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The epsilon expansion meets semiclassics
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… Click to show full abstract
AbstractWe study the scaling dimension $$ {\Delta}_{\phi^n} $$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 $$ {\Delta}_{\phi^n} $$Δϕn can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in$$ {\Delta}_{\phi^n}=\frac{1}{\lambda_{\ast }}{\Delta}_{-1}\left({\lambda}_{\ast }n\right)+{\Delta}_0\left({\lambda}_{\ast }n\right)+{\lambda}_{\ast }{\Delta}_1\left({\lambda}_{\ast }n\right)+\dots $$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.
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