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Are there significant scalar resonances in B--> D(*)(K-K0)-K-0 decays?

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Are there significant scalar resonances in B

À

\D

*

…0

K

À

K

0

decays?

Ron-Chou Hsieh and Chuan-Hung Chen

Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan

共Received 13 December 2003; published 26 March 2004兲

We study the indication of different large branching ratios between B→D(*)0KK0 and B¯

d →D(*)⫹KK0 observed by Belle. Interestingly, the same situation is not found in the decays B

→D(*) KK*0

. If no intermediate resonances exists in the decays B→D(*)0

KK0, a puzzle will arise. We find that the color-suppressed processes B→D(*)0

a0⫺(1450) with a0⫺(1450)→KK 0

could be one of the candidates to enhance the BRs of B→D(*)0KK0. Our conjecture can be examined by the Dalitz plot

technique and the analysis of the angular dependence of the KK0state at B factories.

DOI: 10.1103/PhysRevD.69.057504 PACS number共s兲: 13.25.Hw, 14.40.Cs

Since CP violation was discovered in the K-meson system in 1964 关1兴, the same thing has also been realized by Belle 关2兴 and Babar 关3兴 with a high accuracy in the B system. Although the new observation does not improve our recog-nition of CP violation, it stimulates the development. In ad-dition to the origin of CP violation, B factories also provide the chance to understand or search uncertain states, such as the scalar bosons f0(400⫺1200), f0(980), a0(980) and glueball, etc. Unlike pseudoscalar mesons, the scalar bosons with wide widths are difficult to measure directly via two-body B meson decays. Inevitably, the study of three-two-body decays will become important for extracting the quasi-two-body resonant states from the Dalitz plot and the invariant mass distribution. By the analyses of the Dalitz plot tech-nique, we can recognize whether or not there exist unusual structures in phase space.

Recently, Belle has observed the BRs of the decays B

→D(*)KK*0 to be BR(B→D(*)0KK*0)⫽7.5⫾1.3 ⫾1.1 (15.3⫾3.1⫾2.)⫻10⫺4 and BR(B¯0→D(*)⫹KK*0) ⫽8.8⫾1.1⫾1.5 (12.9⫾2.2⫾2.5)⫻10⫺4, and the decays B →D(*)KK0 to be BR(B¯→D(*)0KK0)⫽5.5⫾1.4 ⫾0.8 (5.2⫾2.7⫾1.2)⫻10⫺4 and BR(B¯0→D(*)⫹KK0) ⫽1.6⫾0.8⫾0.3 (2.0⫾1.5⫾0.4)⫻10⫺4 关4兴. According to

Belle’s analyses, we know that the KK*0 system has the state JP⫽1⫹; also, it is pointed out that the B →D(*)KK*0 decays in the low KK*0 invariant mass re-gion can be fitted well by quasi-two-body decays B

→D(*)a 1

(1260) with a

1

(1260)→KK*0. In the KK0 system of B→D0KK0decays, by adopting the fitting dis-tribution dN/d cosKK⬀关RLcos2␪KK⫹(1⫺RL)sin2␪KK兴 Belle observed the value RL⫽0.97⫾0.08, i.e. the KK0 state pre-fers being JP⫽1⫺. Interestingly, by looking at the data shown above, we find that the BRs of charged B decaying to

KK0 final state are much larger than those of neutral B decays. In terms of central values, the ratios, defined by

R(*)⫽BR(B→D(*)KK0)/BR(B¯0→D(*)⫹KK0), could be estimated roughly to be 3.5 共2.6兲; however, there is no such kind of implication on the KK*0 final state. There-fore, if the data display the correct behavior, something must have happened in B→D(*)KK0 decays. Before discussing the origin of the differences, we need to understand the mechanism to produce KK*0 and KK0 systems in charged and neutral B decays.

