Hadronic decays involving heavy pentaquarks

PHYSICAL REVIEW D 70, 034030 ~2004!

Hadronic decays involving heavy pentaquarks

Xiao-Gang He*Department of Physics, Peking University, Beijing, China

Xue-Qian Li†

Department of Physics, Nankai University, Tianjin, China~Received 29 March 2004; published 31 August 2004!

Recently several experiments have reported evidence for pentaquarkQ1. H1 experiment at HERA-B has

also reported evidence forQc . Q1 is interpreted as a bound state of ans with four other light quarksudud

which is a member of the antidecuplet of flavorSU(3) f , while Qc is a heavy state obtained with thes in Q1

being replaced by ac which is a member of an antisixtet. A heavy pentaquark triplet can also be formed with

a c and four light quarks. Similarly heavy pentaquark antisixtet and triplet can be formed with thec being

replaced by ab. We study decay processes involving at least one heavy pentaquark usingSU(3) f and estimatethe decay widths for some decay modes. We find several relations for heavy pentaquark decays into anotherheavy pentaquark and aB(B* ) or a D(D* ) which can be tested in the future. TheB can also decay throughweak interaction to charmed heavy pentaquarks. We also study someB decay modes with a heavy pentaquarkin the final states. Experiments atB factories will provide important information about the heavy pentaquarkproperties.

DOI: 10.1103/PhysRevD.70.034030 PACS number~s!: 14.20.Lq, 14.20.Mr, 14.40.Lb, 14.40.Nd

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I. INTRODUCTION

Recently several experiments have reported evidencepentaquarksQ1 and other states@1#, although there are alsexperiments reporting null results@2#. The Q1(1540) pen-taquark has strangenessS511 and has quark conten

ududs. This particle is an isosinglet which is a memberthe antidecuplet@3# in flavor SU(3) f symmetry. At presentthere is very limited experimental information on the dtailed properties such as the decay width, spin, and paSeveral models have been proposed to accommodateantidecuplet states@3–13#.

Replacing thes in Q1 by a heavy quark such asc or b, itis also possible to form bound heavy pentaquark sta@5,6,8,10–12,14–18# Qc or Qb . When implementing theminto SU(3) f , in the model of Jaffe and Wilczek@5#, wherepentaquarkQ1 is composed of two (ud) diquarks withspin-0 and ans quark, heavy pentaquarks form a fundametal representation ofSU(3) f triplet Rc,b ~the subindicescandb indicate whether the pentaquark is formed with ac ora b), and an antisixtetSc,b @10#. Discovery of these stateand a study of their properties can provide important infmation about the inner structure of matter. H1 Collaborathas recently reported observation of a narrow resonancRef. @19# D* 2p and D* 1p in inelastic eP collisions atcenter-of-mass energies of 300 and 320 GeV at HERA. Tresonance has a mass of 309963(stat)65(syst) with aGaussian width of 1263 MeV and can be interpreted as evdence forQc . If this observation is confirmed, other relate

*On leave of absence from Physics Department, National TaiUniversity; Electronic address: [email protected]

†Electronic address: [email protected]

1550-7998/2004/70~3!/034030~10!/$22.50 70 0340

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states should exist. In this paper we study some propertiedecay processes involving at least one heavy pentaquarking SU(3) f flavor symmetry.

The tripletRc,b transforms underSU(3) f in a similar wayas the light quark (u,d,s) triplet. To indicate this fact andalso to distinguish the pentaquark triplet from the quark owe useU,D,S to indicate the elements inRc,b . For theantisixtetSc,b , we useTc,b , Nc,b , andQc,b to indicate theisospin triplet, doublet, and singlet members, respectivWe have

Rc5~Rc,i !5~Uc0 ,Dc

2 ,Sc2!,

Sc5~Sci j !5S Tc

22 Tc2/A2 Nc

2/A2

Tc2/A2 Tc

0 Nc0/A2

Nc2/A2 Nc

0/A2 Qc0

D ,

Rb5~Rb,i !5~Ub1 ,Db

0 ,Sb0!,

Sb5~Sbi j !5S Tb

2 Tb0/A2 Nb

0/A2

Tb0/A2 Tb

1 Nb1/A2

Nb0/A2 Nb

1/A2 Qb1

D .

~1!

The quark contents of these particles are

Uc,b0,15~udus!~ c,b!, Dc,b

2,05~udds!~ c,b!,

Sc,b2,05~dsus!~ c,b!,

Tc,b22,25~dsds!~ c,b!, Tc,b

2,05~dsus!~ c,b!,

Tc,b0,15~usus!~ c,b!,

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X.-G. HE AND X.-Q. LI PHYSICAL REVIEW D 70, 034030 ~2004!

