Download Non-equilibrium chiral dynamics by the time

Nonequilibrium chiral dynamics and two-particle
correlations in the time-dependent variational approach
with squeezed states
N. Ikezi (Osaka University)
M. Asakawa (Osaka University)
Y.Tsue (Kochi University)
Contents
1. Introduction
Two-particle correlation function in various state
Disoriented Chiral Condensate (DCC)
2. Method
Time Dependent Variational Approach with squeezed states
O(4) linear sigma model
Initial condition
3. Numerical Result
Domain formation of DCC
Spatial correlation
Growth of quantum fluctuation
Off-diagonal components of Green's function
Two particle correlation function
4. Summary
Two particle correlation function
P2 (k1 , k2 )
C (k1 , k 2 ) 
P1 (k1 ) P1 (k2 )
P2 ( k1 , k 2 ) : two particle distribution, P1 ( k1 ) : single particle distribution
Chaotic source for all kinds
C ( K , q)  1 
d
d
4
4
xS( x, K 2)e
iq x
2
x1S ( x1 , k1 )  d x2 S ( x2 , k 2 )
4
K  k1  k 2  2 , q  k1  k 2
S ( x, k ) : single particle Wigner density of the source
k1 ( k )
k2 (  l )
Hanbury Brown and Twiss effect uses this fact to find source function
from a measurement of two particle correlation function.
Two particle correlation function
Two particle correlation in various states
C2particle(k , l ) 
State
Coherent
Squeezed
Thermal
a†k a†l ak al
†
k
†
l
a ak a al
 1  k ,l
λ
0
-1 < λ< ∞
δkl
In the squeezed states, two particle correlation(TPC) varies widely,
and its off-diagonal components are finite.
→ We study TPC in case of DCC formation by using squeezed states.
Formation of Disoriented Chiral Condensate(DCC)
Relativistic heavy ion collisions
T  Tc
T 0
Sigma model

V  ,  

V  ,  

V  ,  

Chiral
restoration





Disoriented
Chiral
Codensate? 


Classical approximation
Essentially nonequilibrium process ○ Quench × Annealing
Amplification of longwalength modes
Long range correlation
(Formation of DCC domain)
1  2
3
Quantum treatments
Homogeneous (Translationally invariant)
× Long range correlation
Inhomogeneous mean fields
G(k1 , k2 , t )
? Thermalization

~
G (k , t )
translationally invariance
→Green’s function is diagonal in momentum space ! (homogeneous)
The DCC domain formation is not yet observed
We study the dynamics of chiral phase transition in spatially
inhomogeneous systems with mode-mode correlation in the
framework of Time Dependent Variational Approach (TDVA)
with squeezed states.
What will be happen to two particle correlation function?
Time-dependent variational approach with
Y.Tsue,Y.Fujiwara Prog.Theor.Phys. 86 (1991) 469
squeezed states
R.Jackiw, A.K.Kerman Phys.Lett.71A (1979) 158
Squeezed State has large & closed functional space
Mode-mode coupling
Mean Field
C( x, t )  ( t ) φ( x) ( t )
Fluctuation and correlation
G( x, y, t )  (t ) φ( x )φ( y ) (t )  C ( x, t )C ( y, t )
B  B*  0 
 coherent state
C(x,t) : Mean field (condensate)
D(x,t) : Conjugate variable for C(x,t)
G(x,x’,t) : Quantum fluctuation and correlation
Π(x,x’,t) : Conjugate variable for G(x,x’,t)
Time-dependent Variational Principle
O(4) linear sigma model
Hamiltonian




2 22

1  2 1
H   d x  a ( x )  a ( x )a ( x )   a ( x )  v  h0 ( x )
2
2

a  0, 1, 2, 3　  , 1 ,  2 ,  3 
v 2  m 2 4
3
m2
4
Variables
Mean fields(chiral order parameter)


Ca( x , t )  ( t ) a( x ) ( t )
Quantum fluctuations and correlations


 




G aa ( x , y, t )  (t ) a ( x )a ( y ) (t )  Ca ( x , t )Ca ( y, t )
Parameters
m  500 [MeV ]
m  138 [MeV ]
f  93 [MeV ]

Quench initial condition
◇ Mean fields(chiral order parameters)
Randomly distributed according to the Gaussian form with the following
parameters

