The status of the KSS bound and its possible violations (How perfect a fluid can be?)

1. The status of the KSS bound and its possible violations (How perfect a fluid can be?) Antonio Dobado Universidad Complutense de Madrid Ten Years of AdS/CFT Buenos Aires December 2007

2. Holography Viscosity

3. Outline • Introduction • Computation ofh/sfrom the AdS/CFT correspondence • The conjecture by Kovtun, Son, Starinets (KSS) • The RHIC case • Can the KSS bound be violated? • h/s and the phase transition • Conclussions and open questions

4. Viscosity in Relativistic Hydrodynamics: The effective theory desribing the dymnamics of a system (or a QFT) at large distances and time Perfect fluids: Entropy conservation for m = 0 Including dissipation: local rest frame (Landau-Lifshitz) projector conserved currents Fick’s Law

5. Hydrodynamic modes: Look for normal modes of linerized hydrodynamics Examples: difussion law complex because of dissipation pole in the current-current Green function From the energy-momentum tensor equation: Shear modes (transverse) h/s is a good way to characterize the intrinsic ability of a system to relax towards equilibrium Sound mode (longitudinal) speed of sound Notice that for conformal fluids

6. Kinetic theory computation of viscosity: distribtuion function Boltzmann (Uehling-Uhlenbenck) Equation Chapman-Enskog thermal distribution

7. linearized equation energy-momentum tensor in kinetic theory variation of the energy-momentum tensor for fp outside equilibrium

8. When can we apply kinetic theory? The mean free path must be much larger than the interaction distance (two well defined lengh scales) mean free path Typically low density, weak interacting, systems Examples: a) For a NR system Like a hard sphere gas Maxwell’s Law: b) For a relativistic system like massless QFT More Interaction Less Viscosity

9. The Kubo formula for viscosity in QFT: Linear response theory coupling operators to an external source then retarded Green function Consider the energy-momentum tensor as our operator coupled to the metric. for curved space-time by comparison with linear response theory

10. The Kovtun-Son-Starinets holographic computation Consider a QFT with gravitational dual For example for N=4, SU(NC) SYM The dual theory is a QFT at the temperature T equal to the Hawking temperature of the black-brane. natural units

11. Consider a graviton polarized in the x-y direction and propagating perpendiculary to the brane (Klebanov) In the dual QFT the absorption cross-section of the graviton by the brane measures the imaginary part of the retarded Green function of the operator coupled to the metric i.e. the energy-momentum tensor Then we have

12. The absorption cross-section can be computed classically only non-vanishing component, x and y independent Einstein equations Linearized Einstein equations Equation for a minimally coupled scalar

13. Theorem: (Das, Gibbons, Mathur, Emparan) The scalar cross-section is equal to the area of the horizon For but Notice that the result does not depend on the particular form of the metric. It is the same for Dp, M2 and M5 branes. Basically the reason is the universality of the graviton absortion cross-section.

14. Computation of the leading corrections in inverse powers of the ‘t Hooft coupling for N=4, SU(NC) SYM (Buchel, Liu and Starinets) Apéry´s constant 1/(l2ln(1/l))

15. An interesting conjecture: The KSS bound h/s>1/4p For any system described by a sensible* QFT Kovtun, Son and Starinets, PRL111601(2005) *Consistent and UV complete.

16. This bound applies for common laboratory fluids

18. Trapped atomic gas T. Schaefer, cond-mat/0701251

20. RHIC: The largest viscosimeter: (Relativistic Heavy Ion Collider at BNL) Length = 3.834 m ECM = 200 GeV A L = 2 1026 cm-2 s-1 (for Au+Au (A=197)) Experiments: STAR, PHENIX, BRAHMS and PHOBOS

21. Main preliminar results from RHIC Thermochemical models describes well the different particle yields for T=177 MeV, mB = 29 Mev forECM = 200 GeV A From the observed transverse/rapidity distribution the Bjorken model predicts an energy density at t0 = 1 fm of 4 GeV fm-3 whereas the critical density is about 0.7 GeV fm-3, i.e. matter created may be well above the threshold for QGP formation. A surprising amount of collective flow is observed in the outgoing hadrons, both in the single particle transverse momentum distribution (radial flow) and in the asymmetric azimuthal distribution around the beam axis (elliptic flow).

22. Bose-Einstein spectrum as an indication of thermal equilibrium

23. Thermochemical model (STAR data, review by Steinberg nucl-ex/0702020)

24. Main preliminar results from RHIC Thermochemical models describes well the different yields for T=177 MeV, mB = 29 Mev, ECM = 200 GeV A From the observed transverse/rapidity distribution the Bjorken model predicts an energy density at t0 = 1 (0.5) fm of 4 (7 ) GeV fm-3 whereas the critical density is about 0.7 GeV fm-3, i.e. matter created may be well above the threshold for QGP formation. A surprising amount of collective flow is observed in the outgoing hadrons, both in the single particle transverse momentum distribution (radial flow) and in the asymmetric azimuthal distribution around the beam axis (elliptic flow).

