{"id":"810b9cdd-4950-47c1-993f-462e03f926c8","arxiv_id":"2412.02645","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A detailed scan of conformal U(1)' Majoron models finds that supercooled phase transitions produce gravitational waves detectable by LIGO, LISA, and ET, and that current null data already exclude part of the parameter space.","lead":"This paper predicts gravitational wave signals from supercooled first-order phase transitions in a broad class of conformal Majoron models that generate neutrino masses. A null detection at current or future gravitational wave observatories would exclude large regions of this model space, including high-scale seesaw neutrino masses.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exclusion contours (Figs. 13-17) use SNR>10 with GW templates whose efficiency factors are uncertain by orders of magnitude (Sec. 4.2, Fig.","rationale":"Agree with the reader that the template/efficiency-factor uncertainty is the weakest load-bearing link. The central claim—that current LVK data already exclude part of the parameter space and that a null signal at LIGO/ET would disfavor a high-scale seesaw—relies directly on the predicted h²Ω_GW exceeding the experimental sensitivity with SNR > 10. The paper's own Fig. 4 shows that the amplitude can vary by orders of magnitude under plausible changes in κ_SW, the radius distribution, and the wall velocity, yet the exclusion contours are presented as sharp lines. The proposed test—recomputing the LVK contour with κ_SW = 0.01—would empirically settle whether the excluded high-mass region survives a factor-of-100 reduction in the sound-wave efficiency. If it does not, the abstract's rule-out and disfavor statements must be softened. This is not a claim of error in the paper; the authors explicitly acknowledge the uncertainty, making the omission of uncertainty propagation on the contours a fair criticism. A conditional verdict—pending public code/data and an uncertainty treatment—remains appropriate. Therefore the reader's verdict is unchanged.","tokens_in":49831,"tokens_out":15448,"duration_ms":156284,"concrete_test":"Recompute the LVK exclusion region in the (MZ′, gL) plane of Fig. 13 (top-left) with the sound-wave efficiency fixed to the lower envelope of the uncertainty the authors display in Fig. 4a, i.e., set κ_SW = 0.01 for every scan point (instead of Eq. 3.28), keeping all other thermodynamic parameters and templates unchanged. Then check whether any point with MZ′ > 10^13 GeV and gL ≈ 0.3 still satisfies SNR_min > 10. If the contour disappears, the claimed LVK rule-out is not robust to the efficiency-factor uncertainty identified in Section 4.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central 'rule-out' and 'disfavor' claims are quantitative statements built on the predicted SGWB amplitude and the SNR>10 criterion (Eq. 5.3). The amplitude depends on the efficiency factors κ_SW and κ_BC and on the template normalizations (Eqs. 3.16, 3.27, 3.28). Section 4.2 itself demonstrates, in Fig. 4a, that varying κ_SW between 0.01 and 1 changes h²Ω_GW by roughly three orders of magnitude, and panels (b)-(d) show that the bubble radius distribution, wall velocity, and percolation condition also modify the spectra. The exclusion contours in Figs. 13-17 are drawn as sharp boundaries with no uncertainty bands, and the 'minimum SNR' procedure only takes a minimum over the scanned parameters, not over the template uncertainties. In the strongly supercooled regime that dominates the high-frequency excluded region, κ_SW saturates to ~1, but the sound-wave lifetime τ_SW and the bubble-collision efficiency κ_BC (Eq. 3.23) remain model-dependent. If the true efficiency is lower, or if the correct template set yields a smaller amplitude, the SNR values drop below 10 and the claimed exclusion of MZ' > 10^13 GeV (Fig. 17, top-left) and the 'null signal disfavors seesaw above 10^14 GeV' statement would not follow. The paper flags these uncertainties in Section 4.2 but does not propagate them into the headline constraints.