Extended Skyrme Interactions in Nuclear Modeling

• Extended Skyrme Interactions are advanced nuclear models incorporating higher-order momentum and density derivatives to overcome limitations of standard Skyrme parametrizations.
• They include enriched spin, tensor, and isospin channels to better reproduce nuclear excitations, effective masses, and symmetry energies observed in experiments.
• Applications span nuclear structure, heavy-ion collisions, and astrophysical phenomena, enabling precise energy density functional constructions for diverse nuclear systems.

Extended Skyrme Interactions are generalizations of the original Skyrme effective interaction that incorporate higher-order derivatives, momentum and density dependencies, spin and isospin channels, and explicit isospin and charge-symmetry breaking components. These extensions are designed to improve the flexibility and accuracy of energy density functionals (EDFs) for the microscopic description of nuclear matter, finite nuclei, and astrophysical objects such as neutron stars. They also remedy specific deficiencies encountered in standard Skyrme parametrizations, such as incorrect high-density behavior of effective masses, limited control of symmetry energy, inadequate momentum dependence, and unrealistic predictions for spin responses.

1. Formal Structure and Rationale for Extensions

The standard Skyrme interaction is a zero-range (contact) effective force with terms up to second order in relative momenta. In its traditional form, the Skyrme potential is written schematically as:

$$V_{\mathrm{Sk}} = t_0(1 + x_0 P_\sigma)\delta(\mathbf{r}_1-\mathbf{r}_2) + \frac{t_1}{2}(1 + x_1 P_\sigma)[\mathbf{k}'^2\delta(\mathbf{r}_1-\mathbf{r}_2) + \delta(\mathbf{r}_1-\mathbf{r}_2)\mathbf{k}^2] + t_2 (1 + x_2 P_\sigma) \mathbf{k}'\cdot\delta(\mathbf{r}_1-\mathbf{r}_2)\mathbf{k} + \dots$$

where $\mathbf{k},\mathbf{k}'$ are relative momentum operators and $P_\sigma$ is the spin-exchange operator. Extensions of the Skyrme interaction emerge in several directions:

• Higher-order momentum dependence: Terms with derivatives up to 2n-th order (N$n$LO), enabling accurate fits to empirical optical potentials and nucleon dynamics up to several GeV (Wang et al., 12 Dec 2024).
• Refined density dependence: The density-dependent term is expanded in powers of $\rho^{1/3}$, matching the Fermi momentum expansion and allowing systematic control over the equation of state (EOS) and symmetry energy, with parameters such as the symmetry energy slope $L$ varied over a broad range (Wang et al., 2023).
• Spin, tensor, and spin–isospin channels: Additional terms involving spin densities, spin–isospin densities, and explicit tensor contributions, often motivated by the need to describe spin–flip excitations, Landau parameters, and to eliminate unphysical instabilities (e.g., ferromagnetic transitions) (Wen et al., 2014, Duan et al., 6 Jan 2025).
• Isospin and charge–symmetry breaking: Zero-range terms that break isospin symmetry at the level of the strong force, enabling accurate reproduction of mirror and triplet displacement energies in light nuclei (Baczyk et al., 2015).
• Explicit regularization and renormalization: Methodologies to absorb ultraviolet divergences arising from zero-range character, especially when higher-order corrections are included in many-body perturbation theory (Moghrabi et al., 2012, Moghrabi et al., 2013, Moghrabi, 2016).

The driving motive is to ensure that the EDF can consistently describe nuclear matter across a wide density range, finite nuclei (including excitations), astrophysical phenomena (neutron stars, binary mergers), and heavy-ion collisions, while remaining free from artifacts such as superluminal Fermi velocities or unphysical instabilities (Beznogov et al., 28 Mar 2024, Wang et al., 2023, Wang et al., 12 Dec 2024).

2. Higher-Order Momentum and Density Dependencies

The extension to higher derivative (momentum-dependent) terms is formalized by constructing the central Skyrme pseudopotential up to N$n$LO, where terms up to $(\mathbf{k}^2 + \mathbf{k}'^2)^n$ are included. Each higher-order contributes additional flexibility to the single–nucleon and nuclear matter mean field, captured at Hartree–Fock level by generalized Hamiltonian densities:

$$\mathcal{H}(\vec{r}) = \mathcal{H}^{\text{kin}}(\vec{r}) + \mathcal{H}^{\text{loc}}(\vec{r}) + \sum_{n} \frac{1}{2\hbar^{2n}}[C^{[2n]} \mathcal{H}^{\text{md}[2n]}(\vec{r}) + D^{[2n]} \sum_\tau \mathcal{H}_\tau^{\text{md}[2n]}(\vec{r})] + ...$$

with

$$\mathcal{H}^{\text{md}[2n]}(\vec{r}) = \int \frac{d^3p d^3p'}{(2\pi\hbar)^6} (\mathbf{p} - \mathbf{p}')^{2n} f(\vec{r},\mathbf{p}) f(\vec{r},\mathbf{p}')$$

