Institut fr Theoretische Physik GoetheUniversitt Frankfurt am Main

Institut für Theoretische Physik Goethe-Universität Frankfurt am Main Extending the Linear Sigma Model to Nf = 3 Denis Parganlija In collaboration with Francesco Giacosa and Dirk H. Rischke (Frankfurt) [Based on ar. Xiv: 1011. 6104] György Wolf and Péter Kovács (Budapest) Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Motivation l QCD Lagrangian l Chirality Projection Operators Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Motivation l Global Unitary Transformations invariant not invariant Chiral Symmetry Explicit Symmetry Breaking Spontaneously Broken in In addition: Vacuum Chiral U(1)A Anomaly Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Motivation: Linear Sigma Model: l Treats chiral partners on the same footing l Vacuum calculations → calculations at T≠ 0 l Vanishing chiral condensate and degeneration of chiral partners above TC → order parameter for restoration of chiral symmetry l New in our model: Nf = 3 with vector and axialvector mesons extended Linear Sigma Model - e. LSM Vacuum spectroscopy first → Extend to T≠ 0 as a next step Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Sigma Model Lagrangian with Vector and Axial-Vector Mesons (Nf = 3) l Scalars and Pseudoscalars Explicit Symmetry Breaking Chiral Anomaly photon Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Sigma Model Lagrangian with Vector and Axial-Vector Mesons (Nf = 3) l Vectors and Axial-Vectors Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Sigma Model Lagrangian with Vector and Axial-Vector Mesons (Nf = 3) l More (Pseudo)scalar – (Axial-)Vector Interactions Perform Spontaneous Symmetry Breaking (SSB): σ N → σ N + ϕ N, σ S → σ S + ϕ S l SSB leads to mixings: l Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Calculating the Parameters l Shift the (axial-)vector fields: l Canonically normalise pseudoscalars and KS: Perform a fit of all parameters except g 2 (fixed via ρ → ππ) and g 3 (fixed via K* → Kπ) l 12 parameters, none free → fixed via masses l [Parganlija, Giacosa, Rischke in Phys. Rev. D 82: 054024, 2010; ar. Xiv: 1003. 4934] Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Masses Mass Our value [Me. V] 136. 68 524 541. 48 957. 81 775. 51 891. 66 PDG Value [Me. V] 139. 57 493. 68 547. 85 957. 78 775. 49 891. 66 Mass Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France Our value [Me. V] 992. 6 1420. 5 1213. 9 1320. 6 1459 1556. 6 PDG Value [Me. V] 1019. 5 1426. 4 1230 1272 1474 1425

PDG Data on JPC = 0++ Mesons l Five states up to 1. 8 Ge. V State f 0(600) Mass Width [Me. V] 400 - 1200 600 - 1000 f 0(980) 980± 10 40 - 100 f 0(1370) 200 - 500 f 0(1500) 1200 1505± 6 f 0(1710) 1720± 6 135± 8 109± 7 Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Scalar Mesons in Our Model l Mixing between σN and σS → mixing angle θNS ≈ 11° l Two states emerge: Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Predominantly Non-Strange Sigma Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Predominantly Strange Sigma Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Other Results η – η’ mixing angle θη = 40. 3° ↔ KLOE Collaboration: θη = 41. 4° ± 0. 5° l Rho meson mass has two contributions: l ~ Gluon Condensate l Quark Condensates Decays K* → Kπ, φ(1020) → K+ K- agree with data (preliminary) Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Summary l l l Linear Sigma Model Nf = 3 and vector and axialvector mesons as an effective model of QCD Global fit of all masses except the sigmas Obtained masses within several percent w. r. t. experimental data Scalar meson puzzle: structure of f 0(1370) ↔ mostly nonstrange quarkonium and f 0(1710) ↔ mostly strange quarkonium Thus: scalar quarkonia above 1 Ge. V Other results in agreement with other approaches and experiment (preliminary) Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Outlook l l l Lagrangian With Three Flavours + Glueball + Tetraquarks Mixing in the Scalar Sector: Quarkonia, Tetraquarks and Glueball Extension to Non-Zero Temperature: Study Chiral Symmetry Restoration Low Energy Constants of QCD Pion-Pion Scattering Lengths Include Tensor, Pseudotensor Mesons, Baryons (Nucleons) Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Spare Slides Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Motivation: Structure of Scalar Mesons l l l Spontaneous Breaking of Chiral Symmetry → Goldstone Bosons (Nf = 2 → 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) – not preferred by Nf = 2 results Scalar states above 1 Ge. V → f 0(1370), a 0(1450) – preferred by Nf = 2 results [Parganlija, Giacosa, Rischke in Phys. Rev. D 82: 054024, 2010; ar. Xiv: 1003. 4934] Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Note: Nf = 2 Limit l The f 0(600) state not preferred to be quarkonium [Parganlija, Giacosa, Rischke in Phys. Rev. D 82: 054024, 2010; ar. Xiv: 1003. 4934] Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Note: Nf = 2 Limit [Parganlija, Giacosa, Rischke in Phys. Rev. D 82: 054024, 2010; ar. Xiv: 1003. 4934] Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Scenario II (Nf =2): Scattering Lengths l Scattering lengths saturated Additional scalars: tetraquarks, quasi-molecular states l Glueball l Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Scenario II (Nf =2): 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 2011, Les Houches/France

Scenario I (Nf =2): Other Results l l Our Result Experimental Value [KLOE Collaboration, hep-ex/0612029 v 3]: [D. V. Bugg et al. , Phys. Rev. D 50, 4412 (1994)] Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Scenario I (Nf =2): a 1→σπ 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 2011, Les Houches/France

Comparison: the Model with and without Vectors and Axial-Vectors (Nf=2) 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] Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Scenario I (Nf =2): a 1 → ρπ Decay [M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002)] Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Scenario I (Nf =2) : Parameter Determination l Three Independent Parameters: Z, m 1, mσ ~ Gluon Condensate Quark Condensate [S. Janowski (Frankfurt U. ), Diploma Thesis, 2010] Isospin Angular Momentum (s wave) [NA 48/2 Collaboration, 2009] Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France

Lagrangian of a Linear Sigma Model with Vector and Axial-Vector Mesons (Nf =2) l Vectors and Axial-Vectors vectors Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France axialvectors

Lagrangian of a Linear Sigma Model with Vector and Axial-Vector Mesons (Nf =2) l Scalars and Pseudoscalars Explicit Symmetry Breaking scalars Chiral Anomaly pseudoscalars photon Denis Parganlija (Frankfurt U. ) Excited QCD 2011, Les Houches/France