An Elliptic One-Loop Amplitude in Anti-de-Sitter Space

TL;DR

This paper computes the full analytic one-loop four-point amplitude for a conformally coupled scalar in AdS$_4$, corresponding to a boundary operator with scaling dimension $Δ=1$, by exploiting a mapping between Witten diagrams and flat-space Feynman integrals. The result splits into a polylogarithmic part expressible through single-valued MPLs and an elliptic integral, which is evaluated in terms of elliptic multiple polylogarithms and analyzed via its symbol; the elliptic piece is not manifestly single-valued, contrasting with the polylogarithmic part and echoing structures seen in elliptic flat-space amplitudes. The final amplitude has uniform transcendental weight 3 and involves channel-dependent elliptic curves with non-isomorphic $j$-invariants, providing a rare explicit curved-space example written in $e$MPLs. The work motivates further exploration of the AdS–flat-space correspondence for loop amplitudes and the potential generalization to other dimensions, couplings, and higher-point configurations.

Abstract

We present full analytic results for the four-point one-loop amplitude of a conformally coupled scalar in four-dimensional Anti-de-Sitter space dual to a primary operator with scaling dimension 1. The computation is based on an intriguing recent discovery, connecting Witten diagrams and flat-space Feynman integrals, which led to an expression of the amplitude of interest as a pure combination of single-valued multiple polylogarithms and an integral which cannot be reduced to multiple polylogarithms. We explicitly evaluate that integral in terms of elliptic multiple polylogarithms, finding that it is not manifestly single-valued unlike the polylogarithmic contributions to the amplitude. Further we compute the symbol of the integral and observe similar structures as for (elliptic) flat-space amplitudes. The result presented here adds to the relatively short list of explicitly known position space curved-space amplitudes beyond tree level, and constitutes the first curved-space amplitude evaluated in terms of elliptic multiple polylogarithms.