This integral can also be evaluated using Mellin-Barnes techniques. The result is Note that this result agrees with results given for this integral in Ref. [29]. This is the most complicated integral that we will evaluate in this chapter. In the next section we will consider a technique that can be used to construct recurrence relations among loop integrals. Using this approach, all integrals encountered in Chapters 3, 4, 5 and 6 can be evaluated in terms of the integrals given so far in this chapter. Consider the two-loop integral This integral is represented by the Feynman diagram in Fig. 2.6. Note that this is the most complex loop integral that can occur when calculating two-point functions at two-loop level, because at most five independent propagators can be constructed from the two loop momenta k$_{1}$ , k$_{2}$ , and the external momentum q . For this reason, the integral (2.36) is occasionally referred to as the “master” two-loop integral for two-point functions [29]. This integral cannot be evaluated using the methods discussed so far. However, it can be evaluated by appealing to one of the properties of dimensionally regularized momentum 63 F ( d ; n$_{1}$ , m ; n$_{2}$ , m ; n$_{3}$ , 0 ; n$_{4}$ , 0 ; n$_{5}$ , 0) = 1 µ 2( d − 4) ∫ d $^{d}$k$_{1}$ (2 π ) d ∫ d $^{d}$k$_{2}$ (2 π ) d 1 ( 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$_{5}$ . (2.36) J ( d ; n$_{1}$ , m ; n$_{2}$ , m ; n$_{3}$ , 0) = − 1 (4 π ) 4 $^{(}$− m 2 ) 4 − n$_{1}$ − n$_{2}$ − n$_{3}$ [ m 2 4 πµ 2 ] d − 4 Γ ( d 2 − n$_{3}$ $^{)}$Γ $^{(}$n$_{2}$ + n$_{3}$ − d 2 $^{)}$Γ $^{(}$n$_{1}$ + n$_{3}$ − d 2 $^{)}$Γ ( n$_{1}$ + n$_{2}$ + n$_{3}$ − d ) Γ ( d 2 $^{)}$Γ ( n$_{1}$ ) Γ ( n$_{2}$ ) Γ ( n$_{1}$ + n$_{2}$ + 2 n$_{3}$ − d ) $_{4}$F$_{3}$   n$_{1}$ + n$_{2}$ + n$_{3}$ − d , n$_{2}$ + n$_{3}$ − d 2 , n$_{1}$ + n$_{3}$ − d 2 , n$_{3}$ d 2 , n$_{3}$ + 1 2 ( n$_{1}$ + n$_{2}$ − d ) , n$_{3}$ + 1 2 ( n$_{1}$ + n$_{2}$ − d + 1) q 2 4 m 2   . (2.35) 2.4 Integration By Parts