Zee-Babu model in a non-holomorphic modular $A_4$ symmetry and modular stabilization

Abstract

We study a Zee-Babu neutrino model in a non-holomorphic modular $A_4$ symmetry, and we construct a model so that there are minimum free parameters (two complex parameters). 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 study 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.

Figures (4)

• Figure 1: Allowed region in terms of real $\tau$ and imaginary $\tau$. The blue points are localized at the region at nearby $\omega\equiv e^{2\pi i}$ where red point is $\tau=\omega$
• Figure 2: Blue points are our allowed regions of $\Delta m^2_{\rm sol}$(up-left), $s^2_{13}$(up-right), $s^2_{12}$(down-left), and $s^2_{23}$(down-right) in terms of $\sum D_{\nu}$. Each dotted horizontal black line represents experimental upper(lower)-bound at 3$\sigma$ interval where these lines are given by two dimensional $\chi^2$ analysis.
• Figure 3: Allowed region for the neutrinoless double beta decay in terms of sum of neutrino masses. The legend of colors are the same as Fig.1. The shaded regions are allowed by the current experimental data at 3 $\sigma$.
• Figure 4: The scatter plots for the Majorana phase $\alpha$ in terms of Dirac CP phase $\delta_{\rm CP}$, where the red dashed-dot vertical line at -264 [deg] and the black one at 62 [deg] are respectively the lower and upper bound on Nufit 6.0 Esteban:2024eli. The legend of colors are the same as Fig.1.