Deviation from $\mu-\tau$ reflection symmetry under radiative corrections in the minimal seesaw framework

Abstract

The $\mu$-$\tau$ reflection symmetry predicts a maximal atmospheric mixing angle, $\theta_{23}=\pi/4$, and Dirac CP phase, $\delta=\pi/2$ or $3\pi/2$. Recent global analyses indicate small but significant deviations from these predictions, suggesting that the symmetry is approximate and requires breaking. Motivated by this, we study the breaking of $\mu$-$\tau$ reflection symmetry induced by radiative corrections in the minimal seesaw framework, assuming the symmetry to be exact at the high-energy seesaw scale $\Lambda_{\mu\tau}=10^{14}~\mathrm{GeV}$. A distinctive feature of the minimal seesaw is that one light neutrino remains massless, leaving only one physical Majorana CP phase. Starting from the integral solution of the one-loop RGE for the effective Majorana neutrino mass matrix, we derive analytical expressions for the low-energy neutrino parameters at $\Lambda_{\mathrm{EW}}=172.76~\mathrm{GeV}$ in terms of their high-energy counterparts. We then numerically estimate the low-energy parameters within the MSSM, taking $\Lambda_s=1~\mathrm{TeV}$ and $\tan\beta=10$, $30$ and $50$. The analysis is performed separately for Normal Order (NO) and Inverted Order (IO). We find that the predicted neutrino masses and mixing parameters are consistent with current experimental data in both scenarios. The sum of neutrino masses, Jarlskog invariant $J$, and effective Majorana mass $|\langle m\rangle_{ee}|$ also satisfy current experimental constraints. Finally, we estimate the amount of deviations of the low-energy neutrino parameters from their high-energy values and investigate their dependence on $\tan\beta$. We find that the magnitude of these deviations increases with increasing $\tan\beta$ for both NO and IO.