Download Percolation of Color Sources and the determination of the

Percolation of Color Sources and the determination of the
Equation of State of the Quark-Gluon Plasma (QGP) produced in
√
central Au-Au collisions at sN N =200 GeV.
Abstract
The Color String Percolation Model (CSPM) is used to determine
the equation of state of the QGP produced in central Au-Au collisions
√
at sN N = = 200 A GeV. Here the initial density of interacting colored strings exceeds the percolation threshold and forms a single dense
cluster. The event energy E and the cluster density are proportional to
ξ = 2.88 the percolation parameter, which is derived from the particle
momentum spectrum. In CSPM the maximum entropy value of the
temperature T (ξ) is proportional to the interacting color string tension values. The single string tension parameter is determined by the
measured value of the universal critical temperature Tc = 167.7 ± 2.6
MeV at the percolation threshold (ξc =1.2). The experimental value
of the particle multiplicity density 3/2 dNc /dη, the average transverse
momentum < pt > and the QE2 formation time τpro = 1.19 fm/c are
used to convert the energy E to the ideal fluid energy density ε(ξ).
The ξ = 2.88 value determines the initial temperature T (ξ) = 194.0 ±
3.0 MeV, in reasonable agreement with recent experimental data. The
number of the quark-gluon degrees of freedom equals 37.4±3.6 consistent with the formation of a ∼ 2+1 flavor QGP. CSPM constructs
an analytic expression for the equation of state, the sound velocity
Cs2 (ε, T ), of the QGP. The CSPM Cs2 (ε, T ) and the bulk thermodynamic values ε/T 4 and s/T 3 are in excellent agreement with recent
lattice QCD simulations(LQCD) when plotted as a function of T/Tc .
1
Introduction
All high energy soft multi-hadron interactions exhibit thermal patterns of
abundances with the same limiting temperature, independent of the center
of mass energy [1]. Statistical thermal analyses have had success in fitting
the data with a minimum of parameters[1]. In heavy ion collisions it may be
plausible that multiple parton interactions produce a thermalized system. In
1
(p, p), (p, p̄) and (p, A) collisions multiple parton interactions are not likely
to thermalize the system. In (e+ e− ) annihilation where thermal behavior is
observed, the multi-parton mechanism may also have failed [2, 3]. The fundamental question arises; is there a universal mechanism which produces these
thermal distributions? The Schwinger barrier penetration mechanism with
string tension fluctuations was proposed and leads to an early (at birth) thermalized system [4]. The Color String Percolation Model (CSPM) describes
the initial collision of two heavy ions in terms of a dense pattern of interacting colored strings[5, 6]. The two dimensional (2D) percolation threshold
identifies the hadron to quark-gluon connectivity phase transition and the
asociated CSPM temperature. This connected system is considered to be
a perfect fluid which subsequently expands using Bjorken1D ideal hydrodynamics [7]. Both the initial energy density and temperature are determined
by the percolation parameter ξ, which is derived from the particle momentum spectrum. The equation of state Cs2 (ξ) for Quark-Gluon Plasma is due
to the CSPM energy density- temperature relationship.
2
Evidence for the the Universal Temperature Mechanism
The comparative Analysis of Statistical Hadron Production indicates that
the temperatures the same
√ for different initial collision configurations and
independent of energy ( s)[1]. Table 1 shows the average value of temperature obtained from different systems. The average of all the values is 167.7 ±
2.6 MeV. The experimental critical temperature at the percolation threshold
ξc = 1.2 determines the key parameter in CSPM, the single string tension
√
constant h p2t i1 = 207.2 ± 3.3 MeV.
Table 1: Summary of the temperature obtained from high energy p-p, Au-Au
and e+ e− collisions[1].
