$\mu^-\mu^+\to {\nu_\mu}\bar{\nu}_\mu t\bar{t}H$ amplitudes in the Feynman-diagram gauge

Ya-Juan Zheng

REVIEW 3 major objections 5 minor 12 references

$\mu^-\mu^+\to {\nu_\mu}\bar{\nu}_\mu t\bar{t}H$ amplitudes in the Feynman-diagram gauge

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The Feynman-diagram gauge makes weak-boson fusion the clear driver of ttH production at muon colliders.

desk verdict A useful demonstration of FD gauge for a multi-particle SMEFT process at a muon collider, but the missing gauge-equivalence check leaves the central claim unanchored. read the letter →

arxiv 2412.19620 v1 pith:U76LKHXG submitted 2024-12-27 hep-ph

Ya-Juan Zheng This is my paper · ORCID

classification hep-ph

keywords Feynman-diagramgaugeweakbosonfusionmuoncolliderSMEFTCP-violatingtop-HiggscouplingttHproductiondimension-6operatorcancellation

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 studies $\mu^-\mu^+\to \nu_\mu\bar{\nu}_\mu t\bar{t}H$ at a muon collider, with a CP-violating top-Higgs coupling generated by a dimension-6 SMEFT operator. The author's goal is to show that expressing the amplitudes in the Feynman-diagram gauge makes the physics clear: weak-boson fusion dominates the cross section at multi-TeV energies, and at the highest energies a single dimension-6 vertex ($ttHWW$) controls the rate. In the standard unitary gauge the same cross section emerges only after large cancellations between diagram classes, which obscures which subprocess is responsible. The interest is that a muon collider search for new top-Higgs physics would be easier to interpret if the dominant production mechanism is identifiable diagram by diagram. The paper claims the FD gauge achieves this without changing any physical prediction.

What carries the argument

The machinery is the Feynman-diagram gauge, a light-cone gauge with gauge vector $n(q)^\mu_{\rm FD}=(\mathrm{sgn}(q^0),-\vec q/|\vec q|)$, in which the Goldstone boson is promoted to the fifth component of the massive $W$ and $Z$. The corresponding $5\times 5$ propagator and five-component polarization vectors turn Goldstone exchange into an explicit part of the physical weak-boson amplitude. In this gauge the individual Feynman diagrams satisfy naive power-counting, so the large $W$-fusion and single-$W$ contributions that cancel in unitary gauge do not need to cancel; instead the WWF class survives as the visible production mechanism, and the fifth-component $ttHWW$ vertex carries the high-energy growth.

What would settle it

Compute the total cross section at a fixed energy, say $\sqrt{s}=10$ TeV, with $\xi=0.2\pi$, using both unitary-gauge and Feynman-diagram-gauge implementations with identical inputs. Agreement of the total cross section is required; any discrepancy beyond Monte-Carlo integration error, or any dependence of the FD-gauge cross section on the unphysical light-cone vector $n$, would refute the central claim.

Watch

Extended reading notes

Core claim

The central claim is that in the Feynman-diagram gauge the total cross section for $\mu^-\mu^+\to \nu_\mu\bar{\nu}_\mu t\bar{t}H$ receives no unphysical cancellations among amplitudes: the weak-boson-fusion (WWF) diagram class dominates for $\sqrt{s}\gtrsim 3$ TeV, both in the Standard Model and with a CP-violating top-Higgs coupling, and for $\sqrt{s}\gtrsim 100$ TeV the cross section is dominated by the squared amplitude of a single diagram containing the dimension-6 $ttHWW$ vertex. This is stated as a property of the FD gauge, where the Goldstone boson is the fifth component of the massive weak boson; it is not manifest in unitary gauge, where the WWF and single-$W$ fusion classes nearly cancel. The paper presents this as a demonstration that the FD gauge realizes the Goldstone-boson equivalence theorem in a diagram-by-diagram way.

