B and B
sdecays into three pseudoscalar mesons and the determination of the angle
␥
of the unitarity triangle
A. Deandrea,1R. Gatto,2M. Ladisa,1,3G. Nardulli,1,3and P. Santorelli4 1Theory Division, CERN, CH-1211 Gene`ve 23, Switzerland
2De´partement de Physique The´orique, Universite´ de Gene`ve, 24 quai E.-Ansermet, CH-1211 Gene`ve 4, Switzerland 3Dipartimento 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, Via Cintia, I-80126 Napoli, Italy
共Received 6 July 2000; published 30 October 2000兲
We reconsider two classical proposals for the determination of the angle␥ of the unitarity triangle: B⫾ →c0⫾→⫹⫺⫾and Bs→0KS→⫹⫺KS. We point out the relevance, in both cases, of nonresonant
amplitudes, where the ⫹⫺ pair is produced by weak decay of a B*(JP⫽1⫺) or B
0(JP⫽0⫹) off-shell meson. In particular, for the B decay channel, the inclusion of the B0pole completes some previous analyses and confirms their conclusions, provided a suitable cut in the Dalitz plot is performed; for the Bsdecay the
inclusion of the B*, B0amplitudes enhances the role of the tree diagrams as compared to penguin amplitudes, which makes the theoretical uncertainty related to the Bs→0KSdecay process less significant. While the first
method is affected by theoretical uncertainties, the second one is cleaner, but its usefulness will depend on the available number of events to perform the analysis.
PACS number共s兲: 11.30.Er, 12.15.Hh, 13.25.Hw
I. INTRODUCTION
In the next few years dedicated e⫹e⫺ machines at Cor-nell, SLAC, and KEK and hadronic machines such as the CERN Large Hadron Collider 共LHC兲 will explore in depth several aspects of C P violations in the realm of B physics. In particular the three angles ␣, , and ␥ of the unitarity tri-angle will be extensively studied not only to nail down the Cabibbo-Kobayashi-Maskawa 共CKM兲 matrix and its en-coded mechanism for C P violations, but also to examine the possibility of deviations from the pattern expected in the standard model. Some analyses, based on combined Collider Detector at Fermilab 共CDF兲 and ALEPH data 关1,2兴 on sin 2, sin 2⫽0.82⫺⫹0.39, as well as on CLEO results 关3兴 and other constraints on the unitarity triangle, have been al-ready used in关4兴 to get limits on the three angles␣, , and
␥. Although preliminary and based on a number of theoret-ical inputs, these results are worth quoting, as they represent theoretical and phenomenological expectations to be con-firmed or falsified by the experiments to come:1
⫽24.3 ° or 65.7 °, 共1兲
␥⫽55.5°⫺8.5°⫹6.0°, 共2兲
␣⫽180°⫺⫺␥. 共3兲
The first angle to be measured with a reasonable accuracy will be, by the study of the channel B→J/KS, which is free from the theoretical uncertainties related to the evalua-tion of hadronic matrix elements of the weak Hamiltonian. A
few strategies for the determination of ␣ have been also proposed, most notably those based on the study of the chan-nels B→ and B→关5,6兴. For this last channel a recent analysis 关7兴 has stressed the role of non-resonant diagrams where one pseudoscalar meson is emitted by the initial B meson with the production of a B* or a positive parity
B0 (JP⫽0⫹) virtual state followed by the weak decay of these states into a pair of light pseudoscalar mesons.
