preliminary estimate of branching ratios of weak hadronic decays of bottom baryons emitting...

Eur. Phys. J. C (2011) 71: 1538DOI 10.1140/epjc/s10052-011-1538-0

Regular Article - Theoretical Physics

Preliminary estimate of branching ratios of weak hadronic decaysof bottom baryons emitting charmless scalar mesons

Arvind Sharma1,a, Rohit Dhir2,b, R.C. Verma2,c

1Department of Physics, College of Engineering and Management, Kapurthala 144601, India2Department of Physics, Punjabi University, Patiala 147002, India

Received: 13 October 2010 / Published online: 29 January 2011© Springer-Verlag / Società Italiana di Fisica 2011

Abstract We give the first estimate of charmless scalar-meson emitting weak hadronic decays of Λ0

b, Ξ0b and Ξ−

b

bottom baryons employing the pole model and consequentlypredict their branching ratios.

1 Introduction

The heavy baryon mass spectra have become a subject ofgreat interest due to the growing experimental facilities atBelle, BaBar, DELPHI, CLEO, CDF etc. [1–6]. Recently,the lifetime of Λ0

b , Ξ0b and Ξ−

b have been measured [7]. Al-though the experimental data [7] on nonleptonic decays ofcharm (C = 1) baryons have become available in the lastdecade, measurements on weak decays of bottom baryonshave merely begun. On the theoretical side, several authorshave investigated weak decays of charm baryons [8–18],only a few attempts have been made [19–22] to study theweak hadronic decays of bottom baryons, mainly emittings-wave mesons. However, the bottom baryons, being heavy,can also emit p-wave mesons.

In our recent works [17, 18], we have investigated the p-wave meson emitting decays of charmed baryons employingthe factorization scheme and including the pole contribu-tions. It has been shown that such decays emitting scalar andaxial-vector mesons acquire significant branching ratios ofthe order of s-wave meson emitting decays. In this work, westudy the scalar-meson emitting decays of bottom baryons.We have already seen that the factorization contribution isnegligible in comparison to the pole contributions in case ofthe scalar-meson emitting decays of charmed baryons dueto their vanishing decay constants [23]. For the same rea-son, factorizable contributions to the bottom baryon decays

a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

emitting scalar mesons are also expected to be suppressed.Therefore, we present the first estimate of the branching ra-tios of weak nonleptonic decays of Λ0

b , Ξ0b and Ξ−

b emittingscalar mesons in the pole model.

2 General framework

2.1 Kinematics

The matrix element for the baryon Bi(1/2+) → Bf (1/2+)+Sk(0+) decay process can be written as

〈Bf Sk|HW |Bi〉 = iuBf(A + γ5B)uBi

,

where A and B are parity conserving (PC) and parity violat-ing (PV) amplitudes, respectively, uB are Dirac spinors. Thedecay width for Bi(pi) → Bf (pf ) + Sk(q) is given by

Γ = C1[|A|2 + C2|B|2], (1)

and the asymmetry parameter is

α = 2x Re(A ∗ B)

|A|2 + x2|B|2 (2)

where

C1 = |qμ|8π

(mi + mf )2 − m2k

m2i

,

C2 = (mi − mf )2 − m2k

(mi + mf )2 + m2k

,

and x = qμ/(Ef

+ mk). Ef is the energy of the daughterbaryon and four momentum of the scalar meson qμ = (pi −pf )μ is

|qμ| = 1

2mi

√[m2

i − (mf − mk)2][

m2i − (mf + mk)2

],

mailto:[email protected]



of 7 Eur. Phys. J. C (2011) 71: 1538

where mi and mf are the masses of the initial and finalbaryons and mk is the emitted meson mass.

