Three-point functions of higher-spin supercurrents in 4D ${\cal N}=1$ 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 ${\cal N}=1$ superconformal theory. All the constraints imposed by ${\cal N}=1$ superconformal symmetry on the three-point function $\langle J_{\alpha(r_1) \dot{\alpha}(r_1)} J_{\beta(r_2) \dot{\beta}(r_2) }J_{\gamma(r_3) \dot{\gamma}(r_3)}\rangle$ 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 $\langle J_{\alpha(r) \dot{\alpha}(r)} J_{\beta \dot{\beta} } J_{\gamma \dot{\gamma}}\rangle$, 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 $\langle L L J_{\alpha(r) \dot{\alpha}(r)}\rangle$ and $\langle J_{\alpha(r_1) \dot{\alpha}(r_1)} J_{\beta(r_2) \dot{\beta}(r_2)} L \rangle$, where $L$ is the flavour current multiplet.