The epsilon expansion meets semiclassics

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 - epsilon. Even for a perturbatively small fixed point coupling lambda*, standard perturbation theory breaks down for sufficiently large lambdan. Treating lambdan 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) = 1/lambdaDelta(-1)(lambdan)+Delta(0)(lambdan)+lambdaDelta(1)(lambdan)+ ... We explicitly compute the first two orders in the expansion, Delta(-1)(lambdan) and Delta(0)(lambda n). The result, when expanded at small lambdan, perfectly agrees with all available diagrammatic computations. The asymptotic at large lambdan 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.