1. Why is the Quark-Gluon Plasma a “Perfect” Liquid ?
Berndt Mueller – YITP / Duke
RIKEN Workshop
8-9 July 2006
Special thanks to M. Asakawa and S.A. Bass

2. Lecture I What does Lattice QCD tell us about the QGP ?
What do RHIC experiments tell us about the QGP ?
What is a “perfect” fluid ?
What are the origins of viscosity ?

3. What does Lattice QCD tell us about the QGP ?
A: So far, a lot about thermodynamic properties and response to static probes, a little bit about spectral functions, (almost) nothing about transport properties.

4. Lattice - EOS F. Karsch et al. RHIC
Indication of weak or strong coupling?

5. Phase coexistence Study of LQCD at fixed baryon density r = B/V
De Forcrand & Kratochvila
hep-lat/

6. Lattice - susceptibilities
pQGP
R. Gavai & S. Gupta, hep-lat/

7. Heavy quark potentials
Important for insight into medium effects on J/Y, 
Effective coupling as(r,T)
Kaczmarek et al.
Color singlet potential
Kaczmarek et al.

8. Lattice – spectral functions
Spectral functions via analytic continuation using the maximum entropy method: J/Y, etc.
(At present only for quenched QCD.)
Asakawa & Hatsuda
Karsch et al.
J/Y may survive up to 1.5 – 2 Tc !

9. What do RHIC experiments tell us about the QGP ?
A: So far, a lot about transport properties, a little bit about thermodynamic properties, and (almost) nothing about spectral functions and response to static probes.

10. Relativistic Heavy Ion Collider
The real road to the QGP
STAR
…is the
Relativistic Heavy Ion Collider

11. Liberation of saturated low-x glue fields (CGC)
Space-time picture
teq
Pre-equil. phase
Liberation of saturated low-x glue fields (CGC)
Bjorken formula

12. RHIC results Important results from RHIC:
Chemical (flavor) and thermal equilibration
Jet quenching = parton energy loss, high opacity
Elliptic flow = early thermalization, low viscosity
Collective flow pattern related to valence quarks
Strong energy loss of c and b quarks
Charmonium suppression not significantly increased compared with lower (CERN) energies
Photons unaffected by medium at high pT, medium emission at low pT in agreement with models

13. p0 vs. g in Au+Au (vs. p+p) No suppression for photons
Cross section in p+p coll’s
Area density of p+p coll’s in A+A
Yield in A+A
Suppression of hadrons
Without nuclear effects:
RAA = 1.

14. Photons from the medium
Experiment uses internal conversion of g
D’Enterria, Peressounko nucl-th/
t0 = 0.15 fm/c, T = 570 MeV
Turbide, Rapp, Gale PRC (2004)
t0 = 0.33 fm/c, T = 370 MeV
Hard Probes 2006, June 15, 2006 – G. David, BNL

15. Radiative energy loss Radiative energy loss: Lq
Radiative energy loss: q g L
Scattering centers = color charges
Density of scattering centers
Scattering power of the QCD medium:
Range of color force

16. q-hat at RHIC ? RHIC data QGP Pion gas Eskola et al. sQGP? RHIC
Cold nuclear matter
sQGP? ?
RHIC data
“Baier plot”

17. Collision Geometry: Elliptic Flow
Reaction
plane x z y
Bulk evolution described by relativistic fluid dynamics,
F.D assumes that the medium is in local thermal equilibrium,
but no details of how equilibrium was reached.
Input: e(x,ti), P(e), (h,etc.).
Elliptic flow (v2):
Gradients of almond-shape surface will lead to preferential expansion in the reaction plane
Anisotropy of emission is quantified by 2nd Fourier coefficient of angular distribution: v2
prediction of fluid dynamics

