0.4 0.5 0.6 0.7 0.8 0.9 1 (aQ) 2 0 0.005 0.01 2-loop 3-loop 4-loop C$_{0}$(Q$^{2}$, $^{2}$, $_{s}$)-c+1/(2 $^{2}$)ln(aQ)2 FIG. 7: C$_{0}$ ( Q $^{2}$, µ$^{2}$, α$_{s}$ ) − c + 1 / (2 π $^{2}$) ln( aQ ) 2 as a function of ( aQ ) $^{2}$. The perturbative results at two-loop (solid), three-loop (dashed), and four-loop (dotted) calculations are shown. The logarithm at the leading order is subtracted in order to enhance their small differences. As previously discussed, the size of neglected 1 /Q 6 terms in the OPE formula is at most 0.001 for Π$_{J}$( Q ) at the lower end of the fitting range ( aQ ) 2 min = 0 . 463 (see Figure 4). This is less than 1/5 of the estimated discretization effect discussed above. We therefore expect that the impact on α (5) s ( M$_{Z}$ ) is smaller than 0.0001. The uncertainty of charm and bottom quark masses, m$_{c}$ and m$_{b}$ used in a perturbative matching of α (3) s onto α (5) s is +0 . 0001 and − 0 . 0003, which are the maximum and mini- mum values when m$_{b,c}$ are changed within 1 σ in the analysis. The input values ¯ m$_{c}$ ( ¯ m$_{c}$ ) = 1.27( +07 − $_{11}$) GeV and ¯ m$_{b}$ ( ¯ m$_{b}$ ) = 4.20( +17 − $_{07}$) GeV are taken from [2]. The uncertainty of the lattice spacing is the largest source of the systematic error. 17 H. Lattice spacing G. Charm and bottom quark mass F. 1 /Q 2 expansion