Factorization and angular distribution asymmetries in charmful baryonic B decays Y.K. Hsiao and C.Q. Geng Chongqing University of Posts & Telecommunications, Chongqing, 400065, China Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300 6 1 Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 300 0 2 (Dated: February 15, 2016) b e Abstract F 2 We examine the validity of the generalized factorization method and calculate the angular 1 correlations in the charmful three-body baryonic decays of B¯0 Λp¯D(∗)+. With the time- ] → h like baryonic form factors newly extracted from the measured baryonic B decays, we obtain p - p (B¯0 Λp¯D+,Λp¯D∗+) = (1.85 0.30,2.75 0.24) 10−5 to agree with the recent data from the e B → ± ± × h BELLE Collaboration, which demonstrates that the theoretical approach based on the factoriza- [ 2 tionstillworkswell. Fortheangular distributionasymmetries, wefind θ(B¯0 Λp¯D+,Λp¯D∗+) = A → v 4 ( 0.030 0.002,+0.150 0.000), which are consistent with the current measurements. Moreover, 0 − ± ± 8 we predict that (B¯0 pp¯D0,pp¯D∗0)= +0.04 0.01. Future precise explorations of these angu- θ 3 A → ± 0 lar correlations at BELLE and LHCb as well as super-BELLE are important to justify the present . 1 0 factorization approach in the charmful three-body baryonic decays. 6 1 : v i X r a 1 I. INTRODUCTION Recently, the BELLE Collaboration has reported the branching ratios of B¯0 Λp¯D(∗)+ → along with the first angular distribution asymmetries measured in the charmful three-body baryonic B BB¯′M decays, given by [1] c → (B¯0 Λp¯D+) = (25.1 2.6 3.5) 10−6, B → ± ± × (B¯0 Λp¯D∗+) = (33.6 6.3 4.4) 10−6, B → ± ± × (B¯0 Λp¯D−) = 0.08 0.10, θ A → − ± (B¯0 Λp¯D∗−) = +0.55 0.17, (1) θ A → ± withthesubscript θ astheanglebetween p¯andD(∗)− moving directions intheΛp¯rest frame, where ( )/( + ) represents the angular distribution asymmetry, with θ + − + − +(−) A ≡ B −B B B B defined asthebranching ratioof thepositive (negative) cosine value. The datainEq. (1) can be important due to the fact that B¯0 Λp¯D+ and Λp¯D∗+ are two of the few current-type → processes among the richly observed baryonic B decays, connected to the timelike baryonic form factors via the vector and axial-vector quark currents. Note that although B¯0 Λp¯π+ → and B− Λp¯ρ0 are related to the timelike baryonic form factors, they also mix with the → contributions from the scalar and pseudoscalar currents via the penguin diagrams. The decays of B¯0 Λp¯D(∗)− have been previously studied in Ref. [2] with the branching → ratios predicted to be (3.4 0.2) 10−6 and (11.9 2.7) 10−6, respectively, which are ± × ± × obviously much lower than the current data in Eq. (1) and regarded as the failure of the theoretical approach based on the factorization in Ref. [1]. To resolve the problem, in this work we will evaluate the hadronic matrix elements from the observed baryonic B decays directly instead of using the data of e+e− pp¯(nn¯) (pp¯ e+e−) in Ref. [2]. → → Comparedtotheexperimentalresultof (B¯0 Λp¯D−)inEq.