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Register a new userThe 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
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A. L. Kataev
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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.
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This work was carried out in 1985. It was published in Russian in Yad. Fiz. 44, 498 (1986) [English translation Sov. J. Nucl. Phys. 44, 321 (1986)]. None of these publications are available on-line. Submitting this paper to ArXiv will make it accessible. *** A simple method is developed that makes it possible to determine the $k$-loop coefficient of the $beta$-function if the operator product expansion for certain polarization operators in the $(k -1)$ loop is known. The calculation of the two-loop coefficient of the Gell-Mann-Low function becomes trivial -- it reduces to a few algebraic operations on already known expressions. As examples, spinor, scalar, and supersymmetric electrodynamics are considered. Although the respective results for $beta^{(2)}$ are known in the literature, both the method of calculation and certain points pertaining to the construction of the operator product expansion are new.
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