Evidence for a long-range pion emission source in
Au+Au collisions at sNN  200 GeV
Roy Lacey & Paul Chung
Nuclear Chemistry, SUNY, Stony Brook
1
Motivation
Conjecture of collisions at RHIC :
Courtesy S. Bass
initial state
Increased System Entropy
that survives hadronization
hadronic phase
QGP and
hydrodynamic expansion
pre-equilibrium
and freeze-out
hadronization
Expectation:
A strong first order phase transition leads to an emitting
system characterized by a much larger space-time extent
than would be expected from a system which remained in
the hadronic phase
Guiding philosophy in first few years at RHIC = Puzzle ?
Roy Lacey, SUNY Stony Brook
2
Roy Lacey, SUNY Stony Brook
3
What do we know ?
Extrapolation From ET
Distributions

e Bj 
1 1 dET
 R 2 t 0 dy

e
P  ² 





s/ 
Flow
thermalization time
(t0 ~ 0.2 – 1 fm/c)
eBj ~ 5 – 15 GeV/fm3
Roy Lacey, SUNY Stony Brook
4
What do we know ?
PHENIX Preliminary
PHENIX Preliminary
v2 scales with eccentricity
and across system size
Strong Evidence for Thermalization
and hydro scaling
Roy Lacey, SUNY Stony Brook
5
What do we know ?
Scaling breaks
Baryons scale
together
Mesons scale
together
PHENIX preliminary data
Perfect fluid hydro Scaling
holds up to ~ 1 GeV
Strong hydro scaling with
hint of quark degrees of freedom
Roy Lacey, SUNY Stony Brook
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What do we know ?
PHENIX preliminary data
Scaling works
Scaling holds over the
whole range of KET
Compatible with Valence Quark degrees of
freedom
Roy Lacey, SUNY Stony Brook
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What do we know ?
Oh yes - It is Comprehensive !
Roy Lacey, SUNY Stony Brook
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What do we know ?
T. Renk, J. Ruppert
hep-ph/0509036
Away-side peak consistent
with mach-cone scenario
nucl-ex/0507004
nucl-th/0406018 Stoecker
hep-ph/0411315 Casalderrey-Solana, et al
other explanations !
Strong centrality dependent modification
of away-side jet in Au+Au
Implication for viscosity
and sound speed !
Roy Lacey, SUNY Stony Brook
9
A Small digression
High pT particle
12
13

12  13  
Simulated Result
View associated particles in frame
with high pT direction as z-axis
Yes ! We have results
Roy Lacey, SUNY Stony Brook
10
What do we know ?
Sound Speed Estimate
cs ~ 0.35
Soft EOS
F. Karsch, hep-lat/0601013
Compatible
with soft EOS
Sound speed is not zero during an extended
hadronization
period.
Space-time evolution more subtle ?
Roy Lacey, SUNY Stony Brook
11
Extract the full source function
Roy Lacey, SUNY Stony Brook
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Extraction of Source functions
Imaging & Fitting
Moment Expansion
Roy Lacey, SUNY Stony Brook
13
Imaging Technique
Technique Devised by:
D. Brown, P. Danielewicz,
PLB 398:252 (1997).
PRC 57:2474 (1998).
Emitting source

Inversion of Linear integral
equation to obtain source function
1D Koonin Pratt Eqn.
C (q)  1  4  drr 2 K 0 (q, r )S (r )
Encodes FSI
Correlation
function
Source
function
(Distribution of pair
separations)
Inversion of this integral equation
== Source Function
Well established inversion procedure
Roy Lacey, SUNY Stony Brook
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Correlation Fits
[Theoretical correlation function]
convolute source function
with kernel (P. Danielewicz)
Measured correlation function
Minimize Chi-squared
Parameters of the source function
Roy Lacey, SUNY Stony Brook
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Quick Test with simulated source
Input source function recovered
Procedure is Robust !
Roy Lacey, SUNY Stony Brook
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Experimental Results
Roy Lacey, SUNY Stony Brook
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Au  Au
1D Source imaging
snn  200 GeV
Source functions from
spheroid or
Gaussian + Exponential
give good fit.
Source function tail is not due to:
• Kinematics
• Resonance contributions
PHENIX Preliminary
Evidence for long-range source at RHIC
Roy Lacey, SUNY Stony Brook
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PHENIX Preliminary
 r2 

b
S (r ) 
exp


erfi


 
2
8b RT3 a
4
R
2

T 
 1  r
b= 1- 2 
 a  RT
  , a, RT 
kinematics
Centrality dependence also incompatible with resonance decay
Roy Lacey, SUNY Stony Brook
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Pair fractions associated with long- and short-range structures
Core Halo assumption
s = f c2  HBT
l = 2f c f
f
f
l
2  0.1
2
2

