# Presentation: 183763

## Flow Analysis Methods

**Art Poskanzer**

- Color by Roberta Weir

- Exploring the secrets
- of the universe

## First RHIC Elliptic Flow

- 22 k events

- STAR, K.H. Ackermann et al., PRL **86**, 402 (2001)

- First paper from STAR

- Significant that data approach hydro
- for central collisions

- valley / peak = 1 - 4 v2

- dramatic 2 to 1 effect

- 2 independent analyses

## Squeeze-out

- 400 MeV/A Au+Au (MUL 3)

- Plastic Ball, H.H. Gutbrod et al., Phys. Lett. **B216**, 267 (1989)

- Diogene, M. Demoulins et al., Phys. Lett. **B241**, 476 (1990)

- Plastic Ball, H.H. Gutbrod et al., PRC **42**, 640 (1991)

- Negative elliptic flow

- Best ellipsoid

## Prediction of Positive Elliptic Flow

- At a meeting in Jan ‘93, Jean-Yves told me he was predicting in-plane elliptic flow at high beam energies. I responded that we had just discovered out-of-plane elliptic flow

- J.-Y. Ollitrault, PRD **46**, 229 (1992), PRD **48**, 1132 (1993)

- 2-dimensional transverse sphericity analysis: best ellipse

- First Observation: E877, J. Barrette et al., PRC **55, **1420 (1997)

## Transverse Momentum Analysis

**Transverse Momentum Analysis**

- P. Danielewicz and G. Odyniec, Phys. Lett. **157B**, 146 (1985)

- Second to use the transverse plane
- First to define 1st harmonic Q-vector
- First to use weighting
- First to use sub-events
- First to remove auto-correlations

- Mistake in event plane resolution

- unit vector

## Transverse Plane

- Anisotropic Flow as a function of rapidity

- H. Wieman (2005)

- around the beam axis

**self quenching expansion**
**probe of early time**

- H. Sorge, PRL **78**, 2309 (1997)

## Fourier Harmonics

- S. Voloshin and Y. Zhang, hep-ph/940782; Z. Phys. C **70**, 665 (1996)

- First to use Fourier harmonics:

- Event plane resolution correction made for each harmonic

- See also, J.-Y. Ollitrault, arXiv nucl-ex/9711003 (1997)

- First to use the terms directed and elliptic flow for v1 and v2

- Unfiltered theory can be compared to experiment!
- Tremendous stimulus to theoreticians

- First to use mixed harmonics

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

## Flow Vector

- for n=1:

- S. Voloshin and Y. Zhang, Z. Phys. C **70**, 665 (1996)

- wi negative in backward
- hemisphere for odd harmonics

- For each harmonic n:

- Sum of vectors of all the particles

- Q is a 2 vector

## Standard Event Plane Method I

**random subs****	****most affected by non-flow**
**charge subs****	****like-sign less sensitive to neutral ****	****	****	****	****	****resonance decays**
****** ****subs****	****	****suppresses short-range correlations****	****	****	****	****	****FTPC even better**
**Flatten event plane azimuthal distributions in lab**
**Both sub-events and full event Q-vectors**

**Standard Event Plane Method I**

- A.M. Poskanzer and S.A. Voloshin, PRC **58**, 1671 (1998)

## Flattening Methods I

**Flattening Methods I**

**phi weighting** - in constructing the Q-vector one weights with the inverse of the azimuthal distribution of the particles averaged over many events
**recentering** - from each event Q-vector one subtracts the Q-vector averaged over many events
**shifting** - one fits the non-flat azimuthal distribution of the Q-vector angles with a Fourier expansion and calculates the shifts necessary to force a flat distribution

- To remove "Acceptance Correlations” flatten the azimuthal distribution of the event plane

- A.M. Poskanzer and S.A. Voloshin, PRC **58**, 1671 (1998)

## Flattening Methods II

**Mixed events** is not recommended because it has no advantage over phi weights from the same event

**Shifting** is good as a second method when either phi weights or recentering does not produce a flat distribution (e.g. FTPC)

- For elliptic flow, only the **second** harmonic component of the flattened distribution needs to be small !

