{"id":"5087673e-9baf-457b-9054-a210f27a6bad","arxiv_id":"2412.04331","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"In HNL extensions of the Standard Model, mu+e- collisions at muTRISTAN could discover e-tau and mu-tau charged lepton flavour violation with sensitivity orders of magnitude beyond low-energy experiments and FCC-ee.","lead":"This paper calculates how often muon-electron collisions at a proposed muTRISTAN collider would produce forbidden flavour changes like muon-to-tau transitions, through heavy neutrino loops. It finds these high-energy channels could outperform all other planned searches for tau-flavour violation, making muTRISTAN a potentially decisive facility for charged lepton flavour violation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'orders of magnitude' sensitivity gain rests on an unvalidated cut-and-count model: a flat 1% signal efficiency and a single LO background process assumed negligible in Sec. 5.2.","rationale":"The analytic computation is detailed, with explicit Passarino-Veltman decompositions, Fierz identities, and a discussion of finiteness and gauge invariance; nothing in Sections 3-4 or Appendices B-C suggests an internal error in the cross-section calculation. The central quantitative claim, however, is not a statement about cross-sections alone; it is a statement about discovery reach, and the only bridge from sigma to reach is the Sec. 5.2 counting experiment. That bridge has three unvalidated efficiency factors (25% cut efficiency, 40% tau tag, and an additional factor of 0.1 to reach 1% total) and one unquantified set of backgrounds beyond a single LO process. If the real efficiency is lower, or if backgrounds are not negligible, the muTRISTAN sensitivity contours in Figs. 12-13 shift upward in muX; a factor of ten in efficiency or background translates roughly into a factor of a few in muX because the cross-section scales approximately as muX^2. That could reduce, though not necessarily eliminate, the claimed several-orders-of-magnitude advantage. This is exactly the kind of assumption that should be tested before promoting the projection to a robust physics case. I therefore agree with the reader's weakest_assumption and recommend keeping the CONDITIONAL verdict, with the condition being a detector-level validation of the Sec. 5.2 signal and background model.","tokens_in":33261,"tokens_out":7994,"duration_ms":90643,"concrete_test":"Run a fast detector-level simulation for sqrt(s) = 346.4 GeV: generate signal (mu+ e- -> tau+ e-, mu+ e- -> mu+ tau-) and the main backgrounds (mu+ e- -> tau+ e- nu nu, mu+ e- -> mu+ e-, mu+ e- -> tau+ tau- nu nu) with MadGraph5_aMC@NLO + Pythia8 + Delphes using a muTRISTAN detector card with realistic lepton/tau efficiencies versus eta and pT. Apply the Sec. 5.2 cuts (/ET <= 10 GeV, 2.0 <= eta_l <= 4.0) and tau-tagging. Recompute the expected number of signal and background events at 1 ab^-1 and redraw the 1 ab^-1 contours of Figs. 12-13. If the tau-flavour contour shifts upward by more than about one order of magnitude in muX, or if S/sqrt(S+B) drops below 3 for the benchmark points, the headline 'orders of magnitude' claim is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The loop-level cross-sections and analytic expressions appear carefully derived and are not the weak point. The headline claim converts those cross-sections into projected sensitivity contours (Figs. 12-13) using the Sec. 5.2 prescription: apply /ET <= 10 GeV and 2.0 <= eta_l <= 4.0, assume 25% signal efficiency from these cuts, multiply by a 40% tau-tagging efficiency, then impose a further 'conservative' factor so total efficiency is 1%, and require 10 signal events with negligible background (Eq. 58). This is the load-bearing link from sigma to 'several orders of magnitude'. The background estimate is based on one leading-order process (mu+ e- -> tau+ e- nu nu, sigma ~ 21 fb) reduced to 6e-4 fb by cuts; tau fakes from mu+ e- -> mu+ e-, beam-induced backgrounds, and tau misidentification are not quantified. No detector simulation, no tau-decay Monte Carlo, and no eta-dependent tau-ID performance is used. Since the signal cut pushes leptons to 2 <= eta <= 4, and tau/lepton-ID at eta > 2.5-3 is not established at the assumed efficiency, the 25% and 40% factors could be optimistic. A lower total efficiency (or O(10) residual background) shifts the MR-muX contours upward and weakens the comparison with FCC-ee and low-energy tau probes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper computes the one-loop amplitudes, cross-sections, and angular observables for charged lepton flavour violating (cLFV) processes μ+ e− → ℓ+α ℓ−β in Standard Model extensions with heavy neutral