{"id":"2e7dd1c6-e380-4322-b447-db29920eb6c0","arxiv_id":"2412.09681","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Future lepton colliders could probe dark photon and L_mu-L_tau vector portals with one to two orders of magnitude better sensitivity than current constraints across masses from tens of GeV to a few TeV.","lead":"This paper simulates how well future lepton colliders could detect weakly coupled dark photon and L_mu-L_tau Z' bosons, in both visible and invisible decay channels. It finds that the proposed muon collider and FCC-ee could improve sensitivity to these particles by one to two orders of magnitude compared with current experiments.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The projected 1–2 order-of-magnitude improvements and the FCC-ee thermal-relic reach rest on 95% CL bounds computed without a stated likelihood or systematic uncertainties; at FCC-ee luminosities this can shift the claimed reach by more than the quoted improvement.","rationale":"I read the paper as a sensitivity study whose deliverable is the numerical reach in Figs. 6–7. The strongest claim is quantified: 1–2 orders of magnitude improvement for Lμ−Lτ (visible and invisible) and invisible dark photons, and FCC-ee reaching the dark-matter thermal relic target for most of the relevant mass range. The science case is well-motivated and the analysis is transparent about detector benchmarks. However, the paper does not state how the limits are computed, and no systematic uncertainties are included. At the projected luminosities, backgrounds like e+e−→γνν at the Z-pole are so large that the statistical-only limit is far below the systematic floor; a 1–2% background-systematic uncertainty is typical for such channels and would move the contours by a factor comparable to the claimed improvement. The thermal-relic crossing is a binary claim that is especially sensitive to this. I therefore identify the missing statistical/systematic treatment as the load-bearing weakness, rather than the detector-performance parameters emphasized by the reader (though those also matter). A concrete reproducibility check would settle it: recompute with a profile likelihood and nuisance parameters, or at least provide the exact recipe. If the contours move little, the claim stands; if they shift above the relic line, the headline should be softened. This supports the reader's conditional verdict, so I recommend UNCHANGED.","tokens_in":14990,"tokens_out":12483,"duration_ms":144083,"concrete_test":"Ask the authors to provide, or independently recompute, the FCC-ee invisible dark-photon bound of Fig. 6 (left) with a profile-likelihood fit that includes a 2% normalization nuisance for the γνν background and a 0.5% photon energy-scale uncertainty. If the 95% exclusion shifts by more than ~30% in ε, or if it fails to cross the relic target line in the 10–70 GeV window, the headline 'reaching the thermal relic target' is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claims—the 1–2 order-of-magnitude sensitivity gains and the statement that FCC-ee reaches the dark-photon thermal relic target for most of the relevant mass range (Sec. 5.1, Fig. 6)—depend on 95% CL exclusion contours derived from LO MadGraph+Pythia event counts. The paper never states the likelihood or limit-setting recipe: no CLs or profile-likelihood formula, no binning, no treatment of nuisance parameters, and no systematic uncertainties on the dominant backgrounds (e.g., e+e−→γνν for the mono-photon channel, Z→ττ for the exotic Z-decay channel). At FCC-ee luminosity (6×10^12 Z; 5 ab^-1 at ZH), the statistical error on these backgrounds is sub-percent, so the exclusion is controlled by the background normalization and shape. A 1–2% normalization uncertainty or a small photon-energy-scale error directly changes the reconstructed-mass window efficiency and can shift the 95% contours by an amount comparable to the claimed 1–2 orders. In particular, the claim that the dark-photon bound crosses the thermal relic line for most of the 10–70 GeV range is precisely the kind of statement that flips under such systematic shifts. The paper also does not report how the optimized pT and mass cuts are scanned, making the contours irreproducible. This is not a critique of the physics case, but of the evidence supporting the numerical reach.