Exact Results for Scaling Dimensions of Neutral Operators in scalar CFTs

Abstract

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