Correlation functions of spinor current multiplets in N = 1 superconformal theory

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