Selected Papers of K C Chou

The following sections are included:

• Foreword

• Introduction (C. N. Yang)

• Introduction (Y. L. Wu)

• Contents

Recently Gell-Mann¹ has proposed a new theory of the interaction between elementary particles. In this theory all baryons have spin ½, and the same mechanical mass and parity. They form a supermultiplet which is split up only when moderately-strong interactions occur by way of K mesons. The Hamiltonian of the interaction with π mesons is written in the following ways:

In this note we shall give a summary of the angular correlations in the cascades:

If K-mesons, hyperfragments or hyperons are created in the π⁻ + He⁴ reaction near the threshold or in the K⁻+He⁴ reaction at low K⁻-particle energies (but without mesic atom formation), the spins of these unstable particles (and in some cases their parities) can be determined from the correlations between the directions of the momenta of the particles involved in these reactions and the directions of emission of decay products of the unstable particles. Qualitative changes in the correlation formulae for the cascade K⁻ + He⁴ → _ΛH⁴ + π⁰, _ΛH⁴ → He⁴ + π⁻ are indicated which arise if one rejects the assumption that the K-meson spin is zero. The helium experiments together with the previously suggested hydrogen experiments (e.g. π⁻+p → Σ + K or K⁻ + p → Σ + π) are the only ones which are of practical use for the determination of new particle spins from angular correlations.

The cross section for scattering of mesons by nucleons does not change upon application of the phase shift substitution proposed by Minami ¹). The two phase shift sets which are obtained from each other by means of this substitution can be differentiated either by carrying out polarization experiments ^{2, 3}) or by studying the energy dependence of the cross section at small energies. Below we derive similar transformations for nucleon-nucleon scattering…

The cross section for scattering of mesons by nucleons remains invariant under the phase substitution indicated by Minami.¹ The two sets of phase shifts, obtained from one another through this substitution, can only be distinguished either by means of polarization experiments,^2–3 or by analyzing the energy dependence of the cross section at low energies. We obtain below a similar transformation for the case of nucleon-nucleon scattering…

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:

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 the same as the nonrelativistic formulas, if the 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 experiments on double scattering) corrections must be applied to the non-relativistic formal theory. The relativistic changes in the angular correlations are indicated for successive reactions of the type π + p → Y + K, Y → N + π.

Simon and Welton¹ and Shirokov² have obtained the selection rules for a reaction of the type a + b → c + d in the form of relations between the polarization vectors and tensors. They assume that the initial state is not polarized. The present communication gives a derivation of the selection rules for any arbitrarily-polarized initial state. We shall use Shirokov's notation² and assume that all the particles have nonvanishing rest mass…

We give an integral transformation which is related to the decomposition, into irreducible representations of the proper Lorentz group, of the representation according to which the wave function of a particle with mass M and spin s transforms.

This note examines the information which may be derived from measurements of the branching ratios and angular correlation of Σ− and Σ-decays.

The theory of the universal Fermi interaction with (V-A) coupling proposed by Marshak and Sudarshan¹ and by Feynman and Gell-Mann² is supported by all existing experimental evidence on β decay. To explain the equality of the Fermi β-decay constant and the constant for the μ decay, Feynman and Gell-Mann have put forward the hypothesis of a conserved vector current in the weak interactions. This hypothesis leads to the appearance of an anomalous magnetic-moment effect in the β decay (“β magnetism”). Calculations of the effect have been made by Gell-Mann and others³⁻⁵ and have apparently already received experimental confirmation in β decay.⁶…

The extended Lorentz group, including the complete Lorentz group and charge conjugation, is considered. It is shown that the use of irreducible projective representations of this extended group requires the existence of charge multiplets. Charge symmetry and associated production of strange particles follow from the invariance under reflections and charge conjugation and from the conservation laws for the electric and baryonic charges. The Pauli-Gürsey transformation holds for free nucleons. The extension of the condition of invariance under this transformation to the case of interactions leads to isobaric invariance for strong interactions of all particles.

In this paper the group-theoretical point of view is used for the description of the spin states of particles of zero rest mass. Complete sets of operators and of their eigenfunctions for a system of two particles are found in the representation of the momenta and spins. The statistical tensors for the particles produced in a reaction of the type a + b → c + d or a → c + d are obtained in the case in which one of these particles has no rest mass. The most general selection rules are derived for the reaction a + b → c + d in the form of relations between the statistical tensors, under the condition that the space and time parities are conserved. The wave functions are calculated for a system of two identical particles of zero rest mass.

Let us consider the reaction

It is shown that use of the unitary property of the S matrix makes it possible to obtain some information about the scattering of π mesons by Λ and Σ hyperons from an analysis of the data on the interaction of K mesons with nucleons. The possibility of studying the π−Λ and π−Σ interactions by examining peripheral collisions of hyperons with nucleons is discussed.

In the recent experiments by Rosenfeld et al.¹ and Crawford et al.² it was established that the mass of the neutral K meson exceeds that of the charged K⁺ meson by ∼ 4.8 Mev. On the face of it the sign of this mass difference appears to contradict the concept that the K⁺ and K⁰ mesons are spinless particles belonging to the same charge doublet. Indeed, if the K⁰ meson has no electromagnetic interactions and the mass difference is of electromagnetic origin then electromagnetic self-mass of the charged K meson should make it heavier than the neutral one (see, e.g, reference 3). On this basis the above-mentioned authors are inclined to interpret their results as an argument in favor of the Pais hypothesis,⁴ according to which the K⁺ and K⁰ meson do not form a charge doublet and may have different intrinsic parities.

