Remark on Koide’s $Z_3$-symmetric parametrization of quark masses

The charged lepton masses may be parametrized in a $Z_3$-symmetric language appropriate to the discussions of Koide’s formula. The phase parameter ${\delta }_L$ appearing in this parametrization is experimentally indistinguishable from $2/9$. We analyze Koide’s parametrization for the up ($U$) and down ($D$) quarks and argue that the data are suggestive of the low-energy values ${\delta }_U={\delta }_L/3=2/27$ and ${\delta }_D=2{\delta }_L/3=4/27$.

Figure 1

Contour plot of ${\delta }_f(z_1,z_2)$ and the relevant points corresponding to the charged lepton ($L$), up ($U$), and down ($D$) quark sectors, as explained in the text.