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arxiv: 2606.07746 · v2 · pith:GVIX4V6Enew · submitted 2026-06-05 · ✦ hep-ph

Dark Z' at a Muon Collider: Radiative Return versus Vector Boson Fusion

Nicko Angelo Rabang , Philip Tanedo , Marvin Flores This is my paper

Pith reviewed 2026-06-27 21:28 UTC · model grok-4.3

classification ✦ hep-ph
keywords dark Z'muon colliderradiative returnvector boson fusionkinetic mixingmass mixinghidden sector
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The pith

Relative rates of radiative return and vector boson fusion at a muon collider determine dark Z' mixing parameters with polarized-beam precision.

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

This paper studies how a future muon collider can discover a secluded dark Z' gauge boson that couples to the Standard Model through kinetic mixing and Higgs-sector mass mixing. The analysis covers masses from 100 GeV up to the full collider energy for benchmarks reaching 14 TeV. Two production modes are examined: radiative return, tagged by an associated photon, and vector boson fusion of W bosons. The central result is that the ratio of these two rates extracts the relative strength of the two mixing mechanisms. This extraction reaches an accuracy comparable to a left-right asymmetry measured with a fully polarized muon beam, even when soft and collinear photons contribute to the same final state.

Core claim

A dark Z' with mass from 100 GeV up to the collider energy can be discovered at a muon collider through radiative return and vector boson fusion. The relative rates of these processes determine the relative mixing parameters to an accuracy comparable to that obtained from a left-right asymmetry with a fully polarized muon beam, even accounting for contributions from soft and collinear photons in the same final state.

What carries the argument

The ratio of the radiative return cross section to the vector boson fusion cross section, which isolates the relative contributions of kinetic mixing versus mass mixing.

If this is right

  • The discovery reach extends to the full collider energy for the examined benchmarks up to 14 TeV.
  • Sensitivity can be compared directly to current and proposed future colliders.
  • The rate ratio provides an alternative to beam polarization for extracting the mixing parameters.
  • Contributions from soft and collinear photons do not prevent the ratio from yielding information on the mixings.

Where Pith is reading between the lines

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

  • Muon colliders could characterize hidden gauge bosons without requiring advanced beam polarization.
  • Rate-ratio methods might be adapted for other colliders or different hidden-sector particles.
  • If photon tagging acceptance turns out harder to control than modeled, the achievable precision on mixing would be reduced.

Load-bearing premise

The calculation assumes that radiative return and vector boson fusion are the only relevant production mechanisms and that a photon tag can be applied without irreducible backgrounds washing out the rate ratio.

What would settle it

A measured rate ratio that deviates from the predicted dependence on the kinetic and mass mixing parameters, or the appearance of other production channels that alter the observed ratio.

Figures

Figures reproduced from arXiv: 2606.07746 by Marvin Flores, Nicko Angelo Rabang, Philip Tanedo.

