{"id":"d833dc83-9224-4ef6-b093-379c0f02f343","arxiv_id":"2505.07493","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Assuming the Dirac neutrino mass is proportional to the charged lepton mass, the seesaw relation implies heavy sterile neutrino masses of about 81.6κ TeV, 0.634κ EeV, and 31.8κ EeV.","lead":"This paper uses the seesaw mechanism to convert previously estimated tiny active neutrino masses into predicted masses for heavy right-handed 'sterile' neutrinos. The estimates land at roughly 80 TeV, 0.6 EeV, and 32 EeV, suggesting experimental targets and possible dark matter candidates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (12) requires M_D and M_R simultaneously diagonal, which makes the light-neutrino mass matrix diagonal and suppresses the observed PMNS mixing; adding the off-diagonal texture needed for mixing invalidates the one-to-one mass estimates.","rationale":"The paper is a short, transparent phenomenological note; the algebra from Eq. (12) to Eq. (13) is correct, and the authors explicitly flag the inverse procedure and the unknown κ. The reader's conditional verdict is appropriate. The strongest objection is not the arithmetic but the fact that Eq. (12) is not a generic seesaw relation: it relies on a simultaneously diagonal M_D and M_R. The sharper point is that this diagonal texture suppresses lepton mixing, so the model cannot reproduce the oscillation parameters that fix m1, m2, m3; introducing the mixing needed by Eq. (2) breaks the one-to-one relation between m_νi and M_i. This makes the specific mass scales illustrative rather than robust predictions. Because the paper does not overclaim and explicitly labels the result as estimates, the concern does not warrant a change in verdict; CONDITIONAL remains the right assessment.","tokens_in":6061,"tokens_out":11360,"duration_ms":108708,"concrete_test":"Analytical check: derive M_ν from Eq. (8) using M_D = σ diag(m_e, m_μ, m_τ) and M_R = diag(M1, M2, M3); diagonalize M_ν and verify that the resulting PMNS matrix is identity, contradicting Eq. (2). Then repeat the derivation with a minimal non-diagonal M_R, e.g., M_R = [[M1, a, 0], [a, M2, 0], [0, 0, M3]], and show that the light-neutrino eigenvalues are no longer related one-to-one to the M_i by Eq. (12). This would demonstrate that the quoted HSN mass estimates are tied to a texture that cannot accommodate the oscillation data used to set m_i.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (12) is not a generic consequence of the type-I seesaw. The paper obtains it by taking M_D = σ diag(m_e, m_μ, m_τ) and treating M_R as diagonal with eigenvalues M1, M2, M3. Under those assumptions the light-neutrino mass matrix M_ν = -M_D^T M_R^{-1} M_D is also diagonal in the charged-lepton mass basis, so the PMNS matrix is the unit matrix up to phases. This is in tension with the large mixing angles in Eq. (2), which are the same oscillation data used to fix the input masses m1, m2, m3. If, instead, M_R or M_D has the off-diagonal structure needed to produce the observed mixing, then the eigenvalues of M_ν are not given by κ m_li^2/(2 M_i), and the individual estimates M1≈81.6κ TeV, M2≈0.6343κ EeV, M3≈31.8294κ EeV do not follow. The unknown κ also absorbs the entire scale of the Dirac-Yukawa couplings, so \"κ of order unity\" is an assumption, not a consequence of the seesaw. The arithmetic from Eq. (12) to Eq. (13) is correct, but the model texture it rests on is both ad hoc and not the texture required by the observed lepton mixing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper estimates the masses of three heavy right-handed (sterile) neutrinos within a type-I seesaw framework. It assumes that the Dirac neutrino mass matrix is proportional to the charged-lepton mass matrix, M_D = σ diag(m_e, m_μ, m_τ), and that the sterile neutrino mass matrix is diagonal, which leads to the three independent relations m_{νi} = κ m_{li}^2/(2M_i) [Eq. (12)]. Using light neutrino masses m1≈0.0016 eV, m2≈0.0088 eV, and m3≈0.0496 eV taken