Transverse Ward-Takahashi Identity, Anomaly and Schwinger-Dyson Equation

TL;DR

This paper rederives and extends the transverse Ward-Takahashi identities, discusses anomalies, and introduces a new scheme for solving Schwinger-Dyson equations, achieving exact solutions in 2D Abelian gauge theory.

Contribution

It presents a novel approach combining transverse and longitudinal Ward-Takahashi identities to precisely formulate and solve Schwinger-Dyson equations, especially in two-dimensional gauge theories.

Findings

Exact and closed Schwinger-Dyson equation for fermion propagator in 2D Abelian gauge theory.

The scheme yields exact solutions in the chiral limit.

Discussion of anomalies in transverse Ward-Takahashi identities.

Abstract

Based on the path integral formalism, we rederive and extend the transverse Ward-Takahashi identities (which were first derived by Yasushi Takahashi) for the vector and the axial vector currents and simultaneously discuss the possible anomaly for them. Subsequently, we propose a new scheme for writing down and solving the Schwinger-Dyson equation in which the the transverse Ward-Takahashi identity together with the usual (longitudinal) Ward-Takahashi identity are applied to specify the fermion-boson vertex function. Especially, in two dimensional Abelian gauge theory, we show that this scheme leads to the exact and closed Schwinger-Dyson equation for the fermion propagator in the chiral limit (when the bare fermion mass is zero) and that the Schwinger-Dyson equation can be exactly solved.