1. Initial State and saturation Marzia Nardi INFN Torino (Italy) nardi@to.infn.it Quark Matter 2009, Knoxville Student Day

2. WHY ? General interest: Unsolved problems of QCD QCD out of the perturbative regime Looking for universal properties Interest in HIC: Understanding the beginning to understand the end Correct interpretation of experimental data

3. The total hadron-hadron Xsection at high energies is among the unsolved problems of QCD, non- perturbative aspect. Froissart bound (unitarity) : Is this behaviour universal ? as Hadronic interactions at very high energies

4. Leading particles (projectile, target) have rapidity close to the original rapidity. Produced particles populate the region around zero-rapidity. Scaling of rapidity distribution of produced particles. Looking for universal properties…

5. PHOBOS Collab. PRL 91, 052303 (2003)  ’=  beam

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

7. gluon density in hadrons McLerran, hep-ph/0311028

9. gluon density in nuclei +

10. +

11. In a nucleus, the saturation sets in at a smaller scale

12. Color Glass Condensate 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/  s, interactions prevent more gluon occupation

13. Saturation scale in nuclei Boosted nucleus interacting with an external probe Transverse area of a parton ~ 1/Q 2 Cross section :  ~  s /Q 2 Parton density:  = xG(x,Q 2 )/  R A 2 Partons start to overlap when S A ~N A  ~1) The parton density saturates Saturation scale : Q s 2 ~  s (Q s 2 )N A /  R A 2 ~A 1/3 At saturation N parton is proportional to 1/  s Q s 2 is proportional to the density of participating nucleons; larger for heavy nuclei. Q

14. The distribution functions at fixed Q 2 saturate 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.

15. Mathematical formulation of the CGC Effective theory defined below some cutoff X 0 : gluon field in the presence of an external source . The source arises from quarks and gluons with x ≥ X 0 The weight function F [  ] satisfies renormalization group equations (theory independent of X 0 ). The equation for F   JIMWLK  reduces to BFKL and DGLAP evolution equations. Yang Mill eq. : Z =

16. 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 : Q 2 ≤ Q 2 sat (x). The dependence of x has been evaluated: Q s 2 ~(x/X 0 )-  Q s0 2 with ≈ 0.3 [Triantafyllopoulos, Nucl. Phys. B648,293 (2003) A.H.Mueller,Triantafyllopoulos, NPB640,331 (2002)] X 0 must be determined from experiment. A region where the density of gluons is small, high Q 2 (fixed x): perturbative QCD

17. Bibliography on CGC MV Model McLerran, Venugopalan, Phys.Rev. D 49 (1994) 2233, 3352; D50 (1994) 2225 A.H. Mueller, hep-ph/9911289 JIMWLK Equation 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 REVIEW Iancu, Leonidov, McLerran hep-ph/0202270

18. Geometrical scaling In the dense regime (  QCD << p t << Q s (x)) we expect to observe some scaling: p t /Q s (x). Extended scaling region: p t < Q s 2 (x)/  QCD

19. Geometrical scaling at HERA The structure functions depends only upon the scaling variable  = Q 2 /Q s 2 (x) instead of being function of two independent variables : x and Q 2 From the data fit : Q s 2 (x)=Q 0 2 (x)(x 0 /x) with  ~0.3 [ K. Golec-Biernat, Acta Phys. Polon. B33, 2771 (2002) ]

20. Particle multiplicity CGC predicts the ditribution of initial gluons, set free by the interactions. CGC gives the “initial conditions” KLN (Kharzeev, Levin, Nardi) model: PLB 507,121 (2001); PRC 71, 054903 (2005); PLB 523,79(2001) NPA 730,448(2004) Erratum-ibid.A743,329(2004); NPA 747,609(2005)

21. We assume that the number of produced particles is : xG(x, Q s 2 ) ~ 1/  s (Q s 2 ) ~ ln(Q s 2 /  QCD 2 ). The multiplicative constant is fitted to data (PHOBOS,130 GeV, charged multiplicity, Au-Au 6% central ): c = 1.23 ± 0.20 Parton production

22. First comparison to data √s = 130 GeV

23. Energy dependence We assume the same energy dependence used to describe HERA data; at y=0: with  HERA  The same energy dependence was obtained in Nucl.Phys.B 648 (2003) 293; 640 (2002) 331; with   [Triantafyllopoulos, Mueller]

24. PHOBOS PHENIX Energy and centrality dependence / RHIC

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

26. 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)  A is the unintegrated gluon distribution function:

27. Perturbative region:  s / Perturbative region:  s / p T 2 Saturation region: S A /  s Simple form of  A

28. Rapidity dependence in nuclear collisions x 1,2 =longit. fraction of mom. carried by parton of A 1,2 At a given y there are, in general, two saturation scales:

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

30. Predictions for LHC Our main uncertainty : the energy dependence of the saturation scale. Fixed  s : Running  s :

31. Centrality dependence / LHC Solid lines : constant  s dashed lines : running  s Pb-Pb collisions at LHC

32. Elliptic flow Initial anisotropy: [Hirano, Heinz, Kharzeev, Lacey, Nara, nucl-th/0511046]

34. d-Au collisions

35. In AA collisions saturation effects are important, but they are followed by kinetic and chemical equilibration, hadronization... dA (pA) collisions give the opportunity to study initial state effects. Possibly peripheral AA collisions.

36. Deuteron wave function where [Huelthen, Sugawara, “Handbuck der Physik”, vol.39 (1957)]:  is derived from the experimental binding energy:

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

38. p t spectra CGC describes the initial conditions. Hadrons produced in AA have undergone many reinteractions: final momentum spectra can be significantly different from the initial ones. In pA (dA) we do not expect final state interactions to play a dominant role: CGC can explain medium effects responsible for the difference between pA and pp. In AA: CGC calculations are useful to disentangle the final state contributions, centrality dependence.

39. BRAHMS Collab. [nucl-ex/0403005] [Albacete, Armesto, Kovner, Salgado, Wiedemann, Phys.Rev.Lett.92:082001,2004] y=0 y=2

40. Conclusions We have now a picture that is universally applicable to all hadron interactions at high energies, in the whole x,k t plane (except the truly non-perturbative region k t <  QCD ). In the domain of applicability of the CGC picture, the comparison with experimental data are successful. In AA the final state interactions are importsnt, therefore only global observables are preserved from early times to the final state. pA (dA) collisions are the best place to study CGC CGC only provides the initial conditions for the subsequent evolution of the system, leading possibly to the formation of QGP. At LHC the saturation scale will be larger (x=10 -5 -10 -4, Q s =3-4 GeV): even better conditions for CGC.

41. Light-cone coordinates LC time LC longit. coordinate invariant 4-product LC energy LC longit. momentum rapidity