Further study on the production of P-wave doubly heavy baryons from Z-boson decays

In this paper, we carried out a systematic investigation for the excited doubly heavy baryons production in $Z$-boson decays within the NRQCD factorization approach. Our investigation accounts for all the $P$-wave intermediate diquark states, {\it i.e.} $\langle cc\rangle[^1P_1]_{\bar 3}$, $\langle cc\rangle[^3P_J]_{6}$, $\langle bc\rangle[^1P_1]_{\bar 3/6}$, $\langle bc\rangle[^3P_J]_{\bar 3/6}$, $\langle bb\rangle[^1P_1]_{\bar 3}$, and $\langle bb\rangle[^3P_J]_{6}$ with $J = (0, 1, 2)$. The results show that contributions from all diquark states in $P$-wave were $7\%$, $8\%$, and $3\%$ in comparing with $S$-wave for the production of $\Xi_{cc}$, $\Xi_{bc}$ and $\Xi_{bb}$ via $Z$-boson decay, respectively. Based on these results, we predicted about $0.539\times 10^3(10^6)$ events for $\Xi_{cc}$, $1.827\times 10^3(10^6)$ events for $\Xi_{bc}$, and $0.036\times 10^3(10^6)$ events for $\Xi_{bb}$ can be produced annually at the LHC (CEPC). Additionally, we plot the differential decay widths of $\Xi_{cc}$, $\Xi_{bc}$ and $\Xi_{bb}$ as a function of the invariant mass $s_{23}$ and energy function $z$ distributions, and analyze the theoretical uncertainties in decay width arising from the mass parameters of heavy quark.


I. INTRODUCTION
The doubly heavy baryons, contain two heavy quarks and one light quark, are predicted by the quark model theoretically [1][2][3][4].Investigation for the properties of doubly heavy baryons has been an hot topic from last century to the present.Therefore, a thorough study on them is essential for the particle physics.In comparing with other baryons that contain only one or none heavy quark, doubly heavy baryons involve more energy scales such as the heavy quark mass and nonperturbative QCD scale Λ QCD , which lead to more complex dynamics and offer a novel/distinctive platform for investigating strong interactions.
At the same time, theorists have also devoted much spirit to investigating the properties of doubly heavy baryon production.There are numerous studies in the list of literature that focus on the direct production of them .Meanwhile, the indirect production via top quark, Higgs boson, W -boson, and Z-boson decays is theoretically researched in Refs.[51][52][53][54][55][56][57].In which, the process of doubly heavy baryons via W -boson, top quark, and Higgs boson decays are systematically investigated within NRQCD factorization approach by Zheng and Ma [55][56][57].They have explicitly distinguishing the P -wave diquark state's contributions with S-wave ones.Based on the annual production of W -boson and Higgs boson events at the LHC and Circular Electron-Positron Collider (CEPC), the contributions from P -wave diquark states will take the non-negligible contributions in comparing with S-wave diquark states, which can predict a considerable number of doubly heavy baryons events from the excited states.Therefore, one can further illustrate that the P -wave diquark states contributions are significant in detailed calculations.
The doubly heavy baryon production in Z-boson decay provides a good chance to study its relevant mechanisms.On one hand, a large number of Z-boson events can be produced at the LHC (∼ 10 9 per year [62]), and the accumulated Z-boson events will greatly be improved due the increased collision energy (luminosity) at the upgrades of the HE(L)-LHC.On the other hand, the Z-boson events can be produced up to 10 12 -order level per year at the CEPC [59].The large number of Z-boson events that can be generated by the LHC (CEPC) will provides an ideal platform for collecting doubly heavy baryons produced by Z-boson decay, even the excited doubly heavy baryons.Moreover, the research on doubly excited doubly baryons can be considered as a complement to its ground-state, which significantly inspired us to make a detailed research about it.Previously, we make a detailed discussion about the intermediate diquark states in the Swave contributions, which are based on the production of doubly heavy baryons by Z-boson decay [53,54].In this paper, we will study the indirect production of excited doubly heavy baryons via Z-boson decay and present an overall analysis of the P -wave contributions in comparison with S-wave.
The remaining parts of the paper are organized as follows: The detailed calculation technology is demonstrated in Sec.II.Section III presents the phenomenological results and analyses.Finally, a brief summary is provided in Sec.IV.

