On the OPE of 1/2 BPS Short Operators in N=4 SCFT$_4$

TL;DR

This work analyzes the OPE of two 1/2 BPS operators in N=4 SCFT$_4$ by constructing superspace three-point functions with a third operator and exploiting Grassmann-analytic and harmonic analyticity. By focusing on leading SU(4) components in $[0,m,0]\otimes[0,n,0]$ and $[m,0,0]\otimes[0,0,n]$, the authors derive selection rules that classify all protected operators appearing in these OPEs, including BPS multiplets and current-like long multiplets. They provide explicit three-point function structures, identify when harmonic singularities are forbidden, and show how unitarity bounds control the appearance of conserved-like operators in the OPE. The results illuminate non-renormalization properties of extremal and next-to-extremal correlators and have implications for four-point functions in N=4 SYM and the AdS/CFT correspondence via KK-state representations.

Abstract

The content of the OPE of two 1/2 BPS operators in N=4 SCFT$_4$ is given by their superspace three-point functions with a third, a priori long operator. For certain 1/2 BPS short superfields these three-point functions are uniquely determined by superconformal invariance. We focus on the cases where the leading ($θ=0$) components lie in the tensor products $[0,m,0]\otimes[0,n,0]$ and $[m,0,0]\otimes[0,0,n]$ of SU(4). We show that the shortness conditions at the first two points imply selection rules for the supermultiplet at the third point. Our main result is the identification of all possible protected operators in such OPE's. Among them we find not only BPS short multiplets, but also series of special long multiplets which satisfy current-like conservation conditions in superspace.