Singlet-Doublet fermion origin of dark matter, neutrino mass and inverse first-order electroweak phase transition

Debasish Borah; Indrajit Saha; Narendra Sahu; Shashwat Sharma; Sujit Kumar Sahoo

REVIEW 3 major objections 5 minor 123 references

Singlet-Doublet fermion origin of dark matter, neutrino mass and inverse first-order electroweak phase transition

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read One extension with singlet-doublet fermions gives dark matter, neutrino mass, and a two-step electroweak phase transition.

desk verdict An interesting new combination of single-doublet dark matter, radiative Dirac neutrino mass, and a fermion-driven inverse first-order EWPT, but the two-step FOPT and GW predictions look unreliable because the daisy-resummed potential omits the new fermions' large contributions to the Higgs thermal mass. read the letter →

arxiv 2608.12483 v1 pith:7R3P6M7S submitted 2026-08-12 hep-ph astro-ph.COhep-exhep-th

Debasish Borah , Indrajit Saha , Sujit Kumar Sahoo , Narendra Sahu , Shashwat Sharma This is my paper · ORCID

classification hep-phastro-ph.COhep-exhep-th

keywords singlet-doubletfermiondarkmatterradiativeDiracseesawinversefirst-orderelectroweakphasetransitiongravitationalwavesneutrinomassrelicdensityradiationcollidersignatures

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 argues that a single compact extension of the standard model—two generations of singlet-doublet fermions, one complex scalar singlet, and three right-handed neutrinos—can account for the two measured neutrino mass splittings, the thermal dark-matter relic density, and a two-step inverse first-order electroweak phase transition. In that phase history the Universe first tunnels from a broken Higgs phase to a symmetric phase and then, at a lower temperature, tunnels back to the broken phase, giving two first-order transitions instead of the standard crossover. The lightest singlet-doublet admixture, stabilized by the residual $Z_2$ symmetry, is dark matter; the heavier generation couples to the Higgs with order-one Yukawa couplings and drives the transition. The same particle content keeps all existing constraints satisfied and predicts gravitational-wave signals in the band of planned observatories, a dark-radiation shift, and distinct collider signatures.

What carries the argument

The carrying mechanism is the singlet-doublet fermion pair $(\chi_i,\Psi_i)$ and its Yukawa coupling $y_i\,\overline{\Psi_i}\tilde{H}\chi_i$ to the Higgs. After electroweak symmetry breaking this pair mixes by an angle $\theta_i$, producing the light dark-matter state $\chi$ and a heavier state $\psi$; the large coupling $y_2$ feeds the field-dependent fermion masses in the one-loop finite-temperature effective potential, and the competition with the standard-model thermal potential creates the barrier that makes the two-step inverse transition possible. The same two generations run in the one-loop radiative Dirac seesaw, where the soft $Z_4$-breaking term $\mu_\phi^2$ splits the singlet scalar into $\phi_1,\phi_2$ and generates the neutrino mass: each generation contributes the factor $(M_Y-M_X)\sin 2\theta_i$ times a loop function. Thus one portal, the SD fermion-Higgs coupling, drives the phase transition, while the residual $Z_2$ supplies dark-matter stability and the loop seesaw supplies neutrino mass.

What would settle it

Compute the finite-temperature effective potential beyond one-loop, or on the lattice, for the benchmark points of Table 2: if the barrier between the symmetric and broken minima vanishes at $y_2 \simeq 2.8$–$3.1$, the inverse two-step transition is not real. Observationally, a null search for the predicted double-peaked stochastic gravitational-wave background—with the second transition strengths and durations of Table 3 (i.e. $\alpha_2$ between 0.03 and 0.25 and $\beta/H$ between 180 and 1358)—would exclude the two-transition claim.

