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High-energy cLFV at $\mu$TRISTAN: HNL extensions of the Standard Model

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.04331 v2 pith:RLUEOYFO submitted 2024-12-05 hep-ph

J. Kriewald , E. Pinsard , A. M. Teixeira This is my paper · ORCID
classification hep-ph
keywords chargedleptonflavourviolationheavyneutralleptonsmuTRISTANinverseseesawpenguinandboxdiagramsforward-backwardasymmetryfuturecollidersneutrinomassmodels
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

4 major / 5 minor

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.

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 (4)
  1. [Sec. 5.2, Eq. (58), Figs. 12-13] 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.
  2. [Sec. 5.2] 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.
  3. [Sec. 5.2] 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.
  4. [Sec. 6 and Fig. 13] 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.
minor comments (5)
  1. [Eq. (58)] The printed expression 'S = S√S+B' should presumably read S=S/√(S+B); as typeset it is not a valid formula.
  2. [Figs. 12-13] 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.
  3. [Sec. 5.1, Eq. (51)] 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.
  4. [Sec. 5.1 and Fig. 10] 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.
  5. [Sec. 3 and Sec. 5.1] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the HNL benchmark parameters are a priori inputs, the ISS Yukawas are fixed by neutrino data, and the muTRISTAN sensitivity projection uses an explicit, un-fitted counting model.

full rationale

The paper's central derivation is self-contained. The scattering cross-sections and angular observables are computed analytically from the extended lepton mixing matrix and loop functions (Eqs. 5-39), with no target observable fitted. In the ad-hoc '3+2' model, the benchmark mixing angles in Eq. (51) are chosen by hand before any observable is computed; in the ISS(3,3) study, the Yukawa couplings are fixed by neutrino oscillation data through the modified Casas-Ibarra parametrization of Eq. (44), not by any muTRISTAN result. The low-energy cLFV rates and the high-energy cross-sections are independent functions of the same model parameters, so comparing their projected sensitivities is a legitimate model-level comparison, not a prediction equivalent to its input. The 'sensitivity' prescription of Sec. 5.2 (Eq. 58) is an assumed detector model: a 1% total signal efficiency, 10 signal events, and negligible background. These numbers are not derived from the cross-sections or from any fitted parameter, so the projection is not circular, although it is admittedly optimistic and fragile. The paper explicitly acknowledges that 'in the absence of concrete detector designs, a full study of possible backgrounds and systematics is clearly beyond the scope of this work.' The self-citations present in the manuscript ([66] for the finiteness of the Z-penguin, [84] for lepton flavour universality observables, and [71] for CPV phases) are technical references and do not carry the paper's central claim: the muTRISTAN reach and the comparison with FCC-ee and low-energy tau probes rest on the analytic cross-section computation and the explicit counting prescription, not on an unverified self-citation chain. No circular step satisfying the definitional or fitted-input patterns was found.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, or mediators are introduced; the study uses HNLs from the known seesaw landscape. The free parameters are model inputs (mixing angles, masses, mu_X, MR, m0) and analysis choices (efficiency, event threshold). The main axioms are the unitarity of the extended mixing matrix, the massless-external-fermion approximation, and the benchmark choices for the ISS scan.

