1. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Structure of 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 Institut für Theoretische Physik Goethe-Universität Frankfurt am Main [Based on arXiv: 1003.4934 [hep-ph]]

2. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Motivation: Effective Theories of QCD and Linear Sigma Model 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 T C

3. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Motivation: Structure of Scalar Mesons Spontaneous Breaking of Chiral Symmetry → Goldstone Bosons (π) Restoration of Chiral Invariance and Deconfinement ↔ Degeneration of Chiral Partners (π/σ) Nature of scalar mesons Scalar states under 1 GeV → f 0 (600), a 0 (980) Scalar states above 1 GeV → f 0 (1370), a 0 (1450) – additional scalar states under 1 GeV required (tetraquarks?) f 0 (600), „sigma“f 0 (1370)

4. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Scenario I: Scalars under 1 GeV N f = 2 Scalars → Pseudoscalars → Vectors → Axialvectors → scalars pseudoscalars vectors axialvectors

5. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Lagrangian of a Linear Sigma Model with Vector and Axial-Vector Mesons (N f =2) 12 parameters photon

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

7. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Parameter Determination Three Independent Parameters: Z, m 1, m σ Isospin Angular Momentum (s wave) [NA48/2 Collaboration, 2009] ~ Gluon Condensate Quark Condensate

8. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Scenario I – Results Boundaries of m σ ← scattering lengths

9. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Scenario I – Sigma Decay Γ σ→ππ [J. R. Peláez et al.] [H. Leutwyler et al.]

10. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Scenario I - Results Our Result Experimental Value [KLOE Collaboration, hep-ex/0612029v3] [D. V. Bugg et al., Phys. Rev. D 50, 4412 (1994)]

11. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Scenario II: Scalars above 1 GeV N f = 2 Scalars → Pseudoscalars → Vectors → Axialvectors → scalars pseudoscalars vectors axialvectors

12. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Decay Width f 0 (1370) → ππ [D. V. Bugg, arXiv: 0710.4452 [hep-ex]]

13. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland The Rho Mass Contributions

14. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Scenario II – Scattering Lengths Scattering lengths saturated Additional scalars: tetraquarks, quasi- molecular states Glueball

15. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Summary 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 →a o π, 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 GeV required for correct description of pion-pion scattering lengths

16. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Outlook Consequences of Global Invariance up to 4 th Order Chiral Models With Three Flavours Mixing in the Scalar Sector 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)

17. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Spare Slides

18. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Scenario I – Sigma Decay m 1 = 0 → m ρ generated from the quark condensate only; our result: m 1 = 652 MeV a 1 →σπ

19. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Scenario I: a 1 → ρπ Decay [M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002)]

20. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Scenario II: Parameter Determination Masses: Pion Decay Constant Five Parameters: Z, h 1, h 2, g 2, m σ

21. Denis Parganlija (Frankfurt U.) Meson 2010 Workshop, Kraków - Poland Comparison: the Model with and without Vectors and Axial-Vectors