Constraining conformal field theories with a higher spin symmetry in d=4

We study unitary conformal field theories with a unique stress tensor and at least one higher-spin conserved current in four dimensions. We prove that every such theory contains an infinite number of higher-spin conserved currents of arbitrarily high spin, and that Ward identities generated by the conserved charges of these currents suffice to completely fix the correlators of the stress tensor and the conserved currents to be equal to one of three free field theories: the free boson, the free fermion, and the free vector field. This is a generalization of the result proved in three dimensions by Maldacena and Zhiboedov [arXiv:1112.1016].