Correlation functions of spinor current multiplets in ${\mathcal N}=1$ superconformal theory

TL;DR

This paper analyzes three-point correlation functions involving spinor current multiplets in four-dimensional ${ m extbf{N}=1}$ superconformal theories, revealing that additional Grassmann odd multiplets do not necessarily enhance symmetry to ${ m extbf{N}=2}$.

Contribution

It demonstrates that the structure of three-point functions with spinor current multiplets in ${ m extbf{N}=1}$ theories is more complex than previously thought, and does not imply ${ m extbf{N}=2}$ supersymmetry.

Findings

Three independent tensor structures in three-point functions.

Existence of an extra Grassmann odd multiplet does not imply ${ m extbf{N}=2}$ supersymmetry.

Three-point functions are generally not contained in ${ m extbf{N}=2}$ supercurrent correlations.

Abstract

We consider ${\mathcal N}=1$ superconformal field theories in four dimensions possessing an additional conserved spinor current multiplet $S_{\alpha}$ and study three-point functions involving such an operator. A conserved spinor current multiplet naturally exists in superconformal theories with ${\mathcal N}=2$ supersymmetry and contains the current of the second supersymmetry. However, we do not assume ${\mathcal N}=2$ supersymmetry. We show that the three-point function of two spinor current multiplets and the ${\mathcal N}=1$ supercurrent depends on three independent tensor structures and, in general, is not contained in the three-point function of the ${\mathcal N}=2$ supercurrent. It then follows, based on symmetry considerations only, that the existence of one more Grassmann odd current multiplet in ${\mathcal N}=1$ superconformal field theory does not necessarily imply ${\mathcal N}=2$ superconformal symmetry.