Load-bearing premise

The load-bearing premise is that the implemented Feynman-diagram-gauge Feynman rules are exactly equivalent to unitary gauge for this process; the paper interprets the diagram decomposition under that equivalence but does not show a direct numerical cross-check of the total rates in the two gauges.

Editorial extensions

If this is right

At $\sqrt{s}\gtrsim 3$ TeV, the measured $\nu_\mu\bar{\nu}_\mu t\bar{t}H$ cross section at a muon collider can be read directly as weak-boson-fusion production, without subtracting a large interfering background.
At $\sqrt{s}\gtrsim 100$ TeV, the energy growth of the cross section is set by a single dimension-6 $ttHWW$ vertex, so the process offers a clean extraction of that SMEFT coefficient.
A nonzero CP-violating phase $\xi$ changes the high-energy behavior visibly in FD gauge, making the deviation from the Standard Model apparent rather than hidden by cancellations.
The same five-component FD-gauge rules can be applied to other electroweak processes, so the no-cancellation property should extend beyond this one final state.

Reading between the lines

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

Editorial inference: if the FD-gauge decomposition is physically equivalent to unitary gauge, the same technique could be used to identify which single diagram dominates any high-energy electroweak process, turning 'which operator is responsible' questions from a global fit into a per-diagram observable.
The paper does not state it, but the high-energy dominance of a single dimension-6 vertex suggests that differential distributions (for example the $H$ or top transverse-momentum spectrum) in FD gauge may be well approximated by one Feynman diagram; this could be tested by computing the full result and the single-diagram result separately and comparing them.
Another implication not drawn by the paper: because the dominant subprocess is literally $W^+W^-\to t\bar{t}H$, the FD-gauge result may make the effective-$W$ approximation more accurate for this process, and a dedicated study of the approximation error would be a direct test.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Short version: this letter applies the FD gauge to a 2-to-6 electroweak process and shows something useful, but it lacks the one numerical check that would let you trust the result — a direct comparison of the FD-gauge total cross section with the unitary-gauge one.

What's new and worth credit: the specific process (mu−mu+ → nu_mu nu_mu_bar t tbar H with a CP-violating SMEFT ttH coupling) has not been examined in FD gauge before. The paper gives a clear categorization of diagrams (WWF, Wµ+µW, annihilation) and defines R as a measure of cancellation. The figures show that, in FD gauge, WWF visibly dominates over the other classes above a few TeV, and that at very high energies a single diagram with the dimension-6 ttHWW vertex appears to reproduce the total. That last observation is suggestive and potentially practical for identifying which subamplitude drives SMEFT signals.

The soft spots are real, though none of them are disqualifying. Most importantly, the black total curves in the U-gauge and FD-gauge panels are never overlaid, tabulated, or ratioed; the text never states that the two calculations yield the same physical cross section. Because the FD-gauge implementation comes from the same group (ref [6]) and is not benchmarked against an independent calculation in this paper, the claim of 'no unphysical cancellation' is not yet anchored. If the implementation were off by a normalization or a sign, the whole interpretation would collapse.

Secondary issues: the 'naive scaling law' is asserted without derivation; the high-energy single-diagram dominance rests on one green-dashed curve, with no explicit check that this diagram's square alone matches the total; and the paper contains no numerical tables or error estimates. The CP-violating phase setup is fine, and the definition of R is clear.

On balance, the qualitative picture is plausible, but the evidence as presented is incomplete. The missing checks are easy to supply, and the result is worth having in the literature if they are.

I would send this to a serious referee rather than desk-reject. It's a legitimate application of a new tool, and the requested validations are straightforward. My own verdict would be conditional accept after revision.

Referee Report

3 major / 5 minor

Summary. The paper studies the muon-collider process μ⁻μ⁺ → ν_μ ν̄_μ t t̄ H in the SMEFT with a dimension-6 CP-violating top-Higgs operator. It uses the Feynman-diagram (FD) gauge, implemented in MadGraph via new 5-component Feynman rules from ref. [6], to separate the diagrams into weak-boson-fusion (WWF), single-W-fusion (Wμ+μW), and annihilation (anni.) categories. The central claims are that in the FD gauge the WWF-type amplitudes dominate the cross section without the large cancellations seen in unitary gauge, and that at very high energies the total cross section is dominated by a single diagram containing the dimension-6 ttHWW vertex, following a 'naive scaling law'. The paper presents cross-section plots for ξ=0 (SM) and ξ=0.2π in both gauges, but does not provide a numerical demonstration that the FD-gauge total cross section equals the unitary-gauge one for this process.

