On viscosity of Quark Gluon Plasma Defu Hou CCNU , Wuhan RHIC-Star full TOF detector and related physics in China Hangzhou April 27-29 1 Outlines Introduction and motivation Viscosity from Kubo formula Viscosity from kinetic theory (Boltzmann Eq) Viscosity from AdS/CFT Summary 2 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) 3 Motivations Experiments aspect: @ 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 4 Study of dissipative effects on <v2> How sensitive is elliptic flow to finite /s? Viscous Hydro P. Romatschke, PRL99 (07) Dependence on tp relaxation time II0 order expansion with green terms (D. Rischke) Cascade (2<->2,2<->3) Z. Xu & C. Greiner, PRL 101(08) Agreement for s=0.3 – 0.6 /s=0.15 – 0.08 5 Theoretic aspect: • To calculate Trsp. Coefs. in FT in highly nontrivial (nonperturbative ladder resummation) (c around 5) • String theory method: AdS/CFT /s (D.Son et al 2003) = 1/4p . Kinetic theory + uncertainty principle (Gyulassy) 6 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 7 Shear Viscosity Fx v x A y 8 9 10 Viscosity from Kubo formula 11 Nonlinear Response 12 13 S. Jeon, PRD 52; Carrington, Hou, Kobes, PRD61 14 Carrington, Hou, Kobes, PRD64 (2001) 15 Hou, hep-ph/0501284 16 17 Viscosity from kinetics theory 18 19 20 Viscosity of hot QCD at finite density Fluctuation of distribution (s: species) Boltzmann Equation Recast the Boltzmann equation P.Arnold, G.D.Moore and G.Yaffe, JHEP 0011(00)001 21 Shear viscosity With a definition of inner product and expanded distribution functions, where 22 Collision terms Performing the integral over dk’ with the help of Scattering amplitude Distribution function term \chi term 23 Matrix Element 24 Variation method gives Liu, Hou, Li EPJC 45(2006) 25 Computing transport coefficients from AdS/CFT In the regime described by a gravity dual the correlator can be computed using AdS/CFT 26 AdS/CFT at finite temperature Classical Supergravity on AdS-BH×S5 = conjecture Witten ‘98 4dim. Large-Nc strongly coupled SU(Nc) N=4 SYM at finite temperature (in the deconfinement phase). 27 Field Theory = Gauge Theories QCD Gravity Theory Quantum Gravity String theory Holography the large N limit Supersymmetric Yang Mills N large Gravitational theory in 10 dimensions Calculations Correlation functions Quark-antiquark potential 28 AdS/CFT now being applied to RHIC physics Viscosity, /s. EOS Jet quenching “Sound” waves Photon production Friction … Heavy quarkonium Hardron spectrum (ADS/QCD) 29 Universality of shear viscosity in the regime described by gravity duals Graviton’s component obeys equation for a minimally coupled massless scalar. But then . we get Since the entropy (density) is D. Son, P. Kovtun, A.S., hep-th/0405231 30 Shear viscosity in SYM P.Arnold, G.Moore, L.Yaffe, 2001 Correction to : A.Buchel, J.Liu, A.S., hep-th/0406264 31 A viscosity bound conjecture P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231 32 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. 33 Possible Mechanisms for Low viscosity Large cross-section, strong coupling Anomalous viscosity: turbulence M. Asakawa, S.A. Bass, B.M., hep-ph/0603092, PRL See Abe & Niu (1980) for effect in EM plasmas 34 Take moments of p r p D( p ) p f (r , p, t ) C f t E p with pz2 2 2 4 1 g B t N 1 g ln g 1 1 m 2 O 1 2 c O 10 3 Nc 1 sT T3 A C M. Asakawa, S.A. Bass, B.M., hep-ph/0603092 See Abe & Niu (1980) for effect in EM plasmas 35 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) 36 Summary Approches to calculate viscosity Kubo formula: via correlation functions of currents Transport theory: Boltzmann Eqs. (for weak scattering) ADS/CFT(strongly coupled) Lattice calculation (noisy) 37 Thanks 38 Renormalized diffusion 39 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 40 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 41 42 Response function 43 BS Eq. In Diagrams 44 Localization length Itinerant states ---- Localized States 45 II Some applications to N=4 SUSY YM Plasma: Equation of state in strong coupling: Plasma temperature = Hawking temperature Near Schwarzschild horizon z z h (1 2 ) ds2 d 2 1 4 2 2 1 2 dt d x z h2 z h2 Continuating to Euclidean time, t it 2 2d polarcoordinates , t zh To avoid a conic singularity at 0 , the period of t pz h 4 1 ds d 2 2 dt 2 2 dx 2 zh zh 2 2 Recalling the Matsubara formulation 1 T pz h 46 Free energy = temperature X (the gravity action without metric fluctuations) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998), hep-th/9803131. Consider