3.9) i=1,2, .•. , Po=(1,1, ...• 1.1.1 •... ).

IV. Gauge Fields in the Group Space and the 9-vacuum In this section, we consider

anothe~

special case

1

M=d_A+d B+AB+BA=O t, x

(4.1)

J

F=d x A+A 2 ;100 .

In this case the sequence Eqs.(3.3) become

d

d

n°

-

d ~o

f; 2n-2k-1,2k-2-- x"2n-2k ,2k-l

'

n° =-d nO +:] ~ 2n-2k,2k-l x 2n-2k-1,2k 2n-2k,2k

(4.2) J

459 A Simplified Derivation of Chern-Simons Cochain and a Possible origin of a-vacuum Term

where

n2n - 2k ,2k

33

satisfies (4.3)

transfor~5-

This case can also be regarded as a generalized gauge tion.

For example, by choosing B(X,~)=C(;)+U-l(X,;)d>U(x,;)=C(;)+V(x,;), "?

I

(4.4)

we have G=d~B+B:=d~C(;)

...

.'

,

M=d~A+dxV+VA+AV=O

where form

C(~)

1

\ 4.5)

J

,

is Abelian gauge potential only depending on ~, so Chern

~2n-lk,2k

can be written as i.'1

"2n-2k ,2k

0'

P

Tr(F,"1-k Gk)

n!(271)2

=~,,:; n-2:~

2n-2k,2k-1

\

lk-1

• C(" (d,C(

(n_k)!k!(2~)n

: "i . (i

)

n-k. J

T, where T, =A,e 2 /3. The conflict is resolved when we recognize the need for a transform of '" at high temperatures to correctly describe the Lorentz-invariant massive particle. ~asinio

new ground state is given by ,

(1)

p.'

where 9p is related to the dynamical mass m acquired by the quarks tan29 =.!!!.

(2)

p

p

and p here is the magnitude of the momentum. We wiII show in this section how the NIL ground state may be understood as a rotation in chiral space and demonstrate how it affects the Dirac equation. The original massless Dirac field ",(x,O) is taken here to have the expansion at time 1=0:

",(x,O)=

)-V 1: [ [:L;Pp::P,L 1 P.!

R

p,R

11 e'P" . + [ SR(p)b~p,R t -SL(p)b_p,L I

.

== . ~ 1: "'(p)e'P" v V

,

(3)

(4)

p

where S, are helicity eigenstates satisfying

q·PS,(p)=SS.,(p) .

II. NJL VACUUM To appreciate the problem, we review the picture of the chiral-broken vacuum as first formulated by NIL. (We assume that we have integrated over the gluon degrees of freedom in QCD and are here discussing the effective theory involving only fermions.) In the presence of dynamical interactions, the naive vacuum lo} may no longer be the lowest-energy eigenstate of the Hamiltonian. The new ground state of the Hamiltonian Ivac} is the analog of the BCS ground state with quark and antiquark pairings. If ap" and bp " are the annihilation operators for the massless quarks and antiquarks, respectively, with helicities s =± for the R ,L states, then the

II (cos9p - $ sin9p a)"b ~p., )10)

lvac} =

(5)

Note that the Fourier component fields "'(p) are in general time dependent. For the purposes of our discussion here, we focus on the time slice at 1=0. The NIL chiral-broken ground state can be obtained by an infinite chain of SU(2)p chiral rotations around the (parity-conserving) two-axis by angle 29p : IvaC> = '=

II e2i8px2lPllo}

(6)

II RpU:Jp)IO}

(7)

=010) . 596

(8)

@1991 The American Physical Society

492 597

SIGNATURE FOR CHIRAL-SYMMETRY BREAKING AT HIGH ...

For simplicity, we shall refer to this rotation around the two-axis as the chiral-2 rotation. A. SU(l). allebra

(23)

The Xi(p) are the generators X 3 (p)=-i- l:s(o:',op.,+b:"p.,b_ p,,) ,

,

X 2(p)=f

l:s(o:"b~p"-b_p,,op,,), ,

(9) (0)

j,"4=;"0= [0I 0Ij ' [0 -ia 0

(i2)

And if we further take the sum over the momenta and form the global generators

Xi= l:Xi(P)'

(24)

y= ia

They satisfy the SU(2)p algebra at each momentum p: [X,(p},Xj(p'}j=iEilkXk(P)/iP,P' .

Following NIL, we may treat the massive modes as approximate eigenstates of the total Hamiltonian, and consider the time evolution of the massive modes as if they were free particles. With the explicit representation given in Eq. (23) it is straightforward to derive, using the representation

(13)

the free field equation for the chiral rotated spatially nonlocal equation

v' -V 2 +m 2 a [ 1i=Vi "'V+"Oat

q,.

1_\11=0.

It is the

(25)

p

we see that they form the global SU(2) algebra (14)

Note that we are here dealing with the infinitedimensional representation of the global sum algebra.

B. Transforming the Dirac equation

Under the chiral-2 rotation in the Hilbert space, the annihilation operators and the massless Dirac field transformas Qp,s-.A p ,. ,

(is)

bp,s-...".Bp,s ,

(6)

This nonlocal Dirac equation is strange because it appears to show chiral invariance, and yet we know by construction that the chiral-rotated q, describes the massive particle free field. It is also not Lorentz invariant. But in our context of temperature field theory this latter objection plays no role. It is therefore reassuring to find that there is a similarity transformation acting on the components of ifJ that transforms away the nonlocality in the Dirac equation. _ . If we work with the Fourer components of \II In momentum space, and define the similarity transformation ifJ(p,t):ae -i8p Y'il \ll(p,tl,

(26)

then the nonlocal equation for ifJ implies the usual massive Dirac equation for \II:

[Y'V+"O;t +m j\ll(X,tl=o.

(27)

where A p" =:Rp«()p }Op.,:R;I«()p) =cos()pop" +s sin()pb :"p" , Bp,s =:Rp«()p )bp,,:R; I( ()p)

= cos()p bp,s -s sin6po :"p,s ,

(17)

(18)

From Eq. (26), we can show in our representation of Dirac matrices that \II( x, t) has the usual expansion (po=v'p2+m 2)

(9)

(20)

and

+ V _p,sB t_p,se + iPO') e ip·", ,

Ap"lvaC>=Bk,slvac> =0.

The new operators Ap,soBp" describe the excitons with dynamically generated mass m that propagate in the new medium. Under the Hilbert-space chiral-2 rotation, the massless Dirac field of Eq. (3) transforms in an obvious way into .p(x,O}->ifJ(x,O) , where

(28)

(21) where the massive spinors are given explicitly by

(29)

(22)

(30)

493 LAY -NAM CHANG, NGEE-PONG CHANG, AND KUANG-CHAO CHOU

598

sin8pSL(p) 1 [ -cos8pSL(p) ,

V_ p• L

=

V

= [COS8pSR(P) -p.R

1

-sin8ps R (p).

III. HIGH-TEMPERATURE RESULT (31)

(32)

and satisfy the Dirac equations {iy·p-iYo·po+m )Up., =0 ,

(33)

(iY'p-iYo'po-m lVp., =0.

(34)

Equation (26) is in fact an example of a transformation of the generic Foldy-Wouthuysen type (more precisely, a Cini-Touschek transformation).2 The result of this section may therefore be summarized in the transformation law of the massless Dirac field under the Hilbert-space chiral-2 rotation

n~(x,O)n-l =e -/ly.v/"cv;'I'(x,O) ,

(35)

where 8 is here the differential operator in threedimensional space implied by the momentum-space equation (2). The Cini-Touschek similarity transformation is the analog of the :Dati- A) similarity transform of ~ that results from a Hilbert-space Lorentz transformation. Loosely speaking, a Hilbert-space chiral-2 rotation induces on the ~ a Cini-Touschek similarity transform. The nonlocality of the similarity transform reflects the infinite-dimensional nature of the SU(2) representation involved here. The nonlocality of the ijI field equation is also a warning that it is the wrong basis on which to discuss the chirality of the theory. Indeed, Eq. (25) gives the false indication of chiral conservation. It is only after the nonlocal Dirac equation has been straightened out that one can test for chiral-symmetry breaking under the chiral X J rotations. As we shall see, at high temperatures, the thermal radiative corrections lead to a nonlocal Dirac equation for the massive particle pole in the Green's function. Proper physical interpretation of the signature of chiral breaking requires that we do a Cini-Touschek transformation to get rid of the nonlocality in the Dirac equation of the renormalized particle pole in the Green's function. After this similarity transformation, the inherent chiralsymmetry breaking due to temperature effects becomes evident. Before we close this section, we note the equality ~(x,O)='I'(x,O)

.

(36)

This may be verified by direct substitution of the inverse transformation to Eqs. (18) and (20) into the expansion for ~ as given in Eq. (3). At zero temperature, then, the correct signal for chiral-symmetry breaking is to use either ~ or 'I' and calculate the expectation values of ifl/l or iii'l' with respect to the full vacuum. The equality between the two Heisenberg operators at t =0, Eq. (36), guarantees that the vacuum expectations values obtained by the two different I/I's agree: (vaclifl/llvac) = (vacliii\jllvac) .

(37)

In an earlier work, J we had reported the result of a real-time temperature-dependent field-theory calculation of dynamical chiral-symmetry breaking at high temperatures. We found that, for QCD, dynamical symmetry breaking persists at high temperatures. In this section, we present an analysis of the result and point out the close connection between the zero-temperature chiral rotation of the NJL vacuum and the Cini-Touschek transformation needed in the renormalization of the temperature-dependent Fermion two-point function. We perform our calculation in real time.' Our technique is to introduce into the Lagrangian an explicit mass term for the fermion, put the system in a heat !?ath, use renormalization-group analysis to sum over higher loops, and study the critical limit as m, __ 0. If in this limit, the thermal fermion propagator shows a Lorentz-invariant massive particle pole, then we say that dynamical symmetry breaking has occurred. At zero temperature,S we found that the chiral flip part of S,-l(p), for P,J',. in some finite domain, actually survives the critical m,--+O limit, thus signaling the bifurcation in chiral-symmetry breaking. In this section, we look for the temperature dependence of this chiral-symmetry breaking. The result of the real-time thermal field-theory calculation may be put in the form (our results for A ,B agree with the one-loop calculation of Weldon,6 except that we have also included InT /m terms; Weldon does not introduce an explicit mass term, and thus did not look for a perturbative root around the original m, pole)

Si 1 (p2,P5T)=iY'p(1 + A )-iYoPo(1 +B )+m,(1+C) , (38)

where A, B , C are functions of P, Po, and T. In terms of the parameters Ipl::m,sinhs,po::m,cosh s , we have, using the Feynman gauge, and in the limit of T2 /m;» I,

(39)

(40)

(41)

Here we have dropped terms that are of order I as T 2/m; __ oo. Also we have defined >",=g;/(16"/1"2) and the relation T Q • T Q : : C/1. The Lorentz-invariant massive particle pole in the thermal fermion Green's function occurs at Pn=Vp2+.M\ where perturbatively

494 SIGNATURE FOR CHIRAL-SYMMETRY BREAKING AT HIGH ...

m;

. 4] +--41T2T2 .M. 2 = hm m,2 { I+A,Ct [ -6[ I n - - -

'",-0

m;

3

,..,.2

1

T2 -6In--+const + ...

m;

}.

renormalization-group analysis to sum over higher loops . The existence of this particle mass at high temperature is already a good signal that chiral-symmetry breaking persists at high temperatures. What we want to study is whether the traditional signature of chiral-symmetry breaking is still good at high temperatures, viz.,

(42)

The critical limit m,_O is taken using the fixed-point theorem of bifurcation theory. [See the discussion following Eq. (68) in Ref. 3.] As was shown in Ref. 3, this mass survives the critical limit as m, -->0, so that it is a temperature-dependent dynamical mass: ,M.2 _ T-• ..,

2~ ~ 3 In T2 .

(43)

A~ Donoghue and Holstein' considered the case m,~O and also found in the one-loop perturbative calculation the Lorentz-invariant massive pole at high temperatures. Our technique goes beyond one loop by using the

1

Z2 Sii lchiralftip=

2:~om, {I +A,Ct [-3 [In :: -

= lim m,(A,y) -6Cf

599

lim (vacljf!JIlvac)p*0.

(44)

'",-0

A potential conflict with the usual notion of chiralsymmetry breaking arises when we study the chiral flip part of S i 1 [Eq. (38)] and directly take the limit as m,-->O. This would be consistent with the idea that we simply evaluate the (vacljf!JIlvac)/1 in the thermal vacuum without doing any more renormalization than the minimal ones needed at T=O. In our case, our result in Eq. (38) is in Feynman gauge, which explains why even at T=O the coefficient of 'Yol'o is not unity. So we renormalize S 1 by setting the coefficient of 'Yol'o equal to unity when T=O. We do this by multiplying it by the T=O wave-function renormalization Z2:

ii

411 + ... }

(45) (46)

lb ,

m,-O

where

Z2=I-A,Ct !ln : ; -2]+'"

y=.J...+E. A, 2

=E.2

[In T2 A~

[In T2

,.,.2

-~3

-~3

I

(47)

I

(48)

(49)

.

Here y is a renormalization-group invariant. Each term in the perturbative series is valid so long as T2» Since in the end m,_O, it would appear that the series should be valid for all T. However, the positivity requirement for y due to the representation, Eq. (46), shows that we need to impose the condition T> Ace 213 for the validity of the sum. When we now take the critical limit m,_O, y to oneloop renormalization-group accuracy does not depend on m, and does not approach the y = fixed point [see the discussion following Eq. (68) in Ref. 3], and so the chiral flip part does not survive the critical limit. It vanishes for T> Ace2l3. The only exception is when the temperature T is at the critical temperature8 •9

m;.

°

(50)

At

that

point, our one-loop renormalization-group

analysis fails since it diverges even before the critical limit. We need to go to two-loop renormalization-group analysis to study further the order of the phase transition at Te' Based only on the calculation thus far, we would conclude that chiral-symmetry breaking goes away for high temperatures, viz., for T> Te. And yet the same calculation shows a Lorentz-invariant massive particle pole in the thermal fermion Green's function, which is a signal that chiral-symmetry breaking has occurred. To reconcile between the presence of a massive particle and the vanishing of the traditional signature of chiral-symmetry breaking, we proceed with the analysis of the thermal fermion Green's function. The coefficients A ,B, C in Eq. (38) are nonlinear functions of the momenta. At the particle pole Po = v'P 2+,M. 2 the residue of the fermion thermal Green's function is not like the usual

(-i'Y·p+i'YoI'o+.M.) ,

(51)

but instead has the form [-i'Y'p(1 +.A)+i'YnPn(I+'7l)+m,(l +6')] ,

(52)

where .A,.:B, 6' are non polynomial functions of p2= Ipl2, obtai.,d by evaluating A ,B, C on the particle mass shell, Po= p2+.JIt 2. The Dirac equation for the massive particle is then the peculiar one:

495 LAY-NAM CHANG, NGEE-PONG CHANG, AND KUANG-CHAO CHOU

600

[iy·p( I +.A )-iyo"Vp2+.M. 2(1 + 13Hm,( I +@))u=O . (53)

Ifwe naively form the renormalized field operator by

ifi =_I_l:,(u A R

Vv

p.s

p.s

eip.x-nlp2+JI!2r p•.'

+vP•. Btp •. e -iP'x+nlp2+JI!2r) f

f

,

jifiR =0.

We now study the chiral flip part of the renormalized inverse thermal Green's function, S iii, and take the critical limit as mr -->0. If the chiral flip part survives this limit, we then can properly claim that chiral symmetry persists at high temperatures. Perturbatively, the chiral flip part of S fiRI is given by

(54)

we would find that this renormalzed field does not satisfy the usual Dirac equation, but instead obeys the nonlocal equation [(1+.A)y'V+(I+13)YO;t +m,(I+@)

(65)

-I _ { S(3R Ichiralftip-mr I+ArCf

[[

m; 4]

-3 In7-"3

+

2rrT m,2

2 -

3In

Em;

(55)

1

+const + ... }, (66) Based on our experience in the preceding section, it is clear that it would be dangerous to conclude anything about chiral-symmetry breaking based on this ifi R • Thermal radiative corrections have induced some chiral-2 rotation in the vacuum structure and we must find the generalized Cini-Touschek transformation that can remove the associated non locality. If we introduce (56) then the Dirac equation for U is "straightened out" to the usual massive one (iy·p-iY oV p '+.M. 2 +.M.lU=0, p[m,( I + Cl')-.M.(I +.A))

(58)

p'( I +.A )+m,.M.( 1 +@)

The renormalization of the thermal field-theory propagator should thus involve an extra Cini-Touschek transformation on top of the wave-function renormalization factor (59)

where 9 is the differential operator in three-dimensional space as implied by Eq. (58). Accordingly, we have

-z 2(3e iepY'~S(3Re i9pY'~ S (3-

(60)

and as a result Sii=iY'p(l+A')-iYoPo(I+B'Hm,(1+C') ,

I + A '=(1 +':8)-1

[(I + A )cos29 +

I+B'=(I+13)-I(I+B),

p

(67)

(57)

provided

where

so that to first order in renormalization-group analysis, the chiral ftip part of the renormalized inverse Green's function is simply the Lorentz-invariant physical mass of the particle pole,.M.. As shown in our earlier work, this mass survives the critical limit mr-->O. Chiral-symmetry breaking persists at high temperatures. Our conclusion therefore is that at high temperatures the traditional signature of chiral-symmetry breaking reappears only when one uses the transformed '" R field, and we calculate its vacuum expectation value

:r I sin29p

(61)

(62) (63)

I +C'=(1 +13)-I[m r (l +C)cos29p -p( I + A )sin29pl , (64)

so that, at the massive particle pole, the coefficient of 1'0 is properly normalized. Here

Unfortunately, our results at this stage cannot be used to evaluate this thermal vacuum expectation value. We need a study of the thermal propagator in the full complex Po plane. IV. CONCLUSION

In conclusion, we note once again that the traditional signature for chiral-symmetry breaking (vac I~"'I vac ) (3 is an inadequate indicator of chiral-symmetry breaking at high temperatures. Our calculations with dynamical symmetry breaking in QCD at high temperatures show a persistent Lorentz-invariant massive particle pole in the thermal fermion propagator at high temperatures-and this in spite of a vanishing (vacl~"'lvac)(3 at high temperatures. Thermal radiative corrections induce a further chiral-2 rotation of the vacuum structure, and the resulting renormalized Dirac field undergoes a generalized CiniTouschek similarity transformation. In the transformed basis, the traditional signature for chiral-symmetry breaking reappears. ACKNOWLEDGMENTS This work was written up while one of us (N.P.C.) was visiting KEK, Japan, and he wishes to thank Dr. K. Higashijima for some stimulating conversations and the Theory Group for the warm hospitality. This work was supported in part by the NSF U.S.-China Cooperative Program, by grants from the NSF, the Department of Energy, and from PSC-BHE of the City University of New York.

496 SIGNATURE FOR CHIRAL-SVMMETRV BREAKING AT HIGH ... IV. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961). 2M. Cini and B. Touschek, Nuovo Cimento 7, 422 (\958). The Cini~ Touschek transformation is a special case of a genera] class of Foldy-Wouthuysen transformations; see L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950). 3L. N. Chang, N. P. Chang, and K. C. Chou, in Third AsiaPacific Physics Conference, proceedings, Hong Kong, 1988, edited by Y. W. Chan, A. F. Leung, C. N. Yang, and K. Voung (World Scientific, Singapore, 1988). 4For a comprehensive review of the subject, the formalism, as well as citation of earlier work, see Zhou Guang-zhao (K. C. Chou), Su Zhao-bin, Hao Bai-lin, and Vu Lu, Phys. Rep. 118, 1 (1985). For other equivalent approaches, see H. Umezawa, H. Matsumoto, and M. Tachiki, Thermofield Dynamics and Condensed States (North-Holland, Amsterdam, 1982); R. L. Kobes, G. W. Semenoff, and N. Weiss, Z. Phys. C 29, 371 (1985); A. Niemi and G. W. Semenoff, Nucl. Phys. B230, 181 (19841; Ann. Phys. (N.Y.) 152, !O5 (1984).

601

SL. N. Chang and N. P. Chang, Phys. Rev. LeU. 54, 2407 (1985); Phys. Rev. D 29, 312 (1984); see also N. P. Chang and D. X. Li, ibid. 30,790 (1984). 6H. A. Weldon, Phys. Rev. D 26, 2789 (19821. 7J. F. Donoghue and B. R. Holstein, Phys. Rev. D 28, 340 (1983); 29, 3004(El (\984); J. F. Donoghue, B. R. Holstein, and R. W. Robinett, Ann. Phys. (N.V.! 164,233 (19851. See also R. Pisarski, Nucl. Phys. A498, 423C (\ 989). G. Barton, Ann. Phys. (N.V.) 200, 271 (1990), has a very nice discussion of the physical origin of the Lorentz invariance of the massive particle pole. 81t is interesting to note that lattice gauge simulations have found a chiral transition temperature of T, / A.,. of around 2 (for a nucleon mass of 940 MeV) (see Ref. 9), where Ms denotes the modified minimal subtraction scheme. 9For a review, see A. Ukawa, in Lattice '89, proceedings of the International Symposium, Capri, Italy, 1989, edited by R. Petronzio el al. [Nucl. Phys. B (Proc. Suppl.J (in pressl].

497 PHYSICAL REVIEW D

I APRIL 1996

VOLUME 53, NUMBER 7

CP violation, fermion masses and mixings in a predictive 8U8Y 80(10) x .:1(48) x U(I) model with small tanfJ K. C. Chou Chinese Academy of Sciences. Beijing 100864. China

y. L. Wu Department of Physics. Ohio ~iate Unit'"rsity. Colllml>lIs. Ohio 43210 (Received 8 November 1995) Fermion masses and mixing angles are studied in an SUSY SOt 10) x a( 48) X U( I) model with small tan,8. Thirteen parameters involving masses and mixing angles in the quark and charged lepton sector are successfully predicted by a single Yukawa coupling and three ratios of VEV's caused by necessary symmetry breaking. Ten relations among the low energy parameters have been found with four of them free from renormalization modifications. They could be tested directly by low energy experiments. PACS number(s): 12.15.Ff, 11.30.Er, 12.10.Dm, 12.60.Jv

The standard model (SM) is a great success. Eighteen phenomenological paramcters in the SM, which are introduced to describe all the low energy data, have been extracted from various experiments although they are not yet equally well known. Some of them have an accuracy of better than 1%, but some others less than 10%. To improve the accuracy for these parameters and understand them is a hig challenge for particle physics. The mass spectrum and the mixing angles observed remind us that we are in a stage similar to that of atomic spectroscopy before Balmer. Much effort has been made along this direction. The well-known examples are the Fritzsch ansatz [I] and Georgi-Jarlskog texture [2]. A general analysis and review of the previous studies on the texture structure was given by Raby in [3]. Recently, Babu and Barr [4], and Mohapatra [5], and Shafi [6], Hall and Raby [7], Berezhiani [8], Kaplan and Schmaltz [9J, Kusenko and Shrock [I OJ constructed some interesting models with texture zeros based on supersymmetric (SUSY) SO(lO). Anderson, Dimopoulos, Hall, Raby, and Starkman [II J presented a general operator analysis for the quark and charged lepton Yukawa coupling matrices with two zero textures "1\" and "13:' The 13 observablcs in the quark and charged lepton sector were found to be successfully fitted by only six parameters with large tan.8. Along this direction we have shown [12] that the same 13 parameters can be successfully described, in an SUSY SOt 10) X ~(48) X U(I) model with large values oftanf3-m,/m", by only five parameters with three of them determined by the symmetry-breaking scales of U( I), sot I 0), SU(5), and SU(2) L' Ten parameters in the neutrino sector could also be predicted, though not unique, with one additional parameter. In this Rapid Communication wc shall present. based on thc symmctry group SUSY SOt 10) X ~(48) X U( I), an alternative model with small values of tanf3-1 which is of phcnomenological interest in testing the Higgs scctor in the minimum supersymmetric standard model (MSSM) at colliders [13]. The dihedral group ~(48), a subgroup ofSU(3), is taken as the family group. U(I) is family-independent and is introduced to distinguish \'urious fields which belonl! to the same reprcsentations of SOt 10) X M 48). The irr;ducible

representations of ~(48) consisting of five triplets and three singlets are found to be sutlicient to build an interesting texture structure for ferm ion mass matrices. The symmetry ~(48)X U(I) naturally ensures the texture structure with zeros for Yukawa coupling matrices, while the coupling coetlieients of the resulting interaction terms in the superpotential are unconstrained by this symmetry. To reduce the possible free parameters, the universality of coupling constants in the superpotential is assumed; Le., all the coupling coetlicients are assumed to be equal and have the same origins from perhaps a more fundamental theory. We know in general that universality of charges occurs only in gauge interactions due to charge conservation, like the electric charge of different partieles. In the absence of strong interactions, family symmetry could keep the universality of weak interactions in a good approximation after breaking. In our case there are so many heavy fermions above the grand unified theory (GUT) scale and their interactions are taken to be universal in the GUT scale where family symmetries have been broken. It can only be an ansatz at the present moment where we do not know the answer governing the behavior of nature above the GUT scale. As the numerical predictions on the low energy parameters so found are very encouraging and interesting, we believe that there must be a deeper rcason that has to be found in the future. Choosing the structure of the physical vacuum carefully, the Yukawa coupling matrices which determine the masses and mixings of all quarks and Icptons are given by

0556-2821/96/53(7)/3492(4)1$10.00

R3492

o

o (I)

o and '\.'.' 19% The American Physical Society

498 R3493

CP VIOLATION. FERMION MASSES AND MIXINGS IN A ...

- tZiE~

w,,= ~A ){E2 16,(' ~)' "(Au)' ('~) 10 I (~') (A.,) -- (3 N - G - A. u. A. A· v-

3Jjf:~e;d>

..\

".\ I

.\ .

,

forj=d.e, and

_ )20 X

10 , (VIO)(A:)' - ('V - IO)"+I] 16,. (VIO) Ax Ax Us As -

X -

~2

,

7 :! _pcp

(

o

2

- '2 X ,.EC

o

)

(3)

W,.

for Dirac-type neutrino coupling, where the integer II reflects the possible choice of heavy fermion fields above the GUT scale. n = 4 is found to be the best choice in this set of models for a consistent prediction on top and charm quark masses. This is because, for n >4, the resulting value of tanj3 becomes too small, as a consequence, the predicted top quark mass wi II be below the present experimental lower limit. For n =vs diag (

~, ~, ~, ~,

1-) ®

T2 •

~,

;,

+,

-2,

--2) 0T2 and.

The resulting Oebsch facton1 arc: w. =Wd =

w,=w,= 1; x.= -7/9, Xd= -5/27, x.= 1, X.= -1/15; y.=u, Yd=-'y./3=2/27, y.=4/45; z.= 1, -27, z.= -153 = -3375. qJ i!: the physical CI? phase!) arising from the VEVs. The Clebsch factors associated with the ~YTlnnctry breaking directions can be easily read off from etTective operators which err;: obtained when the beavy fermion pairs are integrated out and decoupled z~=z,=

Wn=O AH) ; F.~ 162----r====7==r-6 (~:)(~:) 1O!(~!:)(~; )---;=1+=F.;==:(FA=x-=;:)~6 16

2,

(3)

VIO

where the factor

I!

I +1:~(AX)6 arises from mixing. The

e~

term in the square root is

VIO

negligible for quarks and charged leptons, but it becomes dominant for the neutrinos due to the large Clebsch factor z.. In obtaining the rfG matrices, some small terms arising from 1) We bave rotated away other possible phases by a phase redefinition of the fennion fields.

504 68

SCIENCE IN CHINA (Series A)

Vol. 39

mixings betwc~n the chiral fermion 16i and heavy fermion pairs l/Ij(l/Ij) are neglected. They are expected to change the numerical results no more than a few percent. The factor 1/..[3 associated with the third family is due to the maximum mixing between the third family fermion and heavy fermions. This set of effective operators which lead to the aforementioned YUkaWd coupling matrices rJ is quite unique. Uniqueness of the structure of operator WI2 was first observed by Anderson et al.[4J from the mass ratios of m./m~ and md/m•. The effective operator W33 is also fixed at the GUT scale[7.8.3) in the case of large tanfJ· There is only one candidate for effective operator W22 when the direction of breaking is chosen to be A,. , with Oebsch factors satisfying Y.:Yd:Y. = 0: 1:319] so as to obtain a correct mass ratio m~/m •. The three parameters A.Jl , SG and Sp are determined by the three measUred mass ratios mb/mt' mp/m t and m./mt' Thus, the mass ratio mc/m, and the CKM mixing elements Vcb and V. b put strong constraint to a unique choice of the symmetry bl'eaking direction A. for effective operator W 32 • Unlike many other models in which W 33 is assumed to be a renormalizable interaction before synu'netry breaking, the Yukawa couplings of all the quarks and leptons (both heavy and light) in tbe prese:nt nlQt;ld are generated in the GUT scale after the breakdown of the family gr')up and SO(lO). Therefore, the initial conditions of renormalization group (RG) evo'lIltion will be set in the GUT scale for all the quark and lepton Yukawa .:ouphngs. ConseqLlently, one could avoid the possible Landau pole and flavor ch... mghJg problems encountered in many other models due to RG running of the third famil) Yukawa couplings from the GUT scale to the Planck scale. The hierarchy among the three families is described by the two ratios sG and Sp. Mass splittings between quarks and leptons as well as between the up and down quarks are determined by the Oebsch factors of SO (10). From the GUT scale down to low energies, RG evolution has been taken into account. Top-bottom splitting in the present model is mainly attributed to the hierarchy of the VEVs VI and V2 of the two light Higgs doublets in the weak scale. I) An adjoint 45 Ax and a 16-D representation Higgs field (= Vs diag (0, 0, 0,

AJR

with (A B - L>=vsdiag (

~, ~) ®t"2'

~, ~, ~,

0,

°)®t"2

The 45 A B - L is also necessary for doublet-

triplet mass splitting[l5) in the Higgs lOl' The light neutrino mass matrix is given via see-saw mechanism as follow.;:

1.

ZN

_1_

4

Iz,1

YN

li~ -itp e -3 -ZN 2 -y,- 2

2

Z,

YN

lip

_.:ll. 4

.,

Z.'/

~

Il..

Iz,1

YN

lip

G

0. 73AHe -iO.H6./2

-0.86).4

(11)

1

- -3 2 -E~ Wit. h M04 li p

I - -I -V2 l': • ed to -1-1 V 2 All' Here '1. is the RG evolution Jactor eshmat Z, Z N

'1,

be

VIO

~,~1.35. Diagonalizing the above mass matrix,

we obtain the masses of light Majorana

508 72

SCIENCE IN CHINA (Series .A)

Vol. 39

neutrinos:

m., = 1.- ~ _1_ =0.83 x 10-4, m., 4 Iz.1 YN m. WN - ' =1-3-m., Iz.lzN

X. e~ -J3Iz.1 e

~0.998,

(12)

p

m.. ~Mo~2.U.HeV.

The three heavy Majorana neutrinos have masses

(13)

The CKM-type lepton mixing matrix is predicted to be

v:LEP =vvt= •• (

V•• V•..

V.,P V.,p

CP-violating eiTt'Cts

ht'e

('

Ji~. '

'-0.(>51 \ 0.045

V.~. )~:

V",

O.~'S·76

V•.• \

0.068 0.000 ) 0.748 0.665 . -0.664 0.748

(14)

found to be small in the lepton mixing matrix. As a result we find

(i) a \'~(;p) -il.(V.) short wave-length oscillation with (15)

which is consistent with the LSND experimentl61 (16)

(ii) a vp (vp )

v.(v.) long-wave length oscillation with

-

Am~=m~, -m~. ~(1.6-2.4)

x 1Q-2eV 2,

(17)

which could explain the atmospheric neutrino deficitll7l : &n!.=m~,-m~,~(0.5-2.4) x 1Q-2eV 2,

(18)

with the best fit ll7l (19) However, (vp-v,) oscillation will be beyond the reach of CHORUS/NOMAD and E803. (iii) Two massive neutrinos vp and v, with

.

m.

~m., ~(2.0-2.4)

eV,

which fall in the range required by possible hot dark matte:rl I8J •

(20)

509 No. I

73

LOW ENERGY PHENOMENA

In this case, solar neutrino deficit has to be explained by oscillation between v. and a sterile neutrino[l\lI V,. Since strong bounds on the number of neutrino species both from the invisible ZO-width and from primordial nucleosynthesis[aJ,2JI require the additional neutrino to be sterile (singlet of SU(2) x U(l), or singlet of SO(IO) in the GUT SO (I 0) model). Masses and mixings of the triplet sterile neutrinos can be chosen by introducing an additional singlet scalar with VEV vs~450 GeV. We find

with the mixing angle consistent with the requirement nucleosynthesis[22] given by ref. [20]. The resulting parameters

necessary for

primordial

(22)

are consistent with the values[l9J obtained by fitting the experimental d8.ta: (23)

This scenario can be testl!d by the next generation solar .neutrino experiments in Sudhuray Neutrino Observatc..ry (SNO) and Super-Kamiokanda (Super-K), both planning to start operation in 1996. By measuring neutral current events, one could identify v. - v, or v. - vI' (v since the sterile neutrinos have no weak gauge interactions. By measuring seasonal variation, one can further distinguish the small-angle MSW22] oscillation from vacuum mixing oscillation. t )

3 Superpotential for· fermion Yukawa interactions

Non-Abelian discrete family symmetry ..1 (48) is important in the present model for constructing interesting texture structures of the Yukawa coupling matrices. It originates from the basic considerations that all three families are treated on the same footing at the GUT scale, namely the three families should belong to an irreducible triplet representation of a family symmetry group. Based on the well-known fact that the masses of the three families have a hierarchic structure, the family symmetry group must be a group with at least rank three if the group is a continuous one. However, within the known simple continuous groups, it is difficult to find a rank three group which has irreducible triplet representations. This limitation of the continuous groups is thus avoided by their tinite and discoimected subgroups. A simple example is the finite and disconnected group A (48), a subgroup of SU(3). The generators of the A(3n2 ) group consist of the matrices

510 74

Vol. 39

scmNCE IN ClllNA (Series A)

E(O,O) = (

and

A.(P,q) =

~

c;' ~

o1 o

0 )

(24)

1

0

0

0 . 2
= <X' (2» = (X/ (i) =Vs with (i= I, 2, 3; a =0, I, 2, 3; j= 1, 2, 3), <X (1) > = <X (~) = (x'(I»=<x'(3»=0, =vs~450GeV, (H 2 )=v2 =vsinp with v=Jif.+v~=246GeV.

4 Conclusions It is amazing that nature has allowed us to make predictions in terms of a single Yukawa coupling constant and a set of VEVs determined by the structure of the vacuum and to understand the low energy physics from the Planck scale physics. The present model has provided a consistent picture for the 28 parameters in 8M model with massive neutrinos. The neutrino sector is of special interest for further study. Though the recent LSND experiment, atmospheric neutrino deficit, and hot dark matter could be simultaneously explained in the present model, solar neutrino puzzle can only be understood by introducing an SO (10) singlet sterile neutrino. It is expected that more precise measurements from various low energy experiments in the near future could provide crucial tests .on the present model. Aclmowledgement

Wu Yueliang would like to thank Institute for Theoretical Physics, Chinese Academy

of Sciences. for its hospitality and partial support during his visit.

513 No.1

LOW ENERGY PHENOMENA

77

References 1 Gatto, R .. Sartori, G., Tonin, M., Weak self-masses, Cahbibo angle and Breoken SU(2), Phys., Lett., 1968, B28:128. 2 Weinberg, S .. Rabi, I. 1.. Discrete flavor symmetries and a formula for the Cabbibo angle, Phys. Lett., 1977, 840: 418. Raby, S., Ohio State Univ., Preprint, 0IISTPY-HRI'-·1~9>24. 4 Dimopoulos, S. Anderson, G., Raby. S:, et al .. Predictive ansatz for fermion mass matrires in sypersymmetric grand unified theories, Phys. R.~:., 1992, 045:4192. 5 Hall, L.J., Raby, S., On the generality of rertain predictions for quark mixing, I'hys. Lerr., 1993, D135: 164. 6 Fairbairn, W.M., Fulto, T., Klink, W.H., Finite and diswnnected subgroups of SU(3) and their application to the elementary-particle spectrum, J. Math. Phys., 1964, 5: 1038. 7 Kaplan, D., Schmaltz, M., Flavor unification and discrete non-Abelian symmetries, Phys. Rev., 1994, 049:3741. 8 Ananthanarayan, B., La2llrides, G., Shafi, Q., Top-quark-mass prediction from supersymmetric grand unified theories, Phys. Rev. Lett., 1991, 044: 1613. 9 Hall, L., Rattazzi, R" Sarid, U., The top quark mass in supersymmetric SO(IO) unification, Phys. Rev., 1994, D~{):

7048. 10 Geogi, H., Jarlskog, C., A new lepton-quark mass relation in a unified the('ry, Phvs . .rAt., 19 79, Br-6.297. 11

Partide Data Group, Evidence for top quark production in bar {p} P wllisi.')tlf. al s"rr {s} = 1.8 TeV, Phvs. Rev., 1994, DSO:2966.

12 Abachi, S., ObserV'dtion of the top quark, Phys. Rev. Lett., 1kICLEAR FHYSICS B

PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 52A (1997) 159-162

I-L~EVIr.R

A solution to the puzzles of CP violation, neutrino oscillation, fermion masses and mixings in an SUSY GUT model with small tan,B K.C. Chou

OT1

and Yue-Liang

WU OT1 •

aChinese Academy of Sciences, Beijing 100864, China bDepartment of Physics, Ohio State University, 174 W. 18th Ave., Columbus, OH 43210, USA CP violation, fermion masses and mixing angles including that of neutrinos are studied in an SUSY SO(lO)x.o.(48)x U(l) model with small tanfJ. It is amazing that the model can provide a successful prediction on twenty three observables by only using four parameters. The renormalization group (RG) effects containing those above the GUT scale are considered. Fifteen relations among the low energy parameters are found with nine of them free from RG modifications. They could be tested directly by low energy experiments.

The standard model (SM) is a great success. But it cannot be a fundamental theory. Eighteen phenomenological parameters have been introduced to describe the real world, all of unkown origin. The mass spectrum and the mixing angles observed remind us that we are in a stage similar to that of atomic spectroscopy before Balmer. In this talk, we shall present an interesting model based on the symmetry group SUSY SO(lO) x A(48) x U(l) with small values of tan,B - 1 which is of phenomenological interest in testing the Higgs sector in the minimum supersymmetric standard model (MSSM) at Colliders[l]. For a detailed analysis see ref. [2]. The dihedral group A(48), a subgroup of SU(3), is taken as the family group. U(l) is family-independent and is introduced to distinguish various fields which belong to the same representations of S0(10) x A(48). The irreducible representations of A(4B) consisting of five triplets and three singlets are found to be sufficient to build an interesting texture structure for fermion mass matrices. The symmetry A(48) x U(l) naturally ensures the texture structure with zeros for Yulta.wa coupling matrices, while the coupling coefficients of the resulting interaction terms in the superpotential are unconstrainted by this symme• Supported in part by the US Department of Energy Grant # DOE/ER/0154S-675. Pernu....ent ..ddreso: Institute of Theoretical Phy.ics, Chinese Academy of Sciences, Beijing 100080, China 0920·S632(97)iSI i.OO·n 1997 EIscvi~r Sci~nc" ltV. All righlS reserwd. I'll: SO')20·S632(96)01l553-1

try. To reduce the possible free parameters, the universality of Yukawa coupling constants in the superpotential is assumed, i.e., all the coupling coefficients are assumed to be equal and have the same origins from perhaps a more fundamental theory. Choosing the structure of the physical vacuum carefully, the Yulta.wa coupling matrices which determine the masses and mixings of all quarks and leptons are given at the GUT scale by

r~

= Au

0

~Z~€~

~Zu€~

-3y,,€~ei';

( (

0

0 -~ZIE~ 0

a -

¥-ZJl€~

o

-~:c.. €~ - ~Zl €~

3YJ€~ei'; ~:CJ€~

-

-,{l•.,~ ) w,.