Since B→D(*)KK(*)0 decays are dictated by the b

→cu¯d transition, we describe the effective Hamiltonian as Heff⫽GF

2Vu关C1共␮兲O1⫹C2共兲O2兴 共1兲 with O1⫽d¯u¯cb and O2

(q)⫽d¯

u¯cb, where q¯q

⫽q¯␣␥␮(1⫺␥5)q␤, ␣(␤) are the color indices, Vu

⫽Vud*Vcbare the products of the CKM matrix elements, and

C1,2(␮) are the Wilson coefficients共WCs兲 关5兴. As usual, the effective WCs of a2⫽C1⫹C2/Nc and a1⫽C2⫹C1/Nc, with Nc⫽3 being the color number, are more useful. Accord-ing to the interactions of Eq. 共1兲, we classify the topologies for B→D(*)KK(*)0 decays to be color-allowed共CA兲 and color-suppressed 共CS兲 processes, illustrated by Fig. 1, and also use the decay amplitudes AQ(KK(*)0) to describe their decays, where Q⫽C(N) denote the charged 共neutral兲 B decays. From the figure we see that both CA and CS topolo-gies contribute to charged B decays but neutral B decays are only governed by CA. We note that because we do not con-sider the problem of direct CP asymmetry, the annihilation effects, which can contribute to B¯d→D(*)⫺KK(*)0decays and are usually smaller than the CA contributions, are ne-glected. Although Fig. 1 corresponds to the picture of three-body decay, after removing the ss¯ pair, the figures are related to quasi-two body decays if du¯ could form a possible bound state.

Now, we have to examine that if there exists only a JP ⫽1⫺ state in the KK0 system, the differences between

FIG. 1. Topologies for B→D(*)KK(*)0 decays. 共a兲

Color-allowed with q⫽u and d, and 共b兲 color suppressed. PHYSICAL REVIEW D 69, 057504 共2004兲

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BR(B→D(*)0KK0) and BR(B¯

d→D(*)⫹KK0) are in-significant. To be more understanding of the problem, it is useful to start from the discussion on B→D(*)KK0 pro-cesses. As mentioned before, charged B decays are governed by CA and CS while neutral B decays are only from CA; therefore, the amplitudes for Band B¯0 decays can be writ-ten as AC(KK0)⬀a1M

a

C(KK0)⫹a2M

b

C(KK0) and

AN(KK0)⬀a1MNa(KK0), whereMa(b)Q express the had-ronic transition matrix elements for CA共CS兲. For simplicity, we only consider factorizable parts. As a conjecture, if there exist significant differences in BRs of Band B¯0decays, the source should be from a2Mb

C

(KK0). In the following, we use two examples to show that it is impossible to only in-crease the BRs of B→D(*)KK0 without enhancing the BRs of B→D(*)KK*0. Firstly, if the decays B

→D(*)KK0 are only governed by quasi-two-body decays, say B→D(*)

X with ␳X being arbitrary vector meson, the decay amplitudes can be described by AC(KK0) ⬃a1fXFB→D (*) (0)⫹a2fD(*)FB→X(0) and AN(KK0) ⬃a1fXFB→D (*) (0), where fD(*)and f

X are the decay

con-stants of D(*) and

Xmesons, respectively, and FB→D (*)(

X)

denote the B→D(*)(

X) form factors. However, it is known that兩a2/a1兩⬃0.26 and f

XF

B→D(*)

(0)⬃ fD(*)FB→X(0)关6兴.

If the resonance is the JP⫽1⫺ state, the differences in BR between charged and neutral B decays will be slight. The same argument can be applied to B→D(*)KK*0 decays for the JP⫽1⫹ resonant state. Hence, the consequences are consistent with the observations of B→D(*)KK*0 decays. Secondly, we consider nonresonant mechanism on B

→D(*)KK0. Since the situation corresponds to a three-body phase space, we use the KK0 invariant mass, expressed by ␻, as the variable to describe the behavior of the decay. Hence, the decay amplitudes could be written as AC(KK0)⬃a 1F0→KK(␻2)FB→D (*) (␻2) ⫹a2fD(*)FB→KK(␻2) and AN(KK0) ⬃a1F0→KK(␻2)FB→D (*)

