november 18, 2006@qm2006 shanghai anomalous viscosity of an expanding quark-gluon plasma masayuki...

November 18, 2006 @QM2006 Shanghai

Anomalous Viscosity of an ExpandingAnomalous Viscosity of an Expanding

Quark-Gluon PlasmaQuark-Gluon Plasma

Masayuki ASAKAWA

Department of Physics, Osaka University

S. A. Bass, B. Müller, M.A., Phys. Rev. Lett. 96 (2006) 252301S. A. Bass, B. Müller, M.A., Prog. Theor. Phys. 116 (2006) 725

M. Asakawa (Osaka University)

Success of Hydrodynamics at RHIC

4

3s sT

Teaney PRC 2003

• Small shear viscosity very close to the “universal” lower bound

• Early Thermalization

Two Important (Unexpected) Findings at RHIC

0 (thermalization time scale) 0.6 fm

How are these explained?

Are these related or unrelated physics?


Is QGP strongly coupled or not?

Strong coupling is a natural explanation for the small

• PROS J/survives the deconfinement phase transition

Hatsuda and M.A., PRL 2004

• CONS Near the deconfinement phase transition,degrees of freedom are partons

Success of Recombination

P. Sorensen

22 2 22 32 3

p Bt tt

M ptv p

pp vv

pv

and


Mechanism for Early Thermalization

• 2 → 2 scattering: not sufficient

According to Hydro Models: 0 (thermalization time scale) 0.6 fm

• 2 → 3 processes ?

• Instability of Gauge Field due to Anisotropy (Weibel Instability) ?

Strong Longitudinal Flow

Anisotropy in Momentum Space

pz

py

px

beam


Weibel Instability

Weibel Instability (Weibel 1959)

When particle distribution is anisotropic, instability ( filamentation instability ) exists

Mrówczyński

Exponential growth saturates when

B2 > g2 T4

Arnold, Moore, Yaffe

Turbulent power

spectrum

Arnold and Moore

time


Longitudinal Elongation of Jet?

central

peripheral

STAR, PRC 2006

For Near Side peak, ~ const. (central) (peripheral)

Mrówczyński See, e.g., Majumder, Bass, Müller, hep-ph/0111135


What is viscosity?

1,

3 fnp n s

One of Transport Coefficients

( )

2

3

T pg e p u u

u uu uu u u u

x x x x

ug u u

x

: shear viscosity: bulk viscosity

The more Momentum Transport is prevented, the less viscosities become

• More Collisions Less

• More Deflections Less


Viscosity due to Turbulent Fields

Perturbatively calculated viscosities: Viscosities due to Collisions

Effective in suppressing Momentum Transport

B

B

B

Turbulent Magnetic Field

If this contribution to viscosity is added, total viscosity gets smaller

1 1 1

A C

Has been known as Anomalous Viscosityin plasma physics

A : anomalous viscosity

C : collisional viscosity


Result: How Anomalous?viscous stress/(sT)

shear/T

collisional stress

anomalous stress

Viscous Stress is NOT shear Non-linear response Impossible to obtain on Lattice

g-dependence ~1/g6/5 , while collisional viscosity (perturbative) 1/g4log(1/g)

At large shear and/or in weak coupling, always anomalous viscosity dominates

1 1 1

A C


Theoretical Formulation 1 Transport Equation

We start with the Vlasov equation in the extended phase-space:

( , , , ) 0aa b a a

abc pcv gf Q A Q E v B f r p Q t

x Q

��

a br abc p pc

v gf Q A v f D ft Q

Correlation time/length for the color fields is short compared with thetemporal change of the velocity of a plasma particle (= ultrarelativistic particle)

and 22 1

a b ab

c

CQ Q

N

( , , ) ( , , , )f r p t f r p Q t dQ

By expanding around for weak fieldsand taking ensemble average over the color fields

p 0p

next slide

randomness of parton charge


Theoretical Formulation 2 Field Correlation

2 2

22

( ) ( ) ( )1

el magm m

a a a ap p i j i j p pi j

c i j

g CD p f p E E B B v v f p

N p p

22

2 0( ) ( ( ), ) ( ( ), ( )) ( ( ), )

1

t a ci ac j

c

g CD p F r t t U r t r t F r t t dt

N

Here, ( , ) expx b

ac abcxU x x P f A dx

We assume the field correlations fall offwith correlation time and correlation length

( ) ( )

( ) ( )

