CGC

Color Glass Condensate and saturation: 

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

Introduction to CGC: 

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

Hadronic interactions at very high energies: 

Hadronic interactions at very high energies

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. particle production in hadronic collisions

Deep inelastic scattering: 

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

gluon density in hadrons: 

McLerran, hep-ph/0311028 gluon density in hadrons

Slide7: 

The distribution functions at fixed Q2 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.

Mathematical formulation of the CGC: 

Mathematical formulation of the CGC 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. Z =

Bibliography on CGC: 

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

Experimental “evidence” of CGC: 

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

Geometrical scaling: 

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/L2 instead 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) ]

The Structure Function F2: 

The Structure Function F2 [ H. Abramowitz, A. Caldwell DESY report 98-192 (1998) ]

F2: 

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) µ dk2/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 k2 is in the saturation regime and

F2 : HERA DATA: 

F2 : HERA DATA

Saturation: 

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.

Bjorken frame: 

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~qz can interact with the photon

Parton saturation model: 

Parton saturation model

Slide21: 

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

Saturation scale in nuclei: 

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

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). The multiplicative constant is fitted to data (PHOBOS,130 GeV, charged multiplicity, Au-Au 6% central ): c = 1.23 ± 0.20 Parton production

First comparison to data: 

First comparison to data √s = 130 GeV

Energy dependence: 

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]

Energy and centrality dependence / RHIC: 

PHOBOS PHENIX Energy and centrality dependence / RHIC

Energy dependence : pp and AA: 

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

Rapidity dependence: 

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:

Simple form of jA: 

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

Rapidity dependencein nuclear collisions: 

Rapidity dependence in 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:

Results : rapidity dependence: 

Results : rapidity dependence Au-Au Collisions at RHIC

d-Au collisions: 

d-Au collisions

Deuteron wave function: 

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

Predictions for d-Au: 

Predictions for d-Au Predictions in disagreement with PHOBOS data !!!

Problems and solutions: 

Problems and solutions Present approximation not accurate for deuteron we use Monte Carlo results for Npart proton saturation momentum more uncertain we use the same Qsat as in the Golec-Biernat, Wuesthoff model [dashed line, next plot] CGC not valid in the Au fragmentation region we assume dN/dh=NpartAu dNpp/dh in the Au fragmentation region [solid line, next plot]

d-Au collisions: 

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

Predictions for LHC: 

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

Centrality dependence / LHC: 

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

Other models…: 

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

Light-cone coordinates: 

Light-cone coordinates rapidity