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Quark Matter 2009, Knoxville
Student Day
Initial State and saturation
Marzia Nardi
INFN Torino (Italy)
[email protected]
WHY ?
General interest:
Interest in HIC:
• Unsolved problems of
QCD
• QCD out of the
perturbative regime
• Looking for universal
properties
• Understanding the
beginning to
understand the end
• Correct interpretation
of experimental data
Hadronic interactions at very high energies
The total hadron-hadron
Xsection at high energies is
among the unsolved
problems of QCD, nonperturbative aspect.
Froissart bound (unitarity) :
 ~ ln E
2
as
E 
Is this behaviour universal ?
Looking for universal properties…
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.
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 the
gluon 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
Q
• Transverse area of a parton ~
• Cross section :  ~ as/Q2
• Parton density: r = xG(x,Q2)/pRA2
• Partons start to overlap when SA~NA (r  ~1)
• 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.
1/Q2
• 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
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.
Yang Mill eq. :
• 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:
Qs2~(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 HERA
The 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 :
Qs2(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)
Parton production
We assume that the number of produced particles is :
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
First comparison to data
√s = 130 GeV
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 dependence : pp and AA
D. Kharzeev, E. Levin, M.N.
hep-ph / 0408050
(Nucl. Phys. A)
Rapidity dependence
• Formula for the inclusive production:
4pN c 1 T 2
d
2
2
E 3 = 2
dk
a
j
(
x
,
k
)
j
(
x
,
(
p
k
)
T s A1
1 T
A2
2
T)
2 
d p N c - 1 pT
p
[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
Saturation region: SA/as
Perturbative region: as/pT2
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:
Q -y
x1 =
e
s
Q y
x2 =
e
s
Results : rapidity dependence
Au-Au Collisions at RHIC
PHOBOS
W=200 GeV
Predictions for LHC
• Our main uncertainty : the energy
dependence of the saturation scale.
• Fixed as :
• Running as :
Centrality dependence / LHC
Pb-Pb collisions
at LHC
Solid lines : constant as
dashed lines : running as
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]
y=0
[Albacete, Armesto, Kovner,
Salgado, Wiedemann,
Phys.Rev.Lett.92:082001,2004]
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.