Z-boson hadronic decay width up to ${{\mathcal {O}}}(\alpha _s^4)$-order QCD corrections using the single-scale approach of the principle of maximum conformality

Abstract

In the paper, we study the properties of the Z-boson hadronic decay width by using the $\mathcal {O}(\alpha _s^4)$-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 $\mathcal {O}(\alpha _s^5)$-order terms are highly suppressed, e.g. conservatively, we have $\Delta \Gamma _{\mathrm{Z}}^{\mathrm{had}}|^{{{\mathcal {O}}}(\alpha _s^5)}_{\mathrm{PMC}}\simeq \pm 0.004$ 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 $\Gamma _{\mathrm{Z}}^{\mathrm{had}}=1744.439^{+1.390}_{-1.433}$ MeV, which agrees with the PDG global fit of experimental measurements, $1744.4\pm 2.0$ MeV.

1 Introduction

In quantum chromodynamics (QCD), the Z-boson hadronic decay width plays an important role in determining the strong coupling constant $(\alpha _s)$. The Z-boson hadronic decay width has been measured by various collaborations at the electron–positron colliders such as LEP and SLC [1, 2], which could also be precisely measured in future high luminosity colliders such as the super Z factory [3] or CEPC [4]. Theoretically, the one-loop electroweak (EW) and the mixed EW-QCD contributions to the Z-boson hadronic decay have been investigated in Refs. [5,6,7,8], and the two-loop EW contribution has been given in Refs. [9,10,11]. In large-$m_t$ limit, the higher-loop corrections have been calculated up to $\mathcal {O}(\alpha _t \alpha _s^2)$ [12, 13], $\mathcal {O}(\alpha _t^2 \alpha _s)$, $\mathcal {O}(\alpha _t^3)$ [14, 15], and $\mathcal {O}(\alpha _t \alpha _s^3)$ [16,17,18], respectively, where $\alpha _t\equiv y_t^2/(4\pi )$ with $y_t$ being the top-quark Yukawa coupling constant. The final-state QED radiations have been computed up to $\mathcal {O}(\alpha )$, $\mathcal {O}(\alpha \alpha _s)$, $\mathcal {O}(\alpha ^2)$ in Ref. [19]. The non-factorizable QCD corrections have been estimated in Refs. [8, 20]. The pure perturbative QCD (pQCD) corrections up to $\mathcal {O}(\alpha _s^2)$ [21, 22], $\mathcal {O}(\alpha _s^3)$ [23,24,25,26,27], $\mathcal {O}(\alpha _s^4)$ [28,29,30] have also been performed in the literature. Moreover, the mass corrections to both the vector and axial vector correlators can be found in Refs. [21, 22, 31,32,33,34,35]. Those achievements give us good opportunities for precise determining of $\alpha _s(M_Z)$, e.g. a recent determination of $\alpha _s$ has been given in Ref. [36].

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 $\alpha _s$ 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 high-order 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 $\alpha _s$ for a fixed-order pQCD series by using the renormalization group equation (RGE). The determined effective $\alpha _s$ is independent to any choice of renormalization scale, thus the conventional renormalization scale ambiguity is eliminated. Moreover, since all the scheme-dependent non-conformal $\{\beta _i\}$-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-and-scheme 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 $n!\beta _0^n\alpha _s^n$ or $n!\beta _0^n\alpha _s^{n+1}$ [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 Pad$\acute{e}$ 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. 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.

2 The Z-boson hadronic decay width using the PMC

The hadronic decay width of the Z-boson can be expressed as

$$\begin{aligned} \Gamma _{\mathrm{Z}}^{\mathrm{had}}=\Gamma _0R^{\mathrm{nc}}+\Delta \Gamma _{\mathrm{Z}}^{\mathrm{Extra}}, \end{aligned}$$

