Systematics of High Temperature Perturbation Theory: The Two-Loop Electron Self-Energy in QED

In order to investigate the systematics of the loop expansion in high temperature gauge theories beyond the leading order hard thermal loop (HTL) approximation, we calculate the two-loop electron proper self-energy in high temperature QED. The two-loop bubble diagram contains a linear infrared divergence. Even if regulated with a non-zero photon mass M of order of the Debye mass, this infrared sensitivity implies that the two-loop self-energy contributes terms to the fermion dispersion relation that are comparable to or even larger than the next-to-leading-order (NLO) contributions at one-loop. Additional evidence for the necessity of a systematic restructuring of the loop expansion comes from the explicit gauge parameter dependence of the fermion damping rate at both one and two-loops. The leading terms in the high temperature expansion of the two-loop self-energy for all topologies arise from an explicit hard-soft factorization pattern, in which one of the loop integrals is hard, nested inside a second loop integral which is soft. There are no hard-hard contributions to the two-loop Sigma at leading order at high T. Provided the same factorization pattern holds for arbitrary ell loops, the NLO high temperature contributions to the electron self-energy come from ell-1 hard loops factorized with one soft loop integral. This hard-soft pattern is both a necessary condition for the resummation over ell to coincide with the one-loop self-energy calculated with HTL dressed propagators and vertices, and to yield the complete NLO correction to the self-energy at scales ~eT, which is both infrared finite and gauge invariant. We employ spectral representations and the Gaudin method for evaluating finite temperature Matsubara sums, which facilitates the analysis of multi-loop diagrams at high T.