{"id":"fbb05dbe-2b8e-4c39-bccd-b7adf9d6541d","arxiv_id":"2412.20723","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Three explicit model frameworks correlate tiny neutrino mass with proton longevity, either through a common dark matter source, through proton decay generating neutrino mass, or through neutrino mass mediating proton decay.","lead":"This paper proposes three ways to tie the tiny mass of neutrinos to the extreme longevity of the proton, so one piece of new physics could explain both. It builds explicit models with new particles that connect quarks and leptons, giving experiments a shared target.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The three-loop neutrino mass in Model 2 is never computed, so the claimed correlation for the proton-stability-to-neutrino-mass direction remains unverified; finiteness, dominance over lower-loop topologies, and compatibility with m_nu and tau_p bounds are all outstanding.","rationale":"The reader's CONDITIONAL verdict is appropriate, and my reread does not move it. The strongest structural element is Model 3: explicit decay widths are given in Eqs. (8a,b), lattice matrix elements from Ref. [8] are used, and the kinematic suppression by the heavy sterile neutrino is a concrete mechanism that can make the leading proton decay amplitude proportional to a light neutrino mass. Model 1 is a variant of the published extended scotogenic proton-decay construction of Ref. [9], so it is less novel but not suspect. The load-bearing gap is Model 2: the paper's second causal direction, 'proton stability implies small neutrino mass,' depends on a three-loop Weinberg operator that is never evaluated. This is not merely a missing numerical convenience; finiteness, loop-order dominance, and the absence of competing operators are nontrivial in a model that already contains a dimension-5 scalar interaction and two copies of the same leptoquark with a soft mixing term. The paper's own text defers the correlation study to future work, which is an explicit limitation. I therefore agree with the reader's weakest assumption, and the correct disposition is to keep the CONDITIONAL verdict pending a concrete computation and benchmark scan. I do not see an internal inconsistency that would justify REJECT, and the claim is not established enough for ACCEPT.","tokens_in":11509,"tokens_out":13163,"duration_ms":149072,"concrete_test":"Compute the complete three-loop neutrino-mass amplitude for the diagram in Fig. 2b (all topologies with two external L, two external H, and S3/S'3 internal lines) using pyR@M or FeynArts/FormCalc: first check that the integral is finite and free of subdivergences requiring a lower-order Weinberg counterterm; second compare its coefficient with any two-loop diagrams built from two m3^2 insertions and Y'_1, Y'_2 couplings. Then perform a scan over lambda_HSS/M, m_S3, m_3, and the relevant Yukawa couplings to see whether the resulting m_nu lies in the 0.01-0.1 eV range while the two-body and three-body proton decay widths satisfy the Super-K bounds. If no point exists, or if a lower-loop diagram dominates, the claimed correlation fails as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence claim for the 'neutrino mass from proton stability' direction rests entirely on Model 2, but the key quantity is never computed. Section 3.2 states that tiny radiative neutrino masses are generated by the three-loop Weinberg operator in Fig. 2b, yet no three-loop integral, magnitude estimate, or argument for why this diagram is finite and dominant is provided. The same section explicitly says the correlation between neutrino mass and proton decay, including the relative contribution of the two proton decay channels, 'will be studied elsewhere,' deferring the quantitative core. With the field content and couplings of Eq. (5), alternative closed fermion-loop topologies using the same S3-S'3 mixing and Y'_1/Y'_2 couplings could generate Weinberg-type operators at lower loop order; the paper does not rule these out, nor does it show that subdivergences of the three-loop graph can be absorbed without introducing a lower-order neutrino-mass counterterm that would break the claimed limit m3->0, lambda_HSS->0 => m_nu = 0. Consequently, the paper has not demonstrated that a viable parameter point simultaneously gives m_nu ~ 0.05 eV and tau_p > 10^34 yr, so the model-2 realization of the second causal link remains an assertion rather than a demonstrated existence proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes that the smallness of neutrino masses and the longevity of the proton can share a common dynamical origin, and it presents three UV-complete models corresponding to three causal