{"id":"76e0220c-39ad-4a3c-9162-f69b63d21b0b","arxiv_id":"2603.12385","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"PMNS non-unitarity would make W+W- production cross sections grow anomalously with energy; LEP data already bound δ_e ≲ 0.0135 and future colliders could reach ~10^-5.","lead":"This paper argues that if the PMNS matrix is not perfectly unitary, W-pair production at lepton and hadron colliders should show an anomalous rise with energy, and it converts this into bounds on the non-unitarity parameters. It matters because it offers a collider route to test neutrino-mixing unitarity, including the tau sector that lepton colliders cannot reach.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Heavy-neutrino t-channel exchange is not negligible at the highest probed energies, so the quoted δ bounds implicitly assume M_N^2 ≫ s and are not model-independent.","rationale":"The reader identified the same weakest assumption: the neglect of heavy-neutrino exchange below threshold. My analysis sharpens it by showing the suppression factor is |t|/(|t|+M_N^2), so the effect is reduced by O(1) when M_N is comparable to the probed energies. This directly affects the LEP II bound and the future projections, especially HL-LHC and FCC-hh where m_ℓℓ reaches TeV scales. The central mechanism is physically sound and the amplitude decomposition is standard; the issue is that the numerical bounds are conditional on the heavy mass scale being far above the energy. The paper even states the assumption but does not quantify it, so the appropriate verdict remains CONDITIONAL rather than ACCEPT or REJECT. I found no more load-bearing concern: the garbled Eq. (36) and lack of code/data are secondary and do not undermine the physics. Therefore I keep the reader's CONDITIONAL verdict, i.e., UNCHANGED.","tokens_in":15580,"tokens_out":16583,"duration_ms":158524,"concrete_test":"Compute the exact tree-level e+e- → W+W- cross section in a minimal type-I seesaw with one heavy neutrino of mass M_N and mixing δ, using the full propagator 1/(t-M_N^2) rather than the |t|≪M_N^2 limit. Repeat the LEP II χ2 analysis (Eq. (36)) for M_N = 0.3, 0.5, 1, 5 TeV and compare the resulting 95% CL bound on δ_e with the value 0.0135 in Eq. (37). Also recompute the HL-LHC projection (m_ℓℓ up to 1 TeV) for M_N = 1.2, 2, 5 TeV and compare with Eq. (46). If any bound weakens by more than ~20% when M_N is within a factor of 2 of the maximum probed energy, the model-independent interpretation fails and the mass dependence must be reported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that non-unitarity δ_α produces an anomalous growth of σ(e+e- → W+W-) rests on the approximation (Section III, before Eq. (26)) that t-channel diagrams with heavy neutral fermions are 'effectively negligible' below their mass threshold. This is the load-bearing step: the entire model-independent interpretation of Table II depends on it. In the full theory the t-channel amplitude is M_t = (1-δ) M_t(0) + δ M_t(M_N), where M_t(0) and M_t(M_N) are the amplitudes for massless and heavy neutrino exchange. The difference M_t(0)-M_t(M_N) is suppressed only by a factor ~ |t|/(|t|+M_N^2) (from the propagator), so the effective non-unitarity parameter entering the amplitude is δ_eff ≈ δ M_N^2/(|t|+M_N^2). For √s approaching M_N this factor is O(1), substantially reducing the predicted δ s and δ^2 s^2 growth. The paper does not quantify this mass dependence or marginalize over M_N, so the bounds (e.g., δ_α<1.0×10^-3 at HL-LHC with m_ℓℓ up to 1 TeV) implicitly assume M_N^2 ≫ s at the highest bins, which is not guaranteed in the low-scale seesaw models that motivate the study. The effect is real for M_N^2 ≫ s, but the numerical claims are conditional on an unstated mass assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a collider test of flavor-diagonal PMNS non-unitarity parameters δ_α (α=e,μ,τ) using W-pair production. In the presence of mixing with heavy neutral fermions, the t-channel light-neutrino sum is 1-δ_α, spoiling the standard gauge cancellation and producing an anomalous energy growth in the cross section. The authors decompose the squared amplitude as |M|^2 = |M_SM|^2 + δ_α Δ_1 + δ_α^2 Δ_2, with Δ_2 containing a term ∼ s^2/M_W^4. They use LEP II data to set δ_e ≲ 0.0135, and give projections for FCC-ee, ILC, CLIC, muon colliders, HL-LHC, and FCC-hh, claiming model-independent bounds on δ_α, including