{"id":"59b0faa9-48eb-4e65-a365-05f4ab0d4614","arxiv_id":"2505.02092","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Including one-loop gamma-gamma, gamma-Z, and Z-Z effective vertices can enhance the muon-collider production cross section of the 2HDM pseudoscalar Higgs A by factors of roughly 2 to 20 relative to tree level, with up to about 5 fb in allowed parameter regions.","lead":"Loop-generated couplings of the pseudoscalar Higgs A to photons and Z bosons can roughly double its production rate at a future muon collider in some 2HDM scenarios, and raise it by up to an order of magnitude in others. This matters because A barely couples to muons at tree level, so these loop effects may determine whether a muon collider can see the 2HDM's extra Higgs sector.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"On-shell matching is applied to off-shell VV fusion: the constant-coefficient EFT vertices may artificially generate the claimed high-energy enhancements, so the central quantitative claim is not yet established.","rationale":"The paper asks a well-posed question and the on-shell matching calculation is standard. The concern is not that the matching is wrong, but that the matched constants are inserted into a local dimension-5 operator and used in MadGraph for t-channel VV fusion, where one or both vector bosons are off-shell. All quantitative novelty—the factor ~2 (Type-II), ~10 (Type-X), and especially the up-to-~20 enhancements at high sqrt(s) in Figures 7-9—relies on the momentum growth of the derivative vertex in Eq. (4). The loop form factors are not momentum-independent; Eqs. (13)-(15) are evaluated at on-shell M_Z^2. A constant-coefficient local operator cannot capture the nonlocal form-factor suppression at high virtuality. This is the single most load-bearing assumption because if it fails, the high-energy enhancement disappears and the claim that the muon collider is a feasible alternative for probing the 2HDM pseudoscalar is weakened; the low-mass/low-tan-beta region may remain qualitatively enhanced but at a reduced and uncertain level. I also notice that the abstract, introduction, and figures quote different enhancement factors (2, 2-8, 10, 20), which makes the precise claim harder to evaluate, but that is secondary to the form-factor issue. The proposed test is decisive: recompute one benchmark with full off-shell form factors. If the rate is unchanged, the concern is retired; if it drops significantly, the headline numbers need revision. This matches the reader's identified weakest assumption, and a CONDITIONAL verdict with this specific check is the appropriate posture.","tokens_in":13179,"tokens_out":9277,"duration_ms":125885,"concrete_test":"Recompute the full one-loop amplitude for mu+ mu- -> mu+ mu- A retaining the complete off-shell VV->A form factors (replace the constant g_A VV in Eq. (3) with F_V1V2(q1^2, q2^2, m_A^2) obtained from the Passarino-Veltman integrals for spacelike q_i^2, e.g., using LoopTools/FormCalc or by extending Eq. (12) to arbitrary q1^2, q2^2) and compare with the MadGraph EFT prediction for representative benchmarks mA = 100 GeV, tan beta = 5 and mA = 2 TeV, tan beta = 5 at sqrt(s) = 3 and 10 TeV. If the full result drops by more than ~20% in the region where the claimed enhancement is ~2 or larger, the EFT-based enhancement factors are overestimates and should be replaced by the form-factor-rescaled values.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claims (enhancements of ~2 and ~10, cross sections up to ~1 and ~5 fb, and the statement that the NLO grows at high sqrt(s) because of 'derivative-type couplings') rest on treating the one-loop A VV vertices of Eq. (3) as local, momentum-independent couplings with the tensor structure k1_rho k2_sigma in Eq. (4). These couplings are fixed by matching to on-shell decay amplitudes: Eq. (9) for A->gamma gamma, Eq. (10) for A->gamma Z, and Eqs. (12)-(15) for A->ZZ, where the Passarino-Veltman functions C0, C11, C12 are evaluated at q1^2 = q2^2 = M_Z^2 and P^2 = M_A^2. In the t-channel VV-fusion process mu+ mu- -> mu+ mu- A, the exchanged gamma*/Z* have spacelike q_i^2 that become large at high sqrt(s) and for mA up to 2 TeV. The true one-loop form factors depend on q1^2, q2^2, and P^2; there is no justification for using the on-shell constants throughout the kinematic range. If they fall with increasing |q_i^2|, the linear growth of the EFT vertex in k_i overestimates the cross section, and the factors of ~20 in Figures 7-9, together with much of the claimed NLO enhancement, are artifacts. The paper itself highlights the energy growth of the effective vertex without checking the validity of the constant-coefficient approximation in the off-shell region.