PHYSICAL REVIEW D, VOLUME 61, 114004

QCD-based description of one-particle inclusive B decays Xavier Calmet, Thomas Mannel, and Ingo Schwarze Institut fu¨r Theoretische Teilchenphysik, Universita¨t Karlsruhe, D-76128 Karlsruhe, Germany 共Received 16 July 1999; published 1 May 2000兲 We discuss one-particle inclusive B decays in the limit of heavy b and c quarks. Using the large-N C limit we factorize the nonleptonic matrix elements, and we employ a short distance expansion. Modeling the remaining nonperturbative matrix elements we obtain predictions for various decay channels and compare them with existing data. PACS number共s兲: 13.25.Hw, 12.39.Hg

I. INTRODUCTION

where the semileptonic and nonleptonic pieces are

Over the last ten years, methods have been developed to describe the decays of heavy hadrons within a framework based on the heavy mass expansion of QCD. Using this approach, model dependences have been drastically reduced. The heavy mass expansion has been formulated for exclusive as well as for inclusive decays. While in the former case the heavy mass expansion is the well-known formalism of heavy quark effective theory 共HQET兲 关1兴, the latter case additionally requires a short distance expansion 共SDE兲 关2兴 which is very similar to the operator product expansion 共OPE兲 used for deep inelastic scattering. Both approaches are QCD based methods and have a good theoretical foundation. However, nothing comparable exists for a theoretical description of one-particle inclusive decays such as B→ h D X or h B→ K X. Unlike the fully inclusive case, the expression for the rate always involves a projection on a specific particle, thereby spoiling a straightforward short distance expansion. The semileptonic case has been discussed in 关3兴, and the present paper is devoted to a study of the nonleptonic decays. ¯ X and We shall concentrate on decays of the form B→D B→DX. These decays have already been considered long ago in the context of models 关4兴. Here we aim at a QCD description and exploit the heavy mass limit for both the b and the c quark. Furthermore we shall use the large-N C limit to factorize the hadronic matrix element. In this way we can identify the parts which allow a short distance expansion. The remaining contributions have to be parametrized and we shall discuss simple forms of this parametrization. We first discuss the right charm contribution arising from quark level b→cX decays and the wrong charm contribution ¯ X separately. After recalling some facts arising from b→c about the phenomenology of h D mesons in Sec. IV, we predict rates and spectra for various decay channels in Sec. V. Finally we compare the results with existing data. II. THE EFFECTIVE HAMILTONIAN AND THE RIGHT CHARM CONTRIBUTION

The relevant effective Hamiltonian for the decays B →h D X is given by H e f f ⫽H 共eslf f兲 ⫹H 共enlf f兲 , 0556-2821/2000/61共11兲/114004共11兲/$15.00

共1兲

H 共eslf f兲 ⫽

GF

H 共enlf f兲 ⫽

GF

&

&

V cb 共 ¯b c 兲 V⫺A 共¯l ␯ 兲 V⫺A ⫹H.c.,

冉

兺

q⫽d,s

共2兲

2

* V cb V uq

兺

k⫽1

C k 共 ␮ 兲 O qk

2

* ⫹V cb V cq

兺

k⫽1 6

⫹V tb V * tq

兺

k⫽3

C k 共 ␮ 兲 P qk

冊

C k 共 ␮ 兲 O qk ⫹H.c.

共3兲

O 1,2 and P 1,2 are the current-current operators, and O 3...6 are the QCD penguin operators. For a full list of operators, see, e.g., the review 关5兴. We shall only consider the Cabibbofavored decays and neglect penguin contributions. Hence the operators we need are O d1 ⫽O 1 ⫽ 共 ¯b c 兲 V⫺A 共 ¯u d 兲 V⫺A O d2 ⫽O 2 ⫽ 共 ¯b T a c 兲 V⫺A 共 ¯u T a d 兲 V⫺A P s1 ⫽ P 1 ⫽ 共 ¯b c 兲 V⫺A 共¯c s 兲 V⫺A P s2 ⫽ P 2 ⫽ 共 ¯b T a c 兲 V⫺A 共¯c T a s 兲 V⫺A .

共4兲

C 1 and C 2 are the Wilson coefficients encoding the short distance physics and T a are the generators of color-SU共3兲. The operators O 1 and O 2 as well as the semileptonic Hamiltonian H (sl) e f f contribute to right charm transitions only, while P 1 and P 2 contribute to both right and wrong charm processes. It is well known that in the large-N C limit the matrix elements of the four-fermion operators factorize into products of two current matrix elements 关6兴. Factorization has also been investigated on a phenomenological basis and found to work well for exclusive nonleptonic decays of B mesons 关7兴. The contributions of O 2 and P 2 vanish in the factorization limit since the currents are color octets. Furthermore, the coefficient C 1 differs from unity only through radiative corrections which are very small and will be neglected.

61 114004-1

©2000 The American Physical Society

XAVIER CALMET, THOMAS MANNEL, AND INGO SCHWARZE

PHYSICAL REVIEW D 61 114004

G 1共 M 兲 ⫽ 2

G F2 2

* 兩 2兩 C 1兩 2 兩 V cb V ud

共 2␲ 兲4 兺X 兺 X ⬘

⫻ ␦ 共 M ⫺ p X ⫺ p X ⬘ 兲 具 B 共 p B 兲 兩 „c ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 b… 4

¯ 共 * 兲 共 p D¯ 兲 X 典具 0 兩 „d ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 u…兩 X ⬘ 典 ⫻兩D ¯ 共 * 兲 共 p D¯ 兲 X 兩 ⫻ 具 X ⬘ 兩 „u ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 d…兩 0 典具 D ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 c…兩 B 共 p B 兲 典 . ⫻„b

共9兲

It is convenient to define two tensors FIG. 1. Momenta and velocities used for the description of the right charm contribution.

