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

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 ${}^3{S}_1^{[8]}$ 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.

Comments: 13 pages, 11 figures