1. June 4, 2007@INPC2007 Tokyo Anomalous Viscosity of an Expanding Quark-Gluon Plasma Masayuki ASAKAWA Department of Physics, Osaka University S. A. Bass, B. Müller, M.A., Phys. Rev. Lett. 96 (2006) 252301 S. A. Bass, B. Müller, M.A., Prog. Theor. Phys. 116 (2006) 725

2. M. Asakawa (Osaka University) Success of Hydrodynamics at RHIC Teaney PRC 2003 Small shear viscosity very close to the “universal” lower bound Early Thermalization Two Important (Unexpected) Findings at RHIC  0 (thermalization time scale)  0.6 fm How are these explained? Are these related or unrelated physics?

3. M. Asakawa (Osaka University) Is QGP strongly coupled or not? Strong coupling is a natural explanation for the small  PROS J/  survives the deconfinement phase transition Hatsuda and M.A., PRL 2004 CONSNear the deconfinement phase transition, degrees of freedom are partons Success of Recombination P. Sorensen Possible many body correlation

4. M. Asakawa (Osaka University) Mechanism for Early Thermalization 2 → 2 scattering: not sufficient According to Hydro Models:  0 (thermalization time scale)  0.6 fm 2 → 3 processes ? Instability of Gauge Field due to Anisotropy (Weibel Instability) ? pzpz pypy pxpx beam Strong Longitudinal Flow Anisotropy in Momentum Space

5. M. Asakawa (Osaka University) Weibel Instability Weibel Instability (Weibel 1959) When particle distribution is anisotropic, instability ( filamentation instability ) exists Mrówczyński Exponential growth saturates when B 2 > g 2 T 4 Arnold, Moore, Yaffe Turbulent power spectrum Arnold and Moore time

6. M. Asakawa (Osaka University) What is viscosity? One of Transport Coefficients  : shear viscosity  : bulk viscosity The more Momentum Transport is prevented, the less viscosities become More Collisions Less  More Deflections Less 

7. M. Asakawa (Osaka University) Viscosity due to Turbulent Fields Perturbatively calculated viscosities: Viscosities due to Collisions Effective in suppressing Momentum Transport Turbulent Magnetic Field If this contribution to viscosity is added, total viscosity gets smaller Has been known as Anomalous Viscosity in plasma physics  A : anomalous viscosity  C : collisional viscosity

8. M. Asakawa (Osaka University) Result: How Anomalous? viscous stress/(sT) shear/T collisional stress anomalous stress Viscous Stress is NOT  shear Non-linear response Impossible to obtain on Lattice g-dependence ~1/g 6/5, while collisional viscosity (perturbative)  1/g 4 log(1/g) At large shear and/or in weak coupling, always anomalous viscosity dominates

9. M. Asakawa (Osaka University) Heuristic Understanding 1 For weakly coupled QGP, By expressing collisional f as and using pQCD expression for  tr (transport cross section) the (parametrically correct) pQCD collisional shear viscosity is obtained The anomalous viscosity is estimated similarly Note: Some claim, sudden increase of s is the reason of small  /s but, NOT ON FIRM GROUND Nucl. Phys. A, 2006

10. M. Asakawa (Osaka University) Heuristic Understanding 2 Momentum deflection by a single coherent domain of size r m Mean free path due to the action of the turbulent magnetic field (Anomalous mean free path) This yields the anomalous shear viscosity: Relation between 〈 B 2 〉, r m and anisotropy and shear ?

11. M. Asakawa (Osaka University) Heuristic Understanding 3 The parameter that gives the typical scale to the soft color field modes near thermal equilibrium: Debye mass: Anisotropy in momentum space:  (dimensionless) Wave vector domain of unstable modes: Coherent domain size: On the other hand, exponential growth of soft fields is saturated by nonlinearity of the Yang-Mills equation unlike in QED plasma case (by back reaction to current distribution) Exponential growth saturates when B 2 > g 2 T 4

12. M. Asakawa (Osaka University) Heuristic Understanding 4 When the exponential growth is saturated, Nonlinearity in Yang-Mills equations ~ Gradient term This implies 〈 B 2 〉 is estimated as Thus, anomalous shear viscosity is Then, relation between  and shear ?

13. M. Asakawa (Osaka University) Heuristic Understanding 5 Romatschke and Strickland’s parametrization of anisotropy By calculating stress tensors and comparing with the definition of  for the special case of 1-dim Bjorken flow From this relation and the previously obtained relation between  A and , if

14. M. Asakawa (Osaka University) Result: How Anomalous? viscous stress/(sT) shear/T collisional stress anomalous stress Viscous Stress is NOT  shear Non-linear response Impossible to obtain on Lattice g-dependence ~1/g 6/5, while collisional viscosity (perturbative)  1/g 4 log(1/g) At large shear and/or in weak coupling, always anomalous viscosity dominates n=1.5, 2, 2.5

15. M. Asakawa (Osaka University) Evolution of viscosity Initial state CGC ? QGP and hydrodynamic expansion Hadronization Hadronic phase and freeze-out Cross sections are additive  ~ f ~ 1/   Sum rule for viscosities:  Smaller viscosity dominates in system with two sources of viscosity ! Temperature evolution viscosity: ?  ~  A  ~  C  ~  HG

16. M. Asakawa (Osaka University) Summary and Outlook Do Turbulent Magnetic Fields also contribute to other observables, like Jet Energy Loss? In Plasma Physics, Anomalous Beam Energy Loss is also known We have shown that turbulent color magnetic field leads to anomalous viscosity At large shear and/or in weak coupling, always anomalous viscosity dominates Small viscosity does not necessarily imply strong coupling NOT linear response to shear (velocity gradient) Cannot be calculated on the lattice

17. M. Asakawa (Osaka University) Back Up

18. M. Asakawa (Osaka University) Theoretical Formulation 1 Transport Equation We start with the Vlasov equation in the extended phase-space: Correlation time/length for the color fields is short compared with the temporal change of the velocity of a plasma particle (= ultrarelativistic particle) and By expanding around for weak fields and taking ensemble average over the color fields next slide randomness of parton charge

19. M. Asakawa (Osaka University) Theoretical Formulation 2 Field Correlation Here, We assume the field correlations fall off with correlation time  and correlation length  Introduce the memory time (memory time felt by the parton) Then thermal partons move ultrarelativistically parton

20. M. Asakawa (Osaka University) Theoretical Formulation 3 Linear Response Let us assume a small perturbation of the thermal equilibrium distribution Suppose we are in the local rest frame of fluid, For shear viscosity, we take By calculating T ik with Chapman-Enskog formalism

21. M. Asakawa (Osaka University) Theoretical Formulation 4 Shear Viscosity For simplicity, let us consider the following case here: Color-Magnetic Fields: transverse to the collision axis In the collisionless case, This yields, collisionless: quarks and gluons contribute separately When collisions exist, these viscosities and collisional viscosity couple with each other Color-Electric and Magnetic Fields: correlators are spatially isotropic Another Scenario:

22. M. Asakawa (Osaka University) Theoretical Formulation 5 Anisotropy loop Following Romatschke and Strickland, introduce the anisotropy in momentum distribution:  For Longitudinal Boost-Invariant flow and massless parton gas, On the other hand,

23. M. Asakawa (Osaka University) Theoretical Formulation 6 Result By closing this loop, we obtain viscous stress/(sT) shear/T collisional stress anomalous stress n=1.5, 2, 2.5 The larger the shear is, the smaller the viscosity is !