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Lepton-Trijet and Displaced Vertex Searches for Heavy Neutrinos at Future Electron-Proton Colliders

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper projects that the lepton-flavour-violating trijet channel at electron-proton colliders can exclude $|\theta_e\theta_\mu| \sim 10^{-7}$ at 95% CL, the best reconstructed-level sensitivity claimed for heavy neutrinos in this mass…

desk verdict Solid lepton-trijet projection for sterile neutrinos at ep colliders; the displaced-vertex reach is plausible but rests on an unquantified zero-background assumption. read the letter →

arxiv 1908.02852 v1 pith:22DXR2GO submitted 2019-08-07 hep-ph

Stefan Antusch , Oliver Fischer , A. Hammad This is my paper · ORCID
classification hep-ph
keywords heavyneutrinossterileelectron-protoncollidersLHeCFCC-heleptonflavourviolationdisplacedverticesseesawmechanism
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that future electron-proton colliders would be particularly effective at discovering sterile heavy neutrinos with masses near the electroweak scale. In the symmetry-protected seesaw benchmark model it adopts, the lepton-flavour-violating process $p e^- \to \mu^- + 3j$ has no parton-level Standard Model counterpart, and a reconstruction-level analysis with a boosted decision tree projects 95% CL exclusions of $|\theta_e\theta_\mu|$ down to about $2\times 10^{-7}$ at the LHeC and $10^{-7}$ at the FCC-he for masses of a few hundred GeV. For lighter masses below the $W$ mass, displaced-vertex decays are projected to reach $|\theta_e|^2$ of order $10^{-8}$ at the LHeC and $10^{-9}$ at the FCC-he. If these estimates hold, ep colliders would probe a region of active-sterile mixing that present constraints leave open.

What carries the argument

The analysis is carried by the active-sterile mixing parameters $\theta_\alpha$ and the process chain $p e^- \to N + j$ with $N \to \mu^- W^+ \to \mu^- jj$, whose rate scales as $|\theta_e|^2|\theta_\mu|^2/|\theta|^2$; under the benchmark choice $|\theta_e| = |\theta_\mu|$ this reduces to $|\theta_e\theta_\mu|$. For the prompt search, a boosted decision tree trained on 18 kinematic distributions—including the reconstructed heavy-neutrino invariant mass, muon transverse momentum, missing transverse energy, and angular separations between the $W$, the muon, and the beam jet—separates signal from backgrounds. For the displaced-vertex search, the machinery is the decay probability $P_{\rm dv} = \exp(-x_{\min}/\Delta x_{\rm lab}) - \exp(-x_{\max}/\Delta x_{\rm lab})$, integrated over the full production angular and Lorentz-boost distributions and the asymmetric detector geometry, with a 95% CL exclusion set at $N_{\rm dv} \ge 3.09$ expected decays.

What would settle it

Recompute the displaced-vertex background with explicit veto inefficiencies rather than perfect suppression: if tau-tag inefficiency or B-meson mass-window leakage leaves more than about three background events in the signal region at $1\,{\rm ab}^{-1}$, the claimed 95% CL $|\theta_e|^2$ contours for $m_N$ below $m_W$ would shift upward by roughly an order of magnitude. A direct way to check this is a background-only data sample at an ep detector that records any displaced tau or B decay inside the 40-micrometre window.

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Extended reading notes

Core claim

Working in the symmetry limit of the SPSS benchmark model, where two sterile neutrinos form a pair with opposite charges under a lepton-number-like symmetry and lepton number is conserved, the authors show that two signatures dominate the expected reach at ep colliders. The prompt lepton-trijet channel, $p e^- \to N j \to \mu^- W^+ j \to \mu^- + 3j$, is free of irreducible Standard Model background at parton level; once the dominant backgrounds are included and a boosted decision tree is optimized, the expected 95% CL limits reach $|\theta_e\theta_\mu| \approx 2\times 10^{-7}$ at the LHeC and $\approx 10^{-7}$ at the FCC-he for heavy-neutrino masses of a few hundred GeV. For masses below $m_W$, where the heavy neutrino can travel a macroscopic distance before decaying, the displaced-vertex channel reaches $|\theta_e|^2 \sim 10^{-8}$ at the LHeC and $\sim 10^{-9}$ at the FCC-he. The paper concludes that, in this mass range, the LFV lepton-trijet signature yields the best sensitivity of all currently discussed heavy-neutrino signatures analysed at the reconstructed level.

Load-bearing premise

The displaced-vertex reach assumes that tau leptons, B mesons, and cosmic muons can be completely suppressed by the stated vetoes and mass cuts, and that a displacement of 40 micrometres is enough to identify a secondary vertex; if any of those vetoes leaks events, the projected $|\theta_e|^2$ contours move to larger mixing angles.