The density-dependent term is generalized as: $$V^{\mathrm{DD}}_N = \sum_{n=1,3,5} \frac{1}{6} t_3^{[n]} (1 + x_3^{[n]} P_\sigma) \rho(\vec{R})^{n/3} \delta(\vec{r}_1-\vec{r}_2)$$ ensuring a parametric expansion in $\rho^{1/3}$ (Fermi momentum) up to the desired order (Wang et al., 2023, Wang et al., 12 Dec 2024). This enables nuanced fits to ab initio EOSs for both symmetric matter and pure neutron matter, optimizes nuclear matter incompressibility, and controls higher EOS derivatives (e.g., skewness $J_0$, kurtosis).

3. Spin, Tensor, and Isospin Channels

3.1. Spin and Spin–Isospin

Extended functionals incorporate spin-density and spin–isospin density dependent terms: $$V^{\text{add}}(r_1, r_2) = \frac{1}{6} t_3^s (1 + x_3^s P_\sigma)[\rho_s(\vec{R})]^{\gamma_s} \delta(\vec{r}) + \frac{1}{6} t_3^{st} (1 + x_3^{st} P_\sigma)[\rho_{st}(\vec{R})]^{\gamma_{st}} \delta(\vec{r})$$ where $\rho_s$ and $\rho_{st}$ are spin and spin–isospin densities (Wen et al., 2014). Adjustment of these terms is essential for realistic Landau parameters $G_0$, $G_0'$, and to eliminate spurious magnetic instabilities in symmetric and neutron-rich matter (Duan et al., 6 Jan 2025). Recent EDFs employ independent fitting of spin-channel coefficients to Brueckner–Hartree–Fock (BHF) results, changing the theoretical landscape for collective excitation and neutrino transport calculations.

3.2. Tensor and Central Spin Interactions

Recent analyses have demonstrated the breakdown of the original Skyrme tensor term: its matrix elements either vanish or are absorbed into higher-order corrections, failing to impact ground-state observables at mean-field level (Dong et al., 2020). As a remedy, central spin-dependent ($\bm{\sigma}_1 \cdot \bm{\sigma}_2$) interactions have been introduced: $$V_{\bm{\sigma}}(\vec{r}_1, \vec{r}_2) = \bm{\sigma}_1 \cdot \bm{\sigma}_2 \Big[ \frac{U_s}{4}\left[ \mathbf{k}'^2 \delta(\mathbf{r}) + \delta(\mathbf{r})\mathbf{k}^2\right] + \frac{U_t}{2}\mathbf{k}' \delta(\mathbf{r})\mathbf{k} \Big]$$ to capture shell evolution and single-particle splittings (Dong et al., 2020).

3.3. Isospin and Charge-Symmetry Breaking

Zero-range isospin-symmetry breaking terms are incorporated to match experimental mirror and triplet displacement energies. The class II and III terms are, respectively, charge-independence and charge-symmetry breaking, and are vital for resolving the Nolen–Schiffer anomaly and for reproducing $A=4n$ versus $A=4n+2$ TDE staggering (Baczyk et al., 2015).

4. Regularization, Renormalization, and Beyond-Mean-Field Effects

Zero-range interactions introduce ultraviolet divergences in higher-order (second-order) perturbative corrections. The severity of the divergence scales as $\sim \Lambda^5$ for the full Skyrme force (including velocity-dependent terms) (Moghrabi et al., 2012, Moghrabi et al., 2013). Regularization via a sharp momentum cutoff $\Lambda$ is implemented in all relevant integrals, followed by a simultaneous refit of the Skyrme parameters for each $\Lambda$ to maintain agreement with benchmark (mean-field) equations of state. This renders observable quantities independent of the cutoff, enabling consistent application to finite nuclei and beyond mean-field frameworks (e.g., particle-vibration coupling, higher RPA order, configuration mixing).

Dimensional regularization with minimal subtraction (DR/MS) is used for high-order NLO Skyrme functionals, ensuring the proper removal of power divergences and consistent extraction of renormalized, finite corrections (Moghrabi, 2016). These advances are foundational for tractable, divergence-free extensions of EDFs.