δ_τ.","tokens_in":15942,"tokens_out":8272,"duration_ms":72870,"significance":"The proposed mechanism is physically plausible and the tree-level derivation is clear, with explicit exact expressions collected in the appendices. If the method can be made robust, it would probe PMNS non-unitarity in a way complementary to electroweak precision fits, and it offers a rare handle on δ_τ. The paper is self-contained and cites the analogous CKM study. However, the central 'model-independent' claim is conditional on an unresolved heavy-neutrino-mass assumption, and the hadron-collider projections are signal-only estimates. The core idea is worth publishing after these points are addressed.","major_comments":[{"comment":"The entire method relies on the assumption that heavy-neutrino t-channel exchange is negligible below the mass threshold. The manuscript states this explicitly, but does not quantify the required mass scale. For finite M_N, the effective non-unitarity entering the amplitude is suppressed as δ_eff ≈ δ M_N^2/(|t|+M_N^2) (up to chiral factors). When √s approaches M_N, the predicted δ s and δ^2 s^2 growth is significantly reduced. Since the bounds in Table II and Eqs. (37),(39)-(41),(46),(47) are quoted as numerical numbers, they implicitly assume M_N^2 ≫ s at all probed energies. This is especially problematic because the low-scale seesaw models reviewed in Section II can have TeV-scale M_N, which is comparable to the highest energies at HL-LHC/FCC-hh and to the muon-collider benchmarks. The 'model-independent' wording in the abstract and conclusions is therefore too strong. The authors sho","section":"Section III, before Eq. (26); Eqs. (5), (26); Table II"},{"comment":"The LEP II χ² uses the measured cross sections themselves to rescale the tree-level SM prediction: the theoretical prediction is effectively σ_i × σ_e(s_i)/σSM(s_i). As a result, the SM point δ_e=0 gives χ²=0 by construction, and the analysis assumes that the fractional NLO/QED corrections, experimental cuts, and detector efficiencies are δ-independent and exactly equal to the ratio σ_i/σSM(s_i). This is an ad hoc assumption that is not derived from a full NLO calculation of the δ-dependent terms. The resulting bound δ_e ≲ 0.0135 (Eq. (37)) is therefore not robust. A dedicated NLO simulation, or at least an estimate of the systematic error introduced by this rescaling, is needed before this number is presented as a bound.","section":"Section IV, Eq. (36)"},{"comment":"The hadron-collider projections neglect all backgrounds and detector effects. The authors acknowledge this limitation in the text, but still quote δα < 1.0×10^-3 (HL-LHC) and δα < 4.4×10^-5 (FCC-hh) as 95% CL bounds. In reality, the WW-fusion signal pp → ℓ+ℓ- jj is contaminated by Z+jets, top-pair, and W+jets backgrounds; without a background estimate these numbers represent an optimistic signal-only sensitivity, not a realistic projection. This is particularly important because the hadron-collider channel is the only proposed way to access δ_τ. The quoted limits should be labeled as upper bounds on sensitivity and re-derived with at least a minimal background simulation before being used to support the conclusions.","section":"Section V, Eqs. (46)-(47); Table II"}],"minor_comments":[{"comment":"The printed formula for χ² is ambiguous: the fraction '(σ_i - σ_i σSM(s_i) σ_e(s_i))/ϵ_i' should be restructured, e.g., '[σ_i - σ_i (σ_e(s_i)/σSM(s_i))]^2/ϵ_i^2', to be readable.","section":"Eq. (36)"},{"comment":"Typo: 'unitarity in the the quark mixing matrix' repeats 'the'.","section":"Section IV, text after Eq. (40)"},{"comment":"Notation 's^2_Θ' and 'c^2_Θ' is easy to misread as powers of s; please use sin^2 Θ and cos^2 Θ explicitly.","section":"Section III, Eq. (34) and Appendix A"},{"comment":"The captions should state explicitly that the curves correspond to δ_α = 0.01 and 0.1, and define what 'NP contribution' means in the figure.","section":"Figs. 2 and 3"},{"comment":"Several author names and grant identifiers appear garbled in the source (e.g., 'M¨ u¨ ursepp'); please check that the metadata is correctly encoded.","section":"Author list and affiliations"}],"recommendation":"major_revision","confidential_remarks":"The idea is promising and the tree-level calculation is credible, but the 'model-independent' framing is not supported by the paper's own assumptions. The authors need to address the M_N dependence and the background issue in the hadron-collider projections before the numerical bounds can be taken at face value. I recommend major revision rather than rejection, since these are fixable within the scope of a revised manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is a legitimate extension of the CKM unitarity test to the neutrino sector, and the flavor-specific bounds are new. The core amplitude decomposition is standard and the anomalous growth mechanism works under the stated assumption that heavy neutrino exchange is negligible. I agree with the reader's conditional verdict.