On the Kalb-Ramond field with non-minimal coupling to gravity

We consider a massive Kalb-Ramond field with a general non-minimal coupling to gravity. We first study the theory in flat space-time, taking into account the non-linearities. We show that the coupling with the Ricci scalar gives rise to the strong coupling of the two transverse pseudo-vector degrees of freedom, which are absent in the massless theory. We then show that if the theory is instead coupled to the Ricci tensor or the Riemann tensor, the two tensor modes become strongly coupled in addition to the transverse pseudo-vector modes. We then extend our analysis to homogeneous and isotropic space-time, with vanishing background value of the Kalb-Ramond field. We show that in this case, the couplings with the Ricci and Riemann tensor give rise to the runaway instability. Finally, we discuss the inclusion of the disformal coupling as a possible resolution to this unnatural behavior.