Some chirality-related properties of the 4-D massive Dirac propagator and determinant in an arbitrary gauge field

TL;DR

This paper demonstrates that for a 4-D massive Dirac field in arbitrary gauge fields, the propagator and determinant are fully determined by one chirality sector, extending previous results and linking differences to the chiral anomaly.

Contribution

It generalizes known relations in self-dual backgrounds to arbitrary gauge fields, connecting the determinants of chirality sectors through the chiral anomaly.

Findings

Dirac propagator and determinant determined by one chirality sector

Differences in determinants relate to the integrated chiral anomaly

Extension of previous results to arbitrary gauge fields

Abstract

For a 4-D massive Dirac field in the background of arbitrary gauge fields, we show that the Dirac propagator and functional determinant are completely determined by knowledge of the corresponding quantities for just one of the chirality sectors of the second-order Dirac operator. This generalizes the related, previously known, statements in (anti-)self-dual background gauge fields. The logarithms of the (renormalized) functional determinants from the two chirality sectors are shown to be different only by a term reflecting the integrated chiral anomaly.