Three-point functions of conserved currents in 4D CFT: general formalism for arbitrary spins

TL;DR

This paper develops a general formalism for analyzing three-point functions of conserved higher-spin currents in four-dimensional conformal field theories, providing a classification of possible structures for arbitrary spins.

Contribution

It introduces a comprehensive framework for classifying three-point functions involving conserved currents of any Lorentz representation in 4D CFTs, extending known results to more general cases.

Findings

Number of conserved structures varies for higher-spin currents.

Classification of three-point functions for arbitrary spins.

Deviations from known structures in specific cases.

Abstract

We analyse the general structure of the three-point functions involving conserved higher-spin currents $J_{s} := J_{\alpha(i) \dot{\alpha}(j)}$ belonging to any Lorentz representation in four-dimensional conformal field theory. Using the constraints of conformal symmetry and conservation equations, we computationally analyse the general structure of three-point functions $\langle J^{}_{s_{1}} J'_{s_{2}} J''_{s_{3}} \rangle$ for arbitrary spins and propose a classification of the results. For bosonic vector-like currents with $i=j$, it is known that the number of independent conserved structures is $2 \min (s_{i}) + 1$. For the three-point functions of conserved currents with arbitrarily many dotted and undotted indices, we show that in many cases the number of structures deviates from $2 \min (s_{i}) + 1$.