weak decay of l in nuclei quarks versus mesons.pdf

Ž .Nuclear Physics A 669 2000 331–350

www.elsevier.nlrlocaternpe

Weak decay of L in nuclei: quarks versus mesons

Kenji Sasaki, Takashi Inoue, Makoto Oka

Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551 Japan

Received 21 June 1999; received in revised form 21 September 1999; accepted 8 October 1999

Abstract

Decays of L in nuclei, non-mesonic modes, are studied by using the LN™NN weaktransition potential derived from the meson exchange mechanism and the direct quark mechanism.The decay rates are calculated both for the L in symmetric nuclear matter and light hypernuclei.

Ž ).We consider the exchange of six mesons p , K ,h,r,v, K . The form factor in the mesonexchange mechanism and short-range correlation are carefully studied. q 2000 Elsevier ScienceB.V. All rights reserved.

PACS: 13.30.-a; 21.80.qa; 24.85.qpKeywords: Hypernuclei; Weak interaction; Direct quark

1. Introduction

The L particle is a neutral baryon with spin 1r2 and strangeness y1. It is unstableand has a finite lifetime of about 263 ps. The main decay modes of a free L particle arethe pionic ones,

L™pqpy

™nqp 0 ,

where the final state nucleon has the momentum of about 100 MeVrc. This is one of theDSs1 non-leptonic weak interactions. An important feature of this free L decay is theempirical DIs1r2 rule. The experimental branching ratio of two charge modes,Ž .y 0G rG ,1.78, is close to the theoretical one with the assumption of DIs1r2p p exp

Ž .DIs1r2y 0dominance, G rG s2. This fact tells us that the decay is dominated by thep p theory

DIs1r2 transition. This DIs1r2 dominance is a prominent feature of all theobserved DSs1 non-leptonic weak interactions. However, in the standard theory ofelectro-weak interaction, there are both DIs1r2 and DIs3r2 components. Thus theeffect of strong interaction should be significant for the explanation of enhancement of

w xthe DIs1r2 contribution 1 .

0375-9474r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9474 99 00427-3

( )K. Sasaki et al.rNuclear Physics A 669 2000 331–350332

The nucleus with the L hyperon is called L hypernucleus, and is expressed as A ZL

where Z is the number of proton and A is the mass number. The L hyperon in anucleus falls into the ground state due to the absence of the Pauli blocking effect and,therefore, is regarded as a probe of the nuclear interior. Since the L is not a stableparticle, the hypernucleus also decays via the weak interaction, which is the so-calledweak decay of the hypernucleus. It is known that, when the L hyperon is in the nuclearmedium, the pionic decay mode is strongly suppressed and instead the nucleon induceddecay modes

Lqp™nqp : proton induced decay, 1Ž .

Lqn™nqn: neutron induced decay 2Ž .

become dominant. This can be understood by considering the Pauli principle for thefinal state nucleons, as for the mesonic decay the momentum of the nucleon is less thanthe Fermi momentum in nuclear matter. In the nucleon induced decay the momentum of

Žthe outgoing nucleons is about 400 MeVrc assuming that the relative momentum of the.initial L and N is zero . It is much larger than the Fermi momentum, k ,270 MeVrc.F

The LN™NN transition is the DSs1 reaction analogous to the weak NN interac-tion or the parity violating part of the NN force. Although the weak NN interaction ismasked by the overwhelming strong interaction, the LN™NN transition is inducedpurely by the weak interaction, and therefore it gives us a unique chance to study themechanism of baryon–baryon weak interaction. At present, the direct observation of theLN™NN process is almost impossible because of the lack of a hyperon beam or target.Thus the weak decay of the L hypernucleus will be one of the clues of research for theLN™NN reaction. Many theoretical and experimental efforts have been devoted to the

w xnon-mesonic decays of light and heavy hypernuclei 2–25 . The mechanism of non-mesonic weak decay is still not clear, especially the theoretical prediction of the nrpratio, which is the ratio of the neutron-induced decay to the proton-induced one, is notcompatible with the experimental ration. This is because the theoretical value of theproton-induced rate is much larger than the experimental one, which is measured to agood precision. The value of the nrp ratio will be a key to understanding the mechanismof non-mesonic decay.

The conventional picture of the two-baryon decay process, LN™NN, is theone-pion exchange between the baryons, where LNp vertex is induced by the weak

w x Ž .interaction 2,6–9 see Fig. 1 . In LN™NN, the relative momentum of the final statenucleon is about 400 MeVrc. The nucleon–nucleon interaction at this momentum is

Fig. 1. Non-leptonic decay of the hyperon.

( )K. Sasaki et al.rNuclear Physics A 669 2000 331–350 333

dominated by the short-range repulsion due to heavy meson exchanges andror to quarkexchanges between the nucleons. It is therefore expected that the short-distance interac-tions will contribute to the two-body weak decay as well. Exchanges of K , r, v, K )

mesons and also correlated two pions in the non-mesonic weak decays of hypernucleiw xhave been studied 3–5,10–13,19,20 and it is found that the kaon exchange is

w xsignificant, while the other mesons contribute less 13 .w xSeveral studies have been carried out of the effects of the quark substructure 14–18 .

w xIn our recent analyses 16–18 , we employed an effective weak Hamiltonian for quarks.It was pointed out that the DIs1r2 part of the Hamiltonian is enhanced duringdownscaling of the renormalization point in the renormalization group equation. Yet asizable DIs3r2 component remains in the low energy effective weak Hamiltonian.We proposed to evaluate the effective Hamiltonian in the six-quark wave functions ofthe two baryon systems and derived the ‘‘direct quark’’ weak transition potential for

w x Ž .LN™NN 16–18 . See Fig. 2. Our analysis shows that the direct quark contributionlargely improves the discrepancy between the meson-exchange theory and experimentaldata for the nrp ratios for light hypernuclei. It is also found that the DIs3r2component of the effective Hamiltonian gives a sizable contribution to the Js0

w xtransition amplitudes 15 . Unfortunately, we cannot determine the DIs3r2 amplitudesw xunambiguously from the present experimental data 21–24 .

