International Journal of Scientific & Engineering Research Volume 4, Issue 1, January-2013 1

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Heavy Quark – Light Diquark Approach to a Heavy Baryon in the Heavy Quark Symmetry Limit

S. Jalali F.Karamzadeh A.Haghpeima Z.Emami

.

Abstract— The masses and distance configurations of the ground and excited states of heavy – charmed and bottom baryons JP =1/2+-

,3/2+-,containing a single heavy quark and a light diquark are studied within HQS limit of the HQLD sector of NRQCD framework. We find

how the average distances between the heavy quark and the center of mass of the light diquark are smaller than the size of the light diquark which is in agreement with expectations from QCD sum rules and lattice QCD calculations.

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HEORETICALLY, the study of heavy baryons has always been interesting [1] and these baryons play an important role in our understanding of QCD at the hadronic scale

[2]. There is many theoretical treatments of heavy baryons, including quark models [3] - QCD sum rules [4] - Lattice QCD [5] – Relativistic quark diquark approximation [6] and Non relativistic QCD ( NRQCD ) which has been able to explain the mass spectrum of light baryons and is an effective field theory which is obtained from QCD by integrating out modes of energy of the order of the heavy quark masses [8] for describ- ing baryons made of one or more heavy quarks. The heavy quark light diquark ( HQLD) sector of NRQCD lagrangian is a heavy quark effective theory ( HQET ) . In this effective field theory framework ( EFT ) of heavy baryons where the typical gluon momenta are small compared with the heavy quark mass mQ , QCD dynamics of light diquark is independent of the flavor and spin of heavy quark [9] . For the heavy flavors , this new symmetry called heavy quark symmetry [ HQS ][10] . In fact in this limit of heavy quark mass, low energy QCD dy- namics remains non-perturbative but using HQS one can sep- arate the light quark and gluon dynamics from that of heavy one by systematically expanding the QCD lagrangian in pow- ers of 1/mQ and imposing HQS effects [11] . According to these effects in heavy baryons the light degrees of freedom quantum numbers are well defined up to corrections in the inverse of the mQ . Consequently the heavy quark momentum is close to the kinetic momentum resulting from the hadron motion. Thus the kinetic energy of the internal motion of the

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• *S.Jalali and F.Karamzadeh are currently pursuing masters degree program in Azad University Mashhad Branche , Iran*

• *A.Haghpeima( Corresponding Author ) is currently Associate Professor at*

Faculty of Sciences Islamic Azad University Mashhad Branche , Iran.

E-mail:alirezahaghpeima@gmail.com

• *Z.Emami is currently Associate Professor at Faculty of Sciences I slamic*

heavy baryon system is close to the kinetic energy of the rela- tive motion of the heavy quark and light diquark up to correc- tions of the mL / mQ where L , denotes a light quark.This is one of the basis for treating the light quark subsystem as a diquark in our calculations . The quark-diquark picture of a baryon is the nice approximation used to describe the baryon properties [12] . In this picture we reduce the task of treating a three body system to a two body system which is a success- ful task specially where we approximate the heavy quark mass to be infinity with respect to mass scale in process [13], and hence enormously reduces the complexity of theoretical analysis .The paper is organized as follows . In section 2 we ntroduce HQS effects for heavy baryons and calculate their mass spectrum using this symmetry limit . Finally section 3 devoted to conclusions and results.

Theoretically, the full QCD Lagrangian for a heavy quark ( c , b or t ) is given by

LQ = Q (i µDµ - mQ) Q, ( 1 )

where Dµ � aµ-igsTaAaµ with Ta =Aa/2 .Thus the heavy quark interacts with the light degrees of freedom by ex- changing gluons with the momenta of order AQC D which is much smaller than its mass mQ . In the HQS limit with low energy situations, where the typical gluon momenta is small compared with the heavy quark mass (mQ) , QCD dynam- ics becomes independent of the heavy degrees of free- dom, especially for the flavor and spin of the heavy quark.This means that the hyperfine interaction that involves the heavy quark is suppressed by the mass of the heavy quark. As a consequence, one-gluon exchange HF interac- tion should depend on the interacting light diquark pair, in- dependently of the baryon the pair belongs to. In fact the

