was noticed that the operators ${\ensuremath{\mathcal{O}}}_1 = \bar{\phi}^i\phi_i$ and ${\ensuremath{\mathcal{O}}}_2 = \bar{\psi}^i\psi_i + \frac{4\pi}{k}(\bar{\phi}^i\phi_i)^2$ sit in the same supermultiplet as the $U(1)_f$ symmetry current.[^5] As a consequence, the dimensions of ${\ensuremath{\mathcal{O}}}_1$ and ${\ensuremath{\mathcal{O}}}_2$ are protected to be 1 and 2 respectively. The “double-trace” term in ${\ensuremath{\mathcal{O}}}_2$ does not contribute to the 2-point function $\langle{\ensuremath{\mathcal{O}}}_2(x){\ensuremath{\mathcal{O}}}_2(y)\rangle$ to leading order in $1/N$, and therefore the operator $\bar{\psi}^i\psi_i$ by itself is also protected at large $N$, with dimension $\Delta=2+O(1/N)$.

Let us begin by arguing that the anomalous dimension of ${\ensuremath{\mathcal{O}}}_1$ vanishes at large $N$ also in our non-supersymmetric theory. Consider the diagrams that contribute to $\left< {\ensuremath{\mathcal{O}}}_1 {\ensuremath{\mathcal{O}}}_1 \right>$ and $\left< {\ensuremath{\mathcal{O}}}_2 {\ensuremath{\mathcal{O}}}_2 \right>$ in the ${\ensuremath{\mathcal{N}}}=2$ theory and involve a single matter loop, with possible additional gluon lines. We will denote them collectively as

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   ![image](E1.pdf){height="3.5cm"}   ![image](E2.pdf){height="3.5cm"}
                 (E1)                               (E2)
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For the rest of the section we will keep gluon lines implicit in all diagrams; at large $N$ when we draw the diagrams in double-line notation these lines must sit inside the scalar/fermion loop so that the topology of each scalar/fermion loop is that of a disk. We will show below that in the large $N$ limit the sum of such 