{"id":"bb063613-f443-499a-96f0-242c6f6fcebe","arxiv_id":"1908.09587","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"In the simplest linear seesaw model, astrophysical, unitarity, and LHC constraints force a compressed scalar spectrum and permit the 125 GeV Higgs to decay invisibly into majorons with branching ratios up to about 20%.","lead":"This paper studies a model where neutrinos get their tiny masses from the spontaneous breaking of a symmetry called lepton number, which also creates a new invisible particle, the majoron. It shows that experimental and theoretical constraints force the model's new scalar particles into a narrow mass range, and allow the 125 GeV Higgs boson to decay invisibly up to about 20% of the time.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Conditional on a (nearly) massless majoron: the stellar-cooling bound (5.3) is the load-bearing premise behind small vL, the compressed spectrum, and up-to-20% invisible BR, and Sec. 5.1 concedes it evaporates for heavier majorons; the abstract's categorical 'requires' overstates a scan result.","rationale":"I read the paper as a phenomenological profile of a specific low-scale seesaw variant; the central claim is the emergent scan result stated in Sec. 8. The reader's CONDITIONAL verdict is well calibrated, and my stress-test identifies the same load-bearing premise. Independent checks strengthen confidence in internal consistency: (i) with alpha1 = alpha3 = 0, Eq. (3.25) plus Eq. (3.18) gives lambdaL -> -(MA^2 - M2^2)/(2 vL^2) in the vL << v_phi, v_sigma limit, so the unitarity-driven compression mechanism is analytically real; (ii) the ~20% invisible BR sits at the 3-sigma edge of the 13 TeV ggF->ZZ signal strength in Table 3 (mu ~ kV^2(1 - BR_inv) ~ 0.74 for P2), consistent with the paper's stated 2-sigma maximum of ~10%; (iii) the benchmark spectra and kV values respect the sum rule (5.7). The paper also merits credit for using FeynMaster for amplitudes, making it likely that the Eq. (7.1) typo (O_R^{a2} for O_R^{a3} in the sigma term) is a transcription error rather than a computational one; still, the printed formula is what a reader would implement, and it gives a different width hierarchy. The conditionality concern is not an internal inconsistency: within Eq. (3.3), lepton number is exact, so the massless majoron and Eq. (5.3) are the natural default; escaping to a heavy majoron requires explicit LNV terms absent from the potential, and the paper gives no magnitude estimate. The v_sigma > 1 TeV cut in Sec. 6.1 is not the weak point: lowering v_sigma tightens (5.3), forcing smaller vL and strengthening the compression requirement. This is exactly why the abstract's categorical 'requires' overreaches what is demonstrated, and why REJECT would be too strong (the analysis is coherent within its stated assumption), while ACCEPT would underweight a premise the authors themselves flag. The proposed heavy-majoron re-scan directly tests the scope of the claim. Hence the reader's CONDITIONAL verdict stands unchanged.","tokens_in":22415,"tokens_out":28695,"duration_ms":256052,"concrete_test":"Re-run the constrained scan of Sec. 6.2 after adding an explicit lepton-number-violating majoron mass to Eq. (3.3), e.g., delta (sigma^2 + h.c.) with delta chosen so mJ = 10 keV, 1 MeV, and 100 MeV; drop Eq. (5.3), keep vL free over [10^-6, 10^2] GeV, and re-impose the same unitarity, oblique, and LHC signal-strength constraints (plus the appropriate massive-majoron bounds on g_Jee for mJ above stellar temperatures). If non-compressed spectra with BR_inv(h1) < 1% appear already at mJ = 1 MeV, the abstract's 'requires' is confirmed to hold only for the massless-majoron hypothesis; if the allowed region remains compressed and invisible-rich at all three masses, the conditionality is weakened and a stronger verdict could be considered.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim's logic runs: spontaneous lepton-number violation makes the majoron massless (the beta4 term in Eq. (3.3) conserves lepton number, so this is the model's internal consequence); stellar cooling then requires |<J|phi>| <~ 10^-7, Eq. (5.3), forcing vL <~ 0.5 GeV; at such vL the unitarity bound on lambdaL forces the numerator of Eq. (3.25) to nearly vanish, which the scan finds happens for alpha1 ~ 0 and a compressed spectrum; that is what allows BR(h1 -> JJ) up to ~20%. Each step is internally consistent: for alpha1 = alpha3 = 0 and vL << v_phi, Eq. (3.25) with Eq. (3.18) reduces to lambdaL = -(MA^2 - M2^2)/(2 vL^2), so the compression mechanism is analytically real. But the whole chain hangs on the applicability of (5.3). Sec. 5.1 states the bound 'need not apply' if the majoron is heavier than stellar temperatures, and offers no estimate of mJ from quantum-gravity or other explicit LNV sources; if (5.3) is evaded, vL is no longer forced small, the unitarity argument collapses, and no compression is required. The abstract and Sec. 8 nevertheless state the conclusion categorically. A separate defect compounds this: Eq. (7.1), the master formula for the invisible widths, has the sigma-term mixing element printed as O_R^{a2} instead of O_R^{a3}; since FeynMaster generated the amplitudes, Tables 4-6 may be correct, but the printed equations cannot reproduce them. The load-bearing issue remains the assumed massless majoron; the paper is right to flag it, and the verdict should keep it conditional.