1. Qun Wang/USTC/China1 Shear and Bulk viscosity of QGP in PQCD Qun Wang Univ of Sci & Tech of China Chen, Dong, Ohnishi, QW, Phys. Lett. B685, 277(2010); Chen, Deng, Dong, QW, Phys. Rev. D83, 034031(2011); Chen, Deng, Dong, QW, arXiv:1107.0522 HIC in LHC era, 15-21 July 2012, Quy Nhon, Vietnam 1

2. Qun Wang/USTC/China2 What is shear viscosity (mean free path) x (energy momemtum density) correlation of energy-momemtum tensor in x and y low-momentum behavior of correlator (Kubo formula) 2 Shear & Bulk viscosity of QGP in PQCD

3. Qun Wang/USTC/China3 Shear viscosity in ideal gas and liquid ideal gas, high T liquid, low T lower bound by uncertainty principle Danielewicz, Gyulassy, 1985 Policastro,Son,Starinets, 2001 Frenkel, 1955 3

4. Qun Wang/USTC/China4 η/s around phase transition Lacey et al, PRL98, 092301(2007) Csernai, et al PRL97, 152303(2006) 4 Shear & Bulk viscosity of QGP in PQCD

5. Qun Wang/USTC/China5 What is bulk viscosity (breaking of scale symmetry) X (mean free path) X (energy momemtum density) correlation of trace of energy-momemtum tensor in x and y low-momentum behavior of correlator (Kubo formula) Generated by dilatation 5 Shear & Bulk viscosity of QGP in PQCD

6. Qun Wang/USTC/China6 Scale anomaly in QCD Massless QCD Lagrangian and action The classical action of massless QCD is invariant under scale transformation 6 (1) (2) (3) Shear & Bulk viscosity of QGP in PQCD

7. Qun Wang/USTC/China7 Scale anomaly & bulk viscosity Breaking of scale invariance is proportional to beta-function Beta-function 7 (4) (5) Shear & Bulk viscosity of QGP in PQCD

8. Qun Wang/USTC/China8 ζ/s around phase transition Karsch, Kharzeev, Tuchin, PLB 2008 Noronha *2, Greiner, PRL 2009, Chen, Wang, PRC 2009, Li, Huang, PRD 2008,...... Bernard et al, (MILC) PRD 2007, Cheng et al, (RBC-Bielefeld) PRD 2008, Bazavov et al, (HotQCD), arXiv:0903.4379 8 Shear & Bulk viscosity of QGP in PQCD

9. Qun Wang/USTC/China9 Bulk viscosity at Tc Helium-3 near the critical point ～ a few millions ! Bulk viscosity near T_c diverges in power law, closely related to fluctuation Kogan, Meyer, J.Low.Temp.Phys.110, 899(1998) 3-D Ising model 9 Shear & Bulk viscosity of QGP in PQCD

10. Qun Wang/USTC/China10 Previous works on η,ζ for QGP Shear viscosity: ► PV: Perturbative and Variational approach Danielewicz, Gyulassy, PRD31, 53(1985) ; Arnold, Moore and Yaffe, JHEP 0011, 001 (2000), 0305, 051 (2003). ► Transport model : Xu, Greiner, PRL 100, 172301 (2008). ► Lattice : Meyer, PRD 76, 101701 (2007); NPA 830, 641C (2009). ► Anomalous: Asakawa et al, PRL 96, 252301(2006); Mujumder et al, PRL 99, 192301(2007). Bulk viscosity: ► PV: Perturbative and Variational approach Arnold, Dogan, Moore, PRD74, 085021(2006). ► Sum rule and spectral desity Kharzeev, Tuchin, JHEP 0809,093(2008); Moore, Saremi, PRD JHEP 0809, 015(2008); Romatschke, Son, PRD80, 065021 (2009). ► Lattice Meyer, JHEP 1004, 099 (2010); PRL 100, 162001 (2008). ► Models near T_c Noronha*2, Greiner, PRL103,172302(2009) ; Li, Huang, PRD80,034023(2009). 10 Shear & Bulk viscosity of QGP in PQCD

