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.
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