Transcript Pion Entropy and Phase Space Density at RHIC STAR John G. Cramer Department of Physics University of Washington, Seattle, WA, USA Second Warsaw Meeting on Particle Correlations and.

Pion Entropy and
Phase Space Density at RHIC
STAR
John G. Cramer
Department of Physics
University of Washington, Seattle, WA,
USA
Second Warsaw Meeting on Particle
Correlations and Resonances
in Heavy Ion Collisions
Warsaw University of Technology
October 16, 2003
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 Dp3Dx3 = 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 the
f-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.
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October 16, 2003
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John G. Cramer
Entropy: Calculation & Expectations
Entropy – The pion entropy per particle Sp/Np and the total pion
entropy at midrapidity dSp/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.
A quark-gluon
plasma has a
Entropy is conserved
largeduring
number
of degrees of
hydrodynamic
freedom.
It should
generate a
expansion
and freerelatively
large Thus,
entropy
streaming.
thedensity, up
to 12
to 16 times
than that
entropy
of thelarger
system
of aafter
hadronic
gas. should
freeze-out
be close to the initial
At
RHIC,and
if ashould
QGP phase
entropy
grows
with centrality
provide
a critical we would
expect
the entropy
toearlygrow
constraint
on the
strongly
increasing
stagewith
processes
of thecentrality
and system.
participant number.
hep-ph/0212302
nucl-th/0104023
Can Entropy provide the QGP “Smoking Gun”??
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John G. Cramer
Pion Phase Space Density at Midrapidity
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:
3
2



λ(c π )  1
1
d N
 f (m T ) 



E π  2 π m T dm T dy   R S R O R L  λ
Average phase
space density
Jacobian Momentum Spectrum
to make it
a Lorentz
scalar
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HBT “momentum Pion
volume” Vp
Purity
Correction
John G. Cramer
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 p+p+ and p-p- 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 l, 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.
STAR
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John G. Cramer
RHIC Collisions as Functions of Centrality
At RHIC we can classify
collision events by
impact parameter, based
on charged particle
production.
Frequency of Charged Particles
produced in RHIC Au+Au Collisions
50-80% 30-50% 20-30% 10-20% 5-10% 0-5% of sTotal
Participants
Binary Collisions
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John G. Cramer
Corrected HBT Momentum Volume Vp /l½
50-80%
2000
Centrality
1500
40-50%
Peripheral
GeV3
1000
700
106 Vp
500
30-40%
Fits assuming:
Vp l-½=A0 mT3a
(Sinyukov)
20-30%
10-20%
5-10%
0-5%
300
Central
200
150
0.05
0.1
0.15
0.2
mT - mp (GeV)
STAR
October 16, 2003
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0.25
0.3
Vp
λ
λ (c π )
R SR O R L
John G. Cramer
3
Global Fit to Pion Momentum Spectrum
1000
500
100
d2N 2 mTdmTdy
We make a global fit of the uncorrected
pion spectrum vs. centrality by:
(1) Assuming that the spectrum
has the form of an effective-T
Bose-Einstein distribution:
d2N/mTdmTdy=A/[Exp(E/T) –1]
and
(2) Assuming that A and T have a
quadratic dependence on the
number of participants Np:
50
A(p) = A0+A1Np+A2Np2
T(p) = T0+T1Np+T2Np2
A0
A1
A2
T0
T1
T2
STAR
Value
31.1292
21.9724
-0.019353
0.199336
-9.23515E-06
2.10545E-07
10
Error
14.5507
0.749688
0.003116
0.002373
2.4E-05
6.99E-08
October 16, 2003
5
0.1
0.2
0.3
mT
8
0.4
0.5
0.6
m
John G. Cramer
Interpolated Pion Phase Space Density f at S½ = 130 GeV
HBT points with interpolated spectra
0.4
NA49
f
0.3
0.2
}
Note failure of “universal” PSD
between CERN and RHIC.
Central
0.1
Peripheral
0.1
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October 16, 2003
0.2
mT m
9
0.3
0.4
John G. Cramer
Extrapolated Pion Phase Space Density f at S½ = 130 GeV
Spectrum points with extrapolated HBT Vp/l1/2
Central
0.2
Note that for centralities of 0-40% of sT,
fchanges very little.
f drops only for the lowest 3 centralities.
f
0.1
0.05
0.02
Peripheral
0.01
0.1
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0.2
October 16, 2003
0.3
mT m
10
0.4
0.5
John G. Cramer
0.6
Converting Phase Space Density to Entropy per Particle (1)
Starting from quantum statistical mechanics, we define:
 
f  f ( x , p);
dS6  - f Log ( f ) + ( f + 1) Log ( f + 1)
5
 - f Log ( f ) + f + 12 f - 16 f + 96
f + ...
2
3
4
+0.2%
An estimate of the average
pion entropy per 3particle S/N can be obtained
O(f)
O(f )over the local phase space
O(f4)
from a 6-dimensional space-momentum integral
density f(x,p):
+0.1%
dS6(Series)/dS6 dp 3dx 3dS
S


