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Non linear massive gravity as a gravitational $σ$-model

We show the direct analogy between the ghost-free non-linear formulation of massive gravity and the standard $σ$-models well understood in the literature. This issue explains why there are two non-trivial family of solutions for the spherically symmetric case inside the non-linear massive gravity formulations with two free-parameters $α$ and $β$. In general, the case $β<α^2$ has a single physical vacuum state. On the other hand, the case $β=α^2$ contains a natural vacuum degeneracy. This is in perfect analogy with the $σ$-model for scalar fields where depending on the values taken by the parameters of the theory the vacuum can be single or degenerate.