(␻2), and the F0→KK(FB→KK) are the associated form factors. It is known that at an asymptotic region, in terms of perturbative QCD 共PQCD兲 the behavior of F0→KK has the power law 1/␻2(ln␻2/⌳QCD2 )⫺1关7兴. If we take the behavior of FB→D(*) to be 1/(1⫺␻2/ MX

2

)n, MX is the pole mass and n⫽1 or 2, we clearly see that the decay amplitudes are suppressed at large ␻. Furthermore, accord-ing to the analysis of Ref.关8兴, the dominant region actually is around ␻2⬃⌳¯ mB. Therefore, if a2FB→KK(⌳¯ mB) is the source of the differences in BR between charged and neutral

B decays, the similar effects a2FB→KK*(⌳¯ mB) will also af-fect the decays B→D(*)KK*0. Hence, unless

FB→KK*(⌳¯ mB) is much less than FB→KK(⌳¯ mB), the CS effects should have the similar contributions to B →D(*)KK*0. It is known that with the concept of two-meson wave functions关9兴, the calculations of FB→KK(⌳¯ mB) and FB→KK*(⌳¯ mB) can be associated with the wave func-tions of KK and KK* systems. At the ␻2⬃⌳¯ mB region, two-meson wave functions could be described by two

indi-vidual meson wave functions 关10兴. Refer to the derived K and K* wave functions 关11兴; the value of

FB→KK(⌳¯ mB)/FB→KK*(⌳¯ mB) should be close to

FB→K(0)/FB→K*(0)⬃0.8. As a result, it is hard to imagine that the CS effects on pure three-body decays can make large differences in B→D(*)KK0 processes but not in B

→D(*)KK*0 processes.

Inspired by Belle’s observation, we think the significant differences in the BRs of B→D(*)KK0 should not come from KK0with the JP⫽1⫺state but with the JP⫽0⫹state. This could be easily understood as follows: as discussed be-fore, only CS effects, a2Mb

C(KK0), could make a discrep-ancy in BRs of charged and neutral B decays. Therefore, the mechanism which dominantly contributes to CS topology will be our candidate. If KK0 has the JP⫽0⫹state, due to

F0→KK

KK,0兩u¯

d兩0

, in terms of the equation of

mo-tion, we get that F0→KK is proportional to (md⫺mu). That is, the contribution of the scalar state to CA topology is neg-ligible. On the contrary, there is no such suppression to the CS topology. Now, the problem becomes how to produce

KK0to be a 0⫹state. By searching the Particle Data Group 关12兴, we find that the preference could be the scalar bosons

a0(980) and a0(1450) because both are isovector states and have sizable decay rates for a0→KK. Hence, to realize our thought, we propose that the decays B→D(*)0a

0

(980,1450) with a

0

(980,1450)→KK0 can sat-isfy the required criterion to enhance the BRs of B

→D(*)0KK0.

Since the quark contents of scalar mesons below or near 1 GeV are still obscure in the literature关13兴, for avoiding the difficulty in estimation we only make the explicit calcula-tions on a0(1450) which have a definite composed structure of qq¯ . If a0(980) consists of the qq¯ state, the same estima-tion could also be applied关14兴. In theory, it is known that the serious problem on two-body nonleptonic B decays comes from the calculations of hadronic matrix elements. Since the involving processes are governed by CS topologies, like the well known decays B→J/⌿K(*) and B¯

d→D(*)0␲0, in which nonfactorizable effects play an important role, we adopt the PQCD approach which is described by the convo-lution of a hard amplitude and wave functions关15兴, can deal with the factorizable and nonfactorizable contributions, and can avoid the suffering end-point singularities self-consistently关16兴.

