

 3 Dec 95

LMU-17/95 October, 1995

Possible Effects of D0 \Gamma _D0 Mixing in Weak B Decays

Zhi-zhong Xing 1 Sektion Physik, Theoretische Physik, Universit"at M"unchen,

Theresienstrasse 37, D-80333 M"unchen, Germany

Abstract D0 \Gamma _D0 mixing at a detectable level requires the presence of new physics and may lead to some observable effects in weak decays of B mesons. We show that CP violation induced by D0 \Gamma _D0 mixing can manifest itself in the decay-rate asymmetry of B+u ! DL(H)l+*l vs B\Gamma u ! DL(H)l\Gamma _*l . A rephasing-invariant generalization of the Gronau-Wyler approach is made to determine the weak phase shift in B\Sigma u ! DL(H)K\Sigma , which is only sensitive to the underlying new physics in D0 \Gamma _D0 mixing. We also demonstrate the possible effect of D0 \Gamma _D0 mixing on CP violation in decay modes of the type Bd ! DL(H) + (ss0; ae0; etc). Finally the model of four quark families is taken as an example to illustrate how the new physics affects D0 \Gamma _D0 mixing, B0d \Gamma _B0d mixing and CP asymmetries in the relevant B decays.

1Electronic address: Xing@hep.physik.uni-muenchen.de

1

1 In the standard model the D0 \Gamma _D0 mixing parameter xD is expected to be of the order 10\Gamma 5 \Gamma 10\Gamma 4, well below the current experimental bound xD ! 0:086 [1]. A discovery of D0 \Gamma _D0 mixing at the level of xD , 10\Gamma 2 will definitely imply the existence of new physics [2, 3]. It is known that many extensions of the standard model can accommodate D0 \Gamma _D0 mixing with xD , 10\Gamma 2 and allow significant CP violation in the charm sector. The typical examples of such extensions include the fourth quark family [4], the left-handed SU (2)- singlet up-type quarks [5], and the general two-Higgs doublet model with flavor-changing neutral exchange [6]. In some of them, unitarity of the 3 \Theta 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix is violated [7].

Non-negligible D0 \Gamma _D0 mixing and the associated CP violation can enter those weak B decays with neutral D mesons in the final states. Indeed some kinds of new physics may simultaneously affect D0 \Gamma _D0 mixing, B0 \Gamma _B0 mixing and the loop-induced (penguin) transitions of D and B mesons. In this case, the standard-model predictions to CP asymmetries in B decays are possible to be significantly contaminated, and our natural contemplation is if the direct measurements can provide us some useful information beyond the limit of the standard model. Of course, in the approaches to relate the basic parameters of models to the relevant observables of experiments, one needs to accommodate the possible new physics and to make the approaches themselves independent of any specific model [8].

In this note, we shall carry out an instructive study of the possible effects of D0 \Gamma _D0 mixing in some semileptonic and nonleptonic B-meson transitions. We show that CP

violation of the order 10\Gamma 3 \Gamma 10\Gamma 2 in D0 \Gamma _D0 mixing may lead to an observable signal in the decay-rate asymmetry of B+u ! DL(H)l+*l vs B\Gamma u ! DL(H)l\Gamma _*l . A rephasing-invariant generalization of the Gronau-Wyler approach [9] is made to determine the weak phase shift in B\Sigma u ! DL(H)K\Sigma or in B0d ! DL(H)K\Lambda 0 vs _B0d ! DL(H) _K\Lambda 0, which is only sensitive to the underlying new physics in D0 \Gamma _D0 mixing. We also demonstrate the possible effect of D0 \Gamma _D0 mixing on CP violation in decay modes of the type Bd ! DL(H) + (ss0; ae0; etc). Taking the model of four quark families for example, we finally illustrate how the new physics affects D0 \Gamma _D0 mixing, B0d \Gamma _B0d mixing and CP asymmetries in the relevant weak B decays.

