Tilted Pion Sources and Azimuthal Dependence of HBT Radii
Mike Lisa
• Formalism: R2(f)  spatial correlation tensor
• implicit & explicit f-dependence
• a dominant and overlooked correlation  striking experimental signal
• toy models, RQMD
• pion flow and spatially tilted sources
• preliminary observation of tilted source @ AGS
M.A. Lisa
Ohio State
nucl-th/0003022 MAL, U. Heinz and U.A. Wiedemann
March 2000
1
Extracting & summarizing source information
 d x  S( x, K )  e
4
C(p1, p 2 )  1 
q  p1  p 2
iq  x 2
 d x  S( x, K )
4
2
central Au+Au@2AGeV
y=ycm 0.35 pT=0.1-0.3 GeV/c
K  12 p1  p2 
Gaussian parameterization in
Bertsch-Pratt decomposition

2



 q i q j R ij ( K )
C(q, K )  1  (K )  e
vanish for azimuthally
symmetric sources
M.A. Lisa
Ohio State
 R o2
2  2
R  R os
R 2
 ol
2
2 
R so
R lo
2
2
R s R ls 
R sl2 R l2 
March 2000
2
E895 Collab. PRL 84 (2000)
Space-time interpretation of R2ij for cylindrically symmetric sources
 
 
 
 

~2 
2
~
R l K   z  l t  K 

~ ~
~ 
2
~
R ol K   x   t   z  l t  K 


2
~
K  y K

~2 
2
~
R o K  x   t  K
mass-shell constraint q·K=0:

K   
0
q 
q  q
K0
mixes spatial, temporal information
R s2
d 4 x  S( x, K )  f ( x )

f 
4
 d x  S( x, K )
Q
S
p
1
M.A. Lisa
Ohio State
T
Q
O
p2

implicit K dependence
carries dynamical information
and typically dominates
y (fm)
Q
~
xx x

explicit  dependence
~
separates ~
x and t
beam
towards
viewer
point out: “x” is
“out”,
“y” is
“side”
and “z” is “long” (not “real” x,y,z)
March 2000
3
x (fm)
The next challenge — f-dependent interferometry
side
y
K
Two coordinate systems connected
  by a
pair-dependent rotation f  (K, b)
out
f
reaction
plane
x
b
   
K  
Df 0   0 
K
 
 L
C(K, q BP )  1 
M.A. Lisa
Ohio State

~
x Bertsch
 Pr att
~
~
x out 
x cosf  ~y sin f 



  ~

~
~
~
  x side   Df x    x sin f  y cosf 
~



~z
x


 long 
reaction-plane-fixed
4
d
 x  S( x, K )  e
 2


iq ( D f x  D f  t )
 d x  S( x, K )
4
March 2000
2
 1  e
R ij q i q j
4
Connecting the BP correlations to the source in its natural frame
Spatial correlation tensor S 


K  ~
x~
x

~
~

d x x  x S( x, K )

K 

4
 d x S( x, K )
 
4
R s2  S11 sin 2 f  S22 cos2 f  S12 sin 2f
R o2  S11 cos2 f  S22 sin 2 f  2S00  2S01 cosf  2S02 sin f  S12 sin 2f
R l2  S33  2LS03  2LS00
2
R os
 S12 cos 2f  12 (S22  S11 ) sin 2f  S01 sin f  S02 cosf
2
R ol
 (S13  LS01 ) cosf  (S23  LS02 ) sin f  S03  LS00
R sl2  (S23  LS02 ) cosf  (S13  LS01 ) sin f
U. Wiedemann PRC57 (1998) 266
!
M.A. Lisa
Ohio State
explicit and implicit (S(f)) f dependence
March 2000
5
Constraints on implicit f-dependence S(f)
General symmetry: Mirror
reflection in reaction plane
 
S(x, t, K)  S(x, y, z, t, K x ,K y , Kz )
General symmetry: Point Symmetry @ y (K =0): Point
cm
z
reflection about origin:
reflection about 2D origin:

 

