Production of doubly heavy baryons via Higgs boson decays
Abstract
We systematically analyzed the production of semiinclusive doubly heavy baryons (\(\Xi _{cc}\), \(\Xi _{bc}\) and \(\Xi _{bb}\)) for the process \(H^0 \rightarrow \Xi _{QQ'}+ \bar{Q'} + {\bar{Q}}\) through four main Higgs decay channels within the framework of nonrelativistic QCD. The contributions from the intermediate diquark states, \(\langle cc\rangle [^{1}S_{0}]_{\mathbf {6}}\), \(\langle cc\rangle [^{3}S_{1}]_{\bar{\mathbf {3}}}\), \(\langle bc\rangle [^{3}S_{1}]_{\bar{\mathbf {3}}/ \mathbf {6}}\), \(\langle bc\rangle [^{1}S_{0}]_{\bar{\mathbf {3}}/ \mathbf {6}}\), \(\langle bb\rangle [^{1}S_{0}]_{\mathbf {6}}\) and \(\langle bb\rangle [^{3}S_{1}]_{\bar{\mathbf {3}}}\), have been taken into consideration. The differential distributions and three main sources of the theoretical uncertainties have been discussed. At the High Luminosity Large Hadron Collider, there will be about 0.43\(\times 10^4\) events of \(\Xi _{cc}\), 6.32\(\times 10^4\) events of \(\Xi _{bc}\) and 0.28\(\times 10^4\) events of \(\Xi _{bb}\) produced per year. There are fewer events produced at the Circular Electron Positron Collider and the International Linear Collider, about \(0.26\times 10^{2}\) events of \(\Xi _{cc}\), \(3.83\times 10^{2}\) events of \(\Xi _{bc}\) and \(0.17\times 10^{2}\) events of \(\Xi _{bb}\) in operation.
1 Introduction
Attributed to the first observation of the doubly charm baryon \(\Xi _{cc}^{++}\) [12] by the LHCb collaboration in 2017, the quark model has proved to be a great success [13, 14, 15, 16]. However, there is no explicit evidence of the other doubly heavy baryons \(\Xi _{bc}\) and \(\Xi _{bb}\) so far. To study all possible production mechanisms of doubly heavy baryons shall be helpful for better understanding their properties and shall be a verification of the quark model and nonrelativistic Quantum Chromodynamics (NRQCD) [17, 18]. There were some analyses of the direct/indirect production of doubly heavy baryons through \(e^+~e^\) colliders [19, 20, 21, 22], hadronic production [20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], gamma–gamma production [25, 33], photoproduction [25, 34, 35], heavy ion collisions [36, 37], top quark decays [38], etc.
In this paper, we shall discuss the production of doubly heavy baryons \(\Xi _{QQ'}\) through indirectly Higgs boson decays at the HLLHC and CEPC/ILC. As is well known, the dominant decay channel of Higgs boson is \(H^0 \rightarrow b{\bar{b}}\) and the branching ratio is about 58\(\%\) [39, 40]. For completeness, four main Higgs decay channels, \(H^0\rightarrow b{\bar{b}}\), \(c{\bar{c}}\), \(Z^0 Z^0\), gg, would be taken into consideration. Due to the Yukawa coupling and the perturbative order, the decay channel \(H^0 \rightarrow b{\bar{b}}\) (\(H^0 \rightarrow c{\bar{c}}\)) plays an essential role in the production of \(\Xi _{bc}\) and \(\Xi _{bb}\) (\(\Xi _{cc}\)), but the contributions from the \(H^0\rightarrow Z^0 Z^0/gg\) channel cannot be neglected.
Within the framework of NRQCD, the production of doubly heavy baryons \(\Xi _{QQ^{\prime }}\) can be factorized into the convolution of the perturbative shortdistance coefficient and the nonperturbative longdistance matrix elements. In the amplitude, the gluon is hard enough to produce such a heavy quark–antiquark pair, hence the hard process is perturbatively calculable. The longdistance matrix elements are used to describe the transition probability of the produced diquark state \(\langle QQ^{\prime } \rangle [n]\) binding into doubly heavy baryons \(\Xi _{QQ^{\prime }}\), where [n] stands for the spin and color quantum number for the intermediate diquark state. The spin quantum number of the intermediate diquark state \(\langle QQ^{\prime }\rangle [n]\) can be \([^3S_1]\) or \([^1S_0]\), and the color quantum number is the colorantitriplet \({\bar{3}}\) or the colorsextuplet 6 for the decomposition of \(SU_C(3)\) color group \(3\bigotimes 3={\bar{3}} \bigoplus 6\). All of these intermediate states would be taken into consideration for a sound estimation. Assuming the potential of the binding colorantitriplet \(\langle QQ'\rangle [n]\) state is hydrogenlike, the transition probability \(h_{{\bar{3}}}\) can be approximatively related to the Schrödinger wave function at the origin \(\Psi _{QQ'}(0)\) for the Swave states, where \(\Psi _{QQ'}(0)\) can be obtained by fitting the experimental data or some nonperturbative methods like QCD sum rules [41], lattice QCD [42] or the potential model [43]. As for the transition probability of the colorsextuplet diquark state \(h_{6}\), there is a relatively larger uncertainty, and we would make a detailed discussion about it.
The remaining parts of the paper are arranged as follows: in Sect. 2, the detailed calculation technology, such as the factorization and the color factors, is presented. The numerical results associated with the theoretical uncertainties are given in Sect. 3. And Sect. 4 gives a summary and some conclusions.
2 Calculation technology
2.1 Amplitude
The color factors \(\mathcal {C}^{2}_{ij,k}\) for different channels of Fig. 1
\({\mathcal {C}}^{2}_{ij,k}\)  \(H^0 \rightarrow Q{\bar{Q}}/Q^{\prime }\bar{Q^{\prime }}\)  \(H^0 \rightarrow Z^0 Z^0\)  \(H^0 \rightarrow gg\)  Cross term 1  Cross term 2 

