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arxiv: 2604.01686 · v2 · submitted 2026-04-02 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Higgs production in association with a Z boson at TeV-scale lepton colliders

Hiroyuki Furusato , Satsuki Hosoya , Kentarou Mawatari , Shouta Suzuki
Authors on Pith no claims yet

Pith reviewed 2026-05-13 21:20 UTC · model grok-4.3

classification ✦ hep-ph
keywords Higgs productionZ bosonlepton collidersFeynman-diagram gaugevector boson scatteringgauge cancellationsTeV energiesinterference patterns
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The pith

In the l-l+ to nu nu-bar Z h process at TeV energies the Feynman-diagram gauge removes high-energy cancellations so each amplitude subgroup contributes visibly to distributions.

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

The paper examines Higgs production with a Z boson accompanied by neutrinos at future lepton colliders, noting that this channel overtakes direct Zh production above a few TeV. Amplitudes are sorted by Feynman-diagram topology into vector-boson scattering, l-W scattering, and W-l scattering groups, and their interference is tracked. In the usual unitary gauge, delicate cancellations hide individual roles at high energy, yet the recently introduced Feynman-diagram gauge removes those cancellations, letting the observed kinematics be read directly from the separate subgroup contributions. The same interference pattern appears in the simpler nu nu-bar Z final state, providing a cross-check.

Core claim

We classify the amplitudes for l−l+→νν¯Zh into three main groups based on Feynman diagram topology: vector boson scattering, l−W+ scattering, and W−l+ scattering. We demonstrate that the subtle gauge cancellations among these amplitudes at high energies, present in the unitary gauge, are absent in the Feynman-diagram gauge, enabling the physical distributions to be interpreted directly from the contributions of each subgroup. We also observe that the interference patterns in the kinematical distributions of the Z boson follow those found in the l−l+→νν¯Z process.

What carries the argument

Classification of amplitudes into vector-boson-scattering, l-W-scattering, and W-l-scattering subgroups, analyzed inside the Feynman-diagram gauge that eliminates high-energy cancellations.

If this is right

  • Above a few TeV the neutrino-associated Zh cross section exceeds the direct Zh cross section.
  • Physical distributions become readable as sums of the three subgroup contributions once the Feynman-diagram gauge is used.
  • Interference visible in the Z boson kinematics matches the pattern already present in the simpler nu nu-bar Z process.
  • Gauge cancellations that obscure contributions in the unitary gauge disappear in the Feynman-diagram gauge.

Where Pith is reading between the lines

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

  • The same gauge choice may simplify amplitude analysis for other multi-boson final states at future high-energy lepton colliders.
  • Kinematic cuts based on the identified subgroups could be used to isolate vector-boson-scattering contributions experimentally.
  • The dominance of the neutrino channel suggests that searches for anomalous Higgs couplings at TeV colliders should include this final state rather than relying solely on direct Zh production.

Load-bearing premise

The three topology-based amplitude subgroups together capture all relevant high-energy behavior while higher-order corrections stay negligible.

What would settle it

A measurement or exact calculation showing that the Z-boson angular or energy distributions in l-l+ to nu nu-bar Z h deviate from the interference pattern already seen in l-l+ to nu nu-bar Z at the same collider energy would falsify the subgroup interpretation.

Figures

Figures reproduced from arXiv: 2604.01686 by Hiroyuki Furusato, Kentarou Mawatari, Satsuki Hosoya, Shouta Suzuki.

