1. Productions and Decays of Charmonium Application of perturbative QCD
J.P. Ma , ITP,
Academia Sinica, Beijing
Talk given at
BES-Belle-CLEO-Babar 2007 Joint workshop on Charm Physics

2. Effective theory from QCD for charmonia (NRQCD)
Content:
Effective theory from QCD for charmonia (NRQCD)
Productions of charmonia
2.1 Production at hadron colliders
2.2 Production at electron-positron collider
Decays of charmonia
Outlook
Useful reference: “Heavy quarkonium physics” CERN Yellow Report
hep-ph/
“BES Yellow Book” to appear in ??

3. Effective theory from QCD for charmonia (NRQCD)
Charmonia: A bound state of one charm- and one anticharm quark as a dominant component, plus other possible light dynamical freedoms, i.e., light quarks, gluons.
Charm quarks can be taken as heavy quarks, moving with a small velocity in a charmonium:
The charm quarks move nonrelativistically, one can use
this fact to derive an effective theory from full QCD.

4. NRQCD: An expansion in v :
: QCD lagrangian for gluons and light quarks
for charm quarks
: Pauli spinor fields, two component.
: Corrections to the leading term, of higher orders in v.

5. An important remark:
NRQCD is a quantum field theory for charmonia, not quantum mechanics! In the framework of NRQCD, charmonia are not simply as bound states of a charm- and anticharm quark, they have many components. E.g.,
Taking the first term only leads to the so-called color-singlet model (CSM)

6. Based on the velocity expansion and factorization concept, an
inclusive production rate of a charmonium can be written as:
: process dependent coefficients characterizing productions
of the charm quark pair in various states. They can be
calculated with perturbative QCD
The matrix elements take the form (four quark operator)
and describe how the charm quark pair in various states are hadronized
into a charmonium, they are universal, nonperturbative………
Lepage, Bodwin and Braaten 1995

7. A power-counting in v for those matrix elements can be given to determine the relative importance between them.
Note:
1. The produced charm quark pair does not need to have
the same quantum number as the charmionium, because
the charmonium can have many different components.
2. The factorization has been not proven. It can be
violated at certain level (Nayak, Qiu and Sterman)

8. Inclusive decays into light hadrons:
: process dependent coefficients characterizing decays
of the charm quark pair in possible various states in a charmonium. They can be calculated with perturbative QCD
The matrix elements are of the four-quark operators:
, represent the probability to find the charm quark pair in various states in a charmonium.

9. Again, there is a power-counting in v for those matrix elements can be given to determine the relative importance between them.
Note:
1. The produced charm quark pair does not need to have
the same quantum number as the charmionium, because
the charmonium can have many different components.
2. The factorization has been proven. Unlike the case
of production, here the sum of the final state is
complete, the factorization can be argued to hold at
each loop level.

10. 2. Productions (in nonresonant region)
2.1 Production at hadron colliders (Tevatron)
(Before this century )
At leading order of v,
the differential cross
section was much smaller than measured !
It was realized that the higher order contributions could be important at large pt.
The so-called color-octet channel for production.
This was thought as a triumph of Color-Octet…
At leading order of alpha_s!

11. The signature of color-octet
components of J/\psi has
also been seen clearly in
inclusive production
of two-photon collision.
However…..

12. But, then it was also predicted
that the polarization is
transverse at Tevatron….
The measured polarization
seems longitudinal!
This discrepancy existed
for number of years…
Higher orders at alpha_s?
Importance of color-octet?
Fixed target eperiment
Photo-production…

13. This year, the NLO correction in color single channel appears….
NLO contribution is
important
NNLO is important
too!
Predictions are much
closer to the data in
shape and normalization!
See Maltoni’s talk at QWG07

14. So far, the situation is still unclear, what clear is, without NLO corrections the importance of color-octet channel is overestimated.

15. 2.2 Production at electron-positron collider
Belle in 2002:
Theoretical predictions at leading orders (v and alpha_s) : ??

16. The discrepancies make challenges to the theory…….
Belle in 2004:
Barbar in 2005:
The discrepancies make challenges to the theory…….
Various papers tried to solve them….
Again, higher order corrections seem the most important………

17. Theoretical predictions with the NLO- and relativistic correction
can accommodate… Zhang, Gao, Chao 2006

18. The large corrections come not only from the NLO correction in the perturbative coefficient, but also from the matrix elements used there, which are extracted from the leptonic decay of J/\psi and the two photon decay of eta_c.
The theoretical study is more interesting because one usually does not expect that the factorization with two charmonia in exclusive processes holds.
Now it is also clear that how the factorization with two charmonia holds or violated……….

19. For All these cases tell us that higher order corrections are
Plus one loop correction one has: (Zhang & Chao 2007)
To compare with the measured :
All these cases tell us that higher order corrections are
important for charmonia!

20. 3. Decays of charmonia Leptonic decays:
The wave function at origin can be expressed as a NRQCD matrix element.
The QCD corrections are from matching full QCD to NRQCD and determined
at two-loop level (Beneke et. al )
One may use potential models to determine the wave function and……

21. ???? Good nonperturbative methods are needed to determine
NRQCD matrix elements

22. Inclusive decays into light hadrons:
Many factorization formulas can be given. E.g.
The new factorization solves the old I.R. problem with P-wave charmonia.
But, large corrections from alpha_s and uncertainties of NRQCD matrix elements
prevent from comprehensive predictions………… Uncertainties may be eliminated
by building ratios……… E.g. P-wave..

23. Large QCD corrections, also significant changes in measured values….

24. Exclusive decays into two light hadrons:
One may use the collinear factorization for light hadrons. At leading
twist:
E.g.:
Beside other corrections,
power corrections can be large
because charm quarks are not
heavy enough………
But some general features may
be predicted……

25. Based on the vector coupling of QCD (helicity conservation) one has:
Still, large corrections to the scaling exits!

26. 4. Outlook:
※ Higher order QCD corrections are very important
※ Nonperturbative methods to determine NRQCD matrix elements are needed
※ Precise and stable results from experiment