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REVIEW 3 major objections 4 minor 1 cited by

Seesaw Portal to Super Heavy Dark Matter with $Z_3$ Symmetry

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Promoting the dark-sector symmetry from Z2 to Z3 opens a fast two-body decay of the dark scalar, making the m_N > m_phi mass ordering viable for superheavy freeze-in dark matter.

desk verdict A solid Z3 seesaw-portal dark matter idea with a fixable but load-bearing dimensional error in the printed decay widths. read the letter →

arxiv 2506.16100 v1 pith:IGXBQ4JZ submitted 2025-06-19 hep-ph

Cai-Xia Yang , Zhi-Long Han , Fei Huang , Yi Jin , Honglei Li This is my paper · ORCID
classification hep-ph
keywords freeze-indarkmatterseesawportalright-handedneutrinoZ3symmetrysuperheavyBigBangNucleosynthesisscalardecayleptogenesis
topics Dark Matter
open problems Dark Matter
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 proposes a minimal repair to the seesaw portal for superheavy dark matter, in which right-handed neutrinos explain neutrino masses and mediate freeze-in production of a very heavy dark fermion. In the simplest version with Z2 dark symmetry, the mass ordering with the right-handed neutrino heavier than the dark scalar is nearly ruled out, because the scalar decays only through feeble, phase-space-suppressed channels and lives too long for Big Bang Nucleosynthesis. By changing the dark symmetry to Z3, the model gains the interaction y_chi phi barchi^c chi, which gives the scalar a two-body decay phi -> chi chi. Even with a tiny y_chi this decay is fast enough to keep phi short-lived for m_phi > 2 m_chi, so the previously excluded mass ordering becomes cosmologically allowed. The paper then derives the resulting relic abundance for both mass orderings and shows that the new channel also makes dark matter production more efficient than in the Z2 model.

What carries the argument

The central object is the Z3-symmetric interaction y_chi phi barchi^c chi, allowed when the dark fields transform as chi -> $e^{{2 pi i/3}}$ chi and phi -> $e^{{2 pi i/3}}$ phi. It produces the two-body decay phi -> chi chi with partial width given by Eq. (5), and this decay is the load-bearing mechanism: for m_phi > 2 m_chi it dominates the total width even for y_chi ~ $10^{{-12}}$, pushing tau_phi below $10^{{-2}}$ s and thereby making the m_N > m_phi mass ordering consistent with BBN and CMB constraints. The same width also feeds the Boltzmann equations for Y_chi and Y_phi, raising the final dark matter abundance relative to the Z2 model.

What would settle it

Using a full Boltzmann code that tracks Y_N from a specified reheating history instead of setting Y_N = Y_N^eq would settle the size of the Type B production; if the actual N abundance at T ~ m_N is substantially below equilibrium, the coupling values quoted for the m_N > m_phi benchmarks are underestimated. Separately, detecting a late-time decay of a superheavy scalar with tau_phi > $10^{{-2}}$ s and m_phi > $10^{4}$ GeV into SM particles would directly contradict the claim that the Z3 decay phi -> chi chi always keeps phi short-lived in this regime.

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Extended reading notes

Core claim

The central claim is that a Z3 dark symmetry removes the main cosmological obstruction to the seesaw portal at superheavy masses. With the charge assignment chi -> $e^{{2 pi i/3}}$ chi and phi -> $e^{{2 pi i/3}}$ phi, the interaction y_chi phi barchi^c chi is allowed, and it induces phi -> chi chi with partial width Gamma(phi->chi chi) = $y_chi^{2}$/(4 pi m_phi) (1 - 4 $m_chi^{2}$/$m_phi^{2}$)^{3/2}. For m_phi > 2 m_chi this two-body decay dominates the scalar width even for y_chi ~ $10^{{-12}}$, so tau_phi falls below the $10^{{-2}}$ s BBN limit for m_phi > $10^{4}$ GeV. The paper solves the coupled Boltzmann equations for both mass orderings and finds the final dark matter abundance is enlarged by the factor [2 + BR(phi->chi chi)] relative to the Z2 model, up to 3/2 in the limit where phi->chi chi dominates; in the m_N > m_phi ordering the dominant production is N -> phi chi, while for m_phi > m_N it is the neutrino Yukawa scattering phi chi -> h nu.

