From bulk loops to boundary large-N expansion

TL;DR

We study the analytic structure of loop Witten diagrams in Euclidean AdS through their conformal partial wave decompositions and show that, like in flat space, singularities arise from non-trivial cuts and factorize into products of subdiagram coefficient functions. The authors develop AdS Cutkosky-like rules and demonstrate consistency with large-$N$ boundary bootstrap, illustrating with a detailed one-loop four-point example and outlining extensions to higher loops, spins, and higher-point amplitudes. The results provide a bulk-side justification for key large-$N$ relations on the boundary and offer a unified framework for analyzing loop corrections across AdS/CFT, including higher-spin theories. This approach paves the way for reconstructing full amplitudes from their singular structure and for applying these ideas to non-perturbative or Lorentzian settings in holography.

Abstract

We study the analytic structure of loop Witten diagrams in Euclidean AdS represented by their conformal partial wave expansions. We show that, as in flat space, amplitude's singularities are associated with non-trivial cuts of the diagram and factorize into products of the coefficient functions for the subdiagrams resulting from these cuts. We consider an example of a one-loop four-point diagram in detail and then briefly discuss how the procedure can be extended to more general diagrams. Finally, we show that this analysis reproduces simple relations that follow from the large-N considerations on the boundary.

Figures (2)

• Figure 1: This figure illustrates the key steps of section \ref{['sec:loopfromtree']}. First, we use the split representation for a pair of propagators to represent loop diagram a) in the from b). Blobs L and R refer to generic bulk processes, which, for simplicity, we consider to be tree diagrams. Next, we substitute the conformal partial wave expansions for the subdiagrams into b), which gives c). In c) triangles denote the properly normalized conformally-invariant three-point structures. Finally, we evaluate the bubble integral for the structures highlighted in green, which leads to the conformal partial wave expansion for the original loop diagram d).
• Figure 2: Using representation d) from Fig \ref{['fig:1']} for the loop diagram, we study the analytic structure of the coefficient function of its conformal partial wave expansion in section \ref{['sec:sing']}. We find that this coefficient function has singularities at double-trace locations, which, moreover, factorize into coefficient functions for subdiagrams. On the figure "d.t.s." refers to double-trace singularities.