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Exclusive charmonium productionin hard exclusive processes.
V.V. BragutaInstitute for High Energy Physics
Protvino, Russia
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Content:Content: Introduction Charmonium Distribution Amplitudes
(DA) The properties of DAs Exclusive charmonium production within light cone formalism:
Perspectives
'' ,' ,' / , / cccc JJee
'' ,'/ ,/ / JJJee
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Introduction
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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
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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
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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
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Comparison of LCF and NRQCDThe cross section is double series
1.0~ GeV 6.10s
~
2
2
sMat
sMparameterExpansion
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
MsLogsscorrectionRadiativecLogarithmiLeading
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Relativistic corrections
21
1
1
xx
),( )H(
dT
LCF NRQCD
DA resums relativististic corrections to the amplitude.
nnCT v
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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
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Distribution Amplitudes are the key ingredient
of Light Cone Formalism
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Charmonium Distribution Amplitudes
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Definitions of leading twist DADefinitions of leading twist DA
even- are ),(),,(),,( DAs TLP
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Evolution of DAEvolution of DA DA can be parameterized through the coefficients of conformal expansion an :
Alternative parameterization through the moments:
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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
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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
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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 )
2220 c
z 1 2 01 2
M M kk k , k (x x ) , M2 x x
Brodsky-Huang-Lepage procedure:
c2 M~ ,)k,( kd~),(
2
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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
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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
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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
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Property of DAProperty of DA)( )1(~)( 2
2
2c
2
22
2c'
c
-1M
)-(1M 2)(
M~
DA of nS state has 2n+1 extremums
2
2c
22c2
22c
22c
222
'
1
0
2c2
2
-1M M
-1M )M(
)-(1 2)(
)M1
( d)(2
2
tt
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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
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The moments within The moments within NRQCDNRQCDThe values of <vn> were calculated in paperG. Bodwin, Phys.Rev.D74:014014,206
nn
*
c
bcn
m Mv
42.065.0v states 2SFor
08.025.0v states 1SFor 2
2
The constant can be expressed through the <v2>
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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)(
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The moments of DA within QCD sum The moments of DA within QCD sum rulesrulesAdvantage:The results are free from the uncertainty due to the relativistic correctionsDisadvantage:The results are sensitive to the uncertainties in the sum rules parameters:
QCD sum rules is the most accurate approach
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The details of the calculationThe details of the calculation
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Logitudinally polarized Logitudinally polarized 33SS1 1
mesonsmesons Sum rulesSum rules
Improved Sum rules:Improved Sum rules: Sum rules parameters:Sum rules parameters:
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Numerical analysisNumerical analysis
0 GeV 3.7s '2
0 L 18.0 GeV 3.7s '2
0 L
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The other DAsThe other DAsmesons 0
1S
mesons polarizedly Transverse 13S
These sum rules are less accurate
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The results of the The results of the calculationcalculationThe results for 1S states
The results for 2S states
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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
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The properties of distribution amplitudes
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Relativistic tailRelativistic tail75.0||
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
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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( 4
IV2 dM
),( )( ~ ELogs
The amplitude of meson production
NRQCDLeading logarithmic correctionsare resummed in DA
Leading logarithmic corrections are contained in Wilson coefficients
QCD radiative corrections enhance the role of higher NRQCD operators
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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
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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
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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
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Exclusive charmonium production
within light cone formalism
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The processes:'' ,'/ ,/ / JJJee
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The diagramsThe diagrams
)()()( int sss nonfrfr
Fragmentation diagrams
Nonfragmentation diagrams
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The cross section at NLOThe cross section at NLO
...
issection cross theFormalism ConeLight Within
110 nn ss
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Relativistic and leading logarithmic radiative Relativistic and leading logarithmic radiative correctionscorrections
3
int1,11,00,11,1
1is NLOat section cross The
sOfrfrfr
Interference of fragmentation and nonfragmentation Interference of fragmentation and nonfragmentation diagramsdiagrams
The role of correctionsThe role of corrections
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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
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The processes:'' ,' ,' / , / cccc JJee
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e+e- V(3S1) P(1S0)
This formula was first derived in Bondar, Chernyak, Phys. Lett. B612, 215 (2005)
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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
MsLog
Problem:The scale dependences of some twist-3 DAs are are
unknownunknown
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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)
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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
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Relativistic and radiative Relativistic and radiative correctionscorrections NRQCD formalism
D75 Rev. Phys. al.,et He 7.1Kph/0611002-hep al.,et Bodwin 1.2K
scorrection icRelativist
One loop radiative correctionsK 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 formalismRelativistic correctionsK 1.8-2.1 Leading logarithmic radiative correctionsK 1.9 2.1
fb 25)/( cJee
The amplitude was derived in paper Bondar, Chernyak, Phys.Lett. B612
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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)