 Ca  0
 2
C  Ca

 a
2
2
 Da  0


2
 Da  Da

…..
2

d 2

2
l
Ci Ci+1
  0.19v
◇ Quantum fluctuation and correlation
We assume that the sigma and pion fields are in a coherent state
with the degenerate mass m 0 .
(Diagonal in momentum space)
d 3k 1 ik ( x  y )
Gaa ( x, y,0)  
e
(2 )3 2 k
 aa ( x, y,0)  0
Gab  0,  ab  0

k
 k 2  m02 ; m0  200MeV

Numerical results
To understand the role of the mode-mode correlations, we performed
numerical calculations for the following two cases:
Mode-mode
correlation
Case A
○
Case B
×
Setup
N  64
l  1 fm
(Cut off  1071 MeV)
Dimension
3+1 → 1+1
Time evolution of π3 field
■Case A
■Case B
M-M
Case
A
○
Case
B
×
Space
Time
DCC domain formation continues beyond the time scale
of roling down
Mode-mode correlation plays a key element of DCC
domain formation
M-M
Spatial correlation
    
3
d xC ( x  r , t )  C ( x, t )


Ccorr (r , t )  3   
 
 d x | C( x  r , t ) || C( x, t ) |
■Case A



C  C1 , C2 , C3 
Case
A
○
Case
B
×
■Case B
10 event
10 event
Correlation does not grow without mode-mode correlation
Growth of quantum flctuation
Ga (t )   d 3 x Gaa ( x , x , t ) / V
Case A
Case B
Squeezing!
Coherent state
Amplification of quantum fluctuation by including mode-mode correlation
Duration of amplification of quantum fluctuation is of the same order as
that of the domain formation
Off-diagonal components of Green’s function
Coherent state
t=0 fm
[MeV -4 ]
k'
[MeV -4 ]
t=10 fm
G3 (k , k , t )
k
[MeV ]
[MeV ]
[MeV ]
[MeV ]
[MeV -4 ]
t=40 fm
[MeV ]
[MeV -4 ]
t=60 fm
[MeV ]
[MeV ]
[MeV ]
Off-diagonal component(mode-mode correlation) appears
Lambda parameter in Case A
C2particle(k , k ' ) 
Coherent state
t=10 fm
t=0 fm
a†k a†k ' ak ak '
†
k
†
k' k'
a ak a a
 1  k ,k '
k ,k 
k'
k
[MeV ]
[MeV ]
[MeV ]
[MeV ]
t=40 fm
t=60 fm
[MeV ]
[MeV ]
!
[MeV ]
[MeV ]
domain  coherent
  0
Free pion :
Mπ=138MeV
Two particle correlation approaches thermal value
Lambda parameter in Case B
C2particle(k , k ' ) 
Coherent state
t=0 fm
t=10 fm
a†k a†k ' ak ak '
†
k
†
k' k'
a ak a a
 1  k ,k '
k ,k 
k'
k
t=20 fm
t=60 fm
Free pion :
Mπ=138MeV
Two particle corrlelation is small
Summary
Dynamics of chiral phase transition in spatially inhomogeneous
systems with mode-mode correlation was studied in the framework
of TDVA with squeezed states.
In quantum treatment, the mode-mode correlation plays an important
role in the formation of DCC.
Time evolutions of two particle correlation (TPC) were studied.
In the case with mode-mode correlation, TPC approaches thermal
value in spite of the existence of DCC domain and non-zero modemode correlation.
Future works:
▪ Realistic geometry in relativistic heavy ion collisions
▪ Extension to higher spatial dimensions 1+1 → 1+2, 1+3
▪ Including finite size effect into the analysis of TPC
Energy conservation
Time evolution of sigma field

C0 (t )   d x C0 ( x , t ) / V
3
Two particle correlation function in
squeezed state
C2 particle (k , l ) 

C2 particle 
a†k a†l ak al
†
k
†
l
a ak a al
 1  k , l
Quantum mechanics (single mode)
a† a† aa
aa aa
 1

4  1   
2
2
2
 2 2   cos(2   ) 1  2 
  2  
2
2
2 2
2
Time-dependent Variational Principle

ˆ ( t )  0
  dt (t ) i
H
t
Equations of motion
2
2
(1)
C
(
x
,
t
)

C
(
x
,
t
)

M
( x, t )  0
2
2
t
x

G ( x, y, t )  2 dxG ( x, x, t ) ( x, y, t )   ( x, x, t )G ( x, y, t )
t

1
 ( x, y, t )   dxG 1 ( x, x, t )G 1 ( x, y, t )  2 dx ( x, x, t ) ( x, y, t )
t
8
1 2
1 ( 2)
1




dx G ( x, x , t )G ( x , y, t )  M ( x, t ) ( x  y )
2 
2 x
2
1 2  d n
M [C ]  exp  G 2  n V [  C ]
 2 C  dC
(n)