Neutrino phenomenology and Dark matter in a left-right asymmetric model with non-holomorphic modular $A_{4}$ group

Abstract: We present a model constructed within a non-supersymmetric framework capable of explaining both current neutrino oscillation data and the observed dark matter relic abundance. In this study, the Yukawa couplings are expressed as polyharmonic Maa\ss{} forms, and the non-supersymmetric left-right symmetric model is realized through the $\Gamma_{3}$ modular group, with neutrino masses generated via the Type II seesaw dominance mechanism. The analysis focuses on determining the neutrino oscillation parameters, the effective Majorana mass arising from the standard contribution, and the dark matter relic density. Our results indicate that the model strongly favours the normal mass hierarchy over the inverted one and prefers the lower octant for the mixing angle $\theta_{23}$. Furthermore, the effective Majorana mass is predicted to lie in the range $10^{-3}\,\text{eV}$ to $0.1\,\text{eV}$. In addition, the lightest sterile neutrino present in the model is considered a viable dark matter candidate. A sterile neutrino mass in the range $10~\text{keV}$ to $30~\text{keV}$ is found to yield consistent results for both the relic density and active-sterile mixing angles.