II. CALCULATION TECHNOLOGY
According to the widely accepted hadron production mechanism for doubly heavy baryons Ξ QQ ′ , one can utilize the following two steps to elucidate Ξ QQ ′ production [21,44,50,63].The first step: When utilizing the decomposition of SU c (3) color group 3 ⊗ 3 = 3 ⊕ 6, the diquark state with possible color quantum number colorantitriplet or color-sextet can be perturbative produced.The second step: The diquark fragments and then produces an observed double heavy baryon Ξ QQ ′ q by capturing a light quark from surrounding environment.The fragmentation probability will reach one hundred percent.In convenience, we take the lable Ξ QQ ′ to stand for Ξ QQ ′ q and employ the universal symbol QQ ′ [n] to represent a diquark state with color and spin combinations [n] in this paper.The total "100%" fragmentation probability can further be broken into 43%, 43%, 14% probability for Ξ QQ ′ u , Ξ QQ ′ d , Ξ QQ ′ s respectively [47,64].
Typical Feynman diagrams for the process Z(p 0 ) → QQ ′ [n](p 1 ) + Q(p 2 ) + Q′ (p 3 ) at tree level are presented in Fig. 1, where the Q (′) stand for heavy c, b-quark corresponding to the doubly heavy baryons Ξ cc , Ξ bc and Ξ bb production.To calculate the differential decay width for Z(p 0 ) → QQ ′ [n](p 1 ) + Q(p 2 ) + Q′ (p 3 ) process, one can employ the NRQCD factorization approach [65,66] and to start with the following formula, In for S-wave and P -wave doubly heavy baryons, which are mainly derived from experimental data and different nonperturbative theoretical approaches, such as QCD sum rules, lattice QCD, and potential energy models [66][67][68].
Furthermore, the expression d Γ(Z → QQ ′ [n] + Q + Q′ ) can be formulated as: where m Z refers to Z-boson mass, |M [n]| represent the hard amplitude expressions of the process for the Z-boson decay into S-wave and P -wave doubly heavy baryons, the constant 1/3 is coming from spin average of initial Z-boson, and symbol " " stands for sum of all final particles' color and spin.dΦ 3 is three-body phase space, which is given by A detailed discussion about the three-body phase space calculations can be found in Refs.[69,70].Subsequently, one can revise the Eq.( 2) expression as follows: with the invariant mass s ij = (p i + p j ) 2 .

A.
Amplitudes for the diquark production By applying the charge parity C = −iγ 2 γ 0 , one can derive the hard amplitude expressions M [n] forbaryon production, which can also be obtained from the process of the Z-boson decay into a meson (Q Q′ ) [33,46].Here, we have a notation that the charge parity C can be used to reverse one fermion line, which is represented by Where Γ i with index i = (0, 1, ...) stands for the interaction vertex, S F (q i , m i ) is fermion propagator, s 1,2 is spin index.Then one can obtain: When the axial vector vertices are not included in the fermion line, we can derive: For another case that axial vector vertices include fermion line, amplitudes for baryon production can be derived from the similar meson production.But there should multiply an additional factor (−1) (n+1) or (−1) (n+2) to the pure vector case and the axial vector FIG. 1: The diagrams for Z → QQ ′ [n] + Q + Q′ at leading order, where Q and Q ′ denote the heavy c or b-quark respectively.
case, respectively.Therefore, the S-wave or P -wave amplitude for Z → QQ ′ [n] + Q + Q′ decay process can be expressed as: M i with i = (1, 2, 3, 4) represent S-wave or P -wave amplitude of the similar meson production.Specifically, M a,v i is the axial vector or pure vector part for M i , respectively.In order to obtain the S-wave amplitudes M l [n] with l = (a, b, c, d), one can obey the Feynman rules based on diagrams shown in Fig. 1, which have the following expressions.
In these expressions, the combined factor κ have the relation κ = −Cg 2 s , which C is the abbreviation of color factor C ij,k .The p 11,12 represents momentum for each individual quark in the diquark state.More explicitly, we take p 11 = mc M QQ ′ p 1 + q and p 12 = mc M QQ ′ p 1 − q, which q is the small relative momentum between different heavy quark Q (′) .To maintain gauge invariance, we adopt M QQ ′ ≃ m Q + m Q ′ , and symbol c Q v,a is the vector or axial vector of Z Q Q vertex coupling constant.It can be written as In which, the θ w stand for Weinberg angle.After introducing the spin projector Π [n] p1 , the S-wave amplitude can be further expressed as The spin projector Π p1 appear in Eq. ( 14) can be expressed as [71] Π (11) or which stands for spin-singlet state or spin-triplet state, respectively.
On the other hand, the P -wave diquark states amplitudes can be obtained by making the derivative for S-wave expression either a spin singlet or a spin triplet.
Here we have a notation that the doubly heavy baryons can be excited into P -wave state either in ρ-mode or λmode, which is from the excitation between Q and Q ′ or excitation between QQ ′ and q.In this paper, we solely focus on the effect of ρ-mode to make investigation. and In which, the ǫ l α (q) and ǫ l αβ (q) are the polarized vector and polarized tensor for spin singlet and spin triplet of the P -wave diquark state.After sum over the [ 1 P 1 ] states polarized vectors, one can get the following result In which the symbol l z represents the excited states In the case of excited states [ 3 P J ], the sum over polarized tensors are represented by Jz with the definition

