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REVIEW 2 major objections 4 minor 85 references

Heavy QCD Axions at High-Energy Muon Colliders

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A 10 TeV muon collider could discover heavy QCD axions with decay constants of order TeV and masses up to 10 TeV.

desk verdict Genuinely useful muon-collider projection for heavy QCD axions, but the Fig. 9 model overlay is weaker than claimed: two of four benchmark lines need unargued O(1) electroweak couplings. read the letter →

arxiv 2509.10605 v1 pith:HHWXAWV6 submitted 2025-09-12 hep-ph

Ravneet Bedi , Tony Gherghetta , Soubhik Kumar , Peiran Li , Zhen Liu This is my paper · ORCID
classification hep-ph
keywords heavyQCDaxionmuoncolliderstrongCPproblemqualitydijetresonancesmallinstantonsvectorbosonfusion
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

The paper asks whether a future high-energy muon collider could be the right machine to find heavy QCD axions: axions that solve the strong CP problem but are much heavier than the usual QCD axion because new ultraviolet physics raises their mass. It shows that the same anomaly interactions responsible for the axion mass also produce the axion at a muon collider through electroweak vector-boson fusion and make it decay almost entirely to gluon pairs. Combining event simulation with a background analysis, it concludes that a 10 TeV muon collider with 10 ab$^{-1}$ of integrated luminosity can probe axion decay constants of order TeV and masses up to about 10 TeV, well beyond the reach of existing experiments. Four ultraviolet models that naturally produce such axions are presented, and their predicted parameter space lies inside the projected sensitivity.

What carries the argument

The load-bearing mechanism is the set of dimension-five anomaly operators in Eq. (2.1): $c_3 \frac{\alpha_s}{8\pi f_a} a G\tilde G$, $c_2 \frac{\alpha_2}{8\pi f_a} a W\tilde W$, and $c_1 \frac{\alpha_1}{8\pi f_a} a B\tilde B$. The electroweak operators feed axion production through vector-boson fusion and related channels, while the gluonic operator gives the dominant decay $a\to gg$ with branching ratio above 0.95, so the search is a high-mass dijet resonance. The same small-instanton physics that enhances the axion mass ties $m_a$ to $f_a$ in each ultraviolet benchmark through formulas such as Eqs. (3.12), (3.25)-(3.27), (3.29)-(3.30), and (3.33)-(3.35).

What would settle it

A dedicated high-mass dijet search at a 10 TeV muon collider with 10 ab$^{-1}$, looking for a narrow resonance in the invariant mass of two central jets, would settle the claim: if no excess appears up to $m_a\simeq 10$ TeV where Figure 4 predicts sensitivity for order-one anomaly coefficients, the central projection is falsified.

Watch

Extended reading notes

Core claim

The central claim is that heavy QCD axions, whose masses are raised well above the usual sub-60 meV range by new small-instanton contributions, would be produced at a muon collider through electroweak vector-boson fusion and would decay predominantly to gluons, so they appear as a narrow resonance in the dijet invariant mass spectrum. The paper argues that a 10 TeV muon collider with 10 ab$^{-1}$ can probe axion decay constants $f_a\sim$ TeV and masses up to $m_a\sim 10$ TeV, a region that existing LHC, LEP, and beam-dump experiments cannot cover. The same logic applies to four classes of ultraviolet completions: product groups, extra dimensions with a bulk scalar axion, mirror worlds, and color unification, all of which predict axions inside the projected sensitivity, while the version with an axion from a bulk gauge field does not yield viable muon-collider parameter space.

Load-bearing premise

The projected reach hinges on the heavy axion having couplings to the W and Z bosons that are of order one; if those couplings are suppressed in a given ultraviolet completion, the muon-collider discovery reach shrinks or disappears.

Editorial extensions

If this is right

  • A 10 TeV muon collider with 10 ab$^{-1}$ can probe heavy QCD axions with $f_a\sim$ TeV and masses up to about 10 TeV, a range current LHC, LEP, and beam-dump searches cannot cover.
  • The dominant discovery signature is a high-mass dijet resonance, because the gluonic decay $a\to gg$ has branching ratio above 0.95 for order-one anomaly coefficients.
  • Vector-boson fusion dominates production, while $Za$ associated production and vector-boson scattering add non-negligible sensitivity below about 1 TeV, where backgrounds are smaller.
  • All four ultraviolet benchmark models considered in the paper: product group, extra-dimensional bulk scalar, mirror world, and color unification, have parameter space inside the projected muon-collider reach.
  • The sensitivity scales linearly with the electroweak anomaly coefficients $c_1,c_2$, so the reach holds whenever these couplings are of order one.

