PHYSICAL REVIEW D, VOLUME 62, 096014

One-particle inclusive CP asymmetries Xavier Calmet Ludwig-Maximilians-Universita¨t, Sektion Physik, Theresienstraße 37, D-80333 Mu¨nchen, Germany

Thomas Mannel and Ingo Schwarze Institut fu¨r Theoretische Teilchenphysik, Universita¨t Karlsruhe, D-76128 Karlsruhe, Germany 共Received 5 January 2000; published 12 October 2000兲 One-particle inclusive C P asymmetries in the decays of the type B→ h D ( * ) X are considered in the framework of a QCD-based method to calculate the rates for one-particle inclusive decays. PACS number共s兲: 11.30.Er, 13.25.Hw

I. INTRODUCTION

One of the main goals in B physics is a detailed study of flavor mixing, which is encoded in the Cabibbo-KobayashiMaskawa 共CKM兲 matrix of the standard model. In particular, the violation of the C P symmetry, which the standard model describes by a nontrivial phase in the CKM matrix or equivalently by the angles of the unitarity triangle, will be investigated. Typically C P asymmetries are expected to be large in some of the exclusive nonleptonic B decays which, however, have only small branching ratios. Examples are the determination of ␤ from B→J/ ␺ K S and of ␣ from B→ ␲␲ . In addition, in these exclusive nonleptonic decays it is very hard to obtain a good theoretical control over the hadronic uncertainties, in particular due to the presence of strong phases. On the other hand, inclusive decays have large branching fractions but typically smaller C P asymmetries than exclusive decays 关1兴. One may use parton hadron duality to obtain a good theoretical description. This has been studied by Beneke, Buchalla and Dunietz who set up a theoretically clean method to calculate the C P asymmetries in inclusive B decays 关2兴. They still find sizable C P asymmetries, but their measurement would require to identify charmless final states inclusively, which is not an easy task. One-particle inclusive decays lie somehow between these two cases. This class of decays still has large branching fractions and some of the expected C P asymmetries are sizable. Furthermore, a measurement of these decays is feasible. For one-particle inclusive decays of the type B h → D ( * ) X, a QCD-based description has been developed recently, exploiting factorization and the heavy mass limit for both the b and the c quark 关3兴. Since the expansion parameters are ⌳ QCD /(m b ⫺m c ), 1/N C and ␣ s (m c ), corrections to the leading term could be fairly large, in the worst case of the order of 30%. Using this method, which unfortunately is not completely model independent, we compute mixing induced time-dependent and time-integrated C P asymmetries in the framework of the standard model. In view of the considerable uncertainties due to an unknown strong phase, our method cannot yet be used for a competitive determination of the C P violation parameters, in particular compared to a measurement of sin(2␤) in the 0556-2821/2000/62共9兲/096014共6兲/$15.00

‘‘gold-plated’’ channel B→J/ ␺ K S . However, it can be used as an estimate of the one-particle inclusive C P asymmetries, for which we shall use present central values of the C P angles ␤ and ␥ 关4兴. Compared to fully inclusive methods, the advantage is that we can predict asymmetries for the various spins and charges of the ground-state charmed mesons separately. This is certainly a worthwhile task, in particular since we are not aware of any previous prediction for these asymmetries, not even in the context of quark models. After introducing our notations for B mixing in Sec. II, we calculate the relevant matrix elements in Sec. III and model the form factors in Sec. IV. The numerical results are given in Sec. V. II. CP ASYMMETRIES IN B\ n D „*…X

In Wigner Weisskopf approximation the time evolution of an initially pure B 0 or ¯B 0 , 0 兩 B phys 共 t 兲 典 ⫽g ⫹ 共 t 兲 兩 B 0 典 ⫺

