Simplicity in AdS Perturbative Dynamics

TL;DR

This work develops a comprehensive Mellin-space framework for AdS perturbation theory with scalar non-derivative interactions, introducing a recursive construction of Mellin amplitudes $\mathcal{M}$ and Mellin pre-amplitudes $M$ via vertex insertion and loop formation. It reveals a striking simplicity in pole structures, formulates diagrammatic rules for $M$ that localize to tree-like substructures, and provides a systematic pole-detection algorithm supported by extensive one- and two-loop checks, including generalized bubbles. The authors propose and test conjectures distinguishing minimal and non-minimal poles and show residues factorize into products associated with sub-diagrams, highlighting unitarity-like structure in AdS perturbation theory. Taken together, these results offer a practical, diagrammatic, and recursive approach to loop-level AdS dynamics and deepen links between Mellin space and boundary CFT data, with potential insights toward flat-space limits and holographic correlators. The methods and conjectures illuminate the analytic structure of Witten diagrams and provide a foundation for exploring higher-loop and non-planar AdS processes in a controlled Mellin-space setting.

Abstract

We investigate analytic properties of loop-level perturbative dynamics in pure AdS, with the scalar effective theories with non-derivative couplings as a prototype. Explicit computations reveal certain (perhaps unexpected) simplicity regarding the pole structure of the results, in both the Mellin amplitude and a closely related object that we call Mellin pre-amplitude. Correspondingly we propose a pair of conjectures for arbitrary diagrams at all loops, based on non-trivial evidence up to two loops (and higher loops in a special class of diagrams). We also inspect the structure of residues at poles in the physical channels for several one-loop examples up to a 4-point box, as well as a two-loop double-triangle diagram. These analyses are performed using the recursive construction of Mellin (pre-)amplitudes recently prescribed in arXiv:1710.01361, for which we provide detailed derivation and generalization in this paper. Along the way we derive a set of alternative diagrammatic rules for tree (pre-)amplitudes, which are better suited to our loop construction. On the mathematical aspect we share some new thoughts on improving the contour analysis of multi-dimensional Mellin integrals, which are the essential ingredients that make our approach practical.

Figures (36)

• Figure 1: An example of generalized bubble diagram. The red dashed curves serve as a convenient book-keeping of the operators potentially present in a given OPE of the boundary correlator (which are not the multi-trace operators formed by the boundary operators), or equivalently, potential poles of the Mellin amplitude in the corresponding Mandelstam variable.
• Figure 2: Degeneration of the 4-point triangle into generalized bubbles. (A) original diagram; (B) degeneration in the $S$ channel; (C) degeneration in the $T$ channel. There can be additional tadpoles in the degenerate diagrams (marked green), but these are not relevant for our estimation.
• Figure 3: Vertex insertion to a connected diagram.
• Figure 4: Loop formation.
• Figure 5: General tree diagrams. Each grey blob denotes a bulk interaction vertex, together with possibly any bulk-to-boundary propagators attached to it. The propagators explicitly drawn are bulk-to-bulk propagators only. The newly formed propagator is dotted.

Theorems & Definitions (3)

• Proposition 1
• Proposition 2
• Proposition 3