M.A. Lisa Ohio State Intersections nucl-th/ 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

M.A. Lisa Ohio State Intersections 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

M.A. Lisa Ohio State Intersections 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 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

M.A. Lisa Ohio State Intersections 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) Anisotropy in momentum space (measured flow signature) = anisotropy in configuration space + pressure generated in collision (sensitive to EoS) 3D Hydro “normal flow”

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

M.A. Lisa Ohio State Intersections 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

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

M.A. Lisa Ohio State Intersections 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)

M.A. Lisa Ohio State Intersections 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

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

M.A. Lisa Ohio State Intersections Symmetry simplifies situation at low p T, and y cm, No radii vanish, even at y cm and p T =0 first harmonic 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

M.A. Lisa Ohio State Intersections 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

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

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

M.A. Lisa Ohio State Intersections 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 , 1997) Bass et al [PLB (93), PRC (95)] : pion anisotropies ~ 2 AGeV due to reflection (  N   N) not absorption (  NN  N  NN) experimental handle through tilt?

M.A. Lisa Ohio State Intersections 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:

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

M.A. Lisa Ohio State Intersections GeV cut No-flow toy model GeV cut 20

M.A. Lisa Ohio State Intersections 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

M.A. Lisa Ohio State Intersections 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 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 AGS large and positive (follows protons) measurable (and measured!!) new experimentally-feasible tool to study space-time structure of flow nucl-th/ MAL, U. Heinz and U.A. Wiedemann

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

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

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

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

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

M.A. Lisa Ohio State Intersections 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

M.A. Lisa Ohio State Intersections 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

M.A. Lisa Ohio State Intersections S  measures Gaussian curvature in b-fixed Cartesian system... pair separation distributions Hardtke & Voloshin [PRC (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

M.A. Lisa Ohio State Intersections …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

M.A. Lisa Ohio State Intersections 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)