Fourth-order QCD renormalization group quantities in the V scheme and the relation of the β function to the Gell-Mann-Low function in QED

The semianalytical O(alpha(4)(s)) expression for the renormalization group beta function in the V scheme is obtained in the case of the SU(N-c) gauge group. In the process of calculations we use the existing information about the three-loop perturbative approximation for the QCD static potential, evaluated in the (MS) over bar scheme. The comparison of the numerical values of the third and fourth coefficients for the QCD RG beta functions in the gauge-independent V and (MS) over bar schemes and in the minimal momentum scheme in the Landau gauge is presented. The phenomenologically oriented comparisons for the coefficients of O(alpha(4)(s)) expression for the e(+)e(-) annihilation R-ratio in these schemes are presented. It is shown that taking into account these QCD contributions is of vital importance and leads to a drastic decrease of the scheme-dependence ambiguities of the fourth-order perturbative QCD approximations for the e(+)e(-)-annihilation R-ratio for the number of active flavors, n(f) = 5 in particular. We demonstrate that in the case of QED with N-types of leptons the coefficients of the beta(V) function are closely related to the ones of the Gell-Mann-Low Psi function and emphasize that they start to differ from each other at the fourth order due to the appearance of the extra N-2-contribution in the V scheme. The source of this extra correction is clarified. The general all-order QED relations between the coefficients of the beta(V) and Psi functions are discussed.