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

Abstract

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 $3σ$ NuFIT~5.2 (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.

Figures (12)

• Figure 1: One–loop self–energy diagrams contributing to the CP asymmetries in the visible and dark sectors. The interference between tree–level and loop processes with intermediate triplet exchange induces unequal decay rates into $\ell_{\alpha} H$ and $\chi \phi$ versus their CP–conjugate channels.
• Figure 2: Feynman diagrams contributing to the $2 \!\to\! 2$ scattering processes entering the Boltzmann equations (\ref{['eq:Boltz_L']}) and (\ref{['eq:Boltz_DM']}). The top row shows the $\Delta L = 2$ processes responsible for lepton-number-violating washout, while the bottom row displays the $\Delta L = 0$ transfer processes that mediate the exchange of the asymmetry between the visible and dark sectors.
• Figure 3: Allowed region of the complex modulus $\tau$ in the modular $S_4$ framework. Upper: absolute values of modular Yukawa forms $|Y_i|$ versus $\operatorname{Re}\tau$ (left) and $\operatorname{Im}\tau$ (right). Lower: $\chi^2$ distribution in the $(\operatorname{Re}\tau, \operatorname{Im}\tau)$ plane. All displayed points lie within the NuFIT 5.2 $3\sigma$ ranges.
• Figure 4: Correlations between the total neutrino mass $\sum m_\nu$ and the mixing angles $\sin^2\theta_{12}$ (upper left), $\sin^2\theta_{23}$ (upper right), and $\sin^2\theta_{13}$ (lower panel). Dashed lines denote the $3\sigma$ ranges from NuFIT 5.2, and the shaded band shows the cosmological bound $\sum m_\nu < 0.12~\mathrm{eV}$.
• Figure 5: Left: correlation between $\delta_{\rm CP}$ and $\sin^2\theta_{23}$, showing near-maximal CP violation. Right: predicted Majorana phase structure $(\alpha_{21},\alpha_{31})$ exhibiting discrete clustering near $(0,\pi)$ and $(\pm\pi/2)$, a distinctive imprint of the modular symmetry.