Koide's Z₃-symmetric parametrization, quark masses and mixings

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The sets of charged-lepton (L) and quark (D,U) masses may be parametrized in a Z₃-symmetric language appropriate for the discussion of Koide's formula. Experiment suggests that at the low-energy scale the relevant phase parameters δf take on possibly exact values of δₗ=3δD/2=3δᵤ=2/9. For kf (the other parameter relevant for the pattern of masses), a similarly simple expression (kₗ=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 kD,ᵤ ≈ 1 in the weak basis.