Why is the Quark-Gluon Plasma a

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
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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.
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Lattice - EOS
F. Karsch et al.
Indication of weak or strong coupling?
5
Phase coexistence
Study of LQCD at fixed baryon density r B/V De
Forcrand Kratochvila hep-lat/0602024
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Lattice - susceptibilities
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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.
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Lattice spectral functions
Spectral functions via analytic continuation
using the maximum entropy method J/Y, etc. (At
present only for quenched QCD.)
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.
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The real road to the QGP
is the
Relativistic Heavy Ion Collider
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Space-time picture
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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

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p0 vs. g in AuAu (vs. pp)
No suppression for photons
Suppression of hadrons
Without nuclear effects RAA 1.
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Photons from the medium
Experiment uses internal conversion of g
DEnterria, Peressounko nucl-th/0503054 t0
0.15 fm/c, T 570 MeV
Turbide, Rapp, Gale PRC 69 014903 (2004) t0
0.33 fm/c, T 370 MeV
Hard Probes 2006, June 15, 2006 G. David, BNL
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Radiative energy loss
Radiative energy loss
Scattering centers color charges
Density of scattering centers
Range of color force
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q-hat at RHIC
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Collision Geometry Elliptic Flow

• 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

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Elliptic flow is created early
P. Kolb, J. Sollfrank and U.Heinz, PRC 62 (2000)
054909
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.
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v2(pT) vs. hydrodynamics
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Quark number scaling of v2
In the recombination regime, meson and baryon v2
can be obtained from the quark v2
? Emitting medium is composed of unconfined,
flowing quarks.
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From QGP to hadrons
Nonaka Bass, nucl-th/0510038 Hirano et al.
nucl-th/0511046
Low (no) viscosity
High viscosity
Agreement with data for hQGP 0
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Bounds on h from v2
Relativistic viscous hydrodynamics
D. Teaney
QGP(TTc) sQGP ?
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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

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What is an ideal or perfect liquid ?
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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.

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What is viscosity ?
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Viscosity of plasmas
Shear viscosity of supercooled one-component
plasma fluids
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Lower bound on h/s ?
Argument Kovtun, Son Starinets, PRL 94 (2005)
111601 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
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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 N4 susy SU(Nc) Yang-Mills the bound is
saturated at strong coupling
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Exploring strong coupling

• Ability to perform analytical strong coupling
  calculations in N4 susy SU(Nc) YM and success
  with h have motivated other applications
• Equation of state Gubser, Klebanov, Tseytlin,
  hep-th/9805156
• Spectral densities Teaney, hep-ph/0602044
• Jet quenching parameter Liu, Rajagopal,
  Wiedemann, hep-ph/0605178
• Heavy quark energy loss Herzog, Karch, Kovtun,
  Kozcaz, Yaffe,hep-th/0605158
• Heavy quark diffusion Casalderrey-Solana,
  Teaney, hep-ph/0605199
• Drag force on heavy quark Gubser, hep-th/0605182
  
• and continuing!

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Some results from duality
(31)-D world
r0
horizon
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Lecture II

• Does perfect fluidity imply strong coupling ?
• What is anomalous viscosity ?
• Derivation of the anomalous viscosity
• Manifestations of anomalous QGP transport
  processes

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Today
we ask the question Is strong coupling really
necessary for small h/s ?
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What is viscosity ?
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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) gt 1013  yr! Longer by a
factor of 105 - 106 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) http//grus.berkeley.e
du/jrg/ay202/lectures.html
The solution is String theory?
Unfortunately, NO.
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Anomalous viscosity
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
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Anomalous viscosity A ubiquitous phenomenon
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Anomalous viscosity on the WWW
Google search Results 1 - 10 of about 322,000
for anomalous viscosity. (0.22 seconds) 
Chaotic Dynamics, abstract chao-dyn/9509002 Anoma
lous 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.
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Anomalous viscosity - origins
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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
  http//www.biology-online.org/dictionary)

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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.

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QGP viscosity collisions
Baym, Heiselberg, . Danielewicz Gyulassy,
Phys. Rev. D31, 53 (85)
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QGP viscosity anomalous
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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
In EM case, instabilities saturate due to effect
on charged particles. In YM case, field
nonlinearities lead to saturation (competition
with Nielsen-Olesen instability?)
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Weibel (two-stream) instability
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HTL formalism
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.
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HTL instabilities
Wavelength and growth rate of unstable modes can
be calculated perturbatively
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From instability to turbulence
Kolmogorov-type power spectrum of coherent field
excitations Arnold, Moore, Yaffe, hep-ph/0509226
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Space-time picture
M. Strickland, hep-ph/0511212
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Anomalous viscosity
Formal derivation (Sorry using Chapman-Enskog)
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Expansion ? Anisotropy

• 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.

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Turbulence ? Diffusion
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Shear viscosity
Take moments of
with pz2
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?A - the feedback loop
p?
pz
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Time evolution of viscosity
viscosity
? ?
Smallest viscosity dominates in system with
several sources of viscosity


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Manifestations?

• Possible effects on QGP probes
• 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

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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.
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Additional slides
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Anisotropic HTL modes
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Shear perturbation
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Estimate of hA
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Turbulence ? Diffusion
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Shear viscosity
hep-ph/0603092 PRL (June30)
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Diffusion ? Viscosity
Diffusion corresponds to anomalous viscosity
But recall that ?B2? itself depends on anisotropy
f1(p) viscosity h !
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?B - the feedback loop
p?
pz
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hB vs. hC
Compare anomalous viscosity with HTL (weak
coupling) result for collisional viscosity
? hB indeed dominates at weak coupling !