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 $\lambda \simeq 0.22$, 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: $\delta =\pi \mathrm{/}2$, $\sin^2\theta_{13} \simeq 1/2\lambda^2 \simeq 0.024$($\sin^22\theta_{13} \simeq 0.094$),$\sin^2\theta_{12} \simeq 1/3{3}(1 - 2\lambda^3) \simeq 0.326$and$\sin^2\theta_{23} \simeq 1/2(1 - \lambda^2) \simeq 0.48$, which agree well with the current experimental data. The CP-violating Jarlskog-invariant is obtained to be $J_{CP} \simeq 1/6\lambda(1-\lambda^2/2-\lambda^3)\sin\delta \simeq 0.035$, 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 $m_{\nu_i} \simeq O(\lambda^2) \simeq 0.04\sim 0.06$eVwithatotalmass$\sum m_{\nu} \sim 0.15$ eV, which is testable in future precision astrophysics and cosmology.