{"id":"5f8f575a-5a74-43f5-b92d-16b2737a8e51","arxiv_id":"2412.15055","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Adding one triplet and one singlet scalar leptoquark to the 331RHN model generates one-loop Majorana masses for active neutrinos and two-loop masses for sterile neutrinos, with benchmark parameters consistent with current rare decay limits.","lead":"This physics paper adds two hypothetical leptoquark particles to a 331-extension of the Standard Model and shows they can give tiny masses to ordinary neutrinos in one loop and to heavier sterile neutrinos in two loops. A general reader might care because the same particles could be detected at the LHC and offer a way to explain known neutrino masses and hints of light sterile neutrinos.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The two-loop sterile-neutrino mass formula in Eqs. (18)-(19) is not reproducible as printed: the θ dependence is dropped between the two equations, the h integral is never evaluated, and g13, g23 are introduced without derivation; the quoted 1-50 eV sterile masses are therefore not supported…","rationale":"The reader's weakest_assumption (the M T†ηS term) is indeed load-bearing, but I only partially agree: an explicit LNV seed is expected in any radiative Majorana-mass model, and the paper is transparent about it. The more actionable defect is that the two-loop calculation, which is the part of the central claim that makes sterile neutrinos 'inexorably light', is not reproducible from the text. Eq. (18) contains two terms with U11(θ) and U12(θ); Eq. (19) erases them. Since U11 = cosθ and U12 = -sinθ, the sum is cosθ - sinθ, not 1. The loop function h is never evaluated numerically, and the g13, g23 couplings are presented as 'obtained' without any derivation. The quoted eV-scale sterile masses therefore depend on hidden steps and free inputs; a reader cannot verify the central quantitative prediction. This does not falsify the mechanism, so the verdict remains CONDITIONAL (unchanged): the paper should supply a corrected, reproducible two-loop derivation and numerical evaluation before the claim can be accepted. I credit the one-loop mechanism, which is a standard radiative structure with a sin2θ ln(M1/M2) factor, and the consistency checks against Bs→μ+μ-, B→Kνν, and h→τμ are useful first-order checks. The deferral of cosmological constraints (footnote 4) further supports CONDITIONAL rather than ACCEPT.","tokens_in":18574,"tokens_out":8805,"duration_ms":58457,"concrete_test":"Recompute the two-loop amplitude from the diagram in Fig. 1(b) with standard Feynman rules, keeping the mixing insertions explicit, and evaluate the resulting integral (or h of Appendix A) numerically for the benchmark masses (M = 147 GeV, M2 = 2 TeV, md' = 1 TeV, Mη' = 3 TeV, sin2θ = 10^-2). Check (a) whether Eq. (19) follows from Eq. (18) or should carry a factor (cosθ - sinθ); (b) whether the resulting sterile masses match 1-50 eV when g13 and g23 are varied within the stated values; (c) whether g13, g23 can be constrained by the one-loop active fit, and report the corrected sterile masses if the θ factor changes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that sterile neutrinos are inexorably light at the eV scale rests on the two-loop formula in Eqs. (18)-(21). Three defects make that formula non-reproducible. (1) Eq. (18) has two terms weighted by U11(θ) and U12(θ); after the simplifying assumption that all leptoquark masses equal M2, the two h functions become identical and the θ dependence reduces to (cosθ - sinθ), but Eq. (19) simply drops the U factors, implicitly setting this combination to 1. The two equations are mutually inconsistent unless cosθ - sinθ = 1, which is not the case for sin2θ = 10^-2. (2) The loop function h is defined in Appendix A, but no numerical evaluation is given; the quoted values mR2≈19 eV, mR3≈24 eV (NO) and mR1≈1 eV, mR2≈50 eV (IO) cannot be reproduced by a reader. (3) The couplings g13 and g23 first appear in Eq. (20) as 'we obtain g13 = 0.000119 and g23 = 0.0014'; they are not determined by the active-neutrino fit (which fixes only Y and Ytilde) or by any other constraint, so the eV-scale sterile masses are not predictions of the fit but consequences of hand-picked inputs. The ad hoc M term (Eq. 6) is a legitimate seed for radiative masses, so I do not treat it as the blockage; the blockage is that the quantitative two-loop output built on M is internally inconsistent and uncheckable. Cosmological constraints on these states are deferred to 'elsewhere' (footnote 4), so no external check is offered either.