Equations (1) and (5) work well for lepton pole masses (and not for running masses) to such a high degree of accuracy that it seems appropriate to consider the counterparts of these masses in the quark sector as well. This suggests that the pole masses should be used for the heavy (*b*, *c*, *t*) quarks and the low-energy scale masses should be utilised for the light (*u*, *d*, *s*) quarks.

Furthermore, we observe that Eqs (1) and (5) depend on mass ratios (*z*_{1} = *m*_{1}/*m*_{3}, *z*_{2} = *m*_{2}/*m*_{3}) alone. In order to study the quark sector in detail it is therefore sufficient to estimate the relevant mass ratios in the light and heavy sectors as well as the relative mass scale of the two sectors. Now, the relative mass ratios can be estimated independently in the light and heavy sectors.

In the light quark sector the results of lattice calculations (supplemented with the phenomenological studies of isospin breaking effects) give (in the *MS* scheme at the renormalization scale *μ* = 2 GeV) [10]:

$$\begin{array}{}{\displaystyle {m}_{u}/{m}_{d}\approx 0.46(5)\phantom{\rule{1em}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{1em}{0ex}}{m}_{s}/{m}_{d}\approx 19.9(2).}\end{array}$$(8)

These numbers agree very well with Weinberg’s old low-energy estimates based on the ratios of pion and kaon masses [11]:

$$\begin{array}{}{\displaystyle {m}_{u}/{m}_{d}\approx 0.56,\phantom{\rule{1em}{0ex}}{m}_{s}/{m}_{d}\approx \mathrm{20.2.}}\end{array}$$(9)

We may therefore safely assume that *m*_{u}/*m*_{d} = 0.50 ± 0.05 and that *m*_{s}/*m*_{d} = 20.0.

In the sector of heavy quarks the perturbatively calculated pole masses *m*_{q} are related to the running *m*_{q}(*μ*) masses via [10]:

$$\begin{array}{}{\displaystyle {m}_{q}={\overline{m}}_{q}({\overline{m}}_{q})\left[1+\frac{4{\overline{\alpha}}_{s}({\overline{m}}_{q})}{3\pi}+\dots \right],}\end{array}$$(10)

where the dots symbolize substantial higher order corrections [10]. As these corrections are quite uncertain we treat the relevant independent ratios of heavy quark pole masses (*i*.*e*. *m*_{c}/*m*_{t}, *m*_{b}/*m*_{t}) in two ways.

First, we approximate these ratios by *m*_{c}(*m*_{c})/*m*_{t}(*m*_{t}) and *m*_{b}(*m*_{b})/*m*_{t}(*m*_{t}), where *m*_{q}(*m*_{q}) are the central values of the estimates given in [12] (in GeV):

$$\begin{array}{}{\displaystyle {m}_{c}({m}_{c})={1.29}_{-0.11}^{+0.05},}\\ {\displaystyle {m}_{b}({m}_{b})={4.19}_{-0.16}^{+0.18}\phantom{\rule{1em}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{1em}{0ex}}{m}_{t}({m}_{t})={163.3}_{-1.}^{+1.1}.}\end{array}$$(11)

With *m*_{c}(*m*_{c})/*m*_{t}(*m*_{t}) and *m*_{b}(*m*_{b})/*m*_{t}(*m*_{t}) being pure numbers and *m*_{t}, at around 160 – 170 GeV, setting the absolute scale of heavy quark masses, the relative scale of the light and heavy quark masses may be parametrized by *m*_{s}. The choice of *m*_{s} fixes then the values of the four mass ratios *z*_{1} and *z*_{2} (in the up and down quark sectors). In a previous paper [4] it was argued that the pattern of phases *δ*_{f} is particularly simple for *m*_{s} ≈ 160 *MeV* (*i*.*e*. not for the value of *m*_{s} of the order of 90 – 100 *MeV* which is appropriate for *μ* = 2 *GeV*). It is therefore interesting to see how the values of *λ*_{D} and *λ*_{U} change when *m*_{s} varies from 90 to 160 *MeV* or so. The upper part of shows the relevant *m*_{s}-dependence of *λ*_{D} and *λ*_{U} obtained using (in the estimates of *m*_{c}/*m*_{t}, *m*_{s}/*m*_{b}, etc.) the values given in Eq. (11).

Table 1 Values of *λ*_{D} and *λ*_{U}

Second, instead of employing Eq. (11), we directly use the estimates of pole masses given in [12] (*i*.*e*. (*m*_{c},*m*_{b},*m*_{t}) = (1.84, 4.92, 172.9)) (in GeV). The corresponding variations of *λ*_{D} and *λ*_{U} are presented in the lower part of . We view the differences between the upper and lower parts of as providing an estimate of the errors involved.

We observe that for *m*_{s} ≈ 160 – 170 MeV the values of *λ*_{U,D} in the quark sector are not far from *λ*_{L} = 0 and *δ*_{L} = 2/9. Deviations from these values may be tentatively assigned (as in the case of charged leptons) to some higher order corrections (which, on account of strong interactions being involved, could be larger than in the lepton case). Note that the regularities in question (for *λ*_{U,D} here and for *δ*_{U,D} in [4]) are observed at approximately the same value of *m*_{s} ≈ 160 MeV.

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