One of these diagrams 共the virtual B* graph兲 has been examined also by other authors in the context of the deter-mination of␥ 关8–10兴. It is useful to point out that␥ appears at present to be the most difficult parameter of the unitarity triangle. In recent years several methods have been proposed to measure this angle; some of them are theoretically clean, as they are based on the analysis of pure tree diagrams at quark level, such as b¯→u¯cs¯ and b¯→c¯us¯ transitions. One of the benchmark modes was proposed in关11兴 and employs the decays B⫹→D0K⫹, B⫹→D0K⫹, and B⫹→D⫾0K⫹, where
D⫾0 denotes C P eigenstates of the neutral D meson system with C P eigenvalues⫾1. The difference of the weak phases between the B⫹→D0K⫹and the B⫹→D0K⫹amplitudes is
2␥, which would allow to extract the angle ␥ by drawing two triangles with a common side: one of the triangles has sides equal to A(B⫹→D0K⫹), A(B⫹→D0K⫹), and
冑
2A(B⫹→D⫹0K⫹), respectively, and the other one has sides A(B⫺→D0K⫺)⫽e⫺2i␥A(B⫹→D0K⫹), A(B⫺→D0K⫺)⫽A(B⫹→D0K⫹), and
冑
2A(B⫺→D ⫹0K⫺). Even
though this method is theoretically clean, it is affected by several experimental difficulties 共for a discussion see 关12兴兲. One of these difficulties arises from the need to measure the neutral D meson decays into C P eigenstates, but also the other sides of the triangles present difficult experimental challenges. For example, if a hadronic decay 共e.g., D0 →K⫺⫹) were used to tag the D0 in the decay B⫹
1The fitted value of sin 2, which corresponds to the value 共1兲, is sin 2⫽0.750⫺0.064⫹0.058关4兴.
→D0K⫹, there would be significant interference effects with
the decay chain B⫹→D0K⫹→K⫺⫹K⫹ 共through the
dou-bly Cabibbo suppressed mode D0→K⫺⫹); if, on the other hand, the semileptonic channel D0→l⫹lXswere used to tag the D0, there would be contaminations from the background
B⫹→l⫹lXc.
The other benchmark modes for the determination of ␥ discussed in the recent review prepared for the Large Hadron Collider at CERN 关12兴 have also their own experimental difficulties; for these reasons we consider worthwhile to con-sider other channels, already discussed in the past and some-how now disfavored because of their more intricate theoret-ical status. We are aware of these theorettheoret-ical difficulties and it is the aim of the present paper to discuss them in some detail for two methods proposed for the determination of the angle␥. The first method is based on the idea to analyze the charged B C P-violating asymmetry, which arises from the interference between the resonant 共at the invariant mass
m
c0⫽3.417 GeV) and nonresonant 共the virtual B* graph兲
production of a pair of light pseudoscalar mesons in the de-cay B→3 light mesons. It is an aim of the present work to complete the analyses in关8–10兴 by considering the channel
B→3, including also the contribution of the virtual posi-tive parity B0 (JP⫽0⫹) state and the gluonic penguin opera-tors. We shall therefore analyze the robustness of the conclu-sions in关8,9兴 and 关10兴 once these additional contributions are considered.
The second analysis we consider here is the possible de-termination of ␥ by means of the Bs→0KS decay mode. Also this process has been considered in the past关13兴, but it is presently less emphasized because the tree level contribu-tion, that one hopes to estimate more reliably, is suppressed by the smallness of the Wilson coefficient a1. As we shall
notice below, the non-resonant tree contributions to this de-cay 共i.e., B* and B0) are proportional to the large Wilson coefficient a2 (a2⬇1); therefore we expect that their
inclu-sion can reduce the theoretical uncertainties arising from the penguin terms. This channel could be a second generation experiment provided a sufficient number of events can be collected, once xs, the mixing parameter for the Bs-B¯s sys-tem, and have been determined by other experiments.