K ␮ ␯ 共 p B ,M ,Q 兲 ⫽

The semileptonic case has already been studied in 关3兴, so we focus on the nonleptonic modes. We consider the matrix element

兺X 共 2 ␲ 兲 4 ␦ 4共 M ⫺Q⫺ p X 兲 ¯ 共 * 兲 共 p D¯ 兲 X 典 ⫻ 具 B 共 p B 兲 兩 „c ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 b…兩 D ¯ 共 * 兲 共 p D¯ 兲 X 兩 „b ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 c…兩 B 共 p B 兲 典 ⫻具 D 共10兲

G共 M 兲⫽ 2

兺x

¯ 共 * 兲 共 p D¯ 兲 X 兩 H e f f 兩 B 共 p B 兲 典 兩 2 兩具D

⫻ 共 2 ␲ 兲 4 ␦ 4 共 p B ⫺p D¯ ⫺p X 兲 ,

and 共5兲

where the states 兩X典 form a complete set of momentum eigenstates with momentum p X and H e f f is the relevant part of the weak Hamiltonian. The function G depends on the invariant mass M

2

⫽ 共 p B ⫺p D¯ 兲 2

P ␮␯共 Q 兲 ⫽

¯ ␥ ␮ 共 1⫺ ␥ 5 兲 u…兩 X ⬘ 典 共 2 ␲ 兲 4 ␦ 4 共 Q⫺p X ⬘ 兲 具 0 兩 „d 兺 X ⬘

⫻具 X ⬘ 兩 „u ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 d…兩 0 典 in terms of which we obtain for the rate G 1共 M 2 兲 ⫽

共6兲

1 d⌽ D¯ G 共 M 2 兲 , 2m B

G F2 2

* 兩 2兩 C 1兩 2 兩 V cb V ud

共7兲 P ␮␯共 Q 兲 ⫽

兺X 共 2 ␲ 兲 4 ␦ 4共 M ⫺p X 兲

¯ ␥ ␮ 共 1⫺ ␥ 5 兲 u… ⫻ 具 B 共 p B 兲 兩 „c ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 b…„d ¯ 共 * 兲 共 p D¯ 兲 X 典具 D ¯ 共 * 兲 共 p D¯ 兲 X 兩 „u ⫻兩D ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 d… ⫻ 共 ¯b ␥ ␯ 共 1⫺ ␥ 5 兲 c…兩 B 共 p B 兲 典 .

2

* 兩 2兩 C 1兩 2 兩 V cb V ud

冕

d 4Q K 共 p ,M ,Q 兲 P ␮ ␯ 共 Q 兲 . 共 2 ␲ 兲4 ␮␯ B

共12兲

The tensor P ␮ ␯ involves only light quarks and can be rewritten as

¯ (*) where d⌽ D¯ is the phase space element of the final state D meson. ¯ d and cc ¯ s, there are no Due to the different final states cu interference terms between O 1 and P 1 . The contribution of ¯ d is the channel b→cu

G 1共 M 2 兲 ⫽

G F2

⫻

of the state 兩X典. It is related to the decay rate under consideration by ¯ 共 * 兲X 兲 ⫽ d⌫ 共 B→D

共11兲

共8兲

Using the factorization of the large-N C limit we have 共see Fig. 1兲

冕

¯ 共 x 兲 ␥ ␮ 共 1⫺ ␥ 5 兲 u 共 x 兲 … d 4 x e ⫺iQx 具 0 兩 „d

¯ 共 0 兲 ␥ ␯ 共 1⫺ ␥ 5 兲 d 共 0 兲 …兩 0 典 . ⫻„u

共13兲

For sufficiently large Q this quantity has a short distance expansion in inverse powers of Q. However, the momentum Q is not a measurable kinematical quantity. In particular, it is not equal to the recoil M, but in most of the phase space, it should be of the same order as M, namely O(m b ⫺m c ). Close to the nonrecoil point, M is large, corresponding to either large Q or a large momentum of the gluon depicted in Fig. 1. While in the former case the OPE treatment is justified, the latter case could be treated perturbatively. In other words, the two expansion parameters for the total rate are ⌳ QCD /(m b ⫺m c ) and ␣ s (m c )/ ␲ . However we also consider the momentum spectra of the h D mesons. In that case the expansion parameter depends on the energy of the h D meson since

114004-2

Q 2 ⬇M 2 ⫽ 共 m b ⫺m c 兲 2 ⫺2m b 共 E c ⫺m c 兲 .

共14兲

QCD-BASED DESCRIPTION OF ONE-PARTICLE . . .

PHYSICAL REVIEW D 61 114004

The leading term of the short distance expansion yields the partonic result P ␮␯共 Q 兲 ⫽

NC 共 Q Q ⫺g ␮ ␯ Q 2 兲 ⌰ 共 Q 2 兲 , 3␲ ␮ ␯

共15兲

where we have assumed both light quarks to be massless. For the quantity K ␮ ␯ it is convenient to use the heavy mass limit for both the bottom and the charm quark. To this end we redefine the phases of the quark fields as b 共 x 兲 ⫽b v 共 x 兲 e

⫺im b v x

c 共 x 兲 ⫽c v ⬘ 共 x 兲 e

,

⫺im c v ⬘ x

,

共16兲

where the velocities are defined as p B ⫽m B v and p D¯ ⫽m D¯ v ⬘ . In the following we shall work in the infinite mass limit for the b and the c quark, and we obtain K ␮ ␯ 共 m b v ,M ,Q 兲 ⫽

冕

d 4z

兺X

exp关 ⫺i 共 M ⫺Q 兲 z 兴

later take the mass splitting between the D and D * mesons into account, although formally this is a 1/m c effect. As far as the spinor indices are concerned, the tensor K ␮ ␯ is given by ¯ B 共 v 兲 ␥ ␮ 共 1⫺ ␥ 5 兲 H D¯ 共 兲共 v ⬘ 兲 K ␮ ␯ 共 p B ,M ,Q 兲 ⬀H * ¯ D¯ 共 兲共 v ⬘ 兲 ␥ ␯ 共 1⫺ ␥ 5 兲 H B 共 v 兲 , 丢H *

where the remaining indices are light quark indices which have to be contracted using the most general four-index object. Since there are many possibilities to contract the indices, the discussion of the general case would leave us with a large number of unknown functions, so we have to make a choice. The matrix elements appearing in K ␮ ␯ are identical to the semileptonic case. Therefore we shall use the same ansatz for the nonleptonic as for the semileptonic case 关3兴 and write