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper adds a scalar leptoquark triplet T and a scalar leptoquark singlet S to the 3-3-1 model with right-handed neutrinos (331RHN). A Z2 symmetry and an explicit lepton-number-violating term M T† η S generate S–T mixing. The authors then obtain one-loop Majorana masses for active neutrinos and two-loop Majorana masses for right-handed (sterile) neutrinos, with the right-handed states emerging at the eV scale for their benchmark parameters. Using NuFIT best-fit values, they present benchmark Yukawa matrices Y and \\tilde Y for normal and inverted ordering, and they confront the model with B_s → μ⁺μ⁻, B → K(*)νν, and h → τμ data.","tokens_in":19158,"tokens_out":13571,"duration_ms":87333,"significance":"The model-building idea is interesting and structurally plausible: the one-loop active neutrino mass follows the standard scalar-leptoquark radiative pattern, and the same Yukawa sector feeds a two-loop sterile neutrino mass is a neat feature of the 331RHN framework. If the two-loop computation were made fully reproducible, the claim that sterile neutrinos are unavoidably light would be a valuable result. The paper also makes a genuine effort to connect the benchmark couplings to existing B-meson and Higgs constraints. However, the active neutrino sector is fitted rather than predicted, and the sterile sector currently depends on undetermined inputs and an unevaluated loop integral; in its present form the quantitative support for the central claim is not yet established.","major_comments":[{"comment":"The two-loop sterile neutrino mass computation, which is the central new claim of the paper, is not reproducible as printed. In Eq. (18) the loop contributions are weighted by U11(θ) and U12(θ). After the simplifying assumption that all leptoquark masses are equal to M2, the two h-functions coincide and the θ dependence collapses to U11+U12 = cosθ − sinθ, but Eq. (19) simply omits these factors; for the benchmark sin2θ = 10⁻² this is cosθ − sinθ ≈ 0.995, so the equations are formally inconsistent even if the numerical effect is small. More importantly, the function h defined in Appendix A (Eqs. A1–A7) is never evaluated numerically; no value is quoted for h(m_d′, M_η′0, m_b, M2), so a reader cannot reproduce the quoted masses m_R² ≈ 19 eV, m_R³ ≈ 24 eV (NO) and m_R¹ ≈ 1 eV, m_R² ≈ 50 eV (IO). The two-loop result therefore lacks the quantitative support needed for the paper's main claim that sterile neutrinos are inexorably light.","section":"III.B, Eqs. (18)-(21)"},{"comment":"The couplings g13 and g23 are introduced as numerical inputs immediately after Eq. (21) (\"we obtain g13 = 0.000119 and g23 = 0.0014\"), but their origin is not explained. They are not fixed by the active-neutrino fit of Eqs. (15)–(17), which determines only Y and \\tilde Y; nor are they related to the masses m_d′ or to any symmetry constraint stated in the paper. Because the sterile mass matrix in Eq. (21) is proportional to these couplings, the quoted eV-scale masses are consequences of hand-picked parameters rather than predictions. The paper should either derive these couplings, scan over the allowed range, or explicitly state the criterion used to select them, and should show how the sterile masses vary over that range.","section":"III.B, after Eq. (21)"},{"comment":"The active-neutrino fit is underdetermined as presented. Eq. (15) is a system of six equations for the twelve real entries of Y and \\tilde Y, yet no solution procedure is described: the manuscript does not state an ansatz, a minimization criterion, or a parametrization that selects the matrices displayed in Eqs. (16)–(17). Since the solution space is highly degenerate, the displayed benchmark is one arbitrary point, and the subsequent sterile-mass and B-physics results inherit this arbitrariness. In addition, Eq. (14) contains a misprint: the summation index p also appears as a lepton index in the term \"Y_{2p}\", which should presumably be \"Y_{2a}\", and the expression as a whole needs careful checking.","section":"III.A, Eqs. (14) and (15)"}],"minor_comments":[{"comment":"The text states in Section III.A that the model provides \"eV neutrinos\" while the conclusion refers to \"sub-eV neutrino mass\"; these statements should be reconciled.","section":"III.A and VI"},{"comment":"Cosmological and astrophysical constraints on the light sterile states, which are said to interact through the new 331 gauge bosons and leptoquarks, are deferred to future work; given the eV-scale masses quoted, this leaves the phenomenological viability check