Unitarity and the Holographic S-Matrix

TL;DR

This work derives the unitarity of the holographic S-Matrix from the unitarity of the boundary CFT in the flat-space limit of AdS/CFT. It develops a non-perturbative Conglomeration framework, using Mellin amplitudes to connect OPE data and conformal blocks to bulk scattering, and proves an optical-theorem-like relation by mapping sums over exchanged operators to phase-space integrals. A key advance is the derivative relation between OPE coefficients and anomalous dimensions, together with a complete one-loop example that confirms cutting rules in the holographic setting. The results clarify which bulk states contribute to unitarity (stable multi-particle states) and demonstrate how edge-cut terms affect the real part of the S-Matrix, offering a concrete path toward deriving bulk quantum mechanics from the CFT data and suggesting directions for higher-spin and higher-point extensions.

Abstract

The bulk S-Matrix can be given a non-perturbative definition in terms of the flat space limit of AdS/CFT. We show that the unitarity of the S-Matrix, ie the optical theorem, can be derived by studying the behavior of the OPE and the conformal block decomposition in the flat space limit. When applied to perturbation theory in AdS, this gives a holographic derivation of the cutting rules for Feynman diagrams. To demonstrate these facts we introduce some new techniques for the analysis of conformal field theories. Chief among these is a method for conglomerating local primary operators to extract the contribution of an individual primary in their OPE. This provides a method for isolating the contribution of specific conformal blocks which we use to prove an important relation between certain conformal block coefficients and anomalous dimensions. These techniques make essential use of the simplifications that occur when CFT correlators are expressed in terms of a Mellin amplitude.

Figures (6)

• Figure 1: This figure shows how the AdS$_{d+1}$ cylinder in global coordinates corresponds to the CFT$_d$ in radial quantization. The time translation operator in the bulk of AdS is the dilatation operator in the CFT, so energies in AdS correspond to dimensions in the CFT. A scattering process in the bulk can be set up by acting with smeared CFT operators at an initial and final time that are separated by $\pi R$. In the large $N$ limit, a product of $n$ single-trace CFT operators creates an $n$-particle scattering state in the bulk.
• Figure 2: This figure shows how one can conglomerate $k-2$ CFT operators in an $k$-pt correlation function to obtain a 3-pt function, and then use these 3-pt functions to determine some contributions to the conformal block decomposition of a 4-point correlator. This procedure makes it possible to use one order in perturbation theory to say something about the next; it is precisely analogous to the way that the optical theorem permits the calculation of the imaginary part of the S-Matrix using a phase space integral over the product of lower point scattering amplitudes.
• Figure 3: This figure indicates how the sum over $k$-trace operators with dimension $\Delta$ turns into a phase space integral over $k$-particle states with center of mass energy $\Delta/R$ in the flat spacetime limit of AdS/CFT.
• Figure 4: This figure provides a schematic depiction of how a 1-loop Witten diagram in AdS decomposes via the conformal block decomposition in the dual CFT. For illustrative purposes, the bulk theory has both a $\frac{\lambda}{4} \phi^2 \chi^2$ and a $\frac{g}{4} \chi^2 \psi^2$ interaction. The dashed lines indicate 'cuts'; the central cut, highlighted in purple, provides the familiar imaginary contribution to the optical theorem in the flat space limit. The conformal block decomposition also includes the 'edge cuts' on the left and right, which have no analog in discussions of the cutting rules. These edge cuts are very important in order to obtain the full correlator, but in the flat space limit they only contribute to the real part of the S-Matrix, and so they drop out of the optical theorem.
• Figure 5: This figure depicts how the 'edge cuts', which correspond to the terms in the conformal block decomposition involving free propagation, only contribute to the real part of the bulk S-Matrix, while other operator exchanges contribute to both the real and the imaginary pieces of the S-Matrix in the flat space limit of AdS/CFT.