2 Due to D0 \Gamma _D0 mixing, the mass eigenstates DL and DH are related to the flavor eigenstates D0 and _D0 through:

jDLi = pD jD0i + qD j _D0i ;j

DH i = pD jD0i \Gamma qD j _D0i ; (1)

where jpDj2 + jqDj2 = 1, and CP T invariance has been assumed. Commonly the mixing rate is represented by xD j \Delta mD=\Gamma D and yD j \Delta \Gamma D=(2\Gamma D ), where \Gamma D j (\Gamma L + \Gamma H )=2,

2

\Delta mD j mH \Gamma mL and \Delta \Gamma D j \Gamma H \Gamma \Gamma L (mL;H and \Gamma L;H denote the mass and width of DL;H , respectively). The standard model predicts that both xD and yD are too small to be detectable in experiments [10]. The existence of new physics might enhance xD and make it close to the current experimental bound. Here a reasonable assumption is that any new physics does not affect the direct decays of c quark (via the tree-level W -mediated diagrams) significantly and thus does not affect yD. This implies that the approximation yD !! xD is very reasonable in all reasonable models of new physics contributing to D0 \Gamma _D0 mixing [3]. Consequently jqD=pDj ss 1, i.e., there is not sizeable CP violation in the D0 \Gamma _D0 mixing matrix. In analogy to the notations of the K0 \Gamma _K0 system, one can use fflD j (pD \Gamma qD)=(pD + qD) to express the CP -violating effect induced by D0 \Gamma _D0 mixing. Thenj

qD=pDj ss 1 corresponds to a small Re(fflD). Later on we shall demonstrate that nonvanishing Re(fflD) at the level of 10\Gamma 3 \Gamma 10\Gamma 2 may give rise to an observable decay-rate asymmetry in the semileptonic transitions B\Sigma u ! DL(H)K\Sigma .

Let us consider the decay modes B+u ! DLl+*l and B\Gamma u ! DLl\Gamma _*l , where l = e or _. To lowest order in the standard model, the transition B+u ! _D0l+*l (or B\Gamma u ! D0l\Gamma _*l ) takes place only through the tree-level spectator diagrams with a single weak phase. Since any new physics cannot significantly affect the direct decays of b quark via the tree-level W -mediated graphs [7], only the new physics of D0 \Gamma _D0 mixing is able to enter the processes under consideration. The amplitudes of B+u ! DLl+*l and B\Gamma u ! DLl\Gamma _*l are given as

A(B+u ! DLl+*l ) = q\Lambda D h _D0l+*l jHjB+u i ; A(B\Gamma u ! DLl\Gamma _*l ) = p\Lambda D hD0l\Gamma _*l jHjB\Gamma u i ; (2)

where the \Delta B = \Delta Q rule has been assumed. Between these two decay modes, the CP asymmetry AL is commonly defined as the ratio of the difference to the sum of their decay rates. By use of fflD, one can obtain the explicit expression of AL:

AL = jqDj

2 \Gamma jp

Dj2j

qDj2 + jpDj2 = \Gamma

2 Re(fflD)

1 + jfflDj2 : (3)

In a similar way, the CP -violating signal AH in the transitions B+u ! DH l+*l vs B\Gamma u ! DH l\Gamma _*l can be obtained. The result is nothing but AH = AL. Here it should be noted that the smallness of Re(fflD) does not mean the smallness of Im(fflD). Indeed jIm(fflD)j is likely to be close to unity, thus the jfflDj2 term in AL or AH cannot be neglected.

If 0 ^ jfflDj ^ 1, we find

jRe(fflD)j ^ jALj = jAHj ^ 2 jRe(fflD)j : (4) The current experimental data give Br(B+u ! _D0l+*l ) = (1:6 \Sigma 0:7)% [1]. It is expected that the branching ratios of B+u ! DLl+*l and B+u ! DH l+*l are both at the level of 1%.

