Mellin-(Schwinger) representation of One-loop Witten diagrams in AdS

TL;DR

This work develops a Mellin-(Schwinger) representation for one-loop Witten diagrams in AdS, applied to a scalar theory with interaction $\phi^3+\phi^4$. By using a direct bulk-to-bulk propagator representation and Schwinger parametrization, the authors integrate AdS coordinates and extract Mellin amplitudes for tree-level exchange, bubble, and several one-loop topologies (triangle and box), expressing results as MB integrals with residual Schwinger-parameter integrations. A central challenge identified is the remaining integrals over Schwinger parameters associated with internal lines; the authors propose a contour-integral strategy based on residue calculus and algebraic-geometry techniques (polynomial zero loci, Griffiths residues, and scattering-equations) to systematically evaluate these pieces. The approach clarifies the structure of loop Witten diagrams, aligns with known flat-space S-matrix insights, and lays groundwork for connecting holographic loop computations to the $1/N$ expansion of boundary CFT correlators, with potential links to CHY/AMaPuH behavior in AdS contexts. Future work aims to complete the remaining integrals and to reproduce known one-loop results for boundary correlators within this framework.

Abstract

In this paper we consider Witten diagrams at one loop in AdS space for scalar $φ^3+φ^4$ theory. After using Schwinger parametrization to trivialize the space-time loop integration, we extract the Mellin-Barnes representation for the one-loop corrections to the four-particle scattering up to an integration over the Schwinger parameters corresponding to the propagators of the internal particles running into the loop. We then discuss an approach to deal with those integrals.

Figures (6)

• Figure 1: Scalar Exchange.
• Figure 2: Bubble Witten diagram.
• Figure 3: One loop Vertex correction.
• Figure 4: One loopTriangle.
• Figure 5: One loop box.