The aim of this paper is to combine the long-range meson exchange interactions andthe short-range direct quark transitions coherently in the study of L decay in nuclearmatter and in light hypernuclei. We first consider p and K exchanges and furtherintroduce the heavier mesons. The direct quark mechanism represents the short-rangepart of the transition and may therefore replace the heavy meson exchanges. Wecompare the results of light q heavy meson exchanges with the results with lightmesons q direct quark. For the meson exchange, we particularly study the effects of theform factors on the transition rates. The pion exchange amplitudes are found to besensitive to the choice of the form factors. We propose to use soft pion–baryon formfactors so that the strong tensor transition is suppressed. The direct quark mechanismyields a large neutron-induced amplitude and therefore improves the nrp ratio. How-ever, we find that its enhancement is not large enough to explain the observed large nrpratio.

This paper is organized as follows. In Section 2 we present the transition potential forthe LN™NN transition. In Sections 3 and 4 the transition potential is applied to thedecay in nuclear matter and light hypernuclei, respectively. Discussions and conclusionare given in Section 5.

Fig. 2. LN™ NN transition.


2. Theoretical approach to LN™NN

Since the LN™NN transition involves a large energy transfer, the short-rangereactions may contribute significantly. Thus we expect that the substructure of the

w xbaryons, or the quark degrees of freedom play important roles in this transition 14–18 .In this paper we assume the hybrid mechanism to describe LN™NN decay, meson

Ž . Ž .exchange mechanism ME and direct quark mechanism DQ . Such a combination hasbeen successfully employed in describing the strong baryon–baryon interactions.

2.1. Meson exchange mechanism

Ž .The one-pion exchange OPE process, where the emitted virtual pion from the weakL™Np vertex is absorbed by a nucleon in the nucleus, shown in Fig. 3, is studied to

w xdescribe LN™NN decay 2,6–9 . This mechanism provides the simplest but indispens-able picture because it is based on the main decay mode of the free L. This mechanismyields a long-range transition potential, which is obtained by evaluating the diagramusing the following vertices:

pHH s ig c g tPf c , 3Ž .s NNp N 5 p N

p 2HH s iG m c A qB g tPf c . 4Ž . Ž .w F p N p p 5 p L

Both the strong coupling constant, g , and the weak ones, A and B , are wellNNp p p

determined from experiment. Their values are listed in Table 1. It is known that the OPEtransition potential enhances the Js1 transition due to the strong tensor term andtherefore the amplitude of the proton-induced mode is much larger than the neutron-in-duced one. This tensor dominance causes the nrp ratio problem, namely, the observed

Ž .nrp ratio ,1 is far greater than the OPE prediction.While the OPE is significant for the long-distance baryon–baryon interaction, the

short-range reaction mechanism is also important in LN™NN due to the large energytransfer involved. Within the meson exchange model, shorter-range contributions may

w xcome from the exchange of the heavier mesons 3–5,10–13 and the correlated two-pionw x w xmeson 19,20 . In Ref. 13 the authors consider the exchange of all the octet pseu-

doscalar and vector mesons, p , K , h, r, v, and K ). Although the LNp weakcoupling constant is determined phenomenologically, all the other weak couplings in

w x Ž . w xRef. 13 are estimated theoretically by assuming the SU 6 symmetry 12 , while forw

Fig. 3. Meson exchange diagram for LN™ NN decay. m denotes the weak vertex for the non-strange mesonsand the strong vertex for the strange mesons.


Table1The strong and weak meson baryon coupling constants. The strong coupling constants are taken according to

Ž . w x w xthe Nijmegen potential soft core 26,27 . The weak coupling constants are estimated by Ref. 13 . They are inunits of G m2 s2.21=10y7. The cutoff parameters for the DP form factor are also givenF p

w xMeson Strong c.c. Weak c.c. L MeV

PC PV soft hard

p g s13.3 B sy7.15 A s1.05 800 1300NNp p p

g s12.0LSp

h g s6.40 B sy14.3 A s1.80 1300 1300NNh h h

g sy6.56LLhPC PVK g sy14.1 C sy18.9 C s0.76 1200 1200LNK K KPC PVg s4.28 D s6.63 D s2.09NS K K K

Vr g s3.16 a sy3.50 e s1.09 1400 1400NNr r rTg s13.1 b sy6.11NNr rVg s0LSrTg s11.2LSrVv g s10.5 a sy3.69 e sy1.33 1500 1500NNv v vTg s3.22 b sy8.04NNv vVg s7.11LL vTg sy4.04LL v

) V PC,V PV) ) )K g sy5.47 C sy3.61 C sy4.48 2200 2200LNK K K

T PC,T) )g sy11.9 C sy17.9LNK K

V PC,V PV) ) )g sy3.16 D sy4.89 D s0.60NS K K K

V PC,T) )g sy3.16 D s9.30NS K K

Žthe strong vertices the couplings are taken from the Nijmegen YN potential model soft. w xcore 26,27 .

The strong vertices are given by

KHH s ig c g f c , 5Ž .s L NK N 5 K L

hHH s ig c g f c , 6Ž .s NNh N 5 h N

gTNNrr V m mn rHH sc g g q i s q tPf c , 7Ž .s N NNr n m Nž /2 M

g )

T) )LNKK V m mn K

)HH sc g g q i s q f c , 8Ž .s N L NK n m Lž /2 M

gTNNvv V m mn vHH sc g g q i s q f c 9Ž .s N NNv n m Nž /2 M

while the weak vertices are parameterized as

K 2 PV PC † PV PC †HH s iG m c C qC g f c qc D qD g c f ,Ž . Ž . Ž .Ž .w F p N K K 5 K N N K K 5 N K ss

10Ž .

