Physics Letters B 533 (2002) 271–276

www.elsevier.com/locate/npe

Penguin-induced radiative baryonicB decays

Hai-Yang Chenga,b, Kwei-Chou Yangc

a Institute of Physics, Academia Sinica, Taipei, 115 Taiwan, ROCb C.N. Yang Institute for Theoretical Physics, State University of New York Stony Brook, NY 11794, USA

c Department of Physics, Chung Yuan Christian University, Chung-Li, 320 Taiwan, ROC

Received 9 January 2002; received in revised form 5 March 2002; accepted 21 March 2002

Editor: T. Yanagida

Abstract

Weak radiative baryonicB decaysB → B1�B2γ mediated by the electromagnetic penguin processb → sγ have appreciablerates larger than their two-body counterpartsB →B1 �B2. The branching ratios forB− → Λpγ andB− → Ξ0 �Σ−γ are sizable,falling into the range of(1 ∼ 6) × 10−6 with the value preferred to be on the large side, and not far from the bottom baryonradiative decaysΛb → Λγ andΞb → Ξγ due to the large short-distance enhancement forb → sγ penguin transition and thelarge strong coupling of the anti-triplet bottom baryons with theB meson and the light baryon. These penguin-induced radiativebaryonicB decay modes should be accessible byB factories. 2002 Elsevier Science B.V. All rights reserved.

1. Recently we have presented a systematicalstudy of two-body and three-body charmful and charm-less baryonicB decays [1,2]. Branching ratios forcharmless two-body modes are in general very small,typically less than 10−6, except for the decays witha ∆ resonance in the final state. In contrast, someof charmless three-body final states in which baryon–antibaryon pair production is accompanied by a mesonhave rates larger than their two-body counterparts, forexample,

Γ(�B 0 → pnπ−)

>Γ(B− → np

),

Γ(B− → ppK−)

>Γ(�B 0 → pp

),

and Γ(�B 0 → npπ+)

>Γ(B− → np

).

E-mail address: kcyang@cycu.edu.tw (K.-C. Yang).

In [2] we have explained why these three-body modeshave branching ratios larger than the correspondingtwo-body ones.

In this short Letter, we would like to extendour study to the weak radiative baryonicB decaysB → B1�B2γ . At a first sight, it appears that thebremsstrahlung process will lead to

Γ(B → B1�B2γ

) ∼O(αem)Γ(B → B1�B2

)with αem being an electromagnetic fine-structure con-stant and hence the radiative baryonicB decay is fur-ther suppressed than the two-body counterpart, mak-ing its observation very difficult at the present level ofsensitivity forB factories. However, there is an impor-tant short-distance electromagnetic penguin transitionb → sγ . Owing to the large top quark mass, the am-plitude of b → sγ is neither quark mixing nor loopsuppressed. Moreover, it is largely enhanced by QCDcorrections. As a consequence, the short-distance con-

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)01662-3

272 H.-Y. Cheng, K.-C. Yang / Physics Letters B 533 (2002) 271–276

Fig. 1. Quark and pole diagrams forB− → Λpγ .

tribution due to the electromagnetic penguin diagramdominates over the bremsstrahlung. This phenomenonis quite unique to the bottom hadrons which con-tain a heavyb quark; such a magic short-distance en-hancement does not occur in the systems of charmedand strange hadrons. It has been suggested in [3] thatcharmless baryonicB decays may be more prominentin association with a photon emission. We shall seethat the radiative baryonicB decays proceeded via theb → γ penguin transition indeed can have appreciablerates larger than their two-body counterparts.

It is worth emphasizing that the similar QCDpenguin processb → s + g∗ with g∗ being a virtualgluon has been “observed” in the charmless mesonicB decays, e.g.,�B → Kπ . For the case of baryonicBdecays, the signature of the QCD penguin mechanismmay well manifest in the decay modes accompaniedby the η′ production, for example,B → η′ΛN , asadvocated in [3]. Recall thatB → η′K has the largestdecay rate among the charmless two-body final statesof theB mesons.

