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STAR
Pion Entropy andPhase Space Density at RHIC
Pion Entropy andPhase Space Density at RHIC
John G. CramerDepartment of Physics
University of Washington, Seattle, WA, USA
John G. CramerDepartment of Physics
University of Washington, Seattle, WA, USA
Second Warsaw Meeting on Particle Correlations and
Resonancesin Heavy Ion CollisionsWarsaw University of
TechnologyOctober 16, 2003
Second Warsaw Meeting on Particle Correlations and
Resonancesin Heavy Ion CollisionsWarsaw University of
TechnologyOctober 16, 2003
October 16, 2003 John G. Cramer2STAR
Phase Space Density: Definition & Expectations
Phase Space Density: Definition & Expectations
Phase Space Density - The phase space density f(p,x) plays a fundamental role in quantum statistical mechanics. The local phase space density is the number of pions occupying the phase space cell at (p,x) with 6-dimensional volume p3x3 = h3.
The source-averaged phase space density is f(p)∫[f(p,x)]2 d3x / ∫f(p,x) d3x, i.e., the local phase space density averaged over thef-weighted source volume. Because of Liouville’s Theorem, for free-streaming particles f(p) is a conserved Lorentz scalar.
At RHIC, with about the same HBT source size as at the CERN SPS but with more emitted pions, we expect an increase in the pion phase space density over that observed at the SPS.
October 16, 2003 John G. Cramer3STAR
hep-ph/0212302
Entropy: Calculation & ExpectationsEntropy: Calculation & ExpectationsEntropy – The pion entropy per particle S/N and the total pion entropy at midrapidity dS/dy can be calculated from f(p). The entropy S of a colliding heavy ion system should be produced mainly during the parton phase and should grow only slowly as the system expands and cools.
Entropy is conserved during hydrodynamic expansion and free-streaming. Thus, the entropy of the system after freeze-out should be close to the initial entropy and should provide a critical constraint on the early-stage processes of the system.
nucl-th/0104023 A quark-gluon plasma has a large number of degrees of freedom. It should generate a relatively large entropy density, up to 12 to 16 times larger than that of a hadronic gas.
At RHIC, if a QGP phase grows with centrality we would expect the entropy to grow strongly with increasing centrality and participant number.
Can Entropy provide the QGP “Smoking Gun”??
October 16, 2003 John G. Cramer4STAR
Pion Phase Space Density at Pion Phase Space Density at MidrapidityMidrapidity
Pion Phase Space Density at Pion Phase Space Density at MidrapidityMidrapidity
The source-averaged phase space density f(mT) is the dimensionless number of pions per 6-dimensional phase space cell h3, as averaged over the source. At midrapidity f(mT) is given by the expression:
λ
1
RRR
πλ
ymmπ2
N
E
1)m(
LOS
3
TT
2
πT
)(
c
dd
df
Momentum Spectrum HBT “momentumvolume” Vp
PionPurity
Correction
Jacobianto make ita Lorentz
scalar
Average phasespace density
October 16, 2003 John G. Cramer5STAR
Changes in PSD Analysis since QM-2002
Changes in PSD Analysis since QM-2002
At QM-2002 (Nantes) we presented a poster on our preliminary phase space density analysis, which used the 3D histograms of STAR Year 1 HBT analysis from our PRL. At QM-2002 (see Scott Pratt’s summary talk) we also started our investigation of the entropy implications of the PSD. This analysis was also reported at the INT/RHIC Winter Workshop, January – 2003 (Seattle).
CHANGES: We have reanalyzed the STAR Year 1 data (Snn½ = 130 GeV) into 7
centrality bins for |y| < 0.5, incorporating several improvements :
1. We use 6 KT bins (average pair momentum) rather than 3 pT bins (individual pion momentum) for pair correlations (better large-Q statistics).
2. We limit the vertex z-position to ±55 cm and bin the data in 21 z-bins, performing event mixing only between events in the same z-bin.
3. We do event mixing only for events in ±300 of the same reaction plane.4. We combined and correlations (improved statistics).5. We used the Bowler-Sinyukov-CERES procedure and the Sinyukov analytic
formula to deal with the Coulomb correction.(We note that Bowler Coulomb procedure has the effect of increasing radii and reducing , thus reducing the PSD and increasing entropy vs. QM02.)