h 2HH s iG m c A qB g f c , 11Ž .Ž .w F p N h h 5 h L


s mnqnr 2 m m rHH sG m c a g yb i qe g g tPc c , 12Ž .w F p N r r r 5 m Lž /2 M

) ) )K 2 PC ,V K † m PC ,V m K †) )HH sG m C c f g c qD c g c fŽ . Ž .w F p K N m N K N N ms sž

mn mns q s q)n nPC ,V ) † PC ,V K †

) )yi C c K NqD c c fŽ . Ž .K N m K N N ms s2 M 2 M

) )PV K † m PV m K †) )q C c f g g c qD c g g c f , 13Ž .Ž . Ž .K N m 5 N K N 5 N m s /

s mnqnv 2 m m vHH sG m c a g yb i qe g g f c . 14Ž .w F p N v v v 5 m Lž /2 M

We choose the convention that the three-momentum transfer q is directed towards thestrong vertex, and we assign the spurious isospin state to the L field which behaves like

01 1 N< : Ž .an isospin y . The same isospurion is also employed to c sn andŽ . s2 2 1K Ž 0.f sK . The value of these couplings are given in Table 1. We do not present thes

w xexplicit form of the potential here, because it is given in detail in Ref. 13 .The potential is given by the Fourier transformation of the product of the propagator

and the vertices in the Breit–Fermi reduction. As the baryons and also the mesons havefinite sizes and structures, we need to take a form factor at each meson-baryon vertex

w x Ž 2 .into account 28 . If we assume the same form factor F q for both the strong and weakvertices, the potential is given as

d3q eiqPr2 2V r s OO q F q , 15Ž . Ž . Ž .Ž .H 3 2 2 2q qm yq2pŽ . 0

where OO is a product of the vertex operators and the coupling constants. The operator OO

may include the isospin operator,

y3 Is0² :INt Pt N I s1 2 ½ 1 Is1

In this case we use the initial state which is antisymmetrized in the flavor space.2Ž 2 .A standard choice of F q is the square of the monopole form factor,

22 2L ymDP2 2F q s 16Ž .Ž .DP 2 2ž /L qqDP

Ž .which we call ‘‘double pole’’ DP form factor, while in the literature the simple form

L2 ym2SP2 2F q s 17Ž .Ž .SP 2 2L qqSP

Ž .is often used, which we call ‘‘single pole’’ SP form factor. The cutoff parameter L ischosen for individual mesons independently. These form factors are normalized at the


2 Ž 2 . 2 Ž 2 .on-mass-shell point as F ym sF ym s1. There is another type of formDP SPŽ .factor, ‘‘Gaussian’’ G ,

q22 2F q sexp y , 18Ž .Ž .G 2ž /L

which has the advantage of consistency with the quark structure for the baryon, if thecutoff parameter is taken according to the size of the quark distribution in the baryon.

2Ž .This form factor is normalized as F 0 s1. It should be noted that the SP and DP formG

factors give similar effects. Near the mass shell point, F 2 and F 2 are expanded asSP DP

L2 ym2 q2 qm2SPSPF s s1y , 19Ž .2 2 2 2ž /L qq L qqSP SP

22 2 2 2L ym q qmDPDPF s ,1y2 . 20Ž .2 2 2 2ž / ž /L qq L qqDP DP

'Therefore, the SP and DP form factors are identical if we set L , 2 L byDP SP

assuming L24m2. On the other hand, the G form factor behaves differently at short

distances. Fig. 4 shows the behaviors of the potential for several form factors in the LN:3S –NN: 3D OPE transition at the relative momentum of k s1.97 fmy1. As can be1 1 r

seen clearly, the potential has a node for the G form factor, while the others do notbehave like that. This oscillation suppresses the tensor transition although the long-dis-

Fig. 4. The OPE potentials for the 3S ™3D for various form factors.1 1


tance behavior is similar to the ‘‘ hard ’’ DP form factor or that without the form factor.We test these three form factors and compare the results.

Ž .We cannot neglect the finite energy transfer q in Eq. 15 . In the case of typical0

nucleon–nucleon interaction, energy transfer is small and negligible, but in the case ofLN™NN this energy transfer is not negligible because the mass difference between L

and N causes a large q . If we assume that the initial L–N pair has a low kinetic0Ž .energy, this energy transfer is regarded as constant, q , m ym r2,88 MeV.0 L N

Accordingly we introduce effective masses of the exchanged mesons, m̃ i22(s m y m ym r4 and perform following replacement:Ž .i L N

1 1™ . 21Ž .2 2 2 2 2q qm yq q q m̃0

This effect leads to measurable changes. For example, the pion mass is reduced by about25%, and the range of OPE becomes longer. Therefore the total decay rate is enhancedby about 10%.

2.2. Direct quark mechanism

Studies of nuclear forces in the quark cluster model have revealed that the short-rangerepulsion between nucleons is explained in terms of the quark exchange interactionsw x29–33 . The range of the strong repulsion is about 0.5 fm and corresponds to amomentum transfer of about 400 MeVrc. The same momentum transfer is required inthe LN™NN transition in order to satisfy energy–momentum conservation. It istherefore expected that the quark substructure of the baryons plays a significant role alsoin the short-range part of the LN™NN interaction. Recently, we proposed the direct

Ž .quark DQ mechanism, in which the weak interaction between quarks in baryons causesŽ . w xthe decay without exchanging mesons Fig. 5 16,17 .

The weak interaction among constituent quarks is described as a DSs1 effectiveweak Hamiltonian, which consists of various four-quark weak vertices derived in therenormalization group approach to include QCD corrections to the su™ud transition

w xmediated by the W boson 34–37 . Such an effective weak Hamiltonian has beenapplied to the decays of kaons and hyperons with considerable success. Our aim is toexplain the short-range part of the weak baryonic interaction by using the sameinteraction so that we are able to confirm the consistency between the free hyperondecays and decays of hypernuclei.

Fig. 5. Direct quark mechanism for non-mesonic decay of hyperon. The double line indicates the strange quark,and m stands for the weak vertex.


