Three-body charmless baryonic B¯s0 decays

Physics Letters B 767 (2017) 205–208

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Physics Letters B www.elsevier.com/locate/physletb

Three-body charmless baryonic B¯ 0s decays C.Q. Geng a,b,∗ , Y.K. Hsiao a,b , Eduardo Rodrigues c a b c

Chongqing University of Posts & Telecommunications, Chongqing, 400065, China Department of Physics, National Tsing Hua University, Hsinchu, 300, Taiwan Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA

a r t i c l e

i n f o

Article history: Received 3 January 2017 Accepted 1 February 2017 Available online 3 February 2017 Editor: J. Hisano

a b s t r a c t ¯ M − ), with M = We study for the first time the three-body charmless baryonic decays B¯ 0s → p¯  M + ( p  ¯ K − ) and B¯ 0s → p  ¯ π − are (5.1 ± 1.1) × π , K . We find that the branching ratios of B¯ 0s → ( p¯  K + and p  10−6 and (2.8 ± 1.5) × 10−7 , respectively, which agree with recent experimental results reported by the LHCb collaboration. In addition, we derive the relations B ( B¯ 0s → p¯  K + )  ( f K / f π )2 (τ B 0 /τ B 0 )B ( B¯ 0 →

1. Introduction In contrast with mesonic B decays, the decays of B mesons to baryonic final states have been observed to have unique signa¯ 2 ) formations, which reflect rich tures due to the baryon-pair (B1 B mechanisms for the hadronizations of the spinors. For example, the BaBar and Belle experiments at the B factories [1] reported typical three-body charmless baryonic B decay branching ratios B( B → B1 B¯ 2 M )  O(10−6 ), and provided evidence for prominent peaks around mB1 B¯ 2  mB1 + mB¯ 2 in the baryon–antibaryon spec-

¯ 2 formations tra of baryonic B decays [2], which show that the B1 B ¯ 2, favour the threshold area. However, in two-body decays B → B1 B there is no large energy release from the recoiled meson [3], such ¯ 2 is at the m B scale, which definitely that the total energy of B1 B ¯ 2) deviates from the threshold area [4]. As a result, B ( B → B1 B are seen to be small, around 10−8 –10−7 [5–7]. Furthermore, the angular distribution asymmetry Aθ of B¯ 0 → p¯ π + has been measured to have an unexpectedly large value of (−41 ± 11 ± 3)%, indicating significant interference as a result of the baryonic form factors [9,10]. The same behaviour has been observed in decays to final states with open charm, for example Aθ ( B¯ 0 →  p¯ D ∗+ ) = (55 ± 17)% [8]. ¯ 2 ( M ) decays The aforementioned observations in B¯ 0 / B − → B1 B ¯ 2 ( M ) decays now experimentally acmay also hold for B¯ 0s → B1 B

*

Corresponding author. E-mail address: [email protected] (C.Q. Geng).

s

¯ π − )/B( B¯ 0s → p  ¯ K − )  B( B − → p p¯ π − )/B( B − → p p¯ K − ) to be confronted p¯ π + ) and B ( B¯ 0s → p  ¯ K − , p¯  K + can to future experimental measurements. The fact that all four processes B 0s , B¯ 0s → p  occur opens the possibility of decay-time-dependent CP violation measurements in baryonic B decays, something that had not been realized before. © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

cessible to the LHCb collaboration [11,12]. Nonetheless, baryonic B¯ 0s decays are not trivially related to baryonic B¯ 0 and B − decays. For example, replacing (u¯ , d¯ ) by s¯ in B¯ 0 / B − , one may approximately infer that

B( B¯ 0s → p¯  K + )  B( B¯ 0 → p¯ π + ) ,

¯ π − )  B( B − → p p¯ π − ) , B( B¯ 0s → p  ¯ K − )  B( B − → p p¯ K − ) , B( B¯ 0s → p 

