Scattering versus Forbidden Decay in Dark Matter Freeze-in

It is generically believed that the two-body scattering is suppressed by higher-order weak couplings with respect to the two-body decay. We show that this does not always hold when a heavy particle is produced by forbidden decay in a thermal plasma, where the scattering shares the same order of couplings with the decay. We find that there is a simple and close relation between the forbidden decay and the same-order scattering. To illustrate this point, we consider freeze-in production of heavy dark matter via a light scalar mediator. We point out that, when the Boltzmann (quantum) statistics is used, the forbidden decay can contribute to the dark matter relic density at 5$\%$-24$\%$ (10$\%$-$39\%$) with a weak thermal coupling, while the contribution from the scattering channel can be several orders of magnitude larger than from the forbidden decay if the thermal coupling is much smaller. Such a relative effect between the scattering and the forbidden decay could also exist in other plasma-induced processes, such as the purely thermal generation of the right-handed neutrino dark matter, or of the lepton asymmetry in leptogenesis.