3

If jRe(fflD)j is of the order 10\Gamma 3 \Gamma 10\Gamma 2, then observation of the CP asymmetry AL or AH to 3 standard deviations needs about 107 \Gamma 109 B\Sigma u events. Such a measurement is available at the forthcoming B-meson factories.

3 Within the standard model, Gronau and Wyler have developed an approach to determine the CKM weak phase fl j arg(\Gamma V \Lambda ubV \Lambda cdVcbVud) from B\Sigma u ! D01(2)K\Sigma , where D01(2) = [D0 + (\Gamma ) _D0]=p2 denotes a CP even (odd) state [9]. This approach has not only theoretical simplicity and cleanliness but also several experimental advantages over the other proposed ways for extraction of fl [11]. Its application could only be affected by the new physics of D0 \Gamma _D0 mixing, as we shall see below.

In general, the rephasing-invariant phase shift in B\Sigma u ! DLK\Sigma or DH K\Sigma is given by

' j arg V

\Lambda ubVcs

V \Lambda cbVus \Delta

qD pD ! ; (5)

whose magnitude depends upon the specific mechanism of D0 \Gamma _D0 mixing. For the standard model, we have jDLi ss jD1i and jDH i ss jD2i with qD=pD = (V \Lambda csVus)=(VcsV \Lambda us). Then unitarity of the 3 \Theta 3 CKM matrix allows ' ss fl to a high degree of accuracy [8]. For some non-standard models like those listed in ref. [3], however, the new physics of D0 \Gamma _D0 mixing may influence ' and make it deviate from fl significantly. In this case, the CP asymmetry induced by jqD=pDj 6= 1 is expected to be negligibly small in comparison with that arising from ' and the strong phase shift in B\Sigma u ! DL(H)K\Sigma .

Since the decay modes B+u ! D0K+, B+u ! _D0K+ and their CP -conjugate counterparts are dominated by the tree-level W -mediated quark diagrams (see Fig. 1 for illustration), one can parametrize the transition amplitudes of B\Sigma u ! DL(H)K\Sigma as

A(B+u ! DL(H)K+) = p\Lambda D (V \Lambda ubVcs) Aa eiffia + (\Gamma ) q\Lambda D (V \Lambda cbVus) Ab eiffib ; A(B\Gamma u ! DL(H)K\Gamma ) = p\Lambda D (VcbV \Lambda us) Ab eiffib + (\Gamma ) q\Lambda D (VubV \Lambda cs) Aa eiffia ; (6)

where Aa and Ab are real (positive) hadronic matrix elements, and ffia and ffib are the corresponding strong phases. Unlike ref. [9], here one cannot use a simple triangular relation to describe the above decay amplitudes. Some specific measurements are possible to establish the following (dimensionless) decay-rate asymmetry:

\Delta ij j jA(B

+ u ! DiK+)j2 \Gamma jA(B\Gamma u ! DiK\Gamma )j2j

A(B+u ! D0K+)j \Delta jA(B+u ! _D0K+)j (7)

with i; j = L or H. Denoting the strong phase difference ffib \Gamma ffia j ffi and using the reasonable approximation jqD=pDj ss 1, we explicitly obtain

\Delta LL = 2 sin ' sin ffi ; \Delta HH = \Gamma 2 sin ' sin ffi ; \Delta LH = 2 cos ' cos ffi ; \Delta HL = \Gamma 2 cos ' cos ffi : (8)

4

In experiments, the relations \Delta LL = \Gamma \Delta HH and \Delta LH = \Gamma \Delta HL can be well examined. Note that only the asymmetries \Delta LL and \Delta HH represent CP violation, and they vanish if the weak phase shift ' vanishes.

Obviously eq. (8) can be used to extract '. If the CP asymmetries \Delta LL and \Delta HH were substantially suppressed due to the smallness of ffi, then \Delta LH = \Gamma \Delta HL ss 2 cos ' would be a good approximation. In general, we have

\Delta LL sin ' !

2

+ \Delta LHcos ' !

