We review evolution equations for cut Mellin moments (CMM) of the parton densities. The main finding is that $n$-th cut moment of the parton distribution, $$ q_n(x_{min},Q^2) = \int\limits_{x_{min}}^1 dx \, x^{n-1} q_n(x,Q^2) $$ obeys also the DGLAP equation, but with a rescaled splitting function $P'(n,z)=z^n P(z)$.

This approach allows one to restrict the analysis to the experimentally available Bjorken-x region.The evolution equations for cut moments are valid in each order of the perturbation expansion and can be an useful additional tool in QCD analysis of the unpolarized as well as the polarized nucleon structure functions.

We present generalization of DGLAP equations, based on the evolution of CMM and some applications of CMM approach to analysis of the spin structure functions $g_1$ and $g_2$.

The status of the problem of discrepancy of theoretical predictions and experimental values of the anomalous magnetic moment of muon is discussed. New method of measurement of the value (g-2) of muon based on the deviation from the mass shell is proposed. The contribution to this method is calculated within one-loop approximation in small deviation from the mass shell. In the experiment the deviation from the mass shell occurs when particles of the beam interact with each other.

The cross section of electron-positron annihilation is described in the frames of extended NJL model in the region of energies below 2 GeV. The contributions of the intermediate vector mesons $\rho(770)$ and $\rho(1450)$ are taken into account. The results for the channel of production of $\eta 2\pi$ are in a satisfactory agreement with the experimental data. In case of $\eta (\eta') 2 \pi$ the theoretical predictions are presented and they can be tested in experiments.

Our goal is to develop the methods of solution of Schwinger-Dyson equation for effective models of strong interaction. We establish physically clear and transparent relations between the perturbative QCD to its nonperturbative low energy model by means of normal ordering of fields. We show that in the limit of zero quark masses a solution of the Schwinger-Dyson equation is finite without any regularization and renormalization.

We argue, that if vacuum of Standard Model is stable, and for a Higgs boson mass $M_H \sim 125 GeV$, as suggested by recent Higgs searches at the LHC,

(i) the inclusion of electroweak radiative corrections in the relationship between the pole and $\overline{MS}$ masses of the top quark reduces the difference to about 1 GeV.

(ii) there exists an optimal value of renormalization group scale (IR-point), at which the radiative corrections to the matching condition between parameters of Higgs sector and pole masses is minimal or equal to zero. Within the available accuracy, we find the IR-point to lie in an interval between value of Z-boson mass and twice the value of W-boson mass.

This may be relevant for the scheme dependence of electroweak observables, such as the rho parameter, as well as for the extraction of the top quark mass from experimental data. The value of scale (ii) is relevant for extraction of Higgs self-coupling from cross-section as well as for construction of effective Lagrangian.

During last two decades different types of post-LHC colliders were developed. But none of them was approved since their high cost and indefinite physical scenario. It was expected the rich physics in the region from 200 GeV up to 1 TeV: few Higgs bosons, supersymmetry, dark matter and others phenomena which could be best studied at linear $e^+e^-$ colliders. At the moment only the Higgs boson with mass 125 GeV was found at LHC. The absence of new physics below 1 TeV changed the priorities and lead to the appearance of large (more then 20) number of projects of precision study of Higgs boson with the forthcoming investigtion of higher energies. In this talk we discuss the motivation, specific and limiting possibilities of these projects: proton, linear $e^+e^-$, cyclic $e^+e^-$, photon and muon colliders.

Positivity violations in the ghost and gluon propagators indicate color confinement, i.e. the absence of the corresponding colored particles from the asymptotic state space. In this talk we present the first explicit solutions of the Yang-Mills Dyson-Schwinger equations for complex momenta in a truncation that respects multiplicative renormalizability and transversality. The resulting complex propagators are positivity violating and furthermore analytic apart from distinct cut structures on the real, negative momentum axis. These propagators are the vital input to on-going computations of the glueball spectrum in Landau gauge Yang-Mills using the Bethe-Salpether equation.

On the basis of the investigation of the asymptotic behavior of the correlation functions of the corresponding field currents with the necessary quantum numbers the analytic method for the determination of the mass spectrum and decay constants of mesons consisting of c and b quarks with relativistic corrections is proposed. The dependence of the constituent mass of quarks on the current mass and on the orbital and radial quantum numbers is analytically derived. The mass and wave functions of the mesons are determined via the eigenvalues of nonrelativistic Hamiltonian in which the kinetic energy term is defined by the constituent mass of the bound state forming particles and the potential energy term is determined by the contributions of every possible type of Feynman diagrams with an exchange of gauge field. In the framework of our approach the mass splitting between the singlet and triplet states is determined, and the width of E1 transition rates in the ($\bar{c}c$), ($\bar{b}b$) and ($\bar{b}c$) systems are calculated. The obtained results are satisfactorily agree with the experimental data.

