Initial State and saturation

Marzia NardiINFN Torino (Italy)nardi@to.infn.it

Quark Matter 2009, KnoxvilleStudent Day

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

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 ?

E2ln~ Eas

Hadronic interactions at very high energies

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…

PHOBOS Collab.PRL 91, 052303 (2003)

h’= -h hbeam

Deep inelastic scattering• Hadron = collection of partons with momentum

distribution dN/dx • rapidity : y=yhadron - ln(1/x)

ZEUS data for thegluon distribution inside a proton

small x problem

gluon density in hadrons

McLerran, hep-ph/0311028

gluon density in nuclei

+

gluon density in nuclei

+

gluon density in nuclei

In a nucleus , the saturation sets in at a

smaller scale

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/as ,

interactions prevent more gluon occupation

Saturation scale in nuclei• Boosted nucleus interacting with an external probe• Transverse area of a parton ~ 1/Q2

• Cross section : s ~ as/Q2

• Parton density: r = xG(x,Q2)/pRA2

• Partons start to overlap when SA~NA (s r s ~1)• The parton density saturates• Saturation scale : Qs

2 ~ as(Qs2)NA/pRA

2 ~A1/3

• At saturation Nparton is proportional to 1/as

• Qs2 is proportional to the density of participating nucleons;

larger for heavy nuclei.

Q

• 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

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.

Yang Mill eq. :

Z =

• 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 ≤ Q2

sat(x).

The dependence of x has been evaluated: Qs

2~(x/X0)- l Qs02 with l ≈ 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

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

Geometrical scaling

In the dense regime (LQCD<< pt << Qs(x)) we expect to observe some scaling: pt/Qs(x).

Extended scaling region: pt < Qs2(x)/ LQCD

Geometrical scaling at HERAThe structure functions

depends only upon the scaling variable

= Q2/Qs2(x)

instead of being function of two independent variables : x and Q2

From the data fit : Qs

2(x)=Q02(x)(x0/x)l

with l ~0.3 [ K. Golec-Biernat, Acta Phys. Polon. B33, 2771 (2002) ]

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)

We assume that the number of produced particles is :

xG(x, Qs2) ~ 1/as(Qs

2) ~ ln(Qs2/LQCD

2).

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

√s = 130 GeV

Energy dependence

• We assume the same energy dependence used to describe HERA data;

• at y=0:

• with =0.288 (l HERA)• The same energy dependence was obtained in

Nucl.Phys.B 648 (2003) 293; 640 (2002) 331; with ~ l0.30 [Triantafyllopoulos , Mueller]

Energy dependence : pp and AA

D. Kharzeev, E. Levin, M.N.hep-ph / 0408050(Nucl. Phys. A)

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:

))(,(),(1

1

4 22

21

2223 21 TATA

p

sTTc

c xkxdkpN

N

pdd

ET

kp

Perturbative region: as/pT2

Saturation region: SA/as

Simple form of jA

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:

y

y

es

Qx

es

Qx

2

1

Results : rapidity dependence

PHOBOS W=200 GeV

Au-Au Collisions at RHICAu-Au Collisions at RHIC

Predictions for LHC• Our main uncertainty : the energy

dependence of the saturation scale.• Fixed as :

• Running as :

Centrality dependence / LHC

Solid lines : constant as

dashed lines : running as

Pb-Pb collisionsat LHC

Elliptic flow

• Initial anisotropy:

[Hirano, Heinz, Kharzeev, Lacey, Nara, nucl-th/0511046]

d-Au collisions

• 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.

d-Au collisions

BRAHMS, nucl-ex/0401025 PHOBOS, nucl-ex/0311009

pt 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.

BRAHMS Collab. [nucl-ex/0403005]

[Albacete, Armesto, Kovner, Salgado, Wiedemann, Phys.Rev.Lett.92:082001,2004]

y=0

y=2

Conclusions

We have now a picture that is universally applicable to all hadron interactions at high energies, in the whole x,kt plane (except the truly non-perturbative region

kt<LQCD).

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, Qs=3-4 GeV): even better

conditions for CGC.