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REVIEW 3 major objections 4 minor 93 references

Archimedean Seesaw: Small Neutrino Masses and Large Lepton-number Violation

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A texture-zero seesaw can cancel the active neutrino mass exactly while lepton-number violation in the heavy sector stays large, and small perturbations make the neutrino mass naturally small without reinstating the usual suppression.

desk verdict The tree-level cancelation is real and the paper is worth refereeing, but the 'arbitrarily large LNV' claim is only established at tree level because the accidental symmetry is not a symmetry of the Lagrangian. read the letter →

arxiv 2608.10062 v1 pith:HBVYYEW3 submitted 2026-08-10 hep-ph hep-ex

Tao Han , Alejandro Ibarra , Subhojit Roy , Martina Sabová This is my paper · ORCID
classification hep-phhep-ex
keywords neutrinomassseesawmechanismlepton-numberviolationheavyneutralleptonstexturezerosaccidentalsymmetryMajorananeutrinosArchimedean
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 claims that the familiar seesaw rule, under which observable lepton-number violation is always suppressed by the tiny neutrino mass, is not forced by the seesaw mechanism itself. In a model with one active neutrino and two heavy neutral leptons, a texture-zero (symmetry-enforced vanishing entries) choice for the Yukawa coupling and the heavy Majorana mass matrix leaves one active neutrino exactly massless while the heavy sector still violates lepton number. Small perturbations lift that zero mode and give naturally small neutrino masses that depend on the perturbation parameters $\mu_1,\mu_2$, not on the heavy Majorana mass $M_{22}$. Heavy neutral leptons can therefore have sizeable mixing with the active neutrino and produce observable lepton-number-violating signals while light neutrinos stay sub-eV. The paper argues this would give collider searches for lepton-number violation a concrete target across a broad mass range.

What carries the argument

The mechanism is the texture-zero Archimedean seesaw: a $3\times3$ neutrino mass matrix whose first two rows are linearly dependent, so that one active eigenstate is exactly massless. The load-bearing object is the accidental $U(1)_{\rm acc}$ symmetry generated by the projector $T=|\nu\rangle\langle\nu|$; it leaves the renormalizable mass matrix invariant for all heavy-sector parameters and makes the zero mode exact rather than fine-tuned. Higher-dimensional operators that respect the underlying $U(1)_X$ symmetry but break $U(1)_{\rm acc}$ provide the leading perturbation $\Delta M$, and the active neutrino mass is simply $\langle\nu|\Delta M|\nu\rangle=\mathrm{Tr}(T\Delta M)$, naturally small and independent of $M_{22}$. The paper also gives a lever-arm picture: the relation $m_\nu V_{\ell\nu}^2+M_1 V_{\ell N_1}^2+M_2 V_{\ell N_2}^2=0$ is an Archimedean balance condition, with the heavy masses acting as counterweights that allow large active-sterile mixing without a large neutrino mass.

What would settle it

Compute the three-flavor generalization: if any light neutrino mass acquires a term proportional to $M_{22}$, or if the exact massless mode cannot be maintained while fitting oscillation data, the central decoupling claim collapses. An experimental falsifier would be a precise measurement of same-sign dilepton production that returns $|V_{\ell N}|^2 M_N \simeq m_\nu$ in a parameter region where the Archimedean model predicts a much larger value.

Watch

Extended reading notes

Core claim

The central claim is that the light neutrino mass in a seesaw can be made independent of the heavy Majorana mass that controls lepton-number violation. With the texture-zero choice $m_1=0$ and $M_{11}=0$ for $Y=(0,y_2)$ and $M=(M_{ij})$, the $3\times3$ neutrino mass matrix has an exact zero eigenvalue for arbitrarily large $M_{12}$ and $M_{22}$; the eigenvector $|\nu\rangle=(M_{12},-m_2,0)/\rho$ with $\rho^2=m_2^2+M_{12}^2$ defines a projector $T=|\nu\rangle\langle\nu|$, and $U(\alpha)=e^{i\alpha T}$ is an accidental $U(1)$ symmetry of the renormalizable mass matrix. Lifting the texture zeros by small perturbations $\mu_1,\mu_2$ gives $m_\nu=-2\mu_1 m_2 M_{12}/\rho^2+\mu_2 m_2^2/\rho^2=\mathrm{Tr}(T\Delta M)$, an expression in which $M_{22}$ does not appear. The induced mass is naturally small because the breaking first appears through higher-dimensional operators suppressed by $\langle\phi\rangle/\Lambda$, and the two heavy eigenstates need not form a Dirac pair, so heavy-sector lepton-number violation remains unsuppressed.

