General Three-Point Functions in 4D CFT

TL;DR

This paper classifies and computes all possible three-point functions in 4D conformal field theories using a six-dimensional embedding formalism, providing a systematic way to count tensor structures and analyze constraints like current conservation.

Contribution

It introduces a formalism for classifying three-point functions in 4D CFTs and derives an analytic formula for counting tensor structures, including constraints from conservation laws.

Findings

Derived an analytic formula for tensor structures in three-point functions.

Established a counting method consistent with crossing symmetry.

Provided a systematic approach to analyze 4-point functions using OPE.

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.