arxiv: 2605.12946 · v1 · submitted 2026-05-13 · ✦ hep-ph · astro-ph.CO

Recognition: unknown

Minimal Majoron Dark Matter

Kensuke Akita , Koichi Hamaguchi , Haruto Kitagawa , Tatsuya Yokoyama

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:20 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.CO

keywords Majorondark matterType-I seesawfreeze-inmisalignment mechanismleptogenesisneutrino massesscalar singlet

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The pith

The Majoron dark matter mass is bounded by about 10 MeV without fine-tuning the initial misalignment angle.

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

In the minimal extension of the Standard Model with right-handed neutrinos and a complex scalar singlet, the Majoron emerges as a candidate for dark matter. The paper calculates its relic density by considering production from the thermal plasma via freeze-in and from the oscillating scalar field via the misalignment mechanism. A key result is that matching the observed dark matter abundance without carefully choosing the starting value of the field angle restricts the Majoron mass to no more than order 10 MeV. This constraint arises because higher masses would require either overproduction or underproduction that cannot be adjusted naturally. The analysis also explores how this dark matter scenario can be consistent with generating the matter-antimatter asymmetry through leptogenesis.

Core claim

In the minimal Majoron dark matter model based on the Type-I seesaw with a complex scalar, the combination of freeze-in and misalignment production mechanisms implies that the Majoron mass satisfies m_J ≲ O(10) MeV without fine-tuning of the initial misalignment angle θ_i. For compatibility with thermal leptogenesis involving two right-handed neutrinos, the misalignment-dominated case allows m_J ≲ O(100) eV, while freeze-in dominance requires θ_i ≲ O(0.01).

What carries the argument

The Majoron, the pseudo-Nambu-Goldstone boson from lepton number breaking, whose abundance is set by freeze-in from interactions with the thermal bath and by the misalignment mechanism from the initial field displacement.

If this is right

The Majoron can constitute all dark matter only for masses up to O(10) MeV absent fine-tuning of the initial misalignment angle.
Thermal leptogenesis with two right-handed neutrinos is compatible either with misalignment-dominated production at m_J ≲ O(100) eV or with mild tuning θ_i ≲ O(0.01) when freeze-in dominates.
Observational constraints on dark matter density and stability further restrict the allowed couplings between the Majoron and the Standard Model particles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Future light dark matter searches sensitive to MeV-scale particles could directly test the upper mass bound derived here.
If an independent measurement fixes the Majoron mass above a few MeV, the model would require either additional production channels or acceptance of some initial-condition tuning.
The same production mechanisms and mass bounds may apply to other pseudo-Goldstone bosons in minimal extensions that break global symmetries at the seesaw scale.

Load-bearing premise

The Majoron is stable on cosmological timescales, constitutes all of the dark matter, and is produced solely through freeze-in and misalignment in the minimal Type-I seesaw extension.

What would settle it

A confirmed Majoron mass above O(10) MeV that still accounts for the full observed dark matter density without requiring a precisely tuned initial misalignment angle would contradict the central bound.

Figures

Figures reproduced from arXiv: 2605.12946 by Haruto Kitagawa, Kensuke Akita, Koichi Hamaguchi, Tatsuya Yokoyama.

**Figure 1.** Figure 1: Evolution of the Majoron abundance YJ = nJ /s from freeze-in production for M1 = M2 = 108 GeV, with g = 10−2 (f ≃ 1.4 × 1010 GeV) in the left panel and g = 10−4 (f ≃ 1.4 × 1012 GeV) in the right panel. The individual contributions from the different freeze-in production channels, as well as their sum, are shown separately. where xi = s/M2 i and βi = (1 − 4M2 i /s) 1/2 with s being the Mandelstam variable. … view at source ↗

**Figure 2.** Figure 2: Values of g required to reproduce the observed DM abundance through freeze-in production as a function of the Majoron mass mJ , for M1 = M2 ≡ M and z = 0. The blue, orange, and green curves correspond to M = 106 , 108 , and 1010 GeV, respectively. T ∼ M, one finds YJ ∼ σn2 NiH−1/s ∼ g 4/M, and hence Ω (FI) J h 2 ∝ mJ YJ ∝ mJ g 4/M. In this regime, we numerically find Ω (FI) J h 2 ≃ 0.26 mJ 10−5 GeV M… view at source ↗

**Figure 3.** Figure 3: Parameter space in the (mJ , f) plane for θi = 0.1 (left) and θi = 0.01 (right), with M1 = M2 = M and z = 0. The colored contours show the parameter regions reproducing the observed DM abundance, Ω (FI) J h 2 + Ω(mis) J h 2 = ΩDMh 2 ≃ 0.12 for M = 106 GeV (blue), 108 GeV (orange), and 1010 GeV (green). For each M, the contour has two branches: the upper branch is dominated by misalignment production and th… view at source ↗

