The large charge limit of scalar field theories, and the Wilson-Fisher fixed point at ε=0

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 similar to epsilon. We show that, in the limit g -> 0, when the theory naively approaches the gaussian fixed point, the sector of operators with n -> infinity at fixed g n(2) equivalent to 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 ((phi) over bar phi)(3) potential.