Gravitational waves from color restoration in a leptoquark model of radiative neutrino masses

Ant\'onio P. Morais; Johan Rathsman; M{\aa}rten Bertenstam; Marco Finetti; Roman Pasechnik

REVIEW 3 major objections 5 minor 70 references

Gravitational waves from color restoration in a leptoquark model of radiative neutrino masses

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper argues that a minimal scalar leptoquark model of radiative neutrino masses predicts a stochastic gravitational-wave background from first-order color-restoring phase transitions in the mHz–0.1 Hz band, with leptoquark masses…

desk verdict Solid, candid first numerical study of color-restoring FOPTs in a radiative-neutrino-mass LQ model with a plausible LISA-reachable region, but the headline claim rests on points at the edge of the EFT's validity and the one robustness check does not cover that sector. read the letter →

arxiv 2501.01286 v1 pith:2C4WPKMZ submitted 2025-01-02 hep-ph astro-ph.CO

M{aa}rten Bertenstam , Marco Finetti , António P. Morais , Roman Pasechnik , Johan Rathsman This is my paper · ORCID

classification hep-phastro-ph.CO

keywords gravitationalwavesstochasticwavebackgroundfirst-orderphasetransitioncolorrestorationleptoquarkradiativeneutrinomassdimensionalreductionLISA

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 tries to show that a minimal extension of the Standard Model—one pair of scalar leptoquarks, hypothetical particles that carry both quark and lepton quantum numbers—can do three things at once: radiatively generate neutrino masses, pass existing flavor-physics constraints, and drive a first-order phase transition in the early Universe in which color symmetry is briefly broken and then restored. If that is right, the transition would emit a stochastic gravitational-wave background in the mHz–0.1 Hz band that planned observatories LISA, BBO, and DECIGO could detect. The paper's central, testable correlation is that detectable signals require color restoration and leptoquark masses near 1.5 TeV, putting the scenario within reach of both gravitational-wave astronomy and collider searches. The analysis also presents the first application of a new automated dimensional-reduction pipeline, Dratopi, for phase-transition studies.

What carries the argument

The load-bearing object is the next-to-leading-order, dimensionally reduced finite-temperature effective potential of the three-scalar (Higgs plus two leptoquarks) model, computed through the paper's Dratopi pipeline. Dimensional reduction integrates out heavy high-temperature modes to build a three-dimensional effective theory at the ultrasoft scale; from this potential the paper computes the three-dimensional bounce action, nucleation and percolation temperatures, transition strength $\alpha$, inverse duration $\beta/H$, and the sound-wave gravitational-wave spectrum. A validity check tracks the ratio $m_{\rm US}/\mu$ of the largest ultrasoft boson mass to the hard matching scale, and points with $m_{\rm US}/\mu \lesssim 2$ are used for the detectability claims.

What would settle it

Measure the stochastic gravitational-wave background in the mHz–0.1 Hz band with LISA, BBO, or DECIGO and simultaneously search for scalar leptoquarks near 1.5 TeV at the high-luminosity LHC; null results at the peak amplitudes shown in Figs. 5 and 6 would rule out the paper's detectable parameter region. A dedicated high-order (NNLO) or non-perturbative calculation of the effective potential at $m_{\rm US}/\mu\approx 2$ could also check whether the strong first-order transition survives in that regime.

Watch

Extended reading notes

Core claim

The paper's central claim is that the strongest first-order phase transitions in this leptoquark model are not ordinary electroweak transitions but color-breaking ones: at high temperature one or more of the scalar leptoquark fields develops a vacuum expectation value that breaks $SU(3)_C$, and as the Universe cools the color symmetry is restored. These color-restoring transitions produce gravitational-wave spectra with peak frequencies between roughly $10^{-3}$ and $0.1$ Hz, and a sizable part of the model's parameter space falls within the peak-integrated sensitivity of LISA, BBO, and DECIGO. In the detectable region the three physical leptoquark masses sit near $1.5$ TeV, the trilinear coupling $a_1$ is of order $10^3$ GeV, the mixed quartic couplings are of order $10^{-2}$, and the doublet self-coupling $\lambda_R$ is of order $1$; the ratio $m_{\rm US}/\mu$ characterizing high-temperature perturbativity is close to $2$. The paper also catalogs eleven viable phase-transition patterns, eight of which involve color breaking in the high-temperature phase.