The transition potential is calculated by evaluating the effective Hamiltonian in thew xnon-relativistic valence quark model 16,17 . The valence quark wave function of the

two-baryon system is taken from the quark cluster model, which takes into account theantisymmetrization among the quarks. Then the obtained transition potential has been

w xapplied to the weak decay of light hypernuclei 18 .w xIn Refs. 16,17 , the DQ transition potential is calculated for all the two-body

Ž . Ž X .channels with the initial LN Ls0 and the final NN L s0,1 states. The explicitforms of the transition potential are given in the momentum space representation. In thepresent analysis, we employ the coordinate space formulation, so that realistic nuclearwave functions with short-range correlations can be easily handled. The DQ transitionpotentials in the coordinate space contain non-local terms as a result of the quarkantisymmetrization. It also has terms with a derivative operator acting on the initialrelative coordinate. Thus the general form of the transition potential is

L LX X ² X X < X < :XV r ,r s NN :L S J V r ,r LN :LSJŽ . Ž .DQ SS J

d ryrXd ryrXŽ . Ž .

XsV r qV r E qV r ,r ,Ž . Ž . Ž .loc der r nonyloc2 2r r22Ž .

Ž X. Ž .where r r denotes the relative radial coordinate of the initial final two-baryon state,and E stands for the derivative of the initial LN relative wave function. The explicitr

Ž . Ž . Ž X.forms of V r , V r , and V r,r , are given in Appendix A. It should beloc der nonyloc

noted here that the DQ potential does not contain the tensor terms as we truncated theŽ 2 .non-relativistic expansion at the O p . We expect that the tensor component of the DQ

transition is negligibly small.Ž . Ž .Combining the direct quark DQ and meson exchange ME potentials, we obtain the

total transition potential as

d ryrXŽ .X XME DQV r ,r sV r qV r ,r . 23Ž . Ž . Ž . Ž .2r

The relative phase between ME and DQ is fixed so that the weak quark Hamiltonianw xgives the correct amplitude of the L™Np transition 18 . Note that the relative phases

Ž .among the various meson exchange potentials are determined according to the SU 6 w

symmetry.

3. Nuclear matter calculation

Here we study the LN™NN decay in nuclear matter in order to investigate ourapproach to LN™NN with minimizing ambiguity from the model wave function. Ontop of that, the recent experimental data of life-times of heavy hypernuclei indicate thatthe non-mesonic decay rate of L in heavy hypernuclei is only about 20% larger than the

w xfree L decay rate 8 . This saturation suggests the short-range nature of the LN™NNdecay. We study the L decay in nuclear matter as an approximation to the heavy L

hypernuclei.


Ž .We assume that nuclear matter is symmetric N sN and the L is at rest in it. Then p

Fermi momentum k is chosen as 270 MeVrc. Under this assumption the initial stateF

density is given as

d3kkF 1= = , 24Ž .Ý Ý ÝH 230 2pŽ . p ,nS SN L

where S and S are the spin of nucleon and L hyperon respectively. Similarly, theN L

final state density is

d3k d3k1 2 4 42p d EMC , 25Ž . Ž . Ž .ÝH 3 32p 2pŽ . Ž . S SN N1 2

Ž .where the d EMC stands for the d function coming from energy–momentum conserva-tion. Then the LN™NN decay rate in symmetric nuclear matter becomes

d3k d3k d3kk 1 2F 4 4G s 2p d E. M .CŽ . Ž .H H HNM 3 3 30 2p 2p 2pŽ . Ž . Ž .

=1

2² :N C k ,k NVNC k ,0 N . 26Ž . Ž . Ž .Ý Ý f 1 2 i2 p ,nspin

Ž .:where NC k,0 stands for the initial LN state with the momentum of the nucleon k,iŽ .:and NC k ,k stands for the final two nucleon state with the momenta k and k ,f 1 2 1 2

respectively.We employ a plane wave with short-range correlation for the wave functions of both

the initial and final states. The short-range correlation represents the short-rangerepulsion between two baryons. Therefore the configuration space wave function for thetwo-baryon system with total momentum K and relative momentum k is

1 Sq IiK PR ikPr yik Pr² :R,rNC K,k s e e y y1 e x x f r , 27Ž . Ž . Ž . Ž .Sm IIS 3'2

Ž .where f r is the correlation function and S and I are the spin and isospin of thetwo-baryon system, respectively. We apply the correlation function proposed in Ref.w x13 . For the initial LN, we employ

nr 2 r 2y 2 yf r s 1ye qbr e 28Ž . Ž .a2 c2ž /i

w y2 xwhere as0.5 fm, bs0.25 fm , cs1.28 fm, ns2, and for the final NN

f r s1y j q r , 29Ž . Ž . Ž .f 0 c

where q s3.93 fmy1. The initial state correlation is obtained from a microscopic finitec

nucleus G-matrix calculation, and the final state correlation gives a good description ofthe nucleon pairs in 4 He.

In the present calculation, we only consider the relative Ls0 for the initial L–Nsystem. Then the possible transition channels are those given in Table 2. The decay rateis decomposed as

G sG qG , 30Ž .NM p n


Table2Possible 2 Sq1L channels for the LN™ NN transitions. I stands for the total isospin of the final state. PCJ f

and PV indicate the parity conserving and parity violating channels respectively1 1S ™ S : I s1 a a : PC0 0 f p n1 3S ™ P : I s1 b b : PV0 0 f p n3 3S ™ S : I s0 c : PC1 1 f p3 3S ™ D : I s0 d : PC1 1 f p3 1S ™ P : I s0 e : PV1 1 f p3 3S ™ P : I s1 f f : PV1 1 f p n

where G and G stand for the proton- and neutron-induced decay rates respectively,p n

and are given as

m mkN F 2 2 2 2 2 212 < < < < < < < < < < < <G s k k dk a q b q3 c q3 d q3 e q3 f ,H ž /p 0 p p p p p p45 3 02p mŽ .31Ž .

