{"id":"f31d98f1-0ee2-41f0-8428-94e1d5c68fdd","arxiv_id":"2412.10232","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Adiabatic level crossing between a QCD axion and an ALP can convert ALP dark matter into QCD axion dark matter, and a beat-frequency-based adiabatic condition maps the viable parameter region.","lead":"This paper studies two axions whose masses cross as the universe cools, allowing energy to flow from one axion into the QCD axion. The authors derive a refined condition for this transfer and map out axion masses and couplings that would produce the observed dark matter.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The adiabatic transfer in Eq. (39) is calibrated for a single initial phase (θa=0, θφ=1); for generic O(1) initial angles the transfer efficiency is not proven, so the DM abundance could be phase-dependent.","rationale":"The paper's central claim is that adiabatic level crossing transfers the ALP-dominated heavy-mode number density into the QCD axion, producing dark matter for fa ~ 10^9-10^12 GeV with O(1) initial angles. The mechanism is internally consistent, and the beat-frequency adiabatic condition is a genuine improvement over earlier mass-frequency conditions; the numerical rf^{-2} scaling in Fig. 7 is compelling. The most load-bearing assumption is not the mere order-of-magnitude of the initial misalignment (which is a standard free parameter in the pre-inflationary misalignment scenario), but rather the implicit assumption that the 100% transfer efficiency used in Eq. (39) holds for arbitrary initial conditions. The adiabatic condition (25) is calibrated using a single initial phase—θa=0, θφ=1—and a 10% residual-transfer threshold. For a two-level system undergoing an avoided crossing, the non-adiabatic transition amplitude and the direction of number transfer depend on the relative phase of the two oscillators (Sec. VI, Fig. 8). Therefore, without scanning initial phases, one cannot be certain that YH is conserved at the 10% level (or better) in the white region of Fig. 3. This directly affects the normalization of Eq. (39) and the position of the non-adiabatic boundary. The thermal-friction estimate (Appendix B) is admittedly preliminary, but even the naive (B12) estimate is well below the Hubble rate for fa ≳ 10^9 GeV, so it does not threaten the central abundance claim in the nominal region. The 10% criterion is arbitrary but does not by itself invalidate the existence of a viable region; only a phase-dependent transfer efficiency would. Hence, the most load-bearing concern is the calibration of the adiabatic condition against a single initial phase. An independent numerical scan over initial phases would settle this and is a natural requirement before accepting the quantitative predictions.","tokens_in":22235,"tokens_out":23961,"duration_ms":212026,"concrete_test":"For a benchmark point in the white region (e.g., fa=10^10 GeV, rm=0.1, rf=100), integrate Eqs. (A5)-(A6) with a set of initial conditions spanning the field quadrant, e.g., (θa,i, θφ,i) = (1,1), (1,-1), (-1,1), (-1,-1), (1,0.5), (0.5,1), and compute Rad = |YH(after)−YH(before)|/(YH+YL) as in Eq. (A7). If max Rad over these phases exceeds the 0.1 threshold used to fit Cad, or if the final ΩH varies by more than 10% across phases, then the adiabatic condition as calibrated is insufficient to support the central claim. As a second check, repeat the rf-scan in Fig. 7 with initial condition θa=θφ=1; if the inferred Cad changes by more than ~50%, the fitted coefficient is phase-dependent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central abundance formula Eq. (39) assumes that the comoving number density YH of the heavy mode is transferred to the QCD axion with 100% efficiency whenever the beat-frequency adiabatic condition (25) is satisfied. This condition is calibrated in Sec. VI and Appendix A by integrating the equations of motion (A5)-(A6) with the single initial condition θa(τi)=0, θφ(τi)=1 and defining adiabaticity breaking as Rad=0.1. The actual DM scenario has both θa,i and θφ,i of order unity, so the relative phase of the two modes at the level crossing is arbitrary. Because the two-state system exhibits Landau-Zener-like transitions whose direction and magnitude depend on the relative phase (see Fig. 8 and the discussion in Sec. VI), a calibration run with one initial condition does not establish that the transfer efficiency is ≥90% for all, or even typical, O(1) initial phases. If the residual non-adiabatic transfer exceeds 10% for some phases, the normalization of Eq. (39) and the boundary of the white region in Fig. 3 are not robust.