R-current three-point functions in 4d $\mathcal{N}=1$ superconformal theories

TL;DR

This paper classifies and computes three-point functions involving R-currents in 4d $ abla=1$ superconformal theories, providing a foundation for future bootstrap studies of four R-current correlators.

Contribution

It derives the general form of three-point functions with two Ferrara-Zumino multiplets and a third multiplet, including a counting method for independent tensor structures.

Findings

Identifies conditions for non-trivial three-point functions involving R-currents.

Provides a systematic prescription for tensor structure counting.

Sets the stage for bootstrap analysis of four R-current correlators.

Abstract

In 4d $\mathcal{N}=1$ superconformal field theories (SCFTs) the R-symmetry current, the stress-energy tensor, and the supersymmetry currents are grouped into a single object, the Ferrara-Zumino multiplet. In this work we study the most general form of three-point functions involving two Ferrara-Zumino multiplets and a third generic multiplet. We solve the constraints imposed by conservation in superspace and show that non-trivial solutions can only be found if the third multiplet is R-neutral and transforms in suitable Lorentz representations. In the process we give a prescription for counting independent tensor structures in superconformal three-point functions. Finally, we set the Grassmann coordinates of the Ferrara-Zumino multiplets to zero and extract all three-point functions involving two R-currents and a third conformal primary. Our results pave the way for bootstrapping the correlation function of four R-currents in 4d $\mathcal{N}=1$ SCFTs.