Double charm production in e + e - -annihilation Anatoly Likhoded, IHEP, Protvino The conflict between Theory and Experiment in double charmonium production Beyond the  -approximation Light-cone results Quark-hadron duality for cccc-sector

 Bound states are similar to QED positronium  Simple strong-interactive system  Non-relativistic for less for  small component velocity and small are the expansion parameters in NRQCD  The properties of these bound states, their decay and production channels are a good laboratory for QCD both in perturbative and non- perturbative modes  The simplest system clarifying the transitions particles Introduction

One of the most challenging open problems in heavy quarkonium is: the large discrepancy of the double charmonium production cross sections measured in e + e - annihilation at B factories and the theoretical calculations from NRQCD. Double charmonium production

K. Hagiwara, E. Kou and C.-F.Qiao, A.Berezhnoy, A. Likhoded: E. Braaten, J. Lee: K.-Y. Liu, Z.-G. He and K.-T Chao: The calculated exclusive cross sections are about an order of magnitude smaller than exp. results! The calculated inclusive cross section of J/  is about a factor of 5 smaller than exp. results! NRQCD factorization formalism CS dominant Theoretical calculations (LO) : PLB557 (2003) 45 PLB570 (2003) 39 e+e+ e-e- ** (PRD 67 (2003) ) one of 4 diagrams exclusive double charmonium production via e + e - annihilation to a  *. two charmonium states with opposite C parities

 Experiment Belle BaBar  Trends The Belle cross section has moved down from BaBar cross section is even lower NLO in give an enhancement factor 1.8 (Zhang et al)  Discrepancy still remains

NRQCD predictions for double charmonium production Papers where NRQCD was applied for double charmonium production: E.Braaten, J.Lee, Phys. Rev. D67, (2003) K. Liu, Z. He, K. Chao, Phys.Lett.B557 (2003) G. T. Bodwin, J. Lee, E. Braaten, Phys.Rev.Lett.90, (2003) G. T. Bodwin, J. Lee, E. Braaten, Phys.Rev.D67, (2003) Y. Zhang, Y. Gao, K. Chao, Phys.Rev.Lett.96, (2006)

q e+e+ e-e- ** g x1x1 x2x2   -approximation (Bethe-Heitler) gluon virtuality  In  -approximation for massless quarks

In frame the c-quark distribution over the momentum fraction x is V.G. Kartverishvili, A.K. Likhoded Nucl. Phys. B148 (1979), 400 J/  structure function   is the same in fragmentation function

A proposed solution Bondar and Chernyal have proposed to calculate the cross section in the framework of light-cone formalism. BC result if light-cone distributions  (z-1/2) are taken, the cross section reduces to ~3 fb J.P. Ma, Z.G. Si // Phys.ReV. D70 (2004) A.E. Bondar, V.L. Chernyak // Phys.Lett. B612 (2005) 215 V. Braguta, A. Luchinsky, A. Likhoded // Phys.Rev.D72 (2005)

processes where is defined from Leading asymptotic behavior Light-cone formalism

Light-cone wave functions, that are expressed as the expansion over poorly known wave functions. For example, from the asymptotical Regge behavior e+e+ e-e- ** x1x1 y1y1 In -approximation we have

Light-cone wave functions are unknown. Can be estimated from the wave-functions obtained in the framework of potential models.

Light cone wave functions.

The model for the light cone wave functions: The property of the wave functions: A.E. Bondar, V.L. Chernyak, Phys.Lett.B612:215, (2005)

e + e -  V( 3 S 1 ) P( 1 S 0 )

e + e -  V( 3 S 1 ) S( 3 P 0 )

Numerical results. Conclusion:  Formfactor = NRQCD x Internal Motion  WFs of charmonium are wide  NRQCD is not applicable Uncertainties:  Poor knowledge of the light cone wave functions  Radiative corrections  1/s corrections V.V. Braguta, A.K. Likhoded, A.V. Luchinsky Phys.Rev.D72 (2005) Phys.Lett.B635 (2006) 299

Observation of X(3940) at Belle. K.Abe et al., hep-ex/ Two possibilities :

Hypothesis: X(3940) is one of,,  If is dominant decay mode than X(3940) is  If is seen than must be seen  Decay mode of is We reject this hypothesis

Hypothesis: Our estimation: Experimental result:

Let us consider the process  LO -  tot (4c) ~ O(  s ) 2 depends on  s and m c for m c =1.25 GeV and  s =0.24  tot (4c)=372 fb Quark-hadron duality  OZI-rule forbids transition to and light quark sectors

It is interesting to compare these results with cccc-final state in hadronic Z- decay, where three experimental values are known (L3, OPAL, ALEPH) A.K. Likhoded, A.I.Onishenko Phys.Atom.Nucl, 60 (1997) 623 Than at we have

Inclusive charmonium production In the peak of Z-boson Numerically at  s =0.24 this ratio is Good agreement with L3 result. E. Braaten and K.Cheung Phys.Rev.D48 (1993) 4230 A.L., V. Kiselev, M. Shevlyagin Phys.Lett.B332 (1995) 351

We can compare the sum of the cross sections of inclusive production of J/ ,  c and P-wave sates calculated in the framework of perturbative QCD (LHS), with the cross section of the singlet cc-production in duality interval (RHS) In the duality interval the RHS cross section The sum is LHS cross section consists of perturbative contributions from J/ ,  c and P- wave states B.K.L // Phys.Lett.B 323 (1994) 411 Liu et al // PR D69 (2004) There is a reasonable agreement between these two estimates

If we restrict the mass of pair in the process by the duality interval, we get It should be compared with Belle and BaBar results: We observe a good agreement with the experiment These cross sections include both and

Substracting the cross section of exclusive pair production we obtain This value should be compared with the Belle result It should be noticed, that this result is sufficiently larger, than the absolute bound for production cross section

Using final state we can estimate the upper bound on (ccq) baryon production. Let us assume that it is produced through the production of cc-diquark in color-antitriplet state. Cross section of production in duality interval Double charmed baryon production 3c3c The upper limit for the cross sections of the processes like

First attempt to search doubly charmed baryons and was done in BaBar experiment hep-ex/ Events containing candidates were searched for the processes and Upper limits This is about an order of magnitude higher, than our predictions

Conclusion 1) Light cone partially resolves the discrepancy between theory and experiment The drawback of a light-cone formalism is a large number of unknown wave functions (for critics see G. Bodwin, D. Kang, J. Lee hep-ph/ ) 2) NRQCD (  -approximation) does not work for exclusive processes. In inclusive processes, on the contrary,  -approximation gives a quite good accuracy 3) Quark-hadron duality as an independent way to estimate cross section gives good agreement with experiment (20% accuracy) and allows to predict Double Charm Baryon production cross section. A.P. Martynenko, Phys.ReV. D72 (2005)