m mkN F 2 2 212 < < < < < <G s k k dk a q b q3 f , 32Ž .Ž .Hn 0 n n n45 3 02p mŽ .

k 22k 'm m ym q ,Ž .0 N L N 2m

mLm' ,

m qmN L

where a, b, . . . , f are the matrix elements of the transition potential, for example,

²1 1 :w x w xa ' S : pn NVN S : L p . 33Ž .Is1 sp 0 0

3.1. Form factor dependence

As the L N ™ NN scattering involves a large momentum transfer,2(m m ym c ;400 MeVrc, the amplitudes are rather sensitive to the form factor.Ž .N L N

Here we show the dependence on the form factor in the meson exchange potential. First,we examine the form factor of the OPE potential. Table 3 shows the calculated decayrate when we employ only the OPE-induced LN™NN potential. The SP form has

w xoften been used in the literature 3–5 . McKellar and Gibson employed the SP formfactor with L2 s20m2 , or L ;630 MeV. This form factor is based on theSP p SP

dispersion relation analysis with a semi-pole approximation to the p NN form factor. Itis a very soft form factor, which cuts off the short-range part rather drastically and theresulting decay rate becomes small. The tensor transition, L p: 3S ™pn: 3D , is most1 1

affected by such a form factor. We find that the tensor transition amplitudes are reducedby a factor two or more by the form factor. Therefore, the total decay rate is muchreduced, as shown in Table 3.


Table3Ž Ž y1 2 .y1 .Decay rates of L in nuclear matter in units of G s 263=10 s for various choices of the couplingL

form factors in the OPE mechanism

Total G G G rG PVrPCp n n p

p no-f.f. 2.819 2.571 0.248 0.097 0.358SP L s630 MeV 1.103 0.989 0.114 0.116 0.408p

L s920 MeV 2.332 2.116 0.216 0.102 0.376p

DP L s1300 MeV 2.575 2.354 0.221 0.094 0.337p

L s800 MeV 1.850 1.702 0.148 0.087 0.282p

G L s680 MeV 1.514 1.129 0.386 0.342 0.580p

The DP form factor is popular in meson-exchange potential models of the nuclearw x Ž .force. The Bonn potential 39 , for instance, employs the DP with L p s1300 MeV.DP

This form factor is extremely hard so that the short-range part of the meson exchangepotential becomes relevant. The DP form factor is regarded as the product of themonopole form factors at the two p BB vertices. Lee and Matsuyama carried out an

w xanalysis of the NN™NNp processes and pointed out 40 that a soft form factor, suchas L-750 MeV, is preferable. Recent analyses in the QCD sum rule also suggests a

w xsoft p NN form factor of cutoff L,800 MeV 41 . From the quark substructure pointof view, it is also natural that the cutoff L is of order ;1rR ;400–500 MeV. Tableh

Ž Ž . . Ž3 shows the comparison of the ‘‘ hard ’’ DP, L p s1300 MeV and ‘‘ soft ’’ DP,DP.L s800 MeV pion-exchange form factors for the LN™NN decay rates. We findDP

that the soft form factor reduces the decay rates by about 40%, while the hard onechanges only by (10%.

The G form factor is related to the Gaussian quark wave function of the baryon.Ž .Corresponding to the size parameter b,0.5 fm, we employ L p ,680 MeV becauseG

'the relation between the size and cutoff parameters is Ls 3 rb.The above sensitivity to the choice of the form factor is quite annoying because it is

not easy to judge which of the form factors is the right one. It should also be noted thatthe weak vertex form factor can be different from the strong one, although the poledominance picture of the parity-conserving weak vertex leads to the identical formfactor to the strong one. In the meson-exchange potential models of the nuclear force,there seems to be a tendency to choose a hard form factor because the heavy mesonexchanges are often important for spin-dependent forces. On the other hand, the quarkmodel approach to the short-range nuclear force gives significant spin dependenciescomparable to the heavy meson exchanges and therefore the meson exchanges can becut off rather sharply with a soft form factor. In the present approach to the weakLN™NN interaction, whose short-range part is described by the direct quark mecha-nism, we thus may follow the quark model approach to the nuclear force and take the‘‘ soft ’’ form factor for OPE potential as a standard. And we assume that the heavy

Ž ) .meson K , h, r, v, K exchanges have similar DP form factors with the cutoff givenw xby the Julich potential 38 .¨

3.2. Decay rates

The calculated decay rates of L in nuclear matter for several models are listed inTable 4. We show not only the results of the soft DP form factor but also the results of


Table4Ž . )Non-mesonic decay rates of L in nuclear matter in units of G . The ‘‘all’’ includes p , K , h, r, v, and KL

meson exchanges. Some recent experimental data for medium-heavy hypernuclei are also given

Total G G G rG PVrPCp n n p

p DP hard 2.575 2.354 0.221 0.094 0.337DP soft 1.850 1.702 0.148 0.087 0.282no-f.f. 2.819 2.571 0.248 0.097 0.358

p q K DP hard 1.099 1.075 0.024 0.022 0.631DP soft 0.695 0.674 0.021 0.031 0.632no-f.f. 1.143 1.111 0.032 0.028 0.745

all DP hard 1.672 1.571 0.101 0.064 1.468DP soft 1.270 1.152 0.117 0.101 1.537no-f.f. 1.744 1.704 0.040 0.024 2.952

DQ 0.418 0.202 0.216 1.071 6.759DQ q p DP hard 3.609 2.950 0.658 0.223 0.856

DP soft 2.726 2.202 0.523 0.238 0.896

DQ q p q K DP hard 1.766 1.495 0.271 0.181 1.602DP soft 1.204 0.998 0.206 0.207 1.954

DQ q all DP hard 3.591 3.019 0.572 0.189 3.188DP soft 1.884 1.522 0.361 0.237 3.587

12 q1.12Ž . w xG C EXP 21 1.14 " 0.2 — – 1.33 –L y0.8112 q0.18 q0.32Ž . w xG C EXP 22 0.89 " 0.15 " 0.03 0.31 – 1.87 " 0.59 –L y0.11 y1.0012Ž . w xG C EXP 23 1.14 " 0.08 — – – –L28Ž . w xG Si EXP 23 1.28 " 0.08 — – – –L

Ž . w xG Fe EXP 23 1.22 " 0.08 — – – –L

Ž . w xG Bi EXP 24 1.63 " 0.06 " 0.13 — – – –L

the hard DP form factor. The cutoff parameters for the soft DP form factor and the hardDP form factor are given in Table 3.