(1)

which will be shown to be mostly incorrect, except for the first relation. We will also demonstrate that the recent first observation, made by the LHCb collaboration, of a baryonic B¯ 0s decay, ¯ K − , and the measurement of its branching ranamely B¯ 0s → p  ¯ K− tio [13], combines in reality the branching ratios of B¯ 0s → p  and B¯ 0s → p¯  K + . 2. Formalism The decay B¯ 0 → p¯ π + is flavour specific, unlike the similar mode of the B¯ 0s meson, which can decay to both p¯  K + and ¯ K − final states. The latter three-body baryonic B¯ 0s decays prop ceed through different configurations as demonstrated in the Feynman diagrams in Fig. 1. Specifically, the baryon pairs involve quark currents and B meson transitions as depicted in Figs. 1(a,b) and (c,d), respectively. The amplitudes can be factorized in terms of the effective Hamiltonian at the quark level [14] as [9,15–18]

http://dx.doi.org/10.1016/j.physletb.2017.02.001 0370-2693/© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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C.Q. Geng et al. / Physics Letters B 767 (2017) 205–208

¯ K −. Fig. 1. Feynman diagrams for three-body baryonic B¯ 0s decays, where (a,b) depict B¯ 0s → p¯  K + while (c,d) depict B¯ 0s → p 

A( B¯ 0s → p¯  K + ) =  GF ¯ ) V − A | B¯ 0s  √ α1  p¯ |(¯su ) V − A |0 K + |(ub 2

¯2 with f M the decay constant, the matrix elements of the B → B1 B transition are parameterized as [9,17]

B1 B¯ 2 |¯qγμ b| B  = i u¯ [ g 1 γμ + g 2 i σμν p ν + g 3 p μ + g 4 qμ

 + ¯ 0 ¯ ¯ + α6  p |(¯su ) S + P |0 K |(ub) S − P | B s  ,

+ g 5 ( p B¯ 2 − p B1 )μ ]γ5 v , B1 B¯ 2 |¯qγμ γ5 b| B  = i u¯ [ f 1 γμ + f 2 i σμν p ν + f 3 p μ + f 4 qμ

¯ K −) = A( B¯ 0s → p   GF ¯ ub ¯ ) V − A | B¯ 0s  √ α1  K − |(¯su ) V − A |0 p |( 2

+ f 5 ( p B¯ 2 − p B1 )μ ] v , B1 B¯ 2 |¯qb| B  = i u¯ [ g¯ 1 p/ + g¯ 2 ( E B¯ 2 + E B1 ) + g¯ 3 ( E B¯ 2 − E B1 )]γ5 v ,

 − 0 ¯ ¯ ¯ )S−P |B s  , + α6  K |(¯su ) S + P |0 p |(ub V∗a

∗a V ub V us 1

(2)

∗ 2a , where G is V tb V ts 6 F

with α1 = − V tb ts 4 and α6 = the Fermi constant, V i j are the CKM matrix elements, (¯q1 q2 ) V ( A ) and (¯q1 q2 ) S ( P ) stand for q¯ 1 γμ (γ5 )q2 and q¯ 1 (γ5 )q2 , respectively, ef f

ef f

ef f

and a1(4,6) ≡ c 1(4,6) + c 2(3,5) / N c are composed of the effective ef f ef f Wilson coefficients c i defined in Ref. [14] with N c the effective colour number, ranging between 2 and ∞ to account for the non-factorizable effects in the generalized factorization approach. ¯ π − ) is obtained from A( B¯ 0s → p  ¯ K −) The amplitude A( B¯ 0s → p  of Eq. (2) replacing the strange quark by the down quark. In our calculation, the matrix elements of B¯ 0s → p¯  K + in Eq. (2) are expressed as [15,16]

 M |¯qγ μ b| B  = ( p B + p M )μ F 1B M  B1 B¯ 2 |¯q1 γμ q2 |0 = u¯ F 1 γμ + 

m2 − m2M μ B M + B q ( F 0 − F 1B M ) , t



F2

mB1 + mB¯ 2

B1 B¯ 2 |¯q1 γμ γ5 q2 |0 = u¯ g A γμ +

mB1 + mB¯ 2

 qμ

(4) where g i ( f i ) (i = 1, 2, ..., 5) and g¯ j ( ¯f j ) ( j = 1, 2, 3) are the B → ¯ 2 transition form factors. The form factors in Eqs. (3) and (4) B1 B are momentum dependent. Explicitly, F 0B,M 1 are given by [19]

F 1B M (t ) =

F 0B M (t ) =

fi =

γ5 v ,

¯ , B1 B¯ 2 |q1 γ5 q2 |0 = g P u¯ γ5 v , B1 B¯ 2 |¯q1 q2 |0 = f S uv

F 1B M (0) 2 )(1 − σ112t + σ124t )

t M 2V

(1 −

MV

F 0B M (0) 1−

2 + σ024t

σ01 t M 2V

,

MV

(5)

.

