Neutrinoless double beta decay rates and the $3 + 2$ scenario

Vaisakh Plakkot

REVIEW 3 major objections 5 minor 18 references

Neutrinoless double beta decay rates and the $3 + 2$ scenario

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The standard formula for neutrinoless double beta decay rates may be incomplete, and a 3+2 model with two sterile neutrinos becomes fully testable for inverted mass ordering.

desk verdict A transparent proceedings summary of the author's own prior papers; the physics is in refs [6] and [11], not in this document. read the letter →

arxiv 2505.17750 v1 pith:IRESJLSX submitted 2025-05-23 hep-ph

Vaisakh Plakkot This is my paper · ORCID

classification hep-ph

keywords neutrinolessdoublebetadecayMajorananeutrinossterileheavyneutralleptonsseesawmechanismleptogenesisinvertedmassorderingchiraleffectivefieldtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the standard mass-dependent amplitude used in neutrinoless double $\beta$ decay rates, $A(m_i) = -M(0)\langle p^2\rangle/(\langle p^2\rangle + m_i^2)$, is an oversimplification for light-to-medium neutrinos. By splitting the virtual neutrino momentum into hard, soft, potential, and ultrasoft regions and matching effective field theories at the boundaries, it obtains a parametrisation that can differ significantly from the standard one. These differences change predicted half-lives and therefore matter for interpreting the next generation of experiments. The paper then applies the amplitude to a $3+2$ model with two sterile neutrinos and argues that for inverted neutrino mass ordering, a combination of $0\nu\beta\beta$, collider, and leptogenesis constraints will allow the entire allowed parameter space to be probed.

What carries the argument

The central object is the mass-dependent amplitude $A(m_i)$, and the key move is to compute it region by region—hard, soft, potential, and ultrasoft—using the relevant effective field theory in each region and matching at the boundaries, as summarised in eq. (4). The resulting practical parametrisation replaces the single interpolation formula for masses below 100 MeV, where the ultrasoft piece $A^{(\rm us)}$ and the corrected potential piece $A^{(p,<)}$ matter, while heavier masses are handled separately. For the $3+2$ model, the analysis is driven by the effective amplitude $A_{\rm eff}$ in eq. (7), whose cancellation-and-enhancement structure controls the constraints in the $M$–$U_e^2$ plane.

What would settle it

Compute $A(m_i)$ for a neutrino mass near the pion scale with an independent method, such as lattice QCD or a different nuclear many-body framework, and compare it with both the standard formula $-M(0)\langle p^2\rangle/(\langle p^2\rangle+m_i^2)$ and the decomposition of eq. (4); a clear discrepancy would settle whether the claimed difference is real. Observationally, if next-generation inverted-ordering experiments reach their projected limits and observe nothing, the required cancellation in eq. (7) forces lower bounds on $U_e^2$ that can be confirmed or excluded by future collider and fixed-target searches.

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Extended reading notes

Core claim

The central claim is that the mass-dependent amplitude $A(m_i)$ for neutrinoless double $\beta$ decay depends on the neutrino mass region, and that the standard interpolation in eq. (3) misses the ultrasoft contribution plus the double-counting subtraction in the potential region. As a result, the effective amplitude $A_{\rm eff}=\sum_i U_{ei}^2 m_i A(m_i)$ can differ substantially from the standard prescription in the light-to-medium mass range, changing lifetime predictions. In the $3+2$ model with two nearly degenerate heavy neutral leptons, the decay amplitude contains an interference term with a free phase $\lambda$ that can cancel against the light-neutrino contribution, and once next-generation $0\nu\beta\beta$ limits are combined with leptogenesis and collider or cosmological bounds, the entire inverted-ordering parameter space becomes testable.

Load-bearing premise

The numerical projection collapses if the underlying chiral effective field theory calculation in reference [6], especially the ultrasoft region and the double-counting subtraction, is not correct, because this paper does not reproduce that calculation and uses it for the size of the amplitude differences; the $3+2$ testability result also assumes that this minimal model alone explains the baryon asymmetry and that future experiments reach their projected sensitivities.

