Feynman diagrams and the large charge expansion in $3-\epsilon$ dimensions

In arXiv:1909.01269 it was shown that the scaling dimension of the lightest charge $n$ operator in the $U(1)$ model at the Wilson-Fisher fixed point in $d=4-\epsilon$ can be computed semiclassically for arbitrary values of $\lambda n$, where $\lambda$ is the perturbatively small fixed point coupling. Here we generalize this result to the fixed point of the $U(1)$ model in $3-\epsilon$ dimensions. The result interpolates continuously between diagrammatic calculations and the universal conformal superfluid regime for CFTs at large charge. In particular it reproduces the expectedly universal $O(n^0)$ contribution to the scaling dimension of large charge operators in $3d$ CFTs.