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Catalogues of Cosmologically Self-Consistent Hadronic QCD Axion Models

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper identifies two new cosmologically viable mass windows for hadronic QCD axion dark matter, at f_a ~ 10^12 GeV and f_a ~ 10^14 GeV.

desk verdict A careful, reproducible extension of the KSVZ axion catalogue whose two new model islands are real possibilities, not established targets: the load-bearing caveat is the neglected topological-defect contribution, not the reheating bound the reader uses. read the letter →

arxiv 2412.17896 v2 pith:OK6DS7QF submitted 2024-12-23 hep-ph

Luca Di Luzio , Sebastian Hoof , Coenraad Marinissen , Vaisakh Plakkot This is my paper · ORCID
classification hep-ph
keywords QCDaxionhadronicmodelKSVZearlymatterdominationdarkaxion-photoncouplingcataloguePeccei-Quinnsymmetry
topics 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

The paper extends the catalogue of hadronic (KSVZ) QCD axion models to include heavy, coloured fermions that decay through higher-dimensional operators, up to dimension 9. Such operators make the fermions decay slowly enough to drive a phase of early matter domination, which relaxes the usual lifetime bounds and opens up cosmological viability. The authors find two new 'model islands' of viable axion decay constant $f_a$ in the post-inflationary scenario: a $d=6$ island at $f_a \sim 10^{12}$ GeV and a $d=7$ island at $f_a \sim 10^{14}$ GeV, beyond the standard post-inflationary mass region. These islands give concrete targets for axion searches.

What carries the argument

The central object is the lowest-dimension decay operator for each heavy Peccei-Quinn fermion $Q$, since its dimension $d$ sets the decay rate $\Gamma \sim m_Q (m_Q/\Lambda_{\mathrm{EFT}})^{2(d-4)}$ and hence controls whether the fermions freeze out, drive an early matter domination phase, and decay before Big Bang nucleosynthesis. Models are labelled by their 'dimensional signature'—the multiplicities of the lowest decay-operator dimension for each $Q$—because the cosmology of multiple $Q$s is essentially determined by the largest $d$ among the lowest-dimensional operators. The analysis combines Boltzmann equations for the temperature and fermion number densities with the axion realignment equation solved by the code MiMeS, using the temperature-dependent axion mass from lattice QCD.

What would settle it

Measure the tensor-to-scalar ratio $r$ at the level of about 0.001, which would fix the inflationary Hubble scale at $H_I \approx 6\times 10^{12}$ GeV and the Gibbons-Hawking temperature at $T_{\mathrm{GH}} \approx 10^{12}$ GeV; with $T_{\mathrm{GH}}$ inside the $d=6$ island's range ($f_a$ up to $2\times 10^{12}$ GeV), the post-inflationary interpretation for that island would be ruled out, eliminating the paper's main new prediction.

Watch

Extended reading notes

Core claim

The paper claims that, in the post-inflationary Peccei-Quinn symmetry breaking scenario, hadronic axion models with heavy fermions decaying via dimension-6 and dimension-7 operators form two cosmologically self-consistent 'model islands': the $d=6$ island spanning $f_a \in [0.25, 2.0]\times 10^{12}$ GeV and the $d=7$ island spanning $f_a \in [1.0, 1.7]\times 10^{14}$ GeV. The islands are bounded from below by requiring that the fermions decay before Big Bang nucleosynthesis and from above by the dark matter relic density constraint. The paper also updates the hadronic axion band for the axion-photon coupling and finds that its central region is nearly independent of $f_a$, whereas the maximal coupling grows with $f_a$ because the Landau pole criterion becomes less restrictive at larger $f_a$.

Load-bearing premise

The post-inflationary interpretation requires that the Peccei-Quinn symmetry is unbroken during inflation, meaning $f_a$ must lie below the Gibbons-Hawking temperature $T_{\mathrm{GH}} \approx 6\times 10^{12}$ GeV, a condition that the $d=7$ island at $f_a \sim 10^{14}$ GeV violates.

Editorial extensions

If this is right

  • Haloscope experiments searching for axion masses around 10 µeV gain a motivated window, since the $d=6$ island corresponds to $f_a \sim 10^{12}$ GeV.
  • A detection of an axion inside either island would point to a period of early matter domination and could constrain the effective operator scale $\Lambda_{\mathrm{EFT}}$.
  • The central band of the axion-photon coupling is stable across $f_a$, making the core predictions insensitive to the catalogue's parameter choices.
  • Models with domain wall number $N_{\mathrm{DW}} = 1$, which avoid the domain wall problem, are a subset that yields generally larger couplings $|C_{a\gamma}|$ and are particularly attractive search targets.
  • The publicly released catalogues allow the community to recompute the axion band under different assumptions, for instance including topological defect contributions.

