Abstract
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 $^3S_1^{[8]}$ component, so the decays can be inversely adopted to determine the values of the long-distance matrix elements $\langle {\cal O}^{\eta_{c}}(^{3}S_{1}^{[8]})\rangle$ and $\langle {\cal O}^{\eta_{b}}(^{3}S_{1}^{[8]})\rangle$, respectively. Our numerical results show that at an $e^+e^-$ collider running at the $Z$ pole with a high luminosity around $10^{35}{\rm cm}^{-2}{\rm s}^{-1}$ (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.
Highlights
Heavy quarkonia have attracted a lot of interest since the discovery of the J=ψ meson
When we present the results for Z → ηQ ð3S1 Þ þ qqindividually, we give the contribution from the fragmentation diagrams, which is gauge invariant and counts almost the whole contribution of the NLO decay width of Z → ηc ð3S1 Þ þ g
Z → ηb ð3S1 Þ þ qqwhose decay width is 8.21 keV under the long-distance matrix elements (LDMEs) extracted by Gong et al Similar to the ηc case, the dominant contributions come from the decay channels
Summary
Heavy quarkonia have attracted a lot of interest since the discovery of the J=ψ meson. The nonrelativistic QCD (NRQCD) factorization formalism [1] provides a systematic framework to separate the short-distance and the long-distance effects in the heavy quarkonium production and decay processes. At the LHC or an eþ e− collider running around the Z pole and with an accessible high-luminosity (a super Z factory), the production of the heavy quarkonium ηQ through Z boson decays can provide abundant information. It is interesting to study the production of heavy quarkonia through Z boson decays. The production of heavy quarkonia (J=ψ and Υ etc) through Z decays has been extensively studied at the leading order in αs and vQ [23,24,25,26,27,28,29,30], the typical velocity of the heavy quark in quarkonia.
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