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Fermion-Portal Dark Matter at a High-Energy Muon Collider

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

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Pith's one-line read A 10 TeV muon collider could discover fermion-portal dark-matter mediators up to 3.3–4.7 TeV and would see about 1,000 long-lived events across the full viable parameter space.

desk verdict Solid, useful phenomenology that deserves refereeing; the missing beam-induced background discussion is a real caveat but not a fatal flaw. read the letter →

arxiv 2412.14235 v1 pith:OAOZEJ7K submitted 2024-12-18 hep-ph hep-ex

Pouya Asadi , Samuel Homiller , Aria Radick , Tien-Tien Yu This is my paper · ORCID
classification hep-phhep-ex
keywords fermion-portaldarkmatterfreeze-inmuoncolliderlong-livedparticlesR-hadronsdisappearingtracksPDFM_T2
topics Dark Matter
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 10 TeV muon collider with 10 $ab^{-1}$ of data could discover the mediators of four freeze-in fermion-portal dark matter models—right- and left-handed lepton portals and right- and left-handed quark portals—through simple cut-and-count searches. In the prompt-decay regime, the claimed discovery reach spans mediator masses from about 3.3 TeV to 4.7 TeV, and in the long-lived-particle regime roughly 1,000 signal events land in at least one detector component across the entire viable parameter space. A byproduct of the work is a pipeline for including the muon's own parton distribution inside a muon beam, which changes event kinematics enough to lower the reach that a naive no-PDF estimate would suggest. The paper also computes upper bounds on the dark matter mass from the freeze-in plus superWIMP relic-abundance mechanism, ranging from 2.0 TeV to 6.7 TeV depending on the portal.

What carries the argument

The machinery is a set of four renormalizable fermion-portal models in which a scalar mediator carries the same Standard Model quantum numbers as an SM fermion, plus the freeze-in/superWIMP duet that fixes the tiny Yukawa coupling lambda for each point on the (m_phi, m_chi) plane. The coupling then determines the mediator lifetime and, through it, whether the mediator decays promptly or is long-lived. For collider predictions, the paper adds a muon-component PDF pipeline: the muon PDF is split into a smooth part and a delta-function piece at x=1, with coefficients fixed by muon number and polarization sum rules, and the four pieces are generated separately and recombined. The prompt search uses the transverse-mass variable M_T2, which is bounded by the W mass for W-pair backgrounds, along with invariant-mass cuts to remove Z production and opening-angle, energy, and pseudorapidity cuts to kill ISR/FSR and VBF backgrounds. For LLP signals, the detector geometry (barrel dimensions per subsystem) is used to count displaced vertices, disappearing tracks, and detector-stable charged tracks.

What would settle it

Simulate the proposed signal regions with a full detector response that includes beam-induced background from muon decays in the beam pipe; if the significance at the quoted boundary (mediator mass of 3.3–4.7 TeV) drops below 5 sigma, the paper's reach claim fails.

Watch

Extended reading notes

Core claim

The central claim is that cosmologically motivated fermion-portal dark matter, with dark matter produced by freeze-in and the mediator abundance set by freeze-out followed by decay into dark matter, sits in a parameter region that a 10 TeV muon collider is well placed to explore. For each of the four portals, the paper finds an upper bound on the dark matter mass (3.6, 2.0, 6.7, and 5.4 TeV for the e, L, u, and Q models respectively) beyond which the universe would be overclosed. It then shows that mediator pair production, followed by decay to missing energy plus leptons or jets, can be separated from Standard Model backgrounds with cuts on invariant mass, M_T2, opening angle, energy, and pseudorapidity, giving a 5-$\sigma$ discovery reach of roughly 3.3, 3.4, 4.2, and 4.7 TeV in mediator mass at 10 TeV and 10 $ab^{-1}$. In the long-lived regime, every point of every model's viable parameter space yields about 1,000 events (displaced leptons, disappearing tracks, or R-hadron displaced jets) in at least one detector barrel region. Including the muon parton distribution suppresses the earlier reach estimates, and the paper presents this as the more accurate picture.