Nc,b2,05~udds!~ c,b!, Nc,b

0,15~udus!~ c,b!,

Qc,b0,15~udud!~ c,b!. ~2!

The Uc,b , Dc,b , andNc,b particles haveS521, Sc,b , andTc,b particles haveS522, andQc,b particles haveS50.

In the diquark model of Ref.@5#, Sc,b have positive parity,whereasRc,b have negative parity since there is noP-waveexcitation between the diquarks. In our study we emphaon the flavorSU(3) f properties, the conclusions can be aplied to both parity situations ‘‘1’’ or ‘‘ 2 ’’ for both Rc,bandSc,b .

II. HEAVY PENTAQUARK STRONG DECAY COUPLINGS

Whether heavy pentaquarks can have strong decay mdepends on their masses. WithSU(3) f symmetry, particlesin each multiplet are supposed to have the same mass. Qmodel estimates for the masses of heavy pentaquarksbeen carried out by several groups. In the diquark model,Qc,b masses are estimated to be@5# 2710 and 6050 MeV,respectively, which are below the strongpD and nB decaythreshold. A lattice calculation in Ref.@8#, gives amQc

ofabout 3.5 GeV.

Removing theP-wave excitation energy, which is estmated by using the mass difference ofLc and its excitationLc8 , UP-wave'mL

c82mLc

5310 MeV, fromQc , and adding

a constituent strange quark contributionDs5mJc2mLc

'184 MeV, Ref. @15# obtained 2580 MeV forUc ,Dcmasses. Assuming the sameUP-wave andDs for beauty heavypentaquarks,Ub ,Db masses are estimated to be 5920 M@15#.

The mass degeneracy for the particles in a multiplelifted by quark mass differencesmu , md , andms . The massterms, up to linear corrections in light quark masses,given by

L5m0Rc,bTr~Rc,bRc,b!1am

Rc,bTr@Rc,b~M1M†!Rc,b#

1m0Sc,bTr~Sc,bSc,b!1am

Sc,bTr@Sc,b~M1M†!Sc,b#.

~3!

We have neglected terms of the form Tr@R(S)R(S)#Tr(M )which only rescalem0 . M is the quark mass matrix andgiven by

M5S mu 0 0

0 md 0

0 0 ms

D . ~4!

Neglecting smallmu,d masses, we obtain

03403

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arkvee

s

e

mUc,b5mDc,b

5m0Rc,b , mSc,b

5m0Rc,b12aRc,bms ,

mTc,b5m0

Sc,b , mNc,b5m0

Sc,b1amSc,bms ,

mQc,b5m0

Sc,b12amSc,bms . ~5!

Taking into account theSU(3) f breaking effects via dif-ferences in light quark masses fromDs5mJc

2mLcfor con-

stituent strange quarks, the masses ofSc2 and Sb

0 were esti-mated to be 2770 and 6100 MeV, respectively in Ref.@15#.Making the same assumption we obtain the masses ofNc ,Nb , Tc , and Tb to be 2894, 6236, 3078, and 6420 Merespectively. These values are similar to the estimatestained in Ref.@16#.

There are other model estimates for heavy pentaqumasses which give larger masses. For example, in the mof Karliner and Lipkin, where the pentaquarks are formfrom a triquark and a diquark@6#, the masses are estimatedbe 2985 and 6398 MeV, respectively, which are abovestrongpD and nB decay threshold. And the masses ofNc ,Nb , Tc , andTb are estimated to be 3165, 6570, 3340, a6740 MeV, respectively@16#. Removing theP-wave excita-tion energy@6# dEP-wave'207 MeV between the diquark antriquark fromQc,b and adding the mass difference due to treplacement of a lightu or d quark by ans quark, one obtainsthe masses ofUc

0 ,Dc2 and Ub

1 ,Db0 to be 2858 and 6533

MeV, respectively.Sc2 and Sb

0 are approximately 3028 an6708 MeV. Clearly the above estimates for the massesrather rough and should not be expected to hold withaccuracy better than 50 or even 100 MeV.

If the H1 narrow resonance of mass 3099 MeV is indetheQc particle, both the diquark and triquark-diquark modpredictions for the mass are slightly lower than the daThere is also the possibility that the narrow resonant sobserved at H1 is a chiral partner ofQc . At present theuncertainties involved in the estimates are large so it isearly to make a decisive conclusion. With a mass 3099 Mfor Qc , it is possible for it to decay intoD* 2p and D1p.Similar situation may happen for beauty pentaquarks.therefore will consider processes involving bothD, B andD* , B* .

We now write down the strong decay amplitudes to tleading order using theSU(3) f symmetry for heavy pen-taquark decays with aB or D in the final states. We have

LRbNB5cRbNBRbj Nj

i Bi1H.c.,

LSbNB5cSbNBSb, jkNlj Bie

ikl1H.c.,

LRcND5cRcNDRcj Nj

i D i1H.c.,

LScND5cScNDSc, jkNlj D ie

ikl1H.c. ~6!