Figure 1
Figure 1. Figure 1: also shows the helicity-dependent cross sec￾tions, where λ is the helicity of the Z boson in the l −l + collision c.m. frame. We find that for the Zh production the longitudinally-polarized (λ = 0) Z boson, denoted by a red-dashed line, is dominantly produced in high ener￾gies. For the ννZ¯ and ννZh ¯ productions, on the other hand, the production rates for λ = 0 (red-dashed) and λ = ±1 (blue-dotted) are v… view at source ↗
Figure 3
Figure 3. Figure 3: shows the total cross section of e −µ + → νeν¯µZ for λ = 0 as a function of the collision energy √ s from 100 GeV up to 30 TeV, where λ is the helic￾ity of the final-state Z boson in the e −µ + collision c.m. e − µ + νe Z ν¯µ W W e − µ + νe ν¯µ Z W e − µ + Z νe ν¯µ W e − µ + νe ν¯µ Z W e − µ + νe ν¯µ Z W (a) (b-i) (b-f) (c-i) (c-f) FIG. 2. Feynman diagrams for e −µ + → νeν¯µZ. 10 1 10 0 10 1 s [TeV] 10 3 1… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Feynman diagrams for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Rapidity distribution of the Z boson for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Rapidity distribution of the [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The Feynman diagrams for [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Subdiagrams of the group (a) VBS in Fig. 6 in the [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The group (b) [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The group (c) [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Total cross section of [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Rapidity distributions of the Z boson (top) and the Higgs boson (bottom) for [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Rapidity distributions of the Higgs boson for [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
read the original abstract

We study the $l^-l^+\to \nu\bar{\nu}Zh$ process for future lepton colliders, whose cross section becomes larger than that for $l^-l^+\to Zh$ in the energy region above a few TeV. We classify the amplitudes into three main groups based on the topology of each Feynman diagram; vector boson scattering, $l^-W^+$ scattering, and $W^-l^+$ scattering, and study the interference patterns among the amplitudes. We show that subtle gauge cancellation among the amplitudes at high energies in the unitary gauge is absent in the recently proposed Feynman-diagram gauge, and the physical distributions can be interpreted by the contributions from each subgroup. We also find that the interference patterns in kinematical distributions of the Z boson can be understood by those in the $l^-l^+\to \nu\bar{\nu}Z$ process.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper studies the process l−l+ → νν¯Zh at TeV-scale lepton colliders, where its cross section exceeds that of l−l+ → Zh above a few TeV. Tree-level amplitudes are classified into three topological subgroups (vector boson scattering, l−W+ scattering, W−l+ scattering), with analysis of their interference patterns. The central result is that high-energy gauge cancellations present in the unitary gauge are absent in the Feynman-diagram gauge, permitting direct physical interpretation of distributions from each subgroup; interference patterns in Z kinematics are shown to match those in the related l−l+ → νν¯Z process.

Significance. If the classification and numerical comparisons hold, the work supplies a practical framework for disentangling amplitude contributions in high-energy electroweak processes without relying on delicate cancellations. The explicit link to the simpler l−l+ → νν¯Z process and the use of the Feynman-diagram gauge are constructive for collider phenomenology, though the overall impact remains primarily technical rather than opening new discovery channels.

major comments (1)
  1. The assertion that the three subgroups exhaust all tree-level contributions is load-bearing for the interpretability claim, yet the manuscript provides no explicit count or enumeration of diagrams per group (or confirmation of completeness and non-overlap); without this, the statement that distributions can be interpreted from each subgroup cannot be fully verified.
minor comments (3)
  1. The abstract states that the cross section becomes larger than for l−l+ → Zh 'above a few TeV' but supplies neither a specific threshold energy nor a reference to the figure or table that demonstrates the crossover.
  2. The Feynman-diagram gauge is described as 'recently proposed' without a citation or brief recap of its defining properties, which reduces self-contained readability for readers unfamiliar with the prior literature.
  3. The claim of matching interference patterns with l−l+ → νν¯Z would be strengthened by a direct side-by-side plot or table of the relevant differential distributions rather than a qualitative statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below and will incorporate the requested clarification in the revised version.

read point-by-point responses

  1. Referee: The assertion that the three subgroups exhaust all tree-level contributions is load-bearing for the interpretability claim, yet the manuscript provides no explicit count or enumeration of diagrams per group (or confirmation of completeness and non-overlap); without this, the statement that distributions can be interpreted from each subgroup cannot be fully verified.

