Measurement of Shear Viscosity

in Lattice Gauge Theorywithout Kubo Formula

Masakiyo Kitazawawith

M. Asakawa, B. Muller, C. Nonaka

Lattice2008 Jul. 14, 2008

Transport Coefficients of the QGP Transport Coefficients of the QGP Transport Coefficients of the QGP Transport Coefficients of the QGP

•Success of ideal hydrodynamic models to describe RHIC data.• /s=1/4 from AdS/CFT – lower bound?

Analyses of viscosities on the lattice

12,120

1lim ( , )

0

One of the hottest topics!

Karsch, Wyld 1987; Nakamura, Sakai 1997,2004; Meyer 2007, 2008, talk on Fri. ; Pica talk on Thu.

•based on the Kubo formula - problem in analytic contituation•Is spatial volume large enough?

3 ( , ) ( , ) ( )d xC d K x

Experimental Measurement of Experimental Measurement of

v

F

v

A

A

FL

L

F

v

L

Our idea: Create the spatially inhomogeneous flow on the latticeand measure viscosities “experimentally”.

Spatially inhomogeneous system: Gopie, Ogilvie PRD59,034009(1999)

Velocity Distribution Velocity Distribution

v

F

L

2( ) ( )

3u uT u H

0T

( )p u u Tg pT

0 2 3# # # 0

3(1,0,0 ), )(u u x 3( ) 1u x

x

z

u3(x)

x

31T

31 11

31 0T u u3(x) is linear.

3

31

31

0

1

31( )

T

T

Tp

u

131 3T u 03 3 /( )T pu

:const.

Direct Measurements of Viscosities Direct Measurements of Viscosities

If we can create a static “hydrodynamic” flow on the lattice, transport coefficients can be determined by measuring T’s.

31

031( )

T

Tp

1

4T F F g F F

The energy-momentum tensor

( )p u u Tg pT

directly observed on the lattice

long rangeand course gained

microscopically: hydrodynamic:

Momentum Source Momentum Source

ˆ ˆ HTr eO O

Statistical average of an observable O:

cf.) grand canonical:

H H

Put external sources to Hamiltonian

(0)' )( 2)( /TH H T LH

03( ) ( , , )T x T x yydz zd

Lagrange multiplier

L

0 L/2 L

xy

z

032

0exp (ˆ ) ˆE

E T FFO DA S dx i T d O

Path integral representation

imaginary sign problem

Momentum Flow with Source Momentum Flow with Source

x

yz

The hydro. mode formingthe linear behavior will survive at long range.

03

3

( )

~ ( )

T x

u x

LL/2x

Microscopic dynamicsgoverns short range behavior.

Taylor Expansion Taylor Expansion

0

0

1ˆ ˆ exp EO DUO SZ

(0) (1) (2) (3)03 03 03

2

3 0

3

0 3( ) ( ) ( ) ( ) ( )2 3!

T x T x T x T x T x

(0)03 03 0

( ) ( ) 0T x T x •0th order:

•1st order: 0

(1)03 03

0 0 03 03

(0) ( / 2)

(0) ( / 2

(

( )

) ( )

( ) )

T x T x

T x T x

T T L

T T L

2-point functions ~ .Exe const

•never gives rise to a linear func. at long range•not responsible for the hydrodynamic flow

Teaney, PRD74,045025(2006)Meyer, arXiv:0806.3914

Taylor Expansion Taylor Expansion

(0) (1) (2) (3)03 03 03

2

3 0

3

0 3( ) ( ) ( ) ( ) ( )2 3!

T x T x T x T x T x

•We need higher-order correlation functions including both sources at x=0 and L/2 to create the hydrodynamic modes.

•2nd order term should vanish, since l.h.s. is an odd function of .

•The hydrodynamic mode can appear at least from 3rd order.

3(3)03

03

0

0 0

3 3

030

2

0

2

( ) ( )

( )

(0) ( / 2)

(0) ( / 2)

(0) ( / 2) (0

( )

3 ( ) 3 ( ) ) ( / 2)

T T L

T T L

T T L T

T x T x

T x T x

T x T x T L

Numerical Simulation Numerical Simulation

pure gauge: = 6.499, a = 0.049fm, N=6 (T = 2.5Tc)

•each 20~60 steps of HB+OR4

•on bluegene@KEK (128nodes)•one week simulation

1

4T gF F FF

64x322x6 – Lx= 3.13fm Nconf ~ 20k128x322x6 – Lx= 6.27fm Nconf ~ 27k192x322x6 – Lx= 9.41fm Nconf ~ 13k

parameter determined by Bielefeld group

lattice size:

Clover term for field strength

Numerical results Numerical results 128x322x6 Nconf ~ 27k

1st order

003 3 0( ) (0)TT x

x

L/23.1fm

exp. damping

source

no structure

~0.4fm

2(1) (2) (3)

03 03 03 0

3

33 ~ ( ) ( ) ( ) ( )

2 3!( ) T x T x T x Tu x x

0 0003 33 030(0) ( / 2) ( )( )T x T T LxT

Numerical results Numerical results 128x322x6 Nconf ~ 27k

3rd order

(3)03 ( )T x

2(1) (2) (3)

03 03 03 0

3

33 ~ ( ) ( ) ( ) ( )

2 3!( ) T x T x T x Tu x x

source

•No structure is mesuared except for near the source…

x

L/2

Summary Summary

•Just a problem of statistics?•Microscopic dynamics forbids a generation of hydro. flow?•Is Taylor expansion available for this problem?

•We tried to create a system having hydrodynamic flow by introducing the momentum source to the Hamiltonian.•Evaluating its effect by Taylor expansion up to 3rd order, any signals for the flow is not observed thus far.

What’s wrong? What’s wrong?

LL/2x