Saturation Model for exclusive processesat HERA, DVCS, F2,

including evolution and impact parameter dependence

Henri KowalskiDESY

DIS 2007

Munich, 17th of April 2007

)2

exp(12*21

0

2*

bddzrdptot

2*1

0

22 |)2

exp(12|16

1*

VM

bip

VM dzebdrddt

d

)2/exp(~)(

)exp(~

2 BbbT

tBdt

d diff

NNPZ, GLM, FKS, GBW, MMS DGKP, BGBK, IIM, FSS……KT - Kowalski, TeaneyKMW - Kowalski, Motyka, Watt

proton shape

)(),()( 2222

bTxxgrN s

C

Glauber Mueller

0)t,(WImAW

1σ 2

el2γptot

Dipole Models equivalent to LO perturbative QCD for small dipoles

),( zr

OpticalTheorem

KT

x < 10-2

universal rate of rise of all hadronic cross-sections

tottot xWp )/1(~)(~ 2*

Total *p cross-section

6.520

202

2 )1(1

),( xx

Axxgr

C g

g

KT

KMW

tot

b-CGC

b-Sat

Inclusive Diffraction

-LPS

F2 C

KT

Dipole Model - gluon density convoluted with dipole wave functions simultaneous prediction/description of many reactions

Inclusive Diffraction

BGBK

Diffractive Di-jets Q2 > 5 GeV2

-RapGap

Satrap

H. Kowalski, L. Motyka, G. Watt

Exclusive Vector Meson Production

Effective modification of Fourier Trans by Bartels, Golec-Biernat, Peters

)1( 2

Real part correction

Skewedness correction Martin, Ryskin Teubner

))2/exp(1(2 2

bd

d)(),()( 222

2

bTRxxgrN gs

C

Wave Functions

Boosted Gaussian – NNPZ, FKS, FS Gaussian distribution of quark 3-momentum in the meson rest frame

then boosted to LC

Gauss LC - KTGaussian distribution of quark 2-momentum in LC, factorization of r, z components- strong endpoint suppression in T

)2

exp()1(2

),(2

2

2LL

L R

rzz

R

Nzr

)

2exp()1(

2),(

2

222

2TT

T R

rzz

R

Nzr

Parameters of WF fixed by normalization conditions and the values of mesons decay constant, fV

WF Overlaps

from gluon density convoluted with dipole wave functions we obtain simultaneous prediction/description of many reactions

KMW

Vector Mesons

DVCS

Note: educated guesses for VM wf work surprisingly well

KMW

WVpp ~*

evolution of a saturated gluon density Rates of rise of the VM cross-sections

At EIC it should be possible to reduce the errors by a large factor, O(100), study of t-dependent rate of rise study of b-dependent Pomeron evolution – direct insight into saturation inside the proton or nuclei

KMW properties of wave functions Ration of longitudinal/transverse x-sections

)2/exp(~)( )exp(~ 2GD

diff

BbbTtBdt

d

Exponential fall of t-distributions

gaussian shape of the proton in the impact parameter b

)2/exp(~)( )exp(~ 2GD

diff

BbbTtBdt

d

KMW

BG BG

))1(( rzbibi ee

Description of the size of interaction region BD

Modification by Bartels, Golec-Biernat, Peters

fm 68.03 2 Gp BR

2G

p

GeV 48.6B

fm 008.0870.0R

the gluonic proton radius smaller than the quark radius

t-distributions of DVCS

universal rate of rise of all hadronic cross-sections

tottot xWp )/1(~)(~ 2*

Rate of rise of the *p cross-section In the impact parameter dipole models of DIS with the gaussian proton shape the fit of the rate of rise of *p requires QCD evolution which is DGLAP-like DGLAP-like strong interdependence between and Q2

What have we learnt from HERA about small-x

Large fraction of *p comes from the region of large b where matter density is low

median

40% )2/exp(~)( 2 GMEDIAN BbbT

Saturation scale (a measure of gluon density at which gluon re-scattering starts to be substantial)

HERA RHIC

22

22

22

GeV 3.1

fm 7 1000

1

1

4

S

C

CsS

Q

Rdy

dN

dy

dN

RN

NQ

qSqSF

CgS QQ

C

NQ )(

4

9)()( 222

QSRHIC (x=10-2)

~ QSHERA(x=10-4)

22 2

SS r

Q

))()/,(32

exp(12 ),( 2

022

2

2bTQrCxxgr

bd

rxds

qq

universal rate of rise of all hadronic cross-sections

Is saturated state observed at HERA perturbative?

properties gluon densityRandom walk in the impact parameter

Study of the Glue

t-distributions for exclusive diffractive meson production on proton and nuclei at EIC

first estimate of the expected measurement precision:

pT < 30 MeV, t ~ pT2

t < 0.01 GeV2

for proton and light nuclei

What are the lessons/open questions for EIC from exclusive diffraction at HERA

Precise study of gluon density, tested with dipoles of various sizes,measure gluon diffusion evolution effects

Diffractive vector mesons scattering - an excellent probe of nuclear matter, why is the gluonic radius smaller than the quark radius?? >>>>> Measure t distribution on (polarized) nuclei <<<<<<>>>>> Obtain holographic picture of nuclei !!!! <<<<<<

BGBK – DGLAP evolution & saturation

GBW - saturation only

More about saturation

with impact parameter without impact parameter

properties of the gluon density Rise of the DVCS cross-sections

))(),(

32exp(1

2

),( 222

2bATxxgr

Abd

rxdWSs

Aqq

Smooth Gluon Cloud

Q2 (GeV2) C 0.74 1.20 1.70 Ca 0.60 0.94 1.40

Hard Diffraction – the HERA surprise

Non-Diffractive Event

Diffractive Event expected before HERA <0.01%, seen over 10% at Q2=10 GeV2

Diffraction at HERA is so large because it is a shadow of DIS (i.e. inelastic processes) dipole picture

fm 10001011

qq

xmE p

0)t,(WImAW

1σ 2

el2pγ

tot

*

A glimpse into nuclei

Naïve assumption for T(b): Wood-Saxon like, homogeneous, distribution of nuclear matter

)2/exp(~)(

)exp(~

2 BbbT

tBdt

d diff

DIS on Nuclei