Colorantitriplet \({\bar{3}}\)  4/3  3  1/3  \(\) 2  2/3 
Colorsextuplet 6  2/3  6  1/6  2  1/3 
2.2 Color factor
3 Numerical results
We use FeynArts 3.9 [45] to generate the amplitudes and the modified FormCalc 7.3/LoopTools 2.1 [46] to do the algebraic and numerical calculations. The renormalization scale \(\mu _r\) is set to be \(2m_c\), \(2m_c\) and \(2m_b\) for the production of \(\Xi _{cc}\), \(\Xi _{bc}\) and \(\Xi _{bb}\) correspondingly. Due to the total decay width of the Higgs boson not having been detected so accurately by the experiment, we consider the total decay width of the Higgs boson as 4.2 MeV [47] to estimate the branching ratio and corresponding events for the production of baryons \(\Xi _{cc}\), \(\Xi _{bc}\) and \(\Xi _{bb}\).
3.1 Basic results
The decay widths for the process \(H^0 \rightarrow b{\bar{b}} / c{\bar{c}}/Z^{0}Z^{0} / gg \rightarrow \Xi _{QQ^{\prime }}+ \bar{Q^{\prime }} + {\bar{Q}}\), where Q and \(Q^{\prime }\) denote the heavy c or b quark. Cross term 1 stands for the cross term between \(H^0 \rightarrow Q{\bar{Q}} / Q^{\prime } \bar{Q^{\prime }}\) and \(H^0 \rightarrow Z^{0}Z^{0}\), and Cross term 2 is the cross term between \(H^0 \rightarrow Q{\bar{Q}} / Q^{\prime } \bar{Q^{\prime }}\) and \(H^0 \rightarrow gg\)
\(\Gamma \) (GeV)  \(\Xi _{cc}\)  \(\Xi _{bc}\)  \(\Xi _{bb}\)  

\([^3S_1]_{\bar{3}}\)  \([^1S_0]_{6}\)  \([^3S_1]_{\bar{3}}\)  \([^3S_1]_{6}\)  \([^1S_0]_{\bar{3}}\)  \([^1S_0]_{6}\)  \([^3S_1]_{\bar{3}}\)  \([^1S_0]_{6}\)  
\(H^0\rightarrow b{\bar{b}}\,(\times 10^{7})\)  −  −  5.89  2.95  4.48  2.24  0.41  0.28 
\(H^0\rightarrow c{\bar{c}}\,(\times 10^{7})\)  0.65  0.35  \(1.03\times 10^{2}\)  \(5.16\times 10^{3}\)  \(1.23\times 10^{2}\)  \(6.16\times 10^{3}\)  −  − 
\(H^0\rightarrow Q{\bar{Q}} / Q^{\prime } \bar{Q^{\prime }}\,(\times 10^{7})\)  0.65  0.35  5.87  2.94  4.57  2.29  0.41  0.28 
\(H^0\rightarrow Z^{0}Z^{0}\,(\times 10^{10})\)  0.82  1.63  4.25  8.50  4.32  8.64  0.16  1.09 
\(H^0\rightarrow gg\,(\times 10^{9})\)  3.01  0.47  2.36  1.18  1.00  0.50  0.41  0.11 
Cross term 1 (\(\times 10^{10}\))  0.10  \(\)0.33  2.45  \(\) 2.45  \(\)8.25  8.25  \(\)0.57  \(\)9.16 
Cross term 2 (\(\times 10^{9}\))  0.65  5.64  2.48  1.24  20.58  10.29  0.63  2.91 