Dispersion relations for the scattering of γ quanta by nucleons with one subtraction are considered. For forward scattering six relations have been obtained which do not contain unknown constants or infrared divergencies.

The extended Lorentz group, which includes the complete Lorentz group and the charge conjugation operation, is considered. It is shown that use of irreducible projective representations of the extended group requires the existence of charge multiplets. Charge symmetry and pair production of strange particles follow from invariance under reflections and charge conjugation and from the laws of conservation of electric and baryon charges. The Pauli-Gursey transformation is valid for free nucleons. The requirement of in variance under this transformation in the case of interaction also leads to isobaric invariance for all particles in strong interactions.

An expansion in terms of the irreducible representations of the proper Lorentz group is given for the representation which specifies the transformation of the wave function of a particle of zero mass and of arbitrary spin.

The elastic scattering of γ-ray quanta near the threshold for single meson production is treated by means of dispersion relations. It is shown that when one takes into account meson production in the s state there are appreciable departures from monotonic variation with energy of the scattering amplitudes, cross sections, and other observable quantities near the threshold of the reaction. On definite assumptions about the analysis of photoproduction in the range of γ-ray energies up to 220 Mev, calculations are made of the scattering amplitude and the differential and total cross sections for elastic scattering of polarized and unpolarized γ-rays by protons, and also of the polarization of the recoil protons above the photoproduction threshold.

Present experimental possibilities have allowed a rather close approach to a measurement of the cross section for scattering of a neutrino by an electron.¹ This process is a very important one for testing the theory of the universal weak interaction…

The Hamiltonian describing the π-K system has the form

The experimental data on the probabilities and asymmetry coefficients of the decays of Σ hyperons by various channels evidently satisfy the rule | ΔI | = ½. If the | ΔI | = ½ rule receives final experimental confirmation, it will be necessary to renounce the theory of the universal weak interaction between charged currents.³ At present it is desirable to have more data to test this rule…

The application of dispersion relations to the analysis of the energy dependence of scattering (and reaction) amplitudes near thresholds of new reactions is discussed. General expressions are obtained which characterize the nonmonotonic behavior of forward-scattering amplitudes as functions of the energy. The energy dependence of one of the amplitudes for elastic scattering of γ-ray quanta by deuterons is examined near the threshold for photo-disintegration of the deuteron.

It is shown that non-monotonic energy variations can occur in the energy spectrum of particle a from a reaction of the type A + B → a + C + D in the neighborhood of the threshold for the reaction C + D → E + F. As an example, we analyze the spectrum of K mesons from the reaction N + N → Λ + N + K in the region of the energy of the Λ − N pair close to the threshold for the process Λ + N → Σ + N. For the process p + p → Λ + N + K we find the energy spectrum of the K mesons when the incident nucleons are unpolarized, and the polarization of the baryons when the incident nucleons are polarized.

We discuss the non-monotonic energy variations in the spectra of particles for some other reactions. In the Appendix we analyze the production of Y − K pairs in np collisions and discuss the case of a scalar K particle.

By the use of the analytic properties of a certain matrix element it is shown that the result of Goldberger and Treiman regarding the decay π → µ + ν is valid for wider classes of strong interactions than those found by Feynman, Gell-Mann, and Levy, and in particular for the ordinary pseudoscalar theory with pseudoscalar coupling. A formula is obtained which can be used for an experimental test of the assumptions that are made. Lepton decays of hyperons and K mesons are also discussed.

The energy dependence of the cross section for elastic scattering of γ quanta near the photonuclear threshold is investigated with the help of the dispersion relation for forward scattering. The first peak in the scattering cross section is attributed to dispersion effects. Experiments required for a more detailed analysis are discussed.

Dispersion relations and the conditions for unitarity of the S matrix are used for the analysis of the elastic scattering of γ rays by deuterons below the threshold for pion production. The low-energy limit is examined for the scattering of γ rays by nuclei of arbitrary spin. The energy dependence of elastic _γd scattering is deduced on the basis of the experimental data on the photodisintegration of the deuteron. The result differs decidely from that of the impulse approximation over a wide range of energies. It is found that it is not important to include the influence of photoproduction of pions from deuterons in the range of energy considered.

Quite recently there has been great interest in the question of the existence of a p resonance (isobar) in pion-pion scattering.¹ From a study of the structure of nucleons by the method of dispersion relations, Frazer and Fulco have concluded that an isobar with mass 435 Mev and halfwidth 10 Mev must exist in the p state of pion-pion systems.² Similar results on the presence of a p resonance in pion-pion scattering have also been obtained by other authors.³…

It is shown that when the sign of the γN-scattering pole diagram connected with π⁰-meson decay is correctly chosen, the contribution of the pole to the cross section for the scattering of γ quanta by protons decreases considerably. In order to obtain information on the lifetime of the π⁰ meson, the precision of the experiments must be appreciably improved.

The low energy limit for the γN scattering amplitude is derived with the aid of single-nucleon invariant amplitudes. Subsequent terms in ν for Q² = 0 and the expression for the limiting value of the first derivative in Q² as Q² → 0 can be obtained by taking into account the conditions of crossing symmetry.

Some information on Λ(Σ) + π → Λ (Σ) + γ processes can be obtained by investigating the reaction. A detailed phenomenological analysis of these processes in the s state is performed. The Kroll-Ruderman theorem for photoproduction of pions on hyperons near threshold is considered.

