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

Rel

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isti

c h

eavy

ion

col

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onR

elat

ivis

tic

hea

vy io

n c

olli

sion

(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