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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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  • Introduction

  • NLO Calculations

  • Numerical Results

  • Summary


  • 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.


  • 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)].


  • 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
Our Task

(1) NLO perturbative Calculations for process:

(2) NLO both perturbative and relativistic Calculations for process:

Polarized cross sections
Polarized Cross Sections

Helicity Selection Rule:

v denotes the characteristic velocity of charm quark inside a charmonium.

Slowest asymptotic decrease:

Total cross sections
Total Cross Sections

Parity Invariance:

Total Cross Sections:

Lo results
LO Results

Agree with E. Braaten and J. Lee (2005)

Description of the calculations
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
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:


  • 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.


  • 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


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).