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the production of the CP-odd Higgs A in the process mu+ mu- -> mu+ mu- A at future muon colliders within the Type-II and Type-X 2HDM. It introduces dimension-five effective vertices A gamma gamma, A gamma Z, and A Z Z, with coefficients matched to the known one-loop decay amplitudes for A -> gamma gamma, gamma Z, and Z Z. These effective vertices are implemented in FeynRules/MadGraph to compute the loop-induced VV-fusion contribution (called NLO) and compare it with the tree-level A-strahlung cross section (called LO). The paper reports enhancements up to ~2 in Type-II and ~10 in Type-X, and cross sections up to ~1 fb and ~5 fb in experimentally allowed regions, and concludes that a muon collider is a feasible probe of the 2HDM pseudoscalar sector.","tokens_in":13514,"tokens_out":9896,"duration_ms":117167,"significance":"If the reported cross sections were correct, the loop-induced VV-fusion mechanism would be a leading production mode for a light pseudoscalar A at multi-TeV muon colliders, with discovery potential especially in the weakly constrained Type-X model. The matching of the effective couplings to published exact one-loop amplitudes (Eqs. (5)-(15)) is transparent and correctly implements the top/bottom/tau loop contributions, and the use of external LHC and B->X_s gamma constraints is appropriate. However, the central quantitative claims rest on treating on-shell-matched constants as local couplings for off-shell t-channel vector bosons, an approximation whose validity is not demonstrated and which is likely to fail in the high-energy regime where the claimed enhancements are largest. The paper therefore provides a useful framework and a clear benchmark calculation, but its headline numbers require substantial additional work before they can be considered robust predictions.","major_comments":[{"comment":"The effective couplings g_A^VV are matched to the on-shell decay amplitudes A->VV, with the Passarino-Veltman functions evaluated at q1^2 = q2^2 = M_V^2 and P^2 = M_A^2 (Eqs. (9)-(15)). They are then implemented as constant, momentum-independent local vertex factors in the t-channel vector-boson-fusion process mu+ mu- -> mu+ mu- A, where the exchanged gamma*/Z* have spacelike virtualities that can reach |q^2| ~ s, with s up to (20 TeV)^2. The vertex of Eq. (4) is proportional to k1^rho k2^sigma, so the amplitude grows linearly with the boson momenta; the paper explicitly attributes the high-energy enhancement to this 'derivative-type coupling' (Section III, discussion of Figs. 7 and 9). However, the actual one-loop form factors depend nontrivially on q1^2, q2^2, and P^2, and for off-shell photons the triangle amplitude is known to fall as 1/q^2 at large virtuality rather than to grow. No justification is given for using the on-shell constants throughout the kinematic range, and no check (for example, comparing with the full one-loop matrix element at a test point) is provided. Since the claimed enhancements of ~2 (Type-II) and ~10 (Type-X) and the associated cross-section maxima in Figs. 2-10 are driven by this high-energy growth, the central quantitative claims are not established by the present calculation.","section":"Section II, Eqs. (5)-(10)"},{"comment":"For mA > 2 m_f, the loop functions f(tau) and I2 in Eqs. (7)-(8) and (11) acquire imaginary parts, so the matched coefficients g_A^gamma gamma and g_A^gamma Z are complex in general (for example, for the top loop when mA > 350 GeV). Eq. (5) uses (g_A^gamma gamma)^2 rather than |g_A^gamma gamma|^2, which is only valid for a real coupling, and the effective Lagrangian of Eq. (3) is written with real coefficients. The paper does not explain how the complex couplings are inserted into the FeynRules model, how Hermiticity of the Lagrangian is restored, or how the phases affect the interference between the gamma-gamma, gamma-Z, and Z-Z fusion amplitudes. This is not a purely formal point: for mA = 500 GeV (used in Fig. 8 for Type-X) the imaginary parts from the top loop are sizeable, and the relative phases can change the predicted cross sections. The matching procedure should be restated in terms of a Hermitian Lagrangian with complex coefficients (or with separate real and imaginary parts), and the width formulas should use absolute values.","section":"Section II, Eqs. (5)-(10)"}],"minor_comments":[{"comment":"The text describes the calculation as 'NLO', but only the VV-fusion diagrams with the triangle insertion are computed; the