Axion and Neutrino physics in a $U(1)$-enhanced supersymmetric model

Motivated by the flavored Peccei-Quinn symmetry for unifying the flavor physics and string theory, we construct an explicit model by introducing a $U(1)$ symmetry such that the $U(1)_X$-$[gravity]^2$ anomaly-free condition together with the standard model flavor structure demands additional sterile neutrinos as well as no axionic domain-wall problem. Such additional sterile neutrinos play the role of a realization of baryogenesis via a new Affleck-Dine leptogenesis. We provide grounds for that the $U(1)_X$ symmetry could be interpreted as a fundamental symmetry of nature. The model will resolve rather recent, but fast-growing issues in astro-particle physics, including leptonic mixings and CP violation in neutrino oscillation, high-energy neutrinos, QCD axion, and axion cooling of stars. The QCD axion decay constant, through its connection to the astrophysical constraints of stellar evolution and the SM fermion masses, is shown to be fixed at $F_A=1.30^{+0.66}_{-0.54}\times10^{9}$ GeV (consequently, its mass is $m_a=4.34^{+3.37}_{-1.49}$ meV and axion-photon coupling is $|g_{a\gamma\gamma}|=1.30^{+1.01}_{-0.45}\times10^{-12}\,{\rm GeV}^{-1}$). Interestingly enough, we show that neutrino oscillations at low energies could be connected to astronomical-scale baseline neutrino oscillations. The model predicts non-observational neutrinoless double beta ($0\nu\beta\beta$) decay rate as well as a remarkable pattern between leptonic Dirac CP phase ($\delta_{CP}$) and atmospheric mixing angle ($\theta_{23}$); {\it e.g.} $\delta_{CP}\simeq220^{\circ}-240^{\circ}$, $120^{\circ}-140^{\circ}$ for $\theta_{23}=42.3^{\circ}$ for normal mass ordering, and $\delta_{CP}\simeq283^{\circ},250^{\circ},100^{\circ},70^{\circ}$ for $\theta_{23}=49.5^{\circ}$ for inverted one.