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Resonant Landau-Zener Conversion In Multi-Axion Systems

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Resonant axion conversion is captured by a single analytic formula that predicts paired haloscope targets.

desk verdict First analytic treatment of non-adiabatic axion level crossings with a solid LZ core, but the per-field relic abundances are only O(1) accurate and the abstract overstates what is proven. read the letter →

arxiv 2507.06287 v1 pith:EX7IFH2D submitted 2025-07-08 hep-ph astro-ph.COhep-ex

David I. Dunsky , Claudio Andrea Manzari , Pablo Quílez , Maria Ramos , Philip S{o}rensen This is my paper
classification hep-phastro-ph.COhep-ex PACS 14.80.Va95.35.+d98.80.Cq
keywords QCDaxionaxion-likeparticlesLandau-Zenerconversionlevelcrossingmisalignmentmechanismdarkmatterrelicabundancehaloscopeaxiverse
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 claims that when two axion fields have a temperature-dependent mass matrix, the non-adiabatic resonant conversion between them during the early Universe is captured by the Landau-Zener formula, giving an analytic handle on the final dark matter abundance of each state. The derived conversion probability is $P_{\mathrm{LZ}} = \exp(-\pi\gamma/2)$, with the adiabatic parameter $\gamma = \left|4(m^2_{as})^2/(d m^2_{aa}/dt - d m^2_{ss}/dt) \cdot 1/(m_H+m_L)\right|$ evaluated at the crossing time, and the post-crossing comoving number densities become $n_H = (1-P_{\mathrm{LZ}})n_H^{\mathrm{WKB}} + P_{\mathrm{LZ}} n_L^{\mathrm{WKB}}$ and vice versa. The authors show numerically that this prescription agrees with the full two-field evolution to better than 10% for two concrete mixing potentials, except when the onset of oscillations happens very close to the crossing. The payoff is a quantitative recasting of haloscope bounds and a set of paired discovery targets: a signal off the standard QCD axion line implies specific other masses and couplings to look for.

What carries the argument

The central object is the Landau-Zener adiabatic parameter $\gamma$, equal to the product of the resonance width $\delta t_{\mathrm{res}}$ and the oscillation frequency $\omega_{\mathrm{osc}}(t_\times)$ between the two mass eigenstates at the crossing time $t_\times$, defined by $m^2_{aa}(t_\times)=m^2_{ss}(t_\times)$. It controls the conversion probability $P_{\mathrm{LZ}}=\exp(-\pi\gamma/2)$: large $\gamma$ means adiabatic passage with no conversion, small $\gamma$ means near-total exchange of the comoving number densities. The derivation linearizes the comoving Klein-Gordon equation into a two-state Schr\"odinger equation by separating positive- and negative-frequency modes, and then solves it either by analytically continuing the adiabatic solution around the complex branch point at $z_0 = 2i m^2_{as}/\Delta'$ (Landau's method) or by writing the asymptotic solution in parabolic cylinder functions (Zener's method).

What would settle it

Numerically integrate the two-field equations of motion for the potential studied in Application I with a parameter choice where $d m^2_{as}/dt$ is no longer negligible compared with $d(m^2_{aa}-m^2_{ss})/dt$ across the resonance, and compare the asymptotic survival probability with $\exp(-\pi\gamma/2)$; a mismatch beyond the claimed few percent would show the constant-$m^2_{as}$ assumption is essential. A second, complementary check is to scan the region $T_{\mathrm{osc}}\approx T_\times$ and verify that the deviation of the analytic post-crossing abundances from the full numerical abundances exceeds 10% only there.

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Extended reading notes

Core claim

The paper's central claim is that a two-axion system undergoing a temperature-driven avoided level crossing can be treated as a Landau-Zener transition between the instantaneous mass eigenstates, with the crossing time $t_\times$ defined by $m^2_{aa}(t_\times)=m^2_{ss}(t_\times)$. The survival probability is $|C_1|^2 = 1 - P_{\mathrm{LZ}}$, where $P_{\mathrm{LZ}} = \exp(-\pi\gamma/2)$ and $\gamma = \left|4(m^2_{as})^2/(d m^2_{aa}/dt - d m^2_{ss}/dt) \cdot 1/(m_H+m_L)\right|$ evaluated at $t_\times$. The authors derive this from a linearized Schr\"odinger form of the Klein-Gordon equation using both Landau's complex-contour method and Zener's parabolic-cylinder solution, and they argue that the required linearity of the diagonal mass difference holds generically except when the two derivatives are tuned to cancel. Applying the formula to the WKB comoving number densities gives the post-crossing abundances, and a numerical comparison for two concrete potentials shows agreement to better than 10% except where the onset of oscillations occurs very near the crossing. The paper also uses the result to recast haloscope bounds and to predict paired axion targets in the QCD maxion scenario.

