Chiral Invariance Research Articles

Non-local Yang-Mills theory

The Yang-Mills theory assumes that the parameters of the internal SU(2) group are functions of space-time. This is generalized by making the parameters matrices in space-time. When these are diagonal the author recovers the original theory. Instead of transforming according to just the regular representation, finds that the gauge field contains all representations consistent with angular momentum conservation. A further novelty is a metric introduced to make bilinears invariant and similarities to chiral invariance are pointed out. A suggested physical motivation is the existence of the Planck length.

Chiral-Invariant Regularization and Renormalization of the Axial-Vector Current

The regularization and renormalization procedures for perturbation calculations involving the axial-vector current of the $\ensuremath{\sigma}$ model are considered. It is shown that, even in the absence of electromagnetism, special care must be exercised when introducing fermion regulator fields into the Lagrangian so that the axial-vector Ward-Takahashi identity is not violated in the process. A generalized method of regularization is proposed in which the regulator part of the Lagrangian satisfies chiral invariance and which reduces to the usual one when applied to quantities other than the axial-vector current. By virtue of the chiral-invariant character of this method, a subtraction procedure can be introduced for the axial-vector vertex which eliminates unrenormalizable cutoff-dependent terms. After this subtraction, the regularized axial-vector vertex of the $\ensuremath{\sigma}$ model satisfies the usual Ward-Takahashi identity and is renormalizable by rescaling of the external wave functions.

Neutron Asymmetry in Polarized Nuclear Muon Capture

A general theoretical framework for neutron emissions due to nuclear muon capture is given in terms of relativistic and nonrelativistic kinematics, which can be applied to the cases where the energies of the final nuclear states belong to the continuous spectrum as well as the discrete one. The consequences of chiral invariance and conserved-vector-current (CVC) theory are taken into account in the formulation. The formulas obtained are compared with conventional ones. In particular, the pioneering work due to Primakoff is re-examined from this general framework. The difference between the CVC theory with the impulse approximation and the nuclear CVC theory is discussed. In the limit of small momentum of the emitted neutrino, the present formalism is related to the conventional $\ensuremath{\beta}$-decay formalism. The effect of nuclear collective motions on neutron asymmetry is discussed without assuming any detailed nuclear models.

Hard-meson current algebra analysis of the K [formula omitted]3 decay

A hard-meson calculation of the K l 3 , K ∗ and κ decays is given based upon (i) chiral SU(3) × SU(3) current algebra, (ii) pion and kaon PCAC, (iii) single meson saturation, (iv) approximation of meson vertices by low-order polynomials in momentum. No a priori assumption on the size of SU(3) or chiral breakdown, or type of chiral breakdown, is made. The possible existence of a κ-meson is included by (v) the PCVC condition on the strangeness changing vector current. If ƒ +(0) takes on its SU(3) symmetric value (ƒ +(0) ≈ 1.0) then large breakdown of chiral invariance exists, while the chiral symmetric value (ƒ +(0) ≈ 0.86) implies large breakdown of SU(3). A value of λ + consistent with the K l 3 data and yielding also the correct K ∗ width if one uses the chiral symmetric ƒ +(0) can be obtained, and implies the vanishing of the K ∗-K-π vertex asymptotically. If no κ-meson exists, the theory requires both |ξ| and λ − to be small. This is also generally the case if a κ meson exists and one assumes a (3 ∗, 3) + (3, 3 ∗) form of chiral breakdown. A large |ξ| and small λ − (which is consistent with both the polarization and the most recent branching ratio determinations) can only be achieved if both a κ-meson exists and one violates the (3 ∗, 3) + (3, 3 ∗) assumption. The theory than also predicts a κ width in agreement with recent data. It is further shown that the hard meson assumption are inconsistent with both a large |ξ| and a large λ −, and that the Callan-Treiman relation may be in error by as much as 50% (due to the presence of non-gentle terms in the K l 3 amplitude).

Chiral Dynamics withoutA1

The nonlinear realization method of Weinberg is used to construct a chiral-invariant Lagrangian consisting of $\ensuremath{\pi}$, $N$, ${N}^{*}(1236)$, and $\ensuremath{\rho}$, but not ${A}_{1}$. Chiral invariance can be, of course, maintained without invoking the ${A}_{1}$ meson. In view of the experimental status of ${A}_{1}$, it may be worthwhile studying such a model. The tree-diagram technique is used, with an off-mass-shell ${N}^{*}$ propagator, to calculate pion-nucleon $s$-wave scattering lengths, isobar production parameters, and single-pion photoproduction differential cross sections. A comparison with the current-algebra method is made.

