Neutrino masses and mixed dark matter from doublet and singlet scalars

Hyun Min Lee; Jongkuk Kim; Rojalin Padhan; Seong-Sik Kim

REVIEW 2 major objections 6 minor 1 cited by

Neutrino masses and mixed dark matter from doublet and singlet scalars

T0 review · 2 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A singlet VEV that breaks Z4 to Z2 simultaneously generates one-loop neutrino masses and a dark matter candidate.

desk verdict A solid but incremental scotogenic extension whose central mechanism holds up, with a real gap in the vacuum-stability argument that should be fixed before the benchmark conclusions are quoted. read the letter →

arxiv 2505.00121 v2 pith:PIHJ5DFU submitted 2025-04-30 hep-ph

Jongkuk Kim , Seong-Sik Kim , Hyun Min Lee , Rojalin Padhan This is my paper · ORCID

classification hep-ph

keywords neutrinomassesdarkmatterinertscalardoubletZ4discretesymmetryradiativeseesawsingletdirectdetection

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 single mechanism that produces both tiny neutrino masses and a viable dark matter candidate in one extension of the Standard Model. The model adds an inert scalar doublet, three right-handed neutrinos, and two singlet scalars, with a Z4 symmetry that forbids neutrino masses while unbroken. Once the singlet $\varphi$ acquires a vacuum expectation value, the symmetry is reduced to a Z2 and the neutral parts of the inert doublet mix with the singlet $S$; these mixings split the dark scalars and generate neutrino masses at one loop. The lightest mixed neutral scalar is then the dark matter, and the paper maps four benchmark regimes—no mixing, small mixing, maximal mixing, and a large singlet mass gap—showing which combinations satisfy relic density, direct detection, electroweak precision, and collider constraints.

What carries the argument

The load-bearing object is the $Z_4$ (or gauged $U(1)_X$) symmetry and its breaking by $\langle\varphi\rangle\neq 0$. The VEV generates the $Z_4$-breaking mass parameters $\kappa' = \frac12\lambda'_{S\varphi}v_\varphi$ and $\hat m_S^2 = 2\mu v_\varphi$, producing two $2\times2$ mass matrices for $(H_0,s)$ and $(A_0,a)$; diagonalizing them gives the mixing angles $\theta_s$, $\theta_a$ and the split masses $H_{1,2}$, $A_{1,2}$. These mixings feed the one-loop neutrino mass formula and make the lightest eigenstate a dark matter candidate. The paper uses this machinery to show how small neutrino masses and direct-detection-safe dark matter emerge from small or maximal mixing, with the dark-Higgs annihilation channel $h_2h_2$ providing an extra relic-density route.

What would settle it

Take any benchmark point that passes the paper's four vacuum-stability conditions and minimize the quartic potential as a function of all field directions, not just the four special corners. If some direction gives a negative quartic potential, particularly along the $\lambda'_{S\varphi}$ cross term, that benchmark point is not actually stable and the claimed allowed region would shrink.

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Extended reading notes

Core claim

Neutrino masses are exactly zero in the unbroken-$Z_4$ limit and arise only after $\varphi$ gets a VEV, because the $Z_4\to Z_2$ breaking introduces mixing between the neutral components of the inert doublet $H_2$ and the singlet $S$. In the CP-even sector the off-diagonal mass term is $(\kappa+\kappa')v_H$ and in the CP-odd sector $(\kappa-\kappa')v_H$, so the formerly degenerate doublet scalars split; the one-loop neutrino mass formula weights each dark scalar by the appropriate mixing factors, and in the decoupling limit it reduces to the scotogenic result with an effective $\lambda_{5,\mathrm{eff}}$. The lightest of the mixed scalars is stable under the residual $Z_2$ and is the dark matter. The phenomenological core is a correlation: the same $Z_4$-breaking parameters that set the neutrino mass scale also set the dark-matter mixing angles, and the benchmark scans show that direct detection and electroweak precision data prefer small mixings or near-degenerate dark scalars, which automatically suppresses the neutrino masses.

Load-bearing premise

The benchmark results assume that checking vacuum stability at four special field directions is enough to guarantee the scalar potential is bounded from below in every direction, including the directions where the singlet cross-coupling $\lambda'_{S\varphi}$ enters.

Editorial extensions

If this is right

The same coupling that sets the neutrino scale sets the dark-matter mixings, so neutrino mass and dark-matter observables cannot be adjusted independently: measuring one constrains the other.
In the small-mixing limit the model reproduces scotogenic neutrino masses with an effective $\lambda_5$, and the $h_2h_2$ annihilation channel opens relic-density parameter space where the standard Higgs-portal interaction alone would leave the dark matter overabundant.
Small mixings or nearly degenerate dark scalars are the corners favored by direct detection and electroweak precision data, and these are exactly the corners with small neutrino masses.
The four benchmark scenarios realize distinct dark-matter identities (singlet-like, doublet-like, and mixed), each with different direct-detection and collider signatures that can be probed separately.

