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

TL;DR

This paper computes a detailed one-loop four-point amplitude in Anti-de-Sitter space, revealing complex elliptic structures and advancing the understanding of curved-space quantum field theory beyond tree level.

Contribution

It provides the first explicit calculation of a curved-space amplitude in terms of elliptic multiple polylogarithms, connecting Witten diagrams with advanced polylogarithmic functions.

Findings

Amplitude expressed with single-valued multiple polylogarithms

Integral evaluated using elliptic multiple polylogarithms

Symbol analysis reveals structures similar to flat-space amplitudes

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.