Exclusive charmonium productionwithin
Light Cone Formalism.
V.V. BragutaInstitute for High Energy Physics
Protvino, Russia
Outline:Outline:
Introduction Charmonium Distribution Amplitudes
(DA) Exclusive charmonium production within light cone formalism:
Conclusion
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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
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
),( ),( ),(
E~ ),,( ),( )(
1
0
ch
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
The models of leading twist The models of leading twist DAsDAs
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
V.V. Braguta, A.K. Likhoded, A.V. Luchinsky, Phys.Lett.B646:80-90,2007
V.V. Braguta, Phys.Rev.D75:094016,2007
V.V. Braguta, arXiv:0709.3885 [hep-ph]
The model of DAs within The model of DAs within NRQCDNRQCD
22 v ,1
VELOCITY RELATIVE IN IONAPPROXIMAT ORDERLEADING
n
nn
At leading order approximation is the only parameter
|)|-( 1
)(
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
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
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~~ resumation the toduety Uncertain2.
30%-10%~ :parameters of variation the toduety Uncertain1.
:model theofy Uncertaint
2s
M
sLog
Problem:The scale dependences of some twist-3 DAs are are
unknownunknown
First modification of BC formulaFirst modification of BC formulaPropagators:Propagators:
s
MMs PV
PV
22PV
p2pion approximat LOAt
)(p2p BCpaper In
c2V
2c
2c
m Mk m2s
if
k m2s
if
sy
syy
0 if , :asymptotic Unphysical
sm
~ ),)1(
( ),)(( :paper
~T
:amplitude in the divergence , :coneLight
22
2c22
20
22
x,kq
xxysk
xy
yxsqBC
xyxy
dy dx
sykxysq
Second modification of BC formulaSecond modification of BC formula
2
V
v~ where,M
2 BCpaper In
)(0 CC ),(
:currentTensor
cT
T
mf
ppifpV
Problems:Problems:1. Violation of velocity scaling rules at larger scales2. v2 correction can be large for 1S and 2S states
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 ),(/
(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(
22'22
2/
2'2/
2
22'22
AA
JTJT
LL
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 predictions are much smaller than the experimental results?
a E. Braaten, J. Leeb K.Y. Liu, Z.G. He, K.T. Chao
1. Relativistic corrections K~2.5-62. Leading logarithmic radiative corrections K~1.5-2.5
ConclusionConclusion
The processes considered in the report:
Within the error of the calculation the results are in agreement with the experiments
In hard exclusive processes (e+e- annihilation, bottomonium decays) relativistic and leading logarithmic radiative corrections are very important
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