The biggest decay channel for the production of \(\Xi _{cc}\) (\(\Xi _{bb}\)) is \(H^0 \rightarrow c{\bar{c}}\) (\(H^0 \rightarrow b{\bar{b}}\)). Meanwhile for the production of \(\Xi _{bc}\), the decay width in each diquark state through \(H^0 \rightarrow b{\bar{b}}\) is about two orders of magnitude larger than that through \(H^0 \rightarrow c{\bar{c}}\) mainly for the Yukawa coupling.

From the decay widths through \(H^0 \rightarrow Q{\bar{Q}} / Q^{\prime } \bar{Q^{\prime }}\), it can be seen that the contribution of the cross term between \(H^0 \rightarrow b{\bar{b}}\) and \(H^0 \rightarrow c{\bar{c}}\) is positive for \([^1S_0]_{{\bar{3}}/6}\) states and negative for \([^3S_1]_{{\bar{3}}/6}\) states.

The decay widths through \(H^0 \rightarrow Z^0Z^0/gg\) channels are very small and only a few percent compared to that through \(H^0 \rightarrow Q{\bar{Q}} / Q^{\prime } \bar{Q^{\prime }}\).

The contributions of the cross term between \(H^0 \rightarrow Q{\bar{Q}} / Q^{\prime } \bar{Q^{\prime }}\) and \(H^0 \rightarrow VV~(V=Z^0, g)\) should also be taken into account and the decay width for the production of baryons \(\Xi _{QQ'}\) from these two cross terms are also listed in Table 2.
The decay width and the estimated events through these four Higgs decay channels at the HLLHC and the CEPC/ILC
Fock states  \(\Xi _{cc}\)  \(\Xi _{bc}\)  \(\Xi _{bb}\)  

\([^3S_1]_{\bar{3}}\)  \([^1S_0]_{6}\)  \([^3S_1]_{\bar{3}}\)  \([^3S_1]_{6}\)  \([^1S_0]_{\bar{3}}\)  \([^1S_0]_{6}\)  \([^3S_1]_{\bar{3}}\)  \([^1S_0]_{6}\)  
\(\Gamma ~(\times 10^{7}\)GeV)  0.69  0.41  5.93  2.97  4.78  2.41  0.42  0.30 
HLLHC events (\(\times 10^{4}\))  0.27  0.16  2.33  1.17  1.88  0.95  0.17  0.12 
CEPC/ILC events (\(\times 10^{2}\))  0.16  0.10  1.41  0.71  1.14  0.57  0.10  0.07 
The total decay width, the branching ratio and the estimated events of the doubly heavy baryons \(\Xi _{QQ'}\) by summing up the contribution from each intermediate diquark state
\(\Gamma ~(\times 10^{7}~\mathrm{GeV})\)  Br \((\times 10^{4})\)  HLLHC events  CEPC/ILC events  

\(H^0 \rightarrow \Xi _{cc}\)  1.10  0.26  \(0.43\times 10^{4}\)  \(0.26\times 10^{2}\) 
\(H^0 \rightarrow \Xi _{bc}\)  16.09  3.83  \(6.32\times 10^{4}\)  \(3.83\times 10^{2}\) 
\(H^0 \rightarrow \Xi _{bb}\)  0.72  0.17  \(0.28\times 10^{4}\)  \(0.17\times 10^{2}\) 

The estimated events of \(\Xi _{bc}\) are about one order of magnitude larger than that of \(\Xi _{cc}\) and \(\Xi _{bb}\).

The branching ratio via Higgs boson decays is about \(10^{4}\) for the production of \(\Xi _{bc}\) baryon, and \(10^{5}\) for the production of \(\Xi _{cc}\) and \(\Xi _{bb}\) baryons.

At the HLLHC, there are sizable events of doubly heavy baryons \(\Xi _{QQ'}\), at the order of \(10^4\), produced per year.

