Nouveaux r´ esultats : second sc´ enario

Les nouvelles valeurs des param`etres ajust´es sont alors :

Param`etre Domaine Sc´enario n^o2

γ (deg) [34^◦,82^◦] 81.933 m_s (GeV) [0.085,0.135] 0.085 µ (GeV) [2.1,8.4] 5.971 ρ_A [−1,1] 1.000 φ_A(deg) [−180,180] −87.907 λ_B (GeV) [0.2,0.5] 0.500 f_B (GeV) [0.14,0.22] 0.203 R_u [0.35,0.49] 0.350 R_c [0.018,0.025] 0.018 A^B→ρ₀ [0.3162,0.4278] 0.377 FB→π 1 [0.23,0.33] 0.301 AB→ω 0 [0.25,0.35] 0.326 AB→K^∗ 0 [0.3995,0.5405] 0.469 FB→K 1 [0.28,0.4] 0.280 Re[AP] [−0.01,0.01] 0.00253 Im[AP] [−0.01,0.01] −0.00181 Re[AV] [−0.01,0.01] −0.00187 Im[AV] [−0.01,0.01] 0.00049

Nous notons l`a encore que certains des param`etres prennent une de leurs valeurs extrˆemes. Par ailleurs, les valeurs des termes du mod`ele des pingouins charm´es sont plutˆot petites (principalement `a cause du canal B → ΦK qui est en bon accord avec QCDF1 et donc qui limite ´enorm´ement l’introduction d’un terme de longue distance).

Quant aux rapports d’embranchement, ils sont rassembl´es dans le tableau suivant : Canaux Exp´erience Sc´enario n^o2

Pr´ediction χ²

BR(B⁰ → ρ0π⁰) 2.07 ± 1.88 0.177 1.0

BR(B⁰ → ρ+π⁻) 10.962

BR(B⁰ → ρ−π⁺) 17.429

BR(B⁰ → ρ±π^∓) 25.53 ± 4.32 28.391 0.4

1. Bien que les canaux ´etranges soient g´en´eralement sous-estim´es par QCDF, les canaux s¯ss constituent une exception.

BR(B− → ω π−) 6.22 ± 1.7 5.186 0.4 BR(B− → Φ π−) 0.003 BR(B− → ρ−π0) 11.404 BR(B− → K∗−K0) 0.788 BR(B− → K∗0K⁻) 0.494 BR(B⁰ → ρ0K⁰) 8.893 BR(B⁰ → ω K⁰) 6.34 ± 1.82 5.606 0.2 BR(B⁰ → ρ+K⁻) 15.88 ± 4.65 14.304 0.1 BR(B⁰ → K∗−π⁺) 19.3 ± 5.2 10.787 2.7 BR(B− → K∗−π⁰) 7.1 ± 11.4 8.292 0.0 BR(B⁰ → Φ K⁰) 8.72 ± 1.37 8.898 0.0 BR(B− → K^∗0π⁻) 12.12 ± 3.13 11.080 0.1 BR(B− → ρ0K⁻) 8.92 ± 3.6 5.655 0.8 BR(B− → ρ−K⁰) 14.006 BR(B− → ω K−) 2.92 ± 1.94 6.320 3.1 BR(B− → Φ K−) 8.88 ± 1.24 9.479 0.2 BR(B⁰ → K^∗0η) 16.41 ± 3.21 18.968 0.6 BR(B− → K∗−η) 25.4 ± 5.6 15.543 3.1

et pour les asym´etries CP , nous obtenons :

Exp´erience Sc´enario n^o2 Pr´ediction χ² ∆ C_ρπ 0.38 ± 0.23 0.228 ^         3.9/4 C_ρπ 0.45 ± 0.21 0.092 A^{ρ π}_CP −0.22 ± 0.11 -0.115 A^{ρ K}_CP 0.19 ± 0.18 0.197 Aω π− CP −0.21 ± 0.19 -0.198 0.0 Aω K− CP −0.21 ± 0.28 0.189 2.0 A^{η K}_CP^∗− −0.05 ± 0.3 -0.217 0.3 A^{η K} ∗0 CP 0.17 ± 0.28 -0.158 1.4 A^{φ K}_CP⁻ −0.05 ± 0.2 0.005 0.1 20.8

A^ρ_CP^π −0.11 ± 0.16 ± 0.09 0.04

L’accord avec l’exp´erience est am´elior´e (comme il ´etait pr´evu) et nous avons bien obtenu une augmentation des asym´etries CP . Enfin, le test (( goodness of fit )) nous donne :

L’accord avec l’exp´erience est am´elior´e (comme il ´etait pr´evu) et nous avons bien obtenu une augmentation des asym´etries CP . Enfin, le test !! goodness of fit "" nous donne :

0 5 10 15 20 25 30 35 40 45 50 0 50 100 150 200 250 300 350 400 450 χ2 E x p ´er ie n ce s χ2 data

correspondant `a un niveau de confidence inf´erieur `a 7.7%. Le mod`ele ne peut donc pas ˆetre exclus par les donn´ees exp´erimentales.

