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Leptogenesis with Majoron Dark Matter

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read One seesaw model with a spontaneously broken lepton number explains both dark matter and the baryon asymmetry through the same higher-dimensional operator.

desk verdict Same dimension-five operator drives Majoron freeze-in and resonant leptogenesis; the correlation is partly fitted, but the paper is a solid, referee-able viability study. read the letter →

arxiv 2412.14121 v2 pith:CDI7KZL3 submitted 2024-12-18 hep-ph astro-ph.CO

Stephen F. King , Soumen Kumar Manna , Rishav Roshan , Arunansu Sil This is my paper · ORCID
classification hep-phastro-ph.CO
keywords Majorondarkmatterfreeze-inresonantleptogenesistypeIseesawleptonnumberviolationright-handedneutrinosbaryonasymmetry
topics Matter-Antimatter AsymmetryDark Matter
open problems Dark MatterMatter-Antimatter Asymmetry
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

Two of the biggest unsolved numbers in cosmology—the dark matter abundance and the baryon asymmetry—could come from the same small symmetry-violating sector. This paper aims to show that in a type I seesaw model (light neutrino masses generated by heavy right-handed neutrinos) with two right-handed neutrinos and a spontaneously broken global lepton number, the same dimension-five operators that split the degenerate right-handed neutrino masses (enabling resonant leptogenesis) also mediate the annihilation $N_i N_i \to \chi\chi$ that produces the Majoron dark matter by freeze-in. Solving the coupled Boltzmann equations, the authors identify a parameter region where the observed relic density $\Omega_\chi h^2 \simeq 0.12$ and baryon asymmetry $Y_B \simeq 8.75\times10^{-11}$ are both obtained, with a sub-GeV Majoron and right-handed neutrinos above about $10^6$ GeV. If correct, neutrino mass, dark matter, and baryogenesis are not three independent problems but three outputs of one seesaw sector.

What carries the argument

The load-bearing objects are the dimension-five $U(1)_L$-breaking operators of Eq. (6): $(c_1/2\Lambda)(\Phi^2 + \Phi^{*2})N_1^c N_1$, $(c_2/2\Lambda)(\Phi^2 + \Phi^{*2})N_2^c N_2$, and $y_{\alpha 2}\Lambda^{-1} L_\alpha \tilde H N_2 (\Phi + \Phi^*)$. After $\Phi$ develops a vev, the first two generate the diagonal mass splitting $\kappa = v_\phi^2/\Lambda$ that breaks the degeneracy of the two right-handed neutrinos, while the resulting $\chi^2 N_i N_i$ coupling gives the matrix element $|\mathcal{M}|^2 = (s-4M_i^2)/\Lambda^2$ for $N_iN_i\to\chi\chi$, a UV freeze-in process whose yield scales as $T_{\rm RH}/\Lambda^2$. On the leptogenesis side, the same splitting controls the self-energy CP asymmetry $S_{ij}$, which resonantly enhances $\epsilon_{N_i}$ when $M_i^2 - M_j^2 \sim M_i \Gamma_{N_j}$; the orthogonal-matrix parametrization of the neutrino Yukawa coupling supplies the CP-violating phases through the complex angle $\theta_R = z_R + i z_I$.

What would settle it

A concrete check: measure or exclude the monochromatic neutrino line from $\chi\to\nu\nu$. A detected line fixes $m_\chi$; feeding that mass into the relic-satisfying Boltzmann solutions fixes $\Lambda$, $T_{\rm RH}$, and the right-handed neutrino mass $M_1$ and splitting $\Delta M$, so the observed line energy either lands on the predicted $m_\chi$–$M_1$ track or rules the common region out. Alternatively, a global fit of neutrino oscillation data that leaves no complex $\theta_R$ able to produce $\epsilon_N \gtrsim 7\times10^{-6}$ at the relic-fixed splitting would falsify the leptogenesis side.

