Dynamical Ad SQCD model for lightmesons and baryons

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

Holography - Ad. S/CFT 10 dimensions Gravity Theory Type IIB String Theory on Ad. S 5 x S 5 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.

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

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

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

10 d Type IIB Supergravity Einstein Equation Field Equations

Papadopoulos-Tseytlin ansatz: Coordinates Metric Notation One-forms

Papadopoulos-Tseytlin ansatz: Tensor Fields:

Papadopoulos-Tseytlin ansatz:

PT ansatz: Isometries Lie Derivative Killing Vector Isometries Killing Equations

PT Ansatz: Isometries Killing Vectors

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

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

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

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

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

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

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

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 Ad. S/QCD PRD 79 (2009) 075019 Solve Einstein's equations coupled to a dilaton field. The Ad. S metric is deformed in the IR. UV, z→ 0 scaling behavior IR, z →“large” (confinement) Linear Regge Trajectories for Baryons and Vectors. PLB 693 (2010) 287

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

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

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

Effective Potential

Regge Trajectories

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

Regge Trajectories

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

Backup

Maldacena-Nunez Set to zero by gauge transformation.

Invariant Volume