⫻ 具 B 共 v 兲 兩 „c ¯ v ⬘ 共 z 兲 ␥ ␮ 兲共 1⫺ ␥ 5 兲 b v 共 z 兲 …

K ␮ ␯ 共 p B ,M ,Q 兲 ⫽ 共 2 ␲ 兲 4 ␦ 4 共 M ⫺Q 兲 ␩ 共 vv ⬘ 兲

¯ 共 * 兲 共 v ⬘ 兲 X 典具 D ¯ 共 * 兲共 v ⬘ 兲 X 兩 ⫻兩D

¯ B 共 v 兲 ␥ ␮ 共 1⫺ ␥ 5 兲 H D¯ 共 兲共 v ⬘ 兲兴 ⫻Tr关 H *

¯ v 共 0 兲 ␥ ␯ 共 1⫺ ␥ 5 兲 c v 共 0 兲 …兩 B 共 v 兲 典 , ⫻„b

⬘

Q ␮Q ␯ ␮␯ K 共 m b v ,M ,Q 兲 , Q2

K 2 共 v , v ⬘ ,Q 兲 ⫽K ␮␮ 共 m b v ,M ,Q 兲 ,

¯ D¯ 共 兲共 v ⬘ 兲 ␥ ␯ 共 1⫺ ␥ 5 兲 H B 共 v 兲兴 . ⫻Tr关 H *

共17兲

where in the heavy mass limit M becomes m b v ⫺m c v ⬘ . Using the result in Eq. 共15兲 for P ␮ ␯ we can express the rate in terms of the two quantities K 1 共 v , v ⬘ ,Q 兲 ⫽

共18兲

共23兲 In this way we can model the two functions K 1 and K 2 in terms of the single nonperturbative function ␩ ( vv ⬘ ), the advantage being that spin symmetry relates the rates of B ¯ X and B→D ¯ * X to each other. →D Finally we also have to consider the right charm contribution of the operator P 1 , which is

共19兲 G 2共 M 兲 ⫽ 2

and obtain G 1共 M 2 兲 ⫽

G F2 2

* 兩 2兩 C 1兩 2 兩 V cb V ud

NC 3␲

冕

d 4Q 2 Q ⌰共 Q2兲 共 2␲ 兲4

⫻关 K 1 共 v , v ⬘ ,Q 兲 ⫺K 2 共 v , v ⬘ ,Q 兲兴 .

再

2

* 兩 2兩 C 1兩 2 兩 V cb V cs

兺X 共 2 ␲ 兲 4 ␦ 4共 M ⫺p X 兲

¯ 共 * 兲 共 p D¯ 兲 X 典具 D ¯ 共 * 兲 共 p D¯ 兲 X 兩 „c ⫻兩D ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 s…

共20兲

¯ ␥ ␯ 共 1⫺ ␥ 5 兲 c…兩 B 共 p B 兲 典 . ⫻„b

共24兲

The calculation is exactly the same as before, but the short distance expansion of P ␮ ␯ yields a different result since now a massive charm quark is involved. We have P ␮⬘ ␯ 共 Q 兲 ⫽

1 H B 共 v 兲 ⫽ 冑m B 共 1⫹ v” 兲 ␥ 5 , 2 1 ␥ 共pseudoscalar meson兲, 冑m D 共 1⫹ v” ⬘ 兲 ⑀” 5 共vector meson兲. 2

G F2

⫻ 具 B 共 p B 兲 兩 „c ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 b…„s ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 c…

Not much can be said about the functions K 1 and K 2 . They do not have an obvious short distance expansion due to ¯ meson. The only restricthe projection on the final state D tion we have is from the spin symmetry of the heavy b and c quarks. To implement these symmetries we use the spin projection matrices for the heavy mesons

H D¯ 共 * 兲共 v ⬘ 兲 ⫽

共22兲

¯ ␥ ␮ 共 1⫺ ␥ 5 兲 c…兩 X ⬘ 典 共 2 ␲ 兲 4 ␦ 4 共 Q⫺p X ⬘ 兲 具 0 兩 „s 兺 X ⬘

⫻具 X ⬘ 兩 „c ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 s…兩 0 典

共25兲

from which we have the leading order contribution 共21兲

In the heavy mass limit, D and D * become degenerate and constitute the ground state spin symmetry doublet of the D meson system. For phenomenological applications we shall

P ␮⬘ ␯ 共 Q 兲 ⫽„A 共 Q 2 兲 Q ␮ Q ␯ ⫺B 共 Q 2 兲 Q 2 g ␮ ␯ …⌰ 共 Q 2 ⫺m 2c 兲 , 共26兲 with

114004-3

XAVIER CALMET, THOMAS MANNEL, AND INGO SCHWARZE

PHYSICAL REVIEW D 61 114004

In a similar way as before it is convenient to define two tensors K ␮⬘ ␯ 共 p B ,Q 兲 ⫽

兺X 共 2 ␲ 兲 4 ␦ 4共 p B ⫺ p X ⫺Q 兲 ⫻ 具 B 共 p B 兲 兩 „c ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 b…兩 X 典

FIG. 2. Momenta and velocities used for the description of the wrong charm contribution.

A共 Q2兲⫽

B共 Q2兲⫽

冉 冉

m 2c NC 1⫺ 2 3␲ Q

冊冉 冊冉 2

m 2c 2 2

NC 1⫺ 3␲ Q

1⫹2

1⫹

m 2c Q

2

冊 冊

m 2c 2

1 2 Q

共31兲

and R ␮ ␯ 共 p D ,Q 兲 ⫽

,

共 2 ␲ 兲 4 ␦ 4 共 Q⫺p D ⫺ p X ⬘ 兲 兺 X ⬘

⫻ 具 0 兩 „s ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 c…兩 D 共 * 兲 共 p D 兲 X ⬘ 典 ⫻ 具 D 共 * 兲 共 p D 兲 X ⬘ 兩 „c ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 s…兩 0 典 , 共32兲

共27兲

,

in terms of which the rate becomes

and we get G 2共 M 2 兲 ⫽

¯ ␥ ␯ 共 1⫺ ␥ 5 兲 c…兩 B 共 p B 兲 典 ⫻ 具 X 兩 „b

G F2 2

* 兩 2兩 C 1兩 2 兩 V cb V cs

冕

G 3 共 p B ,p D 兲 ⫽

d 4Q ⌰ 共 Q 2 ⫺m 2c 兲 Q 2 共 2␲ 兲4

2 ⫻

⫻ 关 A 共 Q 兲 K 1 共 v , v ⬘ ,Q 兲 ⫺B 共 Q 兲 K 2 共 v , v ⬘ ,Q 兲兴 . 2

G F2

2

共28兲 Thus the same two nonperturbative functions appear in G 2 . III. THE WRONG CHARM CONTRIBUTION

