Excited QCD 2010, 31 Jan.-6 Feb., 2010

Tatra National Park (Slovakia)

Bum-Hoon Lee

Center for Quantum SpacetimeSogang University

Seoul, Korea

A Dual Geometry and Holographic QCD in medium

I. Introduction – Idea of Holography as a tool for the strongly interacting systems

II. D-branes - Low Energy Dynamics - Gauge Theories - Spacetime Geometry – AdS space

III. AdS/CFT – Holography Principle IV. AdS/QCD

1. Top-down & Bottom up approach2. Dual Geometry at finite temperature3. Dual Geometry at finite density

V. Summary

Contents

• AdS-CFT or Holography 3+1 dim. QFT (large Nc) with running coupling constants <-> 4+1 dim. Effective Gravity description

Ex) 4 d QFT <-> 5d Gravity (on “boundary” D-brane, open string) (in “bulk”,

close string) with (conf. inv.) in Anti-deSitter

Space with (asym. freedom ) in asymptotic

AdS QCD in ??

• AdS-CFT : useful for strongly interacting QFT Ex) (Excited) QCD, Heavy Ion Phys., Condensed Matter

Systems, etc. (* ) AdS/CFT duality applied to QCD is called as AdS/QCD

I. Introduction Parameter in extra “dimension”

of Energy or “radius”

– Idea on Holography

as a tool for the strongly interacting systems

http://cquest.sogang.ac.kr

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1) Low energy dynamics

Strings

3-3 : Aμ, Φ, λ, χ

II. D-Branes

3-7 : Q, ～ , ψ, ～Q 4

0,1,2,3

CPX

(N=2 Vector multiplet)

Adjoint representation

(N=2 hyper multiplet ) Matter in

Fundamental

Still far from QCD !

Yang-Mills in 3+1 dim SU(Nc), #M flavors,N=2 SUSY

Ex) D3-D7

• Low Energy Dynamics of Nc D3 branes (+ others, etc.) -> 3+1 dim. SU(Nc) YM theory (DBI action) + (matter etc.)

• Spacetime Geometry by large Nc D3 branes (+ others, etc.) -> AdS5 x S5 ( etc. … )

( for D-brane )

2) D branes - Spacetime Geometry

(string frame)

Ex) D3 brane :

Dp-brane solution in Supergravity

the radius of = the radius of ( ( )

near horizon limit : reducdes to AdS x S5 geometry

Remark :

AdS5 x S5

= constant (conformal symmetry)

Note : For , we can trust this supergravity solution

Anti-de Sitter (AdS) Geometry : AdS_(p+2)

Metric (Poincare Patch)

Gloabal Geometry as Hyperboloid

Isometry :

SO(p+1,2)

Euclidean AdSy

x

III. AdS/CFT Holography

Extension of the AdS/CFT• the gravity theory on the asymptotically AdS space -> modified boundary quantum field theory (nonconformal, less SUSY, etc.) QCD ?

• the gravity theory on the black hole background -> corresponds to the finite temperature field theory

- Gravity on the 5D bulk - QFT on the 4D boundary SUGRA on AdS5 x S5 N=4 SU(Nc) SYM- Isometries of AdS x S5 - Conformal x R-symmetry SO(4,2) x SO(6) SO(4,2) x SO(6)- the classical gravity theory - strongly coupled QFT

( , ) ( , )

Parameters

Partition function (semi-classial)

boundary value of the bulk field

Generating functional forthe boundary operators

: the source of the boundary operator

• 5D bulk field Operator 5D mass Operator dimesion • 5D gauge symmetry Current (global

symmetry)• small z Large Q • Confinement (IR) cutoff zm (in

Hard Wall)• Kaluza-Klein states Excited, Resonant

spectrum

AdS/CFT DictionaryWitten 98; Gubser,Klebanov,Polyakov 98

: Conformal dimension : mass (squared)

(Operator in QFT) <-> (p-form Field in 5D)

Note : the fluctuation field on the bulk space corresponds to a source for the QCD Operator .

Ex)

Gluon cond . dilaton 0 4 0baryon density vector w/ U(1) 1 3 0

operator of QCD• gluon condensation• chiral condensation

• mesons in the

flavor group

field in gravity• massless dilaton• scalar field with

• m=0 vector field in

the

gauge group

dual

Goal : Try to understand QCD using the 5 dimensional dual gravity theory (AdS/CFT correspondence)

Need the dual geometry of QCD.1. Approaches :

Top-down Approach : rooted in string theory

Find brane config. for the gravity dual

Bottom-up Approach : phenomenological Introduce fields, etc. as needed based on the AdS/CFT * Hard Wall Model - Introduce IR brane for confinement * Soft Wall Model – dilaton running

Light-Front : radial direction of AdS <-> Parton momenta

(Brodsky, de Teramond, 2006)

IV. AdS/QCD

QFT with Asymptotic AdS SUGRA Duals

• N=1* Polchinski & Strassler,hep-th/0003136

• Cascading Gauge Theory Klebanov & Strassler, JHEP 2000

• D3-D7 system Kruczenski ,Mateos, Myers, Winters JHEP 2004

• D4-D8 system Sakai & Sugimoto 2005

-> closely related to QCD

etc.

