Motivated by recent observations of nonzero $\theta${}13 from the Daya Bay and RENO experiments, we propose a renormalizable neutrino model with A4 discrete symmetry accounting for deviations from the tri-bimaximal mixing pattern of the neutrino mixing matrix indicated by neutrino oscillation data. In the model, the light neutrino masses can be generated by radiative corrections, and we show how the light neutrino mass matrix can be diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix whose entries are determined by the current neutrino data, including the Daya Bay result. We show that the origin of the deviations from the tri-bimaximal mixing is nondegeneracy of the neutrino Yukawa coupling constants, and unremovable CP phases in the neutrino Yukawa matrix give rise to both low energy CP violation measurable from neutrino oscillation and high energy CP violation.

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 bilarge 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 charge parity ($CP$) 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 to each other. So the scale of Peccei-Quinn symmetry breaking is shown to be roughly located around the $10^{12}\mathrm{GeV}$ 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 the 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 $m_a\simeq 2.53\times 10^{-5}\mathrm{eV}$ and the axion coupling to photon $g_{a\gamma \gamma }\simeq 1.33\times 10^{-15}{\mathrm{GeV}}^{-1}$. And subsequently the square of the ratio between them is shown to be one or two orders of magnitude lower than that of the conventional axion model.

We propose a new scenario in a radiative seesaw model based on A4 flavor symmetry. In this model, we explore a possibility of linking nonzero $\theta${}13 to dark matter. And we analyze the lepton sector to predict the observed neutrinos and mixings, especially obtaining a lower bound of $\theta${}13$\gtrsim${}3.5\ifmmode^\circ\else\textdegree\fi{}. We show that the nonzero $\theta${}13 is correlated with our heavy Majorana type of dark matter. Also we predict that the mass be O(1\char21{}10)\,\,TeV, as a result of analyzing the Wilkinson-Microwave-Anisotropy-Probe and lepton flavor violation.

We propose a simple renormalizable model for the spontaneous $CP$ violation based on $\mathrm{SU}(2{)}_L\times \mathrm{U}(1{)}_Y\times A_4$ symmetry in a radiative seesaw mechanism, which can be guaranteed by an extra $Z_2$ symmetry. In our model $CP$ is spontaneously broken at high energies, after the breaking of flavor symmetry, by a complex vacuum expectation value of the $A_4$ triplet and gauge-singlet scalar field. We show that the spontaneously generated $CP$ phase could become a natural source of leptogenesis, and also investigate $CP$ violation at low energies in the lepton sector and show how the $CP$ phases in the Pontecorvo-Maki-Nakagawa-Sakata formalism could arise through a spontaneous symmetry-breaking mechanism. As a numerical study, interestingly, we show that the normal mass hierarchy favors relatively large values of ${\theta }_{13}$, large deviations from maximality of ${\theta }_{23}<\pi /4$, and the Dirac-$CP$ phase $0°\le {\delta }_{CP}\le 50°$ and $300°\le {\delta }_{CP}\le 360°$. For the inverted hierarchy case, the experimentally measured values of ${\theta }_{13}$ favors ${\theta }_{23}>\pi /4$ and discrete values of ${\delta }_{CP}$ around 100°, 135°, 255°, and 300°. Finally, with a successful leptogenesis our numerical results give more predictive values on the Dirac $CP$ phase: for the normal mass hierarchy $1°\lesssim {\delta }_{CP}\lesssim 10°$ and for inverted one ${\delta }_{CP}\sim 100°$, 135°, 300°.

To explain the fermion masses and mixings naturally, we introduce a specific flavor symmetry and mass suppression pattern that constrain the flavor structure of the fermion Yukawa couplings. Our model describes why the hierarchy of neutrino masses is milder than the hierarchy of charged fermion masses in terms of successive powers of flavon fields. We investigate $CP$ violation and neutrinoless double beta ($0\nu \beta \beta$) decay and show how they can be predicted and constrained in our model by present and upcoming experimental data. Our model predicts that the atmospheric neutrino mixing angle ${\theta }_{23}$ should be within $\sim 1°$ of 45° for the normal neutrino mass ordering and between $\sim 4°$ and $\sim 8°$ degrees away from 45° (in either direction) for the inverted neutrino mass ordering. For both neutrino mass ordering and inverted neutrino mass ordering, our model predicts that a $0\nu \beta \beta$ Majorana mass in the limited range $0.035\mathrm{eV}<|m_{ee}|\lesssim 0.15\mathrm{eV}$, which can be tested in current experiments. Moreover, our model can successfully accommodate flavorless leptogenesis as the mechanism to generate the baryon asymmetry in the Universe, provided the neutrino mass ordering is normal, $|m_{ee}|\simeq 0.072\pm 0.012\mathrm{eV}$, and either ${\theta }_{23}\simeq 44°$ and the Dirac $CP$-violating phase ${\delta }_{CP}\simeq 20°$ or 60° or ${\theta }_{23}\simeq 46°$ and ${\delta }_{CP}\simeq 205°$ or 245°.

The recent measurement of a nonzero neutrino mixing angle ${$\theta${}}_{13}$ requires a modification of the tri-bimaximal mixing pattern that predicts a zero value for it. We propose a new neutrino mixing pattern based on a spontaneously broken ${A}_{4}$ flavor symmetry and a type-I seesaw mechanism. Our model allows for approximate tri-bimaximal mixing and nonzero ${$\theta${}}_{13}$, and contains a natural way to implement low- and high-energy $CP$ violations in neutrino oscillations, and leptogenesis with a renormalizable Lagrangian. Both normal and inverted mass hierarchies are permitted within $3$\sigma${}$ experimental bounds, with the prediction of small (large) deviations from maximality in the atmospheric mixing angle for the normal (inverted) case. Interestingly, we show that the inverted case is excluded by the global analysis in $1$\sigma${}$ experimental bounds, while the most recent MINOS data seem to favor the inverted case. Our model make predictions for the Dirac $CP$ phase in the normal and inverted hierarchies, which can be tested in near-future neutrino oscillation experiments. Our model also predicts the effective mass $|{m}_{ee}|$ measurable in neutrinoless double beta decay to be in the range $0.04$\lesssim${}|{m}_{ee}|$\lesssim${}0.15\text{ }\text{ }\mathrm{eV}$ for the normal hierarchy and $0.06$\lesssim${}|{m}_{ee}|$\lesssim${}0.11\text{ }\text{ }\mathrm{eV}$ for the inverted hierarchy, both of which are within the sensitivity of the next generation experiments.