Study of B^-\to Λ\bar pη^{(')} and \bar B^0_s\to Λ\barΛη^{(')} decays
aa r X i v : . [ h e p - ph ] M a y Study of B − → Λ ¯ pη ( ′ ) and ¯ B s → Λ ¯Λ η ( ′ ) decays Y.K. Hsiao, C.Q. Geng,
1, 2, 3
Yu Yao, and H.J. Zhao School of Physics and Information Engineering,Shanxi Normal University, Linfen 041004, China Department of Physics, National Tsing Hua University, Hsinchu 300 Chongqing University of Posts & Telecommunications, Chongqing, 400065, China (Dated: May 17, 2019)
Abstract
We study the three-body baryonic B → B ¯B ′ M decays with M representing the η or η ′ meson.Particularly, we predict that B ( B − → Λ¯ pη, Λ¯ pη ′ ) = (5 . ± . , . ± . × − or (4 . ± . , . ± . × − , where the errors arise from the non-factorizable effects as well as the uncertainties inthe 0 → B ¯B ′ and B → B ¯B ′ transition form factors, while the two different results are due to overallrelative signs between the form factors, causing the constructive and destructive interference effects.For the corresponding baryonic ¯ B s decays, we find that B ( ¯ B s → Λ ¯Λ η, Λ ¯Λ η ′ ) = (1 . ± . , . ± . × − or (2 . ± . , . ± . × − with the errors similar to those above. The decays inquestion are accessible to the experiments at BELLE and LHCb. . INTRODUCTION In association with the QCD anomaly, the b and c -hadron decays with η ( ′ ) as the finalstates have drawn lots of theoretical and experimental attentions, where the η and η ′ mesonsare in fact the mixtures of the singlet η and octet η states, with η , being decomposed as η n = ( u ¯ u + d ¯ d ) / √ η s = s ¯ s in the FKS scheme [1]. In addition, the two configurationsof b → sn ¯ n → sη n ( n = u or d ) and b → s ¯ ss → sη s have been found to be the causesof the dramatic interferences between the B → K ( ∗ ) η and B → K ( ∗ ) η ′ decays, that is, B ( B → Kη ) ≪ B ( B → Kη ′ ) and B ( B → K ∗ η ) ≫ B ( B → K ∗ η ′ ) [2]. Note that thetheoretical prediction gives B ( ¯ B s → η ( ′ ) η ′ ) ≫ B ( ¯ B s → ηη ) [3], while the only observation is B ( ¯ B s → η ′ η ′ ) = (3 . ± . × − [4]. On the other hand, with the dominant b → s ¯ ss → sη s transition, the theoretical calculations result in B (Λ b → Λ η ) ≃ B (Λ b → Λ η ′ ) [5, 6], which hasnot been confirmed by the the current data [7]. For the dominant tree-level decay modes, thetheoretical results indicate that B ( B → πη ) ≃ B ( B → πη ′ ) [3] and B (Λ + c → pη ) ≃ B (Λ + c → pη ′ ) [8, 9]. Nonetheless, the observed values of B ( B → πη, πη ′ ) show a slight tension withthe predictions. Experimentally, there are more to-be-measured decays with η ( ′ ) , such asthe B decays of ¯ B s → ηη, ηη ′ and Λ + c decays of Λ + c → pη ′ , Σ + η ′ .Although the charmless three-body baryonic B decays ( B → B ¯B ′ M ) have been abun-dantly measured [2], and well studied with the factorization [10–21], neither theoreticalcalculation nor experimental