from two earlier papers by the same group, the paper obtains M1≈81.6κ TeV, M2≈0.6343κ EeV, and M3≈31.8294κ EeV [Eq. (13)], and also estimates the observable masses m_C≈0.02 eV and m_β≈0.01 eV. The conclusion presents the heavy neutrinos as possible unstable fermionic dark-matter candidates.","tokens_in":6392,"tokens_out":7440,"duration_ms":75791,"significance":"If the derivation were valid, the paper would provide a simple phenomenological link between charged-lepton masses, light neutrino masses, and the scale of heavy right-handed neutrinos, with the interesting qualitative feature that the three heavy masses are hierarchically separated across the TeV–EeV range. Credit is due for the transparent and reproducible algebra: the conversion from Eq. (12) to Eq. (13) is arithmetically correct, and the quoted m_C value follows directly from the input masses. The significance is, however, strongly limited by three structural problems: the flavor-diagonal ansatz is incompatible with the observed large lepton mixing quoted in the same paper, the input light neutrino masses are self-cited without quoted uncertainties, and the overall scale is controlled by an undetermined parameter κ. These issues are load-bearing for the central numerical claims, not presentation details.","major_comments":[{"comment":"The assumed textures M_D = σ diag(m_e, m_μ, m_τ) and M_R = diag(M1, M2, M3) make the seesaw light-neutrino mass matrix M_ν = -M_D^T M_R^{-1} M_D diagonal in the charged-lepton mass basis, so the PMNS matrix is the identity up to phases. This is qualitatively incompatible with the large mixing angles listed in Eq. (2), which are the same oscillation data used to set the input masses. If off-diagonal structure is introduced to reproduce the observed PMNS matrix, the light neutrino eigenvalues are no longer given by κ m_{li}^2/(2M_i), and the individual estimates in Eq. (13) do not follow. The manuscript never addresses this tension, although Eq. (2) appears in the same text.","section":"Section 3, Eqs. (9)–(12)"},{"comment":"The light neutrino masses m1≈0.0016 eV, m2≈0.0088 eV, and m3≈0.0496 eV are taken from self-cited papers (Yudin et al., 2016; Khruschov et al., 2016) without error bars or an independent cross-check, yet Eq. (13) quotes results to several significant figures (e.g., M2≈0.6343κ EeV). Since M_i is inversely proportional to m_i in Eq. (12), the relative uncertainty in each output mass is at least as large as the relative uncertainty in the corresponding input mass. The precision displayed is therefore unjustified unless the input uncertainties are quantified and propagated.","section":"Section 3, Eqs. (12)–(13)"},{"comment":"The coefficient κ is completely undetermined and absorbs the entire scale of the Dirac mass matrix; the statement that κ is 'most likely of the order of unity' is an assumption with no derivation or bound. Consequently Eq. (13) does not provide absolute predictions of the seesaw mechanism, only the ratios M1:M2:M3 fixed by the ansatz. Without a quantitative treatment of κ, the quoted absolute masses cannot be meaningfully compared with collider, astrophysical, or cosmological searches.","section":"Section 3, Eqs. (12)–(13)"}],"minor_comments":[{"comment":"The notation changes from m_{νi} to μ_i without explanation, and the relation of Eq. (11) to the new formula (12) is not derived; a short derivation connecting Eqs. (8)–(10) to Eq. (12) would make the logic easier to follow.","section":"Section 2, Eqs. (10)–(11)"},{"comment":"The notation m^2_β and m_{2β} is visually confusing; please use distinct symbols or names for the kinematic beta-decay mass and the neutrinoless double-beta-decay effective mass.","section":"Section 2, Eqs. (5)–(6)"},{"comment":"The statement that HSNs with the estimated masses 'can decay in time scales shorter than the lifetime of the Universe' is made without a decay-width calculation or a quantitative estimate; if this point is to support the dark-matter discussion, it needs a substantiation or an explicit