Kadanoff-Baym Approach
to Thermalization of Quantum FIelds
Akihiro Nishiyama
Kyoto Sangyo University
Collaboration with
Yoshitaka Hatta
Feb. 9, 2013.
Relativistic Heavy Ion Collision at RHIC and LHC
\sqrt(sNN )=0.2 TeV
Au+Au (RHIC)
\sqrt(sNN )=2.76TeV
Pb+Pb (LHC)
Z=-t
0
=0
Z=t
Formation of Quark-Gluon Plasma (QGP)
Dynamics in an expanding system.
Proper time
Rapidity
Early Thermalization of gluons (0.6-1fm/c)!
(RHIC and LHC).
Comparable to formation time (0.2fm/c).
Semi-classical Boltzmann eq.: (2-3fm/c).
Classical statistical approaches:
Not Bose-Einstein distribution at late time.
Nonequilibrium Quantum Field Theoretical Approach
Application of Kadanoff-Baym equation.
Purpose of this talk
Introduction of time evolution equation for classical field and
Kadanoff-Baym equation for quantum fluctuation in O(N)
scalar model.
To show particle production and equilibration in Numerical
Analyses.
Comparison of quantum and classical dynamics in weakly and
strongly coupled regimes.
O(N) scalar model
In expanding metric
Action of scalar O(N) model
a=1,…,N
g=diag(1, -
2
, -1, …)
All order summation of coupling
by 1/N expansion.
Interaction term
Time Evolution Equation I
< >=Tr ([density matrix]…)
Equation of motion of classical field
a
Or effect of
fluctuations
Damping of classical field for an expanding system
Kadanoff-Baym equation
• Quantum evolution equation of 2-point Green’s
function (fluctuations).
statistical (distribution) and spectral functions
Boson
Memory integral
Self-energies: local
mass shift, nonlocal real
and imaginary part
Merit
• Field-Particle Conversion: Particle production from
classical field.
(Parametric resonance) +
X
• Collision of particles → Bose-Einstein distribution
binary
Finite decay width
• Off-shell effect: Memory effects and
finite spectral width
binary collisions (2-to-2) → Rapid Change of
distribution functions (Lindner and Muller 2006) +
3-to-1 → entropy production + chemical equilibrium.
Demerit
Numerical simulation needs much memory of computers.
Initial condition
Initial condition: Classical field with vacuum
quantum fluctuations (Color Glass Condensate ?)
Weak
Strong
m/
vacuum
We assume homogeneity in space
2+1 dimensions.
0=0
and 0.1
Case I (without collision term)
Reproduction of J. Berges, K. Boguslavski, S. Schlichting, hep-ph 1201.3582.
Evolution of classical field
m/σ0=0
/
φ~τ-1/3
Due to expansion
τ/τ0
0)
Parametric Resonance instability
Berges 2004.
Fluctuation
2(t)~
(t)+…
Flat
p
Curved
exp( g
2/3
)
=2
0
0
Case II (with Collision term)
Berges et. al. Classical Statistical approaches.
Our results (collaboration with Y. Hatta): Quantum collision term.
Y. Hatta and A. N., Phys. Rev. D 86 (2012) 076002.
=
Normal collision term.
x
x
x
x
x
+
x
+・・・
Source induced amplification.
X: Classical field
Summation of Next-to-Leading Order of 1/N expansion.
All covering of F from O(1) to O( 1/
) in evolution.
Classical Statistical Approximation
Gelis et al. (2011)
For example
Berges et al. (2012)
F=
=FF-(1/4)
Neq~T/p-1/2
F~(1/ )n_p, ~1/
n_p~1/ ,at late time
FF>>
(Weakly coupled): Good approximation
~
FF~
(Strongly coupled): Bad. Difference between
classical and quantum evolution.
Berges 2004.