Azad University Mashhad Branche, Iran E-mail: Z.Emami@gmail.com

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QCD hyperfine interaction and the QED electromagnetic hyperfine interaction between i and j quarks are propor-

for the sextet and

c+ c+

tional to1/mimj , where mi, mj are their masses. These interactions contribute to systematic uncertainty of the expe-

A = 1/ V2 (ud - du) c , 3

= 1/ V2 (us - su) c

( 3 )

rimental results and can be ignored in HQS limit, where

one of the quarks is heavy [14] . Indeed we characterize the heavy baryon mass by two widely separated scales: the large heavy quark mass,(mQ), and the low momentum transfer between the heavy and the light quarks of the diquark, which is of order AQCD. In this system the light diquark circle around the nearly static heavy quark and the system behaves as the the QCD analogue of the familiar hy-

drogen bounded by electromagnetic force . In HQS limit, where mQ � oo a good quantum number is the angular momentum of the light degrees of freedom. Thus, heavy quark

baryons belong to either SU(3) antisymmetric 3F or symme- tric 6F representations fig.1. The spin of the light diquark is 0 for 3F, while it is 1 for 6F For the spin of the ground state heavy baryons we have1/2 for 3F , representing the Ah and Sh

heavy baryons , while it can be both 1/2or3/2for 6F, representing Lh, Lh, Sh, Sh, Oh and Oh, where the star and h indicates spin 3/2 c b quarks respectively. The mass differ- ence between states belonging to different representations 3F

and 6F, do contain the dynamics of the light scalar and

Sc0 = 1/ V2 (ds - sd) c

For the antitriplet which are similar to the set of flavor

wave functions for baryons containing b quark

TABLE 1

The s-wave heavy baryons and their quantum numbers.

OQ O

Table 2. shows the experimental masses of the Ground-state charmed and bpttom bar yons [15].

vector diquark subsystem respectively. But the mass splitting between states belonging to same representation is caused by the chromomagnetic interaction at the order1/ mQ and can be ignored in HQS limit. Thus baryons containing a single heavy quark should fall into almost degenerate multiplets .For example the

Lb and Lb doublet will be degenerate in heavy quark

limit approximation. Generally these states have the same

parity as the light component Table 1 .

Fig. 1.SU(3) multiplets of charmed baryons , (a) 3F antisymmetric and

(b) 6F symmetric representations.

The members of the two multiplets of singly charmed ba- ryons have flavor wave functions

Lc++ = uuc, Lc+ = 1/ V2 (ud + du) c , Lc0 = ddc

( 2 )

3c+ = 1/ V2 (us + su) c 3c0 = 1/ V2 (ds + sd) c

Oc = ssc,

TABLE 2

Ground - state charmed bar yons and their SU(3)multiplets

Lattice estimates (†) have been taken from ( Ref [19] ).

Heavy baryon | Mass(GeV) | SU(3)multiplet |

Ac+ Ab+ Lc++,+0 Lb++,+0 Oc0 Ob0 Sc+ Sb+ Sc0 Sb0 S’c+ S’b+ S’c0 S’b0 | 2.285 -5.624 2.455-5.808 2.698-5.990† 2.468-5.793 2.471-5.760† 2.576-5.900† 2.578-5.900† | 3bar 6 6 3bar 3bar 6 6 |

In the limit of HQS, where the heavy quark mass mQ �oo, all states in the 6F representation would be degenerate and this is true for all states in the 3F representation . In this limit without the mQ � oo approximation there is a mass splitting between states belonging to each representation due to differences be- tween the masses of the light diquark sectors of the heavy

baryons.we calculated the light diquark masses by adding the two quarks mass and their binding hyperfine HF energy

.Table .3.

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TABLE 3

Quark and diquark masses and quantum numbers.