2. There are several two-body radiative decays ofbottom hadrons proceeding through the electromag-netic penguin mechanismb → sγ :

Λ0b → Σ0γ, Λ0γ, Ξ0

b → Ξ0γ,

(1)Ξ−b → Ξ−γ, �−

b → �−γ.

The radiative baryonicB decays of interest will be

B− → {Λp,Σ0p,Σ+∆−−,Σ−n,Ξ0 �Σ−,

Ξ− �Λ,Ξ− �Σ 0,�− �Ξ 0}γ,

(2)

�B 0 → {Λn,Σ0n,Σ+p,Σ−∆+,Ξ0Λ,

Ξ0 �Σ 0,Ξ− �Σ +,�− �Ξ +}γ.

Note that in our notation,∆−− denotes the antiparticleof ∆++ and �Σ+ the antiparticle ofΣ−.

Let us first consider the decayB− → Λpγ as anillustration. The short-distanceb → sγ penguin con-tribution is depicted in Fig. 1. Since a direct evalua-tion of this diagram is difficult as it involves an un-known 3-body matrix elementMµν = 〈Λp|sσµν(1 +γ5)b|B−〉 which can be parametrized in terms of sev-eral unknown form factors, we shall instead evaluatethe corresponding diagrams known as pole diagramsat the hadron level (see Fig. 1). In principle, there ex-ist many possible baryon and meson pole contribu-tions. The main assumption of the pole model is thatthe dominant contributions arise from the low-lyingbaryon and meson intermediate states. (For some ear-lier studies of baryonicB decays within the frameworkof the pole model, see [4,5].) The evaluation of thepole diagrams shown in Fig. 1 allows us to determinethe form factors appearing in the aforementioned ma-trix elementMµν . To proceed, we note that the me-sonK∗ pole amplitude is expected to be suppressed asthe intermediateK∗ state is far off mass shell. Con-sequently, the baryon–baryon-K∗ coupling is subjectto a large suppression due to the form-factor effectsat largeq2. Therefore, we will focus on the pole di-agrams with the strong processB− → {

Λ(∗)b ,Σ

0(∗)b

}followed by the radiative transition

{Λ

(∗)b ,Σ

0(∗)b

} →Λγ . Let us consider theΛb pole first. The correspond-ing pole amplitude is then given by

A(B− → Λpγ

)= igΛb→B−p〈Λγ |HW |Λb〉

(3)× 1

(pΛ + k)2 −m2Λb

uΛbγ5vp.

The relevant Hamiltonian for the radiativeb → sγ

transition is

(4)HW = −GF√2V ∗t sVtbc

eff7 O7,

with

(5)O7 = e

8π2mbsσµνFµν(1+ γ5)b.

H.-Y. Cheng, K.-C. Yang / Physics Letters B 533 (2002) 271–276 273

The effective Wilson coefficientceff7 includes the

contributions from the QCD penguin operatorsO5 andO6. Therefore,⟨Λ(pΛ)γ (ε, k)

∣∣HW

∣∣Λb(pΛb)⟩

= −iGF√

2

e

8π2V∗t sVtb2ceff

7 mbεµkν

(6)× 〈Λ|sσµν(1+ γ5)b|Λb〉.In order to evaluate the tensor matrix elements in

Eq. (6), we consider the static heavyb-quark limit sothatγ0uΛb = uΛb . This leads to [6]

(7)〈Λ|siσ0i (1+ γ5)b|Λb〉 = 〈Λ|sγi(1− γ5)b|Λb〉,and

〈Λ|siσ0i (1+ γ5)b|Λb〉ε0ki

(8)= 1

2〈Λ|sγi(1− γ5)b|Λb〉

(ε0ki − εik0).

