Fourth-order QCD renormalization group quantities in theVscheme and the relation of theβfunction to the Gell-Mann–Low function in QED

Abstract

The semianalytical $O({\ensuremath{\alpha}}_{s}^{4})$ expression for the renormalization group $\ensuremath{\beta}$ 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 $\ensuremath{\beta}$ 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 $O({\ensuremath{\alpha}}_{s}^{4})$ expression for the ${e}^{+}{e}^{\ensuremath{-}}$-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 ${e}^{+}{e}^{\ensuremath{-}}$-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 ${\ensuremath{\beta}}^{V}$ function are closely related to the ones of the Gell-Mann--Low $\mathrm{\ensuremath{\Psi}}$ 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 ${\ensuremath{\beta}}^{V}$ and $\mathrm{\ensuremath{\Psi}}$ functions are discussed.