We also found and fixed a bug in our PSD analysis program, which had the effect of systematically reducing <f> for the more peripheral centralities. This bug had no effect on the 0-5% centrality.
October 16, 2003 John G. Cramer6STAR
RHIC Collisions as Functions of Centrality
RHIC Collisions as Functions of Centrality
50-80% 30-50% 20-30% 10-20% 5-10% 0-5%
At RHIC we can classifycollision events by impact parameter, based on charged particle production.
Participants
Binary Collisions
Frequency of Charged Particlesproduced in RHIC Au+Au Collisions
of Total
October 16, 2003 John G. Cramer7STAR
0.05 0.1 0.15 0.2 0.25 0.3
150
200
300
500
700
1000
1500
2000
016
Vp
VeG
3 Corrected HBT Momentum Volume
Vp /½
Corrected HBT Momentum Volume Vp /½
LOS
3
p RRR
πλλV
)( c
STAR Preliminary
Central
Peripheral
mT - m (GeV)
0-5%
5-10%
10-20%
20-30%
30-40%
40-50%
50-80%
Centrality
Fits assuming:
Vp ½=A0 mT3
(Sinyukov)
October 16, 2003 John G. Cramer8STAR
0.1 0.2 0.3 0.4 0.5 0.6mT m
5
10
50
100
500
1000
d2 N2m Tmd
Tyd
Global Fit to Pion Momentum Spectrum
Global Fit to Pion Momentum Spectrum
We make a global fit of the uncorrected pion spectrum vs. centrality by:
(1) Assuming that the spectrumhas the form of an effective-TBose-Einstein distribution:
d2N/mTdmTdy=A/[Exp(E/T) –1]
and
(2) Assuming that A and T have aquadratic dependence on thenumber of participants Np:
A(p) = A0+A1Np+A2Np2
T(p) = T0+T1Np+T2Np2
Value ErrorA0 31.1292 14.5507A1 21.9724 0.749688A2 -0.019353 0.003116T0 0.199336 0.002373T1 -9.23515E-06 2.4E-05T2 2.10545E-07 6.99E-08
STAR Preliminary
October 16, 2003 John G. Cramer9STAR
0.1 0.2 0.3 0.4mTm
0.1
0.2
0.3
0.4
f
Interpolated Pion Phase Space Density f at S½ = 130 GeV
Interpolated Pion Phase Space Density f at S½ = 130 GeV
Central
Peripheral
NA49
STAR Preliminary
Note failure of “universal” PSDbetween CERN and RHIC.}
HBT points with interpolated spectra
October 16, 2003 John G. Cramer10STAR
0.1 0.2 0.3 0.4 0.5 0.6mTm
0.01
0.02
0.05
0.1
0.2
f
Extrapolated Pion Phase Space Density f at S½ = 130 GeV
Extrapolated Pion Phase Space Density f at S½ = 130 GeV
Central
Peripheral
STAR Preliminary
Spectrum points with extrapolated HBT Vp/1/2
Note that for centralities of 0-40% of T, fchanges very little.
f drops only for the lowest 3 centralities.
October 16, 2003 John G. Cramer11STAR
fdxdp
fffffLogfdxdp
xpfdxdp
xpdSdxdp
NS
33
49653
612
2133
33
633 )([
),(
),(
Converting Phase Space Density to Entropy per Particle (1)
Converting Phase Space Density to Entropy per Particle (1)
...)(
)1()1()();,(4
9653
612
21
6
fffffLogf
fLogffLogfdSpxff
Starting from quantum statistical mechanics, we define:
To perform the space integrals, we assume that f(x,p) = f(p) g(x),where g(x) = 23 Exp[x2/2Rx
2y2/2Ry2z2/2Rz
2], i.e., that the source hasa Gaussian shape based on HBT analysis of the system. Further, we make theSinyukov-inspired assumption that the three radii have a momentum dependenceproportional to mT
. Then the space integrals can be performed analytically.This gives the numerator and denominator integrands of the above expressionfactors of RxRyRz = Reff
3mT(For reference, ~½)
An estimate of the average pion entropy per particle S/N can be obtainedfrom a 6-dimensional space-momentum integral over the local phase spacedensity f(x,p):
O(f)
O(f2)
O(f3) O(f4)
f
dS6(Series)/dS6
+0.2%
0.2%
0.1%
0.1%
October 16, 2003 John G. Cramer12STAR
Converting Phase Space Density to Entropy per Particle (2)
Converting Phase Space Density to Entropy per Particle (2)
0
31
0
4
22453
3942
2)8(5
2131
33
4
22453
3942
2)8(5
2133
33
633
][
][
),(
),(
fmpdp
fffffLogfmpdp
fmdp
fffffLogfmdp
xpfdxdp
xpdSdxdp
NS
TTT
LogTTT
T
LogT
The entropy per particle S/N then reduces to a momentum integralof the form:
We obtain from the momentum dependence of Vp-1/2 and performthe momentum integrals numerically using momentum-dependent fits to for fits to Vp-1/2 and the spectra.
