Renormalization and running of quark mass and field in the regularization invariant and (MS)over-bar schemes at three and four loops

We derive explicit transformation formulae relating the renormalized quark mass and field as defined in the <(MS)over bar>-scheme with the correspondi ng quantities defined in any other scheme. By analytically computing the th ree-loop quark propagator in the high-energy limit (that is keeping only ma ssless terms and terms of first order in the quark mass) we find the NNNLO conversion factors transforming the <(MS)over bar> quark mass and the renor malized quark field to those defined in a "Regularization Invariant" (RI) s cheme which is more suitable for lattice QCD calculations. The NNNLO contri bution in the mass conversion factor turns out to be large and comparable t o the previous NNLO contribution at a scale of 2 GeV - the typical normaliz ation scale employed in lattice simulations. Thus, in order to get a precis e prediction for the <(MS)over bar> masses of the light quarks from lattice calculations the latter should use a somewhat higher scale of around, say, 3 GeV where the (apparent) convergence of the perturbative series for the mass conversion factor is better. We also compute two more terms in the high-energy expansion of the <(MS)ove r bar> renormalized quark propagator. The result is then used to discuss th e uncertainty caused by the use of the high energy limit in determining the <(MS)over bar> mass of the charmed quark. As a by-product of our calculati ons we determine the four-loop anomalous dimensions of the quark mass and f ield in the Regularization Invariant scheme. Finally, we discuss some physi cal reasons lying behind the striking absence of zeta(4) in these computed anomalous dimensions. (C) 2000 Elsevier Science B.V. All rights reserved.