Lepton flavor model with modular $A_4$ symmetry in large volume limit

We consider the modular symmetry associated with the compactification of extra dimensions as the flavor symmetry on lepton sector. Especially, we propose a model based on the modular $A_4$ symmetry with three right-handed neutrinos and a gauge singlet Higgs, which works well in the so-called large volume limit of the extra dimensions, i.e., $\mbox{Im}\tau \to \infty$ for a modulus $\tau$. The right-handed neutrinos are introduced to realize the seesaw mechanism for tiny neutrino masses observed in oscillation experiments. The vacuum expectation value of the singlet Higgs gives the Majorana masses for right-handed neutrinos and the $\mu$-term for Higgs fields. The model can explain the observed masses and mixing angles of neutrinos successfully. We find that one of the mixing angles should be in the range $\sin^2 \theta_{23} \ge 0.58$. Importantly, the CP violating phases of neutrinos are predicted in the two restricted regions. One is that the Dirac phase is $\delta_{\rm CP} \simeq - 0.5 \pi$ and the Majorana phases are $\alpha_{21} \simeq 0$ and $\alpha_{31} \simeq \pi$. The other is $\delta_{\rm CP} \simeq + 0.5 \pi$, $\alpha_{21} \simeq 2 \pi$ and $\alpha_{31} \simeq \pi$. The effective neutrino mass in the neutrinoless double beta decay is found to be $m_{\rm eff} = 0.037$--0.047 eV. These predictions will be tested in the future neutrino experiments.