Thermodynamics and phase transition of theO(N)model from the two-loopΦ-derivable approximation

Abstract

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.

Symmetry-improved CJT effective action

“…However, the external propagator for the Higgs particle exhibits erroneous thresholds, since it has no absorptive part below a certain value of the centre-ofmass energy resulting from massive Goldstone bosons in the loops. In the HF approximation, this approach yields a first-order phase transition, which turns into a second-order one once the sunset graphs are included [18]. However, the occurrence of massive Goldstone bosons within loops persists, in spite of going beyond the HF approximation.…”

“…In particular, for the case of global O(N ) symmetries that we will be studying here, a naive truncation of the CJT effective action violates the Goldstone theorem [7,8] by higher-order terms, giving rise to a massive Goldstone boson in the Spontaneous Symmetry Breaking (SSB) phase of the theory [9][10][11][12]. Thus far, several studies have been presented in the literature, attempting to provide a satisfactory solution to this problem [11][12][13][14][15][16][17][18][19][20].…”

On the symmetry improved CJT formalism in the O(4) linear sigma model

“…Any attempts to go beyond the usual Hartree approximation are worthy to do in order to identify their effects in this novel CJT formalism, e.g. the sunset-type diagrams [25][39] or more higher loops. Such an extension is straightforward but technically complicated.…”

“…[7]. As shown in [25][37], in the Hartree-Fock approximation, the CT scheme removes the divergences and the continuum limit can be considered without any obstructions in the chiral limit. However, for higher order truncations, a more general renormalization procedure of the CJT functional needs to be employed in order to remove the obstructions in the chiral limit appeared in Ref.…”

“…Within the CJT formalism, there are many attempts made to restore the Goldstone theorem in chiral limit, nevertheless, the existing approaches still could not provide a satisfactory solution to this long-standing problem [11,[22][23][24][25][26][27]. Very recently, Pilaftsis and Teresi have developed a novel symmetry improved CJT formalism by consistently encoding global symmetries in loop-wise expansions or truncations of the CJT effective action [28].…”