measurement for B → B ¯B ′ η ( ′ ) has been done yet. We notethat the prediction of B ( B − → Λ ¯ pφ ) = (1 . ± . × − [18] based on the factorizationmethod is slightly larger than the recent BELLE data of (0 . ± . ± . × − [22].In B → B ¯B ′ M , the threshold enhancement has been observed as a generic feature [23–28],which is shown as the peak at the threshold area of m B ¯B ′ ≃ m B + m B ′ in the spectrum, with m B ¯B ′ denoted as the invariant mass of the di-baryon. With the threshold effect, one expectsthat B ( B → B ¯B ′ η ( ′ ) ) ∼ − , being accessible to the BELLE and LHCb experiments. Fur-thermore, with b → sn ¯ n → sη n and b → s ¯ ss → sη s , it is worth to explore if B − → Λ ¯ pη ( ′ ) and ¯ B s → Λ ¯Λ η ( ′ ) have the interference effects for the branching ratios, which can be usefulto improve the knowledge of the underlying QCD anomaly for the η − η ′ mixing. In thisreport, we will study the three-body baryonic B decays with one of the final states to bethe η or η ′ meson state, where the possible interference effects from the b → sn ¯ n → sη n and b → s ¯ ss → sη s transitions can be investigated.2 q ¯ q ¯ q B M q ¯ B ′ g g ( a ) qqq ¯ q q ¯ q ¯ q ¯B ′ M q B g g ( b ) ¯ q ¯ q qq B¯B ′ g g q ¯ q M ( c ) ¯ q q q ¯ q ¯ q q FIG. 1. The short-distance pictures for B → B ¯B ′ M through the three quasi-two-body decays,where (a), (b) and (c) correspond to the collinearly moving M B , M ¯B ′ and B ¯B ′ , respectively. B − b ¯ u ¯ u ¯ u Λ ¯p s ¯ d ( a ) W η ( ′ ) u ¯ uud ¯ B s b ¯ u η ( ′ ) Λ u ( b ) W ¯ s s ¯ s ¯ d ¯ s ¯Λ ds B − ( ¯B ) b ¯ u (¯ s ) η ( ′ ) Λ s ( c ) W g,γ ¯ u (¯ s ) u ( s )¯ u (¯ s )¯ u ¯ d ¯p ( ¯Λ ) ud ¯B ¯ s ¯ s Λ ¯Λ s ¯ d ( d ) W η ( ′ ) s ¯ uud g,γ b ¯ s B − ( ¯B ) b ¯ u (¯ s ) ¯ u (¯ s ) s Λ¯p ( ¯Λ ) η ( ′ ) ud ¯ d ¯ uu ¯ u ( e ) W B − ( ¯B ) b ¯ u (¯ s ) ¯ u (¯ s ) s Λ¯p ( ¯Λ ) η ( ′ ) ud ¯ d ¯ us ¯ s ( f ) W g,γ B − ( ¯B ) b ¯ u (¯ s ) ¯ u (¯ s ) s Λ¯p ( ¯Λ ) ud ¯ d ¯ uu ¯ u ( g ) g,γ η ( ′ ) W FIG. 2. Feynman diagrams for B − → Λ¯ pη ( ′ ) and ¯ B s → Λ ¯Λ η ( ′ ) decays through (a,b,c,d) B → η ( ′ ) transitions with 0 → B ¯B ′ productions and (e,f,g) B → B ¯B ′ transitions with the recoiled η ( ′ ) . II. FORMALISM
Unlike the two-body mesonic B → M M decays, the B → B ¯B ′ M decays require twoadditional quark pairs for the B ¯B ′ formation. This is in accordance with the short-distancepictures depicted in Fig. 1 [28, 29], where q ¯ q and q ¯ q are connected by the gluons g , , re-spectively. In Fig. 1a(b), the meson and (anti)baryon move collinearly, with g for a collinearquark pair. By connecting to a back-to-back q ¯ q pair, g is far off the mass shell, such thatit is a hard gluon, resulting in the suppression with the factor of order α s /q . There remainthe resonant contributions observed to be small, which correspond to the suppression dueto the short-distance pictures. For