Small-x physics

3- Saturation phenomenologyat hadron colliders

Cyrille Marquet

Columbia University

Outline of the third lecture

• The hadronic wave functionsummary of what we have learned

• The saturation modelsfrom GBW to the latest ones

• Deep inelastic scattering (DIS)the cleanest way to probe the CGC/saturationallows to fix the model parameters

• Diffractive DIS and other DIS processesthese observables are predicted

• Forward particle production in pA collisionsand the success of the CGC picture at RHIC

The hadronic/nuclearwave function

The hadron wave function in QCDgggggqqqqqqgqqq .........hadron

non-perturbative

regime: soft QCD

1, 1, ~hadron xkxkk QCDTQCDTQCDT

relevant for instance for

the total cross-section in

hadron-hadron collisions

perturbative regime,

dilute system of partons:

hard QCD (leading-twist

approximation)

relevant for instance for

top quark production

S (kT ) << 1

weakly-coupled regime,

dense system of partons (gluons)

non linear QCD

the saturation regime

not relevant to experiments

until the mid 90’s

with HERA and RHIC: recent gain of interest for saturation physics

• one can distinguish three regimes

The dilute regime1, 1, ~hadron xkxkk QCDTQCDTQCDT

1T

QCD

kthe dilute (leading-twist) regime:

hadron =a dilute system of partons which interact incoherently

)Q,/(ˆ)Q,()Q,( 22/

12 xxxdxx Bjapa

apartons x

BjDIS

Bj

for instance, the total cross-section in DIS

partonic cross-sectionparton density

leading-twistregime

1/kT ~ parton transverse size

as kT increases, the hadron gets more dilute

Dokshitzer GribovLipatov Altarelli Parisi

transverse view of the hadron

The saturation regime1, 1, ~hadron xkxkk QCDTQCDTQCDT

the saturation regime of QCD:the weakly-coupled regime that describes the collective behavior of quarks and gluons inside a high-energy hadron

1~)(Q

, 1T

s

T

QCD

kx

kthe saturation regime:

hadron = a dense system of partons which interact coherently

the separation between the dilute and dense

regimes is caracterized by a momentum scale:

the saturation scale Qs(x)

Balitsky Fadin Kuraev Lipatov

as x decreases, the hadron gets denser

N = 1

N << 1

Geometric scaling from BK• what we learned about the transition to saturation:

the amplitude is invariant along anyline parallel to the saturation line

the saturation scale:

traveling wave solutions geometric scaling

the dipole scattering amplitude

• deep inelastic scattering at small xBj :

• particle production at forward rapidities y :

When is saturation relevant ?in processes that are sensitive to the small-x part of the hadron wavefunction

22

2

Q

Q

WxBj

in DIS small x corresponds to high energy

saturation relevant for inclusive,diffractive, exclusive events

pT , y

yT epsx 2

yT epsx 1 in particle production, small x corresponds

to high energy and forward rapidities

saturation relevant for the production ofjets, pions, heavy flavors, photons

at HERA, xBj ~10-4 for Q² = 10 GeV²

at RHIC, x2 ~10-4 for pT ² = 10 GeV²

The dipole models

The GBW parametrization• the original model for the dipole scattering amplitude

Golec-Biernat and Wusthoff (1998)

main problem: the Fourier transform behaves badly at large momenta:

it features geometric scaling:

fitted on F2 data

the saturation scale:

the parameters:

λ consistent with BK + running coupling

• improvement for small dipole sizesBartels, Golec-Biernat and Kowalski (2002)obtained by including DGLAP-like geometric scaling violations

standard leading-twistgluon distribution

this is also what is obtained in the MV model for theCGC wave function, the behavior is recovered

The IIM parametrization

α and β such that N and its derivative are continuous at

• a BK-inspired model with geometric scaling violations

main problem: the Fourier transform features oscillations

Iancu, Itakura and Munier (2004)

the saturation scale:

matching pointsize of scaling violationsquark masses

Soyez (2007)• improvement with the inclusion of heavy quarks

the parameters:

fixed numbers:

originally, this was fixed at the leading-log value

Impact parameter dependencethe impact parameter dependence is not crucial for F2, it only affects the normalization

however for exclusive processes it must be included

• the IPsat model Kowalski and Teaney (2003)

• the b-CGC model

same as beforeimpact parameter profile

Kowalski, Motyka and Watt (2006)

