Figure 2.6: Feynman diagram representing the F -type two-loop master integral. integrals. Specifically, loop integrals satisfy the relation which is proven in Ref. [39]. This identity is commonly referred to as integration by parts. The identity generalizes to multi-loop integrals, the restriction being that v$_{µ}$ must be one of the loop momenta. The technique was developed in Ref. [35] and used to calculate the renormalization group functions in scalar φ 4 theory to four-loop order. In order to illustrate the technique, we will use it to calculate the integral (2.36). Using the identity (2.37), we have where the derivatives can be calculated using the result Explicitly calculating the derivatives, we arrive at the following recurrence relation For convenience we have defined the operators N + and N − that increase and decrease the index n$_{i}$ by one unit when acting on the integral (2.36). Essentially, equation (2.40) is a recurrence relation for integrals of the form (2.36). For instance, consider an integral of the 64 ( d − n$_{1}$ − n$_{3}$ − 2 n$_{5}$ ) F ( d ; n$_{1}$ , m ; n$_{2}$ , m ; n$_{3}$ , 0 ; n$_{4}$ , 0 ; n$_{5}$ , 0) = $^{[}$n$_{1}$ $^{(}$5 − − 2 − $^{)}$1 + + n$_{3}$ $^{(}$5 − − 4 − $^{)}$3 + $^{]}$F ( d ; n$_{1}$ , m ; n$_{2}$ , m ; n$_{3}$ , 0 ; n$_{4}$ , 0 ; n$_{5}$ , 0) . (2.40) ∂ ∂k µ 1 k 2 n = − 2 nk µ k 2( n +1) . (2.39) ∫ d $^{d}$k$_{1}$ (2 π ) d ∫ d $^{d}$k$_{2}$ (2 π ) d ∂ ∂k µ 1 [ ( k$_{1}$ − k$_{2}$ ) µ ( k 2 1 − m $^{2}$) n$_{1}$ ( k 2 − m $^{2}$) n$_{2}$ ( k$_{1}$ − q ) 2 n$_{3}$ ( k$_{2}$ − q ) 2 n$_{4}$ ( k$_{1}$ − k$_{2}$ ) 2 n$_{5}$ ] = 0 , (2.38) 1 µ d − 4 ∫ d $^{d}$k (2 π ) d ∂ ∂v$_{µ}$ [ w$_{µ}$f ( k , q , m )] = 0 , v$_{µ}$ = k$_{µ}$ , w$_{µ}$ ∈ $^{{}$k$_{µ}$ , ( k ± q )$_{µ}$ } , (2.37)