example, one has B ( B − → p Θ(1710) −− , Θ(1710) −− → ¯ pK − ) < . × − and B ( ¯ B → p Θ(1540) − , Θ(1540) − → ¯ pK s ) < × − [2]. Moreover, B − → Λ(1520)¯ p, Λ(1520) → pK − is observed with B ∼ − [30, 31].On the other hand, the baryon pair in Fig. 1c moves collinearly, so that g , are both closeto the mass shell, causing no suppression. Besides, the amplitudes can be factorized as A ∝h B ¯B ′ | J a | ih M | J b | B i and A ∝ h M | J a | ih B ¯B ′ | J b | B i . Accordingly, the Feynman diagrams3or the three-body baryonic B → B ¯B ′ η ( ′ ) decays with the short-distance approximation areshown in Fig. 2. In our calculation, we use the generalized factorization as the theoreticalapproach. The non-factorizable effects are included by the effective Wilson coefficients [32–34]. In terms of the effective Hamiltonian for the b → sq ¯ q transitions [35], the decayamplitudes of B − → Λ ¯ pη ( ′ ) by the factorization can be derived as [11–13, 15, 16, 19, 20, 34] A ( B − → Λ ¯ pη ( ′ ) ) = A ( B − → Λ ¯ pη ( ′ ) ) + A ( B − → Λ ¯ pη ( ′ ) ) , A ( B − → Λ ¯ pη ( ′ ) ) = G F √ (cid:26) α h Λ ¯ p | (¯ sγ µ (1 − γ ) u | ih η ( ′ ) | ¯ uγ µ (1 − γ ) b | B − i + α h Λ ¯ p | ¯ s (1 + γ ) u | ih η ( ′ ) | ¯ u (1 − γ ) b | B − i (cid:27) , A ( B − → Λ ¯ pη ( ′ ) ) = G F √ (cid:26)(cid:20) β h η ( ′ ) | ¯ nγ µ γ n | i + β h η ( ′ ) | ¯ sγ µ γ s | i (cid:21) h Λ ¯ p | ¯ sγ µ (1 − γ ) b | B − i + β h η ( ′ ) | ¯ sγ s | ih Λ ¯ p | ¯ s (1 − γ ) b | B − i (cid:27) , (1)where n = u or d , G F is the Fermi constant, and A and A correspond to the two differentdecaying configurations in Fig. 2. Similarly, the amplitudes of ¯ B s → Λ ¯Λ η ( ′ ) are given by A ( ¯ B s → Λ ¯Λ η ( ′ ) ) = A ( ¯ B s → Λ ¯Λ η ( ′ ) ) + A ( ¯ B s → Λ ¯Λ η ( ′ ) ) , A ( ¯ B s → Λ ¯Λ η ( ′ ) ) = G F √ (cid:26)(cid:20) h Λ ¯Λ | ¯ nγ µ ( α +2 − α − γ ) n | i + h Λ ¯Λ | ¯ sγ µ ( α +3 − α − γ ) s | i (cid:21) × h η ( ′ ) | ¯ sγ µ (1 − γ ) b | ¯ B s i + α s h Λ ¯Λ | ¯ s (1 + γ ) s | ih η ( ′ ) | ¯ s (1 − γ ) b | ¯ B s i (cid:27) , A ( ¯ B s → Λ ¯Λ η ( ′ ) ) = G F √ (cid:26)(cid:20) β h η ( ′ ) | ¯ nγ µ γ n | i + β h η ( ′ ) | ¯ sγ µ γ s | i (cid:21) h Λ ¯Λ | ¯ sγ µ (1 − γ ) b | ¯ B s i + β h η ( ′ ) | ¯ sγ s | ih Λ ¯Λ | ¯ s (1 − γ ) b | ¯ B s i (cid:27) . (2)The parameters α i and β i in Eqs. (1) and (2) are defined as α = V ub V ∗ us a − V tb V ∗ ts ( a + a ) ,α ± = V ub V ∗ us a − V tb V ∗ ts (2 a ± a ± a a ,α ± = − V tb V ∗ ts ( a + a ± a ∓ a − a − a ,α = V tb V ∗ ts a + a ) ,α s = V tb V ∗ ts a − a ,β = − α − , β = − α − , β = α s , (3)where V ij the CKM matrix elements, and a i = c effi + c effi ± /N c for i =odd (even) with N c theeffective color number in the generalized factorization approach, consisting of the effective4ilson coefficients c effi [34]. The matrix elements in Eq. (1) for the η ( ′ ) productions read [36] h η ( ′ ) | ¯ nγ µ γ n | i = − i √ f nη ( ′ ) q µ , h η ( ′ ) | ¯ sγ µ γ s | i = − if sη ( ′ ) q µ , m s h η ( ′ ) | ¯ sγ s | i = − ih sη ( ′ ) , (4)with f n,sη ( ′ ) and h sη ( ′ ) the decay constants and q µ the four-momentum vector. The η and η ′ meson states mix with | η n i = ( | u ¯ u + d ¯ d i ) / √ | η s i = | s ¯ s i [1], in terms of the mixingmatrix: ηη ′ = cos φ − sin φ sin φ cos φ η n η s , (5)with the mixing angle φ = (39 . ± . ◦ . Therefore, f nη ( ′ ) and f sη ( ′ ) actually come from f n and f s for η n and η s , respectively. In addition, h sη ( ′ ) receive the contributions from the QCDanomaly [36]. The matrix elements of the B → η ( ′ ) transitions are parameterized as [37] h η ( ′ ) | ¯ qγ µ b | B i = (cid:20) ( p B + p η ( ′ ) ) µ − m B − m η ( ′ ) t q µ (cid:21) F Bη ( ′ ) ( t ) + m B − m η ( ′ ) t q µ F Bη ( ′ ) ( t ) , (6)with q = p B − p η ( ′ ) = p B + p ¯B ′ and t ≡ q , where the momentum dependences are expressedas [38] F Bη ( ′ ) ( t ) = F Bη ( ′ ) (0)(1 − tM V )(1 − σ tM V + σ t M V ) , F Bη ( ′ ) ( t ) = F Bη ( ′ ) (0)1 − σ tM V + σ t M V . (7)According to the mixing matrix in Eq. (5), one has( F Bη , F Bη ′ ) = ( F Bη n cos φ, F Bη n sin φ ) , ( F B s η , F B s η ′ ) = ( − F B s η s sin φ, F B s η s cos φ ) , (8)for the B − and ¯ B s transitions to η ( ′ ) , respectively, where F Bη ( ′ ) represent F Bη ( ′ ) , (0).The matrix elements in Eq. (1) for the baryon-pair productions are parameterized as [12,13] h B ¯B ′ | (¯ qq ′ ) V | i = ¯ u (cid:20) F γ µ + F m B + m ¯B ′ iσ µν q ν (cid:21) v , h B ¯B ′ | (¯ qq ′ ) A | i = ¯ u (cid:20) g A γ µ + h A m B + m ¯B ′ q µ (cid:21) γ v , h B ¯B ′ | (¯ qq ′ ) S | i = f S ¯ uv , h B ¯B ′ | (¯ qq ′ ) P | i = g P ¯ uγ v , (9)5ith (¯ qq ′ ) V,A,S,P = (¯ qγ µ q ′ , ¯ qγ µ γ q ′ , ¯ qq ′ , ¯ qγ q ′ ), where u ( v ) is the (anti-)baryon spinor, and( F , , g A , h A , f S , g P ) are the timelike baryonic form factors. Meanwhile, the matrix elementsof the B → B ¯B ′ transitions are written to be [11, 15] h B ¯B ′ | (¯ sb ) V | B i = i ¯ u [ g γ µ + g iσ µν p ν + g p µ + g q µ + g ( p ¯B ′ − p B ) µ ] γ v , h B ¯B ′ | (¯ sb ) A | B i = i ¯ u [ f γ µ + f iσ µν p ν + f p µ + f q µ + f ( p ¯B ′ − p B ) µ ] v , h B ¯B ′ | (¯ sb ) S | B i = i ¯ u [¯ g /p + ¯ g ( E ¯B ′ + E B ) + ¯ g ( E ¯B ′ − E B )] γ v , h B ¯B ′ | (¯ sb ) P | B i = i ¯ u [ ¯ f /p + ¯ f ( E ¯B ′ + E B ) + ¯ f ( E ¯B ′ − E B )] v , (10)with p µ = ( p B − q ) µ , where g i ( f i ) ( i = 1 , , ...,
5) and ¯ g j ( ¯ f j ) ( j = 1 , ,
3) are the B → B ¯B ′ transition form factors. The momentum dependences of the baryonic form factorsin Eqs. (9) and (10) depend on the approach of perturbative QCD counting rules, givenby [11, 15, 39, 40], F = ¯ C F t , g A = ¯ C g A t , f S = ¯ C f S t , g P = ¯ C g P t ,f i = D f i t , g i = D g i t , ¯ f i = D ¯ f i t , ¯ g i = D ¯ g i t , (11)where ¯ C i = C i [ln( t/ Λ )] − γ with γ = 2 .
