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REVIEW 2 major objections 4 minor 44 references

Minimal Scale-Invariant Dark Matter

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The minimal scale-invariant singlet model has exactly one viable dark-matter window.

desk verdict Careful one-loop framework for the minimal scale-invariant singlet model, with a plausible 2.6 MeV freeze-in solution, but the uniqueness claim sits on a large-log one-loop/tree cancellation whose higher-order stability is never checked. read the letter →

arxiv 2608.03019 v1 pith:PFN5RQ6F submitted 2026-08-04 hep-ph astro-ph.COhep-th

Alexander Belyaev , Roman Pasechnik , Rishav Roshan , Alfonso Zerwekh This is my paper · ORCID
classification hep-phastro-ph.COhep-th
keywords scale-invariantdarkmatterfreeze-inColeman-WeinbergmechanismGildener-WeinbergHiggsportalrealscalarsingletone-loopeffectivepotentialdirectdetection
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 claims that the minimal classically scale-invariant extension of the Standard Model -- a single real scalar singlet with no bare mass, stabilized by a $\mathbb Z_2$ symmetry -- has exactly one viable dark-matter window. Once the Higgs quartic is required to vanish at the electroweak scale (the Gildener-Weinberg condition) and the observed relic abundance is imposed, the only surviving freeze-in solution is a singlet with $m_S\simeq2.59\text{--}2.68$ MeV and effective portal coupling $\lambda^{\mathrm{eff}}_{HS}\simeq2.4\times10^{-10}$. The authors build a renormalised two-field one-loop effective potential and show the predicted dark-matter-electron cross-section, about $5.5\times10^{-65}\,\mathrm{cm}^2$, lies twenty-eight orders below DAMIC-M sensitivity. If correct, the model connects radiative electroweak symmetry breaking to dark-matter genesis with no free dark-matter mass parameter, while explaining the persistent null results of direct-detection searches.

What carries the argument

The motor of the argument is the renormalised one-loop Coleman-Weinberg effective potential $V_{\mathrm{eff}}(\phi,\varphi)$ for the two-field background, supplemented by finite counterterms fixed at the electroweak vacuum. The modified on-shell scheme preserves $\langle\phi\rangle=v$, the Higgs mass $m_h$, vanishing Higgs-singlet mixing, and the singlet curvature $m_S^2=\tfrac12\lambda_{HS}v^2$ under changes of the renormalisation scale. From derivatives of this potential the paper extracts the effective portal coupling $\lambda^{\mathrm{eff}}_{HS}=\frac1v\,\partial^3V_{\mathrm{eff}}/\partial\phi\,\partial\varphi^2|_{(v,0)}$, which controls freeze-in production via $h\to SS$ and Higgs-mediated scattering, and the effective singlet quartic $\lambda^{\mathrm{eff}}_S$, whose positivity delimits the allowed regions. The relic abundance is then computed with freeze-in dynamics -- the decay $h\to SS$ plus $2\to2$ channels -- and intersected with the theoretically allowed band.

What would settle it

Evaluate the running of $\lambda_H(\mu)$ at two loops and check whether $\lambda_H(v)=0$ for the benchmark couplings: if it is not zero, the Gildener-Weinberg flat-direction condition and the effective potential are not self-consistent and the mass window must be recomputed. A future electron-recoil experiment detecting a signal for a 2.6 MeV dark-matter candidate at a cross-section far above $5.5\times10^{-65}\,\mathrm{cm}^2$ would rule out this freeze-in solution.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the minimal scale-invariant singlet model is not merely a viable dark-matter candidate but a predictive one: theory alone -- perturbativity, vacuum stability and $\lambda^{\mathrm{eff}}_S>0$ -- together with the cosmological relic density reduces the parameter space to a unique freeze-in solution. The light window is bounded by two nearby roots of $\lambda^{\mathrm{eff}}_S=0$; the relic-density contour crosses it at $m_S\simeq2.59$ MeV with $\lambda^{\mathrm{eff}}_{HS}\simeq2.48\times10^{-10}$ and at $m_S\simeq2.68$ MeV with $\lambda^{\mathrm{eff}}_{HS}\simeq2.43\times10^{-10}$. The heavier window, $436\ \mathrm{GeV}\lesssim m_S\lesssim632\ \mathrm{GeV}$, would require a tiny effective portal obtained from a large tree-level $\lambda_{HS}\simeq6.3\text{--}13.2$ through an extreme cancellation, and is excluded by the requirement $\lambda_S>0$ and by loss of perturbative control. The paper argues that the resulting light solution is stable under renormalisation-scale variation within the modified on-shell scheme, and that its direct-detection rate, $\bar\sigma_e\simeq5.5\times10^{-65}\,\mathrm{cm}^2$, is unobservably small for current and foreseeable experiments.

