Exact Results for Scaling Dimensions of Neutral Operators in scalar CFTs

We determine the scaling dimension ${\mathrm{\Delta }}_n$ for the class of composite operators ${\phi }^n$ in the $\lambda {\phi }^4$ theory in $d=4-\epsilon$ taking the double scaling limit $n\to \mathrm{\infty }$ and $\lambda \to 0$ with fixed $\lambda n$ via a semiclassical approach. Our results resum the leading power of $n$ at any loop order. In the small $\lambda n$ regime we reproduce the known diagrammatic results and predict the infinite series of higher-order terms. For intermediate values of $\lambda n$ we find that ${\mathrm{\Delta }}_n\mathrm{/}n$ increases monotonically approaching a $(\lambda n{)}^{1\mathrm{/}3}$ behavior in the $\lambda n\to \mathrm{\infty }$ limit. We further generalize our results to neutral operators in the ${\phi }^4$ in $d=4-\epsilon$, ${\phi }^3$ in $d=6-\epsilon$, and ${\phi }^6$ in $d=3-\epsilon$ theories with $O(N)$ symmetry.