We present the first nucleon-nucleon potential at next-tonext-to-next-to-leading order (fourth order) of chiral perturbation theory. Charge-dependence is included up to nextto-leading order of the isospin-violation scheme. The accuracy for the reproduction of the N N data below 290 MeV lab. energy is comparable to the one of phenomenological high-precision potentials. Since N N 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 reliable N N potential derived from chiral effective Lagrangians. 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).

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We present a charge-dependent nucleon-nucleon (N N ) potential that fits the world proton-proton data below 350 MeV available in the year of 2000 with a χ 2 per datum of 1.01 for 2932 data and the corresponding neutron-proton data with χ 2 /datum = 1.02 for 3058 data. This reproduction of the N N data is more accurate than by any phase-shift analysis and any other N N potential. The charge-dependence of the present potential (that has been dubbed '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 and well-founded 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 review how nuclear forces emerge from low-energy QCD via chiral effective field theory. The presentation is accessible to the non-specialist. At the same time, we also provide considerable detailed information (mostly in appendices) for the benefit of researchers who wish to start working in this field.

After a historical review, I present the progress in the field of realistic NN potentials that we have seen in recent years. A new generation of very quantitative (high-quality/high-precision) NN potentials has emerged. These potentials will serve as reliable input for microscopic nuclear structure calculations and will allow for a systematic investigation of off-shell effects. The issue of three-nucleon forces is also discussed.

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We calculate the triton binding energy with a non-local NN potential that fits the world NN data below 350 MeV with the almost perfect χ 2 /datum of 1.03. The non-locality is derived from relativistic meson field theory. The result obtained in a 34-channel, charge-dependent Faddeev calculation is 8.00 MeV, which is 0.4 MeV above the predictions by local NN potentials. The increase in binding energy can be clearly attributed to the off-shell behavior of the non-local potential. Our result cuts in half the discrepancy between theory and experiment established from local NN potentials. Implications for other areas of microscopic nuclear structure, in which underbinding is a traditional problem, are discussed.

We analyze the results by chiral N N models for the two-nucleon system and calculate the predictions for the nucleon vector analyzing power of elastic nucleon-deuteron (N d) scattering, A y , by these models. Our conclusion is that a quantitative chiral two-nucleon potential does not resolve the N d A y puzzle (when only two-body forces are included).PACS numbers: 21.30.+y, 21.45.+v, 25.10.+s, 27.10+h The term A y puzzle refers to the inability to explain the nucleon vector analyzing power A y in elastic nucleon-deuteron (Nd) scattering below 30 MeV laboratory energy for the incident nucleon by means of three-body calculations in which only two-nucleon forces are applied. The problem showed up as soon as it was possible to conduct three-body continuum calculations with realistic NN potentials. The first such calculation was performed by Stolk and Tjon [1] in 1978 using the Reid soft-core potential [2], and the first calculations with (a separable representation of) the Paris potential [3] were conducted by the Graz-Osaka group in 1987 [4]; in both cases, the A y predictions showed the characteristic problem. Finally, the 'puzzle' became proverbial when rigorous three-nucleon continuum Faddeev calculations using realistic forces were started on a large scale [5]. Over the years, many measurements and calculations of Nd A y were performed (including the pd reaction that involves the Coulomb force [6]) which all confirmed that the problem was real (see Ref.[7] for a review): For energies below 20 MeV, the A y is predicted about 30% too small in the angular region around 120 deg center-of-mass angle where the maximum occurs.There have been many attempts to solve the problem. Already in the very early stages of three-body continuum calculations, when only schematic NN potentials were applied, it was noticed that the Nd A y predictions depend very sensitively on the strength of the input NN potential in the triplet P waves [8,9]-a sensitivity that was confirmed in later calculations using realistic forces [10]. Based upon this experience, Wita la and Glöckle [11] showed in 1991 that small changes in those 3 P wave potentials could remove the discrepancy. This finding gave rise to systematic investigations of the question whether the small variations of the low-energy phase shifts of, particularly, those triplet P waves necessary to explain the Nd A y are consistent with the NN data base. While Tornow and coworkers [12] suggest that the low-energy NN data may leave some lattitude in the NN 3 P waves that could * On leave from University of Salamanca, Spain. Another important observation has been that conventional three-nucleon forces (when added to a realistic two-nucleon potential) change the predictions for Nd A y only slightly and do not improve them [14,6]. Therefore, the general perception in the community has shifted towards the believe that the A y puzzle is the 'smoking gun' for new types of three-nucleon forces [15][16][17][18] or new physics [19].However, very recently, there has been an apparent indicatio...