148 and Λ = 0 . → B ¯B ′ form factors, the B → B ¯B ′ ones have an additional 1 /t , which is for a gluon to speed up theslow spectator quark in B . Due to F = F / ( t ln[ t/ Λ ]) in [41], derived to be much less than F , and h A = C h A /t [42] that corresponds to the smallness of B ( ¯ B → p ¯ p ) ∼ − [43, 44],we neglect F 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]( C F , C g A , C f S , C g P ) = s
32 ( C || , C ∗|| , − ¯ C || , − ¯ C ∗|| ) , (for h Λ ¯ p | (¯ su ) V,A,S,P | i )( C F , C g A , C f S , C g P ) = ( C || , C ∗|| , − ¯ C || , − ¯ C ∗|| ) , (for h Λ ¯Λ | (¯ ss ) V,A,S,P | i )( C F , C g A ) = 12 ( C || + C || , C ∗|| − C ∗|| ) , (for h Λ ¯Λ | (¯ nn ) V,A | i ) (12)with C ∗|| ( || ) ≡ C || ( || ) + δC || ( || ) and ¯ C ∗|| ≡ ¯ C || + δ ¯ C || , where δC || ( || ) and δ ¯ C || are added to accountfor the broken symmetries, indicated by the large and unexpected angular distributions in¯ B → Λ ¯ pπ + and B − → Λ ¯ pπ [24]. With the same symmetries [11, 15, 16, 19, 20], D i arerelated by h Λ ¯ p | (¯ sb ) V,A | B − i : D g = D f = s D || , D g , = − D f , = − s D , || , ABLE I. The values of α i and β i with N c = 2 ,
3, and ∞ . α i ( β i ) N c = 2 N c = 3 N c = ∞ α − . − . i − . − . i − . − . i α +2 − . − . i − . − . i . . i α − ( − β ) 12 . − . i . − . i . . i α +3 − . − . i − . − . i − . − . i α − ( − β ) − . − . i − . − . i − . − . i α . . i . . i . . i α s ( β ) 48 . . i . . i . . i h Λ ¯ p | (¯ sb ) S,P | B − i : D ¯ g = − D ¯ f = s
32 ¯ D || , D ¯ g , = D ¯ f , = − s
32 ¯ D , || , h Λ ¯Λ | (¯ sb ) V,A | ¯ B s i : D g = D f = D || , D g , = − D f , = − D , || , h Λ ¯Λ | (¯ sb ) S,P | ¯ B s i : D ¯ g = − D ¯ f = ¯ D || , D ¯ g , = D ¯ f , = − ¯ D , || , (13)where the ignorances of D g , and D f , correspond to the derivations of f M p µ ¯ u ( σ µν p ν ) v = 0for g ( f ) and f M p µ ¯ up µ v ∝ m M for f ( g ) in the amplitudes. For the integration over thephase space in the three-body decay, we refer the general equation of the decay width in thePDG, given by [2] Γ = Z m Z m π ) | ¯ A| M B dm dm , (14)with m = p B + p ¯B ′ and m = p B + p η ( ′ ) , where | ¯ A| represents the amplitude squared withthe total summations of the baryon spins. On the other hand, we can also study the partialdecay rate in terms of the the angular dependence, given by [15] d Γ d cos θ = Z t β / t λ / t (8 πm B ) | ¯ A| dt , (15)where t ≡ m , β t = 1 − ( m B + m ¯B ′ ) /t , λ t = [( m B + m M ) + t ][( m B − m M ) + t ], and θ isthe angle between the moving directions of B and M .7 II. NUMERICAL ANALYSIS
In the numerical analysis, we use the Wolfenstein parameters for the CKM matrix ele-ments: V ub = Aλ ( ρ − iη ) , V tb = 1 ,V us = λ, V ts = − Aλ , (16)with λ , A , ρ = ¯ ρ/ (1 − λ /
2) and η = ¯ η/ (1 − λ / λ = 0 . ± . , A = 0 . ± . , ¯ ρ = 0 . +0 . − . , ¯ η = 0 . +0 . − . . (17)In the adoption of the effective Wilson coefficients c effi in Ref. [34], the values of α i and β i inEqs. (1), (2) and (3) are given in Table I with N c = (2 , , ∞ ) to estimate the non-factorizableeffects. For the 0 → ( η, η ′ ) productions and B → ( η, η ′ ) transitions, one gets [36, 38, 45]( f nη , f nη ′ , f sη , f sη ′ ) = (0 . , . , − . , . , ( h sη , h sη ′ ) = ( − . , . , ( F Bη n , σ , σ , σ , σ ) = (0 . , . , , . , . , ( F B s η s , σ , σ , σ , σ ) = (0 . , . , . , . , . , (18)with M V = 5 .
