PHYSICAL REVIEW D 98, 036014 (2018)

Heavy P-wave quarkonium production via Higgs decays

† ‡ Qi-Li Liao,1,* Ya Deng,1, Yan Yu, 1, Guang-Chuan Wang,1 and Guo-Ya Xie2 1College of Mobile Telecommunications Chongqing University of Posts and Telecom, Chongqing 401520, People’s Republic of China 2Chongqing University of Posts and Telecom, Chongqing 400065, People’s Republic of China

(Received 6 July 2018; published 21 August 2018)

The production of the heavy quarkonium, i.e., jðcb¯Þ½ni (or jðbc¯Þ½ni), jðcc¯Þ½ni, and jðbb¯Þ½ni- ¯ 0 0 quarkonium [jðQQ Þ½ni-quarkonium for short], through Higgs H semiexclusive decays is evaluated within the nonrelativistic (NRQCD) framework, where [n] stands ¯ 0 1 ¯ 0 3 for the production of the two color-singlet S-wave states, jðQQ Þ½ S01i and jðQQ Þ½ S11i, the production ¯ 0 1 ¯ 0 3 of the four color-singlet P-wave states, i.e., jðQQ Þ½ P01i, jðQQ Þ½ PJ1i (with J ¼½0; 1; 2). Moreover, according to the velocity scaling rule of the NRQCD, the production of the two color-octet components, ¯ 0 1 ¯ 0 3 jðQQ Þg½ S08i and jðQQ Þg½ S18i, are also taken into account. The “improved trace technology” to derive the simplified analytic expressions at the amplitude level is adopted, which shall be useful for dealing with these decay channels. If all higher heavy quarkonium states decay completely to the ground states, 0 ¯ 1 0 1 it should be obtained ΓðH → jðcbÞ½ S01iÞ ¼ 15.14 KeV, ΓðH → jðcc¯Þ½ S01iÞ ¼ 1.547 KeV, and 0 ¯ 1 5 4 ΓðH → jðbbÞ½ S01iÞ ¼ 1.311 KeV. The production of 5.6 × 10 Bc , 4.7 × 10 charmonium meson, and 4.9 × 104 bottomonium meson per year in Higgs decays at the HE/HL-LHC can be obtained.

DOI: 10.1103/PhysRevD.98.036014

I. INTRODUCTION eþe− → H0Z0 and the WþW− fusion process eþe− → ν ν¯ 0 Since the of the (SM) has e eH . The cross section for the Higgs-strahlungpffiffiffi process is dominant at the low energy. For s ¼ 500 GeV, the been found by CMS [1] and ATLAS [2] at the Large þ − Collider (LHC) in July 2012. The lots of experimental cross section for the W W fusion is dominant. The cross 0 0 þ − → þ − 0 results and review papers on the Higgs boson production and section for the Z Z fusion process e e e e H increases significantly with the center-of-mass (c.m.) decay were obtained by the CMS and ATLAS at the LHC 0 0 [3–5]. With the growingly accumulated date, the properties energy increasing,pffiffiffi and can exceeds that of Z H production of a new are consistent with those of Higgs boson around s ¼ 1 TeV. These processes can be well used to predicted by SM [6,7]. Though the LHC offers obvious test the Higgs-gauge couplings. The Higgs self-coupling advantages in proving very high energy and very large rates can be studied through the double Higgs boson production þ − → 0 0 0 þ − → ν ν¯ 0 0 in typical reactions, the measuring precision will be processes e e Z H H and e e e eH H at the restricted due to the complicated background. ILC. The absolute values of the Higgs coupling to , The most precise measurements will be performed in the and heavy can also be measured. When clean environment of the future - collider updated to the Super -Proton Collider (SPPC), for the proposed Higgs factory, like the International Linear researchers can even measure the Higgs self-coupling, Collider (ILC) [8] and the Circular Electron-Positron which is regarded as the holy grail of experimental particle Collider (CEPC) [9]. It is well known that the main physics high luminosity/energy (HL/HE-LHC) scenarios production processes of the Higgs boson in electron- are designed forpffiffiffi the LHC [10,11]. Running at center-of- positron collider collisions are the Higgs-strahlung process mass energy s ¼ 14 TeV, cross-section of the Higgs boson production at the LHC is about 55 pb (-gluon

* fusion process dominates). Given that the integrated lumi- [email protected] −1 8 † 1 65 10 [email protected] nosity is 3 ab , the HL-LHC would produce . × ‡ [email protected] pHiggsffiffiffi events [11]. While at the HE-LHC who runs at s ¼ 33 TeV, the cross-section of the Higgs boson pro- Published by the American Physical Society under the terms of duction would be about 200 pb, hence the Higgs boson the Creative Commons Attribution 4.0 International license. 6 0 108 Further distribution of this work must maintain attribution to events per year can be obtained . × . the author(s) and the published article’s title, journal citation, With the above mentioned excellent platforms, rare and DOI. Funded by SCOAP3. Higgs boson decay processes, like the heavy quarkonium

2470-0010=2018=98(3)=036014(9) 036014-1 Published by the American Physical Society LIAO, DENG, YU, WANG, and XIE PHYS. REV. D 98, 036014 (2018)