In terms of the baryon form factors defined by⟨B1(p1)

∣∣(V −A)µ∣∣B2(p2)

⟩

(9)

= u1(p1)

{fB1B21

(q2)γµ + i

fB1B22 (q2)

m1 +m2σµνq

ν

+ fB1B23 (q2)

m1 +m2qµ

−[gB1B21

(q2)γµ + i

gB1B22 (q2)

m1 +m2σµνq

ν

+ gB1B23 (q2)

m1 +m2qµ

]γ5

}u2(p2),

with q = p2 − p1, we obtain

〈Λ|siσ0i (1+ γ5)b|Λb〉ε0ki

= uΛiσ0iε0ki

[fΛΛb

1 (0)− fΛΛb

2 (0)

(10)

+ gΛΛb

1 (0)γ5 + mΛb −mΛ

mΛb +mΛ

gΛΛb

2 (0)γ5

]uΛb ,

where uses of�pΛb = 0 and the relationki(ε0ki −εik0) = 0 have been made.

Note that there is no contribution from the12−

intermediate stateΛ∗b as the matrix element〈Λ|s(1 −

γ5)b|Λ∗b〉 vanishes. Likewise, the pole statesΣ0

b andΣ0∗

b also do not contribute under the factorization

approximation because the weak transition

〈Λ|b†s bb

∣∣Σ0(∗)b

⟩is prohibited in the quark model as one can easilycheck using the baryon wave functions:

Λ↑b = 1√

6

[(bud − bdu)χA + (12)+ (13)

],

Σ0↑b = 1√

6

[(bud + bdu)χs + (12)+ (13)

],

(11)Λ↑ = 1√6

[(sud − sdu)χA + (12)+ (13)

],

where

abcχs = (2a↓b↑c↑ − a↑b↑c↓ − a↑b↓c↑)

/√

6,

abcχA = (a↑b↑c↓ − a↑b↓c↑)

/√

2,

and (ij) means permutation for the quark in placeiwith the quark in placej . Hence, only the intermediatestateΛb makes contributions to the pole amplitude.

Putting everything together, the pole amplitudereads

A(B− → Λpγ

)= −igΛb→B−puΛ(a + bγ5)σµνε

µkν

(12)× /pΛ + /k +mΛb

(pΛ + k)2 −m2Λb

γ5vp,

with

a = GF√2

e

8π22ceff7 mbVtbV

∗t s

[f

ΛΛb

1 (0)− fΛΛb

2 (0)],

(13)

b = GF√2

e

8π22ceff7 mbVtbV

∗t s

×[gΛΛb

1 (0)+ mΛb −mΛ

mΛb +mΛ

gΛΛb

2 (0)

].

For the heavy-light form factorsf ΛΛb

i and gΛΛb

i ,we will follow [7] to apply the nonrelativistic quarkmodel to evaluate the weak current-induced baryonform factors at zero recoil in the rest frame of theheavy parent baryon, where the quark model is mosttrustworthy. This quark model approach has the meritthat it is applicable to heavy-to-heavy and heavy-to-light baryonic transitions at maximumq2. Following[8] we have

fΛΛb

1

(q2m

) = gΛΛb

1

(q2m

) = 0.64,

274 H.-Y. Cheng, K.-C. Yang / Physics Letters B 533 (2002) 271–276

fΛΛb

2

(q2m

) = gΛΛb

3

(q2m

) = −0.31,

(14)fΛΛb

3

(q2m

) = gΛΛb

2

(q2m

) = −0.10,

for Λb − Λ transition at zero recoilq2m = (mΛb −

mΛ)2. Since the calculation for theq2 dependence of

form factors is beyond the scope of the nonrelativisticquark model, we will follow the conventional practiceto assume a pole dominance for the form-factorq2

behavior:

f(q2) = f

(q2m

)(1− q2m/m

2V

1− q2/m2V

)n

,

(15)g(q2) = g

(q2m

)(1− q2m/m

2A

1− q2/m2A

)n

,

wheremV (mA) is the pole mass of the vector (axial-vector) meson with the same quantum number asthe current under consideration. Conventionally, onlymonopole (n = 1) and dipole (n = 2) q2 dependencefor baryon form factors is considered. A recent calcu-lation of the baryon Isgur–Wise function forΛb −Λc

transition in [9] indicates that a monopoleq2 depen-dence is favored. This is further supported by a recentfirst observation ofB− → ppK− by Belle [10] whichalso favorsn = 1 for the momentum dependence ofbaryon form factors [2].