MV

In perturbative QCD counting rules, the baryonic form factors depend on 1/t n as the leading-order expansion [9,17,20,21], given by

F1 =

i σμν qμ v ,

hA

B1 B¯ 2 |¯qγ5 b| B  = i u¯ [ ¯f 1 p/ + ¯f 2 ( E B¯ 2 + E B1 ) + ¯f 3 ( E B¯ 2 − E B1 )] v ,

C¯ F 1 t2 D fi t3

, gA = , gi =

C¯ g A t2

D gi t3

, fS =

, ¯f i =

C¯ f S t2

D ¯f

i

t3

where C¯ i = C i [ln(t /20 )]−γ with

, gP =

, g¯ i =

C¯ g P

D g¯ i t3

t2

,

,

(6)

γ = 2.148 and 0 = 0.3 GeV.

(3)

with q = p B − p M = p B1 + p B¯ 2 , t ≡ q , p = p B − q, and u (v) the 2

(anti-)baryon spinor, where F 0B,M 1 are the form factors for the B → M transition, and F 1,2 , g A , h A , f S , and g P the timelike baryonic ¯ K − , besides  M |¯q1 γ μ γ5 q2 |0 = −i f M p μ form factors. For B¯ 0s → p  M

3. Numerical results and discussions For the numerical analysis, the theoretical inputs of the CKM matrix elements in the Wolfenstein parameterization are given by [1]

C.Q. Geng et al. / Physics Letters B 767 (2017) 205–208

207

¯ K − ) and (right) B¯ 0s → p  ¯ π −. Fig. 2. Spectra for the three-body baryonic decays (left) B¯ 0s → ( p¯ K + , p 

The effective Wilson coefficients for the B¯ 0s ( B 0s ) decays are given by [14]

V ub = A λ3 (ρ − i η), V tb = 1 − A 2 λ4 /2 , V ud = 1 − λ2 /2, , V td = A λ3 , V us = λ, V ts = − A λ2 + A λ4 [1 + 2(ρ − i η)]/2,

(7)

with (λ, A , ρ , η) = (0.225, 0.814, 0.120 ± 0.022, 0.362 ± 0.013). B K Other parameters are taken to be [19] F 1,s0 (0) = 0.31, (σ11 , σ12 ) = (0.63, 0.33), (σ01 , σ02 ) = (0.93, 0.70), M V = 5.32 GeV, and ( f K , f π ) = (156.2 ± 0.7, 130.4 ± 0.2) MeV [1]. Theoretically, the B¯ 0s → p¯  K + decay is related to B¯ 0 → n p¯ D ∗+ , B¯ 0 →  p¯ D (∗)+ , B¯ 0 → p¯ π + , B − → p¯ (π 0 , ρ 0 ), B¯ 0(s) → p p¯ , and B − → p¯  through the timelike baryonic form factors, which can be connected by the SU (3) flavour and SU (2) spin symmetries [15,20], leading to [10]

 C F1 =

3 2

 C fS = −

 C || , C g A = 3 2

C¯ || , C g P

3

(C || + C 2 ) ,  3 =− (C¯ || + C¯ 2 ) , 2

(8)

2

(C¯ || , C¯ 2 ) = (537.6 ± 28.7, −342.3 ± 61.4) GeV4 ,

(9)

extracted from the data. Here, F 2 = F 1 /(t ln[t /20 ]) [22] and h A = C h A /t 2 have both been neglected. Note that C h A is fitted to be in accordance with B ( B¯ 0 → p p¯ ) = 1.47 × 10−8 [4]. On the other ¯ K − decay corresponds to B¯ 0 → p p¯ D (∗)0 , B − → hand, the B¯ 0s → p  p p¯ ( K (∗)− , π − ), B¯ 0 → p p¯ K¯ (∗)0 , and B − → p p¯ e − ν¯ e [23,24] through ¯ transition form factors, which are related by the same the B → BB symmetries [9,17], given by

