Charming penguins in B ---> K* pi, K(rho, omega, phi) decays

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Charming penguin contributions in B

\K

*

␲

, K

„

␳

,

␻

,

␾

… decays

Claudia Isola

Dipartimento di Fisica, Universita` di Cagliari and INFN, Sezione di Cagliari, Italy and Centre de Physique The´orique, E´ cole Polytechnique, 91128 Palaiseau Cedex, France

Massimo Ladisa

Physics Department, Technion-Israel Institute of Technology, Haifa 32000, Israel

Giuseppe Nardulli

Dipartimento di Fisica, Universita` di Bari, Via Amendola, 173-70126 Bari, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy

Pietro Santorelli

Dipartimento di Scienze Fisiche, Universita` ‘‘Federico II’’ di Napoli, Via Cinthia - 80126, Napoli, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Napoli, Italy

共Received 1 August 2003; published 3 December 2003兲

We evaluate the decays B→K*␲, K(␳,␻,␾) adding the long distance charming penguin contributions to the short distance: tree⫹penguin amplitudes. We estimate the imaginary part of the charming penguin contri-bution by an effective field theory inspired by heavy quark effective theory and parametrize its real part. The final results for branching ratios depend on only two real parameters and show a significant role of the charming penguin contributions. The overall agreement with the available experimental data is satisfactory.

DOI: 10.1103/PhysRevD.68.114001 PACS number共s兲: 13.25.Hw

I. INTRODUCTION

The CLEO II 关1,2兴, the BaBar 关3,4兴 and the Belle 关5兴 Collaborations have recently reported data on the B meson nonleptonic decay channels into a light vector and a pseudo-scalar meson:

B→PV. 共1兲

These data are of the utmost importance for the determina-tion of the angles of the unitarity triangle. In particular, if one of the two mesons in the final state is a strange particle the decays共1兲 offer several channels for the extraction of the␥ angle, thus providing alternatives to the K␲ decay mode. These nonleptonic decays have been proposed long ago

关6–18兴 as a method for the extraction of the␥ angle. By this strategy one relates the data to a theoretical amplitude given as a sum of tree and penguin short distance contributions, whose relative weak phase is indeed␥. The usual computa-tional scheme is the factorization hypothesis, either in the naive or in the QCD based version 关19–26兴. Here one ne-glects nonfactorizable contributions as one proves that they are of the order of⌳/mb and therefore negligible in the mb

→⬁ limit.

There is a class of contributions however that, although suppressed in the infinite heavy quark mass limit, cannot be neglected a priori. These are long distance contributions known in the literature as charming penguin diagrams

关27–32兴. Induced by the nonleptonic Hamiltonian ⬀Vcb*Vcs¯b␥␮(1⫺␥5)c c¯␥␮(1⫺␥5)s⫹Fierz⫹H.c., they

cannot contribute in the vacuum saturation approximation. However, the insertion of at least two charmed particles

共charm⫹anticharm兲 between the currents can produce

siz-able contributions. As a matter of fact the O(⌳⫻m_b⫺1)

sup-pression is compensated by the Cabibbo-Kobayashi-Maskawa enhancement and the actual role of these contributions can be established only as a result of some dynamical calculations. In 关30兴 and 关31兴 we estimated these contributions for the B→PP decay channels and found that for the K␲ case they are indeed relevant. Their imaginary part can be evaluated with some confidence using the heavy meson chiral effective theory corrected for the light meson hard momenta. On the other hand, the real part is less pre-dictable; for example in 关30兴 and 关31兴 the results strongly depend on the cutoff in the loop integrals. Sizable effects were also found in the K⫺␹_c0共J/␺兲 channel 关33兴.

