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SU(3) Gauge Family Symmetry and Prediction for the Lepton-Flavor Mixing and Neutrino Masses with Maximal Spontaneous CP Violation

A model for the lepton-flavor mixing and CP violation is proposed based on the SU$_F$(3) gauge family symmetry and the Majorana feature of neutrinos. A consistent prediction for the lepton-flavor mixing and masses is shown to be resulted from the appropriate vacuum structure of SU$_F$(3) gauge symmetry breaking. By choosing the SU$_F$(3) gauge fixing condition to possess a residual $Z_2$ symmetry and requiring the vacuum structure of spontaneous symmetry breaking to have approximate global U(1) family symmetries, we obtain naturally the tri-bimaximal mixing matrix and largely degenerate neutrino masses in the neutrino sector and the small mixing matrix in the charged-lepton sector. With a simple ansatz that all the smallness due to the approximate global U(1) family symmetries is characterized by a single Wolfenstein parameter $λ\simeq 0.22$, and the charged-lepton mixing matrix has a similar hierarchy structure as the CKM quark mixing matrix, we arrive at a consistent prediction for the MNSP lepton-flavor mixing with a maximal spontaneous CP violation: $δ=π/2$, $\sin^2θ_{13} \simeq 1/2λ^2 \simeq 0.024$ ($\sin^22θ_{13} \simeq 0.094$), $\sin^2θ_{12} \simeq 1/3{3}(1 - 2λ^3) \simeq 0.326$ and $\sin^2θ_{23} \simeq 1/2(1 - λ^2) \simeq 0.48$, which agree well with the current experimental data. The CP-violating Jarlskog-invariant is obtained to be $J_{CP} \simeq 1/6λ(1-λ^2/2-λ^3)\sinδ\simeq 0.035$, which is detectable in next generation neutrino experiment. The largely degenerate neutrino masses with the normal hierarchy and inverse hierarchy are discussed and found be at the order $m_{ν_i} \simeq O(λ^2) \simeq 0.04\sim 0.06$ eV with a total mass $\sum m_ν \sim 0.15$ eV, which is testable in future precision astrophysics and cosmology.