Generalized $\mu$-$\tau$ symmetry and discrete subgroups of O(3)

TL;DR

This paper explores how generalized mu-tau symmetry in leptonic mixing can originate from residual symmetries in finite discrete subgroups of O(3), identifying specific groups like A4, S4, and A5 that can produce realistic neutrino mixing parameters.

Contribution

It demonstrates that residual symmetries in finite discrete subgroups of O(3) can lead to the generalized mu-tau symmetry and identifies A4, S4, and A5 as the groups capable of fixing the leptonic mixing matrix at leading order.

Findings

A4, S4, and A5 can entirely fix the leptonic mixing matrix at leading order.

Only A5 can produce a mixing angle close to the experimental value of θ13.

An explicit A5-based model is constructed with perturbations to achieve realistic neutrino masses and mixing.

Abstract

The generalized $\mu$-$\tau$ interchange symmetry in the leptonic mixing matrix $U$ corresponds to the relations: $|U_{\mu i}|=|U_{\tau i}|$ with $i=1,2,3$. It predicts maximal atmospheric mixing and maximal Dirac CP violation given $\theta_{13} \neq 0$. We show that the generalized $\mu$-$\tau$ symmetry can arise if the charged lepton and neutrino mass matrices are invariant under specific residual symmetries contained in the finite discrete subgroups of $O(3)$. The groups $A_4$, $S_4$ and $A_5$ are the only such groups which can entirely fix $U$ at the leading order. The neutrinos can be (a) non-degenerate or (b) partially degenerate depending on the choice of their residual symmetries. One obtains either vanishing or very large $\theta_{13}$ in case of (a) while only $A_5$ can provide $\theta_{13}$ close to its experimental value in the case (b). We provide an explicit model based on $A_5$ and discuss a class of perturbations which can generate fully realistic neutrino masses and mixing maintaining the generalized $\mu$-$\tau$ symmetry in $U$. Our approach provides generalization of some of the ideas proposed earlier in order to obtain the predictions, $\theta_{23}=\pi/4$ and $\delta_{\rm CP} = \pm \pi/2$.