In the aboveN is the ordinary baryon octet,Di andBi arethe charm and beauty mesons. They are given by

0-2

r

HADRONIC DECAYS INVOLVING HEAVY PENTAQUARKS PHYSICAL REVIEW D70, 034030 ~2004!

TABLE I. Couplings for BNRb(DNRc) in unit cRNB(cRND). The Lorentz structure for the bi-spino

product is of the formNGPR with GP511 andg5 for negative and positive parity forR, respectively. For

B* NRb , the Lorentz structure for the bispinor product should be changed toNgmg5GPRb .

Bu S 1

A6L1

1

2S0D Ub

11S2Db01J2Sb

0 Du S 1

A6L1

1

2S0D Uc

01S2Dc21J2Sc

2

Bd S1Ub11S 1

A6L2

1

A2S0D Db

01J0Sb0 Dd S1Uc

01S 1

A6L2

1

A2S0D Dc

21J0Sc2

Bs pUb11nDb

02A23 LSb

0 Ds pUc01nDc

22A23 LSc

2

llcegotlslejuct

a

ed.e

e toloseareayseter-yinFor

-thte

N5~Nij !5S S0

A21

L

A6S1 p

S22

S0

A21

L

A6n

J2 J02

2L

A6

D .

D5~Di !5~Du ,Dd ,Ds!5~cu,cd,cs!,

B5~Bi !5~Bu ,Bd ,Bs!5~bu,bd,bs!. ~7!

In Tables I and II, we list the couplings forNRb(Sb)B.One can obtain the couplings forNRc(Sc)D by replacingBiby Di and the subindexb to c. Throughout the paper, in athe equations for pentaquark couplings only group indiare properly labeled and all fields in the Lagrangian areing outwards. The Lorentz structures are suppressed inequations, while the proper ones are given in the tables. Awe will assume that heavy pentaquarks are spin-1/2 particTo obtain results for spin-3/2 heavy pentaquarks, one canuse at an appropriate place the Rarita-Schwinger vespinor for the relevant fields.

We would like to point out that the above equations cbe equally applied to processes withB andD replaced byD*

TABLE II. Couplings for BNSb(DNSc) in unit cSNB(cSND).The Lorentz structure is the same as in Table I.

Bu12 @A2J2Tb

012J0Tb12A2S2Nb

02A3LNb11S0Nb

1

22nQb1]

Bd12 @22J2Tb

22A2J0Tb01A2S1Nb

11A3LNb01S0Nb

0

12pQb1]

Bs12 @2S2Tb

222S0Tb022S1Tb

12A2pNb11A2nNb

0#

Du12 @A2J2Tc

212J0Tc02A2S2Nc

22A3LNc01S0Nc

0

22nQc0]

Dd12 @22J2Tc

222A2J0Tc21A2S1Nc

01A3LNc2

1S0Nc2

12pQc0]

Ds12 @2S2Tc

2222S0Tc222S1Tc

02A2pNc01A2nNc

2#

03403

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n

and B* , respectively. We will not distinguish them in thequations in later discussions unless specifically indicate

If the diquark model for pentaquarks is the right one, wsee that all the strong decay modes are forbidden durestriction of phase space. However, if the masses are cto the triquark-diquark model predictions, strong decaysallowed. The H1 data indicate that the above strong decare possible. One can use the allowed decay modes to dmine the parametercabc and then the widths of the heavpentaquarks. Here we give the formula for the couplingsterms of decay widths assuming the decays are allowed.decays withB in the final states we have

cRbNB2 5

16pmSb0G~Sb

0→J2Bu!

@~pmSb01mJ2!22mBu

2 #Ph~mSb0,mJ2,mBu

!,

cSbNB2 5

16pmQb1G~Qb

1→pBd!

@~pmQb1mp!22mBd

2 #Ph~mQb1,mp ,mBd

!,

~8!

wherePh(a,b,c)5A12(b1c)2/a2)(12(b2c)2/a2). p isthe eigenvalue of parity of the heavy pentaquark.

For decays withB* in the final states, we have

cRbNB*2

516pmS

b0G~Sb

0→J2Bu* !

f ~mSb0,mJ2,mB

u*!Ph~mS

b0,mJ2,mB

u*!,

cSbNB*2

516pmQ

b1G~Qb

1→pBd* !

f ~mQb1,mp ,mB

d*!Ph~mQ

b1,mp ,mB

d*!,

~9!

where

f ~a,b,c!5a21b22c22p6ab1@~a22b2!22c4#/c2.

