Thermal field theory with nonuniform chemical potential

We investigate thermal one-loop effective potentials in multiflavor models with chemical potentials. We study four-dimensional models in which each flavor has different global $U(1)$ charges. Accordingly they have different chemical potentials. We call these “nonuniform chemical potentials,” which are organized into a diagonal matrix $\widehat{\mu }$. The mass matrix at a vacuum does not commute with $\widehat{\mu }$. We find that the effective potential is divided into three parts. The first part is the Coleman-Weinberg potential. The UV divergence resides only in this part. The second is the correction to the Coleman-Weinberg potential that is independent of temperature, and the third depends on both temperature and $\widehat{\mu }$. Our result is a generalization of the thermal potentials in previous studies for models with single and multiflavors with (uniform) chemical potentials, and it reproduces all the known results correctly.