1. Kensuke Homma / Hiroshima Univ.
ゆらぎのはなし
本間謙輔
松本大学にて 内容
原始時代の紆余曲折。
こんな簡単なことで臨界現象が議論できるなんて！
我田引水の解釈および現象論屋さんに考えて頂きたいこと。
Kensuke Homma / Hiroshima Univ.

2. Kensuke Homma / Hiroshima Univ.
Why fluctuation ?
Fluctuations carries information in early universe in cosmology despite of the only single Big-Bang event.
Why don’t we use the genuine event-by-event information by getting all phase space information to study evolution of dynamical system in Heavy Ion collisions ?
We can firmly search for interesting fluctuations with more than million times of mini Big-Bangs.
Measure maximum deviation size in homogeneous flux in our method
The Microwave Sky image from the WMAP Mission
Kensuke Homma / Hiroshima Univ.

3. Au+Au √sNN = 200GeV at PHENIX
Using magnetic field-off
Charged Track Drift chamber, Pad chamber1 with BBC vertex
Photon Cluster Electro-magnetic calorimeter
Cluster shower shape
Time of flight
association cuts by tracks
Precisely data quality assurance was necessary to reject detector effect!
Kensuke Homma / Hiroshima Univ.

4. Application of wavelet analysis
High resolution
Low resolution  η
extract this region
Kensuke Homma / Hiroshima Univ.

5. High energy cosmic ray experiment and PHENIX
Can DCC scenario explain these events or something else?
Charged track
Photon cluster
PHENIX 7.24 standard deviation
J. J. Lord and J. Iwai. Int. Conference on High Energy Physics, TX, 1992
○: Photon
+ : Charged Particle
Kensuke Homma / Hiroshima Univ.

6. Maximum differential balance distributions
Work in Progress
δBmax [arbitrary unit]
δBmax distribution (each centrality:10%) with base line fluctuation
black : binomial sample, 100 times larger statistics than real data obtained by hit map
red : data
明らかな離れ孤島は見つからず。
わざわざ大げさな探索しなくたって、
そもそも分布は、二項分布とは
明らかに異なる。
荷電πのみを使用して、熱・統計力学的に
分布を議論するほうが生産的。
Kensuke Homma / Hiroshima Univ.

7. Measuring Multiplicity Fluctuations with Negative Binomial Distributions
Multiplicity distributions in hadronic and nuclear collisions can be well described by the Negative Binomial Distribution.
UA5: sqrt(s)=546 GeV p-pbar, Phys. Rep. 154 (1987) 247.
E802: 14.6A GeV/c O+Cu, Phys. Rev. C52 (1995) 2663.
E802
62 GeV Au+Au
5-10% Central
25-30% Central
dN/Ntracks
dN/Ntracks
PHENIX Preliminary
PHENIX Preliminary
Central 62 GeV Au+Au
Ntracks
Ntracks
Kensuke Homma / Hiroshima Univ.

8. Current understanding of QCD phase diagram
QGP
Experiments see partonic degree of freedom in collected flows at RHIC
RHIC is accessible to
this transition line
2nd order
mq=0
tricritcal end-point
mq=0 mq
<qq>
No quantitative agreement
between theoretical predictions
end-point line
mq!=0
Surface of 1st
order transitions mB
Kensuke Homma / Hiroshima Univ.

9. PHENIX on going analysis
Isothermal compressibility
Heat capacity
Near side azimuthal correlation function
Breaking of v2 scaling
Disappearance of baryon anomaly
Correlation length and susceptibility in longitudinal space
Kensuke Homma / Hiroshima Univ.

10. 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 reflecting basic symmetries and dimensions of the underlying system:
critical exponent (future study after finding Tc)
A simulation based on two dimensional Ising model
from ISBN X C3342l
Kensuke Homma / Hiroshima Univ.

11. 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
Longitudinal multiplicity density fluctuation from the mean density is introduced as an order parameter in the following.
Kensuke Homma / Hiroshima Univ.

12. More exact free potential form
In narrow midrapidity region, cosh(y)~1 and y~h.
Although theorists want to define U(f) as a power term by assuming the system is just on Tc and/or on critical end-point, most of experimentally accessible phase spaces are relatively far from the critical temperature or end-point. It is more natural to use the polynomial expansion for the potential term.
Interpretation.
Kensuke Homma / Hiroshima Univ.

