Measuring B→ρπ decays and the unitarity angle α

(1)

Measuring B

\␳␲

decays and the unitarity angle

␣

A. Deandrea,1R. Gatto,2M. Ladisa,1,3G. Nardulli,1,3and P. Santorelli4

1_{Theory Division, CERN, CH-1211 Gene`ve 23, Switzerland}

2_{De´partement de Physique The´orique, Universite´ de Gene`ve, 24 quai E.-Ansermet, CH-1211 Gene`ve 4, Switzerland} 3_{Dipartimento di Fisica, Universita` di Bari and INFN Bari, via Amendola 173, I-70126 Bari, Italy}

4_{Dipartimento di Scienze Fisiche, Universita` di Napoli ‘‘Federico II’’ and INFN Napoli, Mostra d’Oltremare 20, I-80125 Napoli, Italy} 共Received 4 February 2000; published 15 June 2000兲

The decay mode B→␳␲ is currently studied as a channel allowing us, in principle, to measure without ambiguities the angle␣ of the unitarity triangle. It is also investigated by the CLEO Collaboration where a branching ratio larger than expected for the decay mode B⫾→␳0_␲⫾_{has been found. We investigate the role} that the B*and B0(0⫹) resonances might play in these analyses.

PACS number共s兲: 11.30.Er, 12.15.Hh, 13.25.Hw

I. INTRODUCTION

The measurement of the angle ␣ in the unitarity triangle will be one of the paramount tasks of the future b factories, such as the dedicated e⫹e⫺ machines for the BaBar experi-ment at SLAC关1兴 and the BELLE experiment at KEK 关2兴, or hadron machines such as the Large Hadron Collider 共LHC兲 at CERN, with its program for B physics.1 Differently from the investigation of the ␤ angle, for which the B→J/␺KS

channel has been pinned up 关3兴 and ambiguities can be re-solved 关4兴, the task of determining the angle ␣ is compli-cated by the problem of separating two different weak had-ronic matrix elements, each carrying its own weak phase. The evaluation of these contributions, referred to in the lit-erature as the tree 共T兲 and the penguin 共P兲 contributions, suffers from the common theoretical uncertainties related to the estimate of composite four-quark operators between had-ronic states. For these estimates, only approximate schemes, such as the factorization approximation, exist at the moment, and for this reason several ingenuous schemes have been devised, trying to disentangle T and P contributions. In gen-eral one tries to exploit the fact that in the P amplitudes only the isospin-1/2 part2of the nonleptonic Hamiltonian is active

关5兴; by a complex measurement involving several different

isospin amplitudes, one should be able to separate the two amplitudes and to get rid of the ambiguities arising from the ill-known penguin matrix elements.

One of the favorite proposals involves the study of the reaction B→␳␲, i.e., six channels arising from the neutral B decay: B ¯0_→_␳⫹_␲⫺_, _共1兲 B ¯0_→_␳⫺_␲⫹_, _共2兲 B ¯0_→_␳0_␲0_, _共3兲 together with the three charge-conjugate channels, and the charged decay modes:

B⫺→␳⫺␲0, 共4兲 B⫺→␳0_␲⫺_, _共5兲 with two other charge-conjugate channels. Different strate-gies have been proposed to extract the angle ␣, either in-volving all the decay modes of a B into a␳␲pair as well as three time-asymmetric quantities measurable in the three channels for neutral B decays 关6–8兴, or attempting to mea-sure only the neutral B decay modes by looking at the time-dependent asymmetries in different regions of the Dalitz plot.3

Preliminary to these analyses is the assumption that, using cuts in the three invariant masses for the pion pairs, one can extract the ␳ contribution without significant background contamination. The␳ has spin 1, the␲ spin 0 as well as the initial B, and therefore the ␳ has angular distribution cos2_␪ ₍_␪_{is the angle of one of the}_␳_{decay products with the} other␲ in the␳ rest frame兲. This means that the Dalitz plot is mainly populated at the border, especially the corners, by this decay. Only very few events should be lost by excluding the interior of the Dalitz plot, which is considered a good way to exclude or at least reduce backgrounds. Analyses following these hypotheses were performed by the BaBar working groups关1兴; Monte Carlo simulations, including the background from the f0 resonance, show that, with cuts at m_␲␲⫽m_␳⫾300 MeV, no significant contributions from other sources are obtained. Also the role of excited reso-nances such as the ␳

⬘

and the nonresonant background has been discussed关9兴.

