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

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arXiv:2404.12325v1 Announce Type: new
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.

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