Wrong charm decays can only be mediated by the operators P 1 and P 2 . In the large-N C limit, the contribution of P 2 can be neglected and we have G 3 共 p B ,p D 兲 ⫽

G F2 2

* 兩 2兩 C 1兩 2 兩 V cb V cs

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

共29兲

The quantity K ␮⬘ ␯ is fully inclusive and one may perform a short distance expansion. In the heavy mass limit for the b quark it is convenient to rescale the b quark field as in Eq. 共16兲, and we obtain K ␮⬘ ␯ 共 p B ,Q 兲 ⫽

G 3 共 p B ,p D 兲 ⫽

2

* 兩 兩 C 1兩 兩 V cb V cs 2

2

共2␲兲 兺X 兺 X

冕

d 4 y exp关 ⫺i 共 m b v ⫺Q 兲 y 兴

⫻ 具 B 共 v 兲 兩 „c ¯ 共 y 兲 ␥ ␮ 共 1⫺ ␥ 5 兲 b v 共 y 兲 … ¯ v 共 0 兲 ␥ ␯ 共 1⫺ ␥ 5 兲 c 共 0 兲 …兩 B 共 v 兲 典 . ⫻„b

共34兲

In the region where the momentum m b v ⫺Q is large, we can perform a short distance expansion. As before, m b v ⫺Q is not an observable, but it is of the order of m b ⫺m c in most of the phase space. The leading term is the dimension three operator ¯b v b v , the matrix elements of which are normalized due to heavy quark symmetry. Thus we obtain

Again using the factorization of the large-N C limit we get 共see Fig. 2兲 G F2

冕

d 4Q K ⬘ 共 p ,Q 兲 R ␮ ␯ 共 p D ,Q 兲 . 共 2 ␲ 兲4 ␮␯ B 共33兲

兺X 共 2 ␲ 兲 4 ␦ 4共 p B ⫺p D ⫺ p X 兲

⫻ 具 B 共 p B 兲 兩 „c ¯ ␥ ␮ 共 1⫺ ␥ 5 兲 b…

* 兩 2兩 C 1兩 2 兩 V cb V cs

K ␮⬘ ␯ 共 m b v ,Q 兲 ⫽2 ␲ ␦ „共 m b v ⫺Q 兲 2 ⫺m 2c … ¯ B 共 v 兲 ␥ ␮ 共 1⫺ ␥ 5 兲共 m b v” ⫺Q ⫻Tr关 H ”兲

4

⫻ ␥ ␯ 共 1⫺ ␥ 5 兲 H B 共 v 兲兴 .

⬘

⫻ ␦ 4 共 p B ⫺p D ⫺p X ⫺p X 兲

共35兲

The other factor R ␮ ␯ involves a projection on a D ( * ) meson in the intermediate state and has to be parametrized. Heavy quark spin symmetry for the c quark implies that R ␮ ␯ is of the form

⬘

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

¯ D 共 兲共 v ⬘ 兲 ␥ ␮ 共 1⫺ ␥ 5 兲 R ␮ ␯ 共 m c v ⬘ ,Q 兲 ⬀H *

⫻ 具 D 共 * 兲 共 p D 兲 X ⬘ 兩 „c ¯ ␥ ␯ 共 1⫺ ␥ 5 兲 s…兩 0 典 . 共30兲 114004-4

丢 ␥ ␯ 共 1⫺ ␥ 5 兲 H D 共 * 兲 共 v ⬘ 兲 ,

共36兲

QCD-BASED DESCRIPTION OF ONE-PARTICLE . . .

PHYSICAL REVIEW D 61 114004

where the remaining light quark indices have to be contracted using the most general four-index object. Again there is quite a large number of possibilities to contract the indices, making it useless to discuss the general case. Hence we shall only give two physically motivated ways for modeling this quantity. The first model ansatz corresponds to factorization:

Br共 D * 0 →D 0 Y 兲 ⫽1.

¯ Consequently the total branching ratios to pseudoscalar D mesons are ⫺ X 兲 ⫹0.32 Br共 B→D * ⫺ X 兲 , Br共 B→D ⫺ X 兲 ⫽Br共 B→D dir 0 ¯ 0 X 兲 ⫽Br共 B→D ¯ dir ¯ *0X 兲 X 兲 ⫹Br共 B→D Br共 B→D

R ␮ ␯ 共 m c v ⬘ ,Q 兲 ⫽2F 共 v ⬘ Q,Q 2 兲

⫹0.68 Br共 B→D * ⫺ X 兲 ,

¯ D 共 兲共 v ⬘ 兲 ␥ ␮ 共 1⫺ ␥ 5 兲兴 ⫻Tr关 H * ⫻Tr关 ␥ ␯ 共 1⫺ ␥ 5 兲 H D 共 * 兲共 v ⬘ 兲兴 ,

共37兲

the second one is inspired by the parton model and is defined through

r S⫽

⫻共 Q ” ⫺m c v” ⬘ 兲 ␥ ␯ 共 1⫺ ␥ 5 兲 H D 共 * 兲共 v ⬘ 兲兴 . 共38兲

r Q⫽

We shall discuss the functions F and ˜F using data. IV. EFFECTS OF D * \D DECAYS

The right charm decays of b⫽⫹1 mesons are the ones ¯ 0 or D ⫺ into D ¯ 共 * 兲 0 X, B →D

¯ 共 * 兲 0 X, B →D

B ⫹ →D 共 * 兲 ⫺ X,

B 0 →D 共 * 兲 ⫺ X,

B 0 →D 共 * 兲 0 X,

B ⫹ →D 共 * 兲 ⫹ X,

B 0 →D 共 * 兲 ⫹ X.

