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A simple, physically validated, model of charge-independence and charge-symmetry breaking is proposed for the pion-nucleon coupling constant. Within this model, the pion-nucleon coupling constants are assumed to be in direct proportion to the masses of nucleons and pions involved in the interaction. The charge dependence of the pion-nucleon coupling constants and low-energy parameters of nucleon-nucleon scattering in the 1S0 spin-singlet state is studied on the basis of Yukawa’s meson theory. By using the value of $$f_{pp\pi^0}^2=0.0749(7)$$ established reliably for the pseudovector pion-nucleon coupling constant, which characterizes the strength of the nuclear proton-proton interaction, the values of $$f_c^2=0.0802(7)$$ and $$f_0^2=0.0750(7)$$ were calculated for, respectively, the charged and neutral pion-nucleon coupling constants, along with the value of $$f_{nn\pi^0}^2=0.0751(7)$$, which characterizes the strength of the nuclear neutron-neutron interaction. The values calculated for the low-energy parameters of neutron-proton and neutron-neutron scattering with the aid of the above constants agree well with experimental data.

The relation between quantities that characterize the pion–nucleon and nucleon–nucleon interactions is studied with allowance for the fact that, at low energies, nuclear forces in nucleon–nucleon systems are mediated predominantly by one-pion exchange. On the basis of the values currently recommended for the low-energy parameters of the proton–proton interaction, the charged pion–nucleon coupling constant is evaluated at g π 2 ±/4π = 14.55(13). This value is in perfect agreement with the experimental value of g π 2 ±/4π = 14.52(26) found by the Uppsala Neutron Research Group. At the same time, the value obtained for the charged pion–nucleon coupling constant differs sizably from the value of the pion–nucleon coupling constant for neutral pions, which is g π 2 0/4π = 13.55(13). This is indicative of a substantial charge dependence of the coupling constant.

A relation between quantities that characterize the breaking of the charge symmetry of nuclear forces in systems of two and three nucleons is found on the basis of an analysis of the binding-energy difference between the 3H and 3He mirror nuclei. For the neutron-neutron scattering length and effective range, the values of ann = −18.38(55) fm and rnn = 2.84(4) fm, respectively, were obtained by using purely nuclear parameters of proton-proton scattering in the 1S0 state. The calculated values agree with present-day experimental results.

The effect of the mass difference between the charged and neutral pions on the low-energy parameters of nucleon-nucleon interaction in the 1S0 state is studied in the effective-range approximation. On the basis of experimental values of the singlet parameters of neutron-proton scattering and the experimental value of the virtual-state energy for the neutron-neutron systemin the 1S0 state, the following values were obtained for the neutron-neutron scattering length and effective range: ann = −16.59(117) fm and rnn = 2.83(11) fm. The calculated values agree well with present-day experimental results.

On the basis of the total cross sections for neutron-proton scattering in the region of laboratory energies below 150 keV, the value of σ0 = 20.4288(146) b was obtained for the total cross sections for neutron-proton scattering at zero energy. This value is in very good agreement with the experimental cross sections obtained by Houke and Hurst, but it is at odds with Dilg’s experimental cross section. By using the value that we found for σ0 and the experimental values of the neutron-proton coherent scattering length f, the deuteron binding energy ɛ t , the deuteron effective radius ρ t (−ɛ t , −ɛ t ), and the total cross section in the region of energies below 5 MeV, the following values were found in the shape-parameter approximation for the low-energy parameters of neutron-proton scattering in the spin-triplet and spin-singlet states: a t = 5.4114(27) fm, r 0t = 1.7606(35) fm, v 2t = 0.157 fm3, a s = −23.7154(80) fm, r 0s = 2.706(67) fm, and v 2s = 0.491 fm3.