A signal of possible difficulties for this strategy arises from new results from the CLEO Collaboration recently re-ported at the DPF99 and APS99 Conferences 关10兴:

1_{Opportunities for B physics at the LHC have been recently} dis-cussed at the workshop on Standard Model Physics共and More兲 at the LHC, 1999; copies of transparencies can be found at the site http://home.cern.ch/ mlm/lhc99/oct14ag.html

2_{If one neglects electroweak penguin amplitudes.}

3_{In this way the measurement of a decay mode with two neutral} pions in the final state, Eq.共4兲, can be avoided.

(2)

B共B⫾_→_␳0_␲⫾_{兲⫽共1.5⫾0.5⫾0.4兲⫻10}⫺5_, _共6兲 B共B→␳⫿_␲⫾_兲⫽共3.5 ⫺1.0 ⫹1.1_{⫾0.5兲⫻10}⫺5_, _共7兲 with a ratio R⫽ B共B→␳ ⫿_␲⫾_兲 B共B⫾_→_␳0_␲⫾_兲⫽2.3⫾1.3. 共8兲 As discussed in关10兴, this ratio looks rather small; as a matter of fact, when computed in simple approximation schemes, including factorization with no penguin contributions, one gets, from the Deandrea–Di Bartolomeo–Gatto–Nardulli

共DDGN兲 model of Ref. 关11兴, R⯝13, admittedly with a large

uncertainty; another popular approach, i.e., the Wirbel-Bauer-Stech 共WBS兲 model 关12兴, gives R⯝6 共in both cases we use a₁⫽1.02, a₂⫽0.14). The aim of the present study is to show that a new contribution, not discussed before, is indeed relevant to decay共5兲 and to a lesser extent to decay

共3兲. It arises from the virtual resonant production depicted in

Fig. 1, where the intermediate particle is the B*meson reso-nance or other excited states. The B*resonance, because of phase-space limitations, cannot be produced on the mass shell. Nonetheless the B* contribution might be important, owing to its almost degeneracy in mass with the B meson; therefore its tail may produce sizable effects in some of the decays of B into light particles, also because it is known theoretically that the strong coupling constant between B, B*

and a pion is large关13兴. Concerning other states, we expect their role to decrease with their mass, since there is no en-hancement from the virtual particle propagator; we shall only consider the 0⫹state B0with JP⫽0⫹because its coupling to a pion and the meson B is known theoretically to be uni-formly共in momenta兲 large 关13兴. The plan of the paper is as follows. In Sec. II we list the hadronic quantities that are needed for the computation of the widths; in Sec. III we present the results and finally, in Sec. IV, we give our con-clusions.

II. MATRIX ELEMENTS

The effective weak nonleptonic Hamiltonian for the

兩⌬B兩⫽1 transition is4 H⫽GF

冑

2

再

Vub*Vudk

兺

⫽1 2 Ck共␮兲Qk⫺Vtb*Vtd

兺

k⫽3 6 Ck共␮兲Qk

冎

. 共9兲

The operators relevant to the present analysis are the so-called current-current operators:

Q1⫽共d¯␣u␤兲V⫺A共u¯␤b␣兲V⫺A,

Q2⫽共d¯␣u␣兲V⫺A共u¯␤b␤兲V⫺A, 共10兲

and the QCD penguin operators:

兲V_⫹A, 共11兲

We use the following values of the Wilson coefficients: C1⫽⫺0.226, C2⫽1.100, C3⫽0.012, C4⫽⫺0.029, C5

⫽0.009, C6⫽⫺0.033; they are obtained in the ’t Hooft– Veltmann 共HV兲 scheme 关14兴, with ⌳¯_{M S}

(5)

⫽225 MeV, ␮

⫽m¯

b(mb)⫽4.40 GeV, and mt⫽170 GeV. For the

Cabibbo-Kobayashi-Maskawa 共CKM兲 mixing matrix 关15兴 we use the Wolfenstein parametrization 关16兴 with ␳⫽0.05, ␩⫽0.36, and A⫽0.806 in the approximation accurate to order ␭3 in the real part and ␭5 in the imaginary part, i.e., V_ud⫽ 1⫺␭2_{/2, V}

ub⫽ A␭3关␳⫺i␩(1⫺␭2/2)兴, Vtd⫽A␭3(1

⫺␳⫺i␩), and Vtb⫽1.