Reading between the lines

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

The vacuum-stability criterion used here checks only four special field directions; a full copositivity check over all directions, especially along the $\lambda'_{S\varphi}$ cross term, could remove part of the scanned parameter space even though the central mechanism itself would survive.
Because the same parameters control neutrino mass and dark-matter mixing, future precision measurements of the dark-scalar spectrum, or a direct-detection signal, could test the predicted correlation between the neutrino mass scale and the dark-scalar mass splittings.
A global version of the same $Z_4$-breaking phase could also generate the matter-antimatter asymmetry, a route the paper sets aside; if realized, neutrino masses, dark matter, and baryogenesis would all trace back to the same VEV.

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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

The paper is a serious extension of the authors' own PLB model. The genuinely new content is the treatment of general CP-even and CP-odd mixing between the inert doublet and the singlet S, the one-loop neutrino mass formulas for maximal-mixing cases, and four benchmark scenarios with non-decoupled S. That is not a revolution, but it is a real step beyond Ref. [7].

The one-loop neutrino mass calculation is standard and internally consistent. Recovering the scotogenic limit in Eq. (4.3) is a useful check, and the formulas for bi-maximal mixing and for the case where only one mixing angle is maximal are given cleanly. The phenomenological side is also handled competently: relic density, direct detection, electroweak precision data, LEP and Higgs constraints are all included, and the numerical pipeline uses SARAH/SPheno/micrOmegas. The paper is honest about what it does and does not do.

The main soft spot is the vacuum stability section. Section 3.4 reduces bounded-from-below to the four corner conditions in Eqs. (3.52)-(3.55), but the copositivity condition depends on the intermediate angles and phases. In particular, the lambda'_Sphi cross term vanishes at all four corners, so those corner conditions place no bound on lambda'_Sphi. Since lambda'_Sphi is scanned up to 4pi in Scenarios II and IV, some scan points may violate bounded-from-below even though all four corner conditions hold. This does not invalidate the mechanism, but it does undermine the claim that the chosen parameter space satisfies vacuum stability. The fix is straightforward: impose the full copositivity condition during the scan, or give an analytic argument that intermediate directions cannot be worse. Until that is done, the benchmark and relic-density regions should be treated as provisional.

There are two smaller issues. Eq. (3.16) contains a misprint: the second term should involve lambda_1, not lambda_phi. And the paper makes no quantitative fit to neutrino oscillation data; that is common in this literature, but it limits how strongly one can phrase the neutrino mass claim. The reliance on Ref. [7] is not a problem here; that is the base model and the earlier result is being used appropriately.

Who should read this: model-builders working on radiative neutrino mass and scalar dark matter. It is a useful follow-up, with one real technical gap. A serious referee should see it, and my own verdict would be conditional acceptance after the vacuum stability issue is fixed.

Referee Report

2 major / 6 minor

Summary. The paper studies a scotogenic-type extension of the Standard Model in which an inert doublet H2, a complex singlet S, and a singlet phi plus three right-handed neutrinos are charged under a U(1)_X symmetry containing a Z4 subgroup. It claims that a VEV for phi breaks Z4 to Z2 and induces mixing between the neutral components of H2 and S; this mixing splits the masses of the Z2-odd neutral scalars and generates neutrino masses at one loop, while the lightest mixed scalar is a dark matter candidate. The authors derive the scalar spectrum and mixing structure, impose constraints from vacuum stability, perturbative unitarity, electroweak precision data, LEP/LHC searches, direct detection, and relic density, and then scan four benchmark scenarios ranging from no DM mixing to bi-maximal mixing.

Significance. If correct, the paper would provide a concrete radiative neutrino-mass model with a mixed doublet-singlet scalar dark matter candidate and explicit correlations between neutrino masses and DM observables. The one-loop neutrino mass derivation is internally consistent, and the recovery of the scotogenic limit in Eq. (4.3) is a useful check. The appendices also give complete vertex and loop-function listings, which is a strength. However, the benchmark results rely on theoretical constraints whose derivation is currently incomplete; this affects the reliability of the phenomenological conclusions more than the core mechanism, which appears sound.

major comments (2)

