Vector instabilities and self-acceleration in the decoupling limit of massive gravity

We investigate vector contributions to the Lagrangian of $\Lambda_3-$massive gravity in the decoupling limit, the less explored sector of this theory. The main purpose is to understand the stability of maximally symmetric %self-accelerating vacuum solutions. Around self-accelerating configurations, vector degrees of freedom become strongly coupled since their kinetic terms vanish, so their dynamics is controlled by higher order interactions. Even in the decoupling limit, the vector Lagrangian contains an infinite number of terms. We develop a systematic method to covariantly determine the vector Lagrangian at each order in perturbations, fully manifesting the symmetries of the system. We show that, around self-accelerating solutions, the structure of higher order $p$-form Galileons arise, avoiding the emergence of a sixth BD ghost mode. However, a careful analysis shows that there are directions along which the Hamiltonian is unbounded from below. This instability can be interpreted as one of the available fifth physical modes behaving as a ghost. Therefore, we conclude that self-accelerating configurations, in the decoupling limit of $\Lambda_3$-massive gravity, are generically unstable.