Pion dispersion relation at finite density and temperature

We study the behavior of the pion dispersion relation in a pion medium at finite density and temperature. We introduce a pion chemical potential to describe the finite pion number density and argue that such description is valid during the hadronic phase of a relativistic heavy-ion collision between chemical and thermal freeze-out. We make use of an effective Lagrangian that explicitly respects chiral symmetry through the enforcement of the chiral Ward identities. The pion dispersion relation is computed through the computation of the pion self-energy in a non-perturbative fashion by giving an approximate solution to the Schwinger-Dyson equation for this self-energy. The dispersion relation is described in terms of a density and temperature dependent mass and an index of refraction which is also temperature, density as well as momentum dependent. The index of refraction is larger than unity for all values of the momentum for finite $\mu$ and $T$. We conclude by exploring some of the possible consequences for the propagation of pions through the boundary between the medium and vacuum.