Load-bearing premise

The load-bearing premise is that the realistic three-generation version needed to fit neutrino oscillations keeps the same protective structure, so the heavy mass that controls lepton-number violation stays out of the light neutrino mass formula; the paper shows this only for one active flavor.

Editorial extensions

If this is right

  • If the central claim is right, $|V_{\ell N}|^2 M_N=m_\nu$ is not a universal seesaw constraint, and heavy neutral lepton searches no longer need to be pre-scaled by the tiny neutrino mass.
  • The model maps the observable $(M_N,|V_{\mu N}|^2)$ plane directly onto the heavy-sector scale $M_2$, so current and future exclusions become bounds on $M_2$.
  • Same-sign dilepton signatures, both vector-boson fusion and resonant Drell-Yan $q\bar q\to W\to \ell N_i$, become viable discovery channels at the LHC and future colliders for HNL masses from MeV to TeV.
  • In the limit $M_{22}\to0$ the two heavy states form an approximate Dirac pair and lepton-number violation is partially cancelled, recovering the inverse seesaw; the new physics is the asymmetric regime where this cancellation is absent.

Reading between the lines

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

  • The paper proves the decoupling for a single active flavor; a natural next test is whether a three-flavor texture can protect all three light neutrinos while fitting solar and atmospheric oscillations, a check not carried out here.
  • Because the neutrino mass is proportional to $\mu_1,\mu_2$, generating these parameters radiatively at one loop would make their smallness fully natural without needing a very large $\Lambda$; the paper does not explore this origin.
  • An anomaly-free gauging of $U(1)_X$ would introduce a $Z'$ coupled to the singlet sector, giving an additional search channel through heavy neutral lepton pair production that is not discussed in the paper.
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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 / 4 minor

Summary. The paper proposes a seesaw variant, the "Archimedean seesaw," in which a texture-zero mass matrix for one active and two sterile neutrinos has an exact massless eigenstate even when the heavy Majorana sector violates lepton number by a large amount. Small neutrino masses are then generated by small texture-lifting parameters (mu1, mu2), and the paper claims that the heavy Majorana mass M22 that controls LNV does not enter the light-neutrino mass, thereby evading the conventional seesaw relation U^2 M_N ~ m_nu. The authors present a U(1)_X extension intended to justify the texture zeros, give tree-level formulas for the masses and mixings, and map the model onto experimental HNL searches. The tree-level matrix algebra in Eqs. (8)-(13) is correct, but the paper's broader claims of a natural, symmetry-protected mechanism are not established.

Significance. If the core claim were correct, it would open a qualitatively new avenue for low-scale seesaw phenomenology: heavy neutral leptons with sizeable active-sterile mixing and observable LNV could coexist with sub-eV neutrino masses, in contrast to the standard Type-I seesaw relation. The paper also gives a clear EFT interpretation of the texture-lifting, and the phenomenological projections are concrete and useful. However, the significance is currently conditional: the explicit U(1)_X realization is internally inconsistent, the accidental symmetry used for technical naturalness is not a symmetry of the full Lagrangian, and the three-generation extension is not provided. The tree-level one-flavor result is a valid mathematical observation, but it does not by itself support the advertised class of realistic models.

