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 $\Upsilon(nS)$ 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 $^3S_1^{[8]}$ 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 $^3S_1^{[8]}$ contributions by about $40\%$. Such a remarkable enhancement will to a large extent influence the phenomenological conclusions. For the color-singlet $^3P_J^{[1]}$ state, in addition to $H^{0} \to b\bar{b}[^3P_J^{[1]}]+b+\bar{b}$, the newly introduced light hadrons associated process, $H^{0} \to b\bar{b}[^3P_J^{[1]}]+g+g$, can also provide non-negligible contributions, especially for $^3P_2^{[1]}$. 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.

The color-octet 3 S [8] 1 state related processes consisting of H 0 → bb[ 3 S [8] 1 ] + g and H 0 → bb[ 3 S [8] 1 ] + Q +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 → bb[ 3 S [8] 1 ] + g can enhance its leadingorder result by 3-4 times, subsequently magnifying the total 3 S [8] 1 contributions by about 40%. Such a remarkable enhancement will to a large extent influence the phenomenological conclusions.

I. INTRODUCTION
Bottomonium, as the heaviest bound state, has its own advantages comparing to the charmonium. Due to the large mass of the constituent heavy quarks, both its typical coupling constant α s and relative velocity v are smaller than those of charmonium. As a result, the perturbative results over the expansion of α s and v 2 for bottomonium will be more convergent than the charmonium case, which makes bb mesons an even better place to apply the nonrelativistic QCD (NRQCD) framework [1].
Among the bottomonium family, the Υ and χ b are most studied because the two mesons can be easily detected by hunting their decaying into lepton pairs 1 . Earlier studies of Υ and χ b productions can be found in Refs. [2][3][4][5][6][7][8][9][10] and references therein, where the NRQCD predictions succeeded in explaining almost all the existing experimental measurements. However, considering the fact that the color-octet (CO) long distance matrix elements (LDMEs) that used to well explain the hadroprodution of J/ψ leads to dramatic discrepancies between the theoretical predictions and the measured total cross sections of e + e − → J/ψ + X non−cc from the BABAR and Belle collaborations [11], it is indispensable to take investigations on the Υ(nS) and χ b (nP ) productions in a variety of other processes to further test the validity and universality of the CO LDMEs.
The Higgs boson decay provides a good chance for the studies on Υ and χ b because of the large number of H 0 events at the high energy colliders, e.g., the HL-LHC and HE-LHC can produce 1.65 × 10 8 and 5.78 × 10 8 H 0 events each year, respectively [12]. Although the number of H 0 events at the Circular Electron Positron Collider (CEPC) can only reach up to 1.1 × 10 6 per year [12][13][14], the "clean" background of CEPC comparing to LHC may help us to more easily hunt the heavy quarkonium related processes. Pioneering studies of inclusive Υ and χ b productions via H 0 decay can be found in Refs. [12][13][14]. Qiao et al. studied the direct (no feeddown contributions) inclusive production of Υ(1S) via H 0 decay, including both color-singlet (CS) and CO contributions [13]. Based on the CS mechanism, the investigations on the semi-inclusive productions of Υ and χ b in association with a bb pair, J ] + b +b, were carried out by Liao et al. [14]. Note that, in addition to the processes in [14], the other CS process, H 0 → bb[ 3 P [1] J ] + g + g, 1 The decay of χ b into lepton pair is indirect, might also have remarkable contributions to χ b production. Moreover, we learned from the inclusive productions of heavy quarkonium via the Z boson decay that the 3 S [8] 1 state played a vital role. As shown in our recent work [15], the lowest order process of the 3 S 1 ] + g, could receive a remarkable positive NLO QCD correction, which considerably enhance the NRQCD predictions. It is then natural to wonder whether the NLO QCD corrections to H 0 → bb[ 3 S [8] 1 ] + g can bring a similar significant enhancement on the LO results, so as to influence the phenomenological conclusions markedly. Besides the vital sense in the studies on the production mechanism of the heavy quarkonium, the decay of the Higgs boson into heavy quarkonium is also very helpful for understanding the electroweak breaking mechanism, especially the Yukawa couplings. In view of these points, we use NRQCD to have a complete O(αα 2 s )-order analysis on the inclusive productions of Υ(1S, 2S, 3S) and χ b (1P, 2P, 3P ) via H 0 decay, where all necessary feeddown effects are included.
The rest of the paper is organized as follows: In Sec. II, we give a description on the calculation formalism. In Sec. III, the phenomenological results and discussions are presented. Section IV is reserved as a summary.