2.2 Weak Hamiltonian

For bottom changing �b = 1 decays involving b → c tran-sition, QCD modified current ⊗ current weak Hamiltonianis given below:

HW = GF√2

{VcbV

∗ud

[a1(cb)(du) + a2(db)(cu)

]

+ VcbV∗cs

[a1(cb)(sc) + a2(sb)(cc)

]

+ VcbV∗us

[a1(cb)(su) + a2(sb)(cu)

]

+ VcbV∗cd

[a1(cb)(dc) + a2(db)(cc)

]}, (3)

where (qiqj ) ≡ qiγμ(1 − γ5)qj denotes the weak V –A cur-rent. We follow the convention of large Nc limit to fix QCDcoefficients a1 ≈ c1 and a2 ≈ c2, where [24]:

c1(μ) = 1.12, c2(μ) = −0.26 at μ ≈ m2b. (4)

2.3 Scalar-meson spectroscopy

The identification of the scalar meson family in the stan-dard nonet picture has been a subject of much controversy.The Particle Data Group suggests [7] that there are twosets of scalar mesons nonet, (1) light scalars: two isoscalarsσ(600), f0(980), the isovector a0(980), and the isodou-blet κ(800); (2) heavier scalars: two isoscalars f0(1370),f0(1500)/f0(1710), isovector a0(1450), and isodoubletK∗

0 (1430). In the following, we limit ourselves to lighterscalar-meson emitting decays of Λ0

b , Ξ0b and Ξ−

b .

(i) qq pictureIn the conventional qq picture, isovector and isodou-

blet scalar mesons are given by

a+0 = ud, a0

0 = (uu − dd)/√

2, a−0 = du,

κ+ = us, κ0 = ds, κ0 = sd, κ− = su.

The unitary singlet and octet states,

ε1 = (uu + dd + ss)/√

3,

ε8 = (uu + dd − 2ss)/√

6,(5)

mix to generate the physical states as

σ = cos θSε1 + sin θSε8,

f0 = − sin θSε1 + cos θSε8.(6)

Alternatively, the mixing can also be expressed as

σ =(

uu + dd√2

)cos θ − ss sin θ,

f0 =(

uu + dd√2

)sin θ + ss cos θ,

(7)

where θ = π + (θideal − θS). In case of the ideal mixing,θS = θideal = 35.3◦ [25], the ss component decouples togive

σ = (uu + dd)/√

2, f0 = −ss,

which is supported by the data of D+s → f0π

+ andφ → f0γ implying the copious f0(980) productionvia its ss component. However, there also exists someexperimental evidence indicating that f0(980) is notpurely a ss state. f0–σ mixing has been discussed indetail in [26–31], yielding 25◦ < θ < 40◦, 140◦ < θ <

165◦. In fact, phenomenologically there does not exist aunique mixing angle solution, which may indicate thatσ(600) and f0(980) are not purely qq bound states.

(ii) q2q2 pictureAn alternative and arguably more natural explana-

tion for the masses and decay properties of the lightestscalar mesons is to regard these as exotic q2q2 diquark–antidiquark states. In this picture, scalar mesons aregiven below [25, 31–34]

a+0 = udss, a0

0 = (sdsd − susu)/√

2,

a−0 = duss, κ+ = udsd, κ0 = udsu,

κ0 = suud, κ− = sdud, σ = udud,

and

f0 = (sdsd + susu)/√

2.

This is supported by a lattice calculation [27] and cor-responds to the ideal mixing angle tan θS = −√

2 orθS ≈ −54.8◦ [25]. Similar to the qq scenario, generalmixing can be described as

σ = −ss

(uu + dd√

2

)sin θ + uudd cos θ,

f0 = ss

(uu + dd√

2

)cos θ + uudd sin θ,

(8)

where θ = 174.6◦ ± 3.3◦ [35] indicating a small devia-tion from the ideal mixing angle (θ = 180◦). However,looking at the uncertainty in determining the angle, andfor simplicity, we assume ideal mixing in this work.

Eur. Phys. J. C (2011) 71: 1538 of 7

3 Pole model

In the pole model, one introduces a set of intermediate statesinto the decay process so that the weak and strong verticesbecome separated. In this way, the process under consid-eration passes through certain hadronic intermediate stateswhich can be decomposed into two steps: production ofthese intermediate states in the strong process, followingwhich the intermediate baryon then undergoes a weak tran-sition to the final baryon. A and B are then simply given bythe product of strong- and weak-coupling constants dividedby the mass difference and mass sum, respectively, for A

and B .For Bi(1/2+) → Bf (1/2+)+Sk(0+) decay process in s-

and u-channels, positive-parity intermediate baryon (J P =1/2+) poles give rise to the following terms:

Apole = −∑

n

[gBf BnSk

ani

mi − mn

+ af ngBnBiSk

mf − mn

], (9)

