Production of semi-inclusive doubly heavy baryons via top-quark decays

In the paper, we present a detailed discussion on the semi-inclusive production of doubly heavy baryons ($\Xi_{bc}$ and $\Xi_{bb}$) through top-quark decay channel, $t\rightarrow \Xi_{bQ^{\prime}}+ \bar {Q^{\prime}} + W^+ $, within the framework of non-relativistic QCD (NRQCD). In our calculations, the contributions from the intermediate diquark states, $\langle bc\rangle[^{3}S_{1}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bc\rangle[^{1}S_{0}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bb\rangle[^{1}S_{0}]_{\mathbf{6}}$ and $\langle bb\rangle[^{3}S_{1}]_{\mathbf{\bar 3}}$, have been taken into consideration. Main uncertainties from the heavy quark mass $(m_c,~m_b~\mbox{or}~m_t)$, the renormalization scale $\mu_r$, and the nonperturbative transition probability have been estimated. For a comparison, we also analyze the production of doubly heavy baryons under the approximate fragmentation function approach. Estimated at the LHC or a High Luminosity LHC with $\mathcal{L}$ =$10^{34-36}~\rm{cm}^{-2}~\rm{s}^{-1}$, there will be about $2.25\times10^{4-6}$ events of $\Xi_{bc}$ and $9.49\times10^{2-4}$ events of $\Xi_{bb}$ produced in one operation year through top-quark decays.


I. INTRODUCTION
one is the "direct evolution", which directly set the transition efficiency to be 100%, and the other is the "indirect evolution via fragmentation", which can be estimated by using some phenomenological models. It has been found that the direct evolution approach is of high precision and sufficient enough for studying the production of a doubly heavy baryon [23]. Thus, in the present paper, we shall directly adopt the direct evolution approach to do the calculation.
In addition to the conventional fixed-order calculations, the production of doubly heavy baryon Ξ bQ ′ from the top-quark decays can also be described by using the fragmentation function approach. The fragmentation function approach resums the large logarithms such as ln(M Ξ bQ ′ /E) and thus could provide a more reliable prediction in specific kinematic regions, where E is the energy of the corresponding baryon. Such kinds of large logarithms always come from the collinear emissions for high-order calculations or small p t regions. It is worth mentioning that at leading order in α s , the fragmentation probability 1 0 dzD b→Ξ bQ ′ (z, µ) does not evolve with the scale µ [29][30][31][32]. In the paper, we shall concentrate on the fixed-order calculation with the NRQCD, and shall give a simple discussion on how the fragmentation function approach may change the behaviors of the energy fraction of the produced doubly heavy baryons.
The remaining parts of the paper are organized as follows. In Sec. II, we present the detailed calculation technology for the fixed-order calculation within the framework of NRQCD and the fragmentation function approach. Numerical results and discussions are given in Sec. III. Section IV is reserved for a summary.