1 3 1 3 production in the Higgs boson decays, might be observed for NRQCD, where [n] stands for 1 S0, 1 S1, 1 P0, n PJ the first time. Pioneer investigation on the search of H0 → (J ¼½0; 1; 2). To deal with heavy quarkonium production J=Ψγ and H0 → ϒðnSÞγ has been carried out by ATLAS through H0 semiexclusive decays, one needs to derive the [12]. Theoretically, some related calculations have been pQCD calculable squared amplitudes. But the analytical done [13–17]. Within the nonrelativistic quantum chromo- expression for the usual squared amplitude jΣj2 becomes dynamics (NRQCD) formulism [18] and light-cone meth- too complex and lengthy for more (massive) in the ods [19], both direct and indirect production mechanism and final states and for higher-level Fock states to be generated relativistic corrections to H0 → J=Ψγ and H0 → ϒðnSÞγ for the emergence of massive- lines in the Feynman are studied [16]. In Ref. [20], the Bc (Bc) meson production diagrams, especially to derive the amplitudes of the via Higgs boson decays under the NRQCD [18] is system- P-wave states. To solve the problem, the “improved trace atic investigated. Where both the quantum chromodynamics technology” is suggested and developed in the literature (QCD) and the quantum electrodynamics (QED) contribu- [37–43]; it deals with the process directly at the amplitude tions are included. It is found that the production of Bc (Bc) level. We will continue to adopt improved trace technology meson through the QED contributions is very smaller to derive the analytical expression for all the above- Γð 0 → jð ¯Þ½ i þ ¯ Þ mentioned decay channels. than through the QCD, e.g., H bc n bc QED= 0 −5 The rest of the paper is organized as follows. We ΓðH → jðbc¯Þ½ni þ bc¯ Þ ∼ 10 . In comparison with QCD introduce the calculation techniques for the H0 boson the QCD one, QED contribution is negligible for production ¯ 0 semiexclusive decays to jðQQ Þ½ni-quarkonium within the heavy quarkonium through the Higgs boson decays. So it the NRQCD formulism in Sec. II. In Sec. III, we calculate is only studied the QCD contribution for the Higgs boson the production of jðcb¯Þ½ni (or jðbc¯Þ½ni), jðcc¯Þ½ni, and decays production the heavy quarkonium in this paper. jð ¯Þ½ i 0 The LHCb, ATLAS, and CMS Collaboration experi- bb n -quarkonium through H decay channels, i.e., H0 → jðcb¯Þ½ni þ cb¯ , H0 → jðcc¯Þ½ni þ cc¯ , and H0 → ments have published studies of the Bc meson production ¯ ¯ and of the double J=Ψ production [12,21,22]. Since its jðbbÞ½ni þ bb, with new parameters [37] for the jð ¯ 0Þ½ i discovery by the CDF Collaboration [23], the Bc meson QQ n -quarkonium, an estimation of events at the being the unique “doubly heavy-flavored” meson in the SM HL/HE-LHC. Then we make some discussions on has aroused great interest. The direct hadronic production the theoretical uncertainties of the decays widths by the jð ¯Þ½ i of the Bc meson has been studied systematically in masses of the cb n -quarkonium. The final section is Refs. [24–29]. Therefore, investigation of the heavy quar- reserved for a summary. konium production through H0 decays is worthwhile and meaningful. The heavy quarkonium is presumed to be a II. CALCULATION TECHNIQUES nonrelativistic of the heavy and anti- AND FORMULATION quark. The study of the heavy quarkonium, e.g., jðbc¯Þ½ni The H0 boson decays semiexclusive processes for the (or jðcb¯Þ½ni), jðcc¯Þ½ni, and jðbb¯Þ½ni-quarkonium, can heavy quarkonium production can be analogous dealt with, help us to achieve a deeper understanding of the QCD in 0 → jð ¯Þ½ i þ ¯ 0 → jð ¯Þ½ i þ ¯ 0 → both the perturbative and nonperturbative sectors. A very i.e., H cb n cb (or H bc n bc), H jð ¯Þ½ i þ ¯ 0 → jð ¯Þ½ i þ ¯ practical theoretical tool to deal with the processes involv- cc n cc, and H bb n bb. According to the ing heavy quarkonium is the NRQCD [18], in which the NRQCD factorization formula [18], the square of the low-energy interactions are organized by the expansion in semiexclusive amplitude can be written as the production v, where v stands for the typical relative velocity of the of the perturbatively calculable short-distance coefficients heavy quark and antiquark inside of the heavy quarkonium. and the nonperturbative long-distance factors, the so-called The heavy quarkonium production itself is very useful for nonperturbative NRQCD matrix elements. Its total decay Γ high precision physics in the electroweak sector and testing widths d can be factorized as – the perturbative QCD (pQCD) [30 32]. For compensation, X hOHð Þi 0 ¯ 0 ¯ 0 n it would be helpful to study its indirect production dΓ ¼ dΓˆðH → jðQQ Þ½ni þ Q QÞ ; ð1Þ mechanisms. A systematic study of the heavy quarkonium n Ncol production through W, Z0 boson, and t (or t¯) quark decays can be found in the literature [33–43]. where Ncol refers to the number of colors, n stands for jð ¯ 0Þ½ i ¼ 1 Due to a high collision energy and high luminosity at the the involved state of QQ n -quarkonium. Ncol for ¼ 2 − 1 hOHð Þi HL/HE-LHC, sizable amounts of the heavy quarkonium singlets and Ncol N for octets; n is the events can be produced through H0 decays [10,11].So nonperturbative matrix element which describes the hadro- ¯ 0 these channels may be an important supplement for other nization of a QQ pair into the observable quark hadron measurements at the HL/HE-LHC. In this work, we will state and is proportional to the transition probability of ¯ 0 ¯ 0 study the jðcb¯Þ½ni (or jðbc¯Þ½ni), jðcc¯Þ½ni and jðbb¯Þ½ni- the perturbative state QQ into the bound state jðQQ Þ½ni. quarkonium production in Higgs boson decays under the As for the color-singlet components, the nonperturbative

036014-2 HEAVY P-WAVE QUARKONIUM PRODUCTION VIA … PHYS. REV. D 98, 036014 (2018) matrix elements can be directly related either to the wave functions at the origin for S-wave states or the first derivative of the wave functions at the origin for P-wave states [18], which can be computed via the potential NRQCD (pNRQCD) [31,44,45] and/or lattice QCD [46] and/or the potential models [37,47–51]. Although we do not know the exact values of the two decay color-octet ¯ 0 1 ¯ 0 3 matrix elements, jðQQ Þg½ S08i and jðQQ Þg½ S18i,we know that they are one order in v2 higher than the S-wave color-singlet matrix elements according to NRQCD scale rule. More specifically, based on the velocity scale rule, we have:

¯ 0 1 1 ¯ 0 1 0 hðQQ Þg½ S08jO½ S08jðQQ Þg½ S08i FIG. 1. Feynman diagrams for the process H ðkÞ → jð ¯Þ½ ið Þþ ð Þþ¯ð Þ jð ¯Þ½ i ≃ Δ ð 2Þ hð ¯ 0Þ½1 jO½1 jð ¯ 0Þ½1 i cb n q3 b q2 c q1 , where cb n stands for S v · QQ S0 1 S0 1 QQ S0 1 ; ¯ 1 ¯ 3 ¯ 1 ¯ 3 jðcbÞ½ S01i, jðcbÞ½ S11i, jðcbÞ½ P11i, jðcbÞ½1 PJ1i (with ¯ 0 3 3 ¯ 0 3 ¯ 1 ¯ 3 hðQQ Þg½ S18jO½ S18jðQQ Þg½ S18i J ¼½0; 1; 2), jðcbÞg½ S08i and jðcbÞg½1 S18i quarkonium, 2 ¯ 0 3 3 ¯ 0 3 respectively. ≃ ΔSðv Þ · hðQQ Þ½ S11jO½ S11jðQQ Þ½ S11i: ð2Þ