q g 共 t 兲 兩 ¯B 0 典 , p ⫺

0 兩¯ B phys B 0典 ⫺ 共 t 兲 典 ⫽g ⫹ 共 t 兲 兩 ¯

p g 共 t 兲兩 B 0典 , q ⫺

共1兲

is determined by the time-dependent functions

冋

g ⫹ 共 t 兲 ⫽e ⫺iM t⫺ 共 1/2兲 ⌫t cosh ⫹i sinh g ⫺ 共 t 兲 ⫽e

⌬⌫t ⌬M t cos 4 2

册

⌬M t ⌬⌫t sin , 4 2

⫺iM t⫺ 共 1/2兲 ⌫t

⫹i cosh

冋

⌬M t ⌬⌫t cos sinh 4 2

共2兲

册

⌬⌫t ⌬M t , sin 4 2

where ⌬M ⫽M H ⫺M L ⬎0 and ⌬⌫⫽⌫ H ⫺⌫ L ⬍0 are the mass and width differences between the mass eigenstates 兩 B H 典 ⫽ p 兩 B 0 典 ⫹q 兩 ¯ B 0 典 and 兩 B L 典 ⫽ p 兩 B 0 典 ⫺q 兩 ¯B 0 典 . The quantity q/p is given in terms of the off-diagonal elements of the Hamiltonian H⫽M ⫺i⌫/2 of the neutral B meson system

62 096014-1

©2000 The American Physical Society

XAVIER CALMET, THOMAS MANNEL, AND INGO SCHWARZE

冉

冊

PHYSICAL REVIEW D 62 096014

* M 12 1 q ⌬M ⫺ 共 i/2兲 ⌬⌫ ⫽ 1⫺ a⫹O共 a 2 兲 , ⫽ p 2 关 M 12⫺ 共 i/2兲 ⌫ 12兴 兩 M 12兩 2

再 冎

⌫ 12 a⫽Im . M 12

¯B ¯B

¯ X 兴 ⫽ 兩 g ⫹共 t 兲兩 2⌫ ¯ ⫹ ⌫ 关 ¯B 共 t 兲 →Y Y 共3兲 ⫺2 Re

In fact, ⌫ 12 /M 12⫽O(m 2b /m 2t ) is very small and hence q/p is to a good approximation a phase factor. The time-dependent rate for the decay of a B meson into a set of final states 兩 f 典 ⫽ 兺 i 兩 f i 典 can be written as 1 ⌫关 B共 t 兲→ f 兴⫽ 2m B

兺i

冕

1 2m B

⫽

冕

兺i

冕

d ␾ i 兩 f i 典具 f i 兩

共5兲

⌫ 关 B 0 共 t 兲 → f 兴 ⫺⌫ 关 ¯B 0 共 t 兲 →¯f 兴 ⌫ 关 B 0 共 t 兲 → f 兴 ⫹⌫ 关 ¯B 0 共 t 兲 →¯f 兴

共6兲

which involves the C P conjugate set 兩¯f 典 of final states. Up to here the discussion is completely general. In the following we shall use the above formalism to compute the C P asymmetries for one-particle inclusive final states, for which the projector reads ⌸f⫽

兺X 兩 XY 典具 XY 兩 ,

共7兲

¯ meson. Since the sum runs over where Y can be a D or a D all possible states X, the C P conjugate of the projector is ⌸¯f ⫽

1 2m B

¯ T BB Y ⫽

1 2m B

冕 冕

d 4 x 具 B 兩 H eff共 x 兲 ⌸ Y H eff共 0 兲 兩 B 典 , 共10兲 d 4 x 具 B 兩 H eff共 x 兲 ⌸ Y H eff共 0 兲 兩 ¯B 典 .

¯

is the projector on the set of final states. Note that both an exclusive final state as well as inclusive states can be treated in this way. Even differential distributions can be considered if the phase spaces d ␾ i are not fully integrated. The C P asymmetries we are going to consider are of the type AC P 共 t 兲 ⫽

⌫ BB Y ⫽

兺X 兩 XY¯ 典具 XY¯ 兩 .

共8兲

Inserting the time-dependent states 共1兲 we obtain ⌫ 关 B 共 t 兲 →Y X 兴 ⫽ 兩 g ⫹ 共 t 兲 兩 2 ⌫ BB Y ⫹ ⫺2 Re

再

冏

冏

2 q ¯¯ g ⫺ 共 t 兲 ⌫ BY B p

冎

q ¯ g * g 共 t 兲 T BB , Y p ⫹ ⫺

¯

BB T Y ªT BB Y ⫽共 T Y 兲*.

