Scattering Amplitudes of Kaluza-Klein Strings and Extended Massive Double-Copy

We study the scattering amplitudes of massive Kaluza-Klein (KK) states of open and closed bosonic strings under toroidal compactification. We analyze the structure of vertex operators for the KK strings and derive an extended massive KLT-like relation which connects the $N$-point KK closed-string amplitude to the products of two KK open-string amplitudes at tree level. Taking the low energy field-theory limit of vanishing Regge slope, we derive double-copy construction formula of the $N$-point massive KK graviton amplitude from the sum of proper products of the corresponding KK gauge boson amplitudes. Then, using the string-based massive double-copy formula, we derive the exact tree-level four-point KK gauge boson amplitudes and KK graviton amplitudes, which fully agree with those given by the KK field-theory calculations. With these, we give an explicit prescription on constructing the exact four-point KK graviton amplitudes from the sum of proper products of the corresponding color-ordered KK gauge boson amplitudes. We further analyze the string-based double-copy construction of five-point and six-point scattering amplitudes of massive KK gauge bosons and KK gravitons.