and consequently $^{[}$d $^{d}$x ] = − d . By a similar argument it can be shown that This can be used to determine the dimensions of the fields and parameters appearing in the QCD Lagrangian (1.40). For instance, from the quark term we can show that because the Dirac Gamma matrix is dimensionless. From the gluon term we find that The units of the coupling can be determined from the quark-gluon interaction term: because the Gell-Mann matrix is dimensionless. The renormalization scale has [ µ ] = 1, therefore and hence α is an appropriate expansion parameter in d -dimensions. 106 [ α ] = $^{[}$g $^{2}$µ d − 4 ] = 0 , (A.12) [ g ¯ Q λ a 2 γ $^{µ}$Aa $_{µ}$Q$^{]}$ = d , → [ g ] = 4 − d 2 , (A.11) $^{[}$G µν a G a µν ] = d → $^{[}$G a µν ] = $^{[}$∂$_{µ}$ A a µ ] = d 2 → $^{[}$A a µ ] = d − 2 2 . (A.10) [ ¯ Qγ $^{µ}$∂$_{µ}$ Q ] = d → [ ¯ Q ] = [ Q ] = d − 1 2 , (A.9) exp$^{[}$ i$^{∫}$ d $^{d}$x L$^{]}$ → [∫ d $^{d}$x L$^{]}$ = 0 → [ L ] = d . (A.8)