Investigation of Z-boson decay into and baryons within the NRQCD factorization approach

Z-boson decay provides a good opportunity to search for the baryon because a large number of its events can be collected at high energy colliders. In this paper, we perform a complete investigation of the indirect production of the and baryons via the Z-boson decay process with a quark under the NRQCD factorization approach. After considering the contribution from the diquark states , , , and , the calculated branching fractions are and . Furthermore, the and production events are predicted to be of the order of and at the LHC collider, and the order of and for the CEPC collider. We then estimate the production ratio for with in Z-boson decay, that is, and , respectively. Finally, we present the differential decay widths of and with respect to and z distributions and analyze the uncertainties.


I. INTRODUCTION
Doubly heavy baryons composed by two heavy quarks and one light quark are expected by the quark model [1][2][3][4]. Investigating doubly heavy baryons is significant because it can provide a unique test for perturbative quantum chromodynamics (QCD) and nonrelativistic QCD (NRQCD). In the past few decades, research on doubly heavy baryons has developed rapidly from both the experimental and theoretical perspectives. Experimentally, the doubly charmed baryon was first observed by the LHCb collaboration based on the decay channel [5]. It was subsequently identified via the measurement of by the LHCb collaboration [6,7]. Moreover, the first observation of the doubly charmed baryon was reported from by the SELEX collaboration. Over the past few years, the LHCb collaboration has published their observation of , which is defined as , varying in the region for , and for [8]. These values are lower than the observations ( ) of the SELEX collabor-ation.
has also attracted the attention of researchers owing to its unique nature in the baryon family. In 2020, the LHCb collaboration searched for the doubly heavy baryon via its decay into the final state, although no direct evidence was found [9]. Recently, and were detected via the and decay modes; however, evidence of the signal was not found [10]. is yet to be detected. Overall, there is still no solid signal of the baryon with the heavy quark . To investigate the baryon production properties and further test the NRQCD method, considerable work has been conducted on both direct and indirect production . Compared with direct production, such as hadroproduction, photoproduction, and annihilation, indirect production is also important because of the properties of baryons and their initial particles. baryons can be produced via different channels, such as top quark decay [44], -boson decay [45], and the process [46]. Other than the above channels, they can also be produced from Z-boson decays, for example, the process [47]. events can reach and per year at the LHC and CEPC via Z-boson decays. Mean- while, the branching fraction is also comparable with [48,49]. Ξ bQ 10 9 10 12 Thus, Z-boson decay can provide a good platform to study the baryon based on the large quantity of Z-boson events. Up to and -order Z-boson events are produced per year at the LHC [50] and CEPC [51] colliders. Moreover, the decay channel has advantages over , which offers a new experimental direction in the search for [52]. In this paper, we first focus our attention on the indirect production of via Z-boson decay and then reveal whether a considerable amount of can be collected by Z-boson decay. In addition, we forecast in Z-boson decay using the channel . The rest of the paper is organized as follows: The detailed method is demonstrated in Sec. II, the phenomenological results and analyses are given in Sec. III, and a brief summary is given in Sec. IV.

