  The rates for generating a matter-antimatter asymmetry in extensions of the
Standard Model (SM) containing right-handed neutrinos are the most interesting
and least trivial coefficients in the rate equations for baryogenesis through
thermal leptogenesis. We obtain a relation of these rates to finite-temperature
real-time correlation functions, similar to the Kubo formulas for transport
coefficients. Then we consider the case of hierarchical masses for the sterile
neutrinos. At leading order in their Yukawa couplings we find a simple master
formula which relates the rates to a single finite temperature three-point
spectral function. It is valid to all orders in $g$,where $g$ denotes a SM
gauge or quark Yukawa coupling. We use it to compute the rate for generating a
matter-antimatter asymmetry at next-to-leading order in g in the
non-relativistic regime. The corrections are of order $g ^ 2$, and they amount
to 4% or less.
