PHYSICAL REVIEW D, VOLUME 62, 014027

¯ s X decays One-particle inclusive B s \D Xavier Calmet Ludwigs-Maximilians-Universita¨t, Sektion Physik, Theresienstrasse 37, D-80333 Mu¨nchen, Germany 共Received 21 December 1999; published 7 June 2000兲 ¯ s X decays using a QCD-based method already applied to B We discuss one-particle inclusive B s →D ¯ →D X. A link between the right charm nonperturbative form factors of the semileptonic decays and those of the nonleptonic decays is established. Our results are compatible with current experimental knowledge. PACS number共s兲: 13.25.Hw

I. INTRODUCTION

Some time ago, a QCD-based method was proposed to ¯ l ␯ X decays, which relies on a short distance describe B→D expansion 共SDE兲 and on the heavy quark effective theory 关1兴. The nonperturbative form factors of the singlet operators were parametrized using the Isgur-Wise function. More recently this method was extended to one-particle inclusive nonleptonic B decays 关2兴. In this case, we have to perform a 1/N C expansion, which allows to factorize the matrix elements. One of the goals of this work is to clarify the link between the matrix elements which were encountered in the semileptonic one-particle inclusive B decays 关1兴 and those of the nonleptonic one-particle inclusive B decays encountered in 关2兴. In fact, we prove that these matrix elements are universal. We then apply this method to one-particle inclusive ¯ s X and B s →D s X decays. B s →D It is shown in 关2兴 that the one-particle inclusive decays of a B meson into a vector D meson seem to be, in this framework, well understood whereas decays of a B meson into a pseudoscalar D are troublesome; i.e., the decay widths and ¯ * /D * X admixtures look to be described spectra for B→D ¯ /DX correctly, on the other hand, the predictions for B→D admixture decay widths and spectra do not reproduce the experimental data. Most troublesome is the fact that the spectra are not even described correctly for large transferred momentum. According to our method we expect to describe the experimental data for large transfered momentum particularly well. Keeping in mind that some problems arose in the descrip¯ /DX decays, we apply the method developed tion of B→D ¯ s X and B s →D s X decays. The effor these decays to B s →D fective Hamiltonian is identical in both cases. One-particle ¯ s X decay widths have been measured by inclusive B s →D ALEPH. There are measurements for semileptonic 关3兴 as well as for nonleptonic 关4兴 decays. The decay rates we are computing can be used to study one-particle inclusive C P asymmetries in the B s system 关5兴, which would allow an extraction of the weak angle ␥ which is known to be difficult. This study of B s →D s X decays could also allow us to get a better understanding of the problems encountered in B→DX decays 关2兴. They are also interesting for experimental physics especially in the perspective of B factories as the presently available data on one-particle inclusive B s →D s X decays are sparse. 0556-2821/2000/62共1兲/014027共5兲/$15.00

In the following section, we shall establish the link between the form factors of the semileptonic decays and those ¯ transiof the nonleptonic decays for the right charm ¯b →c tion. II. FROM SEMILEPTONIC TO NONLEPTONIC DECAYS

¯ X, i.e., ¯b →c ¯ tranWe consider right charm decays B→D sitions. The central quantity in the semileptonic case as well as the nonleptonic case is the function G given by G共 M 2兲⫽

兺X 兩 具 B 共 p B 兲 兩 H e f f 兩 D¯ 共 p D¯ 兲 X 典 兩 2共 2 ␲ 兲 4 ⫻ ␦ 4 共 p B ⫺ p D¯ ⫺ p X 兲 ,

共1兲

where 兩 X 典 are momentum eigenstates with momentum p X , H e f f is the relevant part of the weak Hamiltonian, and M 2 ⫽(p B ⫺ p D¯ ) 2 is the invariant mass. The states 兩 X 典 form a complete set, especially 兩 X 典 can be the vacuum in the semi¯ l ␯ contributes to B→D ¯ l ␯ X. This leptonic case, e.g., B→D function G is related to the decay rate under consideration by ¯ X 兲⫽ d⌫ 共 B→D

1 d⌽ D¯ G 共 M 2 兲 , 2m B

共2兲

¯ where d⌽ D¯ is the phase-space element of the final-state D meson. The relevant weak Hamiltonian is given by (nl) H e f f ⫽H (sl) e f f ⫹H e f f ,

