1 On viscosity of Quark Gluon Plasma Defu Hou CCNU, Wuhan RHIC-Star full TOF detector and related physics in China Hangzhou April 27-29

2 Introduction and motivation Viscosity from Kubo formula Viscosity from kinetic theory (Boltzmann Eq) Viscosity from AdS/CFT Summary Outlines

3 QCD under extreme conditions At very High T or density ( deconfined) High T (Early universe, heavy-ion collisions) High density matter ( in the core of neutron stars)

RHIC Robust collective flows, well described by ideal hydro with Lattice-based EoS. This indicates very strong interaction even at early time => sQGP sQGP seems to be the almost perfect fluid known  /s>=.1-.2<<1 Motivations Experiments aspect:

5 Study of dissipative effects on Study of dissipative effects on How sensitive is elliptic flow to finite  /s? Z. Xu & C. Greiner, PRL 101(08) Agreement for  s =0.3 – 0.6  /s=0.15 – 0.08 Viscous HydroCascade ( 2 2,2 3 ) P. Romatschke, PRL99 (07) Dependence on   relaxation time II 0 order expansion with green terms ( D. Rischke )

6 Theoretic aspect: To calculate Trsp. Coefs. in FT in highly nontrivial (nonperturbative ladder resummation) (c around 5) String theory method: AdS/CFT (D.Son et al 2003)  /s = 1/4 . Kinetic theory + uncertainty principle (Gyulassy)

7 Main obstacle for theory QCD in nonperturbative regime (T~200Mev) Pertburb. Expansion of QCD is not well behaved for realistic T For thermodyn.,one can use lattice and resummation techniques Kinetic coefficients are difficult to extract from lattice

8 Shear Viscosity

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11 Viscosity from Kubo formula

12 Nonlinear Response

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14 S. Jeon, PRD 52; Carrington, Hou, Kobes, PRD61

15 Carrington, Hou, Kobes, PRD64 (2001)

16 Hou, hep-ph/

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18 Viscosity from kinetics theory

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21 Boltzmann Equation Fluctuation of distribution(s: species) Recast the Boltzmann equation P.Arnold, G.D.Moore and G.Yaffe, JHEP 0011(00)001 Viscosity of hot QCD at finite density

22 Shear viscosity With a definition of inner product and expanded distribution functions, where

23 Collision terms Performing the integral over dk’ with the help of Scattering amplitude Distribution function term \chi term

24 Matrix Element

25 Variation method gives Liu, Hou, Li EPJC 45(2006)

26 Computing transport coefficients from AdS/CFT In the regime described by a gravity dual the correlator can be computed using AdS/CFT

27 AdS/CFT at finite temperature Classical Supergravity on AdS-BH×S 5 4dim. Large-Nc strongly coupled SU( Nc ) N=4 SYM at finite temperature (in the deconfinement phase). conjecture = Witten ‘98

28 Field Theory Gravity Theory = Gauge Theories QCD Quantum Gravity String theory the large N limit Supersymmetric Yang Mills Gravitational theory in 10 dimensions N large Calculations Correlation functions Quark-antiquark potential Holography

29 AdS/CFT now being applied to RHIC physics Viscosity,  /s. EOS Jet quenching “ Sound ” waves Photon production Friction … Heavy quarkonium Hardron spectrum (ADS/QCD)

30 Universality of shear viscosity in the regime described by gravity duals Graviton’s component obeys equation for a minimally coupled massless scalar. But then. Since the entropy (density) is we get D. Son, P. Kovtun, A.S., hep-th/

31 Shear viscosity in SYM Correction to : A.Buchel, J.Liu, A.S., hep-th/ P.Arnold, G.Moore, L.Yaffe, 2001

32 A viscosity bound conjecture P.Kovtun, D.Son, A.S., hep-th/ , hep-th/

33 Universality of Theorem: For any thermal gauge theory (with zero chemical potential), the ratio of shear viscosity to entropy density is equal to in the regime described by a corresponding dual gravity theory Remark: Gravity dual to QCD (if it exists at all) is currently unknown.

