Determinations of $|V_{cb}|$ and $|V_{ub}|$ from baryonic $\Lambda_b$ decays

We extract the Cabibbo-Kobayashi-Maskawa matrix element $V_{cb}$ from the exclusive decays of $\Lambda_b\to \Lambda_c\ell\bar \nu_\ell$ and $\Lambda_b\to \Lambda_c M_{(c)}$ with $M=(\pi^-,K^-)$ and $M_c=(D^-,D^-_s)$, where the branching ratios of $\Lambda_b\to \Lambda M_{(c)}$ measured with high precisions have not been used in the previous studies. Explicitly, we find $|V_{cb}|=(44.0\pm 3.5)\times 10^{-3}$, which agrees with the value of $(42.11\pm 0.74)\times 10^{-3}$ from the inclusive $B\to X_c\ell\bar \nu_\ell$ decays. Furthermore, based on the most recent ratio of $|V_{ub}|/|V_{cb}|$ from the exclusive $\Lambda_b$ decays, we obtain $|V_{ub}|=(4.2\pm 0.4)\times 10^{-3}$, which is close to the value of $(4.49\pm 0.24)\times 10^{-3}$ from the inclusive $B\to X_u\ell\bar \nu_\ell$ decays. We conclude that our determinations of $|V_{cb}|$ and $|V_{ub}|$ from the exclusive $\Lambda_b$ decays favor the inclusive extractions in the $B$ decays.


I. INTRODUCTION
In the Standard Model (SM), the unitary 3 × 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix elements present the coupling strengths of quark decays, with the unique physical weak phase for CP violation. Being unpredictable by the theory, the matrix elements as the free parameters need the extractions from the experimental data. Nonetheless, there exists a long-standing discrepancy between the determinations of |V cb | based on the exclusive B → D ( * ) ℓν ℓ and inclusive B → X c ℓν ℓ decays, given by [1][2][3]  (1) From the data in Eq. (1), we see that the deviations between the central values of the inclusive and exclusive decays are around (2)(3)σ. For the resolution, the analysis in Ref. [4] suggests that the B → D * transition form factors developed by Caprini, Lellouch and Neubert (CLN) [5] may underestimate the uncertainty that associates with the extraction of |V cb |. Moreover, it has been recently pointed out that the theoretical parameterizations of the B → D ( * ) transitions given by Boyd, Grinstein and Lebed (BGL) [6] are more flexible to reconcile the difference [7,8]. Similar to the data for |V cb | in Eq. (1), there also exists a tension for the determination of |V ub | between the exclusive and inclusive B decays, which has drawn a lot of theoretical attentions to search for the solutions in the SM and beyond [9][10][11][12][13][14].
On the other hand, the baryonic Λ b decays could provide some different theoretical inputs for the CKM matrix elements, which are able to ease the tensions between the exclusive and inclusive determinations. Indeed, to have an accurate determination of |V ub |/|V cb | the LHCb Collaboration has carefully analyzed the ratio of [15] where B denotes the branching fraction and q is the certain range of the integrated energies for the data collection. In Eq. (2), R ub by relating B(Λ b → pµν µ ) to B(Λ b → Λ c µν µ ) reduces the experimental uncertainties, while R F F is a ratio of the Λ b → Λ c and Λ b → p transition form factors, calculated by the lattice QCD (LQCD) model [16] with a less theoretical uncertainty.
In this work, we would like to first explore the possibility to determine |V cb | from the baryonic decays. In particular, we use the observed branching ratios of Λ b → Λ c ℓν ℓ , Λ c → Λℓν ℓ and Λ b → Λ c M (c) with ℓ = e − or µ − , M = (π − , K − ) and M c = (D − , D − s ), which have never been used in the previous studies. The full energy-range measurements of the semileptonic decays are given by [17] where R cb combines the data of B(Λ b → Λ + c ℓν ℓ ) and B(Λ + c → Λℓν ℓ ) to eliminate the uncertainties, similar to R ub in Eq. (2). The decay branching ratios of The above modes in Eq. (4) can be regarded to proceed through the Λ b → Λ c transition together with the recoiled mesons, such that the theoretical estimations give where f M (c) are the meson decay constants and R(M (c) ) are the rates to account for the mass differences from the phase spaces. Note that the ratios in Eq. (5) remarkably agree with (13.6 ± 1.6, 24.0 ± 3.8) from the data in Eq. (4), respectively. This implies that the can be reliable to be involved in the fitting of |V cb |. Particularly, the data in Eq. (4) have the significances of (8-12)σ, which apparently benefit the precise determination of |V cb |. As a result, the extraction of |V cb | from the data in Eqs. (3) and (4) can be an independent one besides those from the B → D ( * ) ℓν ℓ and B → X c ℓν ℓ decays. With the newly extracted |V cb | value, we will be then able to determine

II. FORMALISM
As seen in Fig  In the helicity-based definition, the matrix elements of the Λ b → Λ c transition are given by [16] where q = p − p ′ , s ± = (m Λ b ± m Λc ) 2 − q 2 , and (f 0 , f + , f ⊥ ) and (g 0 , g + , g ⊥ ) are form factors.
The momentum dependences of f = f j and g j (j = 0, +, ⊥) are written as [16] f (t) = 1 where (n max , t + , t 0 ) = (1, (m f pole ) 2 , (m Λ b − m Λc ) 2 ) with m f pole representing the corresponding pole masses. Note that the form factors for the Λ c → Λ transition have similar forms as in Eqs. (7) and (8), given in Ref. [20]. In terms of the equations in Ref. [17], one is able to integrate over the variables of the phase spaces in the two-body and three-body decays for the decay widths.

III. NUMERICAL RESULTS AND DISCUSSIONS
For the numerical analysis, we perform the minimum χ 2 fit with |V cb | being a free parameter to be determined. The parameters a M (c) 1 are able to accommodate the non-factorizable effects, provided that N ef f c is taken as the effective color number to range from 2 to ∞ in accordance with the generalized factorization [19], leading to the initial inputs of a decay constants are given by [17] (|V cd |, |V cs |) = (0.220 ± 0.005, 0.995 ± 0.016) , (|V ud |, |V us |) = (0.97417 ± 0.00021, 0.2248 ± 0.0006) , while the experimental inputs in Eqs. (3) and (4) are accounted to be 6 data points, listed in Table I. Note that the information of the Λ b → Λ c and Λ c → Λ form factors in Eq. (8) are adopted from Refs. [16,20]. Subsequently, we obtain which is consistent with the inclusive result of (4.49 ± 0.24) × 10 −3 from B → X u ℓν ℓ [17] but different from the exclusive one of (3.72 ± 0.19) × 10 −3 from B → πℓν ℓ [17]. To compare our fitting results with different data inputs, we set 4 scenarios: where S0 corresponds to the fitting shown in Eqs. (10) and (11), which gives the lowest uncertainty for |V cb | along with the best value of χ 2 /d.o.f . In Table II, we summarize our results as well as the data from the B decays. As seen from Finally, we remark that if we take the Λ b → Λ c and Λ c → Λ transition form factors in transitions [5].

IV. CONCLUSIONS
In sum, since the extractions of |V cb | showed the (