One-loop Effective Potential for a Fixed Charged Self-interacting Bosonic Model at Finite Temperature with its Related Multiplicative Anomaly

TL;DR

This paper calculates the one-loop effective potential for a charged self-interacting bosonic model at finite temperature, revealing a new vacuum term from the multiplicative anomaly that affects the partition function's invariance.

Contribution

It introduces a previously overlooked vacuum term arising from the multiplicative anomaly in the finite-temperature partition function of a charged bosonic system.

Findings

New vacuum term depends on mass and chemical potential.

The term is crucial for factorization invariance of the partition function.

In the non-interacting case, the Bose-Einstein condensation is revisited with the new term.

Abstract

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 non interacting case, the relativistic Bose-Einstein condensation is revisited. 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, gives actually rise to a non vanishing new contribution in the unbroken phase.