Four-body baryonic decays of $B\to p \bar{p} \pi^+\pi^-(\pi^+K^-)$ and $\Lambda \bar{p} \pi^+\pi^-(K^+K^-)$

We study the four-body baryonic $B\to {\bf B_1 \bar B_2}M_1 M_2$ decays with $\bf B_{1,2}$ ($M_{1,2}$) being charmless baryons (mesons). In accordance with the recent LHCb observations, each decay is considered to proceed through the $B\to M_1 M_2$ transition together with the production of a baryon pair. We obtain that ${\cal B}(B^-\to \Lambda\bar p \pi^+\pi^-)=(3.7^{+1.5}_{-1.0} )\times 10^{-6}$ and ${\cal B}(\bar B^0\to p\bar p \pi^+\pi^-,p\bar p \pi^+ K^-)=(3.0\pm 0.9,6.6\pm 2.4)\times 10^{-6}$, in agreement with the data. We also predict ${\cal B}(B^-\to\Lambda\bar p K^+ K^-)=(3.0^{+1.3}_{-0.9})\times 10^{-6}$, which is accessible to the LHCb and BELLE experiments.


I. INTRODUCTION
One of the main purposes of the B factories and current LHCb is to study CP violation (CPV), which is important for us to understand the puzzle of the matter-antimatter asymmetry in the Universe. As the observables, the (in)direct CP-violating asymmetries (CPAs) require both weak and strong phases [1][2][3], whereas the T-violating triple momentum product correlations (TPCs), such as p 1 · ( p 2 × p 3 ) in a four-body decay, do not necessarily need a strong phase [4,5]. For example, the LHCb Collaboration has provided the first evidence for CPV from the TPCs in Λ b → pπ − π + π − [6], and measured TPCs in Λ b → pK − µ + µ − [7].
As the similar baryonic cases, the four-body baryonic B decays can also provide TPCs.
In this report, we will calculate the four-body baryonic B decays in accordance with the decaying processes in Fig. 1

II. FORMALISM
In terms of the quark-level effective Hamilontion for the charmless b → q 1q2 q 3 transition, the amplitudes of the four-body baryonic B decays by the generalized factorization approach are derived as [10] where G F is the Fermi constant, V ij are the CKM matrix elements, and (q 1 q 2 ) V (A) and (q 1 q 2 ) S(P ) stand forq 1 γ µ (γ 5 )q 2 andq 1 (γ 5 )q 2 , respectively. The parameters α q ξ and β q η in Eq.
For the matrix elements in Eq. (2), the baryon-pair productions from the quark currents are given by [5,12] where where p = p M 2 + p M 1 and (h, r, w ± ) are the form factors. Subsequently, one can also get   [18], which is much less than F 1 , while the small value of we may not consider the effects from F 2 and h A . In addition, by following Ref. [16], we have neglected the terms related to r and w + in Eq. (6) due to the wrong parity [22].
The integration over the phase space of the four-body where X, α B and α M are given by respectively, with λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2bc − 2ca, while the allowed ranges of the five variables are given by
where the added constants for the broken effects have been approved by the excellent agree- . Subsequently, we evaluate the branching ratios of B → B 1B2 M 1 M 2 as shown in Table II, and draw the distributions vs. m B 1B2 in Fig. 3. Table II [30][31][32]. For the measured branching ratio ofB 0 → ppπ + K − + ppπ − K + , it is found that the contribution is mainly from the penguin-level dominantB 0 → ppπ + K − mode. Note that a 3,5 from α s ± ≃ β s ± = −V tb V * ts (a 3 ± a 5 + a 9 ) are also sensitive to the nonfactorizable effects. With N ef f c = 3, we obtain B(B 0 → ppπ + K − ) = (6.6 ± 2.4) × 10 −6 , which suggests that the decay is free from the non-factorizable effects. In Table II we have  We note that the two spectra in Fig. 3 for B − → ΛpM 1 M 2 andB 0 → ppM 1 M 2 present the threshold effects as the peaks around the threshold areas of m Λp ≃ m Λ + mp and m pp ≃ m p + mp, respectively, which are commonly observed in the three and four-body baryonic B decays [9,26].

As seen in
Finally, we remark that we cannot explain the data of B(B 0 s → ppK ± π ∓ , ppK + K − ) = (1.5 ± 0.7, 4.6 ± 0.6) × 10 −6 measured by the LHCb [9] due to the lack of the information for the transition form factors ofB 0 s → (K + π − , K + K − ). This calls for the theoretical and experimental studies of the three-body mesonicB 0 s decays that could proceed with theB 0 s → M 1 M 2 transitions, such as theB 0 s → D * − s π + K 0 ,B 0 s → D * 0 π + K − (K + K − ) and B 0 s → ρ − π + K 0 decays with one of the mesons to be a vector one, in order to extract both (h, w − ) in Eq. (6). On the other hand, the observedB 0 s → D 0 K + π − andB 0 s → D 0 K + K − decays [20] are also important as they relate to w − .