Four-body baryonic $B\to{\bf B_1\bar B'_1 B_2 \bar B'_2}$ decays

LHCb has recently reported the first observation of a four-body baryonic $B\to{\bf B_1\bar B'_1 B_2\bar B'_2}$ decay, where ${\bf B_1\bar B'_1}$ and ${\bf B_2\bar B'_2}$ represent the two pairs of octet baryon states. In our classification, the measured $\bar B^0\to p\bar p p\bar p$ decay is a tree dominated process via internal $W$-boson emission, whose branching fraction is explained as small as $2.2\times 10^{-8}$. We investigate for the first time the phenomenology of other tree and penguin dominated $B\to{\bf B_1\bar B'_1 B_2\bar B'_2}$ decays, and predict the presence of a double threshold effect, manifested as two peaks around $m_{\bf B_1\bar B'_1}\sim m_{\bf B_1}+m_{\bf\bar B'_1}$ and $m_{\bf B_2\bar B'_2}\sim m_{\bf B_2}+m_{\bf\bar B'_2}$ in the invariant mass spectra of ${\bf B_1\bar B'_1}$ and ${\bf B_2\bar B'_2}$, respectively. Moreover, we predict the following branching fractions: ${\cal B}(B^-\to n\bar p p\bar p)=(1.7^{+0.4}_{-0.2}\pm 0.1^{+0.7}_{-0.4})\times 10^{-7}$, ${\cal B}(B^-\to \Lambda\bar p p\bar p)=(7.4^{+0.6}_{-0.2}\pm 0.03^{+3.6}_{-2.6})\times 10^{-7}$, and ${\cal B}(\bar B^0_s\to \Lambda\bar \Lambda p\bar p)=(1.9^{+0.3}_{-0.1}\pm 0.01^{+1.1}_{-0.6})\times 10^{-7}$, which are accessible to experimental facilities.


I. INTRODUCTION
The baryonic B decays have been extensively observed [1].A unique phenomenon in dibaryon formation, known as the threshold effect, has been observed in various decays, including B − → Λpγ [2], B − → ppµ − νµ [3], B → B B′ M [2,4], and B → B B′ MM ′ [5].This effect is manifested as a peak near the threshold region of m B B′ ∼ m B + mB′ in the dibaryon invariant mass spectrum.The threshold effect indicates that B B′ tends to be produced with little extra energy.It can be considered as an enhancing factor [6,7] for B(B − → ppπ − , ppπ − π 0 ) at the level of 10 −6 [1].Conversely, the formation of pp occurring away from the threshold region explains the small branching fraction B( B0 → pp), which is (1.27 ± 0.14) × 10 −8 [8].
The enhancement of branching fractions near the dibaryon spectra threshold needs to be carefully examined.
As the first observed four-body fully baryonic weak decay, B0 → pppp deserves close investigation.It is noteworthy that this decay is a tree-dominated process involving internal W -boson emission, where the exchanges of the two identical particle pairs pp and pp can lead to indistinguishable decay configurations.Additionally, we should consider its counterpart with external W -boson emission, as well as penguin-dominated decay processes, which have not yet been mentioned or measured.
In this paper, we propose to investigate the decays B → B 1 B′ 1 B 2 B′ 2 .Our analysis will focus on interpreting the branching fraction B( B0 → pppp) to demonstrate the validity of our theoretical approach.Additionally, we will study B − → nppp, B − → Λppp, and

