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

TL;DR

This paper investigates the behavior of large charge operators in the $O(2)$ Wilson-Fisher fixed point near four dimensions, revealing exact results for their two-point functions and anomalous dimensions in a non-trivial large charge limit.

Contribution

It provides an exact computation of two-point functions and anomalous dimensions for large charge operators at the Wilson-Fisher fixed point, using resummation and saddle point methods.

Findings

Exact two-point functions for large charge operators are derived.

The anomalous dimensions remain non-trivial in the large charge limit.

Results extend to three-dimensional $O(2)$ theories with $(ar{ield}ield)^3$ potential.

Abstract

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.