# Generalising the matter coupling in massive gravity: a search for new interactions

## TL;DR

This paper explores extending the matter coupling in massive gravity theories, specifically dRGT, by constructing an arbitrary metric from existing degrees of freedom, aiming to avoid introducing unwanted sixth modes.

## Contribution

It introduces a new approach to matter coupling in dRGT massive gravity, identifying conditions to prevent sixth mode contamination and proposing a unique composite coupling extension.

## Key findings

- Identified conditions for safe matter coupling in massive gravity.
- Proposed a unique composite coupling extension up to four-point interactions.
- Provided a new perspective on modifying dRGT without invalidating constraints.

## Abstract

Massive gravity theory introduced by de Rham, Gabadadze, Tolley (dRGT) is restricted by several uniqueness theorems that protect the form of the potential and kinetic terms, as well as the matter coupling. These restrictions arise from the requirement that the degrees of freedom match the expectation from Poincar\'e representations of a spin--2 field. Any modification beyond the dRGT form is known to invalidate a constraint that the theory enjoys and revive a dangerous sixth mode. One loophole is to exploit the effective nature of the theory by pushing the sixth mode beyond the strong coupling scale without completely removing it. In this paper, we search for modifications to dRGT action by coupling the matter sector to an arbitrary metric constructed out of the already existing degrees of freedom in the dRGT action. We formulate the conditions that such an extension should satisfy in order to prevent the sixth mode from contaminating the effective theory. Our approach provides a new perspective for the "composite coupling" which emerges as the unique extension up to four-point interactions.