major comments (3)
  1. [TEXTURE ZEROS FROM AN EXTRA U(1)_X SYMMETRY (Eqs. (15)-(16) and charge table)] The U(1)_X assignments are internally inconsistent under the standard convention that the charge conjugate N_i^c has charge -q_i. With the stated charges q(L)=0, q(N1)=1, q(N2)=0, q(H)=0, q(phi)=-1, the term lambda phi N_1^c N_2 has total charge -2 and the term beta phi^2 N_1^c N_1 has total charge -2, so neither is invariant; only the alpha term is allowed. This invalidates the claim that Eqs. (15)-(16) are the complete set of U(1)_X-invariant operators. More generally, invariance of the beta operator forces q(phi)=0, which together with invariance of the lambda operator forces q1=q2, and then the renormalizable M11 term is allowed, destroying the texture zero. The advertised symmetry protection of the texture-zero structure is therefore not realized by the explicit model.
  2. [End Matter, Eqs. (21)-(25) and main text after Eq. (13)] The accidental symmetry U(1)_acc is only a symmetry of the tree-level mass matrix M0, not of the full Lagrangian. The transformation U(alpha)=exp(i alpha T) mixes nu_L, an SU(2) doublet with U(1)_X charge 0, with N_1, an SU(2) singlet with U(1)_X charge 1, so it does not commute with the gauge-covariant kinetic terms. Consequently, the statement that this symmetry 'assures the zero neutrino mass technically natural' is unsupported. Since the Weinberg operator (LH)(LH) is neutral under U(1)_X, radiative corrections will generically generate the (1,1) entry of the neutrino mass matrix even when mu1=mu2=0, re-introducing M22 into m_nu. The paper does not compute any loop correction, so Eq. (13) and the central conclusion that 'arbitrarily large LNV can coexist with sub-eV neutrino masses' are established only at tree level.
  3. [Footnote 1 and the phenomenology section] The proof of the mechanism is carried out for a single active flavor, and the realistic extension to three generations is deferred to a forthcoming paper. The observed neutrino spectrum requires at least two non-zero masses and a specific flavor mixing pattern, and it is not demonstrated that the texture-zero protection survives the required flavor structure. The Supplemental Material's assertion that the mechanism is 'flavor independent' is not backed by a construction. Without an explicit three-generation model, the abstract's claim of a 'class of seesaw models' that realizes sub-eV neutrino masses with sizeable LNV is not supported.
minor comments (4)
  1. [Section 'ACCIDENTALLY VANISHING NEUTRINO MASSES', after Eq. (6)] The phrase 'without loss of generality' is misleading: the choice m1=0, M11=0 is one particular solution of Eq. (6), not the general solution. The paper is free to adopt this ansatz, but the wording overstates its generality.
  2. [Eq. (13) and the discussion following it] The sign of m_nu is not fixed; a Majorana mass eigenvalue can be negative, but the paper should refer to |m_nu| when comparing to sub-eV bounds and to experimental constraints.
  3. [Charge table in 'TEXTURE ZEROS FROM AN EXTRA U(1)_X SYMMETRY'] The charges of the conjugate fields N_i^c are not listed in the table. Given that the mass terms involve N_i^c, this omission contributes to the inconsistency discussed above and should be clarified.
  4. [Figure captions (Figs. 3 and 4)] The captions refer to 'colored diagonal lines' and shaded regions, but the actual figures are not included in the manuscript text. The captions should be self-contained enough for a reader to interpret the constraints independently.

Circularity Check

1 steps flagged · score 3.0 of 10

Tree-level mass derivation is self-contained, but the claimed U(1)_acc protection of the zero neutrino mass is self-definitional: the symmetry is constructed from the zero-mode projector it is invoked to protect.

  1. self definitional [Supplemental Material, 'Accidental symmetry of the texture-zero solution', Eqs. (21)-(25); main text Section 'Texture zeros from an extra U(1)_X symmetry'.]
    "The renormalizable neutrino mass matrix in Eq. (8) possesses an exact massless eigenstate satisfying M0|ν⟩=0, with normalized eigenvector |ν⟩=1/ρ(M12,−m2,0)... We define the projector onto the exact zero mode, T≡|ν⟩⟨ν|... The transformation U(α)≡e^{iαT}=1+(e^{iα}−1)T leaves the neutrino mass matrix invariant for all α... Therefore, the renormalizable neutrino mass matrix possesses an accidental global U(1) symmetry acting on the exact massless eigenstate, U(1)_acc: |ν⟩ → e^{iα}|ν⟩."

    The 'accidental' U(1)_acc is not an independently motivated symmetry of the Lagrangian; it is defined by T=|ν⟩⟨ν|, the projector onto the very massless eigenstate it is invoked to protect, and its invariance property U^T M0 U = M0 is algebraically equivalent to M0|ν⟩=0 (using T M0 = M0 T = 0). Saying that this symmetry 'assures the zero neutrino mass technically natural' is therefore a restatement of the zero-mode condition, not evidence that radiative corrections vanish. The paper never verifies that U(α) preserves the gauge-covariant kinetic terms or commutes with U(1)_X; since ν_L and N_1 carry different quantum numbers, U mixes fields in a way that is not a symmetry of the full renormalizable action. The tree-level result Eq.

full rationale

The central mass formula Eq. (13) is a straightforward consequence of the explicitly assumed texture-zero structure and the small perturbation matrix ΔM; no parameters are fitted to neutrino data, and the decoupling of M22 from mν is a designed feature of the operator content, not a fitted result. The phenomenology sections recast existing experimental limits rather than deriving predictions from fitted quantities. The paper's self-citations (e.g., refs. [22], [40], [81]) are background references and are not load-bearing for the central mechanism. The one genuine circular element is the technical-naturalness claim: the U(1)_acc symmetry is built entirely from the massless eigenvector it is said to protect, and its invariance of the mass matrix is a tautological consequence of M0|ν⟩=0. Because the paper uses this symmetry to argue that the zero mode is protected against radiative corrections, without demonstrating that U(1)_acc is a symmetry of the full renormalizable Lagrangian (including kinetic and gauge terms), that explanatory step is self-definitional. This does not invalidate the tree-level 'Archimedean seesaw' decoupling, so the overall circularity is mild.

