decays to charmed baryons

PHYSICAL REVIEW D 67, 034008 ~2003!

Hadronic B decays to charmed baryons

Hai-Yang ChengInstitute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China

Kwei-Chou YangDepartment of Physics, Chung Yuan Christian University, Chung-Li, Taiwan 320, Republic of China

~Received 22 October 2002; published 13 February 2003!

We study exclusiveB decays to final states containing a charmed baryon within the pole model framework.

Since the strong coupling forLbBN is larger than that forSbBN, the two-body charmful decayB2→Sc0p has

a rate larger thanB0→Lc1p as the former proceeds via theLb pole while the latter via theSb pole. By the

same token, the three-body decayB0→Sc11pp2 receives less of a baryon-pole contribution thanB2

→Lc1pp2. However, because the important charmed-meson pole diagrams contribute constructively to the

former and destructively to the latter,Sc11pp2 has a rate slightly larger thanLc

1pp2. It is found that

one-quarter of theB2→Lc1pp2 rate comes from the resonant contributions. We discuss the decaysB0

→Sc0pp1 andB2→Sc

0pp0 and stress that they are not color suppressed even though they can only proceedvia an internalW emission.

DOI: 10.1103/PhysRevD.67.034008 PACS number~s!: 13.25.Hw, 14.20.Lq, 14.20.Pt

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I. INTRODUCTION

Previously CLEO has searched for charmful baryonicB

decays in the classB→LcNX. The experimental results ar@1#

B~B0→Lc1pp1p2!5~1.3320.40

10.4660.37!31023,

B~B2→Lc1pp2p0!,3.1231023, ~1.1!

B~B2→Lc1pp2!5~6.222.0

12.361.6!31024,

B~B0→Lc1p!,2.131024.

Recently, Belle@2# and CLEO@3# have reported the measurments of the exclusive decays ofB mesons into final states othe typeBcpn(p), whereBc5Lc ,Lc1 ,Sc(2455),Sc1 @Lc15Lc(2593), Lc(2625) andSc15Sc(2520)] andn is thenumber of the pions in the final state. From Table I we sthat the new measurements ofB2→Lc

1pp2 and B0

→Lc1pp1p2 are consistent with, and much more accur

than, the previous CLEO results~1.1!; however, the new re-sult for the former is somewhat low (1.5s).

In general, CLEO and Belle results are consistent weach other except for the ratio ofSc

11pp2 to Sc0pp1. The

Sc11 decay proceeds via both external and inter

W-emission diagrams, whereas theSc0 decay can only pro-

ceed via an internalW emission. While Belle measuremenimply a sizable suppression for theSc

0 decay~and likewise

for the Sc1 decay!, it is found by CLEO thatSc11pp2,

Sc0pp1 and Sc

0pp0 are of the same order of magnitudTherefore, it is concluded by CLEO that the externalW de-cay diagram does not dominate over the internalW-emissiondiagram in Cabibbo-allowed baryonicB decays. This needto be clarified by the forthcoming improved measuremen

0556-2821/2003/67~3!/034008~14!/$20.00 67 0340

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On the theoretical side, the decaysB0→Lc1p and B2

→Lc1pp2 have been studied by us within the framework

the pole model@4#. We have explained several reasons w

the three-body decay rate ofB2→Lc1pp2 is larger than that

of the two-body oneB0→Lc1p. At the pole-diagram level,

the Sb propagator in the pole amplitude for the latter isorder 1/(mb

22mc2), while the invariant mass of the (Lc

1p2)system can be large enough in the former decay so thapropagator ofLb in the pole diagram is not subject to thsame 1/mb

2 suppression. Moreover, the strong coupling co

stant forLb0→B2p is larger than that forSb

1→B0p, and thissuffices to explain the original CLEO observation.

Since at the pole-diagram level,B0→Sc11pp2 proceeds

through theSb pole, whileB2→Sc0p proceeds through the

Lb pole, it is naively expected thatG(B2→Sc0p).G(B0

→Lc1p) andG(B0→Sc

11pp2),G(B2→Lc1pp2). How-

ever, the latter relation is not borne out by the new measuments of both Belle and CLEO~see Table I!. Indeed, at thequark level, it appears thatSc

11pp2 and Lc1pp2 should

have similar rates as both of them receive exterW-emission contributions.

It turns out that the meson-pole contribution to the threbody baryonicB decays which was originally missed in@4#

is important for the charmfulB decaysB0→Sc11pp2 and

B2→Lc1pp2. Moreover, this meson-pole effect contribut

destructively toLc1pp2 and constructively toSc

11pp2. Aswe shall see, this eventually leads to the explanation of wB(B0→Sc

11pp2)*B(B2→Lc1pp2).

Since B0→Sc0pp1 and B2→Sc

0pp0 can only proceedvia an internalW emission, it is suitable to apply the polmodel to study these two decays. As we shall see later,all the internalW-emission diagrams in baryonic decays asubject to color suppression.

©2003 The American Physical Society08-1

HAI-YANG CHENG AND KWEI-CHOU YANG PHYSICAL REVIEW D 67, 034008 ~2003!

TABLE I. Experimental measurements of the branching ratios~in units of 1024) for the B decay modeswith a charmed baryonLc or Lc15Lc(2593),Lc(2625) orSc(2455) orSc15Sc(2520) in the final state.

Mode Belle@2# CLEO @3#

B2→Lc1pp2p0 18.162.921.6

12.264.7

B0→Lc1pp1p2 11.061.261.962.9 16.761.921.6

11.964.3

B2→Lc1pp2 1.8720.40

10.4360.2860.49 2.460.620.1710.1960.6

B0→Lc1p 0.1220.07

10.1060.0260.03,0.31 ,0.9

B2→Lc11 pp2 ,1.9

B0→Lc11 p ,1.1

B2→Sc11pp2p2 2.860.960.560.7

B2→Sc0pp1p2 4.461.260.561.1

B0→Sc11pp2 2.3820.55

10.6360.4160.62 3.760.860.760.8

B0→Sc0pp1 0.8420.35

10.4260.1460.22,1.59 2.260.660.460.5

B2→Sc0pp0 4.261.360.461.0

B2→Sc0p 0.4520.19

10.2660.0760.12,0.93 ,0.8

B0→Sc111pp2 1.6320.51

10.5760.2860.42

B0→Sc10 pp1 0.4820.40

10.4560.0860.12,1.21

B2→Sc10 p 0.1420.09

10.1560.0260.04,0.46

. I

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The layout of the present paper is organized as followsSec. II we first study the two-body charmful decayB0

→Lc1p, B2→Sc

0p, B0→Sc0n andB2→Lc

1D22. We then

turn to the three-body decaysB0→Lc1p p2, B0

→Sc11pp2, B0→Sc

0pp1 and B2→Sc0pp0 in Sec. III.

