Exclusive charmonium productionin hard exclusive processes.

V.V. BragutaInstitute for High Energy Physics

Protvino, Russia

Content:Content:

Introduction Charmonium Distribution Amplitudes

(DA) The properties of DAs Exclusive charmonium production within light cone formalism:

Perspectives

'' ,' ,' / , / cccc JJee

'' ,'/ ,/ / JJJee

Introduction

Internal structure of charmoniumInternal structure of charmonium

First approximation

Charmonium = nonrelativistic quark-antiquark bound state

Charmonium spectrum

Next approximation: NRQCD

Charmonium = nonrelativistic systemv2~0.3, v~0.5

Production as a probe of the internal structure of charmonium

NRQCD formalismNRQCD formalism

QMq , 1 v

:system isticnonrelativFor

PartSoft

PartHard

qaqqaqq

33 )0(H(0))( dH(0))()H( dT

process theof amplitude The

)0(~0| |~0|)0(| :ionApproximat LO

0|)0(| C T

:formulaion factorizat NRQCD

n

.

n

MOM

OM

LO

PartSoft

n

DistShort

At LO of NRQCD quark-antiquark pair has zero relative momentum

Light Cone Formalism

The amplitude is divided into two parts: Hadronization

Twist-2 2-distribution amplitudesTwist-3 4-distribution amplitudes … …

Light cone formalism is designed to study hard exclusive processes

Comparison of LCF and NRQCDThe cross section is double series

1.0~ GeV 6.10s

~

2

2

sM

at

sM

parameterExpansion

LCFPower corrections:

NRQCDRelativisitic corrections:

Radiative corrections:

sonsS state mefor

sonsS state me for

parameterExpansion

2 50.0~v

1 25.0~v

:

2

2

2.0~)( s ss:correctionRadiative

5.0~)( : 2s

Ms

LogsscorrectionRadiativecLogarithmiLeading

Relativistic corrections

21

1

1

xx

),( )H(

dT

LCF NRQCD

DA resums relativististic corrections to the amplitude.

nnCT v

Leading logarithmic radiative corrections

),( ),( ),(

s~ ),,( ),( )(

1

0

1

0

yDz/yPy

dyzD

zDpddzpd

Mji

jiMi

Mii

iM

Exclusive quarkonium production

Inclusive quarkonium production

DA resums leading logarithmic radiative corrections.

),( ),,V( ),(

s~ ),,( ),H(

1

1

1

1

d

dT

Distribution Amplitudes are the key ingredient

of Light Cone Formalism

Charmonium Distribution Amplitudes

Definitions of leading twist DADefinitions of leading twist DA

even- are ),(),,(),,( DAs TLP

Evolution of DAEvolution of DA DA can be parameterized through the coefficients of conformal expansion an :

Alternative parameterization through the moments:

DA of nonrelativistic systemDA of nonrelativistic system

icrelativist is

1)~( IIRegion point end in themotion The 3.

isticnonerlativ is )v~( IRegion in motion The .2

v~ isDA of width The .1

:Properties

2

22

22

)()()()(

velocityrelativein ion approximatorder leadingAt

TLP

Different approaches to the study of Different approaches to the study of DADA

1. Functional approach

- Bethe-Salpeter equation

2. Operator approach

- NRQCD

- QCD sum rules

Potential modelsPotential models

Solve Schrodinger equation Get wave function in momentum space:

Make the substitution in the wave function:

Integrate over transverse momentum:

2(k )

22

20 cz 1 2 0

1 2

M M kk k , k (x x ) , M

2 x x

Brodsky-Huang-Lepage procedure:

c2 M~ ,)k,( kd~),(

2

Different approaches

(2006) 114028 D74, Rev. Phys. , , ,

)k,( 1

Mk kd~),(

2

2

222

LeeKangBodwin

c

procedure --

)k,( kd~),(

2

2

LepageHuangBrodsky

(2006) 054008 D74, Rev. Phys. , ,

)k,( 1

Mk kd~),(

2

42

222

MartynenkoEbert

c

)O(v 1~)k,( ,)k,( )k,( kd~),(

:as written becan approaches All

22

2

ff

The moments within Potential The moments within Potential ModelsModels

The larger the moment, the larger the contribution of relativistic motion

Only few moment can be calculated

Higher moments contain information about relativistic motion in quarkonium

1

1

n ),( dn

Is there relativistic motion in quarkonium?