directions: (i) a common origin through an extended scotogenic dark sector; (ii) neutrino mass generated from a proton-decay operator through a three-loop Weinberg operator in a model with two S3 leptoquarks and a global U(1)_F symmetry; and (iii) proton decay suppressed by the small neutrino mass through S1bar and R2 leptoquarks together with a seesaw sterile neutrino. For each model the authors give Lagrangians, field content, and expressions for proton decay widths, and they discuss constraints from charged-lepton flavor violation, leptoquark searches, and dark matter. The advertised link is that when the relevant source coupling or mass vanishes, both the neutrino mass and the proton decay width vanish.","tokens_in":11981,"tokens_out":9898,"duration_ms":108688,"significance":"If all three constructions work, the paper would provide an attractive conceptual bridge between neutrino mass generation and proton stability, with model 1 additionally connecting to dark matter and model 3 offering a relatively compact realization. The paper is not circular: no parameters are fitted to observables, and the Lagrangians and width formulas in Eqs. (1), (5), (7), (2), (6), and (8) are explicit enough to be checked. However, the evidence is uneven. Model 2, which is the only realization of the 'neutrino mass from proton stability' direction, lacks the key three-loop computation; no numerical parameter point is checked against proton lifetime or neutrino-mass bounds for any model; and the suppression of R2 contributions in model 3 rests on an unlisted operator. The conceptual idea is valuable, but the manuscript as it stands demonstrates a proof of concept rather than a complete verification of the central claims.","major_comments":[{"comment":"The model-2 realization of the second causal direction depends entirely on the claim that Fig. 2b generates a three-loop Weinberg operator, but no three-loop integral, loop function, or magnitude estimate is given anywhere in the paper. Section 3.2 instead states that the correlation between neutrino mass and proton decay, including the relative contribution of the two decay channels, 'will be studied elsewhere.' Consequently the central statement that in the limit m3 -> 0 and lambda_HSS -> 0 the neutrino mass vanishes is not demonstrated: finiteness, convergence, dominance over other topologies, and consistency with m_nu ~ 0.05 eV are all open.","section":"Sec. 3.2, Fig. 2b"},{"comment":"No operator-counting argument rules out lower-loop neutrino mass contributions from the same field content. With the Yukawa terms Y'_1 Q L S3 and Y'_2 Q Q S'^dagger_3 and the soft mixing m3^2 S3^dagger S'_3, the paper does not show that the lowest-order LLHH operator is necessarily three-loop, nor does it discuss whether subdivergences of Fig. 2b require a neutrino-mass counterterm that would spoil the advertised proportionality to m3 and lambda_HSS. This is load-bearing because model 2 is the only model for the 'neutrino mass from proton stability' direction.","section":"Sec. 3.2, Eq. (5)"},{"comment":"Even the model-2 proton width is not numerically usable: the matrix element W_0(p -> e+ nu nu) is explicitly stated to be 'to be computed on the lattice,' no value is provided, and no parameter point is checked against the proton lifetime bound or against neutrino-mass observables. Section 4 lists leptoquark mass bounds and CLFV constraints qualitatively but does not apply them to any benchmark point, so the paper does not establish that a viable parameter region exists for any of the three models.","section":"Sec. 3.2, Eq. (6a); Sec. 4"},{"comment":"The model-3 claim that R2 does not contribute to proton decay at leading order rests on an unlisted dimension-10 operator lambda M^{-6} R2 R2 R2 (H^dagger)^7. This operator is not present in Eq. (7b), and no derivation is given for why it is the leading R2 contribution or why its combination with the S1bar couplings cannot generate a competing neutrino-mass-independent proton decay amplitude. Because the dominance of the neutrino-mass-proportional channels is the central assertion of this model, this step needs to be either derived explicitly or replaced by a systematic operator analysis.","section":"Sec. 3.3, footnote [11]"}],"minor_comments":[{"comment":"The mass scale M in the dimension-five term (lambda_HSS/M) H^dagger H^dagger S3 S3 S3 is never defined, and the mass dimension of lambda_HSS is therefore ambiguous; please define M and use it consistently in the width formula.","section":"Eqs. (5b), (6a)"},{"comment":"Please clarify whether the factor involving m_nu and M_N is sqrt(m_nu/M_N) or m_nu/M_N; the text says the decay amplitude is proportional to m_nu, in which case