{"id":"1b7dbc96-94f8-4391-94f3-8c4df2ba33ec","arxiv_id":"2501.04132","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Anomalous Z'ZZ, Z'Zγ, and Z'γγ couplings from a muonic U(1)' model are beyond HL-LHC reach but potentially discoverable at CLIC and a 100 TeV pp collider.","lead":"This paper simulates whether future colliders could detect anomalous triple-gauge-boson couplings that arise in a muon-specific extension of the Standard Model. It finds the HL-LHC cannot, while CLIC and a 100 TeV proton collider could probe certain Z' mass ranges.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The discovery reaches hinge on the unconstrained WZW counterterm coefficient: the covariant scheme fixes the anomalous vertices only by convention, so the quoted signal rates and mass windows are not robust predictions of the EFT.","rationale":"The paper's stated goal is to show that anomalous triple-gauge couplings from a muonic U(1)' EFT can be probed at future colliders. The most fundamental ingredient is the magnitude of the anomalous vertex itself. The covariant regularization scheme, together with the Wess-Zumino consistency condition, fixes the SM Ward identities but does not determine the low-energy WZW counterterm coefficient. Appendix A demonstrates that the Ward identities depend on the arbitrary momentum shift (w,z), and that different choices are related by WZW counterterms. The paper selects the covariant scheme (w=-z=1), which sets these counterterms to zero, but this is a convention rather than a prediction from the specified EFT, whose UV completion is not given. Since the signal cross sections scale with the square of combinations of A~1 and A~2, the entire reach analysis is contingent on this choice. A different yet fully consistent regularization would yield different cross sections, and the claimed discovery windows could shrink, move, or disappear. This concern is more load-bearing than the reducible-background and interference issues, because those affect only the experimental analysis, whereas the WZW ambiguity affects the predicted signal itself. The paper is carefully executed and the implementation is nontrivial, including numerical evaluation of the loop integrals and a cross-check with LoopTools, but this does not resolve the scheme ambiguity. The reader's weakest assumption already flagged the covariant-scheme dependence, so the verdict remains CONDITIONAL; no change in verdict is needed, but the WZW dependence should be elevated as the primary condition to verify.","tokens_in":22879,"tokens_out":9949,"duration_ms":103588,"concrete_test":"Recompute the signal cross sections and significances using the consistent-anomaly scheme (e.g., momentum shift w=0, z=0) implemented with the WZW counterterm vertices required to restore the SM Ward identities. Compare S/sqrt(B) after the analysis cuts of Tables 3, 6, and 7 for m_Z' = 200 and 300 GeV in e+e- -> Z'Z -> mu mu jj and e+e- -> Z'gamma -> mu mu gamma at CLIC, and pp -> Z'gamma -> mu mu gamma at the 100 TeV collider. If any of the discovery/exclusion ranges in Figs. 3, 10, or 11 shift by more than ~20%, the central reach claim is scheme-dependent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends entirely on the size of the loop-induced anomalous Z'VV~ couplings. As the paper itself states, there is a one-to-one correspondence between the WZW counterterm coefficients and a momentum shift in the triangle loop (Appendix A, Eqs. (16)-(20)). The covariant scheme (w=-z=1) is chosen so that the SM Ward identities hold without WZW terms, but any other shift with the appropriate WZW counterterms is an equally valid EFT description of the same low-energy content. The physical vertices in Eqs. (21)-(37) are not fixed by anomaly cancellation alone; they depend on this choice. Since the signal cross sections in all three collider studies (e.g., Eqs. (14)-(15) for e+e- -> Z'_L Z, and the pp -> Z'gamma production at 100 TeV) are proportional to combinations of A~1 and A~2, a different but equally consistent regularization/WZW choice can change the predicted cross sections, and hence the required luminosities and the claimed reach ranges (e.g., CLIC discovery [125,225] GeV, 100 TeV discovery [230,330] GeV). The paper acknowledges the freedom but treats the covariant choice as definitive without specifying the UV completion that would select it. This is a load-bearing premise: if the actual WZW coefficient is smaller, the claimed discoveries may not occur.