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Yue-Liang Wu Kavli Institute for Theoretical Physics China Key Laboratory of Frontiers in Theoretical Physics Institute of Theoretical Physics, Chinese Acadeny of Sciences 2010.11.25 Low Energy Dynamics of QCD & Realistic AdS/QCD

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Outline Success of Quantum Field Theory Why Loop Regularization Method Dynamically Generated Spontaneous Chiral Symmetry Breaking Scalars as Composite Higgs and Mass Spectra of Lowest Lying Mesons Why AdS/QCD & Realistic Model Consistent Prediction for the Mass Spectra of Resonance Mesons ( Y.Q.Sui, YLW, Z.F.Xie, Y.B.Yang) Conclusions

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Symmetry & Quantum Field Theory Symmetry has played an important role in physics All known basic forces of nature: electromagnetic, weak, strong & gravitational forces, are governed by U(1)_Y x SU(2)_L x SU(3)_c x SO(1,3) Real world has been found to be successfully described by quantum field theories (QFTs)

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Why Quantum Field Theory So Successful Folk’s theorem by Weinberg: Any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory. Indication: existence in any case a characterizing energy scale (CES) M_c At sufficiently low energy then means: E << M_c QFTs

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Why Quantum Field Theory So Successful Renormalization group by Wilson or Gell-Mann & Low Allow to deal with physical phenomena at any interesting energy scale by integrating out the physics at higher energy scales. To be able to define the renormalized theory at any interesting renormalization scale. Implication: Existence of sliding energy scale (SES) μ_s which is not related to masses of particles. The physical effects above the SES μ_s are integrated in the renormalized couplings and fields.

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How to Avoid Divergence QFTs cannot be defined by a straightforward perturbative expansion due to the presence of ultraviolet divergences. Regularization: Modifying the behavior of field theory at very large momentum Feynman diagrams become well-defined finite quantities String/superstring: Underlying theory might not be a quantum theory of fields, it could be something else.

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Regularization Methods Cut-off regularization Keeping divergent behavior, spoiling gauge symmetry & translational/rotational symmetries Pauli-Villars regularization Modifying propagators, destroying non-abelian gauge symmetry Dimensional regularization: analytic continuation in dimension Gauge invariance, widely used for practical calculations Gamma_5 problem, losing scaling behavior (incorrect gap eq.), problem to chiral theory and super-symmetric theory All the regularizations have their advantages and shortcomings

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Criteria of Consistent Regularization (i) The regularization is rigorous that it can maintain the basic symmetry principles in the original theory, such as: gauge invariance, Lorentz invariance and translational invariance (ii) The regularization is general that it can be applied to both underlying renormalizable QFTs (such as QCD) and effective QFTs (like the gauged Nambu-Jona- Lasinio model and chiral perturbationtheory).

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Criteria of Consistent Regularization (iii) The regularization is also essential in the sense that it can lead to the well-defined Feynman diagrams with maintaining the initial divergent behavior of integrals, so that the regularized theory only needs to make an infinity-free renormalization. (iv) The regularization must be simple that it can provide the practical calculations.

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Symmetry-Preserving Loop Regularization with String Mode Regulators Yue-Liang Wu, SYMMETRY PRINCIPLE PRESERVING AND INFINITY FREE REGULARIZATION AND RENORMALIZATION OF QUANTUM FIELD THEORIES AND THE MASS GAP. Yue-Liang Wu Int.J.Mod.Phys.A18:2003, 5363-5420. Yue-Liang Wu, SYMMETRY PRESERVING LOOP REGULARIZATION AND RENORMALIZATION OF QFTS. Mod.Phys.Lett.A19:2004, 2191-2204. Yue-Liang Wu

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Irreducible Loop Integrals (ILIs)

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Loop Regularization Simple Prescription: in ILIs, make the following replacement With the conditions So that

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Gauge Invariant Consistency Conditions

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Checking Consistency Condition

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Vacuum Polarization Fermion-Loop Contributions

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Gluonic Loop Contributions

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Cut-Off & Dimensional Regularizations Cut-off violates consistency conditions DR satisfies consistency conditions But quadratic behavior is suppressed in DR