B0
s → Λ Λpp as representative decay channels.We will derive the invariant mass spectra of B 1 B′ 1 and B 2 B′ 2 .Our aim through this study is to contribute to the improvement of theoretical understanding regarding baryon-pair hadronization in weak interactions.
2 µ|B in the factorization approach [11], where J 1 µ and J 2 µ are the currents associated with the b → uūd weak decay.Given that the matrix elements for the vacuum (0) → B 1 B′ 1 production and the B → B 2 B′ 2 transition have been studied in other baryonic B decays, the amplitude can be computed.Therefore, it is possible 2 decays are not so complicated as other four-body b-hadron processes.
We find that B − → nppp and B0 → pppp are currently the most measurable treedominated processes, making them the typical decays to explore.Using the effective Hamiltonian for quark-level b decays [12], we derive the amplitudes for B − → nppp and B0 → pppp within the factorization approach [11].They are given by where G F is the Fermi constant, and we define (q 1 q 2 ) V (A) ≡ q1 γ µ (γ 5 )q 2 and (q 1 q 2 ) S(P ) ≡ q1 (γ 5 )q 2 .In dealing with amplitudes involving pairs of identical particles, we follow the studies of π 0 (K L ) → e + e − e + e − [13,14] and four-body leptonic B decays [15,16].For the first time, both 0 → B B′ production and B → B B′ transition occur in a single decay.It is also the first time that each of the V (b) , A (b) , S (b) , and P (b) currents can form its own baryon-pair, which requires considering the interfering effects in the calculation.
The matrix elements of the B → B 2 (p 2 ) B′ 2 (p ′ 2 ) transitions are parameterized like those in the decays of B0 → ppD 0( * ) , B → ppM with M = (π, ρ, K ( * ) ), B − → ppℓν ℓ , and B → B B′ ℓ l, written as [18,[24][25][26][27][28][29][30][31][32][33]] where V b µ (A b µ ) ≡ qγ µ (γ 5 )b, S b (P b ) ≡ q(γ 5 )b, and FB B′ = (g i , f i , ḡj , fj ) with i = 1, 2, ..., 5 and j = 1, 2, 3 are the B → B 2 B′ 2 transition form factors. Inspired by the pQCD counting rules, one obtains FB B′ ∝ 1/t m with t ≡ (p 2 +p ′ 2 ) 2 [18, [24][25][26][27]33], where m = 2 + 1 is in accordance with the fact that there should be 2 gluons for attaching the valence quarks in B 2 B′ 2 and 1 for speeding up the spectator quark in B. We thus present FB B′ as [23,33] Notably, the hard gluon picture of 1/s n and 1/t m has been utilized to explain or predict the threshold effect observed in the invariant dibaryon mass spectra of B − → ppµν µ [3,33], e + e − → pp [38], and Under the SU(2) helicity [SU(2) h ] and SU(3) flavor [SU(3) f ] symmetries, the form factors can be related to one another, and then reduced.For demonstration, we perform the derivation to relate F 1 and g A .Since the timelike form factors can be seen to behave like the spacelike ones according to the crossing symmetry, the relation of F 1 and g A obtained in the spacelike region can be adopted in the timelike region.Accordingly, we recall that [23] where R µ = (V µ +A µ )/2 is the right-handed chiral current, the octet baryons are decomposed as the two chiral states: L with |B ′ R(L) consisting of q R q L q R (q L q R q L ) and their reorderings, and F R,L the baryonic form factors in the chiral representation.
When µ = 0 is fixed in Eq. ( 10), R µ reduces to a right-handed charge density, which acts on a valence quark in |B ′ R+L and transforms B ′ into B. At large energy transfers, such as s = (p 1 + p ′ 1 ) 2 around a few GeV 2 , the chirality R (L) can approximately be taken as the helicity ↑ (↓).Subsequently, q i,R with i = (1, 2, 3) in B ′ R(L) can be illustrated to have the helicity parallel (anti-parallel) [||(||)] to the helicity of B ′ .Thus, the right-handed charge density that acts on q i,R can be more specifically denoted as [22,35], where e R,L ||(||) sum over the weight factors of the baryon states, given by Since As a result, we relate C F )/2 are able to relate (f S , g P ), (f i , g i ) and ( fj , ḡj ), respectively.Here, we list the relations we need in this study, given by [17,22,28]   The large angular asymmetries measured in the decays B0 → Λpπ + and B − → Λpπ 0 indicate that the SU(2) chiral symmetry is broken for F B B′ [2,23].Therefore, in Eq. ( 13 [17,23,28].The chiral-flip form factors F 2 and h A do not have any relations imposed by the SU(2) h symmetry.In the pQCD model [40], F 2 has been calculated as which agrees with the parameterization in Eq. (7).For h A , the SU(3) f symmetry can be applied, such that C h A in Eq. ( 7) is related to the SU(3) f parameters C D and C F .Hence, we obtain [22] [31,32] 2 )/t, and λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2bc − 2ca, where |M| 2 is the squared amplitude with the summation over the spins.For the integration, the allowed ranges of (s, t) and (θ B 1 , θ B 2 , φ) are given by
The effective Wilson coefficients c ef f i can be found in Table I, which take into account the quark rescattering effects in the b decays [11].For F and FB 2 B′ 2 , the constants in Eqs.(13,14) and ( 15) have been extracted in Refs.[23,33] Consequently, we calculate the branching fractions of II.In Fig. 3 we draw the m B 1 B′ 1 and m B 2 B′ 2 invariant mass spectra.To get a i , we have used the generalized edition of the factorization [11,43], where N (ef f ) c , the effective color number from 2 to ∞, estimates the non-factorizable QCD corrections.According to B( B0 → pppp) as measured by LHCb, we determine N ef f c = 2.50 ± 0.06.

