Review of Analysis Methodsfor

Correlations and Fluctuations

Tom Trainor

Firenze

July 7, 2006

Trainor 2

Agenda

• Fluctuations on binned spaces

• Pearson’s normalized covariance

• Scale-dependent variance differences

• Inverting fluctuations to correlations

• Autocorrelations from pair counting

• Relating n and pt fluctuations to physics

• Trompe l’oeil and measure design

Trainor 3

Summary of Charge Fluctuation Measures

CHNQQv 2)(

N

NR

CHch N

QRND

2

2 4

2

2

zN

Z

CHq

2

N

N

N

N

NNstat11

,

dyn

NNQv ,4

1)(

dynNND ,4

222: XXXVariance NNNCH

NNQ

dyn

CH

qN

NN,2

2/32/3

2

2 4CHN

NNz

CHCH

NN

QQZ

2

1

CH

CHCH

NN

QQ

N

)(4 QD

)(Qv

Jeff Mitchell summary

Trainor 4

Summary of Event-by-event <pT> Fluctuation Measures

2.,

2.,, inclpinclppnp TTTT

.,inclp

pp

T

T

TF

22., TTinclp pp

T

pTpT

pT,n

FpT

T

dynp

dynpp pT

TT

||sgn

2,2

,

pT,dyn

N

pTpdynp

T

T

22

,

2

222TTT ppp

Goal of the observables:

State a comparison to the expectation of statistically

independent particle emission.

N

F

pTT

T

p

T

pp

2

Jeff Mitchell summary

Trainor 5

those slides are nice catalogs, but…

• Some similarity relations are violated for relevant conditions (e.g., low multiplicity)

• Some statistical measures exhibit misleading behaviors and/or strong biases

• Random variables in denominators lead to undesirable properties

Trainor 6

Fluctuations on Binned Spaces

x1

x2

a b

a

b

covariance

variancex x

xa b

pt, nmultiple events

single-particle space

two-particle space

particlenumber norscalar pt sumin bin

na-na

n b-n b

b

a

0

0

mixed-pairreference

na-na0

marginal variancesproject

2

2 2

( ) ( )ab a bn n n ns

s sS D

= - -

º -

2 2, ,( )a b a bn ns = -

variances for bins a, b

covariance between a and b

dependence on bin size or scale

bin size = scale

so, back to basics…

Trainor 7

Pearson’s Normalized Covariance

2 2 2

2 22 2[ 1,1]ab

ab

a b

rs s s

s ss sS D

S D

-= = Î -

+

rab = 1 1> 0 0 < 0

covariance relative to marginal variances

a

bcorrelated anticorrelated

Karl Pearson, 1857-1936

uncorrelated

geometric mean of marginal variances

normalized covariance – the basic correlation measure

e.g., forward-backward correlations

1

2

Trainor 8

Variance Difference2

2 2 2ˆ

ˆ ˆ( ) ( ) ( ) ( )~

t

ab a b t t a t t bab

a b a b p a b

n n n n p n p p n pr

n n n n

s

s s s

- - - -= ®

normalized number covariance normalized pt covariance

Poisson Poisson

22 2 2ˆ ˆ: , ˆ /

t t t tp n p p aa t t px r p x n x p n x

normalized (scaled) variances,

covariances: Pearson

22/ / 1n x n x n x n x

a = b

scale-dependent variance difference

2

ˆ

ˆt

t tp

p x n x p

n x

tpΦ δx

omit 2 factor in denominator to

facilitate n vs pt comparisons

ab ab abr r differential measure:CLT

CLT CLT

Trainor 9

Correlations and Fluctuations

runningintegral

a covariance distribution measures correlations

the scale integral of a covariance distribution measures fluctuations

each bin is a mean of normalized covariancesincreasing scale

angular autocorrelations contain averages

of normalized covariances in 4D bins of ()

variance difference

macrobin sizemicrobin separation

autocorrelation

Trainor 10

Fluctuation Inversion2D scale integral

single point

fluctuation excesstp

( , ) ( , )

statisticalreference

data

2D scale inversion

;kernel mn klK

2:

0 0

, 4 , ; , ,tp n

ref

d d K

scale dependence autocorrelation

STAR acceptance

2:

1 1

1/ 2 1/ 2, 4 1 1 ,

t

m n

p nk l ref

k l Am n k l

m n A

Rosetta stone for fluctuation and correlation analysis

Trainor 11

Derivation – Short Form

x1x

x2

x

M0

0

macrobin average

x1,ax

x2,b

x

Mm0

0x

k

microbin average

x1

x2 x x

space invariance?

averageoften true

kx

x1

x2

kxa+k

a

x x

bin method

strip method

k

autocorrelationmethods:

covariancedistribution

variancedifference

integration

mn;kl

k -1/2 l -1/2K = 1 - 1 -

m n

pair counting

Trainor 12

Inversion Precision: ComparisonsPythia

pair counting

inversionCI CDnumber autocorrelations

25 / ref

autocorrelations from pair countingor by inversion of fluctuation scale dependence

Jeff

Duncan

Trainor 13

‘direct’ look at autocorrelation,

computationally expensive:

200 GeV p-p

joint autocorrelations

npt

Autocorrelations from Pair Counting

equivalent ratios for n and pt

k l

)( 2nO

,

,

,

,

,

( , )

( , )

( ) ( )

