The studies on $Z \rightarrow \Upsilon (1S)+g+g$ at the next-to-leading-order QCD accuracy

Abstract

In this paper, we carry out the next-to-leading-order (NLO) studies on $Z \rightarrow \Upsilon (1S)+g+g$ via the color-singlet (CS) $b{\bar{b}}$ state. We find the newly calculated NLO QCD corrections to this process can significantly influence its leading-order (LO) results and greatly improve the dependence on the renormalization scale. By including the considerable feeddown contributions, the branching ratio ${\mathcal {B}}_{Z \rightarrow \Upsilon (1S)+g+g}$ is predicted to be (0.56–0.95) $\times 10^{-6}$, which can reach up to 19–31$\%$ of the LO predictions given by the CS dominant process $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$. Moreover, $Z \rightarrow \Upsilon (1S)+g+g$ also seriously affects the CS predictions on the $\Upsilon (1S)$ energy distributions, especially when z is relatively small. In summary, for the inclusive $\Upsilon (1S)$ productions in Z decay, besides $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$, the gluon radiation process $Z \rightarrow \Upsilon (1S)+g+g$ can provide indispensable contributions as well.

1 Introduction

The L3 group at LEP has released the measurement on the total decay width of $Z \rightarrow \Upsilon (1S)+X$ [1]

$$\begin{aligned} {\mathcal {B}}_{Z \rightarrow \Upsilon (1S)+X} < 4.4 \times 10^{-5}. \end{aligned}$$

The leading-order (LO) color-singlet (CS) predictions obtained by calculating the CS dominant process $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$ are only at the $10^{-6}$ order [2,3,4]. Subsequently Li et al. [4] accomplish the next-to-leading-order (NLO) QCD corrections to this $b{\bar{b}}$ pair associated channel, pointing out the higher-order terms can give rise to a $24\%$ enhancement to the total decay width.

Reviewing the inclusive $J/\psi$ productions via Z decay [2,3,4,5,6,7,8,9,10,11,12,13,14,15], besides the $\alpha \alpha _s^2$-order process $Z \rightarrow J/\psi +c+{\bar{c}}$ that serves as the leading role in the CS predictions, the electromagnetic processes $Z \rightarrow f{\bar{f}}\gamma ^{*}$ with $\gamma ^{*} \rightarrow J/\psi$ $(f=l,u,d,s,c,b)$ and the gluon fragmentation processes $Z \rightarrow f_{q}\bar{f_{q}}g^{*}$ with $g^{*} \rightarrow J/\psi gg$ $(f_{q}=u,d,s,c,b)$ can also provide nonnegligible contributions. Meanwhile, due to the suppression by $\frac{m_c^2}{m_Z^2}$ [3], the total decay width of the other $\alpha \alpha _s^2$-order process $Z \rightarrow J/\psi +g+g$ is only about two orders of magnitudes smaller than that of $Z \rightarrow J/\psi +c+{\bar{c}}$. As for the $\Upsilon$ productions, the situations become just the opposite. The relative significances of the electromagnetic and gluon fragmentation processes are much less important than the $J/\psi$ case, since the value of $m_b$, larger than $m_c$, highly suppresses the denominator of the propagators $\gamma ^{*}(\rightarrow \Upsilon )$ and $g^{*}(\rightarrow \Upsilon gg)$. However, for $Z \rightarrow \Upsilon +g+g$, the suppression effect stated above ($\frac{m_c^2}{m_Z^2}$) will be largely weakened by replacing $m_c$ with $m_b$, subsequently rendering this process indispensable in comparison with $Z \rightarrow \Upsilon +b+{\bar{b}}$ [2]. Moreover, the $\Upsilon$ energy distributions in $Z \rightarrow \Upsilon +g+g$ and $Z \rightarrow \Upsilon +b+{\bar{b}}$ may be thoroughly different. This can be understood by the fact that the former process is seriously suppressed by the factor $\frac{M_{\Upsilon }^2}{E_{\Upsilon }^2}$ for large z [11, 16, 17], so the value of z concerning the maximum $\frac{d\Gamma }{dz}$ should be small; however, as a result of the b quark fragmentation, the dominant contributions of $Z \rightarrow \Upsilon +b+{\bar{b}}$ exist in the region of large z. From these points of view, the process $Z \rightarrow \Upsilon +g+g$ would be crucial for Z decaying to the inclusive $\Upsilon$, deserving a separate investigation.