共4兲

where d ␾ i is the phase space element of the state 兩 f i 典 and ⌸f⫽

冎

¯B B p g * g 共 t 兲 T ¯Y , q ⫹ ⫺

The ⌬B⫽2 transition matrix elements representing the interference between the mixed and the unmixed amplitudes are related by C PT symmetry, such that

d 4 x 具 B 共 t 兲 兩 H eff共 x 兲

⫻⌸ f H eff共 0 兲 兩 B 共 t 兲 典 ,

冏

where the matrix elements are defined by

d ␾ i共 2 ␲ 兲 4␦ 4共 p B⫺ p f i 兲

⫻ 具 B 共 t 兲 兩 H eff兩 f i 典具 f i 兩 H eff兩 B 共 t 兲 典

再

冏

2 p BB g ⫺ 共 t 兲 ⌫ ¯Y q

共11兲

The direct C P asymmetries in these processes are expected to be tiny. In fact, using the method described in Ref. 关3兴, they turn out to be of higher order in the 1/m expansion. Hence we have ¯B ¯B

⌫ Y ª⌫ BB Y ⫽⌫ ¯Y ⫽⌫ 共 B→Y X 兲 ,

BB

⌫ ¯Y ª⌫ ¯Y ,

T Y ⫽T ¯Y .

共12兲 共13兲

Inserting the time-dependent decay rates 共9兲 and neglecting both the width difference and a, such that q/p becomes a phase factor, we obtain for the time-dependent C P asymmetries AC P 共 t 兲 ⫽

sin共 ⌬M t 兲 Im兵 共 q/p 兲 T Y 其 , cos2 共 ⌬M t/2兲 ⌫ Y ⫹sin2 共 ⌬M t/2兲 ⌫ ¯Y

共14兲

from which we get the time-integrated asymmetry AC P ⫽

2x Im兵 共 q/p 兲 T Y 其 , 共 2⫹x 2 兲 ⌫ Y ⫹x 2 ⌫ ¯Y

共15兲

where x⫽⌬M /⌫ is measured to be x⫽0.73 关5兴. III. TRANSITION MATRIX ELEMENTS

In order to compute the C P asymmetries, one has to evaluate the matrix elements 共10兲. The total rates ⌫ Y have already been discussed in Ref. 关3兴, so we only need to calculate the interference term T Y . The relevant pieces of the effective Hamiltonian contrib¯ c) V⫺A and ¯ b) V⫺A (d uting to this interference are (u ¯ u) V⫺A interfering with each other and ¯ b) V⫺A (d (c ¯ c) V⫺A interfering with itself, so T Y is a sum of ¯ b) V⫺A (d (c the two contributions

共9兲 096014-2

T Y ⫽T c ⫹T u ,

共16兲

ONE-PARTICLE INCLUSIVE C P ASYMMETRIES

1 G F2 T q⫽ V V* V V* 兩C 兩2 2m B 2 cb qd qb cd 1

PHYSICAL REVIEW D 62 096014

In principle, all possible contractions of the light quark indices may contribute, giving rise to several form factors. For a first estimate, it is sufficient to use only the simplest one of these contractions,

兺X 共 2 ␲ 兲 4

⫻ ␦ 4 共 p B ⫺p D ⫺p x 兲 具 B 0 兩 共 ¯q b 兲 V⫺A 共 ¯d c 兲 V⫺A 兩 DX 典 ⫻具 DX 兩 共 ¯d q 兲 V⫺A 共¯c b 兲 V⫺A 兩 ¯B 0 典 .

P ␮q ␯ 共 p D ,Q 兲 ⫽2 ␲ ␦ „共 Q⫺p D 兲 2 ⫺m 2q …Tr兵 p” D ␥ ␮ 共 1⫺ ␥ 5 兲

共17兲

¯ b) V⫺A (u ¯ c) V⫺A , Fierzing the operators into the form (d ¯ ¯ ¯ u) V⫺A and (d b) V⫺A (c ¯ c) V⫺A one can reproduce (d b) V⫺A (c the inclusive results of Ref. 关2兴. In order to evaluate the interference term for the one-particle inclusive case, we use the method developed in Ref. 关3兴. It is based on factorization, which holds to leading order in the 1/N C expansion, where N C is the number of QCD colors. Thus we can write the interference terms as products of two tensors

⫻共 Q ” ⫺ p” D 兲 ␥ ␯ 共 1⫺ ␥ 5 兲 其˜f qY ,

corresponding to a replacement of the D ( * ) X final state by a pair of free quarks, rescaled by an operator- and decaychannel-specific form factor ˜f qY , where Y is one of the ground state D mesons. In the following, we call this contraction ‘‘partonic.’’ Using this ansatz and the heavy mass limit, the transition matrix elements read