(1)

where the first term stands for the pure pQCD correction with the leading-order (LO) width $\Gamma _0=\frac{G_F M^3_Z}{24\pi \sqrt{2}}$, and the Fermi coupling constant $G_F=1.166\times 10^{-5} \mathrm{GeV}^{-2}$. The second term $\Delta \Gamma _{\mathrm{Z}}^{\mathrm{Extra}}$ contains four less important corrections, i.e.,

$$\begin{aligned} \Delta \Gamma _{\mathrm{Z}}^{\mathrm{Extra}}= & {} \Delta \Gamma _1+\Delta \Gamma _2+\Delta \Gamma _3+\Delta \Gamma _4 \nonumber \\= & {} -1.577^{+0.183}_{-0.237}+0.695^{+0.000}_{-0.001}+6.577^{+0.560}_{-0.560} \nonumber \\&+0.609^{+0.061}_{-0.049} ~(\mathrm{MeV}) \nonumber \\= & {} 6.304^{+0.804}_{-0.847} ~(\mathrm{MeV}), \end{aligned}$$

(2)

where the central values are for $\mu _r=M_Z$, and the errors are for $\mu _r\in [M_Z/2, 2M_Z]$. Here $\Delta \Gamma _1$ is the b- and t-quark mass corrections to the vector and axial vector correlators [31,32,33,34,35], $\Delta \Gamma _2$ is the quark final-state QED radiation and the mixed QED-QCD correction [19], $\Delta \Gamma _3$ is the electro-weak two-loop corrections and the higher-loop corrections in the large-$m_t$ limit [11], $\Delta \Gamma _4$ 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:

$$\begin{aligned} R^{\mathrm{nc}}=3\bigg [\sum _f v^2_f r^V_{\mathrm{NS}}+\bigg (\sum _f v_f\bigg )^2 r^V_ S+\sum _f a^2_f r^ A_{\mathrm{NS}}+r^A_S\bigg ],\nonumber \\ \end{aligned}$$

(3)

where $v_f\equiv 2I_f-4q_fs^2_W$, $a_f\equiv 2I_f$, $q_f$ is the f-quark electric charge, $s_W$ is the effective weak mixing angle, and $I_f$ is the third component of weak isospin of the left-handed component of f. $r^V_{\mathrm{NS}}=r^A_{\mathrm{NS}}\equiv r_{\mathrm{NS}}$, $r^V_S$, 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

$$\begin{aligned} r_{\mathrm{NS}} = 1+\sum ^n_{i=1}C^{\mathrm{NS}}_ia^i_s, r^V_S =\sum ^n_{i=3}C^{\mathrm{VS}}_{i}a^i_s, ~r^A_S=\sum ^n_{i=2}C^{\mathrm{AS}}_{i}a^i_s , \end{aligned}$$

where $a_s=\alpha _s/(4\pi )$, and the coefficients of $r_{\mathrm{NS}}$, $r^V_S$, 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 $\Gamma _{\mathrm{Z}}^{\mathrm{had}}|^A_S=[-1.725,-1.685]$ MeV for $\mu _r \in [M_Z/2, 2M_Z]$ by using the formulas given by Ref. [28], whose magnitude is quite small in comparison to that of $r_{\mathrm{NS}}$, 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.,

$$\begin{aligned} R^{\mathrm{nc}}= & {} r_0+r_{1,0}a_s(\mu _r) + (r_{2,0}+\beta _{0}r_{2,1})a_{s}^{2}(\mu _r)\nonumber \\&+(r_{3,0}+\beta _{1}r_{2,1}+ 2\beta _{0}r_{3,1}+ \beta _{0}^{2}r_{3,2})a_{s}^{3}(\mu _r)\nonumber \\&+(r_{4,0}+\beta _{2}r_{2,1}+ 2\beta _{1}r_{3,1} + \frac{5}{2}\beta _{1}\beta _{0}r_{3,2} \nonumber \\&+3\beta _{0}r_{4,1}+3\beta _{0}^{2}r_{4,2}+\beta _{0}^{3}r_{4,3}) a_{s}^{4}(\mu _r)+\mathcal {O}(a^5_s),~ \end{aligned}$$

(4)

where $r_0=3(\sum _f v^2_f+\sum _f a^2_f)$, and the coefficients $r_{i,j}$ can be obtained from the known coefficients $C^{\mathrm{NS}}$, $C^{\mathrm{VS}}$, and $C^{\mathrm{AS}}_{i}$ of $r_{\mathrm{NS}}$, $r^V_S$, and $r^A_S$. The coefficients $r_{i,0}$ are $\{\beta _i\}$-independent conformal coefficients, and the $\{\beta _i\}$-dependent non-conformal coefficients $r_{i,j}$ $(j\ne 0)$ are generally functions of $\ln \mu ^{2}_{r} /M_{Z}^{2}$, i.e.,

$$\begin{aligned} r_{i,j}=\sum ^j_{k=0}C^k_j{{\hat{r}}}_{i-k,j-k}\mathrm{ln}^k(\mu _r^2/M_Z^2), \end{aligned}$$