Load-bearing premise

The calculation assumes that over the brief resonance the diagonal mass-squared difference $m^2_{aa}-m^2_{ss}$ changes linearly in time and the off-diagonal element $m^2_{as}$ is effectively constant; if the two diagonal derivatives are tuned to cancel, or if either field begins oscillating only after the crossing temperature, the analytic formula loses accuracy.

Editorial extensions

If this is right

  • For any two-state axion mass matrix satisfying the linearity condition, the final relic abundance of each axion can be written in closed form, so scans over axion parameter space no longer require solving the coupled Klein-Gordon equations.
  • The Landau-Zener region opens new parameter space in which the heavy state, identified with the QCD axion at low temperature, can dominate the dark matter energy density even when initial conditions would make it subdominant.
  • If a single axion signal is found off the QCD mass-coupling line, the dark matter requirement and the mixing model predict a discrete set of partner masses and couplings for haloscope searches, with a two-fold heavy/light ambiguity.
  • In the QCD maxion scenario, maxion pairs remain in the adiabatic regime for $f_a \lesssim 10^{16}$ GeV, so their abundances follow the conserved-comoving-density limit.
  • For the second potential considered, non-adiabatic crossing occurs for $R_f \gg 1$ and $R_m \ll 1$, opening a region far from the QCD axion line where the light field dominates the dark matter.

Reading between the lines

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

  • The same conversion machinery should extend to an $N>2$ axiverse by treating each crossing as an independent two-state Landau-Zener transition; the paper notes this as future work, but the formula makes the sequence concrete.
  • Because $\gamma$ depends on the derivative difference $d m^2_{aa}/dt - d m^2_{ss}/dt$, the precise temperature dependence of the QCD topological susceptibility (the exponent $n$ in the power law) directly controls which regions of parameter space are adiabatic versus non-adiabatic; a different lattice determination of $n$ would shift the LZ region in the parameter-space plot.
  • The phase-dependent error in the subdominant field's abundance, which the paper flags as O(1), means that any experiment targeting the lighter or subdominant member of a pair should be interpreted with a phase-resolved calculation, not with the average LZ survival probability alone.
  • Haloscope reach depends on the local density $\rho_i/\rho_{\mathrm{DM}}$ of each eigenstate; combining the paper's relic-density map with existing astrophysical bounds on the subdominant component could shrink the predicted paired-target locations before any detection.
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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. The manuscript develops an analytic Landau-Zener (LZ) treatment of resonant axion conversion in two-axion systems during misalignment. It derives the conversion probability from the linearized equations of motion (Appendix B), gives an analytic prescription for relic abundances after the crossing (Eq. 36), benchmarks the prescription against full numerical solutions (Figs. 3 and 4), and applies the results to two model potentials, including the maxion scenario and haloscope reach estimates (Figs. 5 and 6). The paper claims that the LZ formalism accurately captures non-adiabatic resonant conversion and permits an analytic description of the relic abundances of each axion field for nearly any arbitrary two-state axion mass matrix.

Significance. The derivation in Appendix B is a genuine contribution: it provides a first-principles path from the axion equations of motion to the LZ probability, with both Landau's complex-plane method and Zener's parabolic-cylinder method shown to agree. The numerical cross-checks in Figs. 3 and 4 are a further strength, and the paper is careful to identify the regions where the approximation fails. If the per-field abundance claim could be established at the claimed accuracy, the paper would be a significant step in multi-axion cosmology and would give useful guidance for haloscope searches. However, the current validation establishes only the total dark-matter density to better than 10%; the individual densities, which are the advertised central result, are not validated at that accuracy, and the paper itself states that the subdominant field abundance is only an O(1) estimate. This gap is load-bearing because the abstract, the conclusions, and the haloscope target plots (Fig. 6) all rely on per-field abundances.