2.6 Conclusions

Au vu de l’´etude pr´ec´edente, il apparaˆıt qu’il est impossible de faire un ajustement correct des donn´ees exp´erimentales avec le mod`ele de factorisation am´elior´e QCDF ; la raison `a l’origine de cet ´echec est une estimation trop faible par QCDF des canaux ´etranges (except´e le canal B → ΦK) ainsi qu’une valeur pour les asym´etries CP directes non-´etranges B → ρ+π− bien plus petite que la valeur exp´erimentale. Par ailleurs, le test !! goodness of fit "" rejette le mod`ele (0.1%) dont les valeurs ajust´ees ont d’ailleurs tendance `a vouloir sortir de leur domaine de variation autoris´e. Un ajustement r´ealis´e sans les canaux K∗ am´eliore, de fait, les r´esultats de mani`ere significante.

Afin d’augmenter les amplitudes P par rapport aux amplitudes T , dans le but de corriger le probl`eme pr´ec´edemment mentionn´e, nous avons rajout´e des termes d’interaction `a longue distance (les pingouins charm´es) ; les modifications introduites ne permettent plus vraiment d’exclure le mod`ele (niveau de confiance du test !! goodness of fit "" de l’ordre de 8%) mais un certain nombre de param`etres prennent encore une de leurs valeurs limites et les contributions des pingouins charm´es ajust´ees sont relativement faibles. Mˆeme si le mod`ele a ´et´e am´elior´e, ce n’est pas encore tr`es convaincant.

En r´esum´e, la situation pour QCDF vis-`a-vis des donn´ees exp´erimentales ne semble pas tr`es bonne et peut-ˆetre faudra-t-il consid´erer l’´evolution des mesures fournies par les exp´erimen-tateurs pour affiner le test ; en effet, la prise en compte de donn´ees plus r´ecentes concernant

correspondant `a un niveau de confiance inf´erieur `a 7.7%. Le mod`ele ne peut donc pas ˆetre exclu par les donn´ees exp´erimentales.

2.6 Conclusions

Au vu de l’´etude pr´ec´edente, il apparaˆıt qu’il est impossible de faire un ajustement correct des donn´ees exp´erimentales avec le mod`ele de factorisation am´elior´e QCDF ; la raison `a l’origine de cet ´echec est une estimation trop faible par QCDF des canaux ´etranges (except´e le canal B → ΦK) ainsi qu’une valeur pour les asym´etries CP directes non-´etranges B → ρ+π⁻ bien plus petite que la valeur exp´erimentale. Par ailleurs, le test (( goodness of fit )) rejette le mod`ele (0.1%) dont les valeurs ajust´ees ont d’ailleurs tendance `a vouloir sortir de leur domaine de variation autoris´e. Un ajustement r´ealis´e sans les canaux K^∗ am´eliore, de fait, les r´esultats de mani`ere significante.

Afin d’augmenter les amplitudes P par rapport aux amplitudes T , dans le but de corriger le probl`eme pr´ec´edemment mentionn´e, nous avons rajout´e des termes d’interaction `a longue distance (les pingouins charm´es) ; les modifications introduites ne permettent plus vraiment d’exclure le mod`ele (niveau de confiance du test (( goodness of fit )) de l’ordre de 8%) mais un certain nombre de param`etres prennent encore une de leurs valeurs limites et les contributions des pingouins charm´es ajust´ees sont relativement faibles. Mˆeme si le mod`ele a ´et´e am´elior´e, ce n’est pas encore tr`es convaincant.

concernant les canaux B → ρπ et B → ρK de la collaboration Babar ont augment´e le niveau de confiance de QCDF jusqu’`a 1% et diminu´e celui du mod´ele(( pingouins charm´es )) `

