  We investigate hard-thermal-loop (HTL) corrections to the CP-asymmetries in
neutrino and, at high temperature, Higgs boson decays in leptogenesis. We pay
special attention to the two leptonic quasiparticles that arise at non-zero
temperature and find that there are four contributions to the CP-asymmetries,
which correspond to the four combinations of the two leptonic quasiparticles in
the loop and in the final states. In two additional cases, we approximate the
full HTL-lepton propagator with a zero-temperature propagator that employs the
thermal lepton mass m_l(T), or the asymptotic thermal lepton mass sqrt{2}
m_l(T). We find that the CP-asymmetries in the one-mode approaches differ by up
to one order of magnitude from the full two-mode treatment in the interesting
temperature regime T \sim M_1. The asymmetry in Higgs boson decays turns out to
be two orders of magnitude larger than the asymmetry in neutrino decays in the
zero-temperature treatment. The effect of HTL corrections on the final lepton
asymmetry are investigated in paper II of this series.
