Infrared behavior of the dispersion relations in high-temperature scalar   QED
  We investigate the infrared properties of the next-to-leading-order dispersion relations in scalar quantum electrodynamics at high temperature in the context of hard-thermal-loop perturbation theory. Specifically, we determine the damping rate and the energy for scalars with ultrasoft momenta. We show by explicit calculations that an early external-momentum expansion, before the Matsubara sum is performed, gives exactly the same result as a late one. The damping rate is obtained up to fourth order included in the ultrasoft momentum and the energy up to second order. The damping rate is found sensitive in the infrared whereas the energy not. 
A. Abada and K. Bouakaz