0 -~;Z:f€~ wf

)

¥-Z~€$

15YJlf~ei';

-~;Z:J1f~

with A.. = 2AH /3, AI = A,,( _1)n+1 /3 n (J = d, e) and Av :::: >'J /5 n +1. Hen the integer n reflects the possible choice of heavy fermion fields above the GUT scale. n 4 is found to be the best choice in this set of models for a consistent prediction on top and charm quark masses. This is because for n > 4, the resulting value of tan,B becomes too small, as a consequence, the predicted top quark

=

515 160

KC. Chou. Y.-L. WulNuciear Physics B (Proc. Suppl.j 52A (1997) 159-162

mass will be below the present experimental lower limit. For n < 4, the values of tanf3 will become larger, the resulting charm quark mass will A.~r3, be above the present upper bound. A.H EG

== (;-:;,)

j?; and

=

Ep

== (it;)

JV; are three pa-

rameters_ Where A.~ is a universal coupling constant expected to be of order one, rI, r2 and T3 denote the ratios of the coupling constants of the superpotential at the GUT scale for the textures '12', '22' ('32') and '33' respectively. They represent the possible renormalilllation group (RG) effects running from the scale Mp to the GUT scale. Note that the RG effects for the textures '22' and '32' are considered to be the same since they are generated from a similar superpotentiai structure after integrating out the heavy fermions and concern the fields which bdong to the same repre5~~tations of the symmetry group, this can be expliCItly seen from their effective operators ~22 and W 32 given below. Mp , VIC, and Vs bemg the vacuum expectation values (VEVs) for U~I) x .!l(4~), SO(10) and SU(5) symmetry br~~kmg respectively. rP is the physical CP phase ~I1smg from the VEVs. The assumption of maxImum CP violation implies that rP 11"/2. :c J, YJ, zJ, and wJ (! u, d, e, II) are the Clebsch factors of 50(10) determined by the directions of symmetry breaking of the adjoints 45's. The Clebsch factors associated with the symmetry breaking directions can be easily read off from the U (1) hypercharges of the adjoints 45's and the rdated effective operators which are obtained when the symmetry 50(10) x .!l(48) x U(I) is broken and heavy fermion pairs are integrated out:

=

=

arises from mixing, and provides a factor of 1/..;3 for the up-type quark. It remains almost unity for the down-type quark and charged lepton as well as neutrino due to the suppression of large Clebsch factors in the second term of the square root. The relative phase (or sign) between the two terms in the operator Wn has been fixed. The three directions of symmetry breaking have been chosen as < Ax >= 2vlo diag.(I, I, I, I, 1)®T2' < A. >= 2vs diag.(-~, -~, -~, -I, -1) ® Ta, < Au >= vs/v'3 diag.(2, 2, 2, 1, 1) ® Ta. The resulting Clebsch factors are w.. Wtl W. WI' = 1, :c u = 5/9, :Ctl = 7/27, :c. -1/3, :c" = 1/5 Yu = 0, Ytl = y./3 = 2/27, y" = 4/225, ~ 1, Ztl Z. -27, z" = -11)3 = -3375, zu = 1 - 5/9 4/9, z~ Ztl + 7/729 ~ Ztl, 3 z~ = Z. - 1/81 ::::: Z., z~ z" + 1/15 ~ Z". An adjoint 45 Ax and a I6-dimensional representation Higgs field ~ (~) are needed for breaking SO(IO) down to SU(5). Another two adjoint 45s A. and Au are needed to break SU(5) further down to the standard model SU(3). x SUL(2) x U(I)y. From the Yukawacoupling matrices given above , the 13 parameters in the SM can be determined by only four parameters: a universal coupling constant A.H and three parameters: EG, Ep and tanf3 V2/Vl. The neutrino masses and mixings cannot be uniquely determined as they rely on the choice of the heavy Majorana neutrino mass matrix. The following texture structure with zeros is found to be interesting for the present model

=

=

(1;) (1;) '1A16a (~;) (1;) 71AI62ei~

W 32

=

A. 2 16371X'7A

W 22

=

A.2 16 271A

W I2

=

A.d6 I [

C~·:

+ (~) Mp 2 71A

=

10 1

r

10 1

'1:" 10 1

(

'1~

Au ) ( A. ) Ax 10 1 Ax '7A]16 2

where A.i A.~ri' '7A =-= (vlO/A x ),,+I, '1~ = (vlo/AX),,-3. The factor 'Ix 1/.)1 + 271!

=

=

=

=

=

=

Mf:

= MR (

~ !zNE~ei6"

A.3 16371X'7A 101'1A'7X 163

W33

= =

= =

Y~ ~ZN1eil") 0

wN4

The corresponding effective operators are

Wta" w~ w~ with

MR = A.HV~OE~E~/ Mp, A.f = A.HV104/ Mp, A.i' = A.f E~E~ and A.f = >.i' E~. It is then not difficult to read off the Clebsch factors YN 9/25,

=

516 161

KC Chou. r·L. /YulNuciear Physics B (Proc. Suppl.) 52A (1997) 159-162 ZN = 4 and WN = 256/27. The CP phase 6" is assumed to be maximal 6" = '11"/2. In obtaining physical masses and mixings, renormalization group (RG) effects should be taken into account. The initial conditions of the RG evolution are set at the GUT scale since all the Yukawa couplings of the quarks and leptons are generated at the GUT scale. As most YUDwa couplings in the present model are much smaller than the top quark Yukawa coupling >.f '" 1, in a good approximation, we will only keep top quark Yukawa coupling terms in the RG equations and neglect all other YUDwa coupling terms. The RG evolution will be described by three kinds of scaling factors. 1/F (F = U, D, E, N) and RI arise from running the YUDwa parameters from the GUT scale down to the SUSY breaking scale Ms which is chosen to be close to the top quark mass, i.e., Ms ~ me ~ 170 GeV. They are de-

n:=l (

cr 121>;

=

fined by 1/F(Ms) = ::~::1 (F UJ DE N)withcr,l (~ 3 ~) cP (.1. I , '18' '3 I , - 15 I 3 I ~) 3 , cf 3, 0), cf == (25,3,0), bi (~, 1, -3), 1 and Rezp[- JlnMs r 1nM"(M!l)2dt] [1 + I 4,..

= m,

)

=

=

=

=

31£:'s) with I(Ms) ==

(>.f)2Klt 1/12 , where KI = J,.~:: "Mt)dt with Ms ~ me

= 170GeV. Other RG scaling factors are derived by running YUDwa couplings below Ms· m.;(m.;) = 7Ji m.;(Ms) for (i == c, b) and m.;(lGeV) = 1/i m.;(Ms) for (i u., d, s). The physical top quark mass is given by Me = me(me) (1 + ~a.~",.)). The scal-

=

ing factor RI or coupling >.f = A.- .jl:::p:;rr ~ is v.n.t Rt determined by the mass ratio of the bottom quark and T lepton. tan (3 is fixed by the T lepton mass • ",_.12 I n numenc . al pre d"lctlOns we via cos fIt:l == ~. 1 take a- (Mz) 127.9, s2(Mz) 0.2319, Mz == 91.187 GeV, a;:l(me) 58.59, a;l(me) 30.02 and al1(Mo) = a;l(MG) = ai1(Mo) ~ 24 with Me '" 2 x 10 16 GeV. For a,(Mz) 0.113, the RG scaling factors have values (""'.d ... 1/c, 1/b, 1/•• ,..T, 1/u, 1/D/1/E == 1/DIE, 1/E, 1/N) = (2.20,2.00, 1.49, 1.02, 3.33, 2.06, 1.58, 1.41). The corre' sponding predictions on fermion masses and mixings thus obtained are found to be remarkable. Our numerical predictions for a,(Mz) = 0.113

=

=

=

= =

are given in table 1 with four input parameters. Where BK and fBv7i in table 1 are two important hadronic parameters and extracted from KO - [(0 and BO - iJo mixing parameters eK and Zd. Re(e' /e) is the direct CP-violating parameter in kaon decays, where large uncertanties mainly arise from the hadronic matrix elements. a, (3 and.., are three angles of the unitarity triangle in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Jcp is the rephase-invariant CPviolating quantity. The light neutrino masses and mixings are obtained via see-saw mechanism M" r;(M~)-1(r;)tvV(2Rt61/~). The predicted values for IV..,I, 1V... I/IVc.I, IVcdl/IVc.I, md/ m ., IV"... I, IV".... I, IV" ... I as well as m.,.I m ",. and m.,,./m,,. are RG-independent. From the results in table 1, we observe the following: 1. a 1I,.(ii,.) .... IIT(iiT) long-wave ~. m~,. ~ length oscillation with ~m!T 1.5 x 1O-3 ey2 and sin2 29,... ~ 0.987 could explain the atmospheric neutrino deficit[3]; 2. Two massive neutrinos II,. and II.. with m.,,. ~ m... ~ 2.45 eV fall in the range required by possible hot dark matter[4]; 3 a short wave-length oscillation with ~m~,. = m!,. - m!. ~ 6 eV 2 an sin 2 28.,. ~ 1.0 x 10- 2 is consistent with the LSND experiment[5]. 4. (II,. - II.. ) oscillation will be beyond the reach of CHORUS/NOMAD and E803. However, (II. - II.. ) oscillation may become interesting as a short wave-length oscillation with ~m2 = ~ - ~ ~ 6 ey2 and Majorana neutrino sin 2 28. T ::: 1.0: 10- 2 ; allows neutrinoless double beta decay ({3{3o,,) [6]. The decay rate is found to be rtltl ~ 1.0 x 10- 61 GeV which is below to the present upper limit; 6. solar neutrino deficit has to be explained by oscillation between II. and a sterile neutrino II, [7](singlet of SU(2)x U(l), or singlet of SO(10) in the GUT SO(10) model). Masses and mixings of the triplet sterile neutrinos can be chosen by introducing an additional singlet scalar with VEV v, ~ 336 GeV. They are found to be mil, = >'HV: /VI0 ~ 2.8 x 1O-3 eV and sin 8.. ~ m"t.".Im.,. == V2EP/(2v,E~) ~ 3.8 X 10- 2 • The resulting parameters ~m~. == ~. - m!. ~ 6.2 x 10- 6 eV 2 and sin 2 28.1 ~ 5.8 X 10- 3 ; are consistent with the values [7] obtained from fitting the

=

=

5.

-

517 K.c.

162

ChUlI, Y.-L. IVu/Nllciear Physics B (I'roc. Supp/.) 52A (1997) 159-162

Table 1 Output observables and model parameters and their predicted values with a.(Mz ) == 0.113 and input parameters: m. == 0.511 eV, m,. == 105.66 MeV, m.,. = 1.777 GeV, and mb(fnb) = 4.25 GeV. Output Output Data[8] Output Output Ml [GeV] 182 180 ± 15 Jcp/IO 2.68 m.:(mc ) [GeV] 1.27 1.27 ± 0.05 a 86.28° 4.31 4.75 ± 1.65 /3 22.11° m,.(lGeV) [MeV] m.(lGeV) [MeV] 156.5 165±65 "y 71.61° m/l(lGeV) [MeV] 6.26 8.5 ± 3.0 m.., [eV] 2.4515 IV... I = ~ 0.22 0.221 ± 0.003 m".. reV] 2.4485 3 ~ 0.083 0.08 ± 0.03 m ... [eV]/101.27 v ••

\~:~

IVcbl = ~f tan/3

A~2

= V2/Vl

fG Ep

BK

fBv'B [MeV] Re(t' /£)/10- 3

0.209 0.0393 1.30 2.33 0.2987 0.0101 0.90 207 1.4 ± 1.0

IV..... I IV".... I IV..,. I IV.... I IV.... I T

0.82 ± 0.10 200 ± 70 1.5 ± 0.8

experimental data. It is amazing that nature has allowed us to make predictions on fermion masses and mixings in terms of a single Yukawa coupling constant and three ratios determined by the structure of the physical vacuum and understand the low energy physics from the GUT scale physics. It has also suggested that nature favors maximal spontaneous CP violation. It is expected that more precise measurements from CP violation, neutrino oscillation and various low energy experiments in the near future could provide a good test on the present model and guide us to a more fundamental model. ACKNOWLEDGEMENTS: YLW would like to thank professor R. Mohapatra for a kind invitation to him to present this work at the 4th SUSY96 conference held at University of Maryland, May 29- June 1, 1996.

REFERENCES 1.

m... [eV]/10- 3

0.24 ± 0.11 0.039 ± 0.005

G. Kane, in this proceedings; see also J. Ellis, talk given at 17th Intern. Symposium on Lepton-Photon Interactions, 10-15 August, 1995, Beijing, China.

MN. [GeV] MN. [GeV]/10 6 MNo [GeV)

2.

3. 4. 5. 6.

7.

8.

2.8 -0.049 0.000 -0.049 -0.707 0.038 ~ 333 1.63 333

K.C. Chou and Y.L. Wu, Phys. Rev. D53 (1996) R3492j hep-ph/9511327 and hepph/9603282, 1996. Y. Fukuda et al., Phys. Lett. 335B, 237 (1994). D.O. Caldwell, in this proceedings; J. Primack et al., Phys. Rev. Lett. 74 (1995) 2160. C. Athanassopoulos et al., Phys. Rev. Lett., (1996) nucl-ex/9504002 (1995). For a recent review see, R.N. Mohapatra, Maryland Univ. Report No. UMD-PP-95-147, hep-ph/9507234. D.O. Caldwell and R.N. Mohapatra, Phys. Rev. D 48, 1993) 3259; J. Peltoniemi, D. Tommasini, and J.W.F. Valle, Phys. Lett. 298B (1993) 383. CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 74, 2626 (1995); DO Collaboration, S. Abachi et aI., Phys. Rev. Lett. 74, 2632 (1995); J. Gasser and H. Leutwyler, Phys. Rep. 87, 17 (1982); H. Leutwyler, Nucl. Phys. B337, 108 (1990); Particle Data Group, Phys. Rev. D50, 1173, (1994).

518 Vol. 41 No.3

SCIENCE 1N CHINA (Series A)

March 1998

A possible unification model for all basic forces * WU Yueliang

c'lUli!5U

(Inslitllte of Theoretical Physics. Chinese Academy of Sciences. Beijing 100080. China)

and ZHOU Guangzhao (K. C. Chou, Jj!;]*t~) (Chinese Academy of Sciences. Beijing 100086, China) Received November 10. 1997

Abstract A unification n"Kldel for strong, electromagnetic, weak and gravitati.ll al forces i5 pmol'sed. The tangent space of ordinary coordinate 4-dimensional spacetime is a submanifold of a 14-dirnensiona. i"ternal spacetime spanned by four frame fields. The unification of the standard IT¥ldel with gravity is governed by gauge symmetry in the internal spacetime. KeyWlrds:

unification, internal spI1cetime, SO( I ,13) , gravity, frame fields.

One of the great theoretical endeavours in this century is to unify gravitational force characterized by the general relativity of Einstein[1 .2] with all other elementary particle forces (strong, electromagnetic and weak) described by Yang-Mills gauge theory[3]. One of the difficulties arises from the no-go theorem[4] which was proved based on a local relativistic quantum field theory in 4dimensional spacetime. Most of the attempts to unify all basic forces involve higher-dimensional spacetime, such as Kaluza-Klein Yang-Mills theories[5.6], supergravity theories[7.MJ and superstring theories[9 -12J. In the Kaluza- Klein Yang-Mills theories, in order to have a standard model gauge group as the isometry group of the manifold, the minimal number of total dimensions has to be II [13J. Even&>, the Kaluza- Klein approach is not rich enough to support the fermionic representations of the standard model due to the requirement of the Atiyah- Hirzebruch index theorem. The maxi mum supergravity has SO (8) symmetry, its action is usually al&> formulated as an N = I supergravity theory in II-dimensional spacetime. Unfortunately, the SO(8) symmetry is too small to include the standard model. Consistent superstring theories have al&> been built based on la-dimensional spacetime. In superstring theories, all the known particle interactions can be reproduced, but millions of vacua have been found. The outstanding problem is to find which one is the true vacuum of the theory. In this paper we will consider an alternative scheme. Firstly, we observe that quarks and leptons in the standard modcl[14·· 16 1 can be unified into a single 16-dimensional representation of complex chiral ~;pinors in SO(IO) 117.IM]. Each complex chiral spinor belongs to a single 4-dimensional rcprcsentatlon of SOC I ,3). In a unified theory, it is an attractive idea to treat these 64 real spinor components on the same footing, i. e. they have to be a single representation of a larger group. It is therefore natural to consider SOC 1,13) asour unified group and the gauge potential of SOC 1.13) as the fundamental interaction that unifies the four basic forces (strong. c1eetromag• "mject supported in part by the Outstanding Young Scientist Fund of China.

519 No.3

A POSSIBLE UNIFICATION MODEL FOR ALL BASIC FORCES

325

netic, weak and gravitational) of nature. Secondly, to avoid the restrictions given by no-go theorem and other problems mentioned above, we consider that the ordinary coordinate spacetime remains to be a 4-dimensional manifold 54 wit h metric gil' (x) . f.l, v = 0 ,I .2 ,3. At each point P: XiI,

there is a d-dimensional flat space M" with d > 4 and signature (I , - I, ... , - I). We as-

sume the tangent space T4 of S4 at point P to be a 4-dimensiomil submanifold of M" spanned by fourvectorse;(x)/I=O,I,2,3; A=O,I,"',d-1 such that gil. (x)

where

77AB

= ei; ( x) e; ( x) f)AB ,

(I)

= diag( I , - I , ... , ... I) can be considered as the metric of the flat ~ace Md. We shall

call ej; (x) the generalized vierbein fields or simply the frame fields. Once the frame fields ei; (x) are given, we can always supplement t hem wit h anot her cf-4 vector fields e;;, (x) -e'~, ( ei; (x)) , III = I ,2, ..., cf-4 such that

ei~ ( x) e~, ( x) 77AB = 0, e;~, ( x) e?, (x) 77AB = gil",' (2) where gil", = diag( . . 1 , ... , . . 1). e'!, (x) can be uniquely determined up to an SO( cf-4) rotation. In the flat manifold M" we can use ei; (x) and e;~, (x) to decompose it into two orthogonal manifolds T4 0Cd- 4, where C d _4 will be considered to be the internal space describing SO( cf-4) internal symmetry besides the spin and is spanned by the cf-4 orthonormal vectors e;;, ( x). In the new frame system of M" the metric tenoor is of the form [

J.

gil. ( x)

0

o

gil",

(3)

With ei~ (x) and e'!, (x) , we can now define the covariant vectors as e:'~ (x) and e.~' (x) satisfying ~ ( x) e; (x)

= g~'

,

e:'~(x)e'~,(x) =0,

e.~' ( x) e'! (x)

//I

gil ,

e:~'(x)e:~(x) =0.

(4)

Under general coordinate transformations and the rotations in M", e;: (x) transforms as a covariant vector in ordinary coordinate spacetime and a vector in the M" rotation, e'~, (x) transforms as a covariant vector in the C d - 4 rotation and a vector in the M" rotation. For a theory to be invariant under both general coordinate transformations and local rotations in the flat space Md, it is necessary to introduce affine connection

r;..( x)

for general coordinate transformations and gauge

potential Q,;B (x) = . . 0:,'" (x) for if-dimensional rotation SO (I , cf-I) in Md. These transformations are connected by the requirement that T4 has to be the submanifold of Md spanned by four vectors ei; (x) at point P and e:~ (x) should be a covariantly constant frame and satisfy the condition (5)

It is then easily verified that (6) (7)

With the above considerations, we can now construct an invariant action under general coordinate transformations in t he ordinary coordinate spacetimc and the local SO (I , if-I) group symmetry in M" with eq. (5) as a constraint. In addition, the action is required to have no dimensional parameters and to be renormalizable in the sense of the power counting. The general form of the action which satisfies these requircments is

520 326

Vol. 41

SCIENCE IN CHINA (Series A)

- 2 ~'I-' 1'"Y(/jeB + 2 ~

0,10, c/>:::! Cp -L 1.1..1 + 4 11 '1-'

+ r;F;:~ 1·~,~)g"I}'I.Ht!~e~ +

01

-L

+

-L

1:.12 .,' B , , ;

a~ f

T'8 ·[D "

Y "

.J'Y

IT

1ns and gluons. that mediate the electromagnetic. weak and strong interactions respectively. are different manifestutions of the guuge potential (.\"1 of the symmetry group SO( 10) [17 .IX]. The cllrvu-

A::

III

ture ten&>r R:,:Y,r and t he Ricci ten!llr R y,T

= R;;YITg~

as well as t he scalar curvat ure R = R ;"g"T are

simply related to the field strength I·~:tl viu R':'Y,r= gur~:tlr!.~euB. R yIT = gl'I'. '- p=. . 1916. 49: 769.

Yang. C. N .. Mills. R. L. . ('onscrvationllf i,.lIopic spin "nd i..:>tllpic g"uge im'[,ri"nce. Ph.,,,.

4 5

6

R,"· .. 1954.96: 191. Re,· .. 1967. 159: 1251. K:,luzll. Th .. 011 the prohkm "I' unity in physics. Sil=. P,,·us.,· . .Hut!. lI"i.,,, .. 1921. KI: 9Cl11. Klein. O .. Quantent h,'!.,ri,' ullli I' ii,rdimcnsionllie Rcillti\'itRtst h,'!.Hie. 7.. Phpik. 1926. 37: N95.

('olem"n. S .. M:lIldula . J .. Ph.n·,

7

Freedm"n. D.. Fcrrllw . S.. V:lIl Nieuwcnhuizcn . I' .. I''''grcss towllrd II t h''!.HY or supcrgr:,,·it)'. Ph,.s. 3214.

X

l>escr. S.. Zumino. B . . Consistcnt supcrgn,,·ity. Ph,'s.

9

(jrccn. M .. Schll'llrtz..I.II .. OHmillnt description of superstrings. Ph.n'.

10

1.1'11,.

R('\' .. 1976. DI3 :

1'176. 62B: 335. 1.1'11 .•

I<JH-I. 1-I9B: 117.

Canddtls. P.. Ikll"u\\'ilz. G. T .. Slromil1g~r. A. c( ,11. . Vacuum cunfigunttiolls I'(H" sLlp~r~trings . .Vnd.

Phy,\" , . IlJK5.

B25H: -16. II

Gross.D .. II "n'cy,.! . . Martincc. E. etal. .lIcteruticstring. Ph...".

I::!

(jr~cll.

Re,·. Lel/ .. IlJH5.5-1:502.

M.B .. Schwartz.J.n .. Wiuen. E .. SHIH.'",""';fI,!:! rl!f.·fII:\,, Cnmhridg.c: Cnmhl'1dgc Uni\'crsily Prcss. 1')X7.

13

Witten. E.. Scarch ((II''' realistic K:liuza-Kicin th''!.H),. 'vuc/.

14

Gi:lsho",. S. L. . Parlial-symmctries of weak intcnlctiuns. 'vue/.

Ph,.s .. IlJ~l. BIK6: -112. PIli'.",. 1, U

+ Iqsl2

wit.h norma.lization Ipsl2 r.oeIkipnls are given hy

qs

q 1 + LlAl

_

1-

p", = ), 1 - LlAJ = 1

AJ2 = PLIM u > -(ILIAfu >

fS

+ '.:;

!!. _ {Ji;; = 1 - tAl P - VIi;; - 1 + EM

=

qL l'L

IPLI 2 +

(4) IqLI 2 = 1. The

= 9..

1 - LlAJ == 1 - f/, 1'l+LlAJ I+FL

(6)

a:nd is given simply hy -t

,,-iIl 111-II ;

If:";; alld CPT-odd lI~j} paTls. i.e' .. 1+)

1l"fJ=Hr.r.r+

1/1-)

(9)

'If

with (10)

Let f denote the final state of the decay and J its charge ronjngat.e st.AI!', The cle'Cfl), amplitudes of Jl.lo Are defined as

9

== < flHeJJ!lIl o >=

~( ~ Ai

i6, + B) i e

T.. == < IIHofflM u >= L(Ci + D,)"i6,

== L(IC;deiq,~' + lJ)ileiq,~')ej6,

( 11)

with .II; 'tnd C; heinp; CPT-r.onservinp; Amplitudes

(7)

',1/-..:.IA/

= '1-EA/.6

CI'T-en~n

< fIH;;}llIlO >== LAiei6, , < JIH~;)IJIlO >==

1

We discuss next severa.l properties related to the symmetries of the sysl,em. The parameters 6Al And Iq/pl are rephasing inva.riant and so are also other parameters defined in trrms of t.hmu. CPT invarimwe reqnin's 11111 = AJ22 and r l1 = r22 , and implies that 6u = O. Thus the difference het.ween 1/8/r,S and (lI-h'l, represent.s a signal of CPT violation. In other words, .6Al different. from zero indica.tes CPT violat.ion. CP invariance requires the dispersive and absorpth'e parts of HI2 and H21 t.o be, respediveiy, "'1ual allli implies q/p = 1. Also if 'I' invariance holds, then independently of CPT synllnetry, the di~persive alld "h~orpth'e pArl.s of 1112 AUe! H21 mnsl, h(' f''1nnlnp 1,0 A 1.01.al relative common pha.se. implying Iq/pl = 1. Therel"orc a Ref,1/ dil~ fe'rcnl: from Z!'ro e!rsf;crihing CPo T (lnd CPT UOl1CCHlSPl"\'ntioll wilil,lt orig;innl,('s ill 1.lw IIlnSS Ilintrix (indirecl). III Ihe lIexl secl ion We' discllss addit jOlln] pnralllC"1 f'rs orig-inatillg; 111 1.111' dl'{"n~' a.mplit.udes (dircn) ns well a.s frolll I'he mixillg b('l.weell IIHlS~ lIIal.rix alld decay alliplihl== LBi ei6 , , < JIH~f)IJIlO >== LD;e iO •. , (13) Here we have used the notation of [lOJ for J;he amplitude g, and have introduced a new alliplitude h.. The secolld amplit.ude is absent. when one considers only h.-meson de(:IIYS a.nd neglect.s possible viola.tion of d8 = LlQ rule a.s was the CAse in [10J. This is heranse l,h!'. K-m,,~on dl'cAYs obey .68 = .6Q rule via. wea.k interactIons 01 the stan(hu'rl model. The' reason is simple since the' st.rnngr ql1ark can only decay t.o the lip qllark. In the case ~f B.-, B:,and D-nH'son sysl,rms both amplitl1(irs .1/ And II. eXIst. vIa I.he lI'-bosoll e~ehange of weak interactions since both bquark and (~'lnark will haw two dilt'crent trall~it.iol~~ duc \.0 CKI\! '1n;1.rk \Ilixings, i.e., b -t c, 'II. and c -t .s. d (lor explicit d,'en)" nlo~ c:i.plllJ u >, thcn IlIJ u >~ r,-'"'IMu >, HI2 ~ e- 2i "'H I2 and H21 ~ e2i .pH21 , as well as h ~ c,-i.ph aud C) ~ c" 2

11IJO(t) >

=

L(ie-i(m,-ir,/2J'llIIi >

(26)

i=l

wit.h ~I = IJ,)(IJSI'I.+IJI.PS) ami (2 = '18/('181'/.+'/1.1'8) for a pur:,e 111 0 state at t = 0 a.s well as (I = ]JL/(qS]JL + q[,]Js) and {~ = -Ps/('1S1'L +'1L1's) for a pure lI~fo state at t = O. Thus t.he delurelllents with scella.rio (i), ill whieh t.he illtlirE'C"i. CP nnd CPT noninvarin.nl. ohsl'rvnhll's a" all a.nd a~ are expected to be determined, one will be able 1·0 extract t.hE' mixed-indllced CP And CPT noninva.riAnt. observahle 0,+rr-fI"l/,) -I- l'{K (t) .. ) 1[+i,·,7!l

530 ICC. Chou et. al.: Searching for rrphil.se-invariallt CP- and CPT-violating obseryables in meSOll d.-cnys

~ - 1/..,:,

'I",

+

-uLlsillhLlJ'1. -I- U~SillLl·'"KI.

coshLlrt + co.sdmJ(t

(:n)

"

( )._ -~~"""''--'----'''':'':'--'--'--'------'r(l?0(t)--'>7I'-I+vl) - J'(K()(t)-'>7I'+/-VI)

ACP+(.'PT /.....

=::

reg ad

(1)--'>71'-/+'1,)

+ 1'(1\'o(/)--,>;r+/-,-;,)

+ 2".(

2 amplit.ndps. The ",lml' deeomposit.ion holds for B( + -) and B(OO) amplihldes 2 . Considering t.he fad t.ha.t w = 1.121/lil n l =:: 1/22 « 1 (hw t.o t.h!' LJ.! = 1/2 nIle, w(' obtain

iI~." )

(38)

whC'r!' t.IlC' din'(t CP-YioIRt.ing pmmnl'1'/'r II," iR (,X],)(,/'.1 I'el t.o be small as t.he final st.ate int.emetions are ele!'.t.romagnetic. It is then clt-ar t.hat lion-zero a~:q1ml!ctl7l A~~P+CPT(t) ·i•• a clean signature of CPT violation. Its t.ime evolution allows us to extract uirect CPT-violatillg ouservahle a c .:, and indirect CPT-violating observables aLl and a~. The

285

=:: (I,'

+ lI.~Ll + OEL! + (J~Ll'

(i~?O) =:: -2a" - :la.:.Ll - :lil,Ll

+ (f~.:,.

(4~)

and -l+-)

a"E+f' ~

0 0'(+(.1

0 + ae+f~...l + (J.e+ 71'+71'- lUal I\O(F:O) --'> 71'071'0 resped.i\'(~ly. whert' II{) and (/.~ (·olTeS])o'H.\ t.o I hI' isospin! 0 ",,,I ! ~

whkh indic-nl.('R t.hnt. h.\· lIl('n.'1u·illg it.~!;') n.nd ri~~~), ontll, V.L. Tekgdi, Phys. Rev. 105, [(j1:$1 (1!J57) 5. R. Christenson, J. Cronin, V.L. Fitch, R. Turlay, Phys. Rev. Lett.. 13, 138 (1964) 6. ,J. Schwinger, PhYR. Rev. 82, 914 (1%1); G. Lueders. Dall~k. Mat. Fy~. Meuu. 28, 17 (H)54); W. Pauli, Niel~ Bohr and the Development of Physics (Pergamon, New York, 1955) 7. For re"ent analyseR soo for example: GO. Dib, R.D. Pe,,cei, Phys. Rev. 0 46, 2265 (1992); C.D. Buchanan, et aI., Phys. Rev. 0 45, 4088 (1992) 8. ,J.S. Hell •. J. Sl.einher!1;~r, Proc. Oxford Inl.. Conf. on ele1I1clItary particles, 1965, p. 195 9. ,l.W. Cronin, Rev. Mod. Phys. 53, 373 (1981) 10. V.V. Barmin et ai., Nud. Phys. B 247, 293 (1984) 11. W.F. Palmer, Y.L. Wu, Phys. Lett.. I3 350, 2,1:' (Hl!lr.) 12. see for example, L. Lavoura, Ann. Phys. 207428 (1991), and references therein 13. '1'.0. Lee, C.S. Wn. Annn. Rev. Nlld. Sci. 16, 4il (1966) 14. E.A. Pa;;chos, R. Zacher, Z. Phys. C 28 521 (1!J!!5); For a review see for example, E.A. Paschos, U. Ti.irke, Phys. Rep. 178 147 (1989) Vi .. J. ElliR, .J.S. Hagelin, D.V. NanopouloR, M. Srednic-hi, :'-iuc!. Phys. B 241, 381 (1984); P. Huet, :\I. E. Peskin, :'-iucl. Phys. B 434, 3 (1995) J. Ellis, J.L. Lopez, N.E. i\lavromatos, D.V. Nanopoulos, Phys. Rev. 0 53, 3846 (IUU(j)

16. L, Wolf..nsteill, Phys.Rev.Lett. 83 911 (1999); L. Lavoura, hep-ph/9911209: A. I. Sanda, hep-ph/9902353: L. Lavonra, .J. 1'. Silva, hep-ph/!l!l0~:l4R: 1'. Hue!., hepph/96074:.15 ; J. Ellis, N.E. I\lavrolllatos, D.V. NauopolIlos, hep-ph/9607434; R. Adler, et al (CPLEAR Collaboration), J. Ellis, J. Lopez, :\:. Mavromatos, D. Nanopoulos. hl'I>-"x/!l5 11001; N. J\.IaVl'OlImtos, T. Rul' (for the eollaboration: J. Ellis, J. Lopez, N. J\.Iavromatos, D. Nanopoulos, and t.he CPLEAR Collabom.t.ion), hep-ph/9506395 17. See for example. L. :\Iaiani, ill 'The Second llA"

.,p" I', 1, m,

~,),

i

rO.)]

r." = 0,

=

1,2, ... ,2',

be these vertex functions, they obey

[ ,.,. aa,.,. +

pO.)

~ al + r ...O.)m ~am

We separate the physical variable

'f

s = 1, 2, ... ,2'.

(4.8)

t; denoting the initial correlation function into two parts:

T/;, which is dimensionless; and So which carries the dimension of a mass. The latter can be the

temperature, chemical potential, etc. For those physical variables with still higher dimensions, we can raise them to a certain power and thus reduce them to physical variables of the type S;. According to dimensional analysis, we have r~"[Kp" ..• ,Kp" K.I&, 1, Km,

7];,

K;;]

=

KDrr~')CPI'" .,p",.,., 1, m,

7];,

SI), (4.9)

where Dr is the canonical dimension of the vertex function r~l. From Eq. (4.9), it can be easily shown that

[K

aa

K

+.It

aa + m ~ + ;; ~ I'

am

a;;

Dr]

r~')CKpl>"

.,Kp" ,.,., 1, m, 1/;, S;) = O. (4.10)

Subtracting Eq. (4.8) from Eq. (4.10) and removing pa lap, we arrive at

rlK ~ + ;i ~ - P(1) ~ +

aK a;; x r~"CKpl'" .,Kp"

al

I',

1,

m,

(1

~ r ... O.))m ~ + rr am

7];, ;;) =

Dr]

o.

(4.11)

Eq. (4.11) is the Callan-Symanzik equation satisfied by the closed-time-path Green's function.

When Eq. (4.11) is solved, it leads to the same results4 obtained by Kislinger and Morley using a finite-temperature Green's function. For example, a non-Abelian gauge field possesses the property of asymptotic freedom not only when the momenta are large but also when S; is large (high temperature or high chemical potential). ID. A. Kirzhnits and A. D. Linde, Pbys. Lett. 41 B, 47111971); D. A. Kirzhnits, JETP Lett. 15, 52911972); S. Weinberg. Phys. Rev. D g, 335711974); L. Dolan and R. Jackiw, Phys. Rev. D 9,3320119741; M. B. Kislinger and P. D. Morley, Phys. Rev. D 13, 2765 (19761. 2M. B. Kislinger and P. D. Morley, Phys. Rev. D 13, 2771 (19761; S. Weinberg, Phys. Rev. D 9.3357 (19741; L. Dolan and R. Jackiw, Phys. Rev. D 9,3320 (19741. 643

Chin. Phys., Vol. 1, No.3, July-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

643

544 'J. Schwinger, J. Math. Phys. 2, 407 (1961); L. V. Ke1dysh JETP 20, 1018 (196S); R. A. Craig, J. Math. Phys. 9, 60S (1968); R. Mills, Propagators/or mtlny panicle systems (Gordon and Breach, New York,1969); V. Korenman, Ann. Phys. (N.Y.) 39, 72 (1966); V. L. Berezinskii, JETP 26,137 (1968); O. Niklasson and A. Sjolander, Ann. Phys. (N. Y.) 49,249 (1968); C. P. Enz, The many body problem (Plenum, New York, 1969); R. Sandstrom, Phys. Status Solidi 38,683 (1970); C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C 4, 916 (1971). 4A. O. Hall, Mol. Phys. 28, 1 (1974); J. Phys. A 8, 214 (1974). 'K. Symanzik, Commun. Math. Phys. 13, 49 (1971); C. O. Callan, Phys. Rev. D 5,3202 (1972). 6Q. 't Hooft, Nue!. Phys. B61, 4SS (1973); J. C. Collins and A. J. Macfarlane, Phys. Rev. D 10, 1210 (1974); S. Weinberg, Phys. Rev. D 8 3497 (1973). Translated by King Yuen Ng Edited by Stanley Wu-Wei Liu

644

Chin. Phys., Vol. 1, No.3, Juiy-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

64

4

545

Dyson equation and Ward-Takahashi identities of the closedtime-path Green's function ZHOU Guang-zhao and SU Zhao-bing Institute o/Theoretical Physics, Academia Sinica. Beijing

(Received 24 January 1979) Phys. Bnerg. Fortis Pbys. Nucl. 3 (3),314-326 (May-June 1979) The Dyson equation satisfied by the closed-time-path Green's function of the order parameters is considered. The transport equation for the number density of the quasiparticles is written down in a general but simple form. With use of the path-integral. formulation for the generating functional of these Green's function, the Ward-Takahashl identites are deduced. PACS numbers: 1l.10.Lm, 1l.10.Bf, 11.10.Np

I. INTRODUCTION In recent years, in the study of high-energy particle physics, many problems concerning collective cooperative phenomena have been raised, e.g., the problem of the phase change of the vacuum state, the problem about the confinement of quarks, the problem of solitons, etc. All of these are not single-particle phenomena, but rather phenomena of collective motion due to the interactions of an infinite number of degrees of freedom. It is expected that, under the interactions of high-energy particles, many degrees of freedom can be excited. When one deals with the motion of these degrees of freedom, the method of nonequilibrium statistical field theory should be used. In our view, the closed-time-path Green's-function formulism developed by Schwinger· and Keldysh· provides an effective method to study nonequilibrium statistical field theory. This method is very similar to the method of Green's function in field theory and, with only small alteration, nearly all methods in field theory can be applied. At the same time, the closed-timepath Green's function contains the statistical correlation for any initial condition; therefore, it can also be used to study the properties of the ground state, the thermal equilibrium state, transport processes, and the properties of stationary states far from equilibrium. In our opinion. the study of nonequilibrium statistical field theory has come under higher and higher demand in the development of high-energy particle physics and astrophysics, and this is a direction that deserves emphasis. Taking solitons as an example, we note that in field theory all solitons are classical solutions of the Euler equation of the primitive Lagrangian. But the solitons observed in solid states physics, like vortex lines in superconductivity, are not classical solutions of the primitive Lagrangian. They are the classical solutions of the equation of an order parameter statistically averaged over all other degrees of freedom. In future particle physics, solitons described by order parameters and not by fundamental fields may also be possible. The main purpose of writing this paper is to arouse the interest of our colleagues in high-energy particle physics in this direction. In Sec. II, the Dyson equation satisfied by the closed-time-path Green's function is studied. This probeIm has already been studied in Ref. 1 and therefore not all of our results are new. However, it is worthwhile writing them out here for our readers because our equations are more general and simpler than those in other references; our transport equation which is in a rather simple and general form deserves special mention. The second-order Green's function in field theory is the Feynman propagator GF ; all the rest are unimportant. Of the second-order closed-time-path Green's functions, there are three independent ones that are very important: They are the second-order retarded Green's function G" the second-order advanced Green's function Ga , and the number density, n, of quasiparticles. In field theory, there is no attenuation when a particle propagates in vacUU!Jl; however, 645

0273-429X/81/010645-14$05.00

@) 1981 American Institute of Physics

645

546 there is in general attenuation in the propagation of quasiparticles. This is why Gr and G are considered as two independent quantities. which contain not only the spectrum of propagation (dispersion) but also attenuation. The Dyson equation of the second-order closed-time-path Green's function can determine not only the propagation and attenuation of quasiparticles. but also" the transportation of the number density. n. of quasiparticles. Q

In Sec. III. the Ward-Takahashi (W-T) identities of the closed-time-path Green's function are discussed. We use the Feynman path-integral method, when the Lagrangian possesses the global symmetry of a Lie group G, .to derive the W -T identities· satisfied by the closed-time-path Green's function. The situation when the symmetry is broken spontaneously2 is also discussed. We have proved that. in general. the equation

lJTjlJQc(x)

= o.

satisfied by the vacuum expectation Q< (x) of the order parameter Q(x) determined by the generating functional of the vertex functions. does not possess stable solutions of the form

Qc(x)

=

Qo(X),,-i"",

W

~

o.

