A Pythagoras-like theorem for CP violation in neutrino oscillations

The probabilities of ${\nu }_{\mu }^{}\to {\nu }_e^{}$ and ${\bar{\nu }}_{\mu }^{}\to {\bar{\nu }}_e^{}$ oscillations in vacuum are determined by the CP-conserving flavor mixing factors ${\mathcal{R}}_{ij}^{}\equiv \mathrm{R}\mathrm{e}(U_{\mu i}^{}{U}_{ej}^{}{U}_{\mu j}^{\ast }{U}_{ei}^{\ast })$ and the universal Jarlskog invariant of CP violation ${\mathcal{J}}_{\nu }^{}\equiv (-1{)}^{i+j}\mathrm{I}\mathrm{m}(U_{\mu i}^{}{U}_{ej}^{}{U}_{\mu j}^{\ast }{U}_{ei}^{\ast })$ (for $i,j=1,2,3$ and $i<j$), where $U$ is the $3\times 3$ Pontecorvo-Maki-Nakagawa-Sakata neutrino mixing matrix. We show that ${\mathcal{J}}_{\nu }^2={\mathcal{R}}_{12}^{}{\mathcal{R}}_{13}^{}+{\mathcal{R}}_{12}^{}{\mathcal{R}}_{23}^{}+{\mathcal{R}}_{13}^{}{\mathcal{R}}_{23}^{}$ holds as a natural consequence of the unitarity of $U$. This Pythagoras-like relation may provide a novel cross-check of the result of ${\mathcal{J}}_{\nu }^{}$ that will be directly measured in the next-generation long-baseline neutrino oscillation experiments. Indirect non-unitarity effects and terrestrial matter effects on ${\mathcal{J}}_{\nu }^{}$ and ${\mathcal{R}}_{ij}^{}$ are also discussed.