Peripheral nucleon-nucleon scattering at fifth order of chiral perturbation theory

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.

Figure 1

Two-pion-exchange contributions at ${\mathrm{N}}^4\mathrm{LO}$. (a) The leading one-loop $\pi N$ amplitude is folded with the chiral $\pi \pi NN$ vertices proportional to $c_i$. (b) The one-loop $\pi N$ amplitude proportional to $c_i$ is folded with the leading-order chiral $\pi N$ amplitude. (c) Relativistic corrections of NNLO diagrams. Solid lines represent nucleons and dashed lines represent pions. Small dots, large solid dots, solid squares, and triangles denote vertices of index $d_i+n_i/2-2=0$, 1, 2, and 3, respectively. Open circles are relativistic $1/M_N$ corrections.

Figure 2

Three-pion-exchange contributions at ${\mathrm{N}}^4\mathrm{LO}$. The classification scheme of Ref. [11] is used. Notation is the same as in Fig. 1.

Figure 3

Effect of individual fifth-order contributions on the neutron-proton phase shifts of some selected peripheral partial waves. The individual contributions are added up successively in the order given in parentheses next to each curve. Curve (1) is ${\mathrm{N}}^3\mathrm{LO}$ and curve (5) is the complete ${\mathrm{N}}^4\mathrm{LO}$. The filled and open circles represent the results from the Nijmegan multi-energy $np$ phase-shift analysis [30] and the VPI-GWU single-energy $np$ analysis SM99 [31], respectively.

Figure 4

Phase shifts of neutron-proton scattering at various orders as denoted. The shaded (colored) bands show the variation of the predictions when the SFR cutoff $\tilde{\Lambda }$ is changed over the range 0.7 to 1.5 GeV. The filled and open circles represent the results from the Nijmegan multi-energy $np$ phase-shift analysis [30] and the VPI-GWU single-energy $np$ analysis SM99 [31], respectively.

Figure 5

Same as Fig. 4, but for $G$ waves.

Figure 6

LO, NLO, and NNLO contributions to the $NN$ interaction. Notation is the same as in Fig. 1.

Figure 7

Two-pion-exchange contributions at ${\mathrm{N}}^3\mathrm{LO}$ with (a) the ${\mathrm{N}}^3\mathrm{LO}$ football diagram, (b) the leading 2PE two-loop contributions, and (c) the relativistic corrections of NLO diagrams. Notation is the same as in Fig. 1.