Top-Down Approach

Observation :• Nc of D3 branes : AdS5 x S5 <-> N=4 SUSY YM• Nc of D3 / orbifold, etc. : AdS5 x X <-> N=2, 1 YM

QFT with Asymptotic AdS SUGRA Duals• N=1* Polchinski & Strassler,hep-th/0003136

• Cascading Gauge Theory Klebanov & Strassler, JHEP 2000

• Nc of D3 + M of D7 system Kruczenski ,Mateos, Myers, Winters 2004

• Nc of D4 + M of D8 system Sakai & Sugimoto 2005

-> closely related to QCD

etc.

D3-D7 system

• large-Nc N=2 SYM with quarks• Flavor branes: Nf D7-branes with flavor symmetry: U(Nf)• Quarks are massive (in general): mq

• Probe approximation (Nc>>Nf) (~ quenched approx. )

No back reaction to the bulk gometry from the flavor branes. • Free energy ~ Flavor-brane action

Kruczenski ,Mateos, Myers, Winters JHEP 2004

D4-D8 system (Sakai-Sugimoto Model)• Type IIA in Nc D4 Background (Witten) + Nf Probe D8 Branes along (x,z) x S4 (Nc >> Nf) as dual description to 4D QCD w/ Nf m=0 quarks• Topology of the background : R(1,3) x R2 x S4• Glueballs from closed strings• Mesons from the open strings on D8• The Effective Action : 5D U(Nf) YM CS theory

Sakai-Sugimoto, 2004

Bottom-Up Approach to AdS/QCD

• Introduce the contents (fields, etc.) as needed based on the AdS/CFT principle

• phenomenological Kaluza-Klein modes - radial excitations of

hadrons identified by the symmetry properties of the modes

• Hard Wall Model Introduce IR brane for confinement

• Soft Wall Model - dilaton

Bottom-Up Approach Hard Wall model

Ex) Hard wall ModelErlich, Katz, Son, Stephanov, PRL (2005)Da Rold, Pomarol, NPB (2005)

Infrared Brane at Confinement

(Polchinski & Strassler ’00)

Metric – Slice of AdS metric

Introduce the contents (fields, etc.) as needed based on the AdS/CFTConfinement realized in * Soft Wall Model – by dilaton running * Hard Wall Model – by introducing IR brane for confinement

1) Low T (confining phase : tAdS (thermal) AdS space, no stable AdS

black hole

2) High T (deconfining phase) : AdS BH Schwarzschild-type AdS black hole

: cosmological

constant

: AdS radius

2. The Dual Geometry for the pure Yang-Mills theory (without quark matters)

Witten ‘98

Transition of bulk geometry at temperature β(=1/T).

Thermal AdS (Low T)

AdS-BH (High T)

“confinement” phase “de-confinement” phase

Hawking-Page transition

This geometry is described by the following action

zOpen string

z=0 (Boundary)

• According to the AdS/CFT correspondence,

the on-shell string action is dual to the

(potential) energy between quark and anti-

quark

• Using the result of the on-shell string

action, obtain the Coulomb potential

• There is no confining potential.

[Maldacena, Phys.Rev.Lett. 80 (1998) 4859 ]

AdS metric :

Wick rotation

the boundary located

at z=0 with the

topology

1) tAdS

tAdS :The periodicity of :

confinement in tAdS

z

Open string

z=0 (Boundary)

IR cut-off

To explain the confinement, we introduce the hard wall ( or IR cut-off) at by hand, which is called `hard wall model’ .

II

I

I

When the inter-quark distance is sufficiently

long,

•In the region I,

the energy is still the Coulomb-like

potential.

• In the region II,

the confining potential appears.

• So, the tAdS geometry in the hard wall

model corresponds to the confining phase of

the boundary gauge theory.

: String tension

In the real QCD at the low temperature, there exists the confinement.

2) AdS BH

z

z=0 (Boundary)

black hole

horizon

• an event horizon at

• The Hawking temperature :

identified with the temperature

of the

boundary gauge theory.

• This black hole geometry

corresponds to

the deconfining phase of the

boundary

gauge theory, since there is no

confining

potential.

black hole

3) the Hawking-Page phase transition

[ Herzog , Phys.Rev.Lett.98:091601,2007 ]

QCD gravity theory

deconfinement phase transition

Hawking-Page transitiondual

The regularized on-shell action

1) for the tAdS,

2) for the AdS BH,

To investigate the Hawking-Page transition, we should calculate

the free energy, which is proportional to the gravity on-shell

action.

: arbitrary

To remove the divergence at

introduce a UV cut-off

the period in the t-direction of tAdS = the period in t-direction of AdS BH at z = . Using this, the difference of two actions :

• The Hawking-Page (or deconfinement) transition occurs at

• At the low temperature

the tAdS space is stable (confining phase) .