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the scalar sector of the simplest linear seesaw extension of the Standard Model, adding a second doublet chi_L and a singlet sigma that carry lepton number. Neutrino masses arise from spontaneous lepton-number violation, producing a majoron. The authors derive the scalar mass matrices, express the quartic couplings in terms of physical masses, vevs, and rotation angles, and impose stability, unitarity, oblique-parameter, astrophysical, and LHC constraints in numerical scans. The central outputs are that the stellar-cooling bound forces v_L below about 0.5 GeV, that perturbative unitarity on lambda_L then selects a compressed scalar spectrum with alpha_1 close to zero, and that the 125 GeV Higgs can acquire an invisible branching ratio into majorons of up to about 20% at the 3-sigma LHC level (about 10% at 2-sigma). Three benchmark points illustrate different invisible-decay patterns. The paper concludes that a consistent electroweak symmetry breaking pattern 'requires' a compressed spectrum with potentially large invisible Higgs decay.","tokens_in":22887,"tokens_out":9168,"duration_ms":83721,"significance":"If the results hold, the paper provides a complete and mostly consistent analysis of the scalar sector of the simplest linear seesaw model, with a concrete and falsifiable consequence: under the massless-majoron assumption, the allowed parameter space is compressed and BR(h1 -> invisible) can approach the current experimental upper bound. The analytical expressions for the quartic couplings in terms of the physical inputs, the explicit treatment of stability and unitarity bounds, and the use of FeynMaster for the decay amplitudes are strengths. The provision of three phenomenologically distinct benchmark points is also valuable. The main caveat is that the predictive chain is conditional on the majoron being effectively massless; the paper itself flags this in Sec. 5.1, but the abstract and conclusions are stated more categorically than the analysis supports. The scan additionally restricts v_sigma > 1 TeV. With appropriate qualification, this is a solid contribution to Higgs phenomenology in low-scale seesaw models.","major_comments":[{"comment":"The conclusion that 'a consistent electroweak symmetry breaking pattern requires a compressed mass spectrum of scalar bosons' is categorical, but it follows only under the assumption of a nearly massless majoron. The astrophysical bound in Eq. (5.3) applies only if the majoron mass is below stellar temperatures; Sec. 5.1 explicitly states that if the majoron is heavier, 'the bound in Eq. (5.3) need not apply,' and no estimate of the majoron mass from explicit lepton-number-violating sources is provided. In that case v_L is no longer forced to be small, the unitarity argument based on Eq. (3.25) collapses, and no compressed spectrum is required. The abstract and conclusions should be rephrased to present the result as conditional, for example: 'Under the assumption of an effectively massless majoron, the scan consistent with all applied constraints exhibits a compressed spectrum...'.","section":"Abstract and Sec. 8, relying on Sec. 5.1"},{"comment":"The master formula for the Higgs-majoron coupling contains a typo in the third term: it is printed with O_R^{a2} multiplying the 1/v_sigma contribution, but the singlet field R3 is the one with the sigma vev, so this factor should be O_R^{a3}. As printed, the formula cannot reproduce the invisible branching ratios in Tables 4-6. This should be corrected, and the text should state clearly that the numerical results were generated with FeynMaster and that the printed formula was verified.","section":"Eq. (7.1)"},{"comment":"The numerical scan imposes the cut v_sigma > 1 TeV 'for technical reasons,' while the text acknowledges that lower values could be possible. Because the conclusion about compressed spectra is derived from the scanned region, either the scan should be extended to lower v_sigma or the conclusion should be explicitly restricted to v_sigma > 1 TeV. The current wording implies broader validity than the sampling supports.","section":"Sec. 6.1"}],"minor_comments":[{"comment":"The denominator in Eq. (5.3) is