incomplete.","section":"III.B and footnote 4"},{"comment":"The phrase \"assuming the current experimental limit()\" has an empty parenthesis and the corresponding citation should be supplied.","section":"IV.A, Eq. (36)"},{"comment":"The couplings y32 and y33 are used in the h → τμ amplitude without being explicitly defined in the Yukawa Lagrangian of Eq. (5); a brief definition would improve readability.","section":"V, Eq. (44)"},{"comment":"The lepton-number-violating parameter M is fixed to 147 GeV without derivation; since all radiatively generated masses vanish in the M → 0 limit, the dependence of the quoted sterile masses on M (or equivalently on sin2θ) should be displayed.","section":"III.B, benchmark"}],"recommendation":"major_revision","confidential_remarks":"The paper is on the boundary between a model-building proposal and a quantitative phenomenological study. The idea is publishable after a substantial revision that makes the two-loop calculation reproducible and clearly separates fitted inputs from predicted outputs. The current version should not be accepted as-is because the central sterile-mass numbers cannot be checked from the information provided."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the idea is worth a serious referee, but the paper's main quantitative claim—eV sterile neutrinos—is not checkable as written. The one-loop active mass mechanism is standard and the benchmark Yukawa matrices do reproduce NuFIT splittings and angles. The novelty is real: previous 331RHN sterile-mass papers used other mechanisms, and this is the first time scalar leptoquarks generate active at one loop and sterile at two loops with shared Yukawas. That correlation is a nice structural point.\n\nWhat the paper does well: it identifies the minimal leptoquark content (triplet T + singlet S), shows the M T-dagger eta S term is what seeds both masses, checks B_s to mu+mu-, B to K(*) nu nu, and h to tau mu against current bounds, and the benchmark points survive those constraints. The Yukawa fits, while underdetermined (12 real unknowns for 6 equations), are at least solved explicitly and give couplings in a plausible range.\n\nSoft spots, in increasing order. Eq. (14) has a misprint: masses p inside the sum where a factor m_p should appear. Minor. More substantial: the active fit is not a prediction; it fits the model matrix to NuFIT with many free Yukawas, so the real test is the sterile sector. There the paper falls short. Eq. (18) has two terms weighted by U_11 and U_12; when the simplifying assumption makes the h functions identical, Eq. (19) drops those factors entirely, which is only valid if cos(theta) - sin(theta) = 1, not the case for sin2theta = 10^-2. The loop function h in Appendix A is defined but never evaluated; no numerical values or code are given, so the quoted sterile masses mR2 ~ 19 eV, mR1 ~ 1 eV cannot be reproduced. And g13, g23 appear in Eqs. (20)-(21) with no derivation—they are inputs, not outputs, of the fit. Cosmological constraints are explicitly deferred to 'elsewhere', so the eV-scale claim is supported neither internally nor externally.\n\nNone of this kills the underlying idea. The M term is a legitimate explicit LNV seed, and the one-loop structure is sound. But the central two-loop result is currently an unverifiable approximation. The authors can fix this fairly easily: carry the theta factors, evaluate h numerically, show how g13/g23 are chosen, and run the cosmological bounds for the eV states.\n\nMy recommendation: send it to peer review. It will need major revision before publication, but the mechanism is plausible, the workload is honest, and the connection between active and sterile radiative masses is worth refereeing properly.","headline":"Plausible radiative mechanism connecting active and sterile neutrino masses in 331RHN, but the headline eV-scale sterile predictions rest on a two-loop formula that is not reproducible as printed.","tokens_in":19640,"tokens_out":1971,"would_cite":false,"duration_ms":17972,"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":"Adding two scalar leptoquarks to the 331RHN model makes both active and sterile neutrinos light through radiative corrections.","keywords":["radiative neutrino mass","leptoquark","3-3-1 model","right-handed neutrinos","sterile neutrinos","Majorana mass","rare B decays","flavor-violating Higgs decay"],"falsifier":"A measurement that the charge $+1/3$ leptoquark states have exactly zero mixing ($\\sin 2\\theta=0$), or an observation of sterile neutrinos with masses far above the eV scale while active neutrinos keep the measured