Transcript Color Strings

Percolation
&
Deconfinement
Brijesh K Srivastava
Department of Physics
Purdue University
USA
ICPAQGP 2010
Goa, Dec. 6-10, 2010
1
Percolation : General
The general formulation of the percolation problem is concerned
with elementary geometrical objects placed at random in a
d-dimensional lattice. The objects have a well defined connectivity
radius λ, and two objects are said to communicate if the distance
between them is less than λ.
One is interested in how many objects can form a cluster of
communication and, especially , when and how the cluster become
infinite. The control parameter is the density of the objects or the
dimensionless filling factor ξ. The percolation threshold ξ = ξc
corresponding to the minimum concentration at which an infinite
cluster spans the space.
Thus the percolation model exhibits two essential features:
Critical behavior
Long range correlations
2
Percolation : General
Rev. Mod. Phys. 64, 961 (1992).
3
Parton Percolation
De-confinement is expected when the density of quarks and gluons becomes so
high that it no longer makes sense to partition them into color-neutral hadrons,
since these would overlap strongly.
We have clusters within which color is not confined -> De-confinement is thus
related to cluster formation.
This is the central topic of percolation theory, and hence a connection between
percolation and de-confinement seems very likely.
Parton distributions in the
transverse plane of nucleus-nucleus
collisions
H. Satz, Rep. Prog. Phys. 63, 1511(2000).
H. Satz , hep-ph/0212046
4
Color Strings
Multiparticle production at high energies is currently described
in terms of color strings stretched between the projectile and target.
Hadronizing these strings produce the observed hadrons.
The no. of strings grow with energy and the no. of participating nuclei
and one expects that interaction between them becomes essential.
This problem acquires even more importance, considering the idea that
at very high energy collisions of heavy nuclei (RHIC) may produce
Quark-gluon Plasma (QGP).
The interaction between strings then has to make the system evolve
towards the QGP state.
5
Color Strings
 At low energies, valence quarks
of nucleons form strings that
then hadronize  wounded
nucleon model.
 At high energies, contribution
of sea quarks and gluons
becomes dominant.
 Additional color strings formed.
1. Dual Parton Model (DPM): A. Capella et al., Phys. Rep. 236, 225 (1994).
2. A. Capella and A. Krzywicki , Phys. Rev. D184,120 (1978).
6
Parton Percolation
In two dimensions, for uniform string density, the percolation
threshold for overlapping discs is:
c  1.18
Satz, hep-ph/0007069
= critical percolation density
The fractional area covered by
discs at the critical threshold is:
1 e
 c
7
Color Strings + Percolation = CSPM
Multiplicity and <pT2 > of particles
produced by a cluster of n strings
Multiplicity (mn)
mn  F ( ) N s m1
F ( ) 
1  e 

Average Transverse Momentum
 pT2  n  pT2 1 / F ( )
= Color suppression factor
(due to overlapping of discs).
ξ is the percolation density parameter.
N s S1

SN
Ns = # of strings
S1 = disc area
SN = total nuclear
overlap area
M. A. Braun and C. Pajares, Eur.Phys. J. C16,349 (2000)
M. A. Braun et al, Phys. Rev. C65, 024907 (2002)
8
CSPM
Using the pT spectrum to calculate ξ
To compute the pT distribution, a parameterization of the pp data
is used:
dN
2
dpt

a
( p0  pt )
n
a, p0 and n are parameters fit to
the data.
This parameterization can be used for nucleus-nucleus
collisions, accounting for percolation by:
 nS1

 S n Au  Au
p0  po 
 nS1

S n pp








1
4
In pp at low energy,
<nS1/Sn>pp = 1 ± 0.1,
due to low string overlap
probability in pp
collisions.
M. A. Braun, et al.
hep-ph/0208182.
9
Percolation and Color Glass Condensate
Both are based on parton coherence phenomena.
Percolation : Clustering of strings
CGC
: Gluon saturation
 Many of the results obtained in the framework of percolation of strings
are very similar to the one obtained in the CGC.
 In particular , very similar scaling laws are obtained for the product and
the ratio of the multiplicities and transverse momentum.
 Both provide explanation for multiplicity suppression and <pt>
scaling w/ dN/dy.
10
Percolation and Color Glass Condensate
Universal relation between dn/dy and <pt>
Percolation:
Phys. Lett. B 581, 156(2004)
 rs  p12 
c   
 R1  n1 
1 dN
 pt  c
2/ 3
N dy
2
A
1
2
CGC:
2
 pt 
1 dN
2/ 3
N dy
Phys. Lett. B 514, 29 ( 2001)
Nucl. Phys. A 705, 494 (2002)
A
Momentum Qs which establishes the scale in CGC with the corresponding
one in percolation of strings
For large value of ξ
hep-ph/1011.1099
2

