Two-loop master integrals for a planar topology contributing to $pp \rightarrow t\bar{t}j$

TL;DR

This paper advances the computation of planar two-loop master integrals for $pp\to t\bar t j$ by constructing a canonical differential-equation system with a uniform transcendental weight basis across pentagon-box, double-box, pentagon-bubble, and box-triangle topologies. It achieves a compact dlog representation with a 71-letter alphabet and demonstrates a semi-analytic solution via generalised power series, using high-precision AMFlow boundary values for numerical continuity. The work provides explicit UT bases for multiple sectors, validates numerical results against AMFlow, and outlines a path toward fully analytic pentagon-functions for top-quark pair production with jets, thereby contributing to NNLO phenomenology. These results underpin potential improvements in theoretical precision for top-quark mass determinations and differential cross sections at hadron colliders, within the planar two-loop limit.

Abstract

We consider the case of a two-loop five-point pentagon-box integral configuration with one internal massive propagator that contributes to top-quark pair production in association with a jet at hadron colliders. We construct the system of differential equations for all the master integrals in a canonical form where the analytic form is reconstructed from numerical evaluations over finite fields. We find that the system can be represented as a sum of d-logarithmic forms using an alphabet of 71 letters. Using high precision boundary values obtained via the auxiliary mass flow method, a numerical solution to the master integrals is provided using generalised power series expansions.

Figures (47)

• Figure 1: The pentagon-box topology contributing to $pp \rightarrow t\bar{t}j$. Black lines denote massless particles and red double-lines denote massive particles.
• Figure 2: The first 30 diagram topologies describing 60 out of 88 master integrals. The label of the individual sub-figures lists the master integrals belonging to the corresponding topology. Massive propagators and massive external momenta are indicated by red double-lines.
• Figure 3: The remaining 23 diagram topologies describing 28 out of 88 master integrals. The label of the individual sub-figures lists the master integrals belonging to the corresponding topology. Massive propagators and massive external momenta are indicated by red double-lines.
• Figure 4: The pentagon-box sector with the master integrals ${\mathcal{I}}_{1}$, ${\mathcal{I}}_{2}$ and ${\mathcal{I}}_{3}$.
• Figure 5: The two five-point double-box topologies, containing the canonical MIs ${\mathcal{I}}_4, \, {\mathcal{I}}_5, \, {\mathcal{I}}_6$, ${\mathcal{I}}_7$, and ${\mathcal{I}}_8, \, {\mathcal{I}}_9$, ${\mathcal{I}}_{10}$ respectively.