But this does not mean that soliton solution or laser-type solutions do not occur in the system; it only says that, because of quantum effects. the wave packet formed by solitons cannot be stable and it must spread and attenuate. Thus in order to search for soliton solutions. we must at first determine the classical equation satisfied by the order parameter Q (x). In the last section. we discuss the W-T identities satisfied by the closed-time-path Green's function when the Lagrangian possesses local gauge in variance. II. CLOSED-TIME-PATH GREEN'S FUNCTION Let Q(x) and p represent. respectively. in the Heisenberg picture. the physical variable and the density matrix designating the"initial conditions of the system. Here Q(x) c~n be a fundamental field; it can also be a composite operator made up of fundamental fields. If Q(x) contains more than one component. we interpret x as {x!, .il for J1. = 0.1.2.3 and i = 1.2 ..... n. with x!' denoting space-time coordinates and i the various components. The closed-time-path Green's function of the physical variable Q(x) is defined as

G,(XI' ''', xz)

=

(-i)l-ltp{T ,((>(XI)" ·Q(x/))p}.

(2.1)

where the subscript p denotes the closed path varying on the time axis from t = - co to t = + co (called the t + branch) and then from t = + co back to - co (called the t _ branch); Tp is the order operator on the closed path p; the time coordinates of xl .... ,xz can be any point on the closed path. When all x 1.... ,xZ are on the t + branch. Tp is the same as the usual time-ordering operator in field theory. and

G,(XI+' "', x/+)

=

(-i)l-ltp{T«(>(x l )" • (>(xz))p}.

(2.2)

The Green's function Gp(XI+ .... ,xZ+ ) is the vacuum expectation of the operator T(Q(xl) .. ·Q(xzl). It differs from the Green's function of ordinary field theory in that the former is an average over any initial condition (described by the density matrix p) while the latter is an average over the vacuum state. When x I .... ,xj lie on the t _ branch and Xj + 1 .... ,xZ on the t + branch. we have

G,(xl_, .••

,Xj_, Xj+I+' ••• ,

x/+)

= (- ;)1-1 tr{r«(>(x

l

) ' ••

(>(xj))T(Q(xj+,) • •• (>(xz))p}

(2.3)

where T is the anti-time-ordering operator (operator of a later time stands to the right).

Let us introduce the generating functional of the closed-time~path GreeQ's functions Z[J,(x)] =tr{Tp(exp{-iLhCx)(>(X)d4x})p}. 646

Chin. Phys., Vol. 1, No.3, July-Sept. 1981

ZHOU Guan9·zhao and SU Zhao-bing

(2.4) 646

547 where Q(x) is the operator in the Heisenberg picture without taking into account of the external source h (x). The integration in Eq. (2.4) is made along the closed path p, the external source h (x) being different when situated on the t + branch and the t _ branch. Differentiating Z [h (x)) with respect to h (x), we get i 8Z [h.(~)] 8h(~)

=

t. {T p(~(~)exp {- i

f

Jp

hey) ~(y)d4y ))p}.

(2.S)

When h (x+) = h (x_~ = h (x),

L

Tp(~(~)exp {- i h(y)~(y)d4y}) = 1J+(t)~(~)U(t)

;:

~A(~)

(2.6)

where

and ~(x) is the operator in the Heisenberg picture when the external source term Jh (y)Q(y)d3y is included in the Hamiltonian. We thus have

(2.7)

Those readers who are not familiar with the closed-time-path Green's functions can learn from Eq. (2.6) why we have to introduce a closed path. Only in this way can we guarantee the operator to be always in the Heisenberg picture; otherwise, on the right-hand side of Eq. (2.7), after application of the trace operator t, there will be an additional factor U (t;), t; being the latest time for x" ...•x,. This factor makes the ordinary Green's functions in any initial state unrelated to the averages of physical variables. Let us introduce the generating functionals of the c1osed-time-path connected Green's functions and vertex functions. W[h(~)l = iln Z[h(~)],

r[Q(~)]

=

Lh(X)Q(~)d4X'

W [hex)] -

(2.8)

where the average of Qh (x) is denoted by Q(~)

= 8W[h(x)] • 8h(x)

Here. we do not require h (x +)

(2.9)

= h (x _), but directly define

QA(~) = T,(Q(~)exp{-i Lh(x)Q(~)d4X}).

(2.10)

When we solve for the observables, we need to take h (x +) = h (x _); then Q (x +) = Q (x _) will be satisfied automatically. Similar to field theory, we have 8T[Q]

8Q(~)

=

=+=

hex).

(2.11)

In the 101l0wing, whenever an equation contains upper and lo~er signs. the upper one applies when Q (x) is a boson operator while the lower one applies when Q (x) is a fermiolN>perator. When 647

Chin. Phys., Vol. 1, No.3, July-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

647

548 Q(x) is a fermion operator, both h (x) and Q (xl are anticommuting c-numbers and all derivatives

operate from the left. Differentiating Eq. (2.9) with respect to Q(Yl and Eq. (2.11) with respect to h (Y), we arrive at two equations,

L

8~(x

G,..(z., y)tl4yTp(Y' z) = -

LTp(X, y)tl4yG pc (Y, z)

= -8~(x

-

z),

(2.12) - z)

where the second-order connected Green's function is Gp«x, y)

=

8h(:;~(Y) == -

i 0, then a quasiparticle with energy (II(k) propagates in the system. The amplitude of the quasiparticle is attenuated' exponentially in propagation, with an attenuation constant

r=

A(k)

(2.29')

aD

ako

.\.=.. (k)

From Eq. (2.22), we can also solve for {j and obtain

~ = -..!.. (T;'*. - *.T;') =..!.. (G.*. - *.G.) 2

where

*. = *. =

(2.30)

2

NfJ - ifJ2

+

fJ3

NfJ - i7]2 - fJ3

= =

7j(N

+

fJ3)'

(2.31)

(N - fJ3)fJ,

and N is a matrix determined by the following equation NT• ...,... TrN

=

2iB,

(2.32)

AN) - 2iB.

(2.32')

or ND - DN

=

i(NA

+

At the poles of Gr and G., there exists in the system a quasiparticle carrying energy (II(k). Its Green's function GF can be expressed in terms of the quasiparticle number density matrix n, (2.33) Comparison of Eqs. (2.33) and (2.30) shows the relation between N and particle density n, N latthe poles of Gr = 1 In terms of n, Eq. (2.32') can be written as

nD - Dn

= i(nA +

An) ± i(A - B)

±

2n,

=

;(nA

(2.34)

+

An) ± T+,

(2.35)

which is in fact the transport equation satisfied by the particle number density n. After neglecting the noncommutative parts of n and A, the right-hand side of Eq. (2.35) can be written as

± (1 ±

n)T+ - nT_

which is just the collision term on the right-hand side of the transport equation, r + directly proportional to the rate of emission (absorption) of quasiparticles.

(r _) being

In the following, we consider the case when Qis of single component. In an approximately uniform system, the matrices D (x,)'), etc., can be written as X

=

"21 (x + y),

III

=

(x - y),

which are slowly varying functions of X. In the momentum representation of z, expanding in terms of X,. up to the lowest order of iJ liJX,., we have, at the poles of Gr , 650

Chin. Phys•• Vol. 1, No.3, July-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

650

551

(aD(k, X) anCk, X) _ aD e.! , X) anCk, X)) ak" ax" ax". ak" aD (..E!!.- + v • 'ilx n + aw an). aka aXa ax;. ak"

(nD - Dn)(k, = -

j

X) = _

j

(2.36)

In writing down Eq. (2.36), we have used the pole condition [Eq. (2.29)] of Gr to obtain the velocity of the quasiparticle, v

= 'il.w == _ 'il!D ~ aD' aka

(2.37)

and

aD aw ax" ax" = - aD' aka From Eqs. (2.36) and (2.35), we can obtain the transport equation satisfied by the quasiparticle number density in an approximately uniform system:

~n +

uXa

v.

'ilxn + aw ~ =....:.L {± iT+(l ± n) - if-n}. ax" ak" aD aka

In a uniform system, both nand becomes zero so that

Q)

(2.~8)

do not change with X,. and the right-hand side of Eq. (2.38)

-l±n - = -T_ -,

(2.39)

± T+

n

which is just the Einstein relation of detailed balance. In order to understand the physical meaning of r + and r _, let us take a scalar field as an example. Let Q(x) be the scalar field $(x), which satisfies the equation of motion, (a Z+ mZ)cpC,,)

= jC,,).

(2.40)

It can be easily proved that the second-order vertex function is

TpCx, y) = ca! +

mZ)8~Cx

y) +

-

~pCx,

y),

(2.41)

where l:p(x,y) is the self-energy part, which can be represented as ~pCx, y)

with (

II.p.1

= -

(2.42)

il r {T p C1C:t)jCY))P} •. P.l.,

denoting the one-particle-irreducible part.

Let In) be a complete set of orthogonal eigenstates of the energy-momentum operators and other operators commuting with them. In a uniform system, because of translational invariance, l:p(x - y) is a function of x - y only, and, in momentum space, it becomes

j~-(k) =

JeiH

r- Y ),.{1(:t)jCy)p} •. p.l.d4(x

....

- y)

= ~1ltp.l.Pftft(2 ...)464Ck - P, +

n)_

(2.43)

When ko> 0, the right-hand side of Eq. (2.43) is directly proportional to the quasiparticles 651

Chin. Phys., Vol. 1, No.3, Ju1y-Sept.1981

ZHOU Guang-zhao and SU Zhao-bing

651

552 absorption cross section at energy-momentum k. To be more precise,

i.E_(k)

2I k lu •boorp'ion(k)

=

= (;~;3 W.bsorption(k),

(2.44)

where W.bsorp'ion (k ) is the probability of quasiparticles absorption per unit time. In the same way it can be shown that, when k o > 0,

..E+ (I.) '\ -

I

where

Wemission (k

Jf

t!

i~·(%-y) tr {~( ) ~( ) 1 Y1 "

-}

P

-

1.1'.1. -

2k W (I.) (2,..)3 emission '\

(2.44')

) is the probability of quasiparticles emission per unit time.

From Eq. (2.41), one readily verifies that T±(r, y)

=

.E±(r, y).

Thus ir ± (k) has the same physical meaning as il: ± (k), and Eq. (2.39) can be written in the form of the ordinary relation of detailed balance:

+

(I

n) ~mission = n Wabsorption •

(2.39')

In the above, we have made rather detailF discussion of the situation of a single-component operator. But we believe that, in the multicomponent case, n still possesses the physical meaning of quasiparticle number density and Eq. (2.35) can be considered as some generalization of the transport equation. Before closing this section, we want to point out that, because of causality, the retarded Green's function G,(x,y), in the momentum representation corresponding to the relative coordinate x - y, should be analytic in the upper half of the ko complex plane. If r> in Eq. (2.29'), the amplitude of the quasiparticle attenuates when moving in a dissipative system, and the demand of analyticity is fulfilled automatically. If r < 0, the quasiparticle propagates in a proliferous system where its amplitude increase, and we have to take a path of integration to pass over the poles of G, in the upper half plane from above in order to guarantee the preservation of causality.

°

III. THE W-T IDENTITIES AND SPONTANEOUS SYMMETRY BREAKING

Suppose that the Lagrangian of the system is globally invariant under a Lie group G; the symmetry leads to a set ofW-T identities satisfied by the c1osed-time-path Green's function. The group G may include the space-time symmetry· as a subgroup. Let fJ (x) be a fundamental field and Q(x), a function of fJ(x), be the order parameter of interest. There are many components on fJ (x) and Q (x), which form a unitary representation of the basis of G. Under an infinitesimal transformation of G, fJ (x) and Q(x) transform as

11"(,,)

11'(") 611'(")

=

=

11'(")

+ 611'(")'

".(;t~O) - r:(,,)8,,)qJ(")

(3.1)

= itrp("n",

and Q(,,) 6Q(,,)

Q'(,,)

=

=

',,(ii.~O)

Q(,,)

-

+

6Q(,,),

r:8,,)Q(")

(3.2)

= iL"QC"n"

ta

where are the no infinitesimal parameters of the group G; j~1 and f~1 are the matrix representation of the generators acting on ({J (x) and Q(x) and are Hermitian. Under the above transformation of G, ~(x) and the space-time point x,. are related by (3.3)

In the following, we let 652

ta be infinitesimal functions of x. It can then be easily proved that

Chin. Phys., Vol. 1, No.3, July-8ept. 1981

ZHOU Guang-zhao and SU Zhao-bing

652

553 the Lagrangian transforms as

where (3.S)

is the current in direction a. If the Lagrangian is globally invariant under the group G, then (3.6) Equation (3.6) indicates that the current.l.!(x) is conserved when rp (x) is a solution of the Euler equation. With use of Eq. (3.6), Eq. (3.4) can be written as st'(cp'(,,))

:i~'

=

q(cp(x-'))

+ i:c,,)all'aC,,).

(3.7)

Equation (3.7) gives the transformation of the Lagrangian under a local transformation when the Lagrangian is only globally invariant under G. We now proceed to derive the W-T identities satisfied by the closed-time-path Green's function, making use of Eq. (3.7). Employing the method used in field theory, one can readily prove that the generating functional of the closed-time-path Green's functions can be w.-i.tten in the form of a Feynman path integral. Introducing the external sources J(x) and h (x) of rp (x) and Q (x), respectively, the generating functional can be represented by Z[J(,,), he,,)]

=

N

1

[dcp(,,)]exp{i

L

[!.f(cpC,,)) -

- h(")Q(,,)]d4x-} are the field operators and density matrix in the Heisenberg representation, index p indicates a closed time path consisting of positive (-00. +oo) and negative (+00. -oo) branches. The time variable I can take values on either branch. T, is the time-ordering operator along the closed time path. The generating functional for the CTPGF's is defined as

as the correlation functional for the initial state. N(J(x» can be expanded into a series of successive cumulants N(J(x» =exp[-iWN(J(x») (2.11)

Z(J(x»=trk,[expl-i LifJ(x}J(X»)]p} ,

W",(J(x»

(2.3)

.h

(2.10)

=

where the integral is taken over the closed time path and the integration variable d 4x is omitted. In Eq. (2.3) the external fields on the positive and negative branches J(x+} and J(x-) are assumed to be different. Taking functional derivatives with respect to J(x} we obtain from Eq. (2.3)

GO· .. n} = ,

i

8"Z (I (x» 8J(I) ... 8J(n)

I

(2.4)

J-O

.

In the interaction representation the generating functional (2.3) can be rewritten as ZU(x»

=trkp[ exp(-i L[v(~,(x» +~ '(X)J(X)))IP} (2.5)

where ~ ,(x) satisfies the Euler equation for the free fields. The interaction term can be taken f.-om behind the trace operator to obtain Z (J(x» =exp[-i

f

p

-I]

vIi _8 8J(x}

x tr{T,[expl-iLcp,(X}J(x)ll;>}

(2.6)

It is easy to show by generalization of the Wick theorem that

T,!expl-i J,, from which it rollows [from Eq. (2.19)] that Qri(X+) = Qr'(x-) = Qi('X,t), where Ji('1.t>, Qi('1.t> are functions defined on the usual time axis (-00. +(0). We next show that Eqs. (4.2) lead to the generalized TOGL equations under the assumption that Qi are smoothly varying runctions or time. Suppose the macrovariables Qi('1. T) to be known at the moment T. At the moment I rollowing closely after T the left-hand side or Eqs. (4.2) can be expanded. If x sits on the positive time branch, we have

(4.3)

where r"ix.y) are two-point retarded vertex functions arter taking Q,.+=Qr-=Q. If xsits on the negative branch or time, the same is true due to Eq. (2.60) r, = r ++ - r +_ = L+ - r __. Since Qj( y.t) in Eq. (4.3) varies smoothly with time, to the first order or (Iy - T) we have (4.4) Substituting Eq. (4.4) back into Eq. (4.3) and taking into account that in the limit I"" Ix

-

T

(4.5)

571 3397

CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

where Cij(X'. f.k o. d are Fourier transforms with respect to (Ix - ty) taken at the average time T = (Ix + ty) == T, we obtain

and .iY. For the moment let

+

I;(X'. T) == BBQr

,,+

(4.6) Here the matrix notation is used and Y ij ( X. Y. T ) are considered to be matrix elements with subscripts i X'

I c+ Q

_Q

c-

(4.7)

_QI,)

and we calculate the functional derivative of I;, considering it as a functional of functions Q (X'. T) with three-dimensional argument X'

where

Thus we obtain BI,(X'. d/BQJ( f. -r) =

r ++ij(X'. Y.ko =0. T) - r +_ij(x. f.ko =O.-r)

where the ko-O components of Fourier transforms appear as in Eq. (4.5). It can be shown in the same way that

where the symmetry properties of

r

following from Eqs. (2.59) and (2.63) are used. The difference

Bli(x. T) BQj(

(4.8)

f. -r)

vanishes due to Eq. (2.77), i.e.,

pative part of the vertex function A = Eq. (2.68c) satisfies the condition

r +_ = exp( -,Bko) r _+ near thermal equilibrium, so that there exists a functional F(Q/(x, with

T»

(4.9) Equation (4.6) can be rewritten as y(-r)aQ(T)/aT=-8F/8Q(-r) +J(-r)

(4.10)

If the macrovariables Q (T) do not change with time in the external field J, i.e, in the stationary state, then BF/BQ =J .

(4.1 \)

Hence F is the effective free energy of the system and Eq. (4.11) is the Ginzburg-Landau equation, determining the stationary distribution of macrovariabies. For systems in stationary states far from equilibrium expression (4.8) is equal to zero only if the dissi-

lim Aij(x. f.ko,t} =0 .

+;(

r _+ - r+_)

(4.12)

to-O

I n this case Ii can also be written as a variational derivative of the free energy or effective potential. Some of the stationary states satisfying the so-called "potential conditions," provided by the detailed balance, as discussed by Graham and Haken,11 must belong to this category. In the vicinity of all stationary states with the potential functions F Eq. (4.10) constitutes the system of time-dependent GL equations, but they are much more general than the TDGL equations in the usual sense since the mode coupling terms are also included. It is usually customary to multiply Eq. (4.10) by the inverse matrix .,,-1 (T) to obtain (4.13)

572 3398

ZHOU, SU, HAO, AND YU

Using the symmetry properties of the vertex functions, following from Eqs. (2.59) and (2.63) t

f(k) = fT( -k) = -uJt*( -k) UJ =-ud· (k)uJ

(4.14) it can be shown that the real part of r r is an even function of ko, while the imaginary part is an odd one, so ')I ( t) is a real matrix according to the definition (4.5l. In accordance with the numeration of the subscripts given at the beginning of this section, the matrix y( T) can be divided into four blocks. Two of them, corresponding to the conserved variables YaT./I'yand ')IaTJV' can be fixed completely by comparison with the WT identities. For the general case, the proper form of the two blocks associated with the order parameters can be determined only by the symmetry considerations. This will be discussed below. It is worthwhile to point out that Eq. (4.4) is equivalent to the Markovian approximation. In principle, the original Eq. (4.2) contains in itself the possibility of considering the memory effects. Under the action of the symmetry group G of the system the conserved variables transform as the generators [a of the group, i.e., (4.\ 5)

where 1 all' are the structure constants of the group and CIl are the infinitesimal parameters of transformation. The order parameters Q, transform as some representation i of group G (4.16) As shown in Sec. II D, if the Lagrangian of the system is invariant under the global symmetry transformations, the WT identities (2.95) are valid on the closed time path. In the present case, Eq. (2.95) can be wrillen as

(iJ"jf{( rp»

-

i(

/SQ~j~X) L;j'Qcj(x) + fa/l, /Sq~~x) q,(x) 1 (4.17)

where, as before, r 5 r(Q.,.qa) is the generating functional of the vertex CTPGF's for the composite operators. PUlling Q 39, 72 (I 966); D. Langrelh, in Lillear alld NOlllilleor EleC'II'O/-;;; Trallsporl ill Solids, edited by J. Devreese and V. Van Doren (Plenum, New York, 1976); A.-M. Tremblay, B. Pallon, P. C. Marlin, and P. Maldaque, Phys. Rev. A 19, 1721 (1979). 4Zhou Guang-zhao and Su Zhao-bin, Progress in Statistical Physics (Kexue, Beijing, 10 be published in Chinese), Chap. 5. 5Zhou Guang-zaho and Su ZhaO-bin, Physica Energiae FOrlis el Physica Nuclearis (Beijing) I, 314 ([979). 6Zhou Guang-zhao and Su ZhaO-bin, Physica Energiae Fnrlis el Physica Nuclearis (Beijing) I, 304 ([ 979). 7Zhnu Guang-zhao and Su ZhaO-bin, ACla Phys. Sin. (in press). sp. C. Marlin, E. D. Siggia, and II. A. Rose, Phys. Rev. A S, 423 ([ 973). 911ao Bai-lin, in I'.rogress in Statistical Physics (Kexue,

(823)

Beijing, to be published in Chinese), Chap. I. lOp. C. 1I0henberg and B. I. Halperin, Rev. Mod. Phys. ~, 435 (1977). 11K. Kawasaki, in Crili",tI Ph,·l/ul/wllt•. ediled by M. S. Green (Academic, New York, 1971). USee for example, B. I. Halperin, P. C. Hohenberg, and S. K. Ma, Phys. Rev. B lQ, 139 ([974); S. K. Ma and G. F. Malenko, ibid. 11,4077 (1975), HH. K. Janssen, Z.Phys. B ll, 377 (976); R. Bausch, H. K. Janssen, and H. Wagner, ibid. .?~, 113 (1976). 14C. De Dominicis and L. Peliti, Phys. Rev. B ]!, 353 (l97S). IlL. Onsager and S. Machlup, Phys. Rev. 'l!.. 1505, 1512 ([953). 16R. Graham, Springer Tracts Mod. Phys. ~, I (]973). 17R. Graham and H. Haken, Z. Phys. 243, 2S9 (1971). IlL. Sasvari, F. Schwabl, and P. Szepfalusy, Physic. (Ulrechl) SI, lOS (1975). 19J. Deker.;.J F. Haake, Phys. Rev. A 11, 2043 (1975). 20E. Brclin, J. C. Le Guillou, and J. Zinn-Juslin, in Phase Trumiiliol1s and C,.ili('ul Phenomena, edited by C. Dumb und M. S. Green (Academic, New York, 1976), Vol. VI. 21Yu Lu and Hao Bai-lin, Wu\i (in press). 2211. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimenlo §" 3 19 (1957>'

582 Commun • .in Theor. Phgs. (Beijing), China)

Vol. 1, No.3 (1982)

29S-30f

ON THEORY OF THE STATISTICAL GENERATING FUNCTIONAL FOR THE ORDER PARAMETER(I) - GENERAL FORMALISM

16:**

ZHOU Guang-zhao ( Jil](;?i ) SU Zhao-bin ( HAO Bai-lin ( ]/!~ff; ) YU Lu ( f ~ ) (Institute of Theoretical Physics. Academia Sinica, Beijing, China) Received December 28, 1981.

Abstract A theoretical scheme using closed

time-pathGreen~

func-

tions is proposed to describe the quantum statistical proper.ties of the order parlllllf!ter .in terms of a generating function-

al.

The dgnamic evolution is generated bg a

while the statistical correlation bg a

source,

driving

fluctuation

source.

The statisdcal causalitg is shown to hold ezplicitlg and to give

rise to a number of i.JlIportant consequences.

The pm.bl.em

of determining the quantum statistical properties for the order parameter is reduced to finding a solution of the

func-

tional equation for it.

I.

Introduct1on

It is well known that special tricks have to be used in the s1:andard Green's function techniques[l] to describe systems with broken symmetry such !'.s superconductivity or Bose-Einstein condensation. On the other hand, a systematic theoretical scheme has been developed in the quantum theory of gauge fields to treat the order parameter. [2] The generating functional formalism developed there has been applied successfully to the classical theory Df static critical phenomena. [3] To treat the time-dependent phenomena Martin,Siggia and Rose(MSR) have constructed a noncommutat1ve classical field theory closely analogous to the quantum field theory. The MSR theory has been reformulated into a Lagrangian formalism[5,6] and has been used extensively in studying dynamiC critical phenomena. [7] But, the physical meaning of the "response field"· introduced in that theory is not clear. There are also some attempts of quantum generalization of Langevin and Fokker-Planck equations for describing order parameter. [8] It seems difficult to use those kinds of semi-phenomenological theories to study nonuniform system~ and composite order parameter. The necessity of constructing a unified quantum theory to describe nonuniform, nonequilibrium systems with broken symmetry is obvious from the current research on condensed matterandlsser phenomena. In some cases (e.g. nonequilibrium superconductivity) the dynamic coupling of the order parameter with the elementary excitations is essential. It is almost impossible to incorporate this kind of coupling into the theoretical schemes mentioned above.

583 296

ZHOU Guangzhao, SU Zhaobi.n, HAC Bailj,n, YU Lu

We have shown previously[9] that a unified microscopic description of both equilibrium and nonequilibrium systems is possible by combining the closed timepath Green's functions (CTPGF) with the techniques of generating functional. The transformation propert ies of CTPGF are studied in Refs. [10,11] and the MSR theory appears to be a "physical" representation of CTPGF in the classical(Rayleigh-Jeans) limit. We prove[12] the existence of a generalized potential functional for the order parameter in a nonequilibrium stationary state for systems with time reversal symmetry. The time-dependent Ginsburg-Landau (TDGL) equations for order parameters and conserved densities are derived[11-13] with both time reversible and irreversible parts, also the Lagrangian formulation in critical dynamics is obtained in a low order approximation of CTPGF. It seems to us that CTPGF formalism is a good candidate for the general quantum theory of nonequilibrium systems discussed above. In the present series of papers we propose to construct a quantum theory of the statistical generating functional for order parameter using CTPGF which is capable of describing both aspects of the Liouville problem: dynamic evolution and statistical correlation. The theoretical scheme is general enough to incorporate nonuniform, nonequilibrium systems with either simple or composite order parameter. We will give a practical prescription for constructing the generating functional being applied to concrete sys~ems. The dynamic coupling of the ord~r parameter with quasi-particles will be ~reated in fu~ure publications. The rest of the first paper in the present series is organized as follows. In Sec. II the generating functional for the order parameter is constructed explicitly. The dynamic evolution is generated by a driving source which might be the.actual external field, while the statistical correlation is generated by a fluctuation source which vanishes on tbe completion of calculation. Thestatistical causality and its consequences are studied in Sec. III. In Sec. IVa statistical functional equat.ion for the order parameter on a generalized manifold which is equivalent to the Langevin equation to certain extent is derived and the problem of dete~ining the statistical properties of the order parameter is reduced therefore to finding a solution of this functional equation. The final section contai·ns a few concluding remarks. Throughout this series of papers we will use the system of units with 11 = C = ka= I, but we might come back to the ordinary units if necessary.

II, The statistical generating functional for order parameter Suppose Qa(X), a = l,2 ••. N, are a set of Hermitian order parameters in Heisenberg picture which might be basic field variables or composite operators and ha(X) are corresponding C-number real external sources. Both sets are defined on a closed t~e-path consisting of a positive branch (-~, +~) and a negative branch (+ .. , - .. ). The CTPGF generating functional for order parameters can be written as[9,10]: (2.1)

584 On Theory or the Statistical Generating FUnctional for the Order Parameter (I) --General Formalism

297

where ~ is the density matrix in Heisenberg picture, Tp is time-ordering operator along the closed time-path. Tr has the usual meaning of trace operation in Bllbert space. Jp d4 X... means 4-dimensional integration along the closed t:imepath. We note in passing that the letter "p" will often indicate quantities defined on closed path or operations along it. We have also omitted the index a for ~a(X) and ha(X) and a summation over a is implicitly understood. Introduce index a for the time branch as: (2.2)

"'r-

Q

(X)

~

==

[(Xv)

I

(2.3)

with 0=+ or -, indicating whether the space-time point is at the positive or at the negative branch. Eq.(2.2) defines two independent C-number functions from a single function defined on the closed path. However, Eq.(2.3) defines two operators QO(X) in the ordinary space-time with different time-ordering properties, but representing the same quantity. Under the action of Tp the operator Q-(X) should always precede the operator Q+(X), the ~+ operators are time-ordered in the usual way among themselves. while the Q- operators are antitime-ordered (denoted by T). After time-ordering operations Q+ and Q- operators are set to be identical. Introducing. furthermore.

~~ ::

and defining

Jr-

{

if r=T if

-1

=I

f"7

r At' Q.c(t)=TS .. 1I. lxl.

A

kc(x)

we have

=+'$.

hlr(x)

•

a-~-

(2.4)

a-=+ au -

'"

h ""0"'

[0-11. (X),

(3.5a)

hAlX)='l.,.hr- cx ) ,

(2.5b)

Q.o.(lC)=

ar-(XJ=J,.Qc ex) + f~/I't(X), (2.6a) I{ClCl =

5,,"hc exl + -}~r- k.o. (X).

(2.6b)

The 4-dimensional integration over the closed path is transformed into the ordinary integration as (2.7) According to Eqs. (2.4)-(2.7) and taking into account that

S h (I) th

(2)

SIt""C/) IIII'C2) -

t; (

1- 2) ,

r

or,

&4

0111

J.,-

,-

, ... "'"

(2.15) with

597 310

ZHOU Guangzhao, SU Zr.aobin, HAO Bailin, YU Lu

(2.16) In Eqs.(2.14) and (2.16), $t(x) and $I (x) are operators in the incoming interaction picture, : ... : means normal product. Since the distinction of time ordering along the positive and negative time branches does not make sense under the normal product,one can easily show that (2.17) in notations of (I), i.e., ""pH (J+, :r)= 'WN

witb

'WNrr.+ r. )=i l' 4 , . . ",,JIll

T [JA , JA]

(2.18)

-'-JJ.r", c:/.md.r .. , dn l7l! n! (2.19)

where (2.20) It is important to note that the functional at the left-band side of Eqs.(2.17) and (2.18) is defined at the closed time-path, while that at the right-hand side is defined in the ordinary space-time. Similarly, Eq.(2.16) is an expression' at the closed time-path and, therefore, each argument can take value of either positive or negative time branch, while Eq. (2 .20) is an expression defined in the usual space-time. Furthermore, taking into account that in the incoming picture the field operators satisfy the free field equation, one can easily find that' • I, I ' " P .. " J.p QiJ'-S-,(",)",Nrll,7II'CI"

.. , III , -n. '"

-,)-0 ,

(2.21) (2.22)

or, equivalently,

(' ,,)-N(II ..... ' j '"J,'---' SOY' 1,1 IN Jeli

' - N ( I I mJ

IV

'

"

-

0

-,

(2.23)

{f"'I"·>t,,n"·T,=,

-..

(J '" >n, 1'1 '"

-

-_.r

..• -:-

i ... I ) 5" ... lL ,

I }

=

a,

(2.24)

Since the moment t = -co is chosen as the starti!!g point of the closed timepath, the initial condi'l:ion for the statistical system is fixed at this moment. Therefore, we are not allowed to integrate by parts in respect to the operator a/at arbitrarily. Tbe correct direction of action is indicated by an arrow in Eqs.(2.2l)-(2.24) to incorporate appropriately the initial condition. Substituting Eq. (2.18) into Eq.(2.11) and taking into account Eq.(2.8), we obtain .l,.['; J+l]=eXr{i (laa(-t i,~, - i.~Jtli .. t(ti ,~, - i1j.J)}

xexp{i. (- J+,s.1+W N (J: .I.d)},

(2.25)

598 on

Theorg or the Statistical Generating Functional ror the Order Parameter (II) --Densitg Hatriz and the Field-Theoretical Structure of ~~e Generating Functional

311

which is the first expression for CTPGF generating functional we derive in this section. Eq.(2.25) specifies the generalized Feynman rules forCTPGF and shows clearly how the density matrix contributes ~o CTPGF in terms of WN[J!.J61 in view of the global structure of the perturbation theory. It tells us that the density matrix affects directly only the correlation functions of the constituent field variables describing the statistical fluctuations (corresponding to oJ6T. 6 ). So far as WN(m,n} satisfy Eqs.(2.23) and (2.24). the contribution of the density matrix can be expressed in terms of the initial conditions(sometimes called boundary conditions) for the statistical Green's functions. Now we derive another expression for the CTPGF generating functional. USing the following equality (up to an unimportant constant factor)

oJ

(2.26)

it is easy to show that

(2.27)

if the path integration is taken by parts.

Taking into accounr. that (2.28)

s.,.-

•&. .,._expCi,/,+S •- '.") =e"p (.''1' +5•-, .,. ) ( St'I -S•-, '"T ) •

(2.29)

Eq.(2.27) can be transformed into

e~p{i (-ItS.rt W'rN[.r+. r)}

= Jp (dt') (df) &JCr{L(J+,/,t"'+J+'I'+S.-',/,)}

(2.30)

Using Eq.(2.3) and the convention of (I) we find that (2.31)

t

S

(2.32)

11".I tl.+,f;C1J?, So,,,. (I· X ) = Jd+; t;(I}S~: (1·X.J,

(2.33)

?cr rt+(xtr )

ft~oc) •

(2.34)

and obtain from Eq.(2.18)

599 312

ZHOU Guangzhao, SU Zha"bin, HAO Ba:i.1in, YU Lu

exp{ i =

w; (: i S~ -

,/,+"$.-' ,

'j

S~. - S.-'t)}

(2.3S)

e"p{iwN[-'I';So;', -S.~ 'l'4J}

and

elC?{ i "'; (-,/,"5;' . -5;''I']} = exr{iwN[-'I';S;;, - 5;~ t~J}

(2.36)

considering 1/1;, 1/1&, as well as lIi • '~t as independent funct iona~ arguments. c Substituting Eqs.(2.3S), (2.36) into Eq.(2.30) and putting the result obtained into Eq.(2.11) we get finally

xeJeP { iW,.N [-'/' +--, S. ,

--, 'I' J}

- S.

(2.37)

as the second expression for the CTPGF generating functional for the order ?arameter-a path integral present:ation. It is easy to rederive Eq, (2.25) from Eq.(2.37), so these two expressions are equivalent to each other. This path integral representation is different from what we obtained previously tor the CTPGF generating functional[3 J , in so far as the contribution of the densit:y y + ,,-1 matrix is expressed bere as an additional term in the action given by W'p [-IV ~o, -S~11/lJ. According to Eq.(2.36) tbis term does depend only upon the field variables 1/1~, 1/11 describing the statistical fluctuat:ion, but not upon the field variables describing the dynamic evolution. On the other hand, it is clear that [-1/I+S'~-!., _~11/l J has nonvanishing contribution to the generating functional only at the initial moment t = -.... Expanding w~ in accord with Eq, (2 .1S) and integrating ~1, S;l by parts we find that only terms corresponding to the complete aifferential contribute, because the expansion coefficients satisfy Eqs.(2.21) and (2.22). Since the functional integral is taken over a closed time-path, starting and ending. at t=_CD, ~[_1jI+S;l, _S~11jl] has nonvanishing contribution only at these end points. Before further analyzing Eqs.(2.2S) and (2.37),the derivation of which is the main subject in this section,we apply first these equations to an important special example: the contribution of the density matrix to the CTPGF generating functional in thermal equilibrium,

W;

III. CTPGF generating functional in thenmal equilibrium As an important special case we will derive an explicit expression for the CTPGF generating functional in thermal equilibrium, i.e., for the density matrix given by (3.1)

600 On Theory of the Statistical Generating Funcr:.i.onal .for the Order Parameter (!::) --Density Matrix and the Field-Theoretical Struc~ure of the Generating Functional

31J

(:1.2)

where H is the total Hamiltonian 0: the system, !Ii-operator of particle number, ).I-chemical potential, exp(-n)-normalization consta'nt, or the inverse oJ the partition function. Substituting Eqs.(2.l7) and (2.18) into Eq.(2.14) we find that

."::;. ~;

where $I'~! at the right-ha~d side are operators in the inco~ir.g pict~re. ~: 1s known for the operator AI(t) in the incoming picture that[5]

It is essential to note that

a in

Eq.(3.4) is the total Ham~ltoni~~.

If an an-

alytic continuation -

i 6

is carried out we find that (3.6)

Taking into account that for complex fields the operator of particle number (S.i)

is a conserved quantity. it is easy to prove that (3.8) where ~

~+

A = + 1,

if

AI(t)

=1jII(x)

,

A = -1,

if

AI(t) = $I (x)

,

A =0 ,

if

AI(t)

(3.9)

is Hermitian.

With Eq.(3.8) we can apply, as done by GaUdin[6 1, the following identity

Tr{ (r A(II t

(±)II

= Tr 0[A (I), A(2) J,AU)" t ...

