Transcript 高強度背景電磁場中の仮想的真空分極の研究（の構想

Study of order parameters through
fluctuation measurements by the
PHENIX detector at RHIC
Kensuke Homma
for the PHENIX collaboration
Hiroshima University
On Aug 11, 2005 at Kromeriz
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XXXV International Symposium
on Multiparticle Dynamics 2005
Motivations
K. Rajagopal and F. Wilczek, hep-ph/0011333
• RHIC experiments probed the state of strongly interacting
dense medium with many properties consistent with partonic
medium. What about the information on the phase transition?
• Is it the first order or second order transition?
• Are there interesting critical phenomena such as tricritical
2
point?
Landau’s treatment
for 2nd order phase transition
Valid in a limit where the fluctuation on order parameter  is negligible even at T~Tc
Gibb’s free energy
G  U  TS
G(T , ) / V  g (T , )  r (T ) 2  u(T ) 4   h
Since order parameter  should disappear at T=Tc, assume r (T )  a(T  Tc) a  0
Susceptibility
  G 
  
2 1



  (2r  12u0 )
 h h0 h  h 
 
g(T,)
T>Tc
T<Tc
1
1
 
,  
2a | T  Tc |
4a | T  Tc |
T>Tc 02=0

Specific heat
(T  Tc) 2
g  a
4u
2
T<Tc 02=a(T-Tc)/2u
T
CH 
V
T   2G 
 S 
     2 
V  T  h
 T  h
 and CH show divergence or discontinuity,
while T varies around Tc.
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Susceptibility and density fluctuations
G(T , ) / V  g (T , )  A( )2  r (T ) 2  u(T ) 4   h




ik r
ik r
h(r )  hk e
 (r )  0  k e
k
2
 (2 Ak 2  2r  12u0 ) 1
hk
1/ 2
1
  (T )


A
k  


   
T>Tc
2
2 2
a
|
T

Tc
|
2( Ak  a | T  Tc |) 1  k  


1/ 2
1
  (T )


A

   
k  

T<Tc
2
2 2
2( Ak  2a | T  Tc |) 1  k  
 2a | T  Tc | 


 
(

(
r
)




)(

(
r
'
)




)

k
T

(
r
 r ')
Fluctuation-dissipation theorem
B
Susceptibility
k 

With Fourier transformation  k   dVe
  
 k ( r  r ')
 
1
k 2   2
k T
)  B  k
V
 (r  r ' ) 
(k   k )( k    k
 


exp( | r  r ' | /  )
Ornstein-Zernike behavior ( (r )   )( (r ' )   )  k BT
 
| r  r '|
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Fluctuation measurements by PHENIX
• Multiplicity fluctuations (density
fluctuations ) as a function rapidity
gap size with as low pt particle as
possible.
 Correlation length  and singular
behavior in correlation function.
• Average pt fluctuations
(temperature fluctuations)
 Specific heat
See PRL. 93 (2004) 092301
• In this talk, I will focus on only
multiplicity fluctuation
measurements.
Geometricall acceptance
Dh  0.7
D  p
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Charged particle multiplicity distributions and
negative binomial distribution (NBD)
DELPHI: Z0 hadronic Decay at LEP
2,3,4-jets events
E802: 16O+Cu 16.4AGeV/c at AGS
most central events
[DELPHI collaboration] Z. Phys. C56 (1992) 63
[E802 collaboration] Phys. Rev. C52 (1995) 2663
Universally, hadron
multiplicity distributions are
well described by NBD. 6
Negative binomial distribution (NBD)
distribution
Pn    n /(1   ) n 1 Bose-Einstein
μ: average multiplicity
n
(k )
n
P
(n  k )   / k 
1



(n  1)(k )  1   / k  1   / k k

1 1
 
2

 k
NBD
2
   n2    n 2
1
2 1
 2   F2 ( )  1
k ( ) 

 n 2   n 
F2 ( ) 
 n 2
NBD correspond to multiple BoseEinstein distribution and the
parameter k corresponds to the
multiplicity of those Bose-Einstein
emission sources.
NBD can be Poisson distribution with
the infinite k value.
F2 : second order normalized factorial moment
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Charged particle multiplicity distributions
in different dh gap
PHENIX: Au+Au √sNN=200GeV
2.16 < φ < 3.73 [rad]
| Z | < 5cm
No magnetic field
Δη<0.7, Δφ<π/2
-0.35 < η < 0.35
The effect of dead areas have been corrected.
δη=
δη=
δη=
δη=
δη=
δη=
δη=
δη=
0.09
0.18
0.35
0.26
0.44
0.53
0.61
0.70
(1/8)
(2/8)
(3/8)
(4/8)
(5/8)
(6/8)
(7/8)
(8/8)
:
:
:
:
:
:
:
:
P(n)
P(n)
P(n)
P(n)
P(n)
P(n)
P(n)
P(n)
x
x
x
x
x
x
x
107
106
105
104
103
102
8 1
10
Relation between k and
integrated two particle correlation function
Normalized
correlation function
R2 ( y1 , y2 ) 
C2 ( y1 , y2 )
 2 ( y1 , y2 )

