Correlation functions in scalar field theory at large charge

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

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