2. Contents Introduction NLO Calculations Numerical Results Summary

3. Introduction  A great amount of interests has been triggered by the obs ervation of several double-charmonium production in two B factories several years ago.  Most famous example is, with large discrepa ncy between experimental data and LO NRQCD predictions.

4. Introduction  One of the important step toward alleviating the discrepan cy in is the discovery of significant and positi ve NLO perturbative corrections (with a K factor of 1.96) [Zhang, Gao, Chao (2005), Gong and Wang (2007)].  Perturbative corrections at NLO plus relativistic correctio ns may bring theory into agreement with experiment [GTB, Chung, Kang, Kim, Lee, Yu (2006), He, Fan, Chao (2007)].

5.  Other double charmonium production processes have also b een measured in the both B factories, notably the proces s:, with disagreements between LO NRQCD predictions and experiment.  Zhang, Ma, Chao (2008): In the cases of, large K factors (> 2.8) may bring theory into agreement wit h experiment. Introduction

6. Our Task (1) NLO perturbative Calculations for process: (2) NLO both perturbative and relativistic Calculatio ns for process:

7. Polarized Cross Sections Helicity Selection Rule: v denotes the characteristic velocity of charm quark inside a charmonium. Slowest asymptotic decrease:

8. Total Cross Sections Parity Invariance: Total Cross Sections:

9. Typical Feynman Diagrams

10. LO Results Agree with E. Braaten and J. Lee (2005)

11. Description of the Calculations  Using FeynArts package to generate all Feynman Diagram s for the partonic process: ( 20 Two-Point, 20 Three-Point, 18 Four-Point, 6 Five-Point Diagrams)  Using FeynCalc to perform DiracGamma and Color matri x trace.  Making expansion in q and projecting out S- or P-Waves.  Aparting the linear-dependent propagators to independe nt ones (Only 1-, 2-, 3-Point Scalar Integrals left).  Using FIRE package to reduce the integrals to Master In tegrals ( MI ).

12. NLO Results

16. Some Observations  The Scaling violation is of the logarithmic form.  For the helicity-conserving channels such as :, the leading behavior of the K function is governed by a single logarithm of r.  For all remaining helicity-suppressed channels, the leading asymptotic b ehaviors of the K functions are all proportional to double logarithm of r.  For the helicity channels:, leading-twist contribution d ominates, one can employ the light-cone approach to efficiently reprod uce the asymptotic expression by resorting to the leading-twist colline ar factorization theorem, like Jia, Wang and Yang (2007).  It remains to be an open challenge for light-cone approach to reproduc e these double logarithms.

17. NLO Results

19. Numerical Results

20. Total Cross Section Plots

21. Comparison with Experiment

22. Calculation at partonic level: Tree-level Result:

23. NLO Results:

24. Factorization at Amplitude Level: Numerical Plot of the finite part: The I.R. divergence will be absorbed into the Matrix Elements: I.R. Safe

25. 20 Diagrams contributing Double Logarithms

26. Numerical Results

27. Numerical Plots

28. Summary  We worked out the NLO corrections to and NLO & relativistic corrections to.  Significant positive NLO perturbative correction was fou nd to.  The impact of NLO corrections to seem s rather modest, even with their signs uncertain.  Detailed study of polarized cross sections, it will be inte resting for the future Super B experiments to test thes e polarization patterns.

29. Summary  Preliminary results on NLO and relativistic corrections to double ch armonium production was obtained.  On the theoretical side, we worked out explicit asymptotic expressi on of all the 10 helicity amplitudes for at lowest o rder in and the one for up to.  Also, the following pattern was further confirmed: The leading twist can only host the single collinear logarithm, while t hose beginning with higher twist are always plagued with double loga rithms.

30. Thank You !

31. Apart Function 3 Propagators: General Cases: If we set

32. FIRE 3 Propagators: generally, the integer l, m and n are larger than 1. and FIRE package will reduce these integer to 1 or 0, i. e. Master Integrals (MI), through Integral By Part (IB P).