New branches of massive gravity

The basic building block for Lorentz-invariant and ghost-free massive gravity is the square root of the combination $g^{-1}\eta$, where $g^{-1}$ is the inverse of the physical metric and $\eta$ is a reference metric. Since the square root of a matrix is not uniquely defined, it is possible to have physically inequivalent potentials corresponding to different branches. We show that around the Minkowski background, the only perturbatively well-defined branch is the potential proposed by de Rham, Gabadadze and Tolley. On the other hand, if Lorentz symmetry is broken spontaneously, other potentials exist with a standard perturbative expansion. We show this explicitly building new Lorentz-invariant, ghost-free massive gravity potentials for theories that in the background preserve rotational invariance but break Lorentz boosts.