Load-bearing premise

The most detectable predictions assume the calculation stays reliable when the relevant particle masses are about twice the scale set by the high-temperature matching procedure, a regime reached only by relaxing the conventional validity condition.

Editorial extensions

If this is right

If the prediction is correct, the stochastic background searched for by LISA, BBO, and DECIGO in the mHz–0.1 Hz band is a direct probe of color restoration in this model, not just of electroweak-scale physics.
A detection would single out leptoquark masses near 1.5 TeV and a particular corner of the scalar potential, giving LHC Run-3 and HL-LHC searches concrete targets.
The same phase transition is tied to the radiative neutrino mass mechanism through the trilinear coupling $a_1$, so a gravitational-wave signal would indirectly support the loop-level origin of neutrino masses in this construction.
Transitions that preserve color throughout are predicted to be too weak to detect ($h^2\Omega_{\rm GW}^{\rm peak}\approx 10^{-21}$), so a future detection would favor color-breaking histories and exclude those color-preserving alternatives.

Reading between the lines

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

If the $m_{\rm US}/\mu\approx 2$ assumption is relaxed, the predicted peak amplitudes in the most sensitive region could shift substantially; a systematic scan that enforces $m_{\rm US}/\mu<1$ would show how much of the LISA-accessible region survives.
The color-restoration mechanism is not tied to the specific $S_1+\tilde R_2$ matter content; other radiative neutrino mass models containing colored scalars could exhibit the same correlation between mHz gravitational waves and TeV-scale scalar masses, making the predicted band a generic target.
An observed stochastic background in this band could act as a coarse 'mass spectrometer' for colored scalars: the peak frequency correlates with the transition temperature, which is set by leptoquark masses, so combined collider exclusions and gravitational-wave observations would tightly constrain the model.

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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: this is a competent first pass at GW predictions from color-restoring FOPTs in the minimal radiative-neutrino-mass leptoquark model, and the central find—LISA/DECIGO/BBO-reachable signals with LQ masses around 1.5 TeV—is plausible but sits outside the strict domain of the EFT calculation in exactly the region that matters.

What's new: the paper applies dimensional reduction (DRalgo) plus CosmoTransitions to a specific LQ model with three VEVs, classifies eleven transition patterns, and identifies the (h,s,r)->0 and (h,s,r)->h channels as the strongest. It is candid: Sec. IV B and V explicitly flag mUS/mu ~ 2 and say the perturbativity criterion is not a strict threshold. That transparency is real and should be credited. The scan is large (~1e5 points) and the GW pipeline follows the LISA Cosmology Working Group templates, so the machinery is sound.

Soft spots: first, the stress-test point is correct. The scale-variation test in Fig. 7 covers two r->h benchmarks, one at mUS/mu ~ 2, but the headline signals are dominated by (h,s,r)->0 points with mUS/mu ~ 2. That leaves the central detectability claim without a direct validity check in the relevant vacuum sector. The line about deviations by a factor of a few being acceptable is an opinion, not a demonstration. Second, Dratopi is unreleased; the numbers cannot be reproduced, though the public pieces (DRalgo, CosmoTransitions) are standard. Third, the statement that every viable FOPT can reproduce neutrino data is asserted, not shown; that is a minor point, but a plot or a footnote with the scan would help. Fourth, the abstract's 'detectable signal' wording is stronger than the body's caveats, which is common but worth tightening.

Net: no fatal flaw. The central argument—that this model has strong color-restoring FOPTs with potentially observable GWs—holds up as a plausible claim, but the precision on the LISA-reachable boundary is not as solid as the abstract suggests. The paper deserves peer review; a referee should ask for either an extended scale-variation test in the (h,s,r)->0 sector or a revised claim that the mUS/mu ~ 2 region is indicative rather than definitive. I would cite it if I worked on BSM phase transitions, and it is a good reading-group case for the dangers of EFT-validity boundaries.

