Two-loop integrals for $t \bar{t} +$jet production at hadron colliders in the leading colour approximation

TL;DR

This work completes the two-loop master-integral basis for pp → t t̄ j in the leading colour approximation by deriving and solving differential equations for the remaining pentagon-box topologies. It distinguishes a canonical d log form for one topology from an elliptic-sector topology in the other, and develops a hybrid analytic-numeric strategy using finite-field IBP and generalized power series expansions to obtain high-precision boundary values via auxiliary mass flow. The authors identify an elliptic curve underlying the elliptic sector and implement a compact one-form representation to manage non-logarithmic contributions, enabling reliable numerical evaluation in the physical s45 region. The results include explicit master-integral bases, their differential equations, alphabet structure, boundary data, and performance benchmarks, laying the groundwork for NNLO predictions of tt̄+jet in hadron colliders.

Abstract

We compute the differential equations for the two remaining integral topologies contributing to the leading colour two-loop amplitudes for $pp \rightarrow t\bar{t}j$. We derive differential equations for the master integrals by solving the integration-by-parts identities over finite fields. Of the two systems of differential equations, one is presented in canonical '${\rm d} \log$' form, while the other is found to have an elliptic sector. For the elliptic topology we identify the relevant elliptic curve, and present the differential equations in a more general form which depends quadratically on $ε$ and contains non-logarithmic one-forms in addition to the canonical ${\rm d} \log$'s. We solve the systems of differential equations numerically using generalised series expansions with the boundary terms obtained using the auxiliary mass flow method. A summary of all one-loop and two-loop planar topologies is presented including the list of alphabet letters for the '${\rm d} \log$' form systems and high-precision boundary values.

Figures (7)

• Figure 1: The three pentagon-box topologies contributing to $pp \rightarrow t\bar{t}j$ in the leading colour limit. Black lines denote massless particles and red double-lines denote massive particles.
• Figure 2: 'Problematic' sectors of topology ${\rm PB}_B$.
• Figure 3: The four five-point double-box topologies covering the MIs ${\mathcal{I}}_{{\rm PB}_B,4}$ -- ${\mathcal{I}}_{{\rm PB}_B,9}$ (a), ${\mathcal{I}}_{{\rm PB}_B,10}$ -- ${\mathcal{I}}_{{\rm PB}_B,12}$ (b), ${\mathcal{I}}_{{\rm PB}_C,5}$ -- ${\mathcal{I}}_{{\rm PB}_C,8}$ (c) and ${\mathcal{I}}_{{\rm PB}_C,15}$ -- ${\mathcal{I}}_{{\rm PB}_C,18}$ (d) respectively.
• Figure 4: The two pentagon-bubble sectors for topologies ${\rm PB}_B$ (sub-figure (a)) and ${\rm PB}_C$ (sub-figure (b)).
• Figure 5: Histogram showing the distribution of the evaluation time per segment for topologies ${\rm PB}_A$, ${\rm PB}_B$ and ${\rm PB}_C$. Note that the number of segments depends on the singularity structure and is therefore different for each topology.