0.3
s
fc
s
0.5
l
Expt  
s
T. Csorgo
M. Csanad
1.0
Contribution from  decay insufficient to account for longrange component.
Full fledge simulation indicate similar conclusion
Roy Lacey, SUNY Stony Brook
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Experimental Results
Roy Lacey, SUNY Stony Brook
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3D Analysis
Basis of Analysis
(Danielewicz and Pratt nucl-th/0501003 (v1) 2005)
Expansion of R(q) and S(r) in Cartesian Harmonic basis
R(q )  
l
S (r )  
l

 
Rl 1 ....l  q  l

 
Sl 1 ....l  r  l
1 .... l
1 .... l
1
1 .... l
.... l
( q )
( r )
(1)
(2)
R(q )  C (q )  1  4  dr 3 K (q , r )S (r ) (3)
3D Koonin
Pratt
Plug in (1) and (2) into (3)
Rl 1 ....l (q)  4  drr 2 Kl (q, r )Sl
1
(1)
(2)
l
R1 ....l
2l  1!!

(q) 

d q
l
....l
(r )
(4)
(q ) R(q ) (4)
l!
4
 2l  1!! d r l ( ) S (r )
Sl 1 ....l (r ) 
r
l !  4 1 ....l
1
....l
Roy Lacey, SUNY Stony Brook
(5)
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Calculation of Correlation Moments:
Fitting with truncated expansion series !
C ( q , ) 

 
Cl 1 ....l (q)l 1 ....l ()
1 .... l
up to order 4
C (q, )  C0 (q ) 0 ()  Cx 2 (q)  x 2 ()  C y 2 (q)  y 2 ()
 C z 2 ( q )  z 2 ( )  C x 4 ( q )  x 4 ( )  C y 4 ( q )  y 4 ( )
Cz 4 (q)  z 4 ()  6Cx 2 y 2 (q)  x 2 y 2 ()  6Cx 2 z 2 ( q)  x 2 z 2 ()
6C y 2 z 2 (q)  y 2 z 2 ()
6 independent
moments
C0 , Cx 2 , C y 2 , Cx 4 , C y 4 , Cz 4   Ci
6
CTh (q , )   fi ()Ci
i=1
(a)
Roy Lacey, SUNY Stony Brook
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A look at the basis
L=0
S
L=2
Axx
Azz
Ayy
Axx  Ayy  Azz  S
Roy Lacey, SUNY Stony Brook
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Strategy
C0 , Cx 2 , Cy 2 , Cx 4 , Cy 4 , Cz 4 
Get values of
CTh (q, )  CExp (q, )
Such that
Fit
CTh (q, ) to CExp (q, ) with moments as fitting parameters.
 
2
C
exp
( )  CTh ( ) 
 ( )
2

Minimize 
 2


2
1

2
C

6
B C
ij
2 
0
Ci
2
j
Exp
 Di

 CTh

for each q.
i=1,..,6

 2CTh
0
Ci
i=1,....,6
j=1
Roy Lacey, SUNY Stony Brook
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Strategy
With
Bij  

Di  

1
2
f i f j
1


C
f
2 Exp i

C j   Bij1 D j
;
f i  f i   
; CExp  CExp   
for each q

Roy Lacey, SUNY Stony Brook
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Simulation tests of the method
input
Procedure
• Generate moments for
source.
• Carryout simultaneous
Fit of all moments
output
Very clear proof of principle
Roy Lacey, SUNY Stony Brook
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Results - moments
C 0  C (qinv )
Very good agreement as it should
Roy Lacey, SUNY Stony Brook
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Results - moments
Sizeable
signals observed
for l = 2
Exquisite/Robust Results
Roy Lacey, SUNY Stony Brook
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Results - moments
l= 4 moments
Exquisite/Robust Results
Roy Lacey, SUNY Stony Brook
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• Extensive study of two-pion source
images and moments in Au+Au collisions at RHIC
• First observation of a long-range source having an
extension in the out direction for pions
Long-range source not due to
kinematics or resonances
Further Studies underway to quantify
A variety of other source functions!
Much more to come !
Roy Lacey, SUNY Stony Brook
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Roy Lacey, SUNY Stony Brook
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Comparison of Source Functions
Source functions from spheroid and Gaussian + Exponential are in
excellent agreement  need 3D info
Roy Lacey, SUNY Stony Brook
33
3D Source imaging
   Au  Au
snn  200 GeV
Origin of deformation
Kinematics ?
or
Time effect
Instantaneous
Freeze-out
PHENIX Preliminary
• LCMS implies kinematics
• PCMS implies time effect
x  out
y  side
z  long
Deformed source in pair cm frame:
Roy Lacey, SUNY Stony Brook
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3D Source imaging
pp
Au  Au
snn  200 GeV
Isotropic emission in the
pair frame
•
PHENIX Preliminary
x  out
y  side
z  long
Spherically symmetric source in pair cm. frame (PCMS)
Roy Lacey, SUNY Stony Brook
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Short and long-range components of the source
RL  a  RT
 r2 

