Planck Scale Symmetry Breaking and Majoron Physics

D. Seckel; I.Z. Rothstein; K.S. Babu

REVIEW

Planck Scale Symmetry Breaking and Majoron Physics

K.S. Babu, I.Z. Rothstein, D. Seckel

High Energy Physics - Phenomenology · Submitted 6 Jan 1993 · 40 references

Pith's verdict: A solid, honest model-building paper: the charge bookkeeping in Sections 3 and 5 is the real contribution, and the cosmology is order-of-magnitude throughout and labeled as such.

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arXiv IDhep-ph/9301213 (version 1)

Pith IDF45XZJXN, Pith's permanent record of this paper and its review

CitationD. Seckel, I.Z. Rothstein, K.S. Babu. "Planck Scale Symmetry Breaking and Majoron Physics." arXiv:hep-ph/9301213 (1993). https://pith.science/paper/F45XZJXN

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Editor's note

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The thing worth knowing: the new content here is the bookkeeping, not the cosmology. Section 3 works out which fractional B-L charges for the second singlet push the lowest gauge-invariant operator that breaks the accidental U(1) to dimension d > 4, and Section 5 works out which charge assignments give a string-wall network with N_w = 1 so it can dissipate. Both are done carefully and both are genuinely new relative to the majoron literature the paper cites. The derivation chain from Eq. 11 through Eq. 14 to Eq. 17 is clean, and the coupling suppression in Eqs. 15-16 follows from the mixing angle without hand-waving. The paper also says up front that Akhmedov et al. were working the same territory, which is the right way to handle it. The cosmology in Sections 4-7 is order-of-magnitude and says so (the 'dropping all dimensionless factors' line in 4.2, the 'we summarize that analysis' in 5). That is appropriate for the question being asked and I would not hold it against them. The g_grav = O(1) assumption the reader flags is real, but it is stated explicitly in 4.1 and it shifts the viable parameter space rather than breaking the construction. I would not treat it as the load-bearing soft spot. On the stress-test's Section 5 concern: I think it is partly wrong. The d = 5 toy model (v1 = v2 = v, q1 = 2, q2 = 1/2) is explicitly not the dark matter case; the paper says in 4.1 that those majorons decay in ~10^4 s and do not constitute dark matter, and that dark matter needs d >= 7. And that toy model's charges give n1 = 1, n2 = 4, so n_a = 1, which is exactly case (b) in Section 5: type-a strings carry a single wall and the network dissipates. The '7 orders of magnitude' line is about the raw single-wall-across-the-horizon estimate, not an unresolvable tension. What does remain is fair but modest: the paper never exhibits one worked point that simultaneously satisfies d >= 7, keV mass, Omega ~ 1, and N_w = 1. It gives the conditions and says the choices are flexible, but a reader who wants to use the model quantitatively has to do that assembly themselves. Minor: the inequality in Eq. 10 is flipped (should be <), and Eq. 26 as printed looks garbled. This is for model builders working on neutrino mass and majoron cosmology, and for anyone thinking about gravitational violation of global symmetries. It deserves referees.

Referee report

Significance

The model-building idea is clean and, as far as I can check, correct: the charge conditions of Eqs. (12)-(13) do forbid all operators of dimension four or less for q2 = 1/2, 4/3, 3, 8, and the mixing formula of Eq. (14) does give the coupling suppression of Eq. (15). The Landau-gauge argument in Section 3 that the Z' and Z-Z' contributions to the majoron-fermion vertex vanish is a genuinely useful observation, and the paper is candid that the majoron mass and coupling are model dependent rather than parameter-free. The phenomenological survey is broad and mostly imports constraints from cited published work, which is appropriate. The paper's value is as a template for how to reconcile a Nambu-Goldstone boson with Planck-suppressed explicit breaking, and on that score it is a worthwhile contribution.

