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The paper claims that a future muon-proton collider at √s = 9.16 TeV and 3000 fb⁻¹ can discover singly produced vector-like T quarks up to about 3.5 TeV, with a 3 TeV quark at 21.86σ (hadronic) and 3.75σ (leptonic).

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A simulated muon-proton collider search claims up to ~3.5 TeV discovery reach for a singly produced vector-like top quark, but its central tables are inconsistent.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection A plausible muon-proton VLQ search idea undermined by inconsistent tables and an unspecified benchmark coupling; the 3.5 TeV claim does not follow from the paper's own numbers. the 5 major comments →

arxiv 2512.11471 v3 pith:7CLJQQPR submitted 2025-12-12 hep-ph

Probing Vector-Like Quarks at a future Muon-Proton Collider

Mudassar Hussain , Ijaz Ahmed , Tayyab Javaid , Haroon Saghir , Jamil Muhammad This is my paper
classification hep-ph PACS 12.60.Fr14.80.Fd
keywords vector-like quarksmuon-proton collidersingle productionT→Wb decaysignal significanceboosted decision treemulti-layer perceptronbeyond Standard Model
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 reading

Vector-like quarks—hypothetical heavy partners of the Standard Model quarks—appear in many extensions of the Standard Model but have not been seen, and current collider searches only exclude pair-produced masses below roughly 1.4–1.5 TeV. This paper argues that a future muon-proton collider would access them through single electroweak production, where the muon's W boson scatters off a quark in the proton to produce a T quark alongside a b quark. Using an effective Lagrangian with couplings consistent with CKM and electroweak constraints, the authors simulate the T → Wb signal and the dominant Standard Model backgrounds, and find that at the highest benchmark energy (√s = 9.16 TeV) with 3000 fb⁻¹, a 3 TeV T quark would be observable at 21.86σ in the hadronic channel and 3.75σ in the leptonic channel. They conclude that masses up to roughly 3.5 TeV are within discovery reach, well beyond the reach of current proton-proton machines. The analysis also shows that a neural-network classifier improves hadronic signal purity by about 2.6 times over a boosted decision tree, hinting that further reach may be gained with more advanced analysis.

Core claim

The central claim is that single production of a vector-like T quark in muon-proton collisions—via t-channel W exchange followed by T → Wb—is an experimentally viable discovery channel, and that a muon-proton collider at √s = 9.16 TeV with 3000 fb⁻¹ can discover such a quark up to m_T ≈ 3.5 TeV. The quantitative evidence: at m_T = 3 TeV, the hadronic channel gives S/B ≈ 0.32 and S/√B ≈ 21.86, while the leptonic channel gives ≈ 3.75σ; at 1.5 TeV the hadronic significance reaches about 660σ. The authors frame this as extending the multi-TeV reach beyond the pair-production limits of current colliders, where single production becomes the dominant mode. All rates scale as (g*)² with the effectiv

What carries the argument

The central object is an effective Lagrangian for single T production and decay, parameterized by mixing matrices and an overall effective coupling g*. Production proceeds through t-channel W exchange (µp → νµ T b), and the cross-section scales as σ ∝ (g*)². The dominant decay T → Wb defines the search topology; a cut-based selection using high-pT central jets, b-tagging, HT > 800 GeV, missing-energy windows, and angular separations isolates the signal from the νtb, νWj, νZj, and νjj backgrounds. The same kinematic variables feed Boosted Decision Tree and Multi-Layer Perceptron classifiers, compared via S/B and S/√(S+B).

Load-bearing premise

The entire projection rests on the value of the effective coupling g* used to generate the cross-sections; the paper never states that number, and if the true coupling is even a factor of two smaller, the 3 TeV significance drops from 21.86σ to roughly 5.5σ and the 3.5 TeV reach is lost.

