On the OPE of 1/2 BPS Short Operators in N=4 SCFT₄

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 ($\theta=0$) 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.