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Wayne Leonardo Silva de Paula Instituto Tecnológico de Aeronáutica

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

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Wayne Leonardo Silva de Paula Instituto Tecnológico de Aeronáutica

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

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

  3. Type IIB String Theory on AdS5 x S5 Holography - AdS/CFT 10 dimensions Gravity Theory 4 dimensions Quantum Field Theory N=4 Super Yang-Mills Maldacena (1998) Low-energy limit of String Theory is Supergravity. For low-curvature regions, String action ~ Classical action. Weak coupling Strong coupling If one can extend to QCD, we would have an analytical tool to study the non-perturbative region.

  4. Holography - AdS/CFT AdS5 x S5 Holographic coordinate Field/Operator correspondence Witten (1998) field theory operators <=> classical fields Operator conformal dimension. small z

  5. Symmetries 4 dimensions Quantum Field Theory 10 dimensions Gravity Theory N=4 Super-Yang-Mills Symmetries AdS5 x S5 Isometries Boschi, Braga (2004) Field Trans.: Conformal Lie Algebra - 15 generators Supersymmetry Trans.: SU(4) group - 15 generators Space-time metric: AdS5 - conformal, 15 Killing Vectors. Internal Space: S5 - 15 Killing Vectors.

  6. 10 dimensions Gravity Theory 4 dimensions Quantum Field Theory AdS5 x S5 N=4 SYM Conformal Klebanov-Strassler attempts to N=1 SYM “QCD-like” Klebanov-Tseytlin Non-conformal Has mass gap Maldacena-Nunez Papadopoulos-Tseytlin ansatz ? QCD

  7. 10d Type IIB Supergravity Einstein Equation Field Equations

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

  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. AdS5 x S5 Isometries N=4 Super-Yang-Mills Symmetries Internal Space: S5 - 15 Killing Vectors. Supersymmetry Trans.- SU(4) group: 15 generators N=1 Super-Yang-Mills PT ansatz Supersymmetry Trans.- SU(2) X U(1) Isometries SU(2) X SU(2) Kiritsis (2007) JHEP 1004 (2010) 113

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

  15. PT ansatz: Vector Fluctuations F3 Eq. of Motion Dynamical Equation Dilaton Equation – ok Einstein Equation - ok

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

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

  18. 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. 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 ( ) Karch, Katz, Son, Stephanov (2006) Polchinski, Strassler (2002) Boschi, Braga (2003)

  19. Hadronic Resonances • 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:

  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 PRD79 (2009) 075019 PLB693 (2010) 287 • Solve Einstein's equationscoupled 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.

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

  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 • Vector field • Sturm-Liouville type eigenvalue problem for vector • Sturm-Liouville Potential

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

  27. Effective Potential

  28. Regge Trajectories

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

  30. Regge Trajectories

  31. Summary and perspectives • 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: • Solution of 5d Einstein's Equation; • Regge Trajectories for Baryons and Vectors; • Future Project: • Nucleon Electromagnetic Form Factors. • Scalars, Pseudoscalars and Higher Spin Mesons.

  32. Backup

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

  34. Invariant Volume

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