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies stochastic gravitational wave backgrounds (SGWB) from strongly supercooled first-order phase transitions in a class of classically scale-invariant U(1)' extensions of the Standard Model with a Majoron-like scalar and a type-I seesaw. The authors build a 4D RG-improved thermal effective potential with one-loop and Daisy corrections, compute the Euclidean bounce action with CosmoTransitions (validated against their own code and against previous results in Refs. [15,21,22]), and scan over U(1)' charges, gauge coupling, the heavy scalar mass, and right-handed neutrino Yukawa couplings. They use the LISA Cosmology Working Group spectral templates to predict SGWB spectra and signal-to-noise ratios for LVK, LIGO-O5, ET, and LISA. The main results are correlations between the seesaw scale and the SGWB peak frequency, and a set of exclusion/projection contours: current LVK data are claimed to exclude MZ' above about 10^13 GeV for small Tr(yσ), a future null result at LIGO/ET would disfavor type-I seesaw scales above roughly 10^14 GeV, and LISA would probe seesaw scales as low as a TeV, even for decoupled right-handed neutrinos.","tokens_in":50098,"tokens_out":9676,"duration_ms":100094,"significance":"If the central predictions hold, the paper is a valuable phenomenological map: it connects neutrino physics to GW observables across a very wide range of scales, provides explicit benchmark points, and makes falsifiable statements that can be tested by current and near-future instruments. The manuscript is unusually transparent about the main sources of uncertainty in Section 4.2, and the numerical workflow is carefully cross-checked against existing codes and published results, including successful reproduction of Refs. [15,21,22]. The novelty relative to earlier B-L studies lies mainly in the generic charge assignment scan and in highlighting how the right-handed neutrino sector shapes high-frequency GW signals. The significance is moderate: the model class is specific, but the methodology and the neutrino-mass/GW connection are of general interest, and the paper provides concrete benchmarks for LISA, LIGO, and ET.","major_comments":[{"comment":"The central exclusion and 'rule-out' statements are built on SNR > 10 computed with fixed spectral templates, but the same section demonstrates that the predicted amplitude is uncertain by orders of magnitude. Fig. 4(a) shows that varying κ_SW from 0.01 to 1 changes h²Ω_GW by about three orders of magnitude, and panels (b)-(d) show comparable sensitivity to the bubble-radius distribution, wall velocity, and percolation condition. The contours in Figs. 13-17 are drawn as sharp boundaries, and the 'minimum SNR' procedure described in Section 5.1.4 minimizes only over the scanned model parameters, not over the template and efficiency uncertainties. Because the headline claims (e.g., the LVK exclusion of MZ' ≳ 10^13 GeV in Fig. 17, and the ET-based disfavoring of seesaw scales above ~10^14 GeV in the abstract) are quantitative, the authors should either propagate the Section 4.2 uncertainties into the SNR contours or explicitly qualify the claims as holding under the fiducial template assumptions. As it stands, the robustness of the main conclusions is not established.","section":"Section 4.2, Figs. 13-17, Eq. (5.3)"},{"comment":"Several points in the scan satisfy the percolation condition only at temperatures below Tp, and the paper itself states that it is 'unclear whether percolation is guaranteed' in these cases. The dashed contours in Figs. 6 and 13-17 mark such points, but the solid SNR>10 exclusion contours appear to be drawn without clearly excluding them. Please clarify whether the solid contours are restricted to points satisfying Eq. (3.9), and if not, state how the possibility of incomplete percolation is accounted for in the 'excluded' and 'disfavored' statements. If the ambiguous points are included, the constraints should be re-evaluated using only the subset with assured percolation.","section":"Section 3, Eq. (3.9); Figs. 6, 13-17"},{"comment":"The 4D RG-improved effective potential is used throughout, with the scale choice µ = max[M_Z'(ϕ), κT]. The discussion in Section 4 argues that the 3D EFT is not reliable in the deeply supercooled regime, but it does not provide a quantitative estimate of the scheme dependence of the 4D calculation for the large-α points that dominate the high-frequency exclusion regions. Since values of α up to 10^20 and percolation temperatures approaching the QCD scale are used, a robustness check (for example, varying κT by a factor of a few for representative benchmark points and showing the resulting shift in the SNR contours) would be needed to support the precision of the exclusion boundaries. This is a second unquantified source of uncertainty in the same load-bearing predictions, even though it may not change the qualitative picture.","section":"Section 4, Eq. (4.15)"}],"minor_comments":[{"comment":"The wording is not uniform: the abstract says strong supercooling 'can be ruled out,' while the Section 6 bullets use 'disfavored' for a null result. Please align the strength of the language with the actual SNR-based procedure.","section":"Abstract and Section 6"},{"comment":"The statement that 'a signal at high frequencies