On the ultimate uncertainty of the top quark pole mass

We combine the known asymptotic behaviour of the QCD perturbation series expansion, which relates the pole mass of a heavy quark to the $\overline{MS}$ mass, with the exact series coefficients up to the four-loop order to determine the ultimate uncertainty of the top-quark pole mass due to the renormalon divergence. We perform extensive tests of our procedure by varying the number of colours and flavours, as well as the scale of the strong coupling and the $\overline{MS}$ mass. Including an estimate of the internal bottom and charm quark mass effect, we conclude that this uncertainty is around 110 MeV. We further estimate the additional contribution to the mass relation from the five-loop correction and beyond to be around 300 MeV