  We present a calculation of the damping of an ultra-energetic (UHE) cosmic
neutrino travelling through the thermal gas of relic neutrinos, using the
formalism of finite-temperature field theory. From the self-energy diagram due
to Z exchange, we obtain the annihilation cross section for an UHE neutrino
interacting with an antineutrino from the background. This method allows us to
derive the full expressions for the UHE neutrino transmission probability,
taking into account the momentum of relic neutrinos. We compare our results
with the approximations in use in the literature. We discuss the effect of
thermal motion on the shape of the absorption dips for different UHE neutrino
fluxes as well as in the context of relic neutrino clustering. We find that for
ratios of the neutrino mass to the relic background temperature $10^2$ or
smaller, the thermal broadening of the absorption lines could significantly
affect the determination of the neutrino mass and of the characteristics of the
population of UHE sources.
