Even in weakly coupled QFTs, perturbation theory breaks down when one considers amplitudes with a large number $n$ of legs. The series cleverly organizes as a double expansion in $g^2$ and $g^2n$. I show how the series in $g^2n$ can be fully captured by a semiclassical expansion around a non-trivial solution. Focussing on $U(1)$ symmetric $|\phi|^4$ theory in $4$ and $4-\epsilon$ dimension, I derive explicit and consistent all order results for the anomalous dimension of the complex operator $\phi^n$. When restricting to the Wilson-Fisher fixed point and working on the cylinder, the dominant trajectory is seen to correspond to a superfluid phase for the conserved U(1). This creates a correspondence between, on one side, the spectrum of operators and fusion coefficients and and on the other the spectrum of hydrodynamics modes and their interactions. The results also nicely match Monte Carlo simulations in 3D, compatibly with the stunt of taking $\epsilon=1$. [please email alejandro.cabo_bizet@kcl.ac.uk for the Zoom link]