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Hadrons in a Dynamical AdS/QCD model

Hadrons in a Dynamical AdS/QCD model. Colaborators: W de Paula (ITA), K Fornazier (ITA), M Beyer (Rostock), H Forkel (Berlin). Tobias Frederico Instituto Tecnológico de Aeronáutica - Brasil. JHEP 07 (2007) 077. PRD 075019 (2009) 79. PLB 693 (2010) 287 . Outline. Regge Phenomenology

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Hadrons in a Dynamical AdS/QCD model

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  1. Hadrons in a Dynamical AdS/QCD model Colaborators: W de Paula (ITA), K Fornazier (ITA), M Beyer (Rostock), H Forkel (Berlin) Tobias Frederico Instituto Tecnológico de Aeronáutica - Brasil JHEP 07 (2007) 077 PRD 075019 (2009) 79 PLB 693 (2010) 287

  2. Outline • Regge Phenomenology • AdS/QCD models • Deformed AdS metric Meson and Baryon spectra • Dynamical AdS/QCD model High spin meson and Scalar spectra I=0 scalar partial decay width • I=1/2 scalar & K-Pi S-wave phase-shift • Summary

  3. Regge trajectories 2S+1LJ M2~W(N+L) W ~ 1GeV2 D.V. Bugg 2004

  4. E. Klempt 2002 Light-hadron spectra N=1 N=2 JHEP 07 (2007) 077

  5. Holography - AdS/CFT AdS5 x S5 Maldacena (1998) Holographic coordinate Field/Operator correspondence Witten (1998) field theory operators <=> classical fields operator dimension scalars small z Mass of the field carries the dimension

  6. AdS/QCD models • Hard IR wall • Polchinski and Strassler, Phys. Rev. Lett.88, 031601 (2002); JHEP 05, 012 (2003) • Boschi and Braga, JHEP 0305, 009 (2003);EPJ C32,529 (2004) • Brodsky and de Téramond, Phys. Rev. Lett. 96, 0201601 (2006). • Katz, Lewandowski, and Schwartz, Phys. Rev. D 74, 086004 (2006). • … • Soft breaking • Karch, Katz, Son and Stephanov, Phys. Rev. D 74, 015005 (2006). • Andreev and Zakharov, arXiv:hep-ph/0703010; Phys. Rev. D 74, 025023 (2006). • Kruczenski, Zayas, Sonnenschein and Vaman, JHEP 06, 046 (2005) • Kuperstein and Sonnenschein, JHEP 11, 026 (2004) • …

  7. Meson and baryon excitations in AdS/QCD(A bottom-up approach) • Hadron phenomenology (light mesons and baryons) • M2 proport. J (L)total (angular) momentum • M2 proport. Nradial excitation • mesons W = 1.25±0.15 GeV2 and 1.14 ± 0.013 GeV2 • baryons W =1.081 ± 0.035 GeV2 • AdS/CFT correspondence and QCD • hard scattering amplitudes ( ) Polchinski and Strassler 2002 • UV ~zΔ, Δdimension of op., z 5th dimension • conformal symmetry broken by confinement M2 = W(L+N), NOT~L+2N etc. almost universal W ~1.1 GeV2

  8. Standard bulk equations with A(z)=0 • Identification of hadrons • lightest string modes↔ leading order twist → low spin hadrons (valence quark states) • orbital excitations of strings↔ fluctuations around the AdS background → higher spin states • Interpolating operators (Brodsky and de Téramond 2005) • fixing the 5-dim effective mass (bottom up, UV phenomenology)

  9. Standard bulk equations with A(z)=0 • String modes in AdS bulk • Sturm Liouville type eigenvalue problem for mesons and baryons • e.g. solve “free” Dirac equation in AdS5 space where A(z)=0 • Choice of states through UV behavior (Polchinsky & Strassler) (scalar) (fermion)

  10. Soft conformal symmetry breaking / Confinement JHEP 07 (2007) 077 • Request phenomenological Regge behavior for • both, orbital AND radial excitations of • both, mesons AND baryons • Confinement • harmonic oscillator type • Soft conformal sym. breaking • universal implementation • only one a priori free scale • simple realisation via • twist dimension • new effective potential

  11. Solutions with this potential • Mesons • Baryons

  12. Regge trajectories Mesons

  13. Delta Regge trajectories

  14. Nucleon Regge trajectories Improvement: Fit of the nucleon spectrum: Brodsky and Teramond (see arXiv:1103.1186)

  15. Holographic encoding, find proper A(z) • The metric of the 10 dim space of strings keep A(z)finite • again calculate bulk equations (Klein Gordan, Dirac, Rarita-Schwinger,…) • Find A(z) that encode the previous potentials • Leads to nonlinear eq. for A(z)

  16. Gravitational potential • Solution baryonic sector • Solutions mesonic sector • numerical only • poles for L=0,1 L=2 L=1 L=3 AM(z) L=0

  17. AdS/QCD Models Soft Wall Model QCD Scale introduced by a dilaton field. AdS + Dilaton is not a solution of Einstein equations The metric does not has Confinement by the Wilson analysis. Has Regge Trajectories for mesons (Barions). • Hard Wall Model • QCD Scale introduced by a boundary condition. • Metric: Slice of AdS. • The metric has Confinement by the Wilson loops area law. • Does not have linear Regge Trajectories. Deformed AdS model QCD Scale introduced by an IR deformation Deformed AdS is not a solution of Einstein Eq. The metric does not has Confinement by the Wilson loops analysis . Has Regge Trajectories for mesons and Barions Forkel, Beyer, Frederico (2007) Karch, Katz, Son, Stephanov (2006) Polchinski, Strassler (2002) Boschi, Braga (2003) Brodsky, Teramond (2008) Brodsky, Teramond (2003) Vega, Schmidt (2008) Maldacena, PRL 80, 4859 (1998); Rey and Yee, EPJC 22, 379 (2001).

