Roy Lacey & Paul Chung Nuclear Chemistry, SUNY, Stony Brook
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Roy Lacey & Paul Chung Nuclear Chemistry, SUNY, Stony Brook
Evidence for a long-range pion emission source inEvidence for a long-range pion emission source inAu+Au collisions atAu+Au collisions at 200 NNs GeV
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initial state
pre-equilibrium
QGP andhydrodynamic expansion
hadronization
hadronic phaseand freeze-out
Conjecture of collisions at RHIC :MotivationMotivation
Courtesy S. BassCourtesy S. Bass
Increased System Entropy Increased System Entropy that survives that survives hadronizationhadronization
Expectation:Expectation:A strong first order phase transition leads to an emitting A strong first order phase transition leads to an emitting system characterized by a much larger space-time extent system characterized by a much larger space-time extent thanthan would be expected from a system which remained in would be expected from a system which remained in
the hadronic phase the hadronic phase
Guiding philosophy in first few years at RHIC =Guiding philosophy in first few years at RHIC = Puzzle ? Puzzle ?
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What do we know now
What do we know now
??
Any Implicatio
ns for
Any Implicatio
ns for
HBT ?HBT ?
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thermalization time (0 ~ 0.2 – 1 fm/c)
Bj~ 5 – 15 GeV/fm3
dy
dE
RT
Bj0
2
11
Extrapolation From EExtrapolation From ETT
DistributionsDistributions
What do we know ?What do we know ?
s/
P ²
FlowFlow
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What do we know ?What do we know ?
PHENIX Preliminary
PHENIX Preliminary
v2 scales with eccentricityand across system size
Strong Evidence for Thermalization Strong Evidence for Thermalization and hydro scalingand hydro scaling
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What do we know ?What do we know ?
Scaling breaks
Perfect fluid hydro Scaling holds up to ~ 1 GeV
Mesons scale together
Baryons scale together
Strong hydro scaling with hint of quark degrees of freedom
PHENIX preliminary data
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What do we know ?What do we know ?
Scaling works
Compatible with Valence Quark degrees of freedom
Scaling holds over the whole range of KET
PHENIX preliminary data
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Oh yes - It is Comprehensive !
What do we know ?What do we know ?
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nucl-ex/0507004
What do we know ?What do we know ?What do we know ?What do we know ?
T. Renk, J. Ruppert hep-ph/0509036
Strong centrality dependent modification Strong centrality dependent modification of away-side jet in Au+Auof away-side jet in Au+Au
Away-side peak consistentAway-side peak consistentwith mach-cone scenariowith mach-cone scenario
nucl-th/0406018 Stoeckerhep-ph/0411315 Casalderrey-Solana, et al
other explanationsother explanations ! !
Implication for viscosityand sound speed !
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View associated particles in frame View associated particles in frame with high pT direction as z-axiswith high pT direction as z-axis
12
High pT particleHigh pT particle
Associated pt
Associated pt
particles
particles
1312 13
Simulated ResultSimulated Result
A Small digressionA Small digression
Yes ! We have resultsYes ! We have results
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Sound Speed Estimate
cs ~ 0.35 Soft EOS F. Karsch, hep-lat/0601013
What do we know ?What do we know ?
Compatible with soft EOSCompatible with soft EOSSound speed is not zero during an extended hadronization period.Sound speed is not zero during an extended hadronization period.Space-time evolution more subtle ?Space-time evolution more subtle ?
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Subtle sig
nals
Subtle sig
nals
require a paradigm
require a paradigm
shiftshift
Extract the full source functionExtract the full source function
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Extraction of Source functionsExtraction of Source functions
Imaging & Fitting Moment Expansion
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Imaging TechniqueImaging Technique
Technique Devised by:
D. Brown, P. Danielewicz,PLB 398:252 (1997). PRC 57:2474 (1998).
Inversion of Linear integral equation to obtain source function
20( ) 1 ) (,4 ( )C K q r S rq drr
Source Source functionfunction
(Distribution of pair separations)
Encodes FSI
CorrelationCorrelationfunctionfunction
Inversion of this integral equation== Source Function
Emitting source
1D Koonin Pratt Eqn.