AlIly JAm.·. A(nl}

'Atlll} ±Tr(f A(2) [A (J),AW)",A (f)'"

Aell)}

A .. ·AOI-I) A ("AII).Alnl:; A ]} (±)11-2 Tr fA fAI21

(3.10)

601 314

ZHOU GUangzhao, SU Zhaobin, HAD Bailin, YU Lu

to the right-hand side of Eq.(3.3) to obtain

Tr{ft~: e xp[Hr;tx +t;J41):}

= Tr {frk} C)

I

I);:

0

(X)=,t"tlC)=O

(3.28)

=:/

. X:... (;< j X" (~) :C)=~(.~) and its complex conjugation. We first consider stationary, homogeneous syscem where X<x), X*(x) are independent of x and Green's functions :lre translational1y invariant. Substituting Eqs.(3.22)-(3.25) into Eq.(3.2d) '.we obtair. the order parameter equation for uniform superconductor in the mean field approximation as

614 On Theory of the Statistical Generating Functional for the Order Parameter (ITI) -- Effective Action Formalism for the Order Parameter

397

(3.29) where

Tr

means trace operation only in the spin space. t.. ()()

Introducing notation

== 3J. ell) •

(3.30) (3.31)

and solving Ga , Gr from Eq.(3.20) in momentum representation, we obtian Gc from FDT given by Eq.(3.26). Substituting Gc thus obtained into Eq.(3.29) yields the famous BCS gap equation as (3.32)

Hereafter in this section we switch onto the usual units. Now we discuss the stationary system with weak inhomogeneity. the scale of space variation obeys the inequality

Assume that

(3.33) where

PF is Fermi momentum of electron. are satisfied by the system

Moreover, the following conditions

(3.34) (3.35) where T = 6-1 definedby[8]

is the temperature of the system, Tc - critical temperature

(3.36) with N(o)=

7IIPF

(3.37)

21C"1; ~

as the density of states at the Fermi surface, wD being Debye frequency of phonon, y- Euler constant. Since Gal is translationally invariant according to Eq.(3.13), the nonuniformity of the system comes from a slow space variation of the order parameter X as follows from Eqs.(3.22)-(3.25). For such a system we can expand the functional arguments X(y), X*(y) in Eq.(3.28) around x(i), x*(i) using O(hl j.tlil IAPI) as small parameter as we did before [7,9]. Taking into account Eqs.(3.33) and (3.35) to neglect quadratic term in 3A(i)/3i, we find that

615 398

ZHOU Guang-zhao, SU Zhao-bin, HAD Bai-lin, YU Lu

(3.38)

. ;;:1, *(-;) . ~iax

o

where (3.39)

(3.40)

Obviously, the first term in Eq.(3.38) is proportional to the left-hand side of Eq.(3.32) with

8

replaced by

8(X).

Considering

6(;) as a small quantity

and keeping its third power we obtain by the well-known technique[8]

& I ..if

&,,:Cxl ;t..•=}:.. =O J&:=~: :X.=J',

(3.41)

substituting Eqs.(3.39) and (3.40) into the four coefficents of Eq.(3.38) with 8

oz

0

in Eq. (3.13) yields (3.42) tmt phenomena for ideal gouIl[ their formalism, it seems hU.l'" :0 "larify the assumptions and approximatior,3 involved, and therefore, difficult to generalize t~eir formalism to a ur.ified theory valid both for ~inite temperatures and stationary states in nonequL1ibrium.

622 670

SU Zhao-bin and CROU Kuang-chao

As a primary check of the "theory of st!ltistical generating functional for the order parameter" [4] , we have rederived the Ginzburg-Landau equation near the critical point [41. In this paper, inspired by Umezawa, Mancini et al., a ,et of interacting weak E.M.F.-order parameter equations for ideal superconductors is derived in the framework of Ref. [4], with its E.M.F.-phase of order parameter part formally identical to that of Ref. [3}. We show that this set of equations is an approximate form of the statistical functional equations for the order parameter f4J • Moreover, the equation for the modulus of the order parameter 1s incorporated into our formalism naturally and statistical information is specified by the parameters of the equations impliCitly in accord with the CTPGF theory. If the intenSity of thp. E.M.F. is not too strong and the superconductor is far from the critical point, these equations are shown to be valid not only for the case of T=O ground state, but also for the case of T~O thermal equilibrium or some nonequilibrium stationary states. All parameters of the equations corresponding to different statistical situations can be calculated by the method given in lief. [4]. In our derivation, a crucial role is played by the gauge symmetry induced Ward-Takahashi CW-Tl identit1es for the vertex generating functional. The general form of the equations follows almost immediately with a transparent physical interprp.tation of our procedure which 1s in some sense the nonrelativistic version of Higgs mechanism for U(l) gauge symmetry wben the Goldstone field has singularities. In Sec.II thp. formulation of the problem is given with the approximations stated explicitly and the general form of the equations for the E.M.F. and the order parameter are derived. In Sec. III a set of coupled E.M.F.~order parameter equations for the ideal superconductor is derived by virtue of the W-T identities generalized to the closed time-path[S]. Finally in Sec.IV, as an example for comparing our results with the known ones and for illustrating the method of evaluating the parameters developed in Ref. r.], we calculate in the long wave-length limit the temperature dependence of the two basic parameters: density of the superconducting electron pairs and phase velocity of the phase excitation of the order parameter.

II. Formulation of the problem and the approximate fo~ of the statistical functIonal equations for the macroscopIc variables For superconductor interacting with the E.M.F., the interesting macroscopic variables are the vector potential of the E.M.F. )l=O. I. 2.3.

(2.1)

and the order parameter of the superconductor

X~X)==.A- (7', (XJti:X, (x I). S IXI exp [-' ® IXI]="Tr{f "'; 01) ~+ O,,} , where

p

is '1:lJenormalized density matrix in Heisenberg picture, AP(x>

(2.2)

being

623 On an Approximate Form of the Coupled Equations of the Order Parameter with the weak Electro~netic Field for the Ideal Superconductor

the Heisenberg operators of the vector potential and electron field operators with up, down spin indices. ventional symbols yields

671

~t(X), ~~(x) - Heisenberg comr .~rison with the con-

A

(2.3)

where d(x) is the energy gap parameter, g the coupling constant of contact interaction for weak coupling superconductor.. We choose gauge condition .; Ca) A/o" == 0

(2.4)

for "the vector potential. In Eq.(2.4), DII(a) is a linear differential operaThe concrete form of DII (3) tor, and "a" is an abbreviation of all = will be determined later. In our paper, we take the metric tensor as

a!1I

;." =

(.!.j.~.; .........), ;

y.,

v= 0, 1,2,3,

(2.5)

-1

. -l and the transformation properties of related variables are defined according to Ref. [6]. There is another variable B(x), the so-called ghost field, needed in the generating functional formalism, which corresponds to a Lagrange multiplier for the gauge condition. In accordance with Ref. [4], introducing the irreducible vertex functional of the macroscopic variables defined along the closed time-path f p =f (A)I(l'J. :::(l'J,@C:r.),8IlLl), we will have the functional equations r satisfied by the macroscopic variables AII(x), X(x), X*(x), B(x)

&Tp(A~.2.®.8JI GA)l,xl

=0

.rre (A~. s.®. 8) G®lXl

I

.t~ .. t-

Ire(A~ s.®. 81 / Ga IX'

srp (A~ s . ® G::\ ,.,

(2.6)

,tt_t-

=

0

(2.7)

=0

(2.8)

0

(2.9)

:t+-.t-

8)

I

;1:'- 1.-

=

Suppose the intensity of the E.M.F. is not too strong and the superconductor is far from the critical pOint. We can neglect the reaction of the E.M.F. to the modulus of the order parameter owing to the finite nonvanishing gap, ::tncl then, assume that 8(1')=

x,t ,

and the equation Eq.(2.9) degenerates into an equation for E.M.F. and with ~(x) taken to be constant

(2.10)

Idl

without the

(2.11)

624 672

SU Zhao-bin and CHOU Kuang-chao

where

=

~.

® . B]

(2.12)

A~= 8=0

$= consl. NOw, the equations for the order parameter interacting with E.M.F. decouple into two parts: the one Eq.(2.11) being the equation for the modulus of the order parameter and the other, Eqs.(2.6)-(2.8), the equations of the E.M.F. vector potential coupled with the phase of the order parameter. We solve the former equation first and then put its solution into the latter one as an input parameter. As the solution of Eq.(2.11) is a problem for the superconductor itself, we will focus on Eqs.(2.6)-(2.8) in the remaining part of this pp.per. Since we have assumed that the intensity of the E.M.F. is not toO strong, we m~y linearize the functional arguments A~(x),~(x) and B(x) linearly in Eqs.(2.6)-(2.8) according to the CTPGF technique given in Ref. [5] as

J-i4-rY {I;.: (x,~)l(V)+I;.; (':':/) C3H~: (X,j'18(t l} = J441 ire: (X"Jl(I Jtf,: rib (~JtfB: (X,YJB(~I} = (ij)

IlI,li

0 ,

(2.13)

0 •

(2.14)

Jd43{r&~ (lq)lclltfb: (lI"J® (1'+f&: (lI"IB(31} =0, where

(2.15)

s'r;cl, ::: . ® .r~ pc... ) .r~

. BJ (I,. )

(2.16 )

'A)I-B=® ... O

s

=s''''t;,"

~

fro

(2;Il)

.t+=t-

are the 2-point retarded vertex functions of the corresponding variables Cij)(x) , B(x) denoted by

(

EC;h

'/~ i IA\I N(O'jdEt1i-LtIt.IEC1! e'(.,! 2

11""[OJ -rls(r]

For T-O as a special case, we have

7

d:C1) \- ECF)

J

(4.14)

(4.15) (4.16)

631 an an Approximate Form of the Coupled Equations of the Order Parameter ~ith the Meak Electromagnetic Field for the Ideal Superconductor

679

J

f.3 ~ rill' nsa= ,;1;3 = 0 -C-21t"";'1I-)T"j

nS [T= Q]

=

11'2[0]\

• =~=_1_1T; 3m

T=D

(4.17) (4.18)

3

where vI" is the electron velocity at Fermi surface. Eqs.(4.17) and (4.18) are identical with tbe known results in literature[7,8] But, we have not found other calculation for the temperature dependence of the phase velocity for tbe Goldstone mode to compare with. It is interesting to note that Eq. (4.15) coincides with the corresponding result derived from Werthamer's extension[9] of Ginzburg-Landau theory. Making the followinr. approximations (4.19) (4.20)

in the sense of dominant contribution to t~e integration jdE(P) diately reobtain the correspondin~ results from BCS tbeory[8]

noS [T]= nso _

'1t/mfl

3

{O~4cl"(_

we imme-

at (f(1'J») dE(PJ

(4.21)

,

where

f (E ci)) =

I -e-Jl""e"",'F=-)-+--

(4.22)

Acknowledgement The autbors wish to thank Prof. CAl Jian-hua and Prof. WU Hang-sheng for calling this problem to theiF attention and are also grateful to Prof. YU Lu and Prof. CHEN Shi-Fang for helpful discussions and a careful reading of the manuscript.

References 1.

V. Ginzburg, L. LaDdau, Zh. B1csp. Tear. Fi.z. 20(19S0), 1064.

2.

P. de aennes, ·Superconductivity of Metals and Alloys·, Benjamin, N.Y. 1966.

3.

See, for example, L. Leplae, F. mancini, B. Umeza~a, Phys. Rep. ~(1974), lSI. If. IfatsWllOto, B. I1IEza~a, Fortschr. Phys. ~(1976), 3S7.

II. Fusco-Gi.rard, F. Hancini, H. Ifari.nuo, Fortschr. Phys. !!..(1980), 3SS. 4.

ZHOU Guan!1'"zhaO, SU Zhao-bin, ~(1982),

5-.

ZHOU Guan!1'"zbao, SU Zhao-bin,

et 081.,

HAD Bai-lin and YU Lu, COJlllllun. in Theor. Phys-., (Beijing).

295,307,389. Ch. S. in ·Progr. in Statistical Physics", eds. Hao Bai-lin

~XUB (Sci.e~ce)Press , Beijing, 1981.

IIcGra~-Hill,

6.

J. Bjorksn, S. Drell, "Relati.vi.stic Quantum Fields",

7.

J. Schriefer, -Theo.zoy of SUperconductivi.tyN, Benjamin, Reading, Hass. 1964, and references

8.

See,

N.Y. 1981.

tbllrei.n. for ezample, A. Fetter, J. Walacka, NQuantum Theory of Ifilny-Particle Systems",

lie Gr__Bi.ll, N.Y. 1911.

9.

N. JII'erthamu', Ph'JS. Rev • .ill,(1963), 663.

N. Wert~r, i.n ·SUpercollductivit'J- Vol.l

filii. R. Parb, lIarcel Dek1Iar, INC., N.Y. 1969.

632 COIIIIIIW2.

ill Theor. Phys. (Beijing, Chilla)

Vol. 2, No.4 (1983)

1181-ll89

ON ADYNAMIC THEORY OF QUENCHED RANDOM SYSTEM SU Zhao-bin ( $~hlc. ), YU Lu (f and ZHOU G1lang-zhao ( }I]!?l )

*)

Illstitute of Theoretical Physics, Academia Sillica, Beijing, China

Received March 10, 1983

Abstract A dynamical theory for quenched random system is developed in the framework of C'l'PGF.

In steady states the

resu.~ts o~

tailled coincide with those following from the quenched average of the free ener!1!l.

The order parameter q, a matrix in general,

becomes an integral part of Lhe second order connected CTPGF. An equation to determine

q

is derived from the Dyson-Schwinger

equation ill t'l.is formalism.

Some general properties of the

C'l'PGF in a quenched random system are discussed.

I. Introduction In quenched random systems, part of th~ degrees of freedom describing impurities are froze·, =nLO a nonequilibrium but random configuration. This could be accomplished by sudden cooling of a sample in thermal equilibr~um to a state with much lower temperature. The impurities are then frozen into a configuration separated by high potential barriers from an equilibrium one. Diffusion throqgh the potential barriers will cause the nonequilibrium state to vary very slowly in time. As pOinted out by Brout[l], the space average of an observable A in a quenched random system can be replaced by the ensemble average over the impurity degrees of freedom J.

A fA (J) p(J)dJ

(1)

where P(J) is the distribution function. Most of the previous workers~] considered quenched ra:ldom systems as if .• they were static. In this approach one has to evaluate quenched average of the free energy which is proportional to the logarithm of the partition function. It is a formidable task and an enormous machinery of n-replica method is introduced. This method has been applied extensively to systems like spin glass[3-l0J. Recently., several authors [11-16] have proposed dynamiC theories of quenched random systems in the study of spin glass b~sed on the MSR statistical field theory [17]. The advantage of the dynamic theory is that it provides means for averaging out the quenched randomness without using the unphysical replica trick. The results obtained so far can be reproduced. by the replica method with special pattern of replica symmetry breaking, which is itself a s;atic theory [18]. Therefore the full content of the Qynamic theory is still

633 llS:J

SU Zhao-bin, YU Lu and ZHOU Guang-zhao

waiting to be uncOvered. The aim of the present paper is to establish a dynamic theory for the quenched random system using the closed time path Green's function method (CTPGF) [19J. CTPGF is a very general method especially suited to study slowly varying nonequilibrium processes. In it are incorporated automatically causality and fluctuation dissipation theorem (FDT). The order parameter q introduced by Edwards and Anderson ~J appears naturally in the second order Green's function. The new result obtianed in the present paper is a DysonSchwinger equation for the order parameter q. For slowly varying processes it is sufficient to use semiclassical approximation, the one employed in the transport equat·ion. In this way a differential equation that describes the time evolution of the order parameter is obtained. In this paper only the general properties of CTPGF and the Dyson-Schwinger equation are studied. Application to long range quenched ISing model will be presented in a subsequent paper. The paper is organized as follows: In Sec.II we introduce CTPGF for a quenched random system. It is proved that as the system approaches equilibrium, there exists a free energy which is the quenched average of the free energy with fixed random degrees of freedom. In Sec.III a Dyson-Schwinger equation for the order parameter q is deduced and simplified in the semiclassical approximation, Sec.IV contains concluding remarks.

II, CTPGF for a Quenched random system We shall use in the following those symbols and the language adopted in the theory of CTPGF without further explanation. The unfamiliar ~eaders are referred to Ref. [19J. Suppose the dynamical field variable of our system is a(x). The action on a closed time path P has the form

1= ~c&d~O"(X) r;o.DtX:'~llfl,)-~d1V((f"(X),Ji) ,

.

1

+ d.~((l't11;(U.) +(flXljlX»+ Ihe«t

rt,SlI.YVOi,..

(2.1)

'

where hex) is the external field; Jl are random variables with given distribution functions. The a(x)j(x) term represents the interaction of the dynamical field with the reservoir conSisting of a set of harmonic oscillators for instance. If there are more than one dynamical f~elds, a(x) should be considered as a vector with many components. We shall use Path integral to evaluate the generating functional of CTPGF. After integrating over the field variables describing heat reservoir, we get the averaged generating functional

~[~(X)]-fptJJ ~(~(XI,J]dJ' with

=< ~['fi,JJ >J

(2.2)

:i:.[RlXI,JJ=J[dtt]e.iIeft ,

Left ....~ 0"(X)r,'(x.,)1tt3)did~ -~ VCQ"lX7. Ji}d~ +~ltlXl~(Xld~

(2.3)

.

(2.4)

634 on a D!/namic 2'heoZ'll of Quenched Random S!/st:em

ll83

The system is supposed to be prepared at time t=to by suddenly cooling to the temperature of the heat reservoir. In Eq.(2.4) the closed time path starts from t=to to t=+m (positive branch) and runs back from t=+m to t=to (negative branch). r~O)(x-y) is the second order vertex function obtained after integrating over the reservoir degrees of freedom, i.e.,

with the self-energy part r.jO) (x, y) determined by the interaction a (x)j (x) with the reservoir. It is easily proved that r p(o) satisfies the FDT

.......(0'li)=Lcth. ~t. IIII rill r ell T

r,

(2.5)

Here and riO) are the correlation and the retarded vertex function, respectiv.ely. Introducing the generating functional for the connected.CTPGF

and

Z [{Ill] = e.x.p{ i W[ il~lJ}

(2.6)

~(!lXl, j] = up{iW[i(:(I, J]}

(2.7)

it is possible to obtain the connected CTPGF by direct differentiation. have the averaged field the connected CTPGF

~IX)= O/LIX) ~?

-Go

(2.8)

S"ijJ

(2.9)

lX, '" ••• XII ) = -=-~:":""":o-r-P I Sllt,I ..... -S(Xnl

and the corresponding ones from

~ (f(:tl=

We

W[h(x),J].

Eq.(2.2) implies that

:r

(2.10)

It is a very important property of Z[h,J} and Zlh(x)} that they are equal to unity in the physical limit when the external field hex) on the positive branch is identified with that on the negative branch. Therefore, the observed field

(j"lX.>=( (JllC.:r):>

(2.11)

J.

satisfying the requirement for a quenched average. Differentiating Eq.(2.10) with respect to hey) in the physical limit, we obtain

Gp (:t. !'+i(fIXlO=(~1 = or

and setting

h(x+)=h(x_)

(GplX.S "T>+i(f(x. J) If(~. :r) \

GplX.,9 1=( G-plX,

~j

J)

>.r+l9.(X:,~)

with the matrix

'l(X, B)=«T"lx.;n(f"(~,J) ~ -

(fIXJ (f'9'

(2.12) (2.13) (2.14)

Edwards and Anderson have defined an order parameter in spin glass

~= Lim «(flO.J"Jlrlt.J"J>

t...,...

(2.15)

635 1184

SU Zhao-bin, YU Lu and ZBOU Guan..,-zi2ao

which is closely related to the matrix q(x,y) deduced in Eq.(2.14). For hermitian field operator o(x) its average a(x,J) is a real function identical on the two tim~ pranches in the physical limit. Hence the matrix q(x,y) is real, symmetrj and equal on the two branches

~olt t'>Ojlt')dtdt'~(LlOi"'lt>+ili(tl6"ilt»dt } +{ J :L

4N

(2.4)

~.I6i(tl OJ'(t)dti~rt'i6j(f)dt' ,..) p p L

The notations used here are essentially those in Ref.[IS]. Howeve~, the bar over the quenched average physical variables has been dropped for simplicity. t+t' Any matrix A(t,t') can be represented by its Fourier transform A(w,-:r-) in the relative time t-t' where (2.5)

In this notation system the low frequency approximation for

,..... f';.lUl, t)= (-Yo+iw/ro )

and

'[ tW, t)=i where

S-l

Co

;-r. " •

ro

has the forms e2.6)

(2.7)

is the temperature of the reservoir. In the infinite range limit with N+~ the matrix Gijet,t') could be approximated by 6i j G(t,t·). In this case the second order vertex function can be calculated with the diagram expansion. It is found that

644 ll94

SU Zhao-bin, YU Lu and ZHOU GUang-zhao

(2.8) where ~ time t.

and r are renormalized quantities that could be functions of the To lowest order perturbation in u we get (2.9)

where J is also renormalized and q(t,t) is the order parameter. In obtaining the second term in Eq.(2.9) we make the approximation that q(w,t) has a sharp peak at waO. In Eq.(2.8) tr(w,t) is the self-energy part with the first two terms in the expansion of w and the term proportional to Or is excluded. They are included in the first three terms in Eq.(2.8). Therefore, we have (2.10)

To the same order of approximation we have calculated the vertex function "...." .. 2.."""'" ~ 2 ,...., «.11, t)= L (lUJ ry{W,t) +LJc. l tt,>'l.-=*fd~ iT se.c.h q.~ (~K, +n.) ~. ",2JT

(5.1)

-00

This equation can also be derived from our formalism. Therefore, the thermodynamic properties predicted by our theory will agree with those by projection hypothesis. Comparing our results with those of Sompolinsky ~OJ it 1S natura! to conjecture that the function q(x), OsxSl corresponds to our q(t,t)· varying along the stability boundary from q1 to qo. Not only does the present theory give a clear physical meaning to the function q(t,t) but also an equation that can be solved explicitly for the time evolution of the order parameter q(t,t). The present formalism can be applied to other quenched random systems, which we hope to present in subsequent publications.

651 1201

A Dynamical Theory of the Infinite Range Random Ising MOdel

References ~

1.

S.F. Edwards and P.W. Anderson, J. Phys.,

2.

D. Sherrington and S. Kirkpatrick, Phgs. Rev. Lett., 35 (1975) 1972.

3.

T.R.L De Almeida and D.J. Thouless, J. Phgs.,

4.

E. Pytte and J. Rudnick, Phys. Rev.,

5.

!!2

(1975) 965. ~

(1978) 983.

(1979) 3603.

G. Parisi, Phys. Lett., A73 (1979) 203; Phgs. Rev. Lett., 43 '1979} 1574; J. Phys., Al3 (1980) lBB7. ~

6.

S.K. Ha and J. Rudnick, Phys. Rev. Lett.,

7.

C. De DOminicis, Phys. Rev.,

B.

J.A. Hertz and R.A. Klemm, Phys. Rev. Lett.,

9.

H. Sampolinsky and A. Zippelius, Phys. Rev. Lett., 47 (19Bl) 359.

edited by C.P. Enz

(S~inger

!!!

(197B) 5B9.

(197B) 4913; Lecture notes in Physics, V.I04 p.253,

Verlag, Berlin, 1979).

.£. '(HB1)

~

10.

H. Sqmpolinsky, Phys. Rev. Lett.,

11.

P.C. Hartin, E.D. Siggia and H.A. Rose, Phys. Rev.,

12.

H.J. Sqmmers, Z. Physik,

~

i!

(197B) 1397;

(19B1) 496.

935.

(197B) 301; ibid

~

~

(1973) 423.

(1979) 173.

13.

C. de DOminicis, M. Gabay and H. Orland, J. Phgs. Lett.,

14.

G. Parisi and G. Toulouse, J. Phys. Lett., 41 (19BO) 361; J. Vannimenns, G. Xbulouse and G. Parisi, J. Physique,

15.

~

~

(19Bl) r.-523.

(HB1) 565.

SU Zhao-bin, YU Lu and ZHOU Guang--zhao, HOn a dynamic theory of quenched random system'·, Commun, in Theor. Phys., this issue.

652 474 A Dynamical Theory of Random Quenched System and Its Application to Infinite-Ranged Ising Model* Su Zhao-bin

Zhou Guang-zhao

Yu Lu

(Institute of Theoretical

Physi~s,

Beijing, China)

Abstract A dynamical thoery for quenched random systems is developed in the framework of the closed time-path Green's functions (CTPGF).

The order parameter q, a matrix in general,

appears naturally as an integral part of the second order connected CTPGF.

An equation to determine q is derived from the

Dyson-Schwinger equation.

The formalism developed is applied

to the study of the long-ran!l'ed random Ising model. dary line is found on the

q-I~I

plane.

A boun-

It is argued that the

spin-glass phase is characterized by the fixed point lying on the stability boundary.

The magnetization is calculated in

perturbation and is found to be in good agreement with those predicted by the

projection hypothesis.

The general validi-

ty of the projection hypothesis is discussed.

653 475 I.

Introduction

Much progress has been made in recent years on the understanding of the spin-glass (SG) phase in magnetic systems with infinite-ranged random exchange.

A mean field theory (MFT) with order parameter being a

continuous function q(x)JO~x~l, has been derived by use of a parti' 1 approach 1-10 · cu 1 ar sch erne 0 f rep 1~ca synunetry b reak'~ng or a d ynam~ca The MFT is free of instabilities although the physical meaning of the order parameter q(x) and the origin of the apparent violation of the fluctuation-dissipation theorem (FDT) are still under intensive investiga11-13 ' t ~on .

On the other hand, a drasti,ally simple extrapolation pro-

cedure projecting physical properties from the marginal stability line onto the SG phase has been shown by Parisi and Toulouse all the nice features coming from the MFT.

~onte-Carlo

14

to reproduce

simulation and the

However, a theoretical explanation of this very simple and elegant

projection hypothesis is still lacking. In this note we investigate properties of the infinite-ranged random Ising model in functions (CTPGF) 15.

the framework of the closed time path Green's The order parameter, a matrix q(t,t') in general,

appears naturally as a part of the second order connected CTPGF which satisfies the Dyson-Schwinger equation.

The FDT is assumed to be sa-

tisfied by the CTPGF before averaging over the random exchange. Fischer's relation is found to be violated in the SG phase

16

The The

validity of Fischer's relation depends on the existence of the FDT and the use of a unsubtracted dispersion relat10n for the retarded CTPGF. It is more likely that a subtracted dispersion relation is necessary on the stability boundary and in the unstable

,

I

reg~on.

We believe, this

subst raction rather than the violation of the FDT is the real cause for the breakdown of the Fis cher' s relation in the SG phase. In the present formalism the dynamical behavior and the static properties of the order parameter q(t,t') are determined completely by the Dyson-Schwinger equation which can be solved approximately for small q in the low freq uency limit. played more clearly on the qceptibility of the system.

The physical picture obtained can be dis-

'X'

plane where:t

is the magnetic sus-

& physical boundary is found on this plane.

654 476 Above Tc,the Fischer relation, a line on the q- I{f

plane, lies comple-

tely inside the stable region and the order parameter q tends exponentially in time to its fixed point q=qo.

Below T c , the Fischer line intersects the boundary at the point q=q, , which is a fixed point only when the external magnetic field h=h c . In case h< he' after reaching q=q. along the Fischer line, the order parameter q will decay further along the boundary down to its fixed point q=q..

In a rough apprixima-

tion the decay along the boundary is found to obey a power law.

In this

formalism the physical boundary on the q-I{1 plane is shown to be temperture independent.

Therefore, the fixed pOint on- the boundary characterizing the

SG phase is temperature independent. As a consequence, the magnetization M(h) is also temperature independent and the entropy is independent of the external magnetic field.

This is just the assumption of the projection hy-

pothesis which follows naturally from our theory. The rest of the paper is organized as follows: In Sec. II the general properties of CTPGF for a random quenched system are presented. The stability condition and the physical boundary are discussed in Sec.III for the infinite-ranged random Ising model.

In Sec.IV the susceptibility

is calculated for small external magnetic field and is compared with what follows of the

from the projection hypothesis.

The dynamical evolution

order parameter is also briefly discussed.

The final Sec.V

contains some concluding remarks. II.

CTPGF for a Quenched random sYstem For the sake of simplicity we study the soft-spin version of the

Edwards-Anderson SG model defined by the Hamiltonian

where the interaction Jij are random Gaussian variables with zero average and mean square fluctuation J2 IN, N being th.e number of the neighbors. The generating functional of

C~PGF

with the interaction kept fixed

can be represented by a path integral in the following form

655

where

1\

and

f

is the density matrix.

from t-=to to t=

00

In Eq. (2.3) the closed time-path starts

(positive branch) and runs back from t= oc

t=to (negative branch).

to

rpo (x-y) is the second order vertex func-

tion obtained after integrating over the reservoir degrees of freedom. It satisfies the FDT with the temperture of the reservoir

(2.4) where

reO) (e

and

,(0)

I

r

are the correlation and the retarded vertex functions

respectively. The advantage of using CTPGF for random systems is that the quenched average can be performed directly on the generating functional instead of its logarithm.

This is possible owing to an important property of the

:.!enerating functional Z[h,J ij] , namely, it equals unity in the physical limit when the external magnetic field hex) on the positive branch is identified with that on the negative branch. Introducing the averaged generating functional (2.5) it is possible to calculate the connected CTPGF by a direct differentiation.

Eq.(2.5) then implies that

(2.6) In the physical limit both Z and

Z equal

unity and the observed magneti-

zation satisfies the requirement of a quenched average (2.7)

656 478 Differentiating Eq.(2.6) w.r.t. hex) and setting h(X+)=h(x-) we obtain (2.8) where the order parameter matrix

is real, symmet ric and equal to each other on the two branches, i. e. , (2.10)

From Eqs.(2.8) and (2.11) follow the retarded, the advanced and the correlated Gre.en's fLlnctions

G-y,j(t,t') -

G(l.0 fc,

t')

< GrrL/(t,t;J'j):;-, O

the matrix Gij(t,t')

In this case the second order vertex

function can be calcul.ated without difficulty in a diagrammatic pansion.

It

where r and

ex-

is found that

tare

and the time t.

renormalized quantities that could depend on q,

(5"

In the limit where J tends to zero r is the inverse of

the magnetic susceptibility which is proportional to the temperature and increases as the magnetization E) increases.

We shall assume that r will

keep this qualitative behavior even in a random spin system.

In Eq.(3.3)

~

~(oJ,t)

sion of

is the self-energy part with the first two terms in the expan~

and the term proportional to Gr being excluded.

Therefore,

we have (3.4)

66Q 482 To the lowest order perturbation in u we find (3.5) where J is also remorma1ized and q=q(t,t) is the order parameter. obtaining the second

,..,

In

term in Eq.(3.5) we have made the approximation

that q (LJ, t) has a sharp peak at

to =0.

To the same order of approximation we find the correlated vertex function

z."-

(3,6)

"'-

-t iJ L!:.fw,t) - LC(W,"t)J where~(tc..),t)

is defined to be "-

LJ(fAJ,t)

( i c,J 'l:

= Jr,l.7:e

r.-

criti-f-)uft-;)

(3.7)

~

which is also sharply peaked atW =0.

In Eq.(3.6)

.Lc is the remaining

self-energy part that does not have a sharp peak at W

=0.

? J~(q)

can be

easily calculated in perturbation to be (3.8)

In the low frequency limit the matrix Q defined in Eq.(2.27) has therefore the form

(3.9) Here only terms that have a sharp peak at ~ =0 are retained. It is noted 2 2 that Jr(q) is guater than Jc(q) for all values of q. This fact is very important in the following. In the zero frequency limit the Dyson-Schwinger equation for the retarded Green's function[Eq.(2.24Uhas the form (3.10)

£;361 483 where

..t =Gr(o,t)

is the susceptibility of the system.

This equation

can be solved to give

(3.11) The susceptibility increases as r decreases and reaches its maximum at -V-I "y r=2J r (q) where Il. =Jr(q)· Further decrease of r will make '\. complex and the system unstable.

Therefore, the stability region is bounded by

the inequality

(3. 12)

It is easily seen from Eq.(3. 11) that in the unstable region

(3.13)

which is ;] curve on the are

q-Ill

plane.

On this plane all stable points

situated in a region bounded from above by the curve [Eq.(3.l3)]

consisting of marginally stable and unstable points. region q and

.{

In the stable

are related by the Fischer relation.

Hence the physical

state of the random system can only evolve either along the Fischer's line when it is stable, or along the boundary Eq. (3,13) when it is margina] l.y stable or lmstable.

Before turning to the next section let us briefly mention the low frequency behavior of the retarded Green's function.

For this purpose

write

where

ot.. (t)

and

J}

are to be determined.

An analysis similar to that

given in Ref.9 shows that )J~ 1/2 i f the state is marginally stable and )} =1 otherwise.

662 484 IV.

Susceptibility and the order parameter q in small external magnetic field In the fOllowing we shall take the value of u to be 1/12 in the

units J=T c =l. From Eqs.(2.28) and (3.7)-(3.9) the static fixed point for q is determined by the equation (4. 1)

where qo=Lim q(t,t). Above Tc , the whole Fischer line lies inside the stable region and we have

(4.2) For small external magnetic field we can solve Eqs.(4.1) and (4.2) for qo

() 2. + ...

(4.3)

The susceptibility is therefore

(4.4)

==

B

r-

-L -/L2.+. 1-(3'"

At the critical temperature Eq.(4.2) is still valid.

In this case

we have

..90 == l5

!L _ .!.J.. .ft

24-

(J l.

+

(4.5)

and

1

(4.6)

663 485 Below Tc. there exists a critical external field h c • above which the static fixed point in still lying in the stable region. The critical field hc can be calculated and is found to be

n: i =

nearTcwherer=

,£:$

(I-t 3

z: + .'. )

(4.7)

1-1j1·

For T p(n _1)2 p (n _2)2 p

for

p

n >2 p

and for all et .

Therefore the free energy F' is smaller than that of Parisi's. Hence we conclude that Parisi's solution

is

not situated at the absolute minimum of the

free energy for n replicated systems with n an even integer greater than 4.

References 1. D.Sherrington and S.Kirkpatrick, Phys. Rev. Lett.,

~(1975)1972;

D.Sherrington and S. Kirkpatrick. Phys. Rev., 817(1978)4385. 2. A.J.Bray and M.A.Moore, Phys. Rev. Lett¥ 3. G.Parisi, J. Phys.,

~(1980)L115;

A13(1980) 1887; Phil os.

~(1978)1068.

J. Phys.,

~(1980)403;

Mag.,~(1980)677;

Phys. ReP., 67(1980) 25. 4. C.De Dominicis and I.Kondor, preprint. 5. F.pytte and T. Rudnik, phys. Rev.,

~(1979)3603.

682 PHYSICS REPORTS (Review Section of Physics Letters) 118, nos. I & 2 (1985) 1-131. North-Holland. Amsterdam

EQUILmRIUM AND NONEQUILmRIUM FORMALISMS MADE UNIFIED Kuang-chao CHOU, Zhao-bin SU, * Bai-lin HAO and Lu YU Institute 0/ Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing, China Received 5 June 1984

COfIIents: 1. Introduction 1.1. Wby closed time-path? 1.2. Few historical remarks 1.3. Outline of the paper 1.4. Notations 2. Basic properties of CfPGF 2.1. Two-point functions 2.2. Generating functionals 2.3. Single time and physical representations 2.4. Normalization and causality 2.5. Lehmann spectral representation 3. Quasiuniform systems 3.1. The Dyson equation 3.2. Systems ncar thermoequilibrium 3.3. Transpon equation 3.4. Muhi-time-scale perturbation 3.5. Time dependent Ginzburg-Landau equation 4. TIme reversal symmetry and nonequilibrium stationary state (NESS) 4.1. TIme inversion and stationarity 4.2. Potential condition and generalized FDT 4.3. Generalized Onsager reciprocity relations 4.4. Symmetry decomposition of the inverse relaxation matrix 5. Theory of nonlinear response 5.1. General expressions for nonlinear response

3

6

7 7 12 18 24 28 31 31

34 39 44 46

48 49 52 53 55 58 58

5.2. General considerations concerning multi-point functions 5.3. Plausible generalization of FDT 6. Path integral representation and symmetry breaking 6.1. Initial correlations 6.2. Order parameter and stability of state 63. Ward-Takahashi identity and Goldstone theorem 6.4. Functional description of fluctuation 7. Practical calculation scheme using CfPGF 7.1. Coupled equations of order parameter and elementary excitations 7.2. Loop expansion for vertex functional 7.3. Generalization of Bogoliubov-de Gennes equation 7.4. Calculation of free energy 8. Quenched random systems 8.1. Dynamic formulation 8.2. Infinite-ranged Ising spin glass 8.3. Disordered electron system 9. Connection with other formalisms 9.1. Imaginary versus real time technique 9.2. Quantum versus fluctuation field theory 9.3. A plausible microscopic derivation of MSR field theory 10. Concluding remarks Note added in proof References

• Current address: Depanment of Physics, City College of New York, New York, NY 10031, U.S.A.

62 67 70 71 76 79 82 Il9

90 92 96 99 103

104 109 114 119 119 123 125 127 128 128

683 KlMlng-chlJO Chou tl aI., Equilibrium and nonequilibrium fonna/isms mcuk unifitd Absll'acl:

In this paper we summarize the work done by our group in developing and applying the closed time-path Green function (CfPGF) formalism, first suggested by J. Schwinger and further elaborated by K.eldysh and others. The generating functional technique and path integral representation are used to discuss the various properties of the CfPGF and to work out a practical calculation scheme. The formalism developed provides a unified framework for describing both equilibrium and nonequilibrium phenomena. It includes the ordinary quantum field theory and the classical tluctuation field theory as its limiting cases. It is well adapted to consider the symmetry breaking with either constituent or composite order parameters. The basic properties of the CfPGF are described, the two-point functions are discussed in some detail with the transport equation and the time dependent Ginzburg-Landau equation derived as illustrations. The implications of the time-reversal symmetry for stationary states are explored to derive the potential condition and to generalize the tluctuation~issipation theorem. A system of coupled equations is derived to determine self-consistently the order parameter as well as the energy spectrum, the dissipation and the particle distribution for elementary excitations. The general formalism and the useful techniques are illustrated by applications to critical dynamics, quenched random systems, theory of nonlinear response, plasma, nuclear many-body problem and so on.

1. Introduction

1.1. Why closed time-path? The field-theoretical technique, introduced into the many-body theory since the late fifties, has proved to be highly successful in studying the ground state, the thermoequilibrium properties and the linear response of the system to the external disturbance [1-3]. However, only limited progress has been made in investigating the non equilibrium properties beyond the linear response by using the fieldtheoretical methods. To appreciate the difficulties encountered here, let us recall some basic ingredients of the field-theoretical approach. The Green function is defined as an average of the time ordered product of Heisenberg field operators over some state which we do not specify for the moment, i.e.,

(1.1) By introducing the interaction picture, (1.1) can be rewritten as

(1.2) where the S matrix is defined as

S == U(C1J, - C1J)::: T exp(-i

J,rt'ln.(t)

dt) ,

(1.3)

with the interacting part of the Hamiltonian 1t'~n.(t) in the interaction picture. If we are interested in the ground-state properties, then

(1.4) where L is a phase factor contributed by the vacuum fluctuations and can be set equal to zero, if the renormalized ground state is considered. Therefore, we can easily get rid of st in (1.2) so that the powerful arsenal of the quantum field theory can be used without major changes in the many-body theory at zero temperature.

