Elliptic Flow Fluctuations with the PHOBOS detector

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Elliptic Flow Fluctuationswith the PHOBOS
detector

• Burak Alver
• Massachusetts Institute of Technology

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PHOBOS Collaboration
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 BROOKHAVEN NATIONAL LABORATORY INSTITU
TE OF NUCLEAR PHYSICS PAN, KRAKOW MASSACHUSETTS
INSTITUTE OF TECHNOLOGY NATIONAL CENTRAL
UNIVERSITY, TAIWAN UNIVERSITY OF ILLINOIS AT
CHICAGO UNIVERSITY OF MARYLAND UNIVERSITY OF
ROCHESTER
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Motivation
High v2 observed in CuCu can be explained by
fluctuations in initial collision region.
Can we test the Participant Eccentricity Model?
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Expected fluctuations

• Assuming v2??part, participant eccentricity model
  predicts
• v2 fluctuations

Expected ?v2 from fluctuations in ?part
Data
MC
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Measuring v2 Fluctuations

• We have considered 3 different methods
• 2 particle correlations ? ltv22gt
• c.f. S. Voloshin nucl-th/0606022
• ?v22 ltv22gt - ltv2gt2
• Do systematic errors cancel?
• 2 particle correlations ? v22 event by event
• Mixed event background generation is possible
• Reduces fit parameters to 1 (no reaction plane)
• Hard to untangle acceptance effects event by
  event
• v2 event by event
• This is the method we are pursuing

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Measuring v2 Fluctuations - Todays Talk

• Measuring v2 event by event
• Ongoing analysis on 200GeV Au-Au
• Today
• How we are planning to make the measurement
• Studies on fully simulated MC events
• Modified Hijing - Flow
• Geant

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Method Overview - Simplified Example
2 possible v2 values
Event by Event measurement
Demonstration
Demonstration
uv2obs.
Kb(u)
Ka(u)
or u v22obs. or u qobs.
V2b
V2a
V2a
V2b
Observed u distribution in a sample
Relative abundance in sample
f1
Demonstration
Demonstration
g(u)
f2
Question What is the relative abundance of 2
v2s in the sample?
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Method Overview - Simplified Example
2 possible v2 values
Event by Event measurement
Demonstration
Demonstration
uv2obs.
Kb(u)
Ka(u)
or u v22obs. or u qobs.
V2b
V2a
V2a
V2b
Measured u distribution in a sample
Relative abundance in sample
f1
Demonstration
Demonstration
g(u)
f2
Question What is the relative abundance of v2a
to v2b in the sample?
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Method Overview - Simplified Example
2 possible v2 values
Event by Event measurement
Demonstration
Demonstration
uv2obs.
Kb(u)
Ka(u)
or u v22obs. or u qobs.
V2b
V2a
V2a
V2b
Extracted v2true distribution from sample
Measured u distribution in a sample
fa
Demonstration
Demonstration
g(u)
fb
V2a
V2b
Question What is the relative abundance of v2a
to v2b in the sample?
g(u)faKa(u) fbKb(u)
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Method Overview
Kernel
In real life v2 can take a continuum of values
uv2obs.
K(u,v2)
Extracted v2true distribution from sample
Measured u distribution in a sample
f(v2)
g(u)
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Method Overview

• 3 Tasks
• Measure u event-by-event g(u)
• Calculate the kernel K(u,v2)
• Extract dynamical fluctuations f(v2)

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PHOBOS Detector

• PHOBOS Multiplicity Array
• -5.4lt?lt5.4 coverage
• -Holes / granularity differences
• Idea Use all available information in event to
  read off single u value

HIJING Geant 15-20 central
dN/d?
Primary particles Hits on detector
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Measuring uv2obs Event by Event I

• Probability Distribution Function (PDF) for hit
  positions

Probability of hit in ?
Probability of hit in ?
PDF
u
demonstration

• Define likelihood of u and ?0 for an event

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Measuring uv2obs Event by Event II

• Maximize likelihood to find most likely value
  of u
• Comparing values of u and ?0
• In an event, p(?i) is same for all u and ?0.
• PDF folded by acceptance must be normalized to
  the same value for different u and ?0s

Acceptance
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Measuring uv2obs Event by Event II

• Maximize likelihood to find most likely value
  of u
• Comparing values of u and ?0
• In an event, p(?i) is same for all u and ?0.
• PDF folded by acceptance must be normalized to
  the same value for different u and ?0s

Acceptance
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Measuring uv2obs Event by Event III
Observed u distribution in a sample
Mean and RMS of u in slices of v2
Error bars show RMS
200 GeV AuAu
Modified HijingGeant
Next Step Construct the Kernel to unfold g(u)
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Calculating the Kernel I

• Simple Measure u distribution in bins of v2
• 2 small complications
• Kernel depends on multiplicity K(u,v2,n)
• n number of hits on the detector
• Measure u distribution in bins of v2 and n.
• Statistics in bins can be combined by fitting
  smooth functions

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Calculating the Kernel II

• In a single bin of v2 and n

u distribution with for fixed v2 and n
(a, b) ? (ltugt,?u)

• Distribution is not Gaussian
• But can be parameterized by ltugt and ?u

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Calculating the Kernel III

• Measure ltugt and ?u in bins of v2 and n
• Fit smooth functions

K(u,v2,n)
K(u,v2,n)
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Calculating the Kernel IV

• Multiplicity dependence can be integrated out

K(u,v2 ,n)
K(u,v2)
N(n) Number hits distribution in sample
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Extracting dynamical fluctuations
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Extracting dynamical fluctuations
Ansatz for f(v2)
ansatz
Ansatz
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Extracting dynamical fluctuations
Ansatz for f(v2)
Expected g(u) for Ansatz
ansatz
Ansatz
integrate
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Extracting dynamical fluctuations
Ansätze for f(v2)
Expected g(u) for Ansätze
ansatz
integrate
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Extracting dynamical fluctuations
Ansätze for f(v2)
Comparison with sample
ansatz
integrate
Compare expected g(u) for Ansatz with
measurement Minimum ?2 ? ltv2gt and ?v2
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Method Summary
K(u,v2 ,n)
MC
MC
Many MC events
integration
N(n)
K(u,v2)
MC
MC
A Small Sample
fin(v2)
measurement
Minimize ?2 in integral
ltv2gt0.05 ?v2 0.02
MC
ltv2gt0.048 ?v2 0.023
MC
measurement
fout(v2)
g(u)
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Verification

• Ran this analysis on Modified Hijing
• v2(?) v2(0) (1-?/6)
• Same as the assumption in our fit
• v2(0) given by a Gaussian distribution in each
  sample
• Same as our Ansatz
• 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

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Verification
ltv2gt 0.020

• Ran this analysis on Modified Hijing
• The input fluctuations are reconstructed
  successfully

?out
ltv2gt 0.050
ltv2gt 0.030
200 GeV AuAu 15-20 central
?out
Modified HijingGeant
ltv2gt 0.040
Only statistical errors shown (from combining
vertex bins)
?out
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Conclusion / Outlook

• A new method to measure elliptic flow
  fluctuations is developed.
• Fluctuations in MC simulations are successfully
  reconstructed.
• Ready to apply the method to extract dynamical
  fluctuations in DATA.
• Important part will be to estimate systematic
  uncertainties due to the MC/DATA differences
• dN/d?(?)
• v2(?)
• Non-flow in data
• Should show up in reaction plane resolutions

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Likelihood Fit Normalization
Acceptance
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Calculating the Kernel. Functions observed to fit
the Kernel