1 G F2 T q⫽ V V* V V* 兩C 兩2 2m B 2 cb qd qb cd 1 ⫻

冕

d 4Q K 共 p ,Q 兲 共 2 ␲ 兲4 ␮␯ B

冕

T c ⫽⫺

with

⫻ 具 B 0 共 p B 兲 兩 共 ¯d ␥ ␮ 共 1⫺ ␥ 5 兲 b 兲 兩 X 典

P q␮ ␯ 共 p D ,Q 兲 ⫽

共19兲

共 2 ␲ 兲 4 ␦ 4 共 Q⫺ p D ⫺p X ⬘ 兲 兺 X

⫻ 具 0 兩 共 ¯q ␥ ␮ 共 1⫺ ␥ 5 兲 c 兲 兩 D 共 * 兲 共 p D 兲 X ⬘ 典 共20兲

The tensor K ␮ ␯ ( p B ,Q) is fully inclusive and one can perform a standard short distance expansion. The resulting ⌬B ⫽2 matrix element can be parameterized by the decay constant f B of the B meson and the bag factors B and B s for the axial vector and the scalar current, respectively. The other tensor P q␮ ␯ (p D ,Q) involves a projection on a one-particle inclusive charmed meson state and hence we cannot perform a short distance expansion. We proceed along the same lines as in Ref. 关3兴, where the rates for wrong charm decays have been modeled. Heavy quark symmetry yields the Dirac matrix structure ¯ D 共 兲共 p D 兲 ␥ ␮ 共 1⫺ ␥ 5 兲 丢 ␥ ␯ 共 1⫺ ␥ 5 兲 H D 共 兲共 p D 兲 , P ␮q ␯ 共 p D ,Q 兲 ⬀H * * 共21兲 where the representation matrices for the charmed mesons are H D ⫽ 冑m D

1⫹ v” D ␥5 , 2

H D * ⫽ 冑m D *

* 兲 2 兩 C 1 兩 2 冑1⫺4z 共 V cb V cd

G F2 m B3 f B2 24␲

1⫹ v” D * ⑀” . 共22兲 2

共24兲

* V ub V cd * 兩 C 1 兩 2 共 1⫺z 兲 2 V cb V ud 共25兲

where z⫽(m c /m b ) 2 and C 1 is the Wilson coefficient of the effective Hamiltonian in the notation of Ref. 关3兴. Equations 共24兲 and 共25兲 correspond to the expression for the width difference of neutral heavy meson systems 关6兴. In the standard CKM parametrization, the phases of the transition matrix elements are

⬘

⫻ 具 D 共 * 兲 共 p D 兲 X ⬘ 兩¯c ␥ ␯ 共 1⫺ ␥ 5 兲 q 兩 0 典 .

24␲

⫻ 关共 1⫺z 兲 B⫹2 共 1⫹2z 兲 B S 兴˜f uY ,

兺X 共 2 ␲ 兲 4 ␦ 4共 p B ⫺p X ⫺Q 兲 ⫻ 具 X 兩 共 ¯d ␥ ␯ 共 1⫺ ␥ 5 兲 b 兲 兩 ¯B 0 共 p B 兲 典 ,

G F2 m B3 f B2

⫻ 关共 1⫺4z 兲 B⫹2 共 1⫹2z 兲 B S 兴˜f cY ,

d ␾ D P ␮q ␯ 共 p D ,Q 兲 共18兲

T u ⫽⫺ K ␮ ␯ 共 p B ,Q 兲 ⫽

共23兲

arg共 T c 兲 ⫽0,

共26兲

arg共 T u 兲 ⫽arg共 ⫺V ub 兲 ⫽⫺ ␥ ,

共27兲

arg共 q/p 兲 ⫽arg共 ⫺V 2td 兲 ⫽⫺2 ␤ ,

共28兲

such that Im

再 冎

q T ⫽sin共 2 ␤ 兲 兩 T c 兩 ⫹sin共 2 ␤ ⫹ ␥ 兲 兩 T u 兩 . p Y

共29兲

IV. MODELING THE FORM FACTORS

We assume that the form factors ˜f qY do not vary strongly over the accessible phase space and hence we approximate them by constants. For the case q⫽c, these constants have been fitted to the wrong charm yield in B decays 关3兴. Operators analogous to the case q⫽u are Cabibbo suppressed when calculating wrong charm rates, so they did not appear in Ref. 关3兴. Assuming that all charm quarks eventually hadronize to D mesons, we use ˜f uD 0 ⫹˜f uD ⫹ ⫽1.

