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A same-sign muon collider could discover the singly-charged scalar of Type-II seesaw through a lepton-flavor-violating process, and the final lepton flavor could distinguish normal from inverted neutrino mass ordering for lightest neutrino mass below 0.02 eV.

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2026-08-04 06:14 UTC pith:6NXWVNND

load-bearing objection New channel and a plausible hierarchy test in Type-II seesaw at muTRISTAN, but the central 5σ claim for m_nu=0.05 eV rests on a Yukawa matrix that likely violates mu->e gamma. the 4 major comments →

arxiv 2601.22000 v2 pith:6NXWVNND submitted 2026-01-29 hep-ph

Discovery prospects of a singly-charged scalar at μTRISTAN

Joseph George , Nobuchika Okada , Dibyashree Sengupta , Sudhir K. Vempati This is my paper
classification hep-ph
keywords modelscalarsingly-chargedtristanarticlediscoveryleptonmass
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies a particle called Δ+, a singly charged scalar that appears in the Type-II seesaw model, one of the standard explanations for why neutrinos have mass. In this model a heavy scalar triplet couples to lepton pairs; the same couplings that generate neutrino masses also allow muons to turn into electrons or taus. The authors propose looking for Δ+ produced together with a W boson in collisions of two positive muons at the proposed μTRISTAN collider, with the Δ+ decaying into an electron or tau plus a neutrino. Because lepton flavor violation is essentially absent in the Standard Model, such events would be very clean.

Using Monte Carlo simulation, the authors compute cross-sections and branching ratios for two benchmark neutrino masses and several Δ+ masses. They find that with 30 inverse femtobarns, the signal would reach 5σ significance over a wide mass range. They also propose a way to determine whether neutrinos follow normal or inverted mass ordering: the relative number of electrons versus taus from the Δ+ decay depends on the ordering, provided the lightest neutrino is below about 0.02 eV. A major caveat is that the benchmark with a 0.05 eV lightest neutrino implies large lepton-flavor-violating Yukawa couplings, and the paper does not check whether those couplings are already excluded by μ→eγ searches. The zero-background assumption also ignores detector fake rates, which the authors themselves mention.

Core claim

In the μ+μ+ mode of μTRISTAN at √s=2 TeV, associated production μ+μ+→Δ+W+ followed by Δ+→e+ν/τ+ν and W+→jj yields a background-free LFV signal with 5σ significance over essentially the entire mass range 101–1901 GeV for mν_lightest=0.05 eV (and fragmented coverage for 0.001 eV); the ratio of electron to tau final states can distinguish Normal from Inverted hierarchy for mν_lightest≲0.02 eV.

Load-bearing premise

The benchmark choice vΔ=10^-9 GeV with mν_lightest=0.05 eV gives Yαβ~mν/(√2 vΔ) ~ O(0.03) (Eq. 2.6). The paper assumes these large Yukawa couplings evade charged-lepton flavor-violation bounds (μ→eγ, μ→3e), citing reviews [18,19] but never enforcing them. If μ→eγ excludes this region, the 5σ discovery claim for the 0.05 eV benchmark collapses, because the signal itself is an LFV process controlled by the same Yukawa matrix.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper studies the process mu+ mu+ -> Delta+ W+ at the muTRISTAN same-sign muon collider (sqrt(s)=2 TeV), in the Type-II seesaw model. The Delta+ is assumed to decay to e+ nu or tau+ nu, giving a charged-lepton-flavor-violating signature with negligible SM background; the W+ is taken to decay to a jet pair. Cross sections and significances are computed for two benchmarks (lightest neutrino mass 0.05 eV and 0.001 eV, v_Delta=10^-9 GeV) with Delta+ masses from 101 to 1901 GeV. The paper claims 5-sigma reach over essentially the whole mass range for m_lightest=0.05 eV and a partial reach for 0.001 eV, and proposes that the electron/tau flavor ratio of the lepton from Delta+ decay can discriminate normal from inverted mass ordering for m_lightest <= 0.02 eV. The simulation chain is standard (MadGraph/Pythia/Delphes), with analytical helicity amplitudes for the 2-to-2 process collected in Appendix C.

