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REVIEW 3 major objections 5 minor 40 references

Estimates of the masses of heavy right-handed neutrino particles using the seesaw mechanism

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Pith's one-line read The seesaw formula, applied family by family, puts the three heavy right-handed neutrinos at roughly 81.6 TeV, 0.63 EeV, and 31.8 EeV, with an unknown factor of order one.

desk verdict A transparent but fragile seesaw estimate: the numbers are new, yet the assumed texture forces zero lepton mixing, contradicting the oscillation data that supply the input masses. read the letter →

arxiv 2505.07493 v1 pith:GPWZMUNJ submitted 2025-05-12 hep-ph

V. V. Khruschov , S. V. Fomichev This is my paper · ORCID
classification hep-ph PACS 14.60.Pq14.60.Lm14.60.St95.35.+d
keywords activeneutrinossterileright-handedseesawmechanismneutrinomassesoscillationsmassobservablesdarkmatter
topics Dark Matter
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

This paper estimates the masses of three heavy right-handed (sterile) neutrinos by turning the seesaw mechanism around: instead of predicting light-neutrino masses from heavy ones, it inserts measured light-neutrino masses into the simple relation $m_{\nu i} = \kappa m_{li}^2/(2M_i)$ and solves for $M_i$. With the normal-ordering values $m_1\approx 0.0016$ eV, $m_2\approx 0.0088$ eV, $m_3\approx 0.0496$ eV, it obtains $M_1\approx 81.6\kappa$ TeV, $M_2\approx 0.6343\kappa$ EeV, and $M_3\approx 31.8294\kappa$ EeV, where $\kappa$ is an unknown coefficient expected to be of order unity. It also derives observable neutrino masses $m_C\approx 0.02$ eV and $m_\beta\approx 0.01$ eV, consistent with existing limits. The result matters because it ties the tiny active-neutrino masses to concrete, widely separated scales for heavy sterile neutrinos, which could be heavy fermionic dark-matter candidates if their decays are slow enough.

What carries the argument

The load-bearing identity is the simplified seesaw formula $m_{\nu i} = \kappa m_{li}^2/(2M_i)$ (Eq. 12), obtained from the standard seesaw relation $M_\nu = -M_D^T M_R^{-1} M_D$ by taking the Dirac neutrino mass matrix proportional to the charged-lepton mass matrix, $M_D = \sigma\,\mathrm{diag}\{m_e,m_\mu,m_\tau\}$, and a diagonal sterile mass matrix with eigenvalues $M_1,M_2,M_3$. This reduction gives one equation per family, so each light-neutrino mass and its corresponding charged-lepton mass determine one heavy sterile scale. The coefficient $\kappa$ absorbs the unknown overall normalization and is expected to be of order unity.

What would settle it

A measurement of the absolute neutrino-mass scale that forces $m_3$ to differ from about $0.0496$ eV while keeping the measured splittings, or a direct search that finds a sterile neutrino far from the three predicted scales ($81.6\kappa$ TeV, $0.6343\kappa$ EeV, $31.8294\kappa$ EeV), would falsify the specific three-way assignment; the same follows if a precise cosmological bound pushes $\sum m_i$ well below $0.06$ eV.

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Extended reading notes

Core claim

The paper's central claim is that the seesaw relation, specialized to a diagonal Dirac mass matrix proportional to the charged-lepton masses, leaves a clean three-family formula $m_{\nu i} = \kappa m_{li}^2/(2M_i)$ with three independent heavy-neutrino masses. Substituting the adopted light-neutrino masses for normal ordering gives $M_1\approx 81.6\kappa$ TeV, $M_2\approx 0.6343\kappa$ EeV, and $M_3\approx 31.8294\kappa$ EeV. These values imply a huge separation among the heavy sterile states, and the corresponding observable masses $m_C\approx 0.02$ eV and $m_\beta\approx 0.01$ eV fall inside current experimental limits. The paper therefore claims that the heavy right-handed neutrinos can be viewed as phenomenologically determined candidates whose masses are tied to the charged-lepton masses and the measured light-neutrino spectrum.

Load-bearing premise

The load-bearing assumption is that each neutrino family's Dirac mass is exactly the same constant times the charged-lepton mass of that family, with one heavy right-handed neutrino per family; if that proportionality is not exact, the three estimated heavy-neutrino masses do not follow.

Editorial extensions

If this is right

  • If the formula is right, the three heavy right-handed neutrinos are separated by many orders of magnitude: one near 81.6 TeV, one near 0.63 EeV, and one near 31.8 EeV, all multiplied by $\kappa$.
  • The paper predicts $m_C\approx 0.02$ eV and $m_\beta\approx 0.01$ eV, and notes that these values do not conflict with current bounds on the neutrino mass observables.
  • Under the assumed structure, the heavy sterile neutrinos can decay on timescales shorter than the age of the Universe, so the paper classifies them as candidates for unstable heavy fermionic dark matter rather than stable dark-matter particles.
  • The total active-neutrino mass is tied to the normal-ordering sum of roughly 0.06 eV, which the paper presents as consistent with recent cosmological limits.