Well established inversion procedureWell established inversion procedure
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Correlation FitsCorrelation Fits
Parameters of the source functionParameters of the source function
Minimize Chi-squared
[Theoretical correlation function]convolute source function convolute source function with kernel with kernel (P. Danielewicz)(P. Danielewicz) Measured correlation function
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Input source function recoveredInput source function recoveredProcedure is Robust !Procedure is Robust !
Quick Test with simulated sourceQuick Test with simulated source
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Experimental ResultsExperimental Results
Gaussian Source
Gaussian Source
functions d
o not provide
functions d
o not provide
good fits
good fits
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Evidence for long-range source at RHICEvidence for long-range source at RHIC
1D Source imaging1D Source imaging
PHENIX Preliminary
200 GeVnnAu Au s
Source functions from Source functions from spheroid or spheroid or Gaussian + Exponential Gaussian + Exponential give good fit.give good fit.
Source function tail is notnot due to:• Kinematics• Resonance contributions
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PHENIX Preliminary
Centrality dependence also incompatible with resonance decay
kinematics
2
3 2
2
( ) exp 8 4 2
1b= 1- , a,
a
T T
TT
r bS r erfi
b R a R
rR
R
2
3 2
2
( ) exp 8 4 2
1b= 1- , a,
a
T T
TT
r bS r erfi
b R a R
rR
R
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Pair fractions associated with long- and short-range structuresPair fractions associated with long- and short-range structuresPair fractions associated with long- and short-range structuresPair fractions associated with long- and short-range structures
T. CsorgoM. Csanad
2s
l
l
s s
=
= 2
2 0.12 2 0.3
0.5
c HBT
c
c
f
f f
f f
f
Core Halo assumption
1.0l
s
Expt
Contribution from decay insufficient to account for long-range component.
Full fledge simulation indicate similar conclusionFull fledge simulation indicate similar conclusion
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Experimental ResultsExperimental Results
A hint of the shape of
A hint of the shape of
things to come
things to come
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3D Analysis3D Analysis
1 11
1 11
.... ........
.... ........
( ) ( ) (1)
( ) ( ) (2)
l ll
l ll
l lq
l
l lr
l
R q R q
S r S r
3( ) ( ) 1 4 ( , ) ( )R q C q dr K q r S r
(3)3D Koonin3D KooninPrattPratt
Plug in (1) and (2) into (3)1 1
2.... ....
( ) 4 ( , ) ( ) (4)l l
l llR q drr K q r S r
1 1
2.... ....
( ) 4 ( , ) ( ) (4)l l
l llR q drr K q r S r
1 1
1 1
.... ....
.... ....
2 1 !!( ) ( ) ( ) (4)
! 42 1 !!
( ) ( ) ( ) (5)! 4
l l
l l
ql lq
l lrr
dlR q R q
ll d
S r S rl
1 1
1 1
.... ....
.... ....
2 1 !!( ) ( ) ( ) (4)
! 42 1 !!
( ) ( ) ( ) (5)! 4
l l
l l
ql lq
l lrr
dlR q R q
ll d
S r S rl
(1)
(2)
Expansion of R(q) and S(r) in Cartesian Harmonic basisExpansion of R(q) and S(r) in Cartesian Harmonic basis
Basis of AnalysisBasis of Analysis
(Danielewicz and Pratt nucl-th/0501003 (v1) 2005)(Danielewicz and Pratt nucl-th/0501003 (v1) 2005)
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Calculation of Correlation Moments:Calculation of Correlation Moments:
1 1
1
.... ........
0 0 2 2 2 2
2 2 4 4 4 4
4 4 2 2 2 2 2 2 2 2
2 2 2
( , ) ( ) ( )
to order 4
( , ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) 6 ( ) ( ) 6 ( ) ( )
6 ( )
l l
l
l l
x x y y
z z x x y y
z z x y x y x z x z
y z y z
C q C q
up
C q C q C q C q
C q C q C q
C q C q C q
C q
2
0 2 2 4 4 4
6
ii=1
( )
C , , , , ,
( , ) f ( )
x y x y z i
Th i
C C C C C C
C q C
Fitting with truncated expansion series !Fitting with truncated expansion series !
6 independent moments
(a)
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A look at the basisA look at the basis
L=0
L=2
xxA yyA zzA
xx yy zzA A A S
S
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StrategyStrategy
Get values of 0 2 2 4 4 4C , , , , ,x y x y zC C C C C
Such that ( , ) ( , )Th ExpC q C q
Fit ( , ) to ( , ) Th ExpC q C q with moments as fitting parameters.