II. B\c0 DECAYS
We consider in this section the decay mode
B⫺→⫹⫺⫺, 共4兲
as well the C P-conjugate mode B⫹→⫺⫹⫹, in the in-variant mass range m⫹⫺⯝m
c0⯝3.417 GeV. For this
de-cay mode we have both a resonant contribution coming from the decay B⫺→c0⫺→⫹⫺⫺ and several non reso-nant contributions. According to the analysis performed in
关8–10兴, this decay mode can be used to determine sin␥ by
looking for the charged B asymmetry arising from two am-plitudes: the resonant production viac0decay and nonreso-nant amplitudes. Among the nonresononreso-nant terms, we have
in-cluded the B* pole, which is the largest among the contributions considered in 关8兴.2 The authors in 关10兴 have considered other decay modes in the same kinematical re-gion, by analyzing the partial width asymmetry in B⫾ →MM¯⫾ decays ( M⫽⫹,K⫹,0,). Spotting the decay
mode B⫺→⫹⫺⫺, they estimate an asymmetry given approximately by 0.33 sin␥, which, however, seems to be sensitive to the choice of the parameters关10兴.
Our interest in this decay channel has been triggered by the study of a different invariant mass region 共i.e., m
⯝m) 关7兴, where also the contribution of the B0 pole (JP ⫽0⫹, with an estimated mass 5.697 GeV兲 was found to be
significant; therefore we include it in the present analysis, which represents an improvement in comparison to previous work. The second improvement we consider is the inclusion of the gluonic penguin operators. We refer to the paper 关7兴 for a full discussion of the formalism and we list here only the relevant contributions A
c0, AB*, and AB0 to the decay
amplitude: A c0⫽K
冉
1 t⫺m c0 2 ⫹im c0⌫c0 ⫹ 1 s⫺m c0 2 ⫹im c0⌫c0冊
, AB*⫽KB*冉
⌸˜ 共t,u兲 t⫺mB2*⫹imB*⌫B* ⫹ ⌸˜ 共s,u兲 s⫺mB2*⫹imB*⌫B*冊
, AB0⫽KB0共mB0 2 ⫺m 2兲冉
1 t⫺mB 0 2 ⫹im B0⌫B0 ⫹ 1 s⫺mB 0 2 ⫹im B0⌫B0冊
, 共5兲 where u⫽共p 1 ⫺⫹p 2 ⫺兲2, s⫽共p⫹⫹p 1 ⫺兲2, t⫽共p⫹⫹p 2 ⫺兲2, ⌸˜ 共x,y兲⫽m2⫺y 2⫹ x共MB2⫺m2⫺x兲 4mB2* . 共6兲In Eq.共5兲 the values of the constants are
K⫽1.52⫻10⫺8 GeV2, 共7兲
2Other less important terms discussed in 关8兴 include a long-distance type diagram, where an intermediate highly off-shell pion is exchanged among the incoming B meson and the outgoing pions, and a short-distance diagram, where the outgoing pions are pro-duced in a pointlike effective interaction by the weak decay of the B meson; we agree with the authors in关8兴 on the smallness of these neglected terms.
KB*⫽⫺4
冑
2gmB 2 A0 B*GF冑
2冋
VubVud*a2 ⫺VtbVtd*冉
a4⫺a6 m2 mq共mb⫹mq兲冊册
, 共8兲 KB 0⫽h冑
mB mB0共mB0 2 ⫺m B 2兲F 0 BGF冑
2冋
VubVud*a2 ⫺VtbVtd*冉
a4⫺a6 m2 mq共mb⫹mq兲冊册
. 共9兲The numerical value in Eq.共7兲 is derived in 关9兴, where the resonance amplitude is given by
R共s兲⫽␣1␣2
冑
⌫c0m c0 s⫺m c0 2 ⫹i⌫ c0mc0 . 共10兲Normalizing the decay rate of B⫹→c0⫹→⫹⫺⫹ by the total B decay rate, the product␣1␣2 in Eq.共10兲 is given
by the product of the corresponding branching ratios: 2␣12␣22⫽Br共B⫹→c0兲⫻Br共c0→⫹⫺兲. 共11兲 In 关9兴 the product of the branching ratios in Eq. 共11兲 is esti-mated to be about 5⫻10⫺7, which gives the numerical value in Eq.共7兲.