First of all, we can see that the OPE model predicts a large total decay rate incomparison with experiment for the heavy hypernuclei, even if we choose the softercutoff. The tensor dominance property leads to a small G rG and PVrPC ratio.n p

One sees that the kaon-exchange contribution reduces G by more than factor two.p

This mainly comes from the cancellation in the channel d . At same time, the kaonp

exchange contribution reduces G . Therefore, the nrp ratio remains small. The decay isn

dominated by the Js1 channels.) Ž .When we include h, r, v, and K ‘‘all’’ mesons, both the proton- and neutron-in-

Ž .duced decays increase but the nrp ratio is still small ,0.1 . It is interesting to see thatthe PVrPC ratio becomes large, when we include heavier mesons.

One sees in Table 4 that the magnitude of DQ itself is small compared to that of p orpqK. However, the DQ has a large G and a large nrp ratio. It is also shown that then

DQ mechanism is dominated by the parity violating channels and thus produces a largePVrPC ratio. The characteristic behaviors of DQ will distinguish it from the othermechanisms.

Now we go to the results of our complete approach, i.e. the meson-exchangemechanism plus the direct quark mechanism. One sees that the pion exchange contribu-tion strongly depends on the choice of the form factor as stated before. The hard form


factor does not suppress the potential at short distances and yields a large OPEcontribution. As we assume that the DQ has little contribution to the tensor channel, Gp

is still too large in the case of DQqp . Again, the K exchange reduces the tensoramplitude and thus G is suppressed by a factor 2. However, G is reduced, at the samep n

time, which results in an nrp ratio ,0.2. This value seems too small compared to theexperimental values for the medium and heavy hypernuclei.

The contribution beyond the p and K mesons does not improve the situation. It isalso questionable whether the DQ mechanism and the vector meson exchanges areindependent and can be superposed. The double counting problem for the vector mesons

w xand DQ contribution in nuclear force is pointed out in Ref. 42 . We thus take the‘‘DQqpqK ’’ with the soft p form factor as our present best model for non-mesonicL decay.

4. Light hypernuclei

The same LN™NN transition potential is applied to the study of non-mesonic weakdecays of light s-shell hypernuclei, 5 He, 4 He, and 4 H. This is an extension of ourL L L

w xprevious work 18 in which we took only the DQ and OPE into account. The formfactor for OPE is also revised. We assume the harmonic oscillator shell model wavefunction for the initial nucleons, while the L single particle wave function calculated

Žwith a realistic L-nuclear potential, which has a repulsion at short distances. See Ref.w x .18 for details. We further consider the short-range correlation, which is multiplied tothe relative two-body wave functions. We employ the same correlation functions asthose used in the nuclear matter calculation.

The Gaussian b parameters are 1.358 fm for As5 and 1.65 fm for As4. For thefinal state, we assume the correlated plane waves for the outgoing two nucleons, andsum up all the residual states.

Table 5 summarizes our results for the p , pqK , and pqK q DQ approaches,compared with available experimental data. One sees that the proton-induced decay rates

Table5Ž . Ž .Non-mesonic decay rates in units of G of light hypernuclei. The DP soft form factor is used for OPEL

Total G G G rGp n n p

5 He p 0.370 0.327 0.044 0.133L

p q K 0.175 0.166 0.014 0.055p q K qDQ 0.261 0.218 0.046 0.195

w xEXP 21 0.41"0.14 0.21"0.07 0.20"0.11 0.93"0.55w xEXP 25 0.50"0.07 0.17"0.04 0.33"0.04 1.97"0.67

4 He p 0.271 0.249 0.022 0.089L

p q K 0.126 0.117 0.010 0.082p q K qDQ 0.155 0.151 0.004 0.024

w xEXP 25 0.19"0.04 0.15"0.02 0.04"0.02 0.27"0.144 H p 0.040 0.011 0.029 2.596L

p q K 0.010 0.005 0.005 1.099p q K qDQ 0.059 0.030 0.030 0.983

w xEXP 25 0.15"0.13 – – –


are fairly well reproduced by the pqK q DQ mechanism, where the soft form factorand the K exchange reduce the large p exchange contribution. It is, however, notpossible to explain the neutron induced rats simultaneously, and therefore the G rGn p

ratio remains small both for the 5 He and 4 He decays. Yet the 4 H decay rates are crucialL L L

in determining the DIs3r2 contribution to the s-shell hypernuclear decays. It is alsonoted that the DQ contribution is most significant for the 4 H, where no Js1 protonL

induced decays are in effect.In this result we find that the most reliable picture in the present paper ‘‘p q K q

DQ’’ gives us good agreement with the experiment for the proton-induced decay rate. Itis due to the large cancellation of the p-exchange contribution by the soft form factorand K exchange, while the nucleon-induced decay rates are much underestimated andthe nrp ratios are still too small.

5. Summary and conclusion

In this paper we study the LN™NN weak transition by combining the ME and DQmechanisms. We use the weak meson–baryon coupling vertices which are evaluated by

Ž .using SU 6 symmetry. The DQ mechanism comes from the quark structure of twow

baryons and is effective at the short range where two baryons overlap each other.Adding these two types of contributions, we calculate the decay rate of L in nuclearmatter and in light hypernuclei. The choice of the form factor for the p exchangeinduced potential is found to be important. With the heavier mesons exchange or DQ forthe short-range part, it seems appropriate to employ the soft form factor for OPE.

