Two-pion exchange contributions to the relativistic chiral nuclear force at N$^3$LO

TL;DR

This work develops a covariant (relativistic) baryon ChEFT description of the NN force, extending two-pion exchange contributions to N$^3$LO and focusing on peripheral partial waves ($J\ge 3$). The authors compute one- and two-loop TPE diagrams using spectral function regularization (SFR) with a Gaussian regulator, and compare the resulting phase shifts to Nijmegen/SAID analyses, finding improved convergence and good agreement relative to non-relativistic approaches. Key contributions include demonstrating faster convergence in the relativistic framework, showing that N$^3$LO corrections are generally modest for many peripheral waves, and highlighting the differences between SFR and dimensional regularization in low-$L$ channels. This approach provides a robust relativistic microscopic input for ab initio nuclear structure and reaction calculations, with planned future work to pin down LECs and assess NN observables within this framework.

Abstract

We present the two-pion exchange contributions to the nucleon-nucleon interaction up to next-to-next-to-next-to leading order (N$^3$LO) in covariant baryon chiral perturbation theory. Both one-loop and two-loop diagrams are calculated with the spectral functional regularization. We show that the phase shifts for partial waves with total angular momentum $3\le J\le 5$ are in better agreement with the partial wave analysis from the Nijmegen or the SAID group than their N$^2$LO counterparts. In addition, the relativistic chiral force exhibits better convergence than its non-relativistic counterpart, suggesting the importance of relativistic corrections.

Figures (10)

• Figure 1: Pion exchange contributions at various orders are shown. The solid and open dots denote the vertices of $\mathcal{L}_{\pi N}^{(2)}$ in Eq. (\ref{['eq:lagpiN2']}) and $\mathcal{L}_{\pi N}^{(3)}$ in Eq. (\ref{['eq:lagpiN3']}), respectively. The gray circles and boxes denote the corresponding one-loop level $\pi\pi$NN and $\pi NN$ interactions depicted in Fig. \ref{['fig:N3LO-part1']} and Fig. \ref{['fig:N3LO-part2']}. All diagrams that occur at each channel are shown, except for the N$^3$LO($\mathcal{O}(p^4)$) part, where the left-right exchanged diagrams and time-reversal symmetric diagrams are not explicitly displayed but included in the calculation.
• Figure 2: One-loop level $\pi N$ interactions of the Weinberg-Tomozawa-like terms.
• Figure 3: One-loop level $\pi N$ interactions of the Born-like terms.
• Figure 4: Football diagram of the LO TPE contributions (left). The right pattern is the corresponding $\bar{N}N\rightarrow \pi\pi \rightarrow \bar{N}N$ annihilation process.
• Figure 5: Phase shifts and mixing angle for the $J=3$ partial waves. The gray, blue, and magenta bands are results from relativistic TPE $NN$ interactions at NLO, N$^2$LO, and N$^3$LO, respectively, with a cutoff in the range $[0.5, 0.8]$ GeV. For comparison, the N$^3$LO non-relativistic results are shown with orange bands (the two $F$-waves from Ref. Entem:2014msa and the $G$-waves from Ref. Entem:2015xwa). The solid and open dots represent the data from the Nijmegen multi-energy neutron-proton (n-p) phase shift analysis Stoks:1993tb and the VPI/GWU single-energy n-p analysis SM99 Arndt:1994br, respectively.