splittings, would eliminate the mechanism; conversely, a precise determination of the two-loop sterile masses near the predicted eV values would support it.","tokens_in":18360,"feed_emoji":"","tokens_out":5615,"duration_ms":55419,"temperature":0.7,"pith_summary":"This paper claims that a minimal addition to the 331RHN model—one triplet and one singlet scalar leptoquark—can generate tiny Majorana masses radiatively for both active and sterile neutrinos. The central result is a loop-level hierarchy: left-handed neutrinos get mass at one loop while right-handed (sterile) neutrinos get mass at two loops, so sterile neutrinos are unavoidably light rather than heavy. With benchmark Yukawa couplings, the model reproduces the measured solar and atmospheric mass splittings and mixing angles for both normal and inverted ordering, and it satisfies current bounds on rare B-meson decays and the flavor-violating Higgs decay $h\\to\\tau\\mu$. The paper therefore offers a radiative, leptoquark-driven alternative to seesaw mechanisms inside the 3-3-1 framework.","feed_headline":"Two leptoquarks make both active and sterile neutrinos light","feed_subtitle":"One-loop active, two-loop sterile masses in the 331RHN model; eV-scale sterile neutrinos follow inevitably.","key_machinery":"The machinery is the minimal leptoquark content: the scalar triplet $T\\sim(3,3,0)$ and the scalar singlet $S\\sim(3,1,1/3)$, added to the 331RHN model. The lepton-number-violating trilinear term $M\\,T^\\dagger\\eta S$ in the scalar potential (Eq. 6) generates a mass-mixing between the charge $+1/3$ leptoquark components; the mixing angle $\\sin 2\\theta = \\sqrt{2}\\,M\\,v_\\eta/(M_1^2-M_2^2)$ (Eq. 9) is the single knob that turns on both mass matrices. That mixing feeds a one-loop diagram (through $d$-quark and leptoquark exchange) that yields the active Majorana mass matrix (Eq. 12), and a two-loop diagram (through $d$, $d'$, leptoquarks and $\\eta'^0$) that yields the sterile Majorana mass matrix (Eq. 18). Because both matrices share the same Yukawa couplings $Y$ and $\\tilde Y$, the hierarchy of loop orders plus the heavy masses $m_{d'}$ and $M$ in the two-loop amplitude produces sterile neutrinos that are inevitably light—heavier than active neutrinos, but far lighter than typical seesaw singlets.","core_discovery":"The paper's discovery is that the same Yukawa couplings that give active neutrinos a one-loop Majorana mass matrix necessarily give the right-handed neutrinos a two-loop Majorana mass matrix, because in the 331RHN model left- and right-handed neutrinos share the same lepton triplet. The mechanism is driven by the mixing between the charge $+1/3$ components of the scalar leptoquarks $T$ and $S$, induced by the lepton-number-violating term $M\\,T^\\dagger\\eta S$ in the scalar potential. When $\\eta^0$ acquires a VEV, the mixing angle obeys $\\sin 2\\theta = \\sqrt{2}\\,M\\,v_\\eta/(M_1^2-M_2^2)$, and every neutrino mass term is proportional to this mixing. With leptoquarks at the TeV scale and Yukawa couplings in the range $10^{-1}$–$10^{-4}$, the active neutrino masses come out at sub-eV scale, while the sterile neutrinos come out at the eV scale (about 1–50 eV in the benchmark points), making them light but heavier than the active ones.","pith_inferences":["If $M$ ultimately originates from a spontaneously broken symmetry rather than being put in by hand at 147 GeV, the same scale would likely control the leptoquark mass splitting and could make the sterile mass pattern more predictive; the paper leaves this connection implicit.","The correlation between the active mass ordering and the sterile spectrum (NO: $m_{R2}\\approx 19$ eV, $m_{R3}\\approx 24$ eV; IO: $m_{R1}\\approx 1$ eV, $m_{R2}\\approx 50$ eV) is a testable linkage: a future determination of the sterile masses, e.g. from cosmology, would discriminate between the two orderings.","Because the two-loop sterile mass scales with $M$ and $m_{d'}$, measuring the new quark mass or the lepton-number-violating parameter in same-sign dilepton searches would quantitatively test the predicted eV-scale sterile masses; the paper does not compute those rates.","The paper does not analyze cosmological constraints on eV-scale sterile neutrinos, yet such states contribute to $N_{\\rm eff}$ and could be excluded or confirmed by CMB-S4-like observations; this is the most direct external check."],"forward_implications":["Sterile neutrinos in this model are unavoidably light (eV scale), not heavy, because their mass is generated at two loops rather than by a seesaw; this is a direct prediction of the mechanism the