full one-loop correction to mu+ mu- -> mu+ mu- A would also include vertex and box corrections to the A-strahlung diagrams. The terminology should be qualified, for example as 'loop-induced VV-fusion contribution', to avoid implying a complete NLO calculation.","section":"Abstract and Sections I, IV"},{"comment":"The exclusion contours are said to come from LHC searches [49] and B->X_s gamma [50-52], but the specific limits used (for example, the numerical value of Br(B->X_s gamma) and the precise LHC analysis) are not given, so the reader cannot reproduce the excluded regions.","section":"Section III, Figs. 2-4"},{"comment":"There is a typo in the conclusion: 'searching for the the pseudoscalar A' should read 'searching for the pseudoscalar A'.","section":"Section IV (Conclusion)"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the off-shell extrapolation. I would ask the authors to repeat the calculation using the exact one-loop form factors evaluated at the off-shell kinematic points, or at least to demonstrate numerically that the constant-coefficient approximation is accurate in the region of parameter space where the enhancement claims are made. If the enhancement disappears when form factors are included, the paper's main claim fails; if it survives, the paper would be considerably strengthened. The complex-coupling issue should also be clarified before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Fair warning up front: the paper's qualitative message—loop-induced VV fusion is a relevant production channel for the pseudoscalar A at a muon collider—is plausible, but the quantitative enhancements it advertises are not yet credible. The matching of gAγγ, gAγZ, and gAZZ to the published one-loop decay amplitudes is done cleanly (Eqs. 5–15), and the FeynRules/MadGraph implementation is a standard way to fold those vertices into a cross section. The parameter scan covers the relevant Type-II/Type-X regions, and they overlay the LHC and B→Xsγ constraints, which kills much of the low-mass Type-II parameter space.\n\nThe soft spot is exactly the one the stress-test note identifies: the effective couplings are fixed at on-shell kinematics and then treated as constant, momentum-independent derivative couplings for the t-channel γ*/Z* in µ+µ−→µ+µ−A. That is an uncontrolled approximation. At √s of several TeV and mA up to 2 TeV, the exchanged virtualities are large, and there is no justification for assuming the one-loop form factors are flat in q1^2,q2^2. If they fall with virtuality, the growth of the vertex as k1·k2 overestimates the cross section at high energy—and the factors of ~20 in Figs. 7–9, together with the claim that the NLO beats the 1/s phase-space suppression, become artifacts. The paper explicitly leans on the derivative-type coupling for that energy growth without a validity check.\n\nThere are also internal inconsistencies: the abstract says ~2 (Type-II) and ~10 (Type-X), the intro says ~2–8 for Type-X, and Figures 7 and 9 show up to ~20. Those numbers need to be reconciled. On novelty, Refs. [37] and [53] already study heavy 2HDM Higgs and the CP-odd A at muon colliders; the present paper cites both but never compares cross sections or states clearly what the loop-vertex treatment adds. That comparison should be included.\n\nIf the form-factor issue is fixed—by computing the full off-shell triangle vertices, or by using EFT with momentum-dependent form factors—the qualitative conclusion that the one-loop contributions are not negligible may survive, at least at low mA and low tanβ. But the claim of a few-fb discovery channel is not yet supported. I would send this to referees because the question is real and the matching is careful, but a referee should insist on seeing the off-shell form-factor behavior and a reconciled set of numbers.","headline":"Clean one-loop matching applied to a muon-collider process, but the advertised high-energy enhancements rest on an uncontrolled off-shell form-factor assumption and need a serious re-derivation.","tokens_in":14089,"tokens_out":3268,"would_cite":false,"duration_ms":34356,"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":"Loop-induced photon and Z fusion can dominate pseudoscalar Higgs production at a muon collider, with rates high enough to probe the extended Higgs sector.","keywords":["two-Higgs-doublet model","pseudoscalar Higgs","muon collider","effective vertices","one-loop corrections","vector-boson fusion","Type-II 2HDM","Type-X 2HDM"],"falsifier":"Compute the exact one-loop amplitude for $\\mu^+\\mu^-\\to\\mu^+\\mu^- A$ with momentum-dependent form factors retained, for