R. Aleksan,^1,*P.-F. Giraud,^1,†V. More´nas,^2,‡O. Pe`ne,^3,§and A. S. Safir^4,储

1DSM/DAPNIA/SPP, CEA/Saclay, F-91191 Gif-sur-Yvette, France

2LPC, Universite´ Blaise Pascal, CNRS/IN2P3 F-63000 Aubie`re Cedex, France

3LPT (Baˆt.210), Universite´ de Paris XI, Centre d’Orsay, 91405 Orsay-Cedex, France

4LMU Mu¨nchen, Sektion Physik, Theresienstraße 37, D-80333 Mu¨nchen, Germany

共Received 26 January 2003; published 23 May 2003兲

We try a global fit of the experimental branching ratios and C P asymmetries of the charmless B→PV

decays according to QCD factorization. We find it impossible to reach satisfactory agreement, the confidence level 共C.L.兲 of the best fit being smaller than 0.1%. The main reason for this failure is the difficulty to accommodate several large experimental branching ratios of the strange channels. Furthermore, experiment was not able to exclude a large direct C P asymmetry in B¯ →␳0 ⫹␲⫺_{, which is predicted to be very small by}

QCD factorization. Trying a fit with QCD factorization complemented by a charming-penguin-diagram-inspired model we reach a best fit which is not excluded by experiment共C.L. of about 8%兲 but is not fully convincing. These negative results must be tempered by the remark that some of the experimental data used are recent and might still evolve significantly.

DOI: 10.1103/PhysRevD.67.094019 PACS number共s兲: 13.25.Hw

I. INTRODUCTION

It is an important theoretical challenge to master the non-leptonic decay amplitudes and particularly B nonnon-leptonic de-cay. It is not only important per se, in view of the many experimental branching ratios which have been measured re-cently with increasing accuracy by BaBar关1–10兴, Belle 关11–

15兴, and CLEO 关16–21兴, but it is also necessary in order to

get control over the measurement of C P violating param-eters and particularly the so-called angle ␣ of the unitarity triangle. It is well known that extracting ␣ from measured indirect C P asymmetries needs a sufficient control of the relative size of the so-called tree共T兲 and penguin 共P兲

ampli-tudes.

However, the theory of nonleptonic weak decays is a dif-ficult issue. Lattice QCD gives predictions for semileptonic or purely leptonic decays but not directly for nonleptonic ones. For a while, one has used what is now called ‘‘naive factorization’’ which replaces the matrix element of a four-fermion operator in a heavy-quark decay by the product of the matrix elements of two currents, one semileptonic matrix element and one purely leptonic. It has been noticed for a while that naive factorization did provide reasonable results, although it was impossible to derive it rigorously from QCD except in the N_c→⬁ limit. It is also well known that the

matrix elements computed via naive factorization have a wrong anomalous dimension.

Recently an important theoretical progress has been per-formed 关22,23兴 which is commonly called ‘‘QCD

factoriza-tion.’’ It is based on the fact that the b quark is heavy

com-pared to the intrinsic scale of strong interactions. This allows one to deduce that nonleptonic decay amplitudes in the heavy-quark limit have a simple structure. It implies that corrections termed ‘‘nonfactorizable,’’ which were thought to be intractable, can be calculated rigorously. The anomalous dimension of the matrix elements is now correct to the order at which the calculation is performed. Unluckily the sublead-ingO(⌳/mb) contributions cannot in general be computed rigorously because of infrared singularities, and some of these which are chirally enhanced are not small, of order

O„m␲^2/mb(m_u⫹md)…, which shows that the inverse mb

power is compensated by m_␲/(m_u⫹md). In the seminal pa-pers of关22,23兴, these contributions are simply bounded

ac-cording to a qualitative argument which could as well justify a significantly larger bound with the risk of seeing these unpredictable terms become dominant. It is then of utmost importance to check experimentally QCD factorization

共QCDF兲.

Since a few years, it has been applied to B→PP 共two

charmless pseudoscalar mesons兲 decays. The general feature

is that the decay to nonstrange final states is predicted to be slightly larger than experiment while the decay to strange final states is significantly underestimated. In 关23兴 it is

claimed that this can be cured by a value of the unitarity-triangle angle␥larger than generally expected, larger maybe than 90 °. Taking also into account various uncertainties the authors conclude positively as for the agreement of QCD factorization with the data. In关24,25兴 it was objected that the

large branching ratios for strange channels argued in favor of the presence of a specific nonperturbative contribution called ‘‘charming penguin diagrams’’ 关25–30兴. We will return to

this approach later.

