Bayesian approach for many-body uncertainties in nuclear structure: Many-body perturbation theory for finite nuclei

TL;DR

The paper develops a Bayesian framework to quantify many-body truncation uncertainties in ab initio nuclear structure, modeling MBPT truncations with a convergence factor $R$ and random coefficients $\gamma_i$ to generate posterior predictive distributions for ground-state energies. By calibrating these hyperparameters against MBPT results for closed-shell nuclei using multiple chiral interactions, the approach yields per-nucleon uncertainty bands that shrink significantly when including MBPT(3) and remain sensitive to interaction softness. The method enables combined uncertainty quantification for both many-body truncations and interaction choices, while highlighting limitations in handling correlations and matter-even regimes, and it provides a pathway toward extending uncertainty quantification to open-shell nuclei and more complex many-body methods. The results show that soft, perturbative interactions lead to rapid convergence and controllable uncertainties, whereas harder interactions and neutron/matter environments require refined modeling and higher-order information for reliable predictions.

Abstract

A comprehensive assessment of theoretical uncertainties defines an important frontier in nuclear structure research. Ideally, theory predictions include uncertainty estimates that take into account truncation effects from both the interactions and the many-body expansion. While the uncertainties from the expansion of the interactions within effective field theories have been studied systematically using Bayesian methods, many-body truncations are usually addressed by expert assessment. In this work we use a Bayesian framework to study many-body uncertainties within many-body perturbation theory applied to finite nuclei. Our framework is applied to a broad range of nuclei across the nuclear chart calculated from two- and three-nucleon interactions based on chiral effective field theory. These developments represent a step towards a more complete and systematic quantification of uncertainties in \emph{ab initio} calculations of nuclei.

Figures (10)

• Figure 1: Ratios $E^{(2)}/E_\textnormal{HF}$ and $E^{(3)}/E^{(2)}$ for our set of nuclei from Ref. Arthuis2024 as well as symmetric nuclear matter at two representative densities from Ref. Alp:2025wjn. $n_0 = 0.16$ fm$^{-3}$ denotes the nuclear saturation density. All shown results have been computed using the 1.8/2.0 (EM) interaction Hebe11fits.
• Figure 2: Conjugate inverse-gamma prior for $\bar{\gamma}^2$ with hyperparameters given by the legend. Our chosen prior is shown as a solid black line. The two alternative priors, shown as gray dashed and dot-dashed lines, are discussed in App. \ref{['app:priors']}.
• Figure 3: Priors (black lines/distributions) and posteriors (purple distributions) for $R$ and $\bar{\gamma}^2$ obtained from the MBPT results with the 1.8/2.0 (EM) interaction. The ranges for around the median of $R$ and $\bar{\gamma}^2$ are given at the 68% credibility level. The details of the inference are given in the text.
• Figure 4: PPDs for the ground-state energy per nucleon of selected closed-shell nuclei at second and third order, MBPT(2) and MBPT(3) respectively, compared to IMSRG(2) results. Results are shown for the 1.8/2.0 (EM) interaction. The dark (light) shaded areas indicate 68% (90%) credibility intervals.
• Figure 5: Empirical coverage ("weather") plot corresponding to the predictions described in Fig. \ref{['fig:ppd_magic']} (left panel), as well as analogous results obtained for the same nuclei calculated from the $\Delta$N$^2$LO$_\text{GO}$ (center panel) and 1.8/2.0 (EM7.5) interactions (right panel). Values above the diagonal indicate too conservative UQ estimates and vice versa, with an ideal result following the diagonal. Results falling within the dark (light) gray area are consistent with the ideal result at the 68% (95%) confidence level given the amount of data we are comparing to (here the IMSRG results).