32 GeV, resulting in ( F Bη , F Bη ′ ) = (0 . , .
21) and ( F B s η , F B s η ′ ) = ( − . , . → B ¯B ′ baryonic form factors, the minimal χ fitting method has beenused to fit with 20 data points, where 11 of them are from the branching ratios of D + s → p ¯ n ,¯ B s ) → p ¯ p , B − → Λ ¯ p , ¯ B → n ¯ pD ∗ + (Λ ¯ pD ( ∗ )+ ), ¯ B ( B − ) → Λ ¯ pπ +(0) , B − → Λ ¯ pρ and B − → Λ ¯Λ K − , 4 the angular distribution asymmetries of ¯ B → Λ ¯ pD ( ∗ )+ , ¯ B ( B − ) → Λ ¯ pπ +(0) and 5 the angular distribution in ¯ B → Λ ¯ pπ + [24]. This presents a reasonable fit with χ /d.o.f ≃ .
3, where d.o.f stands for the degree of freedom. Hence, we adopt the fittedvalues to be [19–21] ( C || , δC || ) = (154 . ± . , . ± .
6) GeV , ( C || , δC || ) = (18 . ± . , − . ± .
0) GeV , ( ¯ C || , δ ¯ C || ) = (537 . ± . , − . ± .
4) GeV . (19)8 ABLE II. Numerical results for B ( B − → Λ¯ pη ( ′ ) ) and B ( ¯ B s → Λ ¯Λ η ( ′ ) ), with B ± = B + B ± B · ,where ( B , B , B · ) are denoted as the partial branching ratios from the amplitudes A , A and theinterferences, while the errors come from the non-factorizable effects and form factors, respectively.branching ratios B + B − B ( B − → Λ¯ pη ) 5 . ± . ± . . ± . ± . B ( B − → Λ¯ pη ′ ) 3 . ± . ± . . ± . ± . B ( ¯ B s → Λ ¯Λ η ) 1 . ± . ± . . ± . ± . B ( ¯ B s → Λ ¯Λ η ′ ) 2 . ± . ± . . ± . ± . Here, we have assumed that the timelike baryonic form factors are real. In general, they canbe complex numbers if some resonances are involved with un-calculable strong phases. How-ever, these phases are believed to be negligible. For the B → B ¯B ′ transition ones, the extrac-tions depend on 28 data points with 7 from the branching ratios of B − → p ¯ p ( K − , π − ), B − → p ¯ pe − ¯ ν e and ¯ B → p ¯ p ( K ( ∗ )0 , D ( ∗ )0 ), 3 the CP violating asymmetries of B − → p ¯ p ( K ( ∗ ) − , π − )and 2 the angular distribution asymmetries of B − → p ¯ p ( K − , π − ), together with 16 datapoints from the angular distributions in B − → p ¯ p ( K − , π − ) [26], resulting in χ /d.o.f ≃ . D i are given by [19, 20] D || = (45 . ± .
8) GeV , ( D || , D || ) = (6 . ± . , − . ± .
3) GeV , ( ¯ D || , ¯ D || , ¯ D || ) = (35 . ± . , − . ± . , . ± .
