Tilted Pion Sources from Azimuthally Sensitive HBT Interferometry
HBT probes freeze-out space-time structure


2
2
~
R s K   x side K 


2
~
~
xx x
R o2 K   ~
x out   t  K 
 d x  S( x, K )  f ( x )
f



2
~
2
~
 d x  S( x, K )
R l K   x long  l t  K 
S(x,K) is the phase-space


distribution at freeze-out
~
~
2
~
~
R ol K   x out   t   x long  l t  K 


~
2
~
~
R os K   x out   t   x side K  vanish for azimuthall y



~
2
R sl K   ~
x side  ~
x long  l t  K   symmetric sources
Ulrich
1
Heinz ,
1
Mike
1
Lisa ,
Urs
2
Wiedemann
The Ohio State University, 2 CERN
What is the space-time structure of the pion freeze-out distribution in heavy ion collisions?
4
4
“View from above”
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
Pair-specific rotation by f  (K,b) connects
collision system to out-side-long system
“View from the front”
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 S
• 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
Elliptic flow
• Special considerations for azimuthally-sensitive interferometry:
• generate separate correlation function for each pairwise cut in f
RQMD Au(2 AGeV)Au
side
x2
K
out

~
x Bertsch
f
reaction
plane
 Pr att
x1
• only mix p from events with “same” reaction plane orientation
~
~
~
x out 
x
cos
f

x 2 sin f 

1

~ 
  x side   Df~
x   ~
x1 sin f  ~
x 2 cosf 


~
~

x3


 x long 
Example analysis of toy model
b
Freezeout distribution:
• tilted Gaussian in space-time
• thermal in momentum
This relates HBT radii to source in “natural” coordinates:
s 2  5 fm
2
2
2
2
R o  S11 cos f  S22 sin f  S00  2S01 cosf  2S02 sin f  S12 sin 2f
2
2
R l  S33  2LS03  LS00
2
R os  S12 cos 2f  12 (S22  S11 ) sin 2f  S01 sin f  S02 cosf
s0  5 fm/c
s1 = 4 fm
 LS01 ) cosf  (S23  LS02 ) sin f  S03  LS00
Analysis of midrapidity pions from semicentral Au (4 AGeV) Au collisions in E895
fpair
x3 Q = 25o
R s2  S11 sin 2 f  S22 cos2 f  S12 sin 2f
2
R ol  (S13
2
R sl  (S23
2D projections of 3D correlation functions
for 8 cuts in f (generated with Pratt’s CRAB)
Extracted spatial correlation tensor
s3 = 7 fm
0.2  1.0
0.0  1.1 
13.4  3.5 0.7  0.9


 0.7  0.9 21.3  0.9  0.1  0.6 4.5  0.6 
S
0.2  1.0  0.1  0.6 25.5  1.0  0.3  0.6 


 0.0  1.1 4.5  0.6  0.3  0.6 22.8  0.9 
x1
(1)
1D projections of the correlation function
for f=45  22.5 (stars), and of the
Gaussian fit (lines)
 LS02 ) cosf  (S13  LS01 ) sin f
Below, the stars are results of Gaussian fits
to the 3D correlation functions…
where the spatial correlation tensor is given by

4 ~ ~


d
x
x
x
S
(
x
,
K
)



S K  ~x  ~x  K 

4
 d x S( x, K )

Nonzero diagonal and S13 elements
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
Symmetry considerations
1) Mirror reflection in reaction plane f-f
 
S( x 0 , x, K)  S( x 0 , x1 ,x 2 , x 3 , K1 ,K 2 , K 3 )
…and the lines are simultaneous fits to the
HBT radii with Eq. (1), yielding the
Extracted spatial correlation tensor:
2) For A=B collisions: Point reflection about origin
 
 
S( x 0 , x, K)  S( x 0 ,x,K)
 y=0: corresponds to ffp
 22.8  4.3 0.7  0.6  0.3  0.7 2.0  0.6 


 0.7  0.6 21.9  0.4  0.1  0.2 12.2  0.3 
S
 0.3  0.7  0.1  0.2 24.7  0.4 0.3  0.2 


 2.0  0.6 12.2  0.3 0.3  0.2 42.9  0.6 
 

S( x 0 , x, K)  S(x 0 ,x,K1 ,K 2 )
necessary
vanishing
points
General constraints
S01
S12
S13
S02
S23
S03
S
~
x 1~
x 0 f   ~
x 1~
x 0  f 
~
x 1~
x 2 f    ~
x 1~
x 2  f  0 ,180 
~
x 1~
x 3 f   ~
x 1~
x 3  f 
~
x0~
x 2 f    ~
x 0~
x 2  f  0 ,180 
~
x ~
x f    ~
x ~
x  f  0 ,180 
2 3
~
x0~
x 3 f  
~
x 2 f  
2 3
~
x0~
x 3  f 
~
x 2  f 