II. CALCULATION TECHNOLOGY
A. Fixed-order calculation within the NRQCD framework Figure 1. Typical Feynman diagrams for the process t → Ξ bQ ′ +Q ′ + W + , where Q ′ denotes the heavy c or b quark for the production of Ξ bc or Ξ bb accordingly.
Typical Feynman diagrams for the process t → Ξ bQ ′ +Q ′ + W + are presented in Fig. 1, where Q ′ stands for the heavy c or b quark, respectively. The decay width of this process can be factorized as the following form [6,7]: to the transition probability from the perturbative quark pair bQ ′ to the heavy baryon Ξ bQ ′ . The nonperturbative matrix elements are unknown yet, which can be approximated by relating it to the Schrödinger wave function at the origin |Ψ bQ ′ (0)| for the S-wave states by assuming the potential of the binding color-antitriplet bQ ′ [n] state is hydrogenlike. The decay widthΓ(t → bQ ′ [n] +Q ′ + W + ) represents the perturbative short-distance coefficients, which can be written aŝ where M is the hard scattering amplitude. means to average over the spin and color of the initial top quark and sum over the colors and spins of all final-state particles. The three-particle phase space dΦ 3 can be represented as After performing the integration over the phase space of this 1 → 3 process, Eq. (2) can be rewritten as where m t is the mass of top quark, and the definitions of invariant mass are s 12 = (q 1 + q 2 ) 2 and s 23 = (q 2 + q 3 ) 2 . The VEGAS [33] program is employed to integrate over the invariant mass s 12 and s 23 . Therefore, not only the total decay width but also the corresponding differential distributions, which are helpful for experimental measurements, can be derived. For useful reference, we shall present the differential distributions in the following discussions, such as dΓ/ds 12 , dΓ/ds 23 , dΓ/dcosθ 13 , and dΓ/dcosθ 12 , where θ 13 is the angle between − → q 1 and − → q 3 and θ 12 is the angle between − → q 1 and − → q 2 . The hard scattering amplitude M for the production of a diquark can be related to the familiar meson production with the action of C parity, which has been proven in Refs. [10,11]. More explicitly, we can obtain the hard scattering amplitude M of the decay channel where there is only a difference between the heavy Q ′ fermion line. Using the charge conjugation matrix C = −iγ 2 γ 0 , the heavy Q ′ fermion line can be reversed with an additional factor (−1) n+1 , where n stands for the number of vector vertices and here n = 1. Hence, we can transform the amplitude of the diquark production to that of the meson production. The square of the hard scattering amplitude |M| 2 = 1 2×3 |A 1 + A 2 | 2 . Here, A 1 (A 2 ) is the amplitude corresponding to the left (right) diagram of Fig. 1, which can be written as in which the projector where ε[ 1 S 0 ] = γ 5 and ε[ 3 S 1 ] = / ε with ε α is the polarization vector of the 3 S 1 diquark state. ε(q 3 ) is the polarization vector of W + , P L = 1−γ 5 2 . M bQ ′ = m b + m Q ′ is adopted to ensure the gauge invariance, q 11 and q 12 are the momenta of those two constituent quarks and take the following forms: where q is the relative momentum between the two constituent quarks of the diquark. On account of the nonrelativistic approximation, q is small enough to be neglected in the amplitude. For the production of Ξ bb , we need to time the squared amplitude by an extra overall factor (2 2 /2!) = 2, where the 1/2! factor is for the identical particles of the bb diquark and the 2 2 factor is because there are two more diagrams coming from the exchange of the two identical quark lines inside the diquark. The overall factor C = gg 2 s C ij,k . Because of the decomposition of the SU(3) C color group 3 3 =3 6, the diquark bQ ′ can be in either the antitriplet3 or the sextuplet 6 color state. According to Fig. 1, the color factor C ij,k can be calculated by where i, j, m, n = 1, 2, 3 are the color indices of the outgoing antiquarkQ ′ , the initial top quark, and the two constituent quarks b and Q ′ of the diquark, correspondingly; a = 1, . . . , 8 denotes as the color index for the gluon; k stands for the color index of the diquark bQ ′ ; and the normalization constant N = 1/2. For the antitriplet3 color state, the function G mnk is equal to the antisymmetric function ε mnk , which satisfies The function G mnk stands for the symmetric function f mnk for the sextuplet 6 color state, which satisfies the relation In the squared amplitude, the final color factor C 2 ij,k equals 4 3 for the color antitriplet diquark production and 2 3 for the color sextuplet diquark production. According to NRQCD theory, the Ξ bQ ′ baryon can be expanded to a series of Fock states which is accounted by the velocity scaling rule, where v is the relative velocity of the constituent heavy quarks in the baryon rest frame.  6 . Here, we use h3 and h 6 to present the transition probability of the color antitriplet diquark and the color sextuplet diquark, correspondingly. In the literature, there are two points of view for the contributions from each Fock state. Following the naive NRQCD power counting approach 1 , it is generally argued that the h 6 should be suppressed by at least v 2 to h3; thus, its contributions can be safely neglected. Another power counting rule indicates that the gluons or light-quarks in the hadron are soft and there is no such v 2 -power suppressions in the color-sextuplet state [9], and thus, those Fock states in Eq. (11) are of same importance, i.e., . At the present considered pQCD level, the matrix elements are overall parameters, and their uncertainties to the decay width can be conventionally obtained when we have their exact values. For convenience, we will adopt the assumption of the transition probability h3 ≃ h 6 = |Ψ bQ ′ (0)| 2 [7,34] in our discussion.

