In this work, we study the scattering amplitudes of massive Kaluza-Klein (KK)
states under toroidal compactification. We present a shifting method to
quantitatively derive the scattering amplitudes of massive KK gauge bosons and
KK gravitons from the corresponding massless amplitudes in the noncompactified
higher dimensional theories. With these we construct the massive KK scattering
amplitudes by extending the double-copy relations of massless scattering
amplitudes within the field theory formulations, including both the BCJ method
and CHY method. We present the massive BCJ-type double-copy construction of the
$N$-point KK gauge boson/graviton scattering amplitudes, and as the
applications we derive explicitly the four-point KK scattering amplitudes as
well as the five-point KK scattering amplitudes. We further study the
nonrelativistic limit of these massive scattering amplitudes with the heavy
external KK states and discuss the impact of the compactified extra dimensions
on the low energy gravitational potential. Finally, we analyze the four-point
and $N$-point mass spectral conditions and newly propose a group theory
approach to prove that {\it only the KK theories} under toroidal
compactification can satisfy these conditions for realizing the massive
double-copy.
