# Presentation: 417817

## Elliptic Flow Fluctuationswith the PHOBOS detector

**Elliptic Flow Fluctuations****with the PHOBOS detector**

**Burak Alver **
**Massachusetts Institute of Technology **

## PHOBOS Collaboration

-  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
- 	INSTITUTE OF NUCLEAR PHYSICS PAN, KRAKOW	MASSACHUSETTS INSTITUTE OF TECHNOLOGY
- 	NATIONAL CENTRAL UNIVERSITY, TAIWAN	UNIVERSITY OF ILLINOIS AT CHICAGO
- 	UNIVERSITY OF MARYLAND	UNIVERSITY OF ROCHESTER

**PHOBOS Collaboration**

## Motivation 

**Can we test the Participant Eccentricity Model?**

**High v****2**** observed in CuCu can be explained by fluctuations in initial collision region.**

## Expected fluctuations

**Assuming v****2********part****, participant eccentricity model predicts**
**v****2**** fluctuations**

**Expected ********v2**** from fluctuations in ********part**** **

**Data**

**MC**

## Measuring v2 Fluctuations

**We have considered 3 different methods**
**2 particle correlations ******** <v****2****2****>**
**c.f. ****S. Voloshin nucl-th/0606022**
******v2****2****= <v****2****2****> - <v****2****>****2**
**Do systematic errors cancel?**
**2 particle correlations ******** v****2****2**** 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**
**v****2 ****event by event**
**This is the method we are pursuing**

## Measuring v2 Fluctuations - 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 simulated MC events**
**Modified Hijing - Flow**
**Geant**

## Method Overview - Simplified Example

**Method Overview - Simplified Example**

**Event by Event measurement**

**Observed u distribution in a sample**

**g(u)**

**K****b****(*****u*****)**

**K****a****(*****u*****)**

**2 possible v****2**** values**

**Relative abundance in sample**

**f****1**

**f****2**

**Demonstration**

**Demonstration**

**Demonstration**

**Demonstration**

**Question: What is the relative abundance of 2 v****2****’s in the sample?**

**V****2a**

**V****2b**

**V****2a**

**V****2b**

***u*****=v****2obs.**

**or *****u***** = v****2****2****obs.**
**or *****u***** = q****obs.**

## Method Overview - Simplified Example

**Method Overview - Simplified Example**

**Event by Event measurement**

**Measured u distribution in a sample**

**g(*****u*****)**

**K****b****(*****u*****)**

**K****a****(*****u*****)**

**2 possible v****2**** values**

**Relative abundance in sample**

**f****1**

**f****2**

**Demonstration**

**Demonstration**

**Demonstration**

**Demonstration**

**Question: What is the relative abundance of v****2a**** to v****2b**** in the sample?**

**V****2a**

**V****2b**

**V****2a**

**V****2b**

***u*****=v****2obs.**

**or *****u***** = v****2****2****obs.**
**or *****u***** = q****obs.**

## Method Overview - Simplified Example

**Method Overview - Simplified Example**

**Event by Event measurement**

**Measured u distribution in a sample**

**g(*****u*****)**

**K****b****(*****u*****)**

**K****a****(*****u*****)**

**2 possible v****2**** values**

**f****a**

**f****b**

**Demonstration**

**Demonstration**

**Demonstration**

**Demonstration**

**V****2a**

**V****2b**

**V****2a**

**V****2b**

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

**V****2a**

**V****2b**

***u*****=v****2obs.**

**or *****u***** = v****2****2****obs.**
**or *****u***** = q****obs.**

**Question: What is the relative abundance of v****2a**** to v****2b**** in the sample?**

**Extracted v****2true**** distribution from sample**

## Method Overview

**Method Overview**

**Kernel**

**Measured *****u***** distribution in a sample**

**In real life v****2**** can take **
**a continuum of values**

***u*****=v****2obs.**

**f(v****2****)**

**g(*****u*****)**

**K(*****u*****,v****2****)**

**Extracted v****2true**** distribution from sample**

## Method Overview

**3 Tasks**
**Measure *****u***** event-by-event ****g(*****u*****)**
**Calculate the kernel ****K(*****u*****,v****2****)**
**Extract dynamical fluctuations ****f(v****2****)**

## PHOBOS Detector

**PHOBOS Multiplicity Array**
**-5.4<********<5.4 coverage**
**-Holes / granularity differences**
**Idea: Use all available information in event to read off single *****u***** value**

