The epsilon expansion meets semiclassics

Abstract

Abstract We study the scaling dimension $$ {\Delta}_{\phi^n} $$ Δ ϕ n of the operator 𝜙 n where 𝜙 is the fundamental complex field of the U(1) model at the Wilson-Fisher fixed point in d = 4 − ε. Even for a perturbatively small fixed point coupling λ ∗, standard perturbation theory breaks down for sufficiently large λ ∗ n. Treating λ ∗ n as fixed for small λ ∗ we show that $$ {\Delta}_{\phi^n} $$ Δ ϕ n can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in $$ {\Delta}_{\phi^n}=\frac{1}{\lambda_{\ast }}{\Delta}_{-1}\left({\lambda}_{\ast }n\right)+{\Delta}_0\left({\lambda}_{\ast }n\right)+{\lambda}_{\ast }{\Delta}_1\left({\lambda}_{\ast }n\right)+\dots $$ Δ ϕ n = 1 λ ∗ Δ − 1 λ ∗ n + Δ 0 λ ∗ n + λ ∗ Δ 1 λ ∗ n + … We explicitly compute the first two orders in the expansion, ∆ −1(λ ∗ n) and ∆0(λ ∗ n). The result, when expanded at small λ ∗ n, perfectly agrees with all available diagrammatic com- putations. The asymptotic at large λ ∗ n reproduces instead the systematic large charge expansion, recently derived in CFT. Comparison with Monte Carlo simulations in d = 3 is compatible with the obvious limitations of taking ε = 1, but encouraging.