As to the numerical values of the constants appearing in
Eqs. 共8兲 and 共9兲, we use the same values adopted in 关7兴: g
⫽0.4, h⫽⫺0.54, mB*⫽mB⫽5.28 GeV, mB0⫽5.697 GeV,
⌫B0⫽0.36 GeV, ⌫B*⫽0.2 keV, mb⫽4.6 GeV, mq⬇mu
⬇md⯝6 MeV, A0
B*⫽0.16, F 0
B⫽⫺0.19. These numerical
estimates agree with results obtained by different methods: QCD sum rules 关14兴, potential models 关15兴, effective La-grangian 关16兴, Nambu–Jona-Lasinio– 共NJL兲-inspired
mod-els关17兴. Moreover, we use the following values of the
Wil-son coefficient: C1⫽⫺0.226, C2⫽1.1, C3⫽0.012, C4
⫽⫺0.029, C5⫽0.009, and C6⫽⫺0.033, with a2⫽C2
⫹C1/3, a1⫽C1⫹C2/3. The Wilson coefficients are obtained
in the ’t Hooft–Veltman 共HV兲 scheme 关18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19兴 we use the Wolfenstein parametrization 关20兴: Vub
⫽A3(⫺i), V
tb⫽1, Vud⫽1⫺2/2, Vtd⫽A3(1⫺
⫺i), Vcb⫽A2, Vcs⫽1⫺2/2, and Vts⫽⫺A2. We take
⫽0.22 and A⫽0.831; moreover, since is better known
than we take it at the value provided by the present analy-ses of the CKM matrix:⫽0.349 关4兴. It follows that will be given, in terms of ␥, by⫽/tan␥.
The asymmetry is given by
A⫽⌫共B⫹→⫺⫹⫹兲⫺⌫共B⫺→⫹⫺⫺兲
⌫共B⫹→⫺⫹⫹兲⫹⌫共B⫺→⫹⫺⫺兲. 共12兲
By introducing only thec0and B*contributions, we repro-duce, within the theoretical uncertainties, the results of关10兴.
However the introduction of the B0 pole contribution dra-matically reduces the asymmetry, because this contribution to the asymmetry is opposite to the B* term. We have ob-served that this cancellation arises from a change of sign around thec0resonance and therefore we change a little bit the procedure by defining a cut in the Dalitz plot. We inte-grate in the region defined by
m c0⫺2⌫c0⭐
冑
s⭐mc0⫹2⌫c0, m c0⭐冑
t, 共13兲 or m c0⫺2⌫c0⭐冑
t⭐mc0⫹2⌫c0, m c0⭐冑
s, 共14兲 where ⌫c0⫽14 MeV. It may be useful to observe that the
integration over the whole available space in the Mandelstam plane around the c0 resonance gives Br(B⫺→⫺⫺⫹)
⯝Br(B⫹→⫺⫹⫹)⫽5.27⫻10⫺7 and therefore the
cut-off procedure introduces a reduction of a factor 5 in the branching ratio.
For the asymmetry we obtain the result in Fig. 1. For ␥
⯝55 °, it can be approximated by Acut⫽0.48 sin␥. In order to assess the relevance of the B0 pole, we report in Table I
the contribution to the branching and to the asymmetry of the different contributions for a particular value of sin␥.
We observe that the inclusion of the next low-lying state
B0 does not alter significantly the conclusions obtained in FIG. 1. Asymmetry as a function␥ for B→c0.
TABLE I. Different contributions to the branching ratio and asymmetry in the decay channel B⫺→⫺⫺⫹. Both branching ratio and asymmetry are cut off according to the rules in Eqs. 共13兲,共14兲 and sin␥⫽0.82.
c0⫹B* c0⫹B*⫹B0 Br(B⫺→⫺⫺⫹)cut 1.18⫻10⫺7 1.06⫻10⫺7
Br(B⫹→⫺⫹⫹)cut 1.48⫻10⫺7 2.54⫻10⫺7
previous works, where basically only the B* nonresonant term was considered; however this conclusion can be ob-tained only if a convenient cut in the Dalitz plot is included. We also observe that the calculations performed in this sec-tion are not sensitive to the inclusion of the gluonic penguin contributions.