example at $\\sqrt{s}=10$ TeV, $m_A=500$ GeV, and $\\tan\\beta=5$ in Type-X; if the resulting ratio of loop-induced to tree-level cross section is far below the claimed factor of about 20, the point-like effective-vertex treatment fails.","tokens_in":12970,"feed_emoji":"⚛️","tokens_out":9368,"duration_ms":89590,"temperature":0.7,"pith_summary":"The paper tries to show that the pseudoscalar Higgs boson $A$ of Type-II and Type-X Two-Higgs-Doublet Models, which is nearly impossible to produce at a muon collider through its tiny direct coupling to muons, can instead be produced through photon and $Z$ boson fusion mediated by one-loop fermion triangles. The triangle diagrams induce effective $\\gamma\\gamma A$, $\\gamma Z A$, and $ZZ A$ vertices, and including them can double the production cross section in Type-II and enhance it by up to an order of magnitude in Type-X, most strongly at low $m_A$ and low $\\tan\\beta$. In the parameter space not yet excluded by experiment, the model predicts cross sections above about 1 fb for Type-II and 5 fb for Type-X at a 3 TeV muon collider, which would make the collider a genuine probe of the extended Higgs sector. If the calculation holds, loop-induced vector-boson fusion is the dominant production mechanism for the pseudoscalar in large parts of the 2HDM parameter space.","feed_headline":"One-loop fusion can dominate pseudoscalar Higgs production","feed_subtitle":"At a 3 TeV muon collider the rate can reach about 1 fb in Type-II and 5 fb in Type-X, enough to probe the pseudoscalar.","key_machinery":"The load-bearing object is the set of one-loop effective vertices $\\gamma\\gamma A$, $\\gamma Z A$, and $ZZ A$, embodied in the effective Lagrangian $\\mathcal{L}_{\\rm eff} \\supset \\frac{g_{A\\gamma\\gamma}}{v} A\\, F_{\\mu\\nu}\\tilde F^{\\mu\\nu} + \\frac{g_{A\\gamma Z}}{v} A\\, F_{\\mu\\nu}\\tilde Z^{\\mu\\nu} + \\frac{g_{AZZ}}{v} A\\, Z_{\\mu\\nu}\\tilde Z^{\\mu\\nu}$. Because $A$ does not couple to $W$ or $Z$ at tree level, only fermion loops contribute; the paper keeps the top, bottom, and tau loops. The coefficients are determined by matching the effective-theory decay widths to the exact one-loop results, so the vertices carry the exact dependence on fermion masses and Yukawa ratios through the standard loop functions. These vertices are then used as local momentum-dependent derivative couplings in the simulation of $t$-channel fusion, and this derivative structure is what produces the energy enhancement at high $\\sqrt{s}$: the amplitude grows with the momentum of the exchanged boson, partially cancelling the phase-space suppression.","core_discovery":"The central claim is that one-loop corrections to the production of the CP-odd Higgs boson $A$ in $\\mu^+\\mu^- \\to \\mu^+\\mu^- A$ are not a small correction: they are comparable to, and in some regions larger than, the tree-level contribution. After integrating out the fermion loops, the paper writes an effective Lagrangian with derivative couplings $\\frac{g_{A VV}}{v} A\\, V_{\\mu\\nu}\\tilde V^{\\mu\\nu}$ and fixes the coefficients by matching the resulting decay widths to the exact one-loop rates for $A\\to\\gamma\\gamma$, $A\\to\\gamma Z$, and $A\\to ZZ$ from the literature. Using these vertices for $t$-channel vector-boson fusion in a simulation, the paper finds enhancements of roughly a factor of 2 for Type-II and up to about 10 for Type-X in the low-$m_A$, low-$\\tan\\beta$ region, with the top-quark loop (plus the bottom-quark loop in Type-X) responsible for the low-$\\tan\\beta$ enhancement. At fixed mass, the loop contribution grows relative to the tree level with collision energy, because the derivative couplings produce an energy-growing amplitude that competes with the $1/s$ phase-space falloff. In the experimentally open regions, the predicted cross sections reach about 1 fb for Type-II and 5 fb for Type-X at $\\sqrt{s}=3$ TeV, which the paper presents as making the muon collider a feasible discovery machine for the pseudoscalar Higgs sector.","pith_inferences":["A natural extension, not pursued in the paper, is to Type-I and Type-Y models: since all quark Yukawa ratios are $\\cot\\beta$ in Type-I, the low-$\\tan\\beta$ enhancement could be even more pronounced, and the absence of a compensating bottom-loop effect might distinguish Type-I from Type-X in the same channel.","The same effective vertices predict $\\gamma\\gamma$, $\\gamma Z$, and $ZZ$ decay signatures for $A$; measuring $A$ in both its production and its loop-induced decays at a muon collider could test the consistency of the derivative-coupling treatment.","The high-energy growth