Reading between the lines

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

  • If the Gibbons-Hawking bound on the post-inflationary scenario is enforced strictly, the $d=7$ island at $f_a \sim 10^{14}$ GeV is ruled out, leaving the $d=6$ island as the only genuinely new post-inflationary target.
  • The islands should be read as an upper bound on viable models, since the neglected topological-defect contribution to the axion relic density can only shrink them or make them disappear.
  • An axion signal at a mass excluded in standard cosmology would be an indirect signature of a self-induced early matter domination phase, a nonstandard cosmic history with no extra new physics.
  • The same combination of operator enumeration, Landau pole preselection, and Boltzmann cosmology could be applied to other long-lived coloured particle scenarios to map their cosmologically consistent parameter regions.
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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

3 major / 5 minor

Summary. This paper extends the catalogue of KSVZ hadronic QCD axion models to include heavy fermion representations compatible with higher-dimensional decay operators up to d≤9. Using DECO for operator enumeration and the Landau-pole criterion as a preselection, the authors identify new Q representations and solve coupled Boltzmann equations that allow multiple Qs to induce a period of early matter domination. They compute the axion realignment abundance with MiMeS and derive constraints from BBN (via the proxy w(T_BBN)>0.3), dark matter overproduction, and hot DM. The main results are an updated axion-photon coupling band whose central region is approximately f_a-independent, and two post-inflationary 'islands' around f_a∼10^12 GeV (d=6) and f_a∼10^14 GeV (d=7). The paper includes extensive supplementary material and is unusually explicit about its assumptions and limitations.

Significance. If the islands survive scrutiny, they provide concrete haloscope targets at ma∼10 μeV and beyond the standard post-inflationary window, tying a possible axion discovery to early matter domination and to the scale Λ_EFT. The paper's strengths are the exhaustive operator enumeration with the public DECO-based pipeline, the public catalogues and analysis scripts, and the stable central band that is robust to many model-building choices. The headline islands, however, are conditional on neglecting the topological-defect contribution to Ω_a h^2 and on the choice Λ_EFT=M_P; the manuscript itself states that the islands could disappear under stronger defect contributions. The significance is therefore moderate and depends on how these caveats are resolved.

major comments (3)
  1. [4.4 (with Eq. (3.2) and Eq. (3.10))] The upper boundaries of both islands are set by the condition Ω_a h^2 < 0.12, but only realignment production is included in the computation. The paper itself notes that 'both the d=7 and the d=6 regions may disappear altogether' once the cosmic-string/domain-wall contribution is included (Section 4.4), and the cited simulations suggest the standard-cosmology upper bound on f_a could drop by O(2)–O(200). Because no quantitative estimate, not even a conservative upper bound, is provided for Ω_strings+DW in the EMD scenario, the existence of the islands is not established by the present analysis. I recommend either adding a quantitative defect estimate, even one with large uncertainties, or explicitly reclassifying the islands as benchmark regions conditional on defect-neglect in the abstract and conclusions.
  2. [3.2.1 and 4.4] The island structure is strongly sensitive to the choice Λ_EFT=M_P in Eq. (3.3). In Section 4.4 the authors state that lowering Λ_EFT to about 0.7M_P merges the d=6 island with the standard region, Λ_EFT∼0.01M_P merges the d=7 island into the d=6 region, and d=8 operators become viable for Λ_EFT≲0.1M_P. Since no theoretical argument uniquely selects Λ_EFT=M_P, the two-island claim is a benchmark-dependent result rather than a robust prediction. The main text should either scan Λ_EFT or present the islands with an explicit confidence statement tied to the cutoff choice.
  3. [3.2.1 and 4.4] The manuscript compares the Gibbons–Hawking temperature T_GH=H_I/2π with f_a and uses this to suggest that the 'limit on the maximum possible reheating temperature... may rule out the post-inflationary scenario for the d=7 island' (Section 4.4). This is not the correct criterion: the maximum reheating temperature after inflation is of order T_RH,max∼(H_I m_P)^{1/2}≈10^16 GeV for H_I≈4×10^13 GeV, which is well above f_a∼10^14 GeV. The de Sitter vacuum temperature T_GH is not the reheating temperature and does not by itself preclude PQ symmetry restoration after inflation. The d=7 island is therefore not internally ruled out by this argument; please correct the text and remove or replace the T_GH-based caveat.
minor comments (5)
  1. [3.2.1] The sentence 'shows a period of EDM' should read 'EMD'.
  2. [4.2] The phrase 'neglected in in ref. [19, Fig. 4]' contains a duplicated 'in'.
  3. [3.2.1] The string 'TBBN = 1 MeVand' is missing a space and should read 'MeV and'.
  4. [3.2.1 (Eq. (3.2))] The neglect of entropy injection from Q annihilations and inverse decays is stated but not quantitatively justified; a short estimate of the size of this effect would make the Boltzmann treatment more robust.
  5. [3.2.1 (Eq. (3.8))] The BBN consistency proxy w(T_BBN)>0.3 is a useful first cut, but since the lower f_a boundaries of the islands are set by this criterion, a dedicated BBN calculation (e.g., with AlterBBN or ACROPOLIS) would strengthen the quantitative boundaries.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the model islands are outputs of the Boltzmann plus misalignment system, with theta_eff calibrated to standard cosmology rather than to the islands.

