Fitted HBT radii versus space-time variances in flow-dominated models

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


Outline

motivation: possible problems in comparing models to data

new formula for “fitting” model calculations

application to two common models

conclusions


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


But neither S(x) nor C(q) is “ever” Gaussian


STAR Phys. Rev. C 71 (2005) 044906

dN

/dx

Retiere & MALPRC70 044907 (2004)

Kisiel, Florkowski, Broniowski, PlutaPRC73 064902 (2006)


The many estimates of length scale


But neither S(x) nor C(q) is “ever” Gaussian



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


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


What do theorists do?cascade

• AMPT R• MPC R-hat• RQMD R• HRM R


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)


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...


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


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:


€

∂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)



rather than one 4x4 set of equations for 4 parameters...

3 sets of 2x2 equations for 6 parameters

similar technique used by Wiedemann, others


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

∑


BW projections - approximately Gaussian

kT=0 kT=0.3 GeV/c

projection of 3D fit

projection of 3D CF

L projection appears least Gaussian


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)


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




qmax (GeV/c)

KT (GeV/c)

Ro Rs RL

Ro Rs RL


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


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




qmax (GeV/c)

Ro Rs RL

KT (GeV/c)

Ro Rs RL


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”


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


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

Documents

Fitted HBT radii versus space-time variances in flow-dominated models