IIM model with the saturation scale is replaced by

• the t-CGC model

the hadron-size parameter is always of order

C.M., Peschanski and Soyez (2007)

the idea is to Fourier transform where is directly related to the measured momentum transfer

The KKT parametrization• build to be used as an unintegrated gluon distribution

the idea is to modify the saturation exponentKovchegov, Kharzeev and Tuchin (2004)

• the DHJ version

• the BUW version

KKT modified to feature exact geometric scaling

Dumitru, Hayashigaki and Jalilian-Marian (2006)

Boer, Utermann and Wessels (2008)

in practice is always replaced by before the Fourier transformation

KKT modified to better account for geometric scaling violations

Deep inelastic scattering (DIS)

Kinematics of DIS

size resolution 1/Q

k

k’

p

lh center-of-mass energyS = (k+p)2

*h center-of-mass energyW2 = (k-k’+p)2

photon virtualityQ2 = - (k-k’)2 > 0

222

22

Q

Q

)'.(2

Q

hMWkkpx

x ~ momentum fraction of the struck parton y ~ W²/S

2

2 /Q

.

)'.(

hMS

x

kp

kkpy

experimental data are often shown in terms of

• the measured cross-section

The virtual photon wave functions

wave function computed from QED at lowest order in em

)();()Q,( 23* kPqkqkkd

• computable from perturbation theory

)();()Q,,( 222* yxyx qqkyxdddk

x : quark transverse coordinate y : antiquark transverse coordinate

• as usual we go to the mixed space

where the interaction with the CGC is diagonal

in DIS we need the overlap function

The dipole factorization

we already computed the dipole-CGC scattering amplitude

• the virtual photon overlap functions

• scattering off the CGC

x

FFc

rbWrbWTrN

bxrN ))2/()2/((1

1),,(

average over the CGC wave functionthen

up to deviations due to quark massesthe geometric scaling implies

at small x, the dipole crosssection is comparable to that

of a pion, even though

r ~ 1/Q << 1/QCD

HERA data and geometric scaling

geometric scaling seen in the data, butscaling violations are essential for a good fit

Stasto, Golec-Biernat and Kwiecinski (2001)

IIM fit (~250 points)

Soyez (2007)

Diffractive DIS

Inclusive diffraction in DIS

k

k’

p

k

k’

p

p’

when the hadronremains intact rapidity gap

some events

are diffractive

22

22

Q

Q

)').('(2

Q

tMkkpp X

momentum fraction of the exchanged object(Pomeron) with respect to the hadron

diffractive mass

MX2 = (p-p’+k-k’)2

• the measured cross-section

momentum transfert = (p-p’)2 < 0

• the contribution

The dipole picturethe diffractive final state is decomposed into contributions

comes from Fourier transform to MX2

overlap ofwavefunctions Fourier transform to t dipole amplitudes

double differential cross-section(proportional to the structure function)

for a given photon polarization:

geometric scaling implies

Hard diffraction and saturation

dipole size r

recall the dipole scattering amplitude• the total cross sections

in DIS

in DDIS

contribution of the different rregions in the hard regime

DIS dominated by relatively hard sizes

DDIS dominated by semi-hard sizes Sr Q1~Sr Q1Q1

22 QQ S

1 )/Qln(Q 1 Q 2S

22 DIS

1 1 Q

1 Q

22 DDIS

• diffraction directly sensitive to saturation

Comparison with HERA data

(~450 points)parameter-free predictionswith IIM model

with proton tagging e p e X p

H1 FPS data (2006) ZEUS LPS data (2004)

without proton tagging e p e X Y

H1 LRG data (2006) MY < 1.6 GeV

ZEUS FPC data (2005) MY < 2.3 GeV

C.M. (2007)

Important features

tot = F2D

contributions of the different final statesto the diffractive structure function:

at small : quark-antiquark-gluon

at intermediate : quark-antiquark (T)

at large : quark-antiquark (L)

• the β dependenceC.M. and Schoeffel (2006)

• geometric scaling

Hard diffraction off nuclei

in diffraction, averaging at the level of the amplitudecorresponds to a final state where the nucleus is intact

averaging at the cross-section levelallows the breakup of the nucleus into nucleons

averaged with the Woods-Saxon distribution

position of the nucleons

• the dipole-nucleus cross-section Kowalski and Teaney (2003)

