Figure 2.4: Feynman diagram representing the V -type two-loop nested integral. loop integrals. Finally, the result for (2.23) is From the definition (2.23) it is clear that this result has the proper mass dimension. Inter- estingly, the result is also valid for any value of n$_{2}$ , including n$_{2}$ = 0. This technique can be extended to integrals containing one massive propagator. For example, consider the integral This integral occurs in Chapter 4, and also in Ref. [66]. Once again the k$_{2}$ integral can be in- tegrated immediately, and the resulting k$_{1}$ integral can be evaluated using (2.22). Expressing the overall scale dependence in terms of the mass m , the result is 59 V ( d ; n$_{1}$ , m ; n$_{2}$ , 0 ; n$_{3}$ , 0 ; n$_{4}$ , 0) = − 1 (4 π ) 4 [ − m 2 1 − z$^{]}$$^{4}$$^{−}$$^{n$_{1}$}$$^{−}$$^{n$_{2}$}$$^{−}$$^{n$_{3}$}$$^{−}$$^{n$_{4}$}$ exp$^{[}$( d − 4)$^{(}$log$^{[}$ m 2 4 πµ 2 ] − log [1 − z ]$^{)]}$ Γ ( d 2 − n$_{3}$ $^{)}$Γ ( d 2 − n$_{4}$ $^{)}$Γ $^{(}$n$_{3}$ + n$_{4}$ − d 2 $^{)}$Γ ( d − n$_{2}$ − n$_{3}$ − n$_{4}$ ) Γ ( n$_{1}$ + n$_{2}$ + n$_{3}$ + n$_{4}$ − d ) Γ ( n$_{1}$ ) Γ ( n$_{3}$ ) Γ ( n$_{4}$ ) Γ ( d 2 $^{)}$Γ ( d − n$_{3}$ − n$_{4}$ ) $_{2}$F$_{1}$$^{[}$ n$_{1}$ + n$_{2}$ + n$_{3}$ + n$_{4}$ − d , d − n$_{2}$ − n$_{3}$ − n$_{4}$ ; d 2 ; z$^{]}$ , (2.26) V ( d ; n$_{1}$ , m ; n$_{2}$ , 0 ; n$_{3}$ , 0 ; n$_{4}$ , 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$_{1}$ − q ) 2 n$_{2}$ ( k$_{2}$ − q ) 2 n$_{3}$ ( k$_{1}$ − k$_{2}$ ) 2 n$_{4}$ . (2.25) V ( d ; n$_{1}$ , 0 ; n$_{2}$ , 0 ; n$_{3}$ , 0 ; n$_{4}$ , 0) = − 1 (4 π ) 4 $^{(}$q 2 ) 4 − n$_{1}$ − n$_{2}$ − n$_{3}$ − n$_{4}$ [ − q 2 4 πµ 2 ] d − 4 Γ ( d 2 − n$_{1}$ $^{)}$Γ ( d 2 − n$_{3}$ $^{)}$Γ ( d 2 − n$_{4}$ ) Γ ( n$_{1}$ ) Γ ( n$_{3}$ ) Γ ( n$_{4}$ ) Γ ( n$_{1}$ + n$_{2}$ + n$_{3}$ + n$_{4}$ − d ) Γ ( d − n$_{2}$ − n$_{3}$ − n$_{4}$ ) Γ $^{(}$n$_{3}$ + n$_{4}$ − d 2 ) Γ $^{(}$n$_{2}$ + n$_{3}$ + n$_{4}$ − d 2 $^{)}$Γ ( 3 d 2 − n$_{1}$ − n$_{2}$ − n$_{3}$ − n$_{4}$ $^{)}$Γ ( d − n$_{3}$ − n$_{4}$ ) . (2.24)