To get an estimate of the dependence of our result on the parameters, we considered the following intervals for the couplings g and h. For h⫽⫺0.54 and g⫽0.4⫾0.1 we obtain
共at sin␥⫽0.82) an asymmetry Acut⫽0.41⫺0.12⫹0.05; for g⫽0.4
and h⫽⫺0.54⫾0.16 we have an asymmetry Acut
⫽0.41⫺0.04⫹0.03. The corresponding variation on␥ is extremely large (30 ° to 150 °) because the asymmetry is rather flat in that region. We conclude that due to the theoretical uncer-tainties inherent to this method, the channelc0can hardly be useful for a precise determination of the angle ␥.
III. Bs\0KSDECAY
In the decay Bs→0KS the final state is a C P eigenstate; in this case one can measure either the time dependent asym-metry R1共t兲⫽ ⌫„Bs共t兲→0KS…⫺⌫„B¯s共t兲→0KS… ⌫„Bs共t兲→ 0K S…⫹⌫„B¯s共t兲→ 0K S… , 共15兲
or the time integrated (t⬎0) asymmetry:
R2⫽
冕
0 ⬁ dt关⌫„Bs共t兲→0KS…⫺⌫„B¯s共t兲→0KS…兴冕
0 ⬁ dt关⌫„Bs共t兲→0K S…⫹⌫„B¯s共t兲→0KS…兴 . 共16兲 Let us define xs⫽ ⌬ms ⌫ , 共17兲where⌬ms is the mass difference between the mass eigen-states and⌫⬇⌫(Bs)⬇⌫(B¯s) and
A⫽A共Bs→0KS兲, A¯⫽A共B¯s→0KS兲, 共18兲
A⫽兩AT兩ei(T⫹␥)⫹兩A
P兩e i(P⫺), 共19兲 A ¯⫽兩A T兩e i(T⫺␥)⫹兩A P兩e i(P⫹). 共20兲
Here T andP are strong phases of the tree and penguin amplitudes,兩AT兩 and 兩AP兩 their absolute values and and␥ the weak phases of the Vtd* and Vub* CKM matrix elements.
The mixing between Bs and B¯s, parametrized by the xs pa-rameter in Eq.共17兲 introduces no weak phase.
Both the0diagram共Fig. 2兲 and the B*, B0non-resonant
diagrams, with a cut in the ⫹, ⫺ pair at m⫽m
⫾2⌫ 共Fig. 3兲 contribute to AT and AP, that are therefore given as follows:
AT⫽兩AT兩ei(T⫹␥)⫽A T⫹A B* T ⫹AB0 T , 共21兲
AP⫽兩AP兩ei(P⫺)⫽A P⫹A B* P ⫹AB0 P . 共22兲
The amplitudes are computed in the factorization approxima-tion from the weak nonleptonic Hamiltonian as given by
关18兴; our approach is similar to the one employed in Ref. 关7兴
where a full description of the method is given. We get (Q
⫽T,P) AQ⫽KQ t⫺t
⬘
u⫺m2⫹im⌫, AB * Q ⫽K B* Q t⫺t⬘
u⫺mB2*⫹imB*⌫B* , AB 0 Q⫽⫺K B0 Q mB0 2 ⫺m Bs 2 u⫺mB0 2 ⫹im B0⌫B0 , 共23兲 where u⫽共p⫺⫹p⫹兲2, t⫽共pK⫹p⫺兲2, t⬘
⫽共pK⫹p⫹兲2. 共24兲In Eq.共23兲 the values of the constants are
FIG. 2. The Feynman diagram for the Bs→KS⫺⫹decay.
