Study of $B^-\to \Lambda\bar p\eta^{(')}$ and $\bar B^0_s\to \Lambda\bar\Lambda\eta^{(')}$ decays

We study the three-body baryonic $B\to {\bf B\bar B'}M$ decays with $M$ representing the $\eta$ or $\eta'$ meson. Particularly, we predict that ${\cal B}(B^-\to\Lambda\bar p\eta,\Lambda\bar p\eta')=(5.3\pm 1.4,3.3\pm 0.7)\times 10^{-6}$ or $(4.0\pm 0.7,4.6\pm 1.1)\times 10^{-6}$, where the errors arise from the non-factorizable effects as well as the uncertainties in the $0\to {\bf B\bar B'}$ and $B\to{\bf B\bar B'}$ transition form factors, while the two different results are due to overall relative signs between the form factors, causing the constructive and destructive interference effects. For the corresponding baryonic $\bar B_s^0$ decays, we find that ${\cal B}(\bar B^0_s\to \Lambda\bar \Lambda \eta,\Lambda\bar \Lambda \eta')=(1.2\pm 0.3,2.6\pm 0.8)\times 10^{-6}$ or $(2.1\pm 0.6,1.5\pm 0.4)\times 10^{-6}$ with the errors similar to those above. The decays in question are accessible to the experiments at BELLE and LHCb.