Significance. If the two gauges are verified to give identical physical cross sections and the FD-gauge decomposition is validated, the paper would offer a practical way to identify new-physics contributions in a complex 2→6 scattering process at a muon collider. The strength of the manuscript is that it leans on a concrete automated implementation (ref. [6]) and gives explicit sample predictions; the weakness is that the load-bearing validation steps are omitted. The claim that the high-energy behaviour is controlled by a single dimension-6 vertex is interesting and falsifiable, but it is not yet quantitatively established. The paper is a short demonstration, so the significance is moderate rather than transformative.

major comments (3)

[Figs. 1 and 2; Eq. (10)] The central claim that FD gauge removes the unphysical cancellations present in unitary gauge presupposes that both gauges produce the same physical total cross section. The black total curves in panels (a) and (b) of Figs. 1 and 2 are never overlaid, tabulated, or ratioed. Please provide a numerical gauge-equivalence check, for example a table of σ_U and σ_FD at several √s values, before interpreting the FD-gauge decomposition. Without this check, the apparent cancellation-free behaviour could be an artifact of the new 5-component Feynman rules rather than a property of the gauge.
[Conclusion; Fig. 2(b)] The conclusion that at √s ≳ 100 TeV the total cross section is dominated by the single diagram with the dimension-6 ttHWW vertex rests on the 'naive scaling law', which is neither defined nor derived anywhere in the manuscript. Please state the scaling law explicitly and support the single-diagram-dominance claim with a quantitative ratio, such as σ(|M_ttHWW|²)/σ_total as a function of √s.
[Figs. 1 and 2; Eq. (10)] The claim that 'the dominance of the WWF type amplitudes is clear' at √s ≳ 3 TeV is based on visual inspection of the cyan dotted versus black curves. Since the manuscript already defines the ratio R in Eq. (10), please report R for the WWF category and for the other categories as a function of √s, so that 'dominance' and 'no unphysical cancellation' are quantified rather than asserted visually.

minor comments (5)

[Abstract and text] The final state is written inconsistently as νμ¯νμt¯tH in the title and abstract and as νμ¯νμ¯ttH in the body; please use one consistent notation, e.g. ν_μ ν̄_μ t t̄ H.
[Figure captions] The TeX expressions in the captions of Figs. 1 and 2 are garbled: 'the sum of the squared of each amplitudes Pall k |Mk|2' should read Σ_k |M_k|², and '|PWWF k Mk|2' should read |Σ_{k∈WWF} M_k|².
[Third paragraph] The sentence 'In this work [6], we extend this prescription...' refers to a separate publication; consider writing 'In ref. [6] we extended...' to avoid confusion about what is new in the present paper.
[Eq. (6)] The placement of parentheses in the SMEFT operator (Q†3 φ̃ t_R)(φ̃† φ̃ − v²/2) is unconventional; please clarify that the second factor is a separate scalar singlet and not contracted with the first factor in a non-trivial Lorentz or color structure.
[Eq. (9)] The relation λ/Λ² = √2(g_SM − g e^{iξ})/v² is stated without explanation of how the SM Yukawa coupling g_SM is separated from the new coupling g; a one-sentence clarification would help the reader connect Eq. (8) to the dimension-6 operator.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the FD-gauge cross sections are computed from a standard SMEFT Lagrangian; reliance on the authors' prior FD-gauge implementation is tooling dependence, not an input-output circularity.

