Institut fr Theoretische Physik GoetheUniversitt Frankfurt am Main

Institut für Theoretische Physik Goethe-Universität Frankfurt am Main Nature of Light Scalar Mesons f 0(600), a 0(980), f 0(1370) and a 0(1450) Denis Parganlija In collaboration with Francesco Giacosa and Dirk H. Rischke Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Motivation: Effective Theories of QCD and Linear Sigma Model l l Description of low-energy hadrons (mesons) Generalisation to T, μ ≠ 0 Linear Sigma Model: Treats chiral partners on the same footing Vacuum calculations → calculations at T≠ 0 Degeneration of chiral partners above TC Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Motivation: Structure of Scalar Mesons l l l Spontaneous Breaking of Chiral Symmetry → Goldstone Bosons (π) Restoration of Chiral Invariance and Deconfinement ↔ Degeneration of Chiral Partners (π/σ) f 0(600), „sigma“ f 0(1370) Nature of scalar mesons Scalar states under 1 Ge. V → f 0(600), a 0(980) Scalar states above 1 Ge. V → f 0(1370), a 0(1450) – additional scalar states under 1 Ge. V required (tetraquarks? ) Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Scenario I: Scalars under 1 Ge. V l Nf =2 Scalars → Pseudoscalars → Vectors → Axialvectors → scalars vectors pseudoscalars Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia axialvectors

Lagrangian of a Linear Sigma Model with Vector and Axialvector Mesons (Nf=2) 12 parameters photon Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Spontaneous Symmetry Breaking (SSB): [S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. 41, 531 (1969)] [R. Pisarski, hep-ph/9503330 (1995)] l Shift: l. Renormalise l Shift (Diagonalise): Pseudoscalar Wave Functions: 11 parameters Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Parameter Determination l Masses: l Pion Decay Constant l Five Parameters: Z, g 2, m 1, h 2, mσ l Three Parameters: Z, m 1, mσ Isospin [NA 48/2 Collaboration, 2009] Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia Angular Momentum (s wave)

Scenario I – Results l. Z = 1. 667 ± 0. 205, mσ = (332 ± 456) Me. V, ~ gluon condensate quark condensate Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Scenario I – Results l Boundaries of mσ ← scattering lengths Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Scenario I – Sigma Decay l Γσ→ππ [H. Leutwyler et al. ] Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia [J. R. Peláez et al. ]

Scenario I - Results l l Our Result Experimental Value [D. V. Bugg et al. , Phys. Rev. D 50, 4412 (1994)] [KLOE Collaboration, hep-ex/0612029 v 3] Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Scenario II: Scalars above 1 Ge. V l Nf =2 Scalars → Pseudoscalars → Vectors → Axialvectors → scalars vectors pseudoscalars Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia axialvectors

Decay Width f 0(1370) → ππ [D. V. Bugg, ar. Xiv: 0710. 4452 [hep-ex]] [S. Janowski (Frankfurt U. ) – Diploma Thesis (preliminary results)] Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

The Rho Mass Contributions Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Scenario II – Scattering Lengths l Scattering lengths saturated Additional scalars: tetraquarks, quasi-molecular states l Glueball l Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Including Baryons [S. Gallas, F. Giacosa and D. H. Rischke, ar. Xiv: 0907. 5084 [hep-ph]] Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Summary l l l LSM with global U(2)R x U(2)L invariance: scattering lengths, low-energy meson decay widths General phenomenology in agreement with experiment (ρ→ππ, a 1→πγ, f 1→aoπ, a 0→ηπ decay, ππ scattering lengths) The f 0(600) → ππ decay width fails to match experiment ↔ quarkonium structure of f 0(600), a 0(980) excluded Quarkonium structure of f 0(1370), a 0(1450) favoured, f 0(1370) → ππ in agreement with experiment Scalar states under 1 Ge. V required for correct description of pion-pion scattering lengths Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Outlook l l l l Consequences of Global Invariance up to 4 th Order Mixing in the Scalar Sector Chiral Models With Three Flavours Extension to Non-Zero Temperature: Study Chiral Symmetry Restoration Low Energy Constants of QCD p, d Wave Scattering Lengths Include Tensor, Pseudotensor Mesons, Baryons (Nucleons) Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Spare Slides Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Scenario I – Sigma Decay m 1 = 0 → mρ generated from the quark condensate only; our result: m 1 = 652 Me. V l a 1→σπ l Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Scenario I: a 1 → ρπ Decay [M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002)] Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Scenario II: Parameter Determination l Masses: l Pion Decay Constant l Five Parameters: Z, h 1, h 2, g 2, mσ Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Comparison: the Model with and without Vectors and Axial-Vectors Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Relevant Vertices for Pion-Pion Scattering l. Feynman Diagrams At Tree Level: Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Tree-Level Scattering Amplitude s-channel isospin I components of the scattering amplitude: partial wave scattering lengths (on the threshold): amplitude Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Results I Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Calculation of Scattering Lengths l. Feynman Diagrams At Tree Level: l. Scattering Length = 0. 233 ± 0. 023 J. R. Batley et al. [NA 48/2 Collaboration], Eur. Phys. J. C 54, 411 (2008) Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Results I: as function of E 865 NA 48/2 Ke 4 Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Calculation of Pion-Pion Scattering Lengths and Decay Widths (Local Invariance) l. Feynman lππ Diagrams At Tree Level: Scattering Lengths: mσ ≈ (320 - 340) Me. V [D. Parganlija, F. Giacosa and D. Rischke, AIP Conf. Proc. 1030, 160 (2008)] 100 Me. V Width at 800 Me. V Mass [H. Leutwyler et al. ] Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Results from the Locally Invariant Lagrangian: ππ Scattering Lengths and Low-Energy Meson Decay Widths l l l Pion-pion scattering lengths correspond to experiment (NA 48/2 data) → mσ = (315 - 345) Me. V f 0(600) decay width: Γσ→ππ < 100 Me. V even at mσ = 800 Me. V [Leutwyler: mσ – iΓσ /2 = (441 -i 272) Me. V] Γρ→ππ = 86. 5 Me. V [PDG: Γρ→ππ = (149. 4 ± 1. 0) Me. V] Decay widths for sigma and rho do not match experiments How to improve decay widths? [D. Parganlija, F. Giacosa and D. Rischke, AIP Conf. Proc. 1030, 160 (2008) ] Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Improvements of the Lagrangian [S. Gasiorowicz and D. A. Geffen (1969), U. G. Meissner (1988), P. Ko and S. Rudaz (1994)] l Strategy 1: Higher-dimension terms (local invariance) l Strategy 2: Non-vanishing meson masses becomes global QCD chiral symmetry: global LSM chiral symmetry global [M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002)] [D. Parganlija, F. Giacosa and D. Rischke, AIP Conf. Proc. 1030, 160 (2008) ] [D. Parganlija, F. Giacosa and D. Rischke, ar. Xiv: 0812. 2183 [hep-ph] ] Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Parameter Determination l k l l Three Parameters: Z, mσ and h 1 Find global minimum of l Look for per D. O. F. at minimum l Parameter errors: diagonal elements of at global minimum for all parameters l Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

s-wave, isospin-two scattering length Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia

Renormalisation Constant Z Denis Parganlija (Frankfurt U. ) Excited QCD 2010, Tatranska Lomnica/Slovakia