There are only about \(10^2\)\(\Xi _{QQ'}\) events produced at the CEPC/ILC, but with a cleaner background. In view of the upgrade of the CEPC/ILC, such as increasing the luminosity to the same level as the HLLHC, there would be 3.75 times the \(\Xi _{QQ'}\) events.
The theoretical uncertainties for the production of baryons \(\Xi _{QQ^{\prime }}\) via Higgs boson decays by varying \(m_c=1.8 \pm 0.3~\mathrm{GeV}\)
\(\Gamma (\times 10^{7}\)GeV)  \(\Xi _{cc}\)  \(\Xi _{bc}\)  \(\Xi _{bb}\)  

\([^3S_1]_{\bar{3}}\)  \([^1S_0]_{6}\)  \([^3S_1]_{\bar{3}}\)  \([^3S_1]_{6}\)  \([^1S_0]_{\bar{3}}\)  \([^1S_0]_{6}\)  \([^3S_1]_{\bar{3}}\)  \([^1S_0]_{6}\)  
\(m_c=1.5~\hbox {GeV}\)  0.85  0.49  10.90  5.46  8.17  4.11  0.42  0.30 
\(m_c=1.8~\hbox {GeV}\)  0.69  0.41  5.93  2.97  4.78  2.41  0.42  0.30 
\(m_c=2.1~\hbox {GeV}\)  0.58  0.35  3.54  1.77  3.07  1.55  0.42  0.30 
The theoretical uncertainties for the production of baryons \(\Xi _{QQ^{\prime }}\) via Higgs boson decays by varying \(m_b=5.1 \pm 0.4~\mathrm{GeV}\)
\(\Gamma (\times 10^{7}\)GeV)  \(\Xi _{cc}\)  \(\Xi _{bc}\)  \(\Xi _{bb}\)  

\([^3S_1]_{\bar{3}}\)  \([^1S_0]_{6}\)  \([^3S_1]_{\bar{3}}\)  \([^3S_1]_{6}\)  \([^1S_0]_{\bar{3}}\)  \([^1S_0]_{6}\)  \([^3S_1]_{\bar{3}}\)  \([^1S_0]_{6}\)  
\(m_b=4.7~\hbox {GeV}\)  0.69  0.41  4.94  2.47  4.11  2.07  0.46  0.33 
\(m_b=5.1~\hbox {GeV}\)  0.69  0.41  5.93  2.97  4.78  2.41  0.42  0.30 
\(m_b=5.5~\hbox {GeV}\)  0.69  0.41  7.02  3.51  5.51  2.78  0.38  0.28 
Concerning the discovery potential of these baryons at the HLLHC and CEPC/ILC, the possible decay channels of \(\Xi _{QQ'}\) is useful. Similar to the observation of \(\Xi _{cc}^{++}\) baryon, the \(\Xi _{bc}\) and \(\Xi _{bb}\) baryons could be observed by cascade decays such as \(\Xi _{bc}^{+}\rightarrow \Xi _{cc}^{++}(\rightarrow p K^\pi ^+ \pi ^+)\pi ^{}\) and \(\Xi ^{0}_{bb}\rightarrow \Xi _{bc}^+ (\rightarrow \Xi ^{++}\pi ^) \pi ^\). At present, many phenomenological models have been suggested to study the decay properties of the doubly heavy baryons, which are at the initial stage for the large nonperturbative effects. An overview of the doubly heavy baryons decay, together with the possibilities of observation may be found in Refs. [49, 50]. As for the detection efficiency in the experiment, the events cannot be 100\(\%\) detected. Compared to the \(\Xi _{cc}^{++}\) events detected by LHCb [12, 27, 28, 29], about \(\mathcal {O}(10)\) doubly heavy baryons \(\Xi _{QQ'}\) events from Higgs boson decays would be detected per year at the HLLHC and there would be \(\mathcal {O}(1)\) events which could be detected at the ILC and CEPC.
3.2 Theoretical uncertainties
In this subsection, the theoretical uncertainties for the production of \(\Xi _{QQ'}\) via the Higgs boson decays would be discussed. There are three main sources of the theoretical uncertainties: the quark mass, the renormalization scale \(\mu _r\) and the transition probability. The likely quark mass uncertainty covers \(m_c\) and \(m_b\) for building the mass of the corresponding doubly heavy baryons \(\Xi _{QQ^{\prime }}\). We shall analyze the caused quark mass uncertainties by varying \(m_c=1.8\pm 0.3~\mathrm{GeV}\) and \(m_b=5.1\pm 0.4~\mathrm{GeV}\), which are listed in Tables 5 and 6, respectively. It is worth mentioning that, discussing the uncertainty caused by one parameter, the others should be fixed to their central values. The decay widths through these four considered decay channels have been summed up for the total decay width. Tables 5 and 6 show that the decay width for the production of \(\Xi _{cc}\) (\(\Xi _{bb}\)) decreases with the increment of \(m_c\) (\(m_b\)) for a suppression of the phase space. Meanwhile, for the production of \(\Xi _{bc}\), the decay width decreases with the increment of \(m_c\) but increases with the increment of \(m_b\).
The theoretical uncertainties for the production of baryons \(\Xi _{QQ^{\prime }}\) via Higgs boson decays by substituting the renormalization scale \(\mu _r=2m_c\), \(\mathrm{M}_{bc}\) or \(2m_b\). The units of the decay widths are \((\times 10^{7}~\hbox {GeV}\))
\(\mu _r\)  \(2m_c\)  \(~\mathrm{M}_{bc}~\)  2\(m_b\) 