Significance. If the central claim survives scrutiny, the paper would be a useful addition to the muTRISTAN physics case: a single LFV process would both discover the singly charged triplet scalar and, for sufficiently small lightest neutrino mass, indicate the mass ordering. The calculation is internally consistent and no circularity was found: the Yukawa matrix is fixed by Eq. (2.6) and the hierarchy discrimination is an output, not an input. The analytical amplitudes in Appendix C are a useful check for the numerical simulation. However, the discovery claim is conditional on benchmark parameters that are not tested against charged-lepton-flavor-violation data; this is the main weakness and must be resolved before the reach statement can be accepted.

major comments (4)
  1. [Section 2, Eq. (2.6); Section 3, Tables I/II] The central benchmark v_Delta=10^-9 GeV with m_lightest=0.05 eV forces Y ~ m_nu/(sqrt(2) v_Delta) ~ O(0.03). The same Yukawa matrix controls the collider LFV signal and one-loop mu->e gamma (and mu->3e) transitions. The paper cites Ref. [18,19] for LFV constraints but never applies them numerically. For the low-mass points in Table I, e.g. m_Delta++=103 GeV, even a modest |(Y^dagger Y)_e mu| ~ 10^-4 yields BR(mu->e gamma) at the 10^-10 to 10^-9 level, well above the MEG bound 3.1e-13. The low-mass benchmark points are therefore not established as allowed, and since they are the ones that make the 'full mass range' 5-sigma claim, that claim is unsupported until a dedicated cLFV calculation is done. Partial cancellations in the loop may exist, but no calculation is shown.
  2. [Section 3, Figs. 3 and 6] The scan starts at m_Delta++=103 GeV and the abstract advertises a 5-sigma reach over 101-1901 GeV. Existing collider searches for doubly charged scalars in the Type-II seesaw with small v_Delta (same-sign dilepton final states at the LHC) already exclude m_H++ below roughly 700-800 GeV for this parameter region. The manuscript mentions existing collider constraints in the introduction (Ref. [16,17]) but does not apply them to its own mass scan. Unless the low-mass portion of the scan is shown to be allowed, the 'entire mass range' statement should be restricted to the not-yet-excluded interval or the low-mass benchmark points must be justified as a future-collider-only projection.
  3. [Section 4, Eq. (4.9)] The significance is computed as S/sqrt(S) for a channel with zero SM background. For a counting experiment with B=0, the Gaussian S/sqrt(S) formula is not the correct frequentist significance, and for the smaller event numbers in Table II (some rows correspond to S~10-20 events) it can be numerically misleading. The footnote acknowledges that Poisson statistics should be used, but the paper still uses S/sqrt(S) for all quoted significances. A proper treatment with an explicit (even small) background estimate and Poisson likelihood would be more appropriate, especially for the 95% CL statements.
  4. [Section 5, Figs. 14-19] The claim that normal and inverted hierarchy can be distinguished for m_lightest <= 0.02 eV is based on a visual comparison of electron and tau pseudorapidity/acceptance histograms. No test statistic, expected significance, or systematic uncertainty (tau identification efficiency, electron fake rate, charge misidentification) is provided. At high Delta+ masses the number of surviving events is small, so the observed flavor asymmetry must be quantified before the proposed discrimination method can be considered established. This is load-bearing for the second advertised result.
minor comments (3)
  1. [Section 3, Tables I and II] The distinction between sigma_Production and sigma (the cross section after decays) is not fully defined. It would help to state explicitly which branching ratios (W+ -> jj, Delta+ -> e+ nu/tau+ nu) are included in sigma and whether tau decays and tau-tagging efficiency are folded in.
  2. [Section 4] The text says the final state should contain exactly one charged lepton, yet many simulated events have zero leptons. This is attributed to pseudorapidity coverage, but it would be clearer to separate generation-level selection from detector acceptance and to show the acceptance times efficiency explicitly.
  3. [Introduction] There is a typo in the Introduction: 'Matter-Antimatter assymmetry' should be 'asymmetry'.

