General Three-Point Functions in 4D CFT

TL;DR

This work provides a complete classification of the most general three-point functions in 4D CFTs for bosonic and fermionic operators in arbitrary Lorentz representations, using a 6D embedding in twistor space with an index-free formalism. It introduces a compact master formula for three-point tensor structures built from SU(2,2) invariants, and demonstrates how current conservation can be implemented covariantly in 6D, with unitarity-bound saturation playing a key role. By linking three-point data to four-point structures through the OPE and verifying crossing symmetry, the authors derive analytic counts for specific cases (notably two symmetric traceless tensors and an arbitrary third operator) and provide a practical framework for constructing higher-point conformal blocks. The results advance the conformal bootstrap program beyond scalars and offer a systematic method to organize and count tensor structures in mixed representations.

Abstract

We classify and compute, by means of the six-dimensional embedding formalism in twistor space, all possible three-point functions in four dimensional conformal field theories involving bosonic or fermionic operators in irreducible representations of the Lorentz group. We show how to impose in this formalism constraints due to conservation of bosonic or fermionic currents. The number of independent tensor structures appearing in any three-point function is obtained by a simple counting. Using the Operator Product Expansion (OPE), we can then determine the number of structures appearing in 4-point functions with arbitrary operators. This procedure is independent of the way we take the OPE between pairs of operators, namely it is consistent with crossing symmetry, as it should be. An analytic formula for the number of tensor structures for three-point correlators with two symmetric and an arbitrary bosonic (non-conserved) operators is found, which in turn allows to analytically determine the number of structures in 4-point functions of symmetric traceless tensors.