(6-D)
(3-D)
(1-D)
October 16, 2003 John G. Cramer13STAR
To integrate over the phase space density, we need a function of pT with some physical plausibility that can put a smooth continuous function through the PSD points. For a static thermal source (no flow), the pion PSD must be a Bose-Einstein distribution:
<f>Static = {Exp[(mTotal )/T0] 1}1. This suggests fitting the PSD with a Bose-Einstein distribution that has been blue-shifted by longitudinal and transverse flow.The form of the local blue-shifted BE distribution is well known.
We can substitute for the local longitudinal and transverse flow rapidities L and T , the average values <L> and <T> to obtain:
Blue-Shifted Bose-Einstein FunctionsBlue-Shifted Bose-Einstein Functions
1
000BlueShift
}][{ 1][][][f T
SinhT
pCoshCosh
T
mExp T
TLT
T
We assume =<L>=0 and consider three models for <T>:
BSBE1: <T> = (i.e., constant average flow, independent of pT)
BSBE2: <T> = (pT/mT) = T (i.e., proportional to pair velocity)
BSBE3: <T> = TT3T
5T7 (minimize S/N)/flow)
1
000Local
}][{ 1][][][f T
SinhT
pCoshCosh
T
mExp T
TLT
T
October 16, 2003 John G. Cramer14STAR
0.05 0.1 0.15 0.2 0.25mTmGeV
0.05
0.1
0.2
0.5
fp
Fits to Interpolated Pion Phase Space Density
Fits to Interpolated Pion Phase Space Density
Central
Peripheral
STAR Preliminary
Warning: PSD in the region measured contributes only about 60% to the average entropy per particle.
HBT points with interpolated spectra
Fitted with BSBE2 function
October 16, 2003 John G. Cramer15STAR
0.1 0.2 0.3 0.4 0.5 0.6mTm
0.001
0.005
0.01
0.05
0.1
0.5
f
Fits to Extrapolated Pion Phase Space Density
Fits to Extrapolated Pion Phase Space Density
Central
Peripheral
STAR Preliminary
Spectrum points with extrapolated HBT Vp/1/2
Each successive centrality reduced by 3/2
Solid = Combined Vp/1/2 and Spectrum fits
Dashed = Fitted with BSBE2 function
October 16, 2003 John G. Cramer16STAR
0.25 0.5 0.75 1 1.25 1.5 1.75 2mTm
0.0001
0.001
0.01
0.1
f
Large-mT behavior of three BSBE Models
Large-mT behavior of three BSBE Models
Solid = BSBE2: T = T
Dotted = BSBE3: 7th order odd polynomial in T
Dashed = BSBE1: T = Constant
Each successive centrality reduced by 3/2
October 16, 2003 John G. Cramer17STAR
Large mT behavior using Radius & Spectrum Fits
Large mT behavior using Radius & Spectrum Fits
0.25 0.5 0.75 1 1.25 1.5 1.75 2mTm
0.0001
0.001
0.01
0.1
f
Solid = fits to spectrum and Vp/1/2
Dashed = BSBE2 fits to extrapolated data
Each successive centrality reduced by 3/2
October 16, 2003 John G. Cramer18STAR
50 100 150 200 250 300 350Npparticipants
3.6
3.8
4
4.2
4.4
4.6
S N
Entropy per Pion from Vp /½ and Spectrum FitsEntropy per Pion from Vp /½ and Spectrum Fits
Central
PeripheralSTAR
Preliminary
Black = Combined fits to spectrum and Vp/1/2
October 16, 2003 John G. Cramer19STAR
50 100 150 200 250 300 350Npparticipants
3.6
3.8
4
4.2
4.4
4.6
S N
Entropy per Pion from BSBE FitsEntropy per Pion from BSBE Fits
Central
PeripheralSTAR
Preliminary
Green = BSBE2: ~ T
Red = BSBE1: Const
Blue = BSBE3: Odd 7th order Polynomial in T
Black = Combined fits to spectrum and Vp/1/2
October 16, 2003 John G. Cramer20STAR
0 0.5 1 1.5 2 2.5 3Tm
2
4
6
8
10
SN
= 0
= m
Thermal Bose-Einstein Entropy per Particle
Thermal Bose-Einstein Entropy per Particle