共30兲

To resolve the spin and charge counting, we first discuss the heavy mass limit where the pseudoscalar and vector

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PHYSICAL REVIEW D 62 096014

FIG. 1. Topology yielding 兩 a 1 兩 2 in Eq. 共33兲.

charmed mesons form a degenerate ground state doublet. The decay of vector to pseudoscalar mesons will be discussed below. In the following, D dir refers to those D mesons that do not result from D * decays, and D ( * ) can be either D dir or D * . As long as the light quark spin indices of the D ( * ) meson representation matrices are contracted with each other, Eq. 共21兲 reproduces the naive spin counting ˜f qD 0 ⫽3˜f qD 0 , * dir

˜f qD ⫹ ⫽3˜f qD ⫹ . * dir

共31兲

Different contractions yield results of comparable size. The experimental spin counting factor appears to be smaller by roughly a factor of 2 关3兴. Since this effect is not yet understood, we treat it as an uncertainty. Concerning charge counting, we argued by isospin symmetry 关3兴 that in the case q⫽c we have ˜f cD 共 兲 0 ⫽˜f cD 共 兲 ⫹ . * *

共32兲

In the case q⫽u, two topologies can contribute to the decay amplitude: the charm quark can either hadronize with the u quark from the weak effective current, in which case the isospin of the state 兩X典 is I X ⫽0, or with a u or d quark from vacuum, which contains both I X ⫽0 and I X ⫽1 contributions. In the case I X ⫽0, both amplitudes can interfere, so there are three contributions to the decay rate ˜f uD 共 兲 0 ⫽ 兩 a 1 ⫹a 2 兩 2 ⫽ 兩 a 1 兩 2 ⫹ 兩 a 2 兩 2 ⫹2 Re兵 a 1* a 2 其 * ˜f uD 共 兲 ⫹ ⫽ 兩 a 2 兩 2 , *

共33兲

see Figs. 1–3. One might doubt whether using the partonic contraction given in Eq. 共23兲 is justified for all the topologies, as it appears to correspond to the topology in Fig. 2, while the topology in Fig. 1 should rather be described by the contraction

FIG. 3. Interference topology for Eq. 共33兲.

¯ D 共 兲共 p D 兲 ␥ ␮ 共 1⫺ ␥ 5 兲 其 P ␮q ␯ 共 p D ,Q 兲 ⬀Tr兵 H * ⫻Tr兵 ␥ ␯ 共 1⫺ ␥ 5 兲 H D 共 * 兲共 p D 兲 其 .

共34兲

This is not a problem for three reasons. First, we do not claim to be able to accurately model the matrix element, but we only give the simplest possible ansatz by rescaling the partonic result. In particular, it is clearly not yet feasible to model particular contributions individually. We only use the three topologies to estimate the integrated relative magnitudes of the two main contributions and to bound the magnitude of their interference term. Secondly, neither the timedependent nor the time-integrated asymmetries depend on the choice of the contraction unless studied differentially in the momentum of the charmed meson, which so far we do not attempt to do. Finally, as noted in Ref. 关3兴, the choice of the wrong charm contraction appeared to have little influence even on differential observables. The topologies in Figs. 1 and 2 also occur in wrong charm production in B decays. Figure 1 corresponds to the process B→D s( * )⫹ X, Fig. 2 to the process B→D ( * ) X, where D ( * ) can be either D ( * )0 or D ( * )⫹ . Both contributions are experimentally known to be of similar size, i.e., (10⫾2.5)% 关5兴 and (7.9⫾2.2)% 关7兴, respectively, such that 兩 a 1 兩 2 ⫽2 兩 a 2 兩 2 .