Discussions and conclusions are given in Sec. IV.

II. TWO-BODY CHARMFUL B DECAYS

In this section we shall study the two-body charmful dcays B0→Lc

1p, B2→Sc0p, B0→Sc

0n and B2→Lc1D22.

Since the former has been discussed in@4#, we will describeit in a somewhat cursory way.

A. B0\Lc¿p

To proceed, we first write down the Hamiltonian relevafor the present paper

Heff5GF

A2VcbVud* @c1

effO11c2effO2#1H.c., ~2.1!

where O15( cb)(du) and O25( cu)(db) with (q1q2)[q1gm(12g5)q2 and the effective coefficientsc1

eff andc2eff

are renormalization scale and scheme independent. In oto ensure that the physical amplitude is renormalization sandg5-scheme independent, we have included vertex cortions to the hadronic matrix elements. This amounts to refining the Wilson coefficientsc1,2(m) into the effective onesc1,2

eff . Numerically we havec1eff51.168 andc2

eff520.365@5#.

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The decay amplitude ofB0→Lc1p consists of factoriz-

able and nonfactorizable parts:

A~B0→Lc1p!5A~B0→Lc

1p! fact1A~B0→Lc1p!nonfact,

~2.2!

with

A~B0→Lc1p! fact5

GF

A2VcbVud* a2^Lcpu~ cu!u0&^0u~ db!uB0&,

~2.3!

where a25c2eff1c1

eff/Nc . The short-distance factorizablcontribution is nothing but theW-exchange diagram. ThisW-exchange contribution has been estimated and is founbe very small and hence can be neglected@6,7#. However, adirect evaluation of nonfactorizable contributions is very dficult. It is customary to assume that the nonfactorizablefect is dominated by the pole diagram with low-lying baryointermediate states; that is, nonfactorizables- and p-waveamplitudes are dominated by12

2 low-lying baryon reso-nances and12

1 ground-state intermediate states, respectiv@8#. For B0→Lc

1p, we consider the strong-interaction pro

cessB0→Sb1(* ) p followed by the weak transitionSb

1(* )

→Lc , whereSb* is a 12

2 baryon resonance~see Fig. 1!. Thepole-diagram amplitude has the form

A~B0→Lc1p!nonfact5uLc

~A1Bg5!v p , ~2.4!

where

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HADRONIC B DECAYS TO CHARMED BARYONS PHYSICAL REVIEW D67, 034008 ~2003!

A52gS

b1* →B0pbS

b1* L

c1

mLc2mS

b*, B5

gSb1→B0paS

b1L

c1

mLc2mSb

,

~2.5!

correspond tos-wave parity-violating ~PV! and p-waveparity-conserving~PC! amplitudes, respectively, and

^Lc1uHeff

PCuSb1&5uLc

aSb1L

c1uSb

,

~2.6!

^Lc1uHeff

PVuSb*1&5 i uLc

bSb1* L

c1uS

b*.

The main task is to evaluate the weak matrix elementsthe strong coupling constants. We shall employ the MIT bmodel @9# to evaluate the baryon matrix elements~see, e.g.@11,12# for the method!. Since the quark-model wave functions best resemble the hadronic states in the frame wboth baryons are static, we thus adopt the static bag appmation for the calculation. Note that because the combtion of the four-quark operatorsO11O2 is symmetric incolor indices, it does not contribute to the baryon-barymatrix element since the baryon-color wave function istally antisymmetric. This leads to the relation^Lc

1uO2uSb1&

52^Lc1uO1uSb

1&. From Eq.~2.1! we obtain the PC matrixelement

aSb1L

c152

GF

A2VcbVud* ~c1

eff2c2eff!

2

A6~X113X2!~4p!,

~2.7!

where1

1For details of the MIT bag model evaluation, see@4,10#. Notethat the bag integralsX1 and X2 given in Eq. ~B4! of @10# aredefined for the operatorO2 rather than forO1.

FIG. 1. Quark and pole diagrams forB0→Lc1p, where the solid

circle denotes the weak vertex.~a! corresponds to a nonfactorizabinternalW emission, while~b! corresponds to aW-exchange contri-bution.

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dg

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n-

X15E0

R

r 2dr@ud~r !vb~r !2vd~r !ub~r !#

3@uc~r !vu~r !2vc~r !uu~r !#,

X25E0

R

r 2dr@ud~r !ub~r !1vd~r !vb~r !#

3@uc~r !uu~r !1vc~r !vu~r !# ~2.8!

are four-quark overlap bag integrals anduq(r ), vq(r ) are thelarge and small components of the quark wave functionsthe ground (1S1/2) state. In principle, one can also follow@11# to tackle the low-lying negative-paritySb* state in thebag model and evaluate the PV matrix elementbS

c* Lc. How-

ever, it is known that the bag model is less successful efor the physical non-charm and non-bottom12

2 resonances@9#, not mentioning the charm or bottom12

2 resonances. Inshort, we know very little about the12

2 state. Therefore, wewill not evaluate the PV matrix elementbS

b* Lcas its calcu-

lation in the bag model is much involved and is far mouncertain than the PC one@11#.

Using the bag wave functions given in the Appendix@4#, we find numerically

X1521.4931025 GeV3, X251.8131024 GeV3.~2.9!

The decay rate ofB→B1B2 is given by

G~B→B1B2!5pc

4p H uAu2~mB1m11m2!2pc

2

~E11m1!~E21m2!mB2

1uBu2@~E11m1!~E21m2!1pc

2#2

~E11m1!~E21m2!mB2 J ,

~2.10!

wherepc is the center-of-mass~c.m.! momentum, andEi andmi are the energy and mass of the baryonBi , respectively.Putting everything together we obtain

B~B0→Lc1p!PC55.031026UgS

b1→B0p

5U2

. ~2.11!

The PV contribution is expected to be smaller. For exampit is found to beGPV/GPC50.59 in @8#. Therefore, we con-clude that

B~B0→Lc1p!&7.931026UgS

b1→B0p

5U2

. ~2.12!

The strong couplinggSb1→B0p has been estimated in@8# using

the 3P0 quark-pair-creation model and it is found to liethe rangeugS

b1→B0pu56;10. At any rate, the prediction

~2.12! is consistent with the current experimental limit3.131025 by Belle@2# and 931025 by CLEO@3#. Note thatall earlier predictions based on the QCD sum rule@13#, the

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pole model@8# or the diquark model@14# are too large com-pared to experiment~see, e.g. Table I of@4#!. In the pole-model calculation in@8#, the weak matrix element is largeloverestimated.