missing a closing parenthesis inside the square root; it should read sqrt((v_phi^2 + v_L^2)(v_phi^2(4 v_L^2 + v_sigma^2) + v_L^2 v_sigma^2)).","section":"Sec. 5.1, Eq. (5.3)"},{"comment":"'Nambu-Golstone' appears in the Introduction; the standard spelling is 'Nambu-Goldstone.'","section":"Sec. 1"},{"comment":"In the sentence before Eq. (4.14), 'potantial' should be 'potential.'","section":"Sec. 4.2"},{"comment":"The sentence 'It just turned out the that the good points have this profile' contains an extra 'the that'; it should read 'It just turned out that the good points have this profile.'","section":"Sec. 6.3"},{"comment":"'Invisibling Higgs decay bosons' should be 'invisible Higgs decay bosons.'","section":"Sec. 7"},{"comment":"The phrase 'For the 13 TeV of Run-2, the data channel' is a fragment; it should be rephrased, e.g., 'For the 13 TeV Run-2 data, the results are shown in Table 3.'","section":"Sec. 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is within JHEP scope and the technical analysis appears careful. My main concern is that the headline conclusion is stated more strongly than the assumptions warrant: the authors' own Sec. 5.1 caveat about a heavier majoron is not carried into the abstract or conclusions. I would ask the authors to qualify the claim and to correct the Eq. (7.1) typo and the v_sigma sampling issue. The paper also draws substantially on the authors' prior framework, but the contained analysis stands on its own."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a careful phenomenology paper, and the main physics claim survives scrutiny—within the assumption of a nearly massless majoron. What's new is the explicit singlet-extended scalar sector for the minimal linear seesaw (no left-right symmetry), and the observation that the combined constraints force a compressed scalar spectrum with α1 ≈ 0, which in turn lets the 125 GeV Higgs decay invisibly to majorons with BR up to ~20% at 3σ (or ~10% at 2σ). The three benchmarks P1–P3 are concrete and usable.\n\nThe paper does a lot right. Mass matrices, minimization conditions, unitarity/stability constraints and the oblique parameters are all written out, and the appendix gives the full coupled-channel matrices. Credit where due: Sec. 5.1 explicitly concedes that the stellar-cooling bound only applies if the majoron is lighter than stellar temperatures, and that a heavier majoron would evade it. The authors know the load-bearing step and say so. The citation pattern is self-heavy, but the prior papers genuinely cover invisible Higgs decays in seesaw variants; this is a legitimate extension, not padding.\n\nSoft spots, in proportion. First, the abstract and conclusions say the compressed spectrum is 'required'. What the scan shows is that it emerges once you assume a nearly massless majoron, impose the astrophysical bound and restrict vσ > 1 TeV (a technical cut the authors admit). That is a conditional result stated categorically. Easy fix: soften the language. Second, Eq. (7.1) has a real typo: the σ-term should couple to the singlet component O_R^{a3}, not O_R^{a2}. The amplitudes came from FeynMaster, so Tables 4–6 may be fine, but as printed the formula cannot reproduce them. Third, the scans are not reproducible from the text: no code, no data, thin sampling details. For a scan-driven headline claim, that is a fair request, though minor.\n\nThe massless-majoron point is the one to stress. The chain—spontaneous LNV gives a massless majoron, stellar cooling bounds its doublet projection, vL is forced small, unitarity on λL forces near-degeneracy—is internally consistent, and the stress-test reduction of Eq. (3.25) checks out. But it all hangs on the majoron staying light. If explicit LNV lifts it, the bound evaporates and the compression argument collapses. The paper flags this; the abstract doesn't. That mismatch, not the physics, is the real weakness.\n\nWho this is for: anyone working on low-scale seesaw or majoron phenomenology, especially invisible Higgs searches. It is a solid benchmark paper, not a paradigm shift. I would send it to review; with the language fixed, the typo corrected and more transparency on the scans, it is publishable. My verdict is conditionally accept: the load-bearing assumption is declared, the math is internally consistent, and the result is concrete and testable.","headline":"A careful, mostly sound scan of the simplest linear seesaw variant—the compressed-spectrum/invisible-BR result holds under the explicitly flagged massless-majoron assumption, but the abstract overstates it and Eq. (7.1) has a typo.","tokens_in":23425,"tokens_out":5326,"would_cite":true,"duration_ms":49057,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In the simplest linear seesaw model, if all known constraints are imposed, the scalar bosons are forced