Extra constraints
at midrapidity
necessary
vanishing
points
~
x 1~
x 0 f    ~
x 1~
x 0 f  p   90 
~
x 1~
x 2 f   ~
x 1~
x 2 f  p 
 90 
~
x 1~
x 3 f   ~
x 1~
x 3 f  p 
~
x ~
x f    ~
x ~
x f  p 
0 2
0 2
2
Ro

1
2
S11  S22   
1
2

S33 S11
2


-80
0
HBT parameters for each of the 8 f cuts (stars),
and a simultaneous fit of the radii to Eq. 1 (lines)
80
Eq. (2)  s = 24.6°  0.6°
Extract tilt angle s , and homogeneity lengths in “natural frame” from diagonal elements of R†(s)S R(s)
• diagonal matrix
• eigenvalues are squared
homogeneity lengths
2

S  s
0.2  1.0
0.4  1.0 
13.4  3.5 0.5  1.0


0  0.9 
 0.5  1.0 15.9  1.0 0.1  0.6
T
R s   S  R s   
0.2  1.0 0.1  0.6 27.0  1.4  0.3  0.6 


0  0.9  0.3  0.6 26.8  0.8 
 0.4  1.0

Phys. Lett. B489, 287 (2000) [nucl-th/0003022]
x2 =x2’
s0 = 3.7 fm/c
s1 = 4.0 fm
s2 = 5.2 fm
s3 = 5.2 fm
4
6
s ( )
47  5
49  2
64  2
“to scale”
x1
x1’
Data
RQMD cs
RQMD mf
Data
RQMD cs
3 .7  0 .5
6 . 4  0. 2
4.0  0.1 5.2  0.1 5.2  0.1 37  4
3.3  0.1 4.2  0.1 4.7  0.1 33  3
RQMD mf
4 .6  0 .3
3.5  0.1 4.7  0.1 4.7  0.1 45  3
Data
RQMD cs
3 . 8  0 .5
5 .9  0 .2
3.5  0.2 4.8  0.2 4.7  0.1 33  6
3.5  0.1 4.3  0.1 4.9  0.1 28  3
RQMD mf
5 . 0  0 .3
3.6  0.1 4.7  0.1 4.5  0.1 48  5
• 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
•No radii vanish, even at ycm and pT=0  always 6 relevant radii
•first harmonic oscillations @ ycm!!!
x3

s 0 (fm/c) s1 (fm) s 2 (fm) s 3 (fm)
3.5  1.0 4.2  0.2 5.8  02 5.4  0.1
6.6  0.2 2.9  0.1 4.4  0.1 4.4  0.1
5.4  0.2 3.1  0.1 5.0  0.1 4.6  0.1
E (AGeV)
x1
x3
(Beam)
Reaction plane
s =
37o
Summary of data and model at 2, 4, 6 AGeV
2
s
S00, S33, and (S22+S11 ) are unaffected
Generally small effect on homogeneity lengths (~0.3 fm)
and tilt angle (~3°)
colored contours: 2D projections of measured 3D correlation functions
black contours: projection of Gaussian fits to correlation function
x2
2
S22  S11 cos 2f  S00
2
2
R l  S33  LS00
2
R os  12 (S22  S11 ) sin 2f
2
R ol  S13 cosf
R sl2  S13 sin f
0

s
 12 S11  S22   12 S22  S11 cos 2f
-80
S13 and diagonal components nonzero
 22.8  4.3  0.2  0.6  0.3  0.7 2.0  0.6 


 0.2  0.6 16.3  0.4  0.2  0.2
0  0.4 

T
S  R s   S  R s   
 0.3  0.7  0.2  0.2 24.7  0.4 0.2  0.2 


0  0.4
0.2  0.2 48.5  0.5 
 2.0  0.6
At low pT (transverse space-momentum correlations small  S ~ f-independent)
5 nonvanishing components S encode all spacetime structure
S: lengths of homogeneity
2
S

1
13
1
S13: correlation between x1 & x3  spatial tilt angle   tan
(2)
2
Rs
0
Rotated spatial correlation tensor:
~
x 2~
x 3 f   ~
x 2~
x 3 f  p   90 
~
x 0~
x 3 f    ~
x 0~
x 3 f  p   90 
~
x 2 f   ~
x 2 f  p 

-80
Second-order oscillations (here quantified by (S22-S11 ))
must be corrected by 1/cos(2Df
• 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
E895 Collab., Phys. Lett. B (in press) [nucl-ex/0007022]