The epsilon expansion meets semiclassics

Abstract

We study the scaling dimension Δϕn of the operator 𝜙n 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 Δϕn can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting inΔϕn=1λ∗Δ−1(λ∗n)+Δ0(λ∗n)+λ∗Δ1(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.