34 Possible Mechanisms for Low viscosity Large cross-section, strong coupling Anomalous viscosity: turbulence M. Asakawa, S.A. Bass, B.M., hep-ph/ , PRL See Abe & Niu (1980) for effect in EM plasmas

35 Take moments of with p z 2 M. Asakawa, S.A. Bass, B.M., hep-ph/ See Abe & Niu (1980) for effect in EM plasmas

36 Low viscosity due to Anderson Local. AL effect renders infinite reduces viscosity significantly even at weak coupling Mechanism:coherent backscattering (CBS) effect Ginaaki, Hou, Ren PRD 77(2008)

37 Summary Kubo formula: via correlation functions of currents Transport theory: Boltzmann Eqs. (for weak scattering) ADS/CFT(strongly coupled) Lattice calculation (noisy) Approches to calculate viscosity

38 Thanks

39 Renormalized diffusion

40 Weak Localization (WL) Anderson proposed (‘58) that electronic diffusion can vanish in a random potential (AL) Experiments detected ( Ishimaru 1984,Wolf Maret 1985) Mechanism:coherent backscattering (CBS) effect after a wave is multiply scattered many times ， its phase coherence is preserved in the backscattering direction ， the probability of back scattering is enhenced via constructive interference

41 Viscosity with random medium System: quasi-particles in random potential Candidate disorder in sQGP ? 1.The islands of heavy state; bound states (Shuryak); 2. The reminiscent of confinement vaccum, say the domain structure of 't Hooft's monopole condensation; 3.The disoriented chiral condensate (DCC); 4. CGC

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43 Response function

44 BS Eq. In Diagrams

45 Localization length Itinerant states ----  Localized States

46 II Some applications to N=4 SUSY YM Plasma: Equation of state in strong coupling: Plasma temperature = Hawking temperature Near Schwarzschild horizon Continuating to Euclidean time, To avoid a conic singularity at, the period of Recalling the Matsubara formulation

47 Free energy = temperature X (the gravity action without metric fluctuations) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998), hep-th/ Consider a 4D Euclidean space of spatial volume V_3 at The EH action of AdS-Schwarzschild: The EH action of plain AdS To eliminate the conic singularity, To match the proper length in Euclidean time Plasma free energy: Plasma entropy:

48 Bekenstein-Hawking entropy: The metric on the horizon ： The gravitational constant of the dual: agree with the entropy extraced from the gravity action. Gubser, Klebanov & Pest, PRD54, 3915 (1996)

49 The ratio 3/4: The plasma entropy density at The free field limit: the contents of N=4 SUSY YM numberentropy density gauge potential1 real scalars6 Weyl spinors4 The lattice QCD yields

50 Shear viscosity in strong coupling: Kubo formula Policastro, Son and Starinets, JHEP09, 043 (2002) where

51 Gravity dual: the coefficient of term of the gravity action

52 The metric fluctuation Substituting into Einstein equation and linearize The Laplace equation of a scalar field

53 Calculation details: Nonzero components of the Christofel (up to symmetris): Nonzero components of the Ricci tensor: Linear expansion:

54 The solution: Heun equation (Fucks equation of 4 canonical singularities) trivial when energy and momentum equatl to zero; low energy-momentum solution can be obtained perturbatively. The boundary condition at horizon: The incoming solution at low energy and zero momentum:

55 Viscosity ratio: Elliptic flow of RHIC: Lattice QCD: noisy V_4 = 4d spacetime volume

56 III. Remarks: N=4 SYM is not QCD, since 1). It is supersymmetric 2). It is conformal ( no confinement ) 3). No fundamental quarks ) and 2) may not be serious issues since sQGP is in the deconfined phase at a nonzero temperature. The supersymmetry of N=4 SYM is broken at a nonzero T ) may be improved, since heavy fundamental quarks may be introduced by adding D7 branes. ( Krach & Katz ) Introducing an infrared cutoff ---- AdS/QCD: Regge behavior of meson spectrum ---- confinement; Rho messon mass gives Lack of string theory support. Karch, Katz, Son & Stephenov

57 Deconfinement phase transition: Herzog, PRL98, (2007) Hadronic phase: Plasma phase: Hawking-Page transition: ---- First order transition with entropy jump ---- Consistent with large N_c QCD because of the liberation of quark-gluon degrees of freedom.

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61 Epilogue AdS/CFT gives insights into physics of thermal gauge theories in the nonperturbative regime Generic hydrodynamic predictions can be used to check validity of AdS/CFT General algorithm exists to compute transport coefficients and the speed of sound in any gravity dual Model-independent statements can presumably be checked experimentally

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64 Mechanisms for Low viscosity Large cross-section, strong coupling Anomalous viscosity: turbulence Soft color fields generate anomalous transport coefficients, which may give the medium the character of a nearly perfect fluid even at moderately weak coupling M. Asakawa, S.A. Bass, B.M., hep-ph/ , PRL See Abe & Niu (1980) for effect in EM plasmas