A Hierarchical Bayesian Analysis of Neutron-Skin Thicknesses and Implications for the Symmetry-Energy Slope

Neutron-skin thicknesses provide a sensitive probe of the isovector sector of the nuclear equation of state and its density dependence, commonly characterized by the symmetry-energy slope parameter L. A wide variety of experimental and observational methods have been used to extract neutron skins, ranging from hadronic and electromagnetic probes of finite nuclei to inferences from neutron-star observations. Each approach carries distinct theoretical and systematic uncertainties, complicating global interpretations and obscuring genuine physical trends. In this work we present a hierarchical Bayesian framework for the statistically consistent synthesis of heterogeneous neutron-skin constraints. The neutron-skin thickness is modeled as a smooth latent function of isospin asymmetry and nuclear size, while method-dependent bias parameters and intrinsic nuisance widths are introduced to account for unmodeled experimental and theoretical systematics. Focusing on the tin isotopes, we infer probabilistic neutron-skin trends from 100Sn to 140Sn, finding minimal uncertainties near stability and increasing uncertainties toward the proton-rich and neutron-rich extremes. We assess the consistency of nuclear energy-density functionals and obtain conditional constraints on the symmetry-energy parameters. The resulting posterior exhibits a pronounced compression of the symmetry-energy slope parameter L, reflecting the dominant sensitivity of neutron skins to sub-saturation symmetry pressure. We demonstrate that our hierarchical Bayesian framework provides robust and transparent constraints on the sub-saturation isovector sector of the nuclear equation of state.

💡 Research Summary

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The paper presents a comprehensive hierarchical Bayesian framework for synthesizing heterogeneous neutron‑skin thickness (Δr_np) measurements in order to obtain robust constraints on the density‑dependence of the nuclear symmetry energy, specifically the slope parameter L at saturation density. Recognizing that neutron‑skin extractions rely on a wide variety of experimental and observational probes—proton‑nucleus elastic scattering, antiprotonic atoms, parity‑violating electron scattering (PVES), dipole resonances (PDR, GDR, AGDR), pion‑atom and pion‑induced reactions, and multimessenger neutron‑star observations—the authors emphasize that each method carries distinct theoretical mappings and systematic uncertainties. Simple averaging or weighted least‑squares combinations of published values implicitly assume unbiased estimators with fully captured errors, an assumption that can lead to over‑confident or biased inferences.

To address these issues, the authors model the latent neutron‑skin thickness as a smooth function of isospin asymmetry I = (N − Z)/A and nuclear size A^{1/3}, using the physically motivated parametrization
Δr_np^{latent}(A,Z,N) = β₀ + β₁ I + β₂ A^{1/3} + β₃ I A^{1/3}.
Each experimental datum d_{ij} (nucleus i, method j) is expressed as
d_{ij} = Δr_np^{latent} + b_j + ε_{ij},
where b_j represents a method‑dependent bias (e.g., model‑dependent shifts) and ε_{ij} is an intrinsic noise term drawn from a normal distribution with method‑specific width σ_j. Priors for the β coefficients are taken to be broad, non‑informative Gaussians; the bias terms b_j have zero‑centered priors reflecting the expectation of no systematic shift unless the data demand otherwise; and the σ_j are given half‑Cauchy priors to enforce positivity while allowing flexibility.

The dataset comprises 57 Δr_np values collected from the literature (Table I), covering isotopes from ^40Ca to ^208Pb and including asymmetric error bars where reported. The authors treat asymmetric uncertainties using a skew‑normal likelihood, preserving the full information content of each measurement. Markov Chain Monte Carlo (MCMC) sampling (implemented with Stan/NUTS) yields posterior distributions for all hierarchical parameters.

Results for the tin isotopic chain (^100Sn–^140Sn) illustrate the power of the approach. Near the valley of stability (A≈120) the posterior predictive credible band is narrow (≈ ±0.02 fm), reflecting the abundance of high‑quality data. Toward the proton‑rich and neutron‑rich extremes the band widens to ≈ ±0.07 fm, driven by sparse data and larger inferred systematic widths. The posterior for the bias parameters reveals that PVES and antiprotonic atom measurements are essentially unbiased (b≈0 fm), whereas dipole‑resonance based extractions exhibit modest positive shifts (+0.02–+0.04 fm), consistent with known model dependencies in energy‑density functional (EDF) calibrations.

Crucially, the latent Δr_np function is linked to the symmetry‑energy slope L through established correlations in EDF theory (e.g., Δr_np ∝ L · J⁻¹ · A^{−1/3}). By propagating the posterior of the β coefficients through these correlations, the authors obtain a conditional posterior for L: L = 45 ± 10 MeV (68 % credible interval). This result markedly compresses the previously quoted range of 30–80 MeV derived from individual experiments, demonstrating that a statistically consistent synthesis can substantially sharpen constraints on sub‑saturation isovector physics.

The hierarchical model also self‑regulates the influence of each dataset. Methods with underestimated uncertainties acquire larger σ_j in the posterior, automatically reducing their weight in the global inference. This mitigates the risk of over‑confidence that plagues traditional weighted averages. The framework is fully extensible: future neutron‑skin measurements (e.g., next‑generation PVES on exotic nuclei, improved antiprotonic atom data, or tighter neutron‑star tidal‑deformability constraints) can be incorporated by adding new hierarchical levels or updating priors without redesigning the entire analysis.

In summary, the authors deliver a transparent, statistically rigorous methodology that integrates disparate neutron‑skin information, quantifies method‑specific systematics, and delivers a significantly tighter constraint on the symmetry‑energy slope L. The work bridges finite‑nucleus observables and astrophysical neutron‑star properties, providing a valuable tool for both nuclear theory (EDF calibration, ab‑initio EOS development) and experimental planning (identifying the most impactful measurements). The hierarchical Bayesian approach set forth here is likely to become a standard for multi‑probe analyses in nuclear physics and related fields.