684 4

KlUJlIg-chao Owu el al., Equilibrium and lIoM/uilibrium formalisms made unified

For systems in thermoequilibrium at different from zero temperature, we cannot relate observable quantities directly to the elements of the S matrix, but the density matrix in this case take the following form:

p= exp[J3(9" -

(1.5)

k)l,

where fJ is the inverse temperature, ~ the free energy, k the Hamiltonian. If we consider fJ as an imaginary time iI, p behaves like an evolution operator exp(-ikl). The well-known Matsubara technique [4-7] has been successfulIy developed by making use of this property. However, it is not easy to handle the st term in (1.2), if a general nonequilibrium state is considered. An intelligent way out was suggested by J. Schwinger in 1961 [8]. Let us imagine a time-path p which goes from -00 to +00 and then returns back from +00 to -00. We can then define a generalized Sp matrix along this closed time-path (as we call it)

Sp == Tp exp{ -i

I ~inl(l)

(1.6)

dt} ,

p

where Tp is the time-ordering operator along this path p. It is identical to the standard T operator on an anti-time-ordering operator on the negative branch the positive branch (-00, +00) and represents (+00, -00). Also, any point at the negative branch is considered as a later instant than any time at the positive branch. Equipped with such generalized Sp matrix we can define the Green function along the closed time-path p as

t-

Gp(lh (2 ) = -i(Tp(A,(tl)B(12»

= -i(Tp(A,I(t l )B1(12 )Sp».

(1.7)

Although for physical observables the time values II, t2 are on the positive branch, both positive and negative branches will come into play at intermediate steps of calculation if a self-consistent formalism is intended. The introduction of the closed time-path appears at the first glance as a purely formal trick to restore the mathematical analogy with the quantum field theory. Actually, it has deeper motivation. In particle physics, people are mostly interested in scattering processes for which the S matrix providing the probability of transition from the in-states to the out-states, is the most suitable framework. In statistical physics, however, we are mainly concerned with the expectation value of physical quantities at finite time I. It is thus natural to introduce the Sp matrix along the closed time-path p going from the state at -00 along t-axis to the +00 state and returning back to the -00 state (see st in (1.22». This way we can establish a direct connection of Sp with observable quantities. As we will see later, the great merits of the closed time-path Green function (CfPGF) formalism more than justify the technical complications occurring due to the introduction of the additional negative time branch.

1.2. Few historical remarks After Schwinger's initiative in 1961 [8], the closed time-path formalism has been elaborated and

685 5

developed further by Keldysh and many others [9-19]. Some people used to call it Keldysh formalism. For the recent 20 years, this technique has been used to attack a number of interesting problems in statistical physics and condensed matter theory such as spin system [20], superconductivity [21-24], laser [25], tunneling and secondary emission [26-32], plasma [33,34], other transport processes [3~38] and so on. For some of these systems like laser, the application of the CfPGF formalism is essential because the standard technique cannot be used directly for far from eqUilibrium situations, whereas for some of the others the CTPGF approach is used mainly due to its technical convenience. It is our impression, however, that the potential advantages of this formalism have not yet been fully exploited, partly because of its apparent technical complexity. For the last few years we have combined the generating functional technique and the path integral representation, widely used in the quantum field theory [39], with the crPGF approach and have developed a unified framework to describe both equilibrium and nonequilibrium systems with symmetry breaking and dynamical coupling between the order parameter and the elementary excitations [40-49]. To check the formalism developed and to explore its potentiality we have applied it to a number of problems including critical dynamics, quenched random systems, nonlinear response theory, superconductivity, laser, plasma, nuclear matter, quasi-one-dimensional conductor, and so on [~57]. Although most of these problems in principle can be also discussed using other techniques, the logical simplicity and the flexibility, the unified approach to eqUilibrium and nonequilibrium processes as well as the deep insight one can get make the CI'PGF formalism promising and encouraging.

1.3. Outline of the paper In this paper we would like to summarize some of the results obtained by our group in developing and applying the crPGF formalism. Because of the limitation of space we will only outline the main features along with some useful techniques of the CfPGF approach and illustrate them by few examples. Since the major part of our papers was published either in Chinese or in not easily accessible English journals, we will attempt to make this article self-contained as much as possible. Nevertheless, we should warn the reader that some part of this review is still descriptive and sketchy. A brief summary of the CfPGF formalism was given by us earlier [58], but this paper is much more extended and complete. Since we are mainly summarizing our own results, the contributions of other authors in developing and applying the CI'PGF approach may not be emphasized as they should be. We apologize to them for any possible omissions or underestimates. To keep the integrity of presentation we will not distinguish carefully what was known before and what is new. The topics to be covered in this review can be seen from the table of contents. We will not repeat them here. Few remarks, however, are in order. Section 2 is mainly tutorial, but the subsection on normalization and causality is important for further discussion. Section 3 is devoted to a detailed discussion of the two-point functions. The differentiation of the micro- and macro-time scales described there is very useful. In section 4 the potential condition and the fluctuation-dissipation theorem (FDT) are discussed from a microscopic point of view. The theory of nonlinear response which may be important for future applications is outlined in section 5. Section 6 is devoted to the consideration of the symmetry breaking and the Ward-Takahashi (WT) identities. We believe that the CTPGF formalism is advantageous in studying systems with broken symmetry. We also mention there the additional way of describing fluctuations available in the CI'PGF approach. The unified framework of treating the dynamical coupling between the order parameter and the elementary excitations mentioned before, is given in section 7. We could start from this formalism at the very beginning, but the present more

686 6

XIIIJng-chao Chou tt al., Equilibrium and nonequilibrium formalisms made unified

inductive exposition is probably more convenient for the reader_ In section 8 we show that the quenched average may be carried out directly on the generating functional in the CfPGF formalism and the replica trick can be thus avoided. The connections with other formalisms are described in section 9. Readers, familiar with them might have a look at this section before the others. An experienced and busy reader could get a rough idea about the CfPGF approach by a quick scanning of sections 2 and 6-9.

1.4. Notations Throughout this paper we will use the units II = kB = C = 1 except for few paragraphs where the Planck constant II is written out explicitly to emphasize the quasiclassical nature of expansion. The metric tensor we use is given by (1.8)

with the scalar product and the d' Alembertian (1.9)

defined correspondingly. The Fourier transformation with respect to the relative coordinates x-y is defined as ] X+ y G(x,y)=G [-2-'x- y =

J(27rY+l

dd+lp.

• (X + Y

)

exp[-IP'(X- y)]G -2-'P ,

(1.10)

where d is the space dimension and p' X = Pot- p' r. The tilde "." will be omitted wherever no confusion occurs. The formalism presented in this paper can be applied to a broad class of fields including non-Abelian gauge fields, but in most cases we will illustrate it by a real boson field, either relativistic or nonrelativistic (e.g. phonons), and a nonrelativistic complex boson or fermion field. The former will be denoted by iP(x), whereas the latter by ~(x) and ~t(x). Wherever a double sign ± or =+= appears, the upper case will always correspond to the boson field, while the lower one corresponds to the fermion field. As a rule, the field operator is not distinguished by the caret "',, which itself is used in some cases to denote a two-component vector or a 2 x 2 matrix. Also, for simplicity we introduce an abbreviated notation for integration Jq;=

J

J(x)q;(x) =

J

ddxdtJ(x)q;(x).

(1.11)

The form at the right is used only in exceptional cases, while the middle one most frequently. The Pauli matrices are defined as (1.12)

687 Kuang·choo C1IOII et al., Equilibrium and noneqllilibrium formalisms made llllified

7

2. Basic properties of CTPGF As mentioned in the Introduction, this section is mainly tutorial. To get familiar with the concepts and notations used in the crPGF formalism we start from two-point functions (section 2.1) in close contact with the ordinary Green functions. We then define the generating functional and discuss the perturbation theory (section 2.2). The single time representation and the physical representation as well as the transformation from one to another are discussed in section 2.3. Furthermore, the consequences of the normalization and the causality are outlined in section 2.4. Finally, the Lehmann spectral representation is described in section 2.5.

2.1. Two-point functions The two-point Green functions are most useful in practical applications and hence their properties have been most thoroughly investigated. In this section we first define the two-point crPGF and then discuss their connection with the ordinary retarded, advanced and correlation functions along with the causality relations. As we will see later, they are special cases of much more general relations following from the normalization condition of the generating functional and the causality. The explicit expressions will be given for free propagators in thermoequilibrium systems.

2.1.1. Definition The two-point CfPGF for a complex field I/I(x) is defined as G(x, y) l5 -i Tr{Tp(I/I(x)I/It(y»p} l5 -i(Tp(I/I(x)I/It(y») ,

(2.1)

where I/I(x),I/It(y) are Heisenberg operators, p the density matrix, Tp the time ordering operator as discussed in the Introduction. Inasmuch as x, y can assume values on either positive or negative time branches, G(x, y) can be presented as a 2 x 2 matrix (2.2) with

Gp(x, y) == -i(T(I/I(x)I/It(y»).

(2.3a)

G+(x, y); +i(I/It(y)",(x»,

(2.3b)

G_(x, Y)l5 -i(I/I(x)I/It(y»,

(2.3c)

GtI(x, y)= -i(T(I/I(x)I/It(y»).

(2.3d)

Here GF is the uSU'al Feynman causal propagator, whereas the other three are new in the CfPGF formalism. Sometimes G,. defined as expectation value of anti-time-ordering product, is called anti-

688 8

Kuang·chao Chou tt al., Equilibrium and lIonequilibrium formaJums made unified

causal propagator. Using the step function 1, 9(x, y) = {0,

if t" > ty otherwise ,

(2.4)

eqs. (2.3a) and (2.3d) can be rewritten as G~x, y) =

-i9(x, y)(I/I(x)I//(Y») + i9(y, x)(I/It(Y)I/I(x») ,

Gr:(x, y) = -i9(y, x)(I/I(x)I/It(y») + i9(x, yXl/lt(Y)I/I(x»).

(2.5)

These four functions are not independent of each other. There is an algebraic identity or

(2.6) G~x, y)+

Gt;(x, y)= G+(x, y)+ G_(x, y),

folIowing from the normalization of the step function 8(x, y)+ 9(y, x) = 1.

(2.7)

In what folIows we will call CfPGFs defined by relations like (2.3) as "single time" representation and denote them by a tensor G ajl ••. p (12 ... n) with Greek subscripts a, /3, ... ,p = ±. As a whole, the tensor itself is written as G.

2.1.2. Physical representation The CfPGFs defined above are most convenient for calculations, but more direct contact with measurable quantities is established via the "physical" representation defined as Gr(x, y)== -i9(x, y)([f/I(x), f/lt(y)].),

(2.8a)

G.(x, y) == i8(y, x)([I/I(x), I/It(y)]:.),

(2.8b)

Gc(x, y) == -i({I/I(x), I/It(y)}) ,

(2.Be)

where Gro G. and Gc are retarded, advanced and correlation functions, correspondingly. In this definition,

It is straightforward to check that these functions are related to the CfPGF in single time representation as follows:

G.= GF - G_= G+- G F ,

(2.9a) (2.9b)

Gc = GF + Gt;= G++ G_.

(2.9c)

Gr = GF - G+ = G_- G1',

689 KlIIIIIg·chGO Chou eJ a/.• Equilibrium and ~librium lonna/isms made unified

9

The inverse relations are given by (2.10) If we introduce two-component vectors

(2.11) (2.10) can be rewritten as

G= W.e71 t +!G.71t"t +~G.t"t"t ,

(2. 12a)

or in components (2. 12b) Sometimes it is convenient to introduce a matrix form for the physical functions

G= (0

Gr

G.). G.

(2.13)

The transformations (2.10) and (2.12) can then be presented as

G= 0- 1 00,

t; =: 0(;0- 1 ,

(2.14)

using the orthogonal matrix (2.15) which was first introduced by Keldysh [9]. In what follows we will call G the CfPGF in physical representation and denote their components by G 1/"" (12 ... n) with Latin subscripts i, j, ... , n = 1, 2. In the case of two-point functions, G11 = !(Gp+ Gp- G+ - G_) = 0,

(2.16a)

G12 = G.= !(Gp - G_ + G+- G,,),

(2.16b)

G21 = Gr=!(Gp - G+ + G_- G,,),

(2.16c)

G22 =G.=~Gp+ G,,+ G++ G_).

(2.16d)

We see thus Gn is always zero and the other equations of (2.16) are identical to those of (2.9) by virtue of the identity (2.6).

690 10

KUIlllg-chao 0.011 eI al.• Equilibrium and lIonequilibrilim formalisms made IInified

It is obvious from definition (2.8) that 012(X,y)==O.(x,y)= 0, if (,>ry;

021(X, y) == Or(X, y) = 0, if ty > tx ,

(2.17)

and also that (2.18) because 9(x, y)9(y, x) = 0.

(2.19)

As will be shown later, almost all that has been said here, e.g., 0 11 = 0, the transformation of single and physical representations, the causality relations (2.17), (2.18), etc., can be easily generalized to the multi-point functions using the generating functional technique. However, before going on to describe this technique itself we give here the explicit expressions for the free propagators.

2.1.3. Free propagators The Lagrangian of the free fermion field is given by !to =

JI/,t(X)(i ~+ at V2 ) 2m

1/1 (X) ,

(2.20)

where m is the particle mass. The single time CfPGFs are defined by (2.3). To distinguish it from the Bose case we will use the letter S instead of O. H the system is in thermoequilibrium, the free propagator can be evaluated immediately from the definition. In Fourier space the CfPGFs tum out to be

SP(p) =

=

1- nCp) + n(p) po - p2/2m + if Po - p2/2m - if

Po-P

1 212

. . + 27TIn(p)8(po-p2/2m) , m +Ie

(2.21a)

S+(p) = 21Tin(p )8(po - p2/2m) ,

(2.21b)

S_(p) = -21Ti(1- n(p»8(po- p2/2m) ,

(2.21c)

Sr{p) = _

n(p) 1- n(p) po- p2/2m + if po- p2/2m - ie -1

. +21Tin(p)8 (Po- p2/2m). po- pm-Ie 212

(2.21d)

691 Kuang-chao Chou et al., Equilibrium and lIOIItI/uilibrium formalisms made unified

11

where 1

n(p)=------

exp[(po --,u. )IT] + 1

(2.22)

is the Fermi distribution with ,u. as the chemical potential. If n(p) is set equal to zero, we recover the propagator for the "pure" vacuum. It is interesting to note that the additional term 27Tin(p)6(pop2/2m) is the same for all components of G. The reason for this will be clear from the next section. In accord with (2.9) we find the physical functions to be 1

Sr(p) =

Po- p

S.(p) =

2/2 +. ,

(2.23a)

1 2/2

(2.23b)

m

Ie

.,

Po- pm-Ie

S.(p) = -27Ti(l- 2n(p»8(po - p2/2m) .

(2.23c)

We note in passing that the retarded and the advanced Green functions S" Sa do not depend on the particle distribution n(p). Similarly, for the Hermitian boson field described by the Lagrangian (2.24) we have (2.25) In the Fourier space the free boson CTPGFs are (2.26a)

d+(p) = -27Ti(6(po) + f(p»6(p~- W2(p» ,

(2.26b)

.!L(p) = -27Ti(6(-po) + f(p»6(p~- W2(p» ,

(2.26c) (2.26d)

where

692 12

Kuang-chao CIIou el al., Equilibrium and nonequilibrium formalisms made unified

1 f{p) = exp[w(p)/T]-1

(2.27)

is the Bose distribution and w(p) = Vp2+ m2

(2.28)

is the particle energy. If f(p) = 0, we recover the standard boson propagator of the quantum field theory [39]. Also, the additional term proportional to f(p) is the same for all components of J. The corresponding retarded, advanced and correlation functions are given by Ll ( )_ r

1

p - pt.:.·,lif(p) + 2iEpo '

---i

(2.29a)

Ll.(p) = p20-(1) 2() P - 2'IEpo ,

(2.29b)

Llc(p) = - 21Ti(1 + 2f(p »8(pij - w2(p» .

(2.29c)

It can be shown [401 that the expressions for fermion and boson propagators (2.21), (2.26) remain the same for inhomogeneous, nonequilibrium systems provided n(p) and f(p) are replaced by their nonequilibrium counterparts - Wigner distributions n(X, p), !(X, p) in the external field, where

X= (x+ y)/2

(2.30)

is the center of mass coordinates.

2.2. Generating functionals For interacting fields we can construct the perturbation expansion in full analogy with the quantum field theory. The Wick theorem can be generalized to the CTPGF case, most conveniently by using the generating functional. For simplicity, we consider real bosons. The extension to other systems is obvious.

2.2.1. Definition of Z[J] The Lagrangian of the system is given by

J

It = Ito(rp) + (J(x)rp(x) - V(rp» ,

(2.31)

where Ito(tp) is given by (2.24), V(tp) the self-interaction and J(x) the external source. The generating functional for CfPGF is defined as

693 13

KUII/lg-chao elwu tl al.• Equilibrium and /lOMquilibrium formalisms made u/lified

Z[J(x)1 == Tr{ Tp [exp(i

JJ(X)~(X»)]p},

(2.32)

p

where the integration path p and the time ordering product along it Tp have been already defined in the Introduction. In general, the external source on the positive branch J+(x) and the negative branch L(x) are assumed to be different. They will be set equal to each other or both to zero at the end of calculation. The n-point CfPGF is defined as

Gp(l'''n)==(-i)n-lTr[Tp(~(l)'''~(n»pl=i(-l)"

8"Z[J(x)]

SJ(I)"'6J(n)

I .

(2.33)

1-0

2.2.2. Generalized Wick theorem In the incoming interaction picture (2.32) can be rewritten as Z[J(x)] = Tr{Tp[exp(-i

f (V(~I(X»-J(X)~I(X»)]p},

(2.34)

p

where the in-field ~I(X) satisfies the free equation of motion. The interaction term can be then taken from behind the trace operator to obtain

(2.35)

It is easy to show by generalizing the Wick theorem that [40]

r,,[exp(i JJ(X)CPI(X»)]

= Zo[J(x)]: exp[i

p

JJ(X)~I(X)l,

(2.36)

p

where: : means normal product and Zo[J(x)] is the generating functional for the free field Zo[J(x)] =exp{ -

iff

J(x)Gop(X - Y)J(y)} ,

(2.37)

p

Gop being the free propagator given by (2.26) with f(p) = O. Substituting (2.36) into (2.35) we obtain

Z[J(x)] = exp[ -i

Jv(T 8J~X»)

p

]Zo[J(X)]N[J(X)] ,

(2.38)

694 14

Kuang-chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

with (2.39) as the correlation functional for the initial state. N[J(x)] can be expanded into a series of successive cumulants

N[J(x)] = exp(iW~[J(x)]), ~

1

W~[J(x)] = L, n "=I

I .. ·I W~(1·"

.p

(2.40)

n)J(1)" ·J(n),

(2.41)

p

where (2.42)

It is worthwhile to note that correlation functions contribute to the propagator only on the mass shell because /PI satisfies the homogeneous equation (2.43) where w(p) is the boson energy (2.28). Also, the definition (2.39) is independent of the time branch, i.e., each CfPGF component will get the same additional term as we have seen in the last section on the example of free propagator. Therefore, the perturbation expansion in the CTPGF approach has a structure identical to that of the quantum field theory except that the time integration is carried out over the closed path consisting of positive and negative branches. Rewritten in single time representation, each n-point function (also the corresponding Feynman diagram) is decomposed into 2" functions (diagrams). The presence of initial correlations W~(1··· n) which vanish for the vacuum state, constitutes another difference from the ordinary field theory. In principle, all orders of correlations can be taken into account, but in most cases we will limit ourselves to the second cumulant. It can be shown quite generally that the counter terms of the quantum field theory at zero temperature are enough to remove all ultraviolet divergences for the CfPGFs under reasonable assumption concerning the initial correlations [41]. We will not elaborate further on this point here, but it should be mentioned that near the phase transition point the infrared singularities have to be separated first so that the ultraviolet renormalization for CfPGFs in this case is different from that of the ordinary field theory (see section 9.2).

2.2.3. Connected and vertex generating functionals The generating functional for the connected CfPGF is defined as

W[J(x)] = -i In Z[J(x}] ,

(2.44)

695 Kuang ·chao Chou el al.• Equilibrium and nonequilibrium formalisms made unified

15

GC(l'" n)=(-l)"-1 8"W[J(x)) \ 8J(1)'''8J(n) J=O p

= (-i)"-I(Tp(c,o(I)' .. c,o(n)))c

(2.45)

where ( )c stands for Tr(·· . fJ) with the connected parts taken only. The n = 1 case corresponds to the expectation value rpc(X) = 8W[J(x))/8J(x) = (c,o(x)J

(2.46)

of the field operator in the presence of the external source. Therefore, rpc(x) is also a functional of J(x). Performing the Legendre transformation upon W[J), we obtain the vertex functional r[rpc(x)) = W[J(x)) -

JJ(x)c,oc(x) ,

(2.47)

p

which depends on rpc(x) explicitly as weB as implicitly via J(x) by eq. (2.46). It follows then from (2.46) and (2.47) that (2.48) This is the basic equation of the CTPGF formalism from which we will derive a number of important consequences. The general n-point vertex function, or one particle irreducible (I PI) function, is defined as (2.49)

2.2.4. Dyson equation As an immediate consequence of the definition for the generating functionals W[J(x)) and r[rpc(x)) we derive here the Dyson equation. Taking functional derivative of (2.48) with respect to J(y) and using (2.45) and (2.46)

we obtain the Dyson equation

J

Gp(y, z)rp(z, x) = c\(x - y).

p

(2.50a)

696 16

KUlIng-chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

Similarly, by varying (2.46) with respect to IPc(Y) we find

JTp(x, Z)Gp(Z, y) = [jp(X - y) .

(2.50b)

p

Here (2.51) is the two-point connected Green function (the subscript c is suppressed), whereas (2.52) is the two-point vertex function containing only 1PI part. The [j·function on the closed path [jp is defined as

I

[jp(X - y)f(y) = f(x) ,

(2.53)

p

In the single time representation

I j dt =

dt+ -

j

dL,

(2.54)

p

where the minus sign in the second term comes from the definition of the closed time-path. The negative branch goes from +00 to -00. To satisfy eq. (2.53), it should be that [j(x - y), if both x, y on positive branch, [jp(x - y) = -[j(x - y), if both x, y on negative branch, {

(2.55)

0, otherwise.

In matrix notation [jp can be written as S(x - y) = [j(x - y)U3 .

(2.56)

The Dyson equation (2.50) in single time representation is thus (2.57) The transformation properties for the two-point connected Green function are the same as those

697 Kuang-cIuw Owu et aI., Equi/ilJrilllfJ and lIOMIJuilibtium formalisms made unified

17

discussed in section 2.1.2, except that a disconnected part proportional to -ilpe(x)lpe(y) should be subtracted from all Gfunctions, whereas a term -2ilpe(x)lpe(Y) must be subtracted only from Ge with Gr and O. remaining the same. Multiplying (2.57) by matrix 0 from left and 0- 1 (see (2.15» from right we obtain (2.58) where

f

=

OtO-l = O(~: ~:)0-1.

(2.59)

We note in passing that the Pauli matrix U3 always accompanies the CfPGF in single ti~e representation G, whereas 01 appears together with the CI'PGF in physical representation 0, f. As seen from eqs. (2.57) and (2.58) UJ'U3 is the inverse of G, while u1ful is that for G. It is more important to point out that all characteristic features of Green's functions 0 discussed in section 2.1 are transmitted to vertex functions f via the Dyson equations (2.57) and (2.58). In particular, we have

t,

(2.60)

f11 == 0, so that

f=

ra),

(0 \rr fe

(2.61)

with (2.62) The inverse transformation from

f

to

t is given by (2.63)

i.e., exactly the same way as Green's function (2.12). Further discussion on the Dyson equation wiD be postponed to section 3.1. Meanwhile, we would like to emphasize that the "transmissibility" of the CfPGF characteristics is an evidence of the logical consistency of the formalism itself. More examples along with some useful computation rules were given before [40, 43, 44]. One more remark concerning the generating functional technique itself. Up to now we have considered only CfPGFs for the constituent field lp(x), but what has been said for it can be repeated almost word for word for any composite operator O[tp(x)]. In the forthcoming discussion we will use the corresponding formulas without repeating their definitions.

698 18

Kuang-chao OIou el al., Equilibrium and nonequilibrium formalisms made unified

2.3. Single time and physical representations In the CfPGF approach we have to deal with three representations which are equivalent to each other, namely, the closed time-path form Gp used for compact writing of formulas, the single time form G and the physical representation O. In section 2.1 we have already discussed these representations and their mutual transformations on the example of two-point functions. In this section we will use the generating functional to consider general n-point functions. The underlying connection here is that different representations of CfPGF are generated by the same generating functional expressed in different functional arguments. The explicit expressions for n-point functions in physical representation will be obtained along with the transformations from G to 0 and vice versa. As we will see later in section 2.4, these formulas are the starting point for discussing the important normalization and causality relations.

2.3.1. Preliminaries To start with, we need to specify some more notations. The multi-point step function

8(1 2 ... n) = { 1, if 11 >: 12 ••• > t,. , ", 0, otherwise,

(2.64)

is a product of two-point step functions

8(1,2, ... , n) = 8(1,2)8(2,3)··· 8(n -1, n).

(2.65)

It can be used to define the time ordered product

(2.66a)

"" or the anti-time-ordered product T(A 1(1)'" An(n» =

L 8(1, ... , n)A,,(n)··· AI(I). ""

(2.66b)

The summation here is carried out over all permutations of n numbers Pn

(:1 :2·..... Iin). These step functions satisfy the normalization condition

L 8(1, ... , n) = 1 , "" and the summation formula

(2.67)

699 KIUI1IK-c/aao Ow.. tt aI., Equilibrium and IIOMqIIilibrium formalisms made llIIifitd

8(1, 2, ... , m) = ~

19

(2.68)

8(1· .. n) ,

".(1"·m)

where Pn(l'" m) means permutations of n numbers with 1 preceding 2, 2 preceding 3, etc., but the order of the rest is arbitrary. In fact, (2.67) is the special case m = 0 of (2.68). The external source term in the generating functional (2.32) can be presented as

.

1= 1

I I(x)~(x)= I dtd"x(J+(x)~+(x)-L(x)~_(x»== I tU3cP, j

(2.69)

p

where

cP = (9'+(x»),

j

~_(x)

= (J+(x»)

\.'-(x) ,

(2.70)

and also as

(2.71) with

I,. == 1/ tj = I+ - L, '1',. == 1/ tcP = 9'+ - '1'_,

Ie =!~tj =!(I+ + L), '1'. == !~t cP = !('I'+ + '1'_) .

(2.72)

Also, we can express j, cP in terms of Ie, I,., 'l'e and '1',. as (2.73) The functional derivatives are related with each other by the following equations: I)

I)

1

I)

8J.. (x) = 2~.. 8Je(x) + 1/0 8J,.{x) , I)

8Je(X) =

I)

I)

~o 8J.. (X) = TJa 8J(x.. ) ,

& 1 I) 1 & I)I,.(x) = 21/.. 8J.. (X) =2~.. 8J(X.. ) ,

(2.74a) (2.74b)

(2.74c)

with a = ± and summation over repeated indices. Here we have introduced a symbolic notation (2.75)

700 20

Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

which is useful for a compact writing of the definition for Green's function as seen from (2.76). A remark concerning the notation fP.(x) is in order. Previously (see (2.46» we have defined fP.(x) as the expectation value of fP(x) on the closed time-path. Hence it is a two-component vector (fP.+(x), fPc-(x», but we do not make the subscripts +, - explicit. Here (see (2.72» fP.(x) is the linear combination of operators fP+(x) and fP-(x), still, in accord with our convention, no caret is put above it. Later on, the same fPc(x) will denote its expectation value. Hopefully, no confusion will occur, since the meaning of fPc(x) is clear from the context and, moreover, fPc+(x) = fPc-(x) = fPc(x) for l+(x) = L(x) as seen from (2.105). The same remark is effective with respect to other functions like Qc(x), I/Ie(x), I/I~(x) and so on, appearing in the future discussion.

2.3.2. "Physical" representation of the generating functional As we said in the introductory remarks to this section, the same generating functional will generate crPGF in different representation provided the external source term is expressed in the corresponding functional arguments. In particular, the generating functional in the form (2.32) will give rise to CfPGF in the closed time-path representation. If, however, the source is written in single time form as given by (2.69), the same generating functional (2.32) can be then expanded as

(2.76) with G a'''p

( ). 8"Z[I+> '-1 1 .. · n == 1(-1)" 8.T(a)'" 8J(p)'

(2.77)

where

and both the space-time coordinates and the dummy indices a, .. _,p should be summed over. Moreover, if the expression (2.71) for the source is used, the same generating functional (2.32) can be expanded as

where

B" ==

(-i),,8"Z[IA , Ie1

8JA(1) .. , 8JA(m )8Je(m + 1)· .. 8Ie(n) = (Tp (fP.(1)· , '({>e(m )fPA(m

+ 1)' , , fPA(n») == i"-12- m + 1GUU(1' .. n), m

n-m

(2.79)

701 KfI4IIg-clulo Chou et al., EquiUbrium and lIonequilibrium formalisms made unified

21

Now we find another expression of the CfPGF in physical representation, namely, in terms of expectation values of nested commutators and anti-commutators. Using the normalization condition for the step function (2.67), eq. (2.79) can be rewritten as

= 2. 6(1'"

;i)2-"'{"'", {"m1/"'''''' ·""'·(Tp(rp,,,(l)··· rp",.. (m)rp", ..+1(m + 1)'" rp".(n»).

(2.80)

"" For convenience we introduce a unified notation for { and 1J aj _

(

-

{f""

if 1 ::;j :5 m , m + 1 :5 i :5 n .

(2,81)

1J '" , if

Since the order of operators under Tp can be changed arbitrarily, (2.80) can be also presented as BII

= 2. 6(1' .. n)2-",(I .. , (II(Tp(rpj(l) ... rpli(;i») ,

(2.82)

""

where the subscript r == a T. Now let us get rid of Tp in (2.82) step by step for each term of permutation PII' As far as the 6-function ensures n to be the earliest moment on the positive branch and the latest one on the negative branch, we have (II(Tp(rp\(i)· ., rp Ii(n») = ~iI(Tp(rpr(l)" . rp,,(n») = (Tp(rp I(l) ... rp n-I(n - l)}rp(n»

= ({Tp(IPr(l)'"

+ (rp(n)Tp(rp 1(1) ... IP II-I (n - 1»)

IPn-l(n -1», rp(n)}) ,

if

or (II(Tp(rp r(I)· .. rp iI(n») = 1/ iI(Tp(rpi(I)· .. rpll(n») = ([Tp(cPI(i)· .. cp n-I(n

-1», rp ,,(n)]) ,

if

m+l:5n:5n.

Such processes may be continued like "a cicada sloughing its skin" in accord with the Chinese saying, up to the last step to get zero if m + 1:5 I :5 n, or the expectation value of nested commutators (and/or anti-commutators).

702 22

Kuang-chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

If we introduce a short writing

( ,cp(I»= {{[ ,cp(I)], when m+lslsn ,cp(l)} when Islsm,

(2.83)

we find finally 6~ ~(1' .. n) == (_i)"-1

L' 6(1' .. ii)«' . - ('1'(1), '1'(2»' . " cp(ii») ,

(2.84)

m n-m

where ~' means that permutations m + 1 s As a special case we find for n == 2 that

1s n should be excluded from the summation.

0 21 (12) = -i9(1, 2)([cp(I), cp(2)]),

0 11 (12) == 0,

Od12) = -i({cp(I), cp(2)}) ,

(2.85)

thus recovering 0 2 1> 0 22 as retarded and correlation functions. Using the symmetry

we obtain the advanced Green function as

0 12(12) = -i9(2, 1)(['1'(2), cp(l)]). Furthermore, for n = 3 case we have 0 111 = 0,

0 211 (123) = (-W

L

9(1ij)([[I, i], j]) ,

pea> 0 221 (123) = H)2

L

(9(ij3)([{i,j}, 3]) + 8(i3j)({[i, 3],j}» ,

pe(:~>

0=(123)= (_i)2

L

(2.86)

6(ijk)({{i,j}, k}).

Here for short we write i,j instead of 1P(i), cpU). Therefore, this way we can exhaust all possible n-point functions. Without resorting to the CfPGF formalism this would be a cumbersome task. The functional expansion (2.78) and the explicit expression for physical CfPGF (2.84) are very important equations from which a number of far-going implications will be extracted in the next section. In the meantime we note only that the functional derivative 8/8Jc will generate CPtJ. which in turn yields a commutator in the Green function, whereas the functional derivative 8/8JtJ. gives rise to IPc leading to an anti-commutator. Moreover, in the summation (2.82) none of the time variables m + 1 s s n can take a value larger than all of the time arguments 1 s Js m, because in that case I should be one of (m + 1, ... ,n) excluded from the summation. This fact will give rise to important causality relations.

r

703 23

2.3.3. Transformation formula Using the definitions of CfPGFs in single time and physical representations as expansion coefficients of the generating functional (2.77) and (2.79), respectively, we can readily find the transformation from one to another. In fact, using (2.79) and (2.74) we have

:=:

2- 1 ~'"

•••

~a"7Ja,,,,

... 11'" 0 """'" (1 ... n) .

(2.87)

In a similar way we find the inverse transformation as given by O QI'·'IIII',. (1 · .. n) == 21-"

'~ " !:ten 1'1 '"

110

!tall

0 it"'''' (1 ... n) ,

(2.88)

where i10 ••. , in == 1, 2 ,

l~ == 11",

(2.89)

Using the orthogonal matrix 0 defined by (2.15) these formulas can be rewritten as 0., ...",(1· .. n) == 2n12-10'ICll" • 0""'00 ClI...... (1· .. n),

(2.90a)

o al'·'GI,. (1", n)== 2

(2.90b)

1 nll -

Oalii .. , QTa"l,. 0 •• "." (1 .. ' n) • T

For the case n == 2 eqs. (2.12) and (2.16) obtained in section 2.1 are recovered. For n == 3 we have 0+++(123) == (-iY(T(123»,

0++_(123) == (-iY(3T(12»,

0+ __ (123) == (-iY(T(23)1),

0 ___ (123) == (-iY(T(123» .

(2.91)

The other functions, i.e., G+_+, 0_++, 0_+_, 0 __ + can be obtained by symmetry. For illustration we also write down some of the transformation formulas such as 0 111 = 0(+) - 0(-) == 0,

0Z22 == 0(+) + 0(-) ,

(2.92)

with 0(+)== 0++++ 0+ __ + 0-+-+ 0 __ +, 0 211 = !(O+ .. + G_ .. ),

(2.93)

with 0_ .. == ~ af30-aIl' CI.IJ-:t:

704 24

Kuang·chao Chou er al., Equilibrium arui nonequilibrium formalisms made unified

2.3.4. "Physical" representation for W[J] and r[ipc] We have discussed above different expansions for Z[J} along with some of their consequences. The same thing can be done toward W[J] and f[ipc]. For example, in the physical representation we have (2.94) (2.95) where (2.96) It follows then from (2.95) and (2.96) that

8f[CPA(X), ipe(x)] == -J ( ) 8CPe(X) A X , 8f[ ip A(X), CPe(X )] == -1. ( ) 8cpA(X) eX.

(2.97)

It is obvious that W[JA , Je] and f[CPA' CPe] defined by (2.94) and (2.95) are identical to Wp[J] and fp[CPc]

as given by (2.44) and (2.47) respectively. We should note, however, that the explicit form of the connected Green function is different from that of the "total" (connected + disconnected) Green function obtained as expansion coefficient of Z[J]. For example, Ob(123) ==

(-W L: 8(ijk )[({{i, j}, k}) -

({i,j})(k) + 2{{(i), V)}, (k)}] .

(2.98)

P3

The only exceptions are the "all retarded" Green functions like 0 21 , 02l1, 02lll etc., for which

Therefore, the transformation from (; to (; for the connected Green function and vice versa should also be correspondingly modified. We note in passing that the "all retarded" functions are nothing but the r-functions used to construct the LSZ field theory [59]. Unlike the zero temperature case, these functions alone are not enough to construct the CTPGF formalism, but they still playa very important role here.

2.4. Normalization and causality As we emphasized in the Introduction, the normalization and causality relations are essential for applications. In fact, they are already implied by the expansion of the generating functional discussed in

705 KIIIUI,-c/ulo Chou et aI., Equilibrium "'"' nOlJtllllilibrium formalisms millie wtlfr«J

25

the last section, but we would like to make them explicit here for future reference. We start from the normalization (section 2.4.1), then indicate the consequences of the causality (section 2.4.2) and wind up with few comments on the two aspects of the Liouville problem - dynamical evolution and statistical correlation - naturally embodied in the CfPGF formalism (section 2.4.3).

2.4.1. Normalization If Ill(x) is set equal to zero, i.e., I+(x) =L(x) in the expansion (2.78), we find 3"Z[lr.,I.] 81.(1) ... 81.(n )

I 1,-0

= 0, for

n 2= 1,

(2.99)

because in accord with (2.84) G ll ... 1(1···n)=0.

(2.100)

Using the normalization condition of the density matrix Tr(p) = 1,

(2.101)

we find (2.102) By definition (2.94) we have (2.103)

or, equivalently,

3" W[lr., I.] 81.(1)" . 3I.(n)

I 1,-0

== 0 for any n 2= 0 .

(2.104)

In particular, (2.105) which leads to (2.106) and (2.107)

706 26

KUfUlg-c/uw Chou et aI., Equilibrium and IIonequilibrium formalisms mode unified

or, equivalently

r

(1·,·n)=0.

(2.108)

l1 ... 1

We see thus the algebraic relations obtained before, such as (2.6), (2.16), (2.60) and (2.92) are special cases of these general conditions following from the normalization. We would like to emphasize here that the normalization condition for the CfPGF generating functional is different from that of the quantum field theory or the standard many-body formalism. In this case we require only the equality of the external source on the positive and negative branches I+(x) =L(x) instead of its vanishing. We can thus incorporate the external field Ic(x) =!(J+(x) + I_(x» into the theoretical framework in a natural way. Moreover, this fundamental property will give rise to a number of important consequences which make the CfPGF formalism advantageous in many cases as we will see later. We note also, that eqs. (2.99), (2.104), (2.107) and (2.108) are valid even in the presence of a finite external field Ic(x).

2.4.2. Causality As mentioned in section 2.3.2, in the functional expansion (2.78) none of the time variables with m + 1 :$ r:$ n can take values greater than the time arguments 1:$ J:$ m, because this would contradict the rule established by (2.84) that terms m + 1 :$ I :$ n should be excluded from the summation. Put in another way, S"Z[IA.JJ 81A(1)·· ·SIA(m)8Jc(m + 1)·· ·8Ic(n)

I

-0

(2.109)

Jl=J,-O-

provided any II> with m + 1 :$ i :$ n is greater than all Ij with 1:$ j :$ m. This is one of the causality relations we consider here. It is obvious that the causality relation for the two-point function (2.17) is a special case of (2.109) for m = 1, n =2, i.e.,

In a sense, the algebraic relation (2.99) or (2.100) is also a special case of (2.109) for m = O. Similarly, under the same condition, i.e., the time argument of any Ie, 'Pc is greater than that of all lA, 'PA, we have for the functional derivatives of W[l],

S"W[IA,J.] 8lA(I) .. . 8lA(m)8J.(m + 1) .. · 81.(n)

I

=0

(2.110)

J,-Jl-O

'

I

= o.

and those of the vertex functional 8"r[cpA, CPc] 8'PA(I)· .. 8'PA(m )8'Pc(m + 1)· .. 8cp.(n)

(2.111)

wO, .,,_.,

In deriving (2.111) we have made use of the relations between the vertex functions and the "amputated" connected Green functions [39,60].

