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### Measuring flow, nonflow, fluctuations

Jean-Yves Ollitrault, Saclay

BNL, April 29, 2008

Workshop on viscous hydrodynamics

and transport models

Outline

- Definition
- Methods & observables
- An improved event-plane method to measure flow without nonflow
- Residual systematic errors on v2, v4
- Flow fluctuations

Definition

Elliptic flow is defined as

v2=<cos(2(φ-φR))>,

where φRis the azimuthal angle of the reaction plane, but we cannot measure φR:

There is always a model underlying flow analyses.

A simple model

In a sample of events with the same centrality and same reaction plane (same geometry), assume

- Symmetry with respect to φR
- No long-range correlation:
- f(p1,p2)-f(p1) f(p2) scales like 1/M (multiplicity), with M»1,
- 3-particle correlations (cumulants) scale like 1/M2, etc.

Bhalerao Borghini JYO nucl-th/0310016

Note that elliptic flow involves only the single-particle distribution f(p): v2=<cos(2(φ-φR))>.

2. implies in particular no fluctuation of elliptic flow.

Then, one can extract v2 from data.

Methods & observables

- 2-particle <cos(2(φ1-φ2))>=v22

Variants: Event-plane method (all experiments at RHIC), scalar-product method (STAR), 2-particle cumulants (PHENIX, STAR)

- 3-particle <cos(2φ1-φ2-φ3)>=v2v12

v2{ZDC-SMD} (STAR)

- ≥ 4-particle «cos(2(φ1+φ2-φ3-φ4))»=v24

4-particle cumulants, Lee-Yang zeroes (STAR)

The event-plane method

Uses an event-by-event estimate of the reaction plane φR, the event plane ψR, defined as the azimuth of the Q vector

Qx=Q cos(2 ψR)=∑ cos(2 φj)

Qy=Q sin(2 ψR)=∑ sin(2 φj)

One then estimates elliptic flow as

v2{EP}=<cos(2(φ-ψR))>/R

Where R is a « resolution » correction.

Comparison between methods

- The event plane method is intuitive, but it amounts to measuring (sums of) 2-particle correlations, which doesn’t mean collective motion into some preferred direction. One measures flow+nonflow.
- Higher-order methods (4-particle cumulants, Lee-Yang zeroes) are able to get rid of nonflow systematically, but they are less intuitive: appear as a « black box » to non experts. They also have larger statistical errors.

Nonflow: should we bother?

- Recent results seem to indicate that differences between methods at RHIC are dominated by flow fluctuations, rather than nonflow effects.
- However, one should remember that a price has been paid for removing nonflow: e.g., rapidity gaps between particle and event plane
- Nonflow is there at high pt. Will be larger at LHC.
- In addition, there are detector-induced nonflow effects: split tracks, detectors with overlapping acceptance.
- A method which is free from nonflow effects guarantees more flexibility in the analysis, and an increased resolution (all pieces of the detector can, and should, be used: we are interested in collective effects).

Autocorrelations & nonflow

In the event-plane method, one must remove the particle under study from the event plane (Danielewicz & Odyniec, 1985)

Qx=Q cos(2 ψ’R)=∑’ cos(2 φj)

Qy=Q sin(2 ψ’R)=∑’ sin(2 φj)

Otherwise there are trivial autocorrelations between φ and ψR. There is not a unique event plane for all particles!

v2 from autocorrelations alone is ~5% at RHIC!

Nonflow effects are qualitatively similar to autocorrelations: a particle in the event plane is correlated (~ collinear) to the particle under study. Unfortunately, there are much harder to remove.

A method which removes nonflow effects will automatically remove autocorrelations as well.

Improving the event-plane method

A. Bilandzic, N. van der Kolk, JYO, R. Snellings, arXiv:0801.3915

A mere reformulation of Lee-Yang zeroes

Event-plane method:

v2{EP}=<cos(2(φ-ψR))>/R

Q

2(ψR-φR)

V2

Reaction plane φR

One uses only ψR, not Q.

We improve the event-plane method by using also Q

This can be done in such a way as to remove nonflow effects !

Event plane and event weight

Instead of

v2{EP}=<cos(2(φ-ψR))>/R,

We define

v2{LYZ}=<WR(Q)cos(2(φ-ψR))>.

WR(Q)=J1(r Q)/Cis the event weight,

where r=2.404/V2,

and C is a normalization constant depending on the resolution

(Simulations: Naomi van der Kolk)

Why a Bessel function?

Test: if there is no flow, the result should be 0.