Experimental data on neutron-proton scattering in the energy range between 0 and 150 keV are analyzed by using various sets of effective-range parameters. It is shown that, in contrast to the parameters corresponding to the phase shifts of a Nijmegen group, the parameters corresponding to the experimental phase shifts reported by a group from George Washington University (GWU group) lead to very good agreement between the calculated cross sections and their experimental counterparts in the energy region under consideration. On the basis of the experimental value of the cross section for neutron—proton scattering at an energy of 2 keV, the total cross section for neutron-proton scattering at zero energy was found to be σ0 = 20.428(16) b, which is in very good agreement with a value of σ0 = 20.423(9) b, which was obtained as the weighted mean of the cross sections presented by Houke and Hurst. It is shown that, in the energy region around several tens of keV units, the effective-range parameters matched with Dilg’s cross-section value of σ0 = 20.491(14) b lead to calculated cross sections whose values are in excess of their experimental counterparts.

The correlation between the root-mean-square matter radius of the deuteron, r m , and its effective radius, ρ, is investigated. A parabolic relationship between these two quantities makes it possible to determine the root-mean-square radius r m to within 0.01% if the effective radius ρ is known. The matter (r m ), structural (r d ), and charge (r ch) radii of the deuteron are found with the aid of modern experimental results for phase shifts from the SAID nucleon-nucleon database, and their values are fully consistent with their counterparts deduced by using the experimental value of the effective deuteron radius due to Borbely and his coauthors. The charge-radius value of 2.124(6) fm, which was obtained with the aid of the SAID nucleon-nucleon database, and the charge-radius value of 2.126(12) fm, which was obtained with the aid of the experimental value of the effective radius ρ, are in very good agreement with the present-day chargeradius value of 2.128(11) fm, which was deduced by Sick and Trautmann by processing world-average experimental data on elastic electron scattering by deuterons with allowance for Coulomb distortions.

On the basis of the Bargmann representation of the S matrix, the pole approximation is obtained for the effective-range function k cot δ. This approximation is optimal for describing the neutron-deuteron system in the doublet spin state. The values of r0 = 412.469 fm and v2 = −35 495.62 fm3 for the doublet low-energy parameters of neutron-deuteron scattering and the value of D = 172.678 fm2 for the respective pole parameter are deduced by using experimental results for the triton binding energy ET, the doublet neutron-deuteron scattering length a2, and van Oers-Seagrave phase shifts at energies below the deuteron-breakup threshold. With these parameters, the pole approximation of the effective-range function provides a highly precise description (the relative error does not exceed 1%) of the doublet phase shift for neutron-deuteron scattering at energies below the deuteron-breakup threshold. Physical properties of the triton in the ground (T) and virtual (v) states are calculated. The results are Bv = 0.608 MeV for the virtuallevel position and CT2 = 2.866 and Cv2 = 0.0586 for the dimensionless asymptotic normalization constants. It is shown that, in the Whiting-Fuda approximation, the values of physical quantities characterizing the triton virtual state are determined to a high precision by one parameter, the doublet neutron-deuteron scattering length a2. The effective triton radii in the ground (ρT = 1.711 fm) and virtual (ρv = 74.184 fm) states are calculated for the first time.

Triplet low-energy parameters of neutron-proton scattering, including high-order shape parameters, are calculated on the basis of approximating the effective-range function k cot δ t by polynomials and rational functions with the aid of the latest experimental data on phase shifts from the SAID nucleon-nucleon database. With the resulting values of the low-energy parameters, a good description of phase shifts over a broad energy interval is obtained by using the effective-range expansion featuring a small number of terms. The properties of the deuteron that were calculated on the basis of the values found for the triplet low-energy parameters of scattering agree very well with experimental values. The triplet low-energy parameters and the properties of the deuteron that were obtained here by using present-day data from the SAID database differ markedly from the analogous results obtained for data of the Nijmegen group. Possible reasons behind this discrepancy are discussed. Highly precise new approximate formulas for determining the shape parameter v 2 are proposed and are shown to be efficient in calculations. The effective-range expansion for the D wave is considered, and preliminary results of calculations of low-energy scattering parameters for this case are obtained.