In the calculations of hadronic effects, the problem is how to determine the wave functions of D(*) and a

0 mesons. For

D(*) meson wave functions, we could model them to fit with the measured BRs of B¯d→D(*)0␲0 decays 关17兴, in which the B meson wave function is chosen to coincide with the observed BRs of B→␲␲ decays while ␲ meson ones are adopted from the derivation of QCD sum rules 关11兴. As to a a0 scalar meson, the spin structures of a0 are required to satisfy

0兩u¯␥␮d兩a0, p3

⫽关(md⫺mu)/ma0兴 f˜p3␮ and

0兩q¯q兩a0

⫽ma

0f˜, where ma0 and f˜ are the mass and decay

constant of a0. In order to satisfy these local current matrix elements, the distribution amplitude for a0 is adopted as

BRIEF REPORTS PHYSICAL REVIEW D 69, 057504 共2004兲

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0 1 dxeix P•z

0兩q¯共0兲 jq共z兲l兩a0

⫽ 1

2Nc关p” 兴l ja0共x兲⫹ma0关1兴l ja0 p 共x兲. 共2兲

By the charge parity invariance and neglecting the effects of the light current quark mass mu(d), we obtain ⌽a0(x)

⬇⫺⌽a0(1⫺x) and ⌽a0 p (x)⫽⌽a 0 p (1⫺x) 关18兴, and their normalizations are 兰01dxa0(x)⫽0 and 兰0

1

dxa

0 p

(x) ⫽ f˜/2

2Nc. Although the vector D*0meson carries the spin degrees of freedom, in the B→D*0a

0

decay only

longitu-dinal polarization is involved. The results should be similar to the D0a0⫺ case. Therefore, we only present the represen-tative formulas for the B→D0a0⫺decay. Hence, the decay amplitude for B→D0a0⫺ is read as

A⫽Vu关 fDFD0a 0 e ⫹MD0a 0 e

whereFe (Me) is the factorized 共non-factorized兲 emission hard amplitudes. According to Eq.共2兲, the typical hard func-tions are expressed as

FD0a 0 e ⫽␩

0 1 dx1dx3

0 ⬁ b1db1b3db3⌽B共x1,b1兲 ⫻兵关共␨1⫹␨2x3兲⌽a0共x3兲⫹ra共1⫺2x3兲⌽a0 p 共x3兲兴 ⫻Ee共te 1兲⫹r a关2⌽a0 p 共x3兲⫺r aa0共x3兲兴Ee共te 2, 共3兲 MD0a 0 e ⫽2␩

0 1 d关x兴

0 ⬁ b1db1b3db3⌽B共x1,b1兲⌽D共x2兲 ⫻„兵⫺关共1⫺ra 2 兲x2⫹共1⫺rD 2 兲x3兴⌽a0共x3兲 ⫹rax3⌽a0 p 共x3兲其Ed 1 共td 1兲⫹ 兵关共␨1⫺ra 2兲共1⫺x2兲 ⫹ra 2 x3⫺rD2x2兴⌽a0共x3兲⫺rax3⌽a0 p 共x3兲E d 2共t d 2兲…, 共4兲 with ␩⫽8␲CFMB 2, r a(D)⫽ma0(D)/ MB, ␨1⫽1⫺ra 2⫺r D 2 , ␨2⫽␨1⫺rD 2 ,Eei(tei)⫽␣s(te i ) a2(te i ) SuB⫹a0(te i ) he(兵x其,兵b其), andEdi(tdi)⫽␣s(td i )关C2(tdi)/Nc兴 Su(td i )B⫹D⫹a0hd(兵x其,兵b其).

te,d1,2, Su and he,d denote the hard scales of B decays, Suda-kov factors, and hard functions which are arisen from the propagators of gluon and internal valence quark, respec-tively. Their explicit expressions can be found in Ref. 关19兴.

