Production of doubly heavy baryons via Higgs boson decays

We systematically analyzed the production of semi-inclusive 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 non-relativistic QCD. The contributions from the intermediate diquark states, $\langle cc\rangle[^{1}S_{0}]_{\mathbf{6}}$, $\langle cc\rangle[^{3}S_{1}]_{\mathbf{\bar 3}}$, $\langle bc\rangle[^{3}S_{1}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bc\rangle[^{1}S_{0}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bb\rangle[^{1}S_{0}]_{\mathbf{6}}$ and $\langle bb\rangle[^{3}S_{1}]_{\mathbf{\bar 3}}$ 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$\times10^4$ events of $\Xi_{cc}$, 6.32$\times10^4$ events of $\Xi_{bc}$ and 0.28$\times10^4$ 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.


I. INTRODUCTION
The Higgs boson, as the last found fundamental particle in the standard model (SM), is of great interest to the experimenter and theorist of particle physics. Some future colliders that can be called "Higgs factory" would generate large amounts of Higgs particles. The High Luminosity LHC (HL-LHC) running at center-of-mass collision energy √ s = 14 TeV with the integrated luminosity of 3 ab −1 would produce about 1.65 × 10 8 events of Higgs boson per year [1]; the Circular Electron Positron Collider (CEPC) would generate more than one million Higgs particles at the center-of-mass energy of 240 GeV with the integrated luminosity of 0.8 ab −1 in 7 years [2]; and the International Linear Collider (ILC) would generate almost the same magnitude of Higgs bosons as the CEPC, about 10 5 to 10 6 at each energy stage [3]. Therefore, the decay of Higgs boson will be a potentially good platform for studying the indirect production mechanism of doubly heavy hadrons. Many pioneer investigations about the production of doubly heavy meson through Higgs boson's decay have been done not only by the experimental groups but also by the theorist, i.e., the production of B c , J/ψ and Υ [4][5][6][7][8][9][10]. And the analysis of Higgs boson decays also provides a platform for seeking the undetected doubly heavy baryons. The doubly heavy baryon contains two heavy quarks and a light quark as valence quarks. For convenience, Ξ QQ ′ is used to stand for the doubly heavy baryons Ξ QQ ′ q l in this paper, where Q and Q ′ represent the heavy quarks (c or b quark) and q l denotes as the light quark (u, d or s quark). A careful study on the production of doubly heavy baryons Ξ QQ ′ through Higgs boson decays shall be helpful for confirming whether enough baryon events could be produced and supporting forward guidance on the experiment research.
In this paper, we shall discuss the production of doubly heavy baryons Ξ QQ ′ through indirectly Higgs boson decays at the HL-LHC and CEPC/ILC. As is well-known, the dominant decay channel of Higgs boson is H 0 → bb and the branching ratio is about 58% [38,39]. For completeness, four main Higgs decay channels, H 0 → bb, cc, Z 0 Z 0 , gg, would be taken into consideration. Due to the Yukawa coupling and the pertubative order, the decay channel H 0 → bb (H 0 → cc) plays an essential role in the production of Ξ bc and Ξ bb (Ξ cc ), but the contributions from the H 0 → Z 0 Z 0 /gg channel can't be neglected.
Within the framework of NRQCD, the production of doubly heavy baryons Ξ QQ ′ can be factorized into the convolution of the perturbative short-distance coefficient and the nonperturbative long-distance 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 long-distance matrix elements are used to describe the transition probability of the produced diquark state QQ , and the color quantum number is the color-antitriplet3 or the color-sextuplet 6 for the decomposition of SU C (3) color group 3 3 =3 6. All of these intermediate states would be taken into consideration for a sound estimation. Assuming the potential of the binding colorantitriplet QQ ′ [n] state is hydrogen-like, the transition probability h3 can be approximatively related to the Schrödinger wave function at the origin |Ψ QQ ′ (0)| for the S-wave states, where |Ψ QQ ′ (0)| can be obtained by fitting the experimental data or some non-perturbative methods like QCD sum rules [40], lattice QCD [41] or the potential model [42]. As for the transition probability of the color-sextuplet 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 Sec. II, 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 Sec. III, and Sec. IV gives a summary and some conclusions.
Typical Feynman diagrams for the process H 0 (p 0 ) → Ξ QQ ′ (p 1 ) +Q ′ (p 2 ) +Q(p 3 ) through four main Higgs decay channels, H 0 → bb/cc/Z 0 Z 0 /gg, are presented in Fig. 1, where Q and Q ′ denote as the heavy c or b quark for the production of Ξ cc , Ξ bc and Ξ bb accordingly. Within the framework of NRQCD [16,17], the decay width for the production of Ξ QQ ′ can be factorized as the following form:  6 and bc [ 1 S 0 ] 6 . All of these Fock states would be taken into consideration for a comprehensive understanding. We shall use h3 and h 6 to describe the transition probability of the color-antitriplet diquark state and the color-sextuplet diquark state, respectively. In addition, the transition probability h3 can be approximatively related to the Schrödinger wave function at the origin |Ψ QQ ′ (0)| for the S-wave states, while there is a relatively larger uncertainty for the transition probability h 6 , which has been analyzed detailedly in Ref. [37]. For convenience, we set h 6 ≃ h3 = |Ψ QQ ′ (0)| 2 [17,42] as an approximate estimate. The decay widthΓ(H 0 → QQ ′ [n]+Q ′ +Q) represents for the perturbative short-distance coefficients which can be written aŝ where m H is the mass of the Higgs boson, M[n] is the hard amplitude, and means to sum over the spin and color of the final-state particles. The three-body phase space dΦ 3 can be expressed as After performing the integration over the phase space dΦ 3 , Eq. (2) can be rewritten as where the definitions of the invariant mass are s ij = (p i + p j ) 2 , (i, j = 1, 2, 3). Therefore, not only the total decay width but also the corresponding differential distributions can be derived, which are helpful for experimental measurements.