R= mq1xq1 + mq2xq2 + mQxh / mq1 +mq2 +m r12= xq ( 6 ) rh=(mq1xq1 + mq2xq2 /mq1 +mq2) – xh

where xq1 , xq2 and xh represent the positions with re- spect to a certain reference frame and r12 and rh are the Jaco-

bian coordinates. Thus we would have for the heavy baryon

Kinetic energy

T(q2Q ) � v 2

/ 2 µ ( 7 )

Where v 2 denotes the Laplacian and µ is the heavy quark-light diquark reduced mass. By using of the Baryon wave function

'-V

B = N[Y

00 ( rh ) exp ( -a2 rh 2

/ 2 ) ] ( 8 )

Now we evaluate the masses of the ground state heavy baryons in the framework of the HQS limit. Thus we can use the mass formula

M = mD + mQ + EL + Er ( 4 )

Here, mD is the light diquark mass, mQ the heavy baryon mass, EL the orbital and Er the radial exciting energies between heavy quark and light diquark respectively. According to table

3 two quarks having a closer mass have more tightly bound which is indicated by the spin-spin interaction, Thus the mass splitting

(ud) - [ud] > (us) - [us] > (uc) - [uc] � 0 ( 5 )

is spected where [ ] ,( ) , denotes scalar and vector diquarks respectively.We have accommodated the ground state ,JP =1/2+ heavy charmed and bottom baryons . These states have no or- bital angular momentum,EL = 0 and the mass splitting between them is indicated by radial exciting energy, Er of each ground

state heavy baryon .By using this exciting energy we have eva-

luated the average distance between heavy quark and the center of mass of the light diquark for each heavy baryon state. We set the Jacobi coordinates for a heavy quark -light diquark descrip-

tion.fig.2.

we would have for the Kinetic energy

Er = < T > '-V � 3a2 / 4µ (9 )

and for the relative distance between havey quark and light di-

quark we have

r0 = < rh > = -Y5 / 2a2 (10)

We have calculated the radial kinetic energy, Er of each ground state heavy baryon listed in table 2 ,using their para-

meters, mD mQ and Er = 0 . Also by using of Eq8-9 we obtained the average distance , r0 between the heavy quark

and the center of mass of light diquark. Table 4 .

The results with QCD sum rule[ 16 ]and lattice QCD calcu- lation [ 17 ] have suggested a clear dominance of the collinear- type configurations ( the heavy quark is close to the center of mass of the light diquark).This results seems to support our calculations based on HQS limit of HQLD picture of heavy baryons. In Ref.[ 18 ],the authors studied the baryon properties using Isgur-Wise function and found the heavy quark is far from the light diquark which is against the HQS approxima- tion of HQLD.

TABLE 4

Ground - state charmed and bottom baryons and their radi- al kinetic energy and relative distance between heavy quark and light diquark center of mass,Experimental masses have been taken from ( Ref [18] ) and Lattice estimates († ) have been taken from ( Ref [19] ) .

FIG.2 : q2Q rest frame.

For the coordinates we consider the following relations

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The average size of a scalar and a vector diquark is 0.0045

MeV -1 and 0.0205 ( 0.0235 ) MeV -1 respectively. According to

Table 4 one sees that the average distance of the heavy quark to the center of mass of the light diquark , r0is smaller than the average size of the light diquark.The picture that emerges

from this analysis is the one depicted in Fig.3, where the heavy quark is too close to the center of mass of the light diquark, which is in agreement with the findings of Ref [20].

FIG.3 :schematic picture of a Ground-state spin 1/2 heavy baryon with a charmed heavy quark (a) , and a bottom heavy quark (b).

This findings based on HQS limit of HQLD approximation shows a dominance of collinear-type configuration,which con- firms the results of QCD sum rules[16]and lattice calcula-

tios[17]We have obtained the average distance, r0* between

the heavy quark and the center of mass of light diquark for

charmed and bottom baryons with spin 3/2 Table .5.One sees

that this average distance for the spin 3/2 state heavy ba-

ryons is smaller than the spin 1/2 states.This distance split-

ting between states belonging to same representation is caused

by the chromomagnetic interaction and usually can be ig- nored in HQS limitd with MQ --+ oo approximation.The picture is depicted in Fig.4.

FIG.4:schematic picture of a Ground-state spin 3/2heavy bar yon with a charmed heavy quark ( a ) and a bottom heavy quark ( b ).

TABLE 5

Charmed and bottom baryons with spin 3/2 ,their masses ( Ref [18-19] their SU(3) multiplets and the relative distance r0* , between heavy quark and light diquark center of mass.

