Three-point functions of higher-spin supercurrents in 4D $\mathcal{N}=1$ SCFT: general formalism for arbitrary superspins

Abstract

We analyse the general structure of the three-point functions involving conserved higher-spin ``vector-like" supercurrents $J_{s}(z) := J_{\alpha(s) \dot{\alpha}(s)}(z)$ in four-dimensional $\mathcal{N}=1$ superconformal field theory. Using the constraints of superconformal symmetry and superfield conservation equations, we utilise a computational approach to analyse the general structure of the three-point function $\langle J^{}_{s_{1}}(z_{1}) \, J'_{s_{2}}(z_{2}) \, J''_{s_{3}}(z_{3}) \rangle$ and provide a general classification of the results. We demonstrate that the three-point function is fixed up to $2 \min (s_{i}) + 2$ independent conserved structures, which we propose to hold for arbitrary superspins. In addition, we show that the conserved structures can be classified as parity-even or parity-odd in superspace based on their transformation properties under superinversion.