\n\nWhat it does well: it spells out the tree-level amplitude with non-unitary light-neutrino sum, derives the δΔ1 + δ^2Δ2 structure, and gives full expressions in the appendices. The LEP II bound δ_e ≤ 0.0135 is consistent with the data table. The hadron collider route to δ_τ is a genuinely useful complement to lepton colliders.\n\nWhere it is soft: the entire interpretation of Table II depends on the assumption that the heavy neutrino t-channel contribution is negligible, i.e. M_N^2 >> s. The stress-test note quantifies this: the effective non-unitarity parameter is δ M_N^2/(|t|+M_N^2), so at the highest energy bins the bounds degrade if M_N is not far above the scale. That is a real caveat, but the paper states the assumption openly, so it is a limitation, not an error. Also, the LEP chi^2 rescaling by σ_i/σ_SM is ad hoc and assumes no δ dependence in NLO corrections, and the hadron collider projections ignore backgrounds and detector effects. The paper acknowledges both; they mean the numbers should be read as idealized sensitivities. One small annoyance: Eq. (36) is garbled in the text, and no code/data are shipped, so reproducing the numbers requires some work.\n\nBottom line: this deserves a serious referee. The mechanism is sound, the application is new, and the bounds are plausibly in the right ballpark, but the paper should be revised to quantify the heavy-mass dependence and to justify or remove the NLO rescaling. I would send it to review rather than desk reject.","headline":"A physically sensible but idealized collider test of PMNS non-unitarity; the numbers need caveats, the mechanism is real.","tokens_in":16448,"tokens_out":1705,"would_cite":true,"duration_ms":17511,"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":"The paper shows that PMNS non-unitarity makes W-boson pair production grow with energy, converting existing LEP II data into a bound δ_e < 0.0135 and projecting future collider sensitivities down to δ ~ 3×10^-5.","keywords":["PMNS unitarity","neutrino mixing non-unitarity","W-boson pair production","collider bounds","seesaw models","anomalous energy growth","LEP II","muon collider"],"falsifier":"Measure the ratio σ(e+e− → W+W−)/σ_SM at two widely separated energies, for example at 350 GeV (FCC-ee) and 3 TeV (CLIC). The paper predicts that for δ_e ~ 0.01 this ratio first dips below 1 and then rises above 1 by an amount growing like δ^2 s^2. If the measured ratios match the SM at both energies within statistical errors, then δ_e is constrained below ~10^-4, refuting the claimed sensitivity for larger δ_e and the specific energy scaling.","tokens_in":15478,"feed_emoji":"⚛️","tokens_out":4301,"duration_ms":39550,"temperature":0.7,"pith_summary":"The paper argues that if the light-neutrino mixing matrix (PMNS) is not exactly unitary—because the light neutrinos mix with heavier states—then the Standard Model cancellation between t-channel neutrino exchange and s-channel gauge-boson exchange in e+e− → W+W− fails. The failure is controlled by the row-deficit parameter δ_α, and it produces a cross-section correction that first slightly suppresses and then anomalously grows with energy, roughly as δ_α^2 s^2, as long as the collision energy stays below the mass of the new heavy neutrinos. Using LEP II measurements, the authors derive δ_e ≲ 0.0135 at 95% CL, and they project that future lepton and hadron colliders could reach δ values as low as ~3×10^-5, far below present electroweak-precision constraints in the electron and tau sectors. If correct, this gives a model-independent, high-energy window into the mechanism behind neutrino masses.","feed_headline":"LEP II data limit PMNS non-unitarity to δ_e < 0.0135","feed_subtitle":"The same energy-growing W-pair signal lets future colliders reach δ ~ 3×10^-5.","key_machinery":"The key object is the deficit δ_α in each row of the PMNS matrix. In the t-channel amplitude, the sum over the three