Sum Rules for Massive Spin-2 Kaluza-Klein Elastic Scattering Amplitudes

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.