⚛️ nuclear theory

Nonlocal nucleon-nucleus optical potentials from chiral effective field theory

This paper investigates nonlocal nucleon-nucleus optical potentials derived from chiral effective field theory using the Perey-Buck ansatz, revealing that spatial nonlocality primarily drives the energy dependence of the real potential while time nonlocality governs the imaginary potential, and presents results for calcium isotopes showing how Woods-Saxon parameters vary with isotopic number.

Original authors: Laina M. Stahulak, Jeremy W. Holt

Published 2026-09-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Laina M. Stahulak, Jeremy W. Holt

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 universe builds its heaviest elements, scientists must first understand how the smallest building blocks of matter stick together. Inside the heart of every atom lies a dense cluster of protons and neutrons, held by a force so powerful it defies simple explanation. When a lone particle, like a neutron or a proton, approaches this cluster, it does not simply bounce off a hard surface. Instead, it interacts with the entire crowd of particles inside, creating a complex, shifting landscape of attraction and repulsion. This interaction is described by something called an optical potential, a mathematical map that predicts how the incoming particle will move and scatter. For decades, scientists have relied on simplified maps that assume the particle feels a force that depends only on where it is and how fast it is moving. However, this view ignores a crucial reality: the particle's experience also depends on where it has been and how its path might be influenced by the quantum nature of the nucleus itself.

This subtle influence, known as nonlocality, suggests that the force acting on a particle at one point is actually shaped by conditions at neighboring points. While this idea has long been part of theoretical physics, it has been difficult to calculate with high precision, especially for unstable, exotic atoms found in the far reaches of the nuclear chart. A recent study by researchers at Texas A&M University has taken a major step forward by creating a new, more accurate map of these forces. By using a sophisticated framework called chiral effective field theory, which treats the nuclear force as a series of interactions between particles, the team has derived optical potentials that account for these nonlocal effects. Their work reveals that the way a particle moves through a nucleus is deeply connected to the quantum "memory" of its path, and that ignoring this connection leads to significant errors in predicting how these particles behave.

The researchers focused on a specific method to translate their complex calculations into a usable form. They started with a theoretical approach that calculates the energy of a particle moving through an infinite sea of nuclear matter, a simplified model that allows for precise mathematical treatment. In this model, the force a particle feels changes depending on its energy and its momentum. The team then applied a technique known as the Perey-Buck ansatz, which acts as a bridge between two different ways of describing the nuclear force. This method allows them to convert a force that changes with energy into a force that stays constant but spreads out over space, effectively capturing the nonlocal nature of the interaction. They tested this approach across a wide range of conditions, simulating how protons and neutrons interact with different types of nuclear matter, from balanced mixtures to those heavily skewed toward one type of particle.

The results of their simulations were revealing. They found that for the real part of the nuclear force—the part that pulls particles together or pushes them apart—the nonlocal approach worked remarkably well. By treating the force as a spread-out influence rather than a sharp, energy-dependent point, they could reproduce the behavior of particles up to energies of about 100 million electron volts. This range covers many of the reactions scientists study in laboratories today. The study showed that the dominant source of energy dependence in their microscopic models actually comes from this spatial nonlocality. In other words, the way the force changes as a particle speeds up is largely because the particle is feeling the influence of a wider area of the nucleus, not because the force itself is fundamentally changing with speed. This finding validates the use of the nonlocal model for a broad spectrum of nuclear reactions.

However, the story was different for the imaginary part of the force, which describes how particles are absorbed or lost from the beam as they interact with the nucleus. The researchers discovered that the nonlocal approach could not fully capture the behavior of this absorption. The energy dependence of the imaginary force appeared to be a genuine feature of time, related to how the nucleus transitions into excited states, rather than a simple spatial effect. This means that while the nonlocal model works well for predicting how particles scatter, it cannot yet fully replace the traditional energy-dependent models for predicting how many particles get absorbed. The team noted that trying to force the imaginary part into the same nonlocal shape would break the fundamental mathematical rules that connect the real and imaginary parts of the force, rules that ensure the physics remains consistent with the laws of causality.