In B → BB ′ M, the threshold enhancement has been observed as a generic feature [23][24][25][26][27][28], which is shown as the peak at the threshold area of m BB ′ ≃ m B + m B ′ in the spectrum, with m BB ′ denoted as the invariant mass of the di-baryon. With the threshold effect, one expects that B(B → BB ′ η (′) ) ∼ 10 −6 , being accessible to the BELLE and LHCb experiments. Fur- andB 0 s → ΛΛη ( ′ ) have the interference effects for the branching ratios, which can be useful to improve the knowledge of the underlying QCD anomaly for the η − η ′ mixing. In this report, we will study the three-body baryonic B decays with one of the final states to be the η or η ′ meson state, where the possible interference effects from the b → snn → sη n and b → sss → sη s transitions can be investigated.
On the other hand, the baryon pair in Fig. 1c moves collinearly, so that g 1,2 are both close to the mass shell, causing no suppression. Besides, the amplitudes can be factorized as Accordingly, the Feynman diagrams for the three-body baryonic B → BB ′ η (′) decays with the short-distance approximation are shown in Fig. 2. In our calculation, we use the generalized factorization as the theoretical approach. The non-factorizable effects are included by the effective Wilson coefficients [32][33][34]. In terms of the effective Hamiltonian for the b → sqq transitions [35], the decay amplitudes of B − → Λpη (′) by the factorization can be derived as [11-13, 15, 16, 19, 20, 34] where n = u or d, G F is the Fermi constant, and A 1 and A 2 correspond to the two different decaying configurations in Fig. 2. Similarly, the amplitudes ofB 0 s → ΛΛη (′) are given by The parameters α i and β i in Eqs. (1) and (2) are defined as where V ij the CKM matrix elements, and a i = c ef f i + c ef f i±1 /N c for i =odd (even) with N c the effective color number in the generalized factorization approach, consisting of the effective 4 Wilson coefficients c ef f i [34]. The matrix elements in Eq. (1) for the η (′) productions read [36] with f n,s η (′) and h s η (′) the decay constants and q µ the four-momentum vector. The η and η ′ meson states mix with |η n = (|uū + dd )/ √ 2 and |η s = |ss [1], in terms of the mixing matrix: with the mixing angle φ = (39.3 ± 1.0) • . Therefore, f n η (′) and f s η (′) actually come from f n and f s for η n and η s , respectively. In addition, h s η (′) receive the contributions from the QCD anomaly [36]. The matrix elements of the B → η (′) transitions are parameterized as [37] with q = p B − p η (′) = p B + pB′ and t ≡ q 2 , where the momentum dependences are expressed as [38] F Bη (′) According to the mixing matrix in Eq. (5), one has . The matrix elements in Eq. (1) for the baryon-pair productions are parameterized as [12,13] with (qq ′ ) V,A,S,P = (qγ µ q ′ ,qγ µ γ 5 q ′ ,qq ′ ,qγ 5 q ′ ), where u(v) is the (anti-)baryon spinor, and (F 1,2 , g A , h A , f S , g P ) are the timelike baryonic form factors. Meanwhile, the matrix elements of the B → BB ′ transitions are written to be [11,15] The momentum dependences of the baryonic form factors in Eqs. (9) and (10) depend on the approach of perturbative QCD counting rules, given by [11,15,39,40], whereC i = C i [ln(t/Λ 2 0 )] −γ with γ = 2.148 and Λ 0 = 0.3 GeV. Compared to the 0 → BB ′ form factors, the B → BB ′ ones have an additional 1/t, which is for a gluon to speed up the slow spectator quark in B. Due to F 2 = F 1 /(tln[t/Λ 2 0 ]) in [41], derived to be much less than F 1 , and h A = C h A /t 2 [42] that corresponds to the smallness of B(B 0 → pp) ∼ 10 −8 [43,44], we neglect F 2 and h A . Under the SU(3) flavor and SU(2) spin symmetries, the constants C i can be related, given by [12,21,39] with C * ||(||) ≡ C ||(||) + δC ||(||) andC * || ≡C || + δC || , where δC ||(||) and δC || are added to account for the broken symmetries, indicated by the large and unexpected angular distributions in B 0 → Λpπ + and B − → Λpπ 0 [24]. With the same symmetries [11,15,16,19,20], D i are related by  Λp|(sb) S,P |B − : where the ignorances of D with m 12 = p B + pB′ and m 23 = p B + p η (′) , where |Ā| 2 represents the amplitude squared with the total summations of the baryon spins. On the other hand, we can also study the partial decay rate in terms of the the angular dependence, given by [15] dΓ d cos where t ≡
In the adoption of the effective Wilson coefficients c ef f i in Ref. [34], the values of α i and β i in Eqs. (1), (2) and (3) are given in Table I With the theoretical inputs in Eqs. (19) and (20), one has well explained the observations of B(B 0 s →pΛK + + pΛK − ) and B(B → ppMM) [20,21]. Since the 0 → BB ′ and B → BB ′ baryonic transition form factors are separately extracted from the data, it is possible to have overall positive or negative signs between C and D in Eqs. (19) and (20), causing two different scenarios for the interferences. In Table II, we present the results for B(B − → Λpη (′) ) and B(B 0 s → ΛΛη (′) ) with B ± ≡ B 1 + B 2 ± B 1·2 , where the notations of "±" are due to the undetermined relative signs, and (B 1 , B 2 , B 1·2 ) are denoted as the partial branching ratios from the amplitudes A 1 , A 2 and the interferences, respectively. Note that the errors in Table II arise from the estimations of the non-factorizable effects in the generalized factorization with N c = 2, 3, ∞ for the parameters in Table I, and the uncertainties in the form factors of the 0 → BB ′ productions and B → BB ′ transitions in Eqs. (19) and (20). On the other hand, the uncertainties from the CKM matrix elements in Eq. (17) have been computed to be negligibly small.
By following Refs. [46,47] 1) and (2) for the di-baryon threshold effect in Region I can be extended to the other regions. In fact, the extension has been demonstrated to be able to describe theB 0 → ppD 0 spectra at different energy ranges [48][49][50], where the data points for the spectrum vs. m Dp are measured at the range of m pp > 2.29 GeV, which correspond to the regions II and III of Fig. 3. In order that the extension of the amplitudes in Eqs. (1) and (2) can be tested by the future observations, we present the spectra versus m BB ′ and m Bη (′) in the three-body B → BB ′ η (′) decays in Fig. 4 for the different kinematic regions in the Dalitz plots. Besides, we show the angular distributions with m BB ′ > 2.5 − 2.7 GeV in Fig. 5, to be compared to the future measurements. Note that cos θ ≃ 0 with θ ≃ 90 • corresponds to the central area of Region II for the Dalitz plots.
Like the B → K ( * ) η (′) decays, it is possible that B → BB ′ η (′) can help to improve the knowledge of the underlying QCD anomaly for the η − η ′ mixing. Provided that the decays of B → BB ′ η (′) are well measured, the experimental values to be inconsistent with the theoretical calculations will hint at some possible additional effects to the η − η ′ mixing, such as the η-η ′ -G mixing with G denoting the pseudoscalar glueball state [51]. Moreover, the gluonic contributions to the B(B 0 s ) → η (′) transition form factors [52] could also lead to visible effects.

IV. CONCLUSIONS
We have studied the three-body baryonic B decays of B − → Λpη (′) andB 0 s → ΛΛη (′) . Due to the interference effects between b → snn → sη n and b → sss → sη s , which can be constructive or destructive, we have predicted that B(B − → Λpη, Λpη ′ ) = (5.  are not considered properly, our results just provide an order of magnitude estimation on branching ratios.