Higher-spin massless S matrices in four dimensions

On-shell, analytic S-matrix elements in massless theories are constructed from a finite set of primitive three-point amplitudes, which are fixed by Poincare invariance up to an overall numerical constant. We classify all such three-point amplitudes in four dimensions. Imposing the simplest incarnation of locality and unitarity on four-particle amplitudes constructed from these three-particle amplitudes rules out all but an extremely small subset of interactions among higher-spin massless states. Notably, the equivalence principle and the Weinberg-Witten theorem are simple corollaries of this principle. Further, no massless states with helicity larger than two may consistently interact with massless gravitons. Chromodynamics, electrodynamics, Yukawa and phi(3) theories are the only marginal and relevant interactions between massless states. Finally, we show that supersymmetry naturally emerges as a consistency condition on four-particle amplitudes involving spin-3/2 states, which must always interact gravitationally.