Four-body baryonic decays of B → p ¯ p π + π − ( π + K − ) and (cid:3) ¯ p π + π − ( K + K − )

We study the four-body baryonic B → B 1 ¯ B 2 M 1 M 2 decays with 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 → M 1 M 2 transition together with the production of a baryon pair. We obtain that B ( B − → (cid:3) ¯ p π + π − ) = ( 3 . 7 + 1 . 5 − 1 . 0 ) × 10 − 6 and B ( ¯ B 0 → p ¯ p π + π − , p ¯ p π + K − ) = ( 3 . 0 ± 0 . 9 , 6 . 6 ± 2 . 4 ) × 10 − 6 , in agreement with the data. We also predict B ( B − → (cid:3) ¯ pK + K − ) = ( 3 . 0 + 1 . 3 − 0 . 9 ) × 10 − 6 , which is accessible to the LHCb and BELLE experiments. © 2017 The Author(s). Published


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 b → pπ − π + π − [6], and measured TPCs in b → pK − μ + μ − [7]. As the similar baryonic cases, the four-body baryonic B decays can also provide TPCs.
where the resonant B(B − → p (ρ 0 , f 2 (1270) →)π + π − ) have been excluded from the data [8]. [9], the decays with B ∼ 10 −6 in Eq. (1) are recognized to have the same theoretical correspondence, where tion along with the B 1B2 production, as depicted in Fig. 1. Note that the B 0 s decays of B 0 s → ppK ± π ∓ and ppK + K − with s being replaced by d in B 0 → ppπ + π − and ppK ∓ π ± have also been found with the branching ratios of order 10 −6 [9], respectively.
In this report, we will calculate the four-body baryonic B decays in accordance with the decaying processes in Fig. 1

Formalism
In terms of the quark-level effective Hamiltonian 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 for q 1 γ μ (γ 5 )q 2 and q 1 (γ 5 )q 2 , respectively. The parameters α q ξ and β q η in Eq. (2) are given by Note that the B 0 → ppπ + K − and B 0 s → pp K + K − decays have the matrix elements of pp|(ss) V ,A,S,P |0 with the ss quark currents, which eventually cause the terms of α s 4,6,10 to give nearly zero contributions due to the OZI suppression of ss → pp [11].
For the matrix elements in Eq. (2), the baryon-pair productions from the quark currents are given by [5,12] where p = p M 2 + p M 1 and (h, r, w ± ) are the form factors. Subsequently, one can also get M 1 M 2 |q 1 (γ 5 )b|B from Eq. (6) based on equations of motion. In terms of the approach of pQCD counting rules, the momentum dependences for the 0 → B 1B2 and B → M 1 M 2 transition form factors are given by [14][15][16][17] [18], which is much less than F 1 , while the small value of B(B 0 → pp) = (1.5 +0.7 −0.5 ) × 10 −8 [19,20] causes a tiny C h A [21] in h A = C h A /t 2 , 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 decay relies on the five kinematic variables, that is, s ≡ p 2 , t and the three angles of θ B , θ M and φ. In Fig. 2  B 1B2 and M 1 M 2 pairs in the B rest frame, respectively. The partial decay width reads [23,24] where X , α B and α M are given by the allowed ranges of the five variables are given by
where the added constants for the broken effects have been approved by the excellent agreement for B(B 0 s → pK + +¯ pK − ) [29]. Subsequently, we evaluate the branching ratios of B → B 1B2 M 1 M 2 as shown in Table 2, and draw the distributions vs. m B 1B2 in Fig. 3. Table 2 With the replacement of B − → π + π − by B − → K + K − , the B − → pπ + π − and pK + K − decays share the same decaying configuration. We hence predict that B(B − → pK + K − ) = (3.0 +1.3 −0.9 ) × For the measured branching ratio of B 0 → ppπ + K − + ppπ − K + , Table 1 The parameters a i with N ef f c = 2, 3, and ∞ to estimate the non-factorizable effects in the generalized factorization.  Table 2 The branching ratios of B → B 1B2 M 1 M 2 , where the errors come from the nonfactorizable effects, CKM matrix elements, and form factors, respectively.

As seen in
Branching ratios Our results Data −0.5 ± 0.1 ± 0.9 5 .9 ± 1.1 it is found that the contribution is mainly from the penguin-level are also sensitive to the non-factorizable 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 nonfactorizable effects. In Table 2 we have included the data to constrain the non-factorizable effects, which results in δN ef f c = 0.06. We note that the two spectra in Fig. 3 for B − → pM 1 M 2 and B 0 → ppM 1 M 2 present the threshold effects as the peaks around the threshold areas of m p m +mp and m pp m p +mp, respectively, which are commonly observed in the three and four-body baryonic B decays [9,26].
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 of B 0 s → (K + π − , K + K − ). This calls for the theoretical and experimental studies of the three-body mesonic B 0 s decays that could proceed with the B 0 s → M 1 M 2 transitions, such as the B 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 observed B 0 s → D 0 K + π − and B 0 s → D 0 K + K − decays [20] are also important as they relate to w − .

Conclusions
In sum, we have studied the charmless four-body baryonic benefits the future test of T violation, as the T-odd triple momentum product correlation of p 1 · ( p 2 × p 3 ) can be directly constructed.