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We compute helicity amplitudes for the one-loop QCD corrections to top-quarkpair production analytically in terms of a set of uniformly transcendentalmaster integrals. We provide corrections up to $O(\epsilon^2)$ in thedimensional regulator for the first time which are relevant at NNLO. Fourindependent pentagon integral topologies appear in the complete description ofthe colour structure for which we provide numerical solutions using canonicalform differential equations and the method of generalised power seriesexpansions. Analytic forms of the boundary values are obtained in all casesexcept one where we find a one-dimensional integral representation.

One-loop QCD helicity amplitudes for $pp\to t\bar{t} j$ to $O(\epsilon^2)$