For the effective-range function $k\cot \delta $, a pole approximation that involves a small number of parameters is derived on the basis of the Bargmann representation of the $S$ matrix. The parameters of this representation, which have a clear physical meaning, are related to the parameters of the Bargmann $S$ matrix by simple equations. By using a polynomial least-squares fit to the function $k\cot \delta $ at low energies, the triplet low-energy parameters of neutron-proton scattering are obtained for the latest experimental data of Arndt et al. on phase shifts. The results are $a_{t}=5.4030 $fm, $r_{t}=1.7494 $fm, and $v_{2}=0.163 $fm$^{3}$. With allowance for the values found for the low-energy scattering parameters and for the pole parameter, the pole approximation of the function $k\cot \delta $ provides an excellent description of the triplet phase shift for neutron-proton scattering over a wide energy range ($T_{\text{lab}}\lesssim 1000 $MeV), substantially improving the description at low energies as well. For the experimental phase shifts of Arndt et al., the triplet shape parameters $v_{n}$ of the effective-range expansion are obtained by using the pole approximation. The description of the phase shift by means of the effective-range expansion featuring values found for the low-energy scattering parameters proves to be fairly accurate over a broad energy region extending to energy values approximately equal to the energy at which this phase shift changes sign, this being indicative of a high accuracy and a considerable value of the effective-range expansion in describing experimental data on nucleon-nucleon scattering. The properties of the deuteron that were calculated by using various approximations of the effective-range function comply well with their experimental values.

The correlation relationship between the deuteron asymptotic normalization constant, $A_{S}$, and the triplet np scattering length, $a_{t}$, is investigated. It is found that 99.7% of the asymptotic constant $A_{S}$ is determined by the scattering length $a_{t}$. It is shown that the linear correlation relationship between the quantities $A_{S}^{-2}$ and $1/a_{t}$ provides a good test of correctness of various models of nucleon-nucleon interaction. It is revealed that, for the normalization constant $A_{S}$ and for the root-mean-square deuteron radius $r_{d}$, the results obtained with the experimental value recommended at present for the triplet scattering length $a_{t}$ are exaggerated with respect to their experimental counterparts. By using the latest experimental phase shifts of Arndt et al., we obtain, for the low-energy scattering parameters ($a_{t}$, $r_{t}$, $P_{t}$) and for the deuteron characteristics ($A_{S}$, $r_{d}$), results that comply well with experimental data.

The scattering of two charged strongly interacting particles is described on the basis of the P-matrix approach. In the P matrix, it is proposed to isolate explicitly the background term corresponding to purely Coulomb interaction, whereby it becomes possible to improve convergence of the expansions used and to obtain a correct asymptotic behavior of observables at high energies. The expressions for the purely Coulomb background P matrix, its poles and residues, and purely Coulomb eigenfunctions in the P-matrix approach are obtained. The nuclear-Coulomb parameters of the low-energy scattering of two charged hadrons are investigated on the basis of this approach combined with the method for isolating the background P matrix. Simple explicit expressions for the nuclear-Coulomb scattering length and effective range in terms of the residual P matrix are derived. For models of short-range strong interaction, these expressions give a general form of nuclear-Coulomb parameters for low-energy scattering. Specific applications of the general expressions derived in this study are exemplified by considering, on the basis of these expressions, some exactly solvable models of strong interaction, including the hard-core model, and, for these models, the nuclear-Coulomb parameters for low-energy scattering at arbitrary values of the orbital angular momentum are found explicitly for the first time. In particular, the nuclear-Coulomb scattering length and effective range are obtained explicitly for the boundary-condition model, the model of a hard-core delta-shell potential, the Margenau model, and the model of square-well hard-core potential.

Low-energy doublet neutron-deuteron scattering is described in terms of the energies of the bound and virtual triton states and the nuclear vertex constants for these states. For the first time, the van Oers-Seagrave formula is derived from the Bargmann representation of the S matrix for a system having two states. The presence of a pole in this formula is shown to be a direct corollary of the existence of a low-energy virtual triton state. Simple explicit expressions for the nd scattering length and for the pole of the function kcotδ are obtained in terms of the parameters of the bound and virtual triton states. The low-energy parameters of nd scattering are calculated numerically, whereby it is shown that these parameters are highly sensitive to variations in the asymptotic normalization constant Cv2 for the virtual state. The Cv2 value fitted in our model to the experimental result for the nd scattering length is 0.0592.