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arxiv: 2605.21657 · v1 · pith:NV3YTPGQnew · submitted 2026-05-20 · ⚛️ nucl-th

Uncertainty Quantification of the ⁷⁶Ge Neutrinoless Double-Beta Decay Nuclear Matrix Element

Mihai Horoi , Andrei Neacsu This is my paper

Pith reviewed 2026-05-22 08:24 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords neutrinoless double-beta decaynuclear matrix element76GeBayesian model averagingshell modeluncertainty quantificationeffective nuclear interaction
0
0 comments X

The pith

The 0νββ nuclear matrix element for 76Ge is determined to be 2.46 with a standard deviation of 0.25 through Bayesian analysis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper quantifies the theoretical uncertainty in the nuclear matrix element for neutrinoless double-beta decay in 76Ge. It does so by applying bounded fluctuations to two-body matrix elements of an effective interaction and using Bayesian model averaging to fit low-energy spectroscopic data. The resulting probability distribution has a central value of 2.46 and standard deviation 0.25. This constrained value helps in interpreting experimental limits on the decay half-life and probing physics beyond the Standard Model.

Core claim

By integrating simulated variations of two-body matrix elements into a Bayesian Model Averaging framework and benchmarking against empirical spectroscopic data, the analysis yields a constrained probability distribution for the 0νββ NME with central value 2.46 and standard deviation 0.25.

What carries the argument

Bayesian Model Averaging of systematic bounded fluctuations applied to the two-body matrix elements of the effective interaction in the interacting shell model.

If this is right

  • The uncertainty in the NME is quantified at the level of 0.25 around the central value of 2.46.
  • Correlation analysis reveals structural dependencies among nuclear observables that can benchmark effective interactions.
  • This distribution provides a more reliable input for deriving limits on the effective Majorana neutrino mass from experimental half-life bounds.
  • Similar statistical protocols can be extended to other candidate isotopes for 0νββ decay.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the derived distribution holds, it suggests that shell-model calculations with this level of uncertainty can be compared across different isotopes to identify systematic trends.
  • Future refinements of effective interactions could be guided by minimizing discrepancies in the correlated observables identified here.
  • An independent ab initio calculation falling outside the 2.21-2.71 range would indicate missing physics in the fluctuation model.

Load-bearing premise

The assumption that systematic bounded fluctuations applied to the two-body matrix elements of an effective interaction previously used for 82Se, when propagated and averaged via Bayesian methods against low-energy spectroscopic data, fully capture the dominant theoretical uncertainty in the NME for 76Ge.

What would settle it

An independent calculation or future experimental constraint on the NME or related observables that yields a value outside the interval from 2.21 to 2.71 would falsify the reported probability distribution.

Figures

Figures reproduced from arXiv: 2605.21657 by Andrei Neacsu, Mihai Horoi.

Figure 1
Figure 1. Figure 1: The heatmap for all 12 observables when using the GCN2850 effective Hamiltonian. Very similar representations are [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Correlation matrix for observables that have correlation factor greater than 0.5, when using the GCN2850 Hamiltonian. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distributions based on experimental data (in red) compared with the KDE (in blue) obtained from the GCN2850 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Similar to Figure [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Similar to Figures 3 and 4, the distributions based on experimental data (in red) compared with the KDE (in blue) obtained from the JJ44b starting Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: PDFs of the 0νββ NME distributions for the JUN45, jj44b and GCN2850 Hamiltonians and their weighted sum (red curve, see text for details). The substantial overlap among all distributions con￾firms that shell-model calculations yield highly consistent 0νββ NME predictions in the jj44 space, irrespective of baseline interaction choice or minor optimization varia￾tions. This contrasts sharply with pn-QRPA res… view at source ↗
read the original abstract

The experimental pursuit of neutrinoless double-beta decay ($0\nu\beta\beta$) constitutes one of the most compelling avenues for probing lepton-number violation and exploring physics beyond the Standard Model. Within this landscape, $^{76}$Ge has consistently ranked among the most promising isotopes for current and next-generation bolometric and liquid-scintillator experiments, notably GERDA and LEGEND. In the present work, we adapt a rigorous statistical protocol previously established for $^{48}$Ca~\cite{Horoi-prc22} and $^{136}$Xe~\cite{Horoi-Xe-2023} to the $^{76}$Ge system, utilizing a valence configuration that aligns with our recent investigation of $^{82}$Se~\cite{Neacsu-Symmetry-2024}. Our methodology introduces systematic, bounded fluctuations to the two-body matrix elements of established effective interactions, subsequently monitoring how these perturbations propagate through a suite of low-energy nuclear observables. Special emphasis is placed on the $0\nu\beta\beta$ nuclear matrix element (NME), whose theoretical uncertainty currently dominates the interpretation of experimental half-life limits. By integrating these simulated variations into a Bayesian Model Averaging framework and benchmarking against empirical spectroscopic data, we derive a constrained probability distribution for the NME. The resulting analysis yields a central value of 2.46 with an associated standard deviation of 0.25, thereby quantifying the intrinsic theoretical spread within the interacting shell model approach. Furthermore, we perform a comprehensive correlation analysis across all computed observables to evaluate internal consistency, identify non-trivial structural dependencies, and establish benchmarks that may guide the refinement of future effective interactions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript adapts a Bayesian Model Averaging protocol, previously applied to 48Ca and 136Xe, to 76Ge. It introduces systematic bounded fluctuations to the two-body matrix elements of an effective interaction previously tuned for 82Se, propagates these through low-energy nuclear observables, and benchmarks against spectroscopic data to derive a probability distribution for the 0νββ nuclear matrix element with central value 2.46 and standard deviation 0.25.

