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

TL;DR

The paper develops a reconstruction framework for $n$-point one-loop parametric conformal integrals in $D$ dimensions by decomposing each integral into a computable, master-function-generated part and a second part recoverable through permutation invariance. A bipartite Mellin-Barnes representation is introduced, where the first bare integral is computed explicitly in terms of multivariate hypergeometric functions (notably Lauricella $F_D^{(N)}$ and Appell-type functions), while the second part is reconstructed via a cyclic symmetry $\mathbb{Z}_n$. Application to box ($n=4$) and pentagon ($n=5$) integrals demonstrates the method: both yield complete bases of $4$ and $10$ basis functions, respectively, generated from a small set of master functions, and reproduce known reductions to lower-point integrals and non-parametric results. For the hexagon ($n=6$), the computable basis is shown to be insufficient, signaling the need for additional master functions and motivating a forthcoming construction of a complete master/basis set. The authors formalize these ideas into a reconstruction conjecture: the full $n$-point integral can be written as a $\mathbb{Z}_n$-invariant sum of basis functions derived from master functions, with explicit combinatorial counts linking $n$ to the number of masters and bases. The framework connects to conformal bootstrap and geometric approaches, offering a structured route to closed-form expressions in certain domains and guiding future extensions to higher-point cases.

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.

Figures (3)

• Figure 1: The parametric conformal integral $I_n^{{\bm a}}({\bm x})$ can be depicted as an $n$-valent vertex. The $i$-th leg denotes the propagator $X_{0i}^{-a_i}$ which is characterized by $i$-th position $x_i$ and propagator power $a_i$, the central dot denotes integration over $x_0$.
• Figure 2: Action of the kinematic group $\mathcal{S}_4^{{\rm kin}}$\ref{['kinematic_4']} on the basis functions \ref{['i4_terms']}. Each node denotes a basis function $\Phi_4^{\langle ijk \rangle}(\bm x)$ which indices are shown inside a node. The color lines represent permutations listed on the right (the identity permutation acts on each node trivially). The diagram also shows the group multiplication law, i.e. each triangle formed by color lines represents a multiplication $g_i \circ g_j = g_k$, where $g_{i,j,k} \in \mathcal{S}_4^{{\rm kin}}$.
• Figure 3: In the 5-point case, the ten basis functions \ref{['i5_basis_345']}-\ref{['i5_basis_134']} can be equivalently obtained by acting with $\widehat{\mathcal{S}}_5^{{\rm kin}}$. These can be grouped into two pentagon diagrams. The identical permutation leaves each node invariant.