Koide's Z_3-symmetric parametrization, quark masses and mixings

Abstract

The sets of charged-lepton (L) and quark (D,U) masses may be parametrized in a Z_3-symmetric language appropriate for the discussion of Koide's formula. Experiment suggests that at the low-energy scale the relevant phase parameters \delta_f take on possibly exact values of \delta_L=3\delta_D/2=3\delta_U=2/9. For k_f (the other parameter relevant for the pattern of masses), a similarly simple expression (k_L=1) is known for charged leptons only. Using the Fritzsch-Xing decomposition of quark-mixing matrices, we show that the suggested pattern of low-energy quark masses is consistent with an earlier conjecture that k_{D,U} \approx 1 in the weak basis.