function (B.9), this integral gives Notice that the integral is divergent, and the divergence is parametrized as a simple pole at ϵ = 0. Ultimately, this divergence can be traced back to the radial integral (2.10). For ℓ ≫ m , the integral goes like ℓ d − 2 $^{n}$, and hence is divergent for d > 2 n . When this is the case, the radial integral is ill-defined. However, the expression in terms of the Gamma function (2.13) uniquely defines the loop integral when d > 2 n . The divergence that arises in (2.57) can be canceled through renormalization, which was discussed in Chapter 1. Recall that in the MS scheme the renormalization constants are defined so that γ$_{E}$ and log (4 π ) terms are canceled in addition to poles at ϵ = 0. A convenient way of partially implementing this to rescale µ 2 → µ $^{2}$e − $^{γ$_{E}$}$/ 4 π . Doing this, all one-loop integrals will contain a factor of where M 2 is the external scale, i.e. the mass m 2 or the external momentum − q $^{2}$. The benefit of this replacement is that it automatically cancels all factors of γ$_{E}$ and log (4 π ) that would otherwise emerge when performing the epsilon expansion. This replacement easily generalizes to multi-loop integrals: an n -loop integral would have n factors of (2.58). All calculations in Chapters 3, 4, 5 and 6 use the MS renormalization scheme, therefore all loop integrals encountered there implicitly include (2.58). Now let us consider a more complicated example. For example, consider the integral 70 F ( d ; 1 , 0 ; 1 , 0 ; 1 , 0 ; 1 , 0 ; 1 , 0) = 1 µ 2( d − 4) ∫ d $^{d}$k$_{1}$ (2 π ) d ∫ d $^{d}$k$_{2}$ (2 π ) d 1 k 2 1 k 2 ( k$_{1}$ − q ) $^{2}$( k$_{2}$ − q ) $^{2}$( k$_{1}$ − k$_{2}$ ) 2 , (2.59) [ M 2 4 πµ 2 ] d − 4 2 ≡ exp$^{(}$ d − 4 2 [ log$^{[}$ M 2 µ 2 ] − γ$_{E}$$^{])}$ , (2.58) A [4 + 2 ϵ ; 1] = − im 2 (4 π ) 2 [ m 2 4 πµ 2 ] ϵ Γ ( − 1 − 2 ϵ ) = − im 2 (4 π ) 2 exp$^{(}$ ϵ log$^{[}$ m 2 4 πµ 2 $^{]) (}$1 ϵ + γ$_{E}$ − 1 + O ( ϵ )$^{)}$ = − im 2 (4 π ) 2 $^{[}$1 ϵ + γ$_{E}$ − log (4 π ) − 1 + log$^{[}$ m 2 µ 2 ]] . (2.57)