On Non-renormalization and OPE in Superconformal Field Theories

TL;DR

The paper analyzes non-renormalization and OPE in superconformal theories by examining superspace three-point functions with two short $N=2$ multiplets and a third long multiplet. Using $N=2$ harmonic superspace and enforcing $G$- and $H$-analyticity, it shows that anomalous dimensions can arise only for an $R$-symmetry singlet, while $R$-symmetry triplets and 5-plets are protected, providing a unified explanation for non-renormalization phenomena observed in correlators, including the four-point function of the $N=4$ stress-tensor multiplet. Extending to $N=4$, the work demonstrates that anomalous dimensions in the OPE are confined to the $SU(4)$ singlet sector, consistent with the structure of four-point functions decomposed into $SU(4)$ irreps and their conformal partial waves. These results offer a selection-rule mechanism for operator contributions in OPEs of short multiplets and motivate further full superspace analyses and generalizations.

Abstract

The OPE of two N=2 R-symmetry current (short) multiplets is determined by the possible superspace three-point functions that two such multiplets can form with a third, a priori long multiplet. We show that the shortness conditions on the former put strong restrictions on the quantum numbers of the latter. In particular, no anomalous dimension is allowed unless the third supermultiplet is an R-symmetry singlet. This phenomenon should explain many known non-renormalization properties of correlation functions, including the one of four stress-tensor multiplets in N=4 SYM_4.