Referee Report

3 major / 5 minor

Summary. This paper studies cosmological first-order phase transitions in a scalar leptoquark extension of the SM (an R2 doublet and an S1 singlet) that radiatively generates Majorana neutrino masses. The authors match the model to the SM, construct the finite-temperature effective potential by NLO dimensional reduction using DRalgo/Dratopi, scan roughly O(10^5) parameter points, and compute the stochastic gravitational wave background from sound waves using the LISA Cosmology Working Group templates. They identify color-breaking 'color-restoration' transitions as the strongest sources and claim that a detectable SGWB in the mHz–0.1 Hz band is correlated with LQ masses near 1.5 TeV. The paper is candid about the main caveats: the high-temperature perturbativity parameter m_US/mu reaches about 2 in the detectable region, the neutrino-data verification is not described, and the bubble wall velocity is fixed at 0.95.

Significance. If the detection claim survives closer scrutiny, the paper is valuable: it gives one of the first quantitative GW analyses of color-restoring FOPTs in a radiative neutrino-mass model and yields concrete, falsifiable correlations between GW observability and LQ masses/couplings. The use of a dimensionally reduced NLO effective potential, a large Monte-Carlo scan, explicit detector PISC curves, and candid acknowledgement of the EFT validity limitation are strengths. The workflow is not circular: the GW parameters are computed from the effective potential after matching to the SM, not fitted to the claimed result. The main risk is not internal inconsistency but extrapolation beyond the demonstrated regime of the EFT, together with two auxiliary assumptions (neutrino-data reproduction and supersonic detonation) that would benefit from dedicated support.

major comments (3)

[Sec. IV B, Eq. (4.11); Sec. IV C, Fig. 11a] The paper's detectability claim is carried by (h,s,r)->0 and (h,s,r)->h transitions, whose strongest points have m_US/mu approximately 2, yet the scale-variation check in Fig. 7 is performed only for r->h benchmarks. Since Eq. (4.11) defines the EFT validity condition and the effective potential (3.5) is computed inside that EFT, the factor-of-two excursion is an extrapolation for exactly the transitions that populate the LISA-reachable region. Please either (i) repeat the scale-variation test on the dominant vacuum channels, ideally with several benchmarks spanning the LISA-reachable region, (ii) estimate the size of the missing NNLO or higher-order terms for those channels, or (iii) explicitly restrict the headline claim to points satisfying Eq. (4.11).
[Sec. III A] The text states 'We have verified that for every viable FOPT... there is at least one solution where the entries of Theta and Omega are small and simultaneously reproduce neutrino oscillation data,' but no procedure, observable definition, or statistical criterion is given, and no reference to a companion calculation is supplied. Because the abstract advertises the model as explaining neutrino oscillation data and the scan accepts or rejects points based on this verification, the claim needs to be reproducible: specify the fit to oscillation parameters, the allowed ranges, and how 'viable FOPT' points were enumerated. Otherwise the model interpretation of the GW predictions is unsupported.
[Sec. IV A and Eq. (3.28)] All GW amplitudes are computed with xi_w fixed to 0.95, which enters both the mean bubble separation R* and the sound-shell width Delta_w. The justification via large alpha and Ref. [52] is plausible, but since the quantitative claim is that a specific region is observable, the authors should show how the peak amplitude and the LISA/BBO/DECIGO reach change when xi_w is varied over a conservative range (e.g., 0.7-0.99) for a few benchmark points in the detection region. This would separate the EFT-validity uncertainty from the hydrodynamic-model uncertainty.

minor comments (5)

