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QCD Axion Dark Matter from level crossing with refined adiabatic condition

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.10232 v1 pith:BTARIKEH submitted 2024-12-13 hep-ph astro-ph.CO

Kai Murai , Yuma Narita , Fuminobu Takahashi , Wen Yin This is my paper · ORCID
classification hep-phastro-ph.CO
keywords QCDaxionaxion-likeparticlelevelcrossingresonantconversionbeatfrequencyadiabaticconditionmisalignmentmechanismdarkmatter
topics Dark Matter
open problems 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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 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.

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 (2)
  1. [Sec. VI and Appendix A; Eqs. (25), (39)] 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.
  2. [Sec. IVB and Eq. (39)] 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.
minor comments (4)
  1. [Sec. IVA, Eq. (16), and Fig. 3] 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.
  2. [Appendix B and Fig. 3] 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.
  3. [Fig. 2] 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.
  4. [Sec. IVB] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

There is no load-bearing circularity: Eq. (39) follows from misalignment production and comoving-number conservation, and the adiabatic condition is calibrated numerically rather than fitted to the abundance.

full rationale

The central abundance formula Eq. (39) is derived from the misalignment initial energy densities in Eqs. (34)-(35), the definition of comoving yields in Eq. (36), and the rf >> 1, rm << 1 mass hierarchy; none of these inputs is fitted to the dark-matter abundance itself. The adiabatic condition Eq. (25) is attributed to the authors' earlier Ref. [19], but it is independently tested in Sec. VI and Appendix A by direct numerical integration of the equations of motion (A5)-(A6), with the coefficient Cad determined from the explicitly stated 10% criterion Rad = 0.1 in Eq. (A7). This is a calibration of a threshold, not a fit of the predicted abundance, and Eq. (39) does not depend on Cad. The basis equivalence in Sec. V is an exact algebraic rotation described by Eqs. (45)-(50), not a renamed empirical result. The thermal-friction estimate in Appendix B is a new chiral-perturbation-theory derivation rather than a cited ansatz. The remaining concern about initial-phase dependence, raised by the θa(τi)=0, θφ(τi)=1 runs used to calibrate the adiabatic threshold, is a robustness or model-validity issue, not a circular one: it does not make Eq. (39) equivalent to its own input by construction. Self-citations to Refs. [16,19,20,39] are frequent, but they are not load-bearing because the beat-frequency criterion is reproduced numerically in this paper rather than assumed uniquely from those references. No equation reduces a prediction to a fitted quantity or to a self-citation chain.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new particles or forces; all quantities are standard axion model parameters. The central calculation rests on a set of cosmological and particle-physics assumptions that are mostly standard but not independently verified here: pre-inflationary homogeneity, radiation domination, the lattice QCD susceptibility power law, O(1) initial angles, and approximate thermal friction. The only fitted numerical constant is Cad, calibrated to the authors' own 10 percent adiabaticity definition, and the level-crossing threshold Delta_alpha_min is chosen by hand. These choices affect the precise boundaries of the viable region but not the qualitative mechanism.

