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REVIEW 4 major objections 5 minor 92 references

Probing the vector-like $X$ quark via the $tW$ channel at future muon-proton colliders

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A future muon–proton collider could exclude the vector-like X quark — a hypothetical +5/3-charge partner of the top quark — for masses up to about 8.3 TeV through its tW decay channel, at couplings far below today's LHC reach.

desk verdict Solid, standard simplified-model projection for VLX at muon-proton colliders; headline reach plausible but the paper omits the luminosity used for its central numbers, and much of the scanned parameter space is already excluded by the APV bound it relaxes. read the letter →

arxiv 2607.18843 v1 pith:PTPK3MV5 submitted 2026-07-21 hep-ph

Liangliang Shang , Shuting Zhao , Bingfang Yang , Stefano Moretti This is my paper · ORCID
classification hep-ph
keywords vector-likequarkXmuon-protoncollidertWchannelsingleVLQproductionboostedfatjetsbeyondStandardModelAsimovsignificance
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

This paper argues that a future muon–proton collider could find or rule out the vector-like X quark, a hypothetical new fermion of electric charge +5/3, produced singly through μ+p → ν̄μ X with X → tW+. Working in a simplified model with three free parameters — the coupling strength g*, the generation-mixing ratio R_L, and the mass m_X — the authors simulate all four decay topologies of the tW final state at collision energies of 5.29, 6.48, and 9.16 TeV, finding the fully hadronic channel the most sensitive: a 2σ exclusion at m_X ≃ 8.3 TeV with g* = 0.009 for R_L = 0.1 at 9.16 TeV. If correct, this means a muon–proton collider could probe a high-mass, weak-coupling regime where current LHC searches lose sensitivity, at masses several TeV beyond present limits. The reach narrows, however, if the atomic parity violation bound — which the paper deliberately sets aside when scanning g* up to 0.5 — is enforced.

What carries the argument

The central object is the effective Lagrangian for the X–W–quark vertex: LX = g* [ √(R_L/(1+R_L)) (g/√2) X̄_L W+_μ γ^μ u_L + √(1/(1+R_L)) (g/√2) X̄_L W+_μ γ^μ t_L ] + h.c., which concentrates all model dependence into the coupling strength g* and the generation-mixing ratio R_L. Detection rides on the charged-current single-production process μ+p → ν̄μ X and the subsequent decay X → tW+, whose final state is partitioned into four channels — fully leptonic, fully hadronic, and two semi-leptonic modes — according to how the two W bosons decay. The fully hadronic channel is reconstructed with large-radius (fat) jets plus a W-mass window, which is what lets it exploit the highly boosted topology

What would settle it

Two checks would settle the matter: re-drawing the exclusion contours with the APV constraint g* √(R_L/(1+R_L)) < 6.5×10⁻² enforced — if the g* > 0.1 regions vanish, the broad-reach claim is falsified; and a 9.16 TeV muon–proton run in the fully hadronic channel with about 100 fb⁻¹, which would either confirm the predicted m_X ≈ 7.3 TeV 5σ discovery or rule out the projection with a null result.

Watch

Extended reading notes

Core claim

The paper's central claim is that single production of the vector-like X quark at a muon–proton collider, followed by X → tW+ decay, is observable over a broad parameter region, with the fully hadronic final state the most powerful of the four channels studied. With an 80% polarized muon beam and fat-jet reconstruction of the boosted hadronic W bosons, the projected 2σ exclusion reaches m_X ≃ 8.3 TeV at √s = 9.16 TeV for g* = 0.009 and R_L = 0.1, and the corresponding 5σ discovery reaches m_X ≃ 7.3 TeV at g* = 0.016. Sensitivity improves monotonically with centre-of-mass energy and with R_L, and the two semi-leptonic modes sit between the hadronic and leptonic extremes. The paper states in S

Load-bearing premise

The claim of a 'broad region' of reachable parameter space rests on relaxing the atomic parity violation bound that constrains the X–u–W coupling; if that low-energy bound is enforced, most of the scanned coupling range g* ≳ 0.1–0.2 is already excluded, and only the low-coupling benchmarks near g* = 0.009 survive.