In the present paper we give an estimate of the charming penguin contributions in the charmless decays 共1兲 for the cases K*␲and K(␳,␻,␾). We do not consider the particles

␩,␩⬘ in the final state because the pseudoscalar SU共3兲 sin-glet is most probably contaminated by the glue and the evaluation of these contributions cannot be reliably done within our theoretical scheme. Because of the above men-tioned difficulties we compute by the effective theory only the imaginary part of the amplitude,I_LD. For the real part of the amplitude, R_LD, we assume a simple parametrization RLD⬀ILD, fixing the two proportionality constants关one for

each of the SU(3)_f multiplets comprising the light vector mesons兴 by a fit. A similar calculation has been recently at-tempted by 关32兴; these authors assume a more phenomeno-logical approach and parametrize the amplitude in the more general way 共two complex numbers兲, using SU共3兲 flavor symmetry to relate the different amplitudes. Therefore our calculation can provide a dynamical test of the model in关32兴. The outline of the paper is as follows. In Sec. II we esti-mate the tree diagram contribution共or short-distance contri-bution兲, computed in the factorization approximation. In Sec.

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III we estimate the absorptive part of the charming penguin contribution. It is evaluated by use of the effective Lagrang-ian for light and heavy mesons based on the heavy quark effective theory共for a review see 关34兴兲. The main uncertainty of this approach is the extrapolation to hard light meson momenta and in this context we use an estimate of form factor given by us in 关30兴. In Sec. IV we present numerical results for the branching ratios and the asymmetry. In our previous papers we estimated the real part of the charming penguin contributions by using a cutoff Cottingham formula; the results were strongly cutoff dependent and for this reason we give here a simple parametrization of the real part and we estimate the relevant two real parameters by comparing with the data. The overall result for both of the branching ratios and the CP-asymmetries is rather satisfying and points to a significant role of the charming penguins in the B →K*␲, K(␳,␻,␾) decay modes.

II. SHORT-DISTANCE NONLEPTONIC WEAK MATRIX ELEMENTS

The effective Hamiltonian for nonleptonic B decays is given by Heff⫽ GF

冑

2

冋

Vub *Vus共c1O1 u_⫹c 2O2 u_兲⫹V cb *Vcs共c1O1 c_⫹c 2O2 c_兲 ⫺Vtb*Vts

冉

兺

i⫽3 10 ciOi⫹cgOg

冊

册

共2兲

where c_iare the Wilson coefficients evaluated at the normal-ization scale␮⫽mb关35–39兴 and next-to-leading order QCD

radiative corrections are included. O1 and O2 are the usual

tree-level operators, Oi(i⫽3, . . . ,10) are the penguin op-erators and Ogis the chromomagnetic gluon operator. The ci in Eq. 共2兲 are as follows 关39兴: c2⫽1.105, c1⫽⫺0.228, c₃⫽0.013, c₄⫽⫺0.029, N c₅⫽0.009, c₆⫽⫺0.033, c₇/␣

⫽0.005, c8/␣⫽0.060, c9/␣⫽⫺1.283, c10/␣⫽0.266.

More-over, for the current quark masses we use the values

mb⫽4.6 GeV mu⫽4 MeV

md⫽8 MeV ms⫽0.150 GeV. 共3兲

_dn_M共B→D s (_*)_D(_*)_兲 ⫻M共Ds (_*)_D(_*)_→K_*_␲_兲, _共10兲

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where the integration is over the solid angle and a sum over polarizations is understood. A similar equation holds for the

K(␳,␻,␾) final state. The amplitudes for the decays B →Ds

(_*)_D(_*) _{are computed by factorization and using}

infor-mation on the Isgur-Wise function共see below兲. Diagrams for the calculation of the amplitudes D_s(*)_D(_*)_→K_*_␲ _{are in}

Fig. 1. A similar diagram can be drawn for the D_s(*) D(*) →K(␳,␻,␾) amplitudes.

The effective Lagrangian to compute these diagrams can be written as follows共see e.g. 关34兴兲:

2共a7⫹a9兲

册

⫹A0

B→␳_共m K 2_兲f KVtb*Vts

冋

册冎

共␧ *•pB兲 B0_→K⫹_␳⫺ ⫺GF

冑

2A0 B→␳_共m K 2_兲f Km␳

冋

冉

冊册

共␧ *•pB兲 B⫹→K⫹␻ ⫺GFm␻兵F1 B→K_共m ␻ 2_兲f ␻关Vub*Vusa1⫺Vtb*Vts„2共a3⫹a5兲⫹ 1 2共a7⫹a9兲兲兴 ⫹A0 B→␻_共m K 2_兲f K

冋

冋

a3⫹a4⫹a5⫺ a7⫹a9⫹a10 2

册

共␧*•pB兲 K * π D D D * _{K *} π D D D K * π D D D K * π D D D K * π D D D K * π D D D a b c d e ( ) ( ) ( ) ( ) ( ) s s s s s * * * * * * * * *

FIG. 1. Diagrams for the calculation of the Ds

(_*)

D(_*)_→K_*_␲ amplitudes.

