Thermal Meson properties within Chiral Perturbation Theory

We report on our recent work about the description of a meson gas below the chiral phase transition within the framework of Chiral Perturbation Theory. As an alternative to the standard treatment, we present a calculation of the quark condensate which combines the virial expansion and the meson-meson scattering data. We have also calculated the full one-loop elastic pion scattering amplitude at finite temperature and we have unitarized the amplitude using the Inverse Amplitude Method in order to reproduce the temperature effects on the mass and width of the ρ and σ resonances. Our results show a clear increase of the thermal ρ width, as expected from previous analysis. The results for the σ are consistent with Chiral Symmetry Restoration. We comment on the relevance of our results within the context of Relativistic Heavy Ion Collisions. INTRODUCTION AND MOTIVATION The recent development of the Relativistic Heavy Ion Collision (RHIC) program is one of the main motivations to study hadronic matter under extreme conditions of temperature T and density. Here we will consider the meson gas formed when the plasma created after one such RHIC has expanded and hadronized, being in a state where the chiral symmetry is restored. There are strong indications that one could observe the medium effects on such a system. For instance, the dilepton spectrum shows an anomalous behaviour for invariant masses near the ρ mass [1]. The flatness of the spectrum is consistent with a modification of the mass and width of the ρ’s which have time to decay inside the plasma, so that their spectral function acquires thermal corrections due to the interaction with the hot and dense hadron gas [2, 3, 4, 5, 6, 7, 8, 9]. In such a system, external momenta and temperature are small compared to the chiral symmetry breaking scale Λχ ≃ 1 GeV. The relevant degrees of freedom are then the lightest mesons and the interactions among them are best described by a low-energy QCD effective theory based on chiral symmetry. The most general framework comprising the QCD Chiral Symmetry Breaking pattern SUL(N f )×SUR(N f ) → SUV (N f ) is Chiral Perturbation Theory (ChPT) [10, 11, 12, 13] where any observable can be calculated as an expansion in p/Λχ , p denoting a meson mass, momenta or the temperature. Thus, ChPT provides model-independent predictions, just by fixing a few low-energy constants (LEC). This program has included the calculation of thermodynamic observables such as the free energy density and the quark condensate, as we will briefly review below. Throughout this paper, we will neglect finite baryon density effects, as ideally in the central rapidity region formed after a RHIC. Since ChPT is intended to provide a systematic low-energy and low-T perturbative expansion, we do not expect it to reproduce resonances. This is strongly related to the fact that ChPT only satisfies unitarity in a perturbative fashion. Over the last few years, there has been a lot of work devoted to enlarge the ChPT applicability range by using unitarization methods, i.e, imposing exact unitarity requirements, like the Inverse Amplitude Method (IAM) [14] or approaches based on Lippmann-Schwinger or BetheSalpeter equations [15]. These methods provide a good agreement with the experimental phase shifts and they are able to generate resonances, like the ρ and the σ for the SU(2) chiral symmetry. In addition, they can be extended to include coupled channels, describing successfully all the meson-meson data and resonances in the SU(3) case, up to 1.2 GeV [16, 17]. What we will show below is that only requiring chiral symmetry and unitarity one can also describe successfully the thermal behaviour of the ρ and σ resonances. For that purpose, one needs first to calculate the thermal pion scattering in ChPT, which has been done in [18]. We shall discuss the main features of such thermal amplitude below. Then, by using the IAM extended to finite T , one can construct a nonperturbative thermal unitarized amplitude which, in particular, describes the behaviour of the thermal ρ and σ [19]. We will present the results for the thermal mass and width of the ρ and σ in that approach, as well as for the T -dependence of the effective gρππ vertex. The main implications of our results in the context of RHIC and chiral symmetry restoration will be also discussed below. THE MESON GAS AND CHIRAL SYMMETRY AT FINITE T WITHIN CHPT For the reasons explained above, it is important to provide an accurate description of the low-T meson gas in thermal equilibrium. For instance, the signature of chiral symmetry restoration at Tc ≃ 150-200 MeV should be observed in the thermal evolution of the order parameter, the quark condensate 〈q̄q〉(T) from below the transition point. As we have just discussed, ChPT provides a model independent description of the thermodynamic observables, based only on chiral symmetry. The only extra ingredient is the temperature, which is treated as an O(p) parameter. The first calculations of the pion gas within ChPT go back to [20], where 〈q̄q〉(T) and the thermal dependence of fπ(T ) were calculated up to O(T 2) (one loop) in the chiral limit. That result already showed a behaviour compatible with chiral symmetry restoration. In [21] a thorough analysis up to O(T 6) was performed, including the free energy, the quark condensate and an estimate of the thermal effects of free kaons and etas. The O(T 4) corrections to fπ(T ) have been analyzed in [22] where it has been shown that beyond O(T 2) one has to consider separately the space and time components of the axial current, so that there are two independent f s,t π , which, in addition, can be complex. In fact, their imaginary part is proportional to the in-medium pion damping rate while their real parts are related to the deviations of the pion velocity from the speed of light. Other analysis of the thermal pion dispersion law can be found in [23, 24]. The analysis of typical nonequilibrium effects such as explosive pion production after a RHIC can be also studied within the ChPT context [25]. It should be pointed out that many of these properties have also been investigated using specific models for lowenergy QCD. The most popular is the O(4) model, which reproduces a critical behaviour already in the mean field approach. Apart from introducing the σ explicitly, conventional perturbation theory in the O(4) model breaks down, which has been dealt with at finite T using different nonperturbative approaches [26, 27]. When calculating thermodynamic quantities such as the pressure or the quark condensate from ChPT, the usual approach is to use the Feynman rules of Thermal Field Theory [28] to the order considered. This is particularly cumbersome in the three flavor case if one wishes to include the full dependence on temperature, quark masses and SU(3) interactions. An alternative [21, 29, 30] is to perform a virial expansion of the pressure as [31] βP = ∑ i Bi(T )ξi +∑ i (