content_image/312061.jpg content_image/312058.jpg content_image/312059.jpg content_image/312060.jpg
invariant nature of loop integrals. A simple example is as follows: 

<img src='content_image/312061.jpg'>

Due to Lorentz invariance, the integral must be proportional to the external momentum q µ and a function of the Lorentz scalar q $^{2}$. Contracting both sides of (2.42) with q$_{µ}$ , the function f ( q $^{2}$) can be determined and hence the integral on the left hand side of (2.42) can be calculated. This technique can be easily generalized to accommodate more complicated tensor structures. Ref. [98] constructs an algorithm to solve this problem in general. However, we will use a far more powerful method to deal with tensor integrals such as (2.42). 

The method that we will utilize was developed in Ref. [117] and relates tensor integrals in d -dimensions to scalar integrals in d + 2 N -dimensions. In Ref. [118] the method was applied to two-point propagator-type integrals, that is, integrals of the type A , B , J , V and F that we have considered so far. This method can be used to find a truly minimal set of basis integrals for the family of two-loop two-point integrals that we have studied so far. The key idea of the method is that it is possible to express an arbitrary tensor integral as 

<img src='content_image/312058.jpg'>

<img src='content_image/312059.jpg'>

<img src='content_image/312060.jpg'>

where we have omitted the renormalization scale for brevity. The tensor operator T µ$_{1}$...µ$_{r}$λ$_{1}$...λ$_{s}$ is a function of the external momentum q $^{µ}$, derivatives with respect to each mass m$_{i}$ , and an operator d + that increases the dimension of any given loop integral by two units, i.e. d $^{+}$I( d ) = I ( d $^{+2)}$. We will now construct the explicit form of this operator. We will follow the derivation given in Ref. [118], although we will retain the loop integral normalization convention that has been used in this chapter. Introducing the auxiliary vectors a$_{1}$ and a$_{2}$ , 

