On the Koide-like Relations for the Running Masses of Charged Leptons,   Neutrinos and Quarks
  Current experimental data indicate that the Koide relation for the pole masses of charged leptons, which can be parametrized as $Q^{pole}_l = 2/3$, is valid up to the accuracy of $O(10^{-5})$. We show that the running masses of charged leptons fail in satisfying the Koide relation (i.e., $Q_l(\mu) \neq 2/3$), but the discrepancy between $Q_l (\mu)$ and $Q^{pole}_l$ is only about 0.2% at $\mu=M_Z$. The Koide-like relations for the running masses of neutrinos ($1/3 < Q_\nu(M_Z) < 0.6$), up-type quarks ($Q_{U}(M_Z) \sim 0.89$) and down-type quarks ($Q_{D}(M_Z) \sim 0.74$) are also examined from $M_Z$ up to the typical seesaw scale $M_R \sim 10^{14}$ GeV, and they are found to be nearly stable against radiative corrections. The approximate stability of $Q_{U}(\mu)$ and $Q_{D}(\mu)$ is mainly attributed to the strong mass hierarchy of quarks, while that of $Q_l(\mu)$ and $Q_\nu(\mu)$ is essentially for the reason that the lepton mass ratios are rather insensitive to radiative corrections. 
Zhi-zhong Xing, He Zhang