To make their findings practical for studying real atoms, the team applied their results to a chain of calcium isotopes, atoms that have the same number of protons but varying numbers of neutrons. They calculated how the nonlocal force changes as the nucleus gets larger and more neutron-rich. They found that the strength of the force and the range over which it acts are not constant; they shift depending on the density of the nucleus and the balance between protons and neutrons. For instance, in neutron-rich nuclei, the force felt by a proton is different from that felt by a neutron, and the "spread" of the force is larger for protons than for neutrons. These differences are subtle but significant, and they become more pronounced as the nucleus moves further away from stability. The researchers also observed that the shape of the force follows a smooth pattern as the number of neutrons increases, with a noticeable change around the isotope with 48 neutrons, a point where the nucleus has a particularly stable, closed structure.

The study concludes that while the nonlocal model provides a powerful new tool for understanding nuclear reactions, it is not a perfect replacement for all existing methods. The team demonstrated that for energies up to 100 million electron volts, the nonlocal description is a valid and accurate way to represent the complex interactions inside a nucleus. This opens the door for more precise calculations of how exotic nuclei behave, which is essential for understanding how heavy elements are forged in the violent environments of exploding stars. By moving beyond simplified, energy-dependent maps to a more nuanced, spatially spread-out view, the researchers have provided a clearer picture of the quantum forces that govern the atomic nucleus. Their work suggests that the future of nuclear physics lies in embracing these nonlocal effects, allowing scientists to predict the behavior of matter in conditions that have never been directly observed.

Technical Summary: Nonlocal Nucleon-Nucleus Optical Potentials from Chiral Effective Field Theory

Problem Statement
Nuclear reaction studies involving exotic isotopes, particularly those far from stability, require theoretical descriptions that extend beyond phenomenological optical model potentials (OMPs). Phenomenological OMPs, while successful near the valley of stability, rely heavily on scattering data that is scarce for exotic nuclei and do not directly reflect underlying microscopic physics. Furthermore, standard microscopic approaches often employ the "on-shell" approximation, which replaces the separate momentum and energy dependence of the nucleon self-energy with a single energy-dependent local potential. This approximation is valid for elastic scattering but fails to capture spatial and temporal nonlocalities essential for inelastic processes, such as transfer and knockout reactions, where wavefunction differences in the nuclear surface and exterior regions significantly impact cross-sections. The central problem addressed is how to rigorously derive nonlocal, energy-independent optical potentials from microscopic chiral effective field theory (ChEFT) to better describe scattering in medium-mass and heavy nuclei, particularly those with high isospin asymmetry.

Methodology
The authors employ the infinite nuclear matter approach combined with the Perey-Buck ansatz to construct nonlocal optical potentials.

Microscopic Foundation: The study utilizes two-body and three-body nuclear forces derived from ChEFT (up to N3LO for two-body and N2LO for three-body forces, with a cutoff $\Lambda = 450$ MeV). The nucleon self-energy is calculated in homogeneous nuclear matter using many-body perturbation theory (MBPT) up to second order.
Perey-Buck Ansatz: To convert the energy-dependent local potentials (derived from the on-shell self-energy) into energy-independent nonlocal potentials, the authors apply the Perey-Buck factorization. This assumes the nonlocal potential $U(\vec{r}, \vec{r}')$ factorizes into a local Woods-Saxon term $U_N$ and a Gaussian nonlocal term $H(|\vec{r} - \vec{r}'|)$ characterized by a nonlocality length scale $\beta$ .
Parameter Extraction: The authors fit the energy dependence of the microscopic real optical potential to the Perey-Buck relation to determine the energy-independent depth $U_N$ and nonlocality range $\beta$ . This is performed across a grid of nuclear densities and isospin asymmetries.
Finite Nuclei Construction: The Improved Local Density Approximation (ILDA) is used to fold the nuclear matter parameters with the density distributions of specific nuclei (specifically the calcium isotopic chain from $^{40}$ Ca to $^{60}$ Ca) to generate radial optical potentials.
Dispersion Relations: The study investigates the consistency of these potentials with causality and the Kramers-Kronig dispersion relations, comparing the real self-energy calculated directly via perturbation theory against that derived from the imaginary part via dispersion integrals.

