A Pythagoras-like theorem for CP violation in neutrino oscillations

Abstract

The probabilities of $\hat{I}{½}_{\hat{I}}^{}¼\to \hat{I}{½}_e^{}$ and $\overline{\hat{I}}{½}_{\hat{I}}^{}¼\to \overline{\hat{I}}{½}_e^{}$ oscillations in vacuum are determined by the CP-conserving flavor mixing factors ${\mathcal{R}}_{ij}^{}\equiv \mathrm{Re}(U_{\hat{I}¼i}^{}{U}_{ej}^{}{U}_{\hat{I}¼j}^{\ast }{U}_{ei}^{\ast })$ and the universal Jarlskog invariant of CP violation ${\mathcal{J}}_{\hat{I}}^{}½\equiv (-1{)}^{i+j}\mathrm{Im}(U_{\hat{I}¼i}^{}{U}_{ej}^{}{U}_{\hat{I}¼j}^{\ast }{U}_{ei}^{\ast })$ (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 ${\mathcal{J}}_{\hat{I}}^2½={\mathcal{R}}_{12}^{}{\mathcal{R}}_{13}^{}+{\mathcal{R}}_{12}^{}{\mathcal{R}}_{23}^{}+{\mathcal{R}}_{13}^{}{\mathcal{R}}_{23}^{}$ holds as a natural consequence of the unitarity of $U$. This Pythagoras-like relation may provide a novel cross-check of the result of ${\mathcal{J}}_{\hat{I}}^{}½$ that will be directly measured in the next-generation long-baseline neutrino oscillation experiments. Indirect non-unitarity effects and terrestrial matter effects on ${\mathcal{J}}_{\hat{I}}^{}½$ and ${\mathcal{R}}_{ij}^{}$ are also discussed.

14 pages, 6 figures. More discussions and references added, accepted for publication in Phys. Lett. B