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The semi-analytical $O(\alpha_s^4)$ expression for the renormalization group$\beta$-function in the ${\rm{V}}$-scheme is obtained in the case of the$SU(N_c)$ gauge group. In the process of calculations we use the existinginformation about the three-loop perturbative approximation for the QCD staticpotential, evaluated in the $\rm{\overline{MS}}$-scheme. The comparison of thenumerical values of the third and fourth coefficients for the QCD RG $\beta$-functions in the gauge-independent ${\rm{V}}$- and $\rm{\overline{MS}}$-schemesand in minimal MOM scheme in the the Landau gauge is presented. Thephenomenologically-oriented comparisons for the coefficients of $O(\alpha_s^4)$expression for the $e^+e^-$-annihilation R-ratio in these schemes arepresented. It is shown, that taking into account of these QCD contributions areof vital importance and lead to the drastic decrease of the scheme-dependenceambiguities of the fourth-order perturbative QCD approximations for the$e^+e^-$ annihilation R-ratio for the number of active flavours,$n_f=5$ inparticular. We demonstrate that in the case of QED with $N$-types of leptonsthe coefficients of the $\beta^{\rm{V}}$-function are closely related to theones of the Gell-Mann--Low $\Psi$-function and emphasise that they start todiffer 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 isclarified. The general all-order QED relations between the coefficients of the$\beta^{\rm{V}}$- and $\Psi$-functions are discussed.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Fourth-order QCD renormalization group quantities in the {\rm{V}}-scheme  and the relation of the $\beta$-function to the Gell-Mann--Low function in  QED