(1),themeasuredvalueof θ A → (B¯0 Λp¯π−) = 0.41 0.11 0.03 [3]asthecharmless counterpartisunexpectedly large. θ A → − ± ± Moreover, the experimental implication of (B¯0 Λp¯π−) (B− Λp¯π0) 3 10−6 [3] B → ∼ B → ∼ × looks mysterious as it breaks the isospin symmetry. Since the decays of B¯0 Λp¯D(∗)− → simply proceed through the (axial)vector currents from the tree contributions, one suspects that (B¯0 Λp¯π−) (B¯0 Λp¯D−) isduetotheadditional(pseudo)scalar currents θ θ |A → | ≫ |A → | from the penguin diagrams in B¯0 Λp¯π−. Likewise, the charmless three-body baryonic → decays of B− pp¯(π−,K−) receive the main contributions from the tree and penguin → 2 s Λ c D(∗)0 ud u¯ d¯ p¯ u¯ b ud p W u¯ W u B¯0 b c u¯ B¯0 D( )+ u¯ p¯ ¯ ¯ ∗ ¯ ¯ d d d d (a) (b) FIG. 1. Feynman diagrams for the three-body baryonic B decays of (a) B¯0 pp¯D(∗)0 and (b) → B¯0 Λp¯D(∗)+. → diagrams, respectively, whichmayresultinthewrongsignof (B− pp¯π−) (B− θ θ A → ≃ −A → pp¯K−) [4, 5]. It is hence expected that B¯0 pp¯D0 from the tree-level diagrams can be more → associated with B− pp¯π−. Clearly, the systematic studies of the angular correlations in → B BB¯′M are needed. c → Most importantly, since the theoretical approach for the three-body baryonic B decays depends on the generalized factorization, according to the comments in Ref. [1], if the calcu- lations fail to explain the data, it will indicate that the model parameters need to be revised and, perhaps, some modificationof thetheoretical framework isrequired. Notethat it isalso commented inRef.[1]thatthefactorizationfailstoprovideasatisfactoryexplanationforthe M-p¯angular correlations in B− pp¯K−, B0 pΛ¯π− and B pp¯D. However, it is clearly → → → misleading as (B− pp¯K−) has been well studied in Ref. [6], whereas (B pp¯D) θ θ A → A → has been neither measured experimentally nor predicted theoretically. In this report, we will study B¯0 pp¯D(∗)0 and B¯0 Λp¯D(∗)− in order to approve the → → factorization approach. In addition, we will calculate their angular distribution asymmetries to have the first theoretical predictions. Moreover, some of these charmful asymmetries will be compared to the charmless counterparts of B− pp¯K−(π−) and B¯0 Λp¯π− → → (B− Λp¯π0). → II. FORMALISM ¯ As shown in Fig. 1, in terms of the effective Hamiltonian for the quark-level b cud(s¯) → transition and the generalized factorization approach [7], the amplitudes of the B BB¯′M c → 3 decays can be written by [2] G (B¯0 pp¯D(∗)0) = FV V∗a D(∗)0 (c¯u) 0 pp¯(d¯b) B¯0 , A → √2 cb ud 2h | V−A| ih | V−A| i G (B¯0 Λp¯D(∗)−) = FV V∗a Λp¯(s¯u) 0 D(∗)− (c¯b) B¯0 , (2) A → √2 cb us 1h | V−A| ih | V−A| i where G is the Fermi constant, V are the CKM matrix elements, (q¯ q ) stands for F ij 1 2 V(A) q¯ γ (γ )q , and a ceff +ceff/Neff is composed of the effective Wilson coefficients ceff 1 µ 5 2 1(2) ≡ 1(2) 2(1) c 1,2 defined in Ref. [7]. In Eq. (2), the matrix elements for the D(∗) meson productions through the c¯u quark currents can be written as hD|c¯γµγ5u|0i = −ifDpµD ,hD∗|c¯γµu|0i = mD∗fD∗εµ∗ , (3) with f the decay constant and pµ (εµ∗) the four-momentum (polarization). The matrix D(∗) D elements of the B D(∗) transitions can be parametrized as [8] → m2 m2 m2 m2 D c¯γµb B = (p +p )µ B − Dqµ FBD(t)+ B − DqµFBD(t), h | | i (cid:20) B D − t (cid:21) 1 t 0 2VBD∗(t) D∗ c¯γ b B = ǫ ε∗νpαpβ 1 , h | µ | i µναβ B D∗mB +mD∗ ε∗ q ε∗ q hD∗|c¯γµγ5b|Bi = i(cid:20)ε∗µ − t· qµ(cid:21)(mB +mD∗)AB1D∗(t)+i t· qµ(2mD∗)AB0D∗(t) m2 m2 ABD∗(t) i (pB +pD∗)µ B − D∗qµ (ε∗ q) 2 , (4) − (cid:20) − t (cid:21) · mB +mD∗ where t ≡ q2 with q = pB − pD(∗) = pB + pB¯′. With the Λp¯ pair produced from the su¯ quark currents, B¯0 Λp¯D(∗)− is classified as the current-type decay, such that the matrix → elements for the baryon pair production are in the forms of F BB¯′ q¯ γ q 0 = u¯ F γ + 2 iσ q v , 1 µ 2 1 µ µν µ h | | i (cid:26) mB +mB¯′ (cid:27) F 2 = u¯ [F1 +F2]γµ + (pB¯′ pB)µ v , (cid:26) mB +mB¯′ − (cid:27) h BB¯′ q¯ γ γ q 0 = u¯ g γ + A q γ v, (5) 1 µ 5 2 A µ µ 5 h | | i (cid:26) mB +mB¯′ (cid:27) where F , g and h are the timelike baryonic form factors, and u(v) is the (anti-)baryon 1,2 A A spinor. Being classified as the transition-type decays, the study of B¯0 pp¯D(∗)0 needs to → know the matrix elements for the B¯0 pp¯ transition, which are parameterized as → BB¯′ q¯′γµb B = iu¯[g1γµ +g2iσµνpν +g3pµ +g4qµ +g5(pB¯′ pB)µ]γ5v, h | | i − BB¯′ q¯′γµγ5b B = iu¯[f1γµ +f2iσµνpν +f3pµ +f4qµ +f5(pB¯′ pB)µ]v, (6) h | | i − 4 where p = p q and g (f ) (i = 1,2,3,4,5) are the B BB¯′ transition form factors. The B i i − → momentum dependences of the B D(∗) transition form factors have been studied in QCD → models, given by [9] f(0) f(t) = , (7) (1 t/M2 )[1 σ t/M2 +σ t2/M4 ] − P(V) − 1 P(V) 2 P(V) for f = FBD(ABD∗,VBD∗) and 1 0 1 f(0) f(t) = , (8) 1 σ t/M2 +σ t2/M4 − 1 V 2 V for f = FBD, ABD∗ and ABD∗, while those of F and g in pQCD counting rules can be 0 1 2 1 A written as [10–12] −γ −γ C t C t F = F1 ln , g = gA ln , (9) 1 t2 (cid:20) (cid:18)Λ2(cid:19)(cid:21) A t2 (cid:20) (cid:18)Λ2(cid:19)(cid:21) 0 0 where γ = 2.148 and Λ = 0.3 GeV. Note that h = C /t2 [13] is in accordance with the 0 A hA violated partial conservation of the axial-vector current, whereas F = F /(tln[t/Λ2]) [14, 15] 2 1 0 is small to besafely neglected. According to the principle of pQCD counting rules, onegluon to speed up the spectator quark within the B meson is required in the B BB¯′ transition, → which causes an additional 1/t to F and g , such that the momentum dependences of f (g ) 1 A i i can be written as [16] D D f (t) = fi , g (t) = gi . (10) i t3 i t3 Furthermore, while the SU(3) flavor symmetry can relate different decay modes, the SU(2) spin symmetry can combine the vector and axialvector currents to be the chiral currents. Consequently, one gets the baryonic form factors to be [2, 10–13, 16, 17] 3 1 C = C = C , C = (C +3C ), F1 gA −r2 || hA −√6 D F 1 2 1 D = D D , D = Dj , g1(f1) 3 || ∓ 3 || gj(fj) ∓3 || 3 3 D = D , D = Dj , (11) g1(f1) −r2 || gj(fj) ∓r2 || with the constants C , C , D , and Dj (j = 2,3,4,5) to be determined. Note that the || D(F) ||(||) || relation for C is simply from the SU(3) symmetry. hA To integrate over the phase space of the three-body B BB¯′M decays, we use [6, 18] c → +1 (mB−mMc)2 β1/2λ1/2 Γ = t t ¯2 dt dcosθ , (12) Z Z (8πm )3|A| −1 (mB+mB¯′)2 B 5 where βt = 1−(mB+mB¯′)2/t, λt = m4B +m4Mc +t2−2m2Mct−2m2Bt−2m2Mcm2B, the angle θ is between B¯′ and M moving directions in the BB¯′ rest frame, and ¯2 is the squared c |A| amplitude of Eq. (2) by summing over all spins. Note that the B(B¯′) energy is given by m2 +t m2 β1/2λ1/2cosθ EB(B¯′) = B − B(4B¯m′) ∓ t t . (13) B From Eq. (12), we define the angular distribution asymmetry: +1 dΓ dcosθ 0 dΓ dcosθ A 0 dcosθ − −1 dcosθ , (14) θ ≡ R+1 dΓ dcosθ +R0 dΓ dcosθ 0 dcosθ −1 dcosθ R R where dΓ/dcosθ is a function of cosθ known as the angular distribution, which presents the M -B¯′ angular correlation in B BB¯′M . c c → III. NUMERICAL ANALYSIS In our numerical analysis, the theoretical inputs of the CKM matrix elements in the Wolfenstein parameterization and the decay constants for D(∗) are given by [19, 20] (V ,V ,V ) = (Aλ2,1 λ2/2,λ), cb ud us − (λ, A, ρ, η) = (0.225, 0.814, 0.120 0.022, 0.362 0.013), ± ± (fD, fD∗) = (204.6 5.0, 252.2 22.7) MeV. (15) ± ± In Table I, we adopt the B D(∗) transition form factors from Ref. [9], in which no → uncertainty has been included. As mentioned early, the decays of B¯0 Λp¯D+ and B¯0 → → Λp¯D∗+ belong to the current-type modes, described by the timelike baryonic form factors via the vector and axial-vector quark currents. Note that B¯0 Λp¯π+ and B− Λp¯ρ0 → → are also connected to the timelike baryonic form factors, but dominated by the additional ones via the scalar and pseudoscalar currents. With the extraction by the data from the TABLE I. The form factors of B D(∗) at t = 0 in Ref. [9] with M M = 6.4 GeV. P V → ≃ B D(∗) FBD FBD VBD∗ ABD∗ ABD∗ ABD∗ → 1 0 1 0 1 2 f(0) 0.67 0.67 0.76 0.69 0.66 0.62 σ 0.57 0.78 0.57 0.58 0.78 1.40 1 σ —– —– —– —– —– 0.41 2 6 current-type baryonic B decays [13], F and g as the timelike baryonic form factors can be 1 A given. Because the B¯0 pp¯ transition form factors in B¯0 pp¯D(∗)0 are related to those → → of the charmless B pp¯M with M = K(∗), π(ρ) and the semileptonic B− pp¯e−ν¯ decay, e → → the extractions of f (g ) are also available [2]. It is hence determined that i i (C , C , C ) = (111.4 14.6, 6.8 2.0, 2.3 0.9) GeV4, || D F ± − ± ± (D ,D ) = (36.9 45.9, 348.2 18.7) GeV5, || || ± − ± (D2,D3,D4,D5) = ( 44.7 30.4, 426.7 182.5,4.3 20.2,135.2 29.4) GeV4. (16) || || || || − ± − ± ± ± In addition, a and a are fitted to be 1 2 a = 1.15 0.04, a = 0.40 0.04. (17) 1 2 ± ± As a result, we can reproduce the branching ratios shown in Table II. It should be pointed out that the main reason for the underestimated breaching ratios of B¯0 Λp¯D(∗)− → in Ref. [2] is due to the small values of F and g extracted from the data of e+e− pp¯(nn¯) 1 A → (pp¯ e+e−), which are in fact related to the electromagnetic form factors of the proton → (neutron) pair without taking into account the timelike axial structures, induced from the weak currents due to W and Z bosons. However, in this work, we take the data from the current-type baryonic B decays as used in Ref. [13], which explains why the data in Eq. (1) of (B¯0 Λp¯D(∗)−) can be explained. With the current precise data for the axialvector B → current already, futurenewdatashouldnotchangeourpresent fittingparametersvery much. TABLE II. The data are from Refs. [1, 20, 21]. decay mode data our results 104 (B¯0 pp¯D0) 1.04 0.07 1.04 0.12 B → ± ± 104 (B¯0 pp¯D∗0) 0.99 0.11 0.99 0.09 B → ± ± 105 (B¯0 Λp¯D−) 2.51 0.44 1.85 0.30 B → ± ± 105 (B¯0 Λp¯D∗−) 3.36 0.77 2.75 0.24 B → ± ± (B¯0 pp¯D0) —– +0.04 0.01 θ A → ± (B¯0 pp¯D∗0) —– +0.04 0.01 θ A → ± (B¯0 Λp¯D−) 0.08 0.10 0.030 0.002 θ A → − ± − ± (B¯0 Λp¯D∗−) +0.55 0.17 +0.150 0.000 θ A → ± ± 7 In the table, we also show our predictions of the angular distribution asymmetries. In particular, our result of (B¯0 Λp¯D−) = 0.030 0.002 is consistent with the data in θ A → − ± Eq. (1) [1], which shows that the unexpected large center number of (B¯0 Λp¯π−) = θ A → 30% is either to be a much small value in the future measurement or due to some unknown − sources through the (pseudo)scalar currents from the penguin diagrams. It is interesting to note that our prediction of (B¯0 Λp¯D∗−) = 0.150 0.000 is large but it is still lower θ A → ± than the data of (55 17)% in Ref. [1]. Note that the small uncertainty of our prediction ± results from the elimination of the timelike form factors by Eq. (14). The reason why the decay of B¯0 Λp¯D∗− can lead to a considerable large 15% is that, being one θ → A ≃ of the B D∗ transition form factors in Eq. (4), the VBD∗ term with ǫ is able to → 1 µναβ relate F and g from different currents, such that VBD∗ABD∗F g (E E ) can arise with 1 A 1 1 1 A p¯− p E E cosθ. It is important to point out that in the future experiments, our prediction p¯ p − ∝ of (B¯0 pp¯D0) = 0.04 0.01 can be used to check if there is a simple relation between θ A → ± B¯0 pp¯D0 and B− pp¯π−, which are both dominated by the tree-level contributions. In → → addition, we remark that our results are based on the form factors in Table I without any uncertainty included. If there are some possible errors, our fitting values for the angular distributions could change. IV. CONCLUSIONS We have revisited the charmful three-body baryonic decays of B¯0 Λp¯D(∗)+. With the → timelikebaryonicformfactorsnewlyextractedfromthebrayonicB decaysinsteadofe+e− → pp¯(nn¯) (pp¯ e+e−), we have found that (B¯0 Λp¯D+,Λp¯D∗+) = (1.85 0.30,2.75 → B → ± ± 0.24) 10−5, which agree with the data in Eq. (1) from the BELLE Collaboration [1]. The × agreement has demonstrated that our theoretical approach based on the factorization is still valid. Clearly, the revision of model parameters and the modification of the factorization approach are not required unlike the statement in Ref. [1]. We have also studied the M -B¯′ angular distribution asymmetries in the charmful bary- c onic B decays of B BB¯′M . Explicitly, we have obtained (B¯0 Λp¯D+,Λp¯D∗+) = c θ → A → ( 0.030 0.002,+0.150 0.000), which are consistent with the current data. In addition, − ± ± we have predicted that (B¯0 pp¯D0,pp¯D∗0) = +0.04 0.01. We believe that the fu- θ A → ± ture precision measurements of (B pp¯D(∗),Λp¯D(∗)) could be used to compare with the θ A → 8 charmless counterparts of (B− pp¯K−(π−)) and (B Λp¯π). It is expected that the θ θ A → A → differences between the charmful and charmless cases, such as (B¯0 Λp¯π−) 41% θ A → ≃ − and (B¯0 Λp¯D−), would be originated from different contributions at tree and penguin θ A → levels. Clearly, it is worthy to have close examinations of (B BB¯′M ) at BELLE and θ c A → LHCb as well as the future super-B facilities. ACKNOWLEDGMENTS The work was supported in part by National Center for Theoretical Science, National Sci- ences Council (NSC-101-2112-M-007-006-MY3), MoST (MoST-104-2112-M-007-003-MY3) and National Tsing Hua University (104N2724E1). [1] Y.Y. Chang et al. [Belle Collaboration], Phys. Rev. Lett. 115, 221803 (2015). [2] C.H. Chen, H.Y. Cheng, C.Q. Geng and Y.K. Hsiao, Phys. Rev. D 78, 054016 (2008). [3] M.Z. Wang et al. [Belle Collaboration], Phys. Rev. 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