Holographic QCD in the medium Chanyong Park (CQUeST) @ LHC Physics Workshop (2010.08.11) Ref. B. H. Lee, CP and S.J. Sin, JHEP 0907 (2009) 087. CP, Phys. Rev. D81, (2010) 045009. 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 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

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

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)

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)

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

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 :

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

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)

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 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.

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

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

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

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

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

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 )

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.

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.

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