This result agrees with an expression for this integral that is given in Ref. [97]. As an additional check, we can verify that this result reproduces (2.19) when m = 0. Setting z = 1 and using the identity (B.17), it is easy to verify that (2.21) is consistent with the massless result (2.19). Note that the mass dimension of the result is carried by the term ( q $^{2}$) 2 − n$_{1}$ − n$_{2}$ and agrees with that of the integral (2.14), as it must. However, for future convenience it will be helpful to recast this mass dependence in terms of the mass m . This can be done using the definition of the dimensionless variable z given in (2.20). Doing so, the final result for the one-loop self-energy integral with one massive propagator is where z is defined as in Eq. (2.20). Note that log [1 − z ] = − log $^{[}$1 + Q 2 m 2 $^{]}$, which has a branch cut on Q 2 ∈ ( −∞ , − m $^{2}$]. It can also be shown that the hypergeometric function has the same branch cut. Therefore the result (2.22) has the branch cut structure that is appropriate for a correlation function of a currents containing one massive quark with mass m . The results above can be used to calculate some two-loop integrals. For instance, consider the following integral which is represented in Fig. 2.4. Note that in (2.23) the momentum k$_{2}$ only appears in the third and fourth propagators, therefore the k$_{2}$ integral can be performed using the result (2.19). Specifically, if we make the change of variables ˜ k$_{2}$ = k$_{2}$ − q , the ˜ k$_{2}$ integral is proportional to ( k$_{1}$ − q ) 2 ( d $_{2}$− n$_{3}$ − n$_{4}$ ) . Because the integrations can be performed in an iterative fashion, these are called nested 58 V ( d ; n$_{1}$ , 0 ; 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 n$_{1}$ 1 ( k$_{1}$ − q ) 2 n$_{2}$ ( k$_{2}$ − q ) 2 n$_{3}$ ( k$_{1}$ − k$_{2}$ ) 2 n$_{4}$ , (2.23) B ( d ; n$_{1}$ , 0 ; n$_{2}$ , m ) = i (4 π ) 2 [ − m 2 1 − z$^{]}$$^{2}$$^{−}$$^{n$_{1}$}$$^{−}$$^{n$_{2}$}$ exp$^{[}$ d − 4 2 ( log$^{[}$ m 2 4 πµ 2 ] − log [1 − z ]$^{)]}$ Γ ( d 2 − n$_{1}$ $^{)}$Γ $^{(}$n$_{1}$ + n$_{2}$ − d 2 ) Γ( n$_{1}$ )Γ( d $_{2}$) $_{2}$F$_{1}$$^{[}$ n$_{1}$ + n$_{2}$ − d 2 , d 2 − n$_{1}$ ; d 2 ; z$^{]}$ , (2.22)