• the Woods-Saxon averaging

Kowalski, Lappi, C.M. and Venugopalan (2008)

• nuclear effects

enhancement at large

suppression at small

Exclusive vector meson production• sensitive to impact parameter

)M,,()Q,,()M,Q,( 2V

22V

2 zrzrdzr V

22V

2.22*

)M,Q,();,(16

1 rexbrTbdrddt

d biqqq

VppVM

the overlap function:instead of

)Q,,( 2zr

)M,,( 2VzrV

lots of data from HERA

)²,Q,(*

txdt

d VppVM

²)Q,(* xVppVM

measurements: rho J/Psi

• success of the dipole models

t-CGC

b-CGC appears to work wellalso but no given

predictions for DVCS are available

Forward particle productionin pA collisions

Forward particle production

),(),( 22

212

2TT

TT kxfkxg

dykd

dk

kT , y

yT eksx 1

transverse momentum kT, rapidity y > 0

yT eksx 2

• forward rapidities probe small values of x

the large-x hadron should be described by

standard leading-twist parton distributions

the small-x hadron/nucleus should be

described by CGC-averaged correlators

values of x probed in the process:

the cross-section:single gluon production

probes only the unintegrated

gluon distribution (2-point function)

RHIC vs LHC

xA xA xp xdLHCRHIC

deuteron dominated by valence quarks

• typical values of x being probed at forward rapidities (y~3)

RHIC

LHC

nucleus dominated by early CGC evolution

on the nucleus side, the CGC

picture would be better tested

the proton description shouldinclude both quarks and gluons

if the emitted particle is a quark, involves

if the emitted particle is a gluon, involves

• how the CGC is being probed

Inclusive gluon production

q : gluon transverse momentum

yq : gluon rapidity

))()((1

11][

~2

zzzz AAc

' W'WTrN

AT

gg dipole scattering amplitude:

with

adjointWilson line

Y'Tzz

~

this derivation is for dipole-CGC scattering

but the result valid for any dilute projectile

hh

• effectively described by a gluonic dipole

the other Wilson lines and (coming

from the interaction of non-mesured partons)

cancel when summing all the diagrams

)(xFW )(yFW

),()(~2.2

222 q

y

i yqrTerdbqdd

dqq

r

rq the transverse momentum spectrum

is obtained from a Fourier

transformation of the dipole size rvery close to the unintegrated

gluon distribution introduced earlier

• the gluon production cross-section

A CGC prediction

in the geometric scaling regime

is peaked around QS(Y)

y

),( yk

• the unintegrated gluon distribution

kdyddN

kdyddN

NR hXpp

hXdA

colldA

2

21

the suppression of RdA was predicted

xA decreases(y increases)

• the suppression of RdA

in the absence of nuclear effects, meaning if the gluons in the nucleus interact incoherently like in A protons

the infrared diffusion problem of the BFKL

solutions has been cured by saturation

RdA and forward pion spectrum

Kharzeev, Kovchegov and Tuchin (2004)

RdA• first comparison to data

qualitative agreement

with KKT parametrization

Dumitru, Hayashigaki and Jalilian-Marian (2006)

shows the importance of both

evolutions: xA (CGC) and xd (DGLAP)

shows the dominance

of the valence quarks

for the pT – spectrum

with the DHJ model

• quantitative agreement

2-particle correlations in pA• inclusive two-particle production

11 , yk 22 , yks

ekekx

yy

p

21 21

s

ekekx

yy

A

21 21

probes 2-, 4- and 6- point functions

final state :

one can test more information about the CGC compared to single particle production

as k2 decreases, it gets closer to QS and thecorrelation in azimuthal angle is suppressed

• some results for azimuthal correlations

obtained by solving BK, not from model

k2 is varied from 1.5 to 3 GeV

C.M. (2007)

at forward rapidities in order to probe small x

What is going on now in this field

• Link with the MLLA ?we would like to understand the differences between the picturessimilar objects have already been identified (triple Pomeron vertex)

• Higher order correctionsrunning coupling corrections to BK are known,but not the full non linear equation at next-to-leading log

• Heavy ion collisionswhat is the system at the time ~1/Qs after the collisioncrucial for the rest of the space-time evolution

• Calculations for RHIC/LHCtotal multiplicities, jets, pions, heavy flavors, photons, dileptons