[Section 3.4, Eqs. (3.48)-(3.55)] The vacuum stability analysis is incomplete. The matrix X in Eqs. (3.49)-(3.51) depends on alpha, beta, chi, zeta, and sigma, and co-positivity requires X11 >= 0, X22 >= 0, and X12 + sqrt(X11 X22) >= 0 for every choice of these variables. The four corner conditions (3.52)-(3.55) only test alpha, beta in {0, pi/2}, and at every corner the lambda'_Sphi term in Eq. (3.51) vanishes; hence these conditions impose no bound on lambda'_Sphi. A concrete counterexample is alpha = beta = pi/4, chi = zeta = 1, cos(sigma) = -1, with lambda1 = lambda2 = lambdaS = lambdaphi = 0.1, lambda3 = lambda4 = 0, lambdaSphi = 0.1, lambdaH1S = lambdaH1phi = lambdaH2S = lambdaH2phi = 0.1, and lambda'_Sphi = 1: the four corner conditions all hold, but X12 = -0.2 while sqrt(X11 X22) = 0.061, so V4 is negative along r = rho and the potential is unbounded from below. Since lambda'_Sphi is scanned to values as large as 4pi in Scenarios II-IV, the statement that the chosen parameter space satisfies vacuum stability is not supported, and some benchmark or relic-density points may be unphysical.
[Section 3.5, Eq. (3.56) and Appendix B] The list of perturbative unitarity eigenvalues is incomplete. The scattering matrix M1 in Eq. (B.1) contains a block spanned by s^2, a^2, rho^2, and eta^2 whose eigenvalues include combinations such as 2(lambdaS + lambdaphi) +/- sqrt(4(lambdaS - lambdaphi)^2 + lambdaSphi^2) in addition to 2lambdaS and 2lambdaphi; these lambdaSphi-dependent eigenvalues do not appear in Eq. (3.56). Because lambdaSphi is scanned up to 4pi and is one of the couplings controlling DM annihilation into h2 h2, the perturbativity constraint used in the scans must be replaced by the full set of eigenvalues before the benchmark results can be regarded as quantitative.

minor comments (6)

[Sections 3.2 and 3.4] The symbol alpha is used both for the Higgs mixing angle in Eq. (3.18) and for the field-direction angle in Eq. (3.48); renaming one of them would remove a source of confusion.
[Eq. (3.56)] The expression contains a stray double comma after 2lambda2, and the notation 2lambda_s is inconsistent with lambdaS used in Eq. (2.3).
[Section 4.4] The first sentence contains the duplicated article in "The the global electroweak fit", which should be corrected.
[Section 4.5] The text says "exclude thw parameter space"; this should read "exclude the parameter space".
[Section 5.4] The second reference to Fig. 8(a) for the branching fractions should refer to Fig. 8(b).
[Section 5.1 and Table 2] The notation such as "m2 2 = 109 GeV" should be typeset as 10^9 GeV or equivalent to avoid ambiguity about whether the exponent is intended.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the one-loop neutrino mass follows from Z4 breaking in the Lagrangian, and benchmark constraints are checked with external codes and data.

full rationale

The derivation chain is self-contained. The scalar potential (2.1)-(2.3) forbids the λ5 term by the Z4 symmetry (Eq. 2.2); when φ acquires a VEV, the effective Z2-invariant potential (2.5) contains the Z4-breaking terms κ′ = λ′_Sφ vφ/2 and m̂_S^2 = 2μvφ, which appear as off-diagonal entries in the dark scalar mass matrices (3.24)-(3.25). The mixing angles θs, θa and mass splittings are then computed from these matrices, and the one-loop neutrino mass formula (4.2) is evaluated with the resulting mass eigenstates. In the unbroken limit θs = θa = 0 and mH1 = mA1, so (4.2) vanishes; after Z4 breaking it is nonzero. No neutrino-mass observable is used as an input anywhere in this chain. The decoupling limit reproduces the known scotogenic formula (4.3), which is a consistency check, not an input. The effective λ5,eff in Eq. (2.6) carries a citation to the authors' earlier Ref. [7], but the same paper re-derives it from the Z2-invariant terms and derives the more general formula (4.2) independently. The benchmark scans are checked against SARAH/SPheno/micrOmegas and external constraints (LZ, EW fit, LEP, LHC), so no fitted parameter is renamed as a prediction. The vacuum-stability corner-condition issue is a correctness risk, not a circularity, and does not enter the neutrino-mass derivation.

