Production pair of higher excited massive quarkonium at super Z factory

Sizable amounts of massive quarkonium events should be produced through $Z^0$ boson decays at a future super $Z$ factory. Such channels will provide a suitable platform to study the precisely properties of massive quarkonium. We show that sizable amounts of events for pair of the higher excited states quarkonia can also be produced at the super Z factory. Therefore, we make a further study on calculate the exclusive production of pair of the higher excited charmonium, pair of the higher excited bottomonium, and pair of the higher excited $B_c$ mesons via $e^+e^-\to Z^0 \to |(Q\bar{Q^\prime})[n]\rangle + |(Q^\prime \bar{Q})[n']\rangle$ ($Q,~Q^\prime=c$- or $b$-quarks) at the future $Z$ factory under the NRQCD, where the $[n]$/$[n^\prime]$ represents the color-singlet states $[n^1S_0], [n^3S_1],~[n^1P_1]$, and $[^3P_J]$ ($[n]=1,2,3;J=0,1,2$). The differential angle distribution of cross sections $d\sigma/dcos\theta$ is given. For a sound estimate, we discus the uncertainties of total cross sections caused by varying quark masses and the renormalization scale $\mu$. The numerical results show that the future super $Z$ factory should be also a good platform to study the exclusive production of pair of the higher excited massive quarkonium.