diagrams at a given order in perturbation theory is finite. However, the diagrams (E1) (with gluon lines running in the loop) are precisely those that contribute to the correlator $\left< {\ensuremath{\mathcal{O}}}_1 {\ensuremath{\mathcal{O}}}_1 \right>$ in our ${\ensuremath{\mathcal{N}}}=0$ model in the large $N$ limit.[^6] Thus, it will follow that the dimension of ${\ensuremath{\mathcal{O}}}_1 = \bar{\phi}^i \phi_i$ in our non-supersymmetric vector model is $1+O(1/N)$ to all orders in planar perturbation theory.

We now prove the finiteness of (E1) and (E2) at large $N$ by induction. At zeroth order in perturbation theory, (E1) and (E2) are single 1-loop diagrams which are finite in our regularization scheme. At the next order the only diagrams contributing to the two-point functions in the ${\ensuremath{\mathcal{N}}}=2$ theory are still of the form (E1) and (E2) (with an extra gluon line), so all divergences in these diagrams must cancel (in fact, it follows from the parity transformation that these diagrams vanish). At higher orders in perturbation theory, there are more general diagrams contributing to $\left< {\ensuremath{\mathcal{O}}}_1 {\ensuremath{\mathcal{O}}}_1 \right>$ at large $N$, which have the general form:

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   ![image](F1.pdf){height="3.5cm"}
                 (F1)
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Again, gluon lines running inside the loops are implicit, and all the other diagrams (not drawn in (F1)) contain tadpole 