Real Time Ultrasoft Longitudinal Photons Self Energy at Next to-leading Order in Hot Scalar QED

We determine a compact analytic expression for the complete next-to-leading contribution to the retarded Longitudinal Photons self-energy with ultrasoft momentum in the framework of hard-thermal-loop (HTL) -summed perturbation of massless Scalar QED at high temperature. The calculation is done using real-time formalism. The real part and the opposite of the imaginary part of the retarded Longitudinal Photons self-energy are related to the next-to-leading order contributions of energy and damping rate respectively.