Mini review on saturation and recent developements

Cyrille MarquetService de Physique Théorique - CEA/Saclay

ICHEP 2006, Moscow, Russia

• Introduction: the saturation regime of QCDweak coupling regime with high gluon densities

• Success of saturationgeometric scaling at HERAhigh-rapidity suppression at RHIC

• Recent developementsPomeron loopsnew scaling laws in the context of

- deep inelastic scattering- particle production in hadron-hadron collisions

• Conclusions

Contents

Introduction

The hadron wavefunction in QCD

light-cone variables:

x and kT : parton kinematics

P+

gggggqqqqqqgqqq .........hadron

non-perturbative

regime: soft QCD

perturbative regime,

dilute system of partons:

leading-twist approximation

hard QCD

perturbative regime,

dense system of partons:

collective phenomena

the saturation regime of QCD

1, 1, ~hadron xkxkk QCDTQCDTQCDT

The saturation scale1, 1, ~hadron xkxkk QCDTQCDTQCDT

The separation between the dilute and dense regimes

is caracterized by a momentum scale:

the saturation scale Qs(x)

The saturation regime of QCD:the perturbative regime that describes the collective behaviorof quarks and gluons inside a hadron

1~)(Q

, 1T

s

T

QCD

kx

ksaturation regime:

a dense system of partons, responsible forstrong color fields and collective phenomena

1)(Q

, 1 T

s

T

QCD

kx

kleading-twist regime:

a dilute system of partons described withparton distributions, collinear factorization …

Qs(x)

saturationregime

leading-twistregime

BalitskyFadin

KuraevLipatov

Dokshitzer GribovLipatov Altarelli Parisi

When is saturation relevant ?

• deep inelastic scattering at small xBj :

• particle production at forward rapidities y :

In processes that are sensitive to the small-x part of the hadron wavefunction

Q2

22

2

Q

Q

WxBj

W 2

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

in DIS small x corresponds to high energy

saturation relevant for inclusive,diffractive, exclusive events

h

pT , y

yT epsx 2

yT epsx 1

in particle production, small x correspondsto high energy and forward rapidities

saturation relevant for the production ofjets, pions, heavy flavours, dileptons

The success of saturation

r

Probing the saturation regime

),( YrT

In DIS, the probe is a dipole with a small transverse size r ~ 1/Q

what the dipole sees:

the physics is invariant along anyline parallel to the saturation line

T = 1

T << 1 )(Q),( 22 YrTYrT S

perturbative scales probe small distances inside the hadrons

the dipole scattering amplitude:

Evolution of with rapidity Y: given by(in the leading logarithmic approximation)the B-JIMWLK equations

Balitsky Kovchegov

Balitsky Jalilian-Marian Iancu McLerran Weigert Leonidov Kovner

Simpler version: the BK equation

),( YrT

22 1~Q r

A. Stasto, K. Golec-Biernat and J. Kwiecinski, Phys. Rev. Lett. 86 (2001) 596

The geometric scaling of DIS(x, Q2)

this is seen in the data with 0.3

saturation models

fit well F2 data:

K. Golec-Biernat and M. Wüsthoff, Phys. Rev. D59 (1999) 014017J. Bartels, K. Golec-Biernat and H. Kowalski, Phys. Rev. D66 (2002) 014001E. Iancu, K. Itakura and S. Munier, Phys. Lett. B590 (2004) 199

update

)(Q),( 22 YrTYrT S

C. M. and L. Schoeffel, Phys. Lett. B, in press, hep-ph/0606079

Geometric scaling in diffraction

scaling also for vector meson production :

)(Q),( 22 YrTYrT S

Saturation at HERAsaturation predictions describe accurately a number of observables at HERA

• F2D

• Deeply virtual Compton scattering

• Diffractive vector-meson productiont integrated

t dependence

• F2c

S. Munier, A. Stasto and A. Mueller, Nucl. Phys. B603 (2001) 427H. Kowalski and D. Teaney, Phys. Rev. D68 (2003) 114005H. Kowalski and D. Teaney and G. Watt, hep-ph/0606272

V. Goncalves and M. Machado, Phys. Rev. Lett. 91 (2003) 202002

K. Golec-Biernat and M. Wüsthoff, Phys. Rev. D60 (1999) 114023J. Forshaw, R. Sandapen and G. Shaw, Phys. Lett. B594 (2004) 283

L. Favart and M. Machado, Eur. Phys. J C29 (2003) 365L. Favart and M. Machado, Eur. Phys. J C34 (2004) 429

E. Gotsman, E. Levin, M. Lublinsky, U. Maor and E. Naftali, Acta Phys. Polon. B34 (2003) 3255

Saturation at RHICsaturation predictions describe accurately a number of observables at RHIC

• High-rapidity suppression of the nuclear modification factor in d-Au

D. Kharzeev, Y. Kovchegov and K. Tuchin, Phys. Lett. B599 (2004) 23D. Kharzeev, E. Levin and M. Nardi, Nucl. Phys. A747 (2005) 609A. Dumitru, A. Hayashigaki and J. Jalilian-Marian, Nucl. Phys. A765 (2006) 464

kdddN

kdddN

NR hXpp

hXdA

colldA

2

21

BRAHMS data

see recent review: J. Jalilian-Marian and Y. Kovchegov, Prog. Part. Nucl. Phys. 56 (2006) 104

