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

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.

37 pages, v2: Mathematica notebook attached, v3: fixed typos and added clarifications in sec 3.4