2

= ` \Delta LLsin ffi '

2

+ ` \Delta LHcos ffi '

2

= 4 : (9)

Note that the angle ' (or ffi) extracted from the above equation has a few ambiguities in its size and sign. This kind of ambiguities can be removed by studying a set of exclusive decay modes like B\Sigma u ! (D0; _D0; DL; DH ) + (K\Sigma ; K\Lambda \Sigma ; K\Sigma ss+ss\Gamma ; etc) [9]. All such processes have a common weak phase shift ', but their strong phase shifts ffi should be different from one another.

In a similar way, one can make a rephasing-invariant generalization of Dunietz's work in ref. [12], so as to extract the weak phase shift ' from the two-body decays B0d ! DL(H)K\Lambda 0 and _B0d ! DL(H) _K\Lambda 0.

4 Now we illustrate the possible effect of significant D0 \Gamma _D0 mixing on CP violation in some neutral B-meson decays. For simplicity, we only consider the transitions B0d ! (DL; DH) + (ss0; ae0; a01; etc) and their CP -conjugate processes. The dominant tree-level W mediated diagrams for this type of decays are depicted in Fig. 2. We observe that the graph amplitudes in Fig. 2(a) are doubly CKM-suppressed with respect to those in Fig. 2(b), and the ratio of their CKM factors is jVcd=Vudj \Delta jVub=Vcbj ss 2%. In discussing the indirect CP violation induced by the interplay of decay and B0d \Gamma _B0d mixing, the contribution of Fig. 2(a) can be safely neglected. By use of this approximation, we obtain the decay amplitudes of B0d ! DL(H)M 0 and _B0d ! DL(H)M 0 (here M 0 = ss0; ae0, etc) as follows:

A(B0d ! DL(H)M 0) ss q\Lambda D (V \Lambda cbVud) Ab eiffib ; A( _B0d ! DL(H)M 0) ss p\Lambda D (VcbV \Lambda ud) Ab eiffib ; (10)

where Ab and ffib are the real hadronic matrix element and the strong phase of Fig. 2(b), respectively. In the above two CP -conjugate amplitudes, we have ignored a possible relative sign arising from CP transformation of the initial and final states [13].

The interplay of decay and B0d \Gamma _B0d mixing gives rise to CP violation in the Bd decays under consideration. This kind of CP -violating signal is characterized by the following

5

rephasing-invariant quantity:

TDM j Im " qBp

B \Delta

A( _B0d ! DL(H)M 0) A(B0d ! DL(H)M 0) # ss Im "

qB pB \Delta

qD pD \Delta

VcbV \Lambda ud V \Lambda cbVud # ; (11)

where qB and pB are the B0d \Gamma _B0d mixing parameters, and jqD=pDj ss 1 has been used. In the standard model, qB=pB = (V \Lambda tbVtd)=(VtbV \Lambda td) and qD=pD = (V \Lambda csVus)=(VcsV \Lambda us). Then we obtain TDM ss \Gamma sin(2fi) with fi j arg(\Gamma V \Lambda cbV \Lambda tdVcdVtb), by use of the unitarity condition V \Lambda udVcd + V \Lambda usVcs = \Gamma V \Lambda ubVcb ss 0. Due to the underlying new physics in D0 \Gamma _D0 mixing (perhaps B0d \Gamma _B0d mixing is affected by the same source of new physics, as illustrated in Sect. 5), what one can extract from TDM is indeed a complicated weak phase shift which is possible to differ from 2fi significantly.