共3兲

where the semileptonic and nonleptonic pieces are given by

62 014027-1

H (sl) ef f ⫽

H (nl) ef f ⫽

GF

冑2

GF

冑2

V cb 共 ¯b c 兲 V⫺A 共¯l ␯ 兲 V⫺A ⫹H.c.,

共4兲

* „共 ¯b c 兲 V⫺A 共 ¯u d 兲 V⫺A V cb V ud

⫹ 共 ¯b T a c 兲 V⫺A 共 ¯u T a d 兲 V⫺A … ⫹

GF

冑2

* „共 ¯b c 兲 V⫺A 共¯c s 兲 V⫺A V cb V cs

⫹ 共 ¯b T a c 兲 V⫺A 共¯c T a s 兲 V⫺A …⫹H.c.,

共5兲

©2000 The American Physical Society

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PHYSICAL REVIEW D 62 014027

where we have neglected the penguins and the Cabibbo suppressed operators. The function G can be written as G共 M 2兲⫽

兺X

冕

¯ 共 p D¯ 兲 X 典 d 4x 具 B共 p B 兲兩 H e f f 共 x 兲兩 D

to be valuable for nonleptonic exclusive B mesons decays 关6兴. In this limit the octet operators vanish. Thus, we obtain G NL 共 M 2 兲 ⫽

¯ 共 p D¯ 兲 X 兩 H e f f 共 0 兲 兩 B 共 p B 兲 典 . ⫻具 D

共6兲

G

共 M 兲⫽ 2

G F2 2

兩 V cb 兩 2

2

兩 V cb V q* q 兩 2 兩 C 1 兩 2 1 2

共 2 ␲ 兲 4 ␦ 4 共 M ⫺p X 兺X 兺 X ⬘

¯ ␥ ␮ 共 1⫺ ␥ 5 兲 c…兩 D ¯ 共 p D¯ 兲 X 典 ⫺ p X ⬘ 兲 具 B 共 p B 兲 兩 „b ¯ 1 ␥ ␮ 共 1⫺ ␥ 5 兲 q 2 …兩 X ⬘ 典 ⫻具 0 兩 „q

In the semileptonic case we can trivially factorize G(M 2 ) and obtain

Lep

G F2

¯ 2 ␥ ␯ 共 1⫺ ␥ 5 兲 q 1 …兩 0 典 ⫻ 具 X ⬘ 兩 „q ¯ 共 p D¯ 兲 X 兩 „c ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 b…兩 B 共 p B 兲 典 , ⫻具D

兺X 共 2 ␲ 兲 4 ␦ 4共 M ⫺p X 兲

⫻ 具 0 兩 „l¯ ␥ ␮ 共 1⫺ ␥ 5 兲 ␯ …„¯␯ ␥ ␯ 共 1⫺ ␥ 5 兲 l…兩 0 典 ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 c…兩 D ¯ 共 p D¯ 兲 X 典 ⫻具 B 共 p B 兲 兩 „b ¯ 共 p D¯ 兲 X 兩 „c ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 b…兩 B 共 p B 兲 典 . ⫻具D

共7兲

The next steps are to insert heavy quark fields in the effective Hamiltonian and considering m b and m c as large scales, to perform a SDE as it has been explained in 关1兴. In the leading order of the SDE, G Lep (M 2 ) reads

where the q i ’s stand for quarks. We see that assuming that X and X ⬘ are disjoint, which is certainly the case in the leading order of the 1/N C limit, we can at once apply the completeness relation for X ⬘ and we just find ourselves in the same situation as in the semileptonic case. ¯ d we have q 1 ⫽u and q 2 For the quark transition b→cu ⫽d, i.e., we have two light quarks whose masses can be neglected just as the one of the leptons in the semileptonic case. We obtain P ␮NL␯ 共 M 兲 ⫽N C P ␮Lep ␯ 共 M 兲,

G Lep 共 M 2 兲 ⫽

G F2 2 ⫻

兩 V cb 兩 2 P ␮Lep ␯ 共M兲

¯ 共 v ⬘ 兲 X 兩 关¯c v ␥ ␯ 共 1⫺ ␥ 5 兲 b v 兴 兩 B 共 v 兲 典 , ⫻具 D ⬘

G NL 共 M 2 兲 ⫽

共8兲

where v is the velocity of the B meson, v ⬘ is the velocity of ¯ meson, and P ␮Lep the D ␯ is a tensor originating from the contraction of the lepton fields in the effective Hamiltonian. This tensor is given by 2 2 2 P ␮Lep ␯ 共 M 兲 ⫽A 共 M 兲共 M g ␮ ␯ ⫺M ␮ M ␯ 兲 ⫹B 共 M 兲 M ␮ M ␯ .

