Double charmonium productionin exclusive processes.

V.V. BragutaInstitute for High Energy Physics

Protvino, Russia

Content:Content:

Introduction Charmonia Distribution Amplitudes Properties of Charmonia Distribution

Amplitudes Double Charmonia Production at B-

factories Conclusion

Introduction

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

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

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

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

Evolution of DAEvolution of DA DA can be parameterized through the coefficients of conformal expansion an :

Alternative parameterization through the moments:

Charmonia 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~),(

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

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

)(

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

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

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

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

Application: Double charmonium production at B-factories

The processes:

'' ,' ,' / , / cccc JJee

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

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

Thank You.Thank You.