2 Our discussion here follows closely the review in Sec. 8 of [21]. 3 Note that the mass of the graviton is stringently constrained by gravitational wave experiments to be m < 1.22 × 10 −22 eV/c 2 [23]. 4 The relative coefficients between the two terms in the mass term are chosen to avoid propagating ghost degrees of freedom. For more details, consult footnote 7.

It has recently been shown explicitly that the high-energy scattering amplitude of the longitudinal modes of massive spin-2 Kaluza Klein states in compactified 5-dimensional gravity, which would naively grow like O(s 5 ), grow only like O(s). Since the individual contributions to these amplitudes do grow like O(s 5 ), the required cancellations between these individual contributions result from intricate relationships between the masses of these states and their couplings. Here we report the explicit form of these sum-rule relationships which ensure the necessary cancellations for elastic scattering of spin-2 Kaluza Klein states in a Randall-Sundrum model. We consider an Anti-de-Sitter space of arbitrary curvature, including the special case of a toroidal compactification in which the curvature vanishes. The sum rules demonstrate that the cancellations at O(s 5 ) and O(s 4 ) are generic for a compact extra dimension, and arise from the Sturm-Liouville structure of the eigenmode system in the internal space. Separately, the sum rules at O(s 3 ) and O(s 2 ) illustrate the essential role of the radion mode of the extra-dimensional metric, which is the dynamical mode related to the size of the internal space. a MSUHEP-19-023

We present the results of the first complete calculation of the tree-level 2 → 2 high-energy scattering amplitudes of the longitudinal modes of massive spin-2 Kaluza-Klein states, both in the case where the internal space is a torus and in the Randall-Sundrum model where the internal space has constant negative curvature. While individual contributions to this amplitude grow as O(s 5 ), we demonstrate explicitly that intricate cancellations occur between different contributions, reducing the growth to O(s). These cancellations require subtle relationships between the masses of the Kaluza-Klein states and their interactions, and reflect the underlying higher-dimensional diffeomorphism invariance. The results illuminate the range of validity of (effective) field theories involving massive spin-2 KK particles.[31] H. Davoudiasl and T. G. Rizzo, Phys. Lett. B512, 100 (2001), arXiv:hep-ph/0104199 [hep-ph].