Star Integrals, Convolutions and Simplices

TL;DR

This paper investigates conformal integrals in flat space, expressing complex multi-loop integrals as simple operators on star integrals related to hyperbolic simplex volumes, and computes specific high-dimensional examples.

Contribution

It introduces a Mellin amplitude approach to relate multi-loop integrals to hyperbolic simplex volumes and explicitly computes high-dimensional integrals using spline technology.

Findings

Explicit computation of the five-dimensional pentagon integral.

Construction of six-dimensional hexagon and eight-dimensional octagon integrals.

Relation of these integrals to double and triple box integrals.

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 Schl\"afli'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.