D g1 = D f 1 = −



D g¯ 1 = D ¯f = − 1

2 3 2

 D || , D g4,5 = − D f 4,5 = −



D || , D g¯ 2,3 = − D ¯f

= 1.168, = −0.365 , = 241.9 ± 3.2η + 1.4ρ + i (31.3 ∓ 1.4η + 3.2ρ ), = −508.7 ∓ 9.6η − 4.2ρ + i (−93.9 ± 4.2η − 9.6ρ ), = 149.4 ± 3.2η + 1.4ρ + i (31.3 ∓ 1.4η + 3.2ρ ), = −645.5 ∓ 9.6η − 4.2ρ + i (−93.9 ± 4.2η − 9.6ρ ). (12)

Integrating over the phase space of the three-body decays [1] ¯ K − ) and B¯ 0s → p  ¯ π− we obtain the spectra for B¯ 0s → ( p¯  K + , p  in Fig. 2, which clearly present the threshold enhancement observed in a multitude of baryonic B¯ 0 and B − decays. The branching ratios are predicted to be

¯ K − ) = (1.31 ± 0.32+0.22 ± 0.01) × 10−6 , B( B¯ 0s → p  −0.10

(C || , C 2 ) = (154.4 ± 12.1, 19.3 ± 21.6) GeV4 ,

3

ef f c2 ef f 104 c 3 ef f 104 c 4 ef f 104 c 5 ef f 104 c 6

0.67 −6 B( B¯ 0s → p¯  K + ) = (3.75 ± 0.81+ , −0.31 ± 0.01) × 10

where



ef f

c1

2,3

=−

3

4,5

D , 2 || 3 2

¯ , D || 2,3

(13)

with the uncertainties from the form factors, non-factorizable effects, and CKM matrix elements in order. The B ( B¯ 0s → p¯  K + ) is calculated to be close to the observed B ( B¯ 0 → p¯ π + ) = (3.14 ± 0.29) × 10−6 [1], which confirms the first relation in Eq. (1). Nonetheless, using the experimental measurements of B ( B − → ¯ M )  0.2 × p p¯ M ) ( M = K − , π − ) [1], we find that B ( B¯ 0s → p  B( B − → p p¯ M ), which disproves the other relations in Eq. (1). The ¯ and B − → p p¯ transitions give reason for this is that the B¯ 0s → p  different contributions. Consequently, we should revise the relations in Eq. (1) to be

B( B¯ 0s → p¯  K + )  ( f K / f π )2 (τ B 0 /τ B 0 )B( B¯ 0 → p¯ π + ) , s

(10)

with the vanishing form factors ( g 2,3 , f 2,3 ) due to the derivations of f M p μ u¯ (σμν p ν ) v = 0 for g 2 ( f 2 ) and f M p μ u¯ p μ v ∝ m2M for f 3 ( g 3 ) in the amplitudes, where

D || = (45.7 ± 33.8) GeV5 ,

( D 4|| , D 5|| ) = (6.5 ± 18.1, −147.1 ± 29.3) GeV4 , ¯ || = (35.2 ± 4.8) GeV5 , D ( D¯ 2|| , D¯ 3|| ) = (−22.3 ± 10.2, 504.5 ± 32.4) GeV4 .

¯ π − ) = (2.79 ± 1.37+0.64 ± 0.17) × 10−7 , B( B¯ 0s → p  −0.30

(11)

¯ π − ) B( B − → p p¯ π − ) B( B¯ 0s → p  .  ¯ K − ) B( B − → p p¯ K − ) B( B¯ 0s → p 

(14)

From an experimental perspective, the measured branching ra¯ K − + B¯ 0s → p  ¯ K − ), given that the flavour of tio is B ( B 0s → p  0 the reconstructed B s meson at production is not determined – the identification of the flavour at production, a procedure known as flavour tagging, requires a decay-time-dependent analysis. As¯ K − + B¯ 0s → p  ¯ K − ) is suming negligible CP violation, B ( B 0s → p  equivalent to the combination of the two branching ratios B ( B¯ 0s → ¯ K − ) = (5.1 ± 1.1) × 10−6 . This calculation agrees p¯  K + + p  ¯ K − + B¯ 0s → well with the experimental measurement, B ( B 0s → p 

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C.Q. Geng et al. / Physics Letters B 767 (2017) 205–208