B. Fragmentation function approach
In the following, we shall adopt the fragmentation function approach to deal with the process t → Ξ bQ ′ +Q ′ +W + . At leading order, the energy fraction distribution of the process can be factorized as where z is the longitudinal momentum fraction of the Ξ bQ ′ relative to the b quark and z = E Ξ bQ ′ /E max Ξ bQ ′ , and µ is the factorization scale. There are large logarithms in the fragmentation function D b→Ξ bQ ′ (z, µ), such as ln(M Ξ bQ ′ /E), due to collinear gluon emission. Those log terms violate the scaling behavior of the fragmentation function, which, however, can be resummed by using the DGLAP equation [35,36], i.e., and the splitting function P b→b (z) is After performing the integration over the energy fraction z for Eq. (12), the fragmentation contribution to the decay rate for the production of Ξ bQ ′ is At leading order in α s , the fragmentation probability 1 0 dzD b→Ξ bQ ′ (z, µ) does not evolve with the factorization scale µ due to the property 1 0 dzP b→b (z, µ) = 0 [31]. Numerically, the fragmentation probability 1 0 dzD b→Ξ bQ ′ (z, µ) can be considered as the branching ratio Br t→Ξ bQ ′ at leading order.
According to the factorization theorem, the heavy baryon fragmentation function (similar to the heavy meson fragmentation function) is independent of the hard processes by which the heavy quark is created. We shall adopt the following fragmentation functions, which are derived by using the Z 0 -boson decays [37,38], to do the numerical calculation, i.e., where

III. NUMERICAL RESULTS
To do the numerical calculation, the input parameters are taken as [14,39] where the first six parameters are the same as that in Ref. [14], m c and m b are the constituent quark mass, which is used to build the mass of the corresponding baryon. |Ψ bc (0)| and |Ψ bb (0)| are the Schrödinger wave functions at the origin, which can be derived from the potential model, and in our calculation, we adopt the ones evaluated by the powerlaw potential model [34]. The remaining adoption parameters come from Particle Data Group [39]. The renormalization scale µ r for the production of Ξ bc (Ξ bb ) is set to be 2m c (2m b ), the same as the factorization scale. With the reference point α s (m Z ) = 0.1181 [39], the strong running coupling α s (2m b ) = 0.178 and α s (2m c ) = 0.239 can be obtained from the solution of the five-loop renormalization group equation [40,41]. To predict the events of the produced Ξ bc and Ξ bb baryons, the total decay width of the top quark is needed. The decay width for the largest decay channel t → bW + is 1.49 GeV, which can be considered as the total decay width of the top quark.

A. Basic results
Based on the input parameters mentioned before, the fixed-order decay widths for all considered spin and color configurations through the process t → Ξ bQ ′ +Q ′ + W + are which indicate that • The total decay width for the production of Ξ bb is about 1 order of magnitude smaller than that of Ξ bc . The main reason is that the mass of the b quark is about three times larger than that of the c quark, leading to a phase space suppression of Γ t→Ξ bb .
• For the production of Ξ bc and Ξ bb , the biggest decay width is from the spin and color state [ 3 S 1 ]3.
• There are four spin and color states for the Ξ bc production. If the transition probability of the color antitriplet diquark bc 3 and color sextuplet diquark bc 6 are considered the same, i.e., h 6 ≃ h3, the decay width in the color antitriplet state shall be about two times of that of the color sextuplet. The corresponding branching ratios Br t→Ξ bQ ′ are provided in Table I, in which the results by using the fragmentation function approach are also presented. Table I shows that the results under those two approaches, the fixed-order calculation and fragmentation function approaches, are consistent with each other for the production of Ξ bc . To be specific, the ratios for the branching ratio Br t→Ξ bc through the fixed-order calculation and fragmentation function approach can be up to 92% for the [ 1 S 0 ]3 /6 state and 91% for the [ 3 S 1 ]3 /6 state. For the case of Ξ bb , the ratios change to 82% for the [ 1 S 0 ] 6 state and 79% for the [ 3 S 1 ]3 state. Roughly, such a difference can be explained by the fact that the mass of the b quark is heavier than the c quark, leading to fewer Ξ bb events going along with the direction of theb quarks. Such an explanation is consistent with that one will find in Fig. 5(b). From Fig. 5(b), a smaller arising trend for the angular distribution between the baryon Ξ bb andb quark can be seen compared to that between the baryon Ξ bc andc quark. We also see the peaks of Ξ bb curves are much slower than that of Ξ bc curves. Table I also shows that the branching ratio Br t→Ξ bQ ′ through the top-quark decays at the LHC is large enough to be detected. Considering that at the LHC running with a high luminosity L =10 34−36 cm −2 s −1 , about N t = 10 8−10 tt pair [27,28] will be produced in one operation year, so the produced Ξ bQ ′ events per year could be roughly estimated by N Ξ bQ ′ = N t Br t→Ξ bQ ′ : • About 2.25 × 10 4−6 Ξ bc events/yr will be produced via the top-quark decays at the LHC. The largest proportion comes from the [ 3 S 1 ]3 state, which is about 8.56 × 10 3−5 events/yr. The events per year for the [ 1 S 0 ]3, [ 1 S 0 ] 6 , and [ 3 S 1 ] 6 states are about 6.46 × 10 3−5 , 3.23 × 10 3−5 , and 4.28 × 10 3−5 , accordingly.