Transition form factors of $\pi^0$, $\eta$ and $\eta'$ mesons are studied by means of the anomaly sum rule (ASR) - an exact nonperturbative relation which is a consequence of the dispersive representation of axial anomaly. In combination with the hypothesis of quark-hadron duality it leads to the relation for the transition form factors at arbitrary virtualities of the photon and is valid even if the QCD factorization is violated. In the isovector channel of the ASR, where a pion contribution dominates, the obtained relation reproduces the Brodsky-Lepage interpolation formula. We compare our results with the experimental data of CELLO, CLEO, BABAR and BELLE collaborations. Possibility of a logarithmic correction to the spectral density, allowing to describe the BABAR data, is investigated. Basing on the general properties of the ASR, we propose a form of such a correction. It allows to describe the available experimental data well. The ASR in the octet channel relates the transition form factors and decay constants of the $\eta$ and $\eta'$ mesons. Mixing parameters of these mesons are determined for different mixing schemes. An estimation of the decay constants in a mixing-scheme-independent way is also done.

Traditional QED radiative corrections are considered in one loop approximation. Virtual, soft and hard photon emission is taken into account. The effects of pion structure is taken into account in frames of Nambu-Jona-Lasinio model. The $\rho$, $\omega$ mesons intermediate states are taken into account.

We extend the known method of "returning to resonance" which is known in electron--positron collisions to hadron collisions. The emission of charged and neutral light vector and pseudoscalar mesons in collinear kinematics is considered. (Anti) neutron beams are discussed. Application to description of pion multiplicities is considered as well.

In this talk we present and discuss the predictions on search for enhanced charm and enhanced strangeness in the proton in hard p-p processes of photon and vector mesons which are accompanied with the production of heavy charmed and bottomed jets. It is demonstrated that the taking into account of internal charm and internal strangeness in the structure function of the proton leads to the significant enhancement of production of charmed and strange hadrons in p-p collisions with large transverse momenta and rapidities.

The measurement of the phase between strong and electromagnetic amplitudes in $J/\psi$ decay using the $J/\psi$ scan data from BES is discussed. The formulae in the fitting of experimental data will be presented.

For a complete description of the physical properties of low-energy QCD, it might be advantageous to first reformulate QCD in terms of gauge-invariant dynamical variables, before applying any approximation schemes.

Using a canonical transformation of the dynamical variables, which Abelianises the non-Abelian Gauss-law constraints to be implemented, such a reformulation can be achieved for QCD. The exact implementation of the Gauss laws reduces the colored spin-1 gluons and spin-1/2 quarks to unconstrained colorless spin-0, spin-1, spin-2 and spin-3 glueball fields and colorless Rarita-Schwinger fields respectively.

The obtained physical Hamiltonian can then be rewritten into a form, which separates the rotational from the scalar degrees of freedom, and admits a systematic strong-coupling expansion in powers of $\lambda = g^{-2/3}$, equivalent to an expansion in the number of spatial derivatives. The leading-order term in this expansion corresponds to non-interacting hybrid-glueballs, whose low-lying masses can be calculated with high accuracy by solving the Schrödinger-equation of the Dirac-Yang-Mills quantum mechanics of spatially constant physical fields (at the moment only for the 2-color case). Due to the presence of classical zero-energy valleys of the chromomagnetic potential for two arbitrarily large classical glueball fields (the unconstrained analogs of the well-known constant Abelian fields), practically all glueball excitation energy is expected to go into the increase of the strengths of these two fields. Higher-order terms in $\lambda$ lead to interactions between the hybrid-glueballs and can be taken into account systematically using perturbation theory in $\lambda$.

A package based on the system "Mathematica" for QCD calculations in the framework of non-power Fractional Analytic Perturbation Theory (FAPT), published in Computer Physics Communications 184 (2013) 183–193, are presented.

The singularity-free, finite couplings $A_\nu(Q^2)$ and $A_\nu(s)$ appear in (F)APT as analytic images of the standard QCD coupling powers $\alpha_s^\nu(Q^2)$ in Euclidean and Minkowski domains correspondingly.

I provide all the procedures in "FAPT package" which are needed to compute these couplings up to 4-loops of renormalization group running and to use it for both schemes: with fixed number of active flavours $n_f$ ($A_\nu(Q^2;n_f)$, $A_\nu(s;n_f)$) and the global one with taking into account all heavy-quark thresholds ($A_\nu^{glob}(Q^2)$, $A_\nu^{glob}(s)$).

Application of this "FAPT package" and some QCD results are considered.

Nonlocal PNJL models allow for a detailed description of chiral quark dynamics with running quark masses and wave function renormalization in accordance with lattice QCD (LQCD) in vacuum. Their generalization to finite temperature $T$ and chemical potential $\mu$ allows to reproduce the $\mu$-dependence of the pseudocritical temperature from LQCD when a nonvanishing vector meson coupling is adjusted [1]. This restricts the region for the critical endpoint in the QCD phase diagram and stiffens the quark matter equation of state (EoS). It is demonstrated that the construction of a hybrid EoS for compact star applications within a two-phase approach employing the nonlocal PNJL EoS and an advanced hadronic EoS leads to the masquerade problem [2]. As a possible solution the construction of a unified quark-hadron matter description is outlined which follows from a beyond meanfield [3] treatment of the model within a generalized Beth-Uhlenbeck approach [4].

[1] G. A. Contrera, A. G. Grunfeld, D. B. Blaschke, arXiv:1207.4890.

[2] M. Alford, M. Braby, M. Paris, S. Reddy, Astrophys. J. 629 (2005) 969.

[3] A.E. Radzhabov, D. Blaschke, M. Buballa, M.K. Volkov, Phys. Rev. D83 (2011) 116004.

[4] D. Zablocki, D. Blaschke, M. Buballa, in preparation.