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arXiv 1507.03547hep-phhep-th
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{ \rm{V} }格式中的四阶QCD重整化群量及QED中$β$ -函数与Gell-Mann-Low函数的关系

Fourth-order QCD renormalization group quantities in the {\rm{V}}-scheme and the relation of the $β$-function to the Gell-Mann--Low function in QED

A. L. Kataev, V. S. Molokoedov

备注：
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
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摘要：

在$SU(N_c)$规范群的情况下，得到了${\rm{V}}$ -方案中重整化群$β$ -函数的半解析$O(α_s^4)$表达式.在计算的过程中，我们使用现有的信息的三圈微扰近似的QCD静态势，评估的$\rm{\overline{MS}}$ -计划。本文比较了在Landau规范下，规范无关的${\rm{V}}$ -和$\rm{\overline{MS}}$ -格式以及最小化格式中QCD RG $β$ -函数的第三和第四系数的数值。从唯象的角度比较了这些方案中$e^+e^-$湮没R比的$O(α_s^4)$表达式的系数。它示出，考虑到这些QCD的贡献是至关重要的，并导致急剧减少的计划依赖的模糊性的四阶微扰QCD近似的$e^+e^-$湮没R-比率的活性风味，特别是$n_f=5$的数量。我们证明，在QED的情况下，与$N$ -类型的轻子的系数的$β^{\rm{V}}$ -函数的Gell-Mann-低$Ψ$ -函数是密切相关的，并强调，他们开始彼此不同的第四阶由于额外的$N^2$ -贡献的出现在V-计划。澄清了这种额外校正的来源。本文讨论了$β^{\rm{V}}$ -和$Ψ$ -函数系数之间的一般全阶QED关系。

英文摘要：

The semi-analytical $O(α_s^4)$ expression for 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 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 $β$- 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 $O(α_s^4)$ 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 $β^{\rm{V}}$-function are closely related to the ones of the Gell-Mann--Low $Ψ$-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 $β^{\rm{V}}$- and $Ψ$-functions are discussed.

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