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

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 ${\varphi }^3$ 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 ${\varphi }^3$ theory.