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

TL;DR

This paper derives the four-loop QCD $eta$-function in the V-scheme, compares it across schemes, and explores its relation to the Gell-Mann--Low function in QED, highlighting scheme dependence and phenomenological implications.

Contribution

It provides the semi-analytical four-loop $eta$-function in the V-scheme for QCD and clarifies its relation to the Gell-Mann--Low function in QED, including scheme comparisons.

Findings

The four-loop $eta$-function in the V-scheme is obtained semi-analytically.

Scheme dependence of the $eta$-function coefficients is significantly reduced when including these contributions.

In QED, the V-scheme $eta$-function coefficients differ from the Gell-Mann--Low function starting at the fourth order.

Abstract

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 existing information about the three-loop perturbative approximation for the QCD static potential, evaluated in the $\rm{\overline{MS}}$-scheme. The comparison of the numerical values of the third and fourth coefficients for the QCD RG $\beta$- functions in the gauge-independent ${\rm{V}}$- and $\rm{\overline{MS}}$-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 $\beta^{\rm{V}}$-function are closely related to the ones of the Gell-Mann--Low $\Psi$-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 $\beta^{\rm{V}}$- and $\Psi$-functions are discussed.