We consider a direct approach to quark mixing based on the discrete family symmetry Delta (6N^2) in which the Cabibbo angle is determined by a residual Z_2 times Z_2 subgroup to be $|V_{us}|=0.222521$, for $N$ being a multiple of 7. We propose a part
icular model in which unequal smaller quark mixing angles and CP phases may occur without breaking the residual Z_2 times Z_2 symmetry. We perform a numerical analysis of the model for $N=14$, where small Z_2 times Z_2 breaking effects of order 3% are allowed by model, allowing perfect agreement within the uncertainties of the experimentally determined best fit quark mixing values.
We propose a first model of quarks based on the discrete family symmetry Delta (6N^2) in which the Cabibbo angle is correctly determined by a residual Z_2 times Z_2 subgroup, and the smaller quark mixing angles may be qualitatively understood from th
e model. The present model of quarks may be regarded as a first step towards formulating a complete model of quarks and leptons based on Delta (6N^2), in which the lepton mixing matrix is fully determined by a Klein subgroup. For example, the choice N=28 provides an accurate determination of both the reactor angle and the Cabibbo angle.
In a new simple application of the non-Abelian discrete symmetry $A_4$ to charged-lepton and neutrino mass matrices, we show that for the current experimental central value of $sin^2 2 theta_{13} simeq 0.1$, leptonic CP violation is necessarily large, i.e. $|tan delta_{CP}| > 1.3$.
In a recently proposed renormalizable model of neutrino mixing using the non-Abelian discrete symmetry T_7 in the context of a supersymmetric extension of the Standard Model with gauged U(1)_{B-L}, a correlation was obtained between theta_{13} and th
eta_{23} in the case where all parameters are real. Here we consider all parameters to be complex, thus allowing for one Dirac CP phase delta_{CP} and two Majorana CP phases alpha_{1,2}. We find a slight modification to this correlation as a function of delta_{CP}. For a given set of input values of Delta m^2_{21}, Delta m^2_{32}, theta_{12}, and theta_{13}, we obtain sin^2 2 theta_{23} and m_{ee} (the effective Majorana neutrino mass in neutrinoless double beta decay) as functions of tan delta_{CP}. We find that the structure of this model always yields small |tan delta_{CP}|.
We study the lepton flavor models with the flavor symmetry (Z_N times Z_N times Z_N)rtimes Z_3. Our models predict non-vanishing discrete values of theta_{13} as well as theta_{12} and theta_{23} depending on N. For certain values, our models realize
the tri-bimaximal mixing angles with theta_{13}=0. For other values, our models provide with discrete deviation from the tri-bimaximal mixing angles.
We propose new equations of motion under the theory of the Brownian motion to connect the states of quantum, diffusion, soliton, and periodic localization. The new equations are nothing but the classical equations of motion with two additional terms
and the one of them can be regarded as the the quantum potential. By choosing a parameter space, various important states are obtained. Further, the equations contain other interesting phenomena such as general dynamics of diffusion process, collapse of the soliton, the nonlinear extension of the Schrdinger equation, and the dynamics of phase transition.
We study lepton flavor models with the $S_4$ flavor symmetry. We construct simple models with smaller numbers of flavon fields and free parameters, such that we have predictions among lepton masses and mixing angles. The model with a $S_4$ triplet fl
avon is not realistic, but we can construct realistic models with two triplet flavons, or one triplet and one doublet flavons.
We discuss slepton mass matrices in the $S_4$ flavor model with SUSY SU(5) GUT. By considering the gravity mediation within the framework of supergravity theory, we estimate the SUSY breaking terms in the slepton mass matrices, which contribute to th
e $mu rightarrow e + gamma$ decay. We obtain a lower bound for the ratio of $murightarrow egamma$ as $10^{-13}$ if $m_{text{SUSY}}$ and $m_{1/2}$ are below 500GeV. The off diagonal terms of slepton mass matrices also contribute to EDM of leptons. The predicted electron EDM is around $10^{-29}-10^{-28}$cm. Our predictions are expected to be tested in the near future experiment.
