Abstract
Different frameworks exist to describe the flat-space limit of AdS/CFT, include momentum space, Mellin space, coordinate space, and partial-wave expansion. We explain the origin of momentum space as the smearing kernel in Poincare AdS, while the origin of latter three is the smearing kernel in global AdS. In Mellin space, we find a Mellin formula that unifies massless and massive flat-space limit, which can be transformed to coordinate space and partial-wave expansion. Furthermore, we also manage to transform momentum space to smearing kernel in global AdS, connecting all existed frameworks. Finally, we go beyond scalar and verify that leftlangle VVmathcal{O}rightrangle maps to photon-photon-massive amplitudes.
Highlights
Including negative cosmological constant, gravity theory coupled to other local fields can be formulated as weakly coupled quantum field theory (QFT) by perturbatively expanding the curvatures around the Anti-de Sitter (AdS) background
We found that the scattering smearing kernel from Poincare AdS is a simple Fourier factor that brings the conformal field theory (CFT) correlator to momentum space
The scattering smearing kernel from global AdS is more nontrivial, and we found that it is served as the unified origin of other known frameworks of flat-space limit: Mellin space, coordinate space, and partial-waves
Summary
Gravity theory coupled to other local fields can be formulated as weakly coupled quantum field theory (QFT) by perturbatively expanding the curvatures around the Anti-de Sitter (AdS) background. Only the scattering smearing kernel constructed from global AdS can be served as the origin of the flat-space limit in Mellin space, coordinate space, and partial-wave expansion; on the other hand, when we construct the scattering smearing kernel from Poincare AdS, we find it performs the Fourier-transform and gives rise to the framework of flat-space limit in momentum space. Using the saddle-points, we find a Mellin formula that applies to both massless scattering and massive scattering We show this Mellin formula gives rise to the flat-space limit in coordinate space, and to the partial-wave/phase-shift formula.
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