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
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
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
