Towards identifying the minimal flavor symmetry behind neutrino oscillations

Current neutrino oscillation data indicate that the $3\times 3$ Pontecorvo-Maki-Nakagawa-Sakata matrix $U$ exhibits a $\mu$-$\tau$ flavor interchange symmetry $\mathrm{\mid }{U}_{\mu i}^{}\mathrm{\mid }=\mathrm{\mid }{U}_{\tau i}^{}\mathrm{\mid }$ (for $i=1,2,3$) as a good approximation. In particular, the T2K measurement implies that the maximal neutrino mixing angle ${\theta }_{23}^{}$ and the CP-violating phase $\delta$ should be close to $\pi \mathrm{/}4$ and $-\pi \mathrm{/}2$, respectively. Behind these observations lies a minimal flavor symmetry – - the effective Majorana neutrino mass term keeps invariant under the transformations ${\nu }_{e\mathrm{L}}^{}\to ({\nu }_{e\mathrm{L}}^{}{)}^c$, ${\nu }_{\mu \mathrm{L}}^{}\to ({\nu }_{\tau \mathrm{L}}^{}{)}^c$ and ${\nu }_{\tau \mathrm{L}}^{}\to ({\nu }_{\mu \mathrm{L}}^{}{)}^c$. Extending this flavor symmetry to the canonical seesaw mechanism, we find that the $R$-matrix describing the strength of weak charged-current interactions of heavy Majorana neutrinos satisfies $\mathrm{\mid }{R}_{\mu i}^{}\mathrm{\mid }=\mathrm{\mid }{R}_{\tau i}^{}\mathrm{\mid }$ as a consequence of $\mathrm{\mid }{U}_{\mu i}^{}\mathrm{\mid }=\mathrm{\mid }{U}_{\tau i}^{}\mathrm{\mid }$. This result can be used to set a new upper bound, which is about three orders of magnitude more stringent than before, on the flavor mixing factor associated with the charged-lepton-flavor-violating decay mode ${\tau }^{-}\to e^{-}+\gamma$.