Download Recent Result at Belle

Refactorizing NRQCD
short-distance coefficients
in exclusive quarkonium production
Graduate University of the CAS
Deshan Yang
Based on Yu Jia & DSY, NPB814, 217 (2009)
2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics
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Outline
1. Introduction & Motivations
2. Refactorization of NRQCD coefficients
3. Example 1 – e  e    b
4. Example 2 – h  Υ
5. Summary
2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics
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Charmonium production at B factories
 Exlusive quarknium production at B-factories
Belle :  [e  e   J / c ]  Bc [ 2]  (25.6  2.8  3.4) fb
BaBar :  [e  e   J / c ]  Bc [ 2]  (17.6  2.8 12..51 ) fb
Abe et al (Belle Collaboration), PRL 2002;
Aubert et al (BaBar Collaboration), PRD 2005;
 Suprises for theorists
 

[
e
e  J /  c ]  (2.3  5.5) fb
LONRQCD:
Braaten and Lee, PRD 2003; PRD 2005;
Liu, He and Chao, PLB 2003;
Hagiwara, Kou and Qiao, PLB 2003;
2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics
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Battlefield for NRQCD guys
 NLO QCD Corrections: K~ 2
Zhang, Gao, Chao, PRL 2006;
Gong, Wang, PRD 2008;
 Relativistic Corrections: K~ 2
Bodwin, Kang, Kim, Lee, Yu, AIP Conf. Proceeding 2007;
He, Fan, Chao, PRD 2007;
 Triumph? Confusions? Questions?
 Perturbation stability
 Non-relativistic vs. Relativistic
2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics
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Salute to those guys
Gong, QWG2008
Non-pert. quantities!
Kill renormalization scale dependence
K ( s / m )   NLO /  LO
2
c
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NRQCD factorization
For single quarkonium production

: NRQCD operator with definite velocity power counting
 multi-scale problem: Q>>m
 stability of the perturbation: large log(Q/m)
may need the resummation.
2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics
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Light-cone framework
Ma and Si, PRD 2004;Bonder and Chernyak, PLB 2005;Braguta, arXiv:0811.2640[hep-ph]
From Ma and Si, PRD 2004;
LCDA: Non-pert. quantities!
Lead to end-point singularity!
Fit to the expr.!
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Motivations
1
2
3
Key
Large logs in NRQCD short-distance
coefficients could ruin the perturbation!
How to harmonize LC factorization and
NRQCD factorization?
Understand the mechanisms to heavy
quarkonium exclusive production!
s  mc  mc v  mc v
2
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Inclusive quarkonium production at hadron collider
J.W.Qiu’s talk at this program
Absent from electron-positron collider!
Less complications!
Perturbative region could be furthur
fined into two regions!
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Exclusive single quarkonium production
2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics
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Refactorization
At the leading power of velocity,
 The hard kernel is the same as the similar process in which the
quarkonium is replaced by a flavor singlet light meson.
 Since
, the LCDA of bounded heavy quark and anti-quark
can be calculated perturbatively.
Ma and Si, PRD 2006; Bell and Feldman 2007;
2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics
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Example-1:
e e   b
 
Amplitude up to leading power of velocity:
The short-distance contribution is parameterized as
The equivalent computation is to calculate the on-shell heavy
quark anti-quark pair with equal momentum and the same
quantum number as the quarkonium. At the tree level,
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One-loop level
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Leading regions
 Hard Region:
 Collinear region:
 Anti-collinear region:
 Potential region:
 Soft region:
 Ultra-soft region:
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Form factor
NRQCD:
Collinear factorization:
 Hard-kernel:
at tree level
 Light-cone distribution amplitude
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RGE for LCDA
Brodsky-Lepage kernel:
Resum the leading logrithms
where
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Resummation in moment space
Both LCDA and Brodsky-Lepage kernel can be expanded in
Gegenbauer polynomial
In moment space, the LL resummed form-factor is
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Truncated LL approximation
In momentum space, solve the RGE recursively, at two-loop
level, we have
which is equivalent to the expressions in moment space
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Comparisons
LL
2-loop LL
1-loop LL
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Example-2:
h  
Amplitude up to the lowest order of velocity:
The short-distance contribution is parameterized as
The factorization formula for the form-factor involves the twist-2
LCDA for transversely polarized quark pair
which corresponding B-L kernel is
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Comparisons
1-loop LL
2-loop LL
LL
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Summary

For the single heavy quarkonium exclussive production at very large
center energy collision, it is possible to derive the refactorization
formula for the relevant NRQCD short-distance coefficients;

In the two examples we showed, the LL resummation does not lead to
large corrections in the accessible ernergy;

However, the refactorization property of the NRQCD short-distance
coefficients may offer an easier way to obtain the higher order
radioactive corrections with price of m/Q suppressed corrections;

For the double charmonium production, double logarithm appears with
the overall m/Q suppression at one-loop level. However we do not know
how to resum such logarithm with the aid of B-L equations.
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2009.9.10 KITPC-EFT in Nuclear Physics
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