From bulk loops to boundary large-N expansion

141

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.

Section: Jhep11(2020)073mentioning

confidence: 99%

“…. , q r 18 The split representation (4.19) is the basis of the Euclidean analysis in [32,71]. There, putting a line on shell corresponds to closing the ω integral on the pole in P (ω, ∆).…”

Section: Ads Transition Amplitudesmentioning

confidence: 99%

CFT unitarity and the AdS Cutkosky rules

Meltzer

Sivaramakrishnan

2020

141

“…The approach outlined above does not make any reference to actual one-loop diagrams of IIB supergravity on AdS 5 ×S 5 , and in fact this computation in the bulk remains very challenging. Instead, scalar theories on AdS at one-loop have been discussed in many references, for example, see [14][15][16][17][18][19]. Our approach here uses CFT techniques to extract data in the dual theory, N = 4 SYM, and it is complemented with an understanding of the possible analytic structure of the one-loop correlators, as functions in position space.…”

Section: Introductionmentioning

confidence: 99%

One-loop amplitudes in AdS5×S5 supergravity from $$ \mathcal{N} $$ = 4 SYM at strong coupling

Aprile

Drummond

Heslop

et al. 2020

171

We explore the structure of maximally supersymmetric Yang-Mills correlators in the supergravity regime. We develop an algorithm to construct one-loop supergravity amplitudes of four arbitrary Kaluza-Klein supergravity states, properly dualised into single-particle operators. We illustrate this algorithm by constructing new explicit results for multi-channel correlation functions, and we show that correlators which are degenerate at tree level become distinguishable at one-loop. The algorithm contains a number of subtle features which have not appeared until now. In particular, we address the presence of non-trivial low twist protected operators in the OPE that are crucial for obtaining the correct oneloop results. Finally, we outline how the differential operators D pqrs and ∆ (8) , which play a role in the context of the hidden 10d conformal symmetry at tree level, can be used to reorganise our one-loop correlators.

“…1 Brute force position space computations of individual loop diagrams were done in [49][50][51]. Cutting rules and unitarity methods in AdS are discussed in [52][53][54]. [55] considered 1-loop polygon diagrams in AdS, e.g the 3-point triangle diagram, the 4-point box diagram, etc.…”

Section: Introductionmentioning

confidence: 99%

Loops in AdS: from the spectral representation to position space

Carmi

2020

We compute a family of scalar loop diagrams in AdS. We use the spectral representation to derive various bulk vertex/propagator identities, and these identities enable to reduce certain loop bubble diagrams to lower loop diagrams, and often to treelevel exchange or contact diagrams. An important example is the computation of the finite coupling 4-point function of the large-N conformal O(N) model on AdS 3. Remarkably, the re-summation of bubble diagrams is equal to a certain contact diagram: theD 1,1, 3 2 , 3 2 (z,z) function. Another example is a scalar with φ 4 or φ 3 coupling in AdS 3 : we compute various 4-point (and higher point) loop bubble diagrams with alternating integer and halfinteger scaling dimensions in terms of a finite sum of contact diagrams and tree-level exchange diagrams. The 4-point function with external scaling dimensions differences obeying ∆ 12 = 0 and ∆ 34 = 1 enjoys significant simplicity which enables us to compute in quite generality. For integer or half-integer scaling dimensions, we show that the M-loop bubble diagram can be written in terms of Lerch transcendent functions of the crossratios z andz. Finally, we compute 2-point bulk bubble diagrams with endpoints in the bulk, and the result can be written in terms of Lerch transcendent functions of the AdS chordal distance. We show that the similarity of the latter two computations is not a coincidence, but arises from a vertex identity between the bulk 2-point function and the double-discontinuity of the boundary 4-point function.