{"id":"a76c185f-540e-4ee2-9290-302510368903","arxiv_id":"2509.10605","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"A 10 TeV muon collider can detect TeV-mass QCD axions through their decay into two jets, extending sensitivity beyond current and HL-LHC experiments.","lead":"High-energy muon colliders could discover heavy QCD axions, particles that solve a long-standing puzzle about time-reversal symmetry in the strong nuclear force, and the paper calculates the expected signal and discovery reach. A 10 TeV muon collider could probe axions with TeV-scale masses, covering parameter space beyond the LHC and testing several theoretical models.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Projected reach assumes O(1) electroweak anomaly coefficients c1,c2; for the product-group and extra-dimension benchmarks these are not predicted and may be loop-suppressed, so the Fig. 9 overlap is conditional.","rationale":"The reader's weakest assumption correctly identifies the O(1) electroweak couplings c1,c2 as the most load-bearing element: the muon collider produces heavy QCD axions through electroweak VBF/VBS processes (Sec. 2.1.1), while the paper's own Fig. 4 lower panel quantifies how strongly the reach degrades when c1 = c2 = 0.01. For the product-group and extra-dimension benchmarks, the manuscript explicitly assumes these couplings rather than deriving them, so the claim that all four benchmark models fall within the projected reach depends on an additional, undocumented model-building choice. The mirror and composite models do naturally generate c1,2 ~ 1, so the general conclusion that a muon collider can probe heavy QCD axions is not overturned; however, the paper should either specify a minimal EW-charged extension for the two affected models or present those benchmarks with the degraded sensitivity. This supports the reader's CONDITIONAL verdict without changing it.","tokens_in":27457,"tokens_out":38068,"duration_ms":313067,"concrete_test":"Take the minimal field content of the product-group benchmark (Sec. 3.1) and of the bulk-scalar extra-dimension benchmark (Sec. 3.2) with the PQ fermions uncharged under SU(2)_L × U(1)_Y, and compute the one-loop-induced c1,c2 (or verify that they vanish). If the resulting coefficients are ≤ 0.01, re-evaluate the benchmark points using the c1 = c2 = 0.01 sensitivity panel of Fig. 4 and determine whether the solid lines in Fig. 9 still lie above the 10 TeV dashed curve. This directly tests whether the central claim requires an undocumented EW-charged extension.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The muon-collider signal is electroweak production of the axion (VBF and VBS, Sec. 2.1.1) followed by gluonic decay, so the reach scales approximately as c_EW^2/f_a^2 with c_EW^2 ~ c1^2 + c2^2. Figure 4 shows that reducing c1 = c2 from 1 to 0.01 lowers the reach in f_a by about two orders of magnitude. In the product-group benchmark (Sec. 3.1) and the bulk-scalar extra-dimension benchmark (Sec. 3.2), c1 and c2 are not determined by the models. 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.' Sec. 3.2 likewise requires ad hoc brane-localized terms to obtain c1,2 ~ c3. If the PQ fermions are EW singlets, the tree-level couplings vanish and loop-induced couplings are typically well below O(1); the low-energy pion-mixing contribution is additionally suppressed for m_a >> Λ_QCD. Hence the Fig. 9 statement that all four benchmark lines lie within the 10 TeV MuC reach rests on an extra assumption. The mirror and composite benchmarks naturally have c1,2 ~ 1, so the central qualitative claim survives for those models, but the 'several UV scenarios' claim is weaker than presented.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":27805,"tokens_out":13765,"duration_ms":132958,"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":[{"comment":"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.","section":"Sec. 3.1, Sec. 3.2, and Fig. 9"},{"comment":"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.","section":"Sec. 4, Fig. 9, and Eq. (3.35)"}],"minor_comments":[{"comment":"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.","section":"Sec. 2.2 and Fig. 4"},{"comment":"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.","section":"Fig. 4 lower panel"},{"comment":"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.","section":"Sec. 3.1 and Figs. 6, 8"},{"comment":"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.","section":"Figs. 10 and 11 and their captions"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically solid and, after revision, suitable for JHEP. The key issue is framing: the conditional nature of the product-group and extra-dimension benchmark lines should be reflected in the abstract, the conclusions, and the Fig. 9 caption. The paper's own Fig. 4 upper and lower panels make the required caveat concrete. I would also ask the authors to address the M_jj < 9800 GeV cut versus the stated 10 TeV mass reach. The self-citation pattern and the explicit documentation of the UFO sign error are not concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth engaging with. It gives a concrete, internally consistent collider analysis of heavy QCD axions at 3 and 10 TeV muon colliders, with the gluon-dominated dijet signature that follows from the axion-gluon coupling. The genuinely new parts are the muon-collider reach curves for the strong-CP-solving heavy axion, the reinterpretation of the 95 GeV diphoton excess constraints, and the overlay of four UV benchmarks in Fig. 9. The fixed-order versus electroweak-PDF cross-check in Appendix A is careful, and documenting the FeynRules UFO sign error is a sign of honest work. The reach calculation itself is not circular: the effective Lagrangian is compared against external constraints, and the model lines are plotted afterward. That part holds up.