the width should scale as m_nu^2, not as m_nu.","section":"Eq. (8b)"},{"comment":"The DiLog and PolyLog approximations are quoted without derivation or a statement of the ranges of x_d, x_nu, x_S, and y in which they are used; please add a validation or cite the relevant formulas for the kinematic region relevant to proton decay.","section":"Appendix D"},{"comment":"Since the detailed study of the extended scotogenic model is already given in Ref. [9], please state explicitly what is new in this section relative to that paper.","section":"Sec. 3.1"},{"comment":"The manuscript repeatedly refers to tables and figures in the supplementary materials without including them in the main text; please ensure that the main text is self-contained for the key definitions, or clearly mark the supplementary material as appendices.","section":"Throughout"},{"comment":"Equation (2) is very difficult to read because of the notation for flavor indices and mass insertions (Y1a, Y^ab_N, Y^b1, etc.); please rewrite with explicit contractions or add a short explanatory sentence.","section":"Eq. (2)"}],"recommendation":"major_revision","confidential_remarks":"This is a borderline case. The central idea is publishable if the authors supply the missing three-loop computation in model 2 and a concrete parameter point satisfying both proton lifetime and neutrino-mass constraints; because these are additive calculations rather than refutations of the concept, I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful part of this paper is the three-way classification of how neutrino mass and proton decay can be correlated, and the model of Section 3.3. Model 1 is already published work (reference [9], by one of the authors), and the paper says so. Model 2 is a sketch: the three-loop Weinberg operator is drawn but never computed, and the claimed correlation for that direction is not demonstrated. What is genuinely new is the organizing idea plus the explicit construction of Model 3.\n\nSection 3.3 is the most complete part. The field content is explicit, both proton decay amplitudes are written out (tree-level via neutrino mixing and a one-loop box), the decay widths and loop functions are given, and the proton decay matrix elements for p -> e+ pi0 and p -> nu pi+ are taken from lattice calculations. The authors also state clearly in footnote [11] why they think the R2 leptoquark contribution is suppressed; that is honest, even if the suppression argument is brief.\n\nNow the soft spots. In Model 2, the three-loop diagram in Fig. 2b is never evaluated. There is no finiteness argument, no check that lower-loop topologies using the same S3-S'3 mixing and Y' couplings do not dominate, and no parameter point that gives m_nu ~ 0.05 eV with tau_p > 10^34 yr. The text itself says the correlation, including the relative contribution of the two proton decay channels, \"will be studied elsewhere.\" That is not a minor omission; it is the quantitative core of the claimed second connection. In Model 3, the uniqueness claim for the S1bar leptoquark — \"no other leptoquark extension of the SM without BSM symmetries\" — is asserted, not proven, and the p -> e+ nu nu matrix element is deferred to future lattice work. There is also no numerical scan anywhere; CLFV and leptoquark mass constraints are discussed only qualitatively.\n\nThe citation pattern looks fair. The paper builds on Gu-Ma-Sarkar and Helo-Hirsch-Ota for loop-level proton decay, and it correctly identifies Model 1 with reference [9]. I do not see a circularity problem; the correlations are built in by construction.\n\nMy take: this deserves a serious referee, not a desk reject. The taxonomy and Model 3 are worth engaging with, and a referee could push for the missing three-loop calculation or a clear re-labeling of Model 2 as a proposal rather than a demonstrated existence proof. I would not cite it in my own work until that calculation exists, but I would bring it to a reading group to see whether the classification holds up.","headline":"A useful taxonomy and one plausible model, with the central new claim in Model 2 still unverified.","tokens_in":12492,"tokens_out":2335,"would_cite":false,"duration_ms":26432,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.60.-i","14.60.Pq","13.30.