Reading between the lines

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

  • A null dijet search would translate into direct lower bounds on $f_a$ and, through the small-instanton mass formulas, on ultraviolet scales such as the product-group scale $M$ or the mirror confinement scale $\Lambda_{\rm QCD'}$, making the muon collider a probe of QCD dynamics in the ultraviolet.
  • If a dijet resonance is discovered, the subdominant decays $a\to ZZ$, $a\to WW$, and $a\to Z\gamma$ could be used as cross-checks and could help measure ratios of the anomaly coefficients $c_1,c_2,c_3$.
  • The reach depends on order-one electroweak couplings, so ultraviolet completions where $c_1,c_2$ arise only through loops would be harder to probe; searching for the associated production of the radial mode or of Peccei-Quinn fermions could extend coverage into that regime.
  • Because both the signal and dominant backgrounds are electroweak, the dijet analysis at a muon collider is comparatively free of the QCD jet-systematics that limit hadron-collider dijet searches.
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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

2 major / 4 minor

Summary. This paper studies the discovery potential of heavy QCD axions at future 3 TeV and 10 TeV muon colliders. It uses the effective interaction Lagrangian of Eq. (2.1), with anomaly coefficients c1, c2, c3, and considers electroweak production channels (VBF, VBS, associated Za) with the dominant hadronic decay a -> gg leading to a dijet resonance. The analysis defines a pre-selection and a mass-window search, enumerates SM backgrounds, and presents 95% CL projections for f_a as a function of m_a in Fig. 4, with existing LEP/LHC/PbPb constraints shown for comparison. The paper then presents four classes of UV models—SU(3)^N product group, flat extra dimension with a bulk scalar, Z2 mirror world, and grand-color/composite axion—and plots benchmark lines in Figs. 8 and 9 to argue that a muon collider can probe each class. Appendix A cross-checks the fixed-order and electroweak-PDF treatments for representative VBF/VBS processes.

Significance. The collider phenomenology is presented with care: signal and background channels are tabulated, the fixed-order and PDF calculations are cross-checked in Appendix A, and the authors document and correct a sign error in the FeynRules UFO output. Public access to data and core codes is a further strength. If the central reach claim is accepted, the paper provides an important physics motivation for a multi-TeV muon collider, going beyond generic ALP studies by tying the dijet search to solutions of the strong CP problem and the axion quality problem. The main caveat is the model-dependence of the electroweak couplings c1 and c2, which controls the production rate; as the paper's own Fig. 4 shows, this assumption is quantitatively significant.

