  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.
