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Measuring flow, nonflow, fluctuations

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Measuring flow, nonflow, fluctuations

Jean-Yves Ollitrault, Saclay

BNL, April 29, 2008

Workshop on viscous hydrodynamics

and transport models

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

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.

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.

- 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)

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.

- 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.

- 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).

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.

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 !

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)

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.

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)

- 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).

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

- 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.

- 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

Au +Au 200 GeV

STAR preliminary

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

PHOBOS, arXiv:0711.3724

The positions of participant nucleons are strongly correlated!

(2 dimensional percolation)

- 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