Seeing that all the existing studies on $Z \rightarrow \Upsilon (1S)+g+g$ are only accurate to the first order in $\alpha _s$, to investigate the effects of the higher-order terms, in this paper we will for the first time carry out the NLO QCD corrections to this process. In general, for the inclusive $\Upsilon (1S)$ productions, the feeddown via the excited states can provide nonnegligible contributions. Therefore, in addition to the direct productions, we will take the feeddown effects via $\Upsilon (2,3S)$ and $\chi _{bJ}(1,2,3P)$ $(J=0,1,2)$ into account as well.

The rest of the paper is organized as follows: in Sect. 2, we give a description on the calculation formalism. In Sect. 3, the phenomenological results and discussions are presented. Section 4 is reserved for a summary.

2 Calculation formalism

Based on the nonrelativistic quantum chromodynamics [18], the decay width of $Z \rightarrow \Upsilon (\chi _{bJ})+g+g$¹ can be factorized as

$$\begin{aligned} \Gamma ={\hat{\Gamma }}_{Z \rightarrow b{\bar{b}}[n]+g+g}\langle {\mathcal {O}} ^{\Upsilon (\chi _{bJ})}(n)\rangle , \end{aligned}$$

(1)

where ${\hat{\Gamma }}_{Z \rightarrow b{\bar{b}}[n]+g+g}$ is the perturbative calculable short distance coefficients, representing the production of a configuration of the $b{\bar{b}}[n]$ intermediate state. The universal nonperturbative long distance matrix elements (LDMEs) $\langle {\mathcal {O}} ^{\Upsilon (\chi _{bJ})}(n)\rangle$ stands for the probability of $b{\bar{b}}[n]$ into $\Upsilon (\chi _{bJ})$. In our studies, we only concentrate on the CS contributions, so $n=^3S_1^{[1]}$ for $\Upsilon$, and $n=^3P_J^{[1]}$ ($J=0,1,2$) for $\chi _{bJ}$. The procedures for dealing with the soft singularities involved in $Z \rightarrow b{\bar{b}}[^3P_J^{[1]}]+g+g$ ($J=0,1,2$) have been described in detail in our previous paper [19, 20], so here we just give a brief presentation on the $b{\bar{b}}[^3S_1^{[1]}]$ related calculations.

Fig. 1

Representative Feynman diagrams for the NLO QCD corrections to $Z \rightarrow b{\bar{b}}[^3S_1^{[1]}]+g+g$. The superscripts “q” and “$u_{g}$” denote the light quarks (u, d, s) and the ghost particles, respectively

Fig. 2

Cancelation of the $\epsilon ^{-2}$- and $\epsilon ^{-1}$-order divergences. The superscripts “(2)” and “(1)” denote the $\epsilon ^{-2}$- and $\epsilon ^{-1}$-order terms, respectively

Fig. 3

The verification of the independence on the cutoff parameters $\delta _s$ and $\delta _c$. The superscript “(0)” denotes the $\epsilon ^{0}$-order terms. In the left diagram, $\delta _c=2 \times 10^{-7}$. In the right diagram, $\delta _s=1 \times 10^{-3}$

The NLO short distance coefficients for $n=^3S_1^{[1]}$ can be written as

$$\begin{aligned} {\hat{\Gamma }}^{NLO}_{Z \rightarrow b{\bar{b}}[^3S_1^{[1]}]+g+g}={\hat{\Gamma }}_{\text {Born}}+{\hat{\Gamma }}_{\text {Virtual}}+{\hat{\Gamma }}_{\text {Real}}, \end{aligned}$$

(2)

where

$$\begin{aligned}&{\hat{\Gamma }}_{\text {Virtual}}={\hat{\Gamma }}_{\text {Loop}}+{\hat{\Gamma }}_{\text {CT}}, \nonumber \\&{\hat{\Gamma }}_{\text {Real}}={\hat{\Gamma }}_{\text {S}}+{\hat{\Gamma }}_{\text {HC}}+{\hat{\Gamma }}_{\text {H}\overline{\text {C}}}. \end{aligned}$$

(3)