Load-bearing premise

The calculation assumes that the running Higgs self-coupling $\lambda_H$ actually vanishes at the scale $\mu_\star=v$ for the chosen $(\lambda_{HS},\lambda_S)$ inputs, and that setting $\lambda_H=0$ inside the one-loop field-dependent masses is a valid power-counting choice; the paper does not verify this numerically.

Editorial extensions

If this is right

  • There is no free dark-matter mass: with the observed relic abundance imposed, the only surviving freeze-in solution has $m_S\simeq2.59\text{--}2.68$ MeV.
  • The singlet never thermalises with the Standard Model bath; the required effective portal is $\lambda^{\mathrm{eff}}_{HS}\sim2.4\times10^{-10}$, far below the freeze-out threshold.
  • The predicted dark-matter-electron reference cross-section is $\bar\sigma_e\simeq5.5\times10^{-65}\,\mathrm{cm}^2$, about twenty-eight orders below DAMIC-M sensitivity, so present direct-detection experiments cannot probe the light solution.
  • The invisible Higgs branching ratio is $\mathrm{BR}(h\to SS)\sim10^{-16}$, far below any collider sensitivity.
  • The heavy $436\text{--}632$ GeV window is excluded or strongly disfavoured because it requires $\lambda_{HS}\sim6.3\text{--}13.2$ with a tiny effective coupling, negative $\lambda_S$ along the relic contour, and loss of perturbative control near the upper edge.

Reading between the lines

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

  • An immediate numerical check of the argument is to evaluate the two-loop running of $\lambda_H$ at $\mu=v$ for the benchmark couplings; the paper does not report this, so the mass window could shift if the flat-direction condition is not satisfied by the full running.
  • The same modified on-shell scheme could be applied to scale-invariant models with hidden gauge dark sectors; each would acquire a similarly sharp freeze-in prediction if the flat-direction condition is imposed at the electroweak scale.
  • If the heavy window is revived by higher-order corrections, it would change the experimental target from electron recoil to collider Higgs-portal signatures; this is a testable consequence the paper leaves open.
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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

2 major / 4 minor

Summary. This paper studies the minimal classically scale-invariant extension of the Standard Model with a Z2-odd real scalar singlet S. The authors construct a two-field one-loop Coleman-Weinberg potential in Landau gauge and introduce a modified on-shell renormalisation scheme that fixes the electroweak vacuum, the Higgs mass, the vanishing Higgs-singlet mixing and the singlet curvature at (v,0). The physical singlet mass and the hSS interaction vertex are extracted from different derivatives of the renormalised potential through effective couplings lambda_eff_HS and lambda_eff_S. Imposing perturbativity, vacuum stability and Omega h^2 = 0.12, the paper finds a unique freeze-in solution at mS around 2.59-2.68 MeV with lambda_eff_HS around 2.43-2.48 x 10^-10, and a predicted DM-electron cross section of about 5.5 x 10^-65 cm^2, roughly twenty-eight orders of magnitude below DAMIC-M sensitivity. A heavy region with mS between about 436 GeV and 632 GeV is identified and argued to be disfavoured.

Significance. If correct, the paper would establish a highly predictive minimal model in which classical scale invariance, vacuum stability and the observed relic abundance single out one light freeze-in mass window. The construction of the renormalised two-field potential, the clear separation between the curvature mass and the effective hSS vertex, the public LanHEP/CalcHEP implementation, and the explicit discussion of caveats such as gauge dependence and residual scale dependence are genuine strengths. However, the uniqueness claim rests on the position of one-loop boundaries in a regime with large logarithms and a one-loop correction comparable to the tree-level term; that perturbative control is not yet demonstrated. The central qualitative conclusion that the freeze-in coupling is tiny and undetectable is robust, but the sharper claim of a unique window at the stated precision needs additional work.

