Electroweak Breaking and Higgs Boson Profile in the Simplest Linear Seesaw Model

Duarte Fontes; Jorge C. Romao; Jose W. F. Valle

REVIEW 3 major objections 6 minor 68 references

Electroweak Breaking and Higgs Boson Profile in the Simplest Linear Seesaw Model

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In the simplest linear seesaw model, if all known constraints are imposed, the scalar bosons are forced into a compressed spectrum and the 125 GeV Higgs can decay invisibly into majorons with branching ratio up to about 20%.

desk verdict A careful, mostly sound scan of the simplest linear seesaw variant—the compressed-spectrum/invisible-BR result holds under the explicitly flagged massless-majoron assumption, but the abstract overstates it and Eq. (7.1) has a typo. read the letter →

arxiv 1908.09587 v2 pith:3NTV7BLY submitted 2019-08-26 hep-ph

Duarte Fontes , Jorge C. Romao , Jose W. F. Valle This is my paper · ORCID

classification hep-ph

keywords linearseesawmajoroninvisibleHiggsdecayspontaneousleptonnumberviolationcompressedscalarspectrumbosonprofilestellarcoolingboundelectroweaksymmetrybreaking

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 examines the simplest linear seesaw model, built from the Standard Model gauge structure plus a nearly inert second scalar doublet and a lepton-number-carrying singlet, and asks what electroweak symmetry breaking looks like once all constraints are imposed. It claims that consistency with the astrophysical bound on the majoron, which forces the lepton-number-breaking vev $v_L \lesssim 0.5$ GeV, requires the neutral scalar spectrum to be compressed, with the three CP-even Higgs bosons nearly degenerate. In this regime the Standard-Model-like 125 GeV Higgs can decay invisibly into a pair of majorons with branching ratio as large as about 20% while satisfying LHC signal-strength data at $3\sigma$ (about 10% at $2\sigma$). The paper maps the resulting Higgs profile and gives benchmark points showing the different invisible-decay patterns among the three neutral scalars.

What carries the argument

The central objects are the pseudo-scalar rotation matrix $O^I$, whose entries give the majoron projection $\langle J|\phi\rangle = 2v_\phi v_L^2 / \sqrt{(v_\phi^2+v_L^2)(v_\phi^2(4v_L^2+v_\sigma^2)+v_L^2v_\sigma^2)}$, and the neutral scalar rotation matrix $O^R(\alpha_1,\alpha_2,\alpha_3)$. The decisive relation is the expression for the quartic coupling $\lambda_L$ in terms of physical masses and angles: since $\lambda_L \propto v_L^{-3}$, a tiny $v_L$ forces near-degenerate CP-even masses and $\alpha_1\approx 0$ to satisfy perturbative unitarity. The invisible-decay coupling $g_{h_a JJ} = -\left(\frac{(O^I_{21})^2}{v_\phi}O^R_{a1}+\frac{(O^I_{22})^2}{v_L}O^R_{a2}+\frac{(O^I_{23})^2}{v_\sigma}O^R_{a2}\right)M_a^2$ then connects the vev hierarchy directly to the observable branching ratio.

What would settle it

A future lepton collider measuring the 125 GeV Higgs invisible width at sub-percent precision: if the measured $\mathrm{BR}(h\to\text{invisible})$ falls below about 1%, the parameter region where this paper finds values up to ~20% under the massless-majoron assumption would be excluded; a null result would push the model into the extreme of parameter space where the invisible width is suppressed.

Watch

Extended reading notes

Core claim

Working in the minimal $SU(3)_c\otimes SU(2)_L\otimes U(1)_Y$ realization of the linear seesaw, the paper shows that spontaneous violation of global lepton number produces a massless majoron whose coupling to electrons is suppressed by the projection $\langle J|\phi\rangle \propto v_L^2$, so stellar cooling forces $v_L \lesssim 0.5$ GeV. It then demonstrates that such a small $v_L$ is compatible with vacuum stability and perturbative unitarity only if the CP-even scalar spectrum is compressed and the mixing angle $\alpha_1$ is near zero; this follows because the quartic coupling $\lambda_L$ of the nearly inert doublet is inversely proportional to $v_L^3$, so its numerator must nearly vanish, which happens for degenerate masses. With this compressed spectrum, the SM-like boson $h_1$ acquires a sizable coupling to two majorons, giving $\mathrm{BR}(h_1\to JJ)$ up to roughly 20% within the $3\sigma$ LHC constraints, and the paper provides three benchmark points P1--P3 covering qualitatively different invisible-decay patterns of the three neutral scalars.

