# The Epsilon Expansion Meets Semiclassics

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Gil Badel, Gabriel Cuomo, Alexander Monin, Riccardo Rattazzi

**3/9/19**Published in : arXiv:1909.01269

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.