reference.","section":"Section 3, final paragraph"},{"comment":"The text contains a Cyrillic character in the units ('эВ2' instead of 'eV^2'); this should be corrected.","section":"Eq. (2c)"},{"comment":"The two central input papers for the light neutrino masses are self-citations; independent determinations or at least explicitly stated uncertainties should be provided when these values are used as inputs.","section":"References"}],"recommendation":"reject","confidential_remarks":"This is a very short paper whose central numerical output is a one-line inversion of Eq. (12). The input masses are self-cited without errors, the parameter κ is unconstrained, and the flavor-diagonal ansatz removes the PMNS mixing that the paper itself quotes. I do not see a result that a general hep-ph readership can use as a reliable estimate of right-handed neutrino masses. The core issue is not stylistic: the relation used for the central claim is not a consequence of the seesaw mechanism once the observed lepton mixing is taken into account, and fixing this would require a different model, not just a revised presentation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a short phenomenological note that plugs an ansatz into the type-I seesaw formula and gets three numbers for heavy sterile neutrino masses. The numbers are new: M1 ≈ 81.6κ TeV, M2 ≈ 0.634κ EeV, M3 ≈ 31.8κ EeV, plus estimates for m_C and m_β. That is the whole contribution, and it is a modest one.\n\nWhat the paper does well: it is transparent. The authors state clearly that they are inverting the seesaw relation using previously estimated active neutrino masses. The algebra from Eq. (12) to Eq. (13) is correct. They also flag the unknown parameter κ and do not oversell the results as predictions. The m_C and m_β values are sensible and within current bounds.\n\nThe soft spots are substantial. The stress-test concern is right: Eq. (12) requires both M_D and M_R to be diagonal in the charged-lepton mass basis. That makes the light-neutrino mass matrix diagonal too, so the PMNS matrix is the identity (up to phases). But the paper uses oscillation data with large mixing angles to fix the input masses m1, m2, m3. In other words, the model that produces the numbers is not the model that fits the oscillation data. If you add the off-diagonal structure needed for real mixing, the simple formula m_νi = κ m_li^2/(2M_i) breaks down, and the individual M_i estimates no longer follow. This is not a minor detail; it is a load-bearing assumption that is in tension with the paper's own inputs.\n\nAlso minor but worth noting: the input masses come from self-cited papers without error bars, κ is undetermined, and no uncertainty is propagated to the final masses. So the estimates are conditional three times over: on the texture, on the value of κ, and on the unstated errors in m_i.\n\nWho is this for? A reader who wants rough mass scales for heavy right-handed neutrinos in a very specific seesaw texture, perhaps as a target for collider or dark-matter searches, might find it useful. But that reader should be warned that the texture is not the one required by neutrino oscillation data.\n\nI would send this to a serious referee rather than desk-reject it, mainly so the texture-mixing tension gets explicitly written down. A good referee will ask the authors to either justify the diagonal texture or clearly delimit the claims to the zero-mixing limit. As it stands, the paper is a fine arithmetic exercise but not a robust estimate.","headline":"A transparent but fragile seesaw estimate: the numbers are new, yet the assumed texture forces zero lepton mixing, contradicting the oscillation data that supply the input masses.","tokens_in":6891,"tokens_out":1780,"would_cite":false,"duration_ms":19178,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["14.60.Pq","14.60.Lm","14.60.St","95.35.