Major objections (3)

[Section 5, Eqs. (38)-(40)] The paper states that the bound m_chi v_eff^2 <~ 10^-9 GeV^3 from Omega_wall <~ 10^-5 is 'violated by ~7 orders of magnitude for the toy model used in presentation of the thermal relics', and then resolves the tension by requiring N_w = 1 for some set of strings. But for that toy model (v1 = v2 = v, q1 = 2, q2 = 1/2, d = 5) the charge condition n1 q1 = n2 q2 gives n1 = 1 and n2 = 4, so n_a = 1 and, by Eq. (40), type-a strings carry a single wall. The toy model therefore already satisfies the N_w = 1 condition, and the Omega_wall bound quoted as violated for it does not apply. The text should say explicitly which string type is supposed to carry multiple walls in this example, or state that the wall bound is evaded there. As written, the reader cannot tell whether the keV dark-matter example of Section 4.1 survives the wall constraint, and that question is load-bearing for the abstract's claim that majorons can play the role of dark matter.
[Section 6, Eqs. (42)-(45)] Eq. (42) requires (Gamma_i/m_i) x 15 MeV < H(5 MeV), which is equivalent to m_i tau_i > 15 MeV/H(5 MeV), i.e. (m_i/MeV)(tau_i/sec) > ~10^-6. Eq. (43) as displayed is (MeV/m_i)(sec/tau_i) > 1.3 x 10^-3, which is the reciprocal of this and points the other way. The direction matters: the physical requirement that inverse decays not be in equilibrium before T = 5 MeV is a lower bound on the coupling, hence a lower bound on v1. Eq. (44) is indeed a lower bound on v1, but Eq. (45) is an upper bound on v for the same physical requirement. Please recheck the algebra and restate Eqs. (43)-(45) with a consistent direction. The qualitative conclusion that BBN is not restrictive for m_nu < 30 eV and v > 100 GeV may well survive, but the quoted bounds as printed are not usable.
[Section 4.1, Eq. (26) and following] The dark-matter window is quoted for g_grav = 1 ('assuming a dimensionless strength of order unity for the explicit symmetry breaking'), and every mass, lifetime, and Omega_chi estimate in Sections 4 through 7 scales with g_grav. The paper does not argue why g_grav should be O(1) rather than 10^-3 or 10^-6, and a smaller value would push m_chi down and lengthen tau_chi, moving the viable (d, v, m_chi) region. Please state the scaling explicitly (for example m_chi proportional to g_grav^{1/2} and tau_chi proportional to g_grav^{-1/2} in the d = 5 case) and indicate how the dark-matter window shifts for smaller g_grav. This is a scaling uncertainty rather than an error, but it is load-bearing for the quantitative dark-matter claim.

Minor objections (10)