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H⫽1⫹v”

2 共⫺P5␥5⫺iP” 兲, 共12兲 and P5, P␮ are annihilation operators of the pseudoscalar P

and vector V heavy mesons normalized as follows:

具

0兩P5兩P

典

⫽

冑

MH,

具

0兩P␮兩V共⑀兲

典

⫽⑀␮

冑

MH. 共13兲 The first term in the Lagrangian 共11兲 contains the kinetic term for the heavy mesons giving the P and P*propagators,

i/(2v•k) and ⫺i(g␮␯⫺v␮v␯)/(2v•k) respectively. How-ever, since the residual momentum is not small we will use the complete form of the propagators instead of their ap-proximate expressions 关30,31兴. The interactions among heavy and light mesons are obtained by the other three terms. In the second term there are interactions among the heavy mesons and an odd number of pions coming from the expan-sionA_␮. HereA_␮⫽1

2(␰

†_⳵

␮␰⫺␰⳵␮␰†). Expanding the field

␰⫽exp(i␲/f) and taking the traces it provides the coupling in the lower vertices of Fig. 1. The last two terms give the upper vertices of Fig. 1. The octet of vector resonances (␳,

K*, etc.兲 is introduced as the gauge multiplet according to the hidden gauge symmetry approach of Ref. 关40兴. We put

␳␮⫽i

gV

冑

2␳

ˆ

␮ 共14兲

where␳ˆ is the usual Hermitian 3⫻3 matrix of flavor SU共3兲 comprising the nonet of light vector mesons and g_Vis deter-mined by vector meson dominance as follows: gV⯝5.8 关40兴. Also the parameter␤ can be fixed by vector meson domi-nance. This corresponds to assuming that, for the heavy pseudoscalar mesons H, the coupling of the electromagnetic current to the light quarks is dominated by the␳,␻,␾vector mesons. This produces the numerical result

␤⫽

冑

2mV

g_Vf_V ⬇0.9. 共15兲

For g we take the recent experimental result from the CLEO Collaboration, obtained by the full width of D*⫹ 关41兴; this

gives a value g⫽0.59⫾0.07⫾0.01.

It can be noted that while these determinations consider soft pions, we are interested in the coupling of a hard pion to

D and D*. This introduces a correction that can be param-etrized by a form factor, see below for a discussion.

MBMD␰共v•v

⬘

兲i关共1⫹v•v

⬘

兲⑀_␮*⫺共⑀*•v兲v_␮

⬘

兴, 共17兲

where␰(v•v

⬘

) is the Isgur-Wise form factor. Note that, by this definition, the choice of phases in Eq. 共17兲 agrees with Eqs.共8兲 and 共9兲. We also write the phenomenological heavy-to-light leading current for coupling to light pseudoscalar mesons of the heavy mesons in the multiplet H

L_a␮⫽iF ˆ 2

具

␥ ␮_共1⫺_␥ 5兲Hb␰ba †

_典

. 共18兲

Here Fˆ is related to leptonic decay constant of the pseudo-scalar heavy meson B appearing in Eq.共6兲 by

fB⫽

Fˆ

冑

MB

. 共19兲

Several numerical analyses based on QCD sum rules have been performed, see for a review e.g. 关34兴. The result we shall use is

Fˆ⫽0.30⫾0.05 GeV3/2 共20兲

which is obtained neglecting radiative corrections since they are neglected also in the evaluation of the Isgur-Wise form factor that the parametrization we use below is based on.

