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In an analytical way of studying matter effects on neutrino oscillations, theNaumov and Toshev relations have been derived to respectively link the Jarlskoginvariant of CP violation and the Dirac phase in the standard parametrizationof the $3\times 3$ flavor mixing matrix to their matter-corrected counterparts.Here we show that there exist similar relations for Dirac neutrinos and chargedleptons evolving with energy scales via the one-loop renormalization-group (RG)equations in the tau-dominance approximation, and for the running behaviors ofup- and down-type quarks in the top-dominance approximation, provided adifferent parametrization is taken into account.

Naumov- and Toshev-like relations in the renormalization-group evolution of quarks and Dirac neutrinos