-

N
1.000

-
 
(
p
, x)
6
 
dp dx f ( p, x )
3



-
dp dx [- f Log ( f ) + f +
3
3

3

-
3
1
2
5
f - 16 f + 96
f +
2
3
4
3
dp dx f
To perform the
space integrals,
we assume
that
f(x,p) = f(p) g(x),
3
2
2
2
2
2
2
where g(x) = 2 Exp[-x /2Rx -y /2Ry -z /2Rz ], i.e., that the source has
-0.1%shape based on HBT analysis of the system. Further, we make the
a Gaussian
Sinyukov-inspired -a
assumption that the three radii have a momentum dependence
proportional to mT . Then the space
can be performed analytically.
O(f2integrals
)
This gives the numerator3 and-3adenominator integrands of the above expression
factors-0.2%
of RxRyRz = Reff mT . (For reference, a~½)
f
STAR
October 16, 2003
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John G. Cramer
Converting Phase Space Density to Entropy per Particle (2)
The entropy per particle S/N then reduces to a momentum integral
of the form:

S
N


 
dp dx dS 6 ( p, x )
3
-


-



3
-
dp mT
3
(6-D)
 
dp dx f ( p, x )
3
-3a
3
[ - f Log f +


3
-



0
dpT pT mT
1- 3a
f +
1
2
dp mT
[ - f Log f +

5 - Log ( 8 )
2
1
2
- 3a
f +

0
dpT pT mT
f
2
-9
4
3
f
3
+
5
f
24 2
4
]
(3-D)
f
5 - Log ( 8 )
2
1- 3a
f
2
- 9 43 f
3
+
5
24 2
f
4
]
f
We obtain a from the momentum dependence of Vpl-1/2 and perform
the momentum
integrals numerically using momentum-dependent fits to f
-1/2
or fits to Vpl and the spectra.
STAR
October 16, 2003
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John G. Cramer
(1-D)
Blue-Shifted Bose-Einstein Functions
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 - mp)/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 hL and hT , the average values <hL> and <hT> to obtain:
m p m p -1
m
pT p
mT
f

{
Exp
[
Cosh
[
h
]
Cosh
[
h
]
Sinh
[
h
]
f Local  { Exp [
hT ] Cosh [ Lh ] Sinh [T h ] - ] - 1]}- 1}
T
BlueShift
TT00
-1
T
T
L
T0 T0
T
T0 T0
We assume mp=<hL>=0 and consider three models for <hT>:
BSBE1: <hT> = a (i.e., constant average flow, independent of pT)
BSBE2: <hT> = a (pT/mT) = abT (i.e., proportional to pair velocity)
BSBE3: <hT> = a1bT +a3bT3+a5bT5+a7bT7 (minimize D(S/N)/Dflow)
STAR
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John G. Cramer
October 16, 2003
Fits to Interpolated Pion Phase Space Density
HBT points with interpolated spectra
Fitted with BSBE2 function
0.5
Central
fp
0.2
0.1
0.05 Warning: PSD in the region measured
contributes only about 60% to the
average entropy per particle.
0.05
STAR
0.1
October 16, 2003
0.15
mT m GeV
14
Peripheral
0.2
0.25
John G. Cramer
Fits to Extrapolated Pion Phase Space Density
0.5
Central
Solid = Combined Vp/l1/2 and Spectrum fits
Dashed = Fitted with BSBE2 function
0.1
f
0.05
0.01
0.005
0.001
Spectrum points with extrapolated HBT Vp/l1/2
Each successive centrality reduced by 3/2
0.1
STAR
0.2
October 16, 2003
0.3
mT m
15
Peripheral
0.4
0.5
0.6
John G. Cramer
Large-mT behavior of three BSBE Models
Solid = BSBE2: hT = abT
Dotted = BSBE3: 7th order odd polynomial in bT
Dashed = BSBE1: hT = Constant
Each successive centrality reduced by 3/2
0.1
f
0.01
0.001
0.0001
0.25
STAR
0.5
0.75
1
1.25
1.5
1.75
2
mT m
October 16, 2003
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John G. Cramer
Large mT behavior using Radius & Spectrum Fits
Solid = fits to spectrum and Vp/l1/2
Dashed = BSBE2 fits to extrapolated data
Each successive centrality reduced by 3/2
0.1
f
0.01
0.001
0.0001
0.25
0.5
0.75
1
1.25
1.5
1.75
2
mT m
STAR
October 16, 2003
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John G. Cramer
Entropy per Pion from Vp /l½ and Spectrum Fits
4.6
Peripheral
4.4
Black = Combined fits to spectrum and Vp/l1/2
S N
4.2
4
3.8
Central
3.6
50
STAR
100
October 16, 2003
150
200
Np participants
18
250
300
350
John G. Cramer
Entropy per Pion from BSBE Fits
4.6
Peripheral
4.4
4
Green = BSBE2: ~ bT
Blue = BSBE3: Odd 7th order Polynomial in bT
S N
4.2
Black = Combined fits to spectrum and Vp/l1/2
Red = BSBE1: Const
3.8
Central
3.6
50
STAR
100
October 16, 2003
150
200
Np participants
19
250
300
350
John G. Cramer
Thermal Bose-Einstein Entropy per Particle
The thermal estimate of the p entropy per particle can be
obtained by integrating a Bose-Einstein distribution over
3D momentum:

