We present a charge-dependent one-boson-exchange nucleon-nucleon $\mathrm{}\left(\mathrm{NN}\right)\mathrm{}$ potential that fits the world proton-proton data below 350 MeV available in the year 2000 with a ${\chi }^2$ per datum of 1.01 for 2932 data and the corresponding neutron-proton data with ${\chi }^2/\mathrm{datum}$ $=1.02\mathrm{}$ for 3058 data. This reproduction of the $\mathrm{NN}$ data is more accurate than by any phase-shift analysis and any other $\mathrm{NN}$ potential. This is achieved by the introduction of two effective $\sigma$ mesons the parameters of which are partial-wave dependent. The charge dependence of the present potential (which we call “CD-Bonn”) is based upon the predictions by the Bonn full model for charge symmetry and charge-independence breaking in all partial waves with $J<~4.$ The potential is represented in terms of the covariant Feynman amplitudes for one-boson exchange which are nonlocal. Therefore, the off-shell behavior of the CD-Bonn potential differs in a characteristic way from commonly used local potentials and leads to larger binding energies in nuclear few- and many-body systems, where underbinding is a persistent problem.

We present the first nucleon-nucleon potential at next-to-next-to-next-to-leading order (fourth order) of chiral perturbation theory. Charge dependence is included up to next-to-leading order of the isospin-violation scheme. The accuracy for the reproduction of the nucleon-nucleon $\left(NN\right)$ data below $290\text{−}\text{MeV}$ lab energy is comparable to the one of phenomenological high-precision potentials. Since $NN$ potentials of order three and less are known to be deficient in quantitative terms, the present work shows that the fourth order is necessary and sufficient for a $NN$ potential reliable up to $290\text{MeV}$. The new potential provides a promising starting point for exact few-body calculations and microscopic nuclear structure theory (including chiral many-body forces derived on the same footing).

We present $\mathit{NN}$ potentials through five orders of chiral effective field theory ranging from leading order (LO) to next-to-next-to-next-to-next-to-leading order (${\mathrm{N}}^4\mathrm{LO}$). The construction may be perceived as consistent in the sense that the same power counting scheme as well as the same cutoff procedures are applied in all orders. Moreover, the long-range parts of these potentials are fixed by the very accurate $\pi N$ low-energy constants (LECs) as determined in the Roy-Steiner equations analysis by Hoferichter, Ruiz de Elvira, and coworkers. In fact, the uncertainties of these LECs are so small that a variation within the errors leads to effects that are essentially negligible, reducing the error budget of predictions considerably. The $\mathit{NN}$ potentials are fit to the world $\mathit{NN}$ data below the pion-production threshold of the year 2016. The potential of the highest order (${\mathrm{N}}^4\mathrm{LO}$) reproduces the world $\mathit{NN}$ data with the outstanding ${\chi }^2$/datum of 1.15, which is the highest precision ever accomplished for any chiral $\mathit{NN}$ potential to date. The $\mathit{NN}$ potentials presented may serve as a solid basis for systematic ab initio calculations of nuclear structure and reactions that allow for a comprehensive error analysis. In particular, the consistent order by order development of the potentials will make possible a reliable determination of the truncation error at each order. Our family of potentials is nonlocal and, generally, of soft character. This feature is reflected in the fact that the predictions for the triton binding energy (from two-body forces only) converges to about 8.1 MeV at the highest orders. This leaves room for three-nucleon-force contributions of moderate size.

We present the two- and three-pion-exchange contributions to the nucleon-nucleon interaction which occur at next-to–next-to–next-to–next-to–leading order (${\mathrm{N}}^4\mathrm{LO}$, fifth order) of chiral effective field theory and calculate nucleon-nucleon scattering in peripheral partial waves with $L\ge 3$ by using low-energy constants that were extracted from $\pi N$ analysis at fourth order. While the net three-pion-exchange contribution is moderate, the two-pion exchanges turn out to be sizable and prevailingly repulsive, thus compensating the excessive attraction characteristic for next-to–next-to–leading order and ${\mathrm{N}}^3\mathrm{LO}$. As a result, the ${\mathrm{N}}^4\mathrm{LO}$ predictions for the phase shifts of peripheral partial waves are in very good agreement with the data (with the only exception being the ${}^1{F}_3$ wave). We also discuss the issue of the order-by-order convergence of the chiral expansion for the $NN$ interaction.

We calculate the impact of the complete set of two-pion exchange contributions at chiral fourth order (also known as next-to-next-to-next-to-leading order) on peripheral partial waves of nucleon-nucleon scattering. Our calculations are based upon the analytical studies by Kaiser. It turns out that the contribution of fourth order is substantially smaller than the one of third order, indicating convergence of the chiral expansion. We compare the prediction from chiral pion exchange with the corresponding one from conventional meson theory as represented by the Bonn full model and find, in general, good agreement. Our calculations provide a sound basis for investigating the issue whether the low-energy constants determined from $\pi N$ lead to reasonable predictions for $\mathrm{NN}.\mathrm{}$

During the past two decades, chiral effective field theory has become a popular tool to derive nuclear forces from first principles. Two-nucleon interactions have been worked out up to sixth order of chiral perturbation theory and three-nucleon forces up to fifth order. Applications of some of these forces have been conducted in nuclear few- and many-body systems—with a certain degree of success. But in spite of these achievements, we are still faced with great challenges. Among them is the issue of a proper uncertainty quantification of predictions obtained when applying these forces in ab initio calculations of nuclear structure and reactions. A related problem is the order by order convergence of the chiral expansion. We start this review with a pedagogical introduction and then present the current status of the field of chiral nuclear forces. This is followed by a discussion of representative examples for the application of chiral two- and three-body forces in the nuclear many-body system including convergence issues.