Perturbation theory in the polaron model at finite temperature

A perturbation theory scheme is constructed for the partition function (and free energy) in the polaron model. The correction of arbitrary order is expressed in terms of multiple integrals which depend parametrically on the temperature. The first two orders are analyzed in detail. In the higher orders, the four-point correlation function, for which an explicit expression is obtained plays a central role. The properties of this correlation function are studied. The analogies with the perturbation theory scheme for the $S$ matrix in quantum field theory are discussed.