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

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/0412158 “BES Yellow Book” to appear in ??

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.

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.

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)

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

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)

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.

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.

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!

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

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…

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

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

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

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………

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

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……….

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!

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. 1997 ) One may use potential models to determine the wave function and……

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

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..

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

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……

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

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