Significance. If the fluctuations and averaging procedure are shown to capture the dominant sources of uncertainty, the result supplies a statistically grounded theoretical error bar on the NME that can be directly used to interpret half-life limits from GERDA and LEGEND. The accompanying correlation analysis across observables is a useful addition for identifying structural dependencies within the shell-model framework.

major comments (2)
  1. [§3.2] §3.2 (fluctuation protocol): the bounded variations are applied only to the TBMEs of the 82Se interaction while the valence-space truncation, single-particle energies, and short-range form of the 0νββ transition operator remain fixed; because the NME is sensitive to high-momentum components not constrained by low-energy spectroscopy, the quoted spread of 0.25 may not represent the full theoretical uncertainty.
  2. [§4] §4 (Bayesian averaging): the weighting of models by agreement with spectroscopic data assumes that the chosen low-energy observables sufficiently constrain the degrees of freedom relevant to the NME; no cross-validation against an independent set of observables or against results from a different model space is presented to test this assumption.
minor comments (2)
  1. [Eq. (7)] The definition of the fluctuation bounds (Eq. (7)) should be accompanied by a brief justification of why the chosen range is appropriate for 76Ge rather than simply inherited from the 82Se study.
  2. [Figure 5] Figure 5 (correlation matrix) would benefit from explicit labeling of which observables are spectroscopic and which are the NME itself to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below, proposing revisions to clarify the scope of our uncertainty quantification.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (fluctuation protocol): the bounded variations are applied only to the TBMEs of the 82Se interaction while the valence-space truncation, single-particle energies, and short-range form of the 0νββ transition operator remain fixed; because the NME is sensitive to high-momentum components not constrained by low-energy spectroscopy, the quoted spread of 0.25 may not represent the full theoretical uncertainty.

    Authors: We concur that the fluctuations are confined to the TBMEs of the 82Se interaction, with the valence-space truncation, single-particle energies, and the short-range form of the 0νββ operator held fixed. This protocol is intended to quantify the uncertainty stemming from variations in the effective two-body interaction while remaining consistent with the low-energy spectroscopic data used for benchmarking. Although the NME receives contributions from high-momentum components that are not fully constrained by low-energy observables, our approach provides a well-defined statistical measure of the spread within the interacting shell-model framework adopted here, in line with our earlier studies. We will revise §3.2 and the conclusions to explicitly delineate the boundaries of this uncertainty estimate and note that a more comprehensive assessment would require exploring additional model variations. revision: partial

  2. Referee: [§4] §4 (Bayesian averaging): the weighting of models by agreement with spectroscopic data assumes that the chosen low-energy observables sufficiently constrain the degrees of freedom relevant to the NME; no cross-validation against an independent set of observables or against results from a different model space is presented to test this assumption.

    Authors: The weighting in the Bayesian model averaging is based on the agreement with a standard set of low-energy spectroscopic observables, as successfully applied in our previous works on 48Ca and 136Xe. The correlation analysis presented in the manuscript demonstrates the internal consistency and highlights how variations in these observables relate to the NME. We recognize that an explicit cross-validation with a held-out set of observables or comparisons to calculations in alternative model spaces is not performed in this study. This would provide additional validation but lies beyond the present scope. We will add a brief discussion in §4 acknowledging this aspect of the methodology and referencing the supporting evidence from the correlation study. revision: partial

Circularity Check

1 steps flagged

Self-cited protocol and prior effective interaction yield NME distribution with moderate dependence on fitted inputs

specific steps
  1. self citation load bearing [Abstract]
    "we adapt a rigorous statistical protocol previously established for ^{48}Ca~[Horoi-prc22] and ^{136}Xe~[Horoi-Xe-2023] to the ^{76}Ge system, utilizing a valence configuration that aligns with our recent investigation of ^{82}Se~[Neacsu-Symmetry-2024]"

    The load-bearing methodology (systematic bounded TBME fluctuations + Bayesian Model Averaging against spectroscopic data) and the valence space are justified solely by citations to the authors' own prior papers rather than re-derived or independently validated in the present work; the final NME probability distribution therefore inherits its structure and bounds from those self-cited frameworks.

full rationale

The derivation applies Bayesian Model Averaging to bounded TBME fluctuations drawn from an effective interaction previously tuned for 82Se, with weights from low-energy spectroscopic data. The central NME value and spread (2.46 ± 0.25) are computed for 76Ge using this imported framework and chosen fluctuation bounds. While the specific numerical result for 76Ge is not identical to prior outputs by construction, the load-bearing statistical protocol, valence configuration, and interaction choice reduce to self-citations without re-derivation or external validation here. This creates moderate circularity but leaves independent computational content in the propagation and averaging steps.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Assessment limited to abstract; the central result rests on the transferability of the fluctuation protocol and the adequacy of the chosen valence space and effective interaction.

free parameters (1)
  • fluctuation bounds on two-body matrix elements
    The paper introduces systematic bounded fluctuations whose specific numerical ranges are not stated in the abstract but are required to generate the quoted NME distribution.
axioms (1)
  • domain assumption The valence configuration and effective interaction calibrated on 82Se are appropriate for 76Ge low-energy structure.
    Invoked when adapting the protocol from prior work without re-deriving the interaction.

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Reference graph

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