Where the second equation comes from the vacuum- only derive the whole decay widths but also obtain saturation approximation. The thickened subscripts of the corresponding differential decay widths that are helpful ð ¯ 0Þ for experimental studies, such as dΓ=ds1, dΓ=ds2, the QQ denote for color indices, 1 for color singlet 2 and 8 for color-octet; the relevant dΓ=d cos θ12, and dΓ=d cos θ23, where s1 ¼ðq1 þ q2Þ , 2 quantum numbers are shown in the parentheses accord- s2 ¼ðq1 þ q3Þ , θ12 is the angle between q⃗1 and q⃗2, ingly. Here v is the relative velocity between the compo- and θ23 between q⃗2 and q⃗3. △ ð 2Þ 2 To better illustrate the Feynman diagrams of the above nents, s v is of the order v or so, and we take it to be 0 ¯ 0 0 ¯ 1 three processes as H ðkÞ → jðQQ Þ½niðq3ÞþQ ðq2Þþ jðbbÞg½ S0 i within the region of 0.10, 0.20, 0.30 for 8 ¯ 0 jð ¯Þ ½3 i jð ¯Þ ½1 i jð ¯Þ ½3 i Qðq1Þ, the Feynman diagrams of the process H ðkÞ → ( bb g S1 8 ), bc g S0 8 ( bc g S1 8 ), and ¯ 1 3 jðcbÞ½niðq3Þþbðq2Þþc¯ðq1Þ is presented in Fig. 1 e.g., jðcc¯Þg½ S08i (jðcc¯Þg½ S18i), respectively; which is in con- sistent with the identification: △ v2 ∼ α ðMvÞ and has where the intermediate gluon should be hard enough to s s ¯ ¯ covered the possible variation due to the different ways to produce a bb pair or cc pair, so the amplitude is pQCD obtain the wave functions at the origin (S-wave) and the calculable. first derivative of the wave functions at the origin (P-wave). These amplitudes can be generally expressed as Γˆ ð 0Þ The short-distance decay width H can be Xm expressed as 0 iM ¼ Cu¯ siðq2Þ Anvs jðq1Þ; ð5Þ n¼1 1 X ˆ 0 ¯ 0 ¯ 0 2 dΓðH → jðQQ Þ½niþQQ Þ¼ jMðnÞj dΦ3; ð3Þ 2mM where m stands for the number of Feynman diagrams, s and s0 are spin states, and i and j are color indices for the outing P ¯ where means that one needs to average over the spin Q-quark and Q-quark, respectively. The overall factor C ¼ C states of the initial particle and to sum over the color and S stands for the specified quarkoniumpffiffiffi in the color-singlet, 2 0 C ¼ δ ð2 0 3Þ C ¼ C spin of all the final particles. In the H rest frame, the three- where S CFggs · ij= mH . Where pffiffiffi O stands particle phase space can be written as C ¼ 2 ð 2 a b aÞ for thep color-octetffiffiffi p states,ffiffiffi where O ggs · T T T ij= ð2 0 2Þ 2 a   mH , here T stands for the color factor of the X3 Y3 d3q⃗ 4 4 f color-octet quarkonium. An’s in the formulas are the dΦ3 ¼ð2πÞ δ k0 − q : ð4Þ f ð2πÞ32 0 0ð Þ → jð ¯ 0Þ½ ið Þþ 0ð Þþ ¯ ð Þ f f¼1 qf amplitudes of H k QQ n q3 Q q2 Q q1 . By using the improved trace technology, one can The process to simplify the 1 → 3 phase space with a sequentially obtain the squared amplitudes, and the numeri- massive quark/antiqark in the final state has been dealt with cal efficiency can also be greatly improved [37–43]. in greater detail in Refs. [38,39]. To shorten the paper, we We adopt the “improved trace technology” to simplify shall not present it here, but the interested reader may turn the amplitudes Mss0 at the amplitude level for the above- A 0ð Þ → to these references for the detailed technology. With the mentioned processes, the amplitudes n of H k ¯ 0 0 ¯ help of the formulas listed in Refs. [38,39], one can not jðQQ Þ½niðq3ÞþQ ðq2ÞþQðq1Þ for the S-wave states are

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 0ðνÞ   0ðνÞ  Π ð Þ ð þ Þþ −ð 1þ Þþ 0 Π ð Þ q3 q =q2 =q3 mQ =q =q3 mQ q3 q A ¼ γ γ ð Þ A ¼ 0 γ γ ð Þ 1 mQ α 2 α 2 2 ; 6 3 mQ 2 2 α 2 α ; 8 ðq2 þ q32Þ ðq2 þ q3Þ − m ðq1 þq3Þ −m 0 ðq1 þq31Þ Q q¼0 Q q¼0    0ðνÞ  0ðνÞ Π ð Þ −ð − Þþ −ð=k−=q32Þþm 0 Π ð Þ q3 q =k =q31 mQ Q q3 q A ¼ γ γ ð Þ A ¼ 0 γ γ ð Þ 2 mQ α 2 2 2 α ; 7 4 mQ α 2 2 2 α : 9 ð þ Þ ð − Þ − ðk−q32Þ −m 0 ðq1 þq31Þ q2 q32 k q31 mQ q¼0 Q q¼0 1 For the P1-wave states, An can be written as

  Π0 ð Þ ð þ Þþ S¼0;L¼1 μ d q3 q =q2 =q3 mQ A ¼ ε ðq Þ m γα γα ; ð Þ 1 l 3 Q ð þ Þ2 ð þ Þ2 − 2 10 dqμ q2 q32 q2 q3 mQ q¼0   Π0 ð Þ −ð − Þþ S¼0;L¼1 μ d q3 q =k =q31 mQ A ¼ ε ðq Þ m γα γα ; ð Þ 2 l 3 Q ð þ Þ2 ð − Þ2 − 2 11 dqμ q2 q32 k q31 mQ q¼0  0ðνÞ  −ð 1 þ Þþ 0 Π ð Þ S¼0;L¼1 μ d =q =q3 mQ q3 q A ¼ ε ð Þ 0 γ γ ð Þ 3 l q3 mQ 2 2 α 2 α ; 12 dqμ ðq1 þ q3Þ − m 0 ðq1 þ q31Þ Q q¼0  0  −ð − Þþ 0 Π ð Þ S¼0;L¼1 μ d =k =q32 mQ q3 q A ¼ ε ð Þ 0 γ γ ð Þ 4 l q3 mQ α 2 2 2 α : 13 dqμ ðk − q32Þ − m 0 ðq1 þ q31Þ Q q¼0 3 and the PJ-wave states (J ¼ 0,1,2)  ν  d Π ðqÞ ð=q2 þ =q3Þþm AS¼1;L¼1 ¼ εJ ð Þ γ q3 γ Q ð Þ 1 μν q3 mQ α ð þ Þ2 α ð þ Þ2 − 2 ; 14 dqμ q2 q32 q2 q3 mQ q¼0  ν  d Π ðqÞ −ð=k − =q31Þþm AS¼1;L¼1 ¼ εJ ð Þ γ q3 Q γ ð Þ 2 μν q3 mQ α ð þ Þ2 ð − Þ2 − 2 α ; 15 dqμ q2 q32 k q31 mQ q¼0  ν  −ð 1 þ Þþ 0 Π ð Þ S¼1;L¼1 d =q =q3 mQ q3 q A ¼ εJ ð Þ 0 γ γ ð Þ 3 μν q3 mQ 2 2 α 2 α ; 16 dqμ ðq1 þ q3Þ − m 0 ðq1 þ q31Þ Q q¼0  ν  −ð − Þþ 0 Π ð Þ S¼1;L¼1 d =k =q32 mQ q3 q A ¼ εJ ð Þ 0 γ γ ð Þ 4 μν q3 mQ α 2 2 2 α : 17 dqμ ðk − q32Þ − m 0 ðq1 þ q31Þ Q q¼0 ¯ 0 Here εsðq3Þ and εlðq3Þ are the polarization vectors relating to the spin and the orbit angular momentum of the jðQQ Þ½ni- J quarkonium, εμνðq3Þ is the polarization tensor for the spin triplet P-wave states (with J ¼½0; 1; 2). The covariant form of the projectors can be conveniently written as pffiffiffiffiffiffiffiffiffiffi − ¯ 0 0 mQQ 1c Π ð Þ¼ ð − 0 Þγ ð þ Þ ⊗ pffiffiffiffiffiffi ð Þ q3 q =q32 mQ 5 =q31 mQ ; 18 4 0 mQmQ Nc and pffiffiffiffiffiffiffiffiffiffi − ¯ 0 ν mQQ 1c Π ð Þ¼ ð − 0 Þγ ð þ Þ ⊗ pffiffiffiffiffiffi ð Þ q3 q =q32 mQ ν =q31 mQ : 19 4 0 mQmQ Nc

