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For the quantum quartic anharmonic oscillator with the Hamiltonian H = (p2+x2)/2+λx4, which is one of the traditional quantum-mechanical and quantum-field-theory models, we study summation of its factorially divergent perturbation series by the proposed method of averaging of the corresponding Padé approximants. Thus, for the first time, we are able to construct the Padé-type approximations that possess correct asymptotic behaviour at infinity with a rise of the coupling constant λ. The approach gives very essential theoretical and applicatory-computational advantages in applications of the given method. We also study convergence of the applied approximations and calculate by the proposed method the ground state energy E0(λ) of the anharmonic oscillator for a wide range of variation of the coupling constant λ.

Quantum chromodynamics (QCD) running coupling constant and its dependence on the energy scale renormalization parameter are studied in the energy range of μ ≲ 25 GeV based on the five-loop calculations according to QCD renormalization group equation. Position of the Landau pole of the dependence (QCD scale parameter ) is calculated in the five-loop order for the first time. Energy dependence of the running coupling constant in the given energy range is very well described by the proposed simple one-pole formula.

Simple relations connecting the strange quark mass to the splittings of the light hyperon masses were obtained on the basis of the phenomenological quark model compatible with the present-day theory of strong interactions, i.e. with quantum chromodynamics (QCD). Strange quark mass ms = 89.5 ±19.5 MeV, calculated in the proposed approach, is in good agreement with the modern evaluations and calculations of this quantity, mainly obtained by the lattice QCD methods.

Simple relations between the masses of the two lightest up and down quarks were obtained on the basis of the simple physically based model compatible with the present-day theory of strong interactions, i.e. with quantum chromodynamics. Relations between the u and d quark masses, on one hand, and nucleon and pion masses, on the other hand, are also established. Thus, the obtained in such a way elementary formula mu/md = 1/(1+√2), relating u and d quark masses, appears to be in excellent agreement with a number of theoretical calculations of the ratio mu/md of the lightest quark masses. The u and d quark masses mu = 1.903 MeV, md = 4.594 MeV, calculated with the help of the obtained relations, are also in very good agreement with the modern evaluations and calculations of these quantities. The average of the u and d quark masses m̅ud = ΔMπ/√2 ≅ 3.248 MeV, obtained in the proposed approach, is in good agreement with previous calculations too.

Charge dependence of the pion-nucleon coupling constants and the 1S0-state low-energy nucleon-nucleon scattering parameters are studied on the basis of the Yukawa meson theory. The charge independence breaking effect in the pion-nucleon coupling constants is entirely explained by the mass difference between the charged and the neutral pions and by the mass difference between the proton and the neutron. Using of well-known pseudovector pion-nucleon coupling constant f 2ppπ0=0.0749(7), which characterize the proton-proton nuclear interaction, we calculate the charged f 2c=0.0802(7) and the neutral f 20=0.0750(7) pion-nucleon coupling constants, and also the pion-nucleon coupling constant f 2nnπ0=0.0751(7), which characterize the neutron-neutron nuclear interaction. With the help of the obtained coupling constants, we also calculate values of the low-energy np- and nn-scattering parameters, which appear to be in good agreement with the experiment.

Charge independence breaking (CIB) in the pion-nucleon coupling constant and the nucleon-nucleon scattering length is considered on the basis of the Yukawa meson theory. CIB effect in these quantities is almost entirely explained by the mass difference between the charged and the neutral pions. Therewith charge splitting of the pion-nucleon coupling constant is almost the same as charge splitting of the pion mass. Calculated difference between the proton-proton and the neutron-proton scattering length in this case comprises ∼90% of the experimental value.

On the basis of the Yukawa potential we study the pion-nucleon coupling constants for the neutral and charged pions assuming that nuclear forces at low energies are mainly determined by the exchange of virtual pions. We obtain the charged pseudovector pion-nucleon coupling constant f2π± = 0.0804(7) by making the use of experimental low-energy scattering parameters for the singlet pp- and np-scattering, and also by use of the neutral pseudovector pion-nucleon coupling constant f2π0 = 0.0749(7). Corresponding value of the charged pseudoscalar pion-nucleon coupling constant g2π0 / 4π = 14.55(13) is also determined. This calculated value of the charged pseudoscalar pion-nucleon coupling constant is in fully agreement with the experimental constant g2π0 / 4π = 14.52(26) obtained by the Uppsala Neutron Research Group. Our results show considerable charge splitting of the pion-nucleon coupling constant.

The influence of the mass difference between charged and neutral -mesons on the low-energy parameters of nucleon-nucleon interaction in the spin-singlet state is studied. Using the experimental singlet neutron-proton scattering parameters and the experimental value of neutron-neutron virtual-state energy we obtain the following values for the neutron-neutron scattering length and effective range: . Calculated values of these quantities appear to be in reasonable agreement with the contempo-rary experimental data.

Singlet low-energy parameters of neutron-proton scattering are calculated on the basis of approximating the singlet effective-range function by polynomials with the help of the latest experimental data on phase shifts of np scattering from the SAID nucleon-nucleon database (data of the GWU group). The obtained values of the singlet low-energy parameters are With the obtained values of the low-energy parameters, precise description of singlet phase shift in the energy interval is received by using the effective-range expansion. Low-energy neutron-proton scattering parameters for the experimental data from the SAID database differ markedly from the analogous results obtained for data of the Nijmegen group. Obtained effective range parameters corresponding to the experimental data from the SAID database lead to an extremely well description of experimental total cross section of neutron-proton scattering, which is in contrast to the set of the effective range parameters of the Nijmegen group.