New Simple A 4 Neutrino Model for Nonzero θ 13 and Large δ CP

In a new simple application of the non-Abelian discrete symmetry A4 to chargedlepton and neutrino mass matrices, we show that for the current experimental central value of sin 2θ13 ≃ 0.1, leptonic CP violation is necessarily large, i.e. | tan δCP | > 1.3. The non-Abelian discrete symmetry A4 was introduced [1, 2, 3] to achieve the seemingly impossible, i.e. the existence of a lepton family symmetry consistent with the three very different charged-lepton masses me, mμ, mτ . It was subsequently shown [4] to be a natural theoretical framework for neutrino tribimaximal mixing, i.e. sin θ23 = 1, tan 2 θ12 = 0.5, and θ13 = 0. This pattern was consistent with experimental data until recently, when the Daya Bay Collaboration reported [5] the first precise measurement of θ13, i.e. sin 2θ13 = 0.092± 0.016(stat)± 0.005(syst), (1) followed shortly [6] by the RENO Collaboration, i.e. sin 2θ13 = 0.113± 0.013(stat)± 0.019(syst). (2) This means that tribimaximal mixing is not a good description, and more importantly, leptonic CP violation is now possible because θ13 6= 0, just as hadronic CP violation in the quark sector is possible because Vub 6= 0. In this paper, we show that A4 is still a good symmetry for understanding this pattern, using a new simple variation of the original idea. As shown below, it predicts a correlation between θ13, θ23, and δCP in such a way that given the experimentally allowed ranges of values for θ13 and θ23, a lower bound on | tan δCP | is obtained. In particular, for the central values of sin 2θ13 = 0.1 and sin 2 2θ12 = 0.87, we find | tan δCP | > 1.3 from sin 2θ23 > 0.92. The most general 3 × 3 Majorana neutrino mass matrix has six complex entries, i.e. twelve parameters. Three are overall phases of the mass eigenstates which are unobservable. The nine others are three masses, three mixing angles, and three phases: one Dirac phase δCP , i.e. the analog of the one complex phase of the 3 × 3 quark mixing matrix, and two relative Majorana phases α1,2 for two of the three mass eigenstates. The existence of nonzero δCP or α1,2 means that CP conservation is violated. It is one of the most important issues of neutrino physics yet to be explored experimentally. 2 Before showing how the A4 model is constructed, consider first the end results. In the A4 basis, the 3× 3 charged-lepton mass matrix is Ml = 1 √ 3 