Transcript Can I show them as work in progress at RHIC&AGS Annual
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 Outline
1. What is the critical behavior ?
2. Order parameter and phase transition 3. Free energy coefficients and correlation function 4. How can we define initial temperature ?
5. PHENIX preliminary results 6. Summary Kensuke Homma / Hiroshima Univ.
1
What is the critical behavior ?
Ordered T=0.995T
c Critical T=T c Disordered T=1.05T
c 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 2
Order parameter and phase transition
In Ginzburg-Landau theory with Ornstein-Zernike picture, free energy density
g
is given as
g
(
T
, ,
h
) external field
h
g
0 1
A
(
T
)( ) 2 2
spatial correlation
1 2
a
(
T
) 2
disappears at Tc
causes deviation of free energy from the equilibrium value
g 0
. Accordingly an order parameter fluctuates spatially.
1 4
b
4
g-g 0
a>0
h
a=0 a<0
In the vicinity of Tc,
a
must vanish, hence (
T
)
a
0 (
T
T c
)
b>0 for 2 nd order
φ 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
G
2 (
y
1 ,
y
2
y
1 ) (
y
2 )
Fourier transformation
G
2 (
y
1 , (
y
2 )
e
ik
(
y
2
y
1 )
dy
1
dy
2
y
1 ) (
y
2 )
e
ik
(
y
2
y
1 )
dy
1
dy
2
Relative distance between two points
y
y
2
y
1
Y
G
2 (
y
)
e
iky dy
(
y
)
e
iky dy
2
k
2 Kensuke Homma / Hiroshima Univ.
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Expectation value of |
k | 2 from free energy deviation
Fourier expression of order parameter (
y
)
k
k e iky
g
/
Y
1
Y
(
g
g
0 )
dy
1 2
k
|
k
| 2 (
a
(
T
)
A
(
T
)
k
2 ) Statistical weight can be obtained from free energy
w
( (
y
))
Ne
g
/
T
k
2 |
k
| 2
w
k
k e iky
d
k
NT Y a
(
T
) 1
A
(
T
)
k
2 Kensuke Homma / Hiroshima Univ.
5
Function form of two point correlation function Fourier transformation of two point correlation of order parameter From
g (up to 2 nd order) due to spatial fluctuation
|
k
| 2 |
k
| 2
Y
G
2 (
y
)
e
ik
(
y
)
dy
NT Y a
(
T
) 1
A
(
T
)
k
2 A function form of correlation function is obtained by inverse Fourier transformation.
G
2 (
y
) 2
Y NT
2
A
(
T
) (
T
)
e
|
y
| / (
T
) (
T
) 2
a
0 (
A
(
T
)
T
T c
) Kensuke Homma / Hiroshima Univ.
6
From two point correlation to two particle correlation
Two point correlation function in 1-D case at fixed
T G
2 (| 1 2 |) ( ( 1 ) )( ( 2 ) )
T A
(
T
) (
T
)
e
| 1 2 | / (
T
) , (
T
) 2
a
0 (
A
(
T
)
T
T c
) Two particle correlation function 1
C
2 ( ) ( 1 , 2 1
inel
)
d d
2 , ( 1 , 2 2 ) ( 1 , 2 ) 1
inel
1 ( 1 ) 1 ( 2 )
d
2
d
1
d
2
C
2 ( 1 , 2 ) / 1 2
e
/
Rapidity independent term is added.
Kensuke Homma / Hiroshima Univ.
7
Number of participants, Np and Centrality peripheral central Participant Np To ZDC b 15-20% 10 To BBC Spectator 0-5% 0-5%
Multiplicity distribution 8
Can Np be related with initial temperature?
Transverse energy E T 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.
9
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 enhance low pt statistics per collision event
n/
Negative Binomial Distribution can describe data very well.
Kensuke Homma / Hiroshima Univ.
10
Relations between N.B.D k and integrated correlation function
Negative Binomial Distribution (Distribution from
k
Bose-Einstein emission sources)
P n
(
k
) (
n
(
n
k
) 1 ) (
k
) 1 /
k
/
k
n
1 1 /
k
k
, 2 2 1 1 ,
k
n
Integrated correlation function can be related with 1/k
k
1 ( ) 0 0
C
2 ( 1 , 2 2 ) / 1 2
d
1
d
2 2 2 ( / 1 2
e
/ ) Kensuke Homma / Hiroshima Univ.
11
N.B.D. k vs. d
1
k
( ) 2 2 ( / 1 2
e
/ )
PHENIX Preliminary 10 % centrality bin width Function can fit the data remarkably well !
5% centrality bin width
PHENIX Preliminary
Kensuke Homma / Hiroshima Univ.
12
Correlation length
and static susceptibility
c
Divergence of correlation length is the indication of a critical temperature.
(
T
)
a
0 (
A
(
T
)
T
T c
)
PHENIX Preliminary Au+Au √sNN=200GeV Divergence of susceptibility is the indication of 2 nd
c
k
h k
order phase transition.
2 (
g
2
k g
0 ) 1 c
k
0
a
0 (
T
1
T c
)( 1
k
2 2 )
a
0 (
T
1
T c
)
T G
2 ( 0 ) c
k
0
T
1 2
T~Tc?
PHENIX Preliminary Au+Au √sNN=200GeV
Kensuke Homma / Hiroshima Univ.
Np
13
Np
What
parameter represents ?
PHENIX Preliminary 10% 5% cent. bin width cent. bin width
can absorb all rapidity independent fluctuations caused by; PHENIX Preliminary Shift to smaller fluctuations
1. finite centrality bin width (initial temperature fluctuations) 2. azimuthal correlations (under investigation)
PHENIX Preliminary
3. Whatever you want.
Our parametrization can produce stable results on and .
Np
14
Summary 1.
2.
3.
Multiplicity density distribution in Au+Au collisions at √S NN =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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