1. Holographic QCD in the medium
Chanyong Park (CQUeST)
@ LHC Physics Workshop ( )
Ref.
B. H. Lee, CP and S.J. Sin, JHEP 0907 (2009) 087.
CP, Phys. Rev. D81, (2010)
K. Jo, B. H. Lee, CP and S.J. Sin, JHEP 1006 (2010) 022.

2. Outline 1. AdS/CFT correspondence 2. Confinement in Holographic QCD
3. Holographic QCD in the medium
4. Conclusion

3. 1. AdS/CFT correspondence
Closed/open string duality
IIB closed string theory
with N D3-brane
Open string theory
on D3-brane
low energy and near horizon limit
Gravity theory on
N=4 supersymmetric
conformal gauge theory
Isometry of
Conformal symmetry on
Isometry of
R-symmetry of N=4 SUSY
large N limit
Classical gravity
Strong coupling region
AdS/CFT correspondence

4. 2. Confinement in Holographic QCD
* Goal
: study the 4-dimensional gauge theory (QCD) in the strong coupling region 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 constant
: AdS radius

5. 1) tAdS AdS metric : tAdS : z
the boundary located at z=0 with the topology
AdS metric :
Wick rotation
The periodicity of :
tAdS :
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 ] z
Open string
z=0 (Boundary)

6. confinement in tAdS
In the real QCD at the low temperature, there exists the confinement.
To explain the confinement, we introduce the hard wall ( or IR cut-off)
at by hand, which is called `hard wall model’ .
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. I II z I
Open string
: String tension
IR cut-off
z=0 (Boundary)

7. 2) AdS BH z 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 deconfining phase of the boundary gauge theory, since there is no confining potential.
black hole z
z=0 (Boundary)
black hole
horizon

8. 3. Holographic QCD in the medium
boundary
bulk
field
dual operator
( quark number density )
Dual geometry for quark matter
5-dimensional action dual to the gauge theory with quark matters
in the Euclidean version ( using )
Ansatz :

9. Equations of motion Note 1) Einstein equation 2) Maxwell equation
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

10. We call it tcAdS (thermal
Solutions
most general solution, which is RNAdS BH (RN AdS black hole)
black hole mass
black charge
quark chemical potential
corresponds to the deconfining phase
( QGP, quark-gluon plasma)
quark number density
What is the dual geometry of the confining (or hadronic) phase ?
find non-black hole solution
baryonic chemical potential
baryon number density
We call it tcAdS (thermal
charged AdS space)

11. 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,

12. 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

13. After adding a boundary term to impose the Neumann B. C
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.
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.

14. 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

15. Hawking-Page transition
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

16. For the fixed chemical potential
Introducing new dimensionless variables
the Hawking-Page transition occurs at
For the fixed chemical potential

17. For the fixed number density
After the Legendre transformation, the Hawking-Page transition in the fixed quark number density case occurs at
For the fixed number density

18. String breaking of the heavy quarkonium
Heavy quarkonium in the QGP
Open string action
Inter-quark distance
Binding energy of the heavy quarkonium z
Open string
z=0 (Boundary)
Insert a hard wall or black hole
where

19. string breaking distance ( )
As the temperature and quark chemical potential increase, the string breaking 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 )

20. Binding energy of the heavy quarkonium
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.

21. 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 chemical 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 breaking, 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.

22. 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 holographic 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