1. Mini review on saturation and recent developements Cyrille Marquet Service de Physique Théorique - CEA/Saclay ICHEP 2006, Moscow, Russia

2. Introduction: the saturation regime of QCD weak coupling regime with high gluon densities Success of saturation geometric scaling at HERA high-rapidity suppression at RHIC Recent developements Pomeron loops new scaling laws in the context of - deep inelastic scattering - particle production in hadron-hadron collisions Conclusions Contents

3. Introduction

4. The hadron wavefunction in QCD light-cone variables: x and k T : parton kinematics P+P+ 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

5. The saturation scale The separation between the dilute and dense regimes is caracterized by a momentum scale: the saturation scale Q s (x) The saturation regime of QCD: the perturbative regime that describes the collective behavior of quarks and gluons inside a hadron saturation regime: a dense system of partons, responsible for strong color fields and collective phenomena leading-twist regime: a dilute system of partons described with parton distributions, collinear factorization … Qs(x)Qs(x) saturation regime leading-twist regime Balitsky Fadin Kuraev Lipatov Dokshitzer Gribov Lipatov Altarelli Parisi

6. When is saturation relevant ? deep inelastic scattering at small x Bj : particle production at forward rapidities y : In processes that are sensitive to the small-x part of the hadron wavefunction Q2Q2 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 p T, y in particle production, small x corresponds to high energy and forward rapidities saturation relevant for the production of jets, pions, heavy flavours, dileptons

7. The success of saturation

8. r Probing the saturation regime In DIS, the probe is a dipole with a small transverse size r ~ 1/Q what the dipole sees: the physics is invariant along any line parallel to the saturation line T = 1 T << 1 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

9. A. Stasto, K. Golec-Biernat and J. Kwiecinski, Phys. Rev. Lett. 86 (2001) 596 The geometric scaling of  DIS (x, Q 2 ) this is seen in the data with  0.3 saturation models fit well F 2 data: K. Golec-Biernat and M. Wüsthoff, Phys. Rev. D59 (1999) 014017 J. Bartels, K. Golec-Biernat and H. Kowalski, Phys. Rev. D66 (2002) 014001 E. Iancu, K. Itakura and S. Munier, Phys. Lett. B590 (2004) 199 update 

10. C. M. and L. Schoeffel, Phys. Lett. B, in press, hep-ph/0606079 Geometric scaling in diffraction  scaling also for vector meson production :

11. Saturation at HERA saturation predictions describe accurately a number of observables at HERA F 2 D Deeply virtual Compton scattering Diffractive vector-meson production t integrated t dependence F 2 c S. Munier, A. Stasto and A. Mueller, Nucl. Phys. B603 (2001) 427 H. Kowalski and D. Teaney, Phys. Rev. D68 (2003) 114005 H. 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) 114023 J. Forshaw, R. Sandapen and G. Shaw, Phys. Lett. B594 (2004) 283 L. Favart and M. Machado, Eur. Phys. J C29 (2003) 365 L. 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

12. Saturation at RHIC saturation 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) 23 D. Kharzeev, E. Levin and M. Nardi, Nucl. Phys. A747 (2005) 609 A. Dumitru, A. Hayashigaki and J. Jalilian-Marian, Nucl. Phys. A765 (2006) 464  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 of back-to-back correlations

13. Recent developements

14. Beyond the B-JIMWLK equations A. Mueller and A. Shoshi, Nucl. Phys. B692 (2004) 175 E. Iancu, A. Mueller and S. Munier, Phys. Lett. B 606 (2005) 342 E. Iancu and D. Triantafyllopoulos, Nucl. Phys. A756 (2005) 419 Several directions: - high-energy effective action - generelized dipole model - reggeon field theory A. Kovner and M. Lublinsky, hep-ph/0512316 A. 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) 423 S. Bondarenko and L. Motyka, hep-ph/0605185 A. Kovner and M. Lublinsky, Phys. Rev. D72 (2005) 074023 C. M., A. Mueller, A. Shoshi and S. Wong, Nucl. Phys. A762 (2005) 252 Y. 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 loops A. Mueller, A. Shoshi and S. Wong, Nucl. Phys. B715 (2005) 440 E. Levin and M. Lublinsky, Nucl. Phys. A763 (2005) 172 E. Iancu and D. Triantafyllopoulos, Phys. Lett. B610 (2005) 253 A. Kovner and M. Lublinsky, Phys. Rev. D71 (2005) 085004 A. Kovner and M. Lublinsky, Phys. Rev. Lett. 94 (2005) 181603 A. Kovner and M. Lublinsky, JHEP 0503 (2005) 001 J.-P. Blaizot, E. Iancu, K. Itakura and D. Triantafyllopoulos, Phys. Lett. B615 (2005) 221 E. Levin, Nucl. Phys. A763 (2005) 140

15. Stochasticity in high energy QCD : related to the average value D : 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 the s-FKPP equation well-known in statistical physics (for ) C. M., G. Soyez and B.-W. Xiao, Phys. Lett. B, in press, hep-ph/0606233 the saturation scale is a stochastic variable distributed according to a Gaussian probability law: corrections to the Gaussian law for improbable fluctuations also known Y r

16. A new scaling law If DY << 1, the diffusion is negligible and with 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: If DY >> 1, the diffusion is important and E. Iancu and D. Triantafyllopoulos, Nucl. Phys. A756 (2005) 419 C. 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:

17. Implications for DIS an intermediate energy regime: geometric scaling HERA it seems that HERA is probing the geometric scaling regime Y. Hatta, E. Iancu, C.M., G. Soyez and D. Triantafyllopoulos, Nucl. Phys. A773 (2006) 95 In the diffusive scaling regime, saturation is the relevant physics up to momenta much higher than the saturation scale at higher energies, a new scaling law: diffusive scaling within the LHC energy range?

18. In the geometric scaling regime is peaked around k ~ Q S (Y) : In forward particle production, the transverse momentum spectrum is obtained from the unintegrated gluon distributionof the small-x hadron Implications for particle production Y important in view of the LHC: large p T, small values of x E. Iancu, C.M. and G. Soyez, hep-ph/0605174 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

19. 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 x Bj in forward particle production in hadron-hadron collisions  HERA and RHIC have initiated strong interest this past decade and 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