A. Amplitude
We made a relatively complete analysis for the production of doubly heavy baryons Ξ QQ ′ through four main Higgs decay channels, H 0 → bb, cc, Z 0 Z 0 , and gg. Subgraphs (a)-(d) in Fig. 1 are specifically represented the channels In subgraphs (f)-(g), q stands for t, b or c quark. According to the Yukawa coupling, the top quark in the triangle loop can make the largest contribution to the decay width through H 0 → gg. After the action of charge parity C = −iγ 2 γ 5 , the hard amplitude M[n] for the production of the intermediate diquark state can be related to the familiar meson production, which has been proved in Ref. [20,21] detailedly. In other words, we could obtain the hard amplitude M[n] of the process H 0 (p 0 ) → QQ ′ [n](p 1 ) +Q ′ (p 2 ) +Q(p 3 ) from the process H 0 (p 0 ) → (QQ ′ )[n](p 1 ) + Q ′ (p 2 ) +Q(p 3 ) with an additional factor (−1) m+1 , where m stands for the number of vector vertexes in the Q ′ fermion line which need to be reversed and here m = 1 for these four decay channels.
It's worth mentioning that there are vector and axial vector contributions in the channel H 0 → Z 0 Z 0 , and m = 0 for the axial vector contribution. Additional, the Z 0 propagators should be considered as Breit-Wigner propagators to avoid the resonance.
According to Fig. 1, the hard amplitude M l [n] (l = a, . . . , g) can be written as in which C stands for the color factor C ij,k , which would be described in detail in subsection II B; θ W is the Weinberg angle; the projector Π p 1 [n] has the form of [43] where is adopted to ensure the gauge invariance; p 11 and p 12 are the specific momentum of these two constituent quarks of the diquark state: where p is the relative momentum between these two constituent quarks and it is small enough to be neglected in the amplitude of S-wave state for the non-relativistic approximation. In Eq. (9), Γ ZQQ and Γ ZQ ′Q′ stand for the vertex of Z 0 boson with quark-antiquark pairs. Because the couplings for Z 0 boson with bb and cc pair are different, we don't state it definitely for the production of Ξ cc , Ξ bc and Ξ bb . In Eqs. (10) and (11), l is the loop momentum that needs to be integrated.

B. Color factor
Given the different topologies in Fig. 1, four considered channels have different color structures, and we would like to take the channel H 0 → QQ as an example to explain how the color factor C ij,k is calculated where i, j, m, n = 1, 2, 3 are the color indices of the outgoing antiquarksQ ′ ,Q and the two constituent quarks Q and Q ′ of the diquark respectively; a = 1, . . . , 8 and k denote as the color indices of the gluon and the diquark state QQ ′ [n]; the normalization constant N = 1/2. For the color-antitriplet3 state, the function G mnk is equal to the anti symmetric function ε mnk , while it will be the symmetric function f mnk for the color-sextuplet 6 state. And the function ε mnk and f mnk satisfy After squared the amplitude through H 0 → QQ, the final color factor C 2 ij,k equals 4 3 for the production of color-antitriplet diquark state and 2 3 for the color-sextuplet diquark state. Due to the different color matrices in subgraphs of Fig. 1, the explicit color factors C 2 ij,k accompanied by the other two different channels are listed in Table I. Cross term 1 stands for the cross term between H 0 → QQ/Q ′Q′ and H 0 → Z 0 Z 0 ; Cross term 2 is the cross term between H 0 → QQ/Q ′Q′ and H 0 → gg.