Relativistic motion can appear only due to the rescatering for a short period of time (v<<1)

Relativistic motion existsv2~0.3 is still large

Property of DAProperty of DA

)( )1(~)( 2

2

2c

2

22

2c'

c

-1M

)-(1

M 2)(

M~

DA of nS state has 2n+1 extremums

2

2c

22c2

22c

22c

222

'

1

0

2c2

2

-1

M M

-1

M )M(

)-(1 2

)(

)M1

( d)(

2

2

tt

The moments within The moments within NRQCDNRQCD

pD i

ionapproximatorder leadingAt

)p( WFhaspair cc Suppose

1v

formula theof Derivation

2

n

n

n

n22n

2n3

v1n

1~)(p )cos( cos

~)(p )( ~

:in vion approximatorder leadingAt

n

zn

pdpd

ppd

The moments within The moments within NRQCDNRQCDThe values of <vn> were calculated in paper

G. Bodwin, Phys.Rev.D74:014014,206

n

n

*

c

bcn

m M

v

42.065.0v states 2SFor

08.025.0v states 1SFor 2

2

The constant can be expressed through the <v2>

The model for DA within The model for DA within NRQCDNRQCD

22 v ,1

VELOCITY RELATIVE IN IONAPPROXIMAT ORDERLEADING

n

nn

At leading order approximation is the only parameter

|)|-( 1

)(

The moments of DA within QCD sum The moments of DA within QCD sum rulesrules

Advantage:The results are free from the uncertainty due to the relativistic corrections

Disadvantage:The results are sensitive to the uncertainties in the sum rules parameters:

QCD sum rules is the most accurate approach

The details of the calculationThe details of the calculation

Logitudinally polarized Logitudinally polarized 33SS1 1

mesonsmesons Sum rulesSum rules

Improved Sum rules:Improved Sum rules: Sum rules parameters:Sum rules parameters:

Numerical analysisNumerical analysis

0 GeV 3.7s '2

0 L 18.0 GeV 3.7s '2

0 L

The other DAsThe other DAsmesons 0

1S

mesons polarizedly Transverse 13S

These sum rules are less accurate

The results of the The results of the calculationcalculationThe results for 1S states

The results for 2S states

The models of DAsThe models of DAs

25.0~1

~v velocity sticcharacteri

,7.08.3

-1-Exp )1(~)~,(

2

22

cm

4.0~1

~v velocity sticcharacteri

,5.2 ,03.0

-1-Exp )( )1(~)~,(

2

2.30.8-

32.003.0

222

cm

1S states 2S states

Borel version sum rules without vacuum condensates

The properties of distribution amplitudes

Relativistic tailRelativistic tail

75.0||

n

2/32 )( )(a )1(~ ),( nn G

cm~

cm

At DA is suppressed in the region

Fine tuning is broken at due to evolution

This suppression can be achieved if there is fine tuning of an

Relativistic tail within Relativistic tail within NRQCDNRQCD

Light Cone

... ...

||0

||0

||0 1

44

22

MD

MD

M

-

-

E~ , ...4!

(0)H2!(0)'H'

H(0) ),( )H( 4IV

2 dM

),( )( ~ ELogs

The amplitude of meson production

NRQCD

Leading logarithmic corrections

are resummed in DA

Leading logarithmic corrections are contained in Wilson coefficients

QCD radiative corrections enhance the role of higher NRQCD operators

The violation of NRQCD scaling The violation of NRQCD scaling rulesrules

At larger scales the fine tuning ofthe coefficients an is broken andNRQCD scaling rules are violated

NRQCD velocity scaling rules are violated in hard processes

Improvement of the model for Improvement of the model for DADA The evolution of the second moment