Similarly one can obtaincRc(Sc)ND(D* )2 by considering

Sc2→Du

0(Du*0)J2 and Qc

0→pDd1(Dd*

1) decays, respectively. At present there is only some information on the widof G(Qc→pDd*

1). Assuming that the narrow resonant staof width 1263 MeV at H1 is theQc particle, we obtain

cScND*2 'H 1.712, p51,

0.167, p521.~10!

0-3

sns

rarettoe

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tr

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ilyctor

by

alls

I.


Using these numbers, the decay widths for decay modeTable II involving D* can be predicted. These predictiocan be tested.

III. WEAK HADRONIC DECAYS OF HEAVYPENTAQUARKS

Heavy pentaquark can also decay through weak intetion. If kinematically the strong decays discussed in the pvious section are not allowed, weak interaction will dominaheavy pentaquark decays. These decays can be semilepor purely hadronic ones. Analysis on some of the heavy ptaquark properties have been carried out@11,12,15–18#. Herewe will concentrate on some two body hadronic heavy ptaquark decays.

A. Rb„Sb…\Rc„Sc…¿P decays

In this subsection we study pentaquark decays of the tRb(Sb)→Rc(Sc)P. HereP represents a meson in the pesdoscalar octet which is given by

P5~P ij !5S p0

A21

h

A6p1 K1

p22

p0

A21

h

A6K0

K2 K0 22h

A6

D . ~11!

The quark level effective Hamiltonian forRb(Sb)→Rc(Sc)P is given by

Heff5GF

A2@Vcb* Vuq~c1O11c2O2!

1VubVcq* ~c1O11c2O2!#, ~12!

O15bgm~12g5!cugm~12g5!q,

O25bgm~12g5!qugm~12g5!c, ~13!

O15bgm~12g5!ucgm~12g5!q,

O25bgm~12g5!qcgm~12g5!u. ~14!

The two operatorsO1,2 in the above can induce decays of ttype Rb(Sb)→Rc(Sc)1P, while the operatorsO1,2 will notcause beauty heavy pentaquark to charmed pentaquarksitions. We write it down here for later discussions.

Under theSU(3) f symmetryO1,2 transforms as an octeH j

i . With proper normalization the nonzero entries ofH ji can

be written as

q5d, H2151; q5s, H3

151. ~15!

The SU(3) f invariant decay amplitudes are

03403

in

c--

enic

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e-

an-

H~Rb→RcP!5Vcb* Vuq@r 81Rb,iRci Pk

j H jk1r 82RbiRc

j P jkHk

i

1r 83Rb,iRcj Pk

i H jk#,

H~Rb→ScP!5Vcb* Vuq@s81RbiSc, j l PkmHm

l e i jk1s82RbiSc, j l

3Pmi Hk

l e jkm1s83Rb,i Sc, j l Pml Hk

i e jkm#.

~16!

The amplitudes forSb→RcP can be obtained by interchanging the indicesb and c, and treating the processes as tcharge conjugated ones in the second equation of Eq.~16!.

The amplitudes forSb→ScP can be written as

H~Sb→ScP!5Vcb* Vuq@s1Sbi j Sc,i j Pk

l Hlk1s2Sb

i j Sc,klPki Hl

j

1s3Sbi j Sc,k jPk

l Hli1s4Sb

i j Sc,k jP liHk

l #. ~17!

We list the results in Tables III, IV, and V. One can easgeneralize the above formulation to the case with the vemeson nonet (r0,6,K* ,0,6,v,f).

B. Rb„Sb…\Rc„Sc…D„D* …

The effective Hamiltonian for these processes is given

Heff5GF

A2Vcb* Vcq~c1O1

c1c2O2c!. ~18!

Here O1c5bgm(12g5)ccgm(12g5)q and O2

c5cgm(1

2g5)cbgm(12g5)q. In the above we have neglected smpenguin contributions. This effective Hamiltonian transformas a triplet 3 with nonzero entries

q5d, H251; q5s, H351. ~19!

The SU(3) f invariant amplitudes can be written as

H@Rb~Sb!→Rc~Sc!D#

5Vcb* Vcq@r 31Rbi Rc,iH jD

j

1r 32Rbi HiRc, jD

j1r e i jk Rbi Sc, j l DkHl

1s31Sb,i j Sci j HkD

k1s32Sb,i j SikHkD

j #.

~20!

The results for individual processes are given in Table V

C. B\NRc„Sc… and B\NRc„Sc…

The conjugate operators ofO1,2 in Eq. ~13! are of theform cbqu and can induce decays of the typeB→N

1Rc(Sc). TheSU(3) f invariant decay amplitudes are

H~Rc!5VcbVuq* @ r 81Bi Rcj Nk

i H jk1 r 82Bi Rc

j NjkHk

i

1 r 83Bi Rci Nk

j H jk#,

H~Sc!5VcbVuq* @ s81Bi Sc j lNkmHm

l e i jk

1 s82Bi Sc j lNmi Hk

l e jkm1 s83BiSc j lNml Hk

i e jkm#. ~21!

