Note on rare $Z$-boson decays to double heavy quarkonia

Within the standard model, we have investigated rare $Z$-boson decays into double heavy quarkonia, $Z\to VV$ and $Z\to VP$ with $V$ denoting vector and $P$ denoting pseudoscalar quarkonia, respectively. It is assumed that the leading-order QCD diagrams would give the dominant contributions to these processes, and the corresponding branching fractions, for instance, ${\cal B}(Z\to J/\Psi J/\Psi)$ has been estimated to be around $10^{-13}$ in the literature. However, these decays could also happen through the electromagnetic transition $Z\to V\gamma^*$ and $Z\to P\gamma^*$, with the virtual photon transforming into $V$. Interestingly, the smallness of the vector quarkonium mass can give rise to a large factor $m_Z^2/m_V^2$, relative to the QCD contributions, which thus counteracts the suppression from the electromagnetic coupling. We systematically include these two types of contributions in our calculation to predict branching fractions for these decays. Particularly, due to the virtual photon effects, it is found that ${\cal B}(Z\to J/\Psi J/\Psi)$ will be significantly enhanced, which could be up to $10^{-10}$.