共40兲

For the semileptonic case the charge of the the lepton tags the b flavor of the decaying B meson. Wrong charm semileptonic decays are suppressed by the large charm mass and will be ignored in our discussion. Off the heavy mass limit the degeneracy between h D and h D * mesons is removed. In fact the mass difference is large enough to allow strong decays into pions. Thus the rate for ¯ mesons is the sum of a direct contribution and decays into D the contribution arising from the decay chain B→ h D *X ⬘ →h D X. In the narrow width approximation, the latter is obtained by weighting the rate for B→ h D * X by the branching ratios D * →DY where Y is either a pion or a photon. While D * ⫾ decays are governed by the isospin ClebschGordan coefficients receiving only tiny corrections from phase space effects and from the radiative process, ⫹

⫹

⫹

,

共44兲

D 0X 兲 Br共 B→ h . Br共 B→D ⫾ X 兲

共45兲

Br共 B→ h D dirX 兲

r Q⫽

while the wrong charm decays are B ⫹ →D 共 * 兲 0 X,

D *X 兲 Br共 B→ h

We assume that h D * decay is the only relevant isospin violating effect. Under this assumption the charge counting ratio is governed by the spin counting ratio:

0

共39兲

Br共 D * →D Y 兲 ⫽0.32, Br共 D * →D Y 兲 ⫽0.68 共41兲 0

in D * 0 decays isospin invariance is maximally broken by phase space effects 关8兴 such that

共43兲

¯ dir refers to the direct contribution, i.e., the contribuwhere D ¯ * meson appears in an intermediate state. tion where no D We shall later consider spin and charge counting in oneparticle inclusive B decays and define the following ratios:

¯ D 共 兲共 v ⬘ 兲 ␥ ␮ 共 1⫺ ␥ 5 兲 R ␮ ␯ 共 m c v ⬘ ,Q 兲 ⫽2 ˜F 共 v ⬘ Q,Q 2 兲 Tr关 H *

⫹

共42兲

1⫹1.68r S . 1⫹0.32r S

共46兲

Assuming equal rates for the pseudoscalar mesons and for each polarization state of the vector mesons, the naive expectations for these two ratios are r S ⫽3 and r Q ⫽3. D meApart from the ground states h D and h D * , excited h sons may be produced in B meson decays. Even in the heavy mass limit a finite splitting to the ground state mesons remains, which means that these states will decay into h D or h * D . In order to analyze decays into higher resonances, one would need to study the state 兩 X 典 , which we take fully inclusive. Thus we may rely on parton hadron duality to obtain a correct description of the h D and h D * production up to leading order in the heavy mass expansion. Charmed mesons can also be produced via charmonium intermediate states. These states show up in the invariant mass distribution d⌫/dm DD¯ as more or less pronounced resonances. The distribution d⌫/dm cc¯ is certainly different, but by global parton hadron duality and the assumption that all charm quarks hadronize into h D mesons, their integrals are equal, so the charmonium contributions are included in our calculation. V. RESULTS

Even after restricting the number of possible form factors by the Ansa¨tze 共23兲, 共37兲, and 共38兲, we are still left with unknown nonperturbative functions. As far as the right charm contributions are concerned we shall first consider the ¯ * mesons. According to Eq. 共23兲, these are decays into D

114004-5

XAVIER CALMET, THOMAS MANNEL, AND INGO SCHWARZE

PHYSICAL REVIEW D 61 114004

described by a single function ␩ ( vv ⬘ ). Spin symmetry re¯ mesons, however, this is lates these decays to the ones into D only true in the heavy quark limit, where again ␩ ( vv ⬘ ) is the nonperturbative input. In reality one has to take into account ¯ * mesons decay into D ¯ ’s, and thus one has to add this that D contribution using Eqs. 共43兲. Still all right charm decays are given in terms of the single function ␩ ( vv ⬘ ). Comparing the present case to the semileptonic one, we shall use the same saturation assumption as in 关3兴 and write

where f and ˜f are constants. This parametrization using a delta function is motivated by the negligible mass of the strange quark against which the two charmed quarks recoil. It can actually be checked by measuring the invariant mass ¯ X ⬘ which m X ⬘ of the additional decay products in B→DD ˜ should turn out to be small. Setting f to one, Eq. 共38兲 reproduces the well-known parton model result. The fits yield

␩ 共 vv ⬘ 兲 ⫽ 兩 ␰ 共 vv ⬘ 兲 兩 2 .

Note that one naively expects ˜f ⫽1/8 when assuming isospin invariance in B→D * X and B→D dirX. Under this assumption, we have two charge states and four spin states, hence in total eight states contributing. For the semileptonic contributions, we use the results of 关3兴,1 but apply the same approximations as for the nonleptonic channels. We neglect the renormalization group improvement and find

共47兲

For numerical calculations we use the measurement 关9兴 of the Isgur-Wise function

␰ 共 vv ⬘ 兲 ⫽1⫺a 共 vv ⬘ ⫺1 兲 ,

a⫽0.84.

共48兲

Since the exclusive semileptonic decays are spectatorlike and since in the limit of factorization, the matrix elements of the heavy quark current are the same in the semi- and nonleptonic cases, dominance of spectatorlike decays is a natural assumption for the nonleptonic right charm case as well. Therefore we use the results of Sec. II for the channels B

⫹ ⫹

0 ¯ dir →D X,

¯ * 0 X, B →D

B

0

f ⫽0.147 GeV

G sl 共 M 2 兲 ⫽

⫺

B →D * X,

1 共 M 2 g ␮ ␯ ⫺M ␮ M ␯ 兲 ⌰ 共 M 2 兲 ␩ 共 vv ⬘ 兲 3␲

¯ D¯ 共 兲共 v ⬘ 兲 ␥ ␯ 共 1⫺ ␥ 5 兲 H B 共 v 兲兴 . ⫻Tr关 H * 共49兲

neglecting possible contributions from nonspectator channels such as B ⫹ →D ⫺ X. Another class of decays allow to obtain a factorizable expression for H eff after a Fierz transformation. However, the color indices need to be rearranged as well, yielding a suppression by one power of 1/N C in the amplitude. Since we are working to leading order in 1/N C , ¯ * 0 X, whereas the channel we find a vanishing rate for B 0 →D ¯ 0 X is fed by the decay chain via the D * ⫺ meson. B 0 →D ¯ s, we For the contribution of the quark level decay b→cc assume the same number of right charm quarks to hadronize ¯ ( * ) as wrong charm quarks hadronize as either D ( * ) or as D ( ) D s* . Using the CLEO wrong charm measurement Br(B →DX)⫽(7.9⫾2.2)% 关10兴 and the Particle Data Group average Br(B→D s⫾ X)⫽(10.0⫾2.5)% 关11兴 and neglecting a possible right charm B→D s⫺ contribution, we obtain a total ¯ s of about 18%. right charm contribution from b→cc It is known that the channel b→cc ¯ s receives large radiative corrections computed in 关12兴. We shall not include these corrections in their detailed form, rather we shall take into account their bulk effect by adjusting the charm quark mass in Eqs. 共26兲–共28兲. Inserting an ‘‘effective’’ mass m eff c ⫽1.0 GeV into the tree level relation, the measured rate is reproduced. Data are sparse for the wrong charm part, therefore we replace the unknown functions in Eqs. 共37兲 and 共38兲 by F 共 v ⬘ Q,Q 2 兲 ⫽ f 2 ␲ ␦ „共 Q⫺m c v ⬘ 兲 2 …, ˜F 共 v ⬘ Q,Q 2 兲 ⫽˜f 2 ␲ ␦ „共 Q⫺m c v ⬘ 兲 2 …,

2

兩 V cb 兩 2

共51兲

¯ B 共 v 兲 ␥ ␮ 共 1⫺ ␥ 5 兲 H D¯ 共 兲共 v ⬘ 兲兴 ⫻Tr关 H *

⫺ →D dir X,

0

G F2

and ˜f ⫽0.121.

