Pion propagation in the linear sigma model at finite temperature

We construct effective one-loop vertices and propagators in the linear sigma model at finite temperature, satisfying the chiral Ward identities and thus respecting chiral symmetry, treating the pion momentum, pion mass and temperature as small compared to the sigma mass. We use these objects to compute the two-loop pion self-energy. We find that the perturbative behavior of physical quantities, such as the temperature dependence of the pion mass, is well defined in this kinematical regime in terms of the parameter m_pi^2/4pi^2f_pi^2 and show that an expansion in terms of this reproduces the dispersion curve obtained by means of chiral perturbation theory at leading order. The temperature dependence of the pion mass is such that the first and second order corrections in the above parameter have the same sign. We also study pion damping both in the elastic and inelastic channels to this order and compute the mean free path and mean collision time for a pion traveling in the medium before forming a sigma resonance and find a very good agreement with the result from chiral perturbation theory when using a value for the sigma mass of 600 MeV.