1. M.A. Lisa Ohio State Intersections 20001 nucl-th/0003022 MAL, U. Heinz and U.A. Wiedemann Probing the Space-time Structure of Anisotropic Flow with HBT (or “How to turn a flow talk into an HBT talk”) Mike Lisa The Ohio State University N.N. Ajitanand, J. Alexander, D. Best, P. Brady, T. Case, B. Caskey, D. Cebra, J. Chance, I. Chemakin, P. Chung, V. Cianciolo, B. Cole, K. Crowe, A. Das, J. Draper, S. Gushue, M. Gilkes, M. Heffner, H. Hiejima, A. Hirsch, E. Hjort, L. Huo, M. Justice, M. Kaplan, J. Klay, D. Keane, J. Kintner, D. Krofcheck, R. Lacey, J. Lauret, E. LeBras, M. Lisa, H. Liu, Y. Liu, B. McGrath, Z. Milosevich, D. Olson, S. Panitkin, C. Pinkenburg N. Porile, G. Rai, H.-G. Ritter, J. Romero, R. Scharenberg, L. Schroeder, R. Sotlz, B. Srivastava, N. Stone, J. Symons, S. Wang, R. Wells, J. Whitfield,T. Wienold, R. Witt, L. Wood, X. Yang, Y. Zhang, W. Zhang Auckland - BNL - CMU - Columbia - UC Davis - Harbin - KSU - LBL - LLNL - St. Mary’s College - OSU - Purdue - Stony Brook (The E895 Collaboration) and U. Heinz and U.A. Wiedemann CERN-TH

2. M.A. Lisa Ohio State Intersections 20002 A fly-by of E895 flow results A new tool to probe the “missing piece”: the geometric structure of flow Formalism: R 2 (  )  spatial correlation tensor implicit & explicit  -dependence a dominant and overlooked correlation  striking experimental signal toy models, RQMD, data pion flow and spatially tilted sources

3. M.A. Lisa Ohio State Intersections 20003 P. Chung Excitation functions of: d 2 N/dp T dy  , p, d, He K 0,  p,  - HBT N ch ~ 200  K p d log(P lab ) dE/dx 0 -2 2 The many possibilities of E895… Au(2,4,6,8 AGeV)Au in “4  ” + event-wise reactionplane: p-space anisotropies (“flow”) x-space anisotropies charged particle PID  + 6 AGeV B. Caskey not background subtracted K0K0 M inv (GeV/c 2 ) P. Chung

4. M.A. Lisa Ohio State Intersections 20004 Directed flow in noncentral heavy ion collisions Au(8 GeV)Au b = 3 fm D. Rischke NPA 610, c88 (‘96) Brachmann et al, PRC61 (2000) 024909 Anisotropy in momentum space (measured flow signature) = anisotropy in configuration space + pressure generated in collision (sensitive to EoS) 3D Hydro “normal flow”

5. M.A. Lisa Ohio State Intersections 20005 Momentum Flow Number Flow H. Liu et al, PRL, in press Proton Sidewards Flow Excitation Function @ AGS Strong sensitivity to medium contributions to pressure No sudden drops in pressure (flow) signaling phase transition No single medium parameterization reproduces flow details

6. M.A. Lisa Ohio State Intersections 20006 Attack from another angle -- low p T pions at the AGS 2 AGeV 4 AGeV 0-2 fm 8-10 fm 4-6 fm big tilts! positive (follow protons) not overly sensitive to b sensitive to dynamics falling with E beam Can we access the tilt of the freezeout configuration in coordinate space through pions? RQMD

7. M.A. Lisa Ohio State Intersections 20007 Extracting & summarizing source information with HBT Gaussian parameterization in Bertsch-Pratt decomposition vanish for azimuthally symmetric sources central Au+Au@2AGeV y=y cm  0.35 p T =0.1-0.3 GeV/c PRL 84, 2798 (2000) emission function

8. M.A. Lisa Ohio State Intersections 20008 Space-time interpretation of R 2 ij for cylindrically symmetric sources 1 p p 2 Q T Q S Q O mass-shell constraint q·K=0: mixes spatial, temporal information x (fm) y (fm) implicit dependence carries dynamical information and typically dominates point out: “x” is “out”, “y” is “side” and “z” is “long” (not “real” x,y,z) beam towards viewer (focus of recent HBT progress)

9. M.A. Lisa Ohio State Intersections 20009 The next challenge —  -dependent interferometry reaction plane out  b KK side x y Two coordinate systems connected by a pair-dependent rotation reaction-plane-fixed

10. M.A. Lisa Ohio State Intersections 200010 Connecting the BP correlations to the source in its natural frame Spatial correlation tensor ! explicit and implicit (S  (  ))  dependence U. Wiedemann PRC57 (1998) 266

11. M.A. Lisa Ohio State Intersections 200011 Symmetry simplifies situation at low p T, and y cm, No radii vanish, even at y cm and p T =0 first harmonic oscillations @ y cm !!! (Beam) Reaction plane x y z ss 5 nonvanishing components S  encode all spacetime geometry driving R 2 ij (  ) S  : lengths of homogeneity S 13 : correlation between x & z  Spatial tilt angle

12. M.A. Lisa Ohio State Intersections 200012 Simple toy case - ellipsoidal source w/ no flow Using CRAB, generate c.f.’s in 8 bins of for a tilted freezeout distribution: x z  z = 7 fm  x = 4 fm  = 25 o results of B-P fits

