3 Roy Lacey, SUNY Stony Brook

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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
6
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
7
What do we know ?
Oh yes - It is Comprehensive !
Roy Lacey, SUNY Stony Brook
8
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
12
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
14
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
17
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
18
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
19
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
20
Experimental Results
Roy Lacey, SUNY Stony Brook
21
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)
22
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
30
• 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
35
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
37
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
38
Short and long-range components of the source
T. Csorgo
M. Csanad
Roy Lacey, SUNY Stony Brook
39
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
40
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
41
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
42
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
43
Cuts
Roy Lacey, SUNY Stony Brook
44
Cuts
Roy Lacey, SUNY Stony Brook
45
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
48
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