Exact Results for Scaling Dimensions of Neutral Operators in scalar CFTs

Abstract

We determine the scaling dimension ${\Delta }_n$ for the class of composite operators ${\varphi }^n$ in the $\lambda {\varphi }^4$ theory in $d=4-ϵ$ taking the double scaling limit $n\to \infty$ 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 ${\Delta }_n/n$ 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 ${\varphi }^4$ in $d=4-ϵ$, ${\varphi }^3$ in $d=6-ϵ$, and ${\varphi }^6$ in $d=3-ϵ$ theories with $O(N)$ symmetry.

Comments: We extended our analysis to O(N) theories in various spacetime dimensions featuring different interactions