Gogny Interactions in Nuclear Physics

• Gogny interactions are finite-range, density-dependent nucleon–nucleon forces that combine Gaussian central terms, density-dependent, and spin–orbit components to mimic effective three-body interactions.
• They are applied in both mean-field and beyond-mean-field frameworks to accurately model nuclear structure, pairing phenomena, and reaction dynamics through refined isovector tuning.
• Recent developments include the integration of ab initio constraints and tensor terms to improve nuclear matter equations of state and support observed properties of neutron stars and shell evolution.

Gogny interactions are a class of finite-range, density-dependent effective nucleon-nucleon forces widely used in mean-field and beyond-mean-field nuclear structure and reaction theory. Originally developed for global nuclear mean-field calculations, Gogny interactions have evolved to address isovector properties, neutron-star physics, and the requirements of consistent structure–reaction unified frameworks. Recent refinements incorporate constraints from ab initio theory, astrophysical observations, and the systematic inclusion of additional operator structures.

1. Mathematical Structure and Parametrization

The canonical Gogny interaction is expressed as the sum of finite-range Gaussian central terms, a zero-range density-dependent term that mimics effective three-body forces, and a zero-range spin-orbit term. In coordinate space, the general form is: $$V(\mathbf{r}_1, \mathbf{r}_2) = \sum_{i=1}^{n_{\mathrm{G}}} [\,W_i + B_i P_\sigma - H_i P_\tau - M_i P_\sigma P_\tau\,] \, e^{-(|\mathbf{r}_1-\mathbf{r}_2|/\mu_i)^2} + t_3 (1 + x_3 P_\sigma) \, [\rho((\mathbf{r}_1 + \mathbf{r}_2)/2)]^\alpha \delta(\mathbf{r}_1-\mathbf{r}_2) + i W_0 (\boldsymbol{\sigma}_1+\boldsymbol{\sigma}_2)\cdot[\nabla\times\delta(\mathbf{r}_1-\mathbf{r}_2)\nabla]$$ where

• $n_{\mathrm{G}}=2$ for standard Gogny D1, D1S, D1N, D1M sets; $n_{\mathrm{G}}=3$ for recent D3G3,
• $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$, $$P_\tau = \frac{1}{2}(1+\boldsymbol{\tau}_1\cdot\boldsymbol{\tau}_2)$$,
• $\rho$ is the total nucleon density,
• parameters $\{W_i,B_i,H_i,M_i,\mu_i\}$ (for each Gaussian), and $\{t_3,x_3,\alpha,W_0\}$ are determined by fit.

Tensors and additional operator structures (e.g., tensor and tensor-isospin finite-range Gaussians) are included in advanced variants (Anguiano et al., 2012).

The most prevalent parameterizations and their properties are listed in the following table for the central part (with all energies in MeV and distances in fm):

Set	μ₁	μ₂	W₁	B₁	H₁	M₁	W₂	B₂	H₂	M₂	t₃	x₃	α	W₀
D1S	0.7	1.2	-1720.30	1300.00	-1813.53	1397.60	103.64	-163.48	162.81	-223.93	1390.6	1.00	1/3	115.0
D1M	0.5	1.0	-12797.6	14048.9	-15144.4	11963.8	490.95	-752.27	675.12	-693.57	1562.22	1.00	1/3	115.36
D1M*	0.5	1.0	-17242.0	19604.4	-20699.9	16408.3	712.27	-982.82	905.67	-878.01	1561.22	1.00	1/3	115.36
D3G3	0.47	0.749	-7543.80	13485.57	-14708.99	6669.46	590.47	-1751.4	1582.84	-909.26	see text	1.00	1/3	115.14

(The D3G3 includes a third Gaussian at $\mu_3=1.967$ fm.) (Batail et al., 2022)

2. Physical Principles and Determination of Ranges

Early parametrizations fixed the Gaussian ranges phenomenologically to reproduce nuclear matter and finite-nucleus properties. Recent developments, such as D3G3 (Batail et al., 2022), physically relate the three central-range Gaussians to underlying meson-exchange mechanisms. The matching is achieved by equating exchange-to-direct ratios calculated for Gaussian and Yukawa form-factors at nuclear saturation density, yielding:

• $\mu_1\simeq0.47\,$fm (short-range, $n_{\mathrm{G}}=2$0 meson scale),
• $n_{\mathrm{G}}=2$1fm (intermediate, $n_{\mathrm{G}}=2$2 meson scale),
• $n_{\mathrm{G}}=2$3fm (long-range, pion Compton wavelength).

