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PHYSICAL REVIEW D 1 JULY 1996VOLUME 54, NUMBER 1

Two-body Cabibbo-suppressed decays of charmed baryons into vector mesons and into photon

Paul Singer and Da-Xin ZhangDepartment of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel

~Received 18 January 1996!

The heavy quark effective theory and the factorization approximation are used to treat the Cabibbo-suppressed decays of charmed baryons to vector mesons,LC→pr0,pv, JC

1,0→S1,0f,S1,0r0,S1,0v, andJC

0→Lf,Lr,Lv. The input from two recent experimental results onLC decays allows the estimation of thebranching ratios for these modes, which turn out to be between 1024 and 1023. The long distance contributionof these transitions via vector meson dominance to the radiative weak processesLC→pg, JC→Sg, andJC

0→Lg leads to quite small branching ratios, 1026–1029; the larger value holds if a sum rule between thecoupling constants of the vector mesons is broken.@S0556-2821~96!02413-7#

PACS number~s!: 13.30.Eg, 12.40.Vv, 14.20.Lq

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The study of the charmed baryon decays has intensiduring the last few years on both the experimental and thretical levels@1#. In particular, the recent measurements wthe CLEO-II detector at the Cornell Electron Storage Ri~CESR! of the form factor ratio in semileptonic decaLC→Le1n @2# and of the branching ratio ofLC→pf, @3#provide information whose usefulness transcends the speprocesses studied in these experiments. In the present wwe focus on a group of Cabibbo-suppressed two-body nleptonic decays of the charmed baryonsLC andJC into alight baryon plus a light-flavor neutral vector meson, whiwere not treated previously, making use of information pvided by the two experiments of Refs.@2,3#.

The Cabibbo-suppressed decays of the charmed hadare described at the quark level by the effective Hamilton

Heff5 (q5d,s

GF

A2Vuq* Vcqa2qgm~12g5!qugm~12g5!c,

~1!

where a2 is a combination of Wilson coefficientsa25c1(m5mc)1c2(m5mc)/Nc @4#. In Eq. ~1! only theleading contribution in the largeNC limit is retained. In orderto evaluate decay amplitudes induced by the effective HamtonianHeff at the hadronic level we shall use the factoriztion approximation@4# and we concentrate now on thosdecays in which (qq) materializes to a vector meson. Foinstance, the decay amplitude ofLC→pf is given by

M~LC→pf!5GF

A2Vus* Vcsa2^fusgm~12g5!su0&

3^puugm~12g5!cuLC&. ~2!

While the matrix element for the vector meson is relatedits decay constant defined by

^fusgmsu0&5 i f fmfef*m , ~3!

the baryonic matrix element can be parametrized by six fofactors f i andgi ~i51,2,3! @5#:

540556-2821/96/54~1!/1225~4!/$10.00

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^puugm~12g5!cuLC&

5up~P2!F S f 1~q2!gm2 if 2~q

2!

mLC

smnqn1

f 3~q2!

mLC

qmD2S g1~q2!gm2 i

g2~q2!

mLC

smnqn1

g3~q2!

mLC

qmD g5G3uLC

~P1! ~4!

with qm5(P12P2)m .We use the heavy quark effective theory, which is consid-

ered to be especially suitable for theLC decays@6,7#. Treat-ing thec as the heavy quark, the matrix element in Eq.~4! isexpanded in 1/mC and in the following we keep only the firstterm of the expansion. Then, only two independent formfactors survive and Eq.~4! is cast into the form

^puugm~12g5!cuLC&

5up~P2!FF1~q2!1F2~q

2!P” 1mLC

Ggm~12g5!uLC~P1!. ~5!

The form factors in Eq.~4! and in Eq.~5! are then related by

f 1~q2!5F1~q

2!1mp

mLC

F2~q2!, f 2~q

2!52F2~q2!,

f 3~q2!5F2~q

2!,

g1~q2!5F1~q

2!1mp

mLC

F2~q2!, g2~q

2!52F2~q2!,

g3~q2!5F2~q

2!. ~6!

The relations in Eq.~6! are expected to hold most likely nearthe zero-recoil pointqm

2[(mLC2mp)

2.1.8 GeV2 of thesemileptonic decays. Since we need here the form factors ithe region of;1 GeV2, we shall assume it is appropriate touse a pole behavior for the extrapolation:

1225 © 1996 The American Physical Society

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1226 54BRIEF REPORTS

f i~q2!5 f i~qm

2 !12qm

2 /MD*2

12q2/MD*2 ,

~7!

gi~q2!5gi~qm

2 !12qm

2 /MD1

2

12q2/MD1

2 ,

where MD*52.007 GeV andMD152.423 GeV are the

masses of the lowest-lying mesons which interpolate the vtor and the axial vector currents, respectively.

