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arxiv: 2606.25486 · v1 · pith:DGO3NJ4Inew · submitted 2026-06-24 · ⚛️ nucl-th · hep-ex· nucl-ex

The renormalization of the shell-model neutrinoless double-beta decay operator starting from effective field theory (I)

L. Coraggio , G. De Gregorio , S. L. Lyu , N. Itaco This is my paper

Pith reviewed 2026-06-25 19:52 UTC · model grok-4.3

classification ⚛️ nucl-th hep-exnucl-ex
keywords neutrinoless double-beta decayshell modelchiral effective field theorymany-body perturbation theorynuclear matrix elements48Ca76Ge82Se
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The pith

The neutrinoless double-beta decay operator is renormalized consistently with the shell-model Hamiltonian using chiral perturbation theory.

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

The authors derive both the effective shell-model Hamiltonian and the transition operators, including those for neutrinoless double-beta decay, from chiral effective field theory by means of many-body perturbation theory. This produces a fully consistent framework applied for the first time to the nuclei 48Ca, 76Ge, and 82Se. Spectroscopic properties, electromagnetic matrix elements, and neutrinoless double-beta decay matrix elements are then computed within the same approach. A dedicated study of perturbative convergence supplies the basis for estimating theoretical uncertainty in the results.

Core claim

The effective shell-model Hamiltonian and all transition operators have been constructed by way of the many-body perturbation theory, and then employed to calculate both spectroscopic properties of the nuclei involved in the decays under our consideration - namely 48Ca, 76Ge, and 82Se -, as well as the nuclear matrix elements of the electromagnetic and neutrinoless double-beta decays.

What carries the argument

Renormalization of the neutrinoless double-beta decay operator through many-body perturbation theory starting from chiral effective field theory.

If this is right

  • Spectroscopic properties of 48Ca, 76Ge, and 82Se can be obtained within the same consistent framework.
  • Nuclear matrix elements for electromagnetic transitions are computed with operators derived identically to the Hamiltonian.
  • Nuclear matrix elements for neutrinoless double-beta decay are obtained after explicit renormalization of the operator.
  • Convergence behavior with perturbative order provides a quantitative handle on theoretical uncertainty.

Where Pith is reading between the lines

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

  • The same construction could be applied to additional candidate nuclei to test consistency across the chart.
  • Extension to three-body forces or higher chiral orders would test how sensitive the matrix elements remain to the input EFT truncation.
  • Direct comparison of these matrix elements with those from other many-body methods could isolate the effect of the consistent operator renormalization.

Load-bearing premise

Many-body perturbation theory converges sufficiently to produce accurate effective Hamiltonians and transition operators for the nuclei 48Ca, 76Ge, and 82Se at the chosen chiral order and cutoff.

What would settle it

A higher-order many-body perturbation theory calculation or a change in cutoff that shifts the neutrinoless double-beta decay matrix elements by more than the estimated uncertainty would show the current results are not yet converged.

Figures

Figures reproduced from arXiv: 2606.25486 by G. De Gregorio, L. Coraggio, N. Itaco, S. L. Lyu.

Figure 1
Figure 1. Figure 1: Experimental and calculated spectra of 48Ca and 48Ti. B(E2) strengths (in e 2 fm4 ) are also reported (see text for details). two-valence nuclei, was reported in Ref. [30], and here we want to extend it to many-valence nucleon systems. It should be noticed that in the present study we have employed the same effective E2 transition operator, the one derived at third order in perturbation theory, to cal￾cula… view at source ↗
Figure 2
Figure 2. Figure 2: Same as in Fig. 1, but for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as in Fig. 1, but for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: M0ν for the decay of the 48Ca ground state to the 48Ti one, as a function of the perturbative order. The green triangles correspond to M0ν F , the blue squares to M0ν GT, the purple diamonds to M0ν T , the brown lower triangles to M0ν S , and the black dots to the full M0ν . A more refined analysis of our results and of the per￾turbative behavior may be obtained performing a decom￾position of M0ν ’s in ter… view at source ↗
Figure 4
Figure 4. Figure 4: M0ν for the 48Ca →48Ti decay as a function of Nmax This convergent behavior appears also for the M0ν ’s calculated for the 76Ge and 82Se decays, as re￾ported in Table II. Table II. M0ν ’s for 76Ge and 82Se 0νββ decay between their ground states, calculated for Nmax = 14, 16, and 18. Decay Nmax = 14 Nmax = 16 Nmax = 18 76Ge→76Se 1.698 1.658 1.661 82Se→82Kr 1.284 1.250 1.253 Now, we shift the focus on the re… view at source ↗
Figure 6
Figure 6. Figure 6: Contributions from pairs of decaying neutrons with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as in Fig. 5, but for the decay of the [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Same as in Fig. 6, but for the decay of the [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
read the original abstract