 
S( x, t, K)  S( x, t,K)
S(x, t, K)  S(x, y, z, t,K x ,K y )
f  f
K determines f
reaction
plane
ff 
Vanishing
points @ ycm
y
b
x
All f-dependence of S due to x-K correlations
M.A. Lisa
Ohio State
March 2000
S01
 90
S12
0 ,90 ,180 
S13

S02
0 ,180 
S23
0 ,90 ,180 
S03
vanishes
S00 , S11 , S22 , S33

6
Neglecting implicit f-dependence
Implicit f-dependence weak compared to explicit f-dependence
Note: usua
is “boring”
something
until now!
• trivially true at K  0
• Negligible if x-K correlations (due to transverse flow, opaqueness)
small compared to thermal smearing
• no restriction on longitudinal flow
Wiedemann
• may be checked via consistency relations 
Heiselberg
L/H/W
Contrast with KT dependence from azimuthally symmetric case
• implicit KT dep. need not vanish @ KT=0
• explicit KT-dependence weighted with (generally small) t
• explicit KT-dependence vanishes at K  0
S(f) = S “Same source viewed from all angles f”
M.A. Lisa
Ohio State
March 2000
7
Neglecting implicit f-dependence, and focussing on ycm
5 nonvanishing components S encode all spacetime geometry driving R2ij(f)
S: lengths of homogeneity

13
S13: correlation between x & z  Spatial tilt angle qs  1 tan 1
2
S33 S11
2S

R s2  12 S11  S22   12 S22  S11 cos 2f
R o2  12 S11  S22   12 S22  S11 cos 2f  2S00
R l2  S33  2LS00
2
R os
 12 (S22  S11 ) sin 2f
2
R ol
 S13 cosf
R sl2
 S13 sin f
M.A. Lisa
Ohio State
y
x
qs
z
(Beam)
Reaction plane
• No radii vanish, even at ycm and pT=0
• first harmonic oscillations @ ycm!!!
• natural consistency
relations to check influence of S(f)8
March 2000
Simple case - ellipsoidal source w/ no flow
2
2
2 
 x '2
y
z
'
t

S( x , K )  e
exp 



2
2
2
2
 2 x 
2 y
2 z 
2 t 

 x   4 fm  t  5 fm/c
E / T
x  x cos  z sin 
z  x sin   z cos
  25
 y  5 fm
z  7 fm T  20 MeV
Using CRAB, generate c.f.’s
 
in 8 bins of f  (K, b)
z
T
S13  12.1  0.6 fm 2 R qs   S  R qs  
x
M.A. Lisa
Ohio State
 26.5  8.3 0.4  1.2  0.4  1.4 1.7  1.5 


 0.4  1.2 16.1  0.7  0.3  0.5
0  0.7 


  0.4  1.4  0.3  0.5 24.9  0.4  0.2  0.5 


March 2000
9
 1.7  1.5
0  0.7
 0.2  0.5 49.6  1.1 

qs  23.5o  1.5o
A series of tilted ellipsoids in relative momentum measure
a single tilted ellipsoid in coordinate space!
emission
angle
z
2D projections of
correlation functions
x
coordinate space
M.A. Lisa
Ohio State
March 2000
10
-80 0 -80 0 -80 0
Adding longitudinal flow
Same model but with additional
boost z   tanh f  z 
(z  z )  v th 
Still “same source(s) viewed
from different f
S13  7.0  0.2 fm 2
qs  33.1o  1.6o
R T qs   S  R qs  
 26.6  4.5  0.8  0.7 0.2  0.7 0.7  0.7 

