John G' Cramer with Gerald A' Miller

1
Radial Sensitivity of the DWEF Model Applied to
RHIC Soft-Sector Data
(DWEF Distorted-WaveEmission Function)

• John G. Cramer (with Gerald A. Miller)
• University of Washington
• Seattle, Washington, USA

(25 min Presentation) WPCF 2006 Sao Paulo,
Brazil September 9, 2006
2
PrimerThe Nuclear Optical Model

• Divide the pions into channels and focus on
  pions (Channel 1) that participate in the BE
  correlation (about 60 of the spectrum pions).
  Omit halo and resonance pions and those
  converted to other particles (Channels 2, 3,
  etc.).
• Solve the time-independent Klein-Gordon equation
  for the wave functions of Channel 1 pions, using
  a complex potential U.
• The complex optical potential U does several
  things(a) absorbs pions, i.e., removes them
  from Channel 1 (opacity)(b) deflects pion
  trajectories (refraction, demagnification)(c)
  steals kinetic energy from the emerging
  pions(d) produces Ramsauer-type resonances in
  the well, which can modulate apparent source size
  and emission intensity.

3
Optical Wave Functions y2r(b)
Imaginary Only
Eikonal Approx.
Full Calculation
KT 25 MeV/c
KT 197 MeV/c
KT 592 MeV/c
4
The DWEF Formalism
Note assumes chaotic pion sources.

• We use the Wigner distribution of the pion source
  current density matrix S0(x,K) (the emission
  function).
• The pions interact with the dense medium,
  producing S(x,K), the distorted wave emission
  function (DWEF)

Distorted Waves
Gyulassy et al., 79
The Ys are distorted (not plane) wave solutions
of , where U is the
optical potential.
Correlationfunction
5
The Hydro-InspiredEmission Function
(space-time locus of emission)
(medium density)
(Bose-Einstein thermal function)
6
A Chiral Symmetry PotentialSon Stephanov
(2002)
Dispersion relation for pions in nuclear matter.
Both v2 and v2m2p(T) 0 near TTc.
p velocity
screening mass
Both terms of U are negative (attractive)
U(b) -(w0w2p2)r(b), w0 is real, w2 is complex.
7
Parameters of the DWEF Model
Thermal T0 (MeV), mp (MeV) Space RWS (fm),
aWS (fm) Time t0 (MeV/c), Dt
(MeV/c) Flow hf (), Dh () Optical
Pot. Re(w0) (fm-2), Re(w2) (), Im(w2) () Wave
Eqn. e ()1 (Kisslinger term)
Note that these parameters describe the initial
in-medium pion emission, not freeze-out (e.g., as
used in the blast-wave model). Red items indicate
parameters not used in Blast Wave model.)
Total number of parameters 12
8
Parameter Ambiguuity 9 Fits
T0 ranges from 173 Mev to 220 MeV
9
c2 vs. Temp for 9 Fits
10
Comparison of Parameters
11
Reduced Parameter Set
Thermal T0 (MeV), mp (MeV)
(fixed) Space RWS (fm), aWS (fm) Time t0
(MeV/c), Dt (MeV/c) Flow hf (), Dh
() Optical Pot. Re(w0) (fm-2), Re(w2) (),
Im(w2) () Wave Eqn. e ()1 (fixed, Kisslinger
term off)
Data fitting has led us to a temperature matching
the Tc estimate from lattice gauge calculations
(QM2005) and chemical potnetial from chiral
symmetry restoration. We therefore set T0 193
MeVTc and mp 139.6 MeV mp.
Total number of parameters 9 (3)
12
Fit F193 to 200 GeV AuAu Radii
T0 193 MeV RWS 11.78 fmaWS 0.91 fm Dt
2.38 fm/c
Curves solid (black)f ull calculation dotted
(green)hf 0 (no flow) dashed (red) ReU0
(no refraction) double-dot-dashed
(violet)substituting Boltzmann for Bose-Einstein
thermal distribution.
13
Fit F193 to 200 GeV AuAu Spectrum
T0 193 MeV RWS 11.78 fmaWS 0.91 fm Dt
2.38 fm/c
Curves solid (black)f ull calculation dotted
(green)hf 0 (no flow) dashed (red) ReU0
(no refraction) dot-dashed (blue)U0 (no
potential), double-dot-dashed (violet)substitutin
g Boltzmann for Bose-Einstein thermal
distribution.
14
F193 Predictions vs. Centrality
AuAu
AuAu Centrality 0-5 5-10 10-20 20-30 30
-50 50-80
Rout
Rside
Scale RWS, aWS and t0 as Npart1/3.
Rlong
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Radial Sensitivity (10 Steps)
T0 193 MeV RWS 11.78 fmaWS 0.91 fm Dt
2.38 fm/c
Rout
Rside
Rlong
Rout/Rside
16
Radial Sensitivity (10 Steps)
Pion Spectrum
T0 193 NeV RWS 11.78 fmaWS 0.91 fm Dt
2.38 fm/c
17
Summary

