System Size and Energy Dependence of Elliptical Flow

1. System Size and Energy Dependence of Elliptical Flow Richard Bindel University of Maryland For the collaboration Division of Nuclear Physics, Maui September 19th, 2005

2. Collaboration (Aug 2005)

3. Motivation For Studying Flow z Reaction plane M. Kaneta y x • Flow probes the very early thermalization of the system • Elliptic flow as a function of: • Pseudorapidity • Momentum • Energy • Centrality Can be used to constrain theory.

4. Two Flow Measurement Methods Hit Based Method Track Based Method

5. Motivation for Comparing Species roughly same number of participants Two similar system sizes! Central CuCu collision Mid-central AuAu collision

6. Motivation for Comparing Species Cu+Cu Preliminary 3-6%, Npart = 100 Au+Au 35-40%, Npart = 99 roughly same number of participants Two similar system sizes! Central CuCu collision Mid-central AuAu collision Some observables scale with number of participants Similar dN/d 200 GeV

7. Motivation for Comparing Species Cu+Cu Preliminary 3-6%, Npart = 100 Au+Au 35-40%, Npart = 99 Two similar system sizes! roughly same number of participants Two similar system sizes! Central CuCu collision Mid-central AuAu collision Some observables scale with number of participants We expect flow to depend on geometry! Similar dN/d 200 GeV

8. Motivation for Comparing Species Cu+Cu Preliminary 3-6%, Npart = 100 Au+Au 35-40%, Npart = 99 Two similar system sizes! roughly same number of participants Two similar system sizes! Central CuCu collision Mid-central AuAu collision Elliptical Some observables scale with number of participants Circular We expect flow to depend on geometry! Similar dN/d 200 GeV Using two species lets us change the geometry while holding the number of participants constant

9. Eccentricity: A measure of “Geometry” Elliptical We need a way to quantify the geometry of the participating nucleons Mid-central AuAu collision The shape is characterized by the eccentricity () Circular Central CuCu collision

10. Eccentricity: A measure of “Geometry” Nucleus A Nucleus B A model of a AuAu collision: Participant Nucleons

11. Eccentricity: A measure of “Geometry” Nucleus A Nucleus B b A model of a AuAu collision: Participant Nucleons Using the impact parameter as the x-axis, we define the standard eccentricity using the widths of the distribution in the x and y directions

12. Eccentricity: A measure of “Geometry” Nucleus A Nucleus B y2 b x2 A model of a AuAu collision: Participant Nucleons Using the impact parameter as the x-axis, we define the standard eccentricity using the widths of the distribution in the x and y directions

13. The Phobos Elliptic Flow Data Set 19.6 GeV 62.4 GeV 130 GeV 200 GeV 0.05 Au+Au 0.05 Cu+Cu Au+Au: PRL 94 122303 (2005) PHOBOS preliminary preliminary from 0% to 40% centrality Sizable v2 for Cu+Cu

14. The Data: Something’s Amiss… || < 1 PHOBOS 200 GeV Hit Based Statistical errors only PHOBOS 200 GeV h± Statistical errors only Au+Au preliminary Cu+Cu preliminary Elliptic flow is high even for the most central bin in Cu+Cu How does it relate to the eccentricity?

15. The Data: Something’s Amiss… || < 1 PHOBOS 200 GeV Hit Based Statistical errors only PHOBOS 200 GeV h± Statistical errors only Au+Au preliminary Cu+Cu preliminary For CuCu, the eccentricity drops to zero by NPart of 100

16. The Data: Something’s Amiss… PHOBOS 200 GeV Statistical errors only PHOBOS 200 Gave h± PHOBOS 200 GeV Statistical errors only Au+Au Cu+Cu preliminary preliminary Au+Au Cu+Cu preliminary v2 near mid-rapidity || < 1 Dividing by the eccentricity shows no connection between the two species.

17. The Data: Something’s Amiss… ? What went wrong? We reexamine our definition of eccentricity

18. Reexamining Eccentricity Eccentricity is not directly measurable We use a Glauber model to relate eccentricity to our NPart bins. What goes into making this plot…?