## Standard Event Plane Method II

- A.M. Poskanzer and S.A. Voloshin, PRC **58**, 1671 (1998)

**Correlate subevent planes            **
**Calculate subevent plane resolution**
**Calculate event plane resolution**

- k = 2 for mixed harmonics
- k > 2 not yet used

- For k = 1:

- modified Bessel functions

## Standard Event Plane Method III

**Correlate particles with the event plane**
**But the event plane not containing the particle**
**(no autocorrelations)**
**Correct for the event plane resolution**
**Average over p****t****, ********, or both (with yield weighting)**
**v****2****(********)****	****	****may need p****t**** extrapolation**
**v****2****(p****t****)****	****	****for specified ******** range**
**v****2 ****vs. centrality****	****called integrated v****2**

**Standard Event Plane Method III**

- A.M. Poskanzer and S.A. Voloshin, PRC **58**, 1671 (1998)

- StFlowAnalysisMaker
- in STAR cvs library

## Elliptic Flow vs. Beam Energy

**Elliptic Flow vs. Beam Energy**

- 25% most central
- mid-rapidity

## Pair-wise Correlations

- PHENIX, K. Adcox et al., PRL **89**, 21301 (2002)

- Streamer Chamber, S. Wang et al., PRC **44**, 1091 (1991)

- no event plane

- acceptance correlations
- removed by mixed events

- initially used by PHENIX

## Scalar Product

** **

- STAR, C. Adler et al., PRC **66**, 034904 (2002)

- similar to standard method
- but weighted by the length
- of the Q-vector

- resolution

- correlation of particles with Q-vector

- centrality

- v2

## -Subs

- Large  gap reduces non-flow due to short-range correlations

- -subs similar to standard method
- but the event plane is from the opposite hemisphere

- but only for the
- peripheral collisions

## FTPC

- Larger  gap reduces non-flow due to short-range correlations

- FTPC similar to standard method
- but the event plane is from the FTPCs

- o v2{EP}

-  v2{FTPC}

- Cu+Cu 200 GeV

- S. Voloshin (2007)

**STAR preliminary**

## ZDC-SMD

- G. Wang, Quark Matter (2005)

- Still larger  gap reduces non-flow due to short-range correlations

- v2{ZDC-SMD} similar to v2{4}

## Cumulants I

- Four-particle correlation subtracts 2-particle nonflow

- N. Borghini, P.M. Dinh, and J.-Y. Ollitrault, PRC **64**, 054901 (2001)
- STAR, C. Adler et al., PRC **66**, 034904 (2002)

- Generating function:

- C{4} term of fit

- Can be calculated directly from

- Voloshin (2002)

- = v2{2}2

- is non-flow

## Cumulants II

- STAR, J. Adams et al., PRC **72**, 014904 (2005)

- v2{6} no better than v2{4}

## q-dist Method I

**q-dist Method I**

- modified Bessel function

- no event plane

- reduced flow vector:

- Bessel-Gaussian distribution of q:

- shifted out by vn2
- n2 = 1/2 from statistical effects
- broadened by non-flow
- and v2 fluctuations

- STAR, C. Adler et al., PRC **66**, 034904 (2002)

## Methods Comparison (2005)

- STAR, J. Adams et al., PRC **72**, 014904 (2005)

- Ratio to the Standard Method:

- Because of nonflow and fluctuations the truth lies
- between the lower band and the mean of the two bands

## Lee-Yang Zeros Method I

**Lee-Yang Zeros Method I**

- All-particle correlation subtracts nonflow to all orders

- R.S. Bhalerao, N. Borghini, and J.-Y. Ollitrault, Nucl. Phys. A **727**, 373 (2003)
- STAR, B.I. Abelev et al, PRC, submitted (2008)