II. CALCULATION TECHNOLOGY
Normally, production of the baryon can be treated in two steps [14,37,43,53]. The first is by producing a bound state, which is also called a diquark , with representing the color-and spin-combinations. Based on the decomposition in the SU group and NRQCD, the quantum color number is only the color-antitriplet and color-sextuplet , and the quantum counts of the diquark state are and . The second step involves turning the diquark fragments into an observable baryon by hunting a light quark from the 'environment' with a fragmentation probability of almost one hundred percent. For convenience, we utilize the label instead of throughout this paper. Among this total " " fragmentation probability, the probability of both and is , and the ratio for is [40,54].
The diagrams for the process at tree level are shown in Fig. 1, where the heavy quark taken as and b represents and , respectively. We can obtain the differential decay width of the process using the NRQCD factorization approach [55,56], Here, the long-distance matrix element describes the hadronization of the diquark state into the doubly heavy baryon . Generally, can be approximately obtained from the original value of the Schrödinger wave function or its derivative. In this paper, we take for the S-wave and P-wave, which are derived from experimental data and non-perturbative theoretical methods, for example, the potential model, lattice QCD, and QCD sum rules [56][57][58].
The differential decay width can be written as where and are the Z-boson mass and hard amplitude, respectively. The constant arises from the spin average of the initial Z-boson, and the symbol " " represents the sum of color and spin for all final particles. The three-body phase space with a massive quark or antiquark in the final state can be written as The calculation for the three-body phase space has been discussed in Refs. [59,60]. Then, Eq. (2) can be rewritten as with . After using charge parity , hard amplitude expressions for baryon production are obtained, which can also be easily gained from the familiar meson production [26,39]. Here, we give brief descriptions.
can be used to reverse one fermion line, which can be written as , where , , , and are the interaction vertex, fermion propagator, spin index, and quantity of interaction vertices in the fermion line, respectively. According to charge parity, we have If the fermion line does not include an axial vector vertex, we can readily obtain the expression Otherwise, the amplitudes of baryon production can be obtained from familiar meson production, except with an additional coefficient for the pure vector case and a factor when including an axial vector case. In other words, the amplitude of can be written as where is the hard amplitude of familiar meson production, and and are the components of the axial vector amplitudes and pure vector amplitudes of , respectively.
Taking the traditional Feynman rules of Fig. 1 into considersion, we can obtain the amplitudes with , which have the following expressions: Here, , with color factor . and are the momenta of the bottom quark and another heavy quark for and production. The vector and axial vector coupling constants of the vertex, that is, and , have the following expressions: Here, is the Weinberg angle. With the help of Eq. (6) and inserting the spin projector , the amplitude can be rewritten as Investigation Z-boson decay into and baryon within NRQCD factorization approach Chin. Phys. C 47, 053102 (2023) 053102-3 , . (10) Here, the spin projector has the following form [61]: Furthermore, the color factor can be easily obtained from Fig. 1, which has the following form: Here, k, , , and represent the diquark color indices, gluon color indices, normalization factor, and two outgoing antiquarks and two constituent quarks in the diquark color indices, respectively. For the state, the function is identical to the antisymmetric function and symmetric function , which obey For color and diquark state production, and , respectively.
Meanwhile, diquark hadronization into a doubly heavy baryon is a non-perturbative procedure, which is factorized into a general coefficient . This coefficient is connected to the wave function at the origin. In this paper, we take the usual assumption that the wave3 6 function of the color state is equal to that of the color state, as discussed in Refs. [26,39,44,45,53] 1) .
The transition probabilities of the color and states are represented by and , respectively. Based on the NRQCD approach, a bound state of two heavy quarks with another light dynamical freedom of QCD can be described by a series of Fock states, where v denotes the small relative velocity between heavy quarks in the rest frame of the diquark. For the color state cases, one of the heavy quarks of the diquark can produce a gluon without altering its spin, which can divide into a light quark pair . Then, the diquark can capture a light quark q to construct a baryon. Regarding the color state, if the baryon is created by , the emitted gluon would alter the spin of the heavy quark, causing suppression of . If the baryon is created from the parts, one of the heavy quarks produces a gluon without altering the spin of the heavy quark. Then, the gluon separates into . Additionally, a light quark q has the ability to produce gluons, which can be used to construct the component with . These contributions are at the same level because a light quark may produce gluons easily, that is, [53]. We can then take the following approximation: for the S-wave, and for P-wave, Xuan Luo, Hai-Bing Fu, Hai-Jiang Tian Chin. Phys. C 47, 053102 (2023) 3 3 1) As reported in Ref. [39], the color 6 state is suppressed to the color state by order v 2 and its contribution can be disregarded, which can be ascribed to the onegluon exchange interaction causes the interaction inside the diquark with the color 6 state to be repulsive rather than attractive. Meanwhile, as pointed out in Refs. [26,44,45,53], the importance of the color 6 and the color states are equal.