Load-bearing premise

The quoted prompt-region reach assumes that the only relevant backgrounds are the Standard Model processes shown in the paper (Z/W pair production, ISR/FSR, and VBF), and that detector-level effects and beam-induced backgrounds from muon decays are negligible or removed by the proposed cuts; if those backgrounds contaminate the signal regions, the 5-sigma reach would be lower.

Editorial extensions

If this is right

  • A 10 TeV muon collider could discover all four freeze-in fermion portals at 5 sigma for mediator masses well above current LHC exclusions, up to 3.3–4.7 TeV.
  • If the mediators are long-lived, the models are testable everywhere: at least about 1,000 events appear in some detector component, so null results would exclude or strongly constrain the entire relic-abundance-viable region.
  • The muon PDF effect is large enough that previous no-PDF reach estimates are misleading; future BSM studies at muon colliders should include it or an equivalent treatment for accurate significances.
  • Distinct signatures such as displaced leptons, disappearing tracks, R-hadron displaced jets, and heavy stable charged tracks motivate dedicated trigger and dE/dx improvements in the tracker and muon system.
  • For lepton-portal models the neutral partner decays invisibly, so a complete search should combine charged-mediator tracks with mono-photon or forward-detector channels.

Reading between the lines

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

  • A detector-level simulation that includes beam-induced backgrounds from muon decays in the beam pipe would be the decisive test; if those backgrounds enter the simple signal regions, the quoted reach could shrink, so the headline numbers should be read as signal-level projections.
  • The same muon-PDF splitting pipeline should apply to other BSM production processes at muon colliders, not just mediator pairs, since its impact on differential distributions is a general lesson rather than a model-specific artifact.
  • The upper bounds on the dark matter mass (2.0–6.7 TeV) suggest a complementary target for direct and indirect detection searches, as a substantial fraction of the cosmologically allowed parameter space lies beyond current probes.
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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

3 major / 5 minor

Summary. The manuscript studies four renormalizable fermion-portal dark matter models in the freeze-in/superWIMP regime at a 10 TeV muon collider. For each model it determines the coupling that reproduces the observed relic abundance, computes the resulting mediator lifetimes, and uses these inputs to propose cut-based searches in the prompt regime and event-count estimates in the long-lived-particle regime. The paper also develops a pipeline for including a helicity-dependent muon component of the muon PDF in MadGraph/LHAPDF and shows that this component has a significant effect on kinematic distributions and, consequently, on reach. The headline results are a 5-sigma mediator mass reach of 3.3-4.7 TeV in the prompt regime and roughly 10^3 signal events in at least one detector component across the entire LLP region of each model's parameter space.

Significance. If the quoted reach is robust, the paper provides a useful and broad study of a well-motivated class of dark matter models at a future muon collider, and it makes a concrete methodological contribution through the muon-PDF implementation pipeline. The authors are candid about the rudimentary nature of their searches and about the absence of detector-level simulation in the LLP estimates. The main quantitative claims, however, are the prompt-region discovery reaches, and those rest on an assumption about backgrounds that is not demonstrated. The value of the paper depends on whether that load-bearing assumption can be supported; if it can, the results would be a valuable input to muon-collider detector design and to future phenomenological work on fermion-portal dark matter.