Load-bearing premise

The entire result stands on the majoron being effectively massless; if the majoron were heavier than stellar temperatures, the astrophysical bound forcing $v_L \lesssim 0.5$ GeV would not apply, and the model would no longer require a compressed spectrum or large invisible branching ratios.

Editorial extensions

If this is right

If the model is right, the 125 GeV Higgs has an invisible branching ratio that current LHC data allow up to about 20% at $3\sigma$ and 10% at $2\sigma$, close to the present experimental upper bound and testable at the HL-LHC.
A consistent electroweak-breaking pattern requires a compressed neutral scalar spectrum, so the model predicts two additional CP-even scalars within a few tens of GeV of each other, along with a nearby charged and pseudoscalar state.
The heavier scalars can be either visible or invisible: benchmark P2 shows $h_2$ with an invisible branching ratio around 13% and a still sizable coupling to vector bosons, while P1 shows only $h_1$ with a large invisible width and the heavier states decaying visibly.
The vector couplings obey the sum rule $\sum_i |k_V(h_i)|^2 = 1$, so fixing the 125 GeV coupling near the SM value limits the production of the heavier scalars and constrains their observability.
The paper notes that future lepton colliders are expected to measure the invisible branching ratio with precision better than 1%, which would sharply constrain or exclude the large-invisible-width region.

Reading between the lines

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

The paper's main conclusion is conditional on the majoron being nearly massless; if higher-dimensional operators give it a mass above stellar temperatures (a possibility the authors mention), the bound $v_L \lesssim 0.5$ GeV evaporates and the compressed-spectrum requirement would not be needed, so the model would open up parameter regions not shown here.
The scan imposes a technical cut $v_\sigma > 1$ TeV that is not physically required; exploring lower $v_\sigma$ values could change the scalar-spectrum correlations and potentially shift the maximum invisible branching ratio.
The same $\lambda_L \propto v_L^{-3}$ mechanism implies a consistency check: measuring the mass splitting $M_3-M_2$ and the mixing angle $\alpha_1$ (via vector couplings) at a future collider could indirectly probe the astrophysical $v_L$ bound without directly observing the majoron.
If a future precision measurement finds $\mathrm{BR}(h\to\text{invisible})$ below about 1%, the model is not dead but is pushed into the extreme of parameter space where the majoron coupling is suppressed; the compressed-spectrum prediction would still remain a distinctive collateral signature.

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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

Quick take: this is a careful phenomenology paper, and the main physics claim survives scrutiny—within the assumption of a nearly massless majoron. What's new is the explicit singlet-extended scalar sector for the minimal linear seesaw (no left-right symmetry), and the observation that the combined constraints force a compressed scalar spectrum with α1 ≈ 0, which in turn lets the 125 GeV Higgs decay invisibly to majorons with BR up to ~20% at 3σ (or ~10% at 2σ). The three benchmarks P1–P3 are concrete and usable.

The paper does a lot right. Mass matrices, minimization conditions, unitarity/stability constraints and the oblique parameters are all written out, and the appendix gives the full coupled-channel matrices. Credit where due: Sec. 5.1 explicitly concedes that the stellar-cooling bound only applies if the majoron is lighter than stellar temperatures, and that a heavier majoron would evade it. The authors know the load-bearing step and say so. The citation pattern is self-heavy, but the prior papers genuinely cover invisible Higgs decays in seesaw variants; this is a legitimate extension, not padding.

