Abstract
We have studied two approaches to predict quark and lepton mixing patterns from the same flavor symmetry group in combination with CP symmetry. The first approach is based on the residual symmetry $Z_2$ in the charged lepton sector and $Z_2\times CP$ in the neutrino sector. All lepton mixing angles and CP violation phases depend on three real parameters $\theta_{l}$, $\delta_{l}$ and $\theta_{\nu}$. This approach is extended to the quark sector, the up and down quark mass matrices are assumed to be invariant under a $Z_2$ subgroup and $Z_2\times CP$. The necessary and sufficient conditions for the equivalence of two mixing patterns are derived. The second approach has an abelian subgroup and a single CP transformation as residual symmetries of the charged lepton and neutrino sectors respectively. The lepton mixing would be determined up to a real orthogonal matrix multiplied from the right hand side. Analogously we assume that a single CP transformation is preserved by the down (or up) quark mass matrix, and the residual symmetry of the up (or down) quark sector is be an abelian subgroup. As an example, we analyze the possible mixing patterns which can be obtained from the breaking of $\Delta(6n^2)$ and CP symmetries. We find $\Delta(294)$ combined with CP is the smallest flavor group which can give a good fit to the experimental data of quark and lepton mixing in both approaches.
Highlights
Over the past few decades, the quark CKM mixing matrix has been measured quite precisely in meson decays [1]
The discovery of neutrino oscillation is a great progress in particle physics and it leads to the 2015 physics Nobel prizes
In the most extensively studied scenario, the flavor and generalized CP symmetries are broken to an Abelian subgroup and Z2 × CP in the charged lepton and neutrino sectors, respectively, the lepton mixing angles and CP phases would be expressed in terms of a single real parameter θ which can take values in the range 0 ≤ θ < π [25,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]
Summary
Over the past few decades, the quark CKM mixing matrix has been measured quite precisely in meson decays [1]. In the most extensively studied scenario, the flavor and generalized CP symmetries are broken to an Abelian subgroup and Z2 × CP in the charged lepton and neutrino sectors, respectively, the lepton mixing angles and CP phases would be expressed in terms of a single real parameter θ which can take values in the range 0 ≤ θ < π [25,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. In order to show concrete examples and find new interesting mixing patterns, we have performed a comprehensive study for the infinite group series Δð6n2Þ which are broken to all possible residual symmetries of the structure indicated above Another three-parameter model is the residual symmetry pattern for which the charged lepton and neutrino mass matrices are invariant under an Abelian subgroup and a single CP transformation, respectively.
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