B. BÀ\Sc0p and B0\Sc

0n

The pole diagrams forB2→Sc0p andB0→Sc

0n consist oftwo poles:Lb

0(* ) andSb0(* ) as depicted in Fig. 2. Proceedin

as before, the parity-conserving amplitudes read2

B~B2→Sc0p!5

gLb0→B2p aL

b0S

c0

mSc2mLb

1gS

b0→B2p aS

b0S

c0

mSc2mSb

,

B~B0→Sc0n!5

gLb0→B0n aL

b0S

c0

mSc2mLb

1gS

b0→B0n aS

b0S

c0

mSc2mSb

,

~2.13!

where

aLb0S

c052

GF

A2VcbVud* ~c1

eff2c2eff!

2

A6~X123X2!~4p!,

aSb0S

c05

GF

A2VcbVud* ~c1

eff2c2eff!

A2

3~X119X2!~4p!.

~2.14!There are two models which can be used to estimate

strong couplings: the3P0 quark-pair-creation model inwhich theqq pair is created from the vacuum with vacuuquantum numbers3P0, and the 3S1 model in which thequark pair is created perturbatively via one gluon exchawith one-gluon quantum numbers3S1. Presumably, the3P0model works in the nonperturbative low energy regime.contrast, in the perturbative high energy region where perbative QCD is applicable, it is expected the3S1 model maybe more relevant as the light baryons produced in two-bcharmless baryonicB decays are very energetic. However,practice it is much simpler to estimate the relative strocoupling strength in the3P0 model @8,15# rather than in the3S1 model where hard gluons arise from four different qualegs and generally involve infrared problems.

In the 3P0 model we have the relations@see Eq.~3.23! of@10##

2It is found in @8# that the parity-violating contribution toB2

→Sc0p is largely suppressed relative to the parity-conserving o

FIG. 2. The internalW-emission diagram and its correspondin

pole diagram forB2→Sc0p, where the solid circle denotes th

weak vertex.

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g

gLb0→B2p53A3gS

b0→B2p523A3

2gS

b1→B0p ,

gLb0→B0n523A3gS

b0→B0n53A3

2gS

b1→B0p .

~2.15!

This leads tougLb0→B2pu518 for ugS

b1→B0pu55. However,

the predicted branching ratio 1.631024 for B2→Sc0p is too

large compared to the experimental limits, 0.9331024 byBelle and 0.831024 by CLEO. This means that the3P0model relation Eq.~2.15! is badly broken. This is not a surprise: As discussed above, the relevant model for energtwo-body baryonicB decays is the3S1 model. Fitting to thecentral value of the measured branching ratio ofB2→Sc

0p,0.4531024 ~see Table I!, we find

gLb0→B2p'1.2A3gS

b0→B2p521.2A3

2gS

b1→B0p .

~2.16!

The isospin relation leads to

gLb0→B0n'21.2A3gS

b0→B0n51.2A3

2gS

b1→B0p .

~2.17!

Hence, ugLb0→B2pu5ugL

b0→B0nu;7 and ugS

b0→B2pu

5ugSb0→B0nu;3.5 for ugS

b1→B0pu55. Since ugL

b0→B2pu

.ugSb0→B2pu, B2→Sc

0p has a larger rate thanB 0→Lc1p.

Note thatB2→Sc0p is thus far the only two-body baryonicB

decay that its evidence has been observed by Belle wisignificance of 3s @2#.

In contrast, the decay rate ofSc0n is quite suppressed,

B~B 0→Sc0n!5631027. ~2.18!

It has something to do with the smallness of the weak trsition for B 0→Sc

0n. SinceX1!X2, to a good approximationwe haveaL

b0S

c0'aS

b0S

c0 /A3 @see Eq.~2.14!#. As gL

b0→B0n'

21.2A3 gSb0→B0n , there is a large cancellation occurred

the PC amplitude, see Eq.~2.13!. Note that the ratioR

[G(B 0→Sc0n)/G(B2→Sc

0p) is predicted to be 1/2 in the3P0 model@8#, whereas it is only of order 1022 in our case.Therefore, a measurement of the ratioR can be used to discriminate between different quark-pair-creation models.

It should be stressed again that the strong couplings arprinciple q2 dependent. Therefore, the values of strong cplings quoted above should be considered as an averagethe allowedq2 region.

C. BÀ\Lc¿DÀÀ

The relevant pole diagram for the decayB2

→Lc1D22 (D22 being the antiparticle ofD11) consists of.

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the intermediate statesSb1(* ) . Since the parity-violating am

plitude vanishes in the3P0 quark-pair-creation model@6,8#,we thus have

C50, D5gS

b1→B2D11 aS

b1L

c1

mLc2mSb

, ~2.19!

corresponding to the parity-violatingp-wave and parity-conservingd-wave amplitudes, respectively, for the dec

B→B1( 12

1)B2( 32

2) with a spin-32 baryon in the final state,

A~B→B1~p1!B2~p2!!5 iqmu1~p1!~C1Dg5!v2m~p2!,

~2.20!

wherevm is the Rarita-Schwinger vector spinor for a spin32

antiparticle andq5p12p2. The decay rate is

G~B→B1~1/21!B2~3/22!!

5pc

3

6p

1

m12 H uCu2

@~E11m1!~E21m2!1pc2#2

~E11m1!~E21m2!mB2

1uDu2~mB1m11m2!2pc

2

~E11m1!~E21m2!mB2J . ~2.21!

In the 3P0 model one has the relation@see, e.g. Eq.~3.32! of@10##

gSb1→B2D1152A6gS

b1→B0p . ~2.22!

As before, this3P0 model relation is also expected to bbadly broken. Indeed, it has been pointed out in@10# thatusing the strong couplinggS

b1→B2D11 extracted from Eq.

~2.22! will lead to B(B2→pD22)55.831026. Because ofthe strong decayD22→ pp2, the resonant contributionfrom D22 to the branching ratio ofB2→ppp2 would be631026. This already exceeds the recent Belle measuremB(B2→ppp2)5(1.920.9

11.060.3)31026 or the upper limit of3.731026 @16#. Therefore, the coupling of theD to the Bmeson and the octet baryon is smaller than what is expefrom Eq. ~2.22!. By applying the same scaling from Eq~2.15! to Eq. ~2.16!, it is natural to have

gSb1→B2D11'0.8A6gS

b1→B0p . ~2.23!

Therefore, gSb1→B2D1159.8 for gS

b1→B0p55, which is

close to the value of 12 employed in@10#. Numerically, weobtain

B~B2→Lc1D22!51.931025, ~2.24!

where use of Eq.~2.7! has been made.

III. THREE-BODY CHARMFUL BARYONIC DECAYS

In this section we shall study the three-body charmbaryonic B decays: B2→Lc

1pp2, B 0→Sc11pp2, B 0

→Sc0pp1 andB2→Sc

0pp 0.