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents projected sensitivities of future lepton colliders—a 3/10 TeV muon collider and the FCC-ee Z-pole and ZH runs—to two weakly coupled vector portals: the dark photon and the L_mu-L_tau gauge boson. Both visible decays into muon pairs and invisible decays into missing energy are considered, using leading-order MadGraph5+Pythia8+Delphes simulations with simple cut-based selections. The channels include associated production with a photon, Z'-bremsstrahlung from muons, and exotic Z decays. The central claims are that future colliders improve sensitivity to L_mu-L_tau bosons and invisibly decaying dark photons by 1--2 orders of magnitude over existing constraints, and that FCC-ee can reach the dark-matter thermal-relic target for most of the relevant mass range for the invisible dark photon. The paper also examines the impact of forward muon detectors at the muon collider, finding only marginal gains.","tokens_in":15331,"tokens_out":3514,"duration_ms":38078,"significance":"If the projections are reliable, the paper provides a useful, reasonably comprehensive map of future lepton-collider sensitivity to vector portals, complementing existing LHC and low-energy probes. Its strengths are that it treats both visible and invisible decays, compares two collider concepts and multiple production channels, and anchors its projections to external constraints (LEP, LHC, EWPT, CCFR) without circularity. The paper is also transparent about several detector-performance assumptions. However, the quantitative reach claims rest on a simplified simulation framework whose statistical and systematic treatment is not fully specified; given that the claimed improvements are large (1--2 orders of magnitude), the missing limit-setting details are load-bearing for the main conclusions.","major_comments":[{"comment":"The 95% CL sensitivity contours are derived from event counts but the paper never states the limit-setting recipe: there is no likelihood or CLs formula, no binning, no treatment of background uncertainties, and no systematic uncertainties. At FCC-ee luminosities (e.g., 6e12 Z bosons, 5 ab^-1) the statistical error on the dominant backgrounds (gamma nu nu for mono-photon, Z->tau tau for exotic Z decays) is sub-percent, so the exclusion is controlled by background normalization and shape. A few-percent uncertainty on the photon energy scale or background rate can shift the contours by an amount comparable to the claimed 1--2 orders of magnitude, and the statement that the FCC-ee dark-photon bound crosses the thermal-relic line (Sec. 5.1) is precisely the kind of claim that can flip under such shifts. Please specify the statistical procedure and quantify the effect of realistic systematic uncertainties on the projected contours.","section":"Secs. 3.1, 3.2 and Fig. 6"},{"comment":"The selection windows for FCC-ee associated production are introduced as '|M_recon - MZ'| < (40 GeV/MZ')^2 8 GeV' and similar, with the text stating only 'We have chosen these values based on Eq. (3.2) and checked its consistency numerically.' The paper does not report how these windows were optimized or scanned, nor the resulting signal and background efficiencies as functions of MZ'. Without this information the contours in Fig. 6 cannot be reproduced, and the sensitivity loss at low masses depends directly on these choices. Please provide the cut-scan procedure and the efficiency tables.","section":"Sec. 3.2"},{"comment":"In the Z'-bremsstrahlung analysis the only stated discriminator after baseline cuts is a minimum transverse momentum of the muon pair, pT(mu+mu-) > 50 GeV, with the text adding that the cut is 'optimized over the MZ' range' without giving the optimized values or the optimization criterion. This matters because the conclusion that forward muon detectors give only minimal improvement depends directly on the pT distribution and the chosen cut. Please report the scan, the resulting pT thresholds as a function of MZ', and the corresponding signal/background acceptances.","section":"Sec. 3.1"},{"comment":"All projections are leading order and omit initial-state radiation; the paper acknowledges this in footnote 4 and argues that ISR events are excluded by the selection criteria. However, for the mono-photon channel at FCC-ee and the associated-production channel at the muon collider, both signal and background are sensitive to QED radiation near thresholds, and the claimed order-of-magnitude improvements rely on precise acceptance calculations. The paper should provide at least an estimate of the size of missing higher-order/ISR corrections to the projected contours, especially for low MZ' values where Eq. (3.2) amplifies energy