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Extended reading notes

Core claim

The central claim is that two vector-like singlet-doublet fermion generations, a complex scalar singlet with a soft $Z_4$-breaking mass term, and three Dirac right-handed neutrinos solve three problems at once. The neutral state $\chi$, a mix of singlet and doublet after electroweak symmetry breaking, provides the observed dark matter relic density; the one-loop diagram with the split scalar singlet and the two fermion generations produces two non-zero Dirac neutrino masses consistent with oscillation data; and the heavier singlet-doublet generation, with Yukawa couplings $y_2 \simeq 2.8$–$3.1$, turns the electroweak transition into an inverse first-order transition with two first-order phase transitions (at $T_{c1} \simeq 215$–$283$ GeV and $T_{c2} \simeq 80$–$89$ GeV for the benchmark points). The resulting gravitational-wave spectrum has two peaks with distinct frequencies and strengths, within reach of future detectors. The paper thereby demonstrates that dark matter, neutrino mass, and a nonstandard electroweak thermal history can share one minimal origin.

Load-bearing premise

The load-bearing premise is that the one-loop finite-temperature effective potential with daisy resummation, evaluated at the benchmark singlet-doublet Yukawa couplings $y_2 \simeq 2.8$–$3.1$, correctly gives the free-energy barrier and minimum ordering; if higher-order or non-perturbative effects destroy that barrier, the inverse first-order transition and its gravitational-wave spectrum disappear.

Editorial extensions

If this is right

If the central claim holds, the same benchmark points that satisfy dark matter and neutrino constraints predict a double-peaked stochastic gravitational-wave background, measurable with planned space- and ground-based detectors.
The two-transition thermal history replaces the single electroweak crossover, so any computation of electroweak-scale out-of-equilibrium processes in the early Universe must be redone in a universe that passes through an intermediate symmetric phase.
Because the neutrinos are Dirac, neutrinoless double beta decay is absent and the right-handed neutrinos contribute an extra radiation component $\Delta N_{\rm eff}$; part of the otherwise allowed dark-matter parameter space is already excluded by current CMB bounds and more will be probed by future CMB experiments.
The heavier charged doublet fermion either decays promptly to multilepton final states (parts now excluded by prompt-decay searches) or is long-lived, producing displaced vertices testable at colliders and at future long-lived-particle detectors; if no such signatures appear, the preferred mixing region shrinks.
When the heavier generation mass exceeds about 2215 GeV with perturbative couplings, the inverse transition no longer occurs, so the phase-transition requirement directly bounds the mass spectrum of the second generation.

Reading between the lines

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

A consequence left implicit is that during the intermediate symmetric phase the electroweak gauge symmetry is restored, so any primordial magnetic fields or baryon-number-violating processes active before the second transition would be reprocessed by the thermal plasma; this is a testable cosmological stamp of the scenario.
The perturbative-control issue noted in the paper (the singlet-doublet Yukawa coupling exceeds the perturbative limit around 4.85 TeV and hits a Landau pole near 26 TeV) suggests the benchmark couplings sit near a strong-dynamics threshold; a UV completion at that scale could alter the free-energy barrier and shift the gravitational-wave frequencies, so the GW prediction doubles as a probe of the
The same two-generation structure could be converted to a Majorana neutrino seesaw by adjusting the symmetry, and the Dirac choice (motivated by dark radiation and the absence of neutrinoless double beta decay) would lose those observational channels while keeping the phase-transition mechanism; comparing the two variants would show how generic the inverse-transition prediction is.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper's core new claim—two-step inverse first-order EWPT from singlet-doublet fermions with y2~3, with double-peaked GW and correlated Delta Neff—rests on an effective potential that omits the new fermion contributions to the Higgs/Goldstone thermal masses. That looks like a load-bearing error. If I'm right, the benchmarks in Tables 2–3 and Figs. 11–12 are not reliable as they stand.

