Improved analysis of rare $Z$-boson decays into a heavy vector quarkonium plus lepton pair

We improve the theoretical predictions for rare $Z$-boson decays, $Z\to V\ell^+\ell^-$ ($\ell=e$ or $\mu$), where $V$ denotes a heavy vector quarkonium including $J/\Psi$, $\Psi (2S)$, and $\Upsilon (nS)$ with $n=1,2,3$. These processes are thought to be dominated by the electromagnetic fragmentation transition, i.e., $Z\to \gamma^*\ell^+\ell^-$ followed by $\gamma^*\rightarrow V$. The present study includes all of the relevant tree-level Feynman diagrams, which contribute to these decays in the standard model. Our analysis shows that, for the charmonium final states, the fragmentation transition almost saturates the whole contribution and the other diagrams can be neglected; while for the bottomonium final states, the inclusion of other diagrams can increase their branching fractions by $4\%\sim 9\%$. Further investigation of the differential distributions, especially the angular distributions, indicates that forward-backward asymmetries for final leptons in these processes would be zero in the standard model. Therefore, in future experimental facilities with large number of $Z$-boson events accumulated, studies of these rare $Z$ decays may help both to test the standard model and to probe its interesting extensions.