Resonant $s$-channel dark matter annihilation at NLO

Studies of dark matter annihilation through an $s$-channel resonance are often based on recipes such as a narrow width approximation or real intermediate state subtraction. We review a recipe-free formalism that can be implemented at the NLO level in the full theory, and ensures the cancellation of mass singularities. Its basic ingredients can be formulated in the relativistic regime, but we show that the procedure simplifies if we go to the non-relativistic one and assume the presence of kinetic equilibrium. The latter case is illustrated for scalar singlet dark matter with $m_{\rm S}^{ } \simeq 60$ GeV, freezing out at $T \simeq (1-3)$ GeV, re-confirming the viability of this scenario with couplings tiny enough to evade experimental constraints.