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Pion Opacity,
Chiral Symmetry Restoration,
and RHIC HBT
John G. Cramer (with Gerald A. Miller)
University of Washington
Seattle, Washington, USA
WPCF 2005
Kromeriz, Czech Republic
August 16, 2005
Primer 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.
August 11, 2005
ISMD 2005, Kromeriz
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Optical Wave Functions [|y|2r(b)]
Imaginary
Only
Full
Calculation
Eikonal
Approx.
Observer
KT =
25 MeV/c
KT =
197 MeV/c
KT =
592 MeV/c
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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:
(  m2  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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ISMD 2005, Kromeriz
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The “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
r (b)
 K  u    (medium density)
exp 
1

T


2
2
2
(Bose-Einstein thermal function)
 t z
tz
1
  2 ln 

tz
August 11, 2005
K  particle momentum 4-vector
u  flow 4-vector
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Primer 2: Chiral Symmetry
Question 1: The up and down “current” quarks have masses of 5 to 10 MeV.
The  (a down + anti-up combination) has a mass of ~140 MeV.
Where does the observed mass come from?
Answer 1: The quarks are more massive in vacuum due to “dressing”. Also
the pair is tightly bound by the color force into a particle so small that
quantum-uncertainty zitterbewegung gives both quarks large average
momenta. Part of the  mass comes from the kinetic energy of the
constituent quarks .
Question 2: What happens when a pion is placed in a hot, dense medium?
Answer 2: Two things happen:
1. The binding is reduced and the pion system expands because of external
color forces, reducing the zitterbewegung and the pion mass.
2. The quarks that were “dressed” in vacuum become “undressed” in
medium, causing up, down, and strange quarks to become more similar
and closer to massless particles, an effect called “chiral symmetry
restoration”. In many theoretical scenarios, chiral symmetry
restoration and the quark-gluon plasma phase usually go together.
Question 3: How can a pion regain its mass when it goes from medium to
vacuum?
Answer 3: It must do work against an average attractive force, losing
kinetic energy while gaining mass. In effect, it must climb out of a
potential well that may be 140 MeV deep.
August 11, 2005
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vacuum
medium
6
A Chiral Symmetry Potential;
[Son & Stephanov, (2002)]
2
2
ˆ
  v [ p  m (T )]
2
2
Dispersion relation for
pions in nuclear matter.
Both v2 and v2m2(T)  0 near T=Tc.
 “velocity”
“screening mass”
2
ˆ
U  (m  v m (T ))  (1  v ) p
2
2
2
2
Both terms of U are negative (attractive)
U(b) = (w0+w2p2)r(b), w0 is real, w2 is complex.
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Parameters of the Model
Thermal:
Space:
Time:
Flow:
Optical Pot.:
Wave Eqn.:
T0 (MeV),
 (MeV)
RWS (fm),
aWS (fm)
Varied for the
0 (MeV/c),  (MeV/c) Cu+Cu prediction.
f (#),
 (#)
Re(w0) (fm-2), Re(w2) (#), Im(w2) (#)
e (#)1 (Kisslinger turned off for now)
Total number of parameters: 11 (+1)
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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. The correlation
function C is calculated at q=30 MeV/c, about half way down the BE
bump. (We do not use the 2nd moment of C, which is unreliable.)
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 ~2.2 and the c2 per degree of freedom is ~3.3. 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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DWEF Fits to STAR 200 GeV
Pion HBT Radii
Full
Calculation
U=0
Boltzmann
Re[U]=0
Non-solid curves
show the effects of
turning off various
parts of the
calculation.
No flow
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DWEF Fit to STAR 200 GeV
Pion Spectrum
Raw Fit
Non-solid curves show
the effects of turning off
various parts of the
calculation
Full
Calculation
U=0
Boltzmann
Re[U]=0
No flow
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Meaning of the Parameters
Temperature: 222 MeV; Chiral PT predicted at ~ 193 MeV
Transverse flow rapidity: 1.6  vmax= 0.93 c, vav= 0.66 c
Mean expansion time: 8.1 fm/c  system expansion at ~ 0.5 c
Pion emission between 5.5 fm/c and 10.8 fm/c  soft EOS .
WS radius: 12.0 fm = R(Au) + 4.6 fm > R @ SPS
WS diffuseness: 0.72 fm (similar to Low Energy NP experience)
Re(U): 0.113 + 0.725 p2  deep well  strong attraction.
Im(U): 0.128 p2  lmfp  8 fm @ KT=1 fm-1  strong absorption 
high density
 Pion chemical potential: m=124 MeV, slightly less than mass()