 z-flow
  0.8  0.7 15.9  0.3  0.1  0.2
0  0.3  effects


 0.2  0.7  0.1  0.2 24.9  0.4 0.3  0.2 


 0.7  0.7
0  0.3
0.3  0.2 31.1  0.4 

M.A. Lisa
Ohio State
March 2000
|pz|<40 MeV/c
11
RQMD@AGS: “Single-pion tilts” (effective source tilts bigger)
4 AGeV
2 AGeV
• big!
• positive!
• not overly sensitive to b
• sensitive to dynamics
• falling with Ebeam
0-2 fm
4-6 fm
8-10 fm
M.A. Lisa
Ohio State
March 2000
12
Opposing average tilts in p, x
and the physics of  flow
RQMD w/ meanfield: Au(2GeV)Au, b=3-7 fm
transport model exhibits weak 
“antiflow,” similar to observation
(Kintner et al. PRL 78 4165, 1997)
• coordinate-space tilt in positive direction
• Antiflow reflects dense low-|z| region
•  from dilute large-|z| show positive flow
Bass et al [PLB 302 381 (93), PRC 51 3343 (95)]:
pion anisotropies ~ 2 AGeV due to
reflection (ND N)
not absorption (NNDN NN)
experimental handle through tilt?
M.A. Lisa
Ohio State
March 2000
13
RQMD w/ meanfield: Au(2GeV)Au, b=2-8 fm, pT<300 MeV/c
long
out
side
positive x-z correlation
negative tilt?
ol
os
sl
S13  3.3  0.4 fm 2 qs  25.5o  3.0o
R T qs   S  R qs  
 27.3  2.8  0.9  0.7  0.5  0.8  0.3  0.8 


  0.9  0.7 20.1  0.5 0.5  0.4
0  0.5 


  0.5  0.8 0.5  0.4 22.4  0.6  0.1  0.4 


  0.3  0.8
0  0.5
 0.1  0.4 11.6  0.7 

M.A. Lisa
March 2000
Ohio State
Clue: oblate source S11>S33
In terms of prolate source:
qs  90o  qs  65o
14
Prolate/oblate sources and large/small tilts
q
z
q z
z
x
x
z
x
Viewed as prolate source

qs  12 tan 1 S
M.A. Lisa
Ohio State
Viewed as oblate source
x
z
q
z >
x
q>0
2S13
33 S11
x
x
z
q + 90o

z <
x
q<0
selects solution with |q| < 45o
March 2000
15
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
RMS y  6 fm
 y  5 fm
Radii reflect Gaussian curvature
near origin in Drout, Drside, Drlong
S22  ( 4.7 fm) 2
Dy / 2
For most sources,
Sx, K  

NK   Sx, K   exp 
 
and S  B1 
M.A. Lisa
Ohio State


~
1~
x
B
x

2

RMSt  10.7 fm/c
 t  5.8 fm/c
S00  (5.2 fm/c)2
March 2000
Dt / 2
16
…but source is tilted in this system + has longitudinal flow effects.
Dz/2
Dz/2
fit to pair distribution in Dz-Dx with tilted ellipsoid
no qz
cut
 x  3.6 fm
M.A. Lisa
Ohio State
Dx / 2
|qz|<25 MeV cut Dx / 2
S11  (4.5 fm) 2
 x  3.4 fm
 z  5.6 fm
S33  (3.4 fm) 2
 z  4.7 fm
q  36
90  qs  65
q  53o
o
o
March 2000
o
17
2 AGeV
Preliminary E895 DATA
S13  5.8  0.9 fm 2
R T qs   S  R qs  
qs  40 o  5o
 12.1  6.9  0.9  1.8 0.2  1.6  2.5  2.0 


  0.9  1.8 29.9  1.2  0.5  0.8
0  1.2 


 0.2  1.6  0.5  0.8 33.1  1.6  0.7  0.9 


  2.5  2.0
0  1.2
 0.7  0.9 18.2  1.2 

90o  qs  50o  5o
4 AGeV
S13  4.0  0.6 fm 2
R T qs   S  R qs  
M.A. Lisa
Ohio State
qs  37 o  4o
0.5  1.0
1.9  1.2 
11.1  3.7 0.3  1.1