The decay modes Bd ! DL(H)M 0 can be measured at an asymmetric B-meson factory running on the \Upsilon (4S) resonance, where the B's are produced in a two-body state (B+u B\Gamma u or B0d _B0d) with odd charge-conjugation parity. Because the two Bd mesons mix coherently until one of them decays, one may use the semileptonic decay of one meson to tag the flavor of the other meson decaying to DL(H)M 0. The (proper) time distribution of the joint decay rates can be given as [14]:

R(l\Gamma ; DL(H)M 0; \Delta t) / e\Gamma \Gamma Bj\Delta tj [1 \Gamma TDM \Delta sin(xB\Gamma B\Delta t)] ; R(l+; DL(H)M 0; \Delta t) / e\Gamma \Gamma Bj\Delta tj [1 + TDM \Delta sin(xB\Gamma B\Delta t)] ; (12)

where \Delta t is the time difference between the semileptonic and nonleptonic decays 2, xB ss 0:71 is a mixing parameter of the B0d \Gamma _B0d system, and \Gamma B is the average width of Bd mass eigenstates. Clearly the CP -violating term TDM is determinable from measurements of the above joint decay rates.

It is worthwhile at this point to emphasize the prospect for detecting the decays Bd ! DL(H)M 0 at the forthcoming facilities of B mesons. With the help of current data Br(B\Gamma u ! D0ae\Gamma ) ss 1:34 \Theta 10\Gamma 2, Br( _B0d ! D+ae\Gamma ) ss 7:8 \Theta 10\Gamma 3 and Br( _B0d ! D0ae0) ! 5:5 \Theta 10\Gamma 4 [1], an isospin analysis shows that there are not significant final-state interactions in B ! Dae and the branching ratio of _B0d ! D0ae0 has a lower bound Br( _B0d ! D0ae0) * 3:8 \Theta 10\Gamma 4 [15]. Therefore, the transitions _B0d ! D0ae0 and Bd ! DL(H)ae0 are able to be measured in the near future. Since all decays of the type Bd ! DL(H)M 0 are governed by the same weak parameters, their branching ratios are expected to be of the same order as Br(Bd ! DL(H)ae0).

5 Finally let us take the model with four families of quarks for example, to look at how the new physics induced by the fourth family (t0; b0) affects D0 \Gamma _D0 mixing, B0d \Gamma _B0d mixing

2Note that the time sum of the semileptonic and nonleptonic decays has been integrated out, since it is

not measured at a B-meson factory [14].

6

and the relevant weak decays of B mesons. Now quark mixings are described by a 4 \Theta 4 unitary matrix, and the unitarity triangles of the 3 \Theta 3 CKM matrix become the unitarity quadrangles in the complex plane [8]. Assuming that the box diagram of D0 \Gamma _D0 mixing is dominated by the heaviest down-type quark b0, one gets

qD pD =

V \Lambda cb0 Vub0 Vcb0 V \Lambda ub0 j e

iOEb0 (13)

to a good degree of accuracy. In contrast, the t0 quark contributes significantly to B0d \Gamma _B0d mixing [16]. The ratio qB=pB can be given as

qB pB =

V \Lambda tbVtd VtbV \Lambda td e

iOEt0 ; (14)

where

OEt0 = arg "1 + 2 V

\Lambda t0bVt0d

V \Lambda tbVtd \Delta

E(t; t0)

E(t; t) +

(V \Lambda t0bVt0d)2

(V \Lambda tbVtd)2 \Delta

E(t0; t0)

E(t; t) # (15)

with E(i; j) being the box-diagram function for the internal i and j quarks [17]. It is clear that jqD=pDj ss 1 and jqB=pBj ss 1 are two safe approximations. However, small deviation ofj

qD=pDj from unity (e.g., at the level of 10\Gamma 3 \Gamma 10\Gamma 2) could lead to a measurable effect in the semileptonic decays B+u ! DL(H)l+*l vs B\Gamma u ! DL(H)l\Gamma _*l , as we have discussed in Sect. 2.