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

14. 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.

15. 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.
In 1-D case
Kensuke Homma / Hiroshima Univ.

16. 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
must be added, since T has a
finite range in experiments.
Kensuke Homma / Hiroshima Univ.

17. Multiplicity density measurements in PHENIX
Δη<0.7 integrated over Δφ<π/2
PHENIX:
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.
p pt down to
100MeV/c in B-OFF
200MeV/c in B-ON
Kensuke Homma / Hiroshima Univ.

18. Relations between N.B.D and integrated correlation function
Negative Binomial Distribution
k →∞ corresponds to Poisson distribution.
k = 1 corresponds to Bose-Einstein distribution.
Intuitively k is the number of Bose-Einstein emission sources.
Integrated correlation function can be related with 1/k
Summarize
x can be obtained from fit to k vs. dh
Kensuke Homma / Hiroshima Univ.

19. Intuitive summary of advocated observation
exponential damping of the number of
waves with a typical correlation length
GL applicable
GL not applicable
Absence of typical correlation
length causes fractal nature
(power law behavior)
Compare many systems by
changing resolutions and
find the increase of correlation
length in the very vicinity of Tc
but not at Tc !
Kensuke Homma / Hiroshima Univ.

20. How to define initial temperature?
Transverse energy ET
Np can be a fine scan on initial temperature T, while collision energy can be a coarse scan. Tc can be investigated with the fine scan.
It is a natural assumption that Np is a monotonic function of initial T.
Kensuke Homma / Hiroshima Univ.

21. 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
dNp depends on
centrality bin width

22. Kensuke Homma / Hiroshima Univ.
N.B.D. k vs. dh
PHENIX Preliminary
k(dh)
10 % centrality
bin width
One correlation length
assumption is reasonable. dh
PHENIX Preliminary
5% centrality
bin width
Kensuke Homma / Hiroshima Univ.

23. PHENIX preliminary results
can absorb all rapidity
independent fluctuations
or offset contributions like;
1. finite centrality bin width
( Np or initial temperature
fluctuations )
2. reaction plain rotations and
elliptic flows due to partial
sampling in azimuth by PHENIX
10% cent. bin width
5% cent. bin width a
PHENIX Preliminary
Shift to smaller fluctuations b
Divergence of correlation length can
be a signature of critical temperature.
PHENIX Preliminary x Np

24. 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.

25. What about relation with <qq>
NA50, Eur. Phys. J. C39 (2005):355
PHENIX dNch/dh
corresponding to Np~90
PRC 71, (2005)
PHENIX eBJ (t=1fm/c)
corresponding to Np~90
PRC 71, (2005)
Accidental coincidence ? Need more simultaneous observations!
Kensuke Homma / Hiroshima Univ.

26. 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. One correlation length assumption is enough accurate.
Absolute scale of correlation length is 0.001~0.01.
The static susceptibility as a function of Np indicate a non monotonic increase at Np~90. The corresponding Bjorken energy density is 2.4GeV/fm3 with t=1.0 fm and transverse area=60fm2
It is interesting to note the coincidence with the energy density at which J/y suppression begins at SPS. Need more simultaneous observations by independent methods in order to conclude whether the non monotonic behavior is related with critical temperature or not.
Kensuke Homma / Hiroshima Univ.

27. Kensuke Homma / Hiroshima Univ.
Questions
What can be an external field to the longitudinal density correlation?
Why the correlation length can be so small ? Is it possible to cause temperature like correlations by using particles each of which occupies 1/1000 of unit rapidity (dNch/dEta ~ 1000 at RHIC)? Is there another type of thermalizaion before hadron formation?
Why the non monotonic behavior of correlation length looks like a peak rather than a step?　Why is the behavior so sensitive to initial energy densities rather than the density at the critical temperature.
How rapid is the longitudinal expansion?
Are there plural formation times? Is there a time for fluctuations are embed before hadron formation time?
How fluctuations in very early stage survive in the final state, while the collision system is rapidly expanding in longitudinal space?
Kensuke Homma / Hiroshima Univ.

28. Kensuke Homma / Hiroshima Univ.
The answer might be Hawking-Unruh effect in background condensed color electric field
Hawing
Unruh
(QED)
(QCD)
Kensuke Homma / Hiroshima Univ.