Unitarity Methods in AdS/CFT

TL;DR

This work develops a comprehensive AdS/CFT unitarity framework that links bulk cutting/gluing of Witten diagrams to boundary CFT unitarity data. Central to the approach is the double discontinuity, which localizes loop amplitudes to internal, physically on-shell states via cuts encoded by $\widehat{\mathbf{Cut}}$ operators, and is reconciled with the Lorentzian inversion formula to reconstruct full correlators. The authors compute explicit one-loop examples (bubble, triangle, box) and several nontrivial reducible diagrams, establishing a precise bulk–boundary dictionary that matches the algebraic holographic unitarity method of Aharony et al. and clarifies the role of multi-trace exchanges, including new poles at mass and vertex renormalization. They also initiate higher-loop and higher-point analyses (e.g., five-point tree and double-ladder), showing how the unitarity method scales recursively with the number of loops and external legs. The work provides a clear, diagrammatic, and conformally manifest path to constructing non-planar AdS/CFT correlators from planar CFT data, with potential extensions to spinning operators and stringy AdS regimes.

Abstract

We develop a systematic unitarity method for loop-level AdS scattering amplitudes, dual to non-planar CFT correlators, from both bulk and boundary perspectives. We identify cut operators acting on bulk amplitudes that put virtual lines on shell, and show how the conformal partial wave decomposition of the amplitudes may be efficiently computed by gluing lower-loop amplitudes. A central role is played by the double discontinuity of the amplitude, which has a direct relation to these cuts. We then exhibit a precise, intuitive map between the diagrammatic approach in the bulk using cutting and gluing, and the algebraic, holographic unitarity method of arXiv:1612.03891 that constructs the non-planar correlator from planar CFT data. Our analysis focuses mostly on four-point, one-loop diagrams -- we compute cuts of the scalar bubble, triangle and box, as well as some one-particle reducible diagrams -- in addition to the five-point tree and four-point double-ladder. Analogies with S-matrix unitarity methods are drawn throughout.

Figures (11)

• Figure 1: General Witten diagram with an internal, double-trace cut.
• Figure 2: A summary of the map between bulk and boundary unitarity methods for one-loop amplitudes in AdS $\phi^3+\phi^4$ theory. The left column shows the off-shell, $s$-channel conformal gluing of Witten diagrams, the $\widehat{{\mathbf{Cut}}}$s of which map to the dDisc of the correlator, $\text{dDisc}_{s}(\mathcal{G}_{\rm 1-loop})$, as explained in the text. The right column shows the CFT data -- namely, the product of tree-level anomalous dimensions -- that are inserted into the $s$-channel expansion of $\text{dDisc}_{s}(\mathcal{G}_{\rm 1-loop})$. The two ways of computing $\text{dDisc}_{s}(\mathcal{G}_{\rm 1-loop})$ match. We have suppressed the tree-level "$(1)$" superscripts on the anomalous dimensions for clarity. The fourth line may be obtained from the third by contraction of the $t$-channel exchange sub-diagram; this is neatly implemented on the CFT side by replacing a $\phi^3$ anomalous dimension with its $\phi^4$ counterpart.
• Figure 3: Outgoing/ingoing lines correspond to ${\cal O}$ and $\widetilde{{\cal O}}$ respectively.
• Figure 4: One-loop corrections $\phi^{3}$ theory. The Witten diagrams on the left and right correspond to renormalization of the bulk-to-boundary and bulk-to-bulk propagator respectively.
• Figure 5: The conformal box amplitude is four CFT three-point functions integrated over the position of internal operators. Outgoing and ingoing arrows denote operators and their shadow, respectively.