Multipoint conformal integrals in $D$ dimensions. Part II: Polygons and basis functions

TL;DR

This paper introduces a diagrammatic algorithm to explicitly construct multivariate hypergeometric series for calculating multipoint one-loop conformal integrals in arbitrary dimensions, focusing on polygons like boxes, pentagons, and hexagons.

Contribution

It presents a novel diagrammatic method to systematically build series representations for multipoint conformal integrals using convex polygons within the Baxter lattice.

Findings

Explicit series for box, pentagon, and hexagon integrals provided.

Algorithm simplifies the calculation of multipoint conformal integrals.

Conjecture on the general applicability of the series to other polygons.

Abstract

We explicitly construct a class of multivariate generalized hypergeometric series which is conjectured in our previous paper [Alkalaev & Mandrygin 2025] to calculate multipoint one-loop parametric conformal integrals in $D$ dimensions. Our approach is based on a simple diagrammatic algorithm which systematically builds both arguments and series coefficients in terms of a convex polygon which is part of the Baxter lattice. The examples of the box, pentagon, and hexagon integrals are considered in detail.