A Short introduction to
HBT
Qinghui Zhang
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Q. H. Zhang
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Heavy Ion Collisions requires
understanding particle distributions
in coordinate and
momentum space
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
GGLP



HBT



Goldhaber, Goldhaber, Lee and Pais
(1960)
First application of “intensity
interferometry” in particle physics
Hanbury-Brown and Twiss (1950’s)
Revolutionary development of
intensity interferometry in
astronomy
B-E


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Bose-Einstein (1920’s)
Role of Bose statistics in
fluctuations
Q. H. Zhang
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1
X
Source
y

2
Neglects
Momentum dependence of source
 Quantum mechanics up to x and y
 Final State Interactions after x and y
C2(q) contains shape information

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 That is, squared Fourier Transforms
measure (only) relative separations of
Sphere vs
source coordinates
hemisphere
 Very different sources r(x)
can give very similar
distributions in
relative separation
qOUT R 
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C 2 (q )  1 | r (q ) | 2 , r (q )   r ( x )e iqx d 4 x



q  p1  p 2  ( E 1  E 2 , p1  p 2 )  (q 0 , q )
 
 q  x  (q 0 t  q  r )
 Fourier Transform provides one time
and three space extensions of source
 BUT:
 2  2
2
2
E1  E 2
p1  p 2
q0  E1  E 2 

E1  E 2
E1  E 2


p1  p 2


 
 ( p1  p 2 ) 
 q  V PAIR
E1  E 2

That is, q0 is not an independent
quantity in the F.T.

(e.g., note that q 0  | q | )
 Fourier Transform has support in only
half of the q0-q plane
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C 2 (q )  1 | r (q ) | 2 , r (q )   r ( x )e iqx d 4 x
 
q 0  q  V PAIR
 
 
  
 q  x  (q  V PAIR t  q  r )  q  ( r  V PAIR t )
  
 
 iq ( r V PAIRt )
 r (q )  r (q ;V PAIR )   r ( x )e
d4x

So Fourier transform (+ two-particle
kinematics) already provides conclusion:
 
 
C 2 (q )  1 | r (q ) |  C 2 (q ; V PAIR )  ( r  V PAIR t ) 2 
2

Additional dynamic effects (expansion,
thermal source, …) will also lead to
systematic dependence of extracted
“radii” on VPAIR (or mT or kT or …)
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So
ur
c
e
q
Y-Axis
P1
K
P2
qSIDE
Beam
Axis
ZAx
is
X-Axis
qLONG

Source
qOUT
There are other decompositions...
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1 1 x 2x 2 y 2y 2 z 2z 2 t 2t 2

r (rr(,rt ,)t~) ~exp[
exp[
  ( ( 2 2  2 2  2 2   2 )]2 )]
2 2RR
RR
RzRz  
x x
y y

1 1 2 2 2 2 2 2 2 2 2 2 2 2     2 22 2

r (rq(q; V; V
)

)

exp[
exp[


(q(xq xRR
q yq yRR
q zq zRzRz  
(q(q V V
) ) )] )]
PAIR
PAIR
x x 
y y 
PAIR
PAIR
22
Simplest possible (yet still very useful)
case:
e
ur
c
So
q
Choose frame so that

x = “Out” (that is, parallel to VPAIR)

y = “Side”

z = “Long”:
Y-Axis

P1
K
P2
qSIDE
Beam
Axis
ZAx
is
X-Axis
z

qLONG
y
x
Source
qOUT

2
2
2
2
2
2
(q x R x  q y R y  q z Rz  (q  V PAIR ) 2  2 )
 q SIDE R SIDE  q LONG R LONG  qOUT ROUT with ROUT  R x  V PAIR  2
2
Symmetry
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2
2
2
2
2
2
2
2
" " R y  R x  ROUT  R x  V PAIR  2  R SIDE  R y
2
Q. H. Zhang
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2
2
10
2

Most experiments with charged
tracking and particle identification
have HBT results:
BNL AGS:
E859, E866
E877
E895

CERN SPS:
NA44
NA49
WA98

RHIC
PHENIX
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STAR
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


The Great Puzzle:
Why so little variation with ECM ?
Why are lifetimes () ~ 0 ??
( especially at RHIC )
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systematic errors in experimental data is
required:



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Definition of C2
Coulomb corrections
Role of l, dependence of R on same
Contributions from resonances
(especially when comparing between
different experiments)