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the low-energy EFT of a muonic U(1)'_mu gauge symmetry with second-generation leptons axially charged, focusing on loop-induced anomalous triple-gauge couplings Z'VV~ where V,V~ are SM electroweak bosons. The vertices are derived in Appendix A in the covariant regularization scheme, implemented in a MadGraph UFO model with form factors checked against LoopTools, and used to estimate discovery reaches at the HL-LHC, a 100 TeV pp collider, a resonant muon collider, and CLIC at sqrt(s)=3 TeV. With g_mu saturating the neutrino trident bound, the paper finds that the HL-LHC cannot probe these couplings, while the 100 TeV collider could reach evidence for m_Z' in [150,800] GeV and discovery in [230,330] GeV at 20/ab, and CLIC could reach 5-sigma discovery for m_Z' in [125,225] GeV at 5/ab. The paper explicitly treats the WZW counterterm freedom by choosing the covariant scheme and limits all analyses to irreducible backgrounds.","tokens_in":23194,"tokens_out":6852,"duration_ms":73385,"significance":"If the central predictions are robust, the paper provides a concrete and falsifiable target: loop-induced anomalous triple-gauge couplings could be observed at a 100 TeV hadron collider or at CLIC even though they are out of reach of the HL-LHC. The strengths of the paper are that the vertex derivation in Appendix A is internally consistent, the numerical amplitudes are checked against LoopTools, the signal predictions are forward predictions with no parameter fitted to the target signals, and the authors explicitly evaluate SM interference in the HL-LHC Z'gamma channel. The main weakness is that the size of the anomalous vertices depends on the WZW counterterm convention, and the projections rely on optimistic background assumptions; these limitations are acknowledged in the text but are load-bearing for the quoted mass windows and should be addressed.","major_comments":[{"comment":"The manuscript correctly states that the WZW coefficients are in one-to-one correspondence with a momentum shift in the triangle loop (Appendix A, Eqs. (16)-(20)), and then fixes the covariant scheme w=-z=1 to set the WZW counterterms to zero. This choice preserves the SM Ward identities, but it does not by itself fix the physical anomalous vertex: a different momentum shift, together with the corresponding WZW counterterms, is an equally valid EFT description of the same low-energy content and can lead to different form factors tilde-A1 and tilde-A2. Since the cross sections used for the reach estimates, e.g. Eqs. (11)-(15) and the pp -> Z'gamma signal in Section 3.2, are proportional to (tilde-A1 - tilde-A2)^2, the quoted mass windows (CLIC discovery [125,225] GeV, 100 TeV discovery [230,330] GeV) are not robust predictions of the EFT unless the WZW coefficient is fixed by a concrete UV completion or by an additional physical criterion. The paper acknowledges the freedom but treats the covariant choice as definitive; this is a load-bearing assumption and should be either justified from a spectator sector or folded into the projections as an explicit parameter.","section":"Appendix A, Eqs. (16)-(20); Section 2, Eq. (9)"},{"comment":"The 100 TeV pp -> Z'gamma -> mu+mu-gamma analysis omits the signal-background interference, although the same final state at the HL-LHC was found to have destructive interference of similar magnitude to the signal (Section 3.1.1) and the Z'Z channel showed order-10% interference (Section 3.1.2). The kinematics at the higher pT thresholds used for the 100 TeV scan (pTgamma ~ 300-1700 GeV) differ from the HL-LHC analysis, but without a dedicated evaluation one cannot exclude a sizable reduction of the signal; the evidence and discovery ranges quoted in Section 3.2 are therefore optimistic. Please compute the interference for the 100 TeV benchmarks or provide a quantitative argument for its smallness.","section":"Section 3.2; Section 3.1.1"},{"comment":"All significances are computed as S/sqrt(B) with only irreducible backgrounds and no systematic uncertainty. The paper explicitly describes this as a first approximation, but it remains load-bearing for the central claims: in the 100 TeV channel the post-cut S/B is about 0.68 (Table 3), so a 10-20% background normalization uncertainty would shift the discovery significance substantially; in the CLIC channels S/B is larger, but the final backgrounds are O(1-10) events after cuts (Tables 6-7), so the Poisson-limited significance also requires care. Please add a scan over a background normalization systematic, or at least state the systematic level at which each claimed discovery window closes.","section":"Sections 2, 3.2, 4.3; Tables 3, 6, 7"}],"minor_comments":[{"comment":"The word 'complimentary' should be 'complementary' in the abstract and in the conclusion.","section":"Abstract