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 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) = 1/lambda*Delta(-1)(lambda*n)+Delta(0)(lambda*n)+lambda*Delta(1)(lambda*n)+ ... 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 epsilon = 1, but encouraging.