Fragmentation fractions of two-body b-baryon decays

We study the fragmentation fractions $(f_{{\bf B}_b})$ of the $b$-quark to b-baryons (${\bf B}_b$). By the assumption of $f_{\Lambda_b}/(f_u+f_d)=0.25\pm 0.15$ in accordance with the measurements by LEP, CDF and LHCb Collaborations, we estimate that $f_{\Lambda_b}=0.175\pm 0.106$ and $f_{\Xi_b^{-,0}}=0.019\pm 0.013$. From these fragmentation fractions, we derive ${\cal B}(\Lambda_b\to J/\psi \Lambda)=(3.3\pm 2.1)\times 10^{-4}$, ${\cal B}(\Xi_b^-\to J/\psi \Xi^-)=(5.3\pm 3.9)\times 10^{-4}$ and ${\cal B}(\Omega_b^-\to J/\psi \Omega^-)>1.9\times 10^{-5}$. The predictions of ${\cal B}(\Lambda_b\to J/\psi \Lambda)$ and ${\cal B}(\Xi_b^-\to J/\psi \Xi^-)$ clearly enable us to test the theoretical models, such as the QCD factorization approach in the $b$-baryon decays.

For example, while the pπ mass distribution in Λ b → J/ψpπ − [2] suggests the existence of the higher-wave baryon, such as N(1520) or N(1535), a peaking data point in the Dp mass distribution in Λ b → D 0 p(K − , π − ) [3] hints at the resonant Σ c (2880) state. On the other hand, it is typical to have the partial observations for the decay branching ratios, given by [4]  and Ω − b [8], f Λ b ≃ f baryon is no longer true. As a result, it is urgent to improve the value of f Λ b and obtain the less known ones of are not in good agreement, given by with the uncertainty related to Br due to the uncertainties on the measured branching ratios, where the first relation given by the CDF Collaboration [9] is obviously two times larger than the world averaged value of the second one [4], dominated by the LEP measurements on Z decays. Moreover, since the recent measurements by the LHCb Collaboration also indicate this inconsistency [10][11][12], it is clear that the values of f Λ b and f Ξ 0,− b can not be experimentally 1 f baryon ∼ 0.1 was also taken in the previous versions of the PDG.
determined yet. In this paper, we will demonstrate the possible range for in accordance with the measurements by LEP, CDF and LHCb Collaborations and give the theoretical estimations of f Λ b and f Ξ 0,− b , which allow us to extract B(Λ b → J/ψΛ), (1). Consequently, we are able to test the theoretical approach based on the factorization ansatz, which have been used to calculate the two-body B b decays [7,[13][14][15][16][17][18][19].
Experimentally, in terms of the specific cases of the charmful Λ b → Λ + c π − andB 0 → D + π − decays or the semileptonic Λ b → Λ + c µ −ν X andB → Dµ −ν X decays detected with the bins of p T and η, where p T is the transverse momentum and η = − ln(tan θ/2) is the pseudorapidity defined by the polar angle θ with respect to the beam direction [9][10][11], the can be related to p T and η. This explains the inconsistency between the results from CDF and LEP with p T = 15 and 45 GeV, respectively. While f s /f u is measured with slightly dependences on p T and η [20] is fitted as the linear form in Ref. [10] with p T = 0 − 14 GeV and the exponential form in Refs. [11,12] with p T = 0 − 50 GeV, respectively, for the certain range of η.
the range of p T = 0 − 14 GeV to be the linear form, given by [11] where Br arises from the absolute scale uncertainty due to the poorly known branching ratio which agrees with the first relation in Eq. (2) given by the CDF Collaboration with p T ≃15 GeV. On the other hand, with the charmful Λ b → Λ + c π − andB 0 → D + π − decays, another analysis by the LHCb Collaboration presents the exponential dependence of f Λ b /f d on p T [11,12]: with the wider range of p T = 0 − 50 GeV. By averaging the value in Eq. (5) with p T = 0 − 50 GeV, we findf with f u = f d due to the isospin symmetry, where the error has combined the uncertainties in Eq. (5). It is interesting to note that, as the relation in Eq. (5) with p T = 0 − 50 GeV overlaps p T ≃ 45 GeV for the second relation from LEP in Eq. (2), its value of form the values in Eqs. (4) and (6), the reanalyzed results by CDF and LHCb Collaborations to be 0.212 ± 0.058 and 0.223 ± 