and Section 5"},{"comment":"The text above Eq. (16) contains a duplicated article ('the the resulting Ward identities'), and Eq. (28) has 'I3(p, q.mf)' with a period instead of a comma.","section":"Appendix A, text above Eq. (16) and Eq. (28)"},{"comment":"Reference [6] is incomplete: it gives the collaboration and title but lacks a journal reference, arXiv identifier, or DOI.","section":"References, [6]"},{"comment":"The relative acceptances in Table 4 are presented without stating explicitly that they are computed with respect to the initial event numbers; please clarify the normalization.","section":"Table 4 caption"}],"recommendation":"major_revision","confidential_remarks":"The WZW convention issue is the key point: the central reach claims depend on a choice that is not fixed by anomaly cancellation alone. I would encourage the editor to send the paper back for a revision in which the authors either justify the covariant choice from a concrete UV completion or present the reach as a function of the WZW coefficient, and also address the omitted 100 TeV interference and background systematics. The paper fits the journal's scope and the technical implementation is a clear strength."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a careful, honest sensitivity study for loop-induced anomalous Z'VV~ couplings in a muonic U(1)' model. The new piece is the collider comparison: the HL-LHC is dead (they even find destructive interference that makes it worse), while a 100 TeV pp collider at 20/ab can get evidence for mZ' in [150,800] GeV and discovery in [230,330] GeV, and CLIC at 3 TeV with 5/ab can reach 5 sigma for mZ' in [125,225] GeV. The muon collider at resonance is complementary, with exclusion/evidence in the 280-850 GeV range. What the paper does well: the vertex derivation in Appendix A is consistent and numerically checked against LoopTools; the model implementation in MadGraph is non-trivial (form factors computed on the fly); and the paper is transparent about using only irreducible backgrounds and about the regularization freedom. The HL-LHC null result is robust because the signal is tiny and the interference is negative. The main soft spot is the one the paper itself exposes: the anomalous Z'VV~ vertices are not uniquely fixed by the anomaly. There is a one-to-one correspondence between the WZW counterterm coefficients and a momentum shift in the triangle loop, and any choice is an equally valid EFT description. The covariant scheme (w=-z=1) is chosen so that SM Ward identities hold without WZW terms, but nothing in the EFT selects that scheme; it corresponds to a particular UV completion (or family of completions). The signal cross sections, and hence all the quoted mass windows and luminosities, are proportional to A~1 and A~2, which depend on this choice. A different but equally consistent WZW coefficient can change the reach, possibly erase the CLIC discovery window. The paper acknowledges the freedom but treats the covariant choice as definitive. This is not fatal if the paper is read as a benchmark for that scheme, but it is load-bearing, and the abstract oversells it as probing anomalous triple-gauge couplings in general. I would like to see either a specific UV completion that fixes the WZW coefficient, or a scan over the allowed WZW coefficients showing how the reach changes. Minor: the 100 TeV study does not check SM interference (the 14 TeV check showed it can be of order the signal), and the lepton collider sections assume only irreducible backgrounds without systematics. These are stated limitations, not hidden ones. Who it is for: people working on anomalous U(1)' models, muon-philic Z' searches, and future collider physics. It deserves a serious referee. I would send it to review and ask for the WZW-sensitivity analysis before publication.","headline":"Solid collider sensitivity study for a muonic U(1)' model with a robust HL-LHC null result, but the quoted future-collider reaches depend on an unconstrained WZW counterterm choice and should be framed as scheme-dependent benchmarks.","tokens_in":23699,"tokens_out":3113,"would_cite":true,"duration_ms":32366,"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":"A gauge-anomalous Z' with muonic couplings could show up at the next colliders as loop-induced diboson signals.","keywords":["anomalous triple-gauge couplings","muonic U(1)'","Z' boson","Wess-Zumino-Witten term","effective field theory","collider phenomenology","neutrino trident","future colliders"],"falsifier":"Reconstruct the $\\mu^+\\mu^-jj$ or $\\mu^+\\mu^-\\gamma$ final state at CLIC with $5~\\mathrm{ab}^{-1}$; for $m_{Z'}=200$ GeV the