Discrete symmetries and mixing of Dirac neutrinos

We study the mixing of the Dirac neutrinos in the residual symmetries approach. The key difference from the Majorana case is that the Dirac mass matrix may have larger symmetries: $G_{\nu }={\mathbf{Z}}_n$ with $n\ge 3$. The symmetry group relations have been generalized to the case of Dirac neutrinos. Using them, we have found all new relations between mixing parameters and corresponding symmetry assignments, which are in agreement with the present data. The viable relations exist only for the charged lepton residual symmetry $G_{\ell }={\mathbf{Z}}_2$. The relations involve elements of the rows of the Pontecorvo-Maki-Nakagawa-Sakata matrix and lead to precise predictions of the 2–3 mixing angle and certain ranges of the $CP$ violation phase. For larger symmetries $G_{\ell }$, an agreement with the data can be achieved if $\sim 10\%$ corrections related to breaking of $G_{\ell }$ and $G_{\nu }$ are included.

Figure 1

The prediction of discrete symmetries for Dirac neutrinos in the $({\sin }^2{\theta }_{23},\delta )$ plane, compared with the experimentally allowed region for normal (a) and inverted (b) hierarchies. The red and green regions correspond to the predictions of the symmetry group $D(3,2,5)$. The blue and brown regions are for $D(4,2,3)$, and the orange region is for $D(5,2,3)$. The solid, dashed, and dotted-dashed curves determine the allowed regions from a global analysis of oscillation data, respectively, at $1\sigma$, $2\sigma$, and $3\sigma$ [14].