Scattering of Massless Particles in Arbitrary Dimensions

We present a compact formula for the complete tree-level $S$-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimensions. The new formula for the scattering of $n$ particles is given by an integral over the positions of $n$ points on a sphere restricted to satisfy a dimension-independent set of equations. The integrand is constructed using the reduced Pfaffian of a $2n\times 2n$ matrix, $\mathrm{\Psi }$, that depends on momenta and polarization vectors. In its simplest form, the gravity integrand is a reduced determinant which is the square of the Pfaffian in the Yang-Mills integrand. Gauge invariance is completely manifest as it follows from a simple property of the Pfaffian.