18. Elliptic flow is created early
spatial
eccentricity
momentum
anisotropy
time evolution of the energy density:
initial energy density distribution:
Anisotropic observables are of special interest as they reflect the dynamics of the earliest stages of the reaction, and thus the dynamics of the hottest phases, eventually the plasma phase that we hope to achieve in heavy ion collisions at RHIC. The overlap region of two nuclei at finite impact parameter is strongly deformed, which you can see here characterized by contours of constant wounded nucleon density in the plane orthogonal to the beam axis.
If the system is driven apart by the pressure in the system, then you see that it will expand faster into the direction of the strongest pressure gradients, and that is in the direction of the reaction plane. Upon building up an anisotropic flow field, the system will therefore get rounder with time, as you can see in those subsequent shots of the transverse plane at different times.
This reduction of the spatial eccentricity can be quantified by averaging the difference of the width in x direction and the width in y direction over this transverse distribution function. You see how this leads to a rapid decrease of the initial large geometric anisotropy over timescales relevant for heavy ion dynamics, once displayed here for a system with a very hard equation of state of a ideal gas, where the transition is very rapid, and once for an equation of state that involves a phase transition from hadronic to partonic degrees of freedom.
While the system becomes isotropic in coordinate space, it builds up anisotropies in momentum space, which you can characterize by the difference of the diagonal components of the energy momentum tensor. And here you see nicely how, as the system gets round in coordinate space, no further momentum anisotropy is built up. Note that for a system with a large mixed phase, the generation of further anisotropies is stalled significantly earlier. The momentum anisotropy you observe is really built up in the plasma stage!
The timescale suggested from all model calculations, no matter if macroscopic or microscopic is generally below 5 fm/c. Anisotropies thus probe the earliest stages of the collision.
P. Kolb, J. Sollfrank and U.Heinz, PRC 62 (2000)
Model calculations suggest that flow anisotropies are generated at the earliest stages of the expansion, on a timescale of ~ 5 fm/c if a QGP EoS is assumed.

19. v2(pT) vs. hydrodynamics
Failure of ideal hydrodynamics - tells us how hadrons form
Mass splitting characteristic property of hydrodynamics

20. Quark number scaling of v2
In the recombination regime, meson and baryon v2 can be obtained from the quark v2 :
T,m,v
 Emitting medium is composed of unconfined, flowing quarks.

21. From QGP to hadrons Agreement with data for hQGP = 0
Full 3-d Hydrodynamics
QGP evolution
Cooper-Frye
formula
UrQMD
t fm/c
hadronic rescattering
Monte Carlo
Hadronization TC
TSW
Nonaka & Bass, nucl-th/
Hirano et al. nucl-th/
Low (no) viscosity
High viscosity
Agreement with data for hQGP = 0

22. Bounds on h from v2 Relativistic viscous hydrodynamics:
Boost invariant hydro with T0t0 ~ 1 requires h/s ~ 0.1.
N=4 SUSY YM theory (g2Nc  1):
h/s = 1/4p (Policastro, Son, Starinets).
Absolute lower bound on h/s ?
h/s = 1/4p implies lf ≈ 0.3 d !
D. Teaney
QGP(T≈Tc) = sQGP ?

23. Ultra-cold Fermi-Gas
Li-atoms released from an optical trap (J. Thomas et al./Duke) exhibit elliptic flow analogous to that observed in relativistic heavy-ion collisions

24. What is an
“ideal” or “perfect”
liquid ?

25. Ideal gas vs. perfect liquid
An ideal gas is one that has strong enough interactions to reach thermal equilibrium (on a reasonable time scale), but weak enough interactions so that their effect on P(n,T) can be neglected.
This ideal can be approached arbitraily by diluting the gas and waiting very patiently (limit t   first, then V  ).
A perfect fluid is one that obeys the Euler equations, i.e. a fluid that has zero viscosity and infinite thermal conductivity.
There is no presumption with regard to the equation of state.
Even an imperfect fluid obeys the Euler equations in the limit of negligible velocity, density, and temperature gradients.

26. What is viscosity ?

27. Viscosity of plasmas
Shear viscosity of supercooled one-component plasma fluids:
Ichimaru
Interaction measure

28. Lower bound on h/s ?
Argument [Kovtun, Son & Starinets, PRL 94 (2005) ] based on duality between thermal QFT and string theory on curved background with D-dimensional black-brane metric, e.g.:
Kubo formula for shear viscosity:
Dominated by absorption of (thermal) gravitons by the black hole:

29. Lower bound on h/s – ctd.
A heuristic argument for (h/s)min is obtained using s ~ 4n :
But the uncertainty relation dictates that tf (e/n)  , and thus:
(It is unclear whether this relation holds in the nonrelativistic domain, where s/n can be much larger than 4. But is obeyed by all known substances.)
For N=4 susy SU(Nc) Yang-Mills the bound is saturated at strong coupling:

30. Exploring strong coupling
Ability to perform analytical strong coupling calculations in N=4 susy SU(Nc) YM and success with h have motivated other applications:
Equation of state [Gubser, Klebanov, Tseytlin, hep-th/ ]
Spectral densities [Teaney, hep-ph/ ]
Jet quenching parameter [Liu, Rajagopal, Wiedemann, hep-ph/ ]
Heavy quark energy loss [Herzog, Karch, Kovtun, Kozcaz, Yaffe,hep-th/ ]
Heavy quark diffusion [Casalderrey-Solana, Teaney, hep-ph/ ]
Drag force on heavy quark [Gubser, hep-th/ ]
…and continuing!