smearing.","section":"Secs. 3.1, 3.2 and footnote 4"}],"minor_comments":[{"comment":"Several mechanical typos appear: 'T able 1' and 'T able 2' in the text, 'the electric charge of the election' in Sec. 2.1, 'as a function M recon' missing 'of' in the Fig. 2 caption, and 'out work' in footnote 4. These should be corrected.","section":"Throughout"},{"comment":"The text says 'we show the 30 TeV CoM energy projections' for invisibly decaying dark photons, but Table 1 and the main text only define 3 and 10 TeV benchmarks. Please clarify whether this is a new benchmark or a typo.","section":"Appendix A"},{"comment":"The thermal-relic target line is defined for fermionic dark matter with m_chi = MZ'/3 and alpha_D = 0.1, and the text notes the dependence on m_chi/MZ'. It would be helpful to state explicitly in the caption of Fig. 6 and in Sec. 5.1 that the 'reaches the thermal relic target' claim refers to this specific benchmark, not to the full model space.","section":"Sec. 5.1"},{"comment":"For the exotic Z-decay analysis the text mentions that the invisible width of the Z' is assumed to be smaller than the experimental MIM spread, corresponding to widths of up to ~5-10%. This assumption should be stated in the model section as well, since it effectively constraints the product gD^2 for the considered parameter range.","section":"Sec. 3.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a standard sensitivity study for a hep-ph journal, and its topic fits the scope. The main issue is the gap between the precision of the quoted numbers (e.g., exclusion contours crossing the thermal-relic line) and the unspecified statistical/systematic treatment. Once the authors supply the limit-setting details and a sensitivity check against systematic shifts, the paper could be acceptable. I do not see a circularity or novelty problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Broadly, this is a useful paper. It fills a real gap by putting dark photon and L_mu-L_tau portals on the same footing at FCC-ee and a muon collider, for both visible and invisible decays, and by including a forward muon detector study. The main findings—1–2 order-of-magnitude improvements for L_mu-L_tau and invisible dark photons, the relative weakness of forward detectors, and the dark-photon thermal-relic reach at FCC-ee—are plausible in broad brush and are benchmarked against LEP, LHC, CCFR, and EWPT results. The simulation chain is standard MadGraph/Pythia/Delphes, the models are cleanly defined, and the paper says plainly where it is leading-order and where ISR is missing. The citation pattern is fair: related muon collider Z' papers are cited, and the forward-muon work from the same groups is acknowledged. For a projection paper, this is above average in clarity.\n\nThe soft spot is real and the stress-test note holds up on reading. Nowhere in the paper is the 95% CL limit-setting defined: no likelihood, no CLs recipe, no binning, no nuisance parameters, and no systematic uncertainties on the backgrounds. At FCC-ee luminosities the background statistical error is negligible, so the exclusion is controlled by background normalization and by the efficiency of the reconstructed-mass windows, which depend on photon energy scale and resolution. A 1–2% normalization or energy-scale error can move the contours by an amount comparable to the claimed improvement. The statement that FCC-ee reaches the thermal relic target for most of the 10–70 GeV range is precisely the kind of claim that can flip under such shifts. The paper also describes 'optimized' pT and mass cuts without reporting the scan. None of this undermines the physics case, but it makes the headline numbers not reproducible as stated.\n\nThe visible dark photon section is more modest—factor-of-two improvement at FCC-ee, little muon collider gain—and that is presented honestly. The forward-detector negative result looks robust given the stated resolutions.\n\nWho is this for? Dark-sector phenomenologists and people working on FCC-ee or muon collider physics cases. It deserves a serious referee. I would recommend acceptance after a revision that states the limit-setting procedure, reports the cut scans, and adds a systematic uncertainty band, ideally as a release of the analysis inputs.