The circle and the box represent, respectively, the strong and the weak interaction vertex.
FIG. 3. The B*, B0 Feynman diagrams for the Bs→KS⫺⫹
decay. The circle and the box represent, respectively, the strong and the weak interaction vertex.
KT⫽ GF 2
冑
2Vud*Vuba1gfF1 BsK , 共25兲 KP⫽ GF 2冑
2Vtd *Vtba4gfF1 BsK , 共26兲 KB * T ⫽4A 0 B*GF冑
2Vud*Vuba2g f fK mB smB*, 共27兲 KBP*⫽⫺4A0B*GF冑
2Vtd *Vtb冉
a4⫺a6 m2 mq共mb⫹mq兲冊
g ⫻ff K mBsmB*, 共28兲 KB 0 T ⫽F˜ 0 B0mB0 2 ⫺m 2 mB0 GF冑
2Vud*Vuba2冑
mB0mBsh f fK , 共29兲 KB 0 P ⫽⫺F˜ 0 B0mB0 2 ⫺m 2 mB0 GF冑
2Vtd*Vtb冉
a4⫺a6 m2 mq共mb⫹mq兲冊
⫻冑
mB0mBs h f fK , 共30兲where g⫽5.8, f⫽0.15 GeV2 关21兴, m⫽770 MeV, ⌫
⫽150 MeV, f⫽130 MeV, fK⫽161 MeV, F˜0
B0
⫽⫺0.19, F1BsK⫽⫺0.19, and m
Bs⫽5.37 GeV 关7兴. From these
equa-tions the parameters appearing in Eqs.共21兲, 共22兲 can be ob-tained. The time integrated asymmetry is
A⫽xs关sin 2␥⫺␣1sin 2⫺2␣2sin共⫺␥兲兴⫺2␣3sin共␥⫹兲
共1⫹xs 2兲关1⫹␣ 1⫹2␣2cos共⫹␥兲兴 . 共31兲 Numerically we obtain ␣1⫽
冕
d⍀兩AP兩2冕
d⍀兩AT兩2 ⫽0.06, ␣2⫽冕
d⍀ cos共T⫺P兲兩APAT兩冕
d⍀兩AT兩2 ⫽⫺0.09, ␣3⫽冕
d⍀ sin共T⫺P兲兩APAT兩冕
d⍀兩AT兩2 ⫽0.015. 共32兲In these equations integrations are performed in a band around the mass: m⫾200 MeV.
For illustrative purposes we consider the value xs⫽23,
⫽65.7 °, and␥⫽55.5 °, corresponding to the central
val-ues in关4兴; one obtains an asymmetry of 3.5%.3
It can be observed that the channel Bs→0KS has been discussed elsewhere in the literature关22兴, but somehow dis-carded for two reasons. First the asymmetry contains a factor
xs/(1⫹xs
2
) which, in view of the large mixing between Bs and B¯s, is rather small. Second, as it is clear from Eq.共30兲, the ratio of the penguin to the tree amplitudes can be large, if one includes only the 0-resonant diagrams;4 indeed the0 contribution is proportional to the Wilson coefficient a1 which is small. As to the first point a small asymmetry can still be useful for determining␥provided a sufficient number of events is available共see below兲; as to the second point the inclusion of the nonresonant contribution B*, B0is of some
help in this context, as the tree contribution is proportional to the Wilson coefficient a2⯝1.0 for these diagrams.
A reliable estimate of the branching ratio is difficult 共be-cause of the uncertainty on the a1parameter兲. The effect on
the asymmetry is to reduce the influence of the penguin op-erator in the final result as can be deduced from Eq.共32兲. In order to assess the validity of the method for the determina-tion of the asymmetry, we varied the penguin contribudetermina-tion by varying the ␣i parameters of Eq.共32兲 by 50%.5Our results for the asymmetry vary by 10% 共assuming ␥⫽55.5 °) and the value of␥ that one can deduce is 55.5⫺5⫹3degrees due to this uncertainty.