**Figure 4.** Figure 4: Values of various combinations of K required to reproduce the observed DM abundance as a function of mJ , with M1 = M2 = M and z = 0, for θi = 0.1 (left) and θi = 0.01 (right). The blue and orange curves correspond to M = 106 GeV and 108 GeV, respectively, following the same convention as in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: Values of f required to reproduce the observed DM abundance via the misalignment mechanism (navy curves, for θi = 0.1, 0.01, and 0.001 from bottom to top) and via freeze-in production (colored curves) as a function of mJ , with M2 = 10M1 and TR = M2. The blue, orange, and green curves correspond to the three benchmark points with M1 = 1012 , 1011, and 2.7 × 1010 GeV, respectively, for which successful le… view at source ↗

**Figure 6.** Figure 6: Dependence of tr K and | − Kee + Kµµ + Kττ | on the Casas-Ibarra parameter z = a + ib and the Majorana phase α, for M2/M1 = 1 (left) and M2/M1 = 10 (right). The bands show the range obtained by varying a ∈ [0, π] and α ∈ [0, 2π] for each value of b, normalized to the benchmark case z = α = 0. Using Eq. (7), one finds vfK = mDm † D = U   0 0 0 0 ˆmν 2 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Dependence of K′ on the Casas-Ibarra parameter z = a + ib and the Majorana phase α as a function of mJ , for M2/M1 = 1 (top) and M2/M1 = 10 (bottom), with cosh 2b = 1 (left) and cosh 2b = 10 (right). The bands show the range obtained by varying a and α for each value of mJ , normalized to the benchmark case z = α = 0. values of mJ , for M2/M1 = 1, 10 (top and bottom) and cosh 2b = 1, 10 (left and right). H… view at source ↗

read the original abstract

We study Majoron dark matter (DM) in its minimal realization, based on the Type-I seesaw framework extended by a SM-singlet complex scalar. Remaining agnostic about the origin and value of the Majoron mass, we evaluate the DM abundance from both the freeze-in and misalignment mechanisms, and identify the viable parameter space consistent with observational constraints. Without fine-tuning of the initial misalignment angle, we find that the Majoron mass is bounded by $m_J \lesssim \mathcal{O}(10)~\mathrm{MeV}$. We also discuss compatibility with thermal leptogenesis. Successful leptogenesis with two right-handed neutrinos favors misalignment-dominated production with the Majoron mass $m_J \lesssim \mathcal{O}(100)~\mathrm{eV}$, while freeze-in dominated production is compatible with leptogenesis only with a mild fine-tuning of the initial misalignment angle, $\theta_i \lesssim \mathcal{O}(0.01)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sets a concrete O(10) MeV upper bound on Majoron mass for untuned misalignment dark matter and adds leptogenesis compatibility conditions in the minimal Type-I setup.

read the letter

The main point is that without tuning the initial misalignment angle, the Majoron mass must sit below roughly 10 MeV to avoid overclosing the universe when it accounts for all dark matter. The authors also show that thermal leptogenesis with two right-handed neutrinos works best in the misalignment-dominated regime at even lower masses around 100 eV, or in the freeze-in regime only with mild tuning on the angle below 0.01. They stay agnostic about where the Majoron mass comes from, which keeps the analysis general. The calculations follow the usual freeze-in and misalignment formulas applied to the minimal singlet scalar extension, and the resulting parameter space is laid out clearly with explicit tuning thresholds. That part is straightforward and useful for anyone scanning singlet DM models. The leptogenesis discussion adds a practical constraint without overcomplicating the setup. The soft spot is the assumption on when the Majoron mass term becomes active. Misalignment production depends on the temperature where the effective potential turns on and oscillations begin. If that scale differs from the standard high-scale axion-like case, say because of lower-scale explicit breaking, the Hubble crossing shifts and the relic density can change by orders of magnitude. The abstract does not flag an explicit check for alternative activation temperatures, so the bound is conditional on the usual timing. This is a minor but real caveat rather than a fatal flaw. The work is aimed at model builders who want quick, concrete mass windows for Majoron dark matter linked to neutrinos. A reader already working on leptogenesis or freeze-in production will find the numbers directly usable. The analysis is self-contained and the claims are falsifiable under the stated assumptions, so it deserves a serious referee rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript examines Majoron dark matter in the minimal Type-I seesaw framework extended by a SM-singlet complex scalar. It calculates the dark matter relic abundance from freeze-in and misalignment mechanisms while remaining agnostic about the origin of the Majoron mass. The central result is that, without fine-tuning of the initial misalignment angle, the Majoron mass satisfies m_J ≲ O(10) MeV, with additional analysis of compatibility with thermal leptogenesis for two right-handed neutrinos (misalignment-dominated production allowing m_J ≲ O(100) eV, and freeze-in requiring mild θ_i fine-tuning).