For explicitness, we adopt the Botella-Chau parametrization for the 4 \Theta 4 quark mixing matrix [18], in which there are three CP -violating phases OE1, OE2 and OE3. The phases OE2 and OE3 are induced by (t0; b0), while OE1 can be regarded as the original phase of the 3 \Theta 3 CKM matrix. It is easy to obtain OEb0 = 2(OE3 \Gamma OE2), but OEt0 appears to be a complicated function of OE1, OE2 and OE3. Assuming a natural hierarchy among the four-family quark mixings [19], we find that in the lowest-order approximation only OE1 enters elements of the 3 \Theta 3 submatrix: Vij (with i; j ! 4). Thus one can still use the denotions fi ss arg(V \Lambda td) and fl ss arg(V \Lambda ub), where fi and fl stand for two angles of the conventional unitarity triangle [1]. From eqs. (5) and (11), we get

' ss fl + OEb0 (16a)

in B\Sigma u ! DL(H)K\Sigma ; and T

DM ss \Gamma sin(2fi \Gamma OEb0 \Gamma OEt0) (16b) for Bd ! DL(H)M 0. Clearly these two types of decay modes can be contaminated by the existence of the fourth quark family.

Following the assumption of a natural hierarchy among jVij j (with i; j = 1; 2; 3; 4), we realize that the approximate triangular relation V \Lambda udVcd + V \Lambda usVcs + V \Lambda ubVcb = \Gamma V \Lambda ub0 Vcb0 ss 0 may hold. As a consequence, fl can be approximately determined from

fl ss arccos jVudVcdj

2 + jVubVcbj2 \Gamma jVusVcsj2

2 jVudVcdj \Delta jVubVcbj ! : (17)

7

All the matrix elements in eq. (17) have been measured in experiments [1], but the precision of their values (in particular, jVcdj, jVcsj and jVubj) need be further improved in order to make the determination of fl available. It is expected that a comparison between the experimental values of fl (from eq. (17)) and ' (from eq. (16a)), if possible, can signify the nonvanishing OEb0 and give a constraint on its magnitude. From this point, one could have got a feeling why D0 \Gamma _D0 mixing is a useful window for probing new physics.

In conclusion, the presence of D0 \Gamma _D0 mixing at a detectable level (xD , 10\Gamma 2) requires new physics and may lead to some observable effects in weak decays of B mesons. A careful study of the decay rates and CP asymmetries for those B transitions with neutral D mesons in the final states is able to shed light on the possible new physics in D0 \Gamma _D0 mixing. Measurements of some such B decay modes are available at the forthcoming B factories and other high-luminosity facilities for B physics.

I would like to thank Prof. H. Fritzsch for his warm hospitality and the Alexander von Humboldt Foundation for its financial support. I am greatly indebted to Prof. D.M. Kaplan for his enlightening and constructive comments on this work. An interesting discussion with Dr. H. Simma in Strasbourg is also acknowledged.

References

[1] Particle Data Group, L. Montanet et al., Phys. Rev. D50 (1994) 1173. [2] L. Wolfenstein, Phys. Rev. Lett. 75 (1995) 2460;

J.L. Hewett, preprint SLAC-PUB-6821 (presented at the Lafex International School on High Energy Physics, Rio de Janeiro, Brazil, February 6 - 22, 1995); G. Burdman, preprint Fermilab-Conf-95/281-T (presented at the Workshop on the Tau/Charm Factory, Argonne, June 21 - 23, 1995); A.L. Yaouanc, L. Oliver, O. P`ene, and J.C. Raynal, preprint LPTHE-Orsay 95/15 (presented at the Journ'ees sur les projects de Physique Hadronique, Soci'et'e, Francaise de Physique, Super-Besse, France, 12 - 14 janvier, 1995).

[3] G. Blaylock, A. Seiden, and Y. Nir, Phys. Lett. B355 (1995) 555;

T.E. Browder and S. Pakvasa, preprint UH 511-828-95 (1995).

[4] See, e.g., K.S. Babu, X.G. He, X.Q. Li, and S. Pakvasa, Phys. Lett. B205 (1988) 540.

