1. Shear viscosity to entropy density ratio below QCD critical temperature Outline: 1)What is the shear viscosity? 2)Background and motivation 3)Shear viscosity/Entropy in Pionic gas 4)Summary and outlook Eiji Nakano Dept. of Physics, National Taiwan Univ. - Checking the viscosity/entropy ratio bound conjectured by string theory- April/21th/2006 at IoP, AS

2. 1) What is the shear viscosity? Shear viscosity (coefficient) is one of transport coefficients in macroscopic hydrodynamic equations for non-equilibrium systems: Where Local collective flow velocity: 2) Number conservation: Energy-momentum conservation: 1) Elementary volume: Basic equations:

3. appears in spatial traceless part (dissipative): Stress pressure (friction) in shear flow ( :coefficient of frictional force) The first term describes Perfect fluid dynamics (dissipationless) :

4. Let’s remember, 1) Isotropic pressure Unit cross-section The number of particle reflected by the cross-section per second: Thus the isotropic pressure becomes

5. 2) Anisotropic(stress) pressure Momentum transfer of x comp. per sec. across unit area normal to y direction: a frictional force facing -x direction Mean-free path: Scattering cross-section Maxwell formula

6. Viscos dynamics e.g., Diffusion equation for transverse momentum : diffusion constant: relaxes transverse fluctuation, in other words, diminishes the velocity gradient (shear flow).

7. Hierarchy in theories for space-time scales, mesoscopic macro micro Kinetic theories Fluid dynamics scales theories Hamiltonian Liouville eq. Linear response theory Boltzmann eq. GL eq. Langevin eq. Fluid eqs, e.g., in Navier-Stokes eq. Our attempt (T<m_pi) Jeon-Yaffe (1996) ~1fm ~100fm ~10^4fm

8. Basic properties of shear viscosity This can be also seen from more microscopic theory, Kubo formula: Auto correlation function of : Keep in mind that large cross section gives small viscosity. by S-G. Jeon (1995) is proportional to the mean-free path : Roughly speaking, (One has to resum infinite number of diagrams to get LO result even for weak coupling theory). Maxwell formula: Scattering cross-section LO

9. 1) A perturbative gravity analysis with a black hole metric corresponding to N=4 supersymmetric gauge field theory in strong coupling (Ads/CFT correspondence) conjectures a lower bound (KSS bound): Shear viscosity/entropy ratio : Kovtun, Son, Starinet, hep-th/0405231 2) Background and motivation

10. 2) Elliptic flow produced just after non-central relativistic heavy ion collisions (RHIC), Hadronic (chiral broken) phase Quark-Gluon Plasma (QGP), suggests that the system is near perfect fluid (small viscosity: ). It implies that expected QGP is in strong coupling regime. RHIC

11. Directed flow y x Elliptic flow QGP Hadrons

12. QCD phase diagram on Density-Temp. plane Karsch & Laermann, hep-lat/0305025 RHIC Tc ?

13. Recent trapped cold atom experiments give an opportunity to investigate strong interacting matter via tunable Feshbach resonance. This dilute and strongly-coupled system of Li6 also behaves hydrodynamically, showing elliptic flow. O ’Hara et al., Science 298, 2179 (2002) Time evolution after trap is turned off Small viscosity is common feature in strongly-coupled systems.

14. ….We investigate how the shear viscosity of QCD (pionic gas) behaves below Tc (chiral / deconfinement transition), with special attentions: a) How the viscosity behaves in Hadronic phase approaching Tc from below, b) How about ? Small or Large? taking the pionic gas…. Motivation:

15. 3) Shear visc./entropy in pionic gas in Kinetic theory Local equilibrium distribution, Small deviation (Dissipationless process) (Dissipative process) is given by as a functional of, which we will obtain from Boltzmann eq.. at local rest frame: Bose distribution function

16. The distribution function is obtained from Boltzmann eq. for, with collision integral ~Scattering cross-section

17. Strategy to obtain f(x,p) from Boltzmann eq. 1.Expand to the 1 st order 2.parametrize 3.Substitute it into Boltzmann eq. 4. Linearize the eq. in terms of 5.Expand using a set of specific polynomials 6. Linearized Bolzmann = Matrix eq. for A polynomial up to Finally, the viscosity is given by, Step Known (by symmetry) unknown

18. Linearized Boltzmann equation for B(p);

19. ChPT: effective theory on the basis of chiral symmetry (low energy limit: Weinberg theorem) Increase with collision energy! Pion-Pion scattering vanishes in massless limit! LO

20. coincide with the behavior in by Jeon,Yaffe, Heinz,Wang, etc… (Low energy limit)

21. From very naïve dimensional analysis, we find a power law in T: Non- monotonic! Universal behavior!

22. Intensive behavior at low T, divergent at T=0 ! But it seems to be typical for pure NG bosons with derivative couplings. This aspect is also seen for CFL phonon by Manuel etal (2004).

23. S: statistical entropy

24. 4) Summary and Outlook We have shown small ratio of the visc./entropy in Chpt approaching Tc of QCD: So we conclude that the small viscosity/entropy ratio <1 is not unique only above Tc, but below Tc. But it suggests discontinuity at Tc (~2times larger than KSS bound). Hadron QGP KSS

25. We are interested in shear visc. behavior in BCS-BEC crossover regime, above and below Tc. Superfluid phonon + …. Quasiparticle with fluctuations This work is close collaboration with Prof. J-W Chen at NTU. As future works Thank you for your attention…

26. Back up files Muroya and Sasaki, PRL(2005) Hadronic gas at finite density 1-2 rho_0

27. Applicability of ChPT ? HadronsQGP Data Melting of Chiral cond.

28. In 1 st Chapman-Enskog expansion, with parametrization Related to bulk viscosity to shear viscosity

29. [CMF] Scatt. Amp. of ChPT

30. with S: statistical entropy