Skyrme-Hartree-Fock-Bogoliubov mass models on a 3D mesh: V. The N2LO extension of the Skyrme EDF

TL;DR

This paper introduces BSkG5, the first large-scale nuclear-structure model based on a next-to-next-to-leading order (N2LO) Skyrme energy density functional with up to four gradients. The authors show that this N2LO formulation can simultaneously deliver high-precision nuclear masses, radii, and fission properties while yielding a stiff equation of state for neutron-rich matter that is compatible with the existence of $2\,M_\odot$ neutron stars, thus reconciling terrestrial nuclear data with astrophysical constraints. They demonstrate the model’s stability in practical 3D coordinate-space HFB calculations and discuss how the N2LO terms influence infinite-matter behavior, single-particle spectra, and spectroscopic observables, as well as its implications for r-process nucleosynthesis in neutron-star mergers. The work also outlines operational considerations, such as an ultraviolet momentum cutoff ($dx=0.4$ fm) to avoid finite-size instabilities and hints at future directions, including adding N2LO spin-orbit terms and exploring finite-temperature nuclear matter.

Abstract

We present BSkG5, the latest entry in the Brussels-Skyrme-on-a-Grid (BSkG) series and the first large-scale nuclear structure model based on next-to-next-to-leading order (N2LO) Skyrme energy density functional (EDF). By extending the traditional Skyrme EDF ansatz with central terms containing up to four gradients, we are able to combine an excellent global description of nuclear ground state properties with a stiff equation of state for pure neutron matter that is consistent with all astronomical observations of neutron stars. More precisely, the new model matches the accuracy of earlier BSkG models but with two parameters less: we achieve root-mean-square deviations of 0.649 MeV for 2457 atomic masses, 0.0267 fm for 810 charge radii, and 0.43 MeV for 45 primary fission barriers of actinide nuclei. We demonstrate that the complexities of N2LO EDFs are not insurmountable, even for demanding many-body calculations.

Figures (10)

• Figure 1: Panel (a): Differences between experimental AME2020 and calculated masses for BSkG5 (red circles) and BSkG4 (blue squares) masses, where positive values indicate overbound nuclei. Panel (b): Mass differences between BSkG5 and BSkG4 for all nuclei with $8\le Z \le 118$ lying between the BSkG5 proton and neutron drip lines.
• Figure 2: Panel (a): FAM photoabsorption strength function of the isoscalar giant monopole resonance of ${}^{208}$Pb calculated with SN2LO1 (dotted green), BSkG4 (dashed blue), and BSkG5 (solid red). The experimental energy-centroid of the resonance is represented by the blue vertical bar patel2013testing. Panel (b): Dynamical moment of inertia of the superderformed band of ${}^{194}$Hg calculated with the same models. Experimental data (empty squares) are those of the SD-1 band in Ref. singh2002.
• Figure 3: Panel (a): Mass-radius curves for families of NS constructed with BSkG5 (solid red), BSkG4 (dashed blue), SN2LO1 (dotted green). Filled contours show the pulsar observations from the NICER telescope, PSR J0614–3329 Mauviard2025, PSR J0030+0451 Vinciguerra2024, PSR J0437–4715 Choudhury2024, and PSR J0740+6620 Salmi2024. The open contour shows the gravitational wave GW170817 observation from LIGO-Virgo interferometers LIGOScientific:2018cki. The gray band marks the mass measurements of PSR J1614$-$2230 Agazie23. Panel (b): Contribution from N2LO terms to the energy per particle of pure neutron matter with respect to baryon density for BSkG5 (solid red), and SN2LO1 (dotted green).
• Figure 4: Panel (a) Final mass fractions $X$ of the material ejected from a 1.2--1.6 $M_\odot$ NS–NS binary system of Ref. Just23, obtained with the BSkG4 and BSkG5 masses. The solar system r-abundance distribution (open circles) from Ref. Goriely99 is shown for comparison and suitably normalised. Panel (b) shows the ratio between the predicted BSkG5 and BSkG4 mass fractions for $X>10^{-6}$.
• Figure 5: Top: Neutron matter energy per particle as a function of the baryon density $\rho_0$, for BSkG5 (solid red), SN2LO1 (solid green), BSkG1 (dashed dark-gray), BSkG2 (dashed light-gray), SLy4 (dashed light-pink), SLy5s1 (dashed dark-pink), and SLy5s8 (dashed purple). Markers show the ab-initio calculations of WFF WFF, APR APR, LS2 LS2, and FP FP. Skyrme N2LO models are shown in solid lines, while standard Skyrme models are shown in dashed lines. We display the contributions from individual terms for the potential energy, LO (second panel), NLO (third panel), and N2LO (bottom panel). The insert plot on the bottom panel shows the $e^{\rm N2LO}_{\rm NeutM}$ at sub-saturation densities.