Nature of f 0 (1370), f 0 (1500) and f 0 (1710) within the eLSM Stanislaus Janowski in collaboration with F. Giacosa, D. Parganlija and D. H. Rischke Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Hadrons Hadrons: `white` particles made of quarks and gluons Well established Hadrons: Baryons: Mesons: Hypothetical Hadrons: Glueballs: e.g. (B=0  Mesons!) Confinement  d.o.f. in the eff. Lagrangian are Hadrons Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Symmetries of eLSM Global chiral symmetry: Explicitly broken by: ▪ axial anomaly  ▪ nonvanishing quark masses in vacuum spontaneously broken  Trace anomaly  nonvanishing gluon condensate  QCD low-energy scale Dilatation symmetry: Discrete symmetries: parity P, charge conjugation C, time reversal T Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Using 3 quark flavours (u, d, s)  9 meson configurations possible, but taking into account angular momentum L and spin S  e.g. 36 mesons Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University Meson Fields (qqbar) of eLSM

Nonets of Meson Fields (qqbar) of eLSM Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University Vector fieldsAxialvector fields

Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University Pseudoscalar fieldsScalar fields Nonets of Meson Fields (qqbar) of eLSM scalars-isoscalars

Motivation Five states listed by PDG: IIs there a Glueball?... which one? Too many scalars-isoscalars to explain their content with quark-antiquark picture only! First Problem: The same QN! Overpoulation in the scalar-isoscalar channel below 2 GeV Glueball? Requirements for glueball decays: flavour-blindness narrow Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Motivation Second Problem: glueball mix with quark-antiquark state IPhenomenological description via decays and comparison with data Treatment of the problems: Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Glueball Spectrum on Lattice QCD [C. Morningstar and M. J. Peardon, Phys. Rev. D60, (1999) arXiv:hep-lat/ ] Quenched approximation: only gluons, no quark loops! Unquenched calculations affect this glueball masses [E. Gregory, A. Irving, B. Lucini, C. McNeile, A. Rago, C. Richards, E. Rinaldi arXiv: [hep-lat]] Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Mixing of the bare Scalars-Isoscalars in the LSM N f = 2  effective model contains 16 mesons and 1 scalar glueball  mixing of two scalar-isoscalar states N f = 3  effective model contains 36 mesons and 1 scalar glueball  mixing of three scalar-isoscalar states Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

(Pseudo)scalar mesons: (Axial)vector mesons: Chiral transformations of the multiplets: Multiplets of qqbar Fields (N f =3) [D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa, D. H. Rischke, Phys. Rev. D87, (2013), arXiv: ]

Effective Dilaton Lagrangian [H. Gomm, P. Jain, R. Johnson and J. Schechter, Phys. Rev. D 33, 801 (1986)] [A. A. Migdal and M. A. Shifman, Phys. Lett. B 114, 445 (1982)] [M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147,385 (1979] [J. Kripfganz, Phys. Lett. B 101, 169 (1981)] Trace anomaly: gluon condensate breaks dilatation invariance Effective Dilaton Lagrangian: Trace of the energy-momentum tensor of the YM Lagrangian: QCD sum rules and QCD lattice Dilatation: : for bosons for fermions Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Effective Model – eLSM with Scalar Glueball [D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa, D. H. Rischke, Phys. Rev. D87, (2013), arXiv: ] [S. Janowski, D. Parganlija, F. Giacosa, D. H. Rischke, Phys. Rev. D84, (2011), arXiv: ] Full mesonic effective Lagrangian: Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Parameters (N f = 2) 10 parameters: fixing to masses and vev of the sigma field  Large N c dependence:

Results - Best Scenario (N f =2) f 0 (1370) consists to 76% of quark-antiquark and to 24% of glueball state f 0 (1500) consists to 76% of glueball and to 24% of quark-antiquark state [S. Janowski, D. Parganlija, F. Giacosa, D. H. Rischke, Phys. Rev. D84, (2011), arXiv: ] The quarkonium-glueball mixing angle: θ = (29.7±3.6)°  Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University Orthonogal transformation:

