  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 $|\kappa_a|$.