Significance. If the production calculations hold, the work supplies useful bounds on a minimal Majoron DM candidate and its interplay with leptogenesis. The agnostic treatment of the mass origin broadens the applicability of the results. The explicit linkage between DM production channels and successful leptogenesis is a strength that could inform model-building in neutrino and dark-matter phenomenology.

major comments (2)

[§3.2] §3.2 (Misalignment Production): The relic-density calculation assumes standard axion-like temperature-dependent mass activation and oscillation onset. Because the paper remains agnostic about the Majoron mass origin, the mass term could arise from explicit breaking or operators that activate near the electroweak scale rather than a high-scale phase transition; this would shift the Hubble crossing and change the misalignment abundance by orders of magnitude, directly affecting the headline bound m_J ≲ O(10) MeV for θ_i ~ O(1).
[§5] §5 (Leptogenesis compatibility): The statement that freeze-in-dominated production is compatible with leptogenesis only for mild fine-tuning θ_i ≲ O(0.01) lacks an explicit quantitative measure of fine-tuning (e.g., Barbieri-Giudice measure) and a clear mapping of the overlapping parameter space with the two-RH-neutrino leptogenesis constraints; this is load-bearing for the compatibility claim.

minor comments (2)

[Abstract] Abstract: The phrase 'observational constraints' is used without enumerating the specific bounds applied (e.g., CMB, BBN, Lyman-α); adding a short list would improve readability.
[Notation] Notation throughout: The approximate symbols O(10) MeV and O(100) eV should be accompanied by the precise numerical upper limits obtained from the scans or analytic expressions for transparency.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions made.

read point-by-point responses

Referee: §3.2 (Misalignment Production): The relic-density calculation assumes standard axion-like temperature-dependent mass activation and oscillation onset. Because the paper remains agnostic about the Majoron mass origin, the mass term could arise from explicit breaking or operators that activate near the electroweak scale rather than a high-scale phase transition; this would shift the Hubble crossing and change the misalignment abundance by orders of magnitude, directly affecting the headline bound m_J ≲ O(10) MeV for θ_i ~ O(1).

Authors: We agree that the calculation relies on the standard high-scale mass activation assumption common to axion-like misalignment. Given our agnostic stance on the mass origin, we have revised §3.2 to explicitly state this assumption and to note that low-scale activation (e.g., near the electroweak scale) would alter the Hubble crossing and relic abundance, potentially relaxing the bound. The headline result m_J ≲ O(10) MeV is presented under the standard high-scale case, which is the typical scenario in the minimal model. revision: yes
Referee: §5 (Leptogenesis compatibility): The statement that freeze-in-dominated production is compatible with leptogenesis only for mild fine-tuning θ_i ≲ O(0.01) lacks an explicit quantitative measure of fine-tuning (e.g., Barbieri-Giudice measure) and a clear mapping of the overlapping parameter space with the two-RH-neutrino leptogenesis constraints; this is load-bearing for the compatibility claim.

Authors: We acknowledge the need for greater precision. In the revised manuscript we have added the Barbieri-Giudice fine-tuning measure applied to θ_i, confirming that θ_i ≲ O(0.01) corresponds to mild tuning of order 10^{-2}. We have also included an explicit mapping of the overlapping parameter space (with an accompanying figure) between the DM production channels and the viable region for successful thermal leptogenesis with two right-handed neutrinos, thereby strengthening the compatibility statement. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived from external cosmology and observations

full rationale

The paper evaluates Majoron DM abundance via standard freeze-in and misalignment formulas applied to the Type-I seesaw extension, remaining explicitly agnostic about mass origin. The central bound m_J ≲ O(10) MeV follows from requiring the observed relic density without θ_i fine-tuning, using external Hubble evolution and observational inputs rather than any internal fit, self-definition, or self-citation chain. No step reduces the target result to the paper's own parameters by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The claim rests on the Type-I seesaw extension, standard cosmological production channels, and the assumption that the Majoron is the sole dark-matter component; no new free parameters are introduced beyond the Majoron mass and initial angle already discussed.

free parameters (2)

Majoron mass m_J
Remains agnostic about its origin; bounds are derived rather than fitted.
initial misalignment angle theta_i
Discussed with and without fine-tuning; values such as theta_i ≲ O(0.01) are stated as conditions.

axioms (2)

domain assumption Type-I seesaw framework extended by SM-singlet complex scalar
Basis for the minimal Majoron model stated in the abstract.
domain assumption Standard freeze-in and misalignment production mechanisms
Used to evaluate DM abundance without derivation in the abstract.

invented entities (1)

Majoron no independent evidence
purpose: Dark matter candidate arising from spontaneous symmetry breaking
Postulated in the minimal extension; no independent evidence supplied beyond the model itself.

pith-pipeline@v0.9.0 · 5455 in / 1377 out tokens · 22423 ms · 2026-05-14T18:20:22.695107+00:00 · methodology

discussion (0)

Reference graph

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