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

OutlineI. Holography - AdS/CFT

II. 10d Type IIB SupergravityIII. Maldacena-Nunez Solution

IV. 5d AdS/QCD modelsV. Dynamical AdS/QCD modelVI. Conclusions

Type IIB String Theory

on AdS5 x S5

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/CFT10 dimensionsGravity Theory

4 dimensionsQuantum 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

AdS5 x S5

Holographic coordinate

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.

N=4 Super-Yang-MillsSymmetries

AdS5 x S5

Isometries

Symmetries10 dimensionsGravity Theory

4 dimensionsQuantum Field Theory

Boschi, Braga (2004)

AdS5 x S

5 N=4 SYM

N=1 SYM“QCD-like”

? QCD

Conformal

Klebanov-StrasslerKlebanov-TseytlinMaldacena-Nunez

Papadopoulos-Tseytlin ansatz

Non-conformalHas mass gap

attempts to

10 dimensionsGravity Theory

4 dimensionsQuantum Field Theory

10d Type IIB Supergravity10d Type IIB Supergravity

Einstein Equation

Field Equations

Papadopoulos-Tseytlin Papadopoulos-Tseytlin ansatz:ansatz:

Metric

One-forms

Notation

Coordinates

Papadopoulos-Tseytlin Papadopoulos-Tseytlin ansatz:ansatz:

Tensor Fields:

Papadopoulos-Tseytlin Papadopoulos-Tseytlin ansatz:ansatz:

PT ansatz: IsometriesPT ansatz: Isometries

Lie Derivative

Killing Vector

Isometries

Killing Equations

PT Ansatz: IsometriesPT Ansatz: IsometriesKilling Vectors

Supersymmetry Trans.-SU(4) group: 15 generators

N=4 Super-Yang-MillsSymmetries

Supersymmetry Trans.-SU(2) X U(1)

N=1 Super-Yang-Mills

AdS5 x S5Isometries

Internal Space:S5 - 15 Killing Vectors.

PT ansatzIsometries

SU(2) X SU(2)

JHEP 1004 (2010) 113

Kiritsis (2007)

PT ansatz: PT ansatz: Vector FluctuationsVector Fluctuations

Dilaton Metric

2-Form

3-Form

PT ansatz: PT ansatz: Vector FluctuationsVector Fluctuations

FF33 Eq. of Motion Eq. of Motion

Dynamical Dynamical EquationEquation

Dilaton Equation – Dilaton Equation – okokEinstein Equation - Einstein Equation - okok

Sturm-Liouville equationSturm-Liouville equation

Effective PotentialEffective Potential

Maldacena-Nunez Maldacena-Nunez Vector FluctuationsVector Fluctuations

goes to a goes to a constantconstant

NoNo mass gap mass gap JHEP 1004 (2010) 113

From 10d to 5d perspective.From 10d to 5d perspective.

Sturm-Liouville equation Sturm-Liouville equation for MN do not depend on for MN do not depend on

the internal space.the internal space.

Phenomenological Phenomenological models in five models in five dimensions.dimensions.

10 dimensions 5 dimensions

AdS/QCD ModelsHard 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.

m 2 ~ n

m2 ~ n2

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 behaviorIR, z →“large” (confinement)

Linear Regge Trajectories for Baryons and Vectors.

PRD79 (2009) 075019

PLB693 (2010) 287

5d Einstein Equations

Also discussed by Csaki and Reece (2007);

Gursoy, Kiristsis, Nitti (2008);Li and Huang (2013).

String Frame

BaryonsFermions in a curved space-time:

Rescaling the fermionic field

We can project

BaryonsWith 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

)770(

)782(w)1400(1)1420(w

)1450()1600(1

)1650(w)1700(

)1710(N

)1440(N

)938(N

Model II• Deformed AdS Metric• Dilaton Field

Soft Wall

Li and Huang (2013)

Regge Trajectories

)770(

)782(w

)1450(

)1400(1)1420(w

)1600(1)1650(w

)1700(

)938(N

)1440(N

)1710(N

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