-a"],"model":"deepseek-v4-flash","headline":"This paper proposes that the tiny size of neutrino masses and the great stability of the proton are two faces of one mechanism, and it builds three explicit particle models in which each effect implies the other.","keywords":["neutrino mass","proton decay","leptoquarks","scotogenic model","radiative mass generation","baryon number violation","seesaw mechanism","beyond Standard Model"],"falsifier":"Compute the three-loop diagram of Fig. 2b in the second model and check convergence and numerical size; if it diverges or is subdominant, the claimed link between proton decay and neutrino mass fails. For the third model, search for a proton decay channel whose amplitude does not vanish as the light neutrino masses go to zero—for instance a $\\bar S_1$-mediated channel with a light neutrino or an $R_2$-mediated dimension-nine operator—since finding such a channel at observable rate would falsify the correlation.","tokens_in":11263,"feed_emoji":"⚛️","tokens_out":9581,"duration_ms":91742,"temperature":0.7,"pith_summary":"The paper asks whether the two smallest numbers in particle physics—the tiny neutrino masses and the very long proton lifetime—could be explained by the same new physics. It argues that they can: the same virtual particles running in loops can generate both effects, so that each quantity is small because the other is small. Three causal directions are realized in explicit models: a common origin, proton decay as the source of neutrino mass, and neutrino mass as the source of proton decay. If this is right, the search for proton decay and the search for neutrino mass become one search, and the absence of one effect would automatically explain the absence of the other.","feed_headline":"One mechanism may explain both tiny neutrinos and proton stability","feed_subtitle":"In the models here, killing one effect kills the other: massless neutrinos mean a stable proton, and vice versa.","key_machinery":"The working parts are scalar leptoquarks—particles that couple quarks to leptons—and the loop-level effective operators they generate. The central identity is the dimension-five Weinberg operator for Majorana neutrino mass, produced radiatively in each model. In the common-origin model, a $Z_2$-odd dark fermion $N$ and the leptoquark $\\tilde R_{2D}$ sit in both the one-loop neutrino-mass diagram and the one-loop proton-decay box, making one source responsible for both. In the proton-decay-as-source model, two $S_3$ leptoquarks with different charges under a softly broken global $U(1)_F$ ($F=3B-L$) close a three-loop Weinberg diagram, so the neutrino mass operator exists only because the proton-decay operator exists. In the neutrino-mass-as-source model, the leptoquark $\\bar S_1$ couples only to neutrino singlets and the leptoquark $R_2$ has only leptoquark couplings, which together force all leading proton-decay channels to be proportional to a Dirac neutrino mass; a seesaw-scale sterile neutrino makes the otherwise dangerous $\\bar S_1$-mediated channel kinematically forbidden.","core_discovery":"On its own terms, the paper's discovery is a family of correlations rather than a single model: both the suppression of proton decay rates and the smallness of neutrino masses originate from the same mediators circulating inside loops. In the first model, an extended scotogenic sector with a dark fermion $N$, a heavy quark $D$, and a leptoquark $\\tilde R_{2D}$ generates the neutrino mass and the $p\\to e^+\\pi^0$, $p\\to \\nu_\\alpha\\pi^+$ operators at one loop, correlating dark-matter mass, neutrino mass, and proton lifetime. In the second model, two copies of the $S_3$ leptoquark with a softly broken global $U(1)_F$ symmetry convert proton decay into a three-loop Weinberg operator for neutrino mass, so switching off proton decay makes neutrinos massless. In the third model, the leptoquarks $\\bar S_1$ and $R_2$ together with seesaw-scale sterile neutrinos make every leading proton decay amplitude proportional to a light neutrino mass, so massless neutrinos imply a stable proton. The paper gives the decay widths for these channels using lattice matrix elements and discusses constraints from lepton flavor violation, leptoquark searches, and dark matter.","pith_inferences":["The paper leaves the three-loop integral of the second model unevaluated; computing it and checking convergence and magnitude would settle whether the neutrino mass is truly dominated by the proton-decay operator.","A natural extension is a global fit that scans each model's parameter space against proton-lifetime bounds, neutrino oscillation data, and lepton-flavor-violation limits at once; the paper gives analytic widths but no such scan.","If the logic is embedded in a grand unified theory (GUT), proton decay and neutrino mass would inherit one common origin from unification, giving a new class of GUT models distinct from the usual $SU(5)$, $SO(10)$, or $E_6$ frameworks.","The three models have distinguishable phenomenology: common-origin ties dark matter to decay, decay-as-source predicts neutrinos only via baryon number violation, and mass-as-source