Watch

Extended reading notes

Core claim

The central claim is that the same lepton-number-violating terms control both sides of the cosmological puzzle. At the renormalizable level the two right-handed neutrinos couple off-diagonally through $f\Phi N_1^c N_2$, producing exactly degenerate masses $M = f v_\phi/\sqrt{2}$; the dimension-five operators $(\Phi^2 + \Phi^{*2}) N_i^c N_i / 2\Lambda$ add diagonal entries that split them, $M_{1,2} = f v_\phi/\sqrt{2} \pm v_\phi^2/\Lambda$, with $\Delta M = 2v_\phi^2/\Lambda$. This same splitting sets the resonance condition for leptogenesis, while the interaction $\chi^2(N_1^c N_1 - N_2^c N_2)/2\Lambda$ opens the annihilation channel $N_iN_i \to \chi\chi$ whose yield $Y_\chi \propto T_{\rm RH}/\Lambda^2$ produces the Majoron. A combined scan over $\Lambda$, $T_{\rm RH}$, $m_\chi$, and the complex angle $\theta_R$ of the neutrino Yukawa parametrization yields a common region in which both the dark matter relic density and the baryon asymmetry are reproduced; the required CP asymmetry is $\epsilon_N \gtrsim 7\times10^{-6}$, and the right-handed neutrino masses fall in the $10^6$–$10^{13}$ GeV range. The authors therefore conclude that the model offers a unified origin for neutrino mass, dark matter, and the baryon asymmetry.

Load-bearing premise

The whole result assumes the lepton-number-breaking scale sits above the reheating temperature, which in turn sits above both right-handed neutrino masses; if reheating were colder than those neutrinos, both the dark matter yield and the leptogenesis calculation would have to be redone.

Editorial extensions

If this is right

  • Matching the observed relic density fixes the right-handed neutrino mass splitting, so in the common region the baryon asymmetry is not an independent input: the same $\Lambda$ and $v_\phi$ set both the freeze-in yield and the leptogenesis resonance.
  • Resonant leptogenesis in this construction happens at high scale, with right-handed neutrino masses from about $10^6$ GeV to $10^{13}$ GeV, rather than at the TeV scale usually invoked for resonant leptogenesis.
  • The successful Majoron is sub-GeV and stable on cosmological timescales, but decays to neutrino pairs; for $m_\chi \gtrsim 4$ MeV it is within reach of monochromatic neutrino searches, and its two-loop decay to photons can be probed by gamma-ray telescopes.
  • The relative weight of self-energy versus vertex contributions to the CP asymmetry changes near $M_1 \sim 10^{11}$ GeV, giving the predicted $m_\chi$–$M_1$ correlation a distinctive shape that could be tested observationally.
  • If the right-handed neutrinos start with zero abundance, inverse decays bring them into equilibrium slightly below $T_{\rm RH}$; the Majoron yield is mildly suppressed and the required $m_\chi$ shifts upward, leaving the overall conclusion intact.

Reading between the lines

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

  • If a monochromatic neutrino line from $\chi\to\nu\nu$ is ever observed, it overdetermines the model: the line fixes $m_\chi$, the relic condition fixes $\Lambda$ and $T_{\rm RH}$, and leptogenesis then predicts $M_1$, so the observed line energy can be checked against the predicted $m_\chi$–$M_1$ track.
  • The high-scale common region evades TeV-scale collider tests but may be reachable through gravitational waves radiated by the heavy right-handed neutrinos during leptogenesis, a signature the paper notes but does not calculate.
  • A flavour-resolved calculation, which the paper drops for simplicity, could modify the washout efficiency and the required $\epsilon_N$; whether the common region survives flavour effects is a natural next check.
  • Relaxing the CP symmetry that forces equal $\Phi$ and $\Phi^*$ couplings introduces additional Majoron decay and production channels, turning the Majoron decay rate into a direct probe of CP violation in the lepton-number-breaking sector.
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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 constructs a type-I seesaw model with two right-handed neutrinos and a global U(1)_L symmetry spontaneously broken by a Z2-odd singlet scalar Phi. The resulting Majoron is proposed as a freeze-in dark matter candidate, produced by right-handed neutrino annihilation through a dimension-five lepton-number-violating operator; the same operator generates the diagonal RHN mass entries that split the otherwise degenerate pair and enables near-degenerate leptogenesis. The authors solve Boltzmann equations for the DM yield and the lepton asymmetry, give a benchmark with M1 = 1.4e10 GeV and CP asymmetry about 6e-6, and scan the TRH-Lambda plane to identify a common region satisfying both the DM relic density and the baryon asymmetry, yielding a correlation between m_chi and M1.