G F2 2 ⫻

* 兩 2 P ␮NL␯ 共 M 兲 兩 V cb V ud

兺X 具 B 共 v 兲 兩 关¯b v ␥ ␮共 1⫺ ␥ 5 兲 c v ⬘ 兴 兩 D¯ 共 v ⬘ 兲 X 典

¯ 共 v ⬘ 兲 X 兩 关¯c v ␥ ␯ 共 1⫺ ␥ 5 兲 b v 兴 兩 B 共 v 兲 典 . ⫻具 D ⬘

共13兲

¯ s can be treated in the same fashion. In The transition b→cc that case the mass of the c quark in the loop cannot be neglected. We obtain

共9兲 P ␮NL␯ 共 M 兲 ⫽A 共 M 2 兲共 M 2 g ␮ ␯ ⫺M ␮ M ␯ 兲 ⫹B 共 M 2 兲 M ␮ M ␯ , 共14兲

Neglecting the lepton masses, we obtain at tree level 1 ⌰ 共 M 2 兲 and B 共 M 2 兲 ⫽0. 3␲

共12兲

where N C is the color number, and

兺X 具 B 共 v 兲 兩 关¯b v ␥ ␮共 1⫺ ␥ 5 兲 c v ⬘ 兴 兩 D¯ 共 v ⬘ 兲 X 典

A 共 M 2 兲 ⫽⫺

共11兲

共10兲

We now consider the nonleptonic case. The nonleptonic case is more complex because two transitions are possible: ¯ transition and the wrong charm transithe right charm ¯b →c tion ¯b →c. The wrong charm transition was treated in 关2兴 and we will not come back to this issue, since this channel is extremely suppressed in the semileptonic case and was neglected in 关1兴 and our aim in this section is strictly to establish the link between the right charm semileptonic and nonleptonic decays. Another difficulty is that factorization can only be performed in the 1/N C limit. This concept is known

where A(M 2 ) and B(M 2 ) are given by

A 共 M 2 兲 ⫽⫺

冉

冊冉 冊 冉 冊

m 2c NC 1⫹ 3␲ 2M 2

m 2c N C m 2c B共 M 兲⫽ 1⫺ 2␲ M 2 M2 2

1⫺

m 2c

M

2

2

⌰ 共 M 2 ⫺m 2c 兲 ,

2

⌰ 共 M 2 ⫺m 2c 兲 ,

共15兲

at tree level. As explained in 关2兴, we set m c ⫽1.0 GeV to parametrize the higher-order QCD corrections to the current ¯ s. b→cc

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We can now establish the connection between the semileptonic and the nonleptonic form factors. The differential decay width for the semileptonic decays is given by

where y⫽ v • v ⬘ and where the invariant mass M 2 is given by 2 ⫺2ym B m D . M 2 ⫽m B2 ⫹m D

G F2

d⌫ 3 冑y 2 ⫺1 关共 m B ⫺m D 兲 2 E S 共 y 兲 ⫽ 兩V 兩2m D dy 12␲ 3 cb ⫹ 共 m B ⫹m D 兲 2 E P 共 y 兲 ⫺M 2 „E V 共 y 兲 ⫹E A 共 y 兲 …兴 ,

共16兲

共17兲

The differential decay width for the right charm nonleptonic decays is then given by

G F2 d⌫ 2 * 兩 2 m D3 冑y 2 ⫺1 关共 m B ⫺m D 兲 2 E S 共 y 兲 ⫹ 共 m B ⫹m D 兲 2 E P 共 y 兲 ⫺M 2 „E V 共 y 兲 ⫹E A 共 y 兲 …兴 ⫽C 1 N C 兩 V cb V ud dy 12␲ 3 ⫹C 21

G F2 4␲2

* 兩 2 m D3 冑y 2 ⫺1 关 „B 共 M 2 兲 ⫺A 共 M 2 兲 …„共 m B ⫺m D 兲 2 E S 共 y 兲 ⫹ 共 m B ⫹m D 兲 2 E P 共 y 兲 … 兩 V cb V cs