13. M.A. Lisa Ohio State Intersections 200013 A series of tilted ellipsoids in relative momentum measure a single tilted ellipsoid in coordinate space! x z -80 0 emission angle coordinate space 2D projections of correlation functions

14. M.A. Lisa Ohio State Intersections 200014 Adding longitudinal flow Same model but with additional boost Still “same source(s) viewed from different  z-flow effects |p z |<40 MeV/c

15. M.A. Lisa Ohio State Intersections 200015 Opposing average tilts in p, x and the physics of  flow RQMD w/ meanfield: Au(2GeV)Au, b=3-7 fm coordinate-space tilt in positive direction Antiflow reflects dense low-|z| region  from dilute large-|z| show positive flow transport model exhibits weak  “antiflow,” similar to observation (Kintner et al. PRL 78 4165, 1997) Bass et al [PLB 302 381 (93), PRC 51 3343 (95)] : pion anisotropies ~ 2 AGeV due to reflection (  N   N) not absorption (  NN  N  NN) experimental handle through tilt?

16. M.A. Lisa Ohio State Intersections 200016 RQMD w/ meanfield: Au(2GeV)Au, b=2-8 fm, p T <300 MeV/c out side long osolsl positive x-z correlation negative tilt? Clue: oblate source S 11 >S 33 In terms of prolate source:

17. M.A. Lisa Ohio State Intersections 200017 Preliminary E895 DATA 4 AGeV 2 AGeV

18. M.A. Lisa Ohio State Intersections 200018 -0.8 0 2 GeV cut 22 -80 0 No-flow toy model -0.8 0 4 GeV cut 20

19. M.A. Lisa Ohio State Intersections 200019 4 AGeV  t = 3.3 fm/c  x’ = 3.9 fm  y = 4.8 fm  z’ = 4.8 fm  s = 37 o y x’ z x Rough idea of extracted shapes of average effective source “to scale” 2 AGeV  t = 3.5 fm/c  x’ = 4.2 fm  y = 5.8 fm  z’ = 5.5 fm  s = 50 o y x’ z x

20. M.A. Lisa Ohio State Intersections 200020 Summary E895’s detailed excitation functions of momentum-space flow signals rigorously challenge collision models - full understanding still eludes us Anisostropic HBT accesses entirely new aspect of flow R 2 ij  spatial correlation tensor S , encodes source information in natural frame symmetry and small transverse flow: 5 nonvanishing components @ y cm encode: 4 spacetime homogeneity lengths [S 22 -S 11 ] (transverse elliptic shape)  2 nd -order oscillations in R o 2, R s 2, R os 2 spatial tilt of source in reaction plane - “the last static component” S 13  striking 1 st -order oscillations in R ol 2, R sl 2 Tilt @ AGS large and positive (follows protons) measurable (and measured!!) new experimentally-feasible tool to study space-time structure of flow nucl-th/0003022 MAL, U. Heinz and U.A. Wiedemann

21. M.A. Lisa Ohio State Intersections 200021 Tilts at RHIC? Theoretically, and experimentally, “challenging” RQMD@RHIC b=3-8 fm c/o N. Xu, R. Snellings But try! RQMD evolves too quickly w/ E Never know what else... @ SPS: R side ~ 30% too big E895, QM99 I.G. Bearden et al (NA44) PRC58, 1656 (1998)

22. M.A. Lisa Ohio State Intersections 200022 (Beam) Reaction plane x y z yy

23. M.A. Lisa Ohio State Intersections 200023 Constraints on S  (  ) General symmetry: Point reflection about origin: General symmetry: Mirror reflection in reaction plane Symmetry @ y cm (K z =0): Point reflection about 2D origin: Vanishing points Harmonic coefficients

24. M.A. Lisa Ohio State Intersections 200024 Prolate/oblate sources and large/small tilts

25. M.A. Lisa Ohio State Intersections 200025 First Harmonics at Midrapidity?? - Ain’t that Impossible? Is “up” different than “down”?

26. M.A. Lisa Ohio State Intersections 200026 Howd’ja get more  -bins? 3) binning in instead of pypy pxpx p1p1 p2p2 K  - boundaries all pairs wind up in some  -bin 1) a bit more statistics 2) tweaked reactionplane resolution (C. Pinkenburg) & HBT cuts binning with  1,  2 also led to increased sensitivity to oscillations in R side as compared to R out pairs straddling  -boundaries never used

27. M.A. Lisa Ohio State Intersections 200027 Prolate/oblate sources and large/small tilts x z zz xx  Viewed as prolate source Viewed as oblate source x z zz xx   z >  x  > 0  z <  x  < 0 xzxz  x  z  + 90 o selects solution with |  | < 45 o

28. M.A. Lisa Ohio State Intersections 200028 S  measures Gaussian curvature in b-fixed Cartesian system... pair separation distributions Hardtke & Voloshin [PRC 61 024905 (2000)] : HBT radii from Gaussian fits sensitive to close pairs Radii reflect Gaussian curvature near origin in  r out,  r side,  r long For most sources, and

29. M.A. Lisa Ohio State Intersections 200029 …but source is tilted in this system + has longitudinal flow effects.  z/  2 fit to pair distribution in  z-  x with tilted ellipsoid no q z cut |q z |<25 MeV cut

30. M.A. Lisa Ohio State Intersections 200030 Approaching from another angle… Can we access the tilt of the freezeout configuration in coordinate space? Yes!! With  HBT measurements… (at AGS, low p T  are “all” from  ’s)