Neutrino energy loss rate in a stellar plasma

We review the purely leptonic neutrino emission processes, contributing to the energy loss rate of the stellar plasma. We perform a complete analysis up to the first order in the electromagnetic coupling constant. In particular the radiative electromagnetic corrections, at order $\alpha$, to the process $e^+ e^- -> \nu \bar{\nu}$ at finite density and temperature have been computed. This process gives one of the main contributions to the cooling of stellar interior in the late stages of star evolution. As a result of the analysis we find that the corrections affect the energy loss rate, computed at tree level, by a factor $(-4 \div 1) %$ in the temperature and density region where the pair annihilation is the most efficient cooling mechanism.