Neutrino phenomenology, W mass anomaly & muon $(g-2)$ in minimal type-III seesaw using $T^\prime$ modular symmetry

In this study, we attempt to introduce a model to illustrate neutrino phenomenology by incorporating two right-handed fermion triplet superfields, i.e., $\Sigma_{R_j}$, in the presence of the modular symmetry $\Gamma_3^\prime \simeq A_4^\prime$, a double cover of the $A_4$ modular symmetry. The motivation in utilizing double cover is, so far only even modular forms were considered for constructing modular invariant models, but, in this case, it is possible to extend the modular invariance approach to general integral weight modular forms, i.e., the odd weight modular forms. Hence, this type of amalgamation between $T^\prime$ modular symmetry and minimally extending the seesaw can correctly explain the neutrino phenomenology. Additionally, we have made an attempt to accommodate the most recent measurement of the $W$ boson mass, published by the CDF-II collaboration and shed some light on the recent results of muon $(g-2)$. Finally, we have discussed lepton flavor violation in order to establish a constraint on the mass of right-handed fermion.