Double Charmonium ProductionDouble Charmonium Production at at

Electron-Positron CollidersElectron-Positron Colliders

J.P. MaJ.P. Ma

ITP, CAS, BeijingITP, CAS, Beijing

Talk given at the third meeting of Quarkonium Working Group

IHEP, Beijing, 12-15. Oct. 2004

Contents:

1. Current Status

2. Another Approach

3. Outlook

1. Current Status

The main topic is:

e+ + e- ——> Charmonium + Charmonium

At the lowest order:

2 pairs of charm and anti-charm forming the two charmonia with possible quantum numbers.Always: s =(10.6GeV)2

Belle and Barbar

For production of two back-to-back quark pairs: perturbative QCD can be used for two cases:

§1: Quarks are heavy………… §2: Quarks are light, but each pair has a small invariant mass.

C-quark case: Formation of pairs into charmonia ?? Non-trivial

Usual way: Taking charmonia as a nonrelativsitic bound state of c and anti-c, using a wave-function…………

Predictions take a factorized form.

called nonrelativistic approach(???)

Predictions from the nonrelativistic approach:

Ref: K.-Y. Liu, Z.-G. He and K.-T Chao, Phys. Lett. B557 (2003) 45, hep-ph/0211181.

E. Braaten and J. Lee, Phys. Rev. D67 (2003) 054007, hep-ph/0211085.

K. Hagiwara, E. Kou and C.-F.Qiao, Phys. Lett. B570 (2003), hep-ph/0305102.

Prediction for the process: 2~5 fb !!!!

Othe processes can also be predicted:

…………………………………

Beyond the leading order of QED, one can have:

Bodwin, Lee and Braaten: Phys.Rev.Lett.90:162001,2003 Phys.Rev.D67:054023,2003

The references listed in this talk is not “exclusive” for those exclusive processes ……………………

Experimental Result:

Belle Collaboration, Phys. Rev. Lett. 89 (2002) 142001

8~10 times larger than the predicted !!!!

Possible explanations: Instead of c

Scalar glueball? Brodsky, Goldhaber and Lee ,Phys.Rev.Lett.91:112001,2003

A new light scalar boson? Cheung and Keung, Phys.Rev.D69:094026,2004

The newest Belle results: (hep-ex/0407009)

≤ at CL 90%

and :

Some drawbacks of the nonrelativistic approach:

§1: Factorization ? Two charmonia in the final state.

One can not have a frame in which they are in

rest. No way to do NRQCD power counting.

§2: Large log: Since c-quarks are massive, perturbative part will contain large logs like ln(s/mc

2 ).

§3: Large relativist corrections??

2. Another approach If S >> mc

2 , one can take the charm quark as a light

quark and neglect its mass.

The same diagram:

The formation of charmonia is parameterized with light-cone wave-functions(LCWF). They are well defined in QCD and classifiedaccording to twist of operators.

Advantage: §1. Factorization can be shown, e.g., at leading

twist….. §2. No large logs in perturbative coefficients, since

we have mass less quarks. They are large logs in LCWF, but they can be resummed by using evolution equations of Efremov-Radyushkin-Brodsky-Lepage

§3. Corrections can be systematically added.

The approach is standard and called as perturbative QCD approach proposed 20 years ago. Ref: Brodsky & Lepage, Phys. Rev. D22 (1980) 2157, Phys, Rev. Lett. 43 (1979) 545.

The process:

is determined by the form factor:

With a generalized power-counting rule [ X.D. Ji, J.P. Ma and F. Yuan, Phys. Rev. Lett. 90 (2003) 241601 ]

One gets:

H1,2,3: Perturbative coefficients.

,

LCWF of

LCWF of

: A soft scale, like mc, ……….

Detailed calculation gives:

Two drawbacks: §1: There are end-point singularities. This may be

killed by using complicated kT factorization. A usual way is to introduce a cut-off:

The cut-off is chosen as:

§2: LCWF’s are unknown at the interested energy scale. For the energy scale being infinity, the form of LCWF’s are known as:

With these LCWF’s

It is interesting to note that LCWF’s of charmonia can take a factorized form in the framework of NRQCD factorization (Work in progress by Ma and Si). At leading orders they all take the form:

The corresponding cross-section is:

These two cases are extreme, one is for infinite large energy scale, another is for the energy scale at mc with possible large corrections. They are not realistic!

LCWF’s of light hadrons are well studied, e.g., those of pion and rho. Taking those for etac and J/psi correspondingly, but with a modfication for the mass

effect, a parameter a is introduced. The results are:

a=1

a=1.5

a=1.75

Details can be found : J.P. Ma and Z.G. Si, hep-ph/0405111

3. Outlook §1: There is a large discrepancy between Belle experiment and theoretical predictions based on the nonrelativistic approach for

It would be good if the discrepancy is or is not confirmed by other experiment like Barbar.

§2: The perturbative QCD approach is a consistent approach for the process and may explain the discrepancy, but a more detailed study is needed.