covariant-scheme model predicts about five signal events after cuts, so a measured yield consistent with background alone would rule out the predicted anomalous $Z'Z$ vertex.","tokens_in":22722,"feed_emoji":"⚛️","tokens_out":7560,"duration_ms":72055,"temperature":0.7,"pith_summary":"The paper tries to show that the loop-induced triple-gauge couplings of a massive $Z'$ boson, forced on the theory by mixed gauge anomalies in a muonic $U(1)'_\\mu$ extension of the Standard Model, can be observed at proposed colliders even though they are invisible at the HL-LHC. Because an anomalous gauge theory is only an effective theory, the anomalous vertices are non-decoupling remnants that carry information about the UV completion. The paper computes these $Z'ZZ$, $Z'Z\\gamma$ and $Z'\\gamma\\gamma$ vertices in the covariant regularization scheme, implements them in event generation, and evaluates signal versus irreducible Standard Model background at four facilities. It finds that a 100 TeV $pp$ collider can gather evidence for $m_{Z'}\\in[150,800]$ GeV and discovery in $[230,330]$ GeV, while CLIC at $\\sqrt{s}=3$ TeV can reach $5\\sigma$ discovery for $m_{Z'}\\in[125,225]$ GeV; a resonant muon collider covers complementary masses. If true, a class of anomalous effective theories usually dismissed as sick would leave observable traces at the next generation of colliders.","feed_headline":"HL-LHC would miss them; a 100 TeV pp collider and CLIC would not","feed_subtitle":"A muonic-U(1)' model predicts anomalous diboson signals that HL-LHC cannot resolve.","key_machinery":"The carrying object is the anomalous triple-gauge vertex $Z'VV$, written in the Rosenberg parametrization with six Lorentz form factors. The convergent form factors $A_3,\\ldots,A_6$ come from one-loop triangle integrals over the muon and the muon neutrino, while the divergent coefficients $\\tilde{A}_1$ and $\\tilde{A}_2$ are fixed by imposing the Standard Model Ward identities, which is the covariant anomaly prescription and corresponds to setting the WZW counterterms to zero. The longitudinal component of the $Z'$ propagator is what feels the anomaly and produces the energy-growing, eventually constant signal cross sections; the model is implemented by exporting the tree-level $Z'$ couplings and then inserting the anomalous vertices with their momentum-dependent form factors so that event generation evaluates them on the fly.","core_discovery":"On the paper's own terms, the claim is that the mixed gauge anomaly of a $U(1)'_\\mu$ under which only second-generation leptons are axially charged generates calculable, loop-suppressed triple-gauge couplings $Z'ZZ$, $Z'Z\\gamma$ and $Z'\\gamma\\gamma$, and that choosing the covariant regularization scheme, together with the Wess-Zumino consistency conditions, fixes the WZW counterterms so that all Standard Model Ward identities hold and the anomaly is concentrated in the $Z'$ vertex. The resulting vertices grow with energy through the longitudinal polarization of the $Z'$, and when the $Z'$--muon coupling is set to the largest value allowed by neutrino trident data, the cross sections become observable: the HL-LHC cannot see them, the 100 TeV $pp$ collider at $20~\\mathrm{ab}^{-1}$ can reach evidence between 150 and 800 GeV and discovery between about 230 and 330 GeV, a muon collider tuned to the $Z'$ resonance can exclude $m_{Z'}\\in[280,850]$ GeV and reach evidence in $[380,700]$ GeV, and CLIC at $\\sqrt{s}=3$ TeV with $5~\\mathrm{ab}^{-1}$ can discover the coupling for $m_{Z'}\\in[125,225]$ GeV while providing evidence up to about 400 GeV.","pith_inferences":["The paper fixes $g_\\mu$ to the neutrino-trident bound for every mass; a natural extension is to recast each luminosity curve as an exclusion or discovery bound in the $(m_{Z'},g_\\mu)$ plane, showing how much coupling below the bound can still be probed.","The same covariant-regularized vertex construction applies to electron- or tau-philic $U(1)'$ models, but the collider story changes: at $e^+e^-$ machines the initial-state electrons would be neutral under such a $Z'$, removing the tree-level $t$-channel contamination that dominates muon-collider production.","Because the $e^+e^-$ signal cross section approaches a constant at high energy, a future higher-energy collider could measure the energy dependence of $Z'V$ production and distinguish the anomalous growth from the $1/s$ falloff expected in a unitary, UV-complete theory.","These searches are also probes of the WZW-scheme ambiguity: if nature realized the consistent-anomaly scheme rather than the covariant one, the reach windows and the high-energy growth pattern would differ, so a