B. Differential decay widths
To make a clear analysis about the distribution that is helpful to the experiments detection, we present the energy fraction distributions for the production of Ξ bc (a) and Ξ bb (b) in Fig. 2, in which both the results for the fixed-order calculation and fragmentation function approach are presented. In doing the calculation, all the parameters are set to be their central values. The fragmentation function approach is at the leading-logarithm (LL) accuracy, the factorization scale of which has been evolved to the order of m t − m W . The contribution from the large logarithms can be extracted by subtracting the fragmentation leading-order (LO) contribution from the fragmentation LL. After matching the fixed-order results with large logarithms contribution from the LL, an overall fixed-order LO+LL result can be given. In drawing the curves, the contributions from different diquark spin and color configurations have been summed up. Figure 2 shows that the behavior of the energy fraction of the produced doubly heavy baryon under the fixed-order calculation is close in shape in comparison to that of the leading-order fragmentation function approach, being without resumming the large logs. After doing the resummation at the LL level, the distributions of doubly heavy baryons Ξ bc and Ξ bb in the low-energy fraction (small z region) estimated by the LL fragmentation function approach are greater than those of the fixed-order calculation. Meanwhile, the invariant mass differential decay width dΓ/ds 13 is displayed in Fig. 3 for the production of Ξ bQ ′ by these two approaches and the combined one. Figure 3 shows that the behavior of the invariant mass distribution is analogous to that of the z distribution. In the fragmentation function approach, the considered process for the production of Ξ bQ ′ is t(p 1 ) → bW + → Ξ bQ ′ (q 1 ) + W + (q 3 ) + X produced heavy flavor baryon Ξ bQ ′ and W + boson, which are back-to-back with an angular separation by θ = π at the LL level. For the angular distribution obtained by the fragmentation function approach, dΓ/dcosθ 13 is represented by a delta function, δ(θ−π). And after the resummation with the DGLAP equation, it does not change the direction of the momentum, and the angular distribution obtained by the fixedorder calculation is not changed after being revised by the fragmentation function approach. The future experimental data may be helpful to test the result of theoretical predictions.
More characteristics of the process t(p 1 ) → Ξ bQ ′ (q 1 )+Q ′ (q 2 )+W + (q 3 ) are obtained by the fixed-order calculation, such as the differential decay widths dΓ/ds 12 , dΓ/ds 23 , dΓ/dcosθ 12 and dΓ/dcosθ 13 which are shown in Figs. 4 and 5. The kinematics parameters s 12 , s 23 , and cos θ are defined in Sec. II. Figure 4(a) shows that the differential decay width monotonously decreases with the increment of s 12 . For smaller and smaller s 12 , the Ξ bQ ′ baryon shall move closer to the  Figure 3. The invariant mass differential decay width dΓ/ds 13 for the production of Ξ bc (a) and Ξ bb (b). The solid black line represents the result obtained by the leading-order fragmentation function approach, the dashed red line denotes that obtained by the leading-logarithm fragmentation function approach, the dotted blue line stands for the fixed-order calculation, and the dash-dotted magenta line is the fixed-order calculation matching with the leading-logarithm fragmentation function approach. All the intermediate diquark states' contributions have been summed up to obtain the total energy fraction distribution for t → Ξ bQ ′ + W + + X.  Figure 4. The differential decay widths dΓ/ds 12 (a) and dΓ/ds 23 (b) for the process t(p 1 ) → Ξ bQ ′ (q 1 ) +Q ′ (q 2 ) + W + (q 3 ). The dashed black, dotted magenta, short dashed blue, dash-dotted red, short dash-dotted purple, and short dotted navy lines represent for the decay width for the production of Ξ bQ ′ in Fock states: bc direction of the heavy quarkQ ′ , leading to a much larger decay width. For the extreme condition, the Ξ bQ ′ baryon and the heavy quarkQ ′ shall move in the same direction, both of which shall move back to back with the W + boson in the rest frame of the top quark. In Fig. 4(b), all the curves are relatively flatter than those in Fig. 4(a), which first increase and then decrease with the increment of s 23 and have maximum values in the small s 23 region. Figure 5(a) shows that when the outgoing Ξ bQ ′ and W + boson move back to back, i.e.,  Figure 5. The differential decay widths dΓ/dcosθ 13 (a) and dΓ/dcosθ 12 (b) for the process θ 13 = 180 o , the differential decay width dΓ/dcosθ 13 can achieve its largest value, which is due to the fact that the W + boson is the heaviest among these three outgoing particles. Figure 5(b) shows that when the outgoing Ξ bQ ′ and antiquarkQ ′ move in the same direction, i.e., θ 12 = 0 o , the differential decay width dΓ/dcosθ 12 can achieve its largest value, which agrees with the result of Fig. 4(a).