leptons (HNLs), considering both a minimal '3+2' ad-hoc model and the ISS(3,3) inverse seesaw realisation. It then uses these results to project the sensitivity of a future μTRISTAN collider and compares it with low-energy cLFV searches and with cLFV Z-pole searches at FCC-ee. The central claim is that, while μ−e flavour violation remains best probed at low energies, eτ and μτ flavour violation searches at μTRISTAN could exceed the prospects of low-energy tau decays and of FCC-ee by several orders of magnitude.","tokens_in":33613,"tokens_out":7916,"duration_ms":87604,"significance":"If the sensitivity projection were reliable, this paper would make a strong case for μTRISTAN as a discovery machine for tau-flavour-violating cLFV in HNL models, with reach far beyond other planned facilities. The analytic part of the work is its main strength: the loop amplitudes are presented in detail, the Fierz and Dirac identities used in the reduction are spelled out in the appendices, the question of gauge invariance of the off-shell Z-penguin contributions is discussed, and the ISS(3,3) implementation is anchored to neutrino oscillation data through the Casas-Ibarra parametrisation. These parts are a useful reference that is largely independent of the collider projection. The sensitivity analysis, however, is currently not at the same standard, and the 'orders of magnitude' claim rests on assumptions that are not yet quantitatively justified.","major_comments":[{"comment":"The projected sensitivity contours are derived from a cut-and-count estimate based on a single leading-order background process, μ+e−→τ+e−νν with σ≈21 fb, reduced to 6×10−4 fb by the cuts, and on a global 1% signal efficiency obtained from a 25% cut efficiency, a 40% tau-tagging efficiency, and an additional ad-hoc suppression factor. No detector simulation, no tau-decay Monte Carlo, no estimate of tau fakes from μ+e−→μ+e−, no beam-induced background, and no tau misidentification rate are provided. Because the headline 'several orders of magnitude' claim derives directly from these contours, this is a load-bearing part of the paper and needs substantially stronger support.","section":"Sec. 5.2, Eq. (58), Figs. 12-13"},{"comment":"The signal process itself contains a tau lepton, whose decay necessarily produces missing neutrinos, but the manuscript does not state how the /ET≤10 GeV cut is applied to signal events. If tau decays are not simulated, the quoted 25% signal efficiency after the basic cuts is not justified, and a significant fraction of signal events could fail the missing-energy cut. This would shift the sensitivity contours in Figs. 12 and 13 upward and could materially weaken the comparison with FCC-ee and low-energy tau probes. The authors should either include tau decays explicitly in the efficiency estimate or restrict the sensitivity claims accordingly.","section":"Sec. 5.2"},{"comment":"The 40% tau-tagging efficiency is taken from an ATLAS high-mass resonance search, while the signal leptons are required to lie in the range 2.0≤ηℓ≤4.0. ATLAS tau identification is not established at such forward pseudorapidities, and a muTRISTAN-specific detector does not yet exist. A scan over plausible tau-tagging efficiencies, or a clear statement of the geometric and kinematic assumptions, is needed before the contours are used for quantitative sensitivity comparisons.","section":"Sec. 5.2"},{"comment":"The claim that μTRISTAN is more sensitive to μ−τ flavour violation than FCC-ee even with 5×10^12 Z bosons is based on an 'irreducible systematic misidentification' of secondary leptons in Z→ττ, but no quantitative estimate of that systematic floor is given. The FCC-ee sensitivities quoted in Table 2 may already include detector assumptions, so without a concrete comparison at the same level of detail the stated outperformance by 'several orders of magnitude' is not established.","section":"Sec. 6 and Fig. 13"}],"minor_comments":[{"comment":"The printed expression 'S = S√S+B' should presumably read S=S/√(S+B); as typeset it is not a valid formula.","section":"Eq. (58)"},{"comment":"The captions of Figs. 12 and 13 do not state the centre-of-mass energy used for the μTRISTAN cross-sections; the text should specify whether √s=346.4 GeV or another configuration is assumed.","section":"Figs. 12-13"},{"comment":"The benchmark in Eq. (51) is a single point in the ad-hoc parameter space; the text should state more explicitly how the conclusions depend on the choice s34=s35=0.1 and on the degenerate-mass assumption.","section":"Sec. 5.1, Eq. (51)"},{"comment":"The comparison in Fig. 10 refers to 'events' without specifying the tau decay channel; since signal and background rates are quoted in