Finite-temperature symmetry breaking as Bose-Einstein condensation

The effects of a net background charge on ideal and interacting relativistic Bose gases are investigated. For a non-Abelian symmetry only chemical potentials that correspond to mutually commuting charges may be introduced. The symmetry-breaking pattern is obtained by computing a μ-dependent functional integral. We find that μ always raises the critical temperature and that below that temperature the existence of a ground-state expectation value for some scalar field produces Bose-Einstein condensation of a finite fraction of the net charge so as to keep the total charge fixed. (In the special, but familiar, case of total charge neutrality, the condensate contains equal numbers of particles and antiparticles). There are four classes of results depending on whether volume or entropy is kept fixed and on whether the quadrative mass term m2 is positive or negative