Transcript Refractive Distortions of HBT

The DWEF Model:
Refractive 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
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 T0=193 MeV to T0=161 MeV. The
need for a very deep and absorptive optical potential remains.
4. Result: The New Improved DWEF Model (DWEF v.2.1).
August 2, 2007
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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-Inspired”
Emission Function
S0 ( x, k )  S0 ( , ) B (b, KT ) /(2 )3
2
2

(   0 )
cosh 
 
S 0 ( , ) 
exp  

2
2
2
2  
2 ( )
 2 
(Space-time function)
1
B (b, KT )  M T
 (b)
 K  u    (medium density)
exp 
1

T


(Bose-Einstein thermal function)
 2  t2  z2
tz
1
  2 ln 

tz
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K  particle momentum 4-vector
u  flow 4-vector
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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
4
d x ' iK 'x ' ( )
1
1
(  )*
S ( x, K )   d K ' S0 ( x, K ') 
e

(
x

x
')

(
x

x ')
p1
p2
4
2
2
(2 )
Gyulassy et al., ‘79
The s are distorted (not plane) wave solutions of:
(  m 2  U )  J , where U is the optical potential.
4

2
d x S ( x, K , q )
Correlation

C ( K , q)  1  4
4
function:
d
x
S
(
x
,
p
)
d
x S ( x, p2 )
1 

4
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(4) Potential Consistent with
Chiral Symmetry Restoration
Son & Stephanov (2002):
v2 and v2m2 (T) approach 0 near T = Tc
Both terms of U are negative (attractive)
U(b)=-(w0+w2p2)(b), w0=real, w2=complex
(We note that this is a low-momentum form of the optical
potential that becomes suspect above p~1-2 fm-1 or so.)
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Parameters of the DWEF Model
Thermal:
Space:
Time:
Flow:
Optical Pot.:
Wave Eqn.:
T0 (MeV),
 (MeV) (fixed at m)
RWS (fm),
aWS (fm)
0 (MeV/c),
 (MeV/c)
f (#),
 (#)
Re(w0) (fm-2), Re(w2) (#), Im(w2) (#)
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  = 139.6 MeV = m. We note that the emission
temperature favored by the fits (T0~162 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 sNN=200 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 LevenbergMarquardt 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 DWEF
Yellow Dots
- Plane wave (W=0, no flow)
Green Short Dash - Re(W2) only, no flow
Aqua Long Dash
- Im(W2) only, no flow
Cyan Dot Dash
- Re(W0) only, no flow
Blue 2-Dot Dash - Flow onlu, W=0
Violet 3-Dot Dash - DWEF with no BE correction
8
RO(fm)
7
6
5
4
100
200
300
400
500
600
1000
KT (MeV/c)
500
8
7
RS(fm)
100
6
5
50
Spectrum
dN2/2MTdMTdy
4
100
200
300
400
500
600
KT (MeV/c)
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200
300
400
KT (MeV/c)
500
600
700
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Optical Wave Functions [|y|2(b)]
Imaginary
Only
Full
Calculation
Eikonal
Approx.
Observer
KT =
25 MeV/c
Wrong!
KT =
197 MeV/c
KT =
592 MeV/c
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Optical Wave Functions [|y|2(b)]
Observer
DWEF
Eikonal
Approx.
KT =
100 MeV/c
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KT =
250 MeV/c
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KT =
600 MeV/c
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DWEF Fits to
STAR 200 GeV Pion HBT Radii
6.5
6
5.5
5
4.5
4
3.5
3
6.5
6
5.5
5
4.5
4
3.5
3
RO(fm)
100
200
300
400
500
600
RL(fm)
100
200
KT (MeV/c)
1.25
7
1.2
6
400
500
600
KT (MeV/c)
8
RS(fm)
300
1.15
RO/RS
1.1
5
1.05
4
1
3
100 200
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300 400 500 600
KT (MeV/c)
0.95
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200
300
400
KT (MeV/c)
500
600
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DWEF Fit to
STAR 200 GeV Pion Spectrum
500
Spectrum
dN2/2MTdMTdy
200
100
50
20
Note: accurate prediction
of spectrum slope involves
subtle cancellations among wave
functions, which puts severe demands
on the numerical accuracy of wave function
computations. => The Numerov algorithm.
100
200
300
400
500
600
700
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 > R @ 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 @ KT=1 fm-1  strong absorption 
high density
 Pion chemical potential: take =m (pions are massless in the well)
We have evidence suggesting a CHIRAL PHASE TRANSITION!








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Centrality: 200 GeV Au+Au
6.5
6
RO(fm)
Au+Au Fit
5.5
5
Space-time parameters RWS, aWS, 0
are scaled by participant number.
Emission duration  is constant.
4.5
7
4
3.5
3
RL(fm)
Au+Au Fit
6
Au+Au Predictions
100
200
300
400
500
600
5
KT (MeV/c)
4
RS(fm)
Au+Au Fit
5
3
Au+Au Predictions
4
100
200
300
400
500
600
KT (MeV/c)
3
2
Red:
Central Collisions
. . .
Indigo: Peripheral Collisions
Au+Au Predictions
100
200
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400
KT (MeV/c)
500
600
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Centrality: 200 GeV Cu+Cu
6
5
Space-time parameters RWS, aWS, 0
are scaled by participant number.
Emission duration  is scaled as A1/3.
RO(fm)
Au+Au Fit
Cu+Cu Predictions
4
7
3
RL(fm)
Au+Au Fit
6
Cu+Cu Predictions
2
5
100
200
300
400
500
600
KT (MeV/c)
4
RS(fm)
Au+Au Fit
5
Cu+Cu Predictions
3
2
4
100
200
300
400
500
600
KT (MeV/c)
3
Red:
Central Collisions
...
Indigo: Peripheral Collisions
2
100
200
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400
KT (MeV/c)
500
600
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Low pT Behavior:
Ramsauer
Resonances
in
Well
40
40
35
30
25
20
15
10
5
RO (fm)
35
30
25
20
15
10
5
10 20 30 40 50 60 70
12
11
10
9
8
7
6
5
RS (fm)
10 20 30 40 50 60 70
5000
RL (fm)
Spectrum
2/2M dM dy
dN

T
T
2000
1000
500
10 20 30 40 50 60 70
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KT (MeV/c)
200
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Phobos 0-6%
10 20 30 40 50 60 70
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
sNN=200 GeV data, simultaneously fitting three
HBT radii and the pT spectrum, and we can use
participant scaling to predict noncentral Au+Au and
Cu+Cu 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