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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Introduction and motivation Viscosity from Kubo formula Viscosity from kinetic theory (Boltzmann
Eq) Viscosity from AdS/CFT Summary
Outlines
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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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@ RHIC
Robust collective flows, well described by ideal hydro with Lattice-based EoS. This indicates very strong interaction even at early time => sQGP
sQGP seems to be the almost perfect fluid known /s>= .1-.2<<1
Motivations
Experiments aspect:
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Study of dissipative effects on <vStudy of dissipative effects on <v22>>How sensitive is elliptic flow to finite /s?
Z. Xu & C. Greiner, PRL 101(08)
Agreement for s=0.3 – 0.6 /s=0.15 – 0.08
Viscous Hydro Cascade (2<->2,2<->3)
P. Romatschke, PRL99 (07)
Dependence on relaxation timeII0 order expansion with green terms (D. Rischke)
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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 (T~200Mev) 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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yv
AF xx
Shear Viscosity
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10
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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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Boltzmann Equation
Fluctuation of distribution (s: species)
Recast the Boltzmann equation
P.Arnold, G.D.Moore and G.Yaffe,
JHEP 0011(00)001
Viscosity of hot QCD at finite density
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Shear viscosity
With a definition of inner product and expanded distribution functions,
where
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Collision terms
Performing the integral over dk’ with the help of
Scattering amplitude
Distribution function term \chi 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-BH×S5
4dim. Large-Nc strongly coupledSU(Nc) N=4 SYM at finite temperature(in the deconfinement phase).
conjecture
=
Witten ‘98
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Field Theory Gravity Theory=
Gauge TheoriesQCD
Quantum GravityString theory
the large N limitSupersymmetric Yang Mills
Gravitational theory in 10 dimensionsN large
Calculations Correlation functionsQuark-antiquark potential
Holography
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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
Graviton’s component obeys equation for a minimally coupled massless scalar. But then .Since the entropy (density) is we get
D. Son, P. Kovtun, A.S., hep-th/0405231
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Shear viscosity in SYM
Correction to : A.Buchel, J.Liu, A.S., hep-th/0406264
P.Arnold, G.Moore, L.Yaffe, 2001
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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 chemicalpotential), the ratio of shear viscosity to entropy density is equal to in the regime describedby a corresponding dual gravity theory
Remark:Gravity dual to QCD (if it exists at all) is currentlyunknown.
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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, PRLSee Abe & Niu (1980) for effect in EM plasmas
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Take moments of
with pz2( ) ( , , )r p p
p
p D p f r p t C ft E
2
23
4
2
12
3
111
101 ln 1mc
c A C
g gOT
g BNON sT
M. Asakawa, S.A. Bass, B.M., hep-ph/0603092See Abe & Niu (1980) for effect in EM plasmas
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Low viscosity due to Anderson Local. AL effect renders infinite reduces viscosity
significantly even at weak coupling
Mechanism:coherent backscattering (CBS) effect
Ginaaki, Hou , Ren PRD 77(2008)
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Summary
Kubo formula: via correlation functions of currents
Transport theory: Boltzmann Eqs. (for weak scattering)
ADS/CFT(strongly coupled) Lattice calculation (noisy)
Approches to calculate viscosity
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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) Mechanism:coherent 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 mediumSystem: quasi-particles in random potential
Candidate disorder in sQGP ?1. The islands of heavy state; bound states
(Shuryak); 2. The reminiscent of confinement vaccum,
say the domain structure of 't Hooft's monopole condensation;
3. 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 N=4 SUSY YM Plasma:
Equation of state in strong coupling: Plasma temperature = Hawking temperature
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222
22
2
141 )1(
xdz
dtz
dds
zz
hh
h
Near Schwarzschild horizon
Continuating to Euclidean time, it
hhh zd
zd
zdds 2 , scoordinatepolar 2d 14 2
222
222 x
To avoid a conic singularity at 0 , the period of hz
Recalling the Matsubara formulation
hzT
1
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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 atThe EH action of AdS-Schwarzschild:
z
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5
3
053
5
118
)1220(16
10h
z
EH zGV
zdzdtV
GI
h
The EH action of plain AdS
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3
053
5
)0( 18
)1220(16
10
GV
zdzdtV
GI EH
----- To eliminate the conic singularity,----- To match the proper length in Euclidean time
nz
00 2
1 )0(4
4
GHGHh
IIz
f
Plasma free energy:
342
2
45
3)0(0 816
00lim1 VTNzG
VIIF c
hEHEH
Plasma entropy:
332
2
23
VTNTFS c
V
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Bekenstein-Hawking entropy:
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areahorizon 41unitsPlanck in measured area)(horizon
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PBH l
S
81
length Planck d10 where GlP
------ The metric on the horizon :
3365
33
25
22
2
) of angle solid the( areahorizon The
1
VTSzV
ddz
ds
h
h
x
------ The gravitational constant of the dual: 2
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10 2 cP N
lG
plasmacBH SVTNS 3322
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agree with the entropy extraced from the gravity action.
Gubser, Klebanov & Pest, PRD54, 3915 (1996)
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The ratio 3/4:
The plasma entropy density at and cN322
3 21/ TNVSs c
The free field limit:
322322
307
24078 TNTN cc
the contents of N=4 SUSY YM number entropy density
gauge potential 1
real scalars 6
Weyl spinors 4
322322
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3016 TNTN cc
322
301 TN c
222)0(
32 TNs c
The lattice QCD yields
75.043
)0( ss
.8.00
ss
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Shear viscosity in strong coupling:
Kubo formula
Policastro, Son and Starinets, JHEP09, 043 (2002)
where
)0(),()(),(
)0,(Im1lim
,
,0
xyxyxitiR
xyxy
Rxyxy
TxTtedtdG
G
qxq
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Gravity dual: the coefficient of term of the gravity action
2xyh
22
2222
2
41 du
fudfdt
uTds x
10 1 22
2
uufzzu
h
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The metric fluctuation
Substituting into Einstein equation
04 gR and linearize
The Laplace equation of a scalar field
dxdxuzthdufu
dfdtuTds ),,(
41 22
2222
2 x
xyxy h
Tuh
xg
xg 22 where 01
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Calculation details:
zyxjiuuf
f
ff
ufTuT
ijiuj
tut
uuuij
uij
utt
,,, 21 1
21
221 2 12 22422
------ Nonzero components of the Christofel (up to symmetris):
fuR
uTRf
uTR uuijijtt 2
2222 1 4 4
------ Nonzero components of the Ricci tensor:
uyxu
xyuz
yxz
xyz
yxt
xyt
uuxyz
zxy
txy ufT
f
,21 ,
21
21
,2 ,21
21
:)symmetries to(up components nonzero with
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Linear expansion:
4,2,2
1
component nonzeroonly the with
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uzzyx
uf
uuu
fu
Tr
rRR
x
ggxgu
fu
uufu
Thr uzzy
xy
x
21,2,
214 3
2
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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: 1u
correlator advanced waveoutgoing )-(1
correlator retarded waveincoming )1(~),,(
)(ˆ2
)(ˆ2
tqzii
tqzii
eu
euuzt
The incoming solution at low energy and zero momentum:
tii
eOuiuuzt
)ˆ(
21ln
2ˆ
1)1(),,( 2ˆ2
Tqq
Teuuuuzt tqzi
i
2ˆ
2ˆ where)()1()1(),,( )(ˆ
21ˆ
2
04ˆˆ
21ˆˆˆ11ˆ)1(1)1(
2222
2
22
uiiq
duduiui
duduu
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32, 8
)0,( TNiG cR
xyxy
32
81 TN c
Viscosity ratio: 08.041
s
Elliptic flow of RHIC:
Lattice QCD: noisy
1.0s
V_4 = 4d spacetime volume
)0,(21
16
lim81
81
)()( of termquadratic The
,432
4
0422
4
1
0
24422
Rxyxyc
ucc
GHEH
GVTNViuu
fTNVuu
fxdduTN
II
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III. Remarks:N=4 SYM is not QCD, since1). It is supersymmetric2). 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 N=4 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:
2222
2
2
4
5
1 fielddilaton thewhere
1216
1
dzddtz
ds
cz
RegxddzG
I EH
x
----- Regge behavior of meson spectrum ---- confinement;----- Rho messon mass gives ----- Lack of string theory support.
MeV; 338c
Karch, Katz, Son & Stephenov
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Deconfinement phase transition: Herzog, PRL98, 091601 (2007)
Hadronic phase:
Plasma phase:
2222
2
4
5hadronic
1with
1216
1 2
dzddtz
ds
RegxddzG
I czEH
x
dzzTddtzTz
ds
RegxddzG
I czEH
1444224442
2
4
5plasma
111with
1216
1 2
x
Hawking-Page transition:
---- First order transition with entropy jump
MeV1914917.0plasmahadronic
cT
II
c
EHEH
2cN
---- 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, PRLSee Abe & Niu (1980) for effect in EM plasmas
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