Multipoint conformal integrals in $D$ dimensions. Part I: Bipartite Mellin-Barnes representation and reconstruction

TL;DR

This paper introduces a systematic method for calculating conformal integrals in D dimensions using bipartite Mellin-Barnes representation, enabling the evaluation of complex n-point integrals through basis functions and master functions.

Contribution

It develops a bipartite Mellin-Barnes representation and a reconstruction procedure to evaluate n-point conformal integrals, including explicit results for pentagon and hexagon cases.

Findings

Successfully evaluated pentagon integrals as sums of basis functions.

Reproduced known box and pentagon integrals using the new method.

Identified limitations for the hexagon integral, indicating need for additional basis functions.

Abstract

We propose a systematic approach to calculating $n$-point one-loop parametric conformal integrals in $D$ dimensions which we call the reconstruction procedure. It relies on decomposing a conformal integral over basis functions which are generated from a set of master functions by acting with the cyclic group $\mathbb{Z}_n$. In order to identify the master functions we introduce a bipartite Mellin-Barnes representation by means of splitting a given conformal integral into two additive parts, one of which can be evaluated explicitly in terms of multivariate generalized hypergeometric series. For the box and pentagon integrals (i.e. $n=4,5$) we show that a computable part of the bipartite representation contains all master functions. In particular, this allows us to evaluate the parametric pentagon integral as a sum of ten basis functions generated from two master functions by the cyclic group $\mathbb{Z}_5$. The resulting expression can be tested in two ways. First, when one of propagator powers is set to zero, the pentagon integral is reduced to the known box integral, which is also rederived through the reconstruction procedure. Second, going to the non-parametric case, we reproduce the known expression for the pentagon integral given in terms of logarithms derived earlier within the geometric approach to calculating conformal integrals. We conclude by considering the hexagon integral ($n=6$) for which we show that those basis functions which follow from the computable part of the bipartite representation are not enough and more basis functions are required. In the second part of our project we will describe a method of constructing a complete set of master/basis functions in the $n$-point case.