Angular asymmetries in $B\to\Lambda\bar p M$ decays

The forward-backward angular asymmetry (${\cal A}_{FB}$) for $\bar B^0\to \Lambda\bar p\pi^+$ measured by Belle has presented an experimental value in the range of $-30\%$ to $-50\%$. In our study, we find that ${\cal A}_{FB}[\bar B^0\to\Lambda\bar p \pi^+(B^-\to \Lambda\bar p\pi^0)]$ can be as large as $(-14.6^{+0.9}_{-1.5}\pm 6.9)\%$. In addition, we present ${\cal A}_{FB}[\bar B^0\to\Lambda\bar p \rho^+(B^-\to \Lambda\bar p\rho^0)] =(4.1^{+2.8}_{-0.7}\pm 2.0)\%$ as the first prediction involving a vector meson in the charmless $B\to{\bf B\bar B'}M$ decays. While ${\cal A}_{FB}(B\to\Lambda\bar p M)$ indicates an angular correlation caused by the rarely studied baryonic form factors in the timelike region, LHCb and Belle~II are capable of performing experimental examinations.

On the other hand, B( B0 → pp) is as small as 10 −8 [10,11], whose suppression reflects the fact that in the two-body baryonic B decays the B B′ formation with m B B′ ∼ m B is away from the threshold area.Theoretically, the baryonic form factors that parameterize the dibaryon formation have been used to describe the threshold effect [12][13][14][15][16][17][18], such that B(B → B B′ M (c) ) can be explained.
The partial branching fraction can be a function of cos θ B( B′ ) , where θ B( B′ ) is the angle between the (anti-)baryon and meson moving directions in the B B′ rest frame.It leads to the forward-backward angular asymmetry: A F B (B → B B′ M (c) ) ≡ (B + − B − )/(B + + B − ), where B + = B(cos θ B( B′ ) > 0) and B − = B(cos θ B( B′ ) < 0).The forward-backward asymmetries have been found in several decay channels [5,6,19,20], of which the interpretations have caused theoretical difficulties [18,21,22].This indicates that the dibaryon production in B → B B′ M has not been fully understood [23].
Compared to the charmful B0 → ΛpD ( * )+ decay channels, where the Λp formation is from the (axial)vector current, the penguin-dominant B → Λpπ decay has an additional contribution from the (pseudo)scalar current.Consequently, there might exist an interference between the (axial)vector and (pseudo)scalar currents, which can cause a possible angular asymmetry.Therefore, we propose to investigate B → Λpπ, along with the rarely studied baryonic form factors in the timelike region.We will also study A F B (B → Λpρ), which can be the first prediction involving a vector meson in the charmless B → B B′ M decays.The isospin relations will be discussed.

