Soft-gluon and Coulomb corrections to hadronic top-quark pair production beyond NNLO

We construct a resummation at partial next-to-next-to-next-to-leading logarithmic accuracy for hadronic top-quark pair production near partonic threshold, including simultaneously soft-gluon and Coulomb corrections, and use this result to obtain approximate next-to-next-to-next-to-leading order predictions for the total top-quark pair-production cross section at the LHC. We generalize a required one-loop potential in non-relativistic QCD to the colour-octet case and estimate the remaining unknown two-loop potentials and three-loop anomalous dimensions. We obtain a moderate correction of 1.5% relative to the next-to-next-to-leading order prediction and observe a reduction of the perturbative uncertainty below +/- 5%.