0 0.2 0.4 MSbar $^{(3)}$ (GeV) -0.02 -0.01 0 < $_{s}$/ GG> (lat. unit) 0.3 0.4 0.5 (aQ)$^{2}$$_{min}$,(aQ)$^{2}$$_{max}$=0.99381 0.007 0.008 c FIG. 3: Dependence of the fit parameters on the lower limit of the fit range. The maximum value is fixed at ( aQ ) 2 ≃ 0 . 994. Open and filled symbols show the results with and without the 1 /Q 4 terms in (14). (Thus, there is no filled symbol in the middle plot.) Strictly speaking, χ $^{2}$/ dof does not have a statistical meaning as the correlation among the data at difference Q 2 is ignored in the fit used here. We discuss on the statistical correlations among the data points in the next section. The limitation of the OPE formula including up to 1 /Q 4 terms can be investigated by looking at its departure from the lattice data at lower values of Q $^{2}$. In Figure 4 we plot the difference of the lattice data from the fit curve including the 1 /Q 4 terms. The Q 2 region in this plot is extended towards the value below ( aQ ) 2 $_{min}$. From this plot, we observe that the next order 1 /Q 6 contribution becomes significant below ( aQ ) 2 ≃ 0.4. We therefore set ( aQ ) 2 min = 0 . 463 in our analysis including the 1 /Q 4 terms. After doing a simultaneous fit of the VPF data at all sea quark masses, the QCD parame- ter is obtained as Λ (3) MS = 0.247(5) GeV. By matching onto four- and then to five-flavor QCD at charm and bottom quark masses respectively, the strong coupling constant is obtained as 11