Gravitational Properties of the Proca Field

We study various properties of a Proca field coupled to gravity through minimal and quadrupole interactions, described by a two-parameter family of Lagrangians. St\"uckelberg decomposition of the effective theory spells out its model-dependent ultraviolet cutoff, parametrically larger than the Proca mass. We present pp-wave solutions that the model admits, consider linear fluctuations on such backgrounds, and thereby constrain the parameter space of the theory by requiring null-energy condition and the absence of negative time delays in high-energy scattering. We briefly discuss the positivity constraints$-$derived from unitarity and analyticity of scattering amplitudes$-$that become ineffective in this regard.