# Renormalization of vector fields with mass-like coupling in curved spacetime

## TL;DR

This paper calculates the divergent part of the effective action for a Proca vector field with a mass-like coupling in curved spacetime, providing explicit local expressions and verifying consistency with Weyl invariance.

## Contribution

It introduces a covariant symbol method to compute divergences for non-minimally coupled vector fields in curved spacetime, extending previous results.

## Key findings

- Derived explicit local expressions for divergences
- Reproduced known special case results
- Confirmed internal consistency with Weyl invariance

## Abstract

Using the method of covariant symbols we compute the divergent part of the effective action of the Proca field with non-minimal mass term. Specifically a quantum Abelian vector field with a non-derivative coupling to an external tensor field in curved spacetime in four dimensions is considered. Relatively explicit expressions are obtained which are manifestly local but non polynomial in the external fields. Our result is shown to reproduce existing ones in all particular cases considered. Internal consistency with Weyl invariance is also verified.