On strong coupling scales in (minimal) massive gravity

Ghost-free massive gravity models generically have a strong coupling scale of ${\mathrm{\Lambda }}_3=(M_{\mathrm{P}\mathrm{l}}{m}^2{)}^{1\mathrm{/}3}$. However, for one of these models - `minimal massive gravity’ - it is not clear what this scale is in the subset of solutions with vanishing vector modes, since there are then no interactions at the scale ${\mathrm{\Lambda }}_3$. We show that there are always scalar-tensor interactions at a scale arbitrarily close to ${\mathrm{\Lambda }}_3$ around the Minkowski vacuum solution. This explicitly confirms and completes previous research establishing that ${\mathrm{\Lambda }}_3$ effectively is the maximal strong coupling scale for all ghost-free (dRGT) massive gravity models (on Minkowski). In the process, we also revisit and clarify the construction of generic Lorentz-invariant massive gravity models, explicitly constructing an improved non-redundant expansion for these models.