Two-pion Source Imaging with
the STAR experiment
Michal Šumbera for the STAR
Collaboration
Nuclear Physics Institute ASCR, Prague
Analysis done by Paul Chung
WPCF 2009,CERN, 14th –17th October, 2009
1
Outline

Overview of 3D source function shape analysis:
Cartesian Spherical Harmonic decomposition & Imaging
Technique

low pT (0.2<pT<0.36 GeV/c) pion moments in central
(cen<20%) Au+Au events @ 200 AGeV

3D source function extraction: Moment Imaging

Comparison to PHENIX published data

kT & centrality dependence of 3D source function
WPCF 2009,CERN, 14th –17th October, 2009
2
Motivation

Model-independent determination of the source shape
(beyond gaussian fits, etc.)

RHIC & SPS heavy ion results:
PHENIX, PRL100:232301,2008: long-range non-Gaussian
tails in out- and long-direction. Model-dependent extracted
proper breakup time τ0 ∼ 9 fm/c and a mean proper
emission duration Δτ∼ 2 fm/c.
NA49, arXiv:0809.1444: need for boost invariance coupled
with “outside-in burning” in transverse direction.

Extraction of π-π scattering length: Are heavy ions
competitive to dedicated experiments – DIRAC, NA48/2,
E865?
M. Bysterský for STAR, WPCF’08.
WPCF 2009,CERN, 14th –17th October, 2009
3
Previous STAR results
WPCF 2009,CERN, 14th –17th October, 2009
4
Imaging
Technique Devised by:
D. Brown, P. Danielewicz,
PLB 398:252 (1997).
PRC 57:2474 (1998).
Emitting source

No Shape assumption for S(r)
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
Extracted S(r) in pair CM frame
Hence Model-independent i.e Kernel independent of freeze-out conditions
WPCF 2009,CERN, 14th –17th October, 2009
5
Imaging : Inversion procedure
C (q)  4  drr 2 K (q, r )S (r )
S (r )   S j  B j (r )
Expansion in B-spline basis
j
CiTh (q)   K ij  S j
j
Kij   dr  K (q, r ) B j (r )
 Expt

 Ci (q )   K ij  S j 
j


2
 
 2Ci (q ) Expt
2
Freeze-out occurs after last scattering
Hence only Coulomb & BE effect included in kernel
WPCF 2009,CERN, 14th –17th October, 2009
6
Imaging C(qinv)
0
[C
moment]
Function restored from image reproduces input data
WPCF 2009,CERN, 14th –17th October, 2009
7
3D Analysis Basics
[Danielewicz and Pratt nucl-th/0501003 (v1)]
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
( q )
(1)
x=out-direction
y=side-direction
( r )
(2)
z=long-direction
l
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
Invert (1)
Invert (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 ) 
 4 1 ....l r
l!
1
....l
WPCF 2009,CERN, 14th –17th October, 2009
(5)
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Cartesian harmonics basis



Cartesian harmonics Aℓα1α2…αℓ are based on the products of unit
vector components, nα1 nα2 ,…, nαℓ .
Due to the normalization identity n2x + n2y + n2z = 1, at a given ℓ ≥ 2,
the different component products are not linearly independent as
functions of spherical angle;
At a given ℓ, the products are spanned by spherical harmonics of
rank ℓ′ ≤ ℓ, with ℓ′ of the same evenness as ℓ.
WPCF 2009,CERN, 14th –17th October, 2009
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1D C(qinv) vs
0
C
moment
3D pion correlation functions obtained with standard STAR track merging & splitting cuts
Close agreement serves as consistency check
WPCF 2009,CERN, 14th –17th October, 2009
10
Calculated Moment
0
C
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
11
Calculated Moment
2
C x2
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
12
Calculated Moment
2
C y2
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
13
Calculated Moment
4
C x4
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
14
Calculated Moment
4
C y4
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
15
Calculated Moment
4
C x2y2
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
16
Calculated Moment
6
C x6
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
17
Calculated Moment
6
C y6
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October,
2009
18
Calculated Moment
6
C x2y4
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
19
Calculated Moment
6
C x4y2
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
20
Calculated Moment
2
C z2
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
21
Calculated Moment
4
C z4
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
22
Calculated Moment
6
C z6
Agreement between Full Field & Reversed Full Field; π+π+ & π-πWPCF 2009,CERN, 14th –17th October, 2009
23
Imaging
2
C x2
&
2
C y2
Moment restored from image reproduces input data
WPCF 2009,CERN, 14th –17th October, 2009
24
Imaging
4
C x4
&
4
C y4
Moment restored from image reproduces input data
WPCF 2009,CERN, 14th –17th October, 2009
25
Imaging
6
C x6
&
6
C y6
Moment restored from image (partially) reproduces input data
WPCF 2009,CERN, 14th –17th October, 2009
26
3D S(r): STAR vs PHENIX
PHENIX: Phys.Rev.Lett.100:232301,2008
Overall agreement, some differences in tails
WPCF 2009,CERN, 14th –17th October, 2009
27
kT dependence of S(r)
Larger source size for higher kT pion pairs in x & z
WPCF 2009,CERN, 14th –17th October, 2009
28
Centrality dependence of S(r)
Larger source size for pion pairs from central collisions
STAR Collaboration Meeting, Berkeley
October 8 2009
29
Conclusions




Reasonable agreement between STAR &
PHENIX S(r), some differences in tails still to
be explored
kT & centrality dependence extraction
possible with STAR Run4 data
For given centrality, higher kT source larger in
x & z directions influence of lifetime effects
For same kT pion pairs, more central events
result in larger source image
WPCF 2009,CERN, 14th –17th October, 2009
30
Thank you !
WPCF 2009,CERN, 14th –17th October, 2009
31
Spherical Harmonics basis
The disadvantage of expansion in the spherical
harmonics Yℓm: connection between the geometric
features of the real source function S(r) and the complex
valued projections Sℓm(r) is not transparent.
The Yℓm harmonics are convenient for analyzing quantum
angular momentum, but are clumsy for expressing
anisotropies of real-valued functions.

WPCF 2009,CERN, 14th –17th October, 2009
32