${\hat{\Gamma }}_{\text {Virtual}}$ is the virtual corrections, consisting of the contributions from the one-loop diagrams (${\hat{\Gamma }}_{\text {Loop}}$) and the counter-terms (${\hat{\Gamma }}_{\text {CT}}$). ${\hat{\Gamma }}_{\text {Real}}$ stands for the real corrections, which include the soft terms (${\hat{\Gamma }}_{S}$), hard-collinear terms $({\hat{\Gamma }}_{\text {HC}})$, and hard-noncollinear terms $({\hat{\Gamma }}_{\text {H}\overline{\text {C}}})$. For ${\hat{\Gamma }}_{\text {Real}}$, three processes are involved:

$$\begin{aligned} Z\rightarrow & {} b{\bar{b}}[^3S_1^{[1]}]+g+g+g, \nonumber \\ Z\rightarrow & {} b{\bar{b}}[^3S_1^{[1]}]+g+q+{\bar{q}}~(q=u,d,s), \nonumber \\ Z\rightarrow & {} b{\bar{b}}[^3S_1^{[1]}]+g+u_g+{\bar{u}}_g~(\text {ghost}). \end{aligned}$$

(4)

There are 177 Feynman diagrams in total, including 6 diagrams for ${\hat{\Gamma }}_{\text {Born}}$, 111 diagrams for ${\hat{\Gamma }}_{\text {Virtual}}$ (30 counter-terms, 12 self-energy, 30 3-points, 27 4-points, 12 5-points), and 60 diagrams for ${\hat{\Gamma }}_{\text {Real}}$ (42 $ggg_\mathrm{{v}}$, 6 $ggg_\mathrm{{av}}$, 6 $gq{\bar{q}}$, and 6 $gu_g{\bar{u}}_g$), as representatively shown in Fig. 1. $ggg_\mathrm{{v}}$ and $ggg_\mathrm{{av}}$ denote the vector and axial-vector parts of $Z \rightarrow b{\bar{b}}[^3S_1^{[1]}]+g+g+g$, respectively.

To isolate the ultraviolet (UV) and infrared (IR) divergences, we adopt the dimensional regularization with $D=4-2\epsilon$. The on-mass-shell (OS) scheme is employed to set the renormalization constants for the heavy quark mass ($Z_m$), heavy quark field ($Z_2$), and gluon field ($Z_3$). The modified minimal-subtraction (${\overline{MS}}$) scheme is used for the QCD gauge coupling ($Z_g$). The renormalization constants are [19,20,21]

$$\begin{aligned} \delta Z_{m}^{OS}= & {} -3 C_{F} \frac{\alpha _s N_{\epsilon }}{4\pi }\left[ \frac{1}{\epsilon _{\text {UV}}}-\gamma _{E}+\text {ln}\frac{4 \pi \mu _r^2}{m_b^2}+\frac{4}{3}\right] , \nonumber \\ \delta Z_{2}^{OS}= & {} - C_{F} \frac{\alpha _s N_{\epsilon }}{4\pi }\left[ \frac{1}{\epsilon _{\text {UV}}}+\frac{2}{\epsilon _{\text {IR}}}-3 \gamma _{E}+3 \text {ln}\frac{4 \pi \mu _r^2}{m_b^2}+4\right] , \nonumber \\ \delta Z_{3}^{OS}= & {} \frac{\alpha _s N_{\epsilon }}{4\pi }\left[ (\beta _{0}^{'}-2 C_{A})\left( \frac{1}{\epsilon _{\text {UV}}} -\frac{1}{\epsilon _{\text {IR}}}\right) \right. \nonumber \\&\left. -\frac{4}{3}T_F\left( \frac{1}{\epsilon _{\text {UV}}} -\gamma _E+\text {ln}\frac{4\pi \mu _r^2}{m_c^2}\right) \right. \nonumber \\&\left. -\frac{4}{3}T_F\left( \frac{1}{\epsilon _{\text {UV}}} -\gamma _E+\text {ln}\frac{4\pi \mu _r^2}{m_b^2}\right) \right] , \nonumber \\ \delta Z_{g}^{{\overline{MS}}}= & {} -\frac{\beta _{0}}{2}\frac{\alpha _s N_{\epsilon }}{4\pi }\left[ \frac{1}{\epsilon _{\text {UV}}}-\gamma _{E}+\text {ln}(4\pi )\right] , \end{aligned}$$

(5)

where $\gamma _E$ is the Euler constant, $N_{\epsilon }= \Gamma [1-\epsilon ] /({4\pi \mu _r^2}/{(4m_b^2)})^{\epsilon }$, $\beta _{0}=\frac{11}{3}C_A-\frac{4}{3}T_Fn_f$ is the one-loop coefficient of the $\beta$-function, and $\beta _{0}^{'}=\frac{11}{3}C_A-\frac{4}{3}T_Fn_{lf}$. $n_f(=5)$ and $n_{lf}(=n_f-2)$ are the number of active quark flavors and light quark flavors, respectively. In $\mathrm{SU}(3)$, the color factors are given by $T_F=\frac{1}{2}$, $C_F=\frac{4}{3}$, and $C_A=3$. The two-cutoff slicing strategy is utilized to subtract the IR divergences in $\Gamma _{\text {Real}}$ [22].