共52兲

The ␶ contribution follows from Eqs. 共26兲–共28兲 replacing V cs , N C and C 1 by one and the charm quark mass by the ␶ mass. Table I shows the predictions for the total rates of these two Ansa¨tze for the wrong charm piece. For the numerical calculation, we used 兩 V cb 兩 ⫽0.04. VI. COMPARISON WITH DATA

Data on one-particle inclusive B decays are available from ⌼(4S) machines and also from the CERN e ⫹ e ⫺ collider LEP. At the ⌼(4S), the total production rates of D ( * )⫹ and ¯ ( * )0 and D ( * )0 are measured on the resonance, D ( * )⫺ or D from which one can only deduce the rates for B admixture 1 ⌫ 共 B→D 共 * 兲 ⫾ X 兲 ⫽ 关 ⌫ 共 B ⫹ →D 共 * 兲 ⫾ X 兲 2 ⫹⌫ 共 B 0 →D 共 * 兲 ⫾ X 兲兴 , 1 D 共 * 兲0X 兲 ⫽ 关 ⌫ 共 B ⫹→ h D 共 * 兲0X 兲 ⌫ 共 B→ h 2 ⫹⌫ 共 B 0 → h D 共 * 兲 0 X 兲 ].

共53兲

Note the sign error afflicting one term in 关3兴, Eq. 共43兲,

1

0¯ 0

共50兲

which should read E VB D (y)⫽⫺ 关 C 11(y,⌳)/C 23 (y,⌳) 兴 Br(D * ⫺ ¯ 0 X) 21 (y 2 ⫺1) 兩 X(y) 兩 2 ⫹C 18(y,⌳)(1/N c ). →D

114004-6

QCD-BASED DESCRIPTION OF ONE-PARTICLE . . .

PHYSICAL REVIEW D 61 114004

TABLE I. Comparison of our results with data. Model 1 uses Eq. 共37兲; model 2 uses Eq. 共38兲. Branching ratios are computed using ␶ B ⫹ ⫽ ␶ B 0 ⫽1.55 ps. Mode

Model 1

Model 2

B→ h DX

68.8%

68.8%

(87.2⫾3.5)%

h D 0 ⫹D ⫾

B→ h D *X

51.8%

52.7%

(48.7⫾3.1)%

h D * 0 ⫹D * ⫾

B→ h D dirX

17.1%

16.1%

(38.5⫾4.7)%

h D ⫺h D*

B→DX

7.9%

7.9%

(7.9⫾2.2)%

B→D * X

5.0%

5.9%

B→ h D 0X

52.0%

52.3%

(63.1⫾2.9)%

关11,14兴

B→D X

16.8%

16.5%

(24.1⫾1.9)%

关11,14兴

B→ h D *0X

25.9%

26.3%

(26.0⫾2.7)%

关11,14兴

25.9%

26.3%

(22.7⫾1.6)%

关11,14,15兴

8.5%

8.1%

(21.7⫾4.1)%

h D 0⫺ h D * 0 ⫺0.68D * ⫾

8.5%

8.1%

(16.8⫾2.9)%

D ⫾ ⫺0.32D * ⫾

⫾

⫾

B→D * X 0 X B→ h D dir ⫾ B→D dirX

Experiment

关10兴 共input兲

B→D ⫺ l ⫹ ␯ l X

2.0%

2.0%

(2.7⫾0.8)%

关11,16兴

¯ 0l ⫹␯ lX B→D

6.5%

6.5%

(7.0⫾1.4)%

关11,16兴

B→D * ⫺ l ⫹ ␯ l X

3.3%

3.3%

(2.8⫾0.4)%

关11,9兴

¯ * l⫹ ␯ l X B→D

3.3%

3.3%

(3.2⫾0.7)%

关11,9兴

B→D ␶ ⫹ ␯ ␶ X

0.6%

0.6%

¯ 0␶ ⫹ ␯ ␶X B→D

2.0%

2.0%

B→D * ␶ ⫹ ␯ ␶ X

1.0%

1.0%

¯ * ␶ ⫹ ␯ ␶X B→D

1.0%

1.0%

31.8%

31.8%

0

⫺

⫺ 0

¯ 0X B 0 →D ⫺

B →D X 0

29.1%

29.1%

¯ 0X B →D

60.9%

60.9%

B ⫹ →D ⫺ X

0%

0%

⫹

B →D X

5.7%

6.0%

B 0 →D ⫹ X

2.2%

1.9%

0

0

B ⫹ →D 0 X

5.7%

6.0%

B ⫹ →D ⫹ X

2.2%

1.9%

¯ *0X B 0 →D

0%

0%

⫺

B →D * X 0

46.8%

46.8%

¯ *0X B →D

46.8%

46.8%

B ⫹ →D * ⫺ X

0%

0%

⫹

B →D * X

2.5%

3.0%

B 0 →D * ⫹ X

2.5%

3.0%

0

0

B ⫹ →D * 0 X

2.5%

3.0%

B ⫹ →D * ⫹ X

2.5%

3.0%

Table I lists the available data. In the fourth row of the table we list the CLEO measurement of wrong charm decays B →DX which is again the average over B ⫹ and B 0 . Although we expect our method to work best near the nonrecoil point vv ⬘ ⫽1, we integrate the spectra in order to obtain total rates, assuming we can still get some insight into the bulk features. Since we calculate the right and wrong

charm contributions separately, we have to add them in order to obtain the number of h D mesons produced in B meson decays. This procedure corresponds to the usual multiplicity definition of the branching ratio 关11, p. 570兴. The first observable one can study is the total number of h D mesons in B decays. This quantity is considerably smaller

