The Epsilon Expansion Meets Semiclassics

TL;DR

This paper develops a semiclassical method to compute the scaling dimensions of operators in the $U(1)$ model at the Wilson-Fisher fixed point, successfully bridging perturbative, large charge, and numerical results.

Contribution

It introduces a semiclassical expansion approach for calculating operator dimensions at large charge, extending beyond standard perturbation theory.

Findings

Explicit computation of the first two orders in the semiclassical expansion.

Agreement with diagrammatic computations at small $\lambda_* n$.

Reproduction of the large charge expansion at large $\lambda_* n$.

Abstract

We study the scaling dimension $\Delta_{\phi^n}$ of the operator $\phi^n$ where $\phi$ is the fundamental complex field of the $U(1)$ model at the Wilson-Fisher fixed point in $d=4-\varepsilon$. Even for a perturbatively small fixed point coupling $\lambda_*$, standard perturbation theory breaks down for sufficiently large $\lambda_*n$. Treating $\lambda_* n$ as fixed for small $\lambda_*$ we show that $\Delta_{\phi^n}$ can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in $$ \Delta_{\phi^n}=\frac{1}{\lambda_*}\Delta_{-1}(\lambda_* n)+\Delta_{0}(\lambda_* n)+\lambda_* \Delta_{1}(\lambda_* n)+\ldots $$ We explicitly compute the first two orders in the expansion, $\Delta_{-1}(\lambda_* n)$ and $\Delta_{0}(\lambda_* n)$. The result, when expanded at small $\lambda_* n$, perfectly agrees with all available diagrammatic computations. The asymptotic at large $\lambda_* n$ 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 $\varepsilon=1$, but encouraging.