Elliptic Flow Fluctuations at PHOBOS

**Elliptic** **Flow** **Fluctu ations**

with the **PHOBOS** detector

Burak Alver

Massachusetts Institute of Technology

v 2

fluctu**at**ions **at** **PHOBOS**

Burak Alver - MIT

**PHOBOS** Collabor**at**ion

Burak Alver, Birger Back, Mark Baker, Maarten Ballintijn, Donald Barton, Russell Betts, Richard Bindel,

Wit Busza (Spokesperson), Zhengwei Chai, Vasundhara Chetluru, Edmundo García, Tomasz Gburek, Kristjan

Gulbrandsen, Clive Halliwell, Joshua Hamblen, Ian Harnarine, Conor Henderson, David Hofman, Richard Hollis,

Roman Holynski, Burt Holzman, Aneta Iordanova, Jay Kane,Piotr Kulinich, Chia Ming Kuo,

Wei Li, Willis Lin, Constantin Loizides, Steven Manly, Alice Mignerey, Gerrit van Nieuwenhuizen,

Rachid Nouicer, Andrzej Olszewski, Robert Pak, Corey Reed, Eric Richardson, Christof Roland,

Gunther Roland, Joe Sagerer, Iouri Sedykh, Chadd Smith, Maciej Stankiewicz, Peter Steinberg,

George Stephans, Andrei Sukhanov, Artur Szostak, Marguerite Belt Tonjes, Adam Trzupek,

Sergei Vaurynovich, Robin Verdier, Gábor Veres, Peter Walters, Edward Wenger, Donald Willhelm,

Frank Wolfs, Barbara Wosiek, Krzysztof Wozniak, Shaun Wyngaardt, Bolek Wyslouch

ARGONNE NATIONAL LABORATORY

INSTITUTE OF NUCLEAR PHYSICS PAN, KRAKOW

NATIONAL CENTRAL UNIVERSITY, TAIWAN

UNIVERSITY OF MARYLAND

BROOKHAVEN NATIONAL LABORATORY

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

UNIVERSITY OF ILLINOIS AT CHICAGO

UNIVERSITY OF ROCHESTER

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

2

Motiv**at**ion

High v 2

observed in

CuCu can be explained

by fluctu**at**ions in

initial collision region.

Can we test the Participant Eccentricity Model?

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

3

Expected fluctu**at**ions

Assuming v 2 ∝ε part , participant

eccentricity model predicts

v 2 fluctu**at**ions

Expected σ v2 from fluctu**at**ions in ε part

D**at**a

Au+Au

MC

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

4

Measuring v 2 **Fluctu ations**

• We have considered 3 different methods

– 2 particle correl**at**ions →

• c.f. S. Voloshin nucl-th/0606022

• σ v22 = - 2

• Do system**at**ic errors cancel?

– 2 particle correl**at**ions → v 2

2

event by event

• Mixed event background gener**at**ion is possible

• Reduces fit parameters to 1 (no reaction plane)

• Hard to untangle acceptance effects event by event

– v 2 event by event

• This is the method we are pursuing

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

5

Measuring v 2 **Fluctu ations** - Today’s Talk

• Measuring v 2 event by event

• Ongoing analysis on 200GeV Au-Au

• Today

– How we are planning to make the measurement

– Studies on fully simul**at**ed MC events

• Modified Hijing - **Flow**

• Geant

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

6

Method Overview - Simplified Example

2 possible v 2 values

Event by Event measurement

Demonstr**at**ion Demonstr**at**ion

K b (u)

K a (u)

V 2b

V 2a

V 2b

V 2a

f 1

Rel**at**ive abundance in sample

Demonstr**at**ion

Observed u distribution in a sample

Demonstr**at**ion

g(u)

u=v 2obs.

or u = v 2

2

obs.

or u = q obs.

Question: Wh**at** is the rel**at**ive abundance of 2 v 2 ’s in the sample?

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

7

Method Overview - Simplified Example

2 possible v 2 values

Event by Event measurement

Demonstr**at**ion Demonstr**at**ion

K b (u)

K a (u)

V 2b

V 2a

V 2b

V 2a

f 1

Rel**at**ive abundance in sample

Demonstr**at**ion

Measured u distribution in a sample

Demonstr**at**ion

g(u)

u=v 2obs.

or u = v 2

2

obs.

or u = q obs.

Question: Wh**at** is the rel**at**ive abundance of v 2a to v 2b in the sample?

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

8

Method Overview - Simplified Example

2 possible v 2 values

Event by Event measurement

u=v 2obs.

V 2b

K b (u)

V 2a

K a (u)

V 2b

V 2a

V 2b

V 2a

Extracted v 2true distribution from sample Measured u distribution in a sample

f a

Demonstr**at**ion

Demonstr**at**ion

g(u)

f b

Demonstr**at**ion Demonstr**at**ion

Question: Wh**at** is the rel**at**ive abundance of v 2a to v 2b in the sample?

g(u)=f a K a (u) + f b K b (u)

or u = v 2

2

obs.

or u = q obs.

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

9

Method Overview

Kernel

In real life v 2 can take

a continuum of values

K(u,v 2 )

200 GeV AuAu

Modified Hijing+Geant

u=v 2obs.