(5)

where the reduced coefficients ${{\hat{r}}}_{i,j}=r_{i,j}|_{\mu _r=M_Z}$, the combination coefficients $C^k_j=j!/[k!(j-k)!]$. We put the known coefficients ${{\hat{r}}}_{i,j}$ up to ${{\mathcal {O}}}(\alpha _s^4)$-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 $\alpha _s(Q_*)$ by absorbing all the non-conformal $\{\beta _i\}$-terms into the running coupling, and the resultant pQCD series becomes the following conformal series,

$$\begin{aligned} R^{\mathrm{nc}}|_{\mathrm{PMC}}= & {} r_0+r_{1,0}a_s(Q_*)+r_{2,0}a^2_s(Q_*)\nonumber \\&+r_{3,0}a^3_s(Q_*)+r_{4,0}a^4_s(Q_*)+\mathcal {O} (a^5_s), \end{aligned}$$

(6)

where $Q_*$ is the PMC scale, which corresponds to the overall effective momentum flow of the process and can be determined up to next-to-next-to-leading log (NNLL) accuracy by using the present known ${{\mathcal {O}}}(\alpha _s^4)$-order pQCD series; i.e., the $\ln {Q^2_*}/{M^2_Z}$ can be expanded as the following perturbative series,

$$\begin{aligned} \ln \frac{Q^2_*}{M^2_Z}=T_0+T_1 a_s(M_Z)+T_2 a^2_s(M_Z)+ {{\mathcal {O}}}(a^3_s), \end{aligned}$$

(7)

where

$$\begin{aligned} T_0=&-\frac{{{\hat{r}}}_{2,1}}{{{\hat{r}}}_{1,0}}, \end{aligned}$$

(8)

$$\begin{aligned} T_1=&\frac{ \beta _0 ({{\hat{r}}}_{2,1}^2-{{\hat{r}}}_{1,0} {{\hat{r}}}_{3,2})}{{{\hat{r}}}_{1,0}^2}+\frac{2 ({{\hat{r}}}_{2,0} {{\hat{r}}}_{2,1}-{{\hat{r}}}_{1,0} {{\hat{r}}}_{3,1})}{{{\hat{r}}}_{1,0}^2}, \end{aligned}$$

(9)

and

$$\begin{aligned} T_2=&\frac{3 \beta _1 ({{\hat{r}}}_{2,1}^2-{{\hat{r}}}_{1,0} {{\hat{r}}}_{3,2})}{2 {{\hat{r}}}_{1,0}^2}\nonumber \\&+\frac{4({{\hat{r}}}_{1,0} {{\hat{r}}}_{2,0} {{\hat{r}}}_{3,1}-{{\hat{r}}}_{2,0}^2 {{\hat{r}}}_{2,1})+3({{\hat{r}}}_{1,0} {{\hat{r}}}_{2,1} {{\hat{r}}}_{3,0}-{{\hat{r}}}_{1,0}^2 {{\hat{r}}}_{4,1})}{ {{\hat{r}}}_{1,0}^3} \nonumber \\&+\frac{ \beta _0 (4 {{\hat{r}}}_{2,1} {{\hat{r}}}_{3,1} {{\hat{r}}}_{1,0}-3 {{\hat{r}}}_{4,2} {{\hat{r}}}_{1,0}^2+2 {{\hat{r}}}_{2,0} {{\hat{r}}}_{3,2} {{\hat{r}}}_{1,0}-3 {{\hat{r}}}_{2,0} {{\hat{r}}}_{2,1}^2)}{ {{\hat{r}}}_{1,0}^3}\nonumber \\&+\frac{ \beta _0^2 (2 {{\hat{r}}}_{1,0} {{\hat{r}}}_{3,2} {{\hat{r}}}_{2,1}- {{\hat{r}}}_{2,1}^3- {{\hat{r}}}_{1,0}^2 {{\hat{r}}}_{4,3})}{ {{\hat{r}}}_{1,0}^3}. \end{aligned}$$

(10)

It can be found that $Q_*$ is exactly free of $\mu _r$, and together with the $\mu _r$-independent conformal coefficients $r_{i,0}$, the conventional renormalization scale ambiguity is eliminated. Therefore, the precision of $R^{\mathrm{nc}}$ can be greatly improved by using the PMC. Moreover, the precision of the predictions depend on the perturbative nature of both the $R^{\mathrm{nc}}$ and the $\ln {Q^2_*} / {M^2_Z}$, which shall be numerically analyzed in the following paragraphs.