For numerical estimations, the B, D(*)0 and a

0 meson wave functions are simply chosen as

B共x,b兲⫽NBx2共1⫺x兲2exp

⫺ 1 2

xmBB

2 ⫺␻B 2 b2 2

, ⌽D(*)共x兲⫽ 3

2NcfD(*)x共1⫺x兲关1⫹0.7共1⫺2x兲兴,a0共x兲⫽ 2

2Nc 关6x共1⫺x兲C13/2 共1⫺2x兲兴,a0 p 共x兲⫽ 2

2Nc 关3共1⫺2x兲2兴, 共5兲

where NB is determined by the normalization of 兰01

dxB(x,0)⫽ fB/(2

2Nc) and C1 3/2

( y ) is the Gegenbauer polynomial. The values of theoretical inputs are set as ␻B

⫽0.4, fB⫽0.19, fD(*)⫽0.20(0.22), f˜⫽0.20, mB⫽5.28, and

mD(*)⫽1.87(2.01) GeV. By the taken values and using Eq.

共3兲 with excluding WC of a2, we immediately obtain the B

→a0(1450) form factor at large recoil to be 0.44. The result is quite close to the value 0.46 which is estimated by the light-cone sum rules关20兴. Hence, the magnitudes of the con-sidered hard functions are given in Table I. Consequently, the BRs of B→D(*)0a(1450) are obtained to be 8.21(9.34) ⫻10⫺4. Since the predictions of PQCD on the BRs of B¯ →D00 关17兴 and B→J/⌿K(*) and the helicity amplitudes of B→J/⌿K* 关21兴 are consistent with the current

experimental data, with the same approach, our results should be reliable. Furthermore, the BR products of Br„B→D(*)0a 0 ⫺(1450)…⫻Br„a0(1450)→KK0 ⬇ 1.81(2.05) Br„B→ D(*)0a 0 ⫺(1450)… ⫻ Br„ a0(1450) →KK0…⬇1.81(2.05)⫻10⫺4 with Br„a0(1450)→KK0 ⬃0.22 关12兴. Clearly, the contributions of quasi-two-body de-cays to B→D(*)0KK0 modes are close to the pure three-body decays B¯→D(*)⫹KK0 关22兴.

In summary, we have investigated that when a proper sca-lar meson is considered, the BRs of B→D(*)0KK0 will deviate from those of B¯d→D(*)⫹KK0. Although we only concentrate on a0(1450), the same discussion is also appli-cable to a0(980). Since our purpose is just to display the importance of a scalar boson on the decays B→D(*)0KK0, we do not consider the theoretical uncertain-ties at this stage. Also, we neglect discussing the interfering effects of resonance and non-resonance. It is worthwhile to mention that by using the powerful Dalitz plot technique, many scalar mesons in charm decays have been observed by the experiments at CLEO关23兴, E791 关24兴, FOCUS 关25兴, and Babar 关26兴. As expected, the significant evidence of scalar productions should also be observed at B factories. In addi-tion, since the scalar meson does not carry spin degrees of freedom, there is no specific direction for KK0 production so that we should see a different angular distribution of

KK0, such as the coefficient associated with the term sin2␪KK in the fitting distribution mentioned early will be TABLE I. Hard functions 共in units of 10⫺2) for B→D(*)0 a0⫺(1450) decays with ␻B⫽0.4, fB⫽0.19, fD(*) ⫽0.2(0.22), and f˜⫽0.20 GeV. fDFD0a 0 e MD0a 0 e fD*FD*0a 0 e MD*0a 0 e ⫺1.42 ⫺2.22⫹i0.97 ⫺1.56 ⫺2.39⫹i1.06

BRIEF REPORTS PHYSICAL REVIEW D 69, 057504 共2004兲

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enhanced. Hence, with more data accumulated, our conjec-ture can be examined by the Dalitz plot and the analysis of angular dependence on KK0 state.

The author would like to thank C.Q. Geng, H.N. Li, and

H.Y. Cheng for their useful discussions. This work was sup-ported in part by the National Science Council of the Repub-lic of China under Grant No. NSC-92-2112-M-006-026 and by the National Center for Theoretical Sciences of R.O.C..

