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We extend our previous investigation of the two-loop $\Phi$-derivableapproximation to finite chemical potential $\mu$ and discuss Bose-Einsteincondensation (BEC) in the case of a charged scalar field with $O(2)$ symmetry.We show that the approximation is renormalizable by means of counterterms whichare independent of both the temperature and the chemical potential. We pointout the presence of an additional skew contribution to the propagator ascompared to the $\mu=0$ case, which comes with its own gap equation (except atHartree level). We solve this equation together with the field equation, andthe usual longitudinal and transversal gap equations to find that thetransition is second order, in agreement with recent lattice results to whichwe compare. We also discuss a general criterion an approximation should obeyfor the so-called Silver Blaze property to hold, and we show that any$\Phi$-derivable approximation at finite temperature and density obeys thiscriterion if one chooses a UV regularization that does not cut off theMatsubara sums.

Bose-Einstein condensation and Silver Blaze property from the two-loop $\Phi$-derivable approximation