The diagram of Fig. 1 describes two processes. For the B*intermediate state there is an emission of a pion by strong interactions, followed by the weak decay of the virtual B*

into two pions; for the ␳ intermediate state there is a weak decay of B→␳␲followed by the strong decay of the␳ reso-nance. We compute these diagrams as Feynman graphs of an effective theory within the factorization approximation, us-ing information from the effective Lagrangian for heavy and light mesons and form factors for the couplings to the weak currents.5

To start with we consider the strong coupling constants. They are defined as

4_{We omit, as usual in these analyses, the electroweak operators}

Qk (k⫽7, 8, 9, 10兲; they are in general small, but for Q9, whose

role might be sizable; its inclusion in the present calculations would be straightforward.

5_{In the second reference of}_{关4兴 a similar approach has been used to} describe the decay mode B0→D⫹D⫺␲0; the main difference is that for B→3␲ we cannot use soft pion theorems and chiral perturba-tion theory, because the pions are in general hard; therefore we have to use information embodied in the semileptonic beauty meson form factors. This is also the main difference with respect to关8兴.

FIG. 1. The polar diagram. For the B resonances (B*

⫽1⫺_{, 0}⫹_{) the strong coupling is on the left and the weak coupling}

(3)

具

B¯0共p

⬘

兲␲⫺共q兲兩B*⫺_共p,⑀_兲

典

⫽gB*B␲_⑀_•q,

具

B⫺共p

⬘

兲␲⫹共q兲兩B¯0₀共p兲

典

⫽GB0B␲共p2兲,

共12兲

具

␲0_共q

_⬘

_兲_␲⫺_共q兲兩_␳⫺_共p,_⑀_兲

_典

_⫽g

␳⑀•共q

⬘

⫺q兲.

In the heavy quark mass limit one has

gB*B␲_⫽2mBg f_␲ , 共13兲 GB0B␲共s兲⫽⫺

冑

mB₀mB 2 s⫺m_B2 mB₀ h f_␲. 共14兲 For g and h we have limited experimental information and we have to use some theoretical inputs. For g and h reason-able ranges of values are g⫽0.3–0.6, ⫺h⫽0.4–0.7 关17兴. These numerical estimates encompass results obtained by different methods: QCD sum rules 关13兴, potential models

关18兴, effective Lagrangian 关19兴, Nambu–Jona-Lasinio– 共NJL-兲 inspired models 关20兴. Moreover g_␳⫽5.8 and f_␲ ⬃130 MeV. This value of g␳is commonly used in the chiral

effective theories including the light vector meson reso-nances and corresponds to ⌫_␳⯝150 MeV; see, for instance

0兩u¯␥␮d兩␳⫺共q,⑀兲

典

⫽ f_␳⑀␮, 共15兲

where f_␳⫽0.15 GeV2 关22兴 and

A₀␲⫽A₀␲共0兲⫽0.16, A₀⫽A₀共0兲⫽0.29, 共16兲

F₁⫽F₁共0兲⫽F₀共0兲⫽0.37, F₀␲⫽F˜₁共0兲⫽F˜₀共0兲⫽⫺0.19.

共17兲

The first three numerical inputs have been obtained by the relativistic potential model; A0 and F1 can be found in关21兴, while A₀␲has been obtained here for the first time, using the same methods. The last figure in 共17兲 concerns F₀␲, for which such an information is not available; for it we used the methods of关17兴 and the strong coupling BB0␲computed in

关23兴.

III. AMPLITUDES AND NUMERICAL RESULTS

For all the channels we consider three different contribu-tions A_␳, AB_*, AB₀, due respectively to the ␳ resonance,

the B*pole, and the B0positive parity 0⫹resonance, whose mass we take6to be 5697 MeV.

