The Planck Scale

for Charged AdS Black Hole

with Baryon Chemical Potential

Holographic Hydrodynamics

June 30th , ’09

Inst. of Theoretical PhysicsUniv. of Wrocław, Poland

Shingo Takeuchi

(APCTP, Korea)

The references :

Sound Modes in Holographic Hydrodynamics for Charged AdS Black Hole.

Yoshinori Matsuo, Sang-Jin Sin, Shingo Takeuchi

Nucl.Phys.B in press, arXiv:0901.0610 [hep-th]

Density Dependence of Transport Coefficients from Holographic Hydrodynamics.

Prog.Theor.Phys.120:833,2008, arXiv:0806.4460 [hep-th]

Xian-Hui Ge, Yoshinori Matsuo, Fu-Wen Shu, Sang-Jin Sin, Takuya.Tsukioka

Takuya.Tsukioka, Chul-Moon Yoo

Introduction

QCD at finite temperature and density

Quantum behavior and non-equilibrium behavior

Lattice QCD

Hydrodynamics

AdS/CFT

quark-gluon-plasma

Density

Temp.

normal nucleus CSC

RHIC, Early universe …

Effective theory Nambu-Jona-Lasinio model …

Chiral sym. breaking, Deconfined trans. …

1/10

Hydrodynamics of dual gravity for strongly coupled QGP

Finite temp. and density QGP

The model

Introduction of D7-branes (D3-D7 system)

Basic AdS/CFT

D7

symmetry symmetry

R-charge

baryon charge = R-charge

baryon charge (Finite density)

R-charge

bulk filling brane

boundary

D7-branes

Back reaction

( dim. Reduction of )

chemical potential

2/10

[Sin ’07]

( → finite temp. )

Background space-time

: Gibbons-Hawking term

[Balasubramanian-Kraus]

Action :

with

chemical potential

: Counter term

3/10

Perturbation

Gauge fixing condition

3 type of perturbations

Fourier expansion

Hydrodynamic approximation

vector type :

tensor type :

scalar type :

scalar tensor

( )

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Sym. for rot. of x-y plane

Effective description of non-equilibrium behavior

: classical perturbation

Einstein eq.

Maxwell eq.

How to solve (I) --- Master eq’s ---

Vector type :

Scalar type :

Tensor type :

Master Variables Vector type :

Scalar type : [Kodama-Ishibashi]

Decoupling of eom

and

and

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Master equations for

How to solve (II) --- Solution ---

Out-going solutionIn-going solution

Dirichlet b.c. at boundary

with and

Restoring

Differential equations are 2nd order Two boundary conditions

Hydro. approximation Truncation at 2nd order

Original variablesMaster variables

where

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Master eq.’s for

Truncation

with and

From the gravity to the dual QGP

GKP-W relation :

( classical sol.’s )

The retarded Green’s function in the field theory

Real-time AdS/CFT

[Son-Starinets], [Herzog-Son]

7/10

Kubo formula for shear viscosity

Transport coefficients

Tensor mode

Universal result

c.f. ) normal water :

Result in RHIC

liquid helium :

0905.2433[nucl-th] where

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• Vector mode

• Tensor mode

• Scalar mode

Vector mode

Pole structure of diffusion eq.

Diffusion constant

Speed of sound

Scalar mode

Pole structure of wave eq.

( Universal value for CFT )

Einstein relation exactly holds :

Charge diffusion coefficient

Electrical conductivity

The charge susceptibility

: Kubo formula

Pole structure of

9/10

: 4d gauge coupling

( )

Summary

( D3D7 system + bulk filling brane )

Decoupling of eq. of motion for the 3 type perturbations

Retarded green function in hydro. approximation

Transport coefficients in the QGP at finite temp. and density

( Kubo formula )

( master variable )

( real time AdS/CFT )

Future direction

( Beyond the hydro. approximation )

Meson physics in the dual gravity

Condensed matter physics

The model with CS term arXiv:0907.????

( topological effect )

The zero temp. limit in the model

( holographical description of Fermi-surface )

arXiv:0909.????

Thank you very much 10/10

Check that GH term is working :

When we take variation, the non-zero surface term appears. GH term cancels that term. As a result , we can obtain the equation of motion.

: extrinsic curvature

is explicitly given in Balasubramanian–Kraus (’99) as

There is divergence at boundary. This can be seen as UV divergence in CFT. Thus, we need to remove this. It is done by this counter term as

( our action )

Gibbons-Hawking term and Counter term

under b.c. :

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: rescaled gauge field

Eq. of motion :

So, by following Ishibashi-Kodama(‘03), we introduce the variable as

Einstein equation

Maxwell equation

3 eqs. w.r.t. first derivative of

1 eq. w.r.t. first derivative of

Variables :

Boundary condition :

As a result, we can get 6 eqs. w.r.t. first derivative of and .

Since num of variable is 6, we need 2 more eqs. w.r.t. first derivative for , .

Dirichlet b.c. for all as

The way to solve eom for perturbation variables : the case for scalar type

total of 6

, total of 4 eqs.

Lastly, we can get solutions around the boundary for and .

① By organizing eq. of motion,

②

(②-1)

(②-2)

(②-3)

③

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