The zero-momentum limit of thermal green functions

The zero momentum limit of thermal self-energies calculated in perturbation theory depends on the order in which the time and the space components of the momentum are taken to zero. We show that this is an artifact of the perturbative calculation, and in fact the limit is well-defined when higher orders of the perturbation expansion are properly taken into account. The existing calculations of thermal self-energy functions using the formalism of Thermal Field Theory yield results that are not defined if the external momentum 4-vector is zero. The classic example is the photon self-energy πμν in an electron gas. It is well known [1] that the result of the one-loop calculation of πμν(p , ~p) for a photon with external momentum p = (p, ~p) is such that lim |~ p|→0 πμν(0, ~p) 6= lim p→0 πμν(p ,~0) , (1)