Editorial extensions

If this is right

Half-life predictions for isotopes can deviate from the standard formula by a substantial factor in the light-to-medium neutrino mass range, so interpreting $0\nu\beta\beta$ limits purely in terms of $m_{\beta\beta}$ may need revision.
In the $3+2$ model with inverted ordering, if the next generation of $0\nu\beta\beta$ experiments sees nothing, the required cancellation forces a floor on $U_e^2$ that collider and fixed-target searches can check.
Combining $0\nu\beta\beta$ limits with the requirement that leptogenesis explains the full baryon asymmetry removes large chunks of the $M$–$U_e^2$ plane.
For normal mass ordering, the same combined constraints still leave an uncovered region that needs further experimental progress.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The region-by-region logic should transfer to other lepton-number-violating observables that integrate over virtual neutrino momenta, such as rare meson or tau decays, where the ultrasoft contribution may also be missing from standard treatments.
If the new amplitude is correct, the effective Majorana mass extracted from a single isotope will not be isotope-independent; comparing $0\nu\beta\beta$ limits across different nuclei could expose the predicted mass dependence.
The free phase $\lambda$ in the $3+2$ amplitude means $0\nu\beta\beta$ can be suppressed even with sizable heavy-neutrino mixing, so an observed signal would not automatically translate into a simple mass measurement.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a workshop proceedings paper, not a research article. It summarizes two of the author's own earlier works: the 0νββ amplitude calculation in ref [6], published in JHEP 2024, and the 3+2 model testability analysis in ref [11], a preprint. The actual physics lives in those references. If you want to evaluate the claims, you need to open them. What the paper does well is give a compact, readable overview of a potentially important point: the standard interpolating formula for the 0νββ amplitude, eq. (3), can fail for light-to-medium mass neutrinos, and a proper momentum-region EFT treatment changes predicted lifetimes by factors to orders of magnitude. The region decomposition in eq. (4) is clear, and the warning about double counting is explicit. The 3+2 section also shows how combining 0νββ with collider and leptogenesis constraints can cover the entire inverted-ordering parameter space. The soft spots are the flip side. There is no derivation here. Eq. (4) is asserted; the exact forms, including the crucial double-counting-subtracted term A^(p,<), are in ref [6]. A reader of this proceedings alone cannot check the boundary matching or scheme independence of that subtraction. That is normal for a proceedings, but it means the central result is not independently verifiable from this text. The 3+2 testability figure is taken from ref [11], which is an unreviewed preprint, so that part of the paper inherits the preprint's status. The typos 'Mev' and 'Gev' are cosmetic. Overall, the summary looks consistent with the cited papers and the physics is plausible. I don't see a load-bearing flaw, just a self-imposed limitation. The paper is for people who want a 10-minute map before going to the source. I would not cite it in my own work; I would cite the originals. For peer review, if this is destined for the proceedings, accept after a quick factual check. If it is being evaluated as a journal research article, it should not be accepted as a new contribution; the results are already in the literature. A serious referee is still warranted, mainly to check that the proceedings accurately represents refs [6] and [11] and does not oversell them.

Referee Report

3 major / 5 minor

Summary. This proceedings contribution summarizes recent work on neutrinoless double beta decay (0νββ) amplitudes and the testability of a 3+2 model with two sterile neutrinos. The paper claims that the standard mass-dependent parametrization A(m_i) = -M(0)<p^2>/(<p^2>+m_i^2) misses ultrasoft and double-counting corrections, and that a proper momentum-region decomposition, Eq. (4), yields significantly different amplitudes for light-to-medium neutrino masses, thereby altering lifetime predictions. For the 3+2 model, the paper claims that combining 0νββ limits, collider searches, and the requirement of successful baryogenesis can probe the entire allowed parameter space for the inverted mass ordering. The amplitude results are taken from ref. [6] and the 3+2 analysis from ref. [11]; the manuscript explicitly refers to these works for exact forms and figures.

Significance. If the amplitude differences advertised in Eq. (4) and Figs. 1–2 are correct, they would warrant a revision of standard 0νββ calculations for sterile-neutrino contributions and affect the inference of HNL parameters from current and future bounds. The claim that the 3+2 model becomes fully testable in the near future for the inverted ordering is also of considerable phenomenological interest. However, since both central results are imported from refs. [6] and [11] without derivation or independent numerical analysis in this manuscript, the significance for the proceedings is primarily that of a summary; the reader must consult the companion papers to assess the underlying assumptions.

major comments (3)

[§2.2, Eq. (4), Figs. 1–2] The central claim that the EFT-based amplitude in Eq. (4) differs significantly from the standard parametrization Eq. (3) relies on the ultrasoft amplitude A^(us) and the double-counting-subtracted potential-region amplitude A^(p,<). Neither object is defined in this paper; the reader is merely directed to ref. [6] for the 'exact form of the amplitudes' and parameter values. As a consequence, the manuscript does not provide the means to verify the matching between the branches at 100 MeV and 2 GeV, the behaviour A(m_i)→A(0) as m_i→0, or the scheme dependence of the subtraction. Because the difference shown in Figs. 1 and 2 is precisely the advertised effect, this missing information is load-bearing and should either be reproduced in an appendix or the claims should be explicitly framed as results imported from ref. [6].
[§3, Eq. (7)] Equation (7) is introduced as 'the relevant combination for 0νββ' without derivation. The paper does not explain how the expansion in 1/M and μ is obtained, why the light-neutrino contribution is proportional to A(0)-A(M), or why the heavy-neutrino contribution has the form e^{iλ} μ U_e^2 M/2 A'(M). These ingredients are essential for the subsequent argument that the HNL contribution can cancel the light-neutrino contribution. Without a derivation or a precise reference to the calculation in ref. [11], the reader cannot assess the validity of the cancellation mechanism.
[§3.1, Fig. 3] The conclusion that the entire allowed parameter space of the 3+2 model for inverted ordering will be probed by next-generation experiments is based entirely on Fig. 3, which is 'taken from ref. [11]'. The assumptions entering this plot (future 0νββ sensitivity, collider reach, cosmological bounds, and the leptogenesis calculation) are not summarized in the text. Since this is one of the two main conclusions of the paper, the manuscript should at least state these assumptions explicitly or provide a self-contained version of the analysis.

