Fitted HBT radii versus space-time
variances in flow-dominated models
Mike Lisa
Ohio State University
Frodermann, Heinz, MAL, PRC73 044908 (2006); nucl-th/0602023
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
1
Outline
 motivation: possible problems in comparing models to data
 new formula for “fitting” model calculations
 application to two common models
 conclusions
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
2
The many estimates of length scale
 HBT radii : parameters of
Gaussian fits
 3D fit to 3D CF R
C(q)  1   eq o R o q s R s q l R l
2
 experimental procedure
 1D fit to projections of
3D CF  R1D (and 3 ’s)
 questionable shortcut
 FWHM of 1D projections R*

 Space-time variances
2
2
2
2
 i  j  k
2
q 2i R1D,i
C(qi;q j  qk  0) 1 i  e
R *i  ln 2 /q*i where C(q*i ;q j  q k  0)  3/2
Rˆ  x˜  2 x˜ o ˜t   ˜t
2
o
2
2
o
R-hat
 ˆ 2
R s  x˜ 2s
 quick to calculate
2
Rˆ  x˜
2
l
 d x  f x,q S x
 d x  S x
4
f P (q) 
2
P
4
P
2
l
x˜   x   x 
if SP(x) Gaussian, then C(q) Gaussian* and R = R1D = R* = R-hat

Sept 2006

* Coulomb
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
ignored throughout3
dN/dx
The many estimates of length scale
STAR Phys. Rev. C 71 (2005) 044906
Retiere & MAL
PRC70 044907 (2004)
Kisiel, Florkowski, Broniowski, Pluta
PRC73 064902 (2006)
if SP(x) Gaussian, then C(q) Gaussian* and R = R1D = R* = R-hat
 But neither S(x) nor C(q) is “ever” Gaussian
Sept 2006
* Coulomb
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
ignored throughout4
What do experimentalists do?
“typical” study from STAR
Ro (fm)
6
Rs (fm)
6
Rl (fm)
Paic and Skowronski J. Phys. G31 1045 (2005)
6
Fit with ad-hoc alternate forms
? what to do with the parameters?
STAR Phys. Rev. C 71 (2005) 044906
4
4
4
0.1
0.2
qmax (GeV/c)
surely the way of the future... imaging
“fit-range study”  syst. err.
if SP(x) Gaussian, then C(q) Gaussian* and R = R1D = R* = R-hat
 But neither S(x) nor C(q) is “ever” Gaussian
Sept 2006
* Coulomb
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
ignored throughout5
What do theorists do?
hydro
• Hirano:
• Soff:
• Zschiesche
• Heinz:
cascade
R1D
R-hat
R*
R-hat
• AMPT
• MPC
• RQMD
• HRM
R
R-hat
R
R
if SP(x) Gaussian, then C(q) Gaussian* and R = R1D = R* = R-hat
 But neither S(x) nor C(q) is “ever” Gaussian
 How much does this (rather than physics) dominate model comparisons?
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
6
It can matter
(how much, is model-dependent)
 AMPT, RQMD, HRM
reproduce HBT radii best.
 Only these use “right” method
 coincidence?
RQMD - some difference
 R-hat
R
AMPT - huge difference
Lin, Ko, Pal PRL89 152301 (2002)
Sept 2006
Hardtke & Voloshin PRC61 024905 (2000)
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
7
Our plan
 Examine two popular models which have published R-hat
 Blast-wave
 Heinz/Kolb B.I. hydro
 Compare R versus R1D versus R-hat
 for fits (R and R1D), perform experimentalist’s “fit-range study”
 But first... an explanation of our “fit” procedure...
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
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The “data” to be “fit”
 Straight-forward to calculate CF
C(q)  1 cosq r   sin q r 
2
2
Blastwave

hydro CE EOS
out
side
long
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
9
Analytic calculation of radii (“fit”) 3D
functional form:
C(q)  1   e
q 2o R 2o q 2s R 2s q 2l R 2l
ln C(q) 1  ln   q2oR 2o  q2s R 2s  q2l R 2l 
• only good for C>1
• not for noisy data


n
F.O.M. to minimize:
2  

ln Cq i  1 ln   q 2oR 2o  q 2s R 2s  q 2l R 2l
i 
2
i1
where 
i 

2
i
Cq i 
is uncertainty on ln Cq i  1
and  i is the uncertainty on Cq i 
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
10
Analytic calculation of radii (“fit”) 3D
 2
0 ;
 ln 

4x4 vector equation :
 2
0
2
R 
 non-homogeneous
 T P  V
linear equations
 invertable to find
parameters P
 ,o,s,l
where P  ln ,R 2o ,R 2s ,R 2l 
n
V  
ln Cq 1
i1
i 
2
n
; V  
q 2 ,i  ln Cq 1
i1
i 
2
 as per data, we take
and T is the symmetric matrix given by
n
n
q 2 ,i
q 2,i  q2 ,i
1
 