What would settle it

Recover the benchmark coupling: compute the leading-order cross-section for µp → νµ T b at √s = 9.16 TeV for m_T = 1.5 TeV from the model Lagrangian using a single explicit g*, and compare with the paper's Table I value of 594.9 fb. If no single g* within CKM bounds reproduces the normalization and mass-dependence of all tabulated cross-sections simultaneously, the significance projections are not reproducible.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the 3.5 TeV reach holds, a muon-proton collider becomes a direct probe of TeV-scale vector-like quarks, complementing and extending the pair-production limits of current hadron colliders.
  • The hadronic channel, not the leptonic one, supplies the primary discovery significance at high mass, so detector design should emphasize high-pT jet reconstruction and b-tagging performance.
  • The leptonic channel, though weaker, provides a clean cross-check signal at the same mass points, allowing confirmation across independent final states.
  • Because all rates scale as (g*)², the same analysis can be inverted: a null result at 3000 fb⁻¹ would translate into upper limits on the product of the effective coupling and the T → Wb branching fraction as a function of m_T.
  • The MLP classifier improves hadronic purity by roughly a factor of 2.6 over the BDT at fixed statistical significance, suggesting that the effective discovery reach could be extended beyond 3.5 TeV with multivariate analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper's headline reach is conditional on an unstated numerical value of g*; a more informative presentation would show the discovery mass as a function of g*, since the reach roughly scales with the square root of the coupling.
  • The analysis uses leading-order event generation and a generic detector simulation; higher-order QCD and electroweak corrections typically soften sharp distributions and could shift both the efficiency tables and the quoted 21.86σ significance.
  • The MLP purity gain implies that combining the hadronic and leptonic channels in a profile-likelihood fit to reconstructed mass or HT might extend the reach beyond the 3.5 TeV quoted for the cut-based analysis—an avenue the authors themselves flag as future work.
  • If the CKM unitarity deficit or related flavor anomalies are eventually traced to a vector-like quark, this single-production search would be a targeted, low-background way to confirm it, because the same mixing that explains those anomalies controls the production rate.
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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

5 major / 6 minor

Summary. The paper studies single production of a vector-like top partner T at a future muon-proton collider via the process μp → ν_μ T b, with T → W b. The analysis uses an effective Lagrangian, MadGraph/Pythia/Delphes simulation, and cut-based plus machine-learning selections in hadronic and leptonic final states at sqrt(s) = 5.29, 6.48, and 9.16 TeV. The central claim is that at sqrt(s) = 9.16 TeV and L = 3000 fb^-1, singly produced T quarks can be discovered up to m_T ≈ 3.5 TeV, with a 3 TeV T giving 21.86σ (hadronic) and 3.75σ (leptonic) significance.

Significance. If the quantitative claims were correct, the paper would provide a useful first estimate of the VLQ discovery potential of a future muon-proton collider and would complement existing LHC and FCC studies. The topic is timely, and the inclusion of both hadronic and leptonic channels plus an ML comparison is commendable. However, the headline result is not supported by the manuscript's own tables, and the effective coupling g* used to generate all signal rates is never specified. Because all cross-sections scale as (g*)^2, the mass reach is a function of an unstated input, and the projections are not independently checkable. The internal inconsistencies in the event counts and significance tables further undermine the central claim.

