Baryonic $B$ meson decays

We review the two and three-body baryonic $B$ decays with the dibaryon (${\bf B\bar B'}$) as the final states. Accordingly, we summarize the experimental data of the branching fractions, angular asymmetries, and $CP$ asymmetries. Using the $W$-boson annihilation (exchange) mechanism, the branching fractions of $B\to {\bf B \bf \bar B'}$ are shown to be interpretable. In the approach of perturbative QCD counting rules, we study the three-body decay channels. In particular, we review the $CP$ asymmetries of $B\to {\bf B\bar B'}M$, which are promising to be measured by the LHCb and Belle~II experiments. Finally, we remark the theoretical challenges in interpreting ${\cal B}(B^-\to p\bar p\rho^-)$ and ${\cal B}(B^-\to p\bar p\mu^-\bar \nu_\mu)$.

The partial branching fraction of B → BB ′ M can be a function of cos θ B , where θ B is the angle between the baryon and meson moving directions in the dibaryon rest frame. One hence defines the forward-backward angular asymmetry, In Table I, A F B (B − → ppπ − , ppK − ) = (−40.9 ± 3.4, 49.5 ± 1.4)% [2] indicate that one of the dibaryon favors to move collinearly with the meson.
We search for the theoretical approach to interpret the threshold effect, branching fractions and angular asymmetries of the baryonic B decays. We find that the factorization approach can be useful [16], where one factorizes (decomposes) the amplitude of the decay as two separate matrix elements. In our case, we present where (qq ′ ) and (qb) stand for the quark currents, and the matrix element of BB ′ |(qq ′ )|0 ( BB ′ |(qb|B ) can be parameterized as the timelike baryonic (B to BB ′ transition) form factors F BB ′ . Moreover, one derives F BB ′ ∝ 1/t n in perturbative QCD (pQCD) counting rules [17][18][19][20][21][22][23][24], where t ≡ (p B + pB′) 2 and n accounts for the number of the gluon propagators that attach to the baryon pair. It results in dB/dm BB ′ ∝ 1/t 2n , which shapes a peak around and then the threshold effect can be interpreted.
In the B → pp transition, there exists the term of (pp − p p ) µū (γ 5 )v for F BB ′ [24], which is reduced as (Ep − E p )ū(γ 5 )v in the pp rest frame. Since (Ep − E p ) ∝ cos θ p , the term for F BB ′ can be used to describe the highly asymmetric A F B (B − → ppπ − , ppK − ). Alternatively, the baryonic B decays is studied with the pole model, where the non-factorizable contributions can be taken into account [25][26][27][28].

