  We study the sector of large charge operators $\phi^n$ ($\phi$ being the
complexified scalar field) in the $O(2)$ Wilson-Fisher fixed point in
$4-\epsilon$ dimensions that emerges when the coupling takes the critical value
$g\sim \epsilon$. We show that, in the limit $g\to 0$, when the theory naively
approaches the gaussian fixed point, the sector of operators with $n\to \infty
$ at fixed $g\,n^2\equiv \lambda$ remains non-trivial. Surprisingly, one can
compute the exact 2-point function and thereby the non-trivial anomalous
dimension of the operator $\phi^n$ by a full resummation of Feynman diagrams.
The same result can be reproduced from a saddle point approximation to the path
integral, which partly explains the existence of the limit. Finally, we extend
these results to the three-dimensional $O(2)$-symmetric theory with
$(\bar{\phi}\,\phi)^3$ potential.