⫹A 共 M 2 兲 M 2 „E V 共 y 兲 ⫹E A 共 y 兲 …兴 ,

共18兲

兺X 具 B 共 v 兲 兩 关¯b v c v ⬘ 兴 兩 D¯ 共 v ⬘ 兲 X 典

ered in 关1兴 since this effect is small. Therefore, we set C 11 ⫽C 3 ⫽1 and C 18⫽0 in the set of nonperturbative form factors given in 关1兴. After the connection between the nonleptonic and the semileptonic case has been established, we consider B s ¯ s X and B s →D s X decays. →D

¯ 共 v ⬘ 兲 X 兩 关¯c v b v 兴 兩 B 共 v 兲 典 , ⫻具D ⬘

¯ s X AND B s \D s X III. THE DECAYS B s \D

where A(M 2 ) and B(M 2 ) are given in Eq. 共15兲. We see that the right charm semileptonic and nonleptonic decay widths are given in terms of the same form factors 4m B m D E S 共 v • v ⬘ 兲 ⫽

⫺4m B m D E P 共 v • v ⬘ 兲 ⫽

As mentioned previously the effective weak Hamiltonian ¯ X case, therefore, Eqs. is identical to the one of the B→D 共16兲 and 共18兲 do also describe the right charm decay of a B s ¯ s meson if one replaces m B by m B and m D by meson into a D s m D s . We have a new set of nonperturbative form factors:

兺X 具 B 共 v 兲 兩 关¯b v ␥ 5 c v ⬘ 兴 兩 D¯ 共 v ⬘ 兲 X 典 ¯ 共 v ⬘ 兲 X 兩 关¯c v ␥ 5 b v 兴 兩 B 共 v 兲 典 , ⫻具D ⬘

4m B m D E V 共 v • v ⬘ 兲 ⫽

兺X 具 B 共 v 兲 兩 关¯b v ␥ ␮ c v ⬘ 兴 兩 D¯ 共 v ⬘ 兲 X 典

4m B s m D s E S 共 v • v ⬘ 兲 ⫽

¯ 共 v ⬘ 兲 X 兩 关¯c v ␥ ␮ b v 兴 兩 B 共 v 兲 典 , ⫻具D ⬘ 4m B m D E A 共 v • v ⬘ 兲 ⫽

兺X 具 B s共 v 兲 兩 关¯b v c v ⬘ 兴 兩 D¯ s共 v ⬘ 兲 X 典 ¯ s 共 v ⬘ 兲 X 兩 关¯c v b v 兴 兩 B s 共 v 兲 典 , ⫻具D ⬘

兺X 具 B 共 v 兲 兩 关¯b v ␥ ␮ ␥ 5 c v ⬘ 兴 兩 D¯ 共 v ⬘ 兲 X 典

⫺4m B s m D s E P 共 v • v ⬘ 兲 ⫽

¯ 共 v ⬘ 兲 X 兩 关¯c v ␥ ␮ ␥ 5 b v 兴 兩 B 共 v 兲 典 . 共19兲 ⫻具 D ⬘ One important point should be stressed. This set 共19兲 of nonperturbative form factors describes a transition from a B meson into a state with a D meson whatever the intermediate state might be. It has been shown in 关1兴 that we can determine these matrix elements in the semileptonic case using constraints from the heavy quark symmetry 共HQS兲 and a saturation assumption. These nonperturbative form factors were given in 关1兴 for each single decay channel. So the non¯ X decays can be deduced from leptonic right charm B→D the semileptonic ones. Note that we have neglected the renormalization-group improvement which had been consid014027-3

兺X 具 B s共 v 兲 兩 关¯b v ␥ 5 c v ⬘ 兴 兩 D¯ s共 v ⬘ 兲 X 典 ¯ s 共 v ⬘ 兲 X 兩 关¯c v ␥ 5 b v 兴 兩 B s 共 v 兲 典 , ⫻具D ⬘

4m B s m D s E V 共 v • v ⬘ 兲 ⫽

兺X 具 B s共 v 兲 兩 关¯b v ␥ ␮ c v ⬘ 兴 兩 D¯ s共 v ⬘ 兲 X 典 ¯ s 共 v ⬘ 兲 X 兩 关¯c v ␥ ␮ b v 兴 兩 B s 共 v 兲 典 , ⫻具D ⬘