\n\nVerdict: solidly in the conditional-accept category; not a game-changer, but a useful reference.","headline":"A solid, clearly-scoped vector-portal sensitivity grid for FCC-ee and muon colliders, with a real reproducibility gap in the limit-setting that could shift the headline contours.","tokens_in":15832,"tokens_out":2744,"would_cite":true,"duration_ms":29328,"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":"Future lepton colliders can probe weakly coupled Z' bosons one to two orders of magnitude beyond current limits.","keywords":["vector portal","dark photon","Lμ − Lτ gauge boson","future lepton colliders","muon collider","FCC-ee","invisible decay","missing energy searches"],"falsifier":"Run the FCC-ee Z-pole and ZH stages at design luminosity and compare the observed mono-photon and exotic-$Z$ constraints with the paper's projected 95% exclusion lines; if the measured limit on the kinetic mixing $\\epsilon$ is appreciably weaker than the projected line, the assumed photon-energy resolution or background model is wrong.","tokens_in":14833,"feed_emoji":"⚛️","tokens_out":11993,"duration_ms":99871,"temperature":0.7,"pith_summary":"This paper asks how well the next generation of lepton colliders could discover weakly coupled vector portals ($Z'$ bosons) that connect the Standard Model to a dark sector. It focuses on the dark photon and the $L_\\mu - L_\\tau$ gauge boson, with masses from tens of GeV to a few TeV, considering both visible decays into Standard Model leptons and invisible decays into dark particles. Using the FCC-ee Z-pole and $ZH$ runs and a 3–10 TeV muon collider as benchmarks, the central result is that both machines would improve sensitivity to $L_\\mu - L_\\tau$ bosons, in both decay modes, and to invisibly decaying dark photons by one to two orders of magnitude across the relevant mass range. For invisibly decaying dark photons, the FCC-ee could reach the thermal-relic target for dark matter over most of the mass range. If these projections are realized, future lepton colliders would turn a broad, currently untested slice of vector-portal parameter space into accessible discovery territory.","feed_headline":"Lepton colliders could find invisible Z' bosons 100x deeper","feed_subtitle":"FCC-ee and a muon collider would push invisible dark-photon and Lμ−Lτ searches 1-2 orders deeper","key_machinery":"The argument is carried by three production channels computed with Monte Carlo simulation: associated production of the $Z'$ with a hard photon, where the photon energy $E_\\gamma = (s - M_{Z'}^2)/(2\\sqrt{s})$ reconstructs the mediator mass; $Z'$-bremsstrahlung from muons in $\\mu^+\\mu^-$ scattering, which is enhanced in the forward region and is analyzed with forward muon detectors using the muon-pair transverse momentum $p_T^{\\mu\\mu}$; and exotic $Z$ decays $Z \\to \\mu^+\\mu^- Z'$ at the FCC-ee Z pole. The sensitivity projections are obtained by applying baseline lepton cuts, reconstructing the mediator mass either from the photon energy or from a missing invariant mass, and computing 95% confidence-level exclusions over the Standard Model backgrounds. A key technical element is the smearing relation $\\Delta M_{Z'}^{\\rm recon}/M_{Z'}^{\\rm recon} \\sim (s/(2M_{Z'}^2))\\,\\Delta E_\\gamma/E_\\gamma$, which controls how much detector resolution matters at low $M_{Z'}$.","core_discovery":"The paper's central claim is that future lepton colliders can probe vector-portal parameter space far beyond existing bounds. For an invisibly decaying dark photon, the FCC-ee's dominant channel is associated production with a photon, $e^+e^- \\to Z'\\gamma$ with $Z' \\to$ invisible, and the reconstruction of $M_{Z'}$ from the photon energy; at the Z pole, exotic decays $Z \\to \\mu^+\\mu^- Z'$ extend the reach to light masses. For the $L_\\mu - L_\\tau$ boson, the muon collider is the natural probe because the new force couples directly to muons, and both associated production and $Z'$-bremsstrahlung in $\\mu^+\\mu^-$ scattering contribute; the FCC-ee probes masses below $M_Z$ through exotic $Z$ decays. The paper reports 95% confidence-level projections in which these channels improve on current leading constraints—LEP mono-photon searches for invisible dark photons, neutrino trident production for $L_\\mu - L_\\tau$, and LHC dilepton and 4-muon searches for visible decays—by one to two orders of magnitude in the relevant coupling. It also finds that forward muon detectors at the muon collider yield only marginal gains, because missing-invariant-mass reconstruction is poor for light $Z'$ bosons.","pith_inferences":["If FCC-ee reaches the thermal-relic line and observes nothing, the absence would disfavor pseudo-Dirac dark matter coupled through a dark photon in that mass window, since the projected reach