In Fig. 4 we report the asymmetry as a function of the angle ␥ 共for xs⫽23 and two values of).
Let us conclude this analysis with a discussion on the reliability of the Bs decay mode for the determination of␥. An estimate of the sensitivity of the method can be obtained
3For the solution⫽24.3 ° and the same values of ␥ and x
sone
gets for the asymmetry again 3.5% as the coefficients␣1, ␣2, ␣3 are small and the asymmetry can roughly be approximated by sin 2␥/xs.
4Without the B* and B
0 contribution the parameters of Eq.共32兲 would be larger:␣1⫽0.26, ␣2⫽⫺0.27, and␣3⫽⫺0.45
5The reason could be a violation of factorization or a variation in the parameters used to estimate the penguin contribution.
by comparing it, as an example, to Bs→J/⌿KS. The branching ratio for Bs→J/⌿KS is expected to be 2.0
⫻10⫺5 关12兴, while the branching B
s→0KS is roughly one order of magnitude smaller.6 The event yield for the Bs →J/⌿KSchannel is estimated to be 4100 event per year by a selection method developed by the CMS Collaboration at the
LHC共with a pT cut⬎1.5 GeV/c on the pions from the KS
decays to suppress the combinatoric background兲. Assuming a similar selection method for Bs→0KS, one could obtain
⯝410 events per year and ⯝2⫻103in 5 years, which would
produce an uncertainty of⫾17 ° on␥ 共assuming xs⫽23 and
␥⯝55 °) to be compared to the estimated error of ⫾9 °
within 3 years at LHC for Bs→J/⌿KS. Therefore even if the mode Bs→0K
S is less competitive than the Bs →J/⌿KSone, it is not dramatically so if the branching ratio is not too small, and could be considered as a complementary analysis for the determination of ␥. The final assessment of the feasibility will be clear as soon as an experimental deter-mination of the branching ratio for Bs→0KS is available.
IV. CONCLUSIONS
In this paper we have reviewed two classical methods proposed in the past few years for the determination of the angle ␥: B⫾→c0⫾→⫹⫺⫾ and Bs→0KS →⫹⫺K
S. For the first decay channel we have included, besides the B* nonresonant diagram, the B0 (JP⫽0⫹)
off-shell meson contribution. This calculation completes previ-ous analyses and confirms their results, provided a suitable cut in the Dalitz plot is performed; however it appears that this method is subject to a large uncertainty on the determi-nation of ␥ coming from the allowed variation in the theo-retical parameters because the asymmetry is rather flat in the region of interest. For the second channel we have pointed out the relevance of the two nonresonant amplitudes, i.e., the mechanism where the ⫹⫺ pair is produced by weak de-cay of a B*(JP⫽1⫺) or B0 (JP⫽0⫹) off-shell meson. The
inclusion of these terms enhances the role of the tree dia-grams as compared to penguin amplitudes, which makes the theoretical uncertainty related to the Bs→0KS decay pro-cess less significant. This method can be considered for a complementary analysis for the determination of␥, provided a sufficient number of events can be gathered.
关1兴 CDF Collaboration, W. Trischuk, FERMILAB-CONF-00-044-E, CDF-PUB-BOTTOM-PUBLIC-5227; CDF Collabora-tion, T. Affolder et al., Phys. Rev. D 61, 072005共2000兲. 关2兴 R. Forty, in Proceedings of the 3rd International Conference
on B Physics and C P Violation共BCONF99兲, Taipei, Taiwan, 1999.
关3兴 Y. Gao and F. Wu¨rthwein, hep-ex/9904008; CLEO Collabora-tion, CLEO CONF 99-13.