Editorial extensions

If this is right

  • In the few-hundred-GeV mass region, the LFV trijet channel is projected to probe $|\theta_e\theta_\mu|$ values an order of magnitude or more below current exclusion limits.
  • The displaced-vertex channel covers the sub-$m_W$ mass range from about 5 GeV upward with $|\theta_e|^2$ reach of $10^{-8}$ to $10^{-9}$, a region where prompt searches lose sensitivity to small mixings.
  • Because the signature rate depends on the flavour combination $2|\theta_e|^2|\theta_\mu|^2/|\theta|^2$, the results transfer to other flavour patterns, such as a $\tau^- jjj$ final state when muon mixing is small.
  • Within the displaced-vertex contour, the lepton-number-conserving framework also allows an anti-lepton version of the trijet final state, whose oscillatory lifetime dependence could reveal heavy-neutrino-antineutrino oscillations and, with enough statistics, a measurement of the mass splitting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's claims, if the displaced-vertex reach holds up under a more realistic background treatment, ep colliders would be the most direct way to test low-scale seesaw masses below $m_W$, where hadron colliders lose sensitivity to small mixings.
  • The strong mass dependence of the reconstructed kinematic distributions suggests that a heavy-neutrino mass could be inferred from shapes alone; the paper notes this only in passing, and a dedicated shape-based measurement would be a natural extension.
  • A testable extension would drop the protective symmetry and repeat the analysis with a single sterile neutrino, restoring lepton-number-violating decays; the change in the trijet reach would quantify how much of the projected sensitivity relies on the symmetry limit.
  • Applying the same search pipeline to a $\tau^-jjj$ final state with realistic tau-tagging efficiencies would test whether the paper's golden-channel claim survives reconstruction losses for taus.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies the discovery and exclusion prospects for sterile (heavy) neutrinos in the symmetry-protected seesaw scenario (SPSS) at the proposed LHeC and FCC-he ep colliders. Two search channels are analysed: (i) a prompt lepton-flavour-violating trijet final state, p e− → μ− + 3j, simulated with MadGraph/Pythia/Delphes and separated from dominant vector-boson-pair backgrounds with a TMVA boosted decision tree (BDT); and (ii) a displaced-vertex search for mN < mW, where the number of signal vertices is estimated with Eqs. (15)–(16) using the full LHeC detector geometry and heavy-neutrino kinematic distributions. The paper reports 95% CL sensitivities |θeθμ| ≈ 10^-7 (FCC-he) and ≈ 2×10^-7 (LHeC) for mN around a few hundred GeV from the trijet channel, and |θe|^2 ≈ O(10^-8) (LHeC) and ≈ O(10^-9) (FCC-he) for mN < mW from displaced vertices. It further claims that, in the considered benchmark model, the LFV lepton-trijet channel gives the best sensitivity of all currently discussed heavy-neutrino signatures analysed at the reconstructed level.

Significance. If the results hold, the paper provides a valuable quantitative benchmark for heavy-neutrino searches at future ep colliders, a channel that is relatively unexplored compared to LHC searches. The prompt-trijet analysis is a genuine improvement over the earlier parton-level estimates of ref. [1]: it includes hadronization, detector simulation, several dominant backgrounds, a BDT with explicit train/test separation, quoted event counts at the working point, and expected 95% CL bands. The displaced-vertex analysis also improves on previous work by incorporating the asymmetric detector geometry and the full angular and boost distributions of the produced heavy neutrinos. The flavor dependence of the trijet limit is given explicitly. The main weakness is that the displaced-vertex reach relies on unvalidated zero-background assumptions, so the sub-mW contours in Fig. 6 are best regarded as optimistic projections until those assumptions are demonstrated or replaced by conservative background estimates.