Circularity Check

0 steps flagged

No significant circularity: the predictions follow from externally fixed Type-II seesaw inputs.

full rationale

The derivation is self-contained. The Yukawa matrix is fixed by the type-II seesaw relation m_nu = sqrt(2) Y v_Delta (Eq. 2.6) from chosen benchmark values of v_Delta and the lightest neutrino mass, via PMNS diagonalization; the Delta+ production cross-section and leptonic branching ratios are then computed from that Y in the standard Type-II seesaw framework. The NH/IH discrimination is genuinely predictive: the branching-ratio flavor asymmetry (more tau for NH, more e for IH) follows from the assumed PMNS structure and is not imposed to fit the signal. No parameter appearing in the claimed 5-sigma reach is fitted to the signal; the LFV final state is the predicted consequence of the same Y matrix. The citations to the Type-II seesaw [12-15], muTRISTAN [26], and LFV reviews [18,19] are external support, not self-referential. The only caveat, that mu->e gamma constraints are cited but not numerically enforced, is a physics-validity/correctness concern for the benchmark, not a circularity, since enforcing those constraints would rescale or exclude an externally fixed input rather than redeclare an output as an input.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

No new particles, forces, or symmetries are introduced; the Δ+ scalar is part of the pre-existing Type-II seesaw model. The load-bearing free choices are vΔ, the lightest neutrino mass, and the triplet mass splitting, all set by hand. The most fragile assumption is that the resulting Yukawa matrix satisfies charged-lepton flavor-violation bounds, which the paper never checks.

free parameters (3)
  • vΔ (triplet VEV) = 1e-9 GeV
    Chosen by hand so that Δ+ decays predominantly leptonically; no scan over vΔ under LFV constraints is provided (Sec. 3).
  • mν_lightest (lightest neutrino mass) = 0.05 eV and 0.001 eV
    Two benchmark inputs; the 0.05 eV case yields Yukawa couplings O(0.03) that may violate μ→eγ bounds.
  • λ4 (mass-ordering parameter) = negative, chosen so m_H0≈m_A0 < m_H+ < m_H++
    Sets the 1–2 GeV triplet mass splittings; the value is introduced ad hoc and affects kinematics, though not the central LFV mechanism.
axioms (5)
  • domain assumption Type-II seesaw scalar sector with one SU(2)_L triplet Δ(Y=1) and the standard scalar potential
    The entire analysis is framed in this model (Sec. 2, Eqs. 2.2–2.5).
  • domain assumption Neutrino mass matrix mν=√2 Y vΔ, diagonalized by the PMNS matrix using standard three-neutrino mixing
    Used to set the Yukawa matrix (Eqs. 2.6–2.7); the paper does not specify the PMNS values it uses beyond the hierarchy.
  • domain assumption Standard Model has no lepton flavor violation, so the e+/τ+ final state has zero SM background
    Used for the S/√S significance calculation (Sec. 4); ignores small neutrino-oscillation-induced LFV and detector fakes.
  • domain assumption The default muon collider Delphes card adequately represents the μTRISTAN detector
    All detector-level distributions rely on this card (Sec. 4); the real detector is not built.
  • domain assumption vΔ≪v limit and mass-eigenstate relations in Eq. (2.8)
    Used to identify H± with the triplet state and to set the small mass splittings; standard in Type-II seesaw literature.

pith-pipeline@v1.3.0-alltime-deepseek · 15835 in / 20102 out tokens · 223551 ms · 2026-08-04T06:14:34.463238+00:00 · methodology

0 comments
read the original abstract

In this article, we study the associated production of a singly-charged ($\Delta^+$) scalar along with a $W^+$ boson in the newly proposed $\mu^+\mu^+$ collider (also known as $\mu$TRISTAN) at $\sqrt{s} = 2~$ TeV. Such a singly-charged scalar is naturally accommodated in an extremely well-motivated neutrino mass model, namely, the Type-II seesaw model. This model, beside providing a viable explanation of neutrino mass generation, also allows for lepton flavor violating (LFV) processes. Since LFV processes are not allowed in the Standard Model (SM), we focus on the discovery prospect of the singly-charged scalar in the Type-II seesaw model at $\mu$TRISTAN through a LFV process, owing to the advantage of this process being free of any SM background. Additionally, this article also proposes a method to indicate if the underlying theory follows a Normal or an Inverted hierarchy depending on the distribution of lepton flavors in the final state.