We have evidence suggesting a CHIRAL PHASE TRANSITION!
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ISMD 2005, Kromeriz
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Potential-Off Radius Re-Fits
Full
Calculation
Out
Side
No Real
Non-solid curves
show the effects of
refitting.
STAR Blast Wave
RO/RS Ratio
Long
No Optical
No Chemical or
Optical Pot.
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ISMD 2005, Kromeriz
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Potential-Off Spectrum Re-Fits
Model
No Optical
No Real
Chi^2
Full Calculation
Chi^2/#data
Chi^2/#dof
69.19
2.16
3.29
905.09
28.28
43.10
No Optical Potential
1003.44
31.36
47.78
No Opt/ Chem Potential
1416.66
44.27
67.46
No Real Potential
Raw Fit
Non-solid curves show
the effects of potentialoff refits.
Full
Calculation
STAR Blast Wave
No Chemical
or Optical Pot.
August 11, 2005
ISMD 2005, Kromeriz
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Low pT Ramsauer Resonances
14
16
12
14
RS 12
(fm)
10
RO 8
(fm)
No flow
10
6
Boltzmann
8
4
2
6
KT (MeV/c)
10
20
30
40
50
60
70
KT (MeV/c)
10
20
30
40
50
Re[U]=0
70
Pion Spectrum
Full Calculation
|y(q, b)|2 r(b) at
KT = 49.3 MeV/c
60
U=0
1.5
1
0.5
1
0
1
0.5
Raw Fit
0.5
0
For fit, would need l=0.41
KT (MeV/c)
0
-0.5
-0.5
-1
August 11, 2005
Phobos 0-6%
(preliminary)
-1
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200 GeV Cu+Cu Predictions
Scale RWS,  by Relect
Scale RWS,  by
Scale RWS, AWS, ,  by Relect
A1/3
Scale RWS, AWS, ,  by A1/3
Conclusion: the 4 space-time parameters scale as A1/3.
STAR
(preliminary)
August 11, 2005
ISMD 2005, Kromeriz
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Correlation Functions (Linear)
Long
100 MeV/c
Calculation
Gaussian Fit
k=0 WS Fit
R from 28 MeV/c
R from 2nd moment
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ISMD 2005, Kromeriz
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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
August 11, 2005
Calculation
Gaussian Fit
k=0 WS Fit
R from 28 MeV/c
R from 2nd moment
ISMD 2005, Kromeriz
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Correlation Functions (Log)
1
0.1
0.01
0.001
Calculation
Gaussian Fit
k=0 WS Fit
R from 28 MeV/c
R from 2nd moment
0.0001
0.00001
1.
10
Out
100 MeV/c
Conclusions:
(1) R from 28 MeV/c
gives a good match to
the full Gaussian fit;
(2) 2nd moment R’s are unreliable;
(3) The k=0 WS gives mixed results
in representing the tail region.
6
0
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20
40
ISMD 2005, Kromeriz
60
80
100
19
Correlation Functions (Log)
KT =
100 MeV/c
Side
Out
Long
KT =
200 MeV/c
KT =
400 MeV/c
KT =
600 MeV/c
August 11, 2005
Calculation
Gaussian Fit
k=0 WS Fit
R from 28 MeV/c
R from 2nd moment
ISMD 2005, Kromeriz
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Summary
 Quantum mechanics has solved the technical problems of
applying opacity to HBT.
 We obtain excellent DWEF fits to STAR sNN=200 GeV
data, simultaneously fitting three HBT radii and the pT
spectrum. The key ingredient is the deep real potential.
 The fit parameters are reasonable and indicate strong
collective flow, significant opacity, and large attraction.
 They describe pion emission in hot, highly dense matter
with a soft pion equation of state.
 We have replaced the RHIC HBT Puzzle with 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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ISMD 2005, Kromeriz
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Outlook
 We may have a new tool for investigating the presence
(or absence) of chiral phase transitions in heavy ion
collision systems.
 Its use requires both high quality pion spectra and
high quality HBT analysis over a region that extends to
fairly low momenta (KT~150 MeV/c).
 We are presently attempting to “track” the CPT
phenomenon to lower collision energies, where the
deep real potential should presumably go away.
 We note that the DWEF distorted wave method
presented here is not limited to the emission function
used in this work, and is applicable to any model that
is capable of producing an analytic emission function or
a multi-dimensional numerical emission function table.
August 11, 2005
ISMD 2005, Kromeriz
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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
Backup Slides
Kurtosis=0 WS Function
1
0.1
0.8
0.6
0.001
0.4
0.00001
0.2
1. 10
0
0
1
2
3
4
5
7
0
1
2
3
4
5
kurtosis  4 0 / 22  3  0
  0.321687
WSFn[ x]  (1  exp[1/  ]) /(1  exp[( x  R WS ) /( R WS )])
RGaussian  0.808158RWS
August 11, 2005
[Note: WSFn(x)=1 at x=0,
however, dWSFn(x)/dx  0 at x=0.]
ISMD 2005, Kromeriz
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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 timedependent 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.
August 11, 2005
ISMD 2005, Kromeriz
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Wave Equation Solutions
We assume an infinitely long Bjorken tube and azimuthal symmetry,
so that the (incoming) waves factorize:
3D  2D(distorted)1D(plane)
We solve the reduced Klein-Gordon wave equation:
Partial wave expansion ! ordinary diff eq
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ISMD 2005, Kromeriz
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The Meaning of U
Im (U) : Opacity, Re (U) :Refraction
Pions lose energy and flux.
-3,

