⚛️ nuclear theory

Data-Driven Statistical Ensembles of Chiral Nuclear Interactions

This paper employs normalizing flows to construct statistical ensembles of chiral nuclear interactions, revealing non-Gaussian correlations among low-energy constants and establishing a framework for systematically constraining nuclear forces with experimental and astrophysical data.

Original authors: Pengsheng Wen, Jeremy W. Holt

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pengsheng Wen, 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

For more than half a century, physicists have struggled to write a single, perfect equation that describes how the tiny particles inside an atomic nucleus stick together. This force, known as the nuclear force, is what holds the protons and neutrons in place, preventing the atom from flying apart. However, unlike the force of gravity, which we can measure directly with a scale, the nuclear force is hidden deep within the quantum world. Scientists cannot observe it directly; they can only infer its strength and shape by watching how particles bounce off one another. To make sense of these observations, researchers use a mathematical framework called chiral effective field theory. This approach breaks the complex nuclear force into two parts: long-range interactions caused by the exchange of particles called pions, and short-range interactions that happen when particles get extremely close. The short-range part is controlled by a set of numbers called low-energy constants. These numbers act like dials on a machine; if you turn them one way, the force gets stronger, and if you turn them another, it gets weaker. For decades, scientists have tried to find the single best setting for these dials that matches experimental data. But this approach has a blind spot. It assumes there is only one correct answer, ignoring the fact that the data itself contains uncertainties and that the "dials" might be connected in complex, unexpected ways.

A team of researchers at Texas A&M University has now moved beyond the search for a single best answer. Instead of trying to pin down one specific set of numbers, they have developed a new method to map out the entire landscape of possible values. They treated the problem not as finding a single point on a map, but as understanding the shape of a cloud of possibilities. To do this, they used a type of advanced computer program known as a normalizing flow. Think of this tool as a sophisticated machine that can learn the shape of a complex, multi-dimensional cloud of data and generate new examples that fit perfectly within that shape. The researchers fed this machine data from neutron-proton scattering experiments, where neutrons and protons are fired at each other and their paths are measured. The machine learned to generate thousands of different sets of the "dial" numbers that would all produce results matching the real-world experiments.

The results of this work reveal a picture of the nuclear force that is far more intricate than previously understood. The researchers found that the numbers controlling the short-range interactions are not independent; they are tightly linked to one another in strong, non-random patterns. When one number changes, the others must shift in a specific way to keep the physics consistent. These connections are not simple or straight; they form complex, curved relationships that evolve as the scale of the interaction changes. The study covered a range of resolution scales from 400 to 550 MeV, a measure of how closely the scientists are looking at the interaction. Across this entire range, the computer-generated ensembles of nuclear forces successfully reproduced the experimental data, including the natural variations and uncertainties found in the measurements. The model did not just match the average values; it captured the full spread of the data, showing that the nuclear force is a statistical entity with a rich internal structure.

One of the most significant findings is that this approach works continuously across different scales. In the past, scientists often had to choose a specific scale and fit their numbers only for that one setting. This new framework allows the probability distributions of the numbers to flow smoothly from one scale to another. The researchers tested their model by generating samples of these numbers and using them to predict how neutrons and protons would scatter at various energy levels, up to 200 MeV. The predictions matched the experimental results with high precision, staying well within the expected margins of error. This success confirms that the complex correlations the model discovered are real and necessary to describe the nuclear force accurately. The study also highlighted that for certain types of particle interactions, the current mathematical models might need more flexibility, as the fit was slightly less perfect for those specific cases. This suggests that while the method is powerful, the underlying theory of the nuclear force may still need refinement in those specific areas.

This work establishes a new way of thinking about the fundamental building blocks of matter. By constructing statistical ensembles of nuclear interactions, the researchers have created a tool that can be systematically improved as new data becomes available. Future experiments involving the properties of atomic nuclei or observations of neutron stars can be fed directly into this framework to further refine the understanding of the nuclear force. The ability to propagate these uncertainties from the microscopic world of nuclear forces to the macroscopic world of neutron stars offers a unified path for connecting laboratory experiments with the cosmos. The study demonstrates that the nuclear force is not a static, fixed set of rules, but a dynamic, probabilistic system that can be mapped with unprecedented clarity. This approach opens the door to a more robust and comprehensive understanding of the universe, grounded in the full complexity of the data rather than a simplified approximation.

Technical Summary: Data-Driven Statistical Ensembles of Chiral Nuclear Interactions

Problem Statement
Developing accurate microscopic models of the nuclear force remains a central challenge in nuclear physics. While Chiral Effective Field Theory ( $\chi$ EFT) provides a systematic framework for describing nuclear forces via long-range pion exchanges and short-range contact interactions parameterized by Low-Energy Constants (LECs), a comprehensive characterization of these forces is incomplete. Traditional methods determine LECs at fixed resolution scales using nonlinear regression, yielding best-fit values but offering limited insight into the full underlying probability distributions. Furthermore, existing approaches often fail to provide a unified characterization of correlated LEC distributions across a continuous range of resolution scales ( $\Lambda$ ). This gap is critical as experimental constraints from nucleon-nucleon scattering, finite nuclei, and neutron-star observations become increasingly diverse, and experimental observables often represent distributions rather than single values.