Significance. If valid, the model offers a compact single-operator link between neutrino mass, Majoron dark matter, and baryogenesis, with a sub-GeV DM candidate and RHN masses above roughly 1e6 GeV. The analysis uses standard Boltzmann equations and the Casas-Ibarra parametrization, provides a concrete benchmark, and explicitly compares self-energy and vertex contributions to the CP asymmetry. At the same time, the headline correlation is partly constructed rather than predicted, because m_chi is set for each scan point to match Omega_chi h^2 and theta_R is chosen to match the baryon asymmetry; several assumptions and observational constraints are not fully quantified.

major comments (3)
  1. [III B] The scale hierarchy v_phi > TRH > M_i introduced in Sec. III B is used in three places: the dimension-five operators in Eq. (6) must be active before reheating, the freeze-in source in Eq. (22) is evaluated with the equilibrium yield Y_N^eq, and the leptogenesis Boltzmann equations (29) and (38) assume radiation domination with M_i < TRH. This ordering is asserted rather than derived, and the thermalization check following Fig. 2 is performed only for the benchmark M1 = 1.4e10 GeV. The combined parameter space extends to M1 around 1e6 GeV (lower panel of Fig. 5), where the Casas-Ibarra Yukawa entries are of order 1e-5. Please show explicitly that Y_N tracks Y_N^eq before the DM production peak for representative low-M1 points, and quantify the shift in m_chi when the RHNs start with zero abundance. If TRH were below M_i, both the Y_chi scaling quoted in Sec. III B and the leptogenesis calculation would require re-initialization with a non-thermal RHN abundance.
  2. [IV, Eq. (36)] The text states that the self-energy CP asymmetry is maximised when M_i^2 - M_j^2 ~ M_i Gamma_Nj, but then says the mass difference is large compared to the decay widths. For the benchmark M1 = 1.4e10 GeV and M2 = 1.36e10 GeV, delta = M2^2 - M1^2 is about -1.1e19 GeV^2, while M_i Gamma_Ni is about 1e14 GeV^2 using Gamma_Ni ~ (y_nu^dagger y_nu)_ii M_i/(8 pi) with diagonal entries of order 1e-5. Thus delta exceeds the width term by about five orders of magnitude, and the self-energy factor in Eq. (36) is S12 ~ M_i/(2 Delta M) ~ 17 rather than the resonant value S ~ M_i/Gamma ~ 1e6. The quoted epsilon ~ 6e-6 therefore arises from off-resonance self-energy enhancement, not from the resonant pole. Please clarify whether 'resonant leptogenesis' is the correct label, and if the resonance condition is imposed, show how Lambda and the correlation plots in Fig. 5 are affected.
  3. [III B and Figs. 5, 7] The monochromatic neutrino line constraint quoted in Sec. III B ('These experiments restrict the Majoron parameter space for m_chi >~ 4 MeV') is not applied in the relic-satisfied parameter space of Fig. 2 or in the combined BAU plots of Figs. 5 and 7. As shown in Fig. 7, the lower-panel successful region reaches m_chi of order 1 GeV, and Fig. 2 includes relic-satisfying points with m_chi up to 10 GeV. Depending on the exact interpretation of the bound, a substantial part of the claimed common parameter region is either excluded or needs to be re-plotted. Please overlay the Borexino/KamLAND/Super-K/IceCube constraint and state clearly which part of the (TRH, Lambda) and (m_chi, M1) planes survives after all observational cuts.
minor comments (4)
  1. [III B and Fig. 2] The text says the viable Majoron mass range is 'O(100) GeV >= m_chi >= 1 keV', but the color bar in Fig. 2 has an upper limit of log10 m_chi = 1, i.e. 10 GeV; please make the caption and text consistent.
  2. [IV, Eq. (37)] In Eq. (37), the numerator contains m_2^2 - m_3^2 while the denominator contains combinations of m_2 and m_3 without squares; please check the dimensions and clarify the notation, since the quantities quoted are mass-squared differences.
  3. [IV, Sec. V] Flavor effects are neglected for leptogenesis, but the BAU patches in Fig. 5 span M1 values from about 1e8 to 1e12 GeV, where the two-flavor and three-flavor regimes are relevant; a short quantitative estimate of the expected uncertainty would strengthen the combined-region claim.
  4. [II B] There is a typographical error in the sentence near Eq. (3): 'as no neglect' should read 'as to neglect'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the common DM–leptogenesis region is a transparent consistency fit, not a prediction reduced by construction.

