Generalization of the Proca Action

TL;DR

The paper constructs a generalized Proca theory for a massive vector field with derivative self-interactions that preserves three propagating degrees of freedom by enforcing Hessian degeneracy. It identifies a special Galileon-like case where the longitudinal mode reproduces scalar Galileon interactions, while the vector sector provides additional intrinsic interactions, increasing the theory's freedom beyond the scalar case. The analysis extends to curved spacetime, yielding a Horndeski-style Proca action with second-order equations of motion. Together, these results connect vector Galileons to Horndeski constructions and expand the landscape of ghost-free, derivative-interacting massive vector theories with potential phenomenological applications.

Abstract

We consider the Lagrangian of a vector field with derivative self-interactions with a priori arbitrary coefficients. Starting with a flat space-time we show that for a special choice of the coefficients of the self-interactions the ghost-like pathologies disappear. For this we use the degeneracy condition of the Hessian. This constitutes the Galileon-type generalization of the Proca action with only three propagating physical degrees of freedom. The longitudinal mode of the vector field is associated to the usual Galileon interactions for a specific choice of the overall functions. In difference to a scalar Galileon theory, the generalized Proca field has more free parameters and purely intrinsic vector interactions. We then extend this analysis to a curved background. The resulting theory is the Horndeski Proca action with second order equations of motion on curved space-times.