will favor U(1)B−L' appears to conflict with the same section's conclusion (and with Fig. 24) that the SGWB is not sensitive to xH and that the xH-induced frequency shift is smaller than the theoretical uncertainties. Please reconcile these statements.","section":"Section 6 and Fig. 24"},{"comment":"The caption defines the percolation condition as I(Tp) = 1 in panels (d), while the text uses I(Tp) = 0.34; please define both conventions explicitly at first use and state which one is used for the main results.","section":"Fig. 4 caption and Section 3"},{"comment":"The displayed expressions for M²_h1 and M²_h2 introduce Σ, Φh, and Φσ, but the final formulae appear to depend on only some of these combinations; please check the definitions and notation for consistency.","section":"Eq. (2.19)"},{"comment":"The acronym 'LVK' is used without expansion; please define LIGO-Virgo-KAGRA at first appearance.","section":"Section 5.1.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for JHEP and the field-theoretic core seems sound. My main concern is that the headline exclusion and disfavoring statements are presented with a precision that the manuscript's own Section 4.2 shows is not supported by the current treatment of template/efficiency uncertainties. If the authors can add uncertainty bands or systematically weaken the claims to 'under the fiducial template assumptions,' the paper would be acceptable. The reliance on a few self-citations for input ingredients (e.g., the thermalization condition and the bubble-size distribution) is not improper, but the editors may wish to ensure those ingredients are independently credible given their role in the conclusions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version: this is the most complete treatment yet of gravitational waves from supercooled phase transitions in conformal U(1)' Majoron models, and it does something genuinely useful by connecting the GW peak frequency to the type-I seesaw scale. The authors go beyond the existing U(1)B-L study [15] in a real way: they scan generic anomaly-free charge assignments, include one-loop RG-improved potential and self-energies, and explicitly account for large Majorana Yukawa couplings. They also validate their bounce calculation against CosmoTransitions and reproduce earlier results, so the numerics look trustworthy. The finding that a high-frequency signal at LIGO/ET would point to heavy right-handed neutrinos, while LISA can probe TeV-scale seesaw, is a nice, concrete takeaway.\n\nThe soft spot is exactly what the stress-test note says. The paper's headline constraints — the LVK exclusion of MZ' above about 10^13 GeV, and the statement that a null LIGO/ET signal disfavors a seesaw scale above 10^14 GeV — are drawn from SNR > 10 contours built on SGWB templates whose efficiency factors are uncertain by orders of magnitude. Section 4.2 and Fig. 4 show that varying kappa_SW from 0.01 to 1 shifts the amplitude by roughly three orders of magnitude, and the bubble radius distribution and wall velocity also move the spectra. The 'minimum SNR' procedure takes a minimum over scanned model parameters only; it does not propagate the template uncertainties. That makes the sharp exclusion boundaries in Figs. 13–17 look over-confident. To be fair, in the strongly supercooled regime kappa_SW tends to saturate, and the paper explicitly acknowledges these uncertainties — but it doesn't quantify their impact on the exclusion regions, which is what the strongest claims need.\n\nThere are minor things too: the percolation condition in Eq. (3.9) fails at Tp for some points (the dashed contours), and those points are borderline by construction; the authors flag this, which is honest. The self-citations for the thermalization condition and bubble-size distribution are contextual, not a red flag.\n\nBottom line: the physics program is sound, the computation is careful, and the paper is a genuine step forward for this model class. It deserves a serious peer review, and I'd recommend a major revision focused on one request: show how the LVK/LIGO/ET exclusion regions shift under the template and efficiency uncertainties, or state the limits as bands. If that's done, the claims will be much more durable. I'd bring it to a reading group and cite it if I worked in this area.","headline":"Solid, careful map of GW signals from conformal U(1)' Majoron models, but the sharp LVK exclusion contours outrun the acknowledged template uncertainties.","tokens_in":50696,"tokens_out":3676,"would_cite":true,"duration_ms":34545,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"If LIGO and ET see no gravitational-wave background, a high-scale seesaw in conformal