A two-component dark matter model with $Z_2 \times Z_4$ symmetry

We consider a two-component dark matter model with $Z_2 \times Z_4$ symmetry, where a singlet scalar $S$ and a Majorana fermion $\chi$ are introduced as dark matter candidates. We also introduce another singlet scalar $S_0$ with a non-zero vacuum expectation value to the SM so that the fermion dark matter can obtain mass after spontaneous symmetry breaking. We have a new Higgs boson in the model and in the case of the decoupling limit, the fermion dark matter production is only determined by $S$ and the new Higgs boson. The mass hierarchy of these new particles can make a difference in the reaction rate of dark matter annihilation processes, contributing to different viable parameter spaces for different mass orderings. We randomly scanned the parameter space with six various cases under relic density constraint and found that when $\chi$ is the lightest among the dark sector, $\chi$ production is generated via the so-called forbidden channels. Moreover, we consider the combined limits arising from Higgs invisible decay, dark matter relic density and direct detection constraints. Within the chosen parameter space, direct detection results put the most stringent constraint, and we have a more flexible value for the scalar dark matter mass when the mass of $\chi$ is not smaller than the new Higgs boson mass.