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Shear and Bulk viscosity of QGP in PQCD

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 .

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Shear and Bulk viscosity of QGP in PQCD

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

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

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

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

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

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

  10. Previous works on η,ζfor 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. Contradicting results on ηfor 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. 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. XG 1) The 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 AMY 12 Shear & Bulk viscosity of QGP in PQCD

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

  15. 1 3 q Matrix elements: gg↔gg (HTL) 2 4 (6) where q is momentum transferred and gluon HTL self-energy Heiselberg, Wang, NPB 462,389(1996) (7)

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

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

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

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

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

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

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

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

  24. Analytic and numerical results 24 Shear & Bulk viscosity of QGP in PQCD

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

  26. N_c scaling (37) (38) 26 Shear & Bulk viscosity of QGP in PQCD

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

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

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

  31. Why different between XG and AMY? • 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, • We can also carry out the integration over the full phase space of • the final state without the symmetry factor (39) (40) (41) 31 Shear & Bulk viscosity of QGP in PQCD

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

  33. Why different between XG and AMY? 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: (42) (43) 33 Shear & Bulk viscosity of QGP in PQCD

  34. Why different between XG and AMY? • Lesson: • Two ways of integration over final state phase space are equivalent: • 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 34 Shear & Bulk viscosity of QGP in PQCD

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

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

  37. Conclusion and outlook ■ We calculate leading order shear and bulk viscosity of gluon plasma in PQCD. EXACT matrix elementfor 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 viscositiesagree withthose of Arnold, Moore and Yaffe (AMY) within errors. ■ The difference between AMY’s and XG’s resultsis 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. THANK YOU ! 38 Shear & Bulk viscosity of QGP in PQCD

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