1. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire1 The simple geometric scaling of flow – perhaps it’s not so simple after all Steven Manly (Univ. of Rochester) For the PHOBOS Collaboration

2. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire2 Collaboration meeting, BNL October 2002 Burak Alver, Birger Back, Mark Baker, Maarten Ballintijn, Donald Barton, Russell Betts, Richard Bindel, Wit Busza (Spokesperson), Zhengwei Chai, Vasundhara Chetluru, Edmundo García, Tomasz Gburek, Kristjan Gulbrandsen, Clive Halliwell, Joshua Hamblen, Ian Harnarine, Conor Henderson, David Hofman, Richard Hollis, Roman Hołyński, Burt Holzman, Aneta Iordanova, Jay Kane,Piotr Kulinich, Chia Ming Kuo, Wei Li, Willis Lin, Constantin Loizides, Steven Manly, Alice Mignerey, Gerrit van Nieuwenhuizen, Rachid Nouicer, Andrzej Olszewski, Robert Pak, Corey Reed, Eric Richardson, Christof Roland, Gunther Roland, Joe Sagerer, Iouri Sedykh, Chadd Smith, Maciej Stankiewicz, Peter Steinberg, George Stephans, Andrei Sukhanov, Artur Szostak, Marguerite Belt Tonjes, Adam Trzupek, Sergei Vaurynovich, Robin Verdier, Gábor Veres, Peter Walters, Edward Wenger, Donald Willhelm, Frank Wolfs, Barbara Wosiek, Krzysztof Woźniak, Shaun Wyngaardt, Bolek Wysłouch ARGONNE NATIONAL LABORATORYBROOKHAVEN NATIONAL LABORATORY INSTITUTE OF NUCLEAR PHYSICS PAN, KRAKOWMASSACHUSETTS INSTITUTE OF TECHNOLOGY NATIONAL CENTRAL UNIVERSITY, TAIWANUNIVERSITY OF ILLINOIS AT CHICAGO UNIVERSITY OF MARYLANDUNIVERSITY OF ROCHESTER Collaboration meeting in Maryland, 2003

3. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire3 Flow in PHOBOS Ring counter Octagon Spectrometer arm Paddle trigger Vertex detector

4. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire4 Correlate reaction plane determined from azimuthal pattern of hits in one part of detector Flow in PHOBOS Subevent A

5. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire5 with azimuthal pattern of hits in another part of the detector Flow in PHOBOS Subevent B

6. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire6 Or with tracks identified in the spectrometer arms Flow in PHOBOS Tracks

7. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire7 Separation of correlated subevents typically large in  Flow in PHOBOS

8. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire8 Differential flow has proven to be a useful probe of heavy ion collisions:  Centrality  p T  Pseudorapidity  Energy  System size  Species Probing collisions with flow

9. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire9 Differential flow has proven to be a useful probe of heavy ion collisions:  Centrality  p T  Pseudorapidity  Energy  System size  Species Elliptic flow – Cu-Cu results

10. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire10 Elliptic flow – Cu-Cu results  Cu flow is large  Track- and hit-based results agree (200 GeV)  ~20-30% rise in v 2 from 62.4 to 200 GeV PHOBOS preliminary Cu-Cu, h ± Hit based 62.4 GeV Hit based 200 GeV Track based 200 GeV S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031

11. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire11 Elliptic flow – Cu-Cu results Cu-Cu v 2 (η) shape reminiscent of Au-Au PHOBOS preliminary Cu-Cu, 62.4 GeV, h± 0-40% centrality PHOBOS preliminary Cu-Cu, 200 GeV, h± 0-40% centrality S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031 Au-Au

12. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire12 Elliptic flow – Cu-Cu results Longitudinal scaling reminiscent of Au-Au PHOBOS preliminary Cu-Cu, h ± v2v2  '=|  |-y beam Cu-Cu collisions also exhibit extended longitudinal scaling statistical errors only Au-Au S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031 PHOBOS Collaboration, Phys. Rev. Lett. 94 (2005) 122303

13. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire13 Bridging experiment and geometry Since experiments cannot measure the underlying geometry directly, models remain a necessary evil. multiplicity, etc. models centrality impact parameter number of participants eccentricity Models are also needed to connect fundamental geometric parameters with each other Experiment Geometry

14. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire14 Modeling Geometry Glauber’s formalism for the scattering of a particle off of a nuclear potential. Historically, this model involved integrating the nuclear overlap function of two nuclei with densities given by the Woods-Saxon distribution. Nucleons proceed in a straight line, undeflected by collisions Irrespective of previous interactions, nucleons interact according to the inelastic cross section measured in pp collisions. Glauber Assumptions

15. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire15 A different application of the Glauber formalism is a Monte Carlo technique, in which the average over many simulated events takes the place of an integration. Au+Au Collisions with the same N part (64 participants) (cross section, shape, impact parameter, number of participating nucleons, etc.) This has been a very successful tool at RHIC in relating various geometric properties

16. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire16 GlauBall is the PHOBOS implementation of a Glauber MC Nucleons are distributed randomly based on an appropriately chosen Woods-Saxon radial density and arbitrary polar coordinates. An internucleon separation can be introduced at this step Subsequently, only the x and y (transverse) nucleon positions are used, so the nuclei can be thought of as 2 dimensional projections

17. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire17 The nuclei are offset by an impact parameter generated randomly from a linear distribution (vanishing small at b=0) Nucleons are treated as hard spheres. Their 2D projections are given an area of  NN (taken from pp inelastic collisions) The nuclei are “thrown” (their x-y projections are overlapped), and opposing nucleons that touch are marked as participants.

18. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire18 Standard eccentricity (  standard ) x System size and eccentricity Expect the geometry, i.e., the eccentricity, of the collision to be important in comparing flow in the Au-Au and Cu-Cu systems Centrality measure  N part   Paddle signal, ZDC, etc. MC simulations What is the relevant eccentricity for driving the azimuthal asymmetry? MC simulations

19. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire19 x2x2 Au-Au collision with Npart =64 y2y2 x2x2 y2y2 Au-Au collision with Npart = 78 x2x2 Eccentricity - a representation of geometrical overlap

20. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire20 Sample of Cu-Cu collisions Cu-Cu collision with Npart = 33Cu-Cu collision with Npart = 28 Yikes! This is a negative eccentricity! y2y2 x2x2 y2y2 x2x2

21. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire21 Cu-Cu collision with Npart = 33Cu-Cu collision with Npart = 28 Gives negative eccentricity Principal axis transformation Maximizes the eccentricity Sample of Cu-Cu collisions y2y2 x2x2 x2x2 y2y2

22. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire22 Fluctuations in eccentricity are important for small A. System size and eccentricity Participant eccentricity (  part ) x Standard eccentricity (  standard ) x Two possibilities

23. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire23 System size and eccentricity Au-Au Cu-Cu PHOBOS-Glauber MC preliminary Mean eccentricity shown in black S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031

24. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire24 Fluctuations in eccentricity are important for the Cu-Cu system. System size and eccentricity Must use care in doing Au-Au to Cu-Cu flow comparisons. Eccentricity scaling depends on definition of eccentricity. S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031

25. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire25 Elliptic flow – v 2 scaling  Expect v 2 /  ~ constant for system at hydro limit.  Note the importance of the eccentricity choice. h ± 1  statistical and systematic errors added in quadrature h ± S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031

26. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire26 Elliptic flow – v 2 scaling h ± 1  statistical and systematic errors added in quadrature h ± Given other similarities between Au-Au and Cu-Cu flow, perhaps this is evidence that  part is (close to) the relevant eccentricity for driving the azimuthal asymmetry S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031

27. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire27 Elliptic flow – v 2 scaling Expectin “low density limit”. S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031 Red is data from Cu-Cu collisions, blue is data from Au-Au collisions

28. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire28 Elliptic flow – v 2 scaling Scaling observed to be similar between systems if participant eccentricity is used.  Caution: we used  part for PHOBOS data. Important for Cu-Cu, less critical for Au-Au.  Scale v 2 (  ) to ~v 2 (y) (10% lower)  Scale dN/d  to be ~dN/dy (15% higher) S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031 Red is data from Cu-Cu collisions, blue is data from Au-Au collisions

29. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire29 Elliptic flow – v 2 scaling Points for STAR, NA49 and E877 data taken from STAR Collaboration, Phys.Rev. C66 (2002) 034904 with no adjustments  Caution: we used  part for PHOBOS data. Important for Cu-Cu, less critical for Au-Au.  Scale v 2 (  ) to ~v 2 (y) (10% lower)  Scale dN/d  to be ~dN/dy (15% higher) Scaling observed to be similar between systems if participant eccentricity is used. S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031 Red is data from Cu-Cu collisions, blue is data from Au-Au collisions

30. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire30 Elliptic flow – system dependence Eccentricity difference is important for same centrality selection. V 2 (p T ) for Cu-Cu is similar to v 2 (p T ) for Au-Au when scaled by  part PHOBOS preliminary h ± 0-50% centrality PHOBOS preliminary h ± 0-50% centrality PHOBOS preliminary h ± 0-50% centrality S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031

31. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire31 v 2 for Cu-Cu is ~20% smaller than v 2 for Au-Au plotted 0-40% centrality. Drops another ~20% if scaled by ratio PHOBOS 62.4 GeV h ± 0-40% centrality Elliptic flow – system dependence preliminary PHOBOS 200 GeV h ± 0-40-% centrality Statistical errors only S. Manly et al., PHOBOS Collaboration, Proc. QM05, nucl-ex/0510031

32. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire32 Conclusions  Cu-Cu elliptic flow large. Similar in shape to Au-Au. PHOBOS preliminary Cu-Cu, h ± Hit based 200 GeV Track based 200 GeV PHOBOS preliminary Cu-Cu, 200 GeV, h± 0-40% centrality

33. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire33 Conclusions  The Cu-Cu systems exhibits extended longitudinal scaling. PHOBOS preliminary Cu-Cu, h ± v2v2  '=|  |-y beam statistical errors only

34. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire34 Conclusions  Eccentricity calculated in standard way from Glauber model is not robust and potentially misleading for small systems.

35. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire35 Conclusions  Eccentricity definition very important for small systems. h ± 1  statistical and systematic errors added in quadrature h ±

36. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire36 Conclusions  Similarity of Au-Au to Cu-Cu flow and the fact that scaling seems to work for  part may imply that  part (or something close to it) is the relevant geometric quantity for generating the azimuthal asymmetry.

37. S. Manly – U. Rochester Gordon Conf. 2006, New London, New Hampshire37 Conclusions  The Cu-Cu systems exhibits extended longitudinal scaling.  Eccentricity definition very important for small systems.  Cu-Cu elliptic flow large. Similar in shape to Au-Au.  Similarity of Au-Au to Cu-Cu flow and the fact that scaling seems to work for  part may imply that  part (or something close to it) is the relevant geometric quantity for generating the azimuthal asymmetry.  Eccentricity calculated in standard way is not robust and potentially misleading for small systems.