**Flow vector projection on arbitrary lab angle, ******

**Generating function for one ******

**Sum Generating Function:**

**Average over ******** to remove acceptance effects**

**Product Generating Function:**

**Better for mixed harmonics but slower**

- First minimum of |G|2
- determines r0

## Lee-Yang Zeros Method II

- STAR, B.I. Abelev et al., PRC, submitted (2008)

- Sum and Prod agree
- Both slightly lower than v2{4}

- Sergei’s Bessel Transform method
- is a simplified version of LYZ sum

## Methods Comparison (2008)

**Methods Comparison (2008)**

- 2-part. methods

- multi-part. methods

- v in the participant plane always
- greater than v in the reaction plane

- very preliminary

- ≈ std

- part

## q-Dist with Nonflow and Fluctuations I

- integrate over  by expansion 2 ways: (for n = 2)

- higher terms different
- they involve the difference between x and y

- Voloshin and Sorensen (2007)

- fluctuations broaden
- non-flow correlations broaden because
-   there are effectively fewer
-   independent particles

- leading term the same

- QM06 left out M
- in front of 

- because x close to y

## q-Dist with Nonflow and Fluctuations II

- Gaussian along PP, but not restricted

- Voloshin and Sorensen (2007)

- Paul sets v2 = 0 for the integration and then smears with v2

- Sergei sets v2x = v2y before integration,
- 	assuming a 2D Gaussian for the fluctuations

- Both depend only on

-     and thus can not separate 2 from v2

- A upper limit on 2 gives lower limit on v2
- or, arrange to have 2 small, as PHOBOS does with an -gap

- No more info from Cumulants since

## Fluctuation Models I

- S.A. Voloshin, A.M. Poskanzer, A. Tang, G. Wang, Phys. Lett. B,  **659**, 537 (2008)

- Paul: 1D Gaussian along participant axis
- 	gives <v2> and v2 directly

- Sergei: 2D Gaussian in reaction plane
- 	gives Bessel-Gaussian in v0 and  along participant axis:

- also, v2{4}  is insensitive to fluctuations

- v2{4} is v2 along the reaction plane axis

- If Paul uses Bessel-Gaussian he gets same result as Sergei

## Fluctuation Models II

- But Sergei must calculate <v2> and v2 along the participant axis:

- S.A. Voloshin, A.M. Poskanzer, A. Tang, G. Wang, Phys. Lett. B,  **659**, 537 (2008)

- Sergei’s 2D Gaussian in RP

- Paul’s 1D Gaussian along PP

- 2 independent methods agree

## q-Dist with Nonflow and Fluctuations III

**STAR preliminary**

- STAR, P. Sorensen, QM08

- See Paul’s talk

- like-sign has less non-flow

## Particle Identification

- both axes scaled by number
- of constituent quarks

- and plotted vs.
- trans. kinetic energy

- STAR, B.I. Abelev et al., PRC, submitted (2008

**Particle Identification**

## Scaling: v2 / nq vs. KEt

- STAR, B.I. Abelev et al, PRC, submitted (2008) ); Yan Lu thesis (2007)

- scaled by part

- scaled by <v2>
- of that particle

- scaled by <v2>
- of charged particles

## Mixed Harmonics

- N. Borghini, P.M. Dinh, and J.-Y. Ollitrault, PRC, **66**, 014905 (2002)

- CERES, S.A. Voloshin, German Physical
- Society meeting (1998) was the first

- Removes nonflow
- Uses best determined
-     2nd har. event plane

- STAR, J. Adams et al., PRC **72**, 014904 (2005)

## Higher Harmonics

- J. Adams et al., PRL **92**, 062301 (2004)

- more details of the event shape
- in momentum space

- vn  v2n/2

## Conclusions

**25 years of flow analysis development**
**Extract parameters independent of acceptance**
**Standard Method is the most efficient of statistics**
**But now systematics are more important than statistics**
**Separation in ******** of particles and plane**
**Multi-particle methods**
**Mixed harmonics**
**Separate nonflow and fluctuations**