1. Exclusive charmonium production in hard exclusive processes. V.V. Braguta Institute for High Energy Physics Protvino, Russia

2. Content: Introduction Charmonium Distribution Amplitudes (DA) The properties of DAs Exclusive charmonium production within light cone formalism: Perspectives

3. Introduction

4. Internal structure of charmonium First approximation Charmonium = nonrelativistic quark-antiquark bound state Charmonium spectrum Next approximation: NRQCD Charmonium = nonrelativistic system v 2 ~0.3, v~0.5 Production as a probe of the internal structure of charmonium

5. NRQCD formalism At LO of NRQCD quark-antiquark pair has zero relative momentum

6. Light Cone Formalism The amplitude is divided into two parts:Hadronization Twist-2 2-distribution amplitudes Twist-3 4-distribution amplitudes … … Light cone formalism is designed to study hard exclusive processes

7. Comparison of LCF and NRQCD The cross section is double series LCF Power corrections: NRQCD Relativisitic corrections: Radiative corrections:

8. Relativistic corrections LCFNRQCD DA resums relativististic corrections to the amplitude.

9. Leading logarithmic radiative corrections Exclusive quarkonium production Inclusive quarkonium production DA resums leading logarithmic radiative corrections.

10. Distribution Amplitudes are the key ingredient of Light Cone Formalism

11. Charmonium Distribution Amplitudes

12. Definitions of leading twist DA

13. Evolution of DA  DA can be parameterized through the coefficients of conformal expansion a n :  Alternative parameterization through the moments:

14. DA of nonrelativistic system

15. Different approaches to the study of DA 1. Functional approach - Bethe-Salpeter equation 2. Operator approach - NRQCD - QCD sum rules

16. Potential models  Solve Schrodinger equation  Get wave function in momentum space:  Make the substitution in the wave function:  Integrate over transverse momentum: Brodsky-Huang-Lepage procedure:

17. Different approaches

18. The moments within Potential Models The larger the moment, the larger the contribution of relativistic motion Only few moment can be calculated Higher moments contain information about relativistic motion in quarkonium

19. Is there relativistic motion in quarkonium? Relativistic motion can appear only due to the rescatering for a short period of time (v<<1) Relativistic motion exists v 2 ~0.3 is still large

20. Property of DA DA of nS state has 2n+1 extremums

21. The moments within NRQCD

22. The values of were calculated in paper G. Bodwin, Phys.Rev.D74:014014,206 The constant can be expressed through the

23. The model for DA within NRQCD At leading order approximation is the only parameter

24. The moments of DA within QCD sum rules Advantage: The results are free from the uncertainty due to the relativistic corrections Disadvantage: The results are sensitive to the uncertainties in the sum rules parameters: QCD sum rules is the most accurate approach

25. The details of the calculation

26. Logitudinally polarized 3 S 1 mesons Sum rules Improved Sum rules: Sum rules parameters:

27. Numerical analysis

28. The other DAs These sum rules are less accurate

29. The results of the calculation The results for 1S states The results for 2S states

30. The models of DAs 1S states 2S states Borel version sum rules without vacuum condensates

31. The properties of distribution amplitudes

32. Relativistic tail  At DA is suppressed in the region  Fine tuning is broken at due to evolution  This suppression can be achieved if there is fine tuning of a n

33. Relativistic tail within NRQCD Light Cone The amplitude of meson production NRQCD Leading logarithmic corrections are resummed in DA Leading logarithmic corrections are contained in Wilson coefficients QCD radiative corrections enhance the role of higher NRQCD operators

34. The violation of NRQCD scaling rules At larger scales the fine tuning of the coefficients a n is broken and NRQCD scaling rules are violated NRQCD velocity scaling rules are violated in hard processes

35. Improvement of the model for DA The evolution of the second moment The accuracy of the model for DA is better at larger scales

36. Models for 2S states At leading order approximation of NRQCD the relative momentum of quark-antiquark pair is zero

37. Exclusive charmonium production within light cone formalism

38. The processes:

39. The diagrams Fragmentation diagrams Nonfragmentation diagrams

40. The cross section at NLO

41. Relativistic and leading logarithmic radiative corrections Interference of fragmentation and nonfragmentation diagrams The role of corrections

42. The results of the calculation a Bodwin, Braaten, Lee, Phys. Rev. D74

43. The processes:

44. e + e -  V( 3 S 1 ) P( 1 S 0 ) This formula was first derived in Bondar, Chernyak, Phys. Lett. B612, 215 (2005)

45. Twist-3 distribution amplitudes are unknown Problem: The scale dependences of some twist-3 DAs are unknown

46. The constants needed in the calculation The values of the constants The values of the constants (preliminary results)

47. The results of the calculation Why LO NRQCD is much smaller than the experimental results? 1. Relativistic corrections K~2.5-6 2. Leading logarithmic radiative corrections K~1.5-2.5 a E. Braaten, J. Lee b K.Y. Liu, Z.G. He, K.T. Chao

48. Relativistic and radiative corrections NRQCD formalism Light cone formalism The amplitude was derived in paper Bondar, Chernyak, Phys.Lett. B612

49. Perspectives: Theoretical LCF can resolve the other problems with exclusive processes? Development of alternative to NRQCD DAs of P-wave and D-wave charmonium mesons New results for bottomonium decays Inclusive charmonium production Experimental LHC (bottomonium decays) B-factories and Super B-factories (exclusive production of mesons and baryons)