major comments (2)
  1. [Sec. 3.3.3, Eqs. (15)–(16), Fig. 6] The displaced-vertex exclusion contours are computed with a zero-background Poisson threshold (N_dv ≥ 3.09), but the background suppression on which this relies is asserted rather than demonstrated. The text assumes that tau leptons can be 'effectively vetoed against by existing tau tags', that B mesons allow 'complete suppression' via B-tag filters and a 5 GeV mass cut, and that cosmic muons 'can be vetoed against effectively' at displacements as small as 40 μm, but no tau-tag efficiency, B-tag fake rate, cosmic-muon rejection factor, or detector-material interaction estimate is provided. This is load-bearing: for example, the tau background alone has σ ≈ 0.34 pb, corresponding to about 3×10^5 events at 1 ab^-1 at the LHeC, and a qualitative statement that all such events can be vetoed is not quantitative. Even one surviving background event changes the 95% CL requirement from 3.09 signal events to roughly 3.7 under a standard Poisson construction, shifting the |θ_e|^2 contours in Fig. 6 upward by a comparable factor. Please either simulate these backgrounds with the same setup used for the prompt analysis or adopt conservative background counts and recompute the contours.
  2. [Sec. 3.3.2] The assumed minimum vertex displacement of 40 μm is not validated for the signal final states. The text cites the LHeC CDR for tracking resolution, but the separation power at 40 μm depends on the track multiplicity, the material budget, and the boost distribution of the heavy neutrino; this quantity enters Eq. (16) through x_min(ϑ) and therefore directly controls the contours in Fig. 6. Please provide a vertexing demonstration based on a full simulation or a conservative scan (e.g., 100 μm) showing how the reach changes. Without this, the sub-mW displaced-vertex reach should be presented as an optimistic sensitivity estimate rather than a demonstrated projection.
minor comments (5)
  1. [Sec. 3.2.3, Fig. 5] Please specify how the 2% log-normal background systematic is implemented in the CLs/profile-likelihood calculation (e.g., as a single nuisance parameter on the total background after the BDT cut) and quantify its effect on the expected limit bands.
  2. [Sec. 3.1, Fig. 1] Please clarify whether the quoted production cross section and the simulated signal samples include Wγ-fusion in addition to t-channel W exchange; Eq. (7) as written describes t-channel exchange only.
  3. [Eq. (15)] The text states that the ~5% invisible N→3ν branching fraction is excluded, but Eq. (15) contains no explicit visible-branching prefactor. Please state where this factor enters the calculation.
  4. [Conclusions, first paragraph] The displaced-vertex sensitivities are quoted as limits on |θ_e θ_μ|, but Fig. 6 and the surrounding text define them as limits on |θ_e|^2 (with θ_μ = θ_τ = 0). Please correct the notation for consistency.
  5. [References and typos] Ref. [38] appears in the bibliography but is not cited in the text; please cite it where relevant or remove it. Please also correct typographical errors: 'Ptyhia6' should be 'Pythia6' in Sec. 3.2.2; 'unless unless' appears in Sec. 3.4; and 'The LHeC makes utilizes' appears in Sec. 3.1.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the projected sensitivities are computed from the model Lagrangian and full event simulation, with self-citations providing benchmark and formalism context rather than fitted inputs.

full rationale

The paper's central predictions are derived by direct calculation and simulation rather than by fitting the target quantities. The lepton-trijet sensitivity follows from MadGraph/Pythia/Delphes event generation of the signal process p e- -> N + j -> mu- + 3j, with the rate proportional to |theta_e|^2 |theta_mu|^2 / |theta|^2 as stated in Section 3.2, and the 95% CL limits are obtained from a frequentist profile-likelihood test on the simulated signal and background event counts after the BDT cut. The displaced-vertex sensitivity is computed with Eq. (15)-(16), which is the standard exponential decay-probability formula multiplied by the production cross section and integrated luminosity; the paper then imposes the Poisson zero-background threshold N_dv >= 3.09 with the full detector geometry. No parameter entering these exclusions is fitted to the exclusion contours themselves. The self-citations to the SPSS benchmark model [7], to the prior parton-level sensitivity survey [1], and to the displaced-vertex formalism [36] supply model assumptions and calculational tools, but the model parameters (mixing angles, mass) are free inputs scanned over, not determined by the paper's target results. The 'best sensitivity' claim is a comparison against existing external bounds from ATLAS, LHCb, DELPHI and MEG, shown in Figure 7, and against the earlier parton-level estimate in [1]; this is a comparative statement, not a derivation that reduces to its own assumptions. The unvalidated zero-background assumption for the displaced-vertex vetoes is a physics-risk concern about background estimates, not an instance of self-definition or fitted-input circularity. Overall, the central derivation chain is self-contained: Lagrangian to cross sections and branching ratios, simulation to event counts, event counts to confidence limits, with no step in which a predicted quantity is identical by construction to an input.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central sensitivity curves are parameter scans over the heavy neutrino mass and the active-sterile mixing angles, with the benchmark flavour choice θe = θμ and θτ = 0. The analysis also relies on assumed detector background vetoes and a flat 2% systematic uncertainty. The paper introduces no new particles; it uses the existing SPSS benchmark model.

