Hierarchical majorana neutrinos from democratic mass matrices

In this paper, we obtain the light neutrino masses and mixings consistent with the experiments, in the democratic texture approach. The essential ansatz is that $\nu_{Ri}$ are assumed to transform as "right-handed fields" $\bf 2_{R} + 1_{R}$ under the $S_{3L} \times S_{3R}$ symmetry. The symmetry breaking terms are assumed to be diagonal and hierarchical. This setup only allows the normal hierarchy of the neutrino mass, and excludes both of inverted hierarchical and degenerated neutrinos. Although the neutrino sector has nine free parameters, several predictions are obtained at the leading order. When we neglect the smallest parameters $\zeta_{\nu}$ and $\zeta_{R}$, all components of the mixing matrix $U_{\rm PMNS}$ are expressed by the masses of light neutrinos and charged leptons. From the consistency between predicted and observed $U_{\rm PMNS}$, we obtain the lightest neutrino masses $m_{1}$ = (1.1 $\to$ 1.4) meV, and the effective mass for the double beta decay \vev{m_{ee}} \simeq 4.5 meV.Comment: 14 pages, 1 table, substantially revised versio

Flavor structure from misalignment of inner products in noncommutative geometry

In this letter, we consider an idea that induces flavor structure from inner products in noncommutative geometry. Assuming proper components of vectors $v_{(L,R) i}$ in enlarged representation space for fermions, we can induce the waterfall texture for Yukawa matrices retaining gauge interactions are universal. The hierarchy of the Yukawa interactions is a consequence of "misalignment" between the vectors $v_{Li}$ and $v_{Rj}$.Comment: 6pages, 1 table, the final version to appear in JHE

Conditions of general Z2Z_{2} symmetry in an arbitrary basis and TM1,2_{1,2} mixing for the minimal type-I seesaw mechanism

In this paper, using a formula for the minimal type-I seesaw mechanism by $LDL^{T}$ decomposition, conditions of general $Z_{2}$-invariance of the neutrino mass matrix $m$ is obtained for Lagrangian parameters in an arbitrary basis. The conditions are found to be $(M_{22} a_{i}^{+} - M_{12} b_{i}^{+}) \, ( M_{22} a_{j}^{-} - M_{12} b_{j}^{-}) = - \det M \, b_{i}^{+} \, b_{j}^{-}$ for the $Z_{2}$-symmetric and -antisymmetric part of a Yukawa matrix $Y_{ij}^{\pm} \equiv (Y \pm T Y )_{ij} /2 \equiv (a_{j}^{\pm}, b_{j}^{\pm})$ and the right-handed neutrino mass matrix $M_{ij}$. In other words, the symmetric and antisymmetric part of $b_{i}$ must be proportional to those of the quantity $(M_{22} a_{i} - M_{12} b_{i})$. Since the eigenvectors of the generator $T$ are orthogonal, we can analyze the eigenvectors and mass eigenvalues of $m m^{\dagger}$ for a $Z_{2}$-symmetric $m$. Two eigenvectors $u_{1,2}$ of $m m^{\dagger}$ coincide with any of those of $T$, and the remaining one is a vector $(u_{1} \times u_{2})^{*}$ orthogonal to them. Furthermore, if the Yukawa matrix does not have the $Z_{2}$ symmetry, two nonzero neutrino masses are concisely represented. These results are applied to three $Z_{2}$ symmetries, the $\mu-\tau$ symmetry, the TM$_{1}$ mixing, and the magic symmetry which predicts the TM$_{2}$ mixing. In particular, for the TM$_{2}$ mixing, the magic (anti-)symmetric Yukawa matrix with $S_{2} Y = \pm Y$ is phenomenologically rejected because it predicts $m_{2}=0$ or $m_{1}, m_{3} = 0$.Comment: 21 page