4m B s m D s E A 共 v • v ⬘ 兲 ⫽

兺X 具 B s共 v 兲 兩 关¯b v ␥ ␮ ␥ 5 c v ⬘ 兴 兩 D¯ s共 v ⬘ 兲 X 典 ¯ s 共 v ⬘ 兲 X 兩 关¯c v ␥ ␮ ␥ 5 b v 兴 兩 B s 共 v 兲 典 . ⫻具 D ⬘ 共20兲

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PHYSICAL REVIEW D 62 014027

Once again we can find a parametrization for these nonperturbative form factors using the semileptonic decays. We consider the s quark as being massless and we can, therefore, use the very same heavy quark symmetry relations as in the ¯ X. As it has been argued in 关1兴, the HQS implies case B→D that at v • v ⬘ ⫽1 the inclusive rate is saturated by the exclu¯s sive decays into the lowest lying spin symmetry doublet D ¯ ¯ ¯ and D s* . The D s* subsequently decays into D s mesons and thus at v • v ⬘ ⫽1 the sum of the exclusive rates for B s ¯ s l ⫹ ␯ and B s →D ¯ s* l ⫹ ␯ is equal to the one-particle inclu→D ¯ s l ⫹ ␯ X. Making use of this assive semileptonic rate B s →D sumption and of the spin projection matrices for the heavy ¯ s( * ) mesons, we obtain B s and D

兩 Tr兵 ␥ 5 共 1⫹ v” 兲 ⌫ i 共 1⫹ v” ⬘ 兲 ⑀” 其 兩 兺 Pol

¯ s* →D ¯ sX 兲, ⫻ 兩 ␰ 共 y 兲 兩 2 Br共 D

0

⫺

1 共 y 兲 ⫽ 共 y⫹1 兲 2 兩 ␰ 共 y 兲 兩 2 , 4

⫺

1 共 y 兲 ⫽ 共 y 2 ⫺1 兲 兩 ␰ 共 y 兲 兩 2 , 4

B Ds

0

B Ds

E Ps

⫺

0

B Ds

E Vs

0

⫺

1 共 y 兲 ⫽⫺ 共 y⫹2 兲共 y⫹1 兲 兩 ␰ 共 y 兲 兩 2 . 2

B s0 →D s⫺ X B s0 →D s⫹ X B s0 →D s⫺ l ⫹ ␯ X B s0 →D s⫺ ␶ ⫹ ␯ ␶ X B s0 →D s* ⫺ X B s0 →D s* ⫹ X B s0 →D s* ⫺ l ⫹ ␯ X B s0 →D s* ⫺ ␶ ⫹ ␯ ␶ X

Br 共theory兲

Br 共data from 关8兴兲

64.9% 3.3% 9.1% 2.7% 49.6% 2.5% 7% 2%

(92⫾33)% (8.1⫾2.5)%

d⌫ 3G F2 C 21 2 冑y ⫺1m D3 s兩 V cb V cs * 兩 2 y 共 M 2 ⫺m D2 兲 2 ⫽ s dy 2 ␲ 3 M 2 ⫻⌰ 共 M 2 ⫺m 2c 兲 F,

共21兲

共23兲

where F is a channel-dependent nonperturbative form factor. We have 0

⫹

F B s D s ⫽ f „1⫹3⌫ 共 D s* →D s X ⬘ 兲 …⫽4 f ,

共24兲

where X ⬘ is a pion or a photon and f is the constant defined in 关2兴; we had f ⫽0.121. Note that the wrong charm decay is being modeled and we have restricted ourselves to the socalled model 2 of 关2兴 since this model seems to yield better results than model 1. IV. DISCUSSION OF THE RESULTS

1 共 y 兲 ⫽ 共 y⫹1 兲共 2⫺y 兲 兩 ␰ 共 y 兲 兩 2 , 2

B Ds

E As

Mode

2

where i stands for S, P, V or A, the sum is over the polarization of the D * meson and ␰ (y)⫽1⫺0.84(y⫺1) is the Isgur-Wise function measured by CLEO 关7兴. The branching ¯ s* →D ¯ s X) is the new input and since a D s* ⫺ alratio Br(D ¯ s* →D ¯ s X)⫽100%. ways decays into a D s⫺ , we have Br(D We then obtain ES s

s

the parton calculation. In the leading order of the 1/N C and of the 1/m B s expansions, the differential decay width reads

1 E i 共 v • v ⬘ 兲 ⫽ 兩 Tr兵 ␥ 5 共 1⫹ v” 兲 ⌫ i 共 1⫹ v” ⬘ 兲 ␥ 5 其 兩 2 兩 ␰ 共 y 兲 兩 2 16 1 ⫹ 16

TABLE I. Comparison of our results with data. To get branching ratios, we used ␶ B 0 ⫽1.55 ps.