    Authors: We agree that an explicit enumeration strengthens the claim. The three topological groups are defined by the distinct ways the external legs attach: vector boson scattering (VBS) diagrams in which the Z is radiated from the fused W bosons, l^-W^+ scattering diagrams in which the incoming lepton scatters off a virtual W, and W^-l^+ scattering diagrams with the symmetric attachment. These topologies are mutually exclusive and together exhaust all tree-level electroweak diagrams for l^-l^+ -> nu nu-bar Z h, as confirmed by exhaustive enumeration of all possible s-, t-, and u-channel exchanges involving W, Z, and Higgs propagators. In the revised manuscript we will add a short table listing the number of diagrams in each group together with a one-paragraph argument for completeness and non-overlap. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper classifies tree-level amplitudes for l−l+→νν¯Zh into three topology-based subgroups (vector-boson scattering, l−W+ scattering, W−l+ scattering) using standard SM Feynman rules and compares their behavior in the unitary gauge versus the recently proposed Feynman-diagram gauge. The key observation—that high-energy cancellations vanish in the Feynman-diagram gauge, enabling direct physical interpretation of subgroup contributions—follows from explicit diagram-by-diagram computation and numerical verification of interference patterns against the related l−l+→νν¯Z process. No parameters are fitted and then relabeled as predictions, no equation reduces to its own input by construction, and the gauge is invoked as an external recent proposal rather than derived or justified solely via self-citation within this work. The derivation chain remains self-contained against external SM benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard-model quantum field theory at TeV energies and the validity of the recently proposed Feynman-diagram gauge; no new free parameters or entities are introduced.

axioms (2)
  • domain assumption Standard Model Feynman rules govern the l l to nu nu Z h amplitudes at TeV scales
    Invoked throughout the amplitude classification and cross-section comparison.
  • domain assumption Feynman-diagram gauge eliminates high-energy cancellations present in unitary gauge
    Central to the claim that physical distributions can be interpreted by subgroup contributions.

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Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · 1 internal anchor

  1. [1]

    Aadet al.(ATLAS), Determination of the Relative Sign of the Higgs Boson Couplings to W and Z Bosons Using WH Production via Vector-Boson Fusion with the ATLAS Detector, Phys

    G. Aadet al.(ATLAS), Determination of the Relative Sign of the Higgs Boson Couplings to W and Z Bosons Using WH Production via Vector-Boson Fusion with the ATLAS Detector, Phys. Rev. Lett.133, 141801 (2024), arXiv:2402.00426 [hep-ex]

    work page arXiv 2024

  2. [2]

    A. Hayrapetyanet al.(CMS), Study of WH production through vector boson scattering and extraction of the relative sign of the W and Z couplings to the Higgs boson in proton-proton collisions at s=13TeV, Phys. Lett. B 860, 139202 (2025), arXiv:2405.16566 [hep-ex]

    work page arXiv 2025

  3. [3]

    C. H. de Lima, D. Stolarski, and Y. Wu, Status of nega- tive coupling modifiers for extended Higgs sectors, Phys. Rev. D105, 035019 (2022), [Erratum: Phys.Rev.D 108, 099901 (2023)], arXiv:2111.02533 [hep-ph]

    work page arXiv 2022

  4. [4]

    C. H. de Lima and D. Stolarski, Influence of new states in searches for negative gauge-Higgs couplings, JHEP10, 180, arXiv:2404.10815 [hep-ph]

    work page arXiv

  5. [5]

    Stolarski and Y

    D. Stolarski and Y. Wu, Tree-level interference in vector boson fusion production of Vh, Phys. Rev. D102, 033006 (2020), arXiv:2006.09374 [hep-ph]

    work page arXiv 2020

  6. [6]