To compute the branching ratio we shall use the ef-fective Wilson coefficientceff

7 (mb) = −0.31 (see, e.g.,[11]), the running quark massmb(mb) =4.4 GeV and the pole massesmV = 5.42 GeV andmA = 5.86 GeV. For quark mixing matrix elements,we use|Vub/Vcb| = 0.085 and the unitarity angleγ =60◦. For the strong couplinggΛb→B−p , we note thata fit to the measured branching ratio for the decayB− → Λcpπ

− implies a strong couplinggΛb→B−p

with the strength in the vicinity of order 16 [1]. Hence,we shall use|gΛb→B−p| = 16 as a benchmarked value.The pole amplitude (12) leads to the branching ratio

(16)

B(B− → Λpγ

) = 5.9× 10−6r(0.7× 10−6r

)

for n = 1 (n = 2), wherer ≡ |gΛb→B−p/16|2. Thedecay rate of�B 0 → Λnγ is the same asB− →Λpγ . As noted in passing, the preferred form-factormomentum dependence is of the monopole formwhich implies thatB− → Λpγ has a magnitude aslarge as 10−5. For comparison, we also show the

branching ratio forΛb → Λγ :

(17)B(Λb → Λγ ) = 1.9× 10−5 (2.3× 10−6)

for n = 1 (n = 2), which is calculated using theformula

(18)

Γ (Λb → Λγ ) = 1

8π

(m2

Λb−m2

Λ

mΛb

)3(|a|2 + |b|2).

The weak radiative decayΛb → Λγ has beendiscussed in [7,12–15] with the predicted branchingratio spanned in the range of(0.2–1.5) × 10−5. Wesee that the magnitude ofB− → Λpγ is close to thatof Λb → Λγ . In [2] we have shown that 2.2×10−7 <

B(B− → Λp) < 4.4 × 10−7 where the upper boundcorresponds toΓPV = ΓPC and the lower bound toΓPV = 0, where the subscript PV (PC) denotes theparity-violating (parity-conserving)decay rate. Hence,it is safe to conclude that

(19)

Γ (Λb → Λγ ) > Γ(B− → Λpγ

)>Γ

(B− → Λp

).

Thus far we have not considered next-to-leadingorder (NLO) radiative corrections and 1/mb powercorrections. It has been pointed out recently that NLOcorrections toB → K∗γ yields an 80% enhancementof its decay rate [11,16]. It is thus interesting to seehow important the NLO correction is for the radiativedecaysΛb → Λγ andB− → Λpγ .

3. Let us study other radiative decays listed in (2).For �B → Σ �Nγ , it receives contributions from theΣb

pole state, while there are two intermediate baryonstates in the pole diagrams for�B → Ξ �Bsγ : the anti-triplet bottom baryonΞb and the sextetΞ ′

b. There aretwo essential unknown physical quantities: strong cou-plings and baryon form factors. For strong couplingswe will follow [5,17] to adopt the3P0 quark-pair-creation model in which theqq pair is created fromthe vacuum with vacuum quantum numbers3P0. Weshall apply this model to estimate the relative strongcoupling strength and choose|gΛb→B−p| = 16 as areference value for the absolute coupling strength. Theresults are (for a calculational detail, see [2])

gΛb→B−p = 3√

3gΣ0b→B−p = 3

√3gΣ0

b→�B0n

H.-Y. Cheng, K.-C. Yang / Physics Letters B 533 (2002) 271–276 275

(20)

= −3

√3

2gΣ+

b →�B0p = −3

√3

2gΣ−

b →B−n,

and

gΛb→B−p = gΞ0b →B−Σ+ = gΞ−

b →�B 0Σ−

= √2gΞ0

b →�B0Σ0 = √2gΞ−

b →B−Σ0

= √6gΞ0

b →�B0Λ = −√6gΞ−

b →B−Λ

= −3√

2gΞ

′0b →�B0Λ

= 3√

2gΞ

′−b →B−Λ

= 3√

3gΞ

′0b →B−Σ+ = 3

√3g

Ξ′−b →�B 0Σ−

(21)

= 3√

6gΞ

′0b →�B0Σ0 = 3

√6g

Ξ′−b →B−Σ0.

