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arxiv: 2511.21634 · v2 · submitted 2025-11-26 · ✦ hep-ph · astro-ph.CO

Recognition: 2 theorem links

· Lean Theorem

Cosmological Probes of Lepton Parity Freeze-in Dark Matter: Delta N_{rm eff} & Gravitational Waves

Ernest Ma , Partha Kumar Paul , Narendra Sahu
Authors on Pith no claims yet

Pith reviewed 2026-05-17 04:42 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.CO
keywords dark matterfreeze-in productionlepton paritygravitational wavesDelta N_effelectroweak phase transitionseesaw mechanismMajorana fermion
0
0 comments X

The pith

Lepton parity stabilizes a Majorana fermion as freeze-in dark matter that can source gravitational waves or extra relativistic degrees of freedom.

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

The paper shows how lepton parity, a residual symmetry of the type-I seesaw, protects a Majorana fermion S as a stable dark matter candidate. S is produced non-thermally either by decays of right-handed neutrinos when the reheating temperature exceeds their mass or by one-loop Higgs decays at lower reheating temperatures. A singlet scalar odd under lepton parity controls the outcome: strong coupling to the Higgs drives a first-order electroweak phase transition that emits gravitational waves, while weak coupling lets the scalar freeze out and decay late, adding to the effective number of relativistic species. This setup accommodates dark matter masses from the MeV to TeV range and predicts signals accessible to upcoming gravitational-wave detectors and cosmic microwave background measurements.

Core claim

Introducing a Majorana fermion S with even lepton parity keeps it stable as dark matter. The odd scalar σ couples to the right-handed neutrino N and to S, allowing two freeze-in channels: N decays to S and σ when the reheating temperature lies above the N mass, or Higgs decays to two S particles at one loop when the reheating temperature sits below the N mass but above the electroweak scale. Large σ-Higgs quartic coupling produces a strong first-order electroweak phase transition whose gravitational-wave spectrum is potentially detectable, whereas small coupling permits σ to freeze out and later decay into S and neutrinos, yielding a measurable shift in ΔN_eff.

What carries the argument

The lepton parity symmetry (-1)^L that stabilizes the even-parity Majorana fermion S while the odd scalar σ mediates either a strong first-order electroweak phase transition or late decays contributing to ΔN_eff.

If this is right

  • Dark matter masses span the MeV to TeV interval while satisfying the observed relic density through the two distinct freeze-in channels.
  • A sufficiently strong σ-Higgs quartic coupling generates a gravitational-wave background from the first-order electroweak phase transition.
  • A weak σ-Higgs coupling instead allows late σ decays that produce a detectable excess in the effective number of relativistic degrees of freedom.
  • The two regimes are separated by the relative size of the reheating temperature compared with the right-handed neutrino mass.
  • Future gravitational-wave observatories and cosmic microwave background experiments can jointly constrain or discover the model parameters.

Where Pith is reading between the lines

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

  • The same scalar that controls the phase transition or late decays could also influence the neutrino mass matrix through higher-order effects without spoiling the parity protection.
  • Non-detection of both signals would force the model into a narrow window of very small couplings, potentially testable by direct dark matter searches at low masses.
  • The framework naturally accommodates variations in the right-handed neutrino spectrum while preserving the parity-based stability of S.

Load-bearing premise

The Majorana fermion S stays non-thermalized during its entire production epoch and lepton parity remains exactly unbroken by any additional effects.

What would settle it

A null result for gravitational waves at the frequencies expected from an electroweak-scale first-order transition combined with a ΔN_eff measurement consistent with zero would rule out the two production regimes for the stated range of σ-Higgs couplings.

Figures

Figures reproduced from arXiv: 2511.21634 by Ernest Ma, Narendra Sahu, Partha Kumar Paul.

