Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions

TL;DR

This work develops an AdS/CFT-compatible eikonal framework by analyzing shock-wave geometries in AdS and relating the resulting two-point functions to the discontinuities of dual CFT four-point functions. It provides explicit integral representations involving transverse space propagators that capture high-energy gravitational exchanges, and it extends the construction to BTZ black holes, offering a nonperturbative lens on Planckian physics in AdS. A concrete $d=2$ example demonstrates the method, including explicit expressions for the amplitude and its discontinuity, and illustrates how Lorentzian analytic continuation encodes the eikonal data. The paper lays groundwork for reconstructing full four-point functions from shock data and for future inclusion of stringy corrections and finite-$N$ CFT checks.

Abstract

We initiate a program to generalize the standard eikonal approximation to compute amplitudes in Anti-de Sitter spacetimes. Inspired by the shock wave derivation of the eikonal amplitude in flat space, we study the two-point function E ~ < O_1 O_1 >_{shock} in the presence of a shock wave in Anti-de Sitter, where O_1 is a scalar primary operator in the dual conformal field theory. At tree level in the gravitational coupling, we relate the shock two-point function E to the discontinuity across a kinematical branch cut of the conformal field theory four-point function A ~ < O_1 O_2 O_1 O_2 >, where O_2 creates the shock geometry in Anti-de Sitter. Finally, we extend the above results by computing E in the presence of shock waves along the horizon of Schwarzschild BTZ black holes. This work gives new tools for the study of Planckian physics in Anti-de Sitter spacetimes.

Figures (10)

• Figure 1: Interaction diagrams in both flat and AdS spaces. In the eikonal regime, free propagation $(a)$ is modified primarily by interactions described by crossed--ladder graphs $(c)$. In flat space and in this regime, the tree level amplitude is dominated by the T--channel graph $(b)$ with maximal spin $j=2$ of the exchanged massless particle. Moreover, the full eikonal amplitude can be computed from diagram $(b)$.
• Figure 2: Analytic continuation of $\mathcal{A}_{1}$ to obtain $\mathcal{A}_{1}^{\circlearrowright}$. The variable $\bar{z}$ is kept fixed and $z$ is transported clockwise around the point at infinity, circling the points $0,1,\bar{z}$.
• Figure 3: Embedding of AdS$_2$ in $\mathbb{M}^{2}\times\mathbb{M}^{1}$. A point $\mathbf{p}$ in the boundary of AdS$_2$ is a null ray in $\mathbb{M}^{2}\times\mathbb{M}^{1}$.
• Figure 4: Poincaré patches of an arbitrary boundary point $\mathbf{p}$, separated by the null surfaces $-2\mathbf{x\cdot p}=0$. Here AdS is represented as a cylinder with boundary $\mathbb{R}\times {S}_{d-1}$. Throughout this paper we shall mostly use a two--dimensional simplification of this picture, as shown in the figure. The point $-\mathbf{p}$ and an image point $\mathbf{p}'$ of $\mathbf{p}$ are also shown.
• Figure 5: Two consecutive Poincaré patches with $x^{-}>0$ and $x^{-}<0$. The shock geometry can be described by specifying gluing conditions on the separating surface $x^{-}=0$, which is parameterized by $x^{+}$ and $x$, with $x\in H_{d-1}$.