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

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

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)

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

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 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)

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

10d Type IIB Supergravity Einstein Equation Field Equations

Papadopoulos-Tseytlin ansatz: Metric One-forms Notation Coordinates

Papadopoulos-Tseytlin ansatz: Tensor Fields:

Papadopoulos-Tseytlin ansatz:

PT ansatz: Isometries Lie Derivative Killing Vector Isometries Killing Equations

PT Ansatz: Isometries Killing Vectors

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 Killing Vectors. PT ansatz Isometries SU(2) X SU(2) JHEP 1004 (2010) 113 Kiritsis (2007)

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

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

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

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

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)

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

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)

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) PLB693 (2010) 287

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

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

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

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

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

Effective Potential

Regge Trajectories

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

Regge Trajectories

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

Backup

Maldacena-Nunez Set to zero by gauge transformation.

Invariant Volume