D. Kharzeev, E. Levin and L. McLerran, Nucl. Phys. A 748 (2005) 627

• Azimuthal correlations

STAR data

suppresion ofback-to-backcorrelations

Recent developements

Beyond the B-JIMWLK equationsA. Mueller and A. Shoshi, Nucl. Phys. B692 (2004) 175E. Iancu, A. Mueller and S. Munier, Phys. Lett. B 606 (2005) 342E. Iancu and D. Triantafyllopoulos, Nucl. Phys. A756 (2005) 419

• Several directions:- high-energy effective action

- generelized dipole model

- reggeon field theoryA. Kovner and M. Lublinsky, hep-ph/0512316A. Kovner and M. Lublinsky, hep-ph/0604085

I. Balistky, Phys. Rev. D72 (2005) 074027 Y. Hatta, E. Iancu, L. McLerran, A. Stasto and D. Triantafyllopoulos, Nucl. Phys. A764 (2006) 423S. Bondarenko and L. Motyka, hep-ph/0605185

A. Kovner and M. Lublinsky, Phys. Rev. D72 (2005) 074023C. M., A. Mueller, A. Shoshi and S. Wong, Nucl. Phys. A762 (2005) 252Y. Hatta, E. Iancu, L. McLerran and A. Stasto, Nucl. Phys. A762 (2005)

272

• Trigerring papers in 2004:

• Then between hep-ph/0501088 and hep-ph/0502243: Pomeron loopsA. Mueller, A. Shoshi and S. Wong, Nucl. Phys. B715 (2005) 440E. Levin and M. Lublinsky, Nucl. Phys. A763 (2005) 172E. Iancu and D. Triantafyllopoulos, Phys. Lett. B610 (2005) 253A. Kovner and M. Lublinsky, Phys. Rev. D71 (2005) 085004A. Kovner and M. Lublinsky, Phys. Rev. Lett. 94 (2005) 181603A. Kovner and M. Lublinsky, JHEP 0503 (2005) 001J.-P. Blaizot, E. Iancu, K. Itakura and D. Triantafyllopoulos, Phys. Lett. B615 (2005) 221E. Levin, Nucl. Phys. A763 (2005) 140

Stochasticity in high energy QCD

: related to the average valueD : dispersion coefficient

E. Iancu, A. Mueller and S. Munier, Phys. Lett. B 606 (2005) 342

Pomeron loops stochasticity in the evolution

similarities between the QCD equation and thes-FKPP equation well-known in statistical physics

(for ) DYSS 22 Q/Qln

C. M., G. Soyez and B.-W. Xiao, Phys. Lett. B, in press, hep-ph/0606233

DYDYP S )Q/Qln²(exp1)Q(ln

22S2

S

the saturation scale is a stochastic variable distributedaccording to a Gaussian probability law:

corrections to the Gaussian law forimprobable fluctuations also known

Y

r

A new scaling law

If DY << 1, the diffusion is negligible and with )(Q),( 22 YrTYrT S

we recover geometric scaling

One obtains the physical dipole amplitude by averaging the event-by-event amplitude which obeys the Langevin equation

we even know the functional form for : DYS 2Qr²ln

DYYrTYrT S )(Qln),( 22

DYYrErfcYrT S )(Qln21),( 22

If DY >> 1, the diffusion is important and

E. Iancu and D. Triantafyllopoulos, Nucl. Phys. A756 (2005) 419C. M., R. Peschanski and G. Soyez, Phys. Rev. D73 (2006) 114005

new regime: diffusive scaling

in the diffusive scaling regime (up to momenta k ~ 1/r much bigger than the saturation scale ):

- cross-sections are dominated by events that feature the hardest fluctuation of the saturation scale - in average the scattering is weak, yet saturation is the relevant physics

New Physics:

),( YrT

Implications for DIS

an intermediate energy regime:geometric scalingHERA

it seems that HERA is probing

the geometric scaling regime

22 1~Q r

Y. Hatta, E. Iancu, C.M., G. Soyez and D. Triantafyllopoulos, Nucl. Phys. A773 (2006) 95

)(Q),( 22 YrTYrT S

In the diffusive scaling regime, saturation is the relevant physics

up to momenta much higher than the saturation scale

at higher energies, a newscaling law: diffusive scaling

within the LHC energy range?

DYYrTYrT S )(Qln),( 22

In the geometric scaling regime

is peaked around k ~ QS(Y) :

In forward particle production, the transverse momentum spectrum is obtained fromthe unintegrated gluon distribution of the small-x hadron

Implications for particle production

),( Yk

Y),( Yk

important in view of the LHC: large pT , small values of x

E. Iancu, C.M. and G. Soyez, hep-ph/0605174

DYk

DYYk S )Q/²ln²(exp1),(

2

In the diffusive scaling regime :

Y

Is diffusive scaling within the LHC energy range?

Hard to tell: theoretically, we have a poor knowledge of the coefficient D

• The saturation regime of QCD:the perturbative regime that describes the small-x part of a hadron wavefunction weak coupling regime with high parton densities

• Sensitivity to the saturation:in deep inelastic scattering at small xBj

in forward particle production in hadron-hadron collisions HERA and RHIC have initiated strong interest this past decadeand saturation has had some success

• Over the past 2 years, new theoretical developements:inclusion of Pomeron loops in the QCD evolution towards high energies several directions for studying the consequences: stochasticity, high-energy effective action, generelized dipole model, reggeon field theory, …for the most part, phenomenology yet to come new scaling laws in the context of DIS and particle production

Conclusions