2
exp22
22
2
2
6
ijj=1
( ) ( ) for each q.
( )
Minimize 0 i=1,..,6
1 2 0
B i=1,....,6
Exp
Th
i
ThTh
i
j i
C C
C
CC C
C
C D
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StrategyStrategy
With
2
2
1
1 ;
1 ; C
for each q
ij i j i i
i Exp i Exp Exp
j ij j
B f f f f
D C f C
C B D
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Simulation tests of the methodSimulation tests of the method
Very clear proof of principleVery clear proof of principle
ProcedureProcedure• Generate moments forsource.
• Carryout simultaneous Fit of all moments
input
output
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Results - momentsResults - moments
0 ( )invC C q
Very good agreement as it should
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Results - momentsResults - moments
Exquisite/Robust ResultsExquisite/Robust Results
Sizeable signals observed
for l = 2
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Results - momentsResults - moments
l= 4 momentsl= 4 moments
Exquisite/Robust ResultsExquisite/Robust Results
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• Extensive study of two-pion source Extensive study of two-pion source images and moments in Au+Au collisions at RHICimages and moments in Au+Au collisions at RHIC
• First observation of a long-range source having an First observation of a long-range source having an extension in the out direction for pionsextension in the out direction for pions
Long-range source not due to Long-range source not due to kinematics or resonanceskinematics or resonances
Further Studies underway to quantify A variety of other source functions!
Much more to come !Much more to come !
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Source functions from spheroid and Gaussian + Exponential are in Source functions from spheroid and Gaussian + Exponential are in excellent agreement excellent agreement need 3D info need 3D info
Comparison of Source FunctionsComparison of Source FunctionsComparison of Source FunctionsComparison of Source Functions
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PHENIX Preliminary
3D Source imaging3D Source imaging
Deformed source in pair cm frame:Deformed source in pair cm frame:
200 GeVnnAu Au s
x out
y side
z long
Origin of deformationKinematics ?
orTime effectTime effect
Instantaneous Freeze-out
• LCMS implies kinematics• PCMS implies time effect
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PHENIX Preliminary
pp3D Source imaging3D Source imaging
Spherically symmetric source in pair cm. frame (PCMS)Spherically symmetric source in pair cm. frame (PCMS)
200 GeVnnAu Au s
x out
y side
z long
Isotropic emission in thepair frame
•
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Short and long-range components of the sourceShort and long-range components of the sourceShort and long-range components of the sourceShort and long-range components of the source
2
3 2
2
( ) exp 8 4 2
1b= 1- , a,
a
L T
T T
TT
R a R
r bS r erfi
b R a R
rR
R
2
3 2
2
( ) exp 8 4 2
1b= 1- , a,
a
L T
T T
TT
R a R
r bS r erfi
b R a R
rR
R
Short-range
Long-range
01.2 4 3.0ls T l T T
s
RR R R R a R R
R
T. CsorgoM. Csanad
1.0l
s
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New 3D AnalysisNew 3D Analysis
1D analysis angle averaged C(q) & S(r) info only• no directional information
Need 3D analysis to access directional informationNeed 3D analysis to access directional information
Correlation and source moment fitting and imagingCorrelation and source moment fitting and imagingCorrelation and source moment fitting and imagingCorrelation and source moment fitting and imaging
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3D Analysis3D Analysis
How to calculate correlation function and Source function in any direction
0 1 2
0 1 2
0 1 2
0 1 2
( ) ( ) ( ) ( ) ...
( ) ( ) ( ) ( ) ...
( ) ( ) ( ) ( ) ...
( ) ( ) ( ) ( ) ...
x x xx
x x xx
y y yy
y y yy
C q C q C q C q
S r S r S r S r
C q C q C q C q
S r S r S r S r
0 1 2
0 1 2
0 1 2
0 1 2
( ) ( ) ( ) ( ) ...
( ) ( ) ( ) ( ) ...
( ) ( ) ( ) ( ) ...