2
Qs  
k  pt 1 F ( )  1  e
2
Qs 
F ( )

11
pp fit parameter
Parametrization of pp UA1 data at 130 GeV
from 200, 500 and 900 GeV
ISR 53 and 23 GeV
QM 2001 PHENIX
p0 = 1.71 and n = 12.42
Ref: Nucl. Phys. A698, 331 (2002).
STAR has also extrapolated UA1 data from
200-900 GeV to 130 GeV
p0 = 1.90 and n = 12.98
Ref: Phys. Rev. C 70, 044901( 2004).
UA1 results at 200 GeV
p0 = 1.80 and n = 12.14
Ref: Nucl. Phys. B335, 261 ( 1990)
12
Percolation density parameter ξ
STAR Collaboration, Nukleonika 51, S109 (2006)
Au+Au
200 GeV
62 GeV
F ( ) 
1  e 

ξc=1.2
STAR Preliminary
Npart
Now the aim is to connect F(ξ) with Temperature and Energy density
13
Relation between Temperature
&
Color Suppression factor F(ξ)
Ref :
1. Fluctuations of the string and transverse mass distribution
A. Bialas, Phys. Lett. B 466 (1999) 301.
2. Percolation of color sources and critical temperature
J. Dias de Deus and C. Pajares, Phys.Lett B 642 (2006) 455
14
Temperature
It is shown that quantum fluctuations of the string tension
can account for the ‘thermal” distributions of hadrons created in the
decay of color string.
The tension of the macroscopic cluster fluctuates around its mean
value because the chromoelectric field is not constant . Assuming a
Gaussian form for these fluctuations one arrives at the probability
distribution of transverse momentum:
2

2
k
P(k )dk 
exp  
2
 2 k2
 k


dk


T 
pt2
1
2 F ( )
which gives rise to thermal distribution

dn
~ exp   p
2

d p

2 
k2 

With temperature
T
k2
2
Average transverse
momentum of a single
string
Average string tension
15
Temperature
T 
pt2
1
2 F ( )
At the critical percolation density  c  1.2
Tc
= 167 MeV
For Au+Au@ 200 GeV 0-10% centrality ξ = 2.88 T = 193 MeV
1. Chemical freeze out temperature
P. Braun-Munzinger et al., Phys. Lett. B596, 61 (2006)
2. Universal chemical freeze out temperature
F. Becattini et al, Eur. Phys. J. C66, 377 (2010).
16
Energy Density
Bjorken Phys. Rev. D 27, 140 (1983)
3 dN c  mt  1
3

GeV / fm
2 dy
A  pro
Proper Time
Transverse overlap area
 pro
is the QED production time for a boson which can be scaled from
QED to QCD and is given by
 pro 
2.405 
 mt 
Introduction to high energy heavy ion collisions
C. Y. Wong
17
Energy Density
STAR Collaboration has published charged particle multiplicities and
transverse momentum of particles in for Au+Au collisions at various energies.
It is found that
 
1.
2.
3.
D. Cebra , STAR Collaboration, QM08
B. I. Abelev et al, STAR Collaboration, Phys. Rev.C79, 34909 (2009)
B. I. Abelev et al, STAR Collaboration, Phys. Rev.C81, 24911(2010)
18
Equation of state

T4
s
T3
Cs2
vs
T
Tc
19
Energy Density
0-10% centrality
Lattice QCD : Bazavov et al, Phys. Rev. D 80, 014504(2009).
20
Thermodynamic Relations
  G (T )
2
30
T4
pC 
Ts  (  p )
2
s
dT
C 
s
d
2
s
21
Sound Velocity
dT
2 
C 
(1  Cs )
d
T
2
s
dT dT dq d

d dq d d
Cs2
C s2

q1/ 2  F ( )
 e 
(1  Cs2 )( 1 / 4)

 1 e


 1


  c
: an analytic function of ξ for the EOS of QGP for T≥ Tc
Finally, we have energy density, temperature and sound velocity
as a function of color suppression factor F(ξ) or percolation density
parameter ξ.
22
Velocity of Sound expressed as Cs2
Lattice QCD : Bazavov et al, Phys. Rev. D 80, 014504(2009).
Physical hadron gas: Castorina et al, arXiv:0906.2289/hep-ph
23
Entropy Density
0-10% centrality
Lattice QCD : Bazavov et al, Phys. Rev. D 80, 014504(2009).
24
Summary
CGC Qs  
Sound velocity C s2
Shear viscosity η ?
Color Suppression
Factor
F(ξ)
Energy density ε/T4
Entropy density s/T3
Temperature
H. Satz : Quantum Field Theory in Extreme Environments,
Paris , April 2009
Clustering and percolation can provide a conceptual basis for the
QCD phase diagram which is more general than symmetry
breaking .