31. Some results from duality
(3+1)-D world r0
horizon

32. Lecture II Does “perfect” fluidity imply “strong coupling” ?
What is “anomalous” viscosity ?
Derivation of the anomalous viscosity
Manifestations of anomalous QGP transport processes

33. Is strong coupling really necessary for small h/s ?
Today…
…we ask the question:
Is strong coupling really necessary for small h/s ?

34. What is viscosity ?

35. Stellar accretion disks
“A complete theory of accretion disks requires a knowledge of the viscosity. Unfortunately, viscous transport processes are not well understood. Molecular viscosity is so small that disk evolution due to this mechanism of angular momentum transport would be far too slow to be of interest. If the only source of viscosity was molecular, then n ~ h/r ~ l vT, where l is the particle mean free path and vT is the thermal velocity. Values appropriate for a disk around a newly formed star might be r ~ 1014 cm, n ~ 1015 cm-3, s ~ 10-16 cm2, so that l ~ 10 cm, and vT ~ 105 cm/s . The viscous accretion time scale would then be r2/(12n) > 1013  yr! Longer by a factor of than the age conventionally ascribed to such disks. Clearly if viscous accretion explains such objects, there must be an anomalous source of viscosity. The same conclusion holds for all the other astronomical objects for which the action of accretion disks have been invoked.”
(From James Graham – Astronomy 202, UC Berkeley)
The solution is: String theory?
Unfortunately, NO.

36. “Anomalous”, i.e. non-collisional viscosity
Anomalous viscosity B
Differentially rotating disc with weak magnetic field B shows an instability (Chandrasekhar)
Spontaneous angular momentum transfer from inner mass to outer mass is amplified by interaction with the rotating disk and leads to instability (Balbus & Hawley – 1991).
“Anomalous”, i.e. non-collisional viscosity

37. A ubiquitous phenomenon
Anomalous viscosity:
A ubiquitous phenomenon

38. Anomalous viscosity on the WWW
Google search:
Results of about 322,000 for anomalous viscosity. (0.22 seconds) 
Chaotic Dynamics, abstract chao-dyn/
Anomalous Viscosity, Resistivity, and Thermal Diffusivity of the Solar Wind Plasma
Authors: Mahendra K. Verma (IIT Kanpur, India)
In this paper we have estimated typical anomalous viscosity, resistivity, and thermal difffusivity of the solar wind plasma. Since the solar wind is collsionless plasma, we have assumed that the dissipation in the solar wind occurs at proton gyro radius through wave-particle interactions. Using this dissipation length-scale and the dissipation rates calculated using MHD turbulence phenomenology [Verma et al., 1995a], we estimate the viscosity and proton thermal diffusivity. The resistivity and electron's thermal diffusivity have also been estimated. We find that all our transport quantities are several orders of magnitude higher than those calculated earlier using classical transport theories of Braginskii. In this paper we have also estimated the eddy turbulent viscosity.

39. Anomalous viscosity - origins

40. Anomalous viscosity - usage
Plasma physics:
A.V. = large viscosity induced in nearly collisionless plasmas by long-range fields generated by plasma instabilities.
Astrophysics - dynamics of accretion disks:
A.V. = large viscosity induced in weakly magnetized, ionized stellar accretion disks by orbital instabilities.
Biophysics:
A.V. = 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 (From:

41. Can the QGP viscosity be anomalous?
Can the extreme opaqueness of the QGP (seen in experiments) be explained without invoking super-strong coupling ?
Answer may lie in the peculiar properties of turbulent plasmas.
Plasma “turbulence” = random, nonthermal excitation of coherent field modes with power spectrum similar to the vorticity spectrum in a turbulent fluid [P(k) ~ 1/k2].
Plasma turbulence arises naturally in plasmas with an anisotropic momentum distribution (Weibel-type instabilities).
Expanding plasmas (such as the QGP at RHIC) always have anisotropic momentum distributions.
Soft color fields generate anomalous transport coefficients, which may give the medium the character of a nearly perfect fluid even at moderately weak coupling.

42. QGP viscosity – collisions
Baym, Heiselberg, ….
Danielewicz & Gyulassy, Phys. Rev. D31, 53 (85)

43. QGP viscosity – anomalous

44. 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 – 1959).
Such conditions are satisfied in HI collisions:
Longitudinal expansion locally “red-shifts” the longitudinal momentum components of released small-x gluon fields (CGC) from initial state: pz py px
beam
In EM case, instabilities saturate due to effect on charged particles. In YM case, field nonlinearities lead to saturation (competition with Nielsen-Olesen instability?)

45. Weibel (two-stream) instabilityv r

46. HTL formalism k ~ gT (gQs) k ~ T (Qs) “soft” “hard”
Nonabelian Vlasov equations describe interaction of “hard” and “soft” color field modes and generate the “hard-thermal loop” effective theory:
k ~ gT (gQs)
k ~ T (Qs)
Effective HTL theory permits systematic study of instabilities of “soft” color fields.

47. HTL instabilities
x=10
q=p/8
Wavelength and growth rate of unstable modes can be calculated perturbatively:
Mrowczynski, Strickland et al., Arnold et al.

48. From instability to “turbulence”
Exponential growth saturates when
B2 > g2 T4.
Quasi-abelian growth
Non-abelian growth
Kolmogorov-type power spectrum of coherent field excitations
[Arnold, Moore, Yaffe, hep-ph/ ]
Turbulent power spectrum

49. Color correlation length
Space-time picture
M. Strickland, hep-ph/
Color correlation length
Time
Length (z)
Quasi-abelian
Non-abelian
Noise

50. (Sorry – using Chapman-Enskog)
Anomalous viscosity
Formal derivation
(Sorry – using Chapman-Enskog)

51. Expansion  Anisotropy
QGP
X-space
QGP
P-space
Anisotropic momentum distributions generate instabilities of soft field modes. Growth rate G ~ f1(p).
Shear flow always results in the formation of soft color fields;
Size controlled by f1(p), i.e. (u) and h/s.

52. Turbulence  Diffusion

53. Shear viscosity Take moments of with pz2
M. Asakawa, S.A. Bass, B.M., hep-ph/
See Abe & Niu (1980) for effect in EM plasmas

54. ηA - the feedback loop Longitudinal flow induces momentum anisotropy:
Anisotropy grows with shear viscosity η
Soft color fields are proportional to Δ:
The anomalous viscosity is inversely proportional
to B2:
Shear viscosity η stabilizes due to: _ pz
Self-consistency

55. Time evolution of viscosity
Initial
state
CGC ?
QGP and
hydrodynamic
expansion
Hadronization
Hadronic phase
and freeze-out
viscosity:
? ?
Smallest viscosity dominates in system with several sources of viscosity

56. Manifestations? Possible effects on QGP probes: Jet
Longitudinal broadening of jet cones (observed – “ridge”)
Anomalous diffusion of charm and bottom quarks (observed)
Synchrotron-style radiation of soft, nonthermal photons ?
Field induced quarkonium dissociation ?
No unstable modes for quarks: quasi-particle picture of QGP is compatible with low viscosity z y x B
Jet
Au+Au 20-30% a b c

57. Summary
Conclusion:
Because the matter created in heavy ion collisions expands rapidly, it forms a turbulent color plasma, which has an anomalously small shear viscosity. Transport phenomena involving quarks and gluons are strongly influenced by the turbulent color fields, especially at early times, when the expansion is rapid.

58. Additional slides

59. Anisotropic HTL modes

60. Shear perturbation

61. Estimate of hA

62. Turbulence  Diffusion

63. Shear viscosity
hep-ph/
PRL (June30)

64. Diffusion  Viscosity Diffusion corresponds to “anomalous” viscosity:
But recall that B2 itself depends on anisotropy f1(p) ~ viscosity h !

65. ηB - the feedback loop Longitudinal flow induces momentum anisotropy:
Anisotropy grows with shear viscosity η
Soft color fields are proportional to Δ:
The anomalous viscosity is inversely proportional
to B2:
Shear viscosity η stabilizes due to: _ p pz
The Münchhausen effect

66.  hB indeed dominates at weak coupling !
hB vs. hC
Compare anomalous viscosity
with HTL (weak coupling) result for collisional viscosity
 hB indeed dominates at weak coupling !
g2T3h g2 T3 g2