Contents Introduction NLO Calculations Numerical Results Summary 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. 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)]. 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 Our Task (1) NLO perturbative Calculations for process: (2) NLO both perturbative and relativistic Calculatio ns for process: Polarized Cross Sections Helicity Selection Rule: v denotes the characteristic velocity of charm quark inside a charmonium. Slowest asymptotic decrease: Total Cross Sections Parity Invariance: Total Cross Sections: Typical Feynman Diagrams LO Results Agree with E. Braaten and J. Lee (2005) 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 ). NLO Results 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. NLO Results Numerical Results Total Cross Section Plots Comparison with Experiment Calculation at partonic level: Tree-level Result: NLO Results: Factorization at Amplitude Level: Numerical Plot of the finite part: The I.R. divergence will be absorbed into the Matrix Elements: I.R. Safe 20 Diagrams contributing Double Logarithms Numerical Results Numerical Plots 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. 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. Thank You ! Apart Function 3 Propagators: General Cases: If we set 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).