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

A Unified Gas-kinetic Scheme for Continuum and Rarefied Flows IV: full Boltzmann and Model Equations

Fluid dynamic equations are valid in their respective modeling scales. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region where it is needed. The central ingredient of the UGKS is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface, where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled. The newly developed UGKS has the asymptotic preserving (AP) property of recovering the NS solutions in the continuum flow regime, and the full Boltzmann solution in the rarefied regime. In the mostly unexplored transition regime, the UGKS itself provides a valuable tool for the flow study in this regime. The mathematical properties of the scheme, such as stability, accuracy, and the asymptotic preserving, will be analyzed in this paper as well

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

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

Radiative Leptonic Decays of BcB_c Meson

To the leading order, the radiative leptonic decays $B_c \to \gamma \ell \bar \nu$ ($\ell=e, \mu$) are studied carefully. In the study, a non-relativistic constituent quark model and the effective Lagrangian for the heavy flavour decays are used. As a result, the branching ratios turn out to be of the orders of $10^{-5}$ for $B_c \to \gamma \mu \bar \nu$ or for $B_c \to \gamma e \bar \nu$. Based on this study, we point out the decays being accessible experimentally at the future LHC, and the possibility to determine the decay constant $f_{B_c}$ through the radiative decays.Comment: 9 pages Latex, including 2 figure