III. NUMERICAL RESULTS
c, b m c = 1.8 GeV To perform the numerical calculation, the following input parameters are taken: The -quark masses are and , respectively. The Z-boson mass 91.1876 GeV and the decay width are from the PDG [62]. For the values of and , we adopt and [15], respectively. The masses of the and baryons are taken as and , respectively. The remaining input parameters are [62] , which denotes the Fermi constant, and the Weinberg angle . The renormalization scale is taken as for the indirect production of .
First, the decay widths of two main Z-boson decay channels for production are given in Table 1. From Table 1, we can see that the state plays a leading role in the production of . The contribution from the state can reach twice that from the state. As for production, the situation is analogous to that of . Moreover, in the case of , the contribution from is significantly smaller than that from , which is only a few percent. To assess the doubly heavy baryon events generated at the LHC (CEPC), the corresponding branching ratio must be obtained from the Z-boson total decay width. At the LHC (CEPC), approximately -bosons can be produced per year [51,63]. Based on the conditions mentioned above, the produced events of the double heavy baryon can be predicted at the LHC (CEPC). We list the total decay width, branching ratios, and events of the and baryons via Z-boson decay in Table 2, where contributions from each diquark state of the Z-boson decay channel are considered for the total decay width. From Table 2, we can reach the following conclusions: • For the production of , the branching ratio is approximately , which is comparable to the results given in Ref. [64].
• The branching ratio of reaches for the production of , which is also comparable to the predictions of [65].  events and order events produced at the LHC. However, the upgraded program of the HE(L)-LHC will significantly improve Z-boson yield events; thus, more and events will be produced.
• Considering the decay rate of the channels [52], [66], and [67], approximately reconstructed events can be collected at the CEPC. These events are comparable to [47], which proves the observability of via Z-boson decay.
Furthermore, the ratio of the production rate , which arises from Z-boson decay to accompanied by , has the following formula: where X denotes all possible particles. First, we use the formula to obtain . This is based on the total decay width, which can be directly related to the branching fraction. The branching fractions of are taken from the PDG, that is, and [68]. The fragmentation fractions of a heavy quark to a particular charmed hadron are and 0.073 [69]. Then, we have Ξ +,(0) bc → Ξ ++ cc + X ≃ 7%(1.5%) Second, according to the decay chains of [52], [66], and [67], we can get the final results Table 1. Predicted decay widths (unit: GeV) with for the and baryons from each Z-boson decay channel.  Fig. 2. Here, the renormalization scale is set to . is one magnitude larger than , which indicates that the decay channel provides key contributions compared with the channel for indirect production. Comparing the predictions of in this study with from our previous work [47] for Z-boson decay, there is a large gap between and of approximately one magnitude. This discrepancy indicates that it will be difficult to collect in experiment collaborations. Moreover, our predictions for via the channel reach the order of in Z-boson decay, which is larger than those for the channel (of the order) [70]. Thus, the observation of via the chan- nel is more feasible than that via the channel.
To further study the production of via these considered decay channels and provide a reference for experimental research, we present the differential decay widths of with respect to the invariant mass and energy fraction z in Figs. 3 and 4, where , and , using the energy and Z-boson energy .  Fig. 3, we find that the state plays the leading role in the cases of and production. The curves have a similar behavior, which increase initially and then decrease with , with the peak located in the small region of .
• As shown in Fig. 4, the behavior of the differential decay widths changes with the energy fraction z-distribution, that is, is similar to , which increases initially and then decreases. In the case of production, the peak of is approximately , and peaks near . As for , the peak of is approximately , and peaks near . Owing to the dominant effect of the quark fragmentation mechanism, the peaks of the differential decay widths for with energy distribution are located in the larger z-region.
Finally, to discuss the theoretical uncertainties for the process precisely, -quark masses of and , and the renormalization scale for and for can be considered. Here, the uncertainties from are not discussed; they are an overall coefficient in calculations and can be computed out easily. The total decay widths within uncertainties arising from the above input parameters are presented in Table 3, which shows that  indicates that the color quantum number is of the diquark state, and "Total" denotes the total decay widths, which means that each diquark state has been summed.   indicates that the color quantum number is of the diquark state, and "Total" denotes the total decay widths, which means that each diquark state has been summed. • For indirect production in Z-boson decay, the decay width decreases as the c-quark mass increases, which is mainly ascribed to the suppression of phase space.
• Owing to the effect of the projector in Eq. (11), an abnormal phenomenon occurs in which the decay width increases as the b-quark mass increases for the state in indirect production via Z-boson decay. The decay width decreases as the b-quark mass increases. Moreover, the uncertainty caused by is larger than that of .
• For the process with the and cases, the decay width decreases as the b-quark mass increases. In this study, we discuss in detail the indirect production of and via Z-boson decay based on the framework of NRQCD. After considering the contributions from the intermediate diquark states, that is, , , , and , the branching ratio is approximately of the order of , and is of the order. There will be events and events produced at the LHC (CEPC). Then, the change in the differential decay widths of and with and z R(Z → Ξ +,0 bc + X)

IV. SUMMARY
In conclusion, at present, studies on the decay properties of doubly heavy baryons are discussed via their decay models. Especially in LHCb collaboration reports [71,72] and theoretical results from Qin [52], decay models and their observational possibilities have been discussed. Inspired by the observation of the doublycharmed baryon , the baryon may be detected by the decay channels , where X represents all possible particles. The advantage of this approach in detecting is that the detection efficiency will be greatly improved because only needs to be reconstructed, as discussed in Ref. [52]. Similar to the baryon, may also be observed via .