Quantum effective action for degenerate vector field theories

TL;DR

This paper develops a method to compute one-loop divergences in curved spacetime for degenerate vector field operators, revealing an effective metric structure and enabling the use of standard heat-kernel techniques.

Contribution

It introduces a novel approach to handle degenerate vector operators by reducing them to scalar operators, uncovering an effective metric in the process.

Findings

Derived the divergent part of the one-loop effective action for degenerate vector operators.

Identified an effective metric structure in the longitudinal sector of the operators.

Expressed divergences using invariants from the effective metric.

Abstract

We calculate the divergent part of the one-loop effective action in curved spacetime for a particular class of second-order vector field operators with a degenerate principal part. The principal symbol of these operators has the structure of a longitudinal projector. In this case, standard heat-kernel techniques are not directly applicable. We present a method which reduces the problem to a nondegenerate scalar operator for which standard heat-kernel techniques are available. Interestingly, this method leads to the identification of an effective metric structure in the longitudinal sector. The one-loop divergences are compactly expressed in terms of invariants constructed from this metric.