full rationale

The derivation chain is self-contained: the dimension-5 operators in Eq. (6) are posited explicitly, and both the RHN mass splitting ΔM = 2 vϕ^2/Λ (Eq. 12) and the freeze-in annihilation cross-section with |M|^2 ∝ 1/Λ^2 (Eq. 26) are computed from that same Lagrangian rather than quoted as inputs. The CP-asymmetry formula and Boltzmann equations are standard external results (Refs. [6,77-82]) and are applied, not fitted. The Majoron mass mχ is indeed assigned point-by-point to match Ωχh^2 and θR = 0.78 + 0.42i is chosen because it maximizes the CP asymmetry; the paper states this openly ("mχ, obtained in order to satisfy the correct relic for each point"), so the resulting TRH–Λ patches and mχ–M1 correlation are consistency regions for a model with free parameters, not predictions that reduce by construction. Self-citations such as [26] and [33] provide context and earlier constraints, but the present computation re-derives the production mechanism, so they are not load-bearing. The asserted hierarchy vϕ > TRH > Mi is an assumption flagged in Sec. III B and the conclusion; an unsupported assumption is a robustness limitation, not circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 3 invented entities

The model introduces a new scalar Φ, two right-handed neutrinos, and a Majoron. The central consistency result depends on several parameters that are scanned or chosen by hand: the Majoron mass, the complex angle θR, the Yukawa coupling f, the TRH/vϕ ratio, and the cutoff Λ. The axioms are the scale hierarchy, thermalization of the RHNs, the imposed CP symmetry, equal coefficients c1=c2, neglect of the Weinberg operator, and neglect of flavor effects in leptogenesis.