11. Qun Wang/USTC/China11 Contradicting results on η for gluon plasma ► PV: Perturbative and Variational approach Danielewicz, Gyulassy, Phys.Rev.D31, 53(1985) Dissipative Phenomena In Quark Gluon Plasmas Arnold, Moore and Yaffe, JHEP 0011, 001 (2000),0305, 051 (2003) Transport coefficients in high temperature gauge theories: (I) Leading-log results (II): Beyond leading log........... ► BAMPS: Boltzmann Approach of MultiParton Scatterings Xu and Greiner, Phys. Rev. Lett. 100, 172301(2008) Shear viscosity in a gluon gas Xu, Greiner and Stoecker, Phys. Rev. Lett. 101, 082302(2008) PQCD calculations of elliptic flow and shear viscosity at RHIC ► Different results of AMY and XG for 2↔3 gluon process: ~ ( 5-10)% (AMY) ~ (70-90)% (XG) η (23) (AMY) >> η (23) (XG)σ(23) (AMY) << σ(23) (XG) 11 Shear & Bulk viscosity of QGP in PQCD

12. Qun Wang/USTC/China12 Difference: AMY vs XG 1) A parton cascade model for solving the Boltzmann equation. Gluons are treated as a Boltzmann gas (i.e. a classical gas). 2) Gunion-Bertsch formula (soft emission) for gg↔ggg process. 12 XG AMY 1) T he Boltzmann equation is solved in a variation method. Gluon as quantum gas. 2) For number changing processes Ng↔ (N+1)g: using collinear splitting g↔gg Shear & Bulk viscosity of QGP in PQCD

13. Qun Wang/USTC/China13 Our goal and strategy Goal: to calculate the shear/bulk viscosity in the leading order using Exact Matrix Element for gg↔ggg process to test previous results Elements: ■ Variational method and Linearized Boltzmann equation for 22+23 processes (AMY) ■ 22: Hard Thermal Loop approximation for 22 (AMY) ■ 23: Exact Matrix Element (new) + Gunion-Bertsch (XG) ■ 23: LPM effects for Exact Matrix Element (new) + Gunion-Bertsch (XG) 13 Shear & Bulk viscosity of QGP in PQCD

14. Qun Wang/USTC/China14 Boltzmann equation for gluon plasma gluon distribution function gg↔gg collision terms gg↔ggg collision terms matrix element delta function EM conservation phase-space measure [ gain - loss ] 14 Shear & Bulk viscosity of QGP in PQCD

15. Qun Wang/USTC/China15 Matrix elements: gg↔gg (HTL) where q is momentum transferred and gluon HTL self-energy q 31 24 Heiselberg, Wang, NPB 462,389(1996) (6) (7)

16. Qun Wang/USTC/China16 Exact matrix element for 23 Exact matrix element in vacumm for massless gluons 1 2 3 4 5 all momenta are incoming or outgoing exact matrix element for massless gluon is invariant for Ellis, Sexton, NPB 269, 445 (1986) 16 (8) Shear & Bulk viscosity of QGP in PQCD

17. Qun Wang/USTC/China17 Exact matrix element to Gunion-Bertsch Taking large s limit (s→ ) and then small y limit (y→0) Gunion-Bertsch formula NOTE (important!!) Gunion-Bertsch formula is only valid for soft region 17 and (13) (14) Shear & Bulk viscosity of QGP in PQCD

18. Qun Wang/USTC/China18 Linearized Boltzmann equation Perturbation in distribution function Right-hand-side or collision part of Boltzmann Eq. Jeon, PRD52, 3591(1995) Jeon, Yaffe, PRD 53,5799(1996) 18 (15) (16)

19. Qun Wang/USTC/China19 Linearized Boltzmann equation Left-hand-side or kinetic part of Boltzmann Eq. Linearized Boltzmann Eq. Collision integral of B(p) shear Collision integral of A(p) bulk 19 (17) (18) Shear & Bulk viscosity of QGP in PQCD

20. Qun Wang/USTC/China20 Shear/bulk viscosity in stress tensor Solve A(p) and B(p) from linearized Boltzmann Eq. → shear/bulk viscosity Stress tensor perturbation 20 (19) (20) Shear & Bulk viscosity of QGP in PQCD

21. Qun Wang/USTC/China21 Constraints ■ Invariance of energy density in local rest frame ■ Speed of sound ■ Mass 21 (27) (28) (29)

22. Qun Wang/USTC/China22 Functional basis for A(p) ■ Our choice ■ ADM’s choice ■ Results do not depend on bases, on a certain basis, 22 (30) (31) (32) Shear & Bulk viscosity of QGP in PQCD

