1

W. HolzmannW. Holzmann

2

Outline

• Introduction (Why are flow measurements important?)Introduction (Why are flow measurements important?)

• (Selected) Methods of flow measurements:(Selected) Methods of flow measurements: - Reaction Plane Method- Reaction Plane Method - Two Particle Correlation Method- Two Particle Correlation Method - (Cumulant Method)- (Cumulant Method) - Lee-Yang Zero Method- Lee-Yang Zero Method

• Recent examples of elliptic flow measurementsRecent examples of elliptic flow measurements

• SummarySummary

3

W. Scheid, H. Muller, and W. Greiner,PRL 32, 741 (1974)

H. Stöcker, J.A. Maruhn, and W. Greiner, PRL 44, 725 (1980)

Ne

Kolb+Heinz nucl-th/0305084, energy density fractional contours

Probing Nuclear Matter Properties via Collective Flow

4

y

x

py

px

coordinate-space-anisotropy momentum-space-anisotropy

€

v2 =px

2 − py2

px2 + py

2

Focus of This Presentation: Elliptic Flow

Elliptic flow strength mostly determined by EOS andElliptic flow strength mostly determined by EOS andinitial eccentricityinitial eccentricity

5

€

Qn cos(nΨn ) = Xn = wi cos(nφi)i

∑

€

Qn sin(nΨn ) =Yn = wi sin(nφi)i

∑

€

tan(nΨnep ) =

YnXn

STAR

€

vnobs = cos n φ − Ψ ep

( )[ ]

€

v2 =px

2 − py2

px2 + py

2

Reaction Plane Method

Obtain estimate of Reaction Plane:The Event Plane

6

The issue of Resolution!

€

vn = vnobs / cos n Ψ − Ψr( )[ ]

€

cos n Ψ a − Ψ b( )[ ] = cos n Ψ a − Ψr( )[ ] × cos n Ψ b − Ψr( )[ ]

Flow if non-flow is small!Flow if non-flow is small!

Obtain resolution from sub-events:

For two subevents w/ equal multiplicity and resolution:

€

cos n Ψ a − Ψr( )[ ] = cos n Ψ a − Ψ b( )[ ]

cos n Ψ − Ψr( )[ ] ≈ 2 cos n Ψ a − Ψr( )[ ]

Approximate formula, better use fit to subevent distributionApproximate formula, better use fit to subevent distribution

7

Resolution from Correlation between Subevents

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

€

dN

dθ=e−χ 2

2

2

π1+ χ 2( ) + z I0(z) + L0(z)[ ] + χ 2 I1(z) + L1(z)[ ]

⎧ ⎨ ⎩

⎫ ⎬ ⎭€

z = χ 2 cos(θ)

€

Qn

Qn

=Qn

/

Qn/

exp(inθ)

only flow!

non-flow!

J-Y. Ollitrault, Nucl Phys A590 561 (1995)

€

cos km Ψm − Ψr( )[ ]

=π

2χ m exp(−χ m

2 /2) × I(k−1)/ 2(χ m2 /2) + I(k+1)/ 2(χ m

2 /2)[ ]

8

Resolution from Correlation between Subevents in Phenix

Very good fits to subevent distribution: non-flow very small!Very good fits to subevent distribution: non-flow very small!

9

What about acceptance?

Remove biases from acceptance correlationsby making distribution of event planes isotropic:Flatening of Event Plane Distribution

a) Recentering of Xn and Yn

b) Fit event plane distribution to Fourrier expansion and shift event-by-event

Strategy:

Many ways( see Poskanzer & Voloshin: PRC 58 1671 (1997)), for example:

This is done in PHENIX

10First Application of the Azimuthal Correlation Technique at RHICFirst Application of the Azimuthal Correlation Technique at RHIC

_

()()

()real

mixedevents

NC

Nφφ

φΔ

Δ =Δ_

()()

()real

mixedevents

NC

Nφφ

φΔ

Δ =Δ

Wang et al., Wang et al., PRC 44, 1091 (1991)PRC 44, 1091 (1991)

Lacey et al. Lacey et al. PRL 70, 1224 (1993)PRL 70, 1224 (1993)

Two Particle Correlation Method

11

• The anisotropy of the correlation function can reflectThe anisotropy of the correlation function can reflectboth flow and Jet contribution both flow and Jet contribution

• The Asymmetry provides crucial Jet InformationThe Asymmetry provides crucial Jet Information

HIJINGHIJING

Jets lead to strong Jets lead to strong anisotropy and an anisotropy and an asymmetryasymmetry

Flow leads to strong Flow leads to strong anisotropy – no asymmetryanisotropy – no asymmetry

Δφ (deg.)

0 20 40 60 80 100 120 140 160 180

C( Δφ)

0.8

0.9

1.0

1.1

1.2

Hydro or TransportHydro or TransportWith large OpacityWith large Opacity

Information Content of Two Particle Correlation Function

12

pT

''2v''2v

pT_ref 0.4 - 0.8

Cent 6 - 11

0.85 < pT < 1.05

0 50 100 150

Delta Phi

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

C

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

pT_ref 0.4 - 0.8

Cent 6 - 11

0.85 < pT < 1.05

0 50 100 150

Delta Phi

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

C

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

))](2cos(21[)()( 2 ϕϕ

ϕΔ++Δ∝

ΔpGauss

ddN

p2(assor.) = v2(pT) x v2’’

p2(assor.)

