matter loops causing them to vanish. We know that in the ${\ensuremath{\mathcal{N}}}=2$ theory the sum of all these (F1) diagrams, with any (odd) number of matter loops, is finite, since ${\ensuremath{\mathcal{O}}}_1$ is not renormalized. Working in momentum space, each (F1) diagram factorizes at large $N$ into a product of sub-diagrams of the form (E1) or (E2). If a given (F1) diagram has more than one matter loop, its sub-diagrams will be of a lower order in perturbation theory. The sum over such sub-diagrams is finite by the induction assumption, and therefore (F1) diagrams with more than one matter loop are finite in total. Since the sum over all (F1) diagrams is also finite, the sum over single-matter-loop diagrams — which are the (E1) diagrams at the order we are in — must be finite. This concludes the induction step for (E1); the step for (E2) is analogous.[^7] In appendix \[adim\] we verify that indeed the anomalous dimension of $\phi^2$ vanishes in the non-supersymmetric theory at two-loop order in the large $N$ limit.

The argument above can be easily generalized to diagrams of the topology (E1), which have three insertions of ${\ensuremath{\mathcal{O}}}_1$ on the scalar loop instead of two. Namely, the 