Thermal Pions and Isospin Chemical Potential Effects

The density corrections, in terms of the isospin chemical potential μI , to the mass of the pions are investigated in the framework of the SU(2) low energy effective chiral invariant lagrangian. As a function of temperature and μI = 0, the mass remains quite stable, starting to grow for very high values of T , confirming previous results. However, the dependence for a non-vanishing chemical potential turns out to be much more dramatic. In particular, there are interesting corrections to the mass when both effects (temperature and chemical potential) are simultaneously present. At zero temperature the π should condensate when μI = ∓mπ. This is not longer valid anymore at finite T . The mass of the π0 acquires also a non trivial dependence on μI at finite T . Pions play a special role in the dynamics of hot hadronic matter since they are the lightest hadrons. Therefore, it is quite important to understand not only the temperature dependence of the pion’s Green functions but also their behavior as function of density. The dependence of the pion mass (and width) on temperature mπ(T ) has been studied in a variety of frameworks, such as thermal QCD-Sum Rules [1], Chiral Perturbation Theory (low temperature expansion) [2], the Linear Sigma Model [3], the Mean Field Approximation [4], the Virial Expansion [5]. In fact the properties of pion propagation at finite temperature have been calculated at two loops in the frame of chiral perturbation theory [6]. There seems to be a reasonable agreement that mπ(T ) is essentially independent of T , except possibly near the critical temperature Tc where mπ(T ) increases with T . Let us proceed in the frame of the SU(2) chiral perturbation theory. The most general chiral invariant expression for a QCD-extended lagrangian, [10, 11] under the presence of external hermitian-matrix auxiliary fields has the form LQCD(s, p, vμ, aμ) = LQCD + q̄γ(vμ + γ5aμ)q − q̄(s− iγ5p)q, (1) where vμ, aμ, s and p are vectorial, axial, scalar and pseudoscalar fields. The vector current is given by J μ = q̄γμτ q. (2) When v, a, p = 0 and s = M , being M the mass matrix, we obtain the usual QCD Lagrangian. The effective action with finite isospin chemical potential is given by LQCD = LQCD(M, 0, 0, 0) + μuJ μ = LQCD(M, 0, μuμ, 0) (3) where μ = (0, 0, μI) is the third isospin component, μ = μτ and uμ is the 4-velocity between the observer and the thermal heat bath. This is required to describe in a covariant way this system, where the Lorentz invariance is broken since the thermal heath bath represents a privileged frame of reference. Proceeding in the same way in the low-energy description, where only the pion degrees of freedom are relevant, let us consider the most general chiral invariant lagrangian. This lagrangian is ordered according to a series in powers of the external momentum. We will start with the O(p2) chiral lagrangian L2 = f 4 Tr [ (DμU) DU + U χ+ χU ]