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

Abstract

We study the sector of large charge operators ${\varphi }^n$ ($\varphi$ 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 ${\varphi }^n$ 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 $(\bar{\varphi }\varphi {)}^3$ potential.

14 pages