Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums

TL;DR

The paper addresses the challenge of evaluating non-planar multiloop Feynman integrals using Mellin-Barnes representations. It presents AMBRE v3.0, an automatic tool capable of generating non-planar MB representations up to two loops and supports GA and hybrid LA+GA strategies. A concrete non-planar massless two-loop box example illustrates how $U(x)$ and $F(x)$ are transformed into a four-dimensional MB integral with Gamma functions, and how AMBRE automates this process. The work then discusses converting MB integrals into infinite sums via MBsums and the computational challenges of contour choices in higher dimensions, outlining a workflow that uses symbolic summation packages to derive iterated integrals. Although full automatic summation remains out of reach, the authors demonstrate progress toward automatic analytic solutions for two- and three-loop massive integrals and connect the approach to differential-equation methods.

Abstract

The construction of Mellin-Barnes (MB) representations for non-planar Feynman diagrams and the summation of multiple series derived from general MB representations are discussed. A basic version of a new package AMBREv.3.0 is supplemented. The ultimate goal of this project is the automatic evaluation of MB representations for multiloop scalar and tensor Feynman integrals through infinite sums, preferably with analytic solutions. We shortly describe a strategy of further algebraic summation.

Figures (4)

• Figure 1: $n$-loop ladder diagram with $k_1,...,k_n$ internal and $p_1,...,p_4$ external momenta.
• Figure 2: The non-planar massless double box.
• Figure 3: Efficient factorization scheme for the $U$ polynomial.
• Figure 4: Hybrid method: first $F$ and $U$ polynomials over the ${k_1}$-subloop are worked out (planar subgraph), then $\{k_2, k_3\}$ are integrated over (non-planar subgraph).