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We study the sector of large charge operators $\phi^n$ ($\phi$ being thecomplexified 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 naivelyapproaches the gaussian fixed point, the sector of operators with $n\to \infty$ at fixed $g\,n^2\equiv \lambda$ remains non-trivial. Surprisingly, one cancompute the exact 2-point function and thereby the non-trivial anomalousdimension 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 pathintegral, which partly explains the existence of the limit. Finally, we extendthese results to the three-dimensional $O(2)$-symmetric theory with$(\bar{\phi}\,\phi)^3$ potential.

The large charge limit of scalar field theories and the Wilson-Fisher fixed point at $\epsilon=0$