Here 1c stands for the unit color matrix with Nc ¼ 3 for the QCD; q stands for the relative momentum between the two ¯ 0 constituent in the jðQQ Þ½ni-quarkonium. q31 and q32 are the momenta of the two constituent quarks, i.e.,

mb mQ q31 ¼ q3 þ q; q32 ¼ q3 − q: ð20Þ ¯ 0 ¯ 0 mQQ mQQ

¯ 0 ¼ þ 0 where mQQ mQ mQ is implicitly adopted to ensure the gauge invariance of the hard scattering amplitude.

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Finally, the decay widths over s1 and s2 can be TABLE I. Mass of the constituent quark and radial wave expressed as functions at the origin under the B.T. potential [37].   ¯ 0 jð ¯Þ½ i ¯ ¯ 3 X hOHð Þi jðQQ Þ½ni cc n jðcbÞ½ni jðbbÞ½ni Γ ¼ j j2 n ð Þ d 3 3 M ds1ds2; 21 ð Þ 1 45 4 85 256π m N mS GeV 1.48 . = . 4.71 H col 2 3 jRj½1Sið0Þj ðGeV Þ 2.458 3.848 16.12 0 m ðGeVÞ 1.75 1.75=4.93 4.94 where mH is the mass of the H boson, the extra factor 3 in P j 0 ð0Þj2ð 5Þ 0.322 0.518 5.874 the numerator comes from the sum of the Q-quark color. Rj½1Pi GeV The color-singlet nonperturbative matrix element hOHðnÞi can be related either to the Schrödinger wave function ψ ¯ 0 ð0Þ þ 0 ðQQ Þ at the origin for the S-wave quarkonium states or mQ mQ . We adopt the values derived in Refs. [37,53] ψ 0 ð0Þ and list them in Table I, since it is noted that the Buchmüller the first derivative of the wave function ð ¯ 0Þ at the QQ and Tye potential (B.T. potential) has the correct two-loop origin for the P-wave quarkonium states: short-distance behavior in QCD [48,55] and the decay H 2 widths are related to the constituent quark mass of the hO ð1SÞi ≃ jψ ¯0 ð0Þj ; jðQQ Þ½1Si jðQQ¯ 0Þ½ni-quarkonium. H 0 2 hO ð1PÞi ≃ jψ ¯0 ð0Þj : ð22Þ jðQQ Þ½1Pi B. Heavy quarkonium production via H0 decays As the spin-splitting effects are small, the difference The decay widths for the jðcb¯Þ½ni (or jðbc¯Þ½ni), between the wave function parameters for the spin-singlet jð ¯Þ½ i jð ¯Þ½ i and spin-triplet states at the same level are not distin- cc n , and bb n -quarkonium states and the 0 0 → guished. The Schrödinger wave function at the origin production channels through H decays, i.e., H jð ¯Þ½ i þ ¯ 0 →jð ¯Þ½ iþ ¯ 0 → jð ¯Þ½ i þ Ψ ¯ 0 ð0Þ and the first derivative of the Schrödinger cb n cb (or H bc n bc), H cc n jQQ Þ½1Si 0 ¯ ¯ 0 cc¯ , and H → jðbbÞ½ni þ bb, are listed in Tables II–IV wave function at the origin Ψ ¯ 0 ð0Þ are related to the jðQQ Þ½1Pi ¯0 ð0Þ radial wave function at the origin RjðQQ Þ½1Si and the first TABLE II. Decay widths and branching fractions for the derivative of the radial wave function at the origin ¯ 0 production of the jðcbÞ½ni-quarkonium through Higgs boson R ¯0 ð0Þ, respectively [18,37]. jðQQ Þ½1Pi semiexclusive decays within the B.T. potential (nf ¼ 3) [37,48]. pffiffiffiffiffiffiffiffiffiffi Ψ ð0Þ¼ 1 4π ð0Þ ΓðH0 → jðcb¯Þ½niÞ jð ¯ 0Þ½1 i = Rjð ¯0Þ½1 i ; 0 QQ S QQ S 0 ¯ ΓðH →jðcb¯Þ½niÞ pffiffiffiffiffiffiffiffiffiffi H → jðcbÞ½ni þ cb¯ ðKeVÞ Γ Ψ0 ð0Þ¼ 3 4π 0 ð0Þ ð Þ H jð ¯0Þ½1 i = Rjð ¯0Þ½1 i : 23 0 ¯ 1 −3 QQ P QQ P H → jðcbÞ½ S01iþcb¯ 5.736 1.37 × 10 0 ¯ 3 −3 H → jðcbÞ½ S11iþcb¯ 7.857 1.87 × 10 ¯ 0 ð0Þ 0 ¯ 1 −5 The radial wave function at the origin RjðQQ Þ½1Si and H → jðcbÞ½ P11iþcb¯ 0.2761 6.57 × 10 0 ¯ 3 −5 the first derivative of the radial wave function at the H → jðcbÞ½ P01iþcb¯ 0.1838 4.38 × 10 0 0 3 −4 origin R ¯ 0 ð0Þ relate to the number of active flavor → jð ¯Þ½ iþ¯ 1 60 10 jðQQ Þ½1Pi H cb P1 1 cb 0.6706 . × ¯ 0 0 → jð ¯Þ½3 iþ¯ 8 38 10−5 n jðQQ Þ½ni H cb P2 1 cb 0.3521 . × quarks f, the constituent quark mass of the - 0 ¯ 1 4 4 −4 H → jðcbÞg½ S08iþcb¯ 0.7170v 1.71v × 10 quarkonium, and the concrete potential models, respec- 0 ¯ 3 4 4 −4 0 H → jðcbÞg½ S18iþcb¯ 0.9821v 2.24v × 10 ¯ 0 ð0Þ ð0Þ tively [37]. Thus, RjðQQ Þ½1Si and RjðQQ¯ Þ½1Pi in this paper are adopted in Ref. [37]. TABLE III. Decay widths and branching fractions for the III. NUMERICAL RESULTS AND DISCUSSIONS production of the jðcc¯Þ½ni-quarkonium through Higgs boson semiexclusive decays within the B.T. potential (n ¼ 3) [37,48]. A. Input parameters f H0 → jðcc¯Þ½ni þ cb¯ ΓðH0 →jðcc¯Þ½niÞðeVÞ ΓðH0→jðcc¯Þ½niÞ The input parameters are adopted as the following values Γ 0 H [52,53]: mH ¼ 125.7 GeV, the Higgs H total decay with 0 1 −4 0 H → jðcc¯Þ½ S01iþcc¯ 646.6 1.54 × 10 ΓðH Þ¼4.2 MeV is adopted [54]. m ¼ 80.399 GeV, 0 3 −4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W H → jðcc¯Þ½ S11iþcc¯ 623.7 1.49 × 10 θ ¼ 0 23119 0 1 −5 W arcsin . is the Weinberg angle. We set the H → jðcc¯Þ½ P11iþcc¯ 69.89 1.66 × 10 ¯ jð ¯Þi 0 3 −5 renormalization scale to be mðcc¯Þ and mðcbÞ of cc and H → jðcc¯Þ½ P01iþcc¯ 101.9 2.43 × 10 ¯ 0 → jð ¯Þ½3 iþ¯ 1 06 10−5 jðcbÞi-quarkonium for leading-order αs running, which H cc P1 1 cc 44.37 . × ¯ 0 → jð ¯Þ½3 iþ¯ 1 10 10−5 leads to α ¼ 0.26 and mð ¯Þ of jðbbÞi-quarkonium for H cc P2 1 cc 46.02 . × s bb 0 → jð ¯Þ ½1 iþ¯ 80 82 4 1 92 4 10−5 α ¼ 0 18 H cc g S0 8 cc . v . v × s . . To ensure the gauge invariance of the hard 0 3 77 96 4 1 86 4 10−5 ¯ 0 H → jðcc¯Þg½ S18iþcc¯ . v . v × amplitude, we set the jðQQ Þ½ni-quarkonium mass M to be

