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