On viscosity of Quark Gluon Plasma

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On viscosity of Quark Gluon Plasma

• Defu Hou
• CCNU , Wuhan

RHIC-Star full TOF detector and related physics
in China Hangzhou April 27-29
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Outlines

• Introduction and motivation
• Viscosity from Kubo formula
• Viscosity from kinetic theory (Boltzmann Eq)
• Viscosity from AdS/CFT
• Summary

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QCD under extreme conditions

• At very High T or density ( deconfined)
• High T (Early universe, heavy-ion collisions)
• High density matter ( in the core of neutron
  stars)

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Motivations
Experiments aspect

• _at_ RHIC
• Robust collective flows, well described by ideal
  hydro with Lattice-based EoS. This indicates very
  strong interaction even at early time gt sQGP
• sQGP seems to be the almost perfect fluid known
  ?/sgt .1-.2ltlt1

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Study of dissipative effects on ltv2gt
How sensitive is elliptic flow to finite ?/s?
Viscous Hydro
Cascade (2lt-gt2,2lt-gt3)
Z. Xu C. Greiner, PRL 101(08)
P. Romatschke, PRL99 (07)
Agreement for ?s0.3 0.6
Dependence on tp relaxation time II0 order
expansion with green terms (D. Rischke)
?/s0.15 0.08
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Theoretic aspect

• To calculate Trsp. Coefs. in FT in highly
  nontrivial
• (nonperturbative ladder resummation) (c
  around 5)
• String theory method AdS/CFT (D.Son et al 2003)
• ?/s 1/4?
• . Kinetic theory uncertainty principle
  (Gyulassy)

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Main obstacle for theory

• QCD in nonperturbative regime (T200Mev)
• Pertburb. Expansion of QCD is not well behaved
  for realistic T
• For thermodyn.,one can use lattice and
  resummation techniques
• Kinetic coefficients are difficult to extract
  from lattice

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Shear Viscosity
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Viscosity from Kubo formula
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Nonlinear Response
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S. Jeon, PRD 52 Carrington, Hou, Kobes,
PRD61
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Carrington, Hou, Kobes, PRD64 (2001)
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Hou, hep-ph/0501284
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Viscosity from kinetics theory
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Viscosity of hot QCD at finite density

• Boltzmann Equation

Recast the Boltzmann equation
P.Arnold, G.D.Moore and G.Yaffe, JHEP 0011(00)001
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Shear viscosity
With a definition of inner product and expanded
distribution functions,
where
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Collision terms
\chi term
Scattering amplitude
Distribution function term
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Matrix Element
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• Variation method gives

Liu, Hou, Li EPJC 45(2006)
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Computing transport coefficients from AdS/CFT
In the regime described by a gravity dual the
correlator can be computed using
AdS/CFT
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AdS/CFT at finite temperature
Classical Supergravity on AdS-BHS5
Witten 98
4dim. Large-Nc strongly coupled SU(Nc) N4 SYM at
finite temperature (in the deconfinement phase).
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Field Theory Gravity
Theory
Gauge Theories QCD
Quantum Gravity String theory
Holography
the large N limit Supersymmetric Yang Mills
N large
Gravitational theory in 10 dimensions
Calculations Correlation functions
Quark-antiquark potential
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AdS/CFT now being applied to RHIC physics

• Viscosity, ?/s.
• EOS
• Jet quenching
• Sound waves
• Photon production
• Friction
• Heavy quarkonium
• Hardron spectrum (ADS/QCD)

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Universality of shear viscosity in the regime
described by gravity duals
Gravitons component obeys equation for a
minimally coupled massless scalar. But then
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Since the entropy (density) is
we get
D. Son, P. Kovtun, A.S., hep-th/0405231
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Shear viscosity in SYM
P.Arnold, G.Moore, L.Yaffe, 2001
Correction to A.Buchel, J.Liu,
A.S., hep-th/0406264
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A viscosity bound conjecture
P.Kovtun, D.Son, A.S., hep-th/0309213,
hep-th/0405231
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Universality of

Theorem
For any thermal gauge theory (with zero
chemical potential), the ratio of shear
viscosity to entropy density is equal to
in the regime described by a corresponding
dual gravity theory
Remark
Gravity dual to QCD (if it exists at all) is
currently unknown.
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Possible Mechanisms for Low viscosity

• Large cross-section, strong coupling
• Anomalous viscosity turbulence

M. Asakawa, S.A. Bass, B.M., hep-ph/0603092,
PRL See Abe Niu (1980) for effect in EM plasmas
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Take moments of
with pz2
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Low viscosity due to Anderson Local.

