Sum rules for massive spin-2 Kaluza-Klein elastic scattering amplitudes

Abstract

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 five-dimensional gravity, which would naively grow like O(s5), grow only like O(s). Since the individual contributions to these amplitudes do grow like O(s5), 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(s5) and O(s4) 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(s3) and O(s2) illustrate the essential role of the radion mode of the extradimensional metric, which is the dynamical mode related to the size of the internal space.