Correlation functions in scalar field theory at large charge

We compute general higher-point functions in the sector of large charge operators $phi^n$, $barphi^n$ at large charge in $O(2)$ $(bar phiphi)^2$ 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 $nto infty $ at fixed $g,n^2equiv lambda$, where $gsimepsilon $ 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.