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3 4 TABLE IV. Decay widths and branching fractions for the jðbc¯Þ ½ S1i, and 1.03 × 10 jðbc¯Þ½1Pi-quarkonium ¯ 1 production of the jðbbÞ½ni-quarkonium through Higgs boson events per year can be obtained. ¼ 4 ¯ 0 semiexclusive decays within the B.T. potential (nf ) [37,48]. (iii) For jðbbÞ½ni-quarkonium production via H boson 1 3 0 ¯ semiexclusive decays, the decay widths for P1, P0, H0 → jðbb¯Þ½ni þ cb¯ ΓðH0 →jðbb¯Þ½niÞðeVÞ ΓðH →jðbbÞ½niÞ Γ 3 3 H P1, and P2-wave states is 3.14% (4.16%), 5.85% 0 ¯ 1 ¯ −4 H → jðbbÞ½ S01iþbb 680.6 1.62 × 10 (7.76%), 5.69% (7.54%), and 2.30% (3.04%) of the 0 ¯ 3 ¯ −4 jð ¯Þ ½1 i jð ¯Þ ½3 i H → jðbbÞ½ S11iþbb 513.4 1.22 × 10 decay width of the bb 1 S0 ( bb 1 S1 ). At the 0 ¯ 1 ¯ −6 4 ¯ 1 4 ¯ 3 H → jðbbÞ½ P11iþbb 21.36 5.09 × 10 LHC, 2.67×10 jðbbÞ1½ S0i, 2.01×10 jðbbÞ1½ S1i, 0 ¯ 3 ¯ −6 3 ¯ H → jðbbÞ½ P01iþbb 39.83 9.48 × 10 and 4.54 × 10 jðbbÞ½1Pi-quarkonium events per 0 ¯ 3 ¯ −6 H → jðbbÞ½ P11iþbb 38.73 9.22 × 10 year can be obtained. 0 ¯ 3 ¯ −6 H → jðbbÞ½ P21iþbb 15.63 3.72 × 10 To better illustrate the relative importance of different 0 ¯ 1 ¯ 85 08 4 2 03 4 10−5 H → jðbbÞg½ S08iþbb . v . v × production channels, we present the differential distribu- 0 → jð ¯Þ ½3 iþ¯ 64 17 4 1 53 4 10−5 H bb g S1 8 bb . v . v × tions dΓ=ds1, dΓ=ds2, dΓ=d cos θ12, and dΓ=d cos θ23 for the H0 → jðcb¯Þ½ni þ cb¯ processes in Figs. 2, 3. These figures show explicitly that the excited Fock states can within the B.T. potential[37], where the Higgs total decay provide sizable contributions in comparison to the lower Γ ¼ 4 20 ¯ 1 ¯ 3 with is adopted H . MeV [54]. If the input param- Fock state jðcbÞ½ S0i or jðcbÞ½ S1i in almost the entire eters of the Ref. [20] is adopted. The results are consistent kinematical region. with the results of this paper for S-wave states. As the As all the excited states decay to the ground state choice of the new model parameters [37], the calculation ¯ 1 jðQQÞ½ S0i with 100% efficiency via electromagnetic or results of our paper and Ref. [20] is different. The main hadronic interactions, we can obtain the total decay width difference of the calculation results comes from the matrix 0 H 0 of H boson decay channels within the B.T. potential: elements hO ðnÞi (wave function). But ΓðH → BcÞ= 0 ΓðH → BcÞ¼2.08 KeV=1.53 KeV ¼ 1.364 in Ref. [20] 0 0 is almost equal to ΓðH → B Þ=ΓðH → B Þ¼7.857 KeV= −8 c c 10 5.736 KeV ¼ 1.370 in our paper. −9 From Tables II–IV, it is noted that, in addition to 10 1 jð ¯ 0Þ½ i

the ground S-level states, the P-states of QQ n - ) −10 quarkonium can also provide sizable contributions to the −1 10

ptotalffiffiffi decay widths. Running at center-of-mass energy (GeV −11 1 10 s ¼ 14 TeV, cross section of the Higgs boson production /ds