Heavy baryon | Mass GeV) | SU(3) multiplet | r0*(MeV -1) |

L:c++,+0* Lb++,+0* Qc0 Ob0 B’c+0* S’b+0* | 2.518-5.833 2.768-6.000 2.646-5.900 | 6 6 6 | 0.00477-0.00190 0.00297-0.00162 0.00350-0.00180 |

We also accommodated the masses of the p-wave charmed baryons, Table.6 these states have orbital angular momen- tum,EL =F 0 between the heavy quark and the center of mass of light diquark. By using of the exciting energy EL we have eva- luated the average distance between heavy quark and the cen- ter of mass of the light diquark for each heavy baryon state.

TABLE 6

Wave charmed baryons and their orbital kinetic energy, EL and relative distance between heavy quark and light di- quark center of mass, Experimental masses have been taken from ( Ref [21] ).

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One sees that this distance, for the p - wave state heavy baryons is smaller than the S -wave states, rp < r0 , r0*.

The picture is depigted in Fig.5

.

FIG 5: :schematic picture of a p- wave heavy baryon with a

charmed heavy quark.

It seems that the higher binding of these bound states is

caused by their higher mass in campare with s-wave states. Because of the unknown JP quantum numbers for most ex-

cited heavy baryons, Ac ( 2765 ) , Ac ( 2880 ) and Ac ( 2940 )

it is not determined if they are excitations of the Ac or L:c

.TABLE 7 shows our predictions for the quantum numbers of

these states.

TABLE 7

S - wave ,P - wave and D - wave charmed baryons and their excitation kinetic energy and relative distance , r , be- tween heavy quark and light diquark center of mass

.Experimental masses have been taken from ( Ref [22] ).

Hea vy- ba- ryo n | J(p) | SU(3) Multip let | Mas s (GeV ) | Or bi- tal | Er (Mev) | Or- bital | EL (Me V) | r (Me v-1) |

Ac L:c Ac L:c | 1/2(+) 3/2(-) 5/2(+) 3/2(+) | 3bar 6 3bar 6 | 2.765 2.765 2.880 2.940 | 2s 2s | 695
617 | 1p 1D ----- | 442 810 ----- | 0.00 207 0.00 217 0.00 212 0.00 184 |

One sees that the relative distance, r , between the heavy quark and the center of mass of light diquark for Ac 5/2 (+)is only 0.00018 MeV-1 smaller in campare with this distance for Ac 3/2 ( - ) state that confirms these states being excitations of the heavy baryons belonging to the same 3 bar SU(3) multip- let. We have considered similar distance splitting for predict- ing other heavy baryon quantum numbers listed in Table .7.

Uunderstanding of low-energy properties of QCD by quark dynamic features in phenomenological models de- pends on the true degrees of freedom of any model.In our model we studied the ground state properties of heavy ba- ryons and extended it to the description of their excited states. Our calculations performed in the framework of the heavy - quark light diquark HQLD sector of NRQCD which is a heavy quark effective theory HQET. Also we used the heavy quark symmetry, HQS ,where QCD dynamics of light diquark is independent of the flavor and spin of heavy quark.Thus we reduced a very complicated three-body problem to a simple two-body problem . For the ground state heavy charmed and bottom baryons we calculated the average distance between heavy quark and the center of mass of light diquark. Here we considered only the kinetic energy of the light diquark with respect to the heavy quark . There are strong indications in QCD sum rules and lattice calculations for a collinear-type configuration for the heavy baryon system,where the heavy quark is too close to the center of mass of light diquark, which is in agreement with our findings.We also accommodated the masses of the s -wave charmed baryons, these states have or-

bital angular momentum EL =F 0 between the heavy quark and

the center of mass of light diquark and the rlative distance

between the heavy quark and the center of mass of light di-

quark is smaller in campare with s –wave states.We find that experimental data for the ground and excited states of heavy baryons can be accommodated in the HQS limit of HQLD sector of NRQCD theory for heavy baryons, by treating

a heavy baryon as the bound state of the heavy quark and light diquark, considering radial and orbitl excita- tions only between these constituents.We emphasize that a combined study of light, heavy and doubly heavy baryons is needed to confirm these achievements.

