The%20DWEF%20Model:%20Refractive%20Distortions%20of%20HBT

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The DWEF ModelRefractive Distortions of HBT

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

Workshop on Particle Correlations Femtoscopy -
2007 Santa Rosa, CA August 2, 2007
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Since WPCF 2006

1. We discovered in November a convergence vs.
   integration step size problem in our calculation
   of optical model wave functions. This had no
   effect on the HBT radii, but had a strong effect
   on the slope of the spectrum. This problem was
   corrected by changing from Runge-Kutta to Numerov
   wave function solutions.
2. We discovered in March that the fugacity from the
   strong pion chemical potential was being applied
   to the spectrum, but not to the variables for the
   HBT radii. This error was corrected.
3. The net result, after refitting, is that the
   ambiguities reported last year are gone, and
   the emission temperature of the model has dropped
   from T0193 MeV to T0161 MeV. The need for a
   very deep and absorptive optical potential
   remains.
4. Result The New Improved DWEF Model (DWEF v.2.1).

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Elements of DWEF Approach(1) The Nuclear
Optical Model

1. 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.).
2. Solve the time-independent Klein-Gordon equation
   for the wave functions of Channel 1 pions, using
   a complex potential U. Im(U) accounts for those
   pions removed from Channel 1.
3. The complex optical potential U does several
   things(a) absorbs pions (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.
   In other words, it quantum-mechanically mocks up
   the effects on pions of passing through the hot
   dense medium of the fireball.

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(2) Hydro-InspiredEmission Function
(Space-time function)
(medium density)
(Bose-Einstein thermal function)
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(3) 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
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(4) Potential Consistent withChiral Symmetry
Restoration
Son Stephanov (2002)
v2 and v2m2p (T) approach 0 near T Tc
Both terms of U are negative (attractive)
U(b)-(w0w2p2)?(b), w0real, w2complex
(We note that this is a low-momentum form of the
optical potential that becomes suspect above
p1-2 fm-1 or so.)
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Parameters of the DWEF Model
Thermal T0 (MeV), mp (MeV) (fixed at
mp) 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 ()0 (fixed, Kisslinger
term off)
Note that these parameters describe pion emission
at chemical freeze-out, not kinematic freeze-out
(e.g., as used in the blast-wave model).
Data fitting has led us to a chemical potential
near the pion mass. We therefore set mp 139.6
MeV mp. We note that the emission temperature
favored by the fits (T0162 MeV) is close to
estimates of the temperature for chemical
freeze-out, but we leave this as a fit variable.
Total number of parameters 10 (2)
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DWEF Fits to STAR Data
We have calculated pion wave functions in a
partial wave expansion, applied them to a
hydro-inspired pion source function, and
calculated the HBT radii and spectrum. This DWEF
model uses 7 pion source parameters and 3 optical
potential parameters, for a total of 10
parameters in the model. The correlation
function C near half-maximum (not the 2nd moment
of C) is calculated. We have fitted STAR
data at ÖsNN200 GeV, simultaneously fitting Ro,
Rs, Rl, and dNp/dy (fitting both magnitude and
shape) at 8 momentum values (i.e., 32 data
points), using a Levenberg-Marquardt fitting
algorithm. In the resulting fit, the c2 per data
point is 3.6 and the c2 per degree of freedom is
4.8. Only statistical (not systematic) errors
are used in calculating c2. We remove
long-lived halo resonance contributions to the
spectrum (which are not included in the model) by
multiplying the uncorrected spectrum by l½ (the
HBT parameter) before fitting, then
un-correcting the predicted spectrum with l-½.
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Components of DWEF Calculations
Red Solid - Full DWEFYellow Dots
- Plane wave (W0, no flow)Green Short Dash
- Re(W2) only, no flowAqua Long Dash -
Im(W2) only, no flowCyan Dot Dash - Re(W0)
only, no flowBlue 2-Dot Dash - Flow onlu,
W0Violet 3-Dot Dash - DWEF with no BE
correction
RO(fm)
KT (MeV/c)
RS(fm)
Spectrum dNp2/2pMTdMTdy
KT (MeV/c)
KT (MeV/c)
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Optical Wave Functions y2r(b)
Imaginary Only
Eikonal Approx.
Full Calculation
KT 25 MeV/c
Wrong!
KT 197 MeV/c
KT 592 MeV/c
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Optical Wave Functions y2r(b)
DWEF
Eikonal Approx.
KT 250 MeV/c
KT 100 MeV/c
KT 600 MeV/c
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DWEF Fits toSTAR 200 GeV Pion HBT Radii
RO(fm)
RL(fm)
KT (MeV/c)
KT (MeV/c)
RO/RS
RS(fm)
KT (MeV/c)
KT (MeV/c)
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DWEF Fit toSTAR 200 GeV Pion Spectrum
Spectrum dNp2/2pMTdMTdy
Note accurate predictionof spectrum slope
involvessubtle cancellations among
wavefunctions, which puts severe demandson the
numerical accuracy of wave functioncomputations.
gt The Numerov algorithm.
KT (MeV/c)
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The Parameters