IV. DISCUSSION AND CONCLUSIONS
Using the amplitude given in Eq. ( 2) and considering the baryonic form factors in Eqs.(6) and (8)    M * dir M ex represents the interference term.It is worth noting that |M ex | 2 integrated over the total phase space is identical to |M dir | 2 integrated over the total phase space, resulting in B dir = B ex .However, calculating B dir×ex can be challenging.The difficulty arises from the momentum dependences of the form factors.For M dir , we have F B B′ ∝ 1/s 2 and FB B′ ∝ 1/t 3 .On the other hand, for M ex , we have F ex B B′ ∝ 1/s ′2 and F ex B B′ ∝ 1/t ′3 with s ′ ≡ (p 2 + p ′ 1 ) 2 and t ′ ≡ (p 1 + p ′ 2 ).Fortunately, the threshold effect observed in baryonic B decays can be utilized to estimate B dir×ex .
When the threshold effect occurs, M dir can be associated with the configurations depicted in Figs.4(a, d), where B 1 B′ 1 (B 2 B′ 2 ) tends to move collinearly.In addition, the quark pairs can be parallel in the direction of motion [7], leading to a stronger association for their hadronization.Consequently, the form factors 6 present in M dir can enhance the branching fraction [30].In the case of M ex , B 1(2) and B′ 1(2) are depicted as moving back-to-back in Figs.4(b, c).In these configurations, each q q pair is anti-parallel in the direction of motion.Since the anti-parallel configuration of the q q pair requires a large energy transfer from the gluon [30], the gluon propagator is suppressed by a factor of 1/s ′ (1/t ′ ) with √ s ′ ( √ t ′ ) being away from the threshold.This mechanism has been previously employed to explain the suppression of B( B0 → pp) to a level of 10 −8 [7], where the valence quark pairs also exhibit an antiparallel configuration in the moving directions.Specifically, we find that F ex B B′ ≃ (1/4) 2 F B B′ and F ex B B′ ≃ (1/4) 3 FB B′ with the occurrence of the threshold effect.Consequently, the contribution from the interference term B dir×ex is estimated to be 0.14 × 10 −8 , which is B 1 + B 2 with (B 1 , B 2 ) = (0.6, 1.3) × 10 −7 .These results demonstrate that the two penguin configurations make compatible contributions.In Fig. 3, we illustrate the double threshold effect using the invariant mass spectra of m B 1 B′ 1 and m B 2 B′ 2 .Additionally, we show the four partial branching fractions as functions of m Λ Λ, m pp , m Λp , and m p Λ for B0 s → Λ Λpp.These distributions can be used to test whether the decay really proceeds through two distinct penguin-level configurations.
As a final remark, once our approach is validated for B decays into four baryons, it opens up possibilities for further investigations into the direct CP asymmetry and triple product asymmetry, extensively studied in various baryonic decay processes [21,[43][44][45][46]. Exploring these observables in the four-body fully baryonic decays would provide valuable insights into baryonic CP violation, which plays a crucial role in understanding the matter-antimatter asymmetry in the universe.
In summary, we have conducted a comprehensive study of four-body fully baryonic B decays, with a particular focus on the recently observed B0 → pppp decay by the LHCb collaboration.We have provided an explanation for its small branching fraction, where we have considered the exchange of identical particles (pp and pp) that leads to indistinguishable configurations.Our analysis has revealed that the tree-dominated decay B − → nppp can be more favorable than B0 → pppp, according to the prediction B(B − → nppp) = (1.7 +0. 4  −0.2 ± 0.1 +0.7 −0.4 ) × 10 −7 .Furthermore, we have investigated penguin-dominated decay channels that have not been measured or studied before.Our calculation has predicted B(B − → Λppp) = (7.4

2 B′ 2 .
Due to flavor conservation in dibaryon formation, we present four different configurations as shown in Fig. 1 for charmless B → B 1 B′ 1 B 2 B′ 2 decays.As an example, the decay depicted in Fig. 1(a) proceeds through the emission of an external W -boson, producing the B 1 B′ 1 pair, along with the B meson transition to B This configuration corresponds to the amplitude 1 and C g A to C || and C || by means of the SU(2) h and SU(3) f symmetries.Similarly, the chiral currents R ≡ (S + P )/2, R b µ ≡ (V b µ + A b µ )/2 and R b ≡ (S b + P b

TABLE I .
The effective Wilson coefficients c ef f i from for the matrix elements of B 1 B′ 1 production and B → B 2 B′ 2 transition, we analyze

TABLE II .
Our calculations for the four-body baryonic B decays, where the errors come from the non-factorizable QCD corrections, CKM matrix elements, and the form factors, in order.