Extracted v 2true distribution from sample

Measured u distribution in a sample

200 GeV AuAu

Modified Hijing+Geant

f(v 2 )

g(u)

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

10

Method Overview

• 3 Tasks

– Measure u event-by-event g(u)

– Calcul**at**e the kernel K(u,v 2 )

– Extract dynamical fluctu**at**ions f(v 2 )

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

11

**PHOBOS** Detector

• **PHOBOS** Multiplicity Array

-5.4

Measuring u=v 2obs Event by Event I

• Probability Distribution Function (PDF) for hit positions:

PDF

Probability of hit in η

Probability of hit in φ

u

demonstr**at**ion

• Define likelihood of u and φ 0 for an event:

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

13

Measuring u=v 2obs Event by Event II

• Maximize likelihood to find “most likely” value of u

• Comparing values of u and φ 0

– In an event, p(η i ) is same for all u and φ 0 .

– PDF folded by acceptance must be normalized to the same

value for different u and φ 0 ’s

Acceptance

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

14

Measuring u=v 2obs Event by Event II

• Maximize likelihood to find “most likely” value of u

• Comparing values of u and φ 0

– In an event, p(η i ) is same for all u and φ 0 .

– PDF folded by acceptance must be normalized to the same

value for different u and φ 0 ’s

Acceptance

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

15

Measuring u=v 2obs Event by Event III

Observed u distribution in a sample

Mean and RMS of u in slices of v 2

200 GeV AuAu

Modified Hijing+Geant

g(u)

Error bars show RMS

200 GeV AuAu

Modified Hijing+Geant

Next Step: Construct the Kernel to unfold g(u)

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

16

Calcul**at**ing the Kernel I

• Simple: Measure u distribution in bins of v 2

• 2 small complic**at**ions

– Kernel depends on multiplicity: K(u,v 2 ,n)

• n = number of hits on the detector

• Measure u distribution in bins of v 2 and n.

– St**at**istics in bins can be combined by fitting

smooth functions

200 GeV AuAu

Modified Hijing+Geant

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

17

Calcul**at**ing the Kernel II

• In a single bin of v 2 and n

u distribution with for fixed v 2 and n

200 GeV AuAu

Modified Hijing+Geant

(a, b) ↔ (,σ u

)

• Distribution is not Gaussian

• But can be parameterized by and σ u

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

18

Calcul**at**ing the Kernel III

• Measure and σ u in bins of v 2 and n

• Fit smooth functions

K(u,v 2 ,n)

200 GeV AuAu

Modified Hijing+Geant

K(u,v 2 ,n)

200 GeV AuAu

Modified Hijing+Geant

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

19

Calcul**at**ing the Kernel IV

• Multiplicity dependence can be integr**at**ed out

K(u,v 2 ,n)

200 GeV AuAu

Modified Hijing+Geant

K(u,v 2 )

200 GeV AuAu

Modified Hijing+Geant

N(n) = Number hits distribution in sample

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

20

Extracting dynamical fluctu**at**ions

known ?

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

21

Extracting dynamical fluctu**at**ions

Ans**at**z with two parameters:

known ?

Ans**at**z for f(v 2 )

ans**at**z

Ans**at**z

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

22

Extracting dynamical fluctu**at**ions

Ans**at**z with two parameters:

known ?

Ans**at**z for f(v 2 )

ans**at**z

Ans**at**z

integr**at**e

Expected g(u) for Ans**at**z

200 GeV AuAu

Modified Hijing+Geant

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

23

Extracting dynamical fluctu**at**ions

Ans**at**z with two parameters:

known ?

Ansätze for f(v 2 )

ans**at**z

Expected g(u) for Ansätze

200 GeV AuAu

integr**at**e

Modified Hijing+Geant

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

24

Extracting dynamical fluctu**at**ions

Ans**at**z with two parameters:

known ?

Ansätze for f(v 2 )

ans**at**z

Comparison with sample

200 GeV AuAu

integr**at**e

Modified Hijing+Geant

Compare expected g(u) for Ans**at**z with measurement

Minimum χ 2 → and σ v2

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

25

Method Summary

K(u,v 2 ,n)

Many MC events

MC

MC

A Small Sample

f in (v 2 )

=0.05

σ v2 =0.02

measurement

N(n) K(u,v 2 )

MC

MC

Minimize χ 2 in integral

measurementintegr**at**ion

MC

MC

=0.048

σ v2 = 0.023

g(u)

f out (v 2 )

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

26

Verific**at**ion

• Ran this analysis on Modified Hijing

– v 2 (η) = v 2 (0) • (1-|η|/6)

• Same as the assumption in our fit

– v 2 (0) given by a Gaussian distribution in each sample

• Same as our Ans**at**z

– Analysis done in 10 collision vertex bins

• Final results are averaged

– 0-40% central events used to construct Kernel

– 15-20% central events used as sample

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

27

Verific**at**ion

• Ran this analysis on Modified Hijing

– The input fluctu**at**ions are reconstructed

successfully

= 0.050

200 GeV AuAu

15-20% central

Modified Hijing+Geant

σ out

= 0.020

0.04

0.02

= 0.030

0.04

σ out

0.02

σ in

0.02 0.04

Only st**at**istical errors shown

(from combining vertex bins)

= 0.040

0.04

σ out

σ in

0.02 0.04

0.02

0.02 0.04

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

28

σ in

Conclusion / Outlook

• A new method to measure elliptic flow

fluctu**at**ions is developed.

• **Fluctu ations** in MC simul

successfully reconstructed.

• Ready to apply the method to extract

dynamical fluctu**at**ions in DATA.

– Important part will be to estim**at**e system**at**ic

uncertainties due to the MC/DATA differences

• dN/dη(η)

• v 2 (η)

• Non-flow in d**at**a

– Should show up in reaction plane resolutions

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

29

Likelihood Fit Normaliz**at**ion

Acceptance

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

30

Calcul**at**ing the Kernel. Functions observed to fit the Kernel

v 2

fluctu**at**ions **at** **PHOBOS** Burak Alver - MIT

31