b
S (r ) 
exp   2   erfi  
3
8b RT a
2
 4 RT 
 1 r
b= 1- 2 
 a  RT
  , a, RT 
T. Csorgo
M. Csanad
Short-range 
Long-range 
Rs  1.2 RT
Rl  R0  a  RT
4  RT 
Rl
Rs
3.0
l
s
Roy Lacey, SUNY Stony Brook
1.0
36
New 3D Analysis
1D analysis  angle averaged C(q) & S(r) info only
• no directional information
Need 3D analysis to access directional information
Correlation and source moment fitting and imaging
Roy Lacey, SUNY Stony Brook
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3D Analysis
How to calculate correlation function and
Source function in any direction
Cx (q )  C 0 (q )  C x1 (q )  C xx2 (q )  ...
S x (r )  S 0 (r )  S 1x (r )  S xx2 (r )  ...
C y (q )  C 0 (q )  C 1y (q)  C yy2 (q)  ...
S y (r )  S 0 (r )  S 1y (r )  S yy2 (r )  ...
Source function/Correlation function obtained via moment
summation
Roy Lacey, SUNY Stony Brook
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Short and long-range components of the source
T. Csorgo
M. Csanad
Roy Lacey, SUNY Stony Brook
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Extraction of Source Parameters
Fit Function (Pratt et al.)
Radii
Pair Fractions
S (r ) 
gaus e
2

r2
2
4 Rgaus
 Rgaus

3
exp
+
e
N (  , Rexp )
r2  2

Rexp
 K 0 ( z ) 2 K1 ( z ) 
N (  , Rexp )  4 


2
z
z


2
Rgaus

Bessel Functions
 =2
, z
Rexp
Rexp
3
This fit function allows extraction of both
the short- and long-range
components of the source image
Roy Lacey, SUNY Stony Brook
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Outline
1. Motivation
2. Brief Review of Apparatus & analysis
technique
3. 1D Results
• Angle averaged correlation function
• Angle averaged source function
4. 3D analysis
• Correlation moments
• Source moments
5. Conclusion/s
Roy Lacey, SUNY Stony Brook
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Imaging
Inversion procedure
C (q)  4  drr 2 K (q, r )S (r )
S (r )   S j  B j (r )
C (q )   K ij  S j
j
Th
i
j
K ij   dr  K (q, r ) B j (r )
 Expt

 Ci (q )   K ij  S j 
j

2  
 2Ci (q ) Expt
Roy Lacey, SUNY Stony Brook
2
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Fitting correlation functions
Kinematics
“Spheroid/Blimp” Ansatz
Brown & Danielewicz PRC 64, 014902 (2001)
 r2 

b
S (r ) 
exp


erfi

 
2 
8b RT3 a
4
R
2

T 
 1  r
b= 1- 2 
 a  RT
  , a, RT 
Roy Lacey, SUNY Stony Brook
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Cuts
Roy Lacey, SUNY Stony Brook
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Cuts
Roy Lacey, SUNY Stony Brook
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Two source fit function
S1 (r ) = sGs  l Gl

 2

2
2
1
x
y
z
 
 

exp


3
s 2
s 2 
 2  Rs 2
s s s
2  Rs Rl Ro
  o   Rs   Rl   

s




 2

2
2
l
 1  x  y  z 
exp
3
l 2
l 2 
 2  Rl 2
l l l
2  Rs Rl Ro
  o   Rs   Rl   


This is the single particle distribution
Roy Lacey, SUNY Stony Brook
46
Two source fit function
S (r ) =  d 3rq S1  r2  r  S 2  r2 

 2

2
2
1
x
y
z
 
 

exp


3
s 2
s 2 
 4  Rs 2
s s s
2  Rs Rl Ro
  o   Rs   Rl   

s2




 2

2
2

 1  x  y  z  
exp
3
l 2
l 2 
 4  Rl 2
l l l
R
R
2  Rs Rl Ro
  o   s   l  

2
l

2s l
2    R    R   R    R   R    R 
3
s 2
s
l 2
s
s 2
l
l 2
l
s 2
o
l
o
2




2
2
2
1
x
y
z

exp   


2
2
2
2
2
2
s
l
s
l
 2  R s  Rl

R

R
R

R












o
s
s
l
l
 o


This is the two particle distribution
Roy Lacey, SUNY Stony Brook
47
Experimental Setup
PHENIX Detector
Several Subsystems
exploited for the
analysis
Excellent Pid is achieved
 TOF ~ 120 ps  /K  2 GeV/c 
 EMC ~ 450 ps  /K  1 GeV/c 
Roy Lacey, SUNY Stony Brook
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