{"id":"eb9eab4b-8b6a-4eec-83d8-a3009bc067e3","arxiv_id":"2603.04394","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For light U(1)X vectors coupled to anomaly-sensitive currents, Wess-Zumino coefficients are fixed by anomaly matching, which lets all missing-energy searches (K, B, D, Z) be mapped onto one (m_X,g_X) plane with minimal UV spectra classified.","lead":"The paper works out how to search for a hypothetical light new particle (an X boson) whose interactions are fixed by quantum anomaly cancellation, rather than chosen by hand. It matters because it converts recent hints at Belle II and NA62 into concrete, testable predictions for existing and future experiments.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4 charge tables are internally inconsistent with the advertised Yukawa terms, so the explicit UV completions do not exist as written; this blocks the UV-side claims until fixed.","rationale":"The reader's stated weakest_assumption is the requirement that anomalon masses arise predominantly from the U(1)_X-breaking VEV. I do not find that to be the most limiting issue: anomaly matching fixes the WZ coefficients independently of whether the mass matrix is dominated by S, provided the heavy fermions decouple. The more concrete and demonstrable problem is that the explicit charge tables in Section 4 are internally inconsistent with the written Yukawa Lagrangians. This is not a matter of convention: for the 2DL model, no choice of the free parameter X makes Eq. (4.9) invariant. It invalidates the advertised UV completions as submitted. The error looks like a sign/notation typo — likely A_R and B_R should carry −X and the S/S* entries should be swapped — so the framework is probably repairable. I therefore agree with the CONDITIONAL verdict, but identify the table inconsistency, rather than the mass-origin assumption, as the load-bearing concern. The central IR claim itself appears sound and should not be rejected without evidence that the anomaly-matching logic fails.","tokens_in":27649,"tokens_out":62727,"duration_ms":567486,"concrete_test":"Recompute the U(1)_X charge of every term in Eqs. (4.4), (4.6), (4.9), and (4.12) using the charge tables in Tables 5 and 6. Identify all terms with nonzero charge and correct the table entries or the S/S* insertions so that all Yukawa terms are gauge invariant. Then verify that the corrected charges still satisfy the anomaly-cancellation conditions (2.2)–(2.6) and the mass-generation criteria (i) of Sec. 2.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central IR claim — WZ coefficients fixed by mixed-anomaly matching — is well supported by Refs. [7,8,11–13] and by anomaly-matching logic. The concrete soft spot is the explicit UV model section. In the 2DL model, Eq. (4.9) with Table 6 requires the four Yukawa entries to be U(1)_X invariant. The printed charges give: y_AA S term has charge (X+1/2)+(X-1/2)+1 = 2X+1; y_AB H† term has charge 2X+1; y_BA H term has charge 2X−1; y_BB S* term has charge 2X−1. No value of X makes both 2X+1=0 and 2X−1=0, so the Lagrangian as written is not gauge-invariant for any X. The correct transcription from Eq. (A.11) appears to require A_R and B_R charges with −X rather than +X, and S/S* in the mass matrix interchanged, but as printed the model is invalid. Similarly, in Table 5 the y_AA A_L S A_R and y_BB B_L S* (B_L)^c terms have nonzero U(1)_X charge using the tabulated entries. Since Section 4 claims 'fully viable benchmark' UV completions and the abstract/conclusion emphasize the minimal anomalon classification, these inconsistencies are load-bearing for the paper's UV-side claims. The IR framework and phenomenological mapping likely survive once the tables are corrected, but the submitted explicit models do not exist as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies light U(1)_X gauge bosons coupled to electroweak-anomalous SM currents. It argues that, when the new fermions ('anomalons') required for anomaly cancellation get their mass predominantly from a SM-singlet VEV, integrating them out yields Wess–Zumino operators whose coefficients are fixed by mixed-anomaly matching. The paper classifies minimal anomalon spectra (N_Ψ ≤ 4), derives the resulting missing-energy phenomenology for K→πE_miss, B→K^(*)E_miss, D→πE_miss, and Z→γE_miss, and presents explicit UV completions as a benchmark for the