Axial-vector Vertices Research Articles

Axial-Vector Vertex in Spinor Electrodynamics

Working within the framework of perturbation theory, we show that the axial-vector vertex in spinor electrodynamics has anomalous properties which disagree with those found by the formal manipulation of field equations. Specifically, because of the presence of closed-loop the divergence of axial-vector current is not the usual expression calculated from the field equations, and the axial-vector current does not satisfy the usual Ward identity. One consequence is that, even after the external-line wave-function renormalizations are made, the axial-vector vertex is still divergent in fourth- (and higher-) order perturbation theory. A corollary is that the radiative corrections to ${\ensuremath{\nu}}_{l}l$ elastic scattering in the local current-current theory diverge in fourth (and higher) order. A second consequence is that, in massless electrodynamics, despite the fact that the theory is invariant under ${\ensuremath{\gamma}}_{5}$ tranformations, the axial-vector current is not conserved. In an Appendix we demonstrate the uniqueness of the triangle diagrams, and discuss a possible connection between our results and the ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ and $\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ decays. In particular, we argue that as a result of triangle diagrams, the equations expressing partial conservation of axial-vector current (PCAC) for the neutral members of the axial-vector-current octet must be modified in a well-defined manner, which completely alters the PCAC predictions for the ${\ensuremath{\pi}}^{0}$ and the $\ensuremath{\eta}$ two-photon decays.

Low-Energy Theorem for the Weak Axial-Vector Vertex

Low-Energy Theorem for the Weak Axial-Vector Vertex

Sidewise Dispersion Relations and the Axial-Vector Coupling Constant

The axial-vector renormalization constant is related to low-energy pion-nucleon scattering amplitude by dispersing the axial-vector vertex in the nucleon mass and using the idea of threshold dominance. The low-energy pion-nucleon amplitude, in turn, can be related to the axial-vector renormalization, giving rise to a consistency condition involving $\frac{{g}_{A}}{{g}_{V}}$ and the pion-nucleon coupling constant. The solution to this consistency condition is in rough agreement with the observed value.

Low-Energy Theorem for the Weak Axial-Vector Vertex

A low-energy theorem is derived for the weak axial-vector vertex. The theorem enables one to calculate from strong or electromagnetic processes the two leading terms in the expansion of the axial-vector vertex in powers of the leptonic four-momentum transfer. Applications to weak pion production, ${K}_{e4}$ decay, and radiative $\ensuremath{\mu}$ capture are discussed. In particular, we express the radiative $\ensuremath{\mu}$-capture matrix element, up to and including contributions linear in the leptonic four-momentum transfer and the photon four-momentum, in terms of the elastic weak form factors and pion photoproduction amplitudes.

Dispersion Relation for the Axial-Vector Vertex and a Sum Rule for the Axial-Vector Coupling-Constant Renormalization

A dispersion relation for the nucleon axial-vector vertex, in which the subtraction term at infinite nucleon mass is given by the unrenormalized vertex, is derived from the axial current-field commutation relations. Using the hypothesis of partially conserved axial-vector current, a sum rule for the axial-vector coupling-constant renormalization ${g}_{A}$ is obtained in terms of the $\ensuremath{\pi}\ensuremath{-}N$ form factor. In the elastic-unitarity approximation, the sum rule involves the integral over the difference of the ${P}_{11}$ and ${S}_{11}$ phase shifts. Certain experimental phase shifts up to 1-GeV pion kinetic energy give ${g}_{A}\ensuremath{\approx}1.1$. However, because of the lack of consistent experimental data beyond 500 MeV, and also because of the lack of reliable theory to estimate the important contribution from inelastic channels, all we can conclude at the moment is that the sum rule is not inconsistent with the experimental data.

The role of τγ5-gauge invariance in a nonlinear model of elementary particles

The role of τγ5-gauge invariance in the theory of elementary particles proposed byNambu andJona-Lasinio (1) has been examined in the framework of the lowest ladder approximation followed by them. It has been shown that the above invariance property is not necessary for proving the vanishing of the pion mass in this approximation. Their integral equation for the axial vector vertex part needs some modification if the solution is to represent the vertex part of a conserved axial vector current.