[Section 3, Eq. (17)] Please show the step from Eq. (11) to Eq. (17). Using the normalization of chi in Eq. (14) and n1 q1 = n2 q2, I obtain a coefficient n2^2/q1^2 rather than n1 n2 q1 q2; the two differ by q1^2, which is a factor of four for q1 = 2. Since all the mass estimates in Section 4 use this coefficient, the displayed form should be checked.
[Section 4.1, Eq. (21)] The displayed product does not reproduce the quoted R_chi^3 = 5.8 x 10^-2; the quoted number corresponds to an entropy-conservation argument with g_*S of order 67 at decoupling. Please write the expression explicitly so the reader can follow the dilution factor.
[Section 4.1, Eq. (26)] As printed, log10(v/1 GeV) ~ 19(d-4) - 12d - 2 gives -43 for d = 5, whereas the numbers quoted in the text (200 GeV, 3 x 10^6 GeV, 10^9 GeV for d = 5, 6, 7) correspond to [19(d-4) - 12]/(d-2). Please correct the displayed form.
[Section 4.1, Eq. (31)] The grouping of g_m, h, m_chi, v_eff, and GeV^{5/2} in Eq. (31) is ambiguous as typeset. Please rewrite with explicit parentheses and state the units of each factor.
[Section 4.2, Eqs. (35)-(37)] These equations should be checked against Eqs. (20) and (31). Eq. (35) appears to require R_chi^{-3} rather than R_chi^{+1}, and the resulting scaling of Eq. (36) differs from what I obtain by combining Eqs. (20) and (31). The conclusion that misalignment matters only for d >= 10 rests on Eq. (32), which I have checked and which is correct, so this is a presentation issue rather than a substantive one.
[Section 4.1, Eq. (20)] With R_chi^3 = 0.058 and h = 0.5, Eq. (20) gives Omega_chi,th ~ 4.6 (m_chi/keV), so Omega_chi = 1 requires m_chi ~ 0.2 keV rather than 1 keV. The d = 7 example ('v = 10^9 GeV and lifetimes longer than the age of the Universe, so majorons can play the role of warm dark matter') would overclose by a factor of about five unless R_chi is larger. Please reconcile the quoted mass with the quoted density.
[Section 2, after Eq. (7)] The sentence 'unless we make the heavy scale v < O(100 GeV) the lifetime of any neutrino nu with mass greater than 30 eV will be too small to avoid the constraint from the age of the universe' should read 'too long' (or 'too large'); a lifetime that is too small is not a problem.
[Section 2, Eq. (10) and following sentence] The red-giant constraint should be g_eechi < 10^-13, not > 10^-13.
[Section 4.1 and Section 7] In Section 4.1, 'Evidentially' should be 'Evidently', and 'v ~ 5 x 10^4' is missing its unit (GeV). In Section 7, 'the diagonal argument (Eq. 48)' should read 'diagonal emission (Eq. 48)'.
[References and figures] There are several typographical errors in the references: 'World-Sceintific' (refs. 5 and 21), 'Berezhaini' (ref. 17), 'ans' for 'and' (ref. 12), and 'in preperation' (ref. 37). Ref. 17 is a preprint; please update it if it has appeared. The captions of Figures 1 and 2 are one line each; Figure 2 in particular should label the tilt epsilon and the barrier Delta V used in the Section 4.2 argument.

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The paper

The paper claims

Gravity can give the majoron a small mass without destroying the neutrino-mass models that need it.

The reading

Majoron models explain small neutrino masses by breaking a global lepton-number symmetry spontaneously, leaving a light Goldstone boson. The trouble is that quantum gravity is widely expected to violate any global symmetry explicitly, which would give the majoron a mass and spoil the story. This paper shows that if the symmetry is really a gauged $U(1)_{B-L}$ and a second singlet scalar is added, the global symmetry becomes an accidental one that can only be broken by operators of dimension $d > 4$, suppressed by powers of the Planck mass. The majoron then gets a small, calculable, model-dependent mass, its coupling to neutrinos can be either ordinary or arbitrarily weak, and much of the usual majoron phenomenology survives.

What carries the argument

The load-bearing device is the accidental global symmetry: a $U(1)$ that is not imposed by hand but emerges because the gauged $B-L$ charges of the two singlets forbid every renormalizable operator that would break it. Choosing $q_2 = 1/2, 4/3, 3,$ or $8$ gives $d = 5$; other choices give larger $d$ and exponentially smaller majoron masses. This single construction does two jobs at once: it suppresses the majoron mass by powers of $M_{pl}$, and it decouples the majoron-neutrino coupling from the neutrino mass, since only $2_1$ couples to right-handed neutrinos while the majoron is mostly the other phase.

Extended reading notes

Core claim

The authors construct models in which $U(1)_{B-L}$ is a local gauge symmetry and two singlet scalars carry $B-L$ charge, with charges $q_1 = 2$ for the field that couples to right-handed neutrinos and a fractional $q_2$ for the second field. If $q_2$ is chosen so that no gauge-invariant operator of dimension four or less breaks the would-be global symmetry, the lowest allowed symmetry-breaking term is $g_{grav} 2^{-n_2} M_{pl}^{-(d-4)}$ (their Eq. 11), and the majoron mass is $m_chi^2 = 2 g_{grav} v_1^{n_1} v_2^{n_2} M_{pl}^{-(d-4)} n_2 n_1 q_2 q_1 / v_{eff}^2$ (Eq. 17). The majoron is the orthogonal combination of the two scalar phases, so its diagonal coupling to neutrinos is reduced by the mixing factor $q_2 v_{eff}/v_1$ (Eq. 15), where $v_{eff}^2 = v_1^2 v_2^2/(q_1^2 v_1^2 + q_2^2 v_2^2)$. The paper then works through the consequences: thermal relic abundance, coherent oscillations, a string-wall network that must be able to dissipate, big-bang nucleosynthesis, and SN 1987A.