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Symmetry–preserving & Infinity-free Loop Regularization With String-mode Regulators Choosing the regulator masses to have the string-mode Reggie trajectory behavior Coefficients are completely determined from the conditions

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Explicit One Loop Feynman Integrals With Two intrinsic mass scales and play the roles of UV- and IR-cut off as well as CES and SES

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Interesting Mathematical Identities which lead the functions to the following simple forms

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Renormalization Constants of Non- Abelian gauge Theory and β Function of QCD in Loop Regularization Lagrangian of gauge theory Possible counter-terms Jian-Wei CuiJian-Wei Cui, Yue-Liang Wu, Int.J.Mod.Phys.A23:2861-2913,2008Yue-Liang Wu

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Ward-Takahaski-Slavnov-Taylor Identities Gauge Invariance Two-point Diagrams

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Three-point Diagrams

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Four-point Diagrams

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Ward-Takahaski-Slavnov-Taylor Identities Renormalization Constants All satisfy Ward-Takahaski-Slavnov-Taylor identities

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Renormalization β Function Gauge Coupling Renormalization which reproduces the well-known QCD β function (GWP)

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Supersymmetry in Loop Regularization Supersymmetry Supersymmetry is a full symmetry of quantum theory Supersymmetry should be Regularization- independent Supersymmetry-preserving regularization J.W. Cui, Y.Tang,Y.L. Wu Phys.Rev.D79:125008,2009

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Massless Wess-Zumino Model Lagrangian Ward identity In momentum space

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Check of Ward Identity Gamma matrix algebra in 4-dimension and translational invariance of integral momentum Loop regularization satisfies these conditions

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Massive Wess-Zumino Model Lagrangian Ward identity

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Check of Ward Identity Gamma matrix algebra in 4-dimension and translational invariance of integral momentum Loop regularization satisfies these conditions

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Triangle Anomaly Amplitudes Using the definition of gamma_5 The trace of gamma matrices gets the most general and unique structure with symmetric Lorentz indices

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Anomaly of Axial Current Explicit calculation based on Loop Regularization with the most general and symmetric Lorentz structure Restore the original theory in the limit which shows that vector currents are automatically conserved, only the axial-vector Ward identity is violated by quantum corrections

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Chiral Anomaly Based on Loop Regularization Including the cross diagram, the final result is Which leads to the well-known anomaly form

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Anomaly Based on Various Regularizations Using the most general and symmetric trace formula for gamma matrices with gamma_5. In unit

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Dynamically Generated Spontaneous Chiral Symmetry Breaking In Chiral Effective Field Theory

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Chiral limit: Taking vanishing quark masses m q → 0. QCD Lagrangian has maximum global Chiral symmetry : QCD Lagrangian and Symmetry

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QCD Lagrangian with massive light quarks

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Effective Lagrangian based on Loop Regularization Y.B. Dai and Y-L. Wu, Euro. Phys. J. C 39 s1 (2004)

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Dynamically Generated Spontaneous Symmetry Breaking

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Composite Higgs Fields

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Scalars as Partner of Pseudoscalars & Lightest Composite Higgs Bosons Scalar mesons: Pseudoscalar mesons :

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Mass Formula Pseudoscalar mesons :

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Mass Formula

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Predictions for Mass Spectra & Mixings

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Predictions

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Chiral SU L (3)XSU R (3) spontaneously broken Goldstone mesons π 0, η 8 Chiral U L (1)XU R (1) breaking Instanton Effect of anomaly Mass of η 0 Flavor SU(3) breaking The mixing of π 0, η and η ׳ Chiral Symmetry Breaking

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Chiral Symmetry Breaking & QCD Confinement in AdS/QCD Models

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Field Theory Gravity theory = Gauge Theories QCD Quantum Gravity String theory Use the field theory to learn about gravity Use the gravity description to learn about the field theory Theories of Field and Gravity (J.M.)

