1. 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
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/ ]
(2)
(Beam)
Reaction plane x1 x2 x3 qs
No radii vanish, even at ycm and pT=0  always 6 relevant radii
first harmonic 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/ ]