From Dirac Neutrino Mass to Dark Radiation: An $A_4$-Symmetric FIMP Dark Matter Framework

We propose a unified $A_4\times Z_2\times U(1{)}_L$ framework for generating small Dirac neutrino masses and address dark matter through feeble interactions in an extended particle physics model. The model extends the Standard Model with a pair of heavy vector-like neutral $A_4$ fermions $N(N_{L,R})$, two additional flavon fields $\eta$ and $\phi$, among which $\eta$ being a $Z_2$-odd inert scalar and a dark-sector vector-like fermion field $\chi$. We numerically explore the Dirac-sector Yukawa couplings and identify the parameter region that is consistent with the {\tt NuFit 6.1} neutrino oscillation data at the $3\sigma$ level. The dark matter candidate $\chi$, stabilized by the $Z_2$ symmetry, is produced non-thermally through the decay $N_L\to \chi +\eta$. By solving the Boltzmann equation, we find that the observed relic abundance, ${\mathrm{\Omega }}_{\mathrm{D}\mathrm{M}}{h}^2\simeq 0.12$, can be reproduced for freeze-in couplings of order $\lambda \sim {10}^{-5.5}$–${10}^{-7.5}$ over the considered dark matter mass range. For the chosen parameter space, we also verify the interaction rate remains below the Hubble expansion rate, confirming $\chi$ never reaches thermal equilibrium. Thus, our results demonstrate that the model provides a consistent realization of Dirac neutrino masses and FIMP dark matter within a common symmetry-based setup. The presence of light Dirac RH neutrinos can contribute to the dark radiation and hence the effective degrees of freedom via thermal and non-thermal processes. We further estimate the contribution of the light right-handed neutrinos to $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}$, finding a maximum characteristic value of $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}\simeq 0.14$ in the benchmark parameter region.