Table 1 The total decay widths of $Z \rightarrow \Upsilon (1S)+g+g$ (in units of $\text {keV}$). K denotes $\Gamma ^{NLO}_{DR}/\Gamma ^{LO}_{DR}$. “NLO” represents the sum of the contribution of the LO terms and that of the QCD corrections. The superscripts “DR” and “FD” denote the direct and feeddown contributions, respectively

The package that has been employed to preform the NLO QCD corrections to several heavy quarkonium related processes [19, 20, 23,24,25] is used to deal with ${\hat{\Gamma }}_{\text {Virtual}}$, ${\hat{\Gamma }}_{{\text {S}}}$, and ${\hat{\Gamma }}_{{\text {HC}}}$. The $\text {FDC}$ package [26] serves as the agent for calculating the hard-noncollinear part ${\hat{\Gamma }}_{\text {H}\overline{\text {C}}}$. Both the cancelation of the $\epsilon ^{-2(-1)}$-order divergences and the independence on the cutoff parameters $\delta _{s,c}$ have been checked carefully, as shown in Figs. 2 and 3.

As a crosscheck, taking the same input parameters, we have reproduced the NLO results of $\sigma _{e^+e^- \rightarrow J/\psi +g+g}$ in Refs. [27, 28].

3 Phenomenological results

In our calculations, the input parameters are taken as

$$\begin{aligned}&\alpha =1/128,~~~m_b=4.9~\text {GeV},~~~m_c=1.5~\text {GeV},\nonumber \\&m_Z=91.1876~\text {GeV},~~~m_{q/{\bar{q}}}=0~(q=u,d,s),\nonumber \\&\sin ^{2}(\theta _W)=0.23116. \end{aligned}$$

(6)

To determine $\langle {\mathcal {O}} ^{\Upsilon (nS)}(^3S_1^{[1]})\rangle$ and $\langle {\mathcal {O}} ^{\chi _{bJ}(mP)}(^3P_J^{[1]})\rangle$, we employ the relations to the radial wave functions at the origin ($n,m=1,2,3$)

$$\begin{aligned} \frac{\langle {\mathcal {O}}^{\Upsilon (nS)}(^3S_1^{[1]}) \rangle }{6N_c}= & {} \frac{1}{4\pi }|R_{\Upsilon (nS)}(0)|^2, \nonumber \\ \frac{\langle {\mathcal {O}}^{\chi _{bJ}(mP)}(^3P_J^{[1]}) \rangle }{2N_c}= & {} (2J+1)\frac{3}{4\pi }|R^{'}_{\chi _b(mP)}(0)|^2, \end{aligned}$$

(7)

where $|R_{\Upsilon (nS)}(0)|^2$ and $|R^{'}_{\chi _b(mP)}(0)|^2$ read [29]

$$\begin{aligned}&|R_{\Upsilon (1S)}(0)|^2=6.477~\text {GeV}^3,~~~|R_{\Upsilon (2S)}(0)|^2=3.234~\text {GeV}^3, \nonumber \\&|R_{\Upsilon (3S)}(0)|^2=2.474~\text {GeV}^3, \nonumber \\&|R^{'}_{\chi _b(1P)}(0)|^2=1.417~\text {GeV}^5,~~~|R^{'}_{\chi _b(2P)}(0)|^2=1.653~\text {GeV}^5, \nonumber \\&|R^{'}_{\chi _b(3P)}(0)|^2=1.794~\text {GeV}^5. \end{aligned}$$

(8)

The branching ratios of $\chi _{bJ}(mP) \rightarrow \Upsilon (nS)$, $\Upsilon (nS) \rightarrow \chi _{bJ}(mP)$, $\Upsilon (3S) \rightarrow \Upsilon (2S)$, $\Upsilon (3S) \rightarrow \Upsilon (1S)$, and $\Upsilon (2S) \rightarrow \Upsilon (1S)$ can be found in Refs. [30,31,32].