共22兲

¯ s X can be calculated using The nonleptonic decays B s →D these nonperturbative form factors. It is clear that this saturation assumption is a crude approximation, but it is well motivated by the heavy quark symmetry at y⫽1 and the available phase space is not very large, so this has to be treated as a theoretical uncertainty due to nonperturbative physics. The results obtained for the semileptonic decays ¯ Xl ␯ 关1兴 give us some confidence in our rates in B→D method. We now consider the wrong charm decays of a B s meson. They are induced by the quark transition ¯b →c. The wrong charm B s0 →D s* ⫹ X decay width can be estimated using the method described in 关2兴, which corresponds to a rescaling of

In Table I, we compare our predictions with the experimental data found in 关8兴. In the semileptonic case the method yields results which agree with the data. Note that we have considered the ␶ lepton as being massive. On the other hand, it is not clear if the nonleptonic decays are problematic; our results are in the experimental error range though at the inferior limit. One should keep in mind that we had estimated in 关2兴 that corrections to our calculation could be fairly large and in the worst case up to 30%. It would be interesting to ¯ s* ⫺ X) to test the agreement bemeasure the rate ⌫(B s →D tween theory and experiment in this channel. Remember that ¯ /DX described in 关2兴, theory and exfor the decays B→D ¯ * /D * X periment looked to be in agreement for the B→D ¯ /DX decays although decays and in disagreement for B→D this could be accidental, for a discussion of this problem see 关2兴. Data are sparse on one-particle inclusive B s decays; especially no spectra are available. It would be instructive to compare the spectra to check if the same discrepancy appears ¯ * /D * X meson as in 关2兴, where the spectra for the B→D decays seemed to be described correctly. On the other hand, ¯ /DX were not compatthe spectra for the decays of a B→D

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ible with the experimental data, especially at the nonrecoil point where the method should work at its best, this effect being therefore very difficult to understand. Although the extension of the method developed for one-particle inclusive B decays to B s decays is trivial, the results we have obtained are interesting especially in the perspective of B factories. These results could also be used to study mixing induced one-particle inclusive C P asymmetries in the B s system 关5兴, and this allows us to determine the weak angle ␥ , which is known to be very difficult. If the problems encountered in the one-particle inclusive B decays 关2兴 were not present in B s decays, one could constrain the kind of diagrammatic topologies contributing to the one-particle inclusive B decays. In B decays as well as in B s decays we have assumed that the dominant diagrammatical topology contributing to the right charm decay rates is spectator like. This study of B s decays once confronted to more precise experimental results could allow one to test the influence of the light spectator quark.

关1兴 C. Balzereit and T. Mannel, Phys. Rev. D 59, 034003 共1999兲. 关2兴 X. Calmet, T. Mannel, and I. Schwarze, Phys. Rev. D 61, 114004 共2000兲. 关3兴 ALEPH Collaboration, D. Buskulic et al., Phys. Lett. B 361, 221 共1995兲. 关4兴 ALEPH Collaboration, D. Buskulic et al., Z. Phys. C 69, 585 共1996兲.

V. CONCLUSIONS

We have clarified the link between the nonperturbative ¯ X. form factors of the semileptonic and nonleptonic B→D We have applied a method described in 关1兴 and 关2兴 to semi¯ s X and B s →D s X decays. leptonic and nonleptonic B s →D This can be done easily by modifying the saturation assumption. It is too early to see if the same problems which were encountered in 关2兴 also appear in our case, the reason being the lack of experimental data. Our results are compatible with current experimental knowledge. ACKNOWLEDGMENTS

The author is grateful to Professor L. Stodolsky for his hospitality at the ‘‘Max-Planck-Institut fu¨r Physik’’ where this work was performed. He would like to thank Z. Z. Xing for reading this manuscript and for his encouragement to publish the present results and A. Leike for his very useful comments.

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