TTP99-43 Renormalization and Running of Quark Mass and Field in the Regularization Invariant and $\msbar$ Schemes at Three and Four Loops

We derive explicit transformation formulae relating the renormalized quark mass and field as defined in the $\msbar$-scheme with the corresponding quantities defined in any other scheme. By analytically computing the three-loop quark propagator in the high-energy limit (that is keeping only massless terms and terms of first order in the quark mass) we find the NNNLO conversion factors transforming the $\msbar$ quark mass and the renormalized quark field to those defined in a ``Regularization Invariant'' (\RI) scheme which is more suitable for lattice QCD calculations. The NNNLO contribution in the mass conversion factor turns out to be large and comparable to the previous NNLO contribution at a scale of 2 GeV — the typical normalization scale employed in lattice simulations. Thus, in order to get a precise prediction for the $\msbar$ masses of the light quarks from lattice calculations the latter should use 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 $\msbar$ renormalized quark propagator. The result is then used to discuss the uncertainty caused by the use of the high energy limit in determining the $\msbar$ mass of the charmed quark. Finally, as a by-product of our calculations we determine the four-loop anomalous dimensions of quark mass and field in the Regularization Invariant scheme.