High-quality two-nucleon potentials up to fifth order of the chiral expansion

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.

Figure 1

Hierarchy of nuclear forces in ChPT. Solid lines represent nucleons and dashed lines represent pions. Small dots, large solid dots, solid squares, triangles, diamonds, and stars denote vertexes of index ${\mathrm{\Delta }}_i=$ 0, 1, 2, 3, 4, and 6, respectively. Further explanations are given in the text.

Figure 2

Chiral expansion of neutron-proton scattering as represented by the phase shifts in $S, P$, and $D$ waves and mixing parameters ${\epsilon }_1$ and ${\epsilon }_2$. Five orders ranging from LO to ${\mathrm{N}}^4\mathrm{LO}$ are shown as denoted. A cutoff $\mathrm{\Lambda }=500$ MeV is applied in all cases. The filled and open circles represent the results from the Nijmegen multienergy $np$ phase-shift analysis [80] and the GWU single-energy $np$ analysis SP07 [102], respectively.

Figure 3

Cutoff variations of the $np$ phase shifts at NNLO (left side, green lines) and ${\mathrm{N}}^4\mathrm{LO}$ (right side, purple lines). Dotted, dashed, and solid lines represent the results obtained with cutoff parameters $\mathrm{\Lambda }=$ 450, 500, and 550 MeV, respectively, as also indicated by the curve labels. Note that, at ${\mathrm{N}}^4\mathrm{LO}$, the cases 500 and 550 MeV cannot be distinguished on the scale of the figures for most partial waves. Filled and open circles as in Fig. 2.