C olor G lass C ondensate and saturation

1. International School Quark Gluon Plasma and Heavy Ion Collisions past, present, future Torino, 11-17 May 2005 Color Glass Condensate and saturation Marzia Nardi Centro Fermi / CERN-TH

2. Introduction to CGC • Hadronic interactions at very high energies are controlled by a new form of matter, a dense condensate of gluons. • Colour: gluons are coloured • Glass: the fields evolve very slowly with respect to the natural time scale and are disordered. • Condensate: very high density ~ 1/as , interactions prevent more gluon occupation • Concepts, examples, applications to HIC

3. as Hadronic interactions at very high energies • unsolved problems of QCD, non-perturbative aspect. • Froissart bound:

4. particle production in hadronic collisions • Leading particles (projectile, target) have rapidity close to the original rapidity. • Produced particles populate the region around zero-rapidity. • Feynman scaling of rapidity distribution of produced particles.

5. Deep inelastic scattering • Hadron = collections of partons with momentum distribution dN/dx • rapidity : y=yhadron - ln(1/x) rapidity distribution of gluons inside a hadron ZEUS data for the gluon distribution inside a proton small x problem

6. pQCD ok ! gluon density in hadrons McLerran, hep-ph/0311028

7. The distribution functions at fixed Q2saturate • The saturation occurs at transverse momenta below some typical scale: • These considerations make sense if • therefore • We are dealing with a weakly coupled and non-perturbative system. • Effective theory : small-x gluons are described as the classical colour fields radiated by colour sources at higher rapidity. • This effective theory describes the saturated gluons (slow partons) as a Coulor Glass Condensate.

8. Mathematical formulation of the CGC Z= Effective theory defined below some cutoff X0 : gluon field in the presence of an external source r. The source arises from quarks and gluons with x ≥ X0 The weight function F[r] satisfies renormalization group equations (theory independent of X0). The equation for F(JIMWLK) reduces to BFKL and DGLAP evolution equations.

9. There are different kinematic regions where one can find solutions of the RGE with different properties. • A region where the density of gluons is very high and the physics is controlled by the CGC. The typical momenta are less than the saturation momenta : Q2 ≤ Q2sat(x). The dependence of x has been evaluated: • Q2~(x/X0)l GeV2 withl≈ 0.3 • [Triantafyllopoulos, Nucl. Phys. B648,293 (2003) • A.H.Mueller,Triantafyllopoulos , NPB640,331 (2002)] • X0 must be determined from experiment. • A region where the density of gluons is small, high Q2 (fixed x): perturbative QCD

10. Bibliography on CGC • MV Model • McLerran, Venugopalan, Phys.Rev. D 49 (1994) 2233, 3352; D50 (1994) 2225 • A.H. Mueller, hep-ph/9911289 • JIMWLKEquation • Jalilian-Marian, Kovner, McLerran, Weigert, Phys. Rev. D 55 (1997) 5414; • Jalilian-Marian, Kovner, Leonidov, Weigert, Nucl. Phys. B 504 (1997) 415; Phys. Rev. D 59 (1999) 014014

11. Experimental “evidence” of CGC • Geometrical scaling • The structure function F2 • Total hadronic cross sections • Heavy ion collisions

12. Geometrical scaling • Experimental data at HERA at x<0.01 show “geometrical scaling”: the structure functions depends only upon the scaling variable t = Q2R2(x) = Q2/L2instead of being function of two independent variables : x and Q2/L2 • From the data fit : R2(x) = xl, with l ~0.3 • Such a scaling behaviour can be explained in the saturation scenario. • But the observed scaling extends to valus of Q2 larger than the estimates of the saturation scale. [ K. Golec-Biernat, Acta Phys. Polon. B33, 2771 (2002) ]

13. The Structure Function F2 • In general, second order DGLAP formalism works very well at HERA in describing F2(x,Q2). However in the region of very small x(≤10-2) and moderate Q2 saturation is needed to fit the data. [ H. Abramowitz, A. Caldwell DESY report 98-192 (1998) ]

