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The perturbative result may be parametrized as Π$_{V}$$_{+}$$_{A}$( Q $^{2}$)latt . pert = c − 1 / (2 π $^{2}$) ln( aQ ) $^{2}$+ 0 . 0062(40)( aQ ) 2 for small a . The logarithmic term is the same as in the continuum per- turbation theory and c is the scheme dependent constant as already noticed. The term +0 . 0062(40)( aQ ) 2 comes from the discretization effect. The error includes a fluctuation of numerical integral as well as the non-smooth behavior due to higher order discretization effects. The non-smooth behavior appears because of different assignments of momentum components aQ$_{µ}$ . In the plots we took several values of ( aQ ) 2 (and so aQ$_{µ}$ ) that also ap- pear in the lattice calculation. We observe that the pattern of non-smooth behavior in the one-loop calculation actually well reproduces that occurring in the numerical simula- tion. It suggests that our estimate of the discretization effect is reasonably realistic. As the plot shows, this error band is taken so that the result of the lattice perturbation theory calculation is covered. 

By subtracting this estimate of the O (( aQ ) $^{2}$) effect from the lattice data, the final result for α $^{(5)}$( M$_{Z}$ ) changes by +0 . 0002(1). We therefore take +0 . 0003 as our estimate of the systematic error from this source, without changing the central value to be conservative. The estimated error in the negative direction is thus taken to be zero. Although the perturbative calculation is done only at the one-loop level, we expect that the higher order effects are 