Load-bearing premise

The majoron mass is set by a single Planck-suppressed operator whose dimensionless coefficient is taken to be of order one; every mass, lifetime, and dark-matter conclusion in the paper scales with that number, and the paper does not argue why it should be near unity.

Editorial extensions

If this is right

Majorons can be stable enough to be warm dark matter for operator dimension $d = 7$ and above, where the lifetime exceeds the age of the universe.
For $d = 5$ the majoron mass sits near a keV and the lifetime near $10^4$ s, so the majorons decay before dominating but leave decay neutrinos only mildly more energetic than the thermal background.
The usual link between neutrino mass and majoron coupling is broken: the coupling can be suppressed by $v_{eff}/v_1$, softening supernova and nucleosynthesis bounds even for light majorons.
Only certain $B-L$ charge assignments are allowed, because a network of strings with more than one wall attached cannot dissipate and would overclose the universe.
Coherent misalignment oscillations matter only for $d \geq 10$; below that the thermal population dominates.

Reading between the lines

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

Because the majoron mass and lifetime both scale with $g_{grav}$, the dark-matter window is really a statement about the unknown coefficient of the Planck-suppressed operator, not about the gauge structure; a modestly small $g_{grav}$ would move the viable dimension down by one or two.
The same accidental-symmetry trick could be applied to other Goldstone-boson problems where a global symmetry is expected to be violated at the Planck scale, provided the gauged charges can be chosen to forbid all operators up to some dimension.
If majorons are the warm dark matter, their free-streaming length at keV masses should leave an imprint on small-scale structure; that is a place where this model could be told apart from a sterile-neutrino or axion account of the same density.
The requirement that the string-wall network dissipate gives a rare handle in which a cosmological relic density argument constrains a particle-physics charge assignment rather than a mass or coupling.

Original abstract

Majoron models provide neutrino masses via the spontaneous breaking of a global $U(1)$ symmetry. However, it may be argued that all global symmetries will be explicitly violated by gravitational effects. We show that it is possible to preserve most of the usual features of majoron models by invoking $U(1)_{B-L}$ to be a gauge symmetry and adding a second singlet scalar field. The majoron gets a small model dependent mass. The couplings of majorons to neutrinos may be of ordinary strength or may be made arbitrarily weak. We discuss the cosmological and astrophysical consequences of majoron models in the context of a model dependent majoron mass and neutrino coupling. For an appropriate choice of parameters majorons can play the role of dark matter.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,465 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.61 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.28 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Majoron models provide neutrino masses via the spontaneous breaking of a global $U(1)$ symmetry. However, it may be argued that all global symmetries will be explicitly violated by gravitational effects. We show that it is possible to preserve most of the usual features of majoron models by invoking $U(1)_{B-L}$ to be a gauge symmetry and adding a second singlet scalar field. The majoron gets a small model dependent mass. The couplings of majorons to neutrinos may be of ordinary strength or may be made arbitrarily weak. We discuss the cosmological and astrophysical consequences of majoron models in the context of a model dependent majoron mass and neutrino coupling. For an appropriate choice of parameters majorons can play the role of dark matter.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Majoron models provide neutrino masses via the spontaneous breaking of a global $U(1)$ symmetry.	background 1.00	0.05	0.04	asserted 1.00
s1	However, it may be argued that all global symmetries will be explicitly violated by gravitational effects.	background 0.92	0.05	0.03	hedged possibility 0.99
s2	We show that it is possible to preserve most of the usual features of majoron models by invoking $U(1)_{B-L}$ to be a gauge symmetry and adding a second singlet scalar field.	contribution 0.81	0.88	0.66	asserted 0.74
s3	The majoron gets a small model dependent mass.	result 0.99	0.71	0.12	asserted 1.00
s4	The couplings of majorons to neutrinos may be of ordinary strength or may be made arbitrarily weak.	result 0.98	0.64	0.11	hedged possibility 0.99
s5	We discuss the cosmological and astrophysical consequences of majoron models in the context of a model dependent majoron mass and neutrino coupling.	contribution 0.97	0.74	0.09	asserted 0.78
s6	For an appropriate choice of parameters majorons can play the role of dark matter.	result 0.50	0.65	0.19	hedged possibility 0.97

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.57.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Leading contribution to geeχ

Figures reproduced from arXiv:hep-ph/9301213 by the authors.