Reading between the lines

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

  • If a future model fixes $\kappa$ independently, the three heavy masses become sharp predictions; a measured $\kappa$ far from unity would shift all scales uniformly without breaking the family-by-family seesaw structure.
  • Applying the same formula under inverted ordering would produce a different set of heavy-neutrino masses; the paper does not compute those, but the relation is ready-made for that comparison.
  • A testable extension is to use the predicted $M_i$ values as targets for leptogenesis, collider searches, or astrophysical probes: finding a sterile neutrino at one of the three scales would support the proportionality assumption, while finding such a state at a different mass would rule out the three-way split.
  • Because $M_i \propto m_{li}^2/m_{\nu i}$, the prediction is most sensitive to the absolute scale of the lightest neutrino; future cosmology and neutrinoless double-beta decay measurements should sharpen that input directly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper estimates the masses of three heavy right-handed (sterile) neutrinos within a type-I seesaw framework. It assumes that the Dirac neutrino mass matrix is proportional to the charged-lepton mass matrix, M_D = σ diag(m_e, m_μ, m_τ), and that the sterile neutrino mass matrix is diagonal, which leads to the three independent relations m_{νi} = κ m_{li}^2/(2M_i) [Eq. (12)]. Using light neutrino masses m1≈0.0016 eV, m2≈0.0088 eV, and m3≈0.0496 eV taken from two earlier papers by the same group, the paper obtains M1≈81.6κ TeV, M2≈0.6343κ EeV, and M3≈31.8294κ EeV [Eq. (13)], and also estimates the observable masses m_C≈0.02 eV and m_β≈0.01 eV. The conclusion presents the heavy neutrinos as possible unstable fermionic dark-matter candidates.

Significance. If the derivation were valid, the paper would provide a simple phenomenological link between charged-lepton masses, light neutrino masses, and the scale of heavy right-handed neutrinos, with the interesting qualitative feature that the three heavy masses are hierarchically separated across the TeV–EeV range. Credit is due for the transparent and reproducible algebra: the conversion from Eq. (12) to Eq. (13) is arithmetically correct, and the quoted m_C value follows directly from the input masses. The significance is, however, strongly limited by three structural problems: the flavor-diagonal ansatz is incompatible with the observed large lepton mixing quoted in the same paper, the input light neutrino masses are self-cited without quoted uncertainties, and the overall scale is controlled by an undetermined parameter κ. These issues are load-bearing for the central numerical claims, not presentation details.

major comments (3)
  1. [Section 3, Eqs. (9)–(12)] The assumed textures M_D = σ diag(m_e, m_μ, m_τ) and M_R = diag(M1, M2, M3) make the seesaw light-neutrino mass matrix M_ν = -M_D^T M_R^{-1} M_D diagonal in the charged-lepton mass basis, so the PMNS matrix is the identity up to phases. This is qualitatively incompatible with the large mixing angles listed in Eq. (2), which are the same oscillation data used to set the input masses. If off-diagonal structure is introduced to reproduce the observed PMNS matrix, the light neutrino eigenvalues are no longer given by κ m_{li}^2/(2M_i), and the individual estimates in Eq. (13) do not follow. The manuscript never addresses this tension, although Eq. (2) appears in the same text.
  2. [Section 3, Eqs. (12)–(13)] The light neutrino masses m1≈0.0016 eV, m2≈0.0088 eV, and m3≈0.0496 eV are taken from self-cited papers (Yudin et al., 2016; Khruschov et al., 2016) without error bars or an independent cross-check, yet Eq. (13) quotes results to several significant figures (e.g., M2≈0.6343κ EeV). Since M_i is inversely proportional to m_i in Eq. (12), the relative uncertainty in each output mass is at least as large as the relative uncertainty in the corresponding input mass. The precision displayed is therefore unjustified unless the input uncertainties are quantified and propagated.
  3. [Section 3, Eqs. (12)–(13)] The coefficient κ is completely undetermined and absorbs the entire scale of the Dirac mass matrix; the statement that κ is 'most likely of the order of unity' is an assumption with no derivation or bound. Consequently Eq. (13) does not provide absolute predictions of the seesaw mechanism, only the ratios M1:M2:M3 fixed by the ansatz. Without a quantitative treatment of κ, the quoted absolute masses cannot be meaningfully compared with collider, astrophysical, or cosmological searches.
minor comments (5)
  1. [Section 2, Eqs. (10)–(11)] The notation changes from m_{νi} to μ_i without explanation, and the relation of Eq. (11) to the new formula (12) is not derived; a short derivation connecting Eqs. (8)–(10) to Eq. (12) would make the logic easier to follow.
  2. [Section 2, Eqs. (5)–(6)] The notation m^2_β and m_{2β} is visually confusing; please use distinct symbols or names for the kinematic beta-decay mass and the neutrinoless double-beta-decay effective mass.
  3. [Section 3, final paragraph] The statement that HSNs with the estimated masses 'can decay in time scales shorter than the lifetime of the Universe' is made without a decay-width calculation or a quantitative estimate; if this point is to support the dark-matter discussion, it needs a substantiation or an explicit reference.
  4. [Eq. (2c)] The text contains a Cyrillic character in the units ('эВ2' instead of 'eV^2'); this should be corrected.
  5. [References] The two central input papers for the light neutrino masses are self-citations; independent determinations or at least explicitly stated uncertainties should be provided when these values are used as inputs.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: Eq. (13) is a transparent algebraic inversion of the explicit seesaw ansatz; the only caveat is that the input light-neutrino masses are inherited from the authors' earlier papers.