• Temperature 162 MeV Chemical freezeout at
  160 MeV
• Transverse flow rapidity 1.215 vmax 0.84 c,
  vav 0.58 c
• Mean expansion time 9.1 fm/c system expands at
  0.47 c
• Pions emitted between 7 fm/c and 11 fm/c soft
  EOS .
• WS radius 11.9 fm R(Au) 4.3 fm gt R _at_ SPS
• WS diffuseness 1.1 fm (a bit larger than LENP
  experience)
• Re(U) 0.49 1.19 p2 very deep well strong
  attraction.
• Im(U) 0.129 p2 lmfp 8 fm _at_ KT1 fm-1
  strong absorption high density
• Pion chemical potential take mpmp (pions are
  massless in the well)
• We have evidence suggesting a CHIRAL PHASE
  TRANSITION!

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Centrality 200 GeV AuAu
Space-time parameters RWS, aWS, t0 are scaled by
participant number. Emission duration Dt is
constant.
RO(fm) AuAu Fit
RL(fm)AuAu Fit
AuAu Predictions
KT (MeV/c)
RS(fm)AuAu Fit
AuAu Predictions
KT (MeV/c)
AuAu Predictions
KT (MeV/c)
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Centrality 200 GeV CuCu
Space-time parameters RWS, aWS, t0 are scaled by
participant number. Emission duration Dt is
scaled as A1/3.
RO(fm) AuAu Fit
CuCu Predictions
RL(fm) AuAu Fit
CuCu Predictions
KT (MeV/c)
RS(fm) AuAu Fit
CuCu Predictions
KT (MeV/c)
KT (MeV/c)
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Low pT BehaviorRamsauer Resonances in Well
RO (fm)
RS (fm)
Spectrum dNp2/2pMTdMTdy
RL (fm)
Phobos 0-6
KT (MeV/c)
KT (MeV/c)
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Summary

• The improved DWEF Model allows good fits to RHIC
  HBT radii and spectrum data at emission
  temperatures of about 162 MeV.
• We obtain excellent DWEF fits to central STAR
  ÖsNN200 GeV data, simultaneously fitting three
  HBT radii and the pT spectrum, and we can use
  participant scaling to predict noncentral AuAu
  and CuCu with the same optical potential
  strengths.
• The fit parameters are reasonable and indicate
  strong collective flow, significant opacity, and
  huge attraction suggesting chiral symmetry
  restoration.
• They describe pion emission in hot, highly dense
  matter with a soft pion equation of state.

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The End
A paper describing this work has been
published in Phys. Rev. Lett. 94, 102302 (2005)
nucl-th/0411031 A longer paper is published
in J. Phys. G nucl-th/0507004