For each of the amplitudes

A⫺⫺⫹⫽A共B⫺→␲⫺␲⫺␲⫹兲, 共18兲 A⫺00⫽A共B⫺→␲⫺␲0_␲0_兲, _共19兲 A⫹⫺0⫽A共B¯0→␲⫹␲⫺␲0兲, 共20兲 we write the general formula7 Ai jk⫽A_␳i jk⫹A_Bi jk_*⫹A_B

0 i jk

. We get, for the process共18兲,

,

6_{We identify the 0}⫹_{state mass with the average mass of the B}_**

states given in关24兴.

7_{We add coherently the three contributions; the relative sign the B} resonances on one side and the␳ contribution on the other is irrel-evant, as the former are dominantly real and the latter is dominantly imaginary. The relative sign between B* and B0 is fixed by the effective Lagrangian for heavy mesons.

(4)

AB0⫺⫺⫹⫽K˜0共mB₀ 2 _⫺m ␲ 2_兲

冋

1 t⫺m_B 0 2 _⫹i⌫ B₀mB₀ ⫹ 1 t

⬘

⫺m_B 0 2 _⫹i⌫ B₀mB₀

册

, 共21兲

where, if p_␲⫺ is the momentum of one of the two negatively charged pions t⫽(p_␲⫺⫹p_␲⫹)2, t

⬘

is obtained by exchang-ing the two identical pions and u is the invariant mass of the two identical negatively charged pions. Clearly one has u

⫹t⫹t

⬘

⫽mB

2_⫹3m

␲

2VtbVtd *g␳

冑

2

c6⫹ c₅ 3

冊

m_␲2 共mb⫹mq兲2mq

册

冎

, K˜0⫽h

冑

mB m_B0F0␲ GF

冑

2共mB0 2 _⫺m B 2_兲

再

VubVud*

冉

c2⫹ c1 3

冊

⫺VtbVtd*

冋

c4⫹ c3 3 ⫺2

冉

c6⫹ c5 3

冊

册

冎

, 共22兲

with mb⫽4.6 GeV, mq⬃mu⬃md⯝6 MeV, ⌫B_*⫽0.2 keV, ⌫B₀⫽0.36 GeV 关17兴. Moreover, for the process 共19兲

⬘

is obtained by

exchanging the two neutral pions and u is their invariant mass 共again we have a relation among the different Mandelstam variables: s⫹s

⬘

冑

2

册

冎

, 共24兲 and ⌸共x,y兲⫽m␲2⫺ y 2⫹ x共m_B2⫺m_␲2⫺x兲 4m_B * 2 , 共25兲

(5)

while K˜0 was given above in共22兲.

Finally, for the neutral B decay 共20兲, we have

A_␳⫹⫺0⫽␩0 u⫺s t⫺m_␳2⫹i⌫_␳m_␳⫹␩ ⫹ s⫺t u⫺m_␳2⫹i⌫_␳m_␳⫹␩ ⫺ t⫺u s⫺m_␳2⫹i⌫_␳m_␳, A_B * ⫹⫺0_⫽K⌸共s,t兲⫹K1⌸共s,u兲 s⫺m_B2_*⫹i⌫B_*mB_* ⫺ K⌸共t,s兲 t⫺m_B2_*⫹i⌫B_*mB_* , A_B 0 ⫹⫺0_⫽

冉

K˜0⫹K˜cc s⫺m_B 0 2 _⫹i⌫ B₀mB₀ ⫹ K˜ 0 t⫺m_B 0 2 _⫹i⌫ B₀mB₀

冊

共mB₀ 2 _⫺m ␲ 2_兲, _共26兲

where s⫽(p_␲⫺⫹p_␲0)2, t⫽(p_␲⫺⫹p_␲⫹)2, u⫽(p_␲⫹⫹p_␲0)2, and s⫹t⫹u⫽m_B2⫹3m_␲2. The constants appearing in these

)⭐m_␳

⫹␦ for B⫺→␲⫺␲⫺␲⫹ and m_␳⫺␦⭐(

冑

s,

冑

s

⬘

)⭐m_␳⫹␦ for B⫺→␲⫺␲0␲0. For␦ we take 300 MeV. This amounts to require that two of the three pions 共those corresponding to the charge of the␳) reconstruct the␳ mass within an interval of 2␦. Numerical uncertainty due to the integration proce-dure is⫾5%.

