Precision Nucleon-Nucleon Potential at Fifth Order in the Chiral Expansion

We present a nucleon-nucleon potential at fifth order in chiral effective field theory. We find a substantial improvement in the description of nucleon-nucleon phase shifts as compared to the fourth-order results utilizing a coordinate-space regularization. This provides clear evidence of the corresponding two-pion exchange contributions with all low-energy constants being determined from pion-nucleon scattering. The fifth-order corrections to nucleon-nucleon observables appear to be of a natural size, which confirms the good convergence of the chiral expansion for nuclear forces. Furthermore, the obtained results provide strong support for the novel way of quantifying the theoretical uncertainty due to the truncation of the chiral expansion proposed by the authors. Our work opens up new perspectives for precision ab initio calculations in few- and many-nucleon systems and is especially relevant for ongoing efforts towards a quantitative understanding of the structure of the three-nucleon force in the framework of chiral effective field theory.

Figure 1

Fifth-order contributions to the TPEP. Solid lines and dashed lines refer to nucleons and pions, respectively. Solid dots denote vertices from the lowest-order $\pi N$ effective Lagrangian. Filled (red) rectangles, (blue) ovals, and gray circles denote the order $Q^4$, order $Q^3$, and order $Q^2$ contributions to $\pi N$ scattering, respectively.

Figure 2

Predictions for the np total cross section based on the improved chiral NN potentials at NLO (orangle filled squares), ${\mathrm{N}}^2\mathrm{LO}$ (green solid diamonds), ${\mathrm{N}}^3\mathrm{LO}$ (blue filled triangles), and ${\mathrm{N}}^4\mathrm{LO}$ (red filled circles) for the different choices of the cutoff: $R_1=0.8\mathrm{fm}$, $R_2=0.9\mathrm{fm}$, $R_3=1.0\mathrm{fm}$, $R_4=1.1\mathrm{fm}$, and $R_5=1.2\mathrm{fm}$. The horizontal band refers to the result of the NPWA with the uncertainty estimated as explained in the text. Also shown are experimental data of Ref. [27].

Figure 3

Results for the np $S$, $P$, and $D$ waves, and the mixing angles ${\epsilon }_1$, ${\epsilon }_2$ up to ${\mathrm{N}}^4\mathrm{LO}$ based on the cutoff of $R=0.9\mathrm{fm}$ in comparison with the NPWA [23] (solid dots) and the single-energy PWA of [28] (open triangles). The bands of increasing width show estimated theoretical uncertainty at ${\mathrm{N}}^4\mathrm{LO}$ (red), ${\mathrm{N}}^3\mathrm{LO}$ (blue), ${\mathrm{N}}^2\mathrm{LO}$ (green), and NLO (yellow).

Figure 4

Predictions for selected np scattering observables at $E_{\mathrm{lab}}=200\mathrm{MeV}$ calculated up to ${\mathrm{N}}^4\mathrm{LO}$ based on the cutoff of $R=0.9\mathrm{fm}$. Open circles refer to the result of the NPWA [23]. For remaining notations, see Fig. 3. For references to data, see [19]. ${\theta }_{\mathrm{C}\mathrm{M}}$ denotes the scattering angle in the center of mass system.