(pT)

If away-side jet correlation is small

V2 From Assorted Correlations

13

PHENIX P

RELIMIN

ARY

V2 From Assorted Correlations - Phenix Example

Year-1 Phnx v2 measurement and 62.4 GeV dataYear-1 Phnx v2 measurement and 62.4 GeV datawere analyzed with two particle correlationswere analyzed with two particle correlations

14

Is there a bias introduced by the finite Phenix acceptance?

€

dN

dΔφ∝ λ ne

inΔφ

n=−∞

n= +∞

∑

€

λn = e inΔφ = e in(φ1 −φ2 )

General Analytic Proof

€

p(φΨ) = A(φ)dN

d(φ − Ψ)

€

A(φ) = aneinφ

n=−∞

n= +∞

∑ ,dN

d(φ − Ψ)=

1

2πvne

in(φ−Ψ )

n=−∞

n= +∞

∑ =1

2π1( + 2vn cos(n(φ − Ψ)))

n=1

+∞

∑

€

Ncor (Δφ) =dφ1

2π∫ dΨ

2πp(φ1 Ψ)p(φ1 − ΔφΨ) =

n=−∞

n= +∞

∑ app=−∞

p= +∞

∑2

vn( )2e i( p+n )Δφ

€

Nmix (Δφ) =dφ1

2π∫ dΨ1

2πp(φ1 Ψ1)

⎛

⎝ ⎜

⎞

⎠ ⎟dΨ2

2πp(φ1 − ΔφΨ)∫

⎛

⎝ ⎜

⎞

⎠ ⎟= ap

p=−∞

p= +∞

∑2

e inΔϕ

€

C(Δφ) =Ncor(Δφ)

Nmix (Δφ)= vn( )

n=−∞

n= +∞

∑2

e inΔφ

acceptance/efficiency function

probability distr. forparticle detection

15

ΔϕΔϕ CorrelationsCorrelations

φΔ

1212() )2 (in innm c

eve ϕϕ ϕ ϕ−− = +

If Flow predominate Multiparticle correlations can be used to If Flow predominate Multiparticle correlations can be used to reduce non-flow contributions reduce non-flow contributions (N. Borghini et al, PRC. C63 (2001) 054906)

12343432 1214()()() () 4()ininin ininneeeee vϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ+ − − − −− −− ≈−−

V2 From Cumulants

€

e in φ1 −φ2( )

m= vn

2 + e in(φ1 −φ2 )

c

if v22 >> e in(φ1 −φ2 )

c: v2 = e in φ1 −φ2( )

m

€

e in(φ1 +φ2 −φ3 −φ4 ) − e in(φ1 −φ3 ) e in(φ2 −φ4 ) − e in(φ1 −φ4 ) e in(φ2 −φ3 ) ≈ vn4

16

PPG047

Comparison of V2 From Different Methods

In 62.4 GeV Au+Au Collisions at RHIC, Reaction PlaneIn 62.4 GeV Au+Au Collisions at RHIC, Reaction Planeand Two Particle Correlation Methods give same answer.and Two Particle Correlation Methods give same answer.

17

An Alternative Approach: Lee-Yang Zeroes

• Removes non-flow correlations from flow measurement, important additional tool for v2 study

• Directly investigates large order behavior of cumulant expansion (no costly computational evaluation of cumulants of specific order n)

18

The Lee-Yang Zero Procedure in a Nutshell

R. S. Bhalerao, N. Borghini, J.-Y. Ollitrault , Nucl. Phys. A727 (2003) 373

1) Calculate Q() the projection of the flow vector on arbitrary angle ,

2) can be done for fixed , but in practice use several values and average

3) define generating function for large number of values of z=ir, w/ r real and positive

and plot mod(G(z)) as a function of r.

1) The integrated flow estimate: w j01=2.405 the 1st root of the

Bessel function J0(x) and r0 the value of r at the first minimum of G(ir).

1) Compute vn: Vn = M vn

In practice, makes more numerical sense to use |G(ir)|2

€

Qθ =Qx cos(nθ) +Qy sin(nθ) = w j cos(n(φ j −θ))j=1

M

∑

€

Gθ (z) = ezQθ

€

Vnθ ∞{ } =

j01

r0θ

19

A working example: Look at int. v2 w/ L.-Y. Zeroes in ATLAS

Use 50 events with fixed 5% v2 from Andrzej’s production at:/usatlas/scratch/olszewsk/data/hijing/flow/

Use all charged tracks (XKalman) w/ unit weights,No track quality cuts, upper pT cut = 4900 MeV.

20

A First Look at int. v2 w/ L.-Y. Zeroes in ATLAS

Input v2 faithfully recovered within statistical errors!

21

22

22

22

yx

yxε

−=

+

R: measure ofsize of system

Bhalerao, Blaizot, Borghini, Ollitrault , nucl-th/0508009

How do you know you are looking at Flow?

23

v2 scales with eccentricityand across system size

Eccentricity scaling of the data

24

We can make an estimate of cs from elliptic flow measurements

Bhalerao, Blaizot, Borghini, Ollitrault , nucl-th/0508009

Definition of v2 in model typically 2 times larger than with usual definition

Can we make an estimate of cs?

M. Issah WWND 2005

25

Equation of state: relation between pressure and

energy density

cs ~ 0.35 ± 0.5(cs

2 ~ 0.12), soft EOSF. Karsch, hep-lat/0601013

v2/ecc for <pT> ~ 0.5 GeV/c

Estimate of cs?

M. Issah WWND 2005

26