B. Color Factor
According to Fig. 1, it is easy to write the expression for color factor C ij,k , with letters k and a are color indices of diquark and gluon, respectively, and N = 1/2 refers to the normalization factor.In addition, those marks i, j, m, and n = (1, 2, 3) are also the color indices, which correspond to two outgoing antiquarks and two constituent active quarks in the diquark state, respectively.For the 3(6) state, the value of G mnk is related to the function ε mjk (f mjk ), by given as The process of diquark states QQ ′ [n] hadronization into the final states Ξ QQ ′ is regarded as a nonperturbative, and its impact is reflected by the overall factors O H (n) .The factors O H (n) , also known as the transition probabilities, can be represented by h3 and h 6 in color 3 and color 6, respectively.Detailed researches and discussions about h3 (h 6 ) can be found in Refs.[33,46,51,55,63].
In accordance with the velocity scaling rule of the NRQCD factorization approach, one can assume that the transition probabilities h3 and h 6 hold an equal horizon [21,44,63].Therefore, the wave function evaluated at the origin can be interconnected with them [44].
for S-wave, and for P -wave The wave function at the origin can be naturally connected to the radial wave function at the origin: Based on the NRQCD factorization approach, Ξ QQ ′ is a bound state of two heavy quarks with other light dynamical freedom of QCD and can be can be described by a series of Fock states: Due to the light quark may produce gluons easily, the constituents c i (v) with i = (1, 2, 3...) in Eq. ( 24) have equivalent significance with each other, e.g.c 1 (v) ∼ c 2 (v) ∼ c 3 (v).In the QQ ′ [ 3 S 1 ]3 state, one of heavy quarks can produce a gluon without altering its spin, and the gluon can divide into a light quark pair q q.Then the heavy diquark state QQ ′ can form a final baryon state by capturing a light quark q.In QQ ′ [ 1 S 0 ] 6 state, the heavy quark spin must be altered by the emitted gluon when a baryon is formed by |(QQ ′ )q .The reason lies in a suppression from h 6 .In the case of the heavy quark's spin remaining unchanged, one of the heavy quarks produces a gluon, and this gluon can separate into a quarkantiquark q q pair.Additionally, the light quark q has the ability to produce a gluon, which can be used to construct the component with qg.Based on the above analysis, the h 6 and h3 hold the same order in v c .We assume h 6 and h3 hold the same order in v c for convenience.
|Ψ cc (0)| = 0.523 GeV  For the production of Ξ cc,bc and Ξ bb baryons, we take the strong coupling constant α s (2m c ) = 0.242 and α s (2m b ) = 0.180 separately.In the following discussions, contributions from S-wave diquark states are also present in Tables I, II and III thorough comparison and analysis.FIG.2: The dΓ/ds23 and dΓ/dz for the Z-boson decay into Ξcc, where 3( 6) is the color quantum number of diquark state.

A. The Ξcc production
At this stage, we present the Z → Ξ cc decay widths, branching ratios, and events at the LHC (CEPC) in Table I, where the branching ratio has the definition as follows: • The total contribution from intermediate P -wave states is about 7% to the S-wave states, which indicate that P -wave states play a significant role in detailed calculations.
In order to take a deep looking at these channels for Ξ cc production, we have plotted the differential decay width curves with respect to s 23 and z in Fig. 2. The s 23 = (p 2 + p 3 ) 2 stand for the invariant mass.The z = 2E 1 /E Z is the energy fraction, where E 1,Z is the energy of Ξ cc or Z-boson.
• Fig. 2(a) provides a clear demonstration that the distributions of various excited states are similar with Ξ cc production.These curves increase with s 23 initially, and then decreased.Their peaks are lie in a small s 23 region.
• Fig. 2(b) illustrates that the dΓ behavior changed with the energy fraction z distribution, i.e., dΓ/dz are similar with dΓ/ds 23 , which are initial increased and then decreased.Their peaks are located at large z region.

B. The Ξ bc production
Secondly, The decay widths, branching ratios, and events at the LHC (CEPC) for Z → Ξ bc are displayed in Table II, which has following notations: • The total intermediate P -wave states contribution for Z-boson decay into excited baryon Ξ bc is about 7% in comparing with S-wave state.
In order to show the behaviors for process Z-boson decay into excited doubly heavy baryon Ξ bc , we plotted the differential decay widths with respect to s 23 and z in Fig. 3.There are eight states totally for Ξ bc baryon, as both color-antitriplet excited state and color-sextet excited state are reasonable for any spin states.To make • The total contribution from intermediate P -wave states is roughly 3% of that from S-wave.
The differential decay widths for Ξ bb with respect to s ij and z are plotted in Fig. 4. From which it can be observed that the characteristics are fundamentally consistent with the production of Ξ cc or Ξ bc .FIG. 4: The dΓ/ds23 and dΓ/dz for the Z-boson decay into Ξ bb , where 3( 6) is the color quantum number of diquark state.

