Contents

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Contents

• Introduction
• NLO Calculations
• Numerical Results
• Summary

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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.

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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).

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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 (gt 2.8) may
  bring theory into agreement with experiment.

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Our Task
(1) NLO perturbative Calculations for process
(2) NLO both perturbative and relativistic
Calculations for process
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Polarized Cross Sections
Helicity Selection Rule
v denotes the characteristic velocity of charm
quark inside a charmonium.
Slowest asymptotic decrease
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Total Cross Sections
Parity Invariance
Total Cross Sections
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Typical Feynman Diagrams
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LO Results
Agree with E. Braaten and J. Lee (2005)
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Description of the Calculations
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NLO Results
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NLO Results
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NLO Results
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NLO Results
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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.

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NLO Results
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NLO Results
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Numerical Results
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Total Cross Section Plots
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Comparison with Experiment
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Calculation at partonic level
Tree-level Result
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NLO Results
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Factorization at Amplitude Level
I.R. Safe
The I.R. divergence will be absorbed into the
Matrix Elements
Numerical Plot of the finite part
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20 Diagrams contributing Double Logarithms
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Numerical Results
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Numerical Plots
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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.

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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
  confirmedThe leading twist can only host the
  single collinear logarithm, while those beginning
  with higher twist are always plagued with double
  logarithms.

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Thank You !
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Apart Function
3 Propagators
General Cases
If we set
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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).