[1] L.A.Copley, N.Isgurand G.Carl, Phy.Rev.D, P.Hasenfratz, R.R.Horgan, J.Kutiand J.M.Richard, Phys. Lett. B 94 , 401 (1980). W.Y.P.Hwang and D.B.Lichtenberg, Phys.Rev.D 35 ,

3526 (1987). C.Itoh, T.Minamikawa, K.Miura andT.Watanabe, Prog.T heor. Phys. 87 ,781 (1992). R.Roncaglia, D.B.Lichtenberg and E.Predazzi, Phys.Rev. D 52 , 1722 (1995). L.Ya.Glozman, D.O.Riska, Nucl.Phy.A 603 (1996) 326. D.Ebert, R.N.Faustov,V.O.Galkin and A.P.Martynenko, Phys. Rev. D 66

,014008 (2002).X.G.He, X.Q.Li, X.Liu and X.Q.Zeng, Eur.Phys.J.C51 , 883 (2007). D.Ebert, R.N.Faustov, and V.O.Galkin, Phys.Lett.B 659 , 612 (2008).

[2] Chien-WenHwang, J.Phys.G: Nucl. Part.Phys.35 ,

075003(2008). H.Garcilazo , J. Phys.G: Nucl. Part. Phys. 34 ,

961-976(2007).S.Migura, D.Merten, B.Metsch and H.R.Petry,

Eur.Phys. J. A 28 , 41 (2006).

[3] S.M.Gerasyuta and D.V.Ivanov, Nuovo Cim.A112 , 261 (1999).D.Ebert, R.N. Faustov, V.O.Galkin and A.P. Martynen- ko, Phys. Rev. D 66 ,014008 (2002). S. Migura, D. Merten, B. Metsch and H.R. Petry, Eur. Phys. J. A 28 , 41 (2006).C. Alber-

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International Journal of Scientific & Engineering Research Volume 4, Issue 1, January-2013 6

ISSN 2229-5518

tus, E. Hernandez, J. Nieves and J. M. Verde-Velasco, Eur. Phys. J. A 32 ,183 (2007).

[4] E. Bagan, M. Chabab, H. G. Dosch and S. Narison, Phys.Lett. B 287 , 176 (1992). D.W.Wang and M. Q. Huang, Phys. Rev. D67 , 074025 (2003). X. Liu, H. X. Chen, Y. R. Liu, A. Hosaka and S. L. Zhu, Phys.Rev. D 77 ,014031 (2008).

[5] A. Ali Khan , Phys. Rev. D 62 ,054505 (2000).R. Lewis, N. Mathur, and R. M. Woloshyn, Phys. Rev D64 , 094509 (2001).J. M. Flynn, F. Mescia and A. S. B. Tariq [UK QCD Col- labora tion], JHEP 0307 , 066 (2003).

[6] S. Capstick and N. Isgur, Phys. Rev. D 34 , 2809 (1986).4 V.E. Lyubovitskij .,Prog. Part. Nucl. Phys. 50,329 (2003) ; M.A. Ivanov, M.P. Locher and V.E. Lyubovit-skij, Phys.Lett.B 408 , 435 (1997); M.A. Ivanov and P.Santorelli, Phys. Lett. B456 , 248 (1999).D. Ebert, R. N. Faus- tov, and V. O. Galkin, Phys. Rev. D 72 , 034026 (2005).D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Lett. B 659 , 612 (2008).

[7] A.Valcarce , Phy. Rev. C.72, 025206 (2005).[8]W. E. Caswell andG. P. Lepage, Phys. Lett. B167 , 437 (1986); G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D51, 1125 (1995)

[9] N. Isgur and M. B. Wise, Phys. Lett. B 232 , 113 (1989); B237 , 527 (1990) H. Georgi, Nucl. Phys. B 240 , 447 (1990).

[10] S. Nussinov, and W.Wetzel, Phys. Rev. D 36, 130 (1987). M. A. Shifman, and M. B.Voloshing, Yad. Fiz. 45,463 (1987 ) [Sov. J. Nucl. Phys. 45, 292 (1987)]. H. D. Politzer, and M. B.Wise, Phys. Lett. B 206, 681 (1988 ); ibid. 208, 504 (1988). N. Isgur, and M. B. Wise, Phys. Lett. B 232, 113 (1989). M. Neubert,Phys.Rep.245,259(1994).