predicts missing-energy channels; a signature-level comparison could tell them apart."],"forward_implications":["In the common-origin model, the dark matter mass, the neutrino masses, and the proton decay width are set by the same couplings, so measuring any two of these observables determines the third.","In the second model, a stable proton implies exactly massless neutrinos, so the observed nonzero neutrino mass would force proton decay to occur at some level.","In the third model, proton decay into neutrino final states must carry missing energy, while charged-antilepton channels are loop-suppressed; this gives a signature that upcoming proton decay searches can look for.","In all three models the new particles are leptoquarks and new fermions, so collider searches for leptoquarks and lepton-flavor-violating processes provide indirect tests of the correlation."],"supporting_citations":[{"why":"Supplies the scotogenic one-loop neutrino mass and dark matter candidate that the common-origin model extends.","marker":"[5]"},{"why":"Establishes the precedents of radiative one-loop proton decay and radiative neutrino mass tied to dark matter.","marker":"[6, 7]"},{"why":"Provides the lattice proton-decay matrix elements $W_0$ and momentum transfer $q^2$ used in the decay-width formulas.","marker":"[8]"},{"why":"Contains the detailed analysis of the extended scotogenic model that realizes the common-origin connection.","marker":"[9]"},{"why":"Classifies leptoquark couplings and supports the claim that $R_2$ has no leading-order diquark couplings and hence no tree-level proton decay.","marker":"[10]"},{"why":"Sets the current experimental bounds on proton lifetime and leptoquark masses used in the constraints section.","marker":"[3]"}],"fun_headline_variants":["Same loop yields tiny neutrino mass and proton stability","Neutrino mass and proton decay share a single origin","Massless neutrinos imply a stable proton","Proton longevity and neutrino lightness from same source","One mediator explains both proton stability and neutrino mass"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the displayed loop diagrams being the complete dominant contributions: no uncalculated operator may generate neutrino mass or proton decay independently, the three-loop integral in the second model must converge and dominate, and no other baryon-number-violating operator in the third model can be lower-order than the neutrino-mass-proportional ones.","fun_headline_variants_meta":{"raw":{"variants":["Same loop yields tiny neutrino mass and proton stability","Neutrino mass and proton decay share a single origin","Massless neutrinos imply a stable proton","Proton longevity and neutrino lightness from same source","One mediator explains both proton stability and neutrino mass"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000669,"raw_usage":{"total_tokens":3016,"prompt_tokens":878,"completion_tokens":2138,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":2066}},"tokens_in":494,"tokens_out":2138,"duration_ms":16222,"temperature":1.0,"reasoning_tokens":2066,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:13:16.716100+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the three-loop diagram of Fig. 2b in the second model and check convergence and numerical size; if it diverges or is subdominant, the claimed link between proton decay and neutrino mass fails. For the third model, search for a proton decay channel whose amplitude does not vanish as the light neutrino masses go to zero—for instance a $\\bar S_1$-mediated channel with a light neutrino or an $R_2$-mediated dimension-nine operator—since finding such a channel at observable rate would falsify the correlation.","supporting_citations":[{"cited_title":"Bridging the gap between quark and lepton sectors naturally calls for leptoquarks, which appear in all three models outlined in the previous sections","cited_arxiv_id":null,"evidence_quote":"Supplies the scotogenic one-loop neutrino mass and dark matter candidate that the common-origin model extends."},{"cited_title":"Georgi, Helen R","cited_arxiv_id":null,"evidence_quote":"Provides the lattice proton-decay matrix elements $W_0$ and momentum transfer $q^2$ used in the decay-width formulas."},{"cited_title":"Proton decay and grand unification","cited_arxiv_id":null,"evidence_quote":"Classifies leptoquark couplings and supports the claim that $R_2$ has no leading-order diquark couplings and hence no tree-level proton decay."},{"cited_title":"1 of the supplementary materials","cited_arxiv_id":null,"evidence_quote":"Sets the current experimental bounds on proton lifetime and leptoquark masses used in the constraints section."}],"review_version":1}