Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension

TL;DR

This work proves the Cachazo–He–Yuan formula for N-point Yang–Mills tree amplitudes in arbitrary spacetime dimensions by first establishing the corresponding result for massless $\phi^3$ theory using BCFW recursions and then extending to gauge theory; it further generalizes the CHY scattering equations to massive external states, enabling massive $\phi^3$ amplitudes. A direct BCFW proof for massless $\phi^3$ is provided, validating the inductive construction, while a Pfaffian-based CHY construction is shown to yield the YM tree amplitudes, satisfying factorization and Möbius invariance. Collectively, the paper connects CHY proposals with standard field-theory amplitudes across dimensions and opens the path to systematic massive extensions within the CHY formalism.

Abstract

A proof is given of the formula, recently proposed by Cachazo, He and Yuan (CHY) for gluon tree amplitudes in pure Yang-Mills theory in arbitrary dimension. The approach is to first establish the corresponding result for massless $φ^3$ theory using the BCFW recurrence relation and then to extend this to the gauge theory case. Additionally, it is shown that the scattering equations introduced by CHY can be generalized to massive particles, enabling the description of tree amplitudes for massive $φ^3$ theory.