Belle II B^+→K^+E_miss excess. The IR framework is standard and the phenomenological survey is broad, but the explicit UV sections contain gauge-invariance inconsistencies in the printed charge tables and Yukawa terms.","tokens_in":28152,"tokens_out":31704,"duration_ms":287372,"significance":"If correct, the paper would provide a valuable model-independent mapping from mixed-anomaly data to rates for rare missing-energy processes, together with a useful classification of minimal anomalon sectors. The central anomaly-matching logic is transparent, the classification in Appendix A is checkable, and the paper makes falsifiable predictions (e.g., B(B→K^*E_miss) at the Belle II best-fit point, Z→γE_miss at FCC-ee). These are real strengths. However, the explicit UV models advertised as 'fully viable benchmarks' in Section 4 are not gauge-invariant as written; the UV-side claims therefore need repair before the paper can be accepted. The IR and phenomenological parts appear sound and should survive the correction.","major_comments":[{"comment":"The Yukawa term y_BB B_L S^* (B_L)^c is not gauge invariant with the charges of Table 5. With X(B_L)=-1/12 and X(S)=1/6, the invariant Majorana combination is B_L S B_L (charge 2·(-1/12)+1/6=0), whereas the printed expression carries charge -1/6. The hypercharge and SU(2) charges are compatible with a chiral ML state, but the displayed operator is not. This invalidates the 1DL–1ML model as written and hence the 'fully viable benchmark' conclusion at the end of Sec. 4.1.","section":"Sec. 4.1, Eq. (4.6) and Table 5"},{"comment":"With the charges of Table 6, the four Yukawa entries in Eq. (4.9) have U(1)_X charges: y_AA S term: 2X+1; y_AB H^† term: 2X+1; y_BA H term: 2X−1; y_BB S^* term: 2X−1. No value of X makes both 2X+1=0 and 2X−1=0, so the Lagrangian is not gauge invariant for any X. The correct transcription from Eq. (A.11) appears to require A_R and B_R charges with −X rather than +X, and S/S^* interchanged in the mass matrix. As printed, the explicit 2DL UV completion does not exist.","section":"Sec. 4.2, Eq. (4.9) and Table 6"}],"minor_comments":[{"comment":"Table 3 is captioned 'one ML and one DL pair', but the text and the table itself describe two DL pairs for N_Ψ=4. Please correct the caption/header.","section":"Sec. 2.3 and Table 3"},{"comment":"The relation A^SM_XYY = -A^SM_XWW = 3 α_{B+L} is easy to misread against Eqs. (2.4)–(2.5), which use a different normalization (factor 1/2 from the anticommutator for the nonabelian trace). Please state explicitly which normalization of the anomaly trace is being used, to avoid a factor-of-2 confusion in the numerical results.","section":"Sec. 3.1, Eq. (3.1)"},{"comment":"The left panel of Fig. 2 refers to 'L-only' and 'L=−4R' models of Ref. [23] without defining them here. A one-sentence definition or reference to the original notation would improve readability.","section":"Sec. 3.2.5 and Fig. 2"},{"comment":"Typos: 'responsable' should be 'responsible'; 'respectevely' should be 'respectively'.","section":"Sec. 4.1"}],"recommendation":"major_revision","confidential_remarks":"The inconsistencies in Section 4 appear to be transcription/sign errors from Appendix A rather than conceptual flaws. The central IR anomaly-matching framework is sound and the phenomenological survey is useful. I recommend asking the authors to correct the charge tables and Yukawa operators, and to re-check the direct-search claims that rely on the explicit models. This should be fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the WZ/infrared part is solid and genuinely useful; the UV model section is not internally consistent as printed, and the Belle II significance is a best-fit rather than an independent prediction.\n\nThe genuinely new thing is the classification of minimal anomalon spectra (NΨ≤4) for vector U(1)X currents, together with the systematic mapping of missing-energy searches—K→π, D→π, B→π/ρ/K, and Z→γ—onto a single (m_X, g_X) plane via anomaly-matched WZ coefficients. The derivation in Section 3 is transparent: the WZ coefficients are fixed by mixed-anomaly matching, and the flavor pattern is a definite, minimal-flavor-violation-like prediction. The finite-naturalness filter and the inclusion of direct anomalon constraints give a concrete IR–UV interplay, and Figure 3 will be useful for interpreting Belle II and NA62 data. The paper is also honest about its main assumptions, e.g., that anomalon masses come predominantly from the U(1)X-breaking VEV; that assumption is stated clearly.