full rationale

The paper's central claims—two post-inflationary 'islands' around fa ~ 10^12 and 10^14 GeV—are obtained by solving the coupled Boltzmann equations (3.1)-(3.2) for the Q abundances and the realignment equation (3.10) via MiMeS, with the LP criterion, Lambda_EFT = M_P, m_Q = f_a, and theta_0 = 2.2 as stated inputs. The theta_0 = 2.2 value is not fitted to the islands: it is calibrated to the standard-cosmology DM condition (f_a ~ 1.9 x 10^11 GeV for all DM, Sec. 3.2.2) and then used as the post-inflationary average. The island boundaries in Sec. 4.2 (f_a in [0.25, 2.0] x 10^12 and [1.0, 1.7] x 10^14 GeV) are root-found outputs of this system, delimited by BBN (w > 0.3) and DM (Omega_c h^2 < 0.12) constraints. Self-citations to refs. [17-19,45] supply the selection criteria and the previous single-Q catalogue, but the multi-Q extension and fa-dependent band are new computations, not imported conclusions. The caveats in Sec. 4.4—topological-defect contributions could remove both islands, and the reheating-temperature limit may rule out the d=7 island—are explicit limitations on robustness, not circular definitions: the realignment-only calculation is presented as conservative, and the caveats identify external physics omitted from the input equations. No equation is defined in terms of a target output, and no fitted parameter is renamed as a prediction. Hence no circular step is present; the low score reflects only the presence of non-load-bearing self-citations for background methodology and criteria.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The main parameters are physical scales (Lambda_EFT, Lambda_thr, m_Q) chosen conservatively rather than fitted; the islands' locations are outputs of the numerical evolution, not fit targets. The assumptions are mostly explicit model-building and cosmology choices; the two most load-bearing are the early matter domination trigger from ref [21] and the post-inflationary consistency bound that the d=7 island seems to violate.

free parameters (6)
  • EFT cutoff Lambda_EFT for decay operators = M_P ~ 1.22 x 10^19 GeV
    Sets the decay widths in Eq. (3.3); lowering it shifts or merges the islands and can make d>=8 operators viable (Section 4.4).
  • Heavy quark mass m_Q (set equal to f_a) = scanned over f_a in [10^8, 10^14] GeV; fixed to 10^17 GeV for LP preselection
    Assumes all Qs degenerate with mass equal to the axion decay constant, i.e. Yukawa coupling y_Q ~ O(1); affects island boundaries (Section 4.4).
  • Landau pole threshold Lambda_thr = 10^18 GeV
    Defines criterion (C4); changing it alters the catalogue size and the LP-allowed representations (Section 3.1).
  • Effective initial misalignment angle theta_eff = 2.2
    Computed with MiMeS for standard cosmology, all dark matter at f_a = 1.9 x 10^11 GeV; used for post-inflationary dark matter bounds, not fitted to the islands (Section 3.2.2).
  • BBN consistency proxy w(T_BBN) > 0.3 = 0.3
    Used in place of a full light-element abundance computation; a rigorous BBN calculation could shift model viability (footnote 4, Section 4.4).
  • Q annihilation coefficient C_ann = ~10, using triplet values c_f = 2/9 and c_g = 220/27
    Used in Eq. (3.4); the O(1) variation due to the Q representation is neglected (Section 3.2.1).
assumptions (8)
  • domain assumption The heavy fermion Q is a Dirac fermion with SM gauge charges and PQ charge +/-1; its mass m_Q is generated by the PQ-breaking vev.
    Standard KSVZ construction, Eq. (2.1).
  • domain assumption The LP criterion (C4) requires no Landau pole below Lambda_thr = 10^18 GeV for a 'preferred' model.
    Phenomenological selection criterion from refs [17,18]; excludes many representations.
  • standard math Decay operators of dimension d > 4 have widths given by Eq. (3.3); only the lowest-dimension operator per Q matters.
    Standard EFT power counting; the suppression of higher-dimension operators by (m_Q / Lambda_EFT)^2 justifies the truncation.
  • domain assumption The heavy quarks thermally decouple, freeze out, and can dominate the energy density, triggering an early matter domination phase as shown in ref [21].
    The mechanism for the nonstandard cosmology is imported from Cheek, Osinski, Roszkowski 2024.
  • ad hoc to paper Entropy injection from Q annihilations and inverse decays is negligible in the Boltzmann equation (3.2).
    Stated simplification in Section 3.2.1; could affect the early matter domination duration.
  • domain assumption Topological defects (axionic strings and domain walls) contribute negligibly to Omega_a h^2.
    Stated as conservative; the paper notes that including defects could shrink or remove the islands (Section 4.4).
  • domain assumption Post-inflationary PQ breaking requires T_RH > 10 m_Q and f_a < T_GH ~ 6 x 10^12 GeV.
    Self-consistency condition of the post-inflationary scenario (Section 3.2.1); violated by the d=7 island, acknowledged in Section 4.4.
  • standard math The temperature-dependent axion mass m_a(T) is taken from lattice QCD [30] and C_{a gamma,0} = 1.92 +/- 0.04 from chiral perturbation theory.
    External inputs used in Eqs. (2.4) and (2.5).