Assumptions & free parameters 9 free parameters · 6 assumptions · 5 invented entities

The model rests on an extended scalar sector and a broken discrete symmetry. Most parameters are unconstrained scan inputs; the paper does not fit any observable, so every benchmark is a chosen point. The main unproved step is the reduction of vacuum stability to corner conditions, and the neutrino Yukawa sector is left essentially arbitrary.

free parameters (9)

vφ = scanned 10^2 to 10^9 GeV in benchmarks
VEV of φ; breaks Z4 to Z2 and sets RH neutrino masses; not fitted to data.
κ = scanned 0 to 10^6 GeV
Trilinear coupling between S, H1, H2; controls DM mixing and neutrino masses.
κ' = scanned 5e-4 to 6.2e9 GeV depending on scenario
Derived from λ'_Sφ vφ/2; enters CP-odd and CP-even mixing.
μ = scanned 0 to 100 GeV
Trilinear coupling in φ†S²; splits CP-even/CP-odd singlet masses.
mS² = scanned over wide ranges
Singlet scalar S mass parameter; controls decoupling limit.
mH0 (or m2²) = scanned over [1, 10^8] GeV² in scenarios II-IV
Inert doublet neutral scalar mass parameter.
λSφ, λH2φ, λH1S, λ3, λ4 = scanned typically [10^-4, 4π]
Quartic couplings that set DM annihilation and mass splittings; not fitted to data.
yN,ij = not specified
Neutrino Yukawa couplings; appear quadratically in one-loop neutrino masses; not fitted to oscillation data.
MN,k = not specified
RH neutrino Majorana masses, set by λN vφ/√2; chosen, not fitted.

assumptions (6)

domain assumption SM gauge symmetry is extended by a local U(1)_X (or a discrete Z4 subgroup) under which only the new fields transform.
Charge assignments in Table 1 determine which couplings are allowed and are the basis of the whole setup.
ad hoc to paper The U(1)_X gauge boson is decoupled, with mass mX = 2gX vφ.
The paper explicitly focuses on the decoupled case to avoid extra gauge interactions, but this is a model choice, not a derived result (Section 2).
domain assumption The scalar potential has a minimum with ⟨H2⟩=⟨S⟩=0 and ⟨φ⟩≠0.
This is the defining inert-doublet plus broken-Z4 vacuum; stability conditions are checked but only via simplified corner inequalities.
domain assumption The Z4 symmetry is exact in the Yukawa and scalar sectors until φ gets a VEV.
Forbids the λ5 term and tree-level Majorana masses; the whole radiative mechanism relies on this symmetry structure.
domain assumption Perturbative unitarity: all 2→2 scalar scattering eigenvalues must be less than 8π.
Used in Section 3.5 to restrict quartic couplings in the numerical scans.
domain assumption The right-handed neutrinos are heavier than the lightest dark scalar.
The paper states this in Section 4 and says the alternative is treated in Ref. [8]; the benchmarks assume this mass ordering.

invented entities (5)

Inert scalar doublet H2
purpose: Provides the loop particles for radiative neutrino masses and part of the DM candidate
No direct observation; constrained indirectly by EW precision and DM direct detection.
Complex singlet scalar S
purpose: Mixes with H2 to split neutral scalar masses and generate neutrino masses
No direct signal predicted with a specific mass; contributes to DM phenomenology.
Singlet scalar φ
purpose: Breaks Z4 to Z2 and generates RH neutrino Majorana masses via its VEV
Its VEV is the source of the neutrino mass mechanism but no independent observable is predicted.
Right-handed neutrinos NR,i
purpose: Complete the one-loop neutrino mass diagrams
No collider or cosmological signature with a sharp predicted mass is given.
U(1)_X gauge boson (decoupled)
purpose: Anomaly-free UV completion; assumed heavy and irrelevant at low energies
The paper explicitly decouples it; no mass or coupling is constrained by data.

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Pith. "Pith review of Neutrino masses and mixed dark matter from doublet and singlet scalars." pith.science (2026). https://pith.science/paper/PIHJ5DFU

@misc{pith2026250500121,
  author       = {Pith},
  title        = {Pith review of: Neutrino masses and mixed dark matter from doublet and singlet scalars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PIHJ5DFU}},
  note         = {Machine review of arXiv:2505.00121}
}

abstract

We consider the extension of the Standard Model with an inert scalar doublet, three right-handed neutrinos, and singlet scalar fields, $\varphi$ and $S$. In this model, neutrino masses are zero in the limit of the unbroken $Z_4$ discrete symmetry. We show that when the singlet scalar field $\varphi$ gets a VEV, the $Z_4$ symmetry is broken to $Z_2$, and neutrino masses are generated at one-loops due to the mixings between the neutral components of the inert scalar doublet and the singlet scalar field $S$. There is a dark matter candidate from the lightest neutral scalar field, which is a mixture of the inert scalar doublet and the singlet scalar field $S$, in general. The $Z_4$ breaking mass terms are constrained by electroweak precision data and direct detection (DD) bounds for dark matter, favoring small mixings or almost degenerate masses for the DM scalars. As a result, we discuss the implications of the results for small neutrino masses and DD-safe dark matter.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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