Collision
pp
Au-Au
e+ e−
Temperature[MeV]
169.8±4.2
168.5±4.0
164.7±0.9
Average relative deviation data vs fit[%]
13.2
12.1
10.8
Average relative error of data [%]
18
10
5.7
2
χ /dof
15.0/8 ≃ 1.9 22.28 ≃2.8 41.5/9 ≃ 4.6
2
3
The 2D Percolation Phase Transition from
Hadronic To Quark-Gluon Matter
Consider the nuclear transverse collision area for Au+Au 0-10% central collisions SN = 143.5 f m2 , which is randomly sprinkled with N colored strings
each with cross-sectional area S1 =πr02 , where r0 is the radius of a single
string. With an increasing number N of strings there is a progression from
isolated individual strings to clusters and then to a large cluster which suddenly spans the area SN . This is the percolation threshold which defines the
onset of a connected system of colored matter, which identifies the connectivity phase transition from hadronic matter (clusters) to the QGP (spanning
cluster). Figure 1 shows the connectivity between the clusetrs.
Figure 1: The connectivity phase transition, in 2D percolation [8].
The percolation parameter is defined as
ξ=
NS1
SN
(1)
At ξ =1 the total string cross-sectional areas S1 equals SN . In two dimensions the critical cluster which spans the collision area, appears for ξc ≥ 1.2
. For ξ > ξc , the fraction of SN covered by the spanning cluster and the
density of the spanning cluster increase.
P
4
The Color string percolation model CSPM
and the intraction of strings
In CSPM, the interactions of the overlapping areas reduces the hadron multiplicity µ and increases the average transverse momentum hp2T iN of these
P
hadrons, so that the total transverse momentum for N strings hpT i1 = N
3
hpT i1 is conserved. Two over-lapping strings are shown in Figure 2. There
are two fractional areas S1 and S3 that do not overlap and one region S2
that is a overlap area[5, 6].
Figure 2: Two discs each with area σ0 [5].
Consider the string areas σ0 . Let ρ be the average color density of the
string in the transverse plane and that Q0 is a color of each string then ρ
= Q0 /σ0 . The color in the overlap area will be a vector sum of the two
overlapping colors ρ S2. In this summation the total color squared should be
′2
conserved: Q22 = ( Qoverlap
+ Q2overlap ). The observables, the multiplicity µ,
and the average transverse momentum squared hp2t i are directly related to
the field strength in the string and thus to the generating color. Let µ0 be the
multiplicity of a single string and hp2t i1 is the average transverse momentum
squared of a single string. For N total strings
µ hp2t i
N=
µ0 hp2t i1
(2)
s
(3)
a direct relationship between multiplicity and transverse momentum. This is
the conservation of the total transverse momentum produced. The evaluation
of the color string damping factor F(ξ), which reduces the independent string
multiplicity from Nµ0 to the interacting string value µ= F(ξ) Nµ0 computed
in thermodynamic limit.
F (ξ) =
1 − e−ξ
ξ
The expected multiplicty using CSPM and from the experiment is shown
in Fig.3. An additional important check of the interactive string approach
was provided by the measurement of the Long Range forward backward multiplicitiy Correlations (LRC) by the STAR group at RHIC[10]. The LRC
were predicted by PSM and CSPM as shown in Fig.4.
4
Figure 3: Comparison of the CSPM multiplicity with experiment.
dashed red lines indicate the experimental values[9].
The
0.8
0-10% Au+Au 200GeV
Data
PSM
0.7
HIJING
FB Correlation Strength b
0.6
0.5
0.4
0.3
0.2
p+p
0.1
0
0.5
1
1.5
2
∆η
Figure 4: Long Range forward backward multiplicity Correlations (LRC)
were predicted by PSM and CSPM[10].
5
5
The empirical value of the percolation parameter ξ for central Au-Au collisions.
To evaluate the percolation density parameter, ξ, a parametrization of p-p
events at 200 GeV is used to compute the pT distribution,
dNc /dp2T = a/(p0 + pT )α
(4)
where a, p0 , and α are parameters to fit the data. This parameterization also can be used for nucleus- nucleus collisions to take into account the
interactions of the strings.
p0 → p0
hnS1 /Sn iAu−Au
hnS1 /Sn ipp
!1/4
(5)
The thermodynamic limit is used to evaluate:
h
nS1
i = 1/F 2 (ξ)
Sn
(6)
Figure 5: The expected Pt spectrum from CSPM compared to PHENIX
experimental data for central (0-5%) Au-Au collisions at 130 GeV [6].
