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 $k_a \cdot k_b$, $\epsilon_a \cdot k_b$, and $\epsilon_a \cdot \epsilon_b$, valid in any number of spacetime dimensions. We use dimensional reduction of higher-dimensional amplitudes of particles with internal momentum $\kappa$ 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 $k_a \cdot k_b$ with $k_a \cdot k_b - \kappa_a \kappa_b$ in the corresponding massless amplitudes, where the masses of the particles are given by $|\kappa_a|$.