1. 有限温度 QCD における 励起モード 理研スパコン研究会 2010/Sep./24 北沢正清 （阪大）

2. Lattice Study of QCD @T>0 Lattice Study of QCD @T>0  EoS  susceptibilities (fluctuations)  order parameters  … macro (bulk) micro staticdynamical  screening mass  potential  wave func.  …

3. Lattice Study of QCD @T>0 Lattice Study of QCD @T>0  EoS  susceptibilities (fluctuations)  order parameters  … macro (bulk) micro staticdynamical  screening mass  potential  wave func.  …  transport coefficients (viscosity)  dynamic critical phenomena?  excitation modes (quasi-particles; hadrons, quarks, gluons, etc.)  mass, decay rates spectral function

4. Spectral Functions at T>0 Spectral Functions at T>0  ( ,p=0)   ( ,p)  T = 0 T > 0 Difficulties at T>0:  continuous spectrum  fixed temporal extent: T=1/aN t 

5. Extracting Spectral Functions Extracting Spectral Functions Ill-posed problem lattice observable discrete and noisy spectral function continuous MEM analysis of  (  ) most probable image estimated by lattice data + prior knowledge Asakawa, Hatsuda, Nakahara, 1999 qualitative structure of  (  ). errors only for average for finite range

6. Dilepton (Photon) Production Rate Dilepton (Photon) Production Rate e+e+ e-e-  Direct probes in heavy ion collisions. Emissions from all stages are superposed PHENIX, PRC,2010 vector channel propagator

7. Quark Number Scaling at RHIC Quark Number Scaling at RHIC v 2 >0 Quark number scaling indicates the existence of quasi-particles having quark quantum number at early stage.

8. Excitation Modes in Hot Medium Excitation Modes in Hot Medium Are there hadronic modes above T c ? fate of charmonia: signal of realization of QGP phase Matsui, Satz, ’86 soft modes of chiral transition in  and  channels. and other excitation modes? glueballs, diquarks, … How do hadrons cease to exist? Hatsuda, Kunihiro, ’85 perturbatively, decay width ~g 2 T How do quarks and gluons disappear? Is quasi-particle picture for quarks and gluons irrelevant in sQGP?

9. Hydrodynamical Models at RHIC Hydrodynamical Models at RHIC Schenke, et al. 1009.3244 Viscous hydrodynamics contains transport coefficients: shear viscosity  bulk viscosity  relaxation times  , … Is the ratio  /s close to the conjectured lower bound 1/4  ?

10. Spectral Functions at T>0 Spectral Functions at T>0 quasi-particle excitation width ~ decay rate  transport coefficients  ( ,p)  peak Kubo formulae slope at the origin shear viscosity : T 12 bulk viscosity : T  electric conductivity : J ii

11. Approximation Approximation So far, almost all studies on spectral functions are performed in quenched approximation. to increase number of data points  larger N t (finer a) to eliminate finite volume effects  larger L > 4/T The largest lattice thus far:N t ~ 48, N x ~128 Because we need  higher statistics Quenched QCD is not the QCD. However, some nonperturbative features would be revealed in the quenched analysis.

12. Charmonium Spectra Charmonium Spectra MEM analysis:  All calculation concludes that J/  survives up to ~1.5T c Asakawa, Hatsuda, 2004 Datta, et al., 2004 Umeda, et al., 2002 Aarts, et al., 2006 Jakovac, et al., 2007 … Asakawa, Hatsuda, 2004 Recent progress: subtract the constant mode Umeda,2007 default model dependence Ding, et al.,2009  Only a few studies on light quarks

13. Transport Coefficients Transport Coefficients shear: Karsch, Wyld, 1987 Nakamura, Sakai, 2005 Meyer, 2007 bulk: Meyer, 2008 electric conductivity: Gupta, 2004 Aarts, et al., 2007 Electric Conductivity: Spectrum in vector channel Aarts, et al., 2007 Several approaches: fitting with Lorentz-type ansatz, ansatz from hydrodynamics at low energy

14. Cut-off Dependence in Vector Channel Cut-off Dependence in Vector Channel Correlator for vector channel for T=1.5T c with different a Kaczmarek, et al., xQCD2010 strong a dependence : free & continuum

15. Spatial Volume Dependence of m T Spatial Volume Dependence of m T m T /T 48 3 x16 64 3 x16 128 3 x16 m T /T=0.725(14) T=3T c Strong spatial volume dependence of m T.

16. Spatial Volume Dependence of m T Spatial Volume Dependence of m T m T /T 48 3 x16 64 3 x16 128 3 x16 m T /T=0.725(14) T=3T c Strong spatial volume dependence of m T. k2nB(Ek)k2nB(Ek) m=0 m=Tm=T k/T N x /N  =4 Discretization of p N x /N  =4  p min ~1.57T N x /N  =8  p min ~0.79T

17. Exploiting Sum Rules Exploiting Sum Rules Spectral function can be constrained by sum-rules. Kharzeev, Tuchin, 2008 Romatschke, Son, 2009 Ellis, Kapusta, Tang, 1998 For bulk sector: Some difficulties:   (  )  T=0 (  ) is not positive semi-definite. MEM is not applicable.  We must know T=0 spectrum. QCD thermodynamics serves as an additional information for the spectrum!

18. Summary Summary 有限温度 QCD の動的な性質を格子 QCD の数値解析により 理解することは、極めて重要かつ興味深い課題である。 スペクトル関数の評価には注意深い解析が必要である。 相関関数の格子間隔および体積依存性 解析の結果得られたスペクトル関数の誤差評価 既存の解析の更新、および新しい物理量の解析のため、 より大きな計算機資源が必要である。 より大きな格子、高統計 フル QCD における解析 解析接続をより確実に行うための手法の開発や、信頼性の 正しい評価等において、更なる理論的進展が望まれる。

19. Spectral Function at T>0 Spectral Function at T>0 nonzero T T = 0