D. Uncertainty analysis
Furthermore, we would like to discuss theoretical uncertainties of the Z-boson decay into Ξ QQ ′ , which arise from three main sources: the mass parameters m Q(Q ′ ) , the transition probability h3 (6) , and the coupling constant.Although the transition probability and the coupling constant might have theoretical uncertainties, they only affect the results as an overall factor.Hence, we will not delve into them here.On the other hand, decay width is greatly influenced by heavy quark masses m Q(Q ′ ) .Therefore, we varied m c = 1.8 ± 0.3 and m b = 5.1 ± 0.3 GeV for doubly heavy baryons with c-quark and b-quark production to discuss the uncertainties, respectively.The results are listed in Tables IV, V, VI and VII.From the    • From Tables.V and VI, it shows that the deviation of m c or m b has a more greater effect on the decay width for Ξ bc production.
The contribution of the P -wave diquark states can be taken as the higher-order contribution of S-wave states.If all these excited P -wave states completely decay into the ground state, we estimate the total decay width Γ(Ξ cc ) = 20.220KeV, Γ(Ξ bc ) = 61.813KeV, Γ(Ξ bb ) = 2.958 KeV.(27) After combined S-wave and P -wave contribution, there will be about 8.104 × 10 3 (10 6 ) Ξ cc events, 24.773 × 10 3 (10 6 ) Ξ bc events and 1.185 × 10 3 (10 6 ) Ξ bb events produced in one operation year at the LHC (CEPC).Meanwhile, the P -wave states events will reach to 0.539× 10 3 (10 6 ), 1.827 × 10 3 (10 6 ), 0.036 × 10 3 (10 6 ) for Ξ cc , Ξ bc , Ξ bb in one year at the LHC (CEPC) respectively.From our detailed calculations with abundant events and significant branching ratios, the P -wave states in doubly heavy baryons make considerable contributions compared with S-wave states.Finally, we present the curves for differential decay width with respect to s 23 and z, i.e dΓ/ds 23 and dΓ/dz, which demonstrate the properties of Z-boson decay into the excited doubly heavy baryons Ξ QQ ′ processes.It is hoped that our predictions can provide assistance to experimental measurements.

TABLE I :
Our results for decay widths (in unit: KeV), branching ratios , and events at the LHC (CEPC) of the process Z-boson decay into Ξcc.States denote the intermediate diquark.

TABLE II :
Our results for decay widths (in unit: KeV), branching ratios , and events at the LHC (CEPC) of the process Z-boson decay into Ξ bc .States denote the intermediate diquark.

TABLE III :
(6) results for decay widths (in unit: KeV), branching ratios, and events at the LHC (CEPC) of the process Z-boson decay into Ξ bb .In which, the states denote the intermediate diquark.thesefiguresmoreclearer,wesumseveralcolored states with same spin state.For example, the line labeled with bc [ 1 P 1 ]3(6)means the sum of the contributions from bc [ 1 P 1 ]3 and bc [ 1 P 1 ] 6 .The similarities between the angular and invariant mass differential widths of Ξ cc and Ξ bc productions via Z-boson decays indicate their similar kinematic behaviors.C.The Ξ bb productionThirdly, decay widths, branching ratios, and events at LHC (CEPC) of the Z-boson decay into Ξ bb are displayed in TableIII, which is similar with Ξ cc and Ξ bc cases.From which we can get the conclusions • In the case of the excited state Ξ bb production via Z-boson decay, the largest contribution among excited states comes from spin and color state [ 1 P 1 ]3.The ratio of bb [ 1 P 1 ]3: bb [ 3 P 0 ] 6 : bb [ 3 P 1 ] 6 : bb [ 3 P 2 ] 6 = 1 : 0.83 : 0.93 : 0.41.

TABLE IV :
Our result for decay widths (in unit: KeV) of the process Z-boson decay into Ξcc by varying mc (in unit: GeV).

TABLE V :
Our result for decay widths (in unit: KeV) of the process Z-boson decay into Ξ bc by varying mc (in unit: GeV).

TABLE VI :
Our result for decay widths (in unit: KeV) of the process Z-boson decay into Ξ bc by varying m b (in unit: GeV).

TABLE VII :
Decay widths (in unit: KeV) for the production of Ξ bb via Z decays by varying m b (in unit: GeV).In comparing TableIVwith Table VII, it can be see that the influence for Z → Ξ cc decay widths from c-quark mass are greatly larger than b-quark mass.