关1兴 J.H. Christenson, J.W. Cronin, V.L. Fitch, and R. Turly, Phys. Rev. Lett. 13, 138共1964兲.

关2兴 Belle Collaboration, A. Abashian et al., Phys. Rev. Lett. 86, 2509共2001兲.

关3兴 Babar Collaboration, B. Aubert et al., Phys. Rev. Lett. 86, 2515共2001兲.

关4兴 Belle Collaboration, A. Drutskoy et al., Phys. Lett. B 542, 171 共2002兲.

关5兴 G. Buchalla, A.J. Buras, and M.E. Lautenbacher, Rev. Mod. Phys. 68, 1230共1996兲.

关6兴 H.Y. Cheng and K.C. Yang, Phys. Rev. D 59, 092004 共1999兲. 关7兴 S.J. Brodsky and G.R. Farrar, Phys. Rev. D 11, 1309 共1975兲. 关8兴 C.H. Chen and H.N. Li, Phys. Lett. B 561, 258 共2003兲. 关9兴 D. Muller et al., Fortschr. Phys. 42, 101 共1994兲; M. Diehl, T.

Gousset, B. Pire, and O. Teryaev, Phys. Rev. Lett. 81, 1782 共1998兲; M.V. Polyakov, Nucl. Phys. B555, 231 共1999兲. 关10兴 M. Diehl, Th. Feldmann, P. Kroll, and C. Vogt, Phys. Rev. D

61, 074029共2000兲; M. Diehl, T. Gousset, and B. Pire, ibid. 62,

073014共2000兲.

关11兴 P. Ball et al., Nucl. Phys. B529, 323 共1998兲; P. Ball, J. High Energy Phys. 01, 010共1999兲.

关12兴 Particle Data Group, K. Hagiwara et al., Phys. Rev. D 66, 010001共2002兲.

关13兴 H.Y. Cheng, Phys. Rev. D 67, 034024 共2003兲. 关14兴 C.H. Chen, Phys. Rev. D 67, 014012 共2003兲.

关15兴 G.P. Lepage and S.J. Brodsky, Phys. Lett. B 87, 359 共1979兲; Phys. Rev. D 22, 2157共1980兲.

关16兴 H.N. Li, Phys. Rev. D 64, 014019 共2001兲.

关17兴 H.N. Li, presented at FPCP, Philadelphia, 2002, hep-ph/0210198; T. Kurimoto et al., Phys. Rev. D 67, 054028 共2003兲; Y.Y. Keum et al., hep-ph/0305335.

关18兴 V.L. Chernyak and A.R. Zhitnitsky, Phys. Rep. 112, 173 共1984兲.

关19兴 C.H. Chen and H.N. Li, Phys. Rev. D 63, 014003 共2001兲. 关20兴 V. Chernyak, Phys. Lett. B 509, 273 共2001兲.

关21兴 C.H. Chen, Phys. Rev. D 67, 094011 共2003兲.

关22兴 C.K. Chua, W.S. Hou, S.Y. Shiau, and S.Y. Tsai, Phys. Rev. D

67, 034012共2003兲; Z.T. Wei, hep-ph/0301174.

关23兴 CLEO Collaboration, H. Muramatsu et al., Phys. Rev. Lett. 89, 251802共2002兲.

关24兴 E791 Collaboration, E.M. Aitala et al., Phys. Rev. Lett. 89, 121801共2002兲.

关25兴 FOCUS Collaboration, J.M. Link et al., Phys. Lett. B 541, 227 共2002兲.

关26兴 Babar Collaboration, B. Aubert et al., hep-ex/0207089.

BRIEF REPORTS PHYSICAL REVIEW D 69, 057504 共2004兲

數據

FIG. 1. Topologies for B →D ( * ) K ⫺ K ( * )0 decays. 共a兲 Color- Color-allowed with q ⫽u and d, and 共b兲 color suppressed.

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