The effective theory approach gives predictions for the form factor V(q2) at high q2 and relates it to the parameter

␭. We can match this result with the theoretical calculations

coming from light cone sum rules关42,43兴共LCSR兲 and lattice QCD 关44–46兴. To do this we compute the form factor at

q2⯝q_max2 where it is dominated by the nearest low-lying vector meson pole. Computing the form factor at q2⯝q_max2

⬅(MB⫺MV)2 and at leading order in 1/mQ, one gets关47兴

V共qmax 2 _兲⫽gV

冑

2␭Fˆ MP⫹MV

冑

M_P 1 MV⫹⌬ 共21兲

where⌬ is the appropriate mass splitting, i.e., for the transi-tion B→K쐓,⌬⫽m_B

s

*⫺mB⯝135 MeV. From the LCSR and lattice QCD analyses for the transition B→K* we infer the value V⫽1.5 at q2_⬅q¯2_{⫽17 GeV}2_{. In order to compare with}

Eq.共21兲 we assume that in the range q2苸(q¯2,q_max2 ) the form factors are dominated by the B_s* simple pole. This gives

V(qmax

2

)⫽1.8 and, as a result,

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We note that to determine ␭ we used a positive sign of

V(q_max2 ). The same sign would be obtained using HQET and by assuming that the strange mass is large, see Eq.共17兲. We also note that a different result for the phase and the magni-tude of␭ was obtained in 关34兴. The difference in magnitude was due to the different methods used. Here we use LCSR and lattice QCD, while in关34兴 an extrapolation is used from the D→K* form factor. As to the phase, it is due to a dif-ferent choice we use here for the phase factor of the heavy vector meson, see Eq.共12兲.

To compute the matrix elements we use the following kinematics: p␮⫽mBv␮⫽共mB,0ជ兲, p␮D(_*)⫽mDv

⬘

␮, q⫽p⫺pD(_*). 共23兲 where ␻*⫽(m_B2⫹m_D2⫺m_D s 2 _)/2m

DmB and the angular inte-gration is over the directions of the vector vជ

⬘

⫽nជ

冑

␻*2_⫺1.

Our results are as follows:

M关B共v兲→Ds共q兲D*共⑀,v

⬘

兲兴⫽⫺K共mB⫹mD兲⑀*•v, 共24兲

M关B共v兲→Ds*共␩,q兲D共v

⬘

兲兴⫽⫺KmD_s␩*•共v⫹v

⬘

兲, 共25兲

M关B共v兲→Ds*共␩,q兲D*共⑀,v

⬘

兲兴⫽⫺iKmD_s␩*␮⑀*␣关i⑀␣␭␮␴v

⬘

␭v␴⫺g␮␣共1⫹␻*兲⫹v␣v␮

⬘

兴, 共26兲

where K⫽(GF/

冑

2)Vcb*Vcsa2

冑

冊

⫺pK␭

冉

p␲␴⫹ v

⬘

•p_␲ mD_* q␴

冊

. 共31兲

Equation共27兲 corresponds to the sum of diagrams 共a兲 and 共b兲 of Fig. 1; Eq. 共28兲 corresponds to diagram 共c兲 and, finally, Eq. 共29兲 corresponds to the sum of diagrams 共d兲 and 共e兲. Similar results hold for the K(␳,␻,␾) final states and we do not reproduce them here for the sake of brevity. F(兩pជ_␲兩) is a form factor taking into account that in the vertex DD*␲the

pion is not soft and therefore the coupling constant should be corrected. Its determination by a quark potential model is discussed in关30兴 using a quark model. The central value we use is F(兩pជ_␲兩)⫽0.065. In the absence of more detailed in-formation, the same form factor is adopted for the upper vertices in Fig. 1 as well.

