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

We consider the FIMPs scenario in a two-component dark matter model with $Z_2 \times Z_4$ symmetry, where a singlet scalar $S$ and a Majorana fermion $χ$ 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. The model admits six free parameters in the decoupling limit: three masses and three dimensionless parameters. Depending on the mass hierarchies between dark matter particles with half of the new Higgs mass, the DM relic density will be determined by different channels, where $χ$ and $S$ production can be generated individually. We numerically study the relic density as a function of the model’s free parameters and determine the regions consistent with the dark matter constraint for four possible cases. Our results show that this scenario is viable over a wide range of couplings and dark matter masses, where the coupling $λ_{ds}$ can be as tiny as $\mathcal{O}(10^{-20})$ level. We stress that even for such tiny couplings, the new Higgs can still play a dominant role in determining dark matter production.

💡 Research Summary

In this work the authors explore a two‑component dark‑matter (DM) framework endowed with a discrete Z₂ × Z₄ symmetry. The particle content beyond the Standard Model (SM) consists of a real singlet scalar S, a Majorana fermion χ, and an additional singlet scalar S₀. The Z₂ charge stabilises S, while the Z₄ charge stabilises χ; S₀ carries a Z₄ charge that allows it to acquire a non‑zero vacuum expectation value (vev) v₀. Through the Yukawa interaction y_sf S₀ χ χ, χ obtains a mass m_χ = y_sf v₀ after spontaneous symmetry breaking. The scalar sector contains the SM Higgs doublet H and S₀, which mix to produce two physical Higgs bosons: the observed 125 GeV state h₁ and a new light scalar h₂. The mixing angle θ is taken to be vanishing (the decoupling limit sin θ → 0) so that h₂ couples only feebly to SM particles and dominates the production of the dark sector.

The model is characterised by six free parameters in this limit: the three masses (m_χ, m_S, m₂) and three dimensionless couplings (λ_ds, λ_dh, y_sf). λ_ds controls the S–S₀ quartic interaction, λ_dh the S–SM Higgs portal, and y_sf the χ–S₀ Yukawa coupling. The authors impose theoretical consistency conditions: perturbativity (|2λ_dh|, |2λ_ds|, |y_sf| < 4π), perturbative unitarity (eigenvalues of the scalar‑scalar scattering matrix < ½), and vacuum stability (copositivity constraints on the quartic coupling matrix). All viable points satisfy these bounds.

Dark‑matter production proceeds via the freeze‑in mechanism: initially the number densities of S and χ are negligible, and they never reach thermal equilibrium because of their feeble interactions. Their abundances are governed by Boltzmann equations that include (i) decays of the new Higgs h₂ (h₂ → SS and h₂ → χχ), (ii) SM‑SM annihilations into S pairs (X X → SS), and (iii) Higgs‑mediated 2→2 processes (h₂ h₂ → SS, h₂ h₂ → χχ). The authors solve these equations numerically with MicrOMEGAs, scanning over the six parameters.

Four qualitatively distinct mass hierarchies are identified:

1. m_χ < m₂/2 < m_S – χ is mainly produced by the decay h₂ → χχ; S production is dominated by h₂ → SS and SM annihilations.
2. m_S < m₂/2 < m_χ – S is chiefly generated by h₂ → SS, while χ arises from the scattering h₂ h₂ → χχ.
3. m₂/2 < m_χ, m_S – Both components are produced primarily through 2→2 processes involving h₂.
4. Special cases where m_S ≈ m₁/2 allow the SM Higgs to decay into S pairs, but these are suppressed in the decoupling limit.

Key findings include:

• The relic density of χ scales as Ω_χ h² ≈ 0.3 (m_χ/0.1 GeV)(1 eV/m₂)(y_sf/10⁻¹⁰)² for temperatures above the h₂ mass. Consequently, Yukawa couplings y_sf in the range 10⁻⁸–10⁻¹² can reproduce the observed DM abundance for a light h₂ (∼ 1–2 eV) and a broad range of χ masses.
• The scalar relic density depends on λ_ds, λ_dh, m_S and the ratio v₀ = m_χ/y_sf. Even extremely tiny portal couplings λ_ds ∼ 10⁻²⁰ are sufficient because the light h₂ provides an efficient production channel. For λ_ds ≈ 10⁻⁸ the decay h₂ → SS dominates; as λ_ds is reduced the contribution from SM annihilations becomes comparable, but the total Ω_S h² remains within the required range.
• The dependence on the new Higgs mass is pronounced: for T > m₂ the production rates rise sharply, while for T < m₂ they are Boltzmann‑suppressed, leading to characteristic “kinks” in the Ω versus mass plots at m_χ ≈ m₂/2 and m_S ≈ m₂/2.
• The combined relic density Ω_χ h² + Ω_S h² can match the Planck value Ω_DM h² ≈ 0.12 over a wide swath of parameter space, demonstrating that a two‑component FIMP scenario is viable without fine‑tuning.

The authors emphasize that the new Higgs h₂, despite its feeble mixing with the SM, plays a dominant role in dark‑matter production. This opens an experimental avenue: indirect constraints from Higgs invisible width measurements, precision Higgs coupling fits, and cosmological observables (CMB, large‑scale structure) could probe the allowed region. Direct detection is essentially impossible due to the ultra‑small portal couplings, but future searches for ultra‑light scalars or deviations in Higgs phenomenology could provide hints.

In summary, the paper presents a consistent, theoretically sound two‑component FIMP model with Z₂ × Z₄ symmetry, shows that both a scalar and a fermion can simultaneously account for the observed dark matter, and highlights that even couplings as tiny as 10⁻²⁰ can be phenomenologically relevant when a light mediator (the new Higgs) is present. This work broadens the landscape of viable dark‑matter models beyond the traditional WIMP paradigm and underscores the importance of multi‑component and feebly interacting scenarios.