Majoron models is disfavored.","keywords":["gravitational waves","stochastic gravitational wave background","supercooled phase transition","conformal Majoron model","type-I seesaw","U(1) gauge extension","LIGO-Virgo-KAGRA constraints","LISA sensitivity"],"falsifier":"Take benchmark point (b), which the paper predicts LIGO-O5 would see with SNR of about 29 after four years, and recompute its SGWB using a dimensionally reduced 3D effective field theory instead of the 4D RG-improved potential; if the resulting peak amplitude drops by more than an order of magnitude, the exclusion contours would not survive. Equivalently, a four-year LIGO-O5 run that places an upper limit below the predicted BP(b) spectrum would refute the claim that this point is detectable.","tokens_in":1937,"feed_emoji":"🌊","tokens_out":9699,"duration_ms":123908,"temperature":0.7,"pith_summary":"The paper argues that current and future gravitational-wave observatories can directly test the mechanism behind neutrino masses. In a wide class of classically scale-invariant U(1)' Majoron-like models, the same symmetry breaking that gives mass to right-handed neutrinos also drives a strongly supercooled first-order phase transition, which would emit a stochastic gravitational-wave background loud enough for LIGO, LISA, and the Einstein Telescope. If those experiments detect nothing, the paper claims that large regions of parameter space, including strong supercooling and type-I seesaw scales above $10^{14}$ GeV, would be disfavored or excluded. Conversely, a detected high-frequency background would point to heavy right-handed neutrinos. This matters because it connects neutrino physics to cosmology at energies far beyond collider reach.","feed_headline":"Null GW search would rule out heavy seesaw in Majoron models","feed_subtitle":"LIGO and ET silence would disfavor seesaw scales above 10^14 GeV; LISA still probes TeV-scale neutrinos.","key_machinery":"The load-bearing object is the renormalization-group-improved four-dimensional thermal effective potential for the Majoron field $\\sigma$, $V_{\\rm eff} = \\frac{1}{4}\\lambda_\\sigma(t) Z_\\sigma^2(t)\\sigma^4 + V_{\\rm CW} + V_T + V_{\\rm Daisy}$, where the Coleman-Weinberg and thermal contributions come from the $Z'$ boson and the three right-handed neutrinos. This potential, combined with the bounce action computed with CosmoTransitions, yields the nucleation, percolation, and reheating temperatures that feed the LISA Cosmology Working Group spectral templates for sound waves and bubble collisions. The key parametric relations are the loop-induced relation $\\lambda_\\sigma \\sim g_L^4$, the minimization condition $V_{\\rm min} \\sim v_\\sigma^4(-96g_L^4 + {\\rm Tr}(y_\\sigma^4))$, and the seesaw relation $M_N \\sim v_\\sigma y_\\sigma/\\sqrt{2}$, which together tie the gravitational-wave peak frequency and amplitude to the U(1)' breaking scale and the neutrino Yukawa sector.","core_discovery":"The central discovery is that a null result from LIGO, LISA, and ET would constitute a meaningful probe of the conformal Majoron parameter space. Concretely, for the U(1)$_{B-L}$ case the paper finds that current LVK data already exclude $M_{Z'} \\sim 10 M_{h_2}$ above $10^{13}$ GeV with $g_L \\sim 0.3$ and $\\mathrm{Tr}(y_\\sigma) < O(0.1)$. A non-observation at LIGO-O5 and ET would disfavor strong supercooling ($g_L \\lesssim 0.4$) for $Z'$ masses above $10^9$ GeV, and would rule out a type-I seesaw scale with $M_{N_i} \\gtrsim 10^{14}$ GeV and $y_\\nu \\sim O(1)$. LISA would remain sensitive to seesaw scales as low as a TeV, even when the right-handed neutrinos are decoupled. The paper also establishes a direct correlation between the peak frequency of the background and the size of the Dirac neutrino Yukawa coupling, so the frequency band of a future detection would identify the seesaw scale.","pith_inferences":["If the spectral templates overestimate the sound-wave efficiency, the exclusion contours would shrink; the paper's own uncertainty analysis shows that varying $\\kappa_{\\rm SW}$ from 0.01 to 1 changes the predicted amplitude by roughly three orders of magnitude, so the exclusion statements carry that theoretical systematic.","The paper's reheating-temperature treatment implies that supercooled models with low percolation temperatures produce backgrounds at higher frequencies than a naive redshift from $T_p$ would suggest, which would make the nanohertz band less accessible to low-scale versions of these models.","A sharper cross-check would be to compare the predicted peak-frequency--seesaw-scale correlation across the U(1)$_{B-L}$ and generic charge assignments; a future detection with a spectrum that