We thus see that the antitriplet bottom baryonsΛb

andΞb have larger couplings than the sextet onesΣb

andΞ ′b. Therefore, the radiative decay�B → Σ �Nγ is

suppressed.Form factors forΣb − Σ , Ξb − Ξ andΞ ′

b − Ξ

transitions at zero recoil can be obtained by usingEq. (22) of [7]. To apply this equation one needs toknow the relevant spin factorη and the flavor fac-tor NB1B2 (see [7] for detail). We findη = −1

3, 1,

−13 andNB1B2 = 1/

√3, 1

√2 and 1/

√6, respectively,

for above-mentioned three heavy-light baryonic tran-sitions. The resultant form factors at zero recoil areexhibited in Table 1 where we have applied the baryonwave functions given in Eq. (A1) of [2]. For bottombaryons, we use the masses:mΛb = 5.624 GeV [18],mΣb = 5.824 GeV,mΞb = 5.807 GeV andmΞ ′

b=

6.038 GeV [19].Repeating the same calculation as before, we obtain

B(B− → Ξ0 �Σ −γ

) = 5.7× 10−6r(0.7× 10−6r

),

B(B− → Ξ−Λγ

) = 1.2× 10−6r(1.4× 10−7r

),

(22)

B(B− → Σ−nγ

) = 2.8× 10−8r(2.4× 10−9r

),

for monopole (dipole) momentum dependence ofbaryon form factors. The decay rates of other modessatisfy the relations

Γ(B− → Σ−nγ

)= Γ

(�B 0 → Σ+pγ) = 2Γ

(B− → Σ0pγ

)= 2Γ

(�B 0 → Σ0nγ),

Γ(B− → Ξ0 �Σ −γ

)= Γ

(�B 0 → Ξ− �Σ+γ) = 2Γ

(B− → Ξ− �Σ 0γ

)(23)= 2Γ

(�B 0 → Ξ0 �Σ 0γ),

and

(24)Γ(B− → Ξ−Λγ

) = Γ(�B 0 → Ξ0Λγ

).

As for the radiative decay of the bottom baryonΞb,we obtain

Γ (Ξb → Ξγ )

(25)= 1.9× 10−17 GeV(2.4× 10−18 GeV

)for n = 1 (n = 2). It follows thatΓ (Ξb → Ξγ ) =1.9Γ (Λb → Λγ ).

4. We have shown that weak radiative baryonicB decaysB → B1�B2γ mediated by the electromag-netic penguin processb → sγ can have rates largerthan their two-body counterpartsB → B1�B2. In par-ticular, the branching ratios forB− → Λpγ andB− → Ξ0 �Σ−γ are sizable, ranging from 1× 10−6 to6 × 10−6 with the value preferred to be on the largeside, and not far from the bottom baryon radiative de-caysΛb → Λγ andΞb → Ξγ . The penguin-inducedradiative baryonicB decays should be detectable byB factories at the present level of sensitivity. This isascribed to the large short-distance enhancement forb → sγ penguin transition and to the large strong cou-pling of the antitriplet bottom baryons with theB me-son and the octet baryon. We conclude that radiative

Table 1Various baryon form factors at zero recoil

Transition f1 f2 f3 g1 g2 g3

Σb–Σ 1.73 2.05 −1.70 −0.21 −0.03 0.11Ξ ′b–Ξ 1.17 1.43 −1.20 −0.16 0.04 0.10

Ξb–Ξ 0.83 −0.50 −0.19 0.83 −0.19 −0.50

276 H.-Y. Cheng, K.-C. Yang / Physics Letters B 533 (2002) 271–276

baryonicB decays are dominated by the short-distanceb → sγ mechanism.

Acknowledgements

One of us (H.Y.C.) wishes to thank C.N. Yang In-stitute for Theoretical Physics at SUNY Stony Brookfor its hospitality. This work was supported in part bythe National Science Council of R.O.C. under GrantNos. NSC90-2112-M-001-047 and NSC90-2112-M-033-004.

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