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.