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 $2\times {10}^{16}\mathrm{GeV}$. 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.

• Received 7 September 2005

DOI:https://doi.org/10.1103/PhysRevD.73.013009