• AL effect renders infinite reduces viscosity
  significantly even at weak coupling
• Mechanismcoherent backscattering (CBS) effect

Ginaaki, Hou , Ren PRD 77(2008)
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Summary
Approches to calculate viscosity

• Kubo formula via correlation functions of
  currents
• Transport theory Boltzmann Eqs. (for weak
  scattering)
• ADS/CFT(strongly coupled)
• Lattice calculation (noisy)

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• Thanks

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Renormalized diffusion
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Weak Localization (WL)

• Anderson proposed (58) that electronic diffusion
  can vanish in a random potential (AL)
• Experiments detected ( Ishimaru 1984,Wolf Maret
  1985)
• Mechanismcoherent backscattering (CBS) effect
• after a wave is multiply scattered many times,
  its phase coherence is preserved in the
  backscattering direction, the probability of back
  scattering is enhenced via constructive
  interference

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Viscosity with random medium

• System quasi-particles in random potential

• Candidate disorder in sQGP ?
• The islands of heavy state bound states
  (Shuryak)
• 2. The reminiscent of confinement vaccum, say
  the domain structure of 't Hooft's monopole
  condensation
• The disoriented chiral condensate (DCC)
• 4. CGC

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Response function
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BS Eq. In Diagrams
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• Localization length

• Itinerant states ----? Localized States

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II Some applications to N4 SUSY YM Plasma
Equation of state in strong coupling
Plasma temperature Hawking temperature
Near Schwarzschild horizon
Continuating to Euclidean time,
To avoid a conic singularity at
, the period of
Recalling the Matsubara formulation
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Free energy temperature X (the gravity action
without metric fluctuations) E.
Witten, Adv. Theor. Math. Phys. 2, 505 (1998),
hep-th/9803131.
Consider a 4D Euclidean space of spatial volume
V_3 at The EH action of AdS-Schwarzschild
The EH action of plain AdS
----- To eliminate the conic singularity, -----
To match the proper length in Euclidean time
Plasma free energy
Plasma entropy
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Bekenstein-Hawking entropy
Gubser, Klebanov Pest, PRD54, 3915 (1996)
------ The metric on the horizon
------ The gravitational constant of the dual
agree with the entropy extraced from the gravity
action.
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The ratio 3/4
The plasma entropy density at
The free field limit
the contents of N4 SUSY YM number entropy density
gauge potential 1
real scalars 6
Weyl spinors 4
The lattice QCD yields
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Shear viscosity in strong coupling
Policastro, Son and Starinets, JHEP09, 043 (2002)
Kubo formula
where
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Gravity dual the coefficient of
term of the gravity action
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The metric fluctuation
Substituting into Einstein equation
and linearize
The Laplace equation of a scalar field
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Calculation details
------ Nonzero components of the Christofel (up
to symmetris)
------ Nonzero components of the Ricci tensor
Linear expansion
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The solution
Heun equation (Fucks equation of 4 canonical
singularities) ------trivial when energy and
momentum equatl to zero ------low
energy-momentum solution can be obtained
perturbatively.
The boundary condition at horizon
The incoming solution at low energy and zero
momentum
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V_4 4d spacetime volume
Viscosity ratio
Elliptic flow of RHIC
Lattice QCD noisy
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III. Remarks
N4 SYM is not QCD, since 1). It is
supersymmetric 2). It is conformal ( no
confinement ) 3). No fundamental quarks ---- 1)
and 2) may not be serious issues since sQGP is
in the deconfined phase at a nonzero
temperature. The supersymmetry of N4 SYM
is broken at a nonzero T. ---- 3) may be
improved, since heavy fundamental quarks
may be introduced by adding D7 branes. ( Krach
Katz)
Introducing an infrared cutoff ---- AdS/QCD
Karch, Katz, Son Stephenov
----- Regge behavior of meson spectrum ----
confinement ----- Rho messon mass gives -----
Lack of string theory support.
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Deconfinement phase transition Herzog, PRL98,
091601 (2007)
Hadronic phase
Plasma phase
Hawking-Page transition
---- First order transition with entropy jump
---- Consistent with large N_c QCD because of the
liberation of quark-gluon degrees of
freedom.
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Epilogue

• AdS/CFT gives insights into physics of thermal
  gauge theories in the nonperturbative regime
• Generic hydrodynamic predictions can be used to
  check validity of AdS/CFT
• General algorithm exists to compute transport
  coefficients and the speed of sound in any
  gravity dual
• Model-independent statements can presumably be
  checked experimentally

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Mechanisms for Low viscosity

• Large cross-section, strong coupling
• Anomalous viscosity turbulence
• Soft color fields generate anomalous
  transport coefficients, which may give the medium
  the character of a nearly perfect fluid even at
  moderately weak coupling

M. Asakawa, S.A. Bass, B.M., hep-ph/0603092,
PRL See Abe Niu (1980) for effect in EM plasmas