25. Expected phase diagram for hadron matter RHIC Ruester, Werth,, Buballa, Shovkovy, Rischke

26. Main preliminar results from RHIC Thermochemical models describes well the different yields for T=177 MeV, mB = 29 Mev, ECM = 200 GeV A From the observed transverse/rapidity distribution the Bjorken model predicts an energy density at t0 = 1 fm of 4 GeV fm-3 whereas the critical density is about 0.7 GeV fm-3, i.e. matter created may be well above the threshold for QGP formation. A surprising amount of collective flow is observed in the outgoing hadrons, both in the single particle transverse momentum distribution (radial flow) and in the asymmetric azimuthal distribution around the beam axis (elliptic flow).

27. Transverse momentum distribution Hydrodynamical model for proton production Kolb and Rapp

28. Central collision: On average, azimuthally symmetric particle emission Elliptic flow Peripheral collision: The region of participant nucleons is almond shaped

29. <Px>2 - <Py>2 <Px>2 +<Py>2 = v2 Peripheral collision: In order to have anisotropy (elliptic flow) the hydrodynamical regime has to be stablished in the overlaping region. Viscosity would smooth the pressure gradient and reduce elliptic flow Found to be much larger than expected at RICH

30. Low multiplicity High multiplicity

31. Hydrodynamical elliptic flow in trapped Li6 at strong coupling

32. Main conclussions from the preliminar results from RHIC Fluid dynamics with very low viscosity reproduces the measurements of radial and elliptic flow up to transverse momenta of 1.5 GeV. Collective flow is probably generated early in the collision probably in the QGP phase before hadronization. The QGP seems to be more strongly interacting than expected on the basis of pQCD and asymptotic freedom (hence low viscosity). Some preliminary estimations of h/sbased on elliptic flow (Teaney, Shuryak) and transverse momentum correlations (Gavin and Abdel-Aziz) seems to be compatible with value close to 0.08 (the KSS bound)

33. If the KSS conjecture is correct there is no perfect fluids in Nature. Is this physically acceptable? In non relativistic fluid dynamics is well known the d’Alembert paradox (an ideal fluid with no boundaries exerts no force on a body moving through it, i.e. there is no lift force. Swimming or flying impossible). More recently, Bekenstein et al pointed out that the accreation of an ideal fluid onto a black hole could violate the Generalized Second Law of Thermodynamics suggesting a possible connection between this law and the KSS bound.

34. Larger s Larger N Is it possible to violate the KSS bound? We have two strategies to break the bound h/s Lower h increase s

35. Increasing the cross-sections is forbidden by unitarity: Example: LSM Kinetic computation Low-energy-approximation (brekas unitarity at higher energies) Reintroducing the Higgs reestablish unitarity and the KSS bound

36. Example: Hadronic matter (mB= 0) Lowest order Quiral Perturbation Theory (Weinberg theorems) Violation of the bound about T = 200 MeV (Cheng and Nakano)

37. Unitarity and Partial Waves: Quiral Perturbation Theory (Momentum and mass expansion)

38. The Inverse Amplitude Method A.Dobado, J.R.Peláez Phys. Rev.D47, 4883 (1993). Phys. Rev.D56 (1997) 3057

39. Chiral Perturbation Theory plus Dispersion Relations Simultaneous description of  and KK up to 800-1000 MeV including resonances The Inverse Amplitude Method Lowest order ChPT (Weinberg Theorems) is only valid at very low energies

40. Unitarity reestablishes the KSS bound!

41. Violation of the KSS bound in non-relativistic highly degenerated system: In principle it could be possible to avoid the KKS bound in a NR system with constant cross section and a large number g of non-identical degenerated particles by increasing the Gibbs mixing entropy (Kovtun, Son, Starinets, Cohen..) However the KKS bound is expected to apply only to systems that can be obtained from a sensible (UV complete) QFT Is it possible to find a non-relativistic system coming from a sensible QFT that violates the KKS bound for large degeneration?

42. To explore this possibility we start from the NLSM coset space coset metric amplitude total cross section De Broglie thermal wave length viscosity hard sphere gas viscosity number density entropy density non relativistic limit

43. Now we can complete the NLSM in at least two different ways The first one just QCD since the NLSM is the lagrangian of CHPT at the lowest order with g=3 Two flavors QCD Low temperature and low density regime QCD beta function for NC=3 different from Quiral coset NLSM coset

44. The second one is the LSM multiplet LSM lagrangian vacuum state potential Higgs field Higgs mass coset space Cross-section viscosity For large enough g KSS violated!

45. Gedanken example: Dopped fullerenes C60 , C82 ... 1) Available in mgrams 2) Sublimation T=750K 3) Caged metallofullerenes 4) BN substitutions 5) Hard sphere gas

46. Dopped fullerenes

47. Large number of isomers !! Can have species: N! (N-M)! M! N M = To reach a factor 20 000 in Gibbs log with two substitutions, need 200 sites three50 four28 five21 However in practice it does not work since the sublimation temperature is too high. It seems that there are not atomic or molecular systems violating the KSS bound in Nature. This applies to superfluid helium too, which has always some viscosity above the KSS bound. (Andronikashvili)

48. Minimun of h/s and phase transition Recently Csernai, Kapusta and McLerran made the observation that in all known systems both happen at the same point.

49. In a gas: As T->¥, less p transport, h->¥

50. In a liquid: (Mixture of clusters and voids) Atoms push the others to fill the voids As T->0, less voids, less p transport h->¥