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.
