The relation between the (MS)over-bar and the on-shell quark mass at orderalpha(3)(s)

The relation between the on-shell and <(MS)over bar> mass can be expressed through scalar and vector part of the quark propagator. In principle these two-point functions have to be evaluated on-shell which is a non-trivial ta sk at three-loop order. Instead, we evaluate the quark self-energy in the l imit of large and small external momentum and use conformal mapping in comb ination with Pade improvement in order to construct a numerical approximati on for the relation. The errors of our final result are conservatively esti mated to be below 3%. The numerical implications of the results are discuss ed in particular in view of top and bottom quark production near threshold. We show that the knowledge of new partial derivative(alpha(s)(3)) correcti on leads to a significant reduction of the theoretical uncertainty in the d etermination of the quark masses, (C) 2000 Elsevier Science B.V. All rights reserved.