On the renormalization of massive vector field theory coupled to scalar in curved space-time

We consider the renormalization of massive vector field interacting with charged scalar field in curved spacetime. Starting with the theory minimally coupled to external gravity and using the formulations with and without St\"uckelberg fields, we show that the longitudinal mode of vector field is completely decoupled and the remaining theory of transverse vector field is renormalizable by power counting. The formal arguments based on the covariance and power counting indicate that multiplicative renormalizability of the interacting theory may require introducing two non-minimal terms linear in Ricci tensor in the vector sector. Nevertheless, a more detailed analysis shows that these non-minimal terms violate the decoupling of the longitudinal mode and are prohibited. As a verification of general arguments, we derive the one-loop divergences in the minimal massive scalar QED, using St\"uckelberg procedure and the heat-kernel technique. The theory without non-minimal terms proves one-loop renormalizable and admits the renormalization group equations for all the running parameters in the scalar and vector sectors. One-loop beta functions do not depend on the gauge fixing and can be used to derive the effective potential.