Non-vanishing $U_{e3}$ and $\cos 2{\theta }_{23}$ from a broken $Z_2$ symmetry

Abstract

It is shown that the neutrino mass matrices in the flavour basis yielding a vanishing $U_{e3}$ are characterized by invariance under a class of $Z_2$ symmetries. A specific $Z_2$ in this class also leads to a maximal atmospheric mixing angle ${\theta }_{23}$. The breaking of that $Z_2$ can be parameterized by two dimensionless quantities, $\text{e}$ and ${\text{e}}^{\prime }$; the effects of $\text{e},{\text{e}}^{\prime }\ne 0$ are studied perturbatively and numerically. The induced value of $\text{ue}3$ strongly depends on the neutrino mass hierarchy. We find that $\text{ue}3$ is less than 0.07 for a normal mass hierarchy, even when $\text{e},{\text{e}}^{\prime }\sim 30$. For an inverted mass hierarchy $\text{ue}3$ tends to be around 0.1 but can be as large as 0.17. In the case of quasi-degenerate neutrinos, $\text{ue}3$ could be close to its experimental upper bound 0.2. In contrast, $|\cos 2{\theta }_{23}|$ can always reach its experimental upper bound 0.28. We propose a specific model, based on electroweak radiative corrections in the MSSM, for $\text{e}$ and ${\text{e}}^{\prime }$. In that model, both $\text{ue}3$ and $|\cos 2{\theta }_{23}|$, could be close to their respective experimental upper bounds if neutrinos are quasi-degenerate.

Comments: 28 pages, 12 figures, Some discussions are revised, references are corrected, Conclusions are unchanged