1
mb,
r
=
1
fm
Im[U0]=-p r0,
0
Im[U0] = .15 fm-2, l = 7 fm
Re(U) must exist:
very strong attraction
chiral phase transition
August 11, 2005
ISMD 2005, Kromeriz
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Compute Correlation Function
C ( K , q)  1 
d
d
4
4
2
x S ( x, K , q )
x S ( x, p1 )  d x S ( x, p2 )
4
Correlation function is not Gaussian;
we evaluate it near the q of experiment.
The R2 values are not the moments of
the emission function S.
August 11, 2005
ISMD 2005, Kromeriz
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Overview of the DWEF Model





The medium is dense and strongly interacting, so the pions must “fight”
their way out to the vacuum. This modifies their wave functions,
producing the distorted waves used in the model.
We explicitly treat the absorption of pions by inelastic processes (e.g.,
quark exchange and rearrangement) as they pass through the medium,
as implemented with the imaginary part of an optical potential.
We explicitly treat the mass-change of pions (e.g., due to chiralsymmetry breaking) as they pass from the hot, dense collision medium
[m()0]) to the outside vacuum [m()140 MeV]. This is
accomplished by solving the Klein-Gordon equation with an optical
potential, the real part of which is a deep, attractive, momentumdependent “mass-type” potential.
We use relativistic quantum mechanics in a cylindrical geometry partial
wave expansion to treat the behavior of pions producing Bose-Einstein
correlations. We note that most RHIC theories are semi-classical,
even though most HBT analyses use pions in the momentum region (p <
600 MeV/c) where quantum wave-mechanical effects should be
important.
The model calculates only the spectrum of pions participating in the BE
correlation (not those contributions to the spectrum from long-lived
“halo” resonances, etc.).
August 11, 2005
ISMD 2005, Kromeriz
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Semi-Classical Eikonal Opacity
b
X
l
+
R
Heiselberg and Vischer
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ISMD 2005, Kromeriz
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Influence of the Real Potential
in the Eikonal Approximation
Factors of  cancel out in the product y (-) ( p1 , b)y *(-) ( p2 , b).
Therefore the real part of U, no matter
how large, has no influence here.
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ISMD 2005, Kromeriz
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Source De-magnification
by the Real Potential Well
Velocity
in well
n=1.33
n=1.00
Velocity
in vacuum
Vcsr = (120 MeV)2
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.
August 11, 2005
ISMD 2005, Kromeriz
Rays bend
closer to
radii
A Fly in a Bubble
33
|y(q, b)|2 r(b) at
KT = 1.000 fm-1 = 197 MeV/c
Observer
1
0.75
0.5
0.25
1
0
1
0.5
0.5
0
0
1
-0.5
0.5
1
-0.5
-1
Imaginary Only
-1
0
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
Wave Function of Full Calculation
Eikonal
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ISMD 2005, Kromeriz
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Primer 1: HBT Interferometry
C(q)-1
1
C(p1,p2) = (p1,p2)/(p1)(p2)
0.8
1. Two identical pions will have
1/R
l
a Bose-Einstein enhancement
BE
enhancement
when their relative momentum
(q) is small.
q (MeV/c)
2. The 3-D momentum width of the BE enhancement 
the 3-D size (R) of the pion “fireball” source.
3. Assuming complete incoherence, the “height” of the
BE bump l tells us the fraction (l½) of pions
participating in the BE enhancement.
4. The “out” radius of the source requires a pion energy
difference E  related to  (emission duration).
0.6
0.4
0.2
0
0
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ISMD 2005, Kromeriz
50
100
150
35
200
HBT Momentum Geometry
Relative momentum q between pions is a vector q  p1  p 2
so we can extract 3D shape information.
Rlong – along beam direction
Rout – along “line of sight”, includes time/energy information.
Rside – ^ “line of sight”, no time/energy information.
K  12 ( p1  p2 )
p1
q
Rside
Rout
Pre-RHIC expectations:
p2
(1) Large Ro,s,l (~12-20 fm)
(2) Rout > Rside by ~4 fm.
August 11, 2005
ISMD 2005, Kromeriz
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The Featureless HBT Landscape
The source radii, as
inferred from HBT
interferometry, are very
similar over almost two
orders of magnitude in
collision energy.
AGS
CERN
RHIC
The ratio of Ro/Rs is
near 1 at all energies,
which naively implies a
“hard” equation of state
and explosive emission
behavior (~0).
August 11, 2005
ISMD 2005, Kromeriz
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