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the level-crossing phenomenon in a two-axion system described by the potential in Eq. (5), in which the QCD axion mixes with an axion-like particle. The authors define a basis-independent level-crossing temperature and timescale, propose the refined adiabatic condition in Eq. (25) involving the beat frequency, calibrate the order-unity coefficient Cad numerically with a 10% criterion, and use conservation of comoving number densities to derive the dark-matter abundance formula in Eq. (39). They present the viable parameter region for QCD axion dark matter in Fig. 3, derive axion-photon coupling predictions in Figs. 4 and 5, prove the equivalence of two bases in Sec. V, and estimate thermal friction from chiral perturbation theory in Appendix B. The central claim is that for rf ≫ 1 and rm ≪ 1 the ALP-dominated heavy mode adiabatically becomes the QCD axion, so that the QCD axion can account for all dark matter with fa ≃ 10^9–10^12 GeV.","tokens_in":22431,"tokens_out":17080,"duration_ms":185190,"significance":"If the central claim holds, this provides a concrete and testable mechanism for enhancing the QCD axion abundance beyond the standard misalignment prediction, opening decay constants around 10^9–10^12 GeV and giving specific correlations between axion mass and photon coupling that can be probed by experiments such as BREAD, MADMAX, and DMRadio. The derivation of the mass eigenvalues, mixing angle, and abundance transfer is standard and internally consistent; the basis equivalence in Sec. V is exact algebra; and the adiabatic-condition analysis correctly identifies the beat frequency as the relevant timescale, improving on earlier criteria. The paper is also commendable for making the numerical validation procedure explicit. The main reservation is that the numerical calibration of the adiabaticity threshold is performed for a single initial field configuration, and the paper does not yet demonstrate that the resulting 10% criterion controls the transfer efficiency for the generic O(1) initial angles assumed in the dark-matter scenario.","major_comments":[{"comment":"The numerical calibration of Cad, which controls the white/gray boundary in Fig. 3, uses a single initial condition, and the two descriptions of that condition are inconsistent. Sec. VI states that the runs set θa = 1, θϕ = 0, 'meaning that only the light mode is present,' while Appendix A states θϕ(τi) = 1, θa(τi) = 0. At temperatures well above the crossing the light mode is approximately a and the heavy mode is approximately ϕ, so these are physically different initial states. Since Fig. 8 and the surrounding discussion in Sec. VI show that the direction and magnitude of non-adiabatic number transfer depend on the relative phase of the two modes, a calibration run that starts with only one mode excited does not by itself bound the transfer efficiency for the generic O(1) initial angles used in Eq. (39). The authors should specify which initial condition was actually used, scan the relative phase θa,i versus θϕ,i (and the oscillation phases at the crossing), and either demonstrate that Rad ≤ 0.1 for all such cases or set Cad conservatively. In addition, the calibrated value of Cad is never quoted in Sec. VI or Fig. 7; this value is needed to apply Eq. (25) and to reproduce Fig. 3.","section":"Sec. VI and Appendix A; Eqs. (25), (39)"},{"comment":"The central abundance formula assumes that both initial misalignment angles are of order unity and scales as θϕ,i²; if the ALP sits near its potential minimum, the enhancement is lost. The paper states this assumption but does not quantify how large θϕ,i must be for the level-crossing mechanism to account for ΩDM h² ≈ 0.12 in the regions shown in Fig. 3, nor does it discuss the tuning cost of this assumption relative to the single-axion case. This is a limitation rather than an inconsistency, but it should be stated prominently because the viability claim is conditional on it.","section":"Sec. IVB and Eq. (39)"}],"minor_comments":[{"comment":"The 'No level crossing' boundary in Fig. 3 depends on an arbitrary threshold Δα_min, with three values π/6, π/4, π/3 shown. Since the level-crossing condition in Eq. (17) is used to define the viable region, the boundaries of that region at moderate rf are not derived from a physical criterion. For the rf ≫ 1 regime of Eq. (39) this is not crucial, but the full parameter-space claim in Fig. 3 would benefit from either a derivation of Δα_min or an explicit statement that the boundary is a convention.","section":"Sec. IVA, Eq. (16), and Fig. 3"},{"comment":"The evaporation boundary uses the thermal-friction estimate Γχ_dis from Eq. (29), which the paper itself describes as