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.
Highlights
From lower-loop on-shell quantities [1, 2].1 Unitarity is a powerful tool for exploring looplevel structure in the S-matrix that is not manifest at the level of the action, including Yangian symmetry [9], color-kinematic duality, and the double-copy property of gravity theories [10, 11]
How do structures uncovered in flat space amplitudes generalize to theories in Anti-de Sitter (AdS) space? While the S-matrix can be recovered from the flat space limit of AdS/conformal field theories (CFTs) [13,14,15,16,17,18], one cannot define an S-matrix in global AdS itself as an overlap of in- and out-states [19]
The cutting rules we explore are directly related to the Lorentzian inversion formula [28], a centerpiece of modern CFT unitarity methods
Summary
We review how unitarity conditions apply to the full correlator and derive (1.3) This identity is useful because it relates the real part of a time-ordered correlator, which can be computed via a set of cutting rules for theories with a weakly-coupled description, to the causal double-commutator, which is the central element of AdS/CFT unitarity methods [28, 32, 69]. Where for convenience we have gone to momentum space and suppressed the other partitions of the external operators This equation is shown graphically in figure 3 and says that summing over all “cuts” of a four-point function vanish. To simplify this equation, it is useful to choose the momentum to lie in the configuration (1.6). The double-commutator is only non-zero for k1 + k2 ∈ V+, so we do not need to relax any conditions on this variable
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