Variational Approach to Non-Equilibrium Gluodynamics
東京大学大学院 総合文化研究科西山陽大
1. Background
• Short Thermalization Time for Partons
• No idea for equilibrium state of thermalized QGP
• Strongly Coupled Quark Gluon Plasma
QGP
nucleus 2-3fm/c (Simulation of Boltzmann eq.)<0.6-1fm/c (Exp.)
CG
CG
lasm
a
thermalization
QG
P
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(Ideal Fluid ?)
PurposePurpose
1. To derive time evolution equations for gluons with gauge invariant density matrix and Liouville equation.
2. To simulate time evolution of gluons after heavy ion collision. (quantum theoretical approach)
OutlineOutline1. Background
2. Variational Approach in the Vacuum
3. Non-equilibrium Gluodynamics
4. Numerical Simulation
5. Conclusion
2. Variational Approach in the Vacuum
Consider a System
Take an expectation value of H
As a result
We can obtain a solution of the vacuum WF and <H> approximately.
Adopt a trial Gaussian wave functional (WF)
Variational MethodVariational Method,
Hamiltonian of pure YM theory
First Gauge Fixing
HamiltonianHamiltonian
Gauss law constraintGauss law constraint
Since for no sources (quarks)
We must select WF and vary <H> under this constraint.WF (Wave functional)
: generator of gauge transformation
Gauge invariance of the WFGauge invariance of the WF
source
Completely gauge fixed formalism, Complicated calculation S. Nojiri (1984), B. Rosenstein and A. Kovner (1986)
Selection of trial WFSelection of trial WF
Examples of Strategies Examples of Strategies
1. Solve (*)
2. Disregard (*)
Evaluate <H> and subtract unphysical contribution A. K. Kerman and D. Vautherin, Ann. Phys. 192, 408 (1989);
C. Heinemann, E. Iancu, C. Martin, and D. Vautherin, Phys. Rev. D61, 116008 (2000)
・・・(*)
3. Average over Gauge
(1995)Kogan-Kovner variational ansatz
which satisfies (*) .
Under Gauge TransformationPotential
V[A] is Periodic in A or NCS[A]
PotentialPotential
Chern Simons Charge
Winding Number proven to be integer
WF and Periodicity of Potential
Wave Functional in Periodic Potential
No second gauge fixing condition
Kogan and Kovner (1995)
5 10 15 20
0.25
0.5
0.75
1
1.25
1.5
1.75
2
5 10 15 20
0.25
0.5
0.75
1
1.25
1.5
1.75
2
V[A
]
Selection of WF is the most significant process.Selection of WF is the most significant process.
02
46
8
10
-4
-2
0
2
4
00.25
0.5
0.75
1
02
46
8
10
Why such a WF ?
Constraint of Gauge Invariance Periodic Potential Unphysical Longitudinal Mode Instanton : Tunneling effect
Constraint of Gauge Invariance Periodic Potential Unphysical Longitudinal Mode Instanton : Tunneling effect
2 4 6 8 10 12 14
0.25
0.5
0.75
1
1.25
1.5
1.75
2
V[A
]
Overlap
Instanton : Tunneling effect
Evaluation of
Saddle PointSaddle Point by use of
2 times larger than instanton action
= Transition Amplitude between topologically distinct sectors
for SU(2)
3. Non-Equilibrium Gluodynamics
Mixing Parameter
Rate of Change
Integration or Average over all the gauge distinctive sectors
No second gauge fixing condition
transverse longitudinal
Gauge Invariant Gaussian Density MatrixGauge Invariant Gaussian Density Matrix
Time dependenceTime dependenceGaussian Size
Assume Translational Invariance in the Coordinate Space
Time Evolution Equations
Mean Field Approximation in interaction terms
Averaged by
Extension of Eboli, Jackiw and Pi, Phys. Rev. D37, 3557 (1988)
Liouville EquationLiouville Equation
Color fieldColor field Topological sectorsTopological sectors
overlapoverlap
AppendixAppendix
Integrate around the particular gauge sector
Integrate around the particular gauge sector
for pure state
Integrate or average over all the gauge sectorsIntegrate or average over all the gauge sectors
Estimate Γas a quadratic form of λ approximately.Estimate Γas a quadratic form of λ approximately.
Quantities to be Varied
Action-like quantity defined by Balian-Veneroni
The VacuumThe Vacuum
Finite TemperatureFinite Temperature
Non-EquilibriumNon-Equilibrium
AppendixAppendix
R. Balian and Veneroni, Phys. Rev. Lett. 47, 1353, (E) 1765 (1981)
4. Numerical Simulation
CGC CGC Expectation Value under our ρ[A,A’]Expectation Value under our ρ[A,A’]
Comparison of electric field for initial condition
Gluon Density Distribution in McLerran Venugopalan (MV) model
Gluon Density Distribution in McLerran Venugopalan (MV) model
(Write in MV model to gauge invariant density matrix.)
Uniform in the coordinate space anisotropic in momentum space
Write in Color Glass Condensate to gauge invariant density matrix.
Write in Color Glass Condensate to gauge invariant density matrix.
5 10 15 20
0.25
0.5
0.75
1
1.25
1.5
1.75
2
-20.00
-10.00
0.00
10.00
20.00
30.00
0.00 0.20 0.40 0.60 0.80
t [fm/c]
KT/Λ
p=0.5GeV
p=0.6GeV
p=0.7GeV
p=0.8GeVQGP
(t)
密度行列
Px=Py=Pz=0.5, 0.6, 0.7 and 0.8 GeV
g=2
β=(0.6GeV)-1
μ=g×0.5GeV
Cutoff 12GeVMomentum dependence of kernel function KT
Tunneling Effect
KT
0.40fm/c
ΔKT<0.001
-30.00
-20.00
-10.00
0.00
10.00
20.00
30.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60
t [fm/c]
KT/Λ
2
1.6
1.3
1.1
t fm/c
KT/Λ
AppendixAppendix
Coupling g=2, 1.6. 1.3, 1.1 for Px=Py=Pz=500MeV
5. Conclusion• We have derived time evolution equations with
respect to gauge invariant density matrix in pure Yang Mills theory (Gluodynamics)
• We adopt an initial condition motivated by Color Glass Condensate and simulate the dynamics of gluons after heavy ion collisions.
• As a result we have seen that the density matrix relaxes to a non-thermal state at a short time due to tunneling effects (instantons).
Remaining Problems
Classical Color Field (z direction) Reformation of KT < 0 by interaction terms Rapid expansion of the system Running Coupling Estimation in τ-η coordinate Expectation value of Other Physical Quantities Non-zero θ vacuum
Quantities to be Varied
Action-like quantity defined by Balian-Veneroni
The VacuumThe Vacuum
Finite TemperatureFinite Temperature
Non-EquilibriumNon-Equilibrium
AppendixAppendix
R. Balian and Veneroni, Phys. Rev. Lett. 47, 1353, (E) 1765 (1981)
for pure state
overlapoverlap
AppendixAppendix
Integrate under the particular gauge
Integrate under the particular gauge
Integrate or average over all the gauge sectorsIntegrate or average over all the gauge sectors
Estimate Γas a quadratic form of λ approximately.Estimate Γas a quadratic form of λ approximately.
AppendixAppendix
Bond State
Antibond state
Ψ1(φ)
Ψ2(φ)
Ψ1(φ) - Ψ2(φ)
Ψ1(φ) + Ψ2(φ)
How to Select Trial Wave Functional
Double-well potential
Double-well potential