fb, the reader should be told whether tau decays are part of the event definition or not.","section":"Sec. 5.1 and Fig. 10"},{"comment":"The use of 'ATLAS-like rapidity coverage |η|≤4' for the signal integration is a useful illustration, but the muTRISTAN detector has not been designed; a brief comment on how this choice affects the quoted cross-sections would improve the presentation.","section":"Sec. 3 and Sec. 5.1"}],"recommendation":"major_revision","confidential_remarks":"The analytic cross-section computation is a solid and useful contribution, and the paper is likely acceptable after a major revision that makes the sensitivity analysis self-consistent (tau decays, backgrounds, detector assumptions) or, failing that, restricts the headline claim to the cross-section results and clearly labels the sensitivity contours as indicative rather than quantitative. I do not recommend rejection, because the central issue is a projection methodology that can in principle be improved within the scope of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the loop-level computation of mu+e- -> l+_alpha l-_beta with HNL penguins and boxes is the real content of this paper, and it looks solid: full amplitude expressions, Fierz identities, gauge-invariance discussion, detailed appendices, and cross-checks with FeynCalc/Package-X. Second, the sensitivity projections, which carry the headline claim, are built on a cut-and-count analysis that is honestly labeled preliminary but is not strong enough to support 'several orders of magnitude' over FCC-ee.\n\nWhat's actually new: first complete treatment of these two-to-two cLFV processes at a muon-electron collider, including the LNV box diagrams and the forward-backward asymmetry and pseudorapidity distributions. That's a useful analytic resource for future detector studies. The ISS(3,3) study uses Casas-Ibarra, so the model is not being fitted to the observables; the comparison with low-energy probes is legitimate as a scan.\n\nWhere it gets soft: Section 5.2. The background is a single LO process (mu+e- -> tau+e- nu nu), reduced to 6e-4 fb with the missing energy and rapidity cuts. One process is a start, not a background estimate. Tau fakes from mu+e- -> mu+e-, beam-induced backgrounds from muon decay in flight, and misidentification in the forward region (2 < eta < 4) are not quantified. The 1% total efficiency is a plausible guess, and the authors say so, but the contours in Figs. 12-13 and the comparison with FCC-ee inherit that guess. The FCC-ee side is also asserted rather than computed: the irreducible systematic on Z -> l tau is mentioned but not modeled. If that systematic is smaller, or the muTRISTAN efficiency is lower than 1%, the 'orders of magnitude' becomes 'competitive' - still interesting, but not the same claim.\n\nThe mathematics of the amplitudes is not the weak point. The weak point is the bridge from cross-section to discovery reach, and the paper itself flags that (absence of concrete detector designs). That is fair, but then the abstract should not be as strong as it is.\n\nWho this is for: phenomenologists working on cLFV and future lepton colliders. They will want the formulas and the qualitative angular features. It deserves a serious referee: the computation is original, the presentation is careful, and the sensitivity question is exactly what a referee should push on. I would accept it with major revision on the projection part, or at least with a heavily qualified abstract.","headline":"A careful, genuinely new one-loop computation of HNL-induced cLFV at muTRISTAN; the cross-sections are the contribution, while the 'orders of magnitude' sensitivity claim outruns the paper's own cut-and-count caveats.","tokens_in":34129,"tokens_out":2638,"would_cite":true,"duration_ms":27558,"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":"The paper argues that muTRISTAN could discover HNL-induced charged lepton flavour violation in e-mu scattering, with tau-flavour channels beating FCC-ee and low-energy probes by orders of magnitude.","keywords":["charged lepton flavour violation","heavy neutral leptons","muTRISTAN","inverse seesaw","penguin and box diagrams","forward-backward asymmetry","future lepton colliders","neutrino mass models"],"falsifier":"A detector-level Monte Carlo of $\\mu^+e^-$ collisions at $\\sqrt{s}=346.4$ GeV that includes $\\tau$ fakes from $\\mu^+e^- \\to \\mu^+e^-$, in-flight muon decays and beam backgrounds would settle the projection: if the background in the signal region with missing transverse energy $\\leq 10$ GeV and lepton rapidities $2 \\leq \\eta \\leq 4$ exceeds roughly 10 events at $1~\\mathrm{ab}^{-1}$, or if the signal efficiency falls below about 