An analysis of elastic scattering of photons with energies up to 300 Mev by protons is carried out by making use of the dispersion relations method. Six dispersion relations are utilized to estimate the real parts of the amplitudes at Q² = 0. Photoproduction of pions is taken into account in a larger energy region than was done previously. Five subtraction constants are determined from the long wavelength limit and expressed in terms of the nucleon charge and magnetic moment. Differential cross sections and polarizations of the recoil nucleons are estimated. Photon-nucleon scattering at high energies is discussed.

In this note we propose an experiment to observe the final state Λ − π resonance and the cusp arising from the near threshold effect of the Σ-production, and from this to determine the spin of and the relative parities of Λ − Σ, and , and . The centre of mass energy of the initial π − P system should be around 1900 MeV (this correspond. to 1305 MeV of the incident pion in the laboratory system). Choose the events in which the energy of the kaon lies below 90 MeV and observe the correlation between resonance and the cusp…

The concept of the pure gauge fields on a coset space is introduced. By using gauge fields on subgroup H, pure gauge fields on coset space G/H and the induced representation, a local gauge invariant Lagrangian theory on group G is constructed. The application of this theory to SU₂ × SU₂ gauge theory, the σ model and the non-trivial topological property of the pure gauge field are discussed.

The soliton-soliton scattering amplitude in three-dimensional space is obtained in the semi-classical approximation. The results obtained are also applicable to the problem of the scattering of ordinary particles and generalize the previous results in this respect.

The non-topological soliton with a non-Abelian internal symmetry in 3 + 1 dimensional space-time is examined. The ease of SU(2) internal symmetry is discussed in detail. Existence and stability of the classical single-solitonl solution are investigated with the example of a concrete model. The quantization of the single-solution solution is carried out with the method of collective coordinates. It is pointed out that the quantized soliton may possess, in addition to ordinary isotopic-spin quantum numbers ℐ and ℐ₃, a new quantum number . A new Lorentz covariant method for the quantization of the moving soliton is proposed.

The topological properties of the vacuum states for the pure gauge fields on coset are studied. When the homotopy group π₃(G/H) of the coset manifold is different from zero, a winding number operator can be constructed. It is possible to introduce a θ_c-vacuum on coset. The constrained conditions restrict the value of θ_c to be zero in physical states.

The helicity mixed representation is used to construct a model of electro-Weak interaction in SU(3) × U(1) gauge theory. It reduces in the low energy region to the simple SU(2) × U(1) model in the limit of sin²θ_W = 1/4. When sin²θ_W is slightly less than 1/4, there is a small correction in the neutral current sector which can be tested.

Ne'eman and Fairlie ^[1,2] have recently attempted to embed the SU(2) × U(1) electro-weak group^[3] into a supersymmetric SU(2|1) gauge theory. Fairlie and others^[2,4] further extended the graded gauge fields over a space time manifold with more than 4 dimensions. Such an embedding would fix the Weinberg angle to be sin²θ_w = 1/4 and if it was performed in an extended space time or super space time manifold. the Higgs mass could thus be determined. The number of extra dimensions is equal to twice the number of lepton triplets. Otherwise, the lepton masses will be the same. The common feature of this kind of theory is such that it contains unphysical particles with unfamiliar statistical properties…

In studying the problem of quark confinement, t′ Hooft introduced a loop dependent disorder parameter B(C′), which depends on loop C′ and is related to the Wilson loop operator A(C) of the non-Abelian gauge field Â_μ by the algebraic equation^[1]

It is well known that the absence of the ABJ anomaly is necessary for the corresponding gauge field theory to be renormalizable¹. 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…

An anomaly-free model of strong and electroweak interactions involving leptons and quarks in the SU(4)_c × SU(3)_f × U(1) gauge theory is constructed. After spontaneous symmetry breaking, it reduces to quantum chromodynamics for strong interactions and a broken SU(3) × 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 sin²θ_W < 1/4 and becomes slightly less than 30° in the limiting case. Below the mass scale of SU(4)_c 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 Weinberg-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 fermions.

By using Faddeev-Popov trick, the path-integral quantization of the pure gauge fields on the coset space is realized, The invariance under B. R. S. transformation of this theory is demonstrated, from which the ward-Slavnov identities are deduced. Since this theory is renormalizable in the gauge ϕ₀ = 1, F_A = 0, it is also renormalizable in other gauges. This is verified by means of the gauge independence of the S-matrix.

It is generally believed that the QCD Lagrangian has 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 L² pseudo-Goldstone 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^[1] and studied subsequently by Wienberg^[2] Kogut and Susskind^[3] and many others^[4–8]…

Two new nonlinear σ models, defined on the symmetric coset spaces GL(n,c) ⊗ GL(n,c)/GL(n,c) and GL(n,c)/U(n) respectively, are formulated in this paper. The latter may be useful in discussing the four-dimensional Yang-Mills fields.

The H_n σ-models. i.e., non-linear σ-models taking values on the symmetric coset space SL(n,c)/SU(n), both in two-dimensional and four-dimensional Euclidean space, are formulated. The relations with self-dual SU(n) Yang-Mills fields are also discussed.

We propose to introduce a strong coupling U(1)_Y axial gauge boson into the rishon 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 10¹ to 10³. Some phenomenological aspects of these exotic particles are discussed.

In this paper we show that the Wilson loop integral corresponds to an external source term in the generating functional for Green's function with a discussion of its effects. This implies that some approximations in the previous derivation of the string-like equation may not be applicable. The calculation of the field strength in the presence of this external source may be more useful in the study of confinement.