II. CALCULATION FORMALISM
Within the NRQCD framework, the decay width of H 0 → Υ(χ b ) + X can be written as: where dΓ n is the perturbative calculable short distance coefficients (SDCs), representing the production of a configuration of the QQ intermediate state with a quantum number J ]. All the involved processes are listed below: 1 ] + g. The superscript "CT" denotes the counterterms.
1 ] + q +q, The label "NLO * " denotes the heavy quark-antiquark pair associated processes, which are free of divergence.
• In the cases of n = 3 S J , the involved channels are ( Typical Feynman diagrams corresponding to Eqs. (2) are presented in Figs. 1, 2, and 3.
J ] + b +b are the same with the first two diagrams of Fig. 3, and the diagrams for H 0 → bb[ 1 S [8] 0 , 3 P In the following, we will briefly present the formalisms for the NLO QCD correc- 1 ] + g as well as the calculations for the tree-level process of The rest processes in Eq. (3) and the NLO * processes are both free of divergence, thus one can take the calculations directly according to the Feynman rules.
To the next-to-leading order in α s , the SDC of the process of H 0 → bb[ 3 S whereΓ Γ Virtual is the virtual corrections, consisting of the contributions from the one-loop diagrams (Γ Loop ) and the counterterms (Γ CT ).Γ Real stands for the real corrections, which includes the soft terms (Γ S ), hard-collinear terms (Γ HC ), and hard-noncollinear terms (Γ HC ). To isolate the ultraviolet (UV) and infrared (IR) divergences, we adopt the dimensional regularization The on-mass-shell (OS) scheme is employed to set the renormalization constants for the heavy quark mass (Z m ), heavy quark filed (Z 2 ), and gluon filed (Z 3 ). The modified minimal-subtraction (MS) scheme is used for the QCD gauge coupling (Z g ). The renormalization constants are [16], where γ E is the Euler's constant, n f and n lf are the number of active quark flavors and light quark flavors, respectively. In SU(3), the color factors are then one can obtain J ]+g+g and dΓ S is the soft part which can be written as where N c is identical to 3 for SU(3) gauge field. E and p denote the energy and 3-momentum of χ b , respectively. δ s is the usual "soft cut" employed to impose an amputation on the energy of the emitted gluon.
1 ) N LO , under the dimensional regularization scheme as is adopted in [18], we have Then the third term in Eq. (8) can be written as where, on the basis of µ Λ -cutoff scheme [18], u c ǫ has the following form µ Λ is the upper bound of the integrated gluon energy, rising from the renormalization of the LDME. Substituting Eqs. (9) and (12) The package MALT@FDC that has been adopted in several heavy quarkonium related processes [15,[20][21][22][23][24][25] is used to deal withΓ Virtual ,Γ S , andΓ HC . To calculate the hard-noncollinear   1 ] + g and calculating other α 2 s −order processes, we employ the two-loop α s running. The one-loop α s running is adopted for the LO cases. The mixed feeddown scheme of χ bJ (3P ) → Υ(nS) in Ref. [7] is used and the value of µ Λ is taken as m b , thus the CO LDMEs in Table 4 where |R Υ(nS) (0)| 2 and |R ′ χ b (mP ) (0)| 2 are taken as [19] |R Υ(1S) (0)| 2 = 6.477 GeV 3 , |R Υ(2S) (0)| 2 = 3.234 GeV 3 , Branching ratios of χ bJ (mP ) → Υ(nS), Υ(nS) → χ bJ (mP ), Υ(3S) → Υ(2S), Υ(3S) → Υ(1S), and Υ(2S) → Υ(1S) can be found in Refs. [6][7][8].   Before presenting the phenomenological results, we first take a look at the effect of the QCD corrections to the process of H 0 → bb[ 3 S [8] 1 ] + g, presented in Table I. We see that the newly calculated NLO terms increase the LO results by about 3-4 times, causing a 40% enhancement on the total 3 S 1 contributions (LO + NLO * bb,cc ). This is consistent with the lesson we learn from Z 0 decay [15]. The 3 S   "FD" denote the direct production processes and feeddown effects, respectively. channels in a wide range of µ r , we provide the predictions at µ r = 2m b and µ r = m H 0 simultaneously. It is noticed that the branching ratios for H 0 → χ bJ (3P, 2P, 1P ) + X are calculated to be on the order of 10 −6 − 10 −5 , indicating the probability of these processes to be observed at the HE-LHC, HL-LHC, and other colliders in near future. In addition to the direct production processes that are dominant, the feeddown effects via the higher excited states, e.g., Υ(2S) and Υ(1S), are also significant, accounting for about 30% of the total decay width, as is shown in Table III and Table IV 1 .