\n\nThe load-bearing soft spot is Section 4. I checked the 2DL model: in Eq. (4.9) with Table 6, the y_AA and y_AB terms carry U(1)X charge 2X+1, while the y_BA and y_BB terms carry 2X−1. There is no value of X that makes both charges vanish, so the Lagrangian is not gauge-invariant as written. Table 5 in the 1DL–1ML model has the same kind of problem: the displayed y_AA and y_BB terms do not have zero U(1)X charge. The text claims these are “fully viable benchmark” UV completions, so this is not cosmetic; it blocks the UV-side claims until the charge assignments and Yukawa terms are reconciled with Appendix A (likely a sign-flip or transposition of S/S*). The IR framework and the phenomenological map should survive a repair, but the paper as submitted has no valid explicit UV model.\n\nA smaller issue: the 4.3σ preference for the τ-flavor benchmark comes from fitting g_X to the Belle II B+→K+Emiss data; it is not an independent prediction. The paper is reasonably clear about this, but it should separate fitted from predicted quantities more sharply, since some readers will otherwise overstate the significance.\n\nOn the strength of the evidence: the algebra in Appendix A is checkable, the constraint list is comprehensive, and the paper is careful about limitations. The central argument holds up; the UV tables need repair, not reinvention. I would send this to a serious referee—a corrected version would be a strong contribution to the light-dark-sector literature.","headline":"The IR/WZ framework and missing-energy map are solid and worth publishing; the printed UV models in Section 4 have gauge-invariance inconsistencies that must be fixed before the UV claims stand.","tokens_in":28646,"tokens_out":4372,"would_cite":true,"duration_ms":51494,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that missing-energy predictions for a light vector coupled to an anomalous Standard-Model current are fixed by anomaly matching once the vector's mass and coupling are chosen.","keywords":["light vectors","U(1)_X gauge bosons","anomaly cancellation","Wess-Zumino terms","missing-energy searches","rare meson decays","anomalons","finite naturalness"],"falsifier":"Measure two missing-energy channels whose ratio the framework fixes, for example B→K E_miss versus B→K*E_miss or K→πE_miss versus Z→γE_miss, with enough precision to extract the two WZ coefficients independently; if the inferred coefficients contradict the mixed-anomaly relation for a known charge assignment, the singlet-VEV-dominated mass assumption is falsified.","tokens_in":27529,"feed_emoji":"⚛️","tokens_out":7898,"duration_ms":73011,"temperature":0.7,"pith_summary":"The paper targets light U(1) gauge bosons whose coupling to Standard-Model fermions is anomalous, requiring new charged fermions (anomalons) for consistency. It argues that if anomalon masses come from the same spontaneous symmetry breaking that gives the vector its mass, then integrating them out leaves Wess-Zumino interactions whose coefficients are set by mixed-anomaly cancellation. These coefficients then determine every missing-energy signal — rare kaon, D, and B decays plus radiative Z decays — with no additional free parameters. The consequence is that a single measurement of one invisible-decay channel predicts all the others, making current and future searches for missing energy a direct probe of the underlying anomaly structure. A reader should care because it turns a zoo of possible models into a parameter-free testable pattern.","feed_headline":"Anomaly matching fixes every missing-energy signal of a light vector","feed_subtitle":"If a new vector boson couples to anomalous weak currents, its invisible K, D, B, and Z decay rates are predicted with no extra parameters.","key_machinery":"The load-bearing object is the set of Wess-Zumino (WZ) operators, of schematic form X(W ∂W + W W W) and X B ∂B. Their coefficients are the SM mixed-anomaly traces, with A^SM_XY Y = −A^SM_XWW = 3α_{B+L}, where α_{B+L} is the combination of baryon and lepton-number charges defining the current. These dimension-4 interactions are generated at one loop when the anomalons are integrated out; they reproduce the anomalous variation of the SM current and, through