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arxiv: 2605.14813 · v2 · pith:TM3JYOSKnew · submitted 2026-05-14 · ⚛️ nucl-th

Skyrme SV density-functional analysis of the 2νββ decay in ⁷⁶Ge

Jan Mi\'skiewicz , Jakub Wysocki , Wojciech Satu{\l}a This is my paper

Pith reviewed 2026-05-25 05:48 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords 2νββ decay76GeSkyrme SV functionalnuclear matrix elementshape coexistencetriaxialityNo-Core Configuration-Interaction
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The pith

The 2νββ matrix element for 76Ge varies by a factor of five depending on g9/2 orbital occupancy and triaxiality of 76Se.

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

This paper computes the two-neutrino double beta decay nuclear matrix element of germanium-76 inside a no-core configuration-interaction model built on the Skyrme SV density functional. Three decay scenarios are distinguished by the fixed number of neutrons and protons placed in the g9/2 intruder orbital together with the triaxial deformation angle of the daughter selenium-76. The energetically lowest configuration gives |M^{2ν}| = 0.069(7) MeV^{-1}. In the higher-occupancy case the result splits further with the two coexisting minima, producing 0.040(4) MeV^{-1} or 0.22(2) MeV^{-1} according to the value of the triaxiality parameter γ. The spread accounts for the order-of-magnitude scatter seen across earlier calculations and shows that the matrix element is acutely sensitive to fine details of nuclear shape and configuration mixing.

Core claim

Within the Skyrme SV No-Core Configuration-Interaction framework the 2νββ matrix element |M^{2ν}| of 76Ge depends strongly on the conserved [n,m] occupancy of the 0g9/2 intruder orbital and on the triaxiality γ of coexisting minima in 76Se, giving |M^{2ν}| = 0.069(7) MeV^{-1} for the favored [4,2] occupancy, |M^{2ν}| = 0.040(4) MeV^{-1} at γ = 17.7° and |M^{2ν}| = 0.22(2) MeV^{-1} at γ = 41.9° for the [6,0] occupancy; the largest value agrees with the empirical result 0.204(14) MeV^{-1}.

What carries the argument

No-Core Configuration-Interaction framework based on the Skyrme SV density functional, with explicit tracking of conserved [n,m] occupancy of the 0g9/2 intruder orbital and triaxial shape coexistence parameterized by γ in 76Se.

If this is right

  • The matrix element can change by nearly an order of magnitude when the assumed occupancy or triaxiality is altered.
  • Precise extraction of |M^{2ν}| requires simultaneous control of shape coexistence and configuration mixing.
  • Structural complexity in the A=76 region explains why different energy-density-functional models produce widely scattered results.

Where Pith is reading between the lines

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

  • If the reported γ dependence is physical, a refined measurement of the quadrupole moments or transition rates that fix γ in 76Se would narrow the theoretical uncertainty on the decay rate.
  • The same occupancy-plus-triaxiality sensitivity may appear in other double-beta candidates near shell closures, suggesting a systematic re-examination of those matrix elements.
  • The calculations imply that future work must include explicit mixing between the two triaxial minima rather than treating them separately.

Load-bearing premise

The [n,m] occupancy of the 0g9/2 orbital stays fixed to high precision and the Skyrme SV functional together with configuration mixing already reproduces the triaxial minima in 76Se without extra tuning.

What would settle it

An independent experimental constraint on the g9/2 occupancy in 76Ge or a measured triaxiality angle γ in the two minima of 76Se that lies outside the calculated range would directly test whether the predicted matrix-element values remain valid.

Figures

Figures reproduced from arXiv: 2605.14813 by Jakub Wysocki, Jan Mi\'skiewicz, Wojciech Satu{\l}a.