matches the template shapes but falls outside the predicted amplitude-frequency region would indicate incorrect efficiency-factor assumptions."],"forward_implications":["Current LVK data already exclude the U(1)$_{B-L}$ model for $M_{Z'} \\sim 10 M_{h_2} > 10^{13}$ GeV, $g_L \\sim 0.3$, and ${\\rm Tr}(y_\\sigma) < O(0.1)$.","A null result at LIGO-O5 and ET would disfavor strong supercooling ($g_L \\lesssim 0.4$) for $Z'$ masses above $10^9$ GeV, and would exclude a type-I seesaw scale with $M_{N_i} \\gtrsim 10^{14}$ GeV and $y_\\nu \\sim O(1)$.","A positive high-frequency signal would be a signature of heavy right-handed neutrinos, because with decoupled right-handed neutrinos the GW spectrum is suppressed below 0.1 Hz.","LISA would test strong supercooling at $Z'$ masses of order 10 TeV and could detect a background even if the right-handed neutrinos are decoupled, probing $y_\\nu$ down to $10^{-6}$.","For a given mass, the gravitational-wave spectrum is almost independent of the Higgs charge $x_H$, so gravitational-wave data alone cannot single out a specific U(1)' model."],"supporting_citations":[{"why":"Supplies the baseline predictions for gravitational waves in the strongly supercooled classically conformal B-L model and the comparison for the one-loop versus tree-level potential treatment.","marker":"[15]"},{"why":"Provides the gravitational-wave energy budget for strongly supercooled transitions, including the bubble-collision efficiency factor and the reheating redshift factors (Gamma_h2/H) used in the spectra.","marker":"[16]"},{"why":"Gives the sound-wave template relation connecting the integrated amplitude to the bubble radius and phase-transition duration, used in the SGWB calculation.","marker":"[17]"},{"why":"Provides the comparison of 4D versus 3D effective-field-theory approaches for supercooled transitions, supporting the paper's choice of the RG-improved 4D potential.","marker":"[22]"},{"why":"Supplies the anomaly-free U(1)' charge assignments for the model class, with x_H and x_sigma as free parameters.","marker":"[26]"},{"why":"Provides the latest neutrino oscillation data fit used to fix the Dirac Yukawa couplings through the seesaw relation.","marker":"[30]"},{"why":"Is the bounce solver used to compute the Euclidean action and nucleation temperatures for the phase transition.","marker":"[44]"},{"why":"Establishes the link between first-order phase transitions in Majoron models and gravitational waves, including the thermalization condition for right-handed neutrinos.","marker":"[46]"},{"why":"Provides the LISA Cosmology Working Group spectral templates for sound waves and bubble collisions that are used to compute signal-to-noise ratios.","marker":"[53]"}],"fun_headline_variants":["GW silence disfavors heavy seesaw in Majoron models","LISA probes TeV-scale seesaw; null GW rules out heavy","Null LIGO/ET disfavors seesaw >10^14 GeV","GW frequency reveals Dirac neutrino Yukawa size","Supercooled transitions: null GW sets new seesaw bounds"],"cache_read_input_tokens":52736,"weakest_assumption_plain":"The predictions rest on the reliability of the gravitational-wave spectral templates and their energy-budget efficiency factors; varying the sound-wave efficiency $\\kappa_{\\rm SW}$ from 0.01 to 1 changes the predicted amplitude by about three orders of magnitude, so if the templates overestimate the emission, the exclusion and disfavor statements would collapse.","fun_headline_variants_meta":{"raw":{"variants":["GW silence disfavors heavy seesaw in Majoron models","LISA probes TeV-scale seesaw; null GW rules out heavy","Null LIGO/ET disfavors seesaw >10^14 GeV","GW frequency reveals Dirac neutrino Yukawa size","Supercooled transitions: null GW sets new seesaw bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000919,"raw_usage":{"total_tokens":3959,"prompt_tokens":976,"completion_tokens":2983,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":2895}},"tokens_in":592,"tokens_out":2983,"duration_ms":22447,"temperature":1.0,"reasoning_tokens":2895,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:13:31.502245+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take benchmark point (b), which the paper predicts LIGO-O5 would see with SNR of about 29 after four years, and recompute its SGWB using a dimensionally reduced 3D effective field theory instead of the 4D RG-improved potential; if the resulting peak amplitude drops by more than an order of magnitude, the exclusion contours would not survive. Equivalently, a four-year LIGO-O5 run that places an upper limit below the predicted BP(b) spectrum would refute the claim that this point is detectable.","supporting_citations":[],"review_version":1}