( ) ( , ) ( )

( ) ( , ) ( )

( ) ( , ) ( ) 0

a b a ai ab j i j

a b a ai ab j i j

a bi ab j

E x U x x E x E E t t x x

B x U x x B x B B t t x x

E x U x x B x

el el

mag mag

B

B

E

Introduce the memory time (memory time felt by the parton)

( ) ( )1( )

2el/mag el/mag el/magm t t v t t dt

Then

thermal partons move ultrarelativistically

parton


Theoretical Formulation 3 Linear Response

Let us assume a small perturbation of the thermal equilibrium distribution

0

1( )

exp( ) 1f p

u p

0 0 0 0 1( , ) ( ) ( , ) ( ) ( )(1 ( )) ( , )f p r f p f p r f p f p f p f p r

Suppose we are in the local rest frame of fluid, 0 1, ( ) 0u u x

For shear viscosity, we take

1 2

( ) 1 1( , )

2 3withi j i j j i ijij ij

p

pf p r p p u u u u u

E T

By calculating Tik with ( , )f p r

3 40

3 2

1( )

15 (2 ) p p

fd p pp

T E E

2 ( )ik ik i k ik ikT P eu u u u

Chapman-Enskog formalism


Theoretical Formulation 4 Shear Viscosity

For simplicity, let us consider the following case here:

21

2

0

a ai j ij iz jz

a ai j

B B B

E E

Color-Magnetic Fields: transverse to the collision axis

In the collisionless case,2 2

2 22

( 1)( )

3 magm

c pN E Tp

C g B

This yields,2 2 6

( )2 2 2

2 6( )

2 2 2

16 (6)( 1)

62 (6)

magm

magm

g cA

c

c fqA

N T

N g B

N N T

g B

collisionless: quarks and gluons contribute separately

B

B

1 2

( ) 1 1( , )

2 3withi j i j j i ijij ij

p

pf p r p p u u u u u

E T

When collisions exist,these viscosities and collisionalviscosity couple with each other


Theoretical Formulation 5 Anisotropy loop

Following Romatschke and Strickland,introduce the anisotropy in momentum distribution:

2

2 20 0 0 0

ˆ( )ˆ( ) ( ) (1 )

2 p

p nf p f p p n f f f

E T

For Longitudinal Boost-Invariant flow and massless parton gas,

1

23 110

2 ij ji

uu u u

s T

On the other hand,

2 2 4 4 1/ 2 10 0( 2) ( )n

mg B b g T n d gT

2 2 3 1/ 20 0 ( ) n

mg B b d gT 2 2 6

( )2 2 2

2 6( )

2 2 2

16 (6)( 1)

62 (6)

magm

magm

g cA

c

c fqA

N T

N g B

N N T

g B

1/ 2

1n


Theoretical Formulation 6 ResultBy closing this loop, we obtain

viscous stress/(sT)

shear/T

collisional stress

anomalous stress

2 1

2 1

0 2

n

nA T

cs g u

C

s

const.

n=1.5, 2, 2.5

The larger the shear is, the smaller the viscosity is !


Evolution of viscosity

Initial stateCGC ?

QGP andhydrodynamic

expansionHadronization

Hadronic phaseand freeze-out

A C A C HG

1 1 1

A C

Cross sections are additive ~ f ~ 1/

Sum rule for viscosities:

Smaller viscosity dominates in system with two sources of viscosity !

Temperatureevolution

viscosity: ? ? ~ A ~ C ~ HG


Summary and Outlook

Do Turbulent (Magnetic) Fields also contribute to other observables, like Jet Energy Loss?

B

B

B

In Plasma Physics,Anomalous Beam Energy Loss is also known

We have shown that turbulent color magnetic and electric fields lead to anomalous viscosity

At large shear and/or in weak coupling, always anomalous viscosity dominates

Small viscosity does not necessarily imply strong coupling

NOT linear response to shear (velocity gradient) Cannot be calculated on the lattice


Back Up


Estimate of rm and field strength

Wave vector domain of Unstable Modes:

2 2const. Dk m 11/ 2

mr gT

The nonlinear term in the Yang-Mills equation ~ The gradient term

Saturation Level of A and B

2 2 4 2 4Dg A k g B k m i.e.

Thus,

2 2 3/ 2 3 3/ 2 3( )m Dg B r m gT

november 18, 2006@qm2006 shanghai anomalous viscosity of an expanding quark-gluon plasma masayuki...

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