The chiral phase transition of the strongly interacting matter is investigated at nonzero temperature and baryon chemical potential (µB) within an extended (2 + 1) flavor Polyakov constituent quark-meson model that incorporates the effect of the vector and axial vector mesons. The effect of the fermionic vacuum and thermal fluctuations computed from the grand potential of the model is taken into account in the curvature masses of the scalar and pseudoscalar mesons. The parameters of the model are determined by comparing masses and tree-level decay widths with experimental values in a χ 2 -minimization procedure that selects between various possible assignments of scalar nonet states to physical particles. We examine the restoration of the chiral symmetry by monitoring the temperature evolution of condensates and the chiral partners' masses and of the mixing angles for the pseudoscalar η −η ′ and the corresponding scalar complex. We calculate the pressure and various thermodynamical observables derived from it and compare them to the continuum extrapolated lattice results of the Wuppertal-Budapest collaboration. We study the T − µB phase diagram of the model and find that a critical endpoint exists for parameters of the model, which give acceptable values of χ 2 . A. Lagrangian of the PQM with (axial) vector mesonsAccording to the considerations above, the Lagrangian we shall use has the following form:

We study the phase transition of a real scalar ϕ 4 theory in the two-loop Φ-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 Φ-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 Φ-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 δ, which relates the field expectation valueφ 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].

We discuss the thermodynamics of the OðNÞ model across the corresponding phase transition using the two-loop È-derivable approximation of the effective potential and compare our results to those obtained in the literature within the Hartree-Fock approximation. In particular, we find that in the chiral limit the transition is of the second order, whereas it was found to be of the first order in the Hartree-Fock case. These features are manifest at the level of the thermodynamical observables. We also compute the thermal sigma and pion masses from the curvature of the effective potential. In the chiral limit, this guarantees that Goldstone's theorem is obeyed in the broken phase. A realistic parametrization of the model in the N ¼ 4 case, based on the vacuum values of the curvature masses, shows that a sigma mass of around 450 MeV can be obtained. The equations are renormalized after extending our previous results for the N ¼ 1 case by means of the general procedure described in Ref. [8]. When restricted to the Hartree-Fock approximation, our approach reveals that certain problems raised in the literature concerning the renormalization are completely lifted. Finally, we introduce a new type of È-derivable approximation in which the gap equation is not solved at the same level of accuracy as the accuracy at which the potential is computed. We discuss the consistency and applicability of these types of ''hybrid'' approximations and illustrate them in the two-loop case by showing that the corresponding effective potential is renormalizable and that the transition remains of the second order.

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