Massive vector field on curved background: Nonminimal coupling, quantization, and divergences

We study the effective action for the massive vector field theory nonminimally coupled to external gravitational field. Such a theory is an interesting model both from the theoretical side and also due to the various phenomenological applications to cosmology and astrophysics. The present work pretends to initiate a systematic study of its properties at the quantum level, by exploring free massive vector coupled to an external symmetric second-rank tensor. Stueckelberg scalar field is used to restore the gauge invariance. After that, by using a special gauge fixing and nonlocal in external fields change of variables, we diagonalize the bilinear form of the action and develop a consistent procedure to study the effective action. As a result we derive a complete nonlinear structure of divergences of the effective action and discuss its properties.