Two-dimensional nonlinear σ-models defined on symmetrical coset spaces are considered. The duality symmetry is used to construct a Bäcklund transformation which depends on a continuous parameter γ. Conserved current depending on γ is obtained. Expanding the current in powers of γ around γ = 1, one gets an infinite number of nonlocal conserved currents. A modified form of conserved current depending on γ is also obtained. When this current is expanded in powers of γ or γ⁻¹, one gets two sets of infinite number of local conserved currents.

On the basis of the formulation given in a preceding paper, we derive an infinitive series of conserved local explicitly for the two-dimensional classical σ-models on the complex Grassmann manifold U(m + n)/U(m)⊗U(n) and for U(N) principal chiral field.

We contend that the recent Weinberg–Witten no-go theorem does not necessarily rule out the possibility of gauge bosons being composite objects. A dynamical model for composite SU(3) gauge bosons is presented.

The spectrum of chiral SU(N) gauge theories in two dimensions is shown to consist of free massless mesons and baryons that may be visualized as composites of free massless quarks and antiquarks. Baryons do not contribute to anomalies which are saturated by the mesons. The singular nature of the chiral limit is recognized.

We propose a renormalization procedure for dynamically generated effective actions. We show that, as expected, it leads to no spontaneous supersymmetry breaking if this is unbroken 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 negative-energy vacuum state.

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 γp 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.

A dual transformation is found for a class of nonlinear σ-model defined on a multidimensional curved space with cylindrical symmetry. The system is invariant under proper combination of the dual transformation and the general coordinate transformation. An infinite number of nonlocal conservation laws as well as the Kac-Moody algebra follow directly from the dual transformation. A Bäcklund transformation that generates new solutions from a given one can also be constructed.

A Darboux transformation depending on single continuous parameter t is constructed for a principal chiral field. The transformation forms a nonlinear representation of the group for any fixed value of t. Part of the kernel in the Riemann-Hilbert transform is shown to be related to the Darboux transformation with its generators forming a Kac–Moody algebra. Conserved currents associated with the Kac–Moody algebra of the linearized equations and the Nöether current for the group transformations with fixed value of t are obtained.

A global anomaly-free condition and a non-abelian gauge invariant Wess–Zumino–Witten effective action with less terms have been found by a systematical method rather than by trial and error. The condition requires the difference between the left-and right-handed Chern–Simon five-forms wrt the gauge group must vanish and it turns out to be the usual condition in the local sense.

Wess and Zumino's effective Lagrangian for the chiral field is derived by a new method which shows the direct relationship between the Lagrangian and the difference of the left handed and right handed Chern-Simons topological invariants. The resulting Lagrangian is explicitly left and right antisymmetric and equal to the one obtained in a previous letter by a different method.

In addition to the recent discovery that the Chern–Simons secondary classes are the topological origins generating special unitary gauge group anomalies, it is shown that some special orthogonal gauge group anomalies can also be generated, from the Pontrjagin and the Euler secondary classes. The common differential geometrical and topological background are explained as well.

An approximate effective Lagrangian of composite operators in superfield formalism for the N=1 pure supersymmetric Yang–Mills model is obtained with the help of renormalization group equations for the generating functional. While the supersymmetry is always kept unbroken by this effective Lagrangian we find that the chiral symmetry may and may not be unbroken.

Based on the Weil homomorphism method, 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 remarkably compact form is given. These properties include the effective action, the skyrmion anomalous current, Abelian anomaly, symmetric and asymmetric non-Abelian chiral anomalies, and anomaly free conditions.

The properties of the effective Lagrangian for the low–energy Goldstone–gauge field system constructed from the Chern–Simons topological invariant are further discussed. Both the symmetric anomaly and the asymmetric anomaly are connected with this Lagrangian.

The classical solutions of the two-dimensional non-linear σ model with Wess-Zumino term are shown to be equivalent to that without the Wess-Zumino term under a suitable transformation.

The following sections are included:

• INTRODUCTION

• PARAMETRIZATION OF THE K–M MATRIX

• PHYSICS OF ε AND ε' IN KAON SYSTEMS

• NEUTRAL PARTICLE-ANTIPARTICLE MIXING AND CP VIOLATION IN SYSTEM

• CP VIOLATION IN PARTIAL DECAY RATES OF PARTICLES AND ANTIPARTICLES

• CONCLUDING REMARKS

• ACKNOWLEDGEMENT

• REFERENCES

• DISCUSSION

To explain CP violation in Kaon system in the light of the recently measured b-lifetime within the framework of three generations of quarks, the top quark mass has to be greater than 50GeV if the current algebra value of the factor B_K is adopted and |ε'/ε| < 0.01. In this letter a fourth generation of quark is considered which can fit the present experimental data on CP violation, and K_L→μ⁺μ⁻ decay rate for top quark mass is around 40GeV. The mass of the new charge 2/3 quark is predicted to be over 100 GeV.

Based upon the properties of the characteristic classes and their Chern-Simons secondary characteristic classes, the “Abelian” anomalies in M²ⁿ⁺², the Euler-Heisenberg effective actions in M²ⁿ⁺¹, as well as the non-Abelian anomalies in M²ⁿ for arbitrary gauge group and its reduction subgroup have been investigated thoroughly and the application to the gravitational anomalies is made. It is shown that the “Abelian” anomalies of such groups are equal to each other, their Euler-Heisenberg actions are also closely related to each other, and their non-Abelian anomalies are also equivalent if their common generating functional can be taken as a counter-term. For the gravitational anomalies we present the common generating functional for both non-Abelian Einstein and Lorentz anomalies in M⁴ⁿ⁺² and show the relationship between them.

