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

The semi-analytical $O({\alpha }_s^4)$ expression for the renormalization group $\beta$-function in the $\mathrm{V}$-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 $\overline{\mathrm{M}\mathrm{S}}$-scheme. The comparison of the numerical values of the third and fourth coefficients for the QCD RG $\beta$- functions in the gauge-independent $\mathrm{V}$- and $\overline{\mathrm{M}\mathrm{S}}$-schemes and in minimal MOM scheme in the the Landau gauge is presented. The phenomenologically-oriented comparisons for the coefficients of $O({\alpha }_s^4)$ expression for the $e^{+}{e}^{-}$-annihilation R-ratio in these schemes are presented. It is shown, that taking into account of these QCD contributions are of vital importance and lead to the 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 flavours,$n_f=5$ in particular. We demonstrate that in the case of QED with $N$-types of leptons the coefficients of the ${\beta }^{\mathrm{V}}$-function are closely related to the ones of the Gell-Mann–Low $\mathrm{\Psi }$-function and emphasise 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 ${\beta }^{\mathrm{V}}$- and $\mathrm{\Psi }$-functions are discussed.