$\gamma_{5}$ algebra ambiguities in Feynman amplitudes: momentum routing invariance and anomalies in $D=4$ and $D=2$

TL;DR

This paper investigates ambiguities in gamma_5 algebra within divergent integrals in chiral theories, proposing a minimal prescription to resolve these ambiguities and demonstrating how momentum routing invariance ensures gauge invariance and consistent anomaly display.

Contribution

It introduces a minimal prescription to uniquely handle gamma_5 ambiguities in divergent integrals and links momentum routing invariance to gauge invariance in chiral Abelian theories.

Findings

A minimal prescription resolves gamma_5 ambiguities.

Momentum routing invariance ensures vector gauge invariance.

Application to chiral Schwinger Model illustrates the approach.

Abstract

We address the subject of chiral anomalies in two and four dimensional theories. Ambiguities associated with the $\gamma_5$ algebra within divergent integrals are identified, even though the physical dimension is not altered in the process of regularization. We present a minimal prescription that leads to unique results and apply it to a series of examples. For the particular case of abelian theories with effective chiral vertices, we show: 1- Its implication on the way to display the anomalies democratically in the Ward identities. 2- The possibility to fix an arbitrary surface term in such a way that a momentum routing independent result emerges. This leads to a reinterpretation of the role of momentum routing in the process of choosing the Ward identity to be satisfied in an anomalous process. 3- Momentum Routing Invariance (MRI) is a necessary and sufficient condition to assure vectorial gauge invariance of effective chiral Abelian gauge theories. We also briefly discuss the case of complete chiral theories, using the Chiral Schwinger Model as an example.