A Pythagoras-like theorem for CP violation in neutrino oscillations

The probabilities of nu mu -> nu e and nu mu -> nu e oscillations in vacuum are determined by the CP-conserving flavor mixing factors Rij equivalent to Re(U mu iUejU*mu jUe*i) and the universal Jarlskog invariant of CP violation J nu equivalent to(-1)i+j Im(U mu iUejU*mu jUe*i) (for i, j = 1, 2, 3 and i < j), where U is the 3 x 3 Pontecorvo-MakiNakagawa-Sakata neutrino mixing matrix. We show that J nu 2=R12R13 + R12R23 +R13R23 holds as a natural consequence of the unitarity of U. This Pythagoras-like relation may provide a novel crosscheck of the result of 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 J nu and Rij are also discussed.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/). Funded by SCOAP(3).