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A. BÀ\Lc¿ppÀ

This decay mode has been studied in@4# by us. However,we have missed an important meson-pole contribution aing from the externalW-emission diagram. As we shall selater, this meson-pole effect dominates the decayB 0

→Sc11pp2.

The decayB2→Lc1pp2 receives resonant and nonres

nant contributions:

G~B2→Lc1pp2!5G~B2→Lc

1pp2!nonr

1G~B2→Sc0p→Lc

1pp2!

1G~B2→Lc1D22→Lc

1pp2!.

~3.1!

As the resonant contributionsB2→Sc0p andB2→Lc

1D22

are discussed in the preceding section, here we will focusthe nonresonant contribution.

The quark diagrams and the corresponding pole diagrfor B2→Lc p p2 are shown in Fig. 3. There exist two distinct internalW emissions and only one of them is factoriable, namely, Fig. 3~b!. The externalW-emission diagramFig. 3~a! is, of course, factorizable. The factorizable amptude reads

A~B2→Lc1pp2! fact5

GF

A2VcbVud* $a1^p

2u~ du!u0&

3^Lc1pu~ cb!uB2&

1a2^p2u~ db!uB2&^Lc

1pu~ cu!u0&%

[A11A2 , ~3.2!

where a15c1eff1c2

eff/Nc . Let us first consider the factorizable amplitudeA2, as shown in Fig. 3~b!, which has theexpression

FIG. 3. Quark and pole diagrams forB2→Lc1pp2, where the

solid circles denote the weak vertex.~a! and~b! correspond to fac-torizable external and internalW-emission contributions, respectively, while ~c! corresponds to nonfactorizable internalW-emissiondiagrams. Note that the charmed-meson pole diagram in~a! is colorallowed, while it is color suppressed in~b!.

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A25GF

A2VudVcb* a2uLc

@~ap” p1b!2~cp” p1d!g5#v p ,

~3.3!

where

a52 f 1Lcp

~ t !F1Bp~ t !12 f 2

Lcp~ t !F1

Bp~ t !,

b5~mLc2mp! f 1

Lcp~ t !FF1

Bp~ t !1~F0Bp~ t !

2F1Bp~ t !!

mB22mp

2

t G22 f 2Lcp

~ t !F1Bp~ t !

3~pLc2pp!•pp /~mLc

1mp!

1 f 3Lcp

~ t !F0Bp~ t !~mB

22mp2 !/~mLc

1mp!,

c52g1Lcp

~ t !F1Bp~ t !12g2

Lcp~ t !F1

Bp~ t !

3~mLc2mp!/~mLc

1mp!, ~3.4!

d5~mLc1mp!g1

Lcp~ t !FF1

Bp~ t !1@F0Bp~ t !

2F1Bp~ t !#

mB22mp

2

t G22g2Lcp

~ t !F1Bp~ t !

3~pLc2pp!•pp /~mLc

1mp!

1g3Lcp

~ t !F0Bp~ t !

3~mB22mp

2 !/~mLc1mp!,

andt[q25(pB2pp)25(pLc1pp)2, and we have employed

the baryonic form factors defined by

^Lc1pu~ cu!u0&5uLcH f 1

Lcp~q2!gm1 i

f 2Lcp

~q2!

mLc1mp

smnqn

1f 3

Lcp~q2!

mLc1mp

qm2Fg1Lcp

~q2!gm

1 ig2

Lcp~q2!

mLc1mp

smnqn

1g3

Lcp~q2!

mLc1mp

qmGg5J v p , ~3.5!

and the mesonic form factors given by@17#

^p2~pp!u~ db!uB2~pB!&5F1Bp~q2!~pB1pp!m1@F0

Bp~q2!

2F1Bp~q2!#

mB22mp

2

q2 qm . ~3.6!

As for the factorizable amplitudeA1, since in practice wedo not know how to evaluate the 3-body hadronic mat

03400

element Lc1pu( cb)uB2& at the quark level, we will instead

evaluate the corresponding two low-lying pole diagramsthe externalW emission as depicted in Fig. 3~a!: ~i! thebaryon pole diagram with strong processB2→Lb

0(* )p fol-lowed by the weak decayLb

0(* )→Lc1p2, and~ii ! the meson

pole diagram with the color-allowed weak processB2

→$D0,D* 0,D10%p2 followed by the strong reaction

$D0,D* 0,D10%→Lc

1p. We consider the baryon pole contrbution first. Its amplitude is given by

A1B52GF

A2VudVcb* gLb→B2pf p a1 uLc

$ f 1LbLc~mp

2 !

3@2pp•pLc1p” p~mLb

2mLc!#g51g1

LbLc~mp2 !

3@2pp•pLc2p” p~mLb

1mLc!#%v p

31

~pLc1pp!22mLb

2, ~3.7!

where we have applied factorization to the weak decayLb0

→Lc1p2. Note that the intermediate statesSb

0 andSb0* also

do not contribute toA1 under the factorization approximatiobecause the weak transition^Lcu( cb)uSb

0(* )& is prohibited asSb andSb* are sextet bottom baryons whereasLc is an anti-triplet charmed baryon.

The meson pole contribution from Fig. 3~a! consists of thepseudoscalar mesonD0, the vector mesonD* 0 and theaxial-vector mesonD1

0(2400). Note that the weak decay processB2→$D0,D* 0,D1

0%p2 in Fig. 3~a! is color allowed,namely, its amplitude is proportional toa1, while the sameprocess in Fig. 3~b!, being proportional toa2, is color sup-pressed. The charmed-meson pole amplitude has the fo

A1M5GF

A2Vud* Vcba1^p

2u~ du!u0&

3H F ^D0u~ cb!uB2&i

q22mD2 gLc→pD0

uLcg5v p

1^D* 0u~ cb!uB2&i

q22mD*2 uLc

i«D*n

3S g1Lc

1→pD* 0

gn1 ig

2Lc

1→pD* 0

mLc1mp

snlqlD v p

1^D10u~ cb!uB2&

i

q22mD1

2uLc

i«D1

n S h1Lc

1→pD10

gn

1 ih

2Lc

1→pD10

mLc1mp

snlqlD g5v pJ , ~3.8!

8-6

tain

on

ent.sum rule


whereq5pB2pp5pLc1pp , andg, g1 , g2 , h1 , h2 are the unknown strong couplings. After some manipulation we ob

A1M52GF

A2Vud* Vcbf pa1H ~mB

22mD2 !F0

BD~mp2 !

gLc→pD 0

q22mD2

uLcg5v p1

2mD*

q22mD*2 A0

BD* ~mp2 !pB

mS 2gmn1qmqn

mD*2 D

3uLcS g

1Lc

1→pD* 0

gn1 ig

2Lc

1→pD* 0

mLc1mp

snlqlD v p12mD1

q22mD1

2V0

BD1~mp2 !pB

mS 2gmn1qmqn

mD1

2 D3uLc

S h1Lc

1→pD10

gn1 ih

2Lc

1→pD10

mLc1mp

snlqlD g5v pJ , ~3.9!

where we have employed the form factors defined by3

^D* 0~pD* ,«!u~ cb!V2A

uB2~pB!&52

mB1mD*emnab«* npD*

a pBbVBD* ~q2!2 i H ~mB1mD* !«m* A1

BD* ~q2!