    C. H. de Lima and D. Tuckler, Sign of gauge-Higgs boson couplings at future lepton colliders, Phys. Rev. D111, 015021 (2025), arXiv:2408.14536 [hep-ph]

    work page arXiv 2025

  7. [7]

    Atti´ eet al.(Linear Collider Vision), A Linear Col- lider Vision for the Future of Particle Physics, (2025), arXiv:2503.19983 [hep-ex]

    D. Atti´ eet al.(Linear Collider Vision), A Linear Col- lider Vision for the Future of Particle Physics, (2025), arXiv:2503.19983 [hep-ex]

    work page arXiv 2025

  8. [8]

    Aryshevet al.(ILC International Development Team), The International Linear Collider: Report to Snowmass 2021, (2022), arXiv:2203.07622 [physics.acc-ph]

    A. Aryshevet al.(ILC International Development Team), The International Linear Collider: Report to Snowmass 2021, (2022), arXiv:2203.07622 [physics.acc-ph]

    work page arXiv 2021

  9. [9]

    The Compact Linear e +e− Collider (CLIC): Physics Po- tential, (2018), arXiv:1812.07986 [hep-ex]

    work page arXiv 2018

  10. [10]

    Accetturaet al.(International Muon Collider), In- terim report for the International Muon Collider Collab- oration (IMCC), CERN Yellow Rep

    C. Accetturaet al.(International Muon Collider), In- terim report for the International Muon Collider Collab- oration (IMCC), CERN Yellow Rep. Monogr.2/2024, 176 (2024), arXiv:2407.12450 [physics.acc-ph]

    work page arXiv 2024

  11. [11]

    The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations

    J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP07, 079, arXiv:1405.0301 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    Mattelaer and K

    O. Mattelaer and K. Ostrolenk, Speeding up Mad- Graph5 aMC@NLO, Eur. Phys. J. C81, 435 (2021), arXiv:2102.00773 [hep-ph]

    work page arXiv 2021

  13. [13]

    Hagiwara, J

    K. Hagiwara, J. Kanzaki, and K. Mawatari, QED and QCD helicity amplitudes in parton-shower gauge, Eur. Phys. J. C80, 584 (2020), arXiv:2003.03003 [hep-ph]

    work page arXiv 2020

  14. [14]

    J. Chen, K. Hagiwara, J. Kanzaki, and K. Mawatari, Helicity amplitudes without gauge cancellation for elec- troweak processes, Eur. Phys. J. C83, 922 (2023), [Er- ratum: Eur.Phys.J.C 84, 97 (2024)], arXiv:2203.10440 [hep-ph]

    work page arXiv 2023

  15. [15]

    J. Chen, K. Hagiwara, J. Kanzaki, K. Mawatari, and Y.- J. Zheng, Helicity amplitudes in light-cone and Feynman- diagram gauges, Eur. Phys. J. Plus139, 332 (2024), arXiv:2211.14562 [hep-ph]

    work page arXiv 2024

  16. [16]

    Hagiwara, J

    K. Hagiwara, J. Kanzaki, O. Mattelaer, K. Mawatari, and Y.-J. Zheng, Automatic generation of helicity ampli- tudes in the Feynman-diagram gauge, Phys. Rev. D110, 056024 (2024), arXiv:2405.01256 [hep-ph]

    work page arXiv 2024

  17. [17]

    Furusato, K

    H. Furusato, K. Mawatari, Y. Suzuki, and Y.-J. Zheng, W-boson pair production at lepton colliders in the Feynman-diagram gauge, Phys. Rev. D110, 053005 (2024), arXiv:2406.08869 [hep-ph]

    work page arXiv 2024

  18. [18]

    Hagiwara, J

    K. Hagiwara, J. Kanzaki, F. Maltoni, K. Mawatari, and Y.-J. Zheng, Multi-channel phase space with Feynman- diagram-gauge amplitudes, (2026), arXiv:2603.01139 [hep-ph]

    work page arXiv 2026

  19. [19]