major comments (2)
  1. [Sec. 3.1, Sec. 3.2, and Fig. 9] The 'Product Group' and 'Extra Dim Bulk' benchmark lines in Fig. 9 depend on an extra assumption about the electroweak anomaly coefficients c1 and c2. These models do not determine O(1) SU(2)_L x U(1)_Y couplings: Sec. 3.1 states 'To make our analysis general, we will also assume SU(2)_L and U(1)_Y couplings of heavy QCD axions,' and Sec. 3.2 obtains c1,2 ~ c3 only if brane-localized electroweak terms are introduced with order-one coefficients; otherwise c1,2 are loop-suppressed. Because the muon-collider signal is electroweak production of the axion (Sec. 2.1.1) and the reach scales roughly as (c1^2 + c2^2)/f_a^2, the lower panel of Fig. 4 shows that c1 = c2 = 0.01 reduces the f_a reach by about two orders of magnitude. The abstract and Sec. 4 claim that several UV scenarios overlap the muon-collider reach; this claim should be qualified as unconditional for the mirror and composite models but conditional for the product-group and extra-dimension benchmarks unless the electroweak couplings are specified or computed in those models.
  2. [Sec. 4, Fig. 9, and Eq. (3.35)] The composite benchmark lies partly in the region where the axion effective field theory used for the collider analysis is not valid. With m_V = 2 TeV, Eq. (3.35) gives m_a^2 ~ 4 pi m_V f_a, so m_a > 4 pi f_a whenever f_a < m_V/(4 pi) ~ 160 GeV. The text acknowledges the m_a > 4 pi f_a region in general, but for this benchmark a substantial part of the line shown in Fig. 9 is in that region, so the projected overlap with the muon-collider reach is not a direct computation from Eq. (2.1) there. The benchmark should be restricted to EFT-valid values of f_a, or the projections in the invalid region should be computed in an explicit UV completion.
minor comments (4)
  1. [Sec. 2.2 and Fig. 4] The pre-selection requires M_jj < 9800 GeV, while the abstract and Fig. 4 claim reach up to m_a ~ 10 TeV; for m_a = 10 TeV the signal peak sits at about 10 TeV and is excluded by this cut. Please clarify whether the actual mass endpoint is 9.8 TeV or justify the cut if the 10 TeV statement is intended.
  2. [Fig. 4 lower panel] Please state explicitly whether the existing-constraint shaded regions in the lower panel are rescaled for c1 = c2 = 0.01 or simply repeated from the upper panel; the text says electroweak-production constraints weaken for small c1,2, and the plot should not visually suggest otherwise.
  3. [Sec. 3.1 and Figs. 6, 8] The notation alpha_1, alpha_2 is overloaded: in Sec. 2 these denote U(1)_Y and SU(2)_L couplings, while in Sec. 3.1 they denote SU(3) gauge couplings. The text notes this once, but a distinct notation for the SU(3) couplings would help readers.
  4. [Figs. 10 and 11 and their captions] The captions refer to '2-to-4(3)' processes in a way that is easy to misread. Please specify in each caption which curve is the 2-to-3 process and which is the 2-to-4 process.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the muon-collider reach is computed from the effective Lagrangian and compared with external constraints; the UV model lines are independent inputs, not recycled predictions.

full rationale

The paper's central derivation is self-contained. The projected reach is obtained from the effective Lagrangian in Eq. (2.1), with production cross sections computed from the stated electroweak couplings and decay widths from Eqs. (2.5)-(2.9). No fitted parameter is recycled as a prediction: the Wilson coefficients c1, c2, c3 are treated as free inputs, and the projected sensitivity curves in Fig. 4 are obtained by applying the same Lagrangian to signal and background simulation, then compared with published experimental constraints from ATLAS, CMS, LEP, and Pb-Pb collisions. The UV model benchmarks are independent inputs: the product-group, extra-dimension, mirror, and composite model lines in Fig. 9 come from previously published mass relations (e.g., Eqs. (3.12), (3.27), (3.29), (3.33), (3.35)), which do not depend on the muon-collider sensitivity calculation. The paper explicitly flags where an additional assumption is needed, e.g., 'To make our analysis general, we will also assume SU(2)_L and U(1)_Y couplings of heavy QCD axions' for the product-group model, and notes that the electroweak couplings may be loop-suppressed in the extra-dimension case; Fig. 4 lower panel transparently shows the resulting loss of sensitivity when c1=c2=0.01. This is an assumption about model parameter space, not a reduction of the prediction to its input. Self-citations to previous model-building papers are used as independent published results for the mass formulas, and the paper does not rely on a self-cited uniqueness theorem to forbid alternatives. The choice of benchmark parameters that overlap the reach region is a presentation choice, not a circular derivation, because the reach is computed before and independently of the benchmark lines. Overall, no equation or projection reduces by construction to a fitted input or to a self-citation.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The projection rests on the effective Lagrangian in Eq. (2.1) and on the assumption that electroweak production is sizeable. The UV benchmarks are derived from small-instanton, extra-dimensional, mirror, and compositeness constructions with parameters chosen by hand; those parameters are listed above. The collider analysis further assumes a clean, parton-level environment with no beam-induced or detector-level backgrounds. No genuinely new particles are introduced.

