On the Eikonal Approximation in AdS Space

TL;DR

This work analyzes high-energy scattering in AdS$_5$ gravity within a conformal, large-$\lambda$ regime relevant to gauge theory via AdS/CFT. It develops three complementary derivations of the eikonal amplitude for graviton exchange: a heuristic scaling argument, a diagrammatic resummation of Witten diagrams with gravitons, and a bulk shock-wave derivation, all in the limit of linearized gravity. The key result is that the eikonal phase is controlled by the $AdS_3$ scalar propagator, $\chi(s,x^\perp - x'^\perp, z, z') = \frac{1}{2} \frac{\kappa_5^2}{R^3} (s z z') G_3(u)$, leading to an amputated amplitude ${\cal M}_{eik} = -2 i s (\frac{z z'}{R^2})^2 [ e^{ i \chi } - 1]$; the same phase is recovered in the shock-wave derivation. The results reveal that high-energy four-dimensional gauge-theory scattering at strong coupling is governed by two-dimensional conformal invariance in the transverse AdS$_3$ and provide a structured framework for extending eikonal methods to finite coupling, confinement, and nonlinear/string effects. Together, the approaches illuminate how eikonalization emerges in AdS spaces and establish a basis for connecting bulk gravity calculations to boundary gauge dynamics in the high-energy limit.

Abstract

We explore the eikonal approximation to graviton exchange in AdS_5 space, as relevant to scattering in gauge theories. We restrict ourselves to the regime where conformal invariance of the dual gauge theory holds, and to large 't Hooft coupling where the computation involves pure gravity. We give a heuristic argument, a direct loop computation, and a shock wave derivation. The scalar propagator in AdS_3 plays a key role, indicating that even at strong coupling, two-dimensional conformal invariance controls high-energy four-dimensional gauge-theory scattering.

Figures (2)

• Figure 1: Ladder and crossed ladder diagrams contributing to the eikonal approximation in the high energy limit.
• Figure 2: Sum of box and cross box diagram is factorized with combinatoric weight $1/2!$.