¯ K − ) = (5.48+0.82 ± 0.60 ± 0.51 ± 0.32) × 10−6 , reported by the p −0.80

¯ π − ) is estimated LHCb collaboration [13]. In contrast, B ( B¯ 0s → p  − 7 to be of order 10 , consistent with its non-observation with the present data sample [13]. ¯ K − , p¯  K + are possible, just as All four processes B 0s , B¯ 0s → p  0 ± ∓ ¯ in the case of the B s → D s K decays [25]. A B-flavour tagged decay-time-dependent analysis of these baryonic decay modes is necessary to disentangle all contributions. As the ratio of the B¯ 0s → p¯  K + and B 0s → p¯  K + branching ratios is predicted to be rather large, cf. Eq. (13), sizeable interference due to B 0s – B¯ 0s mixing is expected, which hints at possibly large time-dependent CP violating asymmetries. Time-dependent analyses require a typical minimum data sample of order 1000 to 1500 signal candidates, see for example the LHCb analysis presented in Ref. [25]. Extrapolating from ¯ K − p¯  K + candidates selected in the rethe 260 ± 21 B 0s , B¯ 0s → p  cent LHCb analysis [13], assuming (as done in LHCb extrapolations) a two-fold increase in the bb¯ production cross-section between the first data taking period of the LHC, and the present second period started in 2015, we conclude that such an analysis will require the full data sample to be collected by 2018. ¯ K − ), it is promisBased on the observation of B¯ 0s → ( p¯  K + , p  ing to study other charmless baryonic B¯ 0s decays such as B¯ 0s → ¯ K ∗− , B¯ 0s → φ ¯  ¯ 0 )φ , B¯ 0s → ¯ , B¯ 0s → ( 0 , ¯ 0, 0 p¯  K ∗+ , p  ¯ 0 K − , and B¯ 0s → p  ¯ 0 π − . The presence of extra resop¯  0 K + , p  nances or neutral particles in the final states of these decay modes makes the experimental searches more demanding, though feasible by both the LHCb experiment and the future Belle II experiment.

by the future experiments at LHCb. The fact that all four pro¯ K − , p¯  K + can occur opens the possibility of cesses B 0s , B¯ 0s → p  decay-time-dependent CP violation measurements in baryonic decays, something that had not been realized before. Acknowledgements The work of C.Q. Geng and Y.K. Hsiao was supported, in part, by National Center for Theoretical Sciences, MoST (MoST104-2112-M-007-003-MY3), and National Science Foundation of China (11547008 and 11675030). The work of E. Rodrigues was supported, in part, by U. S. National Science Foundation award ACI-1450319. References [1] K.A. Olive, et al., Particle Data Group Collaboration, Chin. Phys. C 38 (2014) 090001. [2] K. Abe, et al., Belle Collaboration, Phys. Rev. Lett. 88 (2002) 181803. [3] W.S. Hou, A. Soni, Phys. Rev. Lett. 86 (2001) 4247. [4] Y.K. Hsiao, C.Q. Geng, Phys. Rev. D 91 (2015) 077501. [5] R. Aaij, et al., LHCb Collaboration, J. High Energy Phys. 10 (2013) 005. [6] R. Aaij, et al., LHCb Collaboration, Phys. Rev. Lett. 113 (2014) 141801. [7] R. Aaij, et al., LHCb Collaboration, arXiv:1611.07805 [hep-ex]. [8] Y.Y. Chang, et al., Belle Collaboration, Phys. Rev. Lett. 115 (2015) 221803. [9] [10] [11] [12] [13]

4. Conclusions We have studied the three-body charmless baryonic decays ¯ M − , with M = π , K . We have predicted B¯ 0s → p¯  M + and p  ¯ K − ) to be the combined branching ratio of B¯ 0s → ( p¯  K + and p  (5.1 ± 1.1) × 10−6 , in good agreement with the recently presented experimental result by the LHCb collaboration [13]. We ¯ π − ) = (2.8 ± 1.5) × 10−7 , which is further obtained B ( B¯ 0s → p  below the current experimental sensitivity of the LHCb analysis. We have also presented useful relations between the three-body baryonic decays of B¯ 0s and B¯ 0 / B − , such as B ( B¯ 0s → p¯  K + )  ¯ π − )/B ( B¯ 0s → ( f K / f π )2 (τ B 0 /τ B 0 )B ( B¯ 0 → p¯ π + ) and B ( B¯ 0s → p  s

¯ K − )  B ( B − → p p¯ π − )/B ( B − → p p¯ K − ), which can be tested p

[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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