The B→PV 共charmless pseudoscalar ⫹ vector mesons兲

channels are more numerous and allow a more extensive check. In Ref. 关31兴 it was shown that naive factorization

implied a rather small兩P兩/兩T兩 ratio, for the B¯ →␳0 ⫾␲⫿ de-cay channel, to be compared to the larger one for the B *Email address: aleksan@hep.saclay.cea.fr

†Email address: giraudpf@hep.saclay.cea.fr

‡Email address: morenas@clermont.in2p3.fr

§Email address: pene@th.u-psud.fr 储

Email address: safir@theorie.physik.uni-muenchen.de

→␲⫹␲⫺_{. This prediction is still valid in QCD factorization} where the 兩P兩/兩T兩 ratio is of about 3% 共8%兲 for the B¯0

→␳⫹␲⫺ _(B¯ →␳0 ⫺␲⫹_{) channel against about 20% for the}

B0

¯ →␲⫹␲⫺_{one. If this prediction were reliable, it would put} the B¯ →␳⁰ ⫹␲⫺ _{channel in a good position to measure the} Cabibbo-Kobayashi-Maskawa 共CKM兲 angle ␣ via indirect

C P violation. This remark triggered the present work: we

wanted to check QCD factorization in the B→PV sector to

estimate the chances for a relatively easy determination of the angle␣.

The noncharmed B→PV amplitudes have been computed

in naive factorization关32兴, in some extension of naive

fac-torization including strong phases关33兴, in QCD factorization 关34–36兴, and some of them in so-called perturbative QCD 关37,38兴. In 关39兴, a global fit to B→PP,PV,VV was

investi-gated using QCDF in the heavy-quark limit and a plausible set of soft QCD parameters has been found that, apart from three pseudoscalar vector channels, fit well the experimental branching ratios. Recently关36兴 it was claimed from a global

fit to B→PP,PV that the predictions of QCD factorization

are in good agreement with experiment when one excludes some channels from the global fit. When this paper appeared we had been for some time considering this question and our feeling was significantly less optimistic. This difference shows that the matter is far from trivial mainly because ex-perimental uncertainties can still be open to some discussion. We would like in this paper to understand better the origin of the difference between our unpublished conclusion and the one presented in关36兴 and try to settle the present status of the

comparison of QCD factorization with experiment.

One general remark about QCD factorization is that it yields predictions which do not differ so much from naive factorization ones. This is expected since QCD factorization makes a perturbative expansion the zeroth order of which being naive factorization. As a consequence, QCD factoriza-tion predicts very small direct C P violafactoriza-tion in the nonstrange channels. Naive factorization predicts vanishing direct C P violation. Indeed, direct C P violation needs the occurrence of two distinct strong contributions with a strong phase be-tween them. It vanishes when the subdominant strong con-tribution vanishes and also when the relative strong phase does as is the case in naive factorization. In the case of nonstrange decays, the penguin共P兲 and tree 共T兲 contributions

being at the same order in the Cabibbo angle, the penguin diagram is strongly suppressed because the Wilson coeffi-cients are suppressed by at least one power of the strong coupling constant␣sand the strong phase in QCD factoriza-tion is generated by a O(␣s) correction. Having both P/T and the strong phase small, the direct C P asymmetries are doubly suppressed. Therefore a sizable experimental direct

C P asymmetry in B¯ →␳⁰ ⫹␲⫺ _{which is not excluded by}

ex-periment关9兴 would be at odds with QCD factorization. We

will discuss this later on. Notice that this argument is inde-pendent of the value of the unitarity angle␥, contrarily to arguments based on the value of some branching ratios which depend on␥ 关23兴.

The perturbative QCD共PQCD兲 predicts larger direct CP

asymmetries than QCDF due to the fact that penguin contri-butions to annihilation diagrams, claimed to be calculable in PQCD, contribute to a larger amount to the amplitude and have a large strong phase. In fact, in PQCD, this penguin annihilation diagram is claimed to be of the same order,

O(␣s), than the dominant naive factorization diagram while in QCDF it is also O(␣s) but smaller than the dominant naive factorization which is O(1). Hence, in PQCD, this

large penguin contribution with a large strong phase yields a large C P asymmetry关40–42兴.

