Inclusive production of heavy quarkonium ηQ via Z boson decays within the framework of nonrelativistic QCD

In the paper, the inclusive production of heavy quarkonium ${\ensuremath{\eta}}_{Q}$ ($Q=b$ 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 $Z\ensuremath{\rightarrow}{\ensuremath{\eta}}_{Q}+X$ are presented. It is found that the decays $Z\ensuremath{\rightarrow}{\ensuremath{\eta}}_{Q}+X$ 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 $⟨{\mathcal{O}}^{{\ensuremath{\eta}}_{c}}({^{3}S}_{1}^{[8]})⟩$ and $⟨{\mathcal{O}}^{{\ensuremath{\eta}}_{b}}({^{3}S}_{1}^{[8]})⟩$, respectively. Our numerical results show that at an ${e}^{+}{e}^{\ensuremath{-}}$ collider running at the $Z$ pole with a high luminosity around $1{0}^{35}\text{ }\text{ }{\mathrm{cm}}^{\ensuremath{-}2}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$ (a super $Z$ factory), there are about $4.5\ifmmode\times\else\texttimes\fi{}{10}^{7}$ ${\ensuremath{\eta}}_{c}$ meson events and $6.1\ifmmode\times\else\texttimes\fi{}{10}^{5}$ ${\ensuremath{\eta}}_{b}$ meson events to be produced per operation year, and the inclusive decays may be used for clarifying some problems on the heavy quarkonium ${\ensuremath{\eta}}_{Q}$ and nonrelativistic QCD.