  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.