Load-bearing premise

The argument assumes the heavy right-handed neutrino N remains in thermal equilibrium, with Y_N equal to its equilibrium value, throughout the epoch when dark matter is produced; if N does not fully thermalize for a given y_nu or reheating history, the Type B abundance calculation and the couplings read off from it would shift.

Editorial extensions

If this is right

  • The mass ordering m_N > m_phi, which the minimal Z2 model can only accommodate by fine-tuning the mass spectrum, is open for m_phi > 2 m_chi and m_phi > 10^4 GeV without additional tuning.
  • For m_N > m_phi with high reheating temperature, the observed relic density is obtained from N -> phi chi decay with y_N^2 ~ 4 x 10^{-25} m_N/m_chi, and the required y_N grows with the ratio m_N/m_chi.
  • For m_phi > m_N, production is dominated by phi chi -> h nu scattering, and the required coupling satisfies y_nu^2 y_N^2 ~ 2 x 10^{-22} m_phi/m_chi, so larger m_phi/m_chi demands larger y_N.
  • The Z3 channel makes the scalar-to-dark-matter conversion happen earlier and more efficiently, so the same couplings produce a larger final dark matter abundance than in the Z2 model, by up to a factor 3/2.
  • Low reheating temperatures exponentially suppress production; as T_RH falls below m_phi or m_N, the coupling y_N required for the relic density rises until the perturbativity bound y_N <= sqrt(4 pi) imposes a lower limit on reheating for each mass spectrum.

Reading between the lines

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

  • The same symmetry trick, promoting a Z2 dark parity to a Z3 so that a forbidden two-body decay becomes allowed, could rescue other freeze-in portal models with long-lived mediators; the key requirement is only that the mediator and dark matter carry charges whose product matches the new interaction.
  • Because phi -> chi chi is invisible, the model evades BBN without producing late electromagnetic or hadronic injection; the most direct future test is therefore the reheating temperature inferred from inflationary observables, which decides whether the needed couplings stay below perturbativity.
  • A non-equilibrium treatment of the right-handed neutrino would test the weakest step: if N is not thermalized at T ~ m_N, the Type B couplings shift upward, but the qualitative conclusion that m_N > m_phi is allowed would persist because the phi -> chi chi width depends only on y_chi.
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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-portal model of super-heavy dark matter with an exact Z3 symmetry acting on a dark scalar φ and a dark fermion χ. The new interaction yχ φ \bar{χ}^c χ is claimed to permit the decay φ→χχ, making φ short-lived even for the mass ordering m_N > m_φ and thereby evading BBN constraints. The authors formulate Boltzmann equations for two mass orderings (type A, m_φ > m_N; type B, m_N > m_φ), compute relic densities by freeze-in, and study the impact of low reheating temperature. The paper argues that the Z3 model opens parameter space excluded in the simpler Z2 model.

Significance. If the mechanism were correct, the paper would provide a simple extension of the seesaw portal that rescues the mass ordering m_N > m_φ from BBN constraints, with quantitative predictions for the required Yukawa couplings. The paper uses the standard freeze-in formalism, includes RIS subtraction, and provides numerical Boltzmann solutions with micrOMEGAs, which are appropriate tools for this class of model. However, the central decay φ→χχ is forbidden by the exact Z3 symmetry under the stated charge assignments, and the printed two-body decay widths are dimensionally inconsistent. These issues invalidate the main quantitative results, so the paper in its current form cannot be accepted.