Now, we turn to the experimental information@2# on theform factor ratio in LC→Le1n. Although the absolutevalue ofF1 , F2 defined in Eq.~5! is not measured yet, theiratio is determined to be

R[F2

F1520.2560.16 ~8!

in the semileptonic decay. We remark that in view of tlimited statistics, it was found in@2# that this result is notsensitive to theq2 behavior assumed for the form factors.

Now, we assume that as a consequence of the SU~3!-flavor symmetry for the light quarks the ratio~8! holds alsofor the matrix elements of the decaysLC→p andJC→S,L. Hence, we are now in the position to derive thdecay amplitude~2! in the approximation of Eq.~5! and wearrive at

M~LC→pf!5GF

A2Vus* Vcsa2i f fmfef*

mup~P2!

3@gm~a2bg5!12~x2yg5!P1m#uLC~P1!,

~9!

where

a5 f 1~mf2 !1

mp1mLC

mLC

f 2~mf2 !,

b5g1~mf2 !1

mp2mLC

mLC

g2~mf2 !,

~10!

x521

mLC

f 2~mf2 !,

y521

mLC

g2~mf2 !.

Similiar formulas hold when thef meson is replaced by ar0 or av meson in the final state, where an additional fac1/A2 arises due to the quark content of the mesonr0 or v .In the process ofJC

0→Lf ~or r0,v) there exists anothefactor A1/6 to account for the difference of the flavor-spsuppression for the light quarks@8#.

We proceed now to calculate the decays listed in TablFirst, we use the experimental data of CLEO@3# which mea-sures the branching ratio B(LC→pf)5(1.0660.33)31023 as an input, thus determining the unknowproduct ua2F1(mf

2 )u. This permits one to calculate

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with the model the transitions LC→pr0,pv,JC

1,0→S1,0f,S1,0r0,S1,0v, and JC0→Lf,Lr,Lv.

Since the experimental uncertainty in Eq.~8! is large, wepresent in Table I the values forR520.09, R520.25,R520.41. It turns out that the dependence on the ratioR inthe considered range is weak, the variation in the calculatbranching ratios being less than 10%. The uncertainty in thresults of Table I is then due solely to the precision achievein the determination ofB(LC→pf) @3#. The partial decaywidths are calculated using the helicity representation of thamplitudes@9# and thus we included in Table I also the predictions of the model for transverse/longitudinal ratios inthese decays, which are found to vary about 20% in thrange considered forR. In the calculations we used forf V(V5r,v or f! the values determined from the leptonicdecays of the vector mesons@10#, f r

250.047 GeV2,fv250.038 GeV2, f f

250.055 GeV2.As an alternative, we could have refrained from usin

LC→pf as input, attempting to calculate it as well, by usinga2520.5560.1 from the overall fit@4# to nonleptonicDand Ds decays, and assuming a ‘‘reasonable’’ value fof 1(qm

2 ). With the above value fora2 and taking, for ex-ample, f 1(qm

2 )50.960.1, we find B(LC→pf)5(0.7460.32)31023 ~for R520.25). This is in good agreementwith the measured value@3# and gives strong support to theapproach presented here.

In view of the appropriateness of the model discussehere for decays of the charmed baryons to vector mesons,

TABLE I. Predictions on branching ratios and the ratio of polarized decay ratesGL /GT .

Process R520.09 R520.025 R520.41

LC→pf B 1023~input! 1023~input! 1023~input!GL /GT 1.18 1.29 1.42

LC→pr0 B 0.4231023 0.4431023 0.4731023

GL /GT 2.24 2.50 2.76LC→pv B 0.3431023 0.3631023 0.3831023

GL /GT 2.17 2.42 2.67

JC1→S1f B 1.831023 1.831023 1.731023

GL /GT 1.04 1.14 1.25JC

1→S1r0 B 0.8031023 0.8231023 0.8331023

GL /GT 2.06 2.29 2.55JC

1→S1v B 0.6531023 0.6631023 0.6731023

GL /GT 1.98 2.21 2.46

JC0→S0f B 0.5231023 0.5031023 0.4831023

GL /GT 1.04 1.14 1.25JC

0→S0r0 B 0.2331023 0.2331023 0.2431023

GL /GT 2.06 2.29 2.55JC

0→S0v B 0.1931023 0.1931023 0.1931023

GL /GT 1.98 2.21 2.46

JC0→Lf B 0.9331024 0.9231024 0.9031024

GL /GT 1.19 1.31 1.44JC

0→Lr0 B 0.3931024 0.4031024 0.4131024

GL /GT 2.28 2.54 2.83JC

0→Lv B 0.3131024 0.3231024 0.3431024

GL /GT 2.20 2.46 2.74

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54 1227BRIEF REPORTS

consider also its application to radiative weak decays in cjunction with vector meson dominance~VMD !. Since it isclear now @11# that the short distance contribution fromc→ug to the radiative decays of charmed particles is neggible, it is important to devise reliable models for the londistance one. We adopt here the model which has beenployed recently by Deshpandeet al. @12# and by Eilamet al.@13# to estimate the long distance ‘‘t-channel’’ contributionof the vector mesons to the radiative transitionsb→sg ands→dg ~see also Ref.@14#!. Using now for the charm sectothe derivation steps outlined in@12,13#, the appropriate partof the effective Hamiltonian for the radiative weak decays

HeffVMD52

eGF

A2a2

1

mC@Vus* Vcs~2 1

3 f f2 !1Vud* Vcd

3~2 12 f r0

21 1

6 f v2 !#eg*

mqnusmn~11g5!c. ~11!