C. Theoretical uncertainties
In this subsection, theoretical uncertainties for the production of Ξ bQ ′ through top-quark decays under the fixed-order calculation shall be discussed.    • For the production of Ξ bb , the decay width decreases with the increment of m b . The same as in the case of Ξ bc , its decay width will also increase with the increment of m t . The theoretical uncertainty from m b is bigger than that from m t .
Second, uncertainty comes from the renormalization scale µ r presented in Table V, in which we use three different renormalization scales µ r = 2m c , M bc , and 2m b for the production of Ξ bQ ′ via top-quark decays. And the corresponding running coupling α s is also added in Table V. Obviously, there is a large uncertainty caused by the renormalization scale µ r . Such scale ambiguity could be suppressed by a higher-order perturbative calculation or proper scale-setting methods such as the newly suggested principle of maximum conformal [42][43][44][45], which uses the renormalization group equation to set the optimal behavior of the running coupling at each perturbative order and thus set the optimal value for the renormalization scale.  Finally, uncertainty caused by choices of the nonperturbative transition probability is considered. According to NRQCD, h 6 for the color sextuplet diquark state may be suppressed by v 2 compared to h3 for the color antitriplet diquark state, such as h 6 /v 2 ≃ h3 = |Ψ bQ ′ (0)| 2 . If the contribution from the color sextuplet diquark bQ ′ 6 state can be ignored (h 6 = 0) and only the color antitriplet diquark bQ ′ 3 state is taken into consideration (h3 = |Ψ bQ ′ (0)| 2 ) for the production of Ξ bQ ′ , there are still 1.50 × 10 4−6 events of Ξ bc and 6.29 × 10 2−4 events of Ξ bb produced at the LHC in one operation year. As for the uncertainty of h3, it can be related to the Schrödinger wave function at the origin |Ψ bQ ′ (0)| for the S-wave state. And the wave function at the zero is an overall factor, and its uncertainty can be conventionally discussed when we know its exact values; thus, we directly take the wave function at zero to be the one derived from the power-law potential model [34].

IV. SUMMARY
In this paper, the indirect production of doubly heavy baryons Ξ bc and Ξ bb via topquark decays are discussed under two different approaches: the fixed-order calculation and the fragmentation function approach. In our calculation, all the possible spin and color configurations have been taken into consideration, i.e., bc [ 3 S 1 ]3 /6 , bc [ 1 S 0 ]3 /6 , bb [ 1 S 0 ] 6 and bb [ 3 S 1 ]3. We observe that each spin and color configuration has a sizable contribution to the production of Ξ bQ ′ . By summing up all the intermediate diquark states' contributions, we obtain the total decay width for t → Ξ bQ ′ +Q ′ + W + , Γ t→Ξ bc +c+W + = 0.34 +0. 27 −0.15 MeV, Γ t→Ξ bb +b+W + = 0.014 +0.011 −0.005 MeV, where the uncertainties are squared averages of those from the heavy quark masses (m b , m c and m t ), the renormalization scale µ r , and the nonperturbative transition probability. The decay width for the production of Ξ bc (Ξ bb ) is sensitive to m c (m b ), which is mainly caused by the change of phase space. Because of the running behavior of α s (µ r ), the renormalization scale µ r has a significant impact on the decay width Γ Ξ bQ ′ . Thus, a proper QCD renormalization scale-setting method [46,47] or higher-order perturbative calculation is needed to eliminate this scale ambiguity.