1. Wayne Leonardo Silva de Paula Instituto Tecnológico de Aeronáutica wayne@ita.br Dynamical AdS/QCD model for light-mesons and baryons. Collaborators: Alfredo Vega - Valparaíso Tobias Frederico – ITA Massimo Bianchi – Roma II

2. Outline I. Holography - AdS/CFT II. 10 d Type IIB Supergravity III. Maldacena-Nunez Solution IV. 5 d AdS/QCD models V. Dynamical AdS/QCD model VI. Conclusions

3. Type IIB String Theory on AdS 5 x S 5 N=4 Super Yang-Mills Strong coupling If one can extend to QCD, we would have an analytical tool to study the non-perturbative region. Holography - AdS/CFT 10 dimensions Gravity Theory 4 dimensions Quantum Field Theory Low-energy limit of String Theory is Supergravity. For low-curvature regions, String action ~ Classical action. Weak coupling Maldacena (1998)

4. Field/Operator correspondence field theory operators classical fields Operator conformal dimension. Holography - AdS/CFT Witten (1998) small z AdS 5 x S 5 Holographic coordinate

5. Field Trans.: Conformal Lie Algebra - 15 generators Supersymmetry Trans.: SU(4) group - 15 generators Space-time metric: AdS 5 - conformal, 15 Killing Vectors. Internal Space: S 5 - 15 Killing Vectors. N=4 Super-Yang-Mills Symmetries AdS 5 x S 5 Isometries Symmetries 10 dimensions Gravity Theory 4 dimensions Quantum Field Theory Boschi, Braga (2004)

6. AdS 5 x S 5 N=4 SYM N=1 SYM “QCD-like” ? QCD Conformal Klebanov-Strassler Klebanov-Tseytlin Maldacena-Nunez Papadopoulos-Tseytlin ansatz Non-conformal Has mass gap attempts to 10 dimensions Gravity Theory 4 dimensions Quantum Field Theory

7. 10d Type IIB Supergravity Einstein Equation Field Equations

8. Papadopoulos-Tseytlin ansatz: Metric One-forms Notation Coordinates

9. Papadopoulos-Tseytlin ansatz: Tensor Fields:

10. Papadopoulos-Tseytlin ansatz:

11. PT ansatz: Isometries Lie Derivative Killing Vector Isometries Killing Equations

12. PT Ansatz: Isometries Killing Vectors

13. Supersymmetry Trans.- SU(4) group: 15 generators N=4 Super-Yang-Mills Symmetries Supersymmetry Trans.- SU(2) X U(1) N=1 Super-Yang-Mills AdS 5 x S 5 Isometries Internal Space: S 5 - 15 Killing Vectors. PT ansatz Isometries SU(2) X SU(2) JHEP 1004 (2010) 113 Kiritsis (2007)

14. PT ansatz: Vector Fluctuations Dilaton Metric 2-Form 3-Form

15. PT ansatz: Vector Fluctuations F 3 Eq. of Motion Dynamical Equation Dilaton Equation – ok Einstein Equation - ok

16. Sturm-Liouville equation Effective Potential Maldacena-Nunez Vector Fluctuations goes to a constant No mass gap JHEP 1004 (2010) 113

17. From 10d to 5d perspective. Sturm-Liouville equation for MN do not depend on the internal space. Phenomenological models in five dimensions. 10 dimensions5 dimensions

18. AdS/QCD Models Hard Wall Model QCD Scale introduced by a boundary condition Metric is a Slice of AdS Does not have linear Regge Trajectories ( ) Soft Wall Model QCD Scale introduced by a dilaton field Has Regge Trajectories ( ) The background (AdS + Dilaton) is not a solution of Einstein Equation. The dilaton has no effect in the Dirac Equation. Polchinski, Strassler (2002) Karch, Katz, Son, Stephanov (2006) Boschi, Braga (2003)

19. Holographic Dual model: Hadrons in QCD (4D) correspond to the normalizable modes of 5D fields. These normalizable modes satisfy the linearized equation of motion in the 5D-geometry background. Baryons: Vector Fields: Hadronic Resonances

20. Soft Wall model To overcame this issue, one solution is to introduce a phenomenological potential in the lagrangian. Forkel, Frederico and Beyer (2007) Brodsky and Teramond (2012) Gutsche, Lyubovitskij, Schmidt, Vega (2012)

21. Dynamical AdS/QCD Solve Einstein's equations coupled to a dilaton field. The AdS metric is deformed in the IR. UV, z→0 scaling behavior IR, z → “large” (confinement) Linear Regge Trajectories for Baryons and Vectors. PRD79 (2009) 075019 PLB693 (2010) 287

22. 5d Einstein Equations Also discussed by Csaki and Reece (2007); Gursoy, Kiristsis, Nitti (2008); Li and Huang (2013). String Frame

23. Baryons Fermions in a curved space-time: Rescaling the fermionic field We can project

24. Baryons With the definition: We obtain the Sturm-Liouville Equations: The effective potential

25. Vector states in the Dilaton-Gravity Background Sturm-Liouville type eigenvalue problem for vector Sturm-Liouville Potential Vector field

26. Model I Deformed AdS Metric Dilaton Field Forkel, Frederico and Beyer (2007)

27. Effective Potential

28. Regge Trajectories

29. Model II Deformed AdS Metric Dilaton Field Soft Wall Li and Huang (2013)

30. Regge Trajectories

31. We discussed attempts to QCD-like theories (N=1 SYM): Klebanov-Tseytlin, Klebanov-Strassler and Maldacena-Nunez. i) PT ansatz has SU(2) x SU(2) isometry; ii) MN solution has no mass gap for vector fluctuations. We proposed an Holographic dual model in 5 dimensions: i)Solution of 5d Einstein's Equation; ii)Regge Trajectories for Baryons and Vectors; Future Project: Nucleon Electromagnetic Form Factors. Scalars, Pseudoscalars and Higher Spin Mesons. Summary and perspectives

32. Backup

33. Maldacena-Nunez Set to zero by gauge transformation.

34. Invariant Volume