Study of charmonium distribution amplitudes.

V.V. BragutaInstitute for High Energy Physics

Protvino, Russia

Content:

Introduction Study of 1S and 2S charmonium

distribution amplitudes ( Potential models, NRQCD, QCD sum rules approaches)

Properties of distribution amplitudes Application: double charmonium production

at B-factories

The results were obtained in papers: “The study of leading twist light cone wave functions of eta_c

meson” V.V. Braguta, A.K. Likhoded, A.V. Luchinsky

Phys.Lett.B646:80-90,2007

“The study of leading twist light cone wave functions of J/Psi

meson” V.V. Braguta

Phys.Rev.D75:094016,2007

“The study of leading twist light cone wave functions of 2S state charmonium mesons”

V.V. Braguta, to be published in Phys. Rev. D

Introduction

Light cone formalism

... :section cross The 22

110 nnn sss

The amplitude is divided into two parts:

Hadronization can be describedby distribution amplitudes

'

'

Distribution amplitudes to be studied:

- J/ , mesons leading twist distribution amplitudes

- , mesons leading twist distribution amplitudescc

Definitions of leading twist DADefinitions of leading twist DA

1 2 =

DAs ( , ), ( , ), ( , ) are -evenP L T

x x

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

Study of charmonium distribution amplitudes

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:

2

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

Property of DAProperty of DA

)M1

( d)(

)( )1(~),(

2c2

2

2

tt

)(in extremums two

)k(in zero one 2

DA of nS state has 2n+1 extremums

Model for DA within NRQCDModel for DA within NRQCD

22 v ,1

VELOCITY RELATIVE IN IONAPPROXIMAT ORDERLEADING

n

nn

At leading order approximation is the only parameter

|)|-( 1

)(

Model for DA within QCD sum Model for 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 QCD sum rules parameters:

02 S , , Gmc

QCD sum rules is the most accurate approach

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

The results for 2S states

Models of DAsModels of DAs

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

25.0~1

~v velocity sticcharacteri

,7.08.3

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

2

22

cm

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

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 becomes better at larger scales

increases as decreases in error The

increases as decreases )(a tscoefficien The

2

2

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

Application: Double charmonium production at B-factories

Results of the calculationResults 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

Thank You.Thank You.