free parameters (5)
  • Cad = determined numerically from the 10% adiabaticity criterion in Sec. VI
    The adiabatic condition Eq. (25) uses Cad of order unity calibrated to numerical solutions; the location of the non-adiabatic boundary and therefore the viable region depends on this calibration.
  • Delta_alpha_min = pi/6, pi/4, or pi/3, with pi/4 used for the main boundary
    The level-crossing condition Eq. (16) requires total eigenstate rotation Delta_alpha > Delta_alpha_min; the orange region boundary in Fig. 3 shifts with the chosen threshold.
  • rf = f_phi/(n_phi fa) = scanned over roughly 10^-2 to 10^3; viable enhancement requires rf >> 1
    Dimensionless mixing parameter controlling the rotation angle of the eigenstates; the central enhancement scenario is defined by rf >> 1 and rm << 1.
  • rm = m_phi/ma,0 = scanned over roughly 10^-4 to 10; viable region requires rm < 1 approximately
    Ratio of the ALP mass to the zero-temperature QCD axion mass; determines whether the crossing occurs before the QCD axion mass saturates.
  • initial misalignment angles theta_a,i and theta_phi,i = set to O(1), with theta_phi,i = 1 and theta_a,i = 0 in the numerical validation
    The abundance transfer scales as theta_phi,i^2, so the mechanism and the fa contours in Fig. 3 depend directly on this choice.
assumptions (7)
  • domain assumption Pre-inflationary breaking of the axion symmetries: the axion fields are almost homogeneous and topological defects are absent.
    Stated in Sec. II footnote 1; the entire misalignment-based abundance estimate assumes homogeneous initial angles and neglects strings and domain walls.
  • domain assumption The universe is radiation-dominated with the standard Friedmann equation and entropy conservation.
    Used in Eq. (24) for H(T) and in Eq. (A3) for converting temperature to time; the level-crossing time scale Delta_t_x in Eq. (21) depends on this.
  • domain assumption The QCD topological susceptibility follows the power law chi(T) = chi_0 (T/T_QCD)^(-n) with n = 8.16 above T_QCD and is constant below.
    This temperature dependence drives the level crossing and is adopted in Eq. (7) from the lattice result of Ref. [65].
  • domain assumption The ALP potential is time-independent and has the cosine form m_phi^2 f_phi^2 (1 - cos(phi/f_phi)).
    Assumed in Eq. (5); the authors note in Sec. VII that a time-dependent ALP potential would change the abundance estimates and viable region.
  • domain assumption The initial misalignment angles of both axions are of order unity.
    The enhancement scenario and the fa contours in Fig. 3 assume theta_a,i and theta_phi,i are O(1); small ALP angles would suppress the transferred abundance.
  • domain assumption Comoving number densities of the heavy and light mass eigenstates are separately conserved through an adiabatic level crossing.
    This is the central physical input used in Sec. IVB to derive Eqs. (37)-(38); Sec. VI supports it numerically only under the 10 percent criterion.
  • ad hoc to paper The chiral perturbation theory estimate of thermal friction, Eq. (29), is accurate enough to define the evaporation boundary.
    The main text uses Eq. (29) to draw the green region in Fig. 3, but Appendix B states that the precise estimate is beyond the scope of the paper and one contribution is evaluated 'very naively'.

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Pith. "Pith review of QCD Axion Dark Matter from level crossing with refined adiabatic condition." pith.science (2026). https://pith.science/paper/BTARIKEH

@misc{pith2026241210232,
  author       = {Pith},
  title        = {Pith review of: QCD Axion Dark Matter from level crossing with refined adiabatic condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BTARIKEH}},
  note         = {Machine review of arXiv:2412.10232}
}
read the original abstract

We investigate the level-crossing phenomenon in two-axion systems, where the mass eigenvalues intersect as the mass of one axion increases with the cooling of the universe. This phenomenon can significantly alter the abundance of axions in the early universe. Our study focuses on its impact on the QCD axion and an axion-like particle, identifying viable regions of axion mass and decay constant that explain the observed dark matter. We demonstrate the equivalence of two different bases for describing the axion system in the existing literature. Furthermore, we derive an improved expression for the adiabatic condition that overcomes limitations in earlier formulations. This new formulation is basis-independent, and we numerically validate its effectiveness. Our analysis reveals specific relations between axion masses and axion-photon couplings within the viable region. These relations could potentially serve as a smoking gun signal for this scenario if confirmed experimentally. We also find that, using the chiral perturbation model, the thermal friction on the QCD axion might be significantly larger than previously estimated. Additionally, we show that a simple model with axion mixing can naturally realize either a heavier or lighter QCD axion.

Figures

Figures reproduced from arXiv: 2412.10232 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The contour of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The left panel represents the contour of [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Mass and photon coupling of the heavy and light modes. These panels show the case of [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The same figures as Fig [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The temperature dependence of the mixing angle [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The upper limits of [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The transfer of the number density between the heavy and light fields for non-adiabatic [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The numerical result of the axion oscillations with [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The dissipation rate ( [PITH_FULL_IMAGE:figures/full_fig_p033_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]

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Forward citations

Cited by 6 Pith papers

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  1. Darkly Charged ALPs

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  3. Sign-Flipping Axion Potentials via Kapitza-Type Modulation by Heavy Axions

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  4. Resonant Landau-Zener Conversion In Multi-Axion Systems

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  5. Hierarchical Axiverse

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  6. QCD Axion Dark Matter in the Dark Dimension

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