  We study the phase transition of a real scalar phi^4 theory in the two-loop
Phi-derivable approximation using the imaginary time formalism, extending our
previous (analytical) discussion of the Hartree approximation. We combine Fast
Fourier Transform algorithms and accelerated Matsubara sums in order to achieve
a high accuracy. Our results confirm and complete earlier ones obtained in the
real time formalism [1] but which were less accurate due to the integration in
Minkowski space and the discretization of the spectral density function. We
also provide a complete and explicit discussion of the renormalization of the
two-loop Phi-derivable approximation at finite temperature, both in the
symmetric and in the broken phase, which was already used in the real-time
approach, but never published. Our main result is that the two-loop
Phi-derivable approximation suffices to cure the problem of the Hartree
approximation regarding the order of the transition: the transition is of the
second order type, as expected on general grounds. The corresponding critical
exponents are, however, of the mean-field type. Using a "RG-improved" version
of the approximation, motivated by our renormalization procedure, we find that
the exponents are modified. In particular, the exponent delta, which relates
the field expectation value phi to an external field h, changes from 3 to 5,
getting then closer to its expected value 4.789, obtained from accurate
numerical estimates [2].
