FIMPs in a two-component dark matter model with $Z_2 \times Z_4$ symmetry

We investigate the FIMP-FIMP regime in a two-component dark matter model with a $Z_2\times Z_4$ symmetry, where a singlet scalar $S$ and a Majorana fermion $χ$ serve as the dark matter candidates. A singlet scalar $S_0$ with vacuum expectation value $v_0$ generates the fermion mass through the relation $m_χ=y_{sf}v_0$. We show that the tiny Yukawa coupling $y_{sf}$ needed to reproduce the observed relic abundance naturally leads to a large symmetry-breaking scale $v_0$, which induces an ultra-feeble portal coupling $λ_{ds}$ responsible for the production of $S$. We find that $λ_{ds}$ can reach values of $10^{-25}\lesssimλ_{ds}\lesssim10^{-13}$, while gravitational freeze-in provides an irreducible contribution at extremely small couplings. Our results demonstrate that the relic abundance constraint, combined with symmetry breaking and freeze-in dynamics, naturally drives the portal interaction responsible for scalar dark matter production into the ultra-feeble regime.