1. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page1 Sergei A. Voloshin Wayne State University, Detroit, Michigan Toward energy and system size dependence of anisotropic flow Outline: 1.Flow fluctuations and non-flow: Lee-Yang Zeroes, Fourier Transform, Bessel transform, fitting q-distributions 2.Eccentricity fluctuations 3.Compare to a model and to data… Sorry, no new STAR results…

2. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page2 v2/eps at the time of QM2002/NA49 PRC - uncertainty in the centrality definition - sqrt(s)=130 GeV data: 0.075 < pt < 2.0 GeV/c - sqrt(s)=200 GeV data: 0.15 < pt < 2.0; - the data scaled down by a factor of 1.06 to account for change in (raw) mean pt. - AGS and SPS – no low pt cut - STAR and SPS 160 – 4 th order cumulants - no systematic errors indicated What happened since then? - New data - New methods (e.g. LYZ) - Non-flow and flow fluctuations have been much better understood but the problem has not been resolved… and no new plot yet (note that there is no such a plot in the STAR AuAu 200GeV PRC “flow” paper) Motivation for the plot:

3. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page3 v 2 {2}, v 2 {4}, non-flow and flow fluctuations Correct if v is a constant in the event sample Several reasons for v to fluctuate in a centrality bin: 1)Variation in impact parameter in a centrality bin (taken out in STAR results) 2)Real flow fluctuations (due to fluctuations in initial conditions or in system evolution) Different directions to resolve the problem: - Find method that have direct/different sensitivity to mean v - Estimate flow fluctuations by other means 2 equations, at least 3 unknowns: v, δ, σ Subject of this talk Flow fluctuations and q-distribution method ?

4. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page4 v 2 from q-distributions -- The results are very close to those from 4-particle correlation analysis. -- Difficult to trace the contribution of flow fluctuations. STAR, PRC 66 (2002) 034904

5. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page5 Fourier transform of the distribution in flow vector componenets Due to symmetry (no acceptance effects!) only real part is non-zero General strategy: Let x 01 be the first root of equation J 0 (x 0i )=0. x 01 ~=2.045. Then: v = k 01 /M, where k 01 is the first zero of f(k)

6. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page6 v 2 {LYZ} – flow from Lee Yang Zeroes How accurate is this statement?

7. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page7 Using Bessel transform LYZ == == Fourier transform of distribution in Q x, and/or Q y == Bessel transform of the distribution in Q == Fitting of Q-distribution !?

8. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page8 Error calculation.x qxfA.C("out/ds5_AuAu200.root",4) from qx: v=0.0622787+/-0.00047226 from qy: v=0.0621035+/-0.000531313 root [6].x qqfA.C("out/ds5_AuAu200.root",4) v=0.0628762+/-0.000210269 Error on k 0 is shown in red. Good agreement between results from Fourier Transforms of q x and q y distribution, fitting q-distribution and Bessel transform of q-distribution.

9. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page9 Differential flow From the above expression one can get differential flow in different ways. First way: Alternatively: How accurate is this statement?

10. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page10 Simulations, pure flow 4M events, 400 particle in each event, Case 1 : 50% events with v=0.04 and 50% events with v=0.06 Case 2: 100% with v=0.06 Case 3: 100% with v=0.04 For the case 1, v{2} and v{4} as expected, e.g. v{2}=sqrt(0.04^2+0.06^2). v{BT} is significantly lower than 0.05, close to v{4} !! Case #

11. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page11 Simulations, + non-flow Similar to the previous case +”non-flow”: 300 “direct” particles and 100=50*2 - 50 pairs with the same azimuth. As expected, only v{2} is strongly affected by non-flow.

12. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page12 What is wrong with BT/LYZ? … Nothing really, just the first order approximation mentioned earlier is not good enough. In the graph on the left, the green line shows what one would need to get the correct mean value of v, compared to the black line, what one really gets by transform. One can also track it analytically by expanding Bessel function in the vicinity of the first zero. Summary: BT/LYZ is only slightly better than v 2 {4} in terms of (in)sensitivity to fluctuations

13. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page13 UrQMD calculations Fluctuations are too small to see?

14. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page14 Elliptic flow. Initial eccentricity. Other similar/same quantities: Ollitrault:  s Heiselberg:  Sorge: A 2 Shuryak: s 2 Elliptic flow must vanish if initially the system was created symmetric. Then, at small eccentricities, v 2 ~  “e” -- initialization of energy density; “s” – initialization of entropy density Not important which one to use, but important to use the same!!!

15. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page15 Eccentricity in the optical Glauber model

16. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page16 Fluctuations in eccentricity  fluctuations in v 2 x,y – coordinates of “wounded” nucleons v 2 ~   fluctuations in flow Calculations: R. Snellings and M. Miller One can calculate how cumulants should be affected

17. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page17 Compare to data Fluctuations in initial geometry could explain the entire difference between v 2 {2} and v 2 {4} In fact, using nucleon participants (shown by red line in the plot) generates too much fluctuations, inconsistent with data

18. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page18 UrQMD once more … the paper was not published for a reason… Why these fluctuations are not seen in v 2 {4} compared to real v 2 ?

19. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page19 MC Glauber calculations: “old” and “new” “New” coordinate system – rotated, shifted Idea known for about a year, “went public” : S. Manly’s talk at QM2005

20. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page20 Eccentricity, fluctuations, Monte-Carlo Glauber, Std vs ‘Participant’ Note: -Relative fluctuations are much smaller. - In general, “apparent” (“participant”) eccentricity values are larger compared to “standard”. -In CuCu epsStd{4} fails almost at all centralities - The fluctuations in apparent eccentricity is much smaller than in standard - The difference between standard and apparent is bigger for CuCu than AuAu Monte Carlo Glauber nTuples from J. Gonzales (STAR)

21. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page21

22. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page22 Eccentricity, Monte-Carlo Glauber, all four systems.

23. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page23 But should not we use  {2}, not eps? It could improve the agreement… What about v2{4}/  {4}?

24. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page24 Does it matter, eps, eps{2} or eps{4}? This is just an illustration of an effect of using different eccentricity definitions. Centralities for eccentricity calculations are not correct !

25. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page25 Summary - LYZ method is shown to be ‘identical’ to q-distribution method (and Bessel transform method) - LYZ/BT is close to v2{4} in terms of sensitivity to flow fluctuations - Eps participant is not only different from Eps standard, but fluctuates… can/should we use these fluctuations to estimate flow fluctuations?

26. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page26 EXTRA SLIDES

27. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page27 First hydro calculations In hydro, where the mean free path is by assumption much less than the size of the system, there is no other parameters than the system size (may enter time scales, see below). Then elliptic flow must follow closely the initial eccentricity. J.-Y. Ollitrault, PRD 46 (1992) 229

28. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page28 Low density limit (called “collisionless” in the original paper of Heiselberg and Levy) Below - my own derivation of Heiselberg’s results Change in the particle flux is proportional to the probability for the particle to interact. Integrations over: a) particle emission point b) Over the trajectory of the particle (time) with weight proportional to the density of other particles --“scattering centers” Particle density at time t assuming free streaming Heiselberg & Levy, PRC 59 (1999) 2716

29. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page29 v 2 /  vs particle density, first plot S.V. & A. Poskanzer, PLB 474 (2000) 27 Uncertainties: Hydro limits: slightly depend on initial conditions Data: no systematic errors, shaded area –uncertainty in centrality determinations. Curves: “hand made” E877 NA49 “Cold” deconfinement?

30. S.A. Voloshin Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006page30 “hydro limits” ? RHIC 160 GeV/A SPS SPS 40 GeV/A b (fm) Suppressed scale! v 2 /  Minimum in v2/  due to softening of the EoS at phase transition Q to U. Heinz : Could the solid line in right hand plot be used as HYDRO prediction for v2/eps plot?