major comments (3)
  1. [Sec. 4.1.1; Figs. 14, 16, 18, 20, 23] The prompt discovery reach is computed against only the Standard Model backgrounds of Fig. 14 (Z/W pair production, ISR/FSR, and VBF). A high-energy muon collider has a known, copious beam-induced background from muon decays in the beam pipe, and this is neither simulated nor shown to be rejected by the cuts of Eqs. (11)-(12). The 'tolerable systematics' panels in Figs. 16, 18, 20, and 23 treat a normalization uncertainty on the simulated backgrounds, not an additional source of events, so they do not cover beam-induced background. Since the central 3.3-4.7 TeV discovery claims depend on the signal regions being essentially background-free, this omission is load-bearing. Please either add a beam-induced background estimate (or a quantitative argument that the proposed cuts remove it) or soften the prompt-region discovery claims accordingly.
  2. [Secs. 2.2-2.4; Eqs. (2), (4), (6), (8)] The relic-abundance calculation that fixes the coupling lambda, and through it all mediator lifetimes and collider event counts, is not shown for the three new models. The text states that the Boltzmann equations are solved and refers to Ref. [39] for the e model, but for the L, u, and Q models the reader is given no Boltzmann system, no list of included processes beyond Table 1, and no numerical method. The lambda contours in Figs. 2, 5, and 6, and the upper bounds in Eqs. (2), (4), (6), and (8), are central inputs to every subsequent prediction. The manuscript should present at least the differential equations, the relevant thermally averaged rates, and the numerical procedure, or release the code used to produce the contours, so that these results can be independently checked.
  3. [Eq. (12); Figs. 20 and 23] The invariant-mass cuts in Eq. (12) appear inverted relative to the quoted mediator masses. Signal region SRq1 is assigned to m_phi = 1.6 TeV with m_jj >= 3600 GeV, while SRq2 is assigned to m_phi = 4.2 TeV with m_jj >= 1500 GeV. As written, the higher-mass signal region keeps substantially more Z-pair background, and the lower-mass region may be highly inefficient. Please confirm that these values are not accidentally swapped, and if they are correct, explain the kinematic reasoning and show the sensitivity of Figs. 20 and 23 to this choice.
minor comments (5)
  1. [Abstract and Sec. 2] The abstract states an 'upper bound on the mediator's mass' from the relic abundance calculation, but the bounds derived in Sec. 2, Eqs. (2), (4), (6), and (8), are upper bounds on the dark matter mass m_chi. Please correct the wording in the abstract.
  2. [Sec. 4.1.1] The significance variable sigma is used throughout but never defined. Please state explicitly whether sigma = S/sqrt(B) or includes a systematic term, and how the 'tolerable systematics as a fraction of statistical uncertainties' is computed.
  3. [Sec. 4.1.2] There is a typo in the text: 'the the gap' should read 'the gap'.
  4. [Sec. 4.2.2] The sentence 'alone could discover this model in most of the LLP region' is stronger than the analysis supports, given the paper's own statement that LLP backgrounds from detector response require Geant4-level simulation and are not modeled. Suggest replacing 'could discover' with 'could provide a sensitive signal region' or similar.
  5. [Sec. 3.1, Eq. (10)] The factorization scale Q_f is used in the decomposition of the parton luminosity, but it is not stated whether all terms on the right-hand side of Eq. (10) are evaluated at the same Q_f and how Q_f is chosen in the event generation. A sentence clarifying this would remove ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the relic-density-fixed coupling is a physical input, and the collider reach and LLP event counts are computed from independent matrix-element and background simulations.

full rationale

The paper's derivation chain is not circular. The only externally imposed quantities are the observed DM relic abundance (Planck, Ref. [69]), the reheat-above-mediator assumption, and the SM background processes of Fig. 14. The coupling λ is determined by solving the Boltzmann equations for each model (Figs. 2, 5, 6; the e model reviewed from Ref. [39]), and is then used as an input to compute mediator lifetimes, branching ratios, and collider event rates. No collider observable is fed back into the relic-density calculation, so the reach numbers in Sec. 4 are not fitted parameters renamed as predictions. The prompt-region analyses optimize generic kinematic cuts (m_ll/m_jj, M_T2, θ, E, η) on independently simulated SM backgrounds; the quoted 3.3-4.7 TeV reach follows from the resulting signal significances, and the 'tolerable systematic' curves are derived, not assumed. The LLP event counts are computed from cross sections, branching ratios, and the Table 2 detector geometry; the claim of ~10^3 events in some detector component across the LLP region is a calculated outcome rather than a definitional consequence. The reliance on the authors' earlier Ref. [39] for the e-model baseline is not load-bearing for the new models, and the e-model reach is re-derived with the new muon-PDF pipeline. Concerns about unmodeled beam-induced backgrounds are a correctness/robustness risk (explicitly outside the adopted background list), not a circularity of the derivation.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The analysis rests on the freeze-in/superWIMP cosmology, the LePDF and EVA descriptions of the muon beam, and the simplifying assumption that the listed SM backgrounds are the only relevant ones. The free parameters are the Yukawa coupling lambda, fixed by the relic density, and the delta-function coefficient c_mu in the PDF decomposition. No new physical entities are invented.

