Zee-Babu model in a non-holomorphic modular <i>A</i> 4 symmetry and modular stabilization

Abstract We study a Zee-Babu neutrino model in a non-holomorphic modular A4 symmetry, and we construct a model so that there are minimum free parameters (two complex parameters for neutrino sector). We find only the normal hierarchy is allowed. Moreover, the allowed region to satisfy the neutrino oscillation data is localized at nearby τ = ω. The small absolute deviation plays a crucial role in fitting two mixings of $s^2_{23}$ and $s^2_{12}$. In addition, we obtain several predictions on Majorana and Dirac CP phases, and neutrinoless double beta decay as shown in our chi square numerical analysis. We also discuss modulus stabilization within the framework of non-supersymmetric models. In the end, we compute the expansion of modular forms at nearby τ = ω in the Appendix so that one can apply them for a model and understand its analytical structure.