707 27

Kuang-chao Chou el 01., Equilibrium and nonequi/ibrium formaUsms made unified

It is worthwhile to emphasize that a causality sequence is established by (2.l09H2.11l), namely, the space-time points associated with Jc(x), fPc(x) should precede those of h(x) and tpA(X), since the former is the cause, whereas the latter is the consequence. We indicate here also some useful product relations as a generalization of (2.18) for a two-point function. For example, we have for three-point functions

(2.112) It is easy to see that the general rule is

(2.113) provided iml = ... = im , :;:: 2 and the rest are equal to 1, whereas jml = ... = jm, = 1 but the rest can be either 1 or 2.

2.4.3. Dynamical evolution and statistical correlation Now we discuss the physical meaning of lA, lc,

tpA

and tpc. In addition to (2.105) we have (2.114)

from (2.97) and (2.107), so that the conditions lA = 0 and follows from (2.91), (2.94) and (2.102) that

tpA =

r

= Tr{(texp(-i =L "=0

f I t

OCI

i"

r

lc(y)tp(y»))tp(X)(Texp(i

'1

dl

-00

I

-00

0 are equivalent to each other. Also, it

I

I

lc(y)tp(y»))p}

"'-1

d2 .. ·

dnlc(I)·"lc(n)Tr{[ .. ·[[tp(x),tp(I)],tp(2)] .. ·],tp(n)]p}.

-00

(2.115) We see thus tpc(x) under lA = 0 is the expectation value of the field operator, i.e., the order parameter in the presence of the external field lc, whereas (2.116) is the expectation value that might cause symmetry breaking in the vanishing field. Equation (2.115) is a nonlinear expansion of the order parameter in the external field. A detailed discussion of the nonlinear response will be given in section 5. As we already mentioned in the last section, in accord with (2.78) and (2.84) the functional derivative a/Blc(x) generates the expectation value of the commutator of the field variables describing the

708 28

Kuang-chao Chou el al., Equilibrium and nOMquilibrium formalisms made unified

dynamical evolution in the quantum mechanical sense, whereas 8/8JA(x) generates the expectation value of anti-commutator describing the statistical correlation in the statistical mechanical sense. Although the physical observables are defined on the manifold h(x) = 0 or 'PA(X) = 0, these functional arguments are needed in addition to lc(x) and 'Pc(x) for a complete description of the statistical system. These two complementary aspects of the Liouville problem - dynamical evolution and statistical correlation - have been embodied in the CfPGF formalism in a natural way. It is worthwhile to note that the response and the correlation functions have found their "proper seats" in the CTPGF formalism just because in the external source term (2.71) 'P and J are "twisted", i.e., 'Pc is coupled with h, while 'PA with Ie as follows directly from the definition of the closed time-path. As we will see later in section 9, this is one of the advantages for the CTPGF formalism compared with the others.

2.5. Lehmann spectral representation In this section we study the analytical properties of the Green functions. As in quantum field theory [39] and in the standard Green function technique [1-3], the Lehmann spectral representation is a powerful tool towards this end. We will discuss in some detail the spectral as well as the symmetry properties of Green functions for a nonrelativistic complex (Bose or Fermi) field defined by eqs. (2.3) and (2.8). The modification needed for a real boson field is also briefly mentioned.

2.5.1. Spectral expansion Assume, the inhomogeneity of the system is caused by the nonuniformity of the state, while the evolution of the Heisenberg operators I/I(x),I/It(x) with x is given by the total energy-momentum operator p as I/I(x) = exp(ip' x)I/I(O) exp(-ip' x),

I/It(x) = exp(ip' x)I/It(O)exp(-ip' x).

(2.117)

Let In} be a complete set determined by p... and other operators commuting with P.... According to (2.117) we have for G_(x, y) defined by (2.3c) (2.118) ,.,m,,,·

Set Z

= x - y, X = (x + y)/2 and take Fourier transform with respect to Z, we obtain

L

iG_(k, X) =

(n!I/I(O)!m) (mW(O)!n'}p,..,. exp[i(p,. - p,..) . X](21T)d+18(k - pm + (p,. + p".)/2).

".m,II'

(2.119)

If

p,..,. == (n'!p!n)

IX

6(p" - p,..) ,

(2.120)

G_(k, X) will not depend on X and the system is homogeneous. In the presence of macroscopic inhomogeneity, p,,',. is different from zero only for p" - p", small compared with k, so that the high orders of a/ax can be neglected.

709 KlUJrlg-chao C1wu el aI., EquilIbrium and IIOIIeqllilibrium formalums made unified

29

2.5.2. Sum rule

For non relativistic fields we have the following equal-time (anti-) commutation relation (2.121) which leads to (2.122) Introducing the spectral function p(k, X) == i(G_(k, X) - G+(k, X»,

(2.123)

we rewrite (2.122) as dk o

f 2'17

p(k, X) = 1 .

(2.124)

For a real boson field we have (2.125) from which one can derive dk o

f 2'17

kopek, X) = 1,

aX4

dko

J2'17 p(k, X) =0 ,

(2.126)

with p(k, X} still defined by (2.123). 2.5.3. Lehmann representation

Presenting the retarded Green function G.(x, y) defined by (2.8a) as

we find G (k X) = r

,

Jo

dk pet, kG. X) 2'17 ko kil or It '

(2.127)

which is analytic in the upper half-plane of ko. Similarly, we have for the advanced function

J

G.(k, X) = ~ p(t. k~, X) 2'17 k(j"=:-tO-=-tr

(2.128)

710 30

Kuang-chao Chou et al., Equilibrium and nonequilibrium lonnalisms made unified

which is analytic in the lower half-plane of ko, Presenting G± as

=

'f dk~ , (1 2fT 0,.,(1, ko, X) ko I

1)

(2.129)

k0 - k'0-1£ , ,

'""--'0'£

lion' 1

we find the spectral form of GF and G F ,

oF\Ik, X) = Gr\, Ik X) G (k X) = 'f dk~ (G_(1, k~, X) + +, I 2 -fe fe';.' fT

0

0

G+(1, k~ X») Ie fe" 1£ ,

(2.130)

1£00

Gr:(k, X) = Gr(k, X) + O_(k, X)

= if dk~ (G+(1, ko, &) G_(1, k~, X») 21T

ko - ko + ie

ko - ko - ie

(2.131)

.

Equations (2.127), (2.128), (2.130) and (2.131) are the Lehmann spectral representations we are looking for.

2.5.4. Symmetry relations It is straightforward to check that for nonrelativistic complex (boson or fermion) field we have

G!(x, y) = -G",(y, x),

G~(x,

y) = -G~, x),

G;(x, y) = G.(y, x),

(2.132)

or in Fourier components

G!(k,X) = -G",(k,X),

G~(k,X)=

GHk,X) = -Gr:(k,X),

G.(k,X) ,

(2.133)

whereas for real boson field we have additionally

(2.134) or in Fourier components (2.135) where T means transposition, * means complex conjugation and t Hermitian conjugation.

2.5.5. Two analytic functions It is obvious from (2.133) that G",(k, X) are purely imaginary on the real axis of ko. If they vanish as Ikol-HO , we can define two analytic functions on the complex plane of ko, namely, G (1 Z X)= i 1

,

,

f dkoG _(1, k~X) 2fT Z - ko '

G (1 Z X)=' fdkoG+(k, ko,x). 2 "

I

2fT

Z - ko

(2.136)

711 Kuallg-chao Chou el al., Equilibrium ami lIollequilibrium formalisms made unified

31

In terms of functions 0 1 and O2 we find Or(k, X) = 0 1(1, ko + ie, X) - O2(1, ko + ie, X), Oa(k, X) = Ol(k, ko - ie, X) - 02(k, ko - ie, X) , OF(k, X) = Ol(le, ko + ie, X) - 02(k, ko - ie, X), O,,(k, X)= O2(1, ko + ie, X) - Ol(k, ko - ie, X), O_(k, X)= Ol(k, ko+ ie, X)- 0 1(1, ko- ie, X), O+(k, X) = 02(k, ko + ie, X) - 02(k, ko - ie, X) ,

(2.137)

i.e., all these functions are superpositions of 0 1 and O2 on approaching the real axis from different sides. It follows from (2.133) and (2.136) that 01.2(k, Z, X)*

= 01,2(k, Z*, X).

(2.138)

We see thus to ensure the causality, the retarded Green function should be analytic on the upper half-plane of ko. If a singularity is found on the upper half of ko during the process of solving On it must be located on the second Riemann sheet. The appropriate analytic continuation is to take the integral along a contour in the complex plane of k~ which circulates the singularity from above. 3. Quasiuniform systems In this section we will discuss in some detail further properties of two-point Green functions, mainly concentrating on quasiuniform systems. The starting point is the Dyson equation formally derived from the generating functional in the last section. The quasiuniformity can be realized only near some stationary state, either thermoequilibrium or nonequiIibrium under steady external conditions. We will derive the stability condition from the analytic properties of Green's functions. In section 3.1 the properties of the Dyson equation are further elaborated, especially for a uniform system. The thermoequilibrium situation is then discussed (section 3.2) mainly for the tutorial purpose. Furthermore, the Dyson equation is used to derive the transport equation (section 3.3). Finally, the multi-time-scale perturbation (section 3.4) and the derivation of the time dependent Ginzburg-Landau (TDGL) equation (section 3.5) are briefly described. The separation of micro- and macro-time scales is the common feature of the last three topics.

3.1. The Dyson equation 3.1.1. An alternative derivation The Dyson equation and its equivalent forms (2.50), (2.57) and (2.58) have been derived from the generating functional. Here we give another derivation which will shed some light on the structure of the vertex function. Consider an Hermitian boson field lp(x) described by the Lagrangian density (3.1)

712 32

Kuang·chao Chou el al.• Equilibrium and nonequilibrium formalisms made unified

For simplicity we assume ~c(x) = Tr(rp(x)p) = 0 in zero external field J(x) = O. The field operator satisfies the equation of motion Or~(X) = j(x);: -& V(~(x»/8rp(x),

(3.2)

where (3.3)

and j(x) is sometimes called the internal source of rp(x). The two-point vertex function defined as (3.4)

can be presented as (3.5) where (3.6)

is the vertex function in the tree approximation and Ip(x, y) the self-energy part due to loop corrections. The inverse of rop is Green's function for free field, satisfying OrGop(x, y) = -8:+ 1 (x - y).

(3.7)

Using (3.2) and the commutation relation (2.125) we find

or OrGp(X, y) =- 8:+ 1(x - y) + i

J[Tr(7;,{j(x)j(z»p) + i8:+ (x _ Z)Tr{ 3cp(x}acp(z) 8V p}] 2

1

p

x Gop(z, y) dd+l Z .

(3.8)

Comparing (3.8) with (2.50) we obtain

I

l'p(x, z)Gp(Z, y) dd+1 z

= -i

I p

[Tr(Tp(j(X)j(Z»p)

+ i8:+1(x - Z)Tr{8CP(!;::(Z)

p} ]Gop(Z, y)

dd+1 Z ,

713 Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made uniMd

33

which yields (3.9)

This expression will be used later to discuss the transition probability. 3.1.2. Matrix representation The matrix representation of the Dyson equation as given by (2.57) and (2.58) are very convenient for practical calculations. For example, we find immediately from (2.58) that

(3.10a) (3. lOb)

Using eqs. (2.9) and (2.62) we find the corresponding relations for

G as (3.11)

The symmetry relations (2.132) and (2.133) valid for G can be also transmitted to r to give n(x, y) = -rf:(Y, x),

n(x, y)= -r",(y, x), r!(k) = -r",(k),

n(k) = -rr{.k),

F;(x, y) = r.(y, x),

F;(k) = r.(k).

(3.12a)

For real field we have from (2.134) and (2.135) t(x, y) = tr(y, x) = -O'1t*(x, Y)O'I = -O'l't(y, x)O't.

(3.12b)

t(k) = F(-k) = -O'1t*(-k)O'I = -O'tf't(k)O'I' 3.1.3. Vertex functions

As seen from (3.12a), only three components of t are independent. They can be set as (3.13a)

where A, Band D are real functions in accord with (3.12a). In terms of unity and Pauli matrices, (3.13a) can be rewritten as

t = iB(I +0'1)+ AU2+ DO'3'

(3.13b)

We then find from (3.13) that rr(k) = D(k)+ iA(k),

r.(k) = D(k) - iA(k),

r P(k) = D(k)+ iB(k),

r~k)=

r.(k) = i(B(k) ± A(k» ,

(3.14) -D(k)+iB(k).

714 34

Kuang·chlUl Chou et al.• Equilibrium and nonequilibrium lonnalisms made unified

In what follows we will call D(k) the dispersive part and A(k) the absorptive part of the self-energy in analogy with the quantum field theory.

3.1.4. Green's functions The expressions for Green's functions follow immediately from (3.10) and (3.11) as G(k)r

-

1

D(k)-f iA(k) ,

G Ik)- Qill-iB(k) F\ - D2(k)+A2(k) ,

1

G.(k) = D(k)'::iA(k) , G (k) = ::.l?(~) - iB(k) r: D2(k) + A2(k) .

(3.15)

It follows also from the matrix equation (2.57) and the symmetry relations (2.133) and (3.12) that (3.16) which can be verified directly from (3.14) and (3.15). By virtue of the definition (3.5) we can express functions A(k), B(k) and D(k) in terms of the self-energy part I as A(k) = ~(I_(k) - I+(k », B(k) = ~i(I+(k) + I-(k », 2 2 D(k) = k - m - !(IP(k)- Ir:(k».

(3.17)

It is unlikely that both A(k) and D(k) have zero on the real axis of ko, so the divergence on the mass shell P = m2 can be removed by the renormalization procedure. If there are no zeroes of D(k) + iA(k) in the upper half-plane of ko, then the causality is guaranteed and the pole of Gr in the lower half-plane of ko will describe a quasiparticle moving in a dissipative medium. On the opposite, if there is a pole a in the upper half-plane, then Gr is analytic only for 1m ko > 1m a. This pole will describe a quasiparticle moving in an amplifying medium with growing amplitude of the wavefunction. In such a case the original state is unstable with respect to a new coherent state of quasiparticles like the laser system beyond the threshold.

3.2. Systems near thermoequilibrium The formal solution of the Dyson equation (3.10) and (3.11) as well as the explicit form of the vertex function (3.14) and Green's function (3.15) are valid for any quasiuniform system near eqUilibrium or nonequilibrium stationary state. In this section we consider the thermoequiIibrium system in more detail. The transition probability is first studied (section 3.2.1), the dispersive part is then discussed (section 3.2.2) to show that the thermoequilibrium system is stable and the detailed balance is ensured (section 3.2.3). Furthermore, formulas for nonrelativistic fields are written out explicitly for future reference (section 3.2.4). Finally, the fluctuation-dissipation theorem is derived for the complex boson and fermion field (section 3.2.5).

3.2.1. Transition probability It follows from (3.9) that

715 Kl/IIIIg-chao Otou et Ill., Equilibrium and IIOIIeqIIilibrillllllormalisms made unified

t-(x, y) = -i TrU(x)j(Y)ph P.I.,

t+(X, y) = -i TrU(y)j(x)ph P.I.·

35

(3.18)

As done in section 2.5, the evolution of j(x) under the space-time translation is given by

j(x) = exp(ip· x)j(O) exp(-ip· x).

Substituting this expression into (3.18) and taking Fourier transformation, we obtain it-(k) = ~ 1(1\j(0)lnhp.I.l2p",,(21T)8d+l (k - PI +P.. ), ~

.

it+(k) = ~ l(nV(0)Il}lP.I.1 2 PII(21T)d+18d+ 1(k - PI +P.. ).

(3. 19a) (3.19b)

I...

Here we neglect the off-diagonal elements of the density matrix because the system is uniform. For ko> 0 each term of (3.19a) corresponds to the probability of transition from the state In) to the state II) by absorbing a quasiparticle of momentum k, i.e.,

(3.20a) while each term of (3.19b) corresponds to the probability of emitting a quasiparticle (3.20b) Since EI > E .. for both cases, P"" > PII in thermoequilibrium, so that (3.21a) for ko > 0, i.e., the probability of absorbing a particle is greater than that of emission. Using the relation

following from (3.12b) we find for ko 0 provided (3.74) is satisfied, then 'Pc(x) will grow in time to form a laser-type state with its amplitude being limited by nonlinear term E[rp.(x»). Near the critical point when such instability occurs, borr is a small quantity and 'Pc changes with time rather slowly. Assume that the approximate solution of (3.72) can be presented as Z;1

where E is a small parameter which should be set equal to 1 by the end of the calculation, EX describes the slowly varying part. Set x= EX, assume both borr and E to be of order E, the differentiation with respect to x can be written as iJ.. + eiJ~ so that (3.72) becomes

So far as x is a slow variable, we neglect the difference of x and leading orders we have

y in the last two terms. To the first two

Do(iiJ.. )'P~O)(x, i) = 0,

f

Do(iiJ.. )rp~l)(x, i) + iDo/&(iiJ.. )iJ~'P~O)(x, x) + borr(x, Y)'P~)(y, i) d4 y + E['P~)(x, i)] = 0,

(3.75)

726 46

KlUlIIg-chIW Chou el al., Equilibrium and nonequilibrium formalisms made unified

where

Do .. (k) = aDo(k)/ ak" .

(3.76)

As seen from (3.75), the solution is given by "c(x, x) = "k(X) exp(-ik . x),

(3.77)

where k is determined from (3.74). If we require that 1P~1) does not contain a term proportional to lP~o), then the second equation of (3.75) after Fourier transformation becomes (3.78) where we have also replaced the center-of-mass coordinates !(x + y) in rr(x, y) by x. This is an equation satisfied by the oscillating mode of the mean field. We have used this technique to discuss the laser system coupled with two-energy-level electrons [40,41]. We will not reproduce the calculation here, but it should be mentioned that a stable laser state allowed in the classical theory, is unstable in the quantum case. In the quantum theory we must consider the fluctuation of the photon number. Since the laser system is described by a coherent state with fixed phase, the fluctuation of the photon number diverges. This divergence can be removed by a renormalization procedure which leads to the decay of the laser state. Similarly, the soliton solution of IPc(x) is also unstable due to the quantum fluctuation. It is worthwhile to note that such multi-time-scale perturbation technique is quite useful. In fact, we have already made use of its basic idea in deriving the transport equation in the last section. It is also the key point in obtaining the TOOL equation which we are going to discuss now.

3.5. Time dependent Ginzburg-Landau equation The concept of macrovariableness is very useful in critical dynamics, hydrodynamics, and many other fields [61]. Usually, the set of macrovariables includes both order parameters and conserved quantities. As a rule, their microscopic counterparts are composite operators. In this section we use the equation for the vertex functional (2.48) to derive the TOOL equation [61] for their expectation value. As seen from the later discussion, the term TOOL equation is used here in a much more general sense. Let 0, (x ), i = 1,2, ... , be the set of composite operators corresponding to macrovariables. Without loss of generality, we assume them to be Hermitian Bose operators. The order parameter Oc(x) is determined from the equation for the vertex generating functional (3.79a) Suppose, Oc,(x, T) is known for the moment T. At the time t following T, the left-hand side of (3.78) can be expanded as (3.79b)

727 KlUUIg-chao C7Iou el al.• Equilibrium and nOMquilibrium formalisms made unified

47

which is true for t located either on the positive or negative time branches. So far as 0 varies slowly with time, we can write

Substituting this expression back into (3.79) and taking into account that in the limit I == Ix -+ '1",

(3.80) where fr/i(x,y, ko, '1") is the Fourier transform of frij(x, I",y, Iy) with respect to I" - ty, taken at T = ~(I" + Iy) "" '1", we obtain in the matrix form aO(I) 8f '}'(/)-=-

al

I

80c+

(3.81)

+J(I),

0 •• =0.--0

where we change T for t. For the moment let

I

8f I, (x, t)==--

80c,,"

,

0 .. -0.--0

and calculate the functional derivative of h with respect to O(x, t) as a function of three-dimensional argument M(x, t) 80j (y, t)

I

d d { Z .,Tz

=fFjj(x, y, ko = 0, t) 8lj (y, t)

8Q( I X, t)

Z

2

8f 80k +(z) 8f 80d Z)} 801+(x)8Qk+(Z) 80j (y) 801+(x)80k -(z) 80j {J) f+lj(x, y, ko = 0, I),

fFj,(Y,x,ko=O,I)-f+j,(y,x,ko=O,t)

= fF/i(x, y, -k o = 0, I) -

L,j(x, y, -ko = 0, I),

where in the last step we have used the symmetry relation (3.12b). The difference ~) 8I,Cy, I) . l:>0(y ) l:>Q( )= hm (f-,j(x,y,-k o, I)-f+jj(x,y, ko, I» o 1,1

u,X,1

k(t+O

vanishes due to the relation

(3.82)

728 48

Kuang·chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

L

=

L exp(-pko),

(3.83)

following from (3.29) for a system in equilib,ium. Therefore, a free energy functional such that -'O~/'OOj (x,

Ij(x, I) =

~[O(x,

I)] exists (3.84)

I).

Equation (3.81) can then be rewritten as aO(/) 'O~ y(/)-= --+J(/). al '00(/)

(3.85)

If the macrovariables OCt) do not change with time in the external field J, then 'O~/80 =

(3.86)

J.

Hence ~ is actually the free energy of the system and (3.86) is the Ginzburg-Landau equation to determine the stationary distribution of macrovariables. For non equilibrium systems the potential condition, i.e., the vanishing of (3.84) can be realized if lim A (x, y, ko, t) = 0 ,

(3.87)

ko-O

where A is the absorptive part of Of' In the next section we will show that (3.87) is fulfilled for non equilibrium stationary state (NESS) obeying time reversal symmetry. It is usual to multiply eq. (3.85) by y-l(t) to obtain aO(/)

--=

at

}

y-l(t) {'O~ ---+J(t) . 'OO(t)

(3.88)

This is the generalized TDGL equation we would like to derive. If a random source term is added to the right-hand side of (3.88), it will appear like a Langevin equation. However, there is an important difference. Equation (3.88) includes the renormalization effects. Also, the way of describing the fluctuations in CTPGF formalism is very special as we will see in section 6.

4. Time reversal symmetry and nonequilibrlum stationary state (NESS)

It is well known that the principle of local equilibrium and the On sager reciprocity relations are the two underlying principles on which the thermodynamics of irreversible processes is constructed [62]. This is true near thermoequilibrium. Within the framework of statistical mechanics a successful theory of linear response has been developed by Kubo and others [63-65]. The two fluctuation- 01.{2} = 0 122 - 0 212 + 0 221 , 01.{3} = 0 122 + 0 212 - 0 221 , 02.{2} = -0112 + 0 121 - 0 211 , 02.{3} = 0 112 - 0 121 - 0 211 , O 2.{t} = - 0 112 - 0 121 + 021h OO.{1}

=

OO.{2}

=

OO.13}

(5.61)

O 2.(J}(W) + e302, (J}(-W) + 0222(W) + e30222(-W) =coth(,8w,/2)[ OI.{J}(W) - e30I.{j}(-w») , 02,{j}(W)- e302.{J}(-W) + 0222(W)- e 30222(-W)

= tanh(,8wj/2)[01. (j}(w) + e 0I.{J}(-W»), 3

j

= 1,2,3.

(5.62)

The question whether (5.60) is a correct generalization of FDT should be settled by further studies of nonlinear phenomena. We should mention, however, that Tremblay et al. [38] have considered the heating effects in electric conduction processes using CfPGF formalism. These authors do not discuss the general relations as we do here. In their opinion, the FDT should be model dependent in the nonlinear case. 6. Path integral representation and symmetry breaking The generating functional Z[J(x)] for CfPGF can be presented as a Feynman path integral. In terms of the eigenstate Icp'(x» of the operator cp(x, t = -(0) the density matrix can be written as (6.1) so that the generating functional is given by (6_2) where

U(L

=

-00,

t+

= -oo)=Sp = Tp exp(i

f J(x)cp(x») p

is the evolution operator defined along the closed time-path.

(6.3)

751 KUlIng-chao Chou tl al., Equilibrium and nontquilibrium formalisms made unified

71

It is known in the quantum field theory [39] that

(cpz(x)\ U(tz, t1)\CP1(X») = N

f

'2

[dcp(x)] exp (i

J.P(cp(X» dd+1 X)8(cp(x, tz) - cpz(x»8(cp(x, t1) - CP1(X» , ~

~~

where N is a constant. The path integral representation (6.4) is valid for any t}, t2, so (6.2) can be rewritten as Z[J(x)] = N

f

[dcp(x)]8(cp(x, t+ = -(0)- cp'(x»p",·".8(cp(x, L

x exp[i

J(.P(CP(X» +J(x)cp (X»] ,

= -(0)- cp"(x» (6.5)

p

where the integration in the exponent is carried out over the closed time-path p. Since the functional dependence of p",'",' upon cp'(x) and cp"(x) is rather complicated, in general (6.5) is not very suitable for practical calculation. However, it is useful for discussing the symmetry properties of the generating functional so far as the total Lagrangian of the system appears in the exponent. We will use this representation to discuss the Ward-Takahashi (WT) identities and the Goldstone theorem following from the symmetry (section 6.3). The path integral representation would be well adapted to the practical calculation if the contribution of the density matrix can be expressed in terms of effective Lagrangian in certain simplifying cases. This possibility will be considered in section 6.1. In section 6.2 we briefly discuss the properties of the order parameter and describe two different types of phase transitions. Finally, in section 6.4, the path integral representation is used to consider the fluctuation effects.

6.1. Initial correlations In this section we derive two equivalent expressions for the generating functional to incorporate the effects of the initial correlation in a convenient way [46]. 6.1.1. Model Consider multi-component nonrelativistic field t/lt, t/lb, b = 1,2·· . n which may be either boson or fermion. The action of the system is given by

(6.6) where the free part can be written as

f

1o[t/lt, t/I] = dld2t/1 t (1)Sol(1, 2)t/I(2) == t/l t So 1t/1, p

with

(6.7)

752 72

Kuang-chao Chou el al_. Equilibrium and nonequilibrium formalisms made unified

SOl = !USo/1}t + 1}So.:e + ~Siicln, So/ = So': = [ior + (112m )V2]Sd+I(1- 2),

(6.8)

in accord with (2.23), (2.63) and (3.10)_ 6.1.2. First expression for W ~ The generating functional

(6.9) can be rewritten as (6.10)

in the incoming picture by using the Wick theorem generalized to the CTPGF case as done in section 2.2 for the Hermitian boson field_ Here So is the bare propagator satisfying the equation

(6.11) p

p

and (6.12) Le.,

w~[r, J] =

i +, f dl··· dm d1 -.. dli r(l)' -. r(m) w~m. ")(1· - . m, m.n.

m.n-I

Ii ... I)J(Ii)' .. J(I) ,

p

(6_13)

(6.14)

where 1/1" I/Ir are operators in the incoming picture and: : means normal product. Since the time ordering does not have any effects under normal product, we can rewrite (6.12) as (6.15)

where J~

= J~-J~_

(6_16)

753 KlIIUIg-cIuw 0.011 et III., EquiIibrlIllll tmd 1IOMqllilibrillllllormlllisms millie IUlijied

73

We can then write down an expansion equivalent to (6.13) as

(6.17) with

w(m.n) = i",+n-I Tr{p:IJ1I(m)··· ,J1I(1)t/li(1)'" t/lI(11):}.

(6.18)

We note that W~ and w~m. n) are defined on the closed time-path, whereas WN and w(m. n) are defined on the ordinary time axis. Taking into account that in the incoming picture the field operators satisfy the free field equation, we have

f

di' .5ol (i, i')Wg"·n)(l··· i" .. m, Pi ••• I) = 0,

p

f

di'

w~m.n)(l·· . m, Pi· .• i"" 1)Sol(i', T) = 0,

(6.19)

p

or equivalently,

f f

di' 50/(i, i') w(m. n)(l' .. i' ... m, Pi ••• I) = 0 ,

di' w(m. n)(l' .. m, Pi· .• i" .. 1)So1(i', 1) = O.

(6.20)

As far as the initial condition is fixed at t = -co, we are not allowed to integrate by parts arbitrarily with respect to ill at. The correct direction of acting ill at is indicated by the arrow. Substituting (6.15) into (6.10) we get the first expression for Z[J] we would like to derive as (6.21) from which we can obtain the generalized Feynman rule. We see from (6.21) that the density matrix affects only the correlation generated by 11 and h. So far as W(m. n) satisfy eq. (6.20), the contribution of the density matrix can be expressed in terms of the initial (sometimes called boundary) condition for Green's function.

6.1.3. Second expression fOT W~ Now we derive another expression for the CfPGF generating functional. Using the following identity (up to an unimportant constant)

754 74

Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

f

exp(-irSol) = [dll/)[dl/r] exp{i(I/rtSO'Il/r + P I/r + if/J)},

(6.22)

p

it is easy to show that exp{i(-PSol + W;'[P, J])} =

f

[dl/rt)[dl/r) exp{i(rl/r+ I/rtJ)}exp {iW;' [±i :I/r,i 6~t]}eXp(il/rtSoll/r),

(6.23)

p

if the path integration is taken by parts. Taking into account that

'0: exp(il/rtSillI/r) = exp(il/rtSoll/r) ('O~ ± il/rtSill) , 'O~t exp(il/rtSilll/r) = exp(il/rtSillI/r) ('O~t + iSilll/r ) ,

(6.24)

(6.23) can be transformed into exp{i(-PSol + W;'[P,J])} =

f [dl/rt)[dl/r) exp{iWI/r+

I/rtJ + I/rtSillI/r)}

p

(6.25) Using (6.8) and the convention agreed in (2.69H2.74) we find that (6.26)

f

f f dySiI~(X,Y)I/r4(Y)'

dyI/rt(y)u3Sil 1(y,x),., = dYl/rl(Y)SO'"t(y,x),

,.,t I dySO'l(X,Y)U31/r(Y)=

(6.27)

and obtain from (6.15) that exp{i

w: [± i '0: - I/rtsO'\ i 'O~t - Siill/r ]} = exp{i WN[-I/rlSO'/, - SO'~I/r]} = exp{iW:[-I/rtSO'l, SO'II/r]}.

(6.28)

755 KUlJng·chao Chou et aI.• Equilibrium and nonequilibrium formalisms made unified

75

Substituting (6.28) into (6.25) and the resulting expression into (6.21), we find

J

Zp[Jt, J] == [dq/][dl/l] exp{i(Io[I/It, 1/1] + l in.[ I/It, 1/1] + r 1/1 + I/ItJ)} exp{i W~bl/ SOl, -Soll/l]} p

(6.29) as the second path integral representation for the generating functional. It is easy to rederive (6.21) from (6.29), so these two expressions are equivalent to each other. Note that this expression is different from that given by (6.5) in so far as the contribution of the density matrix appears here as an additional term W;' in the action. According to (6.28), this term does not depend on field variables I/I:,I/Ic describing the dynamical evolution, but does depend on I/Il, 1/111 describing the statistical correlation. It is also obvious that W;'[-I/ItSo\ -So 11/1] has nonvanishing contribution to the path integral only at the end points because of (6.19). 6.1.4. Two-step strategy For a general nonequilibrium process, (6.29) can be rewritten as (6.30) where

z~[r,J]==

J[dl/lt][dl/l] exp{i(Io+l .+rl/l+ I/ItJ)}

(6.31)

in

p

is the generating functional for the ground state. Since z~ has exactly the same structure on the closed time-path as that of the standard quantum field theory, we can first calculate ~ and then "put into" it the statistical information via (6.30). Such "two-step" strategy is well known in solving the Liouville problem in classical statistical mechanics. Many interesting nonequilibrium phenomena can be described by a Gaussian process, i.e., w~m. n)(1

... m, n ... I) == 0 except for

W~· 1)(1, I)

t- 0,

(6.32)

for which the contribution of the density matrix reduces to replacing the bare propagator Sop by Gop (x, y) == Sop(x,y)-

W~I. I)(X,

y).

(6.33)

For the thermoequilibrium case eq. (6.33), after Fourier transformation, is identical to (2.21). A more rigorous derivation of the diagrammatic expansion for thermoequilibrium will be given in section 9.1. Another possibility of simplification comes about when the state is stationary due to the microscopic time reversal invariance, the generalized FDT then holds as shown in section 4.2. As seen from (6.20) and (6.21), w~·n) as solutions of the homogeneous equation can be specified by the FDT.

756 76

Kuang-chao Chou el al.• Equilibrium and nonequilibrium formalisms made unified

To sum up, the determination of the CfPGF generating functional can be divided into two steps: To first "forget" about the density matrix in calculating the generating functional without the statistical information and then "put it into" the generating functional at the second step. In the general case this can be done using (6.30), but a significant simplification results if the initial correlation is Gaussian or a generalized FDT holds. If we are interested in some order parameter Q(x) which is a composite operator of the constituent field, we introduce an additional term h(x)Q(x) 'in the action. The generating functional for the order parameter is given by (6.34) in terms of Zp[Jt, J] for the constituent field. The extension of (6.21), (6.29) and (6.30) to this case is obvious.

6.2. Order parameter and stability of state It is well known that the vertex functional F[ipc] is most suitable for describing the symmetry breaking, inasmuch as it is expressed explicitly in terms of the order parameter rpc. In section 2.2 we have derived an equation (2.48) satisfied by it. Before going on with the discussion of the order parameter, we rewrite this basic equation of the CTPGF formalism in another equivalent form.

6.2.1. Functional form for the vertex equation Let the total action of the system be presented as 1,=1+1.=

I

I

. 0 for some i, then an instability with this k occurs to form a new space-time structure. However, as discussed in section 3.2, in thermoequilibrium we have A(k) =-!iF_(k)[l- exp(-pko)]

= ko Wa(k)[l- exp(-pko)] >0,

and aD! ako > 0 for k o> 0, whereas both of them change sign for ko < 0, so that such instability cannot occur in an eqUilibrium system. In fact, it usually appears in far-from-equilibrium systems under certain special conditions, for example, in a laser system the q>. = solution is unstable above the threshold of pumping. (2) A(k) is not small compared with D(k) as i ... 0. For an eqUilibrium system we can write

°

A(k) = koY,

y

> 0,

D(k) = Do + ako + ....

Up to the first order of ko, the solution of (6.46) is ko = -D(O)/(a + ioy),

with 'Y

1m ko = -2--2 D(O) . a

+ 'Y

(6.49)

759 Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made uniji£d

79

Hence the phase transition occurs at D(O) = O. This is the ordinary second-order phase transition if the nontrivial solution grows continuously from zero. Otherwise, the point D(O) = 0 will correspond to supercooling or superheating temperature.

6.3. Ward-Takahashi (WT) identity and Goldstone theorem In this section we derive the WT identity satisfied by the CfPGF from the invariance of the Lagrangian of the system with respect to global transformations of a Lie group G.

6.3.1, Group transformations Let ip(x) be the constituent field and O(x) the order parameter. Each of them has several components forming by themselves bases of unitary representations. Under the infinitesimal transformation of G ip(x)~

ip'(x) = ip(x) + &ip(x),

&ip(x) = (,,(iI~L X~(x)a,,)ip(x) = iI"ip(x)(",

(6.50)

= O(x) + &O(x), &O(x) = (.. (iL!!') - X ~ (x )19" )O(x) = iL"O(x )(" ,

(6.51)

O(x)~ O'(x)

where ( .. are a total of nG infinitesimal parameters for group G and I~), L~) representation matrices for the generators of G. X~ are associated with the transformation of coordinates

(6.52) It can be easily shown that the Lagrangian function transforms in this case as

(6.53) where

j:;(x) =i

8O~~X) I,,!p(x)- ..'tX~(x)

(6.54)

is the current in the a direction and (.. (x) is an arbitrary infinitesimal function. If the Lagrangian is invariant under the global transformation of G, it then follows that

or equivalently,

o"],,'''() .( &:t' x = I 0" 8O,.!p(x)

&..'t) ()

&ip(x) I"ip x ,

(6.55)

760 80

Kuang·chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

i.e., the current is conserved provided rp(x) is the solution of the Euler-Lagrangian equation. Substituting (6.55) into (6.53) yields (6.56) which is the change of :£ under local transformation of (, if it is invariant under the global action of the same group.

6.3.2. WT identities The path integral representation for the generating functional (6.5) can be written as

Z[J(x), h(x)] = N

f

[drp(x)] exp

{i f (:£(rp(x» + J(x)rp(x) + h(x)Q(x»} p

x (cp(x,

t+ = -

oo)lplcp(x, L = -00» .

(6.5')

Performing a local transformation of rp(x) in (6.5') with (a(x) satisfying the following boundary conditions (a (x, t± = -00) = lim (x, t) = 0,

(6.57)

Ix 1.... 00

and taking into account that the measure [dcp(x)] does not change under unitary transformations, we obtain from the in variance of the generating functional that (6.58)

(a"f~(x» = -iJ(x)IaCPc(x) - ih(x)LaQc(x).

On the other hand

(a,.j~(x» = Z-la,.j~ (rp(x) = -i 8J~X»)Z[J(x), h(x)],

(6.59)

from which it follows that (6.60) by use of (6.38). Using the generating functional W for the connected Green function, the WT identities (6.60) can be rewritten as

a,.Ja',. (8~';)

-.8J(x) _8_) -__ .IJ ()Ia 8J(x) 8 W _ 'h() 8W X La 8h(x)' J

X

I

(6.61)

761 KuolIg-chao Chou el aI., Equilibrium and lIonequilibrium formalisms made unified

81

Taking functional derivatives of (6.61) with respect to J(y) and then setting J(y) = 0, we obtain WT identities satisfied by CTPGFs of different order. In terms of vertex generating functional r[r,o.] (6.61) can be expressed as

(6.62) Here we allow r,o(x) to be either a boson or fermion field. Taking the functional derivative 8/8r,o.(Y) of (6.62) and putting r,o.(y)::: r,odJ, the symmetry breaking in the absence of J(x), we obtain WT identities for different vertex functions.

