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We study the phase transition of a real scalar phi^4 theory in the two-loopPhi-derivable approximation using the imaginary time formalism, extending ourprevious (analytical) discussion of the Hartree approximation. We combine FastFourier Transform algorithms and accelerated Matsubara sums in order to achievea high accuracy. Our results confirm and complete earlier ones obtained in thereal time formalism [1] but which were less accurate due to the integration inMinkowski space and the discretization of the spectral density function. Wealso provide a complete and explicit discussion of the renormalization of thetwo-loop Phi-derivable approximation at finite temperature, both in thesymmetric and in the broken phase, which was already used in the real-timeapproach, but never published. Our main result is that the two-loopPhi-derivable approximation suffices to cure the problem of the Hartreeapproximation regarding the order of the transition: the transition is of thesecond order type, as expected on general grounds. The corresponding criticalexponents are, however, of the mean-field type. Using a "RG-improved" versionof the approximation, motivated by our renormalization procedure, we find thatthe exponents are modified. In particular, the exponent delta, which relatesthe 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 accuratenumerical estimates [2].

Broken phase effective potential in the two-loop Phi-derivable approximation and nature of the phase transition in a scalar theory