III. NUMERICAL RESULTS
In numerical calculation, the input parameters are taken as follows [24,38]: where the quark masses and wave functions are consistent with Ref. [24] and the others can be obtained from the PDG [38]. We use FeynArts 3.9 [44] to generate the amplitudes and the modified FormCalc 7.3/Loop-Tools 2.1 [45] to do the algebraic and numerical calculations. The renormalization scale µ r is set to be 2m c , 2m c and 2m b for the production of Ξ cc , Ξ bc and Ξ bb correspondingly. Due to the total decay width of Higgs boson has not been detected so accurately by the experiment, we consider the total decay width of Higgs boson as 4.2 MeV [46] to estimate the branching ratio and corresponding events for the production of baryons Ξ cc , Ξ bc and Ξ bb .

A. Basic results
Table II. The decay widths for the process H 0 → bb/cc/Z 0 Z 0 /gg → Ξ QQ ′ +Q ′ +Q, where Q and Q ′ denote as the heavy c or b quark. Cross term 1 stands for the cross term between H 0 → QQ/Q ′Q′ and H 0 → Z 0 Z 0 , and Cross term 2 is the cross term between H 0 → QQ/Q ′Q′ and H 0 → gg. Based on the parameters mentioned before, four main Higgs decay channels for the production of Ξ QQ ′ have been analyzed carefully, and the decay width of each channel is presented in Table II. From Table II, we can find that • The biggest decay channel for the production of Ξ cc (Ξ bb ) is H 0 → cc (H 0 → bb). While for the production of Ξ bc , the decay width in each diquark state through H 0 → bb is about two orders of magnitude larger than that through H 0 → cc mainly for the Yukawa coupling.
• From the decay widths through H 0 → QQ/Q ′Q′ , it can be seen that the contribution of the cross term between H 0 → bb and H 0 → cc is positive for [ 1 S 0 ]3 /6 states and negative for [ 3 S 1 ]3 /6 states.
• The decay widths through H 0 → Z 0 Z 0 /gg channels are very small and only a few percent compared to that through H 0 → QQ/Q ′Q′ .
• The contributions of the cross term between H 0 → QQ/Q ′Q′ and H 0 → V V (V = Z 0 , g) should also be taken into account and the decay width for the production of baryons Ξ QQ ′ from these two cross terms are also listed in Table II.
To estimate the events of doubly heavy baryons produced at the "Higgs factory", the total decay width of Higgs boson is needed to obtain the branching ratio correspondingly. But so far, the total decay width of Higgs boson could not be measured so accurately by the experiment and was only given an upper limit of 13 MeV [47]. Here the total decay width of Higgs boson is considered as 4.2 MeV as suggested by Ref. [46]. Running at √ s = 14 TeV with the integrated luminosity of 3 ab −1 , HL-LHC could produce 1.65 × 10 8 Higgs bosons per year [1]; The CEPC, the same as the ILC, would generate more than one million Higgs particles mainly depending on the energy and integrated luminosity in operation [2,3].
With these conditions, we can estimate the produced events of Ξ QQ ′ at the HL-LHC and the CEPC/ILC respectively. The decay width and the estimated events through these four Higgs decay channels are showed in Table III. By summing up the contribution from each intermediate diquark state, the total decay width, the branching ratio and the corresponding estimated events of the doubly heavy baryons Ξ QQ ′ could be obtained, which are given in Table IV.    Table III and IV show that • The estimated events of Ξ bc are about one order of magnitude larger than that of Ξ cc and Ξ bb .
• The branching ratio via Higgs boson decays is about 10 −4 for the production of Ξ bc baryon, and 10 −5 for the production of Ξ cc and Ξ bb baryons.
• At the HL-LHC, there are sizable events of doubly heavy baryons Ξ QQ ′ , at the order of 10 4 , produced per year.
• There are only about 10 2 Ξ 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 as the HL-LHC, there would be 3.75 times the Ξ QQ ′ events.
To make a clear analysis of the distributions for the production of Ξ QQ ′ through these four considered channels and be helpful to the experimental detection, the invariant mass differential decay widths dΓ/ds ij and the angular differential distributions dΓ/dcosθ ij are plotted in Figs. 2 and 3, where the invariant mass s ij = (p i + p j ) 2 and θ ij is the angle between momentum − → p i and − → p j in the Higgs boson rest frame. All the possible spin and color configurations have been taken into consideration, i.e., cc Figs. 2 and 3 show that the behaviors of the differential distributions for the production of baryon Ξ bc are different with that for the production of baryons Ξ cc and Ξ bb . Fig. 2 (a) shows that as s 12 gets smaller and smaller, i.e., p 1 and p 2 are collinear, there is a maximal value of dΓ/ds 12 . From Figs. 3 (a) and (b), one can find that for the production of Ξ bc , dΓ/dcosθ 12 (dΓ/dcosθ 13 ) can achieve to be the largest when cos θ 12 = 1 (cos θ 13 = −1), i.e., the doubly heavy baryons Ξ QQ ′ and the heavy quarkQ ′ (Q) move side by side (back-toback). Fig. 3 (c) illustrates this fact again. While for the production of Ξ cc and Ξ bb , there are similar kinematic behaviors for (a) and (b) in Figs. 2 and 3 for the identical particles in the diquark state.  Concerning the discovery potential of these baryons at the HL-LHC and CEPC/ILC, the possible decay channels of Ξ QQ ′ is useful. Similar to the observation of Ξ ++ cc baryon, the Ξ bc and Ξ bb baryons could be observed by cascade decays such as Ξ + bc → Ξ ++ cc (→ pK − π + π + )π − and Ξ 0 bb → Ξ + bc (→ Ξ ++ π − )π − . 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 non-perturbative effects. An overview of the doubly heavy baryons decay, together with the possibilities of observation could be found in Refs. [48,49]. As for the detection efficiency in the experiment, the events can't be 100% detected. Compared to the Ξ ++ cc events detected by LHCb [11,[26][27][28], about O(10) doubly heavy baryons Ξ QQ ′ events from Higgs boson decays would be detected per year at the HL-LHC and there would be O(1) events which could be detected at the ILC and CEPC.

