Holographic QCDin the medium

Chanyong Park (CQUeST)@ LHC Physics Workshop (2010.08.11)

Ref. 1) B. H. Lee, CP and S.J. Sin, JHEP 0907 (2009) 087.2) CP, Phys. Rev. D81, (2010) 045009. 3) K. Jo, B. H. Lee, CP and S.J. Sin, JHEP 1006 (2010)

022.

Outline

1. AdS/CFT correspondence

2. Confinement in Holographic QCD

3. Holographic QCD in the medium

4. Conclusion

1. AdS/CFT correspondenceIIB closed string theory with N D3-brane

Gravity theory on

low energy and near horizon limit

Open string theory on D3-brane

Closed/open string duality

N=4 supersymmetricconformal gauge theory

large N limit

Strong coupling regionAdS/CFT correspondence

Classical gravity

Isometry of

Isometry of

Conformal symmetry on

R-symmetry of N=4 SUSY

2. Confinement in Holographic QCD

* Goal : study the 4-dimensional gauge theory (QCD) in the strong coupling re-gion using the 5-dimensional dual gravity theory

For this, we should find the dual geometry of QCD.

In the case of the pure Yang-Mills theory (without quark matters),the dual geometry is 1) (thermal) AdS space (tAdS) in the confining phase

2) Schwarzschild-type AdS black hole (AdS BH) in the deconfining phase

This geometry is described by the following action

: cosmological con-

stant

: AdS radius

zOpen string

z=0 (Bound-ary)

• 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, we 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 (Bound-ary)

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 bound-ary gauge theory.

: String tension

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

2) AdS BH

z

z=0 (Bound-ary)

black holehorizon

• There exists an event horizon at

• The Hawking temperature is given by

which can be identified with the temperature of the boundary gauge theory.

• This black hole geometry corresponds to the de-confining phase of the boundary gauge theory, since there is no confining potential.

black hole

3. Holographic QCD in the mediumbulk boundary

field 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

Note1) 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 densitycorresponds 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 (ther-mal

charged AdS space)

RNAdS BH (QGP)• Using the regularity condition of at the black hole horizon, we obtain a relation between and

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

the on-shell action is reduced to

Since the above action diverges, we should renormalize it by subtracting the AdS on-shell action,

• the grand potential ( in micro canonical ensemble )

• Free energy ( in 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 this free energy by imposing the Neumann B.C. at the UV cut-off

After adding 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

which gives the same free energy in the previous slide.

1) The bulk action with the Dirichlet B.C. at the UV cut-off corresponds to the grand potential.2) The bulk action with the Neumann B.C. at the UV cut-off corresponds to the free energy.

tcAdS ( Hadronic phase ) Impose the Dirichlet boundary condition at the IR cut-off

where is an arbitrary constant and will be determined later.

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 find 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

z

Open string

z=0 (Bound-ary)

String breaking of the heavy quarkonium

Insert a hard wall or black hole

1. Heavy quarkonium in the QGP

• Open string action

• Inter-quark distance

• Binding energy of the heavy quarkonium

where

• string breaking distance ( )

• As the temperature and quark chemical potential increase, the string break-ing distance becomes shorter.

• This implies that heavy quarkonium can be broken to two heavy-light mesons more easily at higher temperature and chemical potential due to the (a) thermal and (b) the screening effect of the quarks in QGP ( consistent with our intuition )

2. Heavy quarkonium in the hadronic phase

Here, we use the tcAdS metric instead of one for RN AdS BH

• Inter-quark distance

• Binding energy of the heavy quarkonium

Note that there is no temperature dependence in the confining phase of the holographic QCD model. So we consider the zero temperature case only.

• String breaking length depending on the chemical potential

• As the chemical potential increases, the string breaking distance becomes larger, which means that it is more difficult to break the heavy quarkonium at the higher chem-ical potential.

• Since there is no free quark in the hadronic phase, for the string breaking of the heavy quarkonium we need pair-creation of the light quarks. Therefore, after the string break-ing, the heavy quarkonium is broken to two heavy-light meson bound states.

• As the chemical potential becomes larger, more energy is needed for the pair-creation of light quarks, which makes the string breaking of the heavy quarkonium difficult.

4. Conclusion• We found the dual geometries of the gauge theory with quark matters.

• By studying the Hawking-Page transition between two dual geometries, we investigated the confinement/deconfinement phase transition in the holo-graphic QCD.

• The chemical potential dependence of the string breaking was investigated in the confining and deconfining phases.

Future works

•density dependence of the chiral condesnate• various meson spectra depending on the chiral condensate