$S_4$ Lepton Flavor Model with 3HDM

We prsent a lepton flavor model with $S_4$ and $U(1{)}_{\mathrm{F}\mathrm{N}}$ 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 ${60.12}^{\circ }<\mathrm{\mid }{\delta }_{CP}\mathrm{\mid }<{76.47}^{\circ }$. Our predictions for the lightest neutrino mass $m_{light}\simeq 5.53[\mathrm{m}\mathrm{e}\mathrm{V}]$ and the effective Majorana neutrino mass $m_{ee}\simeq 6.33[\mathrm{m}\mathrm{e}\mathrm{V}]$ from the neutrinoless double beta decay experiments are relatively close to upper limits. Lastly, we obtain the sum of neutrino masses $m_1+m_2+m_3\simeq 66.1[\mathrm{m}\mathrm{e}\mathrm{V}]$, and two Majorana phases ${\eta }_1$ and ${\eta }_2$.