Hadrons in a Dynamical Ad SQCD model Tobias
Hadrons in a Dynamical Ad. S/QCD model Tobias Frederico Instituto Tecnológico de Aeronáutica - Brasil JHEP 07 (2007) 077 PRD 075019 (2009) 79 PLB 693 (2010) 287 Colaborators: W de Paula (ITA), K Fornazier (ITA), M Beyer (Rostock), H Forkel (Berlin)
Outline Regge Phenomenology Ad. S/QCD models Deformed Ad. S metric Meson and Baryon spectra Dynamical Ad. S/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 2 S+1 L J M 2~W(N+L) W ~ 1 Ge. V 2 D. V. Bugg 2004
Light-hadron spectra E. Klempt 2002 N=1 N=2 JHEP 07 (2007) 077
Holography - Ad. S/CFT Ad. S 5 x S 5 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
Ad. S/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 C 32, 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). Kruczenski, Zayas, Sonnenschein and Vaman, JHEP 06, 046 (2005) Andreev and Zakharov, ar. Xiv: hep-ph/0703010; Phys. Rev. D 74, 025023 (2006). Kuperstein and Sonnenschein, JHEP 11, 026 (2004) …
Meson and baryon excitations in Ad. S/QCD (A bottom-up approach) Hadron phenomenology (light mesons and baryons) M 2 proport. J (L) total (angular) momentum M 2 = W(L+N), NOT ~L+2 N etc. 2 M proport. N radial excitation mesons W = 1. 25± 0. 15 Ge. V 2 and 1. 14 ± 0. 013 Ge. V 2 almost universal baryons W =1. 081 ± 0. 035 Ge. V 2 W ~1. 1 Ge. V 2 Ad. S/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
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 Ad. S background → higher spin states Interpolating operators fixing the 5 -dim effective mass (bottom up, UV phenomenology) (Brodsky and de Téramond 2005)
Standard bulk equations with A(z)=0 String modes in Ad. S bulk Sturm Liouville type eigenvalue problem for mesons and baryons e. g. solve “free” Dirac equation in Ad. S 5 space where A(z)=0 (scalar) (fermion) Choice of states through UV behavior (Polchinsky & Strassler)
Soft conformal symmetry breaking / Confinement JHEP 07 (2007) 077 Request phenomenological Regge behavior for universal implementation only one a priori free scale simple realisation via harmonic oscillator type Soft conformal sym. breaking both, mesons AND baryons Confinement both, orbital AND radial excitations of twist dimension new effective potential
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 ar. Xiv: 1103. 1186)
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=2 AM(z) L=0 L=1 L=3
Ad. S/QCD Models Hard Wall Model QCD Scale introduced by a boundary condition. Metric: Slice of Ad. S. The metric has Confinement by the Wilson loops area law. Does not have linear Regge Trajectories. Polchinski, Strassler (2002) Soft Wall Model Deformed Ad. S model QCD Scale introduced by a dilaton field. QCD Scale introduced by an IR deformation Ad. S + Dilaton is not a solution of Einstein equations Deformed Ad. S is not a solution of Einstein Eq. The metric does not has Confinement by the Wilson loops analysis. Has Regge Trajectories for mesons Has Regge Trajectories for and Barions mesons (Barions). Karch, Katz, Son, Stephanov (2006) Boschi, Braga (2003) Brodsky, Teramond (2008) Brodsky, Teramond (2003) Vega, Schmidt (2008) Forkel, Beyer, Frederico (2007) Maldacena, PRL 80, 4859 (1998); Rey and Yee, EPJC 22, 379 (2001).
Dynamical Soft Wall PRD 075019 (2009) 79 Solve Einstein's equations coupled to a dilaton field. The Ad. S metric is deformed in the IR limit. UV, z→ 0 scaling behavior IR, z →“large“ (confinement) Confining Metric Ad. S space with a IR deformation. Regge Trajectories will determine the IR deformation. Background Field Scalar Field (dilaton)
5 d Einstein Equations Dilaton potential Dilaton field Einstein's Equations Dilaton Equation Also discussed by Csaki and Reece (2007); Gursoy, Kiristsis, Nitti (2008).
Solutions of 5 d Einstein Equation For a given warp factor A(z), the above equations give a dilaton field (and its potential) that solves the 5 D Einstein equations.
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 background 5 D-geometry. The eigenvalue corresponding to a normalizable meson mode is its square mass. For Spin S= 1, 2, 3, . . . QCD Operator Karch, Katz, Son and Stephanov, PRD 74, 015005 (2006).
Meson states in the Dilaton-Gravity Background • Sturm-Liouville type eigenvalue problem for mesons • Sturm-Liouville Potential • Deformed Ad. S metric For example
Confinement and Regge Trajectories IR limit Mass Gap It is in agreement with the area law condition Gursoy, Kiritsis and Nitti (2008) Regge Trajectories
Metric Parameters Universal Effective Potential in the IR Limit for all Spins.
PRD 075019 (2009) 79 Vector Meson Experimental Hard Wall Model Soft Wall Model Dynamical Soft Wall Model n
Dilaton Field for Vector Meson
PRD 075019 (2009) 79 Regge Trajectories n=5 n=4 n=3 n=2 Experimental Data Dynamical Soft Wall Model n=1 S
Light Scalar Mesons PLB 693 (2010) 287 Regge slope 0. 5 Ge. V 2
PLB 693 (2010) 287 Light Scalar Meson f 0 Experimental Dynamical Soft Wall Model n
Dilaton Field for Scalar Meson
Pseudoscalar Mesons Universal Effective Potential in the IR Limit Zero mass for the Pion Lagrangian ~ Do not affect the field UV limit as
Scalar Decay Width into two Pions Overlap of WF ~ Transition Amplitude Decay Width
Scalar Wave Functions
PLB 693 (2010) 287 Scalar Decay Width into two Pions
K I=1/2 s-wave phase-shift W. De Paula, T. Frederico, H. Forkel and M. Beyer, Phys. Rev. D 79 (2009) 075019 S-wave K amplitude Ba. Bar parametrization Radial excitations of K*(800) Proposal to interpret the scalar mesons f 0 family as radial excitations of sigma within a Dynamical Ad. S/QCD model
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. , ar. Xiv: 0905. 3615[hep-ex] (2009). With the S-matrix given by:
K I=1/2 s-wave phase-shift K D. Aston et al. , Nucl. Phys. B 296 (1988) 493 D K E. M. Aitala et al. (E 791 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. B 585 (2004) 200, Phys. Lett. B 681 (2009) 38
K I=1/2 s-wave ampl. modulus D K
Baryons Mode Equation To obtain a Regge Trajectories We use the Metric That gives a complex Dilaton field!
Summary 1. Light meson/barion spectra encoded by Gravity/Gauge duality with a soft deformation of the Ad. S metric hadron dependent metric! 2. Dynamical Holographic dual model in 5 dimensions (coupled gravity dilaton) - deformed Ad. S 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 10 d gravity models, e. g. Maldacena-Nunes, and reduction to 5 d to check for hadron metric dependence in 5 d-models. . .
- Slides: 41