6
The pt spectrum from PHENIX experimental data for central 0-5% Au√
Au collsions at sN N =130 GeV (black dots) compared to CSPM predictions(solid line) is shown in Fig.5. The standard expected distribution using
the p-p parametrization and the average value of binary collisions is shown
by the dotted dashed line[6].
The thermodynamic limit means letting SN and N→ ∞ while keeping
ξ fixed[9]. In pp collisions hnS1 /Sn ipp = 1. 0 ± 0.1 due to the low string
overlap probability in pp collisions. F (ξ) is the color compression factor
and is realted to ξ via Eq.3. The factor 1 − e−ξ in Eq. 3 corresponds to
the fractional area of the transverse collision area SN covered by the largest
spanning cluster[11]. The data utilized in this analysis was acquired by the
STAR experiment at RHIC. Only data from within 1 units of pseudo-rapidity
with greater than 10 fit points In the main STAR detector, the Time Projection Chamber, was used. The events studied were minimum bias events, with
standard STAR centrality cuts as defined by the trigger detectors and overall
event multiplicities, which increase with increasing collision centrality. The
longitudinal (z) vertex was constrained to within ± 30 centimeters of the
primary event vertex. The distance of closest approach (d.c.a.) of primary
tracks to the event vertex was less than 3.0 cm. The transverse momentum
of charged particles covreted 0.2 < pt <1.2 GeV [12].
Figure 6 shows a plot of F (ξ) as function of ξ for central Au+Au collisions
at different energies [11]. The fractional area covered by the overlapping
string is shown in Fig.8.
6
Fluctuations about the average value of the
string tension constant and the transverse
mass distribution
The Schwinger Mechanism (SM) for hadron production is:
2
q
dnk /d2pt ∼ e−mt /k
2
(7)
where mt = (m2t + m2 ) and k is string tension constant. However the
universal experimental ”thermal” distribution is
dn/d2 pt ∼ e−mt /T
7
(8)
1
0.95
STAR Preliminary
-ξ
F(ξ)= 1-e
ξ
0.9
F(ξ)
0.85
0.8
Au+Au, 19.6 GeV
0.75
↑
F(ξc)
0.7
Au+Au, 62.4 GeV
0.65
Au+Au, 200 GeV
0.6
0.55
0.5
0
0.5
1
↓
1.5
2
ξ
2.5
3
3.5
4
Figure 6: Color suppression factor F(ξ) vs ξ for different energies from
√
Au+Au collsions ξ=2.88±0.09 for 0-10% Au+Au collisions at sN N =200
GeV [11].
These equations are compatible if the string tension has a Gaussian fluctua2
2
tion about its average value
∼ e(−k /2hk i) then the temperature is related to
q
string tension by T = hk 2 i/2π. Then Schwinger Gaussian distribution is
expressed in therms of p2t
2
dn/dp2t ∼ e−πpt /x
2
(9)
with the average value hpt i. The Gaussian fluctuations transform SM into
the thermal distribution:
√
2
(10)
dn/dp2t ∼ e−pt / 2π/hx i
q
q
with hx2 i = πhp2t i1 /F (ξ) where hp2t i1 = hk 2 /πi = 207.2±5 MeV is determined by the experimental value of the universal critical temperature
Tc =167.7±2.6 at the percolation threshold ξ = 1.2.
T (ξ) =
v
u
u
t
8
hp2t i1
2F (ξ)
(11)
1.2
Fractional area, 1-e
-ξ
1
Au+Au at 200 GeV
0-10%
0.8
ξc
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
ξ
3
3.5
4
Figure 7: Fractional area of the largest ( spanning) cluster vs the string
density parameter ξ. The critical value is at ξ =1.2. The value of 0-10%
central Au+Au collisions at 200 GeV is shown as blue solid circle[11].
The key point is that the percolation parameter ξ determines the cluster
size distribution, the temperature T(ξ) and the total momentum (energy) in
the collision [13]. Above ξc =1.2 the size and density of the spanning cluster
increases. We use, the empirical value of ξ = 2.88 to determine the initial
temperature Ti = 194.0±2.5 MeV of the quark gluon plasma in reasonable
agreement with Ti = 221±stat 19 ± 19sys from the enhanced direct photon
experiment [14].