A better understanding of theoretical
assumptions and uncertainties is
required:
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Resonance contributions
Lorentz effects
Modeling of expansion, velocity profiles
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
Our favorite (charged) bosons never
have pure plane-wave states
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Even our ideal-case Fourier transform
becomes a “Coulomb transform”
This particular case is easy to treat
analytically via “Gamow” or “Coulomb”
corrections:
Q. H. Zhang
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Some indication from both AGS and
SPS that RSide(--) > RSide(++)


E877 (AGS): RSide(--) ~ (1.5 +/0.4)*RSide(++)  R (--) > R (--)
TSide
TOut
NA44 (SPS):  RTSide(++) < RTOut(++)
Ze
Note external Coulomb
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•Systematic in RSIDE
•Cancels in ROUT
-  
-
Q. H. Zhang
Ze
 15

r (q )   r ( x )e iqx d 4 x  r (q  0)  1


 C 2 ( q  0)  2
versus observed value of 1.1-1.5
This is parameterized away via
C2(q) = 1+ l | r(q)|2 , l ~0.1-0.5
“Understood” via assumed resonance
contributions:
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


How do resonances affect l ?
By creating a “halo” around the
source “core”:
Some culprits:
r   (G = 150 MeV  S ~
1.3 fm )
 K*  K (G =
50 MeV  S ~
4
fm )
 w   (G =
8.4 MeV  S ~
20
fm )
,
 h  h (G =
0.2 MeV  S ~
200
fm )
2(1-fCORE )fCORE (1-fCORE )(1-fCORE )
fCORE fCORE
 (and many more…)

Core
2
Effect: C2 develops
structure near
q=0 to describe
the various sources:
1
Momentum
resolution 
pHBT'04
p
Halo
Combination
C2(q)

0
Q. H. Zhang
0
 (~ 1%)  (~ 1%)  p [Ge V/c]
50
q (MeV/c)
100
17
Using Wigner function:
 See for example:
Chao, Gao and Zhang

Phys. Rev. C 49,3224 (1994)
Use ~this in
cascade codes
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Expect C2 to depend on q and K
B.R. Schlei and N. Xu:
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
RQMD vs. Data for HBT
Final state phase space
points xi are weighted
by
1 HBT'04
+ cos[(pa-pb)(xa-xb)]Q. H. Zhang
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
This approach is


Plausible
WRONG
Final state phase space
points xi are weighted
by
1 + cos[(pa-pb)(xa-xb)]
with

Advantage:
 Uses precisely what codes produce

Disadvantage
 Can lead to non-physical oscillations in C2




First noted empirically by Weiner et al and
Later matin et al.
Theoretical analysis by Q.H. Zhang et al.
Phys. Lett. B 407, 33 (1997)
 Quantitatively, is it significant for heavy ion
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Q. H. Zhang
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
Answer 1: No (Empirical)
Study these effects with a source that
explicitly parameterizes x-p
correlations:

Examples for R=5fm, P=200 MeV/c
500
Naive
400
300
Correct
Usual (used in MC's)
2
K
0
p (MeV/c)
100
S=0
C2(q)
200
1
-100
-200
x
-300
0
-400
0
100
q (MeV/c)
-500
-10
-8
-6
-4
-2
0
R (fm)
2
4
6
8
200
10
500
Naive
400
Correct
Usual (used in MC's)
300
2
0
p (MeV/c)
100
S=0.33
C2(q)
200
1
-100
-200
0
-300
0
100
q (MeV/c)
-400
200
-500
-6
-4
-2
0
R (fm)
2
4
6
8
10
Naive
500
400
Correct
Usual (used in MC's)
2
300
S=0.9
200
100
0
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-100
-200
-300
-400
C2(q)
-8
1
p (MeV/c)
-10
Q. H. Zhang
22
0
0
100
q (MeV/c)
200
Q. Can one remove these pathologies
from ~classical predictions?
A. Yes (Q.H. Zhang et al.):

Smear the phase space points (x,p) with
minimum uncertainty wave-packets:
with s ~ 1 fm
Pictorially:
500
5
400
4
300
3
200
2
100
1
0
p (MeV/c)

=
-100
0
-
-200
-
-300
-
-400
-
-500
-8
-6
-4

-2
0
R (fm)
2
4
6
8
-
10
-10
-8
-6
-4
-2
0
R (fm)
2
4
This can be done analytically with
parameterization
on previous slide:
x
500
400
300
200
100
0
p(MeV/c)
-10
-100
-200
-300
-400
-500
-10
-8
-6
-4
-2
R
HBT'04
0
(fm
2
4
6
8
10
)
Q. H. Zhang
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6
8
10

