Multi Component Dark Matter in a Minimal Model

Abstract: We study a minimal model with an imposed $\mathbb{Z}_2$ symmetry incorporating two singlet fermions and a singlet scalar which communicate with the SM particles through a scalar-Higgs portal. We probe regions in the parameter space where stability of the three new particles are guaranteed kinematically, and thus introducing a multi component dark matter (DM) scenario. The regions in the parameter space with predicted total relic density in accordance with the observed DM abundance are found, and the contribution of each species to the total relic density is determined. The elastic DM-nucleon scattering cross section of the two fermions DM are loop suppressed, while that of the scalar DM starts at tree level and thus presumably being dominant. It is found that there exist a viable region in the parameter space that the scalar DM can evade the current direct detection (DD) experimental bounds while it has a minimal contribution to the observed relic density. The DD cross section of the fermion DM being loop suppressed, resides below the lower limit from the {\it neutrino floor}, possessing a large fraction of the total relic density.

• The paper demonstrates that two singlet Dirac fermions and a singlet scalar can form a stable three-component dark matter sector whose coupled freeze-out reproduces Ωh² ≃ 0.12.
• The model’s loop-suppressed fermions can provide most of the relic density while remaining below the neutrino floor, whereas scalar annihilation into Higgs pairs suppresses its abundance for mφ > mh.
• Rescaled direct-detection rates satisfy current XENON limits and leave a potentially discoverable scalar mass window near 125–400 GeV, subject to assumptions about thermal equilibrium and parameter ranges.

Overview

This paper examines a minimal extension of the Standard Model (SM) that realizes a genuine three-component dark matter (DM) scenario. The model adds two gauge-singlet Dirac fermions, $\psi_1$ and $\psi_2$, and a real singlet scalar $\phi$, all stabilized by an exact $\mathbb{Z}_2$ symmetry under which $\psi_1$ and $\phi$ are odd while $\psi_2$ and the SM fields are even. The only renormalizable portal coupling is the quadratic Higgs-portal operator $\lambda \phi^2 H^\dagger H$; the Yukawa interaction $y\,\bar\psi_1\psi_2\phi$ couples the fermions to the scalar mediator. The discrete symmetry forbids linear and cubic scalar terms as well as a bare fermion mixing term, guaranteeing a vanishing vacuum expectation value for $\phi$, no Higgs–scalar mass mixing, and a minimal free-parameter set: three masses plus $\psi_2$0 and $\psi_2$1.

The central question is whether all three new particles can be simultaneously kinematically stable — requiring $\psi_2$2, $\psi_2$3, and $\psi_2$4 — and whether the resulting coupled freeze-out dynamics can reproduce the observed relic abundance $\psi_2$5 while remaining consistent with direct detection (DD) limits.

Model structure and constraints

After electroweak symmetry breaking, the portal operator generates an effective trilinear coupling $\psi_2$6 and shifts the scalar mass. For $\psi_2$7, the model induces invisible Higgs decay with width

$\psi_2$8

constrained by Br($\psi_2$9invisible) $\phi$0 at 95% CL from CMS. Notably, the paper does not impose this bound on its final viable region: since parameter points with $\phi$1 are already excluded by XENON1T direct detection, the invisible-decay constraint is redundant in the surviving window.

Relic density: coupled Boltzmann dynamics

Because all three particles are stable, the number densities $\phi$2, $\phi$3, $\phi$4 evolve via a system of coupled Boltzmann equations including self-annihilation ($\phi$5, $\phi$6SM, $\phi$7), conversion/co-scattering ($\phi$8, $\phi$9), and co-annihilation ($\mathbb{Z}_2$0). This interplay produces freeze-out behavior qualitatively different from single-component WIMP scenarios. The analysis assumes all DM species share the thermal bath temperature.

Using micrOMEGAs 6.0 with its multi-component capability, the author scans $\mathbb{Z}_2$1 and $\mathbb{Z}_2$2. Two results stand out:

• Mass correlation: larger fermion masses predominantly require larger scalar masses to achieve the correct total abundance.
• Suppressed scalar fraction above the Higgs threshold: for $\mathbb{Z}_2$3 GeV, the ratio $\mathbb{Z}_2$4 drops below $\mathbb{Z}_2$5 in part of the parameter space. The mechanism is kinematic: once $\mathbb{Z}_2$6, the annihilation channel $\mathbb{Z}_2$7 opens, enhancing $\mathbb{Z}_2$8 and hence reducing $\mathbb{Z}_2$9 through the generic relation $\psi_1$0.

This last result is the linchpin of the phenomenology: the scalar's relic fraction can be made arbitrarily small precisely where its tree-level DD cross section would otherwise be most dangerous.

Direct detection

The elastic spin-independent DM–proton cross sections differ sharply between components:

Component	Leading order	Cross section
Fermion $\psi_1$1	One loop (triangle with $\psi_1$2, $\psi_1$3, $\psi_1$4)	Loop-suppressed by $\psi_1$5
Scalar $\psi_1$6	Tree level (Higgs exchange)	$\psi_1$7

For the fermions, the effective amplitude in the zero-momentum-transfer limit involves the loop function $\psi_1$8 with $\psi_1$9, yielding $\phi$0 with $\phi$1 encoding the low-energy scalar matrix elements.

After rescaling by relic fractions ($\phi$2) and confronting with XENON1T and XENONnT bounds, the paper finds:

• Fermion DM: the loop-suppressed cross section lies below the neutrino floor over the relevant mass range while carrying a large fraction of the total relic density. These components are therefore effectively undetectable at direct detection experiments.
• Scalar DM: although its tree-level cross section would naively exclude the entire mass range (as in the standard singlet scalar model), the small fraction $\phi$3 pushes $\phi$4 below the XENONnT limit for $\phi$5. A viable detection window remains at $\phi$6–$\phi$7 GeV, sitting above the neutrino floor and thus within reach of future experiments.

The contrast with the two-component study of Bhattacharya et al. is explicit: there, the scalar candidate is excluded over essentially its whole mass range except a resonance region, whereas here the presence of two additional stable fermions absorbing most of the relic density rescues the scalar in a finite window.

Limitations and open questions

Several assumptions and omissions qualify the results. First, the Boltzmann treatment assumes equal temperatures between the DM sectors and the bath throughout freeze-out; deviations could alter the computed fractions. Second, the scan ranges ($\phi$8 TeV, couplings below 2) do not establish completeness of the viable region beyond these bounds. Third, indirect detection signatures — potentially relevant given the sizable $\phi$9-mediated annihilation channels — are not analyzed, nor are collider constraints beyond invisible Higgs decay or perturbative-unitarity considerations on the portal coupling. Finally, the claim that the fermionic signal lies below the neutrino floor depends on the loop function evaluation and the hadronic input $\psi_2$0; improved determinations of either could shift this conclusion. Whether future multi-ton detectors can probe the $\psi_2$1–$\psi_2$2 GeV scalar window before it is closed by refined XENONnT/DARWIN sensitivity remains an open experimental question.

Conclusion

The paper demonstrates that a minimal $\psi_2$3-symmetric extension of the SM — two singlet Dirac fermions, one singlet scalar, and a quadratic Higgs portal — admits a fully stable three-component DM sector whose coupled freeze-out reproduces the observed abundance. Its principal finding is a division of labor among the components: loop-suppressed fermion DM can dominate the relic density while evading direct detection entirely (below the neutrino floor), whereas the tree-level-coupled scalar survives current XENONnT bounds only through its suppressed relic fraction, leaving a concrete, testable window at $\psi_2$4–$\psi_2$5 GeV for next-generation direct detection.