The epsilon expansion meets semiclassics

A bstract We study the scaling dimension 䂞 Δϕn 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 䂞 Δϕn can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in 䂞=1 -₁ (n) +₀ (n) + ₁ (n) + Δϕn=1λ∗Δ−1λ∗n+Δ0λ∗n+λ∗Δ1λ∗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.