23. Qun Wang/USTC/China23 Evaluating bulk viscosity ■ Having A(p), we can evaluate ζ ■ So ζ is proportional to β(g) 23 (33) Shear & Bulk viscosity of QGP in PQCD

24. Qun Wang/USTC/China24 Analytic and numerical results 24 Shear & Bulk viscosity of QGP in PQCD

25. Qun Wang/USTC/China25 Leading-Log result: ζ/s Weinberg, Astrophys. J. 168, 175 (1971) 25 (34) (35) (36) Shear & Bulk viscosity of QGP in PQCD

26. Qun Wang/USTC/China26 N_c scaling 26 (37) (38) Shear & Bulk viscosity of QGP in PQCD

27. Qun Wang/USTC/China27 Numerical results: η/s for 22+23 HTL: hard-thermal-loop AMY: Arnold-Moore-Yaffe gluon mass = m_∞ Chen, Deng, Dong, QW, Phys. Rev. D83, 034031(2011); Erratum: 84, 039902; 27 Shear & Bulk viscosity of QGP in PQCD

28. Qun Wang/USTC/China28 Numerical results: ζ for 22+23 Chen, Deng, Dong, QW, arXiv:1107.0522 28 Shear & Bulk viscosity of QGP in PQCD

29. Qun Wang/USTC/China29 Comparison of ζ and η Chen, Deng, Dong, QW, arXiv:1107.0522 Weinberg, Astrophys. J. 168, 175 (1971) 29 Shear & Bulk viscosity of QGP in PQCD

30. Qun Wang/USTC/China30 Comparison with lattice result and data Chen, Deng, Dong, QW, arXiv:1107.0522 30 Shear & Bulk viscosity of QGP in PQCD

31. Qun Wang/USTC/China31 Why different between XG and AMY? 31 1. For gg ↔ gg: 2. The integration can be done by including the symmetry factor 2 if we constrain the phase space to the region to the t-channel, 3.We can also carry out the integration over the full phase space of the final state without the symmetry factor (39) (40) (41) Shear & Bulk viscosity of QGP in PQCD

32. Qun Wang/USTC/China32 1 2 3 4 1 3 4 2 qT →0, q →0, t-channel qT →0, q → ∞, u-channel Shear & Bulk viscosity of QGP in PQCD

33. Qun Wang/USTC/China33 Why different between XG and AMY? 33 A caveat for gg ↔ ggg: 1. GB holds for constrained phase space in CMF: 2. GB has a symmetry for perm (3,4,5) in CMF, there is a symmetry factor for constrained phase space. On the other hand we can put GB out of summation and obtain the full space expression: (43) (42) Shear & Bulk viscosity of QGP in PQCD

34. Qun Wang/USTC/China34 Why different between XG and AMY? 34 Lesson: Two ways of integration over final state phase space are equivalent: 1. Use symmetry factor × GB for constrained final state phase space in integration 2. Use GB for full phase space for final state gluons in integration equivalent Shear & Bulk viscosity of QGP in PQCD

35. Qun Wang/USTC/China35 Why different between XG and AMY? 35 Introducing the symmetry factor while integrating over full phase space of final state gluons (= without constraining the phase space)  multiple counting ! Shear & Bulk viscosity of QGP in PQCD

36. Qun Wang/USTC/China36 Why GB (soft) ≈ AMY (collinear)? 36 GB: soft emission (in CM frame) AMY: collinear splitting (in heat bath frame) Shear & Bulk viscosity of QGP in PQCD

37. Qun Wang/USTC/China37 Conclusion and outlook ■ We calculate leading order shear and bulk viscosity of gluon plasma in PQCD. EXACT matrix element for 23 process is used for 23 process with m_D as regulator, HTL is used for 22 process.The LPM effects are also included. ■ Our results for shear and bulk viscosities agree with those of Arnold, Moore and Yaffe (AMY) within errors. ■ The difference between AMY’s and XG’s results is clarified. The lesson is to do collisional integration in two equivalent ways: (a) constrained phase space with symmetry factor; (b) full phase space without symmetry factor; otherwise it would lead to multiple counting. ■ The equivalence at LO between AMY’s collinear splitting and GB’s soft gluon bremsstrahlung has been demonstrated. ■ Outlook (undergoing project): (1) Include quark flavor; (2) With chemical potential 37 Shear & Bulk viscosity of QGP in PQCD

38. Qun Wang/USTC/China38 THANK YOU ! 38 Shear & Bulk viscosity of QGP in PQCD