Bpole =∑

n

[gBf BnSk

bni

mi + mn

+ bf ngBnBiSk

mf + mn

], (10)

where gijk are the strong baryon–scalar-meson couplingconstants. Weak baryon–baryon matrix elements aij and bij

are defined as

〈Bi |HW |Bj 〉 = uBi(aij + γ5bij )uBj

. (11)

In addition to the low-lying positive-parity intermediatebaryon poles (J P = 1/2+), negative-parity intermediatebaryon (J P = 1/2−) may also contribute to these processes.Unfortunately, there is no information available about thescalar-meson strong coupling constants for the negative-parity baryons. Further, these contributions are expectedto be relatively suppressed because of their large masses.Therefore, we have restricted to positive-parity intermedi-ate baryon poles in order to obtain the estimate of the polecontributions to the scalar-meson emitting decays of bot-tom baryons. It is well known that the matrix elementsbij vanish for the hyperons in the SU(3) limit [36–41]. Inthe case of the charm decays also, it has been shown [8]that bij � aij , thereby suppressing the PV pole contribu-tions. Assuming the same trend in the bottom sector, PVpole contributions are neglected in the present work. Infact, PV contributions (Bpole) are further suppressed due tothe sum of the baryon masses appearing in the denomina-tor.

3.1 Strong scalar-meson–baryon couplings

In qq picture, Hamiltonian representing the strong cou-plings can be written as

Hstrong = √2gF

(1

2B[a,b]dB[a,b]cSc

d − B[d,a]bB[a,c]bScd

)

+ √2gD

(1

2B[a,b]dB[a,b]cSc

d + B[d,a]bB[a,c]bScd

),

(12)

where B[a,b]c , B[a,b]d and Scd are the baryon, anti-baryon,

and scalar-meson tensors respectively and gD (gF ) are con-ventional D-type and F-type parameters [42–44].

On the experimental side, there is no measurement avail-able for the scalar-meson–baryon coupling constants. Re-cently, G. Erkol et al. [25] have obtained the scalar-meson–baryon coupling constants using QCD sum rules. In theiranalysis, gD and gF have been determined as

gD = 5.4, and gF = 6.6. (13)

Similarly, strong couplings (BBS) have been estimated inq2q2 picture of the scalar mesons in the work [25]. In thiscase the following values have been obtained

gD = 3.8, and gF = 4.7. (14)

The values of strong scalar-meson–baryon coupling con-stants relevant for our calculation have been given in Table 1.

3.2 Weak transitions

In the tensor notation, the weak Hamiltonian (3) for quarklevel process qi + qj → ql + qm can be expressed as

HW = GF√2

VilV∗jm

[c−(mb)H

[l,m][i,j ] + c+(mb)H

(l,m)(i,j)

], (15)

where c− = c1 + c2 and c+ = c1 − c2 and the brackets[,] and (,), respectively, denote the antisymmetrization andsymmetrization among the indices. However, for baryon–baryon weak transitions [45–51], it has been shown that thepart of the Hamiltonian H

(l,m)(i,j) , being symmetric in the color

indices also, does not contribute. Thus, by choosing the ap-propriate indices in the contraction

HW = aW

[B[i,j ]kB[l,m]kH [l,m]

[i,j ]], (16)

we obtain the weak baryon–baryon matrix elements (aij ) for�b = 1, �C = 1, �S = 0 and �b = 1, �C = 1, �S = −1modes which have been given in Table 2. It is worth remark-ing here that since c-quark does not appear as constituent inthe parent baryons (Λ0

b,Ξ0b ,Ξ−

b ) considered here, the de-cays with selection rules �b = 1, �C = 0, �S = −1 and�b = 1, �C = 0, �S = 0 do not acquire pole contribu-tions from the weak Hamiltonian (3). However, these decaymodes may receive contributions through b+u → u+ s andb+u → u+d quark processes, which are highly suppresseddue to the correspondingly small CKM matrix elements.