Figure 1
Figure 1. Figure 1: FIG. 1: (Color online) Depiction of the Fermi golden [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (Color online) The [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (Color online) The lowest-lying mean-field [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (Color online) Shape coexistence in the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (Color online) Schematic illustration of the two [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Hence, in the following we shall consider two pos [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: (Color online) Stability of the [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: (Color online) Stability of the [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: (Color online) Results for the [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: (Color online) A summary of selected NME [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
read the original abstract

We present a theoretical study of the two-neutrino $0^+ \rightarrow 0^+$ double beta decay of $^{76}$Ge within the No-Core Configuration-Interaction framework based on the Skyrme SV density functional. We analyze three allowed decay scenarios distinguished by the $[n,m] \equiv [(\nu g_{9/2})^n, (\pi g_{9/2})^m]$ occupancy of the $0g_{9/2}$ intruder orbital, which remains conserved to high precision, as well as by the triaxiality of the daughter nucleus. The resulting $2\nu\beta\beta$ nuclear matrix element is found to depend strongly on the scenario. For the energetically favored $[4,2]$ occupancy, we obtain $|\mathcal{M}^{2\nu}| = 0.069(7)$~MeV$^{-1}$. For the $[6,0]$ occupancy, the matrix element further depends on the triaxiality parameter $\gamma$ of the two coexisting, closely lying minima in $^{76}$Se, yielding $|\mathcal{M}^{2\nu}| = 0.040(4)$~MeV$^{-1}$ at $\gamma = 17.7^\circ$ and $|\mathcal{M}^{2\nu}| = 0.22(2)$~MeV$^{-1}$ at $\gamma = 41.9^\circ$. The latter result is consistent with the empirical value reported by A. S. Barabash, $|\mathcal{M}^{2\nu}| = 0.204(14)$~MeV$^{-1}$, while the two former results are comparable to existing calculations based on energy-density-functional frameworks. Our calculations reveal challenges in the precise determination of the $|\mathcal{M}^{2\nu}|$ for the $^{76}$Ge decay. The structural complexity, triaxiality, and shape coexistence identified in the analyzed nuclei imply a strong sensitivity to fine details of the interaction and configuration mixing. This, in turn, explains the difficulties in theoretical modeling of the $|\mathcal{M}^{2\nu}|$ matrix elements for the $^{76}$Ge decay, which vary by almost an order of magnitude in the available literature.

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 computes the 2νββ nuclear matrix element |M^{2ν}| for ^{76}Ge in the No-Core Configuration-Interaction framework with the Skyrme SV density functional. It distinguishes three scenarios by the conserved [n,m] occupancy of the 0g_{9/2} intruder orbital and by the triaxiality parameter γ of coexisting minima in ^{76}Se, reporting |M^{2ν}| = 0.069(7) MeV^{-1} for the energetically favored [4,2] case, 0.040(4) MeV^{-1} for [6,0] at γ=17.7°, and 0.22(2) MeV^{-1} for [6,0] at γ=41.9°, with the last value matching the empirical result of Barabash while the others align with prior EDF calculations. The work concludes that structural complexity and shape coexistence cause strong sensitivity to interaction details and explain the order-of-magnitude spread in the literature.

Significance. If the scenario dependence and the identification of the physical configuration are robust, the result supplies a concrete illustration of how triaxial shape coexistence and configuration mixing control the matrix element, thereby accounting for the persistent theoretical scatter. The explicit use of a single Skyrme parametrization together with NCCI configuration mixing is a methodological strength that could be reproduced by other groups.

major comments (2)
  1. [Results section (discussion of [n,m] occupancies and γ dependence)] The central claim that |M^{2ν}| depends strongly on the [n,m] occupancy and on γ rests on the assertion that the occupancy 'remains conserved to high precision' and that the Skyrme SV functional correctly locates the two triaxial minima. No quantitative table or plot in the results section demonstrates the actual occupancy variation (e.g., deviation from integer [4,2] or [6,0] values) when the functional parameters or basis size are varied, nor is an energy ordering or E2-strength comparison to experiment provided to establish which γ minimum is realized. This validation is load-bearing for selecting the physically relevant scenario.
  2. [Numerical results and uncertainty discussion] The reported uncertainties (0.069(7), 0.040(4), 0.22(2)) are presented without an explicit error-propagation formula or sensitivity analysis. It is therefore unclear whether they reflect only statistical sampling within the NCCI space or also variations arising from the Skyrme SV parameters themselves; this directly affects the claim that the γ=41.9° result is consistent with experiment while the others are not.
minor comments (2)
  1. [Abstract and throughout] Notation for the matrix element alternates between |M^{2ν}| and |ℳ^{2ν}|; a single symbol should be used consistently.
  2. [Results] The abstract states that the [6,0] matrix element 'further depends on the triaxiality parameter γ', but the corresponding section would benefit from an explicit plot of |M^{2ν}| versus γ to illustrate the functional dependence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address each major comment below, agreeing where additional clarification or data is warranted and proposing targeted revisions.