measured cross-section shape could identify the regularization."],"forward_implications":["A null result at the HL-LHC is expected and does not disfavor the model; the loop-suppressed signals are too small relative to SM backgrounds.","If the 100 TeV $pp$ collider runs at $20~\\mathrm{ab}^{-1}$, a $Z'\\gamma$ resonance search can establish evidence up to $m_{Z'}\\simeq 800$ GeV and discovery near $m_{Z'}\\simeq 300$ GeV.","CLIC's signal cross section approaches a constant at high energy while the SM background falls, which is what makes $5\\sigma$ discovery possible for light $Z'$ masses.","The resonant muon collider probes a complementary, heavier mass window, and its resonant signal rate is independent of the $Z'$--muon coupling.","Observation of these anomalous couplings would indirectly reveal the presence of the spectator fermions required to render the full UV theory anomaly-free."],"supporting_citations":[{"why":"establishes that gauge-anomalous theories are consistent only as effective theories with a cutoff tied to new fermions.","marker":"[10]"},{"why":"supplies the covariant regularization scheme used to fix the WZW coefficients.","marker":"[12]"},{"why":"applies anomalous $Z'$ diboson vertices at the LHC and underlies the vertex implementation used here.","marker":"[13]"},{"why":"defines the muonic $U(1)'_\\mu$ model, its anomaly-cancellation structure, and the neutrino-trident constraint.","marker":"[14]"},{"why":"gives the Rosenberg parametrization of the triple-gauge vertex.","marker":"[19]"},{"why":"provides the CCFR neutrino-trident measurement bounding $g_\\mu$.","marker":"[27]"},{"why":"updates the neutrino-trident constraint with DUNE prospects, justifying the coupling chosen.","marker":"[28]"},{"why":"specifies the CLIC machine parameters, luminosity, and physics potential used for the $\\sqrt{s}=3$ TeV analysis.","marker":"[40]"}],"fun_headline_variants":["HL-LHC blind to anomalous Z' couplings; 100 TeV and CLIC can see","Anomalous triple-gauge couplings: probe at 100 TeV or CLIC, not HL-LHC","Z' anomaly from muon axial charge: see it at future colliders","For Z' anomaly, HL-LHC says no, but 100 TeV says yes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The projected reaches assume that only irreducible Standard Model backgrounds contribute and that the covariant regularization scheme gives the exact anomalous vertices; if either fails, the significances and mass windows shrink.","fun_headline_variants_meta":{"raw":{"variants":["HL-LHC blind to anomalous Z' couplings; 100 TeV and CLIC can see","Anomalous triple-gauge couplings: probe at 100 TeV or CLIC, not HL-LHC","Z' anomaly from muon axial charge: see it at future colliders","For Z' anomaly, HL-LHC says no, but 100 TeV says yes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000839,"raw_usage":{"total_tokens":3821,"prompt_tokens":1274,"completion_tokens":2547,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":890,"completion_tokens_details":{"reasoning_tokens":2450}},"tokens_in":890,"tokens_out":2547,"duration_ms":16718,"temperature":1.0,"reasoning_tokens":2450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:40:22.412364+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Reconstruct the $\\mu^+\\mu^-jj$ or $\\mu^+\\mu^-\\gamma$ final state at CLIC with $5~\\mathrm{ab}^{-1}$; for $m_{Z'}=200$ GeV the covariant-scheme model predicts about five signal events after cuts, so a measured yield consistent with background alone would rule out the predicted anomalous $Z'Z$ vertex.","supporting_citations":[{"cited_title":"Gauge anomalies in an effective field theory,","cited_arxiv_id":null,"evidence_quote":"establishes that gauge-anomalous theories are consistent only as effective theories with a cutoff tied to new fermions."},{"cited_title":"On Dark Matter Interactions with the Standard Model through an Anomalous $Z'$","cited_arxiv_id":"1707.00709","evidence_quote":"supplies the covariant regularization scheme used to fix the WZW coefficients."},{"cited_title":"Anomalous $Z'$ and Diboson Resonances at the LHC","cited_arxiv_id":"1712.01840","evidence_quote":"applies anomalous $Z'$ diboson vertices at the LHC and underlies the vertex implementation used here."},{"cited_title":"Electromagnetic interactions of neutrinos,","cited_arxiv_id":null,"evidence_quote":"gives the Rosenberg parametrization of the triple-gauge vertex."},{"cited_title":"Recent Electroweak Results from the CCFR Collaboration: Neutrino Tridents and W - Z Interference and the Lorentz Structure of the Weak Current,","cited_arxiv_id":null,"evidence_quote":"provides the CCFR neutrino-trident measurement bounding $g_\\mu$."}],"review_version":1}