( ) ( ) ( ) ( ) ...
x x xx
x x xx
y y yy
y y yy
C q C q C q C q
S r S r S r S r
C q C q C q C q
S r S r S r S r
Source function/Correlation function obtained via moment Source function/Correlation function obtained via moment summationsummation
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Short and long-range components of the sourceShort and long-range components of the sourceShort and long-range components of the sourceShort and long-range components of the source
T. CsorgoM. Csanad
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Extraction of Source ParametersExtraction of Source Parameters
Fit Function Fit Function (Pratt et al.)(Pratt et al.)
2
2 22
exp
4
exp
3exp
3 0 1exp 2
2
exp exp
( ) +( , )2
( ) 2 ( )( , ) 4
=2 ,
gaus
rrRRgaus
gaus
gaus
eS r e
N RR
K z K zN R
z z
Rz
R R
This fit function allows extraction of both This fit function allows extraction of both the short- and long-range the short- and long-range
components of the source imagecomponents of the source image
This fit function allows extraction of both This fit function allows extraction of both the short- and long-range the short- and long-range
components of the source imagecomponents of the source image
Bessel Functions
RadiiPair Fractions
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Outline
1. Motivation2. 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
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Imaging Imaging
Inversion procedure
2( ) 4 ( , ) ( )C q drr K q r S r ( ) ( )j j
j
S r S B r ( )
( , ) ( )
Thi ij j
j
ij j
C q K S
K dr K q r B r
2
22
( )
( )
Expti ij j
j
Expti
C q K S
C q
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43Roy Lacey, SUNY Stony Brook
Fitting correlation functionsFitting correlation functions
KinematicsKinematics““Spheroid/Blimp” AnsatzSpheroid/Blimp” Ansatz
2
3 2
2
( ) exp 8 4 2
1b= 1- , a,
a
T T
TT
r bS r erfi
b R a R
rR
R
2
3 2
2
( ) exp 8 4 2
1b= 1- , a,
a
T T
TT
r bS r erfi
b R a R
rR
R
Brown & Danielewicz PRC 64, 014902 (2001)Brown & Danielewicz PRC 64, 014902 (2001)
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CutsCuts
Dphi (rad) Dz (cm)
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45Roy Lacey, SUNY Stony Brook
CutsCuts
Dz (cm)
Dphi (rad)
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46Roy Lacey, SUNY Stony Brook
Two source fit functionTwo source fit function
1 s
2 2 2
3 2 2 2
2 2 2
3 2 2 2
( ) =
1exp
22
1exp
22
s l l
s
s s ss s so s ls l o
l
l l ll l lo s ls l o
S r G G
x y z
R R RR R R
x y z
R R RR R R
1 s
2 2 2
3 2 2 2
2 2 2
3 2 2 2
( ) =
1exp
22
1exp
22
s l l
s
s s ss s so s ls l o
l
l l ll l lo s ls l o
S r G G
x y z
R R RR R R
x y z
R R RR R R
This is the single particle distribution
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47Roy Lacey, SUNY Stony Brook
Two source fit functionTwo source fit function
31 2 2 2
2 2 2 2
3 2 2 2
2 2 2 2
3 2 2 2
3 2 2 2 2 2 2
2
2 2
( ) = d
1exp
42
1exp
42
2
2
1exp
2
q
s
s s ss s so s ls l o
l
l l ll l lo s ls l o
s l
s l s l s ls s l l o o
s lo o
S r r S r r S r
x y z
R R RR R R
x y z
R R RR R R
R R R R R R
x y
R R
2 2
2 2 2 2s l s ls s l l
z
R R R R
31 2 2 2
2 2 2 2
3 2 2 2
2 2 2 2
3 2 2 2
3 2 2 2 2 2 2
2
2 2
( ) = d
1exp
42
1exp
42
2
2
1exp
2
q
s
s s ss s so s ls l o
l
l l ll l lo s ls l o
s l
s l s l s ls s l l o o
s lo o
S r r S r r S r
x y z
R R RR R R
x y z
R R RR R R
R R R R R R
x y
R R
2 2
2 2 2 2s l s ls s l l
z
R R R R
This is the two particle distribution
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48Roy Lacey, SUNY Stony Brook
Experimental SetupExperimental Setup
PHENIX Detector
Several Subsystems exploited for the
analysis
Excellent Pid is achievedExcellent Pid is achieved
~ 120 ps /K 2 GeV/c
~ 450 ps /K 1 GeV/c
TOF
EMC