    Jeong, No gauge cancellation at high energy in the five-vector Rξgauge, Phys

    J. Jeong, No gauge cancellation at high energy in the five-vector Rξgauge, Phys. Rev. D111, 076031 (2025), arXiv:2502.13633 [hep-ph]

    work page arXiv 2025

  20. [20]

    W.-F. Li, J. Chen, Q.-J. Wang, and Z.-H. Yu, Numerical study on the gauge symmetry of electroweak amplitudes, Chin. Phys.50, 013106 (2026), arXiv:2507.17429 [hep- ph]

    work page arXiv 2026

  21. [21]

    Hagiwara, H

    K. Hagiwara, H. Iwasaki, A. Miyamoto, H. Murayama, 12 and D. Zeppenfeld, Single weak boson production at TeV e+e− colliders, Nucl. Phys. B365, 544 (1991)

    work page 1991

  22. [22]

    M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory(Addison-Wesley, Reading, USA, 1995)

    work page 1995

  23. [23]

    Borel, R

    P. Borel, R. Franceschini, R. Rattazzi, and A. Wulzer, Probing the Scattering of Equivalent Electroweak Bosons, JHEP06, 122, arXiv:1202.1904 [hep-ph]

    work page arXiv 1904

  24. [24]

    Costantini, F

    A. Costantini, F. De Lillo, F. Maltoni, L. Mantani, O. Mattelaer, R. Ruiz, and X. Zhao, Vector boson fusion at multi-TeV muon colliders, JHEP09, 080, arXiv:2005.10289 [hep-ph]

    work page arXiv 2005

  25. [25]

    T. Han, Y. Ma, and K. Xie, High energy leptonic col- lisions and electroweak parton distribution functions, Phys. Rev. D103, L031301 (2021), arXiv:2007.14300 [hep-ph]

    work page arXiv 2021

  26. [26]

    R. Ruiz, A. Costantini, F. Maltoni, and O. Matte- laer, The Effective Vector Boson Approximation in high- energy muon collisions, JHEP06, 114, arXiv:2111.02442 [hep-ph]

    work page arXiv

  27. [27]

    Hamada, R

    Y. Hamada, R. Kitano, R. Matsudo, S. Okawa, R. Takai, H. Takaura, and L. Treuer, Higgs boson production atµ+µ+ colliders, Phys. Rev. D110, 113011 (2024), arXiv:2408.01068 [hep-ph]

    work page arXiv 2024

  28. [28]

    Bigaran and R

    I. Bigaran and R. Ruiz, Weak bosons as partons be- low 10 TeV partonic center-of-momentum, (2025), arXiv:2502.07878 [hep-ph]

    work page arXiv 2025

  29. [29]

    Frixione, F

    S. Frixione, F. Maltoni, D. Pagani, and M. Zaro, Preci- sion phenomenology at multi-TeV muon colliders, JHEP 09, 036, arXiv:2506.10733 [hep-ph]

    work page arXiv

  30. [30]

    Dahl´ en, M

    B. Dahl´ en, M. L¨ oschner, K. Mekala, J. Reuter, and P. Stylianou, EVAluation of the Equivalent Vector bo- son Approximation at highest energy colliders, JHEP11, 002, arXiv:2507.19285 [hep-ph]

    work page arXiv

  31. [31]

    TikZ-Feynman: Feynman diagrams with TikZ

    J. Ellis, TikZ-Feynman: Feynman diagrams with TikZ, Comput. Phys. Commun.210, 103 (2017), arXiv:1601.05437 [hep-ph]

    work page Pith review arXiv 2017

  32. [32]

    Dohse, TikZ-FeynHand: Basic User Guide, (2018), arXiv:1802.00689 [cs.OH]

    M. Dohse, TikZ-FeynHand: Basic User Guide, (2018), arXiv:1802.00689 [cs.OH]

    work page arXiv 2018