Scattering of Massless Particles in Arbitrary Dimension

TL;DR

This work presents a dimension-agnostic, compact construction of tree-level S-matrices for pure Yang–Mills and gravity using scattering equations that tie kinematics to puncture positions on a sphere. The central ingredients are a permutation-invariant measure, a Jacobian ${\det}'\Phi$, and a universal integrand built from the reduced Pfaffian ${\rm Pf}'\Psi$ (with gravity using a product of two such Pfaffians or an equivalent determinant). Gauge invariance emerges transparently from the Pfaffian structure, and the framework passes nontrivial checks including known lower-point results, soft limits, and 4D numeric tests up to $n\le 8$ across helicity sectors. The approach also links to string theory amplitudes and suggests a new interpretation of each scattering-equation solution as a partial amplitude, offering a promising route to a unified, high-dimensional amplitude program.

Abstract

We present a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. The new formula for the scattering of n particles is given by an integral over the position 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 by 2n matrix 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.