Flavored Peccei-Quinn symmetry

In an attempt to uncover any underlying physics in the standard model (SM), we suggest a $\mu$--$\tau$ power law in the lepton sector, such that relatively large 13 mixing angle with bi-large ones can be derived. On the basis of this, we propose a neat and economical model for both the fermion mass hierarchy problem of the SM and a solution to the strong CP problem, in a way that no domain wall problem occurs, based on $A_{4}\times U(1)_{X}$ symmetry in a supersymmetric framework. Here we refer to the global $U(1)_X$ symmetry that can explain the above problems as "flavored Peccei-Quinn symmetry". In the model, a direct coupling of the SM gauge singlet flavon fields responsible for spontaneous symmetry breaking to ordinary quarks and leptons, both of which are charged under $U(1)_X$, comes to pass through Yukawa interactions, and all vacuum expectation values breaking the symmetries are connected each other. So, the scale of Peccei-Quinn symmetry breaking is shown to be roughly located around $10^{12}$ GeV section through its connection to the fermion masses. The model predictions are shown to lie on the testable regions in the very near future through on-going experiments for neutrino oscillation, neutrinoless double beta decay and axion. We examine the model predictions, arisen from the $\mu$--$\tau$ power law, on leptonic $CP$ violation, neutrinoless double beta decay and atmospheric mixing angle, and show that the fermion mass and mixing hierarchies are in good agreement with the present data. Interestingly, we show the model predictions on the axion mass $m_a\simeq2.53\times10^{-5}$ eV and the axion coupling to photon $g_{a\gamma\gamma}\simeq1.33\times10^{-15}~{\rm GeV}^{-1}$. And subsequently the square of the ratio between them is shown to be 1 or 2 orders of magnitude lower than that of the conventional axion model.