3512 )(a

51

22

The accuracy of the model for DA is better at larger scales

decreases in error The

increases as decreases )(a tscoefficien The n

n

19.0 18.0

state 2

005.0123.0 007.0070.0

state 1

3.04.0GeV 10

25.07.0~

2

GeV 102

~2

c

c

m

m

S

S

Models for 2S statesModels for 2S states

p21

~p |~| momentum relative

061.0

21.0

5.2 0

4

2

0~ momentum relative

031.0

12.0

5.2 2.0

4

2

At leading order approximation of NRQCD the relative momentum of quark-antiquark pair is zero

Exclusive charmonium production

within light cone formalism

The processes:

'' ,'/ ,/ / JJJee

The diagramsThe diagrams

)()()( int sss nonfrfr

Fragmentation diagrams

Nonfragmentation diagrams

The cross section at NLOThe cross section at NLO

...

issection cross theFormalism ConeLight Within

110 nn ss

Relativistic and leading logarithmic radiative Relativistic and leading logarithmic radiative correctionscorrections

3

int1,11,00,11,1

1

is NLOat section cross The

sOfrfrfr

Interference of fragmentation and nonfragmentation Interference of fragmentation and nonfragmentation diagramsdiagrams

The role of correctionsThe role of corrections

The results of the calculationThe results of the calculation

CL) % (90 fb 2.5)'()' /(

CL) % (90 fb 1.9)/()/ /(

:(Belle) results alExperiment

2

2

BrJee

JBrJJee

coson xdistributiAngular

a Bodwin, Braaten, Lee, Phys. Rev. D74

The processes:

'' ,' ,' / , / cccc JJee

e+e- V(3S1) P(1S0)

This formula was first derived in Bondar, Chernyak, Phys. Lett. B612, 215 (2005)

Twist-3 distribution amplitudesTwist-3 distribution amplitudes

)(),( 2.

)O(v)m~,()m~,( .1

:known isWhat

3

2c2c3

asymptotictwist

twisttwist

),()(),(

:amplitudeson distributi of model The

2 twistasymptotic

%50~~ of resumation the toduety Uncertain2.

30%-10%~ :parameters of variation the toduety Uncertain1.

:model theofy Uncertaint

2s

Ms

Log

Problem:The scale dependences of some twist-3 DAs are are

unknownunknown

The constants needed in the The constants needed in the calculationcalculation

pifpif

ppifpppifpJ

MfpMfpJ

AcAc

TT

LJL

'5

'5

'

''

/

0 CC 0 CC

)(0 CC ),(' )(0 CC ),(/

0 CC ),(' 0 CC ),(/

70%)(~ GeV )075.0(

(82%) GeV )038.0047.0( (30%) GeV )039.0120.0(

(50%) GeV )038.0076.0()( (24%) GeV )042.0173.0()(

(2.5%) GeV )002.0092.0( %) (2.5 GeV )004.0173.0(

2053.0040.0

2'

22'22

2/

2'2/

2

22'22

A

AA

JTJT

LL

f

ff

MfMf

ff

The values of the constantsThe values of the constants(preliminary results)

The results of the calculationThe results of the calculation

Why LO NRQCD is much smaller than the experimental results?

1. Relativistic corrections K~2.5-62. Leading logarithmic radiative corrections K~1.5-2.5

a E. Braaten, J. Leeb K.Y. Liu, Z.G. He, K.T. Chao

Relativistic and radiative Relativistic and radiative correctionscorrections

NRQCD formalism

D75 Rev. Phys. al.,et He 7.1K

ph/0611002-hep al.,et Bodwin 1.2K

scorrection icRelativist

One loop radiative corrections

K 1.96 Zhang et al., Phys. Rev. Lett. 96

D75 Rev. Phys. al.,et He fb 20 )/(

ph/061102-hep al.,et Bodwin fb 7.55.17)/(

c

c

Jee

Jee

Light cone formalism

Relativistic corrections

K 1.8-2.1

Leading logarithmic radiative corrections

K 1.9 2.1

fb 25)/( cJee

The amplitude was derived in paper Bondar, Chernyak, Phys.Lett. B612

Perspectives:Perspectives:

Theoretical LCF can resolve the other problems with exclusive

processes? Development of alternative to NRQCD DAs of P-wave and D-wave charmonium mesons New results for bottomonium decays Inclusive charmonium production

Experimental LHC (bottomonium decays) B-factories and Super B-factories (exclusive

production of mesons and baryons)