⚛️ nuclear theory

Ab initio-guided optical potentials for semi-magic nuclei from Gorkov self-consistent Green's functions

This paper presents ab initio-guided nonlocal optical potentials for semi-magic nuclei by extending Gorkov self-consistent Green's function theory to open-shell targets and proton projectiles, demonstrating reasonable agreement with scattering data at forward angles while identifying the need for larger model spaces to better reproduce backward-angle absorption.

Original authors: Thibault Demartini, Guillaume Blanchon, Vittorio Somà, Thomas Duguet

Published 2026-10-21

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CC BY 4.0

Original authors: Thibault Demartini, Guillaume Blanchon, Vittorio Somà, Thomas Duguet

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand how the building blocks of matter hold together, physicists often look at the atomic nucleus, a dense cluster of protons and neutrons. When a single particle, such as a neutron or a proton, flies toward this nucleus, it does not simply bounce off like a billiard ball. Instead, it interacts with the entire collection of particles inside, creating a complex wave of influence that can be absorbed, scattered, or deflected. To make sense of these collisions, scientists use a mathematical tool called an optical potential. Think of this as a map that predicts how the incoming particle will behave as it approaches the target, allowing researchers to calculate the likelihood of different outcomes without having to track every single interaction inside the nucleus. For decades, these maps were drawn by adjusting their shapes until they matched experimental data, a method that worked well for known elements but failed when scientists tried to predict how exotic, unstable nuclei would behave. The ultimate goal is to build these maps from the ground up, starting only with the fundamental forces between particles, so that predictions can be made for nuclei that have never been observed in a laboratory.

A team of researchers has taken a significant step toward this goal by developing a new way to calculate these optical potentials for a specific class of nuclei known as semi-magic nuclei. These are atoms where the protons or neutrons fill up complete energy shells, making them relatively stable, yet they possess an open shell of particles that allows for unique internal behaviors like pairing. The scientists focused on calcium isotopes, specifically calcium-40 and calcium-42, which serve as ideal test cases. They used a sophisticated computational framework called the Gorkov self-consistent Green's function method. This approach allows them to treat the nucleus not as a static object, but as a dynamic system where particles are constantly being added and removed, and where pairs of neutrons can form and break apart. By starting with a fundamental theory of nuclear forces known as the chiral interaction, which describes how protons and neutrons talk to each other, they were able to generate a theoretical map of the optical potential without fitting it to experimental results.

The researchers began their work by simulating the behavior of neutrons and protons hitting a calcium-40 nucleus. They calculated the optical potential using a mathematical grid that became finer and finer to ensure accuracy. When they compared their theoretical predictions to real-world measurements of how neutrons scatter off calcium-40, the results were encouraging. At angles close to the direction the particle was originally traveling, the simulation matched the data very well. The team also looked at the total amount of scattering and the amount of energy absorbed by the nucleus. Between energies of 15 and 50 million electron volts, their calculations tracked the experimental data closely, provided they used a sufficiently large mathematical grid. However, when they looked at particles scattering at wide angles, moving backward relative to their original path, the simulation fell short. The model predicted that fewer particles would be absorbed than what was actually observed in experiments, suggesting that the theoretical map was missing some of the complex internal interactions that cause the nucleus to soak up energy.

The challenge became even more apparent when the team turned their attention to protons. Because protons carry a positive electric charge, they experience a repulsive force from the positively charged nucleus that neutrons do not feel. In their initial calculations, the researchers represented this electric force using the same mathematical grid as the nuclear forces. This approach inadvertently cut off the long-range reach of the electric repulsion, leading to incorrect predictions. To fix this, they removed the electric force from the grid calculation and instead applied a standard, long-range electric field model to the final result. This correction restored the proper behavior of the electric force. With this adjustment, the simulation once again matched the experimental data for forward angles, successfully predicting how protons scatter off calcium-40.

The team then applied their method to calcium-42, a slightly heavier isotope that contains two extra neutrons. Unlike calcium-40, which is perfectly symmetric, calcium-42 has these extra neutrons pairing up, a behavior that the new Gorkov method is specifically designed to capture. This was the first time this specific approach had been tested on a nucleus with such pairing effects. The results for proton scattering off calcium-42 mirrored those for calcium-40: the simulation worked well for forward angles but failed to reproduce the absorption seen at backward angles. This consistency across different isotopes confirmed that the method is robust for semi-magic nuclei but highlighted a specific limitation. The inability to fully account for absorption at wide angles suggests that the current level of calculation is missing certain complex correlations between particles, specifically those involving the collective motion of many particles at once.

The researchers conclude that while their approach successfully extends the ability to predict nuclear scattering to open-shell nuclei and proton projectiles, it is not yet a complete solution. The discrepancies at backward angles indicate that the model needs to include even more complex interactions to fully describe how the nucleus absorbs energy. The team plans to refine their calculations by adding more layers of complexity to the mathematical framework, which should help capture the missing absorption. They also note that the current fundamental forces used in the simulation might slightly underestimate the strength of the pairing between neutrons, which could affect low-energy predictions. Despite these limitations, the work represents a major advance in moving from empirical guessing to first-principles prediction, offering a path toward understanding the behavior of exotic nuclei that exist only for fleeting moments in the universe.

