Simultaneous extractions of $|V_{ub}|$ and $|V_{cb}|$ with only the exclusive $\Lambda_b$ decays

We perform the simultaneous $|V_{ub}|$ and $|V_{cb}|$ extractions with only the exclusive $\Lambda_b$ decays of $\Lambda_b\to (p,\Lambda_c^+)\mu\bar \nu_\mu$, $\Lambda_b\to p\pi^-$ and $\Lambda_b\to \Lambda_c^+ (\pi^-, D^-)$. We obtain that $|V_{ub}|=(3.7\pm 0.3)\times 10^{-3}$ and $|V_{cb}|=(45.9\pm 2.7)\times 10^{-3}$. Our value of $|V_{ub}|$ is larger than that of $(3.27\pm 0.15\pm 0.16\pm 0.06)\times 10^{-3}$, previously extracted by the LHC Collaboration from the exclusive $\Lambda_b$ decays also, but nearly identical to $(3.72\pm 0.19)\times 10^{-3}$ from the exclusive $B$ decays. On the other hand, our extracted result of $|V_{cb}|$ favors the value of $(42.2\pm 0.8)\times 10^{-3}$ from the inclusive $B$ decays.


I. INTRODUCTION
The Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix elements, V ub and V cb , in the standard model have been studied extensively in the literature [1]. It is known that there are long-standing discrepancies for their determinations from the exclusive and inclusive B decays. Explicitly, it is given that [1] |V ub | in = (4.49 ± 0.16 +0. 16 −0.18 ) × 10 −3 , |V ub | ex = (3.72 ± 0.19) × 10 −3 , |V cb | in = (42.2 ± 0.8) × 10 −3 , where the subscripts "in" and "ex" stand for the extractions from the inclusive and exclusive B decays, respectively. Clearly, it is important to have some examinations besides the B meson ones, such as those from the inclusive and exclusive Λ b decays. Indeed, with the branching ratios of the Λ b → Λ + c ℓν ℓ and Λ b → Λ + c M (c) decays, where M = (π − , K − ) and M c = (D − , D − s ), |V cb | is extracted to be (44.6±3.2)×10 −3 [2]. The extraction is in accordance with the recent studies, where |V cb | ex has been raised to agree with |V ub | ex [3]. In addition, an extraction of |V ub |/|V cb | from Λ b → pµν µ and Λ b → Λ + c µν µ has been performed by the LHCb Collaboration [4].
In Ref. [4], although the absolute branching ratio of Λ b → pℓν ℓ has not been observed, the ratio of the partial branching fractions of Λ b → pµν µ and Λ b → Λ + c µν µ has been used, which is given by [4] with the selected transferred energy squared q 2 regions of q 2 > 15 GeV 2 and 7 GeV 2 .
According to the theoretical calculation, formulated by with R F F = 0.68±0.07 as the ratio of the Λ b → Λ c and Λ b → p transition form factors, calculated by the lattice QCD (LQCD) [5], it is presented that |V ub |/|V cb | = 0.083 ± 0.004 ± 0.004.
The simplest way to extract |V ub | is by putting the existing value of |V cb | into |V ub |/|V cb |.
The two |V ub | values deviate with each other. Besides, none of them can be claimed to be purely extracted from the exclusive Λ b decays. In this study, we propose to have a complete global fit with the currently existing data in the exclusive Λ b decays, such as R ub/cb and Λ b → pπ − , performing the simultaneous |V ub | and |V cb | determinations. Since R F F will be no longer taken as an independent theoretical input in the extraction, the uncertainties due to the theoretical calculations of the Λ b → (Λ + c , p) form factors can be reduced. In addition, the possible data correlations should be carefully considered. We will also take into account the recently updated B(Λ + c → pK − π + ) data. As a result, we can unambiguously extract |V ub | and |V cb | by fitting with only the exclusive Λ b decays, which are connected by the two ratios in Eqs. (2) and (3), to be regarded as the independent examination other than the B meson ones.
The paper is organized as follows. In Sec. II, we show the formalism. We give our numerical results and discussions in Sec. III. We conclude in Sec. VI.

II. FORMALISM
As seen in Fig. 1, in terms of the quark-level effective Hamiltonian for the semileptonic b → uℓν ℓ and non-leptonic b → uūq transitions, the amplitudes of the Λ b → pℓν ℓ and Λ b → pM decays are found to be [5,7] where G F is the Fermi constant, V ij are the CKM quark mixing matrix elements, and the matrix element of M|qγ µ (1 − γ 5 )u|0 = if M q µ corresponds to the meson production with f M being the decay constant of M. In Eq. (4), the parameter α M (β M ) is given where are similar to a 1 in Eq. (4) but for the M (c) modes. The amplitude of the Λ + c → Λℓν ℓ decay through c → sℓν ℓ can be given by replacing . Note that the extractions with the Λ b → pM and Λ b → Λ c M (c) decays are based on the validity of the factorization approach, which is supported by the recent observations. For example, with the factorization are calculated to be 13.2, 25.1, and 0.84 [7,9], in agreement with the data of 13.6 ± 1.6, 24.0 ± 3.8, and 0.84 ± 0.09 [1,10], respectively. The other justification is from the soft-collinear effective theory [11]. It is proposed that, if the factorization works, , which remarkably agree with the data of (6.2 +1.4 −1.3 ) × 10 −2 and (4.9 ± 0.4) × 10 −3 in the PDG [1], respectively.
The amplitudes in Eqs. (4) and (5) are related to the matrix elements of the and c → s, respectively, given by [12] where q = p − p ′ , s ± = (m 1 ± m 2 ) 2 − q 2 , and f = f j and g j (j = 0, +, ⊥) are the form factors in the helicity-based definition. The momentum dependences of f are written as [5] f (t) = 1 where m f pole are the pole masses and t 0 = (m 1 − m 2 ) 2 , while t + and a f 0,1 have been given in Refs. [5,13] . Consequently, one is able to integrate over the variables of the phase spaces in the two and three-body decays for the decay widths [1].
To demonstrate that the inclusions of B(Λ b → pM, Λ c M (c) ) in the extractions of |V ub | and |V cb | can reduce the theoretical uncertainties from the Λ b → (p, Λ c ) transition form factors due to the LQCD calculations in Eq. (6), we define which leads to that

