Zeros in the magic neutrino mass matrix

We study the phenomenological implications of the presence of two zeros in a magic neutrino mass matrix. We find that only two such patterns of the neutrino mass matrix are experimentally acceptable. We express all the neutrino observables as functions of one unknown phase $\varphi$ and two known parameters $\mathrm{\Delta }{m}_{12}^2$, $r=\mathrm{\Delta }{m}_{12}^2/\mathrm{\Delta }{m}_{23}^2$. In particular, we find ${\sin }^2{\theta }_{13}=(2/3)r/(1+r)$. We also present a mass model for the allowed textures based upon the group $A_4$ using the type $\mathrm{I}+\mathrm{II}$ seesaw mechanism.

Figure 1

The three neutrino masses $m_1$ (dotted line), $m_2$ (dashed line), and $m_3$ (solid line) in eV as functions of $\varphi$ (in degrees).

Figure 2

The neutrino parameters $\sum _{i=1}^3{m}_i$, ${\theta }_{23}$, $\alpha$, $\beta$, $\delta$, and $J_{CP}$ as functions of $\varphi$. All phases and angles are in degrees. The dark (gray) bands depict the $1\sigma$ ($3\sigma$) allowed regions.

Figure 3

The ratio $r=\mathrm{\Delta }{m}_{12}^2/\mathrm{\Delta }{m}_{23}^2$ as a function of ${\theta }_{13}$ (degrees) for a magic mass matrix of type $B_3$.