We can notice that the inclusion of the new diagrams (B resonances in Fig. 1兲 produces practically no effect for the

B⫺→␲⫺␲0_␲0 _{decay mode, while for B}⫺_→_␲⫹_␲⫺_␲⫺ _the effect is significant. For the choice of parameters in Table I the overall effect is an increase of 50% of the branching ratio as compared to the result obtained by the␳ resonance alone. In the case of Table II we obtain an even larger result, i.e., a total branching ratio B(B⫺→␲⫹␲⫺␲⫺) of 0.82⫻10⫺5, in reasonable agreement with the experimental result 共6兲共the contribution of the␳ alone would produce a result smaller by a factor of 2兲. It should be observed that the events arising from the B resonances diagrams represent an irreducible background, as one can see from the sample Dalitz plot de-picted in Fig. 2 for the B⫺→␲⫹␲⫺␲⫺共on the axis the two m_␲2⫹_␲⫺ squared invariant masses兲. The contributions from

the B resonances populate the whole Dalitz plot and, there-fore, cutting around t⬃t

⬘

⬃m_␳ significantly reduces them. Nevertheless their effect can survive the experimental cuts, since there will be enough data at the corners, where the contribution from the␳ dominates. Integrating on the whole Dalitz plot, with no cuts and including all contributions, gives

TABLE I. Effective branching ratios for the charged B decay channels into three pions for the choice of the strong coupling con-stants g⫽0.40 and h⫽⫺0.54. Cuts as indicated in the text.

Channels ␳ ␳⫹B* ␳⫹B*⫹B0

B⫺→␲⫺␲0␲0 1.0⫻10⫺5 1.0⫻10⫺5 1.0⫻10⫺5

B⫺→␲⫹␲⫺␲⫺ 0.41⫻10⫺5 0.58⫻10⫺5 0.63⫻10⫺5

TABLE II. Effective branching ratios for the charged B decay channels into three pions for the choice of the strong coupling con-stants g⫽0.60 and h⫽⫺0.70. Cuts as indicated in the text.

Channels ␳ ␳⫹B* ␳⫹B*⫹B0

B⫺→␲⫺␲0␲0 1.1⫻10⫺5 1.0⫻10⫺5 1.1⫻10⫺5

(6)

X_Br共B⫺→␲⫺␲0_␲0_{兲⫽1.5⫻10}⫺5_,

X_Br共B⫺→␲⫹␲⫺␲⫺兲⫽1.4⫻10⫺5, 共28兲 where the values of the coupling constants are as in Table I. We now turn to the neutral B decay modes. We define effective width integrating the Dalitz plot only in a region around the␳ resonance:

⌫e f f共B¯0→␳⫺␲⫹兲⫽⌫共B¯0→␲⫹␲⫺␲0兲兩m_␳⫺␦⭐冑s⭐m_␳⫹␦, 共29兲 ⌫e f f共B¯0→␳⫹␲⫺兲⫽⌫共B¯0→␲⫹␲⫺␲0兲兩m_␳⫺␦⭐冑u⭐m_␳⫹␦, 共30兲 ⌫e f f共B¯ 0_→_␳0_␲0_{兲⫽⌫共B¯}0_→_␲_⫹_␲_⫺_␲0_兲兩 m_␳⫺␦⭐冑t⭐m_␳⫹␦. 共31兲

The Mandelstam variables have been defined above and again we use␦⫽300 MeV 关25兴. Similar definitions hold for the B0 decay modes. The results in Table III show basically no effect for the B¯0→␳⫾␲⫿decay channels and a moderate effect for the␳0_␲0 _{decay channel. The effect in this channel} is of the order of 20%共resp. 50%兲 for B¯0 共resp. B0) decay, for the choice g⫽0.60, h⫽⫺0.70; for smaller values of the strong coupling constants the effect is reduced. Integration on the whole Dalitz plot, including all contributions, gives

XBr共B¯0→␲⫹␲⫺␲0兲⫽2.6⫻10⫺5 共32兲 confirming again that most of the branching ratio is due to the␳ exchange共the first three lines of the␳ column in Table III sum up to 2.3⫻10⫺5).