full rationale

The paper's claims—that in the FD gauge the WWF-type diagrams dominate and that a single ttHWW diagram dominates at the highest energies—are numerical outputs of a Monte Carlo calculation, not re-statements of the input Lagrangian. The SMEFT operator (6) fixes the ttH and ttHH couplings through Eq. (9), and the cross sections and the cancellation ratio R in Eq. (10) are then evaluated with the Feynman rules (2)-(5). No fitted parameter is renamed as a prediction, and no quantity is defined in terms of the very cross section it is said to predict. The self-citations [3,6] supply the FD-gauge formalism and its MadGraph implementation; this is a normal use of prior tooling, and while an explicit numerical cross-check that the FD-gauge total equals the unitary-gauge total would strengthen the interpretation, its absence is a validation gap, not a circular step. The Goldstone-boson equivalence theorem is cited to independent literature [9,10]. Thus no step in the derivation reduces by construction to its own inputs.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No new physical entities are invented. The main load-bearing assumptions are the correctness of the self-cited FD-gauge implementation and the single-operator SMEFT truncation. The phase xi is a hand-chosen example parameter.

free parameters (1)

CP-violating phase xi = 0.2 pi
Chosen by hand as a numerical example of a non-SM CP-violating ttH coupling. The high-energy ttHWW dominance claim is shown for this value, not for a scan or a derived value.

assumptions (3)

domain assumption The FD-gauge 5-component polarizations and propagators of refs [3,6] correctly reproduce physical unitary-gauge electroweak amplitudes.
The entire diagram decomposition relies on this equivalence. The paper cites prior work but does not show an independent numerical cross-check in this manuscript.
domain assumption SMEFT truncation at a single dimension-6 operator in Eq. (6), with no dimension-8 or double-insertion contributions, is valid at the energies considered.
Standard SMEFT assumption, but the paper shows cross sections up to 100 TeV where the EFT regime is not interrogated.
domain assumption MadGraph and HELAS tree-level amplitude generation are correct for this process.
The numerical results come from these tools and the paper provides no independent derivation or event-level validation.

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Cite this review

Pith. "Pith review of $\mu^-\mu^+\to {\nu_\mu}\bar{\nu}_\mu t\bar{t}H$ amplitudes in the Feynman-diagram gauge." pith.science (2026). https://pith.science/paper/U76LKHXG

@misc{pith2026241219620,
  author       = {Pith},
  title        = {Pith review of: $\mu^-\mu^+\to \nu_\mu\bar\nu_\mu t\bartH$ amplitudes in the Feynman-diagram gauge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U76LKHXG}},
  note         = {Machine review of arXiv:2412.19620}
}

abstract

We study the process $\mu^-\mu^+\to {\nu_\mu}\bar{\nu}_\mu t\bar{t}H$ with complex CP violating $ttH$ couplings in the SMEFT with a dimension-6 operator. When the amplitudes are expressed in the Feynman-Diagram gauge, the dominance of the total cross section via the weak boson fusion diagrams is manifest. The high energy behaviour is dictated by the higher-dimensional vertices in the dimension-6 SMEFT operator. These properties are not manifest in the unitary gauge because of subtle cancellation among diagrams.

Figures

Figures reproduced from arXiv: 2412.19620 by the authors.

**Figure 1.** Total cross section of νµν¯µttH¯ production with ξ = 0 (SM) at a muon collider (black line), the sum of the squared of each amplitudes Pall k |Mk | 2 (red line), the WWF diagrams contribution | PWWF k Mk | 2 (cyan dotted), and the sum of the three annihilation diagrams | Panni. k Mk | 2 (blue-dashed line). in the SMEFT Lagrangian (6) coupling contributes to the process. Its contribution is shown by green-dashed line… view at source ↗

**Figure 2.** Total cross section of νµν¯µttH¯ production with ξ = 0.2π at a muon collider (black line), the sum of the squared of each amplitudes Pall k |Mk | 2 (red line), the WWF diagrams contribution | PWWF k Mk | 2 (cyan dotted). Single diagram contribution of |MttHH| 2 in the U gauge (a) and |MttHWW | 2 in the FD gauge (b) are shown by green dashed curves. Grey line shows the SM as a reference [PITH_FULL_IMAGE:figures/full… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 2 canonical work pages