55 Γ −12 at the LHC is about pb. Given that the integrated d 10 luminosity is 3ab−1, the HL-LHC would be produced −13 1.65 × 108 Higgs boson events per year [11]. 10 jð ¯Þ½ i 0 (i) For cb n -quarkonium production via H boson 2 3 4 1 3 10 10 10 semiexclusive decays, the decay widths for P1, P0, s (GeV2) 3 3 1 P1, and P2-wave states is 4.81% (3.51%), 3.20% −8 (2.34%), 11.69% (8.54%), and 6.14% (4.48%) of the 10 jð ¯Þ ½1 i jð ¯Þ ½3 i decay width of the cb 1 S0 ( cb 1 S1 ). Given −9 −1 10 that the integrated luminositypffiffiffi is 3ab and Running ) ¼ 14 −10 at center-of-mass energy s TeV at the HL/ −1 10 HE-LHC, cross section of Higgs boson production

5 (GeV −11 at the LHC is about 55pb. Then 2.26 × 10 2 10

¯ 1 5 ¯ 3 /ds

jð Þ ½ i 3 08 10 jð Þ ½ i 5 83 Γ cb 1 S0 , . × cb 1 S1 , and . × −12 d 10 104 jðcb¯Þ½1Pi-quarkonium events per year can be −13 obtained at the HL/HE-LHC, where 1P is presented 10 ¯ 1 the summed decays widths of jðcbÞ½ P1i and 3 4 ¯ 3 10 10 jðcbÞ½ P i (J ¼ 0; 1; 2). 2 J s (GeV ) (ii) For jðcc¯Þ½ni-quarkonium production via H0 boson 2 1 3 semiexclusive decays, the decay widths for P1, P0, Γ Γ 3 3 FIG. 2. Differential decay widths d =ds1 and d =ds2 for P1, and P2-wave states is 10.80% (11.21%), H0 → jðcb¯Þ½ni þ bc¯, where the diamond line, the dashed line, 15.76% (16.34%), 6.86% (7.11%), and 7.12% the solid line, the dash-dotted line, the dotted line and the crosses 1 ¯ 1 ¯ 3 ¯ 1 ¯ 3 (7.38%) of the decay width of the jðcc¯Þ1½ S0i line are for jðcbÞ½1 S0i, jðcbÞ½1 S1i, jðcbÞ½1 P1i, jðcbÞ½1 P0i, 3 4 1 4 ¯ 3 ¯ 3 (jðcc¯Þ1½ S1i). 2.54 × 10 jðcc¯Þ1½ S0i, 2.46 × 10 jðcbÞ½1 P1i, and jðcbÞ½1 P2i, respectively.

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3 pffiffiffi 10 Running at the center-of-mass energy S ¼ 14 TeV at 2 34 −2 −1 10 the HL/HE-LHC [10,11] and with luminosity 10 cm s ,

1 8 0 10 one may expect to produce about 1.65 × 10 H boson per ¯ 0 0 jð Þi 10 year. Then we can estimate the event number of QQ - 0

(KeV) −1 quarkonium production through H boson decays, i.e.,

12 10 5 ¯ θ about 5.6 × 10 jðcbÞ½ni (or jðbc¯Þ½ni)-quarkonium events,

/d −2 10 4 4 Γ 4 7 10 jð ¯Þ½ i 4 9 10 jð ¯Þ½ i

d . × cc n -quarkonium events, . × bb n - −3 10 quarkonium events per year. The upgrade HE/HL- 0 −4 10 LHC and the newly purposed H factory with luminosity 36 −2 −1 ¯ 0 10 cm s , the possibility to study jðQQ Þ½ni- −1 −0.5 0 0.5 1 0 cosθ quarkonium via H boson decays at the SPPC, ILC, and 12 CEPC is worth serious consideration. 1 10 C. Uncertainty analysis 0 10 In the subsection, we discuss the uncertainties for the jð ¯Þ½ i −1 cb n -quarkonium production through Higgs boson 10

(KeV) decays. For the present calculation, their main uncertainty 23

θ μ −2 sources include the renormalization scale R, the non-

/d 10 Γ perturbative bound state matrix elements, and the constitu- d

−3 ent quark masses mc and mb. These parameters are the main 10 uncertainty source for estimating the jðcb¯Þ½ni-quarkonium

−4 production. Here we only discuss the decay widths of the 10 ¯ 0 −1 −0.5 0 0.5 1 jðcbÞ½ni-quarkonium production via H decays under cosθ 23 varying the constituent quark masses of the jðcb¯Þ½ni- quarkonium. In the following, we shall concentrate our FIG. 3. Differential decay widths dΓ=d cos θ12 and Γ θ 0 → jð ¯Þ½ i þ ¯ attention on the uncertainties caused by mc and mb, whose d =d cos 23 for H cb n bc, where the diamond line, ¼ 1 45 0 10 ¼ the dashed line, the solid line, the dash-dotted line, the dotted center values are taken as mc . . GeV and mb ¯ 1 ¯ 3 4.85 0.20 GeV for S-states, and m ¼ 1.75 0.10 GeV line and the crosses line are for jðcbÞ½1 S0i, jðcbÞ½1 S1i, c ¯ 1 ¯ 3 ¯ 3 ¯ 3 and m ¼ 4.93 0.20 GeV for P-states. And for clarity, jðcbÞ½1 P1i, jðcbÞ½1 P0i, jðcbÞ½1 P1i, and jðcbÞ½1 P2i, b respectively. when discussing the uncertainty caused by one parameter, the other parameters are fixed to be their center values. In the Tables V and VI, it can be found that sizable 0 ¯ 1 ΓðH → jðcbÞ½1 S0i þ cb¯ Þ¼15.14 KeV; ð24Þ uncertainties for varying mc and mb. The decay width will 0 1 decrease with the increment mass of mc. But the decay ΓðH → jðcc¯Þ½1 S0i þ cc¯ Þ¼1.547 KeV; ð25Þ width will increase with the increment mass of mb. 0 ¯ 1 ¯ ΓðH → jðbbÞ½1 S0i þ bbÞ¼1.311 KeV; ð26Þ Adding all the uncertainties caused by the constituent quark masses mc ¼ 1.45 0.10 GeV and mb ¼ 4.85 2 ¼ 0 20 where v . , 0.30, 0.10 for the color-octet 0.20 GeV for S-states, and mc ¼ 1.75 0.10 GeV and jð ¯Þ ½1 i jð ¯Þ ½3 i jð ¯Þ ½1 i jð ¯Þ ½3 i cb g S0 8 ( cb g S1 8 ), cc g S0 8 ( cc g S1 8 ), mb ¼ 4.93 0.20 GeV for P-states in quadrature for the ¯ 1 ¯ 3 0 ¯ and jðbbÞg½ S08i (jðbbÞg½ S18i) are adopted, respectively. process H → jðcbÞ½ni þ bc¯, we can obtain

0 ¯ TABLE V. Uncertainties for the decay width of the processes H → jðcbÞ½ni þ bc¯ under the B.T. potential (nf ¼ 3) [37,48].

