Energy scale independence of Koide's relation for quark and lepton masses

Abstract

Koide's mass relation of charged leptons has been extended to quarks and neutrinos, and we prove here that this relation is independent of energy scale in a huge energy range from $1 {GeV}$ to $2\times10^{16} {GeV}$. By using the parameters $k_u$, $k_d$ and $k_{\nu}$ to describe the deviations of quarks and neutrinos from the exact Koide's relation, we also check the quark-lepton complementarity of masses such as $k_{l}+k_{d} \approx k_{\nu}+k_{u} \approx 2$, and show that it is also independent (or insensitive) of energy scale.