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We study the effective action for the massive vector field theorynon-minimally coupled to external gravitational field. Such a theory is aninteresting model both from the theoretical side and also due to the variousphenomenological applications to cosmology and astrophysics. The present workpretends to initiate a systematic study of its properties at the quantum level,by exploring free massive vector coupled to an external symmetric second-ranktensor. Stueckelberg scalar field is used to restore the gauge invariance.After that, by using a special gauge fixing and non-local in external fieldschange of variables, we diagonalize the bilinear form of the action and developa consistent procedure to study the effective action. As a result we derive acomplete non-linear structure of divergences of the effective action anddiscuss its properties.

Massive vector field on curved background: non-minimal coupling, quantization and divergences