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Eikonal Scattering in Kaluza-Klein Gravity

Arnau Koemans Collado, Steven Thomas

TL;DR

This work analyzes high-energy 2→2 scattering in a five-dimensional Einstein–gravity setup compactified on a circle, yielding a four-dimensional theory with a KK tower of charged scalars and a massless sector of graviton, photon, and dilaton. By computing the Born and one-loop amplitudes and transforming to impact-parameter space, the authors demonstrate eikonal exponentiation in the KK theory, and they connect the leading eikonal to the time delay of a massless particle in a compactified Aichelburg–Sexl shock-wave background while the large KK-mass limit reproduces the deflection angle of massless probes in Einstein–Maxwell–dilaton black holes with extremal charge-to-mass ratio $Q=2M$. They also show the subleading eikonal vanishes in this KK context due to the extremal relation, and they relate these results to D0–D6 brane dualities and magnetically charged dilatonic black holes, providing a coherent bridge between quantum scattering amplitudes and classical gravitational observables in KK geometries. The findings illuminate how extra dimensions influence gravitational scattering and offer a framework for connecting amplitude techniques to astrophysical and string-theoretic backgrounds.

Abstract

We study eikonal scattering in the context of Kaluza-Klein theory by considering a massless scalar field coupled to Einstein's gravity in 5D compactified to 4D on a manifold $M_4\times S^1 $. We also examine various different kinematic limits of the resulting eikonal. In the ultra-relativistic scattering case we find correspondence with the time delay calculated for a massless particle moving in a compactified version of the Aichelburg-Sexl shock-wave geometry. In the case of a massless Kaluza-Klein scalar scattering off a heavy Kaluza-Klein mode a similar calculation yields the deflection angle of a massless particle in the geometry of an extremal, $Q=2M$, Einstein-Maxwell-dilaton 4D black hole. We also discuss a related case in the scattering of dilatons off a large stack of $D0$-branes or $D6$-branes in dimensionally reduced $D=10$ type IIA string theory.

Eikonal Scattering in Kaluza-Klein Gravity

TL;DR

This work analyzes high-energy 2→2 scattering in a five-dimensional Einstein–gravity setup compactified on a circle, yielding a four-dimensional theory with a KK tower of charged scalars and a massless sector of graviton, photon, and dilaton. By computing the Born and one-loop amplitudes and transforming to impact-parameter space, the authors demonstrate eikonal exponentiation in the KK theory, and they connect the leading eikonal to the time delay of a massless particle in a compactified Aichelburg–Sexl shock-wave background while the large KK-mass limit reproduces the deflection angle of massless probes in Einstein–Maxwell–dilaton black holes with extremal charge-to-mass ratio . They also show the subleading eikonal vanishes in this KK context due to the extremal relation, and they relate these results to D0–D6 brane dualities and magnetically charged dilatonic black holes, providing a coherent bridge between quantum scattering amplitudes and classical gravitational observables in KK geometries. The findings illuminate how extra dimensions influence gravitational scattering and offer a framework for connecting amplitude techniques to astrophysical and string-theoretic backgrounds.

Abstract

We study eikonal scattering in the context of Kaluza-Klein theory by considering a massless scalar field coupled to Einstein's gravity in 5D compactified to 4D on a manifold . We also examine various different kinematic limits of the resulting eikonal. In the ultra-relativistic scattering case we find correspondence with the time delay calculated for a massless particle moving in a compactified version of the Aichelburg-Sexl shock-wave geometry. In the case of a massless Kaluza-Klein scalar scattering off a heavy Kaluza-Klein mode a similar calculation yields the deflection angle of a massless particle in the geometry of an extremal, , Einstein-Maxwell-dilaton 4D black hole. We also discuss a related case in the scattering of dilatons off a large stack of -branes or -branes in dimensionally reduced type IIA string theory.

Paper Structure

This paper contains 16 sections, 106 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Kaluza-Klein Theory
  3. Scattering in the Born approximation
  4. Scattering on ${\mathbb R}^{1,D-1}$
  5. Scattering on ${\mathbb R}^{1,D-2} \times S^1$
  6. Double Exchange Scattering
  7. Scattering on ${\mathbb R}^{1,D-1}$
  8. Scattering on ${\mathbb R}^{1,D-2} \times S^1$
  9. Various Kinematic Limits of the Eikonal
  10. The Leading Eikonal in the Ultra-Relativistic Limit
  11. The Leading Eikonal in the Large Kaluza-Klein Mass Limit
  12. The Subleading Eikonal in the Large Kaluza-Klein Mass Limit
  13. $D0$-$D6$ Brane Duality and Magnetically Charged Dilatonic Black Holes
  14. Discussion
  15. Deflection Angle for Einstein-Maxwell-dilaton Black Hole
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Feynman diagram representation for tree-level scalar scattering with graviton exchange. The solid lines represent scalar states and the wavy lines represent gravitons.
  • Figure 2: The various constituent diagrams found in tree-level scalar scattering with graviton exchange on ${\mathbb R}^{1,D-2} \times S^1$ when decomposed into seperate dilaton, gauge field and dimensionally reduced graviton contributions. The solid lines represent scalar states, the dashed line represent gauge fields and the wavy lines represent gravitons. Note that the internal solid lines represent massless dilaton states.
  • Figure 3: Feynman diagram representation for one-loop scalar scattering with two graviton exchanges. The solid lines represent scalar states and the wavy lines represent gravitons.
  • Figure 4: Feynman diagram representation for one-loop scalar scattering with two dilaton exchanges. The solid lines represent scalar states and the wavy lines represent gravitons. Note that the internal solid lines represent massless dilaton states.