Chiral symmetry and peripheral neutron-$α$ scattering

Abstract

We propose and demonstrate that peripheral neutron-$α$ scattering at low energies can serve as a sensitive and clean probe of the long-range three-nucleon forces. To this aim, we perform {\it ab initio} quantum Monte Carlo calculations using two- and three-nucleon interactions derived in chiral effective field theory up to third expansion order. We show that the longest-range three-nucleon force stemming from the two-pion exchange plays a crucial role in the proper description of the neutron-$α$ $D$-wave phase shifts. Our Letter reveals the predictive power of chiral symmetry in the few-body sector and opens a new direction for probing and constraining three-nucleon forces.

Figures (4)

• Figure 1: Diagrams (a) and (b) show the dominant contributions to low-energy peripheral neutron-$\alpha$ scattering. The OPE process (c) is forbidden by the isospin selection rule for the isospin $T=0$$\alpha$-particle. Iterated OPE (d) is suppressed in the tight-binding limit while short-range interactions (e) are suppressed in peripheral scattering by the centrifugal barrier. Solid and dashed lines denote nucleons and pions, respectively, while green-shaded ellipses denote the corresponding $\pi$N amplitudes.
• Figure 2: (Color online). Phase shifts for neutron-$\alpha$ scattering in the $^2D_{\frac{5}{2}}$ channel as a function of the center-of-mass energy predicted at different orders of the chiral EFT expansion. Empty symbols refer to the GFMC results with error bars denoting the statistical uncertainties. The $1\sigma$ uncertainty bands come from analyzing the GFMC results, which combine the statistical uncertainties and the systematic uncertainties of the BERW formula (see Supplemental Material Supp for details). The stars and diamonds are from the $R$-matrix analyses of the experimental neutron-$\alpha$ elastic scattering data from Bond1977Nucl.Phys.A317 and Hale2025, respectively.
• Figure 3: Phase shifts for neutron-$\alpha$ scattering in the $^2D_{\frac{5}{2}}$ channel as a function of the center-of-mass energy obtained at N$^2$LO with and without the 3NF. The N$^{(4)}$LO predictions show the results based on the effective order-$Q^5$ values of the $\pi$N LECs in the 3NF as explained in the text. The stars are from the $R$-matrix analysis of experimental data Bond1977Nucl.Phys.A317.
• Figure 4: Left panel: chiral EFT predictions for $D$-wave scattering lengths $a_2$ at different expansion orders. Smaller error bars indicate numerical uncertainties, while larger error bars are uncertainties from the truncation of the chiral expansion. The right panel shows the results obtained at N$^2$LO with the full NN$+$3N forces ($v_{ij}+V_{ijk}$), the NN force only ($v_{ij}$), the NN force and the TPE 3NF ($v_{ij}+V_{ijk}^{2\pi}$), and the NN force and the long-range 3NF ($v_{ij}+V_{ijk}^{\rm LR}$). Here, $V_{ijk}^{\rm LR}$ is defined by switching off the 3NF $V$ at nucleons' relative distances $r < 1.5$ fm. The horizontal black band shows the $a_2$ value from the $R$-matrix analysis Bond1977Nucl.Phys.A317.