Fig. 4

The total decay widths of $Z \rightarrow \Upsilon (1S)+g+g$ as a function of the renormalization scale $\mu _r$ with $m_b=4.9$ GeV. “NLO” represents the sum of the contribution of the LO terms and that of the QCD corrections. The superscripts “DR” and “FD” denote the direct and feeddown contributions, respectively

The total decay widths of $Z \rightarrow \Upsilon (1S)+g+g$ are listed in Table 1. To demonstrate the dependence on the renormalization scale $\mu _r$, the results for $\mu _r=2m_b$ and $\mu _r=m_Z$ are presented simultaneously. From the data in this table, one can observe the following.

• (i) For the direct productions, when $\mu _r=2m_b$, the QCD corrections diminish the LO results by about $5\%$, and cause a $60\%$ enhancement for $\mu _r=m_Z$. In addition, incorporating these higher-order terms notably weaken the dependence on $\mu _r$. As is illustrated in Fig. 4, the line referring to “$DR_{NLO}$” decreases much more slowly than that for “$DR_{LO}$” with the increase of $\mu _r$.

• (ii) The decay of $\Upsilon (2,3S)$ and $\chi _{bJ}(1,2,3P)$ can raise the predictions of the direct productions by about $30\%$, manifestly indicating the feeddown significance in the $\Upsilon (1S)$ production.

• (iii) The dependence on the mass of the b quark is mild, e.g., varying $m_b$ by $\pm ~0.2$ GeV from the central value of 4.9 GeV just results in a $10\%$ variation of the total decay width.

Summing up the direct and feeddown contributions, we finally obtain

$$\begin{aligned} {\mathcal {B}}_{Z \rightarrow \Upsilon (1S)+g+g} = (0.56 {-} 0.95) \times 10^{-6}, \end{aligned}$$

(9)

where the theoretical uncertainty is induced by the choices of the values of $\mu _r$ ($2m_b \sim m_Z$) and $m_b$ (4.9–5.1 GeV).

Fig. 5

The $\Upsilon (1S)$ energy distributions in $Z \rightarrow \Upsilon (1S)+g+g$. $z=\frac{2E_{\Upsilon (1S)}}{m_Z}$ and $m_b=4.9$ GeV. “NLO” represents the sum of the contribution of the LO terms and that of the QCD corrections. $\mu _r=2m_b$ in the left diagram, and $\mu _r=m_Z$ in the right diagram. The superscripts “DR” and “FD” denote the direct and feeddown contributions, respectively

Fig. 6

The differential decay widths. $z=\frac{2E_{\Upsilon (1S)}}{m_Z}$ and $m_b=4.9$ GeV. $\mu _r=2m_b$ in the left diagram, and $\mu _r=m_Z$ in the right diagram. The superscripts “$\Upsilon gg$” and “$\Upsilon b{\bar{b}}$” denote the contributions via $Z \rightarrow \Upsilon (1S)+g+g$ and $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$, respectively

In Fig. 5, the $\Upsilon (1S)$ energy distributions in $Z \rightarrow \Upsilon (1S)+g+g$ are presented with z defined as $\frac{2E_{\Upsilon (1S)}}{m_Z}$. For the direct productions, when $\mu _r$ is equal to $2m_b$, the QCD corrections are positive for $z<0.52$, and negative in the remaining scope of z; however, for $\mu _r=m_Z$, these corrections keep always positive for all the available z values. With $\mu _r$ being relatively small, the higher-order terms can significantly enhance the differential decay width. Taking $z=0.3$ as an example,

$$\begin{aligned} \left( \frac{d\Gamma }{dz}\right) _{NLO} / \left( \frac{d\Gamma }{dz}\right) _{LO}= & {} 1.23~\text {for}~\mu _r=2m_b, \nonumber \\ \left( \frac{d\Gamma }{dz}\right) _{NLO} / \left( \frac{d\Gamma }{dz}\right) _{LO}= & {} 1.81~\text {for}~\mu _r=m_Z, \end{aligned}$$

(10)

where “NLO” represents the sum of the contribution of the LO terms and that of the QCD corrections. Including the feeddown contributions can further increase $\frac{d\Gamma }{dz}$ by about 20–30% for most values of z. Note that, phenomenological and theoretical arguments suggest that the NRQCD factorization holds only when the quarkonium is produced at relatively large momentum, thus in Fig. 5 the predicted z distributions for very small z values (close to the left endpoint) may not hold.