a 'very naïve' estimate and states is beyond the scope of the paper to compute precisely. The green region in Fig. 3 should therefore be presented with an uncertainty band or at least with an explicit caveat that the boundary is order-of-magnitude only.","section":"Appendix B and Fig. 3"},{"comment":"The color scale in Fig. 2 is not labeled; the text refers to the 'darkest violet region (left top)' but the reader cannot map the color to the value of α0. Adding a color bar or labeled contours would improve readability.","section":"Fig. 2"},{"comment":"Equation (39) is derived for rf ≫ 1, rm ≪ 1, and g∗ = 60, but the text does not state the range of rf and rm over which this approximation is accurate to, say, 10%. A short quantitative statement would help the reader assess the validity of the analytical contours in Fig. 3.","section":"Sec. IVB"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is well within the scope of the journal and the central idea is attractive. The main technical concern is the numerical calibration of the adiabatic condition: it is based on a single initial condition, the description of that condition is internally inconsistent, and the resulting Cad is not quoted. These issues are fixable by adding a phase scan and reporting the calibration details, so I recommend major revision rather than rejection. I see no grounds for concerns about circularity or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper does what it says. It refines the adiabatic condition for two-axion level crossing, makes it basis-independent, validates it numerically, and maps out a viable dark matter region for QCD axion decay constants below 10^12 GeV. I'd send it to a serious referee.\n\nThe genuinely new pieces are the basis-independent definitions of the level-crossing time and duration (Eqs. 18-20), the demonstration that the beat-frequency condition (25) is the right one — the rf^-2 scaling is confirmed numerically, not just asserted — and the explicit basis-equivalence analysis in Sec V. That last part is particularly nice: it shows that the enhancement scenario is natural in one basis and requires an unnaturally small NA in the other, which explains why earlier papers didn't see it. The abundance estimate, Eq. (39), follows from standard conservation of comoving number density, so it does not depend on the numerical calibration of Cad.\n\nThe weak spots are real but not fatal. The numerical determination of Cad in Sec VI uses a single initial condition, θa=1, θφ=0. The paper itself shows in Sec VI that when the adiabatic condition is only marginally satisfied, the direction and size of non-adiabatic transfer depend on the relative phase of the two modes. That means the 10% criterion calibrated with one initial phase may not bound the transfer error for generic O(1) initial angles, which are the ones relevant for the dark matter abundance. A referee should ask for a scan over initial phases to show the white region in Fig 3 is not overly sensitive. I'd bet the main conclusions survive with modest shifts, but as written it is a gap worth closing.\n\nThe thermal friction estimate in Appendix B is flagged by the authors as preliminary. It is used to carve out the green evaporation region, not the central white region, so it does not undercut the main result, but it deserves a 'handle with care' note.\n\nNo code or data is shipped, which is normal for this subfield, but it means the numerical claims are not fully reproducible without reimplementation.\n\nBottom line: this is a solid paper that moves the phenomenology of axion dark matter forward. The phase-dependence issue should not block peer review; it should be addressed in revision. I'd be happy to cite the basis-independence and beat-frequency results.","headline":"A careful, useful paper on axion level crossing with a refined adiabatic condition; the main results are solid, but the numerical calibration of Cad should be tested against generic initial phases.","tokens_in":23017,"tokens_out":4946,"would_cite":true,"duration_ms":47808,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"An adiabatic level crossing between an ALP and the QCD axion can convert the ALP's dark-matter abundance into the QCD axion, letting a QCD axion with a decay constant near $10^{10}$ GeV account for all dark matter.","keywords":["QCD axion","axion-like particle","level crossing","resonant conversion","beat frequency","adiabatic condition","misalignment mechanism","dark matter"],"falsifier":"Integrate the two-axion equations of motion (A5)–(A6) for, say, $r_f = 10$, $r_m = 0.1$, and $f_a$ a factor of a few above the bound in Eq. (62), starting from $\\theta_\\phi = 1$ and $\\theta_a = 0$: the paper's criterion says the heavy-mode yield $Y_H$ should change by more than 10% across the crossing, while the older $2\\pi/m_L$ condition would predict near-perfect adiabatic transfer, so the numerical outcome settles which adiabaticity claim is right. On the observational side, the scenario is falsified if axion search experiments exclude the predicted bands of Figs. 4–5 — the heavy axion at $|g_{H\\gamma\\gamma}| \\approx \\alpha_{\\rm em}/(2\\pi f_a)$ with mass above $10^{-5}$ eV, or the light axion at the constant-$g_{L\\gamma\\gamma}$ contours — down to the sensitivity required for a subdominant component.","tokens_in":21960,"feed_emoji":"🔄","tokens_out":13266,"duration_ms":122611,"temperature":0.7,"pith_summary":"This paper studies two axions — the QCD axion and a generic axion-like particle (ALP) — whose masses become equal as the universe cools and the QCD axion mass switches on. It argues that if this level crossing happens adiabatically, the comoving number density of the heavier, ALP-dominated mode is transferred wholesale into the QCD axion, so the QCD axion inherits the ALP's abundance. That transfer lets the QCD axion explain the observed dark matter with decay constants around $10^9$–$10^{12}$ GeV, far below the $10^{12}$ GeV needed in the standard single-axion misalignment story. The paper also derives a basis-independent adiabatic condition for the crossing, showing that the beat frequency between the two mass eigenstates, not their individual oscillation frequencies, sets the criterion.","feed_headline":"Level crossing hands dark matter to the QCD axion at 10^10 GeV","feed_subtitle":"When the axion masses cross, the ALP's comoving abundance transfers to the QCD axion, widening the viable dark-matter window.","key_machinery":"The central object is the temperature-dependent $2 \\times 2$ mass matrix of the two axions, whose eigenvalues $m_H(T)$ and $m_L(T)$ cross when the QCD topological susceptibility $\\chi(T)$ turns on. The crossing is characterized by the mixing angle $\\alpha(T)$ and its time derivative at the crossing temperature $T_\\times$ (where $\\alpha$ sits midway between its early- and late-time values): the crossing timescale is $\\Delta t_\\times = |d\\alpha/dt|^{-1}\\big|_{T = T_\\times}$. The load-bearing criterion is the adiabatic condition $\\Delta t_\\times > C_{\\rm ad}\\, \\max(2\\pi/m_L(T_\\times),\\, 2\\pi/[m_H(T_\\times)-m_L(T_\\times)])$, which requires the crossing to last longer than both the light-mode oscillation period and the beat period — the beat term is what earlier conditions missed. Under that condition the comoving number densities of the two eigenstates are separately conserved, and the abundance formula follows from equating the ALP's initial yield to the final QCD axion yield.","core_discovery":"In the limit where the effective ALP coupling is large ($r_f \\gg 1$) and the zero-temperature QCD axion mass exceeds the ALP mass ($r_m \\ll 1$), the heavy mass eigenstate is the ALP before the QCD transition and the QCD axion after it. Because the level crossing is adiabatic, the comoving number densities of the heavy and light eigenstates are separately conserved, so the entire ALP population becomes the QCD axion population. With order-unity initial misalignment angles the ALP stores far more energy than the QCD axion would, and after the transfer the QCD axion relic abundance is given by $\\Omega_{\\rm DM} h^2 \\simeq 0.12\\, n_\\phi^2 \\theta_{\\phi,i}^2 (r_m/0.1)^{-1/2} (r_f/100)^2 (f_a/10^{10}\\,{\\rm GeV})^{3/2}$ (Eq. 39). Thus a QCD axion with $f_a$ around $10^9$–$10^{12}$ GeV can be all the dark matter, and the paper maps the viable region in the ($r_m$, $r_f$) plane, including the axion masses and photon couplings that follow.","pith_inferences":["The same beat-frequency criterion should govern other resonant conversions of oscillating fields, such as axion–dark-photon mixing, where the level-crossing timescale and the beat period compete.","Because the direction of number transfer is set by the relative phase of the two oscillations, a partially non-adiabatic crossing could just as easily enhance as deplete the QCD axion abundance, so the effect's sign is not fixed by the ratio parameters alone.","For $f_a \\sim 10^{10}$ GeV the required $\\theta_{\\phi,i} \\sim 1$ homogeneous field makes the scenario's isocurvature perturbations stronger than in tuned small-angle misalignment, so CMB isocurvature bounds are a direct test of the inflationary part of the setup.","The paper's chiral-perturbation estimate of QCD axion thermal friction may be enhanced by an order of magnitude once the Boltzmann approximation is dropped (Appendix B), which would