1]/)[(
1 where
)]()1()1[(S/N
0
0
TmExpf
fdppm
fLnffLnfdppm
TBE
BETT
BEBEBEBETTT
0. 0.3 0.6 0.90.2 7.37481 5.86225 4.30277 2.431810.4 5.13504 4.33169 3.45065 2.251660.6 4.46843 3.89106 3.23476 2.288370.8 4.16727 3.70431 3.16747 2.369671. 4.00256 3.61107 3.15191 2.458511.2 3.90175 3.56032 3.15728 2.543751.4 3.83522 3.53137 3.17146 2.621951.6 3.78887 3.51456 3.18916 2.692441.8 3.75521 3.50489 3.20786 2.755532. 3.72997 3.49958 3.22638 2.8119
The thermal estimate of the entropy per particle can beobtained by integrating a Bose-Einstein distribution over3D momentum:
/mT/m
Note that the thermal-model entropy per particle usually decreases with increasing temperature T and chemical potential .
October 16, 2003 John G. Cramer21STAR
50 100 150 200 250 300 350Npparticipants3.4
3.6
3.8
4
4.2
4.4
4.6
S N
T90 MeV
T120 MeV
T200 MeV
Landau Limit: m0
BPB
Entropy per Particle S/N with Thermal EstimatesEntropy per Particle S/N with Thermal Estimates
Central
Peripheral STAR Preliminary
Dashed line indicates systematicerror in extracting Vp from HBT.
Dot-dash line shows S/N from BDBE2 fits to f
Solid line and points show S/Nfrom spectrum and Vp/1/2 fits.
For T=110 MeV, S/N impliesa pion chemical potential of=44.4 MeV.
October 16, 2003 John G. Cramer22STAR
50 100 150 200 250 300 350Np
500
1000
1500
2000
2500
Sdyd
Snuc
Total Pion Entropy dS/dyTotal Pion Entropy dS/dy
STAR Preliminary
Dashed line indicates systematicerror in extracting Vp from HBT.
Dot-dash line indicates dS/dy fromBSBEx fits to interpolated <f>.
Solid line is a linear fit through (0,0)with slope = 6.58 entropy unitsper participant
Entropy content ofnucleons + antinucleons
P&P
P&P
Why is dS/dylinear with Np??
October 16, 2003 John G. Cramer23STAR
0 50 100 150 200 250 300 350Npparticipants
20
25
30
35
40
45
Sd ydN p23
Initial collision overlap area is roughlyproportional to Np
2/3
Initial collision entropy is roughlyproportional to freeze-out dS/dy.
Therefore, (dS/dy)/Np2/3
should be proportionalto initial entropydensity, a QGPsignal.
Initial Entropy Density: ~(dS/dy)/Overlap Area
Initial Entropy Density: ~(dS/dy)/Overlap Area
Data indicates that the initialentropy density does grow withcentrality, but not very rapidly.
Solid envelope =Systematic errors in Np
Our QGP “smoking gun” seems to beinhaling the smoke!
STAR Preliminary
October 16, 2003 John G. Cramer24STAR
ConclusionsConclusions1. The source-averaged pion phase space density f is very high, in
the low momentum region roughly 2 that observed at the CERN SPS for Pb+Pb at Snn=17 GeV.
2. The pion entropy per particle S/N is very low, implying a significant pion chemical potential (~44 MeV) at freeze out.
3. The total pion entropy at midrapidity dS/dy grows linearly with initial participant number Np, with a slope of ~6.6 entropy units per participant. (Why?? Is Nature telling us something?)
4. For central collisions at midrapidity, the entropy content of all pions is ~5 greater than that of all nucleons+antinucleons.
5. The initial entropy density increases with centrality, but forms a convex curve that shows no indication of the dramatic increase in entropy density expected with the onset of a quark-gluon plasma.
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