This mapping allows the long-range part to be tuned to one-pion-exchange matrix elements, aligning the phenomenological force with microscopic nucleon-nucleon physics.

For the density-dependent zero-range component, the exponent $n_{\mathrm{G}}=2$4 is universally retained, simulating effective three-body correlations.

3. Nuclear Matter and Astrophysical Constraints

Systematic analyses indicate that most traditional D1-family Gogny sets yield a soft equation of state (EoS) for neutron-rich and pure-neutron matter, producing symmetry-energy slopes $n_{\mathrm{G}}=2$5 at saturation that are significantly below empirical and astrophysical expectations (Sellahewa et al., 2014, Viñas et al., 2018). For standard D1S and D1M, $n_{\mathrm{G}}=2$6–25 MeV, and for D1N $n_{\mathrm{G}}=2$7 MeV—values too low to support observed $n_{\mathrm{G}}=2$8 neutron stars (Viñas et al., 2018, Viñas et al., 2021).

To address this, the D1M* and D1M** parameter sets impose:

• unchanged saturation $n_{\mathrm{G}}=2$9, binding energy $n_{\mathrm{G}}=3$0, incompressibility $n_{\mathrm{G}}=3$1, and effective mass $n_{\mathrm{G}}=3$2,
• fixed $n_{\mathrm{G}}=3$3 (sub-saturation symmetry energy),
• controlled variation of $n_{\mathrm{G}}=3$4 by tuning a single finite-range strength, typically $n_{\mathrm{G}}=3$5,
• a minor readjustment of $n_{\mathrm{G}}=3$6 for nuclear mass rms fit.

As a result, D1M* has $n_{\mathrm{G}}=3$7 MeV, D1M** $n_{\mathrm{G}}=3$8 MeV, and both support $n_{\mathrm{G}}=3$9 neutron stars. D3G3 achieves $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$0–60 MeV, aligning well with astrophysical constraints (Batail et al., 2022).

Key macroscopic properties at saturation for D3G3 (Batail et al., 2022):

• $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$1 fm$$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$2,
• $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$3 MeV,
• $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$4,
• $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$5 MeV,
• $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$6–32 MeV,
• $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$7–60 MeV.

4. Reactions, Optical Potentials, and Structural Applications

Unified structure–reaction frameworks have demonstrated that a single Gogny interaction can underpin both nuclear ground-state structure and nucleon–nucleus scattering observables (Blanchon et al., 2015, Lopez-Moraña et al., 2020, Blanchon et al., 2014, Moraña et al., 2023). In these approaches:

• The real part of the optical potential is generated at the Hartree–Fock level, including exchange,
• The imaginary part arises from second-order (2p–1h) processes, typically via the Brueckner–Hartree–Fock mass operator or RPA polarization,
• Finite-range nonlocality and energy dependence are preserved,
• Local density approximation (LDA) or full folding models map infinite-matter results into finite nuclei using calculated self-consistent neutron and proton densities.

Newer parameterizations (D3G3) improve the description of neutron–nucleus and proton–nucleus scattering along isotopic chains, as well as the volume and surface absorption compared to standard D1S (Batail et al., 2022, Moraña et al., 2023). Renormalized microscopic optical potentials based on Gogny interactions can reach the predictive power of global phenomenological models (e.g., Koning–Delaroche), especially after minimal empirical corrections for the imaginary part (Lopez-Moraña et al., 2020, Moraña et al., 2023).

