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

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.


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”??


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


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.


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


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)


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


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


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.


fdxdp

fffffLogfdxdp

xpfdxdp

xpdSdxdp

NS

33

49653

612

2133

33

633 )([

),(

),(

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%




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)


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


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


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


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



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



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


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


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 .


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.


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??


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


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.


The

End

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

Documents

phase space density

local phase space density

cramer2 phase space

large entropy density

qgp phase

star pion entropy

parton phase

entropy s