major comments (2)
  1. [Sec. III A and Eq. (36)] The per-field abundance claim is not established at the claimed accuracy. Equation (36) converts post-crossing amplitudes into probabilities by replacing the coherent superposition (Eq. 16 or its WKB counterpart) with |C1|^2=1-PLZ and |C2|^2=PLZ, thereby dropping the relative phase of C1 and C2. The paper explicitly states in Sec. III A that the subdominant field can pick up a phase-dependent error across the LZ parameter space and that its abundance should be taken as an O(1) estimate only. Yet the abstract promises 'an analytic description of the relic abundances of each axion field,' and Sec. V uses per-field densities to identify haloscope targets (e.g., the green heavy-field region in Fig. 6). Figure 4 validates the total energy density, not the individual densities; the <10% accuracy statement therefore does not extend to per-field predictions unless the phase issue is resolved or the claims are substantially qualified.
  2. [Sec. III, Eq. (34)] The oscillation-onset condition xi = 1.6 + 0.6 n'_i is introduced as a fit, not derived from the two-axion dynamics. Since the WKB abundances in Eq. (32) depend exponentially on the oscillation temperature through the prefactor mi(T_osc^i) and the amplitude ai(T_osc^i), the overall normalization of the analytic abundances is controlled by this empirical interpolation. The fit is calibrated on single-axion misalignment with a generic power-law mass; its transfer to the two-axion case, where the mass eigenvalues have a nontrivial temperature dependence from mixing, is not demonstrated. The authors should either provide a derivation of Eq. (34) or show explicitly that the final per-field abundances are insensitive to reasonable variations of xi within the parameter space of Fig. 4.
minor comments (4)
  1. [Sec. II, introductory paragraph] There is a typo: 'in a FLR W Universe' should read 'in an FRW Universe.'
  2. [Throughout] The notation 'sin 2 2ξ' is used in several places (e.g., Sec. II and Fig. 1). Since this is easily confused with sin(2) times sin(2ξ), the authors should define it explicitly as sin^2(2ξ) or use a clearer notation.
  3. [Sec. III A, Fig. 4 caption] The 'Seam' and 'Intermediate' regions are identified in the text but the figure itself does not mark their boundaries or report the numerical error scale in those regions; adding contours or shading with the local error would improve the comparison.
  4. [Sec. IV, Fig. 5] The discussion of the example signal at the position marked with a circle states that there are four possible targets, but the figure does not show these four targets. A supplementary panel or a table listing the four targets would make the predictive claim concrete and checkable.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Landau-Zener probability is derived in Appendix B from the axion equations of motion and checked against independent numerics; the disclosed x_i fit and self-citations to the maxion and trapped-misalignment papers are calibration/application steps, not inputs to the central derivation.

full rationale

The central result, the Landau-Zener probability P_LZ = exp(-pi*gamma/2) with gamma from Eq. (18), is derived inside the paper from the two-axion equations of motion: the Feshbach-Villars reduction (Eqs. B5-B13), Landau's complex-contour continuation (Eqs. B20-B32), and Zener's parabolic-cylinder solution (Eqs. B45-B47) agree with each other, and the two LZ assumptions (linear Delta(t), slowly-varying m_as^2) are checked against the paper's own mass matrices in App. B4. gamma contains no fitted parameters: it is built from the mass-matrix elements and their temperature derivatives at t_x, which is defined independently by Eq. (7). The numerical benchmarks in Figs. 3-4 are independent solutions of the full nonlinear equations of motion (App. A), so the reported agreement is external corroboration rather than construction. The x_i = 1.6 + 0.6 n'_i condition (Eq. 34) is disclosed as a fit ('we perform a fit to Eq. 33') and calibrates only the single-field WKB normalization and oscillation onset; the LZ transfer coefficients (1-P_LZ, P_LZ) that constitute the novel effect are not obtained from that fit, so the fitted-input-called-prediction pattern does not apply. Self-citations exist - [35] (maxion) and [46] (trapped misalignment) share two of five current authors - but they are applications, not load-bearing premises: the maxion line never intersects the LZ region for fa below 1e16 GeV (Sec. IV), and the x_i fit is the paper's own rather than imported from [46]. Real caveats remain but are accuracy concerns, not circularity: Eq. (36) drops the relative phase of C1 and C2, and the paper itself states 'the abundance of the subdominant field should be taken as an O(1) estimate only' (Sec. III A), which is in tension with the abstract's 'each axion field' framing and the conclusions' 'below 10%' claim; the phrase 'derived for the first time' for Eq. 17 also overstates novelty relative to the classical LZ formula. None of these steps makes a predicted quantity equivalent to its inputs by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard axion cosmology plus the standard LZ validity conditions. No new particles or forces are introduced. The only fitted ingredient is the oscillation-onset condition Eq. (34); the benchmark initial angles are model choices. The LZ probability itself is derived from the axion equations of motion and is not fitted.

