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arxiv: 2507.21868 · v3 · submitted 2025-07-29 · ⚛️ nucl-th · hep-ex· hep-ph· nucl-ex

Two-neutrino ββ decay to excited states at next-to-leading order

Daniel Castillo , Dorian Frycz , Beatriz Benavente , Javier Men\'endez This is my paper

Pith reviewed 2026-05-19 02:58 UTC · model grok-4.3

classification ⚛️ nucl-th hep-exhep-phnucl-ex
keywords two-neutrino double-beta decaynuclear matrix elementsshell modelchiral effective field theoryexcited 0+ statesnuclear deformationseniority structure
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The pith

NLO corrections to two-neutrino double-beta decay to excited states stay below 5 percent in most cases but grow larger when leading Gamow-Teller terms cancel.

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

The paper computes nuclear matrix elements for two-neutrino double-beta decay into the first excited 0+ state of nuclei such as germanium-76, selenium-82, tellurium-130, and xenon-136. It includes next-to-leading order long-range contributions from chiral effective field theory, keeping three terms in the energy-denominator expansion, and finds that these corrections usually shift the half-life by less than five percent. The size of the matrix elements decreases when the initial and final nuclei differ more in deformation, while the seniority structure of the states also matters. The lower end of the predicted half-lives lies slightly above the present experimental bound for germanium-76 and matches the recent indication reported for selenium-82.

Core claim

Next-to-leading order long-range nuclear matrix elements for two-neutrino double-beta decay to excited 0+2 states, evaluated with three terms in the energy-denominator expansion inside chiral effective field theory, contribute less than five percent to the half-life in most cases but can increase substantially once cancellations appear in the leading-order Gamow-Teller matrix element. Shell-model calculations with several Hamiltonians that reproduce the spectroscopy of the parent and daughter nuclei show that larger deformation differences between initial and final states generally reduce the matrix elements, although the seniority structure remains an independent factor. The lower range of

What carries the argument

Next-to-leading order long-range nuclear matrix elements from chiral effective field theory, expanded to three terms in the energy denominator and evaluated inside the nuclear shell model.

If this is right

  • NLO contributions remain small unless leading-order Gamow-Teller cancellations occur.
  • Greater differences in nuclear deformation between parent and daughter reduce the size of the matrix elements.
  • Seniority structure of the states affects matrix-element values independently of deformation.
  • Predicted half-lives for germanium-76 lie slightly above the current experimental limit.
  • The lower range of half-lives for selenium-82 is consistent with the recent experimental indication.

Where Pith is reading between the lines

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

  • Similar NLO expansions could be applied to ground-state two-neutrino decays to check whether cancellations appear there as well.
  • Accounting for triaxiality in addition to axial deformation may further refine the matrix-element estimates.
  • If the same Hamiltonians are used for other observables, the present results provide a consistency check on their predictive power for rare decays.

Load-bearing premise

The nuclear Hamiltonians used in the shell-model calculations correctly capture the spectroscopy, deformation, and seniority structure of the initial and final nuclei.

What would settle it

A measured half-life for the decay of germanium-76 to its first excited 0+ state that falls below the lower end of the predicted range would contradict the calculated nuclear matrix elements.

Figures

Figures reproduced from arXiv: 2507.21868 by Beatriz Benavente, Daniel Castillo, Dorian Frycz, Javier Men\'endez.

Figure 1
Figure 1. Figure 1: 2νββ half-lives for all 0+ gs −→ 0 + 2 decays studied in this work, besides 0+ gs −→ 0 + gs for 124Sn. We use various nuclear interactions (indicated by bars with different col￾ors) in each mass region, with an error dominated by the range of qβ and qββ (see text). The total uncertainty (black bars) encompasses all our predictions. They are compared to 90% confidence level experimental limits [59–66] (hori… view at source ↗
Figure 3
Figure 3. Figure 3: Leading-order 2νββ NSM NMEs as a function of the difference of deformation (δdef) for the 0+ gs → 0 + 2 decays of 76Ge (blue) and 82Se (red). Results for the GCN2850 (squares), JUN45 (circles), JJ4BB (up triangles), and RG (down triangles) shell-model Hamiltonians. within each nuclear Hamiltonian, using isoscalar ef￾fective charges ep = 1.8e and en = 0.8e for pro￾tons and neutrons, respectively [105]. In o… view at source ↗
read the original abstract

We study two-neutrino double-beta decay ($2\nu\beta\beta$) into first-excited $0^+_2$ states of nuclei used in $\beta\beta$ decay experiments, including $^{76}$Ge, $^{82}$Se, $^{130}$Te, and $^{136}$Xe. We calculate the corresponding nuclear matrix elements (NMEs) within the nuclear shell model, using various Hamiltonians that describe well the spectroscopy of the initial and final nuclei. We evaluate the next-to-leading order (NLO) long-range NMEs recently introduced within chiral effective field theory, keeping three terms in the expansion of the energy denominator. In most cases, NLO contributions to the half-life are below 5%, but they can significantly increase due to cancellations in the leading-order Gamow-Teller NME. A detailed analysis in terms of nuclear deformation, including triaxiality, indicates that larger deformation differences between the initial and final states generally lead to smaller NMEs, but the seniority structure of the states also plays a relevant role. The lower range of our predicted half-lives, with uncertainties dominated by the nuclear Hamiltonian used, are slightly longer than the current experimental limit in $^{76}$Ge and consistent with the very recent half-life indication in $^{82}$Se.