free parameters (5)
  • Majoron mass mχ = scanned; e.g., 2.5e-5 GeV for the benchmark point
    Introduced as a soft-breaking parameter in Eq. 5; in the relic scan, mχ is chosen for each (TRH, Λ) point to satisfy Ωχh²=0.12, so it is fitted to the data.
  • Complex angle θR = 0.78 + 0.42i
    Appears in the Casas-Ibarra R matrix of Eq. 14; the benchmark value is chosen because it maximizes the CP asymmetry F_y ≈ 0.7 and thus helps satisfy BAU.
  • Yukawa coupling f = 0.1 and 0.005
    Coupling of Φ to the right-handed neutrinos in Eq. 1; two representative values are chosen by hand, setting the heavy neutrino mass scale.
  • TRH/vϕ hierarchy ratio = 0.5 or 0.1
    The reheating temperature is fixed as a fraction of the lepton-number breaking scale rather than derived; this ratio partially determines the allowed parameter space.
  • Cutoff scale Λ = scanned 1e11 to 1e18 GeV; benchmark 1.5e14 GeV
    Suppression scale of the dimension-five operators in Eq. 6; it is scanned and selected to satisfy the relic density and BAU constraints.
assumptions (6)
  • domain assumption Global U(1)L is spontaneously broken before reheating, with vϕ > TRH.
    Assumed in Sec. III B to ensure ϕ decoupled and Majoron production from RHN annihilation dominates; the entire parameter scan uses this hierarchy.
  • domain assumption The two right-handed neutrinos are in thermal equilibrium at TRH, or quickly reach equilibrium via inverse decay.
    Used in Secs. III B and IV to compute freeze-in yield from equilibrium RHN abundances and to justify out-of-equilibrium decay for leptogenesis.
  • ad hoc to paper A CP symmetry under which Φ goes to Φ* ensures equal couplings of Φ and Φ* in the dimension-five Yukawa term.
    Invoked before Eq. 6 and in Appendix A; without it the Majoron can decay into neutrinos through new channels, though the widths are argued to be subdominant.
  • ad hoc to paper The dimension-five coefficients for the two RHN diagonal operators are taken equal (c1 = c2 = 1).
    Stated in Sec. II B: 'Considering c1 = c2 = 1 for simplification purpose.' This sets the mass splitting ΔM = 2 vϕ²/Λ; unequal coefficients would shift the parameter mapping, though the authors argue it is a redefinition.
  • domain assumption The Weinberg operator δij Li LH jH is negligibly small.
    Stated in Sec. II A: 'assuming δij is sufficiently small, we exclude this contribution from the analysis without any loss of generality.'
  • domain assumption Flavor effects in leptogenesis are neglected.
    Stated in Sec. IV: 'Here we do not incorporate the flavor effects on leptogenesis for simplicity.' This can modify washout and the final asymmetry.
invented entities (3)
  • Pseudoscalar Majoron χ independent evidence
    purpose: Freeze-in dark matter candidate whose decay into neutrino pairs is a search target
    The model gives a definite mass range (keV to GeV) and decay channels (χ to νν, χ to γγ) that existing neutrino and gamma-ray experiments can test, so there is a falsifiable handle outside the paper.
  • Heavy right-handed neutrinos N1 and N2
    purpose: Generate light neutrino masses via seesaw and the baryon asymmetry via resonant leptogenesis
    Their masses are around 1e6 to 1e13 GeV, far beyond direct search reach, and their only indirect evidence is the consistency of neutrino masses and leptogenesis within the model.
  • Lepton-number-carrying scalar Φ
    purpose: Spontaneously breaks global U(1)L, gives the Majoron and heavy neutrino masses
    No direct observable is predicted for the CP-even component ϕ, which is assumed heavy and decoupled from the thermal bath.

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Cite this review

Pith. "Pith review of Leptogenesis with Majoron Dark Matter." pith.science (2026). https://pith.science/paper/CDI7KZL3

@misc{pith2026241214121,
  author       = {Pith},
  title        = {Pith review of: Leptogenesis with Majoron Dark Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CDI7KZL3}},
  note         = {Machine review of arXiv:2412.14121}
}
abstract

We discuss a model of neutrino mass based on the type I seesaw mechanism embedded in a spontaneously broken global lepton number framework with a $Z_2$ symmetry. We show that the resulting Majoron is a viable freeze-in dark matter candidate. Two right-handed neutrinos are assumed to have dominant off-diagonal masses suggesting resonant leptogenesis as the origin of baryon asymmetry of the Universe. Explicit higher dimensional lepton number violating operators, are shown to play a crucial role in simultaneously controlling both the Majoron production in the early Universe and the right handed neutrino mass splitting relevant for resonant leptogenesis. We perform a combined analysis of Majoron dark matter and leptogenesis, discussing the relative importance of self energy and vertex contributions to CP asymmetry, and explore the parameter space, leading to an intricate relation between neutrino mass, dark matter and baryon asymmetry.

Figures

Figures reproduced from arXiv: 2412.14121 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic diagram for generation of the light neutrino [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Relic satisfied parameter space represented by colored [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Relic satisfied parameter space represented by colored [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Evolution of the yields of RHNs and lepton asymme [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The correct BAU satisfied parameter spaces are in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Plot showing the relative importance between the self [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Correlation between masses of Majoron and RHNs, [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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