Eikonal Approximation in AdS/CFT: Resumming the Gravitational Loop Expansion

TL;DR

This work derives a gravity-dominated eikonal approximation for high-energy AdS scattering and shows how it resums ladder-type diagrams to all orders in the gravitational coupling $G$. Using AdS/CFT, the bulk eikonal phase maps to the Lorentzian limit of the CFT four-point function, yielding a universal relation between the phase shift and anomalous dimensions of large-dimension double-trace operators, with graviton exchange providing the leading contribution. The analysis introduces a transverse propagator on the hyperboloid $H_{d-1}$, an impact-parameter framework, and a precise analytic continuation between Euclidean and Lorentzian correlators, enabling explicit predictions for anomalous dimensions such as $2\Gamma(h,\bar{h}) \sim -16 G h\bar{h} \Pi_\perp(h/\bar{h})$ in the gravity limit. These results hold at strong coupling and to all orders in $1/N$, and set the stage for including string effects, Reggeization, and connections to weak-coupling Pomeron physics in future work.

Abstract

We derive an eikonal approximation to high energy interactions in Anti-de Sitter spacetime, by generalizing a position space derivation of the eikonal amplitude in flat space. We are able to resum, in terms of a generalized phase shift, ladder and cross ladder graphs associated to the exchange of a spin j field, to all orders in the coupling constant. Using the AdS/CFT correspondence, the resulting amplitude determines the behavior of the dual conformal field theory four point function < O_1 O_2 O_1 O_2 > for small values of the cross ratios, in a Lorentzian regime. Finally we show that the phase shift is dominated by graviton exchange and computes, in the dual CFT, the anomalous dimension of the double trace primary operators O_1 \partial ... \partial O_2 of large dimension and spin, corresponding to the relative motion of the two interacting particles. The results are valid at strong t'Hooft coupling and are exact in the 1/N expansion.

Figures (10)

• Figure 1: Classical null trajectories of two incoming particles moving in AdS$_{d+1}$ with total energy $E$ and relative angular momentum $J$. They reach a minimal impact parameter $r$ is given by $\tanh \left( r/2\right) =J/E$.
• Figure 2: The crossed--ladder graphs describing the T--channel exchange of many soft particles dominate the scattering amplitude in the eikonal regime.
• Figure 3: (a) A generic null hypersurface ${\bf k} \cdot{\bf y} =0$ in conformally compactified AdS. (b) The two null hypersurfaces ${\bf k}_1 \cdot{\bf y} =0$ and ${\bf k}_2 \cdot{\bf y} =0$. Their intersection is the transverse hyperboloid $H_{d-1}$ containing the reference point ${\bf x}_0$. We shall see in section \ref{['sectCFT']} that the null vectors ${\bf k}_i$ and $-{\bf k}_i$ can be thought of as points in the AdS conformal boundary.
• Figure 4: The coordinates $\{u,v\}$ and $\{\bar{u},\bar{v}\}$ for the simplest case of AdS$_2$. In general, the wave function of particle 1 is independent of the coordinate $u$, while that of particle 2 is independent of the coordinate $\bar{v}$.
• Figure 5: The null geodesics with constant $\bar{u}=-4/v$ are the reflection in the AdS conformal boundary of the null geodesics with constant $v$.