The four-loop renormalization group QCD and QED $β$-functions in the $V$-scheme and their analogy with the Gell-Mann--Low function in QED and QCD

The semi-analytical expression for the forth coefficient of the renormalization group $β$-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 three-loop perturbative approximation for the QCD static potential, evaluated in the $\rm{\overline{MS}}$-scheme. The importance of getting more detailed expressions for the $n_f$-independent three-loop contribution to the static potential,obtained at present by two groups, is emphasised. The comparison of the numerical structure of the four-loop approximations for the RG $β$- function of QCD in the gauge-independent ${\rm{V}}$- and $\rm{\overline{MS}}$-schemes and in the minimal MOM scheme in the Landau gauge are presented. Considering the limit of QED with $N$-types of leptons we discover that the $β^{\rm{V}}$-function is starting to differ from the Gell-Mann--Low function $Ψ(α_{\rm{MOM}})$ at the level of the forth-order perturbative corrections, receiving the proportional to $N^2$ additional term. Taking this feature into account, we propose to consider the $β^{\rm{V}}$-function as the most theoretically substantiated analog of the Gell-Man--Low function in QCD.

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