The epsilon expansion meets semiclassics

Abstract

A bstract We study the scaling dimension $$ {\Delta}_{\phi^n} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Δ</mml:mi><mml:msup><mml:mi>ϕ</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:msub></mml:math> 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 $$ {\Delta}_{\phi^n} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Δ</mml:mi><mml:msup><mml:mi>ϕ</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:msub></mml:math> can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in $$ {\Delta}_{\phi^n}=\frac{1}{\lambda_{\ast }}{\Delta}_{-1}\left({\lambda}_{\ast }n\right)+{\Delta}_0\left({\lambda}_{\ast }n\right)+{\lambda}_{\ast }{\Delta}_1\left({\lambda}_{\ast }n\right)+\dots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Δ</mml:mi><mml:msup><mml:mi>ϕ</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msub><mml:mi>λ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mfrac><mml:msub><mml:mi>Δ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mfenced><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mi>n</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:msub><mml:mi>Δ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mfenced><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mi>n</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:msub><mml:mi>Δ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mfenced><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mi>n</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mo>…</mml:mo></mml:math> 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.