The canonical quantization is carried out for the two dimensional non-linear σ model with Wess-Zumino term. It is shown that the currents in this model satisfy a Kac-Moody algebra for arbitrary values of the coupling constant.

A 2-cocycle is constructed which gives the anomalous term in the Virasoro algebra.

The nonlinear sigma model describes Goldstone bosons originating from spontaneous symmetry breaking. A set of local counterterms is found to shift the anomaly of the nonlinear sigma model to that of the original model with fermions interacting with external gauge fields. The 't Hooft consistency conditions are matched automatically.

In this paper, it is shown that the cohomology of generalized secondary classes, the Feddeev type cohomology and he generalized gauga transformation can be easily obtained by expanding the Chern form according to the degree of the forms in its submanifolds and using the closed property of the Chern form. It is also shown that a θ-vacuum term in the effective Lagrangian arises when gauge field in the group manifold is present.

Simple derivation of the general Chern-Simons characteristic classes is presented. Application to construct current conservation anomalies and effective action is discussed in detail. The effective action of nonlinear σ model where the fermions are interacting with external gauge fields is obtained. The central term of Virasoro algebra is also derived by topological method. Possible existence of a gauge potential in the group manifold and its relation to θ-vacuum is indicated.

In this paper we study the temperature dependence of chiral-symmetry breaking. Our real-time calculation shows that the thermal fermion propagator has a Lorentz-invariant massive particle pole even as vanishes for T > Λ_ce^2/3. The traditional signature of chiral-symmetry breaking is found only after transforming the Dirac field to the new chiral basis at high temperatures.

Fermion masses and mixing angles are studied in an SUSY SO(10)×Δ(48)×U(1) model with small tanβ. 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.

Fermion masses and mixing angles including that of neutrinos are studied in a model with symmetry group SUSY SO(10)×Δ(48)×U(1). Universality of Yukawa coupling of superfields is assumed. The resulting texture of mass matrices in the low energy region depends only on a single coupling constant and VEVs caued by necessary symmetry breaking. 13 parameters involving masses and mixing angles in the quark and charged lepton sector are successfully described by only five parameters with two of them determined by the scales of U(1), SO(10) and SU(5) symmetry breaking compatible with the requirement of grand unification and proton decay. The neutrino masses and mixing angles in the leptonic sector are also determined with the addition of a Majorana coupling term. It is found that LSND events, atmospheric neutrino deficit and the mass limit put by hot dark matter can be naturally explained. Solar neutrino puzzle can be solved only by introducing sterile neutrino with one additional parameter. More precise measurements of α_s(M_z), V_cb, V_ub/V_cb, m_b, m_t, as well as various CP violation and neutrino oscillation experiments will provide crucial tests of the present model.

CP violation, fermion masses and mixing angles including that of neutrinos are studied in an SUSY SO(10)×Δ(48)×U(1) model with small tanβ. 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.

A unification model for strong, electromagnetic, weak and gravitational forces is proposed. The tangent space of ordinary coordinate 4-dimensional spacetime is a submanifold of a 14-dimensional internal spacetime spanned by four frame fields. The unification of the standard model with gravity is governed by gauge symmetry in the internal spacetime.

We present a general model-independent and rephase-invariant formalism that cleanly relates CP and CPT noninvariant observables to the fundamental parameters. Different types of CP and CPT violations in the K⁰-, B⁰-, and D⁰-systems are explicitly defined. Their importance for interpreting experimental measurements of CP and CPT violations is emphasized. In particular, we show that the time-dependent measurements allow one to extract a clean signature of CPT violation.

The problem of renormalization of the closed–time-path Green's function in nonequilibrium statistical field theory is studied. Under some reasonable assumptions on the high-energy behavior of the initial correlation functions, it is found that the same counterterms which eliminate the ultraviolet divergences in the usual field theory can also make the closed–time-path Green's functions free of ultraviolet divergences. The renormalization-group equation satisfied by the closed–time-path vertex functions is obtained and the Callan-Symanzik coefficient functions are shown to be the same as in the usual field theory.

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-Takahashi identites are deduced.

The closed time path Green's function (CTPGF) formalism is applied to the critical dynamics. The related results for the CTPGF approach are briefly reviewed. Three different forms of CTPGF's are defined, transformations from one to another form and other useful computation rules are given. The path integral presentation of the generating functional for CTPGF's is used to derive the Ward-Takahashi identities under both linear and nonlinear transformations of field variables. The generalized Langevin equations for the order parameters and conserved variables are derived from the vertex functional on the closed time path. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, is determined according to the Ward-Takahashi identities and the linear response theory. All existing dynamic models are reobtained by assuming the corresponding symmetry properties for the system. The effective action for the order parameters is deduced by averaging over the random external field. The Lagrangian formulation of the statistical field theory is obtained if the random field one-loop approximation and the second-order approximation of order-parameter nuctuations on different time branches are both taken. The various possibilities of improving the current theory of critical dynamics within the framework of CTPGP's are discussed. The problem of renormalization for the finite-temperature field theory is considered. The whole theoretical framework is also applicable to systems near the stationary states far from equilibrium, whenever there exists an analog of the potential function (“free energy”).

A theoretical scheme using closed time-path Green's functions is proposed to describe the quantum statistical properties of the order parameter in terms of a generating functional. The dynamic evolution is generated by a driving source, while the statistical correlation by a fluctuation source. The statistical causality is shown to hold explicitly and to give rise to a number of important consequences. The problem of determining the quantum statistical properties for the order parameter is reduced to finding a solution of the functional equation for it.

