Amplitudes for massive vector and scalar bosons in spontaneously-broken gauge theory from the CHY representation

In the formulation of Cachazo, He, and Yuan, tree-level amplitudes for massless particles in gauge theory and gravity can be expressed as rational functions of the Lorentz invariants ka · kb, ϵa · kb, and ϵa · ϵb, valid in any number of spacetime dimensions. We use dimensional reduction of higher-dimensional amplitudes of particles with internal momentum κ to obtain amplitudes for massive particles in lower dimensions. In the case of gauge theory, we argue that these massive amplitudes belong to a theory in which the gauge symmetry is spontaneously broken by an adjoint Higgs field. Consequently, we show that tree-level n-point amplitudes containing massive vector and scalar bosons in this theory can be obtained by simply replacing ka · kb with ka · kb − κaκb in the corresponding massless amplitudes, where the masses of the particles are given by |κa|.