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TopDownTopDown

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Most SUSY QCD SU(N) String theory on AdS x S 5 5 = Radius of curvature (J.M.) Duality: g 2 N is small perturbation theory is easy – gravity is bad g 2 N is large gravity is good – perturbation theory is hard Strings made with gluons become fundamental strings. Particle Theory Gravity Theory N colors N = magnetic flux through S 5

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AdS/CFT Dictionary N=4 SYM U(Nc) = g N SO(2,4) superconformal group SO(6) flavour (R) symmetry RG scale Sources and operators Glueballs Type IIB strings in AdS5xS5 Only gauge invariant operators R ’ = 4 π g N SO(2,4) metric isometries SO(6) S5 isometries Radial coordinate Constants of integration in SUGRA field solutions Regular linearized fluctuations of dilaton s 22

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AdS/CFT Qualitative Similarities to QCD Real QCD, full string construction? X AdS 5 S5S5 Bulk Space N D-branes SU(N) Yang-Mills Symmetry More D-branes For flavor Deviations from AdS for finite N Top down string-model approach appears to be far away Difficult to find reasonable supergravity background & brane configurations Top Down

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Quantum ChromoDynamics QCD colors (charges) They interact exchanging gluons Electrodynamics QED Chromodynamics (QCD) electron photon gluon g g g g Gauge group U(1)SU(3) 3 x 3 matrices Gluons carry color charge, so they interact among themselves Bottom upBottom up

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QCD Strings & Gravity Gluon: color and anti-color Closed strings glueballs Open strings mesons At distances larger than the typical size of the string Gravity theory Radius of curvature >> string length gravity is a good approximation lsls R Gauge Theory + Large N_c String Theory Gravity Theory Large N_c

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Dual Theory of QCD In the UV regime: highly nonlocal, corresponding to asymptotic freedom. In the IR regime: local, corresponding to the strongly correlated QCD. QCD Strings to the gravitational dual as a local theory.

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Holographic QCD Sum over all geometries that have an AdS boundary. Large N typically one geometry dominant contribution Determined by the boundary conditions Holographic QCD Holographic QCD is a gravitational theory of gauge invariant fields in 5 dimensions. The 5 th dimension play the role of the energy scale. + + ….

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AdS/CFT(QCD) A scale invariant (conformal) field theory in 1+3 dimensions has symmetry group SO(2,4) Classical AdS has an SO(2,4) symmetry group. (Such a symmetry is analogous to Lorentz symmetry, in the infinity limit of the curvature radius, it becomes the Poincare group ) It is the same as symmetries of 1+4 dimensional Anti-de-Sitter space = the simplest and most symmetric negatively curved spacetime Quantum gravity in AdS is the same as a conformal field theory on the boundary This symmetry is preserved by the quantization, in the sense that the dual field theory has the full conformal symmetry.

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AdS/CFT(QCD) Dictionary 4D CFT QCD 5D AdS operators 5D bulk fields global symmetries local gauge symmetries correlation functions Resonance hadrons KK mode states 4D generating functional 5D (classical) effective action Chiral symmetry breaking 5D bulk VEV Linear Confinement IR boundary condition of Dilaton IR brane, QCD confinement Bulk fields JmJm

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AdS/CFT(QCD) Bottom Up Bottom up approach is directly related to QCD data & fit to QCD Works around the conformal limit, check consistency as an effective field theory Carries out calculations for non-perturbative quantites Predicts mass spectra, form factors, hadronic matrix elements Insights into chiral dynamics, vector meson dominance, quark models, instantons Understands chiral symmetry breaking & linear confinement Other insights into QCD: Chiral symmetry breaking Linear confinement mass p r a1a1 f0f0 h’h’ KK modes mass spectra IR Brane σ 5D bulk

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AdS/CFT Correspondence AdS/QCD Model Klebanov and Witten 1999

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It has constant negative curvature, with a radius of curvature given by R. ds 2 = R 2 (dx 2 3+1 + dz 2 ) z 2 Boundary R4R4 AdS 5 z = 0 z z = infinity Gravitational potential w(z) z Anti-de-Sitter space Solution of Einstein’s equations with negative cosmological constant

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Hard-Wall AdS/QCD Model Global SU(3) L x SU(3) R symmetry in QCD 5D Gauge fields A L and A R 4D Operators 5D Bulk fields X ij A L, A R, X ij Mass term is determined by the scaling dimension Xij has dimensionΔ= 3 and form p=0, AL & AR have dimension Δ= 3 and form p=1 Hard-Wall AdS/QCD Lagrangian SU(3) L XSU(3) R gauge symmetry in AdS 5 4D Operators