free parameters (2)
  • DM Yukawa coupling lambda = Set by Omega h^2 = 0.12; values roughly 10^-11 to 10^-7 depending on m_chi and m_phi (Figs. 2, 5, 6)
    The coupling is fixed by requiring the freeze-in plus superWIMP relic abundance to match Planck data. It directly sets the mediator lifetime and therefore every collider signal rate and reach number in the paper.
  • Muon PDF delta-function coefficient c_mu(Q^2) = Determined by sum rules and polarization matching for each Q^2; values shown in Fig. 9
    The split of the muon PDF into continuum plus c_mu delta(1-x) is normalized to the total muon number and polarization. The coefficient varies with Q^2 and the regulator epsilon; the paper checks stability at the percent level. This is a technical normalization choice on which the reach estimates depend.
assumptions (6)
  • domain assumption The mediator is in thermal equilibrium with the SM bath and the reheat temperature exceeds the mediator mass (Sec. 2, paragraph 2).
    This is required for the superWIMP contribution to the relic abundance; if not satisfied, the lambda normalization and the upper bounds on m_chi change.
  • domain assumption Freeze-in production of DM is dominated by the 2-body decay of the mediator; 2-to-2 processes contribute negligibly to the DM yield (Sec. 2 and Table 1).
    The lambda-relic relation and mediator lifetimes are computed under this assumption, which is standard for freeze-in but not derived in this paper.
  • domain assumption The muon PDF grids from LePDF (Ref [48]) correctly describe the muon component of a 10 TeV muon beam, including helicity splitting (Sec. 3).
    The pipeline builds on these external grids; any error or limitation in LePDF propagates into the kinematic distributions and reach.
  • domain assumption The effective vector approximation with a sqrt(s-hat) >= 1 TeV cutoff gives a valid description of VBF production (Sec. 3 and Fig. 10).
    All vector-boson-initiated signals and backgrounds are generated with EVA; the cutoff produces a visible kink in cross sections and is an approximation to the full electroweak PDF treatment that is not implemented.
  • domain assumption The only backgrounds relevant to the prompt searches are the SM processes in Fig. 14 (Z/W pair production, ISR/FSR, VBF); beam-induced and detector-level backgrounds are negligible for the signal regions (Sec. 4).
    This assumption is load-bearing for the 5-sigma reach claims. Muon colliders have substantial beam-induced backgrounds from muon decays, which are not modeled. If they contaminate the signal regions, the reach would decrease.
  • domain assumption The Boltzmann solver and mass-splitting formulas from Ref [39] and Ref [81] extend to the new models without modification (Sec. 2.2-2.4 and App. A).
    The paper does not present the coupled Boltzmann equations for the L, u, and Q models; it reports results from solving them. The mass splitting follows the one-loop formula of Ref [81].