To allow the measurement of ␣, the experimental pro-grams will consider the asymmetries arising from the time-dependent amplitude: A共t兲⫽e⫺⌫/2t

冉

_cos⌬mt 2 A ⫹⫺0_{⫾i sin}⌬mt 2 A¯ ⫹⫺0

冊

_, _共33兲

where one chooses the⫾ sign according to the flavor of the B, and ⌬m is the mass difference between the two mass eigenstates in the neutral B system. Here A¯⫹⫺0is the charge-conjugate amplitude. We have performed asymmetric inte-grations over the Dalitz plot for three variables: R₁, R₂, and R3, which multiply, in the time-dependent asymmetry, re-spectively, 1, cos⌬mt, and sin ⌬mt. We have found no sig-nificant effect due to the B*or the B0resonance for R1 and R3. On the other hand these effects are present in R2, but R2 is likely to be too small to be accurately measurable.

IV. CONCLUSIONS

In conclusion our analysis shows that the effect of includ-ing B resonance polar diagrams is significant for the B⫺

→␲⫺_␲⫺_␲⫹ _{and negligible for the other charged B decay}

mode. This result is of some help in explaining the recent results from the CLEO Collaboration, since we obtain

R⫽3.5⫾0.8, 共34兲

to be compared with the experimental result in Eq.共8兲. The␳ resonance alone would produce a result up to a factor of 2 higher. Therefore we conclude that the polar diagrams exam-ined in this paper are certainly relevant in the study of the charged B decay into three pions.

In the case of neutral B decays we have found that, as far as the branching ratios are concerned, the only decay mode where the contribution from the fake ␳’s 共production of a pion and the B* or the B0 resonance兲 may be significant is the neutral ␳0␲0 decay channel. As for the time-dependent asymmetry no significant effect is found. Therefore the B

→␲␲␲decay channel allows an unambiguous measurement

TABLE III. Effective branching ratios for the neutral B decay channels into ␳␲ (g⫽0.60, h⫽⫺0.70). Cuts as indicated in the text. Channels ␳ ␳⫹B* ␳⫹B*⫹B0 B ¯0_→_␳⫺_␲⫹ _0.50⫻10⫺5 _0.52⫻10⫺5 _0.49⫻10⫺5 B ¯0_→_␳⫹_␲⫺ 1.7⫻10⫺5 1.7⫻10⫺5 1.7⫻10⫺5 B ¯0_→_␳0_␲0 _0.10⫻10⫺5 _0.15⫻10⫺5 _0.12⫻10⫺5 B0→␳⫹␲⫺ 0.49⫻10⫺5 0.51⫻10⫺5 0.48⫻10⫺5 B0→␳⫺␲⫹ 1.7⫻10⫺5 1.7⫻10⫺5 1.7⫻10⫺5 B0→␳0␲0 0.11⫻10⫺5 0.17⫻10⫺5 0.15⫻10⫺5

FIG. 2. Sample Dalitz plot for the decay B⫺→␲⫹␲⫺␲⫺. In order to show the mass distribution of the B resonance diagrams, only their contribution is taken into account for this plot.

(7)

of␣, with two provisos:共1兲 only the neutral B decay modes are considered; 共2兲 the ␳0␲0 final state can be disregarded from the analysis.

ACKNOWLEDGMENTS

We thank J. Charles, Y. Gao, J. Libby, A. D. Polosa, and S. Stone for discussions.

关1兴 The BaBar Physics Book, SLAC-R-504 共1998兲, http://www.slac.stanford.edu/pubs/slacreports/slac-r-504.html. 关2兴 BELLE Technical Design Report, KEK-Report 95-1,

http://bsunsrv1.kek.jp/bdocs/tdr.html; F. Takasaki, 1999, hep-ex/9912004.