0.022 with the averaged p T ≃ 13 and 7 GeV, respectively [12]. We hence make the assumption of to cover the possible range in accordance with the measurements from the three Collaborations of LEP, CDF and LHCb, which will be used to estimate the values of In principle, when the ratios of and it was once given that where the first relation from Refs. [8,21] requires the assumption of [11], while the second one from Ref. [22] uses 1 from the naive Cabibbo factors. However, we note that the theoretical calculations provide us with more understanding of b-baryon decays, such as the difference between the Λ b → Λ and Ξ − b → Ξ − transitions, based on the SU(3) flavor and SU(2) spin symmetries. As a result, the assumption of R 1 = R 2 ≃ 1 might be too naive. Since the theoretical approach with the factorization ansatz well explains B(Λ b → pπ − ) and B(Λ b → pK − ), and particularly the ratio of B(Λ b → pπ − )/B(Λ b → pK − ) ∼ 0.84 [23], it can be reliable to determine Theoretically, we use the factorization approach to calculate the two-body b-baryon decay, such that the amplitude corresponds to the decaying process of the B b → B n transition with the recoiled meson. Explicitly, as shown in Fig. 1, where the W -boson emission is internal, the amplitude via the quark-level b → ccs transition can be factorized as where the parameter a 2 is given by [24,25] with the effective Wilson coefficients (c ef f 1 , c ef f 2 ) = (1.168, −0.365). Note that the color number N c originally being equal to 3 in the naive factorization, which gives a 2 = 0.024 in Eq. (11), should be taken as a floating number from 2 → ∞ to account for the nonfactorizable effects in the generalized factorization. The matrix element for the J/ψ production is given by J/ψ|cγ µ c|0 = m J/ψ f J/ψ ε * µ with m J/ψ , f J/ψ , and ε * µ as the mass, decay constant, and polarization vector, respectively. The matrix elements of the B b → B n baryon transition in Eq. (10) have the general forms: where f j (g j ) (j = 1, 2, 3) are the form factors, with f 2,3 = 0 and g 2,3 = 0 due to the helicity conservation, as derived in Refs. [7,14,26]. It is interesting to note that, as the helicity-flip terms, the theoretical calculations from the loop contributions to f 2,3 (g 2,3 ) indeed result in the values to be one order of magnitude smaller than that to f 1 (g 1 ), which can be safely neglected. In the double-pole momentum dependences, one has that [23] F We are able to relate different B b → B n transition form factors based on SU(3) flavor and SU(2) spin symmetries, which have been used to connect the space-like B n → B ′ n transition form factors in the neutron decays [27], and the time-like 0 → B nB ′ n baryonic as well as B → B nB ′ n transition form factors in the baryonic B decays [28][29][30][31][32]. As a result, we obtain (f 1 (0), g 1 (0)) = (C, C), (− 2/3C, − 2/3C), and (0, 0) with C a constant for p|ūγ µ (γ 5 )b|Λ b , Λ|sγ µ (γ 5 )b|Λ b , and Σ 0 |sγ µ (γ 5 )b|Λ b , which are the same as those in Ref. [26] based on the heavy-quark and large-energy symmetries for the Λ b → (p, Λ, Σ 0 ) transitions, respectively. In addition, we have f 1 (0) = g 1 (0) = C for Ξ − |sγ µ (γ 5 )b|Ξ − b . To obtain the branching ratio for the two-body decays, the equation is given by [4] with . As a result, we obtain [33]. We note that, theoretically, R 1 = 1.63 apparently deviates by 63% from R 1 = 1 in the simple assumption. To Eq. (16) to (1) to give which is different from the numbers in Eq. (9).
To explain the branching ratios of Λ b → J/ψΛ and Ξ − b → J/ψΞ − in Eq. (18), the floating color number N c is evaluated to be N c = 2.15 ± 0.17 , which corresponds to a 2 = 0.18 ± 0.04, in comparison with a 2 = 0.024 for N c = 3. Note that since N c = 2.15 in Eq. (20) is not far from 3, we conclude that the non-factorizable effects are controllable. As a result, the theoretical approach based on the factorization ansatz is demonstrated to be reliable to explain the two-body B b decays.

III. CONCLUSIONS
In sum, we made the assumption of