Search for critical temperature by
measuring spatial correlation length
via multiplicity density fluctuations
Kensuke Homma / Hiroshima Univ.
from PHENIX collaboration
July 3, 2006 in Florence (Italy) at the Galileo Galilei Institute
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6.
Outline
What is the critical behavior ?
Order parameter and phase transition
Free energy coefficients and correlation function
How can we define initial temperature ?
PHENIX preliminary results
Summary
Kensuke Homma / Hiroshima Univ.
1
What is the critical behavior ?
Scale transformation
Ordered T=0.995Tc
Critical T=Tc
Disordered T=1.05Tc
Spatial pattern of
ordered state
Black
Black & White
Various sizes
from small to large
Gray
 Large fluctuations of correlation sizes on order parameters:
 critical temperature (focus of this talk)
 Universality (power law behavior) around Tc caused
by basic symmetries and dimensions of an underlying system:
 critical exponent
Kensuke Homma / Hiroshima Univ.
A simulation based on two dimensional Ising model
from ISBN4-563-02435-X C3342l
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Order parameter and phase transition
In Ginzburg-Landau theory with Ornstein-Zernike picture,
free energy density g is given as
1
1
1 4
2
2
g (T ,  , h)  g 0  A(T )( )  a(T )  b    h
2
2
4
spatial correlation
disappears
at Tc
g-g0
external field h causes deviation of free energy
from the equilibrium value g0. Accordingly an order
parameter  fluctuates spatially.
In the vicinity of Tc,  must vanish, hence
a(T )  a0 (T  Tc )
a>0
a=0
a<0
φ
b>0 for 2nd order
1-D spatial multiplicity density fluctuation from the mean density is
introduced as an order parameter in the following.
 ( )   ( )  
Kensuke Homma / Hiroshima Univ.
3
Two point correlation function
& Fourier transformation
Two point correlation
G2 ( y1 , y2 )   ( y1 ) ( y2 )
Fourier transformation
ik ( y 2  y1 )
G
(
y
,
y
)
e
dy1dy2
 2 1 2

  ( y ) 
1

 ( y ) 
2
 e ik ( y
2  y1 )
dy1dy2
Relative distance between two points
y  y2  y1
Y  G2 ( y ) e
iky
dy 
  ( y) 
 e
iky
Kensuke Homma / Hiroshima Univ.
2
dy
 k
2
4
Expectation value of |k|2
from free energy deviation
Fourier expression of order parameter
 ( y )   k eiky
k
1
1
g / Y   ( g  g 0 )dy   | k |2 (a(T )  A(T )k 2 )
Y
2 k
Statistical weight can be obtained from free energy
w( ( y ))  Ne  g / T
k
2
NT
1

iky 
  | k | w  k e dk 
2

Y
a
(
T
)

A
(
T
)
k
 k


2
Kensuke Homma / Hiroshima Univ.
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Function form of
two point correlation function
Fourier transformation
of two point correlation
of order parameter
| k |2  Y  G2 ( y )e  ik ( y ) dy
From g (up to 2nd order)
due to spatial fluctuation
NT
1
| k | 
Y a (T )  A(T )k 2
2
A function form of correlation function is obtained by
inverse Fourier transformation.
NT
| y| /  (T )
G2 ( y )  2
 (T )e
2Y A(T )
A(T )
2
 (T ) 
a0 (T  Tc )
Kensuke Homma / Hiroshima Univ.
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From two point correlation to two particle correlation
Two point correlation function in 1-D case at fixed T
G2 (| 1   2 |)  (  (1 )   )(  ( 2 )   )
T
A(T )
|1  2 |/  (T )
2

 (T )e
,  (T ) 
A(T )
a0 (T  Tc )
Two particle correlation function
1 d
1 d
1 ( ) 
,  2 (1 , 2 ) 
 inel d
 inel d1d 2
2
C2 (1 , 2 )   2 (1 , 2 )  1 (1 ) 1 ( 2 )
C2 (1 , 2 ) /   e
2
1
 / 

Rapidity independent
term is added.
Kensuke Homma / Hiroshima Univ.
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Number of participants, Np and Centrality
peripheral
central
Participant Np
b
To ZDC
To BBC
Spectator
15-20%
1015%5-10%
0-5%
0-5%
Multiplicity distribution
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Can Np be related with initial temperature?
Transverse energy ET
Np scan may be a fine scan on the initial temperature T, while collision
energy is a coarse scan (?). Tc should be rather investigated with fine scan.
Let’s suppose that Np can be a monotonic function of T.
Kensuke Homma / Hiroshima Univ.
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Probability (A.U.)
Multiplicity density measurements in PHENIX
Δη<0.7 integrated over Δφ<π/2
PHENIX: Au+Au √sNN=200GeV
PHENIX Preliminary
small 
large 
Zero magnetic field to
n/<n>
enhance low pt statistics
per collision event
Negative Binomial Distribution
can describe data very well.
Kensuke Homma / Hiroshima Univ.
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Relations between
N.B.D k and integrated correlation function
Negative Binomial Distribution
(Distribution from k Bose-Einstein emission sources)
n
Pn( k )
( n  k )   / k 
1
2 1 1



,
  ,   n 
k
2
(n  1)(k )  1   / k  1   / k 

 k
Integrated correlation function can be related with 1/k
k
1


0
0


() 

C2 (1 , 2 ) / 12 d1d 2
 2
2 2 ( /   1  e  /  )

2
Kensuke Homma / Hiroshima Univ.

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N.B.D. k vs. d
1
k ()

2 2 ( /   1  e  /  )

2

k()
PHENIX Preliminary
10 % centrality
bin width
Function can fit the data
remarkably well !

PHENIX Preliminary
5% centrality
bin width
Kensuke Homma / Hiroshima Univ.
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Correlation length 
Correlation length  and static susceptibility c
Divergence of correlation length is the
indication of a critical temperature.
A(T )
 (T ) 
a0 (T  Tc )
Divergence of susceptibility is the
indication of 2nd order phase transition.
c k 0
Au+Au √sNN=200GeV
T~Tc?
1
1


 G2 (0)
a 0 (T  Tc ) T
Np
PHENIX Preliminary
Au+Au √sNN=200GeV
c k=0 * T
k   ( g  g 0 ) 

ck 
 
2
h   k 
1

a 0 (T  Tc )(1  k 2 2 )
2
PHENIX Preliminary
c k 0T  12
Np
Kensuke Homma / Hiroshima Univ.
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What  parameter represents ?
 can absorb all rapidity
10% cent. bin width independent fluctuations
5% cent. bin width caused by;

PHENIX Preliminary
PHENIX Preliminary

Shift to smaller fluctuations
1. finite centrality bin width
(initial temperature fluctuations)
2. azimuthal correlations
(under investigation)
3. Whatever you want.

PHENIX Preliminary
Our parametrization
can produce stable results
on  and .
Np
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Summary
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2.
3.
Multiplicity density distribution in Au+Au collisions at
√SNN=200 GeV can be well approximated by N.B.D..
Two point correlation lengths have been extracted based
on the function form by relating pseudo rapidity density
fluctuations to the GL theory up to the second order term
in the free energy. The lengths as a function of Np
indicates non monotonic increase at Np~100.
The product of the static susceptibility and the
corresponding temperature shows no obvious
discontinuity within the large systematic errors at the
same Np where the correlation length is increased.
Kensuke Homma / Hiroshima Univ.
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