shrink the viable region at $f_a$ below $10^9$ GeV; this is an open correction rather than a settled result."],"forward_implications":["A QCD axion with decay constant between roughly $10^9$ and $10^{12}$ GeV can be all of the dark matter when an ALP of higher bare mass mixes with it, provided the ALP starts with an order-one misalignment angle.","The adiabaticity of axion level crossing is controlled by the beat frequency $m_H - m_L$, so previous estimates based only on the light-mode frequency $2\\pi/m_L$ overstate the allowed parameter space.","The two potential forms used in the literature for axion mixing are exactly equivalent, but the mixing parameter that looks natural in one basis (order-one $n_\\phi$) requires extreme smallness in the other ($N_A \\ll 1$), which explains why the enhancement and suppression scenarios favour different ultraviolet embeddings.","In the viable region the heavy axion's mass and photon coupling track the standard QCD axion band for $m > 10^{-5}$ eV while the light axion can be much lighter and more weakly coupled, giving concrete targets for axion-photon searches.","Thermal friction on the QCD axion from pion interactions may be larger than previous estimates, which could alter the abundance for decay constants well below $10^9$ GeV."],"supporting_citations":[{"why":"Establishes that the comoving axion number density, not energy density, is the adiabatic invariant at level crossing—the method the abundance estimates rely on.","marker":"[16]"},{"why":"Studies the two-axion viable parameter space for suppressing the QCD axion abundance and first notes that the beat frequency enters the adiabatic condition.","marker":"[19]"},{"why":"Demonstrates that the QCD axion abundance can be enhanced by level crossing to explain dark matter at small $f_a$, the scenario this paper refines with a corrected adiabatic condition.","marker":"[21]"},{"why":"Provides the lattice-QCD topological susceptibility $\\chi(T)$ ($\\chi_0 = (75.6\\,{\\rm MeV})^4$, $T_{\\rm QCD} = 153\\,{\\rm MeV}$, $n = 8.16$) that drives the temperature-dependent level crossing.","marker":"[65]"},{"why":"Supplies the standard single-axion misalignment abundance formula (Eq. 1) against which the level-crossing enhancement is compared.","marker":"[62]"},{"why":"Gives the observed dark-matter density $\\Omega_{\\rm DM} h^2 \\simeq 0.12$ that the viable parameter region is required to match.","marker":"[63]"}],"fun_headline_variants":["Axion crossing hands dark matter to QCD axion","QCD axion dark matter emerges from level crossing","Level crossing makes QCD axion all the dark matter","How axion level crossing widens the QCD dark matter window","QCD axion dark matter via refined adiabatic crossing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire enhancement rests on the ALP field beginning with an order-one initial misalignment angle in a homogeneous pre-inflationary patch; if the ALP starts near its potential minimum, none of the transferred abundance exists and the dark-matter claim collapses.","fun_headline_variants_meta":{"raw":{"variants":["Axion crossing hands dark matter to QCD axion","QCD axion dark matter emerges from level crossing","Level crossing makes QCD axion all the dark matter","How axion level crossing widens the QCD dark matter window","QCD axion dark matter via refined adiabatic crossing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000731,"raw_usage":{"total_tokens":3312,"prompt_tokens":1023,"completion_tokens":2289,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":639,"completion_tokens_details":{"reasoning_tokens":2207}},"tokens_in":639,"tokens_out":2289,"duration_ms":17334,"temperature":1.0,"reasoning_tokens":2207,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:12:24.940180+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Integrate the two-axion equations of motion (A5)–(A6) for, say, $r_f = 10$, $r_m = 0.1$, and $f_a$ a factor of a few above the bound in Eq. (62), starting from $\\theta_\\phi = 1$ and $\\theta_a = 0$: the paper's criterion says the heavy-mode yield $Y_H$ should change by more than 10% across the crossing, while the older $2\\pi/m_L$ condition would predict near-perfect adiabatic transfer, so the numerical outcome settles which adiabaticity claim is right. On the observational side, the scenario is falsified if axion search experiments exclude the predicted bands of Figs. 4–5 — the heavy axion at $|g_{H\\gamma\\gamma}| \\approx \\alpha_{\\rm em}/(2\\pi f_a)$ with mass above $10^{-5}$ eV, or the light axion at the constant-$g_{L\\gamma\\gamma}$ contours — down to the sensitivity required for a subdominant component.","supporting_citations":[],"review_version":1}