3 Numerical results

To do the numerical calculation, we adopt the Z-boson mass $M_{Z}=91.1876\pm 0.0021$ GeV and top-quark pole mass $M_{t}=172.9$ GeV [71]. We use the four-loop $\alpha _s$-running behavior [52] to analyse the ${{\mathcal {O}}}(\alpha _s^4)$-order QCD corrections. i.e.,

$$\begin{aligned} \alpha _s(\mu _r)\simeq & {} \frac{1}{\beta _0 t}-\frac{b_1\ln t}{(\beta _0 t)^2}+\frac{b_1^2(\ln ^2 t-\ln t-1)+b_2}{(\beta _0 t)^3}\\&+\frac{1}{(\beta _0 t)^4}\bigg [b_1^3\left( -\ln ^3 t+\frac{5}{2}\ln ^2 t+2\ln t-\frac{1}{2}\right) \\&-3b_1 b_2\ln t+\frac{b_3}{2}\bigg ]+\mathcal {O}\left( \frac{1}{(\beta _0 t)^5}\right) , \end{aligned}$$

Where $t=\ln (\mu _r^2/\Lambda _{\mathrm{QCD}}^2)$, $b_i=\beta _i/\beta _0$, and the $\beta _i(i=0,1,2,3)$-functions have been calculated in Refs. [72,73,74,75,76,77,78,79,80]. Taking $\alpha _s(M_{Z})=0.1181$ [71], we obtain $\Lambda ^{(n_f=5)}_{\mathrm{QCD}}=209.5$ MeV.

Fig. 1

The Z-boson hadronic decay width $\Gamma _{\mathrm{Z}}^{\mathrm{had}}$ under the conventional scale-setting approach versus the renormalization scale $\mu _r$. The dotted, the dashed-dot, the dashed and the solid lines are for the predictions up to ${{\mathcal {O}}}(\alpha _s)$, ${{\mathcal {O}}}(\alpha _s^2)$, ${{\mathcal {O}}}(\alpha _s^3)$, ${{\mathcal {O}}}(\alpha _s^4)$ order levels, respectively

First, by setting all input parameters to be their central values, we present the Z-boson hadronic decay width $\Gamma _{\mathrm{Z}}^{\mathrm{had}}$ up to different known $\alpha _s$-orders under conventional scale-setting approach in Fig. 1. It shows that in agreement of the conventional wisdom, the renormalization scale dependence becomes small when we have known more loop terms. For examples, we obtain $\Gamma _{\mathrm{Z}}^{\mathrm{had}}|_{\mathrm{Conv.}} =[1744.378,1744.587]$ MeV for $\mu _r \in [M_Z/2, 2M_Z]$, and $\Gamma _{\mathrm{Z}}^{\mathrm{had}}|_{\mathrm{Conv.}} =[1744.378,1745.008]$ MeV for $\mu _r \in [M_Z/3, 3M_Z]$; e.g., the net scale errors are only $\sim 0.01\%$, and $\sim 0.04\%$, respectively. We should point out that as has been mentioned in the Introduction, such small net scale dependence for the $\mathcal {O}(\alpha _s^4)$-order prediction is due to good convergence of the perturbative series, e.g., the relative magnitudes of the $\alpha _s$-terms: $\alpha _s^2$-terms: $\alpha _s^3$-terms: $\alpha _s^4$-terms=1: $2.9\%$: $-2.2\%$: $-0.4\%$ for the case of $\mu _r=M_Z$; 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 $\Gamma _{\mathrm{Z}}^{\mathrm{had}}$ has the following perturbative feature up to $\mathcal {O}(\alpha _s^4)$-order:

$$\begin{aligned} \Gamma _{\mathrm{Z}}^{\mathrm{had}}|_{\mathrm{Conv.}}= & {} 1681.262+62.966^{-5.925}_{+4.268}+1.802^{+4.838}_{-4.078}\nonumber \\&-1.382^{+1.311}_{-0.505}-0.230^{-0.055}_{+0.275} \nonumber \\= & {} 1744.418^{+0.169}_{-0.040}~(\mathrm{MeV}), \end{aligned}$$

