Almost general analysis of $\mu$-$\tau$ reflection symmetry perturbed by charged leptons and its testability by DUNE and T2HK

Abstract

In this paper, we generally analyze the $\mu - \tau$ reflection symmetry modified by small mixings of charged leptons and how will future experiments verify deviations from the predictions of the symmetry. As an approximation, the left-handed diagonalization $U_{e}$ of charged leptons is assumed to have a similar magnitude as the CKM matrix. In other words, the 1-3 mixing is neglected and the 1-2 and 2-3 mixing are to be approximately $O(0.1)$. The Dirac CP phase $\delta$ of the MNS matrix is evaluated in such parameter regions. As a result, deviations from the predictions $\sin \theta_{23} = \pi/4$ and $\delta = \pm \pi/2$ depend on relative CP phases between $U_{e}$ and diagonalization of neutrinos $U_{\nu}$. While phases of the second and third generations cause only about $\pm 10^{\circ}$ deviations for the Dirac phase $\delta$, the phase of the first generation can cause up to $\pm 30^{\circ}$. This flavor dependence is distinguished to some extent by the next-generation experiments. On the other hand, if $\delta$ is not observed, such a scenario is excluded by about 5 years of observation.