# Scattering Amplitudes of Massive Spin-2 Kaluza-Klein States with Matter

**Authors:** R. Sekhar Chivukula, Joshua A. Gill, Kirtimaan A. Mohan, Dipan Sengupta, Elizabeth H. Simmons, Xing Wang  
**arXiv:** [2311.00770](https://arxiv.org/abs/2311.00770)  
**Date:** 2023-11-01  
**Summary page:** https://www.summarizepaper.com/en/paper/2311.00770v2/

## Key points

- Comprehensive analysis of scattering of matter and gravitational Kaluza-Klein (KK) modes in five-dimensional gravity theories
- Investigation of behavior of scalars, fermions, and vectors localized on a brane and in the bulk of an extra dimension
- Analysis performed in an arbitrary warped background
- Cancellations between contributions result in total amplitude growing no faster than O(s)
- Cancellations occur due to sum-rule relations between couplings and masses of modes
- Relations can be proven from properties of mode equations tied to diffeomorphism invariance
- Generalization of results when considering stabilization of size of extra dimension via Goldberger-Wise mechanism
- Implications for freeze-out and freeze-in relic abundance calculations in dark matter models involving spin-2 portal from five-dimensional theory
- Provides insights into scattering amplitudes involving matter and gravitational KK modes in five-dimensional gravity theories
- Highlights cancellations between contributions and sheds light on sum rule relations associated with diffeomorphism invariance

## Summary

In this study, the authors conduct a comprehensive analysis of the scattering of matter and gravitational Kaluza-Klein (KK) modes in five-dimensional gravity theories. They investigate the behavior of scalars, fermions, and vectors localized on a brane as well as in the bulk of an extra dimension. The analysis is performed in an arbitrary warped background. Naive power-counting suggests that there are scattering amplitudes that grow as fast as ${\cal O}(s^3)$, where $s$ represents the center-of-mass scattering energy-squared. However, the authors demonstrate that cancellations between various contributions result in a total amplitude that grows no faster than ${\cal O}(s)$. This finding challenges the initial expectation based on power-counting arguments. Building upon previous work on self-interactions of gravitational KK modes, the authors show that these cancellations occur due to sum-rule relations between the couplings and masses of the modes. These relations can be proven from properties of mode equations describing gravity and matter wavefunctions which are tied to the underlying diffeomorphism invariance of the five-dimensional theory. The authors also discuss how their results generalize when considering stabilization of the size of the extra dimension via the Goldberger-Wise mechanism which has implications for freeze-out and freeze-in relic abundance calculations in dark matter models involving a spin-2 portal arising from a five dimensional theory. Overall, this study provides valuable insights into scattering amplitudes involving matter and gravitational KK modes in five dimensional gravity theories, highlighting important cancellations between contributions and shedding light on sum rule relations associated with diffeomorphism invariance which have implications for understanding dark matter models and more accurate relic abundance calculations.