Dark matter freeze-in via a light fermion mediator: forbidden decay and scattering

The connection between a hidden nonthermal sector and a thermal plasma can be established by a light thermal fermion mediator. When the fermion mediator is much lighter than the hidden species, kinematically forbidden decay of the mediator can be opened at finite temperatures to produce the hidden species. Unlike bosons having quartic couplings, renormalizable forbidden fermion decay generically shares the same order of couplings with the scattering. We present a dedicated investigation into the freeze-in dark matter production via a thermal fermion mediator. We demonstrate that the plasma-induced decay rate differs from that calculated via the tree-level amplitude, but the former can be obtained from the latter via constant rescaling. Furthermore, we find that the relative effect of the forbidden decay and the scattering on the dark matter relic density can be simply estimated via the thermal coupling between the plasma and the mediator. Applying to different thermal interactions, we show that the forbidden decay contribution can reach the level of $4\%-45\%$ for a thermal coupling at $0.1-1$.