Editorial extensions

If this is right

  • At the highest energy considered (√s = 9.16 TeV), a null result in the fully hadronic channel would exclude X-quark masses up to about 8.3 TeV at 2σ for R_L = 0.1, several TeV beyond present LHC bounds.
  • A 5σ discovery in the same channel would be possible up to m_X ≃ 7.3 TeV, opening a multi-TeV window that pair-production searches cannot reach for weakly coupled states.
  • Because the reach extends down to g* ≈ 0.006–0.016, the analysis covers couplings far below the κ > 0.16–0.2 range probed by current LHC single-production searches.
  • Sensitivity scales strongly with centre-of-mass energy and with R_L, so the same analysis design transfers directly to any future upgrade of the muon beam energy.
  • The fully leptonic mode, despite its clean signature, has the weakest mass reach, so a complete programme would need all four channels to cover the parameter space.

Reading between the lines

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

  • Enforcing the atomic parity violation bound instead of relaxing it would not destroy the headline benchmark — g* = 0.009, R_L = 0.1 sits safely below the limit — but it would erase most of the g* ≳ 0.1 exclusion contours, so the 'broad region' claim should be read as conditional on that relaxation.
  • The analysis uses sequential cuts; replacing them with a multivariate discriminator over the same boosted-jet observables would likely push the reach beyond what any single cut optimization achieves here.
  • The same effective-Lagrangian structure and four-channel final-state decomposition transfers to the vector-like T and B quarks and to the (X,T) doublet representation, making this a template for a broader vector-like quark programme at a muon–proton collider.
  • Since production is mediated by a W boson from the muon, the attainable mass limit inherits the proton PDF behaviour; a higher-energy proton beam, or a larger R_L, would push the exclusion boundary toward 10 TeV, as Tables VI–IX already hint.
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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. The paper studies single production of a vector-like X quark (charge +5/3) in a simplified triplet model at future muon-proton colliders, via the process μ+p → ν̄_μ X, X → tW+. The model parameters are the coupling g*, the generation-mixing parameter R_L, and the mass m_X. The analysis simulates four final-state channels (FL, FH, SL1, SL2) with MadGraph5_aMC@NLO, Pythia 8, and Delphes, uses fat-jet reconstruction and 80% muon beam polarization, and computes Asimov exclusion/discovery significances at √s = 5.29, 6.48, and 9.16 TeV. The central claim is that the FH channel is the most sensitive, reaching a 2σ exclusion of m_X ≃ 8.3 TeV at g* = 0.009, R_L = 0.1, and √s = 9.16 TeV. The paper explicitly relaxes the APV constraint and scans g* up to 0.5.

Significance. Future μp collider projections for a vector-like X quark are timely, and the paper offers a genuine phenomenological study with a full simulation chain, external LHC/EWPO comparisons, and a four-channel analysis. If the quantitative claims are correct, the proposed search would substantially extend the current and HL-LHC reach. However, the numerical results are not currently reproducible or fully verifiable because the integrated luminosity and reference g* used in the reach calculation are not stated, and at least one quoted mass point lies at the kinematic boundary. The APV relaxation is disclosed, but the abstract's 'broad region' wording overstates what remains allowed under existing constraints.