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Gauge coupling determined from correlation functions In QCD, correlation function of vector current is Leading order of quark loop In AdS, correlation function of source fields on UV brane Bulk to boundary propagator solution to the equation of motion with V(0)=1

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Chiral Symmetry Breaking in QCD Nonvanishing breaks chiral symmetry to diagonal subgroup JJ J Goldstones

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Hard-Wall AdS/QCD with/without Back-Reacted Effects Quark masses Explicit chiral breaking Relevant in the UV Spontaneous chiral breaking Relevant in the IR Quark condensate just solve equations of motion! =

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Results from hard-wall AdS/QCD J.P. Shock, F.Wu,YLW, Z.F. Xie, JHEP 0703:064,2007

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Soft-Wall AdS/QCD Solving equations of motion for vector field Linear trajectory for mass spectra of vector mesons Dilaton field

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Achievements & Challenges Hard-wall AdS/QCD models contain the chiral symmetry breaking, the resulting mass spectra for the excited mesons are contrary to the experimental data Soft-wall AdS/QCD models describe the linear confinement and desired mass spectra for the excited vector mesons, while the chiral symmetry breaking can't consistently be realized. A quartic interaction in the bulk scalar potential was introduced to incorporate linear confinement and chiral symmetry breaking. While it causes an instability of the scalar potential and a negative mass for the lowest lying scalar meson state. How to naturally incorporate two important features into a single AdS/QCD model and obtain the consistent mass spectra.

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Modified Soft-Wall AdS/QCD by Y.Q.Sui, YLW, Z.F.Xie, Y.B.Yang Deformed 5D Metric in IR Region & Quartic Interaction Minimal condition for the bulk vacuum UV & IR boundary conditions of the bulk vacuum Solutions for the dilaton field at the UV & IR boundary PRD arXiv:0909.3887

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Various Modified Soft-wall AdS/QCD Models Some Exact Forms of bulk VEV in Models: I, II, III Two IR boundary conditions of the bulk VEV Ia, IIa, IIIa: Ib, IIb, IIIb :

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Behaviors of VEV & Dilaton

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Determination of Model Parameters Two Energy Scales as Input Parameters

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Fitted Parameters Without Quartic Interaction Effective IR Cut-off Scale in Soft-Wall AdS/QCD

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Fitted Parameters With Quartic Interaction of bulk scalar

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Solutions via Solving Equations of Motion Pseudoscalar Sector Equation of Motion

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Mass Spectra of Pseudoscalar Mesons

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Resonance States of Pseudoscalars

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Solutions via Solving Equations of Motion Scalar Sector Equation of Motion IR & UV Boundary Condition

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Mass Spectra of Scalar Mesons

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Resonance States of Scalars

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Wave Functions of Resonance Scalars

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Solutions via Solving Equations of Motion Vector Sector Equation of Motion IR & UV Boundary Condition

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Mass Spectra of Vector Mesons

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Resonance States of Vectors

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Wave Functions of Resonance Vectors

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Solutions via Solving Equations of Motion Axial-vector Sector Equation of Motion IR & UV Boundary Condition

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Mass Spectra of Axial-vector Mesons

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Resonance States of Axial-vectors

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Vector Coupling & Pion Form Factor

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Structure of Pion Form Factor

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The PrimEx Experimental Project @ JLab Experimental program Precision measurements of: Two-Photon Decay Widths: Γ( 0 → ), Γ( → ), Γ( ’ → ) Transition Form Factors at low Q 2 (0.001-0.5 GeV 2 /c 2 ): F( * → 0 ), F( * → ), F( * → ) Test of Chiral Symmetry and Anomalies via the Primakoff Effect

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Conclusions Why such a simply modified soft-wall AdS/QCD model works so well How to understand dynamical origin of the metric induced conformal symmetry breaking in the IR region. What is dynamics of the dilaton and gravity beyond as the background The important role of the dilaton field and the effect from the back-reacted geometry. The possible higher order interaction terms and their effects on the mass spectra and form factors. Extend to the three flavor case and consider the SU(3) breaking and instanton effects. Can lessons from AdS/QCD be applied to other gauge theories and symmetry breaking systems

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THANKS

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