is designed to cover the relic target.","The mass-smearing relation implies that improving the FCC-ee calorimeter's photon energy resolution may be more valuable for low-mass dark photons than increasing luminosity.","The same strategy could be applied to other electron-positron machines running at similar energies; their reach would track their assumed luminosity and detector resolution rather than the machine's center-of-mass energy."],"forward_implications":["For invisibly decaying dark photons, the FCC-ee would probe kinetic mixing $\\epsilon$ down to the thermal-relic target for pseudo-Dirac dark matter over most of the 10 GeV to hundreds-of-GeV mass range.","For $L_\\mu - L_\\tau$ bosons, the FCC-ee improves sensitivity below $M_Z$ and the muon collider extends reach to TeV masses, together surpassing current bounds by more than an order of magnitude for both visible and invisible decays.","The muon collider's forward muon detectors add only a mild improvement, so the main discovery potential is already captured by the central detector.","Because the results depend only on the electron and muon charges of the new vector, they transfer directly to any leptophilic $Z'$ portal, not just the two benchmark models."],"supporting_citations":[{"why":"Supplies the FCC-ee energies, integrated luminosities, and detector resolutions used for the $e^+e^-$ projections.","marker":"[1]"},{"why":"Supplies the muon collider energies and the luminosity scaling used for the $\\mu$ projections.","marker":"[6]"},{"why":"Provides the forward-muon detection strategy and the detector-resolution assumptions used in the $Z'$-bremsstrahlung analysis.","marker":"[16]"},{"why":"Gives the LEP mono-photon constraints that the invisible-dark-photon projections are compared against and improve upon.","marker":"[33]"},{"why":"Documents the forward muon detector design whose coverage and performance enter the muon-collider projections.","marker":"[34]"},{"why":"Provides the electroweak-precision constraints on dark photons and the kinetic-mixing framework in which the dark photon is defined.","marker":"[38]"},{"why":"Gives the LHC dimuon resonance limits that the visible-dark-photon projections must beat.","marker":"[45, 46]"},{"why":"Supplies the neutrino-trident bound that the invisible $L_\\mu - L_\\tau$ projections surpass.","marker":"[52]"},{"why":"Gives the LHC 4-muon limits that the visible $L_\\mu - L_\\tau$ projections improve upon.","marker":"[53, 54]"}],"fun_headline_variants":["Lepton colliders boost Z' search sensitivity up to 100x","Invisible dark photon and L_mu-L_tau probes gain 10-100x","Muon collider and FCC-ee set to sharpen Z' limits","Z' vector portal reach extends 1-2 orders at lepton colliders","Future lepton colliders promise 100x better Z' detection"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The projections assume the benchmark luminosities and detector resolutions quoted from the design reports, such as 4% photon energy resolution at FCC-ee and a forward muon detector with 10% energy resolution and 5 mrad angular resolution, and if any of those are not achieved, the claimed improvements shrink.","fun_headline_variants_meta":{"raw":{"variants":["Lepton colliders boost Z' search sensitivity up to 100x","Invisible dark photon and L_mu-L_tau probes gain 10-100x","Muon collider and FCC-ee set to sharpen Z' limits","Z' vector portal reach extends 1-2 orders at lepton colliders","Future lepton colliders promise 100x better Z' detection"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001109,"raw_usage":{"total_tokens":4656,"prompt_tokens":1011,"completion_tokens":3645,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":3558}},"tokens_in":627,"tokens_out":3645,"duration_ms":29222,"temperature":1.0,"reasoning_tokens":3558,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:50:17.105434+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the FCC-ee Z-pole and ZH stages at design luminosity and compare the observed mono-photon and exotic-$Z$ constraints with the paper's projected 95% exclusion lines; if the measured limit on the kinetic mixing $\\epsilon$ is appreciably weaker than the projected line, the assumed photon-energy resolution or background model is wrong.","supporting_citations":[{"cited_title":"Abada et al., FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2 , Eur","cited_arxiv_id":null,"evidence_quote":"Supplies the FCC-ee energies, integrated luminosities, and detector resolutions used for the $e^+e^-$ projections."}],"review_version":1}