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

Abstract

The semi-analytical $O({\alpha }_s^4)$ expression for the renormalization group $\beta$-function in the $\mathrm{V}$-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 $\beta$- functions in the gauge-independent $\mathrm{V}$- and $\overline{\mathrm{MS}}$-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 ${\beta }^{\mathrm{V}}$-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 ${\beta }^{\mathrm{V}}$- and $\Psi$-functions are discussed.

The text is in accordance with the one, published in Phys.Rev.D92 (2015), 5, 054008; title a bit modified, 1 reference removed in accordance with new style of references presntation in PRD, Editorial misprints in Eq.(4.17) and above Eq.(5.6) are corrected

References in corpus (8)

• Hadronic Z- and tau-Decays in Order alpha_s^4
• Fermionic contributions to the three-loop static potential
• The bottom quark mass from the ${\rm Y}(1S)$ system at NNNLO
• The relation between the QED charge renormalized in MSbar and on-shell schemes at four loops, the QED on-shell beta-function at five loops and asymptotic contributions to the muon anomaly at five and six loops
• Full Formula for Heavy Quarkonium Energy Levels at Next-to-next-to-next-to-leading Order
• Banks-Zaks fixed point analysis in momentum subtraction schemes
• Momentum subtraction and the R-ratio
• Peculiarities in the Spectrum of the Adjoint Scalar Kinetic Operator in Yang-Mills Theory

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