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

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 $T_{mn}$. 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 $$\mathcal{N}$ = 1$ superconformal field theories. However, we show that in general, the three-point correlation functions $⟨QQT⟩$, $\langle \bar{Q} Q V\rangle $and$\langle \bar{Q} Q T\rangle $arenotconsistentwith$$\mathcal{N}$=1$ supersymmetry