minor comments (5)

[§2.2, Eq. (4)] In Eq. (4), '100 Mev' should read '100 MeV'.
[§2.2, Eq. (4)] The notation 'A(9)' in Eq. (4) is inconsistent with the text; it should be written as A^(9) or A_9 to match the superscript notation used for the other amplitudes.
[§2.1, Eqs. (2)–(3)] The relation between M(0) in Eq. (3) and the components of A(0) in Eq. (2) is not stated; using the same symbol M for a generic nuclear matrix element and for the specific combination in Eq. (2) may confuse readers.
[Abstract] The phrase 'mass-dependent matrix elements' is imprecise; the mass dependence in Eq. (1) enters through the amplitude A(m_i), not the matrix elements themselves.
[Abstract] The abstract states that 'a fresh look at the different momentum regions leads us to an effective practical parametrisation', but the derivation is not present in this manuscript. Consider rewording to indicate that this is a summary of work presented in ref. [6].

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified: the paper summarizes previously published EFT results; no target quantity is defined in terms of the conclusion, and no fit is relabeled as a prediction.

full rationale

This manuscript is a proceedings summary. Its two central results—the momentum-region decomposition of A(m_i) and the 3+2 testability map—are explicitly imported from refs. [6] and [11], both of which list the author as a co-author. For example, after Eq. (4) the text states 'The exact form of the amplitudes, and the values of parameters involved, can be found in ref. [6],' and Fig. 3 is 'Taken from ref. [11].' That is self-citation, and it makes the manuscript uninformative about the actual derivation, but it is not circular in the sense required here: the paper nowhere defines A^(p), A^(p,<), A^(us), or the 3+2 effective amplitude in terms of the conclusions it draws, and no fitted parameter from this paper is later relabeled as a prediction. The EFT calculation in ref. [6] is a separate published computation with stated assumptions; it is not shown to be equivalent to Eq. (3) by construction. Likewise Eq. (7) is a Taylor-expanded seesaw identity from ref. [11], not an input defined by the testability claim. Without access to the cited papers, no equation-level reduction (Eq. X = Eq. Y by construction) can be exhibited. Concerns about the numerical validity of A^(p,<) are correctness/verifiability concerns, not circularity. Therefore the circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The paper introduces these model parameters to define the 3+2 seesaw scenario and to produce the testability plot. The physical axioms are standard BSM assumptions, while the key load-bearing assumption for the amplitude claim is that the author's own ref. [6] is correct. The two sterile neutrinos are borrowed from the well-known seesaw literature, not invented here, but they carry the model's experimental predictions.

free parameters (4)

Average Majorana mass M
Defines the scale of the two heavy right-handed neutrinos in eq. (6). The testability projection in fig. 3 scans over M; the allowed region and its reachable bounds depend on this parameter.
Degeneracy-breaking parameter mu
Splits the two heavy masses in eq. (6). The paper states that the unconstrained region will push mu to values resolvable at experiments, so the completeness of the test depends on the mu range considered.
Electron-flavor mixing U_e^2
Defined as the sum of squared HNL mixings with the electron flavour. It is scanned in fig. 3 and constrained by 0nu beta beta, collider searches, and cosmology; the predicted lower bounds depend on it.
Free phase lambda
Appears in eq. (7) and is described as 'completely free'. It controls whether the HNL term interferes constructively or destructively with the light neutrino contribution, so the resulting limits on U_e^2 depend on its value.

assumptions (5)

domain assumption Neutrinos are Majorana particles and the Weinberg operator is the leading source of neutrino mass.
The whole 0νββ framework assumes lepton-number violation via Majorana masses (Introduction).
domain assumption The two-right-handed-neutrino (3+2) model can simultaneously explain light neutrino masses via seesaw and the baryon asymmetry via leptogenesis.
Used in section 3 to restrict the parameter space; if leptogenesis fails within this minimal model, the 'complete testability' projection is void.
domain assumption The chiral EFT momentum-region decomposition and matching in ref. [6] is correct.
The paper does not derive the amplitude; it relies on ref. [6] for the exact forms.
domain assumption Future experimental sensitivities (DUNE, SHiP, HL-LHC, improved 0νββ) will be reached as projected.
The claim that the remaining parameter space is 'readily testable' assumes the projected reaches.
standard math Standard quantum field theory and effective field theory techniques.
Used throughout for decay rate and mass matrix calculations.

