Identifying a minimal flavor symmetry of the seesaw mechanism behind neutrino oscillations

In the canonical seesaw framework flavor mixing and CP violation in weak charged-current interactions of light and heavy Majorana neutrinos are correlated with each other and described respectively by the $3\times 3$ matrices $U$ and $R$. We show that the very possibility of $\big|U^{}_{\mu i}\big| = \big|U^{}_{\tau i}\big|$ (for $i = 1, 2, 3$), which is strongly indicated by current neutrino oscillation data, automatically leads to a novel prediction $\big|R^{}_{\mu i}\big| = \big|R^{}_{\tau i}\big|$ (for $i = 1, 2, 3$). We prove that behind these two sets of equalities and the experimental evidence for leptonic CP violation lies a minimal flavor symmetry -- the overall neutrino mass term keeps invariant when the left-handed neutrino fields transform as $\nu^{}_{e \rm L} \to (\nu^{}_{e \rm L})^c$, $\nu^{}_{\mu \rm L} \to (\nu^{}_{\tau \rm L})^c$, $\nu^{}_{\tau \rm L} \to (\nu^{}_{\mu \rm L})^c$ and the right-handed neutrino fields undergo an arbitrary unitary CP transformation. Such a generalized $\mu$-$\tau$ reflection symmetry may help constrain the flavor textures of active and sterile neutrinos to some extent in the seesaw mechanism.