6.3.3. Goldstone theorem Now we use the WT identity to discuss the symmetry breaking after phase transition. Suppose the equations

have solutions r,o.(x) = 0, O.+(x) = O.-(x) "I- O. Differentiating (6.62) with respect to O.(y), setting J(x) = h(x) = 0 and integrating over x, we obtain

which can be rewritten in the single time representation as (6.63a) In matrix form (6.63a) appears as (6.63b) i.e., LaO.(x) is the eigenvector of rr with zero eigenvalue. Assume O.(x) is invariant under a subgroup H of G with nH as its dimension. Therefore, a

= 1, ... , nH

if a belongs to the generators of H. On the contrary, if a belongs to the coset G/H, LaO. "I- 0, then (6.63b) shows that rr has no - nH eigenvectors with zero eigenvalue. Suppose the representation to be

762 82

Kuang·chao Chou el aI., Equilibrium and nonequilibrium formalisms made unified

real, then taking complex conjugation of (6.63b) we obtain (6.64)

due to the orthogonality of La. Separating the real and imaginary parts we find OcLa . A = A . LaOc "" 0 ,

(6.65)

i.e., LaOc are zero eigenvalue eigenstates for both D and A. It follows from the Dyson equation (3.10) that the retarded Green function Gr has no - nH non dissipative elementary excitations called Goldstone modes. If Oc does not depend on coordinates, in Fourier representation of x - y such excitation occurs at zero energy and momentum and is called Goldstone particle. 6.3.4. Applications The Goldstone bosons considered above have important consequences in the symmetry broken state. For example, in laser system the U(l) symmetry is broken, so the corresponding Goldstone boson leads to the divergence of the fluctuation which in turn makes the classical solution unstable. This phenomenon in the CTPGF approach was observed by Korenman [25] and was analyzed by us in [40,52]. The WT identity is used to derive a generalized Goldstone theorem in a slowly varying in time system. As its consequence the pole of the Green function splits into two with equal weight, equal energy but different dissipation. Combined with the order parameter (average value of the vector potential) these two quanta (one of which is the Goldstone boson) provide a complete description of the order-disorder transition of the phase symmetry in the saturation state of the laser. We have also used the WT identity in combination with order parameter expansion (cL section 3.5) to derive the generalized TDGL equation [43,47]. We will apply the same identity to discuss the localization problem in section 8.3 [56]. We should mention that the transformations given by (6.50) and (6.51) are linear. We can consider nonlinear transformations under C as we did in [43,47]. In that case we need to take into account the Jacobian of transformation for the path integral. The result thus obtained turns out to be the same as the nonlinear mode-mode coupling introduced phenomenologically by Kawasaki [67]. 6.4. Functional description of fluctuation 6.4.1. Stochastic functional It is known that the Gaussian stochastic process [I (t) appearing iii the Langevin equation (cf. (4.33))

ao;/at = K,(Q)+ [jet)

(6.66)

can be presented by a stochastic integral [86]. Equation (6.66) can then be considered as a nonlinear mapping of the Gaussian process on to a more complicated process O/(t). Realization of such mapping actually results in a functional description of O/(t) [72]. Nevertheless, such functional description can be achieved by a more straightforward way [87-89]. Consider the normalization of the 5-function under path integration (6.67)

763 Kuang·cilao CIIou et aI., Equilibrium and lIonequiJibrium formalisms made llllijied

83

where the Jacobian .1 (0) appears because the argument of the 8-function is not 0 itself but a rather complicated expression. Neglecting multiplicative factors, .1 (0) turns out to be [72] .1(0) = exp

{-! I 8K(0)/80} .

(6.68)

Using the integral representation of 8-function (6.69) (6.67) can be rewritten as (6.70) Inserting the source term

one obtains the generating functional

Zf[J,J] =

I[dO] [~~] exp {I [iO (a~ - K(O)-~) -~:~ + i(JO +JO)]} ,

(6.71)

with the normalization condition

Zf[O,O] = 1. Averaging over the random noise distribution W[~] oc exp(-!t'o'-l~)

(6.72)

with u as the diffusion matrix, one obtains Lagrangian formulation of the generating functional for the statistical fluctuation

Z[J,J] =

I [dO][~~]

exp

The Gaussian integration over Z[J]=

{f [-!OuO+iO (aa~ - K(O») -!:~ +i(JO +JO)]} .

(6.73)

0 can be carried out to yield

18K . ]} . a,-K(O)-JA) u- l(ao a,-K(O)-JA) -28Q+IJO I [dO]exp {I[ -21 (ao

(6.74)

764 84

Kuang·chao Chou et al., Eqwlibrium and noneqwlibrium formalisms made unified

Historically, the theory of noncommutative classical field was first suggested by Martin, Siggia and Rose (MSR hereafter) [90]. This theory has been extensively applied to critical dynamics [61] and has been later reformulated in terms of a Lagrangian field theory [88,89] as presented by (6.73) and (6.74).

6.4.2. Effective action We will show now that such description occurs within the CfPGF formalism in a natural way [43,47], postponing the comparison with the MSR field theory to section 9.3. Let Q/(x) be composite operators of the constituent fields ipJ(x). Both of them are taken to be Hermitian Bose operators. Assuming the randomness of the initial phase, the density matrix is diagonal at moment 1= 10, i.e., (ip'(x, to)lplip"(x, (0

»= P(ip'(x), to)8(ip"(x, to) - ip'(x, to» .

(6.75)

The initial distribution of the macrovariables Q1(x) is then given by P(Q/(x), to) = Tr(8(Q/(x) - Qi(ip(X»P}

f

=

[dip(x)]8(Q/(x)- Q/(cp(x»P(cp(x), to).

(6.76)

The generating functional for Q/ can be written as

Z[J(x)] = exp(i W[J(x)]) =Tr{

Tp[ exp (i f J(x)Q(ip(X»)]p} p

(6.77) where 8(cp+ - 1,0-) ==

f

dcp'(x) 8(ip(x, t+ = to)- ip'(x»8(cp(x, L

= (0)- ip'(x»P(ip'(x), to).

(6.78)

MUltiplying (6.77) by the normalization factor of the 8-function

f [dQ]8(Q+ -

Q_)6(Q(x)- Q(ip(x») = 1,

(6.79)

changing the order of integration to replace Q(ip(x» by Q(x) and using the 8-function representation (6.69) with 6 changed for I, we can rewrite (6.77) as Z[J] =N

f [dQ] exp (i

Sell +

if JQ )8(Q+ - Q-), p

(6.80)

765 K/lQIIg-chao Chou et aI., EquilibriwtJ and nonequilibrium formalisms malk unified

85

where exp(iSefI[O]) =

f

[dl/21T1 exp (i W[I1 - i

f 10) .

(6.81)

p

Here we have performed direct and inverse Fourier transformations of the path integral. So far as a continuous integration is taken over l(x), W[I] can be considered as a generating functional in the random external field. Calculating the functional integral in the one loop approximation which is equivalent to the Gaussian average, we can obtain the effective action Seff[ 0] for O(x). So far we have discussed the case when macrovariables are composite operators. The same is true if part or all of macrovariables are constituent fields themselves. A new "macro" field can be also introduced by use of the 6-function. However, one should carry out the path integration simultaneously in spite of the fact that the initial correlations are multiplicative, because in general the Lagrangian itself is not additive in terms of these variables. Before going on to calculate the integral (6.81) we first discuss the basic properties of the effective action Seff[ 0]. It is ready to check that apart from the normalization condition (2.103) the generating functional for the Hermitian boson field also satisfies the relation (6.82) It then follows from (6.81) that

(6.83) Hence Sell is purely imaginary for O+(x) = O_(x). Setting O,.(x) = 0 + flO± and taking successive functional derivatives of (6.83) near 0, we obtain (6.84) SFi/(X, y) = SFjl(Y, x) = -S;/i(Y, x),

5,,,,/ (x, y) = S"/I (y, x) = -S:/I(Y, X),

(6.85)

where

We see that S,j respect the same symmetry as the two-point Green functions (cf. (2.134» and vertex functions (3.12b). It the system is invariant under a symmetry group G, i.e., both the Lagrangian and the initial distribution do not change under

then

766 86

Kuang-chao Chou et al., Equilibrium and nonequi/ibrium formalisms made unified

WW(x»)

If = ~(X)V;I(g),

= W[I(x»),

Sell[QB(X»)=Sef([Q(x)],

Qf= V;j(g)Qj(q»_

The above said is true if the effective action Sell is calculated exactly. However, the symmetry properties of Sell, being related to those of the Lagrangian, may be different from the latter due to the average procedure. If the lowest order of WKB, i.e., the tree approximation is taken in (6.81) we find that (6.86)

Q = 6W/6I,

(6.87) In this case, Sell inherits all properties of the generating functional r[Q], e.g.,

(6.88) (6.89)

(6.90) (6.91) In accord with (6.87), (3.5) and (3.19) we have (6.92)

-is,,(k) >0

after taking Fourier transformation. Near thermoequilibrium we find from (3.40) S_;j -

S+ij~-

fikoS-I/(k).

(6.93)

kct-+O

6.4.3. Saddle point approximation Up to now we have discussed only the general properties of Sell[ Q). In principle, Sell can be derived from the microscopic generating functional W[I] by averaging over the random external field, it can be also constructed phenomenologically in accord with the symmetry properties required. We now calculate (6.81) in the one-loop approximation. Near the saddle point given by (6.86) we expand the exponential factor in (6.81)

E=W-

I QI=r-~I MG(2)t:.I=r-!I t:.Pui';u~J, p

p

(6.94)

767 Kuallg·chao Chou el al.• Equilibrium and lIonequilibrium fonnalisms made unified

87

where (] is the two-point connected functions, tJ.jT = (tJ.I+, tJ.L). Up to a numerical constant, the result of the Gaussian integration is (6.95) It then follows from the Dyson equation (2.57) that iSefJ[Q] = ir[Q] + !Tr In

t.

(6.96)

By use of the transformation formula (2.59) we have Idet tl = Idet tl = Idet r.lldet f.1 = Idet frl2 , where 82 f fr(x, y) = 8QA(X)8Qc(Y) . As shown in section 3.5, (6.97) Comparing (6.97) with (6.67) we find that 82rt8QA(X)8Qc(Y) is just the transformation matrix up to the numerical factor ,),-1. Therefore, we can calculate the Jacobian in the same way to get

J

iSefJ[ Q] = if[Q]-! 8K/8Q,

(6.98)

with K = -')'-18~/8Q.

In the path integral (6.80) the most plausible path is given by 8SefJ[Q]l8Q(x ±) = -l:t(x) ,

(6.99)

= to) = Q(x, L

(6.100)

Q(x, t+

= to) .

In the tree approximation of (6.81) 8SefJ /8Q

= -J = -,),iJQliJt -

8~/8Q,

which is nothing but the TDGL equation derived in section 3.5 (cf. (3.85».

(6.101)

768 88

KUQIIg-chao Chou et aI., Equilibrium and nonequilibrium formalisms made unified

6.4.4. Role of fluctuations We now discuss fluctuations around the most plausible path. In the CTPGF approach there is an additional way of describing the fluctuation: To aJlow field variables to take different values on the positive and negative time branches. Changing variables in (6.80) to Q = Qe = ~(Q+ + Q_), QtJ. = Q+ - Q-, the effective action can be expanded as

Seff[Q+(X), Q_(x)] = S.f![Q, Q] +!

f

[6S.f!/I)Q(x+)+ 6Se f!/6Q(x- }]QtJ.(x)

f

+ 1 QtJ.(x}(S++ + s__ + S+_ + S_+}(x, y)QtJ.(y) + .... Denoting

!i(S++ + s__ + S+_ + S_+ )(x, y) == - y(x )u(x, y)y(y)

(6.102)

and using (6.88), (6.98), (6.101), we obtain eiW[J(x») =

f

[dQ(x)][dQtJ.(x)] exp [ -!

- i

f

(y(x)

f

QtJ.(x)y(x)u(x, y)y(y)QtJ.(y)

f

aa~ + :~) QtJ.(x)- ~ :~ + i

f

(ftJ.Q + feQtJ.)] 6(QtJ.(x».

(6.103)

If we take ftJ. = f and y(x)QtJ.(x) -+ 6, fey-I -+ j the stochastic generating functional (6.73) is retrieved. Carrying out integration over QtJ.(x) wiIllead to an equation identical to (6.74). It is important to note here, that in the CfPGF approach, fe, the counterpart of j in MSR theory, is the physical external field, whereas ftJ. = f+ - L, the counter part of J in MSR theory is the fictitious field. It is clear by comparing (6.103) and (6.73) that u(x, y) is the correlation matrix for the random force. If Q is a smooth function of x, u can be taken as a constant (6.104) which reduces to (6.105) by virtue of (6.87) and (6.90). This is the expression we have used in section 4 (cf. (4.35}) to discuss the symmetry properties of the kinetic coefficients, if it is generalized to the multi-component case. According to the definition of y given by (3.81) (6.106) Comparison of (6.106) with (6.105) yields the Einstein relation (FDT)

769 KUiIIIg-cl!ao Ow" et al., Equilibrium and nOMqUilibrium formalisms mtule IUliMd

u = 2/Py

89

(6.107)

for the diffusion coefficient in the case of single macrovariable. For simplicity we consider only one component order parameter in this section, but we consciously write some of formulas in such a way, so that the generalization to the mUlti-component case is obvious_ To sum up, the MSR field theory of stochastic functional is retrieved in the CfPGF approach if the one-loop approximation in the random field integration and the second cumulant expansion in Q6(X) are taken. The possibility to go beyond such approximation is apparent. 7. Practical calculation scheme using CTPGF As we have seen, the crPGF provides us with a unified approach to both equilibrium and nonequilibrium systems. However, to make it practically useful we need a unified, flexible enough calculation scheme. Such scheme has been already worked out by us [48,49]. In fact, most of the calculations carried out by us so far using CfPGF [40,46,52-57] can be cast into this framework. Consider a typical situation when fermions t/I(x), t/lt(x) are coupled to the order parameter Q(x) which might be a constituent field like the vector potential A .. (x) in the laser case, or a composite operator like x(x) = t/I dX)t/I ~ (x)

in the theory of superconductivity, or S= t/lt(x)!ut/l(x) in the case of itinerant ferromagnetism, where u are Pauli matrices. The boson field Q(x) via which the fermions interact with each other, may be nonpropagating at the tree level like the Coulomb field. However, the radiative correction will in general make Q(x) a dynamical variable and the fluctuations around the mean field Qc(x) will propagate and form collective excitations. Therefore, the system is characterized by the mean field Qc(x) and the two kinds of quasiparticles - constituent fermions and collective excitations with their own energy spectrum, dissipation and distribution. Such a way of description has been found useful in condensed matter physics [1-3,21-24,91], plasma physics [33,34] as well as in the nuclear many-body theory [92-96]. In this section we first (section 7.1) derive a system of coupled equations satisfied by the order parameter and the two kinds of Green's functions using the generating functional with two-point source terms. Next (section 7.2), the technique of Cornwall, Jackiw and Tomboulius (CIT) [97], developed in the quantum field theory to calculate the effective potential for composite operators is generalized to the CfPGF case and is used as a systematic way of computing the self-energy part by a loop expansion. In thermoequilibrium when the dissipation is negligible, the mean field Qc(x) and the energy spectrum of the fermion field are determined to the first approximation by the Bogoliubov-de Gennes (BdeG) [98] equation, in which the single particle fermion wavefunction satisfies the Hartree type self-consistent equation without Fock exchange term. In section 7.3 we discuss the generalization of the BdeG equation in the four-fermion problem, whereas the free energy in various approximations is calculated explicitly in section 7.4 by directly integrating the functional equation for it. Some problems related to those discussed in this section were considered by Kleinert using the functional integral approach [99].

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Kwollg-chao Chou et al., Equilibrium alld lIonequilibrium formalisms made unified

7.1. Coupled equations of order parameter and elementary excitations 7.1.1. Model Consider a fermion field I/I(x) interacting via a boson field O(x) with the action given by (7.1)

where 10[I/It, 1/1] =

J

I/It(X)SOl(X, y)I/I(Y) ,

(7_2)

p

10 [0]

=!

I

O(X).1ol(X, y)O(Y) ,

(7.3)

P

with So I (x, y), .1 OI(X, y) as inverse fermion and boson propagators respectively, For a system with four-fermion interaction only, we can use the Hubbard-Stratonovich (HS) transformation [100} to introduce the effective fermion-boson interaction. Let the generating functional for the order parameter O(I/It(x), I/I(x» be defined as

Using the Gaussian integral identity, i.e. the HS transformation, eq. (7.4) can be presented as (7.5)

Zp[h] =

I

[dl/lt][dl/l][dO] exp{i[/o[I/It, 1/1] + lo[ OJ + lint[I/It, 1/1, OJ + hOl},

(7.6)

p

where

lint[I/It,I/I,O] being the nonlinear interaction. It is important to note that Mo,.1(j1 are two-point functions, independent of either field variables or external source. Therefore, up to an additive constant Mo the Green functions of the original system are the same as those of the effective system described by Z. The formal ambiguity in defining the [dO(x)] integration [96] can be avoided by imposing the condition

~\ Sh(x)

_ Si,.[h] \ Sh(x) h-O

hzO -

(7.7)

771 KUlJIIg·chao Chou et aI., EAjuilibrium and lIOIIeI/uilibrium formalisms made UIIifitd

91

We see thus the four-fermion interaction can be considered on an equal footing with system of fermions interacting through a constituent boson field Q(x).

7.1.2. Two-point source The generating functional with a two-point source is defined as

Zp[h, r, J, M, K]

J

= [dl//][drfr][dQ] exp{i[Io[t/lt, t/I] + 10[Q] + 1int [t/lt, t/I, Q] p

(7.8)

where W~[t/lt, t/I, Q] takes care of contribution from the density matrix as discussed in section 6.1. Here we adopt the abbreviated notation and M(x, y), K(x, y) are external sources to generate the second-order CfPGFs. Introducing the generating functional for the connected CfPGF as usual W[h, r, J, M, K) = -i In Z[h, r, J, M, K] ,

(7.9)

it then follows that 8Wp/8h (x) = Qe(X) ,

(7.10a)

8 Wp/8r(x) = t/le(X) ,

(7JOb)

8Wp/8J(x) = -t/l~(x),

(7.10c)

y», 8Wp/8K(y, x) = -(t/le(X)t/I~(y)+ iG(x, y».

(7.10d)

8Wp/8M(y, x) = ~Qe(X)Qe(y)+ iJ(x,

(7J0e)

In case of vanishing sources Qe(X), t/le(x) and t/I:(x) become expectation values of the corresponding fields Q(x), t/I(x) and t/lt(x), whereas J(x, y), G(x, y) are the second-order CfPGF for the boson field Q(x) and the fermion field t/I(x), t/lt(x) respectively. 7.1.3. Coupled equations The generating functional for the vertex CfPGF is defined as the Legendre transform of Wp , rp[Qe, t/I:, t/le, J, G) = W[h, r, J, M, K) - hQe- t/I!] - r t/le -! Tr[M(QeQe + iJ)] - Tr[K(t/I.t/I! + iG)] ,

where

(7.11)

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Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

Tr[M(QeQe+ i.1)] ==

f

M(x, y)(Qe(y)Qe(X)+ i.1(y, X»,

p

Tr[K(I/Iel/l~ + iG)] ==

f

K(x, y)(!{Ie(y)I/Ic(x) + iG(y, x».

p

Using (7.10) it is straightforward to deduce from (7.11) that afp = -h(x)aQc(x)

f

M(x, y)Qe(y),

(7.12a)

K(x, Y)I/Ic(y) ,

(7.12b)

p

afp al/l~(x) = -J(x) -

f p

f

afp _ t() t a!{lc(x) - J X + I/Ie(y)K(y, x),

(7.12c)

p

~ 1 a.d(x, y) = 2i M(y, x),

(7.l2d)

8fp 'K(y) 8G(x, y) = I ,x.

(7.l2e)

Equations (7.12) form a set of self-consistent equations to determine the order parameters Qc(x), !{I~(x), I/Ic(x) as well as the second-order CTPGF .1 (x, y) and G(x, y) provided fp is known as a functional of these arguments. In almost all cases of practical interest the condensation of the fermion field is forbidden in the absence of the external source, i.e., I/I! = !{Ie = 0 for = J = O. On the other hand, the condensation of the boson field (elementary or composite) is described by the order parameter Qc(x). Since the energy spectrum, the dissipation and the particle number distribution are determined by the second-order CTPGF, eqs. (7.12a, d, e) are just those equations we are looking for. The only question remaining is how to construct the vertex functional rp. In the next section a systematic loop expansion will be developed for this purpose.

r

7.2_ Loop expansion for vertex functional 7.2.1. CJT rule [97] Without loss of generality in what follows we will set r

= J = 1/1: = !{Ie = O. The vertex functional

773 Kua/lg-chao Chou et al., Equilibrium and /lonequilibrium fomralisms made u/lifted

93

is the generating functional in 0 for the two-particle irreducible (2 PI) Green's functions expressed in terms of propagators .1 and G. To derive a series expansion for rp we note that after absorbing into the effective action, i.e.,

W:

(7.13) the only difference remaining between the CTPGFs and the ordinary Green functions in the quantum field theory is the range of the time axis. For CfPGF the time integration is taken for both positive and negative branches. Hence the loop expansion technique for the vertex functional and its justification developed by eJT [97] in the quantum field theory can be easily extended to the CTPGF formalism provided the difference in the definition of the time axis is properly taken into account. Here we shall simply state the result as

rp[ 0 0 ,.1, G] = I[ 0 0 ] - Wi Tr{ln[.1 ol.1]- L1 01 .1 + I}

+ iii Tr{ln[SoIG] -

GOIG + I} + r 2p [ Oc,.1, G] ,

(7.14)

where

I[Oo] =IetM/, 1/1, O]I",~",t_o '

(7.15a)

Q~Qc

(7.15b)

(7.1Sc)

(7. 15d) Note that G OI is different from SOl and

(7.16)

r

The quantity 2p[ 00'.1, GJ appearing in (7.14) is computed as follows. First shift the field O(x) in the effective action IetI[I/It, 1/1, 0] by Oc(x) and keep only terms cubic and higher in I/It, 1/1 and 0 as interaction vertices which depend on Oc. The 2p is then calculated as a sum of all 2PI vacuum diagrams constructed by vertices described above with ..1 (x, y), G(x, y) as propagators_ In (7.14) the trace, the 1 products..1 0 .1, etc., as well as the logarithm are taken in the functional sense with both internal indices and space-time coordinates summed over.

r

7.2.2. Coupled equations The self-energy parts for the fermion and the boson propagators are defined to be

774 94

K/IIJIIg·chao Chou el 01., Equilibrium and nonequilibrium jormalisms made unified

-i ~ !(X, Y) == h &G(y, X)'

(7.17a)

(7.17b) Hereafter in section 7 we restore the Planck constant h in formulas to show explicitly the order of magnitude. The equations for the order parameter O.(x) and the second-order CTPGF .:1 (x, y) and G(x, y) for a physical system can be obtained from eqs. (7.12) by switching off the external sources. We have thus: l

&fp

&O.(x)

=&I[O.]_iIiTr{&Gii G}+ Sf2p =0

SO.(x)

&O.(x)

&O.(x)

(7.18)

,

2i Sfp _ -I( -I _ Ii M(Y,x)-.:1 x,y)-.:1 o (x,y)+l1(x,y)-O,

(7.19)

(7.20) Rewritten in the ordinary time variable in accord with the rule set in section 2.3, eq. (7.18) becomes the generalized TDGL equation for the order parameter, whereas (7.19) and (7.20) are the Dyson equations for the retarded and advanced Green functions along with the transport equation for the quasiparticle distribution. Equation (7.18) can be rewritten in a symmetric form as

(7.18') The retarded, advanced and correlation Green functions are related to the matrix

A as (7.21)

in accord with (2.12). Therefore, the Dyson equations for the retarded propagators take the form (7.22a)

(7.22b) The corresponding equations for the advanced functions can be obtained by taking the Hermitian conjugation of (7.22), whereas equations for correlation functions appear in the matrix form as

775 KlIIIIIg-chao Chou et aI.• Equilibrium and 1I000uiUbrium formalisms made uni/ild

95

.de =-.dr(.d ilc1-lIe).d a ,

(7.23)

Ge = -G.(GoJ - te)G •.

(7.24)

As shown in section 3.3, the latter equations reduce to transport equations for the quasiparticle distribution. 7.2.3. Summary To sum up, we have derived seven equations to determine seven functions Qc(x), .d" .d., .dc, Gn G. and Ge , from which the order parameter as well as the energy spectrum, the dissipation and the distribution function for the corresponding quasiparticles can be calculated. Up to now we have not yet made any approximations. As is well known [97], the loop expansion is actually a series expansion in fl. Therefore, for systems which can be described by quasi classical approximation one needs only the first few terms of this expansion. In fact, one recovers the mean field result if the contribution from r2p is neglected altogether. In some other cases like in the theory of critical phenomena, one needs to partially resume the most divergent diagrams. For most cases of practical interest including thermoequiIibrium, the initial correlations expressed in terms of W~ are Gaussian. As shown in section 6.1, in such cases the statistical information can be included in the free propagators .do, So by FDT, so that W;-' term drops out from the effective action. Hence the analogy with the quantum field theory can be carried through even further for such systems. As seen from the derivation, this calculation scheme can be applied to both equilibrium and nonequilibrium systems. It is particularly useful when the dynamical coupling between the order parameter and the elementary excitations is essential. We note in passing that the logical simplicity of the present formalism comes partly from introducing the two-point sources M(x, y), K(x, y) and performing Legendre transformation with respect to them. 7.2.4. Comparison with earlier formalism To make contact with the generating functional introduced before (marked by a prime), we note that Z~h]

= Zp[h, M, K]IM_K=O,

(7.25a) (7.25b)

W;[h] = Wp[h, M, KlIM-K-o, r~ Qe]

= rp[ Qe,.d, G]lar"lM-ar"laa=o,

8r~ Qe] 8Qe(x) =

[8rp[ Qe, .1, G] 8Qc(x)

I ]I 4.a

ar"lM-ar"laa-o'

(7.25c) (7.25d)

Previously, an effective action method was introduced by us in the third paper of reference [46] to calculate r; explicitly. The disadvantage of that technique compared with the present formalism lies in tile difficulties connected with fermion renormalization when the fermion degrees of freedom were integrated out at the very beginning. 7.2.5. Applications We have already applied the present formalism to study the weak electromagnetic field in super-

776 96

Kuang-chao Chou et al.• Equilibrium tuuJ nonequilibrium formalisms matk unified

conduction [53] as well as the nonequilibrium superconductivity in general [54], the dynamical behaviour of quenched random systems and a long-ranged spin glass model in particular [55], the quantum fluctuations in the quasi-one-dimensional conductors [57] and the exchange correction in systems with four-fermion interaction [49]. The last topic will be discussed in the following two sections, whereas the random systems are considered in section 8.

7.3. Generalization of Bogoliubov-de Gennes (BdeG) equation As mentioned in the introductory remarks to this section, the self-consistent equations for the order parameter Qc(x) and the complete set of single-particle fermion wavefunctions I/In(x) are known as the BdeG equations which are of Hartree-type without exchange effects being accounted for. There have been some attempts of extending these equations to include the correlation effects with limited success [96]. These authors emphasize the nonuniqueness of the HS [100] transformation and make use of it to derive various approximations. As mentioned in section 7.1, such ambiguity can be avoided by using the generating functional technique with given definition of the order parameter. As we will show in this section, the successive approximations can be derived in a systematic way using the formalism developed in the preceding two sections.

7.3.1. Model The effective action of the system is given by

where i, j are indices of internal degrees of freedom, Oile matrix in this space and .1 o(x - y) = ~(tx - ty)V(x - y).

(7.27)

Using the fermion commutation relation and the matrix notation for Olk, (7.26) can be rewritten as
Te, the whole Fischer line (8.26) is located inside the stable region, so for small field h we have (8.67)

(8.68) At the critical point the Fischer relation (8.26) still holds to yield (8.69) (8.70) Below Te, there exists a critical magnetic field he above which the static fixed point is still sitting in the stable region. The value of he turns out to be (8.71) near Te with T = 1- p-l. For T < Te and h < he, the intersection of the Fischer line with the fixed point equation (8.66) is outside the physical boundary, so it can never be reached. This false fixed point is just what was found before [104, 105] to yield negative entropy at low temperatures. The only plausible fixed points are those located on the stability boundary (8.56) which in new units appears as (8.72) Solving (8.66) and (8.72) yields (8.73) (8.74) All results obtained in eqs. (8.67)-(8.71) as well as in eqs. (8.73) and (8.74) agree with those predicted by the projection hypothesis [125, 126].

8.2.6. Summary To sum up, we have found from a systematic analysis using the CTPGF formalism that a physical boundary exists on the plane q-Ixl. Above Te, the Fischer line is lying entirely in the stable region and the order parameter approaches the fixed point at this line exponentially in time. Below Te and he there are no fixed points on the Fischer line inside the stable region. In this case the fixed point is located on the stability boundary. In the presence of a persistent magnetic field h the order parameter will first decay exponentially to the intersection point ql and then decreases further along the boundary down to

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KUlJng-chao Chou el ai. Equilibrium and nonequilibn"um formalisms made unified

the fixed point qo. The magnetization calculated for the state qo agrees with what follows from the projection hypothesis. In the absence of magnetic field, qo = O. If the w = 0 component of the order parameter q(t, t) constitutes a finite part of q, say qEA, the system will finally reach a steady state with q = qEA' The value of qEA depends on the dynamic processes with infinite long relaxation time in the N ~oo limit. It is worthwhile to note that the boundary line on the q-IXI plane, and hence the fixed point, is temperature independent. As a consequence, the magnetization is also temperature independent, while the entropy does not vary with magnetic field. This is just the assumption of the projection hypothesis [125,126] which follows from the CTPGF formalism in a natural way. Comparing the results obtained here with those of Parisi [110] as well as Sompolinsky and Zippelius [117-119J, it is natural to conjecture that Parisi's function q(x), D5x 51, corresponds to our q(t, t) varying along the boundary from ql to qo. The present formalism not only elucidates the physical meaning of the order parameter but also provides us with an equation to solve its time evolution.

8.3. Disordered electron system Anderson first showed in 1958 [128] that if the electron site energies in a solid were sufficiently random, some of the energy eigenstates became localized instead of being extended as they would be in a regular crystal. Such localized states will not contribute to the electric conduction. The nature of the state depends on whether the energy value is located on the "localized" or "extended" side of the "mobility edge" (129]. The drastic change in the behaviour of the wavefunction with energy is reminiscent of phase transition. The great success in applying the field-theoretical technique and the renormalization group to critical phenomena [60] encouraged similar attempts in the localization problem. Wegner [130] suggested a nonlinear u-model to study the scaling properties of the noninteracting disordered electron system near the mobility edge [131] with conductance playing the role of coupling constant. This model was later derived from the field theory [132,133] and was further studied by other authors [134-136]. Recently, there is a revival of interest in this problem due to the discovery of the quantized Hall effect in two-dimensional electron system [137,138]. This is a challenge to the theory since according to the existing theory all electron states in disordered two-dimensional system should be localized in the absence of magnetic field [131], whereas extended states are certainly needed for explaining the observed quantized Hall effect. The extension of the field-theoretical approach to include the external magnetic field was made very recently by Pruisken et al. [139,140]. However, almost all studies of field-theoretic approach in the localization problem were based on the replica trick. With n-replicated system all O(n+, n_) or U(2n) symmetry is used to construct the nonlinear u-model. The critical behaviour from the extended state side of the mobility edge is described by a Goldstone mode resulting by virtue of the spontaneous breaking of the replica symmetry. The difficulty of the replica trick is the necessity of continuing n, originally defined for integers, to zero to get the physical result. Such process might not be unique as in the case of spin glass. It turns out that the CfPGF formalism can be also applied to the localization problem without resorting to the replica trick [56]. In this section we describe the symmetry of the model and derive the corresponding WT identities. The order parameter and the symmetry breaking pattern are also briefly sketched.

795 Kuang-choo a.ou el al., Equilibrium and nonequilibrium formalisms made unified

115

8.3.1. Model We are concerned with the effect of disorder on Green's functions of a noninteracting electron gas moving in external fields. The Lagrangian of the system is given by 2=

JI//(X)(i :t-Lo- V) cfr(x) ,

(8.75)

where 1

.

(8.76)

Lo = 2m (-IV - eA(x)f+ erp(x).

and V the random potential. In the CfPGF formalism the generating functional Z[l(x), V] is specified by the effective action which in the single-time representation can be written as (8.77) where cfr(x), lex) are two-component vectors as usual. The vertex function in the tree approximation is given by (8.78) where e is a positive infinitesimal constant. To convinve ourselves in the validity of (8.78) we note that for noninteracting fermions

rOr= E-e(k)+ie,

r

Oa

= E - e(k) - ie,

rOc = 2ie,

(8.79)

as follows from (2.23) and the Dyson equation (3.10). Equation (8.78) is then a direct consequence of (2.63). The generating functional can be thus rewritten as Z[l, V]=

J[dcfr][dl//] exp {i J[cfrt(X)((i :t -Lo-

V)0'3

+ ie(J - 171 + i0'2») cfr(x) + cfrtO'J + ]f0'3cfr ]}.

(8.80)

As shown in section 8.1, the quenched average of random potential can be carried out directly on the generating functional. It is, however, more convenient to work in the energy representation for the localization problem. After Fourier transformation the effective action is given by

f

Soft = d"x dE {cfrt(x, E)[0'3(E - Lo- V) + ie(I - 0'1 + i0'2)]cfr(x, E) + cfrtO'~(x, E) + ]f0'31/1(X, E)}. (8.81)

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KUlmg·chIUJ OIou el al., Equilibrium and nonequilibrium formalisms made unified

We restrict ourselves to the Wegner model [130] where the correlation of the random potential between different energy shells vanishes, i.e., (V(x, E»

= 0,

(V(X, E) V(y, E'» = 18(x - y)8(E - E') .

(8.82)

After averaging over the random potential V the generating functional

I

Z[l] = [dV]Z[l,

V]exp(-2~I V2(X)dx)

(8.83)

is determined by a new effective action S which differs from Self (8.81) by dropping V in the single-particle Lagrangian and a new term (8.84) The generating functionals W[l] and f[ r/lc] are defined in the standard manner so that there is no need to write them down explicitly. We would like to mention, nevertheless, that to avoid the possible confusion with sign of the Grassman variables we adopt the convention that 8/8l(x), 8/8r/1c(x) act from the right, whereas 8/8r(x), 8/8r/1~(x) act from the left.

8.3.2. Symmetry properties Before discussing the specific model under consideration we would like to make a general remark concerning the symmetry properties in the CfPGF formalism. It is well known that the U(l) symmetry of a complex field

corresponds to the charge conservation. In the CfPGF approach we deal with an action defined on the closed time-path which in single time representation appears as

I '"

S=

dt [. A2, A3 are group parameters corresponding to the three generators. The term proportional E does not respect this symmetry. Like a small magnetic field determines the direction of magnetization in an 0(3) ferromagnet, the E term can be considered as a small external field inducing the breakdown of the Sp(2) symmetry. Actually, the Sp(2) symmetry is spontaneously broken by dynamic generation of the imaginary part of the retarded (advanced) Green functions. 8.3.3. WT identities If we make an infinitesimal transformation with group parameters Aj(E) for functions r/lt(x), r/I(x) in the path integral of Z[J] we obtain the following three WT identities corresponding to the three generators of the Sp(2) group:

-2iE =

I I

dd

X

SP{(1 + (71)

ddx [r(x,

(-j 8J\t, ~~(x, E) + 8J~(:'E)8J~:E)]}

E)1718J~(:'E) +8J~:E) utl(x, E)] ,

(8.9Oa)

(8.90b)

(8.9Oc) The WT identities for various Green functions can be obtained by differentiating (8.90) with respect to r(x, E), J(x, E) and then setting r = J = O. As an example, we show how the dynamic generation of the imaginary part breaks the Sp(2) symmetry. Using eq. (2.12) we find that

798 118

Kuang-chQ(} alOu el al.• Equilibrium and nonequilibrium formalisms made unified

(8.91) Taking the functional derivative 'fl/'OP(y, E)'OJ(y, E) of both sides of (8.90b) we obtain

(8.92) As is well known, 1m Or(y, y, E) is proportional to the density of states p(E). It is different from zero certainly for extended states and possibly for localized states as E ~ O. Therefore, the Sp(2) symmetry is broken for both cases. McKane and Stone [133] pointed out that there are two ways to satisfy the WT identities. Although their interpretation is given in an entirely different theory based on the replica trick, we expect it applicable to our case as well. For extended states, the dynamic generation of the imaginary part for the retarded Green function caused by the breakdown of the Sp(2) symmetry leads to Goldstone mode characterized by long-range correlation and governing the critical behaviour from the "extended" side. On the other hand, there is no Goldstone mode with vanishing momentum for localized states, so the integrand on the right-hand side of (8.92) must diverge as E ~ 0 before integration to satisfy the WT identities in this case.

8.3.4. Order parameter and nonlinear u-model As said before, the order parameter breaking the Sp(2) symmetry is proportional to the imaginary part of the retarded Green function. A Goldstone mode is therefore generated. To describe this mode it is convenient to introduce a composite matrix field q(x, E) = ",(x, E)", t(x, E) ,

(8.93)

the vacuum expectation value of which is connected to the second-order CTPGF. Under the Sp(2) group, the field q transforms as (8.94) where U is given by (8.88). The vacuum expectation value of q is (8.95) where the diagonal part of the first term describes the imaginary part of OF and OF, whereas the second term describes their real part. As seen from (8.89) the b term does not break the Sp(2) symmetry, while the a term does. Hence Goldstone modes will be dynamically generated by the condensation of the q field.

799 Kuang-chao Clwu el al.• Eqllilibrilllfl tJNJ nonequilibrilllfl lonnalisms made IDIified

119

In analogy with the earlier work [133], we can derive a nonlinear q-model describing such Goldstone modes and carry out the renormalization procedure to study the scaling behaviour near the mobility edge. However, we will not elaborate further on such discussion here. To summarize section 8, we note that the theoretical framework outlined here for quenched random systems is quite general as well as flexible. It is based on the dynamics of the system itself without resorting to replica trick, so the approximation involved are well under control. Apart from spin glass and disordered electron system discussed in this section, the present formalism can be applied to other quenched random systems as well. In particular, we discussed [141] the controversial problem of the lower critical dimension for an Ising spin system in a random magnetic field [142]. We hope the CTPGF formalism is helpful in solving some of the difficult problems still remaining open in this field. 9. Connection with other formalisms To save space in this paper we attempted to avoid as much as possible digressing from the main subject and comparing the CfPGF approach with other formalisms in passing. We would like such comparison to be concentrated here. Although not so much new information will be presented to experts, hopefully, this section will help the newcomer entering this field to orient himself in the forest of diversified formalisms. We will mainly discuss the connection of the CfPGF approach with the Matsubara technique (section 9.1), quantum and fluctuation theories as low and high temperature limits of the CTPGF formalism (section 9.2) and also the CTPGF formalism as a plausible microscopic justification of the MSR field theory (section 9.3). There are still many related papers not covered in this review for which we apologize to their authors once again.

9.1. Imaginary versus real time technique The Matsubara technique [1-7] using the imaginary time for thermoequilibrium is well developed and highly successful. However, there are two difficulties from the technical point of view. One is associated with the fact that in this technique Green's functions are defined on a finite section of the imaginary time axis (0, -i/3) so the Fourier series expansion in frequency is used instead of Fourier integral. The frequency summation is sometimes quite cumbersome. Another difficulty is connected with the analytiC continuation of the frequency (or time) variable in the final answer. Usually, such a process is rather delicate. We see thus in spite of the great success of the Matsubara technique, a convenient real time formalism would be highly desirable. The CTPGF formalism is one of the possible candidates for this purpose.

