$A_4 \times U(1)_{PQ}$ Model for the Lepton Flavor Structure and the Strong $CP$ Problem

We present a model with $A_4 \times U(1)_{PQ}$ lepton flavor symmetry which explains the origin of the lepton flavor structure and also solves the strong $CP$ problem. Standard model gauge singlet fields, so-called "flavons", charged under the $A_4 \times U(1)_{PQ}$ symmetry are introduced and are coupled with the lepton and the Higgs sectors. The flavon vacuum expectation values (VEVs) trigger spontaneous breaking of the $A_4 \times U(1)_{PQ}$ symmetry. The breaking pattern of the $A_4$ accounts for the tri-bimaximal neutrino mixing and the deviation from it due to the non-zero $\theta_{13}$ angle, and the breaking of the $U(1)_{PQ}$ gives rise to a pseudo-Nambu-Goldstone boson, axion, whose VEV cancels the QCD $\theta$ term. We investigate the breaking of the $A_4 \times U(1)_{PQ}$ symmetry through an analysis on the scalar potential and further discuss the properties of the axion in the model, including its decay constant, mass and coupling with photons. It is shown that the axion decay constant is related with the right-handed neutrino mass through the flavon VEVs. Experimental constraints on the axion and their implications are also studied.