Upload
ira-osborne
View
215
Download
0
Tags:
Embed Size (px)
Citation preview
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
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