AdS/CFT Unitarity at Higher Loops: High-Energy String Scattering

TL;DR

The paper investigates the space of weakly-coupled gravitational theories with higher-spin content using the Regge limit in large $N$, $4d$ CFTs and finds bulk amplitudes that match string-theory predictions to all orders in $1/N$ for large but finite coupling, providing evidence that UV completions are stringy. It develops a unitarity-based AdS/CFT bootstrap leveraging the Lorentzian inversion formula to extract loop corrections from tree-level data, and demonstrates that the leading Regge (Pomeron) exchange exponentiates in the bulk eikonal limit, with corrections suppressed by a large $ riangle_{ ext{gap}}$. The work shows that inelastic effects such as long-string creation and tidal excitations are subleading at large $ riangle_{ ext{gap}}$, and that certain one-loop sum rules can enforce exact exponentiation of the tree-level phase, linking boundary CFT data to bulk stringy dynamics. Together, these results connect causality, unitarity, and Regge physics in AdS to the emergence of string-like behavior in the dual bulk theory and outline paths for including broader inelastic effects.

Abstract

What is the space of weakly-coupled, gravitational theories which contain massive, higher-spin particles? This class of theories is highly constrained and it is conjectured their ultraviolet completion must be string theory. We provide more evidence for this conjecture by studying the Regge limit in large $N$, $4d$ CFTs with single-trace operators of unbounded spin. We show that in the Regge limit, these theories have bulk scattering amplitudes which are consistent with the string theory prediction to all orders in $1/N$ for large, but finite, coupling. In the language of Regge theory, we show Pomeron exchange naturally exponentiates in the $1/N$ expansion. To do this, we solve the bootstrap equations at tree-level and then use the Lorentzian inversion formula to find the one-loop correlator in the Regge limit. This is a unitarity method for AdS/CFT which can be repeated iteratively to make all orders statements. We also explain under what conditions the tree-level result exponentiates in the $1/N$ expansion at arbitrary coupling. Finally, we comment on further inelastic effects and show they give subleading contributions at large coupling. As a consistency check, we recover results from bulk Einstein gravity in the limit where all higher-spin particles decouple.

Figures (6)

• Figure 1: The Regge limit is defined by sending the four operators to null infinity while keeping the spacelike distance between the pairs of identical operators fixed.
• Figure 2: The red contour $\Gamma$ comes from the original, partial wave decomposition. We deform it to the blue contour $\Gamma'$ and in the process pick up a pole, $j(\nu)$, which gives the leading Regge growth. We have suppressed other possible non-analyticities, such as branch cuts.
• Figure 3: High-energy, fixed impact parameter scattering in AdS. The boundary operators are centered around $P_i$, the centers of different Poincaré patches. The impact parameter space is the spatial $H_{d-1}$ slice, drawn in blue.
• Figure 4: Exchange Witten diagram where we label the bulk field by its dual CFT operator.
• Figure 5: We glue two, tree-level Witten diagrams along the $\phi$ and $\psi$ lines to create a box diagram.