0-4

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TABLE III. SU(3) decay amplitudes forRb→Rc(Sc)P. The

Lorentz structure of the bispinor product is of the formRc(Sc)(11bg5)Rb . Hereb is a parameter.

Ub1 decay DS50 DS51

Uc0p1 r 811r 83 Uc

0K1 r 811r 83

Tc0K1 s812s82 Nc

0K1 1

A2~s812s82!

Nc0p1

21

A2~s812s82! Qc

0p1 2(s812s82)

Db0 decay

Uc0K0 1

A2~r 822r 83! Uc

0p0 r 83

Uc0h

1

A6~r 821r 83! Dc

2K1 r 81

Dc2p1 r 811r 82 Nc

2K12

1

A2s81

Sc2K1 r 82 Nc

0K02

1

A2s82

Tc2K1

21

A2~s811s83! Qc

0p0 1

A2~s812s82!

Tc0K0 2(s821s83) Qc

0h1

A6~s811s82!

Nc2p1 1

A2s83

Nc0p0 1

2 (s812s822s83)

Nc0h

1

2A3~s811s8213s83!

Qc0K0 s83

Sb0 decay

Uc0K0 r 83 Uc

0p0 1

A2r 82

Sc2p1 r 81 Uc

0h1

A6~2r 822r 83!

Tc2p1 1

A2s81

Dc2p1 r 82

Tc0p0

21

A2s81

Sc2K1 r 811r 82

Tc0h 2

1

A6~s8122s82! Tc

2K12

1

A2s83

Nc0K0 1

A2s82

Tc0K0 2s83

Nc2p1 1

A2~s811s83!

Nc0p0

212~s811s83!

Nc0h 2

1

2A3~s8122s8223s83!

Qc0K0 s821s83

03403

The conjugate operatorsO1,2 in Eq. ~14! are of the formqbuc. These operators can induceB→N1Rc(Sc). It con-tains anSU(3) triplet and an antisixtet. The nonzero entriare

q5d, H~3c!351, H~6c!125H~6c!2151;

q5s, H~3c!2521, H~6c!135H~6c!3151.~22!

One can write downSU(3) f decay amplitudes forB→NRc(Sc) as

H~Rc!5VubVcq* @ r 31BiRc jNliHke

j lk1 r 32BiRc jNljHke

i lk

1 r 61BiRc jNki H jk1 r 62BiRc jNk

j Hik#,

H~Sc!5VubVcq* @ s31BiSci j N j

kHk1 s32BiScjkNj

i Hk

1 s61BiSci j Nl

kHlme jkm1 s62BiScjkNk

l Hime j lm#.

~23!

The hadronic parametersr i j are expected to be similarG@B→NRc(Sc)# would be smaller by a factor ouVcbVuq* u2/uVubVcq* u2 compared withG(B→NRc ,(Sc)). Thedetails are listed in Tables VII, VIII, and IX.

IV. DISCUSSIONS AND CONCLUSIONS

If the recently discovered stateQ1 is interpreted as apentaquark bound state with ans and four light quarks,heavy pentaquarks with thes replaced by ab or a c shouldexist. H1 experiment at HERA-B has obtained some edence forQc . These states formSU(3) f triplets Rc,b andantisixtetsSc,b . Rc and Sc can also be produced fromBdecays atB factories. If pentaquarksRc,b andSc,b are kine-matically allowed to decay through strong interactions, ocan use Tables I and II to relate different decay widths, ato test the model.

At present there is only some evidence forQc→D* 2p.Using the decay widthG512 MeV obtained from H1, wedeterminecSND*

2 to be 1.712 and 0.167 forQc with positiveand negative parities, respectively. One expectscSNB*

2 to besimilar to cSND*

2 . Using Table II, we can obtain other decawidths which can be tested in the future. We have nothmuch to say about the size of the couplings except thatexpect the couplingscRbNB andcSbND to be about the same

ascRcND andcScND , respectively. If one extendsSU(3) f to

SU(4) f , the strong couplings can be related in principlethe ones involving just light pentaquarks@12#. We will notconsider this possibility here.

We have parametrized some of the two body hadroweak decays of heavy pentaquarks in terms ofSU(3) f in-variant amplitudes. From the tables obtained we seethere are several relations among different decay modes.example, forDS50 processes of the typeSb→ScP, fromTable V we obtain

0-5

r


TABLE IV. SU(3) decay amplitudes forSb→ScP with DS50. The Lorentz structure is similato Table III.

