Real time ultrasoft fermion self energy at next to-leading order in hot QED

Subsequent studies of the behavior of the gluon and quark damping rates in the imaginary-time formalism have indicated that there are difficulties in the infrared sector [1, 2, 3, 4, 5, 6, 7]. To look further into the infrared behavior, we propose to calculate the next-to-leading order dispersion relations for slow-moving Fermions at high-temperature quantum electrodynamics (QED) in real-time formalism. We determine a compact analytic expression for the complete next-to-leading contribution to the retarded fermion self-energy with ultrasoft momentum in the framework of hard-thermal-loop (HTL)-summed perturbation of massless QED at high temperature. The calculation is done using real-time formalism. The next-to-leading order fermion self-energy is written in terms of three and four HTL-dressed vertex functions. The real part and the opposite of the imaginary part of the retarded fermion self-energy are related to the next-to-leading order contributions of energy and damping rate respectively.