Technical Summary: Ab Initio-Guided Optical Potentials for Semi-Magic Nuclei from Gorkov Self-Consistent Green's Functions

Problem and Motivation
The optical potential is the standard mechanism for reducing nucleon-nucleus scattering to an effective one-body problem. Traditionally, these potentials are constructed empirically by fitting functional forms to experimental data. While successful within calibration domains, such phenomenological approaches lack predictive power for exotic nuclei far from stability and offer no systematic path for uncertainty propagation from underlying nuclear forces. Microscopic approaches, such as the Dispersive Optical Model (DOM), have made progress but often remain constrained by data fitting. The goal of this work is to compute the optical potential (identified as the irreducible self-energy) directly from the nuclear Hamiltonian using ab initio many-body methods, thereby extending these capabilities to open-shell (semi-magic) targets and proton projectiles.

Methodology
The authors employ the Self-Consistent Green's Function (SCGF) theory, specifically utilizing the Gorkov extension to handle open-shell systems.

Formalism: The Gorkov formalism breaks U(1) symmetry associated with particle-number conservation to capture pairing correlations, allowing access to semi-magic nuclei. The optical potential is identified with the Gorkov-ADC(2) (Algebraic Diagrammatic Construction to second order) self-energy.
Interaction: The calculation starts from the NNLOsat chiral interaction. The three-nucleon (3N) component is treated as an effective two-body interaction by averaging over the one-body density.
Basis and Transformation: The self-energy is computed in a harmonic-oscillator (HO) basis with truncation parameters $e_{max} = 7$ and $11$. It is then transformed to coordinate space using HO radial functions.
Scattering Code: The transformed self-energy is utilized in the scattering code SIDES to compute elastic scattering observables.
Imaginary Part and "Ab Initio-Guided" Label: In a finite HO basis, the continuum is discretized, leading to a self-energy with isolated poles and an imaginary part consisting of delta functions. To obtain a smooth potential suitable for scattering, the infinitesimal broadening parameter $\eta$ is replaced by an energy-dependent phenomenological width $\Gamma(E)$ . Because this imaginary part relies on a phenomenological width to mimic spreading and escape widths (configurations beyond ADC(2) and continuum coupling), the authors label the resulting potentials "ab initio-guided" rather than strictly ab initio.
Coulomb Correction: For proton scattering, the direct Coulomb term in the truncated HO basis lacks the correct long-range $1/r$ tail. The authors implement a minimal correction: the direct Coulomb term from the HO basis is subtracted from the static self-energy, and the correct long-range behavior is restored using a local Coulomb potential (uniformly charged sphere) within the SIDES code.

Key Contributions

Extension to Open-Shell Targets: The work lifts the restriction of previous Dyson-ADC approaches to closed-shell (doubly-magic) targets, applying the Gorkov-ADC(2) framework to semi-magic nuclei (specifically $^{40}$ Ca and $^{42}$ Ca).
Proton Projectile Inclusion: The method is extended to treat proton projectiles, addressing the specific challenges of Coulomb interactions in a truncated basis.
Validation of Gorkov-ADC(2): The study demonstrates the feasibility of computing nonlocal optical potentials for semi-magic systems using this specific truncation of the many-body expansion.

Results
The study compares differential, total, and reaction cross sections for $n+p+^{40}$ Ca and $p+^{42}$ Ca against experimental data.

Neutron Scattering ( $n+^{40}$ Ca):
- At incident energies of 21.7 and 40 MeV, the $e_{max}=11$ calculation reproduces differential cross sections well at forward angles.
- Backward angles are overestimated by up to an order of magnitude, indicating insufficient absorption. This is attributed to missing correlations (e.g., resummation of ring and ladder diagrams) and the limitations of the ADC(2) truncation.
- Total and reaction cross sections are well reproduced between 15 and 50 MeV with $e_{max}=11$ . The $e_{max}=7$ truncation fails at higher energies due to partial-wave truncation effects.
Proton Scattering ( $p+^{40}$ Ca):
- Without the Coulomb correction, the partial cross sections fail to vanish at large angular momenta ( $l$ ), violating physical expectations.
- With the Coulomb correction, the angular distributions are well reproduced up to $\approx 60^\circ$ , including the first diffraction minimum. However, the second minimum and backward-angle oscillations are not fully reproduced, again pointing to insufficient absorption.
Semi-Magic Test ( $p+^{42}$ Ca):
- Results for $p+^{42}$ Ca (where the proton shell is closed but neutrons are paired) show similar behavior: forward angles are well reproduced, while backward angles are overestimated. This confirms the method's applicability to systems with pairing correlations, although the pairing effects enter the proton potential only indirectly via intermediate configurations.

Significance and Limitations
The paper claims that this work represents a first application of Gorkov self-consistent Green's function self-energies to nucleon scattering off semi-magic targets. The primary significance lies in establishing a systematic, ab initio-guided route to optical potentials that can, in principle, be extended to exotic nuclei.

However, the authors are modest regarding the current predictive power:

Absorption Deficit: The consistent overestimation of backward-angle cross sections suggests that the ADC(2) truncation misses significant absorption mechanisms (likely requiring higher-order resummations like ring and ladder diagrams).
Basis Limitations: The accessible incident energies are limited by the HO basis truncation ( $e_{max}$ ).
Pairing Strength: The authors note that current realistic chiral interactions (like NNLOsat) may produce pairing correlations that are too weak, potentially leading to an underestimation of pairing effects in low-energy scattering.
Future Work: The authors plan to advance to Gorkov-ADC(2)-X (incorporating first-order interactions among 2p1h and 2h1p configurations) and eventually Gorkov-ADC(3) to recover missing correlations and improve absorption. They also intend to quantify dependencies on the HO basis and the chiral interaction.

Technical Summary: Ab Initio-Guided Optical Potentials for Semi-Magic Nuclei from Gorkov Self-Consistent Green's Functions

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