Nonleptonic Decays ofKMesons in Chiral Dynamics and the Proof of the ΔI=1/2 Rule

We construct weak interactions from the nonlinear chiral invariant Lagrangian, and dis­ cuss the nonleptonic decays of K mesons in the chiral vector meson model. Correction to this model due to the current-current interact jon and the K*- and KA-poles violates the AI=1/2 rule, but the violation is quite small and consistent with experiment. Moreover we assume the intermediate boson mass to be 12 Be V in the model to explain the K 1°-'>2rr: decay rate. 1 ) has derived an effective Lagrangian for soft-pion interac­ tion, which reproduces the results of current algebra in a simple way. On the basis of the (J model, he has employed an SU(2) chiral transformation. Schwinger2) has also derived an effective Lagrangian, and has introduced the vector and axial-vector mesons as gauge fields. The concept of chiral invariance plays ail important role in understanding the low energy phenomena. The purpose of this paper is to show that chiral invariance is also useful for the problem in weak interactions. Moreover we discuss the nonleptonic decays of K mesons, and prove the validity of the LJI = 1/2 rule on the basis of the chiral invariant Lagrangian. In§ 2 a generalization of the SU (2) chiral dynamics to SU (3) is formulated ll1 the quark model. S ) The vector and axial-vector mesons are introduced as gauge fields. The Lagrangian written by actual fields is invariant under the nonlinear chiral transformation. In § 3 the Lagrangian for the weak interaction is constructed from the Lagrangian for the strong interaction obtained in § 2. In § 4 the nonleptonic decays of K mesons and the LJI = 1/2 rule are discussed. It can be shown that the LJI = 1/2 rule holds, though at first sight the weak Lagrangian (17) seems to violate the 111=1/2 rule. In a previous paper 4 ) we considered the K-72n decay and proved the 111=1/2 rule for this process on the basis of the quark model. In this paper we consider the K1D decay by the self­ energy type diagram due to the vector mesons, the axial-vector mesons and the intermediate boson without introduction of quark. We find that this model (chiral vector meson model) also explains the 111=1/2 rule, and that the result of this model is the same as current algebra with a slight difference in coefficient. The contributions of the current-current interaction and the K* -pole and KA-pole;· 1402

Nonlinear Realizations of Chiral Symmetry

We explore possible realizations of chiral symmetry, based on isotopic multiplets of fields whose transformation rules involve only isotopic-spin matrices and the pion field. The transformation rules are unique, up to possible redefinitions of the pion field. Chiral-invariant Lagrangians can be constructed by forming isotopic-spin-conserving functions of a covariant pion derivative, plus other fields and their covariant derivatives. The resulting models are essentially equivalent to those that have been derived by treating chirality as an ordinary linear symmetry broken by the vacuum, except that we do not have to commit ourselves as to the grouping of hadrons into chiral multiplets; as a result, the unrenormalized value of $\frac{{g}_{A}}{{g}_{V}}$ need not be unity. We classify the possible choices of the chiral-symmetry-breaking term in the Lagrangian according to their chiral transformation properties, and give the values of the pion-pion scattering lengths for each choice. If the symmetry-breaking term has the simplest possible transformation properties, then the scattering lengths are those previously derived from current algebra. An alternative method of constructing chiral-invariant Lagrangians, using $\ensuremath{\rho}$ mesons to form covariant derivatives, is also presented. In this formalism, $\ensuremath{\rho}$ dominance is automatic, and the current-algebra result from the $\ensuremath{\rho}$-meson coupling constant arises from the independent assumption that $\ensuremath{\rho}$ mesons couple universally to pions and other particles. Including $\ensuremath{\rho}$ mesons in the Lagrangian has no effect on the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths, because chiral invariance requires that we also include direct pion self-couplings which cancel the $\ensuremath{\rho}$-exchange diagrams for pion energies near threshold.

Chirality and the structure of the pion-nucleon interaction

The concept of chirality invariance is extensively applied to the direct four-fermion interaction of the nucleon-antinucleon system. It is shown that if we extend the chirality concept to the isotopic spinor,P-invariance is derived in a natural way, and if we further extend this concept (according to Watanabe-Tanikawa) to the tensor quantities appearing in the interaction Lagrangian, (parity conserving)V+A orV−A is suggested as the type of basic interaction. Based on the Fermi-Yang model, effective pion-nucleon couplings are derived by the use of the chain approximation, and it is found that theV+A andV−A correspond to the pseudoscalar and pseudovector pion-nucleon couplings respectively. Some remarks concerning the results are made, in view of the symmetry of strong interactions.

On the symmetries of strong and weak interactions

A model of strong interactions is proposed which admits a group (G 4×H) in the limit of a suitably defined doublet approximation,G 4 being a 4-dimensional extension of the isotopic spin group andH the hypercharge gauge group. Weak interactions having phenomenological chirality invariance properties or obeying the ΔI=1/2 rule are shown to be invariant under an unitary subgroup of the group (G 4×H).

Note on possible parity non-conservation in strong interactions

As experimental tests of the conservation of parity in strong interactions have not been completed, it may be of interest to observe that a principle which has already been successfully applied to the theory of weak interactions may allow a distinction to be made between the strong interaction terms of the Lagrangian which do, and those which do not conserve parity. The principle of chirality invariance, or its equivalent expression in terms of the two-component fermion theory, is considered. (W.D.M.)

Discrete symmetries and chirality invariance in quantum field dynamics

The aim of the present paper is to discuss systematically the discrete symmetry operations on a quantized field in interaction; and to base the introduction of the new quantum number “chirality” for spinor fields on these symmetry properties. In the course of this investigation, several general results on the group of symmetry operations are proved and relation between certain sets of discrete symmetry operations and the spinor representation of the rotation group in 3 and 4 dimensions is established. An attempt has been made to present clearly the connection between additive and multiplicative quantum numbers, gauge transformations, unitary transformations and invariance laws. The chirality invariance of spinor fields in interaction is discussed in some detail. The emphasis throughout is on the systematic development rather than on details of application. The paper is divided into two parts, the first dealing with the general theory of discrete symmetry operations and the second concerned with chirality invariance for spinor fields.