Three-point functions of higher-spin supercurrents in 4D $\mathcal{N}=1$ superconformal field theory

We develop a general formalism to study the three-point correlation functions of conserved higher-spin supercurrent multiplets $J_{\alpha (r)\dot{\alpha }(r)}$ in 4D $\mathcal{N}=1$ superconformal theory. All the constraints imposed by $\mathcal{N}=1$ superconformal symmetry on the three-point function $⟨J_{\alpha (r_1)\dot{\alpha }(r_1)}{J}_{\beta (r_2)\dot{\beta }(r_2)}{J}_{\gamma (r_3)\dot{\gamma }(r_3)}⟩$ are systematically derived for arbitrary $r_1,r_2,r_3$, thus reducing the problem mostly to computational and combinatorial. As an illustrative example, we explicitly work out the allowed tensor structures contained in $⟨J_{\alpha (r)\dot{\alpha }(r)}{J}_{\beta \dot{\beta }}{J}_{\gamma \dot{\gamma }}⟩$, where $J_{\alpha \dot{\alpha }}$ 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 $⟨LL{J}_{\alpha (r)\dot{\alpha }(r)}⟩$ and $⟨J_{\alpha (r_1)\dot{\alpha }(r_1)}{J}_{\beta (r_2)\dot{\beta }(r_2)}L⟩$, where $L$ is the flavour current multiplet.