Soft spots, in proportion. First, the abstract and conclusions say the compressed spectrum is 'required'. What the scan shows is that it emerges once you assume a nearly massless majoron, impose the astrophysical bound and restrict vσ > 1 TeV (a technical cut the authors admit). That is a conditional result stated categorically. Easy fix: soften the language. Second, Eq. (7.1) has a real typo: the σ-term should couple to the singlet component O_R^{a3}, not O_R^{a2}. The amplitudes came from FeynMaster, so Tables 4–6 may be fine, but as printed the formula cannot reproduce them. Third, the scans are not reproducible from the text: no code, no data, thin sampling details. For a scan-driven headline claim, that is a fair request, though minor.

The massless-majoron point is the one to stress. The chain—spontaneous LNV gives a massless majoron, stellar cooling bounds its doublet projection, vL is forced small, unitarity on λL forces near-degeneracy—is internally consistent, and the stress-test reduction of Eq. (3.25) checks out. But it all hangs on the majoron staying light. If explicit LNV lifts it, the bound evaporates and the compression argument collapses. The paper flags this; the abstract doesn't. That mismatch, not the physics, is the real weakness.

Who this is for: anyone working on low-scale seesaw or majoron phenomenology, especially invisible Higgs searches. It is a solid benchmark paper, not a paradigm shift. I would send it to review; with the language fixed, the typo corrected and more transparency on the scans, it is publishable. My verdict is conditionally accept: the load-bearing assumption is declared, the math is internally consistent, and the result is concrete and testable.

Referee Report

3 major / 6 minor

Summary. This paper studies the scalar sector of the simplest linear seesaw extension of the Standard Model, adding a second doublet chi_L and a singlet sigma that carry lepton number. Neutrino masses arise from spontaneous lepton-number violation, producing a majoron. The authors derive the scalar mass matrices, express the quartic couplings in terms of physical masses, vevs, and rotation angles, and impose stability, unitarity, oblique-parameter, astrophysical, and LHC constraints in numerical scans. The central outputs are that the stellar-cooling bound forces v_L below about 0.5 GeV, that perturbative unitarity on lambda_L then selects a compressed scalar spectrum with alpha_1 close to zero, and that the 125 GeV Higgs can acquire an invisible branching ratio into majorons of up to about 20% at the 3-sigma LHC level (about 10% at 2-sigma). Three benchmark points illustrate different invisible-decay patterns. The paper concludes that a consistent electroweak symmetry breaking pattern 'requires' a compressed spectrum with potentially large invisible Higgs decay.

Significance. If the results hold, the paper provides a complete and mostly consistent analysis of the scalar sector of the simplest linear seesaw model, with a concrete and falsifiable consequence: under the massless-majoron assumption, the allowed parameter space is compressed and BR(h1 -> invisible) can approach the current experimental upper bound. The analytical expressions for the quartic couplings in terms of the physical inputs, the explicit treatment of stability and unitarity bounds, and the use of FeynMaster for the decay amplitudes are strengths. The provision of three phenomenologically distinct benchmark points is also valuable. The main caveat is that the predictive chain is conditional on the majoron being effectively massless; the paper itself flags this in Sec. 5.1, but the abstract and conclusions are stated more categorically than the analysis supports. The scan additionally restricts v_sigma > 1 TeV. With appropriate qualification, this is a solid contribution to Higgs phenomenology in low-scale seesaw models.

major comments (3)

[Abstract and Sec. 8, relying on Sec. 5.1] The conclusion that 'a consistent electroweak symmetry breaking pattern requires a compressed mass spectrum of scalar bosons' is categorical, but it follows only under the assumption of a nearly massless majoron. The astrophysical bound in Eq. (5.3) applies only if the majoron mass is below stellar temperatures; Sec. 5.1 explicitly states that if the majoron is heavier, 'the bound in Eq. (5.3) need not apply,' and no estimate of the majoron mass from explicit lepton-number-violating sources is provided. In that case v_L is no longer forced to be small, the unitarity argument based on Eq. (3.25) collapses, and no compressed spectrum is required. The abstract and conclusions should be rephrased to present the result as conditional, for example: 'Under the assumption of an effectively massless majoron, the scan consistent with all applied constraints exhibits a compressed spectrum...'.
[Eq. (7.1)] The master formula for the Higgs-majoron coupling contains a typo in the third term: it is printed with O_R^{a2} multiplying the 1/v_sigma contribution, but the singlet field R3 is the one with the sigma vev, so this factor should be O_R^{a3}. As printed, the formula cannot reproduce the invisible branching ratios in Tables 4-6. This should be corrected, and the text should state clearly that the numerical results were generated with FeynMaster and that the printed formula was verified.
[Sec. 6.1] The numerical scan imposes the cut v_sigma > 1 TeV 'for technical reasons,' while the text acknowledges that lower values could be possible. Because the conclusion about compressed spectra is derived from the scanned region, either the scan should be extended to lower v_sigma or the conclusion should be explicitly restricted to v_sigma > 1 TeV. The current wording implies broader validity than the sampling supports.

