Excited QCD 2010, 31 Jan.-6 Feb., 2010 Tatra National Park (Slovakia) A Dual Geometry and Holographic QCD in medium Bum-Hoon Lee Center for Quantum Spacetime Sogang University Seoul, Korea 제목 Excited QCD 2010, 31 Jan.-6 Feb., 2010 Tatra National Park (Slovakia)

Contents 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 approach 2. Dual Geometry at finite temperature 3. Dual Geometry at finite density V. Summary

I. Introduction AdS-CFT or Holography – Idea on Holography as a tool for the strongly interacting systems Parameter in extra “dimension” of Energy or “radius” 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

Adjoint representation (N=2 hyper multiplet ) Matter in Fundamental II. D-Branes 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. … ) 1 2 3 4 5 6 7 8 9 #N D3 ㅡ X #M D7 Ex) D3-D7 1) Low energy dynamics Yang-Mills in 3+1 dim SU(Nc), #M flavors,N=2 SUSY 89 CPX Strings 4567 3-3 : Aμ, Φ, λ, χ 123 #N 0,1,2,3 (N=2 Vector multiplet) Adjoint representation #M 3-7 : Q, ～, ψ, ～ Q 4 Still far from QCD ! (N=2 hyper multiplet ) Matter in Fundamental http://cquest.sogang.ac.kr

2) D branes - Spacetime Geometry Remark : Dp-brane solution in Supergravity ( for D-brane ) (string frame) Ex) D3 brane : = constant (conformal symmetry) near horizon limit : reducdes to AdS x S5 geometry AdS5 x S5 the radius of = the radius of ( ( ) Note : For , we can trust this supergravity solution

Anti-de Sitter (AdS) Geometry : AdS_(p+2) Gloabal Geometry as Hyperboloid Isometry : SO(p+1,2) Metric (Poincare Patch) Euclidean AdS y x

III. AdS/CFT Holography - 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 ( , ) ( , ) 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

AdS/CFT Dictionary Partition function (semi-classial) Witten 98; Gubser,Klebanov,Polyakov 98 Partition function (semi-classial) boundary value of the bulk field Generating functional for the 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

(Operator in QFT) <-> (p-form Field in 5D) : Conformal dimension : mass (squared) Note : the fluctuation field on the bulk space corresponds to a source for the QCD Operator . Ex) Gluon cond . dilaton 0 4 0 baryon density vector w/ U(1) 1 3 0 field in gravity massless dilaton scalar field with m=0 vector field in the gauge group operator of QCD gluon condensation chiral condensation mesons in the flavor group dual

IV. AdS/QCD 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)

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 : QFT with Asymptotic AdS SUGRA Duals 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.

D4-D8 system (Sakai-Sugimoto Model) Kruczenski ,Mateos, Myers, Winters JHEP 2004 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 D4-D8 system (Sakai-Sugimoto Model) Sakai-Sugimoto, 2004 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

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 Introduce the contents (fields, etc.) as needed based on the AdS/CFT Confinement realized in * Soft Wall Model – by dilaton running * Hard Wall Model – by introducing IR brane for confinement (Polchinski & Strassler ’00) Erlich, Katz, Son, Stephanov, PRL (2005) Da Rold, Pomarol, NPB (2005) Ex) Hard wall Model Infrared Brane at Confinement Metric – Slice of AdS metric

2. The Dual Geometry for the pure Yang-Mills theory (without quark matters) Witten ‘98 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 This geometry is described by the following action : cosmological constant : AdS radius 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

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, 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 The Hawking temperature : 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 z z=0 (Boundary) black hole horizon

deconfinement phase transition Hawking-Page transition 3) the Hawking-Page phase transition [ Herzog , Phys.Rev.Lett.98:091601,2007 ] QCD gravity theory deconfinement phase transition Hawking-Page transition dual To investigate the Hawking-Page transition, we should calculate the free energy, which is proportional to the gravity on-shell action. The regularized on-shell action : arbitrary 1) for the tAdS, 2) for the AdS BH, 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

Meson spectra (in Hard Wall) According to the AdS/CFT correspondence, Meson spectra (in Hard Wall) Action for light meson spectra [ EKSS , Phys.Rev.Lett.95:261602,2005 ] According to the AdS/CFT correspondence, field in gravity scalar field with gauge group of local symmetry dual operator of QCD chiral condensation chiral group of global flavor symmetry dual Introduce new variables Complex scalar field vector meson axial vector meson : break the chiral symmetry : pseudoscalar meson

Vector meson ( ) In the gauge , 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 In the gauge , equations of motion for the axial vector meson and pion : transverse part of the axial vector field : pseudoscalar meson (pion)

Pion Axial vector meson Boundary conditions at the UV cut-off and at the IR cut-off We obtain numerically. Axial vector meson Boundary conditions at the UV cut-off and at the IR cut-off After numerical calculation,

3. AdS/QCD with quark matters 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 1) Einstein equation 2) Maxwell equation Note 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) 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 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 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

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

Light meson spectra in the hadronic phase Turn 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.

Thank You !