• For the CS cases, the processes of H 0 → bb[ 3 P [1] J ] + b+b ("bb") serve as the leading role in the total CS prediction due to the b-quark fragmentation mechanism. However, the light hadrons associated process H 0 → bb[ 3 P [1] J ] + g + g ("gg") can also provide non- The decay widths of H 0 → χ bJ (1P ) + X (in units of ev). The superscripts "DR" and "FD" denote the direct production processes and feeddown effects, respectively. 1 states, the contribution of the "gg" channel enhance the "bb" cases by about 5% and 7%, respectively. Moreover, for the 3 P 2 case, the "gg" contribution can surprisingly reach up to about 81% of the "bb" contribution. Therefore, to achieve a sound estimate, besides H 0 → bb[ 3 P J ] + g + g must be also taken into consideration.
• Regarding the CO cases, including the 3 S contributions account for about 31%, 56%, and 77% of Γ DR , corresponding to J = 0, 1, and 2, respectively. As for the χ b (2P ) and χ b (1P ) cases, the proportions are about 28%, 52%, 74% and 26%, 49%, 72%, respectively. The ratios of Γ χ b2 /Γ χ b0 and Γ χ b2 /Γ χ b1 . "CS" denotes the sum of the CS direct ( 3 P  In addition to the large contributions to the total decay width, the 3 S 1 state also has crucial effect on the ratios of Γ χ b2 /Γ χ b0 and Γ χ b2 /Γ χ b1 , as shown in Table V, where the feeddown effects have been incorporated. Since the dependence of the CS channels, "gg" and "bb", on µ r is only in the strong coupling constants α s , varying µ r of course does not affect the ratios. However, for the CO cases, due to the NLO corrections to H 0 → bb[ 3 S [8] 1 ]+g, the form of the dependence on µ r is not only α s . Although varying µ r in [2m b , m H 0 ] greatly influence the total decay widths, the ratios of Γ χ b2 /Γ χ b0 and Γ χ b2 /Γ χ b1 are quite insensitive to the choice of µ r . Taking Γχ b2 Γχ b0 | 3P for example, when µ r is varied from 2m b (9.8 GeV) to m H 0 (125 GeV), the ratios just increase by about 4%. In addition, the differences between the CS and NRQCD results are rather conspicuous, which can be regarded as an outstanding probe to distinguish between the two heavy quarkonium production mechanism.
In addition to the total decay width, we also calculate the ratios of Γ Υ(2S) /Γ Υ(3S) and where "CS" denotes the sum of the CS direct ( 3 P where the four columns are the uncertainties caused by µ r , m H 0 , m b , and the CO LDMEs, respectively. The center values in Eqs. (18) and (19) are calculated at m H 0 = 125 GeV, m b = 4.9 GeV, and µ r = m H 0 /2, with the LDMEs taken as the center values in Table 4 of Ref. [8].

IV. SUMMARY
In this paper, we used NRQCD factorization to investigate the inclusive productions of the Υ(1S, 2S, 3S) and χ b (1P, 2P, 3P ) via the Standard Model Higgs boson decay up to O(αα 2 s ) order. It is found that the CO states, especially 3 S [8] 1 , provide remarkable contributions, leading to vital effect on the predictions on the total decay widths. The newly calculated NLO QCD corrections to the lowest order process of 3 S [8] 1 , H 0 → bb[ 3 S [8] 1 ]+g, can significantly (3-4 times) enhance the LO results, subsequently enlarging the total 3 S [8] 1 contributions by about 40%. In addition to the crucial effect on the total decay widths of Υ(nS) and χ b (nP ), including the CO states also influence the ratios of J ] + b +b process, the newly introduced light hadrons associated process, H 0 → bb[ 3 P [1] J ] + g + g, can also provide non-negligible contributions, especially for J = 2. The feeddown contributions via the decay of the higher excited states are found to be substantial, significantly influencing the NRQCD predictions.
In the end, the branching ratios of H 0 → Υ(nS) + X and H 0 → χ b (nP ) + X are predicted to be on the order of 10 −5 − 10 −4 and 10 −6 − 10 −5 , reflecting the great potential of these processes to be detected at high energy colliders. As a conclusion, the decay of Higgs boson into Υ(nS) and χ b (nP ) can be considered as an ideal laboratory not only to study the heavy quarkonium production mechanism, but also to understand the electroweak breaking mechanism especially the Yukawa couplings.