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We investigate the Majorana neutrino mass matrix $M_{ u}$ with one parameter in the context of two texture zeros and its symmetry realization by non-Abelian discrete symmetry. From numerical calculation, we confirm that the textures $(M_{ u})_{11,12}=0$ and $(M_{ u})_{11,13}=0$ are consistent with the current experimental constraints, and show the correlations between non-zero elements of $M_{ u}$. The ratios of non-zero elements of $M_{ u}$ are constrain in small regions, and we find simple examples of $M_{ u}$ with one real mass parameter. We also discuss symmetry realization of the mass matrix by the type-II seesaw mechanism based on the binary icosahedral symmetry $A_5$ .
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Using the seesaw mechanism, we construct a model for the light-neutrino Majorana mass matrix which yields trimaximal lepton mixing together with maximal CP violation and maximal atmospheric-neutrino mixing. We demonstrate that, in our model, the light-neutrino mass matrix retains its form under the one-loop renormalization-group evolution. With our neutrino mass matrix, the absolute neutrino mass scale is a function of |U_e3| and of the atmospheric mass-squared difference. We study the effective mass in neutrinoless double beta-decay as a function of |U_e3|, showing that it contains a fourfold ambiguity.
The $mu$-$tau$ exchange symmetry in the neutrino mass matrix and its breaking as a perturbation are discussed. The exact $mu$-$tau$ symmetry restricts the 2-3 and 1-3 neutrino mixing angles as $theta_{23} = pi/4$ and $theta_{13} = 0$ at a zeroth order level. We claim that the $mu$-$tau$ symmetry breaking prefers a large CP violation to realize the observed value of $theta_{13}$ and to keep $theta_{23}$ nearly maximal, though an artificial choice of the $mu$-$tau$ breaking can tune $theta_{23}$, irrespective of the CP phase. We exhibit several relations among the deviation of $theta_{23}$ from $pi/4$, $theta_{13}$ and Dirac CP phase $delta$, which are useful to test the $mu$-$tau$ breaking models in the near future experiments. We also propose a concrete model to break the $mu$-$tau$ exchange symmetry spontaneously and its breaking is mediated by the gauge interactions radiatively in the framework of the extended gauge model with $B-L$ and $L_mu - L_tau$ symmetries. As a result of the gauge mediated $mu$-$tau$ breaking in the neutrino mass matrix, the artificial choice is unlikely, and a large Dirac CP phase is preferable.
The residual symmetry approach, along with a complex extension for some flavor invariance, is a powerful tool to uncover the flavor structure of the $3times3$ neutrino Majorana mass matrix $M_ u$ towards gaining insights into neutrino mixing. We utilize this to propose a complex extension of the real scaling ansatz for $M_ u$ which was introduced some years ago. Unlike the latter, our proposal allows a nonzero mass for each of the three light neutrinos as well as a nonvanishing $theta_{13}$. A major result of this scheme is that leptonic Dirac CP-violation must be maximal while atmospheric neutrino mixing need not be exactly maximal. Moreover, each of the two allowed Majorana phases, to be probed by the search for nuclear $0 u betabeta$ decay, has to be at one of its two CP-conserving values. There are other interesting consequences such as the allowed occurrence of a normal mass ordering which is not favored by the real scaling ansatz. Our predictions will be tested in ongoing and future neutrino oscillation experiments at T2K, NO$ u$A and DUNE.
Neutrino mass sum rules have recently gained again more attention as a powerful tool to discriminate and test various flavour models in the near future. A related question which was not yet discussed fully satisfactorily was the origin of these sum rules and if they are related to any residual or accidental symmetry. We will address this open issue here systematically and find previous statements confirmed. Namely, that the sum rules are not related to any enhanced symmetry of the Lagrangian after family symmetry breaking but that they are simply the result of a reduction of free parameters due to skillful model building.
Using the residual symmetry approach, we propose a complex extension of the scaling ansatz on $M_ u$ which allows a nonzero mass for each of the three light neutrinos as well as a nonvanishing $theta_{13}$. Leptonic Dirac CP violation must be maximal while atmospheric neutrino mixing need not to be exactly maximal. Each of the two Majorana phases, to be probed by the search for $0 u betabeta$ decay, has to be zero or $pi$ and a normal neutrino mass hierarchy is allowed.
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