1. Measuring flow, nonflow, fluctuations Jean-Yves Ollitrault, Saclay BNL, April 29, 2008 Workshop on viscous hydrodynamics and transport models

2. Outline Definition Methods & observables An improved event-plane method to measure flow without nonflow Residual systematic errors on v 2, v 4 Flow fluctuations

3. Definition Elliptic flow is defined as v 2 =, where φ R is the azimuthal angle of the reaction plane, but we cannot measure φ R : There is always a model underlying flow analyses.

4. A simple model In a sample of events with the same centrality and same reaction plane (same geometry), assume 1.Symmetry with respect to φ R 2.No long-range correlation: –f(p 1,p 2 )-f(p 1 ) f(p 2 ) scales like 1/M (multiplicity), with M»1, –3-particle correlations (cumulants) scale like 1/M 2, etc. Bhalerao Borghini JYO nucl-th/0310016 Note that elliptic flow involves only the single-particle distribution f(p): v 2 =. 2. implies in particular no fluctuation of elliptic flow. Then, one can extract v 2 from data.

5. Methods & observables 2-particle =v 2 2 Variants: Event-plane method (all experiments at RHIC), scalar-product method (STAR), 2-particle cumulants (PHENIX, STAR) 3-particle =v 2 v 1 2 v 2 {ZDC-SMD} (STAR) ≥ 4-particle « cos(2(φ 1 +φ 2 -φ 3 -φ 4 ))»=v 2 4 4-particle cumulants, Lee-Yang zeroes (STAR)

6. The event-plane method Uses an event-by-event estimate of the reaction plane φ R, the event plane ψ R, defined as the azimuth of the Q vector Q x =Q cos(2 ψ R )=∑ cos(2 φ j ) Q y =Q sin(2 ψ R )=∑ sin(2 φ j ) One then estimates elliptic flow as v 2 {EP}= /R Where R is a « resolution » correction.

7. 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.

8. 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).

9. Autocorrelations & nonflow In the event-plane method, one must remove the particle under study from the event plane (Danielewicz & Odyniec, 1985) Q x =Q cos(2 ψ’ R )=∑’ cos(2 φ j ) Q y =Q sin(2 ψ’ R )=∑’ sin(2 φ j ) Otherwise there are trivial autocorrelations between φ and ψ R. There is not a unique event plane for all particles! v 2 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.

10. 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: v 2 {EP}= /R One uses only ψ R, 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 ! Q 2(ψ R -φ R ) V2V2 Reaction plane φ R

11. Event plane and event weight Instead of v 2 {EP}= /R, We define v 2 {LYZ}=. W R (Q)=J 1 (r Q)/C is the event weight, where r=2.404/V 2, and C is a normalization constant depending on the resolution (Simulations: Naomi van der Kolk)

12. Why a Bessel function? Test: if there is no flow, the result should be 0. J 1 (rQ)cos(2(φ-ψ R ))=(-i/2π)∫dθ exp(irQ θ ) cos(2(φ-θ)), where Q θ ≡Q cos(2(ψ R -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 = x r is defined such that =0 (Lee-Yang zero). Test OK: nonflow & autocorrelations removed.

13. Simulations for ALICE Input v 2 (p t ) : linear below 2 GeV, constant above Resolution: χ=1, corresponding to R= =0.71 in the standard event-plane analysis Top: flow only Bottom: flow+nonflow, simulated by embedding collinear pairs of particles, irrespective of p t. Simulations: Ante Bilandzic (cumulants) and Naomi van der Kolk (Lee-Yang zeroes)

14. Technical issues Statistical errors are much larger with Lee-Yang zeroes if the resolution is too low. As a rule of thumb, one needs χ 2 ≡ ∑v 2 2 ≥ 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 v 2 is only 1%, and this 1% can be corrected. The improved event-plane method works for v 2 only, not for v 4 (the original Lee-Yang zeroes method does both).

15. Systematic uncertainties There are residual systematic uncertainties due to Non-gaussian fluctuations of the Q- vector (higher-order terms in the central limit expansion) : δv 2 / v 2 ~ 1/M 2 v 2 2, where M is the multiplicity of detected particles Non-isotropic fluctuations of the Q vector. δv 2 / v 2 ~ 1/M+v 4 /Mv 2 2 (cf talk by P. Sorensen) This must be compared to the error from nonflow effects in the standard method δv 2 / v 2 ~ 1/Mv 2 2, a factor M~400 larger The higher harmonic v 4 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

16. What are flow fluctuations? v 2 can be defined event by event if φ R is known Even if φ R is not known, one can define an event v 2 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 φ R, no long-range correlation for fixed φ R ), with its own v 2 We are interested in the dynamical fluctuations, i.e., the fluctuations of v 2 from one sample to the other.

17. Effect of fluctuations on flow estimates 2-particle : v 2 {2} 2 = 2 +δv 2 2 +nonflow 4-cumulant, Lee-Yang zeroes: v 2 {4} 2 = 2 -δv 2 2 v 2 {2} 2 - v 2 {4} 2 =nonflow+2 δv 2 2 : 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

18. No symmetry with respect to φ R ! PHOBOS collaboration, nucl-ex/0510031 The ellipse defined by participant nucleons, which defines the direction where elliptic flow develops may be tilted relative to φ R We should think of fluctuations of v 2 as 2- dimensional. If fluctuations are gaussian, v 2 {4} is the center of the gaussian, i.e., the standard eccentricity, and v 2 {SMD-ZDC}=v 2 {4} Voloshin Poskanzer Tang Wang arXiv:0708.0800 Bhalerao JYO nucl-th/0607009 Au +Au 200 GeV STAR prelimin ary

19. Eccentricity fluctuations are not gaussian PHOBOS, arXiv:0711.3724 The positions of participant nucleons are strongly correlated! (2 dimensional percolation)

20. 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