2«* •pB

mB1mD*~pB1pD* !mA2

BD* ~q2!22mD1

«* •pB

q2 qm@A3BD* ~q2!2A0

BD* ~q2!#J ,

~3.10!

with

A3BD* ~q2!5

mB1mD*

2mD*A1

BD* ~q2!2mB2mD*

2mD*A2

BD* ~q2!, ~3.11!

and

^D10~pD1

,«!u~ cb!V2A

uB2~pB!&52

mB1mD1

emnab«* npD1

a pBbABD1~q2!2 i H ~mB1mD1

!«m* V1BD1~q2!

2«* •pB

mB1mD1

~pB1pD1!mV2

BD1~q2!22mD1

«* •pB

q2 qm@V3BD1~q2!2V0

BD1~q2!#J ,

~3.12!

with

V3BD1~q2!5

mB1mD1

2mVV1

BD1~q2!2mB2mD1

2mD1

V2BD1~q2! ~3.13!

andA3BD* (0)5A0

BD* (0) as well asV3BD1(0)5V0

BD1(0). Theabove amplitude can be further simplified by applying the Gorddecomposition as

3Our definition forB→D* form factors is the same as@17# except for a sign difference for the matrix elements of the axial-vector currThis sign change is required in order to ensure positive form factors as one can check via heavy quark symmetry or the QCDanalysis.

034008-7


A1M5GF

A2Vud* Vcbf pa1H 2~mB

22mD2 !F0

BD~mp2 !

gLc→pD 0

q22mD2

uLcg5v p1

2mD*

q22mD*2 A0

BD* ~mp2 !uLcF ~g1

Lc→pD* 01g2

Lc→pD* 0

!p” B

2g1Lc→pD* 0 ~pB•q!~mLc

2mp!

mD*2 2g2

Lc→pD* 0 pB•~pLc2pp!

mLc1mp

Gv p12mD1

q22mD1

2V0

BD1~mp2 !uLcF S h

1Lc→pD1

0

1mLc

2mp

mLc1mp

h2Lc→pD1

0D p” B2h1Lc→pD1

0 ~pB•q!~mLc1mp!

mD1

22h

2Lc→pD1

0 pB•~pLc2pp!

mLc1mp

Gg5v pJ . ~3.14!

beth

athe

t

en

hatwer

n.

ac-enthis

toat

let

-

In order to compute the nonresonant decay rate forB2

→Lc1pp2 we need to know the strong couplingsg, g1 , g2 ,

h1 , h2 and their q2 dependence. Fortunately, this canachieved by considering the meson-pole contributions tofactorizable internalW emission as depicted in Fig. 3~b!. Inthe pole model description, the relevant intermediate stareD0, D* 0 andD1

0(2400) as shown in the same figure. T

matrix element Lc1pu(V2A)mu0& then reads

^Lc1pu~V2A!mu0&pole

5uLcH f D* mD*

q22mD*2 Fg1

Lc→pD* gm1 ig2

Lc→pD*

mLc1mp

smnqnG2

f D1mD1

q22mD1

2 Fh1Lc→pD1gm1 i

h2Lc→pD*

mLc1mp

smnqnGg5

2F f DgLc→pD

q22mD2

2f D1

g1Lc→pD1

q22mD1

2

mLc1mp

mD1Gqmg5J v p ,

~3.15!

where the decay constants are defined by

^D~q!uAmu0&52 i f Dqm ,

^D* ~q,«!uVmu0&5 f D* mD* «m* , ~3.16!

^D1~q,«!uAmu0&5 f D1mD1

«m* .

Comparing this with Eq.~3.5! we see that theD* meson isresponsible for the strong couplingsg1 and g2 , D1(2400)for h1 andh2, andD for the couplingg. More precisely,

g1Lc→pD*

~q2!5q22mD*

2

f D* mD*f 1

Lcp~q2!,

g2Lc→pD*

~q2!5q22mD*

2

f D* mD*f 2

Lcp~q2!,

03400

e

es

h1Lc→pD1~q2!5

q22mD1

2

f D1mD1

g1Lcp

~q2!,

h2Lc→pD1~q2!5

q22mD1

2

f D1mD1

g2Lcp

~q2!, ~3.17!

gLc→pD~q2!5q22mD

2

f D~mLc1mp!

g3Lcp

~q2!,

where theD1 pole contribution tog3Lcp can be neglected a

the q2 range of interest.The form factorsf i and gi for the heavy-to-heavy and

heavy-to-light baryonic transitions at zero recoil have becomputed using the non-relativistic quark model@18#. Inprinciple, heavy quark effective theory~HQET! puts someconstraints on these form factors. However, it is clear tHQET is not adequate for our purposes: the predictive poof HQET for the baryon form factors at order 1/mQ is limitedonly to the antitriplet-to-antitriplet heavy baryonic transitioHence, we will follow@18# to apply the nonrelativistic quarkmodel to evaluate the weak current-induced baryon form ftors at zero recoil in the rest frame of the heavy parbaryon, where the quark model is most trustworthy. Tquark model approach has the merit that it is applicableheavy-to-heavy and heavy-to-light baryonic transitionsmaximumq2. It has been shown in@18# that the quark modepredictions agree with HQET for the antitriplet-to-antitripl~e.g. Lb→Lc , Jb→Jc) form factors to order 1/mQ . ForsextetSb→Sc andVb→Vc transitions, the quark-model results are also in accord with the HQET predictions~for de-tails see@19#!. Numerically we have@19#

f 1LbLc~qm

2 !5g1LbLc~qm

2 !51.02,

f 2LbLc~qm

2 !5g3LbLc~qm

2 !520.23, ~3.18!

f 3LbLc~qm

2 !5g2LbLc~qm

2 !520.03,

8-8

-ele

nd

wui

dern.

rel

of

ame

ling

e-

-

ne-

e-

he


for the Lb→Lc transition at zero recoilqm2 5(mLb

2mLc)2,

and4 @18#

f 1Lcp

~qm2 !5g1

Lcp~qm

2 !50.79,

f 2Lcp

~qm2 !5g3

Lcp~qm

2 !520.69, ~3.19!

f 3Lcp

~qm2 !5g2

Lcp~qm

2 !520.20,

for the Lc→p transition atqm2 5(mLc

2mp)2.