114004-7

XAVIER CALMET, THOMAS MANNEL, AND INGO SCHWARZE

PHYSICAL REVIEW D 61 114004

TABLE II. Predicted and measured spin counting ratio r S . For the last line and for the fit, the relation 共46兲 between charge and spin counting has been used as a constraint. For the column ‘‘all channels,’’ the error of the fit has been scaled by 冑␹ 2 /N f , where ␹ 2 ⫽4.2 and N f ⫽2. D *X 兲 Br共 B→ h rS⫽ Br共 B→ h D dirX 兲

All channels

Naive spin counting Model 1 关see Eq. 共37兲兴 Model 2 关see Eq. 共38兲兴 Fit all data using h D (*)0, D (*)⫾

Data using h D * and h D dir only 0 h Data using D and D ⫾ only

Semileptonic

3 3.04 3.26 1.39⫾0.27

3

FIG. 3. Non-factorizing isospin-violating topology, supplying a ¯ 0 rate. possible explanation for an enhanced D

3.36 1.58⫾0.70

1.26⫾0.22

1.62⫾0.91

1.92⫾0.46

1.87⫾1.74

than the usually performed charm counting since it does not include the production of D s mesons, charmed baryons and charmonium, the sum of which amounts to about 20% of the total charm production in B decays. Using the saturation assumption in Eq. 共47兲, our model yields a total h D rate about 20% below the experimental result. However, Eq. 共47兲 was taken from the analysis of the semileptonic decays where the ¯ and D ¯ * exclusive final states saturate only (71⫾13)% of D ¯ l ⫹ ␯ l X. Thus Eq. 共47兲 should be rethe inclusive rate B→D placed by

␩ 共 vv ⬘ 兲 ⫽ 兩 ␰ 共 vv ⬘ 兲 兩 2 ⫹ ␦␩ 共 vv ⬘ 兲 ,

共54兲

where ␦␩ accounts for the remaining contributions. Because we use factorization, this would enhance the nonleptonic channels by a similar amount and hence improve the predic¯ mesons. tion for the total number of D Keeping this in mind, we still stick to the saturation assumption 共47兲 since ␦␩ is unknown and the naive use of Eq. 共47兲 reproduces the data on B→ h D * X decays. This could be accidental, as the real problem is spin counting. The ratio of vector to direct pseudoscalar mesons r S defined in Eq. 共44兲 is expected to be about three, but experiment yields a ratio barely above one 共see Table II兲. Even in the semileptonic case, the experimental value is slightly lower than expected, although errors are large. Since the rates of pseudoscalar and TABLE III. Predicted and measured charge counting ratios r Q . Using the measured value r S ⬇3/2, naive charge counting yields r Q ⫽7/3 instead of 3. Channel

Model:

Br(B→ h D * 0 X)/Br(B→D * ⫾ X) 0 ⫾ X)/Br(B→D dir X) Br(B→ h D dir

Br(B→ h D 0 X)/Br(B→D ⫾ X) Br(B→ h D * 0 l ␯ X)/Br(B→D * ⫾ l ␯ X) 0 ⫾ l ␯ X)/Br(B→D dir l ␯ X) Br(B→ h D dir 0 ⫾ Br(B→ h D l ␯ X)/Br(B→D l ␯ X)

Naive

1

2

Data

1

1

1

1.15⫾0.14

1

1

1.29⫾0.17

1 3

3.09 3.17 2.62⫾0.24

1

1

1.14⫾0.30

1

1

1.05⫾0.89

3

3.20

2.59⫾0.93

vector h D mesons are connected by heavy quark spin symmetry, this large discrepancy is difficult to understand. Although the effect is less pronounced than the spin counting problem, neutral h D mesons tend to occur slightly more often than expected, see Table III. Assuming the measured value for the spin counting ratio r S , the measured charge counting ratios r Q are generally one or two standard deviations above expectations. If confirmed, this kind of effect would point to an additional contribution which could for instance arise from a nonfactorizing topology as depicted in Fig. 3. Nevertheless, current data are still consistent with the relation 共46兲 between charge and spin counting, see the fit in Table II. Another piece of information for these decays are the h D momentum spectra. These have been measured by ARGUS 关13兴 and CLEO 关14兴, we use the data from CLEO which is more recent and more precise. The spectra are momentum distributions in the rest frame of the ⌼(4S). However, the effect of the motion of the B mesons produces only a negligible smearing of these spectra, and we can safely ignore this effect here. In Fig. 4 we compare the data obtained by CLEO 关14兴 with our theoretical prediction. The spectra show that there is indeed a problem with the ratio of pseudoscalar to vector mesons. Using the saturation assumption in Eq. 共47兲, the spectra of the vector mesons are described within experimental uncertainties and the problem appears with the low momentum region of the decay spectra for the pseudoscalar mesons. Our theoretical ansatz, especially the SDE, should work best in this region of small h D momentum. In addition, the shape of the spectra in this region is mainly determined by phase space, which yields a 2 for small p D and constant matrix behavior proportional to p D element. The steep rise of the momentum spectra for the pseudoscalar h D mesons is thus difficult to understand. An investigation of slow direct pseudoscalar h D mesons, i.e., those that do not originate from intermediate h D * vector mesons, would be desirable. This problem appears to be limited to nonleptonic decays. ¯ meson momentum In Fig. 5 we compare the semileptonic D spectrum measured by CLEO 关10兴 with the theoretical predictions from 关3兴 as well as with the sum of the first six contributing exclusive channels 关17兴 and find no significant disagreement. There is also a measurement of the wrong charm D spectrum by CLEO 关10兴. Unfortunately only four bins could be

114004-8

QCD-BASED DESCRIPTION OF ONE-PARTICLE . . .

