1 Anomalous Viscosity of the Quark-Gluon Plasma Berndt Mueller – Duke University Workshop on Early Time Dynamics in Heavy Ion Collisions McGill University, 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  /s ? A heuristic argument for (  /s) min is obtained using s  4n : The uncertainty relation dictates that  f (  /n)  , and thus: All known materials obey this condition! For N=4 SU(N c ) SYM theory the bound is saturated at strong coupling:

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

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

7 Part II Anomalous Viscosity

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

9 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). pzpz pypy pxpx beam 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 Spontaneous color fields Color correlation length Time Length (z) Quasi- abelian Non- abelian Noise M. Strickland, hep- ph/

11 Anomalous viscosity - heuristic

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

13 Expansion  Anisotropy Anisotropic momentum distributions generate instabilities of soft field modes. Shear viscosity  and growth rate  controlled by f 1 (p). QGP X-space QGP P-space

14 Random fields  Diffusion

15 Shear viscosity Take moments of with p z 2 M. Asakawa, S.A. Bass, B.M., PRL 96: (2006) Prog Theo Phys 116:725 (2007)

16 Part III Formalities

17 Vlasov-Boltzmann eq. for partons parton distribution functions: color Lorentz force: 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 Φ el and Φ mag : Explicit form of Vlasov diffusion term: with the memory time:

19 Example: Transverse B a 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[f 1 ]: Following AMY, make the variational ansatz:

21 Parametric dependence Romatschke & Strickland convention: Perturbation of equilibrium distribution: Unstable modes: k inst 2 ≈  m D 2 Saturation condition: g|A| ≈ k inst

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

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. With  p  ~ 3T,  s^  ~ 18T 2 and s  4  one finds: Then the transport cross section is: Majumder, BM, Wang, hep- ph/

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

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

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

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.