content_image/313821.jpg content_image/313820.jpg content_image/313819.jpg content_image/313818.jpg content_image/313817.jpg
Integration variables may be rescaled, leading to the definition 

<img src='content_image/313821.jpg'>

which is consistent with the d -dimensional definition of the Jacobian. The integration oper- ation also respects translation invariance: 

<img src='content_image/313820.jpg'>

The normalization of the integrals is fixed through the Gaussian integral 

<img src='content_image/313819.jpg'>

An important property is that integrals that do not involve a mass or external momenta are identically zero, that is 

<img src='content_image/313818.jpg'>

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. 

## 2.2 Integrals with at most one massive propagator 

The most basic loop integral is the one-loop massive tadpole, which is given by 

<img src='content_image/313817.jpg'>

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 