**dN/d******

**Primary particles**
**Hits on detector**

**HIJING + Geant **
**15-20% central**

## Measuring u=v2obs Event by Event I

**Measuring u=v****2obs**** Event by Event I**

**Probability Distribution Function (PDF) for hit positions:**

**PDF**

**u**

**Define likelihood of u and ********0**** for an event:**

**demonstration**

**Probability of hit in ******

**Probability of hit in ******

## 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 0’s

**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 ********0****’s**

**Measuring u=v****2obs**** Event by Event II**

**Acceptance**

## 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 0’s

**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 ********0****’s**

**Measuring u=v****2obs**** Event by Event II**

**Acceptance**

## Measuring u=v2obs Event by Event III

**Observed *****u***** distribution in a sample**

**Error bars show RMS**

**Mean and RMS of *****u***** in slices of v****2**

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

**Modified Hijing+Geant**

**200 GeV AuAu**

**Measuring u=v****2obs**** Event by Event III**

## Calculating the Kernel I

**Calculating the Kernel I**

**Simple: Measure *****u***** distribution in bins of v****2**
**2 small complications**
** ****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*****.**
**Statistics in bins can be combined by fitting smooth functions**

## Calculating the Kernel II

**In a single bin of v****2**** and *****n***

**(a, b) ******** (<*****u*****>,*********u*****)**

***u***** distribution with for fixed v****2**** and *****n***

**Distribution is not Gaussian **
**But can be parameterized by <*****u*****> and *********u***

## Calculating the Kernel III

** ****Measure <*****u*****> and *********u***** in bins of v****2**** and n**
** ****Fit smooth functions**

**K(*****u*****,v****2****,*****n*****)**

**K(*****u*****,v****2****,*****n*****)**

## Calculating the Kernel IV

**Calculating the Kernel IV**

**Multiplicity dependence can be integrated out**

**N(*****n*****) = Number hits distribution in sample **

**K(*****u*****,v****2 ****,*****n*****)**

**K(*****u*****,v****2****)**

## Extracting dynamical fluctuations

**Extracting dynamical fluctuations**

## Extracting dynamical fluctuations

**Ansatz**

**Ansatz for f(v****2****)**

**ansatz**

## Extracting dynamical fluctuations

**Ansatz**

**Expected g(*****u*****) for Ansatz**

**Ansatz for f(v****2****)**

**ansatz**

**integrate**

## Extracting dynamical fluctuations

**Extracting dynamical fluctuations**

**Expected g(*****u*****) for Ansätze**

**Ansätze for f(v****2****)**

**ansatz**

**integrate**

## Extracting dynamical fluctuations

**Extracting dynamical fluctuations**

**Comparison with sample**

**ansatz**

**integrate**

**Ansätze for f(v****2****)**

**Compare expected g(u) for Ansatz with measurement**
**Minimum ********2**** ******** <v****2****> and ********v2**

## Method Summary

**Method Summary**

**Many MC events **

**K(*****u*****,v****2 ****,*****n*****)**

**MC**

**A Small Sample**

**<v****2****>=0.05**
******v2**** =0.02**

**f****in****(v****2****)**

**<v****2****>=0.048**
******v2**** = 0.023**

**f****out****(v****2****)**

**g(*****u*****)**

**N(*****n*****)**

**K(*****u*****,v****2****)**

**MC**

**MC**

**MC**

**MC**

**MC**

**measurement**

**measurement**

**integration**

**Minimize ********2**** in integral**

## Verification

**Ran this analysis on Modified Hijing**
**v****2****(********) = ****v****2****(0) •**** (1-|********|/6)**
**Same as the assumption in our fit**
**v****2****(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**

## Verification

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

**Modified Hijing+Geant**

**200 GeV AuAu **
**15-20% central**

**<v****2****> = 0.050**

**Only statistical errors shown**
**(from combining vertex bins)**

**<v****2****> = 0.020**

**<v****2****> = 0.030**

******out**

******out**

******out**

**<v****2****> = 0.040**

## 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********(********)**
**v****2****(********)**
**Non-flow in data**
**Should show up in reaction plane resolutions**

## Likelihood Fit Normalization

**Acceptance**

## Calculating the Kernel. Functions observed to fit the Kernel