of 7 Eur. Phys. J. C (2011) 71: 1538

Table 1 Scalar-meson–baryon strong coupling constants

B → BS qq picture q2q2 picture

Ξ0c → Λ+

c κ−0 −4.2 −3.1

Ξ0c → Σo

c κ00 6.2 4.4

Ξ0c → Ξ0

c a00 −3.0 2.2

Ξ0c → Ξ0

c σ −4.3 3.1

Ξ0c → Ξ0

c f0 −7.0 6.5

Ξ0c → Ξ ′0

c a00 3.1 −2.2

Ξ0c → Ξ ′0

c σ 4.4 −3.1

Ξ0c → Ξ ′0

c f0 −3.1 2.2

Ξ0c → Ω0

c κ00 −6.2 −4.4

Ξ ′0c → Λ+

c κ−0 −4.4 −3.1

Ξ ′0c → Σ0

c κ00 13.2 9.4

Ξ ′0c → Ξ ′0

c a00 −6.6 4.7

Ξ ′0c → Ξ ′0

c σ 6.6 6.6

Ξ ′0c → Ξ ′0

c f0 −9.3 14.1

Ξ ′0c → Ω0

c κ00 13.2 9.4

Σ0c → Λ+

c a−0 −6.2 −4.4

Σ0c → Σ0

c a00 −13.2 9.4

Σ0c → Σ0

c σ 13.2 13.3

Σ0c → Σ0

c f0 −13.2 9.4

Σ0c → Ξ0

c κ00 6.2 4.4

Ω0c → Ξ+

c κ−0 −6.2 −4.4

Ω0c → Ξ ′+

c κ−0 13.2 9.4

Ω0c → Ω0

c a00 0 0

Ω0c → Ω0

c σ 0 0

Ω0c → Ω0

c f0 18.6 18.8

Λ0b → Λ0

ba00 0 0

Λ0b → Λ0

bσ 6.0 6.1

Λ0b → Λ0

bf0 0 4.3

Λ0b → Σ0

b a00 −6.2 4.4

Λ0b → Σ0

b σ 0 0

Λ0b → Σ0

b f 00 0 0

Λ0b → Ξ0

b κ00 4.2 3.0

Λ0b → Ξ ′0

b κ−0 4.4 3.1

Ξ0b → Σ0

b κ00 4.4 3.1

Ξ0b → Ξ0

b a00 3.0 −2.2

Ξ0b → Ξ0

b σ 3.0 3.1

Ξ0b → Ξ0

b f0 −4.2 6.5

Ξ0b → Ξ ′0

b a00 −3.1 2.2

Ξ0b → Ξ ′0

b σ 4.4 −3.1

Ξ0b → Ξ ′0

b f 00 −3.1 2.2

Ξ−b → Λ0

bκ−0 −4.2 −3.0

Ξ−b → Σ0

b κ−0 4.4 3.1

Ξ−b → Ξ0

b a−0 4.2 3.1

Ξ−b → Ξ ′0

b a−0 −4.4 −3.1

Table 2 Weak baryon–baryon transition amplitudes

Weak transition Transition amplitude (×aW )

�b = 1, �C = 1, �S = 0

Λ0b → Σ0

c

√3/2

Σ+b → Λ+

c −√3/2

Σ+b → Σ+

c 3/√

2

Σ0b → Σ0

c 3/√

2

Ξ0b → Ξ0

c 1/2

Ξ0b → Ξ ′0

c

√3/2

Ξ ′0b → Ξ0

c

√3/2

Ξ ′0b → Ξ ′0

c 3/2

�b = 1, �C = 1, �S = −1

Λ0b → Ξ0

c −1/2

Λ0b → Ξ ′0

c

√3/2

Σ+b → Ξ+

c −√3/2

Σ+b → Ξ ′+

c 3/√

2

Σ0b → Ξ0

c −√3/2

Σ0b → Ξ ′0

c 3/2

Ξ0b → Ω0

c

√3/2

Ξ ′0b → Ω0

c 3/√

2

4 Numerical results: discussion and conclusion

We compute the pole contributions using (9) for �b = 1,�C = 1, �S = 0 and �b = 1, �C = 1, �S = −1 modes. Itmay be noted that weak baryon–baryon transitions appear-ing in pseudoscalar or scalar-meson emitting decays of bot-tom baryons are the same. Sinha et al. [20] have already esti-mated the weak transition amplitudes by quark model calcu-lations as aΛ0

b→Σ0c

is related to aΣ+→p (= 1.2 × 10−7 GeV)through the following relation:

〈Σ0c |HPC

W |Λ0b〉 = 1√

6

Vcb

Vus

〈p|HPCW |Σ+〉. (17)

