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 determine the baryon spectrum of 1 + 1 + 1-flavor QCD in the presence of strong background magnetic fields using lattice simulations at physical quark masses for the first time. Our results show a splitting within multiplets according to the electric charge of the baryons and reveal, in particular, a reduction of the nucleon masses for strong magnetic fields. This first-principles input is used to define constituent quark masses and is employed to set the free parameters of the Polyakov loop-extended Nambu-Jona-Lasinio (PNJL) model in a magnetic field-dependent manner. The so constructed model is shown to exhibit inverse magnetic catalysis at high temperatures and a reduction of the transition temperature as the magnetic field grows -in line with non-perturbative lattice results. This is contrary to the naive variant of this model, which gives incorrect results for this fundamental phase diagram. Our findings demonstrate that the magnetic field dependence of the PNJL model can be reconciled with the lattice findings in a systematic way, employing solely zero-temperature first-principles input.

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.

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.

The SUð2Þ L Â SUð2Þ R chiral quark model consisting of the ð;Þ meson multiplet and the constituent quarks propagating on the homogeneous background of a temporal gauge field is solved at finite temperature and quark baryon chemical potential q using an expansion in the number of flavors N f , both in the chiral limit and for the physical value of the pion mass. Keeping the fermion propagator at its tree level, several approximations to the pion propagator are investigated. These approximations correspond to different partial resummations of the perturbative series. Comparing their solution with a diagrammatically formulated resummation relying on a strict large-N f expansion of the perturbative series, one concludes that only when the local part of the approximated pion propagator resums infinitely many orders in 1=N f of fermionic contributions a sufficiently rapid crossover transition at q ¼ 0 is achieved allowing for the existence of a tricritical point or a critical end point in the q À T phase diagram. The renormalization and the possibility of determining the counterterms in the resummation provided by a strict large-N f expansion are investigated.