a 4D Euclidean space of spatial volume V_3 at z The EH action of AdS-Schwarzschild: zh 1 dz V3 1 1 I EH 0 V3 (20 12) dt 5 4 4 16pG5 8pG5 zh 0 z The EH action of plain AdS I ( 0) EH 1 dz V3 1 0 V3 (20 12) dt 5 4 16pG5 8 p G z 5 0 ----- To eliminate the conic singularity, pz n ----- To match the proper length in Euclidean time 4 ( 0) 0 f 1 4 I GH 0 I GH 2 z h Plasma free energy: F 1 lim I 0 I ( 0) EH V3 p2 2 4 0 N c T V3 8 16pG5 z h4 0 Plasma entropy: p2 2 3 F S N c T V3 2 T V3 EH 47 Bekenstein-Hawking entropy: Gubser, Klebanov & Pest, PRD54, 3915 (1996) S BH 1 1 horizonarea (horizonarea) measuredin Planckunits 8 4 4 l P 1 where l P 10d PlancklengthG 8 ------ The metric on the horizon: ds2 1 2 2 d x d 5 z h2 T hehorizonarea V3 ( thesolid angle of S 5 ) p 6T 3V3 3 zh ------ The gravitational constant of the dual: S BH 1 2 2 3 p N c T V3 S plasma 2 G10 l 8 P p4 2 N c2 agree with the entropy extraced from the gravity action. 48 The ratio 3/4: The plasma entropy density at N c and s S / V3 1 2 2 3 Nc p T 2 The free field limit: the contents of N=4 SUSY YM number gauge potential 1 real scalars 6 6 1 2 2 3 1 2 2 3 Nc p T Nc p T 30 5 Weyl spinors 4 8 7 7 2 2 3 N c2p 2T 3 Nc p T 240 30 entropy density 1 2 2 3 Nc p T 30 2 2 2 2 Nc p T 3 s 3 0.75 (0) s 4 s (0) s 0 .8. The lattice QCD yields s0 49 Shear viscosity in strong coupling: Policastro, Son and Starinets, JHEP09, 043 (2002) Kubo formula 1 R lim 0 Im G xy, xy ( ,0) G R xy, xy ( , q) dtdxe i t i q x (t ) Txy ( x),Txy (0) where 50 Gravity dual: the coefficient of term of the gravity action hxy2 pT 1 2 2 2 ds fdt dx 2 du u 4u f 2 2 2 2 z u 2 zh f 1 u 2 0 u 1 51 The metric fluctuation pT 1 2 2 2 ds fdt dx 2 du h (t, z, u)dx dx u 4u f 2 2 2 Substituting into Einstein equation R 4g 0 g 0 x g x 1 and linearize u where h 2 2 hxy p T x y The Laplace equation of a scalar field 52 Calculation details: ------ Nonzero components of the Christofel (up to symmetris): ttu 2p 2T 2 u 4 1 utt 1 2 f uuu 2u f iju 2p 2T 2 f ij 1 f 1 2 f u uji 1 ij 2u i , j x, y , z ------ Nonzero components of the Ricci tensor: 4p 2T 2 Rtt f u 4p 2T 2 Rij ij u Ruu 1 u2 f Linear expansion: xyt withnonzerocomponents(up to symmetries) : 1 2f 1 2 xyz , z 1 2 1 2 ytx xty yzx xzy , z R R r rxy r x y 4h x y 1 2pT 1 2pT 2 2 xyu 2p 2T 2 f u , u 1 2 yux xuy , u with the only nonzero component u f u , zz 2u 3 , u 4 u u f u f 1 u , zz 2u 3 , u g g f u u x x 2 g 53 The solution: (t , z, u) (1 u) i ˆ 2 (1 u ) 1 ˆ 2 (u)e i ( qz t ) where ˆ q qˆ 2pT 2pT 2 d 2 1 i ˆ 2 2 d 2 u (1 u ) 2 1 (1 i)ˆ u 1 1 i ˆ u ˆ qˆ ˆ i du 2 4 du 2 u 0 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: u 1 i ˆ i ( qz t ) (1 u ) 2 e (t , z, u ) ~ i (1 - u ) 2 ˆ e i ( qz t ) incomingwave retardedcorrelator outgoingwave advancedcorrelator The incoming solution at low energy and zero momentum: (t , z, u) (1 u) i ˆ 2 ˆ 1 u 1 i ln O(ˆ 2 ) e it 2 2 54 T hequadratic termof I EH ( ) I GH ( ) 1 1 f N c2p 2T 4 du d 4 x 8 u 0 1 f 2 2 4 V N p T lim 4 c u 0 u 8 u u 2 i 1 V4 N c2pT 3 V4 G xyR , xy ( ,0) 16 2 i G xyR , xy ( ,0) N c2pT 3 8 1 8 N c2pT 3 Viscosity ratio: V_4 = 4d spacetime volume s Elliptic flow of RHIC: 1 0.08 4p s 0.1 Lattice QCD: noisy 55 III. Remarks: N=4 SYM is not QCD, since 1). It is supersymmetric 2). It is conformal ( no confinement ) 3). No fundamental quarks ---- 1) 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. ---- 3) may be improved, since heavy fundamental quarks may be introduced by adding D7 branes. ( Krach & Katz) Introducing an infrared cutoff ---- AdS/QCD: I EH 1 dz d 4 x g e R 12 16pG5 where thedilatonfield cz 2 Karch, Katz, Son & Stephenov 1 dt 2 dx 2 dz 2 2 z ----- Regge behavior of meson spectrum ---- confinement; ----- Rho messon mass gives c 338MeV; ----- Lack of string theory support. ds2 56 Deconfinement phase transition: Hadronic phase: I EH hadronic with ds 2 Plasma phase: I EH plasma wit h ds 2 Herzog, PRL98, 091601 (2007) 1 4 cz 2 R 12 dz d x g e 16pG5 1 2 2 2 dt d x dz z2 1 4 cz 2 R 12 dz d x g e 16pG5 1 4 4 4 2 2 4 4 4 1 p T z dt d x 1 p T z 2 z 1 dz Hawking-Page transition: I EH hadronic I EH plasma Tc 0.4917 c 191MeV 2 ---- First order transition with entropy jump N c ---- Consistent with large N_c QCD because of the liberation of quark-gluon degrees of freedom. 57 58 59 60 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 61 62 63 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/0603092, PRL See Abe & Niu (1980) for effect in EM plasmas 64