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

Abstract

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: $\delta =\pi /2$, ${\sin }^2{\theta }_{13}\simeq 1/2{\lambda }^2\simeq 0.024$ (${\sin }^{2}2{\theta }_{13}\simeq 0.094$), ${\sin }^2{\theta }_{12}\simeq 1/33(1-2{\lambda }^3)\simeq 0.326$ 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 $m_{{\nu }_i}\simeq O({\lambda }^2)\simeq 0.04\sim 0.06$ eV with a total mass $\sum m_{\nu }\sim 0.15$ eV, which is testable in future precision astrophysics and cosmology.

Comments: 14 pages, it is explicitly shown that the smallness for both the charged-lepton mixing and neutrino masses with the standard seesaw mechanism can naturally be explained by the approximate global U(1) family symmetries of vacuum structure in the SU(3) gauge family model, references added, published version in PLB