is the place where the shortcut is most vulnerable: if the full one-loop form factors for off-shell $V^*V^*\\to A$ flatten or fall at large virtuality, the advertised enhancement factors, especially the factor of about 20 at high $\\sqrt{s}$, would be an artifact of the local vertex approximation.","Because the tree-level process is governed by the muon Yukawa coupling, the same method could be applied to other weakly coupled new scalars at muon colliders, turning loop-induced fusion into a general search strategy for CP-odd states."],"forward_implications":["In Type-II, the one-loop $\\gamma\\gamma$, $\\gamma Z$, and $ZZ$ fusion contributions can double the production cross section of $A$ in the low-$m_A$, low-$\\tan\\beta$ region, where the top-quark loop dominates.","In Type-X, the same loop contributions can enhance the cross section by up to about an order of magnitude, making loop-induced vector-boson fusion the dominant production channel there.","At a 3 TeV muon collider, the experimentally allowed parameter space can yield cross sections of about 1 fb in Type-II and 5 fb in Type-X, rates high enough for a dedicated search for the pseudoscalar Higgs.","Raising the collision energy does not simply suppress the loop-induced signal: because the effective vertices are derivative couplings, the NLO-to-LO ratio grows with $\\sqrt{s}$, reaching factors of up to about 20 for Type-X with $m_A=500$ GeV at low $\\tan\\beta$.","The largest absolute cross sections occur at high $\\tan\\beta$, while the largest relative enhancements occur at low $\\tan\\beta$, a separation that could help discriminate the 2HDM type."],"supporting_citations":[{"why":"Supplies the exact one-loop A-to-gamma-gamma and A-to-gamma-Z decay amplitudes used to fix the EFT vertex coefficients by matching.","marker":"[41]"},{"why":"Provides one of the derivations of the AZZ effective coupling used for the ZZ fusion contribution.","marker":"[42]"},{"why":"Gives the AZZ vertex expression with the three-point loop integrals that the paper adopts for its Z Z A effective vertex.","marker":"[43]"},{"why":"Establishes the muon collider as a capable probe of 2HDM Higgs types, the context this production study extends.","marker":"[37]"},{"why":"The event generator used to compute the tree-level and effective-vertex cross sections presented in the figures.","marker":"[48]"},{"why":"Supplies the collider exclusion contour that defines the experimentally open region for Type-II.","marker":"[49]"},{"why":"Supplies the B-to-X_s-gamma constraint used to exclude the low-mass Type-II region.","marker":"[50]"}],"fun_headline_variants":["Pseudoscalar Higgs: one-loop vertices boost rates up to 10x","Loop corrections can multiply pseudoscalar Higgs rate by up to 10","One-loop fusion lifts pseudoscalar Higgs yields at muon colliders","Muon collider: loop effects make pseudoscalar Higgs reachable","One-loop vertices: tenfold pseudoscalar Higgs boost at muon colliders"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes that the one-loop gamma/Z fusion vertices, fixed by matching on-shell decay amplitudes, remain accurate as local derivative couplings when the exchanged bosons are virtual and far off their mass shells in the t-channel fusion process; if the true quantum amplitudes flatten or fall with virtual momentum instead of growing, the predicted enhancement is an overestimate.","fun_headline_variants_meta":{"raw":{"variants":["Pseudoscalar Higgs: one-loop vertices boost rates up to 10x","Loop corrections can multiply pseudoscalar Higgs rate by up to 10","One-loop fusion lifts pseudoscalar Higgs yields at muon colliders","Muon collider: loop effects make pseudoscalar Higgs reachable","One-loop vertices: tenfold pseudoscalar Higgs boost at muon colliders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001792,"raw_usage":{"total_tokens":7147,"prompt_tokens":1121,"completion_tokens":6026,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":737,"completion_tokens_details":{"reasoning_tokens":5922}},"tokens_in":737,"tokens_out":6026,"duration_ms":43250,"temperature":1.0,"reasoning_tokens":5922,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T01:03:53.244520+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact one-loop amplitude for $\\mu^+\\mu^-\\to\\mu^+\\mu^- A$ with momentum-dependent form factors retained, for example at $\\sqrt{s}=10$ TeV, $m_A=500$ GeV, and $\\tan\\beta=5$ in Type-X; if the resulting ratio of loop-induced to tree-level cross section is far below the claimed factor of about 20, the point-like effective-vertex treatment fails.","supporting_citations":[],"review_version":1}