Sources Estimated error in α (5) s ( M$_{z}$ ) Uncorrelated fit ± 0.0003 Lattice artifact ( O ( a $^{2}$) effect) +0.0003 Δ V + A µν ± 0.0002 Quark condensate ± 0.0001 Z$_{m}$ ± 0.0001 Perturbative expansion ± 0.0003 1 /Q 2 expansion < 0.0001 m$_{c,b}$ +0 . 0001 − 0 . 0003 Lattice spacing +0 . 0013 − 0 . 0010 Total (in quadrature) +0 . 0014 − 0 . 0012 TABLE I: Summary of systematic errors in α (5) s ( M$_{Z}$ ). V. CONCLUSIONS Determination of the strong coupling constant α$_{s}$ ( µ ) may be achieved through a pertur- bative expansion of any physical quantity in terms of α$_{s}$ ( µ ) at a given scale µ . Experimental determination typically uses a perturbative amplitude of quarks at high energy. Comparison with the lattice QCD calculation provides a highly non-trivial test of QCD, as lattice uses 19 This result is consistent with other recent lattice QCD results 0.1174(12) [9], 0.1183(8) [31], 0.1192(11) [32], 0.1205(8)(5)( + 0 − $_{17}$)[33], and with the world average 0.1184(7) including various high-energy experiments [1, 2] (updated on-line in 2010). Here, the first error is statistical error and the second is a sum of the various systematic errors in quadrature. Table I shows a summary of our estimate of the systematic errors in our determination of α$_{s}$ ( M$_{Z}$ ). We quote final result of the strong coupling constant at the Z boson mass scale as α (5) s ( M$_{Z}$ ) = 0 . 1181(3)( +14 − $_{12}$) . (17) I. Final result