Sergei A. Voloshin Wayne State University, Detroit, Michigan

1. Toward energy and system size dependence of anisotropic flow Sergei A. Voloshin Wayne State University, Detroit, Michigan Sorry, no new STAR results… • Outline: • Flow fluctuations and non-flow:Lee-Yang Zeroes, Fourier Transform, Bessel transform, fitting q-distributions • Eccentricity fluctuations • 3. Compare to a model and to data… page

2. v2/eps at the time of QM2002/NA49 PRC Motivation for the plot: • What happened since then? • New data • New methods (e.g. LYZ) • Non-flow and flow fluctuations havebeen much better understood but the problemhas not been resolved… andno new plot yet (note that there is no such a plotin the STAR AuAu 200GeV PRC “flow” paper) • 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 – 4th order cumulants • no systematic errors indicated page

3. v2{2}, v2{4}, non-flow and flow fluctuations Subject of this talk ? Flow fluctuations and q-distribution method • Several reasons for v to fluctuate in a centrality bin: • Variation in impact parameter in a centrality bin (taken out in STAR results) • Real flow fluctuations (due to fluctuations in initial conditions or in system evolution) Correct if v is a constant in the event sample 2 equations, at least 3 unknowns: v, δ, σ • Different directions to resolve the problem: • Find method that have direct/different sensitivity to mean v • Estimate flow fluctuations by other means page

4. v2 from q-distributions STAR, PRC 66 (2002) 034904 -- The results are very close to those from 4-particle correlation analysis. -- Difficult to trace the contribution of flow fluctuations. page

5. Fourier transform of the distribution in flow vector componenets Due to symmetry (no acceptanceeffects!) only real part is non-zero General strategy: Let x01 be the first root of equation J0(x0i)=0. x01~=2.045. Then: v = k01/M, where k01 is the first zero of f(k) page

6. v2{LYZ} – flow from Lee Yang Zeroes How accurate is this statement? page

7. Using Bessel transform LYZ == == Fourier transform of distribution in Qx, and/or Qy == Bessel transform of the distribution in Q == Fitting of Q-distribution !? page

8. Error calculation Error on k0 is shown in red. Good agreement between results from Fourier Transforms of qx and qy distribution, fitting q-distribution and Bessel transform of q-distribution. root [6] .x qqfA.C("out/ds5_AuAu200.root",4) v=0.0628762+/-0.000210269 .x qxfA.C("out/ds5_AuAu200.root",4) from qx: v=0.0622787+/-0.00047226 from qy: v=0.0621035+/-0.000531313 page

9. Differential flow How accurate is this statement? From the above expression one can get differential flow in different ways. First way: Alternatively: page

10. Simulations, pure flow Case # 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} !! page

11. 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. page

12. 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 analyticallyby expanding Bessel function in the vicinity of the first zero. Summary: BT/LYZ is only slightly better than v2{4} in terms of (in)sensitivity to fluctuations page

13. UrQMD calculations Fluctuations are too small to see? page

14. Elliptic flow must vanish if initially the system was created symmetric. Then, at small eccentricities, v2~ Elliptic flow. Initial eccentricity. “e” -- initialization of energy density; “s” – initialization ofentropy density Other similar/same quantities: Ollitrault: s Heiselberg:  Sorge: A2 Shuryak: s2 Not important which one to use, but important to use the same!!! page

15. Eccentricity in the optical Glauber model page

16. Fluctuations in eccentricity  fluctuations in v2 x,y – coordinates of “wounded” nucleons Calculations: R. Snellings and M. Miller v2 ~  fluctuations in flow One can calculate how cumulants should be affected page

17. Compare to data Fluctuations in initial geometrycould explain the entire differencebetween v2{2} and v2{4} In fact, using nucleon participants (shown by red line in the plot)generates too much fluctuations, inconsistent with data page

18. UrQMD once more … the paper was not published for a reason… Why these fluctuations are not seen in v2{4} compared to real v2? page

19. 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 page

20. Eccentricity, fluctuations, Monte-Carlo Glauber, Std vs ‘Participant’ • Note: • Relative fluctuations are much smaller. • In general, “apparent” (“participant”) eccentricity values are larger comparedto “standard”. • In CuCu epsStd{4} failsalmost at all centralities • The fluctuations in apparenteccentricity is much smaller than in standard • The difference betweenstandard and apparent isbigger for CuCu than AuAu Monte Carlo Glauber nTuples from J. Gonzales (STAR) page

21. page

22. Eccentricity, Monte-Carlo Glauber, all four systems. page

23. But should not we use {2}, not eps? It could improve the agreement… What about v2{4}/{4}? page

24. 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 ! page

25. Summary - LYZ method is shown to be ‘identical’ toq-distribution method (and Bessel transform method) - LYZ/BT is close to v2{4} in terms of sensitivity to flow fluctuations - Epsparticipant is not only different from Epsstandard, but fluctuates… can/should we use these fluctuations to estimate flow fluctuations? page

26. EXTRA SLIDES page

27. First hydro calculations J.-Y. Ollitrault, PRD 46 (1992) 229 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. page

28. Low density limit (called “collisionless” in the original paper of Heiselberg and Levy) Below - my own derivation of Heiselberg’s results Heiselberg & Levy, PRC 59 (1999) 2716 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 page

29. v2/ 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” “Cold” deconfinement? E877 NA49 page

30. “hydro limits” ? v2 /  SPS 40 GeV/A SPS RHIC 160 GeV/A b (fm) Suppressed scale! 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? page