invented entities (1)

Two heavy neutral leptons (sterile neutrinos) independent evidence
purpose: Generate light neutrino masses via seesaw and explain BAU via leptogenesis in the 3+2 model
Not new particles, but the specific 3+2 realization is the subject of the testability claim. They are falsifiable via 0νββ, DUNE, SHiP, and HL-LHC searches, so independent evidence exists.

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Cite this review

Pith. "Pith review of Neutrinoless double beta decay rates and the $3 + 2$ scenario." pith.science (2026). https://pith.science/paper/IRESJLSX

@misc{pith2026250517750,
  author       = {Pith},
  title        = {Pith review of: Neutrinoless double beta decay rates and the $3 + 2$ scenario},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRESJLSX}},
  note         = {Machine review of arXiv:2505.17750}
}

abstract

The possible Majorana nature of neutrinos leads to lepton-number-violating effects such as neutrinoless double beta decay. The standard study of this process involves mass-dependent matrix elements which, although easy to use, might be missing important effects, especially in the light neutrino regime where the ultrasoft contributions become important. A fresh look at the different momentum regions leads us to an effective practical parametrisation for the decay amplitude that can show significant differences in the light-to-medium mass range of neutrinos compared to the standard parametrisation. As a concrete realisation of a UV model leading to Majorana neutrinos, the testability of a $3 + 2$ model with two sterile neutrinos is discussed.

Figures

Figures reproduced from arXiv: 2505.17750 by the authors.

**Figure 1.** Contribution to the 0𝜈𝛽𝛽 amplitude from a single neutrino with mass 𝑚𝑖 . Figure adapted from ref. [6]. 0.01 0.10 1 10 100 1000 1030 1035 1040 1045 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 3.** Future prospects for the 3 + 2 model in the inverted mass ordering of neutrinos. With two orders of magnitude improvement in 0𝜈𝛽𝛽 limits, the hatched region can be ruled out if the model is to explain the entire BAU. The current limits from experiments and cosmology are shown in grey. Future experimental programmes, shown with dotted lines, will be able to probe all of the remaining allowed parameter space. Taken fr… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [6]

Dekens, J

W. Dekens, J. de Vries, D. Castillo, J. Menéndez, E. Mereghetti, V. Plakkot et al., Neutrinoless double beta decay rates in the presence of light sterile neutrinos , JHEP 09 (2024) 201 [2402.07993]

arXiv 2024

[2] [11]

de Vries, M

J. de Vries, M. Drewes, Y. Georis, J. Klarić and V. Plakkot,Confronting the low-scale seesaw and leptogenesis with neutrinoless double beta decay , 2407.10560

arXiv

[3] [1]

Super-Kamiokandecollaboration, Evidence for oscillation of atmospheric neutrinos, Phys. Rev. Lett. 81(1998) 1562 [hep-ex/9807003]

arXiv 1998

[4] [2]

LEGEND collaboration, The Large Enriched Germanium Experiment for Neutrinoless Double Beta Decay (LEGEND), AIP Conf. Proc. 1894(2017) 020027 [1709.01980]

arXiv 2017

[5] [3]

nEXOcollaboration, Sensitivity and Discovery Potential of nEXO to Neutrinoless Double Beta Decay, Phys. Rev. C 97(2018) 065503 [1710.05075]

arXiv 2018

[6] [4]

Cirigliano, W

V. Cirigliano, W. Dekens, J. De Vries, M.L. Graesser, E. Mereghetti, S. Pastore et al.,New Leading Contribution to Neutrinoless Double-𝛽 Decay, Phys. Rev. Lett. 120 (2018) 202001 [1802.10097]

arXiv 2018

[7] [5]

Dekens, J

W. Dekens, J. de Vries, E. Mereghetti, J. Menéndez, P. Soriano and G. Zhou,Neutrinoless double-𝛽 decay in the neutrino-extended standard model, Phys. Rev. C 108 (2023) 045501 [2303.04168]. 6 0𝜈𝛽𝛽 and 2 HNLs Vaisakh Plakkot

arXiv 2023

[8] [7]

Drewes,The Phenomenology of Right Handed Neutrinos, Int

M. Drewes,The Phenomenology of Right Handed Neutrinos, Int. J. Mod. Phys. E 22(2013) 1330019 [1303.6912]

arXiv 2013