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