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

TL;DR

This paper derives the four-loop $eta$-function in the V-scheme for QCD and QED, compares it with other schemes, and discusses its analogy with the Gell-Mann--Low function, highlighting new theoretical insights.

Contribution

It provides the semi-analytical four-loop $eta$-function in the V-scheme for QCD, and analyzes its relation to the Gell-Mann--Low function in QED, offering a new perspective on scheme dependence.

Findings

Derived the four-loop $eta$-function in the V-scheme for QCD.

Compared the structure of $eta$-functions across different schemes.

Identified the $N^2$-proportional difference between the V-scheme $eta$-function and the Gell-Mann--Low function in QED.

Abstract

The semi-analytical expression for the forth coefficient of 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 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 $\beta$- 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 $\beta^{\rm{V}}$-function is starting to differ from the Gell-Mann--Low function $\Psi(\alpha_{\rm{MOM}})$ 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 $\beta^{\rm{V}}$-function as the most theoretically substantiated analog of the Gell-Man--Low function in QCD.