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.

No abstract

Non-perturbative renormalisation of a general class of scalar field theories is performed at the Hartree level truncation of the 2PI effective action in the broken symmetry regime. Renormalised equations are explicitly constructed for the one-and two-point functions. The non-perturbative counterterms are deduced from the conditions for the cancellation of the overall and the subdivergences in the complete Hartree-Dyson-Schwinger equations, with a transparent method. The procedure proposed in the present paper is shown to be equivalent to the iterative renormalisation method of Blaizot et al.[1]. MotivationOne of the most popular approximation techniques in many-body quantum theory is the Hartree approximation. In quantum field theory it corresponds to the momentum independent two-loop truncation of the two-particle irreducible effective action. It is used extensively both in equilibrium [2,3,4] and out-of-equilibrium [5,6,7,8] non-perturbative investigations of phase transition phenomena. Its non-perturbative renormalisability was demonstrated as particular case of the general proof of renormalisability of the physical quantities computed in various 2PI approximations [9,1,10]. These proofs are rather involved especially in the broken symmetry phase. For this reason in many practical applications the renormalised equations are not constructed explicitly. For instance, investigations of the finite temperature phase transitions in strongly interacting matter frequently either omit zero temperature quantum corrections in the 2PI approximate equations of the relevant 1-and 2-point functions [3,4,11] or take into account vacuum fluctuations by applying some cut-off [12].The exact generating 2PI-functional Γ[Φ, G] fulfils generalised Ward-Takahashi identities reflecting global internal symmetries of the models. As a consequence the 1PI effective potential Γ[Φ, G(Φ)]

We study the effective potential of a real scalar ' 4 theory as a function of the temperature T within the simplest È-derivable approximation, namely, the Hartree approximation. We apply renormalization at a ''high'' temperature T ? where the theory is required to be in its symmetric phase and study how the effective potential evolves as the temperature is lowered down to T ¼ 0. In particular, we prove analytically that no second order phase transition can occur in this particular approximation of the theory, in agreement with earlier studies based on the numerical evaluation or the high temperature expansion of the effective potential. This work is also an opportunity to illustrate certain issues on the renormalization of È-derivable approximations at finite temperature and nonvanishing field expectation value and to introduce new computational techniques which might also prove useful when dealing with higher order approximations.

By assuming certain analytic properties of the propagator, it is shown that universal features of the spectral function including threshold enhancement arise if a pole describing a particle at high temperature approaches in the complex energy plane the threshold position of its two-body decay with the variation of T. The case is considered, when one can disregard any other decay processes. The quality of the proposed description is demonstrated by comparing it with the detailed large N solution of the linear sigma model around the pole-threshold coincidence.Comment: 4 pages, 2 figure

We discuss various aspects of the O(N )-model in the vacuum and at finite temperature within the Φ-derivable expansion scheme to order λ 2 . In continuation of an earlier work, we look for a physical parametrization in the N = 4 case that allows us to accommodate the lightest mesons. Using zero-momentum curvature masses to approximate the physical masses, we find that, in the parameter range where a relatively large sigma mass is obtained, the scale of the Landau pole is lower compared to that obtained in the two-loop truncation. This jeopardizes the insensitivity of the observables to the ultraviolet regulator and could hinder the predictivity of the model. Both in the N = 1 and N = 4 cases, we also find that, when approaching the chiral limit, the (iterative) solution to the Φ-derivable equations is lost in an interval around the would-be transition temperature. In particular, it is not possible to conclude at this order of truncation on the order of the transition in the chiral limit. Because the same issue could be present in other approaches, we investigate it thoroughly by considering a localized version of the Φ-derivable equations, whose solution displays the same qualitative features, but allows for a more analytical understanding of the problem. In particular, our analysis reveals the existence of unphysical branches of solutions which can coalesce with the physical one at some temperatures, with the effect of opening up a gap in the admissible values for the condensate. Depending on its rate of growth with the temperature, this gap can eventually engulf the physical solution.