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Cite this review

Pith. "Pith review of Fermion-Portal Dark Matter at a High-Energy Muon Collider." pith.science (2026). https://pith.science/paper/OAOZEJ7K

@misc{pith2026241214235,
  author       = {Pith},
  title        = {Pith review of: Fermion-Portal Dark Matter at a High-Energy Muon Collider},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OAOZEJ7K}},
  note         = {Machine review of arXiv:2412.14235}
}
abstract

In this work, we provide a comprehensive study of fermion-portal dark matter models in the freeze-in regime at a future muon collider. For different possible non-singlet fermion portals, we calculate the upper bound on the mediator's mass arising from the relic abundance calculation and discuss the reach of a future muon collider in probing their viable parameter space in prompt and long-lived particle search strategies. In particular, we develop rudimentary search strategies in the prompt region and show that cuts on the invariant dilepton or dijet masses, the missing transverse mass $M_{T2}$, pseudorapidity and energy of leptons or jets, and the opening angle between the lepton or the jet pair can be employed to subtract the Standard Model background. In the long-lived particle regime, we discuss the signals of each model and calculate their event counts. In this region, the lepton-(quark-)portal model signal consists of charged tracks ($R$-hadrons) that either decay in the detector to give rise to a displaced lepton (jet) signature, or are detector stable and give rise to heavy stable charged track signals. As a byproduct, a pipeline is developed for including the non-trivial parton distribution function of a muon component inside a muon beam; it is shown that this leads to non-trivial effects on the kinematic distributions and attainable significances. We also highlight phenomenological features of all models unique to a muon collider and hope our results, for this motivated and broad class of dark matter models, inform the design of a future muon collider detector. We also speculate on suggestions for improving the sensitivity of a muon collider detector to long-lived particle signals in fermion-portal models.

Figures

Figures reproduced from arXiv: 2412.14235 by the authors.