major comments (2)
  1. [Sec. IIIA, Figs. 2-3, Eq. (A55)] The lambda_eff_S = 0 boundary that defines the light window is not under perturbative control. In the light region lambda_HS is about 2.2 x 10^-10, so log(lambda_HS/2) is about -23 and Eq. (A55) reduces to lambda_eff_S approximately lambda_S (1 - 1.31 lambda_S). The upper root of this boundary is at lambda_S about 0.76, where the one-loop term is equal and opposite to the tree-level term. The condition |lambda_S| < 4 pi in Eq. (31) does not suppress the effective expansion parameter lambda_S^2 |log(lambda_HS/2)| / (16 pi^2) of order 0.6, and the paper's own discard criterion in Sec. IIE (one-loop correction comparable to the leading contribution) is violated on the boundary itself. Since the relic-density contour crosses the allowed interval precisely at these roots (Eq. (42)), a two-loop or scheme-variation shift of the boundary of order the interval width could move or remove the claimed unique solution. Please provide a two-loop estimate, an explicit mu-variation scan, or an alternative resummation demonstrating the stability of this boundary.
  2. [Sec. IIF and Appendix A4, Eqs. (A50)-(A55)] The effective couplings controlling both the relic abundance and the stability window are asserted without derivation. lambda_eff_HS is defined in Eq. (34) as 1/v times the third derivative of Veff at (v,0), i.e. a zero-momentum derivative coupling, but it is then used in the on-shell decay width in Eq. (39) and in direct detection; the relation between these quantities is not shown. Similarly, Eq. (A55) is stated as the one-loop expression used without displaying the derivatives of Veff, the treatment of Goldstone contributions, or the finite-scheme dependence. This is not merely a presentation issue: with lambda_S about 0.76, the one-loop shift in Eq. (A50) is about 9.5 x 10^-12, i.e. several per cent of lambda_eff_HS about 2.46 x 10^-10, while the separation of the two crossing points in Eq. (42) is only about 5 x 10^-12. Please derive Eqs. (A50)-(A55) or show that the momentum-dependent and field-renormalisation corrections are negligible at the required precision.
minor comments (4)
  1. [Eq. (1)] The transformation is written as 'S- -> -S, H- -> H', which appears to contain typographical artifacts; the intended transformation is S -> -S and H -> H.
  2. [Fig. 4 caption] The caption says 'two black dots mark the boundaries', but the displayed figure appears to have one black dot at the lower lambda_eff_S = 0 crossing and a grey dashed vertical line at mS = 632 GeV for the upper boundary; please clarify what the two black dots denote.
  3. [Eq. (A55)] The argument log(lambda_HS/2) is dimensionless only because the renormalisation scale is fixed to mu = v; the text should state explicitly that the general expression is log(lambda_HS v^2 / (2 mu^2)) and specify the scheme in which Eq. (A55) is derived.
  4. [Figs. 2-4] The label 'h2 = 0.12' in the figures should be 'Omega_S h^2 = 0.12' for consistency with Eq. (40).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the relic abundance is an external input, and the claimed MeV-scale window is a genuine intersection of theory-defined stability boundaries with that input.

full rationale

The paper's central derivation chain is not circular. The observed relic abundance (Omega_S h^2 = 0.12, Eq. 40) is an external cosmological input imposed on the model, not a quantity derived from the model's own assumptions. For each input pair (lambda_HS, lambda_S), the paper computes m_S^2 = lambda_HS v^2/2, the effective portal coupling lambda_eff_HS (Eqs. 34 and A50), and the effective singlet quartic lambda_eff_S (Eq. A55); the theoretically allowed region is then selected by lambda_eff_S > 0, perturbativity, and global vacuum stability, conditions that do not use the relic-density measurement. The light solution (Eq. 42) is the intersection of the externally imposed relic contour with these theory-defined boundaries, which is standard inverse-model logic rather than a self-referential construction. The direct-detection cross section (Eq. 54) uses the same lambda_eff_HS fixed by the relic condition, but applied to a different observable (electron scattering); this is the model's predictive content, not circular fitting to the same data. The self-citations to micrOMEGAs 7 (Ref. 31) and to the general derivative formalism (Ref. 37) are to publicly available, independently maintained computational tools or standard formulas, and they are not load-bearing in the sense of a self-citation chain. The unverified identification lambda_H(mu*)=0 at mu*=v is an explicitly stated scheme assumption and, while it may be a correctness risk, it is not a circular reduction of the dark-matter prediction to its inputs. The concern that the lambda_eff_S = 0 boundary is controlled by a large logarithmic one-loop term is a perturbative-reliability/correctness question, not a circularity, and does not affect this score.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The model has two free dimensionless couplings (lambda_HS, lambda_S). The singlet mass is defined as mS^2 = lambda_HS v^2/2, so it is not an independent input but is not predicted until lambda_HS is fixed. The relic abundance fixes a combination; theory inequalities cut the remaining freedom to a narrow window. Everything rests on the Gildener-Weinberg one-loop machinery and freeze-in assumptions listed below.

