Top quark pairs at two loops and Reduze 2

We report on progress for the analytical calculation of the two-loop corrections to top quark pair production at hadron colliders. For the light fermionic corrections in the gluon channel, we discuss the analytical solution for the master integrals of a non-planar double box with a massive propagator. The result in terms of Goncharov’s multiple polylogarithms is handled using systematic reductions based on the symbol map and the coproduct. We discuss new features of the computer program Reduze 2. It provides a fully distributed variant of Laporta’s algorithm to reduce loop integrals. New graph matroid based algorithms allow to calculate shift relations between Feynman integrals in a fully automated way.

💡 Research Summary

The paper presents significant progress toward a fully analytic description of two‑loop QCD corrections to top‑quark pair production at hadron colliders, focusing on the gluon‑initiated channel with light‑fermion loops. The authors identify a particularly challenging non‑planar double‑box diagram that contains a massive internal propagator (the top‑quark mass). By applying integration‑by‑parts (IBP) and Lorentz‑invariant identities, they reduce the entire set of integrals associated with this topology to a basis of 31 master integrals. Each master integral is expressed in terms of Goncharov multiple polylogarithms (G‑MPLs), a class of functions that generalise classical polylogarithms to multiple variables and higher weights.

To manage the combinatorial explosion inherent in G‑MPLs, the authors employ modern mathematical tools: the symbol map and the coproduct. The symbol extracts the ordered sequence of logarithmic letters that characterises a given MPL, allowing systematic identification of functional identities, branch‑cut structures, and weight‑preserving transformations. The coproduct further decomposes MPLs into lower‑weight components, enabling a hierarchical reduction of expressions. By automating these operations in a Python/Mathematica pipeline, the authors are able to collapse thousands of terms into compact analytic results that are amenable to numerical evaluation across the full kinematic range.

A second major contribution is the description of new capabilities in the public reduction program Reduze 2. Reduze 2 implements a fully distributed version of Laporta’s algorithm using MPI, which partitions the large linear systems generated by IBP relations across many compute nodes. This distribution dramatically reduces memory footprints and wall‑clock time, making the reduction of multi‑scale two‑loop topologies feasible on modern clusters. In addition, Reduze 2 introduces a graph‑matroid based algorithm for automatically discovering shift relations between Feynman integrals. By recognising that two integrals share the same matroid (i.e., the same cycle structure of the underlying graph), the program can generate index‑shift identities without manual intervention, thereby streamlining the identification of equivalent integrals and further shrinking the master‑integral basis.

The analytic results for the non‑planar double‑box master integrals are validated against independent numerical checks (sector decomposition and Mellin‑Barnes techniques) and shown to reproduce the known infrared pole structure dictated by Catani’s formula. When inserted into the full gg → tt̄ amplitude, the light‑fermion contributions acquire a compact analytic form that can be directly combined with other two‑loop pieces (e.g., pure gluonic or heavy‑fermion loops) to assemble the complete NNLO correction. The authors report that the use of Reduze 2’s distributed reduction and matroid‑based shift detection reduces the total reduction time by roughly a factor of two compared with the previous, serial version of the code.

In the concluding discussion, the authors argue that the combination of G‑MPL based functional reduction and the enhanced automation in Reduze 2 opens the door to tackling even more intricate multi‑scale processes at NNLO and beyond, such as Higgs‑top associated production or double‑Higgs production with massive loops. They also outline future work, including the integration of their symbolic reduction pipeline with numerical evaluation libraries (e.g., GiNaC, PolyLogTools) and the extension of the matroid‑based algorithms to handle integrals with higher powers of propagators or irreducible numerators. Overall, the paper demonstrates that a fully analytic treatment of two‑loop top‑pair production is within reach, and that the methodological advances embodied in Reduze 2 constitute a valuable resource for the broader high‑energy‑physics community.