2 ; T ,   
2 ; T ,  
2












i1
i1
i1
i
i
i
n
T,
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
 = fixed (not ´)
 (its value does
not matter)
11
Analytic calculation of radii (“fit”) 1D
Similarly, for R1D...
 rather than one 4x4 set
2
q 2 R1D,

C(q ;q    0) 1   e
ln  
X 2, Y2,  X 0, Y4,
Y2,2   Y0, Y4,
; R
2
1D, 

of equations for 4
parameters...
 3 sets of 2x2 equations
for 6 parameters
X 2, Y0,  X 0, Y2,
Y2,2   Y0, Y4,
 similar technique used
where


n
X n,
i1
Sept 2006

ln Cq ,i ;q    01  q n
 
1D,i
2
by Wiedemann, others
n
; Yn,  
q n
 
i1 1D,i
2
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
12
BW projections - approximately Gaussian
kT=0
kT=0.3 GeV/c
projection of 3D CF
projection of 3D fit
L projection appears least Gaussian
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
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Ro
pT=0.1
Rs
RL
BW - 1D studies
o
•Transverse radii: R1D  R-hat
•Longitudinal
• R1D  R-hat
• signif. fit-range systematic
s
L
pT=0.9
qmax (GeV/c)
“HBT radii” from variances
R 2o  x˜ 2  2 T x˜  ˜t   T2 ˜t 2
RS2  y˜ 2
Ro
Rs
RL
o
; R 2L  z˜ 2
s
radii from ‘fit’ using
various q-ranges
L
STAR Au+Au @ 200 GeV 0-5%
Phys. Rev. C 71 (2005) 044906
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian
effects
KT (GeV/c)
14
Ro
Rs
RL
BW - 3D studies

•-coupling / 3D structure
Ro fit range systematic
•still, BW agreement
w/data persists
qmax (GeV/c)
“HBT radii” from variances
R 2o  x˜ 2  2 T x˜  ˜t   T2 ˜t 2
RS2  y˜ 2
Ro
Rs
RL

; R 2L  z˜ 2
radii from ‘fit’ using
various q-ranges
STAR Au+Au @ 200 GeV 0-5%
Phys. Rev. C 71 (2005) 044906
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian
effects
KT (GeV/c)
15
CE Hydro projections - Gaussian fits “look bad”
kT=0.3 GeV/c
kT=0.6 GeV/c
• CF projections appear Gaussian
• projections of 3D Gaussian fit match poorly
 (unseen) 3D q structure of CF drives fit
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
16
CE Hydro - 3D studies
Ro
Rs
RL

larger fit-range systematic
(side is least affected, despite
“looking” worst in projections)
significant difference b/t R, R-hat
“fitted” R agree better with data
qmax (GeV/c)
“HBT radii” from variances
R 2o  x˜ 2  2 T x˜  ˜t   T2 ˜t 2
RS2  y˜ 2
Ro
Rs
RL

; R 2L  z˜ 2
radii from ‘fit’ using
various q-ranges
STAR Au+Au @ 200 GeV 0-5%
Phys. Rev. C 71 (2005) 044906
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian
effects
KT (GeV/c)
17
Hydro using 2 EoS
“CE” EoS assuming Chem. Equilib until FO
- original publications -
Ro
Rs
RL

More realistic “NCE” EoS
Ro
KT (GeV/c)
Rs
RL

KT (GeV/c)
 similar non-Gaussian effects
 NCE always compared better to data,
 STAR data
 Variance
 3D “fit”
for R-hat and (by construction) for yields.
 apples::apples comparison further improves agreement
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
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“CE” EoS
BW & Hydro
Ro
Rs
RL

Blast-wave
Ro
Rs
RL

KT (GeV/c)
“NCE” EoS
Ro
Rs
RL

KT (GeV/c)
 Qualitatively sim non-Gauss effects
 magnitude much smaller for BW
 conclusions about BW agreement ~same
(still “good” but  will increase)
 hydro agreement (for Ro, Rl) improves
in apples::apples comparison
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
KT (GeV/c)
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Summary / Conclusions
 Variety of length-scale estimators are compared to experimental
HBT radii
 danger of apples::oranges comparison
 magnitude of difference is model-dependent
 analytic calculation of “fit” parameters in models
 R versus R1D versus R-hat
 non-Gaussian features generate differences, fit-range systematic
 R≠R1D : importance of global 3D fit (as experimentally done)
 R < R-hat in temporal components (long & out)
 agreement w/hydro much improved in apples::apples
 impact on “puzzles”
 effect significantly smaller for BW
Sept 2006
WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects
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