Star Integrals, Convolutions and Simplices

TL;DR

The paper develops a Mellin-space framework to organize single- and multi-loop conformal integrals in dual conformal theories, identifying one-loop $n$-gon star integrals in $n$ dimensions with volumes of hyperbolic simplices and explicitly computing the $d=5$ pentagon via Schläfli's formula. It then uses spline methods to construct the $d=6$ hexagon and $d=8$ octagon in $2d$ kinematics, relating these to fully massive double and triple box integrals, and discusses the likely appearance of elliptic and higher functions beyond generalized polylogarithms in general kinematics. The work provides concrete analytic results in specialized kinematics, clarifies how higher-loop integrals can be built from star integrals via integro-differential operations, and highlights the function spaces required to express these integrals across kinematics. Overall, it advances understanding of the analytic structure of multi-loop conformal integrals and the transition from polylogarithmic to elliptic regimes as loop order and dimensionality increase.

Abstract

We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space. Using Mellin amplitudes, a large class of higher loop integrals can be written as simple integro-differential operators on star integrals: one-loop $n$-gon integrals in $n$ dimensions. These are known to be given by volumes of hyperbolic simplices. We explicitly compute the five-dimensional pentagon integral in full generality using Schläfli's formula. Then, as a first step to understanding higher loops, we use spline technology to construct explicitly the $6d$ hexagon and $8d$ octagon integrals in two-dimensional kinematics. The fully massive hexagon and octagon integrals are then related to the double box and triple box integrals respectively. We comment on the classes of functions needed to express these integrals in general kinematics, involving elliptic functions and beyond.

Figures (2)

• Figure 1: The one-, two- and three-loop ladder diagrams (black) and their corresponding dual tree diagrams (blue). The external faces of the former, or equivalently the external vertices of the latter, are labeled $x_1,x_2,\ldots$ clockwise starting from $x_1$ as indicated.
• Figure 2: The 'star' graphs for $n=4,6,8$, in blue, correspond to the one-loop box, hexagon, and octagon integrals in $d=4,6,8$ respectively. These are the basic building blocks for many integrals relevant to multi-loop scattering amplitudes in SYM theory since each one is simply $M=1$ in Mellin space.