0.1%, the claimed $e\\tau$/$\\mu\\tau$ sensitivity contours in the $(M_R,\\mu_X)$ plane would weaken and the quoted advantage over FCC-ee would not hold.","tokens_in":33077,"feed_emoji":"⚛️","tokens_out":10816,"duration_ms":103103,"temperature":0.7,"pith_summary":"The paper argues that an asymmetric muon-electron collider, $\\mu$TRISTAN, could discover charged-lepton flavour violation (cLFV) induced by heavy neutral leptons (HNLs) in $\\mu^+ e^- \\to \\ell_\\alpha^+ \\ell_\\beta^-$ scattering, and that for the $e\\tau$ and $\\mu\\tau$ flavour pairs its reach would exceed dedicated low-energy searches and $Z$-pole searches at FCC-ee by several orders of magnitude. The authors compute full one-loop cross sections and angular observables for two HNL frameworks, a minimal ad-hoc \"3+2\" extension and the Inverse Seesaw realisation ISS(3,3). The $t$-channel penguin-dominated processes $\\mu^+ e^- \\to \\tau^+ e^-$ and $\\mu^+ e^- \\to \\mu^+ \\tau^-$ have nearly energy-independent cross sections and can yield tens to hundreds of thousands of signal events in large parameter regions, while the estimated SM background can be cut to less than one event per $\\mathrm{ab}^{-1}$. A conservative cut-and-count estimate with a total signal efficiency of 1% and a 10-event requirement maps the sensitivity in the $(M_R, \\mu_X)$ plane, where $\\mu$TRISTAN probes $e\\tau$/$\\mu\\tau$ flavour violation several orders of magnitude beyond future Belle II and FCC-ee sensitivities. The projected reach depends on the assumed 1% efficiency and on a background estimated from one leading-order process.","feed_headline":"MuTRISTAN could beat FCC-ee on tau-flavour violation","feed_subtitle":"HNL-driven e-tau and mu-tau scattering could exceed low-energy probes and Z-pole searches by orders of magnitude.","key_machinery":"The load-bearing object is the complete one-loop amplitude for $\\mu^+e^- \\to \\ell_\\alpha^+\\ell_\\beta^-$ in the limit of massless external fermions, written after Fierz rearrangement in terms of three chiral amplitudes $A_{LL}$, $A_{LR}$ and $A_{RL}$ that combine photon- and $Z$-penguin form factors $F_\\gamma^{\\alpha\\beta}(q^2)$ and $F_Z^{\\alpha\\beta}(q^2)$ with four box-diagram functions $F_1$ and $F_2$; all flavour violation enters through the extended leptonic mixing matrix $U$. From these amplitudes the authors derive the differential cross section, the pseudo-rapidity distributions $d\\sigma/d\\eta$ and the forward-backward asymmetry $A_{FB}$. This machinery carries the argument because it connects the high-energy scattering rates to the same form factors that control low-energy cLFV decays and $Z\\to\\ell_\\alpha\\ell_\\beta$, and it allows the angular cuts ($|\\eta|\\leq 4$, $2\\leq \\eta_\\ell \\leq 4$) that reduce the SM background to sub-event levels.","core_discovery":"In the paper's own terms, HNLs with non-negligible mixings to active neutrinos generate cLFV at one loop, and this is enough to make $\\mu^+e^- \\to \\ell_\\alpha^+\\ell_\\beta^-$ scattering observable at $\\mu$TRISTAN. The central quantitative claim is that, in both a minimal ad-hoc extension with two sterile states and in the Inverse Seesaw ISS(3,3), the cross sections for $\\mu^+e^- \\to \\tau^+e^-$ and $\\mu^+e^- \\to \\mu^+\\tau^-$ remain large over a wide range of HNL masses (roughly $100$ GeV to several TeV), so that even with a deliberately conservative detector efficiency of order 1% the expected event counts reach the thousands. In the $(M_R,\\mu_X)$ plane of the ISS(3,3), the resulting sensitivity contours for $e\\tau$ and $\\mu\\tau$ flavour violation lie several orders of magnitude beyond the future sensitivities of searches for $\\tau\\to \\ell\\gamma$, $\\tau\\to 3\\ell$ and $Z\\to \\ell\\tau$, while for $\\mu e$ flavour violation low-energy dedicated experiments remain superior. The paper further claims that the forward-backward asymmetry of the final-state leptons is sensitive to which amplitude topology dominates and to CP-violating phases of the generalised lepton mixing matrix.","pith_inferences":["If real detector performance gives a signal efficiency below the assumed 1% or a non-negligible background from $\\tau$ fakes in $\\mu^+e^- \\to \\mu^+e^-$, the sensitivity contours in the $(M_R,\\mu_X)$ plane would shift; the 'orders of magnitude' statements should be read as contingent on the background and efficiency model.","The same form-factor calculation could be extended to $\\mu^+e^- \\to q\\bar q$, which would connect high-energy cLFV to the $\\mu$-$e$ conversion process that currently gives the strongest low-energy bounds; the authors mention this as future work.","The strong dependence of $A_{FB}$ on the CP phase $\\delta_{24}$ suggests that