+d"],"model":"deepseek-v4-flash","headline":"The seesaw formula, applied family by family, puts the three heavy right-handed neutrinos at roughly 81.6 TeV, 0.63 EeV, and 31.8 EeV, with an unknown factor of order one.","keywords":["active neutrinos","sterile neutrinos","right-handed neutrinos","seesaw mechanism","neutrino masses","neutrino oscillations","neutrino mass observables","dark matter"],"falsifier":"A measurement of the absolute neutrino-mass scale that forces $m_3$ to differ from about $0.0496$ eV while keeping the measured splittings, or a direct search that finds a sterile neutrino far from the three predicted scales ($81.6\\kappa$ TeV, $0.6343\\kappa$ EeV, $31.8294\\kappa$ EeV), would falsify the specific three-way assignment; the same follows if a precise cosmological bound pushes $\\sum m_i$ well below $0.06$ eV.","tokens_in":5867,"feed_emoji":"⚛️","tokens_out":8745,"duration_ms":75154,"temperature":0.7,"pith_summary":"This paper estimates the masses of three heavy right-handed (sterile) neutrinos by turning the seesaw mechanism around: instead of predicting light-neutrino masses from heavy ones, it inserts measured light-neutrino masses into the simple relation $m_{\\nu i} = \\kappa m_{li}^2/(2M_i)$ and solves for $M_i$. With the normal-ordering values $m_1\\approx 0.0016$ eV, $m_2\\approx 0.0088$ eV, $m_3\\approx 0.0496$ eV, it obtains $M_1\\approx 81.6\\kappa$ TeV, $M_2\\approx 0.6343\\kappa$ EeV, and $M_3\\approx 31.8294\\kappa$ EeV, where $\\kappa$ is an unknown coefficient expected to be of order unity. It also derives observable neutrino masses $m_C\\approx 0.02$ eV and $m_\\beta\\approx 0.01$ eV, consistent with existing limits. The result matters because it ties the tiny active-neutrino masses to concrete, widely separated scales for heavy sterile neutrinos, which could be heavy fermionic dark-matter candidates if their decays are slow enough.","feed_headline":"Seesaw formula fixes heavy neutrinos at 81.6 TeV, 0.63 EeV, 31.8 EeV","feed_subtitle":"The predicted heavy partners are separated by five orders of magnitude and could be unstable dark matter.","key_machinery":"The load-bearing identity is the simplified seesaw formula $m_{\\nu i} = \\kappa m_{li}^2/(2M_i)$ (Eq. 12), obtained from the standard seesaw relation $M_\\nu = -M_D^T M_R^{-1} M_D$ by taking the Dirac neutrino mass matrix proportional to the charged-lepton mass matrix, $M_D = \\sigma\\,\\mathrm{diag}\\{m_e,m_\\mu,m_\\tau\\}$, and a diagonal sterile mass matrix with eigenvalues $M_1,M_2,M_3$. This reduction gives one equation per family, so each light-neutrino mass and its corresponding charged-lepton mass determine one heavy sterile scale. The coefficient $\\kappa$ absorbs the unknown overall normalization and is expected to be of order unity.","core_discovery":"The paper's central claim is that the seesaw relation, specialized to a diagonal Dirac mass matrix proportional to the charged-lepton masses, leaves a clean three-family formula $m_{\\nu i} = \\kappa m_{li}^2/(2M_i)$ with three independent heavy-neutrino masses. Substituting the adopted light-neutrino masses for normal ordering gives $M_1\\approx 81.6\\kappa$ TeV, $M_2\\approx 0.6343\\kappa$ EeV, and $M_3\\approx 31.8294\\kappa$ EeV. These values imply a huge separation among the heavy sterile states, and the corresponding observable masses $m_C\\approx 0.02$ eV and $m_\\beta\\approx 0.01$ eV fall inside current experimental limits. The paper therefore claims that the heavy right-handed neutrinos can be viewed as phenomenologically determined candidates whose masses are tied to the charged-lepton masses and the measured light-neutrino spectrum.","pith_inferences":["If a future model fixes $\\kappa$ independently, the three heavy masses become sharp predictions; a measured $\\kappa$ far from unity would shift all scales uniformly without breaking the family-by-family seesaw structure.","Applying the same formula under inverted ordering would produce a different set of heavy-neutrino masses; the paper does not compute those, but the relation is ready-made for that comparison.","A testable extension is to use the predicted $M_i$ values as targets for leptogenesis, collider