The Epsilon Expansion Meets Semiclassics

Abstract

We study the scaling dimension ${\Delta }_{{\varphi }^n}$ of the operator ${\varphi }^n$ where $\varphi$ is the fundamental complex field of the $U(1)$ model at the Wilson-Fisher fixed point in $d=4-\epsilon$. Even for a perturbatively small fixed point coupling ${\lambda }_{\ast }$, standard perturbation theory breaks down for sufficiently large ${\lambda }_{\ast }n$. Treating ${\lambda }_{\ast }n$ as fixed for small ${\lambda }_{\ast }$ we show that ${\Delta }_{{\varphi }^n}$ can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in $${\Delta }_{{\varphi }^n}=\frac{1}{{\lambda }_{\ast }}{\Delta }_{-1}({\lambda }_{\ast }n)+{\Delta }_0({\lambda }_{\ast }n)+{\lambda }_{\ast }{\Delta }_1({\lambda }_{\ast }n)+\dots$$ We explicitly compute the first two orders in the expansion, ${\Delta }_{-1}({\lambda }_{\ast }n)$ and ${\Delta }_0({\lambda }_{\ast }n)$. The result, when expanded at small ${\lambda }_{\ast }n$, perfectly agrees with all available diagrammatic computations. The asymptotic at large ${\lambda }_{\ast }n$ 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 $\epsilon =1$, but encouraging.

Comments: 24 pages + appendices, 2 figures v2 fixed typos