9.1.1. Real time diagrammatic technique We have mentioned in section 6.1 that the diagrammatic expansion for thermoequilibrium system at finite temperature is similar to that of the quantum field theory provided the free propagator is given by (6.33). Here we would like to justify this statement using expressions for the effective action derived in section 6.1. The correlation functional W N defined by (6.15) becomes in this case (9.1) where

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lith = exp(-n -

(9.2)

/3('Je - J.'N»

where n is the thermodynamic potential, whereas known that for the in-field we have [39]

.pi, .pI are operators in the incoming picture.

It is

(9.3) where '!( is the total Hamiltonian. This point is essential for our derivation. By analytic continuation 7'-+ -i/3 we find (9.4) Taking into account that for a complex field the operator of particle number

is a conserved quantity, it is easy to show that (9.5) where

,\ = 0, if AI(t) is Hermitian. (9.6) Using (9.5) we can apply as done by Gaudin [7] the following identity Tr{(pA(l) + (±)"A(1)p)A(2)-" A(n)} =

Tr{p[A(l), A (2)J,.A(3) - .. A(n )}± {pA(2)[A(1), A(3)J,.A(4)· .. A(n)}

+ _... ,. (±)"-2Tr{pA(2)' .. A(n -l)[A(l), A(n)],.}

(9.7)

to the right-hand side of (9.1) to obtain (9.8) where (9.9)

In deriving (9.8) the properties of the normal product and the particle number conservation are taken

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into account properly. Note that for nonrelativistic complex fields the operator ",,(x) contains only positive frequencies whereas ",i(x) only negative ones, so that (9.10)

where So-+ is given by (6.8). Substituting (9.10) into (9.9) we find that

F, = ±i

f

8"':(X) Sl;" (x, y)

8"'~(y)

(9.11)

with

a

Sl;"(X,y)= (1+exP

(iP axo -PIL))

-1

So-+(x,y).

(9.12)

It then follows from (9.8) by using (9.11) that

(9.13)

In accord with (6.15) the correlation functional defined on the closed time-path is

(9.14) p

where (9.15)

Equations (9.13) and (9.14) are the contribution of the density matrix to the generating functional. We see thus the initial correlation is actually a Gaussian process described by the two-point correlation function Sl;" (x, y). Substituting (9.14) back into (6.10) and (6.29) we obtain the following expressions for the CfPGF generating functional in thermal equilibrium: (9.16) (9.17) p

where

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(9.18) Gol (l, 2) = Sol(l, 2) -

I

dl d2 501(1, 1)Si;'p(12) So 1(2, 2).

(9.19)

p

It is ready to check that Go and G OI defined by (9.18) and (9.19) are reciprocal to each other and that

the FDT (3.43) is satisfied. Since the Green functions in thermoequilibrium are time translationally invariant, the only role of the second term in (9.19) is to produce Si;'p(l, 2) term in Go(I,2) and cannot appear independently in the final result. Hence we can ignore the difference of SOl and SOl from the very beginning and take (9.20) instead of (9.19). In that case the first and the second terms of (9.18) can be considered as superposition of solutions for inhomogeneous equation SOlp = 1 and homogeneous equation SOlp = O. The numerical coefficient of the latter is determined by the FDT. We see thuG (9.16) and (9.17) are the generalization of the Matsubara technique to real time axis. The advantage of using real time variables in some cases more than justifies the technical complications owing to the matrix representation of the propagator.

9.1.2. Other real time formalisms Several authors previously considered the possible generalization of the Feynman-Wick expansions for the Matsubara functions [143,144]. However, some of these attempts ended up with very involved formalism, whereas the others were difficult to justifiy. We believe the incoming picture adopted here is helpful in avoiding these difficulties. Very recently, Niemi and Semenoff [145] proposed a version of real time technique to study the finite temperature field theory. Their work is close to ours but is still different. The time-path in the complex plane they adopt consists of four pieces (-00, +00), (+00, +00if3/2), (+00 - if3/2, -00 - if312) and (-00 - if3/2, -00 - i(3). Their free boson propagator is given by (9.21) where A

A=

(COSh (J sinh (J ) sinh (J cosh (J

,

cosh 2 (J = exp(f3lkol)/[exp(f3lkol) - 1] .

(9.22)

Obviously, (9.22) is different from that given by (2.26) in our formalism. A detailed comparison of these two versions has to be made by future studies.

9.1.3. Thermo field dynamics For the last ten years Umezawa and coworkers [91] have developed the "thermo field dynamics" and applied it to a number of interesting problems in condensed matter physics. They have adopted a great

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many ideas and techniques from the arsenal of the quantum field theory, especially the operator transformations. So far we have not yet studied all the topics they have covered, and it is hard to make a thorough analysis of the merits as well as the shortcomings of each formalism. It s~ems to us, however, that the extensive use of the generating functional technique, especially the vertex functional in the CfPGF formalism, is advantageous. As we found from the study of the weak electromagnetic field coupled to the superconductor [53] the ambiguity connected with the dynamiC mal!ping and boson transformation occurring in thermo field dynamics [146], can be avoided in the CfPGF formalism. Another merit of the latter is the unified approach to both eqUilibrium and nonequilibrium phenomena, whereas the thermo field dynamics is limited to equilibrium systems up to now.

9.1.4. Kadanoff-Baym formalism We should also mention the Green function formalism developed by Kadanoff and Baym (KB) [147]. These authors do not use the closed time-path, but rather start from the original paper by Martin and Schwinger (83). There are still many common features of these two formalisms. In fact, the G> and G< functions appearing in the KB technique are nothing but G_+ and G+_ in the CTPGF approach. There are many papers applying theKB formalism to different problems in both eqUilibrium and nonequilibrium systems [148]. We will not go on to compare these two formalisms in further detail. The interested readers are referred to their excellent book (147]. 9.2. Quantum versus fluctuation field theory In this section we consider the low and high temperature limits of the CfPGF formalism for thermoequilibrium. It is natural to expect that in the zero temperature limit the standard quantum field theory or its equivalent in the many-body systems should be recovered. In fact, if the boson density is set equal to zero in (2.26), .:h becomes the usual Feynman propagator. Of course, there is no need to duplicate the time axis in this limit. 9.2.1. Critical phenomena Now consider the high temperature limit. As seen from the Bose distribution 1 n(p) = exp[e(p)/T»)-1

for particle number nonconserving system or near the critical point (where the chemical potential II- = 0), the quasiparticle density

n "" T/e(p)~ 1,

(9.23)

e(p)/T~

(9.24)

if 1.

In the ordinary units (9.24) can be rewritten as A ~ k/V2mkBT,

(9.25)

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Kuang·chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

i.e., the characteristic wavelength of elementary excitation is much greater than the thermal wavelength. Hence near the critical point the thermal fluctuations dominate, whereas the quantum fluctuations are irrelevant. This is the basis of the modern theory of critical phenomena. It is worthwhile noting that this classical limit is not described by the Boltzmann distribution which holds when exp[(E(p)- p,(T)/T] ~ 1. Therefore, it is more appropriate to call this limit "super Bose" distribution, as far as the expectation value of at a is of order n ~ 1, so that the noncommutativity of Bose operators can be ignored. We have thus two types of field theory: quantum field theory at zero temperature and classical field theory near the critical temperature. They have many features in common but differ from each other in some essential aspect.

9.2.2. Finite temperature field theory For the recent years many authors study the finite temperature field theory and possible phase transitions in such systems [149]. As mentioned before, we have shown [41] that the counter terms introduced in the quantum field theory for T = 0 K are enough to remove all ultraviolet divergences for CTPGF at any finite temperature, even in nonequilibrium situation. This has been shown also by other authors [149] for thermoequilibrium without resorting to CfPGF. This result is understandable from a physical point of view since the statistical average does not change the properties of the system at very short distance and hence does not contribute new ultraviolet divergences. What we should like to emphasize is that in considering phase transition-like phenomena one must first separate the leading infrared divergent term and then carry out the ultraviolet renormalization which is different from that for the ordinary quantum field theory. 9.2.3. Leading infrared divergence To be specific, let us consider the real relativistic scalar boson theory the free propagator of which is given by (2.26). Nea~ the transition point, the mass m vanishes so the energy spectrum is given by w(k) = \k\ (cf. (2.28». Since terms proportional to the particle density n appear together with the .5-function, i.e., on the mass shell, the integration over frequencies can be carried out immediately to give stronger infrared divergence (k- 2 ) than the other terms (k- 1). Therefore, the marginal dimension of renormalizability for ip4- theory is de = 4, whereas for quantum field theory at T = 0 K the marginal space dimension is de = 4 - 1. This is what is usually meant by saying "quantum system in d dimensions corresponds to classical system in d + 1 dimensions". If we keep only the most infrared divergent terms, then all components of G become (P+ m2t\ i.e., exactly the same as that used in the current theory of critical phenomena [60]. What has been said above can be checked explicitly by calculating the primitive divergent diagrams for mass, vertex and wavefunction renormalization, carrying out the frequency integration and taking the high temperature limit T ~ w(k). The results obtained turned out to be identical to those resulting from the theory of critical phenomena. For example, in 1,04 theory, the primitive mass and wavefunction correction diagrams have quadratic divergence, whereas the vertex correction term diverges logarithmically in four-space dimensions. We know that in the quantum field theory such divergences occur for three-space and one-time dimensions. 9.2.4. High temperature limit in Matsubara formalism The high temperature limit can be easily taken in the Matsubara technique. For example, the free

805 Kuallg-chao CIIou et aI.• Equilibrium and nonequilibrium Jonnalisms made unified

125

propagator for non relativistic complex boson field is given by (9.26) where w" := 21fnT, m2- T - Te. Since T ~ k 2+ m2, in the frequency summation to be carried out later we need to keep only the w" := 0 term. Hence the propagator (9.26) reduces to minus the correlation function in the theory of critical phenomena. This fact seemed to be first realized by Landau [150]. Some investigators of finite temperature field theory in the early stage of their work incorrectly used the renormalization constants for T = 0 K to study phase transition related phenomena. As far as the high frequency limit, or, equivalently, the leading infrared divergent terms are picked up, both relativistic and quantum effects are irrelevant. The only exception is the phase transition near T = 0 K when both thermal and quantum fluctuations are essential so a special consideration is needed. Otherwise, the field-theoretic models (including non-Abelian gauge models) cannot provide anything new beyond the current theory of critical phenomena as far as the phase transition is concerned, i.e., they are classified into the same universality classes as their classical counterparts. 9.3. A plaUSible microscopic derivation of MSR field theory

We have mentioned already in section 6.4, that Martin, Siggia and Rose (MSR) [90] proposed a field theory to describe the classical fluctuations. There are several peculiar features of this theory: (i) Being a classical field theory, it deals with noncommutative quantities; (ii) A response field tP is introduced in addition to the ordinary field lP; (iii) Some components of the Green functions should be zero along with their counterparts - vertex functions. Nevertheless, the general structure of this theory is very close to that of the quantum field theory. These authors originally proposed their theory to consider nonequilibrium fluctuations such as those in hydrodynamics, but it has been extensively used in critical dynamics near thermoequilibrium [61,77]. In spite of the great success, its microscopic foundation especially the motivation for using noncommutative variables to describe classical fields, was poorly understood. A few years later, the MSR theory has been reformulated in terms of stochastic functionals as a Lagrangian field theory [87-89]. The noncommutativity of field variables was thus obscured by the continuum integration, whereas the calculation procedure was significantly simplified. Nonetheless, the physical meaning remains not sufficiently clarified. We would like to note that the CTPGF formalism provides us with a plausible microscopic justification for the MSR field theory. In a sense, the MSR theory is nothing but the physical representation, i.e., in terms of retarded, advanced and correlation functions, of the CTPGFs in the quasiclassical (the low frequency) limit. 9.3.1. Noncommutativity First consider the operator nature of the field variable. As we discussed in the last section, in the high temperature limit the noncommutativity of operators can be ignored altogether for static critical phenomena, which implies that all components of the Green functions are replaced by correlation functions. This is no more true for dynamic phenomena. The first term in G++ and G __ (2.26) i.e., (k~ - k 2- m2tl comes from the inhomogeneous term of the Green function equation which in turn is determined by the commutator of operators. If only leading infrared divergent terms are retained, the retarded function

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Kuang-chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

Therefore, to describe the response of the system to the external disturbance we need to keep the next to leading order of infrared divergence. Put another way, the response is less infrared divergent than the correlation function. In the sense of "super Bose" distribution, the commutator of order 1 can be neglected in the leading order of n but not in the next to leading order. This is the physical interpretation for the noncommutativity of a "purely classical" field variable. We would like to emphasize here that the classical field should be considered as condensation of bosons. This is why the "quantum" or wave nature of the classical field comes into play. The deep analogy of the quantum and fluctuation field theories is then understandable. Such parallelism is particularly evident in the functional formulation. The classical path in the quantum field theory corresponds to the mean field, or TDGL orbit in the fluctuation theory.

9.3.2. Doubling of degrees of freedom Next consider the necessity of doubling the degrees of freedom. It has been realized for a long time that to describe the time-dependent phenomena one needs both response and correlation functions. This was, probably, the motivation of introducing the response field Ij; and putting together the response and correlation functions into a matrix function by MSR [90]. In the CfPGF formalism we introduce an extra negative time axis, so also double the degrees of freedom, i.e., to use ifi+, ifi- instead of one ifi. In fact, the MSR response field Ij; = ifi", == ifi+ - ifi- ,

whereas their physical field

The CfPGF formalism is constructed on the functional manifold ifi+ and ifi-, or, equivalently, ifi", and ifie, but in the final answer one should put ifi+ = ifi- to get the physical result. As mentioned before, this is an additional way of describing fluctuations, the physical content of which should be further uncovered. In using the generating functional technique the following external source terms are introduced in the MSR theory r87-891

f

Is = (Iifi + JIj;), where I is the usual source, J the response source. This is rather similar to our generating functional, but with an important difference. As naturally follows from the definition of the closed time path we should set (see eq. (2_71)) Is =

f

(I"'ific +Icifi"').

As seen before, such "twisted" combination is most natural. In fact, the physical source Ie generates the dynamic response in terms of ifi", functional, whereas the fluctuation source I", generates the

807 Kuong-choo Chou el aI•• Equilibrium and nOlU!lJuilibrium fonnalisms made unified

127

statistical correlation in terms of ({Jc functional. Another advantage of such "twisted" combination is that we do not need to introduce any extra physical field in the Hamiltonian like in the MSR theory [87-89). In the CfPGF approach lc is the physical field built in the formalism itself.

9.3.3. Constraints Finally, a remark concerning various constraints imposed on propagators and vertex functions. In the original formulation of MSR [90] such constraints appeared rather difficult to understand. They became more systematic in the latter Lagrangian formulation [89] but remained not so transparent. Within the CfPGF formalism, as shown in section 2.4, they are natural consequences of the normalization for the generating functional and the causality. While in the MSR theory one needs to explore the implications of the causality order by order [151), within the CfPGF framework it is ensured from the right beginning, so that causality violating terms can never occur. As seen before, in the low frequency limit when the MSR theory holds, the CfPGF formalism yields the same results in rather low approximations. We believe, therefore, the CfPGF formalism provides us with a plausible microscopic justification for the MSR theory and indicates how to go beyond it. 10. Concluding remarks It is time now to summarize what has been achieved and what has to be done. (1) The CfPGF formalism is a rather general as well as flexible theoretical framework to study the field theory and many-body systems. It describes the equilibrium and nonequilibrium phenomena on a unified basis. The ordinary quantum field theory and the classical fluctuation field theory are included in this formalism as different limits. The two aspects of the Liouville problem, i.e., the dynamic evolution and the statistical correlation are incorporated into it in a natural way. The formalism is well adapted to consider systems with symmetry breaking described by either constituent or composite order parameters. If different space-time variation scales can be distinguished, a macroscopic or mesoscopic description can be provided for inhomogeneous systems from the first principles. (2) The powerful machinery of the quantum field theory including the generating functional technique and the path integral representation can be transplanted and further developed in the CfPGF formalism to study the general structure of the theory. The implications of the normalization condition for the generating functional and the causality are explored. The consequences of the time reversal symmetry such as the potential condition, the generalized fluctuation-dissipation theorem and reciprocity relations for kinetic coefficients are derived. The role of the initial correlation is clarified. The symmetry properties ,of the system under consideration are studied to derive the Ward-Takahashi identities. Also, a general theory of nonlinear response is worked out. (3) A practical calculation scheme is worked out which derives a system of coupled, self-consistent equations to determine the order parameter along with the energy spectrum, the dissipation and the particle number distribution for both constituent fermions and collective excitations. A systematic loop expansion is developed to calculate the self-energy parts. The Bogoliubov-de Gennes equation is generalized to include the exchange and correlation effects. A way of computing the free energy by a straightforward integration of the functional equation is found. (4) The general formalism is applied to a number of physical problems such as critical dynamics, superconductivity, spin system, plasma, laser, quenched random systems like spin glass and disordered electron system, quasi-one-dimensional conductors and so on. Although most of these problems can be and have been discussed using other formalisms, but, as far as we know, the CTPGF approach is the

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KUIJIIg-chao Chou el al., Equilibrium and IWnequilibrium fonna/isms mcuk unified

only one to consider them within a unified framework. Moreover, new results, new insight or significant simplifications are always found by using the CfPGF approach. (5) In general, systems in stationary state or near it have been studied more thoroughly, whereas the transient processes need further investigation. As far as the formalism itself is concerned, the two-point functions are well under control, but the properties of multi-point functions must be explored further in the future. Our general impression, or our partiality, is that the potentiality of the CfPGF formalism is still great. One has to overcome the "potential barrier" occurring due to its apparent technical complexity to appreciate its logical simplicity and power. It is certainly not a piece of virgin soil, but the efforts of a dedicated explorer will be more than justified. We hope that in applying this formalism to attack more difficult problems in condensed matter, plasma, nuclear physics as well as particle physics and cosmology, its beauty and power will be uncovered to a greater extent. Acknowledgements It is a great pleasure for us to sincerely thank our coworkers, Jiancheng Lin, Zhongheng Lin, Yu Shen, Waiyong Wang, Yaxin Wang and, especially, Prof. Royce K.P. Zia for a fruitful and enjoyable colIaboration. A great many people were helpful to us by their enlightening comments, interesting discussions as welI as sending us preprints prior to publication. Their names are too numerous to enumerate, but we would like to particularly mention Profs. Shi-gang Chen, B.1. Halperin, Tso-hsiu Ho, Yu-ping Huo, P.C. Martin, Huanwu Peng, Chien-hua Tsai and Hang-sheng Wu. To all of them we are deeply grateful.

Note added in proof

After our manuscript was submitted we became aware of some more references [152-158] where the CTPGF formalism was applied to various problems. We would appreciate other colIeagues to inform us of their work using this approach. References [I] A.A. Abrikosov, L.P. Gorkov and I.E. Dzaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliffs, New Jeney, 1963). [2] A.L. Feller and J.D. Walecka, Quantum Theory of Many-Particle Physics (McGraw-Hill, New York, 1971). [3] G.D. Mahan, Many-Particle Physics (Plenum, New York. 1981). [4] T. Matsubara. Progr. Theor. Phys. (Kyoto) 14 (1955) 351. [51 A.A. Abrikosov. L.P. Gorkov and I.E. Dzaloahinskii, Zh. Eks. Teor. Fiz. 36 (1959) 900. [6) E.S. Fradkin. Zh. Eks. Teor. Fiz. 36 (1959) 1286. [7] M. Gaudin. Nucl. Phys. 15 (1960) 89. [8] J. Schwinger. J. Math. Phys. 2 (1961) 407. (9) L.V. Keldysh, JETP 20 (1965) 1018. [101 R. Craig. J. Math. Phys. 9 (1968) 60S. [U) K. Korenman. J. Math. Phys. 10 (1969) 1387. [12] R. MiDs. Propqators for Many-Particle Systems (Gordon and Breach. New York. 1969). [13) A.G. Hall. Molec. Pbys. 28 (1974) 1; J. Pbys. AS (1975) 214; Phys. Let!. 55B (1975) 31.

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[59) (60) (61) (62) (63) (64) (65)

129

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[122) [123) [124) [125) [126) [127) [128) [129) [130) [131) [132) [133) [134) [135) [136) [137)

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131

812

34

Progress of Theoretical Physics Supplement No. 86, 1986

Spontaneous Symmetry Breaking and Nambu-Goldstone Mode in a Non-Equilibrium Dissipative System Kuang-Chao CHOU and Zhao-Bin Su

Institute of Theoretical Physics, Academia Sinica p. 0. Box 2735, Beijing (Received December 5, 1985) In a non-equilibrium stationary state with space-time structure a N ambu-Goldstone mode with dissipation arises as a consequence of the spontaneous symmetry breaking. A laser type model is considered as an illustration. From the Ward identity in the scheme of the closed time path Green's function (CTPGF) it is shown that the Nambu-Goldstone mode splits into two waves with different dissipations.

Spontaneous symmetry breaking and the N ambu-Goldstone mode playa very important role in contemporary physics. It is with feeling of high appreciation to write an article to honor one of the founders of these fundamental concepts, Professor N ambu's sixty-fifth birthday. Over the past thirty years our understanding of the micro world is much richer because of the valuable contribution of Professor Nambu who initiated many important research directions in particle and solid state physics. In a stable stationary state of a nonequilibrium open system, variation of the external parameters sometimes causes bifurcation with spontaneous symmetry breaking and the generation of Nambu-Goldstone mode. Typical example of this kind is the laser system, where the vector potential develops a non-vanishing vacuum expectation value when the inversion of electron occupation number exceeds a critical limit. It breaks spontaneously the phase symmetry and generates a N ambu-Goldstone mode. As far as we know most laser theories deal with the coherent wave by semiclassical method without paying attention to the existence of the N ambu-Goldstone mode which in its turn is also a light wave of the same frequency with a tiny width caused by the dissipation of the laser system. It is the aim of the present paper to study the properties of the N ambu-Goldstone mode generated in a nonequilibrium steady state of a laser-like system. The field theory applicable to nonequilibrium system was first developed by Schwinger in the early sixties and by Keldysh and many others later. 1 ),2) A recent review of this method which is called closed time path Green's function (CTPGF) was published in Physics Reports. 3 ) We shall use the formalism developed there without detailed explanation. For further reference the reader should consult the original literature. In the first part a model Lagrangian of a laser-like two-level system is studied and the U(l) phase symmetry is considered to be broken spontaneously by the variation of the occupation number of the two levels. AWard identity is written down which insures the existence of a N ambu-Goldstone mode accompanied by the spontaneous symmetry breaking. In §2 the two-point retarded Green's function is solved and a split of the N ambuGoldstone mode into two waves with equal half probability and different dissipations is found. In §3 it is shown how the broken symmetry is restored and the coherent wave goes

813 Spontaneous Symmetry Breaking and Nambu-Goldstone Mode

35

gradually into one of the N ambu-Goldstone modes with less dissipation. Conclusions and speculations are presented in the last section. Some of the contents of the present paper were published by the authors in Chinese in Ref. 4). § 1.

Model Lagrangian and the symmetry breaking

Consider a two-level system interacting with a scalar field aex). The Lagrangian is

.l=rlh*(i ~ -EI)r/J1+1h*(i ~ -Ez)r/Jz+a*(i ~ -ko)a+g(r/Jz*r/Jla*+r/JI*r/Jza).

(1)

The Lagrangian (1) has a U (1) symmetry

(2)

For simplicity we have neglected the space dependence of the fields involved. In general there are random interactions of the system with the surrounding causing dissipation. To a first approximation we shall include them in the energies E •• i =1.2 and ko. widths rio i=I, 2 and r. respectively. The state of the system is described by a density matrix p and the dynamics can be formulated in terms of a path integral along a closed time path P starting from t = - 00 to t= +00 (positive branch) and back from t= +00 to t= -00 (negative branch). The generating functional is ZU(x}] = eiW[J(ZI) =![dr/J ] [dr/J*][da] eiJp[.£(ZI+JO(zla(zl+aO(z)J(zIWZ

x -0 ~ 0 with E(O) = -1. In this case, we may live in one world, and do not interact wi th the time-reversal counterpart because the barrier between the two worlds is infinitely high. Then we can always assume the density matrix satisfies Eq. (8) in the study of dynamical systems living in our world. The equal to

average

value

of

Do (t; J, A)

=

the

operators

cc

tr(c(A)Q(t,J)~

a

t.

J)

is

(9)

which will be called order parameters in the Following. It is easil y deduced from Eqs. (5) and (8) for a time reversal invariant system that

The correlation functions are

Co 1 .••

=

t Q.

0Q.

tr(p(A)D

;

J, A)

(t 1 ,J)···

01

Q (tn, on x.

J)1.(11)

Time reversal invariance requires that

••• 0Q.

=

t Q. ;

(t 1 ,'"

e (0 1) •• , e (0 Q. )

C

0Q.

J, >.)

... ° 1(- tn, x. ( 12)

From Eqs. (10) and (12) we see that time reversal invariance relates physical quantities in one world to those in the time reversal counterparts if time reversal symmetry is spontaneously broken.

120.

833 2. Time-dependent Ginsburg-Landau Equation (TDGL) In this section we are going to derive TDGL from the field theory of CTPGF and prove the existence of generalized free energy for time reversal invariant systems. Since the relation obtained in the previous section is independent of the particular initial time chosen, we shall take the initial time to minus infinity in the following. Consider the external source term JaOa(JQ) to be, a perturbation adiabatically swi,tched on. Any operator 00 in the Heisenberg picture of HO will change to (13 )

5

where

is the S-matrix co

S

=

T(exp

f

-i

J(-;-) 0(-:- ) d-

~ )

.

(14 )

-co

'+

The S on the left in Eq. (13) is also necessary to guarantee the causality of the interaction. Equation (13) can be expressed as 6 (15 ) \~here

,

Sp

::

Tp f exp f -i .fp J(d Q( .;) dT)}.

(16 )

The path of integration P is a closed path starting from T = -CD to T :: +CD (positive branch) and back from T = +CD to T = -0:) (negative branch). T p is an ordering operator along the whale pa~h P. It is easily.seen that the positive branch of Sp cqrresponds to S in Eq. (13) and the negative branch to S+. In statistical mechanics we are interested in average values of physical variables rather than transition amplitudes. The average value is equal to a trace of a density matrix at an initial time to and some Heisenberg operators at the time t 1, ..• tn which consists of amplitudes propagating from to to the time of the operators and back. This is why we need an S-matrix along a closed path in statistical physics. The generating functional for CTPGF is defined to be Z[J] = exp {-iw[J]l

= trf

P Tp(exp {-i

fp J(T) OCT) dT }) 1.

121.

(17)

834 The external sources Ja(t), a =l, .•• n are taken to be different on positive (J+ a (t» and negative (J -a (t» branches. They will be put equal at the final step when physical results are evaluated. wCJ is the generating functional for the connected Green's functions whose first derivative gives the order parameters

J

(18 )

When the external sources on the two branches equal to the physical external source, we have

=

=

Qa

ph.

(t)

are

(19 )

,

Qa(t~

which is the average value df the physical variable From the definition follows easily that

W [J +

of the generating

J-JIJ:J + -

Let

r

=

1/2(J+ + J

J/:;.

=

J

J

-

+

functional

it

(20)

0

=

set

'}

J

(21)

Equation ( 20) can be rewritten as (22) By differentiation with respect to J I: we obtain a series of equalities among Green's functions from Eq. (22). The first equality is

&W[Jr, J/:;.=O]

Q

& lYa (t) coinciding with Eq.

+0

(t) - Q

-a

(t)

o

(23)

(19).

Second order connected tained by differentiation

Green's

functions

OJa{t) OJ~{tl)

122.

can

be

ob-

(24)

835 In the single time formalism ther!' ,"'" four second order Green's functions for each pair ( 1 : , \,f which only three are independent. They are the re t ,I"" ".1

G

ra~

(t, t' ; J, X)

-i9(t-t') tr {p(X)

[0 ., \ t.

J),

Q

~

(t', J)]l

(25)

the advanced

G

oa~

(t, t'; J, X)

and the correlation Green's functi""

G

ca

B (t, t'; J, X)

=

( ) ~J

SJll.a t



~(~

( ")

A

-i tdp(X) {Qa(t, J), ("r~(t"

The vertex functional is defi.r. f ,.:

where aa(t, J) is determined "/ from Eqs. (18) and (28) that

sr[Q] 6 Q (t) a

123.

=

1,0

:'l.

-'• ' . I

.

J)} -;.1

(27)

be

(18).

It

follows

(29)

836 In terms of the variables ( 30)

and Q we have from Eqs.

l1a

==

Q

+0

- Q

-0

,

(31 )

o

( 32)

(22) and (28)

and

(33 )

Q ==0 11

The vertex functional can be calculated by summing all I-particle irreducible diagrams. Once this is done, the physical order parameter Q fa (t) can be determined from Eq. (33). We shall show in the following that TDGL is an approximation of Eq. (33) for systems near stationary states where the motion is slow. There are four second order vertex functions for each pair a ~ in single time formalism, of which three are independent in the physical region where Q 11 = O. They are the retarded ==

==

- D

a~

- iA

a~

(34 )

the advanced

r aat-'It (t, t'

; Q , ).)

==

==

- D

a~

and the correlation vertex function

124.

+ iA

(.I

a,..

(35)

837

r cal"'R (t,

t' ;

a , X)

5 a ~a (t) 5 a ~ ~ (t')

(36)

In Eqs. (34) and (35) Da~ is called the dispersive part and Aa~ the absorptive part of the vertex. One can easil y deduce the Dyson-Schwinger equations for the second order Green I s functions. In matrix form they are

rr

G

r

G

rr

- 1 ,

(37)

G

r

- 1

( 38)

r

r a Ga

a a

and

rc

r R Gc r a

=

(39)

Now we are prepared to derive TDGL for a system near stationary states. In the following we shall assume the physical external source Jr. to be constant in time and the equation

or[a]

- J

5a~ (t)

a~ = 0

r

(40)

0 [. Then the system is conto have constant solution The question is sidered to be in a stationary state. whether or not there exists a generalized free energy such that 0 r. is the solution of the equation F [0

[J

aF

= - J La

(41)

If this is possible we must have

aF

aa ra

=

125.

cSr (42)

838 Then

J

dt' - - - - - - oar~ (t') OaAO(t) t"

J dt' r

LJ,

~ (t' rI"'o

- t ;

a6.0:: 0, ar(3(t)=aI~

a,).) .

(43)

Here we have used the time displacement invariance when the external sources are time independent. A well-defined funct ion F can be obtained by integrating Eq. (42) if the order of differentiation can be changed in Eq. (43). Therefore the condition for the existence of the generalized free energy is

J dt'

r r~o (t'

-

t .

a ).)

J dt'

I'

r rot"'(.\ (t' - t ; a, ).)

or in Fourier transform

r rof3 (w=o

a,).)

=

r ro~(w=o;

a,).)

r

a,).) .

6(w=0; 00

(44)

The last equality in Eq. (44) follows from the relation

r

~(w; a,).)

ro~.

=

r

~ (-w; a,).) .

(45)

0t" 0

Equation (44) can also be written in the form ..J~

~o~

(w=O; a,).) = o.

(46)

Therefore, vanishing of the zero frequency component of the absorptive part of the vertex function is a sufficient condition for the existence of a generalized free energy.

126.

839 Our next task is to show that Eq. (46) follows from time reversal invariance. For constant external sources, the order parameters are time independent. We have from Eqs. (10) and (12) that Qa(J,~)

=

E (a)

Q a (EJ

, E

(47)

~)

and

From Eq. (47) we obtain by differentiation with respect to constant Jr~ the following relation

Gr~a ( w =

a; J ,

~) =

E

(a)

E

(8) Gr~a (w = 0,

E

J,

E

(49 )

~)

The Fourier transform of Eq. (48) reads (50)

From Eqs. (49) and (50) we get ( 51 )

Since - rr and - ra are the inverse 0 f Grand shown in Eqs. (37) and (38) we obtain finally r~ (w=O;J,~) =

r"a

r

~ (w=O;J,~)

aI-a

Ga

as

( 52 )

•

Equation (52) is just the condition (46) required. Hence we have shown that generalized free energy exists for time reversal invariant sy~tems in stationary states. Near the stationary state the order parameters vary slowly with time. We can expand Eq. (29) around the stationary point at time t

or

or

oQ6.a(t)

+

J dt"

Q6.a=O,

2 6 r

- - - - - - (Q

127.

r~

Qr.~(t)=Qr~

(t')-Q

r~

(t}+ .. ••

(53)

840 To first order approximation in time dependence we neglect higher order terms in the expansion and put (t' - t)

aQL(3

(54)

at

Then Eq. (29) becomes - J

(55)

ta

where !dt'(t'-t)

r r~a (t'-t,

arr~a ( 101 , Q a101

, X)

Q, X)

I

(56)

101=0

Equat ion (55) is the TDGL equation for the order par amet er s Q to Ti mer eve r sal in v a ria n c e imp 1 i e s also a reciprocity relation for the relaxation matrix ~~ ( 57)

This is one kind of Onsager relation generalized to systems near nonequilibrium stationary states. It

was

proved

in

Ref.

2

that

r c yo.,.{L1=0,

Q, X)

both

the

di ffusion

matrix a

a~

-1 = Yoy

r- 1 y 6(3

(58)

and the response matrix

L

a~

=

101=0

(59)

satisfy the reciprocity relations (60)

and

128.

841 Lo ~ (J , >")

=

E (0) E

(~) L~ 0 (e J, E >.. )

(61)

~or"systems near stationary states. We shall not discuss It"l~ detail here. The interested reader can consult the

orlglnal literature.

3. Generalized fluctuation-dissipation Theorem We have shown in Ref. 5 that the correlation function Gc can be written in the form (62)

G = GN-NG era

where ticle state

N is a hermitian matrix related to the quasi pardensity distribution. In the thermal equilbrium

(63)

where

T

is the temperature. lim N (w, Q, >..) w-O a~

Substituting Eq. (39) we obtain

rc

(62)

= 5

into

r r N - N ra

In the low frequency limit 2T a~

the

(64)

w Dyson-Schwinger

= - ON + NO - i (AN + NA)

equation

,

(65)

where D and A are the dispersive part and the absorptive part respectively. Equation (65) is the transport equation for the particle density N in a slightly inhomogeneous system. In a stationary state where the vertex functions depend only on the time difference, Eq. (65) can be written in frequency representation in the following form

-0 (w)N ~(w)+N (w)O ~(w) oy Yl"' oy Yt'

- i(A

or (w)

129.

N A(w)+ N Y\"'

0

(101) A ~(w» • (66)

Y

Y"t'

842 It is easily proved that

2:: r caa (w)

i

a

is the quantum fluctuation which has to be positive definite. Therefore we have

>

tr (A( w) N ( w»

0

( 67)

From time reversal invariance we have already shown in Eq. (46) that A

a~

(w=O)

=

0

In the low frequency limit we may put A

a~

wa

(w)

(68)

a~

where caB is the real part of the relaxation matrix Ya j3 • For a statile stationary state the eigenvalues of the matrix caB have to be positive definite. Since Eq. (67) holds also at zero frequency, the density matrix N must have a pole at w = O. Hence

N

a~

for small frequency. into Eq. (66) we get

o

ay

2 T eff

(w)

w

(69)

a~

Substituting Eqs. (68) and (69) back

(w = 0) T eff = T eff 0 y~

ay

y~

(w = 0)

(70)

and tr

crc (w = 0»

= - 4i

tr(c(w

=0) Teff)

.

(71)

This is the generalized fluctuation-dissipation theorem. In the literature of critical dynamics it is always assumed that T eff =

(72 )

a~

130.

843 In this case we have

r caB (1.1 = 0)

-

. Teff 4I a

(73)

aB

Substituting Eq. (69) into Eq. (62), we obtain form of the fluctuation-dissipation theorem

aGe

at

another

2 (G Teff _ Teff G ) r a (74)

References 1.

T. D. Lee, (1974) .

Phys. Rev. D.!! (1973) 1226; Physics Reports

2.

K.C. Chou and Z.B. 164, 401.

3.

N. Van Kampen, Physica 23 (1957) 707, 816; U. Uhlhorn, Arkiv. foerFysik 17 (1960) 361; R. Graham and H. Haken, Z.PhYS. 243 (1971) (1971) 141. -

Su,

Acta

Physica Sinica

30

(1981)

289;

4.

G. Agarwal, Z.Phys. 252 (1972) 25; J. Deker and F. Haak~Phys.Rev. All (1975) 2043; S. K. Ma and G. Mazenko, Phys. Rev. Bll (1975) 4077.

5.

K.C. Chou, Z.B. Su, B.L. Hao and L. Yu, (1980) 3385; Commun. in Theor. Phys • 307, 389.

6.

j. Schwinger, J.Math.Phys. 2 (1961) 407; L. Ke1dysh, Zh.Eksp.Teor.Fiz. !:2 (1964) 1515.

131.

..!

.2.f.

245

Phys.Rev. 822 (1982) 2'95,'"

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Part III ~ i~, ~ T 49lJ 1Il:3=~ 49lJ 1I

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847

~

11

~ if,

4 ltIl

1955

~f-

7 J=]

56t*

aRB

if *3t!=p-{'\(;·fJI~:m:mf~jJ *lMftfj~, wurJ y~&"t.f·.:rfiflt~~f{-=f-(-1'.Jtit3H.

JlIJ~ lltT5Rfft1} atf1:/f].

2H, JH, 3He, ~He ~~:M[ .:r-~(j{jf.ff·frn~,

iit~rfli.R;f-fI=PJCdJf.Jj(: •

= 23.68

= 2.65

~v:g1W~IH~II1J·l'ffj~iit E = 0 II".J:fJ{J S

!l§(}ii7riWtifflM.1:

'~1JIIji:if yUIH~it~

flifj~~Il:~1t G.

fij, tIt Wiitt f:lT y,( ~ffl(; :

1f=f~ . :fj1f!~ Cl~:'l[ .L.,~~ ~ rt~, S

1 .pJ=-u(r)X;

(3)

I'

:1t=~ 1: 1it;f;flf.t*~, X 1.~gI:ME'&IiI}f~1DE~i§tt,~1'-~,*i~-€i-fi~t.mH<J~;:R.

E = 0 1Iij:, u(r)

'it

~1:i--;":I.w-~t-:

u(O)=o, r II{r) = 1 - -

'.ltT~OO,

a

} (5)

pTJ..!;(;JffIJ,iu;~~*~~tiA:ff:tU (4) *.fHffi1~~~1~#,1:tIJm.ifi I fI~a<JM

liJ u i@l1:i--i1Ul/J 1i~ (7)

H(r)

=

1-

r

-

a

_~.-f3',

;1t:~ {1 ~iH1-~!1:1; -kDe.~n1rrfjl5sm)~~tt G1 &. G 2

*iil.:t

a 'flI r.o

;tAli, «iJJ:jzl!;iHt·~i:iJ ~ffl

:miE:J.\(.~'~l't.Jr..~~',*~H!:ll3t.:i&H!EllOO1mc a fU re ~Ji::1±l G1

III

849 -4 WJ Gz

301

*.

j/t!j:,'f:J~-tr'Jn'iM~iii --;J;!W.ff~~*, 1& a ~;)i~"et~Q-{tfi,MA: (5) 1!,}:tI/

n%fi, :Ji.H: A.7it'f:,w.

~JH

I =

o:~ ~~ =

0 rj 1 1!I["ilf P-l:;,a