A Thermal Field Theory with Non-uniform Chemical Potential

We investigate thermal one-loop effective potentials in multi-flavor models with chemical potentials. We study four-dimensional models in which each flavor have different global U(1) charges. Accordingly they have different chemical potentials. We call these "non-uniform chemical potentials," which are organized into a diagonal matrix \mu. The mass matrix at a vacuum does not commute with \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 \mu. Our result is a generalization of the thermal potentials in previous studies for models with single and multi-flavors with (uniform) chemical potentials and reproduces all the known results correctly.