The Axial Anomaly Revisited

TL;DR

This paper revisits the axial anomaly in gauge theories with chiral fermions, proposing a method to construct a gauge-invariant, renormalizable theory by adding specific terms, akin to a gauge-invariant regularization.

Contribution

It introduces a novel approach to handle abelian anomalies by modifying the action to achieve gauge invariance and renormalizability, extending the abelian Wess-Zumino model.

Findings

Constructed a gauge-invariant, renormalizable theory with abelian anomalies.

Explicitly worked out the case of a massless Dirac fermion with two abelian gauge fields.

Demonstrated the approach's similarity to gauge-invariant regularization methods.

Abstract

We consider theories with gauged chiral fermions in which there are abelian anomalies, and no nonabelian anomalies (but there may be nonabelian gauge fields present). We construct an associated theory that is gauge-invariant, renormalizable, and with the same particle content, by adding a finite number of terms to the action. Alternatively one can view the new theory as arising from the original theory by using another regularization, one that is gauge-invariant. The situation is reminiscent of the mechanism of adding Fadeev-Popov ghosts to an unsatisfactory gauge theory, to arrive at the usual quantization procedure. The models developed herein are much like the abelian Wess-Zumino model (an abelian effective theory with a Wess-Zumino counterterm), but unlike the W-Z model are renormalizable! Details of the approach are worked out explicitly for the special case of a single massless Dirac fermion, for which we couple one abelian gauge field to the vector current and another abelian gauge field to the axial current.