free parameters (9)
  • Anomaly coefficients c1, c2, c3 = c1=c2=c3=1 (benchmark); c1=c2=0.01 (weak-coupling case)
    Set by hand in Eq. (2.1); the projected reach scales linearly with c1 and c2 (Sec. 2.2, Fig. 4).
  • Product-group breaking scale M = 5 x 10^12 GeV
    Chosen in Sec. 3.1 to satisfy the CP-odd operator constraint (2 pi M / M_Pl)^2 less than about 10^-10 and to give a heavy axion near the TeV scale.
  • Product-group gauge coupling alpha1(M) and decay constants = alpha1(M)=0.12, fa1=500 GeV, fa2=fa3=M, lambda1=lambda2=0.5
    Chosen so the heaviest product-group axion has ma1 about 250 GeV (Sec. 3.1, Fig. 6), within muon collider reach.
  • Extra-dimension compactification scale 1/R = 2.5 x 10^6 GeV (benchmark)
    Chosen for the bulk-scalar axion benchmark in Fig. 9 and Eq. (3.27) to give ma about 285 GeV.
  • Extra-dimension perturbativity parameter epsilon = 0.36
    Maximizes the 5D small-instanton enhancement while keeping the theory perturbative (Sec. 3.2).
  • 5D cutoff scale M5 and symmetry-breaking scale f_hat_a = M5 about 10^2/R, f_hat_a about 60 GeV
    Chosen so the volume factor gives fa about TeV in the bulk-scalar extra-dimension model (Sec. 3.2).
  • Mirror QCD confinement scale Lambda_QCD' = 500 GeV
    Benchmark for the mirror model in Fig. 9, chosen below the dimension-six operator bound Lambda_QCD' less than about 2 TeV.
  • Vector-like quark mass m_V (composite axion) = 2 TeV
    Benchmark for the color-unification composite axion line in Eq. (3.35) and Fig. 9.
  • W/Z reconstruction efficiency = 0.80
    Assumed in Sec. 2.2 for VBS and associated production channels; directly affects the projected reach.
assumptions (6)
  • domain assumption The dilute instanton gas approximation reliably computes the heavy-axion mass from small instantons in product-group and extra-dimensional models.
    Used in Sec. 3.1 (Eqs. 3.6-3.12) and Sec. 3.2 (Eqs. 3.19-3.27) to derive the benchmark mass relations; the paper restricts to parameter regions where the relevant gauge coupling remains perturbative.
  • ad hoc to paper The heavy QCD axion couples to electroweak gauge bosons with O(1) anomaly coefficients c1 and c2.
    Needed for VBF and VBS production at a muon collider; for product-group and extra-dimension models this is assumed, as stated in Sec. 3.1 and Sec. 3.2.
  • domain assumption The muon collider environment is clean at parton level, with backgrounds limited to electroweak processes and no significant beam-induced or detector-level background.
    Backgrounds in Table 1 and Fig. 3 contain only EW VBS and Drell-Yan-like processes; no simulation of beam-induced muon-decay backgrounds is included (Sec. 2.2).
  • domain assumption Planck-scale suppressed operators with O(1) coefficients parameterize the axion quality problem (Eq. 3.3).
    Standard assumption used to define the 'Axion Quality Dim-5' boundary in Fig. 9 and to constrain M and Lambda_5 in Sec. 3.
  • standard math The standard QCD axion mass relation m_a = sqrt(mu md)/(mu+md) m_pi f_pi / f_a holds for the minimal QCD contribution.
    Eq. 3.2, cited to chiral Lagrangian literature; used as the baseline that heavy-axion models modify.
  • domain assumption The EFT description of the axion is valid only for ma less than 4 pi fa.
    Used to draw the gray 'ma > 4 pi fa' boundary in Fig. 9 and to exclude some parameter space from the projection.

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Cite this review

Pith. "Pith review of Heavy QCD Axions at High-Energy Muon Colliders." pith.science (2026). https://pith.science/paper/HHWXAWV6

@misc{pith2026250910605,
  author       = {Pith},
  title        = {Pith review of: Heavy QCD Axions at High-Energy Muon Colliders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HHWXAWV6}},
  note         = {Machine review of arXiv:2509.10605}
}
abstract

We study the physics potential of heavy QCD axions at high-energy muon colliders. Unlike typical axion-like particles, heavy QCD axions solve the strong CP problem with phenomenology driven by the anomalous gluon ($aG\widetilde G$) couplings. Several ultraviolet scenarios are presented in which QCD axions with TeV-scale masses and decay constants arise consistently with a solution to both the strong CP problem and the axion quality problem. We perform a detailed collider analysis for both a 3 and 10~TeV muon collider, focusing on hadronic axion decays that gives rise to a dijet-resonance signature. Our projections for the axion discovery reach in the multi-TeV mass range demonstrate that a muon collider can significantly extend sensitivity to heavy QCD axions compared to existing experiments.

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