Figure 1
Figure 1. Figure 1: Branching ratio of dark Z ′ as a function of its mass for different values of mass mixing parameter (κ = 0, 5×10−3 , 1×10−2 , 1×10−1 ) at fixed kinetic mixing parameter, ε = 5×10−3 . The branching ratios are approximately constant for mZ′ ≳ 400 GeV. The convergence of the Z ′ → Zh and Z ′ → W+W− branching ratios at large mZ′ reflects the Goldstone equivalence theorem. The lepton (ℓ +ℓ −) and neutrino νν¯ l… view at source ↗
Figure 2
Figure 2. Figure 2: Representative non-VBF diagrams for the process µ −µ + → Z ′νµν¯µ. 3 Dark Z ′ at a Muon Collider 3.1 Collider Benchmarks A standard benchmark scenario is a 10 TeV muon collider with 10 ab−1 of integrated luminosity for which one expects percent level precision for Standard Model electroweak processes [28]. This machine may be preceded by a 3 TeV low-energy phase. We thus examine the following benchmark col… view at source ↗
Figure 3
Figure 3. Figure 3: Representative Feynman diagrams for the VBF contribution to µ −µ + → Z ′νµν¯µ (left panel) and the radiative return channel µ −µ + → Z ′γ (middle panel). On the right is an example of the combined VBF–radiative return processes that we do not include in our analysis because they are always subdominant. where αW is the weak coupling constant, mW is the mass of the W boson, and µf ∼ mZ′ is the factorization … view at source ↗
Figure 4
Figure 4. Figure 4: Vector boson fusion cross section as a function of the collider energy corresponding to the process for three values of the dark Z ′ mass and benchmark mixing parameters. Radiative Return Radiative return [68–72] is the process ℓ +ℓ − → Z ′γ by which one of the incoming leptons emits a hard photon to carry away the ‘excess’ energy so that the Z ′ may be produced on shell [13, 31, 47, 73–81]. The 2 → 1 proc… view at source ↗
Figure 5
Figure 5. Figure 5: Parton-level dark Z ′ production cross sections for radiative return (dashed) and vector boson fusion (solid) using methods in Section 4.2. We separate the µPDF cross section (dot–dashed) for radiative return topologies with a soft/collinear photon—these contribute to the VBF signal. In each plot we highlight the range where the radiative return (dielectron plus visible photon) signal dominates for one of … view at source ↗
Figure 6
Figure 6. Figure 6: Normalized distributions of the the photon energy Eγ (left) and dielectron invariant mass mee (middle) in the radiative return channel for a set of benchmark dark Z ′ masses at a 3 TeV muon collider. Normalized distributions of the dielectron invariant mass mee in the VBF channel (right) including the soft-photon limit of radiative return events (µPDF). The shape of the latter events is described in Append… view at source ↗
Figure 7
Figure 7. Figure 7: Resolutions for reconstructed dielectron invariant mass (left) and photon energy (middle) in the radiative return channel. Resolutions for reconstructed dielectron invariant mass (right panel) in the VBF channel. These are shown as a function of the dark Z ′ mass mZ′ for benchmark center-of-mass energies of 3 TeV, 10 TeV, and 14 TeV. Ref. [1] assumes that a dark Z ′ in the mass range accessible to a future… view at source ↗
Figure 8
Figure 8. Figure 8: 95% confidence level exclusion limits on kinetic mixing ε as function of dark Z ′ mass mZ′ for the radiative return (left) and VBF (right, including µPDF contributions to VBF) channels. We show the results for a set of muon collider collision energies (colors) and mass mixings κ (line dashing). The insets zoom into the low-mass region. The true VBF cross section for masses heavier than mZ′ ≳ TeV have uncer… view at source ↗
Figure 9
Figure 9. Figure 9: 95% confidence level exclusion limits on kinetic mixing ε as function of dark Z ′ mass mZ′ for two illustrative values of mass mixing, κ = 0 (left) and κ = 10−2 (right). We show the results for a set of muon collider collision energies (colors) and overlay the radiative return, ‘true’ VBF, and µPDF channels (line dashing). The µPDF lines are degenerate for all κ. Horizontal bars masses indicate the range w… view at source ↗
Figure 10
Figure 10. Figure 10: Combined 2σ reach for the dark Z ′ model for a benchmark mass mixing parameter and a set of three muon collider energies (solid, colored lines). Dotted lines show the reach if the µPDF events are not included. These are compared to present exclusion (gray region) from a recast of the LHC dark photon searches (gray line) [12,59–61,102] (via Ref. [1]) and electroweak precision observables (black line) [11].… view at source ↗
Figure 11
Figure 11. Figure 11: Left: Confidence intervals for two benchmark models with respect to the unpolarized cross sections for radiative return (horizontal band), VBF (wavy band), and the left– right asymmetry (ALR, pizza-slices from the origin). We chose an extreme value of ∆χ 2 = 10−3 so that the ALR bands fit in the plot. The ALR is comparable to the other observables for the unrealistic case of 100% polarization, but is sign… view at source ↗
Figure 12
Figure 12. Figure 12: ∆χ 2 = 0.1 contours for the two benchmark points in (5.3) (BP1 in blue dashed, BP2 in red) with respect to the radiative return (north-east hash) and VBF (south-west hash) channels. The grid of plots corresponds to three choices of collider energy (rows) and dark Z ′ masses (columns). The virtual Sudakov uncertainty on the VBF band width is under 10% for each mass. Dotted lines show the VBF band without µ… view at source ↗
Figure 13
Figure 13. Figure 13: ∆χ 2 = 0.1 (solid), 3 (dashed), 5.99 (dotted) contours for different dark Z ′ masses and collider energies for two benchmark points in (5.3) (BP1 in blue, BP2 in red). The outermost contour corresponds to a 2σ confidence. ALR data is included but the effect is negligible. The grid of plots corresponds to three choices of collider energy (rows) and dark Z ′ masses (columns). This accesses an alternative pr… view at source ↗
read the original abstract

A secluded, massive Abelian gauge boson called a dark Z' may interact with the Standard Model through kinetic mixing and mass mixing in the Higgs sector. We determine the sensitivity of a future high-energy muon collider to discover such a particle and determine its mixing parameters. We examine a dark Z' with mass from 100 GeV up to the collider energy for a set of collider benchmarks up to 14 TeV. We show the discovery reach and compare to the current and proposed future colliders. A muon collider is sensitive to two complementary production modes: radiative return (muon fusion with an associated photon), and vector boson fusion of W bosons. An observable photon distinguishes these production modes and the relative rates of these processes allows one to determine the relative mixing. Soft and collinear photons in the radiative return diagram contribute to the same final state as vector boson fusion. We show that these relative rates alone can determine the mixing to an accuracy comparable to that of a fully polarized muon beam using a left-right asymmetry.

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

2 major / 2 minor

Summary. The manuscript examines a secluded dark Z' boson interacting with the SM via kinetic and mass mixing. It computes the discovery reach at a future muon collider (benchmarks up to 14 TeV) for masses 100 GeV to the collider energy, focusing on two production channels—radiative return (muon fusion + photon) and vector-boson fusion of W bosons—that yield the same visible final state. The central claim is that the ratio of these rates, distinguished by an observable photon tag, determines the relative mixing parameters to an accuracy comparable to a left-right asymmetry measurement with a fully polarized muon beam.