7
The Determination of Initial Energy Density for Au-Au ( 0-10%) central collisions.
The initial energy density in is given by Bjorken equation[7]:
dN
3 dηc hpt i
εi =
2 SN τpro
9
(12)
where τpro is the QE2(τ ,z) quantum electrodynamics production time for a
boson which can be scaled from electrodynamics to quantum chromodynamics[15].
c
The initial energy density εi in |η| ≤0.5 from STAR data [12]. (3/2)( dN
)hpt iN
dη
=(3/2)(639±45)(0.40±0.0026)= 383.7±27 GeV. The transverse overlap area
SN = 143.25±3.5 f m2 ,τpro = 2.405h̄
= 1.19 ±0.009 fm/c [15, 16].
hpt i
The initial energy density εi = 383.7
τ
= 2.25±0.16 GeV/f m3 . The
SN pro
number of degree of freedom (DOF) is related to the energy denisty by the
expression:
G(T )π 2 Ti4
(13)
εi =
30(h̄c)3
This gives 37.4±3.6 DOF at Ti = 194.0±3 MeV.
8
Equation of State
Equation of state (EOS) is one of the most important inputs of the hydrodynamical models. Determining the EOS of hot, strongly interacting matter is
one of the main challanges of strong interaction physics[17]. Recently, LQCD
presented high statistics results for the bulk thermodynamic observables, e.g.
pressure energy density , entropy density etc. along with the sound velocity.
CSPM has been explored to calculate these quantities to compare with the
LQCD results.
8.1
The Sound Velocity Cs2 determination using CSPM
Energy Density ε(ξ) and Temperature T(ξ)[17, 15]
.
The QGP according to CSPM is born in local thermal equilibrium because
the temperature is determined at the string level. After τpro a ”perfect fluid”
may evolve according to the laws of hydrodynamics.
1 dT
= −Cs2 /τ
T dτ
dT dε
dT
=
dτ
dǫ dτ
dǫ
= −T S/τ
dτ
10
(14)
(15)
(16)
dT
s = Cs2
dǫ
(17)
where τ is the proper time and s is the entropy density.
Cs2 = (1 + Cs2 )
G(T )π 2Ti3
ε
= (1 + Cs2 )
3
30(h̄c)
T
(18)
The entropy density is s = (1 + Cs2 )(ξ/1.278)/T , since ξ = 2.88/2.25 =
1.278ε. Hence:
Cs2
dT
(1 + Cs2 )(ξ/2.278) dT
= (s)
=
dε
T
dε
(19)
Let q 1/2 = F (ξ)
dT
dT dq dξ
=
dǫ
dq dξ dǫ
(20)
ξe−ξ
− 1)
(21)
1 − e−ξ
for ξ ≥ 1.2. An analytic function fo ξ for the equation of state of the QGP
for T ≥ Tc Since a critical point occurs at a critical density and temperature
we set the density parameter e−ξ to the critical value e−1.2 and compare Cs2
for T ≤ Tc with the Physical Hadron Resonance Gas(PHRG) values (mass
cut off M ≤ 2.5 GeV)[18].
Cs2 = (1 + Cs2 )(−0.25)(
Cs2 = (1 + Cs2 )(0.25)(
e−ξ
− 1)
F (ξ)2
(22)
Figure 8 shows the comparson of sound velocity from CSPM and LQCD.
The LQCD values are obtained using the EOS of 2+1 flavor QCD at finite
temperature with physicsal strange quark mass and almost physical light
quark masses [17].
8.2
Bulk thermodynamics Comparison of CSPM and
LGS
The bulk thermodynamics quantities are represented by energy and entropy
density ε/T 4 and s/T 3 respectively. Figures 9 and 10 show these two quantities as obtained from CSPM along with the LQCD calculations [17].