major comments (5)
  1. [Section IX vs. Tables III and VI] The conclusion states that 'the 5σ discovery threshold is maintained for VLQ masses up to approximately 3.5 TeV.' This is contradicted by the paper's own numbers: Table III gives, for m_T = 3500 GeV in the hadronic channel, S/√B = 2.80 and S/√(S+B) = 2.99, and Table VI gives 0.48 in the leptonic channel. Both are far below 5σ. The reach claim therefore cannot be derived from the displayed analysis; either the luminosity, the combination procedure, or the significance definition must be different and should be stated explicitly.
  2. [Section II, Eq. (1) and Fig. 2(b)] The effective coupling g* is introduced but its numerical value is never given. The text only says it is 'within the perturbative regime and consistent with... |V_Lb| ≲ 0.1.' Since the cross-section is stated to scale as σ ∝ (g*)^2, every event count, significance, and the 3.5 TeV reach depend quadratically on this unreported input. Without the actual g* (and the κ_W,Z,H values) used in the UFO implementation, the quantitative projections are not reproducible or falsifiable. Please state the benchmark values and show the resulting cross-section for at least one mass point.
  3. [Tables I, II, and III] The central significance table is internally inconsistent. For m_T = 1500 GeV, Table I lists 204,256 hadronic signal events, while Table III lists 43,518. The background events in Table II sum to 88.5 + 370.5 + 20.4 + 6.2 ≈ 485.6, whereas Table III quotes B = 3338.5. Furthermore, the S/√B value in Table III for this row (660.0) is not equal to 43,518/√3338.5 ≈ 753. These discrepancies invalidate the significance columns and the conclusions drawn from them.
  4. [Table VII and Section IX] Table VII is titled 'Cross sections (in fb),' but its 5.29 TeV hadronic row reproduces exactly the S/√(S+B) significance column of Table III (195.04, 130.70, 66.66, ...). The text then refers to '195.04 fb' as a cross-section. This mislabeling of significance as cross-section makes the energy comparison in Table VII unreliable and the statement in the conclusion about 'production cross-section reaches 594.93 fb in the hadronic channel' ambiguous.
  5. [Section X, Tables IX–XII] The ML results are incompatible with the cut-based results on the same signal and background samples. For m_T = 3000 GeV at √s = 9.16 TeV, Table III gives S/B = 0.316 and S/√(S+B) = 18.67 at L = 3000 fb^-1. Table IX reports BDT S/B = 15.14 and S/√(S+B) = 270.54 at the same mass and luminosity. A classifier cannot increase S/B by a factor of ~48 on the same dataset; these values imply a different normalization or a different definition of S and B. The ML comparison is therefore not a meaningful extension of the cut-based analysis.
minor comments (6)
  1. [Figure/table cross-referencing] Several figure references are wrong: Section II refers to 'Figure 1 (Left)' for cross-sections, but Fig. 1 is the Feynman diagram; the σ plot is Fig. 2(a). Section IV refers to 'Figure 2 (Left)' for branching ratios, but the BR plot is Fig. 3(a). Please renumber or fix the in-text references throughout.
  2. [Table IV] The 'Total Events' row in Table IV contains values around 100,000 for every mass point, which cannot be scaled event counts for L = 3000 fb^-1. This row appears to be a formatting or labeling error; it should be removed or replaced with the actual cumulative efficiency.
  3. [Table V] The muon efficiency after the PT(l) > 100 GeV cut is 0.0297 while the electron efficiency is 0.348. The text states the channels are 'comparable performance (approx 30-34% for main cuts).' This large discrepancy needs to be explained or corrected.
  4. [Section II vs. Section IV] The branching-fraction normalization is inconsistent: Section II says with κ_W = κ_Z = κ_H = 1 the T has 50% W, 25% Z, and 25% H decays at high mass, while Section IV and Fig. 3(a) report BR(T→Wb) ≈ 35%. Please clarify which branching fractions are used in the simulation.
  5. [Abstract and Section II] The phrase 'production cross sections peaking at 9.16 TeV and decreasing above 3 TeV' is confusing: at fixed √s, σ decreases monotonically with m_T (Table I). It should be rephrased to describe the m_T dependence or the √s dependence separately.
  6. [Typos and notation] There are numerous formatting issues: 'M ET≤ 700 GeV' in the text, inconsistent use of 'S/(S+B)' versus 'S/√(S+B)', and 'Tb' labels in figures. Please proofread carefully.

Circularity Check

3 steps flagged

The 3.5 TeV discovery claim is set by an unreported coupling g* (σ ∝ (g*)^2) and conflicts with the paper's own significance tables.

specific steps
  1. fitted input called prediction [Section II (THE VLQ MODEL AND SINGLE PRODUCTION), paragraph defining g*; Section IX (CONCLUSION)]
    "Specifically, for single production, the cross-section is expected to scale as σ ∝ (g∗)2. We have carefully selected our benchmark coupling values to be within the perturbative regime and consistent with the most recent constraints from electroweak precision observables and CKM unitarity, which typically require |VLb| ≲ 0.1 for heavy vector-like states."

