Universal correlation functions in rank 1 SCFTs

Abstract

Carrying to higher precision the large-$\mathcal{J}$ expansion of Hellerman and Maeda, we calculate to all orders in $1/\mathcal{J}$ the power-law corrections to the two-point functions $\mathcal{Y}_n \equiv |x - y|^{2n\Delta_{\mathcal{O}}} \langle {\mathcal{O}}_n(x) \bar{\mathcal{O}}_n(y) \rangle$ for generators $\mathcal{O}$ of Coulomb branch chiral rings in four-dimensional $\mathcal{N} =2$ superconformal field theories. We show these correlators have the universal large-$n$ expansion \[ \log(\mathcal{Y}_n) \simeq \mathcal{J} \mathbf{A} + \mathbf{B} + \log(\Gamma( \mathcal{J} + \alpha + 1)) , \] where $\mathcal{J} \equiv 2 n \Delta_{\mathcal{O}}$ is the total $R$-charge of $\mathcal{O}_n$, the $\mathbf{A}$ and $\mathbf{B}$ are theory-dependent coefficients, $\alpha$ is the coefficient of the Wess-Zumino term for the Weyl $a$-anomaly, and the $\simeq$ denotes equality up to terms exponentially small in $\mathcal{J}$. Our methods combine the structure of the Coulomb-branch effective field theory (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 $\mathcal{N} = 2$ 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. 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 $|x - y|$.