What's genuinely new: earlier work (refs [35,56,58]) did the same dark sector and Dirac seesaw, and refs [66,68] did fermion-driven inverse FOPT, but the joint analysis with two FOPTs, GW spectra, Delta Neff, and collider constraints is new. The model is clearly laid out, the neutrino mass parametrization via Casas-Ibarra is standard, relic density and direct detection are handled with micrOMEGAs, and the authors are upfront about the Landau pole at ~26 TeV in Appendix A. That honesty is real.

The problem: Eq. (4.12) for Pi_{h,g} contains only SM contributions—Higgs, Z, W, top—even though the heavy singlet-doublet fermions carry Yukawa couplings y2~2.8–3.1 to the same Higgs doublet. Their thermal self-energy contribution scales like y^2 T^2, which for these benchmark points is several times larger than the SM terms. In the Arnold-Espinosa daisy resummation, leaving that out can shift the minima and plausibly remove the barrier that the two FOPTs rely on. You cannot compensate with V_T and V_CW; the ring-resummed zero-mode part is missing a term of the same parametric size. This is independent of the Landau pole worry. If the omitted terms restore the usual high-T symmetric phase, the GW signals disappear.

Other soft spots are secondary: no code or data shipped, so the micrOMEGAs and FindBounce numbers are not independently reproducible; and the O(1) Yukawas with a Landau pole at ~26 TeV mean the effective potential itself is on shaky ground without a UV completion. But the missing daisy term is the one that decides the verdict.

Who it's for: model-builders working on radiative seesaw plus GW phenomenology will want to know about this combination, but they should treat the phase-transition predictions as needing a corrected calculation. It deserves a serious referee—the error is a straightforward fix and the question is worth asking—but I'd only accept after the authors recompute with the full thermal masses. My own verdict is conditional, leaning no on the central claim until it's fixed.

Referee Report

3 major / 5 minor

Summary. The paper extends the singlet-doublet fermion dark matter framework by adding two generations of vector-like singlet-doublet fermions, one complex scalar singlet, and three right-handed neutrinos, with the aim of simultaneously explaining radiative Dirac neutrino masses, the observed dark matter relic density, and an inverse two-step first-order electroweak phase transition. The authors reconstruct the neutrino-portal Yukawa couplings from oscillation data via a Casas-Ibarra style parameterization, impose dark matter relic and direct-detection constraints, and then compute the finite-temperature effective potential with singlet-doublet fermions to obtain three benchmark points with two first-order phase transitions and gravitational-wave spectra that they project to be observable by future experiments. The paper also discusses \Delta N_eff and collider signatures, and it acknowledges in Appendix A that the large fermion-Higgs couplings hit a Landau pole near 26 TeV.

Significance. If the phase-transition calculation survives scrutiny, the model would be a compact and phenomenologically rich framework connecting dark matter, radiative neutrino mass, and gravitational-wave astronomy, with falsifiable predictions for cLFV, \Delta N_eff, displaced-vertex searches, and direct detection. The paper uses standard tools (micrOMEGAs, FindBounce, SARAH) and the neutrino Yukawa reconstruction makes the parameter scan predictive rather than ad hoc. However, the central FOPT and GW claims rest on a one-loop effective potential with a load-bearing technical gap: the daisy-resummed Debye masses omit the singlet-doublet fermion contributions, and the mass-eigenvalue formulas contain an internal factor inconsistency. These issues must be fixed before the benchmark results can be considered reliable.

major comments (3)