5. Tensor and Beyond-Standard Terms

Incorporation of explicit tensor and tensor–isospin components into the Gogny interaction allows for independent control of like-nucleon and proton–neutron tensor strengths. Finite-range tensor terms of the form: $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$8 enable the proper reproduction of observed trends in neutron shell gaps (e.g., $$P_\sigma = \frac{1}{2}(1+\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2)$$9 in Ca, $$P_\tau = \frac{1}{2}(1+\boldsymbol{\tau}_1\cdot\boldsymbol{\tau}_2)$$0 in O, $$P_\tau = \frac{1}{2}(1+\boldsymbol{\tau}_1\cdot\boldsymbol{\tau}_2)$$1 in Sn), which standard D1S fails to replicate. Parameterizations such as D1ST2a and D1ST2b demonstrate sensitivity of gap evolution to the like-nucleon tensor strength, indicating the necessity of such terms for global shell evolution (Anguiano et al., 2012).

6. Performance in Finite Nuclei, Stability, and Recent Developments

D1M*, D1M**, and D3G3 maintain the essential mean-field descriptions (masses, radii, shell structure) of their predecessors (D1M, D1S), with only marginal increases in rms mass deviation or charge radii when subjected to extensive surveys (e.g., $$P_\tau = \frac{1}{2}(1+\boldsymbol{\tau}_1\cdot\boldsymbol{\tau}_2)$$2–1.4 MeV for masses). Triaxial and shape-evolution calculations (Pt isotopes, etc.) are almost unaffected by the isovector-tuned density term (Rodriguez-Guzman et al., 2010).

However, attention to spurious finite-size instabilities is critical when adjusting isovector properties. For instance, D1M* was shown to exhibit instabilities in the scalar-isovector channel under certain conditions, with unphysical oscillations in neutron–proton density difference developing during self-consistent HF(B) iterations (Martini et al., 2018). Stability criteria based on linear response in infinite matter (RPA) are now frequently imposed in the fitting protocols of new Gogny parameterizations (Batail et al., 2022).

For semi-infinite nuclear matter, solutions with Lagrange-mesh methods validate that accurate surface energies ($$P_\tau = \frac{1}{2}(1+\boldsymbol{\tau}_1\cdot\boldsymbol{\tau}_2)$$3) are obtained only with parameter sets such as D1S, D1M, D1M*, and D3G3 ($$P_\tau = \frac{1}{2}(1+\boldsymbol{\tau}_1\cdot\boldsymbol{\tau}_2)$$4–18 MeV, including spin–orbit), in agreement with empirical constraints from fission barriers (Davesne et al., 12 Nov 2025).

7. Beyond Mean-Field, Pairing, and Transport Modeling

Gogny interactions are systematically employed in Hartree–Fock–Bogoliubov (HFB) and Quasiparticle Random Phase Approximation (QRPA) calculations for both ground and excited states. The finite-range nature yields:

• Realistic pairing fields without external momentum cutoffs,
• Accurate reproduction of BCS–BEC crossover features in the $$P_\tau = \frac{1}{2}(1+\boldsymbol{\tau}_1\cdot\boldsymbol{\tau}_2)$$5 pairing channel; all standard D1-family sets yield pairing gaps below those of realistic bare forces at low density and support a smooth BCS–BEC crossover in $$P_\tau = \frac{1}{2}(1+\boldsymbol{\tau}_1\cdot\boldsymbol{\tau}_2)$$6 (Sun et al., 2013),
• QRPA E1 strength distributions for deformed nuclei, with systematic centroid energy shifts between D1S and D1M, and requirements for phenomenological folding to match experimental GDR energies (Martini et al., 2016).

Extended Gogny-type interactions have recently been adapted for quantum molecular dynamics (QMD)-like models, introducing a direct mapping of the operator structure from the nuclear-structure version to transport settings. This enables controlled momentum dependencies of mean fields, accessible symmetry-energy slopes, and the reproduction of both monotonic and non-monotonic momentum-dependent symmetry potentials—capabilities not found in Skyrme-based dynamics (Sun et al., 20 Apr 2025).

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References:

(Batail et al., 2022, Viñas et al., 2018, Sellahewa et al., 2014, Martini et al., 2018, Blanchon et al., 2015, Lopez-Moraña et al., 2020, Moraña et al., 2023, Davesne et al., 12 Nov 2025, Rodriguez-Guzman et al., 2010, Sun et al., 2013, Martini et al., 2016, Viñas et al., 2021, Anguiano et al., 2012, Sun et al., 20 Apr 2025)