If QCD factorization is concluded to be unable to describe the present data satisfactorily, while there is to our knowl-edge no theoretical argument against it, we have to incrimi-nate nonperturbative contributions which are larger than ex-pected. One could simply enlarge the allowed bound for those contributions which are formally subleading but might be large. However, a simple factor of 2 on these bounds makes these unpredictable contributions comparable in size with the predictable ones, if not larger. This spoils the pre-dictivity of the whole program.

A second line is to make some model about the nonper-turbative contribution. The ‘‘charming penguin diagram’’ ap-proach 关27,30兴 starts from noticing the underestimate of

strange-channel branching ratios by the factorization ap-proaches. This will be shown to apply to the PV channels as well as to the P P ones. This has triggered us to try a charming-penguin-diagram-inspired approach. It is assumed that some hadronic contribution to the penguin loop is non-perturbative, in other words that weak interactions create a charm-anticharm intermediate state which turns into non-charmed final states by strong rescattering. In order to make the model as predictive as possible we will use not more than two unkown complex numbers and use flavor symmetry in strong rescattering.

In Sec. II we will recall the weak-interaction effective Hamiltonian. In Sec. III we will recall QCD factorization. In Sec. IV we will compare QCD factorization with experimen-tal branching ratios and direct C P asymmetries. In Sec. V we will propose a model for nonperturbative contribution and compare it to experiment. We will then conclude.

II. EFFECTIVE HAMILTONIAN

The effective weak Hamiltonian for charmless hadronic B decays consists of a sum of local operators Q_i multiplied by short-distance coefficients C_i given in Table I and products of elements of the quark mixing matrix,␭p⫽VpbV_ps* or␭p⬘

⫽VpbV_pd* . Below we will focus on B→PV decays, where P

and V hold for pseudoscalar and vector mesons, respectively. Using the unitarity relation⫺␭t⫽␭u⫹␭c, we write

Heff⫽^GF

冑

2 _p

兺

_⫽u,c ␭p

冉

C₁Q₁^p⫹C2Q₂^p⫹_{i⫽3, . . . ,10}

兺

C_iQ_i

⫹C7␥^Q7␥⫹C8gQ_8g

冊

⫹H.c., 共1兲

where Q_1,2^p are the left-handed current–current operators arising from W-boson exchange, Q_{3, . . . ,6} and Q_{7, . . . ,10} are

QCD and electroweak penguin operators, and Q₇␥ ^{and Q}8g

are the electromagnetic and chromomagnetic dipole opera-tors. They are given by

Q₁^p⫽共p¯b兲V⫺A共s¯p兲V⫺A^{, Q}2 p⫽共p¯ib_j兲V⫺A共s¯jp_i兲V⫺A^, Q₃⫽共s¯b兲V⫺A

兺

_q 共q¯q兲V⫺A^, Q₄⫽共s¯ib_j兲V⫺A

兺

_q 共q¯jq_i兲V⫺A^, Q₅⫽共s¯b兲V⫺A

兺

_q 共q¯q兲V⫹A^, Q₆⫽共s¯ib_j兲V⫺A

兺

_q 共q¯jq_i兲V⫹A^, Q₇⫽共s¯b兲V⫺A

兺

_q 3 2e_q共q¯q兲V⫹A^, Q₈⫽共s¯ib_j兲V⫺A

兺

_q 3 2e_q共q¯jq_i兲V⫹A^, Q₉⫽共s¯b兲V⫺A

兺

_q 3 2 e_q共q¯q兲V⫺A^, Q₁₀⫽共s¯ib_j兲V⫺A

兺

_q 3 2 e_q共q¯jq_i兲V⫺A^, Q₇␥⫽^⫺e 8␲2m_b^¯s␴_␮␯共1⫹␥5兲F␮␯_b, Q_8g⫽^⫺gs 8␲2m_b^¯s␴_␮␯共1⫹␥5兲G␮␯_b, 共2兲

where (q^¯₁q₂)_V⫾A⫽q¯1␥␮⁽¹⫾␥5)q₂, i, j are color indices, e_q are the electric charges of the quarks in units of兩e兩, and a

summation over q⫽u,d,s,c,b is implied. The definition of

the dipole operators Q₇␥ ^{and Q}8g corresponds to the sign convention iD␮⫽i⳵␮⫹gsA_a␮_t

a for the gauge-covariant de-rivative. The Wilson coefficients are calculated at a high scale␮⬃MW and evolved down to a characteristic scale␮

⬃mb using next-to-leading order renormalization-group equations. The essential problem obstructing the calculation of nonleptonic decay amplitudes resides in the evaluation of the hadronic matrix elements of the local operators contained in the effective Hamiltonian.

