Abstract
Abstract We derive a simple relation between the Mellin amplitude for AdS/CFT correlation functions and the bulk S-Matrix in the flat spacetime limit, proving a conjecture of Penedones. As a consequence of the Operator Product Expansion, the Mellin amplitude for any unitary CFT must be a meromorphic function with simple poles on the real axis. This provides a powerful and suggestive handle on the locality vis-a-vis analyticity properties of the S-Matrix. We begin to explore analyticity by showing how the familiar poles and branch cuts of scattering amplitudes arise from the holographic description. For this purpose we compute examples of Mellin amplitudes corresponding to 1-loop and 2-loop Witten diagrams in AdS. We also examine the flat spacetime limit of conformal blocks, implicitly relating the S-Matrix program to the Bootstrap program for CFTs. We use this connection to show how the existence of small black holes in AdS leads to a universal prediction for the conformal block decomposition of the dual CFT.
Highlights
That the Mellin representation [1, 4, 5] of Conformal Field Theory (CFT) correlation functions defines a dual bulk S-Matrix via the vanishing-curvature limit of the AdS/CFT correspondence [6,7,8]
The reason is that operators are “exchanged” in correlation functions whenever they appear in the operator product expansion (OPE) of the external operators, e.g. for scalars
Where k and n − k are the number of operators to the left and right, respectively of the scalar propagator, and MkL+1, MnR−k+1 are the Mellin amplitudes for the corresponding (k + 1)- and (n − k + 1)-point correlation functions obtained by cutting that propagator
Summary
The Mellin representation of CFT correlation functions is best understood by thinking of CFT correlators in analogy with flat space scattering amplitudes The basis of this analogy is the Hilbert space of states at large N. Of states, which in the limit of weak interactions are the single- and multi-particle states and form a Fock space This space has the essential physical property that the energy of a multi-particle state is the sum of the energies of the individual particles. Rather than having a definite frequency, eigenstates of the “Hamiltonian” have a definite scaling dimension Such eigenstates no longer correspond to particles per se, which anyway do not exist due to the scale-invariance of the theory, but rather to operators, through the state-operator correspondence of radial quantization. Annihilation operators of the theory. We will review how it is that Mellin space makes this structure manifest in the correlation functions
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