19. Reexamining Eccentricity AuAu collisions with same NPart • An eccentricity distribution is built up for each NPart AuAu • The black line shows the average eccentricity • Glauber collisions are modeled over a range of • impact parameters and are sorted by the number • of participants.

20. Reexamining Eccentricity Au-Au Cu-Cu • When we examine the eccentricity distribution for CuCu, • it looks much broader than AuAu • Also, notice that there are many more events with negative • eccentricity.

21. Meaning of Negative Eccentricity y2 b x2 Here we revisit the standard definition of eccentricity applied to a Gluaber model.

22. Meaning of Negative Eccentricity y2 b x2 Here we revisit the standard definition of eccentricity applied to a Gluaber model. Negative eccentricity results when x2 > y2, apparently due to fluctuations in the positions of the nucleons. Because of its smaller size, CuCu is more susceptible to fluctuations

23. Redefining Eccentricity x Nucleus 1 y Nucleus 2 b Participant Region One reasonable method is to realign the coordinate system to maximize the ellipsoidal shape (a principal axis transformation) “Participant” eccentricity Opposed to “standard” eccentricity

24. Standard and Participant Eccentricity Mean eccentricity shown in black Au-Au Cu-Cu Standard Eccentricity Cu-Cu Au-Au Participant Eccentricity

25. Impact of Eccentricity Fluctuations Fluctuations in eccentricity are important for the Cu-Cu system. Must use care in doing Au-Au to Cu-Cu flow comparisons. Eccentricity scaling depends on definition of eccentricity.

26. Elliptic Flow Puzzle Solved? Standard Eccentricity Participant Eccentricity PHOBOS 200 GeV PHOBOS 200 GeV Cu+Cu preliminary Cu+Cu preliminary Au+Au Au+Au “Participant Eccentricity” allows v2 scaling from Cu+Cu to Au+Au

27. <dN/dy> / <S> scaling Standard Eccentricity Participant Eccentricity Cu+Cu Au+Au Cu+Cu Au+Au Overlap Area <dN/dy> / <S> scaling: STAR, PRC 66 034904 (2002) Voloshin, Poskanzer, PLB 474 27 (2000) Heiselberg, Levy, PRC 59 2716, (1999) Caveat: dNch/d corrected to dNch/dy

28. Conclusions • Phobos has continued to expand its extensive flow data set using two techniques • Studying CuCu compared to AuAu allows us to vary the geometry while holding Npart constant • The expectation that elliptic flow scales with eccentricity continues to seem reasonable • Careful consideration is needed when using eccentricity

29. Back-up Slides

30. Limiting Fragmentation (Au+Au)  - ybeam 19.6 GeV 62.4 GeV 130 GeV 200 GeV PHOBOS PHOBOS preliminary Au+Au 0-6% 200GeV 130GeV 62.4 GeV (prel) 19.6 GeV Au+Au 0-40% Au+Au 0-40% preliminary preliminary preliminary preliminary preliminary “Extended Longitudinal Scaling” of all longitudinal distributions

31. Limiting Fragmentation (Cu+Cu)  - ybeam 62.4 GeV 200 GeV preliminary preliminary PHOBOS PHOBOS Cu+Cu 0-6% 200GeV 62.4GeV preliminary preliminary preliminary Cu+Cu 0-40% preliminary ‘Extended Longitudinal Scaling’ also seen in Cu+Cu Persists from p+p to Au+Au over large range in ′

32. Compared to JAM Model preliminary 200 GeV 15-25% Cu-Cu preliminary 200 GeV Cu-Cu Statistical errors only Cu-Cu more like Hydro than JAM hadron string cascade model Here JAM uses a 1 fm/c formation time. Hydro (160) has kinetic freezeout temperature at 160 MeV

33. Two Flow Measurement Methods Hit based 62.4 GeV PHOBOS preliminary Cu-Cu, h± Hit based 200 GeV Good agreement between methods Track based 200 GeV PHOBOS preliminary Cu-Cu, h± || < 1