Exact results for scaling dimensions of neutral operators in scalar conformal field theories

We determine the scaling dimension Δn for the class of composite operators φn in the λφ4 theory in d=4-ϵ taking the double scaling limit n→∞ and λ→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 Δn/n increases monotonically approaching a (λn)1/3 behavior in the λn→∞ limit. We further generalize our results to neutral operators in the φ4 in d=4-ϵ, φ3 in d=6-ϵ, and φ6 in d=3-ϵ theories with O(N) symmetry.