S-states mc (GeV) 1.35 1.45 1.65 0 ¯ 1 ΓðH → jðcbÞ½ S01iÞ (KeV) 7.108 5.736 4.697 0 ¯ 3 ΓðH → jðcbÞ½ S11iÞ (KeV) 9.981 7.857 6.283

mc (GeV) 1.65 1.75 1.85 0 ¯ 1 P-states ΓðH → jðcbÞ½ P11iÞ (KeV) 0.3763 0.2761 0.2145 0 ¯ 3 ΓðH → jðcbÞ½ P01iÞ (KeV) 0.2326 0.1838 0.1478 0 ¯ 3 ΓðH → jðcbÞ½ P11iÞ (KeV) 0.8879 0.6706 0.5143 0 ¯ 3 ΓðH → jðcbÞ½ P21iÞ (KeV) 0.4820 0.3521 0.2615

Color-octet S-states mc (GeV) 1.35 1.45 1.65 0 ¯ 1 4 4 4 ΓðH → jðcbÞg½ S08iÞ (KeV) 0.889v 0.717v 0.587v 0 ¯ 3 4 4 4 ΓðH → jðcbÞg½ S18iÞ (KeV) 1.248v 0.982v 0.785v

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0 ¯ ¯ TABLE VI. Uncertainties for the decay width of the processes H → jðcbÞ½ni þ bc under the B.T. potential (nf ¼ 3) [37,48].

S-states mb (GeV) 4.65 4.85 5.05 0 ¯ 1 ΓðH → jðcbÞ½ S01iÞ (KeV) 5.280 5.736 6.210 0 ¯ 3 ΓðH → jðcbÞ½ S11iÞ (KeV) 7.141 7.857 8.611

P-states mb (GeV) 4.73 4.93 5.13 0 ¯ 1 ΓðH → jðcbÞ½ P11iÞ (KeV) 0.2614 0.2761 0.3016 0 ¯ 3 ΓðH → jðcbÞ½ P01iÞ (KeV) 0.1771 0.1838 0.1907 0 ¯ 3 ΓðH → jðcbÞ½ P11iÞ (KeV) 0.6244 0.6706 0.7175 0 ¯ 3 ΓðH → jðcbÞ½ P21iÞ (KeV) 0.3209 0.3521 0.3845

Color-octet S-states mb (GeV) 4.65 4.85 5.05 0 ¯ 1 4 4 4 ΓðH → jðcbÞg½ S08iÞ (KeV) 0.660v 0.717v 0.776v 0 ¯ 3 4 4 4 ΓðH → jðcbÞg½ S18iÞ (KeV) 0.893v 0.982v 1.076v

¯ 0 Γð 0 → jð ¯Þ½11 iþ ¯ ¼ 5 736þ1.452 P jðQQ Þ½ni H cb S0 1 bc . −1.135 KeV; Numerical results show that -states of - 1S Γð 0 → jð ¯Þ½13 iþ ¯ ¼ 7 857þ2.254 quarkonium in addition to the ground wave states H cb S1 1 bc . −1.729 KeV; can also provide sizable contributions to the heavy quar- Γ 0 → jð ¯Þ½1 iþ ¯ ¼ 1 483þ0.508 ð Þ 0 H cb P 1 bc . −0.359 KeV: 27 konium production through H boson decays, so one needs to take the excited wave states into consideration for a If the excited jðcb¯Þ½ni-quarkonium states decay to the sound estimation. If all of the excited heavy quarkonium 1P ¯ 1 ground spin-singlet S-wave state jðcbÞ½1 S0i with 100% Fock states almost decay to the ground spin-singlet S wave ¯ 0 1 efficiency via hadronic interactions or electromagnetic, we state jðQQ Þ½1 S0i via electromagnetic or hadronic inter- can obtain the total decay width of the Higgs boson decay actions, we obtain the total decay width the total decay ¯ 0 0 channels under the B.T. potential. width for jðQQ Þi-quarkonium production through H boson decays as shown by Eqs. ((22)–(26) ). Atpffiffiffi the Γð 0 → jð ¯Þ½11 i þ ¯Þ¼15 14þ2.735 ð Þ LHC at running with center-of-mass energy S ¼ H cb S0 bc . −2.105 KeV: 28 14 TeV with the luminosity L ∝ 1034 cm−2 s−1, due to the high collision energy and high luminosity, sizable IV. CONCLUSIONS heavy quarkonium events can be produced through 0 5 ¯ In this paper, we have made a detailed study on H boson decays; i.e., about 5.6 × 10 of ðcbÞ (or ðbc¯Þ) 4 4 ¯ the jðcb¯Þ½ni (or jðbc¯Þ½ni), jðcc¯Þ½ni, and jðbb¯Þ½ni- meson, 4.7 × 10 of ðcc¯Þ meson, and 4.9 × 10 of ðbbÞ quarkonium via H0 boson semiexclusive decays meson events per year can be obtained. At the under the NRQCD framework, i.e., H0 → jðbc¯Þ½ni þ bc¯ newly purposed H factory with the high luminosity 36 −2 −1 ¯ 0 0 (or H0 → jðcb¯Þ½ni þ cb¯ ), H0 → jðcc¯Þ½ni þ cc¯ , and L ∝ 10 cm s , the jðQQ Þi-quarkonium through H 0 ¯ ¯ 1 3 H → jðbbÞ½ni þ bb, where [n] stands for ½ S0, ½ S1, boson decays will be more abundantly produced. 1 3 Therefore, one needs to take these P-states into consid- ½ P1, and ½ PJ,(J ¼½0; 1; 2). To provide the analytical expressions as simply as possible, we have adopted the eration for a sound evaluation. improved trace technology to derive Lorentz-invariant 0 ACKNOWLEDGMENTS expressions for H boson decay processes at the amplitude level. Such a calculation technology will be very helpful for We thank Professor Xing-Gang Wu for useful dealing with processes with massive spinors. discussions.

[1] S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B [4] The ATLAS and CMS Collaborations, Reports No. ATLAS- 716, 30 (2012). CONF-2015-044, CMS-PAS-HIG-15-002. [2] G. Aad et al. (ATLAS Collaboration), Phys. Lett. B 716,1 [5] C. Kourkoumelis (ATLAS and CMS Collaborations), Proc. (2012). Sci., Charged 2014 (2015) 001. [3] K. Mochizuki (ATLAS Collaboration), J. Phys. Conf. Ser. [6] The ATLAS Collaboration, Report No. ATLAS-CONF- 623, 012020 (2015). 2013-040; Phys. Lett. B 726, 120 (2013).

