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

TL;DR

This paper advances the calculation of two-loop integrals for top-antitop plus jet production at hadron colliders by deriving and solving differential equations, including elliptic sectors, to improve precision in theoretical predictions.

Contribution

It introduces a method to derive and solve differential equations for complex two-loop integrals, including elliptic sectors, using finite field techniques and numerical solutions.

Findings

Derived differential equations for all relevant integral topologies.

Identified elliptic curve structures in the integrals.

Provided numerical solutions and boundary conditions for the integrals.

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 $\epsilon$ 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.