Transcript スライド 1

Longitudinal density correlations
in Au+Au collisions at √sNN = 200 GeV
Tomoaki Nakamura
(KEK - High Energy Accelerator Research Organization)
for the PHENIX collaboration
6/25/2007
Tomoaki Nakamura - KEK
1
Pressure [Atm]
Phase diagram of He4
Tricritical point
Normal
liquid phase
Solid phase
Super fluidity
phase
Temperature [K]
[J. H. Vignos and H. A. Fairbank, Phys. Rev. Lett. 6, 265 (1961)]
6/25/2007
Tomoaki Nakamura - KEK
2
Phase transition to the Super fluidity
Phase diagram He4
Cs [J/gK]
Pressure [Atm]
critical phase
boundary
Specific heat Cs
[K]
Temperature [K]
This behavior around Tc
can be a robust signature
to indicate the phase
boundary.
6/25/2007
[mK]
|T-Tc|
[μK]
[W. M. Fairbank and M. J. Buckingam,
Int. Conf. on Low Temp. Phys. (1957)]
Tomoaki Nakamura - KEK
3
Second order phase transition
• The second order phase transition have a
close relationship on the global symmetry of
a system, rely on the universality hypothesis.
Imaginary QCD phase diagram
• If QGP phase transition corresponds to the
chiral phase transition, it would be the
second order phase transition.
• Surveying the second order derivative of
free energy i.e. Susceptibility,
  2G 
  
      2  ,
 h T
 h T
is a standard tactic to identify the second
order phase boundary, like the specific heat
in He4,
  2G 
Ch  T  2  .
 T  h
6/25/2007
K. Rajagopal,
Acta. Phys. Polon. B, 3021 (2000)
However, there is no reliable
and quantitative indication on
phase boundary in both
theoretically and experimentally.
Tomoaki Nakamura - KEK
4
Ginzburg and Landau theory
Concerning Ginzburg-Landau theory, free energy density f can be
expressed as this expansion,
1
1
1 4
2
2
f (T ,  , h)  f 0 (T )  A(T )(  )  a(T )  b      h
2
2
4
spatial fluctuation
φ: scalar order parameter, h: external field
f-f0
a>0
a=0
a<0
φ
One dimensional density fluctuation from the mean density is introduced
as an order parameter.
 ( y)   ( y)  
a(T )  a0 (T  Tc )
6/25/2007
Tomoaki Nakamura - KEK
5
Density correlation among
hydrodynamical sub elements
• Density fluctuation can be
measured in the final state
particle density by
introducing proper time
frame.
time
hadrons
(1)
hadron phase
(2) (3)
τc
• Differential length dz
among hydrodynamical sub
elements (1), (2), (3)…, at a
common proper time τ can
be expressed as
QGP
phase
τｆ
dz    cosh( y )dy.
• In the case of limiting the
space region of interest only for
the mid-rapidity, cosh(y) ~
1, then,
nucleus
nucleus
dz ~   dy.
6/25/2007
Tomoaki Nakamura - KEK
6
The case in heavy-ion collision
χ, ξ
The susceptibility can be derived by
introducing correlation length ξ as
1
  h 

  
  k 
1

a0 | T  Tc | (1  k 2 (T ) 2 )
 2 f
 k   2
 h
hadron phase
In the long wave length limit, k = 0,
1
 k 0 
  (T )G2 (0).
a0 | T  Tc |
QGP phase
no phase transition
critical phase boundary
6/25/2007
Tomoaki Nakamura - KEK
T, ε
7
Correlation length
The correlation length is useful not only for searching the phase
transition but also for the visualization of phase order.
[H. Nishimura, 2D Ising model (2005)]
a) T < Tc
Ordered phase
→ indicate long
correlation length
6/25/2007
b) T = Tc
c) T > Tc
At critical temperature
Disordered phase
→ coexistence of various
→ indicate short
correlation length
correlation length
→ diverged measured value
Tomoaki Nakamura - KEK
8
Charged track reconstruction in PHENIX
• We use straight line tracks measured at
no magnetic field condition to optimize
low momentum charged particles.
• Minimum pT threshold.
π: 0.1 GeV/c
 K : 0.25 GeV/c
 p : 0.35 GeV/c
• Particle composition.
 π : K : p = 94 : 4 : 2
• Acceptance: Δη < 0.7, Δφ < π/2
• Track identification: Using vector of
charged track (DC) associated with beam
vertex (BBC) and two hit points in wire
chamber (PC1, PC3). Clusters in EMC are
used for reference of association.
6/25/2007
• Mean pT for π = 0.57 GeV/c.
• For inclusive charged particle, maximum
3 % difference at η = 0.35 for the
conversion of rapidity to pseudo rapidity.
Tomoaki Nakamura - KEK
9
Spectator neutrons are counted
as the energy sun in ZDC.
Centrality and Npart and εBj
• Centrality class is determined
by forward detectors.
[PHENIX, arXiv:0704.2894 (nucl-ex)]
• Transverse total energy is measured
by central arm EM calorimeter.
[PHENIX, Phys. Rev. C71, 34908 (2005)]
Relative number of fragment particles
from participant nucleons are measured by
charge sum in BBC.
Number of participants
• Number of participant nucleons • Bjorken energy density is
calculated as
are obtained by the Glauber
1 dET
 Bj 
.
model calculation for each
cAT dy
centrality class.
6/25/2007
Tomoaki Nakamura - KEK
10
Multiplicity fluctuation and correlation
[PHENIX, arXiv:0704.2894 (nucl-ex)]
Negative Binomial distribution.
n
Pn( k )
(n  k )   / k 
1