ˆ ˆ( ) ( )

kl t

kl ref t

ab a k b l

ab a k b l ab

t t ab t t a k b l

ab a k b l ab

A n p

A n p

n n n n

n n

p n p p n p

n n

+ +

+ +

+ +

+ +

D

ì ü- -ï ïº í ý

ï ïî þ

ì ü- -ï ïº í ý

ï ïî þ

average over a,bon kth, lth diagonals

STAR preliminary

√ r

ef

√ r

ef (G

eV2 /

c2 )

autocorrelations the hard way

Trainor 14

Projecting Two-point Momentum Space(yt1,yt2) correlations () correlations

√ r

ef√ r

ef

angular autocorrelations

(yt,)1(yt,)2

not an autocorrelation

n - number

pt

with ‘hard’pt cut

away-side

same-side

unlike-sign pairs

rapidity correlations

√ r

ef

√ r

ef

√ r

ef (G

eV2 /

c2 )intra-jet

inter-jet

Trainor 15

Two Correlation Types

na-na0

0

na-na0

na-na0

0

n b-n b

commonevents

exceptionalevents

soft

hard

rare events

relativefrequency

amplitude

correlated and anticorrelated bins

mixedreference

Au-Aup-p

log

n b-n b

mixed reference

√ r

ef (G

eV2 /

c2 )

√ r

ef

Trainor 16

The Physics Behind n Fluctuations

an

bn Q chn

an

bn

CLT

fluctuation scale (bin size) dependence integrates these two-particle correlations

p-pAu-Aup-p

Au-Au

CD (negative)

CI (positive)

varies withbin size ,

( ) ( )/ | ab cd

ref abcd

ab cd

n n n n

n n

1 2 1 2on ( , , , )

4D hypercube: (a,b,)

HBT

Trainor 17

The Physics Behind pt Fluctuations

fluctuation scale (bin size) dependence integrates these two-particle correlations

p-p

Au-Aup-p

Au-Au

pt

nvaries withbin size x

CLT

ˆ ( )tp n n

ˆt tp np0

0

ˆ ˆ( ) ( )/ | t t ab t t cd

ref abcd

ab cd

p np p np

n n

1 2 1 2on ( , , , )

t

t

p

n - p covariance :

what motivated Φ

Trainor 18

Trompe l’Oeil

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v2 and Elliptic Flow

conventional abscissa and ordinate

total variancedifference

multipoles extracted from pt (not number) autocorrelations

1/Npart

1/Npart

t

2p ,dynamical

ti tj

σ

δp δp

tpΣ

v2

quadrupole

2 2: :2 / 2 /

t tpart p n part ch p nN N N

= mean participant path length

pt correlations number correlations

same 11 points

elliptic flow compared directlyto minijet correlations

per-pairper-particle

1

8

8

1

1/Npart

v2 gives a misleading impression of flow centrality and pt trends

Trainor 20

The Balance Function

true widthvariation

fixed amplitude

autocorrelation balance function BF rms width

acceptancefactor

acceptance width

p-p 200 GeV Au-Au 130 GeV

project

HI

peak volumevariation

acceptance

invert

0 2

CD angular autocorrelations

STAR

STAR

HBT

the BF width measures amplitudes, not correlation lengths

rms width

y

Trainor 21

pt: the real DaVinci Code

2 22 2

ˆ2ˆ 2

ˆ 2 2ˆ2

t

t

ppartt t ttp

part part t t

np np p ndp dnn d n dn p n np

2: ( , , )

tp n NNs 2 ( , , )tp NNs

sNN

sNN

8-12

CERES

STAR

running average

integralinversion

2: 2

,

( , )

( , )t

t

p np dynamn

2.5

2.5

SSC:

~3

acceptance dependence

pt

sNN

r

ef

direct 2-point correlations

2 2ˆ

2

ˆ1ˆ ( 1) 1

tpt tp np

p n n n

A

B

BA

centrality

STAR estruct CERES

ebye

2 20/T T claim

130 GeV

130 GeV

2

C

C

assumes globalthermalization

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Summary

• Pearson’s normalized covariance provides the basis for fluctuation/correlation measure design

• An integral equation connects fluctuation scale dependence to angular autocorrelations

• Fluctuations are physically interpretable by means of two-particle correlations

• Optimal projections of two-particle momentum space to lower-dimensional spaces are defined

• Some F/C measures can be misleading

Trainor 23

2 Dynamical – .

22 2

ˆ:

ˆ( )t t

t tp n p

p n p

n

2

2ˆ

ˆ( ) 1 1( 1)

1 1t

t tti tj pi j

p n p n n nn p p

n n n n

fluctuationinversion

correlations

fluctuations

t

2p :n

biasedmeasure

n = 30

Hijing 200 GeV Au-Au 65-85% central

doesn’t toleratelow multiplicities

physical

unphysical

scale dependence

auto-correlations

r

ef**

(G

eV/c

)2

r

ef (

GeV

/c)2

unphysical

ti tjδp δp

ti tjδp δp

,(n-1)t

2p dynamical

Trainor 24

Fragmentation

a new view of fragmentation

p-p 200 GeV

e+-e 91 GeV

p-p 200 GeV

Trainor 25

0 2

r

ef CERES

STAR

running average

sNNC

Trainor 26

Fussing about Fragmentation

a new view of fragmentation

p-p 200 GeV

e+-e 91 GeV

p-p 200 GeV