Several equivalent expressions for the generating functional of the order parameter are derived in this paper to elucidate how the statistical properties are incorporated in its field-theoretical structure. It is shown that in the process of determining the generating the dynamic evolution and the statistical information can be seperated to a certain extent which facilitates solving the problem. The whole procedure is greatly simplified if the statistical correlation is Gaussian or a generalized fluctuation-dissipaticn theorem (FDT) holds. As an example, the Gaussian character of the thermal equilibrium distribution is justified and the Matsubara technique along the imaginary time axis is extended to the real axis to provide a complete perturbation scheme for the closed time-path Green's functions (CTPGF).

A practical scheme is proposed in this paper to determine the quantum statistical properties of the order parameter in the framework of the closed time-path Green's functions (CTPGF). As a microscopic quantum theory, this formalism is applicable in principle to both equilibrium and nonequilibrium phenomena and is capable of dealing with statistical systems both above and below their phase transition point. This formalism can be used to discuss the statistical properties of simple as well as composite order parameters in either uniform or nonuniform systems. As simple illustrations and check for the theory, the proposed prescription is applied to the contact-interaction model for superconductivity and a unimode laser system coupled with two-energy-level bound electrons.

A simplified derivation of the macroscopic electrodynamic equations of Umezawa, Mancini et al. for superconductors is given in the framework of the closed time path Green's functions (CTPGF) using generalized Ward-Takahashi identities. It is shown that the forms of the equations obtained are the same for both thermoequilibrium and nonequilibrium stationary states provided the eletromagnetic field is weak and its effect on the modulus of the order parameter can be neglected. The statistical behavior of the states is completely specified in the equations by parameters which can be calculated by the method of CTPGF.

A dynamical theory for quenched random system is developed in the framework of CTPGF. In steady states the results obtained coincide with those following from the quenched average of the free energy. The order parameter q, a matrix in general, becomes an integral part of the second order connected CTPGF. An equation to determine q is derived from the Dyson-Schwinger equation in this formalism. Some general properties of the CTPGF in a quenched random system are discussed.

The dynamics of spin glass is studied in the framework of CTPGF. A marginal stability line is found on the q-X plane. Below T_C with h < h_C, the time evolution of the order parameter follows Fischer's line exponentially to the stability boundary and then decreases in power law along the boundary to its fixed point. The Langevin equation for the spin σ(t) is no longer valid along the stability boundary. The susceptibility is calculated in perturbation theory and found to be in good agreement with those predicted by the projection hypothesis. The general validity of the projection hypothesis is justified in the present formalism.

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-ranged random Ising model. A boundary line is found on the q-|χ| plane. 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 validity of the projection hypothesis is discussed.

We study the field theory approach to Anderson localization in the framework of closed time path Green's function (CTPGF). The theory is found to be invariant under an Sp(2) group. Ward identities related to this symmetry are derived. A non-linear σ model arises as a consequence of the dynamical symmetry breaking caused by the imaginary part of the retarded Green's function.

In the spin glass phase near the critical temperature we find in a particular replica symmetry breaking pattern an order parameter with free energy greater than that of the Parisi's.

In this paper we summarize the work done by our group in developing and applying the closed time-path Green function (CTPGF) formalism, first suggested by J. Schwinger and further elaborated by Keldysh and others. The generating functional technique and path integral representation are used to discuss the various properties of the CTPGF 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 fluctuation 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 CTPGF 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 fluctuation–dissipation 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.

In a non-equilibrium stationary state with space-time structure a Nambu-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.

In this note, by introducing the canonical description which is one dimensional in nature, we establish the Bohr-Sommerfeld quantization condition for the fractional quantum Hall effect system. The generic and model independent physical implications of this condition are explored. The guiding center representations of the N-body Schrödinger equation for the system is also derived.

In this Brief Report, we explicitly show the equivalence of the Feynman-Vernon influence-functional approach and the path-integral formulation of the Schwinger-Keldysh closed-time-path Green's function. The latter simplifies the practical calcuiations considerably with a systematic diagrammatic technique and is expected to have a broader application.

Generalized free energy of order parameters near nonequilibrium stationary states is shown to exist for systems obeying time reversal invariance. A low frequency fluctuation-dissipation theorem similar in form to that in thermoequilibrium is obtained in this case.

The binding energies of nuclei H³, He³, He⁴, the low energy n-p scattering length and effective range are calculated by using the standard variational methods. A two-range central Yukawa potential is considered in the first two sections. The longer range corresponds to the π meson mass. The smaller range corresponds either to the heavier meson or to the higher order field interactions. No repulsive core appears, when the force parameters are chosen to fit the low energy scattering and deuteron data. The calculated binding energies of nuclei H³, He³ and He⁴ are too high. This result is in agreement with most of the previous calculations. Tensor force of the Schwinger mixed type is considered in the third and fourth sections. The force parameters are chosen to fit the low energy two body data. They are not uniquely determined and are given for a set of possible D-percentage values. The adding of the tensor force reduces considerably. the calculated binding energies of nuclei H³, He³, He⁴. But still, the calculated values increase too fast with the mass number. It does not fit the triton and helium binding energies simultaneously. The possibility of adding many body forces is discussed at the end of the paper.

