Introduction eta/s in field theory Quasiparticle systems Conclusions On the lower bound of the viscosity to entropy denstiy ratio Antal Jakov´ ac BME Technical University Budapest A. Jakovac and D. Nogradi, arXiv:0810.4181 A. Jakovac, arXiv:0901.2802 A. Jakovac, PhysRevD.81.045020 [arXiv:0911.3248] Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Outlines Introduction 1 Transport in plasma Lower bound for η/ s eta/s in field theory 2 Generic formulae Transport coefficient Entropy density Quasiparticle systems 3 Small width case Wave function renormalization High temperature effects System with zero mass excitations Conclusions 4 Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Outlines Introduction 1 Transport in plasma Lower bound for η/ s eta/s in field theory 2 Generic formulae Transport coefficient Entropy density Quasiparticle systems 3 Small width case Wave function renormalization High temperature effects System with zero mass excitations Conclusions 4 Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Transport in plasma Outlines Introduction 1 Transport in plasma Lower bound for η/ s eta/s in field theory 2 Generic formulae Transport coefficient Entropy density Quasiparticle systems 3 Small width case Wave function renormalization High temperature effects System with zero mass excitations Conclusions 4 Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Transport in plasma Gas vs. fluid Inhomogeneous distribution of conserved charge density N ⇒ way to equilibrium if elementary excitations are weakly interacting mean free path is large independent smoothing of charge excess at each point homogenization before equilibration ballistic regime (gases) strongly interacting systems mean free path is small induced currents depend on the local environment ˙ ⇒ linear response theory J = − D ∇ N ⇒ N = D △ N equilibration before homogenization diffusive regime (fluids) Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Transport in plasma Diffusion constant The order of magnitude of the diffusion constant D : ( τ lifetime, ℓ mean free path, v velocity, σ cross section) δ N = D δ N D ∼ ℓ v ∼ v 2 τ ∼ v ˙ N = D △ N ⇒ ⇒ ℓ 2 τ n σ D is large in weakly, small in strongly interacting systems D ∼ ( g 4 ln g ) − 1 in PT. description sensible only in strongly interacting (fluid) systems Determination of D in QFT from linear response C ( x ) = � [ J i ( x ) , J i (0)] � C ( k = 0 , ω ) ⇒ D = lim ω ω → 0 Boltzmann equations Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Transport in plasma Viscosity transport coefficient in momentum transport ⇒ viscosity η ∼ ̺ v 2 τ ∼ ǫτ v ∼ η △ v (Navier-Stokes) ⇒ ̺ ˙ ( ǫ energy density) damping rate of small perturbations: Γ = 4 k 2 η s . 3 T Typical values of η/ s : water at room temperature ∼ 30 superfluid 4 He at λ -point ∼ 0 . 8 smallest at the phase transition point ⇓ what is the smallest value for η/ s ? (R.A.Lacey, A. Taranenko, PoSCFRNC2006:021 (2006), [arXiv:nucl-ex/0610029v3]) Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Lower bound for η/ s Outlines Introduction 1 Transport in plasma Lower bound for η/ s eta/s in field theory 2 Generic formulae Transport coefficient Entropy density Quasiparticle systems 3 Small width case Wave function renormalization High temperature effects System with zero mass excitations Conclusions 4 Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Lower bound for η/ s η argumentation: η ∼ ǫτ, s ∼ n ⇒ s ∼ E τ ∼ > � P. Danielewicz, M. Gyulassy, PRD 31, 53 (1985); P. Kovtun, D.T. Son, A.O. Starinets PRL 94, 111601 (2005). calculation: for N = 4 SYM theory at N c ≫ 1 , λ = g 2 N c ≫ 1 from graviton absorbtion in the dual 5D AdS gravity: η s = 1 4 π (P. Kovtun, D.T. Son, A.O. Starinets JHEP 0310, (2003) 064.) for