4) GeV . (20)With the theoretical inputs in Eqs. (19) and (20), one has well explained the observationsof B ( ¯ B s → ¯ p Λ K + + p ¯Λ K − ) and B ( B → p ¯ pM M ) [20, 21].Since the 0 → B ¯B ′ and B → B ¯B ′ baryonic transition form factors are separately ex-tracted 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 − → Λ ¯ pη ( ′ ) ) and B ( ¯ B s → Λ ¯Λ η ( ′ ) ) with B ± ≡ B + B ± B · ,where the notations of “ ± ” are due to the undetermined relative signs, and ( B , B , B · ) aredenoted as the partial branching ratios from the amplitudes A , A and the interferences, re-spectively. Note that the errors in Table II arise from the estimations of the non-factorizableeffects in the generalized factorization with N c = 2 , , ∞ for the parameters in Table I, andthe uncertainties in the form factors of the 0 → B ¯B ′ productions and B → B ¯B ′ transitionsin Eqs. (19) and (20). On the other hand, the uncertainties from the CKM matrix elements9 LΗ = L p = @ I D @ II D @
IIIa D@ IIIb D m L p H GeV L m L Η H G e V L FIG. 3. The kinematical allowed regions of I, II, IIIa and IIIb (left panel) and Dalitz plotdistribution (right panel) in the plane of m Λ¯ p and m Λ η for B − → Λ¯ pη . in Eq. (17) have been computed to be negligibly small.In Table II, we have used the central values of ( B , B , B · ) = (2 . , . , . × − and ( B ′ , B ′ , B ′ · ) = (1 . , . , − . × − for B − → Λ ¯ pη ( ′ ) . Clearly, the results of |B ( ′ )1 · | ∼ O (10 − ) indicate sizable interferences. As shown in the table, we find that B ( ′ )1 · causes a constructive (destructive) interfering effect in B + ( B − → Λ ¯ pη ( ′ ) ), and a destructive(constructive) interfering one in B − ( B − → Λ ¯ pη ( ′ ) ). Besides, the inequalities of B > B ′ and B < B ′ are due to F Bη > F Bη ′ and | h sη | < | h sη ′ | , respectively. Similarly, one hasthat ( B s , B s , B s · s ) = (1 . , . , − . × − for ¯ B s → Λ ¯Λ η and ( B ′ s , B ′ s , B ′ s · s ) =(1 . , . , . × − for ¯ B s → Λ ¯Λ η ′ , which present that B ( ′ ) s · s has a destructive (construc-tive) interfering effect in B + ( ¯ B s → Λ ¯Λ η ( ′ ) ), and a constructive (destructive) interfering one in B − ( ¯ B s → Λ ¯Λ η ( ′ ) ). As a result, in terms of |B ( ′ )( s )1 · ( s )2 | ∼ O (10 − ) being traced back to the in-terferences between the two decaying configurations in Fig. 2, we conclude that B − → Λ ¯ pη ( ′ ) and ¯ B s → Λ ¯Λ η ( ′ ) are like B → K ( ∗ ) η ( ′ ) to have large values from the interferences, in com-parison with B (Λ b → Λ η ) ≃ B (Λ b → Λ η ′ ) [5, 6] and B (Λ + c → pη ) ≃ B (Λ + c → pη ′ ) [8, 9],which show less important interferences.By following Refs. [46, 47], we present the kinematical allowed regions and Dalitz plotdistribution in the plane of m Λ¯ p and m Λ η for B − → Λ ¯ pη in Fig. 3 to illustrate the genericfeatures in B → B ¯B ′ M . As shown in the left panel in the figure, the allowed area can bedivided into four different regions, denoted as I, II, IIIa and IIIb, respectively. In Region I, B and ¯B ′ can move collinearly, with the recoiled M in the opposite direction. In Region II, B , ¯B ′ and M all have large energies, so that none of any two final states can be back-to-back.10n Region IIIa(b), M and B ( ¯B ′ ) move collinearly, with ¯B ′ ( B ) being energetic and separatedfrom the meson-(anti)baryon system. Since the collinear moving di-baryon in Region I andmeson-(anti)baryon in Region IIIa(b) cause different kinds of quasi-two-body decays, the t and s ( u )-channel contributions should be dominant, respectively, where t ≡ ( p B + p ¯B ′ ) , s ≡ ( p B + p M ) and u ≡ ( p ¯B ′ + p M ) are the Mandelstam variables. In Region II, the threechannels are supposed to contribute with ( s, t, u ) ∼ m B / B → p ¯ pD spectra at different energy ranges [48–50], where the data points for thespectrum vs. m Dp are measured at the range of m p ¯ p > .
