Shear viscosity to entropy density ratio below QCD critical temperature

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

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

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

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

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

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

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

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

2) Background and motivation 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

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

Directed flow Elliptic flow y x QGP Hadrons

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

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. Time evolution after trap is turned off O ’Hara et al., Science 298, 2179 (2002) Small viscosity is common feature in strongly-coupled systems.

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

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

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

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

Linearized Boltzmann equation for B(p);

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

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

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

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).

S: statistical entropy

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). QGP Hadron KSS

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

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

Applicability of ChPT Melting of Chiral cond. Hadrons QGP ? Data

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

[CMF] Scatt. Amp. of ChPT

S: statistical entropy with

Shear viscosity to entropy density ratio below QCD critical temperature