In our calculations we find that the choice of ‘‘pqK q DQ’’ with the soft DP formfactor for OPE reproduces the current experimental data rather well. The OPE contribu-tion dominates the decay. The K contribution is also large and interferes destructivelywith the p contribution. These two contributions yield the long-range part of thisprocess. In the ‘‘pqK q DQ’’ picture it is assumed that the DQ mechanism replacesthe role of the heavier mesons.

Although the present analysis gives the total decay rates both in nuclear matter and inlight hypernuclei fairly well, there remains a difficult problem. That is, the neutron-in-duced decay rate is still too small compared to the experimental one and thus we predicta small nrp ratio. The ratio is improved from the pqK exchange prediction due to the

Ž .DQ contribution, in which the nrp ratio is about unity. It is, however, still small ,0.2for ‘‘pqKqDQ’’. The experimental numbers are not completely fixed, but theysuggest a value around 1 for heavy hypernuclei. The situation is similar for 5 He. It isL

thus urgent and important to find out what causes this discrepancy.

Acknowledgements

The authors acknowledge Drs. A. Ramos, A. Parreno, C. Bennhold for valuable˜discussions. They also thank Profs. T. Motoba and K. Itonaga for helping us carry outthe finite nucleus calculation and Prof. H. Bhang and Dr. H. Outa for information ontheir experiments. T.I. thanks to JSPS Research Fellow for their generous financial


assistance. This work is supported in part by the Grant-in-Aid for Scientific ResearchŽ .Ž . Ž .Ž . Ž .C 2 08640356, C 2 11640261, and Priority Areas Strangeness Nuclear Physics ofthe Ministry of Education, Science, Sports and Culture of Japan.

Appendix A

In this appendix we present the explicit forms of the DQ transition potential in thew xcoordinate space. This is based on its momentum representation given in Refs. 16,17

Ž Ž . .referred to as I hereafter . The potential is expressed in terms of three sets of sevenX Ž X. Ž X. Ž X. Ž .functions of k and k , f k,k , g k,k and h k,k , given in Tables 6, 7 and 8 of I .i i i

Ž . Ž .The complete transition potential is given by Eq. 24 of I as a combination of thefunctions, f , g, and h, and the coefficients, which depends on the initial and finalorbital angular momenta, spins and isospin. Those coefficients, denoted by V ’s and W,i

Ž .are explicitly given in Tables 2, 3, 4, and 5 in I , so that the whole transition potentialwithout any further information can be reconstructed.

The purpose of this appendix is to replace the functions f , g and h, by their Fouriertransformed counterparts, which are convenient for hypernuclear decays with sophisti-cated nuclear wave functions. In the coordinate space, the potential becomes non-localdue to the quark antisymmetrization effect, and also contains a derivative term, whichgenerates a transition from Ls0 to LX s1. Such derivative terms appear for thefunctions g and h. The complete potential is now given as a non-local form as

7GX FX X X XL L f g hXV r ,r sy =W V f r ,r qV g r ,r qV h r ,r ,Ž . Ž . Ž . Ž .� 4ÝDQ SS J i i i i i i'2 is1

A.1Ž .

Ž X. Ž .where r r stands for the radial part of the relative coordinate in the initial final state.We only need to give the forms of the radial functions f , g, and h as the coefficients Vi

Ž . Žand W are the same ones as in I . However, we call the reader’s attention to the factŽ .that the transition potential given in I is for the antisymmetrized initial states. They are

'Ž .antisymmetrized so that the flavor antisymmetric states are defined as pLyL p r 2'Ž .or nLyLn r 2 . Thus they have an opposite sign to those commonly used in the

literature for the meson exchange potential. It is necessary to take care of this differencein the convention when we superpose the DQ potential with that from the mesonexchange. If one adds our transition potential to the OPE in the conventional definition,we need to change the signs of the coefficients for the flavor-antisymmetric initial

.states.Ž .The local parts with a derivative on the initial radial coordinate are given for is1,

6, and 7, by

6 d ryrXŽ .y3r2X 2f r ,r s 1P1P 2p b ,Ž . Ž .1 2N r

6 1 d ryrXŽ .y3r2X 2g r ,r s 1P P 2p b yi E , A.2Ž . Ž . Ž . Ž .1 r2'N r3 m


6 1 d ryrXŽ .y3r2X 2h r ,r s 1P P 2p b yi E ,Ž . Ž . Ž .1 r2'N r3 m

9 d ryrXŽ .2y3r2X yr r4 Af r ,r s 1P1P 4p A e ,Ž . Ž .6 2N r9 1 d ryrXŽ .2y3r2X yr r4 Ag r ,r s 1P P 4p A e yi E , A.3Ž . Ž . Ž . Ž .6 r2'N r3 m

X9 1 d ryr 1Ž .2y3r2X yr r4 Ah r ,r s 1P P 4p A e yi E q r ,Ž . Ž . Ž .6 r2'N 2 Ar3 m1

X Xf r ,r sy f r ,r ,Ž . Ž .7 631

X Xg r ,r sy g r ,r , A.4Ž . Ž . Ž .7 631

X Xh r ,r sy h r ,r ,Ž . Ž .7 63where A'b2r3, the normalization factor N is taken as Ns1 and E is the derivativer

on the initial radial coordinate.Among the above local potentials, the f part contains a constant term, which1

survives at rsrX™`. This term comes from the internal weak conversion of L into a

neutron and is to be suppressed due to the mismatch of the initial and final wavefunctions. However, in the application to hypernuclear decays we do not necessarilyemploy wave functions which satisfy the orthogonality condition of the bound andscattering states. Thus the f term may survive in the actual calculation. As this term is1

spurious in most of the cases, we omit the f term in the actual calculation.1

The non-local parts are given in terms of the functions

CrrX yBX rX 2 yBr 2y3r22G ' 4p D 4p i exp ,Ž .0 0 ž / ž /D D

CrrX yBX rX 2 yBr 2y3r22G ' 4p D 4p i exp , A.5Ž . Ž .1 1 ž / ž /D D

where i is the ll th modified spherical Bessel function.ll

For is2, the constants B, BX, C, and D are given by

5b2 b2 4b4XBsB s , Cs , Ds A.6Ž .

