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

TL;DR

This paper extends the Naumov and Toshev relations, originally for matter effects in neutrino oscillations, to the renormalization-group evolution of quarks and Dirac neutrinos, revealing similar invariant relations under different energy scale approximations.

Contribution

It demonstrates that analogous relations to Naumov and Toshev exist for Dirac neutrinos, charged leptons, and quarks under RG evolution with specific dominance approximations and parametrizations.

Findings

Derived relations for Dirac neutrinos and charged leptons under RG evolution.

Established similar relations for quarks in top-dominance approximation.

Showed these relations depend on the chosen parametrization.

Abstract

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