searches, or astrophysical probes: finding a sterile neutrino at one of the three scales would support the proportionality assumption, while finding such a state at a different mass would rule out the three-way split.","Because $M_i \\propto m_{li}^2/m_{\\nu i}$, the prediction is most sensitive to the absolute scale of the lightest neutrino; future cosmology and neutrinoless double-beta decay measurements should sharpen that input directly."],"forward_implications":["If the formula is right, the three heavy right-handed neutrinos are separated by many orders of magnitude: one near 81.6 TeV, one near 0.63 EeV, and one near 31.8 EeV, all multiplied by $\\kappa$.","The paper predicts $m_C\\approx 0.02$ eV and $m_\\beta\\approx 0.01$ eV, and notes that these values do not conflict with current bounds on the neutrino mass observables.","Under the assumed structure, the heavy sterile neutrinos can decay on timescales shorter than the age of the Universe, so the paper classifies them as candidates for unstable heavy fermionic dark matter rather than stable dark-matter particles.","The total active-neutrino mass is tied to the normal-ordering sum of roughly 0.06 eV, which the paper presents as consistent with recent cosmological limits."],"supporting_citations":[{"why":"Supplies the adopted light-neutrino mass values $m_1$, $m_2$, and $m_3$ for normal ordering that are substituted into Eq. (12).","marker":"Yudin et al., 2016"},{"why":"Provides the neutrino mass values and normal-ordering spectrum used together with Yudin et al. as the phenomenological inputs.","marker":"Khruschov et al., 2016"},{"why":"Cited as the source for the seesaw-mechanism treatment and the choice $M_D = \\sigma m_l$.","marker":"Borisov and Isaev, 2024"},{"why":"Cited for the same proportional Dirac-mass assumption in the seesaw framework.","marker":"Vysotsky, 2011"},{"why":"Earlier phenomenological formula from which the simplified Eq. (12) is developed.","marker":"Khruschov, 2011"},{"why":"Provides the oscillation parameters and the experimental limits used to constrain the light-neutrino masses and the observable neutrino masses.","marker":"Esteban et al., 2024"}],"fun_headline_variants":["Seesaw ties heavy neutrino masses to charged leptons","Heavy neutrino masses span five orders via seesaw","Seesaw predicts heavy neutrinos at TeV to EeV scales","Seesaw heavy neutrino masses: 81.6 TeV to 31.8 EeV","Seesaw suggests heavy neutrinos as unstable dark matter"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that each neutrino family's Dirac mass is exactly the same constant times the charged-lepton mass of that family, with one heavy right-handed neutrino per family; if that proportionality is not exact, the three estimated heavy-neutrino masses do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Seesaw ties heavy neutrino masses to charged leptons","Heavy neutrino masses span five orders via seesaw","Seesaw predicts heavy neutrinos at TeV to EeV scales","Seesaw heavy neutrino masses: 81.6 TeV to 31.8 EeV","Seesaw suggests heavy neutrinos as unstable dark matter"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001767,"raw_usage":{"total_tokens":6918,"prompt_tokens":837,"completion_tokens":6081,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":453,"completion_tokens_details":{"reasoning_tokens":5989}},"tokens_in":453,"tokens_out":6081,"duration_ms":42089,"temperature":1.0,"reasoning_tokens":5989,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:15:36.961698+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A measurement of the absolute neutrino-mass scale that forces $m_3$ to differ from about $0.0496$ eV while keeping the measured splittings, or a direct search that finds a sterile neutrino far from the three predicted scales ($81.6\\kappa$ TeV, $0.6343\\kappa$ EeV, $31.8294\\kappa$ EeV), would falsify the specific three-way assignment; the same follows if a precise cosmological bound pushes $\\sum m_i$ well below $0.06$ eV.","supporting_citations":[],"review_version":1}