However, this estimate is not reliable due to the badly bro-ken SU(5) and ignores the difference in QCD enhancementsand flavor dependent baryon overlap function, |ψ(0)|2 ap-pearing in the baryon to baryon weak transitions. Therefore,we follow the quark model analysis of [9, 13, 52, 53], whichexpress

〈Σ0c |HPC

W |Λ0b〉 = 2

√2

3

GF√2

c−(mb)VcbV∗ud |ψ(0)|2b, (18)

where |ψ(0)|2b ≡ 〈ψΣ0c|δ3( r)|ψΛ0

b〉. Similarly, we obtain

〈p|HPCW |Σ+〉 = −3

GF√2

c−(ms)VusV∗ud |ψ(0)|2s , (19)

where |ψ(0)|2s ≡ 〈ψp|δ3( r)|ψΣ+〉. The QCD enhancementdue to the hard gluon exchange in the bottom sector

Eur. Phys. J. C (2011) 71: 1538 of 7

Table 3 Branching ratio for Λ0b decays (qq picture)

Decay Branching ratio (%)without |ψ(0)|2variation

Branching ratio (%)with |ψ(0)|2variation

�b = 1, �C = 1, �S = 0

Λ0b → Λ+

c a−0 4.52 × 10−5 2.34 × 10−4

Λ0b → Ξ0

c κ00 3.50 × 10−4 1.81 × 10−3

Λ0b → Σ+

c a−0 1.03 × 10−3 5.33 × 10−3

Λ0b → Σ0

c a00 1.03 × 10−3 5.32 × 10−3

Λ0b → Σ0

c σ 6.13 × 10−3 3.17 × 10−2

Λ0b → Ξ ′0

c κ00 3.08 × 10−3 1.59 × 10−2

�b = 1, �C = 1, �S = −1

Λ0b → Λ+

c κ−0 1.34 × 10−5 6.96 × 10−5

Λ0b → Ξ+

c a−0 4.91 × 10−6 2.53 × 10−5

Λ0b → Ξ0

c a00 2.48 × 10−6 1.28 × 10−5

Λ0b → Ξ0

c σ 6.97 × 10−6 3.60 × 10−5

Λ0b → Ξ0

c f0 1.33 × 10−5 6.88 × 10−5

Λ0b → Σ+

c κ−0 1.53 × 10−4 7.94 × 10−4

Λ0b → Σ0

c κ00 3.07 × 10−4 1.58 × 10−3

Λ0b → Ξ ′+

c a−0 2.08 × 10−5 1.07 × 10−4

Λ0b → Ξ ′0

c a00 1.05 × 10−5 5.43 × 10−5

Λ0b → Ξ ′0

c σ 1.83 × 10−5 9.48 × 10−5

Λ0b → Ξ ′0

c f0 1.51 × 10−4 7.80 × 10−4

Ξ0b → Ω0

c κ00 1.48 × 10−6 7.69 × 10−6

c−(mb) = 1.38 is lower than that in the charm and hyperonsector with c−(mc) = 1.77 and c−(ms) = 2.23 respectively.

Further, |ψ(0)|2, being a dimensional quantity, may alsoshow variation with flavor [9, 13]. Already, in the study ofweak hadronic decays of charmed baryons, this has been es-timated through the Σc − Λc hyperfine splitting. Similarly,using the constituent quark model [9, 13, 54, 55], the follow-ing ratio of the hyperfine splitting in the strange and bottomsectors:

Σb − Λb

Σ − Λ= αs(mb)

αs(ms)

ms(mb − mu)|ψ(0)|2bmb(ms − mu)|ψ(0)|2s

, (20)

yields

R ≡ |ψ(0)|2b|ψ(0)|2s

≈ 2.27, (21)

for the choice αs(mb)/αs(ms) ≈ 0.40. Finally branching ra-tios are evaluated without and with |ψ(0)|2 variation, whichare presented in Tables 3, 4, 5 and Tables 6, 7, 8 in both qq

and q2q2 pictures respectively. We observe the following.

(1) In both the qq and q2q2 pictures, the dominant decaysmodes are Λ0

b → Σ+c a−

0 /Σ0c a0

0/Σ0c σ/Ξ ′0

c κ00 , Ξ0

b →Ξ ′+

c a−0 /Ξ ′0

c σ/Ω0c κ0

0 and Ξ0b → Σ0

c κ−0 with branching

ratios of the order of 10−3–10−4, hopefully within thereach of experimental observation.

