Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions

TL;DR

The paper develops an AdS/CFT version of the flat-space eikonal approximation for four-point CFT correlators, introducing an impact-parameter representation that captures the large-spin high-energy regime of exchanged primaries. It shows that graviton exchange in the T-channel drives the leading high-spin behavior and obtains anomalous dimensions for high-spin double-trace composites, connecting these to the AdS transverse propagator. By relating the T-channel discontinuity to CPW data, the authors extract the high-spin content of tree-level Witten diagrams and demonstrate graviton dominance across channels, with explicit checks in two dimensions. A conjectured all-orders resummation formula for crossed-ladder Witten diagrams is proposed, offering a path to finite-N information from the eikonal regime and suggesting potential generalizations to AdS/CFT setups with internal symmetries and strings.

Abstract

We introduce the impact-parameter representation for conformal field theory correlators of the form A ~ < O_1 O_2 O_1 O_2 >. This representation is appropriate in the eikonal kinematical regime, and approximates the conformal partial-wave decomposition in the limit of large spin and dimension of the exchanged primary. Using recent results on the two-point function < O_1 O_1 >_{shock} in the presence of a shock wave in Anti-de Sitter, and its relation to the discontinuity of the four-point amplitude A across a kinematical branch-cut, we find the high spin and dimension conformal partial- wave decomposition of all tree-level Anti-de Sitter Witten diagrams. We show that, as in flat space, the eikonal kinematical regime is dominated by the T-channel exchange of the massless particle with highest spin (graviton dominance). We also compute the anomalous dimensions of the high-spin O_1 O_2 composites. Finally, we conjecture a formula re-summing crossed-ladder Witten diagrams to all orders in the gravitational coupling.

Figures (8)

• Figure 1: In the eikonal regime, free propagation $\left( a\right)$ is modified primarily by interactions described by crossed--ladder graphs $\left( b\right)$.
• Figure 2: Some possible interactions at tree--level, both in flat space and in $\mathrm{AdS}$. When decomposing the amplitude in the $S$--channel, only graph $(a)$, with maximal spin $j=2$, dominates the dynamics at large intermediate spin and energy.
• Figure 3: Intermediate primaries of dimension $E=h+\bar{h}$ and integer spin $J=h-\bar{h}\ge 0$ can exist in the white region along the dashed lines. The light grey region is excluded due to the unitarity constraint, with the vacuum at $h=\bar{h}=0$ being the unique exception. Also shown are the impact parameter labels $s=4h\bar{h}$ and $r=\ln\left( h/\bar{h}\right)$.
• Figure 4: Collision of two particles in AdS. The trajectories are shown both in the full space $\tau,\rho,\theta$ as well as projected onto a spatial slice $\rho,\theta$. The full configuration has total energy $E$ and spin $J$, with the particles reaching a minimum geodesic distance $r$ given by $\tanh\left( r/2\right)=J/E$.
• Figure 5: Partial wave decomposition of the amplitude $\mathcal{A}_{0}=1$ corresponding to free propagation in AdS in figure $1(a)$. In the T--channel decomposition only the vacuum contributes, whereas the S--channel decomposition receives contributions from a full lattice of $\mathcal{O}_{1}\mathcal{\partial}^{N}\mathcal{O}_{2}$ composites of dimensions $h,\bar{h}\in\Delta+\mathbb{N}_{0}$.