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

In this paper, we carry out the complete $\mathcal{O}(\alpha {\alpha }_s^2)$-order study on the inclusive productions of $\mathrm{\Upsilon }(nS)$ 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 $H^0\to b\overline{b}[{{}^{3}S}_1^{[8]}]+g$ and $H^0\to b\overline{b}[{{}^{3}S}_1^{[8]}]+Q+\overline{Q}$ ($Q=c$, $b$) play a vital role in the predictions on the decay widths. Moreover, our newly calculated next-to-leading order QCD corrections to $H^0\to b\overline{b}[{{}^{3}S}_1^{[8]}]+g$ can enhance its leading-order result by 3-4 times, subsequently magnifying the total ${{}^{3}S}_1^{[8]}$ contributions by about 40%. 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 $H^0\to b\overline{b}[{{}^{3}P}_J^{[1]}]+b+\overline{b}$, the newly introduced light hadrons associated process, $H^0\to b\overline{b}[{{}^{3}P}_J^{[1]}]+g+g$, 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 ${\mathcal{B}}_{H^0\to \mathrm{\Upsilon }(nS)+X}\sim {10}^{-5}–{10}^{-4}$, which meets marginally nowadays LHC experimental data and can help in understanding the heavy quarkonium production mechanism as well as the Yukawa couplings.

Figure 1

Typical Feynman diagrams for the virtual corrections to the process of $H^0\to b\overline{b}[{{}^{3}S}_1^{[8]}]+g$. The superscript “CT” denotes the counterterms.

Figure 2

Typical Feynman diagrams for the real corrections to the process of $H^0\to b\overline{b}[{{}^{3}S}_1^{[8]}]+g$.

Figure 3

Typical Feynman diagrams for the ${\mathrm{NLO}}^{*}$ processes of ${{}^{3}S}_1^{[8]}$, $H^0\to b\overline{b}[{{}^{3}S}_1^{[8]}]+Q+\overline{Q}$, where $Q=c$, $b$. For $Q=c$, the first two diagrams are excluded.

Figure 4

The verification of the independence on the cutoff parameters of ${\delta }_{s,c}$ for the SDCs of ${{}^{3}S}_1^{[8]}$ (the upper two diagrams) and ${{}^{3}P}_0^{[1,8]}$ (the lower two diagrams), respectively. The superscript “(0)” denotes the ${\epsilon }^0$-order terms.