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.) CGC Glasma thermalization QGP Relativistic heavy ion collision (Ideal Fluid ?)

Purpose 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) Outline 1. 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 approximately. Adopt a trial Gaussian wave functional (WF) Variational Method,

Hamiltonian of pure YM theory First Gauge Fixing Hamiltonian Gauss law constraint Since for no sources (quarks) We must select WF and vary under this constraint. WF (Wave functional) : generator of gauge transformation Gauge invariance of the WF source

Completely gauge fixed formalism, Complicated calculation S. Nojiri (1984), B. Rosenstein and A. Kovner (1986) Selection of trial WF Examples of Strategies 1. Solve （＊） 2. Disregard （＊） Evaluate 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, (2000) ・・・ （＊） 3. Average over Gauge (1995) Kogan-Kovner variational ansatz which satisfies （＊）.

Under Gauge Transformation Potential V[A] is Periodic in A or N CS [A] Potential 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) V[A] Selection of WF is the most significant process.

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

V[A] Overlap Instanton : Tunneling effect

Evaluation of Saddle 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 transverselongitudinal Gauge Invariant Gaussian Density Matrix Time dependence Gaussian 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 Equation Color field Topological sectors

overlap Appendix Integrate around the particular gauge sector for pure state Integrate or average over all the gauge sectors Estimate Γas a quadratic form of λ approximately.

Quantities to be Varied Action-like quantity defined by Balian-Veneroni The Vacuum Finite Temperature Non-Equilibrium Appendix R. Balian and Veneroni, Phys. Rev. Lett. 47, 1353, (E) 1765 (1981)

4. Numerical Simulation CGC Expectation Value under our ρ[A,A’] Comparison of electric field for initial condition 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.

QGP (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 12GeV Momentum dependence of kernel function K T

Tunneling Effect

KTKT 0.40fm/c ΔK T <0.001 t fm/c K T /Λ Appendix Coupling g=2, , 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 K T < 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 Vacuum Finite Temperature Non-Equilibrium Appendix R. Balian and Veneroni, Phys. Rev. Lett. 47, 1353, (E) 1765 (1981)

for pure state overlap Appendix Integrate under the particular gauge Integrate or average over all the gauge sectors Estimate Γas a quadratic form of λ approximately.

Appendix

Bond State Antibond state Ψ 1 (φ) Ψ 2 (φ) Ψ 1 (φ) － Ψ 2 (φ) Ψ 1 (φ) ＋ Ψ 2 (φ) How to Select Trial Wave Functional Double-well potential