CFT unitarity and the AdS Cutkosky rules

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.

Opacity from Loops in AdS

“…(4.15) by taking the first term of the series. The scaling in p is consistent with the one found in [42]. For the last equality we used…”

“…Unitarity methods in AdS have recently been investigated in [3,42]. These methods typically aim to compute the double discontinuity of the dual CFT correlators from the AdS side [107], and therefore focus on full Witten diagrams.…”

“…In [42], unitarity cuts in the momentum spectral representation have been explored. Following this reference, we can evaluate a cut of the Π bubble instead of evaluating its imaginary part.…”

“…With such substitution in eq. (2.29), one can evaluate the integral inp, which gives rise to the AdS Wightman propagator [42] given schematically by 14…”

“…Momentum space is also the natural language to study the loop summation and to connect with flat space QFT knowledge and tools. Among the above references, [40,41] and [42] involve calculations in momentum space -the use of momentum space proves to be instrumental in these works. Momentum space has also been used in diverse studies of tree-level amplitudes and in the calculation of cosmological observables [66][67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82].…”