At last, we simply compare the contributions of $Z \rightarrow \Upsilon (1S)+g+g$ with that of the dominant channel $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$. With the help of the FDC package, taking the parameters in Eq. (6), we have

$$\begin{aligned} \Gamma ^{LO,\mu _r=2m_b}_{Z \rightarrow \Upsilon (1S)+b+{\bar{b}}}= & {} 11.5~keV, \nonumber \\ \Gamma ^{LO,\mu _r=m_Z}_{Z \rightarrow \Upsilon (1S)+b+{\bar{b}}}= & {} 4.98~keV. \end{aligned}$$

(11)

Comparing to the results with $m_b=4.9$ GeV in Table 1, one can obtain

$$\begin{aligned} \frac{\Gamma _{Z \rightarrow \Upsilon (1S)+g+g}}{\Gamma ^{LO}_{Z \rightarrow \Upsilon (1S)+b+{\bar{b}}}} = 19{-}31\%, \end{aligned}$$

(12)

where the uncertainty arises from the variation of $\mu _r$ in $\left[ 2m_b,m_Z\right]$. This ratio suggests that, for the total decay width, the contributions via $Z \rightarrow \Upsilon (1S)+g+g$ is comparable with the magnitude of the QCD corrections to $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$. In Fig. 6, the comparison of the $\Upsilon (1S)$ energy distributions in $Z \rightarrow \Upsilon (1S)+g+g$ and $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$ are presented, where “$\Upsilon gg$” denotes the contributions via the former process up to the NLO accuracy in $\alpha _s$, including the feeddown effects; “$\Upsilon b{\bar{b}}$” stands for the direct contributions of the $b{\bar{b}}$ pair associated process at the LO QCD accuracy. One can find, for relatively small z, that adding the $\Upsilon gg$ contributions can increase the $\Upsilon b{\bar{b}}$ predictions to a surprisingly large extent. Such a remarkable enhancement on $\frac{d\Gamma }{dz}$ is almost the same in size as the QCD corrections to $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$. All these points obviously reveal the phenomenological importance of $Z \rightarrow \Upsilon (1S)+g+g$ for Z decaying to the inclusive $\Upsilon (1S)$.

Note that, within the NRQCD framework, for the $\Upsilon$ production in association with final gluon(s) via Z decay, in addition to the CS process $Z \rightarrow b{\bar{b}}[^3S_1^{[1]}]+g+g$, which we focus on in our present paper, the color-octet (CO) channels $Z \rightarrow b{\bar{b}}[^1S_0^{[8]},^3S_1^{[8]},^3P_J^{[8]}]+g$ are also allowed. Generally speaking, according to the velocity-scaling rule of NRQCD, the CO LDMEs are suppressed by a power of $v^{4}$ ($v^2_{b} \simeq 0.1$), where v denotes the relative velocity between the constituent quark and anti-quark of the heavy quarkonium. However, the smallness of $\alpha _s$ in $Z \rightarrow b{\bar{b}}[^1S_0^{[8]},^3S_1^{[8]},^3P_J^{[8]}]+g$ comparing to $Z \rightarrow b{\bar{b}}[^3S_1^{[1]}]+g+g$, together with the potential enhancement via the NLO corrections to these CO channels, may partly compensate for the suppression caused by LDMEs, subsequently making the CO contributions nonnegligible. Of course, whether this is indeed the case depends on the future rigorous NLO calculations for these CO production channels.

4 Summary

In this manuscript, we for the first time perform the complete NLO studies on the process $Z \rightarrow \Upsilon (1S)+g+g$ via the CS $b{\bar{b}}$ states. We find that the impacts of the QCD corrections on the LO results are significant, including both the total and differential decay widths. In addition, these higher-order terms markedly weaken the dependence of the theoretical predictions on $\mu _r$. By incorporating the substantial feeddown effects via $\Upsilon (2,3S)$ and $\chi _{bJ}(1,2,3P)$, ${\mathcal {B}}_{Z \rightarrow \Upsilon (1S)+g+g}$ is scattered in the range $(0.56 {-} 0.95) \times 10^{-6}$. This value is about $19{-} 31\%$ of the LO predictions given by $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$, which is responsible for the main contributions in the CS predictions. Moreover, for small $E_{\Upsilon (1S)}$, $Z \rightarrow \Upsilon (1S)+g+g$ has vital influence on the $\Upsilon (1S)$ energy distributions. In view of these points, for Z decaying to the inclusive $\Upsilon (1S)$, in addition to $Z \rightarrow \Upsilon (1S)+b+{\bar{b}}$, the process $Z \rightarrow \Upsilon (1S)+g+g$ is also phenomenologically important.