angular observables at lepton colliders could serve as CP-violation probes in the sterile-neutrino sector, an application the paper does not develop.","Because the comparison with FCC-ee is driven by irreducible $\\tau$ misidentification at the $Z$ pole, the real $\\tau$-tagging performance at $\\mu$TRISTAN will decide whether the claimed advantage survives; the underlying cross-section advantage alone does not guarantee it."],"forward_implications":["For $e\\tau$ and $\\mu\\tau$ flavour pairs, $\\mu$TRISTAN would probe HNL parameter space in the ISS(3,3) several orders of magnitude deeper than future $\\tau\\to \\ell\\gamma$, $\\tau\\to 3\\ell$ and $Z\\to\\ell\\tau$ searches, making it a discovery machine for tau-flavoured cLFV.","For $\\mu e$ flavour violation, low-energy experiments such as $\\mu\\to e$ conversion remain more sensitive, so $\\mu$TRISTAN and dedicated high-intensity facilities are complementary rather than competing.","A measurement of $A_{FB}$ for $\\mu^+e^- \\to e^+e^-$ and $\\mu^+e^- \\to \\mu^+\\mu^-$ could distinguish $s$-channel from $t$-channel (or box) dominance and give information on the HNL mass scale.","Even if no low-energy cLFV decay or $Z$-pole signal is ever seen, $\\mu$TRISTAN could still observe $\\tau$-flavoured cLFV events in large regions of the allowed parameter space."],"supporting_citations":[{"why":"The muTRISTAN design paper supplying the beam-energy configurations and the integrated luminosity (100 fb^-1 per year, 1 ab^-1 total) used for the sensitivity projections.","marker":"[51]"},{"why":"The physics case for an electron-muon collider, providing the broader motivation and additional beam configurations considered.","marker":"[52]"},{"why":"Earlier charged-lepton flavour violation analysis at muon-electron colliders that the paper extends to HNL loop-induced processes.","marker":"[53]"},{"why":"A prior heavy neutral lepton study at muTRISTAN with jets and missing energy, whose two-lepton final states this analysis complements.","marker":"[58]"},{"why":"The global neutrino oscillation fit supplying the PMNS parameters used for the benchmark points.","marker":"[72]"},{"why":"The modified Casas-Ibarra parametrisation used to generate Yukawa couplings in the ISS(3,3) benchmarks.","marker":"[77]"},{"why":"The detector study supplying the |eta|<=4 rapidity coverage adopted for angular integration and event selection.","marker":"[112]"},{"why":"The detector study supplying the 40% tau-tagging efficiency used in the overall signal-efficiency estimate.","marker":"[113]"}],"fun_headline_variants":["MuTRISTAN could see e-tau and mu-tau flavour violation beyond FCC-ee","For tau-flavour violation, MuTRISTAN outclasses FCC-ee by orders","MuTRISTAN's HNL searches promise thousands of tau-flavour events","Even 1% efficiency yields thousands of tau-flavour events at MuTRISTAN","MuTRISTAN's e-tau and mu-tau signals dwarf FCC-ee's reach"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reach projections depend on the assumption that a basic cut-and-count selection leaves the SM background below one event per $ab^{-1}$ while keeping a total signal efficiency of about 1%, and that ten signal events then constitute a discovery-level sensitivity.","fun_headline_variants_meta":{"raw":{"variants":["MuTRISTAN could see e-tau and mu-tau flavour violation beyond FCC-ee","For tau-flavour violation, MuTRISTAN outclasses FCC-ee by orders","MuTRISTAN's HNL searches promise thousands of tau-flavour events","Even 1% efficiency yields thousands of tau-flavour events at MuTRISTAN","MuTRISTAN's e-tau and mu-tau signals dwarf FCC-ee's reach"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00078,"raw_usage":{"total_tokens":3469,"prompt_tokens":993,"completion_tokens":2476,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":609,"completion_tokens_details":{"reasoning_tokens":2363}},"tokens_in":609,"tokens_out":2476,"duration_ms":18882,"temperature":1.0,"reasoning_tokens":2363,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:31:31.110537+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A detector-level Monte Carlo of $\\mu^+e^-$ collisions at $\\sqrt{s}=346.4$ GeV that includes $\\tau$ fakes from $\\mu^+e^- \\to \\mu^+e^-$, in-flight muon decays and beam backgrounds would settle the projection: if the background in the signal region with missing transverse energy $\\leq 10$ GeV and lepton rapidities $2 \\leq \\eta \\leq 4$ exceeds roughly 10 events at $1~\\mathrm{ab}^{-1}$, or if the signal efficiency falls below about 0.1%, the claimed $e\\tau$/$\\mu\\tau$ sensitivity contours in the $(M_R,\\mu_X)$ plane would weaken and the quoted advantage over FCC-ee would not hold.","supporting_citations":[],"review_version":1}