Eikonal Methods in AdS/CFT: Regge Theory and Multi-Reggeon Exchange

TL;DR

This work extends Regge theory to conformal field theories by developing a Lorentzian Regge analysis of four-point functions and an impact-parameter (eikonal) framework in AdS/CFT. It identifies Regge poles with trajectories $J=j(\nu)$ and constructs composite partial waves that resum high-spin exchanges, enabling a controlled high-energy expansion via a Sommerfeld-Watson transform. In theories with a string dual, the tree-level gravi-reggeon exchange dominates in the Regge limit and can be resummed to yield a phase shift $\Gamma$ that unitarizes the amplitude at large impact parameter. As a concrete example, the paper analyzes $\mathcal{N}=4$ SYM at large 't Hooft coupling, deriving leading string corrections to graviton exchange and illustrating how multi-reggeon exchanges are captured in the eikonal regime; this provides a bridge between AdS/CFT and high-energy Regge phenomenology, with future work aimed at weak coupling and BFKL-like limits.

Abstract

We analyze conformal field theory 4-point functions of the form A ~ O_1(x_1) O_2(x_2) O_1(x_3) O_2(x_4), where the operators O_i are scalar primaries. We show that, in the Lorentzian regime, the limit x_1 -> x_3 is dominated by the exchange of conformal partial waves of highest spin. When partial waves of arbitrary spin contribute to A, the behavior of the Lorentzian amplitude for x_1 -> x_3 must be analyzed using complex-spin techniques, leading to a generalized Regge theory for CFT's. Whenever the CFT is dual to a string theory, the string tree-level contribution A_tree to the amplitude A presents a Regge pole corresponding the a gravi-reggeon exchange. In this case, we apply the impact parameter representation for CFT amplitudes, previously developed, to analyze multiple reggeon exchanges in the eikonal limit. As an example, we apply these general techniques to N=4 super-Yang-Mills theory in d=4 in the limit of large 't Hooft coupling, including the leading string corrections to pure graviton exchange.

Figures (6)

• Figure 1: Basic Lorentzian kinematics for the correlator $A\left( \mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3},\mathbf{x}_{4}\right)$. The relevant light cones are shown in (a) and (b). The gray region in (b) with $x^{+}>0$ and $x^{-}<0$ is mapped, under the conformal maps $\mathbf{\pi ,\bar{\pi}}$, to the future Milne cone $\mathrm{M}$.
• Figure 2: Relevant analytic continuation in $z,\bar{z}$ for the Lorentzian amplitude $\hat{\mathcal{A}}$ starting from the Euclidean amplitude $\mathcal{A}$ with $\bar{z}=z^{\star}$. In (b) we show the same paths under the map $z\rightarrow z/(z-1)$ which exchanges the roles of particles $2$ and $4$.
• Figure 3: Boundary of AdS, with global time running vertically. The CFT amplitude is originally defined on the Poincarè patch defined by the regions a,b,c,d. On the other hand, it is more convenient to parameterize insertions of the operators $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ at the points $3,2$ with coordinates $\mathbf{x},\mathbf{\bar{x}}$ relative to the left and right Poincarè patches. These are defined by regions a,b,e and b,d,f, respectively, and are related to the original Poincarè patch by the conformal transformations $\mathbf{\pi},\mathbf{\bar{\pi}}$.
• Figure 4: The path $\Gamma$ in the complex $J$--plane, which gives the usual sum over integral spins, is deformed to analyze the $\sigma\rightarrow0$ limit of the Lorentzian amplitude. Contributions come from Regge poles and cuts (a pole is shown at $J=j\left( \nu\right)$), together with subdominant shadow contributions coming from the dashed contours. This includes the usual path at $\operatorname{Re}J=-1$ together with the contributions of the spurious poles at $J=\pm i\nu+m$ with $m$ a positive odd integer, which have an effective spin $-1\pm i\nu$.
• Figure 5: Spurious poles of $\mathcal{T}_{E,J}$ in the complex $E$--plane for $\operatorname{Re}E>0$. They are located at $E=J-m$, with $m$ a positive odd integer, and have residues proportional to $\mathcal{T}_{1+J,\,J-1-m}$.