Since the calculation for theq2 dependence of form factors is beyond the scope of the non-relativistic quark modwe will follow the conventional practice to assume a podominance for the form-factorq2 behavior:

f ~q2!5 f ~qm2 !S 12qm

2 /mV2

12q2/mV2 D n

,

~3.20!

g~q2!5g~qm2 !S 12qm

2 /mA2

12q2/mA2 D n

,

wheremV(mA) is the pole mass of the vector~axial-vector!meson with the same quantum number as the current uconsideration. The function

G~q2!5S 12qm2 /mpole

2

12q2/mpole2 D n

~3.21!

plays the role of the baryon Isgur-Wise functionz(v) for theLQ→LQ8 transition, namely,G51 at q25qm

2 . However,whether theq2 dependence is monopole (n51) or dipole(n52) for heavy-to-heavy transitions is not clear. Henceshall use both monopole and dipole dependence in enscalculations. Moreover, one should bear in mind that theq2

behavior of form factors is probably more complicated anis likely that a simple pole dominance only applies to a ctain q2 region, especially for the heavy-to-light transitioWe will use the pole massesmV52.01 GeV and mA52.42 GeV for theLc→p transition andmV56.34 GeV,mA56.73 GeV forLb→Lc andSb→Sc transitions.

For the form factorsF0,1Bp(q2) we consider the Melikhov-

Stech ~MS! model based on the constituent quark pictu@20#. Although the form factorq2 dependence is in generamodel dependent, it should be stressed thatF1

Bp(q2) in-creases withq2 more rapidly thanF0

Bp(q2) as required byheavy quark symmetry.

4The Lc→p form factors f 2 , f 3 ,g2 ,g3 given in Eq. ~3.19! aredifferent from that in@4# owing to a different definition of theseform factors.

03400

l,

er

eng

it-

The total decay rate for the processB2(pB)→Lc(p1)1 p(p2)1p2(p3) is computed by

G51

~2p!3

1

32mB3E uAu2dm12

2 dm232 , ~3.22!

wheremi j2 5(pi1pj )

2 with p35pp . Under naive factoriza-tion, the parametera2 appearing in Eq.~3.2! is numericallyequal to 0.024, which is very small compared to the valuea250.4020.55 extracted fromB0→D0(* )p 0 decays@21#and ua2u50.2660.02 from theB→J/cK decay@22#. Sincea2 may receive sizable contributions from the pole diagrFig. 3~c!, we will thus treata2 as a free parameter and taka250.30 as an illustration.

Collecting everything together we obtain numerically5

B~B2→Lc1pp2!nonr5H 1.831024 for n51,

2.631024 for n52,~3.23!

where we have usedgLb→B2p527 ~or gSb1→B0p55), f D

5200 MeV, f D* 5 f D15230 MeV and V0

BD1(0)50.37. Itshould be stressed that the sign of the strong coupgLb→B2p must be negative and hencegS

b1→B0p has to be

positive@see Eq.~2.16!# so that the interference between mson and baryon pole contributions is destructive forn51 andconstructive forn52. Indeed, ifgLb→B2p57 is employed,

one will have a branching ratio of order 9.631024 for n51 and 4.731025 for n52, in disagreement with experiment. We shall see below that the prediction of theSc

11pp2

rate is consistent with experiment only forn51. Adding theresonant contributions fromSc

0p and Lc1D22 as discussed

in Sec. II, we are led to

B~B2→Lc1pp2!'2.431024. ~3.24!

Note that the resonant contributions account for about oquarter of the total decay rate.

In @4# we obtained a branching ratio of order (4.9;9.2)31024 for ugLb→B2pu516. Our present results~3.23! aresmaller for two reasons:~i! The large strong couplingugLb→B2pu516 will lead to a too largeB2→Sc

0p which is

ruled out by experiment. Therefore we useugLb→B2pu57,

obtained by fitting to the observed central value ofB2

→Sc0p. The branching ratio due to the baryon poles b

comes 1.731024 for n51 and 1.131024 for n52. ~ii ! Thecolor-allowed charmed-meson pole contribution to tbranching ratio is 3.631024 for n51 and 0.531024 for n52. It has a destructive~constructive! interference withbaryon pole contributions forn51 (n52).

5If we use ugLb→B2pu516 as in @4#, then we will haveB(B2

→Lc1pp2)nonr55.031024 for n51 and 8.331024 for n52.

8-9

:

ad

w

en-

andibu-


B. B0\Sc¿¿ppÀ

The three-body modeB0→Lc11 p p2 does receive fac-

torizable internalW-emission andW-exchange contributions

A~B0→Sc11pp2! fact

5GF

A2VcbVud* $a1^p

2u~ du!u0&^Sc11pu~ cb!uB0&

1a2^Sc11pp2u~ cu!u0&^0u~ db!uB0&%. ~3.25!

As before, we do not know how to evaluate the 3-body hronic matrix element Sc

11pu( cb)uB0& at the quark level.Thus we will instead evaluate the corresponding two lolying pole diagrams for the externalW-emission: ~i! thebaryon pole diagram with strong processB0→Sb

(* )1p fol-lowed by the weak decaySb

(* )1→Sc11p2 @see Fig. 4~a!#,

and ~ii ! the meson pole diagram with the weak processB0

→$D1,D* 1,D11%p2 followed by the strong reaction

$D1,D* 1,D11%→Sc

11p @Fig. 4~a!#. We consider the baryonpole contribution first. Its amplitude is given by

A~B0→Sc11pp2!B

52GF

A2VudVcb* gS

b1→B0pf pa1uSc

$ f 1SbSc~mp

2 !

3@2pp•pSc1p” p~mSb

2mSc!#g51g1

SbSc~mp2 !

3@2pp•pSc2p” p~mSb

1mSc!#%v p

31

~pSc1pp!22mSb

2, ~3.26!

where we have applied factorization to the weak decaySb1

→Sc11p2.

The heavy-to-heavy transitionSb→Sc at zero recoil ispredicted by HQET to be~see, e.g.@23#!

03400

-

- f 1SbSc~qm

2 !521

3 F12~mSb1mSc

!S 1

mSb

11

mScD G ,

f 2SbSc~qm

2 !51

3 S 1

mSb

11

mScD ~mSb

1mSc!,

~3.27!

f 3SbSc~qm

2 !51

3 S 1

mSb

21

mScD ~mSb

1mSc!,

g1SbSc~qm

2 !521

3, g2

SbSc~qm2 !5g3

SbSc~qm2 !50,

where qm2 5(mSb

2mSc)2. Numerically we find a small

branching ratio

B~B0→Sc11pp2!B5H 0.4531024, n51,

0.2431024, n52~3.28!

arising from the baryon poles. Comparing to the experimtal value ~see Table I!, it is obvious that the baryon polecontribution alone is not adequate to account for the datait is necessary to take into account the meson pole contrtion.

