Mellin Amplitudes for Dual Conformal Integrals

TL;DR

This paper explores the use of Mellin space to represent dual conformal integrals in super-Yang-Mills theory, providing new methods for representing and relating complex integrals, especially fully massive ones.

Contribution

It develops Feynman-like rules for Mellin amplitudes, offers explicit representations for various integrals, and demonstrates Mellin space's utility in relating multi-loop integrals.

Findings

Mellin space simplifies representation of massive integrals.

Explicit formulas for dual conformal integrals in Mellin space.

Higher-loop integrals can be expressed as operators on simpler integrals.

Abstract

Motivated by recent work on the utility of Mellin space for representing conformal correlators in $AdS$/CFT, we study its suitability for representing dual conformal integrals of the type which appear in perturbative scattering amplitudes in super-Yang-Mills theory. We discuss Feynman-like rules for writing Mellin amplitudes for a large class of integrals in any dimension, and find explicit representations for several familiar toy integrals. However we show that the power of Mellin space is that it provides simple representations even for fully massive integrals, which except for the single case of the 4-mass box have not yet been computed by any available technology. Mellin space is also useful for exhibiting differential relations between various multi-loop integrals, and we show that certain higher-loop integrals may be written as integral operators acting on the fully massive scalar $n$-gon in $n$ dimensions, whose Mellin amplitude is exactly 1. Our chief example is a very simple formula expressing the 6-mass double box as a single integral of the 6-mass scalar hexagon in 6 dimensions.