sept 2006wpcf 2006, sao paulo brazil - lisa non-gaussian effects1 fitted hbt radii versus space-time...
TRANSCRIPT
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 1
Fitted HBT radii versus space-time variances in flow-dominated models
Mike LisaOhio State University
Frodermann, Heinz, MAL, PRC73 044908 (2006); nucl-th/0602023
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 2
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 3
The many estimates of length scale
HBT radii : parameters of Gaussian fits 3D fit to 3D CF R
experimental procedure
1D fit to projections of 3D CF R1D (and 3 ’s)questionable shortcut
FWHM of 1D projections R*
Space-time variances R-hat quick to calculate
€
C(r q ) =1+ λ ⋅e−q o
2 R o2 −q s
2R s2 −q l
2R l2
€
C(qi;q j = qk = 0) ≅1+ λ i ⋅e−q i
2R1D,i2
i ≠ j ≠ k( )
€
ˆ R o2 = ˜ x o
2 − 2β ˜ x o˜ t + β 2 ˜ t 2
ˆ R s2 = ˜ x s
2 ˆ R l2 = ˜ x l
2
€
f r P (q) ≡
d4x ⋅f x,q( ) ⋅Sr P
x( )∫d4x ⋅Sr
P x( )∫
˜ x μ ≡ xμ − xμ
if SP(x) Gaussian, then C(q) Gaussian* and R = R1D = R* = R-hat
* Coulomb ignored throughout
€
R i* = ln2 /qi
* where C(qi*;q j = qk = 0) = 3/2
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 4
But neither S(x) nor C(q) is “ever” Gaussian
* Coulomb ignored throughout
STAR Phys. Rev. C 71 (2005) 044906
dN
/dx
Retiere & MALPRC70 044907 (2004)
Kisiel, Florkowski, Broniowski, PlutaPRC73 064902 (2006)
if SP(x) Gaussian, then C(q) Gaussian* and R = R1D = R* = R-hat
The many estimates of length scale
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 5
But neither S(x) nor C(q) is “ever” Gaussian
* Coulomb ignored throughout
if SP(x) Gaussian, then C(q) Gaussian* and R = R1D = R* = R-hat
What do experimentalists do?
STAR Phys. Rev. C 71 (2005) 044906
Fit with ad-hoc alternate forms? what to do with the parameters?
Paic and Skowronski J. Phys. G31 1045 (2005)
Ro (
fm)
4
6
Rs
(fm
)
4
6
Rl (
fm)
4
6
qmax (GeV/c)0.1 0.2
“fit-range study” syst. err.
“typical” study from STAR
surely the way of the future... imaging
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 6
But neither S(x) nor C(q) is “ever” Gaussian How much does this (rather than physics) dominate model comparisons?
hydro
• Hirano: R1D
• Soff: R-hat• Zschiesche R*• Heinz: R-hat
if SP(x) Gaussian, then C(q) Gaussian* and R = R1D = R* = R-hat
What do theorists do?cascade
• AMPT R• MPC R-hat• RQMD R• HRM R
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 7
It can matter(how much, is model-
dependent) AMPT, RQMD, HRM
reproduce HBT radii best. Only these use “right”
method coincidence?
Hardtke & Voloshin PRC61 024905 (2000)
RQMD - some difference
R-hat R
AMPT - huge difference
Lin, Ko, Pal PRL89 152301 (2002)
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 8
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 9
The “data” to be “fit”
Straight-forward to calculate CF
€
C(r q ) =1+ cos
r ′ q ⋅
r ′ r ( )
2+ sin
r ′ q ⋅
r ′ r ( )
2
out side
long
hydro CE EOSBlastwave
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 10
Analytic calculation of radii (“fit”) 3D
€
χ 2 ≡ln C
r q i( ) −1( ) − lnλ + qo
2Ro2 + qs
2R s2 + ql
2R l2
[ ]2
′ σ i( )2
i=1
n
∑
where ′ σ i =σ i
Cr q i( )
is uncertainty on ln Cr q i( ) −1( )
and σ i is the uncertainty on Cr q i( )
€
C(r q ) =1+ λ ⋅e−q o
2 R o2 −q s
2R s2 −q l
2R l2
€
ln C(r q ) −1( ) = ln λ − qo
2Ro2 + qs
2R s2 + ql
2R l2
( )
functional form:
• only good for C>1• not for noisy data
F.O.M. to minimize:
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 11
€
∂χ 2
∂ lnλ= 0 ;
∂χ 2
∂Rμ2
= 0
€
4x4 vector equation : Tαβ Pα
α =∅ ,o,s,l
∑ = Vβ
where P = ln λ ,Ro2,R s
2,R l2
( )
V∅ = −ln C
r q ( ) −1( )
′ σ i( )2
i=1
n
∑ ; Vμ = +qμ ,i
2 ⋅ln Cr q ( ) −1( )
′ σ i( )2
i=1
n
∑
and T is the symmetric matrix given by
T∅ ,∅ = −1
′ σ i( )2
i=1
n
∑ ; Tμ ,∅ = +qμ ,i
2
′ σ i( )2
i=1
n
∑ ; Tμ ,ν = −qμ ,i
2 ⋅qν ,i2
′ σ i( )2
i=1
n
∑
non-homogeneous linear equations
invertable to find parameters P
as per data, we take = fixed (not ´) (its value does
not matter)
Analytic calculation of radii (“fit”) 3D
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 12
rather than one 4x4 set of equations for 4 parameters...
