Resolving the octant of ${\theta }_{23}$ via radiative $\mu -\tau$ symmetry breaking

We point out that the observed neutrino mixing pattern at low energies is very likely to originate from the $3\times 3$ Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix $U$ which possesses the exact $\mu$-$\tau$ permutation symmetry $|U_{\mu i}|=|U_{\tau i}|$ (for $i=1$, 2, 3) at a superhigh energy scale ${\mathrm{\Lambda }}_{\mu \tau }\sim 10^{14}\mathrm{GeV}$. The deviation of ${\theta }_{23}$ from 45° and that of $\delta$ from 270° in the standard parametrization of $U$ are therefore a natural consequence of small PMNS $\mu$-$\tau$ symmetry breaking via the renormalization-group equations (RGEs) running from ${\mathrm{\Lambda }}_{\mu \tau }$ down to the electroweak scale ${\mathrm{\Lambda }}_{\text{EW}}\sim 10^2\mathrm{GeV}$. We find that the RGE-corrected value of ${\theta }_{23}$ is uniquely correlated with the neutrino mass ordering. In the minimal supersymmetric standard model, the best-fit results ${\theta }_{23}\simeq 42.4°$ reported by Capozzi et al. (or ${\theta }_{23}\simeq 48.9°$ reported by Forero et al.) at ${\mathrm{\Lambda }}_{\text{EW}}$ can arise from ${\theta }_{23}=45°$ at ${\mathrm{\Lambda }}_{\mu \tau }$ if the neutrino mass ordering is inverted (or normal). Accordingly, the preliminary best-fit results of $\delta$ at ${\mathrm{\Lambda }}_{\text{EW}}$ can also evolve from $\delta =270°$ at ${\mathrm{\Lambda }}_{\mu \tau }$ no matter whether the massive neutrinos are Dirac or Majorana particles.

Figure 1

The RGE-triggered $\mu$-$\tau$ symmetry breaking effects for Dirac neutrinos running from ${\mathrm{\Delta }}_i=0$ at ${\mathrm{\Lambda }}_{\mu \tau }\sim 10^{14}\mathrm{GeV}$ down to ${\mathrm{\Lambda }}_{\text{EW}}\sim 10^2\mathrm{GeV}$ in the MSSM with $\tan \beta =31$.

Figure 2

The RGE-triggered $\mu$-$\tau$ symmetry breaking effects for Majorana neutrinos running from ${\mathrm{\Delta }}_i=0$ at ${\mathrm{\Lambda }}_{\mu \tau }\sim 10^{14}\mathrm{GeV}$ down to ${\mathrm{\Lambda }}_{\text{EW}}\sim 10^2\mathrm{GeV}$ in the MSSM with $\tan \beta =30$.