S/N 

0
mT pT dpT [( f BE + 1) Ln( f BE + 1) - f BE Ln( f BE )]


0
0.2
0.4
0.6
0.8
1.
1.2
1.4
1.6
1.8
2.
mT pT dp f BE
0.3
5.86225
4.33169
3.89106
3.70431
3.61107
3.56032
3.53137
3.51456
3.50489
3.49958
8
0.6
4.30277
3.45065
3.23476
3.16747
3.15191
3.15728
3.17146
3.18916
3.20786
3.22638
0.9
2.43181
2.25166
2.28837
2.36967
2.45851
2.54375
2.62195
2.69244
2.75553
2.8119
mp = 0
6
4
2
mp = mp
0
STAR
Exp[( mT - mp ) / T ] - 1
10
mp/mp
0.
7.37481
5.13504
4.46843
4.16727
4.00256
3.90175
3.83522
3.78887
3.75521
3.72997
1
SN
T/mp
where f BE 
0.5
1
Note that the thermal-model entropy per
particle usually decreases with increasing
temperature T and chemical potential mp.
October 16, 2003
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1.5
Tm
2
2.5
John G. Cramer
3
Entropy per Particle S/N with Thermal Estimates
4.6
4.4
Peripheral
BPB
T 90 MeV
Solid line and points show
S/N
from spectrum and Vp/l1/2 fits.
4.2
S N
T 120 MeV
For T=110 MeV, S/N implies
a pion chemical potential of
mp=44.4 MeV.
4
Dashed line indicates systematic
error in extracting Vp from HBT.
T 200 MeV
3.8
3.6
Central
Landau Limit: m 0
Dot-dash line shows S/N from BDBE2 fits to f
3.4
50
STAR
100
October 16, 2003
150
200
N p participants
21
250
300
John G. Cramer
350
Total Pion Entropy dSp/dy
2500
Dashed line indicates systematic
error in extracting Vp from HBT.
2000
dS dy
1500
P&P
Why is dSp/dy
linear with Np??
Solid line is a linear fit through (0,0)
with slope = 6.58 entropy units
per participant
Dot-dash line indicates dS/dy from
BSBEx fits to interpolated <f>.
1000
P&P
500
Snuc
Entropy content of
nucleons + antinucleons
50
STAR
100
150
200
250
300
Np
October 16, 2003
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John G. Cramer
350
Initial Entropy Density: ~(dSp/dy)/Overlap Area
45
dS dy Np2 3
40
35
30
Initial collision overlap
area is roughly
2/3
proportional to Np
Initial collision entropy is roughly
proportional to freeze-out dSp/dy.
Therefore, (dSp/dy)/Np2/3
should be proportional
to initial entropy
density, a QGP
signal.
Solid envelope =
Systematic errors in Np
Data indicates that the initial
entropy density does grow with
centrality, but not very rapidly.
Our QGP “smoking gun” seems to be
inhaling the smoke!
25
20
0
STAR
50
100
October 16, 2003
150
200
Np participants
23
250
300
John G. Cramer
350
Conclusions
1. 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 Sp/Np is very low, implying a
significant pion chemical potential (mp~44 MeV) at freeze out.
3. The total pion entropy at midrapidity dSp/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 quarkgluon plasma.
STAR
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John G. Cramer
October 16, 2003
The
End
STAR
October 16, 2003
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John G. Cramer