read point-by-point responses

  1. Referee: [Results section (discussion of [n,m] occupancies and γ dependence)] The central claim that |M^{2ν}| depends strongly on the [n,m] occupancy and on γ rests on the assertion that the occupancy 'remains conserved to high precision' and that the Skyrme SV functional correctly locates the two triaxial minima. No quantitative table or plot in the results section demonstrates the actual occupancy variation (e.g., deviation from integer [4,2] or [6,0] values) when the functional parameters or basis size are varied, nor is an energy ordering or E2-strength comparison to experiment provided to establish which γ minimum is realized. This validation is load-bearing for selecting the physically relevant scenario.

    Authors: We agree that explicit quantitative support for occupancy conservation strengthens the central claim. In the revised manuscript we will add a table (or supplementary plot) in the results section documenting the computed g_{9/2} occupancies and their deviations from the nominal integer values [4,2] and [6,0] across the basis sizes employed. The energy ordering that identifies the [4,2] configuration as favored is already stated in the text; for the [6,0] case both triaxial minima are presented precisely to demonstrate the sensitivity to γ. An E2-strength comparison to experiment is not part of the present NCCI study, which focuses on the 2νββ matrix element; we will expand the discussion to make this scope limitation explicit while retaining the energy-based identification of scenarios. revision: partial

  2. Referee: [Numerical results and uncertainty discussion] The reported uncertainties (0.069(7), 0.040(4), 0.22(2)) are presented without an explicit error-propagation formula or sensitivity analysis. It is therefore unclear whether they reflect only statistical sampling within the NCCI space or also variations arising from the Skyrme SV parameters themselves; this directly affects the claim that the γ=41.9° result is consistent with experiment while the others are not.

    Authors: The quoted uncertainties originate exclusively from statistical sampling and convergence diagnostics within the NCCI model space for the fixed Skyrme SV parametrization; no variation over Skyrme parameters is performed or included. We will add an explicit paragraph (and, where appropriate, the underlying formula) describing how these uncertainties are obtained, thereby clarifying that they do not incorporate functional-parameter sensitivity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results are direct computations

full rationale

The paper applies the No-Core Configuration-Interaction method with the Skyrme SV functional to compute |M^{2ν}| for selected [n,m] occupancies and γ values in 76Se. These matrix elements are numerical outputs of the framework rather than quantities fitted to the decay data or reduced by definition to the inputs. The abstract states that occupancies 'remain conserved to high precision' as an observed feature of the calculations, and the functional is an external standard input. No equations, self-citations, or ansatzes are shown that would make any reported value equivalent to its inputs by construction. This is a standard application of an existing EDF+NCCI approach to a new observable, qualifying as self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central results rest on the Skyrme SV functional parameters taken from prior literature, the assumption of high-precision conservation of g9/2 occupancy, and the modeling of triaxial minima without additional validation data. No new entities are introduced.

free parameters (2)
  • Skyrme SV parameters
    Taken from earlier global fits to nuclear data; not re-derived here.
  • Triaxiality angle γ
    Values 17.7° and 41.9° are reported as outputs but the underlying energy minimization depends on the functional and configuration space.
axioms (2)
  • domain assumption Occupancy of 0g9/2 orbital is conserved to high precision in the three scenarios
    Stated in abstract as basis for distinguishing decay scenarios.
  • domain assumption No-Core Configuration-Interaction with Skyrme SV captures shape coexistence in 76Se
    Implicit in the choice of framework and reported minima.

pith-pipeline@v0.9.0 · 5947 in / 1665 out tokens · 23065 ms · 2026-05-25T05:48:11.376503+00:00 · methodology

discussion (0)

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Reference graph

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