Dominant contributions to the nucleon-nucleon interaction at sixth order of chiral perturbation theory

We present the dominant two- and three-pion-exchange contributions to the nucleon-nucleon interaction at sixth order (next-to-next-to-next-to-next-to-next-to-leading order, ${\mathrm{N}}^5\mathrm{LO}$) of chiral perturbation theory. Phase shifts with orbital angular momentum $L\ge 4$ are given parameter free at this order and allow for a systematic investigation of the convergence of the chiral expansion. The ${\mathrm{N}}^5\mathrm{LO}$ contribution is prevailingly repulsive and considerably smaller than the ${\mathrm{N}}^4\mathrm{LO}$ one, thus, showing the desired trend towards convergence. Using low-energy constants that were extracted from an analysis of $\pi N$ scattering at fourth order, the predictions at ${\mathrm{N}}^5\mathrm{LO}$ are in excellent agreement with the empirical phase shifts of peripheral partial waves.

Figure 1

Two-pion-exchange contributions to the $NN$ interaction at ${\mathrm{N}}^5\mathrm{LO}$. (a) The subleading one-loop $\pi N$ amplitude is folded with the chiral $\pi \pi NN$ vertices proportional to $c_i$. (b) The leading one-loop $\pi N$ amplitude is folded with itself. (c) The leading two-loop $\pi N$ amplitude is folded with the tree-level $\pi N$ amplitude. Solid lines represent nucleons and dashed lines pions. Small dots and large solid dots denote vertices of chiral order 1 and 2, respectively. Shaded ovals represent complete $\pi N$-scattering amplitudes with their order specified by the number in the oval.

Figure 2

Relativistic $1/M_N^2$ corrections to $2\pi$-exchange diagrams that are counted as order 6. Notation as in Fig. 1. Open circles represent $1/M_N$ corrections.

Figure 3

Three-pion exchange contributions at ${\mathrm{N}}^5\mathrm{LO}$. (a) Diagrams proportional to $c_i^2$. (b) Diagrams involving the one-loop $\pi N$ amplitude. Roman numerals refer to subclasses following the scheme introduced in Refs. [7, 15]. Notation as in Fig. 1.

Figure 4

Effect of individual sixth-order contributions on the neutron-proton phase shifts of two $G$ waves. The individual contributions are added up successively in the order given in parentheses next to each curve. Curve (1) is ${\mathrm{N}}^4\mathrm{LO}$ and curve (6) contains all ${\mathrm{N}}^5\mathrm{LO}$ contributions calculated in this work. A SFR cutoff $\tilde{\mathrm{\Lambda }}=800$ MeV is applied. The filled and open circles represent the results from the Nijmegen multienergy $np$ phase-shift analysis [28] and the GWU $np$ analysis SP07 [29], respectively.

Figure 5

Phase-shifts of neutron-proton scattering in $G$ and $H$ waves at various orders as denoted. The shaded (colored) bands show the variations of the predictions when the SFR cutoff $\tilde{\mathrm{\Lambda }}$ is changed over the range 700 to 900 MeV. Empirical phase shifts are as in Fig. 4.

Figure 6

Phase-shifts of neutron-proton scattering in $G$ and $H$ waves at all orders from LO to ${\mathrm{N}}^5\mathrm{LO}$. A SFR cutoff $\tilde{\mathrm{\Lambda }}=800$ MeV is used. Empirical phase shifts are as in Fig. 4.

Figure 7

Mixing angles for neutron-proton scattering for $J=4,5$ at all orders from LO to ${\mathrm{N}}^5\mathrm{LO}$. A SFR cutoff $\tilde{\mathrm{\Lambda }}=800$ MeV is used. Filled and open circles are as in Fig. 4.