[Sec. 4, Eqs. (4.11)-(4.15)] The daisy-resummed thermal masses for the Higgs and Goldstone bosons contain only SM contributions, yet the singlet-doublet fermions with y2 \approx 2.8-3.1 couple to the Higgs with O(1) Yukawa couplings and contribute a leading high-T term to Pi_{h,g} of order y2^2 T^2 times an O(1) group-theory coefficient. For the benchmark values this contribution is at least comparable to, and plausibly larger than, the entire SM contribution in Eq. (4.12), and it is simply absent from the calculation. Because the daisy correction enters as (m_i^2 + Pi_i)^{3/2}, omitting this term can change the shape of V_eff, remove or create the barrier, and therefore alter the two-step inverse-FOPT structure in Figs. 11-12 and the GW spectra in Fig. 12. The authors should include the singlet-doublet fermion contributions to the Debye masses, or demonstrate numerically that the benchmark transitions survive their inclusion.
[Sec. 2 and Sec. 4, Eqs. (2.7), (2.8), (4.6)] The mass-matrix formulas are mutually inconsistent. From the interaction -y_i \bar Psi_i \tilde H chi_i, the off-diagonal entry after EWSB is y_i v / sqrt(2), which gives eigenvalues containing sqrt((M_{Psi_i}-M_{chi_i})^2 + 2 y_i^2 v^2) and tan(2 theta_i) = sqrt(2) y_i v / (M_{Psi_i}-M_{chi_i}). This matches Eq. (2.7), but Eqs. (2.8) and (4.6) instead contain 4 y_i^2 h^2 inside the square root, a factor-of-two error. Since the field-dependent masses m_X(h) and m_Y(h) enter V_CW and V_T and hence V_eff in Eq. (4.16), the FOPT results and the benchmark points in Tables 2-3 can shift. The authors must correct the factor or explicitly clarify the definition of y_i used in the numerical calculation.
[Appendix A and Sec. 7] The paper acknowledges that the singlet-doublet Yukawa coupling exceeds the perturbative limit at about 4.85 TeV and develops a Landau pole near 26 TeV. Although the phase transition occurs at temperatures below 300 GeV, the effective potential is computed at one-loop order with y2 \approx 3, so higher-order fermionic corrections to V_eff and to the thermal self-energies are not parametrically small. The central claim of an inverse FOPT with detectable GW signals should be accompanied by a quantitative estimate of the sensitivity of the barrier and of the transition parameters to the truncation of the perturbative expansion, for example by varying the renormalization scale or by estimating the size of two-loop contributions.

minor comments (5)

[Eqs. (2.18) and (2.19)] The sums are written over i=1 to 3, but there are only two generations of singlet-doublet fermions; the sums should run over i=1,2 unless a third generation is intended.
[Table 2 and related text] The quantities M_chi, Delta M_1, M_chi', and M_psi' should be explicitly labeled as physical mass-eigenstate masses or as Lagrangian masses; the text appears to use both notions, and the distinction matters for Eqs. (2.7)-(2.8).
[References [84] and [85]] The DarkSide reference is rendered as 'Darkside binom.' and the DARWIN reference as 'DAR WIN'; these should be corrected to 'DarkSide-50' and 'DARWIN'.
[Fig. 12] The legend entries for the benchmark points are garbled; they should read BP1, BP2, BP3.
[General] There are several typographical errors, including 'natural occurance' in the introduction and 'equilibrium comiving densities' in Section 3; these should be corrected in a final proofreading.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: relic, Delta_Neff, and GW outputs do not reduce to inputs; only non-load-bearing self-citations.

