Overview of saturation

Yoshitaka Hatta

(Saclay)

Low-x meeting, 2007, Helsinki

Outline

BFKL and beyond Saturation momentum The BK-JIMWLK equation Fluctuations & Correlations

Regge limit of QCD

E

E

,...,,2 TpMtEs

One of the most challenging problems in QCD is the understanding of its high energy limit.

Can we compute hadron-hadron (nucleus) cross sections in this regime from first principles?

Deep Inelastic Scattering

2Q

xY

1ln

DGLAP

BF

KL

p

222

22

2 QMM

Q

qP

Qx

pX

X

s

Q2

)1( x

The BFKL Pomeron

1x2x

nx

x

p

n

s

x

xn

n

x

ns xnx

dx

x

dx n

1ln

!

111

1

1

ss sxptot

2ln42ln4*

…

xxx n 11

Rapidity ordering

Balitsky, Fadin, Kuraev, Lipatov, `75~`78

Ways to go — beyond BFKL

LLA BFKL predicts indefinite growth of the gluon number.

Infrared diffusion invalidates perturbative treatment in the middle of the ladder.

Many reggeized gluon exchange à la BKP

Next to leading-log approximation (NLLA) Nonlinear equations from gluon saturation

Must be tamed (How?)

12 3

N 4……

High energy QCD as an integrable modelLipatov, `93 Faddeev & Korchemsky, `94

The BKP Hamiltonian is identical to that of an exactly solvable system.Eigenvalue (energy dependence ) obtained by the Bethe ansatz

HHHBKP

s

Heisenberg spin chain0s

1

23

4N

…..

BKP equation at large Nc

Ways to go — beyond BFKL

LLA BFKL predicts indefinite growth of the gluon number.

Infrared diffusion invalidates perturbative treatment in the middle of the ladder.

Many reggeized gluon exchange à la BKP

Next to leading-log approximation (NLLA) Nonlinear equations from gluon saturation

Must be tamed (How?)

Saturation criterion : hadron

)(22 xQQ s

hR

222

),( hs RQxxG

Q

Collinear or kt factorization : One parton from each hadron scatters. Partons inside a hadron do not interact. exponential growth in number

recombinationcross section

# of gluonscomputed in BFKL

When gluons begin to overlap…

A large nucleus ( ) is a dense system from the beginning.

Saturation criterion : nucleus

3/12 AQs

AR22

2),( A

s RQxAxGQ

# of gluons computed in lowest order perturbation theory

At the same time, multiple scattering becomes unsuppressed. (Glauber—Mueller)

A

McLerran & Venugopalan `94

3/1A

The saturation momentum at LLA

Find the line of constant amplitude in the BFKL solution.

Gribov, Levin, Ryskin, `83

21 1

)(

37.0

22 ),( QQxxG

}ln)1()(exp{ 22 QYd

increase with decrease with 2QY

YQCDs

seYQ 9.422 )(

Calculations at (improved) NLO available Triantafyllopoulos `03 Khoze et al. `04

YQCDe 8.92

(fixed coupling)

(running coupling)

The new phase of QCD

2Q

sY ln

DGLAP

BF

KL

)(2 YQs

extended scaling region

Evolution in the presence of saturation

A

dressed propagator

Incoming hadrons (nuclei) are replaced by non-Abelian Weizsacker-Williams fields of strength ~ g1

A

gA gANeed to sum all orders in , or , or both in a single step of evolution

BK-JIMWLK (1996~2001)

BFKL (1975~78)

A

A

Pomeron loop (~20XX)

The long and winding road to unitarity…

A

A

A

A

A

A

BKP (1978~80)

A

A

NLO BFKL (1996~98)

NLO BK(Coming soon !)

The BK-JIMWLK equation

xy

xxA

Yyxc

Y VVN

yxT ][Tr1

1),(

22

22

)()(

)(

2),(

yzzx

yxzdyxT

Ys

Y

YyzTzxTyxTyzTzxT }),(),(),(),(),({

Dipole-nucleus scattering (subprocess of DIS)

BFKL gluon recombination

),(),(),(),( yzTzxTyzTzxT `Mean field’ approximation

)},(exp{

xxAdxigPVx

1

)(rTY

2

1ln

r122 ))(( YQr s

Y`front’

|| yxr : dipole size

The geometric scaling

Mapping onto the traveling wave solution of the FKPP equation

22 ffff xt Munier & Peschanski, `03

Some indication in the HERA data Stasto, Golec-Biernat & Kwiecinski, `01 Marquet & Schoeffel, `06 (diffraction)

Iancu, Itakura & McLerran, `02The scaling persists even when 22sQQ

Application to other processes

Diffractive DIS

pA (and AA) collisions at RHIC

Odderon

The BLV solution : constant in energy Saturation effects tend to suppress the odderon amplitude.

Saturation prediction 2ln1 Qtotdiff Independent of ! x

Saturation models confront RHIC data, doing well.

Multiplicity, pt distribution, heavy quark, , limiting fragmentation, etc. pAR

Inclusive & exclusive Talk by Marquet

Beyond BK—JIMWLK

What’s missing in the BK-JIMWLK?

A kind of mean field approximation, neglecting fluctuations and correlationsin the target wavefunction.

)1(),(),(),(),( 2cNOwzTyxTwzTyxT

xy

zw

Not symmetric w.r.t. the target and projectile,Saturation effect of the projectile missing.

A

A

The gluon number fluctuation

boost

Salam, `96

QCD dipole model

xzy large Nc

Significant consequences in the approach to unitarity.Deep connection to the stochastic FKPP equation in statistical physics.

zdYdyzzx

yxdP s

222

2

)()(

)(

fffff xt22

Universal behavior of the sFKPP equation

1

),( YrT

21ln r

Front position (saturation scale ) becomes a random variable2sQ

Each event shows geometric scaling, but the average does not !

Observed amplitude is obtained after averaging over events.

Y

Iancu, Mueller, Munier, `04

TTTT Factorization `maximally’ violated

Caveats: Requires enormous energy for the fluctuation to become significant. Running coupling may be (very) important.

Talks by Beuf, Kozlov, Soyez

HERA ?

LHC, cosmic ray ??

Updating the phase diagram

Figure from Iancu, Marquet, Soyez `06

ax

bx

abx

Power—law correlation in the transverse plane

01x

Y.H. & Mueller, `07

Small-x gluons are correlated because they come from a common ancestor.

L

abxncorrelatio

1)(zdYd

yzzx

yxdP s

222

2

)()(

)(

Determination of the power

Breakdown of factorizationfrom BFKL. Explicit !

)2(2 0

1)()()()(

abbaba x

xTxTxTxT

ax

bx

abx

Peschanski, `97 Braun & Vacca, `97

Dipole pair density

Quest for unitarity : a difficult but fascinating problem

Continual efforts & progresses, still many open questions

BK-JIMWLK equation : the best ‘simple’ equation including nonlinear effects.

Beyond the BK-JIMWLK : Pomeron loops.

Hadron wavefunction teems with correlations and fluctuations.

Summary

Important theoretical problems

Full inclusion of Pomeron loops, and its physical consequences.

Gluon production in AA collision, quantum evolution.

NLO BK phenomenology

Gelis, Lappi, Venugopalan `07

Talk by Y.H.

Saturation in AdS/CFT ?

Balitsky `06, Kovchegov, Weigert `06, Balitsky, Chirilli, to appear

Talks by Levin and Lublinsky