1
1 ( y1 ) 1 ( y2 ) 1 ( y1 ) 1 ( y2 )
1 ( y ) :
 2 ( y1 , y2 ) :
inclusive single particle density
inclusive two-particle density
C2 ( y1 , y2 ) : two-particle correlation function
Relation with NBD k
1
k (h )
 F2  1  K 2 


h
h
C2 ( y1 , y2 ) dy1dy2
1 ( y1 ) 1 ( y2 )dy1dy2
Candidates of function forms with two particle correlation length 
Most general form: many trials failed.
e | y1  y2 |/ 
R2  R0
| y1  y2 |
HBT type correlation in E802 : failed to describe data
R2  R0e | y1  y2 |/ 
Empirical two component model with R0=1.0
R2  e | y1  y2 |/   b
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E802 type function can not describe the data
Correlation function used in E802
NBD k vs. δη at E802
P. Carruthers and Isa Sarcevic,
Phys. Rev. Lett. 63 (1989) 1562
R2  R0e | y1  y2 |/ 
1
h / 2
k (h ) 
R0 [1  ( / h )(1  eh /  )]
Phys. Rev. C52 (1995) 2663
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Empirical two component fit
 dependent part +  independent part with R0 =1
R2  e
| y1  y2 |/ 
2 2 [h /   1  e h /  ] b
b :
 F2  1 

2
k (h )
h
2
1
PHENIX: Au+Au √sNN=200GeV, Δη<0.7, Δφ<π/2
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Participants dependence of ξ and b
PHENIX: Au+Au √sNN=200GeV
Two particle correlation length
Correlation strength of what?
12
What is the origin of the two components?
Go back to Ornstein-Zernike’s theory (see Introduction to
Phase Transitions and Critical Phenomena by H.E.Stanley)
which explains the growth of forward scattering amplitude of light
interacting with targets at the phase transition temperature.
N
Density of fluid element at r
n(r )    (r  ri ),  n(r )  N / V  n
i 1
C2 (r, r' )  (n(r)  n(r) )(n(r' )  n(r' ) )



r
(r-r’)
N
N
i 1
j 1
C2 (r  r ' )    (r  ri )  (r  rj )  n 2


r’
C2 (r  r ' )  n (r  r ' )  n2(r  r ' )
R2 (r  r ' )   (r  r ' ) / n  (r  r ' )

b
Self interaction
renormalizing
singular part ?
 exp( | r  r ' | /  )
Long range correlation
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ξ vs. number of participants
PHENIX: Au+Au √sNN=200GeV
In the case of thermalized ideal gas,
Two particle correlation length
  dN / dy  N part  T 3
  N part  | T  Tc |3
log( )   log(N part )
One slope fit gives
α = -0.72 ± 0.03
Linear behavior of the correlation length as a function of the number of
participants has been obtained in the logarithmic scale.
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Conclusions
• Multiplicity distributions measured in Au+Au collisions at
√SNN=200GeV can be described by the negative binomial
distributions.
• Two particle correlation length has been measured based on
the empirical two component model from the multiplicity
fluctuations, which can fit k vs. dh in all centralities remarkably
well.
• Extracted correlation length behaves linearly as a function of
number of participants in logarithmic scales. Assuming one
slope component, the exponent was obtained as -0.72±0.03.
• The interpretation of b parameter is still ambiguous. Any
criticize or different view points are more than welcome.
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Backup Slide
16
Uncorrected Npart*b vs. Npart
Centrality k-Map in 10% bins
450
350
Bias on NBD k
due to finite bin size of centrality
300
"0-10"
250
"10-20"
200
"20-30"
150
"30-40"
100
50
0
50
100
200
500
1000
infinity
intrinsic k
Intrinsic k
Npart*b
Observed
observed k k
400
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Npart
Important HI jargon : Participants (Centrality)
peripheral
Participant
b
central
To ZDC
To BBC
Spectator
Whether AA is a trivial sum of NN
or
something nontrivial ?
Relate them to Npart and Nbinary (Ncoll )
using Glauber model.
–
Straight-line nucleon trajectories
–
Constant NN=(40 ± 5)mb.
–
Woods-Saxon nuclear density:
15-20%
1015%5-10%
0-5%
0-5%
1
 (r )   o  distribution Nch
Multiplicity
rR
1  exp

d


R  1.19A1/ 3  1.61A1/ 3
 (6.65  0.03) fm
d  (0.54  0.01) fm
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Why not observing fluctuations ?
The Microwave Sky image from the WMAP Mission
http://map.gsfc.nasa.gov/m_mm.html
• Fluctuation carries
information in early
universe in cosmology
despite of the only single
Big-Bang event.
• Why don’t we use the
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.
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