11
0.4
Color string percolation model
Limiting value
0.35
0.3
Lattice QCD simulation
2+1 flavor
0.25
cs2
0.2
0.15
0.1
c2s =0.14
Physical hadron resonance gas
0.05
0
0.6
0.8
1
1.2
1.4
1.6
T/Tc
1.8
2
Figure 8: The speed of sound from CSPM and the physical hadron gas versus
T /TcCSP M and Lattice QCD speed of sound versus T /TcLGS [17].
25
Color string percolation model
sSB/T
3
20
15
ξ = 2.88
s/T
3
Limiting value
ξc = 1.2
10
5
0
Lattice QCD simulation
2+1 flavor
0.5
1
1.5
T/Tc
2
2.5
Figure 9: The entropy density from CSPM versus T /TcCSP M and Lattice
QCD entropy density vs T /TcLGS [17].
12
Upper limiting values for ε/T 4 and s/T 3 for ξ ≫ ξc ∼ ξ/(ξ 1/4 )4 = 1.
For ξ = 2.88 ,ε = 2.25GeV=0.7813 ξ and T= 0.1463ξ 1/4 ε/T 4 = 13.04±1.2
and G(T )L40.4±3.6 DOF consistent with ∼2+1 flavor QGP. s/T 3 = (1 +
Cs2 )ε/T 4 =17.34±1.6. Lower limit for G(T )LL = 3(pions). ε/T 4 = 0.987 and
s/T 3 =1.16
18
16
14
ε/T 4
12
3p/T
Limiting value
4
ξ = 2.88
10
ε/T 4
εSB/T 4
Color string percolation model
8
ξc = 1.2
6
Lattice QCD simulation
2+1 flavor
4
2
0
0.5
1
1.5
T/Tc
2
2.5
Figure 10: The energy density from CSPM versus T /TcCSP M and Lattice
QCD energy density vs T /TcLGS [17].
9
Discussion
The initial success of the Color String Percolation model was to correctly
describe the hadron multiplicities and momentum spectra. The prediction
and observation of the long range forward-backward multiplicity correlations
lent further support to this approach. CSPM at ξ > ξc exhibits gluon saturation effects similar to the Color Glass Condensation
Model (CGC) [19]. The
q
saturation scale Qs in CGC corresponds to hpt i1 /F (ξ) in CSPM. CSPM
constructs analytic function for the equation of state of the QGP, which is in
agreement with lattice QCD, when the the results are plotted with respect
to T /TcCSP M T /TcLQCD . The critical value of the sound velocity Cs2 =0.14 in
13
agreement with the physical resonance gas value. The high temperature limit
for Cs2 = 0.33 is the expected clasical value. The percolation critical point is
considered to be a continuous phase transition. In central Au-Au collisions
√
at sN N = 200 GeV the QCD to hadron phase transition for baryon density µB =0 is believed to be a cross-over transition which also does not have
a latent heat [20]. The CSPM EOS should correctly describe the QCD to
hadron cross-over transition.
10
Open Questions
Two fundamental questions remain. Multiple constituent interactions lead
to thermal systems in atomic and molecular thermodynamics. Multiple parton interactions are expected to lead to thermal systems in all high energy
collisions. In heavy ion collisions it may be plausible that multiple parton
interactions produce a thermalized system. In (p, p), (p, p̄) and (p, A) collisions multiple parton interactions are not likely to thermalize system. In e+ e−
annihilation where thermal behavior is observed, the mutiparton mechanism
may also have failed. The fundamental question arises : is there a universal
mechanism which produces these thermal distributions? Color confinement
could form an event horizon for the quantum mechanical tunneling of quarks
and gluons in analogy to the Hawking-Unruh black hole radiation which produces a thermal spectrum. In CSPM the Schwinger quantum mechanical
tunneling mechanism combined with the Gaussian string tenstion fluctuations also produces a thermal spectrum. This remains an open question !
The CSPM model where the percolation strings in the initial collision combined with the Schwinger mechanism indicates that the QGP is formed in
√
central Au-Au collisions at sN N = 200 GeV. It also suggests that the QGP
is produced in all high multiplicity high energy collisions when string density
exceeds the connectivity percolation transition.
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15