B. Theoretical uncertainties
In this subsection, the theoretical uncertainties for the production of Ξ QQ ′ via Higgs boson decays would be discussed. There are three main sources of the theoretical uncertainties: the quark mass, the renormalization scale µ 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 Ξ QQ ′ . We shall analyze the caused quark mass uncertainties by varying m c = 1.8±0.3 GeV and m b = 5.1±0.4 GeV, which are listed in Table V and VI respectively. It is worth mentioning that when 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. Table V and VI show that the decay width for the production of Ξ cc (Ξ bb ) decreases with the increment of m c (m b ) for a suppression of the phase space. While for the production of Ξ bc , the decay width decreases with the increment of m c but increases with the increment of m b .   Due to the QCD running coupling, the renormalization scale µ r would make a significant contribution to the decay width. And we could obtain the uncertainties by substituting three different renormalization scales, i.e., µ r = 2m c , M bc or 2m b , which are presented in Table VII. As a supplement, the QCD running coupling α s (µ r ) is also added into Table VII. In fact, such scale ambiguity could be suppressed by a higher-order perturbative calculation or proper scale-setting methods such as the Principle of Maximum Conformal (PMC) [50][51][52][53]. Finally the theoretical uncertainty caused by the non-perturbative 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 color-antitriplet diquark state QQ ′ 3 and colorsextuplet diquark state QQ ′ 6 to the heavy baryon Ξ QQ ′ have been considered as the same, i.e., h 6 ≃ h3 = |Ψ QQ ′ (0)| 2 [17,42], where the wave functions at the origin |Ψ QQ ′ (0)| are derived from the power-low potential model. However there is a larger uncertainty for h 6 compared to h3. Within the framework of NRQCD, the intermediate diquark state QQ ′ [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 ≃ h3. In addition to this view, there is another view that the color-sextuplet state would be suppressed by v 2 compared to the color-antitriplet state, i.e., h 6 /v 2 ≃ h3 = |Ψ QQ ′ (0)| 2 . Even if the contribution of color-sextuplet diquark (QQ ′ ) 6 state can be ignored (h 6 = 0) and only the color-antitriplet diquark (QQ ′ )3 state is taken into consideration (h3 = |Ψ QQ ′ (0)| 2 ), there are still 0.27×10 4 events of Ξ cc , 4.21 ×10 4 events of Ξ bc and 0.17 ×10 4 events of Ξ bb produced per year at the HL-LHC. However, there are fewer events produced at the CEPC/ILC, only 0.16×10 2 events of Ξ cc , 2.55 ×10 2 events of Ξ bc and 0.10 ×10 2 events of Ξ bb .