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.

February 18, 2004 John G. Cramer18STAR

The

End