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Hagiwara, J

K. Hagiwara, J. Kanzaki, O. Mattelaer, K. Mawatari, Y.J. Zheng, Automatic generation of helicity amplitudes in the Feynman-diagram gauge , Phys. Rev. D 110, 056024 (2024), 2405.01256 . 10.1103/PhysRevD.110.056024

arXiv 2024
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J. Chen, K. Hagiwara, J. Kanzaki, K. Mawatari, Y.J. Zheng, Helicity amplitudes in light-cone and Feynman-diagram gauges , Eur. Phys. J. Plus 139, 332 (2024), 2211.14562 . 10.1140/epjp/s13360-024-05067-5

arXiv 2024
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Hagiwara, H

K. Hagiwara, H. Murayama, I. Watanabe, Search for the Yukawa interaction in the process e+ e- --- > t anti-t Z at TeV linear colliders , Nucl. Phys. B 367, 257 (1991). 10.1016/0550-3213(91)90017-R

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J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H.S. Shao, T. Stelzer, P. Torrielli, M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations , JHEP 07, 079 (2014), 1405.0301 . 10.1007/JHEP07(2014)079

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V. Barger, K. Hagiwara, Y.J. Zheng, CP-violating top-Higgs coupling in SMEFT , Phys. Lett. B 850, 138547 (2024), 2310.10852 . 10.1016/j.physletb.2024.138547

arXiv 2024

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[12]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [6]

Hagiwara, J

K. Hagiwara, J. Kanzaki, O. Mattelaer, K. Mawatari, Y.J. Zheng, Automatic generation of helicity amplitudes in the Feynman-diagram gauge , Phys. Rev. D 110, 056024 (2024), 2405.01256 . 10.1103/PhysRevD.110.056024

arXiv 2024

[2] [1]

Hagiwara, J

K. Hagiwara, J. Kanzaki, K. Mawatari, QED and QCD helicity amplitudes in parton-shower gauge , Eur. Phys. J. C 80, 584 (2020), 2003.03003 . 10.1140/epjc/s10052-020-8154-9

arXiv 2020

[3] [2]

J. Chen, K. Hagiwara, J. Kanzaki, K. Mawatari, Helicity amplitudes without gauge cancellation for electroweak processes , Eur. Phys. J. C 83, 922 (2023), [Erratum: Eur.Phys.J.C 84, 97 (2024)], 2203.10440 . 10.1140/epjc/s10052-023-12093-7

arXiv 2023

[4] [3]

J. Chen, K. Hagiwara, J. Kanzaki, K. Mawatari, Y.J. Zheng, Helicity amplitudes in light-cone and Feynman-diagram gauges , Eur. Phys. J. Plus 139, 332 (2024), 2211.14562 . 10.1140/epjp/s13360-024-05067-5

arXiv 2024

[5] [4]

Hagiwara, H

K. Hagiwara, H. Murayama, I. Watanabe, Search for the Yukawa interaction in the process e+ e- --- > t anti-t Z at TeV linear colliders , Nucl. Phys. B 367, 257 (1991). 10.1016/0550-3213(91)90017-R

work page doi:10.1016/0550-3213(91)90017-r 1991

[6] [5]

Murayama, I

H. Murayama, I. Watanabe, K. Hagiwara, HELAS: HELicity amplitude subroutines for Feynman diagram evaluations , KEK-91-11 (1992)

work page 1992

[7] [7]

Alwall, R

J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H.S. Shao, T. Stelzer, P. Torrielli, M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations , JHEP 07, 079 (2014), 1405.0301 . 10.1007/JHEP07(2014)079

arXiv 2014

[8] [8]

Barger, K

V. Barger, K. Hagiwara, Y.J. Zheng, CP-violating top-Higgs coupling in SMEFT , Phys. Lett. B 850, 138547 (2024), 2310.10852 . 10.1016/j.physletb.2024.138547

arXiv 2024