II. FORMALISM
To review the two-body baryonic B decays, we takeB 0 → pp as our example. According to Fig. 1,B 0 → pp is regarded as an W -boson exchange process [29,[39][40][41]. In the factorization, we derive the amplitude as [29] M where G F is the Fermi constant, and V where f B is the decay constant and q µ the four-momentum. For the pp production, the matrix elements read [22,23] with the (axial-)vector current V (A) µ =qγ µ (γ 5 )q ′ , where F 1,2 , g A and h A are the timelike baryonic form factors.
As V µ and A µ are combined as the right or left-handed chiral current, that is, With the right-handed current, the matrix elements can be written as [19,29] where |B R+L = |B R + |B L , and F R,L are the chiral form factors. With q i (i = 1, 2, 3) denoting one of the valence quarks in B, Q ≡ J R µ=0 known as the chiral charge is able to change the flavor for q i , such that B is transformed as B ′ . Note that the chirality is regarded as the helicity at Q 2 → ∞. Since the helicity of q i can be (anti-)parallel [||(||)] to the helicity of B, we define Q ||(||) (i) that is responsible for acting on q i . Thus, the approach of pQCD counting rules lead to [19,29] (2) spin symmetries are both respected. In the crossing symmetry, the spacelike form factors behave as the timelike ones, such that one can relate F 1 and g A with the chiral form factors in Eqs. (6,7) derived in the spacelike region, leading In addition to the momentum dependence of Eq. (5), F 1 and g A are presented as [22,23,29] where γ = 2 + 4/(3β) = 2.148.
To describe the three-body baryonic B decays, we take B → ppV with V = ρ −(0) or K * − (K * 0 ) as our examples. According to Fig. 2, the amplitudes are given by [30][31][32] with q = (s, d) and q ′ = u for B − → pp(K * − , ρ − ), and q = (s, d) and q ′ = d forB 0 → pp(K * 0 , ρ 0 ). For α V , we define In Eq. (12), one presents that V |qγ µ (1 − γ 5 )q ′ |0 = f V m V ε * µ , where f V and ε * µ are, respectively, the decay constant and polarization four-vector of the vector meson. The amplitudes are both associated with the matrix elements of the B → BB ′ transition, and we parameterize them as [24] where , and (f i , g i ) (i = 1, 2, ..., 5) are the B → BB ′ transition form factors. Inspired by pQCD counting rules [17,18,21,23,24], the momentum dependences for f i and g i are given by with D f i (g i ) to be extracted by the data. According to the gluon lines in Fig. 2, n in 1/t n should be 2+1, which accounts for two gluon propagators attaching to the valence quarks in BB ′ and an additional one for kicking (speeding up) the spectator quark in B [23]. For the gluon kicking, it is similar to the meson transition form factor derived as F M ∝ 1/q 2 in pQCD counting rules [20,51], where 1/q 2 is for a hard gluon to transfer the momentum to the spectator quark in the meson.
Like the case of F 1 and g A , we relate (f i , g i ) to the B → BB ′ chiral form factors, which is in terms of [23,24] where |B q ∼ |bγ 5 q|0 has been used. In addition, we obtain G R(L) = e for pp|(ūb)|B − and pp|(db)|B 0 , respectively. We also review the direct CP asymmetry, defined by where Γ denotes the decay width, andB →BB ′M the anti-particle decay.

Decay mode
Exp't data [1,2,12] Theory [31] given by (f B , f ρ , f K * ) = (0.19, 0.21, 0.22) GeV [1,54]. With the global fit to the data, we obtain [29,37] ( We take a i ≡ c ef f i + c ef f i±1 /N c for i =odd (even) with N c the color number, where the effective Wilson coefficients c i come from Ref. [16]. For a 2 inB 0 → ppρ 0 andB 0 → pp, we use N c = 2 to take into account the non-factorizable QCD corrections. We can hence present our theoretical calculations for B → pp(V ) in Table II, along with the other B → BB ′ M results. Besides, we present our studies of the angular and CP asymmetries.

IV. DISCUSSIONS AND CONCLUSIONS
The theoretical results in Table II can agree well with the data, which is based on the factorization, pQCD counting rules, and baryonic form factors. In particular, several CP containing a charm quark, which are compared with the experimental data.
As a consequence, (B, A F B ) can also be expalined (see Table III where the amplitude of B − → ppµ −ν µ is given by Since B → ppρ and B − → ppµ −ν µ are seen to be associated with the B → pp transition form factors, which are inferred to cause the overestimations. Besides, B(B − → ppµ −ν µ ) inconsistent with the data can be partly due to the inconsistent determination of |V ub | between the inclusive and exclusive B decays.
As the final remark, since the predictions of B(B − → ppρ − ) and B(B − → ppµ −ν µ ) are shown to deviate from the observations by the factors of 6 and 20, respectively, the theoretical approach is facing some difficulties. Therefore, the re-examination should be performed elsewhere.
In summary, to review the baryonic B decays, we have summarized the experimental data, which includes branching fractions, angular and CP asymmetries. We have taken B 0 → pp and B → ppV with V = ρ −(0) or K * − (K * 0 ) for theoretical illustration. We have also reviewed the CP asymmetries of B → ppM, which can be used to compare with future measurements by LHCb and Belle II. With the theoretical results listed in the tables, we have demonstrated that the theoretical approach can be used to interpret most observations. Finally, we have also remarked that the theoretical approach has currently encountered some challenges in interpreting B(B − → ppρ − ) and B(B − → ppµ −ν µ ).