ar X iv : h ep - p h / 04 11 29 3 v 1 22 N ov 2 00 4 Chiral Lagrangians at finite temperature and the Polyakov Loop

Heat kernel expansions at finite temperature of massless QCD and chiral quark models generate effective actions relevant for both low and high temperature QCD. The key relevance of the Polyakov Loop to maintain the large and non-perturbative gauge invariance at finite temperature is stressed. In the imaginary time formulation of quantum field theory [1], finite temperature is introduced by imposing periodic or anti-periodic boundary conditions for bosons and fermions respectively. This approach necessarily breaks the Lorentz invariance of the starting Lagrangian since the heat bath is a static infinitely heavy system which is assumed to be at rest and communicates energy at no expense with the fundamental degrees of freedom. This fact generates an unpleasant plethora of possible Lorentz contributions to Feynman diagrams which fortunately are ultraviolet finite due to the presence of suppressing kinetic Boltzmann factors. As a gratifying consequence, the finite temperature renormalizability of a theory relies indeed on its renormalizability at zero temperature and provides any fundamental theory with definite predictive power, if the parameters entering the Lagrangian are assumed to be temperature independent. This standard procedure might be called minimal thermal coupling of the heat bath since, at least in perturbation theory, finite temperature (and hence Lorentz breaking) effects are indeed quantum effects. For an effective field theory of composite particles the situation may be not that simple since the corresponding Low Energy Constants (LEC's) encode the microscopic information of the underlying substructure. Actually, the constituents are also heated up and it is clear that LEC's inherit a finite temperature dependence. Moreover, genuinely Lorentz breaking and temperature dependent terms might be further added to the effective Lagrangian. The previous discussion is actually relevant to QCD at finite temperature [2, 3]. At low energies and temperatures, one uses Chiral Perturbation Theory (ChPT) [4] and assumes that in the effective theory the LEC's are temperature independent; the T dependence is generated through thermal pion loops [5]. This assumption relies strongly on the existence of a mass gap in the physical hadronic spectrum due to the presence of would-be massless pseudo-scalar Goldstone bosons, so one expects the leading temperature effects, well below the phase transition, to be O(e −M π /T), whereas the next corrections