  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
$d=4-\varepsilon$ 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
$3-\varepsilon$ 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.
