Fragmentation-fraction ratio $f_{\Xi_b}/f_{\Lambda_b}$ in $b$- and $c$-baryon decays

We study the ratio of fragmentation fractions, $f_{\Xi_b}/f_{\Lambda_b}$, from the measurement of $\Xi_b^0\to \Xi_c^+\pi^-$ and $\Lambda_b^0\to \Lambda_c^+\pi^-$ with $\Xi_{c}^{+}/\Lambda_{c}^{+}\to p K^-\pi^+$. With the branching fraction $\mathcal{B}(\Xi_c^+\to pK^-\pi^+)=(2.2\pm0.8)\%$ obtained under the U-spin symmetry, the fragmentation ratio is determined as $f_{\Xi_b}/f_{\Lambda_b}$=$0.054\pm0.020$. To reduce the above uncertainties, we suggest to measure the branching fractions of $\Xi_c^+\to p \overline K^{*0}$ and $\Lambda_c^+\to \Sigma^+ K^{*0}$ at BESIII, Belle(II) and LHCb.


I. INTRODUCTION
Bottom quarks can be produced at the high energy colliders, such as LHC and Tevatron, and then hadronized into B mesons and b-baryons. The probability of a bottom quark fragments into a certain weakly decaying b-hadron is called the fragmentation fractions, i.e. f u,d,s ≡ B(b → . As non-perturbative effects, the fragmentation fractions can only be determined by experiments in some phenomenological approaches.
The B-meson fragmentation fractions have been measured by LEP, Tevatron and LHC with a relatively high precision [1,2]. However, the current understanding of b-baryon productions is still a challenge. The total fragmentation fraction of b-baryon is the sum of all the weakly-decaying b-baryons, where the isospin symmetry is assumed as is the correction from f Λ b to f baryon . The averages of the total baryon production fractions are [2] f baryon =      0.084 ± 0.011, at LEP, 0.196 ± 0.046, at Tevatron, (2) which are inconsistent with each other, and of large uncertainties.
The total fraction of b-baryons has not been determined by LHCb because of its lack of measurements on Ξ 0,− b and Ω − b . It has been found that the ratio f Λ b /f d depends on the p T of the final states [3][4][5][6]. At LHCb, the kinematic averaging ratio is [5] f It is required for the information of f Ξ b and f Ω b to determine the other fragmentation fractions at LHCb due to the constraint of Since the production of Ω − b is suppressed compared to those of Ξ 0,− b by the production of an additional strange quark, the determination of f Ξ b /f Λ b is essential to understand the productions of b-baryons and B mesons.
So far, only Refs. [7,8] have predicted the ratio f Ξ b /f Λ b , both based on the processes of 0.29 ± 0.10 (MIT bag model) [11] 0.08 ± 0.03 (diquark model) [11] 0.054 ± 0.020 (this work) of Λ b and Ξ b have been measured by the heavy-flavor-conserving process of Ξ − b → Λ 0 b π − [9], and the charm-baryon involving decays of Ξ . All the results are listed in Table. I. The production with the charm-baryon involving method is of the most high precision. The ratio f Ξ b /f Λ b can be obtained as long as the related branching fractions are determined. Among them, the absolute branching fraction of Ξ + c → pK − π + has never been measured [1], thus is of the largest ambiguity. In this work, This article is organized as follows. In Sec. II, we introduce the status of f Ξ b /f Λ b . The branching fraction of Ξ + c → pK − π + and f Ξ b /f Λ b are obtained in Sec. III and IV, respectively. Sec. V is the summary.
In some literatures, it is usually assumed that the difference between the productions of Ξ b and Λ b is from the strange quark and up or down quarks [10,12], However, since the fragmentation fractions are non-perturbative effects, they can only be extracted from experimental data. In this section, we introduce the status of f Ξ b /f Λ b by means of the relevant measurements.
So far, the only theoretical analysis on f Ξ b /f Λ b are performed in Refs. [7,8] based on the experimental data of Ξ − b → J/ψΞ − and Λ 0 b → J/ψΛ. In Ref. [1], the relevant results averaging the measurements by CDF and D0 [13][14][15][16], are given as The fragmentation fraction ratio of f Ξ b /f Λ b can be obtained unless the ratio of branching fractions respectively. Therefore, the two processes are related to each other under the flavor SU (3) symmetry. The width relation of is given by Voloshin [7]. Using the experimental data in (6), the ratio of the fragmentation fractions can then be obtained as [7] f  8), B. Heavy-flavor-conserving decay The LHCb collaboration has observed the first heavy-flavor-conserving ∆S = 1 hadronic weak In Ref. [9], f Ξ b /f Λ b is assumed to be bounded between 0.1 and 0.3, and then obtain the branching In Ref. [11], the branching fraction of Ξ − b → Λ 0 b π − is calculated in the MIT bag model and the diquark model, Subsequently, we can obtain the ratio of fragmentation fractions according to Eq.(10) as, In the above two methods, the experimental measurements are of large uncertainties, as seen in Eqs. (6) and (10). In the decay of Ξ − b → J/ΨΞ − , the efficiency of reconstruction of Ξ − with Ξ − → Λπ − and Λ → pπ − , is very small in the hadron colliders [13,14]. On the other hand, the branching fraction of Ξ − b → Λ 0 b π − is expected to be very small. Compared to the above processes, the relative production ratio between Ξ 0 b → Ξ + c π − and Λ 0 b → Λ + c π − has been measured by LHCb with much higher precision [10], As long as the branching fractions of the relevant b-and c-baryon decays are known, Eq. (14) could provide a good determination of f Ξ b /f Λ b . In Ref. [10], with naively expected values of The branching fraction of Ξ 0 b → Ξ + c π − has not been directly measured in experiment.
