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Contents. Introduction NLO Calculations Numerical Results Summary. Introduction. A great amount of interests has been triggered by the observation of several double-charmonium production in two B factories several years ago.
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Contents • Introduction • NLO Calculations • Numerical Results • Summary
Introduction • A great amount of interests has been triggered by the observation of several double-charmonium production in two B factories several years ago. • Most famous example is , with large discrepancy between experimental data and LO NRQCD predictions.
Introduction • One of the important step toward alleviating the discrepancy in is the discovery of significant and positive NLO perturbative corrections (with a K factor of 1.96) [Zhang, Gao, Chao (2005), Gong and Wang (2007)]. • Perturbative corrections at NLO plus relativistic corrections may bring theory into agreement with experiment [GTB, Chung, Kang, Kim, Lee, Yu (2006), He, Fan, Chao (2007)].
Introduction • Other double charmonium production processes have also been measured in the both B factories, notably the process: , 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 with experiment.
Our Task (1) NLO perturbative Calculations for process: (2) NLO both perturbative and relativistic Calculations 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:
LO Results Agree with E. Braaten and J. Lee (2005)
Description of the Calculations • Using FeynArtspackage to generate all Feynman Diagrams for the partonic process: • ( 20 Two-Point, 20 Three-Point, 18 Four-Point, 6 Five-Point Diagrams) • Using FeynCalc to perform DiracGamma and Color matrix trace. • Making expansion in q and projecting out S- or P-Waves. • Aparting the linear-dependent propagators to independent ones (Only 1-, 2-, 3-Point Scalar Integrals left). • Using FIRE package to reduce the integrals to Master Integrals (MI).
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 behaviors of the K functions are all proportional to double logarithm of r. • For the helicity channels: , leading-twist contribution dominates, one can employ the light-cone approach to efficiently reproduce the asymptotic expression by resorting to the leading-twist collinear factorization theorem, like Jia, Wang and Yang (2007). • It remains to be an open challenge for light-cone approach to reproduce these double logarithms.
Calculation at partonic level: Tree-level Result:
Factorization at Amplitude Level: I.R. Safe The I.R. divergence will be absorbed into the Matrix Elements: Numerical Plot of the finite part:
Summary • We worked out the NLO corrections to and NLO & relativistic corrections to . • Significant positive NLO perturbative correction was found to . • The impact of NLO corrections to seems rather modest, even with their signs uncertain. • Detailed study of polarized cross sections, it will be interesting for the future Super B experiments to test these polarization patterns.
Summary • Preliminary results on NLO and relativistic corrections to double charmonium production was obtained. • On the theoretical side, we worked out explicit asymptotic expression of all the 10 helicity amplitudes for at lowest order 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 those beginning with higher twist are always plagued with double logarithms.
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 (IBP).
