[Paper Review] Scattering Amplitudes in Theories of Compactified Gravity

Dennis Foren|arXiv (Cornell University)|Aug 23, 2020

Black Holes and Theoretical Physics34 references1 citations

TL;DR

This dissertation calculates 2-to-2 scattering amplitudes of massive spin-2 Kaluza-Klein (KK) modes in the Randall-Sundrum 1 (RS1) model and five-dimensional orbifolded torus (5DOT) compactifications. It demonstrates that despite individual diagrams diverging as O(s⁵), total amplitudes grow only as O(s) due to intricate cancellations across infinite towers of KK modes, requiring exact sum rules for consistency. The work derives these sum rules and computes the 5D strong coupling scale as Λπ ≡ MPl e⁻ᵏʳᶜπ.

ABSTRACT

In this dissertation we discuss the properties of matrix elements describing the scattering of massive spin-2 particles in theories of compactified gravity. Our primary result is the calculation of 2-to-2 massive spin-2 Kaluza-Klein (KK) mode scattering matrix elements in the Randall-Sundrum 1 (RS1) model and the demonstration that those matrix elements grow no faster than $\mathcal{O}(s)$ irrespective of the KK mode numbers and helicities considered. Because this calculation requires summing infinitely-many spin-2 mediated diagrams which each diverge like $\mathcal{O}(s^{5})$, overall $\mathcal{O}(s)$ growth is only attained through cancellations between these diagrams. This in turn requires intricate cancellations between infinitely-many KK mode masses and couplings. We derive these sum rules, including their generalization to fully inelastic processes. We also consider these matrix elements in the five-dimensional orbifolded torus (5DOT) and large $kr_{c}$ limits, investigate the impact of including only finitely-many diagrams in the calculation (as measured via truncation error), and calculate the five-dimensional strong coupling scale $Λ_π \equiv M_{ ext{Pl}}\, e^{-kr_{c}π}$ via the four-dimensional scattering calculation.

Motivation & Objective

• To understand the high-energy behavior of massive spin-2 KK modes in extra-dimensional gravity models.
• To resolve the apparent conflict between individual O(s⁵) divergent diagrams and the observed O(s) growth of total scattering amplitudes.
• To derive exact sum rules for KK masses and couplings that ensure unitarity and consistency at high energies.
• To compute the 5D strong coupling scale Λπ = MPl e⁻ᵏʳᶜπ from 4D scattering amplitudes.
• To analyze truncation errors when only finitely many KK modes are included in calculations.

Proposed method

• Derives the 4D effective field theory of massive spin-2 KK modes from the 5D RS1 and 5DOT models via Kaluza-Klein decomposition.
• Constructs the full set of 2-to-2 scattering matrix elements using helicity eigenstates and Lorentz-invariant phase space integrals.
• Computes infinite sums over KK modes using contour integration and residue theorems to handle divergent diagrams.
• Applies the optical theorem and unitarity constraints to derive sum rules linking KK masses and couplings.
• Performs explicit calculations in the center-of-momentum frame using Wigner D-matrices and partial wave decomposition.
• Evaluates the strong coupling scale Λπ via matching 4D scattering amplitudes to the 5D effective theory.

Experimental results

Research questions

• RQ1 How do 2-to-2 scattering amplitudes of massive spin-2 KK modes behave at high energies in the RS1 model?
• RQ2 Why does the total amplitude grow only as O(s) despite individual diagrams diverging as O(s⁵)?
• RQ3 What sum rules must KK masses and couplings satisfy to ensure O(s) growth and unitarity?
• RQ4 How does the inclusion of only finitely many KK modes affect the accuracy of scattering amplitude calculations?
• RQ5 What is the 5D strong coupling scale Λπ in terms of the 4D Planck scale and the curvature scale in the RS1 model?

Key findings

• The total 2-to-2 scattering amplitude for massive spin-2 KK modes in the RS1 model grows at most as O(s), not faster, due to cancellations across infinite KK mode contributions.
• Each individual diagram involving a single KK mode diverges as O(s⁵), but the sum over all KK modes converges to O(s) growth.
• The cancellation mechanism requires exact sum rules relating the masses and couplings of all KK modes, which are derived explicitly.
• These sum rules generalize to fully inelastic processes, ensuring unitarity across all channels.
• The 5D strong coupling scale is computed as Λπ = MPl e⁻ᵏʳᶜπ, consistent with known holographic expectations.
• Truncation errors from including only finitely many KK modes are quantified, showing slow convergence without the full sum rule.