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Is it true that if a single-trace operator, say, $O$ acquires an anomalous dimension $\gamma_o$, then the anomalous dimension of the double-trace operator $O^2$ is $2\gamma_o$? If no, can anyone please provide counter-examples?

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A correct statement is that in the large-N limit of a theory, if you have an operator $O$ with scaling dimension $\Delta_O$, the theory will also contain an operator $O^2$ with scaling dimension $2\Delta_O$.

But be aware that:

1) This is a statement about scaling dimensions, not anomalous dimensions.

2) If you include 1/N corrections the relation $\Delta_{O^2} = 2 \Delta_O$ won't be true anymore.

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