One-loop amplitudes for ttj and tt productions at the LHC through O (²)

We present analytic expressions for the one-loop QCD helicity amplitudes contributing to top-quark pair production in association with a photon or a jet at the Large Hadron Collider (LHC), evaluated through O (²) 𝒪 (ϵ 2) in the dimensional regularisation parameter, ϵ. These amplitudes are required to construct the two-loop hard functions that enter the NNLO QCD computation. The helicity amplitudes are expressed as linear combinations of algebraically independent components of the ϵ -expanded master integrals–known as pentagon function–with the corresponding rational coefficients written in terms of momentum-twistor variables. We derive differential equations for the pentagon functions and solve them numerically using the generalised power series expansion method.