[11] H. Georgi, Nucl. Phys. B 240 , 447 (1990). M. Neubert, Phys. Rep.245, 261 (1994). T. Mannel, W. Roberts and Z. Ry- zak, Nucl. Phys. B 368 ,204 (1992). C. V. Efimov, M. A. Ivanov, N. B. Kulimanova, and V. E. Lyubo vitskij, Z. Phys.C 54, 349 (1992). X.- H. Guo, and P. Kroll, Z. Phys. C 59, 567 (1993). X.- H. Guo, and T. Muta, Phys. Rev. D 54, 4629 (1996), Mod. Phys. Lett. A 11, 1523 (1996). B. K¨onig, J. G. K¨orner, M. Kr¨amer, and P. Kroll, Phys. Rev. D 56, 4282 (1999). H . Kaur, and M. P. Khanna, J. Phys. G: Nucl. Part. Phys. 25, 1149 (1999). X.- H. Guo, A. W. Thomas, and A. G. Williams, Phys. Rev.D5 9,

116007 (1999); ibid. 64, 096004 (2001). D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Rev. D 72, 034026 (2005). D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Rev. D 73,094002 (2006); Phys. Lett. B 659, 612 (2007). X.- H. Guo, k.-W.Wei, and X.- H.Wu, arXiv:0710.1474 [he p-ph].D. Ebert,T.Feldmann, C. Kettner, and H. Reinhardt, Phys. C 71,329 ( 1996); Int. J. Mod. Phys. A 13,1091(1998).

[12] M.Anselminoetal., Rev. Mod. Phys. 65 , 1199 (1993). D. Ebert, H.Reinhard tand M. K. Volkov, Prog. Part. Nucl.Phys. 33 1 (1994). J. G. K¨orner, M. Kr¨amer and D. Pirjol, Prog. Part. Nucl. Phys. 33 , 787 (1994). R. L. Jaffe, Phys.

Rep.409 , 1 (2005). F. Wilczek, hep - ph / 0409168.

[13] J. G. Korner , M. Kramer and D. Pirjol, Prog. Part. Nucl. Phys.33, 787 (1994).

[14] T.Altonen ( CDF Collaboration ), Phys. Rev. Lett. 99 ,

202001 (2007). C. Amsler (Particle Data Group), Phys. Lett.B

667, 1 (2008). J. L. Rosner, Phys. Rev. D 75 ,013009 (2007).

[15] CDFCollaboration,T.Aaltonen , Phys. Rev. Lett.99 ,

052002 (2007). D0Collaboration,V.Abazov , Phys. Rev. Lett.99 ,

052001 (2007).

[16] A.G.Grozin and O. I. Yakovlev, Phys. Lett. B 291,

441(1 992); O .I. Yakovlev in Proc. ofthe IIIGerman - Russian-

Workshopon Heavy Quark Physics, Dubna, Russia, May 20-

22, 1996, hepph/9608248.

[17] UKQCDCollaboration,K.C.Bowler , Phys. Rev. D 57,

6948 (1998).

[18] Particle Data Group, W. - M. Yao , J. Phys. G: Nucl. Part. Phys. 33, 1 (2006). BABAR Collaboration, B. Aubertet ., Phys. Rev. Lett. 97, 232001 (2006). CDF Collaboration, T. Aal- tonen , Phys. Rev. Lett. 99, 202001 (2007).

[19] UK QCD Collaboration, K. C. Bowler , Phys. Rev. D

54, 3619 (1996).

[20] F. Cardare lli, S. Simula, Phys. Lett. B 421, 295 (1998). [21] Particle Data Group, J. Phys. G33 , 1 (2006). Belle Col-

laboration, R. Mizuk , Phys. Rev. Lett.94 , 122002 (2005). Belle

Collaboration, R. Chistov , Phys. Rev. Lett.97 , 162001 (2006). P. Cho, Phys. Rev. D 50 , 3295 (1994).

[22] B.Aubert .[BABAR Collaboration], Phys. Rev. Let9.8 ,

012001 (2007). R.Mizuk .[BELLE Collaboration], arXiv:hep-

ex/0608043.

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