Anomalous Chiral Action from the Path-Integral

TL;DR

This paper generalizes the Fujikawa approach within the path-integral formalism to derive the anomalous chiral action, linking it to the Chern-Simons term and exploring regularization methods for consistent anomalies.

Contribution

It introduces a method to derive the full anomalous chiral action from the fermion measure variation, connecting it to five-dimensional Chern-Simons terms and regularization techniques.

Findings

Derived the anomalous chiral action via path-integral measure variation

Connected the action to five-dimensional Chern-Simons terms

Explored regularization methods for consistent vs. covariant anomalies

Abstract

By generalizing the Fujikawa approach, we show in the path-integral formalism: (1) how the infinitesimal variation of the fermion measure can be integrated to obtain the full anomalous chiral action; (2) how the action derived in this way can be identified as the Chern-Simons term in five dimensions, if the anomaly is consistent; (3) how the regularization can be carried out, so as to lead to the consistent anomaly and not to the covariant anomaly. Our method uses Schwinger's ``proper-time'' representation of the Green's function and the gauge invariant point-splitting technique. We find that the consistency requirement and the point-splitting technique allow both an anomalous and a non-anomalous action. In the end, the nature of the vacuum determines whether we have an anomalous theory, or, a non-anomalous theory