Generalised CP and Trimaximal TM1_1 Lepton Mixing in S4S_4 Family Symmetry

We construct two flavor models based on $S_4$ family symmetry and generalised CP symmetry. In both models, the $S_4$ family symmetry is broken down to the $Z^{SU}_2$ subgroup in the neutrino sector, as a consequence, the trimaximal $\text{TM}_1$ lepton mixing is produced. Depending on the free parameters in the flavon potential, the Dirac CP is predicted to be either conserved or maximally broken, and the Majorana CP phases are trivial. The two models differ in the neutrino sector. The flavon fields are involved in the Dirac mass terms at leading order in the first model, and the neutrino mass matrix contains three real parameters such that the absolute neutrino masses are fixed. Nevertheless, the flavon fields enter into the Majorana mass terms at leading order in the second model. The leading order lepton mixing is of the tri-bimaximal form which is broken down to $\text{TM}_1$ by the next to leading order contributions.Comment: 28 page

Deviation from Bimaximal Mixing and Leptonic CP Phases in S4S_4 Family Symmetry and Generalized CP

The lepton flavor mixing matrix having one row or one column in common with the bimaximal mixing up to permutations is still compatible with the present neutrino oscillation data. We provide a thorough exploration of generating such a mixing matrix from $S_4$ family symmetry and generalized CP symmetry $H_{CP}$. Supposing that $S_4\rtimes H_{CP}$ is broken down to $Z^{ST^2SU}_2\times H^{\nu}_{CP}$ in the neutrino sector and $Z^{TST^{2}U}_4\rtimes H^{l}_{CP}$ in the charged lepton sector, one column of the PMNS matrix would be of the form $\left(1/2, 1/\sqrt{2}, 1/2\right)^{T}$ up to permutations, both Dirac CP phase and Majorana CP phases are trivial in order to accommodate the observed lepton mixing angles. The phenomenological implications of the remnant symmetry $K^{(TST^2, T^2U)}_4\rtimes H^{\nu}_{CP}$ in the neutrino sector and $Z^{SU}_{2}\times H^{l}_{CP}$ in the charged lepton sector are studied. One row of PMNS matrix is determined to be $\left(1/2, 1/2, -i/\sqrt{2}\right)$, and all the three leptonic CP phases can only be trivial to fit the measured values of the mixing angles. Two models based on $S_4$ family symmetry and generalized CP are constructed to implement these model independent predictions enforced by remnant symmetry. The correct mass hierarchy among the charged leptons is achieved. The vacuum alignment and higher order corrections are discussed.Comment: 44 pages, 7 figure

Toward a unified interpretation of quark and lepton mixing from flavor and CP symmetries

We discussed the scenario that a discrete flavor group combined with CP symmetry is broken to $Z_2\times CP$ in both neutrino and charged lepton sectors. All lepton mixing angles and CP violation phases are predicted to depend on two free parameters $\theta_{l}$ and $\theta_{\nu}$ varying in the range of $[0, \pi)$. As an example, we comprehensively study the lepton mixing patterns which can be derived from the flavor group $\Delta(6n^2)$ and CP symmetry. Three kinds of phenomenologically viable lepton mixing matrices are obtained up to row and column permutations. We further extend this approach to the quark sector. The precisely measured quark mixing angles and CP invariant can be accommodated for certain values of the free parameters $\theta_{u}$ and $\theta_{d}$. A simultaneous description of quark and lepton flavor mixing structures can be achieved from a common flavor group $\Delta(6n^2)$ and CP, and accordingly the smallest value of the group index $n$ is $n=7$.Comment: 40 pages, 8 figure

