Integration variables may be rescaled, leading to the definition which is consistent with the d -dimensional definition of the Jacobian. The integration oper- ation also respects translation invariance: The normalization of the integrals is fixed through the Gaussian integral An important property is that integrals that do not involve a mass or external momenta are identically zero, that is for all values of n . This result is proven in Ref. [39]. Integrals such as (2.5) that do not depend on any external scales are called massless tadpoles. The most basic loop integral is the one-loop massive tadpole, which is given by where µ denotes the renormalization scale. A Feynman diagram representing this loop in- tegral is shown in Fig. 2.1. The momentum of the particle with mass m that circulates the loop is k , as such k is called the loop momentum. There is a correspondence between the topology of a Feynman diagram and the structure of the loop integral that it represents. In particular, the flow of momenta through a diagram is closely related to the form of the 53 A ( d ; n , m ) = lim η → 0 + 1 µ d − 4 ∫ d $^{d}$k (2 π ) d 1 [ k 2 − m 2 + iη ] n . (2.6) ∫ d $^{d}$k k 2 n = 0 , (2.5) ∫ d $^{d}$k exp$^{[}$ − k 2 a 2 ] = a $^{d}$π d 2 , a 2 > 0 . (2.4) ∫ d $^{d}$kf ( k + q ) = ∫ d $^{d}$kf ( k ) . (2.3) ∫ d $^{d}$kf ( ak ) = a − d ∫ d $^{d}$kf ( k ) , (2.2) 2.2 Integrals with at most one massive propagator