major comments (4)
  1. [Section V, Eqs. (8)-(9); Tables VI-IX and II-V] The Asimov significances require event yields s and b, which depend on the integrated luminosity L. The paper never states the L used for the reach curves; Sec. I only mentions 'target integrated luminosities ranging from 10 to 100 fb−1'. In addition, the cut-flow tables quote only 'Signal(fb)' without the reference g* value. For example, the 23.99 fb after all cuts for X2500 at √s = 9.16 TeV in Table XIV cannot be mapped to the g* axis of Tables VI-IX without knowing the reference coupling. Without specifying L and g*_ref, the g* lower bounds in Tables VI-IX cannot be verified. Please provide these inputs explicitly, or report absolute event yields.
  2. [Table VII, FH row at √s = 9.16 TeV] For R_L = 0.3 and 0.5, the 2σ exclusion upper limit is quoted as m_X = 9160 GeV, equal to the nominal √s. At m_X = √s, the subprocess requires the proton parton momentum fraction x ≈ 1, where the PDF vanishes; the production cross section is zero. Thus the reach cannot extend to 9160 GeV. The threshold and PDF suppression should be handled explicitly, and the kinematic edge must be reflected in the reported mass limits.
  3. [Sec. II, Fig. 1 and Figs. 11-14; Tables VI-IX] The APV constraint g* sqrt(R_L/(1+R_L)) < 6.5×10^-2 is relaxed, as stated in Sec. II. For R_L = 0.5 this bound gives g* ≲ 0.11, and for R_L = 0.1 it gives g* ≲ 0.22. Consequently, large parts of the displayed parameter space with g* up to 0.5 are excluded under that bound, and the abstract's 'broad region' claim depends on the relaxation. Please overlay the APV-excluded region in the reach plots or explicitly qualify that the displayed regions above the bound are only valid once the APV constraint is dropped.
  4. [Sec. IV.B and Appendix Tables XI-XVIII] The text states that 'the same event selection is adopted' for the higher center-of-mass energies, but the appendix cut-flow tables use different selection criteria. For example, the FH selection at 9.16 TeV (Table XIV) uses HT > 800 GeV and p_bT > 500 GeV, whereas at 5.29 TeV (Table III) it uses HT > 900 GeV and p_bT > 400 GeV. Similar differences appear in the FL pseudorapidity windows and the SL1 η_b window. Please correct the text or document the exact selection used at each energy.
minor comments (5)
  1. [Section V, Eqs. (8)-(9)] The Asimov significances include only statistical uncertainties. No background systematic uncertainty is incorporated. Please state explicitly that the contours are statistical-only, or include a nuisance-parameter treatment if systematics are intended.
  2. [Throughout] 'Branch ratio' should be 'branching ratio' in several places (e.g., Sec. I and Table captions).
  3. [Table VII] In the 5.29 TeV, R_L = 0.5 row, the 2σ upper edge of g* is 0.29, which is lower than the R_L = 0.3 value. Please check whether this is a scan-range clipping or a physical NWA/width cut.
  4. [Sec. III] A summary of the Delphes detector parameters (b-tagging efficiency, jet radius, jet energy scale) would improve reproducibility.
  5. [Figs. 11-15] The Γ_X/m_X contours are used to justify the NWA, but the actual contour values are not stated in the text. Please provide the relevant Γ_X/m_X values for the benchmark points.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the reach projections are forward MC calculations over scanned model parameters, not fits to their own outputs.

full rationale

The derivation chain is a standard forward calculation: the simplified Lagrangian Eq. (6) defines the coupling structure with scanned inputs g*, R_L, and m_X; the width is computed analytically in Eq. (7); signal and background cross sections are obtained from MadGraph/Pythia/Delphes (Section III); cut flows are given in Tables II–V and XI–XVIII; and the Asimov significances in Eqs. (8)–(9) convert the resulting yields s and b into the reach contours and Tables VI–IX. No target quantity is fitted from the data to produce another target quantity: the coupling lower bounds and mass reaches are outputs of the significance calculation, not inputs to it. The explicit relaxation of the APV bound ('we relax this strict APV bound in our collider simulations for the sake of exploring full phenomenological projections, and scan the broader parameter space of g* ≤ 0.5 and R_L ≤ 1', Section II) is a stated modeling choice that weakens an external constraint; it is not circular. The self-citations [63] and [84] are used only to draw shaded existing-constraint regions (EWPO and LHC single-production) in the comparison figures and are not load-bearing for the headline FH reach, which is computed from the MC yields. The NWA validity check overlaying Gamma_X/m_X contours uses the authors' own computed width, but this is a self-consistency check, not a derivation of the reach. The main caveat is reproducibility rather than circularity: the integrated luminosity used for the reach curves is never stated, and the reference g* value entering the signal cross sections in the cut-flow tables is not given, so the numerical values in Tables VI–IX cannot be independently regenerated from the paper alone. This is an omitted-input/reproducibility issue, not an equivalence between prediction and input.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The reach predictions are computed from an assumed simplified Lagrangian with three scanned parameters. The main external anchors are LHC and EWPO constraints, but one important external constraint (APV) is consciously relaxed, so the large-g* scan is a projection rather than a prediction of a derived model.