authors present.","The model reproduces the measured solar and atmospheric mass splittings and mixing angles for both normal and inverted ordering, with Yukawa couplings of order $10^{-1}$–$10^{-4}$ and TeV-scale leptoquarks.","The sterile neutrinos do not mix with active ones, so they escape short-baseline experimental constraints; their new interactions are through 331 gauge bosons and leptoquarks, leaving cosmology ($N_{\\rm eff}$, BBN, CMB) as the natural probe.","Benchmark points satisfy the current bounds on $B_s\\to\\mu^+\\mu^-$, the Belle II $B\\to K\\nu\\nu$ correlation, and the ATLAS limit on $h\\to\\tau\\mu$; the predicted $\\text{Br}(h\\to\\tau\\mu)$ is about $1.56\\times 10^{-4}\\%$.","Leptoquarks with masses around 2 TeV are the signature of the scenario and are being probed at the LHC."],"supporting_citations":[{"why":"Establishes the 331RHN framework with the leptonic triplet containing right-handed neutrinos, the model in which neutrinos are originally massless.","marker":"[28]"},{"why":"Provides the original 331 model with right-handed neutrinos that defines the gauge and fermion content used here.","marker":"[29]"},{"why":"Supplies the one-loop radiative neutrino mass formula that the paper adapts for the active neutrino mass matrix.","marker":"[50]"},{"why":"Gives the two-loop radiative mass framework for sterile neutrinos that the paper uses for the right-handed neutrino mass matrix.","marker":"[52]"},{"why":"First introduced leptoquarks in the 331RHN model and provides the Yukawa structure for leptoquark couplings used here.","marker":"[34]"},{"why":"Provides the NuFIT best-fit oscillation parameters (mass splittings and mixing angles) used to constrain the active neutrino mass matrix.","marker":"[4]"},{"why":"Reports the Belle II measurement of $B^+\\to K^+$+invisible that motivates the $B\\to K\\nu\\nu$ analysis and correlation plots.","marker":"[65]"},{"why":"Provides the correlation formula between $B\\to K\\nu\\nu$ and $B\\to K^*\\nu\\nu$ used to test the model against Belle II and Belle bounds.","marker":"[68]"}],"fun_headline_variants":["Leptoquark mixing yields active neutrinos at one loop, sterile at two","Same leptoquark coupling gives active and sterile neutrinos mass","TeV leptoquarks produce sub-eV active and eV sterile neutrinos","One-loop active, two-loop sterile neutrino masses from leptoquarks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire mass generation rests on the lepton-number-violating coupling $M$ in the scalar potential term $M\\,T^\\dagger\\eta S$: if $M$ were zero, the $S$–$T$ mixing angle vanishes and both the one-loop active and two-loop sterile neutrino mass matrices vanish; the paper fixes $M=147$ GeV as an input without deriving it from a more fundamental scale.","fun_headline_variants_meta":{"raw":{"variants":["Leptoquark mixing yields active neutrinos at one loop, sterile at two","Same leptoquark coupling gives active and sterile neutrinos mass","TeV leptoquarks produce sub-eV active and eV sterile neutrinos","One-loop active, two-loop sterile neutrino masses from leptoquarks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00026,"raw_usage":{"total_tokens":1563,"prompt_tokens":894,"completion_tokens":669,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":590}},"tokens_in":510,"tokens_out":669,"duration_ms":5167,"temperature":1.0,"reasoning_tokens":590,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:40:42.702548+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A measurement that the charge $+1/3$ leptoquark states have exactly zero mixing ($\\sin 2\\theta=0$), or an observation of sterile neutrinos with masses far above the eV scale while active neutrinos keep the measured splittings, would eliminate the mechanism; conversely, a precise determination of the two-loop sterile masses near the predicted eV values would support it.","supporting_citations":[{"cited_title":"An extended 3-3-1 model with radiative linear seesaw mechanism","cited_arxiv_id":"2105.01731","evidence_quote":"Supplies the one-loop radiative neutrino mass formula that the paper adapts for the active neutrino mass matrix."},{"cited_title":"Phenomenology of 3-3-1 models with radiative inverse seesaw mechanism","cited_arxiv_id":"2404.13373","evidence_quote":"Gives the two-loop radiative mass framework for sterile neutrinos that the paper uses for the right-handed neutrino mass matrix."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the correlation formula between $B\\to K\\nu\\nu$ and $B\\to K^*\\nu\\nu$ used to test the model against Belle II and Belle bounds."}],"review_version":1}