major comments (3)
  1. [Section II.A, Eqs. (1) and (5)] The decay φ→χχ, which is the central mechanism for evading the BBN bound, is forbidden by the exact Z3 symmetry with the charge assignments stated in the Introduction (χ → e^{i2π/3}χ and φ → e^{i2π/3}φ). The operator yχ φ \bar{χ}^c χ is Z3-invariant because the total charge of the operator is 3 ≡ 0 mod 3. However, the initial state φ carries charge +1, while the final state χχ carries charge +2 mod 3. A Z3-invariant Hamiltonian cannot connect two states in different charge sectors, so the amplitude for φ→χχ vanishes identically. Thus the BBN-avoidance argument based on this decay is invalid from the outset.
  2. [Section III, Eq. (12)] The decay N→φχ is likewise forbidden by the Z3 selection rule. The right-handed neutrino N is neutral under Z3, while φ and χ each carry charge +1, so the final state N→φχ has total charge +2 mod 3, which cannot equal the initial charge 0. The invariant interaction y_N φ \bar{χ} N actually connects N to a φ and an anti-χ, not to a φ and a χ. Consequently the Type B production mechanism, which relies on the decay N→φχ and the resulting abundance formula in Eq. (25), does not follow. This is a second, independent instance where the charge assignments are inconsistent with the processes used.
  3. [Section II.A, Eqs. (4), (5), and (12)] The printed partial decay widths have mass dimension -1 instead of +1, because the right-hand sides are proportional to 1/m_φ or 1/m_N rather than m_φ or m_N. A decay width must have mass dimension +1. Even setting aside the symmetry selection-rule problem, the lifetime curves in Fig. 1, the branching ratios used in Eqs. (17), (21), and (25), and the y_N–y_χ relations in Figs. 3 and 5 are not reproducible from the displayed formulas. The authors should correct these expressions and verify that their numerical code uses the dimensionally correct widths.
minor comments (4)
  1. [Throughout] The name 'Planck' is misspelled as 'Plank' in several places, including the text near Eq. (10) and Fig. 4.
  2. [Section IV.A, Eq. (20)] The numerical prefactor in Eq. (20) is printed in a garbled form ('214π^5'); the expression should be checked and simplified so that the coefficient is unambiguous.
  3. [Section III] The statement that the right-handed neutrino N remains in thermal equilibrium because y_ν is 'large enough' is plausible for m_N above the electroweak scale, but the quantitative condition (e.g., Γ/H at T = m_N) is not given; since the Type B calculation assumes Y_N = Y_N^{eq}, the criterion for this assumption should be stated explicitly.
  4. [Section IV] The perturbativity constraint y_N ≤ sqrt(4π) is applied to y_N, but no analogous perturbativity constraint on y_χ is discussed; for consistency, the same bound should be applied to both dark-sector Yukawa couplings.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Z3 decay channel and BBN-avoidance claim are evaluated against external BBN constraints, and the relic-density matching is a parameter fit, not a prediction.

full rationale

The paper's derivation chain is not circular. The central claim—that the Z3-allowed interaction y_chi phi barchi^c chi opens phi -> chi chi and makes phi short-lived when m_N > m_phi—is evaluated by computing the partial width from the Lagrangian and comparing the resulting lifetime with the external BBN bound tau_phi < 10^-2 s (refs. 46 and 47). This comparison does not use the measured relic density as an input; the dark-sector couplings are treated as free parameters. The relic-density section then takes Omega_DM h^2 = 0.120 as an external constraint and solves for y_N and y_chi; these are parameter determinations for the model, not predictions of a measured quantity, so no 'fit renamed as prediction' step occurs. Self-citations to the authors' earlier Z3 models (refs. 41 and 42) merely supply precedent for the Z3 charge assignment and the existence of the y_chi interaction; they are not invoked as load-bearing uniqueness theorems, and an independent non-self citation (ref. 40) accompanies them. The paper is otherwise self-contained against external benchmarks (BBN lifetime limits, Planck relic density, micrOMEGAs). A separate correctness concern, not a circularity concern, is that the printed two-body widths in Eqs. (4), (5), and (12) have mass dimension -1 instead of +1; if taken literally this would affect quantitative lifetimes and branching ratios, but the qualitative mechanism remains externally constrained and is not circular.

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

The central model rests on several free mass and coupling parameters that are scanned or fitted to the observed relic density, plus standard freeze-in assumptions. The most consequential internal assumptions are thermal equilibrium of N, zero initial abundances, and vanishing Higgs portal and trilinear couplings. The invented dark sector particles have no independent observational evidence.