full rationale

The paper's derivation chain is self-contained and the central claims do not reduce to their inputs by construction. The neutrino Yukawa matrices lambda_psi and lambda_chi are reconstructed from observed neutrino masses and the PMNS matrix via the Casas-Ibarra-like parameterization in Eq. (2.16), with R a general complex matrix and V_R set to unity; they are then used as inputs to compute DM annihilation, direct detection, cLFV, and Delta_Neff. No observable that is later called a prediction is used to fix these couplings. The DM relic density is obtained by solving the coupled Boltzmann equations in Section 3 with micrOMEGAs, and the measured relic is imposed as a constraint; the surviving parameter regions in Figs. 6-8 are outputs of the scan, not fitted quantities renamed as predictions. Similarly, Delta_Neff follows from Eq. (5.3) once lambda_chi is fixed by neutrino data; there is no equation defining lambda_chi from Delta_Neff. The IFOEWPT and gravitational-wave results are genuine outputs of the finite-temperature effective potential in Eq. (4.16), built from Eqs. (4.2)-(4.7) and standard thermal corrections, with S3 computed by FindBounce. The benchmark values of y2 in Table 2 are scanned model parameters; the critical temperatures, strengths alpha, and inverse durations beta/H in Table 3 are computed outputs, and the GW spectra in Fig. 12 are derived from them. No equation in the paper defines y2, M_chi', or M_psi' in terms of alpha, beta/H, or Omega_GW, so the predictions are not forced by construction. The self-citations [35,56,57,58] supply the model template and computational conventions, but the paper displays the relevant formulas (loop neutrino mass, Boltzmann equations, effective potential), and the main physics steps are supported by external references and external codes [62,66,68,92,109]. These self-citations are therefore not load-bearing. The paper itself explicitly flags a limitation in Appendix A and Section 7: y_chi exceeds the perturbative limit near 4.85 TeV and develops a Landau pole near 26 TeV; this is a validity and correctness concern, not a circularity. A separate physics concern not affecting the circularity verdict is that the daisy thermal masses in Eq. (4.12) are SM-only even though the singlet-doublet fermions have y2 ~ 3; that omission could change the barrier shape, but it is an internal-consistency issue rather than a reduction of outputs to inputs.

Assumptions & free parameters 9 free parameters · 7 assumptions · 4 invented entities

The model introduces four new field types and more than a dozen free mass and coupling parameters. Most are chosen by hand or scanned, and the strongest phenomenological outputs (two FOPTs, GW spectra, Delta Neff) depend on benchmark values rather than being parameter-free predictions. The neutrino mass matrix is imposed through Casas-Ibarra reconstruction, not predicted.

free parameters (9)

Mchi (lightest singlet-doublet fermion, DM mass) = 40.96, 294.84, 724.59 GeV (BP1-3)
Hand-picked benchmark values that set the DM mass and scan the observable relic and direct-detection plane.
Delta M1 = Mpsi - Mchi = 284.46, 61.70, 14.82 GeV (BP1-3)
Chosen in scans; controls coannihilation and the Higgs-Yukawa coupling y1 through Eq. (2.7).
sin theta1 (singlet-doublet mixing, first generation) = 0.00407, 0.01264, 0.00972 (BP1-3)
Free parameter set by benchmark choices; determines DM annihilation, direct detection, and charged fermion decay length.
Mchi' (heavier singlet fermion mass) = 400, 600, 800 GeV (BP1-3)
Free dark-sector mass chosen for the benchmark points and mass hierarchies.
Mpsi' (heavier doublet fermion mass) = 1661, 1965, 2225 GeV (BP1-3)
Free mass parameter; sets the scale of the heavy generation that drives the inverse FOPT through y2.
y2 (heavy generation Higgs-Yukawa coupling) = 2.763, 2.991, 3.123 (BP1-3)
Chosen large, of order 1, to generate the inverse FOPT; this is the coupling that runs non-perturbative near 4.85 TeV (Appendix A).
Mphi (singlet scalar mass) = 189.10, 594.12, 783.01 GeV (BP1-3)
Free scalar mass; with a soft Z4-breaking term it splits into phi1 and phi2 and closes the neutrino mass loop.
lambda_phi H (Higgs portal coupling)
Assumed small so the scalar singlet does not contribute to the phase transition, but the value is not quoted or constrained.
Casas-Ibarra R matrix entries
Free 3x3 complex matrix in Eq. (2.16) used to reconstruct lambda_psi and lambda_chi from assumed neutrino masses; not fully specified.

