Ghosts and matter couplings in massive gravity, bigravity and multigravity

Recently, several works have investigated the coupling to matter in ghost-free massive bigravity and multigravity and a new effective coupling to matter has been proposed. In this paper we clarify some confusion on the existence and the implications of a ghost above the strong coupling scale. We confirm that the standard constraint which is otherwise typically present in this type of theories disappears on generic backgrounds as soon as this new coupling is considered. This implies the reemergence of the Boulware-Deser ghost. Nevertheless the absence of ghost in the decoupling limit implies that the cutoff scale (if identified with the scale at which the ghost enters) is higher than the strong coupling scale. Therefore there is a valid interesting region of applicability for these couplings at scales below the cutoff.