共35兲

The relative phase of the two contributions is unknown. Therefore, although it may be large, we have to treat the interference part as a theoretical uncertainty. This is acceptable since the q⫽u contribution is smaller than the q⫽c contribution according to

冏冏冏 Tu Tc

⫽

冏

V ub V ud 共 1⫺z 兲 2 共 1⫹z 兲 ˜f u V cb V cd

冑1⫺4z

˜f c

⬀

冏 冏

V ub 共 1⫹z 兲 V cb 兩 V cd 兩

⬇0.4. 共36兲

Off the heavy mass limit, D * →D decay has to be taken into account. In the same way as in Ref. 关3兴, we get ˜f qD ⫹ ⫽˜f qD ⫹ ⫹Br共 D * ⫹ →D ⫹ X 兲˜f qD ⫹ , * dir

共37兲 ˜f qD 0 ⫽˜f qD 0 ⫹˜f qD 0 ⫹Br共 D * ⫹ →D 0 X 兲˜f qD ⫹ . * * dir

FIG. 2. Topology yielding 兩 a 2 兩 2 in Eq. 共33兲.

The coefficients obtained from Eqs. 共30兲–共37兲 and Ref. 关3兴 are summarized in Table I. The ranges given result from varying the spin counting factor in Eq. 共31兲 from 3 down to 096014-4

ONE-PARTICLE INCLUSIVE C P ASYMMETRIES

PHYSICAL REVIEW D 62 096014 TABLE II. Branching ratios, integrated C P asymmetries and numbers of necessary tagged B 0 decays for the one-particle inclu¯ * 0 , see the text. sive B 0 → h D ( * ) X decay channels. Concerning D

TABLE I. Operator- and channel-specific form factors.

Decay channel B 0 →D ⫺ X ¯ 0X B 0 →D

B 0 →D ⫹ X B 0 →D 0 X B 0 →D * ⫺ X ¯ *0X B 0 →D

and the interference in Eq. 共33兲 from the central value of vanishing interference to full constructive and destructive interference. 3 2

V. RESULTS

We have computed the parameters for the time-dependent C P asymmetries as well as the time-integrated asymmetries. We have inserted recent values for sin 2␤⫽0.75 and ␥ ⫽68° 关4兴. In addition, we use V cb ⫽0.04, V ub ⫽0.08 V cb , z ⫽0.09, x⫽0.73, f B ⫽180 MeV, Br(D * ⫹ →D 0 Y )⫽1 ⫺Br(D * ⫹ →D ⫹ Y )⫽0.683 and C 1 ⫽B⫽B S ⫽1. The results of the calculations can be found in Fig. 4 and Table II. To assess the uncertainties involved in Fig. 4, note that according to Eq. 共14兲 the shapes of the time-dependent asymmetries are determined by the ratios of the wrong to right charm rates ⌫ ¯Y /⌫ Y . We checked numerically that the shapes would hardly change even if these ratios were off by 30%. The dominant contribution to the uncertainty of the amplitudes arises from the transition matrix elements T Y and

B 0 →D * ⫹ X B 0 →D * 0 X

Br 关3兴 共%兲

A 共%兲

A range 共%兲

29.1 31.8

0.16 0.58

0.15–0.29 0.59–0.46

6.000.000 400.000

2.2 5.7 46.8 共0兲

0.58 1.53 0.16 共20兲

0.54–1.04 1.56–1.23 0.12–0.23 共21兲–共10兲

6.000.000 350.000 4.000.000 ⬎80.000

2.5 2.5

0.61 4.17

0.45–0.89 4.40–2.20

5.000.000 100.000

necessary B 0 decays

is directly proportional to the uncertainties of the timeintegrated asymmetries given in Table II. Suppose N perfectly tagged B 0 decays are recorded in an experiment. In order to establish the asymmetry in a channel with a branching ratio b on the 3␴ level, A 1 ⭓⌬A⫽ 3 冑2bN

共38兲

has to be satisfied. The necessary numbers of tagged B 0 decays are given in the last column of Table II. Since the asymmetry tends to be roughly inversely proportional to the branching ratio by Eq. 共15兲, we obtain from Eq. 共38兲 N⬀

1 ⬀b, A2 b

共39兲

such that rare channels are advantageous for observing oneparticle inclusive asymmetries. ¯ * 0 X deserves a further comment. The channel B 0 →D Looking at Fig. 4, there is an obvious problem at small proper decay times. The reason for this problem is that we have discussed all the rates only to leading order in the combined 1/N C and 1/m Q expansions. However, this leading ¯ * 0 X and thus subleadterm vanishes for the channel B 0 →D ing terms become relevant. On the other hand, the numerator

FIG. 4. Time-dependent C P asymmetries in B 0 → h D X for pseudoscalar 共above兲, vector 共below兲, charged 共left兲, neutral 共right兲, right charm 共solid兲, and wrong charm 共dashed兲 h D mesons.