References

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hep-ph 2018-07 accept novelty 6.0 of 10

A sub-MeV thermal dark sector that equilibrates with Standard Model neutrinos after neutrino-photon decoupling evades Neff bounds, and a lepton-number-breaking majoron model realizes this with dark matter near the MeV scale.
Neutrinoless double-beta decay with massive scalar emission
hep-ph 2018-02 accept novelty 6.0 of 10

For a scalar of mass near the double-beta Q-value, the estimated 0ν2βφ limit on the scalar-neutrino coupling weakens by up to an order of magnitude, driven by phase space suppression and a degraded signal-to-background ratio.
Cold keV dark matter from decays and scatterings
hep-ph 2017-06 accept novelty 6.0 of 10

Phase-space-suppressed freeze-in decays and near-resonant scatterings can produce keV-scale dark matter cold enough to satisfy Lyman-alpha constraints while keeping the parent particles in thermal equilibrium with the...
Neutrino Lines from Majoron Dark Matter
hep-ph 2017-01 conditional novelty 6.0 of 10

Majoron dark matter decaying to neutrinos gives monochromatic lines whose electron fraction is about 0.03 for normal hierarchy and 0.48 for inverted, a flavor signature no standard astrophysical source reproduces.
Soft Breaking Naturalness in Majorana Fermion Loop induced Scalar and Majoron Amplitudes
hep-ph 2026-10 conditional novelty 5.0 of 10

At one loop, every symmetry-violating scalar and Majoron amplitude scales as a positive power of the small Majorana mass M when evaluated at kinematic points set by m_phi or the scalar VEV, restoring 't Hooft naturalness.
Probing Scalar Non-Standard Neutrino Interactions using High-Energy Astrophysical Neutrinos
hep-ph 2026-06 unverdicted novelty 5.0 of 10

IceCube astrophysical neutrino data is analyzed for flavor ratios and spectral shapes to constrain scalar non-standard neutrino interactions via induced pseudo-Dirac behavior.
Gravitational Wave Imprints of a High-Quality Axion and the Origin of Flavor Hierarchies
hep-ph 2026-06 unverdicted novelty 5.0 of 10

Gauged U(1)_F flavor symmetries shield the axion from quantum gravity corrections, yielding unit domain wall number and a plateau-valley GW spectrum from flavonic and axionic strings as a probe of flavored axion dark matter.
The Majoron Cosmological Window: Dark Matter and Thermal Leptogenesis
hep-ph 2026-05 unverdicted novelty 5.0 of 10

The minimal majoron framework permits simultaneous majoron dark matter and thermal leptogenesis in a constrained cosmological window set by freeze-in production, warm dark matter bounds, and indirect detection limits.
High-Quality Axion Dark Matter at Gravitational Wave Interferometers
hep-ph 2025-09 unverdicted novelty 5.0 of 10

In gauged U(1) completions enabling high-quality axion dark matter, cosmic string loops generate a stochastic gravitational wave background with an infrared break frequency that exceeds foregrounds above 10^14 GeV bre...
Did we hear the sound of the Universe boiling? Analysis using the full fluid velocity profiles and NANOGrav 15-year data
astro-ph.HE 2023-07 conditional novelty 5.0 of 10

Using the full sound-shell velocity profile instead of a broken power law gives a modestly better fit to NANOGrav 15-year data, with a lower wall velocity and a phase transition near the QCD scale.
The Hubble Tension as a Hint of Leptogenesis and Neutrino Mass Generation
hep-ph 2021-03 conditional novelty 5.0 of 10

