A Compact Formula for Conserved Three-Point Tensor Structures in 4D CFT

TL;DR

This paper presents a compact, analytic formula for all conformally invariant tensor structures in three-point functions of conserved operators in 4D CFT, using a novel superspace approach.

Contribution

It introduces a unified superspace framework to derive a complete basis of tensor structures, simplifying conservation constraints and linking structure counting to SU(2n) representation theory.

Findings

Derived a compact formula for three-point tensor structures in 4D CFT.

Connected tensor structure counting to Littlewood-Richardson coefficients.

Applicable to superconformal theories with N=2 and N=4.

Abstract

We derive a compact analytic formula for a complete basis of conformally invariant tensor structures for three-point functions of conserved operators in arbitrary 4D Lorentz representations. The construction follows directly from a novel constraint equivalent to applying conservation conditions at each point: the leading terms in all OPE limits appear as symmetric traceless tensors. We derive this by lifting to a unified $\mathrm{SU}(m,m|2n)$ analytic superspace framework, where the conservation conditions are automatically solved and then reducing back to 4D CFT. The same method is also used for cases involving one non-conserved operator. This formalism further reveals a map of the counting of CFT tensor structures to that of finite-dimensional $\mathrm{SU}(2n)$ representations, solved by Littlewood-Richardson coefficients. All results can be directly re-interpreted as three-point $\mathcal{N}=2$ and $\mathcal{N}=4$ superconformal tensor structures via the unified analytic superspace.