(n  1)(k )  1   / k  1   / k k
δη=0.09
2 1 1
2
2


,



n



n

2  k
Second order factorial moment as a
representation of two particle correlation.
 n(n  1)   n 2    n 
F2 

2
n
 n 2
 2   n 2   n 
2 1

 1 2 
2
n
 
1
 1
k
6/25/2007
δη=0.7
• Uncorrected charged particle
multiplicity distribution in various
pseudo rapidity gap and NBD fits
for most central (10%) events in
Au+Au collisions at √sNN=200GeV.
• Accuracy of fits : 80% C.L.
Tomoaki Nakamura - KEK
11
Extraction of the correlation
Using Ornstein-Zernike formula, 1D two
particle correlation function,
[PHENIX, arXiv:0704.2894 (nucl-ex)]
C2 (1 , 2 )   2 (1 , 2 )  1 (1 ) 1 ( 2 ),
can be parameterized as
C2 (1 , 2 )
 / 


e
 .
2
1
α: correlation strength
ξ: correlation length
β: constant.
Using relation with NBD k:
k 1 ( )  F2  1
 
 

0

0
C2 (1 , 2 )d1d 2
 2 12
2 ( /   1  e
6/25/2007
2

2
 / 
)

k ( ) 
Tomoaki Nakamura - KEK
1
2 /   
(   )
99% C. L.
12
αξ, β vs. Npart
[PHENIX, arXiv:0704.2894 (nucl-ex)]
• β absorb effects on the finite
resolution of centrality binning i.e.
the fluctuation of Npart.
β
●5%
○10%
• αξ product, which is
monotonically related with χk=0
indicates the non-monotonic
behavior around Npart ~ 90.
  1
2
1
| 1  Tc / T |
αξ
1
 k 0 
| T  Tc |
●5%
○10%
• The difference of data points
between 5% and 10% binning can
be understood the smearing
5% binning
effects around the peak.
10% binning
6/25/2007
Tomoaki Nakamura - KEK
13
Other correlation sources
 Pseudorapidity independent correlations are all absorbed by the
constant term β. e.g. elliptic flow etc.
 Npart fluctuations (residual effect) are also absorbed owing to the β.
 Trivial particle correlations originating from charged track
reconstructions in tracking detectors have been suppressed a priory.
 Effects from weak decay particles (Λ, Ks) were estimated for the
NBD k by the MC calculation. It is less than 1% for each.
 Effects from photon conversion electrons is about 10-3%, which was
obtained by GEANT MC simulation.
 Effects from knock on electron in detector material is about 10-5%.
 Above contribution is negligible as compared to total error on k.
6/25/2007
Tomoaki Nakamura - KEK
14
Evaluation of the non-monotonicity
χ2/NDF = 2.76 : 0.60
[PHENIX, arXiv:0704.2894 (nucl-ex)]
5%
χ2/NDF = 2.10 : 1.23
5%
Power law
Linear
Power law
+ Gaussian
Linear
+ Gaussian
10%
10%
χ2/NDF = 1.23 : 0.79
χ2/NDF = 1.23 : 0.79
Power law + Gaussian:
Linear + Gaussian:
6/25/2007
3.98 σ (5%), 3.21 σ (10%)
1.24 σ (5%), 1.69 σ (10%)
Tomoaki Nakamura - KEK
15
On the relation with HBT effect
• If all correlations are originated
in HBT effect,
α corresponds to the
chaoticity parameter λ
ξ corresponds to the radius
parameter R
used in HBT analysis.
Au+Au √sNN=200GeV
• However, λ is constant as a
function of Npart, and R
monotonically increases with
increasing Npart.
• Therefore, known HBT effects
cannot explain the non-monotonic
behavior of αξ.
6/25/2007
One dimensional radius parameters.
[A. Enokizono, Ph. D. thesis,
Hiroshima Univ.]
Tomoaki Nakamura - KEK
16
Conclusion I
• The charged particle multiplicity distributions for the
various pseudorapidity gap, δη < 0.35, in Au+Au
collisions at √sNN = 200 GeV are found to be well
described by NBD as well as the other collision system.
• We found the constant β parameter is necessary to
avoid the residual effects in the measurement for the
extraction of correlations from the integrated
correlation function.
• Upper limit of correlation length over all centrality
bins is less than 0.035, which is obtained by the free
parameter fits.
6/25/2007
Tomoaki Nakamura - KEK
17
Conclusion II
• αξ product, which monotonically related to
susceptibility in the long wavelength limit, χk=0, show a
non-monotonic behavior as a function of the number of
participant nucleons, Npart.
• A possible indication of a local maximum or critical
behavior is seen at Npart ~ 90 and the corresponding
energy density is εBjτ ~ 2.4GeV/(fm2c).
6/25/2007
Tomoaki Nakamura - KEK
18
Backup
6/25/2007
Tomoaki Nakamura - KEK
19
Systematic check on αξ
Fit range:
0.066 < δη < 0.306
6/25/2007
Tomoaki Nakamura - KEK
20
Correlation length vs. temperature
Cube of the equilibrium
correlation length (thin
lines) and nonequilibrium correlation
length (thick lines) as a
function of temperature.
[C. Nonaka and M. Asakawa, Phys. Rev. C71, 44904 (2005)]
6/25/2007
Tomoaki Nakamura - KEK
21
Two point correlation function
G2 ( y1 , y2 )   ( y1 )    ( y2 )  
dy dy
 G ( y , y )e
   ( y )    ( y )   e

