# Generalized Proca and its Constraint Algebra

## TL;DR

This paper revisits the construction of derivative self-interactions for a massive Proca field, ensuring the theory propagates at most three degrees of freedom and exploring the structure of the constraints algebra.

## Contribution

It introduces a systematic approach to constructing ghost-free Proca interactions using degenerate Hessian conditions and analyzes the constraint algebra's role in these theories.

## Key findings

- The sixth order Lagrangian is topological and irrelevant.
- New classes of nonlinear interactions satisfying the constraints are identified.
- The constraint algebra structure is crucial for ghost-free Proca theories.

## Abstract

We reconsider the construction of general derivative self-interactions for a massive Proca field. The constructed Lagrangian is such that the vector field propagates at most three degrees of freedom, thus avoiding the ghostly nature of a fourth polarisation. The construction makes use of the well-known condition for constrained systems of having a degenerate Hessian. We briefly discuss the casuistry according to the nature of the existing constraints algebra. We also explore various classes of interesting new interactions that have been recently raised in the literature. For the sixth order Lagrangian that satisfies the constraints by itself we prove its topological character, making such a term irrelevant. There is however a window of opportunity for exploring other classes of fully-nonlinear interactions that satisfy the constraint algebra by mixing terms of various order.