Superspace invariants and correlators in 4d $\mathcal{N}=1$ superconformal field theories

TL;DR

The paper develops a systematic framework to classify and construct 3-point correlators of spinning operators in four-dimensional ${\cal N}=1$ superconformal field theories. Building on the well-established 4d CFT invariants constructed with polarization spinors, the authors extend to superspace, introducing superinversion and a complete basis of superconformal invariants (including novel Grassmannian structures) to express any spinning 3-point SCFT correlator. They provide a detailed, case-by-case analysis of correlators with varying spins and parity, including numerous explicit forms and the impact of conservation (shortening) constraints that fix correlators up to a small set of coefficients. The results reproduce and extend known Osborn/Buchbinder-type constructions, and offer a comprehensive toolkit for analyzing SCFT correlators with both conserved and non-conserved operators, with potential applications to higher supersymmetry and mixed-symmetry tensors. The work thereby clarifies the structure of 3-point functions in 4d SCFTs and sets the stage for exploring higher-point functions and extended multiplets.

Abstract

Using polarization spinor methods in conjunction with the superspace formalism, we construct 3-point superconformal invariants that are used to determine the form of 3-point correlators of spinning superfield operators in $\mathcal{N}=1$ superconformal field theories (SCFTs) in 4-dimensions. We enumerate the structural form of various spinning 3-point correlators using these invariants and find additional constraints on their form when the operators are conserved supercurrents. For these purposes, we first construct the invariants and 3-point correlators in non-supersymmetric $4d$ CFTs which are then extended using superspace methods to $4d$ SCFTs.