 0.3  1.1 15.0  0.8  0.3  0.6
0  0.8 


 0.5  1.0  0.3  0.6 22.9  1.0 0.4  0.6 



0  0.8
0.4  0.6 18
23.3  0.8 
March 2000 1.9  1.2
2 GeV cut 22
M.A. Lisa
Ohio
State
-0.8
0 -0.8 0 -0.8 0
4 GeV cut 20
March
20000
-0.8 0 -0.8
0 -0.8
No-flow toy model
-80 0 -80 0 -80 0
19
Rough idea of extracted shapes of average effective source
“to scale”
2 AGeV
z
y
t = 3.5 fm/c
x’ = 4.2 fm
y = 5.8 fm
z’ = 5.5 fm
qs = 50o
x
4 AGeV
z
x’
y
t = 3.3 fm/c
x’ = 3.9 fm
y = 4.8 fm
z’ = 4.8 fm
qs = 37o
M.A. Lisa
Ohio State
x
March 2000
20
x’
Summary
• R2ijf  spatial correlation tensor S, encodes source information in natural frame
• implicit dependence Sf due to transverse x-p correlations (flow, opaqueness…)
• complicate relation between R2 and S: higher harmonics to both
• presence of significant implicit dependence checkable via consistency relations
• explicit f-dependence (due to spacetime geometry) dominates implicit @ low pT
• Sf  S: 5 nonvanishing components @ ycm encode:
• 4 spacetime homogeneity lengths
• S22-S11 (transverse elliptic shape)  2nd-order oscillations in Ro2, Rs2, Ros2
• spatial tilt of source in reaction plane - “the last static component”
• S13  striking 1st-order oscillations in Rol2, Rsl2
• Tilt @ AGS
• large and positive
• measurable (and measured!!)
• experimental handle on physics of flow dynamics (rescatt., meanfield)
M.A. Lisa
Ohio State
March 2000
21
Tilts at RHIC? Theoretically, and experimentally, “challenging”
But try!
• RQMD evolves too quickly w/ E
• Never know what else...
RQMD@RHIC b=3-8 fm
c/o N. Xu, R. Snellings
@ SPS:
Rside ~ 30% too big
I.G. Bearden et al (NA44)
PRC58, 1656 (1998)
M.A. Lisa
Ohio QM99
State
E895,
March 2000
22
y
x
y
z
(Beam)
Reaction plane
M.A. Lisa
Ohio State
March 2000
23
Vanishing
points
Constraints on S(f)
General symmetry: Mirror
reflection in reaction plane
 
S(x, t, K)  S(x, y, z, t, K x ,K y , Kz )
f  f
General symmetry: Point
reflection about origin:

 

S( x, t, K)  S( x, t,K)
Symmetry @ ycm (Kz=0): Point
reflection about 2D origin:
 
S(x, t, K)  S(x, y, z, t,K x ,K y )
ff 
f   Dn cosnf  Dn sin nf
n
M.A. Lisa
Ohio State
S01
S12
S13
S02
S23
S03
 90
0 ,180 
 90

S22
S33
~
~
~
x t f   ~
x t  f 
Dn  0
~
~
~
x t f    ~
x t f    D n  0 n even
~
x~y f    ~
x~y  f 
Dn  0
~
x~y f   ~
x~y f    Dn  0 n odd
~
x~z f   ~
x~z  f 
D  0
0 ,180 
~
x~z f   ~
x~z f   
~y~t f    ~y~t  f 
0 ,180 
~y~t f    ~y~t f   
~y~z f    ~y~z  f 
 90

vanishes
S00
S11
Harmonic
coefficients

March 2000
~y~z f   ~y~z f   
~z ~t f   ~z ~t  f 
n
D n  0 n odd
Dn  0
Dn  0 n even
Dn  0
Dn  0 n odd
Dn  0
 0 @ midrapidit y
~
a 2 f   ~
a 2  f 
Dn  0
~
a 2 f   ~
a 2 f   
D n  0 n odd
24
Prolate/oblate sources and large/small tilts
M.A. Lisa
Ohio State
March 2000
25
First Harmonics at Midrapidity?? - Ain’t that Impossible?
Is “up” different than “down”?
M.A. Lisa
Ohio State
March 2000
26
Howd’ja get more f-bins?
1) a bit more statistics
2) tweaked reactionplane resolution (C. Pinkenburg) & HBT cuts
 
 
3) binning in f  (K, b) instead of f1,2  (p1, 2 , b)
all pairs wind up
in some f-bin
py
K
p1
p2
pairs straddling
f-boundaries
never used
px
f - boundaries
binning with f1, f2 also led to increased sensitivity
to oscillations in Rside as compared to Rout
M.A. Lisa
Ohio State
March 2000
27