In this work, we approach for the first time the task to perform a shell-model calculation of the matrix element for the neutrinoless double-beta decay, within a fully-consistent framework where the expressions of the nuclear Hamiltonian and of the decay operators have been derived through chiral perturbation theory. More precisely, the effective shell-model Hamiltonian and all transition operators have been constructed by way of the many-body perturbation theory, and then employed to calculate both spectroscopic properties of the nuclei involved in the decays under our consideration - namely 48Ca, 76Ge, and 82Se -, as well as the nuclear matrix elements of the electromagnetic and neutrinoless double-beta decays. We also present a study of the convergence properties of the calculated matrix elements in order to provide the elements for an estimate of the theoretical uncertainty.

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

1 major / 1 minor

Summary. The manuscript develops a consistent framework for shell-model calculations of neutrinoless double-beta decay matrix elements by deriving both the effective Hamiltonian and all transition operators from chiral effective field theory via many-body perturbation theory. It applies the method to 48Ca, 76Ge, and 82Se, computes spectroscopic properties together with electromagnetic and 0νββ matrix elements, and presents a convergence study to support an uncertainty estimate.

Significance. If the MBPT truncation errors for the two- and three-body operators prove small, the work supplies the first fully EFT-derived, parameter-free effective operators for 0νββ shell-model calculations, allowing systematic uncertainty quantification that is currently unavailable in phenomenological approaches.

major comments (1)
  1. [Convergence study section] The central claim that the framework is fully consistent and that the convergence study supplies a reliable uncertainty estimate rests on the unverified assumption that MBPT truncation errors remain small for the 0νββ operators at the chosen chiral order and cutoff. No order-by-order comparison, coupled-cluster benchmark, or exact-diagonalization cross-check for the transition operators in 48Ca, 76Ge, or 82Se is described that would quantify residual MBPT error.
minor comments (1)
  1. [Abstract] The abstract asserts the calculation is performed 'for the first time' in a fully consistent framework but does not cite the specific prior shell-model or EFT works whose consistency is being improved upon.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance. We address the single major comment below.

read point-by-point responses

  1. Referee: [Convergence study section] The central claim that the framework is fully consistent and that the convergence study supplies a reliable uncertainty estimate rests on the unverified assumption that MBPT truncation errors remain small for the 0νββ operators at the chosen chiral order and cutoff. No order-by-order comparison, coupled-cluster benchmark, or exact-diagonalization cross-check for the transition operators in 48Ca, 76Ge, or 82Se is described that would quantify residual MBPT error.

    Authors: We agree that direct, operator-specific benchmarks (order-by-order comparisons, coupled-cluster cross-checks, or exact diagonalization in 48Ca) would provide stronger validation of the MBPT truncation error for the 0νββ operators themselves. Our convergence analysis is performed on the final nuclear matrix elements, which incorporate both the effective Hamiltonian and the transition operators derived at the same chiral order and cutoff via MBPT. This study examines the stability of the matrix elements under changes in the MBPT truncation level and other parameters, thereby supplying quantitative elements for an uncertainty estimate as stated in the abstract. We do not claim that the present analysis fully quantifies the residual MBPT error on the operators in isolation. In the revised manuscript we will add an explicit paragraph in the convergence section acknowledging this limitation and clarifying that the reported uncertainty estimate is based on the observed stability of the matrix elements rather than on separate operator benchmarks. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation starts from chiral EFT + MBPT inputs

full rationale

The paper constructs the effective shell-model Hamiltonian and all transition operators via many-body perturbation theory applied to chiral EFT expressions, then computes spectroscopic properties and 0νββ matrix elements for 48Ca, 76Ge, and 82Se while studying convergence. No quoted step reduces the target matrix element to a fitted parameter by construction, invokes a self-citation as the sole justification for uniqueness, or renames a known result; the central claim remains independent of the final numerical output.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no specific free parameters, new entities, or detailed axioms beyond the general reliance on chiral EFT and MBPT are identifiable.

axioms (1)
  • domain assumption Chiral effective field theory supplies a valid low-energy description of nuclear forces and currents
    The entire construction begins from chiral perturbation theory as stated in the abstract.

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