Mixed Freeze-In and Freeze-Out Histories and Dark-Sector Decays in a $\mathbb{Z}_4$ Two-Scalar Model

Abstract

We present a systematic non-equilibrium analysis of a renormalisable $\mathbb{Z}_4$ Higgs-portal dark sector with a complex scalar $S_A$ and a real scalar $S_B$, including conversion, semi-annihilation, and, when kinematically allowed, the dark-sector decay $S_B\to S_A S_A$. We impose theoretical consistency, Higgs invisible-decay limits, relic-fraction-rescaled LZ bounds, and an a posteriori thermalisation check of the WIMP/FIMP assignments. Allowing one or both dark states to remain out of equilibrium qualitatively enlarges the set of viable cosmological histories. In the stable regime, FIMP--FIMP and mixed WIMP--FIMP configurations can share the observed relic density between the two components, while the WIMP-like state remains subject to the usual Higgs-portal correlation between freeze-out and spin-independent scattering and the FIMP-like state can provide a substantial abundance with a negligible direct-detection rate. For $M_{S_B}>2M_{S_A}$, the decay of the unstable $S_B$ opens three distinct abundance-transfer mechanisms: SuperWIMP production, injection-assisted freeze-out, and sequential freeze-in. Because the decay proceeds entirely within the dark sector, the usual electromagnetic and hadronic BBN energy-injection bounds do not apply; we instead track the parent energy fraction and use an equivalent thermal-WDM mass to diagnose the warmness of the decay-produced daughter population. The minimal $\mathbb{Z}_4$ Higgs portal thus supports thermal, mixed, and fully non-thermal relic histories within the same renormalisable framework, with dark-sector decay linking relic-density production to small-scale structure.