Writing CFT correlation functions as AdS scattering amplitudes

By J. Penedones. 2010

619 citations recorded by the source corpus.

We explore the Mellin representation of conformal correlation functions recently proposed by Mack. Examples in the AdS/CFT context reinforce the analogy between Mellin amplitudes and scattering amplitudes. We conjecture a simple formula relating the bulk scattering amplitudes to the asymptotic behavior of Mellin amplitudes and show that previous results on the flat space limit of AdS follow from our new formula. We find that the Mellin amplitudes are particularly useful in the case of conformal gauge theories in the planar limit. In this case, the four point Mellin amplitudes are meromorphic functions whose poles and their residues are entirely determined by two and three point functions of single-trace operators. This makes the Mellin amplitudes the ideal objects to attempt the conformal bootstrap program in higher dimensions.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/2d484c78c6213114e843e591811aeaaa0a418d50

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Citations in the corpus

Unitarity and the holographic S-Matrix (2011, methodology): “The introduction of the Mellin amplitude [4, 5] for CFT correlation functions has led to progress [6, 7, 8, 9, 10] along these lines, and in particular, we recently argued [9] that bulk locality can be understood by showing how the meromorphy of the Mellin amplitude(1) leads to an analytic S-Matrix.”
A Mellin space approach to cosmological correlators (2019, methodology): “8 In Euclidean anti-de Sitter space, such a representation for propagators is often referred to in the AdS/CFT literature as the “split representation”, which has proven to be an invaluable tool in the evaluation of Witten diagrams [34–36, 66–82].”
Tailoring three-point functions and integrability (2010, background): “455824 (1,4) [0, 7] (2,7) −51/4 [3, 6] (1,8) [1, 4] −1.”
From correlation functions to event shapes (2013, background): “The Mellin representation is a natural language for the description of CFT correlators and came into a widespread use in recent years (see, e.g., [40, 41, 42, 43]).”
Virasoro conformal blocks and thermality from classical background fields (2015, positive): “This phenomenon can be generalized far beyond the class of theories with a sparse spectrum by using Mellin amplitudes [55, 58{61] to represent CFT correlators, as we will discuss in a forthcoming paper [62].”
Quartic AdS interactions in higher-spin gravity from Conformal Field Theory (2015, background): “In [79] it was shown that the Mellin amplitude M(δij , l) for the vertex (6.”
Universality of long-distance AdS physics from the CFT bootstrap (2014, methodology): “In the appendices of [22, 30] it was shown how to decompose a primary operator [O1O2]n,` in a generalized free theory6 into the descendants of O1 and O2.”
Analyticity and the holographic S-matrix (2011, background): “Penedones provided many pieces of evidence for equation (27), showing that it works for tree-level and one-loop 4-pt amplitudes, and that it accords with earlier observations [12] about a certain singularity in CFT correlators connected with flat space scattering amplitudes.”
Emergent spacetime and holographic CFTs (2011, background): “enjoy an alternate OPE expansion that manifestly solves crossing symmetry (perhaps not unrelated to the proposal of [104](73) or that of [13]).”
Semiclassical Virasoro blocks from AdS3 gravity (2015, background): “They feature prominently in many applications of CFT, including in the conformal bootstrap program [10, 11] and in the study of the emergence of bulk locality from CFT [12–14].”
How to succeed at holographic correlators without really trying (2017, background): “A precise prescription for relating the massless17 flat-space scattering amplitude T (Ki) to the asymptotic form of the holographic Mellin amplitude was given in [41] and justified in [71], M(δij) ≈ Rn(1−d)/2+d+1 Γ(12 ∑ i ∆i − d 2) ∫ ∞”
Spinning conformal blocks (2011, background): “Additional progress has been made at understanding the constraints of unitarity in this context [9], as well as in exploring the recently found analogy between CFT correlation functions in the Mellin representation and scattering amplitudes [10–13].”
Causality constraints in conformal field theory (2015, background): “Equivalently, for each flat-space scalar S-matrix, there is a corresponding crossing-symmetric CFT correlator in Mellin space [29, 31, 37, 38].”
Towards Feynman rules for Mellin amplitudes in AdS/CFT (2011, methodology): “The Mellin-Barnes integral can be exactly evaluated in terms of a hypergeometric 3F2 function [21]:”