looking through the “veil of hadronization”: pion entropy & psd at rhic
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Looking Through the “Veil of Hadronization”: Pion Entropy & PSD at RHIC. 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. - PowerPoint PPT PresentationTRANSCRIPT
STAR
Looking Through the“Veil of Hadronization”:Pion Entropy & PSD at
RHIC
Looking Through the“Veil of Hadronization”:Pion Entropy & PSD at
RHIC
John G. CramerDepartment of Physics
University of Washington, Seattle, WA, USA
John G. CramerDepartment of Physics
University of Washington, Seattle, WA, USA
STAR Collaboration MeetingCalifornia Institute of
TechnologyFebruary 18, 2004
STAR Collaboration MeetingCalifornia Institute of
TechnologyFebruary 18, 2004
February 18, 2004 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 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.
February 18, 2004 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. It never decreases (2nd Law of Thermodynamics.)
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.
Entropy penetrates the “veil of hadronization”.
February 18, 2004 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
February 18, 2004 John G. Cramer5STAR
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
February 18, 2004 John G. Cramer6STAR
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)
130 GeV/nucleon
February 18, 2004 John G. Cramer7STAR
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
130 GeV/nucleon
February 18, 2004 John G. Cramer8STAR
0.1 0.2 0.3 0.4mTm
0.1
0.2
0.3
0.4
f
Interpolated Phase Space Density f at S½ = 130 GeV
Interpolated 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
February 18, 2004 John G. Cramer9STAR
0.1 0.2 0.3 0.4 0.5 0.6mTm
0.01
0.02
0.05
0.1
0.2
f
Extrapolated Phase Space Density f at S½ = 130 GeV
Extrapolated 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.
February 18, 2004 John G. Cramer10STAR
fdxdp
fffffLogfdxdp
xpfdxdp
xpdSdxdp
NS
33
49653
612
2133
33
633 )([
),(
),(
Converting f to Entropy per Particle (1)Converting f 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%
February 18, 2004 John G. Cramer11STAR
Converting f to Entropy per Particle (2)
Converting f 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)
February 18, 2004 John G. Cramer12STAR
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 Fits
Entropy per Pion from Vp /½ and Spectrum Fits
Central
PeripheralSTAR
Preliminary
Line = Combined fits to spectrum and Vp/1/2
February 18, 2004 John G. Cramer13STAR
0 0.3 0.6 0.90.2 7.80625 6.29571 4.74597 2.943680.4 5.48443 4.69072 3.83487 2.741680.6 4.75415 4.19131 3.56733 2.740290.8 4.40528 3.95892 3.45659 2.780181. 4.2043 3.82985 3.40494 2.829251.2 4.07531 3.75054 3.38033 2.878171.4 3.98644 3.69848 3.36949 2.923971.6 3.92204 3.6627 3.36614 2.965841.8 3.87358 3.63726 3.36702 3.003782. 3.83602 3.61869 3.37031 3.03804
Thermal Bose-Einstein Entropy per Particle
Thermal Bose-Einstein Entropy per Particle
2
0
2
0
[( 1) ( 1) ( )] 1S/N where
[( ) / ] 1
T T BE BE BE BE
BETT T BE
p dp f Ln f f Ln ff
Exp m Tp dp f
0 0.5 1 1.5 2Tm
2
4
6
8
10
SN
= 0
= m
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 .
February 18, 2004 John G. Cramer14STAR
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
Entropy per Particle S/N with Thermal Estimates
Entropy per Particle S/N with Thermal Estimates
Central
Peripheral
STAR Preliminary
Dashed line indicates systematicerror in extracting Vp from HBT.
Solid line and points show S/Nfrom spectrum and Vp/1/2 fits.
For T=120 MeV, S/N impliesa pion chemical potential of=63 MeV.
February 18, 2004 John G. Cramer15STAR
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>.
Entropy content ofnucleons + antinucleons
P&P
P&P
Why is dS/dylinear with Np??
February 18, 2004 John G. Cramer16STAR
50 100 150 200 250 300 350Np
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Sd ydN p
Total Pion Entropy per Participant (dS/dy)/Np
Total Pion Entropy per Participant (dS/dy)/Np
Central
Peripheral
Average
February 18, 2004 John G. Cramer17STAR
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 (~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 dS/dy grows linearly with initial participant number Np. (Why?? Is Nature telling us something?)
5. The pion entropy per participant (dS/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 quark-gluon plasma.
6. Our next priority is to obtain similar estimates of (dS/dy)/ Np for the d+Au and p+p systems at RHIC.