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[7] S. Chatrchyan et al. (CMS Collaboration), Phys. Rev. Lett. and X. G. Wu, Comput. Phys. Commun. 174, 241 (2006); 110, 081803 (2013). 175, 624 (2006); X. Y. Wang and X. G. Wu, Comput. Phys. [8] H. Baer et al., arXiv:1306.6352. Commun. 183, 442 (2012). [9] (CEPC-SPPC Study Group Collaboration), CEPC-SPPC [30] N. Brambilla et al. (Quarkonium Working Group), CERN Preliminary Conceptual Design Report. 1. Physics and Yellow Report; arXiv:hep-ph/0412158. Detector, Reports No. IHEP-CEPC-DR-2015-01, IHEP- [31] N. Brambilla et al. (Quarkonium Working Group), Eur. TH-2015-01, HEP-EP-2015-01; CEPC-SPPC Preliminary Phys. J. C 71, 1534 (2011). Conceptual Design Report. 2. Accelerator, Reports [32] The CMS Collaboration, J. Phys. G 34, 995 (2007). No. IHEP-CEPC-DR-2015-01, IHEP-AC-2015-01. [33] V. Barger, K. Cheung, and W. Y. Keung, Phys. Rev. D 41, [10] E. Todesco, M. Lamont, and L. Rossi, Report No. C12-05- 1541 (1990). 28.4. [34] E. Braaten, K. Cheung, and T. C. Yuan, Phys. Rev. D 48, [11] LHC Higgs Cross Section Working Group, https://twiki 4230 (1993). .cern.ch/twiki/bin/view/LHCPhysics/ HiggsEuropeanStrat- [35] C. F. Qiao, C. S. Li, and K. T. Chao, Phys. Rev. D 54, 5606 egy and SM Higgs production cross se AN2. (1996). [12] G. Aad et al. (ATLAS Collaboration), Phys. Rev. Lett. 113, [36] C. F. Qiao, L. P. Sun, D. S. Yang, and R. L. Zhu, Eur. Phys. 212004 (2014); Phys. Rev. Lett. 114, 121801 (2015). J. C 71, 1766 (2011). [13] V. Kartvelishvili, A. V. Luchinsky, and A. A. Novoselov, [37] Q. L. Liao and G. Y. Xie, Phys. Rev. D 90, 054007 (2014). Phys. Rev. D 79, 114015 (2009). [38] C. H. Chang, J. X. Wang, and X. G. Wu, Phys.Rev.D [14] A. Djouadi, Phys. Rep. 457, 1 (2008); 459, 1(2008). 77, 014022 (2008);X.G.Wu,Phys. Lett. B 671,318 [15] N. N. Achasov and V. K. Besprozvannykh, Sov. J. Nucl. (2009). Phys. 55, 1072 (1992). [39] L. C. Deng, X. G. Wu, Z. Yang, Z. Y. Fang, and Q. L. Liao, [16] G. T. Bodwin, F. Petriello, S. Stoynev, and M. Velasco, Eur. Phys. J. C 70, 113 (2010). Phys. Rev. D 88, 053003 (2013); G. T. Bodwin, H. S. [40] C. H. Chang and Y. Q. Chen, Phys. Rev. D 46, 3845 (1992). Chung, J. H. Ee, J. Lee, and F. Petriello, Phys. Rev. D [41] Z. Yang, X. G. Wu, L. C. Deng, J. W. Zhang, and G. Chen, 90, 113010 (2014). Eur. Phys. J. C 71, 1563 (2011). [17] C. F. Qiao, F. Yuan, and K. T. Chao, J. Phys. G 24, 1219 [42] Q. L. Liao, X. G. Wu, J. Jiang, Z. Yang, and Z. Y. Fang, (1998). Phys. Rev. D 85, 014032 (2012); Report No. SLAC-PUB- [18] G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D 14983. 51, 1125 (1995); 55, 5853(E) (1997). [43] Q. L. Liao, X. G. Wu, J. Jiang, Z. Yang, and J. W. Zhang, [19] G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 Phys. Rev. D 86, 014031 (2012). (1980). [44] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Nucl. Phys. [20] J. Jiang and C. F Qiao, Phys. Rev. D 93, 054031 (2016). B566, 275 (2000). [21] R. Aaij et al. (LHCb Collaboration) Phys. Rev. Lett. 108, [45] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Rev. Mod. 251802 (2012); Phys. Lett. B 707, 52 (2012). Phys. 77, 1423 (2005). [22] The CMS Collaboration, Report No. CMS-PAS-BPH-11- [46] G. T. Bodwin, D. K. Sinclair, and S. Kim, Phys. Rev. Lett. 021, 2013. 77, 2376 (1996). [23] F. Abe et al. (CDF Collaboration), Phys. Rev. D 58, 112004 [47] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. M. (1998); A. Abulencia et al. (CDF Collaboration), Phys. Rev. Yan, Phys. Rev. D 17, 3090 (1978); 21, 313(E) (1980); 21, Lett. 96, 082002 (2006); A. Abulencia et al. Phys. Rev. Lett. 203 (1980); E. Eichten and F. Finberg, Phys. Rev. D 23, 97, 012002 (2006). 2724 (1981). [24] K. Kolodziej, A. Leike, and R. Ruckl, Phys. Lett. B 355, 337 [48] W. Buchmüller and S.-H. H. Tye, Phys. Rev. D 24, 132 (1995). (1981). [25] S. P. Baranov, Phys. At. Nucl. 60, 1322 (1997). [49] A. Martin, Phys. Lett. 93B, 338 (1980). [26] C. H. Chang and Y. Q. Chen, Phys. Rev. D 48, 4086 (1993); [50] C. Quigg and J. L. Rosner, Phys. Lett. 71B, 153 (1977). C. H. Chang, Y. Q. Chen, G. P. Han, and H. T. Jiang, Phys. [51] Y. Q. Chen and Y. P. Kuang, Phys. Rev. D 46, 1165 (1992); Lett. B 364, 78 (1995); C. H. Chang and X. G. Wu, Eur. Y. Q. Chen and Y. P. Kuang, Phys. Rev. D 47, 350(E) Phys. J. C 38, 267 (2004). (1993). [27] A. V. Berezhnoi, A. K. Likhoded, and M. V. Shevlyagin, [52] J. Alcaraz et al., arXiv:0911.2604. arXiv:hep-ph/9408284. [53] J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, [28] C. H. Chang, J. X. Wang, and X. G. Wu, Phys. Rev. D 70, 010001 (2012). 114019 (2004); C. H. Chang, C. F. Qiao, J. X. Wang, and X. [54] S. Heinemeyer et al. (LHC Higgs Cross Section Working G. Wu, Phys. Rev. D 71, 074012 (2005); 72, 114009 (2005). Group Collaboration), arXiv:1307.1347. [29] C. H. Chang, C. Driouich, P. Eerola, and X. G. Wu, Comput. [55] W. Fritsch, R. Lipperheide, and U. Wille, Phys. Lett. 45B, Phys. Commun. 159, 192 (2004); C. H. Chang, J. X. Wang, 103 (1973).

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