(11)

where the central values are for $\mu _r=M_Z$, and the errors are obtained by varying $\mu _r \in [M_Z/2, 2M_Z]$. It shows that the absolute scale errors are $16\%$, $495\%$, $131\%$, and $143\%$ for the $\alpha _s$-terms, $\alpha _s^2$-terms, $\alpha _s^3$-terms, and $\alpha _s^4$-terms, respectively; and there do have large scale cancellations among different orders.

Fig. 2

The Z-boson hadronic decay width $\Gamma _{\mathrm{Z}}^{\mathrm{had}}$ under the PMC scale-setting approach versus the renormalization scale $\mu _r$. The dotted, the dashed-dot, the dashed and the solid lines are for the predictions up to ${{\mathcal {O}}}(\alpha _s)$, ${{\mathcal {O}}}(\alpha _s^2)$, ${{\mathcal {O}}}(\alpha _s^3)$, ${{\mathcal {O}}}(\alpha _s^4)$ order levels, respectively

Second, we present the Z-boson hadronic decay width $\Gamma _{\mathrm{Z}}^{\mathrm{had}}$ up to different known $\alpha _s$-orders under the PMC scale-setting approach in Fig. 2. At the $\mathcal {O}(\alpha _s)$-order level, the perturbative series of $\Gamma _{\mathrm{Z}}^{\mathrm{had}}$ does not have $\{\beta _i\}$-terms to fix the $\alpha _s$ value, thus the prediction of $\Gamma _{\mathrm{Z}}^{\mathrm{had}}|^{\mathcal {O}(\alpha _s)}_{\mathrm{PMC}}$ is the same as the conventional one. The PMC starts to work at the $\mathcal {O}(\alpha ^2_s)$ 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 $\alpha _s$-terms: $\alpha _s^2$-terms: $\alpha _s^3$-terms of the pQCD series changes 1: $4.34\%$: $-0.49\%$ by applying the PMC scale-setting to the perturbative series up to $\mathcal {O}(\alpha _s^3)$, whose PMC scale $Q_*=113.0$ GeV is fixed up to the NLL accuracy. The relative magnitudes of the $\alpha _s$-terms: $\alpha _s^2$-terms: $\alpha _s^3$-terms: $\alpha _s^4$-terms of the pQCD series becomes to 1: $4.33\%$: $-0.49\%$: $0.01\%$ by applying the PMC up to $\mathcal {O}(\alpha _s^4)$, whose PMC scale $Q_*=114.9$ GeV is fixed up to NNLL accuracy. And there is no renormalization scale dependence for $\Gamma _{\mathrm{Z}}^{\mathrm{had}}$ at any fixed order, i.e.,

$$\begin{aligned} \Gamma _{\mathrm{Z}}^{\mathrm{had}}|_{\mathrm{PMC}}= & {} 1681.262+60.838+2.634-0.299\nonumber \\&+0.004 \nonumber \\= & {} 1744.439~(\mathrm{MeV}), \end{aligned}$$

(12)

where each perturbative terms and the net total decay width are unchanged for any choice of $\mu _r$. This behavior is consistent with that of the previous PMC multi-scale approach analysis on $R^{\mathrm{nc}}$ [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 $Q_*=114.9$ GeV $\sim 1.3 M_Z$, which can be fixed up to NNLL accuracy by using the present known ${{\mathcal {O}}}(\alpha _s^4)$-order pQCD series, i.e.,

$$\begin{aligned} \ln \frac{Q^2_*}{M^2_Z}&=0.2249+21.7363a_s(M_Z)+376.287a^2_s(M_Z) \nonumber \\&=0.2249+0.2043+0.0332. \end{aligned}$$

(13)