Tb2 Tc

22p1 Tc2p0 Tc

2h Tc0p2

Nc2K0 Nc

0K2

s11s312~s22s31s4!

1

2A3~s21s31s4! s2

1

A2s3

1

A2s2

Tb0 Tc

2p1 Tc0p0 Tc

0h Nc0K0

12 (s11s21s4) 2

12 s2

1

2A3s2

12 s2

Tb1 Tc

0p1

s11s4

Nb0 Tc

2K1 Tc0K0 Nc

2p1 Nc0p0 Nc

0h Qc0K0

12 (s21s4)

1

A2s2

12 (2s11s3) 2

1

2A2(s32s4) 2

1

A6(s22s32s4)

1

A2s3

Nb1 Tc

0K1 Nc0p1

1

A2s4

12 (2s11s4)

Qb1 Nc

0K1 Qc0p1

1

A2s4

s1

ele

aveout

eryind to

to

G~Tb2→Tc

0p2!52G~Tb2→Nc

0K2!

54G~Tb0→Tc

0p0!512G~Tb0→Tc

0h!

54G~Tb0→Nc

0K0!52G~Nb0→Tc

0K0!,

G~Nb1→Tc

0K1!5G~Qb1→Nc

0K1!. ~24!

More relations can be read off from the tables. These rtions can be used to study the properties of heavy p

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taquarks and test the model provided that the decays hsubstantial branching ratios which require knowledge abthe size of theSU(3) f invariant amplitudes.

Theoretical calculations of the decay amplitudes are vdifficult since multiquarks are involved. However, for certadecays, the structure is very simple and can be relateexperimentally measured modes, for example,Qb

1→Qc0p1

can be related toLb→Lc1p2.

In Qb1→Qc

0p1 decay, the main contributions is due

n
TABLE V. SU(3) decay amplitudes forSb→ScP with DS51. The Lorentz structure is similar to that iTable III.
Tb2 Tc

22K1 Tc2K0 Nc

2p0 Nc2h Nc

2p2 Qc0K2

s11s31

A2s3

12 (s21s4)

1

2A3(s22s31s4)

1

A2s2 s2

Tb0 Tc

2K1 Nc2p1 Nc

0p0 Nc0h Qc

0K0

12 s1

12 (s21s4) 2

1

2A2s2

1

2A6s2

1

A2s2

Tb1 Tc

0K1 Nc0p1

s11

A2s4

Nb0 Nc

2K1 Nc0K0 Qc

0p0 Qc0h

12 (2s11s21s31s4) 1

2 (s21s3) 12 s4 2

1

A3(s21s32s4)

Nb1 Nc

0K1 Qc0p1

12 (2s11s4) 1

A2s4

Qb1 Qc

0K1

s11s4

0-6

I.


TABLE VI. SU(3) decay amplitudes forSb→ScD. The Lorentz structure is similar to that in Table II

For Sb→ScD* , the Lorentz structure of the bi-spinor product is of the formScgm(11bg5)Sb .

DS50 DS51

Ub1 r 31Uc

0Dd r 31Uc0Ds

Db0 (r 311r 32)Dc

2Dd1r 32Uc0Du1r 32Sc

2Ds r 31Dc2Ds

Sb0 r 31Sc

2Dd (r 311r 32)Sc2Ds1r 32Uc

0Du1r 31Dc2Dd

Ub1

rTc0Ds2

1

A2rNc

0Dd

1

A2rNc

0Ds2rQc0Dd

Db0 1

A2rNc

0Du21

A2rTc

2Ds 21

A2rNc

2Ds1rQc0Du

Sb0 1

A2rTc

2Dd2rTc0Du

1

A2rNc

2Dd21

A2rNc

0Du

Tb2

s31Tc22Dd1

1

A2s32Tc

2Du s31Tc22Ds1

1

A2s32Nc

2Du

Tb0

~s31112 s32!Tc

2Dd11

A2s32Tc

0Du

s31Tc2Ds1

12 s32Nc

0Du112 s32Nc

2Dd

Tb1 (s311s32)Tc

0Dd s31Tc0Ds1

1

A2s32Nc

0Dd

Nb0 s31Nc

2Dd112 s32Nc

0Du112 s32Tc

2Ds (s31112 s32)Nc

2Ds11

A2s32Nc

0Du

Nb1

(s31112 s32)Nc

0Dd11

A2s32Tc

0Ds ~s31112 s32!Nc

0Ds11

A2s32Qc

0Dd

Qb1

s31Qc0Dd1

1

A2s32Nc

0Ds(s311s32)Qc

0Ds

of

thtiv

cra

er

e

-ul

s

w

re-

vy

be

ase

the factorized matrix elements wherep is emitted from twolight quarks in the effective Hamiltonian, the transitionQb

1 to Qc0 is due to the transition of ab quark to ac quark,

and the structure of the rest of the four light quarks inpentaquarks is basically preserved. Based on this intuipicture, Ref.@18# relatesQb

1→Qc0p1 to Lb→Lc

1p2 usingheavy quark effective theory, and concludes that the braning ratios for these two processes are similar. The decayG(Qb

1→Qc0p1) is estimated to be about 2.5G(B0

→D2p2). This prediction can be tested at future collidexperiments.