Table 4 Branching ratio for Ξ0b decays (qq picture)



�b = 1, �C = 1, �S = 0

Ξ0b → Λ+

c κ−0 9.00 × 10−5 4.64 × 10−4

Ξ0b → Ξ+

c a−0 2.35 × 10−4 1.21 × 10−3

Ξ0b → Ξ0

c a00 3.96 × 10−4 2.05 × 10−3

Ξ0b → Ξ0

c σ 3.30 × 10−6 1.71 × 10−5

Ξ0b → Ξ0

c f0 6.34 × 10−6 3.27 × 10−5

Ξ0b → Σ+

c κ−0 4.51 × 10−4 2.34 × 10−3

Ξ0b → Σ0

c κ00 4.19 × 10−6 2.16 × 10−5

Ξ0b → Ξ ′+

c a−0 2.68 × 10−3 1.39 × 10−2

Ξ0b → Ξ ′0

c a00 4.17 × 10−4 2.15 × 10−3

Ξ0b → Ξ ′0

c σ 2.91 × 10−3 1.51 × 10−2

Ξ0b → Ξ ′0

c f0 4.23 × 10−6 2.19 × 10−5

Ξ0b → Ω0

c κ00 5.47 × 10−3 2.83 × 10−2

�b = 1, �C = 1, �S = −1

Ξ0b → Ξ+

c κ−0 1.42 × 10−6 7.34 × 10−6

Ξ0b → Ξ0

c κ00 1.75 × 10−5 9.09 × 10−5

Ξ0b → Ξ ′+

c κ−0 5.39 × 10−5 2.78 × 10−4

Ξ0b → Ξ ′0

c κ00 1.64 × 10−4 8.35 × 10−4

Ξ0b → Ω0

c a00 2.92 × 10−5 1.51 × 10−4

Ξ0b → Ω0

c σ 3.07 × 10−5 1.59 × 10−4

Ξ0b → Ω0

c f0 3.27 × 10−4 1.69 × 10−3

Table 5 Branching ratio for Ξ−b decays (qq picture)



�b = 1, �C = 1, �S = 0

Ξ−b → Ξ0

c a−0 1.64 × 10−4 8.50 × 10−4

Ξ−b → Σ0

c κ−0 1.03 × 10−3 5.35 × 10−3

Ξ−b → Ξ ′0

c a−0 5.24 × 10−4 2.71 × 10−3

�b = 1, �C = 1, �S = −1

Ξ−b → Ξ0

c κ−0 9.08 × 10−6 4.69 × 10−5

Ξ−b → Ξ ′0

c κ−0 2.90 × 10−5 1.50 × 10−4

Ξ−b → Ω0

c a−0 5.84 × 10−5 3.02 × 10−4

(2) However, the decay Λ0b → Σ0

c f0 forbidden in qq pic-ture of scalar mesons acquire a non-zero branching ra-tio around 2.89 × 10−3 in q2q2 picture. This provides auseful test for the 4-quark picture of the scalar mesons.

(3) All decays of �b = 1, �C = 1, �S = −1 mode aresuppressed in comparison to �b = 1, �C = 1, �S = 0mode due to the small value of CKM matrix elements

(4) Asymmetry parameters for all decays vanish due to sup-pressed weak PV transition amplitudes bij ’s.

of 7 Eur. Phys. J. C (2011) 71: 1538

Table 6 Branching ratio for Λ0b decays (q2q2 picture)



