  In the paper, we discuss the possibility of observation of heavy quarkoniums
via the processes involving flavor changing neutral currents (FCNC). More
explicitly, we systematically calculate the production of heavy charmonium and
$(c\bar{b})$-quarkonium through the top quark semi-exclusive rare FCNC decays
in the framework of the non-relativistic QCD (NRQCD) factorization theory. Our
results 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, $\Gamma_{t\to
B_c}=2.06^{+0.17+0.91}_{-0.17-0.54}\times 10^{-18}$ GeV, and $\Gamma_{t\to
B^*_c}=6.27^{+0.63+2.78}_{-0.62-1.64}\times 10^{-18}$ GeV, where the
uncertainties are from variation of quark masses and renormalization scales.
Even though the decay widths are small, it is important to make a systematic
study on the production of charmonium and $(c\bar{b})$-quarkonium through the
top-quark decays via FCNC in the Standard Model, which will provide useful
guidance for future new physics research from the heavy quarkonium involved
processes.