Tri-Direct CP in the Littlest Seesaw Playground

We discuss spontaneously broken CP symmetry in two right-handed neutrino models based on the idea of having a {\it different residual flavour symmetry}, together with a {\it different residual CP symmetry}, associated with each of the two right-handed neutrinos. The charged lepton sector also has a {\it different residual flavour symmetry}. In such a {\it tri-direct CP approach}, we show that the combination of the three residual flavour and two residual CP symmetries provides a new way of fixing the parameters. To illustrate the approach, we revisit the Littlest Seesaw (LSS) model based on $S_4$ and then propose new variants which have not so far appeared in the literature, with different predictions for each variant. We analyse numerically the predictions of the new variants, and then propose an explicit model which can realise one of the successful benchmark points, based on the atmospheric flavon vacuum alignment $(1, \omega^2 , \omega)$ and the solar flavon vacuum alignment $(1, -7/2, -7/2 )$.Comment: 31 pages, 4 figure

Golden Littlest Seesaw

We propose and analyse a new class of Littlest Seesaw models, with two right-handed neutrinos in their diagonal mass basis, based on preserving the first column of the Golden Ratio mixing matrix. We perform an exhaustive analysis of all possible remnant symmetries of the group $A_5$ which can be used to enforce various vacuum alignments for the flavon controlling solar mixing, for two simple cases of the atmospheric flavon vacuum alignment. The solar and atmospheric flavon vacuum alignments are enforced by {\em different} remnant symmetries. We examine the phenomenological viability of each of the possible Littlest Seesaw alignments in $A_5$, which preserve the first column of the Golden ratio mixing matrix, using figures and extensive tables of benchmark points and comparing our predictions to a recent global analysis of neutrino data. We also repeat the analysis for an alternative form of Golden Ratio mixing matrix.Comment: 32 pages, 7 figure

Eclectic flavor group Δ(27)⋊S3\Delta(27)\rtimes S_3 and lepton model building

We have performed a systematical study of the eclectic flavor group $\Delta(27)\rtimes S_3$ which is the extension of the traditional flavor symmetry $\Delta(27)$ by the modular symmetry group $S_3$. Consistency between $\Delta(27)$ and $S_3$ requires that the eight nontrivial singlet representations of $\Delta(27)$ should be arranged into four reducible doublets. The modular transformation matrices are determined for various $\Delta(27)$ multiplets, and the generalized CP symmetry compatible with $\Delta(27)\rtimes S_3$ are discussed. We study the general form of the K\"ahler potential and superpotential invariant under $\Delta(27)\rtimes S_3$, and the corresponding fermion mass matrices are presented. We propose a bottom-up model for lepton masses and mixing based on $\Delta(27)\rtimes S_{3}$, a numerical analysis is performed and the experimental data can be accommodated.Comment: 33 pages, 2 figure

Lepton Mixing Predictions from S4S_4 in the Tri-Direct CP approach to Two Right-handed Neutrino Models

We perform an exhaustive analysis of all possible breaking patterns arising from $S_4\rtimes H_{CP}$ in a new {\it tri-direct CP approach} to the minimal seesaw model with two right-handed neutrinos, and construct a realistic flavour model along these lines. According to this approach, separate residual flavour and CP symmetries persist in the charged lepton, `atmospheric' and `solar' right-handed neutrino sectors, i.e. we have {\it three} symmetry sectors rather than the usual two of the {\it semi-direct CP approach} (charged leptons and neutrinos). Following the {\it tri-direct CP approach}, we find twenty-six kinds of independent phenomenologically interesting mixing patterns. Eight of them predict a normal ordering (NO) neutrino mass spectrum and the other eighteen predict an inverted ordering (IO) neutrino mass spectrum. For each phenomenologically interesting mixing pattern, the corresponding predictions for the PMNS matrix, the lepton mixing parameters, the neutrino masses and the effective mass in neutrinoless double beta decay are given in a model independent way. One breaking pattern with NO spectrum and two breaking patterns with IO spectrum corresponds to form dominance. We find that the lepton mixing matrices of three kinds of breaking patterns with NO spectrum and one form dominance breaking pattern with IO spectrum preserve the first column of the tri-bimaximal (TB) mixing matrix, i.e. yield a TM1 mixing matrix.Comment: 70 pages, 2 figure