Transcript Looking Through the “Veil of Hadronization”: Pion Entropy & PSD at RHIC STAR John G.

Looking Through the
“Veil of Hadronization”:
Pion Entropy & PSD at RHIC
STAR
John G. Cramer
Department of Physics
University of Washington, Seattle, WA,
USA
STAR Collaboration Meeting
California Institute of Technology
February 18, 2004
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.
Sinyukov has recently asserted that f(p) is also approximately
conserved from the initial collision to freeze out.
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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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. It never decreases (2nd Law of Thermodynamics.)
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
Entropy penetrates the “veil of hadronization”.
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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
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Jacobian Momentum Spectrum
to make it
a Lorentz
scalar
February 18, 2004
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HBT “momentum Pion
volume” Vp
Purity
Correction
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
1500
Centrality
130 GeV/nucleon
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)
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0.25
0.3
Vp
λ (c π )3
λ
R SR O R L
John G. Cramer
Global Fit to Pion Momentum Spectrum
1000
130 GeV/nucleon
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
February 18, 2004
5
0.1
0.2
0.3
mT
7
0.4
0.5
0.6
m
John G. Cramer
Interpolated 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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February 18, 2004
0.2
mT m
8
0.3
0.4
John G. Cramer
Extrapolated 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
STAR
0.2
February 18, 2004
0.3
mT m
9
0.4
0.5
John G. Cramer
0.6
Converting f 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)
 - f Log ( f )  f  12 f 2 - 16 f 3  965 f 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 ( p , x )
6

-
S


N1.000  3 3  
 dp dx f ( p, x )


-
dp 3dx 3[- f Log ( f )  f  12 f 2 - 16 f 3  965 f 4  

-

-
dp 3dx 3 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
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John G. Cramer
Converting f to Entropy per Particle (2)
The entropy per particle S/N then reduces to a momentum integral
of the form:
 
dp dx dS 6 ( p, x )

-
S
 
 
N
3
3
 dp dx f ( p, x )

3
3
(6-D)
-



3
-
dp mT
-3a
[- f Log f 

f 
1
2

3
-



0
dpT pT mT
1-3a
5- Log ( 8 )
2
dp mT
-3a
f
2

0
dpT pT mT
3
f
3

5
24 2
f
4
]
(3-D)
f
(8)
[- f Log f  12 f  5- Log
f
2

-9
4
1-3a
2
- 9 43 f
3
 245 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.
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John G. Cramer
(1-D)
Entropy per Pion from Vp /l½ and Spectrum Fits
4.6
Peripheral
4.4
Line = Combined fits to spectrum and Vp/l1/2
S N
4.2
4
3.8
Central
3.6
50
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100
February 18, 2004
150
200
Np participants
12
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
pT 2 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.
p/mp
0
7.80625
5.48443
4.75415
4.40528
4.2043
4.07531
3.98644
3.92204
3.87358
3.83602
0.3
6.29571
4.69072
4.19131
3.95892
3.82985
3.75054
3.69848
3.6627
3.63726
3.61869
1
Exp[(mT - p ) / T ] - 1
10
0.6
4.74597
3.83487
3.56733
3.45659
3.40494
3.38033
3.36949
3.36614
3.36702
3.37031
0.9
2.94368
2.74168
2.74029
2.78018
2.82925
2.87817
2.92397
2.96584
3.00378
3.03804
8
p = 0
6
SN
T/mp
pT 2 dpT f BE
where f BE 
4
2
p = mp
0
0.5
Note that the thermal-model entropy per
particle usually decreases with increasing
temperature T and chemical potential p.
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1
Tm
1.5
John G. Cramer
2
Entropy per Particle S/N with Thermal Estimates
T 90 MeV
4.6
Peripheral
S N
4.4
T 120 MeV
Solid line and points show
S/N
from spectrum and Vp/l1/2 fits.
4.2
T 200 MeV
4
For T=120 MeV, S/N implies
a pion chemical potential of
p=63 MeV.
Dashed line indicates systematic
error in extracting Vp from HBT.
3.8
3.6
Landau Limit: m 0
Central
3.4
50
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100
February 18, 2004
200
150
Np participants
14
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??
Dot-dash line indicates dS/dy from
BSBEx fits to interpolated <f>.
1000
P&P
500
Snuc
Entropy content of
nucleons + antinucleons
50
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100
150
200
250
300
Np
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John G. Cramer
350
Total Pion Entropy per Participant (dSp/dy)/Np
Central
6.7
dS dy Np
6.6
6.5
Average
6.4
6.3
6.2
6.1
Peripheral
50
100
150
200
250
300
350
Np
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John G. Cramer
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 (p~63 MeV) at freeze out.
3. For central collisions at midrapidity, the entropy content of all pions is
~5 greater than that of all nucleons+antinucleons.
4. The total pion entropy at midrapidity dSp/dy grows linearly with initial
participant number Np. (Why?? Is Nature telling us something?)
5. The pion entropy per participant (dSp/dy)/ Np , which should penetrate
the “ veil of hadronization”, has a roughly constant value of 6.5 and
shows no indication of the increase expected with the onset of a quarkgluon plasma.
6. Our next priority is to obtain similar estimates of (dSp/dy)/ Np for the
d+Au and p+p systems at RHIC.
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The
End
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