weaker coupling we expect larger ratio: indeed, first λ, N c corrections are positive (R.C. Myers, M.F. Paulos, A. Sinha, arXiv:0806.2156) universal for a wide class of theories (A. Buchel, R.C. Myers, M.F. Paulos, A. Sinha, Phys.Lett.B669:364-370,2008.; M. Haack, A. Yarom, arXiv:0811.1794) so far we did not find counterexamples experimentally ⇒ commonly accepted lower bound for η/ s Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Lower bound for η/ s MC studies In principle exact method to measure η/ s . . . measure � T 12 ( x ) T 12 (0) � correlator on lattice ⇒ Euclidean discrete time we need the spectral function, which is related to the correlator as ∞ π C ( ω, k = 0) cosh( ω ( β/ 2 − τ )) � d 3 x � T 12 ( τ, x ) T 12 (0) � = � d ω . sinh βτ/ 2 0 invert this relation with the prior knowledge C ( ω > 0) > 0 Maximal Entropy Method, or ad hoc solutions too little sensitivity to small ω regime ⇒ large systematical uncertainties; additional assumptions are needed best estimates η/ s = 0 . 102 (56) at T = 1 . 24 T c (H. B. Meyer, Phys. Rev. D 76, 101701 (2007)) ⇒ needs analytic control! Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Lower bound for η/ s RHIC data Non-central heavy ion collisions have initial anisotropy. Time evolution of anisotropy: the larger the viscosity, the more extent the initial anisotropy is washed out 25 ideal 0.08 ideal η /s=0.03 η /s=0.03 20 η /s=0.08 η /s=0.08 η /s=0.16 η /s=0.16 0.06 PHOBOS STAR v 2 (percent) 15 v 2 0.04 10 0.02 5 0 0 0 100 200 300 400 0 1 2 3 4 N Part p T [GeV] (P. Romatschke, U. Romatschke, Phys.Rev.Lett.99:172301,2007.) upper bound: η < 0 . 16 ⇒ is there a lower bound? s ∼ Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Lower bound for η/ s RHIC data: is there a lower bound? 25 ideal η /s=0.03 20 η /s=0.08 ⇒ η /s=0.16 STAR v 2 (percent) 15 10 5 0 0 1 2 3 4 p T [GeV] P. Romatschke, U. Romatschke, PRL.99:172301,2007. R.A. Lacey, A. Taranenko, R. Wei, arXiv:0905.4368 Quadratic correction in p T is expected (D.A. Teaney, arXiv:0905.2433) � Statistically η s < 1 4 π is not excluded (favored: η � ≈ 0 . 9 ± 0 . 07 ) � s � � p T � =0 4 πη/ s = 1 only at infinite coupling and N c ! RHIC seriously challenges the lower bound! invalidate? Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Lower bound for η/ s Theoretical caveats N = 4 SYM theory is not QCD higher curvature and dilaton corrections to 5D AdS actions η s < 1 ⇒ 4 π is conceivable dual theory? unitarity? Counterexample: N non-interacting species ⇒ η is not η 1 changed, s ∼ ln N (mixing entropy) ⇒ s ∼ ln N (A. Cherman, T. D. Cohen, and P. M. Hohler, JHEP 02, 026 (2008), 0708.4201.) metastable system? what is the case with limited N ? Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Lower bound for η/ s Theoretical caveats E τ ∼ > � ? In fact ∆ E τ ∼ > � ! First is true only if E > ∆ E Condition for (small width) quasiparticle system ⇒ the argumentation is applicable only in quasiparticle systems in QFT there is always a continuum – effect on η/ s ? 1.0 pure hadron gas vs. hadron 0.8 ⇒ gas with continuum 0.6 Η � s considerable difference 0.4 0.2 0.0 0.15 0.16 0.17 0.18 0.19 T � GeV � (J. Noronha-Hostler, J. Noronha and C. Greiner, PRL 103, 172302 (2009), 0811.1571) Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
Introduction eta/s in field theory Quasiparticle systems Conclusions Lower bound for η/ s We need exact statements about η s from first principles! Result: there is a non-universal lower bound at finite entropy density; for smal s : � η s � ∼ N Q LT 4 , � s � min where N Q : number of relevant quantum channels (species) L : interaction range Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, H´ ev´ ız
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