One-loop effective potential for a fixed charged self-interacting bosonic model at finite temperature with its related multiplicative anomaly

The one-loop partition function for a charged self-interacting Bose gas at finite temperature in D-dimensional spacetime is evaluated within a path integral approach making use of zeta-function regularization. For D even, a new additional vacuum term—overlooked in all previous treatments and coming from the multiplicative anomaly related to functional determinants—is found and its dependence on the mass and chemical potential is obtained. The presence of the new term is shown to be crucial for having the factorization invariance of the regularized partition function. In the noninteracting case, the relativistic Bose-Einstein condensation is reexamined. By means of a suitable charge renormalization, for D=4 the symmetry breaking phase is shown to be unaffected by the new term, which, however, actually gives rise to a nonvanishing new contribution in the unbroken phase.