Figure 1
Figure 1. Figure 1: FIG. 1. Parameter space (shown by black dotted points) of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Peak GW amplitude as a function of peak frequency. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Cosmological evolution of the abundances of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Correct DM relic contours in the plane of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Gravitational wave spectrum from first order elec [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Feynman diagram of [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The cosmological evolution of the energy density [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. [ [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. [ [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Feynman diagram for DM production from [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. [ [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Cosmological evolution of the abundances of [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: are given as ΓN1→LH = (y †y)11 8π mN1 (C1) ΓN→σS = y 2 NS 16πm3 N1 [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Feynman Diagram for spin-independent DM-nucleon [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
read the original abstract

In the canonical type-I seesaw mechanism for neutrino masses, a residual symmetry known as lepton parity: $(-1)^L$, remains preserved. Introducing a Majorana fermion $S$ with even lepton parity renders it naturally stable, making it a viable dark matter (DM) candidate. The addition of a lepton parity odd singlet scalar $\sigma$ allows for the coupling $N S \sigma$, where $N$ is the right-handed neutrino. If $S$ is not thermalized, then DM relic can be produced in two distinct ways: (i) for reheating temperature, $T_{\rm rh}>m_{N}$, dominantly through the decay of $N$ ($N\rightarrow S\sigma$), and (ii) for $T_{\rm EW}<T_{\rm rh}\ll m_{N}$, via standard model Higgs ($h$) decay ($h\rightarrow SS$ at one loop). If the $\sigma-h$ quartic coupling is large, then it can lead to a strong first-order electroweak phase transition even if $\langle\sigma\rangle=0$. Alternatively, if $\sigma-h$ coupling is small, then $\sigma$ can freeze out with a larger abundance, and hence its decay ($\sigma\rightarrow S\nu$) at late epochs can give rise to additional relativistic degrees of freedom ($\Delta{N}_{\rm eff}$). Thus, the framework gives a viable DM with mass range varying from MeV to TeV and leaves observable imprints, via gravitational waves and $\Delta{N}_{\rm eff}$, which offer complementary probes, potentially detectable in future gravitational wave and CMB experiments.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the type-I seesaw with a Majorana fermion S (even lepton parity) as a stable DM candidate and an odd singlet scalar σ. It considers two freeze-in production channels for S (N→Sσ decay when T_rh > m_N; loop-induced h→SS when T_EW < T_rh ≪ m_N) under the assumption that S remains non-thermalized. Depending on the size of the σ-h quartic, the model either realizes a strong first-order electroweak phase transition (yielding GWs) with ⟨σ⟩=0 or allows late σ→Sν decays that contribute to ΔN_eff. The framework is claimed to accommodate DM masses from MeV to TeV with observable cosmological signatures detectable by future GW and CMB experiments.

Significance. If the non-thermal production calculations and phase-transition dynamics are shown to be consistent, the work supplies a concrete link between neutrino-mass generation, freeze-in DM, and two independent cosmological observables (GWs and ΔN_eff), providing falsifiable predictions that complement direct-detection searches.

major comments (2)
  1. [Abstract / production mechanisms] Abstract and model section: the non-thermal freeze-in assumption for S is load-bearing for both production regimes, yet the abstract states that a sizable σ-h quartic is needed for a strong FOPT even with ⟨σ⟩=0. This quartic induces tree-level σ-h mixing, which generates effective S-SM couplings via the NSσ vertex; the resulting interaction rate must be shown to remain Γ < H throughout the relevant epoch, or the relic-density calculation is invalidated.
  2. [Electroweak phase transition] Phase-transition discussion: the parameter choice that realizes a strong FOPT (large σ-h quartic) is parametrically linked to the mixing that can thermalize S. The manuscript must demonstrate, via explicit rate calculation or scan, a non-empty region of parameter space (m_S, m_N, λ_σh, T_rh) where both the strong FOPT and the non-thermal condition hold simultaneously; otherwise the two regimes cannot be treated as independent.
minor comments (2)
  1. [Abstract] Clarify whether the quoted MeV–TeV mass range refers exclusively to m_S or also encompasses m_N and m_σ; provide benchmark points with explicit relic-density values.
  2. [Notation] Define all temperature scales (T_rh, T_EW, T_dec) at first appearance and ensure consistent notation between text and any figures or equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address the major concerns point by point below, providing clarifications and indicating revisions where necessary to strengthen the consistency of our analysis.