The meson pole contribution from Fig. 4~a! is

FIG. 4. Quark and pole diagrams forB 0→Sc11pp2 where the

solid circle denotes the weak vertex.~a! and ~b! correspond to theexternalW-emission andW-exchange contributions, respectively.

A~B0→Sc11pp2!M5

GF

A2Vud* Vcba1^p

2u~ du!u0&H F ^D1u~ cb!uB0&i

q22mD2 gSc

11→pD1uSc

g5v p

1^D* 1u~ cb!uB0&i

q22mD*2 uSc

i«D*n S g

1Sc

11→pD* 1

gn1 ig

2Sc

11→pD* 1

mSc1mp

snlqlD v p

1^D11u~ cb!uB0&

i

q22mD1

2uSc

i«D1

n S h1Sc

11→pD11

gn1 ih

2Sc

11→pD11

mSc1mp

snlqlD v pJ . ~3.29!

After some manipulation we obtain

8-10


A~B0→Sc11pp2!M5

GF

A2Vud* Vcbf pa1H 2~mB

22mD2 !F0

BD~mp2 !

gSc11→pD1

q22mD2

uScg5v p

12mD*

q22mD*2 A0

BD* ~mp2 !uScF ~g

1Sc

11→pD* 1

1g2Sc

11→pD* 1

!p” B2g1Sc

11→pD* 1 ~pB•q!~mSc2mp!

mD*2

2g2Sc

11→pD* 1 pB•~pSc2pp!

mSc1mp

Gv p12mD1

q22mD1

2V0

BD1~mp2 !uScF S h

1Sc

11→pD11

1mSc

2mp

mSc1mp

h2Sc

11→pD11D p” B

2h1Sc

11→pD11 ~pB•q!~mSc

1mp!

mD1

22h

2Sc

11→pD11 pB•~pSc

2pp!

mSc1mp

Gg5v pJ . ~3.30!

-

yq

o

O

ela-rat

h,o

the

thee

loryon-

There exist five unknown strong couplingsgSc11→pD1

,

g1,2

Sc11→pD* 1

andh1,2

Sc11→pD1

1

which areq2 dependent. To de

termine these couplings we apply the3P0 quark-pair-creation model to obtain

g1,2

Sc11→pD* ~q2!5A3

2g

1,2

Lc1→pD* ~q2!,

h1,2

Sc11→pD1~q2!5A3

2h

1,2

Lc1→pD1~q2!, ~3.31!

gSc11→pD~q2!523A3

2gLc

1→pD~q2!.

As noted in passing, the3P0 model is perhaps reliable onlin the low energy regime. Nevertheless, we will use E~3.31! for an estimation. We obtain, numerically,

B~B0→Sc11pp2!5H 4.531024 for n51,

4.331026 for n52,~3.32!

for gSb1→B0p55. Note that the interference between mes

and baryon pole contributions is constructive~destructive!for n51 (n52), opposite to the case ofLc

1pp2. Evidently,n51 is favored by the measurements of Belle and CLE

FIG. 5. Quark and pole diagrams forB 0→Sc0pp1, where the

solid circle denotes the weak vertex.

03400

.

n

.

Therefore, we conclude thatB 0→Sc11pp2 has a rate

slightly larger thanB2→Lc1pp2.

C. B 0\Sc0pp¿ and BÀ\Sc

0pp 0

The decaysB 0→Sc0pp1 and B2→Sc

0pp 0 proceed viathe nonfactorizable internalW emission~see Figs. 5 and 6!.Naively one may argue that they are color suppressed rtive to B 0→Sc

11pp2. However, it may not be the case fobaryonicB decays. To demonstrate this, let us take a lookFig. 3~b! which proceeds via an internalW emission. Thisdiagram is color suppressed because in order to form thep2

and Lc1p, the color of the spectatoru quark has to be

matched with that of thed quark created from theb quarkdecay, and similarly the color of thec quark has tobe matched with that of theu quark created from theb quark decay. In the effective Hamiltonian approacthe factorizable amplitude is proportional ta2^p

2u(db)uB2&^Lc1pu( cu)u0&, where the coefficienta2 is

equivalent to 1/3 in the absence of strong interactions. Oncontrary, the other internalW-emission diagram Fig. 3~c! isnot color suppressed because the color wave function ofbaryon is totally antisymmetric and hence the color of thcquark must be different from that of thed quark created fromthe b quark decay. Likewise, Figs. 5 and 6 are not cosuppressed. Indeed, as shown in Sec. II, the weak barbaryon transition is found to be proportional toc12c2 ratherthan toc21c1 /Nc .

FIG. 6. Same as Fig. 5 except forB2→Sc0pp 0.

8-11

u

rshe

a

toed

q.

rheor

-derm

ca

i-

isle

atecou-

isith

actiteds

ti-e a

-

-le

d to


Theoretically, it is not easy to estimate the pole contribtions as the weak decay processes, for exampleB0→Sc

0nandSb

1→Sc0p1 in Fig. 5, are not factorizable. This rende

the calculation difficult. Among the pole diagrams, only tintermediaten state in Fig. 5~a! and thep state in Fig. 6 canbe reliably estimated since the involved weak transitionsalready discussed in Sec. II and, moreover, the strongpNNcoupling can be related to the nucleon-nucleon form facg3

np(q2) and hence itsq2 dependence can be determin@24#.

Consider the decayB0→Sc0pp1 first. The decay ampli-

tude of then pole diagram reads

A~B0→Sc0pp1! n-pole5A2gpNN~q2!B~B0→Sc

0n!

3uScp” pv p

1

q22mN2 , ~3.33!

where B is the parity-conserving amplitude given in E~2.13! andq25(pp1pp)2. In @24# we have shown that

gpNN~q2!5q22mp

2

2A2 f pmN

g3np~q2!, ~3.34!

whereg3np is one of the form factors defined by

^n~pn!u~V2A!mu p~pp!&

5 v n~pn!H f 1np~q2!gm1 i

f 2np~q2!

2mNsmnqn1

f 3np~q2!

2mNqm

2Fg1np~q2!gm1 i

g2np~q2!

2mNsmnqn

1g3

np~q2!

2mNqmGg5J v p~pp!, ~3.35!

with q5pn2pp . As shown in@10#, the induced pseudoscalaform factorg3 corresponds to a pion pole contribution to tn2p axial matrix element and it is related to the form factg1 via

g3np~ t !52

4mN2

t2mp2 g1

np~ t !. ~3.36!

