Thermal self-energies at zero momentum

In general 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 nonanalyticity of the one-loop self-energy disappears when it is calculated with improved vertices and/or improved propagators that incorporate the imaginary part of the self-energy