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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 $\big|U^{}_{μi}\big| = \big|U^{}_{τi}\big|$ (for $i = 1, 2, 3$), which is strongly indicated by current neutrino oscillation data, automatically leads to a novel prediction $\big|R^{}_{μi}\big| = \big|R^{}_{τi}\big|$ (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 $ν^{}_{e \rm L} \to (ν^{}_{e \rm L})^c$, $ν^{}_{μ\rm L} \to (ν^{}_{τ\rm L})^c$, $ν^{}_{τ\rm L} \to (ν^{}_{μ\rm L})^c$ and the right-handed neutrino fields undergo an arbitrary unitary CP transformation. Such a generalized $μ$-$τ$ reflection symmetry may help constrain the flavor textures of active and sterile neutrinos to some extent in the seesaw mechanism.