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

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.

56 pages, 16 figures, v2: motivation for the diagrammatic algorithm is reinforced, new subsections concerning reduction and convergence of the polygonal functions are added, the discussion throughout the text is expanded, typos are corrected. Journal version