Figure 1
Figure 1. e model: (left) The DM Yukawa coupling, λ, that gives rise to the correct relic abun￾dance today. Here, we have taken the coupling λi ≡ λ to be the same for all fermion flavors. The interplay of the freeze-in and superWIMP mechanisms gives rise to an upper bound on DM mass. The top right gray region is excluded from overclosure constraints, while the bottom right gray region is kinematically not allowed in our scena… view at source ↗
Figure 2
Figure 2. L model: The DM Yukawa coupling, λ, that gives rise to the correct relic abundance today. Here, we have taken the coupling λi ≡ λ to be the same for all fermion flavors. The top right gray region is excluded from overclosure constraints, while the bottom right gray region is kinematically not allowed in our scenario. Strong astrophysical bounds rule out mχ below about ∼10 keV [70], giving rise to a completely-bounde… view at source ↗
Figure 3
Figure 3. L model: The lifetime of the neutral (left) and the charged (right) mediator. For every point in the parameter space we assume the freeze-in coupling λ is set to the value that gives rise to the observed DM abundance today, see [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: L model: Branching ratio of the charged mediator ϕ − L to different channels: DM and a charged lepton (left), neutral mediator ϕ 0 L accompanied by a neutrino and a soft electron (middle) or muon (right). Note the difference between the latter two channels arises from …
Figure 5
Figure 5. Figure 5: u model: (left) The required freeze-in coupling λ from Eq. (5) of the RH quark-portal model to explain the observed DM relic abundance from a combination of the freeze-in and the superWIMP mechanisms. The interplay of these mechanisms gives rise to an upper bound of on…
Figure 6
Figure 6. Figure 6: Q model: The required freeze-in coupling λ from Eq. (7) of the LH quark-portal model to explain the observed DM relic abundance from a combination of the freeze-in and the superWIMP mechanisms. The interplay of these mechanisms gives rise to an upper bound on DM mass. …
Figure 7
Figure 7. Figure 7: Q model: The lifetime of the lighter (left) and the heavier (right) charged mediators in the LH quark-portal model; for every point in the parameter space we assume the freeze-in coupling λ is set to the value that gives rise to the observed DM abundance today, see [P…
Figure 8
Figure 8. Figure 8: Q model: Branching ratio of the heavier, charge-2/3 mediator ϕ 2/3 Q to different channels: DM and a light up-type quark (left), or a lighter charge-1/3 mediator accompanied by a neutrino and a soft electron (right). The analogous 3-body decay with a soft muon is vanis…
Figure 9
Figure 9. Figure 9: Values of the coefficients cµL,µR as a function of the factorization scale Q, for different values of the width ε used in eliminating the x → 1 limit of the muon component of the PDFs, as described in the text. The solid (dashed) lines show the values for the L (R) hel…
Figure 10
Figure 10. Figure 10: Cross section for mediator pair production for lepton-portal (left) or quark-portal (right) models. The solid (dashed) lines show muon (VBF) initial states, including the muon parton PDF (EVA). The dotted lines shows muon initial states with no PDF. Note that the kink…
Figure 11
Figure 11. Figure 11: e model: Histograms of βγ of the ϕ final states. The left (right) figure shows the distribution of βγ without (with) muon parton PDFs. The colors correspond to three different mass points (blue = 500 GeV, orange = 1 TeV, red = 4 TeV). As expected, including the muon c…
Figure 12
Figure 12. Figure 12: e model: Achievable discovery significance of the e model for two signal regions (SRs) used in Ref. [39]. Solid (dashed) lines show the result with(out) the muon component of the PDFs. Including this PDF drastically diminishes the efficacy of these signal regions to b…
Figure 13
Figure 13. Figure 13: e model: Number of expected displaced leptons in two regions of the detector for a 10 TeV MuC, assuming the tentative design of [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: Background topologies at a future muon collider for all fermion-portal models under study. They include a pair production of electroweak gauge bosons Z or W and their subsequent decay to SM fermions (left), a muon annihilation process with an FSR (middle) or an ISR (n…
Figure 15
Figure 15. Figure 15: e model: Distribution of events for the opening angle between charge leptons in the final state (left columns), rapidity of the final leptons (middle columns), and MT2 (right columns) for two benchmark mediator masses 0.7 TeV (top row) and 2.8 TeV (bottom row). The si…
Figure 16
Figure 16. Figure 16: e model: The best achievable discovery significance with zero systematic uncertainty (left) and the tolerable uncertainty that still allows a 5σ discovery of the mediator (right) for the signal regions defined in Eq. (11). The dashed blue line on the left plot marks t…
Figure 17
Figure 17. Figure 17: e model: Number of expected displaced leptons in different barrel regions of the detector, assuming the tentative design of [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: L model: The best achievable discovery significance with zero systematic uncertainty (left) and the tolerable uncertainty that still allows a 5σ discovery of the mediator (right) for the signal regions defined in Eq. (11). The dashed blue line on the left plot marks t…
Figure 19
Figure 19. Figure 19: L model: Number of expected disappearing tracks and displaced leptons in different barrel regions of the detector, assuming the tentative design of [PITH_FULL_IMAGE:figures/full_fig_p032_19.png]
Figure 20
Figure 20. Figure 20: u model: The best achievable discovery significance with zero systematic uncertainty (left) and the tolerable uncertainty that still allows a 5σ discovery of the mediator (right) for the signal regions defined in Eq. (12). The dashed blue line on the left plot marks t…
Figure 21
Figure 21. Figure 21: The β (left) and βγ (right) distribution of events for three different mediator masses (blue = 500 GeV, orange = 1 TeV, red = 4 TeV) in the u model (top) and the Q model (bottom). These distributions are used in determining various signals of LLP signals, e.g. the tim…
Figure 22
Figure 22. Figure 22: u model: Number of expected displaced vertices in different barrel regions of the detector assuming the tentative design of [PITH_FULL_IMAGE:figures/full_fig_p037_22.png]
Figure 23
Figure 23. Figure 23: Q model: The best achievable discovery significance with zero systematic uncertainty (left) and the tolerable uncertainty that still allows a 5σ discovery of the mediator (right) for the signal regions defined in Eq. (12). The dashed blue line on the left plot marks t…
Figure 24
Figure 24. Figure 24: Q model: Number of expected displaced vertices in different barrel regions of the detector assuming the tentative design of [PITH_FULL_IMAGE:figures/full_fig_p040_24.png]

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Pith tools

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