12 2 9and we find

'18 1 3 6Xf r ,r sy P1P G ,Ž .2 0N 3 4

'18 1 1 3 6 2 B CX Xg r ,r sy P P i y rG q r G , A.7Ž . Ž . Ž .2 1 0'N 3 4 D D3 m

X'18 1 1 3 6 2 B CX Xh r ,r sy P P yi y r G q rG .Ž . Ž .2 0 1'N 3 4 D D3 m


Fig. 6. The transition potentials for the proton-induced decay.

For is4,

'36 1 3 3Xf r ,r sy P1P G ,Ž .4 0N 3 8


X'36 1 1 3 3 2 B CX Xh r ,r sy P P yi y r G q rGŽ . Ž .4 0 1'N 3 8 D D3 m


with

b2 b4XBsB s , Cs0, Ds . A.9Ž .

6 9For is3, and is5, we find

'36 1 24 33Xf r ,r sy P1P G ,Ž .3 0N 3 121


X'36 1 1 24 33 2 B CX Xh r ,r sy P P yi y r G q rGŽ . Ž .3 0 1'N 3 121 D D3 m

with for is3,

13b2 7b2 12b2 20b4XBs , B s , Cs , Ds A.11Ž .

33 33 33 99and for is5,

7b2 13b2 12b2 20b4XBs , B s , Cs , Ds . A.12Ž .

33 33 33 99The local parts of the DQ, p-, K-exchange potentials are illustrated in Fig. 6 for the

proton-induced transitions and Fig. 7 for the neutron-induced transition. The meson

Fig. 7. The transition potentials for the neutron-induced decay.


Žexchange potentials are multiplied by the form factors. We take the soft L s800p

.MeV DP form factor for the OPE.

References

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Nuclear Physics A 678 (2000) 455–456www.elsevier.nl/locate/npe

Erratum

Erratum to “Weak decay ofΛ hypernuclei:quark versus mesons”

[Nucl. Phys. A 669 (2000) 331–350]✩

Kenji Sasaki∗, Takashi Inoue, Makoto Oka

We have found an error in the K exchange potential, so that the K exchange amplitudes,fp andfn (3S1→ 3P1) change the sign. Accordingly, Tables 1 and 2 of the original papershould be replaced by the new ones. The main consequence is that theπ and K exchangesinterfere constructively and thus enhancefp andfn.

In the nuclear matter calculation we see that

Γp = 1.431ΓΛ(0.674),

Γn = 1.025ΓΛ(0.021),

Γn/Γp = 0.716(0.031)

for the “π (soft)+K +DQ” mechanism. The values in the parentheses are original valueswith error. This indicates thatΓn, Γn/Γp , and PV/PC ratio are all enhanced strongly.Note that we also see that the total decay rate (Γtot ' 2.5ΓΛ) is larger than the heavyhypernucleus data even with the soft form factor for theπ exchange. It is possible that theshort rangeΛN andNN correlations in the nuclear medium are not sufficiently strong inthe present calculation.

For light hypernuclei, similarly to nuclear matter, we see large enhancement of the totaldecay rates and theΓn/Γp ratios. Now the results using the soft form factor for the pionexchange are all consistent with the current light hypernuclear data. Remember that the oldvalue of theΓn/Γp ratio for 5

ΛHe is about 0.2.In conclusion, we see a great improvement of theΓn/Γp ratio due to the contribution of

the K exchange and the DQ process. The K exchange reduces the strong tensor transitionof theπ exchange, reducingΓp , while it enhancesΓn throughfn. DQ enhances bothΓpandΓn resulting “π + K + DQ” results are in good agreement with experimental data forlight hypernuclei.

We acknowledge Dr. Assumpta Parreño for communications regarding this error.

✩ PII of original article: S0375-9474(99)00427-3∗ Corresponding author.

E-mail address:[email protected] (K. Sasaki).

0375-9474/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0375-9474(00)00356-0

456 K. Sasaki et al. / Nuclear Physics A 678 (2000) 455–456

Table 1Nonmesonic decay rates ofΛ in nuclear matter with realistic short-range correlation (in units ofΓΛ).The “all” includesπ , K , η, ρ, ω, andK∗ meson exchange

Total Γp Γn Γn/Γp PV/PC

π +K DP hard 1.759 1.300 0.459 0.353 1.594DP soft 1.216 0.850 0.365 0.430 1.841no-f.f. 2.068 1.424 0.644 0.452 2.140

all DP hard 2.444 1.828 0.616 0.337 2.618DP soft 1.906 1.363 0.543 0.398 2.818no-f.f. 2.863 2.077 0.786 0.379 5.538

DQ+ π +K DP hard 3.145 1.970 1.176 0.597 3.578DP soft 2.456 1.431 1.025 0.716 4.811

DQ+ all DP hard 3.974 2.570 1.404 0.546 5.206DP soft 3.288 2.016 1.273 0.631 6.312

Table 2Nonmesonic decay rates of each hypernuclei. DP (soft) is used

Total Γp Γn Γn/Γp

5ΛHe π +K 0.302 0.209 0.094 0.450

π +K +DQ 0.519 0.305 0.214 0.701

EXP[21] 0.41± 0.14 0.21± 0.07 0.20± 0.11 0.93± 0.55EXP[25] 0.50± 0.07 0.17± 0.04 0.33± 0.04 1.97± 0.67

4ΛHe π +K 0.156 0.147 0.009 0.065

π +K +DQ 0.218 0.214 0.004 0.017

EXP[25] 0.19± 0.04 0.15± 0.02 0.04± 0.02 0.27± 0.14

4ΛH π +K 0.069 0.005 0.065 13.508

π +K +DQ 0.184 0.030 0.154 5.088

EXP[25] 0.15±0.13 – – –

weak decay of l in nuclei quarks versus mesons.pdf

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