Study of charmonium distribution amplitudes.
description
Transcript of Study of charmonium distribution amplitudes.
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.