From Table IV, we see thatQb1→Qc

0p1 is proportionalto theSU(3) f invariant amplitudes1, one therefore can usthe estimate in Ref.@18# to obtain an estimate fors1. Theinvariant amplitudess2,3,4 involve more complicated topology and are much harder to estimate. Although it is difficto know all decay amplitudes, knowings1 one can makesome useful predictions. The branching ratios for procesin Tables IV and V which just depend ons1 are thereforeknown. Up to mass splitting corrections in phase space,obtain

03403

ee

h-te

t

es

e

G~Tb0→Tc

2K1!;1

4

uVusu2

uVudu2G~Qb

1→Qc0p1!,

G~Tb1→Tc

0K1!;uVusu2

uVudu2G~Qb

1→Qc0p1!.

~25!

Using the heavy quark effective theory, one can alsolate several otherSU(3) f invariant amplitudes toLb

→Lcp by realizing the fact that all amplitudes for the heapentaquark transitions of the formRb,iRc

i , Sb,i j Sci j have simi-

lar factorization structure, and their strength should alsosimilar. We therefore expect thats1;r 81;r 31;s31.

One then has, up to mass splitting corrections in phspace, the following relations:

0-7


TABLE VII. SU(3) decay amplitudes forB→NRc(Sc). The

Lorentz structure should be understood to beN(11bg5)Rc(Sc)since weak interaction can haveS- andP-wave amplitudes.

Bu decay DS50 DS521

Uc0S2 r 811 r 83 Uc

0J2 r 811 r 83

Tc0J2 s812 s82 Nc

0J2 1

A2( s812 s82)

Nc0S2

21

A2( s812 s82) Qc

0S2 2( s812 s82)

Bd decay

Uc0S0

21

A2( r 812 r 82)

Uc0J0 r 81

Uc0L 1

A6( r 811 r 82)

Dc2J2 r 83

Dc2S2 r 821 r 83 Nc

2J22

1

A2s81

Sc2J2 r 82 Nc

0J02

1

A2s82

Tc2J2

21

A2( s811 s83)

Qc0S0 1

A2( s812 s82)

Tc0J0 2( s821 s83) Qc

0L 1

A6( s811 s82)

Nc2S2 1

A2s83

Nc0S0 1

2 ( s812 s822 s83)

Nc0L 1

2A3( s811 s8213s83)

Qc0n s83

Bs decay

Uc0n r81 Uc

0S0 1

A2r 82

Sc2S2 r 83 Uc

0L 21

A6(2r 812 r 82)

Tc2S2 1

A2s81

Dc2S2 r 82

Tc0S0

21

A2s81

Sc2J2 r 821 r 83

Tc0L 2

1

A6( s8122s82)

Tc2J2

21

A2s83

Nc0n 1

A2s82

Tc0J0 2 s83

Nc2S2 1

A2( s811 s83)

Nc0S0 2

12 ( s811 s83)

Nc0L 2

1

2A3( s8122s8223s83)

Qc0n s821 s83

03403

TABLE VIII. SU(3) decay amplitudes forB→N(Rc ,Sc) with

DS50. The Lorentz structure is of the formRc(Sc)(11bg5)N.

Bu S1Uc0 ( r 311 r 321 r 611 r 62)

S0Dc2

21

A2( r 311 r 322 r 611 r 62)

LDc2

21

A6( r 312 r 3212r 332 r 612 r 62)

J0Sc2 r 322 r 331 r 62

J2Tc22 s312 s612 s62

J0Tc2 1

A2( s311 s612 s62)

S0Nc2

12 ( s3212s611 s62)

LNc2

21

2A3(2s312 s3223s62)

S1Nc0 1

A2( s321 s62)

pQc0 s321 s62

Bd S0Uc0

21

A2( r 311 r 321 r 612 r 62)

LUc0 1

A6( r 312 r 3212r 331 r 611 r 62)

S2Dc2 2( r 311 r 322 r 612 r 62)

J2Sc2 2( r 322 r 331 r 62)

J2Tc22 1

A2( s312 s611 s62)

J0Tc0 s311 s611 s62

S2Nc2 1

A2( s322 s62)

LNc0

21

2A3(2s312 s3213s62)

S0Nc0

212 ( s3222s612 s62)

nQc0 s322 s62

Bs J0Uc0 r 311 r 331 r 61

J2Dc2 2( r 311 r 332 r 61)

J2Nc2 1

A2( s311 s322 s61)

J0Nc0 1

A2( s311 s321 s61)

S0Qc0 A2s61

LQc0

2A 23 ( s311 s32)

0-8

g

ral

rela-m-

ate,

ach-

e


TABLE IX. SU(3) decay amplitudes forB→N(Rc ,Sc) withDS521. The Lorentz structure is the same as Table VIII.

