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We study relativistic Bose-Einstein condensation at finite density andtemperature using the linear sigma model in the one-particle-irreducible1/N-expansion. We derive the effective potential to next-to-leading (NLO) orderand show that it can be renormalized in a temperature-independent manner. As aparticular application, we study the thermodynamics of the pion gas in thechiral limit as well as with explicit symmetry breaking. At nonzero temperaturewe solve the NLO gap equation and show that the results describe thechiral-symmetry-restoring second-order phase transition in agreement withgeneral universality arguments. However, due to nontrivial regularizationissues, we are not able to extend the NLO analysis to nonzero chemicalpotential.

Linear sigma model at finite density in the 1/N expansion to next-to-leading order