Asymmetric dark matter from leptogenesis in type-III seesaw framework with modular $S_4$ symmetry

We present a unified framework for neutrino masses, baryogenesis, and dark matter based on a modular $S_4$ symmetry combined with a type-III seesaw mechanism. All Yukawa couplings, CP phases, and flavor textures originate from a single complex modulus $τ$, whose vacuum expectation value controls both visible and dark sector dynamics. The same modular parameter fixes the neutrino mass matrix, determines the CP asymmetries driving resonant leptogenesis, and correlates the resulting baryon and dark matter abundances. A detailed numerical analysis shows that the model reproduces all neutrino oscillation data within the $3σ$ NuFIT5.2 (2024) ranges for normal ordering, predicting $δ_{\rm CP} \simeq \pm (150^\circ-180^\circ)$, $\sum m_ν\simeq(0.06-0.08)\mathrm{eV}$, and an effective Majorana mass $m_{ββ} \simeq (8 - 18)\times 10^{-3}\mathrm{eV}$, testable in next-generation neutrinoless double-beta decay experiments. The same modular Yukawas yield resonantly enhanced CP asymmetries $|ε_{L,χ}| \sim 10^{-9}-10^{-6}$ at $M_Σ\sim 10^{7}\mathrm{GeV}$, successfully generating the observed baryon asymmetry $η_B\simeq6\times10^{-10}$ and dark relic density $Ω_χh^2\simeq0.12$ without additional free parameters. The predicted correlation $Ω_χ/Ω_B\simeq5.4$ fixes the dark matter mass to $m_χ\simeq0.1-2~\mathrm{GeV}$, consistent with all current constraints. This framework therefore realizes a fully predictive baryon$-$dark matter co-genesis, where the geometry of the modular symmetry links the origin of flavor, CP violation, and the cosmic matter asymmetry.

💡 Research Summary

The paper proposes a unified framework that simultaneously addresses three of the most pressing puzzles in particle physics and cosmology: the origin of neutrino masses, the baryon asymmetry of the Universe (BAU), and the nature of dark matter (DM). The authors employ a modular S₄ flavor symmetry at level 3, wherein all Yukawa couplings, CP‑violating phases, and flavor textures are generated as modular forms of a single complex modulus τ. The vacuum expectation value (VEV) of τ, taken near the self‑dual point ω = e^{2πi/3}, fixes the numerical values of the modular forms Y^{(k)}(τ) and therefore determines the entire lepton sector.

Neutrino masses arise via a type‑III seesaw mechanism: two SU(2)L fermionic triplets Σ₁ and Σ₂, forming an S₄ doublet, acquire a common mass scale M ≈ 10⁷ GeV with a tiny splitting ΔM induced by a modular‑symmetry breaking parameter M_ε. The Dirac mass matrix M_D and the heavy‑triplet mass matrix M_Σ are both functions of the same modular forms, leading to the light‑neutrino mass matrix M_ν = −M_D M_Σ⁻¹ M_Dᵀ. By scanning the complex τ, the model reproduces the full set of neutrino oscillation parameters within the 3σ ranges of NuFIT 5.2 (2024), predicting a normal ordering, a Dirac CP phase δ_CP ≈ ±(150°–180°), a summed mass Σ m_ν ≈ 0.06–0.08 eV, and an effective Majorana mass m{ββ} ≈ (8–18) meV, all within reach of upcoming neutrinoless double‑beta decay experiments.

The same triplets mediate a co‑genesis of the visible and dark sectors. Their decays proceed through two channels: Σ → L H_u (producing a lepton asymmetry) and Σ → χ ϕ (producing a dark‑sector asymmetry). The dark sector contains a Z₂‑odd fermion χ (the DM candidate) and a scalar ϕ; χ is stabilized by the discrete symmetry. Because the Yukawa matrices Y_ν (visible) and Y_χ (dark) are both modular functions of τ, the CP‑violating phases that drive the asymmetries are identical in origin. In the resonant regime (ΔM ≈ Γ), self‑energy loop diagrams dominate the CP asymmetries, yielding ε_L and ε_χ in the range 10⁻⁹–10⁻⁶. The total decay width includes both channels, and the branching ratios are controlled by the modular‑induced effective masses \tilde m_1 and \tilde m_{DM}.

Coupled Boltzmann equations are solved for the number densities of Σ, the lepton asymmetry ΔL, and the dark asymmetry Δχ. Sphaleron processes convert ΔL into a baryon asymmetry Y_B = (28/79) ΔL. The final relic densities satisfy Ω_χ/Ω_B = (m_χ/m_p) |Δχ/ΔL| ≈ 5.4, reproducing the observed ratio of dark‑to‑baryonic matter. This fixes the DM mass to m_χ ≈ 0.1–2 GeV, a range compatible with current direct‑detection limits, CMB constraints, and big‑bang nucleosynthesis. Importantly, the model achieves this without introducing any extra free parameters beyond the overall scale of the modular forms; all phenomenology is dictated by the single complex modulus τ.

The authors provide extensive numerical scans. They demonstrate that for τ values yielding realistic neutrino mixing, the resonant leptogenesis condition is naturally satisfied, and the required CP asymmetries emerge automatically. The dark‑Yukawa coupling y_DM can be O(1) without spoiling the relic density, confirming that the portal interaction is fully controlled by the modular structure. Benchmark points are presented, showing simultaneous agreement with neutrino data, the BAU (η_B ≈ 6 × 10⁻¹⁰), and the DM relic density (Ω_χ h² ≈ 0.12).

Phenomenologically, the model offers several testable predictions. The predicted δ_CP and m_{ββ} lie within the sensitivity of DUNE, Hyper‑K, and next‑generation 0νββ experiments. The fermionic triplets Σ couple to electroweak gauge bosons, implying possible signatures at the LHC or future 100 TeV colliders, such as multilepton final states or displaced vertices if the mass splitting is small. The light GeV‑scale DM candidate could be probed by low‑threshold direct‑detection experiments (e.g., CRESST‑III, SuperCDMS) and by indirect searches for annihilation or decay products. Gravitational‑wave signatures from a possible first‑order phase transition associated with τ stabilization are also mentioned as a speculative avenue.

In conclusion, the paper demonstrates that a modular S₄ symmetry with a single complex modulus can simultaneously generate realistic neutrino masses, resonantly enhanced leptogenesis, and asymmetric dark matter, thereby providing a highly predictive and economical solution to three major open questions in fundamental physics. The framework is robust, minimal in parameters, and offers clear experimental targets for the next decade.