An étude of momentum space scalar amplitudes in AdS

TL;DR

The authors extend momentum-space techniques to AdS scalar amplitudes by adapting the Arkani-Hamed–Benincasa–Postnikov algorithm to AdS transition amplitudes, enabling algebraic, bulk-integral-free computations at tree and loop levels. They first treat minimally coupled scalars in AdS$_4$ to illustrate the standard momentum-space approach, then develop a robust, conformal-algebraic framework for conformally coupled scalars that either simplifies or eliminates bulk integrations. A generalized algorithm using Laplace-transformed couplings extends the method to all conformally coupled scalars, and they demonstrate loop extensions with explicit one-loop calculations. The work provides a systematic, dimension-independent pathway to AdS amplitudes in momentum space, with potential applications to de Sitter correlators and higher-point/spinful processes, and opens avenues for a polytopic or crossing-symmetric understanding of holographic amplitudes.

Abstract

In this paper, we explore momentum space approach to computing scalar amplitudes in Anti-de Sitter space. We show that the algorithm derived by Arkani-Hamed, Benincasa, and Postnikov for cosmological wavefunctions can be straightforwardly adopted for AdS transition amplitudes in momentum space, allowing one to bypass bulk point integrations. We demonstrate the utility of this approach in AdS by presenting several explicit results both at tree and loop level.

Figures (5)

• Figure 1: A five point tree level Witten diagram, labeled as $W_{5,1}$ below.
• Figure 2: Various tree level Witten diagrams that will be of interest below. From left to right, we label them as $W_{3,0}$, $W_{4,1}$, $W_{5,2}$, $W_{6,1}$, $W_{8,2}$, $W_{10,1}$, $W_{14,2}$, and $W_{6,3}$.
• Figure 3: Diagrammatic illustration of the algorithm
• Figure 4: One loop correction to the propagator and four point interaction in AdS
• Figure 5: The application of the algorithm to one loop two vertex diagram