free parameters (2)
  • lambda_HS (tree-level Higgs-portal coupling) = around 2.3e-10 for the light solution (mS around 2.6 MeV)
    Free dimensionless input; mS is defined by mS^2 = lambda_HS v^2/2. Its value is adjusted so the derived lambda_eff_HS produces Omega h^2 = 0.12; not predicted independently.
  • lambda_S (tree-level singlet self-coupling) = O(0.1-0.8) within the lambda_eff_S > 0 window, set by Eq. (A51)
    The second dimensionless input; fixed by the one-loop consistency relation between lambda_eff_HS and lambda_HS together with relic-density and stability requirements.
assumptions (6)
  • domain assumption Classical scale invariance: no explicit mass terms in the scalar potential (Eq. 2).
    The entire dimensional-transmutation mechanism depends on the absence of bare masses; this is a model-building postulate, not a derived property.
  • domain assumption Exact Z2 symmetry S to -S, H to H (Eq. 1).
    Stabilises S as DM and forbids mixing at the Z2-preserving vacuum.
  • domain assumption Gildener-Weinberg flat-direction condition lambda_H(mu*)=0 with mu* set equal to v; lambda_H is O(hbar) and is set to zero inside NLO masses (Eqs. 13, A9-A10).
    Provides the flat direction and power counting for the two-field effective potential; the identification mu* = v is a defining choice of the modified scheme.
  • standard math One-loop MS effective potential in Landau gauge is a sufficient approximation for vacuum selection and DM observables (Eq. 14 and Sec. IIB).
    Coleman-Weinberg potential is standard; gauge dependence is deferred to the Nielsen identity and not computed.
  • domain assumption The renormalised potential has no deeper stationary point than (v,0) (Eq. 29).
    The global vacuum-selection requirement is imposed numerically, not proved for all parameters.
  • domain assumption Freeze-in conditions: negligible initial S abundance and Gamma_{S<->SM} < H over production temperatures (Eq. 46).
    The calculation assumes the singlet never thermalises; this is checked a posteriori, not guaranteed by the model.
invented entities (1)
  • Z2-odd real scalar singlet S
    purpose: Dark matter candidate; its mass emerges from lambda_HS v^2/2 and its couplings are controlled by lambda_eff_HS.
    No independent detection; the only predicted signals (invisible Higgs decay around 1e-16, DM-electron cross section around 5e-65 cm^2) are far below current and near-future sensitivity.

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Pith. "Pith review of Minimal Scale-Invariant Dark Matter." pith.science (2026). https://pith.science/paper/PFN5RQ6F

@misc{pith2026260803019,
  author       = {Pith},
  title        = {Pith review of: Minimal Scale-Invariant Dark Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFN5RQ6F}},
  note         = {Machine review of arXiv:2608.03019}
}
abstract

We study the minimal classically scale-invariant extension of the Standard Model, containing a single $\mathbb Z_2$-stabilised real scalar singlet whose mass is not an independent input but is generated dynamically through the quantum effective potential and linked to radiative electroweak symmetry breaking through the Higgs portal. We construct the full two-field one-loop effective potential and introduce a modified on-shell renormalisation scheme that fixes the electroweak vacuum, the Higgs mass, the vanishing Higgs--singlet mixing and the singlet curvature at the physical point, yielding predictions stable under renormalisation-scale variation. Since thermal freeze-out is excluded by direct-detection limits, we identify a highly predictive freeze-in realisation of this model. Imposing perturbativity, vacuum stability and the observed relic abundance leads to a freeze-in solution with a dark-matter mass of around $2~{\rm MeV}$. The predicted electron-scattering cross section for this solution lies far below the current sensitivity of DAMIC-M. Current direct-detection experiments therefore do not constrain this scenario. The minimal scale-invariant singlet model thus provides a robust and highly predictive framework connecting radiative electroweak symmetry breaking and freeze-in dark-matter genesis.

Figures

Figures reproduced from arXiv: 2608.03019 by the authors.

Figure 1
Figure 1. FIG. 1. The renormalised effective potential [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Relic-density condition [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Light-mass region. The two blue curves enclose the narrow interval where [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Heavy-mass region. The blue line marks the root [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

Discussion (0). Continue with ORCID to comment.

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