14. F2 In the dipole picture the virtual photon breaks up into the quark-antiquark pair of relative momentum 2k which scatters, inelastically, on the proton. Where dPr(k2)µdk2/Q2 is the probability that the virtual photon breaks up into the quark-antiquark pair in the momentum range dk2 In perturbation theory and but if k2isin the saturation regime and

15. F2 :HERADATA

16. Saturation • At low values of x, QCD evolution predicts a fast increase of the parton densities which violate unitarity constraints. • QCD evolution equations neglect interactions between partons in the parton cascade: this approximation can not be valid when the parton density is high. We expect that these interactions will contribute to saturate the parton densities, at a typical value of the average transverse momentum.

17. Bjorken frame • In this frame the fast hadron decays into a system of partons with • longit. mom. pi,z=xiP • Uncertainty principle: • Dzi~1/(xiP) • only partons with xiP~qzcan interact with the photon

18. The applicability of BFKL is limited : when the density of produced partons is very high the recombination process (~asr2) competes with the emission of new partons (~r) and their number saturates. • Saturation condition : r ~ 1/as • The packing factor is defined as : k= sdipole/pR2

20. Parton saturation model

21. There is a critical momentum scale Qs which separates the two regimes : Saturation scale • For pT < Qs the gluon density is very high, they can not interact independently, their number saturates • For pT > Qs the gluon density is smaller than the critical one, perturbative region • The CGC approach is justified in the limit Qs >>LQCD : • ok at LHC • ~ ok at RHIC

22. Saturation scale in nuclei • Boosted nucleus interacting with an external probe • Transverse area of a parton ~ 1/Q2 • Cross section parton-probe : s~ as/Q2 • Partons start to overlap when SA~NAs • The parton density saturates • Saturation scale : Qs2 ~ as(Qs2)NA/pRA2 ~A1/3 • At saturation Nparton is proportional to 1/as • Qs2 is proportional to the density of participating nucleons; larger for heavy nuclei. Q

23. centrality dependence ! Parton production • We assume that the number of produced particles is : • or • c is the “parton liberation coefficient”; • xG(x, Qs2) ~ 1/as(Qs2) ~ ln(Qs2/LQCD2). • Themultiplicative constant is fitted to data (PHOBOS,130 GeV, charged multiplicity, Au-Au 6% central ): c = 1.23 ± 0.20

24. First comparison to data √s = 130 GeV

25. Energy dependence • We assume the same energy dependence used to describe HERA data; • at y=0: • with l=0.288 (HERA) • The same energy dependence was obtained in Nucl.Phys.B 648 (2003) 293; 640 (2002) 331; with l ~ 0.30 [Triantafyllopoulos , Mueller]

26. Energy dependence : pp and AA D. Kharzeev, E. Levin, M.N. hep-ph / 0408050 (Nucl. Phys. A)

27. Rapidity dependence • Formula for the inclusive production: • [Gribov, Levin, Ryskin, Phys. Rep.100 (1983),1] • Multiplicity distribution: • S is the inelastic cross section for min.bias mult. (or a fraction corresponding to a specific centrality cut) • jA is the unintegrated gluon • distribution function:

28. Simple form of jA Saturation region: SA/as Perturbative region: as/pT2

29. Rapidity dependencein nuclear collisions • x1,2=longit. fraction of mom. carried by parton of A1,2 • At a given y there are, in general, two saturation scales:

30. W=200 GeV PHOBOS Results : rapidity dependence Au-Au Collisions at RHIC

31. d-Au collisions

32. d-Au collisions BRAHMS, nucl-ex/0401025 PHOBOS, nucl-ex/0311009

33. Predictions for LHC • Our main uncertainty : the energy dependence of the saturation scale. • Fixed as : • Running as : • we give results for both cases…

34. Centrality dependence / LHC Pb-Pb collisions at LHC Solid lines : constant as dashed lines : running as

35. Other models… Central Pb-Pb collisions at LHC energy from: N. Armesto, C.Pajares, Int.J.Mod.Phys. A15(2000)2019