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

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