1. Kensuke Homma / Hiroshima Univ. from PHENIX collaboration
Search for critical temperature by measuring spatial correlation length via multiplicity density fluctuations
Kensuke Homma / Hiroshima Univ.
from PHENIX collaboration
July 3, in Florence (Italy) at the Galileo Galilei Institute
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.

2. What is the critical behavior ?
Ordered T=0.995Tc
Critical T=Tc
Disordered T=1.05Tc
Spatial pattern of
ordered state
Scale transformation
Black
Black & White
Various sizes
from small to large
Gray
Spatial pattern of spin correlation. Black is the aligned state of spins.
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
A simulation based on two dimensional Ising model
from ISBN X C3342l
Kensuke Homma / Hiroshima Univ.

3. Order parameter and phase transition
In Ginzburg-Landau theory with Ornstein-Zernike picture,
free energy density g is given as φ
g-g0
spatial correlation
disappears
at Tc
external field h causes deviation of free energy
from the equilibrium value g0. Accordingly an order
parameter f fluctuates spatially.
a>0
a=0
a<0
Interpretation.
In the vicinity of Tc, f must vanish, hence
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.

4. Two point correlation function & Fourier transformation.
Relative distance between two points
Kensuke Homma / Hiroshima Univ.

5. Expectation value of |fk|2 from free energy deviation
Fourier expression of order parameter
Statistical weight can be obtained from free energy
Kensuke Homma / Hiroshima Univ.

6. Function form of two point correlation function
Fourier transformation
of two point correlation
of order parameter
From Dg (up to 2nd order)
due to spatial fluctuation
A function form of correlation function is obtained by inverse Fourier transformation.
Kensuke Homma / Hiroshima Univ.

7. From two point correlation to two particle correlation
Two point correlation function in 1-D case at fixed T
Two particle correlation function
Summarize
Rapidity independent
term is added.
Kensuke Homma / Hiroshima Univ.

8. Number of participants, Np and Centrality
peripheral
central
0-5%
15-20%
10-15%
5-10% b
To ZDC
To BBC
Spectator
Participant Np
Multiplicity distribution

9. 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.

10. Multiplicity density measurements in PHENIX
Δη<0.7 integrated over Δφ<π/2
PHENIX: Au+Au √sNN=200GeV
Probability (A.U.)
PHENIX Preliminary
small dh
large dh
Zero magnetic field to
enhance low pt statistics
per collision event
n/<n>
Negative Binomial Distribution
can describe data very well.
Kensuke Homma / Hiroshima Univ.

11. Relations between N.B.D k and integrated correlation function
Negative Binomial Distribution
(Distribution from k Bose-Einstein emission sources)
Integrated correlation function can be related with 1/k
Summarize
Kensuke Homma / Hiroshima Univ.

12. Kensuke Homma / Hiroshima Univ.
N.B.D. k vs. dh
PHENIX Preliminary
k(dh)
10 % centrality
bin width
Function can fit the data
remarkably well ! dh
PHENIX Preliminary
5% centrality
bin width
Kensuke Homma / Hiroshima Univ.

13. Correlation length x and static susceptibility c
Divergence of correlation length is the
indication of a critical temperature.
PHENIX Preliminary
Au+Au √sNN=200GeV
Correlation length x(h)
Divergence of susceptibility is the
indication of 2nd order phase transition.
T~Tc? Np
PHENIX Preliminary
Au+Au √sNN=200GeV
c k=0 * T Np
Kensuke Homma / Hiroshima Univ.

14. What b parameter represents ?x
PHENIX Preliminary
10% cent. bin width
5% cent. bin width
Shift to smaller fluctuations
can absorb all rapidity
independent fluctuations
caused by;
1. finite centrality bin width
(initial temperature fluctuations)
2. azimuthal correlations
(under investigation)
3. Whatever you want.
Our parametrization
can produce stable results
on a and x. Np

15. Kensuke Homma / Hiroshima Univ.
Summary
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.