Z-boson hadronic decay width up to O(αs4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}(\alpha _s^4)$$\end{document}-order QCD corrections using the single-scale approach of the principle of maximum conformality

In the paper, we study the properties of the Z-boson hadronic decay width by using the O(αs4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(\alpha _s^4)$$\end{document}-order quantum chromodynamics (QCD) corrections with the help of the principle of maximum conformality (PMC). By using the PMC single-scale approach, we obtain an accurate renormalization scale-and-scheme independent perturbative QCD (pQCD) correction for the Z-boson hadronic decay width, which is independent to any choice of renormalization scale. After applying the PMC, a more convergent pQCD series has been obtained; and the contributions from the unknown O(αs5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(\alpha _s^5)$$\end{document}-order terms are highly suppressed, e.g. conservatively, we have ΔΓZhad|PMCO(αs5)≃±0.004\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \Gamma _{\mathrm{Z}}^{\mathrm{had}}|^{{{\mathcal {O}}}(\alpha _s^5)}_{\mathrm{PMC}}\simeq \pm 0.004$$\end{document} MeV. In combination with the known electro-weak (EW) corrections, QED corrections, EW–QCD mixed corrections, and QED–QCD mixed corrections, our final prediction of the hadronic Z decay width is ΓZhad=1744.439-1.433+1.390\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\mathrm{Z}}^{\mathrm{had}}=1744.439^{+1.390}_{-1.433}$$\end{document} MeV, which agrees with the PDG global fit of experimental measurements, 1744.4±2.0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1744.4\pm 2.0$$\end{document} MeV.

Following the renormalization group invariance, the physical observable should be independent to theoretical conventions, such as the choices of the renormalization scale and scheme, which is ensured by mutual cancellation of the scale and scheme dependence among different orders for an infinite-order pQCD prediction. However, for a fixed-order pQCD prediction, if the perturbative coefficients and the corresponding α s do not match properly, the pQCD series may have large scale and scheme ambiguities [37]. Conventionally, the renormalization scale is taken as the "guessed" momentum flow of the process, as well as the one to eliminate the large logs or to minimize the contributions from highorder terms or to achieve the prediction in agreement with the experimental data. Those naive treatment directly causes the mismatching between the strong coupling constant and its coefficients and resulting in conventional renormalization scale and scheme ambiguities [38,39]. Such guessing treatment decreases the predictive power of pQCD. In fact, predictions based on conventional scale setting are even incorrect for Abelian theory -Quantum Electrodynamics (QED); the renormalization scale of the QED coupling constant can be set unambiguously by using the Gell-Mann-Low method [40].
A correct renormalization scale-setting approach is thus important for achieving an accurate fixed-order pQCD prediction. Many ways have been suggested in the literature, most of them such as the renormalization group improved effective coupling method (FAC) [41,42] and the principle of minimum sensitivity (PMS) [43][44][45] are designed to find an optimal renormalization scale of the process. On the contrary, the principle of maximum conformality (PMC) [46][47][48][49][50] provides a rigorous idea, whose purpose is not to find an optimal scale, but to determine the effective magnitude of α s for a fixed-order pQCD series by using the renormalization group equation (RGE). The determined effective α s is independent to any choice of renormalization scale, thus the conventional renormalization scale ambiguity is eliminated. Moreover, since all the scheme-dependent non-conformal {β i }-terms have been eliminated, the resultant pQCD series becomes scheme independent conformal series; thus the conventional renormalization scheme ambiguity is simultaneously eliminated.
A recent demonstration of renormalization scale-andscheme independence of PMC prediction has been given in Refs. [51,52], where by using the C-scheme coupling [53], it has been proven that the PMC prediction is independent of the choice of renormalization scale and scheme up to any fixed order. Generally, the convergence of the pQCD series can be improved due to the elimination of the divergent renormalon terms like n!β n 0 α n s or n!β n 0 α n+1 s [54][55][56]. 1 The PMC accurate renormalization scale-and-scheme independent conformal series is helpful not only for achieving precise pQCD predictions but also for a reliable prediction on the contributions of unknown higher-orders; some applications can be found in Refs. [57][58][59][60][61] which are estimated by using the Padé resummation approach [62][63][64]. In the present paper, we shall adopt the PMC single-scale approach [65] 2 to analyze the Z -boson hadronic decay width. 1 The β n (n ≥ 1)-term could be estimated by using the approximation, β n ≈ β n+1 0 , which can be adopted for transforming the {β i }-series at each order into β 0 -power series. 2 In the original PMC multi-scale approach [46,49], different types of {β i }-terms are absorbed into α s via an order-by-order manner, and distinct PMC scales are determined at each order. Such PMC multiscale approach has thus two kinds of residual scale dependence due to the unknown perturbative terms, which has already been pointed out in year 2013 [66]. It should be pointed out that the recently so-called ambiguities of PMC given in Ref. [67] are not the default of PMC, It is noted that the original PMC multi-scale approach [46][47][48][49][50] and single-scale approach are equivalent to each other in sense of perturbative theory [65], but the residual scale dependence emerged in PMC multi-scale method can be greatly suppressed by applying the single-scale approach.