    The numerical value of g* is never reported, yet every signal rate and significance in the paper scales as (g*)^2 via the paper's own statement σ ∝ (g*)^2 (also Fig. 2b). The headline quantities — 21.86σ (hadronic) and 3.75σ (leptonic) at 3 TeV, and 'discovery ... up to approximately 3.5 TeV' — are therefore the chosen benchmark normalization expressed as a reach: with g* hidden, the 'prediction' is a free input renamed as a collider capability, not a result derived from fixed external inputs. The quoted σ ≈ 425 fb at 3 TeV is also incompatible with the stated |VLb| ≲ 0.1 bound if g* ≈ V, since σ ∝ (g*)^2 would then be ~100× smaller, so the numbers cannot be checked against the paper's own constraint.

  2. other [Section IX CONCLUSION vs Table III (hadronic significance table)]
    "As illustrated in the significance curves of Figure 7 (Right), the 5 σ discovery threshold is maintained for VLQ masses up to approximately 3.5 TeV. (Table III at 3500 GeV: S/sqrt(B) = 2.7962, S/sqrt(S+B) = 2.985)"

    The paper's own Table III gives at m_T = 3500 GeV, hadronic S/sqrt(B) = 2.80 and S/sqrt(S+B) = 2.99, and Table VI gives 0.48 in the leptonic channel — all below the 5σ threshold the text says is maintained. The claimed ~3.5 TeV reach is therefore not derivable from the displayed significance calculation; the 5σ crossing of the tabulated points lies near 3.2 TeV at most, so the conclusion's reach value is asserted rather than obtained from the stated analysis.

  3. other [Section VI HADRONIC ANALYSIS, Table I vs Table III (same mass, sqrt(s), luminosity)]
    "Table I: 'T otal Events 204256 189445 163674 123806 89550 54805 27065 8081'; Table III: '3000 1355.6 4282.5 0.3165 21.86502'"

    For the same point (m_T = 3000 GeV, sqrt(s) = 9.16 TeV, L = 3000 fb^-1), Table I's chain (σ = 425 fb × 3000 fb^-1 × cumulative efficiency 0.0971) yields ~123,806 signal events, whereas Table III lists 1,355.6 signal events — a factor ~91 discrepancy. The quoted significance of 21.86σ is thus not reproducible from the paper's own event-count derivation; the headline 'prediction' is not a well-defined function of the displayed inputs, so the result cannot be independently re-derived from the paper.

full rationale

The paper's quantitative chain is g* -> σ ∝ (g*)^2 -> event rates -> S/sqrt(B) -> reach. The background simulation, detector efficiencies, and cut optimization are genuine, independent computations, so the analysis has real content. However, the absolute normalization is set by g*, whose numerical value is never given: the text defines g*, states the quadratic scaling, gives only the constraint |VLb| ≲ 0.1, and then quotes specific cross-sections, significances, and the 3.5 TeV reach. Because every headline number scales as (g*)^2, the discovery claim is the chosen (undisclosed) benchmark coupling re-expressed as a property of the collider — the 'fitted input called prediction' pattern, with the parameter selected rather than fitted but with the same effect: the prediction is forced by the input normalization. This is compounded by two internal failures of derivability: (a) the conclusion's '5σ up to ~3.5 TeV' contradicts Table III's 2.80σ at 3500 GeV, and (b) Table I and Table III disagree by ~90× on the same signal yield, so the significance cannot even be recovered from the paper's own tables. There is no self-citation chain, imported uniqueness theorem, or ansatz-smuggling; the circularity is of the parameter-normalization kind and is partial — the shape of the reach curve is computed, but the reported reach value reduces to the unreported coupling input. Score 6.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The simulation rests on a standard but unvalidated chain: assumed VLQ Lagrangian with unspecified g*, LO event generation, generic detector simulation, hand-tuned cuts, and pure-statistical significance. No experimental data are used, so all quoted sensitivities are conditional outputs of these inputs.

