Worldline Path Integrals for Fermions with Scalar, Pseudoscalar and Vector Couplings

TL;DR

This paper systematically derives worldline path integrals for Dirac fermions interacting with scalar, pseudoscalar, and vector fields, introducing new interaction terms and a novel regularization, and computes the leading order chiral anomaly contribution.

Contribution

It provides a comprehensive derivation of worldline path integrals for fermions with general background fields, including new interaction terms and a heat-kernel-like regularization for the phase.

Findings

Derived worldline path integrals for real and imaginary parts of the effective action.

Introduced a new regularization method for the phase of fermion determinants.

Calculated the leading order contribution to the chiral anomaly.

Abstract

A systematic derivation is given of the worldline path integrals for the effective action of a multiplet of Dirac fermions interacting with general matrix-valued classical background scalar, pseudoscalar, and vector gauge fields. The first path integral involves worldline fermions with antiperiodic boundary conditions on the worldline loop and generates the real part of the one loop (Euclidean) effective action. The second path integral involves worldline fermions with periodic boundary conditions and generates the imaginary part of the (Euclidean) effective action, i.e. the phase of the fermion functional determinant. Here we also introduce a new regularization for the phase of functional determinants resembling a heat-kernel regularization. Compared to the known special cases, our worldline Lagrangians have a number of new interaction terms; the validity of some of these terms is checked in perturbation theory. In particular, we obtain the leading order contribution (in the heavy mass expansion) to the Wess-Zumino-Witten term, which generates the chiral anomaly.