The remaining parts of the paper are organized as follows. In Sect. 2, we will give the detailed PMC treatment for a precise determination of the Z -boson hadronic decay width. In Sect. 3, we will give the numerical results. Section 4 is reserved for a summary.

The Z-boson hadronic decay width using the PMC
The hadronic decay width of the Z -boson can be expressed as where the first term stands for the pure pQCD correction with the leading-order (LO) width 0 = , and the Fermi where the central values are for μ r = M Z , and the errors are for μ r ∈ [M Z /2, 2M Z ]. Here 1 is the b-and t-quark mass corrections to the vector and axial vector correlators [31][32][33][34][35], 2 is the quark final-state QED radiation and the mixed QED-QCD correction [19], 3 is the electro-weak two-loop corrections and the higher-loop corrections in the large-m t limit [11], 4 is the mixed EW-QCD correction and nonfactorizable QCD correction [6][7][8]20].
Our main concern is the perturbative QCD corrections to the dominant correlator of the neutral current, which can be divided as the following four parts: where s W is the effective weak mixing angle, and Footnote 2 continued but the residual scale dependence due to unknown perturbative terms. Such residual scale dependence generally suffer from both the α s -power suppression and the exponential suppression, but could be large due to possibly poor pQCD convergence for the perturbative series of either the PMC scale or the pQCD approximant [52].
and r A S stand for the non-singlet, the vector-singlet, and the axial-singlet part, respectively. Those contributions can be further expressed as where a s = α s /(4π), and the coefficients of r NS , r V S , and r A S can be obtained from Refs. [28][29][30]68,69]. As for r A S , we adopt conventional scale setting approach to perform our analysis, 3 and numerically, we obtain had by using the formulas given by Ref. [28], whose magnitude is quite small in comparison to that of r NS , thus fortunately, this approximate treatment will not affect our final conclusions.
The R-ratio can be rewritten as the following perturbative form by using the degeneracy relations [49,50,70], i.e., where the reduced coefficientsr i, j = r i, j | μ r =M Z , the combination coefficients C k j = j!/[k!( j − k)!]. We put the known coefficientsr i, j up to O(α 4 s )-level in the Appendix. Following the standard PMC single-scale procedures as described in detail in Ref. [65], with the help of RGE, one can determine an effective coupling α s (Q * ) by absorbing all the non-conformal {β i }-terms into the running coupling, and the resultant pQCD series becomes the following conformal series, where Q * is the PMC scale, which corresponds to the overall effective momentum flow of the process and can be deter- 3 From the known O(α 4 s )-order expressions, we cannot derive the exact RG-dependent n f -series for r A S , which is however very important for using the PMC scale-setting; so we have to take this approximation. mined up to next-to-next-to-leading log (NNLL) accuracy by using the present known O(α 4 s )-order pQCD series; i.e., the ln Q 2 * /M 2 Z can be expanded as the following perturbative series, where and It can be found that Q * is exactly free of μ r , and together with the μ r -independent conformal coefficients r i,0 , the conventional renormalization scale ambiguity is eliminated. Therefore, the precision of R nc can be greatly improved by using the PMC. Moreover, the precision of the predictions depend on the perturbative nature of both the R nc and the ln Q 2 * /M 2 Z , which shall be numerically analyzed in the following paragraphs.

