Universal correlation functions in rank 1 SCFTs

Abstract

A bstract Carrying to higher precision the large- $$ \mathcal{J} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> </mml:math> expansion of [1], we calculate to all orders in $$ 1/\mathcal{J} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>J</mml:mi> </mml:math> the power-law corrections to the two-point functions $$ {\mathcal{Y}}_n\equiv {\left|x-y\right|}^{2n{\varDelta}_{\mathcal{O}}}\cdot \le \left\langle {\mathcal{O}}^n(x){\overline{\mathcal{O}}}^n(y)\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>≡</mml:mo> <mml:msup> <mml:mfenced> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> </mml:mfenced> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>O</mml:mi> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>⋅</mml:mo> <mml:mo>≤</mml:mo> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mi>O</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mfenced> <mml:mi>x</mml:mi> </mml:mfenced> <mml:msup> <mml:mover> <mml:mi>O</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>n</mml:mi> </mml:msup> <mml:mfenced> <mml:mi>y</mml:mi> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> for generators $$ \mathcal{O} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> </mml:math> of Coulomb branch chiral rings in four-dimensional $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> superconformal field theories. We show these correlators have the universal large- n expansion $$ \log \left({\mathcal{Y}}_n\right)\simeq \mathcal{J}\mathbf{A}+\mathbf{B}+\log \left(\Gamma \left(\mathcal{J}+\alpha +1\right)\right), $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>log</mml:mo> <mml:mfenced> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mfenced> <mml:mo>≃</mml:mo> <mml:mi>J</mml:mi> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> <mml:mo>+</mml:mo> <mml:mo>log</mml:mo> <mml:mfenced> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mfenced> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>+</mml:mo> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> </mml:math> where $$ \mathcal{J}\equiv n{\Delta}_{\mathcal{O}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> <mml:mo>≡</mml:mo> <mml:mi>n</mml:mi> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>O</mml:mi> </mml:msub> </mml:math> is the total R -charge of $$ \mathcal{O} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> </mml:math> n , the A and B are theory-dependent coefficients, α is the coefficient of the Wess-Zumino term for the Weyl a -anomaly, and the ≃ denotes equality up to terms exponentially small in $$ \mathcal{J} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> </mml:math> . Our methods combine the structure of the Coulomb-branch eft with the supersymmetric recursion relations. However, our results constrain the power-law corrections to all orders, even for non-Lagrangian theories to which the recursion relations do not apply. For the case of $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> sqcd , we also comment on the nature of the exponentially small corrections, which can be calculated to high precision in the double-scaling limit recently discussed by Bourget et al. in [2]. We show the exponentially small correction is consistent with the interpretation of the eft breaking down due to the propagation of massive bps particles over distances of order of the infrared scale | x − y |.