the equivalence theorem for the longitudinal mode (X_μ → ∂_μ ξ/m_X), generate the axion-like couplings that drive rare meson decays. The classification of minimal anomalon spectra (one Majorana-like multiplet plus one Dirac pair, or two Dir","core_discovery":"The central claim is that below the U(1)_X-breaking scale the nontrivial physics of a light vector X coupled to an electroweak-anomalous Standard-Model current is captured by two Wess-Zumino operators, X W ∂W and X B ∂B. Their coefficients are not free: matching the cancellation of the [SU(2)^2 U(1)_X] and [U(1)_Y^2 U(1)_X] anomalies fixes them to ∓3α_{B+L} times gauge couplings. From these two operators the paper derives a predictive flavor-changing structure for s→d, c→u, b→d, and b→s transitions, via the longitudinal-mode equivalence theorem, and a tree-level Z→γX amplitude. Thus K→πE_miss, D→πE_miss, B→π/ρ/K^(*)E_miss, and Z→γE_miss are predicted once m_X and g_X are chosen. The paper al","pith_inferences":["If the central assumption holds, the same anomaly-matching logic could be applied to any future anomalous light vector, giving a general dictionary between gauge charges and missing-energy rates.","A natural next test is to use the predicted relation between Z→γE_miss and meson decays to check the framework against future measurements; a deviation would point toward explicit anomalon mass terms or additional light states rather than a failure of the missing-energy classification itself.","The finite-naturalness bound suggests that if no anomalons appear below a few TeV, the viable parameter space shrinks to tuned or non-minimal UV completions; this could be sharpened by extending the classification beyond N_Ψ=4 or to colored anomalons."],"forward_implications":["Once m_X and g_X are chosen, the framework predicts the rates of K→πE_miss, D→πE_miss, B→π/ρ/K^(*)E_miss, and Z→γE_miss with no further parameters.","The WZ mechanism imposes a minimal-flavor-violation pattern, so the ratio of B→K*E_miss to B→K E_miss is fixed; future data can discriminate this class from scalar or axion explanations.","In the gauged τ-flavor benchmark with m_X below the τ-pair threshold, X decays invisibly, and the framework can accommodate the recent B→K missing-energy excess while predicting an observable B→K* missing-energy rate.","Z→γE_miss is the most powerful single probe; a future high-luminosity Z factory can improve sensitivity by two to three orders of magnitude and test the framework down to per-mille fine-tuning.","Finite naturalness plus current collider searches restrict minimal anomalon spectra to masses around a few hundred GeV to a few TeV, making direct anomalon searches a complementary test."],"fun_headline_variants":["Anomaly matching dictates invisible decays of light vectors","Invisible signals of anomalous light vectors are fully predictable","Predicting missing-energy decays from anomaly cancellation","Anomaly cancellation predicts all missing-energy signals"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The framework hinges on the new fermions getting their mass from the same scalar that breaks the new U(1), not from the Standard-Model Higgs or from explicit mass terms; if that fails, the anomaly-matching prediction collapses.","fun_headline_variants_meta":{"raw":{"variants":["Anomaly matching dictates invisible decays of light vectors","Invisible signals of anomalous light vectors are fully predictable","Predicting missing-energy decays from anomaly cancellation","Anomaly cancellation predicts all missing-energy signals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000488,"raw_usage":{"total_tokens":2270,"prompt_tokens":804,"completion_tokens":1466,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":1415}},"tokens_in":548,"tokens_out":1466,"duration_ms":11465,"temperature":1.0,"reasoning_tokens":1415,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T18:51:25.868013+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure two missing-energy channels whose ratio the framework fixes, for example B→K E_miss versus B→K*E_miss or K→πE_miss versus Z→γE_miss, with enough precision to extract the two WZ coefficients independently; if the inferred coefficients contradict the mixed-anomaly relation for a known charge assignment, the singlet-VEV-dominated mass assumption is falsified.","supporting_citations":[],"review_version":1}