[Fig. 10, Sec. IV C] The caption of Fig. 10(b) describes the example as an (r,h)->h transition, while Sec. IV C cites (r,h)->0 as a typical color-breaking example; please make the notation consistent.
[Eq. (3.28)] The numerical factor appears as '10.^-5' in Eq. (3.28); it should read 10^{-5}.
[Sec. IV A] The text states 'beta/H is less than or similar to 105' where the upper limit should evidently read 10^5; please correct the superscript to match the subsequent discussion.
[Sec. IV, Ref. [49]] Dratopi is cited as 'To appear' [49], and the numerical results depend on this package; the authors should release it or provide a public artifact (e.g., a repository with the model export and scan scripts) before or at acceptance to make the analysis reproducible.
[Appendix B, Eq. (B1)] The Monte-Carlo improvement weights a and b are set to 10 without a convergence or bias check; since the histograms in Fig. 11 are based on this scan distribution, a brief statement that the scan is exploratory and not intended as a Bayesian posterior would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the gravitational-wave predictions are computed from the model's effective potential after dimensional reduction, not fitted to the claimed signal or forced by self-citation.

full rationale

The derivation chain is self-contained: the scalar potential (2.1) is matched to the SM (2.8)-(2.9), dimensionally reduced at NLO with DRalgo to the effective potential (3.5), and then used to compute the bounce action, alpha (3.17), beta/H (3.18), and the GW spectrum (3.26)-(3.28), which is compared to detector sensitivity curves. No step uses the LISA/BBO/DECIGO reach as a fitting target, and the 'detectable signal near 1.5 TeV' statement is an output of the parameter scan, not an input. The reliance on the authors' earlier model paper [16] supplies the model and flavor constraints but does not determine the phase-transition or GW predictions; the unreleased code [49] is an implementation tool, not an imported result that the argument reduces to. The high-temperature perturbativity caveat (mUS/mu approximately 2 for the strongest (h,s,r)->0 transitions, with scale-variation tests only on r->h benchmarks, Secs. IV B and IV C) is a validity and extrapolation concern about the EFT calculation, not a circularity: the predictions are still derived from the same effective potential rather than being definitionally equal to their inputs. The Monte Carlo improvement function (B1) biases sampling toward lower mUS/mu and quartic couplings, but it does not fit any output quantity and does not force the gravitational-wave amplitudes. No self-definitional, fitted-input-as-prediction, self-citation-chain, uniqueness-import, or renamed-known-result circularity is present.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the scanned LQ scalar potential parameters and on the validity of the high-temperature EFT. The model parameters (lambda, g, mass, a1, theta) are scanned over hand-chosen ranges, so the 'predictions' for 1.5 TeV and coupling patterns are conditioned on those priors. The EFT validity assumption mUS/mu < 1 is relaxed to about 2 in the signal region. The claim that neutrino oscillation data can be reproduced for every viable point is stated without an explicit scan. No new physical entities are introduced.

free parameters (8)

Scalar self-couplings lambda_S, lambda_R = Scanned in (1e-3, 2)
Hand-chosen scan range; the conclusion that lambda_R ~ O(1) in the detectability region depends on this prior.
Mixed quartic couplings g_HS, g_HR, g'_HR, g_RS = Scanned in +/- (1e-3, 2)
Ranges and random signs chosen by hand; sign preferences in the signal region are outcomes of the scan.
LQ mass parameter m_S1/3_2 = Scanned in (0.8, 3) TeV
The quoted 1.5 TeV detectability mass is an emergent cluster within this range, so the range itself shapes the prediction.
Trilinear coupling a1 = Scanned in (1e-2 TeV, a1^max), with a1^max from Eq (4.6)
Controls neutrino mass generation and phase transition strength; detectability region has a1 ~ 1e3 GeV.
Mixing angle theta = Scanned in (0, pi/2)
Sets the LQ mass splitting via Eq (2.4); affects the dynamics.
Bubble wall velocity xi_w = 0.95
Fixed by hand assuming supersonic detonations; directly scales GW amplitudes.
Monte Carlo improvement weights a and b = 10, 10
Arbitrary parameters in the walker's acceptance function (Eq B1); bias exploration toward low mUS/mu and low |lambda|.
High-temperature perturbativity threshold = mUS/mu < 1, relaxed to ~2 in signal region
The strict bound defines 'high T OK' points; the signal region uses points deviating by a factor of about 2.