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IV. NUMERICAL RESULTS

Let us first consider the short distance contribution. Us-ing, for the CKM matrix elements, ␳⫽0.2296, ␩⫽0.3249,

A⫽0.819, and for the involved form factors the value

col-lected in Table IV of Ref. 关48兴共a different determination based on QCD sum rules is in 关49兴兲, one gets the results reported in Table II. These values correspond to the follow-ing value of the angle␥ of the unitarity triangle:

␥⫽arctan

冉

␩_␳

冊

⯝54.8°. 共32兲

Next we consider the long distance absorptive part. For its parameters we assume the values of the previous section. For the Dsdecay constant we use fD_s⫽Fˆ/

冑

mD_s, with Fˆ given in Eq. 共20兲; for the Isgur-Wise form factor we use the param-etrization

␰共v•v

⬘

兲⫽

冉

2

1⫹v•v

⬘

冊

␳ˆ2

共33兲

with␳ˆ2⯝1. This parametrization agrees with the results for the b→c exclusive decays obtained by the CLEO and Belle Collaborations. It is a fit to the results obtained by the QCD sum rules method, see关34兴.

The numerical results for the imaginary part ofM_LDare given in Table II. Typical sizes of the different contributions to B⫺→K¯*0_␲⫺ _{are as follows. From the two terms in Eq.} 共27兲: Im M_LDa,b_{⫽⫺1.32⫻10}_⫺8

; from the terms in Eq.共28兲: ImM_LDc ⫽⫺0.16⫻10⫺8; from the terms in Eq. 共29兲: ImM_LDd,e⫽⫺0.66⫻10⫺8. For the B⫺→K0_␳⫺ _{channel we}

find: ImM_LDa,b⫽⫺1.52⫻10⫺8; ImM_LDc ⫽⫺0.70⫻10⫺8; ImM_LDd,e⫽⫺0.52⫻10⫺8. It can be noted that the phase of MSDis purely weak while the phase inMLDis only due to

strong interactions.

A. Branching ratios

From the results of Table II we can compute the branch-ing ratios (B) and the CP asymmetries. As explained in the

Introduction, we have not presented an evaluation of the real part of A_LD. Attempts for the K␲ channel can be found in

关30,31兴. Typical results are that the real part is of the same

order of the imaginary part, but the uncertainties are large. To estimate the CP averaged branching ratios we add there-fore to the imaginary part a real part as follows:

MLD⫽RLD⫹iILD 共34兲

and consider the values R_LD⫽0,⫾I_LD. The results are re-ported in Table III for the three values ofR_LD, together with the results obtained considering only the short distance am-plitude. They show the relevance of the long-distance contri-bution. The ‘‘best value’’ ofR_LDmight be obtained by a fit to the available experimental data; these data are reported in Table IV and a comparison with the results of the previous table shows that the preferred values, except for the channels with ␾ in the final state, satisfy R_LD⬇⫺I_LD. Instead, the decay in K␾ prefersR_LD⬇⫹I_LD. To find a ‘‘best value’’ of RLDwe defined a ␹2 as ␹2_⬅

兺

i

冉

Br_ith⫺Br_iex p ␴共Bri兲

冊

2 共35兲

where the Br_iex p are reported in Table IV and the␴(Br_i) are obtained summing quadratically the statistic and systematic errors in the same table. When the experiment provides an asymmetric error _␴

2

␴1 _{a conservative error was assumed:} ␴ ⫽max(␴1,␴2). For the decay B¯0→K¯*0␲0 we put Brex p ⫽0 and the corresponding error was fixed to be equal to the

experimental upper limit ␴⫽3.6⫻10⫺6. We have first at-tempted a fit with only one real parameter r⫽R_LD/I_LD, which corresponds to the use of the SU共3兲 nonet symmetry for the light vector mesons. The minimum value of

␹2₍_{⫽12.1) is obtained for r⫽1.207. Since however we do}

not expect the validity of the nonet hypothesis we have also TABLE II. Theoretical values forM_SD,M_LD. Units are GeV.