关3兴 Y. Azimov, Phys. Rev. D 42, 3705 共1990兲; B. Kayser, hep-ph/9709382; A. Dighe, I. Dunietz, and R. Fleischer, Phys. Lett. B 433, 147共1998兲.

关4兴 J. Charles, A. Le Yaouanc, L. Oliver, O. Pe`ne, and J. C. Raynal, Phys. Lett. B 425, 375共1998兲; 433, 441共E兲共1998兲; P. Colangelo, F. De Fazio, G. Nardulli, N. Paver, and Riazuddin, Phys. Rev. D 60, 033002共1999兲.

关5兴 M. Gronau and D. London, Phys. Rev. Lett. 65, 3381 共1990兲. 关6兴 H. J. Lipkin, Y. Nir, H. R. Quinn, and A. E. Snyder, Phys.

Rev. D 44, 1454共1991兲.

关7兴 A. E. Snyder and H. R. Quinn, Phys. Rev. D 48, 2139 共1993兲; H. R. Quinn and J. P. Silva, hep-ph/0001290.

关8兴 N. G. Deshpande, G. Eilam, Xiao-Gang He, and J. Trampetic, Phys. Rev. D 52, 5354共1995兲; B. Bajc, S. Fajfer, R. J. Oakes, T. N. Pham, and S. Prelovsek, Phys. Lett. B 447, 313共1999兲. 关9兴 J. Charles, Ph.D. thesis 共in French兲, http://qcd.th.u-psud.fr/ preprints_labo/1999.ps/99-31.ps.gz; S. Versille, Ph.D. thesis 共in French兲, http://www-lpnhep.in2p3.fr/babar/public/versille/ Thesis/

关10兴 Y. Gao and F. Wu¨rthwein, hep-ex/9904008; CLEO Collabora-tion, CLEO CONF 99-13.

关11兴 A. Deandrea, N. Di Bartolomeo, R. Gatto, and G. Nardulli, Phys. Lett. B 318, 549共1993兲.

关12兴 M. Bauer, B. Stech, and M. Wirbel, Z. Phys. C 34, 103 共1987兲. 关13兴 P. Colangelo, G. Nardulli, A. Deandrea, N. Di Bartolomeo, R.

Gatto, and F. Feruglio, Phys. Lett. B 339, 151共1994兲; P. Co-langelo, F. De Fazio, G. Nardulli, N. Di Bartolomeo, and R. Gatto, Phys. Rev. D 52, 6422共1995兲.

关14兴 A. J. Buras, in Probing the Standard Model of Particle

Inter-actions, edited by F. David and R. Gupta 共Elsevier Science,

Amsterdam, in press兲.

关15兴 N. Cabibbo, Phys. Rev. Lett. 10, 531 共1963兲; M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652共1973兲. 关16兴 L. Wolfenstein, Phys. Rev. Lett. 51, 1945 共1983兲.

关17兴 R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, and G. Nardulli, Phys. Rep. 281, 145共1997兲. 关18兴 P. Colangelo, F. De Fazio, and G. Nardulli, Phys. Lett. B 334,

175共1994兲.

关19兴 R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, and G. Nardulli, Phys. Lett. B 299, 139共1993兲. 关20兴 A. Deandrea, R. Gatto, G. Nardulli, and A. D. Polosa, J. High

Energy Phys. 021, 9902共1999兲.

关21兴 P. Colangelo, F. De Fazio, M. Ladisa, G. Nardulli, P. San-torelli, and A. Tricarico, Eur. Phys. J. C 8, 81 共1999兲; M. Ladisa, G. Nardulli, and P. Santorelli, Phys. Lett. B 455, 283 共1999兲.

关22兴 P. Ball, J.-M. Fre`re, and M. Tytgat, Phys. Lett. B 365, 367 共1996兲.

关23兴 A. D. Polosa, in Proceedings of 11th Rencontres de Blois: Frontiers of Matter, Chateau de Blois, France, hep-ph/9909371.

关24兴 Particle Data Group, C. Caso et al., Eur. Phys. J. C 3, 1 共1998兲, and 1999 partial update for edition 2000 共pdg.lbl.gov兲. 关25兴 See, for example, J. Libby, talk given at the workshop on