Associated Production of Charmonia-Bottomonia with Color-Octet Channels at the Z Factory, CEPC and FCC-ee

Abstract

Within the nonrelativistic QCD (NRQCD) framework, we investigate the associated production of charmonia+bottomonia at future super $Z$ factory and at the CEPC/FCC-ee. The color-octet(CO) channels in the $\gamma^*/Z^0$-propagated process are considered besides the color-singlet(CS) channels. We find that the contributions of the CO states to the total cross section are dominant for almost all processes from the production threshold to the $Z^0$ mass. Thus, the comparison between the theoretical results and future data may give constraints to the CO matrix elements. In addition, we calculate the relativistic corrections to both the CS and CO channels, which decrease the cross sections significantly, with the $K$ factor $\simeq0.5$. The predicted events of ($J/\psi+\Upsilon$, $\Upsilon+\eta_c$) production, with $J/\psi, \Upsilon$ reconstructed by lepton pair and $\eta_c$ reconstructed by hadronic decays, are (1,1) and (13,10) at the CEPC(2-yr) and FCC-ee(4-yr), respectively.