GeV-scale sterile neutrinos in a singlet majoron model generate Delta N_eff around 0.4 at BBN, which raises the CMB-inferred H0 to 70.2 +/- 0.6 km/s/Mpc and makes the SH0ES value about ten times more likely than in La...
Cold light dark matter in extended seesaw models
hep-ph 2017-09 conditional novelty 5.0 of 10

Freeze-in dark matter from decays of quasi-degenerate heavy particles has a cold enough momentum spectrum to survive Lyman-alpha bounds at keV masses, and extended seesaw models naturally provide the required quasi-de...
Supernova Constraints on Massive (Pseudo)Scalar Coupling to Neutrinos
hep-ph 2016-09 conditional novelty 5.0 of 10

Supernova 1987A neutrino observations bound a massive scalar coupled to neutrinos at roughly 10^-12 to 10^-5 for the dimension-4 coupling and 10^-6 to 10^-2 for a dimension-5 operator at 1 TeV, over scalar masses from...
Real versus Complex Fermion Bare Mass and Multi-Wall String Defect Annihilation
hep-ph 2026-06 unverdicted novelty 4.0 of 10

Small bare fermion masses generate a bias via radiative corrections that triggers annihilation of multi-wall cosmic string networks in gravitationally broken global U(1) models, illustrated in a majoron framework with...
Majoron Dark Matter, High-Scale Seesaw, and Leptogenesis
hep-ph 2026-06 unverdicted novelty 4.0 of 10

Majoron dark matter is viable for sub-MeV masses in high-scale seesaw models with thermal leptogenesis, produced via misalignment and cosmic strings in pre- and post-inflationary scenarios and constrained by CMB, X-ra...
Minimal Majoron Dark Matter
hep-ph 2026-05 unverdicted novelty 4.0 of 10

In the minimal Majoron model the particle can explain all dark matter with mass below about 10 MeV from misalignment or freeze-in, and remains compatible with thermal leptogenesis when misalignment dominates or with m...
The Poker Face of the Majoron Dark Matter Model: LUX to keV Line
hep-ph 2014-04 conditional novelty 4.0 of 10

Thermal Majoron dark matter coupled through the Higgs portal is excluded by LUX below 225 GeV except at the Higgs resonance, while a keV-scale triplet-Higgs variant can be tuned to the 3.5 keV cluster line.
Updated CMB, X- and gamma-ray constraints on majoron dark matter
astro-ph.HE 2013-03 conditional novelty 4.0 of 10

Majoron dark matter must live longer than about 50 Gyr and, for triplet vacuum expectation values above a few MeV, decay to two photons faster than X-ray and gamma-ray line searches allow.
Mass Generation for Axion Like Particles
hep-ph 2026-09 accept novelty 3.0 of 10

A conference proceedings reviewing the MMM model, where the Majoron mass arises from lepton number breaking and is correlated with active neutrino masses.

How the literature uses it

57 arXiv papers cite this in 82 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 60 sentences without a confident label

Compares with it 1 paper

Our approach differs from that of Ref. [ref] in that we do not allow gravitational symmetry violations, unless they are mediated by “instanton” effects, making such violations exponentially suppressed.

Gravitational Interactions and Neutrino Masses · Introduction · 0.94

Cites it as background 20 papers

Majoron cosmology was discussed long ago in the context of Planck-suppressed breaking of global lepton number, with overclosure constraints already emphasized in Refs. [ref] and the majoron as a DM candidate considered in Refs. [ref] .

Majoron Dark Matter, High-Scale Seesaw, and Leptogenesis · Introduction · 0.99
The production mechanisms of majoron DM with mass of $keV–10 TeV$ has been proposed in Ref. [ref] .

Updated Constraints and Future Prospects on Majoron Dark Matter · Singlet majoron model · 0.87
Assuming these effects are suppressed by powers of $M_P$, they can be used to generate a Majoron mass at the keV scale [ref] , which has been extensively studied as DM candidate [ref] .

Sub-GeV dark matter as pseudo-Goldstone from the seesaw scale · Approximate symmetries of the seesaw sector · 0.86

Discussion

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