Three-point functions of fermionic higher-spin currents in 4D conformal field theory

TL;DR

This paper analyzes three-point functions involving fermionic higher-spin currents in 4D conformal field theories, revealing that these functions generally do not align with $ ext{N}=1$ supersymmetry, and explores their tensor structures.

Contribution

It provides a detailed computational study of the tensor structures of three-point functions involving fermionic higher-spin currents and shows their incompatibility with $ ext{N}=1$ supersymmetry.

Findings

Three-point functions have multiple independent tensor structures.

Fermionic higher-spin currents with $k=1$ resemble supersymmetry currents.

These correlation functions are generally inconsistent with $ ext{N}=1$ supersymmetry.

Abstract

We investigate the properties of a four-dimensional conformal field theory possessing a fermionic higher-spin current $Q_{\alpha(2k) \dot{\alpha}}$. Using a computational approach, we examine the number of independent tensor structures contained in the three-point correlation functions of two fermionic higher-spin currents with the conserved vector current $V_{m}$, and with the energy-momentum tensor $T_{m n}$. In particular, the $k = 1$ case corresponds to a "supersymmetry-like" current, that is, a fermionic conserved current with identical properties to the supersymmetry current which appears in $\mathcal{N} = 1$ superconformal field theories. However, we show that in general, the three-point correlation functions $\langle Q Q T\rangle $, $\langle \bar{Q} Q V\rangle $ and $\langle \bar{Q} Q T\rangle $ are not consistent with $\mathcal{N}=1$ supersymmetry