Significance. If the rate-ratio method can be shown to survive realistic detector effects, the result would supply a polarization-independent handle on the two mixing angles, which is valuable for muon-collider phenomenology where beam polarization is technically demanding. The comparative sensitivity plots versus other proposed colliders are also useful for planning.

major comments (2)
  1. [Abstract] Abstract and the paragraph following Eq. (the rate-ratio expression): the assertion that the relative rates alone determine the mixing parameters to an accuracy comparable to polarized-beam left-right asymmetry is load-bearing for the paper’s main result, yet no numerical study of photon-tagging efficiency, angular/energy resolution migration, or irreducible soft/collinear-photon background is presented. Without these, the claimed discriminating power cannot be verified.
  2. [Production mechanisms] The section discussing the two production mechanisms: the text states that soft and collinear photons from radiative return populate the same final state as VBF, but provides no cut-flow table, efficiency curves, or purity estimate after a photon-tag requirement. This omission directly affects whether the extracted ratio retains the advertised sensitivity.
minor comments (2)
  1. [Results] The collider-energy benchmarks (3, 10, 14 TeV) and the mass range are clearly stated, but the luminosity assumptions used for the reach plots should be tabulated explicitly for reproducibility.
  2. [Introduction] Notation for the kinetic-mixing parameter and the mass-mixing angle is introduced without a dedicated equation; a single defining equation would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. The comments correctly identify that our parton-level analysis leaves open questions about experimental realism. We address each point below and will revise the manuscript to clarify the scope of our results while preserving the core theoretical observation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the paragraph following Eq. (the rate-ratio expression): the assertion that the relative rates alone determine the mixing parameters to an accuracy comparable to polarized-beam left-right asymmetry is load-bearing for the paper’s main result, yet no numerical study of photon-tagging efficiency, angular/energy resolution migration, or irreducible soft/collinear-photon background is presented. Without these, the claimed discriminating power cannot be verified.

    Authors: We agree that the absence of a detector-level study limits the strength of the claim as written. Our calculation demonstrates that, at the parton level, the ratio of the two channels is sensitive to the relative size of kinetic and mass mixing. We will revise the abstract and the paragraph after the rate-ratio equation to state explicitly that this is a parton-level result and that a full simulation including tagging efficiency, resolution smearing, and background rejection is required to confirm the quoted accuracy. We will also add a short paragraph noting that standard photon isolation and pT thresholds used in similar muon-collider studies typically retain >70% efficiency for the signal photon while suppressing soft/collinear radiation, but we will not claim this has been verified in our analysis. revision: partial

  2. Referee: [Production mechanisms] The section discussing the two production mechanisms: the text states that soft and collinear photons from radiative return populate the same final state as VBF, but provides no cut-flow table, efficiency curves, or purity estimate after a photon-tag requirement. This omission directly affects whether the extracted ratio retains the advertised sensitivity.

    Authors: We accept this criticism. The manuscript currently presents only the parton-level cross sections without applying explicit selection cuts or reporting efficiencies. In the revised version we will add a cut-flow table for both channels that includes a minimum photon pT requirement, isolation criterion, and basic acceptance cuts. We will also provide a brief estimate of the purity after the photon tag, based on the kinematic separation between the hard photon in radiative return and the softer radiation in VBF. These additions will be presented as illustrative rather than exhaustive, and we will note that a dedicated experimental study is needed for a definitive assessment. revision: yes

Circularity Check

0 steps flagged

No circularity: rate-ratio extraction follows from explicit cross-section calculations

full rationale

The central claim rests on computing the radiative-return and VBF cross sections for a dark Z' with kinetic and mass mixing, then showing that their ratio distinguishes the two mixing parameters at a level comparable to a left-right asymmetry. No equation defines one mixing parameter in terms of the other or renames a fitted quantity as a prediction; the photon-tag separation is an input assumption whose validity is external to the derivation itself. No self-citation is invoked as a uniqueness theorem or load-bearing premise. The result is therefore self-contained against collider kinematics and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard model plus a minimal dark Z' extension; no additional free parameters are introduced beyond the two mixing angles that the analysis aims to measure.

axioms (2)
  • standard math Standard Model gauge interactions and particle content govern the vector boson fusion and radiative return processes
    Used to compute the production cross sections at muon-collider energies.
  • domain assumption Kinetic mixing and Higgs-sector mass mixing are the only portals between the dark Z' and the Standard Model
    Defines the model space explored in the sensitivity study.
invented entities (1)
  • dark Z' no independent evidence
    purpose: Secluded Abelian gauge boson that mixes with the SM via kinetic and mass mixing
    Postulated new particle whose discovery reach is being calculated

pith-pipeline@v0.9.1-grok · 5709 in / 1310 out tokens · 17521 ms · 2026-06-27T21:28:38.301885+00:00 · methodology

discussion (0)

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Reference graph

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