F=O(
=O(
-1)
-1)
Non-perturbative regime: quasistationary evolution
-2/3)
x
F=O(
-1/2
0)
=O(
time
F=O(
)
x
=O(
0)
Nonlinear regime: source induced amplifacation
F=O(
0),
=
F~exp(g
(2/3)
)
Linear regime: Parametric resonance
Quantum and Classical evolution
(Weakly coupled =10-4).
Evolution of classical field
(
m/
0=0.1
)/
pT=0
F
pT=pi/8
pT=pi/4
Damping of classical field
x
+
Parametric resonance
>1
x
+
+
>25
…
>40
Late time statistical function
(Weakly coupled)
Time evolution
O(1) to O(1/
F
)
=190
pT
We cannot fit the
above function.
Lack of difference indicates that the system is far from equilibrium
at late time at
=190.
Quantum and Classical evolution
(Strongly coupled
).
F
Expanding
pT=0
(
)/
-1
pT=pi/8
Quantum
Classical
pT=pi/4
+ Parametric resonance
x
+
x
…
Late time distribution function
(Strongly coupled)
=150
np
Classical statistical approximation: Neq~ T/p-1/2. It is not true
thermalization. This is a problem in classical evolution.
Summary
• We have proposed the Kadanoff-Baym approach to
thermalization of O(N) scalar fields from initial
background classical field with longitudinal expansion
in 2+1 dimensions. Only 1/N covers all evolution of
instability (F: O(1) to O(1/
) by summing all order of
coupling.
• In weakly coupled regimes, no difference between
quantum and classical evolution appears. No hint of
quantum equilibration.
• In strongly coupled regimes, the late-time Bose-Einstein
distribution is realized only in quantum evolution. In
classical statistical approximation, it is not true
thermalization. Need for quantum evolution equation.
Remaining problems
• Application to non-Abelian gauge theories in
expanding system.
• Initial condition in an expanding system
(Color Glass Condensate with vacuum
fluctuations).
• Renormalization procedure in an expanding
system.
• Tuning of program codes.
Thank you.
Berges 2004.
(Flat metric)
Relativistic Heavy Ion Collison
at RHIC and LHC
Proper time
Rapidity
Freeze-out
Hadron gas
Mixed phase
Quark-Gluon Plasma
τeq=0.6-1fm/c
Nonequilibrium
phase
√sNN=0.2 TeV
√sNN=2.76TeV
Au+Au (RHIC)
Pb+Pb (LHC)
Evolution Equation of Classical fields.
Phi^2: 1/la to 0
F: 1 to 1/la/tau
F, Weakly coupled.
Phi (strongly coupled).
Phi(t): Nonexpanding
Tau^(1/3)phi(tau): Expanding
Dilite : due to small \int k F
F, strongly coupled.
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Pη
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Pη
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Pη
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Pη
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Pη
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Pη
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Pη
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Pη
Px
Evolution of Green’s functions F
Quantum evolution
Nonexpanding
Expanding
σ0
)
)
Pz
Px
Pη
Px
Pz=Pη/τ
pη2/τ2
Strongly coupled.
Late-time distribution function
(Quantum evolution)
Nonexpanding
t/t0=150
np
εp
Expanding
τ/τ0=150
Time irreversibility
Symmetric phase 〈Φ〉=0
λΦ4
Exact 2PI (no
truncation)
×
NLO of λ
O(N)
SU(N)
×
×
NLO of 1/N
LO of g2
Truncation
LO of
Gradient
expansion
H-theorem
△
△
△(TAG)
○
○
○ (TAG)
Numerical Simulation for KB eq.
Symmetric phase 〈Φ〉=0
λΦ4
O(N)
Truncation
NLO of λ
Others’
Code
1+1 dim
2+1 dim
3+1 dim
1+1 dim
Our Code
1+1 dim
2+1 dim
3+1 dim
1+1 dim
2+1 dim
3+1 dim
NLO of 1/N
SU(N)
LO of g2
?
3+1 dim
2+1 dim
3+1 dim