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

TL;DR

This work develops a diagrammatic, polygon-based method to construct a class of multivariate generalized hypergeometric functions that encode multipoint one-loop conformal integrals in $D$ dimensions. By associating each $n$-point integral with a conformal polygon on a Baxter lattice and a corresponding set of cross-ratios, the authors define basis functions $oldsymbol{ m abla}_n^{raket{ijk}}$ and master functions that assemble $I_n^{m a}({m x})$ via a reconstruction formula. The framework is explicitly demonstrated for box, pentagon, and hexagon integrals, with reductions to lower-point cases and non-parametric limits providing nontrivial consistency checks. The polygonal functions offer a geometric, invariant structure that may illuminate connections to Yangian symmetry and GKZ systems, and suggest pathways to generalized, canonical representations of conformal integrals. Overall, the paper advances a geometric, hypergeometric approach to systematic analytic expressions for multipoint conformal integrals in arbitrary dimensions.

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.

Figures (13)

• Figure 1: The conformal integral $I_n^{{\bm a}}({\bm x})$ represented as the labelled $n$-valent vertex. The $i$-th leg depicts the propagator $X_{0i}^{-a_i}$ which is characterized by the position $x_i$ and the propagator power $a_i$; the vertex denotes integration over $x_0$.
• Figure 2: The Baxter lattice and the conformal $n$-valent vertex drawn in the central polygon. The angle $\alpha$ is related to the propagator power \ref{['angle']}. Note that compared to the conformal graph in fig. \ref{['fig:vertex']} the emphasis here is shifted to the polygon as such, and, in particular, to its angles which encode the conformality condition.
• Figure 3: Left: Irregular conformal polygon from the Baxter lattice in fig. \ref{['fig:baxter']} shown here for simplicity as the regular $n$-gon. The sum of angles $(n-2)\pi$ reproduces the conformality constraint. Right: The triangle $\triangle_{n}^{\langle ijk \rangle}\subset P_n$ (shown in red) represents the ordered index triple $\langle ijk \rangle \in {\rm R}_n$.
• Figure 4: Left: The superposition of two Baxter lattices leads to the notion of chambers (shown in grey). The first Baxter lattice is the same as in fig. \ref{['fig:baxter']}, the second Baxter lattice is given by three red lines. Right: Open chambers are shown in grey. Adding red boundaries yields closed chambers.
• Figure 5: The quadratic (a) and cubic (b) cross-ratio diagrams. The numerators and denominators are colored with green and orange, respectively.