1. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Non-Strange and Strange Scalar Quarkonia Denis Parganlija In collaboration with Francesco Giacosa and Dirk H. Rischke Institut für Theoretische Physik Goethe-Universität Frankfurt am Main [Based on Phys. Rev. D 82: 054024, 2010; arXiv: 1003.4934]

2. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Motivation QCD Lagrangian Chirality Projection Operators

3. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Motivation Global Unitary Transformations invariantnot invariant Chiral SymmetryExplicit Symmetry Breaking Spontaneously Broken in Vacuum

4. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Motivation: Effective Theories of QCD and Linear Sigma Model Linear and non-linear sigma models 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 → order parameter for restoration of chiral symmetry

5. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon 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)

6. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon N f = 2 Scalars → Pseudoscalars → Vectors → Axialvectors → scalars pseudoscalars vectors axialvectors Scenario I Scenario II

7. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Lagrangian of a Linear Sigma Model with Vector and Axial-Vector Mesons (N f =2) 12 parameters photon

8. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon 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

9. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Scenario I: Parameter Determination Three Independent Parameters: Z, m 1, m σ Isospin Angular Momentum (s wave) [NA48/2 Collaboration, 2009] ~ Gluon Condensate Quark Condensate [S. Janowski (Frankfurt U.), Diploma Thesis, 2010]

10. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Scenario I: f 0 (600) is predominantly quarkonium [J. R. Peláez et al.]

11. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Comparison: the Model with and without Vectors and Axial-Vectors Note: other observables (ππ scattering lengths, a 0 (980)→ηπ decay amplitude, phenomonology of a 1, and others) are fine [Parganlija, Giacosa, Rischke, Phys. Rev. D 82: 054024, 2010]

12. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Scenario II: Scalars above 1 GeV N f = 2 Scalars → Pseudoscalars → Vectors → Axialvectors → scalars pseudoscalars vectors axialvectors

13. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon N f =2, Scenario II: f 0 (1370) is predominantly quarkonium ~ Gluon Condensate in m ρ

14. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Including Strange Degrees of Freedom Established resonances in the validity region of the model; experimental data more precise No free parameters: fixed via meson masses and decay widths Important role in the chiral transition

15. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Scalar Mesons in the N f = 3 Case Mixing between σ N and σ S Preliminary: Note: full calculation, mixing with tetraquark and glueball needed → f 0 (1370)? → f 0 (1710)?

16. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Summary Linear Sigma Model with N f = 2 and N f = 3 as effective model of QCD General phenomenology in agreement with experiment (ρ→ππ, a 1 →πγ, f 1 →a o π, a 0 →ηπ decay, ππ scattering lengths) Scalar meson puzzle: structure of f 0 (600) and f 0 (1370) (→ chiral partner of the pion → correct order parameter for restoration of chiral invariance) The f 0 (600) → ππ decay width fails to match experiment ↔ quarkonium structure of f 0 (600), a 0 (980) not favoured Quarkonium structure of f 0 (1370), a 0 (1450) favoured, f 0 (1370) → ππ in agreement with experiment Appears to be confirmed by preliminary N f = 3 results

17. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Outlook Lagrangian With Three Flavours + Glueball Mixing in the Scalar Sector: Quarkonia, Tetraquarks and Glueball 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)

18. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Spare Slides

19. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Scenario I: Other Results Our Result Experimental Value [KLOE Collaboration, hep-ex/0612029v3]: [D. V. Bugg et al., Phys. Rev. D 50, 4412 (1994)]

20. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Scenario I: a 1 →σπ Decay m 1 = 0 → m ρ generated from the quark condensate only; our result: m 1 = 652 MeV a 1 →σπ

21. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Scenario I: a 1 → ρπ Decay [M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002)]

22. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Scenario II: Parameter Determination Masses: Pion Decay Constant Five Parameters: Z, h 1, h 2, g 2, m σ

23. Denis Parganlija (Frankfurt U.) Finite-Temperature QCD Workshop, IST Lisbon Scenario II: Scattering Lengths Scattering lengths saturated Additional scalars: tetraquarks, quasi- molecular states Glueball