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 ${{\mathcal {O}}}(\alpha ^4_s)$-order term is only $0.01\%$ of ${{\mathcal {O}}}(\alpha _s)$-order term, it is reasonable to take the magnitude of the last known term $\pm |r_{4,0} a^4_s(Q_*)|$ as a conservative prediction of the uncalculated higher-order terms [39]. By further taking the variation of $\Delta Q_* \simeq \pm 1.9$ 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

$$\begin{aligned} \Delta \Gamma _{\mathrm{Z}}^{\mathrm{had}}|^{\text {High order}}_{\mathrm{PMC}}\simeq \pm 0.004 ~(\mathrm{MeV}). \end{aligned}$$

(14)

Finally, after eliminating the renormalization scale uncertainty by applying the PMC, we still have uncertainties from the $\alpha _s$ fixed-point error $\Delta \alpha _s(M_Z)$ and the Z-boson mass error $\Delta M_Z$. As for the $\alpha _s$ fixed-point error, by using $\Delta \alpha _s(M_Z) =0.0011$ [71] together with the four-loop $\alpha _s$-running behavior, we obtain $\Lambda _{\mathrm{QCD}}^{n_f=5}=209.5^{+13.2}_{-12.6}$ MeV and

$$\begin{aligned} \Delta \Gamma _{\mathrm{Z}}^{\mathrm{had}}|^{\Delta \alpha _s(M_Z)}_{\mathrm{PMC}}&=\pm 0.574 ~(\mathrm MeV). \end{aligned}$$

(15)

And for the error of Z-boson mass $\Delta M_Z=\pm 0.0021 \mathrm {GeV}$, we obtain

$$\begin{aligned} \Delta \Gamma _{\mathrm{Z}}^{\mathrm{had}}|^{\Delta M_Z}_{\mathrm{PMC}}&=\pm 0.120 ~(\mathrm MeV). \end{aligned}$$

(16)

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, $\pm 0.586$ MeV, to the PMC prediction of the total decay width $\Gamma _{\mathrm{Z}}^{\mathrm{had}}$, among which the magnitude of $\Delta \alpha _s(M_Z)$ dominates the error sources. Thus more precise measurements on the reference point $\alpha _s(M_Z)$ is important for a more precise pQCD prediction.

4 Summary

Under conventional scale-setting approach, the fixed-order scale-setting ambiguity could be softened by including enough higher-order loop terms due to large cancelation among different orders; for the present considered decay width up to ${{\mathcal {O}}}(\alpha _s^4)$-order, the net scale uncertainty is $\left( ^{+0.169}_{-0.040}\right)$ MeV for $\mu _r \in [M_Z/2, 2M_Z]$; and by further including the mentioned other error sources, we have $\Gamma _{\mathrm{Z}}^{\mathrm{had}}|_{\mathrm{Conv.}}=1744.418^{+1.595}_{-1.621}$ (MeV).

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] collaborations

In the paper, we have presented an accurate prediction on the Z-boson hadronic decay width by applying the PMC single-scale approach to eliminate the conventional renormalization scale ambiguity. We obtain

$$\begin{aligned} \Gamma _{\mathrm{Z}}^{\mathrm{had}}|_{\mathrm{PMC}} = 1744.439^{+1.390}_{-1.433} ~(\mathrm MeV). \end{aligned}$$

(17)

where the errors are the sum of two parts, one is the squared average of those from $\Delta \alpha _s(M_Z)$, $\Delta M_Z$, and the uncalculated higher-order terms, another is the error from $\Delta \Gamma _{\mathrm{Z}}^{\mathrm{Extra}}$ 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.

Appendix: The PMC reduced perturbative coefficients \({{\hat{r}}}_{i,j}\)

In this appendix, we give the required PMC reduced coefficients ${{\hat{r}}}_{i,j}$ for the perturbative series of the Z-boson hadronic decay width, which can be obtained from Refs. [28,29,30, 68, 69] with proper transformations, i.e.,

$$\begin{aligned} {{\hat{r}}}_{1,0}= & {} 9\gamma ^{NS}_1\bigg (\sum _f a^2_f+\sum _f v^2_f\bigg ), \end{aligned}$$

(18)

$$\begin{aligned} {{\hat{r}}}_{2,0}= & {} 4\left[ 9\gamma ^{NS}_2\bigg (\sum _f a^2_f+\sum _f v^2_f\bigg )-37+12\ln {\frac{M_Z^2}{M_t^2}} \right] , \end{aligned}$$