into a compressed spectrum and the 125 GeV Higgs can decay invisibly into majorons with branching ratio up to about 20%.","keywords":["linear seesaw","majoron","invisible Higgs decay","spontaneous lepton number violation","compressed scalar spectrum","Higgs boson profile","stellar cooling bound","electroweak symmetry breaking"],"falsifier":"A future lepton collider measuring the 125 GeV Higgs invisible width at sub-percent precision: if the measured $\\mathrm{BR}(h\\to\\text{invisible})$ falls below about 1%, the parameter region where this paper finds values up to ~20% under the massless-majoron assumption would be excluded; a null result would push the model into the extreme of parameter space where the invisible width is suppressed.","tokens_in":22200,"feed_emoji":"👻","tokens_out":8738,"duration_ms":82614,"temperature":0.7,"pith_summary":"This paper examines the simplest linear seesaw model, built from the Standard Model gauge structure plus a nearly inert second scalar doublet and a lepton-number-carrying singlet, and asks what electroweak symmetry breaking looks like once all constraints are imposed. It claims that consistency with the astrophysical bound on the majoron, which forces the lepton-number-breaking vev $v_L \\lesssim 0.5$ GeV, requires the neutral scalar spectrum to be compressed, with the three CP-even Higgs bosons nearly degenerate. In this regime the Standard-Model-like 125 GeV Higgs can decay invisibly into a pair of majorons with branching ratio as large as about 20% while satisfying LHC signal-strength data at $3\\sigma$ (about 10% at $2\\sigma$). The paper maps the resulting Higgs profile and gives benchmark points showing the different invisible-decay patterns among the three neutral scalars.","feed_headline":"Seesaw Higgs can vanish 20% of the time","feed_subtitle":"Stellar cooling forces a near-massless majoron and a compressed scalar spectrum, enabling invisible Higgs decay.","key_machinery":"The central objects are the pseudo-scalar rotation matrix $O^I$, whose entries give the majoron projection $\\langle J|\\phi\\rangle = 2v_\\phi v_L^2 / \\sqrt{(v_\\phi^2+v_L^2)(v_\\phi^2(4v_L^2+v_\\sigma^2)+v_L^2v_\\sigma^2)}$, and the neutral scalar rotation matrix $O^R(\\alpha_1,\\alpha_2,\\alpha_3)$. The decisive relation is the expression for the quartic coupling $\\lambda_L$ in terms of physical masses and angles: since $\\lambda_L \\propto v_L^{-3}$, a tiny $v_L$ forces near-degenerate CP-even masses and $\\alpha_1\\approx 0$ to satisfy perturbative unitarity. The invisible-decay coupling $g_{h_a JJ} = -\\left(\\frac{(O^I_{21})^2}{v_\\phi}O^R_{a1}+\\frac{(O^I_{22})^2}{v_L}O^R_{a2}+\\frac{(O^I_{23})^2}{v_\\sigma}O^R_{a2}\\right)M_a^2$ then connects the vev hierarchy directly to the observable branching ratio.","core_discovery":"Working in the minimal $SU(3)_c\\otimes SU(2)_L\\otimes U(1)_Y$ realization of the linear seesaw, the paper shows that spontaneous violation of global lepton number produces a massless majoron whose coupling to electrons is suppressed by the projection $\\langle J|\\phi\\rangle \\propto v_L^2$, so stellar cooling forces $v_L \\lesssim 0.5$ GeV. It then demonstrates that such a small $v_L$ is compatible with vacuum stability and perturbative unitarity only if the CP-even scalar spectrum is compressed and the mixing angle $\\alpha_1$ is near zero; this follows because the quartic coupling $\\lambda_L$ of the nearly inert doublet is inversely proportional to $v_L^3$, so its numerator must nearly vanish, which happens for degenerate masses. With this compressed spectrum, the SM-like boson $h_1$ acquires a sizable coupling to two majorons, giving $\\mathrm{BR}(h_1\\to JJ)$ up to roughly 20% within the $3\\sigma$ LHC constraints, and the paper provides three benchmark points P1--P3 covering qualitatively different invisible-decay patterns of the three neutral scalars.","pith_inferences":["The paper's main conclusion is conditional on the majoron being nearly massless; if higher-dimensional operators give it a mass above stellar temperatures (a possibility the authors mention), the bound $v_L \\lesssim 0.5$ GeV evaporates and the compressed-spectrum requirement would not be needed, so the model would open up parameter regions not shown here.","The scan imposes a technical cut $v_\\sigma > 1$ TeV that is not physically required; exploring lower $v_\\sigma$ values could change the scalar-spectrum correlations and potentially shift the maximum invisible branching ratio.","The same $\\lambda_L \\propto v_L^{-3}$ mechanism implies a consistency check: measuring the mass splitting $M_3-M_2$ and the mixing angle $\\alpha_1$ (via vector couplings) at a future collider could indirectly probe the astrophysical $v_L$ bound without directly observing the majoron.","If a future precision measurement finds $\\mathrm{BR}(h\\to\\text{invisible})$ below about 1%, the model is not dead but is pushed into the extreme of parameter space where the majoron coupling is suppressed; the compressed-spectrum prediction would still remain a distinctive collateral signature."],"forward_implications":["If the model is right, the 125 GeV Higgs has an invisible branching ratio that current LHC data allow up to about 20% at $3\\sigma$ and 10% at $2\\sigma$, close to the present experimental upper bound and testable at the HL-LHC.","A consistent electroweak-breaking pattern requires a compressed neutral scalar spectrum, so the model predicts two additional CP-even scalars within a few tens of GeV of each other, along with a nearby charged and pseudoscalar state.","The heavier scalars can be either visible or invisible: benchmark P2 shows $h_2$ with an invisible branching ratio around 13% and a still sizable coupling to vector bosons, while P1 shows only $h_1$ with a large invisible width and the heavier states decaying visibly.","The vector couplings obey the sum rule $\\sum_i |k_V(h_i)|^2 = 1$, so fixing the 125 GeV coupling near the SM value limits the production of the heavier scalars and constrains their observability.","The paper notes that future lepton colliders are expected to measure the invisible branching ratio with precision better than 1%, which would sharply constrain or exclude the large-invisible-width region."],"supporting_citations":[{"why":"Derives the majoron-electron coupling via Noether's theorem and gives the model-independent projection that underlies the stellar-cooling bound in Eq. (5.3).","marker":"[7]"},{"why":"Established that spontaneously broken lepton number generically leads to invisible Higgs decays, which is the paper's central observable.","marker":"[8]"},{"why":"Supplies the bounded-from-below stability conditions for the scalar potential that are imposed on the parameter scan.","marker":"[53]"},{"why":"Provides the perturbative unitarity bounds for multi-Higgs-doublet models and the sum rules for vector-boson couplings used throughout the Higgs profile analysis.","marker":"[54]"},{"why":"The ATLAS and CMS measurements of the 125 GeV Higgs signal strengths at 8 and 13 TeV are used to constrain the model at $2\\sigma$ and $3\\sigma$.","marker":"[59,60]"},{"why":"Implements the LEP, Tevatron and LHC exclusion bounds on the additional neutral and charged scalars in the scan.","marker":"[61]"},{"why":"Gives the multi-Higgs-doublet formula for the vector couplings $k_V(h_i)$ and the $h\\to\\gamma\\gamma$ / $h\\to Z\\gamma$ decay expressions used to compute the branching ratios.","marker":"[62]"},{"why":"Current LHC searches for invisible Higgs decays set the upper bound against which the predicted branching ratio up to 20% is compared.","marker":"[28,29]"}],"fun_headline_variants":["Invisible Higgs decay up to 20% in linear seesaw","Seesaw majoron triggers 20% invisible Higgs decays","Stellar cooling forces 20% invisible Higgs decay","Compressed scalars yield 20% invisible Higgs","Majoron from seesaw up to 20% invisible Higgs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire result stands on the majoron being effectively massless; if the majoron were heavier than stellar temperatures, the astrophysical bound forcing $v_L \\lesssim 0.5$ GeV would not apply, and the model would no longer require a compressed spectrum or large invisible branching ratios.","fun_headline_variants_meta":{"raw":{"variants":["Invisible Higgs decay up to 20% in linear seesaw","Seesaw majoron triggers 20% invisible Higgs decays","Stellar cooling forces 20% invisible Higgs decay","Compressed scalars yield 20% invisible Higgs","Majoron from seesaw up to 20% invisible Higgs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000494,"raw_usage":{"total_tokens":2416,"prompt_tokens":928,"completion_tokens":1488,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":1405}},"tokens_in":544,"tokens_out":1488,"duration_ms":10759,"temperature":1.0,"reasoning_tokens":1405,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:07:51.405773+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A future lepton collider measuring the 125 GeV Higgs invisible width at sub-percent precision: if the measured $\\mathrm{BR}(h\\to\\text{invisible})$ falls below about 1%, the parameter region where this paper finds values up to ~20% under the massless-majoron assumption would be excluded; a null result would push the model into the extreme of parameter space where the invisible width is suppressed.","supporting_citations":[{"cited_title":"On Necessary and Sufficient Conditions for Some Higgs Potentials to Be Bounded From Below,","cited_arxiv_id":null,"evidence_quote":"Supplies the bounded-from-below stability conditions for the scalar potential that are imposed on the parameter scan."}],"review_version":1}