Methodology
To address these limitations, the authors employ normalizing flows (nflows), a class of expressive generative machine learning models, to infer the joint probability distribution of two-body LECs in $\chi$ EFT continuously across resolution scales.

Generative Framework: The model utilizes a chain of transformation layers to map samples from a simple latent distribution (chosen as a uniform distribution) to a complex target distribution of LECs. The transformation is conditioned on an external variable $z$ , which in this work corresponds to the resolution scale $\Lambda$ .
Transformation Function: The authors utilize rational quadratic splines as the transformation function. These split the sampling space into bins parameterized by trainable bin widths, heights, and knot derivatives, all of which depend on the conditioning variable $\Lambda$ .
Forward Model and Simulation: Calculating observables (neutron-proton scattering phase shifts) from LECs is computationally intensive, involving the construction of nuclear potentials, partial-wave decomposition, and solving the Lippmann–Schwinger (LS) equation. To facilitate training, a separate neural network, $NN^{(L,S,J)}_{\delta}$ , was trained as a phase-shift simulator to approximate the mapping from LECs and $\Lambda$ to phase shifts at specific laboratory energies ( $E_{lab}$ ) and angular momentum channels.
Optimization Strategy: Instead of minimizing the deviation of a single predicted value from an experimental mean (as in traditional regression), the framework minimizes the discrepancy between the entire theoretical probability distribution and the experimental distribution.
- Theoretical phase-shift distributions are estimated using a Gaussian Kernel Density Estimator (KDE) on samples generated by the normalizing flow.
- The loss function is defined by the reverse Kullback-Leibler (KL) divergence between the theoretical PDF ( $p_{theor}$ ) and the experimental target PDF ( $p_{exp}$ ).
- The optimization sums this loss over all considered laboratory energies (up to 200 MeV) and resolution scales ( $\Lambda$ from 400 to 550 MeV).

Key Results
The framework was applied to Next-to-Next-to-Next-to-Leading Order (N3LO) $\chi$ EFT potentials, focusing on neutron-proton scattering phase shifts.

Distribution Learning: The normalizing flow successfully learned concentrated, strongly correlated distributions for LECs across the specified range of $\Lambda$ . For the $^3P_0$ channel, the initially uniform samples collapsed into distinct clusters associated with different $\Lambda$ values, effectively filtering out non-physical regions of the parameter space.
High-Dimensional Correlations: In the more complex $^1S_0$ channel (a four-dimensional parameter space involving $\tilde{C}_{1S0}, C_{1S0}, D_{1S0}, \hat{D}_{1S0}$ ), the model identified narrow, physically meaningful regions. The results revealed strong, non-Gaussian correlations among the LECs that evolve continuously with the resolution scale.
Validation against Experiment:
- The ensembles of nuclear interactions generated by the inferred LEC distributions accurately reproduced the central values and experimental uncertainties of neutron-proton phase shifts across $E_{lab} \leq 200$ MeV.
- Normalized deviations of calculated phase shifts from experimental means rarely exceeded three standard deviations.
- The framework was extended to coupled channels ( $J=1, 2, 3$ ). While the mean predictions generally agreed well with experimental data, the authors noted that channels with fewer fitting parameters (e.g., $^3F_2$ ) showed weaker dependence on the fitted LECs and stronger dependence on the choice of resolution scale, suggesting limitations in the N3LO interaction structure rather than the fitting procedure.

Significance and Claims
The paper establishes a general framework for constructing statistical ensembles of nuclear interactions that are systematically constrained by experimental data. The authors claim that this approach offers several distinct advantages over traditional methods:

Full Probabilistic Characterization: It moves beyond identifying single optimal sets of LECs to characterizing the full joint probability distributions, capturing non-Gaussian correlations and their evolution with resolution scale.
Uncertainty Quantification: By learning the distribution conditioned on $\Lambda$ , the framework provides a unified method for comprehensive uncertainty quantification in $\chi$ EFT, allowing for the propagation of uncertainties from microscopic interactions to macroscopic observables.
Scalability and Flexibility: The generative nature of the model allows for the simultaneous optimization of distributions across all relevant conditions (resolution scales). The authors posit that this framework can be extended to incorporate diverse constraints, including few- and many-body observables (e.g., binding energies, neutron-skin thicknesses) and three-nucleon interactions, ultimately providing a unified connection between microscopic nuclear interactions, laboratory experiments, and astrophysical observations.

Technical Summary: Data-Driven Statistical Ensembles of Chiral Nuclear Interactions

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