order as the estimate from F V + A 11 ( Q ). 

Adding or subtracting this amount of systematic effect from the lattice data, we repeat the whole analysis to estimate the systematic error due to the Lorentz (or WT) violating terms, which gives ± 0 . 0002 in α (5) s ( M$_{Z}$ ). 

D. Quark condensate and Z$_{m}$ 

The uncertainty due to the quark condensate is estimated as ± 0 . 0001 for α (5) s ( M$_{Z}$ ) by varying the input value from − [0 . 220 GeV] 3 to − [0 . 265 GeV] $^{3}$, which correspond to the lower and upper limits of the estimate of 〈 ¯ qq 〉 in [23]. 

The uncertainty due to Z$_{m}$ is also estimated as ± 0 . 0001 for α (5) s ( M$_{Z}$ ) by varying Z$_{m}$ within its estimated error (from 0.777 to 0.832). 

E. Perturbative expansion 

The truncation effect of the perturbative expansion can be estimated by comparing the results with different orders of the perturbative expansion. Fortunately, the four-loop cal- culation is known for C$_{0}$ ( Q $^{2}$, µ$^{2}$, α$_{s}$ ) [28, 29], and we can explicitly estimate the effect of O ( α 3 $_{s}$). 

A comparison of two-, three- and four-loop calculations of C$_{0}$ ( Q $^{2}$, µ$^{2}$, α$_{s}$ ) is shown in Figure 7. They correspond to O ( α$_{s}$ ), O ( α 2 $_{s}$), and O ( α 3 $_{s}$) calculations, respectively. We observe that the difference between three-loop and four-loop is of order of 0.0001 for Π$_{J}$( Q ), which is much smaller than other systematic effect. 

Strictly speaking, the smallness of the four-loop contribution does not guarantee that the unknown higher orders are even smaller. We therefore attempted to fit the data with a formula including unknown O ( α 4 $_{s}$) term c (4) 0 α 4 $_{s}$( Q ). The fit gives c (4) 0 ∼ O (10) with a shift of resulting α (5) s ( M$_{Z}$ ) by +0 . 0003. We therefore put a conservative systematic error from the truncation of the perturbation series as ± 0 . 0003. This can only be reduced by including the data at higher Q 2 values, which needs finer lattice spacings. 