8

[5] G.C. Branco, P.A. Parada, and M.N. Rebelo, preprint CFIF-IST-4/94 (1994). [6] See, e.g., Y.L. Wu and L. Wolfenstein, Phys. Rev. Lett. 73 (1994) 1762. [7] Y. Nir and D. Silverman, Nucl. Phys. B345 (1990) 301;

Y. Nir and H.R. Qiunn, in B Physics, edited by S. Stone (World Scientific, Singapore, 1992), p. 362.

[8] Z.Z. Xing, preprint LMU-13/95 (presented at the Conference on Production and Decay

of Hyperons, Charm and Beauty Hadrons, Strasbourg, France, September 5 - 8, 1995).

[9] M. Gronau and D. Wyler, Phys. Lett. B265 (1991) 172. [10] D.M. Kaplan, preprint IIT-HEP-95/3 (presented at the Workshop on the Tau/Charm

Factory, Argonne National Laboratory, June 21 - 23, 1995); preprint IIT-HEP-95/4 (presented at the Four Scas Conference, SISSA, Trieste, Italy, June 26 - July 1, 1995); T. Liu, Princeton preprint HEP/95-6 (presented at the Workshop on the Tau/Charm Factory, Argonne National Laboratory, June 21 - 23, 1995); J.L. Hewett, preprint SLAC-PUB-6695 (to appear in The Tau-Charm Factory in the Era of B Factories and CESR, Stanford, CA, August 15 - 16, 1994).

[11] See, e.g., S. Stone, Nucl. Inst. & Meth. A333 (1993) 15. [12] I. Dunietz, Phys. Lett. B270 (1991) 75. [13] For a detailed clarification of CP parities in the final states, see: I. Dunietz and A.

Snyder, Phys. Rev. D43 (1991) 1593.

[14] Z.Z. Xing, preprint LMU-22/94 (to appear in Phys. Rev. D53, Vol. 1, 1 January 1996);

Z.Z. Xing, Phys. Rev. D50 (1994) R2957; H. Fritzsch, D.D. Wu, and Z.Z. Xing, Phys. Lett. B328 (1994) 477.

[15] Z.Z. Xing, Phys. Lett. B364 (1995) 55. [16] See, e.g., W.S. Hou and A. Soni, Phys. Lett. B196 (1987) 92. [17] T. Inami and C.S. Lim, Prog. Theor. Phys. 65 (1981) 297; 1772 (E). [18] F.J. Botella and L.L. Chau, Phys. Lett. B168 (1986) 97. [19] C. Hamzaoui, A.I. Sanda, and A. Soni, Phys. Rev. Lett 63 (1989) 128.

9

_b

u B+u

_u

u c

_s K+

D0 =) DL;H*_ oe

oe -

oe

(a\Gamma 1)

_b

u B+u

_s

c u

_u D0 =

) DL;H

K+,ss '`'&oe -

oe

oe

^^^^

(a\Gamma 2)

_b

u B+u

_c

u

_s u

_D0 =) DL;H

K+ff\Omega oe

oe oe

- -

(b\Gamma 1)

_b

u B+u

_c

u u

_s K+

_D0 =) DL;H*_ oe

oe -

oe

(b\Gamma 2)

Figure 1: The dominant tree-level W -mediated diagrams for B+u ! DLK+ or DH K+.

_b

d B0d

_u

d c

_d M 0

D0 =) DL;H*_ oe

oe -

oe

(a\Gamma 1)

*_ oe

oe -

oe

?? ?? ?? ??

_b

d B0d

_u

c u

_u D0 =) D

L;H

M 0

(a\Gamma 2) _b

d B0d

_c

d u

_d M 0

_D0 =) DL;H*_ oe

oe -

oe

(b\Gamma 1)

*_ oe

oe -

oe

?? ?? ?? ??

_b

d B0d

_c

u u

_u

_D0 =) DL;H

M 0 (b\Gamma 2)

Figure 2: The dominant tree-level W -mediated diagrams for B0d ! DLM 0 or DH M 0, where M 0 represents the meson ss0; ae0; a01, etc.

10