To treat the newly encountered matrix elementsmn(11g5), we use the heavy quark effective scheme wthec quark only treated as heavy. Then, similiarly to Cheet al. @15#, we obtain

^puusmn~11g5!cuLC&5up~P2!FF1~q2!1F2~q

2!P” 1mLC

G3smn~11g5!uLC

~P1!. ~12!

Next, we use again the measured result ofB(LC→pf)@3# and ofR, @2# in order to calculate the VMD contribution@12,13,16# to the processesLC→pg, JC

1,0→S1,0g, andJC

0→Lg. We are still faced, however, with the questionthe q2 dependence of thef V(q

2) couplings. It is customaryto assume that theq2 variation of thef r , f v is small and onemay take for the vector-meson-photon couplings atq250,f r(mr

2). f r(0), f v(mv2 ). f v(0). We may reasonably as-

sume also f f(mf2 ). f f(0). Since in Eq. ~11! we use

Vud* Vcd.2Vus* Vcs , the amplitude for the processes consiered is proportional to the quantityCVMD8 [2 1

3f f21 1

2f r02

216f v

2 . It has already been pointed out in Ref.@13# that byusing the values determined in the leptonic decaysf V(0), anear cancellation occurs inCVMD8 . This cancellationis due to the combination of SU(3)-flavor symmetry with theGlashow-Iliopoulos-Maiani~GIM! relation and occurs at alevel below 10%. As a result, all the considered radiatdecays are reduced by more than two orders of the matude. Denoting the suppressed branching ratios obtafrom the combined contributions off, v, r by the subscriptSR ~sum rule!, we find

B~LC→pg!SR51.831029,2.331029,3.131029, ~13!

B~JC1→S1g!SR53.531029,4.531029,6.231029,

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B~JC0→S0g!SR51.031029,1.331029,1.831029,

B~JC0→Lg!SR50.1631029,0.2131029,0.2831029

for R520.09, 20.25, or20.41. On the other hand, thepossibility exists that a certain variation occurs inf V(q

2)betweenq250 and q25mV

2 . For instance, there is strongevidence thatf c(q

2) varies considerably betweenq25mc

andq250, f c2 being reduced in this range by a factor of

@12,13#. Hence one should also consider the possibility ththe above-mentioned cancellation is avoided. Since thereno accurate model for thef V(q

2) variation, we take as alter-natives the rates resulting if only ther↔g is considered.This leads to

B~LC→pg!r50.7331026,0.9331026,1.331026,

B~JC1→S1g!r51.431026,1.731026,2.531026,

~14!

B~JC0→S0g!r50.4131026,0.5331026,0.7331026,

B~JC0→Lg!r50.6531027,0.9731027,1.131027

for R520.09, 20.25, or 20.41. These figures may becompared with a previous calculation@17# using bremsstrah-lung fromW-exchange diagrams on the quark level. Brancing ratios of the order of 1025 were obtained for these pro-cesses, however, as the authors pointed out, launcertainties are involved.

In summary, we have presented the first calculatiof the Cabibbo-suppressed processesLC→pr0,pv,JC

1,0→S1,0f,S1,0r0,S1,0v, andJC0→Lf,Lr,Lv by us-

ing the factorization approximation, heavy quark effectivtheory, and experimental input from recent experiments. Wfound branching ratios between 1024 and 1023 for thesemodes, which bring their detection into the realm of feasibity in the near future. The model accounts correctly for thobservedLC→pf. Then, using vector meson dominance fothe long distance contribution, we calculated the radiatiprocessesLC→pg, JC

1,0→S1,0g, and JC→Lg in thesame model. If a certain GIM-type sum rule holds for thvector-meson-photon couplings, these transitions astrongly suppressed to the level 1028–1029. Otherwise, in-dividual vector mesons contribute branching ratios of thorder of 1026. Detection or limits on these modes woulthus test the validity of interesting theoretical models.

This research was supported in part by Grant No. 5423-96 from the Ministry of Science and the Arts of Israel. Thwork of P.S. has also been supported in part by the FundPromotion of Research at the Technion. We thank ProfesG. Eilam for helpful remarks.

ys.

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1228 54BRIEF REPORTS

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