Correlation functions in scalar field theory at large charge

TL;DR

This paper calculates higher-point correlation functions involving large charge operators in an $O(2)$ scalar field theory, revealing simplified forms for extremal correlators in a double-scaling limit and providing explicit formulas for complex four-point functions.

Contribution

It introduces a new approach to compute large charge correlators, especially extremal ones, in $O(2)$ theories, with explicit formulas and analysis of their structure in a specific double-scaling limit.

Findings

Extremal correlators have a simple form in the large charge double-scaling limit.

Non-extremal correlators can also be computed in this limit.

Explicit formula provided for a four-point correlator involving large charge operators.

Abstract

We compute general higher-point functions in the sector of large charge operators $\phi^n$, $\bar\phi^n$ at large charge in $O(2)$ $(\bar \phi\phi)^2$ theory. We find that there is a special class of "extremal" correlators having only one insertion of $\bar \phi^n$ that have a remarkably simple form in the double-scaling limit $n\to \infty $ at fixed $g\,n^2\equiv \lambda$, where $g\sim\epsilon $ is the coupling at the $O(2)$ Wilson-Fisher fixed point in $4-\epsilon$ dimensions. In this limit, also non-extremal correlators can be computed. As an example, we give the complete formula for $ \langle \phi(x_1)^{n}\,\phi(x_2)^{n}\,\bar{\phi}(x_3)^{n}\,\bar{\phi}(x_4)^{n}\rangle$, which reveals an interesting structure.