CFT Unitarity and the AdS Cutkosky Rules

TL;DR

This paper develops diagrammatic Cutkosky rules for conformal field theories at both weak and strong coupling, providing a unified method to compute double-commutators and analyze unitarity in holographic contexts.

Contribution

It introduces a generalized set of Cutkosky rules for CFTs, applicable to both flat space and holographic theories, linking them to the optical theorem and unitarity.

Findings

Cutkosky rules factorize loop diagrams into on-shell sub-diagrams.

Rules are consistent with the OPE limit and flat space S-matrix limits.

Provides a holographic unitarity method to reconstruct Witten diagrams from cuts.

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.