assumptions (7)

domain assumption The one-loop finite-temperature effective potential with Arnold-Espinosa daisy resummation (Eq. 4.16) is adequate for ychi ~ O(1).
Used in Section 4 to compute IFOEWPT; large Yukawas sit at the edge of perturbativity (Appendix A).
domain assumption An unbroken global U(1) lepton-number-like symmetry forbids all Majorana mass terms, making neutrinos Dirac.
Stated in the footnote of Section 2; required for the Dirac seesaw and Delta Neff predictions.
domain assumption A residual Z2 from the Z4 symmetry stabilizes the lightest neutral SD fermion as dark matter.
Section 2; assumes no competing decay channels and relies on the imposed mass hierarchy.
ad hoc to paper The mass hierarchy Mchi < Mpsi, Mphi1,2 < Mchi', Mpsi' is imposed.
Section 2 and 3; needed for single-component DM and for decays of heavier dark sector states.
ad hoc to paper The scalar singlet phi does not contribute to the phase transition because lambda_phi H is chosen small.
Section 4; this choice is not independently constrained by data.
domain assumption The right-handed neutrinos interact only through lambda_chi phi nuR, with electron-type lambda_chi entries set to zero.
Section 5; controls thermalization and sets N_nuR = 2 in the Delta Neff calculation.
domain assumption Perturbative running is valid up to the few-TeV scale; the Landau pole near 26 TeV is accepted as an effective-theory cutoff.
Appendix A; self-identified limitation in Section 7.

invented entities (4)

Two generations of vector-like SU(2)L singlet fermions chi_i
purpose: Provide radiative-seesaw partners and, for the lightest state, the dark matter candidate.
No external observation supports their existence; only prospective collider, direct-detection, and GW signals are discussed.
Two generations of vector-like SU(2)L doublet fermions Psi_i
purpose: Close the one-loop neutrino mass diagram and, through the heavy generation, drive the inverse first-order electroweak phase transition.
No external evidence; their large Higgs couplings are chosen for the phase transition and are not independently observed.
Complex scalar singlet phi
purpose: Generate radiative Dirac neutrino masses via the loop with SD fermions and split into phi1 and phi2 through a soft Z4-breaking term.
New scalar field with no independent detection; its Higgs portal is left weakly coupled by assumption.
Three right-handed neutrinos nuR
purpose: Provide the right-chiral components of light Dirac neutrinos and contribute to Delta Neff through their thermal history.
Postulated light Dirac partners; while Delta Neff is a falsifiable handle, there is no current observational evidence for their existence.

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Cite this review

Pith. "Pith review of Singlet-Doublet fermion origin of dark matter, neutrino mass and inverse first-order electroweak phase transition." pith.science (2026). https://pith.science/paper/7R3P6M7S

@misc{pith2026260812483,
  author       = {Pith},
  title        = {Pith review of: Singlet-Doublet fermion origin of dark matter, neutrino mass and inverse first-order electroweak phase transition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7R3P6M7S}},
  note         = {Machine review of arXiv:2608.12483}
}

abstract

We study the possibility of an inverse first-order electroweak phase transition (IFOEWPT) and observable gravitational waves (GW) in a radiative neutrino mass model of scotogenic type where singlet-doublet (SD) fermions, the lightest of whom is the dark matter (DM) candidate, generate the necessary seesaw at one-loop level. Considering the possibility of light neutrinos being Dirac for simplicity and additional detection prospects, we extend the standard model (SM) with two generations of $SU(2)_L$ singlet and doublet fermions, one singlet scalar, and three right-handed neutrinos (RHNs). While RHNs provide the right chiral parts of light Dirac neutrinos, the SD fermions and the scalar singlet facilitate the one-loop neutrino mass diagram. The neutral component of the lighter SD fermion, stabilized under a residual $Z_2$ symmetry plays the role of DM while the heavier SD fermions strongly couple to the Higgs leading to an IFOEWPT where the Universe undergoes two different first-order phase transition as it goes from the symmetric to the final broken Higgs phase. We constrain the parameter space from the requirements of generating the correct neutrino mass, DM relic as well as IFOEWPT while incorporating the existing constraints from different experiments. The final allowed parameter space of the model can be probed at collider, direct-detection, GW and cosmic microwave background (CMB) experiments in near future.

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