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 PRD (2009) 79 JHEP 07 (2007) 077 PLB 693 (2010) 287

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

Regge trajectories 2S+1 L J M 2 ~W(N+L) W ~ 1GeV 2 D.V. Bugg 2004

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

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

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

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

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)

 String modes in AdS bulk  Sturm Liouville type eigenvalue problem for mesons and baryons  e.g. solve “free” Dirac equation in AdS 5 space where A(z)=0  Choice of states through UV behavior (Polchinsky & Strassler) (scalar) (fermion) Standard bulk equations with A(z)=0

Soft conformal symmetry breaking / Confinement  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 JHEP 07 (2007) 077

Solutions with this potential  Mesons  Baryons

Regge trajectories Mesons

Delta Regge trajectories

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

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)

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

AdS/QCD Models 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. 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). Polchinski, Strassler (2002) Karch, Katz, Son, Stephanov (2006) 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) Boschi, Braga (2003) Brodsky, Teramond (2003) Brodsky, Teramond (2008) Vega, Schmidt (2008) Maldacena, PRL 80, 4859 (1998); Rey and Yee, EPJC 22, 379 (2001).

Dynamical Soft Wall Solve Einstein's equations coupled 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) PRD (2009) 79

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

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

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. Hadronic Resonances QCD Operator For Spin S= 1, 2, 3,... Karch, Katz, Son and Stephanov, PRD74, (2006).

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

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

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

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

Dilaton Field for Vector Meson

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

Light Scalar Mesons Regge slope 0.5 GeV 2 PLB 693 (2010) 287

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

Dilaton Field for Scalar Meson

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

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

Scalar Wave Functions

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

K  I=1/2 s-wave phase-shift S-wave K  amplitude  BaBar parametrization Proposal to interpret the scalar mesons f 0 family as radial excitations of sigma within a Dynamical AdS/QCD model Radial excitations of K*(800) W. De Paula, T. Frederico, H. Forkel and M. Beyer, Phys. Rev. D 79 (2009)

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

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

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

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

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 Next step: - Fields in 10d gravity models, e.g. Maldacena-Nunes, and reduction to 5d to check for hadron metric dependence in 5d-models... Summary