CFT Unitarity and the AdS Cutkosky Rules

TL;DR

The paper extends unitarity methods to conformal field theories by deriving Cutkosky-like rules for both weakly coupled CFTs in flat space and strongly coupled holographic CFTs, recasting the double-commutator in the Lorentzian inversion formula as arising from on-shell cuts. It formulates explicit cutting rules in Lorentzian momentum space and connects them to AdS transition amplitudes via a split representation, providing a holographic unitarity framework for reconstructing Witten diagrams from their cuts. The approach is validated through tree- and loop-level Witten-diagram examples, demonstrating correct OPE behavior and matching to the S-matrix discontinuity in the flat-space limit. Together with the CFT dispersion formula, these results offer a practical, locality-respecting method to access loop-level AdS/CFT physics from tree-level data, with broad potential for higher-point, Mellin-space, and cosmological applications.

Abstract

We derive the Cutkosky rules for conformal field theories (CFTs) at weak and strong coupling. These rules give a simple, diagrammatic method to compute the double-commutator that appears in the Lorentzian inversion formula. We first revisit weakly-coupled CFTs in flat space, where the cuts are performed on Feynman diagrams. We then generalize these rules to strongly-coupled holographic CFTs, where the cuts are performed on the Witten diagrams of the dual theory. In both cases, Cutkosky rules factorize loop diagrams into on-shell sub-diagrams and generalize the standard S-matrix cutting rules. These rules are naturally formulated and derived in Lorentzian momentum space, where the double-commutator is manifestly related to the CFT optical theorem. Finally, we study the AdS cutting rules in explicit examples at tree level and one loop. In these examples, we confirm that the rules are consistent with the OPE limit and that we recover the S-matrix optical theorem in the flat space limit. The AdS cutting rules and the CFT dispersion formula together form a holographic unitarity method to reconstruct Witten diagrams from their cuts.

Figures (5)

• Figure 1: The map from the cutting to coloring rules for the AdS box diagram.
• Figure 2: A cut bubble is the on-shell gluing of contact diagrams: the undotted lines are Feynman propagators and the dotted lines are Wightman, or on-shell, propagators.
• Figure 3: Optical theorem for a QFT four-point function. The grey disk represents a general correlation function and not a Feynman diagram. The external lines label the momentum of the external operator. Operators to the (right) left of the blue line are (anti-)time-ordered in the four-point functions given in \ref{['eq:CombCor']}.
• Figure 4: Feynman tree theorem in AdS for a one-loop correction to the propagator in $\Phi^3$ theory. The arrow indicates the flow of positive energy across the cut.
• Figure 5: Timefolded contour used for $\langle\overline{T}[\phi_3\phi_4]T[\phi_1\phi_2]\rangle$. The arrows indicate the flow of time and the labels $R$ and $L$ are used to distinguish the two contours by the direction in which time flows.