III. NUMERICAL RESULTS AND DISCUSSIONS
For the numerical analysis, we perform the minimum χ 2 fit with the experimental inputs given in Table I, where |V ub | and |V cb | are treated as the free parameters to be determined.
The theoretical inputs for the CKM matrix elements and decay constants are given by [1] (|V tb |, 10 3 |V td |, In addition, we use and ranging from 2 to ∞ [8] for the error to account for the non-factorizable effects in the generalized factorization. Since Λ b → pM have been tested to be insensitive to the nonfactorizable effects [7], we adopt the values of a 1,4,6 from Ref. [8] with N ef f c = 3. Note that the initial inputs for the Λ b → (p, Λ c ) and Λ c → Λ form factors defined in Eq. (7) are chosen from Refs. [5,13].
There can be two issues for the simultaneous extractions of |V ub | and |V cb | in the exclusive Λ b decays. First, when the non-leptonic and semileptonic decays are all included in the  [1] global fit, there are some possible uncertainties from the data correlations, which should be avoided or estimated. According to the "CONSTRAINED FIT INFORMATION" in the PDG [1], B(Λ b → Λ + c π − , Λ + c K − ) and B(Λ b → pπ − , pK − ) are 94% and 83% correlated, respectively. Moreover, B(Λ b → Λ + c ℓν ℓ ) has 14% correlations with the individual value of As a result, we adopt B(Λ b → Λ + c π − ) = (4.57 +0.31 −0.30 ± 0.23)×10 −3 observed in Ref. [15] and rescaled in the PDG [1], instead of the weighted average one with other data, to minimize its correlation with B(Λ b → Λ + c ℓν ℓ ). We also use three different scenarios with or without including B(Λ b → Λ + c K − , pK − ) in the fit to estimate the uncertainties. Second, the recently updated data for B(Λ + c → pK − π + ) would help to data fitting as Λ + c is one of the final states. In Table I, we have used the revised R ub/cb value from Ref. [14], in which the correction is around 5%. Although B(Λ b → Λ + c K − ) has an unknown correction from B(Λ + c → pK − π + ) [16], it has been excluded in one of the fitting scenarios. On the other hand, it is found that , whose examinations rely on the measurements in Refs. [10,15,17]. Accordingly, the nine data points in Table I are classified into three types of inputs, being denoted as I1, I2, and I3, where I1 is for the semileptonic decays, while I2 and I3 for the non-leptonic ones with q = (d, s). There can be three scenarios for the extractions. In the first scenario (S1), we perform the global fit with the six data points in I1 and I2 of Table I, such that there is no correlation in the calculation, leading to (a M 1 , a Mc 1 ) = (1.14 ± 0.07, 0.98 ± 0.06) , Our fitting results for S1 are summarized in Table II, where we have also shown those from LHCb and LQCD.
As shown in Table II, we obtain |V cb | = (45.9 ± 2.7) × 10 −3 in S1, which agrees with the value of (42. LHCb [4]. Compared to |V cb | = (39.5 ± 0.8) × 10 −3 from the exclusive B decays, adopted by LHCb, our extracted value of |V cb | has a lager uncertainty. Nonetheless, we still get |V ub | with the error compatible to that of LHCb. This is due to the fact that the measured ) are involved in the fitting, which reduce the theoretical uncertainties from Λ b → (p, Λ c ) transition form factors to be 2 times smaller than the value of 0.68 ± 0.07 in Refs. [4,5]. It can be demonstrated by R F F = (0.67 ± 0.03, 0.68 ± 0.07) from the fitting and the initial LQCD inputs, respectively, where the nearly identical values of R F F show that LQCD calculation is also suitable for the two-body Λ b decays that proceed at the low q 2 regions, which have never been tested previously. It is interesting to note that the connection of the fitted values of |V ub | and |V cb | causes |V ub |/|V cb | = 0.081 ± 0.008, being nearly the same as the value from the LHCb extraction. We also predict B(Λ b → pµν µ ) = (5.2 ± 1.1) × 10 −4 , which is slightly lager than (4.1 ± 1.0) × 10 −4 by the extrapolation from the data at q 2 > 15 In the second scenario (S2), we fit with all data points from I1, I2 and I3 in Table I with the correlations. Note that B(Λ b → Λ c K − ) also mixes with the unknown contribution from B(Λ + c → pK − π + ). As a result, we are able to test possible deviations caused by the correlations as well as the new result from Λ + c → pK − π + . Furthermore, we only use the three data points from the semileptonic processes in I1 as the third scenario (S3). In this scenario,  there is no need to introduce the factorization. However, from Table II we find that the fitted results in S3 are very close to those in S1, indicating that the correlations and the effect of Λ + c → pK − π + are insensitive to the fit. We also see that, even without the factorization assumption, the central value of |V ub | in S3 is almost the same as those in S1 and S2, except the larger errors. This implies that the global fit with the additional non-leptonic decays reduces the uncertainties but without violating the outcome of the factorization.