Three-Point Functions of Higher-Spin Supercurrents in 4D N=1${{\cal N}}=1$ Superconformal Field Theory

We develop a general formalism to study the three-point correlation functions of conserved higher-spin supercurrent multiplets  J𝜶(r)̇𝜶(r) in 4D N=1 superconformal theory. All the constraints imposed by N=1 superconformal symmetry on the three-point function ⟨J𝜶(r₁)̇𝜶(r₁)J𝜷(r₂)̇𝜷(r₂)J𝜸(r₃)̇𝜸(r₃)⟩ are systematically derived for arbitrary r₁,r₂,r₃, thus reducing the problem mostly to computational and combinatorial. As an illustrative example, we explicitly work out the allowed tensor structures contained in ⟨J𝜶(r)̇𝜶(r)J𝜷̇𝜷J𝜸̇𝜸⟩, where J𝜶̇𝜶 is the supercurrent. We find that this three-point function depends on two independent tensor structures, though the precise form of the correlator depends on whether r is even or odd. The case r=1 reproduces the three-point function of the ordinary supercurrent derived by Osborn. Additionally, we present the most general structure of mixed correlators of the form ⟨LLJ𝜶(r)̇𝜶(r)⟩and ⟨J𝜶(r₁)̇𝜶(r₁)J𝜷(r₂)̇𝜷(r₂)L⟩, where L is the flavour current multiplet.