minor comments (6)

[Sec. 5.1, Eq. (5.3)] The denominator in Eq. (5.3) is missing a closing parenthesis inside the square root; it should read sqrt((v_phi^2 + v_L^2)(v_phi^2(4 v_L^2 + v_sigma^2) + v_L^2 v_sigma^2)).
[Sec. 1] 'Nambu-Golstone' appears in the Introduction; the standard spelling is 'Nambu-Goldstone.'
[Sec. 4.2] In the sentence before Eq. (4.14), 'potantial' should be 'potential.'
[Sec. 6.3] The sentence 'It just turned out the that the good points have this profile' contains an extra 'the that'; it should read 'It just turned out that the good points have this profile.'
[Sec. 7] 'Invisibling Higgs decay bosons' should be 'invisible Higgs decay bosons.'
[Sec. 5.2] The phrase 'For the 13 TeV of Run-2, the data channel' is a fragment; it should be rephrased, e.g., 'For the 13 TeV Run-2 data, the results are shown in Table 3.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the compressed-spectrum and invisible-BR results are scan outputs of a self-contained model under external constraints, with the massless-majoron condition explicitly stated.

full rationale

Walking the derivation chain: the paper starts from the scalar potential (Eq. 3.3), derives minimization conditions, mass matrices, and quartic-parameter relations (Eqs. 3.25-3.29). The compression argument is an algebraic consequence: Eq. (3.25) has lambda_L inversely proportional to v_L^3, so the unitarity bound forces the numerator small for small v_L; the scan finds this happens near alpha_1~0 with a compressed spectrum, and the paper explicitly shows the numerator vanishes for alpha_1=alpha_3=0 and equal masses without imposing that in the scan. No fitted input is renamed as a prediction: the astrophysical bound (Eq. 5.3), unitarity eigenvalues (Appendix A), oblique parameters, and LHC signal strengths (Tables 2-3) are external inputs, while the invisible branching ratios in Figs. 6-8 and benchmarks P1-P3 are computed outputs. The majoron is massless by the model's own spontaneous lepton-number breaking, and the paper candidly states in Sec. 5.1 that Eq. (5.3) need not apply if the majoron is heavier than stellar temperatures; this is an explicit condition/caveat, not a circular input-output equivalence. Self-citations such as [7], [8], [14], [26], and [27] provide historical context or independently checkable foundations; the majoron projection is verified explicitly in Eq. (3.12), and the invisible-decay amplitudes are computed here with FeynMaster. There is no uniqueness theorem imported from the authors and no ansatz smuggled in via citation. The typo in Eq. (7.1) (O_R^{a2} instead of O_R^{a3}) is a reproducibility issue, not a circularity.

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

The central claim rests primarily on the scalar sector parameters that are scanned rather than derived. The v_L smallness is imposed by the astrophysical bound, and the compressed spectrum follows from unitarity. The neutrino sector parameters are left free but do not affect the Higgs profile analysis. No parameter is fitted to data to force the main result; the scan ranges and benchmark choices are the only hand-set inputs.