full rationale

The paper's central estimate of HSN masses is obtained by inverting Eq. (12), m_νi = κ m_li^2/(2M_i), for M_i after inserting m_νi = m_i from the authors' prior work (Yudin et al., 2016; Khruschov et al., 2016). This is an open inverse calculation, not a fit of the target HSN masses to data. The assumed texture M_D = σ m_l with diagonal M_R is stated explicitly as a phenomenological choice ('Let us take a value for M_D...'), and the relation is not derived from the HSN masses themselves, so no equation reduces to its own input. The light-neutrino masses m1, m2, m3 are cited from previous papers by overlapping authors, but those values are constrained by external oscillation data and the cosmological sum bound (Σm_i ≈ 0.06 eV), so the self-citation is not definitionally circular. The m_C and m_β values in Eq. (14) are likewise just the averages implied by the same input masses. No self-definitional, fitted-input-called-prediction, uniqueness-imported, ansatz-smuggling, or renaming step occurs. The known tension with observed PMNS mixing would be a correctness/model-validity concern, not a circularity one.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard seesaw relation plus the specific proportionality M_D∝m_l. The undetermined coefficient κ and the adopted light-neutrino masses are inputs that directly control the output masses; no new entities are introduced.

free parameters (4)
  • κ = unknown, assumed O(1)
    Introduced in Eq. (12) as an undetermined coefficient; all HSN mass estimates scale linearly with it.
  • m1 (lightest active neutrino mass, NO) = 0.0016 eV
    Input from self-cited prior fits; sets M1 via M1 = κ m_e^2/(2 m1).
  • m2 = 0.0088 eV
    Input from self-cited prior fits; sets M2 via M2 = κ m_μ^2/(2 m2).
  • m3 = 0.0496 eV
    Input from self-cited prior fits; sets M3 via M3 = κ m_τ^2/(2 m3).
assumptions (4)
  • domain assumption Seesaw mechanism: active neutrino masses are generated by heavy right-handed Majorana neutrinos via M_ν'' = -M_D^T M_R^{-1} M_D.
    Invoked in Section 3 (Eq. 8) as the framework for relating active and sterile masses.
  • ad hoc to paper M_D is diagonal and proportional to the charged lepton mass matrix, M_D = σ m_l.
    This proportionality is the key modeling assumption that makes Eq. (12) separable; it is a choice, not derived within the paper.
  • domain assumption The sterile neutrino mass matrix M_R is diagonal with eigenvalues M1, M2, M3.
    Assumed so that each active neutrino can be paired with a single heavy sterile neutrino in Eq. (12).
  • domain assumption The active neutrino masses are m1≈0.0016 eV, m2≈0.0088 eV, m3≈0.0496 eV in normal ordering.
    Adopted from the authors' earlier papers; the current paper does not re-derive them or provide uncertainties.

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Pith. "Pith review of Estimates of the masses of heavy right-handed neutrino particles using the seesaw mechanism." pith.science (2026). https://pith.science/paper/GPWZMUNJ

@misc{pith2026250507493,
  author       = {Pith},
  title        = {Pith review of: Estimates of the masses of heavy right-handed neutrino particles using the seesaw mechanism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GPWZMUNJ}},
  note         = {Machine review of arXiv:2505.07493}
}
read the original abstract

The mass estimates of active and sterile (right-handed) neutrinos are considered at the phenomenological level using the seesaw mechanism. It is assumed that the neutrino mass values depend on three characteristic scales, and the sum of the active neutrino mass values is limited from above by 0.06 eV. The neutrino mass values are estimated, as well as the observable neutrino mass values, namely, the average mass m_C of active neutrinos and the kinematic neutrino mass m_{\beta} of beta-decay.

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