5. Applications: Nuclear Structure, Heavy-Ion Collisions, and Neutron Stars

5.1. Nuclear Structure and Excitations

Extended Skyrme EDFs accurately reproduce bulk observables (binding energies, charge radii, isoscalar monopole energies) and collective excitations. Spin-density and spin–isospin extensions yield improved predictions for M1 and Gamow–Teller (GT) excitations, with repulsive residual interactions shifting centroid energies upward and reducing spurious instability (Wen et al., 2014, Duan et al., 6 Jan 2025). Inclusion of higher-order derivatives further increases the accuracy of radii, densities, and excitation spectra in large-scale Hartree–Fock–Bogoliubov calculations (Becker et al., 2017).

5.2. Heavy-Ion Collisions and Transport Models

N$n$LO Skyrme pseudopotentials, with polynomial momentum structure up to $p^{10}$, are uniquely suited for transport simulations (BUU, QMD, lattice BUU), as they enable a realistic, saturating momentum dependence that matches empirical nucleon optical potentials up to at least 2 GeV (Wang et al., 12 Dec 2024, Wang et al., 2018). The polynomial expansion ensures computational efficiency with Gaussian wave packets and allows direct factorization of the momentum dependence (Yang et al., 2023). Extended interactions support benchmarking of collective flows (v₁–v₄) in Au+Au collisions, matching HADES data across rapidity and transverse momentum (Wang et al., 12 Dec 2024).

5.3. Neutron Stars, EOS, and Astrophysical Observables

The Fermi momentum expansion and extended density-dependent terms provide fine control over the symmetry energy $E_{\mathrm{sym}}(\rho)$, its slope $L$, curvature, and higher derivatives, allowing EOSs to be constructed that satisfy neutron matter constraints, support heavy neutron stars ($>2\,M_\odot$), and yield tidal deformabilities $\Lambda_{1.4}$ compatible with gravitational-wave results (Wang et al., 2023, Beznogov et al., 28 Mar 2024, Wang et al., 12 Dec 2024). Bayesian inference incorporating physical constraints, such as subluminal neutron Fermi velocities, further refines the EOS and narrows uncertainties in astrophysical predictions (Beznogov et al., 28 Mar 2024).

The thermal response of extended Skyrme EOSs is governed by the density dependence of the effective mass, which may display a nonmonotonic (U-shaped) behavior consistent with ab initio calculations. In extreme cases, negative thermal pressures occur at high densities, implying that heated compact stars can support less mass than cold configurations—a reversal of standard assumptions, with significant impact on supernova and post-merger collapse thresholds (Raduta et al., 28 Sep 2025).

6. Current Challenges and Ongoing Developments

Recent work emphasizes:

• Decoupling spin–dependent and spin–independent channels: Realistic adjustment of EDF Landau parameters, enabling the elimination of unphysical ferromagnetic instabilities and improved description of spin excitation modes and neutrino opacities in neutron-star matter (Duan et al., 6 Jan 2025, Wen et al., 2014).
• Construction of EDFs with variable $L$ allowing coverage of the experimental and theoretical uncertainty in the symmetry energy slope, and the impact of effective-mass splitting on isospin-sensitive heavy-ion observables (Wang et al., 2023, Yang et al., 2023, Wang et al., 12 Dec 2024).
• Handling of tensor and central spin-dependent terms to address shell evolution and known deficiencies in conventional EDFs (Dong et al., 2020).
• Bayesian frameworks for EOS construction that enforce physical constraints (e.g., no superluminal neutron Fermi velocity), ab initio matching, and systematic error quantification (Beznogov et al., 28 Mar 2024).

Several unresolved issues persist, notably the reconciliation of spin–isospin response with ab initio constraints across all densities, the quantification of three–body and finite–range effects beyond the current polynomial expansion, robust regularization schemes for finite nuclei beyond mean field, and the credible extrapolation to high densities relevant for mergers and collapsars.

7. Summary Table: Key Structural Elements in Extended Skyrme Interactions

Extension	Purpose/Achievement	Illustrative Reference(s)
N$n$LO momentum terms	Realistic, saturating high-energy optical potentials	(Wang et al., 12 Dec 2024, Wang et al., 2018)
Fermi momentum density exp	Flexible EOS, symmetry energy tuning	(Wang et al., 2023, Marsh, 2015)
Spin/Spin–Isospin terms	Landau parameters, GT/M1 response, stability	(Wen et al., 2014, Duan et al., 6 Jan 2025)
Zero-range tensor, central	Shell structure, single-particle level evolution	(Dong et al., 2020)
ISB (Class II/III) terms	MDE, TDE, Nolen–Schiffer anomaly	(Baczyk et al., 2015)
Regularization/renorm.	Consistent beyond-mean-field corrections	(Moghrabi et al., 2012, Moghrabi, 2016)

These elements define the current paradigm for extended Skyrme interactions, placing emphasis on flexibility, physical consistency, and broad applicability across nuclear structure, reactions, and astrophysics.