free parameters (4)

v_L (vev of the second doublet chi_L) = constrained to < 0.5 GeV; benchmark values 0.13, 0.06, 0.44 GeV
This vev sets the lepton-number breaking scale and enters the majoron-electron coupling; its smallness is required by the stellar cooling bound Eq. (5.3).
v_sigma (vev of the singlet sigma) = scanned in [10^3, 1.2e4] GeV with a hand-imposed lower cut > 1 TeV
The singlet vev sets the scale of the new scalar sector and affects the pseudo-scalar and charged Higgs masses; the scan lower bound is a technical choice.
Scalar masses M2, M3, MA, MH+ = scanned in [125, 800] GeV (general scan) and [125, 600] GeV (constrained scan)
These are the physical masses of the new scalars; the compressed spectrum emerges from unitarity constraints on these and on the quartic couplings.
Mixing angles alpha1, alpha2, alpha3 = scanned in [-pi/2, pi/2]
Rotation angles diagonalizing the CP-even scalar mass matrix; they enter all couplings and the compressed-spectrum condition.

assumptions (5)

domain assumption Lepton number is an exact global symmetry of the Lagrangian, spontaneously broken by the scalar vacuum expectation values.
This is the defining setup of the model (Sec. 2-3). It implies the existence of a massless majoron, which is central to both the astrophysical constraint and the invisible decay signature.
ad hoc to paper The most general renormalizable scalar potential with the stated field content and lepton number assignments, with all couplings real.
Reality of the quartic and trilinear couplings is assumed 'for definiteness' (Sec. 3). This excludes CP-violating phases in the scalar sector and simplifies the mass matrices.
domain assumption Perturbative unitarity requires all coupled-channel eigenvalues |Lambda| < 8*pi, from Ref. [54].
Standard constraint in multi-Higgs models; the paper implements it on the matrices listed in Appendix A.
domain assumption The stellar cooling bound on the majoron-electron coupling |g_Jee| <~ 1e-13 from Refs. [57,58] applies, and the majoron is effectively massless.
This external constraint drives the smallness of v_L. The paper explicitly acknowledges that it would not apply if the majoron were heavier (Sec. 5.1).
domain assumption The heavy neutral leptons implementing the seesaw do not affect the scalar sector and are not constrained in the scans.
The neutrino Yukawa couplings h, f, and the mass matrix M enter only the neutrino mass formula Eq. (2.5), not the Higgs potential, so the analysis is performed in the scalar sector alone.

invented entities (3)

Majoron J independent evidence
purpose: Gives an invisible decay channel for the 125 GeV Higgs and for the heavier scalars; in some variants it can be dark matter.
The majoron is a pseudo-Nambu-Goldstone boson. Its couplings produce stellar cooling and invisible Higgs decays that are testable; the LHC bound on invisible Higgs BR is the handle.
Second scalar doublet chi_L and singlet sigma independent evidence
purpose: Carry lepton number, break it spontaneously, and generate the linear seesaw term.
These scalars give physical states (H+, A, h2, h3) whose production is constrained by LEP/LHC via HiggsBounds-4 and by the S,T,U parameters.
Heavy neutral leptons nu^c_i and psi_i
purpose: Mediate the linear seesaw and generate neutrino masses.
These fermions realize the linear seesaw but the paper does not analyze their collider signatures or provide specific testable predictions; they are only cited (Refs. [44-52]).

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Cite this review

Pith. "Pith review of Electroweak Breaking and Higgs Boson Profile in the Simplest Linear Seesaw Model." pith.science (2026). https://pith.science/paper/3NTV7BLY

@misc{pith2026190809587,
  author       = {Pith},
  title        = {Pith review of: Electroweak Breaking and Higgs Boson Profile in the Simplest Linear Seesaw Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3NTV7BLY}},
  note         = {Machine review of arXiv:1908.09587}
}

read the original abstract

We examine the simplest realization of the linear seesaw mechanism within the Standard Model gauge structure. Besides the standard scalar doublet, there are two lepton-number-carrying scalars, a nearly inert SU2 doublet and a singlet. Neutrino masses result from the spontaneous violation of lepton number, implying the existence of a Nambu-Goldstone boson. Such "majoron" would be copiously produced in stars, leading to stringent astrophysical constraints. We study the profile of the Higgs bosons in this model, including their effective couplings to the vector bosons and their invisible decay branching ratios. A consistent electroweak symmetry breaking pattern emerges with a compressed spectrum of scalars in which the "Standard Model" Higgs boson can have a sizeable invisible decay into the invisible majorons.

Discussion (0). Continue with ORCID to comment.

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Reviewed August 14, 2026 · model on record in the stance chip above.

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