The semianalytical $O({\alpha }_s^4)$ 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 $\overline{\mathrm{MS}}$ 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 $\overline{\mathrm{MS}}$ schemes and in the minimal momentum scheme in 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 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 $\mathrm{\Psi }$ 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 $\mathrm{\Psi }$ functions are discussed.

• Received 29 April 2015

DOI:https://doi.org/10.1103/PhysRevD.92.054008