free parameters (4)
  • g*
    Overall coupling strength in the effective Lagrangian; scanned up to 0.5; no model prediction for its value.
  • R_L
    Generation-mixing ratio between first- and third-generation couplings; scanned over 0.1–0.5 to favor X→tW.
  • m_X
    VLX mass scanned from 1000 to 9160 GeV; the upper scan edge appears kinematically questionable.
  • P_eff (muon beam polarization) = 80%
    Chosen to improve signal-to-background; the reach numbers depend on this assumed beam performance.
assumptions (5)
  • ad hoc to paper Effective Lagrangian Eq. (6) with only X-u-W and X-t-W couplings and no other decay modes.
    Sets the production and decay chain; all cross sections and branching ratios follow from this assumed structure.
  • ad hoc to paper The atomic parity violation constraint g* sqrt(R_L/(1+R_L)) < 6.5e-2 is relaxed in the simulations.
    Stated in Section II; allows g* up to 0.5 despite APV excluding g* > ~0.22 (R_L=0.1) and > ~0.11 (R_L=0.5).
  • domain assumption Narrow-width approximation is valid; off-shell and interference effects are neglected.
    Width-to-mass contours are shown, but full off-shell/interference effects are not included; NWA references are cited.
  • domain assumption DELPHES generic detector response approximates a future muon–proton detector.
    No detector design exists; acceptance, tagging, and jet reconstruction are modeled with a generic simulation.
  • domain assumption Leading-order cross sections with default PDFs and dynamic renormalization/factorization scales.
    No NLO QCD corrections or PDF/scale uncertainties are applied, which affects absolute yields and reach.
invented entities (1)
  • Vector-like X quark (VLX) independent evidence
    purpose: Signal particle with charge +5/3 in an (X,T,B) triplet; produced via W exchange and decaying as X→tW.
    The paper gives falsifiable reach/exclusion contours in the (g*, R_L, m_X) parameter space, and LHC constraints are used as external benchmarks. The VLX is assumed from prior literature, not newly introduced.

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Pith. "Pith review of Probing the vector-like $X$ quark via the $tW$ channel at future muon-proton colliders." pith.science (2026). https://pith.science/paper/PTPK3MV5

@misc{pith2026260718843,
  author       = {Pith},
  title        = {Pith review of: Probing the vector-like $X$ quark via the $tW$ channel at future muon-proton colliders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTPK3MV5}},
  note         = {Machine review of arXiv:2607.18843}
}
abstract

We investigate the discovery potential for the vector-like $X$-quark (VLX) at future muon--proton ($\mu p$) colliders through the process $\mu^+ p \to \bar{\nu}_\mu X \to \bar{\nu}_\mu t W^+$. A simplified effective model is adopted in which the production and decay of the VLX are governed by the coupling strength $g^{*}$, the generation-mixing parameter $R_{L}$, and the VLX mass $m_X$. A comprehensive Monte Carlo analysis is performed at $\sqrt{s}=5.29$, $6.48$, and $9.16\ \mathrm{TeV}$, considering four complementary decay channels: the Fully Leptonic (FL), Fully Hadronic (FH), and two Semi-Leptonic (SL1 and SL2) modes. An $80\%$ polarized muon beam together with boosted-object reconstruction based on fat-jet techniques is employed to improve the signal sensitivity. The expected exclusion and discovery reaches are evaluated using the Asimov significance. We find that the sensitivity can be improved substantially with increasing center-of-mass energy and larger values of $R_L$. Among the four channels, the FH mode provides the strongest sensitivity, reaching a $2\sigma$ exclusion limit of $m_X\simeq8.3\ \mathrm{TeV}$ with $g^* = 0.009 $ for $R_L=0.1$ at $\sqrt{s}=9.16\ \mathrm{TeV}$, whereas the FL mode gives the weakest reach because of its smallest branch ratio. These results demonstrate that future $\mu p$ colliders can offer significant sensitivity to heavy VLX over a broad region of parameter space.

Figures

Figures reproduced from arXiv: 2607.18843 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
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Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
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Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
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Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
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Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
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Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
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Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
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Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p024_12.png]
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Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
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Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
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Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]

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Reference graph

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Pith tools

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