Process M_SD⫻108 ImM_LD⫻108 B⫹→K*0_␲⫹ _⫹1.45 _⫺2.14 B⫹→K*⫹␲0 _{⫹1.02⫺0.79i} _⫺1.52 B0_→K_*0_␲0 _{⫺0.60⫺0.08i} _⫹1.52 B0_→K_*⫹_␲⫺ _{⫹0.85⫺1.01i} _⫺2.14 B⫹→K0_␳⫹ _⫹0.17 _⫺2.74 B⫹→K⫹␳0 _{⫹0.39⫺0.64i} _⫺1.94 B0_→K0_␳0 _{⫹0.48⫺0.11i} _⫹1.94 B0_→K⫹_␳⫺ _{⫺0.28⫺0.74i} _⫺2.74 B⫹→K⫹␻ ⫺0.27⫺0.63i ⫺1.94 B0_→K0_␻ _{⫹0.03⫺0.10i} _⫺1.94 B⫹→K⫹␾ 1.30 ⫺2.58 B0_→K0_␾ _1.30 _⫺2.58

TABLE III. Branching ratiosB 共units 10⫺6). In the second and third columns, theoretical values computed using only the short distance amplitude and the full amplitude 共i.e. short distance and long distance兲; in the latter case the three values correspond respec-tively toR_LD⫽(⫹1,0,⫺1)⫻I_LD, see text.

Process B (SD only兲 B (SD⫹LD) B⫹→K*0_␲⫹ _1.96 _{(4.71,6.22,16.3)} B⫹→K*⫹_␲0 _1.56 _{(5.20,5.95,11.0)} B0_→K_*0_␲0 _0.32 _{(2.51,2.10,5.66)} B0_→K_*⫹_␲⫺ _1.50 _{(9.94,9.13,16.2)} B⫹→K0_␳⫹ _0.03 _{(13.1,6.99,14.8)} B⫹→K⫹␳0 _0.52 _{(8.42,6.32,11.2)} B0_→K0_␳0 _0.21 _{(7.88,3.07,4.71)} B0_→K⫹_␳⫺ _0.54 _{(18.1,10.4,15.6)} B⫹→K⫹␻ 0.44 (10.7,6.21,8.73) B0→K0␻ 0.01 (6.70,3.58,6.91) B⫹→K⫹␾ 1.55 (7.58,7.63,19.8) B0→K0␾ 1.42 (6.98,7.02,18.3)

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tried a two parameter fit, corresponding to the two multiplets

共octet⫹singlet兲 of the light vector mesons. This gives as a

result R_LD⫽⫺0.954I_LDfor the (␳, K*, ␻) set of particles and two solutions, R_LD⫽⫺0.201I_LD and R_LD

⫽⫹1.21ILD, for the␾ with the same␹2⫽7.8. The former

solution would be preferred because it represents a less im-portant deviation of the SU共3兲 nonet symmetry. Even if the two solutions produce the same values of the branching ra-tios they would produce different CP asymmetries. The re-sults we find are in Table IV. Besides our data, we also present two model calculations presented in 关32兴, where an analysis of all the PV 共not only strange兲 final states is per-formed. The first model 共called in that paper scenario 1兲 contains short distance terms 共QCD factorization兲 and un-constrained ␥ angle. Other theoretical determinations based on QCD factorization are in Ref. 关50兴, e.g. B(B0→K⫹␳⫺)

⫽12.1. The authors of 关32兴 do not agree with the

conclu-sions of Du et al., Ref.关50兴, because, differently from them, they include the K*␲ channels. The second model共scenario 2兲 uses QCD factorization and charming penguin contribu-tions as in the present paper, but differently from this paper, where only two real free parameters are used, they employ two universal complex amplitudes multiplied by a computed Clebsh-Gordan coefficient. It is worthwhile to note that a significant agreement with the data is obtained with our simple hypothesis. We also note that the value␥ of Eq.共32兲 obtained by a global fit to the CKM matrix 关51兴 is compat-ible with the B→K*␲, K(␳,␻,␾) branching ratios only if the charming penguin diagrams are included.

