Two-loop neutrino mass model with modular $S_4$ symmetry

We propose a two-loop radiative neutrino mass model based on the modular $\Gamma_4 \simeq S_4$ flavour symmetry supplemented by a discrete $Z_3$ symmetry. After spontaneous modular symmetry breaking, a remnant $Z_2$ symmetry guarantees both the radiative origin of active neutrino masses and stabilizes dark matter candidates. The model successfully reproduces charged lepton masses and neutrino oscillation data for normal ordering. It also predicts observable rates for charged lepton flavour violation (LFV). Due to the singlet-doublet mixing the model provides a viable scalar dark matter candidate. A fermionic dark matter candidate, strongly linked to LFV, is also present. We identify parameter regions consistent with relic density, LFV constraints, and direct detection limits, providing testable benchmark configurations.