J1(rQ)cos(2(φ-ψR))=(-i/2π)∫dθ exp(irQθ) cos(2(φ-θ)),

whereQθ≡Q cos(2(ψR-θ)) is the projection of the Q vector onto the direction 2θ.

Separate the Q vector into flow and nonflow parts.

Average over events: flow and nonflow are uncorrelated

<exp(irQθ)cos(2(φ-θ))> = <exp(irQflowθ)> x

<exp(irQnonflowθ) cos(2(φ-θ))>

r is defined such that <exp(irQflowθ)>=0 (Lee-Yang zero).

Test OK: nonflow & autocorrelations removed.

Simulations for ALICE

Input v2(pt) : linear below 2 GeV, constant above

Resolution: χ=1, corresponding to

R=<cos(2 ΔψR)>=0.71 in the standard event-plane analysis

Top: flow only

Bottom: flow+nonflow, simulated by embedding collinear pairs of particles, irrespective of pt.

Simulations: Ante Bilandzic (cumulants) and Naomi van der Kolk (Lee-Yang zeroes)

Technical issues

- Statistical errors are much larger with Lee-Yang zeroes if the resolution is too low. As a rule of thumb, one needs

χ2 ≡ ∑v22≥ 1 (typically 400 particles seen at RHIC)

Use all detectors!

- Lee-Yang zeroes do better than the standard event-plane method if the detector lacks azimuthal symmetry. No flattening procedure is required, because one projects the flow vector onto a fixed direction θ (Selyuzhenkhov & Voloshin, arxiv:0707.4672) With a 60 degrees dead sector in the detector, the relative error on v2 is only 1%, and this 1% can be corrected.
- The improved event-plane method works for v2 only, not for v4 (the original Lee-Yang zeroes method does both).

Systematic uncertainties

There are residual systematic uncertainties due to

- Non-gaussian fluctuations of the Q- vector (higher-order terms in the central limit expansion) : δv2/ v2~ 1/M2v22, where M is the multiplicity of detected particles
- Non-isotropic fluctuations of the Q vector.δv2/ v2~ 1/M+v4/Mv22(cf talk by P. Sorensen)

This must be compared to the error from nonflow effects in the standard method δv2/ v2~ 1/Mv22, a factor M~400 larger

The higher harmonic v4has a systematic uncertainty of (absolute) order 1/M, due to an interference between flow and nonflow, which no method is presently able to correct.

Borghini Bhalerao JYO nucl-th/0310016

STAR nucl-ex/0310029

What are flow fluctuations?

- v2 can be defined event by event if φR is known
- Even if φR is not known, one can define an event v2 from the ellipse formed by outgoing particles

Both quantities are dominated by trivial statistical fluctuations ~1/√M~5%. Not interesting!

Consider a superposition of several samples of events, each sample as defined above (symmetry with respect to φR, no long-range correlation for fixed φR), with its own v2

We are interested in the dynamical fluctuations, i.e., the fluctuations of v2 from one sample to the other.

Effect of fluctuations on flow estimates

- 2-particle : v2{2}2=<v2>2+δv22+nonflow
- 4-cumulant, Lee-Yang zeroes: v2{4}2=<v2>2-δv22
- v2{2}2- v2{4}2=nonflow+2 δv22: we always see the sum of fluctuations and nonflow, because fluctuations and correlations really are the same thing.
- One possibility to disentangle nonflow from fluctuations is to use the reaction plane from directed flow

Wang Keane Tang Voloshin nucl-ex/0611001

Nonflow only

Reaction plane

Nonflow+fluctuations

STAR preliminary

No symmetry with respect to φR !PHOBOS collaboration, nucl-ex/0510031

The ellipse defined by participant nucleons, which defines the direction where elliptic flow develops may be tilted relative to φR

We should think of fluctuations of v2 as 2-dimensional.

If fluctuations are gaussian, v2{4} is the center of the gaussian, i.e., the standard eccentricity, and

v2{SMD-ZDC}=v2{4}

Voloshin Poskanzer Tang Wang arXiv:0708.0800

Bhalerao JYO nucl-th/0607009

Eccentricity fluctuations are not gaussian

PHOBOS, arXiv:0711.3724

The positions of participant nucleons are strongly correlated!

(2 dimensional percolation)

Conclusions

- We are able to eliminate nonflow correlations. This requires to weight events depending on the length of the flow vector.
- In order to match theory with experiment, we must improve our quantitative understanding of eccentricity, and eccentricity fluctuations

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