$S_4$ Lepton Flavor Model with 3HDM

We prsent a lepton flavor model with $S_4$ and $U(1)_{\mathrm{FN}}$ symmetry. The left-handed leptons are assigned as a triplet under the $S_4$ symmetry, the right-handed electron and muon as $S_4$ doublets, and the right-handed tauon as an $S_4$ singlet. We introduce three right-handed Majorana neutrinos that are charged under the $S_4$ symmetry and two scalar fields. Additionally, the three Higgs doublets are assigned as an $S_4$ triplet, and we analyze the scalar potential to get the conditions for the Higgs VEVs. We numerically calculate the PMNS matrix, and give strong predictions for the mixing angle $0.486<\sin\theta_{23}<0.603$ and the Dirac CP phase $60.12^\circ<|\delta_{CP}|<76.47^\circ$. Our predictions for the lightest neutrino mass $m_{light}\simeq5.53~[\mathrm{meV}]$ and the effective Majorana neutrino mass $m_{ee}\simeq6.33~[\mathrm{meV}]$ from the neutrinoless double beta decay experiments are relatively close to upper limits. Lastly, we obtain the sum of neutrino masses $m_1+m_2+m_3\simeq66.1~[\mathrm{meV}]$, and two Majorana phases $\eta_1$ and $\eta_2$.