(19)

$$\begin{aligned} {{\hat{r}}}_{2,1}= & {} 9\Pi ^{NS}_1\bigg (\sum _f a^2_f+\sum _f v^2_f\bigg ), \end{aligned}$$

(20)

$$\begin{aligned} {{\hat{r}}}_{3,0}= & {} 144\left[ \gamma ^{NS}_3+\frac{\gamma ^S_3\bigg (\sum _f q_f\bigg )^2}{\sum _f q^2_f}\right] \bigg (\sum _f a^2_f+\sum _f v^2_f\bigg )\nonumber \\&+\bigg (\sum _f v_f\bigg )^2\left( \frac{440}{9}-\frac{320\zeta _3}{3}\right) +\frac{2144}{3}\ln {\frac{M_Z^2}{M_t^2}}\nonumber \\&+368\ln ^2{\frac{M_Z^2}{M_t^2}}+192\zeta _3+\frac{368\pi ^2}{3}-\frac{40600}{9}, \end{aligned}$$

(21)

$$\begin{aligned} {{\hat{r}}}_{3,1}= & {} 36\Pi ^{NS}_2\bigg (\sum _f a^2_f+\sum _f v^2_f\bigg ), \end{aligned}$$

(22)

$$\begin{aligned} {{\hat{r}}}_{3,2}= & {} -3\pi ^2\gamma ^{NS}_1\bigg (\sum _f a^2_f+\sum _f v^2_f\bigg ), \end{aligned}$$

(23)

$$\begin{aligned} {{\hat{r}}}_{4,0}= & {} 576\left[ \gamma ^{NS}_4+\frac{\gamma ^S_4\bigg (\sum _f q_f\bigg )^2}{\sum _f q^2_f}\right] \bigg (\sum _f a^2_f+\sum _f v^2_f\bigg )\nonumber \\&+\bigg (\sum _f v_f\bigg )^2\left( \frac{3980}{9}-\frac{28960\zeta _3}{9}+\frac{11200\zeta _5}{9}\right) \nonumber \\&+\left( \frac{356000}{9}-44192\zeta _3\right) \ln {\frac{M_Z^2}{M_t^2}}+\frac{33776}{3}\ln ^2{\frac{M_Z^2}{M_t^2}} \nonumber \\&+\frac{8464}{3}\ln ^3{\frac{M_Z^2}{M_t^2}}-\frac{13083735979}{56700}+\frac{35934334\zeta _3}{525} \nonumber \\&+\frac{12328\zeta _5}{9}-\frac{170272\ln ^4{2}}{405}+\frac{512\ln ^5{2}}{45} \nonumber \\&+\left( 11748+\frac{512\ln {2}}{3}+\frac{170272\ln ^2{2}}{405}-\frac{512\ln ^3{2}}{27}\right) \pi ^2\nonumber \\&+\left( \frac{4425752}{6075}+\frac{1424\ln {2}}{135}\right) \pi ^4, \end{aligned}$$

(24)

$$\begin{aligned} {{\hat{r}}}_{4,1}= & {} 144\left[ \Pi ^{NS}_3+\frac{\Pi ^S_3\bigg (\sum _f q_f\bigg )^2}{\sum _f q^2_f}\right] \bigg (\sum _f a^2_f+\sum _f v^2_f\bigg )\nonumber \\&+\bigg (\sum _f v_f\bigg )^2\left( \frac{5960}{27}-\frac{1040\zeta _3}{3}-\frac{320\zeta _3^2}{3}+\frac{800\zeta _5}{3}\right) , \end{aligned}$$

(25)

$$\begin{aligned} {{\hat{r}}}_{4,2}= & {} -12\pi ^2\gamma ^{NS}_2\bigg (\sum _f a^2_f+\sum _f v^2_f\bigg ), \end{aligned}$$

(26)

$$\begin{aligned} {{\hat{r}}}_{4,3}= & {} -9\pi ^2\Pi ^{NS}_1\bigg (\sum _f a^2_f+\sum _f v^2_f\bigg ), \end{aligned}$$

(27)

where the expressions for the coefficients $\gamma _i$ and $\Pi _i$ can be found in Refs. [69, 86].