\(\alpha _s\)  0.242  0.198  0.180 
\(\Gamma _{\Xi _{cc}[^3 S_1]_{\bar{3}}}\)  0.69  0.45  0.37 
\(\Gamma _{\Xi _{cc}[^1 S_0]_{6}}\)  0.41  0.27  0.22 
\(\Gamma _{\Xi _{bc}[^3 S_1]_{\bar{3}}}\)  5.93  3.94  3.24 
\(\Gamma _{\Xi _{bc}[^3 S_1]_{6}} \)  2.97  1.97  1.63 
\(\Gamma _{\Xi _{bc}[^1 S_0]_{\bar{3}}}\)  4.78  3.16  2.59 
\(\Gamma _{\Xi _{bc}[^1 S_0]_{6}}\)  2.41  1.60  1.31 
\(\Gamma _{\Xi _{bb}[^3 S_1]_{\bar{3}}}\)  0.77  0.51  0.42 
\(\Gamma _{\Xi _{bb}[^1 S_0]_{6}}\)  0.57  0.37  0.30 
Finally, the theoretical uncertainty caused by the nonperturbative transition probability is considered. Considering that the transition probability is proportional to the decay width, its uncertainty can be conventionally obtained when we know its exact value. Throughout the paper, the transition probability of the colorantitriplet diquark state \(\langle QQ^{\prime }\rangle _{\bar{3}}\) and the colorsextuplet diquark state \(\langle QQ^{\prime }\rangle _6\) to the heavy baryon \(\Xi _{QQ^{\prime }}\) have been considered as the same, i.e., \(h_{6} \simeq h_{{\bar{3}}}=\Psi _{QQ^{\prime }}(0)^2\) [18, 43], where the wave functions at the origin \(\Psi _{QQ^{\prime }}(0)\) are derived from the powerlaw potential model. However, there is a larger uncertainty for \(h_{6}\) than for \(h_{{\bar{3}}}\). Within the framework of NRQCD, the intermediate diquark state \(\langle QQ^{\prime }\rangle [n]\) can be expanded into a series of Fock states with the relative velocity (v) and, according to the NRQCD power counting rule, each Fock state is of the same importance, which is the main reason why we took \(h_{6} \simeq h_{{\bar{3}}}\). In addition to this point of view, there is another point of view: namely, that the colorsextuplet state would be suppressed by \(v^2\) compared to the colorantitriplet state, i.e., \(h_{6}/v^2 \simeq h_{{\bar{3}}}=\Psi _{QQ^{\prime }}(0)^2\). Even if the contribution of the colorsextuplet diquark \((QQ^{\prime })_6\) state can be ignored (\(h_{6}=0\)) and only the colorantitriplet diquark \((QQ^{\prime })_{\bar{3}}\) state is taken into consideration (\(h_{{\bar{3}}}=\Psi _{QQ^{\prime }}(0)^2\)), there are still 0.27\(\times 10^{4}\) events of \(\Xi _{cc}\), 4.21 \(\times 10^{4}\) events of \(\Xi _{bc}\) and 0.17 \(\times 10^{4}\) events of \(\Xi _{bb}\) produced per year at the HLLHC. However, there are fewer events produced at the CEPC/ILC, only 0.16\(\times 10^{2}\) events of \(\Xi _{cc}\), 2.55 \(\times 10^{2}\) events of \(\Xi _{bc}\) and 0.10 \(\times 10^{2}\) events of \(\Xi _{bb}\).
4 Summary
Notes
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (nos. 11375008, 11647307, 11625520, 11847301). This research was also supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).
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