3 sets of 2x2 equations for 6 parameters
similar technique used by Wiedemann, others
Analytic calculation of radii (“fit”) 1D
Similarly, for R1D...
€
C(qμ ;qν ≠μ = 0) ≅1+ λ μ ⋅e−q μ
2 R1D,μ2
€
lnλ μ =X2,μ Y2,μ − X0,μ Y4,μ
Y2,μ2 − Y0,μ Y4,μ
; R1D,μ2 =
X2,μ Y0,μ − X0,μ Y2,μ
Y2,μ2 − Y0,μ Y4,μ
where
Xn,μ ≡ln C qμ ,i;qν ≠μ = 0( ) −1( ) ⋅qμ
n
′ σ 1D,i( )2
i=1
n
∑ ; Yn,μ ≡qμ
n
′ σ 1D,i( )2
i=1
n
∑
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 13
BW projections - approximately Gaussian
kT=0 kT=0.3 GeV/c
projection of 3D fit
projection of 3D CF
L projection appears least Gaussian
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 14
BW - 1D studies
•Transverse radii: R1D R-hat
•Longitudinal
• R1D R-hat
• signif. fit-range systematic
pT=0.1
pT=0.9
€
Ro2 = ˜ x 2 − 2βT ˜ x ⋅ ˜ t + β T
2 ˜ t 2
RS2 = ˜ y 2 ; RL
2 = ˜ z 2
“HBT radii” from variances
radii from ‘fit’ usingvarious q-ranges
STAR Au+Au @ 200 GeV 0-5%Phys. Rev. C 71 (2005) 044906
Ro Rs RL o
s
L
Ro Rs RL o
s
L
qmax (GeV/c)
KT (GeV/c)
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 15
BW - 3D studies
-coupling / 3D structure Ro fit range systematic
•still, BW agreement w/data persists
€
Ro2 = ˜ x 2 − 2βT ˜ x ⋅ ˜ t + β T
2 ˜ t 2
RS2 = ˜ y 2 ; RL
2 = ˜ z 2
“HBT radii” from variances
radii from ‘fit’ usingvarious q-ranges
STAR Au+Au @ 200 GeV 0-5%Phys. Rev. C 71 (2005) 044906
qmax (GeV/c)
KT (GeV/c)
Ro Rs RL
Ro Rs RL
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 16
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 17
CE Hydro - 3D studies
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
€
Ro2 = ˜ x 2 − 2βT ˜ x ⋅ ˜ t + β T
2 ˜ t 2
RS2 = ˜ y 2 ; RL
2 = ˜ z 2
“HBT radii” from variances
radii from ‘fit’ usingvarious q-ranges
STAR Au+Au @ 200 GeV 0-5%Phys. Rev. C 71 (2005) 044906
qmax (GeV/c)
Ro Rs RL
KT (GeV/c)
Ro Rs RL
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 18
Hydro using 2 EoS
similar non-Gaussian effects NCE always compared better to data,
for R-hat and (by construction) for yields.
apples::apples comparison further improves agreement
KT (GeV/c)
Ro Rs RL
KT (GeV/c)
Ro Rs RL
“CE” EoS assuming Chem. Equilib until FO- original publications - More realistic “NCE” EoS
STAR data Variance 3D “fit”
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 19
BW & Hydro
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
KT (GeV/c)
Ro Rs RL
“CE” EoS
KT (GeV/c)
Ro Rs RL
“NCE” EoS
Blast-wave
KT (GeV/c)
Ro Rs RL
Sept 2006 WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects 20
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