![Page 1: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/1.jpg)
Double charmonium productionin exclusive processes.
V.V. BragutaInstitute for High Energy Physics
Protvino, Russia
![Page 2: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/2.jpg)
Content:Content:
Introduction Charmonia Distribution Amplitudes Properties of Charmonia Distribution
Amplitudes Double Charmonia Production at B-
factories Conclusion
![Page 3: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/3.jpg)
Introduction
![Page 4: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/4.jpg)
Processes to be discussed
...J/ ,J/ :factories-Bat production Charmonia 2.
...J/ J/ ,J/ , :decays Bottomonia .1
:processes Exclusive
c0
0221b
c
bccc
ee
Υ
10
1~~ : parameter Expansion
E~GeV 3~ GeV 10~s ,~E
:featureCommon
2
2
2
softhard
s
M
M
M
MM
cc
bb
cc
ccbb
![Page 5: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/5.jpg)
FactorizationFactorization
effects) bative(nonpertur 0|| :oncontributi distance Large 2.
QCD) ive(perturbat C :oncontributi distanceShort 1.
n
:onsContributiDifferent
nOM
0|)0(| C T
:formulaion Factorizat
n
.
n
PartSoft
n
DistShort
OM
![Page 6: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/6.jpg)
Nonperturbative effectsNonperturbative effects
q...DD q q,D q q, q
...; qDDD q q,DD q q,D q q, q
...; qDDD q q,DD q q,D q q, q
0
211
321211
321211
555
5555
production tocontribute that Operators
GGG
On
... :section cross The 22
110 nnn sss
At given order approximation in 1/s expansion some operators can be omitted
![Page 7: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/7.jpg)
The leading twist distribution The leading twist distribution amplitudeamplitude
Properties of distribution amplitudes
Resume infinite series of operators Resume leading logarithmic radiative corrections
function. meson wave a as considered becan )( amplitudeon Distributi
,0z ),( (pz) ~z...zz 0|q...DD q|M(p)
:ionapproximatorder leading at the contribute that Operators
212
1
1
n1n5
1
n1
xxdn
hard
1
1
E~ ), ,( ),H(
dT
![Page 8: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/8.jpg)
Evolution of DAEvolution of DA DA can be parameterized through the coefficients of conformal expansion an :
Alternative parameterization through the moments:
![Page 9: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/9.jpg)
Charmonia Distribution Amplitudes
![Page 10: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/10.jpg)
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
![Page 11: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/11.jpg)
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~),(
![Page 12: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/12.jpg)
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
![Page 13: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/13.jpg)
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
)(
![Page 14: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/14.jpg)
DA within QCD sum rulesDA within QCD sum rules
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
![Page 15: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/15.jpg)
The results of the The results of the calculationcalculationThe results for 1S states
The results for 2S states
![Page 16: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/16.jpg)
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
![Page 17: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/17.jpg)
Properties of distribution amplitudes
![Page 18: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/18.jpg)
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
![Page 19: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/19.jpg)
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
![Page 20: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/20.jpg)
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
![Page 21: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/21.jpg)
Pion distribution amplitudePion distribution amplitude
Distinction from pion distribution amplitude
Much better knowledge of DAs (even for higher twist and excited states) Improvement of the accuracy of models
Rather accurate predictions for exclusive charmonia production
![Page 22: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/22.jpg)
Application: Double charmonium production at B-factories
![Page 23: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/23.jpg)
The processes:
'' ,' ,' / , / cccc JJee
![Page 24: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/24.jpg)
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
![Page 25: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/25.jpg)
ConclusionConclusion
One can get rather good knowledge of charmonia distribution amplitudes
One can get rather good description of hard exclusive processes
In hard exclusive processes (e+e- annihilation, bottomonium decays) relativistic and leading logarithmic radiative corrections are very important
![Page 26: Double charmonium production in exclusive processes](https://reader036.vdocument.in/reader036/viewer/2022062500/56815219550346895dc05ab0/html5/thumbnails/26.jpg)
Thank You.Thank You.