Assumptions & free parameters 5 free parameters · 4 assumptions · 3 invented entities

The central claim rests on a specific choice of charges and operator content; no parameters are fitted to data, but several mass scales and couplings are free benchmarks. The naturalness of the small neutrino mass is inherited from the EFT expansion epsilon = <phi>/Lambda, which is an assumption about the UV completion.

free parameters (5)
  • y2 = benchmark 0.1, also 1e-2 and 1e-5
    The active-sterile mixing and LNV rates are proportional to y2^2; it is chosen by hand in the collider plots.
  • M22 = benchmark 1e4 to 1e12 GeV in Fig. 4
    The heavy Majorana mass controlling LNV; its absence from m_nu is the central decoupling feature.
  • M12
    Off-diagonal heavy Majorana mass generated by lambda <phi>; determines the massless eigenvector and HNL masses.
  • mu1, mu2 = not numerically fit
    Texture-lifting parameters from higher-dimensional operators; they set m_nu but are not constrained by the paper.
  • epsilon = <phi>/Lambda
    The EFT expansion parameter assumed to make mu1 and mu2 small relative to the heavy scales.
assumptions (4)
  • domain assumption Type-I seesaw with two right-handed neutrinos and SM gauge content
    The Lagrangian in Eq. (3) assumes this framework and the seesaw relation Eq. (5).
  • domain assumption One-lepton-generation simplification
    The paper sets the model in a single active flavor and defers the three-generation extension to a forthcoming paper (footnote 1).
  • ad hoc to paper U(1)_X charge assignments and the operator selection in Eqs. (15)-(16)
    The texture zeros are enforced by this specific global symmetry and scalar content; no independent evidence for these charges is given.
  • ad hoc to paper Higher-dimensional operators are the leading breaking of U(1)_acc and are controlled by one scale Lambda
    The smallness of m_nu follows from assuming no other operators or radiative corrections dominate the EFT expansion.
invented entities (3)
  • Global U(1)_X symmetry
    purpose: Enforces the texture zeros m1 = 0 and M11 = 0 at the renormalizable level
    The symmetry is imposed to realize the desired texture; no independent observable is attached to it.
  • Singlet scalar phi
    purpose: Generates M12 through its VEV and the mu1, mu2 terms through higher-dimensional operators
    No dedicated signatures for phi are discussed; spontaneous breaking may also produce a Goldstone boson that is not analyzed.
  • Accidental U(1)_acc symmetry
    purpose: Protects the exact massless neutrino in the texture-zero limit
    An emergent symmetry of the mass matrix, not directly observable and not a symmetry of the full gauge-kinetic sector.

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Cite this review

Pith. "Pith review of Archimedean Seesaw: Small Neutrino Masses and Large Lepton-number Violation." pith.science (2026). https://pith.science/paper/HBVYYEW3

@misc{pith2026260810062,
  author       = {Pith},
  title        = {Pith review of: Archimedean Seesaw: Small Neutrino Masses and Large Lepton-number Violation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HBVYYEW3}},
  note         = {Machine review of arXiv:2608.10062}
}
read the original abstract

Contrary to the common lore that observable lepton-number violation (LNV) is inevitably suppressed by tiny neutrino masses, we identify a class of seesaw models in which arbitrarily large LNV can naturally coexist with sub-eV neutrino masses. We construct a symmetry-protected texture-zero structure in the neutrino Yukawa couplings and heavy Majorana mass matrix that gives rise to the required accidental symmetry, thereby protecting the light neutrinos from acquiring mass even in the presence of arbitrarily large LNV in the heavy sector. Small neutrino masses arise naturally from lifting the texture-zero structure while preserving the underlying symmetry. The resulting framework offers a rich and experimentally accessible phenomenology, predicting Heavy Neutral Leptons with sizeable active-sterile mixing over a broad range of experimentally accessible masses, giving rise to observable LNV signatures at collider and intensity-frontier experiments.

Figures

Figures reproduced from arXiv: 2608.10062 by the authors.

Figure 1
Figure 1. FIG. 1. Pictorial illustration of the standard Type-I seesaw [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Representative Feynman diagrams for the LNV pro [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Left: ATLAS exclusion limit in ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Current experimental constraints (gray shaded regions) and projected sensitivities of future experiments (colored [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

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