Unveiling the inconsistency of the Proca theory with non-minimal coupling to gravity

We study the degrees of freedom of the Proca theory, non-minimally coupled to gravity. In the Minkowski background, this theory propagates five degrees of freedom -- a massive longitudinal mode, two massive vector ones, and two massless tensor modes. At first sight, the non-linear coupling between the metric perturbations and the vector field indicates that both longitudinal and tensor modes become strongly coupled, at the same scale. This would imply that no matter how small the photon mass is if non-minimal coupling is taken into account, gravitational waves would necessarily be strongly coupled. We show that the way out of this inconsistency is through the introduction of the disformal coupling to the metric perturbations that resemble the vector-type disformal transformations. This way, the unphysical coupling between the two types of modes can be avoided, rendering the model consistent. As a result, we show that only the longitudinal modes enter a strong coupling regime, while both tensor and transverse modes remain weakly coupled at all scales up to the Planck length. Finally, using the same form of the disformal transformation, we introduce a disformal frame in which the recently reported runaway modes are absent.