The vector form factorsf i(q2) can be related to the nucle

on’s electromagnetic form factors which are customarilyscribed in terms of the electric and magnetic Sachs fofactorsGE

N(t) andGMN (t). A recent phenomenological fit to

the experimental data of nucleon form factors has beenried out in @25# using the following parametrization:

uGMp ~ t !u5S x1

t21x2

t3 1x3

t4 1x4

t5 1x5

t6 D F lnt

Q02G2g

,

~3.37!

uGMn ~ t !u5S y1

t2 1y2

t3 D F lnt

Q02G2g

,

03400

-

re

r

-

r-

where Q05LQCD'300 MeV and g521(4/3b)52.148.For our purposes, we just need the best fit values ofx1 andy1,

x15420.96 GeV4, y15236.69 GeV4, ~3.38!

extracted from the neutron data@26#. For the axial form fac-tor g1

np(t), we shall follow@27# to assume that it has a simlar expression asGM

n (t)

g1np~ t !5S d1

t2 1d2

t3 D F lnt

Q02G2g

, ~3.39!

where the coefficientd1 is related tox1 andy1 by consider-ing the asymptotic behavior of Sachs form factorsGM

p andGM

n @see Eq.~3.37!#

d155

3x12y1 . ~3.40!

For d2 we shall use the value of22370 GeV6 obtained byfitting to the data ofB0→D0pp @4#.

Collecting all the inputs, we finally obtain

B~B0→Sc0pp1! n2pole

PC51.331024. ~3.41!

It is interesting to note that this pole contribution aloneconsistent with both Belle and CLEO. The remaining podiagrams in Fig. 5 are more difficult to get a reliable estimas the strength and momentum dependence of the strongplings is unknown. Therefore, whether or notSc

0pp1 is sub-

stantially suppressed relative toSc11pp2 is unknown. Nev-

ertheless, as noted in passing, even if the formersuppressed relative to the latter, it has nothing to do wcolor suppression.

Likewise, we find thep pole diagram in Fig. 6 gives

B~B2→Sc0pp0! p2pole54.831023. ~3.42!

This enormously large branching ratio comes from the fthat Sc

0p@Sc0n as discussed in Sec. II B. At first sight,

appears that this prediction is ruled out as it already excethe measurement by CLEO~see Table I!. However, the decayamplitude of Fig. 6~b! has a sign opposite to that of Fig. 6~a!owing to thep0 wave functionp05(uu2dd)/A2. Hence,there exists a destructive interference between Fig. 6~a! andFig. 6~b!. Unfortunately, as we do not have a reliable esmate of other pole diagrams in Fig. 6, we cannot makreliable prediction of the branching ratio forB2→Sc

0pp0.

Nevertheless, it is very conceivable thatSc0pp0 has a larger

rate thanSc0pp1. Recall that the CLEO measurements im

ply that Sc0pp0*Sc

11pp2*Sc0pp1 @3#.

IV. CONCLUSIONS

We have studied exclusiveB decays to final states containing a charmed baryon within the framework of the pomodel. We first draw some conclusions and then proceediscuss some sources of theoretical uncertainties.

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~1! In the pole model, the two-body baryonicB decayamplitudes are expressed in terms of strong couplingsweak baryon-baryon transition matrix elements. We apthe bag model to evaluate the baryon matrix elements. Sthe strong coupling forLb

0→B2p is larger than that forSb0

→B2p, the two-body charmful decayB2→Sc0p has a rate

larger thanB0→Lc1p as the former proceeds via theLb pole

while the latter via theSb pole. However, the relative coupling strength predicted by the quark-antiquark creation3P0

model, namely,gLb0→B2p53A3gS

b0→B2p will lead to a large

rate for B2→Sc0p that already exceeds the present expe

ment limit. Likewise, the 3P0 relation gSb1→B2D11

52A6 gSb1→B0p will lead to too largeB2→pD22 andB2

→Lc1D22. Our best values for strong couplings a

ugSb1→B2D11u;10, ugL

b0→B2pu;7 andugS

b0→B2pu;3.5. The

inconsistency of the3P0 model’s predictions with experiment may imply that the relevant one is the3S0 model forquark pair creation.

~2! The ratio ofR[G(B0→Sc0n)/G(B2→Sc

0p) also pro-vides a nice test on the3P0 model. WhileR is predicted tobe 1/2 in the3P0 model, it is of order only 0.01 in our case

~3! At the quark level, B0→Sc11pp2 and B2

→Lc1pp2 are expected to have similar rates as they b

receive externalW-emission contributions. By the same tken as two-body decays, the three-body decayB0

→Sc11pp2 receives less baryon-pole contribution th

B2→Lc1pp2 at the pole-diagram level. However, becau

the important charmed-meson pole diagrams contribute cstructively to the former and destructively to the latter,B0

→Sc11pp2 has a rate slightly larger thanB2→Lc

1pp2.

~4! B2→Lc1pp2 also receives the resonant contributio

from B2→Sc0p andB2→Lc

1D22. The nonresonant contribution to the branching ratio is smaller than our previoestimate mainly because the strong couplinggL

b0→B2p be-

comes smaller as implied by the data ofB2→Sc0p. We

ev

e

03400

ndyce

i-

h

n-

s

found that resonant contributions account for about oquarter of theB2→Lc

1pp2 rate.~5! The decaysB0→Sc

0pp1 and B2→Sc0pp0 that can

only proceed via an internalW emission are not color suppressed. In the pole model, it is easily seen that the wbaryon-baryon transition vertex in the pole diagram is pportional to (c12c2) rather thana2. We have estimated theneutron pole contribution toB0→Sc

0pp1 and the proton

pole contribution toB2→Sc0pp0. Because of the lack o

information of the momentum dependence of strong cplings, we cannot have a definite prediction for the decrates of these two modes, though it is conceivable tSc

0pp0.Sc0pp1. If these two decays are found to be su

pressed relative toSc11pp2, it has nothing to do with color

suppression and must arise from some other dynamic coneration.

The calculation of baryonicB decays is rather complicated and very much involved and hence it suffers frmany possible theoretical uncertainties. Many of them hbeen discussed in detail in@10#. For the present work, wewould like to mention three uncertainties. First, the charmmeson-pole amplitude is sensitive to the form factorV0

BD1(0)which we have taken it to be 0.37. Hence, a model calcution of the form factorsV0,1,2

BD1(0) is urgent. Second, a reliablestimate of the strong couplings forSc

11→pD(D* ,D1) andtheir q2 dependence is needed in order to calculate the ratB0→Sc

11pp2 more accurately. Third, final-state interations may contribute sizably to the charmful two-body baonic B decays. For example, the color- and Cabibbo-allowdecay B0→D1p2 followed by the rescattering ofD1p2

→Lc1p may introduce a significant contribution toB0

→Lc1p. This deserves further study.

ACKNOWLEDGMENT

This work was supported in part by the National ScienCouncil of R.O.C. under Grant Nos. NSC91-2112-M-00038 and NSC91-2112-M-033-013.

J.

e

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decays to charmed baryons