assumptions (6)

domain assumption High-temperature expansion m << T holds for the relevant LQ masses.
Sec III A, Eq (3.3); the dimensional reduction relies on this hierarchy, and points near the boundary are dealt with by the perturbativity check.
domain assumption Debye masses are larger than the phase transition scale, so temporal scalar modes can be integrated out.
Appendix A states 'this is an assumption we make throughout the present project'.
ad hoc to paper The 3d EFT remains quantitatively reliable for mUS/mu up to about 2.
Sec IV B: deviations 'by a factor of a few above unity are considered acceptable'; the strongest GW points have mUS/mu around 2.
ad hoc to paper For every accepted phase transition point there exists a choice of LQ Yukawa matrices with entries < 0.01 reproducing neutrino oscillation data.
Sec III A: 'We have verified...' with no explicit scan or calculation shown.
ad hoc to paper Bubble walls expand as supersonic detonations with xi_w = 0.95 for all transitions of interest.
Sec IV A: justified by alpha > 1e-2 following Ref [52]; no dynamical wall velocity computation.
standard math Standard thermal field theory and dimensional reduction formalism (Matsubara sums, hard/soft/ultrasoft matching) are valid.
Appendix A; implemented via DRalgo at NLO; this is accepted background methodology.

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Pith. "Pith review of Gravitational waves from color restoration in a leptoquark model of radiative neutrino masses." pith.science (2026). https://pith.science/paper/2C4WPKMZ

@misc{pith2026250101286,
  author       = {Pith},
  title        = {Pith review of: Gravitational waves from color restoration in a leptoquark model of radiative neutrino masses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2C4WPKMZ}},
  note         = {Machine review of arXiv:2501.01286}
}

abstract

We study the first-order phase transitions and the emerging stochastic gravitational wave spectrum in a minimal leptoquark extension of the Standard Model that explains active neutrino oscillation data while satisfying current flavor physics constraints. This model exhibits diverse phase transition patterns, including color symmetry-breaking scenarios in the early Universe. Strong correlations between model parameters and gravitational-wave signals yield testable predictions for future experiments such as LISA, BBO, and DECIGO. Specifically, a detectable signal in the mHz$\unicode{x2013}$0.1~Hz frequency range features color-restoration and leptoquark masses near $1.5~\mathrm{TeV}$. With this article, we also present the first application in the literature of \texttt{Dratopi}. This is a soon-to-be-released tool for phase transition analysis using the dimensional reduction formalism, that interfaces the \texttt{DRalgo} package with \texttt{Python} and a slightly modified version of \texttt{CosmoTransitions}.

Figures

Figures reproduced from arXiv: 2501.01286 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p030_12.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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[2] [1]

The extension of the SM through the inclusion of LQs allows, via modifications to the Higgs sector, for first-order EWPTs. These transitions can be either from a symmetric state to one where the Higgs field acquires a non-zero VEV (0 → h) or from a state where the Higgs field already has a non-zero VEV to another state with a different Higgs VEV, which we...

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To clarify our notation, a phase transition is denoted as F → T, where F refers to the false vacuum configuration and T refers to the true vacuum configuration

In addition, the model allows for a variety of phase transitions in which the SU(3) C color symmetry is broken in one or both of the phases involved in the transition. To clarify our notation, a phase transition is denoted as F → T, where F refers to the false vacuum configuration and T refers to the true vacuum configuration. The labels h, r, and s are u...

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[4] [3]

Define the model by specifying its 4d parameters, collectively denoted p4d, at some given reference energy scale µ0

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[5] [4]

Using the beta functions, solve the renormalization group (RG) equations, to obtain an interpolated solution of p4d(µ) as a function of the RG scale/energy scale µ (over some specified range)

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[6] [5]

Set the hard matching scale to µ4d = C4d · T , where C4d is a prefactor that defaults to π. 15

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[7] [6]

Construct the soft 3d EFT, by matching the 4d theory to the 3d EFT, at the scale µ4d

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[8] [7]

Here, mD,min is equal to the smallest Debye mass, at temperature T

Set the soft-to-ultrasoft matching scale to µUS 3d = C3d · mD,min. Here, mD,min is equal to the smallest Debye mass, at temperature T . Further, C3d is a prefactor which defaults to 1. 16

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