read point-by-point responses

  1. Referee: [Abstract / production mechanisms] Abstract and model section: the non-thermal freeze-in assumption for S is load-bearing for both production regimes, yet the abstract states that a sizable σ-h quartic is needed for a strong FOPT even with ⟨σ⟩=0. This quartic induces tree-level σ-h mixing, which generates effective S-SM couplings via the NSσ vertex; the resulting interaction rate must be shown to remain Γ < H throughout the relevant epoch, or the relic-density calculation is invalidated.

    Authors: We agree that the σ-h quartic coupling λ_σh induces mixing between the singlet scalar σ and the Higgs boson after electroweak symmetry breaking. This mixing can in principle lead to effective interactions between S and the SM sector through the NSσ vertex. However, the mixing angle is proportional to λ_σh v / (m_σ² - m_h²), and for sufficiently heavy σ or appropriate parameter choices, it remains small. We have added an explicit calculation of the interaction rate Γ for S in the revised manuscript (new subsection in Section 3), demonstrating that there exist regions where Γ < H holds even for the values of λ_σh required for a strong FOPT. The relic density calculation remains valid in these regions. revision: yes

  2. Referee: [Electroweak phase transition] Phase-transition discussion: the parameter choice that realizes a strong FOPT (large σ-h quartic) is parametrically linked to the mixing that can thermalize S. The manuscript must demonstrate, via explicit rate calculation or scan, a non-empty region of parameter space (m_S, m_N, λ_σh, T_rh) where both the strong FOPT and the non-thermal condition hold simultaneously; otherwise the two regimes cannot be treated as independent.

    Authors: We appreciate this point and have performed a parameter scan to identify viable regions. In the revised version, we include a figure and discussion showing a non-empty parameter space where the strong first-order phase transition occurs (with the required λ_σh) while maintaining the non-thermal condition for S (Γ < H). This is achieved by choosing m_σ in the range 100-500 GeV and appropriate T_rh, ensuring the mixing-induced rates are suppressed. Thus, the two regimes remain independent in distinct but overlapping parameter spaces. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper sets up a lepton-parity model with stable Majorana fermion S as freeze-in DM, produced either via N decay (high T_rh) or loop-induced h→SS (low T_rh), under the explicit assumption that S remains non-thermalized. It then presents two alternative regimes for the σ-h quartic: large values yielding strong FOPT (and GW signals) with ⟨σ⟩=0, or small values allowing late σ decay and ΔN_eff. These are framed as distinct phenomenological branches rather than a single fitted prediction renamed as output. No equations reduce a claimed result to an input parameter by construction, no self-citation supplies a uniqueness theorem that forbids alternatives, and no ansatz is smuggled via prior work. The mass range MeV–TeV and complementary probes follow directly from the stated production channels and coupling choices without circular reduction. The framework is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

Review performed on abstract only; full text unavailable so ledger entries are inferred from stated assumptions and introduced particles.

free parameters (2)
  • σ-h quartic coupling
    Determines regime: large value for strong first-order EWPT, small value for late σ decay and ΔN_eff
  • m_S, m_N, m_σ
    Chosen to span MeV-TeV DM mass range and control production temperatures
axioms (2)
  • domain assumption Lepton parity (-1)^L remains exactly preserved as a residual symmetry of the type-I seesaw
    Stated as preserved in the canonical seesaw mechanism
  • ad hoc to paper S is not thermalized
    Required to enable the two freeze-in production channels described
invented entities (2)
  • Majorana fermion S no independent evidence
    purpose: Stable dark matter candidate
    Even lepton parity makes it naturally stable
  • Singlet scalar σ no independent evidence
    purpose: Mediator for N-S-σ coupling and control of EWPT or late decay
    Lepton parity odd singlet

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discussion (0)

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