• Quantum mechanics has solved the technical
  problems of applying opacity to HBT.
• We obtain excellent DWEF fits to STAR ÖsNN200
  GeV data, simultaneously fitting three HBT radii
  and the pT spectrum. The key ingredient is the
  deep real potential.
• If this deep real potential is present, ALL
  models of RHIC collisions in the soft sector,
  e.g. r mass shift, etc., should take the presence
  of this potential into account.
• The model parameters describe pion emission in
  hot, highly dense matter with a soft pion
  equation of state.
• We have found evidence suggesting a chiral phase
  transition in RHIC collisions.
• We note that in most quark-matter scenarios, the
  QGP phase transition is usually accompanied by a
  chiral phase transition at about the same
  critical temperature.

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Outlook

• l We have a new tool for investigating the
  presence (or absence) of chiral phase transitions
  in heavy ion collision systems.
• l DWEF needs both high quality pion spectra and
  high quality HBT analysis over a region that
  extends to fairly low momenta (KT150 MeV/c).
• l We are presently attempting to track the CPT
  phenomenon to lower collision energies, where the
  deep real potential should presumably go away.
  (NA49 HBT and spectrum analysis is in the works.)
• l We would like to replace the empirical
  emission function with a relativistic
  hydrodynamic calculation of the multidimensional
  phase space density.(DWEF DWRHD)

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The End
A short paper (with erratum) describing this
work has been published in Phys. Rev. Lett. 94,
102302 (2005) see ArXiv nucl-th/0411031
A longer paper has been submitted to Phys. Rev.
C see ArXiv nucl-th/0507004
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Backup Slides
21
Predicted Correlation/Gaussian
22
Ratio of 8 Fits to F193
23
9 Fits at Low Momentum
24
Time-Independence,Resonances, and Freeze-Out

• We note that our use of a time-independent
  optical potential does not invoke the mean field
  approximation and is formally correct according
  to quantum scattering theory. (The
  semi-classical mind-set can be misleading.)
• While the optical potential is not
  time-dependent, some time-dependent effects can
  be manifested in the energy-dependence of the
  potential . (Time and energy are conjugate
  quantum variables.)
• An optical potential can implicitly include the
  effects of resonances, including heavy ones.
  Therefore, our present treatment implicitly
  includes resonances produced within the hot,
  dense medium.
• We note that more detailed quantum
  coupled-channels calculations could be done, in
  which selected resonances were treated as
  explicit channels coupled through interactions.
  Describing the present STAR data apparently does
  not require this kind of elaboration.

25
Wave Equation Solutions

• We assume an infinitely long Bjorken tube and
  azimuthal symmetry, so that the (incoming) waves
  factorize3D 2D(distorted)1D(plane)

We solve the reduced Klein-Gordon wave equation
Partial wave expansion ! ordinary diff eq
26
The Meaning of U
Im (U) Opacity, Re (U) Refraction
Pions lose energy and flux.
ImU0-p? ?0,

• 1 mb, ?0 1 fm-3,ImU0 -.15 fm-2, ? 7 fm

Re(U) must exist very strong attraction chiral
phase transition
27
Source De-magnificationby the Real Potential
Well
n1.33
n1.00
Rays bend closer to radii
Because of the mass loss in the potential
well, the pions move faster there (red) than in
vacuum (blue). This de-magnifies the image of
the source, so that it will appear to be smaller
in HBT measurements. This effect is largest at
low momentum.
A Fly in a Bubble
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Correlation Functions (linear)
Out
Side
Long
KT 100 MeV/c
KT 200 MeV/c
KT 400 MeV/c
KT 600 MeV/c
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Correlation Functions (log)