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 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.
Highlights
Scattering amplitudes of massive spin-2 particles have been investigated in a variety of contexts throughout the last few decades
The growth of the elastic scattering amplitude of massive spin-2 states in the FP theory has been computed using a variety of approaches, including direct calculation [2], the Stueckelberg formalism [3,4], and via deconstruction [5,6]
We have described here only the scattering of the longitudinal helicity states, we have verified that these relations are sufficient to ensure that the elastic scattering of all helicity states cancels down to OðsÞ as well
Summary
Scattering amplitudes of massive spin-2 particles have been investigated in a variety of contexts throughout the last few decades. We demonstrate that the relationships needed to ensure the elastic scattering amplitudes grow no faster than Oðs3Þ arise from the Sturm-Liouville form of the KK mode expansion in the internal space and will naturally generalize to KK theories with internal dimensions with arbitrary internal structure. These relations are closely related to coupling relationships which arise in compactified gauge theories [18].1. We conclude with a discussion of the properties of an effective theory in which the tower of KK modes is truncated and discuss questions for future investigation
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