B. CP asymmetries

From previous results we can also compute the CP asym-metries for the various channels. The Belle Collaboration关5兴 reported the result

AC P

K⫺␻_⫽B共B⫺→K⫺␻兲⫺B共B⫹→K⫹␻兲

B共B⫺_→K⫺_␻_兲⫹B共B⫹_→K⫹_␻_兲

⫽⫺0.21⫾0.28⫾0.03. 共36兲

The result we find agrees with the data: AC P K⫺␻ ⫽⫺0.37 共theory兲. 共37兲 We also get AC P K¯0␻_⫽B共B¯ 0_→_␻_K_¯0_兲⫺B共B0_→_␻_K0_兲 B共B¯0_→_␻_K_¯0_兲⫹B共B0_→_␻_K0_兲⫽⫺0.05. 共38兲

We can similarly compute the asymmetries for the K*␲,K␳ channels, defined by A_␲⫹0_⫽B共B⫹→K*⫹␲0兲⫺B共B⫺→K*⫺␲0兲 B共B⫹_→K_*⫹_␲0_兲⫹B共B⫺_→K_*⫺_␲0_兲, A_␲0⫹_⫽B共B ⫹_→K_*0_␲⫹_兲⫺B共B⫺_→K¯_*0_␲⫺_兲 B共B⫹_→K_*0_␲⫹_兲⫹B共B⫺_→K¯_*0_␲⫺_兲, A_␲⫹⫺_⫽B共B0→K*⫹␲⫺兲⫺B共B¯0→K*⫺␲⫹兲 B共B0_→K_*_⫹_␲_⫺_{兲⫹B共B¯}0_→K_*_⫺_␲_⫹_兲, A_␲00_⫽B共B 0_→K_*0_␲0_{兲⫺B共B¯}0_→K¯_*0_␲0_兲 B共B0_→K_*0_␲0_{兲⫹B共B¯}0_→K¯_*0_␲0_兲, 共39兲

TABLE V. Theoretical values for the asymmetries; for the defi-nitions see Eq.共39兲.

Asymmetry A_␲⫹0 A_␲0⫹ A_␲⫹⫺ A00_␲ A_␳⫹0 A_␳0⫹ A_␳⫹⫺ A_␳00

AC P 0.27 0 0.31 ⫺0.04 0.27 0 0.30 ⫺0.08

TABLE IV. CP averaged branching ratiosB 共units 10⫺6). In the second column theoretical values computed using the present model. In the third column and fourth column theoretical computations based on Ref.关30兴: Scenario 共Sc.兲 1 refers to QCD with factorization and free

␥; scenario 2 refers to QCD⫹charming penguin contributions with constrained ␥ 共see text兲. Experimental data are from CLEO 关1,2兴, BaBar 关3,4兴, and Belle 关5兴 or averages from these data.

Process B 共this paper兲 B共关32兴, Sc. 1兲 B共关32兴, Sc. 2兲 B 共Expt.兲

B⫹→K*0_␲⫹ _15.6 _7.889 _11.080 _12.1_{⫾3.1 共av.兲} B⫺→K*⫺␲0 8.44 7.303 8.292 7.1_⫺7.1⫹11.4⫾1.0(⬍31) 共CLEO兲 B ¯0_→K¯_*0_␲0 _5.61 ⬍3.6 共CLEO兲 B0→K*⫹␲⫺ 12.0 9.760 10.787 19.3⫾5.2 共av.兲 B⫹→K0␳⫹ 14.1 7.140 14.006 B⫹→K⫹␳0 8.56 1.882 5.665 8.9⫾3.6 共av.兲 B→K0␳0 4.86 5.865 8.893 B0_→K⫹_␳⫺ _11.6 _6.531 _14.304 _15.9_{⫾4.7 共av.兲}

B⫹→K⫹␻ 6.19 2.398 6.320 9.2_⫺2.3⫹2.6⫾1.0 共CLEO兲; ⬍4 共BaBar兲; ⬍7.9 共Belle兲

B0→K0␻ 6.27 2.318 5.606 6.3⫾1.8 共av.兲

B⫹→K⫹␾ 9.11 8.941 9.479 8.9⫾1.2 共av.兲

(8)

with analogous definitions for the K␳ channel, with A_␲ →A␳ and the changes K*→K and␲→␳.

The results are presented in Table V. They have a peculiar pattern and will therefore provide a crucial test of the present model when future experimental data for these asymmetries are available.

ACKNOWLEDGMENTS

We thank T.N. Pham for valuable collaboration at an early stage of this work. C.I. is partially supported by MIUR under COFIN PRIN-2001.

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