free parameters (4)
  • g* (effective VLQ-SM coupling) = not stated
    Signal cross-sections and all reported significances scale as (g*)^2; the paper only says g* satisfies perturbativity and CKM bounds, so the 3.5 TeV reach is not reproducible without this value.
  • Benchmark coupling normalizations κW=κZ=κH=1 = 1
    Sets the T→Wb branching ratio and relative decay widths; changing these changes rates and reach.
  • Integrated luminosity L=3000 fb^-1 = 3000 fb^-1
    All event counts scale linearly with L and significances as √L; this is an assumed future machine parameter.
  • Hadronic/leptonic selection cuts = e.g., pT(j)>300 GeV, HT≥800 GeV, ΔR(b1,b2)∈(2.5,4.0); leptonic pT(l)>100 GeV, MET>200 GeV
    Optimized by hand on simulated samples; no optimization scan or closure test is shown, and the quoted significances depend on these thresholds.
axioms (5)
  • domain assumption The singlet vector-like T quark and its couplings are described by Eq. (1), with mixing matrices V4i and κV=1.
    The model is assumed as the BSM framework; not derived or constrained by data in this paper.
  • domain assumption Leading-order matrix elements with the Narrow Width Approximation correctly describe signal and backgrounds.
    No NLO QCD/EW corrections or off-shell effects are included; quoted cross-sections could shift.
  • domain assumption The Delphes detector card for a generic high-energy lepton-hadron environment approximates the future µp detector, including b-tagging and jet reconstruction.
    No real detector exists; the card is not provided, so the acceptance numbers are unverifiable.
  • domain assumption The listed SM background processes (νtb, νWj, νZj, νjj) with the listed cross-sections are complete and correctly normalized.
    Table II cross-sections look like round benchmarks; there is no validation against data or higher-order calculations.
  • domain assumption The asymptotic formula for significance with no systematic uncertainties applies.
    Used to convert event counts into σ values; neglecting systematics overstates sensitivity.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Probing Vector-Like Quarks at a future Muon-Proton Collider." pith.science (2026). https://pith.science/paper/7CLJQQPR

@misc{pith2026251211471,
  author       = {Pith},
  title        = {Pith review of: Probing Vector-Like Quarks at a future Muon-Proton Collider},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CLJQQPR}},
  note         = {Machine review of arXiv:2512.11471}
}
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read the original abstract

This study investigates the discovery potential of a singly produced vector-like top quark ($T$) at a future muon-proton collider with center-of-mass energies of 5.29, 6.48, and 9.16~TeV using a model-independent effective Lagrangian consistent with CKM and electroweak constraints. The $T$ quark predominantly decays into $Wb$, with production cross sections peaking at 9.16~TeV and decreasing above 3~TeV due to parton distribution functions (PDFs) and phase-space suppression. Sensitivity is enhanced through optimized kinematic selections, with the hadronic channel providing higher event rates due to the larger hadronic branching fraction of the $W$ boson, while the leptonic channel offers a cleaner background environment. At an integrated luminosity of 3000~fb$^{-1}$, a 3~TeV $T$ quark can be observed with statistical significances of $21.86\sigma$ and $3.75\sigma$ in the hadronic and leptonic channels, respectively. A machine-learning analysis employing a Boosted Decision Tree (BDT) and a Multi-Layer Perceptron (MLP) is performed at 9.16~TeV for $m_T = 3000$~GeV using $S/B$ and $S/\sqrt{S+B}$ as performance metrics. The MLP consistently outperforms the BDT, achieving a hadronic purity gain of approximately 2.62 while maintaining stable performance across all luminosities. These results demonstrate that a future muon-proton collider can probe vector-like $T$ quark masses up to approximately 3.5~TeV, significantly extending the search for physics beyond the Standard Model.

Figures

Figures reproduced from arXiv: 2512.11471 by Haroon Saghir, Ijaz Ahmed, Jamil Muhammad, Mudassar Hussain, Tayyab Javaid.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (Left) Production cross-section [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (Left) Branching ratios of the vector-like [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Representative Feynman diagrams for the dominant Standard Model background [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The transverse momentum ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Kinematic distributions for the hadronic channel at [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: (a) Missing transverse energy ( [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: a) Reconstructed invariant mass [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Leptonic channel kinematics: (a) Lepton [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Missing transverse energy ( [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: B-tagging performance in the leptonic channel: (a) Number of b-tagged jets. (b) [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Angular separation ∆ [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: A view of the [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.