Mellin Amplitudes for Dual Conformal Integrals

TL;DR

The paper proposes and tests a Mellin-space framework for dual conformal integrals arising in planar SYM, introducing simple Mellin Feynman-like rules and showing that even fully massive loop integrals admit compact representations. By computing explicit examples (polygons, pentagons, hexagons, and ladder-type configurations) it demonstrates that Mellin amplitudes can be as simple as constants or simple rational functions, and that higher-loop integrals can be expressed as integral operators acting on lower-dimensional, fully massive polygons. It also reveals that certain numerators (magic numerators) induce contour-cancellation effects in Mellin space, producing compact analytic results, and that chiral numerator structures map to differential operators relating lower- and higher-dimensional instances of conformal integrals. The work suggests that Mellin space could serve as an efficient intermediary between integrands and integrals in SYM, enabling systematic derivations of differential relations and potentially guiding symbol-level analyses for polylogarithmic structures.

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.

Figures (5)

• Figure 1: The one-loop 8-point four-mass integral, labeled according to the usual amplitude convention in (a) and according to our streamlined notation in (b). In (a) it is implicit in the notation that each $x_i$ should be null-separated from its neighbors $x_{i-1}$ and $x_{i+1}$. In contrast the $x_i$ in (b) and (c) are arbitrary, and (a) is recovered by a simple relabeling. This integral corresponds in the dual (Mellin momentum) space to a tree-level contact interaction (c).
• Figure 2: The two-loop four-mass double box integral (a) is a particular limit of the 'fully massive' double box (b), computed in Mellin space as an exchange diagram contribution to a tree-level 6-point correlation function (in blue). The integral (a) is recovered from (b) by taking the limit $x_4 \to x_3$, $x_6 \to x_1$ and then relabeling $x_5 \to x_4$. We define the integral (b) to include the overall factor $x_{14}^2 x_{25}^2 x_{36}^2$ in order to provide dual conformal invariance. This reduces to the factor $x_{13}^4 x_{24}^2$ for integral (a).
• Figure 3: 'Magic' numerator factors are denoted graphically by a red line crossing an internal face.
• Figure 4: The chiral pentagon (a) and hexagon (b) integrals under consideration in this paper.
• Figure 5: The simplest example of a dual conformal 'window' diagram (black) whose dual diagram (blue) has a loop in $x$-space. Here we have shown the fully massive version of the integral; the fully massless version contributes to the four-loop four-particle MHV amplitude in SYM theory.