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Evaluation of Israel-Stewart parameters in lattice gauge theory

Evaluation of Israel-Stewart parameters in lattice gauge theory. KEK 理論センター研究会 『 原子核・ハドロン物理 』 Aug 11-13, 2009. Yasuhiro Kohno (Osaka University) M. Asakawa 1 , M. Kitazawa 1 , C. Nonaka 2 , S. Pratt 3 1 Osaka Univ. 2 Nagoya Univ. 3 Michigan State Univ. Contents. 1. Introduction

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Evaluation of Israel-Stewart parameters in lattice gauge theory

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  1. Evaluation of Israel-Stewart parameters in lattice gauge theory KEK理論センター研究会 『原子核・ハドロン物理』 Aug 11-13, 2009 Yasuhiro Kohno (Osaka University) M. Asakawa1, M. Kitazawa1, C. Nonaka2, S. Pratt3 1Osaka Univ. 2Nagoya Univ. 3Michigan State Univ.

  2. Contents 1. Introduction 2. Strategy 3. Numerical Results 4. Summary

  3. Contents 1. Introduction 2. Strategy 3. Numerical Results 4. Summary

  4. クォーク・グルーオン・プラズマ（QGP）の 物性・時空発展および非平衡現象 • クォーク・ハドロンの世界

  5. RHIC Scientists Serve Up “Perfect” Liquid New state of matter more remarkable than predicted -- raising many new questions April 18,2005 • 重イオン衝突実験＠RHIC QGP（near TC）≒完全流体（？） QGPの時空発展は相対論的流体力学で記述できる 強結合QGP 強相関QGP Lattice QCD　○ 摂動論　× 輸送係数（粘性係数etc.）に着目 Lattice QCDで輸送係数を数値計算

  6. 相対論的流体力学 • 1st order theories for dissipative fluid (by Eckart or Landau & Lifshitz) ⇒散逸の効果を１次まで取り入れる entropy current s: entropy density , uμ: 4-velocity , T: temprature 散逸量の輸送方程式⇒因果律 × 輸送係数:ζ(bulk viscosity),κ(heat conductivity), η(shear viscosity) 散逸項（qμ: 熱流） C. Eckart, Phys. Rev. 58, 919 (1940) L. D .Landau and E. M. Lifshitz, Fluid Mechanics (1959)

  7. 相対論的流体力学 • 2nd order theory for dissipative fluid (by Muller or Israel & Stewart) ⇒1st order theoryに緩和時間を導入 entropy current 緩和時間(τi→0で2nd order→1st order) 散逸量の輸送方程式⇒ 因果律○（ただし例外有り） 輸送係数 : ζ, κ, η, α0, α1, β0, β1, β2 散逸項 I.Muller, Z. Phys. 198, 329 (1967) W.Israel and J.M.Stewart, Ann. Phys. (N.Y.) 118, 341 (1979)

  8. 先行研究 • Using Kubo formula with ansatz for spectral function. But the validity remains questionable. Analytic continuation Lattice QCD Viscosities Kubo formula Imaginary time correlator Real time correlator ? F. Karsch and H. W. Wyld, Phys. Rev. D35, 2518(1987) A. Nakamura and S. Sakai, Phys. Rev. Lett. 94, 072305(2005) H. B. Meyer, Phys. Rev. D76, 101701(2007)

  9. Evaluation of the ratios of the viscosities to the relaxation times of Israel-Stewart (IS) theory in SU(3) lattice QCD. • 研究方針 Reduce the number of IS parameters We try to obtainsecond order coefficients β0&β2.

  10. Contents 1. Introduction 2. Strategy 3. Numerical Results 4. Summary

  11. Israel-Stewart entropy usingthese relations and ・・・(1) • Israel-Stewart entropy uμ:4-velocity of particles Seq: equilibrium entropy , qμ: heat flux Π : bulk viscous pressure , πμν: shear viscous pressure

  12. 　平衡状態におけるゆらぎの確率分布はBoltzmann-Einsteinの原理に従う　平衡状態におけるゆらぎの確率分布はBoltzmann-Einsteinの原理に従う 　状態変数a=a0の状態が実現される確率は Equation (1) と (2)より • Boltzmann-Einsteinの原理 c.f. S=logW ・・・(2) A. Muronga, Eur. Phys. J. ST 155:107-113(2008) S. Pratt, Phys. Rev. C77, 024910(2008)

  13. 期待される分布 • Lattice QCDでやること • π13のゆらぎの確率分布を数値計算する • π13の分布とequation (3)を比較してβ2を得る ・・・(3) π13 BE principle Probability of fluctuations The ratios between IS parameters IS entropy Lattice QCD

  14. イメージ • ４次元Euclid空間の格子 ・・・ Configuration = 微視状態 ・・・確率 1/6 π13 ・・・確率∝ exp[-Vβ2π132/2T]

  15. Contents 1. Introduction 2. Strategy 3. Numerical Results 4. Summary

  16. Lattice parameters • SU(3) pure gauge theory (gluon only) • 3 isotropic lattice boxes • 10,000 configurations for each box • Blue Gene @ KEK β = 2NC/g2 a: lattice spacing TC: critical temperature (~300MeV) Nτ: number of sites in spatial direction NS: number of sites in temporal direction

  17. Result (Probability distribution with box1)

  18. Result (Probability distribution with box1)

  19. Result (The ratio β2) • Our present result with box1 • Characteristic velocity of dissipative flow From AdS/CFT From our result R. Baier, P. Robatschke, D. T. Son, A. O. Starinets and M. A. Stephanov, JHEP 0804:100 (2008) ⇒因果律○ ε : energy density P : pressure ⇒因果律 ×

  20. Contents 1. Introduction 2. Strategy 3. Numerical Results 4. Summary

  21. Summary • Lattice QCDによる散逸量（π13）のゆらぎの確率分布の数値計算を行った。 • Boltzmann-Einsteinの原理に基づき、Israel-Stewart (2nd order)理論の枠組み内で粘性係数と緩和時間の比（β2）を導出した。 • Lattice QCDからの結果からは、Israel-Stewart(2nd order)理論は因果律を破る（？） ⇒AdS/CFTからの結果と矛盾… • Future plan • β0=τΠ/Πの導出（box1） • その他のLattice（box2,box3）のデータの解析 • AdS/CFTとの矛盾を議論

  22. ありがとうございました。

  23. Appendix

  24. Result (Spatial correlation) Lattice spacing dependence of π13

  25. Shear viscosity from perturbation theory In high temperature region P. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0011 001 (2000)

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