Modular $A_4$ symmetry in 3+1 active-sterile neutrino masses and mixings

Motivated by the significance of modular symmetry in generating neutrino masses and flavor mixings, we apply the modular $A_4$ symmetry in a 3+1 scheme of active-sterile neutrino mixings. Neutrino oscillation observables in the 3$\sigma$ range are successfully reproduced through the vacuum expectation value of the modulus $\tau$. We also study phenomenologies related to the effective neutrino masses $m_{\beta }$ in tritium beta decay and $m_{\beta \beta }$ in neutrinoless double beta decay. Mixings between active neutrinos and eV scale sterile neutrino are analyzed in detail. The model also predicts the Dirac CP-violating phase ${\delta }_{CP}$ and Majonara phases $\alpha$ and $\beta$. The best-fit values of the neutrino mixing angles and ratios of two mass-squared differences are determined using minimum ${\chi }^2$ analysis. The best-fit values of the neutrino oscillation observables are predicted as ${\sin }^2{\theta }_{23}=0.573,$ ${\sin }^2{\theta }_{12}=0.300,$ ${\sin }^2{\theta }_{13}=0.022$ and $r=0.172$ for NH while ${\sin }^2{\theta }_{23}=0.550,$ ${\sin }^2{\theta }_{12}=0.303,$ ${\sin }^2{\theta }_{13}=0.022$ and $r=0.171$ for IH. We also observe that the predictions of effective neutrino mass parameters in the 3+1 scheme are significantly different from the three neutrino paradigm.