Applications of the light-cone distribution amplitudes for quakonia

Applications of the light-cone distribution amplitudes for

quakonia

Applications of the light-cone distribution amplitudes for

quakonia

2010.4.18 南昌大学第八届全国高能物理大会 11

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

Graduate University of the CASDeshan Yang

OutlineOutline


1. Introduction & Motivation

2. Example 1 –

3. Example 2 –

4. Summary

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Exclusive QCD ProcessesExclusive QCD Processes


History: in 1979-81, Brodsky et al

when , the form-factor can be factorized

Where the light-cone distribution amplitude (LCDA) can be defined as

The same LCDA can appear in the factorization formulae in other exclusive processes, such as pion electro-magnetic form-factor and B exclusive hadronic

decays.

0 2

. .( ) [ ( ) (0)] 0 ( )v

em em F Tp T j x j iF Q p q

12 2

0( ) ( , ) ( , ; )F FF Q dx x T x Q

2 2 2QCDQ q

105 0

( ) ( ) [ ,0] (0)] 0 ( , )iup xFp q x x q if p due u

* 0( ) ( ) ( )q p p

The LCDA for quarkoniumThe LCDA for quarkonium


110 5 0

[ ( )]( ) ( ) [ ,0] (0)] 0 ( , )iup xH H FH QQ S p Q x x Q if p due u

Definition: 1981, Chernyak & Zhintnisky

Could be used in procecesses Different from the LCDAs of light mesons, it can be calculated perturbatively. The RGE running is governed by the Brodsky-Lepage equation

* ,H B HM

Charmonium production at B factories

Charmonium production at B factories


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


Salute to those guysSalute to those guys


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.


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;


Fit to the expr.!

LCDA: Non-pert. quantities!

Lead to end-point singularity!

MotivationsMotivations


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

Exclusive single quarkonium productionExclusive single quarkonium production


RefactorizationRefactorization


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 Feldmann, JHEP 2007;

Example-1:Example-1:


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NRQCD factorization 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


Sang, Chen, arXiv:0910.4071; Li, He, Chao arXiv:0910.4155

Leading regionsLeading regions


Hard Region:

Collinear region:

Anti-collinear region:

Potential region:

Soft region:

Ultra-soft region:

2( , ) ~ (1, , ),n k k n k s ~ / ,Qm s 2 2~ Qk m

2( , ) ~ ( , ,1),n k k n k s

~ ,k s 2 ~k s

2 2~ Qk m

NRQCD regionsNon-perturbative

Form factorForm factor


NRQCD:

Collinear factorization:

Hard-kernel:

at tree level

Light-cone distribution amplitude


RGE for LCDARGE for LCDA


Brodsky-Lepage kernel:

Resum the leading logrithms

where

Resummation in moment spaceResummation in moment space


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


In momentum space, solve the RGE recursively, at two-looplevel, we have (neglecting mixing)

which is consistent with the NRQCD calculation

Actually, we even got the correct NLO results by summing upthe hard part and collinear part. (see below).

QQ gg

Sang, Chen, arXiv:0910.4071; Li, He, Chao arXiv:0910.415524 /r m s

Braaten, PRD 1981;Ma and Si, PRD 2006; Bell and Feldmann, JHEP 2007;

ComparisonsComparisons


LL2-loop LL

1-loop LL

NLO results (preliminary)NLO results (preliminary)


Braaten, PRD 1981;


Hard Part

Collinear Part

Total Results

2(1) 2 ln

( ) (3 2ln ) ln ln 9 ( 1 )4 1S FC x x

T x x x x xx s i x

Sang, Chen, arXiv:0910.4071; Li, He, Chao arXiv:0910.4155

2(1) (0) (1) 2 2(1/ 2) ((9 6ln 2) ln 9ln 2 3ln 2 27 )

mT T

s

Example-2:Example-2:


NRQCD factorization 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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ComparisonsComparisons


1-loop LL

2-loop LL

LL

SummarySummary


The LCDA can be used to refactoriz NRQCD short-distance coefficients in the single heavy quarkonium exclusive production at very large center energy collision;

The RG running of LCDA can be used to re-sum the large single logarithm;

In the two examples we showed, the LL resummation does not lead to large corrections in the accessible energy;

However, the refactorization property of the NRQCD short-distance coefficients may offer an easier way to obtain the higher order radiative corrections with loss of m/Q suppressed corrections;

252010.4.18 南昌大学第八届全国高能物

理大会

Applications of the light-cone distribution amplitudes for quakonia

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