free parameters (3)
  • x_i intercept in oscillation-onset condition = 1.6
    Eq. (34) sets T_osc_i through m_i(T_osc_i)=x_i H(T_osc_i); the intercept is fit to single-axion numerical abundances, so it is an input to the analytic relic-density predictions, not derived from first principles.
  • x_i slope in oscillation-onset condition = 0.6
    Eq. (34) adds 0.6 n'_i to interpolate between zero-temperature-mass and QCD-like scalings; fitted rather than derived and affects the normalization of every abundance prediction.
  • Benchmark initial misalignment angles = theta_a=1.25, theta_s=0.75
    Used for Figs. 5 and 6. These O(1) angles are chosen by hand and set the absolute dark matter density; the qualitative conclusions do not depend on the exact values, but the target maps do.
assumptions (6)
  • domain assumption The QCD topological susceptibility follows the power law m_a^2(T)=m_a,0^2 max(1,(T/T_QCD)^(-2n)) with n=3.34.
    Equation (21) is the temperature dependence that creates the level crossing; the value n is taken from lattice/DIGA literature, not derived in this paper.
  • domain assumption The universe is radiation dominated with standard entropy conservation, so dT/dt=-H T and H(T) follows the usual Friedmann relation.
    Used to convert crossing times to temperatures and to bound the Taylor expansion of the mass difference in App. B4.
  • domain assumption Axion fields are spatially homogeneous with negligible momentum, appropriate for pre-inflationary misalignment.
    Gradient term is dropped in Eq. (4) and p is set to zero in the Feshbach-Villars reduction (App. B1).
  • domain assumption During the resonance, Delta(t)=m_aa^2(t)-m_ss^2(t) is linear in time and m_as^2 is effectively constant.
    Core validity condition for the parabolic-cylinder/Landau solution; App. B4 argues it holds unless Delta' is tuned small.
  • domain assumption The sum of comoving number densities n_H+n_L is conserved during the brief resonance.
    Equation (B18) drops mass-variation terms to infer |C1|^2=1-|C2|^2; this underlies the abundance mixing formula Eq. (36).
  • domain assumption The axion potential is truncated to quadratic order for the mass matrix, while the numerical validation uses the full cosine potential.
    Equation (2) expands the cosine potential to second order; anharmonic corrections for O(1) misalignment angles are not included analytically.

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Pith. "Pith review of Resonant Landau-Zener Conversion In Multi-Axion Systems." pith.science (2026). https://pith.science/paper/EX7IFH2D

@misc{pith2026250706287,
  author       = {Pith},
  title        = {Pith review of: Resonant Landau-Zener Conversion In Multi-Axion Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EX7IFH2D}},
  note         = {Machine review of arXiv:2507.06287}
}
read the original abstract

Multiple axions may emerge in the low-energy effective theory of Nature. Generically, the potentials describing these axion fields are non-diagonal, leading to mass mixing between axion states which can be temperature-dependent due to QCD instanton effects. As the temperature of the Universe drops, level crossing can occur, causing resonant conversion between axion states. In this work, we present an analytic study of the cosmological evolution of multi-axion systems including adiabatic and non-adiabatic resonant conversion from one axion state into another during the misalignment process. We show how the Landau-Zener formalism accurately captures the non-adiabatic resonant conversion, permitting an analytic description of the relic abundances of each axion field for nearly any arbitrary two-state axion mass matrix. As an application, we study the mixing of a QCD axion with an axion-like-particle for specific potentials to identify the predictions for haloscope experiments. We conclude that the detection of an axion off the expected QCD mass-coupling line predicts other haloscope targets if it mixes with the QCD axion.

Figures

Figures reproduced from arXiv: 2507.06287 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Regions where level crossing can take place, assuming the potential in Eq. 19 with [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Comparison of the analytical prediction of the LZ survival probability, 1 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Evaluation of the precision of the analytic prescription. We show the ratio of total DM abundance [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Analytic continuation of adiabatic evolution. [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

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Forward citations

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