II. FORMALISM
According to Fig. 1, where the decay process is drawn with the B meson transition to a meson, along with the dibaryon production, the amplitude of B → ΛpM can be factorized as [21,24,[27][28][29] with q = d and u for B0 → Λpπ + (ρ + ) and B − → Λpπ 0 (ρ 0 ), respectively, where G F is the Fermi constant, (q i q j ) V −A = qi γ µ (1 − γ 5 )q j , (q i q j ) S±P = qi (1 ± γ 5 )q j , and |0 in Λp|(su)|0 denotes the vacuum state.We define , where V q i q j are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and the parameters a 1,4,6 consist of the effective Wilson coefficients c ef f i [30] , given by with N c the color number.
In Eq. ( 2), the matrix elements of the B to π(ρ) transition can be written as [31] π|(ūb with and ε * µ defined as the polarization four-vector of the ρ meson, where are the mesonic form factors.Using the equation of motion, we obtain π|(ūb) The form factor F A(B) can be given in the three-parameter representation [31]: where F A(B) (0) is the form factor at the zero momentum transfer squared (t = 0), σ 1,2 the parameters, and M A(B) the pole mass.One has determined F A(B) (0), σ 1,2 and M A(B) with the results of the model calculation, such that the momentum transfer squared dependences for F A(B) can be described.
Using the SU(2) helicity [SU(2) h ] symmetry, F 1 and g A can be related.To this end, we parameterize B R+L |J R µ |B ′ R+L in the spacelike region as [13,32] where L , and F R,L the chiral form factors. Furthermore, we define Q ≡ J R 0 as the chiral charge to act on the valence quark q i in B ′ (q 1 q 2 q 3 ), such that one transforms B ′ into B. With the chirality that is regarded as the helicity at t → ∞, the helicity of q i can be (anti-)parallel [||(||)] to the helicity of B ′ , such that we denote the chiral charge for q i by Q ||(||) (i) (i = 1, 2, 3).Thus, we obtain [13,32] where In the crossing symmetry, since the spacelike form factors can be seen to behave as the timelike ones, the derivation can be applied to those in Eqs.(6,7).Similarly, we relate f S and g P .
It is hence obtained that with and C * || ≡ C|| + δ C|| , where the second line for np|( du) V,A |0 is to include more data in the numerical analysis.Note that our derivation depends on the SU(2) helicity [SU (2) h ] symmetry at large t (t → ∞), where quarks can be seen as massless particles.In the baryonic B decay processes, since t is ranging from (m B + mB′) 2 to (m B − m M ) 2 , instead of t → ∞, the fact that the quarks are no longer massless can induce the SU(2) h symmetry breaking.Hence, δC ||(||) and δ C|| are added in Eq. (10) to estimate the possible broken symmetry effect.One has derived F 2 = F 1 /(tln[t/Λ 2 0 ]) in the pQCD model [34], which verifies the parameterization in Eq. (7).By contrast, the model calculation for h A has not been available yet.In the SU(3) flavor [SU(3) f ] symmetry, C h A can be related as [32] C for Λp|(su) A |0 and np|( du) A |0 , respectively.
To integrate over the phase space in the three-body B → B B′ M decays, we adopt the equation as [18,22,35] where and Γ represents the decay width.Moreover, | M| 2 denotes the squared amplitude summed over the baryon spins.We choose θ as the angle between B′ and M moving directions in the B B′ rest frame.Accordingly, the (anti-)baryon energy can be a function of cos θ, given by We reduce | M | 2 (B → Λpπ) as with reduced form is for a simple presentation; however, no approximation is made in the real calculation.Similarly, we obtain For the angular asymmetry, we define where dΓ/d cos θ is the angular distribution.
From Ref. [30], with N c = (2, 3, ∞), where N c is taken as a floating number, in order that the nonfactorizable QCD corrections can be estimated in the generalized edition of the factorization [30].In Eqs.(10) and (11), there are totally eight constants that correspond to the baryonic form factors: We perform the minimum χ 2 -fit to extract the constants, which includes the experimental inputs from B( B0 → Λpπ + ), B(B − → Λpπ 0 (ρ 0 )), A F B ( B0 → Λpπ + ), and A F B (B − → Λpπ 0 ) in Table III, and the angular distribution of B0 → Λpπ + in Fig. 2; the branching fractions [38,39], and A F B ( B0 → ΛpD ( * )+ ) [19] are also included.We thus present the results of the global fit in Table II.
Since we get C g P = (0.38 ± 0.37)C f S different from C g P = C f S in the SU(2) helicity symmetry at t → ∞, it clearly indicates a broken symmetry effect with δ C|| .Currently, δ C|| is determined with a large uncertainty, which reflects the fact that A F B (B → Λpπ) and the angular distribution in Fig. 2 have not been precisely measured.Without a model calculation, we obtain C h A from the fit.It is found that C h A = −798.6GeV 6 for Λp|(su) A |0 gives 2.8% of B( B0 → Λpπ + ) and 3.4% of A F B ( B0 → Λpπ + ).
The angular asymmetry of B0 → Λpπ + decay was once studied in Ref. [21], where A F B ( B0 → Λpπ + ) ≃ 0 is not verified by the observation.The cause is that g P = f S as a strong relation has been used, such that g P with 2|α 6 | 2 [f 2 S (t − 4m 2 p ) + g 2 P t] in the a term becomes the dominant form factor in the branching fraction.By contrast, f S turns out to be a less important form factor both in the a and b terms, leading to A F B ( B0 → Λpπ + ) ≃ 0.
For the first time, we study the angular asymmetry of the charmless B → B B′ M decay

FIG. 2 .
FIG.2.The angular distribution of B0 → Λpπ + with the solid (dotted) line for the central value (error), where the data points are adopted from Ref.[20].

TABLE II .
Fit results of the constants (C i ) derived from the baryonic form factors, along with the χ 2 value; n.d.f denotes the number of degrees of freedom.We study the penguin-dominant B → Λpπ decay with the branching fraction and angular asymmetry.With χ 2 /n.d.f ≃ 2.7 calculated from TableII, it demonstrates that the theoretical study can accommodate the experimental data.Particularly, we find that ∆χ 2 = 14.3 that comes from A F B (B → Λpπ) and five data points of dB( B0 → Λpπ + )/d cos θ gives sizeable contribution to the total χ 2 value, indicating that more accurate measurements of

TABLE III .
Branching fractions and angular asymmetries of the baryonic decay channels, where the first error of our results estimates the non-factorizable effects, while the second one combines the uncertainties from CKM matrix elements and the hadronic parameters.