Refactorizing NR Q C D short-distance coefficients in exclusive quarkonium production

Refactorizing NRQCD short-distance coefficients

in exclusive quarkonium production

Refactorizing NRQCD short-distance coefficients

in exclusive quarkonium production

2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics

11

Based on Yu Jia & DSY, NPB814, 217 (2009)

Graduate University of the CASDeshan Yang

OutlineOutline


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1. Introduction & Motivations

2. Refactorization of NRQCD coefficients

3. Example 1 –

4. Example 2 –

5. Summary

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Υh

Charmonium production at B factories

Charmonium production at B factories


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Exlusive quarknium production at B-factories

Abe et al (Belle Collaboration), PRL 2002; Aubert et al (BaBar Collaboration), PRD 2005;

Suprises for theorists LONRQCD:

Braaten and Lee, PRD 2003; PRD 2005;Liu, He and Chao, PLB 2003;

Hagiwara, Kou and Qiao, PLB 2003;

fbBJee

fbBJeec

c

c

c

)8.26.17(]2[]/[ :BaBar

)4.38.26.25(]2[]/[ : Belle5.11.2

fbJee c )5.53.2(]/[

Battlefield for NRQCD guys

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


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Salute to those guysSalute to those guys


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Non-pert. quantities!

Kill renormalization scale dependence

LONLOcmsK /)/( 2

Gong, QWG2008

NRQCD factorizationNRQCD factorizationFor 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.


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Light-cone frameworkLight-cone framework Ma and Si, PRD 2004;Bonder and Chernyak, PLB 2005;Braguta, arXiv:0811.2640[hep-ph]

From Ma and Si, PRD 2004;


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Fit to the expr.!

LCDA: Non-pert. quantities!

Lead to end-point singularity!

MotivationsMotivations


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Large logs in NRQCD short-distance coefficients could ruin the perturbation!

Large logs in NRQCD short-distance coefficients could ruin the perturbation!

1

How to harmonize LC factorization and NRQCD factorization?

How to harmonize LC factorization and NRQCD factorization?

2

Understand the mechanisms to heavy quarkonium exclusive production!

Understand the mechanisms to heavy quarkonium exclusive production!

3

Key 2vmvmms ccc

Inclusive quarkonium production at hadron collider

Inclusive quarkonium production at hadron collider


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J.W.Qiu’s talk at this program

Absent from electron-positron collider!Less complications!

Perturbative region could be furthurfined into two regions!

Exclusive single quarkonium productionExclusive single quarkonium production


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RefactorizationRefactorization


1111

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;

Example-1:Example-1:


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Amplitude up to leading power of velocity:

The short-distance contribution is parameterized as

The equivalent computation is to calculate the on-shell heavyquark anti-quark pair with equal momentum and the samequantum number as the quarkonium. At the tree level,

One-loop levelOne-loop level


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Leading regionsLeading regions


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Hard Region:

Collinear region:

Anti-collinear region:

Potential region:

Soft region:

Ultra-soft region:

Form factorForm factor


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NRQCD:

Collinear factorization:

Hard-kernel:

at tree level

Light-cone distribution amplitude

RGE for LCDARGE for LCDA


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Brodsky-Lepage kernel:

Resum the leading logrithms

where

Resummation in moment spaceResummation in moment space


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Both LCDA and Brodsky-Lepage kernel can be expanded in Gegenbauer polynomial

In moment space, the LL resummed form-factor is

Truncated LL approximationTruncated LL approximation


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In momentum space, solve the RGE recursively, at two-looplevel, we have

which is equivalent to the expressions in moment space

ComparisonsComparisons


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LL2-loop LL

1-loop LL

Example-2:Example-2:


2020

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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

ComparisonsComparisons


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1-loop LL

2-loop LL

LL

SummarySummary


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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.

232009.9.10 KITPC-EFT in Nuclear Physics

& Particle Physics

Refactorizing NR Q C D short-distance coefficients in exclusive quarkonium production

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