free parameters (4)
  • Heavy neutrino mass MN = scanned over 5-1000 GeV across benchmark points
    Sets the kinematics and lifetime; the sensitivity curves are parameterized in MN.
  • Active-sterile mixing angles |θe| and |θμ| = benchmark: |θe| = |θμ| = 0.01, |θτ| = 0; scanned for limits
    The flavor structure is chosen for the LFV trijet analysis; the paper explains how to generalize via |θeθμ| to 2|θe|^2|θμ|^2/|θ|^2.
  • Background systematic uncertainty = 2% log-normal
    Introduced in Section 3.2.3 to account for unknown systematics when setting 95% CL limits; affects the width of the expected limit bands.
  • Minimum vertex displacement = 40 μm
    Assumed in Section 3.3.2 as the smallest displacement for secondary vertex identification; directly sets the acceptance for displaced vertex decays.
assumptions (5)
  • domain assumption The SPSS benchmark model in the exact symmetry limit is a valid proxy for low-scale seesaw phenomenology.
    Used in Section 2 for all signal simulations; assumes lepton number conservation and mass-degenerate heavy neutrinos. The paper acknowledges that LNV effects can change some signatures (Section 3.4).
  • domain assumption The t-channel W-exchange production cross section (Eq. 7) is accurate at leading order.
    Heavy neutrino production is computed with this process only; NLO corrections are not applied, which could shift normalization and kinematics.
  • domain assumption The four background processes in Table 1 dominate the lepton-trijet background.
    Other backgrounds, such as triboson production, are argued to be negligible in Section 3.2.1.
  • ad hoc to paper Tau, B-meson, and cosmic-muon backgrounds can be completely suppressed for displaced vertex searches.
    Stated in Section 3.3.3 without dedicated simulation; the analysis assumes the vetoes and mass cuts are fully efficient.
  • ad hoc to paper A 40 μm displacement is sufficient to distinguish a secondary vertex from the primary vertex.
    Assumed in Section 3.3.2 based on tracking resolution (~8 μm) and interaction region spread; this threshold defines the minimum accepted decay length.

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Cite this review

Pith. "Pith review of Lepton-Trijet and Displaced Vertex Searches for Heavy Neutrinos at Future Electron-Proton Colliders." pith.science (2026). https://pith.science/paper/22DXR2GO

@misc{pith2026190802852,
  author       = {Pith},
  title        = {Pith review of: Lepton-Trijet and Displaced Vertex Searches for Heavy Neutrinos at Future Electron-Proton Colliders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/22DXR2GO}},
  note         = {Machine review of arXiv:1908.02852}
}
read the original abstract

Electron proton (ep) colliders could provide particle collisions at TeV energies with large data rates while maintaining the clean and pile~up-free environment of lepton colliders, which makes them very attractive for heavy neutrino searches. Heavy (mainly sterile) neutrinos with masses around the electroweak scale are proposed in low scale seesaw models for neutrino mass generation. In this paper, we analyse two of the most promising signatures of heavy neutrinos at ep colliders, the lepton-flavour violating (LFV) lepton-trijet signature and the displaced vertex signature. In the considered benchmark model, we find that for heavy neutrino masses around a few hundred GeV, the LFV lepton-trijet signature at ep colliders yields the best sensitivity of all currently discussed heavy neutrino signatures (analysed at the reconstructed level) up to now.

Figures

Figures reproduced from arXiv: 1908.02852 by the authors.

Figure 1
Figure 1. Left: Feynman diagram representing the leading order production channel for heavy neutrinos in electron￾proton scattering. Right: Cross section for heavy neutrino production in electron-proton collisions, divided by the active-sterile mixing paramter |θe| 2 . The latter channel, though suppressed by the parton distribution function of the photon within the proton, becomes increasingly important for larger centre-of-… view at source ↗
Figure 2
Figure 2. Kinematics of the heavy neutrino produced in electron-proton collisions at the LHeC (upper row) and at [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Kinematical distributions for the 6 signal benchmark points and all the backgrounds summed at the LHeC. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: All the plots use MN = 400 GeV and θe = θµ = 0.01, |θτ | = 0. Upper left: BDT distribution at the LHeC for both train and test samples superimposed. Bottom left: BDT distribution at the FCC-he for both train (black dotted distributions) and test (filled blue and red di…
Figure 5
Figure 5. Figure 5: Left: Expected limit on the production section times branching ratio of σ(pe− → Nj) × BR(N → µ−jj) when testing the signal hypotheses (for |θe| = |θµ| and |θτ | = 0) at LHeC (up) and FCChe(down). Right: Corresponding expected limit on the mixing parameters |θeθµ| when …
Figure 6
Figure 6. Figure 6: Parameter space giving rise to N = 3, , 10, 100 heavy neutrino decays with a displaced secondary vertex at the LHeC (left) and the FCC-he (right). The gray area denotes the best exclusion limits from the experiments from ATLAS [6], LHCb [36], LEP [2], and MEG [37]. In …
Figure 7
Figure 7. Figure 7: Sensitivity of the LFV lepton-trijet searches (at 95% C.L.) and the displaced vertex searches (at 95% [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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    1902 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.