Renormalization of massive vector field theory coupled to scalar in curved spacetime

We consider the renormalization of a massive vector field interacting with a charged scalar field in curved spacetime. Starting with the theory minimally coupled to external gravity and using the formulations with and without St & uuml;ckelberg fields, we show that the longitudinal mode of the vector field is completely decoupled and the remaining theory of the 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 nonminimal terms linear in the Ricci tensor in the vector sector. Nevertheless, a more detailed analysis shows that these nonminimal 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 the St & uuml;ckelberg procedure and the heat-kernel technique. The theory without nonminimal 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.