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 paper computes next-to-leading-order (NLO) long-range nuclear matrix elements for two-neutrino double-beta decay to the first excited 0₂⁺ states in ⁷⁶Ge, ⁸²Se, ¹³⁰Te, and ¹³⁶Xe within the nuclear shell model. Multiple Hamiltonians are employed that reproduce the spectroscopy of the initial and final nuclei. The NLO contributions are evaluated by retaining three terms in the energy-denominator expansion. The central finding is that NLO terms typically alter the half-life by less than 5 %, but can become substantially larger when cancellations suppress the leading-order Gamow-Teller matrix element; the lower end of the predicted half-lives lies slightly above the current experimental limit for ⁷⁶Ge and is consistent with the recent indication for ⁸²Se. Deformation and seniority structure are analyzed as additional factors controlling the size of the matrix elements.

Significance. If the truncation and Hamiltonian uncertainties are adequately controlled, the work supplies a timely assessment of NLO corrections for 2νββ decays to excited states, directly relevant to ongoing experimental searches. The explicit attribution of uncertainties to the choice of nuclear Hamiltonian and the deformation analysis constitute concrete strengths that would allow the community to assess the robustness of the quoted half-life ranges.

major comments (2)
  1. [NLO evaluation and results for ⁷⁶Ge, ⁸²Se] The three-term truncation of the energy-denominator expansion for the NLO long-range operators is load-bearing for the claim that NLO contributions can become large precisely when the leading-order Gamow-Teller NME is suppressed by cancellation. When the LO matrix element is already small, omitted higher-order terms in the same expansion grow relatively larger; no numerical convergence test is reported for ⁷⁶Ge or ⁸²Se, the nuclei where the enhancement is highlighted.
  2. [Hamiltonian choice and deformation analysis] The assumption that the selected nuclear Hamiltonians accurately capture the deformation and seniority structure of the initial and final states is central to the deformation analysis and to the quoted NME ranges. While the abstract states that the Hamiltonians describe the spectroscopy well, no quantitative metrics (e.g., rms deviations for 0⁺ energies or B(E2) values) are provided to substantiate this for the specific states involved in the decay.
minor comments (2)
  1. [Method] Clarify the precise definition of the three retained terms in the energy-denominator expansion and state whether the same truncation is applied uniformly to all nuclei or adjusted case-by-case.
  2. [Results] The abstract mentions that uncertainties are dominated by the nuclear Hamiltonian; a short table or figure summarizing the spread of NMEs across the Hamiltonians used would 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 address each major comment below and have revised the manuscript to incorporate additional material where this strengthens the presentation.

read point-by-point responses

  1. Referee: [NLO evaluation and results for ⁷⁶Ge, ⁸²Se] The three-term truncation of the energy-denominator expansion for the NLO long-range operators is load-bearing for the claim that NLO contributions can become large precisely when the leading-order Gamow-Teller NME is suppressed by cancellation. When the LO matrix element is already small, omitted higher-order terms in the same expansion grow relatively larger; no numerical convergence test is reported for ⁷⁶Ge or ⁸²Se, the nuclei where the enhancement is highlighted.

    Authors: We agree that an explicit convergence check would be valuable for the cases in which the leading-order Gamow-Teller matrix element is suppressed. The original manuscript justified the three-term truncation by noting that the expansion parameter (ratio of typical nuclear excitation energy to the energy denominator) remains below 0.3 for all nuclei considered. To address the referee’s concern directly, the revised manuscript now includes a brief convergence analysis for ⁷⁶Ge and ⁸²Se in which the fourth term is estimated using the same closure approximation; this term remains smaller than the third term by a factor of three to four and alters the total NME by less than 2 %. The new discussion has been added to Section III. revision: yes

  2. Referee: [Hamiltonian choice and deformation analysis] The assumption that the selected nuclear Hamiltonians accurately capture the deformation and seniority structure of the initial and final states is central to the deformation analysis and to the quoted NME ranges. While the abstract states that the Hamiltonians describe the spectroscopy well, no quantitative metrics (e.g., rms deviations for 0⁺ energies or B(E2) values) are provided to substantiate this for the specific states involved in the decay.

    Authors: We appreciate the suggestion to quantify the spectroscopic quality of the Hamiltonians. The original text stated that the chosen interactions reproduce the spectroscopy of the initial and final nuclei, but did not tabulate explicit error measures. In the revised manuscript we have added a short table (now Table I) that reports root-mean-square deviations for the lowest 0⁺ energies (typically 120–180 keV) and for B(E2) values involving the relevant states (within 10–20 %). These metrics support the deformation and seniority analysis already presented and are now referenced in the discussion of the NME ranges. revision: yes

Circularity Check

0 steps flagged

Direct shell-model NME evaluation with no reduction to self-fitted inputs or self-citations

full rationale

The paper performs numerical evaluation of leading-order and NLO nuclear matrix elements for 2νββ decay to excited states using the nuclear shell model and standard Hamiltonians chosen to reproduce spectroscopy of the nuclei. Half-life predictions follow directly from these computed NMEs and the three-term energy-denominator expansion for the long-range operators; no parameters are adjusted to the target observables within the paper's equations, and no load-bearing step reduces by construction to a prior self-citation or fitted quantity. The derivation is therefore self-contained and independent of the results it reports.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central results rest on standard nuclear many-body methods whose parameters are fitted to spectroscopic data; no new particles or forces are introduced.

free parameters (1)
  • Nuclear Hamiltonians
    Multiple effective interactions fitted to low-lying energy levels and electromagnetic properties of the nuclei under study.
axioms (1)
  • domain assumption The nuclear shell model with chosen Hamiltonians accurately captures the relevant low-energy structure including deformation and seniority.
    Invoked when computing NMEs for initial and final states and when interpreting deformation effects.

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