  In this paper, we will apply the Goldstone equivalence gauge to calculate the
$1 \leftrightarrow 2$ processes of a sterile neutrino in the thermal plasma
below the standard model (SM) critical temperature $T_c \approx 160 \text{
GeV}$. The sterile neutrino's mass is around the electroweak scale $50 \text{
GeV} \leq m_N \leq 200 \text{ GeV}$, and the acquired thermal averaged
effective width $\bar{\Gamma}_{\text{tot}}$ is continuous around the
cross-over. We will also apply our results to perform a preliminary calculation
of the leptogenesis.