light neutrino mass eigenstates gives a factor Σ|U_ν^{αi}|^2 = 1 − δ_α, so the unitarity-violating part survives and prevents the exact cancellation between t- and s-channel diagrams that gauge invariance would otherwise enforce. The surviving terms, Δ_1 and Δ_2, carry the energy-growing behavior and are the basis of all the collider bounds derived in the paper.","core_discovery":"The central claim is that the flavor-diagonal non-unitarity parameters δ_α (defined by δ_α = 1 − Σ_i |U_ν^{αi}|^2 for α = e, μ, τ) make the squared amplitude for ℓ_α^+ ℓ_α^- → W+W− decompose as |M_α|^2 = |M_SM|^2 + δ_α Δ_1 + δ_α^2 Δ_2, with Δ_1 negative and roughly linear in s while Δ_2 is positive and grows like s^2/M_W^4. Consequently the total cross section first dips below the SM prediction and then rises steeply above it with increasing center-of-mass energy, a distinctive signature that persists until the heavy-neutrino threshold is reached. The authors use this mechanism to extract a LEP II bound δ_e < 0.0135 and to project sensitivities for FCC-ee, ILC, CLIC, a muon collider, HL-LHC,","pith_inferences":["The same amplitude decomposition could be applied to other neutrino t-channel processes, such as single-W or Z production, where similar energy-growing unitarity-violating terms would appear and could serve as independent cross-checks.","If heavy neutrinos are kinematically accessible, the anomalous growth is expected to stop and turn over; tracing the energy where the deviation peaks could therefore give a direct handle on the heavy mass scale.","The method is complementary to low-energy searches for lepton-flavor violation: flavor-diagonal δ_α are hard to constrain at low energies, so high-energy colliders may offer the cleanest probe of these particular parameters.","The angular distribution, which is suppressed near cos Θ = 0 in the SM but enhanced by the new terms, suggests that angular cuts could be used to isolate the non-unitarity contribution at CLIC or FCC-ee."],"forward_implications":["Existing LEP II measurements already constrain the electron-sector non-unitarity to δ_e < 0.0135, a model-independent limit from a single process.","A future FCC-ee run at 350 GeV with 1.8 ab^-1 could reach δ_e < 1.6×10^-4, while ILC and CLIC push to ~1×10^-4.","A 10 TeV muon collider is projected to reach δ_μ < 3.1×10^-5, roughly an order of magnitude beyond current precision fits.","Hadron colliders can probe all flavors, including δ_τ, with HL-LHC reaching δ < 1×10^-3 and FCC-hh reaching δ < 4.4×10^-5.","For δ_α above ~0.01, the cross-section ratio σ/σ_SM shows a distinctive turn-on at high energies, providing a clear experimental signature."],"fun_headline_variants":["Energy-growing W pairs test neutrino mixing unitarity","PMNS non-unitarity exposed by W-pair production","LEP II and future colliders probe PMNS unitarity","W boson pair production reveals neutrino mixing flaws"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire derivation assumes that, below the heavy-neutrino mass threshold, the only effect of the heavy states is to reduce the light-neutrino t-channel sum by the factor (1−δ_α); if heavy-neutrino exchange or effective operators from integrating them out contribute at order δ_α, the predicted energy growth and all derived bounds would change.","fun_headline_variants_meta":{"raw":{"variants":["Energy-growing W pairs test neutrino mixing unitarity","PMNS non-unitarity exposed by W-pair production","LEP II and future colliders probe PMNS unitarity","W boson pair production reveals neutrino mixing flaws"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000265,"raw_usage":{"total_tokens":1462,"prompt_tokens":781,"completion_tokens":681,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":616}},"tokens_in":525,"tokens_out":681,"duration_ms":6895,"temperature":1.0,"reasoning_tokens":616,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T18:18:40.673533+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the ratio σ(e+e− → W+W−)/σ_SM at two widely separated energies, for example at 350 GeV (FCC-ee) and 3 TeV (CLIC). The paper predicts that for δ_e ~ 0.01 this ratio first dips below 1 and then rises above 1 by an amount growing like δ^2 s^2. If the measured ratios match the SM at both energies within statistical errors, then δ_e is constrained below ~10^-4, refuting the claimed sensitivity for larger δ_e and the specific energy scaling.","supporting_citations":[],"review_version":1}