III. QCD FACTORIZATION IN B\PV DECAYS

When the QCDF method is applied to the decays B

→PV, the hadronic matrix elements of the local effective

operators can be written as 具PV兩Oi兩B典⫽F1 B→P共0兲TV,i I 쐓 fV⌽V⫹A0 B→V共0兲TP,i I 쐓 fP⌽P ⫹Ti II쐓 fB⌽B쐓 fV⌽V쐓 fP⌽P, 共3兲

where ⌽M are leading-twist light-cone distribution ampli-tudes, and the쐓-products imply an integration over the

light-cone momentum fractions of the constituent quarks inside the mesons. A graphical representation of this result is shown in Fig. 1.

Here F₁^B→P and A₀^B→V denote the form factors for B

→P and B→V transitions, respectively. ⌽B(␰), ⌽V(x), and

⌽P(y ) are the light-cone distribution amplitudes共LCDAs兲 of

parameters are ⌳MS (5)

⫽0.225 GeV, mt(m_t)⫽167 GeV, mb(m_b) ⫽4.2 GeV, MW⫽80.4 GeV, ␣⫽1/129, and sin2␪W⫽0.23 关23兴. NLO C₁ C₂ C₃ C₄ C₅ C₆ ␮⫽mb/2 1.137 ⫺0.295 0.021 ⫺0.051 0.010 ⫺0.065 ␮⫽mb 1.081 ⫺0.190 0.014 ⫺0.036 0.009 ⫺0.042 ␮⫽2mb 1.045 ⫺0.113 0.009 ⫺0.025 0.007 ⫺0.027 C7/␣ C8/␣ C9/␣ C10/␣ C₇^eff_␥ C_8g^eff ␮⫽mb/2 ⫺0.024 0.096 ⫺1.325 0.331 — — ␮⫽mb ⫺0.011 0.060 ⫺1.254 0.223 — — ␮⫽2mb 0.011 0.039 ⫺1.195 0.144 — — LO C₁ C₂ C₃ C₄ C₅ C₆ ␮⫽mb/2 1.185 ⫺0.387 0.018 ⫺0.038 0.010 ⫺0.053 ␮⫽mb 1.117 ⫺0.268 0.012 ⫺0.027 0.008 ⫺0.034 ␮⫽2mb 1.074 ⫺0.181 0.008 ⫺0.019 0.006 ⫺0.022 C₇/␣ C₈/␣ C₉/␣ C₁₀/␣ C₇^eff_␥ C_8g^eff ␮⫽mb/2 ⫺0.012 0.045 ⫺1.358 0.418 ⫺0.364 ⫺0.169 ␮⫽mb ⫺0.001 0.029 ⫺1.276 0.288 ⫺0.318 ⫺0.151 ␮⫽2mb 0.018 0.019 ⫺1.212 0.193 ⫺0.281 ⫺0.136

FIG. 1. Graphical representation of the factor-ization formula. Only one of the two form-factor terms in Eq.共3兲 is shown for simplicity.

valence quark Fock states for B, vector, and pseudoscalar mesons, respectively. T_i^I,IIdenote the hard-scattering kernels, which are dominated by hard gluon exchange when the power suppressed O(⌳QCD/m_b) terms are neglected. So they are calculable order by order in perturbation theory. The leading terms of T^Icome from the tree level and correspond to the naive factorization共NF兲 approximation. The order of

␣sterms of T^Ican be depicted by vertex-correction diagrams Figs. 2共a兲–2共d兲 and penguin-correction diagrams Figs.

2共e兲,共f兲. TIIdescribes the hard interactions between the spec-tator quark and the emitted meson M₂when the gluon virtu-ality is large. Its lowest order terms areO(␣s) and can be depicted by hard-spectator-scattering diagrams, Figs. 2共g兲–

2共h兲. One of the most interesting results of the QCDF

ap-proach is that, in the heavy-quark limit, the strong phases arise naturally from the hard-scattering kernels at the order of␣s. As for the nonperturbative part, they are, as already mentioned, taken into account by the form factors and the LCDAs of mesons up to corrections which are power sup-pressed in 1/m_b.

With the above discussions on the effective Hamiltonian of B decays Eq.共1兲 and the QCDF expressions of hadronic

matrix elements, Eq.共3兲, the decay amplitudes for B→PV in

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