关4兴 F. Caravaglios, F. Parodi, P. Roudeau, and A. Stocchi, hep-ph/0002171.
关5兴 M. Gronau and D. London, Phys. Rev. Lett. 65, 3381 共1990兲. 关6兴 A. E. Snyder and H. R. Quinn, Phys. Rev. D 48, 2139 共1993兲;
H. R. Quinn and J. P. Silva, ibid. 62, 054002共2000兲. 关7兴 A. Deandrea, R. Gatto, M. Ladisa, G. Nardulli, and P.
San-torelli, Phys. Rev. D 62, 036001共2000兲.
关8兴 N. G. Deshpande, G. Eilam, Xiao-Gang He, and J. Trampetic, Phys. Rev. D 52, 5354共1995兲.
关9兴 G. Eilam, M. Gronau, and R. R. Mendel, Phys. Rev. Lett. 74, 4984共1995兲.
关10兴 B. Bajc, S. Fajfer, R. J. Oakes, T. N. Pham, and S. Prelovsek,
Phys. Lett. B 447, 313共1999兲; S. Fajfer, R. J. Oakes, and T. N. Pham, Phys. Rev. D 60, 054029共1999兲.
关11兴 M. Gronau and D. Wyler, Phys. Lett. B 265, 172 共1991兲. 关12兴 P. Ball et al. 共convenors兲, ‘‘B decays at the LHC,’’
CERN-TH/2000-101.
关13兴 The BaBar Physics Book, Chap. 7, SLAC-R-504 共1998兲. 关14兴 P. Colangelo et al., Phys. Lett. B 339, 151 共1994兲; V. M.
Be-lyaev, V. M. Braun, A. Khodjamirian, and R. Ruckl, Phys. Rev. D 51, 6177 共1995兲; P. Colangelo, F. De Fazio, N. Di Bartolomeo, R. Gatto, and G. Nardulli, ibid. 52, 6422共1995兲; P. Ball and V. M. Braun, ibid. 58, 094016共1998兲.
关15兴 M. Wirbel, B. Stech, and M. Bauer, Z. Phys. C 29, 637 共1985兲; N. Isgur, D. Scora, B. Grinstein, and M. B. Wise, Phys. Rev. D 39, 799 共1989兲; P. Colangelo, F. De Fazio, and G. Nardulli, Phys. Lett. B 334, 175共1994兲; P. Colangelo et al., 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16兴 R. Casalbuoni et al., Phys. Lett. B 299, 139 共1993兲; Phys. Rep. 281, 145共1997兲.
关17兴 A. Deandrea, R. Gatto, G. Nardulli, and A. D. Polosa, J. High Energy Phys. 02, 021共1999兲.
6The precise value critically depends on the parameter a
1which is the result of the partial cancellation of the Wilson coefficient c1and
c2and on the validity of the factorization approximation. In关23兴 an estimate of (1⫾0.5)⫻10⫺6is given; with the values adopted in the present paper we get 2⫻10⫺7 because a much smaller value of a2 is used. Note however that the asymmetry is largely independent of the precise values of the parameters used to obtain the branching ratio.
FIG. 4. The relevant asymmetry in the decay channel Bs →KS⫹⫺ as a function of ␥. The solid line corresponds to 
关18兴 A. J. Buras, in Probing the Standard Model of Particle
Inter-actions, edited by F. David and R. Gupta 共Elsevier Science,
Amsterdam, 1998兲.
关19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20兴 L. Wolfenstein, Phys. Rev. Lett. 51, 1945 共1983兲.
关21兴 P. Ball, J.-M. Fre´re, and M. Tytgat, Phys. Lett. B 365, 367 共1996兲.
关22兴 R. Fleischer, Int. J. Mod. Phys. A 12, 2459 共1997兲.
关23兴 A. Deandrea, N. Di Bartolomeo, R. Gatto, and G. Nardulli, Phys. Lett. B 318, 549共1993兲.
