which upto contact terms gives We can calculate exactly the same quantity using Feynman diagrams [115, 116]. The leading-order diagram is the quark bubble And so comparing (5.10) and (5.11) we can fix g$_{5}$ as The matching we have just performed is a little naive. One should match the gravitational theory to QCD only at the point where the QCD coupling becomes non-perturbative, for this is the point where the gravitational dual will be weakly coupled. And of course, when the QCD coupling is non- perturbative, gluonic contributions become important. In all of the analysis to follow, we keep g$_{5}$ fixed as in equation (5.12). However we have performed an analysis that treats g$_{5}$ as a free parameter: this indicated that the optimal value of g$_{5}$ is 5.19, rather than the 3.54 that (5.12) suggests: this is a discrepancy of 32%! So for the rest of this section we add the disclaimer that non-perturbative effects could have a significant effect on our results. We also remind the reader that AdS/CFT correspondence is a large N$_{c}$ , large ‘t Hooft limit ( g 2 $_{Y M}$N ≡ λ ≫ 1). In calculating g$_{5}$ we have just assumed N$_{c}$ = 3; this is a further source of error. We return to the matter of g$_{5}$ in chapter 6. 76 A cautionary note g 2 5 = 12 π 2 N$_{c}$ (5.12) Π$_{V}$ ( Q $^{2}$) = − N$_{c}$ 24 π 2 ln( Q $^{2}$) (5.11) Π$_{V}$ ( Q $^{2}$) = − 1 2 g 2 5 ln( Q $^{2}$) (5.10)