Inclusive productions of $\mathrm{{\rm Y}}(1S,2S,3S)$ and ${\chi }_b(1P,2P,3P)$ via the Higgs boson decay

In this paper, we carry out the complete $\mathcal O(\alpha\alpha_s^{2})$-orderstudyontheinclusiveproductionsof$\Upsilon(nS)$and$\chi_b(nP)$($n=1,2,3$) via the Standard Model Higgs boson decay, within the framework of nonrelativistic QCD. The feeddown effects via the higher excited states are found to be substantial. The color-octet ${}^3{S}_1^{[8]}$ state related processes consisting of $H^0 \to b\bar{b}[^3S_1^{[8]}]+g$and$H^0 \to b\bar{b}[^3S_1^{[8]}]+Q+\bar{Q}$($Q=c,b$) play a vital role in the predictions on the decay widths. Moreover, our newly calculated next-to-leading order QCD corrections to $H^0 \to b\bar{b}[^3S_1^{[8]}]+g$ can enhance its leading-order result by 3-4 times, subsequently magnifying the total ${}^3{S}_1^{[8]}$ contributions by about $40\mathrm{\%}$. Such a remarkable enhancement will to a large extent influence the phenomenological conclusions. For the color-singlet ${}^3{P}_J^{[1]}$ state, in addition to $H^0\to b\bar{b}{[}^3{P}_J^{[1]}]+b+\bar{b}$, the newly introduced light hadrons associated process, $H^0\to b\bar{b}{[}^3{P}_J^{[1]}]+g+g$, can also provide non-negligible contributions, especially for ${}^3{P}_2^{[1]}$. Summing up all the contributions, we have $\mathcal B_{H^0 \to \chi_b(nP)+X} \sim 10^{-6}-10^{-5}$and$\mathcal B_{H^0 \to \Upsilon(nS)+X} \sim 10^{-5}-10^{-4}$, which meets marginally nowadays LHC experimental data and can help in understanding the heavy quarkonium production mechanism as well as the Yukawa couplings.