# The Maxwell-Proca theory: definition and construction

## TL;DR

This paper systematically constructs the most general ghost-free Maxwell-Proca theory with multiple interacting fields, including non-linear self-interactions, ensuring correct degrees of freedom and clarifying previous misconceptions about multi-Proca interactions.

## Contribution

It provides a formal definition and explicit construction of a general, ghost-free Maxwell-Proca theory with multiple interacting fields, including derivative self-interactions and necessary differential relations.

## Key findings

- Demonstrates ghost-freedom via constraint algebra
- Identifies differential relations for multi-Proca interactions
- Clarifies previous multi-Proca ghost issues

## Abstract

We present a systematic construction of the most general first order Lagrangian describing an arbitrary number of interacting Maxwell and Proca fields on Minkowski spacetime. To this aim, we first formalize the notion of a Proca field, in analogy to the well known Maxwell field. Our definition allows for a non-linear realization of the Proca mass, in the form of derivative self-interactions. Consequently, we consider so-called generalized Proca/vector Galileons. We explicitly demonstrate the ghost-freedom of this complete Maxwell-Proca theory by obtaining its constraint algebra. We find that, when multiple Proca fields are present, their interactions must fulfill non-trivial differential relations in order to ensure the propagation of the correct number of degrees of freedom. These relations had so far been overlooked, which means previous multi-Proca proposals generically contain ghosts. This is a companion paper to arXiv:1905.06968 [hep-th]. It puts on a solid footing the theory there introduced.