\n\nThe soft spots are real but not fatal. The most load-bearing one is the assumption about c1 and c2. In the product-group benchmark (Sec. 3.1), the paper explicitly says it will \"also assume\" SU(2)_L and U(1)_Y couplings, and in the extra-dimension bulk-scalar case (Sec. 3.2) it says c1,2 receive only loop contributions unless brane-localized terms with O(1) couplings are added. Since the muon-collider signal is electroweak production followed by gluonic decay, the reach in f_a scales roughly as c_EW^2, and the paper's own Fig. 4 shows that dropping c1=c2 to 0.01 kills the reach. So the Fig. 9 statement that all four benchmark lines lie within the 10 TeV MuC reach is conditional: it is solid for the mirror and composite models, which naturally give c1,2 ~ 1, but for the product-group and extra-dimension lines it rests on an extra assumption that is not derived. That should be stated more carefully than the current text does. Minor issues: no systematic uncertainties on the jet backgrounds, beam-induced backgrounds are not addressed (for a parton-level projection that is acceptable but should be said), and the GitHub availability claim is unverified but not important.\n\nNet: this is a useful phenomenological study with a clear central claim and transparent limitations. The physics case for a muon collider as a probe of heavy QCD axions is strengthened, at least for models where the electroweak couplings are O(1). The paper deserves a serious referee. Send it to review; ask the authors to sharpen the model-coverage claim, justify or relax the c1,c2 assumption for the product-group and extra-dimension benchmarks, and add a paragraph on systematic uncertainties.","headline":"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.","tokens_in":28384,"tokens_out":1454,"would_cite":true,"duration_ms":350103,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A 10 TeV muon collider could discover heavy QCD axions with decay constants of order TeV and masses up to 10 TeV.","keywords":["heavy QCD axion","muon collider","strong CP problem","axion quality problem","dijet resonance","small instantons","vector boson fusion"],"falsifier":"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.","tokens_in":27221,"feed_emoji":"⚛️","tokens_out":9442,"duration_ms":80642,"temperature":0.7,"pith_summary":"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.","feed_headline":"A 10 TeV muon collider could see heavy QCD axions up to 10 TeV","feed_subtitle":"The axion's gluon-decay signature is a clean dijet bump, giving reach well beyond today's LHC searches.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the product-group SU(3)^N construction whose small-instanton contributions raise the axion mass into the TeV range.","marker":"[77–79]"},{"why":"Provides the extra-dimensional frameworks, including the bulk scalar axion, used to derive the 'Extra Dim Bulk' benchmark.","marker":"[92, 93]"},{"why":"Gives the mirror-world and high-quality axion constructions that yield the mirror-model benchmark and earlier LHC targets.","marker":"[35, 94–97]"},{"why":"Presents color-unification models that produce the composite and grand-color axion benchmarks.","marker":"[99–101]"},{"why":"Sets the LHC diphoton constraints that define the existing exclusion regions compared against the muon-collider reach.","marker":"[21, 22, 34]"},{"why":"Supplies the high-mass diphoton search limits used as baseline constraints at larger axion masses.","marker":"[51, 52]"},{"why":"Underlies the electroweak parton distribution treatment used to compute vector-boson-fusion and vector-boson-scattering production rates.","marker":"[61–64]"},{"why":"The event generator used to simulate signal and background processes for the sensitivity projections.","marker":"[65]"}],"fun_headline_variants":["Muon collider can spot heavy QCD axions up to 10 TeV","Dijet resonance lets muon collider probe TeV-scale QCD axions","Heavy QCD axions produce dijet bumps at muon colliders","10 TeV muon collider probes QCD axions via dijet resonance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Muon collider can spot heavy QCD axions up to 10 TeV","Dijet resonance lets muon collider probe TeV-scale QCD axions","Heavy QCD axions produce dijet bumps at muon colliders","10 TeV muon collider probes QCD axions via dijet resonance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000702,"raw_usage":{"total_tokens":3147,"prompt_tokens":900,"completion_tokens":2247,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":2159}},"tokens_in":516,"tokens_out":2247,"duration_ms":14257,"temperature":1.0,"reasoning_tokens":2159,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:55:43.343762+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}