PHYSICAL REVIEW D 61 114004

FIG. 4. Momentum spectra 1/⌫ tot d⌫/dpD of B→ h D X in comparison with theoretical predictions. The solid line is model 2, the dashed one model 1. The columns refer to h D, h D * and direct h D mesons, the rows to a charge sum, neutral and charged h D ( * ) mesons.

measured having still substantial uncertainties. In Fig. 6 we see that the parton-model inspired model 2 fits the data better than model 1, although evidence is not yet conclusive due to the quality of the data. VII. CONCLUSIONS

In this paper we have developed a QCD based description ¯ X and B of one-particle inclusive decays of the type B→D →DX. The method we suggest is based on the large-N c limit of QCD, allowing us to factorize certain matrix elements. Once factorization has been performed, one can identify pieces in the rates which can be treated by a short distance expansion, assuming the bottom and the charm quarks to be heavy. This yields a series in inverse powers of the parameter m b ⫺m c . The numerator of the expansion param-

eter is a typical QCD scale for the light degrees of freedom. Thus the expansion parameters are ⌳ QCD /(m b ⫺m c ), 1/N C and ␣ s (m c ) and hence corrections to our calculation could be fairly large, in the worst case of the order of 30%. We have studied the leading term of this expansion which still contains a number of unknown nonperturbative functions. These functions have to be parametrized. In the same way as in the semileptonic case, we reduce the number of functions appearing in the right charm contributions to a single nonperturbative function which can be related to the Isgur Wise function, once we assume that most of the rate is saturated by the decays into the two ground state mesons h D h and D * . For the wrong charm case we suggest two models corresponding to two different ways of contracting the spinor indices. Both models have a single nonperturbative form factor, which we adjust to the experimental wrong charm yield.

114004-9

XAVIER CALMET, THOMAS MANNEL, AND INGO SCHWARZE

¯ momentum specFIG. 5. One-particle inclusive semileptonic D ¯ l ⫹ ␯ l X measured by CLEO 关10兴. The solid line is the trum in B→D prediction of 关3兴; the dashed lines are sums of predictions for exclusive channels following 关17兴.

PHYSICAL REVIEW D 61 114004

FIG. 6. One-particle inclusive wrong charm D momentum spectrum in B→DX measured by CLEO 关10兴 in comparison with theoretical predictions. The solid line is model 2; the dashed one model 1.

perimental rates are above the theoretical ones, it would be interesting to investigate which exclusive channels contribute in the small momentum region.

Although our method should work best close to the nonrecoil point, we also calculate total rates in order to discuss the total number of h D mesons in B decays and the spin and charge counting. While the well-known problem of spin counting, see, e.g., the review 关18兴, is not solved by our ansatz, charge counting seems to work well, once we properly take the D * →D decays into account. The shapes of the decay spectra into vector mesons are already described quite satisfactorily, in particular in the low momentum region, while the h D mesons spectra are off in this region. Furthermore, our model reproduces the normalization of the h D * spectra, such that the spin counting problem manifests itself in a deficit of h D mesons. Since the ex-

The authors thank U. Nierste for discussions concerning the method, I. Bigi for valuable criticism and L. Gibbons for clarifications concerning the CLEO data. We also thank C. Balzereit who participated in the early stages of this work and M. Feindt for discussions on the experimental prospects. This work was supported by the DFG Graduiertenkolleg ‘‘Elementarteilchenphysik an Beschleunigern’’ and by the DFG Forschergruppe ‘‘Quantenfeldtheorie, Computeralgebra und Monte-Carlo-Simulation.’’

关1兴 See, e.g., the following reviews: M. Neubert, in Effective Theories and Fundamental Interactions, Proc. Erice 1996, edited by A. Zichichi 共World Scientific, Singapore, 1997兲, p. 98, hep-ph/9610266; T. Mannel, Rep. Prog. Phys. 60, 1113 共1997兲; M. Shifman, in QCD and Beyond, Proc. TASI ’95, edited by D. E. Soper 共World Scientific, Singapore, 1996兲, p. 409, hep-ph/9510377; H. Georgi, in Perspectives in the Standard Model, Proc. TASI ’91, edited by R. K. Ellis, C. T. Hill, and J. D. Lykken 共World Scientific, Singapore, 1992兲, p. 589; N. Isgur and M. B. Wise, in B Decays, 2nd ed., edited by S. Stone 共World Scientific, Singapore, 1994兲, p. 231. 关2兴 I. I. Bigi, N. G. Uraltsev, and A. I. Vainshtein, Phys. Lett. B 293, 430 共1992兲; I. I. Bigi, M. Shifman, N. G. Uraltsev, and A. Vainshtein, Phys. Rev. Lett. 71, 496 共1993兲; J. Chay, H. Georgi, and B. Grinstein, Phys. Lett. B 247, 399 共1990兲; A. V. Manohar and M. B. Wise, Phys. Rev. D 49, 1310 共1994兲; T. Mannel, Nucl. Phys. B413, 396 共1994兲. 关3兴 C. Balzereit and T. Mannel, Phys. Rev. D 59, 034003 共1999兲.

关4兴 M. Wirbel and Y. L. Wu, Phys. Lett. B 228, 430 共1989兲. 关5兴 G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 共1996兲. 关6兴 G. ’t Hooft, Nucl. Phys. B72, 461 共1974兲; B75, 461 共1974兲. 关7兴 M. Bauer, B. Stech, and M. Wirbel, Z. Phys. C 84, 103 共1987兲. 关8兴 Mark I Collaboration, G. Goldhaber et al., Phys. Lett. 69B, 503 共1977兲. 关9兴 CLEO Collaboration, B. Barish et al., Phys. Rev. D 51, 1014 共1995兲. 关10兴 CLEO Collaboration, T. E. Coan et al., Phys. Rev. Lett. 80, 1150 共1998兲. 关11兴 Particle Data Group, C. Caso et al., Eur. Phys. J. C 3, 1 共1998兲. 关12兴 E. Bagan, P. Ball, B. Fiol, and P. Gosdzinsky, Phys. Lett. B 351, 546 共1995兲. 关13兴 ARGUS Collaboration, H. Albrecht et al., Z. Phys. C 52, 353 共1991兲. 关14兴 CLEO Collaboration, L. Gibbons et al., Phys. Rev. D 56, 3783

ACKNOWLEDGMENTS

114004-10

QCD-BASED DESCRIPTION OF ONE-PARTICLE . . .

PHYSICAL REVIEW D 61 114004

共1997兲. 关15兴 ARGUS Collaboration, H. Albrecht et al., Phys. Lett. B 374, 256 共1996兲. 关16兴 CLEO Collaboration, R. Fulton et al., Phys. Rev. D 43, 651 共1991兲.

关17兴 A. K. Leibovich, Z. Ligeti, I. W. Stewart, and M. B. Wise, Phys. Rev. D 57, 308 共1998兲. 关18兴 ARGUS Collaboration, H. Albrecht et al., Phys. Rep. 276, 223 共1996兲.

114004-11