content_image/323915.jpg content_image/323914.jpg content_image/323913.jpg
function (B.9), this integral gives 

<img src='content_image/323915.jpg'>

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 

<img src='content_image/323914.jpg'>

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 

<img src='content_image/323913.jpg'>

