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In the paper, we discuss the possibility of observation of heavy quarkoniumsvia the processes involving flavor changing neutral currents (FCNC). Moreexplicitly, we systematically calculate the production of heavy charmonium and$(c\bar{b})$-quarkonium through the top quark semi-exclusive rare FCNC decaysin the framework of the non-relativistic QCD (NRQCD) factorization theory. Ourresults show that the total decay widths $\Gamma_{t\to \eta_c}=1.20^{+1.04+1.14}_{-0.51-0.45}\times 10^{-16}$ GeV, $\Gamma_{t\to J/\psi}=1.37^{+1.03+1.30}_{-0.51-0.51}\times 10^{-16}$ GeV, $\Gamma_{t\toB_c}=2.06^{+0.17+0.91}_{-0.17-0.54}\times 10^{-18}$ GeV, and $\Gamma_{t\toB^*_c}=6.27^{+0.63+2.78}_{-0.62-1.64}\times 10^{-18}$ GeV, where theuncertainties are from variation of quark masses and renormalization scales.Even though the decay widths are small, it is important to make a systematicstudy on the production of charmonium and $(c\bar{b})$-quarkonium through thetop-quark decays via FCNC in the Standard Model, which will provide usefulguidance for future new physics research from the heavy quarkonium involvedprocesses.

Heavy quarkonium production through the top quark rare decays via the channels involving flavor changing neutral currents