The following sections are included:

• 引言

• 場的方程式同正則變換

• 核子間的三体力位能

• 膺标介子膺向量耦合与对称型标量介子标量耦合的静三体力位能

• 静三体力位能对 ³H, ³He 及 ⁴He 諸核基态結合能的貢献

• 参考文献

The space time average method, used in the derivation of macroscopic maxwell equations, is elaborated in a systematic way. The technique of Dirac δ–function is used which greatly simplifies the calculation. The results are identical with those obtained by Mazur and Nijiboer^[4], who use the ensemble average method. The idea involved in the present paper is much simplier than that of Mazur and Nijiboer and the calculation is straightforward. The present method can be extended to the derivation of hydrodynamical equations and easily generalized to the quantum ease.

The following sections are included:

• 引言

• 質量算符

• 虚光子与核子散射的色散关系

• 單核子中間态对吸收部 Λ_μ 的貢献

• 其他中間态对核子电磁質量的貢献

• 結論

• 参考文献

The following sections are included:

• 引言

• μ介子在原子核上俘获的普遍公式

• μ^–介子在質子上的俘获

• μ^–介子在氘核上的俘获

• μ^–介子在原子核上的俘获

• 参考文献

• 附录 I

• 附录 II

The rate of the muon capture reaction μ⁻ + He³→H³ + ν, the angular distribution and the polarization of the final nucleus H³ are calculated basing on the universal V – A coupling with induced pseudoscular term and the weak magnetic current included. A general initial condition with polarized muon and polarized He³ nucleus in different hyperfine states is assumed.

The ground state wave function is assumed to be a pure S-state. Thus we neglected the contribution of other states, which may be caused by the presence of tensor force or other velocity dependent forces. Meson exchange current is also neglected. Under these assumptions we have proved that the capture amplitude contains only one unknown nuclear matrix element, which is the Fourier transform of the nuclear density function. To determine this nuclear matrix element by the scattering of electron (or muon) with He³ (H³) nucleus is finally proposed.

Recent experiments seem to show that the sign of the effective pseudoscalar term in μ–capture reactions is different from the theoretical result of Goldberger and Treiman based on V – A theory and dispersion relation technique. A possible explanation of this difference is proposed in the present note. It is shown that a very small p. s. tem in the HamiHonian (of the order 1/100 of the effective p. s. term) can change the sign of the effective p. s. term. If this small term is transformed and added as a part of the universal axial vector current, it will produce practically no effect in β–decay and π → μ + ν decay. Finally the inclusion of hyperon pairs is considered.

The correction of the recoil to the potential in the scattering of μ-mesons (or electrons) by zero-spin nuclei is derived by means of a simple method. The potential obtained is corrected to first order in the velocity of the nuclei but without restriction on the velocity of the μ-mesons (or electrons). The effect of recoil on the phase-shift and cross section is discussed. The results are compared with those of Foldy, Ford and Yennie. It is pointed out that the method used in the latter's work is not perfect, because it makes use of Breit's interaction which is valid only for small velocities.

Dispersion relations for pion production in pion-nucleon collissions are discussed. The scattering amplitude is written in a form, in which the operators of nucleon field are extracted from the state vectors of initial and final stale. The kinematies and symmetry, properties are worked out in detail and the contribution of bound state is evaluated.

An Experiment designed to detect the p-resonance in π – π scattering process is suggested.

In this paper an experiment for the reaction π + p → Λ + π + K is proposed. The total energy in the centre of mass system is fixed at 1900 MeV. The kinetic energy of the final K-meson is then in the range 0—90 MeV. We propose to observe the final state Λ-π resonance and the cusp at the Σπk threshold. From the correlation between the resonance and the cusp and from the angular distribution of the resonance, the spin of Y₁^* and the relative parities among Y₁^*, Λ and Σ may be determined. Based on the diagonal representation of T-matrix in the channel space, a general phenomenological description of the resonance effect, the near threshold effect and the final state interaction is developed, which takes into account especially the correlation of the cusp with a nearby resonance. The general theory is applied to the present experiment as a special case.

The electro-weak model in SU(3) × U(1) is extended to include strong interaction. A possible exchange symmetry between the flavour and the colour gauge interactions is discussed.

In this paper a unified electro-weak model for leptons based on the SU(3) gauge group is suggested by means of four kinds of realization for the generators of the group. For all low energy electro-weak processes, this model predicts the same results as the conventional Weinberg-Salam model does. The Weinberg angle is shown to be sin³θ_w = ¼ in a natural way. When the Higgs self potential respects a discrete symmetry Φ → − Φ, a new conserved quantum number called weak strangeness emerges from the model after spontaneous symmetry breaking. In the present model there exist another four heavy vector gauge bosons V^± and U^±± together with some heavy fermions and Higgs scalars, which have non vanishing weak strangeness quantum numbers. These weak strange particles have no direct couplings with leptons. Their existence will not influence the low energy electro-weak processes. Nevertheless, they can be produced in pairs in high energy collisions and the lightest of them should be stable if the conservation of weak strangeness is exact. The experimental implications and the possibility of violation of the conservation of weak strangeness are also discussed.

本文給出了非綫性拉氏函数路径积分量子化的一般形式。在泛函积分中出現的等效拉氏函数, 除原始拉氏函数以及李、楊的δ(0)項外, 还有一个修正項, 这个修正項比例于δ(0), 而且是δ(0)的冪級数的对数。这种形式为非綫性場論的量子化提供了理論基础。

In this paper we discussed the following problems in the SU(3)×U(1) unified model proposed earlier: (i) Two possible choices of the Higgs fields and their comparison with the Weinberg-Salam model; (ii) The form of the Higgs self potential and the realization of the spontaneous symmetry breaking; (iii) The relation between the mass spectrum and the generalized Cabibbo mixing angles in a model with several generations of fermions.

