A Pythagoras-like theorem for CP violation in neutrino oscillations

It is well known that the golden appearance channels of neutrino oscillations are $\nu^{}_{\mu} \to \nu^{}_{e}$ and $\overline{\nu}^{}_{\mu} \to \overline{\nu}^{}_{e}$, and their probabilities in vacuum are determined by three CP-conserving flavor mixing factors ${\cal R}^{}_{ij} \equiv {\rm Re} (U^{}_{\mu i} U^{}_{e j} U^{*}_{\mu j} U^{*}_{e i})$ and the universal Jarlskog invariant of CP violation ${\cal J}^{}_{\nu} \equiv (-1)^{i+j} {\rm Im} (U^{}_{\mu i} U^{}_{e j} U^{*}_{\mu j} U^{*}_{e i})$ (for $i, j = 1, 2, 3$ and $i < j$) with $U$ being the $3 \times 3$ Pontecorvo-Maki-Nakagawa-Sakata matrix. We show that the magnitude of ${\cal J}^{}_{\nu}$ can be calculated from ${\cal J}^{2}_{\nu} = {\cal R}^{}_{12} {\cal R}^{}_{13} + {\cal R}^{}_{12} {\cal R}^{}_{23} + {\cal R}^{}_{13} {\cal R}^{}_{23}$ which holds as a natural consequence of the unitarity of $U$, and this Pythagoras-like relation provides a novel cross-check of the result of ${\cal J}^{}_{\nu}$ that will be directly measured in the next-generation $\nu^{}_{\mu} \to \nu^{}_{e}$ and $\overline{\nu}^{}_{\mu} \to \overline{\nu}^{}_{e}$ oscillation experiments. Terrestrial matter effects on ${\cal J}^{}_{\nu}$ and ${\cal R}^{}_{ij}$ are also discussed.