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Quantum Opacity, RHIC HBT
Puzzle, and the Chiral Phase
Transition
Gerald Miller and John Cramer, UW
• RHIC Physics, HBT and RHIC HBT Puzzle
• Formalism
• Quantum mech. treatment of optical
potential, U (Chiral symmetry)
• Reproducing data, wave function
• Summary
The RHIC HBT Puzzle
Data from the first five years of RHIC
Some evidence supports the presence of QGP formed in early
stages of Au+Au collisions:
 Relativistic hydrodynamics describes the low and medium
energy dynamic collision products
Elliptic flow data implies very high initial pressure and
collectivity

Most energetic pions, produced early, strongly
suppressed
 Strong suppression of back-to-back jets.
D Au vs
Au Au, central vs peripheral
Hydrodynamics works
BUT NOT FOR HBT
HBT- 2 particle interferometry
p1
qside
q
q=p1-p2
p2
p2
Rside
+
qout
p2
Rout
p1
qlong
Quantum mechanical interference-space time
separation of source
K=(p +p )/2
1
2
Hydrodynamics predicts big RO/RS,
Data
RO/R
1 HBT
C(q,K)
=s(p
,pS )about
/(s(p )s(p
))-1 ~puzzle
1
2
1
2
λ(1-q2L R2L-q2S R2S –q2O R2O )
Time extent of source R2o >>R2s
j=w1 t1 +w2 t2
Expect R2o >>R2s
Rs =Ro
A highlight from this week
Burt Holzman, PHOBOS
Old Formalism
 source current density =J
Chaotic sources, Shuryak ‘74 S0~<J J*>
σ(p1)
Source Properties
hydrodynamics inspired source function of Wiedemann
Heinz et al
Bjorken tube model-boost invariant
S0(x,K) ~freeze out surface
but π emission allowed
everywhere
ρ(b) medium density
radial flow
Overview of Our Model
 Allow pions to be emitted anywhere in
medium, not only at freeze-out surface
 Pions interact with the matter on their way
out.
 Pion absorption implemented via
imaginary part of optical potential.
 Real part of optical must exist, acts as mass
and velocity change of pions due to chiralsymmetry breaking as they pass from the hot,
dense collision medium to the outside
vacuum
Formalism
• Pions interact U with dense medium
Gyulassy et al ‘79

is distorted (not plane) wave
DWEF- distorted wave emission function
Wave Equation Solutions
Matter is infinitely long Bjorken tube and azimuthal
symmetry, wave functions factorize: 3D 
2D(distorted)1D(plane)
We solve the reduced Klein-Gordon wave equation for yp:
Partial wave expansion ! ordinary diff eq
Son & Stephanov 2002
v2, v2 m2 (T)approach 0 near T = Tc
=ω2-m2π
Both terms of U are negative (attractive)
Fit STAR Data
6 source, 3 optical potential parameters
Fit central STAR data at sNN=200 GeV
reproduce Ro, Rs, Rl
reproduce dN/dy (both magnitude and shape)
8 momentum values (i.e., 32 data points)
Fit to 200 GeV Au+Au Radii
U=0
Re U=0
Potential Effects
200 GeV Au+Au Spectrum
U=0
noBE
no flow
Meaning of the Parameters
• Temperature: 193 MeV fixed at phase transition temperature
S. Katz, QM05
• Transverse flow rapidity: 1.5  vmax=0.85 c, vav=0.6 c
• Pion emission between 6.2 fm/c and 11 fm/c soft EOS .
• WS radius: 11.8 fm = R (Au) + 4.4 fm > R @ SPS
• Re(U): 0.14 + 0.85 p2  deep well strong attraction.
• Im(U): 0.12 p2 mfp 8 fm @ KT=1 fm-1 strong
absorption  high density
• Pion chemical potential: m= mass()
Consistent with CHIRAL PHASE TRANSITION!
Wave Functions |y(q,
DWEF
(Full)
KT=
100 MeV/c
KT=
250 MeV/c
KT=
600 MeV/c
DWEF
(Im Pot only)
2
b)| r(b)
Eikonal
Approx.
Centrality & Nuclear Dependence
Cu+Cu
Au+Au
Rout
Rside
Rlong
Au+Au
Centrality:
Rout
Cu+Cu
Centrality:
0-5%
0-10%
5-10%
10-20%
10-20%
20-30%
20-30%
Rside
30-40%
30-50%
40-50%
50-80%
50-60%
Rlong
Summary
 Quantum mechanics solves technical problems of applying
opacity to HBT.
 Excellent fits sNN=200 GeV data: three radii, pT spectrum.
 Fit parameters: indicate strong collective flow, significant
opacity, and huge attraction. Describe pion emission in hot,
highly dense matter (a soft pion equation of state) .
 Replace the RHIC HBT Puzzle with evidence for a chiral
phase transition. In most scenarios, the QGP phase transition
is accompanied by chiral phase transition at about same
critical temperature.
• Phys.Rev.Lett.94:102302,2005, nucl-th/0507004
The End
SPARES FOLLOW
For details see:
Phys.Rev.Lett.94:102302,20
05
and a newer preprint:
nucl-th/0507004 ,
submitted PRC
The End
Source Properties
S0 ( x, k ) = S0 ( , ) B (b, KT ) /(2 )3
2
2

(   0 )
cosh 
 
S 0 ( , ) =
exp  

2
2
2
2  
2 ( )
 2 
(“hydrodynamics inspired” source function of Heinz & collaborators)
1
B (b, KT ) = M T
r (b)
 K  u  m  (medium density)
exp 
1

T
2
2
2


 =t z
(Bose-Einstein thermal function)
tz
1
 = 2 ln 

tz
K = particle momentum 4-vector
u = trasverse flow 4-vector
S0(x,K) ~freeze out surface
Correlation/Gaussian Fit
Eikonal Magnitude of wave function
b
l
q
b/R
RO = R/4.48 HV
Correlation Functions
9 Fits: 200 GeV/A Au+Au
c2 vs. Temp for 9 Fits