It is shown that in order to introduce the gauge invariant Wess-Zumino-Witten effective action a global anomalyfree condition should be satisfied by the gauged subgroup of SU(3)_L × SU(3)_R. The condition requires that the left handed and the right handed Chern-Simons 5-forms with respect to the gauge group be equal to each other and it turns out in the local sense to be the usual perturbative anomaly-free condition. It is also constructed a gauge invariant effective action under the anomaly-free condition by means of a systematic method rather than the trial and error Noether method. In the non-abelian case, the gauge invariant effective action presented here contains less terms than the one obtained by Witten. The case of pure gauge is discussed in the present note as well.

Both the symmetrical anomaly and the unsymmetrical anomaly, are derived from an effective Lagrangian recently constructed on the basis of Chern-Simons topological invariants.

General expression for gauge covariant anomalous current is obtained from the difference of left handed and right handed Abelian anomaly in 2n+2 dimensions by the method of Weil homomorphism. The general form of symmetric and asymmetric anomalies, gauge invariant Wess-Zumino effective action and anomaly free condition in 2n dimensions are summarized in one closed formula, showing the deep connections among all these topological properties of gauge fields and pseudascalar Goldstone fields in 2n and 2n+é dimensions.

Some general Chern-Simons cochains are easily obtained by expanding the Chern form according to the degree of the form in its submanifold and using the closed property of the Chern form. The recurrent relatuns of Chern form between submanifolds are discussed under some constraints. We also consider their application.

In this note new co-boundary operators are defined in the product form. The associative composition law of spatial translation group field is discussed using these new operators. The quantization condition of monopoles in SU(2) and SU(3) gauge theories follows easily from the new formalism.

In consequence of the spontaneous symmetry breaking, non-zero energy Goldstone modes with dissipation are excited in a non-equilibrium stationary state with space-time structure. In this paper as a specific example, the Ward-Takahashi identities formulated in the close time path Green's function method is applied to the saturation state of a single mode laser. A generalized Goldstone theorem in a weak inhomogeneous dissipative system is established and the physical interpretation of the Goldstone mode is discussed. As a result of the Goldstone theorem, the pole in the Green's function of the laser light splits into two with equal weights, each corresponding to a quanta with the same frequency but different dissipation. Together with the order parameter (the average value of the vector potential), these two kinds or quanta (one of which is the Goldstone mode) give a complete description of the order-disorder transition of the phase symmetry in the saturation state of the laser. A detailed discussion on the restoration of the spontaneously broken symmtry of the phase is given.

In this article, we derived the transformations among three sets of closed time path Green's functions and some calculation rules, from which a general definition of arbitrary multipoint retarded and advanced Green's functions follows naturally. Some algebraic identities among multipoint functions have shown to be the consequences of the property W[J₊,J₋]|_J+=J−=J = 0 for the generating functional on the closed time path.

Starting from the equations satisfied by the vertex functions on the closed time path, we derived the generalized Langevin equations for the order parameters and the conserved variables. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, was determined from the Ward-Taka-hashi identities and the linear response theory. All existing dynamic models were recovered by assuming the corresponding symmetry properties of the system. The whole theoretical framework is also applicable for describing the systems near steady states far from equilibrium.

The expression of the effective action for the order parameters is derived from the continuous integral representation for the generating functional on the closed time path in the one loop approximation for the Fourier transforms. The Lagrangian formulation of the critical dynamics is recovered in the second order approximation of fluctuations in the closed time path continuous integral. The various possibilities of improving the existing theory of critical dynamics are considered.

This is the first part of our work on time reversal symmetry applied to non-equilibrium statistical stationary states from a unified microscopic quantum statistical point of view. In this paper, a formalism for time reversal symmetry is constructed in the framework of the Closed Time Path Green's Functions (CTPGF), which can be applied both, to equilibrium and non-equilibrium stationary states. By using the generating functional technique of the CTPGF. symmetry relations for the statistical Green's functions and vertex runctions are derived for systems invariant under time reversal.

This is the 2nd part of our discussion on time reversal symmetry applied to the non-equilibrium statistical stationary states (NESS) from a microscopic quantum statistical point of view. With the application of the main results of I. a systematic investigation on the general properties of the NESS is given. For systems invariant under time reversal, the existence of a generalized potential and the fluctuation-dissipation theorem in the low frequency limit are established in the NESS. The Onsager's reciprocity relations for the local thermodynamical equilibrium systems are also generalized to that for the NESS invariant under time reversal symmetry. Finally, the time dependent Ginzburg-Landan equations for order parameters and conserved densities have been expressed in a general form with time irreversible and reversible parts similar to that met in the literatures studying critical dynamics.

The set of coupled, self-consistent equations for the order parameter, the fermion field and the collective excitations developed previously in the framework of the closed time-path Green's functions (CTPGF) is applied to systems with four-fermion interaction. The Hartree, the Hartree-Fock (HF) and the random phase approximations (RPA) for the order parameter and its fluctuations are derived in a systematic way. The thermodynamical potential for the fermion system is calculated explicitly in these approximations. The formalism presented here can be applied to nuclear as well as other many fermion systems in both equilibrium and nonequilibrium states.

A set of coupled equations for a quantum statistical system that determine self-consistently the order parameter, the energy spectrum, the dissipation and the distribution for both fermion field and collective excitation is suggested with a loop expansion formalism. They are applicable to non-equilibrium as well as equilibrium systems.