free parameters (6)
  • ad-hoc benchmark mixing angles = s14=s15=5e-4, s24=s25=0.01, s34=s35=0.1
    Chosen by hand as an illustrative flavour pattern for the '3+2' model (Eq. 51); the headline event-rate claims in Sec. 5.1 are produced at this point.
  • HNL masses m4=m5 = 0.1, 0.5, 1, 5 TeV
    Degenerate masses assumed for the benchmark; Figs. 4-10 show cross-sections as a function of this choice.
  • signal efficiency = 1%
    Ad hoc conservative estimate combining tau-tagging (40%), geometric cuts (25%), and additional losses (Sec. 5.2); directly scales the projected sensitivity contours in Figs. 12-13.
  • minimum required signal events = 10
    Chosen threshold for sensitivity (S approximately 3 sigma, Eq. 58); determines the reach lines.
  • ISS(3,3) parameters MR, mu_X = MR in [100, 10^4] GeV, mu_X in [10^-10, 10^-4] GeV
    Scanned parameters of the inverse seesaw model (Sec. 5.2); the plotted sensitivity regions are functions of this plane.
  • lightest neutrino mass m0 = 10^-5 eV
    Input for the Casas-Ibarra parametrization, keeping normal ordering; adopted from oscillation fits.
assumptions (6)
  • domain assumption Leptonic mixing matrix U is (semi-)unitary; the would-be PMNS is non-unitary due to HNL mixing.
    Used to cancel divergences in B0/C00 combinations (Sec. 3, Eqs. 7-10); the unitarity of the 5x5 or larger U is assumed.
  • domain assumption External fermions are massless in the one-loop computation.
    Allows only three Lorentz structures (Eq. 2) and enables Fierz identities in Appendix C; valid for sqrt(s) much larger than m_l but neglects tau mass effects near threshold.
  • standard math SM gauge structure and Feynman rules extended with HNL vertices.
    Charged and neutral currents from Appendix A follow from the standard gauge Lagrangian with Majorana fermions, taken from ref. [68].
  • domain assumption The ad-hoc '3+2' model is a valid effective description with no mechanism for neutrino mass.
    The paper explicitly assumes no origin for the sterile masses (Sec. 4.1); the results are illustrative for generic HNL frameworks.
  • ad hoc to paper Casas-Ibarra parametrization with R=1 and degenerate MR reproduces neutrino oscillation data.
    Sec. 5.2; the choice R=1 is not required by data, and the assumption of universal MR and mu_X restricts the flavour structure scanned.
  • domain assumption Perturbative unitarity bound Gamma(Ni)/m_Ni < 1/2.
    Used to constrain allowed parameter space (Eq. 46), taken as a standard requirement for HNL masses above the weak scale.

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Cite this review

Pith. "Pith review of High-energy cLFV at $\mu$TRISTAN: HNL extensions of the Standard Model." pith.science (2026). https://pith.science/paper/RLUEOYFO

@misc{pith2026241204331,
  author       = {Pith},
  title        = {Pith review of: High-energy cLFV at $\mu$TRISTAN: HNL extensions of the Standard Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLUEOYFO}},
  note         = {Machine review of arXiv:2412.04331}
}
abstract

Within the context of heavy neutral lepton (HNL) extensions of the Standard Model, we compute the cross-sections for $\mu^+ e^-\to \ell_\alpha^+\ell_\beta^-$ scattering, as well as several angular observables. In particular, we investigate the future sensitivity of a $\mu$TRISTAN collider in discovering such charged lepton flavour violating processes and the potential constraining power of these searches on the parameter space of HNL models. Our results show that while low-energy probes of $\mu-e$ flavour violation do offer the most promising potential, the prospects for $e\tau$ and $\mu\tau$ flavour violation searches at $\mu$TRISTAN can exceed those of related low-energy probes (as well as flavour violating $Z$-pole processes at FCC-ee) by several orders of magnitude.

Figures

Figures reproduced from arXiv: 2412.04331 by the authors.

Figure 1
Figure 1. Feynman diagrams contributing to µ +e − → τ +e −, µ +e − → µ +τ − and µ +e − → ℓ +ℓ −. The shaded “blob” denotes the inclusion of the loop diagrams depicted in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. One-loop interactions contributing to the penguin-diagrams under consideration (see Fig. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Box diagrams contributing to µ +e − → ℓ + α ℓ − β . The upper row only contains lepton number conserving interactions, while the lower row also contains lepton violating vertices. B αβ 2 (s, t) = g 4 128π 2 X i,j U ∗ αi Uβj U ∗ ej Uµi F1(s, t), (18) B αβ 3 (s, u) = g 4 64π 2 X i,j U ∗ αj Uβi U ∗ ej Uµi F2(s, u), (19) B αβ 4 (t, u) = g 4 64π 2 X i,j U ∗ αj Uβi U ∗ ej Uµi F2(t, u), (20) F1(q 2 1 , q2 2 ) = 1 m4 W  q … view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Behaviour of the different cross-sections for [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Behaviour of the different cross-sections [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Behaviour of the differential distribution [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Forward-backward asymmetry with respect to the c.o.m energy for two choices of the HNL [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Forward-backward asymmetry with respect to the HNL masses for two choices of the c.o.m [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: Variation of the forward-backward asymmetries for [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Comparative prospects for cLFV processes: [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Comparative prospects for cLFV processes: [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Prospects for ISS(3,3) cLFV searches depicted in the ( [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Prospects for ISS(3,3) cLFV searches depicted in the ( [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Taming flavour violation in the Inverse Seesaw

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    New eta-based parametrisations of the ISS(3,3) show that Z-penguin flavour violation can reveal non-degenerate heavy sterile mixing even when radiative decays are absent.

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