Determination of $|V_{ub}|$ from exclusive baryonic $B$ decays

We use the exclusive baryonic $B$ decays to determine the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $V_{ub}$. From the relation $|V_{ub}|^2/|V_{cb}|^2=({\cal B}_\pi/{\cal B}_D){\cal R}_{ff}$ based on $B^-\to p\bar p \pi^-$ and $\bar B^0\to p\bar p D^0$ decays, where $|V_{cb}|$ and ${\cal B}_\pi/{\cal B}_D\equiv{\cal B}(B^-\to p\bar p \pi^-)/{\cal B}(\bar B^0\to p\bar p D^0)$ are the data input parameters, while ${\cal R}_{ff}$ is the one fixed by the $B\to p\bar p$ transition matrix elements, we find $|V_{ub}|=(3.48^{+0.87}_{-0.63}\pm 0.40\pm 0.07)\times 10^{-3}$ with the errors corresponding to the uncertainties from ${\cal R}_{ff}$, ${\cal B}_\pi/{\cal B}_D$ and $|V_{cb}|$, respectively. Being independent of the previous results, our determination of $|V_{ub}|$ has the central value close to those from the exclusive $\bar B\to \pi\ell\bar \nu_\ell$ and $\Lambda_b\to p\mu^-\bar \nu_\mu$ decays, but overlaps the one from the inclusive $\bar B\to X_u \ell\bar \nu_\ell$ with the current uncertainties. The extraction of $|V_{ub}|$ in the baryonic $B$ decays is clearly very useful for the complete determination of the CKM matrix elements as well as the exploration of new physics.


I. INTRODUCTION
In the standard model (SM), the unique physical phase in the 3 × 3 unitary Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] provides the only source for CP violation. However, it is known that this CP phase is not sufficient to solve the mystery of the matterantimatter asymmetry in the universe. To test the SM and look for other CP violation mechanisms, many CP violating processes, proceeding through the CKM matrix element V ub = |V ub |e −iγ with γ the CP phase, have been extensively explored by both experimental and theoretical studies. Nonetheless, with γ more precisely analyzed from the present data [3], the |V ub | determination is not conclusive. In particular, the experiments in the inclusiveB → X u ℓν ℓ and exclusiveB → πℓν ℓ decays give [3] respectively, where the first result has 3σ deviation from the second one. This is the wellknown long-standing tension between V ub measured by inclusive and exclusive decays at the B-factories, which triggers the theoretical studies in the SM [4,5].
To resolve the problem, it has been proposed that there exists some new physics, such as the right-handed quark current with the form ofūγ µ (1 + γ 5 )b inB → X u ℓν ℓ [6, 7], but not supported by the test of B → ρℓν ℓ [8]. It is also not sustained by the recent measurement of |V ub | = (3.27 ± 0.15 ± 0.17 ± 0.06) × 10 −3 [9] in the exclusive baryonic decay of Λ b → pµ −ν µ , which contains both contributions from the vector and axial-vector quark currents asB → X u ℓν ℓ . Clearly, the resolution of the dual nature of |V ub | in Eqs. (1) and (2) is one of the most important tasks in particle physics and it would lead to physics beyond the SM.
The exclusive baryonic B decays is worthwhile to have its own version for the extraction of |V ub |, which can be independent of the previous ones fromB → πℓν ℓ and Λ [9] with R F F as the ratio of the Λ b → p and Λ b → Λ c transition form factors calculated in the lattice QCD [10]. Likewise, by connecting B − → ppπ − andB 0 → ppD 0 decays that proceed through b → uūd and b → cūd at the quark level, and R f f the parameter related to the hadronic effects including those from the B → pp transition matrix elements. Note that the momentum dependences of these transition elements have been well studied in the literature [11][12][13][14] to explain the threshold effect in the baryonic B decays with fully accounted theoretical uncertainties. On the other hand, the decay ofB → πℓν ℓ can not be isolated from the uncertainty caused by the momentum dependences of the form factors in theB → π transition, calculated in different QCD models [17,18]. Besides, since |V ub | 2 /|V cb | 2 = (B π /B D )R f f receives the contributions from both vector and axial vector currents, it can also be used to test new physics in the form of the axial vector current. It is clear that once R f f is obtained in the baryonic B decays, one which is independent of the previous cases.
Subsequently, in terms of the amplitudes in Eq. (3), we derive |V ub | 2 /|V cb | 2 as where B π /B D ≡ B(B − → ppπ − )/B(B 0 → ppD 0 ), and R f f is given by with m 2 ij = (p i + p j ) 2 , in which the allowed ranges over the phase space can be referred in the PDG [3].

B. The |V ub | extraction
Since the relation in Eq. (7) can be used to extract |V ub |, we adopt the data from PDG as the experimental inputs, given by [3] (f D , f π ) = (204. For the theoretical inputs, we use the new extraction for the B → pp transition form factors, which includes the new observation of B(B − → ppe −ν e ) [19], such that the overestimation in Ref. [20] can be fixed. The results from the global fitting with all available data of B − → ppe −ν e and B → ppM (c) (M = π, K and K * and M c = D ( * ) ) are given by [15,20,21] (  [13][14][15][16]. In the generalized version of the factorization, one is able to float N c from 2 to ∞ to estimate the non-factorizable effects, resulting in a 1 = 1.05 ± 0.12. Since the parameter a 2 forB 0 → ppD 0 is sensitive to the non-factorizable effects, the fitting with the all available data gives a 2 = 0.42±0.04 [15,21]. We then estimate R f f in Eq. (8) to be which leads to |V ub |/|V cb | = 0.088 +0.022 −0.016 ± 0.010 with the errors from R f f and B π /B D , respectively. Since |V cb | has been well measured, with |V cb | = (39.5 ± 0.8) × 10 −3 [3], we obtain |V ub | = (3.48 +0.87 −0.63 ± 0.40 ± 0.07) × 10 −3 , with the third error for |V cb |.

III. DISCUSSIONS AND CONCLUSIONS
Our result in Eq. (12) is close to the exclusiveB → πℓν ℓ and Λ b → pµ −ν µ cases; particularly, nearly the same as |V ub | ≃ |Aλ 3 (ρ − iη)| ≃ 3.56 × 10 −3 in the Wolfenstein parameterization [3]. Nonetheless, the complete estimation of the theoretical uncertainties from the B → pp transitions gives the biggest error of 0.87 × 10 −3 , such that our result also overlaps the inclusive value in Eq. (1). While the tension between the exclusive and inclusive extractions in Eqs. (1) and (2) is suspected to be due to the underestimated theoretical uncertainties [8], in our case the range of |V ub | = (2.73 − 4.43) × 10 −3 seems to reconcile the difference.