pole prescription in the propagators in Eq. (1.41). In addition, we have ignored the leftmost and rightmost quark propagators with momentum q in Fig. 1.5. It is customary to remove (or amputate) external propagators in Feynman diagrams. The integral in Eq. (1.41) can be evaluated in spherical coordinates [100]. However, the result is surprising: the integral diverges at large values of the gluon momentum k . Integrals that diverge in this way are called ultraviolet divergent. In order to extract meaningful physical information from the integral (1.41), the ultravi- olet divergence must first be brought under control, or regulated. In order to do this, we will utilize dimensional regularization [115, 25]. With this method, integrals in four dimensional Minkowski space are reinterpreted as integrals in d -dimensions. For example, the integral above is reinterpreted as In dimensional regularization the number of dimensions, d , is best thought of as a parameter that can be adjusted such that the integral (1.42) converges. Integrals that are formally divergent in a certain number of dimensions can be uniquely defined through analytic con- tinuation in the parameter d . In four dimensions, the coupling g is dimensionless and hence is suitable to be used as an expansion parameter. However, in d -dimensions the combination g $^{2}$µ d − 4 is dimensionless, where µ is the renormalization scale. This can be used to define the d -dimensional expansion parameter The remaining factor of the renormalization scale in the denominator of Eq. (1.42) ensures that the d -dimensional integral has the same dimensions as the original four dimensional integral (1.41). The integral (1.42) can be evaluated in d -dimensions using the methods de- scribed in Chapter 2. The result naturally depends on d , and we may examine the behaviour of the integral near four dimensions by setting d = 4 + 2 ϵ and expanding around ϵ = 0. The methods used to perform this expansion are discussed in Section 2.6. For the integral (1.42), 24 α = g $^{2}$µ d − 4 4 π . (1.43) Σ ( q ) ≃ $^{(}$g $^{2}$µ d − 4 ) 1 µ d − 4 ∫ d $^{d}$k (2 π ) d 1 k $^{2}$[ ( k + q ) 2 − m 2 ] . (1.42)