Exploring active-sterile neutrino mixings models in MES mechanism using modular $S_3$ symmetry

We study the minimal extended seesaw mechanism with one sterile neutrino in a 3+1 framework using modular $S_3$ symmetry. The active-sterile neutrino models are classified based on the assignments of $S_3$ representations and modular weights of the left-handed lepton doublets, triplet right-handed neutrino, and sterile neutrino. No scalar flavons are considered, and the flavor symmetry is broken by the vacuum expectation value (vev) of the modulus $\tau$. For a particular set of representations of the Leptons and Higgs field, we obtain eleven (11) different models based on different modular weights of charged lepton $(k_L)$ and right-handed neutrino ($k_N$). Out of these, we consider two models, which are discriminated by carrying out the numerical analysis so that the parameter space in each model can fit the latest neutrino oscillation data at 3$\sigma$. The Planck cosmological bound on the upper limit of the sum of the active neutrino masses $\sum m_i <0.12$eV is also considered. Finally, the best-fit parameters of the neutrino observables and model predictions are evaluated using the minimum $\chi^2$ analysis.