Volume 449 -
The European Physical Society Conference on High Energy Physics (EPS-HEP2023) -
T08 Flavour Physics and CP Violation

A Pythagoras-like theorem for the Jarlskog invariant of CP violation

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Pre-published on:
January 11, 2024

Published on:
March 21, 2024

Abstract

The $\nu^{}_\mu \to \nu^{}_e$ oscillation probability is fully determined by the CP-conserving quantities ${\cal R}^{}_{ij} \equiv {\rm Re} (U^{}_{\mu i} U^{}_{e j} U^*_{\mu j} U^*_{e i})$ and the Jarlskog invariant of CP violation ${\cal J}^{}_\nu \equiv (-1)^{i+j} \hspace{0.05cm} {\rm Im} (U^{}_{\mu i} U^{}_{e j} U^*_{\mu j} U^*_{e i})$ (for $i, j = 1, 2, 3$ and $i < j$), where $U$ is the unitary PMNS lepton flavor mixing matrix. We find that a Pythagoras-like relation ${\cal J}^{2}_\nu = {\cal R}^{}_{12} {\cal R}^{}_{13} + {\cal R}^{}_{12} {\cal R}^{}_{23} + {\cal R}^{}_{13} {\cal R}^{}_{23}$ holds, and it may hopefully offer a novel cross-check of the result of ${\cal J}^{}_\nu$ that will be directly measured in the next-generation long-baseline neutrino oscillation experiments. Terrestrial matter effects are briefly discussed.

DOI: https://doi.org/10.22323/1.449.0329

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