Key Contributions and Results

Validation of the Perey-Buck Ansatz for Real Potentials: The study finds that the dominant source of energy dependence in the microscopic real optical potential arises from spatial nonlocalities. The Perey-Buck ansatz adequately captures the energy dependence of the real part of the self-energy up to projectile energies of $E \sim 100$ MeV. Beyond this energy, the ansatz breaks down.
Distinction Between Real and Imaginary Nonlocality: A critical finding is that while the real potential's energy dependence can be mapped to spatial nonlocality, the energy dependence of the imaginary optical potential cannot be fitted to the Perey-Buck form. The imaginary part exhibits a "genuine time nonlocality" that does not conform to the Gaussian spatial nonlocality assumption, as the energy dependence of the imaginary part does not scale in a way that allows the exponential term in the Perey-Buck relation to remain constant.
Isospin and Density Dependence: The derived nonlocal parameters ( $U_N$ $U_{N}$ and $\beta$ $β$ ) exhibit systematic dependencies on nuclear density and isospin asymmetry ( $\delta_{np}$ $δ_{n p}$ ) that are not present in standard phenomenological models.
- The nonlocal potential depth $U_N$ increases with density.
- The nonlocality range parameter $\beta$ generally increases with density in first-order calculations but shows a weaker dependence (and even a decrease for neutrons) when second-order contributions are included.
- Significant isovector effects are observed: in neutron-rich matter, protons experience a stronger nonlocal potential and a larger nonlocality range ( $\beta$ ) than neutrons.
Calcium Isotopic Chain: For the calcium chain ( $^{40}$ Ca to $^{60}$ Ca), the authors present radial profiles for the nonlocal potentials. They observe a smooth dependence of Woods-Saxon shape parameters on the mass number $A$ , with an inflection point around $A=48$ attributed to shell effects. The proton and neutron potential depths diverge symmetrically starting from $^{40}$ Ca due to Coulomb repulsion and neutron excess.
Dispersion Consistency: The authors demonstrate that the real self-energy calculated explicitly in second-order perturbation theory agrees well with results obtained via the Kramers-Kronig dispersion relation, confirming the dispersive nature of the microscopic approach. However, they note that constructing a fully dispersive nonlocal potential requires treating the real and imaginary parts consistently in coordinate space, which is complicated by the failure of the Perey-Buck ansatz for the imaginary part.

Significance and Claims
The paper claims to provide a rigorous microscopic derivation of nonlocal optical potentials that bridges the gap between ab initio many-body theory and practical reaction modeling. By validating the Perey-Buck ansatz for the real part of the potential up to 100 MeV, the work suggests that spatial nonlocality is the primary mechanism driving the energy dependence of the real optical potential in this energy regime. This allows for the construction of energy-independent nonlocal potentials suitable for reaction codes that require nonlocal inputs (e.g., for transfer reactions).

The authors modestly state that their results offer a foundation for future improvements in phenomenological modeling by providing microscopic constraints on the density and isospin dependence of nonlocal parameters. They explicitly identify that the inability to fit the imaginary part to the Perey-Buck form represents a fundamental physical inconsistency in current hybrid models (nonlocal real + local imaginary) and highlights the need for fully dispersive nonlocal models. The work does not claim to solve all reaction theory problems but establishes a method to derive nonlocal potentials from ChEFT that are consistent with microscopic many-body theory up to second order, offering a path toward more reliable predictions for exotic nuclei where experimental data is lacking.

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