29 GeV, which correspond to theregions 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 B ¯B ′ and m B η ( ′ ) inthe three-body B → B ¯B ′ η ( ′ ) decays in Fig. 4 for the different kinematic regions in the Dalitzplots. Besides, we show the angular distributions with m B ¯B ′ > . − . θ ≃ θ ≃ ◦ corresponds tothe central area of Region II for the Dalitz plots.Like the B → K ( ∗ ) η ( ′ ) decays, it is possible that B → B ¯B ′ η ( ′ ) can help to improve theknowledge of the underlying QCD anomaly for the η − η ′ mixing. Provided that the decaysof B → B ¯B ′ η ( ′ ) are well measured, the experimental values to be inconsistent with thetheoretical 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 s ) → η ( ′ ) transition form factors [52] could also lead tovisible effects. IV. CONCLUSIONS
We have studied the three-body baryonic B decays of B − → Λ ¯ pη ( ′ ) and ¯ B s → Λ ¯Λ η ( ′ ) .Due to the interference effects between b → sn ¯ n → sη n and b → s ¯ ss → sη s , which can beconstructive or destructive, we have predicted that B ( B − → Λ ¯ pη,
Λ ¯ pη ′ ) = (5 . ± . , . ± . × − or (4 . ± . , . ± . × − , to be compared to the searching results by LHCband BELLE. We have also found that B ( ¯ B s → Λ ¯Λ η, Λ ¯Λ η ′ ) = (1 . ± . , . ± . × − - ®L p Η @ I D @ II D@ III D m L p = m L p H GeV L d B (cid:144) d m L p H - (cid:144) G e V L @ I + II D B - ®L p Η @ IIIa
D @
IIIb D m LΗ = m LΗ H GeV L d B (cid:144) d m L Η H - (cid:144) G e V L B - ®L p Η ¢ @ I D @ II D@ III D m L p = m L p H GeV L d B (cid:144) d m L p H - (cid:144) G e V L @ I + II D B - ®L p Η ¢ @ IIIa
D @
IIIb D m LΗ ¢ = m LΗ ¢ H GeV L d B (cid:144) d m L Η ¢ H - (cid:144) G e V L B s0 ®LLΗ @ I D @ II D@ III D m L L = m L L H GeV L d B (cid:144) d m LL H - (cid:144) G e V L @ I + II D B s0 ®LLΗ @ IIIa
D @
IIIb D m LΗ = m LΗ H GeV L d B (cid:144) d m L Η H - (cid:144) G e V L B s0 ®LLΗ ¢ @ I D @ II D@ III D m L L = m L L H GeV L d B (cid:144) d m LL H - (cid:144) G e V L @ I + II D B s0 ®LLΗ ¢ @ IIIa
D @
IIIb D m LΗ ¢ = m LΗ ¢ H GeV L d B (cid:144) d m L Η ¢ H - (cid:144) G e V L FIG. 4. Decay spectra versus m B ¯B ′ (left) and m B η ( ′ ) (right) of the three-body B → B ¯B ′ η ( ′ ) decays,where the solid (dash) curves for B − → Λ¯ pη with the constructive (destructive) interfering effectscorrespond to the kinematical regions in the left panel of Fig. 3, while those for B − → Λ¯ pη ′ and¯ B s → Λ ¯Λ η ( ′ ) are similarly presented. - ®L p Η- - Θ H m L p > L d B (cid:144) d c o s Θ H - L B - ®L p Η ¢ - - Θ H m L p > L d B (cid:144) d c o s Θ H - L B s0 ®LLΗ- - Θ H m LL > L d B (cid:144) d c o s Θ H - L B s0 ®LLΗ ¢ - - Θ H m LL > L d B (cid:144) d c o s Θ H - L FIG. 5. Angular distributions of B → B ¯B ′ η ( ′ ) versus cos θ with θ being the angle between thebaryon and meson moving directions, where the solid (dash) curves correspond to the constructive(destructive) interference effects. or (2 . ± . , . ± . × − . In our calculations, the errors came from the estimations ofthe non-factorizable effects in the generalized factorization, together with the uncertaintiesfrom the form factors of the 0 → B ¯B ′ productions and B → B ¯B ′ transitions, which aredue to the fit with the existing data for the baryonic B decays. Due to the fact that thecontributions from Region II and the resonant meson-baryon pairs in Regions III in Fig. 3are not considered properly, our results just provide an order of magnitude estimation onbranching ratios. ACKNOWLEDGMENTS
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