Inclusive production of heavy quarkonium ${\eta }_Q$ via $Z$ boson decays within the framework of nonrelativistic QCD

In the paper, the inclusive production of heavy quarkonium ${\eta }_Q$ ($Q=b$ or $c$) via $Z$ boson decays within the framework of nonrelativistic QCD effective theory are studied. The contributions from the leading color-singlet and color-octet Fock states are considered. Total and differential decay widths for the inclusive decays $Z\to {\eta }_Q+X$ are presented. It is found that the decays $Z\to {\eta }_Q+X$ are dominated by the ${{}^{3}S}_1^{[8]}$ component, so the decays can be inversely adopted to determine the values of the long-distance matrix elements $⟨{\mathcal{O}}^{{\eta }_c}({{}^{3}S}_1^{[8]})⟩$ and $⟨{\mathcal{O}}^{{\eta }_b}({{}^{3}S}_1^{[8]})⟩$, respectively. Our numerical results show that at an $e^{+}{e}^{-}$ collider running at the $Z$ pole with a high luminosity around $10^{35}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$ (a super $Z$ factory), there are about $4.5\times {10}^7$ ${\eta }_c$ meson events and $6.1\times {10}^5$ ${\eta }_b$ meson events to be produced per operation year, and the inclusive decays may be used for clarifying some problems on the heavy quarkonium ${\eta }_Q$ and nonrelativistic QCD.

Figure 1

Feynman diagrams for $Z\to (Q\overline{Q})[{{}^{2S+1}L}_J^{[8]}]+g$.

Figure 2

Four sample Feynman diagrams for the virtual corrections to $Z\to (Q\overline{Q})[{{}^{2S+1}L}_J^{[8]}]+g$.

Figure 3

Six sample Feynman diagrams for the real corrections to $Z\to (Q\overline{Q})[{{}^{2S+1}L}_J^{[8]}]+g$.

Figure 4

Four of the Feynman diagrams for $Z\to (Q\overline{Q})[{{}^{2S+1}L}_J^{[1,8]}]+Q\overline{Q}$.

Figure 5

Two of the Feynman diagrams for $Z\to (Q\overline{Q})[{{}^{2S+1}L}_J^{[8]}]+Q\overline{Q}$.

Figure 6

Two of the Feynman diagrams for $Z\to (Q\overline{Q})[{{}^{3}S}_1^{[8]}]+Q\overline{Q}$.

Figure 7

Two of the Feynman diagrams for $Z\to (Q\overline{Q})[{{}^{2S+1}L}_J^{[8]}]+Q^{\prime }\overline{Q^{\prime }}$.

Figure 8

Two of the Feynman diagrams for $Z\to (Q\overline{Q})[{{}^{3}S}_1^{[8]}]+Q^{\prime }\overline{Q^{\prime }}$.

Figure 9

Half of the Feynman diagrams for $Z\to (Q\overline{Q})[{{}^{1}S}_0^{[1]}]+gg$.

Figure 10

The differential decay widths $d\mathrm{\Gamma }/dz$ for $Z\to {\eta }_c+X$, where ${\mu }_R=2m_c$. The top one shows the distributions based on the LDMEs of Butenschoen et al. [4], the middle one shows the distributions based the LDMEs of Chao et al. [5], and the bottom one shows the distributions based the LDMEs of Gong et al. [6].

Figure 11

The differential decay widths $d\mathrm{\Gamma }/dz$ for $Z\to {\eta }_c+X$, where ${\mu }_R=2m_b$. The top one shows the distributions based on the LDMEs of Gong et al. [48], and the bottom one shows the distributions based the LDMEs of Feng et al. [49].