Figures

Figures reproduced from arXiv: 2601.22000 by Dibyashree Sengupta, Joseph George, Nobuchika Okada, Sudhir K. Vempati.

Figure 1
Figure 1. Figure 1: Generic Feynman diagram for signal If the lepton arising from the decay of the ∆+ is a µ +, then the process in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Feynman diagram for lepton flavor violating (LFV) signal processes chosen to get [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cross-Section for the process µ +µ + −→ ∆+W+ for masses of ∆+ varying between 101 and 1901 GeV. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Branching Ratio of ∆+ for masses of ∆+ varying between 101 and 1901 GeV. (a) Normal Hierarchy (b) Inverted Hierarchy [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Branching Ratio of ∆+ for masses of the lightest neutrino varying between 0.05 and 0.001 eV for a fixed m∆+ = 101 GeV. 4 Signal analysis and Mass reach of ∆+ All benchmark points have been generated using the UFO model [40] and simulated us￾ing Madgraph [41] followed by showering in Pythia [42] and detector level analysis in Delphes [43] using the default Muon collider delphes card. The total cross-section… view at source ↗
Figure 6
Figure 6. Figure 6: Plot of (a) Cross-section and (b) Significance of the signal process (shown in Fig. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Distribution for a) Electrons in the final state and b) Taus in the final state for [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Distribution for a) Electrons in the final state and b) Taus in the final state for [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution for a) Electrons in the final state and b) Taus in the final state for [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution for a) Electrons in the final state and b) Taus in the final state for [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Distributions of Electrons and Taus for mνlightest = 0.05 eV and m∆+ = 101 GeV assuming a) Normal Hierarchy and b) Inverted Hierarchy (a) Normal Hierarchy (b) Inverted Hierarchy [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Distributions of Electrons and Taus for mνlightest = 0.05 eV and m∆+ = 1001 GeV assuming a) Normal Hierarchy and b) Inverted Hierarchy (a) Normal Hierarchy (b) Inverted Hierarchy [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Distributions of Electrons and Taus for mνlightest = 0.05 eV and m∆+ = 1901 GeV assuming a) Normal Hierarchy and b) Inverted Hierarchy evident from [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Distributions of Electrons and Taus for mνlightest = 0.02 eV and m∆+ = 101 GeV assuming a) Normal Hierarchy and b) Inverted Hierarchy (a) Normal Hierarchy (b) Inverted Hierarchy [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Distributions of Electrons and Taus for mνlightest = 0.02 eV and m∆+ = 1001 GeV assuming a) Normal Hierarchy and b) Inverted Hierarchy 6 Summary and Outlook Although hadron colliders such as the LHC are extremely advantageous in looking for new BSM particles produced via direct production, owing to much higher c.o.m energy as com￾pared to the c.o.m energy in lepton colliders, a lepton collider enjoys a mu… view at source ↗
Figure 16
Figure 16. Figure 16: Distributions of Electrons and Taus for mνlightest = 0.02 eV and m∆+ = 1901 GeV assuming a) Normal Hierarchy and b) Inverted Hierarchy (a) Normal Hierarchy (b) Inverted Hierarchy [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Distributions of Electrons and Taus for mνlightest = 0.001 eV and m∆+ = 101 GeV assuming a) Normal Hierarchy and b)Inverted Hierarchy (a) Normal Hierarchy (b) Inverted Hierarchy [PITH_FULL_IMAGE:figures/full_fig_p015_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Distributions of Electrons and Taus for mνlightest = 0.001 eV and m∆+ = 1001 GeV assuming a) Normal Hierarchy and b)Inverted Hierarchy energy. Several lepton colliders such as e +e − collider, µ +µ − collider, µ +µ + collider, µ +e − 15 [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Distributions of Electrons and Taus for mνlightest = 0.001 eV and m∆+ = 1901 GeV assuming a) Normal Hierarchy and b)Inverted Hierarchy collider have been proposed to build. However, in this article we focus on the µ +µ + mode of the µTRISTAN [26] that is planned to operate at √ s = 2 TeV. The reasons behind choosing this particular setup among several proposed lepton colliders are two fold: i) Muon collid… view at source ↗

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