Results - Best Scenario (N f =2) Using the experimental data we obtain with the χ 2 analysis following parameters: Total decay width: Experiment: Our value: consistent with lattice QCD Consequences and predictions: χ 2 /d.o.f. = 0.29 Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University [S. Janowski, D. Parganlija, F. Giacosa, D. H. Rischke, Phys. Rev. D84, (2011), arXiv: ]

Results - Second Best Scenario (N f =2) Using the experimental data we obtain with the χ 2 analysis following parameters: Consequences and predictions: χ 2 /d.o.f. = 1.72 Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University The quarkonium-glueball mixing angle: θ = (37.2 ± 21.4)°  unexpectedly large, reverse ordering is possible! Large decay width, but experimentally not seen ! [S. Janowski, D. Parganlija, F. Giacosa, D. H. Rischke, Phys. Rev. D84, (2011), arXiv: ]

Parameters of the eLSM (N f =3) Global fit of eLSM without scalar glueball We determine the parameters m G, Λ and λ 1 by using the mass equations of the scalar-isoscalar states Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University Condition for SSB of CS : [D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa, D. H. Rischke, Phys. Rev. D87, (2013), arXiv: ] Talk on Fr. by Peter Kovacs

Potential of the Scalars-Isoscalars (N f =3) [D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa, D. H. Rischke arXiv: [hep-ph] (2012)] Mixing parameter: Bare mass of the scalar glueball and mixing parameters: Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University Bare masses of the nonstrange and strange mesons:

Mixing Matrix (N f = 3) Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University Admixtures of the f 0 resonances:

Decays of Scalars-Isoscalars in the eLSM Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University Decays of the scalars-isoscalars into ππ and KK depend on the Large-N c suppressed parameter h 1 Simultaneously description of these decay processes within a small range of h 1 failed  Fit including h 1 necessary!

Status of the Fit of eLSM with Scalar Glueball Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University Fit of the eLSM including the scalar glueball does not jet work satisfactorily  The Large-N c suppressed parameter h 1 is still to large! M f0(1370), Γ f0(1370)→ππ/KK/ηη only poorly known! The resonance f 0 (1790) can affect the mixing behaviour, but is not included in the fit Possible reasons: Approach to solving this problem(s)  technical improvement of the fit (iteration method)  including f 0 (1790)...?  new global fit

Conclusions Global chiral invariant eLSM including (axial)vector mesons and scalar glueball was presented N f =2  Best scenario: f 0 (1500) predominantly scalar glueball state and f 0 (1370) predominantly quark-antiquark state Second best scenario: f 0 (1710) predominantly scalar glueball state and f 0 (1370) predominantly quark- antiquark state N f =3  f 0 (1710) predominantly scalar glueball, f 0 (1370) predominantly nonstrange quark-antiquark state and f 0 (1500) predominatly strange quark-antiquark state  at the mass level! Fit of the eLSM including a scalar glueball requires technical improvements Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Outlook Continue the calculations of the model with three quark flavours (u,d,s) and eventually including of a nonet of tetraquark states in order to obtain a general mixing scenario of five scalars-isoscalars. Thank you for your attention! Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Pseudoscalar Glueball Lattice QCD: BESIII: [W. I. Eshraim, S. Janowski, F. Giacosa, D. H. Rischke, arXiv: [hep-ph] (2012)] [C. Morningstar and M. J. Peardon, Phys. Rev. D60, (1999) arXiv:hep-lat/ ] [BES III Collaboration, Phys. Rev. Lett (2011)] Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University

Branching Ratios Pseudoscalar Glueball (N f = 3) Dominant channels: [W. I. Eshraim, S. Janowski, F. Giacosa, D. H. Rischke, arXiv: [hep-ph] (2012)] Stanislaus Janowski FAIRNESS, Berlin 2013 Goethe-University