What are the Lorentz properties of
C2(q)?
d 6n
3
3
E1 E 2
d 6n
3
3
dp dp
dp1 dp 2
C 2 ( p1 , p 2 )  3 1 23 
 Lorentz Invariant
d n d n
d 3n
d 3n

E1
 E2
3
3
3
3
dp1 dp1
dp1
dp1


How to write our favorite practice Gaussian
in an explicitly Lorentz invariant way?
Answered in
F.B. Yano and S.E. Koonin, Phys. Lett. B78, 556 (1978).
1 r2
t2

r ( r , t ) ~ exp[ ( 2  2 )]
2 R

1  2 2
 
 
 r (q ; V PAIR )  exp{ [ | q | R  (q  V PAIR ) 2  2 ) ]}
2
so
 
 2 2
 
2
C 2 ( p1 , p 2 )  1 | r (q ; V PAIR ) |  1  exp{ | q | R  (q  V PAIR ) 2  2 }
i.e. , not explicitly Lorentz covariant

Fix by introducin g source four - velocity u   s (1, v s )

2
q  q  q0  | q |2 ,
 
q  u   s (q 0  q  v s )  q 0 (in source rest frame)
Then
CHBT'04
qH.
) 2 Zhang
R  (q  u) 2 [ R 2   2 ]}
2 ( p1 , p 2 )  1  exp{( q Q.
2
i.e. , explicitly Lorentz covariant
24


Can Lorentz effects “distort”
information?
Apply Yano-Koonin-Podgoretzsky
p1
result to another toy model:
VPAIR
uS
Beam
axis

p2
Apply this to ~1-d motion in “Out” direction:
Study argument of exponentia l
ARG  (q  q ) 2 R  (q  u) 2 [ R 2   2 ] for various cases :

 Source at rest v S  0 :

2
 ARG  | q | 2 [ R 2  V PAIR  2 ]
2
 ROUT  R 2  V PAIR  2

 Pair at rest V PAIR  0 :

*2
*2
 ARG  | q | 2  s [ R 2  v S  2 ]
2
2
 ROUT   s [ R 2  v S  2 ]


 Numerical example : | V PAIR |  1 , | v S |  0.7
2
*2
*2
 ROUT  (0.4) 2 [ R 2   2 ]
2

Nota Bene: This last case allows ROUT<RSIDE
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even for  ~ RSIDE Q. H. Zhang


The chief lesson from last 10-15
years of (theoretical) HBT work:
The “radii” are a (potentially)
complicated mixture of



Space-time distribution
Spatial flow gradients
Temperature gradients
 
This average   ( r  V PAIR t ) 2  is over

space, time, flow profile, temperatur e, .....

More correct terminology:
“radii”  “lengths of homogeneity”

In the presence of source dynamics,
“radii” depend on mean pair
(energy, momentum, transverse mass, …)
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
But :
ROUT/RSIDE
trend
completely
eliminates
one “hydro”
calculation
HBT'04
Soff, Bass, Dumitru,
Phys. Rev. Lett. 86, 3981 (2001)
Q. H. Zhang
27

Plotted versus KT, there is an amazing
consistency between data from AGS,
SPS, and RHIC, spanning a factor of 100
in CM energy(!):
RHIC
AGS
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SPS
Q. H. Zhang
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

There is now an extensive
experimental and theoretical
literature built on 20+ years of
“HBT” studies
We’re now ready to start doing
things right:

Experimentally




Theoretically



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Measure resonance contributions
Measure like and unlike particle effects
Understand systematic errors
Cascade codes for RHIC
Right parametrization
Coulomb systematics
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Multipion correlations
Each events has large number of
pions, therefore, multi-pion
correlation will affect
Two-pion interferometry
(1): If the pion correlation formula
is right?
(2): If it is not right, which kind of
formula should we use!
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Seems right!
Zhang, Phys. Rev. C58, 22 (1998)
 Proved that for a class model, the
present pion correlation function
formula is right.
However generally, the formula is
not right!
Zhang, Phys. Rev. C59,1646 (1999).

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31
The formula is :
C(p1,p2)=

A(p1,p2)[1+R(p1,p2)]
A(p1,p2) is a function of p1,p2.
R(p1,p2) can be written as the same
as above.
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Three-pions correlations
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Some of slides taken from
W. A. Zajc, R. Wilson.

Thank you!
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Q. H. Zhang
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