�b = 1, �C = 1, �S = 0

Λ0b → Λ+

c a−0 2.24 × 10−5 1.15 × 10−4

Λ0b → Ξ0

c κ00 1.64 × 10−4 8.52 × 10−4

Λ0b → Σ+

c a−0 5.66 × 10−4 2.92 × 10−3

Λ0b → Σ0

c a00 5.65 × 10−4 2.92 × 10−3

Λ0b → Σ0

c σ 6.06 × 10−3 3.14 × 10−2

Λ0b → Σ0

c f0 2.89 × 10−3 1.49 × 10−2

Λ0b → Ξ ′0

c κ00 1.57 × 10−3 8.14 × 10−3

�b = 1, �C = 1, �S = −1

Λ0b → Λ+

c κ−0 6.26 × 10−6 3.23 × 10−5

Λ0b → Ξ+

c a−0 2.19 × 10−6 1.13 × 10−5

Λ0b → Ξ0

c a00 1.11 × 10−6 5.74 × 10−6

Λ0b → Ξ0

c σ 6.49 × 10−6 3.35 × 10−5

Λ0b → Ξ0

c f0 3.09 × 10−6 1.60 × 10−5

Λ0b → Σ+

c κ−0 7.85 × 10−5 4.06 × 10−4

Λ0b → Σ0

c κ00 1.57 × 10−4 8.12 × 10−4

Λ0b → Ξ ′+

c a−0 9.44 × 10−6 4.88 × 10−5

Λ0b → Ξ ′0

c a00 4.76 × 10−6 2.46 × 10−5

Λ0b → Ξ ′0

c σ 1.69 × 10−5 8.76 × 10−5

Λ0b → Ξ ′0

c f0 2.33 × 10−4 1.20 × 10−3

Ξ0b → Ω0

c κ00 7.42 × 10−7 3.84 × 10−6

(5) Branching ratios of all the decays, in both the pictures,get enhanced by a factor of five due to the possible flavordependence of |ψ(0)|2 appearing in the baryon-baryonweak transition amplitudes.

(6) It is also noted that the decays with selection rules�b = 1, �C = 0, �S = −1 and �b = 1, �C = 0,�S = 0 do not acquire pole contributions as the c-quarkdoes not appear as constituent in the parent baryons (Λ0

b ,Ξ0

b , Ξ−b ) considered in this work. However, these decay

modes may receive contributions through b+u → u+ s

and b + u → u + d quark processes, which are highlysuppressed due to the correspondingly small CKM ma-trix elements.

References

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Table 7 Branching ratio for Ξ0b decays (q2q2 picture)



�b = 1, �C = 1, �S = 0

Ξ0b → Λ+

c κ−0 4.01 × 10−5 2.07 × 10−4

Ξ0b → Ξ+

c a−0 1.09 × 10−4 5.65 × 10−4

Ξ0b → Ξ0

c a00 1.83 × 10−4 9.48 × 10−4

Ξ0b → Ξ0

c σ 3.27 × 10−6 1.69 × 10−5

Ξ0b → Ξ0

c f0 1.57 × 10−6 8.11 × 10−6

Ξ0b → Σ+

c κ−0 2.06 × 10−4 1.06 × 10−3

Ξ0b → Σ0

c κ00 2.06 × 10−6 1.06 × 10−5

Ξ0b → Ξ ′+

c a−0 1.37 × 10−3 7.08 × 10−3

Ξ0b → Ξ ′0

c a00 2.31 × 10−4 1.19 × 10−3

Ξ0b → Ξ ′0

c σ 2.88 × 10−3 1.49 × 10−2

Ξ0b → Ξ ′0

c f0 1.53 × 10−3 7.95 × 10−3

Ξ0b → Ω0

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Ξ0b → Ξ0

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Ξ0b → Ξ ′+

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Ξ0b → Ξ ′0

c κ00 8.25 × 10−5 4.26 × 10−4

Ξ0b → Ω0

c a00 1.34 × 10−5 6.95 × 10−5

Ξ0b → Ω0

c σ 2.82 × 10−5 1.46 × 10−4

Ξ0b → Ω0

c f0 4.82 × 10−4 2.49 × 10−3

Table 8 Branching ratio for Ξ−b decays (q2q2 picture)



�b = 1, �C = 1, �S = 0

Ξ−b → Ξ0

c a−0 7.55×10−5 3.90 × 10−4

Ξ−b → Σ0

c κ−0 4.74 × 10−4 2.45 × 10−3

Ξ−b → Ξ ′0

c a−0 2.40 × 10−4 1.24 × 10−3

�b = 1, �C = 1, �S = −1

Ξ−b → Ξ0

c κ−0 4.16 × 10−6 2.15 × 10−5

Ξ−b → Ξ ′0

c κ−0 1.33 × 10−5 6.88 × 10−5

Ξ−b → Ω0

c a−0 2.69 × 10−5 1.39 × 10−4

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preliminary estimate of branching ratios of weak hadronic decays of bottom baryons emitting...

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