Looking Through the Veil of Hadronization: Pion Entropy

1
Looking Through theVeil of HadronizationPio
n Entropy PSD at RHIC

• John G. CramerDepartment of PhysicsUniversity
  of Washington, Seattle, WA, USA

STAR Collaboration MeetingCalifornia Institute
of Technology February 18, 2004
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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
  áf(p)ñ º ?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 Liouvilles
  Theorem, for free-streaming particles áf(p)ñ is a
  conserved Lorentz scalar. Sinyukov has recently
  asserted that áf(p)ñ 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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Entropy Calculation Expectations

• Entropy The pion entropy per particle Sp/Np and
  the total pion entropy at midrapidity dSp/dy can
  be calculated from áf(p)ñ. 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 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 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.
hep-ph/0212302
nucl-th/0104023
Entropy penetrates the veil of hadronization.
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Pion Phase Space Density at Midrapidity
The source-averaged phase space density áf(mT)ñ
is the dimensionless number of pions per
6-dimensional phase space cell h3, as averaged
over the source. At midrapidity áf(mT)ñ is
given by the expression
Average phasespace density
HBT momentumvolume Vp
PionPurityCorrection
Momentum Spectrum
Jacobianto make ita Lorentzscalar
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RHIC Collisions as Functions of Centrality
Frequency of Charged Particlesproduced in RHIC
AuAu Collisions
At RHIC we can classifycollision events by
impact parameter, based on charged particle
production.
of sTotal
Participants
Binary Collisions
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Corrected HBT Momentum Volume Vp /l½
50-80
Centrality
130 GeV/nucleon
40-50
Peripheral
30-40
Fits assuming Vp l-½A0 mT3a (Sinyukov)
20-30
10-20
5-10
0-5
Central
STAR Preliminary
mT - mp (GeV)
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Global Fit to Pion Momentum Spectrum
130 GeV/nucleon

• We make a global fit of the uncorrected pion
  spectrum vs. centrality by
• Assuming that the spectrumhas the form of an
  effective-TBose-Einstein distribution
• d2N/mTdmTdyA/Exp(E/T) 1
• and
• Assuming that A and T have aquadratic dependence
  on thenumber of participants NpA(p)
  A0A1NpA2Np2T(p) T0T1NpT2Np2

STAR Preliminary
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Interpolated Phase Space Density áfñ at S½ 130
GeV
HBT points with interpolated spectra
Note failure of universal PSDbetween CERN and
RHIC.

NA49


Central

STAR Preliminary

Peripheral
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Extrapolated Phase Space Density áfñ at S½ 130
GeV
Spectrum points with extrapolated HBT Vp/l1/2
Central
Note that for centralities of 0-40 of sT, áfñ
changes very little. áfñ drops only for the
lowest 3 centralities.
STAR Preliminary
Peripheral
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Converting áfñ to Entropy per Particle (1)
Starting from quantum statistical mechanics, we
define
0.2
An estimate of the average pion entropy per
particle áS/Nñ can be obtainedfrom a
6-dimensional space-momentum integral over the
local phase spacedensity f(x,p)
O(f)
O(f3)
O(f4)
0.1
dS6(Series)/dS6
1.000
To perform the space integrals, we assume
that f(x,p) áf(p)ñ g(x),where g(x) Ö23
Exp-x2/2Rx2-y2/2Ry2-z2/2Rz2, 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-a. Then the space integrals can be performed
analytically.This gives the numerator and
denominator integrands of the above
expressionfactors of RxRyRz Reff3mT-3a. (For
reference, a½)
-0.1
O(f2)
-0.2
f
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Converting áfñ to Entropy per Particle (2)
The entropy per particle áS/Nñ then reduces
to a momentum integralof the form
(6-D)
(3-D)
(1-D)
We obtain a from the momentum dependence of
Vpl-1/2 and performthe momentum integrals
numerically using momentum-dependent fits to
áfñor fits to Vpl-1/2 and the spectra.
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Entropy per Pion from Vp /l½ and Spectrum Fits
Peripheral
STAR Preliminary
Line Combined fits to spectrum and Vp/l1/2
Central
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Thermal Bose-Einstein Entropy per Particle
The thermal estimate of the p entropy per
particle can beobtained by integrating a
Bose-Einstein distribution over3D momentum
mp/mp
T/mp
mp 0
mp mp
Note that the thermal-model entropy per particle
usually decreases with increasing temperature T
and chemical potential mp.
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Entropy per Particle S/N with Thermal Estimates
STAR Preliminary
Peripheral
Solid line and points show S/Nfrom spectrum and
Vp/l1/2 fits.
For T120 MeV, S/N impliesa pion chemical
potential ofmp63 MeV.
Dashed line indicates systematicerror in
extracting Vp from HBT.
Central
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Total Pion Entropy dSp/dy
STAR Preliminary
Dashed line indicates systematicerror in
extracting Vp from HBT.

PP
Why is dSp/dylinear with Np??
Dot-dash line indicates dS/dy fromBSBEx fits to
interpolated ltfgt.
PP


Entropy content ofnucleons antinucleons
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Total Pion Entropy per Participant (dSp/dy)/Np
Central
Average
Peripheral
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Conclusions

• The source-averaged pion phase space density áfñ
  is very high, in the low momentum region roughly
  2 that observed at the CERN SPS for PbPb at
  ÖSnn17 GeV.
• The pion entropy per particle Sp/Np is very low,
  implying a significant pion chemical potential
  (mp63 MeV) at freeze out.
• For central collisions at midrapidity, the
  entropy content of all pions is 5 greater than
  that of all nucleonsantinucleons.
• The total pion entropy at midrapidity dSp/dy
  grows linearly with initial participant number
  Np. (Why?? Is Nature telling us something?)
• 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 quark-gluon plasma.
• Our next priority is to obtain similar estimates
  of (dSp/dy)/ Np for the dAu and pp systems at
  RHIC.

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The End