Exact Results for Scaling Dimensions of Neutral Operators in scalar CFTs

We determine the scaling dimension $\Delta_n$ for the class of composite operators $\phi^n$ in the $\lambda \phi^4$ theory in $d=4-\epsilon$ taking the double scaling limit $n\rightarrow \infty$ and $\lambda \rightarrow 0$ 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 $\Delta_n/n$ increases monotonically approaching a $(\lambda n)^{1/3}$ behavior in the $\lambda n \to \infty$ 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.