Example analysis of toy model Tilted Pion Sources from Azimuthally Sensitive HBT Interferometry Ulrich Heinz1, Mike Lisa1, Urs Wiedemann2 1 The Ohio State University, 2 CERN HBT probes freeze-out space-time structure What is the space-time structure of the pion freeze-out distribution in heavy ion collisions? S(x,K) is the phase-space distribution at freeze-out “View from above” Elliptic flow “View from the front” Coordinate space tilt... vs. momentum-space tilt reveals nature of p directed flow; RQMD predicts opposite tilts, contrary to hydro scenario p from D-decays reflect coordinate-space structure of proton directed flow Transverse deformation... initial vs. final shape sensitive to history of elliptic flow coordinate space vs. momentum space deformation sensitve to nature of elliptic flow exotica (“nutcrackers”…) E895: a TPC-based AGS experiment capable of extracting Smn eventwise determination of the magnitude and orientation of b from multiplicity and momenta of charged particles measured in ~4p good PID and momentum resolution allow excellent interferometry no beampipe  measure to pT=0 Special considerations for azimuthally-sensitive interferometry: generate separate correlation function for each pairwise cut in f only mix p from events with “same” reaction plane orientation Pair-specific rotation by f = (K,b) connects collision system to out-side-long system RQMD Au(2 AGeV)Au reaction plane out f b K side x1 x2 Example analysis of toy model 2D projections of 3D correlation functions for 8 cuts in f (generated with Pratt’s CRAB) Analysis of midrapidity pions from semicentral Au (4 AGeV) Au collisions in E895 Freezeout distribution: tilted Gaussian in space-time thermal in momentum -80 0 -80 0 80 fpair This relates HBT radii to source in “natural” coordinates: x3 s1 = 4 fm Q = 25o x1 Extracted spatial correlation tensor Nonzero diagonal and S13 elements s3 = 7 fm (1) 1D projections of the correlation function for f=45  22.5 (stars), and of the Gaussian fit (lines) Below, the stars are results of Gaussian fits to the 3D correlation functions… HBT parameters for each of the 8 f cuts (stars), and a simultaneous fit of the radii to Eq. 1 (lines) where the spatial correlation tensor is given by Experimental Reality - finite reaction plane resolution Measured reaction plane differs statistically from true one by some angle Df, resulting in reduced oscillations. Similar to flow studies, first-order oscillations (here quantified by S13 ) must be corrected by 1/cos(Df) Second-order oscillations (here quantified by (S22-S11 )) must be corrected by 1/cos(2Df) S00, S33, and (S22+S11 ) are unaffected Generally small effect on homogeneity lengths (~0.3 fm) and tilt angle (~3°) Symmetry considerations 1) Mirror reflection in reaction plane f-f 2) For A=B collisions: Point reflection about origin y=0: corresponds to ff+p …and the lines are simultaneous fits to the HBT radii with Eq. (1), yielding the Extracted spatial correlation tensor: S13 and diagonal components nonzero Eq. (2)  qs = 24.6°  0.6° colored contours: 2D projections of measured 3D correlation functions black contours: projection of Gaussian fits to correlation function necessary vanishing points necessary vanishing points Extra constraints at midrapidity General constraints Extract tilt angle qs , and homogeneity lengths in “natural frame” from diagonal elements of R†(qs)Smn R(qs) Rotated spatial correlation tensor: diagonal matrix eigenvalues are squared homogeneity lengths x2 =x2’ qs = 37o s0 = 3.7 fm/c s1 = 4.0 fm s2 = 5.2 fm s3 = 5.2 fm x3  “to scale” x1 x1’ At low pT (transverse space-momentum correlations small  Smn ~ f-independent) 5 nonvanishing components Smn encode all spacetime structure Smm: lengths of homogeneity S13: correlation between x1 & x3  spatial tilt angle Summary of data and model at 2, 4, 6 AGeV large and positive (follows protons) measurable (and measured!!) sensitive to dynamics in transport model RQMD better describes momentum-space component of proton directed flow when meanfield is included [E895, PRL 84, 5488 (2000)] However, the coordinate-space component (tilt) is better described in cascade mode more stringent probe of dynamical details of flow new experimentally-feasible tool to study space-time structure of flow E895 Collab., Phys. Lett. B (in press) [nucl-ex/0007022] (2) (Beam) Reaction plane x1 x2 x3 qs No radii vanish, even at ycm and pT=0  always 6 relevant radii first harmonic oscillations @ ycm!!! Clearly observable new HBT effect! Spatial tilt of source in reaction plane - “the last static component” Measures geometrical aspects of anisotropic flow N.B. for studies at RHIC: need first-order reaction plane for tilt Phys. Lett. B489, 287 (2000) [nucl-th/0003022]