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The production of the heavy quarkonium, i.e., $|(c\bar{b})[n]\rangle$ (or$|(b\bar{c})[n]\rangle$), $|(c\bar{c})[n]\rangle$, and $|(b\bar{b})[n]\rangle$-quarkonium [$|(Q\bar{Q'})[n]\rangle$-quarkonium for short], through Higgs$H^{0}$ boson semiexclusive decays is evaluated within the NRQCD framework,where $[n]$ stands for the production of the two color-singlet $S$-wave states,$|(Q\bar{Q'})[^1S_0]_{\textbf{1}} \rangle$ and$|(Q\bar{Q'})[^3S_1]_{\textbf{1}} \rangle$, the production of the fourcolor-singlet $P$-wave states, i.e., $|(Q\bar{Q'})[^1P_0]_{\textbf{1}}\rangle$,$|(Q\bar{Q'})[^3P_J]_{\textbf{1}}\rangle$ (with $J =[0, 1, 2]$). Moreover,according to the velocity scaling rule of the NRQCD, the production of the twocolor-octet components, $|(Q\bar{Q'})g[^1S_0]_{\textbf{8}} \rangle$ and$|(Q\bar{Q'})g[^3S_1]_{\textbf{8}} \rangle$, are also taken into account. The"improved trace technology" to derive the simplified analytic expressions atthe amplitude level is adopted, which shall be useful for dealing with thesedecay channels. If all higher heavy quarkonium states decay completely to theground states, it should be obtained $\Gamma{(H^0\to|(c\bar{b})[^1S_0]_{\textbf{1}}\rangle)}=15.14$ KeV, $\Gamma{(H^0\to|(c\bar{c})[^1S_0]_{\textbf{1}}\rangle)}=1.547$ KeV, and $\Gamma{(H^0\to|(b\bar{b})[^1S_0]_{\textbf{1}}\rangle)}=1.311$ KeV. The production of$5.6\times10^{5}$ Bc meson, $4.7\times10^{4}$ charmonium meson, and$4.9\times10^{4}$ bottomonium meson per year in Higgs decays at the HE/HL-LHCcan be obtained.

Heavy $P$-wave quarkonium production via Higgs decays