free parameters (5)
  • y_N = varied; roughly 1e-11 to 1e-2 depending on spectrum and TRH
    Chosen to reproduce the observed relic density Omega h^2 ~ 0.120; see Figs. 3, 5, and 6.
  • y_chi = varied; often set equal to y_N
    Controls the phi to chi chi width and branching ratio; chosen to satisfy relic density and BBN constraints.
  • m_chi = benchmarks from 1e5 to 1e13 GeV
    Scanned dark matter mass; not predicted by the model.
  • m_phi = benchmarks from 1e7 to 1e15 GeV
    Scanned mediator mass; determines which decay channels are open.
  • m_N = benchmarks from 1e9 to 1e14 GeV
    Scanned right-handed neutrino mass; subject to the leptogenesis lower bound.
assumptions (6)
  • domain assumption The right-handed neutrino N is in thermal equilibrium, Y_N = Y_N^eq, throughout the production epoch.
    Stated in Section III; used to set the source term for N to phi chi production in Type B. If N is not thermal, the relic density calculation changes.
  • domain assumption Initial dark sector abundances vanish at reheating: Y_phi = Y_chi = 0 at T = TRH.
    Freeze-in boundary condition introduced in Section III; non-thermal production during reheating is not modeled.
  • ad hoc to paper The Higgs portal coupling lambda_Hphi and the trilinear coupling mu are set to zero.
    Section II states this is assumed for simplicity. Nonzero values would introduce additional production and decay channels, changing relic density and lifetimes.
  • domain assumption The BBN bound requires the dark scalar lifetime to be below about 1e-2 s.
    Imported from refs. [46,47] and used to exclude long-lived phi in the Z2 model and to validate the Z3 model.
  • standard math The Z3 charge assignments allow the y_chi phi barchi^c chi operator.
    Group-theoretic assignment introduced in Section II and borrowed from refs. [40-42]; if the charges were different, the key decay would be forbidden.
  • domain assumption Only hierarchical mass spectra with phi to chi chi open are considered; compressed spectra are excluded.
    Section IV restricts attention to m_phi >> m_chi; the BBN-avoidance via phi to chi chi requires m_phi > 2m_chi.
invented entities (3)
  • Dark scalar phi
    purpose: Mediates the neutrino-dark matter interaction and provides the short-lived source via phi to chi chi.
    No direct detection or collider handle is provided; its mass and couplings are free parameters.
  • Dark fermion chi
    purpose: Superheavy FIMP dark matter candidate.
    Its only role in the paper is to match relic density for chosen masses; no independent observable is identified.
  • Z3 discrete symmetry
    purpose: Stabilizes the dark sector and permits the y_chi operator while forbidding Z2-parity-breaking decays.
    A common model-building device, but no empirical evidence for this specific charge assignment is presented.

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Cite this review

Pith. "Pith review of Seesaw Portal to Super Heavy Dark Matter with $Z_3$ Symmetry." pith.science (2026). https://pith.science/paper/IGXBQ4JZ

@misc{pith2026250616100,
  author       = {Pith},
  title        = {Pith review of: Seesaw Portal to Super Heavy Dark Matter with $Z_3$ Symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IGXBQ4JZ}},
  note         = {Machine review of arXiv:2506.16100}
}
abstract

Right-handed neutrinos $N$ are introduced to explain the origin of the tiny neutrino masses via the seesaw mechanism. Required by relatively large Yukawa coupling and leptogenesis, masses of right-handed neutrinos are beyond $10^{9}$ GeV. Such heavy right-handed neutrino can mediate the production of super heavy dark matter $\chi$ via the freeze-in mechanism. In the minimal $Z_2$ symmetric model, the right-hand neutrino portal interaction is $y_N \phi \bar{\chi} N$ with the dark scalar $\phi$. One drawback of the $Z_2$ symmetric model is that the mass ordering $m_N>m_\phi$ with long-lived $\phi$ is almost ruled out by Big Bang Nucleosynthesis. In this paper, we propose that by extending the dark symmetry to $Z_3$, one additional interaction $y_\chi \phi \bar{\chi}^c \chi$ is further allowed. In this way, the new decay mode $\phi\to \chi\chi$ would lead to the dark scalar $\phi$ being short-lived even with a feeble $y_\chi$, thus it is allowed by the cosmological constraints. The phenomenology of the $Z_3$ symmetric super heavy dark matter model is also studied in this paper.

Figures

Figures reproduced from arXiv: 2506.16100 by the authors.

Figure 1
Figure 1. FIG. 1. The lifetime of dark scalar in scenario with [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Evolution of dark sector for ordering type A. The solid (dashed) blue and green lines are the results for dark [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The dependence of Yukawa couplings for ordering type A. Panel (a): The required values of the Yukawa [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Evolution of dark sector for ordering type B. The solid (dashed) blue and green lines are the results for dark [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The dependence of Yukawa couplings for ordering type B. Panel (a): The required values of the Yukawa [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The required values of the Yukawa coupling [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The relic density required value of the right-handed neutrino coupling [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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