• At the high temperature the AdS BH is more stable (deconfining phase)

or

Introduce new variables

Complex scalar field

Action for light meson spectra

vector meson axial vector

meson : break the chiral symmetry

: pseudoscalar meson

Meson spectra (in Hard Wall) [ EKSS ,

Phys.Rev.Lett.95:261602,2005 ]

dual operator of QCD• chiral condensation

• chiral

group

of global flavor symmetry

field in gravity• scalar field with

• gauge

group

of local symmetry

dual

According to the AdS/CFT correspondence,

Vector meson ( )

In the gauge , equations of motion for the axial vector meson and pion

• Boundary conditions

- at the UV cut-off , impose the Dirichlet BC - at the IR cut-off , impose the Neumann BC

• If requiring that the rho meson mass becomes 776 MeV,

we obtain

• : transverse part of

the

axial vector field

• : pseudoscalar meson

(pion)

Axial vector meson

• Boundary conditions

at the UV cut-off and at the IR cut-off

• After numerical calculation,

Pion

• Boundary conditions

at the UV cut-off and at the IR

cut-off

•

• We obtain numerically.

3. AdS/QCD with quark matters

bulk boundaryfield dual operator

Dual geometry for quark matter

( quark number density )

5-dimensional action dual to the gauge theory with quark matters

in the Euclidean version ( using )Ansatz :

Equations of motion1) Einstein equation

2) Maxwell equation

Note

1)The value of at the boundary ( )

corresponds to the

quark chemical potential of QCD.

2) The dual operator of is denoted by , which

is the

quark (or baryon) number density operator.

3) We use

most general solution, which is RNAdS BH (RN AdS black hole)

Solutions

black hole mass

black charge

quark chemical potential quark number density

corresponds to the deconfining phase

( QGP, quark-gluon plasma)

What is the dual geometry of the confining (or hadronic) phase ?

find non-black hole solution

• baryonic chemical potential

• baryon number densityWe call it tcAdS (thermal

charged AdS space)

RNAdS BH (QGP)

• black hole horizon :

• Hawking temperature

• For the norm of at the black hole horizon

to be regular, we should impose the Dirichlet boundary condition

• From this, we can obtain a relation between and

• Using these relations, we can rewrite as a function of and

• After imposing the Dirichlet boundary condition at the UV cut-off

the on-shell action is reduced to

• Note that the above action has a divergent term as

• So, we need to renormalize the above action.

• By subtracting the AdS on-shell action,

we can obtain the renormalized on-shell RN AdS BH action

• the grand potential ( micro canonical ensemble )

Free energy ( canonical ensemble)

• For describing the quark density dependence in this system, we

should find

the free energy by using the Legendre transformation

where

• As a result, the thermodynamical free energy is

We can reproduce free energy by imposing the Neumann B.C. at the

UV cut-off

Note that we should add a boundary term to impose the Neumann

B.C. at the

UV cut-off .

• The renormalized action with the Neunmann B.C. becomes

with the boundary action

• Using the unit normal vector and

the boundary term becomes

• Therefore, the free energy is the same as the previous

one.

We can conclude that

1)The bulk action with the Dirichlet B.C. at the UV cut-off

corresponds to

the grand potential, which is a function of the chemical potential.

2) The bulk action with the Neumann B.C. at the UV cut-off

corresponds to

the free energy, which is a function of the number density.

tcAdS ( Hadronic phase )

Impose the Dirichlet boundary condition at the IR cut-off

where is an arbitrary constant and will be determined later.

Using this, we can find the relation between

After imposing the Dirichlet B.C at the UV cut-off, the

renormalized on-shell action for the tcAdS

From this renormalized action, the particle number is reduced to

Using the Legendre transformation, should satisfy the

following relation

where the boundary action for the tcAdS is given by

So, we see that should be

Then, the renormalized on-shell action for the tcAdS

with

Hawking-Page transition

The difference of the on-shell actions for RN AdS BH and tcAdS

When , Hawking-Page transition occurs

Suppose that at a critical point

1) For deconfining

phase 2) For , tcAdS is stable. confining

phase

Introducing new dimensionless variables

the Hawking-Page transition occurs at

For the fixed chemical potential

After the Legendre transformation, the Hawking-Page transition in

the fixed quark number density case occurs at

For the fixed number density

Light meson spectra in the hadronic phaseTurn on the fluctuation in bulk corresponding the meson spectra in QCD

Using the same method previously mentioned, we can investigate the meson spectra depending on the quark number density.

1. Vector meson

2. Axial vector meson

3. pion

V. Summary• Holographic Principles :

(d+1 dim.) (classical) Sugra ↔ (d dim.) (quantum) YM

theories

• AdS/QCD : can be a powerful tool for QCD

- Top-down Approach & Bottom-up Approach

• Holographic QCD using dual Geometry

- confinement/deconfinement phase transition

by the Hawking-Page transition between thermal AdS

↔ AdS BH

• Dual Geometry of YM theories in Dense matter

- U(1) chemical potential baryon density

- In the hadronic phase, the quark density dependence

of the light

meson masses has been investigated.