Correlation functions in scalar field theory at large charge
G. Arias-Tamargo, D. Rodriguez-Gomez, J. G. Russo
Abstract
A bstract We compute general higher-point functions in the sector of large charge operators ϕ n , $$ {\overline{\phi}}^n $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>n</mml:mi> </mml:msup> </mml:math> at large charge in O(2) $$ {\left(\overline{\phi}\phi \right)}^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mfenced> <mml:mrow> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:mfenced> <mml:mn>2</mml:mn> </mml:msup> </mml:math> theory. We find that there is a special class of “extremal” correlators having only one insertion of $$ {\overline{\phi}}^n $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>n</mml:mi> </mml:msup> </mml:math> that have a remarkably simple form in the double-scaling limit n →∞ at fixed g n 2 ≡ λ, where g ~ ϵ 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 $$ \left\langle \phi {\left({x}_1\right)}^n\phi {\left({x}_2\right)}^n\overline{\phi}{\left({x}_3\right)}^n\overline{\phi}{\left({x}_4\right)}^n\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:msup> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mfenced> <mml:mi>n</mml:mi> </mml:msup> <mml:mi>ϕ</mml:mi> <mml:msup> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mfenced> <mml:mi>n</mml:mi> </mml:msup> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:msup> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mfenced> <mml:mi>n</mml:mi> </mml:msup> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:msup> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mfenced> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> , which reveals an interesting structure.