Bu pUc0 r 311 r 321 r 611 r 62

nDc2 r 322 r 331 r 62

S0Sc2

21

A2( r 311 r 332 r 61)

LSc2

21

A6( r 3112r 322 r 332 r 6112r 62)

S2Tc22 2( s312 s612 s62)

S0Tc2 2

12 ( s312 s322 s6122s62)

LTc2

21

2A3( s311 s3213s61)

S1Tc0 2( s321 s62)

nNc2

21

A2( s311 s612 s62)

pNc0

21

A2( s321 s62)

Bd nUc0

( r 311 r 331 r 61)

S2Sc2 2( r 311 r 332 r 61)

S2Tc2

21

A2( s311 s322 s61)

S0Tc0 1

A2( s311 s322 s61)

LTc0

21

A6( s311 s3213s61)

nNc0

21

A2( s311 s321 s61)

Bs S0Uc0

21

A2( r 322 r 332 r 62)

LUc0

21

A6(2r 311 r 321r 3312r 612 r 62)

S2Dc2 2( r 322 r 332 r 62)

J2Sc2 2( r 311 r 322 r 612 r 62)

J2Tc2

21

A2( s322 s62)

J0Tc0

2( s322 s62)

S0Nc0 1

2 ( s312 s611 s62)

LNc0

21

2A3( s3122s3213s6113s62)

S2Nc2

21

A2( s312 s611 s62)

nQc0 2( s311 s611 s62)

03403

G~Qb1→Qc

0p1!;G~Sb0→Sc

2p1!;uVudu2

uVusu2G~Db

0→D02K1!

;uVudu2

uVcdu2@G~Ub

1→Uc0Dd!,

G~Sb0→Sc

2Dd!];uVudu2

uVcsu2@G~Ub

1→Uc0Ds!,

G~Ub1→Dc

2Ds!, G~Sb0→Dc

2Dd!]

;uVudu2

uVcdu2@G~Tb

2→Tc22Dd!,

G~Nb0→Nc

2Dd!, G~Qb1→Qc

0Dd!]

;uVudu2

uVcsu2@G~Tb

2→Tc22Ds!,

G~Tb0→Tc

2Ds!, G~Tb1→Tc

0Ds!]. ~26!

The above relations also hold for processes with replacinDby D* .

Pentaquark properties can also be studied atB factories.We have studiedB→NRc(Sc) and B→NRc(Sc) decays.From Tables VII, VIII, and IX, we see that there are severelations. For example,

G~Bu→Uc0S2!5

uVudu2

uVusu2G~Bu→UcJ

2!,

G~Bu→ pUc0!5

uVudu2

uVusu2G~Bu→S1Uc

0!. ~27!

Should the heavy pentaquarks be discovered, thesetions can also provide important information. The decay aplitudes for B→NRc(Sc) and B→NRc(Sc) are difficult toestimate. We are not able to provide any reliable estimexcept that we expect them to be smaller thanB→NLc am-plitudes.

Using the same formulation, one can also studyB decaysinto an ordinary baryonN and a light pentaquark, such asQ1 in the antidecuplet. We, however, expect that the braning ratios are smaller thanB→NN. Since B→NN havesmall branching ratios, it may be difficult to studyB→NQ1 experimentally. This situation may change if onstudies three body decays ofB with a light pentaquark in thefinal states, such asB→Dp(n)Q. TheB→DK decay has abranching ratio of order a few times 1024. The K has astrong coupling top(n)Q1 which can be determined fromthe Q1 decay. WithGQ of Q1 being about one MeV, onecan obtain a branching ratio forB→Dp(n)Q as large as1026 which is within the reach of near futureB factories.

0-9

ru

alsoans.

nd


ACKNOWLEDGMENTS

We would like to thank H.-Y. Cheng and Shi-Lin Zhu fouseful discussions. We thank H.-Y. Cheng for informingof a paper by Cheng, Chua, and Hwang@20# where some of

-

-

cz

.

.

3

gh

03403

s

the heavy pentaquark decays discussed in this paper areconsidered before its publication. We also thank M.-L. Yfor bringing his early work on pentaquark to our attentionThis work is partially supported by Grants from NSC aNNSF.

ys..,

na,,

. B

hys.

D.

,

2.

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Hadronic decays involving heavy pentaquarks