ik ( y 2  y1 )
2
1
2
1
1
2
ik ( y 2  y1 )
2
dy1dy2
y  y2  y1
Y  G2 ( y )e
 k
6/25/2007
iky
dy 
  ( y) 
 e
iky
2
dy
2
Tomoaki Nakamura - KEK
22
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 agree with NBD in
high energy collisions.
6/25/2007
Tomoaki Nakamura - KEK
23
NBD and NFM
Bose-Einstein distribution
Pn   /(1   )
n
n 1
Negative binomial distribution
(k )
n
P
n
(n  k )   / k 
1



k
(n  1)(k )  1   / k  1   / k 
2 1 1
2
2


,



n



n

2

 k
σ: standard deviation
NBD (k→∞) = Poisson distribution
NBD (k<0) = Binomial distribution
μ: average multiplicity
 n(n  1)   n 2    n 
F2 

2
2
n
n
 2   n 2   n 
2 1 1

 1 2   1
2
n
  k
6/25/2007
Tomoaki Nakamura - KEK
24
Integration of correlation function
R2 ( y1 , y2 )  ae| y1  y2 |/   b
F2


0
0


1 
(ae| y1  y2 |/   b)dy1dy2
 2
0  y1  
0  y2  
6/25/2007
Tomoaki Nakamura - KEK
25
Centrality determination
•
•
•
Event characterization in terms of impact
participants
go into BBC
parameter (b) in Au+Au collisions.
– Large : peripheral collision
– Small : central collision
Coincidence between BBC and ZDC.
go into ZDC
spectator
– Determine collision centrality.
– 93 % of inelastic cross section can be seen.
Extract variables using Glauber Model
– Number of participants (N_part).
• Number of nucleons participate in a
collision.
• Represents centrality.
• Related with soft physics.
15-20%
– Number of binary collisions (N_binary).
1015%5-10%
• Number of Nucleon-Nucleon collisions.
• Related with hard physics.
0-5%
• Incoherent sum of N-N collisions
becomes a baseline for A-A collisions.
0-5%
6/25/2007
Tomoaki Nakamura - KEK
26
Glauber model, BBC and ZDC
Dencity profile
1
 (r )   0 
rR
1  exp

 a 
R  6.38 fm, a  0.54 fm,  NN  42 mb
Glauber R. J., Phys. Rev. 100 242 (1955); in: Lectures
in the theoretical physics, ed. W. E. Brittin, L. G.
Dunham, Interscience, N. Y., 1959, v. 1, p. 315.
BBC
6/25/2007
Tomoaki Nakamura - KEK
27