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Pith. "Pith review of Catalogues of Cosmologically Self-Consistent Hadronic QCD Axion Models." pith.science (2026). https://pith.science/paper/OK6DS7QF

@misc{pith2026241217896,
  author       = {Pith},
  title        = {Pith review of: Catalogues of Cosmologically Self-Consistent Hadronic QCD Axion Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OK6DS7QF}},
  note         = {Machine review of arXiv:2412.17896}
}
abstract

We extend the catalogue of "phenomenologically preferred" hadronic axion models to include heavy fermion representations associated with higher-dimensional decay operators. The latter have recently been shown to self-consistently trigger a period of early matter domination, making the underlying axion models cosmologically viable. After identifying all possible representations up to decay operator dimension $d \leq 9$, we update the hadronic axion band for the axion-photon coupling. The central regions of the axion band are similar to those found previously and approximately independent of the axion decay constant $f_a$, suggesting that they are robust predictions and targets for future axion searches. Moreover, we find that $d = 6$ and $d = 7$ operators can lead to two new viable "model islands" around $f_a \sim 10^{12}$ GeV and $f_a \sim 10^{14}$ GeV, i.e., beyond the standard post-inflationary mass region.

Figures

Figures reproduced from arXiv: 2412.17896 by the authors.

Figure 1
Figure 1. Example for the cosmological evolution of two Qs with mQ = 2×1014 GeV and associated with d = 6 and d = 7 decay operators, respectively. The top panel shows the evolution of their comoving energy densities (solid and dashed red lines) and the total commoving energy density. The bottom panel shows the effective equation of state as a function of temperature. around TQCD and demonstrates that, while self-consistent, t… view at source ↗
Figure 2
Figure 2. Axion relic density Ωah 2 for different models. We show the range θ0 ∈ [0.5, π/√ 3] for standard cosmology (grey shading), NQ = 1 with decay operators of dimensions d = 6, 7, 8 (coloured shading), and NQ = 2 where the second Q is associated with decay operators with d = 5, 6, 7 (coloured shading, dotted outline). The BBN bound excludes cosmologies with an EOS of w < 0.3 at TBBN = 1 MeV, while the DM bound is Ωch 2 <… view at source ↗
Figure 3
Figure 3. Self-consistent 68% (green) and 95% (yellow) central regions of the hadronic axion band in the post-inflationary PQ symmetry breaking scenario for |Caγ|. We show the band as obtained from the full catalogues (left), and for models with NDW = 1 (right). The dashed line marks the original KSVZ model (E/N = 0), while the solid lines respectively delimit the minimal and maximal value of |Caγ| = |E/N − 1.92|, i.e., ignor… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Self-consistent 68% (green) and 95% (yellow) central regions of the hadronic axion band in the post-inflationary PQ symmetry breaking scenario for |gaγ|. The dashed and solid lines mark the typical location of models with E/N = 0 and the largest possible value of |E/N …
Figure 5
Figure 5. Figure 5: The maximum value of the sum of Dynkin indices of the additional Qs allowed by the LP criterion from the one-loop beta functions. We show the results for the three SM gauge groups as different lines. C Analytical approximation for the allowed representation sizes The L…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Accidental Peccei-Quinn Symmetry From Gauged U(1) and a High Quality Axion

    hep-ph 2024-12 conditional novelty 6.0 of 10

    Three explicit axion models are built where a gauged axial U(1) makes the Peccei-Quinn symmetry accidental, preserving axion quality against quantum gravity and giving domain wall number one.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.