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The f 0 mesons are studied in a combined analysis of data on isoscalar S-wave processes π π → π π , K K ¯ , η η and on decays J / ψ → ϕ ( π π , K K ¯ ) , ψ ( 2 S ) → J / ψ ( π π ) and ϒ ( 2 S ) → ϒ ( 1 S ) π π from the Argus, Crystal Ball, CLEO, CUSB, DM2, Mark II, Mark III, and BES III Collaborations. The method of analysis is based on analyticity and unitarity and uses an uniformization procedure. Some spectroscopic implications from results of the analysis are discussed.

The basic shape of di-pion mass spectra in the two-pion transitions of both charmonia and bottomonia states is explained by an unified mechanism based on contributions of the $\pi\pi$, $K\overline{K}$ and $\eta\eta$ coupled channels including their interference. The role of the individual $f_0$ resonances in shaping the di-pion mass distributions in the charmonia and bottomonia decays is considered., Comment: 8 pages, 4 figures, Talk at the XXIII International Baldin Seminar on High Energy Physics Problems "Relativistic Nuclear Physics \& Quantum Chromodynamics" (Dubna, Russia, 19--24 September 2016). arXiv admin note: text overlap with arXiv:1610.07898

The f 0 mesons and ππ -scattering length a 0 0 are studyed in a model-independent analysis of data on the isoscalar S-wave processes π π → π π , K K ¯ including the very precise NA48/2-Collaboration ππ -data and taking the alternative ππ -scattering data by B. Hyams et al.(1973) and by R. Kaminski et al.(2002).

In a model-independent approach the data on ππ → ππ, K\( \bar K \), ηη, ηη′ in the IGJPC = 0+0++ channel and on the Kπ scattering in the \( I\left( {J^P } \right) = \frac{1} {2}\left( {0^ + } \right) \) channel are analyzed jointly for studying the status and QCD nature of the f0- and the K*0-mesons. It is shown that in the 1500-MeV region, there are two states, wide (interpreted as a glueball) and narrow (q\( \bar q \)). In the Kπ-scattering data analysis, the proof for the K*0(900) is given.

In the analysis a status and parameters of the scalar, vector, and tensor mesonic resonances are obtained and compared with other results. Possible classification of the resonance states in terms of the SU (3) multiplets is discussed.

In approach, based on analyticity and unitarity, experimental data on the isoscalar S- and D-waves of processes ππ → ππ, [Formula: see text], ηη, ηη′ have been analyzed for studying the status and QCD nature of the f0- and f2-mesons below 2.3 GeV. Assignment of scalar and tensor mesons to lower nonets is proposed.

Two-and three-channel analyses of experimental data on the coupled processes ππ → ππ, k, ηη in the channel with I G J PC = 0+0++ are carried out and discussed. In a model-independent approach, confirmation of the σ-meson below 1GeV and definite indications of the QCD nature of other f0 resonances are obtained. An assignment of the scalar mesons below 1.9 GeV to lower nonets is proposed.

In a combined analysis of the experimental data on the coupled processes ππ↦ππ, K¯K in the channel with IGJPC = 0+0+ +, the various scenarios of these reactions (with different numbers of resonances) are considered. In a model-independent approach, based only on analyticity and unitarity, a resonance is represented by a pole cluster (poles on the Riemann surface) of the definite type that is defined by the state nature. The best scenario contains the resonances f0(665) (with properties of the σ-meson), f0(980) (with a dominant s¯s component), f0(1500) (with a dominant flavour-singlet, e.g., glueball component) and the f0(1710) (with a considerable s¯s component). If the f0(1370) exists, it has a dominant s¯s component. The coupling constants of the observed states with the considered channels and the ππ and K¯K scattering lengths are obtained. The conclusion on the linear realization of chiral symmetry is drawn.

In the combined analysis data on multi-channel ππ scattering and J/ψ decays, a status and parameters of the scalar mesonic resonances are obtained.

/                                            � �                

In the framework of the generalized Hamiltonian formalism by Dirac, local symmetries of dynamical systems with first- and second-class constraints are investigated. The method of constructing the generator of local-symmetry transformations is presented both for theories with an algebra of constraints of a special form (a majority of the physically interesting theories) and in the general case without restrictions on the algebra of constraints. It is proven that second-class constraints do not contribute to the transformation law of the local symmetry entirely stipulated by all the first-class constraints. A mechanism of the occurrence of higher derivatives of coordinates and group parameters in the symmetry transformation law in Noether's second theorem is elucidated. In the latter case it is shown that the obtained transformations of symmetry are canonical in the extended (by Ostrogradsky) phase space. It is thereby shown that in the general case the degeneracy of theories with first- and second-class constraints is due to their invariance under local-symmetry transformations.

In the generalized Hamiltonian formalism by Dirac, the method of constructing the generator of local-symmetry transformations for systems with first- and second-class constraints (without restrictions on the algebra of constraints) is obtained from the requirement for them to map the solutions of the Hamiltonian equations of motion into the solutions of the same equations. It is proved that second-class constraints do not contribute to the transformation law of the local symmetry entirely stipulated by all the first-class constraints (and only by them). A mechanism of occurrence of higher derivatives of coordinates and group parameters in the symmetry transformation law in the Noether second theorem is elucidated. It is shown that the obtained transformations of symmetry are canonical in the extended (by Ostrogradsky) phase space. An application of the method in theories with higher derivatives is demonstrated with an example of the spinor Christ -- Lee model.

Analysis of the isoscalar S- and D-waves of processes ππ→ππ, [Formula: see text], ηη is carried out aimed at studying the QCD nature of scalar and tensor mesons below 2 GeV. An assignment of the scalar mesons below to lower nonets is proposed.