  18. Dynamical Soft Wall PRD 075019 (2009) 79 • Solve Einstein's equationscoupled to a dilaton field. • The AdS metric is deformed in the IR limit. • UV, z→0 scaling behavior • IR, z →“large“ (confinement) Confining Metric • AdS space with a IR deformation. Regge Trajectories will determine the IR deformation. Background Field • Scalar Field (dilaton)

  19. 5d Einstein Equations Dilaton potential Dilaton field Einstein's Equations Dilaton Equation Also discussed by Csaki and Reece (2007); Gursoy, Kiristsis, Nitti (2008).

  20. For a given warp factor A(z), the above equations give a dilaton field (and its potential) that solves the 5D Einstein equations. Solutions of 5d Einstein Equation

  21. 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 background 5D-geometry. • The eigenvalue corresponding to a normalizable meson mode is its square mass. QCD Operator For Spin S= 1, 2, 3, ... • Karch, Katz, Son and Stephanov, PRD74, 015005 (2006).

  22. Meson states in the Dilaton-Gravity Background • Sturm-Liouville type eigenvalue problem for mesons • Sturm-Liouville Potential • Deformed AdS metric For example

  23. IR limit Mass Gap Regge Trajectories Confinement and Regge Trajectories It is in agreement with the area law condition Gursoy, Kiritsis and Nitti (2008)

  24. Metric Parameters Universal Effective Potential in the IR Limit for all Spins.

  25. PRD 075019 (2009) 79 Vector Meson Experimental Hard Wall Model Soft Wall Model Dynamical Soft Wall Model n

  26. Dilaton Field for Vector Meson

  27. PRD 075019 (2009) 79 Regge Trajectories n = 5 n = 4 n = 3 Experimental Data Dynamical Soft Wall Model n = 2 n = 1 S

  28. Light Scalar Mesons PLB 693 (2010) 287 Regge slope 0.5 GeV2

  29. PLB 693 (2010) 287 Light Scalar Meson f0 Experimental Dynamical Soft Wall Model n

  30. Dilaton Field for Scalar Meson

  31. Pseudoscalar Mesons Universal Effective Potential in the IR Limit Do not affect the field UV limit as Zero mass for the Pion Lagrangian ~

  32. Scalar Decay Width into two Pions Overlap of WF ~ Transition Amplitude Decay Width

  33. Scalar Wave Functions

  34. Scalar Decay Width into two Pions PLB 693 (2010) 287

  35. K I=1/2 s-wave phase-shift S-wave K amplitude BaBar parametrization W. De Paula, T. Frederico, H. Forkel and M. Beyer, Phys. Rev. D 79 (2009) 075019 Radial excitations of K*(800) Proposal to interpret the scalar mesons f0 family as radial excitations of sigma within a Dynamical AdS/QCD model

  36. K I=1/2 s-wave phase-shift We introduce K*(1630) and K*(1950) extending the parametrization given in the BABAR Collaboration: B. Aubert, et al., arXiv:0905.3615[hep-ex] (2009). With the S-matrix given by:

  37. K I=1/2 s-wave phase-shift K  D. Aston et al., Nucl. Phys. B296 (1988) 493 DK   E. M. Aitala et al. (E791 Collaboration), Phys. Rev. Lett. 86 (2001) 765; Phys. Rev. Lett. 86 (2001) 770; Phys. Rev. Lett. 89 (2002) 121801. J. M. Link et al. (FOCUS Collaboration) Phys. Lett. B585 (2004) 200, Phys. Lett. B681 (2009) 38 38

  38. K I=1/2 s-wave ampl. modulus DK  

  39. Baryons Mode Equation To obtain a Regge Trajectories We use the Metric That gives a complex Dilaton field!

  40. Summary • 1. Light meson/barion spectra encoded by Gravity/Gauge duality • with a soft deformation of the AdS metric  hadron dependent metric! • 2. Dynamical Holographic dual model in 5 dimensions • (coupled gravitydilaton) • - deformed AdS metric (confinement and consistency with area law ) • - Regge trajectories for light mesons S > 0 • - I=0 scalars and radial excitations & SPP decay width • - I=1/2 scalars and S-wave K-Pi scattering • meson dependent metric! • nucleon does not fit into the model... • 3. Next step: • - Fields in 10d gravity models, e.g. Maldacena-Nunes, and reduction to 5d to check for hadron metric dependence in 5d-models...

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