Modular $S_4$ Scotogenic Model with Flavored Resonant Leptogenesis

Abstract

We construct a radiative neutrino mass model that combines the scotogenic mechanism with modular $S_4$ flavour symmetry. The entire lepton flavour structure is governed by holomorphic modular forms of a single complex modulus $τ$, eliminating the need for flavon fields. Beyond the Standard Model, the particle content consists of two right-handed Majorana fermions assigned to the $S_4$ doublet representation and an inert scalar doublet odd under a $\mathbb{Z}_2$ parity. Neutrino masses emerge at one loop through the scotogenic mechanism, and the lightest $\mathbb{Z}_2$-odd state serves as a dark matter candidate. A comprehensive scan of the parameter space demonstrates consistency with all five neutrino oscillation observables at the $3σ$ level. Having exactly two right-handed neutrinos forces the light neutrino mass matrix to rank two, leaving one neutrino massless and selecting normal ordering as the only viable option. The framework predicts a total neutrino mass in the narrow window $Σm_ν\simeq 0.059$--$0.06\,\mathrm{eV}$, well within current cosmological bounds, and an effective Majorana mass $m_{ββ} \simeq (1.3$--$3.5)\times 10^{-3}\,\mathrm{eV}$ relevant for neutrinoless double beta decay searches. The modular structure of the right-handed Majorana mass matrix intrinsically produces a quasi-degenerate heavy neutrino spectrum, enabling flavoured resonant leptogenesis at $M_1 \sim 10^5\,\mathrm{GeV}$ without any fine-tuning. Integration of the full three-flavour Boltzmann equations confirms that the observed baryon asymmetry is reproduced, establishing that neutrino masses, leptonic mixing, and the baryon asymmetry of the Universe all find a common explanation within this framework.