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Unitarity Check in Gravitational Higgs Mechanism

The effective field theory of massive gravity had long been formulated in a generally covariant way arXiv:hep-th/0210184. Using this formalism, it has been found recently that there exists a class of massive nonlinear theories that are free of the Boulware-Deser ghosts, at least in the decoupling limit arXiv:1007.0443. In this work we study other recently proposed models that go under the name of "gravitational Higgs theories" arXiv:1002.3877, arXiv:1008.5132. We show that these models, although seemingly different from the effective field theories of massive gravity, are in fact equivalent to them. Furthermore, based on the results obtained in the effective field theory approach, we conclude that the gravitational Higgs theories need the same adjustment of the Lagrangian to avoid the ghosts. We also show the equivalence between the noncovariant mode decomposition used in the Higgs theories, and the covariant Stückelbergization adopted in the effective field theories, thus, proving that the presence or absence of the ghost is independent of the parametrization used in either theory.