are equal to each other in the heavy quark limit and the flavor SU (3) symmetry. In literatures, only Refs. [17] and [18] calculate both the branching fractions With the transition form factors in the non-relativistic quark model, the ratio of branching fractions involving the factorizable contribution can be obtained in [17]: where the difference in the lifetimes is neglected since τ (Ξ 0 b )/τ (Λ 0 b ) = 1.006 ± 0.021, and a 1 = C 1 + C 2 /3 is the effective Wilson coefficient. The deviation from unity results from the mass difference between m Ξ b +m Ξc and m Λ b +m Λc , i.e. the SU (3) breaking effect. In the soft-collinear effective theory, the non-factorizable contributions from the color-commensurate and the W -exchange diagrams are suppressed by O(Λ QCD /m b ) [19]. In Ref. [18] in a relativistic three-quark model, it is found that the non-factorizable contributions amount up to 30% of the factorizable ones, with 25. Therefore, even without a reliable study in a QCD-based approach, it can still be expected that the deviation of the ratio from unity is under control.
The decays of Ξ + c → pK * 0 and Λ + c → Σ + K * 0 are both singly Cabibbo-suppressed modes, with the transition of c → (ss − dd)u where the minus sign between ss and dd comes from the Cabibbo-Kobayashi-Maskawa matrix elements, V * cd V ud = −V * cs V us . Note that the U -spin doublets are (|d , |s ) and (|s , −|d ). The effective Hamiltonian of c → (ss − dd)u changes the U -spin by ∆U = 1, ∆U 3 = 0, i.e. |H eff = √ 2|1, 0 . Ξ + c and Λ + c form a U -spin doublet of (Λ + c , Ξ + c ). We have The U -spin representations of the |pK * 0 and |Σ + K * 0 states are The decay amplitudes are then where A 3/2 and A 1/2 are the amplitudes of U -spin of 3/2 and 1/2, respectively. It is clear that the amplitudes satisfy This relation can also be seen from the topological diagrams in FIG.1.
Due to the larger lifetime and phase space, the branching fraction of Ξ + c → pK * 0 is then at the order of percent, (1.2 ± 0.4)%.
The understanding of the dynamics of charmed baryon decays is still a challenge at the current stage. Recent theoretical studies are mostly based on the flavor SU (3) analysis [26][27][28][29][30] and the current algebra [31]. They have not yet been applied to the singly Cabibbo-suppressed charmed baryon decays into a light baryon and a vector meson. Therefore, it is not available to estimate the U -spin breaking effects in the above analysis of Eq. (26). In the D-meson decays, the U -spin breaking effects, or say the SU (3) breaking effects, are mainly from the transition form factors and decay constants in the factorizable amplitudes, the difference between uū, dd and ss produced from vacuum in the W -exchange and W -annihilation amplitudes, and the Glauber strong phase with pion in the non-factorizable contributions [32,33]. In Fig. 1, both amplitudes in the Ξ + c → pK * 0 and Λ + c → Σ + K * 0 decay are non-factorizable. The vacuum production of dd and ss in the Wexchange diagrams would be a main source of U -spin breaking. In the modes involving a vector meson and a pseudoscalar meson in the final states of D-meson decays, the difference between dd and ss production in the W -exchange diagrams can be seen from χ E d e iφ E d = (0.49 ± 0.03)e i(92±4) • and χ E s e iφ E s = (0.54 ± 0.03)e i(128±5) • [34] where χ and φ are the magnitude and strong phase of the non-perturbative parameters in the W -exchange diagrams, and the subscripts d and s denotes the quark flavor of qq produced from the vacuum. It can be found that the U -spin breaking effects are It is equivalent that the correction δ in Eq. (1) is δ = 0.11 ± 0.04, which is smaller than the estimation of δ = 0.25 ± 0.10 in Ref. [38].
With the result of f Ξ b /f Λ b in Eq. (27), the branching fraction of Ξ − b → Λ 0 b π − can be determined from Eq. (10),