FIG. 5. Time-integrated asymmetry in B 0 → h D * 0 X as a function 0 0 ¯ * X). of Br(B →D

096014-5

XAVIER CALMET, THOMAS MANNEL, AND INGO SCHWARZE

PHYSICAL REVIEW D 62 096014 VI. CONCLUSION

FIG. 6. Necessary number of tagged B 0 events in B 0 → h D * 0 X as 0 0 ¯ * X). a function of Br(B →D

T Y of the C P asymmetries is given by a matrix element of a dimension six operator and hence is suppressed compared to the leading terms of most of the rates. In other words, while in most of the rates the asymmetries are of subleading order ¯ * 0 X. f B2 /m B2 , this is not the case for the channel B 0 →D Unfortunately we cannot compute this possibly large asymmetry, since this would involve to compute subleading terms for the decay rate. Hence we try to estimate the asym¯ * 0 X) in Eq. 共15兲 and show the metry by varying Br(B 0 →D reaction of the asymmetry in Fig. 5 and of the necessary number of tagged B 0 events in Fig. 6. The wrong charm ¯ * 0 X) since asymmetry is practically unaffected by Br(B 0 →D the pole occurs near four average lifetimes where most of the B mesons have already decayed, but the right charm asymmetry turns out to be extremely sensitive. Therefore we cannot predict the latter quantitatively, but it can be as large as several percent, and it will be measurable with a few 100 000 tagged B 0 events.

关1兴 I. Dunietz, Eur. Phys. J. C 7, 197 共1999兲. 关2兴 M. Beneke, G. Buchalla, and I. Dunietz, Phys. Lett. B 393, 132 共1997兲. 关3兴 X. Calmet, T. Mannel, and I. Schwarze, Phys. Rev. D 61, 114004 共2000兲. 关4兴 M. Ciuchini, E. Franco, L. Giusti, V. Lubicz, and G. Marti-

Motivated by the work on fully inclusive C P asymmetries and the question how to measure them, we studied oneparticle inclusive C P asymmetries. In the final state only a h D ( * ) meson has to be identified and thus they are experimentally more easily accessible than the fully inclusive C P asymmetries. We have used a similar method as in Ref. 关3兴 to calculate the time-dependent and time-integrated C P asymmetries for one-particle inclusive B→ h D ( * ) X decays. It turns out that, as in Ref. 关3兴, one cannot avoid to introduce some model dependence. Furthermore, there is also some dependence on an unknown relative phase, which we treat as an uncertainty. Due to these uncertainties we cannot expect our method to compete with proposed methods using ‘‘gold-plated’’ channels for determining CKM parameters, but we can still give estimates for the expected C P asymmetries of the different ground state h D mesons. For most of the asymmetries we find results of a few 10⫺3 , but some are expected to be as large as several percent. These effects should be observable at the B factories. The channels involving right and wrong charm neutral vector mesons turn out to be most promising: they are expected to have the largest asymmetries, and the theoretical method yields the best results for the production rates and spectra of the vector mesons 关3兴. ACKNOWLEDGMENTS

The authors thank Thomas Gehrmann for fruitful discussions. This work 共X.C. during his time in Karlsruhe, T.M. and I.S.兲 was supported by the DFG Graduiertenkolleg ‘‘Elementarteilchenphysik an Beschleunigern’’ and by the DFG Forschergruppe ‘‘Quantenfeldtheorie, Computeralgebra und Monte-Carlo-Simulation.’’

nelli, Nucl. Phys. B573, 201 共2000兲. 关5兴 Particle Data Group, C. Caso et al., Eur. Phys. J. C 3, 1 共1998兲. 关6兴 J. S. Hagelin, Nucl. Phys. B193, 123 共1981兲. 关7兴 CLEO Collaboration, T. E. Coan et al., Phys. Rev. Lett. 80, 1150 共1998兲.

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