Numerical results
To do the numerical calculation, we adopt the Z -boson mass M Z = 91.1876 ± 0.0021 GeV and top-quark pole mass M t = 172.9 GeV [71]. We use the four-loop α s -running behavior [52] to analyse the O(α 4 s )-order QCD corrections. i.e., Where t = ln(μ 2 r / 2 QCD ), b i = β i /β 0 , and the β i (i = 0, 1, 2, 3)-functions have been calculated in Refs. [72][73][74][75][76][77][78][79][80]. Taking α s (M Z ) = 0.1181 [71], we obtain  s )-order prediction is due to good convergence of the perturbative series, e.g., the relative magnitudes of the α s -terms: α 2 s -terms: α 3 s -terms: α 4 s -terms=1: 2.9%: −2.2%: −0.4% for the case of μ r = M Z ; and also due to the cancellation of the scale dependence among different orders. 4 The scale errors for each order term remain unchanged and large, e.g. the had Z has the following perturbative feature up to O(α 4 s )-order: where the central values are for μ r = M Z , and the errors are obtained by varying μ r ∈ [M Z /2, 2M Z ]. It shows that the absolute scale errors are 16%, 495%, 131%, and 143% for the α s -terms, α 2 s -terms, α 3 s -terms, and α 4 s -terms, respectively; 4 In cases when each perturbative terms varies synchronously with the changes of μ r , there will have no cancellations of scale dependence among different orders and the pQCD approximant could be still large even for higher-orders. A recent example can be found in a two-loop QCD correction for γ + η c production in electron-positron collisions [61]. PMC is the same as the conventional one. The PMC starts to work at the O(α 2 s ) and higher order levels. It shows that after applying the PMC, the pQCD convergence can be greatly improved, e.g., the relative magnitudes of the α s -terms: α 2 s -terms: α 3 s -terms of the pQCD series changes 1: 4.34%: −0.49% by applying the PMC scale-setting to the perturbative series up to O(α 3 s ), whose PMC scale Q * = 113.0 GeV is fixed up to the NLL accuracy. The relative magnitudes of the α s -terms: α 2 s -terms: α 3 s -terms: α 4 s -terms of the pQCD series becomes to 1: 4.33%: −0.49%: 0.01% by applying the PMC up to O(α 4 s ), whose PMC scale Q * = 114.9 GeV is fixed up to NNLL accuracy. And there is no renormalization scale dependence for had Z at any fixed order, i.e., where each perturbative terms and the net total decay width are unchanged for any choice of μ r . This behavior is consistent with that of the previous PMC multi-scale approach analysis on R nc [81]. The PMC single scale Q * is an effective scale which effectively replaces the individual PMC scales introduced in the PMC multi-scale approach in the sense of a mean value theorem, which can be regarded as the overall effective momentum flow of the process; it shows stability and convergence with increasing order in pQCD via the pQCD approximates. More explicitly, we obtain Q * = 114.9 One may observe that the relative magnitudes of each order terms in Q * perturbative series are 1 : 91% : 15%, which also shows a good convergence behavior. Third, it is helpful to predict the magnitude of the "unknown" higher-order pQCD corrections. The renormalization scale independent PMC series is helpful for such purpose. Because the PMC series has a good pertubative convergence, e.g., the magnitude of O(α 4 s )-order term is only 0.01% of O(α s )-order term, it is reasonable to take the magnitude of the last known term ±|r 4,0 a 4 s (Q * )| as a conservative prediction of the uncalculated higher-order terms [39]. By further taking the variation of Q * ±1.9 GeV, which is the difference between the NLL and NNLL PMC scales, as the magnitude of its unknown NNNLL term into consideration, 5 we obtain Here, when discussing one uncertainty, the other input parameters shall be set as their central values. As a whole, the squared average of the above mentioned three errors leads to a net error, ±0.586 MeV, to the PMC prediction of the total decay width had Z , among which the magnitude of α s (M Z ) dominates the error sources. Thus more precise measurements on the reference point α s (M Z ) is important for a more precise pQCD prediction. Fig. 3 The PMC prediction of the Z -boson hadronic decay in comparison with the experimental values given by the PDG global fit of experimental data [71], and by the OPAL [82], DELPHI [83], L3 [84], and ALEPH [85] where the errors are the sum of two parts, one is the squared average of those from α s (M Z ), M Z , and the uncalculated higher-order terms, another is the error from Extra Z as given in Eq. (2). After applying the PMC single-scale approach, the pQCD series becomes scale independent and more convergent, thus a reliable pQCD prediction can be achieved. Due to the perturbative terms have been known up to enough high-orders, the predictions under the PMC and conventional scale-setting approaches are consistent with each other. We present the PMC prediction of the Z -boson hadronic decay width in Fig. 3, where the experimental data are presented as a comparison. The PMC prediction agrees with the PDG global fit of the experimental measurements. Thus, one obtains optimal fixed-order predictions for the Z -boson hadronic decay width by applying the PMC, enabling high precision test of the Standard Model. China under Grant no. 11625520, no. 11905056, no. 11975187, no. 11947406, and the Fundamental Research Funds for the Central Universities under Grant No.2020CQJQY-Z003.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: All the figures and the numerical predictions can be derived from the formulas presented in the paper, so the data do not need to be deposited.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .