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Anomalous Viscosity of theQuark-Gluon Plasma
Berndt Mueller – Duke University
Workshop on Early Time Dynamicsin Heavy Ion Collisions
McGill University, 16-19 July 2007
Special credits to M.
Asakawa S.A. Bass
and A. Majumder
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Part I
Viscosity
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What is viscosity ?
( ) ( )23
and are defined as coefficients in the expansion of the stress tensor in gradients of the velocity fie
viscosityld
Shear b k:
ul
ik i k i k k i ik ikik i kT u u P u uu u u uε δ ς δη δ= + + ∇ + +∇ − ∇⋅ ∇ ⋅−
13
tr
3
tr 2
Microscopically, is given by the rate of momentum transport:
Unitarity limit on cross sections suggests that has a lower bound: 3
412
fpnp
pp
λσ
π
η
η
η
πησ
≈ =
≤ ≥⇒
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Lower bound on η/s ?
A heuristic argument for (η/s)min is obtained using s 4n :
( )1 13 12v
vf
fn p sn
λ εη τ⎛ ⎞ ⎛ ⎞≈ ≈⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
The uncertainty relation dictates that f (/n) , and thus:
12 4s sη
π≥ ≈
h h
All known materials obey this condition!
For N=4 SU(Nc) SYM theory the bound is saturated at strong coupling:
( )3/ 22
135 (3)14 8 c
s
g N
ςηπ
⎡ ⎤⎢ ⎥= + +⎢ ⎥⎣ ⎦
L
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QGP viscosity – collisions
( )
( )
( ) ( )
13
1( )tr
2tr 2
Classical expression for shear viscosity:
Collisional mean free path in medium:
Transport cross section in QCD medium:
Collisional shear
5
11
viscos
2 l
i
2
t
n 1
f
Cf
s s
s ss
np
n
Ip
I
η λ
λ σ
πσ α α
α αα
−
≈
=
≈
⎛ ⎞= + + −⎜ ⎟
⎝ ⎠
2 1tr
9y of QGP
100 ln
:
Cs s
T sησ πα α −≈ ≈
Baym, Gavin, HeiselbergDanielewicz & GyulassyArnold, Moore & Yaffe
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What can cause the very low η/s ratio for the matter produced in nuclear collisions at RHIC?
There are two logical possibilities:(i) The quark-gluon plasma is a strongly coupled
state, not without well defined quasiparticle excitations;
(ii) There is a non-collisional (i.e. anomalous) mechanism responsible for lowering the shear viscosity.
Low T (Prakash et al.)using experimental data for 2-body interactions.
1/4π
High T (Yaffe et al.)using perturbative QCD
RHIC data
QCD matter
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Part II
Anomalous Viscosity
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A ubiquitous concept
Google search:Results 1 - 10 of about 571,000 for
anomalous viscosity. (0.24 seconds)
From Biology-Online.org Dictionary:
anomalous viscosity
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.
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Color instabilitiesSpontaneous generation of color fields requires infrared instabilities. Unstable modes in plasmas occur generally when the momentum distribution of a plasma is anisotropic (Weibel instabilities).
( ) ( ) ( )22 22 1forx y s zs
p p Q pQ
τΔ = Δ = Δ? ?pz
py
px
beam
Such conditions are satisfied in HI collisions: Longitudinal expansion locally “red-shifts” the longitudinal momentum components of small-x gluon fields released from initial state:
In EM case, instabilities saturate due to effect on charged particles. In YM case, field nonlinearities lead to saturation.
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Spontaneous color fields
Color correlation
lengthTime
Length (z)
Quasi-abelian
Non-abelian
Noise
M. Strickland, hep-ph/0511212
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Anomalous viscosity - heuristic
p
pΔaB
mr
( )
13
2 2( )
2 2 2 2
Classical expression for shear viscosity:
Momentum change in one coherent domain:
Anomalous mean free path in medium:
Anomalous viscosity due to random color fie
f
a am
Af m
m
np
p gQ B r
p prg Q B rp
η λ
λ
≈
Δ ≈
≈ ≈Δ
33 94
2 2 2 2 2 2
lds:
3Am m
sTnpg Q B r g Q B r
η ≈ ≈
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The Logic
(Longitudinal) expansion
Momentum anisotropy
QGPlasma instabilities
Anomalous viscosity
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Expansion Anisotropy
Perturbed equilibrium distribution:
f ( p) = f0(π) 1+ f1(π) 1± f0(π)( )⎡⎣ ⎤⎦f0(π)=xπ[−uμ π
μ / T]
For σηar fλow of uλrarλa. fλuid:
f1(π)=−Δ(π)EπT
2πiπ j−1
3d ij( ) ∇u( )ij
∇u( )ij= 1
2∇iuj+∇jui( )−1
3d ij∇⋅u
η =−1
15Td 3π(2π)3
π4
Eπ2
∂f0∂Eπ
∫ Δ(π)
Anisotropic momentum distributions generate instabilities of soft field modes. Shear viscosity η and growth rate controlled by f1(p).
QGPX-space
QGP
P-space
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Random fields Diffusion
[ ]
( )
Vlasov-Boltzmann transport of thermal partons:
with Lorentz force
Assuming , random Fokker-Planck eq:
w
( , , )
v
( ,
ith f
)( ) ,
di
p p
r pp
a a a
rp
p F f r p t C ft E
F gQ E B
p f r p t C ft E
D p
E B
⎡ ⎤∂ + ⋅∇ + ⋅∇ =⎢ ⎥∂⎢ ⎥⎣ ⎦
= + ×
⎡ ⎤∂ ⎡ ⎤+ ⋅∇ − =⎢ ⎥ ⎣ ⎦∂⎢ ⎥⎣ ⎦
⇒
∇ ⋅ ⋅∇
( ) ( )( )
fusion coefficient
.' ( '), ' ,i i jj
t
dt F r t t F rp tD−∞
= ∫
If diffusion is dominated bychromo-magnetic fields:
dt ' B(t ')B(t)∫ ≡ B2 μ
( )r t r=
,a aE B
( ')r t
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Shear viscosity
Take moments of
with pz2( , , )( )r p p
p
D pp f r p t C ft E
⎡ ⎤∂ ⎡ ⎤+ ⋅∇ −∇ ⋅ ⋅∇ =⎢ ⎥ ⎣ ⎦∂⎢ ⎥⎣ ⎦
( ) ( )22
1
3
2
3
4 ln 111 111
0mc
A Cc
FNON
gsT
gOT
τ
η ηη
−−= ≡ +
−+
M. Asakawa, S.A. Bass, B.M.,
PRL 96:252301 (2006) Prog Theo Phys 116:725
(2007)
Dij ( p) = d' Fia r('),'( )U ab(r,r)Fj
b r,( )−∞
∫
rFa =g
rEa + rv×
rBa( ) = coλor forc
dt ' Fi+(')F+i()∫ = F2 μ ≡q
= j quncηing πaraμ r !!!
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Part III
Formalities
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Vlasov-Boltzmann eq. for partons
vμ ∂
∂xμ f r,π,( )+ gFa ⋅∇π fa r,π,( )+C f⎡⎣ ⎤⎦=0
f a p, x( )=−igC2
Nc2 −1
d 4k
2π( )4∫ d 4 ′x U ab x, ′x( )
ik⋅x− ′x( )
v⋅k+ iFb ′x( )⋅∇π f π( )∫
gFa x( )⋅∇π fa π,x( )=−ig
C2
Nc2 −1
Fa x( )⋅∇π
d 4k
2π( )4∫ d 4 ′x U ab x, ′x( )
ik⋅x− ′x( )
v⋅k+ iFb ′x( )⋅∇π f π( )∫
( ) ( )( ) ( ), , , ,
, , ,
,
, ,a af
f
t dQQ f Q t
t dQ f Q t
=
=
∫∫
r p r
r r
p
p p
%%
a a a= + ×F vE B
parton distribution functions: color Lorentz force:
Perturbative solution for octet distribution:
yielding a diffusive Vlasov term:
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Random (turbulent) color fields
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ,ab aba b a b
i j i jfU x x UF x F x F x F x fx x′ ′=′ ′p p
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )
(el)
(ma
(el)
(magg) )
,
,
, 0
a b a ai j i j
a b a ai j i j
a bi j
ab
ab
ab
U x x
U x x
U
x x
x x
x
t t
t
x
t
x x
σ
σ
τ
τ
′ ′′ ΦΦ −
′Φ −
−
′Φ −
=
′ =
′ =
′
′
′
x x
x x
%%
E E E EB B B BE B
Assumption of color chaos:
Short-range, Gaussian correlations of fields with functions Φel and Φmag :
Explicit form of Vlasov diffusion term:
with the memory time:
( ) ( )( ) ( )
( ) ( )
2 2el mag2
2 1p m ma a a a a
i j i j p pi jc i j
a
p p
x fg CgN p p
f
D f
τ τ⎡ ⎤∂⋅∇ = − + ×∇ ×∇⎢ ⎥
− ∂ ∂⎢ ⎥⎣ ⎦≡ −∇ ⋅ ∇
v vF p
pp
E E B B
( ) ( )( )(el/mae (l/mag el/m g) g)a12m tt t t td τ στ
∞
−∞′ ′Φ −′= Φ −∫ v%
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Example: Transverse Ba only
( )1 ; 02
a a a ai j ij iz jz i jδ δ δ= − ≈2B B B E E
−∇π ⋅Δ π( )∇π f π( )=
3C2Δ g2 B2 μμ ag
Nc2 −1( )EπT
2f0 1± f0( ) πiπj ∇u( )ij
( ) ( )2 2
2 mag2
1
3c p
m
N E T
gp
C τ
−=Δ 2B
( )( )(gluo6
an)
2
2 m g
16 6 1c
c mA
N TN gπ
ης
τ−
= 2B( ) 2
(quark)6
2 2 mag
62 6 f
mA
cN N Tg
ης
π τ= 2B
Additional assumption: (satisfied at early times)
Diffusive Vlasov term:
Balance between drift and Vlasov term gives:
Anomalous viscosities for gluons and quarks:
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Complete Shear Viscosity
W f1⎡⎣ ⎤⎦≡
d 3π
2π( )3f1 π( ) vμ
∂f0 π( )∂xμ +
12
−∇π ⋅Δ ⋅∇πd f π( )+ IC f1⎡⎣ ⎤⎦( )⎡
⎣⎢⎢
⎤
⎦⎥⎥∫ =μ in
Δ π( )=A
π
T, A=
Aq
Ag
⎛⎝⎜
⎞⎠⎟ ⇒ aA + aC( )A=r
r =32ς 5( )3π 2
Nc2 −1
158NcN f
⎛
⎝⎜⎜
⎞
⎠⎟⎟
aA =
32ς 4( )5π 2
g2 B2 μμ ag
T 3
Nc 0
078N f
⎛
⎝⎜⎜
⎞
⎠⎟⎟
aC =π Nc
2 −1( )45
g4 λng−1 29
2Nc + N f( )Nc 0
078N f
⎛
⎝⎜⎜
⎞
⎠⎟⎟+
π 2N f Nc2 −1( )
128Nc
1 −1−1 1
⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
Minimization of full Vlasov-Boltzmann functional W[f1]:
Following AMY, make the
variational ansatz:
η =−
115T
d 3π(2π)3
π4
Eπ2
∂f0∂Eπ
∫ Δ(π)
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Parametric dependence
( )2
21 2 3z
p
pf pE Tξ ⎛ ⎞
= − −⎜ ⎟⎝ ⎠
p
ηA
σ≈
T 3
g2 B2 μ
≈Tσ
g2 ∇uηA
⎛⎝⎜
⎞⎠⎟3/ 2
⇒ηA
σ≈
Tg2 ∇u
⎛⎝⎜
⎞⎠⎟3/5
Romatschke & Strickland convention:
Perturbation of equilibrium distribution: x =2Δ
∇uT
=10ησ∇uT
⇒ g2 B2 μ ≈kinσ3 ≈x3/ 2(gT)3Unstable modes: kinst
2 ≈ xmD2
Saturation condition: g|A| ≈ kinst
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Who wins?
Smallest viscosity dominates in system with several sources of viscosity
Anomalous viscosity
ηA
σ:
Tg2 ∇u
⎛⎝⎜
⎞⎠⎟3/5
Collisional viscosityηC
σ≈
36π50g4 λng−1
∇u : τ −1 ⇒ Anomalous viscosity wins out at small g and τ
Interestingly, a (magnetic) gauge field expectation value also arises in the linearly expanding N=4 SYM solution (hep-th/0703243):
Time dependence of turbulent color field strength:
F2 : x2(gT)4 : ∇u / T( )4 /5T 4 : −28/15
tr F2 : −10/3
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Part IV
Is the QGP weakly or strongly
coupled ?What exactly do we mean by this
statement?
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Connecting η with q^
Hard partons probe the medium via the density of colored scattering centers:
q =r q2dq2 dσ / dq2( )∫
If kinetic theory applies, the same is true for thermal quasi-particles.
Assumptions:- thermal QP have the same interactions as hard partons;- interactions are dominated by small angle scattering.
η ≈13
ρ p λ f (p) =13
pσ tr (p)
σ tr (p) ≈4s
dq⊥2 q⊥
2 dσdq⊥
2 ≈∫4qsρ
With p ~ 3T, s^ ~ 18T2 and s 4r one finds:
ηs
≈54
T 3
q
Then the transport cross section is:
Majumder, BM, Wang,
hep-ph/0703085
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Examples
From RHIC data:
T0 ≈335 Mς , q0 ≈1−2 ς 2 /fμ ⇒54T0
3
q0
≈0.15 −0.30
ηs
≈54
T 3
q
q=
8πaσNc
3 Nc −1( )2E2 +B2 μ ηA =
16ς(6) Nc −1( )2
πNc
T 6
g2 E2 + B2 μ
Turbulent gluon plasma:
Perturbative gluon plasma:
q =8ς(3)Nc
2
πT 3aσ
2 λn 1 /aσ( ) ηLL =3.81 Nc
2 −1( )π 2Nc
2
T 3
aσ2 λn 1 /aσ( )
s =4π 2 Nc
2 −1( )45
T 3
?
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Counter-examples ηs
≠54
T 3
q
Strongly coupled N=4 SYM:
q =
π 3/2(3 / 4)(5 / 4)
g2Nc T3 ⇒
54T 3
q≈
0.166
g2Nc
=ησ=
14π
Chiral limit of QCD for T << Tc (pion gas):
q ≈
4π 2C2aσ
Nc2 −1
rπ xπ (x)[ ]x→ 0⇒
54T 3
q≈
0.235aσ xπ (x)[ ]x→ 0
=ησ≈
15 fπ4
16πT 4 From RHIC data:
T0 ≈335 Mς , q0 ≈5−15 ς 2 /fμ ⇒54T0
3
q0
≈0.02 −0.06 ?
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Strong vs. weak coupling
At strong coupling, is a more faithful measure of medium opacity.
T 3
q
η/s
5T 3
4q
ln T / Tc( )1
(ln T)-4
(fπ/T)4
ln 1 / λ( )~1
λ−2
λ−1/2
4π( )−1
xGπ x( )x→ 0
QCD N=4 SYM
strong strong
weak weak
RHIC data:
q0 ≈1−2 ς 2 /fμ or q0 ≈5−15 ς 2 /fμ ?
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Conclusion
Summary:The matter created in heavy ion collisions forms a highly (color) opaque plasma, which has an extremely small shear viscosity. The question remains whether the matter is a strongly coupled plasma without any quasiparticle degrees of freedom, or whether it is a marginal quasiparticulate liquid with an anomalously low shear viscosity due to the presence of turbulent color fields, especially at early times, when the expansion is most rapid.Jets constitute the best probe to ascertain the structure of this medium. The extended dynamic range of RHIC II and LHC will be essential to the success of this exploration, but so will be sophisticated 3-D models and simulations of the collision dynamics and their application to jet quenching.