A method is proposed for constructing infinitesimal gauge transformations for an arbitrary degenerate Lagrangian without restrictions on the constraint algebra. It is thereby shown that in the general case degeneracy of a theory is due to its gauge invariance.

A mechanism of occurrence of higher derivatives of the coordinates in the symmetry transformation law in Noether's second theorem is elucidated. It is shown that the corresponding transformations in the Hamiltonian formalism are canonical in an extended phase space.

The coupled processes -- the pion-pion scattering and pi pi --> K antiK in the I^GJ^{PC}=0^+0^{++} channel -- are analyzed (both separately and combined) in a model-independent approach based on analyticity and unitarity and using a uniformization procedure. It is shown: 1) a structure of the Riemann surface of the S-matrix for considered coupled processes must be allowed for calculating both amplitudes and resonance parameters, such as the mass and width; 2) the combined analysis of coupled processes is needed as the analysis of only pi-pi channel does not give correct values of resonance parameters even if the Riemann surface structure is included.

Peculiarities of obtaining parameters for broad multichannel resonances from data are discussed analyzing the experimental data on processes ππ → ππ, KK¯${m{Kar K}}$, in the scalar channel in a model-independent approach based on analyticity and unitarity and using an uniformization procedure. It is shown that one can obtain a good description of the ππ scattering data from the threshold up to 1.89 GeV with values of resonance parameters cited in the PDG tables as preferred. However, in this case the representation of the ππ background is not satisfactory and the data on the coupled process ππ → KK¯${m{Kar K}}$ are not well described, above 1.15 GeV even qualitatively. The combined analysis of the considered coupled processes is needed, which is also carried out satisfactorily. Then both above-indicated deficiencies related to the one-channel analysis are removed. The most important change concerns parameters of the f0(600) which are now in some accordance with the prediction by Weinberg on the basis of mended symmetry. The obtained ππ- scattering length agrees well with the values extracted from the data on Ke4 decay and DIRAC experiment and also with results from ChPT.

The status and parameters of the scalar, vector, and tensor mesonic resonances are obtained in the multichannel analysis of processes {pi}{pi}{yields}{pi}{pi}, KK, {eta}{eta}, and {eta}{eta}{sup '} and compared with other results. For example, we obtained the f{sub 0}(600) meson with mass 774{+-}15 MeV and width 988{+-}36 MeV (the pole position on sheet II is 596-i494 MeV) and the first {rho}-like meson with mass 1275{+-}32 MeV and total width 304{+-}24 MeV, which differ significantly from the mean value 1459{+-}11 MeV cited in the Particle Data Group tables. Spectroscopic implications from results of these analyses and possible classification of the resonance states in terms of the SU(3) multiplets are also discussed.

It is shown that the basic shape of dipion mass distributions in the two-pion transitions of both charmonia and bottomonia states are explained by an unified mechanism based on the contribution of the $\pi\pi$, $K\overline{K}$ and $\eta\eta$ coupled channels including their interference., Comment: 5 pages, 3 figures, talk at the14th International Workshop on Meson Production, Properties and Interaction (Krakow, Poland, 2nd - 7th June 2016)

Analysis of the isoscalar S- and D-waves of processes pi-pi-> pi-pi, K-Kbar, eta-eta is carried out aimed at studying the status and QCD nature of scalar and tensor mesons below 2 GeV and 2.3 GeV, respectively. Assignment of these mesons to lower scalar and tensor nonets is proposed.

When simultaneously analysing experimental data on the processes ππ → ππ, KK, ηη in the channel with the vacuum quantum numbers, definite indications of the QCD nature of the f0 resonances below 1.9 GeV are obtained. An assignment of the scalar mesons below 1.9 GeV to lower nonets is proposed. An approach, using only first principles (analyticity and unitarity) and the uniformizing variable, is applied.

Parameters of ρ-like resonances are obtained in the multichannel analysis of the ππ-scattering data below 1.9 GeV and a possible classification of the resonance states is discussed. In the 3-channel analysis, we confirmed the existence of the ρ(1250) meson and showed that the simultaneous existence of the ρ(1450) and ρ(1250) mesons does not contradict the data. The analysis indicates an importance of presence of some effective channel with the threshold at 1512 MeV, which might be interpreted as the ρσ channel.

TheN-channel formalism (in more detail, 2- and 3-channel) is presented which is based on such general principles, as analyticity and unitarity, and realizes an idea of the consistent account of the nearest (to the considered physical region) singularities on all sheets of the Riemann surface of theS-matrix, thus giving a chance to obtain model-independent information on multichannel resonances from the analysis of data on the coupled processes. The resonance representations by pairs of complex-conjugate clusters of poles and zeros on the Riemann surface and the conception of standard clusters as a model-independent test of the resonance presence are discussed. The role of closed channels in forming resonances and fictitious states is investigated. The method is exemplified with the isoscalars-wave channel of coupled processes ππ→ππ, K\(\overline K \), ππ and with analysing ππ scattering up to 1.9 GeV and scalar resonances.

The s-wave pion scattering amplitude is analysed with the aim to clarify the mass spectrum of scalar mesons and to find evidence of lightest glueball. The S-matrix ππ and K¯K coupled channel formalism is used. The existence of scalar mesons S* and ɛ is implied by the data. The production ππ → K¯K and the elastic K¯K→K¯K coupled amplitudes are predicted from the ππ →ππ scattering data. The couplings c f S* to ππ and K¯K states are determined.

The method is formulated for constructing the generators of gauge transformations in the phase space for a given singular Lagrangian. These generators are used to impose the necessary and sufficient conditions on gauge functions. The results are analyzed for particular examples.