measuring flow, nonflow, fluctuations
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Measuring flow, nonflow, fluctuations
Jean-Yves Ollitrault, SaclayBNL, 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 φR is 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
1. Symmetry with respect to φR
2. 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
One uses only ψR, not Q. We improve the event-plane method by using also QThis can be done in such a way as to remove nonflow effects !
Q2(ψR-φR)
V2 Reaction plane φR
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)/C is 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)),
where Qθ≡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 ALICEInput 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 onlyBottom: 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/M2v2
2, 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/Mv2
2, a factor M~400 larger
The higher harmonic v4 has 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 particlesBoth 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
Reaction plane
Nonflow+fluctuations
Nonflow only
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
Au +Au 200 GeV
STAR
prelimi
nary
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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