Anomalous Viscosity of the Quark-Gluon Plasma

1. Anomalous Viscosity of theQuark-Gluon Plasma Berndt Mueller – Duke University Workshop on Early Time Dynamics in Heavy Ion Collisions McGill University, 16-19 July 2007 Special credits to M. Asakawa S.A. Bass and A. Majumder

2. Part I Viscosity

3. What is viscosity ?

4. Lower bound on h/s ? A heuristic argument for (h/s)min is obtained using s 4n : The uncertainty relation dictates that tf (e/n), and thus: All known materials obey this condition! For N=4 SU(Nc) SYM theory the bound is saturated at strong coupling:

5. QGP viscosity – collisions Baym, Gavin, Heiselberg Danielewicz & Gyulassy Arnold, Moore & Yaffe

6. Low T (Prakash et al.) using experimental data for 2-body interactions. High T (Yaffe et al.) using perturbative QCD RHIC data 1/4 QCD matter What can cause the very low /s ratio for the matter produced in nuclear collisions at RHIC? There are two logical possibilities: The quark-gluon plasma is a strongly coupled state, not without well defined quasiparticle excitations; There is a non-collisional (i.e. anomalous) mechanism responsible for lowering the shear viscosity.

7. Part II Anomalous Viscosity

8. A ubiquitous concept Google search: Results 1 - 10 of about 571,000 for anomalousviscosity. (0.24 seconds)  From Biology-Online.org Dictionary: anomalous viscosity The viscousbehaviour of nonhomogenous fluids or suspensions, e.g., blood, in which the apparent viscosityincreases as flow or shear ratedecreases toward zero.

9. py px beam pz Color instabilities Spontaneous generation of color fields requires infrared instabilities. Unstable modes in plasmas occur generally when the momentum distribution of a plasma is anisotropic (Weibel instabilities). Such conditions are satisfied in HI collisions: Longitudinal expansion locally “red-shifts” the longitudinal momentum components of small-x gluon fields released from initial state: In EM case, instabilities saturate due to effect on charged particles. In YM case, field nonlinearities lead to saturation.

10. Color correlation length Time Non-abelian Quasi-abelian Noise Length (z) Spontaneous color fields M. Strickland, hep-ph/0511212

11. Anomalous viscosity - heuristic

12. (Longitudinal) expansion Momentum anisotropy QGPlasma instabilities Anomalous viscosity The Logic

13. QGP X-space QGP P-space Expansion  Anisotropy Anisotropic momentum distributions generate instabilities of soft field modes. Shear viscosity  and growth rate Gcontrolled byf1(p).

14. Random fields  Diffusion

15. Shear viscosity Take moments of with pz2 M. Asakawa, S.A. Bass, B.M., PRL 96:252301 (2006) Prog Theo Phys 116:725 (2007)

16. Part III Formalities

17. parton distribution functions:color Lorentz force: Vlasov-Boltzmann eq. for partons Perturbative solution for octet distribution: yielding a diffusive Vlasov term:

18. Random (turbulent) color fields Assumption of color chaos: • Short-range, Gaussian correlations of fields with functions Φel and Φmag : • Explicit form of Vlasov diffusion term: with the memory time:

19. Example: Transverse Ba only Additional assumption: (satisfied at early times) Diffusive Vlasov term: Balance between drift and Vlasov term gives: Anomalous viscosities for gluons and quarks:

20. Complete Shear Viscosity Minimization of full Vlasov-Boltzmann functional W[f1]: Following AMY, make the variational ansatz:

21. Parametric dependence • Romatschke & Strickland convention: • Perturbation of equilibrium distribution: Unstable modes: kinst2 ≈ mD2 Saturation condition: g|A| ≈ kinst

22. Who wins? Smallest viscosity dominates in system with several sources of viscosity Time dependence of turbulent color field strength: Interestingly, a (magnetic) gauge field expectation value also arises in the linearly expanding N=4SYM solution (hep-th/0703243):

23. Part IV Is the QGP weakly or strongly coupled ? What exactly do we mean by this statement?

24. Connecting  with q^ Hard partons probe the medium via the density of colored scattering centers: If kinetic theory applies, the same is true for thermal quasi-particles. Assumptions: - thermal QP have the same interactions as hard partons; - interactions are dominated by small angle scattering. Then the transport cross section is: With p ~ 3T, s^ ~ 18T2 and s  4 one finds: Majumder, BM, Wang, hep-ph/0703085

25. Perturbative gluon plasma: Turbulent gluon plasma: From RHIC data: Examples   ?

26. Strongly coupled N=4 SYM: Chiral limit of QCD for T << Tc (pion gas): From RHIC data: Counter-examples    ?

27. At strong coupling, is a more faithful measure of medium opacity. QCD N=4 SYM (fT)4 (ln T)-4 /s ~1 1 ? RHIC data: Strong vs. weak coupling strong strong weak weak

28. Conclusion Summary: The matter created in heavy ion collisions forms a highly (color) opaque plasma, which has an extremely small shear viscosity. The question remains whether the matter is a strongly coupled plasma without any quasiparticle degrees of freedom, or whether it is a marginal quasiparticulate liquid with an anomalously low shear viscosity due to the presence of turbulent color fields, especially at early times, when the expansion is most rapid. Jets constitute the best probe to ascertain the structure of this medium. The extended dynamic range of RHIC II and LHC will be essential to the success of this exploration, but so will be sophisticated 3-D models and simulations of the collision dynamics and their application to jet quenching.